Spin-diffusion coefficients

It is found that the relaxation parameter T p as a function of temperature does not follow an increase with chain length, as the square of the number of methylene carbons. Nor is it linear with N, the number of methylene carbons, which should be true if relaxation to the lattice were rate controlling. Rather, it shows a temperature-induced increase of the minimum value of Tjp with about the 1.6 of N. So, both spin diffusion and spin lattice coupling are reflected. For a spin diffusion coefficient D of approximately 2 x 10 12 cm.2/sec., the mean square distance for diffusion of spin energy in a time t is the ft1 = 200/T A, or about 15A on a Tjp time scale. 

By chosing short values of x, the effect of diffusion may be eliminated, because the diffusion is effective in reducing the echo intensity only during the 2x period. However, there is a modified version of spin-echo pulse sequences which provides the measurement of the spin diffusion coefficients (D) by NMR, as it will be shown later. 

As reviewed in the previous section, measurements of Ti and Tip can provide an estimation of the length scale of miscibility of polymer blends. Compared with such kinds of experiments, the results of the spin-diffusion experiments are more quantitative and straightforward. The accuracy of the results of spin-diffusion experiments relies, to a large extent, on the values of spin-diffusion coefficients (7)) employed in calculation of the constituent phase components. Despite efforts that have been made, there still lacks a suitably applicable method of directly measuring the spin-diffusion coefficients, at least for polymers. For rigid polymer below Tg, 0.8 nm /ms has been turned out to be a reliable value of spin-diffusion coefficient. The difficulty left then concerns how to determine the coefficient of the mobile phase, which is very sample dependent. Recently, through studies on diblock copolymers and blend samples with known domain sizes, Mellinger et al established empirical relations between the T2 and D as follows ... 

In this equation, fis the number of orthogonal directions relevant for the spin diffusion process. Its value depends on morphology and is 1 for lamellar block copolymers, 2 for phases with a cylinder-like morphology in a matrix, and 3 for discrete phases (for example spheres in a matrix). The remaining parameter in Eq. 1 is Deff. the effective spin diffusion coefficient. It can be calculated according to Eq. 2. 

From the T2 values the diffusion coefficient of the mobile phase could be determined. The corresponding D values are listed in Table 1 as well. Using Eq. 2, the effective spin diffusion coefficient Deff can be calculated (see Table 1). 

Spin diffusion measurements of poly(phthalainide)/poly(diniethylsiloxane) block copolymers having separated phases (rigid poly(phthalamide) part, mobile PDMS part) showed different domain sizes depending on the length of the PDMS chain. The shorter the chain, the smaller is the domain size of the separated PDMS phase. To determine the spin diffusion coefficient of the mobile phase spin-spin, relaxation time T2 was measured. It showed the big mobility difference between the sample with the short chain (PAioPDMSn) and the other two samples (PA10PDMS50 and PAioPDMSioo) — one order of magnitude. Therefore, the PDMS chain with 11 links is fairly restricted in its mobility compared with the other two chains with 50 and 100 links. 

White and coworkers have recently described an experimental approach in which intramonomer spin-diffusion is used to quantitatively define upper limits on the value of spin-diffusion coefficients D in mobile and rigid homopolymers, as well as in copolymers and blends [61-63]. The independent determination of the diffusion coefficient using only NMR data would be possible if a unique, invariant reference volume or distance existed in the polymer sample that could be used to quantitatively define the diffusive length scale. In other words, an internal distance calibration on the sample itself would eliminate the need for independent... 

Figure 7.6 One-dimensional three-region model used in the computer simulation of relaxation in a heterogeneous system. The overall repeat distance is indicated. The calculations are carried out over the unit indicated in the lower half of the figure, comprising half of the regions A and C and the whole of B for which the dimensions, spin diffusion coefficients and relaxation rates are indicated. Redrawn and adapted with permission from [79].

Q. Chen, K. Schmidt-Rohr, Measurement of the local 1H spin-diffusion coefficient in polymers. Solid State Nucl. Magn. Reson. 29 (2006) 142—152. 

---