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).

Contents

Preface...................................................................................................xi

Useful Equations and Definitions.......................................................xv

Definitions.......................................................................................... xxi

1 Some Remarks on the Gas State

1.1 Equation of State (EOS) of Real Gases................................................................ 1

1.1.1 The Van der Waals Equation................................................................. 1

1.1.2 The Virial Equation.................................................................................2

1.2 Order in the Gas State............................................................................................3

1.3 Heat Capacity of Gases......................................................................................... 4

1.3.1 How Well Does This Model Work?...................................................... 4

1.4 Vibrational Raman Spectroscopy of Gases........................................................6

1.5 Viscosity of Gases...................................................................................................8

1.6 Why Are Liquids so Difficult?............................................................................ 10

1.6.1 Molecular Dynamics (MD)................................................................. 10

1.6.2 The Fundamental EOS (Section 3.3)....................................................11

1.6.3 Treat the Fluid as Gas-Like.................................................................. 12

1.6.4 Treat the Fluid as Solid-Like................................................................ 12

References.......................................................................................................................... 13

2 The Vapour Pressure Curve and the Liquid State Close to

the Vapour Pressure Curve

2.1 Classical Versus Quantum Liquids.................................................................... 15

2.2 The Transition Across the Vapour Pressure Curve......................................... 17

2.3 The Clausius-Clapeyron Equation.....................................................................19

2.3.1 Validity of the Clausius-Clapeyron Equation.................................. 20

2.4 The Critical Point................................................................................................. 20

2.4.1 Critical Constants and the Van Der Waals

Equation of State....................................................................................25

2.5 Summary............................................................................................................... 29

References......................................................................................................................... 30

3 Equations of State for Fluids

3.1 Cubic EOS Based on the Van der Waals Equation..........................................32

3.1.1 Volume Translation of Cubic EOS..................................................... 34

3.2 The Carnahan-Starling EOS...............................................................................35

3.3 The Fundamental EOS........................................................................................ 36

3.3.1 Ideal Gas Component of the Helmholtz Function.......................... 36

3.3.2 Residual Component of the Helmholtz Function............................39

3.3.3 Fitting the Helmholtz Function to the

Experimental Data................................................................................39

3.4 Conclusions...........................................................................................................41

3.4.1 For What Fluids Is a Fundamental EOS Available?.........................41

3.4.2 How Can We Test the Validity of an EOS?........................................41

3.4.3 What Is the Best Way to Implement Your

Chosen EOS? ............................................................................................................... 44

References......................................................................................................................... 46

4 The Liquid State Close to the Melting Curve (I):

Static Properties

4.1 Density and Bulk Modulus of Fluids Close to the Melting Curve............... 47

4.1.1 Density of Fluid Ar Close to the Melting Curve.............................. 48

4.1.2 Density and Bulk Modulus of Fluid N2 Close to

the Melting Curve................................................................................ 49

4.2 Elastic Neutron and X-ray Diffraction from Liquids Close to the

Melting Curve....................................................................................................... 51

4.2.1 Distinctions Between X-ray and Neutron

Diffraction Experiments......................................................................53

4.2.2 Fourier Transform of Fluid Diffraction Data

to Obtain g (r) ....................................................................................55

4.2.3 Fourier Transform of Modified Fluid Diffraction Data

to Obtain g (r) ..................................................................................................58

4.2.4 Comparison of Diffraction Data to Simulated Fluid

Structures in Reciprocal Space............................................................61

4.2.5 Relation Between g (r), the Partition Function, Internal

Energy, and Pressure.............................................................................63

4.2.6 Relation Between g (r) and Entropy....................................................65

4.2.7 Relation Between g (r) and Co-ordination Number (CN)............. 66

4.3 Short-Range Order and Phase Transitions in Fluids Close to the

Melting Curve...................................................................................................... 67

4.3.1 Co-ordination Number....................................................................... 67

4.3.2 Liquid-Liquid Phase Transitions........................................................ 67

4.4 Equations to Fit the Melting Curve on the P,T Phase Diagram................... 69

4.5 What Happens to the Melting Curve in the High P,T Limit?........................72

4.6 Summary................................................................................................................74

References..........................................................................................................................77

5 The Liquid State Close to the Melting Curve (II):

Dynamic Properties

5.1 Phonon Theory of Liquids...................................................................................79

5.1.1 Frenkel and Maxwell Models..............................................................79

5.1.2 Prediction of Liquid Heat Capacity....................................................82

5.2 Raman Spectroscopy of Liquids and Supercritical Fluids

Close to the Melting Curve................................................................................ 88

5.2.1 Grüneisen Model for Vibrational Raman

Peak Position ........................................................................................................................90

5.2.2 Hard Sphere Fluid Theory of Vibrational

Raman Peak Positions.......................................................................... 91

5.2.3 Peak Position of Rotational Raman Spectra.....................................93

5.2.4 Peak Intensity and Linewidth of Fluid Raman Spectra..................93

5.2.5 Prediction of Fluid Raman Spectra Using MD............................... 94

5.3 Brillouin Spectroscopy of Liquids Close to the Melting Curve................... 96

5.4 Inelastic Neutron and X-ray Scattering from Liquids Close to

the Melting Curve................................................................................................ 98

5.4.1 Distinction Between Neutron and X-ray Scattering....................... 98

5.4.2 The Scattered Intensity........................................................................101

5.4.3 What Can We Learn from Inelastic Neutron and X-ray

Scattering from Liquids?.................................................................... 103

5.5 Summary and Outlook......................................................................................107

References........................................................................................................................108

6 Beyond the Critical Point

6.1 The Widom Lines................................................................................................. 111

6.1.1 A Simple Phenomenological Fitting Procedure for

the Widom Lines..................................................................................114

6.1.2 Some Examples of Widom Line Paths..............................................117

6.1.3 The Widom Lines as a Function of

Reduced...