Correlation radii, length

Table 5. Scaling predictions for the mean square radius of gyration and the mean square correlation lengths < 2> in the different regimes (see Fig. 38) of polymer solutions [102-104]...

Figure 8.35. The homogeneous sphere of radius R. Radial correlation function, ys (r), distance distribution function (DDF) ps (r) and chord length distribution (CLD) gs (r)...

For example, the volume of a cube is perfectly correlated to the length of each side as V = s Likewise the volume of a sphere is perfectly correlated to its radius as V = 4/377r However, the mass of such objects will be highly correlated to, v or r only when the density (d) of the materials used to form the shapes are identical, since d = mass/volume. There is no correlation of mass to s or r when vastly different densities of material are used for comparison. Thus a first-order approximation for s and r vs. mass for widely different materials would lead one to believe that there is not a relationship between volume and mass. Conversely, when working with the same material one would find that volume and mass are perfectly correlated and that there is a direct relationship between volume and mass irrespective of shape. This simple example points to the requirements for a deeper understanding of the underlying phenomena in order to draw conclusions regarding cause and effect based on correlation. 

The Dependencies of Radius of Gyration Rg, Static Correlation Length Hydrodynamic Screening Length Viscosity r, Self-Translational Diffusion Coefficient D, Cooperative Diffusion Coefficient Dc, Coupled Diffusion Coefficient Df, and Electrophoretic Mobility p on c and N for Various Regimes of Polyelectrolyte and Salt Concentrations... 

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. 

The experimental diffusion parameters, D /r., at 30°C. are presented in Table II for all the coals. Clearly, no correlation exists between diffusion parameter and rank. If r<> is taken as the average particle radius for the 200 X 325 mesh samples, an upper limit to the values of diffusion coefficient, D, is obtained. The diffusion coefficient ranges from 1.92 X 10 9 sq. cm./sec. for Kelley coal to 1.41 X 10"8 sk. cm./sec. for the Dorrance anthracite. Our previous studies on the change of D /n with particle size suggested that n is not necessarily the particle radius (7) but is a smaller distance related to the average length of the micropores in the particles. That is, the calculated... 

The reaction radius ro does not enter the asymptotic relation (5.2.1). Its absence at t —> oo in the spirit of Section 5.1 could be interpreted as emergence of a new spatial scale - the correlation length . It arises in (5.2.11) through the terms u>mgm which play the role of the correlation sources. 

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