Distribution relaxation

G. J. Hirasaki 2003, (Diffusion-relaxation distribution functions of sedimentary rocks in different saturation states), Magn. Reson. Imaging 21, 305-310. 

In Section 4.1.4.1, we develop the estimation of the relaxation distribution functions from NMR data. These are used to determine porosity and saturation distributions. In Section 4.1.4.2, we develop the estimation of permeability distri-... 

We represent the NMR relaxation distribution by the continuous number density function, P( t), of characteristic relaxation time t. Our measurements correspond to a series of CPMG echoes, represented by... 

Fig. 4.1.2 The estimated normalized T2 relaxation distribution for the selected voxel in the Bentheimer sample.

L. D. Grandine, and E. R. Fitzgerald The relaxation distribution function of polyisobutylene in the transit ion from rubber-like to glass-like behavior. J. Appl. Phys. 24, 911 (1953). 

J. R. Macdonald, "Linear Relaxation Distributions, Thermal Activation, Structure, and Ambiguity," Journal of Applied Physics, 62 (1987) R51-R62. 

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

Fig. 11 The dynamic structure factor C(, r) of polybutadiene star 12880 (nominally f = 128, Ma = 80kgmol ) in cyclohexane at ci = 0.016gmL and q = 0.035nm , along with the fit (solid line) from the ILT analysis. The corresponding relaxation distribution function L(ln(T)) (shown here for f i and q = 0.023gmL ) embraces the cooperative diffusion (1), the collective apparent diffusion (2), and the self-diffusion (3). The slowing-down of the middle structural mode (2) and the increase of its intensity with q are shown in the upper inset whereas the lower cartoon illustrates the liquid-like ordering [43,189]. The core regions are drawn out of scale (larger) for clarity...

Figures 7 and 8 are the master curves at 25 °C for elastic modulus G and loss modulus G". The relaxation distribution function has also...

Figure 9. Relaxation distribution function using the Roesler and Twyman method. = 80-100 shell asphalt. = 30-70 S/A emulsion.

Fig. 2. The nascent vibrotational product distribution in the F-l-H2- HF(c, J) + H reaction (full lines). The curves within each c represent X,(J v). The areas under these curves are proportional to X (v). The dashed lines describe thermal rotational populations Xg(/lv). The doubly peaked (dotted and dash-dotted) curves correspond to partly relaxed distributions.

The molecular domains in amorphous structure behave like an ensemble of autonomous substates, each following unique relaxation kinetics during annealing (Kawakami and Pikal 2005). This relaxation distribution is often expressed using an empirical Kohlrausch-Williams-Watts (KWW) equation (Eq. 14.3) ... 

Thus, T is expressed via the parameters describing the interaction with the heat bath at equilibrium. This is what permits a rather simple estimate of t without solving the kinetic equations. Note, however, that in view of the approximation adopted the relaxation time given by Eq. (8.41) characterizes the relaxation distribution only on the average. 

For high M and high L values, the common instabilities occur, because of the strong coupling between equations in neighbouring points. A moderate under-relaxation factor (of 4) was used In the spike region only, to avoid these Instabilities. However, to penetrate this region, a new type of relaxation (distributed relaxation) has to be developed. 

The positive semidefinite function A(F) is the relaxation distribution function. 

Hence, the parameter q i indicates the relative distance between the particle surface and the next fluid node in ot-direction (cf. Fig. 1). Note that for finishing the propagation step of the fluid at xp, the post-relaxation distribution function at the solid node f i(xs, t) is needed and defined in the following way ... 

FIGURE 8.29 Real and imaginary part of relative permittivity with conductivity C — 0. Solid line Debye model with single relaxation dashed line Cole-Cole model with relaxation distribution with a = 0.5. After Chelidze and Gueguen (1999). 

The HCl vibrational-rotational arrested relaxation distributions reported by ref. 82b are in close agreement with those given in this table for S2CI2 and SOCI2 but not for SCI2. 

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