Premelting at the ice–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¨
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 water 3 2.1 Importance ...... 3 2.2 The H2O molecule and the hydrogen bond ...... 4 2.3 Anomalies and mysteries ...... 5 2.4 The quest for the water structure ...... 6 2.5 Ice Ih ...... 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–solid interfaces ...... 26 3.4.5 Further aspects ...... 29 3.5 Consequences of ice premelting ...... 30 3.5.1 Permafrost ...... 30 3.5.2 Glacier 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 biosphere, 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 properties of water 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 glaciers sculpt the landscape. The climate crucially depends on the presence of ice, water, and vapor. Snow, 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 oxygen (large sphere) and two hydrogens (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 hail, often causing considerable damage. Ice is of great relevance for buildings and infrastructure (frost heave, ice on power lines, freezing pipes, avalanches), traffic (slippery and snow covered roads, ice on airplane wings, icebergs 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, icicles, and frost patterns on windows. We enjoy skiing and ice skating, 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 hydrogen bond
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 crystal 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 oxygens, 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 amorphous ice (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 solids, 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 triple point, 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 crystals 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