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A spin-1 Ising model to describe amphiphile monolayer phase transition
A Spin-l king Model to Describe Amphiphile Monolayer Phase Transition JEAN-FRANCOIS
BARET
AND
JEAN-LUC
FIRPO
Dkpartement de Physique des Liquides, Universitk de Provence, 13331 Marseille Cedex 3, France Received July 8, 1982; accepted January 11, 1983 The liquid expanded-liquid condensed phase transition observed with many amphiphile monolayers is a second-order transition since there is a compressibility discontinuity on the isotherm. To describe such a transition two molecular states and the vacancies are included in a model isomorphic to a spin- 1 king model. The computed isotherms compare well with the data only if the two molecular states are symmetrical (i.e., when the interaction energies J AA and Jan are equal). Therefore, it may be concluded that the internal configuration of the molecule is the same in both molecular states, only the orientation of the molecule is involved in the process underlying this transition.
I. INTRODUCTION
Amphiphile substances spread on top of water to form monolayers. When compressed in two dimensions these monolayers display phase transitions which in general do not bear the essential feature of first-order transitions, viz., a density discontinuity at the transition pressure (1) (here, surface pressure). It seems that a true first-order transition has been evidenced for pentadecanoic acid at very low density corresponding to a two-dimensional gas-liquid transition (2, 3). But, beside this example all the monolayer transitions observed with fatty alcohols (4), acids (5) or amines (6), with phospholipids (7) or diglycerides or monoglycerides (8), or with substances with two polar groups show a single singularity along the isotherms, i.e., no density gap. To describe these transitions several mechanisms have been proposed. The most widely described in the literature is the so-called chain-melting process which is deduced from what is observed on X-rays diffraction patterns of bilayers and membranes when they are heated (9). Indeed below a given temperature, 4 1“C for dipalmitoyl lecithin, the pattern shows well-defined rings correspond-
ing to the regular lattice made by the wellordered layers of zig-zag hydrocarbon chains, then above this transition temperature, a diffuse ring, similar to those obtained from liquids, is observed. For many authors (10-14) the process underlying the monolayer phase transitions would be very similar to what occurs in bilayers: the monolayer would go from a well-ordered lattice of parallel alltrans chains at low temperature-high surface pressure, to a disordered state in which the chains contain many gauche bonds. This transition would be therefore first-order since the chain melting process yields a latent heat. Because this is at variance with the apparent shape of the isotherms some authors (15- 19) have proposed alternative explanations relying on the idea that the transition is dominated by an intermolecular ordering process in which the intramolecular conformational changes play no or little part. Indeed, Landau (20) has shown that a second-order phase transition to occur needs a change in the symmetry described by means of an order parameter which is zero in the more symmetric phase and different of zero in the less symmetric one. This kind of situation is achieved for instance when two molecular states like lying on the interface “in direction 487 0021-9797183 $3.00
Journal o,fCollord and Inrer..ce Science, Vol. 94, No. 2, August 1983
Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
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x” or “in direction y” or like “lying down” and “standing up” are assumed to be accessible for a molecule in the monolayer, the order parameter is zero when as many molecules occupy both states and it is different of zero when the two states become occupied unequally. Ising models or more precisely their lattice gas isomorphisms have been used as well to describe a first-order transition resulting of a chain-melting process than to describe a collective ordering which yields a second-order transition. Thus, Ohki (21) has applied a spin- l/2 Ising model to a two-dimensional lattice gas. In this model each lattice sites may be either full or empty (a spin- l/2 may have only two states: “up” or “down”) and no ordering can occur, the only parameter is the density. The surface pressure-area isotherms obtained have the discontinuity in area which characterizes the first-order transition. CaillC et al. (22) has also used a spinl/2 Ising model but in a different way, they consider that a site may be occupied by a molecule either in state cyor in state /?, there is no vacant sites; an ff molecule occupies one site whereas a /3 molecule occupies m sites. As it could be expected they find a coexistence curve corresponding to phase separation between the a! and the 0 molecules, here again the transition is first-order. In order to describe a collective ordering of a monolayer spin-l Ising models have to be used, in these models beside the empty state, state “0,” there are two ways of occupying a site either with a molecule in state “1” or in state “2.” These two states can be two orientations of the molecules in the plane of the interface as proposed by Bell et al. (15) or Firpo et al. (17-19) or they could be the a! and /l states of Caille et al. (to transform Caille’s model in spin-l model vacant sites should be added). II.
SPIN-1
ISING
MODEL
The spin- I Ising model or lattice-gas model cannot be solved exactly, even in the simpler Journn/o~Col/oidandInlerfaceSc~ence.
Vol. 94, No. 2, August 1983
FIRPO
case of spin- l/2 Ising model an exact solution has been obtained only in the two-dimensional case (Onsager). Therefore one has to use approximation, the simplest one is the molecular-field approximation or mean-field approximation (which is known in physical chemistry as the Bragg-Williams or regular solution approximation), in which the local order is supposed to be identical with the global order. This means that the number of nearest neighbors of one species is assumed to depend only on the number of molecules of this species and not on their interactions. The basic idea (25) underlying the spin-l Ising model is to attribute to each site of the lattice (here a plane lattice which can be square or hexagonal) a first number Si which can be + 1, - 1, or 0 and then to introduced a second number Qi = (Si)* which takes the values 1 or 0. When the site is empty Si = Qi = 0, when it is filled by a type A molecule Si = + 1 and Qi = + 1, when it is a type B molecule Si = - 1 and Qi = + 1. Therefore, Q is a parameter characterizing the “fullness” of the lattice sites, its average value over the whole lattice Q = (Qj) is the density Q = NA + N,/N = l/a (a being the area occupied by one molecule). Whereas Si indicates which species is on a given site, its average value M = (Si) is an order parameter which indicates the difference between the concentrations of the two species
(JV = total number of sites). If the molecules A interact with each other with an energy JAA, the molecules B with an energy JBB, the couples A-B with an energy JAB, the Hamiltonian of the system is then written
+ J&Pf’PB
+ PFPf)]
- c (PAP4+ Pd?>,
PI
AMPHIPHILE
MONOLAYER
where PA and & are the chemical potential of the two species, i.e., the free energy per molecule which should be introduced since the number of molecules is not fixed. Pf and PF are
So, for each site Pp (or PF) equals one if the site is occupied by an A (or B) molecule, and is zero otherwise. The Hamiltonian of the system becomes 2 = -K C QiQj - J C: SiSj i,j i,j I
-L C (SiQj+ QS’j>, [Al Li where JBB
+ ~JAB),
J = a (JAa + JBB - ~JAB),
1 L = 4 (JAAD
=
&A
+
JBB),
YB),
H
=
+ (PA
-
PB)
The molecular-field approximation leads to the following expression for the surface pressure II = -KQ2 - JM2 - 2LMQ - kTln
CASE OF MODELS
In a phase of maximum no hole) Eq. [6] becomes
SPIN-l/2
density (Q = 1,
H’ = ; (WA- PB)+ ; (JAA-
Pr = i (Qi - Sl).
(JAA+
PARTICULAR ISING
[31 with
L
K = a
III.
489
TRANSITION
M = tanh [(H’ + 2JM)/kT]
P; = ; (Qi + S,)
-DCQi-ZZSi i
PHASE
(1 - Q)
[5]
in which M and Q are related by M = Q tanh [(2JM + 2LQ + H)/kT].
[6]
(the factor l/2 z, where z is the coordination number of the lattice, have been implicity included in the above equations).
JBB).
This equation is in fact a mean-field version of an argument given by Bell and Fairbairn (34). This is the Lee-Yang lattice gas result obtained in a case where the usual “filled” and “empty” sites are replaced by sites occupied either by A or by B molecules. This kind of spin- l/2 Ising model has been used to describe monolayer behavior by Ohki (21) for one molecular species and holes and by Caille et al. (22) with two molecular species a and 0. Caille et al. consider only one type of interaction as it should be in a Lee-Yang model but they incorrectly assume that it is JAA instead of J = 1/4(JAA + JBB - 2JAB). This energy describes the interchange of one molecule A from the bulk of phase A into the bulk of phase B with one molecule of phase B. It is this energy which intervenes in the theory of regular solution, it represents the departure from ideality of the solution since it is zero for an ideal solution. Moreover, these authors have not constrained molecules to occupy only one site since molecules p occupy an area which is m times the area occupied by an LYmolecule, m being not necessarily an integer. Whereas the possibility for a molecule to occupy several sites seems to be easily acceptable, the fact that a molecule may occupy a fractional number of site appears at first sight in contradiction with the discrete nature of a lattice model. In fact the size of the unit cell of the lattice should not be chosen as the area of the smaller molecular species but rather such as it makes all the numbers of occupied sites integer, e.g. if m = 1.5 the unit site is taken
490
BARET
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as half of the molecule LYarea thus three sites are occupied by a molecule p. IV. PARTICULAR
CASE WITH
L=OANDH=O
If the interactions between A molecules are identical to these between B molecules (JAA = JBB and ,.&A= /LB) L = 0 and H = 0. Then Eqs. [5] and [6] become II=-KQ2-JM2-kTln(l-Q)
[5’]
the density l/2 for classical Ising models to a value which depends on the molecular asymmetry p. Kaye (35) studied a model intermediate between those of Refs. (17- 19) and that of Ref. ( 15), in which the asymmetry of the molecule is p = 2 but the ordering of the elements is promoted by energetic as well as steric effect. Dunne and Bell (36) have incorporated configurational effects by including an extended (dimer) state as well as the two symmetrical orientations.
with M = Q tanh (2JM/KT).
[@I
Bell et al. ( 15) have described lecithin monolayer data with a model very similar to this one. Indeed, they consider beside the vacancy two orientational states 1 and 2. The interaction energy 11 is of course equal to the interaction 22. To calculate the degeneracy of a lattice state they use the quasi-chemical (Bethe) approximation (23) rather than the mean-field approximation but this is not changing much the results. Firpo et al. (1719) have used a model in which the monolayer molecules may exhibit two orientational states in the interface plane, either lying along the x direction or along the y direction. In their model the molecule occupies p lattice sites, whereas in Bell et al. model it occupies only one, therefore they have to compute the number of possible configurations corresponding to a given number of molecules in each orientational state by taking into account the fact that no crossing of the molecules is possible. This calculation has been done in the general three-dimensional case by DiMarzio (24) and it is the restriction to the planar case which is used by these authors. The results are qualitatively similar to those obtain by other spin-l Ising models, like those of Bell et al. or those described in the next section, i.e., there are two transitions one corresponding to the condensation (first order) and the other one to the orientation (second order). The most important difference is the displacement of the condensation critical point which moves from Journal of Colloid and lnterfice Snence, Vol. 94, No. 2,
August 1983
V. GENERAL
CASE
In its general form the spin-l Ising model leads to Eqs. [5] and [6] in which the interactions between A-A molecules, B-B molecules, and A-B molecules intervene as parameters. This means such a model may be useful to describe a process in which, beside a variation of density (taken into account by the presence of holes, order parameter Q), the relative concentration of the two species A and B may vary (order parameter M). These two molecular species may be simply two different chemical compounds and then the ordering transition corresponding to the order parameter M is a demixtion; or these species are two different physical states of the same chemical, like for instance the two orientational states considered in the previous section. Other double states could be considered: upright and lying-down molecules, alltrans and gauche molecules (13), solid and liquid molecules (25), but as far as the model is concerned what matters is not the qualitative nature of the states but whether JAA is different or not of JBB, in the latter case H = L = 0 and the M transition is second-order as described above, the former case is going to be described in details now. In order to draw the isotherms from Eq. [5] we need first to solve Eq. [6] with respect to M, i.e., to find the intersection between a y = tanh (f(M)) curve and a y = M/Q straight line. Differently to what occurs in the L = H = 0 case the argument of the hyperbolic tangent is not zero at the origin and
AMPHIPHILE
MONOLAYER
therefore A4 = 0 is no longer a solution. It results from this that II-Q plane is not anymore divided in two half-plane, one corresponding to M = 0 and the other one to A4 # 0, the border line being the second-order transition line as it was the case when L = H = 0 (26). But instead there is a competition between the increasing -kT In (1 - Q) term and the decreasing -(JM2 + 2LMQ) term, leading to the classical sigmoidal behavior with a maximum and a minimum from which are constructed the first-order transition plateaus. To make short, one may say that differentiation between the two intra-species interactions supresses the ordering transition. This result is not surprising. Since for fixed Q a change of variables (e.g., M/Q, M) transforms the Eq. [6] into the first equation of Section III. It is wellknown that the introduction of a constant
PHASE
491
TRANSITION
external field H’ into the spin-l/2 Ising model, described by the latter equation, removes the higher order transition. When H = 0 and L = 0 the abscissa M of the intercept jumps from a finite value to zero for a critical value of the slope of the straight lines, this value corresponds to the crossing of the second-order transition line. On the contrary, as shown on Fig. 1, when H or L are different from zero the abscissa of the intercept is monotonously decreasing toward the origin, and there is no more transition. The way variations in L are affecting the isotherms is evidenced on Figs. 2 and 3: the break point corresponding to the transition is rounded up significantly, even with values of L 2 orders of magnitude lower than K. When these computed curves are compared with the experimental isotherms it should be concluded that the interaction differences
1.6 -
1.2 -
Y
-
FIG. 1. y = tanh (2JA4 + 2LQ + H)/kT kT = 1.3, and l/Q = a = 1.05 (b), 1.25 (c),
(Curve a) and y = M/Q, 1.45 (d), 1.65 (e), 1.85 (f). Journal ofcolloid
for J = (minimum
1, H = 0.1, L = 0, area @ = 1)
and Inter/ace Scmce, Vol. 94, No. 2, August 1983
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n [dyne /cm I 30 -
25 -
20 -
15 -
10 -
I
25I
30I
351
40I
451
a [@I FIG. 2. Surface pressure (II) vs. area per molecule (a); k = 1.38 X lo-l6 erg OK-‘, minimum area a0 = 20 A’, J = 4 X IO-“’ erg, K = 10mL4erg, L = IO-l6 erg, H = 0, T = 280 (a), 300 (b), 320 (c), 340 (d) “K.
n [dyne km 1 30 -
25 -
20 -
15 -
10 -
25
30
35
40
45
a #I FIG. 3. Same as in Fig. 2 but L = 5 X lo-l6 erg. Journal of Colloid and Inlerjoce Science, Vol. 94, No. 2, August 1983
AMPHIPHILE
MONOLAYER
between species A and B must be zero or at least very small. Accordingly, the two species, A and B, should be either identical or at least very similar. Therefore, a model in which these two species are merely two symmetrical states like the two directions of the interface plane or “lying down” and “standing up”is more realistic than a model considering two really different species-like “trans” and “gauche” chains-for which the intra-species interactions are likely to be different. On Fig. 4 and 5 we have drawn isotherms for L = H = 0 showing how K controls the shapes of the curves. One notes that the abscissa of the transition points remains unchanged whereas the ordinate decreases when K increases. On the other hand, a change in J displaces both coordinates of the transition points, when J increases the ordinates II, decreases and the abscissa a, = 2J/aoT increases. Jis the parameter measuring the ease of the ordering transition, for large J the transition occurs at low surface pressure, large area. Rather, K scaled the surface pressures and the temperatures, it determines the crit-
PHASE
493
TRANSITION
ical parameters of the first-order condensation transition occuring at low pressure and large area that we do not consider here. VI.
COMPARISON MONOLAYER
WITH DATA
On the experimental isotherms obtain with fatty acids (27, 28) it appears that a chain length increase of one CH2 from Cl5 to C,6 (see Figs. 2 and 3 of Ref. (27)) displaces, for instance, the transition point of the 30°C isotherm from 13 dyn/cm, 31 A2 to 3.5 dyn/cm, 40 A2, or a chain length increase from C,4 to CL5 (see Fig. 1 from Ref. (29)) displaces the transition point at 25°C from 2 1 dyn/cm, 26 A2 to 9 dyn/cm, 33 A2. This behavior is similar to what has been seen to occur on computed isotherms when J increases. This is not surprising since J is the difference between intra-species and interspecies interactions, a measure of the departure from ideality, indeed if the two species A and B correspond to two orientational states, JAA = JBB is the interaction between
I
/
25
30
35
40
45 a
FIG. 4. Same
as in Fig.
THE
2 with
L = 0, J = 3 X 10-l“
erg, and K = 2 X lo-l4
[@I
erg.
Journal of CoNoid and Infer/ace Science. Vol. 94, No. 2, August 1983
494
BARET
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FIRPO
I 25
30
35
40
45
a [PI FIG.
5. Same
as in Fig. 2 with
L = 0, J = 3 X lo-l4
parallel molecules, whereas JAB is the interaction between perpendicular molecules. As the chain length increases the parallel interaction increases and the perpendicular one should remain more or less constant. There is another very striking similarity between experimental data and calculated isotherms, it is the superposition of the liquid expanded regions for fatty acids of different chain lengths at the same temperature. Thus, on Fig. 1 of Ref. (29) the expanded portion of the isotherms of pentadecylic and myristic acids are exactly the same, this can be related to the description of the expanded portion by an equation which does not contain J, since in this phase A4 = 0. Therefore, once again we are found supposing that J is a parameter linked to the chain length which determines the behavior of the condensed part of the isotherm. Differently, the expanded part is independent of the chain length, only the parameter K may change its shape. K plays also a role in the determination of the condensed region since it scales pressures and temperatures. It is likely that K expresses the interaction energy between the polar groups. Journal of CoNoid and Inferface Science, Vol. 94, No. 2. August 1983
erg,
K = 4 X lo-l4
erg.
This is consistent with the assumption that the gas-liquid-expanded transition is a condensation of a Coulombic gas (30). VII.
CONCLUSIONS
The sharp break point observed on experimental isotherms of many amphiphilic substances looks more like a singular point occurring at the crossing of a second-order transition line than like the onset of a condensation plateau since there is neither flat portion nor a second singular point corresponding to the coexistence zone. Therefore, it is worth looking for models yielding second-order transitions, spin- 1 Ising model includes a broad class of lattice systems which display, beside a first-order condensation transition, an ordering transition which privileges one of the species A(spin +l) or B(spin - 1). The results computed with such a model are closed to the experimental data when the two species are really symmetrical, i.e., when the JaA and JBB interactions are identical. This leads to the conclusion that the liquid-condensed phase differs from the
AMPHIPHILE
MONOLAYER
liquid expanded not by an internal state of the molecules but by a global order of the monolayer. The individual molecules in both phases are more or less in the same isomeric rotational state (same average number of gauche bonds) but the collective organization of the phases is different, the liquid expanded is an isotropic phase, whereas in the liquid condensed a privileged direction has appeared on the interface plane along which the molecules tend to line up. The liquid expanded-liquid condensed transition is then a two-dimensional (2-d) equivalent to the isotropic-nematic transition observed in bulk (3-d) liquid crystal. The reduction of the space dimensionality from 3-d to 2-d changes the nematic transition from a weak first-order to a second-order transition (31). Obviously, the converging features of the experimental isotherms and the isotherms computed from a spin- 1 Ising model are not sufficient to assert that the condensed phase is an oriented phase. Direct evidences are needed to identify clearly this phase. The main difficulty to show whether or not a nematic phase appears in the monolayer, is the limited range of the correlation length which could create oriented clusters of no more than some hundred molecules (32). Therefore, before any attempt to analyze the anisotropy of the monolayer one should first orient the autonomous clusters with respect to one another, this may be done with a field. A magnetic field may be used if the molecular susceptibility of the amphiphile is large enough. Similarly, global orientation of the clusters by an electric field is thinkable if the geometric sum of the dipole moments of the molecules belonging to the same oriented cluster is sufficiently large to make the field necessary to move the cluster against the thermal motion not innaccesible. The problem here is to obtain a water substrate very pure to minimize the current. Another way to orient the clusters could be to apply a shear flow by forcing the monolayer into a canal, this to operate requires that the clusters have an elongated shape. Once the monolayer has
495
PHASE TRANSITION
been globally oriented by one of the abovementioned fields then the anisotropy should be detected by a probe. Since the two directions of the interface plane correspond now to two different electron densities an electromagnetic radiation should be absorbed differently whether the polarizing plane corresponds to one or the other of these directions. Because of the minute amount of matter involved in a monolayer, in order to realize this dichroic experiment a device allowing several crossings of the interface should be set up. Evidence of the monolayer ordering in the liquid-condensed region may be obtained from a completely different point of view. Instead of a previous orientation of the clusters with respect to one another by an external field, a direct measurement of the fluctuations of the molecular orientations can be made by detecting the depolarization of a scattered beam (33). Indeed, the orientation fluctuation of a molecule in the expanded phase should be large and therefore the incident orientation of the polarization plane is averaged out in the scattered beam, whereas in the condensed phase the molecules keeping approximately the nematic direction, the polarization plane is conserved in the scattered beam. Of course, these experiments may yield negative results, i.e., the condensed-liquid phase may not be an anisotropic phase, then the whole problematics of the liquid expanded-liquid condensed transition should be raised in new terms: how a transition which is not an orientational transition may display a second-order discontinuity? ACKNOWLEDGMENT The authors thank Genevi&ve Danoy for her technical assistance. REFERENCES 1. Baret, J. F., Progr. Surf:Membr. Sci. 14,291 (1981). 2. Kim, M. W., and Cannell, D. S., Phys. Rev. A 13, 411 (1976). Journal of Co/hid and Inter$m
Science. Vol. 94, No. 2. August 1983
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3. Hawkins, G. A., and Benedek, G. B., Phys. Rev. Lett. 32, 524 (1974). 4. Harkins, W. D., and Copeland, L. E., J. Chem. Ph$s. 10, 272 (1942). 5. Hard, S., and Neuman, R. D., J. Colloid Interface Sci. 83, 315 (1981). 6. Adam, N. K., Proc. Roy. Sot. (London) A 126,526 (1930). I. Albrecht, O., Gruler, H., and Sackmann, E., J. Phys. 39, 301 (1978). 8. Phillips, M. C., and Hauser, H., J. Colloid Interface Sci. 49, 31 (1974). 9. Albon, N., J. Chem. Phys. 78, 4676 (1983). 10. Scott, H. L., J. Theor. Biol. 46,241 (1974); .I. Chem. Phys. 62, 1347 (1975). 11. MarEelja, S., Biochim. Biophys. Acta 367, 165 (1974). 12. Jacobs, R. E., Hudson, B., and Anderson, H. C., Proc. Nat. Acad. Sci. USA 72, 3993 (1975). 13. Caille, A., and Agren, G., Canad. J. Phys. 53, 2369 (1975). 14. Nagle, J. F., J. Chem. Phys. 63, 1255 (1975); J. Membr. Biol. 27, 233 (1976). 15. Bell, G. M., Mingins, J., and Taylor, J. A. G., J. Chem. Sot. Faraday II 74, 223 (1978). 16. Kaye, R. D., and Burley, D. M., J. Phys. A 7, 1303 (1974). 17. Firpo, J. L., Dupin, J. J., Albinet, G., Bois, A. G., Casalta, L., and Baret, J. F., J. Chem. Phys. 68, 1369 (1978). 18. Dupin, J. J., Firpo, J. L., Albinet, G., Bois, A. G., Casalta, L., and Baret, J. F., J. Chem. Phys. 70, 2357 (1979).
Journal of Co/bid and Interfnce Science. Vol. 94, No. 2. August 1983
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19. Firpo, J. L., Dupin, J. J., Albinet, G., Baret, J. F., and Caillt, A., J. Chem. Phys. 74, 2569 (1981). 20. Landau, L., and Lifchitz, E., in “Statistical Physics,” 2nd ed. Pergamon, New York, 1968. 21. Ohki, S., J. Theor. Biol. 15, 346 (1967). 22. Caillt, A., Rapini, A., Zuckermann, M. J., Cros, A., and Doniach, S., Canad. J. Phys. 56,348 (1978). 23. Hill, T. L., in “Statistical Thermodynamics,” 2nd ed. Addison-Wesley, Reading, Massachusetts, 1962. 24. DiMarzio, E. A., J. Chem. Phys. 35, 658 (1961). 25. Lajzerowicz, J., and Sivardiere, J., Phys. Rev. A 11, 2079 (1975). 26. Sivardiere, J., Phys. Stat. Sol. B 103, 823 (1981). 27. Harkins, W. D., Young, T. F., and Boyd, E., J. Chem. Phys. 8, 954 (1940). 28. Adam, N. K., and Jessop, G., Proc. Roy. Sot. (London)
A112,
362 (1926).
29. Nutting, G. C., and Harkins, W. D., J. Amer. Chem. Sot. 61, 1180 (1939). 30. Albon, N., and Baret, J. F., J. Colloid Znterjkce Sci. 92, 545 (1983). 31. Baret, J. F., Bois, A. G., Dupin, J. J., and Firpo, J. L., J. Colloid Interface Sci. 86, 370 (1982). 32. de Gennes, P. G., Symp. Faraday Sot. 5, 16 (1971). 33. Langevin, D., private communication. 34. Bell, G. M., and Fairbaim, W. M., Mol. Phys. 4, 481 (1961). 35. Kaye, R. D., Physica 18A, 347 (1981). 36. Dunne, L. J., and Bell, G. M., J. Chem. Sot. Faraday Trans. ZZ 76,43 1 (1980).
BARET
AND
JEAN-LUC
FIRPO
Dkpartement de Physique des Liquides, Universitk de Provence, 13331 Marseille Cedex 3, France Received July 8, 1982; accepted January 11, 1983 The liquid expanded-liquid condensed phase transition observed with many amphiphile monolayers is a second-order transition since there is a compressibility discontinuity on the isotherm. To describe such a transition two molecular states and the vacancies are included in a model isomorphic to a spin- 1 king model. The computed isotherms compare well with the data only if the two molecular states are symmetrical (i.e., when the interaction energies J AA and Jan are equal). Therefore, it may be concluded that the internal configuration of the molecule is the same in both molecular states, only the orientation of the molecule is involved in the process underlying this transition.
I. INTRODUCTION
Amphiphile substances spread on top of water to form monolayers. When compressed in two dimensions these monolayers display phase transitions which in general do not bear the essential feature of first-order transitions, viz., a density discontinuity at the transition pressure (1) (here, surface pressure). It seems that a true first-order transition has been evidenced for pentadecanoic acid at very low density corresponding to a two-dimensional gas-liquid transition (2, 3). But, beside this example all the monolayer transitions observed with fatty alcohols (4), acids (5) or amines (6), with phospholipids (7) or diglycerides or monoglycerides (8), or with substances with two polar groups show a single singularity along the isotherms, i.e., no density gap. To describe these transitions several mechanisms have been proposed. The most widely described in the literature is the so-called chain-melting process which is deduced from what is observed on X-rays diffraction patterns of bilayers and membranes when they are heated (9). Indeed below a given temperature, 4 1“C for dipalmitoyl lecithin, the pattern shows well-defined rings correspond-
ing to the regular lattice made by the wellordered layers of zig-zag hydrocarbon chains, then above this transition temperature, a diffuse ring, similar to those obtained from liquids, is observed. For many authors (10-14) the process underlying the monolayer phase transitions would be very similar to what occurs in bilayers: the monolayer would go from a well-ordered lattice of parallel alltrans chains at low temperature-high surface pressure, to a disordered state in which the chains contain many gauche bonds. This transition would be therefore first-order since the chain melting process yields a latent heat. Because this is at variance with the apparent shape of the isotherms some authors (15- 19) have proposed alternative explanations relying on the idea that the transition is dominated by an intermolecular ordering process in which the intramolecular conformational changes play no or little part. Indeed, Landau (20) has shown that a second-order phase transition to occur needs a change in the symmetry described by means of an order parameter which is zero in the more symmetric phase and different of zero in the less symmetric one. This kind of situation is achieved for instance when two molecular states like lying on the interface “in direction 487 0021-9797183 $3.00
Journal o,fCollord and Inrer..ce Science, Vol. 94, No. 2, August 1983
Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
488
BARET
AND
x” or “in direction y” or like “lying down” and “standing up” are assumed to be accessible for a molecule in the monolayer, the order parameter is zero when as many molecules occupy both states and it is different of zero when the two states become occupied unequally. Ising models or more precisely their lattice gas isomorphisms have been used as well to describe a first-order transition resulting of a chain-melting process than to describe a collective ordering which yields a second-order transition. Thus, Ohki (21) has applied a spin- l/2 Ising model to a two-dimensional lattice gas. In this model each lattice sites may be either full or empty (a spin- l/2 may have only two states: “up” or “down”) and no ordering can occur, the only parameter is the density. The surface pressure-area isotherms obtained have the discontinuity in area which characterizes the first-order transition. CaillC et al. (22) has also used a spinl/2 Ising model but in a different way, they consider that a site may be occupied by a molecule either in state cyor in state /?, there is no vacant sites; an ff molecule occupies one site whereas a /3 molecule occupies m sites. As it could be expected they find a coexistence curve corresponding to phase separation between the a! and the 0 molecules, here again the transition is first-order. In order to describe a collective ordering of a monolayer spin-l Ising models have to be used, in these models beside the empty state, state “0,” there are two ways of occupying a site either with a molecule in state “1” or in state “2.” These two states can be two orientations of the molecules in the plane of the interface as proposed by Bell et al. (15) or Firpo et al. (17-19) or they could be the a! and /l states of Caille et al. (to transform Caille’s model in spin-l model vacant sites should be added). II.
SPIN-1
ISING
MODEL
The spin- I Ising model or lattice-gas model cannot be solved exactly, even in the simpler Journn/o~Col/oidandInlerfaceSc~ence.
Vol. 94, No. 2, August 1983
FIRPO
case of spin- l/2 Ising model an exact solution has been obtained only in the two-dimensional case (Onsager). Therefore one has to use approximation, the simplest one is the molecular-field approximation or mean-field approximation (which is known in physical chemistry as the Bragg-Williams or regular solution approximation), in which the local order is supposed to be identical with the global order. This means that the number of nearest neighbors of one species is assumed to depend only on the number of molecules of this species and not on their interactions. The basic idea (25) underlying the spin-l Ising model is to attribute to each site of the lattice (here a plane lattice which can be square or hexagonal) a first number Si which can be + 1, - 1, or 0 and then to introduced a second number Qi = (Si)* which takes the values 1 or 0. When the site is empty Si = Qi = 0, when it is filled by a type A molecule Si = + 1 and Qi = + 1, when it is a type B molecule Si = - 1 and Qi = + 1. Therefore, Q is a parameter characterizing the “fullness” of the lattice sites, its average value over the whole lattice Q = (Qj) is the density Q = NA + N,/N = l/a (a being the area occupied by one molecule). Whereas Si indicates which species is on a given site, its average value M = (Si) is an order parameter which indicates the difference between the concentrations of the two species
(JV = total number of sites). If the molecules A interact with each other with an energy JAA, the molecules B with an energy JBB, the couples A-B with an energy JAB, the Hamiltonian of the system is then written
+ J&Pf’PB
+ PFPf)]
- c (PAP4+ Pd?>,
PI
AMPHIPHILE
MONOLAYER
where PA and & are the chemical potential of the two species, i.e., the free energy per molecule which should be introduced since the number of molecules is not fixed. Pf and PF are
So, for each site Pp (or PF) equals one if the site is occupied by an A (or B) molecule, and is zero otherwise. The Hamiltonian of the system becomes 2 = -K C QiQj - J C: SiSj i,j i,j I
-L C (SiQj+ QS’j>, [Al Li where JBB
+ ~JAB),
J = a (JAa + JBB - ~JAB),
1 L = 4 (JAAD
=
&A
+
JBB),
YB),
H
=
+ (PA
-
PB)
The molecular-field approximation leads to the following expression for the surface pressure II = -KQ2 - JM2 - 2LMQ - kTln
CASE OF MODELS
In a phase of maximum no hole) Eq. [6] becomes
SPIN-l/2
density (Q = 1,
H’ = ; (WA- PB)+ ; (JAA-
Pr = i (Qi - Sl).
(JAA+
PARTICULAR ISING
[31 with
L
K = a
III.
489
TRANSITION
M = tanh [(H’ + 2JM)/kT]
P; = ; (Qi + S,)
-DCQi-ZZSi i
PHASE
(1 - Q)
[5]
in which M and Q are related by M = Q tanh [(2JM + 2LQ + H)/kT].
[6]
(the factor l/2 z, where z is the coordination number of the lattice, have been implicity included in the above equations).
JBB).
This equation is in fact a mean-field version of an argument given by Bell and Fairbairn (34). This is the Lee-Yang lattice gas result obtained in a case where the usual “filled” and “empty” sites are replaced by sites occupied either by A or by B molecules. This kind of spin- l/2 Ising model has been used to describe monolayer behavior by Ohki (21) for one molecular species and holes and by Caille et al. (22) with two molecular species a and 0. Caille et al. consider only one type of interaction as it should be in a Lee-Yang model but they incorrectly assume that it is JAA instead of J = 1/4(JAA + JBB - 2JAB). This energy describes the interchange of one molecule A from the bulk of phase A into the bulk of phase B with one molecule of phase B. It is this energy which intervenes in the theory of regular solution, it represents the departure from ideality of the solution since it is zero for an ideal solution. Moreover, these authors have not constrained molecules to occupy only one site since molecules p occupy an area which is m times the area occupied by an LYmolecule, m being not necessarily an integer. Whereas the possibility for a molecule to occupy several sites seems to be easily acceptable, the fact that a molecule may occupy a fractional number of site appears at first sight in contradiction with the discrete nature of a lattice model. In fact the size of the unit cell of the lattice should not be chosen as the area of the smaller molecular species but rather such as it makes all the numbers of occupied sites integer, e.g. if m = 1.5 the unit site is taken
490
BARET
AND FIRPO
as half of the molecule LYarea thus three sites are occupied by a molecule p. IV. PARTICULAR
CASE WITH
L=OANDH=O
If the interactions between A molecules are identical to these between B molecules (JAA = JBB and ,.&A= /LB) L = 0 and H = 0. Then Eqs. [5] and [6] become II=-KQ2-JM2-kTln(l-Q)
[5’]
the density l/2 for classical Ising models to a value which depends on the molecular asymmetry p. Kaye (35) studied a model intermediate between those of Refs. (17- 19) and that of Ref. ( 15), in which the asymmetry of the molecule is p = 2 but the ordering of the elements is promoted by energetic as well as steric effect. Dunne and Bell (36) have incorporated configurational effects by including an extended (dimer) state as well as the two symmetrical orientations.
with M = Q tanh (2JM/KT).
[@I
Bell et al. ( 15) have described lecithin monolayer data with a model very similar to this one. Indeed, they consider beside the vacancy two orientational states 1 and 2. The interaction energy 11 is of course equal to the interaction 22. To calculate the degeneracy of a lattice state they use the quasi-chemical (Bethe) approximation (23) rather than the mean-field approximation but this is not changing much the results. Firpo et al. (1719) have used a model in which the monolayer molecules may exhibit two orientational states in the interface plane, either lying along the x direction or along the y direction. In their model the molecule occupies p lattice sites, whereas in Bell et al. model it occupies only one, therefore they have to compute the number of possible configurations corresponding to a given number of molecules in each orientational state by taking into account the fact that no crossing of the molecules is possible. This calculation has been done in the general three-dimensional case by DiMarzio (24) and it is the restriction to the planar case which is used by these authors. The results are qualitatively similar to those obtain by other spin-l Ising models, like those of Bell et al. or those described in the next section, i.e., there are two transitions one corresponding to the condensation (first order) and the other one to the orientation (second order). The most important difference is the displacement of the condensation critical point which moves from Journal of Colloid and lnterfice Snence, Vol. 94, No. 2,
August 1983
V. GENERAL
CASE
In its general form the spin-l Ising model leads to Eqs. [5] and [6] in which the interactions between A-A molecules, B-B molecules, and A-B molecules intervene as parameters. This means such a model may be useful to describe a process in which, beside a variation of density (taken into account by the presence of holes, order parameter Q), the relative concentration of the two species A and B may vary (order parameter M). These two molecular species may be simply two different chemical compounds and then the ordering transition corresponding to the order parameter M is a demixtion; or these species are two different physical states of the same chemical, like for instance the two orientational states considered in the previous section. Other double states could be considered: upright and lying-down molecules, alltrans and gauche molecules (13), solid and liquid molecules (25), but as far as the model is concerned what matters is not the qualitative nature of the states but whether JAA is different or not of JBB, in the latter case H = L = 0 and the M transition is second-order as described above, the former case is going to be described in details now. In order to draw the isotherms from Eq. [5] we need first to solve Eq. [6] with respect to M, i.e., to find the intersection between a y = tanh (f(M)) curve and a y = M/Q straight line. Differently to what occurs in the L = H = 0 case the argument of the hyperbolic tangent is not zero at the origin and
AMPHIPHILE
MONOLAYER
therefore A4 = 0 is no longer a solution. It results from this that II-Q plane is not anymore divided in two half-plane, one corresponding to M = 0 and the other one to A4 # 0, the border line being the second-order transition line as it was the case when L = H = 0 (26). But instead there is a competition between the increasing -kT In (1 - Q) term and the decreasing -(JM2 + 2LMQ) term, leading to the classical sigmoidal behavior with a maximum and a minimum from which are constructed the first-order transition plateaus. To make short, one may say that differentiation between the two intra-species interactions supresses the ordering transition. This result is not surprising. Since for fixed Q a change of variables (e.g., M/Q, M) transforms the Eq. [6] into the first equation of Section III. It is wellknown that the introduction of a constant
PHASE
491
TRANSITION
external field H’ into the spin-l/2 Ising model, described by the latter equation, removes the higher order transition. When H = 0 and L = 0 the abscissa M of the intercept jumps from a finite value to zero for a critical value of the slope of the straight lines, this value corresponds to the crossing of the second-order transition line. On the contrary, as shown on Fig. 1, when H or L are different from zero the abscissa of the intercept is monotonously decreasing toward the origin, and there is no more transition. The way variations in L are affecting the isotherms is evidenced on Figs. 2 and 3: the break point corresponding to the transition is rounded up significantly, even with values of L 2 orders of magnitude lower than K. When these computed curves are compared with the experimental isotherms it should be concluded that the interaction differences
1.6 -
1.2 -
Y
-
FIG. 1. y = tanh (2JA4 + 2LQ + H)/kT kT = 1.3, and l/Q = a = 1.05 (b), 1.25 (c),
(Curve a) and y = M/Q, 1.45 (d), 1.65 (e), 1.85 (f). Journal ofcolloid
for J = (minimum
1, H = 0.1, L = 0, area @ = 1)
and Inter/ace Scmce, Vol. 94, No. 2, August 1983
492
BARET
AND
HRPO
n [dyne /cm I 30 -
25 -
20 -
15 -
10 -
I
25I
30I
351
40I
451
a [@I FIG. 2. Surface pressure (II) vs. area per molecule (a); k = 1.38 X lo-l6 erg OK-‘, minimum area a0 = 20 A’, J = 4 X IO-“’ erg, K = 10mL4erg, L = IO-l6 erg, H = 0, T = 280 (a), 300 (b), 320 (c), 340 (d) “K.
n [dyne km 1 30 -
25 -
20 -
15 -
10 -
25
30
35
40
45
a #I FIG. 3. Same as in Fig. 2 but L = 5 X lo-l6 erg. Journal of Colloid and Inlerjoce Science, Vol. 94, No. 2, August 1983
AMPHIPHILE
MONOLAYER
between species A and B must be zero or at least very small. Accordingly, the two species, A and B, should be either identical or at least very similar. Therefore, a model in which these two species are merely two symmetrical states like the two directions of the interface plane or “lying down” and “standing up”is more realistic than a model considering two really different species-like “trans” and “gauche” chains-for which the intra-species interactions are likely to be different. On Fig. 4 and 5 we have drawn isotherms for L = H = 0 showing how K controls the shapes of the curves. One notes that the abscissa of the transition points remains unchanged whereas the ordinate decreases when K increases. On the other hand, a change in J displaces both coordinates of the transition points, when J increases the ordinates II, decreases and the abscissa a, = 2J/aoT increases. Jis the parameter measuring the ease of the ordering transition, for large J the transition occurs at low surface pressure, large area. Rather, K scaled the surface pressures and the temperatures, it determines the crit-
PHASE
493
TRANSITION
ical parameters of the first-order condensation transition occuring at low pressure and large area that we do not consider here. VI.
COMPARISON MONOLAYER
WITH DATA
On the experimental isotherms obtain with fatty acids (27, 28) it appears that a chain length increase of one CH2 from Cl5 to C,6 (see Figs. 2 and 3 of Ref. (27)) displaces, for instance, the transition point of the 30°C isotherm from 13 dyn/cm, 31 A2 to 3.5 dyn/cm, 40 A2, or a chain length increase from C,4 to CL5 (see Fig. 1 from Ref. (29)) displaces the transition point at 25°C from 2 1 dyn/cm, 26 A2 to 9 dyn/cm, 33 A2. This behavior is similar to what has been seen to occur on computed isotherms when J increases. This is not surprising since J is the difference between intra-species and interspecies interactions, a measure of the departure from ideality, indeed if the two species A and B correspond to two orientational states, JAA = JBB is the interaction between
I
/
25
30
35
40
45 a
FIG. 4. Same
as in Fig.
THE
2 with
L = 0, J = 3 X 10-l“
erg, and K = 2 X lo-l4
[@I
erg.
Journal of CoNoid and Infer/ace Science. Vol. 94, No. 2, August 1983
494
BARET
AND
FIRPO
I 25
30
35
40
45
a [PI FIG.
5. Same
as in Fig. 2 with
L = 0, J = 3 X lo-l4
parallel molecules, whereas JAB is the interaction between perpendicular molecules. As the chain length increases the parallel interaction increases and the perpendicular one should remain more or less constant. There is another very striking similarity between experimental data and calculated isotherms, it is the superposition of the liquid expanded regions for fatty acids of different chain lengths at the same temperature. Thus, on Fig. 1 of Ref. (29) the expanded portion of the isotherms of pentadecylic and myristic acids are exactly the same, this can be related to the description of the expanded portion by an equation which does not contain J, since in this phase A4 = 0. Therefore, once again we are found supposing that J is a parameter linked to the chain length which determines the behavior of the condensed part of the isotherm. Differently, the expanded part is independent of the chain length, only the parameter K may change its shape. K plays also a role in the determination of the condensed region since it scales pressures and temperatures. It is likely that K expresses the interaction energy between the polar groups. Journal of CoNoid and Inferface Science, Vol. 94, No. 2. August 1983
erg,
K = 4 X lo-l4
erg.
This is consistent with the assumption that the gas-liquid-expanded transition is a condensation of a Coulombic gas (30). VII.
CONCLUSIONS
The sharp break point observed on experimental isotherms of many amphiphilic substances looks more like a singular point occurring at the crossing of a second-order transition line than like the onset of a condensation plateau since there is neither flat portion nor a second singular point corresponding to the coexistence zone. Therefore, it is worth looking for models yielding second-order transitions, spin- 1 Ising model includes a broad class of lattice systems which display, beside a first-order condensation transition, an ordering transition which privileges one of the species A(spin +l) or B(spin - 1). The results computed with such a model are closed to the experimental data when the two species are really symmetrical, i.e., when the JaA and JBB interactions are identical. This leads to the conclusion that the liquid-condensed phase differs from the
AMPHIPHILE
MONOLAYER
liquid expanded not by an internal state of the molecules but by a global order of the monolayer. The individual molecules in both phases are more or less in the same isomeric rotational state (same average number of gauche bonds) but the collective organization of the phases is different, the liquid expanded is an isotropic phase, whereas in the liquid condensed a privileged direction has appeared on the interface plane along which the molecules tend to line up. The liquid expanded-liquid condensed transition is then a two-dimensional (2-d) equivalent to the isotropic-nematic transition observed in bulk (3-d) liquid crystal. The reduction of the space dimensionality from 3-d to 2-d changes the nematic transition from a weak first-order to a second-order transition (31). Obviously, the converging features of the experimental isotherms and the isotherms computed from a spin- 1 Ising model are not sufficient to assert that the condensed phase is an oriented phase. Direct evidences are needed to identify clearly this phase. The main difficulty to show whether or not a nematic phase appears in the monolayer, is the limited range of the correlation length which could create oriented clusters of no more than some hundred molecules (32). Therefore, before any attempt to analyze the anisotropy of the monolayer one should first orient the autonomous clusters with respect to one another, this may be done with a field. A magnetic field may be used if the molecular susceptibility of the amphiphile is large enough. Similarly, global orientation of the clusters by an electric field is thinkable if the geometric sum of the dipole moments of the molecules belonging to the same oriented cluster is sufficiently large to make the field necessary to move the cluster against the thermal motion not innaccesible. The problem here is to obtain a water substrate very pure to minimize the current. Another way to orient the clusters could be to apply a shear flow by forcing the monolayer into a canal, this to operate requires that the clusters have an elongated shape. Once the monolayer has
495
PHASE TRANSITION
been globally oriented by one of the abovementioned fields then the anisotropy should be detected by a probe. Since the two directions of the interface plane correspond now to two different electron densities an electromagnetic radiation should be absorbed differently whether the polarizing plane corresponds to one or the other of these directions. Because of the minute amount of matter involved in a monolayer, in order to realize this dichroic experiment a device allowing several crossings of the interface should be set up. Evidence of the monolayer ordering in the liquid-condensed region may be obtained from a completely different point of view. Instead of a previous orientation of the clusters with respect to one another by an external field, a direct measurement of the fluctuations of the molecular orientations can be made by detecting the depolarization of a scattered beam (33). Indeed, the orientation fluctuation of a molecule in the expanded phase should be large and therefore the incident orientation of the polarization plane is averaged out in the scattered beam, whereas in the condensed phase the molecules keeping approximately the nematic direction, the polarization plane is conserved in the scattered beam. Of course, these experiments may yield negative results, i.e., the condensed-liquid phase may not be an anisotropic phase, then the whole problematics of the liquid expanded-liquid condensed transition should be raised in new terms: how a transition which is not an orientational transition may display a second-order discontinuity? ACKNOWLEDGMENT The authors thank Genevi&ve Danoy for her technical assistance. REFERENCES 1. Baret, J. F., Progr. Surf:Membr. Sci. 14,291 (1981). 2. Kim, M. W., and Cannell, D. S., Phys. Rev. A 13, 411 (1976). Journal of Co/hid and Inter$m
Science. Vol. 94, No. 2. August 1983
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AND
3. Hawkins, G. A., and Benedek, G. B., Phys. Rev. Lett. 32, 524 (1974). 4. Harkins, W. D., and Copeland, L. E., J. Chem. Ph$s. 10, 272 (1942). 5. Hard, S., and Neuman, R. D., J. Colloid Interface Sci. 83, 315 (1981). 6. Adam, N. K., Proc. Roy. Sot. (London) A 126,526 (1930). I. Albrecht, O., Gruler, H., and Sackmann, E., J. Phys. 39, 301 (1978). 8. Phillips, M. C., and Hauser, H., J. Colloid Interface Sci. 49, 31 (1974). 9. Albon, N., J. Chem. Phys. 78, 4676 (1983). 10. Scott, H. L., J. Theor. Biol. 46,241 (1974); .I. Chem. Phys. 62, 1347 (1975). 11. MarEelja, S., Biochim. Biophys. Acta 367, 165 (1974). 12. Jacobs, R. E., Hudson, B., and Anderson, H. C., Proc. Nat. Acad. Sci. USA 72, 3993 (1975). 13. Caille, A., and Agren, G., Canad. J. Phys. 53, 2369 (1975). 14. Nagle, J. F., J. Chem. Phys. 63, 1255 (1975); J. Membr. Biol. 27, 233 (1976). 15. Bell, G. M., Mingins, J., and Taylor, J. A. G., J. Chem. Sot. Faraday II 74, 223 (1978). 16. Kaye, R. D., and Burley, D. M., J. Phys. A 7, 1303 (1974). 17. Firpo, J. L., Dupin, J. J., Albinet, G., Bois, A. G., Casalta, L., and Baret, J. F., J. Chem. Phys. 68, 1369 (1978). 18. Dupin, J. J., Firpo, J. L., Albinet, G., Bois, A. G., Casalta, L., and Baret, J. F., J. Chem. Phys. 70, 2357 (1979).
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19. Firpo, J. L., Dupin, J. J., Albinet, G., Baret, J. F., and Caillt, A., J. Chem. Phys. 74, 2569 (1981). 20. Landau, L., and Lifchitz, E., in “Statistical Physics,” 2nd ed. Pergamon, New York, 1968. 21. Ohki, S., J. Theor. Biol. 15, 346 (1967). 22. Caillt, A., Rapini, A., Zuckermann, M. J., Cros, A., and Doniach, S., Canad. J. Phys. 56,348 (1978). 23. Hill, T. L., in “Statistical Thermodynamics,” 2nd ed. Addison-Wesley, Reading, Massachusetts, 1962. 24. DiMarzio, E. A., J. Chem. Phys. 35, 658 (1961). 25. Lajzerowicz, J., and Sivardiere, J., Phys. Rev. A 11, 2079 (1975). 26. Sivardiere, J., Phys. Stat. Sol. B 103, 823 (1981). 27. Harkins, W. D., Young, T. F., and Boyd, E., J. Chem. Phys. 8, 954 (1940). 28. Adam, N. K., and Jessop, G., Proc. Roy. Sot. (London)
A112,
362 (1926).
29. Nutting, G. C., and Harkins, W. D., J. Amer. Chem. Sot. 61, 1180 (1939). 30. Albon, N., and Baret, J. F., J. Colloid Znterjkce Sci. 92, 545 (1983). 31. Baret, J. F., Bois, A. G., Dupin, J. J., and Firpo, J. L., J. Colloid Interface Sci. 86, 370 (1982). 32. de Gennes, P. G., Symp. Faraday Sot. 5, 16 (1971). 33. Langevin, D., private communication. 34. Bell, G. M., and Fairbaim, W. M., Mol. Phys. 4, 481 (1961). 35. Kaye, R. D., Physica 18A, 347 (1981). 36. Dunne, L. J., and Bell, G. M., J. Chem. Sot. Faraday Trans. ZZ 76,43 1 (1980).