Correlation length definition

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

This estimate should be made more precise. To do it, let us use some results of the numerical solution of a set of the kinetic equations derived in the superposition approximation. The definition of the correlation length o in the linear approximation was based on an analysis of the time development of the correlation function Y(r,t) as it is noted in Section 5.1. Its solution is obtained neglecting the indirect mechanism of spatial correlation formation in a system of interacting particles, i.e., omitting integral terms in equations (5.1.14) to (5.1.16). Taking now into account such indirect interaction mechanism, the dissimilar correlation function, obtained as a solution of the complete set of equations in the superposition approximation... 

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients, Da = D, in which case the similar correlation functions just coincide, Xv r),T) = X(t),t). Taking into account the definition of correlation length Id = VDt, where D = Da + D = 2D a, as well as time-dependence of new variables r) and r, one gets from (5.1.2) to (5.1.4) a set of equations... 

According to this definition dilute solutions of long macromolecules are critical. The role of the correlation length is played by the radius of gyration Rg rsj Nu — oo N — oo, and by virtue of the chain structure a polymer coil shows density fluctuations on all scales r < Rg. Indeed, a blob of size r is just a correlated fluctuation of the density. 

By definition, criticality is reached if the correlation length diverges ... 

It is however possible to discuss several special cases analytically. The zero temperature correlation length can still be observed as long as this is smaller than the thermal de Broglie wave length At which can be rewritten for K not too close to Ku as t < f/y Kt(, KtK" 1 with tu Lpl, where we defined tk via = jj-, analogously to the definition of At, and used (30). We call this domain the quantum disordered region. 

Let us consider two more important aspects of nanofiller particles aggregation within the frameworks of the model [31]. Some features of the indicated process are defined by nanoparticle diffusion at nanocomposite processing. Specifically, length scale, connected with diffusible nanoparticle, is correlation length of diffusion. By definition, the growth phenomena in sites, remote more than are statistically independent. Such definition allows to connect the value with the mean distance between nanofiller particle aggregates L. The value can be calculated according to the equation as in what follows [31] ... 

The definition of this apparent correlation length can be extended to dilute regimes, in which case app gives g/V3 (see eqn (2.72)). The behaviour of app is entirely different in dilute solutions and concentrated solutions. In dilute solutions, lapp dincreases with the molecular weight and the excluded volume, while in concentrated solutions app — I is independent of the molecular weight and decreases as a function of concentration and excluded volume (see eqn (5.36)). The reason can be easily understood from Fig. 5.1 once polymers overlap each other, the excluded volume interaction tends to make the concentration homogeneous. 

Fig. 8.17 Definition of self-similarity, correlation length, and average molecular weights.

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