Surface spinodal

Linear Dynamics of Surface Mode Surface Spinodal Decomposition.114... 

Fig. 32. Phase diagram of the surface plotted in terms of the scaled variables h jy and g/y. For g/y < —2 one observes critical wetting and for g/y > —2 one observes first-order wetting. In the latter regime, mean field theory predicts melaslable wet and non-wet regions limited by the two surface spinodal lines ft, and ft( respectively. Also two quenching experiments arc indicated where starting at a rescaled time r = 0 from a stable state in the non-wet region one suddenly brings the system by a change of fti into the metastahle wet or unstable non-wet region, respectively. From Schmidt and Binder (1987).

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],...

The unstable situation caused when a spread him begins to dewet the surface has been studied [32, 33]. IDewetting generally proceeds from hole formation or retraction of the him edge [32] and hole formation can be a nucleation process or spinodal decomposition [34]. Brochart-Wyart and de Gennes provide a nice... 

In figure A3.3.9 the early-time results of the interface fonnation are shown for = 0.48. The classical spinodal corresponds to 0.58. Interface motion can be simply monitored by defining the domain boundary as the location where i = 0. Surface tension smooths the domain boundaries as time increases. Large interconnected clusters begin to break apart into small circular droplets around t = 160. This is because the quadratic nonlinearity eventually outpaces the cubic one when off-criticality is large, as is the case here. 

Phase transitions in two-dimensional (adsorbed) layers have been reviewed. For the multicomponent Widom-Rowlinson model the minimum number of components was found that is necessary to stabilize the non-trivial crystal phase. The effect of elastic interaction on the structures of an alloy during the process of spinodal decomposition is analyzed and results in configurations similar to those found in experiments. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are layers of H2, D2, N2, and CO molecules on graphite substrates. We review the PIMC approach, to such phenomena, clarify certain experimentally observed anomahes in H2 and D2 layers and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are also analyzed via PIMC. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions, where quantum effects play a role. 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

Now the plait point is on the spinodal curve, and any two corresponding points of the connodal curve adjacent to the plait point are on a part of the surface which is convex in every direction, and for which therefore... 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

Figure 17. Different stages of the spinodal decomposition in a symmetric mixture (4>0 = 0.5) r is the dimensionless time. The Euler characteristic is negative, which indicates that the surfaces are bicontinuous. The Euler characteristic increases with dimensionless time. This indicates that the surface connectivity decreases.

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

K is positive, representing the "surface free energy at the boundary between emergent phases. Thus, if (3 f/3c ) > 0 the solution is stable to the small fluctuations applicable to eqn. 9 and phase separation by a random nucleation and growth mechanism can only be initiated by a finite, thermally driven fluctuation. The limit of this metastability (i.e., the spinodal) occurs at (3 f/3c ) 0 and the solution becomes unstable whenever (3 f/3c ) is negative. The... 

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