Tracer diffusion matrix

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

For nuclei that are coherent with the surrounding crystal, the lattice is continuous across the a//3 interface. The jumps controlling the /3C frequency factor will then be essentially matrix-crystal jumps and /3C will be equal to the product of the number of solute atoms surrounding the nucleus in the matrix, zcXg, and the solute atom jump rate, T, in the a crystal. The jump frequency can reasonably be approximated by T Di/a2 (see Eq. 7.52, where D is the solute tracer diffusivity and a is the jump distance). Therefore,... 

Theory for the self- and tracer-diffusion of a diblock copolymer in a weakly ordered lamellar phase was developed by Fredrickson and Milner (1990). They modelled the interactions between the matrix chains and a labelled tracer molecule as a static, sinusoidal, chemical potential field and considered the Brownian dynamics of the tracer for small-amplitude fields. For a macroscopically-oriented lamellar phase, they were able to account for the anisotropy of the tracer diffusion observed experimentally. The diffusion parallel and perpendicular to the lamellae was found to be sensitive to the mechanism assumed for the Brownian dynamics of the tracer. If the tracer has sufficiently low molecular weight to be unentangled with the matrix, then its motion can be described by a Rouse model, with an added term representing the periodic potential (Fredrickson and Bates 1996) (see Fig. 2.50). In this case, motion parallel to the lamellae does not change the potential on the chains, and Dy is unaffected by... 

The obtained value is smaller by about four orders of magnitude from experimental values15,16 of the chemical diffusion coefficient for oxygen in the ab-plane of the 123 matrix (D0(i23)ab), and higher by five orders from the experimental value15 of the tracer diffusion coefficient for copper in the ab-plane of the 123 matrix (D Cu(i23)ab), Table 1 the latter can be of the same order or by one order lower than the chemical diffusion coefficient15,16. 

Tracer diffusion is a measure of the ease and frequency with which radioactive or tagged atoms are diffusing in a matrix. It can be shown that = /cor- ion where for is a correlation coefficient that depends... 

The diffusion coefficients used to describe multi-component diffusion are mutual diffusion coefficients. In the multi-component system, mutual diffusion coefficients are defined by Equation 4-13 the matrix of diffusion coefficients depends on the concentration of individual components. The diffusion coefficients used in the earlier sections of the chapter, however, describe solute molecules diffusing in a medium at infinite dilution. The isolated molecule is called a tracer these tracer diffusion coefficients are defined by the physics of random walk processes, as described in Chapter 3. The self-diffusion coefficient, used in Equation 4-11, is a tracer diffusion coefficient in the situation where all of the molecules in the system are identical. The self-diffusion coefficient, T>aa is defined by (recall Equation 3-12) [62] ... 

Fig. 10.12 The molecular-weight dependence of the tracer diffusion constant obtained for the nearly monodisperse polystyrene samples in a polystyrene matrix of molecular weight P = 2 X 10 at 174°C. The solid line represents Dq = 0.008M . Reproduced, by permission, from Ref. 28.

The crucial assumption of the Doi-Edwards theory is that the primitive chain reptates in a tube fixed in space. However, in monodisperse solutions, all chains are wriggling simultaneously, so that the tube around each chain is never fixed but successively renewed by different chains. Hence, the Doi-Edwards theory is not self-consistent. This fact has given rise to recent measurements of the tracer diffusion coefficient as a function of the molecular weight and concentration of the matrix component. 

The literature on self-diffusion of polymers in solution, and on tracer diffusion of probe polymers through solutions of matrix polymers, is systematically reviewed. Virtually the entirety of the published experimental data on the concentration dependence of polymer self— and probe- diffusion is represented well by a single functional form. This form is the stretched exponential exp(—ac"), where c is polymer concentration, a is a scaling prefactor, and is a scaling exponent. 

If Dp depends significantly on Cp, extrapolation to Cp 0 must be performed. The initial slope of the dependence of Dp on probe concentration, and the slope s dependence on matrix concentration, have been measured in some systems and should be accessible to theoretical analysis. In this review If the probe and matrix polymers differ appreciably in molecular weight or chemical nature, the phrase probe diffusion coefficient is applied. If the probe and matrix polymers differ primarily in that the probes are labelled, the phrase self diffusion coefficient is applied. The tracer diffusion coefficient is a single-particle diffusion coefficient, including both the self cind probe diffusion coefficients as special cases. The interdiffusion and cooperative diffusion coefficients characterize the relaxation times in a ternary system in which neither m lcrocomponent is dilute. 

Jones [20] used a Smoluchowski approach to examine interacting spherical polymers. Jones predicted that, if one polymer species is dilute and labelled, the measured diffusion coefficient from QELSS is determined only by hydrodynamic interactions of the tagged polymers and their untagged matrix neighbors, and is the single-particle diffusion coefficient. The hydrodynamic approach culminated in analyses of Carter, et al. [21] and Phillies [22] of mutual and tracer diffusion coefficients, including hydrodynamic and direct interactions and reference frame issues. 

Wheeler, et al. [124] studied tracer diffusion of linear polystyrenes having molecular weights 65, 179, 422, and 1050 kDa (with M /M < 1.1) through a 1.3 MDa polyvinyl-methylether matrix polymer, 1.6, in orthofluorotoluene at concentrations 1-100... 

Although cannot be measured in DLS, a closely related tracer diffusion coefficient Dj can be measured. In the tracer diffusion, the motion of a labeled solute called a probe or a tracer is traced selectively. A second solute called a matrix is added to the solution and its concentration is varied, whereas the concentration of the probe molecules is held low. The matrix must be invisible, and the probe must be visible. We can give a large contrast between the matrix and probe by choosing a pair of solvent and matrix that are nearly isorefractive, i.e., having the same refractive index. Then, the light scattering will look at the probe molecules only. For instance, we can follow the tracer diffusion of polystyrene in a matrix solution of poly(dimethyl siloxane) in tetrahydrofuran. 

Figure 4.41. Tracer diffusion coefficient Di of polystyrene in solutions of matrix polymer, poly(vinyl methyl ether), in o-fluorotoluene at various concentrations of the matrix polymer, plotted as a function of molecular weight Mps of polystyrene. The solvent is isorefractive with the matrix polymer. The concentration of the matrix polymer and the slope obtained in the best fitting by a power law (straight line) are indicated adjacent to each plot. (From Ref. 53.)...

Several authors have explored tracer diffusion for the case that M and P are very different. By examining diffusion of photolabeled polymers through unlabeled homologous polymers, questions of thermodynamic incompatibility between the tracer and matrix chains may be almost entirely eliminated. Forced Rayleigh scattering was employed by Nemoto and collaborators to examine labeled tracer polystyrenes in polystyrene dibutylphthalate mixtures(48,49). Nemoto, et al. 

The objective here is to identify features characteristic of single-chain diffusion by an ideal polymer in solution, following which it becomes possible to identify specific chemical effects in particular series of measurements. As discussed below first, the functional forms of the concentration and molecular weight dependences of the self- and tracer diffusion coefficients are found. Second, having found that Ds almost always follows a particular functional form, correlations of the function s phenomenological parameters with other polymer properties are examined. Third, for papers in which diffusion coefficients were reported for a series of homologous polymers, a joint function of matrix concentration and matrix and probe molecular weights is found to describe Ds. Fourth, a few exceptional cases are considered. These cases show that power-law behavior can be identified when it is actually present. Finally, correlations between Ds, rj, and Cp are noted. In more detail ... 

This chapter examines the diffusion of mesoscopic rigid probe particles through polymer solutions. These measurements form a valuable complement to studies of polymer self- and tracer diffusion, and to studies of self- and tracer diffusion in colloid suspensions. Any properties that are common to probe diffusion and polymer self-diffusion cannot arise from the flexibility of the polymer probes or from their ability to be interpenetrated by neighboring matrix chains. Any properties that are common to probe diffusion and to colloid diffusion cannot arise from the flexibility of the matrix polymers or from the ability of matrix chains to interpenetrate each other. Conversely, phenomena that require that the probe and matrix macromolecules be able to change shape or to interpenetrate each other will reveal themselves in the differences between probe diffusion, single-chain diffusion, and colloid single-particle diffusion. 

Phillies, etal. (77) re-examined results of Brown and Zhou(78) and Zhou and Brown (79) on probe diffusion by silica spheres and tracer diffusion of polyisobutylene chains through polyisobutylene chloroform solutions. These comparisons are the most precise available in the literature, in the sense that all measurements were made in the same laboratory using exactly the same matrix polymer samples, and were in part targeted at supporting the comparison made by Phillies, et al.(Jl). Comparisons were made between silica sphere probes and polymer chains having similar Dp and Dt in the absence of polyisobutylene. For each probe sphere and probe chain, the concentration dependence of the single-particle diffusion coefficient is accurately described by a stretched exponential in c. For large probes (160 nm silica spheres, 4.9 MDa polyisobutylene) in solutions of a small (610 kDa) polyisobutylene, Dp c)/Dt(c) remains very nearly independent of c as Dp c) falls 100-fold from its dilute solution limit. 

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