Square gradient approximation

In two classic papers [18, 46], Calm and Flilliard developed a field theoretic extension of early theories of micleation by considering a spatially inliomogeneous system. Their free energy fiinctional, equations (A3.3.52). has already been discussed at length in section A3.3.3. They considered a two-component incompressible fluid. The square gradient approximation implied a slow variation of the concentration on the... 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

Here H is the Heaviside function and the sum is over sites in the (real space) lattice. In the interface region, p and Q are functions of z, with a ramped shape used. Working again in the square gradient approximation, Klupsch finds an interface of width 3.592a (where a is now the Lennard-Jones parameter) and an interfacial free energy of 0.968e/a. ... 

In the opposite limit of strong segregation (SSL) x 2, the free energy can also be expressed in terms of a square gradient functional, and one obtains qcR = 6 xN - 2). At intermediate segregation, which is most relevant to experiments, the square-gradient approximation breaks down and there is... 

Once the equilibrium phases have been determined, a parametrization of the Interfacial p(r) is constructed by allowing the order parameters used in the equilibrium calculation to vary with z, the coordinate perpendicular to the interface. The shape of these z-dependent order parameter profiles, as well as the interfacial excess free energy, can then be determined from the minimization condition, together with the boundary conditions [the order parameters must tend toward their crystal (liquid) equilibrium values as z goes to -foo (—oo)]. Many authors also assume that the Fourier components vary slowly across the interface, allowing a square gradient approximation to be used. 

Being a mean-field approach, the Flory-Huggins theory neglects fluctuations which are important in the vicinity of the critical point or the spinodal (cf. Sect. 2), i.e., just where the square gradient approximation is useful. 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

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... 

Figure 1, Composition of the critical cluster (bubble of critical size), Xgas, thermodynamic driving force of critical bubble formation, CJ,radius of the critical bubble, Rc, and work of critical bubble formation, [J Gc, computed for the case of boiling in binary liquid-gas solutions in dependence on supersturation here expressed via the density of the liquid puq (for the details see Ref 21). By the number (1), the results are shown computed via the classical Gibbs approach employing the capillarity approximation, number (2) refers to computations via the generalized Gibbs approach and number (3) to computations via the van der Waals square gradient density functional method.

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.

A detailed derivation of the square gradient term relies on the use of the random phase approximation, which is discussed in the appendix to this chapter. By this approach we find that we can write the free energy per segment of an inhomogeneous system as... 

Spatial inhomogeneities are only captured by a square gradient term in the free energy functional [3]. While this is appropriate in the vicinity of the critical point (or the spinodal), where the width of interfaces grows very large [146], this approximation becomes less accurate away from the critical point, where the intrinsic width (i.e., without accounting for capillary waves [100, 147]) of the interface is on the order of the interparticle distance in the liquid. 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

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