Theory of simple liquids

419 Pages·2006·2.499 MB·English

by  Jean-Pierre Hansen, I.R. McDonald

Preface to the Third Edition
Atthetimewhenthesecondeditionofthisbookwaspublishedthestudyoftheliquidstate
was a rapidlyexpandingfieldof research. In the twentyyears since then,the subjecthas
maturedboththeoreticallyandexperimentallytoapointwherearealunderstandingexists
of the behaviour of “simple” liquids at the microscopic level. Although there has been a
shift in emphasis towards more complex systems, there remains in our view a place for
a book that deals with the principles of liquid-state theory, covering both statics and dy-
namics.Thus,inpreparingathirdedition,wehaveresistedthetemptationtobroadentoo
far the scope of the book, and the focus remains firmly on simple systems, though many
of the methods we describe continue to find a wider range of application. Nonetheless,
somereorganisationofthebookhasbeenrequiredinordertogiveproperweighttomore
recent developments. The most obvious change is in the space devoted to the theory of
inhomogeneousfluids,anareainwhichconsiderableprogresshasbeenmadesince1986.
Othermajoradditionsaresectionsonthepropertiesofsupercooledliquids,whichinclude
adiscussionofthemode-couplingtheoryofthekineticglasstransition,ontheoriesofcon-
densation and freezing and on the electric double layer. To make way for this and other
newmaterial,somesectionsfromthesecondeditionhaveeitherbeenreducedinlengthor
omittedaltogether.Inparticular,wenolongerseeaneedtoincludeacompletechapteron
molecularsimulation,thepublicationofseveralexcellenttextsonthesubjecthavingfilled
what was previously a serious gap in the literature. Our aim has been to emphasise what
seemstoustobeworkoflastinginterest.Suchjudgementsareinevitablysomewhatsub-
jectiveand,asbefore,thechoiceoftopicsiscolouredbyourownexperienceandtastes.We
makenoattempttoprovideanexhaustivelistofreferences,limitingourselvestowhatwe
considertobethefundamentalpapersindifferentareas,alongwithselectedapplications.
We are grateful to a number of colleagues who have helped us in different ways: Dor
Ben-Amotz, Teresa Head-Gordon, David Heyes, David Grier, Bill Jorgensen, Gerhard
Kahl,PeterMonson,AnnaOleksy,AlbertReiner,PhilSalmon,IljaSiepmann,AlanSoper,
GeorgeStellandJens-BoieSuck.BobEvansmademanyhelpfulsuggestionsconcerning
themuchrevisedchapteronionicliquids,GeorgeJacksonactedasourguidetothelitera-
tureonthetheoryofassociatingliquids,AlbertoParolaprovidedavaluablesetofnoteson
hierarchicalreferencetheory,andJean-JacquesWeisundertookonourbehalfnewMonte
Carlo calculations of the dielectric constant of dipolar hard spheres. Our task could not
have been completed without the support, encouragement and advice of these and other
colleagues,toallofwhomwegiveourthanks.Finally,wethanktherespectivepublishers
for permission to reproduce figures from Journal of Chemical Physics, Journal of Non-
CrystallineSolids,PhysicalReviewandPhysicalReviewLetters.
November2005 J.P.HANSEN
I.R.MCDONALD
v
Preface to the Second Edition
The first edition of this book was written in the wake of an unprecedented advance in
ourunderstandingofthemicroscopicstructureanddynamicsofsimpleliquids.Therapid
progress made in a number of different experimental and theoretical areas had led to a
rather clear and complete picture of the properties of simple atomic liquids. In the ten
yearsthathavepassedsincethen,interestintheliquidstatehasremainedveryactive,and
the methods described in our book have been successfully generalised and applied to a
varietyofmorecomplicatedsystems.Importantdevelopmentshavethereforebeenseenin
thetheoryofionic,molecularandpolarliquids,ofliquidmetals,andoftheliquidsurface,
whilethequantitativereliabilityoftheoriesofatomicfluidshasalsoimproved.
Inanattempttogiveabalancedaccountbothofthebasictheoryandoftheadvancesof
thepastdecade,thisneweditionhasbeenrearrangedandconsiderablyexpandedrelativeto
theearlierone.Everychapterhasbeencompletelyrewritten,andthreenewchaptershave
beenadded,devotedtoionic,metallicandmolecularliquids,togetherwithsubstantialnew
sectionsonthetheoryofinhomogeneousfluids.ThematerialcontainedinChapter10of
thefirstedition,whichdealtwithphasetransitions,hasbeenomitted,sinceitprovedim-
possible to do justice to such a large field in the limited space available. Although many
excellentreviewarticlesandmonographshaveappearedinrecentyears,acomprehensive
and up-to-date treatment of the theory of “simple” liquids appears to be lacking, and we
hope that the new edition of our book will fill this gap. The choice of material again re-
flectsourowntastes,butwehaveaimedatpresentingthemainideasofmodernliquid-state
theoryinawaythatisbothpedagogicaland,sofaraspossible,self-contained.Thebook
shouldbeaccessibletograduatestudentsandresearchworkers,bothexperimentalistsand
theorists, who have a good background in elementary statistical mechanics. We are well
aware,however,thatcertainsections,notablyinChapters4,6,9and12requiremorecon-
centrationfromthereaderthanothers.Althoughthebookisnotintendedtobeexhaustive,
we give many references to material that is not covered in depth in the text. Even at this
level,itisimpossibletoincludealltherelevantwork.Omissionsmayreflectourignorance
oralackofgoodjudgement,butweconsiderthatourgoalwillhavebeenachievedifthe
bookservesasanintroductionandguidetoacontinuouslygrowingfield.
Whilepreparingthenewedition,wehavebenefitedfromtheadvice,criticismandhelp
of many colleagues. We give our sincere thanks to all. There are, alas, too many names
to listindividually,butwewishtoacknowledgeourparticulardebtto MarcBaus, David
Chandler,GiovanniCiccotti,BobEvans,PaulMaddenandDominicTildesley,whohave
readlargepartsofthemanuscript;toSusanO’Gorman,forherhelpwithChapter4;and
to Eduardo Waisman, who wrote the first (and almost final) version of Appendix B. We
vi
PREFACETOTHESECONDEDITION vii
arealsogratefultothosecolleagueswhohavesuppliedreferences,preprints,andmaterial
forfiguresandtables,andtoauthorsandpublishersforpermissiontoreproducediagrams
from published papers. The last stages of the work were carried out at the Institut Laue-
Langevin in Grenoble, and we thank Philippe Nozières for the invitations that made our
visits possible. Finally, we are greatly indebted to Martine Hansen, Christiane Lanceron,
RehdaMazighiandSusanO’Gormanfortheirhelpandpatienceinthepreparationofthe
manuscriptandfigures.
May1986 J.P.HANSEN
I.R.MCDONALD
Preface to the First Edition
Thepasttenyearsorsohaveseenaremarkablegrowthinourunderstandingofthestatisti-
calmechanicsofsimpleliquids.Manyoftheseadvanceshavenotyetbeentreatedfullyin
anybookandthepresentworkisaimedatfillingthisgapatalevelsimilartothatofEgel-
staff’s “The Liquid State”, though with a greater emphasis on theoretical developments.
We discuss both static and dynamic properties, but no attempt is made at completeness
and the choice of topics naturally reflects our own interests. The emphasis throughout is
placed on theories which have been brought to a stage at which numerical comparison
with experiment can be made. We have attempted to make the book as self-contained as
possible,assumingonlyaknowledgeofstatisticalmechanicsatafinal-yearundergraduate
level.Wehavealsoincludedalargenumberofreferencestoworkwhichlackofspacehas
prevented us from discussing in detail. Our hope is that the book will prove useful to all
thoseinterestedinthephysicsandchemistryofliquids.
Ourthanksgotomanyfriendsfortheirhelpandencouragement.Wewish,inparticular,
to express our gratitude to Loup Verlet for allowing us to make unlimited use of his un-
published lecture notes on the theory of liquids. He, together with Dominique Levesque,
KonradSingerandGeorgeStell,havereadseveralpartsofthemanuscriptandmadesug-
gestions for its improvement. We are also greatly indebted to Jean-Jacques Weis for his
help with the section on molecular liquids. The work was completed during a summer
spentasvisitorstotheChemistryDivisionoftheNationalResearchCouncilofCanada;it
is a pleasure to have this opportunity to thank Mike Klein for his hospitality at that time
andformakingthevisitpossible.ThanksgofinallytoSusanO’Gormanforherhelpwith
mathematicalproblemsandforcheckingthereferences;toJohnCopley,JanSengersand
Sidney Yip for sending us useful material; and to Mrs K.L. Hales for so patiently typing
themanydrafts.
Anumberoffiguresandtableshavebeenreproduced,withpermission,fromThePhys-
ical Review, Journal of Chemical Physics, Molecular Physics and Physica; detailed ac-
knowledgementsaremadeatappropriatepointsinthetext.
June1976 J.P.HANSEN
I.R.MCDONALD
viii
CHAPTER 1
Introduction
1.1 THELIQUIDSTATE
Theliquidstateofmatterisintuitivelyperceivedasbeingintermediateinnaturebetween
agasandasolid.Thusanaturalstartingpointfordiscussionofthepropertiesofanygiven
substance is the relationship between pressure P, number density ρ and temperature T
in the different phases, summarised in the equation of state f(P,ρ,T)=0. The phase
diagraminthe ρ–T planetypicalofasimple,one-componentsystemissketchedinFig-
ure1.1.Theregionofexistenceoftheliquidphaseisboundedabovebythecriticalpoint
(subscript c) and below by the triple point (subscript t). Above the critical point there is
onlyasinglefluidphase,soacontinuouspathexistsfromliquidtofluidtovapour;thisis
nottrueofthetransitionfromliquidtosolid,becausethesolid–fluidcoexistenceline,or
meltingcurve,doesnotterminateatacriticalpoint.Inmanyrespectsthepropertiesofthe
dense, supercritical fluid are not very different from those of the liquid, and much of the
theorywedevelopinlaterchaptersappliesequallywelltothetwocases.
Weshallbeconcernedinthisbookalmostexclusivelywithclassicalliquids.Foratomic
systemsasimpletestoftheclassicalhypothesisisprovidedbythevalueofthedeBroglie
thermalwavelengthΛ,definedas
(cid:2) (cid:3)
2πβh¯2 1/2
Λ= (1.1.1)
m
wheremisthemassofanatomandβ=1/k T.Tojustifyaclassicaltreatmentofstatic
B
propertiesitisnecessarythatΛbemuchlessthana,wherea≈ρ−1/3isthemeannearest-
neighbour separation. In the case of molecules we require, in addition, that Θ (cid:5)T,
rot
where Θ =h¯2/2Ik is a characteristic rotational temperature (I is the molecular mo-
rot B
ment of inertia). Some typical results are shown in Table 1.1, from which we see that
quantumeffectsshouldbesmallforallthesystemslisted,withtheexceptionsofhydrogen
andneon.
Useoftheclassicalapproximationleadstoanimportantsimplification,namelythatthe
contributionstothermodynamicpropertieswhicharisefromthermalmotioncanbesepa-
rated from those due to interactions between particles. The separation of kinetic and po-
tentialtermssuggestsasimplemeansofcharacterisingtheliquidstate.LetV bethetotal
N
potentialenergyofasystem,whereN isthenumberofparticles,andletK bethetotal
N
1
2 INTRODUCTION
F
critical point
e
r
u T
t c
a
re V L S
p
m
e
t
T
t
triple point
c t
density
FIG.1.1. Schematicphasediagramofatypicalmonatomicsubstance,showingtheboundariesbetweensolid(S),
liquid(L)andvapour(V)orfluid(F)phases.
TABLE1.1. Testoftheclassicalhypothesis
Liquid Tt/K Λ/Å Λ/a Θrot/Tt
H2 14.1 3.3 0.97 6.1
Ne 24.5 0.78 0.26
CH4 91 0.46 0.12 0.083
N2 63 0.42 0.11 0.046
Li 454 0.31 0.11
Ar 84 0.30 0.083
HCl 159 0.23 0.063 0.094
Na 371 0.19 0.054
Kr 116 0.18 0.046
CCl4 250 0.09 0.017 0.001
ΛisthedeBrogliethermalwavelengthatT =Ttanda=(V/N)1/3.
kineticenergy.ThenintheliquidstatewefindthatK /|V |≈1,whereasK /|V |(cid:6)1
N N N N
correspondstothedilutegasandK /|V |(cid:5)1tothelow-temperaturesolid.Alternatively,
N N
if we characterise a given system by a length σ and an energy ε, corresponding roughly
to the range and strength of the intermolecular forces, we find that in the liquid region
ofthephasediagramthereducednumberdensityρ∗=Nσ3/V andreducedtemperature
T∗=k T/ε arebothoforderunity.Liquidsanddensefluidsarealsodistinguishedfrom
B
dilutegasesbythegreaterimportanceofcollisionalprocessesandshort-range,positional
correlations, and from solids by the lack of long-range order; their structure is in many
INTERMOLECULARFORCESANDMODELPOTENTIALS 3
TABLE1.2. Selectedpropertiesoftypicalsimpleliquids
Property Ar Na N2
Tt/K 84 371 63
Tb/K(P =1atm) 87 1155 77
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 =isothermal compressibility, c=speed of sound, γ =surface tension, D=self-diffusion
coefficient,η=shearviscosityandλ=thermalconductivity,allatT =Tt;Lvap=heatofvapor-
isationatT =Tb.
casesdominatedbythe“excluded-volume”effectassociatedwiththepackingtogetherof
particleswithhardcores.
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
othersystems,notablyinthevaluesofthethermalconductivity,isothermalcompressibility,
surfacetension,heatofvaporisationandtheratioofcriticaltotriple-pointtemperatures;the
source of these differences should become clear in Chapter 10. The quantity k T/2πDη
B
inthetableprovidesaStokes-lawestimateoftheparticlediameter.
1.2 INTERMOLECULARFORCESANDMODELPOTENTIALS
Themostimportantfeatureofthepairpotentialbetweenatomsormoleculesistheharsh
repulsionthatappearsatshortrangeandhasitsoriginintheoverlapoftheouterelectron
shells.Theeffectofthesestronglyrepulsiveforcesistocreatetheshort-rangeorderthatis
characteristicoftheliquidstate.Theattractiveforces,whichactatlongrange,varymuch
more smoothly with the distance between particles and play only a minor role in deter-
miningthestructureoftheliquid.Theyprovide,instead,anessentiallyuniform,attractive
backgroundandgiverisetothecohesiveenergythatisrequiredtostabilisetheliquid.This
separationoftheeffectsofrepulsiveandattractiveforcesisaveryold-establishedconcept.
ItliesattheheartoftheideasofvanderWaals,whichinturnformthebasisofthevery
successfulperturbationtheoriesoftheliquidstatethatwediscussinChapter5.
4 INTRODUCTION
The simplest model of a fluid is a system of hard spheres, for which the pair potential
v(r)ataseparationr is
v(r)=∞, r<d,
(1.2.1)
=0, r>d
whered isthehard-spherediameter.Thissimplepotentialisideallysuitedtothestudyof
phenomenainwhichthehardcoreofthepotentialisthedominantfactor.Muchofourun-
derstandingofthepropertiesofthehard-spheremodelcomefromcomputersimulations.
Suchcalculationshaverevealedveryclearlythatthestructureofahard-spherefluiddoes
notdifferinanysignificantwayfromthatcorrespondingtomorecomplicatedinteratomic
potentials,atleastunderconditionsclosetocrystallisation.Themodelalsohassomerele-
vancetoreal,physicalsystems.Forexample,theosmoticequationofstateofasuspension
ofmicron-sizedsilicaspheresinanorganicsolventmatchesalmostexactlythatofahard-
sphere fluid.1 However, although simulations show that the hard-sphere fluid undergoes
a freezing transition at ρ∗ (=ρd3)≈0.945, the absence of attractive forces means that
thereisonlyonefluidphase.Asimplemodelthatcandescribeatrueliquidisobtainedby
supplementingthehard-spherepotentialwithasquare-wellattraction,asillustratedinFig-
ure1.2(a).Thisintroducestwoadditionalparameters:ε,thewelldepth,and(γ −1),the
widthofthewellinunitsofd,whereγ typicallyhasavalueofabout1.5.Analternative
tothesquare-wellpotentialwithfeaturesthatareofparticularinteresttheoreticallyisthe
hard-coreYukawapotential,givenby
v(r)=∞, r∗<1,
(cid:4) (cid:5)
ε (1.2.2)
=− exp −λ(r∗−1) , r∗>1
r∗
wherer∗=r/d andtheparameterλmeasurestheinverserangeoftheattractivetailinthe
potential.ThetwoexamplesplottedinFigure1.2(b)aredrawnforvaluesofλappropriate
either to the interaction between rare-gas atoms (λ=2) or to the short-range, attractive
forces2 characteristicofcertaincolloidalsystems(λ=8).
Amorerealisticpotentialforneutralatomscanbeconstructedbyadetailedquantum-
mechanical calculation. At large separations the dominant contribution to the potential
comesfromthemultipolardispersioninteractionsbetweentheinstantaneouselectricmo-
mentsononeatom,createdbyspontaneousfluctuationsintheelectronicchargedistribu-
tion,andmomentsinducedintheother.Alltermsinthemultipoleseriesrepresentattractive
contributions to the potential. The leading term, varying as r−6, describes the dipole–
dipole interaction. Higher-order terms represent dipole–quadrupole (r−8), quadrupole–
quadrupole (r−10) interactions, and so on, but these are generally small in comparison
withtheterminr−6.
Arigorouscalculationoftheshort-rangeinteractionpresentsgreaterdifficulty,butover
relativelysmallrangesofr itcanbeadequatelyrepresentedbyanexponentialfunctionof
theformexp(−r/r ),wherer isarangeparameter.Thisapproximationmustbesupple-
0 0
mented by requiring that v(r)→∞ for r less than some arbitrarily chosen, small value.
Inpractice,largelyforreasonsofmathematicalconvenience,itismoreusualtorepresent
theshort-rangerepulsionbyaninverse-powerlaw,i.e. r−n,with n lyinggenerallyinthe
INTERMOLECULARFORCESANDMODELPOTENTIALS 5
1
(a) square-well potential
/ ) 0
r
(
v
-1
( - 1)d
0.5 1.0 1.5 2.0 2.5
1
(b) Yukawa potential
/ ) 0
r
(
v
= 2
= 8
-1
0.5 1.0 1.5 2.0 2.5
r / d
FIG.1.2. Simplepairpotentialsformonatomicsystems.Seetextfordetails.
range9to15.Thebehaviourofv(r)inthelimitingcasesr→∞andr→0maytherefore
beincorporatedinasimplepotentialfunctionoftheform
(cid:4) (cid:5)
v(r)=4ε (σ/r)12−(σ/r)6 (1.2.3)
whichisthefamous12-6potentialofLennard-Jones.Equation(1.2.3)involvestwopara-
meters:thecollisiondiameterσ,whichistheseparationoftheparticleswhere v(r)=0;
andε,thedepthofthepotentialwellattheminimuminv(r).TheLennard-Jonespotential
provides a fair description of the interaction between pairs of rare-gas atoms and also of
quasi-spherical molecules such as methane. Computer simulations3 have shown that the
triplepointoftheLennard-Jonesfluidisatρ∗≈0.85,T∗≈0.68.
Experimental information on the pair interaction can be extracted from a study of any
process that involves collisions between particles.4 The most direct method involves the
measurementofatom–atomscatteringcross-sectionsasafunctionofincidentenergyand
scatteringangle;inversionofthedataallows,inprinciple,adeterminationofthepairpo-
6 INTRODUCTION
tentialatallseparations.Asimplerprocedureistoassumeaspecificformforthepotential
anddeterminetheparametersbyfittingtotheresultsofgas-phasemeasurementsofquan-
titiessuchasthesecondvirialcoefficient(seeChapter3)ortheshearviscosity.Inthisway,
forexample,theparametersεandσ intheLennard-Jonespotentialhavebeendetermined
foralargenumberofgases.
Thetheoreticalandexperimentalmethodswehavementionedallrelatetotheproperties
of an isolated pair of molecules. The use of the resulting pair potentials in calculations
for the liquid state involves the neglect of many-body forces, an approximation that is
difficult to justify. In the rare-gas liquids, the three-body, triple-dipole dispersion term is
themostimportantmany-bodyinteraction;theneteffectoftriple-dipoleforcesisrepulsive,
amounting in the case of liquid argon to a few percent of the total potential energy due
to pair interactions. Moreover, careful measurements, particularly those of second virial
coefficientsatlowtemperatures,haveshownthatthetruepairpotentialforrare-gasatoms
isnotoftheLennard-Jonesform,buthasadeeperbowlandaweakertail,asillustratedby
thecurvesplottedinFigure1.3.ApparentlythesuccessoftheLennard-Jonespotentialin
accountingformanyofthemacroscopicpropertiesofargon-likeliquidsistheconsequence
ofafortuitouscancellationoferrors.Anumberofmoreaccuratepairpotentialshavebeen
developedfortheraregases,buttheiruseinthecalculationofcondensed-phaseproperties
requirestheexplicitincorporationofthree-bodyinteractions.
Althoughthetruepairpotentialforrare-gasatomsisnotthesameastheeffectivepair
potential used in liquid-state work, the difference is a relatively minor, quantitative one.
The situation in the case of liquid metals is different, because the form of the effective
ion–ion interaction is strongly influenced by the presence of a degenerate gas of con-
duction electrons that does not exist before the liquid is formed. The calculation of the
ion–ion interaction is a complicated problem, as we shall see in Chapter 10. The ion–
electroninteractionisfirstdescribedintermsofa“pseudopotential”thatincorporatesboth
the coulombic attraction and the repulsion due to the Pauli exclusion principle. Account
200
Ar-Ar potentials
100
K
/ ) 0
r
(
v
-100
-200
3 4 5 6 7
r / Å
FIG.1.3. Pairpotentialsforargonintemperatureunits.Fullcurve:theLennard-Jonespotentialwithparameter
valuesε/kB=120K,σ=3.4Å,whichisagoodeffectivepotentialfortheliquid;dashes:apotentialbasedon
gas-phasedata.5