Closure relationships

The RSOZ equations Eqs. (7.41) and (7.42) still involve both the total and the direct correlation functions. Therefore, appropriate closure expressions relating the correlation functions to the pair potentials are needed to calculate the correlation functions at given densities and temperatures. Typically, one uses standard closure expreasions familiar from bulk liquid state theory [30]. One should note, however, that the performance of these closures for disordered fluids can clearly not be taken for granted. Instead, they need to be tested for each new model system under consideration. 

In the following discussion we present as an example closure expressions appropriate for systems where both fluid and matrix particles are spherical and have fixed diameters ( hard cores ) (Xf (fluid) and (matrix). The corresponding fluid fluid, fluid matrix, and matrix matrix interactions then contain (apart from other contributions) the hard-sphere(HS) potential 

Following the strategy describ(xl at the beginning of Seetion 7.5, we start by considering closure relations for the replicated system, emplojdng the notation introduced in Eq. (7.36). The (exact) hard-core conditions can be written as 

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies 

Note the appearance of the quenching temperature To (instead of T) for the matrix correlations [first member of Eq. (7.46)]. This is a consequence of 

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This form follows directly from the closure relationship... 

Either of these two types of coupled function can be expressed in terms of the other using the closure relationship ... 

The spin term is straightforward to evaluate by the Wigner-Eckart theorem but the rotational term requires further consideration. Let us introduce the projection operator onto the complete set of rotational functions between the operators Pp (J) and 2) (< >) (the closure relationship) ... 

More sophisticated approaches to the evaluation of wfr) exist. Some of them are the technique based on the Yoon-Flory expansion, variations of the Gaussian approximation, and an implementation of the Koyama distribution. Freely jointed chains and rotational isomeric chains have also been used as models of chains. Research is actively pursued on the subject of formulating more realistic closure relationships. In considering the choice of an intramolecular structure factor, a paper by McCoy et al. on model polyethylene melts will be useful. 

Using the closure relationships Exj. (G.19) and (G.20) for separations below and above the hard core, we can now calculate the new direct correlation functions (r). We stress that the closure can be solved independentlj- for each index combination a/ . Furthermore, note that the HNC closure (r > a p) involves the total correlation functions (f), which can bo approximated by the current estimates (r) -... 

Using the closure relationship for any operators A and B allows us to account for the contributions from all final states of the system (that conserve energy and momentum). 

Models of this type are beginning to show rather good agreement with experimental data, but they still involve a wide variety of assumptions with regard to closure relationships. 

Each block is modeled as linear, isotropic, homogeneous and elastic medium and subdivided with a mesh of constant-strain triangle finite-difference elements. Key factors affecting the hydraulic behaviour of fractures such as opening, closure, sliding and dilation of fractures are modeled by an elasto-perfectly plastic constitutive model of a fracture. A step-wise non-linear normal stress-normal closure relationship is adopted with a linear Mohr-Coulomb failure for shear (Figure 3). 

The orientational averages required in Eq. (23) are facilitated by a well-known addition theorem (also called the closure relationship) for the Wigner polynomials ... 

In the special case (b) for co = ir/2 (Fig. 1), the reduced linear dichroism reduces to LD = 5. By concatenating a series of rotational transformations (protein - membrane -> cell) and applying the closure relationship, it can be shown that... 

Equation (41) represents a closure relationship for resonant states and Eqs. (42) and (43) represent, respectively, sum rules obeyed by resonant states. 

In general, the exact bridge functions BJr) are represented as an infinite series of integrals over high-order correlation functions and are therefore practically incomputable, which makes it necessary to incorporate some approximations [82, 83, 89]. The most commonly used closure relationship is the KH closure proposed by Kovalenko and Hirata [91], which was designed to improve convergence rates and to prevent possible divergence of the numerical solution of the RISM equations [91] ... 

We can identify the weights, A , if we equate them as the range, or width, about the points. We can also use eq. (8.1) to replace Xk the range (c - AJ2, + AJ2). Using the exact subspace closure relationship we thus write... 

It is obvious that even higher TDB moment distribution balances can be constructed, but we will restrict ourselves to the ones developed for up to the second moments. When applying the TDB moment model, in principle two solution strategies are possible one using the zeroth and first TDB moments only and the second with zeroth, first, and second TDB moments. In the first case we have to find a closure relationship for the second moment, and in the second case, one for the third moment. Below, we show that the system becomes simpler in the case of a maximum of one TDB per chain. 

Here the functions D and D(, are in fact polydispersities of the branching moment distributions and in principle are to be determined as functions of chain length n. Inserting these closure relationships in Eqs. (65), (66), and (70), (71), reduces them to ... 

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