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THERMODYNAMIC PROPERTIES OF HELIUM IN POROUS MEDIA

A Dissertation in

Physics

Zhigang Cheng

 2013 Zhigang Cheng

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

May 2013 ii

The dissertation of Zhigang Cheng was reviewed and approved* by the following:

Moses H. W. Chan Evan Pugh Professor of Physics Dissertation Advisor Chair of Committee

Jainendra K. Jain Erwin W. Mueller Professor of Physics Evan Pugh Professor of Physics

Thomas E. Mallouk Evan Pugh Professor, DuPont Professor of Materials Chemistry and Physics Professor of Biochemistry and Molecular Biology

Julian D. Maynard Distinguished Professor of Physics

Jorge O. Sofo Professor of Physics Professor of Materials Science and Engineering

Nitin Samarth Professor of Physics George A. and Margaret M. Downsbrough Department Head

*Signatures are on file in the Graduate School

iii

ABSTRACT

The first credible experimental hint of non-classical rotational inertia (NCRI) or supersolidity was reported in 2004 in a torsional oscillator experiment of solid 4He confined in Vycor. Since then numerous studies on the possible novel state have been carried out. While it was shown very recently that the observed drop in the resonant period of the torsional oscillators housing solid helium which was interpreted as a signature of NCRI is more likely a mechanical phenomenon other than a real phase transition, a number of interesting properties of solid 4He have been observed during the last decade in many laboratories. These newly discovered results include a shear modulus anomaly, dc flow through solid helium and a heat capacity peak. Most of these studies focus on bulk crystalline solid 4He. This dissertation focuses on the study of thermodynamic properties of helium in porous media.

We have measured the heat capacity of solid 4He grown in aerogel and Vycor. For solid 4He in aerogel, the dependences of heat capacity on pressure and 3He concentration have been systematically studied. We found evidence that 3He atoms tend to reside in the vicinity of silica strands as temperature is decreased forming a 3He rich region.

We have also carried out measurements of thermal conductivity of solid 4He embedded in Vycor. The thermal conductivity of Vycor is not significantly changed with the infusion of solid helium. Interestingly, the infusion of liquid 4He in Vycor pores results in a three-fold reduction in thermal conductivity below 0.5 K. The introduction of superfluid 4He films and liquid 3He into the Vycor pores also results in the reduction of thermal conductivity. We propose a model suggesting the origin of the reduction is the

iv presence of hydrodynamic slow sound modes in liquid 4He, as well as in superfluid 4He films and liquid 3He. The slow sound modes facilitate the quantum tunneling of two-level systems (TLS) in silica and dramatically increase the TLS-phonon scattering. The more modest reduction in solid helium-Vycor composite is caused by the presence of phonon excitations in solid helium which also facilitate TLS tunneling in silica.

TABLE OF CONTENTS

List of Figures ...... vii

List of Tables ...... xi

Acknowledgements ...... xii

Chapter 1 Introduction to Superfluid ...... 1

1.1 Quantum Phenomena at Low Temperatures ...... 1 1.2 Superfluidity ...... 2 1.2.1 Two-fluid Model ...... 3 1.2.2 Landau’s Theory ...... 6 1.2.3 Sound Modes in Liquid 4He ...... 8

Chapter 2 Introduction to Supersolidity ...... 11

2.1 Solid 4He ...... 11 2.2 Experiments after 2004 ...... 14 2.2.1 Torsional Oscillator Experiments ...... 14 2.2.2 Shear Modulus Measurements ...... 17 2.2.3 Shear Modulus Effect on Torsional Oscillators ...... 18 2.2.4 Other Experiments on Solid 4He ...... 20

Chapter 3 Heat Capacity of Solid 4He in Porous Media ...... 22

3.1 Experimental Details ...... 22 3.1.1 AC Calorimetry ...... 22 3.1.2 Calorimeters ...... 25 3.1.3 Thermometers...... 28 3.1.4 Heater ...... 30 3.1.5 Weak Thermal Link ...... 31 3.1.6 Electronics ...... 31 3.1.7 Gas Handling System for Sample Growth ...... 34 3.1.8 Sample Growth Methods ...... 35 3.2 Silica Aerogel ...... 37 3.3 Solid Helium in Aerogel ...... 38 3.4 Results ...... 41 3.4.1 Pressure Dependence ...... 41 3.4.2 3He Concentration Dependence ...... 46 3.5 Measurements of solid 4He in Vycor ...... 51 3.6 Review of Heat Capacity Experiments of Bulk Solid 4He ...... 54

Chapter 4 Thermal Conductivity of Vycor with Helium ...... 68

4.1 Two-Level Systems (TLS) ...... 68 4.2 Thermal Conductivity of Solids ...... 69

4.2.1 Thermal Conductivity of Crystalline Solids ...... 69 4.2.2 Thermal Conductivity of Glass ...... 71 4.2.3 Thermal Conductivity of Porous Vycor Glass ...... 72 4.3 Experimental Setup ...... 74 4.4 Results and Analysis ...... 77 4.4.1 Thermal Conductivity of Empty Vycor Glass ...... 77 4.4.2 Thermal Conductivity of Vycor Infused with Helium ...... 77 4.5 Conclusion ...... 85 4.6 Prospective Experiments ...... 85 4.6.1 Vycor Filled with Liquid 3He-4He Mixture ...... 86 4.6.2 Porous Glass with Smaller Pores...... 86 4.6.3 Vycor Filled with Liquid Nitrogen at 77 K ...... 87

Appendix: Heat Capacity of Helium Film in Porous Media ...... 88

A.1 Introduction to Helium Films ...... 88 A.1.1 Kosterlitz-Thouless Theory ...... 88 A.1.2 Helium Film in Porous Media ...... 90 A.2 Heat Capacity of Helium Films ...... 98 A.2.1 Vycor vs. Silica Gel ...... 100 A.2.2 Localized BEC ...... 101 A.3 Some Comments on Torsional Oscillator Results ...... 106 A.4 Conclusion...... 106

vii

LIST OF FIGURES

Figure 1-1 Heat capacity of liquid 4He at different pressures: triangle - vapor pressure; square -19 bar; circle - 25 bar [Keesom 1935]...... 3

Figure 1-2 Top: torsional oscillator used to measure superfluid density by Andronikashvili. Bottom: superfluid and normal fluid faction as a function of temperature (picture from [Carusotto 2010])...... 5

Figure 1-3 Neutron scattering results showing the dispersion relation for superfluid 4He including phonon and roton branches [Henshaw 1961]...... 8

Figure 1-4 Velocity of different sound modes in superfluid liquid 4He. The velocity of third sound is strongly dependent on the film thickness and not plotted...... 10

Figure 2-1 Phase diagram of 4He ...... 12

Figure 2-2 (a) Schematic drawing of torsional oscillator. (b) Resonant period vs. temperature for solid 4He in Vycor [Kim 2004b]...... 15

Figure 2-3 Top: Schematic drawing of dislocation lines and 3He atoms in solid 4He. Bottom: Shear modulus and NCRI of solid 4He with different 3He concentrations. Both data are normalized by the lowest temperature values [Day 2007]...... 17

Figure 2-4 (a) Torsional oscillator with “bare Vycor”; (b) Measured period drop of solid 4He grown in the “bare” Vycor torsional oscillator [Kim 2012] compared with that in previous torsional oscillator [Kim 2004b]...... 20

Figure 3-1 AC calorimetry: diagram of a sample with a heater and a thermometer coupled to the thermal bath...... 23

Figure 3-2 Typical frequency scan of a calorimeter ...... 25

Figure 3-3 Sapphire calorimeter with aerogel piece...... 28

Figure 3-4 Temperature dependence of RuO2 and germanium thermometers ...... 30

Figure 3-5 Block diagram of the thermometer circuits for heat capacity measurements...... 32

Figure 3-6 Gas handling system for sample growth in heat capacity experiment...... 35

Figure 3-7 Solid 4He growth in blocked capillary method...... 36

Figure 3-8 Transmission electron microscope graph of silica aerogel. The scanned aerogel is thick so that the fractural structure is overlapped. (Courtesy of the National Center for Electron Microscopy at Berkeley Lab) ...... 38

Figure 3-9 Top:X-ray diffraction pattern for solid helium in aerogel. 5 rings indicate reflections of an hcp structure; bottom: NCRI deduced from the measured period drop of

viii torsional oscillator of solid 4He grown in aerogel. The Bulk solid 4He data have been scaled down by a factor of 3 for comparison [Mulders 2008]...... 39

Figure 3-10 Diffraction peak for the (101) reflection. The peak width indicates grain size of ~100 nm [Mulders 2008]...... 40

Figure 3-11 Heat capacity of solid and liquid 4He in aerogel ...... 42

Figure 3-12 Fitting of heat capacity data using . Data have been vertically shifted for clarity...... 42

Figure 3-13 Anomalous heat capacity bump after subtracting polynomial fittings...... 45

Figure 3-14 Pressure dependence of peak temperature (left scale) and peak size (right scale). Bulk solid samples and solid-liquid coexistent samples (open symbols) are also plotted for comparison...... 46

Figure 3-15 Heat capacity of solid 4He grown in aerogel with different concentrations of 3He impurities...... 48

Figure 3-16 Onset temperature of solid 4He heat capacity enhancement by 3He compared with theoretical [Edwards 1962] and experimental [Lin 2009] phase separation temperature of bulk solid helium...... 49

Figure 3-17 (a) Heat capacity peak with high 3He concentrations: (b) zoom-in only showing the smaller peaks with lower 3He concentrations...... 50

Figure 3-18 Peak size as a function of 3He concentration...... 51

Figure 3-19 Heat capacity of the empty calorimeter with Vycor and with solid 4He grown in Vycor. For the heat capacity of empty calorimeter, the deviation from the fitting has the possible origin from surface excitation contribution from Vycor pores [Tait 1975]...... 53

Figure 3-20 Heat capacity “bump” after subtracting the polynomial fitting...... 53

Figure 3-21 Heat capacity of solid 4He (top) and the heat capacity peak after subtracting Debye contributions (bottom)...... 56

Figure 3-22 Extra heat capacity due to phase separation with high 3He concentration in solid helium is superimposed on the heat capacity peak...... 57

Figure 3-23 Measured heat capacities of empty calorimeters [Lin 2007, Lin 2009, West 2009]...... 60

Figure 3-24 Measured heat capacity of empty calorimeter and estimated heat capacity contributions from different components [Lin 2009]. It shows that the measured heat capacity is in consistent with the estimation...... 61

Figure 3-25 Frequency scan of the calorimeter with solid helium sample reported in [Lin 2009]...... 62

Figure 3-26 Schematic thermal contact profile of calorimeter with solid helium: (a) heater and thermometers are attached on the outer surface of calorimeter; (b) heater and thermometers are attached on the inner surface of calorimeter...... 64

Figure 4-1 Schematic drawing of an asymmetric double-well potential that forms a two- level system. is the asymmetry of the double-well potential, is the tunneling barrier and is the spatial distance between two potential minima...... 69

Figure 4-2 Thermal conductivity of crystalline quartz and vitreous silica [Zeller 1971]...... 71

Figure 4-3 Thermal conductivity of crystalline quartz, vitreous silica and porous Vycor [Zeller 1971, Zaitlin 1975]...... 74

Figure 4-4 Experimental setup of thermal conductivity measurements...... 76

Figure 4-5 Thermal conductivity of empty Vycor and Vycor with atomically thin helium films (a) and solid helium (b) of both isotopes...... 80

Figure 4-6 Thermal conductivity of empty Vycorand Vycor with liquid 4He, liquid 3He and superfluid 4He films with superfluid transition temperature well above 1 K...... 81

Figure A-1 Top: Superfluid fraction of liquid 4He confined in Vycor as a function of temperature. Solid curve represents superfluid fraction of bulk liquid 4He. Bottom: Superfluid fraction as a function of reduced temperature [Kiewiet 1975]...... 93

Figure A-2 Top: period drop plotted as a function of temperature for different thickness of 4He films adsorbed on Vycor. Bottom: period drop vs reduced temperature [Reppy 1992]...... 94

Figure A-3 Period drop at as a function of surface coverage of 4He films on the substrate of Vycor [Crooker 1983]. The relation between and coverage is nonlinear at low coverage...... 95

Figure A-4 Specific heat of 4He adsorbed on Vycor [Brewer 1970]...... 96

Figure A-5 Coverage dependence of TC or T0 [Tait 1979b]. Dashed line indicates the superfluid onset temperature determined by Berthold et al. [Berthold 1977]...... 97

Figure A-6 Heat capacity per unit area for various coverage of 4He film adsorbed on Vycor [Finotello 1988]. Arrows label the heat capacity peaks at the superfluid transition. ... 98

Figure A-7 Heat capacity plotted as a function of coverage at fixed temperatures for helium films adsorbed on Vycor. Solid curves are for guiding eyes [Chan 1991]...... 99

Figure A-8 Heat capacity plotted as a function of coverage at fixed temperatures for helium films adsorbed on silica gel [Finotello unpublished]...... 100

Figure A-9 Schematic diagram of the structure of Vycor (left) and silica gel (right). Black color represents silica and white color represents empty space...... 101

Figure A-10 Schematic drawing of localized BEC of helium adsorbed silica surface ...... 103

Figure A-11 (a) Quantum tunneling between localized BECs without phase coherence when potential wells are not completely filled; (b) quantum tunneling is suppressed when phase coherence realized by completely filling potential...... 104

Figure A-12 Heat capacity isotherm of helium films adsorbed on silica gel at 0.3 K. The dotted line is a horizontal extrapolation of the heat capacity at high coverage...... 105

LIST OF TABLES

Table 3-1 Fitting parameters for heat capacity of solid 4He in aerogel...... 44

Table 3-2 Components of empty calorimeters for heat capacity measurements...... 60

xii

ACKNOWLEDGEMENTS

I would like to thank my research advisor Dr. Moses Chan. He provides me not only the fantastic knowledge of physics, well-equipped laboratory, but a great model of extraordinary physicist as well. Besides pushing me to do my best on researches, he also generously gives me all kinds of supports in academics and personal life. Most importantly, the critical thinking in physics and confidence in making things work that I learned from him prove to be the most precious assets in my research.

I would also like to thank the other members of my committee: Jainedra Jain, Thomas

Mallouk, Julian Maynard and Jorge Sofo. Their enlightening guidance and advices made my research a lot more fruitful. I would also thank many other people at Penn State that offered me great help over these years. I apologize that I can only list a few of these great people here:

Milton Cole, Dezhe Jin, Xia Hong, Neal Staley, Tim Bowmaster, Barry Dutrow, Megan

Meinecke, John Passaneau.

It is a great honor for me to collaborate with some fantastic people in the community of low temperature physics. Special thank to Norbert Mulders who always gave me brilliant advice.

It was also a happy and wonderful time to work with Clement Burns and Lawrence Lurio at

Argonne National Laboratory.

I cannot be more grateful to Anthony Clark and Xi Lin who taught me all the techniques and skills to survive through these years in the lab. Even though they left Penn State several years ago, all those moments that they taught me running experiments seem to happen yesterday.

I would also thank Mingliang Tian, Josh West, Tyler Engstrom, Duk-Young Kim, Meenkshi

Singh, Samhita Banavar, Stefan Omelchenko. It is so lucky to work in the lab with them.

I can never do enough to thank my parents. They have done their best to provide me the best education, the best life and the best love. They miss me more than I miss them when I am

xiii thousands miles away. And they always give me the strongest support even though they do not understand what I am working on (my mother is always worrying whether I feel cold when doing low temperature experiments).

Last, I would deliver my special gratitude to my wife. She is the original power that keeps me moving forward. She comforts me when I am depressed, encourages me when I make progress and is always next to me when I need her.

Chapter 1 Introduction to Superfluid

1.1 Quantum Phenomena at Low Temperatures

The study of quantum mechanics can be traced back the problem of black body radiation described by Ludwig Boltzmann and Max Planck as the energy is radiated and absorbed in discrete “quanta” in late 19th century. The foundation of quantum mechanics was established in late 19th to early 20th century. Although quantum theory had a glorious development, quantum phenomena were still hard to observe, because they only manifest themselves microscopically at room temperature. Macroscopic quantum phenomena require relatively low temperature. The success of liquefying helium by Kamerligh Onnes

[Onnes 1911] brought down the lowest possible temperature that mankind could achieved at that time below 4 K. It made the observation of macroscopic quantum phenomena possible. One of the famous macroscopic quantum phenomena is superfluidity of liquid helium itself: liquid helium can flow with zero viscosity. This phenomenon happens below 2.176 K.

Quantum phenomena are basically governed by two classes of particles that obey different statistics: bosons and fermions. Bosons are with integer spin and like to occupy the same state together whereas fermions are with half integer spin and each state can only be occupied by one fermion. Superfluidity happens in a bosonic system where all bosons are condensed in the ground state establishing macroscopic phase coherence. 3He atoms and electrons, even though fermions, can form pairs and undergo bosonic

2 condensation. Therefore there also exist superfluid 3He and superconductivity. This dissertation will only discuss superfluid phenomena of 4He condensation.

1.2 Superfluidity

Helium was first liquefied on July 10th, 1908 by Dutch physicist Kamerlingh

Onnes [Onnes 1911]. In 1938, Kaptiza, and separately Allen and Misener found that the viscosity of liquid helium vanishes when cools down below 2.176 K [Allen 1938,

Kapitza 1938]. Kapitza gave the name “superfluid” to this phenomenon. Other than vanishing viscosity, liquid 4He in superfluid state also exhibits some other intriguing properties, including that thermal conductivity becomes too large to sustain any temperature gradient, liquid 4He tends to go to lowest energy state, etc. Investigators

4 therefore distinguished liquid He when as “superfluid” state (He II), different from “normal fluid” state (He I) when .

There are also exotic thermodynamic properties at the superfluid transition. A significantly sharp heat capacity peak is at [Keesom 1935]. The singularity of heat capacity indicates that the superfluid onset is a genuine phase transition. For , heat capacity of superfluid helium show additional contribution due to roton in addition to the phonon contribution [Greywall 1978]. Thermal conductivity of liquid 4He diverges as temperature approaches to from above [Ahlers 1968]. The superior thermal conductivity of liquid 4He below 1 K is due to the counter-flow of normal fluid.

Figure 1-1 Heat capacity of liquid 4He at different pressures: triangle - vapor pressure; square -19 bar; circle - 25 bar [Keesom 1935].

1.2.1 Two-fluid Model

Superfluid was related to Bose-Einstein Condensation (BEC) by London in 1938

[London 1938]. The superfluid plays a role corresponding to the ideal boson gas that is condensed into ground state. At BEC state, bosons occupy the lowest quantum state. At this point, quantum effects become obvious on macroscopic scale and quantum coherence is achievable. For non-interacting boson gas, BEC occurs at

Eq. 1.1

where  is reduced Planck’s constant, is Boltzmann constant, m is the mass atom and

4 n is the number density. Eq. 1.1 gives for ideal He gas, close to of liquid 4He. Two-fluid model was proposed based on this analogy to BEC by Tisza [Tisza

4 1938]: liquid He below is considered to be composed of two interpenetrating fluids.

One is called “normal fluid” with its density and its velocity representing the part that is not condensed. The other part is called “superfluid” with density and velocity

, representing the condensed part. The total density .

Two-fluid model provides an appropriate method to understand many exotic behaviors of superfluid. To measure the superfluid fraction is actually to measure its companion: normal fluid. Unlike superfluid with zero viscosity, normal fluid has finite viscosity. Therefore, variation of normal fluid density can be used to deduce the variation of the superfluid density. This idea was put in practice in 1946 by Andronikashvili who made a torsional oscillator composed with flat discs with separation smaller than the viscous penetration depth of normal fluid (~1 m) [Andronikashvili 1946]. Normal fluid would move together with discs while superfluid is out of this motion. The period drop as temperature is cooled below , corresponding to a mass decoupling, indicates the emergence of superfluid. The superfluid onsets at 2.176 K and its density rapidly increases as temperature decreases and would reach 100% at . This torsional oscillator technique is used to determine superfluid fraction in various systems, including studying superfluid 4He films and searching for superflow in solid 4He.

Figure 1-2 Top: torsional oscillator used to measure superfluid density by Andronikashvili. Bottom: superfluid and normal fluid faction as a function of temperature (picture from [Carusotto 2010]).

Liquid 4He is certainly not ideal boson gas due to the interaction between 4He atoms. Leggett considered the interaction between 4He atoms and stated that superfluidity in liquid 4He is characterized by “generalized BEC” [Leggett 1970]. Due to the

6 interaction between particles, the fraction of condensate is depleted and is ~10% at even though the superfluid fraction is 100%. According to Leggett’s “generalized BEC” assumption, the macroscopic superfluid can be described by a single wave function with a coherent phase:

Eq. 1.2 where and are amplitude and phase, respectively. Applying the momentum operator

, we find

Eq. 1.3

It is straightforward to see that , implying that the contour integral is always zero in single connected region. However, some experimental results [Osborne 1950] showing made the theory to be amended by introducing vortex lines, i.e. singular line at the rotation axis. It is found that the integral around vortex (vortices) is quantized:

Eq. 1.4

where is integer, representing the number of vortices inside the contour. This is named as the Onsager-Feynman quantization condition and is called “quantum circulation”.

1.2.2 Landau’s Theory

Landau proposed a theory for superfluid 4He focusing on the excitations in liquid

4He. Consider an object with mass moving through superfluid with velocity . After

7 creating some excitation in liquid 4He, the new object velocity is . Energy and momentum conservations give the following relations

Eq. 1.5

where and are the energy and momentum of the quasiparticle associating with the excitation. With Eq. 1.5, we have

Eq. 1.6

The approximate equation is based on the assumption that is sufficiently large that the last term on the left hand side can be neglected. Therefore

and Eq. 1.7

Eq. 1.7 must be satisfied to create an excitation. Therefore, is the Landau critical velocity. If the object is moving with velocity lower than , it would not generate excitations and can move without friction.

Landau distinguished two types of excitations in liquid 4He: phonons and rotons

[Landau 1947]. In Landau’s theory, the energy spectrum of phonon is linear

Eq. 1.8 where c is the sound speed and p is the momentum. And the roton spectrum is

Eq. 1.9 where is the roton gap, is the roton minimum momentum and is effective mass

[Landau 1947] (as shown in Fig. 1-3). The values of the parameters of roton are

, and . Based on this model, the critical

8 velocity should be . However, the measured critical velocity is much lower than this value, and is dependent on the local dimensions.

Figure 1-3 Neutron scattering results showing the dispersion relation for superfluid 4He including phonon and roton branches [Henshaw 1961].

1.2.3 Sound Modes in Liquid 4He

Liquid 4He is capable of supporting multiple kinds of wave motions. Two-fluid model is helpful to illustrate the physical picture of wave motions in liquid 4He. For bulk liquid 4He, the in-phase motion of superfluid and normal fluid components results in an

9 adiabatic density variation. In this case, liquid 4He behaves like ordinary liquid in which sound waves propagate with velocity .

4 Tisza predicted that liquid He below should be able to transmit a wave motion different from the first sound [Tisza 1938]. Instead of density variation, temperature variation is involved in this mode. Within this mode, the net motion of liquid is zero:

, therefore the superfluid and normal fluid components make out-of- phase motions. This sound mode is named “second sound” with a velocity

, where is entrope and is specific heat at constant pressure.

Atkins predicted another two sound modes in liquid 4He, called “third sound” and

“fourth sound” respectively [Atkins 1959]. The third sound is a surface wave on helium film. Such wave can be produced by locally and periodically heating helium film. Such heating causes the oscillation of the local superfluid density. Since normal fluid component is locked to the substrate due to its finite viscosity, superfluid component is involved in the oscillatory motion on the surface under the superfluid density gradient.

Therefore the heating causes oscillation of films thickness effectively, which propagate as

a surface wave. The velocity of third sound is where is substrate-

dependent van der Waals binding constant and is the thickness of helium film, is latent heat and is entropy.

“Fourth sound” refers to the compression wave propagating inside pores or slits where normal fluid component is viscously locked ( ). Since normal component is immobile, fourth sound involves only the oscillation of density, but also of pressure,

10 temperature and superfluid density. Due to this property, fourth sound can be excited by the application of oscillatory pressure (like the first sound) and temperature (like the

second sound). Its velocity is given by . We can see that is close

to near when and . Under certain condition (pressure-released condition), only the second sound contribution can be observed. This is also referred to as

“fifth sound” [Rudnick 1979].

Figure 1-4 Velocity of different sound modes in superfluid liquid 4He. The velocity of third sound is strongly dependent on the film thickness and not plotted.

Chapter 2 Introduction to Supersolidity

Supersolid has been an extremely interesting topic for both experimentalists and theorists. Recent experiments from our lab at Penn State, the place where a suggestion of supersolidity was observed, announce the disappointing conclusion that supersolid is yet to be discovered. However, the process of studying it is profitable. Therefore, it is worthwhile to re-visit the chronicle of this journey.

2.1 Solid 4He

Unlike most of the substances, helium remains liquid instead of solidifying as approaches to absolute zero unless it is pressurized to beyond 25 bar. Solid 4He has hexagon closed pack (hcp) lattice structure for most of the case except a narrow strip region on the phase diagram (Fig. 2-1) where the lattice structure is body center cubic

(bcc).

Figure 2-1 Phase diagram of 4He

The possibility of superflow in solid 4He has been discussed since the discovery of superfluid. Solid 4He is the best candidate to exhibit superfluid properties, if there is any, because it has a very high degree of “quantumness”. 4He atoms in solid state still have large zero-point kinetic energy at their ground state. The zero-point kinetic energy compared with the confinement potential can be represented by DeBoer parameter

Eq. 2.1

where is Planck’s constant, is the atom mass, and are effective radius and depth parameters that describe the Lennard-Jones potential. Larger means stronger zero-point motion. equals to 2.6 for solid 4He, larger than any other solid composed of bosons.

Lindeman parameter is another measure of the quantumness of a system. It is given by

Eq. 2.2

which represents the ratio of quantum fluctuation amplitude to the interatomic distance

. It is usually used to predict the melting of a solid. For ordinary solid, melting happens

4 when . However, for solid He [Burns 1994, Glyde 1994], much larger than the threshold value of melting. Both DeBoer and Lindaman parameters indicate that solid 4He is a possible candidate to exhibit superflow in solid.

The term “supersolid” simply means “super” plus “solid”. “Solid” means it exhibits diagonal long-range order (DLRO), i.e. it has crystalline structure. And “super” comes from “superfluid” meaning the existence of superflow in solid. It is described by off-diagonal long-range order (ODLRO) [Yang 1962]. Penrose and Onsager [Penrose

1956] thought BEC is forbidden in commensurate crystalline solids. Andreev and Lifshitz stated that defects in solids experience BEC [Andreev 1969b]. Chester pointed out that

BEC can exist in incommensurate solids and it can exhibit both crystalline order as well as superfluidity [Chester 1970]. In the same year, Leggett suggested looking for non- classical rotational inertia (NCRI) as the evidence of superfluidity in solid, and estimated that the superfluid fraction is less than 10-4 [Leggett 1970].

At the same time, various experiments were also trying to search for evidence of supersolid. In plastic flow experiments, a steel ball was pulled through solid 4He to detect abnormal mass flow [Andreev 1969a]. Greywall made solid 4He with difference pressures in two chambers connected with capillary and tried to observed the pressure relaxation due to the mass flow across the capillary [Greywall 1977]. Bishop et al. carried out a torsional oscillator experiment to search for NCRI mentioned by Leggett

14 [Bishop 1981b]. Bonfait moved the solid samples to U-tube and looked for height variation [Bonfait 1989]. All these searches failed to find strong evidence to prove the existence of supersolid. Recently, Kim and Chan [Kim 2004b, a] found a period drop similar as that observed by Andronikashvili using torsional oscillator technique. They interpreted it as the probable evidence of the supersolid state and triggered enormous interests of both experimental and theoretical physicists.

2.2 Experiments after 2004

2.2.1 Torsional Oscillator Experiments

In 2004, Kim and Chan reported the observation of resonant period drop for solid

4He in porous Vycor glass [Kim 2004b] using torsional oscillator technique. The resonant period starts to drop at ~200 mK, and saturates near 40 mK (as shown in Fig. 2-

2). If assuming the period drop is due to the loss of moment of inertia of solid 4He, this phenomenon is consistent with the non-classical rotational inertia (NCRI) described by

Leggett. If this is a real NCRI, the ratio of the period drop to the total period increase caused by the presence of solid sample represents the fraction of the solid that exhibits superflow. The measured period drop indicates a supersolid fraction of 2.5% at lowest temperature (~20 mK). A dissipation peak is also observed accompanying the period drop. The period drop is also dependent on velocity of the oscillation (usually represented by the rim velocity of torsional oscillator ). It starts to decrease when

. It is also found that 3He impurities push the onset of the period drop to

15 higher temperature but suppress its amplitude. The later publication [Kim 2004a] reports the same phenomenon observed for bulk solid 4He in annular geometry. The indicated supersolid fraction is 0.7~1.7% for samples with different pressures. The control experiment by inserting a block in the annulus shows a significantly reduced period drop.

Figure 2-2 (a) Schematic drawing of torsional oscillator. (b) Resonant period vs. temperature for solid 4He in Vycor [Kim 2004b].

The period drop in torsional oscillator experiments have been repeated and confirmed by many other groups. Rittner and Reppy [Rittner 2006] observed that period drop and dissipation peak can be reduced below their resolution after the solid sample is

“annealed” (staying at a temperature slightly lower than the melting temperature for a period of time). The annealing process is effective to release strain and reduce disorder density within solid. On the other hand, Rittner and Reppy used very thin annular

16 geometry (0.15 mm) to increase disorder density, and the measured period drop indicates a supersolid fraction of ~20% [Rittner 2007]. This gives the clue that disorder in solid

4He plays an important role for the period drop. Clark et al. grew solid samples in an open cylindrical geometry using various methods (blocked capillary method and constant pressure/temperature method) to qualitatively control the disorder density [Clark 2007].

Solids grown in blocked capillary method experience much larger pressure variation during the freezing process and are supposed to be polycrystalline while solids grown in constant pressure/temperature method are single crystalline with much fewer disorder.

They observed smaller period drop and lower onset temperature for single crystal samples and larger period drop and higher onset temperature for polycrystalline. Similar annealing effect as Rittner and Reppy was reported for some but not all of the polycrystalline samples.

A number of torsional oscillator experiments from other groups have also been reported. I cannot list all of them but just some: Hunt and Pratt performed torsional oscillator experiments on an ultra-stable cryostat [Hunt 2009, Pratt 2011]. They claimed their data are consistent with a glassy transition. On the other hand, Choi et al. performed the experiment on a rotational cryostat [Choi 2010]. They observed that dc rotation suppresses the NCRI fraction. They claimed this as the evidence of real supersolid state because it can be suppressed by dc rotation which creates vortices.

17 2.2.2 Shear Modulus Measurements

A shear modulus increase of solid 4He with decreasing temperature was reported by Day and Beamish from University of Alberta [Day 2007]. This increase has similar temperature dependence as the period drop observed in torsional oscillator experiments.

Together with the shear modulus increase about 20%, similar dissipation peak and 3He dependence were also observed as in torsional oscillator. They concluded that the shear modulus increase and dissipation peak are due to thermally activated relaxation process of disordered system. Specifically, 3He atoms pin down the dislocation lines and reduce their mobility at low temperatures. Therefore, shear modulus increases by 20%.

18 Other than the Alberta group, the anomalous shear modulus increase was also observed by Rojas et al. from ENS in Paris [Rojas 2010]. They grew high quality single crystalline solid samples and the reported shear modulus increase is as high as 86%. This actually fits the dislocation line model mentioned above: compared with polycrystalline samples, single crystal samples have much less dislocation density. Therefore, the entanglement of dislocation lines themselves is not as strong as for polycrystalline samples after 3He atoms unpin dislocations lines at high temperature. The contrast of the single crystal sample between low and high temperature is much larger.

2.2.3 Shear Modulus Effect on Torsional Oscillators

For torsional oscillator experiments, the solid 4He shear modulus increase leads to an increase of resonant frequency and therefore a period drop. Clark et al. used finite element method (FEM) to simulate the period drop contribution by 20% shear modulus increase [Clark 2008]. The simulation is performed with the assumption that the torsional oscillator is perfectly rigid and no slippage or gaps exist on the interfaces between two attached object. For most torsional oscillators, the resonant period is ~1 ms.

The 20% increase of shear modulus will cause a period shift (drop) by 10-7, i.e. ~0.1 ns.

This is ~10 times smaller than some observations of several nanoseconds period drop.

However the FEM simulations are done with the assumption that the torsional oscillators are perfectly rigid. This is not necessarily true in real case. Almost all torsional oscillators with a few exceptions are assembled and sealed by epoxy resin. For those with

Vycor or annular geometry, Vycor or central cylinders are also attached to the torsional

19 bob by epoxy. Epoxy itself, even less than 1 mm thick, may not be rigid compared with solid helium. When solid 4He sample is grown, it helps epoxy to “glue” different parts to make the torsional oscillator more rigid. This leads to a period drop. Therefore, the shear modulus effect is amplified. Moreover, for some torsional oscillators, a hole is drilled at the center of the torsion rod for helium input. The hole along the torsion rod is inevitably filled with solid helium when a sample is made. Therefore the torsion spring constant is increased by the solid helium. It has been shown that this effect can account for the observation of period drop in several torsional oscillators [Beamish 2012], including the one that was claimed as the evidence of “superglass” [Hunt 2009, Pratt 2011].

Two recent experiments are performed to distinguish period drops contributions by possible supersolid from that by shear modulus effect. Mi and Reppy [Mi 2012] designed a compound torsional oscillator with Vycor. It consists of two moments of inertia (one is the sample space and the other is a dummy mass) and two similar torsion rods. Therefore it can be driven in two torsional modes at different frequencies. The real supersolid fraction should be independent on the resonant frequency while the contribution of shear modulus effect is dependent. Therefore it is possible to distinguish the two effects. The measured period drop is strongly frequency dependent, indicating significant mechanical effect. After excluding the mechanical contribution, the estimated supersolid fraction is smaller than the experimental resolution. Therefore it suggests that the NCRI is not detected with their resolution. Another experiment is to minimize the shear modulus effect [Kim 2012]. A bare Vycor is sealed by painting a thin layer of epoxy. This is enough to hold pressure up to 90 bar below 4 K. This design eliminates the possibility of bulk space between torsional bob and Vycor that can help to “glue” the

20 torsional oscillator. As shown in Fig. 2-4, quite different from previous experiment [Kim

2004b], there is no period drop below 200 mK. The significant difference between these two experiments suggests that the period drop observed previously does not originate from helium inside Vycor. It is highly possible that there exist gaps between Vycor and torsional bob and the “glue effect” of the bulk solid helium in the gaps mimics NCRI in the previous design (Fig. 2-2(a)). Both the compound torsional oscillator and the “bare

Vycor” torsional oscillator suggest that the period drop is a mechanical effect.

2.2.4 Other Experiments on Solid 4He

Before the recent TO experiments showing the observed period drop can be completely accounted for by shear modulus effect, there were some other experiments

21 carried out to prove or disprove the existence of supersolid. This includes dc flow experiments [Day 2005, Day 2006, Ray 2010b], melting curve measurements

[Todoshchenko 2006], heat capacity measurements [Lin 2007, Lin 2009, West 2009].

Two of these experiments show anomalous phenomena. One is the mass flow through solid 4He driven by chemical potential [Ray 2010b] and the other is a heat capacity peak at similar temperature as the period drop [Lin 2007, Lin 2009, West 2009]. The previous one is recently interpreted as Luttinger-liquid behavior [Vekhov 2012]. Since the period drop is proved to be a pure mechanical effect, the origin of the heat capacity peak is yet unknown. A detailed review on heat capacity experiments will be in the next chapter.

22 Chapter 3 Heat Capacity of Solid 4He in Porous Media

3.1 Experimental Details

3.1.1 AC Calorimetry

Several methods of precise heat capacity measurements have been used for decades at low temperature. Lots of the heat capacity measurements of helium were conducted in adiabatic calorimetry technique. Alternatively, an AC calorimetry was firstly proposed by Sullivan and Seidel [Sullivan 1968]. Compared with the adiabatic calorimetry, AC calorimetry has better resolution especially on samples with small heat capacity. In the ac calorimetry, one detects the steady temperature variation caused by ac power to determine heat capacity. As shown in Fig. 3-1, a sample is in contact with thermal bath with a weak thermal link . Heater with heat capacity and thermometer with heat capacity are attached to the sample with heat capacity through thermal conductance and , respectively. The heater generates a sinusoidal heat

, and leads to a temperature oscillation through out the sample. The

oscillation is detected by the thermometer and the heat capacity of the system can be determined by the following equations:

Eq. 3.1

Figure 3-1 AC calorimetry: diagram of a sample with a heater and a thermometer coupled to the thermal bath.

Here the subscripts , , , stand for heater, thermometer, sample and thermal bath; ,

, stand for heat capacity, thermal conductance and temperature. Heat capacities and thermal conductivities are constant with the assumption that the temperature oscillation is sufficiently small. The steady state solution for Eq. 3.1 is

Eq. 3.2

where , is a phase shift between detected temperature oscillation and heat, and is correction factor related to the system. Assuming that the heat capacity of heater and thermometer is negligible compared with the sample, . Intuitively, the

24 sample should be completely thermally isolated from thermal bath so that the measured heat capacity is exactly that of the sample. However, to cool down and adjust the temperature, the sample has to be thermally linked to the thermal bath. This consequently introduces some correction factor to the measured heat capacity. On the other hand, the sample should ideally be excellent thermal conductor to achieve uniform temperature oscillation through the entire sample. This is certainly not always the case. Therefore, the correction factor δ is related to the thermal link to the bath and the thermal conductivity of the sample κ. If the whole system is in the steady state of ac temperature oscillation with a frequency ,

Eq. 3.3

where , representing the time constant for the heat to be drained away from the sample to the thermal bath, , representing the time constant for the sample to reach thermal equilibrium. Therefore,

Eq. 3.4

where is the thermal conductance of the sample between heater an the thermometer.

Under the condition that , and ,

Eq. 3.5

To make the correction factor , it requires that the thermal conductivity of the sample should be sufficiently good and the thermal link between the sample and the thermal bath should be sufficiently weak. Thus, the measured heat capacity is

Eq. 3.6

For the non-ideal condition,

Eq. 3.7

Thus, the measured heat capacity is a function of the heat frequency. A typical frequency scan is shown in Fig. 3-2. The upturn at low frequency end is because frequency is close to , and the upturn at high frequency end is because frequency is close to .

1.05

1.04

s 1.03

1.02 measure

C 1.01

1.00

10-2 10-1 100 101

Frequency (Hz) Figure 3-2 Typical frequency scan of a calorimeter

3.1.2 Calorimeters

Background heat capacity of the empty calorimeter is a hurdle for a lot of heat capacity measurements on solid 4He. The calorimeter has to be mechanically strong to hold high pressure required for solidifying 4He. A variety of materials have been used to

26 construct the calorimeter, including copper, aluminum, etc. The drawback of these materials is the high heat capacity below 0.5 K compared with that of solid 4He. This is because metallic materials usually have free electrons that contribute a heat capacity that is linear in which overwhelms the heat capacity of solid 4He at low temperature.

What is worth to mention is that Clark et al. did the measurements with an aluminum calorimeter [Clark 2005]. Making use of the superconducting transition of aluminum above 1 K, they successfully excluded most of the electron heat capacity below 0.5 K.

However, another property of superconductors was the disadvantage of this measurement: poor thermal conductivity leads to long , especially below 0.1 K, and the experiment could not be cooled down below 85 mK.

Lin et al. [Lin 2007, Lin 2009] successfully found an excellent material to make the calorimeter: silicon. Undoped single crystal silicon is insulating, non-magnetic, and only has heat capacity from phonon contribution. On the other hand, it has superior thermal conductivity and the measurements could be extended down to ~30 mK. There is only one flaw of this choice, if we are trying to be picky: under the high pressure (~70 bar) where liquid 4He needs to be pressurized to before freezing into solid samples, the chances for liquid 4He to leak through the silicon is relatively high (12 out of 15 calorimeters ended with leaking), even though the thickness of the calorimeter is already

0.5 cm thick.

Therefore, for the work described in this dissertation, we decided to switch to single crystal sapphire (Al2O3). Sapphire is another material with low heat capacity and extremely high thermal conductivity. The heat capacity of the empty calorimeter is at least 10 times smaller than that of solid 4He contained inside. And the thermal contraction

27 coefficient is very small so that the chance of leaking is reduced. The sapphire was machined by Imetra Inc., Elmsford, NY. As shown in Fig. 3-3, the sapphire calorimeter was made with two tight-fitting pieces glued together with Stycast 1266. A hole

(diameter = 0.79 mm) is drilled at the center of the top sapphire piece (“cap”) and a Cu-

Ni capillary (outer diameter = 0.76 mm, inner diameter=0.46 mm) guides helium into the calorimeter through the hole. A small and thin aluminum cap is glued on top of the sapphire cap with Stycast 1266 to seal any possible gaps. Heater and thermometers are glued to the calorimeter on flat surfaces with GE varnish. A copper wire is used as a weak thermal link between the calorimeter and the mixing chamber stage as a thermal bath. An aerogel piece resides in the cavity of the sapphire calorimeter. It occupies all empty space in the calorimeter except the volume in the capillary ( ).

Figure 3-3 Sapphire calorimeter with aerogel piece.

3.1.3 Thermometers

Two thermometers are attached to the sapphire calorimeter with GE varnish. One is RuO2 2200, whose resistance is ~2200 ohm at room temperature and ~30 kohm at 20 mK. Another is germanium thermometer, made by AdSem Inc, Mountain View, CA. Its resistance is ~1 ohm at room temperature, ~30 ohm at 0.3 K, and ~80 kohm at 20 mK.

Obviously, germanium thermometer is much more sensitive than RuO2 thermometer, especially below 0.3 K. Both thermometers are calibrated against the main thermometer on mixing chamber for every cooling down of the experiments. The main thermometer on

29 mixing chamber is a carbon thermometer from Leiden Cryogenics, the manufacturer of the dilution refrigerator that is used to operate the experiments.

RuO2 and germanium thermometers are part of two separate homemade

Wheatstone bridge circuits to precisely monitor the resistance change due to temperature variation. Details of the circuits will be discussed in later section. The standard resistor for RuO2 thermometer bridge circuit is carbon film resistor and is placed at room temperature. Due to the small resistance (~30 ohm) of the germanium thermometer resistance even below 1 K, the standard resistor for germanium thermometer bridge circuit has to be very stable so that the high sensitivity of the germanium thermometer will not be submerged by noise. To satisfy this requirement, the standard resistor is made with gold film with a thickness of ~8 nm, deposited on single crystal undoped silicon wafer with 7.5 nm thick chromium film between as a glue. The standard resistor is glued to the mixing chamber with GE varnish. The resistance shift of the gold film from 1 K to

20 mK is less than 0.5%. To eliminate this resistance shift, the calibrations for two thermometers are operated in the following procedure. Firstly, mixing chamber temperature is controlled and RuO2 thermometer is calibrated against the main thermometer on the mixing chamber. Secondly, mixing chamber temperature maintains at

20 mK and different amount of heat is applied on the heater to raise the temperature of the calorimeter recorded by RuO2 thermometer. Therefore, the power curve of the heater is available. Lastly, the same set of heat is applied and germanium thermometer is calibrated against the heater power curve.

Figure 3-4 Temperature dependence of RuO2 and germanium thermometers

3.1.4 Heater

The heater is made in the same method with the same thickness of the gold film as the standard resistor for germanium thermometer bridge circuit, except that the aspect ratio of the gold film strip is different. The usual resistance of heater is ~1000 ohm, and similar as the standard resistor of germanium thermometer, the resistance shift between 1

K and 20 mK is within 0.5%. Superconducting wires are used to make the leads and a section of the copper matrix is dissolved with nitride acid to make sure that the heat leak through leads is negligible compared with the heat flow through the calorimeter.

31 3.1.5 Weak Thermal Link

The weak thermal link between calorimeter and thermal bath is through copper wire (0.5 mm diameter, 5 cm long). One end of the wire is squeezed to copper foil. The foil is glued to the calorimeter with GE varnish. The other end of the wire is tightly pressed by a screw onto mixing chamber stage. The weak thermal link determines the cooling power of the calorimeter, and therefore the time period of the sample growth, thermal equilibrium time constant, etc.

3.1.6 Electronics

A homemade ac-Wheatstone bridge is built to measure the temperature oscillation of the calorimeter. The schematic diagram for the germanium thermometer circuit is shown in Fig. 3-5. For RuO2 thermometer circuit, the setup is similar except that the standard resistor is at room temperature outside of the cryostat. Stanford Research

Systems DS345 (Function Generator 1) generates 700 Hz sine wave signal as the excitation of the circuit. The excitation signal goes through the 1:1 ST-248 transformer and is divided into two paths. One path goes into cryostat to reach thermometer and standard resistor, and the other goes to the ac standard ratio. The ac standard ratio is set at

, to make the resistance on two side of the ground identical. Stanford SR810 lock- in amplifier is triggered by the reference signal from DS345 and measures the potential of the point between thermometer and standard resistor relative to the ground. This potential is dependent on the ratio of thermometer resistance to standard resistor, and therefore measures the temperature. The output of lock-in amplifier is sent to Agilent 33401A

32 voltmeter. It records 500 readings to form the wave form and sends it to computer. The amplitude of the oscillation and the average are calculated by fast Fourier transform via

LabView code. Agilent 33401A is triggered by the reference signal from another function generator (Function Generator 2 in Fig. 3-5). The triggering (or sampling) frequency is usually 10 Hz. This frequency is chosen to make sure there are enough oscillations within the 500 readings in order to get reliable Fourier transform results. It is an alternative method to the conventional two lock-in amplifier technique. The advantage is that it requires less equipment and also reduces the electronic noise influence.

Figure 3-5 Block diagram of the thermometer circuits for heat capacity measurements.

33 Due to the wide range of the germanium thermometer resistance, two bridge circuits are built to cover the whole temperature range of our interest. Above 0.3 K, the germanium resistance is less than 30 ohm. Hence the standard resistor is chosen to be 20 ohm to optimize the sensitivity. From 0.3 to 0.02 K, the germanium resistance shoots up from 30 ohm to 80 kohm, and the standard resistor is chosen to be 200 ohm. For RuO2 thermometer, the resistance ranges from 3 to 30 kohm below 1 K. Only one standard resistor of 10 kohm is used at room temperature.

Electronic noise is the largest problem that we need to overcome in the heat capacity measurements. Using ac excitation instead of dc excitation is for the purpose of reducing noise and interference. The signal goes through the ac transformer to eliminate any dc noise or bias. RG108A/U cables and TWINAX connectors are found to be more efficient to attenuate noise compared with regular coaxial cables and BNC connectors, because RG108A/U cables have an extra copper ground shield to screen out external electric and magnetic waves. Moreover, Spectrum Control 1293-001 low-pass RF filters are used for every lead. The filters are placed both at room temperature and mixing chamber temperature. The filters at mixing chamber stage are connected with homemade thermocoax as extra filtering for high frequency noise. The thermocoax has NiCr as inner conductor, stainless steel as outer conductor and MgO powder as dielectric material between. Very careful shielding and grounding everywhere are required for this experiment.

The heater circuit is much simpler. HP 3325B function generator is used to apply voltage across the heater. The signal is usually below 1 Hz and also passes filter box

34 before reaching the heater. Four-lead method is used to measure the actual voltage across the heater and precisely determine the generated heat power.

3.1.7 Gas Handling System for Sample Growth

The gas handling systems for the heat capacity experiment is described in details here. Fig. 3-7 is the schematic diagram of the gas handling system and capillary inside cryostat. To cut down the heat leak from upper stage to the calorimeter, long and thin capillaries are used. The total length of capillary within cryostat is ~7 m and inner diameter of most of the capillary is 0.3 mm with 0.18 mm diameter stainless steel wire inserted or 0.18 mm with 0.14 mm wire inserted to cut down the heat leak due to the solid/liquid helium inside capillary. Moreover, capillaries are carefully heat sunk at different stages. Heat exchangers are made by winding 1 m long Cu-Ni capillary around a

1 cm diameter copper tube and attached on 1 K pot, still and 50 mK stages. Two heat exchangers at mixing chamber stage contain tightly packed silver sinter and a third one is open space. Every heat exchanger except the last one has thermometer and heater and the temperatures can be controlled.

Figure 3-6 Gas handling system for sample growth in heat capacity experiment.

3.1.8 Sample Growth Methods

All of the solid samples studied in this thesis are made using blocked capillary method, also called constant density method. In this method, the whole capillary system together with the fridge is warmed up to high temperature in order to pressurize liquid with sufficient density into the cell. Then a certain section of the capillary is cooled down below the freezing temperature. The helium inside freezes and forms a plug. Therefore, no more helium can be supplied to the cell and the liquid below the plug freezes into solid samples with the same average density. Once the temperature of the sample cell hits

36 the freezing point on the melting curve (as shown in Fig. 3-7), the helium in the cell starts to freeze. Because the density of solid is larger than liquid by ~10%, the system follows the melting curve with pressure decreasing. Once all liquid freezes into solid, the pressure ceases to drop. The system departs the melting curve and continues to cool down.

Figure 3-7 Solid 4He growth in blocked capillary method.

Liquid samples are usually made by keeping the gas handling system at desired pressure. Helium gas within the cell and the capillary system will be automatically absorbed and condensed into liquid. The heat leak along the liquid 4He in the capillary to the experimental cell is the major problem of the heat capacity measurements, since the thermal conductivity of liquid 4He is much larger than that of solid 4He. Therefore, thin and long capillary and strong heat exchangers as described above should be used. The lowest temperature of liquid sample is 40 mK for the present heat capacity measurements. However, there are some tricks to cut down the heat leak along the

37 capillary when making the liquid sample in Vycor. The melting curve of 4He in Vycor is elevated to ~40 bar. This makes a region between 25 and 40 bar that 4He remains liquid in Vycor but freezes into solid in bulk space in the capillary. Therefore the heat leak along the capillary is greatly reduced.

3.2 Silica Aerogel

Silica aerogel is a highly porous material derived from gel with liquid component leached out. It is composed of thin strands of silica which only occupies 2~10% of the volume and the rest is open space. The typical diameter of silica strands is ~5 nm and separation between strands ranges from 2.5 to 100 nm [Fricke 1988]. The density of silica strands is about . A transmission electronic microscopic (TEM) picture of aerogel is shown in Fig. 3-8. Compared with Vycor glass, it has much larger porosity, much smaller density and large specific surface area per unit volume. These unique properties make it widely used in studying helium physics. Aerogel as a quenched impurity has shown profound effect on 4He superfluid transition. Superfluidity of 4He films made in aerogel has been observed to have quite different superfluid critical exponent from bulk-like superfluidity, indicating a different universality [Chan 1988].

Compared with the well-known lamda heat capacity peak at bulk superfluid transition, heat capacity of liquid 4He in aerogel at saturated vapor pressure locates at slightly lower temperature and loses the logarithmic singularity [Wong 1990]. Aerogel also modifies the phase diagram on 3He-4He mixture liquid [Kim 1993]. The 3He-4He coexistent region is found to detach from superfluid transition and gives rise to a superfluid state at 3He

38 rich and low temperature region. Aerogel is also found to strongly suppress the superfluid fraction and transition temperature of superfluid 3He [Porto 1995].

3.3 Solid Helium in Aerogel

Because of the amorphous silica and disordered fractal structure, solid helium grown in aerogel has much higher disorder density compared with bulk crystalline solids

(the density of silica strands ( ) is much higher than the density of dislocations in bulk solid 4He ( )). The first several layers in contact with silica strands are amorphous under strong and irregular van der Waals potential. Higher layers experience weaker and smoother interaction and exhibit periodicity mostly limited in the plane that conforms to the curvature of silica strands. These layers form two-dimensional

39 lattice. Three-dimensional crystallite grains form at the region remote from silica strands.

Other than the influence from amorphous silica, the fractal structure brings in large quantity of lattice mismatches, dislocations, grain boundaries, etc., leading to high density of disorders. X-ray experiment shows that solid 4He grown in aerogel is highly polycrystalline with grain size of ~100 nm [Mulders 2008]. The grain size is larger than the average separation between silica strands. It is believed that typical crystallite has silica strands embedded in, which means amorphous state and two-dimensional solid or liquid exist in grains.

Figure 3-9 Top:X-ray diffraction pattern for solid helium in aerogel. 5 rings indicate reflections of an hcp structure; bottom: NCRI deduced from the measured period drop of torsional oscillator of solid 4He grown in aerogel. The Bulk solid 4He data have been scaled down by a factor of 3 for comparison [Mulders 2008].

Figure 3-10 Diffraction peak for the (101) reflection. The peak width indicates grain size of ~100 nm [Mulders 2008].

As shown in previous experiments [Lin 2009], one significant feature of heat capacity peak of bulk solid helium is its sample quality dependence: sample prepared in 4 hours, assumed to have more disorder, shows larger peak locating at higher temperature than the sample prepared in 20 hours; solid-liquid coexistent samples, assumed to have the least disorder, show smaller peaks at the lowest temperature. Similar dependence is also observed in shear modulus anomaly. It is reasonable to postulate that the origin of heat capacity peak is related to disorders in solids. There are two methods to test this assumption: preparing as perfect crystal as possible to see whether the peak vanishes, or preparing sample extremely disordered to see whether the peak becomes larger. Even though solid-liquid coexistent samples have been made to release stress in solids, it is

41 almost impossible to guarantee a perfect crystal. The second way seems easier: aerogel can be used to introduce disorders to solid samples.

3.4 Results

3.4.1 Pressure Dependence

The calorimeter is made of single crystal sapphire with a cylindrical piece of aerogel (diameter=1.43 cm, height=0.76 cm, porosity=95%) inside to occupy the inner space except the hole (diameter=0.64 mm) for filling helium (as shown in Fig.3-3). Solid samples are grown with blocked capillary method using commercial ultra high purity 4He with a 3He concentration of 0.3 ppm. The typical solid samples growth time is ~4 hours.

Final pressures of the solid samples are 28, 32, 42, 47 and 50 bar with uncertainty of ±1 bar. These pressures are estimated based on the initial liquid pressures before freezing.

There is possibility that liquid might exist in the 28 bar solid sample because it is close to the elevated melting pressure by aerogel [Molz 1995]. The pressure of liquid sample is

23.5 bar.

10-3 23.5 bar liquid ------28 bar (possible solid+liquid) 32 bar 42 bar 10-4 47 bar

50 bar

C[J/K] 10-5

10-6 0.04 0.06 0.080.1 0.2 0.4 0.6 T [K]

Figure 3-11 Heat capacity of solid and liquid 4He in aerogel

10-1 23.5 bar liquid ------10-2 28 bar 32 bar 42 bar -3 47 bar 10 50 bar

10-4 C[J/K] 10-5

10-6

0.04 0.06 0.080.1 0.2 0.4 0.6 T [K]

Figure 3-12 Fitting of heat capacity data using . Data have been vertically shifted for clarity.

43 Fig. 3-11 shows the heat capacity of four solid samples and one liquid samples in aerogel. Quite different from their bulk counterparts, heat capacity no longer obeys dependence at high temperature. This originates from the presence of amorphous and two-dimensional helium near silica strands. Amorphous helium contributes heat capacity that scales with and two-dimensional helium contributes heat capacity that scales with

. The crystalline helium far away from silica strands contribute heat capacity that scales with . Considering all contributions from helium in different states, we can use

the polynomial function to fit the measured heat capacity data, as shown in Fig. 3-12. The liquid heat capacity can be fitted very well by the polynomial term below 0.4 K. The deviation above 0.4 K is attributed to roton contribution [Greywall 1978]. However, none of the solid heat capacity data can be fitted by a uniform polynomial function for the entire temperature range with satisfactory.

There is always an extra bump in addition to the fitted heat capacity and the size and position of the bump are pressure dependent. The fitting parameters are listed in Table 4-

1. The term contribution is on the same order as the result in heat capacity measurement of full pore liquid 4He in Vycor pore at standard vapor pressure [Tait

1979a]. The discrepancy might originate from the different structure of aerogel and

Vycor and different pressure. We also note that the term contribution is larger than the

Debye contribution in bulk solids. This is because those crystalline grains still have much higher disorder density than bulk solid even though they are far away from silica strands.

(It is known that heavily disordered solid contributes a much higher heat capacity than

44 Debye contribution.) In our case, the high density disorders come from the randomness of silica strands that make crystal growth in random orientation.

Table 3-1 Fitting parameters for heat capacity of solid 4He in aerogel.

2 3 4 Pressure (bar) A1 (J/K ) A2 (J/K ) A3 (J/K )

23.5 (liquid) 50 260 3050

28 (possible 80 1500 7500 solid+liquid)

32 38 1500 5250

42 20 1200 5200

47 20 930 4900

50 20 800 4800

The anomalous extra heat capacity peak in addition to the regular terms of polynomial fitting is more interesting as shown in Fig. 3-13 after subtracting the fitting terms. The peak is normalized by the total amount of 4He. It shows obvious pressure dependence for both temperature and in peak size. With increasing pressure, the temperature of the peak center ranges from 75 to 220 mK and the peak size ranges from

30 to . The pressure dependence of peak temperature and size are plotted in Fig. 3-14. It is clear to see that the heat capacity peak of bulk solid and liquid-solid coexistent samples fit into this pressure dependence reasonably well.

250 28 bar (possible solid+liquid) 32 bar 200 42 bar 47 bar 50 bar 150

100

J/K-mole]

 [

peak C 0

0.0 0.1 0.2 0.3 0.4 T [K]

Figure 3-13 Anomalous heat capacity bump after subtracting polynomial fittings.

0.25 200

T aerogel peak T bulk 0.20 peak 150

amplitude aerogel amplitude bulk

0.15 100 J/K-mole]

[K]  peak

T S+L BC 4h 0.10 50

BC 20h PeakSize [ 0.05 0 25 30 35 40 45 50 Pressure [bar]

Figure 3-14 Pressure dependence of peak temperature (left scale) and peak size (right scale). Bulk solid samples and solid-liquid coexistent samples (open symbols) are also plotted for comparison.

3.4.2 3He Concentration Dependence

Previous heat capacity of bulk solid 4He shows that extra heat capacity due to phase separation overrides the heat capacity peak when [Lin 2009]. All heat capacities collapse onto identical curve before the phase separation onset, which means that the heat capacity peak is independent of . This feature is different from what has been observed in NCRI and shear modulus anomaly, where 3He impurities push the onset of the anomaly to higher temperature. Therefore, if heat capacity peak is related with the shear modulus anomaly (the binding of 3He on dislocation lines), 3He should

4 saturate the dislocation lines before reaches 1 ppb. On the other hand, solid He in aerogel presumably is much more disordered. The effect of 3He on heat capacity should be different from the bulk solid case.

3He concentration dependence of heat capacity is studied for solid 4He in aerogel.

The 3He concentrations that have been studied include 1 ppb, 0.3, 6, 20, 60, 300, 2000,

10000, 14000 ppm. The 1 ppb sample was made with isotopically pure 4He gas, and all the other samples are made by doping the required amount of 3He gas into empty

4 calorimeter first and filling it with ultra-high purity He with . The actual amount of 3He eventually staying in the cell is not precisely predictable due to the surface area of the long and thin capillary and porous heat sink that the 3He has to go through before reaching the cell. But it is estimated be within 30% from the labeled concentration.

All samples are made with final pressure at 50±1 bar. The heat capacity of 1 ppb sample is identical with that of 0.3 ppm sample. This is consistent with bulk solid samples [Lin

2007, Lin 2009]. However, higher concentrations of 3He impurities have profound effect in enhancing the heat capacity at low temperature but do not affect heat capacity at high temperature (as shown in Fig. 3-15). 6 ppm 3He enhances heat capacity below 0.3 K by a factor of 2, while 1.4% 3He enhances it by ~15 times. The onset temperature of the enhancement also increases with , shifting from 0.3 K with to 0.6 K with

10-3

1.4% 1% 10-4 0.2%

0.12%

300ppm [J/K] v 60ppm C 20ppm 10-5 6ppm 0.3ppm 1ppb

0.03 0.06 0.1 0.2 0.4 0.6 0.8 1 T [K]

Figure 3-15 Heat capacity of solid 4He grown in aerogel with different concentrations of 3He impurities.

Previous measurements on bulk samples [Lin 2009] observed extra heat capacity due to three-dimensional phase separation of 3He-4He mixture. The phase separation onsets below 0.2 K for . The separation temperature is in agreement with the model proposed by Edward et al. [Edwards 1962] and by Edwards and Balibar

[Edwards 1989]. For the solid 4He in aerogel, the onset temperatures of the heat capacity enhancements are too high to agree with the three-dimensional phase separation (as shown in Fig. 3-16). It must have some origin unique to aerogel. As mentioned above, aerogel is a highly porous medium with a large area of interface between silica and solid helium. Path Integral Monte Carlo simulation [Khairallah 2005] has shown that for the case of solid helium grown in silica porous media, 3He atoms tend to concentrate in the vicinity of the silica strands as temperature is decreased and deposit on the second or the

49 third monolayers. This behavior of 3He can be seen as a kind or phase separation: 3He atoms aggregate onto the interfaces near silica strands instead of forming 3D aggregations. The simulations show the onset temperature for 3He to deposit on the interfaces is higher than 0.2 K. This is consistent with the onset temperature of the heat capacity enhancement we observe.

solid in aerogel 0.6 bulk solid (Lin 2009) Theory (Edwards 1962)

0.4

T [K] T 0.2

0.0 10-5 10-4 10-3 10-2 X 3 Figure 3-16 Onset temperature of solid 4He heat capacity enhancement by 3He compared with theoretical [Edwards 1962] and experimental [Lin 2009] phase separation temperature of bulk solid helium.

The extra heat capacity due to the 3He dosage is plotted in Fig. 3-17 after subtracting the heat capacity of 50 bar sample with . A broad peak is shown with the peak position moving to higher temperature as is increased. The peak size also increases with . The peak size is plotted as a function of in Fig. 3-18 and

shown to have a rough dependence.

1.5x10-3 1.4% 1% 0.2% (a) 0.12%

-3 300ppm 1.0x10 60ppm 20ppm

6ppm

[J/K]

-4

peak 5.0x10 C

0.0 0.0 0.1 0.2 0.3 0.4 0.5 T [K]

1.5x10-4 300ppm 60ppm 20ppm (b) 6ppm

1.0x10-4

[J/K]

-5

peak 5.0x10 C

0.0 0.0 0.1 0.2 0.3 0.4 T [K]

Figure 3-17 (a) Heat capacity peak with high 3He concentrations: (b) zoom-in only showing the smaller peaks with lower 3He concentrations.

103

1/2

J/K] C ~X

peak 3



[ 102

peak C

101 10-5 10-4 10-3 10-2 X 3 Figure 3-18 Peak size as a function of 3He concentration.

3.5 Measurements of solid 4He in Vycor

The measurement of heat capacity of solid 4He in Vycor was carried out with the same sapphire calorimeter that was used for the measurement with aerogel. The aerogel piece was substituted by a piece of Vycor with the same dimensions. The Vycor piece was glued to the internal surface of the cavity with Stycast 2850FT at the bottom and side. Epoxy was applied on the top surface for helium to enter Vycor.

Different from silica aerogel, Vycor is a much more compact porous material.

Therefore the embedded solid 4He is much less interconnected compared with that grown in aerogel. The average pore size of Vycor is characterized as ~7 nm in various methods

[Wiltzius 1987, Wallacher 2005] and the porosity is only 28%. Therefore, the

52 propagation of thermal phonons in solid 4He is strongly suppressed by the tortuosity and the pore confinement. This leads to the inability of solid 4He to conduct heat. Details of the heat conduction in Vycor-helium composites will be systematic discussed in Chapter

4. Since the presence of solid 4He makes a huge contribution to heat capacity but negligibly to the thermal conductivity, the internal time constant is significantly increased

(up to ~1 sec). Since heater and thermometer do not directly contact with Vycor-helium composite, the thermal penetration depth inside the Vycor-helium composite is smaller than the length scale of the Vycor dimension. Effectively only the “outer shell” of the composite close to the calorimeter is measured. The penetration depth is inversely proportional to the frequency of the heat. Therefore low frequency is preferred. However, the part of the composite that is measured is estimated to be less than 30% even at the lowest frequency limit (0.02 Hz).

Even though measurement of heat capacity of solid 4He in Vycor is not quantitatively reliable, it still provides qualitative results. As shown in Fig. 3-19, the measured heat capacity is increased by one order of magnitude. This indicates that the contribution from solid 4He dominates. It does not show dependence at high temperature as bulk solid 4He, which suggests that the structure of the micro solid 4He pockets have significant percentage of amorphous or non-crystalline states. This is consistent with the neutron scattering study [Wallacher 2005] that shows 70% of the helium in pores is in non-crystalline state while the other 30% is in bcc state.

10-4 48 bar solid empty calorimeter with Vycor

Fit: 16T+388T2+725T3 J/K

10-5

C[J/K]

10-6 Fit: 3.6T+45T2+20T3J/K

0.05 0.1 0.2 0.3 0.4 T [K]

-6 6.0x10 48 bar solid

4.0x10-6

-6

[J/K-mol] 2.0x10

bump C 0.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 T [K]

Figure 3-20 Heat capacity “bump” after subtracting the polynomial fitting.

The contributions from non-crystalline and crystalline should be able to fitted by

polynomial . However, the measured heat capacity cannot be

54 satisfactorily fitted in such way by a single function across the whole temperature range.

Some anomalous “bump” or broad peak appears between 0.1 and 0.3 K, with center at

~0.22 K.

3.6 Review of Heat Capacity Experiments of Bulk Solid 4He

Liquid 4He has a famous lambda heat capacity peak at the superfluid transition.

Therefore, a thermodynamic signature should be observed if the normal-supersolid transition is real. Lin et al. observed a heat capacity peak in addition to the Debye contribution [Lin 2007, Lin 2009] (Fig. 3-21). The peak locates at the same temperature as the onset of the period drop in torsional oscillator experiments and shares similar sample quality dependence. The observed peaks of solid samples prepared in 4 hours, presumably with higher density of defects, are larger than that of samples prepared in 20 hours. The solid-liquid coexistent samples, presumably with high crystalline quality because stress in solid can be easily released, have the smallest peak amplitude. The

amplitudes of the peaks correspond to . On the other hand, if this

3 3 peak is attributed to He impurities, each He atom contributes (since the peak amplitudes of 1ppb and 0.3ppm samples are identical, 1ppb 3He should saturates the

3 effect). The He concentration ( ) of the samples ranges from 1 ppb to 500 ppm. The heat capacity peak is independent except extra heat capacity due to phase separation of solid 3He-4He superimposes onto the heat capacity peak (Fig. 3-22).

As mentioned above, it is accepted that period drop is more likely to be a mechanical “glue effect” by solid helium. This leads to a series of questions that have to

55 be answered. Is the observed heat capacity peak real? If it is real, what is the origin of the heat capacity peak? Why does it appear in the same temperature as the period drop and has similar sample quality dependence? Why is it independent of 3He concentration? To answer these questions, we have to review the heat capacity experiments carefully.

Figure 3-21 Heat capacity of solid 4He (top) and the heat capacity peak after subtracting Debye contributions (bottom).

Figure 3-22 Extra heat capacity due to phase separation with high 3He concentration in solid helium is superimposed on the heat capacity peak.

Question 1: Is the heat capacity peak real?

Heat capacity measurements with AC calorimetry method have a set of high standard requirements. There are several major concerns that we need to be careful about.

These include heat leak to the calorimeter, precision of temperature and heat power measurements, heat capacity of empty calorimeter background and most importantly, time constants of thermal equilibrium between calorimeter and thermal bath and within calorimeter.

In our measurements, long and thin Cu-Ni capillary is used to introduce helium from top of the cryostat at room temperature to the calorimeter at low temperature. The

58 capillary system is carefully heat-sunk at different stages, including 1 K pot heat exchanger at 1.8 K, still heat exchanger at 0.7 K, 50 mK stage heat exchanger at 0.1 K and two silver powder heat sinters and one open volume heat sinter at mixing chamber stage. All of the heat exchangers are made by winding 1 meter long capillary around copper post to increase the contacting area. The estimated surface area in each silver powder heat sinter is 10 m2. The capillary length between neighboring heat exchangers and sinters is between 0.4 and 1.1 m. To cut down the heat leak due to solid helium in capillary after a sample is made, stainless steel wire is inserted into all Cu-Ni capillaries to reduce 70% of the volume along the capillary. The last section of Cu-Ni capillary connecting to the calorimeter is 0.5 m long, with 0.3 mm outer diameter and 0.18 mm inner diameter, and with a stainless wire (0.14 mm in diameter) inserted. Based on the dimensions, the heat conduction along capillaries between neighboring heat exchangers is three decades lower than that between the heat exchangers and corresponding stages.

This makes the heat sunk properly at each stage and the heat leak to the calorimeter along the capillary is negligible. On the other hand, it is difficult to see how the heat leak from the capillary can contribute extra heat capacity in the form of a peak.

To measure heat capacity, the heat power applied on calorimeter has to be precisely measured. The heater we use is gold film deposited on undoped single crystal silicon wafer. The gold film resistance variation is less than 0.1% within the temperature range from 20 mK to 0.7 K. Four-lead measurement is used to measure voltage across the gold film and therefore the generated heat is precisely known. The leads are superconducting wires with a section of 5 cm long etched by acid to get rid of copper matrix to cut down its thermal conductance. The estimated thermal conductance through

59 superconducting leads is ~ , four decades smaller than the copper wire thermal link. It guarantees that all the heat is applied on the calorimeter and no heat can be drained through leads. Heater and thermometers are attached to the calorimeter on flat surfaces by GE varnish. Both the heater and the germanium thermometer are made on silicon wafer. The GE varnish layer is less than 0.03 mm thick and the contact area is

0.2~0.5 cm2. These make the heater and thermometers strongly thermally attached to the calorimeter. However, there is a possibility that the thermal contacts between calorimeter and heater/thermometer become worse as temperature is decreased below 0.1 K. This would leads to a longer internal time constant and measuring larger heat capacity than real value. To overcome this, putting heater and thermometers inside the calorimeter is suggested. With heater and thermometer in direct contact with helium sample, the heat flow will bypass the thermal boundary on the calorimeter and directly goes through the sample.

Before the heat capacity measurements at Penn State [Lin 2007, Lin 2009, West

2009], there are many other experiments measuring solid 4He heat capacity. However, they used either metallic or epoxy calorimeters. Too large background heat capacity is the common disadvantage of these calorimeters, often a decade higher than that of solid 4He below 1 K. The large background reduces the accuracy of measuring solid helium heat capacity. Heat capacity of single crystal silicon or sapphire is 10 times smaller than solid

4He contained inside. The empty calorimeter heat capacities are plotted in Fig. 3-23. The empty background measurements are mainly dominated by epoxy and aluminum cap used to glue and seal the calorimeter (as shown Fig. 3-24). To reduce the background,

60 minimum amount of epoxy and aluminum are used and the background is less than 5% of the total heat capacity with sample.

Table 3-2 Components of empty calorimeters for heat capacity measurements

Mass of Material Reference Epoxy (gram) Al (gram) (gram) [Lin 2007] 16 (silicon) ~0.6 (Stycast 2850) ~0.4

[Lin 2009] 38.2 (silicon) 0.39g (Stycast 2850) 0.25

[West 2009] 43 (sapphire) 0.5g (Stycast 1266) 0.25

Lin 2007, Silicon Lin 2009, Silicon 10-5 West 2009, Sapphire

10-6 C[J/K]

10-7

0.03 0.06 0.1 0.2 0.4 0.6 0.8 T[K]

Figure 3-23 Measured heat capacities of empty calorimeters [Lin 2007, Lin 2009, West 2009].

Lin 2009, Silicon -5 Aluminum 10 Stycast 2850 Silicon

10-6 C[J/K]

10-7

0.03 0.06 0.1 0.2 0.4 0.6 0.8 T[K]

To carry out heat capacity measurements, should be satisfied, i.e. there should be enough time for the sinusoidal heat with a specific frequency to transport through the entire calorimeter and make it thermal equilibrated before it is drained through thermal link to thermal bath. If is too low, the heat will be drained by thermal link faster than a complete cycle. If is too high, the thermal penetration depth is too small that the calorimeter does not have a uniform temperature distribution.

Therefore, appropriate frequency should be chosen to avoid these two cases. However, if the external and internal time constants are too close, it is impossible to choose a proper frequency. We can roughly estimate the proper frequency range by measuring heat capacity at fixed temperature with scanning frequency. In frequency scans, proper

62 frequency range can be located where measured heat capacity is independent of frequency. Details will be discussed in the next chapter.

Figure 3-25 Frequency scan of the calorimeter with solid helium sample reported in [Lin 2009].

Fig. 3-25 shows the frequency scan of the calorimeter with solid helium sample grown inside. For the scans at low temperatures, upturn appears at low frequency end because the condition is no longer satisfied. Here , where is heat capacity and is thermal conductance of copper wire. Because and ,

and becomes smaller at low temperature. The upturn gradually appears below

0.4 Hz. To avoid this influence, all the measurements are carried out above 0.4 Hz.

However, we also notice that the measured heat capacity is not completely flat above 0.4 Hz. There is a further small and gradual drop above 0.5 Hz. This means there is a second time constant involved. This is the internal time constant for thermal

63 equilibrium throughout the whole sample. The reason for effect of internal time constant to appear as a drop instead of an upturn is that heater and thermometers are not direction contacting the helium sample directly. In the current experiment setup, thermal boundary resistance between calorimeter and solid helium greatly dominates the internal time constant. Its shrinks the proper frequency range for the measurement. Because both the heater and the thermometer are attached on the outer surface of calorimeter, heat generated by heater has to go through silicon/sapphire and interface of silicon/sapphire and solid helium to reach solid helium. Even though silicon/sapphire and solid helium have high thermal conductivity, the interface can hinder the heat propagation because of lattice mismatch and can be viewed as a weak thermal link. However, if heater and thermometers are located inside the calorimeter, they will contact solid helium directly and the heat path does not need to go through the interface (Fig. 3-26). Therefore, the internal time can be shortened and the proper frequency range for measurement is broader.

As analyzed above, the reported heat capacity measurements [Lin 2007, Lin

2009, West 2009] were carefully measured, considering the possible influences from heat leaking, calibration, signal-background ratio and thermal equilibrium process and trying to exclude any possible issues that can harm the reliability of the data. The influences from heat leak and calibration are on negligible level. The signal-background ratio has been greatly improved compared with previous experiments. It is because of the high signal-background ratio that a heat capacity anomalous peak is resolved below 0.1 K. The thermal equilibrium process is not perfectly optimized due to the thermal boundary resistance between solid helium and calorimeter. The imperfect thermal equilibrium process might leads to some possible error in the measured heat capacity by a few percent. The way to further optimize the thermal equilibrium process and improve the measurements is to repeat the experiments with heater and thermometers inside the

65 calorimeter. Heat flow will therefore bypass calorimeter-solid helium interface and directly go through solid helium. This experiment is currently in process.

Question 2: What is the physics behind the heat capacity peak if it is real?

The observed heat capacity peak has following properties. Firstly it appears to have sample quality dependence. The freezing time is assumed to be related to the sample crystalline quality: sample prepared within shorter freezing time is more disordered. The liquid-solid coexistent sample is assumed to have the least disorder because liquid can release stress. The observed peak size decreases from 4-hour sample to 20-hour sample and to liquid-solid coexistence sample. This sample quality dependence is consistent with that observed in torsional oscillator and shear modulus experiment. This suggests that the heat capacity peak might be related to disorder in solid helium. Secondly, heat capacity

3 peak is independent of He concentration . As shown in Fig. 3-21, samples with

or 0.3 ppm show the same heat capacity peaks. Higher brings extra heat capacity due to phase separation in solid mixture. But this extra contribution is superimposed onto the heat capacity peak. The heat capacity collapses with the low data before the phase separation onset. There are two possible reasons to account for the

3 independence of : 1ppb concentration of He impurities already saturates the effect or the heat capacity peak is unrelated with 3He impurities. Let us firstly consider the case the

1ppb concentration of 3He already saturates the effect. Assuming the average separation between 3He atoms along dislocation line is 0.3 nm, the dislocation density has to be lower than . Corboz et al. [Corboz 2008] proposed a model of 3He impurity binding to screw dislocation in solid 4He. The first-principle simulations determine the binding energy is 0.8±0.1 K. The consequent heat capacity peak centers at ~0.06 K and

3 the peak size is per He atom. Even the peak temperature is close to the experiment, the peak size is about 20 times smaller than the measured peak size

( ).

If 3He is not involved in the effect, the heat capacity peak may originate from the defects in 4He. One reasonable assumption is that the peak is due to the entangling of dislocation lines. Assuming a high density of dislocation and the separation of nodes is 100 nm, the density of dislocation nodes is ~ . According to the peak

3 size, the binding energy of each node is . Different from He atoms, the entangling of dislocation lines at one node involves multiple atoms. Therefore this value is possible. Another model is proposed by the theory group from Los Alamos National

Lab. They claims that there exists glass phase in solid helium [Su 2010]. The local excitations in amorphous glass state, called two-level systems (TLS) gives rise to the heat capacity anomaly. In their model, the TLS distribution is cut off at high energy, instead of the constant distribution in standard glass model [Anderson 1972, Phillips 1972]. The cutoff characterizes the largest tunneling barrier. The cutoff used to fit data is 0.017 ~

0.033 meV for different samples, corresponding to 0.2 ~ 0.38 K. However, this theory cannot generate a peak-shape heat capacity anomaly.

In addition to heat capacity, melting curve measurements have been done as another thermodynamic signature search [Todoshchenko 2006, Todoshchenko 2007].

According to Clausius-Clapeyron relation, solid 4He with pure T3 dependent specific heat should give melting curve relation . Additional heat capacity anomaly should be accompanied by melting curve anomaly. However, the melting curve is observed to have

dependence down to 10 mK without any anomaly. This is not consistent with heat

67 capacity results. The absence of anomaly in the melting curve results is possibly because of the experimental resolution. Based on the heat capacity peak size and assuming reasonable peak temperature range, the pressure anomaly is estimated to be , smaller than the resolution (5 bar) of the melting curve measurements.

68 Chapter 4 Thermal Conductivity of Vycor with Helium

Liquid 4He shows a diverging thermal conductivity as approaches to -point from the high temperature side. Inspired by this fascinating phenomenon, measuring thermal conductivity of solid 4He can be a feasible way to search for any superflow in solid. Due to the superior thermal conductivity of bulk solid 4He, it is relatively difficult to resolve any anomaly. However, porous Vycor glass provides a system to reduce the thermal conductivity of sample. Therefore, this motivates us to start the experiment of measuring the thermal conductivity of Vycor with helium in the pores.

4.1 Two-Level Systems (TLS)

Different from their crystalline counterparts, amorphous solids have heavily disordered structure. The lack of long-range order leads to quite distinguished physical properties, especially at low temperature. Two-level system (TLS) model [Anderson

1972, Jäckle 1972, Phillips 1972] is proposed to explain these anomalous properties.

The atoms or molecules in a glassy material have multiple potential minima, and the thermodynamic properties of the material below 1 K are dominated by quantum mechanical tunneling of the constituent atoms or molecules between the two adjacent accessible energy levels (as shown in Fig. 4-1). The predictions of the TLS model, including a specific heat that scales linearly with temperature and a thermal conductivity that scales with have been confirmed by experiments [Zeller 1971,

Stephens 1973, Lasjaunias 1975].

4.2 Thermal Conductivity of Solids

4.2.1 Thermal Conductivity of Crystalline Solids

Thermal conductivity of crystalline solids has been understood well. In a crystal, heat is transported by phonons. The thermal conductivity is given by

Eq. 4.1

where is phonon contribution to specific heat, is phonon velocity, is phonon mean free path, is Debye frequency. Dominant phonon approximation simplifies the problem by exclusively considering phonons that make the dominant

70 contribution [Zeller 1971]. The spectrum of thermal phonons obeys the Planck’s distribution which shows a peak at the characteristic or the dominant wavelength:

Eq. 4.2

The half maxima of the distribution are found at and . Here is Planck’s constant, the sound velocity, the Boltzmann constant. The corresponding dominant frequency is . Based on dominant phonon approximation, thermal conductivity expression is simplified as

Eq. 4.3

At high temperature, Umklapp process (phonon-phonon scattering) dominates the total phonon scattering mechanism. As temperature is decreased, phonon-phonon interaction becomes less and mean free path increases. Therefore thermal conductivity increases.

Below 1 K, the major phonon scattering happens at boundaries (ballistic propagation).

The mean free path approaches to constant and thermal conductivity scales with

because and is weakly temperature dependent.

Figure 4-2 Thermal conductivity of crystalline quartz and vitreous silica [Zeller 1971].

4.2.2 Thermal Conductivity of Glass

With the same chemical constituents but disordered structure, thermal conductivity of amorphous solids is much lower than crystalline quartz. Moreover, it scales with instead of below 1 K. The temperature dependence implies that the phonon scattering at the sample boundary is no longer important and extra phonon scattering mechanism dominates. According to TLS model, TLS interact with and effectively scatter phonons. TLS scatter phonon by absorbing a phonon to tunnel to the excited state and re-emitting the phonon to tunnel back to the ground state. The re- emitted or scattered phonon is generally propagating in a different direction from the

72 incident phonon. This indirect scattering of the thermal phonons gives rise to the highly attenuated thermal conductivity in a glassy material as compared with its crystalline counterpart. This interaction is characterized by the quantum tunneling rate of TLS, .

The value of characterizes how efficiently TLS exchanges energy with its surrounding. The mean free path of phonon scattering by TLS are given by

Eq. 4.4

where is TLS energy separation, the deformation potential of the medium where

TLS emit phonons to, the density, , is TLS density [Jäckle

1972]. Considering the resonant condition: , we have and therefore

. The magnitude of thermal conductivity depends on TLS density ( ), and the properties of the medium where phonon is released to ( , , ). If there are mechanisms that can enhance the TLS tunneling rate , more phonons will be scattered per unit time and will be reduced further. We show in this paper that the infusion of liquid helium into Vycor pores dramatically reduces .

4.2.3 Thermal Conductivity of Porous Vycor Glass

In addition to the TLS-phonon scattering, phonons are also scattered by the porous structure when propagating in a porous glass like Vycor. Nitrogen adsorption isotherm measurements at show Vycor has a porosity of 28% and a specific surface area of about [Levitz 1991, Wallacher 1998]. Transmission electron microscopy (TEM) revealed the pore space to be a network of multiply connected

73 cylindrical channels [Levitz 1991]. The diameter of the channel and the thickness of the silica “walls”, based on TEM and adsorption studies, are both found to be ~7 nm [Levitz

1991, Wallacher 1998]. The spatial correlation of the silica structure (also the pore structure) in Vycor, , is found to be distributed between 5 to 60 nm by small angle X-ray and neutron scattering and TEM studies [Schaefer 1987, Wiltzius 1987, Levitz 1991].

Therefore, phonons with wavelengths near or shorter than 60 nm are expected to be strongly scattered by the Vycor porous structure because of the acoustic mismatch at the silica-pore interface. Phonons with wavelength much longer than 60 nm, on the other hand, will be insensitive to and will not be scattered by the porous structure. They are scattered solely by TLS. Since the temperature range of this experiment is between 0.06 and 0.5 K, the characteristic frequency of thermal phonons, is between

5 and 40 GHz. Taking the transverse sound velocity in Vycor, to be

[Mulders 1993], is found to be 420, 170, 50 nm at , respectively. At low temperature when is much longer than , phonons are mainly scattered by TLS. As a consequence should scale with , similar as solid vitreous silica. With increasing temperature, decreases towards and the scattering effect by the porous structure becomes prominent. This will reduce and result in the sub- quadratic temperature dependence. This behavior has been confirmed as shown in Fig. 4-

3 [Stephens 1974, Tait 1975, Hsieh 1981].

Figure 4-3 Thermal conductivity of crystalline quartz, vitreous silica and porous Vycor [Zeller 1971, Zaitlin 1975].

4.3 Experimental Setup

Fig. 4-4 shows our experimental setup. The Vycor rod is 3 mm in diameter and

22.8 mm long. It is secured mechanically and thermally into a copper base with epoxy resin (Stycast 2850FT) and attached to the mixing chamber of a dilution refrigerator cryostat. Outside the copper base, the Vycor rod is sealed only by a thin layer of epoxy resin less than 0.07 mm thick painted on its outer surface. The thin epoxy layer, impregnating the pores on the surface, is able to hold pressure up to at least 85 bar below

4 K but does not contribute measurably to the thermal conductance of the sample. Such an experimental configuration was recently used in studies of supersolidity in solid

75 helium [Ray 2010a, Kim 2012]. Helium is introduced into Vycor through the copper base with a thin Cu-Ni capillary. A heater is attached at the top of the Vycor rod. A germanium thermometer is secured 12.7 mm away from the copper base reading and another thermometer is attached directly onto the copper base reading . The copper base holding the Vycor piece is at a uniform temperature because the thermal conductivity of copper is times higher than Vycor [Lounasmaa 1974]. Thermal conductivity of the Vycor rod is given by , where is the distance between the germanium thermometer and the copper base, is cross section area of Vycor, is the steady state heat power and . Measurements are made by imposing a dc power between 0.1 and 1.6 nW to maintain a small but experimentally significant temperature difference (typically 2 mK) along the Vycor rod.

The germanium thermometer and heater are made in the same way as those used in the heat capacity measurement. All of the leads for heater and thermometer are superconducting wire and a section of copper matrix is dissolved to make sure that heat leaks along leads are negligible. All leads go through the low-pass filters that are the same type as in heat capacity measurements.

Figure 4-4 Experimental setup of thermal conductivity measurements.

Linear Research 700 resistant bridge is used to monitor the thermometer resistance. Optimized excitation voltage is applied so that the noise level is small and self-heating is negligible. Germanium thermometer is calibrated against the main thermometer on mixing chamber by sweeping the mixing chamber temperature without power applied on the heat on the top of Vycor. Measurements were made by imposing a dc power between 0.1 and 1.6 nW. The specific power level was chosen to maintain a small but experimentally significant temperature difference (typically ) along the Vycor rod. The noise is on the level of 0.05 mK.

77 4.4 Results and Analysis

4.4.1 Thermal Conductivity of Empty Vycor Glass

We firstly measured the thermal conductivity of empty Vycor. As shown in Fig.

4-5, the deviation below the dependence for is clearly evident. Our results are consistent with previous experiments within a few percent [Stephens 1974, Tait

1975, Hsieh 1981].

4.4.2 Thermal Conductivity of Vycor Infused with Helium

After the measurement of empty Vycor, solid, liquid and adsorbed films of helium of both isotopes were introduced into the Vycor pores. For ease of description, these samples will be identified as solid, liquid and film samples. For clarity, we plot of the solid and atomically thin film samples in Fig. 4-5 and of liquid and superfluid film samples in Fig. 4-6. When Vycor is infused with liquid or solid helium, the samples can be considered as a composite consisting of the two intertwining (silica and helium) networks. However, the heat is still conducted primarily by the silica network. This is the case because the sound velocities (and hence phonon wavelength) in liquid and solid helium are ~10 times smaller than that in silica. The phonon wavelength in helium is always shorter than 60 nm for . As a result, phonon propagation along the helium network is scattered by the porous structure much more strongly than that along the silica network.

78 As shown in the inset of Fig. 4-5, of the atomically thin 4He and 3He film samples are slightly (~3%) higher than that of the empty Vycor over the entire temperature range. The helium atoms in these thin films are immobile and tightly bound to the random silica surface with a typical adsorption potential higher than 25 K for the

4 3 He film (surface coverage ) and 3 K for the He film (

) [Sabisky 1973]. The 4He film does not show superfluidity even at

. In our temperature range of interest, the adsorbed atoms are therefore “inert” and merely serve to slightly thicken the silica network. These tightly bound atoms soften the acoustic mismatch at the silica-pore interface and leads to the slight enhancement in as compared with the empty Vycor. A more noticeable enhancement in is observed for the solid helium samples in the low temperature limit. However, as is increased above 0.2 and 0.36 K, of the solid 3He and 4He samples respectively crossover from being higher to being lower than the empty Vycor value. We attribute the reduction of to the coupling between the thermal phonons in solid helium and the TLS in silica. For empty

Vycor, phonons absorbed by TLS in silica can only be re-emitted to the silica structure.

But for Vycor filled with solid helium, TLS has a second channel to release energy by exciting phonons in solid helium via modulating the van der Waals interaction at the silica-helium interface [Kinder 1981, Beamish 1984]. The TLS-helium coupling facilitates the TLS tunneling and enhances the tunneling rate . As a result TLS- phonon scattering in silica becomes stronger and is reduced. However phonon excitation in solid helium is strongly suppressed in the low temperature limit when the phonon wavelength exceeds the pore dimension. This is why an enhancement of is

79 seen in the solid samples for due to the softening of acoustic mismatch across the silica-solid helium interface. With increasing temperature, decreases to become comparable and then shorter than the pore dimension thus opening the TLS-helium coupling channel and resulting in a reduction of . The difference in the “crossover” temperatures of the solid 3He (0.2 K) and solid 4He (0.36 K) samples reflects the different transverse sound velocities and hence of the two solid helium samples. of 50

3 bar solid He is 180 m/s. This translates to a of ~10 nm near 0.2 K matching the pore

4 dimension of Vycor. In comparison, of 50 bar solid He is 270 m/s which translates to a of ~10 nm near 0.36 K. The modest difference in (<20%) between the solid samples and the empty Vycor over the full temperature range of the experiment confirms the reasoning stated above that heat conduction is primarily along the silica network and perturbed by the presence of helium.

2 (a) ~T

]

W/K-cm



[ 

 empty 1 4 2 He film 13.6 mol/m 3 2 He film 22.0 mol/m

0.1 0.2 0.3 0.4 0.5 T [K]

(b) ~T 2

]

W/K-cm

 [

 empty  1 50 bar solid 4He 47 bar solid 4He 50 bar solid 3He

0.1 0.2 0.3 0.4 0.5 T [K]

Figure 4-5 Thermal conductivity of empty Vycor and Vycor with atomically thin helium films (a) and solid helium (b) of both isotopes.

81 Although the density and (first) sound velocity of liquid helium are similar as those of solid helium, the measured of the liquid samples are dramatically different.

Instead of a modest change, of full pore liquid 4He sample is reduced three fold as compared with empty Vycor over the entire temperature range as shown in Fig. 4-6. A smaller but still sizable (a factor of 1.5) reduction of is also seen in unsaturated

4 superfluid He film samples ( ). Interestingly, a two-fold reduction of is found when the pores are filled with non-superfluid liquid 3He. This suggests the dramatic reduction of is a consequence of fluidity and not superfluidity of liquid helium.

empty Vycor 4He film 38.1 mol/m2 10 4He film 51.7 mol/m2 4He film 65.3 mol/m2 liquid 3He at 3 bar 4

] liquid He at v.p.

W/K-cm

 1

[

 

0.1 0.2 0.3 0.4 0.5 T [K]

Figure 4-6 Thermal conductivity of empty Vycorand Vycor with liquid 4He, liquid 3He and superfluid 4He films with superfluid transition temperature well above 1 K.

82 The fluidity of the liquid helium enables hydrodynamic sound mode to be excited in the porous structure. Biot showed that such excitation requires the viscous penetration depth of the liquid to be smaller than the pore radius , so that a fraction of liquid can be decoupled from and in relative motion with respect to the solid (in our case the silica) matrix [Biot 1956b, a]. Here and are viscosity and density of the liquid, is the sound frequency. Similar to phonons in solid helium, the hydrodynamic sound is also excited by TLS via the modulation of van der Waals interaction at the silica-helium interface. But in contrast to thermal phonons in solid helium in the pores which must satisfy the boundary condition that their wavelengths should be comparable or shorter than the pore dimensions, there is no such wavelength (and hence frequency) restriction for the hydrodynamic sound mode. The sound mode conforms to and also extends to the entire multiply connected pore space. The frequency spectrum of this hydrodynamic sound is continuous and is given by the Planck’s distribution of thermal

phonons in silica with the additional Biot condition: . Therefore the hydrodynamic sound is much more efficient in enhancing of TLS in the silica over the whole temperature range as compared with phonons in solid helium. This is the key reason for the dramatic reduction of in the liquid samples. The hydrodynamic sound is often named as “slow wave” or “slow sound” [Johnson 1980, Plona 1980, Johnson

1982, Beamish 1987, Kwon 2010] because its velocity is slower than the sound velocity in the solid matrix. Johnson pointed out that the slow sound in superfluid 4He confined in porous media is the well-known fourth sound [Johnson 1980]. Fourth sound can be excited at any finite frequency within the Planck’s distribution because the zero viscosity

83 of superfluid gives . In the case of unsaturated superfluid films where viscosity is also zero, third sound plays the role of slow sound in enhancing . The fourth and third sounds themselves do not contribute to thermal conductivity because superfluid carries no entropy.

Slow sound can also be excited in liquid 3He with a finite viscosity as long as

3 3 . The density of liquid He at 3 bar (pressure of our sample) is 0.089 g/cm .

Between 0.15 K and 0.5 K, the viscosity ranges from 150 to 50 [Bertinat

1972]. As a result ranges from 4.4 to 1.5 GHz, substantially lower than the characteristic frequency of the slow sound, equivalently in silica, ranging from 12 to

40 GHz in the same temperature range. Therefore, there is a significant fraction of liquid

3He in the pores of Vycor that supports the slow sound to enhance . The slow sound in liquid 3He is expected to vanish below 0.07 K when the viscosity of liquid 3He is large enough to completely lock the liquid to the silica matrix. Unfortunately, the large heat capacity of 3He and the low thermal conductivity of the Vycor prevent us from cooling the composite below 0.15 K.

The TLS tunneling rate in an amorphous system has been shown to be [Jäckle

1972, Kinder 1981]

Eq. 4.5

The deformation potential characterizes the coupling strength between TLS tunneling and the resultant strain in the medium. For empty Vycor, the deformation occurs at the

Si-O bonds and is on the order of 1 eV [Jäckle 1972]. In this case the TLS

tunneling rate is labeled as . With helium in the pores, the TLS on the pore surface

84 also couple with helium via modulating the silica-helium van der Waals interaction and

is ~2 meV [Kinder 1981]. In this case the total TLS tunneling rate is given by

. Since the density and sound velocity of liquid helium are ~10

times smaller than those of silica, Eq. 4.5 shows that is larger than by a

factor of 4. As a result, the total tunneling rate, is enhanced by a factor of 5. Since the physical dimension of the SiO4 tetrahedra groups that make up the TLS more than 1 nm and the thickness of silica “walls” is ~7 nm, about 30% of the TLS reside on the pore surface and couple with helium. The average of the entire sample is therefore enhanced by ~2.2 times, close to the three-fold reduction in found in full pore liquid

4He sample. This agreement supports the model we are proposing for the observed reduction of . Superfluid helium films cause less reduction of than full pore liquid 4He because less helium are coupled to TLS. In the liquid 3He sample, the slow sound propagates with finite damping because of the finite viscosity. The damping may dissipate energy to the silica network in the form of phonons. This may explain that liquid 3He causes less reduction in than liquid 4He.

Beamish et al. [Beamish 1984] and Mulders et al. [Mulders 1993] also reported an enhancement of TLS tunneling rate in Vycor due to the adsorption of liquid and solid in their ultrasound experiments between 0.08 and 5 K. However, these experiments were not probing thermal phonons since the frequencies of the ultrasound correspond to characteristic temperature that ranges between 0.3 and 10 mK. . Schubert et al. observed that the transmission of 25-GHz phonons from quartz to amorphous paraffin film is enhanced by the adsorption of liquid helium film on the paraffin at 1.8 K [Schubert

85 1982]. They attributed this to the enhancement of the tunneling of TLS in the amorphous paraffin film.

4.5 Conclusion

In summary, we observe a dramatic phenomenon where the infusion of liquid 4He in Vycor pores results in a three-fold reduction in thermal conductivity. The origin of the reduction is the presence of slow sound mode in liquid 4He, as well as in superfluid 4He films and liquid 3He, which facilitates the quantum tunneling of TLS in silica and dramatically enhances the TLS-phonon scattering. A more modest reduction is observed for solid helium-Vycor composites at high temperature. The reduction is caused by the presence of phonon excitations in solid helium which also facilitate TLS tunneling in silica.

4.6 Prospective Experiments

The results presented above describe a phenomenon that is counter-intuitive at the first glance. However, the model involving the coupling between TLS in silica and hydrodynamic slow sound modes in liquid helium gives semi-quantitative explanation for the observations. More systematic experiments are necessary to further understand this phenomenon.

86 4.6.1 Vycor Filled with Liquid 3He-4He Mixture

The presence of liquid 3He in Vycor leads to a twofold reduction of thermal conductivity, less than the threefold reduction by liquid 4He. We attribute this to the damping of slow sound in liquid 3He due to its finite viscosity. To further study the dependence of thermal conductivity reduction on the liquid viscosity, liquid 3He can be substituted by liquid 3He-4He mixture. The viscosity of the liquid mixture increases with increasing 3He concentration until the phase separation onsets. Therefore, by controlling the 3He concentration, the viscosity of the liquid mixture can be adjusted [Bertinat

1972]. It is expected to see the reduction of thermal conductivity by filling liquid 3He-4He mixture lies between two and three fold and decreases with increasing 3He concentration.

4.6.2 Porous Glass with Smaller Pores

As noted above, it is very interesting to observe the disappearance of the reduction of thermal conductivity in the Vycor-liquid 3He sample below 0.07 K due to the increasing viscosity. However, it is unfortunate that we could not cool down the sample below 0.15 K because the heat capacity of liquid 3He rapidly increases below 0.2

K and the thermal conductivity of the sample is low. In order to observe the reduction of thermal conductivity to disappear, a different porous glassy medium with smaller pore dimension can be chosen. To push this disappearance above 0.15 K, the pore dimension should be smaller than 4 nm. This may be realized by pre-plating nitrogen on the pore suface.

87 4.6.3 Vycor Filled with Liquid Nitrogen at 77 K

We attribute the dramatic reduction of thermal conductivity to the coupling between TLS in silica and the hydrodynamic sound modes in liquid helium. At 77 K, the thermal energy is large enough to overcome the tunneling barrier and quantum tunneling between to adjacent energy levels does not exist. On the other hand, the characteristic phonon wavelength in silica is less than 1 nm at 77 K, much smaller than the thickness of the silica strands. The phonon propagation is hereby confined in the silica strands and is strongly scattered by the disordered lattice structure of amorphous silica and the porous structure of Vycor. In this sense, the presence of liquid nitrogen should reduce the acoustic mismatch at the silica-pore interface. Therefore, the thermal conductivity of

Vycor-liquid nitrogen should be enhanced as compared with the empty Vycor. If this prediction is right, it provides further evidence to justify that the reduction of thermal conductivity below 0.5 K by the presence of liquid helium is a consequence of TLS- hydrodynamic slow sound coupling.

88 Appendix: Heat Capacity of Helium Film in Porous Media

In this appendix, we review some data of heat capacity of thin 4He films adsorbed on the pore surface of Vycor and Xerogel [Finotello 1988, Chan 1991, Finotello unpublished]. The heat capacity as a function of surface coverage at constant temperature shows two peaks: the first one near the completion of the first monolayer and the second one at the superfluid onset. We propose a localized BEC model to explain these peaks. This model is also compatible with previous torsional oscillator experiments of superfluid 4He films absorbed on Vycor surface.

A.1 Introduction to Helium Films

Helium films provide a platform to study novel two-dimensional physics. The adsorption on the substrate modifies the interaction between helium atoms. The roughness of the surface and the amorphous structure of the substrate bring disorders to the bosonic system. All these make the research of helium films interesting.

A.1.1 Kosterlitz-Thouless Theory

Reduced dimensionality on phase transitions and critical phenomena has been of great interests to theorists and experimentalists for a long time. Two-dimensional problems receive special attentions. Kosterlitz-Thouless (KT) theory was proposed for

89 two-dimensional XY model [Kosterlitz 1972, 1973]. In the 2D XY model, classic spins of unit length rotate in two-dimensional lattice. The Hamiltonian describing this system is

Eq. A.1

where is the spin coupling coefficient, and the summation includes neighboring spins i and j, is the angle of the spin relative to a specific orientation. Vortices play the central role in determining the phase transition in 2D XY model. According to the KT theory, superfluid in 2D contains vortices and antivortices, with opposite flow direction around the vortex lines. The bound state of vortex-antivortex pair emerges at low temperature to lower the energy. The pairs maintain the phase coherence and therefore superfluid exists.

As temperature is increased above , the pairs disassociate. Therefore the phase coherence is destroyed and the liquid is no longer superfluid. Superfluid density serves as the order parameter in the superfluid film system. As the temperature approaches from high temperature side, jumps from zero to a finite value:

Eq. A.2

4 where is the He atom mass, is Boltzmann constant, is reduced Planck’s constant.

The relation between and was firstly experimentally demonstrated by Bishop and

Reppy [Bishop 1980] in torsional oscillator experiment of helium film on mylar. A more thorough experimental study of helium film on planar substrate was done by Agnolet et al. [Agnolet 1989].

90 A.1.2 Helium Film in Porous Media

Behaviors of helium film on planar substrate can be explained by KT theory successfully. On the other hand, behaviors of helium films adsorbed on the surfaces of porous amorphous media are not in satisfactory agreement with KT theory because the theory does not take into account the effect of random potential exerted by the amorphous substrate on the helium film. The amorphous substrates include Vycor, aerogel, xerogel and porous gold. For a wide range of substrates, approximately the first two atomic layers

4 of He atoms adsorbed (corresponding to surface coverage ) are tightly bound to substrate and do not exhibit macroscopic superfluidity even at , forming the so- called “inert layer”. For the porous media that have random porous structure, the concept

“atomic layer” only has statistical meaning due to the strong randomness of inner structures. For most of the case, the inert layer can be viewed as localized solid layer. 4He layer on top of “inert layer” shows superfluidity at zero temperature. As temperature rises

4 beyond the critical temperature , He film becomes normal. is a function of helium surface coverage . The superfluid transition for film with larger happens at higher .

The inert layer does not participate in macroscopic superfluidity and therefore cannot be detected in torsional oscillator experiments. The measured superfluid density is consistent with the density of helium above the inert layer for a wide range of surface coverage. However, there are evidences showing that inert layer is not really inert. Heat capacity results of 4He films show that superfluid film adsorbed in Vycor has smaller heat capacity than that of the inert layer film with coverage right below [Finotello 1988]. It suggests the existence of some excitations in inert layer.

91 Vycor is an appropriate substrate to study helium in porous media. It was firstly used as a superleak by Atkins et al. [Atkins 1956]. They discovered that the superfluid onset temperature is lower than that of the bulk superfluid when helium is confined in

Vycor. Superfluid density of full pore liquid 4He and films in Vycor has later been studied in various methods, including torsional oscillator [Berthold 1977, Bishop 1981a,

Crooker 1983, Chan 1988, Murphy 1990], persistent current [Henkel 1969, Chan

1974], third and fourth sound [Kiewiet 1975, Bishop 1981a], etc. On the other hand, heat capacity measurements have also been performed of helium films adsorbed on Vycor

[Tait 1979b, Finotello 1988, Murphy 1990].

Kiewiet et al. used fourth sound technique to determine temperature dependence of superfluid fraction [Kiewiet 1975]. In addition to the suppression of , they

observed the power law relation near with , close to the value obtained from bulk superfluid 0.67 (as shown in Fig. A-1). They concluded that the superfluid transition of liquid 4He in Vycor belongs to the same universality as the bulk superfluid transition. Later torsoinal oscillator experiments confirmed the same critical exponents for both thin superfluid film and full pore liquid

4He in Vycor [Chan 1988].

In addition to full pore liquid, Vycor also has profound effects on properties of adsorbed films. Berthold et al. used torsional oscillator technique to measure superfluid density of thin 4He films adsorbed on Vycor [Berthold 1977]. It is discovered that the transition temperature is suppressed to be lower temperature compared with full pore liquid and decreases with coverage. The period drop, , proportional to superfluid density, obeys the power law with similar critical exponent as full pore liquid near as

shown in Fig. A-2. It also shows that scales linearly with coverage for

all films with with . The finite interception suggests an

“inert layer” that does not participate in superfluidity even at zero temperature. However, thin helium films with show non-linear relation between and

[Crooker 1983] (Fig. A-3). Superfluidity starts to emerge from the coverage

. It implies that the “inert layer” model that simply divides helium films into inert part and superfluid part is not accurate.

Figure A-2 Top: period drop plotted as a function of temperature for different thickness of 4He films adsorbed on Vycor. Bottom: period drop vs reduced temperature [Reppy 1992].

Figure A-3 Period drop at as a function of surface coverage of 4He films on the substrate of Vycor [Crooker 1983]. The relation between and coverage is nonlinear at low coverage.

Heat capacity of liquid 4He is also modified by Vycor significantly. Compared with the well-known sharp lambda peak at the bulk superfluid transition, there is a rounded and broad peak for helium films at lower temperature. The peak magnitude is significantly reduced (as shown in Fig. A-4). However, specific heat of liquid 4He in

Vycor is larger than that of bulk liquid 4He below ~1 K. This implies that some excitations are introduced due to the presence of silica substrate.

Figure A-4 Specific heat of 4He adsorbed on Vycor [Brewer 1970].

Along with full pore liquid helium, Brewer also measured the specific heat of helium films adsorbed on Vycor [Brewer 1970]. The rounded lamda peak continues shrinking in size and shifting to lower temperature as the film thickness decreases. In addition, as shown in Fig. A-4, all the rounded peaks are at higher temperature than the superfluid onset determined by the torsional oscillator experiments. However, Tait and

Reppy did a series of measurements of films with different surface coverage from sub- monolayer to full pore [Tait 1979b]. They were able to identify characteristic temperatures for superfluid films at which the heat capacity changes the temperature dependence. is close to that of superfluid onset temperature (Fig. A-5). More interestingly, Finotello et al. observed a sharp heat capacity peak right at

[Finotello 1988] (Fig. A-6). The peak is superimposed onto the broad rounded peak

97 mentioned above. The observation of sharp heat capacity peak at superfluid onset temperature together with the critical exponent obtained from torsional oscillator experiments supports the statement that superfluid transition of helium film is a genuine critical phase transition.

Although the sharp heat capacity peak is a strong evidence of critical behavior, there remain some questions to be answered. As can be seen in Fig. A-6, heat capacity decreases as coverage increases when and . Therefore, at , all the films with have smaller heat capacity than the “inert” film with [Finotello

1988]. This conclusion is contradictory with the “inert layer” model that completely separates the “inert layer” from the superfluid layer. Obviously, there are excitations in

“inert layer” that are suppressed by the superfluid layer overlaying above.

Figure A-5 Coverage dependence of TC or T0 [Tait 1979b]. Dashed line indicates the superfluid onset temperature determined by Berthold et al. [Berthold 1977].

100

2 29.58 mol/m 80 2 30.89 mol/m 2 33.54 mol/m 2 ] 35.97 mol/m

2 60

J/K-m 40

 C[ 20

0 0.0 0.2 0.4 0.6 0.8 1.0 T [K]

Figure A-6 Heat capacity per unit area for various coverage of 4He film adsorbed on Vycor [Finotello 1988]. Arrows label the heat capacity peaks at the superfluid transition.

A.2 Heat Capacity of Helium Films

The existence of excitations in “inert layer” is supported by the heat capacity anomaly of “inert layer” observed by Finotello et al.. Fig. A-7 shows the heat capacity isotherm as a function of coverage for 4He film adsorbed on Vycor at .

The isotherm shows two peaks at all temperatures. The position of first peak is temperature independent and is at (equivalent to 1.4 monolayer). The position of the second peak is temperature dependent, indicating the critical coverage for the emergence of superfluid at finite temperature. Similar measurements have been done for films adsorbed on silica gel as shown in Fig. A-8. The results shear the same features with that of helium-Vycor system.

99 Although the heat capacity isotherms with different substrates share the same features, the peak sizes and temperatures are still different, even though the chemical components of the substrates are both silica. At , as compared with the films on Vycor, the size of the first peak for films on silica gel is 2~3 times larger. The size of the second peak is also larger by 10~20%.

Figure A-7 Heat capacity plotted as a function of coverage at fixed temperatures for helium films adsorbed on Vycor. Solid curves are for guiding eyes [Chan 1991].

100

Figure A-8 Heat capacity plotted as a function of coverage at fixed temperatures for helium films adsorbed on silica gel [Finotello unpublished].

A.2.1 Vycor vs. Silica Gel

Both Vycor and silica gel are composed of amorphous silica. Vycor is made from spinodal decomposition of a boron-rich glass-forming melt into boron-rich and silica-rich phases on cooling process. The boron-rich phase is leached out, leaving interconnected pore structure in silica glass matrix. Transmission electron microscopy and small angle

X-ray and neutron scattering studies reveal a broad distribution of correlation length from

5 to 60 nm, centering at 20 nm [Schaefer 1987, Wiltzius 1987]. Its porosity is ~28% and typical pore size is 7 nm assuming pores are in cylindrical geometry. On the other hand, silica gel is produced by removing liquid out of colloidal silica. It is reported to have self- similar structure for length scale up to tens of nanometers [Schaefer 1986]. The silica gel piece discussed here has a porosity of 60%.

101 Different manufacturing procedures lead to different structure between Vycor and silica gel. As shown in A-9, silica molecules are compact and most silica molecules are tightly bound to neighbors in Vycor. With the assumption that pores are in cylindrical,

~9% of silica molecules are on the pore surface. On the other hand, silica molecules in silica gel are loosely connected. The percentage of silica molecules on surface is more than 30%. The irregularity and potential depth at silica gel surface is larger than that of

Vycor.

Figure A-9 Schematic diagram of the structure of Vycor (left) and silica gel (right). Black color represents silica and white color represents empty space.

A.2.2 Localized BEC

As shown in Fig. A-7 and A-8, the first peak in the non-superfluid region indicates that the so-called “inert layer” is thermodynamically active. The excitation in

102 inert layer does not show macroscopic superfluidity and therefore cannot be observed in torsional oscillator experiments.

The first monolayer of helium atoms are tightly bound on the silica and completely classical. Due to the irregular potential of silica, the adsorbed helium film is amorphous and two-level systems (TLS) exist in the amorphous helium. TLS gives large contribution to heat capacity in addition to the Debye contribution. With the screening effect of the adsorbed atoms, helium atoms residing on top of the first layer experience a weaker and smoother potential. The dynamics of these atoms are controlled by two competing mechanisms: localization by the substrate and the tendency to bose- condensate. As the number of helium atoms increases, localization by the substrate becomes weaker and bose-condensate effect becomes more obvious. Beyond certain coverage, BEC emerges in the potential well. However, these BEC are spatially localized in the potential wells because there are potential barriers that are too high to prevent the macroscopic phase coherence. Therefore, the adsorbed film does not show macroscopic superfluidity at this point. As the density of helium atoms increases, more amorphous helium transits to the condensation state and joins the localized BEC. Therefore TLS density decreases and its heat capacity contribution decreases. Fig. A-10 schematically shows the localized BEC in potential wells. In silica gel, where the localization is stronger than in Vycor, the first peak appears at higher coverage because more amorphous helium is formed to screen the localization from the silica before BEC appears. Therefore the size of the first peak in silica gel is larger.

103

Figure A-10 Schematic drawing of localized BEC of helium adsorbed silica surface

When localized BEC forms in discrete potential wells, the phase for each BEC patch is not coherent. The phase gradient between two neighboring localized BECs drives quantum tunneling between different regions (as shown in Fig. A-11 (a)). The tunneling costs energy and therefore contributes to heat capacity. As film coverage is increased, the tunneling barrier becomes smaller and the tunneling rate becomes larger. Therefore the heat capacity contributed by tunneling rapidly increases. This leads to an increase of heat capacity as the observed the low-coverage side of the second heat capacity peak. To achieve phase coherence between potential wells, different localized BEC need to be connected by superfluid helium. This is realized when all potential wells for . For temperature higher than the superfluid transition temperature, unpaired vortices destroy superfluidity. Therefore, the position of the second peak is temperature dependent. Once

104 superfluid helium connects two potential wells, quantum tunneling between the two wells is suppressed due to the phase coherence. Eventually the tunneling disappears when all potential wells are filled. This corresponds to the rapid heat capacity drop on the high- coverage side of the second peak.

It is interesting to observe that the rapid heat capacity drop on the superfluid side saturated at high coverage and the saturation value is close to the minimum between the two peaks but higher than that of bare substrate (Fig. A-8). It suggests that part of the helium on the pore surface remains in amorphous state even when the porous medium is saturated by liquid helium. The strong interaction from silica prevents them from becoming superfluid. To deduce the amount of this amorphous part in the silica gel- helium system, the heat capacity of the high coverage film is horizontally extrapolated to

105 intercept with the low-coverage region (as shown in Fig. A-12) where only amorphous helium contributes to heat capacity. The intercepted value corresponds to a coverage of

, equivalent to 1.27 atomic layer ( for one atomic layer).

Finotello et al. only measured the heat capacity of 4He films in Vycor below

[Finotello 1988], not reaching the saturation region. However, earlier experiment by

Tait and Reppy [Tait 1979b] extended the measurement till the Vycor is completely saturated. Their data also show that heat capacity of full pore liquid helium is equal to that of 1.3 monolayer of helium adsorbed on Vycor. Both results point to the conclusion that the first 1.3 layer of helium adsorbed on silica stays in amorphous state and never participates in superfluidity.

2.0x10-5 Silica gel T=0.3 K

1.5x10-5

] 2

1.0x10-5

C[J/K-m 5.0x10-6

0.0 0 10 20 30 40 50 2 n [mol/m ]

Figure A-12 Heat capacity isotherm of helium films adsorbed on silica gel at 0.3 K. The dotted line is a horizontal extrapolation of the heat capacity at high coverage.

106

A.3 Some Comments on Torsional Oscillator Results

Previous torsional oscillator experiments can be explained by inert layer model because the measured superfluid density is linearly dependent on helium coverage when the superfluid onset temperature . However, a nonlinear dependence is observed for thin films with [Crooker 1983, Crowell 1997]. The nonlinear dependence makes the superfluid onset critical coverage smaller than the

value predicted by the inert model based on the linear dependence at high coverage.

To observe the macroscopic superfluidity in the helium films, not all localized BEC are needed to be connected. It only requires sufficient localized BEC patches to be connected to form a continuous path. As is further increased, newly added helium not only becomes superfluid itself, but also connects more localized BEC patches. Therefore the measured macroscopic superfluid density grows faster than linearly with the coverage,

leading to the upward curvature. indicates the coverage at which all BEC patches are

connected. Therefore for , no more localized BEC join the macroscopic superfluidity and the superfluid density increases linearly with coverage.

A.4 Conclusion

In this chapter, we show results of heat capacity isotherm of helium films adsorbed on Vycor and silica gel. Some heat capacity feature is discovered within the so-

107 called inert layer region. A model is proposed to explain the data. This model proposes that part of amorphous helium adsorbed on silica transits into localized BEC state. Extra helium connects these localized BECs and makes macroscopic superfluid state with long- range phase coherence. This model is different from previous inert layer model by stating that the layer that does not participate in superfluid near the superfluid onset is thermodynamically active. It can explain the nonlinear relation between superfluid density and helium film coverage for thin films which cannot be explained by inert layer model.

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VITA

Zhigang Cheng

Zhigang Cheng was born to Guoqiang Cheng and Guoqin Wang in Wuhan, China in 1983. He graduated from Wuhan No. 2 Middle School in 2002. This is also the place he first met his wife. After high school, he was enrolled in the Department of Physics,

Wuhan University in September 2002 and received the Degree of Bachelor in Science in

June 2006.

In August 2006, Zhigang moved to United State to attend graduate school in physics at Pennsylvania State University. He was trained as an experimental physicist, focusing on low temperature physics.