Probe diffusion coefficient

The apparent diffusion coefficient, Da in Eq. (11) is a mole fraction-weighted average of the probe diffusion coefficient in the continuous phase and the microemulsion (or micelle) diffusion coefficient. It replaces D in the current-concentration relationships where total probe concentration is used. Both the zero-kinetics and fast-kinetics expressions require knowledge of the partition coefficient and the continuous-phase diffusion coefficient for the probe. Texter et al. [57] showed that finite exchange kinetics for electroactive probes results in zero-kinetics estimates of partitioning equilibrium constants that are lower bounds to the actual equilibrium constants. The fast-kinetics limit and Eq. (11) have generally been considered as a consequence of a local equilibrium assumption. This use is more or less axiomatic, since existing analytical derivations of effective diffusion coefficients from reaction-diffusion equations are approximate. 

When is a probe species dilute Dp is the probe diffusion coefficient at near-zero probe concentration. If Dp is substantially independent of Cp, no extrapolation 0 is needed. 

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. 

Molecular weight dependence of the self and probe diffusion coefficients ) and Dp for molecular weight P probes in solutions of matrix polymers at a fixed concentration c. The fits are to stretched exponentials Z oexp(—aM ) in matrix molecular weight M. The Table gives the best-fit parameters, the percent root-mean-squaxe fractional fit error %RMS, the system, and the reference. Square brackets [ ] denote paxameters that were fixed rather than floated. Abbreviations as per previous Tables, and DBP-dibutylphthalate. 

When is a probe species dilute The careful experimenter makes an adequate control study of the effect of probe concentration Cp. The ideal Dp is the extrapolation Cp 0 of the measured probe diffusion coefficient. In many systems Dp for small Cp is substantially independent of Cp, and no extrapolation is needed. In other systems Dp depends significantly on Cp, and an extrapolation Cp O must be performed. Several authors have measured the initial slope kp of the dependence of Dp on probe concentration, and the effect of matrix concentration on kp. 

Figure 9.28 Probe diffusion coefficients for (a) 350 kDa polymethylmethacrylate in good solvents tetrahydrofuran (O) and N,N-dimethylformamide (0), and the Theta solvent dioxane water ( ), after Gold, et al.(45), and (b) 67 nm polystyrene spheres in aqueous 139 kDa HPC at (O) 10 °C (good solvent conditions) and ( ) 41 (pseudo-theta point), after Phillies and Clomenil(46).

Results on probe diffusion fall into three phenomenological classes. In the first two classes, diffusion is usefully characterized by a single relaxation time and hence a well-defined probe diffusion coefficient In the third class, light scattering spectra are more complex. The classes are ... 

We begin with systems having a well-defined probe diffusion coefficient. 

Chapter 9 considers the experimental literature on diffusion of mesoscopic probe particles through polymer solutions. A coherent description was obtained, namely that the probe diffusion coefficient generally depends on polymer concentration as Dp = Dpoe p(—ac ). The parameters a and v both depend strongly on polymer molecular weight, with a M > and y close to 1. Parameter u tends toward 1.0 -with much variation - at small matrix M, but reached v 0.5 at large M. 

Comparison may be made with literature data on solution viscosities. The probe diffusion coefficient does not simply track the solution viscosity instead. Dpi] typically increases markedly with increasing polymer concentration and molecular weight. The probe rotational diffusion coefficient, for which there are Umited sets of experiments, either has the same concentration dependence as the viscosity or, in other systems, has a concentration dependence that is significantly weaker than the concentration dependence of the viscosity or the probe translational diffusion coefficient. 

In a few systems, Dp c) shows re-entrant behavior, in which the probe diffusion coefficient increases with increasing c, and then perhaps falls at still larger c. Re-entrance is prominent in systems having multimodal spectra there is a need for more systematic smdy. The Dp (c) also sometimes has a plateau at very low concentrations. The interpretations of re-entrance and plateau behavior are uncertain. 

In the small-molecule probe diffusion measurements, the observations are more controversial. FRAP measurements have shown that the relationship of the dye-probe diffusion coefficient in the thin film and in the bulk is temperature dependent below a certain temperature, the diffusivity in the thin film is higher than the bulk, and above that temperature, the opposite is found. NRET measurements also reveal a complicated response of the small molecule s mobility in polymer thin films. On the other hand, simple mass uptake and film thickness measurement show that the water diflftjsivity in polymer thin films is substantially smaller than that in the bulk. " ... 

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