SCGLE

The present chapter is aimed at reviewing the development and the specific applications of the SCGLE theory of colloid dynamics and of dynamic arrest. Thus, it is not aimed at reviewing the state-of-the-art in either of these research areas, for which excellent reviews are available [2,3,43-45]. We must also say that notable topics in both fields are barely or never mentioned here. This includes, for example, the structural, mechanical, and rheological properties, and the effects of hydrodynamic interactions. Instead, we focus on the treatment of the effects of direct conservative interactions in simple colloidal systems. Thus, here we shall primarily deal with monodisperse suspensions of spherical particles in the absence of hydrodynamic interactions, although the extension to multi-component systems will also be an important aspect of this review. 

This section deals with the fundamental basis of the SCGLE theory. We first describe what is understood here for the GLE and then illustrate its use in the derivation of exact result for the time-dependent friction function A (t), and for the collective and self intermediate scattering functions. In addition, we discuss two additional approximations that convert these exact results into a closed self-consistent system of equations. 

The first such relation involving the irreducible memory functions is based on a physically intuitive notion Brownian motion and diffusion are two intimately related concepts we might say that collective diffusion is the macroscopic superposition of the Brownian motion of many individual colloidal particles. It is then natural to expect that collective diffusion should be related in a simple manner to self-diffusion. In the original proposal of the SCGLE theory [18], such connections were made at the level of the memory functions. Two main possibilities were then considered, referred to as the additive and the multiplicative Vineyard-like approximations. The first approximates the difference [C(k, z) - O Kk, z)], and the second the ratio [C k, z)IO k, z)], of the memory functions, by their exact short-time limits, using the fact that the exact short-time values, C P(fe, t) and (35)SEXP( 0, of these memory functions are known in terms of equilibrium structural properties [18]. The label SEXP refers to the single exponential time dependence of these memory functions. 

In reference [19], a systematic comparison between the predictions of the SCGLE theory and the corresponding computer simulation data for four idealized model systems was reported. The first two were two-dimensional systems with power law pair interaction, u(r) = Air", with n = 50 (i.e., strongly repulsive, almost hard-disk like) and with n = 3 (long-range dipole-dipole interaction). The third one was the three-dimensional weakly screened repulsive Yukawa potential (whose two-dimensional version had been studied in reference [18]). The last system considered involved short-ranged, soft-core repulsive interactions, whose dynamic equivalence with the strictly hard-sphere system allowed discussion of the properties of the latter reference system. For all these systems G(r, f) and/or F(k, f) were calculated from the self-consistent theory, and Brownian dynamics simulations (without hydrodynamic interactions) were performed to carry out extensive quantitative comparisons. In all those cases, the static structural information [i.e., g(r) and 5(A )] needed as an input in the dynamic theories was provided by the simnlations. The aim of that exercise was to... 

The dynamics of colloidal mixtures provided by the previous extension of the SCGLE theory was illustrated in reference [21] with its application to a binary mixture of particles interacting through a hard-core pair potential of diameter a (assumed to be the same for both species), and a repulsive Yukawa tail of the form... 

The dimensionless parameters that define the thermodynamic state of this system are the total volume fraction ( ) = (nl6)n (with n being the total number concentration, n = n, + n, the relative concentrations = njn, and the potential parameters K2, and z. The free-diffusion coefficients D° are also assumed identical for both species, that is, D = D = D°. Explicit values of the parameters a and D are not needed, since the dimensionless dynamic properties, such as t), only depend on the dimensionless parameters specified above, when expressed in terms of the scaled variables ka and t/to, where to = a /D . Besides solving the SCGLE scheme, in reference [21] Brownian dynamics simulations were generated for the static and dynamic properties of the system above. 

The main conclusion of the previous comparisons is that, except for very short times, in reality there is no practical reason for preferring the multiplicative version of the SCGLE theory over its additive counterpart, particularly if we are interested in intermediate and long times. Since the additive approximation is numerically simpler to implement, we shall no longer refer to the multiplicative approximation. Still, one of the remaining practical difficulties of the SCGLE theory is the involvement of the SEXP irreducible memory functions z) and z) the need to previously... 

It is not difficult to see that the original self-consistent set of equations (involving Equations 1.33 and 1.34) shares the same long-time asymptotic solutions as its simplified version. It is then natural to ask what the consequences would be of replacing Equations 1.33 and 1.34 of the full SCGLE set of equations by the simpler approximations in Equations 1.37 and 1.38, that no longer contain the functions t) and t). The proposal of a simplified version of the SCGLE theory... 

To illustrate these ideas let us summarize the general system of equations that constitute the SCGLE theory. In principle, these are the exact results for A (f), F k, f), and t) in Equations 1.20,1.23, and 1.24, complemented with the simplified Vineyard approximation in Equation 1.37 and the simplified interpolating closure in Equation 1.38. This set of equations define the SCGLE theory of colloid dynamics. Its full solution also yields the value of the long-time self-diffusion coefficient which is the order parameter appropriate to detect the glass transition from the fluid side. This is, however, not the only method to detect dynamic arrest transition, as we now explain. 

FIGURE 1.7 (a) Theoretical fit (solid line) of the static structure factor, and (b) SCGLE theoretical predictions (solid line) for the nonergodicity parameter/(A ) of the mono-disperse charged sphere system of reference [33], modeled by the pair potential of Equation 1.36 at ( ) = 0.27, z = 3.1587, and K = 11.66. The symbols correspond to the experimental data. The inset in (b) enlarges the region where experimental data for/(A) are available. (From Yeomans-Reyna, L. et al. 2007. Phys. Rev. E 76 041504. With permission.)... 

The multicomponent extension of the SCGLE theory [21] may be summarized, in its simplified form [59], by Equations 1.30 through 1.32, 1.37, and 1.38. It was applied rather recently [72] to the description of dynamic arrest in two simple model colloidal mixtures, namely, the hard-sphere and the repulsive Yukawa binary mixtures. The main contribution of reference [72], however, is the extension to mixtures of Equation 1.39. Thus, the resulting equation for y , the localization length squared of particles of species a, was shown to be... 

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