Premelting at the –SiO2 interface A high-energy x-ray microbeam diffraction study

Von der Fakult¨at fur¨ Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von Simon Christoph Engemann

aus Stuttgart

Hauptberichter: Prof. Dr. Helmut Dosch Mitberichter: Prof. Dr. Clemens Bechinger

Eingereicht am: 7. Oktober 2004 Tag der mundlichen¨ Prufung:¨ 4. Februar 2005

Institut fur¨ Theoretische und Angewandte Physik der Universit¨at Stuttgart, Max-Planck-Institut fur¨ Metallforschung in Stuttgart

2005 Bibliografische Information Der Deutschen Bibliothek: Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet uber¨ abrufbar.

Bibliographic information published by Die Deutsche Bibliothek: Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de.

Engemann, Simon: Premelting at the ice–SiO2 interface, a high-energy x-ray microbeam diffraction study Download at http://www.ice-premelting.net/diss/.

c 2005 Simon Engemann Herstellung und Verlag: Books on Demand GmbH, Norderstedt.

ISBN 3-8334-3980-7 Contents

Contents vi

Deutsche Zusammenfassung vii 0.1 Eis und Wasser ...... vii 0.2 Grenzfl¨achenschmelzen von Eis ...... viii 0.3 Messprinzip ...... ix 0.4 Probenpr¨aparation und Probenumgebung ...... x 0.5 Ergebnisse und Diskussion ...... xi 0.5.1 Morphologie der Substrate ...... xi 0.5.2 Wachstum der quasiflussigen¨ Schicht ...... xi 0.5.3 Struktur der quasiflussigen¨ Schicht ...... xii 0.5.4 Weitere Experimente ...... xiii 0.5.5 Bedeutung der Ergebnisse ...... xiii 0.6 Ausblick ...... xiii

1 Introduction 1

2 Ice and 3 2.1 Importance ...... 3 2.2 The H2O molecule and the bond ...... 4 2.3 Anomalies and mysteries ...... 5 2.4 The quest for the water structure ...... 6 2.5 ...... 8

3 Interface melting 11 3.1 The melting transition ...... 12 3.2 Theory of interface melting ...... 13 3.2.1 Phenomenological description ...... 14 3.2.2 Landau-Ginzburg models ...... 16 3.2.3 Density functional theory ...... 19 3.2.4 Lattice theory ...... 19 3.2.5 Other approaches ...... 19 3.2.6 Molecular dynamics simulations ...... 20

iii iv CONTENTS

3.2.7 Interfacial melting and substrate roughness ...... 20 3.2.8 Further aspects ...... 21 3.3 Experimental evidence for interface melting ...... 21 3.4 Interface melting of ice ...... 22 3.4.1 Theory and simulations ...... 23 3.4.2 The free surface of ice ...... 23 3.4.3 Ice in porous media ...... 24 3.4.4 Ice– interfaces ...... 26 3.4.5 Further aspects ...... 29 3.5 Consequences of ice premelting ...... 30 3.5.1 Permafrost ...... 30 3.5.2 motion ...... 30 3.5.3 Thunderstorms and atmospheric chemistry ...... 31 3.5.4 Friction ...... 31 3.6 Summary and conclusions ...... 31

4 Theory of x-ray reflectivity 33 4.1 Index of refraction for x-rays ...... 33 4.2 Reflection at an ideal interface ...... 34 4.3 Reflection at multiple interfaces ...... 35 4.4 Arbitrary dispersion profiles ...... 37 4.5 The kinematical approximation ...... 37 4.6 Data analysis and phase inversion ...... 38 4.7 Description of rough interfaces ...... 39 4.8 Reflectivity from rough interfaces ...... 40 4.8.1 Specular reflectivity ...... 42 4.8.2 Integrated diffuse intensity ...... 44 4.8.3 Off-specular reflectivity ...... 44 4.9 Further remarks ...... 45

5 High-energy x-ray-reflectivity experiments 47 5.1 Principle ...... 47 5.2 Application to the interface melting of ice ...... 50 5.3 Experimental and instrumental considerations ...... 51 5.3.1 Source and optics ...... 51 5.3.2 Sample stage ...... 54 5.3.3 Detector ...... 55 5.3.4 Scattering geometry ...... 56 5.3.5 Resolution ...... 56 5.3.6 Integration by the detector ...... 61 5.3.7 Illumination of the sample ...... 61 5.3.8 Coherence ...... 66 5.3.9 Data correction ...... 68 CONTENTS v

6 Sample preparation and environment 71 6.1 The substrates ...... 71 6.2 The ice samples ...... 75 6.3 The cold room ...... 76 6.4 Interface preparation ...... 76 6.5 The in situ chamber ...... 78 6.6 Temperature stability and accuracy ...... 80

7 Results and discussion 85 7.1 Overview of the main experiments ...... 86 7.2 Density profiles ...... 92 7.2.1 Raw data analysis ...... 92 7.2.2 Reconstruction of density profiles ...... 93 7.2.3 Reliability of the fits ...... 99 7.3 Substrate morphology ...... 103 7.3.1 Smooth substrate ...... 103 7.3.2 Rough substrate ...... 105 7.3.3 AFM measurements ...... 112 7.3.4 Conclusion ...... 113 7.4 Growth law ...... 114 7.4.1 What is expected from theory? ...... 114 7.4.2 Experimentally observed growth law ...... 115 7.4.3 Onset ...... 116 7.4.4 Growth amplitude ...... 120 7.4.5 Influence of roughness ...... 121 7.4.6 Comparison with surface melting ...... 122 7.4.7 Influence of temperature error ...... 122 7.5 Density and structure of the quasiliquid ...... 125 7.5.1 Experimentally observed density ...... 125 7.5.2 Conclusions about the structure ...... 127 7.6 Si wafer as substrate ...... 130 7.6.1 Sample ...... 130 7.6.2 Experimental setup ...... 132 7.6.3 Results ...... 132 7.7 Neutron reflectivity ...... 135 7.7.1 Sample ...... 136 7.7.2 Experimental setup ...... 136 7.7.3 Results ...... 136 7.8 Substrate termination and radiation effects ...... 138 7.9 Implications ...... 140 vi CONTENTS

8 Outlook 143 8.1 Influence of the substrate material and the confinement ...... 143 8.2 Surface melting ...... 145 8.3 Influence of the substrate morphology ...... 145 8.4 Influence of impurities ...... 145 8.5 Growth law ...... 146 8.6 Structure of the quasiliquid ...... 146 8.7 Influence of electric fields ...... 147

9 Summary 149

List of acronyms 155

List of figures 159

List of tables 161

Bibliography 163

Acknowledgements 179 Deutsche Zusammenfassung

Im Rahmen dieser Doktorarbeit wurde das Grenzfl¨achenschmelzen von Eis mit- tels einer neuen R¨ontgenstreumethode basierend auf hochenergetischen Mikro- strahlen untersucht. Dieses Kapitel stellt eine deutsche Zusammenfassung der in Englisch verfassten Dissertation (folgende Kapitel) dar. Eine kurzer¨ gefasste englischsprachige Zusammenfassung findet sich in Kapitel 9.

0.1 Eis und Wasser

Eis und Wasser bedecken einen Großteil unseres Planeten und haben seine Ober- fl¨ache uber¨ Jahrmillionen hinweg gepr¨agt. Das Gleichgewicht von Eis, flussigem¨ Wasser und Wasserdampf ist entscheidend fur¨ das Klima. Wasser bildet die Grundlage unseres Lebens und ist in seinen verschiedenen Erscheinungsformen Teil unseres Alltags. Obwohl der Aufbau des Wassermolekuls¨ sehr einfach erscheint, weist Wasser eine Reihe von außergew¨ohnlichen Eigenschaften und Anomalien auf. Oftmals sind gerade diese Anomalien von entscheidender Bedeutung bei der Rolle, die das Wasser in der Natur und fur¨ unser Leben spielt. Trotzdem sind diese Anomalien bis heute nicht vollst¨andig verstanden [1]. Anomalien treten u.a. in den Antwortfunktionen des Wassers, wie der iso- thermen Kompressibilit¨at, der isobaren W¨armekapazit¨at und dem W¨armeaus- dehnungskoefizienten auf. Der Betrag dieser Funktionen steigt beim Abkuhlen¨ stark an, beim Unterkuhlen¨ wird dieser Anstieg noch st¨arker. Die Antwortfunk- tionen scheinen bei einer singul¨aren Temperatur TS = 228 K zu divergieren [2]. In einer normalen Flussigkeit¨ wurden¨ die genannten Gr¨oßen mit sinkender Tem- peratur langsam abfallen. Eine weitere Besonderheit ist die so genannte Dichteanomalie des Wassers. Wasser weist die gr¨oßte Dichte bei +4◦C auf, bei weiterem Abkuhlen¨ sinkt die Dichte wieder. Dies ist equivalent zu einem Vorzeichenwechsel des W¨armeaus- dehnungskoefizienten bei +4◦C. Ferner ist die feste Phase Eis weniger dicht als flussiges¨ Wasser. Auch beim Fest-Flussig-¨ Ubergang,¨ d.h. dem Schmelzen von Eis, treten Beson- derheiten zu Tage. So fuhrt¨ der schon genannte Dichteunterschied zwischen Eis und Wasser dazu, dass sich Eis durch Druck verflussigen¨ l¨asst. Außerdem zeigt

vii viii DEUTSCHE ZUSAMMENFASSUNG

Eis eine ausgepr¨agte Tendenz zum Oberfl¨achenschmelzen [3], welches im n¨achsten Abschnitt erl¨autert werden soll.

0.2 Grenz߬achenschmelzen von Eis

Befindet sich ein Festk¨orper s in Kontakt mit einem anderen Medium b, so kann sich an der Grenzfl¨ache s–b eine dunne¨ Schicht des Festk¨orpers schon unter- halb der Schmelztemperatur Tm im Volumen des Materials verflussigen.¨ Dieses Ph¨anomen nennt man Grenzfl¨achenschmelzen“. Handelt es sich bei dem Medi- ” um b um Vakuum, Luft oder die Gasphase von s, so bezeichnet man die Grenz- fl¨ache ublicherweise¨ als Oberfl¨ache von s und das eben genannte Ph¨anomen als Oberfl¨achenschmelzen“. Handelt es sich hingegen bei dem Medium b um einen ” anderen Festk¨orper (im folgenden auch Substrat genannt), mithin um eine Fest– Fest-Grenzfl¨ache, so spricht man von Grenzfl¨achenschmelzen im engeren Sinne. Man bezeichnet die geschmolzene Schicht an der Grenzfl¨ache normalerweise als quasiflussig“¨ (englisch abgekurzt¨ mit qll), da im allgemeinen nicht zu erwarten ” ist, dass sie die gleiche Struktur wie die Flussigkeit¨ im Volumen hat. Man erwar- tet vielmehr eine durch den Kontakt mit dem darunter liegenden Festk¨orper s und dem daruber¨ liegenden Medium b modifizierte Struktur. Das Grenzfl¨achenschmelzen wird vom Prinzip der Minimierung der freien Energie getrieben. In einem ph¨anomenologischen Modell (siehe z.B. [4]) l¨asst sich die Anderung¨ der freien Energie durch das Auftreten einer dunnen¨ quasiflussigen¨ Schicht der Dicke L folgendermaßen beschreiben:

Tm − T ∆F (L) = ρqllQmL + ∆γ (L) . (1) Tm

Hierbei bezeichnet T die Temperatur, ρqll die Dichte der Quasiflussigkeit¨ und Qm die Schmelzw¨arme. ∆γ (L) ist der Unterschied in den Grenzfl¨achenenergien, der durch das Auftreten der quasiflussigen¨ Schicht erzeugt wird. Aufgrund der Wechselwirkung zwischen den beiden Grenzfl¨achen s–qll und qll–b h¨angt ∆γ von deren Abstand, d.h. der Dicke der quasiflussigen¨ Schicht, ab. Abh¨angig vom Typ der Wechselwirkungen ergibt die Minimierung der frei- en Energie unterschiedliche Wachstumsgesetze fur¨ die Dicke der quasiflussigen¨ Schicht als Funktion der Temperatur. Im Falle kurzreichweitiger exponentiell zerfallender Wechselwirkungen erh¨alt man ein logarithmisches Wachstumsgesetz der Form   Tm − T0 L (T ) = L0 ln . (2) Tm − T

Das Grenzfl¨achenschmelzen setzt bei der Temperatur T0 ein. Die Amplitude L0 kann mit der Korrelationsl¨ange der Quasiflussigkeit¨ identifiziert werden. 0.3. MESSPRINZIP ix

Im Falle langreichweitiger Van-der-Waals-Wechselwirkungen ergibt sich ein Potenzgesetz der From

T − T −1/(n+1) L (T ) ∝ m . (3) Tm

Hierbei gilt n = 2 fur¨ nicht-retardierte und n = 3 fur¨ retardierte Wechselwirkun- gen. Die bisherige Beschreibung trifft im Prinzip auf das Oberfl¨achenschmelzen wie auch auf das Grenzfl¨achenschmelzen an Fest–Fest-Grenzfl¨achen zu. Im letzteren Fall h¨angen die Grenzfl¨achenenergien und damit das Schmelzverhalten naturlich¨ vom Material des Substrats ab. Ferner k¨onnte die Morphologie des Substrats eine wichtige Rolle spielen, was in der bisherigen Betrachtung nicht berucksichtigt¨ ist. Das Grenzfl¨achenschmelzen ist von fundamentaler Bedeutung fur¨ den gesam- ten Schmelzprozess, da die quasiflussige¨ Schicht an der Grenzfl¨ache als Nukleati- onskeim fur¨ das Schmelzen im Volumen dienen kann. Außerdem hat es speziell im Fall von Eis wichtige Konsesquenzen fur¨ eine Reihe von Prozessen in Natur und Technik [5], wie z.B. Gletscherbewegung, Stabilit¨at von Permafrost und Vereisung von Flugzeugtragfl¨achen. Oberfl¨achenschmelzen wurde an einer Vielzahl von Materialien beobachtet, besonders ausgepr¨agt ist es bei Eis. Auch fur¨ das Auftreten von Grenzfl¨achen- schmelzen an Eis–Festk¨orper-Grenzfl¨achen gibt es viele Hinweise. Allerdings feh- len hier schlussige¨ mikroskopische Experimente an wohl definierten Grenzfl¨achen. Dies liegt daran, dass es kaum geeignete Methoden gibt, um solche vergrabenen Grenzfl¨achen zu untersuchen.

0.3 Reflektivit¨atsmessungen mit hochenergeti- scher R¨ontgenstrahlung

Zur Untersuchung von Grenzfl¨achen wurde in unserer Arbeitsgruppe ein neues Verfahren entwickelt [6]. Der Aufbau dazu wurde am Strahlrohr ID15A der ESRF (European Synchrotron Radiation Facility) in Grenoble installiert. Das Verfahren beruht auf der Verwendung brillianter und hochenergetischer (in dieser Arbeit ca. 70 keV) R¨ontgenstrahlung, die an modernen Synchrotronstrahlungsquellen erzeugt werden kann. Hochenergetische R¨ontgenstrahlung kann mehrere Millimeter oder gar Zen- timeter in Materialien eindringen und daher auch tief vergrabene Grenzfl¨achen erreichen. Mit den herk¨ommlichen R¨ontgenstreumethoden dagegen k¨onnen nur Grenzfl¨achen von dunnen¨ Schichten untersucht werden. Dies fuhrt¨ insbesonde- re bei Flussigkeiten¨ zu der experimentellen Schwierigkeit, die Schichtdicke kon- stant zu halten. Zum anderen interferiert bei dunnen¨ Schichten das Streusi- gnal der Grenzfl¨ache mit dem Streusignal der Oberfl¨ache. Mit hochenergetischer x DEUTSCHE ZUSAMMENFASSUNG

R¨ontgenstrahlung lassen sich auch tief vergrabene Grenzfl¨achen untersuchen und damit die geschilderten Probleme vermeiden. Aufgrund der kleinen Streuwinkel bei hochenergetischer R¨ontgenstrahlung werden allerdings sehr hohe Anforderungen an die Genauigkeit und Stabilit¨at des gesamten Aufbaus gestellt (besser als ±10 µrad bzw. ±1 µm). Ferner er- fordert der Einsatz von hochenergetischen R¨ontgenstrahlen sehr kleine Strahl- durchmesser am Probenort, da bei den kleinen Streuwinkeln die Projektion der Grenzfl¨ache senkrecht zum Strahl sehr klein ist. Dies wurde in den Experimen- ten fur¨ diese Arbeit mittels Brechungslinsen fur¨ R¨ontgenstrahlen erreicht, die es erlauben, R¨ontgenstrahlen auf einen Durchmesser von wenigen Mikrometern zu fokussieren. Dies hat zudem den Vorteil, dass sich durch entsprechend kleine Blenden¨offnungen der Streuuntergrund weitgehend unterdrucken¨ l¨asst. Dadurch ergibt sich ein sehr großer dynamischer Bereich der Messungen von bis zu 10 Gr¨oßenordnungen. Die in dieser Arbeit angewandten R¨ontgenreflektivit¨atsmessungen [7] erlauben die Rekonstruktion des Dichteprofils senkrecht zur Grenzfl¨ache mit einer Aufl¨o- sung bis in den atomaren Bereich. Dies erm¨oglicht, etwaiges Grenzfl¨achenschmel- zen am Erscheinen einer zus¨atzlichen Schicht im Dichteprofil zu erkennen. Es lassen sich sowohl die Dicke als auch die Dichte der Schicht temperaturabh¨angig verfolgen. Die Experimente wurden in einem Temperaturbereich von −30 bis −0.022◦C durchgefuhrt.¨

0.4 Probenpr¨aparation und Probenumgebung

In dieser Arbeit wurde das Grenzfl¨achenschmelzen von Eis an Eis–SiO2–Si Grenz- fl¨achen untersucht. Diese k¨onnen als Modell fur¨ Eis–Mineral-Grenzfl¨achen, die in der Natur vorkommen, betrachtet werden.

Fur¨ die SiO2–Si Substrate wurde einkristallines Silizium verwendet, das che- momechanisch poliert und anschließend aufwendig chemisch gereinigt wurde. An Luft bildet Silizium dann ein natives amorphes Oxid von ca. 1–2 nm Dicke. Die Substrate sind ursprunglich¨ hydrophob, unter Bestrahlung mit hochenergetischer R¨ontgenstrahlung und im Kontakt mit H2O bildet sich allerdings eine hydrophile Terminierung. Die Hauptexperimente wurden mit zwei Substraten durchgefuhrt,¨ die eine unterschiedliche Oberfl¨achenmorphologie aufweisen, ein glattes“ Sub- ” strat und ein raues“ Substrat (Details s.u.). Damit sollte der Einfluss der Rau- ” igkeit auf das Grenzfl¨achenschmelzen untersucht werden. Eis-Einkristalle wurden von Prof. Bilgram (ETH Zurich)¨ aus hochreinem Was- ser gezuchtet¨ [8]. Mittels eines Zweikreis-Diffraktometers wurden die Eis-Kristalle mit der c-Achse senkrecht zum Substrat ausgerichtet. Beim Kontaktieren mit dem Substrat wurde durch kurzzeitiges Erw¨armen des Substrats die Eisoberfl¨ache auf- geschmolzen. Beim anschließenden Abkuhlen¨ wurde das Substrat st¨andig nach- gefahren, so dass die flussige¨ Schicht immer sehr dunn¨ war. Dadurch wurden et- 0.5. ERGEBNISSE UND DISKUSSION xi waige Verunreinigungen ausgeschwemmt und Lufteinschlusse¨ vermieden, so dass nach dem langsamen Rekristallisieren eine homogene und glatte Eis–Substrat Grenzfl¨ache vorlag. Die gesamte Probenpr¨aparation wurde in einem begehbaren Kuhlraum¨ durchgefuhrt.¨ Fur¨ die eigentlichen R¨ontgenstreuexperimente am Synchrotron wurde eine mobile Probenumgebung konstruiert. Sie erm¨oglicht eine sehr stabile Tempera- turregelung uber¨ Peltier-Elemente, die an zwei Seiten der Probe angebracht sind.

0.5 Ergebnisse und Diskussion

0.5.1 Morphologie der Substrate Die Morphologie der Substrate wurde sowohl anhand der R¨ontgenstreudaten als auch erg¨anzend mittels Rasterkraftmikroskopie untersucht. Der quadratische Mittelwert σ der H¨ohenabweichung fur¨ das glatte Substrat betr¨agt nach den R¨ontgenmessungen (2.7±0.4) A.˚ Die Rauigkeit ist lateral nur schwach korreliert. Beim rauen Substrat dagegen weist die Rauigkeit ein selbst-affines Verhalten auf, wie es oft als Folge von Wachstums- oder Atzprozessen¨ beobachtet wird. Die Messung der diffusen R¨ontgenreflektivit¨at erm¨oglicht es, die lateralen Kor- relationen der Rauigkeit zu bestimmen: g(R) = 0.11 · R2·0.34. In diesem Fall ist σ unbestimmt. Die Analyse der Rasterkraftbilder best¨atigt die Ergebnisse der R¨ontgenmessungen.

0.5.2 Wachstum der quasiflussigen¨ Schicht Sowohl am rauen als auch am glatten Substrat konnte eindeutig das Auftreten von Grenzfl¨achenschmelzen nachgewiesen werden. Das Wachstum der quasiflussigen¨ Schicht kann uber¨ einen weiten Bereich der Temperatur T durch ein logarithmisches Wachstumsgesetz beschrieben werden (s.o.). Im Falle des glatten Substrats ist die Ubereinstimmung¨ mit einem loga- rithmischen Wachstum außerordentlich gut. Beim rauen Substrat dagegen zeigen sich Abweichungen von einem logarithmischen Wachstum. Die Amplitude L0 des Wachstumsgesetzes betr¨agt (3.7±0.3) A˚ beim glatten Substrat und (8.2±0.4) A˚ beim rauen Substrat. Diese Werte liegen im Bereich der Literaturwerte fur¨ die Korrelationsl¨ange von Wasser. Beim Vergleich des glatten und des rauen Substrats ergibt sich eine gute Ubereinstimmung¨ im Bereich tiefer Temperaturen bis ca. −0.7◦C. Danach steigt die Schichtdicke am rauen Substrat st¨arker als am glatten Substrat. Die h¨ochste Schichtdicke betr¨agt 55 A˚ bei −0.036◦C am rauen Substrat gegenuber¨ 27.5 A˚ bei −0.022◦C am glatten Substrat. Das Schichtwachstum am rauen Substrat l¨asst sich auch gut durch ein Potenzgesetz beschreiben. Der Exponent liegt hierbei nahe bei einem Wert von −1/3, wie es fur¨ nicht-retardierte Van-der-Waals-Wech- xii DEUTSCHE ZUSAMMENFASSUNG selwirkungen zu erwarten ist. Das Wachstum am glatten Substrat hingegen l¨asst sich nicht durch ein Potenzgesetz beschreiben. Diese Ergebnisse lassen sich so interpretieren, dass es durch die Rauigkeit zu einem fruheren¨ Ubergang¨ von einem logarithmischen Gesetz zu einem Potenz- gesetz kommt. Fur¨ sehr große Schichtdicken wurde¨ man solch einen Ubergang¨ in jedem Fall erwarten, da dann die langreichweitigen Van-der-Waals-Wechsel- wirkungen dominieren. Wenn das logarithmische Wachstumsgesetz fur¨ das raue Substrat nicht gultig¨ ist, kann die daraus bestimmte Amplitude naturlich¨ nicht mehr mit der Korrelationsl¨ange verglichen werden. Ein besonderes Verhalten zeigt sich bei sehr dunnen¨ Schichtdicken (d.h. bei sehr tiefen Temperaturen). Hier scheint eine dunne¨ Schicht auch noch bei tieferen Temperaturen flussig¨ zu bleiben, als man aus der Extrapolation des logarithmi- schen Wachstumsgesetzes erwarten wurde.¨ Dies ist insofern nicht verwunderlich, als im Bereich sehr dunner¨ Schichtdicken das Kontinuumsmodell, das zur Herlei- tung der Wachstumsgesetze verwendet wurde, nicht mehr gultig¨ ist. Die beobachteten Schichtdicken der quasiflussigen¨ Schicht liegen bei gleichen Temperaturen deutlich unter den beim Oberfl¨achenschmelzen gemessenen Wer- ten.

0.5.3 Struktur der quasiflussigen¨ Schicht

Die R¨ontgenreflektivit¨atsmessungen erlauben auch, die mittlere Dichte ρqll der quasiflussigen¨ Schicht zu bestimmen. Die Werte von 1.20 g/cm3 am rauen und 1.19 g/cm3 am glatten Substrat stimmen gut uberein¨ und sind deutlich h¨oher 3 als die Dichte ρl=1.0 g/cm von normalem Wasser. Es stellt sich also die Frage, wie die Struktur dieser hochdichten quasiflussigen¨ Schicht aussehen k¨onnte. Beim Vergleich mit anderen Wasserphasen zeigt sich, dass die ermittelte Dichte nahe 3 bei der Dichte ρHDA=1.17–1.19 g/cm von hochdichtem amorphen Eis [9, 10] (englisch abgekurzt¨ mit HDA) bei Atmosph¨arendruck liegt. Dies legt eine strukturelle Verwandtschaft mit HDA nahe. Allerdings han- delt es sich beim Grenzfl¨achenschmelzen um ein Gleichgewichtsph¨anomen, was das Auftreten einer metastabilen Struktur fraglich erscheinen l¨asst. In aktuel- len Theorien zur Struktur des Wassers wird jedoch eine der HDA entsprechende hochdichte flussige¨ Form von Wasser (englisch abgekurzt¨ mit HDL) postuliert [1]. Diesen Theorien zufolge werden die (oftmals anomalen) Eigenschaften von Wasser durch Fluktuationen dieser hochdichten und einer ebenfalls postulierten niedrig- dichten (englisch abgekurzt¨ LDL) Form von Wasser bestimmt. Es k¨onnten an der Grenzfl¨ache also Fluktuationen in die postulierte hochdichte Form von Was- ser stabilisiert werden. Ein derartiges Ph¨anomen legen auch andere Experimente an Wasser-Grenzfl¨achen nahe [11, 12]. 0.6. AUSBLICK xiii

0.5.4 Weitere Experimente Ein weiteres Experiment wurden mit einem sehr glatten Silizium-Wafer als Sub- strat durchgefuhrt.¨ Bei diesem Experiment wurde ein anderes Reinigungsverfah- ren fur¨ das Substrat verwendet, das direkt zu einer hydrophilen Terminierung mit einer dicken Oxidschicht fuhrt.¨ Leider machte eine Krummung¨ des nur etwa 0.6 mm dicken Wafers die Messung vollst¨andiger Reflektivit¨atskurven unm¨oglich. Trotzdem konnte das Auftreten von Grenzfl¨achenschmelzen auch bei diesem Sub- strat best¨atigt werden. Andere Experimente wurden mittels Neutronenreflektivit¨atsmessungen, der Standardmethode fur¨ tief vergrabene Grenzfl¨achen, durchgefuhrt.¨ Aufgrund des im Vergleich zu Synchrotronstrahlungsquellen geringen Flusses der Neutronen- quellen sind allerdings nur kleine Impulsubertr¨ ¨age bei Reflektivit¨atsmessungen zug¨anglich. Dadurch ist die erreichbare Aufl¨osung im Realraum zu sehr begrenzt.

0.5.5 Bedeutung der Ergebnisse Die Ergebnisse dieser Arbeit haben Bedeutung fur¨ das Verst¨andnis wichtiger Ph¨anomene in der Natur (s.o.). Die Auswirkungen des beobachteten Grenzfl¨a- chenschmelzens h¨angen von den bisher unbekannten Eigenschaften der quasi- flussigen¨ Schicht ab. In diesem Zusammenhang ist auch die Beobachtung einer hochdichten Form von Wasser bedeutsam, da dies auch Unterschiede in den ande- ren Eigenschaften, wie z.B. der Viskosit¨at oder der L¨oslichkeit von Verunreinigun- gen, nahe legt. Da in der Natur vorkommende Grenzfl¨achen normalerweise rau sind, ist der beobachtete Einfluss der Substratmorphologie wichtig fur¨ Schluss- folgerungen uber¨ reale“ Systeme. Außerdem k¨onnen Grenzfl¨achenexperimente ” wie sie in dieser Arbeit durchgefuhrt¨ wurden, neue Einblicke in die Struktur von Wasser liefern, insbesondere, wenn tats¨achlich die postulierten Wasserformen an Grenzfl¨achen stabilisiert werden k¨onnen.

0.6 Ausblick

Aus dieser Arbeit ergeben sich weitere Fragen und m¨ogliche Forschungsthemen. Darunter befinden sich die Frage nach dem Verhalten von Eis und Wasser im Kontakt mit weiteren Materialien und die Frage nach dem Einfluss von Verun- reinigungen. Dazu geh¨ort auch die Frage, ob die beobachtete hochdichte Form von Wasser nur in dem dunnen¨ Spalt zwischen Eis und dem verwendeten Sub- strat stabilisiert werden kann, oder ob dies auch an der Grenzfl¨ache zu flussigem¨ Wasser m¨oglich ist. Zukunftige¨ Experimente k¨onnten auch einen gr¨oßeren Temperaturbereich ab- decken. Dann k¨onnte das Verhalten nahe des Schmelzpunktes und ein m¨oglicher Wechsel des Schichtwachstums zu einem Potenzgesetz untersucht werden. Dies wurde¨ allerdings eine Verbesserung der Temperaturstabilit¨at voraussetzen. Zum xiv DEUTSCHE ZUSAMMENFASSUNG anderen k¨onnte das Einsetzen des Grenzfl¨achenschmelzens bei tiefen Temperatu- ren genauer untersucht werden. Die wohl spannendste aber zugleich schwierigste Aufgabe wird darin beste- hen, die Struktur der quasiflussigen¨ Schicht zu bestimmen. Hierzu reichen Reflek- tivit¨atsmessungen nicht aus, da sie prinzipiell nur erlauben, die lateral gemittelte Struktur zu untersuchen. Eine M¨oglichkeit bietet die evaneszente Braggstreuung [13], allerdings w¨are in dem vorliegenden Fall eine Trennung der Streusignale von der Quasiflussigkeit¨ und der amorphen Oxidschicht kaum m¨oglich. Hier musste¨ ein kristallines Substrat verwendet werden. Selbst dann bleibt eine Messung mit hochenergetischer R¨ontgenstreuung aufgrund der extrem kleinen Streuwinkel sehr schwierig. Chapter 1

Introduction

Much of the current research in condensed matter physics is not devoted to the the understanding of bulk materials, but matter in confinement and reduced di- mensions. This includes nanoparticles, thin films, and all kinds of interfaces. In all of these situations, the properties of the materials involved can differ drasti- cally from the bulk. The special case of the free surface has been studied in detail over the last decades, and x-ray scattering techniques have made a significant contribution to the understanding of surface structures. The center of interest has now moved to solid–solid, solid–liquid, and liquid–liquid interfaces. Such interfaces are of great technological interest (for example electrode–electrolyte interfaces) and play an important role in other disciplines (for example biology). Deeply buried interfaces, however, are difficult to probe experimentally. The ideal probe would allow non-destructive in situ measurements with a resolution on the atomic scale. X-rays meet all these requirements, but usually lack the necessary penetration depth. In this work, a recently developed scheme has been used, which exploits the properties of high-energy x-ray microbeams. It allows to apply the established surface-sensitive x-ray scattering techniques to study deeply buried interfaces. Only modern Synchrotron Radiation sources can provide beams with the brilliance and stability required by this scheme. Water and ice are not only of paramount importance for the , but also exhibit a large number of surprising properties, which are still not fully understood. At the free surface of ice, a phenomenon called ‘surface melting’ oc- curs. It is the formation of a liquid-like layer below the bulk melting temperature. There are many indications that an analogous effect exists at ice–solid interfaces. Because of the experimental difficulties in probing such interfaces, little is known about this ‘interface melting’, which might depend on the morphology of the interface, for example.

In this work, model interfaces of ice in contact with SiO2 have been stud- ied. Measurements of the x-ray reflectivity reveal the density profile across the interface. These measurements allow to observe premelting layers on nanoscopic

1 2 CHAPTER 1. INTRODUCTION length scales, and to determine their thickness and density as a function of tem- perature. SiO2 substrates with different morphology have been used to determine the influence of the roughness. Other projects in the context of this thesis include ordering and segregation at CuPd-surfaces1 and Neutron Compton Scattering experiments on ice2. These projects have some relation to the work on interface melting, but are beyond the scope of this dissertation. For the bigger part, the organization of this dissertation should be self-ex- planatory, but sometimes a justification is given where I deemed it helpful. Each chapter starts with a short overview of its content.

1in collaboration with H. Reichert, C. Mocuta, W. Schweika, and H. Dosch 2in collaboration with H. Reichert, J. Mayers, G. Reiter, J. Bilgram, and H. Dosch Chapter 2

Ice and water

The purpose of this chapter is to provide a short overview of the and ice. It will concentrate on aspects which are relevant for this work. Sec. 2.1 explains the importance of water and ice, Sec. 2.2 introduces the H2O molecule as the building block of the water and ice structure, and Sec. 2.3 highlights the anomalous properties of H2O. The complex phase behavior of H2O is part of the mystery and in the focus of current water theories trying to explain the anomalous properties of H2O. These issues will be discussed in Sec. 2.4; for more information see [1] and references therein. The chapter closes with a description of the ice structure in Sec. 2.5. A more detailed description, including crystallographic data, can be found in [14]. For a general overview of ice physics and chemistry, the reader is referred to the work of Petrenko and Whitworth [14], Hobbs [15], and Whalley [16]. More information about water can be found in the books of Franks [17] and Ball [18].

2.1 Importance

Water in its various forms is virtually omnipresent on the surface of the Earth, it touches nearly all aspects of our everyday life, and without water, life would not even exist. Ice and water shape the surface of our planet. 70% of the Earth is covered by oceans, 10% of the land mass is currently covered by ice (up to 30% were covered during the Earth’s history), and around 5% of the oceans are covered with ice, depending on the season. Retreating oceans have left their sediments, rivers cut deep valleys and sculpt the landscape. The climate crucially depends on the presence of ice, water, and vapor. , ice, and cloud cover determine the balance of radiation received and reflected/emitted by the Earth. Evaporation of water, snowfall, and subsequent flow (see Sec. 3.5.2) of polar ice back into the oceans form another delicate equi- librium. Ice in the atmosphere is important for the production of rain and for the

3 4 CHAPTER 2. ICE AND WATER

0.9572 Å

104.52°

Figure 2.1: Free water molecule made up of one (large sphere) and two (small spheres). Note the bent shape, which gives rise to an electric dipole moment. The H—O—H bond angle (104.52◦) is close to, but not exactly the tetrahedral angle (109.47◦). scavenging of atmospheric pollutants. It affects the chemistry of the atmosphere, as for example the reactions responsible for ozone depletion (see Sec. 3.5.3). Ice particles also play a great role in the electrification of thunderstorms (see also Sec. 3.5.3). Under certain conditions, atmospheric ice reaches the ground in the form of , often causing considerable damage. Ice is of great relevance for buildings and infrastructure ( heave, ice on power lines, freezing pipes, avalanches), traffic (slippery and snow covered roads, ice on airplane wings, and pack ice disturbing shipping traffic), and agriculture (freeze damage). The effects on constructions and infrastructure es- pecially concern the permafrost regions (see Sec. 3.5.1), which cover some 20% of the land mass on the northern hemisphere. Whereas the effects of ice often cause problems, we admire its beauty in the form of snowflakes, , and frost patterns on windows. We enjoy skiing and , and we use ice to preserve food and to cool drinks. Before the advent of refrigerators and freezers, ice was an important trading good [19, 20, 21]. Ice can also be found throughout the universe. Tiny particles in cold areas of interstellar space are covered by thin ice layers. In the solar system, ice is present on moons and comets. For many of the aspects mentioned here, interface melting of ice is essential (see Sec. 3.5).

2.2 The H2O molecule and the

Water and ice are made up of H2O molecules bound together by hydrogen bonds. The arrangement of the oxygen and the two hydrogens is shown in Fig. 2.1. For a free molecule, the O—H distance is (0.9572±0.003) A˚ and the H—O—H angle is (104.52±0.05)◦ The bent shape is a consequence of the ground state. It gives rise to an electric dipole moment of (6.186±0.001)×10−30 Cm. It also defines the possible arrangements of molecules in the structures. Hydrogen bonds are a distinct type of chemical bond where a hydrogen atom 2.3. ANOMALIES AND MYSTERIES 5 sits between two highly electronegative atoms (F, O, N). If the highly electroneg- ative atoms are , the bond can be represented as O—H··· O. The hy- drogen atom stays covalently bound to one of the oxygen atoms (O—H), the proton ‘donor’. The distance to the proton ‘acceptor’ is much larger (O··· ). The strength of the hydrogen bond is between that of covalent bonds and Van der Waals interactions. Hydrogen bonds are crucial for the properties of water, but difficult to account for in calculations. As every H2O molecule can act as a proton donor for two hydrogen bonds and as a proton acceptor for two additional bonds, complex networks of hydrogen bonds can form. They also play an important role in biochemistry, where nearly all processes take place in aqueous environments.

2.3 Anomalies and mysteries

Water in its various forms has always evoked interest and fascination in many fields, and has been in the focus of scientific research since its beginnings. Despite the importance of water and the amount of research dedicated to its understand- ing, water still holds unsolved mysteries [1]. The elusive simplicity of the water molecule (see Sec. 2.2) contrasts with the complex behavior of water, its unusual and all too often counterintuitive proper- ties, and its large number of solid phases (see Sec. 2.4) Therefore, the scientific interest in water stems not only from its relevance in nature and technology, but also from the fundamental questions it poses for condensed matter physics. The anomalous properties of water are still not fully understood, but it is those very anomalies that are responsible for the importance of water. One of the anomalous characteristics of water is the behavior of its response functions (see Fig. 2.2) like the isothermal compressibility KT , the isobaric heat capacity CT , and the thermal expansion coefficient αT . Their magnitude increases sharply upon cooling (from a certain point on). When water is cooled further and eventually supercooled, the increase becomes even more pronounced. When extrapolated, the response functions seem to diverge at a singular temperature TS = 228 K [2]. In a typical liquid, all the mentioned response functions would decrease slightly upon cooling (see Fig. 2.2). Adding to these oddities, the co- efficient of thermal expansion in water changes its sign at 4◦C, which expresses the anomalous and well known density maximum at 4◦C. Furthermore, the vis- cosity of water decreases and its diffusivity increases upon compression, again in contrast to typical liquids. Water can only be supercooled down to about TH = 231 K, the homogenous nucleation temperature, before it crystallizes to ice Ih. Therefore, its behavior at TS cannot be probed directly by experiments. 6 CHAPTER 2. ICE AND WATER

T

C

P

T

a

K Tm

Isobaric

Isothermal 277 K

Coefficient of

compressibility

heat capacity Temperature

Tm 319 K Tm 308 K thermal expansion Temperature Temperature Water normal liquid

Figure 2.2: The anomalous response functions of water. The response functions of water (solid lines) increase strongly in magnitude upon cooling and supercooling. The response functions for a ‘normal’ liquid are shown for comparison (dashed lines). The coefficient of thermal expansion of water changes its sign from negative to positive at 277 K (4◦C), which is equivalent to a density maximum at this temperature.

2.4 The quest for the water structure

There are at least 13 different crystalline forms of water of which 9 are stable (see Fig. 2.3). Only the ‘ordinary’ ice Ih is stable at atmospheric pressure. In addition, there are several amorphous forms, a behavior called polyamorphism. Glassy water was first produced by depositing water vapor onto a cold metal plate [22]. Direct vitrification of the liquid by rapid cooling (hyperquenching) was later achieved by Br¨uggeler and Mayer [23, 24]. After annealing, those forms of glassy water relax to low-density (LDA). When ice is compressed to about 11 kbar at 77 K, high-density amorphous ice (HDA) is formed [9, 10]. Both LDA and HDA can be recovered at atmospheric pressure with the densities 3 3 ρLDA = 0.94 g/cm and ρHDA = 1.17–1.19 g/cm respectively. LDA and HDA have distinct structures manifested by their large density difference. What distin- guishes them from common glassy states, is a sharp and reversible transformation between the two forms at about 2 kbar and 135 K [25], which is characteristic for a thermodynamic phase transition, as is the large change in density. There are currently two different conjectures for a coherent theory of water which could explain its anomalous properties and their relation to the amorphous states: the liquid-liquid phase transition hypothesis [26] and the singularity-free scenarios [27]. Both share the idea that LDA and HDA are the vitreous counter- parts of two different forms of liquid water, a low-density liquid phase (LDL) and a high-density liquid phase (HDL). According to the liquid-liquid phase transition hypothesis, the transition between the two liquids is of first order and terminates at a critical point (Tc ≈ 220 K, pc ≈ 1 kbar, see Fig. 2.3a) below the bulk melting point. At higher temperatures, the two phases become indistinguishable, and the 2.4. THE QUEST FOR THE WATER STRUCTURE 7

310 ab 290 Liquid 270 600 250 Ih III V VI 230 (0.92) Liquid C 210 II VII 400 190 III V 170 LDL X 150 HDL VI

Temperature (K) Temperature (K) Ih 200 130 II VIII 110 LDA HDA 90 XI ? (0.94) (1.17) 0 0 200 400 600 800 0.1 1.0 10 100 Pressure (MPa) Pressure (GPa)

Figure 2.3: H2O phase diagram compiled from [14, 31]. (a) Phase diagram for moderate pressures showing the stable phases, the metastable amorphous forms HDA (high-density amorphous) ice and LDA (low-density amorphous) ice, as well as the postulated corresponding liquid phases HDL (high-density liquid) water and LDL (low-density liquid) water. HDL and LDL terminate at a second critical point C (see the text). The numbers in brackets denote the density at ambient pressure in g/cm3. (b) Phase diagram for very high pressures showing the stable phases. The dotted green box marks the region shown in a. characteristics of water are determined by the coexisting fluctuations of these two states. In the singularity free scenarios, the transition between the two liquids is continuous, and the response functions show strong maxima, but do not di- verge. Owing to the supercooling limit mentioned before, the two suggestions are difficult to probe directly. There are experiments in support of the liquid-liquid phase transition hypothesis [28, 29] as well as the singularity-free theories [30], and no consensus has emerged so far. Up to now, the LDL and HDL phase of water have not been observed experimentally. It has also been suggested that there is more than one liquid-liquid transition in water [32] and that HDA is not even a glass in the sense of a quenched liquid, but rather a poorly crystalline form of ice, so the quenched liquid might look different [33]. The role of the recently discovered very-high-density amorphous 3 ice (VHDA, ρVHDA = 1.25 g/cm at atmospheric pressure) [34, 35] is not yet clear either. Although the possibility of different forms of liquid water may sound rather speculative, liquid-liquid transitions have been experimentally observed in other materials, for example phosphorus [36]. Recent theoretical work indicates that liquid-liquid transitions might actually be a rather generic phenomenon [37]. 8 CHAPTER 2. ICE AND WATER

[112 0]

[0001] [1 100]

Figure 2.4: The structure of ice Ih. View along the [0001]-direction (c-axis). The oxygen atoms are represented by large spheres, the hydrogen atoms by small spheres. The connecting lines represent the hydrogen bonds. The oxygen atoms follow the ‘wurtzite’ structure. Note the puckering of the hexagonal rings and the open channels formed along the c-axis. Note also the disorder in the distribution of the hydrogen atoms, which follows the ice rules (see the text).

It is commonly accepted that the highly directional (tetrahedral) hydrogen bonding is to a large extent responsible for the behavior of water. (It causes the tendency of the ordered states to have a higher specific volume, e.g.) Model calculations were in fact able to exhibit both scenarios, the liquid-liquid transition and the singularity-free scenario, by altering the geometrical constraints of the bonding [38].

2.5 Ice Ih

The structure of ice Ih is illustrated in Fig. 2.4. The oxygen atoms are arranged on a hexagonal lattice. The lattice parameters are a = 4.519 A˚ and c = 7.357 A˚ at −20◦C [39]. The arrangement of the oxygen atoms follows the ‘wurtzite’ struc- ture. Each oxygen atom is tetragonally (O—O—O angle of 109.47◦) surrounded by 4 nearest neighbors. They form layers of puckered hexagonal rings perpen- dicular to the c-axis. The stacking of these layers has the sequence ABABAB... known from hexagonal close-packed metals. The arrangement of the hydrogen atoms follows the model proposed by Paul- ing [40]. There are two possible hydrogen sites on each line between neighboring oxygen atoms, and the distribution of the hydrogen atoms satisfies the two ice rules: 1. There are two hydrogens adjacent to each oxygen. 2.5. ICE IH 9

2. There is only one hydrogen between two neighboring oxygens.

The crucial point in Pauling’s model is the lack of long-range order in the occu- pation of the two possible hydrogen sites on each bond, therefore, the structure of ice Ih has an intrinsic disorder (see Fig. 2.4). This hydrogen disorder leads to an excess entropy [41]. On average, each hydrogen site is occupied by half a hydro- gen atom. The space group for this average structure is P63/mmc. The oxygen atoms are covalently bonded to their two adjacent hydrogen atoms forming H2O molecules (see rule 1). These molecules are connected via hydrogen bonds. Each line between two oxygen atoms represents such a hydrogen bond, and either of the two H2O molecules can provide the hydrogen (see rule 2). The disorder in the occupation of the hydrogen sites can thus also be seen as a disorder in the orientation of the H2O molecules. The lattice parameters were first determined correctly by Dennison [42] using x-ray diffraction. Based on theoretical considerations, Bragg [43] then proposed a structure with the correct position of the oxygen atoms and the hydrogen atoms halfway between the oxygen atoms. But already the question was raised whether the hydrogen atoms were shifted from the center (thus destroying the symmetry of the system). At this time, however, x-ray scattering experiments were not able to resolve the position of the hydrogen atoms due to their weak contribution to the scattering signal. Single crystal diffraction experiments by Barnes [44] affirmed the arrangement of the oxygen atoms proposed by Bragg. Bernal and Fowler [45] suggested that the water molecule as shown in Fig. 2.1 would stay intact in ice, which was backed by the similarity of the Raman spectra of water, ice, and vapor. Such molecules, however, cannot be arranged on the sites of the observed unit cell without destroying the hexagonal symmetry. The simplest structure retaining the hexagonal symmetry requires a unit cell 3 times larger as the one proposed by Barnes. Such a structure would be polar, but this was deemed more probable than the smallest non-polar structure, which would have an extremely complicated and large (96 molecules) unit cell. After Giauque and Ashley [41] found out by experiments that ice had an excess entropy, Pauling finally proposed his model (presented above) supposing that no particular ordering of the H2O molecules was stabilized (at least at ordi- nary temperatures). Pauling also calculated the entropy of its proposed structure which is in good agreement with the experimental value of Giauque and Ashley. Neutron diffraction allowed the first crystallographic study of the hydrogen posi- tions. Powder diffraction experiments by Wollan et al. [46] agreed with Pauling’s model and ruled out several others. Final confirmation came from a single crys- tal neutron diffraction experiment by Peterson and Levy [47]. An equivalent x-ray experiment, which was later performed by Goto et al. [48], agreed with the neutron experiment. X-rays, however, are sensitive to the electron density distribution, whereas neutrons are sensitive to the distribution of the nuclei. It was thus possible to 10 CHAPTER 2. ICE AND WATER detect the deviation between the position of the hydrogen nuclei and the center of the electron distribution, which is shifted towards the oxygen atoms. Chapter 3

Interface melting

Several scenarios can lead to premelting, i.e. the formation of a (quasi)liquid equilibrium phase in the solid region of the bulk phase diagram. Among those scenarios are interfaces (as the surface, solid–solid interfaces, and grain bound- aries) and more complex confinement situations (like in small particles and porous media). The effect of an interface may depend on its chemical composition, cur- vature, and roughness. It is difficult to separate the contribution of these various mechanisms in the more complicated situations like porous media. Supercooled liquids do not fall in the category of premelting, since they are in a metastable state. Also, premelting does not include the reduction of the bulk melting temperature due to dissolved impurities or change of pressure. In the case of premelting caused by the influence of an interface, the effect is called interface melting, or also interfacial melting. If this interface is the solid– vapor, solid–vacuum or solid–air interface, the effect is called surface melting (see Fig. 3.1a). Surface melting is thus a special case of interface melting and the underlying theory is the same. Heterogenous solid–solid interfaces (see Fig. 3.1b), which are the focus of this work, represent another class of interfaces. The term interface melting in the narrower sense refers to these interfaces. Experiments on interface melting at well-defined solid–solid interfaces are very rare due to the difficulties in probing deeply-buried interfaces. Premelting occurs in all types of materials and is quite pronounced in the case of ice, where it has also important implications for many environmental and technical processes (see Sec. 3.5). A short section (3.1) in this chapter deals with the melting process in general. The following sections present the theory (3.2) and experiments (3.3) related to interface melting of various materials, whereas a separate section (3.4) is dedi- cated to the interface melting of ice. The vast majority of the literature concerns surface melting, and only a part of the work can be presented here. The emphasis will be on the much smaller number of studies on interface melting at solid–solid interfaces. The last section in this chapter (3.5) deals with the consequences of ice premelting.

11 12 CHAPTER 3. INTERFACE MELTING

a surface meltingb interface meltingc permafrost

vapor solid quasiliquid qll quasiliquid qll qll ice ice

Figure 3.1: Interface melting scenarios for ice. (a) Surface melting of ice. (b) Interface melting of ice at a heterogenous ice–solid interface (interface melting in the narrower sense). (c) Permafrost shows an abundance of such ice–solid interfaces.

This chapter cannot cover all aspects of interface melting and the related lit- erature. For a review and more details, see for example [49, 4] (focus on surface melting experiments), [50], [51], [52, 53] (focus on theory, metal surfaces and in- terplay with other surface phenomena), [54, 55] (surface melting and roughening), [5] (premelting of ice and environmental consequences).

3.1 The melting transition

Melting and the reverse process of freezing are among the most prominent and dramatic phase transitions. The melting of ice may be the most important phase transition on Earth. Thermodynamics provides a description in terms of the Gibbs free energy G(p, T ). It is a continuous function of p and T during the transition, whereas other thermodynamic quantities such as the volume V or the entropy S undergo discontinuous changes. Nearly all materials expand upon melting (∆V > 0) with a few exceptions, among them Sb, Bi, Ga, silica—and ice. With the only exception of He, the entropy increases upon melting (∆S > 0), melting is thus a disordering transition. The relevant order parameters for the melting transition are the Fourier com- ponents of the density, measured directly by the Bragg scattering intensities. Melting is a first-order transition, i.e. the order parameter changes discontinu- ously at the transition. It is characterized by latent heat and coexistence of the solid and the liquid phase at the transition. Thermodynamics provides little information about the mechanism of melting and its kinetics. Several theoretical approaches have been developed to gain a microscopic understanding of the melting process. Most of them start from either the liquid, or the solid phase, but melting involves both. In the liquid-based approach, density functional theory is used to describe 3.2. THEORY OF INTERFACE MELTING 13 freezing as a condensation of liquid density modes. Whereas this approach has allowed to gain microscopic insight into the freezing process, the construction of the functional is ad hoc and the positional order of the solid not a result of the calculation, but an input. The properties of the solid are not exactly reproduced, either. Solid-based theories focusing on lattice stability provide some useful phe- nomenological criteria for melting. The Lindemann criterion [56] states that melting sets in, when the root-mean-square displacement ph(∆r)2i reaches about 15% of the interparticle distance. The underlying model describes melting in terms of individual atomic properties and ignores the cooperative character of this phase transition. Nevertheless, it provides a quasi-universal empirical esti- mate for the melting transition. It has later been generalized by Ross [57]. The Born criterion [58] links the melting transition to the decrease of the shear elastic moduli, which finally leads to a mechanical instability of the solid structure. It was later modified to reach better agreement with experimental data [59] and to incorporate contributions from external stress [60]. Another class of solid-based theories concentrates on structural defects like vacancies [61, 62, 63] or dislocations [64, 65], but today it seems clear that defect generation is not the mechanism for bulk melting. Despite its ubiquity, the microscopic mechanism of melting is still not fully understood and subject of current research (for example [66]). It is now commonly accepted that surfaces and interfaces play a great role for the melting process, as will be explained in the next section.

3.2 Theory of interface melting

We consider the case of an interface between a solid s and another medium b. When the temperature of the system approaches the bulk melting point Tm of the solid s, a thin premelting layer of thickness L can intervene between s and b (see Fig. 3.2). This phenomenon is called interface melting. The structure of the strongly confined premelting layer may differ from the bulk liquid phase of s, therefore, it is usually referred to as the quasiliquid layer (qll). This ‘liquid embryo’ may serve as a nucleation site for the bulk melting. The interface would act as a large natural defect initiating the melting process. It has been argued that this could be the reason for the difficulties in superheating , while most liquids can be supercooled (every real solid has at least one interface, its surface). This idea is supported by the fact that solids can be superheated under certain circumstances, namely when special coatings are applied to change the surface properties. Interface melting starts at a certain onset temperature T0. Further increase of the temperature leads to the growth of the quasiliquid layer thickness L. As the bulk melting point is reached, L diverges: L → ∞ for T → Tm. Interface 14 CHAPTER 3. INTERFACE MELTING

abTnTm T.Tm

medium b medium b L quasiliquid qll solid s solid s

Figure 3.2: Interface melting of a solid s in contact with another medium b. (a) Shows the situation far below the bulk melting temperature Tm. (b) Close to, but still below the bulk melting temperature Tm, a liquid-like (quasiliquid) layer qll might intervene at the interface between the solid s and the medium b. The thickness L(T ) of this layer is determined by the competition between the possible reduction of the interfacial free energies γ and the energy needed to transform the layer from the solid to the quasiliquid state. melting can be considered as a special case of a wetting transition, where a solid is wetted by its own melt. There are, however, cases of incomplete wetting or blocked interface melting, where L remains finite up to the melting point: L → Lm for T → Tm. The driving force for interface melting is the minimization of the free energy of the system.

3.2.1 Phenomenological description The simplest approach to interface melting is a phenomenological thermodynamic model. The description presented in this section is based on a continuum model. It is not applicable for very thin quasiliquid films of less than a few molecular layers. We calculate the free energy per unit area for the system shown in Fig. 3.2 with an intervening quasiliquid layer of thickness L at the temperature T :

 Tm − T  ρqllQmL + γs−qll + γqll−b + P (L) for L > 0,  T F (L) = m (3.1)  γs−b for L = 0.

Here, ρqll is the density of the quasiliquid and Qm denotes the latent heat of melting. The interfacial energies γs−qll, γqll−b, and γs−b are not known, in general, and difficult to determine experimentally. P (L) represents the inter- action between the two interfaces (s–qll and qll–b) and can be considered as a thickness-dependent correction to the interfacial energies. 3.2. THEORY OF INTERFACE MELTING 15

Interfacial melting occurs, if the free energy F has its minimum at a finite thickness L > 0 at a temperature T < Tm. Minimizing F with respect to L yields the growth law L (T ) of the quasiliquid layer. In order to minimize F , the term P (L), which depends on the nature of the molecular interactions, has to be known. If we assume that exponentially decaying short-range forces are dominating, the phenomenological expression

P (L) = −Ae−L/L0 (3.2) can be used, where L0 is a correlation length of the quasiliquid, and A defines the strength of the interactions.1  ∂(∆F )  Minimization of the free energy ∂L = 0 yields the equilibrium thickness of the quasiliquid layer   TmA L (T ) = L0 ln , (3.3) ρqllQmL0 (Tm − T ) which can be rewritten as   Tm − T0 L (T ) = L0 ln (3.4) Tm − T with the onset temperature

 A  T0 = Tm 1 − . (3.5) ρqllQmL0

For T < T0, where the argument of the logarithm in Eq. 3.4 is negative, the quasiliquid layer is unstable and L = 0. The logarithmic growth law (Eq. 3.4) is characteristic for short-range forces. For dominating long-range Van der Waals type dispersion forces, P (L) has the form W (3.6) Ln with n = 2 for non-retarded and n = 3 for retarded Van der Waals forces. W is the Hamaker constant. Minimization of the free energy then yields an algebraic growth law (power law) for W > 0:

 1/(n+1) nW Tm p L (T ) = ∝ (Tm − T ) (3.7) ρqllQm (Tm − T )

1 Assuming F to be continuous at L = 0 implies A = −γs−qll − γqll−b + γs−b, see for example [49]. But for L → 0, i.e. layers with a thickness of about the molecular diameter, the continuum approach presented in this section is not valid anyway. 16 CHAPTER 3. INTERFACE MELTING with the exponent p = −1/(n + 1). If W is negative, the Van der Waals inter- actions lead to blocked melting, since long-range forces dominate from a certain layer thickness on. The Hamaker constant W is a measure for the strength of the Van der Waals interaction. In the case of surface melting (at the free surface), the Hamaker constant can be approximated by [67] π W = λ6 (ρ − ρ ) ρ (3.8) 12 s l l for non-retarded pair interactions decaying as − (r/λ)−6. The densities of the solid and liquid phase are ρs and ρl, respectively. The Hamaker constant at the free surface has the same sign as the density difference between the solid and the liquid phase. This implies that blocked surface melting should occur when the liquid phase has a higher density then the solid phase. It should be noted that Eq. 3.6 is no longer valid for small L, i.e. W is not a constant anymore. A detailed discussion of dispersion forces and the Hamaker constant can be found in [68]. As the thickness of the quasiliquid layer increases with temperature, short- range forces get damped, while Van der Waals forces become more important. This can lead to a cross-over from a logarithmic to an algebraic growth law (as observed for the premelting of Ne films [69]). For even larger values of L, there can be another cross-over from non-retarded to retarded Van der Waals forces. Fig. 3.3 shows model calculations of the free energy for different types of interactions and the associated growth laws. The various contributions to the free energy are illustrated in Fig. 3.4. Interfaces can also induce layering in liquids. Such layering has been observed at liquid surfaces (see for example [70]) as well as at solid-liquid interfaces [71, 72], and is expected to play a role in interfacial melting, since the quasiliquid is strongly confined between two solids. In order to include such layering effects, P (L) has to be complemented by terms of the form

−L/a b cos (k1L) e . (3.9) This expresses the preference for layer thicknesses which are a multiple of the nearest-neighbor distance 2π/k1 of the particles in the quasiliquid.

3.2.2 Landau-Ginzburg models The expressions for the interfacial free energy as presented in Sec. 3.2.1 can also be derived by considering Landau-Ginzburg models. These models are still phe- nomenological in the sense that they do not provide a microscopic description which would allow to calculate the interfacial free energies in Eq. 3.1. In the framework of Landau theory, interface melting is a special case of an interface- induced disordering transition in a semi-infinite system with a first-order bulk 3.2. THEORY OF INTERFACE MELTING 17

a growth-laws b free energy

100 AW 1 C A: =1, =0 4x10 C AW A short-range forces 80 B: =0, =2 AW B VdW forces A C: =1, =2 1 60 D:AW =1, =-1 10 C cross-over D blocked melting (Å) (arb. units) L 40 D 0

F 10 A 20 B D 0 -2 B 10 10-4 10-3 10-2 10-1 100 101 102 0 1020304050 T -T L m (K) (Å)

Figure 3.3: Free energy calculations and growth laws for different types of interactions with ρqllQm = 1 (energies in arbitrary units), Tm = 273.15 K, γs−qll + γqll−b = 0, L0 = 5 A.˚ Several cases are considered here. A: only short- range forces, A = 1, W = 0, which leads to a logarithmic growth law. B: only Van der Waals forces, A = 0, W = 2, which leads to a power law growth. C: both short-range and Van der Waals forces, A = 1, W = 2, which leads to a cross-over from a logarithmic to a power law. D: both short-range and Van der Waals forces, but a negative Hamaker constant, A = 1, W = −1, which leads to blocked melting. (a) Growth law L (T ). (b) Free energy F (L) for Tm − T = 1 K. In case D the calculated free energy goes to −∞ for L → 0, which does not correspond to reality, of course. The approach is simply not valid for very thin films (see the text). 18 CHAPTER 3. INTERFACE MELTING

2 0.5 10 a b 101 melting short-range forces 0.4 melting 100 VdW forces short-range forces 0.3 VdW forces 10-1 -2 0.2 (arb. units) (arb. units) 10 F F 10-3 0.1 10-4 0.0 0 1020304050 0 1020304050 L (Å) L (Å)

Figure 3.4: Contributions to the free energy from melting (solid line), short- range forces (short dashed line), and Van der Waals forces (long dashed line). The parameters for this calculation are the same as in Fig. 3.3, case C. (a) Logarithmic plot, the Van der Waals forces can be seen to dominate over the exponentially decaying short-range forces for large L. (b) Linear plot. transition. This theoretical approach was introduced by Lipowsky [73] and fur- ther developed in numerous papers [74, 75, 67, 76, 77, 78, 79, 80]. Most of the models were initially developed for explaining surface phenomena, this is why the established terms refer to surfaces, like ‘surface induced order’ for example. Although the models are here applied to interfaces in general, the established terms will be used. The presence of an interface strongly influences the phase behavior of a phys- ical system. A system which undergoes a first-order bulk transition can show several types of interface transitions, in particular ‘surface induced order’ (SIO) and ‘surface induced disorder’ (SID). In the case of SIO, the interface remains ordered up to an interface transition temperature TSIO>Tbulk. In the case of SID, the order parameter at the interface vanishes continuously on approaching Tbulk. Interface melting is a special case of SID, which should in principle be described by a multi-component order parameter [77, 79]. A layer of the (nearly) disordered phase grows from the interface into the ordered bulk. The thickness of this layer follows the growth laws presented in the previous section. In the case of dominating short-range forces this is the logarithmic growth law from Eq. 3.4. The prefactor L0 in this equation is also called ‘growth amplitude’ and related to a decay length a of the system. This can be [78, 81]

• a decay length aOP of an order parameter (OP) density, in this case a L = OP , (3.10) 0 2 3.2. THEORY OF INTERFACE MELTING 19

• a decay length aNO of a non-ordering (NO) density, in this case 1 L = a (without the factor ). (3.11) 0 NO 2 This decay length can then be compared with a corresponding correlation length of the system.

3.2.3 Density functional theory For a Lennard-Jones (LJ) system near the , surface melting could be observed with density functional theory (DFT) [82]. Such a LJ system may serve as a model for rare gases. The density functional used in this first study, however, does not provide a phase diagram in agreement with simulations. A subsequent study applied a better weighted-density approximation (WDA) functional [83]. It showed complete surface melting for the fcc-crystal–gas interface of LJ sys- tems. The thickness of the premelting layer varied with surface orientation. Its temperature dependence followed a logarithmic growth law.

3.2.4 Lattice theory Trayanov and Tosatti have developed a microscopic lattice theory of surface melt- ing [84, 85, 86]. It is based on the minimization of the free energy with respect to density and ‘crystallinity’ as spatially varying order parameters. By introducing a discrete reference lattice and applying mean-field and free-volume approxima- tions, their approach allows to calculate the partition function of the system. It was applied to the case of the (100) and (110) LJ crystal surfaces. The density and crystallinity profiles show a rather abrupt jump at the interface between the solid and the quasiliquid. It might, however, be due to the mean-field (MF) approximations which suppress fluctuations. The layer thickness of the quasiliq- uid was calculated from the density profiles and follows an algebraic growth law with the exponent −1/3. A change from long-range to short-range interactions changes the growth law from algebraic to logarithmic. Switching the long-range tail of the interactions from attractive to repulsive leads to blocked surface melt- ing. The difference between different surface orientations (anisotropy) was shown to diminish with temperature.

3.2.5 Other approaches Other approaches include phonon theory [87]. In this approach, it was found that the surface becomes unstable with respect to melting before the bulk, and that this instability then proceeds into the bulk. The approach was applied to the case of copper [88, 89], where it was also used to explain the dependence on the surface orientation. 20 CHAPTER 3. INTERFACE MELTING

3.2.6 Molecular dynamics simulations

Molecular dynamics simulations were used extensively to study surface and inter- face melting, as well as melting in other confinement situations like nano-particles. The first simulations were performed for LJ systems [90], for a review see [91]. They were then extended to metals, where a lot of experimental data was avail- able. In this case, additional difficulties for the simulations arise due to the presence of many-body interactions. The simulations on metals include Au [92], Cu [93], Ni [94, 95] and Al [96, 97]. Semiconductors (for example Si [98]) and oxides (for example Cr2O3 [99]) have been investigated as well.

3.2.7 Interfacial melting and substrate roughness

The influence of substrate roughness and curvature on wetting phenomena has been studied in detail, see for example [100, 101, 102, 103, 104, 105] and references therein. The question is not yet completely solved, which is illustrated by the number of recent publications. The effect of roughness can be dramatic and even change a non-wetting to a wetting scenario (roughness-induced wetting [103]). In general, roughness affects the wetting behavior in quite different ways. Its first consequence is the increase of the effective interface area. This obvious ef- fect can be expected to amplify the general tendency with respect to interfacial melting. If the interfacial energy is lowered by an intervening quasiliquid layer, it is so even more when the interface area is larger. However, the roughness of the substrate is not necessarily replicated at the interfaces between different layers on top of the substrate. In the case of interfacial melting of ice, this is the ice–quasiliquid interface. The morphology of this interface is itself a conse- quence of the minimization of the free energy. This does not only complicate the calculations, but also influences the net roughness effect. The second consequence of the roughness concerns the interface potential denoted with P (L) in Eq. 3.1, which depends on the molecular interactions. Since the roughness changes the distances between particles in the various layers, it modifies P (L) (see for example [103]). This effect could be called a change of the effective layer thickness, although it does not appear as a simple correction to the layer thickness in the calculations. A third effect concerns only solid wetting films. In this case, the bending energy of the solid film picking up the substrate roughness must be taken into account [104, 105]. The effect of roughness on wetting phenomena thus depends on the specific sit- uation. This includes the type of roughness (self-affine roughness, mound rough- ness, periodic structures, ...) and its length scales, the type of interactions (Van der Waals interactions, exponentially decaying interactions, ...), and the wetting layers involved (solid or liquid). 3.3. EXPERIMENTAL EVIDENCE FOR INTERFACE MELTING 21

3.2.8 Further aspects At surfaces, other phenomena can occur, which can either be superimposed to premelting, or lead to an interplay with premelting. Such phenomena are rough- ening, pre-roughening and faceting. Pre-roughening [106, 107] starts at a tem- perature Tpr where the free energy cost for step formation vanishes. It produces a roughly half-filled outermost layer by the formation of surface vacancies and adatoms, which associate to form islands and holes. When the step free energy vanishes with an essential singularity, this gives rise to a roughening [108, 109] transition at the temperature Tr. Islands form on top of other islands and holes inside other holes, which causes the surface width to diverge. As the surface free energy depends on the crystal orientation, the macroscopic orientation of crys- tals can be unstable with respect to faceting. Such form large low-energy facets while retaining the average orientation. The surface phenomena described here can also appear at solid–(quasi)liquid interfaces. They might thus also play a role at solid–solid interfaces once premelting has set in. As the surface free energy varies with crystal orientation, the surface melting behavior can also be anisotropic. An example is the surface behavior of aluminum. Whereas the relatively open Al(110) surface shows surface melting, the close- packed Al(111) surface remains stable up to the bulk melting point [110]. The description in the sections 3.2.1 and 3.2.2 obtained in the framework of Landau theory underestimates the effect of fluctuations [73]. Fluctuations of the interface, capillary waves at surfaces, e.g., can modify the behavior.

3.3 Experimental evidence for interface melting

Surface melting has been studied with a large variety of techniques and has been observed in many classes of materials. Only examples will be cited here. The reader is referred to the reviews mentioned above (page 12). The first microscopic studies have been performed on metals including Pb [111], Al [110], Au [112], Ni [113], and Ga [114, 115]. The general tendency of metals is to show complete surface melting on the relatively open faces such as fcc(110), no surface melting for the densely packed faces like fcc(111), and in- complete surface melting on faces with intermediate packing density like fcc(100). The growth law usually shows a cross-over from a logarithmic to a power law. Other studies have been performed on the rare gases Ne and Ar [116, 69], where again a cross-over in the growth law was observed. Experimental evidence for (at least incomplete) surface melting has also been found for the semiconductors Ge [117] and Si [118]. Finally, surface melting can also occur in organic substances, as has been shown for caprolactam (C6H11ON) [119] and methane (CH4) [120]. For the surface melting of ice see Sec. 3.4. Surface melting is a well-established phenomenon that occurs in a wide range 22 CHAPTER 3. INTERFACE MELTING of materials. For solid–solid interfaces however, few experimental results are available. Most studies on interface melting were done on ice (see Sec. 3.4). However, some experimental observations attributed to surface melting might actually be due to interface melting. An example is the work of Zhu et al. [116, 69] cited above. They attributed heat capacity anomalies of adsorbed Ne and Ar films to surface melting and roughening. As their measurements do not provide any spatial information, they could be explained by surface melting, or interface melting, or a combination of both. In another case, Chernov and Yakovlev [121] observed premelting of biphenyl in contact with glass by ellipsometry. They called their observation surface melting, although the premelting occurred at a solid–solid interface. The distinction is very important, as no premelting was observed on the free surface of biphenyl in a study from another group using x-ray reflectivity [122]. Other evidence for interface melting was found for Ar in porous Vycor glass by heat capacity and vapor pressure measurements [123]. High-resolution transmis- sion microscopy observations of the interface between Al and amorphous Al2O3 in nanoparticles were also attributed to interface melting of Al [124]. For nano-crystals embedded in the matrix of another material, superheating instead of premelting has been observed (for example Pb embedded in Zn [125]). The experimental results suggest that in these cases, the size and shape of the particles play a crucial role.

3.4 Interface melting of ice

The melting of ice is part of our everyday life. The melting point of ice at standard atmospheric pressure, 273.15 K, is the zero of the Celsius scale, 0◦C. However, the anomalous properties of water (see Sec. 2.3) also show up in the melting of ice. Two particular features are linked to its melting behavior. One is pressure melting, caused by the anomalous density increase upon melting. The 3 solid phase ice has a lower density (ρs = 0.92 g/cm ) than the liquid phase water 3 (ρl = 1.0 g/cm ), in contrast to most other materials. Clausius-Clapeyron’s relation then implies a negative slope of the melting curve Tm(p). The other feature is surface melting (see below), which also occurs in other materials, but is especially pronounced in the case of ice. This is striking, as the negative density difference ∆ρ = ρs − ρl between the solid and the liquid phase renders the Hamaker constant negative and surface melting energetically unfavorable (see Sec. 3.2.1). The quasiliquid, however, may have a different density than the bulk liquid, thus modifying the value of the Hamaker constant. In the case of interface melting, the Hamaker argument does no longer hold in its original form and the Hamaker constant depends on the specific interface (see below). 3.4. INTERFACE MELTING OF ICE 23

3.4.1 Theory and simulations In principle, the theoretical considerations presented in Sec. 3.2 apply to ice interfaces as well. Interface melting is driven by the minimization of the free energy. In the case of ice with its network forming hydrogen bonds, this is more complicated than in other materials. In an early theory Fletcher [126] (see also [127, 128]) evaluated the free energy at the free surface of ice. Taking into account electrostatic effects and the dipole and quadrupole moment of the water molecule, he concluded that the free energy gain due to surface polarization is sufficient to induce surface melting at about −5◦C with a quasiliquid layer thickness reaching 10 to 40 A˚ close to 0◦C. Elbaum and Schick [129] applied the theory of dispersion forces to the surface of ice. They found that electromagnetic interactions result in incomplete surface melting with a maximum layer thickness of about 30 A.˚ According to the authors, fluctuations could lead to larger layer thicknesses but would not lead to complete melting. In a subsequent study Wilen et al. [130] evaluated the contribution of Van der Waals forces to interface melting at various ice–solid interfaces. Taken alone, Van der Waals interactions can lead to complete or incomplete interface melt- ing, depending on the substrate. However, additional (short-range) interactions can change the overall behavior. An estimate shows that electrical interactions can indeed be dominant if present. Layer thicknesses for various substrates are estimated to be of the order of 10 A˚ at −0.1◦C. A recent theory of Ryzhkin and Petrenko [131] links surface melting to pro- ton disorder and supports the idea of two overlapping surface regions, a proton disordered region and a second region where the oxygen lattice breaks down. Several molecular dynamics studies have been performed on surface melting of ice. In an early work by Weber and Stillinger [132], surface melting was seen in a simulation of ice crystallite melting. Later studies by Kroes [133] and Nada and Furukawa [134] (among others) directly addressed the problem of melting at ice surfaces. Both used the TIP4P potential and observed the formation of quasiliquid structures at the surface. Kroes only investigated the basal face, whereas Nada and Furukawa studied basal and prismatic faces and observed anisotropic behavior with the basal face exhibiting thicker quasiliquid layers. Wettlaufer [135] has investigated the effect of impurities on surface and in- terface melting of ice by calculating Van der Waals and Coulombic interactions in contaminated interfacial films. His results suggest that impurities can have a dramatic influence on surface and interface melting.

3.4.2 The free surface of ice Surface melting of ice has been studied in a large number of experiments with var- ious techniques, among them photoemission (Nason and Fletcher [136]), proton 24 CHAPTER 3. INTERFACE MELTING backscattering (Golecki and Jaccard [137]), ellipsometry (Beaglehole and Na- son [138], Furukawa et al. [139] ), laser reflection (Elbaum et al. [140]), sum- frequency vibrational spectroscopy (Wei et al. [141]), and photoelectron spec- troscopy (Bluhm et al. [142]). While practically all experiments confirm the presence of a quasiliquid layer at the surface of ice, there are large discrepancies in the layer thicknesses and onset temperatures reported, which cannot be discussed in detail here. Part of the discrepancies might be due to the fact that smooth and clean ice surfaces are difficult to prepare and even more difficult to maintain in the same state for the duration of the experiment. The high vapor pressure of ice causes problems for many standard surface techniques which require UHV (ultra-high vacuum). The vapor pressure has to be controlled very precisely, oth- erwise the surface morphology changes rapidly due to sublimation/resublimation. Another problem might be to avoid contamination of the surface by impurities, which can have a significant influence on the melting process (see above). Fur- thermore, care has to be taken when comparing different experiments, as the surfaces of ice against air and against pure water vapor as well as different crys- tal orientations do not behave in the same way. Some experiments were actually performed on thin films, where the strong confinement as well as the interface with the underlying substrate may significantly influence the experimental re- sults. Another part of the discrepancies might stem from the fact that different physical properties are probed by the various experimental techniques. Experi- mental support for this idea comes from experiments by Dosch et al. [143], who observed that a loss in long-range coherence of the hydrogen network occurs in a deep-ranging surface layer prior to actual surface melting. The relevant order parameters for the solid–liquid transition are the Fourier components of the solid density-density correlation function measured as Bragg scattering intensities. Lied et al. [144, 3, 145, 146, 143] performed a series of surface-sensitive x-ray-diffraction experiments in order to directly probe this order parameter. They studied several surfaces with high-symmetry orientations and found onset temperatures between −13.5◦ and −12.5◦C. The layer thickness could be best fitted with a logarithmic growth law, with deviations towards a higher thickness for temperatures above −1◦C. The amplitude of the growth law varied between 37 A˚ and 84 A.˚ These findings will later be discussed in comparison with this work (see Chapt. 7.4).

3.4.3 Ice in porous media While the experimental results which will be described in this section all give evidence of premelting, they have two important shortcomings: First, the inter- pretation of the physical origin is difficult, as powders, porous media, and soils are not well characterized with respect to size distribution of grains and pores, curvature, roughness, surface termination, impurities, ice crystallinity and orien- tation, etc. Second, these experiments only indicate the existence of a premelting 3.4. INTERFACE MELTING OF ICE 25 liquid. Neither do they allow to locate this liquid, nor do they permit to observe variations of the properties within the layer. It might be possible that premelting only occurs in the smallest pores, for example, but the experiments presented in this section only allow to estimate an average layer thickness. In order to address the first problem, experiments on better defined solid–solid interfaces have been performed, which will be presented in the next section.

The advantage of porous media is the abundance of interfaces. If interfacial melting occurs, this leads to a macroscopic quantity of quasiliquid material. A signal from this quasiliquid can then be detected with common bulk methods. Porous media can also serve as a more realistic model for permafrost soil.

Maruyama et al. [147] have studied H2O-saturated powders of graphitized carbon black and talc by quasi-elastic neutron scattering (QENS). They observed unfrozen water down to temperatures below −30◦C. The temperature dependence of the calculated liquid fraction is linked to size effects. The translational diffusion coefficient of the liquid fraction differs from supercooled water. A later analysis of the results by Cahn et al. [148] incorporated separate terms for interfacial and curvature melting and found good agreement with the measurements.

A later study by Gay et al. [149] was aimed at reducing the curvature effects in fine powders. Therefore, melting of D2O in exfoliated graphite with a laminar structure was studied by neutron diffraction. The liquid fraction grows according to a power law with exponent −0.54, significantly larger than the value −1/3 expected for long-range forces. The discrepancy is again attributed to size effects.

Bellissent-Funel and Lai [150] performed neutron scattering experiments of D2O confined in porous Vycor glass. They observed the formation of cubic ice and the persistence of a small liquid fraction down to −40◦C.

Ishizaki et al. [151] used pulsed NMR to study ice in porous silica in a temper- ature range of −30 to 0◦C. They deduced from their measurements a (average!) thickness for the quasiliquid layer at the ice–silica interface as a function of tem- perature. At −30◦C, this layer is still ≈10 A˚ thick. It seems to diverge at a depressed melting point, which the authors explain by effects from the pore cur- vature. The behavior strongly depends on the pore size. The growth of the quasiliquid layer follows a power law with exponent −0.60.

Ordered porous silica materials with cylindrical pores of uniform (and tunable) size have become more readily available. They enable studies of the premelting as a function of the pore diameter (in the range of a few nanometers). These experiments (see [152] and references therein) agree with the predicted linear dependence of the melting point depression on the inverse pore radius. They also hint to a thin (≈4 A)˚ layer of interfacial water remaining liquid down to very low temperatures. 26 CHAPTER 3. INTERFACE MELTING

3.4.4 Ice–solid interfaces Ellipsometry Furukawa and Ishikawa [153] have studied ice single crystals of unspecified orien- tation in contact with BK 7 optical glass by ellipsometry. Ellipsometry exploits the change of polarization when light is reflected. It can be characterized by the ellipticity R ρ = p = tan ψ exp (i∆) , (3.12) Rs

where Rp and Rs are the Fresnel coefficients for the p- and s-components of the polarized light. The polarization change is sensitive to the profile of the refractive index n across the interface. Monochromatic ellipsometry employed by Furukawa and Ishikawa yields only 2 numbers (tan ψ and ∆) which can be used to reconstruct the profile of the refractive index. Therefore, strong assumptions have to be made. Furukawa and Ishikawa used the measured ellipticity values to deduce the refractive index and the thickness of a quasiliquid premelting layer between ice and glass. The values were different from sample to sample, but a systematic change with temperature could be observed. The 10 nm microroughness of the glass sample produces a smearing of the observed profile of the refractive index. For this reason, an apparent layer with 10 nm thickness and a refractive index between water and glass is already observed at −5◦C. This layer has no physical meaning but limits the resolution to about 10 nm. At temperatures above −1◦, the authors see a change in the ellipticity which they attribute to a quasiliquid layer with the refractive index of bulk water reaching a thickness of more than 100 nm. It is not clear how the microroughness of the glass is included in their model in this regime. Optical anisotropy is not considered in their analysis. In a subsequent study, Beaglehole and Wilson [154] applied ellipsometry to ice in contact with different glass surfaces: smooth and clean glass, roughened glass, glass with surface impurities, and roughened glass with a hydrophobic coating. They tried to take into account as a sort of background the ellipticity change induced by the roughness. Therefore, they performed reference measurements on the various glass substrates in contact with water. For ice in contact with the clean and smooth, or with the hydrophobic glass, they saw no significant change in the ellipticity up to about −0.1 to −0.05◦C. For the roughened glass, they obtained water thicknesses reaching up to about 200 nm at −0.2◦C. They did not evaluate the refractive index (density) of this layer. In the case of the glass substrate with impurities, significant influence of the impurities on the refractive index are expected. Rather than measuring the refractive index, the authors per- formed an estimate based on literature values and calculated the layer thickness using these assumed values of the refractive index. Their plot of the tempera- ture dependence of the layer thickness contains three data points. The deduced exponent for the temperature variation is −1. In conclusion, Beaglehole and Wil- 3.4. INTERFACE MELTING OF ICE 27 son show clear differences for the different glass substrates. However, the same limitations as for the measurements of Furukawa and Ishikawa apply:

• The profile of the refractive index has to be deduced from only 2 numbers, therefore strong assumptions have to be made.

• It is difficult to distinguish apparent layers due to substrate roughness from actual layers of a different material.

• The spatial resolution in the measurement of the layer thickness appears to be on the order of 10 nm.

For the study of interfaces by ellipsometry, one of the two materials has to be transparent for the wavelength used (typically in the optical regime). In princi- ple, this could be the ice. But due to the high vapor pressure of ice, it is difficult to prepare and preserve a smooth ice surface through which the light beam could penetrate. This limits the range of other ice–solid interfaces accessible to ellip- sometry measurements.

Sum-frequency vibrational spectroscopy Wei, Shen et al. [155, 156, 157, 158, 159] reported results from sum-frequency vibrational spectroscopy (SFVS) measurements at various ice and water inter- faces. This method yields information about the various bond modes at the interface which could not be obtained with other experimental techniques. But the method is only sensitive to the topmost layer. It does not allow any depth profiling across the quasiliquid layer, nor does it provide information about its thickness [159]. The most detailed analysis has been performed for the ice and water surface (ice–air, water–air). The measurements allow to deduce the ori- entational order and maximum tilt angle for the free OH bonds at the surface. Surface disordering of ice sets in at 200 K. This temperature should not be iden- tified with the onset temperature T0 from Sec. 3.2. The onset temperature T0 presumes that a quasiliquid layer intervenes at the interface replacing it by two new interfaces. The surface disordering here refers to a partial disordering of the first monolayer, which might set in well below the actual onset of surface melting as described in Sec. 3.2. Therefore, this finding is not in disagreement with other studies reporting much higher onset temperatures. Another interesting feature of the results from Wei et al. is the observation of strong differences between the quasiliquid surface of ice and the surface of water (even if supercooled) which shows that the structure of the quasiliquid indeed differs from bulk water. Further measurements have been performed on hydrophilic ice–silica inter- faces and hydrophobic ice–OTS–silica interfaces (silica coated with a hydropho- bic octadecyltrichlorosilane monolayer). These measurements are more difficult to interpret than those for the free surface [159]. The authors conclude that at 28 CHAPTER 3. INTERFACE MELTING the hydrophobic interface, the dangling OH bonds are highly disordered in ori- entation, regardless of the temperature (this refers to the first monolayer at the interface, the method is not sensitive to deeper layers). As for the free surface, the hydrogen-bonded OH-peak decreases with temperature. At the hydrophilic interface, no peak for dangling OH bonds is observed due to hydrogen bonding to the SiOH (silanol) groups of the silica surface. The hydrogen-bonded OH-peak remains quite strong up to −1◦C. Measurements have been performed across the melting transition, and the authors conclude from these measurements that the net orientation of water molecules at the hydrophilic interface flips upon melting. They also suggest that interface melting at the hydrophilic interface only occurs very close to the bulk melting temperature.

Atomic force microscopy A common method in surface analysis is atomic force microscopy (AFM), which has also been applied to study ice surfaces. Where the microscope tip is in contact with the surface, however, there is no surface in the strict sense anymore, but an interface between the sample and the tip. This is important, as the presence of the tip might have an effect on the behavior of the sample. In the case of surface melting studies with AFM, the question is whether an eventually observed quasiliquid layer is to be attributed to surface melting, or interface melting induced by the contact with the microscope tip. Petrenko [160] comes to the conclusion that thermal equilibrium between tip and ice always occurs, but Bluhm et al. [161] conclude that in their lateral force microscopy experiments, the quasiliquid layer between tip and ice is squeezed out. The AFM measurements are presented in this subsection about ice–solid in- terfaces, as an influence of the tip cannot be excluded a priori. The interface ice–tip, however, is not as well defined as the interfaces studied in the other ex- periments presented in this subsection (based on methods like ellipsometry and sum-frequency vibrational spectroscopy). First of all, the tip obviously is not flat, but has a radius of curvature in the range of 10 to 100 nm. Then, it also has a (unknown) microscopic roughness. Furthermore, there might be heating due to the laser beam used to detect the deflection of the cantilever. Eastman and Zhu [162] estimated this effect to be significant, therefore special precautions had been taken in the studies presented here to minimize the heating of the cantilever. If on the other hand the temperature of the cantilever is too low, condensation of water vapor can also disturb the measurements. This is why Pittenger et al. [163] keep the tip at a temperature about 0.1–0.3◦C above the sample temperature. Several groups have conducted AFM studies of ice surfaces, among them Pe- trenko [160], Pittenger, Fain, Slaughterbeck et al. [164, 165, 163], D¨oppenschmidt, Butt et al. [166, 167, 168], and Bluhm et al. [161]. In general, such experiments seem to be quite tricky. The results of the various studies show large differences, and even results obtained in the same study under identical experimental con- 3.4. INTERFACE MELTING OF ICE 29

103

102 (Å) L 101

100 0.1 1 10 T -T (K) m

Figure 3.5: Growth laws for interfacial melting of ice from the literature. From [154], determined with ellipsometry: ice against roughened float glass (squares), ice against float glass with impurities (triangles). From [151], determined with NMR: ice in porous silica with 500 A˚ pore diameter (circles). From [163], de- termined with AFM: ice against uncoated AFM tip (diamonds), ice against hy- drophobic AFM tip (crosses). ditions scatter strongly [160, 167]. Most of the studies agree on the presence of a quasiliquid layer. The thickness of this layer is calculated from measured force curves. The interpretation of such force curves is not trivial, as adhesion, capillary fores, elastic and plastic deformation of the ice, flow of the quasiliq- uid under the tip, as well as electrostatic and Van der Waals forces can play a role. The models describing these contributions often have to include strong assumptions, as for example for the viscosity of the quasiliquid [163]. The most recent and most convincing study from Pittenger et al. [163] covers temperatures between −1 and −10◦C, where the calculated thickness of the quasiliquid layer −0.68 grows like L ≈ 1.1 nm (Tm − T ) for a silicon tip. This leads to a layer thick- ness of about 1 nm at −1◦C. When the tip has a hydrophobic coating, the layer thicknesses are slightly smaller. Petrenko [160] obtains 2–16 nm at −10.7◦C, D¨oppenschmidt and Butt [167] about 30–50 nm at −1◦C and about 5–15 nm at −10◦C. In a later analysis, Butt et al. [168] explained their measurements with plastic deformation of the ice rather than a quasiliquid layer.

3.4.5 Further aspects Premelting of ice is a vast topic and cannot be presented here in its full scope and depth. One of the aspects that should not be omitted is the premelting at grain boundaries (see [169, 170]). The macroscopic manifestations of premelting are another issue. Some of them are discussed in Sec. 3.5. An example is the 30 CHAPTER 3. INTERFACE MELTING work of Jellinek [171] on the adhesive properties of ice. For snow-ice sandwiched between stainless steel, he observed a breakdown of the adhesive strength starting at about −15◦C, the temperature where interface and surface melting typically set in. In the interface melting scenario, the quasiliquid layer is strongly confined between two solids. Therefore, other studies of liquids in confinement might be relevant. The structure of the quasiliquid may have similarities with the structure of water in nanopores, for example.

3.5 Consequences of ice premelting

The environmental consequences of ice premelting have been reviewed by Dash et al. [5] (see also [172]). Another good source is [173]. A few issues will be presented in this section.

3.5.1 Permafrost Permafrost is a composite structure of rock or soil remaining at or below 0◦C for two or more years. It contains in many cases over 30% ice, and hence abundant ice–mineral interfaces (see Fig. 3.1c). Permafrost covers about 20% of the land mass on the northern hemisphere and can be up to several hundred meters thick. Permafrost can be an ideal terrain to build on, but any internal melting process can turn it into a slurry-like material with disastrous consequences for buildings and infrastructure. Damage can also occur through frost heave (expansion of the soil). Permafrost is characterized by massive transport of water (and solutes), which—due to premelting phenomena—is present in soils even at temperatures well below 0◦C (see for example Williams [174]).

3.5.2 Glacier motion There are three mechanisms of glacier motion: ice deformation, bed deformation, and basal sliding [175]. The first process is usually dominant, but rather slow (some 10 meters per year). Basal sliding spans a much greater range of velocities and can change over periods of hours. It strongly depends on the presence of water at the base of the ice sheet. One possible mechanism for a reduction of the melting temperature at the glacier bed is pressure melting (see Sec. 3.4). As dTm the negative slope of the melting curve is very small, dp = −7.15 mK/atm, the ice melting temperature at the base of a 1000 m thick ice sheet is only lowered to −0.7◦C (assuming uniform distribution of the weight). Any premelting at the ice–rock interface significantly below the bulk melting point would therefore be a dominant factor for the basal sliding of glaciers. The observed sliding of polar glaciers has been attributed to interfacial melting of ice [176]. 3.6. SUMMARY AND CONCLUSIONS 31

3.5.3 Thunderstorms and atmospheric chemistry Much is known about the structure of thunderstorms and their formation, but the microscopic mechanism responsible for the charging remains unclear. It is generally agreed that charging occurs in collisions of tiny ice particles. The liquid generated by surface melting of these particles was suggested to play a key role in the charge transfer [177, 178]. Ice particles play a similarly important role for atmospheric chemistry. It is known, for example, that polar stratospheric clouds are relevant for ozone depletion. Surface melting may be involved in these processes. The quasiliquid layer on ice particles can serve as a reservoir and reaction site for the chemicals (HCl, e.g.) implicated in atmospheric reactions [179].

3.5.4 Friction The coefficient of friction of ice can be very low compared to other common ma- terials, which makes ice skating and skiing possible. Several effects contribute to the friction of ice and snow: dry friction, lubricated friction, ploughing, and capillary forces. Lubrication comes from a thin liquid film, the debate is about the origin of this film. Three mechanisms have been proposed: pressure melting, frictional heating, and interface melting. In many physics textbooks (for exam- ple [180]), ice skating is still explained by pressure melting, although a simple calculation shows that its contribution is negligible for ice skaters with a mass smaller than a few tons. For ice skating, frictional heating seems to be a domi- nant factor [181, 182], but interface melting also contributes to lubrication [183]. The respective contributions depend on factors like temperature, skating speed, and the properties of the skate.

3.6 Summary and conclusions

Surface melting is a well-established phenomenon and occurs in a wide range of materials (see Secs. 3.2 and 3.3). There is a lot of experimental evidence for interface melting of ice in contact with other solids, but microscopic observa- tions at well-defined interfaces are scarce, owing to the difficulties in probing deeply buried interfaces (see Sec. 3.4). Interface melting of ice has important consequences for processes in nature and technology (see Sec. 3.5) and is also of great interest from a purely scientific point of view. It might be related to the anomalies of water and the unsolved puzzle of the water structure (see Chapt. 2). Although the relevant order parameters for the solid–liquid transition are the Fourier components of the solid density-density correlation function, the average density plays an important role for the melting of ice (see Sec. 3.4). 32 CHAPTER 3. INTERFACE MELTING Chapter 4

Theory of x-ray reflectivity

The aim of this chapter is to acquaint the reader with the basic principles of x-ray reflectivity needed for the understanding of this work. An extensive review of x-ray reflectivity techniques can be found in [7], the basic principles are also covered in [184]. Other references are given in the text.

4.1 Index of refraction for x-rays

We consider a homogeneous material consisting of N different atom types with the respective number densities ηj. For x-rays, the index of refraction of this material can be written as n = 1 − δ + iβ (4.1) with the dispersion and absorption terms

N λ2 X δ = r η f 0 + f 0 (λ) (4.2) 2π e j j j j=1 N λ2 X λ and β = r η f 00 (λ) = µ . (4.3) 2π e j j 4π j=1

Here λ denotes the x-ray wavelength, re the classical electron radius, and µ the linear absorption coefficient. The x-ray form factor is

0 0 00 fj = fj + fj (λ) + ifj (λ) , (4.4)

0 00 where fj (λ) and fj (λ) account for dispersion and absorption corrections, re- 0 spectively. fj depends on the momentum transfer q, but can be considered to be constant over the q-range typically covered by x-ray reflectivity measurements. A 0 good approximation (far from absorption edges) is fj ≈ Zj (Zj being the atomic number), and thus λ2 δ = r σ , (4.5) 2π e e 33 34 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

ki kf

n1 a a =a i f i interface at n2

kt

Figure 4.1: Reflection and refraction of a plane electromagnetic wave. An incident plane wave with wave vector ki hits an interface at an incident angle αi. It splits into a reflected (αf = αi) wave with wave vector kf and a refracted wave with wave vector kt transmitted at an angle αt. For incident angles αi < αc, total reflection occurs (see the text).

where σe denotes the electron density. Tabulated values for the form factor can be found in [185].

4.2 Reflection at an ideal interface

0 A plane electromagnetic wave Ei (r) = Ei exp (iki · r) with wave vector ki and 0 amplitude Ei impinging on an interface at an incident angle αi splits into a 0 reflected wave (αf = αi, amplitude Ef ) and a refracted wave transmitted at the angle αt (see Fig. 4.1). The angle of the refracted wave is linked to the angle of the incident wave by Snell’s law:

cos α n i = 2 , (4.6) cos αt n1 where n1, n2 denote the respective indices of refraction for the media of the in- coming wave and the transmitted wave. If the incident angle is smaller than the critical angle   1 − δ2 p αc = arccos ≈ 2 (δ2 − δ1) , (4.7) 1 − δ1 total external reflection occurs: No transmitted wave is created, and only an evanescent wave field is induced in the second medium. Apart from small losses due to absorption, all incoming radiation is reflected. The critical angle is only defined for δ2 > δ1. This means that total reflection can only occur, when the medium on the side of the incoming wave is optically denser (Re(n1) − Re(n2) = δ2 − δ1 > 0). 4.3. REFLECTION AT MULTIPLE INTERFACES 35

The reflection coefficients r for the components of the electric field parallel k and perpendicular ⊥ to the interface are1

0 Efk rk = 0 , (4.8) Eik 0 Ef⊥ r⊥ = 0 . (4.9) Ei⊥ They can be calculated using the fact that the tangential components of the electric and magnetic field have to be continuous at the interface. In the case of x-rays, where n is close to unity, there is practically no difference between the different polarizations and

n2 sin αi − sin αt ki,z − kt,z n r = = 1 , (4.10) k + k n2 i,z t,z sin αi + sin αt n1

n2/n1 where ki,z = k sin αi and kt,z = k sin αt are the z-components of the wave vector of the incident and transmitted wave, respectively. The intensity of the reflected wave, the so-called Fresnel reflectivity, is

2 RF = |r| . (4.11)

For αi & 3αc, the Fresnel reflectivity can be well approximated by √ 4  4 ! αc k 2∆δ RF ≈ ≈ . (4.12) 2αi qz

For the connection between angles and the momentum transfer, see Sec. 5.3.4.

4.3 Reflection at multiple interfaces

Now we consider the case of a multilayer system, where the reflections from all interfaces contribute to the total reflection. A sketch of a system consisting of N + 1 layers is shown in Fig. 4.2. The layer j (j = 1 ...N + 1) has the refractive index nj and the thickness dj. The layers 1 and N + 1 are semi-infinite (d1 = d∞ = ∞). The system has N interfaces at the positions zj. Two waves are created at each interface: a ‘reflected’ wave (Rj) propagating in the layer j and a ‘transmitted’ wave (Tj+1) propagating in the layer j + 1. But unlike at a single interface, there are also two incoming waves at each interface: the transmitted wave from the interface j −1 (Tj) and the reflected wave from the

1The components parallel and perpendicular to the interface are independent, see [184]. 36 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

T=1 R z 1 1 layer 1 n1 ai af=ai x z1=0 T R 2 layer 2 2 n2 d2 R2 T2 z2 T n R3 3 3

...... T R j-1 j-1 n z j-1 j-1 T Rj j

layer j n d R j j Tj j

zj

Tj+1 nj+1 Rj+1

...... RN-1 TN-1 nN-1 zN-1 R N TN layer N nN dN

TN RN z N T n layerN +1 N+1 N+1

Figure 4.2: Reflection and refraction of a plane wave at a system of multiple interfaces. The system shown in this figure consists of N +1 layers with refractive indices nj and thicknesses dj separated by N interfaces. A recursive approach allows to calculate the reflectivity (see the text). 4.4. ARBITRARY DISPERSION PROFILES 37

interface j + 1 (Rj+1). The amplitude of the incoming wave in the semi-infinite layer 1 is normalized to unity, T1 = 1, and no reflected wave is propagating through the last layer, RN+1 = 0. The Parratt formalism [186] connects the Rj and Tj:

Rj rj,j+1 + Xj+1 exp (2ikz,j+1zj) Xj = = exp (−2ikz,jzj) , (4.13) Tj 1 + rj,j+1Xj+1 exp (2ikz,j+1zj) where kz,j − kz,j+1 rj,j+1 = (4.14) kz,j + kz,j+1 is the Fresnel coefficient of the interface between layer j and layer j + 1, and kz,j denotes the z-component of the wave vector in layer j. Recursive application of Eq. 4.13 with RN+1 = XN+1 = 0 as the start of the recursion yields after N iterations the reflectivity

2 2 R = |X1| = |R1| . (4.15)

4.4 Arbitrary dispersion profiles

Arbitrary (continuous) dispersion profiles can be treated with the Parratt for- malism by slicing the profile into a large number of thin layers of thickness ε with constant δ in each layer and sharp interfaces. If the dispersion profile is described with subatomic resolution (ε  1 A,˚ taking into account the electron distribution of the atoms), the Parratt formalism allows to calculate the intensity distribution on the specular rod over the whole momentum transfer range from q = 0 A˚−1 up to the Bragg peaks. Such a scheme has recently been used by Schweika et al. [187] in a study of surface segregation and ordering in CuAu. Arbitrary dispersion profiles can also be calculated in the kinematical approx- imation (see the next section).

4.5 The kinematical approximation

In the kinematical approximation, multiple scattering effects are neglected. It is only valid, when the scattering cross section is small (‘weak scattering’). For x-rays, the kinematical approximation can often be used and allows to analyze reflectivity data in a straight-forward way (in contrast to electron diffraction, e.g., where a dynamical scattering theory is always necessary). In the kinematical approximation, the reflectivity is the Fourier transform of the derivative of the dispersion profile δ (z) multiplied by the Fresnel reflectivity RF (qz) (a derivation can be found in Sec. 4.8.1): Z 2 1 dδ (z) R (qz) = RF (qz) exp (iqzz) dz , (4.16) δ−∞ − δ+∞ dz 38 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

where δ−∞ and δ+∞ denote the dispersion at z = −∞ and z = +∞, respectively. This so-called ‘Master formula’ is in very good agreement with the exact treatment except for the region around and below the critical angle. A better agreement in the vicinity of the critical angle can be reached by replacing q   z 0 p 2 2 in the Fourier transform of Eq. 4.16 by qz = 2k sin αi − αc , where αc = p 2 (δ−∞ − δ+∞) (see Eq. 4.7). The advantage of the kinematical approximation is the closed form expres- sion for the reflectivity. This provides good insight into the relation between the dispersion profile and the reflectivity, and allows to some extent a qualitative interpretation of reflectivity curves. Since it permits an effective numerical cal- culation of the reflectivity, the kinematical approximation is also often the basis for advanced data analysis techniques like phase inversion.

4.6 Data analysis and phase inversion

A basic problem, not only for reflectivity measurements, but also for other diffrac- tion techniques, is the loss of the information about the phase of the scattered wave, as only the square modulus (intensity) of the complex wave amplitude can be measured. Consequently, a direct Fourier back transformation of Eq. 4.16 is not possible. The usual approach for analyzing reflectivity data is thus to assume a model of the dispersion profile (incorporating all knowledge about the system) and fit the free parameters of the model to the measured data. Several techniques have been used to overcome ambiguities in the analysis of diffraction data, among them

• adding a known reference layer to an unknown system (see for example [188]),

• exploiting different polarizations of the incoming and reflected beam (neu- tron reflectivity),

• using anomalous scattering.

While these methods have specific experimental requirements, other tech- niques only concern the data analysis. For x-ray reflectivity measurements, phase- guessing methods have been used with success. They are based on the fact that the phase, although unknown, is not arbitrary. For more information about these methods, see [7, 189] and references therein. 4.7. DESCRIPTION OF ROUGH INTERFACES 39

z

z()R z()0

R=(xy , ) y x

Figure 4.3: Sketch of an interface contour z (R).

4.7 Description of rough interfaces

A single interface without overhangs can be defined by a contour function (see Fig. 4.3) z (R) with R = (x, y) . (4.17) We introduce the mean height

0 z = hz (R )iR0 , (4.18) where the angle brackets denote a spatial average. Assuming ergodicity, this corresponds to an ensemble average. The interface can also be described by the height fluctuations h (R) around the mean height z:

h (R) = z (R) − z. (4.19)

The interface can be characterized by statistical properties, such as:

• the root mean square (rms) roughness σ, D E σ2 = [h (R0)]2 , (4.20) R0

• the height-height correlation function

0 0 C (R) = hh (R ) h (R + R)iR0 , (4.21) 40 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

• the height-difference correlation function D E g (R) = [h (R0) − h (R0 + R)]2 = 2σ2 − 2C (R) . (4.22) R0 From Eq. 4.21, it also follows that σ2 = C (0) . (4.23) For isotropic surfaces g and C only depend on R = |R|. Many isotropic solid surfaces have a self-affine character and can be described by ( " #) R2h g (R) = 2σ2 1 − exp − , or equivalently by ξ " # (4.24) R2h C (R) = σ2 exp − , ξ where ξ is the cutoff length and h the ‘Hurst parameter’ with 0 < h ≤ 1. Small values of h lead to jagged surfaces, while values close to 1 correspond to a surface with smaller gradients. For R  ξ, the surface is self-affine rough, g (R) ∼ R2h, while for R  ξ, the surface appears to be smooth, and g(R) saturates at 2σ2. For an ideally self-affine surface with no cut-off, g(R) does not saturate for R → ∞, and g (R) = BR2h. (4.25)

4.8 Reflectivity from rough interfaces in the kinematical approximation

This section is based on work from Sinha et al. [190] and Rauscher et al. [191]. In the kinematical approximation, an incident plane wave

Ei (rE) = exp (iki · rE) (4.26) creates a scattered wave Z exp (ikrE) 3 E (rE) = − d r exp (−iq · r) ρ (r) , (4.27) 4πrE where q denotes the momentum transfer kf − ki and r = (x, y, z). The density ρ used here is defined as ρ = k2 1 − n2 , (4.28) which can be written using Eq. 4.1 and δ = bNλ2/2π as ρ ≈ k22δ = 4πNb, (4.29) 4.8. REFLECTIVITY FROM ROUGH INTERFACES 41

z

interface contour

x

Figure 4.4: Illustration of an interface contour with a fixed (conformal) density profile. The density is represented by the color. where bN is the scattering length density. We now consider the case of a rough interface described by an interface contour h (R) (see Sec. 4.7) with a density profile across the interface which only depends on the distance z − h (R) from the interface along the z-direction (see Fig. 4.4):

ρ (r) = ρ (z − h (R)) . (4.30) For such a system, Eq. 4.27 can be written as Z Z ∞ exp (ikrE) 2 E (rE) = − d R exp (−iQ · R) dz exp (−iqzz) ρ (z − h (R)) , 4πrE −∞ (4.31) where Q = (qx, qy) is the momentum transfer parallel to the interface and R = (x, y). By substituting z0 = z − h (R), one obtains Z Z ∞ exp (ikrE) 2 0 0 0 E (rE) = − d R exp (−iQ · R − iqzh (R)) dz exp (−iqzz ) ρ (z ) 4πrE −∞ Z exp (ikrE) 2 = − d R exp [−iQ · R − iqzh (R)]ρ ˜(qz) 4πrE (4.32) with Z ∞ 0 0 0 ρ˜(qz) = dz exp (−iqzz ) ρ (z ) (4.33) −∞ being the Fourier transform of the density profile. 42 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

From this, the differential scattering cross section is obtained: dσ = r2 |E|2 dΩ E 1 ZZ = d2Rd2R0 exp [−iQ · (R − R0)] exp [−iq (h (R) − h (R0))] |ρ˜(q )|2 . (4π)2 z z (4.34)

Assuming that [h (R) − h (R0)] is a Gaussian random variable and that the x-ray coherence length is large compared to the correlation length of h(R00), this yields with the substitution R00 = R − R0: dσ A (q) Z  q2  = d2R00 exp (−iQ · R00) exp − z g (R00) |ρ˜(q )|2 , (4.35) dΩ (4π)2 2 z where A (Q) is the illuminated interface area. Usually, g (R00) saturates at 2σ2 for R00 → ∞ (see Eq. 4.24). The differential cross section then splits into a specular and a diffuse (off-specular) part (see Fig. 4.5):   dσ A (q) 2 2 2 = 2 exp −qz σ δ (Q) |ρ˜(qz)| , (4.36) dΩ spec (4π)   Z dσ A (q) 2 2 2 00   00 2 00 2 = 2 exp −qz σ d R exp C (R ) qz − 1 exp (Q · R ) |ρ˜(qz)| . dΩ diff (4π) (4.37)

The measured intensity I is the integral of the differential cross section over the solid angle covered by the detector,

Z  dσ  I = dΩ , (4.38) dΩ detector which is calculated in Sec. 5.3.6. The illuminated interface area A (q) is calculated in Sec. 5.3.7. It depends on the vertical and the horizontal size of the beam, wz and wy, respectively. In the following, we assume that the intensity has been measured by a so- called ‘rocking scan’ (see Sec. 5.3.6) and that the full incident beam illuminates the sample (see Fig. 5.12c+d).

4.8.1 Specular reflectivity With Eqs. 4.36, 5.15, and 5.22, the specular part of the reflected intensity is

Z   2 2 dσ wzwy exp (−σ qz ) 2 dΩ = 2 |ρ˜(qz)| . (4.39) dΩ spec 2 qz 4.8. REFLECTIVITY FROM ROUGH INTERFACES 43

q (Å-1) x -3.0x10-4 0.0 3.0x10-4 100 A

10-1

10-2 C B B C 10-3 Intensity (arb. units)

10-4 0.000 0.001 0.002 0.003 0.004 ? (rad) i

Figure 4.5: Schematic graph of (A) Specular and (B+C) diffuse reflectivity in a so-called ‘rocking scan’ (αi + αf = const). The ‘Yoneda wings’ (C) can only be explained by dynamical scattering, whereas the regions (A+B) are covered by the kinematical scattering theory described in this chapter.

The reflectivity is defined as the reflected intensity divided by the incident intensity (measured in the same way), which leads to

2 2 exp (−σ qz ) 2 Rspec = 2 |ρ˜(qz)| qz 2 (4.40) exp (−σ2q2) 1 gdρ = z , 2 2 qz qz dz

gdρ  where dz denotes the Fourier transform of the derivative of the density profile. One can see that the specular reflectivity is the same, if the interface is treated as smooth, but the laterally averaged density profile is taken. For Gaussian roughness, this corresponds to a convolution of ρ (z0) with an error function of width σ. In comparison with the reflectivity from a perfectly smooth (σ = 0) interface, the reflectivity from the rough interface decays much faster with qz due 2 2 to the damping factor exp (−σ qz ) in Eq. 4.40. If we use Eqs. 4.28 and 4.12, the specular reflectivity can be rewritten as

(k2∆δ)2 1 Z ∞ dδ 2 0 0 (4.41) Rspec = 4 0 exp (−iqzz ) dz qz ∆δ −∞ dz with ∆δ = δ−∞ − δ+∞. 44 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

(k2∆δ)2 The prefactor 4 is ≈ RF for qz  k. We can replace it with RF to qz obtain a better approximation (refraction phenomena have been neglected in the derivation so far):

1 Z ∞ dδ 2 0 0 (4.42) Rspec = RF 0 exp (−iqzz ) dz . ∆δ −∞ dz

This is the Master formula (Eq. 4.16) already presented in Sec. 4.5. The same result is obtained for the specular reflectivity measured with a finite detector slit opening instead of a rocking scan.

4.8.2 Integrated diffuse intensity In this section, we calculate the measured intensity if the contributions from both the specular and the diffuse part of the reflected intensity are completely integrated in a rocking scan. We start from Eq. 4.35 containing specular and diffuse contributions and use Eqs. 5.14 and 5.22 as in Sec. 4.8.1, which yields2

Z Z Z  2  dσ wzwy 2 2 00 00 qz 00 2 dΩ = 2 d Q d R exp (−iQ · R ) exp − g (R ) |ρ˜(qz)| dΩ 2qz 2 wzwy 2 = 2 |ρ˜(qz)| . 2qz (4.43)

The same expression as for the specular part of the reflected intensity (Eq. 4.39) 2 2 is found, but without the damping factor exp (−σ qz ). This can be understood intuitively, as the damping factor accounts for intensity ‘lost’ due to scattering from roughness in off-specular directions. This intensity is regained via the inte- gration.

4.8.3 Off-specular reflectivity Here, only isotropic rough interfaces will be considered and two specific cases discussed:

• ideally self-affine rough surfaces with no cutoff described by Eq. 4.25,

• self-affine rough surfaces with cutoff described by Eq. 4.24.

2 2 dσ qz Eq. 5.22 can only be used if dΩ (qx) ≈ 0 for |qx| > 2k , i.e. if the scattered intensity is restricted to the vicinity of the specular rod. Otherwise, the exact expression for the illuminated area has to be used to correct variations of the illuminated area during a rocking scan. 4.9. FURTHER REMARKS 45

Self-affine rough surfaces with no cutoff In this case, the scattering does not split into specular and diffuse parts. We have to start from Eq. 4.35, which may now be expressed using Eq. 4.25 as

dσ A Z  q2  = dR00R00 exp − z g (R00) J (QR00) |ρ˜(q )|2 , (4.44) dΩ 8π 2 0 z where J0 denotes the Bessel function.

Self-affine rough surfaces with cutoff In this case, the scattering splits into specular and diffuse parts. The specular part is given by Eq. 4.36. The diffuse part given by Eq. 4.37 may be expressed with Eq. 4.24 as

  Z dσ A 2 2 00 = exp −qz σ dR dΩ diff 8π (4.45) 00 n h 2 2  00 2hi o 00 2 R exp qz σ exp − (R /ξ) − 1 J0 (QR ) |ρ˜(qz)| .

4.9 Further remarks

Reflectivity from a rough interface can also be calculated in the Distorted-Wave Born Approximation (DWBA), see for example [190, 192]. The main differences in the formula for the diffuse scattering are

• the transmission functions, which lead to an enhancement of the density for αi or αf close to αc (the so-called ‘Yoneda-wings’), and

t • the appearance of qz as the wave vector transfer in the medium of the transmitted wave instead of qz. The formulae derived in this section refer to a rough interface with a fixed density profile. If this density profile contains several layers, the interfaces be- tween the layers completely replicate the roughness (conformal roughness). This assumption is not always fulfilled. This is probably not a problem, if just uncor- related roughness is added between the other layers, as this will simply lead to an additional broad diffuse background. The situation will be more complicated for partial correlations between layers. For a discussion of these questions, see [7] and references therein. 46 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY Chapter 5

High-energy x-ray-reflectivity experiments

A number of surface-sensitive x-ray diffraction techniques, such as x-ray reflec- tivity, evanescent x-ray diffraction, and crystal truncation rod diffraction have been developed in the past. They allow detailed and non-destructive structure determination and have been applied with great success to the study of surfaces, interfaces, and multilayers. Much of our current understanding of surface struc- tures is due to surface-sensitive x-ray scattering experiments. The free surface is a special (and idealized) case of an interface. Many phe- nomena in nature and technology occur at more complex, deeply-buried solid– solid, solid–liquid or liquid–liquid interfaces. Examples of such phenomena are lubrication, electrochemistry, processes at membranes—and interfacial melting. Unfortunately, the surface sensitive x-ray scattering techniques mentioned before are not well suited for the study of such interfaces. A new x-ray transmission-reflection scheme for the study of deeply-buried interfaces using high-energy microbeams has been recently developed in our group [6]. The inherent limits of the conventional techniques and the principle of our new x-ray scattering scheme will be discussed in Sec. 5.1. The use of this method for the study of melting at ice–solid interfaces will be shortly presented in Sec. 5.2. The experimental and instrumental details will be explained in Sec. 5.3.

5.1 Principle

Typical x-ray energies for conventional x-ray scattering techniques are in the range of 10–20 keV. Because of the high attenuation coefficient for x-rays in this energy range (see Fig. 5.1), the penetration depth is usually in the µm-regime. Therefore, deeply-buried interfaces are generally not accessible to these methods. For the study of interfaces, this limitation is usually circumvented by placing the interface as close as possible to the surface, so that it is covered with a film

47 48 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

103

102 Pb 101

µ(1/cm) Cu 100 Si 10-1 water 20 40 60 80 100 120 140 E (keV)

Figure 5.1: The linear attenuation coefficient µ decreases rapidly with the x-ray energy E (data from [185]). thin enough to be penetrated by the x-ray beam (see Fig. 5.2a, [7]). This method has several drawbacks. • The in situ study of real devices with deeply-buried interfaces is not possi- ble.

• The interface beneath the thin film might not behave like the deeply-buried interface.

• The measured x-ray signal is a complicated interference pattern created by contribution from all interfaces passed by the x-ray beam, including the surface (see Fig. 5.2a).

• The thin film covering the interface of interest has to be stable throughout the experiment. This is not trivial in the case of liquid films, where a slight change in the temperature and thus the vapor pressure can have a dramatic influence on the film thickness. The same holds true for ice films. In addition, sublimation and condensation can also change the morphology of the ice surface.

• Due to background scattering, the dynamic range is usually limited to 8–9 orders of magnitude, especially if the x-ray beam has to penetrate liquids or amorphous materials. A new x-ray transmission reflection scheme for studying deeply-buried in- terfaces [6] has recently been developed in our group. It is based on the use of modern Synchrotron Radiation (SR) sources and compound refractive lenses (CRL), which allow to produce high-energy (≥70 keV) x-ray microbeams. As 5.1. PRINCIPLE 49

a b transmission-reflection scheme conventional scheme with high-energy microbeam

R1 R 2 up to cm

R1 <µm buried interface up to cm

Figure 5.2: Comparison of (a) conventional surface scattering scheme and (b) transmission-reflection scheme with high-energy microbeam. In the conventional scheme waves reflected from all interfaces create a complicated interference signal and the interface of interest can only be covered with a thin film. the attenuation coefficient decreases rapidly with the x-ray energy (see Fig. 5.1), high-energy x-rays can penetrate materials with a macroscopic thickness of up to several centimeters. In the transmission-reflection scheme, a high-energy x-ray microbeam is used to penetrate the sample from the side (thereby avoiding reflections from the sur- face and other interfaces) and probe the buried interface of interest (see Fig. 5.2b). In principle, the whole spectrum of x-ray techniques used at surfaces (see above) can be applied at buried interfaces if high-energy x-rays are used. In this work x-ray reflectivity (see Chapt. 4 and Sec. 5.2), has been used. This technique is sensitive to the (electron) density profile perpendicular to the interface. In an- other work, we have already used evanescent x-ray diffraction to study the lateral structure of liquid lead at the interface with silicon [193]. Compound refractive lenses made of aluminum are used to produce highly- collimated microbeams with a small divergence (≈30 µrad) and a small beam size (≈10 µm). There are two reasons why this is necessary: 1. On its long path through the material the x-ray beam produces diffuse scattering background, which would normally limit the measurement to a small dynamic range. The small beam size and divergence allow to strongly limit the x-ray phase space covered by the detector and thereby minimize the diffuse background. 2. The incident angles for measurements at high x-ray energies are very small (the critical angle for an ice–Si interface at 70 keV is ≈0.3 mrad, see Tab. 7.1). Consequently, the projection of the interface perpendicular to 50 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

the beam is very small (≈10 µm at the critical angle for a sample length of 30 mm). The small beam size ensures that even at these small incident angles the entire beam hits the interface (see Sec. 5.3). This maximizes the scattering signal for reflectivity measurements and is absolutely necessary for evanescent scattering to avoid bulk scattering.

The small incident angles, and, as a consequence, the small projection of the interface perpendicular to the beam are the reasons for the major difficulties associated with this method. A high angular resolution (< 10 µrad) is needed for defining such small angles. A high resolution (< µm) in the height adjustment is needed to place the interface into the beam (see Fig. 5.12). Similar requirements concern the beam and sample stability, as well as the sample quality (curvature). The possibility of radiation damage is always a concern with experiments making use of highly brilliant Synchrotron Radiation. High-energy x-rays interact less strongly with matter (see Fig. 5.1) than x-rays with lower energies , but the small beam size used in these experiments can be a problem (see Sec. 7.8).

5.2 Application to the interface melting of ice

The main reason why experimental data on interface melting of ice is quite scarce is that adequate methods have been lacking so far (see Sec. 3.4). Conventional surface-sensitive x-ray scattering techniques in conjunction with thin films of ice (see Sec. 5.1) are not suitable because of the high vapor pressure of ice and the important changes in the surface morphology due to sublimation/resublimation. The high-energy transmission-reflection scheme presented in this chapter is the ideal probe for investigating ice–solid interfaces. It has already been successfully applied to study metal(liquid)–semiconductor(solid) interfaces [193, 194].

A schematical sketch of the setup for the study of an ice–SiO2–Si interface (see Chapt. 6) is shown in Fig. 5.3. A high-energy (≈70 keV) x-ray beam from a Synchrotron Radiation (SR) source is focused by a compound refractive lense (CRL). It penetrates the sample through the ice and hits the interface at a (small) incident angle. As can be seen from Fig. 5.1, the attenuation coefficient for water is very low at these energies. The reflected intensity is measured as a function of the momentum transfer qz perpendicular to the interface. The reflectivity profiles obtained in this way allow to deduce the profile of the (laterally averaged) density across the interface (see Chapt. 4). If a quasiliquid layer (qll) with different density emerges due to interfacial melting, interference fringes will appear on the reflectivity curve (see inset of Fig. 5.3). The distance of these fringes is related to the thickness L of the layer. Although the relevant order parameters for the melting transition are the Fourier components of the density, the average density probed by the reflectivity measurements plays an important role for the melting of ice (see Sec. 3.4). 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 51

70 keV qz

ice CRL SR source qll SiO 2 »2p/L »p2/Lqll Si sample cell TT. m Intensity

TTn m q q c z

Figure 5.3: Schematical sketch of the setup for the study of ice–solid interfaces with high-energy x-ray microbeams (see the text).

5.3 Experimental and instrumental considera- tions

Results from several experiments are presented in this dissertation (see Chapt. 7). The two main experiments of this work were carried out using the high-energy x- ray microbeam transmission-reflection scheme. The basic concept of this scheme has been presented in Sec. 5.1 of this chapter. In this section the experimental details of this scheme as used in the two main experiments are discussed. The experimental parameters of these two experiments were almost identical. Addi- tional experiments have been performed, some of them with completely different methods (neutron reflectivity). Since these experiments are of minor importance for this work, their experimental aspects are not discussed in the same detail. For the sake of clarity, the experimental aspects of these experiments are presented together with their results in separate sections of Chapt. 7. A sketch of the setup for the high-energy microbeam transmission-reflection scheme is shown in Fig. 5.5. The principal components are the source, the monochromator, the focusing device, the sample stage, and the detector. These components are presented in the following subsections. Some important aspects like resolution, coherence, and correction factors will also be discussed. The setup presented here has been installed at the high-energy beamline ID15A of the ESRF (a schematic layout of the beamline is shown in Fig. 5.4). For each experiment, the whole setup had to be reinstalled. A permanent setup will be available in the near future.

5.3.1 Source and optics The source of ID15A is a seven period permanent magnet asymmetric wiggler located at the center of the straight section. It has a critical energy of 44.1 keV 52 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

experimental setup

AMPW SW BL A PS MP MS ID15 A SS MB ID15 A hutch P hutch B

ID15 B hutch S

0m 26 38 42 42

Figure 5.4: Schematic layout of the high-energy beamlines ID15A with hutches P (port) and B (bow), and ID15B with hutch S (starboard) [195]. AMPW: Asymmetric multipole wiggler, SW: shield wall, BL: beam limiter, A: attenuator, PS: primary slits, SS: secondary slits, MP: monochromator for experimental hutch P (Laue), MS: monochromator for hutch S (Bragg/Laue), MB: monochromator for hutch B (for this experiment: double Laue) at a gap of 20.3 mm. The first element of the beamline is a 4 mm thick aluminum absorber which filters out x-ray energies below 40 keV (see Fig. 5.4). The experi- ments were carried out at an x-ray energy of around 71 keV (wavelength 0.175 A,˚ see Tab. 7.1). The energy was calibrated by measuring the scattering angle of a specific Si or ice Bragg reflection. For monochromatization of the beam, two asymmetrically cut and bent Si crystals in a fixed exit Laue geometry were used (see Fig. 5.5) [196, 197]. The first crystal creates a virtual source for the second crystal, which allows to compensate the divergence of the primary beam in one direction (horizontal). The energy bandpass of the monochromator crystals is determined by the asymmetric cut to the (111) reflection. A 37.74◦ cut for the first crystal and a 37.76◦ cut for the second crystal were used, which leads to a relatively large energy band pass of 165 eV, corresponding to an energy reso- lution ∆E/E ≈ 2 × 10−3. As the monochromator crystals are only exposed to the high-energy part of the x-ray spectrum, direct cooling of the first crystal in an In-Ga eutectic is sufficient and the crystals can be kept in air. Due to the simplicity of the setup and the low thermal load on the monochromator crystals, a high stability of the monochromatized beam is achieved. The beam is focused on the sample position (alternatively to the detector position) by a compound refractive lense1 [198, 199, 200] with an effective aperture of about 0.35 mm (at the x-ray energy of 71 keV). It consists of around 230 single aluminum lenses and produces a spot size of about 5 µm (vertical) × 20 µm (horizontal) at a focal distance of about 4.5 m (for exact numbers see Tab. 7.1). The beam profiles at the sample position are shown in Fig. 5.6. The long focal

1provided by A. Snigirev (ESRF) 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 53

a top view monochromator CRL sample detector MB tower table

2q mono2 »230 || mono1 detector

virtual source

0.8 m 2 m 4m 1.3 m

b side view 2qz

c1

c2 z y x

ai

Figure 5.5: Sketch of the setup for the high-energy microbeam transmission- reflection scheme. (a) Top view. (b) Side view. Two asymmetrically cut and bent Si crystals in a fixed exit Laue geometry are used for monochromatization (71 keV, bandwidth 165 eV). A compound refractive lense (CRL) serves to focus the beam on the sample position (spot size 5×20 µm2). A specially built diffractometer carries the sample chamber and defines the sample position and sample tilt. The detector is placed on a table which allows to set the horizontal and vertical scattering angle. 54 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS a vertical b horizontal

20

-15 10

-10 0 (µm) z (arb. units) -5 -10

dI/dx 0 -20 10 0 -10 -20 -30 -40 -50 -60 -50 0 50 dI/dz(arb. units) x (µm)

Figure 5.6: Beam profiles at the sample position. The measured intensity profile (derivative of a ‘knife-edge’ scan, open circles) is shown together with Gaussian fits (solid line). (a) Vertical profile, FWHM = 5.6 µm. (b) Horizontal profile, FWHM = 19.2 µm. distance leads to a small divergence of about 30 µrad. A flux of 2×1010 photons/s at a storage ring current of 200 mA can be reached in the focused beam. In comparison with the previously used [193] microbeam collimator, the lenses lead to a strongly enhanced stability and a flux which is about 7 times higher (at the same vertical spot size and a horizontal spot size of 20 µm for the lenses and 350 µm for the collimator).

5.3.2 Sample stage

The sample chamber (see Chapt. 6.5) is mounted on a specially designed diffrac- tometer in vertical axis geometry (see Fig. 5.7), which defines the sample position and tilt. It allows to control the vertical position z of the sample with an ac- curacy of 0.2 µm and the incident angle αi with an accuracy of 5 µrad. The surface normal of the sample is aligned with a double tilt station (χ1 and χ2) and the lateral position of the sample (x and y) can be adjusted with a double translation stage. The sample can also be rotated around the surface normal (important for measurements of anisotropies in the lateral structure as in [193]). The diffractometer can carry a load of up to 120 kg which allows to use heavy UHV chambers. For stability reasons, the whole diffractometer is mounted on a 2 ton granite block. 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 55

Figure 5.7: Diffractometer with sample chamber.

5.3.3 Detector

The scattering angle is defined by a set of detector slits made of tungsten. An identical slit system is used as scatter slits which define the field of view for the detector. Even when focusing on the sample position, the beam size at the detector position is as small as 50 µm vertical × 100 µ horizontal. This allows to work with small slit gaps leading to a very small solid angle covered by the detector on the order of 10−9 (10−10 when the beam is focused onto the detector). Consequently, the diffuse scattering background is reduced to a minimum. Together with careful shielding, this leads to a background level of only about 1 count/s in the detector at a primary beam flux of the order of 1010 photons/s, which allows to access a dynamic range of up to 10 orders of magnitude. As x-ray detectors, a Si-PIN-diode was used for the highest intensities and a scintillation counter for lower intensities. Due to the limited dynamic range of the scintillation counter (≈2×104), beam attenuators had to be used. An additional PIN-diode placed behind the CRL was used for monitoring the incident beam flux. The detector and slit system were mounted on a separate table decoupled from the sample tower. The vertical scattering angle (2θ⊥) is set by two independent translations integrated in the detector table, the horizontal scattering angle (2θk) by an additional translation and rotation. 56 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

5.3.4 Scattering geometry The scattering geometry is illustrated in Fig. 5.8. The momentum transfer is controlled by varying the angles αi, αf and β. Its components are given by

qx = k (cos αf cos β − cos αi) ,

qy = k cos αi sin β, (5.1)

qz = k (sin αi + sin αf ) .

In the case of in-plane scattering (used to measure specular reflectivity, for example) β = 0 and

qx = k (cos αf − cos αi) ,

qy = 0, (5.2)

qz = k (sin αi + sin αf ) .

In the case of specular reflectivity αf = αi and

qx = 0,

qy = 0, (5.3)

qz = k2α. with α = αi = αf . In this work, a so-called ‘rocking-scan refers to a rotation of the sample around the y-axis in the in-plane scattering geometry (see Fig. 5.9). During a rocking scan 2α = αi + αf = const. Such a rocking scan corresponds to a transverse momentum scan along qx. The perpendicular momentum transfer qz stays prac- tically constant during the scan. All the angles are usually quite small in reflectivity experiments, especially in experiments with high-energy x-rays. In this work, the largest angles are about 20 mrad. Therefore, the following approximations can often be used:

sin η ≈ η ≈ 0, (5.4) cos η ≈ 1 − η2/2 ≈ 1, where η is a small angle.

5.3.5 Resolution Calculating the total differentials from Eq. 5.2 and using the approximations in Eq. 5.4 yields the momentum transfer resolution for in-plane scattering geometry (see also Fig. 5.8): 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 57

detector Dv detector slits q Daf

k a side view f Dai qz z a ki f a x i interface qx

b top view Db b q k i x x i k q y f y

Dbf Dh z q k f interface c steric view

ki b a a x f i

y

Figure 5.8: Scattering geometry. The incident and scattered wave vectors are ki and kf , the incident and exit angles αi and αf , β. The momentum transfer is q = kf − ki with the components qz perpendicular and qx, qy parallel to the interface. The resolution is determined by the divergence (∆αi vertical, ∆βi horizontal), the acceptance (∆αf vertical, ∆βf horizontal) set by the detector slit openings (∆v, ∆h), and the wavelength spread. The latter leads to a spread of the wave vector length not shown in this illustration. 58 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

q q x

k k z f i qz af ai x interface y 2a

Figure 5.9: Illustration of a rocking scan. The sample is rotated around the y- axis. 2α = αi + αf is constant. The momentum transfer is fixed in the reference frame of the laboratory, but its components with respect to the interface change.

∆λ δq = q + k (α ∆α + α ∆α ) , x λ x i i f f δqy = k∆β = k (∆βi + ∆βf ) , (5.5) ∆λ δq = q + k (∆α + ∆α ) . z λ z i f

The resolution of the momentum transfer qz perpendicular to the interface is most relevant for reflectivity measurements. The resolution in qy is usually relaxed, leading to an effective integration over qy. The wavelength spread ∆λ/λ can be calculated from the energy resolution (bandwidth) ∆E/E:

hc λ = , (5.6) E ∆λ ∆E = . (5.7) λ E

∆αi and ∆βi are the angular beam divergence in the vertical and horizontal direction, respectively. ∆αf and ∆βf are the acceptance of the detector in the vertical and horizontal direction, respectively. They are defined by the opening of the detector slits, ∆v in the vertical and ∆h in the horizontal direction (see Fig. 5.8):

∆v ∆αf ≈ l , (5.8) ∆h ∆βf ≈ l , (5.9) where l is the distance between the detector slits and the sample. Additional contributions from the finite size of the illuminated area can be neglected. The 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 59 a rough substrate b smooth substrate

0.020 0.020 total Da 0.015 f 0.015 ) ) -1 -1 total (Å (Å 0.010 0.010 z z q q Da

? f ? 0.005 0.005 Da Dl/l Dai Dl/l i 0.000 0.000 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 q -1 q -1 z (Å ) z (Å )

Figure 5.10: Momentum transfer resolution δqz perpendicular to the interface as a function of qz. The contributions from the divergence ∆αi, acceptance ∆αf , and wavelength spread ∆λ/λ are plotted. (a) Experiment with rough substrate. The jump in the momentum resolution is due to a change of the slit settings. (b) Experiment with smooth substrate. The accessible qz-range is larger, since the small roughness leads to a slower decay of the reflected intensity. same applies to corrections due to the tilt of the detector slit gap and the change of the projected distance between detector and slits. The horizontal detector slit opening used in this work leads to an effective integration over qy (see above). The vertical detector slit opening leads to a moderate resolution in qz. In the experiments presented in this work, we are dealing with thin layers (≈ x nm), which lead to rather well separated (≈(0.6/x) A˚−1) and broad features on the reflectivity curve with modest requirements for the resolution. Several other considerations favor a larger resolution element:

• increased intensity,

• less sensitivity to misalignment,

• less sensitivity to sample imperfections (e.g., curvature), especially at small incident angles.

Therefore, the choice of the resolution (i.e. the vertical detector slit opening) is a compromise and the smallest possible resolution element δqz is not necessarily the best option. The resolution is shown Figs. 5.10, 5.11 and summarized in Tab. 5.3.5. The values have been calculated with the experimental parameters from Tab. 7.1. 60 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

a on specular path b during rockings scan

1.5x10-4 1.5x10-4 q =0Å-1 q = 0.3 Å-1 x rough substrate z 1.0x10-4 1.0x10-4 ) rough substrate ) -1 -1 Å ( Å ( x x q q ? -5 -5 5.0x10 ? 5.0x10 smooth substrate smooth substrate 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 -1.0x10-3 0.0 1.0x10-3 q -1 q -1 z (Å ) x (Å )

Figure 5.11: Momentum transfer resolution δqx parallel to the interface. (a) As a function of qz (at specular condition). (b) As a function of qx (at fixed qz and −1 qy = 0 A˚ ).

Table 5.1: Momentum transfer resolution (mean values). For q-dependence, see Figs. 5.10 and 5.11. substrate δqx δqy δqz (A˚−1)(A˚−1)(A˚−1) rough substrate 7 × 10−5 0.06 0.02 smooth substrate 5.5 × 10−5 0.01 0.01 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 61

5.3.6 Integration by the detector For the determination of the density profile from diffuse reflectivity, integrated rocking scans are used in this work (see Sec. 4.8.2). Integration of a rocking scan corresponds to the integration over a certain solid angle with the condition αi + αf = 2α = const.: Z

dαf dβ .

αi+αf =2α=const But even without explicit integration, the finite size of the detector slit open- ing always leads to an integration: Z dαf dβ.

The integration in angular space leads to an effective integration in momentum space. In this section, the connection between the integration over the solid angle R R dαf dβ and the integration over the lateral momentum transfer dqxdqy will be calculated for the case of a rocking scan and for the case of a finite detector slit opening (without rocking the sample). The partial derivatives are calculated from Eq. 5.1 (using the approximations from Eq. 5.4):   ∂ (qx, qy) −β −2α Jrocking = ≈ k , (5.10) ∂ (β, αf ) 1 −βαf αi+αf =2α=const.   ∂ (qx, qy) −β −αf Jdetector = ≈ k . (5.11) ∂ (β, αf ) 1 −βαf From this, we get

2 2  2 det (Jrocking) = k β αf + 2α ≈ k 2α ≈ kqz, (5.12) kq det (J ) = k2 β2α + α ≈ k2α ≈ z . (5.13) detector f f f 2 The last approximation in Eq. 5.13 is only valid close to the specular condition. We finally obtain Z Z 1 dαf dβ = dqxdqy, (5.14) rocking kqz Z Z 2 dαf dβ = dqxdqy. (5.15) detector kqz 5.3.7 Illumination of the sample The illuminated interface area and the proportion of the beam hitting the in- terface depend on the incident angle αi. The different situations are depicted in 62 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

0 Fig. 5.12. For αi < α0 = arcsin (wz/a), only a proportion fbeam = wz/wz of the beam hits the interface, and thus the reflected intensity is reduced by the same ratio. This geometric effect has to be accounted for, either by correcting the measured data, or by including it in the calculation of the reflected intensities:

−1 Rcorrected = fbeamRmeasured (5.16) or Rcalculated = fbeamRideal, (5.17) where   a sin (αi)  for αi < α0 = arcsin (wz/a) , wz fbeam = (5.18)   1 otherwise.

The angle of total illumination, α0, decreases with the sample length a. Therefore, a smaller part of the reflectivity curve is affected by the illumination correction for longer samples. With the experimental parameters for this work (see Tab. 7.1), α0 ≈ 0.24 mrad. Eq. 5.18 is valid when the beam has a box profile (in the vertical direction). The measured beam profile, however, is a Gaussian (see Fig. 5.6). For an arbitrary beam profile I(z), the incident intensity I0 calculates as

Z z+ I0 = I(z)dz, (5.19) z− where z± = ±a sin (αi) denotes the extension of the interface projected perpen- dicular to the beam (see Fig. 5.13). The correction factor is then z 0 R + I(z)dz I z− fbeam = = R +∞ . (5.20) I0 −∞ I(z)dz Graphs of the correction factor for a box profile and a Gaussian profile, as well as corrected and uncorrected reflectivity curves are shown in Fig. 5.14. However, the area correction often does not work as well as in the example shown in the graph. This is in part caused by the extreme sensitivity on the alignment. If the beam does not exactly hit the center of the sample, the formulae for the area correction are not strictly valid anymore. Furthermore, the effective sample length is in most cases smaller than the substrate length. One reason is the curvature of the sample (see Sec. 6.1). Another reason is that over the duration of a beamtime, some of the ice evaporates preferentially at the edges of the sample. 0 Once the whole beam illuminates the interface (αi = α0), its footprint a on the interface (see Fig. 5.12) rapidly decreases with αi (qz), see Fig. 5.15:

 wz for α > α = arcsin (w /a) ,  sin (α ) i 0 z a0 = i (5.21)   a otherwise. 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 63

a ai=0 b aai< 0

ai interface beam a’=a wz w’

a

c ai=a0 d aai> 0

a i ai a’=a a’

Figure 5.12: Illumination of the interface. αi incident angle, a length of the 0 0 sample, a illuminated part of the interface, wz beam height, wz part of the beam height which illuminates the interface. (a) αi = 0. (b) αi < α0. The whole length of the interface is illuminated (a0 = a), but not all of the beam hits the interface 0 (h < h), thus the reflected intensity is reduced. (c) αi = α0. At this angle, the 0 0 projection of the beam exactly covers the sample length (a = a, wz = wz). (d) 0 αi > α0. At larger angles, the whole beam hits the interface (wz = wz), but its footprint is shorter than the sample length (a0 < a). 64 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

beam profile z

ai

z+ I’

z-

I(z) a

Figure 5.13: Illumination of the interface with an arbitrary beam profile. The incident intensity I0 is the integral of the intensity profile I(z) over the projection of the interface onto the z-axis.

a correction factor b measured and corrected data

1.0 100 measured data corrected data: 0.8 box profile box profile Gaussian profile 0.6 Gaussian profile 10-1 beam f

1/ 0.4

0.2 -2 aa 10 i 0 i 0 Intensity (arb. units) 0.0 0.00 0.02 0.04 0.06 0.02 0.04 0.06 0.08 q (Å-1) q (Å-1) z z

Figure 5.14: Illumination correction. (a) Correction factor as a function of qz for a box-profile (Eq. 5.18 ) and a Gaussian profile (Eq. 5.20). (b) Open cir- cles: measured (uncorrected) reflectivity data , solid line: corrected with box profile, dashed line: corrected with Gaussian profile. The Gaussian profile allows a slightly better correction (note the small dip on the edge of total reflection for the correction with the box profile). Renormalization of the data after correction leads to a small difference even at large qz, where the correction factor is 1 for both beam profiles. 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 65

30 experiment 1 a =24mm,w = 5.6 µm z 20 (mm) experiment 2 a' a =30mm,w = 7.1 µm z 10 Footprint

0 0.0 0.2 0.4 0.6 0.8 1.0 q (Å-1) z

Figure 5.15: Length a0 of the beam footprint on the interface for the rough substrate (solid line) and the smooth substrate (dashed line) calculated from Eq. 5.21. a: sample length, wz vertical beam size (FWHM).

0 For a Gaussian profile, wz denotes the FWHM of the beam profile and a the FWHM of the beam projected onto the interface.

The small footprint for larger values of qz has several advantages:

• less sensitivity to sample curvature,

• less sensitivity to sample inhomogeneities (in this case: temperature gradi- ents),

• no influence from the edges of the sample.

For small values of qz, however, the footprint is large. While temperature gradi- ents are nonetheless small (see Sec. 6.6), scattering from the edges of the sample can be a problem.

According to Eq. 5.21 the illuminated area A is for αi > α0 w w A = z y sin αi 2kw w (5.22) ≈ z y close to the specular condition, qz wy denotes the horizontal width of the beam profile. This equation is used in Chapt. 4 where the diffuse reflectivity is discussed. Unlike for specular reflectivity, the intensity of the diffuse reflectivity is proportional to the illuminated area. 66 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

a b c

z wz x t x x t ki xz x t a l a i r kf x x ai a t b f

Figure 5.16: Coherence in x-ray scattering. (a) Illustration of longitudinal (ξl) and transverse (ξt) coherence. (b) Projections of the transverse coherence. (c) Path length difference a − b for scattering from two points separated by r.

5.3.8 Coherence A wave field is called coherent, when it is able to produce interference phenom- ena. This requires a spatial and temporal relation between the phases of the wave field. The degree of coherence is usually characterized by a longitudinal and a transverse coherence length, ξl and ξt, respectively (see Fig. 5.16a). The longitudinal coherence is a measure of the monochromaticity of the source,

λ2 ξ = , (5.23) l ∆λ where λ denotes the wavelength and ∆λ/λ the wavelength spread. The longitu- dinal coherence length is associated with a coherence time ξ τ = l , (5.24) c c being the speed of light. The transverse coherence length is related to the finite size s of the source, λd ξ = , (5.25) t s where d denotes the distance from the source. Note that the definitions for the longitudinal and transverse coherence length that can be found in the literature sometimes contain an additional factor of 1/2 or 1/π. In the case of high-energy x-rays the wavelength is small, which leads to a low degree of coherence. In this work, only a moderate energy resolution ∆λ/λ has been used, which further decreases the longitudinal coherence length (for the values see Tab. 5.2). 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 67

The degree of coherence is crucial for the interpretation of the x-ray scattering signal. In most cases, an intermediate degree of coherence is desired. If the scat- tering were completely incoherent, no interference would be possible. If, on the other hand, the scattering is completely coherent (the effective coherence length reaches the dimensions of the sample), the scattering signal does not correspond to an average structure anymore. In this case, a complicated speckle pattern is produced, which contains information about the whole sample. Some techniques exploit the coherent regime, but in most cases it is better to avoid this situation in order to obtain an averaged information. The longitudinal and transverse coherence length can be used to calculate whether scattered waves from two points of the sample separated by the vector r can interfere. First, the lateral coherence length ξl has to be larger than the path length difference l for a given momentum transfer q (see Fig. 5.16c), r · q ξ > l = b − a = r (cos α − cos α ) + r (sin α + sin α ) = . (5.26) l x f i z f i k If we define ξ k ξx = l , l q x (5.27) z ξlk ξl = , qz the condition expressed by Eq. 5.26 is fulfilled for

x (ξl > rx and rz = 0) or z (5.28) (ξl > rz and rx = 0).

The second condition for interference is that the transverse coherence length projected on r is larger than r. This can also be written as (see Fig. 5.16b)

ξ ξx = t > r and t sin α x i (5.29) z ξt ξt = > rz. cos αi If the limiting coherence length is smaller than the linear dimensions of the scattering volume, the scattering signal corresponds to an averaging over this volume. In the experiments of this work, the coherence is large enough to produce interference from the density profile perpendicular to the interface and from the roughness of the interface (see Tab. 5.2). But at the same time, the transverse coherence length is smaller than the beam size, and thus the transverse coher- ence length projected on the interface smaller than the illuminated length. This 68 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

Table 5.2: Coherence parameters. rough smooth substrate substrate λ (A)˚ wavelength 0.1740 0.1736 ∆λ λ (%) wavelength spread 0.23 0.23 d (mm) distance source (lenses) to sample 4380 4530 s (µm) source size (effective aperture) 350 350 ξl (A)˚ longitudinal coherence length 76 75 ξt (A)˚ transverse coherence length 2200 2200 max ˚−1 −4 qx (A ) max. parallel 0.003 10 momentum transfer max ˚−1 qz (A ) max. perpendicular 0.57 1.0 momentum transfer x ˚ max 5 7 ξl (A) lower limit (at qx = qx ) 9.1×10 2.7×10 z ˚ max ξl (A) lower limit (at qz = qz ) 4800 2700 x ˚ max 4 4 ξt (A) lower limit (at qz = qz ) 2.8×10 1.6×10 z ˚ ξt (A) lower limit (at αi = 0) 2200 2200 wz (µm) beam size vertical 5.6 7.1 wy (µm) beam size horizontal 19.2 ≈24 means that the scattering signal corresponds to an (incoherent) average over the illuminated sample area as assumed in Sec. 4.8. The reader is referred to the literature for more details on coherence in x- ray physics in general [201], and coherence in x-ray scattering experiments in particular [202].

5.3.9 Data correction The correction due to different illumination of the interface has already been discussed in detail in the last section. As mentioned above, a PIN diode has been used to monitor the intensity of the incident beam. This intensity varies due to the decrease of the storage ring current between injections, temperature changes of the monochromators and other reasons. Therefore, all measured intensities have been normalized to the incident beam intensity. Several types of background show up in the data. One part is due to scattering from the air, optical elements, and the bulk of the sample. These contributions can be strongly reduced with shielding and small slit openings. When specular reflectivity is measured, the diffuse reflectivity from the roughness of the inter- face represents another type of ‘background’. Separation and analysis of the background will be discussed in Sec. 7.3. 5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 69

As mentioned before, absorbers had to be used due to the limited dynamic range of the detectors. Different parts of the reflectivity curves were thus mea- sured with different absorber settings and had to be adjusted (see Sec. 7.2.1). Finally, the data had to be scaled appropriately (see Chapt. 7) as the measure- ments do not yield directly the absolute reflectivity, but only a scattered intensity proportional to the reflectivity. 70 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS Chapter 6

Sample preparation and environment

In this chapter, the samples, preparation techniques, and the sample environment are described. The chapter includes sections about the substrates and their sur- face preparation (Sec. 6.1), the ice single crystals (Sec. 6.2), and the preparation of well-defined ice–substrate interfaces (Sec. 6.4). Another section describes the in situ chamber designed for the ice experiments and the temperature control sys- tem (Sec. 6.5). The last section discusses the temperature stability and accuracy (Sec. 6.6).

6.1 The substrates

The aim of this work was to investigate the melting behavior at the heterogeneous interface between ice and a solid ‘substrate’. There were two main considerations for the choice of SiO2 as the substrate material: 1. the relevance of this particular ice–substrate interface, 2. the suitability of the substrate for the sample preparation and the reflec- tivity measurements. A number of interfaces could serve as model systems for important ice inter- faces in nature and technology. The ice–SiO2 interface can be regarded as an idealized model system for the situations where ice is in contact with a mineral like at glacier beds or in permafrost. As discussed before, the high-energy x-ray transmission-reflection scheme (see Chapt. 5) used for the experiments puts high demands on the sample quality. The sample has to be smooth on an atomic scale, must have a very low waviness and curvature (which has been characterized by interferometry1, see Fig. 6.1), and should be long in the direction of the beam. 1in collaboration with A. Weißhardt (MPI f¨urMetallforschung)

71 72 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT a rough substrate b smooth substrate

0.5 x direction, 0.5 x (long) direction, 0.0 y=0 1mm 0.0 y=0 1mm m) m) -0.5 ? ? ( ( -0.5 z z -1.0 y (short) direction, y direction, -1.0 x=0 -1.5 x=0

-2.0 -1.5 -10 -5 0 5 10 -15 -10 -5 0 5 10 15 x y x (mm),y (mm) (mm), (mm)

Figure 6.1: Figure error of the substrates. (a) Rough substrate. (b) Smooth substrate. In both cases, the figure error is about 1 µm over the whole sample, which corresponds to an angular spread of about 50-200 µrad.

A SiO2 substrate which fulfills these requirements can be prepared from a sil- icon block. Large single crystals of silicon are readily available and can be easily polished. After cleaning, a native amorphous oxide forms at the surface when silicon is kept in air. Compared to quartz glass, an oxidized silicon block also has the advantage of a high thermal conductivity (about two orders of magnitude larger), which is important for the sample preparation (see Sec. 6.4). Moreover, a substrate with an amorphous termination comes closer to an ideal hard wall. Unlike a crystalline substrate, it does not impose a specific structure in the adja- cent quasiliquid and, therefore, is thought to reveal more intrinsic properties of water. Several parameters can have an influence on interface melting (see Chapt. 8), like • the substrate material, • substrate morphology, • crystal orientation, • impurities.

In this work, interface melting at the ice–SiO2 interface was studied. In addition, the influence of one of the aforementioned parameters, namely the substrate morphology, was tested. Therefore, two substrates with different roughness were prepared, a ‘rough’ substrate and a ‘smooth’ substrate. See Sec. 7.1 for an overview of the experiments and the substrates used. For a detailed discussion of the substrate morphology, see Sec. 7.3. 6.1. THE SUBSTRATES 73

Figure 6.2: Photograph of a polished Si block used as substrate (here: the smooth substrate). The top face is contacted with the ice single crystal after cleaning.

The dimensions l × w × h of the substrates in mm (with the length l along the beam and the height h perpendicular to the interface) were 24 × 24 × 10 for the rough substrate and 30 × 24 × 8 for the smooth substrate. The surface cut was (111) for the rough and (100) for the smooth substrate. The chemo- mechanical polishing of the substrates was performed by H. Wendel (MPI f¨ur Festk¨orperforschung) for the rough sample and by the crystal laboratory group of the ESRF for the smooth sample. A photograph of a substrate is shown in Fig. 6.2 and a typical reflectivity measurement on a substrate in Fig. 6.3. After polishing the substrates were thoroughly cleaned in several steps using some standard recipes:

1. ‘Piranha clean’

2 • H2SO4 (80%) and H2O2 (30%), volume ratio 3:1 (gets hot ), 5 minutes, removes organic contaminants and chemically oxidizes the surface

• rinsing with H2O, 3 minutes • HF dip (5%), 2 minutes, removes the oxide layer

• rinsing with H2O

2. RCA clean

◦ • H2O, NH3 (25%) and H2O2 (30%), 4:1:1, 70 C, 10 minutes, also re- moves organic contaminants

• rinsing with H2O • HF dip (5%), 1 minute

2Contact with residues of organic solvents must be avoided, explosive reactions! 74 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT a reflectivity b transverse momentum scan 100 4.0x10-5 -1 -1 q = 0.171 Å 10 z 10-2 3.0x10-5 see b 10-3 -5 10-4 2.0x10 10-5 1.0x10-5 10-6 Intensity (arb. units) -7 Intensity (arb. units) 10 0.0 0.0 0.1 0.2 0.3 0.4 0.5 -2.0x10-4 0.0 2.0x10-4 q (Å-1) q (Å-1) z x

Figure 6.3: Reflectivity measurement of a substrate with the diffractometer of the cold room (Cu-Kα x-ray radiation). (a) Reflectivity curve. Open circles: measured points. Filled squares: integrated transverse momentum scans. Solid curve: fit with oxide layer thickness of 15 A˚ and rms roughness of 3.9 A.˚ (b) Example of a transverse momentum scan (filled squares) and fit (solid line) with a Gaussian (representing the resolution function).

• rinsing with H2O

◦ • H2O, HCl (30%) and H2O2 (30%), 8:1:1, 70 C, 10 minutes, removes metallic contaminants, oxidizes the surface

• rinsing with H2O

• HF dip (5%), 1 minute, removes the oxide

The silicon sample was then kept in air which leads to the formation of a thin native amorphous oxide layer. The thickness of this oxide layer is 20 A˚ for the rough substrate and 16 A˚ for the smooth substrate. The contact angle of the substrates depends on the chemical treatment used for the surface preparation (see for example [203]. The method used in this work initially leads to a hydrophobic surface (due to the HF dip at the end). The oxide-covered surface is usually hydrophilic, but can become hydrophobic due to contaminations with hydrocarbons from the air. When exposed to intensive high-energy x-ray radiation in the presence of water, the surface becomes strongly hydrophilic again (see Sec. 7.8). An additional experiment was performed with Si wafers and a different chemi- cal treatment was used. For details see Sec. 7.6. For more information on Si/SiO2 surfaces and preparation techniques, see [204, 205, 203, 206, 207]. 6.2. THE ICE SAMPLES 75

a b

hot wire

I

ice

Figure 6.4: Cutting of . (a) Schematic view: A resistance wire grid is pulled through the ice. (b) Photograph after cutting: A short disk has just been cut from the large cylindrical ice single crystal.

6.2 The ice samples

Ice single crystals were provided by J. Bilgram (ETH Z¨urich) [8]. The single crystals were grown with a zone-melting technique from high-purity water (con- ductance < 10−7 S/cm). They are of very high quality with a dislocation density of about 103/cm2 (determined from Lang topographs) and a mosaicity of around 0.3 mrad (determined from x-ray rocking scans). The crystals have a cylindrical shape with a diameter of about 50 mm and a length of about 300 mm. For the crystals used in this work, the cylinder axis was oriented along the [0001] direc- tion (c-axis), but other orientations are possible. Smaller pieces with a length of about 15 mm were ‘thermally’ cut from the large crystal using hot wires that were pulled through the ice (see Fig. 6.4). In principle, mechanical sawing is also possible below ≈−15◦C, but leads to more crystal defects [208]. A walk-in cold room was installed and operated for handling the ice sam- ples (see Sec. 6.3). Due to the high vapor pressure of ice, the samples cannot be kept in air for long periods of time. Instead, they were kept in heptane (n- 3 ◦ heptane C7H16 ‘for analysis’, density 0.68 g/cm , melting temperature −90.6 , vapor pressure 48 hPa), which is practically insoluble in ice. It prevents sublima- tion/resublimation of ice that would otherwise lead to a visible deterioration of the ice single crystals [208, 14]. When taken from the heptane, the ice samples were kept in air for some time before further use. This allowed the remaining traces of the highly volatile heptane to evaporate. For the transport of ice sam- ples, a mobile freezer was used, which can also be operated in a car using the 76 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT on-board electrical system.

6.3 The cold room

A walk-in cold room (≈20 m3)was installed for handling the ice samples (storage, characterization, preparation, and mounting). It consists of a main room (about −17◦C) and an entry room (about −14◦C), which is the only access to the main room. This reduces air exchange between the main room and the exterior and leads to low humidity levels in the main room despite of its small size. For stability reasons, a rigid metal frame, which all the experimental tables are bolted to, has been mounted in the cold room. Besides the ice samples, the cold room contains all the sample preparation tools (see Secs. 6.2 and 6.4) and a two-circle diffractometer for orienting ice samples prior to contacting. The diffractometer also allows to check the crystallinity of the samples and to measure the reflectivity of the substrates. The x-ray tube (Cu target, operational up to 60 kV and 1.5 kW), high-voltage generator, motor controls, and control computers are placed outside the cold room. The x-ray beam is fed into the cold room via a beam tube, and the cold room also serves as radiation shielding. The distance between source and sample is about 0.6 m. A focusing Ge(111) monochromator and two slit systems are used, a pair (horizontal and vertical) of entry slits and a pair of detector slits. A NaI scintillation counter is used as x-ray detector. Sev- eral feedthroughs, cameras, a temperature monitoring system, and an intercom system have also been installed.

6.4 Interface preparation

Samples which had been cut from the large crystals were then mounted on a goniometer head. Using the two-circle diffractometer, a specific crystal axis (here the [0001]-direction) could be aligned parallel to the axis of the goniometer. The goniometer head was then mounted on an optical bench preserving the orientation of the ice sample. The substrate was mounted on the same optical bench with its surface normal aligned along the crystal axis of the ice sample (see Fig. 6.5). The subsequent contacting of the two surfaces (see below) lead to an oriented ice–SiO2 interface. The interface normal deviated from the desired crystal axis by less than 0.1◦ (miscut). For the first experiment (rough sample), however, the two-circle diffractometer was not yet operational, therefore the miscut was about 1◦. The temperature of the substrate during the preparation process was con- trolled via a Peltier element. The substrate was heated to about +4◦C and then brought into contact with the ice sample leading to the formation of a molten zone between the ice and the substrate. The temperature was then slowly re- 6.4. INTERFACE PREPARATION 77

a Peltier b c

+4°C Si ice -5°C

goniometer head

optical bench

Figure 6.5: Interface preparation. (a) Sketch of the setup for contacting ice with the substrate (side view). The desired ice crystal axis (here the c-axis) is aligned with a goniometer head. The Si substrate is heated with a Peltier element and moved against the ice creating a thin molten layer. Subsequent cooling leads to recrystallization, if the right temperature gradient is maintained. (b) Photograph of the sample after contacting (still mounted on the preparation stage). The protruding parts of the ice still need to be molten away. 78 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT

abc

Figure 6.6: Photographs of the sample. (a) Protruding parts of the ice have been molten away. (b) A temperature sensor has been molten into the ice. (c) The whole sample has been wrapped in aluminum foil. duced while continuously moving the substrate against the ice (with a microme- ter screw) and thus keeping the molten layer very thin. This prevents inclusion of air bubbles in the molten layer and flushes out contaminants (which are much more soluble in water than in ice). The temperature at the substrate was always kept higher than the temperature at the ice, which leads to epitaxial recrystal- lization of the molten layer at the ice–water interface instead of the formation of polycristalline ice at the water–substrate interface. The ice part of the sample was then brought into the right form by carefully melting away protruding parts, so that the whole sample finally had the shape of a parallelepiped (see Fig. 6.6a). In the next step, a calibrated Pt100 resis- tive temperature sensor was molten into the ice (see Fig. 6.6b). The sample was thereafter sealed in high-purity aluminum foil to prevent contamination and sub- limation/resublimation of the ice (see Fig. 6.6c). The sample was then ready to be mounted in the sample chamber used for the actual experiments (see Sec. 6.5).

6.5 The in situ chamber

A special sample chamber (see Fig. 6.7) has been designed for experiments with ice samples and high-energy x-ray radiation. Only a small fraction of the high- energy x-ray radiation is absorbed by the aluminum windows of 0.1 mm thickness. Since the windows are also practically transparent for neutrons, the chamber is suited for neutron experiments as well. The inner part of the chamber can be exchanged. A different inner setup has been designed for Neutron Compton Scattering experiments on ice (see Chapt. 8). For the interface experiments, the sample was mounted on a copper holder. The temperature was controlled via two Peltier elements (lapped and sealed) situated on two sides of the sample holder, thereby minimizing temperature gra- 6.5. THE IN SITU CHAMBER 79

X-ray beam a b

1 3 5 2 4

100 mm c d

1

e f

6

Figure 6.7: In situ chamber for x-ray and neutron scattering experiments with ice providing stable control of the sample temperature (see the text). (a)+(b) Top view of open chamber. (c)+(d) Side view of open chamber, photograph). (e) Side view of closed chamber. (f) Look into open chamber (rendered image). (1) Sample (sealed with aluminum foil, calibrated sensor frozen into the ice). (2) Peltier element. (3) Heat exchanger. (4) Tube for the cooling liquid. (5) Electrical feedthrough for sensor and Peltier leads. (6) Aluminum window. 80 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT dients in the sample. A cooling liquid (about −15 to −5◦C) was used to remove heat from the backside of the Peltier elements via heat exchangers. A refrigerated bath/circulator (Haake DC10-K15 [209], temperature accuracy ±0.02 K) served for keeping the temperature of the cooling liquid constant. For measurements close to the melting point, instead of raising the temperature of the cooling liq- uid, the polarity of the current for the Peltier elements was reversed. In this mode, they were heating ‘against’ the cooling liquid. This has two advantages: First, the cooling liquid prevents the sample from melting, even if the Peltier elements fail. Second, the temperature stability is better when the temperature of the cooling liquid remains constant. On each side of the sample, a Pt100 temperature sensor (Heraeus M-FK 222, DIN 1/3 B [210]) was placed on the sample holder. Together with the two Peltier elements, they formed two independent control loops. A factory calibrated Pt100 sensor (Lakeshore Model Pt-111 [211]) was molten into the ice for measuring the sample temperature. Additional Pt100 sensors were installed to monitor the temperature in other parts of the chamber (on the heat exchangers, e.g.). These sensors had been calibrated using the factory calibrated sensor which was later molten into the ice. All sensor measurements were performed with a 4-lead technique to eliminate the effect of lead resistance. Thin manganin wires with low thermal conductivity were used for the wiring in order to minimize heat transfer along the leads. A Lakeshore Model 340 temperature controller [211] was used for temperature measurement and control. The sensor excitation was 1 mA. Two DC power supplies (HP/Agilent 6553A [212]) delivered the current for the Peltier elements. They were programmed by the Lakeshore controller and had to be connected to the latter via a voltage converter specifically designed for this purpose. A small programme was written for logging the temperatures on a personal computer connected to the Lakeshore controller via a serial connection. A schematic view of the control setup is shown in Fig. 6.8.

6.6 Temperature stability and accuracy

The temperature stability was in the mK regime as can be seen from Fig. 6.9 show- ing part of a temperature log. Temperature stabilities of ±1 mK were reached, which is better than the specification (at 300 K) of the Lakeshore 340 temperature controller (±5 mK). One has to distinguish between temperature stability and accuracy though. The specified accuracy of the controller is ±30 mK. For the measurement of the sample temperature, a factory calibrated Lakeshore Model Pt-111 was used (see above). This sensor was also used for calibrating the remaining sensors. The error of this calibration was estimated by performing two calibrations, one before the experiment and one after the experiment. The differences were on the order of ±10 mK. 6.6. TEMPERATURE STABILITY AND ACCURACY 81

loop 1

Tsample sample holder loop 2

\ \b \r\f \n \b \b\n \b \b

\n\b \n \b \ !\b

\n\b \f \f\ %& \n !\b\b \r\n \

\ \f \b !\b\b \b \f heat \n \ \f\ ice exchanger temperature Si controller computer sensor

Peltier signal converter cooling liquid power bath supplies

Figure 6.8: Schematic view of the control setup. The temperature at two sides of the sample is controlled via independent control loops. A temperature con- troller reads out the Pt100 sensors and programs the power supplies providing the current for the Peltier elements, which can be either used as cooling or heating devices. The temperature on the backside of the Peltier elements is defined by a cooling liquid circuit. The temperature measurements are logged on a computer.

-0.030

-0.032

-0.034

-0.036

-0.038 Temperature (°C) -0.040 00:00 02:00 04:00 06:00 Time (h)

Figure 6.9: Part of temperature log (from first experiment with the rough sub- strate). The sample temperature −0.036◦C varies by only ±1 mK over the du- ration of a measurement (≈ 6 hours). 82 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT

Another source of errors are temperature gradients. The effect of these gra- dients is two-fold:

1. The temperature of the sensor is not exactly the temperature of the interface area probed by the x-ray beam.

2. Owing to the finite size of the x-ray beam the measured signal corresponds to an average over a certain temperature range.

In order to estimate effects of temperature gradients, measurements of the temperature distribution have been performed. Therefore 8 sensors were used (2 were used for the control loops and placed on the sample holder, 6 were molten into the ice, see Fig. 6.10a). The variation ∆T of the temperature over the whole sample gives an upper limit of the error caused by temperature gradients. It is plotted in Fig. 6.10b as a function of temperature. It decreases on approach- ing 0◦C, as the temperature difference between the sample and the surrounding gets smaller. As this is exactly where the measurements are most temperature sensitive (due to the divergence of the quasiliquid layer thickness), the following considerations refer to ≈0◦C. At this temperature the ∆T is as small as ±15 mK. Temperature gradients along the different directions are plotted in Fig. 6.10c. The gradient along the z-direction (perpendicular to the interface) is largest (1.9 mK/mm). As the sensor used for measuring the sample temperature during the experiments is placed approximately 7 mm above the interface, it may yield a temperature which is ≈13 mK higher than the actual interface temperature (for a discussion of the effect on the results, see Sec. 7.4). The gradients in the x and y-direction (parallel to the interface) cause an averaging over a certain temperature range. The gradient in the y-direction (per- pendicular to the copper holder) is 0.92 mK/mm. This direction is perpendicular to the beam and the temperature variation over the width (≈0.02 mm) of the beam is negligible. However, the beam might not be exactly perpendicular to the y-direction, and therefore cover a larger y-range. But even in the case of a 3◦ tilt, the temperature range covered by the beam is only ±0.5 mK at full illumi- nation of the sample length. The temperature distribution in the y-direction is not symmetric which indicates that the thermal contact is not the same on both sides of the sample. The assessment of the effect of gradients in the x-direction (along the beam) is more complicated, as the beam footprint on the interface changes with the mo- mentum transfer qz (see Sec. 5.3.7). Fig. 6.10d shows the temperature range along the x-direction covered by the beam footprint as a function of the momentum transfer qz. It starts with a maximum value of 15 mK and rapidly decreases with −1 qz. At qz = 0.1 A˚ , where the first features appear on the reflectivity curves, it is as small as 2.6 mK. The temperature distribution along the x-direction is symmetric as would be expected. To summarize this subsection: 6.6. TEMPERATURE STABILITY AND ACCURACY 83

a 250 S6 S7 S8 b S1 z 200 S5 150 ice S4

Si T(mK) 100

y D X-rays 50 x S2 S3 0 -20 -15 -10 -5 0 on sample holder Temperature (°C)

20 18 c 16 d 10-1 T x d /d 14 15 dT/dy -3 12 10 dT/dz Reflectivity 10 10 -5

(mK) 10

x 8 T

D -7 5 6 10 grad(T) (mk/mm) 4 DT x -9 2 10 Intensity (arb. units) 0 0 0.0 0.2 0.4 0.6 0.8 1.0 -20 -15 -10 -5 0 -1 q (Å ) Temperature (°C) z

Figure 6.10: Temperature distribution in the sample. (a) Setup for measuring the temperature distribution. Sn: Pt100 temperature sensors. S1: factory cali- brated sensor. S2, S3: control loop sensors on sample holder, S4-S8: sensors for measuring the temperature distribution (together with S1). The x-ray beam is roughly parallel to the x-direction. (b) Variation ∆T of the temperature over the whole sample as a function of the nominal temperature. With increasing temperature, the temperature difference between the sample and the surround- ing decreases. This results in a more uniform temperature distribution over the sample. (c) Temperature gradients along different directions. Gradients parallel to the interface (x and y) lead to an averaging of the temperature ‘seen’ by the x-ray beam (see d). The gradient perpendicular to the interface (z) may induce an offset between the measured temperature and the true interface temperature. (d) Temperature range ∆Tx covered by the footprint of the x-ray beam along x (solid line, experimental parameters from the smooth substrate). A measured reflectivity curve at T = −0.034◦C is shown in the same graph (dashed line). At −1 qz = 0.1 A˚ , where the first features appear on the reflectivity curve, ∆Tx is as small as 2.6 mK. 84 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT

• The temperature stability is of the order of ±1 mK (see Fig. 6.9).

• The temperature error is between ≈ +40 and ≈ −20 mK (specified accuracy plus offset between sensor and interface).

• The temperature range, over which the beam averages, is <3 mK for the relevant part of the reflectivity curve and never larger than 15 mK.

Suggestions for improving the temperature accuracy are discussed in Sec. 8.5. Chapter 7

Results and discussion

The results obtained in this work will be presented and discussed in this chapter. The main results were obtained from two high-energy x-ray reflectivity experi- ments. The theory (Chapt. 4) and experimental details (Chapt. 5) of high-energy x-ray reflectivity have been presented in previous chapters. In the two main ex- periments, well-defined ice–solid model interfaces (see Chapt. 6) have been stud- ied as a function of temperature, and interfacial melting has been observed. Two substrates with different morphology were used, which are referred to as the rough and the smooth substrate. The morphology of the substrates is characterized in Sec. 7.3. An overview and a qualitative interpretation of the reflectivity measurements from the two main experiments is given in Sec. 7.1. Quantitative results can be obtained by reconstructing density profiles from the reflectivity measurements using the methods presented in Chapt. 4. This is done in Sec. 7.2 and the reliability of the results is discussed. The reconstruction of the density profiles yields the growth law of the quasiliquid layer, which is presented and discussed in Sec. 7.4. An intriguing aspect of this work is the extraordinarily high density observed in the quasiliquid. Sec. 7.5 highlights this point and offers a possible explanation. Sec. 7.8 deals with the effect of high-energy x-rays on the substrate termination and the consequences for interfacial melting. Sec. 7.9 finally discusses implications of this work and its relation to other studies. Further evidence for interfacial melting of ice has been obtained from two other experiments, one with high-energy x-ray reflectivity (using a slightly modified setup) and a silicon wafer as the substrate (Sec. 7.6), and the other with neutron reflectivity (Sec. 7.7). The results from these two experiments are presented in separate sections of this chapter together with short subsections about the respective samples and methods. This division has been chosen because these two experiments represent only a minor contribution to the overall results of this work, but use a different experimental approach (at least in the case of the neutron experiment). Considering the significance of the results, a more detailed discussion of these experiments and the methods does not seem appropriate.

85 86 CHAPTER 7. RESULTS AND DISCUSSION

Table 7.1: Overview of experimental parameters. rough substrate smooth substrate beamtime 08/2002 and 08/2003 12/2002 x-ray energy (keV) 71.26 71.44 bandwidth (%) 0.23 0.23 x-ray wavelength (A)˚ 0.1740 0.1736 critical angle αc (mrad) 0.3266 0.3257 −1 corresponding qc (A˚ ) 0.02359 0.02358 no. of lenses in CRL 232 223 focal distance of CRL (mm) 4380 4530 beam size∗ vertical (µm) 5.6 7.1 beam size∗ horizontal (µm) 19.2 ≈24 detector slit vert. (mm) 0.06–0.6 0.25 detector slit horiz. (mm) 2 0.3 divergence vertical (µrad) 30 30 divergence horizontal (µrad) 54 54 −1 −3 −2 −3 −2 resolution δqz (A˚ ) 4.3×10 –2.1×10 9.6×10 –1.2×10 temperature range (◦C) −30 to −0.036 −30 to −0.02 max. momentum transfer (A˚−1) 0.57 1.0 size of substrate (mm) 24 × 24 × 10 30 × 24 × 8 orientation of substrate (111) (001) orientation of ice (0001) (0001) miscut of ice 1◦ 0.1◦ rms roughness of substrate (A)˚ -† (2.7 ± 0.4) A˚ ∗ FWHM † self-affine roughness, see Sec. 7.3

7.1 Overview of the main experiments

The main results of this work have been obtained from two experiments using two substrates of different morphology. An overview of the experimental parameters of these two experiments is given in Tab. 7.1. On each substrate, reflectivity measurements have been performed at various temperatures, ranging from far below (−30◦C) up to very close (≈ −0.022◦C) to the bulk melting temperature. In order to test the reversibility with temperature, measurements have been done during alternating cooling and heating cycles, and repeated measurements have been performed for some temperatures. Figs. 7.1 and 7.2 show ‘time lines’ for the experiments indicating the temperature and chronology of the individual reflectivity measurements. For the rough substrate, some additional measurements have been performed during another beamtime in order to verify the observations. These measurements are also included in Fig. 7.1. 7.1. OVERVIEW OF THE MAIN EXPERIMENTS 87

1E-4 1st 2nd beamtime beamtime 1E-3 T15 substrate 0.01 (mm) (K) T12 -0.036 y -T m T 0.1 T11 -0.114 T14 -0.175 T10 -0.25 T02 -0.5 T09 -0.5

1 T08 -1.0 T07 -1.7 T06 -3.1 T05 -7.7

10 T01 -12.7 T04 -14.7 T13 -25.0 T03 -30.0 60 number of measurement

Figure 7.1: Time line of the experiments with the rough substrate. The tem- perature (left axis, bottom curve) and the position (right axis, top curve, only schematical) are plotted against the time (represented by the number of the mea- surement). Each measurement is consecutively numbered (Tnn, where nn denotes the number) and labelled with the temperature in ◦C. Measurements with the rough substrate were performed during two different beamtimes. During the sec- ond beamtime, a measurement on the bare substrate (denoted T15 substrate) was also performed after removing the ice.

In the case of the smooth substrate, measurements have also been performed at different positions on the interface. This is also indicated in Fig. 7.2. The measured reflectivity curves together with a fit to the data (see Sec. 7.2.2) are shown in Figs. 7.3 and 7.4. With increasing temperature, an additional signal appears at high momentum transfers qz and gets gradually shifted towards smaller qz. With a further increase of the temperature, this signal develops into interference fringes, causing a modulation of the reflectivity curve. The distance between interference fringes decreases with temperature. This observation unambiguously shows that a layer with different density emerges (giving rise to interference fringes) and that the layer thickness (inversely related to the distance between the interference fringes) grows with temperature. The qualitative result from this observation is: Interfacial melting indeed occurs at the ice–solid interface investigated. 88 CHAPTER 7. RESULTS AND DISCUSSION

0.1

1E-4 0.0

1E-3 -0.1

0.01 -0.2 T32 -0.02 (mm) (K) T20 -0.034 T21 -0.035 T22 -0.035 y T31 -0.052 -T T19 -0.072 m T23 -0.098 T

0.1 T16 -0.126 T17 -0.125 T18 -0.126 T04 -0.137 -0.3 T03 -0.253 T24 -0.368 T05 -0.518 T25 -0.657 T06 -0.819 T02 -1.00 T07 -1.02 1 T15 -1.02 -0.4 T14 -1.37 T08 -1.72 T13 -2.12 T12 -2.62 T09 -3.12 T26 -3.12

T27 -6.23 -0.5

10 T28 -12.4 T10 -14.7 T29 -20.0 T01 -24.7 T11 -25.0 T30 -30.0 60 -0.6 5 1015202530 number of measurement

Figure 7.2: Time line of the experiment with the smooth substrate. The tem- perature (left axis, bottom curve) and the position (right axis, top curve) are plotted against the time (represented by the number of the measurement). Each measurement is consecutively numbered (Tnn, where nn denotes the number) and labelled with the temperature in ◦C. The measurements which are not performed on the initial position y = 0 are highlighted with a dashed pattern. 7.1. OVERVIEW OF THE MAIN EXPERIMENTS 89

101

10-2

10-5

T12 -0.036°C 10-8 T11 -0.114°C T14 -0.175°C 10-11 T10 -0.25°C T09 -0.5°C 10-14 T02 -0.5°C T08 -1.0°C 10-17 T07 -1.7°C T06 -3.1°C -20

Intensity (arb.10 units) T05 -7.7°C T01 -12.7°C 10-23 T04 -14.7°C T13 -25°C -26 10 T03 -30°C

10-29 T15 substrate 10-32 0.0 0.1 0.2 0.3 0.4 0.5 0.6 q (Å-1) z

Figure 7.3: Reflectivity measurements for ice in contact with the rough substrate. The measurements show that an additional layer appears and grows in thickness with temperature. The solid lines are fits to the data. The curves are displaced by factors of 30. 90 CHAPTER 7. RESULTS AND DISCUSSION

10-1

10-4

10-7 T32 -0.022°C 10-10 T20 -0.034°C 10-13 T31 -0.052°C T19 -0.072°C 10-16 T23 -0.098°C T16 -0.126°C 10-19 T04 -0.137°C T03 -0.25°C 10-22 T24 -0.37°C T05 -0.52°C -25 10 T25 -0.66°C

Intensity (arb. units) T06 -0.82°C 10-28 T02 -1.00°C 10-31 T07 -1.02°C T08 -1.72°C 10-34 T26 -3.12°C T27 -6.23°C 10-37 T28 -12.4°C T29 -20.0°C 10-40 T01 -24.7°C T30 -30.0°C 10-43 0.0 0.2 0.4 0.6 0.8 1.0 1.2 q (Å-1) z

Figure 7.4: Reflectivity measurements for ice in contact with the smooth sub- strate. The measurements show that an additional layer appears and grows in thickness with temperature. The solid lines are fits to the data. The curves are displaced by factors of 45. 7.1. OVERVIEW OF THE MAIN EXPERIMENTS 91

100

10-3

10-6

10-9 T09 -3.12°C T10 -14.7°C 10-12 T11 -25.0°C T12 -2.62°C 10-15 T13 -2.12°C Intensity (arb. units) 10-18 T14 -1.37°C T15 -1.02°C 10-21 T17 -0.125°C T18 -0.126°C 10-24 T21 -0.035°C T22 -0.035°C 10-27 0.0 0.2 0.4 0.6 0.8 1.0 q (Å-1) z

Figure 7.5: Reflectivity measurements on different positions of the sample (smooth substrate, compare Fig. 7.2) that were not irradiated long enough to change the termination from hydrophobic to hydrophilic. This is thought to be the reason why the interfacial melting is not pronounced in this case. The curves are displaced by factors of 45. 92 CHAPTER 7. RESULTS AND DISCUSSION

The data in Figs. 7.3 and 7.4 also demonstrate that this behavior is reversible (compare the timeline in Figs. 7.1 and 7.2), with the exception of the lowest temperatures. The reflectivity measurements shown in Fig. 7.5, however, do not display this evolution with temperature. They have been measured during the experiment with the smooth substrate like those from Fig. 7.4, but at another position on the sample. Only later did the reason for this observation become clear: Under the influence of the high-energy x-ray microbeam and in the presence of water, the substrate termination changes from hydrophobic to hydrophilic, which strongly influences the interface melting behavior. This observation is discussed in Sec. 7.8. This radiation-induced change of the substrate termination requires some time. When the measurements shown in Fig. 7.5 where performed, the substrate was still hydrophobic at this position, since it had not been illuminated long enough. The measurements shown in Figs. 7.3 and 7.4, in contrast, have been performed after significant irradiation (setup and alignment time), rendering that position hydrophilic. The effect of the radiation might also explain why the low-temperature mea- surements are not fully reversible. For both experiments, the low-temperature curves measured at the beginning of a beamtime do not match those measured later. The reflectivity of the bare substrates (without ice) has also been measured in air. In the case of the rough substrate, it has been measured at the end of a beam- time after removing the ice (lowest curve in Fig. 7.3). For the smooth substrate, only measurements with a Cu-Kα sealed tube are available (see Fig. 6.3).

7.2 Density profiles

7.2.1 Raw data analysis

As described in Sec. 5.3.9, all measured intensities have been normalized to the monitored primary intensity. Parts of reflectivity curves measured with different absorbers have been adjusted. Rocking scans (transverse momentum scans, see Sec. 5.3.4) have been measured in order to determine the background and evaluate the morphology of the interface (see Sec. 7.3). In the case of the rough substrate, integrated intensities have been obtained from rocking scans (Secs. 7.3 and 4.8). The illumination correction has been performed on the data from the smooth substrate. For the data from the rough substrate, the illumination correction has been included in the fitting routine. 7.2. DENSITY PROFILES 93

-5 -5 2.4 ) 2.0x10 a 2.0x10 b )

0.7 3 2.2 3 (e/Å

d ? ?

d

e 2.0 -5 0.6 -5 (g/cm ?

1.6x10 1.6x10 ? 1.8 0.5 HO2 HO2 1.6 SiO2 Si 1.2x10-5 1.2x10-5 (ice) (qll) 1.4 dispersion 0.4 dispersion 1.2

-6 0.3 -6 1.0 mass density 8.0x10 electron density 8.0x10 -40 -20 0 20 40 -40 -20 0 20 40 z (Å) z (Å)

Figure 7.6: Dispersion and density profiles. (a) Dispersion/electron density pro- file. (b) Dispersion profile (solid line) and corresponding mass density profile (dashed line) assuming the composition indicated on the graph.

7.2.2 Reconstruction of density profiles The (specular) x-ray reflectivity is determined by the laterally averaged dispersion profile (see Sec. 4). For a large x-ray energy range, the dispersion is proportional to the electron density (see Eq. 4.5). The mass density profile can be calculated from the dispersion or electron density profile if the the chemical composition is known (see Fig. 7.6). The dispersion profiles perpendicular to the interface have been reconstructed by fitting a model to the measured data. For this purpose, the program ‘Winfit’ written by A. R¨uhmhas been used [213]. It is based on the Parratt formalism described in Sec. 4.3. The density/dispersion profile is described with a model consisting of 4 lay- ers (see Fig. 7.7). The first layer represents the ice, it is semi-infinite1 and has the dispersion of ice. The second layer represents the premelting quasiliquid. Its thickness and density are the parameters which should be determined as a function of temperature. The third and fourth layer represent a thin SiO2 layer covering a semi-infinite2 Si block (taking literature values for the dispersion). Each interface between two layers is characterized by a parameter for the rms roughness. The thickness of the SiO2 layer and the associated roughness param- eters are in principle free parameters. They should, however, be consistent for all measurements at different temperatures, since the SiO2 layer is not supposed to change with temperature. In the case of the lowest temperatures, where the interfacial melting has not yet set in, the second layer describing the quasiliquid can be omitted. This is the simplest model which can be used to adequately describe the

1The incoming wave enters the ice through its side and no reflection occurs at the top of the ice. It can therefore be treated as semi-infinite. 2 Due to the small incident angles and the large thickness of the Si–SiO2 substrate, the transmitted wave leaves the substrate through its side and does not get reflected from the bottom. The substrate therefore appears to be semi-infinite. 94 CHAPTER 7. RESULTS AND DISCUSSION

ice rice s L quasiliquid r ice qll s SiO2 LSiO SiO rSiO 2 2 2 sSi

Si rSi

Figure 7.7: Model for fitting the reflectivity data. The densities are denoted with ρ, the rms roughness parameters with σ. The densities ρice, ρSiO2 and ρSi are fixed to the nominal values. The thickness LSiO2 of the SiO2 layer and the roughness parameters σSiO2 , σSi of the substrate have to be consistent over all temperatures. This leaves the thickness L of the quasiliquid layer, its density ρqll, and the roughness σice of the ice–quasiliquid interface as temperature dependent parameters. interface. It contains the lowest possible number of layers and free parameters. It does not account for a variation of the density within the quasiliquid layer. Reflectivity curves calculated from the fitted model are shown in Figs. 7.3 and 7.4 (solid lines). The simple model allows to reproduce quite well the measured data. The remaining differences might indeed be due to an internal structure of the quasiliquid layer. But since the differences are small, they cannot be used to unambiguously deduce the internal profile of the quasiliquid layer. In the case of the rough substrate, the reflectivity has been obtained from integrated intensities of rocking scans (see Sec. 4.8.2). By doing so, the informa- tion about the roughness gets lost. The roughness parameters obtained from the fitting only describe a ‘local’ roughness on small length scales (see Sec. 7.3, where the morphology of the rough substrate is discussed). The reconstructed density profiles are shown in Figs. 7.8 and 7.9, the corre- sponding fit parameters are listed in Tabs. 7.2 and 7.3. Not listed in the table are a scaling factor, a constant background, and the angle of complete illumination in the case of the rough substrate (for the smooth substrate, the correction for the illuminated area has been applied to the data, not to the calculated reflectivity). For the lowest temperatures the density profile simply shows ice in contact with the substrate consisting of a thin SiO2 layer and the underlying silicon block. At higher temperatures, a layer with a different density intervenes and increases in thickness with temperature. An atomic scale illustration representing the interface model is shown in Fig. 7.10. The data has been analyzed with the dynamical Parratt formalism. However, the kinematical approximation shows practically no differences for typical reflec- tivity curves from this work, as can be seen from Fig. 7.11. Differences would 7.2. DENSITY PROFILES 95

7 T12 -0.036°C T11 -0.114°C T14 -0.175°C 6 T10 -0.25°C T09 -0.5°C 5 T02 -0.5°C T08 -1.0°C ) 3 T07 -1.7°C 4 T06 -3.1°C

(g/cm T05 -7.7°C ? 3 T04 -14.7°C T13 -25°C T03 -30°C 2

1

-50 0 50 z (Å)

Figure 7.8: Reconstructed density profiles for the rough substrate. The dashed lines indicate the positions of the layers. The curves are displaced by vertical offsets of 0.4 (g/cm3). 96 CHAPTER 7. RESULTS AND DISCUSSION

T32 -0.022°C 10 T20 -0.034°C T31 -0.052°C 9 T19 -0.072°C T23 -0.098°C T16 -0.126°C 8 T04 -0.137°C T03 -0.25°C 7 T24 -0.37°C T05 -0.52°C )

3 T25 -0.66°C 6 T06 -0.82°C T02 -1.00°C

(g/cm T07 -1.02°C

? 5 T08 -1.72°C T26 -3.12°C 4 T27 -6.23°C T28 -12.4°C 3 T29 -20.0°C T01 -24.7°C T30 -30.0°C 2

1 -50 0 50 z (Å)

Figure 7.9: Reconstructed density profiles for the smooth substrate. The dashed lines indicate the positions of the layers. The curves are displaced by vertical offsets of 0.4 (g/cm3). 7.2. DENSITY PROFILES 97

Table 7.2: Fit parameters for the rough substrate. ∗ ∗ ∗ label Tm − T L ρqll LSiO2 σice σSiO2 σSi (K) (A)˚ (g/cm3)(A)˚ (A)˚ (A)˚ (A)˚ T12 0.036 55.0 1.12 18.5 0 3.46 0 T11 0.114 45.5 1.13 20.1 0 3.89 0 T14† 0.175 35.9 1.13 19.6 0 5.20 0 T10 0.25 36.3 1.13 21.5 0 4.43 0 T09 0.5 26.7 1.25 18.9 0 6.45 0 T02 0.5 23.7 1.22 19.7 0 4.80 3.73 T08 1.0 17.2 1.18 22.1 0 4.25 0 T07 1.7 14.1 1.17 22.0 0 4.51 0 T06 3.1 11.6 1.26 23.3 0 3.73 0 T05 7.7 9.93 1.26 21.5 0 3.68 0 T01 12.7 −‡ −‡ −‡ −‡ −‡ −‡ T04 14.7 7.46 1.21 22.9 0 4.53 5.33 T13† 25.0 0 − 21.4 0 5.93 0 T03 30.0 7.96 1.25 19.1 0 4.23 3.15 T15 -§ (10.6)k (1.00)k 15.2 0 6.92 0 average 1.20 20.4 0 4.7 0.9 sd¶ 0.05 2.1 0 1.1 1.8 ∗ fit parameters do not reflect the real roughness (see the text) † second beamtime ‡ no significant fit possible due to small q-range of measurements § bare substrate k the bare substrate (measured in air) is covered with an adsorbed water layer ¶ standard deviation 98 CHAPTER 7. RESULTS AND DISCUSSION

Table 7.3: Fit parameters for the smooth substrate.

label Tm − T L ρqll LSiO2 σice σSiO2 σSi (K) (A)˚ (g/cm3)(A)˚ (A)˚ (A)˚ (A)˚ T32 0.022 27.5 1.16 19.1 1.75 3.00 0 T20 0.034 26.5 1.20 17.0 0.85 3.02 0 T31 0.052 26.3 1.13 17.9 2.01 2.62 0 T19 0.072 23.8 1.18 16.6 2.18 3.00 0 T23 0.098 23.9 1.23 16.4 1.56 2.99 0 T16 0.126 23.1 1.18 15.7 3.42 2.43 0 T04 0.137 20.8 1.31 14.5 2.79 2.31 0 T03 0.25 18.5 1.24 14.4 2.11 1.86 0 T24 0.37 19.3 1.24 15.2 1.94 2.77 0 T05 0.52 14.6 1.19 13.5 0 3.11 2.12 T25 0.66 17.3 1.18 13.8 1.63 2.51 0 T06 0.82 13.2 1.16 13.5 0 3.01 0.44 T02 1.00 13.2 1.08 12.3 0 2.32 0 T07 1.02 12.6 1.12 13.0 0 3.13 0.90 T08 1.72 10.9 1.10 13.7 0 3.17 0 T26 3.12 8.84 1.24 15.7 2.88 2.53 0 T27 6.23 7.66 1.27 15.4 2.27 2.50 0 T28 12.4 6.97 1.27 15.1 1.82 2.79 0 T29 20.0 4.19 1.12 20.2 3.65 2.96 0 T01 24.6 0 − 13.5 0 2.35 0 T30 30.0 4.51 1.27 19.0 2.23 2.23 0 average 1.19 15.5 1.58 2.70 0.22 sd∗ 0.06 2.2 1.19 0.4 0.43 ∗ standard deviation 7.2. DENSITY PROFILES 99

T = -1°C

iceI h

quasiliquid 1.7 nm layer

amorphous 2.1 nm SiO2

Si

Figure 7.10: Illustration of the model for the ice–SiO2–Si interface. The length scales and the densities correspond to the experimentally determined values for the rough substrate at −1◦C. For clarity, the atomic radii are not on scale. only become apparent for large layer thicknesses, when features on the reflectivity curve appear in the vicinity of the critical angle.

7.2.3 Reliability of the fits The density profiles are reconstructed by fitting a model to the measured data. This raises two question:

• Are other density profiles capable of explaining the measurements?

• How accurate can the model parameters be determined by fitting?

In order to address these issues, several tests have been performed. Fig. 7.12 shows calculated curves for ice–SiO2–Si interfaces with intervening layers of different thickness and density. It can be seen that the layer thickness determines the distance between interference fringes (compare Eq. 4.16) whereas the density is linked to the amplitude of the modulation. The layer thickness is, therefore, a very robust parameter, whereas the density cannot be determined with the same precision, which is why the fitted density values scatter more strongly (compare Figs. 7.33 and 7.29). Nevertheless, the measured reflectivity is sensitive to both the thickness and the density (as well as the roughness). In order to verify the reliability of the deduced parameter values, fitting runs have been performed, where one of the parameters (the density of the quasiliquid, 100 CHAPTER 7. RESULTS AND DISCUSSION

-1 10 dynamical 10-3 kinematical 10-5 10-7

Intensity (arb.10 units) -9

0.0 0.2 0.4 0.6 0.8 1.0 q (Å-1) z

Figure 7.11: Comparison between kinematical and dynamical calculation of re- flectivity. Practically no differences can be seen for typical reflectivity curves from this work. for example) was fixed and the other parameters were free. These series of fitting runs show that the quality of the fit effectively depends on the value of the fixed parameter (see Fig. 7.13). Such tests show that the fits are not very sensitive to the roughness at the SiO2–Si interface (see Fig. 7.14). In most cases, the best fit is obtained with ˚ σSi = 0 A, but a finite value on the order of σSiO2 is much more realistic and matches the measured data almost as well (see Fig. 7.14). Due to the special way of measuring the reflectivity in the case of the rough substrate, the roughness parameters in this case are not very meaningful anyhow (see Sec. 7.3). This explains why the roughness obtained at the ice–quasiliquid interface is zero in the case of the rough substrate. The sensitivity of the reflectivity on the model parameters and the degree to which different fits of the same reflectivity curve scatter are used to estimate the error bars of the fit parameters as plotted in Figs. 7.29 and 7.33. The errors are not necessarily the same for all measurements. Large layer thicknesses can be determined with better accuracy than small ones because they are fixed by a higher number of interference fringes. For individual reflectivity curves, several simple density profiles may exist which fit well the measured data. But the model should allow to explain all reflec- tivity curves measured at different temperatures simultaneously. This eliminates most ambiguities for a single reflectivity measurement, since many parameters are not completely free, but need to be consistent when comparing measurements at different temperatures. The results have also been verified with phase inversion (see Sec. 4.6). 7.2. DENSITY PROFILES 101

ab

0 0 10 10 r L = 20.3 Å = 0.98 g/cm³ -2 -2 r 10 L = 18.8 Å 10 = 1.08 g/cm³ r = 1.18 g/cm³ -4 L = 17.3 Å -4 10 L 10 r = 1.28 g/cm³ = 15.8 Å r -6 L -6 = 1.38 g/cm³ 10 = 14.3 Å 10 10-8 10-8

-10 -10 Intensity (arb. units) 10 Intensity (arb.10 units)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 q -1 q -1 z (Å ) z (Å ) c

10-1 s = 1.5 Å s = 2.0 Å -3 10 s = 2.5 Å -5 s = 3.0 Å 10 s = 3.5 Å 10-7

-9

Intensity (arb. units) 10 10-11 0.0 0.2 0.4 0.6 0.8 1.0 q -1 z (Å )

Figure 7.12: Calculated reflectivity curves showing the influence of model pa- rameters. The black line corresponds to the fit results for the smooth substrate at −0.657◦C. The other lines are calculated with different values for (a) the quasiliquid layer thickness L, (b) the density ρ of the quasiliquid, and (c) the rms roughness σ of the substrate. 102 CHAPTER 7. RESULTS AND DISCUSSION

ab 70 0.07 60 0.06 T = -1°C T 0.05 = -0.036°C 50 T 0.04 40 = -0.036°C 2 (Å) T c = -1°C 0.03 L 30 0.02 20 0.01 10 0.00 0 0.8 1.0 1.2 1.4 1.6 0.8 1.0 1.2 1.4 1.6 r (g/cm³) r qll qll (g/cm³)

Figure 7.13: Reliability of the fits. (a) Quality of the fit (measured by the mean square deviation χ2) as a function of the model parameter ρ denoting the density of the quasiliquid (rough substrate). The other parameters were free except for the dispersion of ice and silicon, which were fixed to the nominal values. It can be seen that χ2 strongly depends on the density. In particular, good fits cannot be achieved with the density of water (1 g/cm3). The open boxes mark the results that were finally obtained from the fitting. These are not necessarily the fits with the smallest χ2, as the retained fits have to satisfy certain constraints: The results need to be physical and consistent over all temperatures. (b) Fit parameter L denoting the thickness of the quasiliquid layer as a function of the density (same fitting runs as in (b)). This plot shows that the thickness and the density which result from the fits are practically uncorrelated. Only a strongly different density leads to a significant change of the thickness.

0.1 1E-3 1E-5 1E-7 ? (Å) Si 1E-9 0 1E-11 0.5

Intensity (arb. units) 1.0 1E-13 1.5 1E-15 2.0 0.0 0.2 0.4 0.6 0.8 1.0 q (Å-1) z

Figure 7.14: Reliability of the roughness parameter σSi. Reflectivity curves cal- culated with different values for this parameter show that the measurements are not very sensitive to the roughness at the SiO2–Si interface. 7.3. SUBSTRATE MORPHOLOGY 103 abDv = 0.25 mm FWHM = 87 µrad Dv = 0.1 mm FWHM = 60 µrad 20 10

15 8 6 10 4 5 2 Intensity (arb. units) Intensity (arb. units) 0 0 -1.0x10-5 0.0 1.0x10-5 -1.0x10-5 0.0 1.0x10-5 q (Å-1) q (Å-1) x x

−1 Figure 7.15: Rocking scans at qz = 0.063 A˚ for the smooth substrate with different resolutions defined by the vertical detector slit gap. The width of the specular peak is limited by the resolution.

7.3 Substrate morphology

Two substrates with different morphology have been mainly used in this work. They are referred to as the smooth substrate and the rough substrate. The morphology does not only have an influence on the feasibility of reflectivity mea- surements and their interpretation (see Chapts. 4 and 5), but is also one of the parameters determining the interface melting behavior (see Chapt. 3). Therefore, a characterization of the morphology is needed. This has been done by analyzing the reflectivity data itself, in particular the off-specular scattering. As a com- plementary approach, AFM images of the bare substrate have been taken3 and analyzed.

7.3.1 Smooth substrate Rocking scans (see Sec. 5.3.4) on the smooth substrate show a narrow resolution- limited peak. It is the convolution of the specular reflectivity, characterized by a δ function (see Sec. 4.8), with the instrumental resolution and the curvature of the sample. Consequently, the peak width decreases when the resolution is improved by choosing a smaller vertical detector slit gap (see Fig. 7.15).

The peak width changes with qz like the resolution function and hence in- creases linearly with qz in momentum space (see Fig. 7.16). In angular space, it remains constant. The width is slightly smaller than the calculated resolution, which indicates that the effective detector slit gap and/or beam divergence are somewhat smaller than the nominal values used for calculating the resolution. In fact, the peak width of the specular reflectivity can be regarded as a measurement

3in collaboration with U. T¨affner(MPI f¨urMetallforschung) 104 CHAPTER 7. RESULTS AND DISCUSSION

measurement 6.0x10-5 linear fit resolution (nominal) 4.0x10-5 ) -1 (Å

w 2.0x10-5

0.0 0.0 0.1 0.2 0.3 0.4 0.5 q (Å-1) z

Figure 7.16: Peak width in a rocking scan as a function of qz (smooth substrate). The resolution is shown as a dashed line.

-1 -1 -1 qz =Å0.19 qz =Å0.32 qz = 0.50 Å 2.5 0.8 1.6 2.0 0.6 1.2 back- 1.5 ground 0.4 0.8 1.0 0.4 0.5 0.2 Intensity (arb. units) Intensity (arb. units) 0.0 Intensity (arb.0.0 units) 0.0 -3.0x10-50.0 3.0x10 -5-6.0x10 -50.0 6.0x10 -5 -8.0x10-50.0 8.0x10 -5 q -1 q -1 q -1 x (Å ) x (Å ) x (Å )

Figure 7.17: Rocking scans with the smooth substrate for different values of qz. The peak always has a Gaussian line shape. of the resolution.

The line shape of the rocking scans does not change with qz (see Fig. 7.17) and can be well described by a Gaussian. The background contains the diffuse non-specular reflectivity (and some other contributions like air scattering), and is orders of magnitude smaller. This shows that the roughness must be rather weakly correlated. The background has been subtracted from the peak intensity measured at qx = 0 A˚ in order to obtain the true specular reflectivity. The specular reflectivity allows to reconstruct the laterally averaged dispersion profile (see Sec. 7.2.2). These profiles contain the rms roughness of the substrate. Its value is (2.7±0.4) A˚ (see Tab. 7.3). 7.3. SUBSTRATE MORPHOLOGY 105

7.3.2 Rough substrate In the case of the rough substrate, a specular component cannot be recognized in the rocking scans (see Figs. 7.18 and 7.23). In Sec. 4.8, it has been shown, that the density profile can nevertheless be reconstructed from the integrated in- tensity of the rocking scans. Such measurements are very time-consuming, since a complete rocking scan profile has to be measured for each value of qz (each point on the reflectivity curve). If the line shape of the rocking scans does not change with qz, however, the peak intensity at qx = 0 A˚ is a measure of the integrated intensity. It only needs to be multiplied by a qz-dependent correction factor accounting for the change of the peak width (see Fig. 7.18). However, the integrated intensity does not contain any information about the roughness any- more. The reflectivity corresponds to that of a perfectly smooth interface. Even so, fitting the reflectivity curves obtained in this way does not always yield zero roughness (see Tab. 7.3). The reason is the finite range of integration in momen- tum space, which allows only partial integration of the diffuse reflectivity. The roughness on small lateral length scales, therefore, still shows up in the reflectiv- ity profiles. This ‘local roughness’ also cannot be expected to be fully conformal. The rms value of this local roughness obtained from fitting the reflectivity curves (integrated intensities) is (4.7 ± 1.1) A.˚ The fact that even at small qz no specular component can be seen in the rocking scans shows that the correlation length of the roughness is very large. In the case of self-affine roughness, there is no cutoff of the self-affine behavior up to the length scales probed by the x-rays. The height-difference correlation function g (R) is then given by Eq. 4.25:

g (R) = BR2h.

The information about correlations of the roughness is contained in the dif- fuse reflectivity, this means the transverse momentum scans (rocking scans). The theory for this diffuse reflectivity is discussed in Sec. 4.8. For self-affine roughness without cutoff, it is described by Eq. 4.44. In general, this cannot be calculated analytically, except for the special cases h = 0.5 and h = 1. Since the resolu- tion in qy is usually relaxed (see Sec. 5.3.5), the measurement corresponds to an integration of Eq. 4.44 over qy. The line shape of the rocking scans depends on the Hurst parameter h, while the amplitude B only influences the peak width. Fig. 7.19 shows calculated rocking scans for different h. For the special cases h = 0.5 and h = 1 the line shape is Lorentzian and Gaussian, respectively. For values of h close to 0.5, the line shape is not very sensitive to h. The qz-dependence of its FWHM, however, strongly depends on h and B. A fit of the measured width w as a function of qz yields h =0.34 ± 0.01 and B =0.107 ± 0.006 (see Fig. 7.20). Fig. 7.22 compares calculated rocking scan profiles with the measurement. The measured data has been corrected for the variation of the illuminated area 106 CHAPTER 7. RESULTS AND DISCUSSION

a rocking scan b reflectivity 101

-1 -1 qz = 0.16 Å 3 10 integrated -3 corrected 10 uncorrected 2 10-5 10-7 1 -9 Intensity (arb. units)

Intensity (arb. units) 10 0 10-11 0.000 0.002 0.004 0.0 0.1 0.2 0.3 0.4 0.5 ? (rad) q -1 i z (Å )

Figure 7.18: Measurement of integrated intensity (rough substrate). (a) Rocking scan. The central peak is diffuse reflectivity, the other two are the so-called Yoneda wings (compare Fig. 4.5). The asymmetry is explained in the text. (b) Reflectivity. The solid circles are the integrated intensity obtained from rocking scans (corresponding to the shaded area in a). The peak intensity has been −1 measured at qx = 0 A˚ (solid square in a). The background has been obtained from two points (open squares in a) between the diffuse reflectivity peak and the Yoneda wings. Substraction of the background yields the lower curve in b (triangles). This curve represents the peak height, not the integrated intensity. Therefore, it has to be corrected using the intensities obtained from integrating rocking scans, yielding the upper curve in b (open circles). 7.3. SUBSTRATE MORPHOLOGY 107 a h = 0.1 b h = 0.25 c h = 0.34 5 7 8 4 6 5 6 3 4 2 3 4 2 1 2 1 0 0

Intensity (arb. units) 0 Intensity (arb. units) Intensity (arb. units) -4.0x10-80.0 4.0x10 -8 -4.0x10-50.0 4.0x10 -5 -2.0x10-40.0 2.0x10 -4 q (Å-1) q (Å-1) q (Å-1) x x x d h = 0.5 (Lorentzian) e h = 1 (Gaussian) 8 6 6 4 4 2 2 0 0 Intensity (arb. units) Intensity (arb. units) -4.0x10-40.0 4.0x10 -4 -2.0x10-40.0 2.0x10 -4 q (Å-1) q (Å-1) x x

−1 Figure 7.19: Calculated rocking scans at qz = 0.16 A˚ for different values of the Hurst parameter h. The amplitude B has been obtained from a fit of the qz-dependence of the peak width (see Fig. 7.20). This fit yields h = 0.34, the corresponding profile is shown in c. For intermediate values of h (b-d) the rocking scans are not very sensitive to h.

(see Sec. 5.3.7). This removes most of the asymmetry present in the raw data (compare to Fig. 7.18a). Since the calculation is based on a kinematical theory, the Yoneda wings cannot be reproduced. In Fig. 7.22a, where h and B have been fixed to the values determined from fitting w(qz), only a constant background and a scaling factor have been used to adjust the calculation to the measurement. Fig. 7.22b+c show fits of the measurement with a Lorentzian and a Gaussian line shape, corresponding to h = 0.5 and h = 1, respectively. For these two cases, the width of the peak has been fitted to the data. This corresponds to a fit of B. While a Gaussian line shape (h = 1) can clearly be excluded, the distinction between h = 0.34 and h = 0.5 cannot be made from a single rocking scan (hence the fit of w(qz)). With the roughness parameters h and B obtained from fitting w(qz), all rock- ing scan profiles (at different qz) can be calculated. They fit very well to the mea- surements, illustrated by Fig. 7.23. Again, only a scaling factor and a constant background have been used to adjust the calculated profiles to the measurements. The remaining asymmetry in the profiles measured at large qz (Fig. 7.23e) is due to the variation of the resolution with qx (compare Fig. 5.11). The effect is more pronounced at large qz, since here the peak width, and hence the qx-range of the profile, is larger. The variation of the resolution is also demonstrated by the fact that the width of the two Yoneda wings is different (Fig. 7.23e). At very 108 CHAPTER 7. RESULTS AND DISCUSSION

a differenth, fit of B (linear plot) b differenth, fit of B (log-log plot) 3.0x10-3 h -2 =0.25 10 2.5x10-3 h -3 =0.1 h=0.34 2.0x10 10-3 ) )

-1 -3 1.5x10 -1 (Å (Å -4 h=1 -3 10 w 1.0x10 h=0.5 w h=0.5 -4 -5 5.0x10 h=1 10 0.0 h=0.34 h=0.25 h=0.1 10-6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.05 0.1 0.5 q -1 q -1 z (Å ) z (Å )

c differentBh , =0.34 d comparision with resolution -3 3.0x10 -2 10 2.5x10-3 B=0.11 -3 -3 2.0x10 10 ) ) -1 -1 -3 B 1.5x10 =0.2

(Å -4 (Å -3 10 w w 1.0x10 -4 -5 5.0x10 10 B=0.05 resolution 0.0 -6 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.05 0.1 0.5 q -1 q -1 z (Å ) z (Å )

Figure 7.20: Width w of the diffuse reflectivity peak (FWHM) in a rocking scan as a function of qz. (a) Measurement (solid squares) and fits (linear plot). The best fit (bold line) is obtained with h =0.34 ± 0.01 and B =0.107 ± 0.006. Fits with fixed h are shown for comparison (thin lines). (b) Same as a (log-log plot). (c) Best fit (solid line) and calculations for different values of B (fixed h = 0.34). (d) Comparison with the resolution function. The measured peak width is always larger than the resolution function. 7.3. SUBSTRATE MORPHOLOGY 109

ice

quasiliquid

substrate

Figure 7.21: Roughness replication at substrate–quasiliquid–ice interface. The roughness at the substrate–quasiliquid interface is replicated at the quasiliquid– ice interface (conformal roughness), except for small length scales.

small qz (Fig. 7.23a), the calculated peak profile has a smaller width than the measured one, even if convoluted with the resolution function. The discrepancy can be explained with broadening due to sample curvature. At small qz, the beam footprint is approximately as long as the interface (24 mm), therefore, the measurement is very sensitive to the curvature. With a figure error of around 1 µm (see Sec. 6.1) a broadening of around 40–80 µrad can be expected. The angular width of the measured peak shown in Fig. 7.23a is 65 µrad. At larger qz, the curvature has no visible effect, since the intrinsic width of the peak is larger and at the same time the beam footprint is smaller. The rocking scan profiles are virtually identical (apart from the height, of course) for all temperatures, and for measurements with the bare substrate. This indicates that the roughness is basically conformal. The morphology at the substrate–quasiliquid interface must be replicated on the quasiliquid–ice in- terface, except for roughness on very small length scales (see Fig. 7.21). This seems quite plausible, as long as the interface is not too jagged. It justifies the analysis of the integrated diffuse reflectivity as presented in Sec. 4.8.2. 110 CHAPTER 7. RESULTS AND DISCUSSION

8 7 a h = 0.34 6 5 4 Yoneda background 3 wing 2

Intensity (arb.1 units) 0 -3.0x10-40.0 3.0x10 -4 q -1 x (Å ) 8 8 7 b h = 0.5 7 c h =1 6 (Lorentzian) 6 (Gaussian) 5 5 4 4 3 3 2 2 Intensity (arb. units) 1 Intensity (arb.1 units) 0 0 -3.0x10-40.0 3.0x10 -4 -3.0x10-40.0 3.0x10 -4 q -1 q -1 x (Å ) x (Å )

Figure 7.22: Comparison of calculated (open circles) and measured (solid lines) rocking scans for different h. The measured data has been corrected for the illuminated area. In (a), h and B from the fit of w(qz) (see Fig. 7.20) have been used, only a scaling factor and a constant background have been fitted to the rocking scan. In (b) and (c) h has been fixed, but B fitted to the rocking scans. While h = 1 can clearly be excluded, the distinction between h = 0.34 and h = 0.5 cannot be made from a single rocking scan. 7.3. SUBSTRATE MORPHOLOGY 111

-1 7 -1 a qz = 0.03 Å b qz = 0.07 Å 15 6 5 10 4 3 5 2 Intensity (arb. units) Intensity (arb. units) 1 0 0 -1.0x10-50.0 1.0x10 -5 -5.0x10-50.0 5.0x10 -5 q (Å-1) q (Å-1) x x

8 -1 -1 7 c qz = 0.16 Å 5 d qz = 0.25 Å 6 4 5 4 3 3 2 2 1 Intensity (arb. units) 1 Intensity (arb. units) 0 0 -3.0x10-40.0 3.0x10 -4 -1.0x10-30.0 1.0x10 -3 q (Å-1) q (Å-1) x x

7 q = 0.35 Å-1 6 e z 5 4 3 2

Intensity (arb.1 units) 0 -1.0x10-30.0 1.0x10 -3 q (Å-1) x

Figure 7.23: Comparison of calculated (solid lines) and measured (open circles) rocking scans for different qz. The values for h and B have been obtained from the fit of w(qz). Only a scaling factor and constant offset (background) have been fitted to the individual rocking scans. The calculated curves fit very well to the measurement, except for very small qz (see a). In this case, the calculation yields a much smaller peak width (dashed line). A better match is found when the resolution is included (solid line). For a more detailed discussion, see the text. 112 CHAPTER 7. RESULTS AND DISCUSSION a b +32 Å

-32 Å

1mm 1mm

Figure 7.24: AFM images of (a) the smooth substrate and (b) the rough substrate. The same scale and color coding have been used. The AFM picture of the rough substrate shows larger height variations and a stronger ‘texture’. More details are revealed by a statistical analysis (see the text and Fig. 7.25).

7.3.3 AFM measurements AFM measurements of the substrates provide real-space images of the topogra- phy and thereby allow a ‘visual’ comparison of the substrates (see Fig. 7.24). The AFM images directly show that the rough substrate has larger and more structured height variations. Unlike x-ray diffraction, AFM does not directly yield statistical information about the substrate. Such information has to be calculated from the measured height values. This has been done for the height-difference correlation function g(R) (see Fig. 7.25). This information is still local in the sense that it is calcu- lated from a single AFM image taken from one spot on the substrate. A more representative result could be obtained by averaging images from different spots on the substrate. For intermediate length scales, the substrate morphology shows the charac- teristics of self-affine roughness. In this regime, g(R) appears as a straight line in a log-log plot (see Fig. 7.25). A fit yields h = 0.38 and B = 0.098 for the rough substrate, which agrees well with the x-ray data. For the smooth substrate4, h = 0.56 and B = 6.5 × 10−4 is obtained. Compared to the rough substrate, h is larger, which corresponds to a less jagged surface. The parameter B setting the

4In order to preserve the ice–substrate interface, the AFM measurements were performed on another identical substrate. The substrate belongs to the same batch as the smooth substrate used for the x-ray measurements. Both substrates have been polished and cleaned together. 7.3. SUBSTRATE MORPHOLOGY 113

smooth substrate rough substrate 1000 fits ) 2 )(Å R (

g 100

103 104 105 106 R (Å)

Figure 7.25: Height difference correlation functions g(R) determined from AFM measurements. The solid lines are fits assuming self-affine roughness (see the text). scale of the self-affine roughness is about two orders of magnitude smaller. Owing to the finite size of the image, the number of points entering the calculation of g(R) decreases with R. Therefore, the calculated g(R) is not valid for large R. Similarly, the calculation is affected by the size of the AFM tip (here around 100–500 A)˚ at small length scales [214]. A number of other factors can possibly lead to errors in the interpretation of the AFM images. Since the measurements were performed in air, small dust particles are found on some parts of the images. Sometimes, these particles are dragged by the AFM tip. Areas affected by dust particles were excluded from the analysis. As the sample surface does not coincide with the z = 0 plane of the instrument, the measured heights have to be corrected for the sample tilt. Otherwise, the tilt of the sample leads to an artificial increase of g(R). Furthermore, consecutive lines of the AFM image can have a large offset in the height. Therefore, the images shown in Fig. 7.24 have been ‘flattened’ to remove this offset. In order to avoid any artifacts caused by the offset or the flattening, only points within the same line have been considered for the calculation of g(R).

7.3.4 Conclusion The morphology of the rough and the smooth substrate differs strongly. This applies to the amplitude of the roughness as well as its lateral correlations. The roughness has been characterized using the x-ray data and AFM measurements, which are in good agreement. The characteristics of the two substrates are sum- marized in Tab. 7.4. 114 CHAPTER 7. RESULTS AND DISCUSSION

Table 7.4: Morphology of the substrates. ‘rough’ substrate ‘smooth’ substrate Si orientation (111) (001) ∗ SiO2 layer thickness 20 A˚ 16 A˚ figure error† ≈ 1 µm ≈ 1 µm concave convex rms roughness∗ not defined‡ (2.7 ± 0.4) A˚ lateral correlations g(R) = 0.107 · R2·0.34 weak AFM g(R) = 0.098 · R2·0.38 g(R) = 6.5 × 10−4 · R2·0.56 ∗ see Tabs. 7.2 and 7.3 † see Sec. 6.1 ‡ no cut-off of self-affine roughness (see the text), ‘local’ rms roughness (4.7 ± 1.1) A˚

In the case of the smooth substrate, the correlations of the roughness are weak (low level of diffuse reflectivity). Detailed measurements of the diffuse reflectivity would allow to better characterize these correlations, like in the case of the rough substrate, but would require much more time. In the case of the rough substrate, no cut-off of the self-affine behavior can be detected on the length scales probed by the x-rays. This means that the rms roughness does not saturate, and hence the ‘amplitude’ of the roughness cannot be determined from the x-ray data (besides a ‘local’ rms roughness, see above). Various growth mechanisms can lead to self-affine roughness [215]. It has also been observed on etched Si surfaces [216, 217]. It is thus no surprise to find self-affine roughness on the substrates used in this work, which have undergone several cycles of etching and oxide growth during the sample preparation (see Sec. 6.1). The chemical treatment was the same for both substrates, but the initial chemomechanical polishing was performed in different laboratories. The difference in the Si orientation might also play a role, since the etching rate depends on the orientation.

7.4 Growth law

7.4.1 What is expected from theory? The theory for interface melting has been discussed in Sec. 3.2. In the case of dominating short-range interactions, a logarithmic growth law is expected from Landau theory. For large layer thicknesses L, long-range interactions will domi- nate and cause a cross-over to a power-law (the cross-over thickness depends on the ratio of the Hamaker constant and the strength of the exponentially decaying interactions). For very small layer thicknesses on the order of a few molecu- lar layers, the mean-field approach is not valid anymore. Therefore, deviations 7.4. GROWTH LAW 115

6 5 4

(nm) 3 L 2 power logarithmic layering 1 law law 0 0.1 1 10 T T m- (K)

Figure 7.26: Growth law for interfacial melting as expected by theory. For large layer thicknesses, long-range forces dominate and the growth follows a power law. Short-range interactions at smaller layer thicknesses lead to a logarithmic growth law. For very small layer thicknesses, a layer-by-layer growth may occur. from the continuous logarithmic law may occur, as for example layer-by-layer growth. It can also be expected that the very first layer in contact with the substrate shows a specific behavior like premelting at much lower temperatures or substrate-induced ordering (as for example observed for water films adsorbed on certain metal substrates, see [218] and references therein). The theoretically expected growth law can thus be divided into three regions: power-law for large L, logarithmic law for intermediate L, deviations like layer-by-layer growth for small L. This is illustrated in Fig. 7.26.

7.4.2 Experimentally observed growth law In this work, the thickness of the quasiliquid layer has been obtained (see Sec. 7.2) from reflectivity measurements. The values are plotted as a function of temper- ature in Figs. 7.27 and 7.28. The individual points are labelled with the corre- sponding reflectivity measurement in Figs. 7.27a and 7.28a. The data can be fitted with a logarithmic growth law appearing as a straight line in a plot where the scale of the relative temperature Tm − T is logarithmic (see Figs. 7.27b and 7.28b). The fit is much better in the case of the smooth substrate, whereas the growth law in the case of the rough substrate seems to be more complicated and might have to be separated into several regimes (discussed in Sec. 7.4.5). Due to the limited number of data points for the rough substrate, one could only speculate about deviations from a pure logarithmic growth law at first. Such deviations became only clear once it was able to compare the data from the rough and the smooth substrate (see Fig. 7.29). In the case of the 116 CHAPTER 7. RESULTS AND DISCUSSION smooth substrate, a logarithmic growth law fits the experimental data extremely well over the whole temperature range of the experiment. Deviations from a logarithmic growth law can be observed for both substrates in the low-temperature region, which will be discussed in the next section. Fits with a power law have also been performed, appearing as straight lines in a log-log plot (see Figs. 7.27c and 7.28c). The power law is clearly not valid in the case of the smooth substrate, where it strongly deviates from the measurement at high temperatures. In the case of the rough substrate, however, the power law matches quite well the data, especially in the high temperature regime (see Fig. 7.27f). This might indicate a cross-over from a logarithmic to a power law (see Sec. 7.4.5). The fits with a power law yield an exponent of −(0.33 ± 0.03) for the rough substrate, which is very close to the exponent −1/3 expected for non-retarded Van der Waals forces. The Hamaker constant W (defined as in Sec. 3.2.1) can also be determined from the fit with a power law, which gives (6.6± 1.3) × 10−21 J for the rough substrate. This is the typical order of magnitude for Hamaker constants (see for example [130]). In the case of the smooth substrate, only the low temperature data can be fitted to a power law, which yields an exponent of −(0.29±0.03) (corresponding to n = 2.4, i.e. more strongly retarded Van der Waals interactions) and a small value of (2.6 ± 0.4) × 10−25 J for the Hamaker constant.

7.4.3 Onset Extrapolating the logarithmic growth law to zero layer thickness yields the on- ◦ set temperature T0 with the values (−19 ± 3) C for the rough substrate and (−47 ± 16)◦C for the smooth substrate. However, the fit is not very sensitive to the value of the onset temperature. Furthermore, the mean-field approximation behind the logarithmic growth law is not valid anymore in the regime of small layer thicknesses. The real on-set for interfacial melting might be different. The experimental values for the layer thickness shown in Fig. 7.29 do not seem to give a coherent picture. Some values are actually zero and would be consistent with T0 as determined from the fit with a logarithmic growth law. These are values from measurements performed at the beginning of a beam-time (compare Figs. 7.1 and 7.2). Only later, we realized that after a certain time, a quasiliquid layer with finite thickness seems to persist down to temperatures that are even lower than the initial temperature. This can be explained with the observed radiation-induced change of the substrate termination from hydrophobic to hy- drophilic. This means that the data points showing a finite layer thickness at low temperatures are the ones corresponding to the same substrate termination as the rest of the data. For the hydrophilic termination, the ‘real’ onset of inter- face melting is thus considerably lower than the T0 deduced from extrapolating the logarithmic growth law. This is consistent with other studies showing the presence of a quasiliquid at very low temperatures (see Sec. 3.4). 7.4. GROWTH LAW 117

60 60 T12 a b

50 T11 50

40 T14 T10 40

(Å) 30 T09 (Å) 30 L L T02 20 T08 20 T07 T06 T05 10 T04T03 10 0 T13 0 0.1 1 10 0.1 1 10 T -T (K) T -T (K) m m 100 c 60 d 50 40 (Å) (Å) 30 L 10 L 20

T13 10 c 0 0.1 1 10 0.1 1 10 T -T (K) T -T (K) m m

60 e 60 f 50 50 40 40 (Å) 30 (Å) 30 L L 20 20 10 10 0 0 0.1 1 10 0.1 1 10 T -T (K) T -T (K) m m

Figure 7.27: Growth law for the rough substrate. (a) Data from first experiment (filled symbols) and second experiment (open symbols) and labels. (b) Fit with ◦ logarithmic growth law, which yields L0=(8.2 ± 0.4) A˚ and T0=(−19 ± 3) C. (c) Fit with a power law yielding an exponent of −(0.33±0.03). (d) Possible layering at low temperatures. The solid line indicates the c lattice unit of ice. (e) Fit with separate logarithmic growth laws for the high temperature (solid squares) and low temperature (crosses) parts, see the text. (f) Fit with a power law for the high temperature (solid squares) part and a logarithmic growth law for the low temperature (crosses) part, see the text. 118 CHAPTER 7. RESULTS AND DISCUSSION

ab30 30 T32 T31 T23 T04 T20 T24 20 T19 T25 20 T16 T02 T08

(Å) T03 (Å)

L T05 T27 L 10 T29T30 T06T07 10 T26 0 T28 T01 0 0.1 1 10 0.1 1 10 T -T (K) T -T (K) m m

c 30 d

20

(Å) 10 (Å) L L 10 c T01 0 0.1 1 10 0.1 1 10 T -T (K) T -T (K) m m

Figure 7.28: Growth law for the smooth substrate. (a) Data points and labels. (b) Fit with logarithmic growth law, which yields L0=(3.7±0.3) A˚ and T0=(−47± 16)◦C. (c) Inadequate fit with a power law yielding an exponent of −(0.29±0.03). (d) Possible layering at low temperatures. The solid line indicates the c lattice unit of ice. 7.4. GROWTH LAW 119

60 rough substrate 50 smooth substrate 40

(Å) 30 L 20 10 c 0 0.1 1 10 T T m- (K)

Figure 7.29: Influence of roughness on the growth law. Comparison of the rough and the smooth substrate. The solid lines indicate fits with a logarithmic growth law and the c lattice unit of ice. The logarithmic growth law fits very well the data of the smooth substrate. The growth behavior at the rough substrate corresponds well to the one at the smooth substrate up to about −0.7◦C. From this point on the layer thickness at the rough substrate increases much faster (discussed in the text).

As the mean-field approach is not valid for very thin layers, it is not surprising to observe indications for some sort of layering (see Figs. 7.27d and 7.28d) instead of a continuous growth at low temperatures. In the case of the smooth substrate, two jumps in the layer thickness can be recognized. The two jumps together correspond approximately to the lattice unit c of the ice crystal perpendicular to the interface and thus to about two molecular layers. In the case of the rough substrate, the observed interfacial melting directly sets in with this layer thickness, but this is probably only due to the limited spatial resolution. (In the case of the rough substrate, the qz range of the reflectivity measurements is smaller, thereby limiting the spatial resolution perpendicular to the interface.) Further experiments could address the question of the onset, but it might be difficult to answer, as the reflectivity has to be measured up to high momentum transfers in order to reach atomic resolution. Moreover, interfacial roughness might mask a very thin layer by smearing the density profile.

It has to be noted that apart from the data points at very low temperatures, the observations are completely reversible (see the ‘timeline’ in Figs. 7.1 and 7.2 for the data shown in Fig. 7.29). The reversibility is also illustrated by Fig. 7.30 showing reflectivity curves measured at approximately the same temperature, but during different heating/cooling cycles. 120 CHAPTER 7. RESULTS AND DISCUSSION

a rough substrate 100 100 100 T02 -0.5°CT04 -14.7°C T09 -0.5°C 10-2 10-2 10-2 10-4 10-4 10-4 10-6 10-6 10-6

-8 Intensity (arb. units) -8 -8 Intensity (arb. units) Intensity10 (arb. units) 10 10 0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4 q -1 q -1 q -1 z (Å ) z (Å ) z (Å ) b smooth substrate 100 100 100 10-2 T02 -1.0°C10-2 T04 -0.137°C 10-2 T07 -1.0°C 10-4 10-4 10-4 range of T02 10-6 10-6 10-6 10-8 10-8 10-8 Intensity (arb. units) Intensity (arb. units) Intensity10 (arb. units) -10 10-10 10-10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 q -1 q -1 q -1 z (Å ) z (Å ) z (Å )

Figure 7.30: Reversibility of the quasiliquid layer formation. Illustrated by reflec- tivity measurements for (a) the rough substrate and (b) the smooth substrate.

7.4.4 Growth amplitude

5 The fit with a logarithmic growth law yields the growth amplitude L0 =(8.2 ± 0.4) A˚ for the rough substrate and L0 =(3.7 ± 0.3) A˚ for the smooth substrate. The fit is extremely good for the smooth substrate, but does not match too well the data for the rough substrate. As discussed in Sec. 3.2, the growth amplitude can be compared to the correlation length of the quasiliquid. The experimentally determined values for the correlation length in bulk water range from 4.5 A˚ [219] to 8 A˚ [220]. This signifies that the correlation length of the quasiliquid layer is of the same order as in bulk water. In the case of the smooth substrate, the value lies at the lower end of the water data, the correlations in the quasiliquid seems to be less pronounced. In the case of the rough substrate, the conclusion would be that the quasiliquid has stronger correlations. But as a single logarithmic growth low does not fit too well to the data, the reason for the faster growth of the layer thickness is more likely a difference in the growth law (see Sec. 7.4.5). This interpretation is supported by the fact that the growth law for the rough substrate in the region of smaller layer thickness follows quite well the one for the smooth substrate and deviates only at higher temperatures. Otherwise, it would remain to explain why the quasiliquid on a rough substrate should have a larger

5the amplitude defines the slope of the logarithmic growth law in a log-linear plot as for example in Fig. 7.29 7.4. GROWTH LAW 121 correlation length. This might not be be impossible. Effects of the substrate morphology on the density of an adjacent liquid have been discussed recently [221]. But in our case, the observed density of the quasiliquid in contact with both substrates is the same (see Sec. 7.5), which does not support the idea of a strong structural difference.

7.4.5 Influence of roughness Comparing the layer thickness for the rough and for the smooth substrate, one realizes that there is a good match for low temperatures up to about −0.7◦C (or a corresponding layer thickness of about 16 A).˚ The layer thickness for the rough substrate is systematically higher, but the difference is within the error bars. The thickness might also appear larger because the density profile is more strongly smeared out by the larger roughness. Up to this point, and with the exception of the very low temperatures, the interface melting at the rough and at the smooth substrate follows a logarithmic growth law with a growth amplitude on the order of the bulk correlation length of water. Fitting this part of the growth law for the rough substrate with a logarithmic law yields a growth amplitude of 3.5 A˚ (see Fig. 7.27e), which is close to the value obtained for the smooth substrate. For temperatures above −0.7◦C, the quasiliquid layer at the rough substrate grows much faster with temperature. At the highest temperatures covered by the experiment, its thickness is about twice as large at the rough substrate than at the smooth substrate (55 A˚ compared to 27 A).˚ This is approximately the ratio of the values for the rms roughness (‘local’ roughness in the case of the rough substrate, see Sec. 7.3) of the two substrates. The high temperature part of the growth law for the rough substrate can be fitted with a different logarithmic growth law (see Fig. 7.27e), but this would be difficult to explain theoretically. Fitting this part with a power law (see Fig. 7.27f) yields the exponent −(0.31 ± 0.03), still very close to the value expected for non-retarded Van der Waals forces. A possible interpretation is that the roughness shifts the cross-over from logarithmic to power law growth. For the smooth substrate, the necessary layer thickness is larger than the values reached in the experiment, therefore only a logarithmic growth law is observed. For the rough substrate, in contrast, the cross-over thickness is lowered by the morphology. This leads to the observation of an apparent cross-over in the growth law. An alternative interpretation could be that the interface melting, at both the rough and the smooth substrate, in principle follows a power law. In the case of the smooth substrate, the interfacial melting is blocked, therefore the growth law deviates from the power law at high temperatures. The roughness switches the behavior to complete melting, therefore a power law fits the data of the rough substrate over the whole temperature range. Roughness effects have been discussed in the context of other wetting phe- nomena (see Sec. 3.2.7). Netz and Andelman [103] investigated theoretically the 122 CHAPTER 7. RESULTS AND DISCUSSION influence of roughness for the case of interfacial melting. They consider the case of Van der Waals type interactions and a simple sinusoidal roughness. In contrast, the substrates used in this work exhibit self-affine roughness (see Sec. 7.3), and the interfacial melting appears to be governed by exponentially decaying inter- actions for small L. It should be possible, however, to use Netz and Andelman’s approach with the type of roughness and interactions found in this work. Other experiments have also shown that curvature effects as they occur for example in porous media have an influence on ice premelting (see Sec. 3.4.3). Beaglehole and Wilson [154] also concluded from their ellipsometry measurements that the roughness of a glass substrate favors the interface melting of ice (see Sec. 3.4.4).

7.4.6 Comparison with surface melting There is a large number of experiments on the surface melting of ice whose results scatter quite strongly (see Sec. 3.4.2). For comparison, the work of Lied et al. (see references in Sec. 3.4.2) is presented. They studied the surface melting of ice using evanescent x-ray diffraction [13]. This allows to directly probe the relevant order parameter of the melting transition contained in the Bragg intensities. Lied et al. have investigated three different high-symmetry faces of ice, among them the basal orientation used in this work. The observed layer thickness as a function of temperature for various orientations is shown in Fig. 7.31c. The temperature range covered by those experiments is smaller than in this work. All curves can be approximated by logarithmic growth laws. Deviations appear at high temperatures, but unlike in this work, no deviations are observed for low temperatures. The onset temperature obtained from the surface melting experiments is be- ◦ ◦ tween −12.4 C and −13.5 C, and thus higher than the T0 obtained from extrap- olating the logarithmic growth law at the interface. A difference in the onset temperature is no surprise, as T0 is determined by the interfacial free energies according to mean-field theory (see Eq. 3.5). The interfacial energies for an ice surface and an ice–solid interface are different, of course. More striking is the large growth amplitude for surface melting between 36 A˚ and 84 A,˚ and a quasiliquid layer thickness of up to 500 A˚ at around −0.3◦C. This implies that either the quasiliquid at the surface of ice displays much stronger correlations, or there is another effect enhancing the surface melting, which is not accounted for in the model predicting the logarithmic growth law.

7.4.7 Influence of temperature error Due to the logarithmic or power law growth of the quasiliquid layer, errors in the temperature measurement are of minor importance for low-temperatures, but can have a strong influence on the observed growth law for temperatures close 7.4. GROWTH LAW 123

a surface/interface melting, basal face b surface/interface melting, basal face 600 1000 surface 500 rough 400 interface 100 smooth (Å) (Å) 300 interface L L 10 200 surface 100 rough interface 1 smooth interface 0 0.1 1 10 0.1 1 10 T T T T m- (K) m- (K)

c surface melting, different faces 600 (0001) basal 500 (1100) 400 (1000) prism 300 (Å) L 200 100 0 0.1 1 10 T T m- (K)

Figure 7.31: Growth law for surface melting of ice. (a) Comparison of surface [144] and interface melting (this work), basal orientation. (b) Same in a log-log plot. (c) Surface melting of ice on different faces, from [144]. 124 CHAPTER 7. RESULTS AND DISCUSSION

100 DT = +20 mK 80 DT =0mK DT = -13 mK 60 DT = -40 mK (Å) L 40

20

0 0.01 0.1 1 10 T -T (K) T m measured smooth T rough

Figure 7.32: Effect of temperature error ∆T = Treal−Tmeasured on observed growth law. The long dashed lines correspond to the maximum errors (see Sec. 6.6). The green short dashed line corresponds to the error expected from the measured temperature distribution in the sample. The effect on the growth law is larger for positive ∆T , as in this case the real temperature is closer to the bulk melting point Tm where the layer thickness diverges. The layer thickness then already diverges at temperatures that are nominally below Tm (upper dashed line), but such a behavior is not observed in the experiments. The maximum temperature reached in the experiments with the rough and the smooth substrate (Trough and Tsmooth, respectively) is indicated.

to the melting point. Fig. 7.32 shows the calculated temperature dependence of the layer thickness that would be observed for various offsets in the temperature measurement (the real growth law is always the same). The effect is strongest for a positive offset ∆T = Treal − Tmeasured, which causes a positive curvature of the observed L(T ) close to Tm. Such a behavior is not found in this work, and according to Sec. 6.6, ∆T is expected to be about −13 mK (which corresponds to the green short dashed line in Fig. 7.32). The real layer thickness is, therefore, expected to be slightly larger than the observed one, but with no significant influence on the type of growth law observed. For measurements even closer to the bulk melting point, however, the absolute temperature accuracy has to be improved (see Sec. 8.5). 7.5. DENSITY AND STRUCTURE OF THE QUASILIQUID 125 a rough substrate b smooth substrate 1.8 1.8

1.6 1.6

1.4 1.4 (g/cm³) (g/cm³)

qll 1.2 qll 1.2 ? ? 1.0 1.0

0.8 0.8 0.1 1 10 0.1 1 10 T T T T m- (K) m- (K) c both substrates d both substrates (as a function ofL ) 1.8 1.8 rough substrate rough substrate 1.6 smooth substrate 1.6 smooth substrate 1.4 1.4 (g/cm³) (g/cm³) qll qll 1.2 HDA 1.2 HDA ? ? 1.0 water 1.0 water ice Ih ice Ih 0.8 0.8 0.1 1 10 0 102030405060 T T L m- (K) (Å)

Figure 7.33: Density of quasiliquid layer. The solid lines indicate the trend. (a) Rough substrate. (b) Smooth substrate. (c) Comparison of the rough and the smooth substrate. The densities of ice Ih, bulk water, and HDA are indicated by dashed lines. (d) Density as a function of the layer thickness. The solid line is a linear fit (in this log-lin plot) and indicates the decay of the average density towards the bulk value as the layer thickness increases.

7.5 Density and structure of the quasiliquid

7.5.1 Experimentally observed density

The density of the quasiliquid layer is part of the model describing the dispersion profile of the interface used to fit the reflectivity data. In the beginning, it 3 was fixed to the density ρwater = 1.0 g/cm of bulk water. But it soon became apparent that a layer with the density of water would not be able to reproduce the measured data. The density of the quasiliquid has then be used as a free parameter. The values obtained from the fitting are shown in Fig. 7.33 as a function of temperature for both the rough and the smooth substrate. The values scatter quite strongly, but are distinctly higher than the density of bulk water. The error bars are upper bound uncertainties as they include 126 CHAPTER 7. RESULTS AND DISCUSSION systematic errors due to the nature of the fitting process used. The mean values are 1.20 g/cm3 for the rough substrate and 1.19 g/cm3 for the smooth substrate. The density of the quasiliquid is thus about 20%√ higher than in bulk water. This corresponds to a compression by a factor of 3 1.2 ≈ 1.063. The agreement of the observed densities at the smooth substrate and the rough substrate shows that the density enhancment is not a roughness-induced artifact from the fitting procedure. It also shows that the enhanced density is not caused by the specific substrate morphology. The strongly enhanced density of the quasiliquid layer is an experimental fact. The reliability of the model has been extensively verified (see Sec. 7.2.3). Only a strongly enhanced density can consistently explain the measured reflectivity curves (see for example Fig. 7.13). In the following, the origin of this experimen- tally observed density enhancement will be discussed. In principle, impurities dissolved in the quasiliquid layer can lead to an in- creased density. Impurities were found to be the origin of the anomalous prop- erties of the so-called ‘polywater’ reported in the 1960s, see for example [222]. There are several arguments against impurities being the reason of the density increase observed in this work: • The sample preparation process renders significant contaminations very un- likely: The ice single crystals are grown from high-purity water, and most contaminations have a very low solubility in ice. The substrates have been thoroughly cleaned (see Sec. 6.1). When contacting the ice with the sub- strate, a molten layer is created and squeezed out, which flushes away im- purities (see Sec. 6.4). • A strong degree of contamination would be needed to obtain the observed density increase. • If contaminations were responsible for the density increase, the density would decrease strongly with the thickness of the quasiliquid layer due to dilution. • It would be unlikely to find the same degree of contamination in two differ- ent samples (rough substrate and smooth substrate). Dissolution of the substrate also has to be considered as a source for the increased density. The solubility of SiO2 in water is practically zero [223]; for amorphous silica, the reported values are around 100 ppm [204]. The solubility increases with temperature, pressure, and pH (for pH values larger than 8). At a pH of 11, the solubility is as high as 1000 ppm [204]. The radiation-induced production of OH− could have a similar effect, but the amount of dissolved ma- terial would still be far too low. Moreover, the dissolution of SiO2 would lead to a decrease of the SiO2 layer thickness. The data analysis, which also yields the thickness of the SiO2 layer, does not show such a decrease (see Fig. 7.34). 7.5. DENSITY AND STRUCTURE OF THE QUASILIQUID 127

25

20

15 (Å) SiO2

d 10 rough substrate 5 smooth substrate 0 01020 number of measurement

Figure 7.34: Thickness of the SiO2 layer as a function of the temperature. Since the measurements are not very sensitive to the thickness of the SiO2 layer, the values scatter considerably, but do not indicate a significant decrease.

7.5.2 Conclusions about the structure One is forced to conclude that the high density of the quasiliquid comes from structural differences compared to bulk water, i.e. a different local arrangement of the H2O molecules. Structural differences are not unexpected. They are the reason why the premelting layer is called quasiliquid instead of simply liquid. In the experiments reported in this work, the structural differences must be very pronounced. The main question is how the structure of the quasiliquid may be related to the other known structures of H2O. Are there actually any possible local arrange- ments of H2O molecules yielding such a high density? In order to answer these questions, the experimentally determined density of the quasiliquid has been compared with the density of other forms of H2O (see Tab. 7.5) . Many of these forms can only be produced under high pressure, but can also be recovered at atmospheric pressure, if the temperature is low enough (usually, the temperature of liquid nitrogen, 77 K, is sufficient). Tab. 7.5 includes the densities measured at atmospheric pressure, which were compared to the density of the quasiliquid. For most of the structures listed in Tab. 7.5, temperature dependent data is not available. Regular ice Ih expands by 1.7% upon heating from 85 K to 265 K. This can be used as an estimate for the other H2O structures. This relatively small effect due to thermal expansion is neglected in the following. Comparing the densities one realizes that the density of the quasiliquid is very close to the one of high-density amorphous (HDA) ice, which points to a close structural relationship. There are also crystalline H2O structures with sim- ilar densities (ice II, V, and IX, see Tab. 7.5), but the density of HDA matches 128 CHAPTER 7. RESULTS AND DISCUSSION

Table 7.5: Solid forms of H2O. Densities are given for the temperature T and pressure p stated. Ice Crystal Proton T p Density Reference system order (K) (GPa) (g/cm3) Ih Hexagonal No 250 0 0.920 [14] Ic Cubic No 78 0 0.931 [14] II Rhombohedral Yes 123 0∗ 1.170 [14] III Tetragonal No 250 0.28 1.165 [14] 0† 1.14 [224] IV Rhombohedral No 260 0.5 1.292 [14] 110 0∗ 1.272 [14] V Monoclinic No 223 0.53 1.283 [14] 98 0∗ 1.231 [14] VI Tetragonal No 225 1.1 1.373 [14] 0† 1.31 [224] VII Cubic No 295 2.4 1.599 [14] 0† 1.50 [224] VIII Tetragonal Yes 10 2.4 1.628 [14] 0† 1.46 [224] IX Tetragonal Yes 165 0.28 1.194 [14] 0† 1.16 [224] X Cubic Symmetric 300 62 2.79 [14] 0† 2.51 [224] XI Orthorhombic Yes 5 0 0.934 [14] XII Tetragonal No 260 0.50 1.292 [14] 0† 1.29 [224] LDA Amorphous n/a 77 0∗ 0.94 [9] HDA Amorphous n/a 77 1.0 1.31 [9] 77 0∗ 1.17-1.19 [9, 10] VHDA Amorphous n/a 77 0∗ 1.25 [34] ∗samples recovered at low temperatures †see [224] 7.5. DENSITY AND STRUCTURE OF THE QUASILIQUID 129 best. Several arguments can be used to argue against the formation of a dif- ferent crystalline structure at the interface upon approaching the bulk melting temperature.

• Interfacial melting is considered to be a disordering transition (like surface- induced disorder), with

• the quasiliquid having liquid properties as shown by measurements of ad- hesion [171] and friction, as well as experiments in porous media (see Sec. 3.4.3) showing the presence of a liquid fraction.

• The peculiar features of the possible crystalline phases: Ice II and ice IX show proton ordering, ice IX has no region of stability and an ex- tremely complicated structure with 28 molecules in the unit cell. The large unit cell makes a continuous growth of the layer difficult.

As the observed interfacial melting is an equilibrium phenomenon, the forma- tion of the metastable HDA would be quite odd. It could not fully account for the liquid-like properties either. In current theories of the water structure HDA ice is the glassy counterpart of a postulated high-density liquid (HDL) form of water (see Sec. 2.4). The idea that the quasiliquid might be governed by fluctuations into the hypothetical HDL phase is therefore very appealing. According to the hypothesis of a liquid-liquid transition, bulk water consist of fluctuations of HDL and LDL (low-density liquid), see Fig. 7.35a. Interfaces might then display a preference for one of the two liquid phases leading to an effective pinning of part of the density fluctuation spectrum present in water, in this case the high-density fluctuations (see Fig. 7.35b). This may lead to an effective stabilization of a thin HDL (or LDL) layer at interfaces and explain the observation of an enhanced density in the quasiliquid. The stabilization of one of the liquid forms might also be influenced by the nano-confinement between the two solids SiO2 and ice (see Fig. 7.35c). This picture is supported by a number of other experiments aimed at under- standing the interface between liquid water and a hydrophobic substrate [11, 12]. These experiments consistently show an interfacial water layer with a density de- pletion of around 10%. Although these results have been interpreted differently so far, they fit well to the idea that different water fluctuations can be stabilized at interfaces. They also give a hint to what leads to the preference of either HDL or LDL fluctuations. It seems that hydrophobic interface potentials stabilize a low-density form of water, whereas hydrophilic interface potentials created by the particular interface used in this work stabilize a high-density form of water. In other experimental studies of the Au(111)–electrolyte interface [225] and the Ag(111)–electrolyte interface [226], a strong density increase in the interfacial water has been observed as well. The interface effect is restricted to the first molecular layers. In the case of [226], a layering of the water molecules leading 130 CHAPTER 7. RESULTS AND DISCUSSION a bulk b interface c confinement

Figure 7.35: Schematic illustration of fluctuations in water. Black: High-density fluctuations. White: Low-density fluctuations. (a) In bulk water, both high- density and low-density fluctuations are present and lead to the average density of 1 g/cm3. (b) Interface potentials may lead to an effective stabilization of part of the fluctuation spectrum. This effect is restricted to an interfacial layer, while the bulk situation is found further away from the interface. (c) A strongly confined layer can be completely governed by one part of the fluctuation spectrum. to an oscillation of the density profile has been found. The ‘areal density’ within the first layer is increased by a factor of four. Although the density oscillates, there is a net density excess in the interface region (see Fig. 7.36). The experiments that we carried out in this work lack the resolution to resolve unambiguously individual layers of water molecules (possible layering within the quasiliquid). Instead, they provide an average density and the overall thickness of the quasiliquid.

7.6 Si wafer as substrate

As already mentioned in Sec. 7.1, an additional experiment has been performed using a thin and ultra-smooth silicon wafer as substrate. The cleaning procedure for the wafer (see Sec. 7.6.1) was different from the one used for the other ex- periments (compare Sec. 6.1). The same experimental technique was used (see Chapt. 5), but the setup was slightly different (see Sec. 7.6.2).

7.6.1 Sample Si(001) wafers with a thickness of ≈ 0.6 mm were kindly provided by P. Dreier (Siltronic AG). The rms roughness of these wafers was extremely small (σ ≈ 2 A).˚ The experiment with one of these wafers aimed at obtaining more information 7.6. SI WAFER AS SUBSTRATE 131

5 data running average (3 Å) 4 running average (6 Å) 3

2

Relative density 1

0 0 5 10 15 20 z (Å)

Figure 7.36: Water density at Ag(111)/electrolyte interface at +0.52 V of the potential of zero charge (normalized to the value far from the interface). Data from Toney et al. [226] (solid line) and running averages (dashed lines).

wafer a>amax

a

b

Figure 7.37: Reflection from the backside of thin samples can occur for incident angles larger than αmax. about the influence of roughness on interface melting. Furthermore, a low rough- ness leads to a slow decay of the reflectivity curve, which allows measurements up to high momentum transfers, corresponding to a high resolution in real space. Despite the small thickness of the wafers, reflection of the beam from the bottom side of the wafer does not occur, because the incident angles are sufficiently small. The maximum angle αmax without reflection from the back side of the wafer is (see Fig. 7.37) t α = arctan = 24 mrad, (7.1) max l where t denotes the thickness of the wafer and l its length. This formula applies to the case of full illumination of the interface. In practice, the footprint at the maximum angle is very small and located in the center of the sample. Then l can be replaced by l/2 and the maximum angle is even larger. But already the smaller value corresponds to a momentum transfer of 2 A˚−1 at the x-ray energy used. ◦ The wafer was first cleaned in concentrated HNO3 at ≈ 70 C for several 132 CHAPTER 7. RESULTS AND DISCUSSION hours. This also leads to the formation of a thick oxide layer with a strongly hydrophilic termination. This step was succeeded by the RCA standard clean 1 (H2O2-NH4OH-H2O, 1:1:5). After cleaning, the wafer was placed on a clean block of pure aluminum and contacted with an ice single crystal as described in Sec. 6.4.

7.6.2 Experimental setup The wafer experiment was carried out at beamline ID15B of the ESRF, unlike the other x-ray experiments described in this work. For monochromatization, a bent Laue-type Si(511) monochromator was used at an energy of 78.2 keV (cor- responding to a wavelength of 0.159 A).˚ The energy bandwidth was 0.3%, defined by the secondary slits in this setup. The monochromator also demagnifies the beam in the horizontal direction to about 50 µm. For focusing in the vertical direction (which is more relevant for this experiment, see Chapt. 5) we commis- sioned a new bent multilayer (W/B4C, 150 layers, period 23.73 A,˚ 5.5% d-spacing gradient along the length of 240 mm, slope error ≈ 1 µrad). It allows to focus the x-ray beam to a vertical beam size of less than 4 µm at a distance of 2200 mm. For a comparable focal size, a focal distance of ≈ 4000 mm is needed with the CRL used in the other experiments. For smaller focal distances, the number of single lenses in the CRL has to be increased, which considerably reduces the transmission. Therefore, the setup with the multilayer is also of great interest for other experiments where limited space is an issue. Like the optics setup presented in Chapt. 5, it allows a good control of the various beam parameters such as focal size, x-ray energy, and bandwidth, at a comparable or even higher flux. It is more sensitive to beam instabilities, but can nevertheless be used for high-energy x-ray reflectivity experiments. The sample tower and detector setup is identical to the one described in Chapt. 5. A schematic view of the setup is shown in Fig. 7.38.

7.6.3 Results Unfortunately, complete reflectivity curves could not be measured for this sam- ple. The reason is that the thin wafer is also rather flexible, which causes a slight bending of the wafer (probably during the preparation of the ice–substrate in- terface). As a consequence, the incident angle is not well defined (see Fig. 7.39). The local incident angle actually varies on the interface. This can be seen by moving the interface across the beam (see Fig. 7.40). This leads to • a change of the incident angle by minor movements of the beam relative to the sample, since this also considerably moves the beam footprint on the interface (compare Fig. 7.40), • an averaging over a range of incident angles by the beam footprint. 7.6. SI WAFER AS SUBSTRATE 133

a top view

monochromator multilayer sample detector tower table secondary slits 2q||

detector

42 m 5 m 4.4 m 2.2 m 1.3 m b side view

2qML

2qz c1

c2 z y x

ai

Figure 7.38: Sketch of the high-energy setup with focusing multilayer. (a) Top view. (b) Side view. The focusing in the horizontal direction is achieved by a bent monochromator and decoupled from the focusing in the vertical direction obtained by the bent multilayer. The reflection angle 2θML of the multilayer is largely exaggerated in the sketch. The real angle is close to 35 mrad . 134 CHAPTER 7. RESULTS AND DISCUSSION

z

bent substrate X-ray beam a2

a1 x

Figure 7.39: Reflection from bent substrate. The incident angle depends on where the beam hits the substrate.

x (mm) -15 -10 -5 0 5 10 15 1.0 0.8 0.6 0.4 0.2 (mrad) 0.0 -0.2 -0.4 -0.6 -60 -40 -20 0 20 40 60 z (µm)

Figure 7.40: Bending of the wafer. Measured by the difference between actual and nominal (3.5 mrad) incident angle as a function of the sample position z. The movement of the sample along z causes a movement of the beam footprint along x (see also Fig. 7.39). 7.7. NEUTRON REFLECTIVITY 135

0 -0.32°C 10 -0.79°C -15.60°C 10-2

10-4 Intensity (arb. units)

10-6 0.2 0.4 0.6 0.8 1.0 q (Å-1) z

Figure 7.41: Reflectivity measurements for ice in contact with a Si wafer. The appearance of interference fringes with increasing temperature indicates interfa- cial melting. Unfortunately, measurement of complete reflectivity curves was not possible due to bending of the wafer.

As the size of the beam footprint on the interface changes with the momentum transfer (see Sec. 5.3), the averaging range also changes. The range is smaller for large momentum transfers (as are the movements of the footprint along the interface). Therefore, parts of the reflectivity curves could still be measured at large momentum transfers. A few measurements for different temperatures are shown in Fig. 7.41. It can be seen that the measurements qualitatively behave like the ones performed with the silicon blocks as substrates (see Sec. 7.1): With in- creasing temperature, interference fringes with decreasing intervals appear. This means that a layer with different density and increasing thickness forms. The qualitative conclusion from this observation is that interfacial melting occurs for the wafer substrate as well. This observation is important, since the substrate in this case has undergone a different chemical treatment leading to a thick oxide layer that is hydrophilic from the beginning (see also Sec. 7.8).

7.7 Neutron reflectivity

First attempts to study interface melting with neutron reflectivity have been undertaken by Lied [144]. In the first experiment of this work, neutron reflectivity was used as well, and the results of this experiment will be presented here. The formalism for neutron reflectivity is analogous to x-ray reflectivity (de- scribed in Chapt. 4) when the materials involved are not magnetic. However, the scattering length density for neutrons is not connected to the electron density. The neutrons interact mainly with the nuclei, and the scattering length den- 136 CHAPTER 7. RESULTS AND DISCUSSION sity depends crucially on the isotopes involved. As the typical extinction length for neutrons is large, they are often the method of choice for studying buried interfaces. There are some drawbacks though: • the low flux compared to Synchrotron radiation sources, which limits the dynamic range of reflectivity measurements to about 6 orders of magnitude, • the low resolution in q-space, • the low spatial resolution in real space (large beam size and limited flux)

• the necessity to use D2O, since H2O produces strong background scattering due to the large incoherent neutron scattering cross section of H (see [227]).

7.7.1 Sample As in most of the x-ray experiments, a silicon block covered with a native oxide layer was used as substrate. The chemical preparation of this substrate and the preparation of the interface were as described in Sec. 6.1. The ice single crystals were made of D2O instead of H2O.

7.7.2 Experimental setup The neutron reflectivity experiments were carried out at the evanescent-wave reflecto-diffractometer EVA [228] of the Institut Laue-Langevin (ILL) in Greno- ble. The instrument uses cold neutrons delivered from a reactor source. A py- rolytic graphite (002) monochromator provides a neutron beam with a wavelength of 5.5 A˚ (corresponding to an energy of 2.7 meV) and a bandwidth of 1.2%. A cooled Be filter suppresses higher harmonics. The neutron flux at the sample po- sition is about 1 × 106 n/s. The EVA diffractometer resembles the one described in Sec. 5.3, but does not have the same level of mechanical accuracy (which is not needed for the neutron reflectivity experiments, as the typical angles of inci- dence are much larger). The same sample chamber as in the x-ray experiments (see Sec. 6.5) was used. The aluminum windows are suitable for experiments with neutrons and high-energy x-rays as well. For the neutron reflectivity ex- periment, however, the beam penetrates the sample through the silicon side and gets reflected from the ice interface (see Fig. 7.43), as D2O has a higher neu- tron scattering length density than silicon. A 3He linear wire counter serves as a 1-dimensional position-sensitive detector (PSD) for measuring the scattered intensity (see Fig. 7.42).

7.7.3 Results Measured neutron reflectivity curves for two different temperatures are shown in Fig. 7.44a in comparison with measurements from Lied (Fig. 7.44b) and calcula- 7.7. NEUTRON REFLECTIVITY 137

1200 primary ? = 10 mrad beam i 800 measurement Gaussian fit 400 reflected

Intensity (arb. units) beam 0 100 200 300 400 500 channel number

Figure 7.42: Example of PSD recording showing the reflected and primary neu- tron beam.

qz

n Si SiO 2 qll PSD ice (D2 O)

sample cell

Figure 7.43: Schematic geometry for neutron reflectivity measurements at ice– SiO2–Si interfaces. 138 CHAPTER 7. RESULTS AND DISCUSSION tions (Fig. 7.44c). At high momentum transfers, the reflectivity curves measured ◦ close to the bulk melting point (Tm = 3.8 C for D2O [229]) lie all above the curves measured far below. The effect is small, but consistent with the measurements performed by Lied. It indicates that there is actually a change in the density profile as the temperature approaches the melting point. The calculations show that the formation of a layer with increased density can explain the change of the reflectivity. Unfortunately, the density profile cannot be resolved due to the limited q-range accessible in the neutron reflectivity measurements.

7.8 Substrate termination and radiation effects

As discussed in Sec. 7.1, long exposure to high-energy x-rays apparently causes the substrate to change from hydrophobic to hydrophilic. Further experiments have been undertaken in order to investigate this effect. First, the assumption of a radiation-induced change of the substrate termi- nation has been verified by controlled irradiation. Therefore, a substrate was prepared as described in Sec. 6.1, which was initially hydrophobic. One sam- ple was placed in a small plastic container half-filled with milli-Q water. The sample was then exposed to high-energy x-ray radiation from an x-ray tube with a W target (Kα1 59.3 keV). In order to maximize the flux, no monochromator was used, but the low-energy part of the spectrum was filtered out by a 2 mm thick aluminum absorber. The part of the sample which was immersed in water changed from hydrophobic to hydrophilic where it was exposed to radiation (see Fig. 7.45). The part of the sample which had stayed in air showed this change only on tiny isolated spots (where water droplets may have condensed during the exposure). This shows that the radiation damage is apparently mediated by + − radicals created by photolysis of H2O molecules (H2O → H + OH ). This fits to the fact that these radicals are also used in Si cleaning procedures leading to a hydrophilic termination (for example treatment with KOH solution or HNO3). In order to characterize the change of the substrate termination, XPS mea- surements have been performed6 on irradiated hydrophilic and non-irradiated hydrophobic parts of the substrate. Fig. 7.46 shows a comparison of the spectra for silicon and carbon. The carbon peak comes from hydrocarbon contamina- tions which cannot be completely avoided when the sample is kept in air for an extended period of time. For the Si spectrum, an additional component can be distinguished on the hydrophobic part when compared to the hydrophilic one. A scan across the irradiated part shows that this feature also occurs at a point about 3 mm from the hydrophilic zone, which is probably one of the hydrophilic spots outside the water-immersed area. The difference in the silicon spectrum coincides with an energy shift of the carbon peak indicating a different bonding of the hydrocarbons in the hydrophilic and the hydrophobic region.

6in collaboration with L. Jeurgens and M. Wieland (both MPI f¨urMetallforschung) 7.8. SUBSTRATE TERMINATION AND RADIATION EFFECTS 139

a this work b previous measurements (Lied) 100 100 10-1 10-1 3.5°C 10-2 3.5°C 10-2 -22.2°C -15.0°C 10-3 10-3 10-4 10-4 10-5 10-5 Intensity (arb. units) Intensity (arb. units) 10-6 10-6 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 q (Å-1) q (Å-1) z z b simulation

100 10-1 L =32Å 10-2 L =0Å 10-3 10-4

-5 Intensity (arb.10 units) 10-6 0.00 0.05 0.10 0.15 q (Å-1) z

Figure 7.44: Neutron reflectivity from D2O(ice)–SiO2–Si interfaces. (a) Measure- ments from this work. (b) Measurements from Lied [144]. (c) Calculated neutron reflectivity curves for different values of the quasiliquid layer thickness L (thick- ness of SiO2 layer 20 A,˚ rms roughness 5 A).˚ The measurements close the melting ◦ point (Tm=3.8 C) show increased reflectivity (see a+b), which can be explained by the formation of a quasiliquid layer (see c). 140 CHAPTER 7. RESULTS AND DISCUSSION

hydrophilic

Figure 7.45: Photograph of substrate showing radiation induced change of the hydrophilicity. The central part of the sample has been exposed to high-energy x-ray radiation for several days. It was half-way immersed in water during this time. The irradiation has produced a strongly hydrophilic termination where the sample was in contact with water (arrow). The irradiated area exposed to air only shows a few hydrophilic spots (above).

As mentioned in Sec. 7.1, the reflectivity measurements on briefly irradiated parts of the sample do not clearly indicate interfacial melting. This means that either the hydrophobic termination of the substrate does not lead to interfacial melting, or it leads to a low-density quasiliquid as suggested by other experiments (see Sec. 7.5.2). In the latter case, the density contrast between ice and the quasiliquid might be too small for detecting the quasiliquid layer (see Fig. 8.1).

7.9 Implications

The main conclusion from this work is that intrinsic interface melting of ice is indeed possible, which has been controversially debated in literature. Interfacial melting has been observed at the interface between ice and SiO2. The growth law is in good agreement with expectations from theory and other experiments. The effect strongly depends on the hydrophilicity of the SiO2. Further experiments are necessary to investigate the influence of the substrate chemistry. A distinct influence of the substrate morphology was also observed, as suggested by previous experiments and theory. An intriguing property of the observed quasiliquid layer is its surprisingly large density, which is about 20% higher than the density of bulk water. This suggests that the quasiliquid resembles the postulated HDL form of water. Therefore, the results of this work do not only have implications for the theory of interface melting, but may also have implications for our understanding of water. The 7.9. IMPLICATIONS 141

5 a b 8 Si 2p metallic 4 6 y =0mm C 1s peak 1 Si 2p extra 4 3 Si 2p oxide 2 2 Intensity (arb. units) Intensity (arb. units) 0 1 98 100 102 104 106 280 285 290 295 Energy (eV) Energy (eV) 5 8 c Si 2p metallic d 4 6 y =7mm C 1s peak 1 4 3 Si 2p oxide 2 2 Intensity (arb. units) Intensity (arb. units) 0 1 98 100 102 104 106 280 285 290 295 Energy (eV) Energy (eV)

286.2 35 e Si 2p extra f C 1s peak 1 30 286.0 25 20 285.8 15 285.6

10 Energy (eV)

Peak area (arb. units) 5 285.4 0 -2 0 2 4 6 8 10 12 14 -202468101214 Position y (mm) Position y (mm)

Figure 7.46: XPS spectra of partially irradiated substrate. First column: silicon peaks. Second column: carbon peaks. (a)+(b) Non-irradiated part showing an additional component in the silicon spectrum. (c)+(d) Irradiated part. (e)+(f) Scan across the irradiated part showing the additional component in the silicon spectra and a shift of the carbon peaks. 142 CHAPTER 7. RESULTS AND DISCUSSION question is how the observation of a dense form of water depends on the chemistry of the interface and on the nano-confinement between ice and SiO2. If these questions are answered, systems might be tailored to stabilize thin films of either HDL or LDL and study its properties. The results of this work also have important ramifications for processes in na- ture and technology, some of which have been mentioned in Sec. 2.1. They might offer a better understanding of phenomena like glacier motion and permafrost, for example. It is also conceivable that further research could be technologically exploited, for example for the development of coatings capable of preventing the icing of airplane wings. The observation of a dense form of water with hitherto unknown properties is important in this context. The physical and chemical properties of the quasiliquid determine the impact of interfacial melting in the aforementioned situations. The postulated HDL form of water is expected to have a lower viscosity and a larger solubility for impurities than bulk water. A quasiliquid with similar properties would promote glacier motion better than normal bulk water. The high solubility for impurities could imply that enrichment of impurities in the quasiliquid lead to a self-amplifying process, where the impurities cause a true reduction of the melting point, which further increases the thickness of the premelting layer. In a wider context, the results of this work affect all situations where water is in confinement or at interfaces. If the modification of the water structure in such situations is indeed a generic phenomenon, it will be important for a much wider range of issues, from electrochemistry to biological systems. A more detailed discussion of the questions raised by this work follows in Chapt. 8. Chapter 8

Outlook

This chapter summarizes open questions and prospects for further research.

8.1 Influence of the substrate material and the confinement

Measurements with different substrates, especially substrates with different hy- drophilicty/hydrophobicity, could address some important open questions:

1. How does interface melting depend on the substrate termination? (In the case of SiO2, our experiments suggest that interface melting might only occur when the termination is hydrophilic.)

2. How does the observation of a high-density liquid depend on the sub- strate? (Our observation in conjunction with other experiments suggests that hydrophilic substrates favor the high-density liquid and hydrophobic substrates favor the low-density liquid, see Sec. 7.5.)

Another question is how the formation of the high-density and low-density liquid depends on the nano-confinement between ice and the substrate. Therefore, experiments should be performed for ice in contact with a substrate (interface melting scenario), and for bulk liquid water in contact with the same substrate. A problem for future experiments is that if the LDL form of water is stabilized 3 upon interfacial melting of ice, the contrast between ice (ρice = 0.92 g/cm ) and 3 LDL (≈ ρLDA = 0.94 g/cm ) would be very low. In this case, it might be difficult to observe a LDL layer in reflectivity measurements (see Fig. 8.1). If the ice is replaced by liquid water, the density contrast is larger, and it is thus easier to observe a LDL layer at the water–substrate interface than in the interface melting scenario. From our observations, we have concluded that interfacial melting occurs when ice is in contact with hydrophilic SiO2. But unfortunately we only have complete

143 144 CHAPTER 8. OUTLOOK

0.1 1E-3 1E-5 1E-7 1E-9 1E-11 without quasiliquid layer r 3 qll=0.94 g/cm (LDL) Intensity (arb.1E-13 units) r 3 1E-15 qll=1.00 g/cm (water) r 3 1E-17 qll=1.18 g/cm (HDL) 0.0 0.2 0.4 0.6 0.8 1.0 q -1 z (Å )

Figure 8.1: Calculated reflectivity profiles for various densities of the quasiliq- uid. The curve for a quasiliquid layer with the density of LDL is practically indistinguishable from the curve without a quasiliquid layer. Parameters for this ˚ ˚ ˚ ˚ ˚ calculation: L = 25 A, LSiO2 = 15 A, σice = 1.5 A, σSiO2 = 3 A, σSi = 1 A, x-ray energy 71 keV. data sets for the substrates that were rendered hydrophilic by irradiation. For the thin wafers, which were hydrophilic from the beginning, complete reflectivity curves could not be measured due to bending of the wafers (see Sec. 7.6). It would, therefore, be interesting to repeat these measurements with thick substrates. Silicon samples with a hydrophobic coating are a good choice for a hydropho- bic substrate, since smooth samples can be easily prepared. The problem here is to find a coating which is stable against radiation damage. We have performed test experiments with coatings of trimethylchlorosilane (TMCS), but the coating was removed after exposure times of just a few seconds. If possible, substrates with a hydrophilic and a hydrophobic part should be prepared, or even with parts having different degrees of hydrophilicity. In this way, the influence of the hydrophilicity could be tested without the need for exchanging the sample and without altering the experimental conditions. For a more fundamental understanding, experiments with well-controlled hy- drophobic and hydrophilic substrates are most interesting. But one can also look for substrates that are of specific interest for processes in nature and technology. The SiO2 substrates used in this work may serve as a model for ice–mineral inter- faces as they occur in nature. The interface between ice and stainless steel may represent the contact of an ice skate with the rink, and the interface between ice and rubber might be investigated in order to improve the contact between tires and icy roads. Many other materials are of interest, however, the experimental technique used in this work requires that smooth surfaces can be prepared from the substrate material. Also for some substrates, the large density compared to 8.2. SURFACE MELTING 145 ice, or an internal density profile, can make it difficult to resolve a thin quasiliquid layer in the density profile. Further problems may arise for the preparation of the ice–substrate interface, if the heat conductivity of the substrate is small. Another question is the difference between amorphous and crystalline sub- strates (with the same composition). For the substrates used in this work, this means replacing the amorphous SiO2 with a quartz single crystal. A crystalline substrate is more likely to induce a specific structure in the interfacial water, as for example seen in the adsorption of water on Pt(111) [218]. The amorphous substrates used in this work come closer to an ideal hard wall and, therefore, are thought to reveal intrinsic water properties like the fluctuations in HDL and LDL. An advantage of crystalline substrates is the possibility to compare the experimental results to theoretical calculations, whereas amorphous substrates are still very difficult to calculate. For the interface between quartz and water, results from calculations can already be found in the literature [230, 231].

8.2 Surface melting

Some of the experiments on surface melting of ice are not sensitive to the average density. In the analysis of other experiments, the density of the quasiliquid is often assumed to be the same as in bulk water. It might be worthwhile reexamining some of these results to see whether there are any indications for a change in the average density at the free surface as well.

8.3 Influence of the substrate morphology

This work has shown that the substrate morphology can have a pronounced influence on interface melting. Further experiments could help to understand this influence. In any case, this influence has to be taken into account when comparing results from different substrates.

8.4 Influence of impurities

Impurities can cause a drastic reduction of the ice melting temperature. Accord- ing to theory [135] and experiments [154], impurities should also have a great influence on surface and interface melting. This influence is of great interest for ‘real’ interfaces in nature which are practically never free of impurities. It is of special interest for atmospheric chemistry (see Sec. 3.5.3), since ice particles in the atmosphere serve as a reservoir and reaction site for chemicals. The experimental challenge consists in controlling the concentration of impu- rities at the interface. One way might be by doping ice crystals. The advantage of this method is that the doping with certain chemicals (like HF or NH3) is 146 CHAPTER 8. OUTLOOK possible after contacting with the substrate1. Otherwise, the distribution of im- purities will most probably change during the interface preparation (see Sec. 6.4), as it involves the melting of the ice surface.

8.5 Growth law

As discussed in Sec. 7.4, deviations from a logarithmic growth law are observed at low temperatures, where a thin quasiliquid layer seems to persist below the onset temperature obtained by extrapolating the logarithmic growth law. This behavior should be explored by measurements at low temperatures. Several factors define the lower temperature limit that can be reached with the current setup:

• the maximum temperature difference between back side and front side of the Peltier elements (about 65 K),

• the cooling power of the Peltier elements,

• the cooling power of the circulator for the cooling liquid,

• the admissible temperature range of the hoses used for the cooling liquid (currently −20◦C).

With the current setup, sample temperatures down to about −50 or −60◦C should be possible. However, reflectivity measurements up to very high momentum transfers of about 1.8 A˚−1 are required to achieve the real space resolution necessary to observe molecular monolayers. The second question concerning the growth law is the behavior very close to the melting point. From theory, a cross-over to a power-law would be expected as already suggested by the experiment with the rough substrate. In order to con- duct reliable measurements at temperatures even closer to the bulk melting point, the temperature accuracy and uniformity has to be improved. This presumably requires replacing the sample holder by an additional inner chamber thermally decoupled from the exterior by an insulating vacuum in the outer chamber.

8.6 Structure of the quasiliquid

While the density of the interfacial quasiliquid hints to a close structural relation- ship with the HDA ice (and might actually be identified with the postulated HDL water), definite proof is missing. Therefore, diffraction experiments are necessary to resolve the structure of the quasiliquid.

1However, the use of such chemicals in a cold-room would certainly be a safety issue. 8.7. INFLUENCE OF ELECTRIC FIELDS 147

This can in principle be achieved with evanescent x-ray diffraction, success- fully applied to the study of Pb(liq.)–Si interfaces [193]. Unfortunately, this method cannot be applied to the system studied in this work, since the signal from the amorphous SiO2 layer cannot be effectively separated from the signal of the quasiliquid. One would need a crystalline substrate to do that. Also, a substrate with a lower electron density than ice would be a great advantage. In this case, the evanescent wave can be induced in the quasiliquid instead of the substrate, which leads to an enhancement of the signal. Due to the low elec- el ˚−1 tron density of ice (ρice = 0.31 A ), a suitable substrate with lower electron density is difficult to find. Beryllium, for example, has an electron density of el ˚−3 el ˚−3 ρBe = 0.49 A , Kapton (polyamide) ρKapton = 0.44 A . Another possibility for probing the structure of the quasiliquid would be to use a sample containing a large number of parallel ice–substrate interfaces If the fraction of the quasiliquid in the sample is large enough, bulk scattering techniques could be used.

8.7 Influence of electric fields

One possible microscopic mechanism for interfacial melting is the preference of a certain orientation of the water molecules with respect to the interface (‘interface polarization’, see the work of Fletcher mentioned in Sec. 3.4.1). Such an alignment would violate the ice rules and could in this way break the hydrogen bond network in ice. Due to the dipole moment of the water molecules, a similar effect could be achieved with electric fields. It would, therefore, be interesting to study the effect of electric fields on interfacial melting. However, large electric fields are needed to induce an appreciable alignment of the water molecules. The ratio r of molecules with parallel and antiparallel alignment with respect to an electric µE  −30 field E behaves as r = exp kT , where µ = 6.186 × 10 Cm is the dipole moment of a water molecule (see Sec. 2.2), k = 1.381 × 10−23 J/K the Boltzmann constant, and T the temperature. For a 10% orientation (r =55/45) and at the bulk melting point T = 273.15 K, an electric field of about 1.2 × 108 V/m would then be required. A possibility might be to use a tip-shaped electrode, which allows to produce locally very strong fields. 148 CHAPTER 8. OUTLOOK Chapter 9

Summary

In this work the interfacial melting of ice has been investigated with a novel high-energy x-ray diffraction scheme. Ice and water are abundant on Earth, of paramount importance for the bio- sphere, and part of our everyday life. Despite the apparent simplicity of the water molecule, H2O has astonishing and often anomalous properties of which many are still not completely understood. (See Chapt. 2.) Among these phenomena is the melting of ice which exhibits two special fea- tures. The first is pressure melting, i.e. the possibility to melt ice by applying pressure. The second is surface melting, i.e. the appearance of a thin (quasi)liquid layer at the surface of ice at temperatures below the bulk melting point. Surface melting is observed in other materials, but appears to be particularly strong in the case of ice. The phenomenological explanation of this effect is the minimization of the free energy of the system by introducing a quasiliquid layer. The quasiliquid at the surface may serve as a nucleation site for the bulk melting, and is thus of fundamental importance for understanding the melting process in general. (See Chapt. 3.) Surface melting is well established and in the case of ice, it has important consequences for environmental processes. In most cases, however, ice is in con- tact with other materials, and the question arises whether a similar effect also occurs at ice–solid interfaces. There are many indications that such interface melting is possible. But it may depend on many parameters like the material of the solid substrate and its surface morphology. So far, investigations of well- defined ice–solid interfaces on nanoscopic length scales have been missing. (See Chapt. 3.) This is mainly due to the lack of suitable experimental techniques for prob- ing in situ deeply buried interfaces with adequate resolution. A recently devel- oped high-energy x-ray transmission-reflection scheme is an ideal probe for such cases. This technique is based on the use of high-energy x-ray (here ≈70 keV) mi- crobeams, which can only be produced at modern Synchrotron Radiation sources. At the energies used in this scheme, x-rays can penetrate up to several centimeters

149 150 CHAPTER 9. SUMMARY of material. Compound refractive lenses (CRLs) allow to focus the x-ray beam down to a spot size of a few microns. Among the advantages of this scheme are the large dynamic range and the low background level for diffraction experiments. However, very high angular and positional accuracy are required. Therefore, a special diffractometer with selected components was used for this work. The whole setup was installed at beamline ID15A of the ESRF. (See Chapt. 5.) The high-energy x-ray scheme allows to apply established x-ray surface tech- niques at buried interfaces. In this work, x-ray reflectivity measurements have been used. These measurements are sensitive to the (electron) density profile per- pendicular to the interface. If a quasiliquid layer forms due to interfacial melting, it gives rise to interference fringes in the measured reflectivity profiles. Complete density profiles can be reconstructed with the Parratt formalism. (See Chapt. 4.)

In this work, interfacial melting of ice at ice–SiO2–Si interfaces has been stud- ied. This particular interface might serve as a model for ice–mineral interfaces as they occur in nature. The SiO2–Si substrates were prepared from polished Si single crystals. After cleaning, a native amorphous oxide of 1–2 nm thickness forms at the Si surface in air. The substrates become strongly hydrophilic when exposed to high-energy x-ray radiation in the presence of water. Single crystals of ice were provided by J. Bilgram (ETH Z¨urich). Smooth and homogeneous interfaces were prepared by melting and subsequent recrystallization of the ice surface in contact with the substrate. The ice crystals were oriented with their c-axis [0001] perpendicular to the interface. The sample preparation was carried out in a walk-in cold room. For the Synchrotron experiments, a mobile sample chamber was constructed. It allows a precise control of the sample temperature, which is necessary for measurements close to the melting point. (See Chapt. 6.) Two substrates with different morphology have been used in the experiments. The roughness has been analyzed with x-ray and AFM measurements. The ‘smooth’ substrate exhibits an rms roughness of ≈ (2.7 ± 0.4) A˚ and only weak lateral correlations. The ‘rough’ substrate can be described by means of a height- difference correlation function g(R) = 0.11 · R2·0.34 (from the analysis of the x-ray data). (See Sec. 7.3.) The x-ray reflectivity measurements clearly show that interfacial melting of ice occurs in contact with both the rough and the smooth substrate. Growth of the interfacial quasiliquid layer sets in at about −20◦C, but a very thin layer seems to persist down to lower temperatures. The thickness of the quasiliquid layer increases with temperature and reaches 55 A˚ at −0.036◦C for the rough substrate and 27 A˚ at −0.022◦C for the smooth substrate. Theory predicts a logarithmic growth of the quasiliquid layer thickness L as a function of the temperature T when exponentially decaying (short-ranged) interactions dominate:   Tm − T0 L (T ) = L0 ln , (9.1) Tm − T 151

where Tm is the bulk melting temperature. A fit with a logarithmic growth law matches extremely well the measured data for the smooth substrate. It yields the growth amplitude L0 =(3.7 ± 0.3) A,˚ which is slightly smaller than the reported bulk correlation length of water (4.5 A˚ to 8 A).˚ In the case of the rough substrate, the fit with a logarithmic growth law does not match the measured data too well, although the growth amplitude L0 =(8.2 ± 0.4) A˚ is still close to the reported correlation length of water. (See Sec. 7.4.) The quasiliquid layer thickness at the rough substrate agrees with the smooth substrate at low temperatures up to about −0.7◦C, corresponding to a layer thickness of about 16 A.˚ From then on, the quasiliquid layer at the rough substrate grows much faster. The growth law for the rough substrate can also be described by a power law p L (T ) ∝ (Tm − T ) (9.2) with an exponent p close to −1/3 expected from theory for dominating (non- retarded) Van der Waals type dispersion forces (see Tab. 9). Since these in- teractions are long-ranged, a cross-over from a logarithmic to a power law can occur. The observed difference in the growth laws for the rough and the smooth substrate may indicate a shift of the cross-over thickness due to the roughness. The mechanism for the roughness effect in this particular situation is not yet fully understood, but studies of roughness effects in other wetting scenarios offer promising routes for a theoretical study of this aspect. (See Sec. 7.4.5.) For both the rough and the smooth substrate, deviations from the logarithmic growth law are also visible at very low temperatures (corresponding to very small layer thicknesses). A thin layer seems to remain molten at temperatures below the onset expected from the fit of the layer thickness to a logarithmic growth law. In this regime, however, the continuum model leading to the logarithmic growth law is no longer valid. On the molecular scale, one might rather expect some sort of layering as suggested by the experimental observations. (See Sec. 7.4.3.)

The x-ray-reflectivity measurements also reveal the density ρqll of the inter- facial quasiliquid, 1.20 g/cm3 for the rough substrate, and 1.19 g/cm3 for the 3 smooth substrate. This is much higher than the density ρl=1.0 g/cm of bulk 3 water, but close to the density ρHDA=1.17–1.19 g/cm of high-density amorphous (HDA) ice at atmospheric pressure. This points to a close structural relationship between the quasiliquid and HDA. (See Sec. 7.5.) In the postulated two-phase scenario (see Sec. 2.4), the properties of water are explained by fluctuations into a high-density liquid (HDL) and a low-density liquid (LDL) form of water, terminating at a second critical point around 220 K (below the supercooling limit). The high-density and low-density amorphous ice (HDA and LDA) are the vitreous counterparts of the postulated liquid forms. The fact that interfacial melting is an equilibrium phenomenon suggests that the quasiliquid is governed by fluctuations into the postulated (liquid) HDL in- stead of the metastable (solid) HDA. The fluctuations into HDL might be sta- 152 CHAPTER 9. SUMMARY

Table 9.1: Summary of the results. ‘rough’ substrate ‘smooth’ substrate rms roughness not defined∗ (2.7 ± 0.4) A˚ lateral correlations g(R) = 0.11 · R2·0.34 weak logarithmic growth law fit good for low T good amplitude (8.2 ± 0.4) A˚ (3.7 ± 0.3) A˚ power law fit good strong deviations for high T exponent −(0.33 ± 0.03) −(0.29 ± 0.03) average density 1.20 g/cm3 1.19 g/cm3 largest layer thickness 55 A˚ at −0.036◦C 27.5 A˚ at −0.022◦C ∗ ‘local’ rms roughness (4.7 ± 1.1) A˚ bilized by the particular (hydrophilic) interface and by the confinement between the ice and the substrate. A similar effect has recently been reported for water in contact with hydrophobic substrates. In this case, a low-density form of water has been observed at the interface. (See Sec. 7.5.) In this work, additional experiments were performed using an ultra-smooth Si wafer as substrate. An alternative chemical cleaning procedure has been used in this case. It leads to a thick oxide layer with a hydrophilic termination (even without irradiation). Unfortunately, measurements of complete reflectivity curves were not possible due to bending of the thin wafer. The measurements never- theless show that interfacial melting occurs with this substrate as well. (See Sec. 7.6.) Other experiments have initially been performed with neutron reflectivity, the standard technique for probing buried interfaces. However, due to the small flux of neutron sources, the range in momentum space accessible to reflectivity measurements and hence the resolution in real space are too limited. This demon- strates the necessity for and the potential of the new high-energy x-ray scheme. (See Sec. 7.7.) The results of this work have important implications. The presence of a quasiliquid layer at ice–SiO2 interfaces has ramifications for many phenomena in nature, like the motion of glaciers and the stability of permafrost. These consequences depend on the so far unknown properties of the quasiliquid. In this context, the observation of a high-density form of water is important, since it suggests differences in other properties, like the viscosity and the solubility for impurities. As ice interfaces in nature are usually rough, the observed influence of the substrate morphology is relevant for conclusions about ‘real’ systems. Finally, interfaces or confinement situations as provided by interface melting can reveal new information about the intrinsic properties of water and may eventually allow 153 to study the postulated forms of water. (See Sec. 7.9.) A number of important questions is raised by this work. This includes the influence of the specific substrate material, the role of the confinement, the effect of the substrate morphology, and the influence of impurities. Future experiments could also cover a larger temperature range to study the transition from inter- face melting to bulk melting (which requires a better control of the temperature) and the initial stage of interface melting at low temperatures. The most intrigu- ing (and most challenging) task will certainly be to resolve the structure of the interfacial quasiliquid. (See Chapt. 8.) 154 CHAPTER 9. SUMMARY List of acronyms

AFM atomic force microscopy/microscope CRL compound refractive lense DFT density functional theory DWBA Distorted-Wave Born Approximation ESRF European Synchrotron Radiation Facility EVA evanescent wave diffractometer (instrument at the ILL) fcc face-centered cubic FWHM full width at half maximum HDA high-density amorphous ice HDL high-density liquid water ILL Institut Laue-Langevin LDA low-density amorphous ice LDL low-density liquid water LJ Lennard-Jones MF mean-field NCS Neutron Compton Scattering NMR nuclear magnetic resonance PSD position-sensitive detector QENS quasi-elastic neutron sscattering qll quasiliquid layer rms root mean square SID surface induced disorder SIO surface induced order SFVS sum-frequency vibrational spectroscopy SR Synchrotron Radiation UHV ultra-high vacuum XPS x-ray photoelectron spectroscopy

155 156 LIST OF ACRONYMS List of figures

2.1 Free water molecule...... 4 2.2 Response functions of water...... 6

2.3 H2O phase diagram for moderate and high pressures...... 7 2.4 The structure of ice Ih...... 8

3.1 Interface melting scenarios for ice...... 12 3.2 Interface melting of a solid s in contact with another medium b. . 14 3.3 Free energy calculations and growth laws for different types of interactions...... 17 3.4 Contributions to the free energy...... 18 3.5 Growth laws for interfacial melting of ice from the literature . . . 29

4.1 Reflection and refraction of a plane wave...... 34 4.2 Reflection and refraction of a plane wave at multiple interfaces. . 36 4.3 Sketch of an interface contour z (R)...... 39 4.4 Illustration of an interface contour with a fixed (conformal) density profile...... 41 4.5 Schematic graph of specular and diffuse reflectivity...... 43

5.1 Linear attenuation coefficient µ as a function of the x-ray energy. 48 5.2 Comparison of conventional x-ray surface scattering scheme and the transmission-reflection scheme...... 49 5.3 Schematical sketch of the setup for the study of ice–solid interfaces with high-energy x-ray microbeams...... 51 5.4 Schematic layout of the high-energy beamlines ID15A/B...... 52 5.5 Sketch of the setup for the high-energy microbeam transmission- reflection scheme...... 53 5.6 Beam profiles...... 54 5.7 Diffractometer with sample chamber...... 55 5.8 Scattering geometry...... 57 5.9 Illustration of a rocking scan...... 58

5.10 Momentum transfer resolution δqz...... 59 5.11 Momentum transfer resolution δqx...... 60

157 158 LIST OF FIGURES

5.12 Illumination of the interface...... 63 5.13 Illumination of the interface with an arbitrary beam profile. . . . 64 5.14 Illumination correction...... 64 5.15 Footprint of the beam on the interface...... 65 5.16 Coherence in x-ray scattering...... 66

6.1 Figure error of the substrates...... 72 6.2 Photograph of a Si block used as substrate...... 73 6.3 Reflectivity measurement of a substrate...... 74 6.4 Cutting of ice crystals...... 75 6.5 Interface preparation...... 77 6.6 Photographs of the sample...... 78 6.7 In situ chamber for x-ray and neutron scattering experiments with ice ...... 79 6.8 Schematic view of the control setup...... 81 6.9 Part of temperature log...... 81 6.10 Temperature distribution in the sample...... 83

7.1 Time line of the experiment with the rough substrate...... 87 7.2 Time line of the experiment with the smooth substrate...... 88 7.3 Reflectivity measurements for ice in contact with the rough substrate. 89 7.4 Reflectivity measurements for ice in contact with the smooth sub- strate...... 90 7.5 Reflectivity measurements on different positions of the sample. . . 91 7.6 Dispersion and density profiles...... 93 7.7 Model for fitting the reflectivity data...... 94 7.8 Reconstructed density profiles for the rough substrate...... 95 7.9 Reconstructed density profiles for the smooth substrate...... 96 7.10 Illustration of the model for the ice–SiO2–Si interface...... 99 7.11 Comparison between kinematical and dynamical calculation of re- flectivity...... 100 7.12 Calculated reflectivity curves showing the influence of model pa- rameters...... 101 7.13 Reliability of the fits...... 102 7.14 Reliability of the roughness parameter σSi...... 102 7.15 Rocking scans for the smooth substrate with different resolutions. 103 7.16 Peak width in a rocking scan for the smooth substrate...... 104 7.17 Rocking scans with the smooth substrate for different qz...... 104 7.18 Measurement of integrated intensity...... 106 7.19 Calculated rocking scan profiles for different values of the Hurst parameter h...... 107 7.20 Width of the diffuse reflectivity peak as a function of qz...... 108 7.21 Roughness replication at substrate–quasiliquid–ice interface. . . . 109 LIST OF FIGURES 159

7.22 Comparison of calculated and measured rocking scans for different h...... 110 7.23 Comparison of calculated and measured rocking scans for different qz...... 111 7.24 AFM images of the substrates...... 112 7.25 Height-difference correlation functions g(R) determined from AFM measurements...... 113 7.26 Growth law for interfacial melting as expected by theory...... 115 7.27 Growth law for the rough substrate...... 117 7.28 Growth law for the smooth substrate...... 118 7.29 Influence of roughness on the growth law...... 119 7.30 Reversibility of the quasiliquid layer formation...... 120 7.31 Growth law for surface melting of ice...... 123 7.32 Effect of temperature error on observed growth law...... 124 7.33 Density of the quasiliquid layer...... 125 7.34 Thickness of the SiO2 layer...... 127 7.35 Schematic illustration of fluctuations in water...... 130 7.36 Water density at Ag(111)/electrolyte interface...... 131 7.37 Reflection from the backside of thin samples...... 131 7.38 Sketch of the high-energy setup with focusing multilayer...... 133 7.39 Reflection from bent substrate...... 134 7.40 Bending of the wafer...... 134 7.41 Reflectivity measurements for ice in contact with a Si wafer. . . . 135 7.42 Example of PSD recording for neutron reflectivity...... 137 7.43 Scattering geometry for neutron reflectivity ...... 137 7.44 Neutron reflectivity from D2O(ice)–SiO2–Si interfaces...... 139 7.45 Photograph of substrate showing radiation induced change of the hydrophilicity...... 140 7.46 XPS spectra of irradiated substrate...... 141

8.1 Calculated reflectivity profiles for various densities of the quasiliquid.144 160 LIST OF FIGURES List of tables

5.1 Momentum transfer resolution...... 60 5.2 Coherence parameters...... 68

7.1 Overview of experimental parameters...... 86 7.2 Fit parameters for the rough substrate...... 97 7.3 Fit parameters for the smooth substrate...... 98 7.4 Morphology of the substrates...... 114 7.5 Solid forms of H2O...... 128 9.1 Summary of the results...... 152

161 162 LIST OF TABLES Bibliography

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Hauptberichter: Helmut Dosch Mitberichter: Clemens Bechinger Betreuer: Harald Reichert

Special thanks to

J¨org Bilgram, Matthias Denk, Veijo Honkim¨aki, Oliver Klein, Markus Rauscher, Sebastian Sch¨oder and

Frank Adams, Stephanie Adelhelm, Esther Barrena, Jean-Fran¸coisChemin, P. Dreier, Alwin Engemann, Christa Engemann, Felix Engemann, Helen Engemann, Dimas Garcia de Oteyza Feldermann, Robert Fendt, Ulrich Gebhardt, Ernst G¨unther, Christian Gutt, Rolf Henes, Lars Jeurgens, Peter Keppler, Anne-C´ecile Lacroix, Klaus Mecke, Bert Nickel, Ben Ocko, Walter Plenert, Craig Priest, Ingo Ramsteiner, Adrian R¨uhm,Werner Schweika, Andreas Sch¨ops,Udo Seifert, Ana- toly Snigirev, David Snoswell, Michael Sprung, Eugene Stanley, Reinhard Streitel, Ulrike T¨affner, Metin Tolan, Philippe Villermet, Alexei Vorobiev, Peter Weiss, Annette Weißhardt, Helmut Wendel, G¨unther Wiederoder, Michaela Wieland, Martin Zimmermann, the MPI machine shops, the OZ-team,¨ the low-temperature service group, those I forgot.

This work has been funded by the Deutsche Forschungsgemeinschaft in the pri- ority program on ”wetting and structure formation at interfaces”.

179