Vineyard-Like Approximations

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. 

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Either of these Vineyard-like approximations, along with an additional closure relation, will allow the exact results for A (t), F(k, t), and F %k, t) to constitute a closed set of equations. The closure relation consists of an independent approximate determination of the self irreducible memory function O Kk, t). One inmitive notion behind the proposed closure relation is the expectation that the -dependent self-diffusion properties, such as F k, t) itself or its memory function O k, t), should... 

We now have all the elements needed to define a self-consistent system of equations to describe the full dynamic properties of a colloidal dispersion in the absence of hydrodynamic interactions. In this section we summarize the relevant equations for both, mono-disperse and multicomponent suspensions, and review some illustrative applications. The general results for A (t), F(k, t), and F k, t) in Equations 1.20,1.23, and 1.24, complemented by either one of the Vineyard-like approximations in Equations 1.25 and 1.26, and with the closure relation in Equation 1.27, constitute the full self-consistent GLE theory of colloid dynamics for monodisperse systems. Besides the unknown dynamic properties, it involves the properties SQi), t), and t), assumed to be deter-... 

Yeomans-Reyna, L., Acnna-Campa, H., and Medina-Noyola, M. 2000. Vineyard-like approximations for coUoid dynamics. Phys. Rev. E 62 3395. 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

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