7 A complex molecular system: surfactant micelle

Transcription

1 7 A complex molecular system: surfactat micelle formatio 7.1 Itroductio I this chapter we give a real life example of a theorectical study of a complex molecular system i the form of self-assemblig surfactats. Amphiphillic surfactat molecules cosistig of a hydrophilic head-group ad a hydrophobic tail ca aggregate ito micelles 11. Micelle formatio of such surfactats (see figure 9) is importat for trasportig oily molecules i washig processes, ad has received much attetio from the idustry. The formatio of a micellar cluster is favored by the hydrophobic tails, but a surface eergy pealty has to be paid leadig to a ucleatio barrier. I additio, the cluster size is kept fiite by the decrease i cofiguratioal etropy due to the micro-phase separatio. The result is that the free eergy of formatio of a sigle micelle shows a barrier ad a well defied miimum aroud the average cluster size. A cosequece of the geeral shape of the free eergy profile is that the micellar size distributio is fiite ad reasoably sharply peaked. Moreover, it also follows that amphiphiles oly aggregate whe the drivig force is large eough: whe their cocetratio is above the so-called critical micelle cocetratio (CMC). The CMC is a importat quatity i experimetal ad theoretical research of amphiphillic phase behavior, ad ca i priciple be estimated from a probability distributio of the micellar size usig molecular simulatio. However, have to be performed far above the CMC where there is o barrier for micelle formatio. To make a accurate estimate of the CMC oe has to be able to perform simulatios at cocetratios below or aroud the CMC. Usig straightforward Mote Carlo i combiatio with atomistic force fields this would require huge system sizes, because may micelles have to be formed i 11 This chapter is based o the article by R. Pool ad P.G. Bolhuis, J. Phys. Chem. B, 19, (14), , (25) Figure 9: Cartoo of micelle formatio of amphiphiles. Surfactats are represeted by hydrophilic heads(gray circles) with hydrophobic tails(lies). Solvet molecules are depicted by black circles. 158

2 order to get a reasoable size distributio. Such a large system size is accompaied by log equilibratio time scales, makig this simulatio approach beyod curret computer capabilities. A more efficiet approach for low CMC s therefore is to determie the micelle sizedistributiofromtheexactshapeofthefreeeergyasafuctiooftheumber of surfactats i a micelle. This free eergy ca be obtaied by computer simulatios of much smaller systems cotaiig oly oe sigle micelle. A calculatio of the CMC usig all-atom forcefield simulatios is still very demadig, eve whe usig this approach. Therefore, we istead calculate the CMC ad the free eergy of micelle formatio for a off-lattice coarse-graied model for surfactat ad solvet based o the Leard-Joes (LJ) potetial. This coarsegraied potetial takes ito accout excluded volume, va der Waals attractio ad hydrophobic iteractio of all compoets. To obtai the free eergy profile as a fuctio of micelle size, we employ the isobaric semi-grad caoical MC (SGCMC) samplig method i combiatio with umbrella samplig. Imposig a chemical potetial differece µ betwee surfactats ad solvet, the SGCMC method allows for the trasformatio of solvet molecules ito surfactats ad vice-versa. The combiatio with umbrella samplig eables us to sample the etire size probability distributio of a sigle micelle. Efficiecy is further icreased by makig use of the cofiguratioal bias Mote Carlo techique to grow chai molecules i dese solutios [17]. 7.2 Theory of micelle formatio The atural way to view a micelle solutio is as a multi phase coexistece i which micelles are i equilibrium with a solutio of sigle surfactats. If we cosider a almost ideal solutio, the equilibrium cocetratios are oly determied by the free eergydifferece G betwee a micelle of surfactats ad a solutio of sigle surfactats exp[ β G ] = [ρ σ 3 ]/[ρ 1 σ 3 ], (7.1) where β = 1/k B T is the iverse temperature with k B Boltzma s costat ad T the temperature, ρ is the desityofclustersofsizead σ is amolecularsize parameter. The crossover betwee sigle surfactats ad micelles, (the CMC) takes place whe [18] l[ρ 1 σ 3 ] β G /. (7.2) The free eergy differece G = f f 1 betwee a micelle of surfactats (f ) ad a solutio ofsiglesurfactats(f 1 ) is giveby the sum ofthree terms. G = µ tras +4.8γ(δσ 2 ) 2/3 +.75(σ/δ) 4/3 5/3 k B T. (7.3) The first is the free eergy µ tras eeded to trasfer a surfactat from the core of the micelle to the solvet. The secod cotributio is the iterfacial free eergy 4πγR 2 4.8γ(δσ 2 ) 2/3 where γ is the oil-water surface tesio ad R is the 159

3 radius of the micelle, δ is the typical legth over which hydrophobic ad hydrophilic parts are separated. The last part is the loss of etropy coected with the fact that the hydrophilic head-groups are restrait to the micelle surface. This term is estimated S.75(σ/δ) 4/3 5/3 k B [18]. The first cotributio i Eq. 7.3 is egative whereas the latter two are positive. The sum of these cotributios gives rise to a miimum i the free eergy of micelle formatio. The optimal micelle size will be reached at a surfactat umber that miimizes the free eergy per surfactat G /. For more details o this approach we refer the reader to Ref. [18]. 7.3 Simulatio methods To estimate the free eergy profile for micelle formatio as a fuctio of micelle size by computer simulatio we employ a isobaric semi-grad scheme with cofiguratioal bias Mote Carlo swap moves i combiatio with umbrella samplig. I the ext few paragraphs we explai the several techiques i more detail Semi-grad esemble We could estimate the micelle free eergy as a fuctio of cluster size by computig the chemical potetial of addig a surfactat to a existig micelle for every cluster size. However, this is very expesive because may simulatios have to be performed. A better optio is the isobaric grad caoical esemble i which the umber of surfactats is allowed to vary but the umber of solvet molecules is kept costat. Whe the simulatio is restricted to a sigle micellar cluster i the system, oe ca determie the micelle distributio at a particular chemical potetial. The use of biasig techiques such as umbrella samplig allows visitig the ulikely regios i the size probability distributio. However, usig grad caoical simulatios, the probability of addig a surfactat to a deseliquidisstillverylow. Toicreasethe isertioprobabilitywemakeuseof the semi-grad caoical MC (SGCMC) [17]. I the SGCMC techique oe imposes a chemical potetial differece µ betwee differet species, istead of the absolute chemical potetials. The isobaric semigrad potetial Y(N, µ, p, T) of a biary mixture of surfactat chais (c) ad solvets (s) is defied by the Legedre trasform betwee the variables (umber of surfactat molecules) ad µ = µ c µ s of the usual Gibbs free eergy Y = G µ = µ s N, (7.4) dy = SdT +Vdp d µ+µ s dn, (7.5) where N = + s deotes the total umber of particles with s the umber of solvet particles, ad V is the volume. The correspodig partitio fuctio is give by Ξ SG = βp N!Λ 3N dv exp[ βpv]v N idetities exp[β µ] ds N exp [ βu(s N ) ]. 16

4 where p is the pressure. The sum is over the particle idetities, rather tha all compositios. I a simulatio we ca impose the chemical potetial differece µ betwee surfactat ad solvet by choosig a molecule radomly ad tryig to chage its idetity, e.g. a solvet molecule s to a surfactat molecule c. The Metropolis acceptace rule for radomly selectig a particle ad chagig it from a solvet to a surfactat is P acc (s c) = mi[1,exp{ β U s c +β µ}], (7.6) where U s c is the eergy differece ivolved i chagig species. The mi fuctio returs the smaller of its argumets. The Mote Carlo move for chagig species, i priciple greatly speeds up the samplig of the phase space Cofiguratioal Bias MC Because we are tryig to isert a chai molecule, the isertio probability decreases expoetially with the umber of beads i the chai. We ca icrease the isertio acceptace probability usig the Cofiguratioal Biased MC (CBMC) method, which was especially devised for polymers [17]. I ordiary CBMC the first polymer segmet is iserted radomly. The subsequet segmets are grow i a biased fashio, with each segmet selected from a set of trial positios with a probability proportioal to the Boltzma distributio, favorig the positio havig the lowest eergy. The bias has to be corrected after the isertio to obey detailed balace, but the acceptace probability of the ew cofiguratio is much higher tha just some radomly iserted cofiguratio. I our approach the first segmet (the surfactats head-group) comes i the place of the selected solvet molecule. The tail of legth l is grow i the usual way by geeratig a set of k trial positios. The acceptace rule for a Mote Carlo move ivolvig the chage from a solvet molecule s to a surfactat chai c is [ P acc (s c) = mi 1,exp{β µ+β U s h }W ()], (7.7) where U s h = U h U s is the eergy differece ivolved i chagig the solvet to a head-group h. W () = l i=1 w i is the socalled Rosebluth factor for the ewly iserted tail, with w i = k j=1, where U e βuj j is the total eergy of the j th trial prositio of the i th bead. A trial tail-bead is selected with a probability p i = e βui /w i. Similarly, a chage from surfactat to solvet is accepted accordig [ P acc (c s) = mi 1,exp{ β µ β U s h } 1 ], (7.8) W (o) with W (o) the Rosebluth factor for the old chai cofiguratio, calculated i the same way as W () but with oly k 1 radom trial positios, ad icludig the old positio of the i th bead as the k th positio [17]. 161

5 7.3.3 Isobaric MC I additio to a efficiet isertio algorithm to esure a fixed chemical potetial differece, proper samplig also demads thermal equilibrium. I a Mote Carlo simulatio, this is esured by particle displacemets. Additioally, we require a costat pressure to allow the micelle to grow without raisig the pressure. Therefore, durig the samplig we occasioally, at radom itervals, perform volume chage MC moves [17] Determiatio of the CMC I priciple, by usig the CBMC semigrad algorithm, the micellar size distributio P(, µ) ca be obtaied as a fuctio of micellar cluster size. However, biasig techiques such as umbrella samplig have to be ivoked i order to obtai sufficiet statistics i the ulikely regios of the size distributio (see Sectio 6.6). We obtaied the etire distributio at a certai µ sim by restrictig the semigrad simulatio to widows with a hard boudary ad by applyig a biasig fuctio B(, µ sim ), tued to flatte the measured distributio P sample (, µ sim ) durig the samplig. I the aalysis of the results this biasig fuctio is added agai to the size distributios, so that the fial size distributio for µ sim is P() = P sample (, µ sim )exp [ B(, µ sim ) ]. (7.9) From this distributio the sigle micelle free eergy profile this G() ca be calculated by G() = k B T lp(, µ sim )+ µ sim +costat. (7.1) Oce this G() is kow, we ca calculate the distributios for every value of µ by simply ivertig Eq. (7.1) P(, µ) = exp[ βg()+β µ]/q, (7.11) where Q = exp[ βg()+β µ] is a ormalizig costat. To estimate the CMC from this distributio requires a proper defiitio of the CMC. Here we defie the CMC as the surfactat cocetratio at which half of the surfactats is isolated ad the other half is i aggregated form: P(1, µ) = M ip(i, µ), (7.12) i=2 where M is the aggregatio umber above which the micelle breaks up ito two aggregates. Although this defiitio is slightly arbitrary, the CMC is ot very sesitive to this defiitio. Solvig Eq yields µ CMC which i tur leads to the surfactat umber desity at the CMC ρ CMC by [ ] ρ CMC k B T l = µ CMC µ ex. (7.13) ρ s 162

6 Figure1: Cartooofsurfactath 1 t 4. Thehydrophobictail beads(darkcircles) ad the hydrophilic head (light circle) are coected by harmoic sprigs. The excess chemical potetial differece µ ex betwee surfactat ad solvet has to be determied i a separate simulatio at costat NVT, by meas of the Widom particle isertio techique. Also here we make use of the CBMC algorithm[17]. 7.4 Surfactat Model We base our surfactat model o the trucated ad shifted Leard-Joes potetial. To allow variety i the stregth of attractio betwee two particles of species i ad j we use a species depedet LJ potetial [ (σij ) 12 ( ) ] 6 φ LJ σij (r ij,ǫ) = 4 ǫ, (7.14) where r ij is the distace betwee two particles of species i ad j, σ ij is the rage of iteractio of the potetial for two species i ad j, ad ǫ is the depth. The iteractio rage parameters σ ij were calculated from the stadard mixig rule σ ij = 1 2 (σ i + σ j ), where σ i ad σ j are the particle sizes of species i ad j, respectively. To avoid tail correctios, we trucate the potetial at a cutoff r c = 2.5σ ij, ad shift it to obtai a smooth potetial at the cutoff { φ φ TS (r ij,ǫ) = LJ (r ij,ǫ) φ LJ (r c,ǫ) r ij r c, (7.15) r ij > r c Here we use reduced uits. The reduced uit of eergy is ǫ, of legth σ, of desity σ 3, of pressure σ 3 ǫ ad of temperature ǫ/k B. The solvet molecules s are modeled by stadard (ǫ s = ǫ,σ s = σ) LJ particles. The surfactats c cosist of a chai of LJ particles (a head group h ad tail beads t) liked together by a harmoic potetial (see Fig. 1). r ij φ harm (r ij ) = 1 2 k harm(r ij r eq ) 2, (7.16) where r eq is the average equilibrium distace betwee beads ad k harm is the sprig costat. The harmoic equilibrium distace was set to r eq = σ ij ad the sprig costat was set to k harm = 5, esurig that 98% of the bod legths were withi 2% of the average value of r eq. May values for the iteractio parameters i the LJ model will iduce surfactat behavior, as log as the tail-beads behave oil-like ad the headbeads water-like. Oe requiremet is that the surfactat tail should dislike the solvet ad the head group. This ca be reproduced, for istace, by the r ij 163

7 settig the cutoff radius r c = 2 1/6 σ ij, the miimum i the potetial, thus oly retaiig the repulsive part of the LJ potetial. However, the parameters should also be such that spherical micelle formatio is favored. I particular, the size of the head group should be large eough so that spherical micelles are preferred istead of other topologies [19]. Table 1: Iteractio parameters for the model surfactats. Top: Values for the cutoff radius r c i terms of the size parameter σ ij = (σ i +σ i )/2 for the solvet (s), head group (h) ad tail bead (t). Trucatig ad shiftig at r c = 2 1/6 σ ij leaves oly the repulsive part of the LJ potetial, at r c = 2.5σ ij the attractive part is cosidered as well. Bottom: Size parameters for surfactats A, B, C, D ad E. All values i reduced uits. r c /σ ij s h t s /6 h /6 2 1/6 t 2 1/6 2 1/6 2.5 surfactat ame σ s σ h σ t A h 1 t B h 1 t C h 1 t D h 1 t E h 1 t ITable1wegivether c iteractioparametersusedithiswork. Ther c values effectively iduce hydrophillicity of the surfactat head ad hydrophobicity of the tail. To obtai spherical micelles the head group eeds to be sigificatly larger i size tha the tail beads [19]. We achieve just that by chagig the σ h /σ t ratio. The head group size of our surfactats should be about 1.5 to 2. times as largeas a tail-bead. We modeled a h 1 t 4 surfactat with oe head group ad four tail beads, usig four differet head ad tail particle sizes (deoted A, B, C ad D) ad a h 1 t 5 surfactat with five tail groups deoted E. (see Fig. 1 ad table 1.) For surfactat h 1 t 4 we obtai spherical micelles oly whe usig bulky head group, for istace, whe σ h = 2σ t. The choice for the iteractio rage parameters σ ij has cosequeces for the mappig of the coarse-graied potetial o realistic systems. Although the diameter of the tail bead i surfactat B is oly 75% of that of the solvet, its volume is about 2.4 times as small. Assumig that oe tail bead cosists of three CH 2 groups (which is reasoable if we wat to compare to experimetal surfactats), we ca say that there are approximately seve waters i oe solvet bead, which is rather high. Surfactat B is simulated at <ρ s >=.7σs 3 (see Sec. 7.5), resultig i a molecular legth scale of σ s m, give that water has a desity of ρ H2O = mol m 3 at room temperature. The average separatio betwee a carbo atom at positio m ad a carbo atom 164

8 at positio m +4 i a liear alkae is 3.8 Å. This distace should correspod to the bead separatio i the tails σ t =.75σ s 4. Å. Hece, the coarsegraiig of surfactat B is reasoable. The tail particles of surfactats C, D ad E have the same size as the solvet particles. For these surfactats we also use a coarse-graiig mappig of three CH 2 groups per tail bead ad for the solvet three waters per bead. The surfactats C,D ad E were simulated at <ρ s >=.6σs 3 (see Sec. 7.5), givig a legth scale of σ s = σ t 3.8 Å, also a very reasoable value. 7.5 CMC For surfactats A ad B, the simulatios were carried out at T = 1., N = 288 ad at p =.6. This pressure correspods to a solvet desity of ρ s =.7, well withi the liquid regio of the trucated ad shifted LJ phase diagram [17]. For surfactat C, D ad E, we imposed a pressure of p =.13 with a correspodig desity of ρ s =.6. This desity is still withi the liquid regio of the trucated ad shifted LJ phase diagram. The micellar size distributio was sampled i the semi-grad caoical isobaric esemble usig umbrella samplig, as described i Sectio 7.3. Dividig the etire rage of micellar sizes = 6 ito several widows of aroud 1 uits, we performed idepedet samplig for each widow. The applicatio of a biasig potetial i each widow esures that all micelle sizes i the widow were equally frequetly visited, which ehaces the statistics sigificatly. Afterward, this bias was removed by reweightig the observed distributio usig Eq The biasig fuctio itself ca be obtaied by a iterative procedure. First, we performed a ubiased simulatio ad obtaied a estimate for the free eergy from the resultig distributio usig Eq The egative of this free eergy fuctio was the used as a biasig potetial i the ext iteratio. To obtai good statistics, we repeated this procedure util the samplig distributio was almost uiform. Fially, the data sets for all widows were fitted simultaeously to the same 6 th order polyomial usig the additive costat of Eq. 7.1 as a additioal fittig parameter. Theexcesschemicalpotetialdifferece µ ex betweesurfactatadsolvet was determied i a separate simulatio at costat NVT usig the Widom isertio method [17]. Because the desity of the solvet is that of a liquid, we ca expect that the surfactat solubility is low. The sigle micelle free eergies as a fuctio of aggregatio umber for all surfactats at the computed µ CMC are plotted i Fig. 11. These curves all show a barrier betwee the sigle surfactat ad the micelle. Figure 11 also shows the correspodig size distributios, which have shapes typical for micelles. For surfactats A ad B the most probable micelle size is 2 to 25 surfactats. This size is 2 ad 3 for surfactats D ad E respectively. The optimal micelle size of surfactat D is lower tha that of A ad B, although the head to tail size ratio is the same. While this differece could be caused by the differet imposed pressures, it is more likely that the relatively smaller solvet particles peetrate deeper ito the micelle, ad effectively icrease the headgroup-headgroup repulsio, thus givig rise to smaller optimal micelle sizes. 165

9 G() [k B T] P() G() [k B T] P() G() [k B T] P() G() [k B T] P() G() [k B T] P() Figure 11: Left: Free eergy profiles of a sigle micelle at the CMC as a fuctio of umber of surfactats (aggregatio umber). Right: a zoom i o the correspodig micellar size distributio. From top to bottom the curves correspod to surfactat A, B, C, D ad E. The solid lies are the oliear fit to the theory by Chadler et al. [18].

10 Table 2: Chemical potetial differeces at the CMC ad correspodig CMC s for the surfactats studied. Usig the Widom isertio method, the solvet excess chemical potetials were estimated βµ ex s = 2. ad βµ ex s = 2.5 for the LJ fluid at ρ s =.7σs 3 ad ρ s =.6σs 3 respectively. The error i the last decimal is give i paretheses. surfactat ame σ h σ t µ ex µ CMC ρ CMC CMC (M) A h 1 t (1).2 5.3(5) (4) 1 4 B h 1 t (6).8 7(4) 1 7 6(3) 1 5 C h 1 t (1) (3) (3) 1 4 D h 1 t (1) (5) (5) 1 4 E h 1 t (2) (1) 1 7 5(1) 1 5 Itable2welistthecalculatedCMC sforallsurfactatsaswellastheexcess chemical potetial differeces µ ex betwee the solvet ad the surfactats. Iterestigly, the calculated CMC s lie withi the experimetal rage of oioic surfactats, provided oe assumes three CH 2 groups per tail bead. At first sight it seems strage that µ ex for surfactat D is lower tha that of surfactatcas the headgroupis larger. However, ifσ h is icreased, ot oly the repulsive rage of the iteractio potetial is icreased but also the attractive part, leadig to a lower value of µ ex. The computatio of µ ex is proe to sigificat statistical errors due to the difficulty of isertios durig the Widom procedure. It is this statistical error that largely determies the error i the CMC, rather tha that of the free eergy profile. The CMC s of surfactats A ad B were determied from simulatio data at the same pressure ad thus at almost the same solvet desity. Surfactat B is larger tha A ad should therefore have lower solubility i the LJ fluid. As a result its CMC should also be lower tha that of surfactat A. This is cofirmed by the simulatio results: surfactat B has a 7-fold lower CMC tha surfactat A. Oe would expect the eve larger surfactat C to have a lower CMC tha B. However, surfactat C was simulated at a lower pressure(desity) ad has therefore a higher solubility. The higher CMC of about 5 times that of surfactat B, is probably caused by this effect. Comparig the CMC values for surfactat C, D ad E (which were determied uder the same coditios), we ca coclude that icreasig the head size of the surfactat results i a higher CMC, whereas icreasig the tail legth lowers the CMC. These treds are also observed i experimets. Huibers et al. discussed experimetal CMC s of 77 oioic surfactats [2], of which we list a small selectio i Table 3. These liear surfactat have a relatively small but bulky head group, ad as such mostly resemble our model surfactats. Assumig three CH 2 groups per tail bead we ca map the h 1 t 4 surfactat o a C12 surfactat, ad the h 1 t 5 o a C15 surfactat. The simulated CMC s lie all withi the experimetal rage. If we cosider icreasig tail legth by goig from C12 to 167

11 Table 3: Experimetal CMC s at T = 298 K. surfactat CMC (M) Ref. Liear alkylethoxylates decyl triethylee oxide (C1E3) dodecyl diethylee oxide (C12E2) dodecyl triethylee oxide (C12E3) dodecyl octaethoxylate (C12E8) petadecyl octaethoxylate (C15E8) Alkae diols 1,2-decae diol (C1DIOL) ,2-dodecae diol (C12DIOL) ,3-petadecae diol (C15DIOL) Alkyl glucose ethers -dodecyl-β-d-glucoside (C12GLUC) sucrose moolaurate(c12sucr) C15 while keepig the same headgroup we see a decrease i the CMC, which is reproduced by our results. Icreasig the size of the headgroup from C12E3 to C12E8 icreases the CMC, a tred that is also see i our simulatios. We are aware that comparig simulatio results to these experimetal CMC s is dagerous, as our coarse-graied potetials ad coditios ot ecessarily reproduce all relevat properties of the experimetal surfactat. I particular, we caot expect the simple LJ solvet at our reduced temperatures ad pressure to reproduce liquid water at ambiet coditios. Also, it is ot clear that the average micellar size, or aggregatio umbers we obtai i the simulatios are preset i experimetal micelles. Nevertheless, this exercise shows that simulated surfactat CMC s ca i priciple be mapped o real surfactats. 7.6 Coclusio I summary, we have show that the CMC ca be determied accurately for coarse-graied LJ-type surfactats i a solvet, by computig the free eergy of a sigle micelle as a fuctio of aggregatio umber employig the isobaric SGCMC esemble i combiatio with CBMC ad umbrella samplig. We also obtaied a micellar size distributio from the free eergy of micelle formatio. The calculated CMC s are withi the rage of experimetal values. Icreasig the head group size gives a higher CMC ad icreasig surfactat tail legth lowers the CMC, two treds that are also see i experimet. 168

12 Refereces [1] D. Chadler, Itroductio to Moder Statistical Mechaics (Oxford Uiversity Press, New York, 1987). [2] J.-L. Barrat ad J.-P. Hase, Basic Cocepts for Simple ad Complex Liquids (Cambridge U.P., Cambridge, 23). [3] P.M. Chaiki ad T.C.Lubesky, Priciples of Codesed Matter Physics (Cambridge U.P., Cambridge, 2). [4] J. L. Lebowitz, J. K. Percus, ad L. Verlet, Phys. Rev. 153, 25 (1967). [5] J. I. Siepma, I. R. McDoald, ad D. Frekel, Joural of Physics Codesed Matter 4, 679 (1992). [6] W. W. Wood, J. Chem. Phys. 48, 415 (1968). [7] V. F. Weisskopf, Tras NY Acad. Sci. Ser. II 38, 22 (1977). [8] H.C. Loguet-Higgis ad B. Widom, Mol. Phys. 8, 549 (1964). [9] P.C. Hemmer ad J.L. Lebowitz, i Critical Pheomea ad Phase Trasitios 5b, edited by C. Domb ad M. Gree (Academic Press, New York, 1976). [1] P. Bolhuis ad D. Frekel, Phys. Rev. Lett. 72, 2211 (1994). [11] J. P. Hase ad I. R. McDoald, Theory of Simple Liquids (Academic Press, (2d editio), Lodo, 1986). [12] R. P. Feyma, Statistical Mechaics (Bejami, Readig (Mass.), 1972). [13] J.S.Rowliso, i Studies i Statistical Mechaics, Vol. XIV, edited by J.L.Lebowitz (North Hollad, Amsterdam, 1988). [14] P. Ehrefest, Proc. Amsterdam Acad. 36, 153 (1933). [15] L.D. Ladau, Z. Phys. Sovjetuio 11, 26 (1937). [16] L.D. Ladau, Z. Phys. Sovjetuio 11, 545 (1937). [17] Frekel, D.; Smit, B. Uderstadig Molecular Simulatio; Academic Press: Sa Diego, CA, secod ed.; 22. [18] Maibaum, L.; Dier, A.R.; Chadler, D. J. Phys. Chem. B 24, 18, [19] Isrealachvilli, J.N. Itermolecular ad Surface Forces; Academic Press: Lodo (UK), secod ed.; [2] Huibers, P.D.T.; Lobaov, V.S.; Katritzky, A.R.; Shah, D.O.; Karelso, M. Lagmuir 1996, 12,