Maxwell-type construction

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).

These equations immediately yield 48 different interrelations of the Maxwell type which are listed in Table 5.2.1. They are found by the standard technique of taking each function of state and carrying out a double differentiation in either order with respect to two of its four appropriate independent variables. That is, one must construct relationships of the form d /dqidqj - d2Z/d q dqit where Z is any of the eight functions introduced on the left side of Eq. (5.2.3), and dq dqj are two of the four differentials appearing on the right side of the equation in question. Only a small fraction of these interrelations is actually useful these are introduced later as needed. 

A Maxwell-type element is constructed of the usual viscous dashpot but with an ideal rubber band in place of the usual Hookean spring. In a creep experiment with a constant load of 1000 Pa, and at a temperature of 27 C, the total deformation in tensile strain units after 3 h is e = 2.00. When the... 

Since gels which lack hydrophobic interactions are thought to show a thermo-swelling type of the transition, we applied our model to such systems to confirm its applicability. Calculated results are plotted against the reduced temperature (7 ) at various ft values in Fig. IS. The transition points were determined from the Maxwell construction. The calculated swelling curves reveal the thermoswelling behavior. An increase in fi enhances the ma tude of the volume change and lowers the transition temperature. 

Remark 4.7 In the case e = 1 (Maxwell models), system (16)-(17) is not always of evolution type. (See section 2.1.) Indeed, Renardy et al. [49] have constructed initial data in the hyperbolic domain, with steep gradients, such that the velocity and the stress develop singularities in their first space derivatives in finite time. The idea is to reduce the system, by a clever change of variables, to a degenerate system of three nonlinear hyperbolic equations. 

To make it applicable to other transition types as well, it is attractive to extend this flatness idea by replacing the Maxwell construction with a more general principle, the principle of least sensitivity. This is a weaker condition, but it allows us to investigate first-and higher-order transitions by a microcanonical analysis more systematically and in much more detail [61],... 

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