Correlation length index

Wun and Prins (32) have used a Gaussian form for the correlation function, and then defined a non-random index in terms of a correlation volume rather than in terms of correlation length. 

Hence, we obtain the critical index for correlation length ... 

In this range, the connecting set is a fractal that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length and the P s of the connecting set (the upper index oo means that the limit / —> oo is taken) should scale with distance from the critical point (i.e., percolation threshold pc = p ) as... 

A good quality fit to experimental data can be achieved by assuming polydisperse dynamical fractal clusters with a fractal dimension df = 2.5 and a polydispersity index T = 2.2. From the fits we can deduce the short-range correlation length < o the radius of the cluster elements / (. We obtain ( o = 1.3 0.1 nm and / i = 5 1 nm. When none of the parameters is fixed, the best fit to the data leads to df = 2.5 0.1 and r = 2.3 0.1. These values are in good agreement with the theoretical estimates to within experimental error and lead to critical index values very close to the universal ones. The two other fitted parameters are ( o = 1.2 0.1 nm and / i = 6 1 nm. 

Equation (6) is obtained by using Debye s theory (14,IS). Here a (A) is called the correlation length and is a measure of the size of the heterogeneous structure inside the polymer. The symbol denotes the mean-square average of the fluctuation of all dielectric constants, n is the refractive index of medium, and A is the wavelength of light in a medium. 

As p approaches Pc = I, the correlation length grows to infinity. If we introduce the critical index v by the form... 

At short distances this coincides with the pair correlation for a single, ideal chain. At larger distances r > it is reduced. The correlation length depends on concentration but is independent of the polymerization index. Scaling suggests... 

Two example applications are described in detail, which address such issues as the convergence of the reliability index for the two methods of random field discretization, the effect of correlation length on the convergence, the influence of correlation length on reliability, reliability sensitivity with respect to distribution parameters, and multiplicity of failure modes. These examples clearly demonstrate the applicability and usefulness of the first>order reliability approach in conjunction with the finite element method for reliability analysis of complex structures. 

Fig. 7.41. The variation of the correlation length, , as a function of T — Tc for polystyrene in cyclohexane (a) and PDMS in supercritical CO2 (b). The slopes give the values of the critical index, v. (Reproduced with permission from [239]. Copyright 1999 American Chemical Society.)...

As one can see, the data for both considered polymer lie on one straight line, that allows to determine the value Vp, which is equal to 0.76. This magnitude corresponds well to classical value of correlation length percolation index Vp, which is equal to -0.80 [14]. 

The Debye-Bueche analysis (5,6) was employed to obtain the correlation length, and the mean-square refractive index fluctuation, of the gels from... 

Figure 4. Correlation length and mean-square fluctuation of refractive index versus network density.

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