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VANDERWAALSFORCES

SPECIALCHARACTERISTICSINLIPID-WATER

SYSTEMSANDAGENERALMETHODOF

CALCULATIONBASEDONTHELIFSHITZTHEORY

B.W.NINHAMandV.A.PARSEGIAN

FromthePhysicalSciencesLaboratory,DivisionofComputerResearchandTechnology, NationalInstitutesofHealth,Bethesda,Maryland20014.Dr.Ninham'spermanentaddress istheDepartmentofAppliedMathematics,UniversityofNewSouthWales,

Kensington,NewSouthWales,Australia2033.

ABsTRAcTApracticalmethodforexaminingandcalculatingvanderWaalsforcesisderivedfromLifshitz'theory.RatherthantreatthetotalvanderWaalsenergyasasumofpairwiseinteractionsbetweenatoms,theLifshitztheorytreatscomponentmaterialsascontinuainwhichthereareelectromagneticfluctuationsatallfrequenciesovertheentirebody.Itisnecessaryinprincipletousetotalmacroscopicdielectricdatafromcomponentsubstancestoanalyzethepermittedfluctuations;inpracticeitispossibletouseonlypartialinformationtoperformsatisfactorycalculations.Thebiologicallyinterestingcaseoflipid-watersystemsisconsideredindetailforillustration.ThemethodgivesgoodagreementwithmeasuredvanderWaalsenergyofinteractionacrossalipidfilm.Itappearsthatfluctuationsatinfraredfrequenciesandmicrowavefrequenciesareveryimportantalthoughtheseareusuallyignoredinpreferencetouvcontributions."Retarda-tioneffects"aresuchas todampouthighfrequencyfluctuationcontributions;ifinteractionspecificityisduetouvspectra,thiswillberevealedonlyatinteractions

across<200angstrom(A).DependenceofvanderWaalsforcesonmaterialelectricpropertiesisdiscussedintermsofillustrativenumericalcalculations.

INTRODUCTION

Thepurposeofthisandasucceedingpaper(forapreliminaryreport,seereference1) istodevelopinsomedetailageneralmethodforcalculatingvanderWaalsforces insituationsofbiologicalinterest.Fordefinitenessweshallreferinmostparttothe interactionofwateracrossathinhydrocarbonfilm.Theenergyofsuchasystem wasrecentlymeasuredbyHaydonandTaylor(2).Weshallshowthatvander Waals

forcesinthissystemcanbeanalyzedeasilyandaccuratelybythemacro-scopictheoryofLifshitz(3).Further,severalimportantqualitativefeaturesof

theseforces,unnoticedinearliertreatments,arerevealedbythisanalysis.The approachelaboratedhereallowspredictionofattractiveenergiesinothersystems,

646brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by Elsevier - Publisher Connector

aswellaspointedexaminationoftheprobableroleoftheseforcesinbiological processes.Inparticular,weshallfindthatacarefulanalysisprovidesastrong hintconcerningtheproblemofattractivespecificity.

Theoutlineofthepaperisasfollows:

(a)WedefinepreciselywhatwemeanbyavanderWaalsforceand,toputour investigationintoperspective,brieflysummarizeearlierworkontheproblem. (b)WegiveadescriptionoftheformulaeofLifshitzapplicabletothinfilms betweensemi-infinitemedia,anddiscussausefulrepresentationofdielectricsus- ceptibilitybymeansofwhichspectroscopicinformationmaybeadaptedforcal- culation. (c)Estimatesofthedispersionforceinlipid-watersystemsareobtainedand comparedwiththemeasuredvalues. (d)Adetailedexaminationismadeofthehighfrequencyspectrumofcontribu- tionstotheenergyinordertodevelopanintuitiveunderstandingofthebehavior ofvanderWaalsforcesacrosslipidlayers,andoftheexperimentalvariableswhich affectforces.Thisisimportantinasmuchasthisstudythrowssomelightonthe physicaloriginofspecificityinbiologicalsystems.Itwillbeshownthatthereisas greatacontributiontotheenergyfrominfraredfrequenciesasfromtheuvand thecontributionfromthemicrowaveregionisasgreatasinfraredanduvcombined.' Further,theso-calledretardationeffectsduetothefinitevelocityofsignalpropaga- tionemergeasaprogressivedampingofhigherfrequencycontributionsasfilm thicknessisincreased. (e)Finallywesummarizesomequalitativeimplicationsconcerningbiological systemswhichcanbedrawnfromadetailedstudyofvanderWaalsforcesby

Lifshitztheory.

THEVANDERWAALSFORCE

ThevanderWaalsforceisfoundedontherecognitionthatspontaneous,transient

electricpolarizationcanariseatacenterduetothemotionofelectrons,moleculardistortion,ormolecularorientation.Thispolarizationwillactonthesurrounding

regiontoperturbspontaneousfluctuationselsewhere.Theinteractionresulting fromthisperturbationissuchastolowertheenergy.Theclassesofsuchinteraction havebeenextensivelyreviewedbyKauzmann(4)andJehle(5). Theresultingattractive"dispersion"forceforelectronicfluctuationswasfirst calculatedbyF.London(6)inhisfamous1930paper.Heshowedthattheforce

betweentwoisolatedatomsisproportionaltotheinverseseventhpowerofdistance,andtheproductoftheirpolarizabilities.London'stheoryprovidedthe

basisforsubsequenttheoreticalestimates(7)ofdispersionforcesbetweencon-

1Thecharacteristicsofthemicrowave-frequencyfluctuationsaresufficientlydistinctfromthoseofhigherfrequenciessothattheyareconsideredseparatelyinthesucceedingpaperreferredtoas"two";Parsegian,V.A.,andB.W.Ninham.1970.Biophys.J.10:664.

NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems647 densedmedia.Theseestimateshavebeenseverelylimitedbyseveraladhocassump- tionswhichare: (a)theassumptionofpairwiseadditivityofindividualinteratomicinteractions inacondensedmedium; (b)theapproximationthatcontributionscenteredaroundasingledominant frequencyoftheelectromagneticfieldintheultravioletareimportant;and (c)thatthedifficultyofdealingwithanintermediatesubstance(e.g.hydrocarbon betweenaqueousregions)canbehandledbytheinsertionofanarbitrary"dielec- tricconstant"correctionatasinglefrequency.Inaddition,detailedinformation aboutatomicpolarizabilitiesandrelaxationfrequencieswasrequiredforcom- putations.Asemphasizedbylaterworkers(3)acalculationofvanderWaals forcesbasedontheseassumptions(validfordilutegases)isintrinsicallyunsound. Thismethodobscuresalmosttotallythemostinterestingqualitativefeaturesof vanderWaalsforces. In1955Lifshitz(8)andlaterheandhiscoworkers(3)developedatheorybased

onideasofCasimirandPolder(9)whichovercomesthesedifficulties.Histheoryincludesallmany-bodyforcesthroughacontinuumpicture,retainscontributions

fromallinteractionfrequencies,anddealscorrectlywiththeeffectsofintermediate substances.Moreimportant,theinformationrequiredforcalculationsiscontained inthedielectricpropertiesofcomponentmaterials-informationavailableinprin- ciplefromindependentspectroscopicmeasurements. Foracondensedmedium,wheretherangeofstronginteractionexceedsthe distancebetweenatomiccenters,Lifshitzregardstheentiresetoflocalspontaneous electricfieldfluctuationsasanelectromagneticfieldwhichextendsoverthewhole system.Thistime-varyingfieldcanbefrequencyanalyzed;thestrengthofafield ofagivenfrequencyisdirectlydependentontheresponseofthematerialtoan appliedfieldofthatfrequency,i.e.,itsdielectricsusceptibilitye(W).Boundary surfacesbetweenunlikematerialswillaffecttheseelectricfieldsandconsequently theelectromagneticenergyofthesystem.Thisapproachtodispersionforces examinesthechangeinenergyofasystemwithmovementoftheboundarysur- facesbetweenunlikeregions. Itwasthought(10)thatthetheoryofLifshitzcouldnotbeappliedtothecalcu- lationofdispersionforcesintheabsenceofcompletespectralinformation.Wefind thatsuchcompletedataareunnecessary,particularlywhencomponentsubstances

areofsimilarweightdensity.Infact,itispossibletomakeseveralsimplifyingassumptionsregardingspectra.Allthatappearsnecessaryforthecalculationofvan

derWaalsforcesinthepresentthinhydrocarbonfilmsystemaresingleaverage absorptionfrequenciesintheinfraredanduvforwater,oneaveragefrequencyin theuvforhydrocarbon,indicesofrefractionatvisiblefrequencies,alimiting

valueforthedielectricsusceptibilityofwaterbetweenmicrowaveandinfraredfrequencies,andthesimplestformforthedielectricdispersionofliquidwaterinthemicrowaveregion.

BIOPHYSICALJOURNALVOLUME101970648

FORMULARY

LifshitzExpression

Thegeneralformulafortheattractiveforceperunitareabetweentwosemi-infinite mediaofsubstance1acrossaplanarslabofsubstance2ofthicknessI(Fig.1)(3)is

F(l)=0CE823n$f2{[(S+p)exp(2Pne2)]-1

F(1)~[(e-pl(=C~ex]+[(SE2+;::)2ex2P~nIE2)f}d/21

wherekisBoltzmann'sconstant,Tabsolutetemperature,cthevelocityoflightin vacuum,qandE2thedielectricsusceptibilitiesevaluatedontheimaginaryfre- quencyaxisatco=itn, s=vi2_17/2,(2) and 2irkT with2wrhPlanck'sconstant.Thesumistakenoverintegralnandtheprime(')on thesumsignindicatesthatthen=0termbemultipliedbyY2. Byintegrationoftheforcewithrespectto1,thecorrespondingfreeenergyof interactionperunitarea(takingG=0atI=X)is

G(l,T)kTEI((n1)(4)870ln=O

waterhydrocarbonwater

FIGURE1Twosemi-infinitemediaofsubstance1separatedbyaplanarslab,thickness1,ofsubstance2.Inthetextweconsiderthecasewhere1iswaterand2ishydrocarbon.

NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems649 where

1(n1)p(2nle2)f{[1-i2exp(2ptnIE2

+In[-A2expe2)]}dp(5) with

Se2-=l-P(s6)

SE2+pC-1s+p

Toconformwiththenotationofcolloidchemistry,itwillbeconvenientinthe followingtodefinea"Hamakerfunction,"A=A(l,T),suchthat

G(l,T)=A(1,)7

Comparingequation7withequation4,then

00

A(1,T)=1.5kTElI(n21).(8)n==O

LimitingCases.Lowtemperature:whenkT<ForI<

IU(n0)=fxln[1-(::)2e~ld(10)

(Inthiscasespsincevaluesofp>>1contributetotheintegralinequation5.)SometimesA2<<1and /2o0)'(2-61

InthedoublelimitI->0,T->0

A(O)_3h4JolxIun[1_(2'Ele-dxdt

,3hlo(f2-fld(112

BIOPHYSICALJOURNALVOLUME101970650

DielectricDispersionA

Inordertousetheaboveformulae,weneedaconvenientrepresentationforthe dielectricsusceptibility. Thefunctione(co)isacomplexfunctionE!+iE"ofacomplexfrequencyco= (wR+it.Thedispersionenergydependsonlyonthevaluesofeontheimaginary frequencyaxise=e(iQ)and,fromequations4and5,aknowledgeofE(i#)isim- portantonlyforthosevaluesoftforwhichthereisadifferencebetweenthedi- electricsusceptibilitiesofthetwomaterialsatagivenfrequency.Weseekasuitable representationfore(it)whichwillincorporateexperimentaldataaswellassatisfy generalconstraintswhichmustbesatisfiedbyanydielectricsusceptibility.These constraintsarethatE(it)beapurerealquantitymonotonicdecreasingwithi, andthatthefunctione(w)(C=WR+it)havenozerosorpolesintheupperhalfW planeinordertosatisfytheKramers-Kronigrelations(11).Asuitablerepresenta- tionis e(X)1+1-ixjx+E1-(co/c)2+i(*(12') ThisrepresentationincludessimpleDebyerelaxationformicrowavefrequencies plustheclassicalformofLorentzelectrondispersionforinfraredthroughmid-uv

frequencies.WewilldescribebelowhowthenecessaryconstantsCmwandCy(proportionalto"oscillatorstrengths")andresonancefrequenciescomw,WXmay

bedeterminedfrommeasurementsmadeontherealfrequencyaxisco=WR.On theimaginaryaxis(co=it),wehavefromequation11 +1mwCE1j(13)1+~/cmwjtc)Y Nowthedampingterminyjcoinequation12issignificantonlywhencoCojsince

bandwidthsarealwaysmuchlessthanabsorptionfrequencies.Thecontributionofthisterminequation13willbenegligible(1+(/Coj)2>>yjy)sothatwemay

take

CmwC,e(it)=1+1+/jm+E1+(/coj)(14)

Clearlythen,onlytheoscillatorstrengthsandabsorptionfrequenciesofcom- ponentmaterialswilldeterminedispersionenergies. Atveryhighfrequenciesthedielectricdispersionhasthelimitingform(11) (O)1-_4xA2(15) Hereeandmareelectronchargeandmass,andNisthenumberofelectrons/cc. NINHAMANDPARSEGIANVanderWaalsForces:CakulationforLipidWaterSystems651

Ontheimaginaryaxisequation15becomes

=1+42Ne(16)mi Thisformholdsformaterialsmadeoflightelementsatfrequenciesinandabove thefar-uv.Itisimportanttoemphasizethatmaterialsoflightelementsandsimilar weightdensitywillhavesimilarsusceptibilitiesatthesehighfrequencies,andcon- sequentlygiveaverysmallcontributiontotheforceandenergyintegrals.Thisis theusualsituationinsystemsofbiologicalinterest. Betweenthelow-tomid-uvregiondescribedbyequation14andthefar-uvto

X-rayregiondescribedbyequation16,thesimplestprocedureistoconstructaninterpolationformulaforE(it).GiventheconditionthatE(it)bemonotonicde-

creasing,thereislittleambiguityinsodoing. However,becauseofthesimilarityofthesusceptibilitiesofcomponentsubstances inthemid-uv,theintegrandsorsumtermsoftheforceandenergyexpressions tendtozeroveryrapidlyinthisregion,andgivelittlecontribution.Inpractice therefore,itisusuallysufficienttousetheformofequation14forsusceptibilities, andnointerpolationisrequired.Indeed,inspectionofthefullexpressionsequations

1,4,and5fortheforceandenergyshowsthattheeffectoftheexponen-tialsexp(-2ptle2112/c)istodiminishcontributionstotheintegrandseverelyforfrequenciest.>c/(2lV\e2).Forexample,andtypicallywhenI150A,t,>(3X

1016)~10166,frequenciesintheuvbandsatisfy10el.8.,1016.8sothatfrequencies

atandabovethefar-uvwillbeinanyeventunimportant.Thispointisillustrated indetailinthesectionAnalysisofIntegralswherewecarryoutananalysisofthe integralofequation9.

DATAANDCALCULATIONS

SpectroscopicExperimentalData

Forwaterandhydrocarbonsthefollowingdielectricdataareknown. Water.Fromaudiothroughmicrowavefrequencieswaterexhibitssimple

Debyerelaxationfromitsstaticvalueof80.4downto5.2withacharacteristicrelaxationwavelength1.78cm.Thus,Cmw=(80.4-5.2)=75.2,andwmw=27rC/XmW=(1.06X1011)=1011.026radians/sec.(Datahere(12)areforT=

20°C.)Theinfraredspectrumisdominatedbythreecloselyspacedabsorption

peaks(13)atcir=3.0X1014,6.89X1014,and7.08X1014,afterwhichere- laxestovaluesobservedasthesquareoftheindexofrefractionntwIwhere

2nw=1.78.WeshalluseCir=(5.2-1.78)=3.42andapproximatethevibrationfrequencybyanaveragevaluewir=[U3(3+6.89+7.08)X1014]=1014.75radians/

sec.Insensitivityofthecalculatedenergytothisapproximationischeckedbelow. Inthenear-tomid-uv,water(14)showsaweakabsorptionatX=1650Aor

BIOPHYSICALJOURNALVOLUME101970652

wuv=(1.14Xl01)=1016.058andanapparentlymuchstrongerabsorptionat aboutX=1250Aorwuv=(1.507X101w)=1016.178.However,therelativestrength ofthesepeaksandthepresenceofotherabsorptioninthefurtheruvarenotknown.

Becauseoftheslowvariationofe(it)withtonemayuseanaveragefrequencyw,uvforthenear-tomid-ultravioletregion.Acommonapproximationistousea

frequencyequivalenttothefirstionizationpotential(4).Inthiscase12.62ev-hcouv,orcouv=(1.906X101")=1016.28rad/sec.(Throughoutpaperradusedfor

radian.)Itwillbeshownthatuseoftheionizationpotentialvalueorthestronger absorptionbandvaluehaslittleeffectontheestimatedenergy.ThevalueofCu, forthisrelaxationis(n2-1)=(1.78-1)=0.78.Wehavethen

Ewater=ew(i)=1+1+7mW+1+C+1+(17)

wheretheconstantsaregivenabove. Hydrocarbons.Thereappearstobelittledielectricrelaxationshownby

hydrocarbonliquidsbetweenaudioandvisiblefrequencies(12)where(ha=nhe(n=indexofrefraction).Byexaminingthereflectionoflaserbeamsfromathin

lipidfilm,CherryandChapman(15)havemadeaprecisemeasurementofan anisotropicindexofrefraction:nhe=1.486forpolarizationperpendiculartotheplaneofthefilmandnho-

1.464paralleltothefilm.ThesegiveEhe=2.208and2.143,respectively.Other

2possibleestimatesatopticalfrequenciesareEhc=nhc=1.89forn-hexane(12)

andehe=nhc=2.0,anaveragevalueoftenusedastypicalforhydrocarbons.

Eachofthesenumberswillbeconsideredbelow.Againwewillsummarizerelaxationasoccurringatanaveragefrequencycor-

respondingtotheionizationpotential.Anappropriatevalueofthisforathin

hydrocarbonfilmisnoteasilydeterminedfromavailabledata.Valuesfordecane(16),10.19evorCuv=1.54X1016,andforethane(16,14),11.65evorcouv=

1.76X1016,willsufficeinthepresentinstance.Usingthesevalueswehave

2-1Ehe(lt)=1+1he+v)2

TheoreticalEstimates

WefirstcalculatethevanderWaalsenergyofwateractingacrossalipidfilm (Fig.1)50Athickat20°C.Usingequations17and18fore.(it)andfhc(it)with anaverageinfraredabsorptionfrequencyforwaterandaverageuvabsorption

derivedfromionizationpotentialsofwateranddecaneorethane,wefindA(50A)byequation8forfourvaluesofn20(TableI).Forthesedata,valuesofA(=50A,

T=20°C)rangefrom5.5to7.1X1014erg.WenotethatAisnotsimplymono- tonicinnhhc NINHAMANDPARSEIGANVanderWaalsForces:CalculationforLipidWaterSystems653

Theseestimatesareslightlyhigherthanthoseinferredfromexperimentsonthinlipidfilms(2)(4.7X10-14erg)butmuchlowerthanthosederivedfromex-

perimentsonsuspensionsofparaffinsinaqueoussuspension(17)(r1.6X10-'1 erg).Inprinciplebothexperimentalestimatesshouldbeapproximatelythesame sothattheoreticalestimatesherearelessambiguousthanexperimentsbutwithin thecorrectranges. TableIIAsuggeststhatAisrelativelyinsensitivetotheassumedcUs,aslong astheyarechoseninaconsistentwayforthetwomaterials.Usingstrongestnear- uvabsorptionpeaksorionizationpotentialsforbothmaterialsgiveAestimates

6.1-7.1X10-14forn2=2.208thataredistinctfromthevalueobtainedbyusing

thesecond-strongestwaterpeak(9X10-14).Also,(TableIIB)itmakeslittle differenceifoneaverage(Dirisusedforwaterorifthethreemaininfraredpeaksare equallyweighted,andseparatelycontributetoew(it).

Acarefulattempttousee(it)thatsatisfybothlowandhighfrequencybehavior(equations14and16)makesnegligibledifferencetotheestimatedA.Bystraight-

lineinterpolationbetweenformsfornear-uvandX-rayregionsandusingthe

TABLEI

ESTIMATESOFTHEHAMAKERFUNCTIONAUSINGTHEGENERALEQUATION7

Takelr=5.66X1014rad/sec,C1°r=3.4cow=1.06X1011rad/sec,C;mw=75.2.2.Wuv=1.906X10"rad/sec,n2=1.78.I=50A,T=20°C.

2A(erg),wuv=1.54X1016A(erg),Wuv=1.76X1016nho(decaneI.P.*)(ethaneI.P.)

ergerg

1.9(n-hexane)5.8X10-145.8X10-14

2.0("typical"value)5.55.82.143(laserexpt.)5.76.52.208(laserexpt.)6.17.1

*I.P.=ionizationpotentials.

TABLEIIA

DEPENDENCEOFAONWufv

Takenhw=2.208,otherwiseasinTableI.

uhoX10-16-vX10-16AX1014Sourceofw01 rad/secrad/secerg

1.41.5076.6Strongestabsorption

1.41.149.2ndstrongestH20absorption

1.761.9067.1Ionizationpotentials(ethaneforhc)

1.541.9066.1Ionizationpotentials(decaneforhc)

BIOPHYSICALJOURNALVOLUME101970654

TABLEIIB

EFFECTOFAVERAGINGVS.EQUALWEIGHTINGOFTHREEWATERINFRAREDFREQUENCIES

Dataasabove.

WjyA,(couh=1.54X1016)A,(coh"=1.76X10")

ergerg

3.0,6.89,7.88X1014rad/secequally5.9X10-147.0X1-14weighted=5.66X1014md/sec6.17.1

TABLEIII

A(l)VS.I

Usen2hO=2.208,V=1.54X1016andasinTableI.

I(A)A(l)X1014A(l)/A(O)A(l)/A(O)oldtheory(18)

erg

06.3656.351.1.106.331.0.95506.090.960.841005.780.910.725004.740.750.2810004.430.70.1550003.650.570.03100003.330.520.015500003.160.50.003

electrondensityofmaterialhaving1g/ccweightdensity,wefindthattheeffectis alwayslessthan6%.(Themethodofinterpolationisdescribedindetailelsewhere forthecaseofsoapfilmsinair[18].) AasaFunctionofThickness1.Byvirtueofequations8and10thevalue ofAas1->0convergesto

A(I=0,T)=l.5kT>2'fxIn[1-(2)e-x]dx.

nO0O2+C-1 DeviationsofAfromthislimitingformwhenI>0arecalled"retardationeffects" sincetheyarecausedbythefinitetraveltimeofelectromagneticradiationacross thegap1. InTableIIIwehavelistedcalculationofA(L)forI=0to50,000A.There-

tardationfactorofA(1)/A(0)isalsolistedforcomparisonwiththefunctionusedincolloidchemistry(19).Clearlytheneglectofnon-uvfrequencyfluctuations,typical

oftheoldertheory,givesvastlylowerestimatesatlargeseparationdistances. NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems655 Theproperanalysisofretardationeffectsisdiscussedinthefollowingsectionand intheAppendix.AtverylargedistancesthecalculationofAisdominatedbythe n=0terminequation8;thistermisthesubjectofthesucceedingpaper(two).

ANALYSISOFINTEGRALS

SpectralContributions

Inordertoclarifythedependenceofthedispersionenergyonfluctuationfrequencies suchthathwkT,weexaminetheintegrandsoftheintegralswhichoccurin equations7and9.Theseareoftheform

JI(t,1)dt,(19)

whereI(,1)definedbyequations5and10measuresthecontributiontotheenergy

1-infrared-l1-.UV--.1

6-n2n=2.208

5I(C;0)x)

Q:0

1j21031415161718

x=logioCP

FIGURE2SpectrumofrelativeinfraredanduvcontributionstovanderWaalsenergyinthelimitI=0.Useequations17,18,and10withdataforwaterandethane.Datafordecanegivesharplyreduceduvpeak.Notelargeinfraredcontribution,sharpdecreasein

uvcontributionwhenn2h,isreducedfrom2.208to2.0tobringitclosertontO2=1.78for water.ThefactortmultiplyingI(Q,I=0)intheordinateistocorrectforthelogarithmicscaleintheabscissa.

BIOPHYSICALJOURNALVOLUME101970656

offrequenciesintherange(Q,t+dc).Changingvariablestox=logio(Q),the relativecontributionfromthespectralrange(x,x+dx)isthen2.303(IQ,1)dx. Fig.2showsplotsofI(Q,I=0)fortwovaluesofn2,2.208and2.0withI=

0.(OtherdataareasinTableI.)Thefinitefrequencycontributiontothetotal

energyiscleanlydividedintoaninfraredandanuvpeak.Therelativemagnitude ofthesetwopeaksobviouslydependsonthevalueofnh.Forexample,withnh=

2.208,35%oftheintegralcomesfromtheinfrared;butwithnhe=2.0,80%of

theenergyintegralisduetoelectromagneticfluctuationsatinfraredfrequencies! Itisremarkablethattheroleoftheinfraredspectrumofwaterhasbeenignoredin previousdiscussionsofvanderWaalsforces.Similarly,asalreadyemphasized, thereisanimportantcontributionfromthemicrowaveregionwhosequalitative featuresaresufficientlyuniquetobeconsideredseparately(two).

ForfixedIandfixedwatercomposition,theonlytwoparameterswhichcan2-hoaltersignificantlywithhydrocarboncompositionarenhoand&Dhc.TheplotsofI(Q,1)inFig.2alsoshowthattheeffectofchangingn20istoaltertheentirespectrum

ofcontributingfrequencies.Similarly,variationinabsorptionfrequenciescO, willshiftthepositionofmaximaofthespectrum,andaltertheheightofpeaks. Forthepresentexamplec,>Elhiintheinfrared,butc,RetardationEffects ThecaseI=0consideredaboveshowsthenatureofspectralcontributionstothe energywithvariationofmaterialpropertiesofcomponentsubstances.Thereis anotherconditionimposedontheallowedcorrelationsinelectromagneticfluc- tuationsacrossagapduetothefinitevelocityoflight.Whenthetraveltimeofa

fluctuationsignalacrossthegapiscomparabletoafluctuationfrequency(21E2'12)/c1/i,thecorrelationinfluctuationsbetweeneithersideisdiminished.Thein-

tegrandI(Q,I)satisfiestheunequalityI(Q,I)1=0,50Aand500A,andfornh0=2.208and2.0;theyillustratetheprogressive removalofhighfrequencymodeswithincreasing1.Onthesameabscissawehave alsoplotted(Fig.3c)theratioI(Q,I)/I(I,0)<1.Thisrathercomplicatedfunction isweaklydependentonmaterialpropertiesandisthemultiplicativefactorfordampingthesehighfrequencymodes. Thefrequencyregimeoverwhichtheintegrandintisdampedoutisaboutonedec- NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems657 -infrared--

II(C;i)xC

x10-13 nh2a2.208n.2=1.78A

Xc-I0910-

FIGuRE3aTheinfluenceofthefinitevelocityofpropagationonthespectrumofinfraredanduvcontributionstovanderWaalsenergies.n2hc=2.208,dataforethaneandwater.LargeuvcontributionatI=0andsignificantdecreaseofAwith1.

IF-*---infrared-1-UV---

tI(C;l)4X

Xto-s3

n2L2.0 n2=1.78

FIGURE3bTheinfluenceofthefinitevelocityofpropagationonthespectrumofinfraredanduvcontributionstovanderWaalsenergies.nfhc=2.0,dataforethaneandwater.SmalluvcontributionatI=0andweakdependenceofAon1.

BIOPHYSICALJOURNALVOLUME101970658

1. 0. 0. 0

0..0CA=__QOA

I(C;O).4%Q=50A

.2Q-500A\\

I~II}_I_

FIGURE3cTheinfluenceofthefinitevelocityofpropagationonthespectrumofinfraredanduvcontributionstovanderWaalsenergies.Retardationfactorfordampingcontribu-tionsatdifferentfrequencies.ArrowsIindicatefrequencyatwhichE=c/21.NotethatretardationdampingiseffectivelyashiftofthecurveI(t;I)/I(I;0)tothelefttocutouthigherfrequencies.Therangeofdampingisapproximatelyonedecadewideandcenteredaboutt=c/21.

adewideandroughlycenteredabout=c/(21E'12)-'(c/21)whereeho--s1.Byvirtue ofthelargeinfraredcontributiontotheenergy,evenat500A(where,10155< uvfrequencies),almosthalftheoriginalintegralisintactwhennh.,=2.208(Fig.

3a).Fornh.2=2,(Fig.3b)some78%oftheoriginalcontributiontotheintegral

remains.ItwouldappearthenthatvanderWaalsinteractionsinhydrocarbon- watersystemsarepeculiarlylongrangebecauseoftheinfraredandmicrowave spectrumofwater.Theeffectofdampingouthighfrequencymodesbyretardation

isintimatelyconnectedwiththemagnitudesofoscillatorstrengthsaswellasab-sorptionfrequencies;bothmaterialpropertiesandgeometricfactorssetsimul-

taneousconditionsontheenergyspectrumandmustbeconsideredtogetherin addingupthemodalenergies.

CONCLUSIONS

Thetimeislongoverdueformakingsystematicinquiryintothenatureofvander Waalsforcesinbiologicalsystems.ThepresentmethodderivedfromtheLifshitz theoryallowsonetolearnthequalitativefeaturesoftheseinteractionsaswellas makesatisfactoryestimatesoftheirmagnitude.Inadditionitmakesclearthede- pendenceofvanderWaalsforcesonbothdielectricpropertiesanddimensionsofa system. Inprincipleitisnecessarytoknowthedielectricdispersionofallcomponent substancesforallfrequencies;inpracticeitisimportantonlytoknowoscillator strengthsandmeanabsorptionfrequenciestogetagoodideaoftheelectromagnetic NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems659

forces.WefindthatfittingthesimpleformE(ic)=1+(Cmw/[l+(Q/cmmw)])+2(Cj/[I+(/lwj)2])givesanadequaterepresentationforthedielectricdispersion

asitisneededhere. Thereisaverystrongcontributiontothetotaldispersionenergyfromthein- fraredspectrumofwater.Thisfeature,certaintoholdinbiologicalsituations,is usuallyignoredinpreferencetotheuvcontributionsandisimportantincalculating thelong-distancebehaviorofthevanderWaalsforce. Ontheotherhand,twophenomenamilitateagainstimportantcontributions fromthefar-uvfluctuationfrequenciesinbiologicalsystems.First,materialsof similarweightdensitywillhavesimilardielectricsusceptibilitiesathighfrequencies (i.e.far-uvtoX-rayregion).Sincetheinteractingsurfacesareelectromagnetically definedonlywhentherearedifferencesindielectricsusceptibilityofinteracting speciesandinterveningmedium,therewillnotbeimportantbehaviorfromthe far-uv.Thisisqualitativelydifferentfromthecasewherebodiesinteractatshort distanceacrossavacuum. Second,thefinitevelocityoflight,c,causesretardationdampingacrossgaps1 atandabovefrequenciestsuchthat1/t<21/corwavelengthsX<4irl.This saysthatatlongdistance,e.g.I=50m,u=500A,thematerialpropertiesonlyfor X<500m,u=5000Aareneeded.Atshortdistancesshorterwavelengthsbegin tocontribute. Itisclearthatsubstancesthatchangetheindexofrefractionofwatercanchange thedispersionforces.Wehavecalculatedelsewhere2bythepresentmethod thatproteinorsaccharidematerialsonthebiologicalcellsurfacecangreatlyin- creaseintercellattractiveforces. Ifthecoatingmaterialsondifferentkindsofcells havedifferentuvspectra,weexpectspecificityinattractiveforcestoappearfor intercellulardistanceslessthan200A.(Itwouldbemostfortunateifthedistinctive spectrawereintheeasilyaccessible150m,u oo)limit.Inotherpapers(18)wehavederivedformulaefortriplefilms(e.g.soap bubblesinair),multilayers3(e.g.myelinfiguresandnervemyelin),andalarge numberofconfigurations2appropriatetotheinteractionofbiologicalcells witheachotherorwithasemi-infinitesubstratum. Two,therepresentationforeandnumericalintegrationasusedhereareapractical

andsimpleprocedureforexaminingvanderWaalsforcesquantitatively.Webelieveitwillbeusefulinmanyconnections.

ItisourhopethattheLifshitztheorycanbeusedforausefultheoreticalandexperimentalframeworkinwhichtoviewelectromagneticforcesinawidevariety

2Ninham,B.W.,andV.A.Parsegian.Manuscriptsubmittedforpublication.Ninham,B.W.,andV.A.Parsegian.Manuscriptsubmittedforpublication.

BIOPHYSICALJOURNALVOLUME101970660

ofbiologicalproblems.Betterspectraldatamayalterthenumericalestimatesmade inthispaperbutshouldnotinvalidatethemethodofcalculationintroducedhere, ortheprincipalqualitativefeatureswhicharealreadyapparent.

APPENDIX

RetardationEffects:ExplicitAnalysis

TheprecisenatureofthedampingduetoretardationdiscussedinAnalysisofIntegralscanbemadeexplicitbyacarefulanalysisoftheintegralI(t,1)whichgivesthespectrumoffrequencycontributions.Werecalleq.(8)intheform

I(t;l)=-(t/28)2fpdp{-(5P)e-(In'I)P]

+InLl(SE2+P)e-')PJ).X(A) where

2=e212]=V(1/e2)-1±p2.(A2)

Intermsofthisintegral,theHamakerfunctionandenergyaregivenas

A(l)=4I(;1)d{;E(l)=-A/(12X12).(A3)

ToagoodapproximationthelogarithmswhichoccurinequationA1canbereplacedbytheirleadingterms,andweconsiderfirst

Iapprox=(t/t*)2Jpdp(SE2e-E8)P(A4)

Forafixedfrequencyi,thefunction([SE2-PEL]/[SE2+pei])isslowlyvaryingasafunctionofp.Sincetheexponentialisadecreasingfunctionofpwhilepisincreasing,theintegrandofequationA4hasamaximumatp=pogivenby

dcdp(Inp-[p/l.81)=O;POrl%W/=2tl-,*/e2(A5)

Thismeansthatforafixedvalueof1,andsmalli,themajorcontributiontothepintegralcomesfromlargevaluesofp.Ontheotherhandforthesamegivenvalueof1,forlarget,Po<1andthemaincontributiontothepintegralcomesfromtheregionp-1.Two

casesthenarise (a)t2v1.

ThiscorrespondstothecaseofsmalldistancesdiscussedbyLifshitz(3).Forthesevaluesoftwemayputptsintheintegrandofequation19.Thefirsttermdisappears.Inthe

NINHAMANDPARSEGIANVanderWaalsForces:CalculationforLipidWaterSystems661 remainingintegrandthelowerlimitcanbereplacedbyzeroandwehave

J(t;1)(<)2dpapiE2-Ele-tPlt8'V~~~~~~~-E2+'El/

=-fxdxln-1_)2e-]

5m=::).(A6)E2+'El

Thecorrectiontermsareoftheorderof(Q,/t)2.

AtI=50A,t8t3X1016whichhesinthemid-uv.Fort>3X1016thefunction(1C2-1lJ/[E2+El])2isalreadyverysmallcomparedwithitsmaximumvalue,sothateffectsofretardationarealsosmall.OntheotherhandforI=500A,t8liesinthenear-uvandweexpectthesehighfrequencycontributionstobediminished.

(b)t/A="<<1

ConsiderfirstthesecondintegralofequationA1

Jqt;1)(>)(Z/Z8)2pdp(5E2ElPC-pQlt.).A7)

Writingx=p-1,wehave~(~,~j2f[si(+X)(El/E2)I(t;l)(>)-"t/JoL5+(1++X))(E12)Jexp[-(t/t,)(I+x)]dx.(A8)

Afterafurtherchangeofvariabletoxt/l8=y,notingthatthemajorcontributioncomesfromtheregiony=0,wecanexpandtheintegrandinascendingpowersofytoget

00(;)>)t0)exp(/8)adyo~~~~~~~~~

Xexp(-Y)(_23_/)(1+O(YAk/ID))

(tt)exp(-2/_VAsimilarcontributioncomesfromtheremainingintegrandofequationA1.Thusfort>t,(I;1)hasaformessentialyequivalenttoequationA6butwithexponentialdamp-ing.Completeasymptoticexpansionswhichconvergerapidlyforcomputationcaneasilybeconstructed.However,thereappearstobelittlepointinsodoing-theexpansionsarerathercomplicated,anditissimplertocalculatethespectrumnumericallyforeachcase.

RequestsforreprintsshouldbeaddressedtoDr.Parsegian. Receivedforpublication14November1969andinrevisedform3April1970.

BIOPHYSICALJOURNALVOLUME101970662

REFERENCES

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