Two-dimensional colloids. The linear thermodynamics of monolayers

Colloids and Surfaces A: Physicochemical and Engineering Aspects 88 (1994) 147-155

Two-dimensional

COiLOIDS AND SURFACES

A

colloids. The linear thermodynamics of monolayers B.A. Pethica

Langmuir Center for Colloids and Interfaces, 911 S. W. Mudd Building, Columbia University, New York, N Y10027, USA

(Received 30 April 1993; accepted 3 February 1994)

Abstract A variety of two-dimensional microscopic structures resembling emulsions and other colloidal forms have been reported in spread and adsorbed monolayers at fluid interfaces. These two-dimensional colloids may be described as the mutual dispersion of first-order co-existing surface phases associated with the linear tension between the phases, in a manner analogous to the role of surface tension in the dispersion of three-dimensional phases. A formal description of linear tension in monolayers is given, and several relationships, notably the one-dimensional analogues of the Gibbs adsorption isotherm and Kelvin equation, are discussed in the context of potential new experiments. The discussion is primarily for spread insoluble monolayers. The limitations of the utility of the linear tension concept are also considered. Keywords: Linear thermodynamics;

Monolayers; Two-dimensional

1. Introduction

In a system of three co-existing bulk fluid phases, the free energy includes so-called excess terms associated with the three interfaces. The excess free energy per unit area of each interface is dimensionally equivalent to the surface tension of that interface, and the angles at the triple phase junction are given in terms of the three interfacial tensions by Neumann’s triangle. There is another excess free energy associated with the triple junction itself, usually neglected and frequently negligible. This excess free energy per unit length of triple junction is dimensionally equivalent to the line tension at the junction (see Szleifer and Widom [l] for a recent extensive account of the excess functions at a three-phase junction). The line tension will influence the contact angle for very small drops in contact with two other phases [2], and corre0927-7757/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0927-7757(94)02831-C

colloids

spondingly affect drop condensation from supersaturated vapors and similar related phenomena [3]. Both positive and negative line tensions are reported for solid and liquid particles at the interface between two other bulk phases [4]. In the case of co-existing phases in a monolayer at a fluid interface, the complications in defining the excess functions at the linear junction are simplified compared with those for the line between three bulk phases. For spread insoluble monolayers in particular, the thermodynamic description of the contact between two surface phases can, with certain constraints, usefully treat the system as two-dimensional (surface) when the components of the bulk fluid phases - usually aqueous solutions in contact with oil or gas layers - are at fixed chemical potentials for a given temperature and (three-dimensional) pressure and not altered by changes in monolayer composition or surface

148

B.A. PethicaJColloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

pressure. The situation for co-existing phases in films formed by adsorption is more complex since the monolayer properties depend directly on the solution activity of the adsorbed species [ 51. Similarly, description of the spread monolayer case will require modification when species such as counterions or soluble surfactants are shared between bulk and surface. For the sake of simplicity, the following discussion is primarily for uncharged insoluble two-phase monolayers in contact with bulk phases of fixed composition. Extensions to more general cases are briefly indicated, as are the limitations of the use of the linear tension concept for two-dimensional colloids.

2. Linear tension for two co-existing macroscopic surface phases Many phases have been described in spread monolayers [6], primarily at the air/water interface. Recent studies have demonstrated the existence of microscopic structures in insoluble monolayers, and estimates of the related line tensions have been made by Benvegnu and McConnell from dynamic relaxation measurements on fluid two-dimensional drops [7]. Line tensions in monolayers have been discussed qualitatively by several authors, for example as related to the thinning of two-dimensional foams [ 81, but direct measurements of the line tension independent of non-thermodynamic assumptions have not yet been made. The discussion here will be for fluid surface phases, for which the mechanical relationships involving linear tensions are simple. These fluid phases include the two-dimensional gas to liquid (g/l) transition and the so-called liquid expanded to liquid condensed (le/lc) transition at which many of the surface colloids so far reported have been observed [ 91. In strictly one-component monolayers, the g/l and le/lc transitions have been shown for several lipids at the air/water interface to be of simple classical first-order form [lo]. The degenerate transitions often observed are associated with inadequate humidity control and with the presence of other components, as adventitious impurities or by deliberate addition (e.g. fluorescent markers or mixed lipid films) [ 101. The degenerate

behavior is characterized by a non-constant surface pressure across the phase transition region and is associated with the appearance of microscopic (colloidal) structures. This contrasts with the (twodimensional) macroscopic regions which are found, as to be expected, for classical one-component first-order transitions showing a constant transition pressure [ 111. The thermodynamics of two co-existing macroscopic surface phases may be followed by reference to Fig. 1, which describes a surface system with a junction line between the phases such that the area (A) of the surface (or of the individual phases as appropriate) and the length of the junction line may be varied independently. The junction line is taken as straight, so that the surface pressures of the two phases are equal. Reversible work done by the system due to area changes is given by l7dA, where 17 is the surface pressure (lowering of the film-free surface tension). Similarly, the work done on the system by extending the junction line length (L) will be ldL, where 1 is the line tension. For the whole surface system bounded by the edges of the trough, the movable barriers and the line extension device, and neglecting line tension contributions at the edges of the balance and slides, changes in the Helmholtz free energy are given by dF=-SdT_PdT/-17dA+~dL+C~idni

(1)

where S is the entropy, V the volume and ni the number of moles of species i. P and T are the pressure and temperature and pi represents the chemical potentials. Variations in the Gibbs free energy are given by dG= -SdT+VdP+Adl7_Ld13.+Cpidni I

(2) where G = C mini

(3)

From Eqs. (2) and (3) it follows that 0 = SdT-

VdP - Ad17+ Ldl + 1 nidpi

(4)

At constant T and P O= -Adi7+LdA+Cnidpi

I

(5)

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B.A. PethicajColloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

SURFACE PHASE 1

i L

SURFACE PHASE 2

I

n, A2 3

(

% Fig. 1. Schematic representation of two co-existing surface phases at equilibrium. The phases are separated into two regions of area A, and AZ, making contact at the line junction of length I,. The areas are controlled by slides exerting a surface pressure l7. The length L is controlled by a thin slide exerting a force 1.

Since the pi for species present in the bulk phases are taken as constant, Eq. (5) refers expressly to the defined surface and the monolayer components only of the surface system between the two given bulk phases. Defining co) and c(Z) as the surface densities of i in surface phases (1) and (2) and neglecting linear adsorption at the geometric boundaries of the surface balance and slides we have

where AI and AZ (which sum to A) are the areas of the two phases, bounded in each by the geometry of the surface balance and by the line of tension at the surface phase junction. /li is the linear excess concentration of i per unit length of the junction, expressing the “line activity” of component i [ 121. Since at constant T and P we have dli’= Xi&:dpi for each of the two-dimensional phases, it follows that Eq. (5) reduces to (7)

This equation is the linear analogue to the Gibbs adsorption isotherm. An equation of the same form applies to adsorption at the line junction of three bulk phases [ 131 and to co-existing surface phases formed by adsorption [ 51. Since Eq. (6) supposes knowledge of the position of the line of tension, it is useful to express Eq. (7) in terms of linear excess adsorption with reference to zero adsorption for one component to give a “Gibbs-zero convention” form without reference to the position of the unknown line of tension. This destroys the symmetry of Eq. (7) but avoids the need to locate the line of tension [14]. Choosing component j as reference, and using the Gibbs-Duhem relation for the components in either two-dimensional phase to eliminate d~j we have by choosing phase (1)

= 1 Aidpi i#j

(8)

B.A. PethicalColloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

150

where _4:is the relative linear excess concentration of i. This formulation is equivalent to defining ni for a convention that nj is zero.

3. The line of tension By direct analogy with the well-known quasithermodynamic description of surface tension and surface excess quantities, it is readily seen that the surface pressure perpendicular to the line of tension is everywhere constant and equal to the measured 17 for a straight line junction, whereas the surface pressure parallel to the line of tension (ITX)has a value which depends, in the inhomogeneous region close to the line, on the distance (x) from the line of tension such that cc

J

(n, - 17)dx = /I

(9)

-0z

and m

J

(l7, - ZI)(x)dx = 0

(10)

-cc

according to l17-Ilc=~/R

(12)

where subscripts c and d denote the continuous and dispersed phases respectively and R is the radius of the two-dimensional drop or bubble. The pressure in the continuous phase, II,, is the measured surface pressure. The line tension and ZI, are not taken as independent variables, but are related to the measured 17, and R by Eq. (12). This assumes that a measured R is identical to the radius of the line of tension, which is only exact for large radii. An optically observed R may be assumed as sufficiently close to the radius of the line of tension for structures of the order of a micron or more in size. The uncertainties increase for smaller sizes, and the line tension becomes less useful as a thermodynamic parameter. The first example considered is the metastable equilibrium of a drop and a supersaturated onecomponent ideal two-dimensional gas. With the subscripts c and d now referring to the continuous gas phase and the dispersed liquid phase, respectively, we have pC= pd for the monolayer substance at local equilibrium with d,uLd = AddI& and

where Eq. (10) defines the position of the line of tension. The /ii may be given formally as

dpC = kTd In lI,

(13)

where Ad is the molar area in the liquid phase and k is the Boltzmann constant. From Eqs. (13) --m

0

(11) where l& is the local surface density of i at a distance x from the line of tension and x is taken as negative in phase 1.

4. Surface dispersions and colloids The case of the mutual dispersion of the surface phases is considered next. For simplicity, the geometry of the dispersed region is assumed circular. The surface pressure differs on each side of the line

and (12) it follows that if Lro is the measured surface pressure when R is infinite (i.e. the equilibrium first-order transition pressure) (14) This equation is the two-dimensional Kelvin equation for supersaturated drops. A, may be approximated to the molar area of the monolayer substance at the dense end of the gas-liquid transition. Non-ideal gas phase behavior is readily accommodated in Eq. (14) by substituting the surface fugacity obtained from the experimental pressure-area isotherms in the gaseous region. The more general case of the co-existence of

B.A. Pethica/Colloids Surfaces A: Physicochem. Eng. ‘Aspects 88 (1994) 147-155

dispersions of two multicomponent fluid phases in a spread monolayer follows from the equivalence of the chemical potential for any component i in the dispersed and continuous phases as composition or area are changed at constant T and P to give d/+, = dpi(d) = dpi

(15)

Since for each surface phase dl7 = ~i~d~i, we have for constant 1, using Eq. (12),

and --

(17)

where AC = & - I&. For the equilibrium between liquid drops and an ideal multi-component gas phase, Eq. (17) becomes

(18) where the fl* terms indicate partial pressures in the continuous gas phase. Equation (18) reduces to Eq. (14) for the single component case since &A, >>CCC,. Equation (17) also applies to the formation of a two-dimensional gas bubble or foam with liquid as the continuous phase, and to the formation of less dense liquid drops (liquid-expanded) within a liquid-condensed phase. For a single component monolayer, in either case, AT is negative and the measured pressure increases as the bubble or low density liquid drop grows (A positive). The formation of gas bubbles or liquid-expanded drops on expanding a liquid monolayer into a transition region should therefore show a minimum in the surface pressure before attaining the equilibrium transition value. The same conclusion will apply typically to multi-component monolayers since most or all of the A& terms will be negative. Lucassen et al. [8] have observed time-dependent two-dimensional foams by expanding a liquid monolayer of a fluorescent molecule with some impurities. The authors did not measure the associ-

151

ated pressure changes, which are predicted to show an increase as the foam bubbles grow and break. The foregoing arguments take 1 constant as R varies, which will not be generally correct. The assumption is likely to fail particularly for colloidal systems containing low concentrations of very lineactive molecules, for which the concentration in the two-dimensional phases will fall off as the total line length increases. The total line length in these circumstances may well be set by the availability of the line-active component, in the same way that the surface area of some three-dimensional emulsion systems is related to the close-packed area of the available molecules of the surface active compound. Progress for all but the simplest two-dimensional colloids will depend on the development of new methods, particularly for the composition of the co-existing phases.

5. Effect of temperature The variation of 1 with temperature at constant V, A and ni gives, from Eq. (l), the excess entropy of formation of the line junction per unit length. For dispersed systems, the effect of temperature on A, R and the distribution of components between the surface phases is complex, and the terms nidCLi of Eq. (4) include components from the bulk phase, unlike Eq. (5). No obviously useful relationships are suggested in the absence of further experimental data.

6. Linear contact angles and the disjoining force If three or more surface phases co-exist in an insoluble monolayer, a surface contact angle may be exhibited. The horizontal resolution of the three il terms by a two-dimensional Neumann triangle will describe such a phenomenon. Similarly a Young-Dupre analogue equation will describe the contact angle formed, for example, at a rigid boundary to a surface monolayer region when each phase contacts the boundary. It is easily shown that the pressure registered by the sensor barrier is still lTC when each phase contacts the barrier and the contribution of the line tension is

B.A. PethicalColloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

152

included for circular geometries. For two-phase monolayers, contact angles are in principle observable for thin strips contacting two-dimensional plateau borders, as described by Lucassen et al. [S] for two-dimensional foams. Although not presently resolved optically, these strip contact angles are potentially valuable in giving the difference in il between strip and border and the one-dimensional disjoining force. By direct analogy with the well-known Derjaguin method [ 151, the disjoining force per unit length (f) will depend on 1 in a metastable parallel linear strip according to f=-2

0

7. Nucleation kinetics In the generation of one separated fluid phase within another, the initial step is usually the formation of nuclei. The free energy of nucleus formation includes the transfer of molecules to the nucleus and formation of the nuclear line according to AF = nR2 C ~(d,A~i + 271R~ (21) I where A,ui is the difference in chemical potential for component i between the (dispersed) nucleus phase and the continuous phase. For the simple case of a one-component ideal gas monolayer forming a liquid nucleus

g

(22)

Pi

where h is the strip thickness - the distance between the two lines of tension. Correspondingly, the disjoining force and linear adsorptions are related by

($),= +gPi

(20)

Equation (20) will also describe the forces between a pair of two-dimensional colloid particles if f signifies the force between two particles at separation h and /ii is the total linear adsorption per particle (see Hall [ 161 for the corresponding discussion for the force between three-dimensional colloid particles). It should be noted that when interaction between the dispersed regions occurs, the assumption of circular curvature fails and the surface pressure in the continuous phase varies from point to point. In these circumstances, the measured pressure at a sensor barrier no longer gives l7=. The situation can be analyzed by methods similar to those used for three-dimensional foams [ 17,181, although formulations for the measured pressures at walls or barriers contacting the dispersions are not yet developed in a form appropriate for designing experiments to calculate 1. Correspondingly, the several relationships for il given here involving Z7, as the measured pressure are to be regarded as giving the dilute dispersion limits of more general relationships.

where 17R is the surface pressure for a nucleus of radius R as given by the Kelvin equation (Eq. ( 14)) and l7, is the experimental continuous phase pressure, typically supersaturated. The critical nucleus is defined by tYF/BR = 0 at a given l7,, leading to (23)

and 3 Id2 AFcrit= - - ~ 4 GA/bit assuming I and rd constant. These equations have physical meaning when l7, > Il,, giving Ap negative. They are difficult to use and the conventional simplification, as applied for three-dimensional nucleation [ 191 is to assume that Ap is invariant with R, and n, and l7, in Eq. (22) be replaced by & and f10 respectively, where n, is the equilibrium transition pressure. The rate of nucleation then becomes -

J=Kexp

AFcrit kT

- 7d2

=Kexprd(kT)21n(17J170) (25)

where K is a constant. The diffusion rates for gas to liquid nucleus formulation will necessarily be much lower in the monolayer than in an ordinary gas due to the drag of the bulk phase fluids in

B.A. PethicaJColloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

contact with the monolayer. Experiments on nucleation rates will be correspondingly difficult.

8. Measurement of the line tension The direct measurement of line tension in monolayers will undoubtedly be difficult. Forces in monolayers have been measured to less than lo-* N, but the problems of reproducibility in relation to impurities are severe [lo]. Szleifer and Widom [l] consider that the line tension at the junction of three bulk phases increases rapidly as the contact angle goes to zero. A monolayer phase junction line can be regarded as one limit of zero contact angle and the line tension may well be correspondingly larger than the values reported for the triple line junction. The existence of macroscopic two-dimensional phase regions in monolayers also indicates large positive line tensions. Indirect estimates of the line tension in mixed cholesterol-phospholipid monolayers have been made by Benvegnu and McConnell 173 from the shape relaxation of dispersed structures after shearing at the surface. Their results suggest that 1 varies from 1.6 x lo-‘N at low surface pressures to zero at about 10 mNm-‘. Although these results are based on a variety of assumptions, the magnitudes should encourage attempts at direct measurements of /2. The schematic in Fig. 1 for two monolayer phases of macroscopic area could be realized in practice for measuring 1 in several ways. For example, a semicircular flexible thread attached at one end to a side of the trough and at the other end to a vertical rod (mounted on a force balance) could be arranged at the surface to separate the condensed and dilute phases such that the open line junction extends from the rod to the second side of the trough. The position of the line junction could be visualized by Brewster angle microscopy [20]. With this geometry the sum of the line tensions between the thread and the two surface phases acts at right angles to the direction of the tension to be measured. Another arrangement would be to have two threadless floating horizontal barriers mounted on torsion balances such that

153

one phase is confined between the barriers to allow the linear tension to pull them together [ 171. Perhaps the most practical method would be the analogue of the bubble-pressure method for surface tensions. The line junction should be made at the end of a narrow channel between the two phases and two-dimensional drops expelled under a small measured surface pressure difference. With a Iz of lo-’ N, a difference in II of 0.1 mN m-l would be seen for drops emerging from a channel of radius 0.1 mm. Such experiments are feasible. A major experimental difficulty to be anticipated is how to reduce the effects of contact angles between the monolayer-covered surface and the channel walls. When one phase is dispersed in another, less direct methods for I will be required. If the radius of curvature of the two-dimensional drop is estimated by optical means, Eq. (5) can be used to interpret measurements of the supersaturation associated with the presence of the dispersion in a one-component monolayer. For 1= lo-* N, detectable pressure changes of 10e2 mN m-l would be observed for drops of 1 l.trn radius. The kinetics of nucleation, as discussed above, constitute another potential route.

9. Discussion The line tension arguments may be extended to ionized insoluble monolayers by including the conditions for equilibrium of the counterions between surface and bulk phases. For a fully ionized monolayer, variations of density or composition occur at constant chemical potential of the counterions for a given bulk phase composition. The situation is more complex for partially ionized films for which the degree of ionization depends on both surface density and composition as well as the acidity and electrolyte composition of the bulk phase equilibrium [21]. It will be instructive to observe phase behavior and surface colloid formation in fatty acid monolayers, in particular, as a function of pH and salt composition. Octadecyl sulfate forms fully ionized insoluble monolayers which show degenerate phase behavior and should repay further investigation [ 223.

154

B.A. PethicafColloiak Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

Similar arguments have been applied to phase transitions in adsorbed surfactant monolayers [ 51. Despite claims to have demonstrated first-order transitions in adsorbed surfactants at fluid interfaces, no unambiguous demonstration has yet been given. The basic experimental proof would be a sharp discontinuity in the surface tension as a function of the solution activity of the adsorbing species. In practice, the discontinuities observed may be drawn as sharp, but experimental error is so large that the transition may more fairly be represented as degenerate [23]. The observation of surface colloid structures at adsorbed films has been reported [24], and estimates of 1 may perhaps be obtained from the geometries of adsorbed surface colloids in conjunction with surface tension measurements. Surface phases have been reported for insoluble monolayers at the oil/water (o/w) interface. Despite considerable efforts in refining the methods and purification of the monolayer substances, an unambiguous first-order transition at the o/w interface has not been observed to date. Indeed, the evidence for the phospholipids suggests that the transitions observed correspond to the formation of aggregates or surface micelles with aggregation numbers dependent on chain length and temperature [ZS]. The concept of line tension is not useful in these circumstances, and a better description may well follow small system thermodynamics via the subdivision potential as applied to micelles or by conventional mass action models [26]. Unless an unambiguous demonstration can be made of firstorder transitions in lipid monolayers at the o/w interface, it may be concluded that the intermolecular forces at the o/w interface are so attenuated by the screening of the oil molecules (at least for hydrocarbon liquids) that the line tension is too small to allow first-order transitions. The pertinence of the line tension to the description of two-dimensional colloidal dispersions is clearest for fluid phases with particles large enough to allow an adequate determination of curvature. Many of the dispersions observed by fluorescent microscopy show non-circular, sometimes dentate, structures which will be solid or liquid crystals in two dimensions. The arguments developed

here are not directly useful for describing these structures.

10. Conclusion For many two-dimensional dispersions (colloids) formed in an insoluble monolayer at a fluid interface, the concept of linear tension in the phase boundaries is useful for representing the thermodynamic properties of the dispersions. Basic relations for the line tensions are given and linear analogues to the Kelvin equation and the Gibbs adsorption isotherm are derived. The effect of temperature on the line tension is formally stated. Methods appropriate to the measurement of line tension are discussed for co-existing monolayer phases with or without dispersion of one phase in the other.

Acknowledgment The comments of Professor much appreciated.

D.G. Hall were

Note added in proof A valuable recent paper by P. Muller and F. Gallet [ 271 reports measurements of the density of needle-shaped 200 urn particles in a monolayer of a fluorescent labelled stearic acid as a function of pressures in excess of the phase transition pressure. Using an argument for the free energy of nucleation similar to that given here, replacing the second term on the right-hand side of Eq. (21) by an integral over the perimeter of the non-circular seed, and developing an explicit expression for the nucleation rate based on the concept of nucleation sites, these authors find A values of the order of lo-l2 N. They also report the first observation of line activity, with i decreasing on addition of stearic acid to the monolayer.

B.A. Pethicaf Colloids Surfaces A: Physicochem. Eng. Aspects 88 (1994) 147-155

References Cl1 I. Szleifer and B. Widom, Mol. Phys., 75 (1992) 925. PI B.A. Pethica, Rep. Prog. Appl. Chem., 46 (1961) 14. B.A. Pethica, J. Colloid Interface Sci., 62 (1977) 567. c31 A. Sheludko, V. Chakarov and B. Toshev, J. Colloid Interface Sci., 83 (1981) 82. R.D. Gretz, Surface Sci., 5 (1966) 239. A. Sheludko, B. Toshev and D. Platikanov, God. Sofii. Univ., Khim Fak., 71 (1980) 1. I.B. Ivanov, P.A. Kralchevsky and A.D. Nikolov, J. Colloid Interface Sci., 112 (1986) 97, 108, 122. c41 A. Sheludko, Colloids Surfaces, 1 (1980) 191. J. Mingins and A. Sheludko, J. Chem. Sot. Faraday Trans. 1, 75 (1979) 1. P.A. Kralchevsky, A.D. Nikolov and I.B. Ivanov, J. Colloid Interface Sci., 112 (1986) 132. [51 I. E. Lane, Trans. Faraday Sot., 64 (1968) 221. C61N.K. Adam, Proc. R. Sot. London Ser. A, 103 (1923) 687. W.D. Harkins and L.E. Copeland, .I. Chem. Phys., 10, (1942) 272. S. Stallberg-Stenhagen and E. Stenhagen, Nature, 156 (1945) 239. R.M. Kenn, C. Bohm, A.M. Bibo, I.R. Petersen and H. Mohwald, J. Phys. Chem., 95 (1991) 2092. c71 D.J. Benvegnu and H.M. McConnell, J. Phys. Chem., 96 (1992) 6820. 181J. Lucassen, S. Akamatsu and F. Rondelez, J. Colloid Interface Sci., 144 (1991) 434. c91 H.M. McConnell, L.K. Tann and R.M. Weiss, Proc. Natl. Acad. Sci. U.S.A., 81 (1984) 3249. B. Moore, C.M. Knobler, D. Broseta and F. Rondelez, J. Chem. Sot. Faraday Trans. 2, 82 (1986) 1753. M. Losche and H. Mohwald, Eur. Biophys. J., 11 (1984) 35.

[lo]

155

S.R. Middleton and B.A. Pethica, Faraday Symp., 16 (1981) 109. S.R. Middleton, M. Iwahaashi, N.R. Pallas and B.A. Pethica, Proc. R. Sot. London Ser. A, 396 (1984) 143. Cl11 N.R. Pallas and B.A. Pethica, Langmuir, 1 (1985) 509. N.R. Pallas and B.A. Pethica, Langmuir, 9 (1993) 361. Cl21 K.J. Mysels, Langmuir, 4 (1988) 1305. Cl31 J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, International Series of Monographs on Chemistry 8, Oxford Scientific Publishers, 1986, Oxford, p. 236. Cl41 E.A. Guggenheim, Thermodynamics, 5th edn., North Holland, Amsterdam, 1967. Cl51 B.V. Dejaguin, Kolloid Z., 69 (1934) 155. [W D.G. Hall, J. Chem. Sot. Faraday Trans. 2,68 (1972) 2169. Cl71 D.G. Hall, personal communication, 1993. CW T.L. Crowley and D.G. Hall, Langmuir, 9 (1993) 101. H.M. Princen, Langmuir, 4 (1988) 164. [I91 J.T. Davies and E.K. Rideal, Interfacial Phenomena, Academic Press, New York, 1963. cm S. Henon and J. Meunier, Rev. Sci. Instrum., 62 (1991) 936. WI J.J. Betts and B.A. Pethica, Trans. Faraday Sot., 52 (1956) 1581. cm A.V. Few and B.A. Pethica, Discuss. Faraday Sot., 18 (1954) 258. c231N.R. Pallas and B.A. Pethica, Prog. Colloid Polym. Sci., 6 (1983) 221. [241 S. Henon and J. Meunier, Thin Solid Films, 210/211 (1992) 121. c251J. Mingins, J.A. Taylor, B.A. Pethica, C.M. Jackson and B.Y.T. Yue, J. Chem. Sot. Faraday Trans. 1,78 (1982) 323. P61 D.G. Hall and B.A. Pethica, in M. Schick (Ed.), Non-Ionic Surfactants, Edward Arnold, New York, 1966. c271P. Muller and F. Gallet, Phys. Rev. Lett., 67 (1991) 1106.