Square gradient theory

Once the free energy of an inhomogeneous system is given, one can calculate by standard methods the properties of the interface—for example, the interfacial tension or the density profile perpendicular the interface [285]. Weiss and Schroer compared the various approximations within square-gradient theory discussed earlier in Section IV.F for studying the interfacial properties for pure DH and FL theory [241, 242], In theories based on local density approximations the interfacial thickness and the interfacial tension were found to differ by up to a factor of four in the various approximations. This contrasts with nonionic fluids, where the density profiles and interfacial... 

On the basis of Cahn-Hilliard s work, Sanchez [1983] derived the square-gradient theory of surface tension relations between surface tension and compressibility. 

As we expect, as the temperature increases the surface tension decreases. This is because as a density gradient is introduced at the surface there is an extra entropy associated with the surface that reduces the free energy. As expected, at the critical point the surface tension goes to zero. At the critical point the difference between liquid and gas phases disappears, and the system is in a sense all interface. In fact, as one approaches the critical point square gradient theory predicts that the swface tension goes to zero according to a power law ... 

Figure 2.16. The surface tension of a lattice gas as a function of the temperature, assuming that the surface is the (111) face of a FCC lattice, with units chosen so that the lattice spacing is imity. Boltzmann s constant is unity and the interaction energy e = — 1. In these units the critical temperature is 3. The solid line is the prediction of square gradient theory, whereas the points are the predictions of an analogous mean-field theory in which no small-gradient approximation is made.

To apply this theory to calculate surface tensions one simply uses the free energy density function derived from equation (2.5.8) in conjunction with the square gradient theory of section 2.4. As usual this leads to a smoothly var3dng density profile between the liquid and vapour phase, which may be visualised in terms of the FOV model as a variation in cell size through the stnface, as shown schematically in figure 2.24. 

We now turn to the question of what we can say about the structure of the polymer surface at the microscopic level. As we have seen, in addition to predicting the surface tension, square gradient theories also predict the density profile at the surface of the polymer melt. Typically, the density goes smoothly from the melt density to zero (the density of the vapour phase being vanishingly small for high polymers) over a few angstroni imits, in very much the same way as it does at the surface of a small-molecule liquid. 

Figure 4.9. (a) Interfacial widths as a ftinction of the interaction parameter % calculated by square gradient theory for polymer mixtures with equal chain lengths N. The dashed line is the incompatible limit of equation (4.2.10). (b) Interfacial widths normalised by the chain dimensions a N, plotted against the inverse degree of incompatibility l/(%iV). These are the same results as those shown in (a) (+, N = 3000 O, N = 1000 and X, N = 300) the solid line is the prediction of equation (4.2.10). 

The simplest theories of polymer adsorption are mean-field theories. In fact, it is possible simply to apply the square gradient theory of surface segregation developed in section 5.1 to polymer solutions by setting the degree of polymerisation of one of the components to unity. An analogous theory was developed by de Gennes (1981) and applied to adsorption from theta solvents (under which circumstances mean-field theory can be expected to be reasonably accurate) by Klein and Pincus (1982). 

How can we reconcile this with om microscopic picture of the wetting layer derived from the square gradient theory To do this we must go back to... 

Calculations in the framework of square gradient theory (Fredrickson 1987) and self-consistent field theory (Shull 1992) confirm this expectation. Figure 6.28 shows the results of such calculations for three values of the degree of incompatibility %N, for a symmetrical block copolymer. For a value of = 5, well below the critical value for microphase separation of — 10.495,... 

Already in the eighteenth century it was realized that the capillary effect of fluids must arise from attractive forces between the constituents of matter, the molecules. This realization led to the idea that examination of the capillaiy effects could tell something about the attractive forces and possibly also about the molecules. Also modem physicists are interested in the explanation of the capillaiy phenomena in terms of intermolecular forces. This chapter highlights some of the applications of the square gradient theory of van der Waals [1] in modeling the behaviour of fluids near interfaces. For a more extensive discussion of this theoiy we refer to Rowlinson and Widom [2]. 

Traditional square-gradient theories [1], as introduced by van der Waals, assume a free energy density/depending in a unique way on the local density p and added to a free energy density proportional to the gradient Vp. Thus... 

Region I (Critical Region L FQ. According to the square gradient theory first proposed by Cahn and Hilliard,... 

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