Liquid and Supercritical Fluid States of Matter

John E. Proctor is a senior lecturer in physics at the University of Salford and is head of the Materials and Physics Research Group. He specialises in condensed matter physics, particularly the study of fluids and solids under extreme pressure and temperature, principally through X-ray and neutron diffraction along with optical spectroscopy. His research is regularly published in leading international peer-reviewed journals. He completed his Ph.D. (2007) from the University of Manchester and his M.Phys. (2004) from the University of Oxford. He is one of the authors of An Introduction to Graphene and Carbon Nanotubes (CRC Press, 2017).

• ContentsPreface...................................................................................................xiUseful Equations and Definitions.......................................................xvDefinitions.......................................................................................... xxi1 Some Remarks on the Gas State1.1 Equation of State (EOS) of Real Gases................................................................ 11.1.1 The Van der Waals Equation................................................................. 11.1.2 The Virial Equation.................................................................................21.2 Order in the Gas State............................................................................................31.3 Heat Capacity of Gases......................................................................................... 41.3.1 How Well Does This Model Work?...................................................... 41.4 Vibrational Raman Spectroscopy of Gases........................................................61.5 Viscosity of Gases...................................................................................................81.6 Why Are Liquids so Difficult?............................................................................ 101.6.1 Molecular Dynamics (MD)................................................................. 101.6.2 The Fundamental EOS (Section 3.3)....................................................111.6.3 Treat the Fluid as Gas-Like.................................................................. 121.6.4 Treat the Fluid as Solid-Like................................................................ 12References.......................................................................................................................... 132 The Vapour Pressure Curve and the Liquid State Close tothe Vapour Pressure Curve2.1 Classical Versus Quantum Liquids.................................................................... 152.2 The Transition Across the Vapour Pressure Curve......................................... 172.3 The Clausius-Clapeyron Equation.....................................................................192.3.1 Validity of the Clausius-Clapeyron Equation.................................. 202.4 The Critical Point................................................................................................. 202.4.1 Critical Constants and the Van Der WaalsEquation of State....................................................................................252.5 Summary............................................................................................................... 29References......................................................................................................................... 303 Equations of State for Fluids3.1 Cubic EOS Based on the Van der Waals Equation..........................................323.1.1 Volume Translation of Cubic EOS..................................................... 343.2 The Carnahan-Starling EOS...............................................................................353.3 The Fundamental EOS........................................................................................ 363.3.1 Ideal Gas Component of the Helmholtz Function.......................... 363.3.2 Residual Component of the Helmholtz Function............................393.3.3 Fitting the Helmholtz Function to theExperimental Data................................................................................393.4 Conclusions...........................................................................................................413.4.1 For What Fluids Is a Fundamental EOS Available?.........................413.4.2 How Can We Test the Validity of an EOS?........................................413.4.3 What Is the Best Way to Implement YourChosen EOS? ............................................................................................................... 44References......................................................................................................................... 464 The Liquid State Close to the Melting Curve (I):Static Properties4.1 Density and Bulk Modulus of Fluids Close to the Melting Curve............... 474.1.1 Density of Fluid Ar Close to the Melting Curve.............................. 484.1.2 Density and Bulk Modulus of Fluid N2 Close tothe Melting Curve................................................................................ 494.2 Elastic Neutron and X-ray Diffraction from Liquids Close to theMelting Curve....................................................................................................... 514.2.1 Distinctions Between X-ray and NeutronDiffraction Experiments......................................................................534.2.2 Fourier Transform of Fluid Diffraction Datato Obtain g (r) ....................................................................................554.2.3 Fourier Transform of Modified Fluid Diffraction Datato Obtain g (r) ..................................................................................................584.2.4 Comparison of Diffraction Data to Simulated FluidStructures in Reciprocal Space............................................................614.2.5 Relation Between g (r), the Partition Function, InternalEnergy, and Pressure.............................................................................634.2.6 Relation Between g (r) and Entropy....................................................654.2.7 Relation Between g (r) and Co-ordination Number (CN)............. 664.3 Short-Range Order and Phase Transitions in Fluids Close to theMelting Curve...................................................................................................... 674.3.1 Co-ordination Number....................................................................... 674.3.2 Liquid-Liquid Phase Transitions........................................................ 674.4 Equations to Fit the Melting Curve on the P,T Phase Diagram................... 694.5 What Happens to the Melting Curve in the High P,T Limit?........................724.6 Summary................................................................................................................74References..........................................................................................................................775 The Liquid State Close to the Melting Curve (II):Dynamic Properties5.1 Phonon Theory of Liquids...................................................................................795.1.1 Frenkel and Maxwell Models..............................................................795.1.2 Prediction of Liquid Heat Capacity....................................................825.2 Raman Spectroscopy of Liquids and Supercritical FluidsClose to the Melting Curve................................................................................ 885.2.1 Grüneisen Model for Vibrational RamanPeak Position ........................................................................................................................905.2.2 Hard Sphere Fluid Theory of VibrationalRaman Peak Positions.......................................................................... 915.2.3 Peak Position of Rotational Raman Spectra.....................................935.2.4 Peak Intensity and Linewidth of Fluid Raman Spectra..................935.2.5 Prediction of Fluid Raman Spectra Using MD............................... 945.3 Brillouin Spectroscopy of Liquids Close to the Melting Curve................... 965.4 Inelastic Neutron and X-ray Scattering from Liquids Close tothe Melting Curve................................................................................................ 985.4.1 Distinction Between Neutron and X-ray Scattering....................... 985.4.2 The Scattered Intensity........................................................................1015.4.3 What Can We Learn from Inelastic Neutron and X-rayScattering from Liquids?.................................................................... 1035.5 Summary and Outlook......................................................................................107References........................................................................................................................1086 Beyond the Critical Point6.1 The Widom Lines................................................................................................. 1116.1.1 A Simple Phenomenological Fitting Procedure forthe Widom Lines..................................................................................1146.1.2 Some Examples of Widom Line Paths..............................................1176.1.3 The Widom Lines as a Function ofReduced Temperature......................................................................... 1196.1.4 The Widom Lines in Relation to the VapourPressure Curve ..............................................................................................1206.1.5 The Widom Lines as a Function of Density.....................................1216.2 The Fisher-Widom Line......................................................................................1216.3 The Joule-Thomson Inversion Curve...............................................................1236.4 A General Approach to Inversion Curves......................................................1276.4.1 First Order Inversion Curves: Definitions.......................................1286.4.2 First Order Inversion Curves: Path on the P,TPhase Diagram ........................................................................................................................1326.4.3 Zeroth and First Order Inversion Curves: Can WeMeasure Them? Do We Need to Measure Them?...........................1346.4.4 Use of Zeroth Order and First Order Inversion Curves toVerify Equations of State....................................................................1366.5 The Frenkel Line.................................................................................................1386.5.1 Definitions of the Frenkel Line.........................................................1386.5.2 The Frenkel Line and the Widom Lines........................................... 1476.5.3 Positive Sound Dispersion Above TC................................................1486.5.4 Termination of the Frenkel Line.......................................................1506.6 Conclusions.........................................................................................................150References.........................................................................................................................1517 Miscibility in the Liquid and Supercritical Fluid States7.1 Introduction........................................................................................................ 1537.2 Raoult’s Law, Henry’s Law, and the Lever Rule.............................................1547.2.1 Raoult’s Law and Henry’s Law..........................................................1547.2.2 Change in Gibbs Function on Mixing ofRaoultian Liquids.................................................................................. 1567.2.3 Phase Equilibria in Miscible Fluids: The Lever Rule.....................1587.3 Hildebrand Theory of Mixing..........................................................................1587.3.1 Internal Energy of Fluid Mixtures UsingHildebrand Theory................................................................................ 1587.3.2 P, V, T EOS for Mixtures Using Hildebrand Theory.....................1607.4 Application of the Fundamental EOS to Mixtures....................................... 1627.5 Some Comments on Experimental Study of Supercritical FluidMixtures............................................................................................................... 1637.5.1 Preparation of Fluid Mixtures in the DiamondAnvil Cell (DAC)................................................................................. 1637.5.2 Raman Spectra of Fluid Mixtures; CohesiveEnergy Density ....................................................................................................... 1647.6 Open Questions in the Study of Dense Fluid Mixtures............................... 1657.6.1 Is Hydrophobicity an Absolute Property?....................................... 1657.6.2 Miscibility in the Supercritical Fluid State......................................166References........................................................................................................................ 1678 Applications of Supercritical Fluids8.1 Applications of Supercritical Fluids in Power Generation Cycles..............1698.1.1 Efficiency of Thermodynamic Cycles...............................................1698.1.2 Use of Supercritical H2O in Power Generation.............................. 1708.1.3 Use of Supercritical CO2 in Power Generation................................1718.1.4 Use of Supercritical N2 in Power Generation.................................. 1738.2 Use of Supercritical Fluids in Food Processing............................................. 1758.2.1 Decaffeination...................................................................................... 1758.2.2 Other Food Processing Applications............................................... 1758.3 Supercritical CO2 Cleaning and Drying......................................................... 1758.4 Chromatography................................................................................................. 1768.5 Crystal and Nanoparticle Growth................................................................... 1768.6 Exfoliation of Layered Materials......................................................................177References........................................................................................................................ 1789 Supercritical Fluids in Planetary Environments9.1 Introduction.........................................................................................................1819.2 Mineral and Material Processes with Supercritical Fluids.......................... 1829.2.1 Dissolution of Minerals...................................................................... 1829.2.2 Mineral Reactions...............................................................................1849.2.3 Partition of Elements.......................................................................... 1859.3 Supercritical Fluids within Surface and Subsurface Environments........... 1859.3.1 Earth......................................................................................................1869.3.2 Other Terrestrial Planets.................................................................... 1879.3.3 Dwarf Planets and Icy Satellites........................................................1909.4 Supercritical Fluids within Planetary Interiors.............................................1909.4.1 Jupiter and Saturn................................................................................1919.4.2 Uranus and Neptune...........................................................................1929.4.3 Transitions in the Supercritical Fluids; Effecton the Gas Giants................................................................................1949.5 Summary..............................................................................................................194References........................................................................................................................194Appendix A: Reference Data on Selected Atomic FluidsA.1 Table of Phase Change Properties for He, Ne, and Ar..................................199A.2 Phase Diagram of He.........................................................................................199A.3 Phase Diagram of Ne........................................................................................ 205A.4 Phase Diagram of Ar......................................................................................... 208References........................................................................................................................210Appendix B: Reference Data on Selected Molecular FluidsB.1 Table of Phase Change Properties for CH4, CO2,H2, H2O, and N2...................................................................................................211B.2 Phase Diagram of CH4........................................................................................211B.3 Phase Diagram of CO2....................................................................................... 217B.4 Phase Diagram of H2......................................................................................... 220B.5 Phase Diagram of H2O...................................................................................... 226B.6 Phase Diagram of N2.......................................................................................... 231References....................................................................................................................... 234Appendix C: Some Thermodynamic and Diffraction DerivationsC.1 Thermodynamic Quantities..............................................................................237C.1.1 Application of the First Law of Thermodynamics.........................237C.1.2 Adiabatic Changes; Enthalpy........................................................... 238C.1.3 Isothermal Changes; Helmholtz Function..................................... 238C.1.4 Isobaric and Isothermal Changes; Gibbs Function...................... 239C.1.5 Constraints on the P, V, T EOS of the Ideal Fluid and theCondensing Fluid (Brown’s Conditions)........................................ 239C.2 Fourier Transform Treatment of Diffraction................................................ 242Appendix D: The Diamond Anvil Cell (DAC)D.1 Design of the DAC............................................................................................. 245D.2 Loading of Fluid and Fluid Mixture Samples into the DAC...................... 247D.2.1 Pure Fluids........................................................................................... 247D.2.2 Fluid Mixtures.................................................................................... 249D.3 High Temperatures in the DAC...................................................................... 249D.3.1 Resistive Heating Experiments in the DAC................................... 250D.3.2 Laser Heating in the DAC.................................................................. 251D.4 Pressure Measurement in the DAC..................................................................252References....................................................................................................................... 254Appendix E: Code for Selected Computational ProblemsE.1 Boiling Transition in the van der Waals Fluid...............................................255E.1.1 Estimate of Pb...................................................................................... 256E.1.2 Evaluation of ΔG..................................................................................257E.1.3 Octave Code for van der Waals’ Boiling Transition......................257E.2 Prediction of Fluid Heat Capacity.................................................................. 260E.2.1 Octave Code for Heat Capacity Calculations................................ 260Bibliography...................................................................................... 263Index................................................................................................... 267