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ENERGY OF INTERACTION BETWEEN SOLID SURFACES AND LIQUIDS 8 Henri GOUIN 0 0 2 L. M. M. T. Box 322, University of Aix-Marseille n Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20 France a J 9 E-mail: [email protected] 2 ] h p Abstract - m We consider the wetting transition on a planar surface in contact with a e h semi-infinite fluid. In the classicalapproach,the surface is assumedto be solid, c and when interaction between solid and fluid is sufficiently short-range, the . s contribution of the fluid can be represented by a surface free energy with a c density of the form Φ(ρ ), where ρ is the limiting density of the fluid at the i S S s surface. y h Inthepresentpaperweproposeamorepreciserepresentationofthe surface p energythattakesintoaccountnotonlythevalueofρ butalsothecontribution S [ fromthe whole densityprofileρ(z)ofthe fluid, wherez is coordinatenormalto 1 the surface. v The specific value of the functional of ρ at the surface is expressed in mean- S 1 field approximation through the potentials of intermolecular interaction and 8 some other parameters of the fluid and the solid wall. 4 4 An extension to the case of fluid mixtures in contact with a solid surface is . 1 proposed. 0 8 0 : v i X 1. Introduction r a The phenomenonofsurfacewettingis asubjectofmanyexperiments. They have already been used to determine many important properties of the wetting behavior for liquids on low-energy solid surfaces 1. To gain a theoretical ex- planation of the phenomenon of wetting, a generalized van der Waals model is often used 2. Inthe 1950’sZisman3 developedanexperimentalmethodofcharacterizingthe free energy of a solid surface by the measurement of the contact angle with respect to the liquid-vapor surface tension and by changing the test liquid. 1 More recent measurements done by Li and all 4,5 improved the understanding of this problem. While the contact angle and surface tension are macroscopic quantities, they have their origin in molecular interactions. In the following, we use a mean-field theory to investigate how the surface energy is related to molecular interactions. The approximation of mean field theory is too simple to be quantitatively accurate. However it does provide a qualitativeunderstandingandallowsone to calculateexplicitely the magnitude of the coefficients in our model. In 1977, John Cahn1 gave simple illuminating arguments to describe the interaction between solids and liquids. His model is based on a generalized van der Waals theory of fluids treatedas attracting hardspheres6. It entailed assigning tothe solidsurfaceanenergythatwasafunctionalofthe liquiddensity”atthe γ surface”; the particular form Φ(ρ ) = −γ ρ + 2 ρ2 of this energy, where S 1 S 2 S ρ is the fluid density at the solid wall, is now widely known in the literature S and is due to Nakanishi and Fisher 7. It was thoroughly examined in a review paper by de Gennes 8. To account for the wetting behavior of liquids on solid surfaces, one needs to know Φ(ρ ); a major object of the present paper is to S obtainexplicitformulasfor the coefficientsγ andγ , expressingthem interms 1 2 of the parameters in assumed microscopic interaction potentials. Three hypotheses are implicit in Cahn’s picture. i) For the liquid density to be taken to be a smooth function ρ(z) of the distance z from the solid surface, that surface is assumed to be flat on the scaleofmolecularsizesandthecorrelationlengthisassumedtobegreaterthan intermolecular distances (as is the case, for example, when the temperature T is not far from the critical temperature T ). c ii) The forces between solid and liquid are of short range with respect to intermolecular distances. iii) The fluid is considered in the framework of a mean-field theory. This means, in particular, that the free energy of the fluid is a classical so-called ”gradient square functional”. The pointofview thatthe fluidin the interfacialregionmaybe treatedas bulk phase with a local free-energy density and an additional contribution arising from the nonuniformity which may be approximated by a gradient expansion truncated at the second order is most likely to be successful and perhaps even quantitatively accurate near the critical point 6. Some numerical calculations based on Cahn’s model and comparison with experiments can be found in the literature 2,9. The aim of this paper, as was stated, is to obtain the values of the different coefficientsoftheexpressionofenergyduetotheinteractionbetweenasolidwall and a liquid bulk. These values are associated with intermolecular potentials of the liquid and the solid wall. They take into account the molecular sizes and finite range of interactions between the molecules of the liquid and the solid. Thevalueofcoefficientsγ andγ arepositive. Consequentlytheγ term 1 2 1 2 describes an attraction of the liquid by the solid, and the γ term a reduction 2 of the attractive interactions near the surface. In fact, the calculation leads to a more complex functional dependence than the one given by Cahn, Nakanishi andFisherordeGennes1,7,8. Theenergyofthewalltakesintoaccountnotonly the value of the density at the wall but also the gradient of density normal to the wall. This should be more accurate when the variation of density is strong withrespecttomolecularsizes. Thedensityoftheliquidν ischosendepending 1 on three coordinates x,y,z. The direction to the wall is associated with z, and near the wall, the density may be chosen to be a function only of z. In fact such an assumption does not simplify the calculations. In our expression, the energy smoothly depends on x,y and allows continuous variations of density alongthe wall. TheconnectionwithNakanishi-Fisherexpressionisnotaffected by this more general hypothesis. We denote by E the new expression of the S wall energy. Thecalculationmaybeextendedtomorecomplexcases: forexampletothecase ofaliquidmixtureincontactwithasolidwall. Weproposeageneralexpression of the energy density at the solid surface. We note that the distribution of the concentrations of the components may be influenced by a solid wall effect. 2. Description of the solid-fluid interaction in mean field approximation In regions of a fluid where the density ρ in nonuniform, the intermolecular potentialsproduceaforceonagivenmoleculethatmaygeneratesurfacetension effects 6,10. Using classical molecular theory, it is possible to obtain a system ofpressureandcapillarytensionthatis mathematicallyequivalentto the stress tensor of a continuum model 6,11. Such a description does not take into account the possibility of interaction of the liquid with the solid wall (when the distributionofdensityvariesstronglyinthedirectionnormaltoasolidsurface). These effects constitute the subject of the present paper. TheeffectsareillustratedinFigure1andaredescribedbelow. Weconsideraflat solid wall. The approach is aimed to describe the interaction of a macroscopic surfacewithabulkfluid. Theso-called”hardspherediameter”ofthemolecules is denoted by σ for the fluid and τ for the solid. Then, the minimal distance 1 between solid and liquid molecules is δ = (σ+τ). 2 Inmean-fieldtheory,werepresentby ψ(r) theintermolecularpotentialbetween two molecules of the fluid and respectively χ(r) the potential between a fluid molecule and a wall molecule at separation r. The density of molecules at a point depends on its coordinates x, y, z, but the masses m and m 1 2 of each molecule of the liquid and solid are assumed to be fixed. The energy correspondingtotheactionofallmoleculesoftheliquidandthesolidonagiven 3 molecule 1 located in 0 (see Figure 1) is W = m2 ψ(r )+ m m χ(r ). (1) 0 1 i 1 2 j i j X X Figure 1: Molecular layer between the liquid and the solid surface. The first summation (over i) is over fluid molecules (except for molecule 1), and the second summation (over j) is over the wall molecules. Molecule 1 is in the fluid and r ,r are distances from molecule i or j to molecule 1. i j Denoting by ν dω the number of molecules of fluid in the volume element 1 1 dω and ν dω the number of molecules of solid in volume element dω , the 1 2 2 2 potential energy resulting from the action of all molecules in the medium on molecule 1 located in 0 may be expressed in a continuous way: W = m2 ψ(r)ν dω + m m χ(r)ν dω (2) 0 1 1 1 1 2 2 2 ZZZΩ1 ZZZΩ2 where Ω and Ω are the domains occupied by the liquid and the solid. 1 2 Potential energy W will be summed over all the molecules of the liquid. 0 So, the first integral is counted twice in the previous integral over the domain Ω occupied by the liquid. In such a condition, we have to consider only the 1 expression 1 m2 ψ(r)ν dω + m m χ(r)ν dω (3) 2 1 1 1 1 2 2 2 ZZZΩ1 ZZZΩ2 for the potential energy W . 0 4 Let ν (x,y,z) be an analytic function in each point M(x,y,z) of the liquid. 1 Inthe next derivationwe replace atthe point (0,0,0)ν with its Taylorexpan- 1 sioninvariablesx, y, z limitedtothesecondorder. Suchanassumptionmeans that the forces between the solid and the liquid are of a short range. This is similar to the expansion given by Rocard11. This means that ψ(r) is a rapidly decreasing function of r. Then, it is only necessary to know the distribution of molecules at a short distance from molecule 1. This case reflects the reality when the main force potentials decrease as r−6 with the distance. So, the potentials decrease very rapidly from the solid wall. The distribution of density inside the solid is assumed to be uniform. We write this expansion: ∂ν ∂ν ∂ν ν (x,y,z) = ν +x 10 +y 10 +z 10 + (4) 1 10 ∂x ∂y ∂z 1 ∂2ν ∂2ν ∂2ν ∂2ν ∂2ν ∂2ν x2 10 +y2 10 +z2 10 +2xy 10 +2xz 10 +2yz 10 +... 2 ∂x2 ∂y2 ∂z2 ∂x∂y ∂x∂z ∂y∂z (cid:18) (cid:19) ∂ν ∂2ν where ν , 10 , 10 ,... represent the values of ν and its derivatives 10 ∂x ∂x2 1 at point (0,0,0) . If we note ρ = m ν and ρ = m ν the densities in the liquid at point 1 1 10 2 2 2 (0,0,0) and inside the solid wall, we obtain ∞ ∞ W = 2π m ρ r2ψ(r)dr+π m ρ r(h−r)ψ(r)dr + 0 1 1 1 1 Zσ Zh ∞ ∞ π ∂ρ 2π m ρ r(r−h)χ(r)dr+ m 1 r(h2−r2)ψ(r)dr + 1 2 2 1 ∂z Zh Zh ∞ ∞ π π m ∆ρ r4ψ(r)dr+ m ∆ ρ (−2r4+3hr3−rh3)ψ(r)dr + 1 1 1 T 1 3 12 Zσ Zh π ∂2ρ ∞ m 1 r(h3−r3)ψ(r)dr (5) 6 1 ∂z2 Zh In Eq. (5), h represents the distance between the molecule 1 and the solid wall, ∆ is the Laplace operatorand ∆ is the Laplace operator tangentialto T the wall. Details of the calculation are given in Appendix 1. Notice that the last two terms may also be written in the form π ∞ π ∂2ρ ∞ m ∆ρ (−2r4+3hr3−rh3)ψ(r)dr+ m 1 rh(h2−r2)ψ(r)dr 12 1 1 4 1 ∂z2 Zh Zh (6) The energy density per unit volume at point 0 is E = ν W . o 10 0 In fact, this result is independant of the reference point. If we denote E the energy per unit volume at any point M in the liquid, we obtain 5 ∞ ∞ E = 2π ρ2 r2ψ(r)dr+π ρ2 r(h−r)ψ(r)dr + 1 1 Zσ Zh ∞ ∞ π ∂ρ 2π ρ ρ r(r−h)χ(r)dr+ ρ 1 r(h2−r2)ψ(r)dr + 1 2 2 1 ∂z Zh Zh ∞ ∞ π π ρ ∆ρ r4ψ(r)dr+ ρ ∆ ρ (−2r4+3hr3−rh3)ψ(r)dr + 1 1 1 T 1 3 12 Zσ Zh π ∂2ρ ∞ ρ 1 r(h3−r3)ψ(r)dr (7) 6 1 ∂z2 Zh Expression(7) yields ∂ρ E = Kρ2+α(h)ρ ρ + β(h)ρ2 + γ(h)ρ 1 + Kb2ρ ∆ρ + B (8) 1 1 2 1 1 ∂h 1 1 with ∞ ∞ π K = 2π r2ψ(r)dr , Kb2 = r4ψ(r)dr 3 Zσ Zσ ∞ ∞ α(h) = 2π r(r−h)χ(r)dr , β(h) = π r(h−r)ψ(r)dr Zh Zh ∞ π γ(h) = − r(h2−r2)ψ(r)dr 2 Zh π ∞ π ∂2ρ ∞ B = ρ ∆ ρ (−2r4+3hr3−rh3)ψ(r)dr+ ρ 1 r(h3−r3)ψ(r)dr 12 1 T 1 6 1 ∂z2 Zh Zh Theconstant b appearingintheρ ∆ρ termdenotesthecovolumeoftheliquid 1 1 as in the van der Waals equation 11,12 . The corresponding energy of the fluid is W = E dω (9) 1 ZZZΩ1 where E is given by Eq. (8) . To take into account the kinetic effects like in Rocard 11 , p. 392,we must add the terms corresponding to kinetic pressure to the value of W. The first term Kρ2 yields the internal pressure. 1 Now, we calculate the different values of coefficients in Eq. 8 in a special case of London forces. 3. Calculations of the energy of interaction in the case of London forces It is now possible to calculatethe value of W forthe particularinteraction potentials. For example, one can take (see Appendix 2) k µ ψ(r) = − and χ(r) = − (10) rn−1 rn−1 6 In the case of London forces one has 10 n=7. In fact this rough approximation is valid only at short range from the wall (hypothesis ii). Following the calculations in Appendix 2, we obtain that the 1 two integrals in the B term are of the order of ; moreover, h µπ kπ kπ α(h) = − , β(h) = , γ(h) = − (11) 6h3 12h3 8h2 Integralsassociatedwith Eq. (9) are takenoveran interval [δ, d] , where d is therangeofmolecularforcesintheliquidandδ istheminimaldistancebetween solid and liquid molecules. ∂ρ ∂2ρ For the same reasons as in Eq. (4), the expansion of ρ , 1 , 1 at the 1 ∂h ∂h2 solid wall is taken in the form ∂ρ h2 ∂2ρ S S ρ = ρ +h + + ··· 1 S ∂h 2 ∂h2   ∂∂ρh1 = ∂∂ρhS + h ∂∂2hρ2S + ··· (12) ∂2ρ ∂2ρ 1 S = + ··· ∂ρ ∂2ρ∂h2 ∂h2 S S Terms ρ , , denote values of ρ and its normalderivativesat the S ∂h ∂h2 1 solid wall. ∂ρ ∂ρ ∂2ρ ∂2ρ This means ρ =(ρ ) , S =( 1) , S =( 1) . S 1 S ∂h ∂h S ∂h2 ∂h2 S Now we make the important assumption that the variations of ρ take into 1 δ account several molecular ranges. Hence, can be considered as a small pa- d rameter. It implies that the first derivative of ρ with respect to h is on the 1 ρ ρ order of 1, the second derivative of ρ is on the order of 1. Then, the B d 1 d2 term can be removed from the integration of E. Consequently, in the calculation of the surface energy in Eq. (4), the second derivative terms may be removed (but not in the calculation of the bulk energy of the liquid associated with K(ρ2+b2ρ ∆ρ )). This result is in agreement with some considerations 1 1 1 by de Gennes 13. Keeping terms up to the first order in Eq. (8) yields ∂ρ ∂ρ ∂ρ E =K(ρ2+b2ρ ∆ρ )+α(h)ρ ρ +h S +β(h) ρ2+2hρ S +γ(h)ρ S 1 1 1 2 S ∂h S S ∂h S ∂h (cid:18) (cid:19) (cid:18) (cid:19) or ∂ρ 2hβ(h)+γ(h) ∂ρ2 E =K(ρ2+b2ρ ∆ρ )+α(h)ρ ρ +β(h)ρ2+hα(h)ρ S + S 1 1 1 2 S S 2 ∂h 2 ∂h 7 Then, the energy of the fluid is W =W +W , with 1 2 W = K(ρ2+b2ρ ∆ρ )dω 1 1 1 1 1 ZZZΩ1 W = ρ ρ α(h)dhdxdy + ρ2 β(h)dhdxdy 2 2 S S ZZZΩ1 ZZZΩ1 ∂ρ ∂ρ2 2hβ(h)+γ(h) +ρ S hα(h)dhdxdy+ S dhdxdy 2 ∂h ∂h 2 ZZZΩ1 ZZZΩ1 with +∞ µπ +∞ kπ α(h)dh = − , β(h)dh = 12δ2 24δ2 Zδ Zδ +∞ µπ +∞ kπ hα(h)dh = − , hβ(h)dh = 6δ 12δ Zδ Zδ +∞ γ(h) kπ dh = − (13) 2 6δ Zδ −−→ −−→ −−→ We note that ρ ∆ρ = ρ div(grad ρ ) = div(ρ grad ρ )−(grad ρ )2 1 1 1 1 1 1 1 and the Stokes formula yields W = K(ρ2−b2(−gr−a→d ρ )2)dω + Kb2ρ ∂ρS ds (14) 1 1 1 1 S ∂h ZZZΩ1 ZZΣ Here Σ notes the surface of the solid wall corresponding to the boundary of Ω . As it was said in paragraph 2, to take into account the kinetic effects, we 1 must add to energy W the terms corresponding to kinetic pressure. 1 The first term Kρ2 yields the internal pressure, and consequently 1 Kb2 −−→ ε = α(ρ ,η) − (grad ρ )2 1 ρ 1 1 denotestheinternalspecificenergy. Term ρ α(ρ ,η) isthebulkinternalvolume 1 1 energy as a function of ρ and of the specific entropy η in the liquid. 1 Let us note that ∂ρ kπ ∂ρ2 Kb2ρ S ds = − S ds S ∂h 6σ ∂h ZZΣ ZZΣ Then, a straighforward calculation yields the energy of the liquid in the well known form −−→ W = ρ α(ρ ,η)−Kb2 (gradρ )2 dω + E ds 1 1 1 1 S ZZZΩ1 ( ) ZZΣ 8 but with γ ∂ρ ∂ρ2 E = −γ ρ + 2 ρ2 −γ S −γ S (15) S 1 S 2 S 3 ∂h 4 ∂h ∂ρ2 ∂ρ Here, γ S means 2ρ γ S, and 4 ∂h S 4 ∂h µπ kπ γ = ρ , γ = (16) 1 12 δ2 2 2 12 δ2 µπ kπ 1 1 γ = ρ , γ = − 3 6 δ 2 4 6 σ 8 δ (cid:18) (cid:19) E is the form of the special energy to be added at the solid surface to obtain S the total energy of the liquid. In expression (15), the γ term (favoring large 1 ρ )describesanattractionoftheliquidbythesolid. The γ termrepresentsa S 2 reduction of the liquid/liquid attractive interactions near the surface : a liquid moleculelyingdirectlyonthesoliddoesnothavethesamenumberofneighbors thatitwouldhaveinthebulk. Thetermswiththe coefficients γ and γ also 3 4 describe a reduction of the liquid/liquid attractive interactions due to the lack ∂ρ S of molecules of the liquid near the wall (in the case where is positive). ∂h Expressions (15) and (16) do generalize the results by Nakanishi and Fisher7: Expression(15)containsterms(thefirsttwo)similartotheexpressiongivenby NakanishiandFisher,butwefindadditionaltermsassociatedwith γ and γ . 3 4 Additional terms are the correction of the two first terms. Our main interest is to obtain values of coefficients of the energy of the wall as a function of the propertiesofmoleculesandtotakeintoaccountvariationsofdensityatthewall strong enough with respect to molecular sizes. 4. Extension to the case of liquid mixtures in contact with a solid wall Hereweproposeanextensionoftheabovetheorytothecaseofliquidmixtures. Thisexampleisforabinarymixture,butthereisnoreasononecouldnotinclude more species in the mixture. The hypotheses are the same as in the case of one-component liquid. The only difference will come from the interaction of molecules of the two liquids. An adaptation of the previous calculations yields the following results. The potential energy resulting from the action of all molecules in the medium on molecule 1 of liquid 1 located in 0 is W = m2 ψ (r)ν dω 10 1 1 1 1 ZZZΩ1 + m m ψ (r)ν dω + m m χ (r)ν dω (17) 1 2 3 2 1 1 3 1 3 2 ZZZΩ1 ZZZΩ2 9 Thisenergyisonlyforonespeciesintheliquid. Todeterminethewholeenergy, one must first sum over all molecules of species 1 and then do an analogous summation over the molecules of species 2. Potential energy W will be 10 summed over all molecules of the liquid mixture. In this way the contribution of the liquid-liquid integral, W , will be counted twice in the previous integral 10 over the domain Ω . 1 Wehavedenotedby m ,themassofthemoleculeoffluid i(i∈{1,2}), ψ i 1 and ψ are the potentials of interaction between molecules of fluid 1 with 2 themselves and among molecules of fluid 2, ψ is the potential of molecular 3 interaction between the two fluids, χ (i ∈ {1,2}) are the intermolecular po- i tentialsoffluids withthe solidwall,andν (i∈{1,2,3})denote the number of i molecules in liquids and solid wall per unit volume. Then, following the procedure of section 2, we obtain : ∞ E = ν W = 2π ρ2 r2ψ (r)dr 10 10 10 1 1 Zσ1 ∞ ∞ + π ρ2 r(h−r) ψ (r)dr+ 2πρ ρ r(r−h)χ (r)dr 1 1 1 3 1 Zh Zh ∞ ∞ π ∂ρ π + ρ 1 r(h2−r2) ψ (r)dr+ ρ ∆ρ r4ψ (r)dr 2 1 ∂z 1 3 1 1 1 Zh Zσ1 ∞ π + ρ ∆ ρ (−2r4+3hr3−rh3)ψ (r)dr 1 T 1 1 12 Zh π ∂2ρ ∞ + ρ 1 r(h3−r3)ψ (r)dr 6 1 ∂z2 1 Zh ∞ ∞ + 2 πρ ρ r2ψ (r)dr+ πρ ρ r(h−r)ψ (r)dr 1 2 3 1 2 3 Zσ2 Zh ∞ ∞ π ∂ρ π + ρ 2 r(h2−r2) ψ (r)dr+ ρ ∆ρ r4ψ (r)dr 2 1 ∂z 3 3 1 2 3 Zh Zσ2 ∞ π + ρ ∆ ρ (−2r4+3hr3−rh3)ψ (r)dr 1 T 2 3 12 Zh π ∂2ρ ∞ + ρ 2 r(h3−r3)ψ (r)dr 6 1 ∂z2 3 Zh where σ (i∈{1,2}) is the diameter of molecule of liquid i. i Wemayalsorepeatthesamecalculationfor E associatedwiththesecond 20 component of the mixture. As for a simple fluid, we may take into account the kinetic effects and the internal energy of a nonhomogeneous mixture as in Fleming et al 14. Then, we obtain the following additional energy at the solid surface for the liquid mixture in the form 1 E = −γ ρ −γ ρ + ( γ ρ2 +γ ρ2 +2γ ρ ρ ) S 11 1S 21 2S 2 12 1S 22 2S 32 1S 2S 10