hal , version 1-29 Jan 2008

Transcription

1 ENERGY OF INTERACTION BETWEEN SOLID SURFACES AND LIQUIDS Henri GOUIN L. M. M. T. Box 3, University of Aix-Marseille Avenue Escadrille Normandie-Niemen, 3397 Marseille Cedex 0 France enri.gouin@univ-cezanne.fr Abstract al-0080, version - 9 Jan 008 We consider te wetting transition on a planar surface in contact wit a semi-infinite fluid. In te classical approac, te surface is assumed to be solid, and wen interaction between solid and fluid is sufficiently sort-range, te contribution of te fluid can be represented by a surface free energy wit a density of te form Φ(ρ S ), were ρ S is te limiting density of te fluid at te surface. In te present paper we propose a more precise representation of te surface energy tat takes into account not only te value of ρ S but also te contribution from te wole density profile ρ(z) of te fluid, were z is coordinate normal to te surface. Te specific value of te functional of ρ S at te surface is expressed in meanfield approximation troug te potentials of intermolecular interaction and some oter parameters of te fluid and te solid wall. An extension to te case of fluid mixtures in contact wit a solid surface is proposed.. Introduction Te penomenon of surface wetting is a subject of many experiments. Tey ave already been used to determine many important properties of te wetting beavior for liquids on low-energy solid surfaces. To gain a teoretical explanation of te penomenon of wetting, a generalized van der Waals model is often used. In te 950 s Zisman 3 developed an experimental metod of caracterizing te free energy of a solid surface by te measurement of te contact angle wit respect to te liquid-vapor surface tension and by canging te test liquid.

2 More recent measurements done by Li and all 4,5 improved te understanding of tis problem. Wile te contact angle and surface tension are macroscopic quantities, tey ave teir origin in molecular interactions. In te following, we use a mean-field teory to investigate ow te surface energy is related to molecular interactions. Te approximation of mean field teory is too simple to be quantitatively accurate. However it does provide a qualitative understanding and allows one to calculate explicitely te magnitude of te coefficients in our model. In 977, Jon Can gave simple illuminating arguments to describe te interaction between solids and liquids. His model is based on a generalized van der Waals teory of fluids treated as attracting ard speres 6. It entailed assigning to te solid surface an energy tat was a functional of te liquid density at te surface ; te particular form Φ(ρ S ) = γ ρ S + γ of tis energy, were ρ S ρ S is te fluid density at te solid wall, is now widely known in te literature and is due to Nakanisi and Fiser 7. It was torougly examined in a review paper by de Gennes 8. To account for te wetting beavior of liquids on solid surfaces, one needs to know Φ(ρ S ); a major object of te present paper is to obtain explicit formulas for te coefficients γ and γ, expressing tem in terms of te parameters in assumed microscopic interaction potentials. Tree ypoteses are implicit in Can s picture. i) For te liquid density to be taken to be a smoot function ρ(z) of te distance z from te solid surface, tat surface is assumed to be flat on te scale of molecular sizes and te correlation lengt is assumed to be greater tan intermolecular distances (as is te case, for example, wen te temperature T is not far from te critical temperature T c ). ii) Te forces between solid and liquid are of sort range wit respect to intermolecular distances. iii) Te fluid is considered in te framework of a mean-field teory. Tis means, in particular, tat te free energy of te fluid is a classical so-called gradient square functional. Te point of view tat te fluid in te interfacial region may be treated as bulk pase wit a local free-energy density and an additional contribution arising from te nonuniformity wic may be approximated by a gradient expansion truncated at te second order is most likely to be successful and peraps even quantitatively accurate near te critical point 6. Some numerical calculations based on Can s model and comparison wit experiments can be found in te literature,9. Te aim of tis paper, as was stated, is to obtain te values of te different coefficients of te expression of energy due to te interaction between a solid wall and a liquid bulk. Tese values are associated wit intermolecular potentials of te liquid and te solid wall. Tey take into account te molecular sizes and finite range of interactions between te molecules of te liquid and te solid. Te value of coefficients γ and γ are positive. Consequently te γ term

3 describes an attraction of te liquid by te solid, and te γ term a reduction of te attractive interactions near te surface. In fact, te calculation leads to a more complex functional dependence tan te one given by Can, Nakanisi and Fiser or de Gennes,7,8. Te energy of te wall takes into account not only te value of te density at te wall but also te gradient of density normal to te wall. Tis sould be more accurate wen te variation of density is strong wit respect to molecular sizes. Te density of te liquid ν is cosen depending on tree coordinates x, y, z. Te direction to te wall is associated wit z, and near te wall, te density may be cosen to be a function only of z. In fact suc an assumption does not simplify te calculations. In our expression, te energy smootly depends on x, y and allows continuous variations of density along te wall. Te connection wit Nakanisi-Fiser expression is not affected by tis more general ypotesis. We denote by E S te new expression of te wall energy. Te calculation may be extended to more complex cases: for example to te case of a liquid mixture in contact wit a solid wall. We propose a general expression of te energy density at te solid surface. We note tat te distribution of te concentrations of te components may be influenced by a solid wall effect.. Description of te solid-fluid interaction in mean field approximation In regions of a fluid were te density ρ in nonuniform, te intermolecular potentials produce a force on a given molecule tat may generate surface tension effects 6,0. Using classical molecular teory, it is possible to obtain a system of pressure and capillary tension tat is matematically equivalent to te stress tensor of a continuum model 6,. Suc a description does not take into account te possibility of interaction of te liquid wit te solid wall (wen te distribution of density varies strongly in te direction normal to a solid surface). Tese effects constitute te subject of te present paper. Te effects are illustrated in Figure and are described below. We consider a flat solid wall. Te approac is aimed to describe te interaction of a macroscopic surface wit a bulk fluid. Te so-called ard spere diameter of te molecules is denoted by for te fluid and τ for te solid. Ten, te minimal distance between solid and liquid molecules is δ = ( + τ). In mean-field teory, we represent by ψ(r) te intermolecular potential between two molecules of te fluid and respectively χ(r) te potential between a fluid molecule and a wall molecule at separation r. Te density of molecules at a point depends on its coordinates x, y, z, but te masses m and m of eac molecule of te liquid and solid are assumed to be fixed. Te energy corresponding to te action of all molecules of te liquid and te solid on a given 3

4 molecule located in 0 (see Figure ) is W 0 = i m ψ(r i ) + j m m χ(r j ). () Figure : Molecular layer between te liquid and te solid surface. Te first summation (over i) is over fluid molecules (except for molecule ), and te second summation (over j) is over te wall molecules. Molecule is in te fluid and r i, r j are distances from molecule i or j to molecule. Denoting by ν dω te number of molecules of fluid in te volume element dω and ν dω te number of molecules of solid in volume element dω, te potential energy resulting from te action of all molecules in te medium on molecule located in 0 may be expressed in a continuous way: W 0 = m ψ(r)ν dω + Ω m m χ(r)ν dω Ω () were Ω and Ω are te domains occupied by te liquid and te solid. Potential energy W 0 will be summed over all te molecules of te liquid. So, te first integral is counted twice in te previous integral over te domain Ω occupied by te liquid. In suc a condition, we ave to consider only te expression for te potential energy W 0. Ω m ψ(r)ν dω + Ω m m χ(r)ν dω (3) 4

5 Let ν (x, y, z) be an analytic function in eac point M(x, y, z) of te liquid. In te next derivation we replace at te point (0, 0, 0) ν wit its Taylor expansion in variables x, y, z limited to te second order. Suc an assumption means tat te forces between te solid and te liquid are of a sort range. Tis is similar to te expansion given by Rocard. Tis means tat ψ(r) is a rapidly decreasing function of r. Ten, it is only necessary to know te distribution of molecules at a sort distance from molecule. Tis case reflects te reality wen te main force potentials decrease as r 6 wit te distance. So, te potentials decrease very rapidly from te solid wall. Te distribution of density inside te solid is assumed to be uniform. We write tis expansion: ν (x, y, z) = ν 0 + x ν 0 x + y ν 0 y + z ν 0 z + (4) ( ) x ν 0 x +y ν 0 y +z ν 0 z + xy ν 0 x y + xz ν 0 x z + yz ν y z ν 0 were ν 0, x, ν 0 x,... represent te values of ν and its derivatives at point (0, 0, 0). If we note ρ = m ν 0 and ρ = m ν te densities in te liquid at point (0, 0, 0) and inside te solid wall, we obtain W 0 = π m ρ π m ρ π 3 m ρ r ψ(r)dr + π m ρ r(r )χ(r)dr + π m ρ z r( r)ψ(r)dr + r( r )ψ(r)dr + r 4 ψ(r)dr + π m T ρ ( r 4 + 3r 3 r 3 )ψ(r)dr + π 6 m ρ z r( 3 r 3 )ψ(r)dr (5) In Eq. (5), represents te distance between te molecule and te solid wall, is te Laplace operator and T is te Laplace operator tangential to te wall. Details of te calculation are given in Appendix. Notice tat te last two terms may also be written in te form π m ρ ( r 4 +3r 3 r 3 )ψ(r)dr+ π 4 m ρ z Te energy density per unit volume at point 0 is E o = ν 0 W 0. r( r )ψ(r)dr In fact, tis result is independant of te reference point. If we denote E te energy per unit volume at any point M in te liquid, we obtain (6) 5

6 E = π ρ π ρ ρ π 3 ρ ρ Expression (7) yields wit r ψ(r)dr + π ρ r(r )χ(r)dr + π ρ ρ z r( r)ψ(r)dr + r( r )ψ(r)dr + r 4 ψ(r)dr + π ρ T ρ ( r 4 + 3r 3 r 3 )ψ(r)dr + π 6 ρ ρ z E = Kρ + α()ρ ρ + β()ρ + γ()ρ K = π α() = π r( 3 r 3 )ψ(r)dr (7) r ψ(r)dr, Kb = π 3 r(r )χ(r)dr, β() = π γ() = π ρ + Kb ρ ρ + B (8) r( r )ψ(r)dr B = π ρ T ρ ( r 4 +3r 3 r 3 )ψ(r)dr+ π 6 ρ ρ z r 4 ψ(r)dr r( r)ψ(r)dr r( 3 r 3 )ψ(r)dr Te constant b appearing in te ρ ρ term denotes te covolume of te liquid as in te van der Waals equation,. Te corresponding energy of te fluid is W = E dω (9) Ω were E is given by Eq. (8). To take into account te kinetic effects like in Rocard, p. 39, we must add te terms corresponding to kinetic pressure to te value of W. Te first term Kρ yields te internal pressure. Now, we calculate te different values of coefficients in Eq. 8 in a special case of London forces. 3. Calculations of te energy of interaction in te case of London forces It is now possible to calculate te value of W for te particular interaction potentials. For example, one can take (see Appendix ) ψ(r) = k r n and χ(r) = µ r n (0) 6

7 In te case of London forces one as 0 n=7. In fact tis roug approximation is valid only at sort range from te wall (ypotesis ii). Following te calculations in Appendix, we obtain tat te two integrals in te B term are of te order of ; moreover, α() = µπ 6 3, β() = kπ kπ, γ() = 3 8 () Integrals associated wit Eq. (9) are taken over an interval [δ, d], were d is te range of molecular forces in te liquid and δ is te minimal distance between solid and liquid molecules. For te same reasons as in Eq. (4), te expansion of ρ, solid wall is taken in te form ρ = ρ S + ρ S + ρ = ρ S + ρ S + ρ = ρ S + ρ S + ρ, ρ at te () ρ S Terms ρ S,, ρ S denote values of ρ and its normal derivatives at te solid wall. ρ S Tis means ρ S = (ρ ) S, = ( ρ ) ρ S S, = ( ρ ) S. Now we make te important assumption tat te variations of ρ take into δ account several molecular ranges. Hence, can be considered as a small parameter. It implies tat te first derivative of ρ wit respect to is on te d order of ρ d, te second derivative of ρ is on te order of ρ. Ten, te B d term can be removed from te integration of E. Consequently, in te calculation of te surface energy in Eq. (4), te second derivative terms may be removed (but not in te calculation of te bulk energy of te liquid associated wit K(ρ + b ρ ρ )). Tis result is in agreement wit some considerations by de Gennes 3. Keeping terms up to te first order in Eq. (8) yields ( E = K(ρ +b ρ ρ )+α()ρ ρ S + ρ ) ( ) S +β() ρ S +ρ ρ S ρ S S +γ()ρ S or E = K(ρ +b ρ ρ )+α()ρ ρ S +β()ρ S +α()ρ ρ S + β() + γ() ρ S 7

8 Ten, te energy of te fluid is W = W = W + W, Ω K(ρ + b ρ ρ )dω W = ρ ρ S α()ddxdy + ρ Ω S ρ S + ρ α()ddxdy+ ρ S Ω wit β()ddxdy Ω β() + γ() ddxdy Ω wit + δ α()d = µπ δ, + δ β()d = kπ 4δ + δ α()d = µπ 6δ, + δ β()d = kπ δ + δ γ() d = kπ 6δ We note tat ρ ρ = ρ div( grad ρ ) = div(ρ grad ρ ) ( grad ρ ) and te Stokes formula yields W = K(ρ b ( grad ρ ) )dω + Ω Σ Kb ρ S ρ S (3) ds (4) Here Σ notes te surface of te solid wall corresponding to te boundary of Ω. As it was said in paragrap, to take into account te kinetic effects, we must add to energy W te terms corresponding to kinetic pressure. Te first term Kρ yields te internal pressure, and consequently ε = α(ρ, η) Kb ρ ( grad ρ ) denotes te internal specific energy. Term ρ α(ρ, η) is te bulk internal volume energy as a function of ρ and of te specific entropy η in te liquid. Let us note tat Σ Kb ρ S ρ S ds = Σ kπ 6 ρ S ds Ten, a straigforward calculation yields te energy of te liquid in te well known form { W = ρ α(ρ, η) Kb ( grad ρ ) }dω + E S ds Σ Ω 8

9 but wit E S = γ ρ S + γ ρ S γ 3 ρ S γ 4 ρ S (5) Here, γ 4 ρ S means ρ Sγ 4 ρ S, and γ = µπ δ ρ, γ = kπ δ (6) γ 3 = µπ 6 δ ρ, γ 4 = kπ 6 ( 8 δ E S is te form of te special energy to be added at te solid surface to obtain te total energy of te liquid. In expression (5), te γ term (favoring large ρ S ) describes an attraction of te liquid by te solid. Te γ term represents a reduction of te liquid/liquid attractive interactions near te surface : a liquid molecule lying directly on te solid does not ave te same number of neigbors tat it would ave in te bulk. Te terms wit te coefficients γ 3 and γ 4 also describe a reduction of te liquid/liquid attractive interactions due to te lack ρ S of molecules of te liquid near te wall (in te case were is positive). Expressions (5) and (6) do generalize te results by Nakanisi and Fiser 7 : Expression (5) contains terms (te first two) similar to te expression given by Nakanisi and Fiser, but we find additional terms associated wit γ 3 and γ 4. Additional terms are te correction of te two first terms. Our main interest is to obtain values of coefficients of te energy of te wall as a function of te properties of molecules and to take into account variations of density at te wall strong enoug wit respect to molecular sizes. 4. Extension to te case of liquid mixtures in contact wit a solid wall Here we propose an extension of te above teory to te case of liquid mixtures. Tis example is for a binary mixture, but tere is no reason one could not include more species in te mixture. Te ypoteses are te same as in te case of one-component liquid. Te only difference will come from te interaction of molecules of te two liquids. An adaptation of te previous calculations yields te following results. Te potential energy resulting from te action of all molecules in te medium on molecule of liquid located in 0 is W 0 = m ψ (r)ν dω Ω + m m ψ 3 (r)ν dω + m m 3 χ (r)ν 3 dω (7) Ω Ω ) 9

10 Tis energy is only for one species in te liquid. To determine te wole energy, one must first sum over all molecules of species and ten do an analogous summation over te molecules of species. Potential energy W 0 will be summed over all molecules of te liquid mixture. In tis way te contribution of te liquid-liquid integral, W 0, will be counted twice in te previous integral over te domain Ω. We ave denoted by m i, te mass of te molecule of fluid i (i {, } ), ψ and ψ are te potentials of interaction between molecules of fluid wit temselves and among molecules of fluid, ψ 3 is te potential of molecular interaction between te two fluids, χ i (i {, }) are te intermolecular potentials of fluids wit te solid wall, and ν i (i {,, 3} ) denote te number of molecules in liquids and solid wall per unit volume. Ten, following te procedure of section, we obtain : + π ρ + π ρ + π ρ ρ z E 0 = ν 0 W 0 = π ρ r( r) ψ (r)dr + πρ ρ 3 r ψ (r)dr r( r ) ψ (r)dr + π 3 ρ ρ r(r )χ (r)dr + π ρ T ρ ( r 4 + 3r 3 r 3 )ψ (r)dr + π 6 ρ + πρ ρ ρ z ρ z r ψ 3 (r)dr + πρ ρ r( 3 r 3 )ψ (r)dr r 4 ψ (r)dr r( r)ψ 3 (r)dr r( r ) ψ 3 (r)dr + π 3 ρ ρ + π ρ T ρ ( r 4 + 3r 3 r 3 )ψ 3 (r)dr + π 6 ρ ρ z r( 3 r 3 )ψ 3 (r)dr were i (i {, }) is te diameter of molecule of liquid i. r 4 ψ 3 (r)dr We may also repeat te same calculation for E 0 associated wit te second component of te mixture. As for a simple fluid, we may take into account te kinetic effects and te internal energy of a nonomogeneous mixture as in Fleming et al 4. Ten, we obtain te following additional energy at te solid surface for te liquid mixture in te form E S = γ ρ S γ ρ S + ( γ ρ S + γ ρ S + γ 3 ρ S ρ S ) 0

11 γ 3 ρ S γ 3 ρ S ( ρ S γ 4 + γ ρ S 4 + γ 34 ρ S ρ S ) (8) All te coefficients γ ij can be calculated explicitly after te particular form of interaction potentials was cosen. For example in te case of London forces, te values of coefficients related to te densities of te two fluids at te surface are γ = µ π δ γ = k π δ ρ 3, γ = µ π δ, γ = k π δ ρ 3 (9) γ 3 = k 3π 4 ( δ + δ ) were ρ 3 is te density of te solid, µ i are te coefficients associated wit potentials χ i, i {, }, k i is associated wit potentials ψ i, i {,, 3} and δ i = ( i + τ), i {, } are te minimal distances between te solid and molecules of te two species of te mixture. Suc an expression allows one to estimate te influence of a solid wall on eac component of a fluid mixture. Depending on te values and signs of different coefficients γ ij, one can estimate te magnitude of te attraction or repulsion effects due to te wall. Placing te fluid mixture in contact wit specially cosen solid walls may be an efficient way to separate constituents of a molecular mixture. 5. Conclusion Te energy of a fluid in contact wit a solid wall contains a contribution from te solid wic may be represented by a surface density function. For a flat wall we obtain tis expression by taking into account not only te density of te liquid at te surface but also its normal derivative. All te numerical values of te coefficients of te surface energy functional are calculated in terms of te parameters of molecules in solid and liquid. Tis energy caracterizes te beavior of te surface in contact wit te fluid in te wetting transition. Te metod may be extended to te case of nonflat solid surfaces wic is important in catalysis cemistry. Appendix. Calculation of te value of E in Eq. (8) To obtain te formula for E given in Eq. (8), we ave to calculate te two integrals:

12 m ψ(r)ν dω and m m χ(r)ν dω Ω Ω a) Calculation of m ψ(r)ν dω Ω Tis integral is associated wit te energy of interaction between molecules in te liquid. Let us denote by S(0, ), te domain occupied by te spere centered at (0, 0, 0) and wit radius (see Figure ). We introduce (r, θ, ϕ) te sperical coordinates associated wit te center of te spere. Ten, m ψ(r) ν dω = Ω m ψ(r) ν dω + m ψ(r) ν dω S(0,) Ω S(0,) Let us note tan for any integers p, q, r, and any boundary Σ of a spere centered at (0, 0, 0) and radius r, x p+ y q z r ds = 0 and x ds = y ds = z ds = 4π 3 r4 Σ Ten, m ψ(r)ν dω = πm ν 0 S(0,) and Ω S(0,) m ψ(r)ν dω = π 0 Σ Σ Σ r ψ(r)dr + π 3 m ν 0 ( + ( π Arccos r r 4 ψ(r)dr ) ) m ψ(r)ν sin ϕdϕ r dr dθ Te expansion up to te second order of ν (see Eq. (4)), yields m ψ(r) ν dω = π ν 0 m r ( + r) ψ(r) dr + π Ω S(0,) ( + π ν 0 x + ν 0 y + π 6 wic is te desired result. ν 0 z m ) m ν 0 z m r ( r ) ψ(r) dr ( r 4 + 3r 3 r 3) ψ(r) dr r ( 3 + r 3 ) ψ(r) dr

13 Figure : Representation of te canged variable associated wit te molecular layer at te solid surface. b) Calculation of m m χ(r) ν dω Ω Tis term corresponds to te energy wit respect to te solid wall. = m m ν π wic gives relation (7). 0 ( Ω m m χ(r) ν dω χ(r) = π m m ν ( Arccos r 0 r (r ) χ(r) dr ) ) sin ϕ dϕ r dr dθ Now, we cange variable as in Figure. Te origin of te tird coordinate z is placed at 0. Consequently z = 0 0 wit 0 te position of molecule. If we take now te origin of te z-axis at te solid wall and cange te orientation in suc a way tat 00 =, we obtain tat z + is a constant for a given molecule. Hence, we get Ten, E = πρ +πρ ρ ρ = ρ z, ρ = ρ z r ψ(r) dr + πρ r(r ) χ(r) dr + π ρ ρ r( r) ψ(r) dr r( r ) ψ(r) dr 3

14 + π 3 ρ ρ r 4 ψ(r) dr + π ρ T ρ + π 6 ρ ρ ( r 4 + 3r 3 r 3) ψ(r) dr r( 3 r 3 ) ψ(r) dr (we note similarly te Laplace operator in new coordinates and in old coordinates). So, we obtain Eq. (8). Appendix. Some remarks about te potential associated wit coesive forces In molecular teory it is proved by te virial metod tan te so-called coefficient a of te van der Waals equation can be obtained from ϕ(r) 4πr 3 dr = 6a N (0) Te term ϕ(r) is te magnitude of te attractive force between te two molecules of te fluid at te distance r, and N is te Avogadro number. Te potential ψ (r) = ϕ(r) dr is assumed to be zero at infinity. ten, Take te forces in te form and Eq. (0) yields ϕ(r) = A r n, ψ (r) = 4π A A = A (n )r n r 3 n dr = 4π A n 4 n 4 3(n 4)a n 4 π N () Let us introduce ψ(r) suc tat m ψ(r) = ψ (r). Te term ψ gives te potential per unit mass. Ten, ψ(r) = 3(n 4) a n 4 (n ) π N m rn Here, Nm = M denotes te molar mass of te fluid. In case n = 7 (London forces), we get: 3a 3 ψ(r) = 4π M r 6 In te following, we use: ψ(r) = k r 6 wit k = 3a 3 4π M 4

15 Consequence: Calculation of E in Eq. (8) for London forces. Now, using direct integration, we obtain te values of te coefficients given in Eq. (8). Since now we obtain ψ(r) = k µ, χ(r) =, and n = 7 rn rn π r ψ(r) dr = k π 3 3 π 3 r 4 ψ(r) dr = k π 3 π r (r ) χ(r) dr = µ π 6 3 π r ( r) ψ(r) dr = k π 3 π r ( r ) ψ(r) dr = k π 8 π ( r r 3 r 3 ) ψ(r) dr = k π 6 π 6 r ( 3 r 3 ) ψ(r) dr = k π 8 Acknowledgements: I ave greatly benefited from generous discussions and correspondence wit Professor B. Widom and D r A. E. van Giessen. Witout teir interest and elp, tis paper would be never publised. Te support of PRC/GdR CNES-CNRS 85 is gratefully acknowledged. 5

16 References () Can J. W., J. Cem. Pys., 977, 66, () van Giessen A. E., Bukman D. J., Widom B., J. Colloid and Interface Science, 997, 9, 57. (3) Zisman W. A., In Contact angle, wettability and adesion : Advances in Cemistry Series, 43 (Gould R. F., ed.), A.C.S., Wasington D. C., 964,. (4) Li D., Neuman A. W., Langmuir 993, 9, 378. (5) Kwok D. Y., Li D., Neuman A. W., Colloids surfaces, 994, 89, 8. (6) Rowlinson J. S., Widom B., Molecular teory of capillarity, Clarendon Press, Oxford, 984. (7) Nakanisi H., Fiser M.E., Pys. Rev. Lett., 98, 49, 565. (8) de Gennes P. G., Review of Modern Pysics, 985, 57, 3, 87. (9) Snook I., van Megen W., J. Cem. Pys., 979, 70, (0) Israelacvili J., Intermolecular and surface forces, Academic Press, London, 99. () Rocard Y., Termodynamique, Masson, Paris, 967, Capter 5. () Gouin H., Comptes Rendus Acad. Sci. Paris, 988, 306, II, 755. (3) de Gennes P. G., J. Pys. (Paris) Lett. 98, 4, 377. (4) Fleming P. D., Yang A. J. M., Gibbs J. H. J., Cem. Pys., 976, 65, 7. 6