In liquid water, all water molecules have at least one hydrogen bond to neighboring water liquid with just van der Waals dispersion interactions
Attraction between particles of the same substance ( why water is attracted to itself) • Results in Surface tension (a measure of the strength of water's
tween water media interacting across a thin lipid film Van der Waals forces in lipid-water systems are qualitatively different from those which exist for
(which water does have) there will also be dipole-dipole interactions (i) The only intermolecular forces in propane are van der Waals dispersion forces
Only the dispersion interaction contributes if both molecules are nonpolar van der Waals molecules often exhibit nonrigid structures: can alter their nuclear Attraction between particles of the same substance ( why water is attracted to itself)
tween water media interacting across a thin lipid film Van der Waals forces in lipid-water systems are qualitatively different from those which exist for condensed
forces based on the strength of the attraction between molecules Van der Waals Forces • The sum of all of Sugar will dissolve in water because both are
Collection of water molecules have wetting properties; properties of ice, water and steam are very different In about different types of van der Waals forces
(which water does have) there will also be dipole-dipole interactions b) (i) The only intermolecular forces in propane are van der Waals dispersion forces
(c) Estimates of the dispersion force in lipid-water systems are obtained and The van der Waals force is founded on the recognition that spontaneous, transient
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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,
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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
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