Mutual diffusion coefficients, polymer transport

Various diffusion coefficients have appeared in the polymer literature. The diffusion coefficient D that appears in Eq. (3) is termed the mutual diffusion coefficient in the mixture. By its very nature, it is a measure of the ability of the system to dissipate a concentration gradient rather than a measure of the intrinsic mobility of the diffusing molecules. In fact, it has been demonstrated that there is a bulk flow of the more slowly diffusing component during the diffusion process [4], The mutual diffusion coefficient thus includes the effect of this bulk flow. An intrinsic diffusion coefficient, Df, also has been defined in terms of the rate of transport across a section where no bulk flow occurs. It can be shown that these quantities are related to the mutual diffusion coefficient by... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

Experimental diffusion coefficients, as obtained from time-lag measiu ements, report a transport diffusion coefficient which carmot be obtained from equilibrium MD simulation. Comparisons made in the simulation literatme are typically between time-lag diffusion coefficients (even calculated for glassy polymers without correction for dual-mode contributions and self-diffusion coefficients. As discussed above, mutual diffusion coefficients can be obtained directly from equilibrium MD simulation but simulation of transport diffusion coefficients require the use of NEMD methods, that are less commonly available and more computationally expensive [117]. 

A discussion of the charge transfer reaction on the polymer-modified electrode should consider not only the interaction of the mediator with the electrode and a solution species (as with chemically modified electrodes), but also the transport processes across the film. Let us assume that a solution species S reacts with the mediator Red/Ox couple as depicted in Fig. 5.32. Besides the simple charge transfer reaction with the mediator at the interface film/solution, we have also to include diffusion of species S in the polymer film (the diffusion coefficient DSp, which is usually much lower than in solution), and also charge propagation via immobilized redox centres in the film. This can formally be described by a diffusion coefficient Dp which is dependent on the concentration of the redox sites and their mutual distance (cf. Eq. (2.6.33). 

In accordance with theoretical predictions of the dynamic properties of networks, the critical concentration of dextran appears to be independent of the molecular weight of the flexible polymeric diffusant although some differences are noted when the behaviour of the flexible polymers used is compared e.g. the critical dextran concentrations are lower for PEG than for PVP and PVA. For ternary polymer systems, as studied here, the requirement of a critical concentration that corresponds to the molecular dimensions of the dextran matrix is an experimental feature which appears to be critical for the onset of rapid polymer transport. It is noteworthy that an unambiguous experimental identification of a critical concentration associated with the transport of these types of polymers in solution in relation to the onset of polymer network formation has not been reported so far. Indeed, our studies on the diffusion of dextran in binary (polymer/solvent) systems demonstrated that both its mutual and intradiffusion coefficients vary continuously with increasing concentration 2. ... 

Separation from mixtures is achieved because the membrane transports one component more readily than the others, even if the driving forces are equal. The effectiveness of pervaporation is measured by two parameters, namely flux, which determines the rate of permeation and selectivity, which measures the separation efficiency of the membrane (controlled by the intrinsic properties of the polymer used to construct it). The coupling of fluxes affecting the permeability of a mixture component can be divided into two parts, namely a thermodynamic part expressed as solubility, and a kinetic part expressed as diffusivity. In the thermodynamic part, the concentration change of one component in the membrane due to the presence of another is caused by mutual interactions between the permeates in the membrane in addition to interactions between the individual components and the membrane material. On the other hand, kinetic coupling arises from the dependence of the concentration on the diffusion coefficients of the permeates in the polymers [155]. 

In a miscible polymer blend, mutual diffusion measurements provide a very sensitive method for probing the thermodynamics of interaction between two chemically different segments. From the relationship between the transport coefficient Dt and the D s of the polymer components one can extract the Flory parameter X from the measured values of D and the D s. Upon rewriting Eq. 1, x is given by... 

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