Bulk aspects of the Gibbs surface excess parameters

COLLOIDS Colloids and Surfaces A: Physicochemical and Engineering Aspects 102 (1995) 21-29

ELSEVIER

AND A SURFACES

Bulk aspects of the Gibbs surface excess parameters I.R. P e t e r s o n * lnstitut Curie PC-PSI, 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France

Received 19 September 1994; accepted 1 May 1995

Abstract A simple model of an amphiphilic monolayer at an air water interface is used to demonstrate that the Gibbs formalism for surface excess parameters correctly takes into account the volume work performed by the interface during the course of a thermodynamic process. An expression for the surface excess free energy is derived explicitly in terms of the bulk thermodynamic parameters of the system. It is found to have the form of a Legendre transform. A modification to the Gibbs formalism is proposed, in which the need to choose a reference substance is replaced by taking a symmetrically weighted mean of the quantities of all bulk substances instead, removing the arbitrariness of Gibbs' definition. In one experimentally important case, the surface excess quantities derived using this modified definition differ minimally from the Gibbs values. The expression for the surface excess free energy in terms of the bulk parameters retains the Legendre relationship of the simple system. Keywords: Air-water interface; Gibbs surface excess parameters

1. Introduction Almost a century ago, Gibbs demonstrated an objective method for assigning extensive thermodynamic parameters to interfaces, in which the much larger quantities in the adjoining bulk phases are cancelled out by a well-defined procedure. For any particular extensive parameter q~ (for example, the internal energy U, the free energies F and G. the entropy S, the volume V, etc.), the surface excess value qss of that parameter is defined as rb~= (/5 - ( 7 @ t + fiq52)

(1)

where q~l and q52 are the values of that parameter in interface-free samples of the two bulk phases 1 and 2 at the same pressure and temperature as the real system, and the factors ~ and fi, the same for * Present address: Nima Technology, University of Warwick Science Park, Coventry CV4 7EZ, UK. 092%7757/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0927-7757(95)03248-7

all the extensive parameters of the system, are chosen to bring to zero the surface excess volume Vs and the surface excess number of molecules nit of one of the system substances R selected as reference [1 3]. V - (~V, + flY2) = 0

(2a)

nR -- (anR1 + flnR2) = 0

(2b)

Since the value (~q~l + flq~2) substracted from q~ in this explicit definition is typically equal to q~ within a few parts per 10 s, the parameters are almost never determined in this way. Instead, intensive parameters are measured using surface sensitive techniques, and the extensive parameters are deduced using analytical procedures justified by the original definition and by thermodynamical arguments. For example, the surface tension 7, which is the derivative of the surface excess free energy with respect to the surface area, is measured

22

LR. Peterson/Colloids Surfaces A." Physicochem. Eng. Aspects 102 (1995) 21-29

mechanically by balancing two forces against one another. The surface excess concentration F of a particular system substance is then obtained from the behaviour of 1' using the Gibbs adsorption isotherm. Unfortunately, Gibbs accompanied his paper by an illustration which has been a source of confusion ever since. He described the value (~q~l + fl~2) subtracted from ~b in Eq. (1) as being the value of q~ for a fictitious system. By Eq. (2), the fictitious system has the same volume as the actual system, and the same amount of reference substance. Instead of the physical interface, however, the two bulk phases are separated by a mathematical plane of zero thickness, the "Gibbs dividing surface", across which the phases do not interact. Many authors [4-6] have expressed doubts as to whether the formalism is capable of treating cases where the interface has an appreciable volume which can change in the course of a thermodynamic process. The Gibbs formalism is important [7] for the adsorption at interfaces of amphiphilic or surfactant monolayers, which certainly do have a finite thickness. These systems are of considerable current interest, not only in connection with detergency, but also because of their biological relevance [8,9] and the possibility of technological applications [10,11]. A recurring theme in the work over the past three decades is the possibility of an equivalence between water-surface monolayers and the bilayer scaffolding of biological membranes [12-14]. Other authors have suggested an equivalence between water-surface monolayers and bulk lamellar systems [15-18]. The motivation for the present work was to find the thermodynamically correct correspondence between surface excess and bulk parameters.

2. The interfaciai volume paradox Distinct differences of bulk density, and hence interfacial volume, have been well documented for different states of a number of lipids 1-19-20]. However, in the formula for conservation of energy expressed in terms of the Gibbs surface excess

parameters: dF s = ~ dA - Ss d T + ~ ~i dn~

(3)

there is apparently no term P dV which might correspond to the mechanical work performed by a monolayer when its volume changes. We analyse here the simple model system shown in Fig. 1, consisting of water (W), nitrogen (N), and an insoluble, involatile amphiphile (M). To represent the essentials of the system without adding to the mathematical complexity, the small concentration of nitrogen in the liquid phase will be ignored, as will the small concentration of water vapour in the gas phase. The monolayer, of thickness t, will be assumed to exclude both water and nitrogen. The reference substance will be chosen, conventionally, to be water. Since the surface excess of water is then by definition zero, the Gibbs dividing surface must be located at the monolayer-water interface, i.e. in the plane of the amphiphilic headgroups. The concentration of the amphiphile in both bulk phases is zero, so that the surface excess of amphiphile n~ is equal to the total amount nM, independent of the monolayer thickness. However, the surface excess number of nitrogen molecules n~ does depend on the monolayer thickness, because the monolayer excludes nitrogen. As a result, nh is negative, its magnitude being equal to the number of molecules excluded by the monolayer thickness t over its area A:

n~ = ArN

]

= -- Ac N t ?

(4)

|

=--cNV ) where CN is the number density of nitrogen molecules. Note that the negative value of the surface excess concentration of nitrogen has no absolute significance. It is a result of the fact that the reference substance was chosen to the water. If nitrogen had been chosen as the reference substance, its surface excess would have been zero. The Gibbs dividing surface would pass through the hydrophobic ends of the amphiphilic chains, and the surface excess concentration of the water would then be negative.

23

I.R. Peterson/Colloids Surfaces A." Physicochem. Eng. Aspects 102 (1995) 21-29

f J -

j

Nitrogen

J

I

J

I

/

~ " ~

Amphiphile

~

t Water

~

'

/

'

~

//jZ s J

Fig. 1. The thermodynamicsystemfor analysingthe interfacialvolumeparadox. The third possible choice of reference, the amphiphile, would lead to completely different values. Since its concentrations in both bulk phases are very small, the Gibbs dividing surface would perhaps be millimetres distant from the physical monolayer, leading to enormous values, one positive and the other negative, for the surface excess concentrations of nitrogen and water. If the monolayer now undergoes a process in which its volume V = tA varies, there is a term in Eq. (3) of the correct form to represent the volume work:

~N dn~ = --~NCN dV

(5)

corresponding to the nitrogen. Note that the attribution of the volume work to the nitrogen is an "artifact" of the formalism. The above discussion indicates that the work could equally well have been ascribed to the water. The differential of Eq. (5) is not equal to - P dV. The two are related, because the chemical potential VN of the nitrogen is in a sense equal to the free energy per molecule, so that #NCN is a free energy per unit volume, as is the pressure P. The correspondence is not precise, as both quantities are partial derivatives of the free energy rather than ratios, and the auxiliary quantities being kept constant are different. However, there is no theoretical requirement for the volume work term to be

- P d V, as can be seen by inspecting the equation of energy conservation involving the bulk Gibbs free energy: dG= VdP-

SdT+

~#idni

(6)

The important point is that Eq. (3) for the conservation of surface energy takes into account changes of interface volume. The lack of interfacial volume and interaction in the fictitious system used by Gibbs to illustrate the reference value eq~l + fltJ~2 for the extensive parameters in no way means that this is the case for the real thermodynamic system with the total value ~b. In the light of its rigorous derivation from fundamental thermodynamic principles, the formalism is perfectly valid and quite capable of handling the case of amphiphilic monolayers.

3. The surface excess free energy of an amphiphilic monolayer

The coefficients ~ and fl used in Eq. (1) to define the Gibbs surface excess parameters are defined in Eq. (2), and are the same for all parameters. Since all the defining equations are linear in e and fl, it is not difficult to eliminate the latter and obtain an explicit expression for the surface excess parameters in terms of extensive thermodynamic parame-

24

L R. Peterson/Colloids Surfaces A: Physicochem. Eng. Aspects 102 (1995) 21~9

ters only. However, in most cases this gives rise to a rather messy nonlinear expression which does not give any insight into their nature. To the authors' knowledge, no explicit expression of this sort has ever been published. Instead, Eqs. (1) and (2) are essentially always handled in differential form. There is one surface excess parameter which is more useful than all the rest. This is the surface excess free energy. It might be considered that there are two such parameters, corresponding to the bulk Helmholtz and Gibbs free energies, F and G respectively. However, the two surface excess parameters derived from them are related by

ence substance is still taken to be water (W). Eqs. (2) can be solved to express ~ and fl explicitly: Vnw2 -/'/w V2 Vl nw2 - nwx V~ (8)

fl=

V n w ~ - nw V1 v2 n w l - nw2 v~

from which the following expression for F ~ is obtained: nw1F2 F ~= F - V Flnw2 V~ nw2 - n w l V~

(7)

G S = Fs + p v ~

-

Since V~ is by definition zero, the two are identical. In the following, the symbol F ~ will be used. The utility of the surface excess free energy is related to the fact that most other surface excess parameters can be expressed in terms of this quantity and its derivatives. The surface excess free energy is derived for the system of Fig. 2, simplified with respect to Fig. 1 by removing the nitrogen. Since the presence of air is not normally considered to affect the process of monolayer formation, Fig. 2 retains the essential features of Fig. 1, and will be seen to yield a worthwhile mathematical simplification. The refer-

Liquid W a t e r

(9)

It is possible to use the fact that the liquid and vapour phases 1 and 2 are in equilibrium to simplify Eq. (9). From the Duhem relationship for each individual phase, which contains pure water: FI = nwx/~w - PV1 "~

(lO)

F2 = nw2~w - PV2 it follows that (11)

F ~ = F + P V - n w#w

It can be seen that, in spite of its derivation from the bulk Helmholtz free energy, the surface

fJ jf

F1 V2 - Vl F2 nw 111nw2 - nwa V2

J- ~ / f

~'~'-

Water Vapour

jf/j"

. 1/~j

JJ

Fig. 2. The thermodynamic system for derivation of the surface excess free energy.

L R. Peterson/Colloids Surfaces A: Physicochem. Eng. Aspects 102 (1995) 21-29

excess free energy is more closely related to the bulk Gibbs free energy G. Moreover, it is related to G in the same way that G is to F, and F in turn is to the internal energy U, as shown in Table 1. Each step represents a Legendre transformation, in which a term is subtracted that is equal to the product of an extensive variable and the derivative of the potential with respect to that variable. The condition for equilibrium is in each case that the particular potential should be a minimum, but the boundary conditions are different. Each step is appropriate for equilibrium in which one additional quantity is freely exchanged between the system and its environment: in order, heat, volume and water.

4. Redefinition of the surface excess parameters

Eq. (11) is of great interest, because information is available about the surface excess free energy of monolayers of fatty acids [ 17,22], and it would be useful to compare these systems to bulk systems of lamellar compounds of similar structure, both pure and lyotropic. Eq.(11) provides a simple, explicit link between the surface excess and bulk parameters, which lends itself to further analysis. It is important to know whether its validity can be extended to other, more practical, systems. However, if the system is made more realistic by admitting other substances, and allowing the amphiphile to be both soluble and volatile, the Table 1 The series of Legendre transformations between the various thermodynamic potentials. In each case, system equilibrium corresponds to a minimum of the potential, but keeping different system parameters constant. Minimise

While keeping constant

U

S, V, riw, riM, (A)

F=U

gU

S-8S

T,V, nw, nM,(A)

8F G = F-- V-~V

T, P, nw, riM, (A)

~G F ~ = G -- nw Onw

T, P, Pw, nM, (A)

25

elegance of Eq. (11) is not preserved. Eqs. (8) and (9) remain valid, but the Duhem relationships (10) now contain terms involving other substances which do not cancel. An essential element in the derivation of Eq. (11) was that both bulk phases should contain only the reference substance. It has proved worthwhile to investigate whether the problem lies not with Eq. (11), but instead with the original definition of Eqs.(2). Others have previously attempted to improve the Gibbs definition: Hansen [6], for example, demonstrated another system of surface excess parameters in which Eq. (2a) is changed. Since this equation sets the surface excess volume by definition to zero, it leads to the elimination of the differential dP of bulk pressure in the equation of conservation of surface energy. Hansen argued that it was an unfortunate choice for the majority of experiments in which P is kept constant. His substitute Eq. (2a) sets to zero by definition the surface excess quantity of a second reference substance. In Hansen's scheme, the volume is no longer one of the quantities whose surface excess is zero, so that the corresponding equations contain a term P d V s explicitly representing the volume work. Of course, it has just been demonstrated that the absence of such an explicit term is not a problem, and unfortunately Hansen's scheme does not remove the arbitrariness of the Gibbs surface excess parameters, but instead compounds it. This arbitrariness has been commented on by many authors and is apparent in the analysis of the simple model of Fig. 1. They are arbitrary because one particular substance is singled out for special treatment, and with Hansen's system the quantities are doubly so. In a general system containing many substances, there are many possible choices and hence many possible sets of surface excess parameters. There cannot be significant variations in the surface excess quantity of any substance which is strongly attracted to an interface. However, as shown in the example, the differences may be important for substances which are only weakly bound. This arbitrariness can be removed by choosing, as the second parameter whose surface excess is by definition zero, one which is symmetric in the system substances. Surface excess entropy is such

26

I.R. Peterson/Colloids Surfaces A: Physicochem. Eng. Aspects 102 (1995) 2 1 - 2 9

a choice. Unfortunately, since the Duhem equations do not involve the entropy, its choice does not lead to a simple expression for F s even with the system of Fig. 2, let alone one with the elegance of Eq.(ll). In a system containing many substances, what is required is a new Eq. (2b) which leads to a surface excess free energy of the form G e = G - ~ lain i

(12)

The superscript e is chosen to distinguish this surface excess quantity from the normal Gibbs quantity. It turns out that this is possible. The required parameter is simply the weighted mean fi of the quantities of the N substances. The mean is taken over all substances present, and the weight of each substance is proportional to its chemical potential:

= Y ini/Z

(13)

It is clear that this new definition differs from the Gibbs definition in the simple case of Fig. 2 only in that surface-bound amphiphiles are included in the mean. The relationship is also close in more general systems. Since the definition retains the Eq. (2a), the new surface excess quantity of the substance i, symbolized nT, is related to the Gibbs parameter n7 by the same general equation as that relating two Gibbs parameters with different reference substances: (14)

n ei - rl si - - K A c i

Aci = c i l - ci2 is the difference between the concentrations of the substance in the two bulk phases, and its prefactor K is chosen to satisfy the second defining equation of the set, where ~° = 0, giving n e -~- gl~ - - z]C i E

'rn~r ~/rACr

(15)

Surface chemistry is often carried out using a solvent as the main bulk substance, with the concentrations of the others kept very low to avoid self-association effects. The solvent would then normally be chosen as the Gibbs reference substance. Aci is orders of magnitude larger for the solvent than for all the other substance, while /~i are typically of the same order of magnitude.

Hence, the prefactor K, which has the dimensions of volume, is of the order of magnitude of the nominal volume (thickness times area) of the adsorbed layer. For substances which interact with the interface, the differences in the numerical values using the present revised definition for the surface excess quantities of substances are minimal for all but the solvent. As in the standard derivation for the Gibbs surface parameters, the equation for conservation of energy is derived by combining the bulk equations for the system with interfaces and the two for the interface-free reference systems: dG=ydA-

dG1 =

S dT-

VdP + ~,#idni

(16a)

d T - V1 dP + ~ #i dnil

(16b)

dG2 = - $ 2 d T - 1/2 dP + ~/~i dni2

(16c)

G e = G - ~G l - f i G 2

(16d)

0 = Z I.tini- ~ Z ltinil -- fl Z ,uiniz

(16e)

S° = S - c~S1-/3S2

(16f)

0= V-TV~-flV2

(16g)

-S

1

As in the usual Gibbs derivation, the coefficients and /~ may be taken to be fixed (even unity), because any sizes of the reference systems necessary to keep the surface excess volume and reference amount equal to zero can be achieved by adding or subtracting substances, and this is mathematically simpler than allowing z and /~ to vary. However, here the different substances must be added in ratios depending on their chemical potentials/~i, which do not necessarily remain constant if the temperature, pressure and relative amounts of the different substances change. Because the coefficients of n in Eq. (16e) are the same as those of the dn in Eqs. (16a)-(16c), the terms in dn cancel completely. However, because of the possibility of changes of the coefficients, they are replaced by terms in dp: dGe=ydA-

Se d T - ~ nT d~i

(17)

When there is no interface, Ge vanishes, so this equation may be integrated keeping the pressure, temperature and chemical potentials constant, i.e. all the extensive system parameters increase in

I.R. Peterson/Colloids Surfaces A: Physicochem. Eng. Aspects 102 (1995) 21-29

relationship is also a Legendre transformation:

proportion to A:

,18,

Ge = ~:A

The "natural" independent variables for the Helmholtz free energy F, in terms of which the thermodynamic equations have the simplest form, are V, T, and the substance quantities n. After Legendre transformation with respect to V to the Gibbs free energy G, T and n remain as natural independent variables, while V is exchanged for P. In exactly the same way, Eq. (17) shows that the natural independent variables for Ge are P, T, and the chemical potentials #i of the substances. In reactions involving the adsorption of bulk substances at interfaces, the change of bulk concentration during the reaction is usually negligible, so that the chemical potentials /~i stay essentially constant, and this form is well adapted to the experimental boundary conditions. This section has demonstrated that there is a simple variant to Gibbs' system for deriving surface excess parameters. The reference quantity whose surface excess must be zero does not have to be the amount of a single substance, but may be a weighted mean, and the weights for each substance may be chosen to simplify the form of the surface excess free energy.

5. Discussion In general, Legendre transforms are reversible. For example, if the defining relationship for G given in Table 1 is partially differentiated with respect to P keeping T constant, two terms cancel, giving

~G)

r=~

(F-PV)

c~F ~V - OV ~.P

= - V

27

0V P ~p - V

(19)

Hence, F may be recovered from G, and the inverse

F = G- P

(20) T

However, this cannot be the case for Ge. Adding to the system a sample of either bulk phase free of interfaces leaves Ge unchanged, so that it is not possible to recover from Ge any information about the total free energy G. The reason why the analogue to Eq. (20) cannot be used to recover G is that the independent variables cannot be chosen independently of each other. In particular, the state of reference system 1 is not completely defined by P, T and the N chemical potentials Pi, because they are all intensive variables, so that the set leaves the volume V~ undefined. Since the total of N + 2 degrees of freedom of the system includes V1, the N + 2 intensive variables cannot be independent, but must be linked by a relationship RI(pl,

~/2 . . . . .

~N, P, T) = 0

(21)

If the system is brought into contact with a reservoir imposing values of/~ which do not satisfy this relationship, the hyperspace point defined by p must lie either above or below the hypersurface of Eq. (21). If the point lies above the hypersurface, the system grows until it is comparable in size with the reservoir, at which point the latter can no longer maintain the given potentials. If it lies below, then the phase shrinks and finally vanishes. The physical system of current interest has two bulk phases in equilibrium, and phase 2 also restricts the possible values of the intensive variables: R2(/~1,122

.....

fiN, P, T) = 0

(22)

giving a total of N degrees of freedom. These two relationships embody the Gibbs phase rule, restricting the chemical potentials, pressure P and temperature T at which the two phases can be in equilibrium. Since the aim of the present exercise is to find a formulation which is symmetric in the system substances, the independent variables must be taken to be p, and Eqs. (21) and (22) can be

I.R. Peterson/Colloids Surfaces A: Physicochem. Eng. Aspects 102 (1995)21-29

28

solved to express P and T in terms of them: P

= P 1 2 ( # l , ]A2. . . . , ]AN) ]

(

(23)

T = Tlz(]Aa,]A2..... ]AN)) Subscripts 12 indicate that these functions depend upon the choice of the bulk phases 1 and 2 which are in equilibrium. There is a second paradox linked to this mutual dependence of the intensive parameters. Eq. (17) gives the derivatives of G~ explicitly in terms of other surface excess quantities:

OpiJ

~Pi

(24)

Eq. (18) leads immediately to Gibbs' adsorption equation:

It has already been noted that the quantity G° does not correspond exactly to the value F s given in Eq. ( 11 ) for the simple system of Fig. 2, because in the latter equation only terms involving water were subtracted from the bulk free energy, while the corresponding terms for the amphiphile were not. Just as the Helmholtz free energy is more convenient than the Gibbs free energy for analysing constant volume systems, it may be more convenient to limit the sum term in Eqs. (12) and (13) to those substances which are present in larger quantities in the bulk than at the interface. The resulting surface excess parameters are to this extent arbitrary and are clearly not symmetric in the system substances, although the choice is made on physical grounds and reflects an objective asymmetry between the substances.

Se

d? = -- ~ d T - • F~ d]Ai

(25)

In surface chemistry experiments it is usual to keep the external pressure P and temperature T constant. This justifies the elimination of the term in dT from Eq. (25) to give the Gibbs surface isotherm. However, if this procedure is repeated for the present parameters, then exactly the same equation results. Although it has been shown that the surface excess quantities ne and hence the surface excess densities F e often differ only minimally from the corresponding Gibbs surface excess quantities, this iden.tity of the adsorption isotherms for the two families of parameter appears impossible. The paradox is resolved by noting that, in general, P and T cannot be kept fixed while at the same time allowing # to vary independently. Even if side 2 of the interface is a solid which is not in equilibrium with the liquid phase 1, Eq. (21) still holds. Eq. (14) is the transformation appropriate for a change in the choice of reference substance, and the Gibbs equation (25) is valid whichever is chosen, i.e. for N different values of the parameter ~c. d? is independent of the choice, but varies linearly with K. Hence, the coefficient is zero, and any transformation of the surface excess quantities of the form given in Eq. (14) must leave the righthand side of Eq. (25) unchanged.

6. Conclusions The Gibbs surface energy equation is seen to account correctly for processes involving a change of interracial volume, even if the volume work is ascribed counter-intuitively to "bystander" substances. The surface excess free energy for a simple model of an amphiphilic monolayer on a free liquidvapour interface of water has been derived explicitly in terms of the bulk thermodynamic parameters of the system. It is seen that the Gibbs procedure in this very simple case is exactly equivalent to Legendre transformation of the Gibbs free energy to take into account free exchange of water with the environment. The redefinition of the surface parameters in terms of the Legendre relationship beween the bulk and surface excess free energies can be maintained for more complex systems. This change of definition makes very little difference to the numerical values in practical cases, and it removes the arbitrariness of the original Gibbs definition.

Acknowledgements The author would like to acknowledge financial support from the Fonds Henri de Rothschild, to

LR. Peterson/Colloids Surfaces A. Physicochem. Eng. Aspects 102 (1995)21 29

thank Professor C.M. Knobler for useful discussions, and to thank Dr. F. Rondelez for providing a stimulating environment.

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29

[10] G.G. Roberts (Ed.), Langmuir Blodgett Films, Plenum, New York, 1990. [11] I.R. Peterson, in W. G/~pel and C. Ziegler (Eds.), Nanostructures Based on Molecular Materials, VCH, Weinheim, 1992, pp. 195 208. [12] K. Kimelberg and D. Papahadjopoulos, Biochim. Biophys. Acta, 233 (1971) 805. [13] O. Albrecht, H. Gruler and E. Sackmann, J. Phys. (France), 39 (1978) 301. [14] D.A. Cadenhead, in G. Benga (Ed.), Structures and Properties of Cell Membranes, CRC Press, Boca Raton, FL, 1985. [15] M. Lundquist, Chem. Scr., 1(5)(1971) 197. [,16] M.C. Shih, T.M. Bohanon, J.M. Mikrut, P. Zschack and P. Dutta, Phys. Rev. A, 45 (1992) 5734. [,17] I.R. Peterson, V. Brzezinski, R.M. Kenn and R. Steitz, Langmuir, 8 (1992) 2995. [18] I.R Peterson and R.M. Kenn, Langmuir, 10 (1994)4645. [19] R.M. Richardson and S.J. Roser, Liq. Cryst., 2 (1987) 797. [20] C.A. Helm, H. M6hwald, K. Kjaer and J. Als-Nielsen, Europhys. Lett., 4 (1987) 697. [21] K. Kjaer, J. Als-Nielsen, C.A. Helm, P. Tippmann-Krayer and M. M6hwald, J. Phys. Chem., 93 (19891 3200. [22] N.R. Pallas and B.A Pethica, Langmuir, 8 (1992) 599.