The Pennsylvania State University The Graduate School Department of Physics THERMODYNAMIC PROPERTIES OF HELIUM IN POROUS MEDIA A Dissertation in Physics by 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. v 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 vi 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 ..............................................................................................