Second-order wetting

The Cahn approach describing simple fluid mixtures has been adopted by a mean field theory developed for polymer mixtures by Nakanishi and Pincus [61] and Schmidt and Binder [15] and is presented in the next section. The mean field theory and its various extensions [7] have been successfully used to describe much of the experimental segregation isotherm z (())<, ) data obtained so far [16, 92,120,145,165-167,169-175]. It allows us not only to distinguish isotherms z (J characteristic for partial and complete (first and second order) wetting but also to determine surface free energy parameters useful in predicting surface phase diagrams for the studied mixtures. 

In recent work Jerry and Dutta [176] reanalyzed, with the mean field approach, conditions [8] of the second order wetting transition. They have found that critical wetting transition must be accompanied by a prerequisite phenomenon of an enrichment-depletion duality it is expected that the surface is enriched in the given component when bulk composition ( )M is below a certain value Q and is depleted in the same component for >Q. Such an effect, easily predicted by simple lattice theory [177] and observed in Monte Carlo simulations [178, 179], has been very recently determined by us for a real polymer blend [175] (see Sect. 3.1.2.4). 

The graphical Cahn technique [8,61,153,176] and its numerical analogs [15, 159,162,187] have been used to study the conditions for first and second order wetting transition. In these calculations two simplifying assumptions, not valid in real polymer blends [16,120,167,170,175] (see the following sections), have been imposed on the bare surface contribution fs it is taken to be temperature independent and is approximated by the first two terms of an expansion in the volume fraction at the surface ( )s ... 

Fig. 51. Plol of — t)(iiZ)/i)Z z=u versus /j(0) for the cases of a second-order wetting transition (a) and a first-order wetting transition (b). The solution consistent with the boundary condition always is found by intersection of the curve /x2(0) - 1 with the straight line [h + gq.(0)]/y. In case (a) this solution is unique for all choices of k /y (keeping the order parameter g/y fixed). Critical wetting occurs for the case where the solution (denoted by a dot) occurs for y (0) = +1, where then ) i(Z)/dZ z u = I) and hence the interface is an infinite distance away from the surface. For ft] > ftlc the surface is non-wet while for fti > ft C the surface is wet. In case (b) the solution is unique for ft] < ft (only a non-wet state of the surface occurs) and for ft > ft (only a wet state of the surface occurs). For ft > ft] > ft three intersections (denoted by A, B, C in the figure) occur, B being always unstable, while A is stable and C metastable for ft]c > ft > ft and A is metastable and C stable for ft " > ft > ftic. At ftic where the exchange of stability between A and C occurs (i.e., the first-order wetting transition) the shaded areas in fig. 51b are equal. This construction is the surface counterpart of the Maxwell-type construction of the first-order transition in the bulk lsing model (cf. fig. 37). From Schmidt and Binder (1987).

One can show that the second-order wetting transition is characterized by a divergence of the susceptibility xi > in this mean field theory of critical wetting... 

Fig. 54. Schematic phase diagrams for wetting and capillary condensation in the plane of variables temperature and chemical potential difference, (a) Refers to a case in which the semi-infinite system at gas-liquid condensation (ftaKX — d = 0) undergoes a second-order wetting transition at T = 7V The dash-dotted curves show the first-order (gas-liquid) capillary condensation at p = jt(I), T) which ends at a capillary critical point T v, for two choices of the thickness D. For all finite D the wetting transition then is rounded off. (b), (c) refer to a case where a first-order wetting transition exists, which means that ps remains finite as T - T and there jumps discontinuous towards infinity. Then for /iaKX - /i > 0 a transition may occur during which the thickness of the layer condensed at the wall(s) jumps from a small value to a larger value ( prewelting ). For thick capillaries, this transition also exists (c) but not for thin capillaries because then /Jcnn - (D,T) simply is loo large.

Fig. 55. Surface phase diagram in the plane of variables g, and <)> for three values of g. The region where the surface is non-wet (at small gj is separated from the wet region by a phase boundary which describes the wetting transition. For > (second-order wetting) this is just the straight line giril = — g(l—) The region of first order wetting is shown for symmetrical mixtures with Na = NB = N = 10 and N = 100, respectively, and the first-order transitions are shown by dash-dotted curves. In this regime metastable wet and non wet phases are possible up to the stability limits ( surface spinodals ) denoted by dashed curves. Assuming that g, and g are essentially independent of temperature T, variation of T essentially means variation of <)>, . From Schmidt and Binder [125],...

It turns out [125] that the wetting transition separating the nonwet state of the surface is always first order when g > 0, while it may be second order for g < 0. Figure 55 shows the phase diagram [12 resulting from this simple mean field theory of wetting in polymer mixtures. A second order wetting transition occurs when the solution of Eq. (213) reaches the value The surface... 

Figure 4.6 Plot of the local free energy, U, as a function of = zo/. the local, dimensionless height of the film. This plot is for the case > 0 and represents a second-order wetting transition for the thickness of the film as a goes through zero the minimum at finite moves continuously to infinity.

Figure 4.8 The profile of the wedge and the thin film determined by the theory, for a system with a second-order wetting transition. Here AH = h x) — zq)/ so that AH = 0 is the top of the thin film whose thickness is zo. In the figure, we define X = KX. n these units, the contact angle in the figure is always finite, but when one takes the scalings into account, it is seen that the observable contact angle is equal to K, which vanishes at the transition.

In the particular case where we have a second-order wetting transition, an analytical form for the profile can be derived. We consider Eq. (4.62) with a small and positive, > 0 and neglect the term proportional to A since we focus on small values of and the free energy is already stabilized by > 0. The equilibrium value of the film thickness is thus given by zq = — log(a/ ) (this is equivalent to Eq. (4.55)) and the minimization of Eq. (4.65) with respect lo h x) yields... 

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