Table Of ContentChapter 1
Introduction
1.1 THE LIQUID STATE
Theliquidstateofmatterisintuitivelyperceivedasonethatisintermediatein
nature between a gas and a solid. Given that point of view, a natural starting
point for discussion of the properties of a given substance is the relationship
betweenpressureP,numberdensityρandtemperatureT initsdifferentphases,
summarised in the equation of state f(P,ρ,T) = 0. The phase diagram in
the density-temperature plane typical of a simple, one-component system is
sketchedinFigure1.1.Theregionofexistenceoftheliquidphaseisbounded
abovebythecriticalpoint(subscriptc)andbelowbythetriplepoint(subscript
t).Abovethecriticalpointthereisonlyasinglefluidphase,soacontinuouspath
existsfromliquidtofluidtovapour.Thisisnottrueofthetransitionfromliquid
to solid because the solid-fluid coexistence line (the melting curve) does not
endatacriticalpoint.Inmanyrespectsthepropertiesofthedense,supercritical
fluidarenotverydifferentfromthoseoftheliquidandmuchofthetheorywe
developinlaterchaptersappliesequallywelltothetwocases.
We shall be concerned in this book almost exclusively with classical
liquids,thatistosaywithliquidsthatcantoagoodapproximationbetreated
theoretically by the methods of classical statistical mechanics. A simple test
oftheclassicalhypothesisisprovidedbythevalueofthedeBrogliethermal
wavelengthΛ,definedforaparticleofmassm as
(cid:2) (cid:3)
2πβ(cid:2)2 1/2
Λ= (1.1.1)
m
with β = 1/k T, where k is the Boltzmann constant. To justify a classical
B B
treatmentofstaticpropertiesΛmustbemuchsmallerthana,wherea ≈ρ−1/3
isthemeannearest-neighbourseparation.Someresultsforavarietyofatomic
andsimplemolecularliquidsareshowninTable1.1;hydrogenandneonapart,
quantumeffectsshouldbesmallforallthesystemslisted.Inthecaseoftime-
dependentprocessesitisnecessaryinadditionthatthetimescaleinvolvedbe
muchlongerthanβ(cid:2),whichatroomtemperature,forexample,meansfortimes
TheoryofSimpleLiquids,FourthEdition.http://dx.doi.org/10.1016/B978-0-12-387032-2.00001-5
©2013,2006,1990ElsevierLtd.Allrightsreserved. 1
2 TheoryofSimpleLiquids
FIGURE1.1 Schematicphasediagramofatypicalmonatomicsubstance,showingtheboundaries
betweensolid(S),liquid(L)andvapour(V)orfluid(F)phases.
(cid:2) (cid:4)
TABLE1.1 Testoftheclassicalhypothesis.
Liquid Tt(K) Λ(Å) Λ/a
H2 14.1 3.3 0.97
Ne 24.5 0.78 0.26
CH4 91 0.46 0.12
N2 63 0.42 0.11
Li 454 0.31 0.11
Ar 84 0.30 0.083
HCl 159 0.23 0.063
Na 371 0.19 0.054
Kr 116 0.18 0.046
CCl4 250 0.09 0.017
(cid:3) (cid:5)
t (cid:3)10−14 s.Thissecondconditionissomewhatmorerestrictivethanthefirst,
butwheretranslational motionisconcernedtheproblemisagainsevereonly
inextremecasessuchashydrogen.
Use of the classical approximation leads to an important simplification
insofarasthecontributionstothermodynamicpropertiesarisingfromthermal
motion can be separated from those due to interactions between particles.
CHAPTER | 1 Introduction 3
The separation of kinetic and potential terms suggests a simple means of
characterisingtheliquidstate.LetV bethetotalpotentialenergyofasystem,
N
where N is the number of particles, and let K be the total kinetic energy.
N
Then in the liquid state we find that K /|V | ≈ 1, whereas K /|V | (cid:3) 1
N N N N
correspondstothedilutegasand K /|V | (cid:4) 1tothelow-temperaturesolid.
N N
Alternatively,ifwecharacteriseagivensystembyalengthσ andanenergy(cid:7),
correspondingroughlytotherangeandstrengthoftheintermolecularforces,we
findthatintheliquidregionofthephasediagramthereducednumberdensity
ρ∗ = Nσ3/V,where V isthevolume,andreducedtemperature T∗=k T/(cid:7)
B
are both of order unity. Liquids and dense fluids are also distinguished from
dilutegasesbythegreaterimportanceofcollisionalprocessesandshort-range,
positionalcorrelations,andfromcrystallinesolidsbytheabsenceofthelong-
rangeorderassociatedwithaperiodiclattice;theirstructureisinmanycases
dominatedbythe‘excludedvolume’effectassociatedwiththepackingtogether
ofparticleswithhardcores.
Selected properties of a simple monatomic liquid (argon), a simple
molecular liquid (nitrogen) and a simple liquid metal (sodium) are listed in
Table 1.2. Not unexpectedly, the properties of the liquid metal are in certain
respects very different from those of the other systems, notably in the values
ofthethermalconductivity,isothermalcompressibility,surfacetension,heatof
vaporisationandtheratioofcriticaltotriple-pointtemperatures;thesourceof
(cid:2) (cid:4)
TABLE1.2 Selectedpropertiesoftypicalsimpleliquids.
Property Ar Na N2
Tt/K 84 371 63
T /K(P =1atm) 87 1155 77
b
Tc/K 151 2600 126
Tc/Tt 1.8 7.0 2.0
ρt/nm−3 21 24 19
cP/cV 2.2 1.1 1.6
Lvap/kJmol−1 6.5 99 5.6
χT/10−12cm2dyn−1 200 19 180
c/ms−1 863 2250 995
γ/dyncm−1 13 191 12
D/10−5cm2s−1 1.6 4.3 1.0
η/mgcm−1s−1 2.8 7.0 3.8
λ/mWcm−1K−1 1.3 8800 1.6
(kBT/2πDη)/Å 4.1 2.7 3.6
χT =isothermalcompressibility,c=speedofsound,γ=surfacetension,
D=self-diffusioncoefficient,η=shearviscosityandλ=thermalconductivity,allatT=Tt;
(cid:3)Lvap=heatofvaporisationatT=Tb. (cid:5)
4 TheoryofSimpleLiquids
thesedifferenceswillbecomeclearinChapter10.Thequantityk T/2πDηin
B
thetableprovidesaStokes-lawestimateoftheparticlediameter.
1.2 INTERMOLECULAR FORCESAND MODEL POTENTIALS
Themostimportantfeatureofthepairpotentialbetweenatomsormoleculesis
theharshrepulsionthatappearsatshortrangeandhasitsoriginintheoverlap
of the outer electron shells. The effect of these strongly repulsive forces is
tocreatetheshort-rangeordercharacteristicoftheliquidstate.Theattractive
forces, which act at long range, vary much more smoothly with the distance
betweenparticlesandplayonlyaminorroleindeterminingthestructureofthe
liquid.Theyprovide,instead,anessentiallyuniform,attractivebackgroundthat
givesrisetothecohesiveenergyrequiredtostabilisetheliquid.Thisseparation
oftheeffectsofrepulsiveandattractiveforcesisaveryold-establishedconcept.
ItliesattheheartoftheideasofvanderWaals,whichinturnformthebasisofthe
verysuccessfulperturbationtheoriesoftheliquidstatediscussedinChapter5.
Thesimplestmodelofafluidisasystemofhardspheres,forwhichthepair
potentialv(r)ataseparationr is
v(r) =∞, r <d
=0, r >d (1.2.1)
whered isthehard-spherediameter.Thissimplepotentialisideallysuitedto
thestudyofphenomenainwhichthehardcoreofthepotentialisthedominant
factor.Muchofourunderstandingofthepropertiesofthehard-spheremodel
comesfromcomputersimulations.Suchcalculationshaverevealedveryclearly
thatthestructureofahard-spherefluiddoesnotdifferinanysignificantway
from that corresponding to more complicated interatomic potentials, at least
under conditions close to crystallisation. The model also has some relevance
to real, physical systems. For example, the osmotic equation of state of a
suspensionofmicron-sizedsilicaspheresinanorganicsolventmatchesalmost
exactlythatofahard-spherefluid.1 However,althoughsimulationsshowthat
thehard-spherefluidundergoesafreezingtransitionatρ∗(=ρd3)≈0.945,the
absenceofattractiveforcesmeansthatthereisonlyonefluidphase.Amodel
that can describe a true liquid is obtained by supplementing the hard-sphere
potential with a square-well attraction, as illustrated in the left-hand panel of
Figure1.2.Thisintroducestwoadditionalparameters,(cid:7)andγ;(cid:7)isthedepthof
thewelland(γ −1)d isthewidth,whereγ typicallyhasavalueofabout1.5.
An alternative to the square-well potential with features that are of particular
interesttheoreticallyisthehard-coreYukawapotential,givenby
v(r)=∞, r <d
(cid:7)d
=− exp[−λ(r/d−1)], r >d (1.2.2)
r
CHAPTER | 1 Introduction 5
FIGURE1.2 Simplepotentialmodelsformonatomicsystems.Seetextfordetails.
where the parameter λ measures the inverse range of the attractive tail in the
potential. The two examples plotted in the right-hand panel of the figure are
drawn for values of λ appropriate either to the interaction between rare-gas
atoms(λ = 2)ortotheshort-range,attractiveforces2 characteristicofcertain
colloidalsystems(λ= 8).Thelimitinwhichtherangeoftheattractiontends
tozerowhilst the well depth goestoinfinity corresponds toa ‘sticky sphere’
model, an early version of which was introduced by Baxter.3 Models of this
type have proved useful in studies of the clustering of colloidal particles and
theformationofgels.
Amorerealisticpotentialforneutralatomscanbeconstructedbyadetailed
quantum-mechanicalcalculation.Atlargeseparationsthedominantcontribu-
tiontothepotentialcomesfromthemultipolardispersioninteractionsbetween
theinstantaneouselectricmomentsononeatom,createdbyspontaneousfluc-
tuationsintheelectronicchargedistribution,andmomentsinducedintheother.
Alltermsinthemultipoleseriesrepresentattractivecontributionstothepoten-
tial.Theleadingterm,varyingasr−6,describesthedipole-dipoleinteraction.
Higher-ordertermsrepresentdipole-quadrupole(r−8),quadrupole-quadrupole
(r−10)interactions,andsoon,butthesearegenerallysmallincomparisonwith
theterminr−6.
Arigorouscalculationoftheshort-rangeinteractionpresentsgreaterdiffi-
culty,butoverrelativelysmallrangesofr itcanbeadequatelyrepresentedby
anexponentialfunctionoftheformexp(−r/r ),wherer isarangeparameter.
0 0
Thisapproximationmustbesupplementedbyrequiringthatv(r) → ∞forr
lessthansomearbitrarilychosen,smallvalue.Inpractice,largelyforreasonsof
mathematicalconvenience,itismoreusualtorepresenttheshort-rangerepul-
sionbyaninverse-powerlaw,i.e.r−n,whereforclosed-shellatomsnliesinthe
rangefromabout9to15.Thebehaviourofv(r)inthelimitingcasesr → ∞
6 TheoryofSimpleLiquids
andr →0maythereforebeincorporatedinapotentialfunctionoftheform
(cid:4) (cid:5)
v(r)=4(cid:7) (σ/r)12−(σ/r)6 (1.2.3)
whichisthefamous12–6potentialofLennard-Jones.Equation(1.2.3)involves
two parameters: the collision diameter σ, which is the separation of the par-
ticleswherev(r) = 0;and(cid:7),thedepthofthepotential wellattheminimum
in v(r). The Lennard-Jones potential provides a fair description of the inter-
actionbetweenpairsofrare-gasatomsandofquasi-sphericalmoleculessuch
as methane. Computer simulations4 have shown that the triple point of the
Lennard-Jonesfluidisatρ∗ ≈0.85,T∗ ≈0.68.
Experimental information on the pair interaction can be extracted from a
studyofanyphenomenonthatinvolvescollisionsbetweenparticles.Themost
directmethodinvolvesthemeasurementofatom-atomscatteringcross-sections
as a function of incident energy and scattering angle; inversion of the data
allows,inprinciple,adeterminationofthepairpotentialatallseparations.A
simplerprocedureistoassumeaspecificformforthepotentialanddetermine
theparametersbyfittingtotheresultsofgasphasemeasurementsofquantities
suchasthesecondvirialcoefficient(seeChapter3)orshearviscosity.5 Inthis
way,forexample,theparameters(cid:7) andσ intheLennard-Jonespotentialhave
beendeterminedforalargenumberofgases.
Thetheoreticalandexperimentalmethodswehavementionedallrelateto
thepropertiesofanisolatedpairofmolecules.Useoftheresultingpairpotentials
incalculationsfortheliquidstateinvolvestheneglectofmany-bodyforces,an
approximationthatisdifficulttojustify.Intherare-gasliquidsthethree-body,
triple-dipoledispersiontermisthemostimportantmany-bodyinteraction;the
net effect of triple-dipole forces is repulsive, amounting in the case of liquid
argontoasmallpercentageofthetotalpotentialenergyduetopairinteractions.
Moreover,carefulmeasurements,particularlythoseofsecondvirialcoefficients
atlowtemperatures,haveshownthatthetruepairpotentialforrare-gasatoms6
isnotoftheLennard-Jonesform,buthasadeeperbowlandaweakertail,as
illustrated by the curves plotted in Figure 1.3. Apparently the success of the
Lennard-Jonespotentialinaccountingformanyofthemacroscopicproperties
ofargon-likeliquidsistheconsequenceofafortuitouscancellationoferrors.
A number of more accurate pair potentials have been developed for the rare
gases,buttheiruseinthecalculationofpropertiestheliquidorsolidrequires
theexplicitincorporationofthree-bodyinteractions.
Although the true pair potential for rare-gas atoms is not the same as the
effectivepairpotentialusedinliquidstatetheory,thedifferenceisarelatively
minor, quantitative one. The situation in the case of liquid metals is different
becausetheformoftheeffectiveion-ioninteractionisstronglyinfluencedbythe
presenceofadegenerategasofconductionelectronsthatdoesnotexistbefore
theliquidisformed.Thecalculationoftheion-ioninteractionisacomplicated
problem, as we shall see in Chapter 10. The ion-electron interaction is first
CHAPTER | 1 Introduction 7
FIGURE1.3 Pairpotentialsforargonintemperatureunits.Fullcurve:theLennard-Jonespotential
withparametervalues(cid:7)/kB=120K,σ =3.4Å,whichisagoodeffectivepotentialfortheliquid;
dashes:apotentialbasedongasphasedata.7
describedintermsofa‘pseudopotential’thatincorporatesboththecoulombic
attraction and the repulsion due to the Pauli exclusion principle. Account
must then be taken of the way in which the pseudopotential is modified by
interactionbetweentheconductionelectrons.Theendresultisapotentialwhich
representstheinteractionbetweenscreened,electricallyneutral‘pseudoatoms’.
Irrespectiveofthedetailedassumptionsmade,themainfeaturesofthepotential
arealwaysthesame:asoftrepulsion,adeepattractivewellandalong-range
oscillatorytail.Thepotential,andinparticularthedepthofthewell,arestrongly
densitydependentbutonlyweaklydependentontemperature.Figure1.4shows
aneffectivepotentialforliquidpotassium.Thedifferencescomparedwiththe
potentialsforargonareclear,bothatlongrangeandinthecoreregion.
Formoltensaltsandotherionicliquidsinwhichthereisnoshieldingofthe
electrostaticforcesofthetypefoundinliquidmetals,thecoulombicinteraction
provides the dominant contribution to the interionic potential. There must, in
addition,beashort-rangerepulsionbetweenionsofoppositecharge,without
whichthesystemwouldcollapse,butthedetailedwayinwhichtherepulsive
forcesaretreatedisofminorimportance.Polarisationoftheionsbytheinternal
electric field also plays a role, but such effects are essentially many body in
natureandcannotbeadequatelyrepresentedbyanadditionalterminthepair
potential.
Description of the interaction between two molecules poses greater
problems than that between spherical particles because the pair potential is
afunctionofboththeseparationofthemoleculesandtheirmutualorientation.
8 TheoryofSimpleLiquids
FIGURE1.4 Mainfigure:effectiveion-ionpotential(intemperatureunits)forliquidpotassium
athighdensity.8Inset:comparisononalogarithmicscaleofpotentialsforargonandpotassiumin
thecoreregion.
Themodelpotentialsdiscussedinthisbookmostlyfallintooneoftwoclasses.
Thefirstconsistsofidealisedmodelsofpolarliquidsinwhichapointdipole-
dipoleinteractionissuperimposedonasphericallysymmetricpotential.Inthis
casethepairpotentialforparticleslabelled1and2hasthegeneralform
v(1,2)=v0(R)−µ1·T(R)·µ2 (1.2.4)
where R is the vector separation of the molecular centres, v (R) is the
0
spherically symmetric term, µ is the dipole moment vector of particle i and
i
T(R)isthedipole-dipoleinteractiontensor:
T(R)=3RR/R5−I/R3 (1.2.5)
whereIistheunittensor.
Twoexamplesof(1.2.4)thatareofparticularinterestarethoseofdipolar
hard spheres, where v (R) is the hard-sphere potential, and the Stockmayer
0
potential, where v (R) takes the Lennard-Jones form. Both these models,
0
together with extensions that include, for example, dipole-quadrupole and
quadrupole-quadrupoleterms,havereceivedmuchattentionfromtheoreticians.
Theirmainlimitationasmodelsofrealmoleculesisthefactthattheyignorethe
anisotropyoftheshort-rangeforces.Onewaytotakeaccountofsucheffects
is through the use of potentials of the second main type with which we shall
CHAPTER | 1 Introduction 9
beconcerned.Thesearemodelsinwhichthemoleculeisrepresentedbyaset
of discrete interaction sites that are commonly, but not invariably, located at
thesitesoftheatomicnuclei.Thetotalpotentialenergyoftwointeraction-site
molecules is then obtained as the sum of spherically symmetric, interaction-
sitepotentials.Letriα bethecoordinatesofsiteαinmoleculei andletrjβ be
thecoordinatesofsiteβ inmolecule j.Thenthetotalintermolecularpotential
energyis
(cid:6)(cid:6)
1
v(1,2)= vαβ(|r2β −r1α|) (1.2.6)
2
α β
where vαβ(r) is a site-site potential and the sums on α and β run over all
interactionsitesintherespectivemolecules.Electrostaticinteractionsareeasily
allowedforbyinclusionofcoulombictermsinthesite-sitepotentials.
Letustakeasanexampleoftheinteraction-siteapproachthesimplecaseof
ahomonucleardiatomic,suchasthatpicturedinFigure1.5.Acrudeinteraction-
sitemodelwouldbethatofa‘harddumb-bell’,consistingoftwooverlapping
hardspheresofdiameterd withtheircentresseparatedbyadistance L < 2d.
Thisshouldbeadequatetodescribethemainstructuralfeaturesofaliquidsuch
as nitrogen. An obvious improvement would be to replace the hard spheres
by two Lennard-Jones interaction sites, with potential parameters chosen to
fit, say, the experimentally determined equation of state. Some homonuclear
diatomicsalsohavealargequadrupolemoment,whichcanplayasignificant
roleindeterminingtheshort-rangeangularcorrelationsintheliquid.Themodel
couldinthatcasebefurtherrefinedbyplacingpointchargesq attheLennard-
Jonessites,togetherwithacompensatingcharge−2q atthemid-pointofthe
internuclear bond; such a charge distribution has zero dipole moment but a
non-vanishingquadrupolemomentproportionaltoqL2.Modelsofthisgeneral
typehaveprovedremarkablysuccessfulindescribingthepropertiesofawide
varietyofmolecularliquids,bothsimpleandcomplicated.
FIGURE1.5 Aninteraction-sitemodelofahomonucleardiatomic.
10 TheoryofSimpleLiquids
1.3 EXPERIMENTAL METHODS
Theexperimentalmethodsavailableforstudyingthepropertiesofsimpleliq-
uidsfallintooneoftwobroadcategories,dependingonwhethertheyarecon-
cernedwithmeasurementsonthemacroscopicormicroscopicscale.Ingeneral,
valuesobtainedtheoreticallyformicroscopicpropertiesaremoresensitiveto
theapproximationsmadeandtheassumedformoftheinterparticlepotentials,
butmacroscopicpropertiescanusuallybemeasuredwithconsiderablygreater
accuracy.Thetwoclassesofexperimentarethereforecomplementary,eachpro-
vidinginformationthatisusefulinthedevelopmentofastatisticalmechanical
theoryoftheliquidstate.
The classic macroscopic measurements are those of thermodynamic
properties,particularlyoftheequationofstate.Integrationofaccurate P-ρ-T
data yields information on other thermodynamic quantities, which can be
supplemented by calorimetric measurements. For most liquids the pressure
is known as a function of temperature and density only in the vicinity of the
liquid-vapourequilibriumline,butforcertainsystemsofparticulartheoretical
interest experiments have been carried out at much higher pressures; the low
compressibility of a liquid near its triple point means that highly specialised
techniquesarerequired.
The second main class of macroscopic measurements are those relating
to transport coefficients. A variety of experimental methods are used. The
shear viscosity, for example, can be determined from the observed damping
oftorsionaloscillationsorfromcapillaryflowexperiments,whilstthethermal
conductivitycanbeobtainedfromasteady-statemeasurementofthetransferof
heatbetweenacentralfilamentandasurroundingcylinderorbetweenparallel
plates.Adirectmethodofdeterminingthecoefficientofself-diffusioninvolves
the use of radioactive tracers, which places it in the category of microscopic
measurements; in favourable cases the diffusion coefficient can be measured
bynuclearmagneticresonance(NMR).NMRandotherspectroscopicmethods
(infraredandRaman)arealsousefulinthestudyofreorientationalmotionin
molecularliquids,whilstdielectricresponsemeasurementsprovideinformation
ontheslow,structuralrelaxationinsupercooledliquidsneartheglasstransition.
Much the most important class of microscopic measurements, at least
from the theoretical point of view, are the radiation scattering experiments.
ElasticscatteringofneutronsorX-rays,inwhichthescatteringcross-section
is measured as a function of momentum transfer between the radiation and
thesample,isthesourceofourexperimentalknowledgeofthestaticstructure
ofafluid.Inthecaseofinelasticscatteringthecross-sectionismeasuredasa
functionofbothmomentumandenergytransfer.Itistherebypossibletoextract
informationonwavenumberandfrequency-dependentfluctuationsinliquidsat
wavelengths comparable with the spacing between particles. This provides a
very powerful method of studying microscopic time-dependent processes in
liquids.Inelasticlightscatteringexperimentsprovidesimilarinformation,but
