Table Of ContentSoft and Biological Matter
Paola Gallo
Mauro Rovere
Physics
of Liquid
Matter
Soft and Biological Matter
SeriesEditors
DavidAndelman,SchoolofPhysicsandAstronomy,TelAvivUniversity,
TelAviv,Israel
WenbingHu,SchoolofChemistryandChemicalEngineering,Departmentof
PolymerScienceandEngineering,NanjingUniversity,Nanjing,China
ShigeyukiKomura,DepartmentofChemistry,GraduateSchoolofScienceand
Engineering,TokyoMetropolitanUniversity,Tokyo,Japan
RolandNetz,DepartmentofPhysics,FreeUniversityofBerlin,Berlin,Berlin,
Germany
RobertoPiazza,DepartmentofChemistry,MaterialsScience,andChemical
Engineering“G.Natta”,PolytechnicUniversityofMilan,Milan,Italy
PeterSchall,VanderWaals-ZeemanInstitute,UniversityofAmsterdam,
Amsterdam,Noord-Holland,TheNetherlands
GerardWong,DepartmentofBioengineering,CaliforniaNanoSystemsInstitute,
UCLA,LosAngeles,CA,USA
“SoftandBiologicalMatter”isaseriesofauthoritativebookscoveringestablished
andemergentareasintherealmofsoftmatterscience,includingbiologicalsystems
spanningallrelevantlengthscalesfromthemoleculartothemesoscale.Itaimsto
serve a broad interdisciplinary community of students and researchers in physics,
chemistry,biophysicsandmaterialsscience.
Pure research monographs in the series, as well as those of more pedagogi-
cal nature, will emphasize topics in fundamental physics, synthesis and design,
characterization and new prospective applications of soft and biological matter
systems. The series will encompass experimental, theoretical and computational
approaches. Topics in the scope of this series include but are not limited to: poly-
mers, biopolymers, polyelectrolytes, liquids, glasses, water, solutions, emulsions,
foams,gels,ionicliquids,liquidcrystals,colloids,granularmatter,complexfluids,
microfluidics,nanofluidics,membranesandinterfaces,activematter,cellmechanics
andbiophysics.
Bothauthoredandeditedvolumeswillbeconsidered.
Moreinformationaboutthisseriesathttp://www.springer.com/series/10783
Paola Gallo (cid:129) Mauro Rovere
Physics of Liquid Matter
PaolaGallo MauroRovere
DepartmentofMathematicsandPhysics DepartmentofMathematicsandPhysics
RomaTreUniversity RomaTreUniversity
Roma,Italy Roma,Italy
ISSN2213-1736 ISSN2213-1744 (electronic)
SoftandBiologicalMatter
ISBN978-3-030-68348-1 ISBN978-3-030-68349-8 (eBook)
https://doi.org/10.1007/978-3-030-68349-8
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Preface
Thisbookintroducesthereadertothefascinatingfieldofliquidstatephysics.The
liquid state is located in a restricted zone of the phase diagram in coexistence
with the solid and the gas phases. Apart for the special case of helium, the
conditions of density and temperature are such that the liquids can be under
most circumstances considered as classical melts. The determination of the liquid
propertiesstartingfrommicroscopicmodelsisachallengeforstatisticalmechanics
due to the disordered arrangement of atoms and high density. An impressive
progress in the understanding of the liquid state has been achieved in recent years
mainly with the improvement of the experimental techniques and the large use
of computer simulations, made possible by the continuous improvement of the
computationaltechniques.Thenumericalsimulationisinfactnotanymoreasimple
support to theoretical studies, and it can be considered a third relevant column
besides approximate theoretical methods and experiments. Computer simulation
hasbecomeapredictivemethodologystimulatingexperimentalwork.Animportant
example is the study of region of the phase space where metastable states are
present.
Theexperimental,theoreticalandcomputationalmethodsdevelopedforstudying
theliquidstatehaverecentlybeenextendedtotheresearchinthefieldofmaterials
with large technological relevance like colloids, biomolecules and amorphous
solids.
Our opinion is that we are at a turning point in the research in this field. At
variance with the old times when it was possible to treat only very simplified
microscopicmodel,inthefuturemoresophisticatedmodelswillbeintroducedand
studied with more refined techniques. For this reason we believe that liquid state
physics will become even more of interest for a large community of researchers
in physics, chemistry and other related disciplines. Our intention with this book
is to provide an introduction to the liquid state physics for graduate students and
researchers and fill the gap between books that contain basic skills in statistical
mechanicsandthermodynamicsandveryspecializedbooks.
v
vi Preface
Inthefirstchapter,wegiveanintroductiontotheliquidstatephysicswithsome
example of phase diagrams. We discuss the conditions for the classical limit; we
describethefeaturesofsometypeoffluidsandthemethodsusedintheexperimental
andtheoreticalstudiesinthefield.
Basicthermodynamicsconceptsarerecalledinthesecondchapterwherethevan
derWaalstheoryisdescribed.Thestatisticalmechanicsensemblesareintroduced.
Inthethirdchapter,weintroducethedistributionfunctionsoftheliquidsandthe
experimental methods used for determining the structure of liquids. Examples of
simpleandmolecularliquidsarepresented.
Thefourthchapterisdevotedtothetheoreticalapproachesbasedontheclassical
densityfunctionaltheory.
In Chap. 5, we present the computer simulation methods, molecular dynamics
and Monte Carlo. We discuss in particular how the phase diagram can be studied
insimulationwiththeuseofdifferentmethodsderivedfromtheso-calledumbrella
sampling. The problem of studying critical phenomena in finite size simulation is
alsointroduced.
Starting from Chap. 6, we consider the dynamics of liquids. In particular
in Chap. 6, the dynamical correlation functions are defined; the linear response
theoryandthefluctuation-dissipationtheoremarepresented.Thedensitycorrelation
functionsareconsideredwiththeexperimentalmethodsusedtostudythem.
InChap.7,wepresentthethermalmotioninliquids.Weconsiderthetheoryof
the Brownian motion, the dynamics of liquids in the hydrodynamics limit and the
visco-elasticregime.Thememoryfunctionsarealsointroduced.
In Chap. 8, we consider the important area of supercooled liquids, the glass
transitionandtherelaxationdynamicsuponsupercooling.Thecomplexbehaviour
of supercooled liquids in approaching the glassy state is still a challenge in the
condensedmatterresearch.Inthisrespectthemodecouplingtheoryintroducedin
thischapterprovidestheonlytheoreticaldetailedinterpretationofthedynamicsof
supercooledliquids.
In the last chapter as a particularly important subject in the research on
supercooled liquids, we consider the behaviour of supercooled water, a topic to
which we gave our contribution. The anomalies of supercooled water are still an
openfieldofresearchforexperimentsandstatisticalmechanics.
WewishtothankthestaffoftheSpringereditorialboardfortheirprecioushelp
duringtheprocessofpublicationofthisbook.
Roma,Italy PaolaGallo
Roma,Italy MauroRovere
November2020
Contents
1 AnIntroductiontotheLiquidStateofMatter ........................... 1
1.1 LiquidStateofMatter................................................. 1
1.1.1 ExamplesofPhaseDiagramofPureSubstances:
CO andWater............................................... 3
2
1.1.2 PhaseDiagramofBinaryMixtures ......................... 5
1.2 Structure and Dynamics of Liquids: Experiments
andCorrelationFunctions............................................. 8
1.3 MicroscopicModelsforLiquids...................................... 9
1.3.1 ClassicalApproximation .................................... 10
1.3.2 DifferentModels............................................. 11
1.4 PotentialEnergyLandscape........................................... 12
1.5 ApproximateTheoriesandComputerSimulation ................... 13
1.6 WaterandHydrogenBond............................................ 14
1.7 MetastableStatesandDisorderedSolidMatter ..................... 16
1.8 SoftMatter............................................................. 16
1.8.1 Colloids....................................................... 17
1.8.2 Biomolecules................................................. 18
References..................................................................... 20
2 ThermodynamicsandStatisticalMechanicsofFluidStates............ 23
2.1 ExtensiveandIntensiveFunctions.................................... 23
2.2 EnergyandEntropy ................................................... 24
2.3 Gibbs-DuhemRelation................................................ 26
2.4 EquilibriumConditions ............................................... 27
2.5 EquilibriumConditionsandIntensiveQuantities ................... 29
2.6 MacroscopicResponseFunctionsandStabilityConditions......... 29
2.7 LegendreTransformsandThermodynamicPotentials............... 31
2.7.1 HelmholtzFreeEnergy ..................................... 32
2.7.2 GibbsFreeEnergy........................................... 33
2.7.3 Enthalpy...................................................... 35
2.7.4 GrandCanonicalPotential................................... 36
2.7.5 TabulatedThermodynamicPotentials....................... 37
vii
viii Contents
2.8 StabilityConditionsforThermodynamicPotentials................. 37
2.9 CoexistenceandPhaseTransitions ................................... 38
2.10 PhaseTransitionsandTheirClassifications.......................... 39
2.11 VanderWaalsEquation............................................... 41
2.12 General Form of the Van der Waals Equation
andCorrespondingStates ............................................. 46
2.13 CriticalBehaviouroftheVanderWaalsEquation................... 47
2.14 EnsemblesinStatisticalMechanics................................... 50
2.14.1 MicrocanonicalEnsemble................................... 51
2.14.2 CanonicalEnsemble ......................................... 52
2.14.3 GrandCanonicalEnsemble.................................. 54
2.14.4 Isobaric-IsothermalEnsemble............................... 56
2.15 FluctuationsandThermodynamics ................................... 57
References..................................................................... 60
3 MicroscopicForcesandStructureofLiquids............................. 61
3.1 ForceFieldforAtomsinLiquids..................................... 61
3.2 LocalStructureofaLiquid............................................ 64
3.3 DistributionFunctionsintheCanonicalEnsemble .................. 65
3.4 RelationoftheRDFwithThermodynamics.......................... 66
3.4.1 Energy........................................................ 66
3.4.2 PressurefromtheVirial...................................... 67
3.5 DistributionFunctionsintheGrandCanonicalEnsemble........... 68
3.6 HierarchicalEquations ................................................ 71
3.7 QualitativeBehaviouroftheRadialDistributionFunction.......... 72
3.8 ExperimentalDeterminationoftheStructureofLiquids............ 73
3.9 NeutronScatteringonLiquids........................................ 75
3.10 StaticLimitandtheStructureofLiquid.............................. 79
3.11 TheStaticStructureFactor............................................ 81
3.12 TheStructureFactorandtheRDFofLiquidArgon ................. 82
3.13 TheStructureFactorClosetoaCriticalPoint........................ 84
3.14 StructureofMulticomponentLiquids ................................ 85
3.14.1 PartialStructureFactorofMulticomponentLiquids....... 86
3.14.2 IsotopicSubstitution......................................... 86
3.14.3 AnExample:MoltenSalts................................... 87
3.15 StructureofMolecularLiquids ....................................... 88
3.15.1 StructureofLiquidWater.................................... 90
References..................................................................... 94
4 TheoreticalStudiesoftheStructureofLiquids .......................... 95
4.1 VirialExpansionintheCanonicalEnsemble......................... 95
4.1.1 FromHardSpherestotheVanderWaalsEquation........ 97
4.2 TheMeanForcePotential............................................. 99
4.3 KirkwoodApproximation............................................. 100
4.4 RadialDistributionFunctionfromtheExcessFreeEnergy......... 101
4.5 DensityDistributionsfromtheGrandPartitionFunction............ 101
4.6 GrandPotentialasGeneratingFunctional............................ 103
Contents ix
4.7 ClassicalDensityFunctionalTheory ................................. 104
4.7.1 EquilibriumConditions...................................... 105
4.7.2 TheOrnstein-ZernikeEquation ............................. 106
4.7.3 TheOrnstein-ZernikeEquationink-Space................. 108
4.7.4 FreeEnergyCalculation..................................... 109
4.7.5 ExpansionfromtheHomogeneousSystem................. 110
4.8 ClosureRelationsfromtheDensityFunctionalTheory ............. 112
4.9 AnExactEquationfortheg(r)....................................... 114
4.10 HNCandPercus-YevickApproximations............................ 115
4.10.1 RPAandMSA................................................ 116
4.11 PropertiesoftheHardSphereFluid .................................. 117
4.12 EquationofStateandLiquid-SolidTransitionofHardSpheres .... 119
4.13 Percus-YevickfortheHardSphereFluid............................. 121
4.14 EquationofStateandThermodynamicInconsistency............... 123
4.15 RoutestoConsistency:ModifiedHNCandReferenceHNC........ 124
4.16 PerturbationTheories:OptimizedRPA............................... 125
4.17 ModelsforColloids ................................................... 126
References..................................................................... 128
5 MethodsofComputerSimulation.......................................... 131
5.1 MolecularDynamicsMethods........................................ 132
5.1.1 MolecularDynamicsandStatisticalMechanics............ 132
5.1.2 AlgorithmsfortheTimeEvolution ......................... 134
5.1.3 Predictor/Corrector........................................... 134
5.1.4 VerletAlgorithms............................................ 135
5.1.5 CalculationoftheForces ................................... 138
5.1.6 InitialConfiguration ......................................... 139
5.1.7 TemperatureintheMicrocanonicalEnsemble ............. 139
5.1.8 EquilibrationProcedure...................................... 140
5.1.9 ThermodynamicandStructure .............................. 143
5.1.10 Long-RangeCorrections..................................... 143
5.1.11 EwaldMethod................................................ 144
5.2 MonteCarloSimulation............................................... 149
5.2.1 MonteCarloIntegrationandImportanceSampling........ 149
5.2.2 IntegralsinStatisticalMechanics ........................... 150
5.2.3 ImportanceSamplinginStatisticalMechanics............. 151
5.2.4 MarkovProcesses............................................ 152
5.2.5 ErgodicityandDetailedBalance............................ 153
5.2.6 MetropolisMethod........................................... 154
5.2.7 AveragingonMonteCarloSteps............................ 156
5.2.8 MCSamplinginOtherEnsembles.......................... 157
5.2.9 MCintheGibbsEnsemble.................................. 160
5.3 MDinDifferentEnsembles........................................... 160
5.3.1 Controlling the Temperature: MD in the
CanonicalEnsemble ......................................... 161
5.3.2 PressureControl.............................................. 166
