Correlation functions blocking

Finally, C tld /,.[ = cdd = c Jm is the Fourier transform of the replica-replica direct correlation function (blocking function), and the connected function is defined as usual by cc = cdd — cdd, and similarly for hc. Let us recall that the replicated particles are the dipolar hard spheres, i.e. the annealed fluid in the partly quenched mixture. 

Then, performing a disorder average in Eq. (19), and using Eq. (18) we can obtain the following two relations for the connected and blocking correlation functions... 

We proceed with cluster series which yield the integral equations. Evidently the correlation functions presented above can be defined by their diagrammatic expansions. In particular, the blocking correlation function is the subset of graphs of h rx2), such that all paths between... 

However, in the case of the ROZ equations, in addition to these closures, we also must use the closure for the blocking part of the direct correlation function. The HNC closure for this function reads... 

Finally, the closure relations for the inhomogeneous pair functions must be chosen. The PY approximation for the fluid-fluid direct correlation function presumes that its blocking part vanishes. This implies that c, ii(/,y) = 0, and... 

Here q is a wavevector (eqn 1.6), ip(q) is the Fourier transform of />(r), and S(q) is the structure factor (Fourier transform of the two-point correlation function). The cubic term, ft, is zero for a symmetric system and otherwise may be chosen to be positive. The quartic term, y, is then positive to ensure stability. For block copolymers, these coefficients may be expressed in terms of vertex functions calculated in the random phase approximation (RPA) by Leibler (1980). The structure factor is given by... 

Figures 7.2 and 7.3 show the relevant correlation functions. In three dimensions at the dimensionless time pvot = 10 the steady-state is already nearly achieved (the deviation from the unity is seen only at r < lOro). Since the correlation length at large t is finite, microscopic defect segregation takes place for d = 3. Quite contrary, for low (J < 2) dimensions the correlation functions are no longer stationary. Similarly to the recombination decay kinetics treated in [14], the accumulation kinetics demonstrates also an infinite increase in time of the correlation length (defined by a coordinate where X (r, f) 1 or F(r, t)

The next correlation function we consider is characteristic for a quenched-annealed system in the sense that it vanishes for conventional, fully annealed fluids. This is the so-called blocked correlation function hb(q, q ) defined by... 

For coiiveiitioual fluids the outer (disorder) average of the first term on the right side is absent and each thermal average equals the singlet density. Thus, hb = 0 for systems without quenched disorder. In the presence of disorder, on the other hand, the blocked correlation function is usually nonzero, because the singlet density for a particular realization, (9 ) can be... 

The total and the blocked correlation function introduced above Jire already sufficient to describe the structure within the adsorbed fluid. However, to describe thermal fluctuations we need to introducx two additional eorrev lation functions. The first one is the response function Gg (q, q ) defined as... 

Figure 7,4 Top Replica HNC predictions for the stability limits of the homogeneous isotropic phase of Stockmayer fluids adsorbed to disordered DHS matrices of density p,n = 0.1. Curves are labeled according to the reduced matrix dipole moment fJ m/ sTocr (the pure HS matrix corresponds to = 0). Bottom Dielectric constant of a dense adsorbed fluid as a function of the matrix dipole moment T = 0.5, f) = 0.7, Pm = 0.1). The inset shows the integrated blocking part of the dipole dipole correlation function.

To derive Eq. (7.23), which relates the blocked correlation function, hb, to the replicated system, we start from the statistical physical definition of the blocked function given in Eq. (7.22). On the right side of this equation, the double average over one of the pair terms, that is, for example, the term with = 1, / = 2, can he written as... 

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