Solubilities and Diffusion Coefficients

A further problem is possible if the reinforcements are very small. Coarsening of the particles or whiskers may occur driven by Ostwald ripening, in which large particles grow through diffusional transport at the expense of smaller ones. This can be minimized by choosing matrices in which the reinforcement elements have very low solid solubilities and diffusion coefficients. Platelets, however, have been shown to be more resistant to coarsening than particles or whiskers. 

By inspection, the flux is directly proportional to the solubility to the first power and directly proportional to the diffusion coefficient to the two-thirds power. If, for example, the proposed study involves mass transport measurements for series of compounds in which the solubility and diffusion coefficient change incrementally, then the flux is expected to follow this relationship when the viscosity and stirring rate are held constant. This model allows the investigator to simulate the flux under a variety of conditions, which may be useful in planning experiments or in estimating the impact of complexation, self-association, and other physicochemical phenomena on mass transport. 

To maximize the current limit that could be shunted by redox additives so that the occurrence of such irreversible processes due to overflowing current could be more efficiently suppressed, the redox additive apparently should be present in the electrolyte at high concentrations, and both its oxidized and reduced forms should be very mobile species. Where the criteria for selecting potential redox additives are concerned, these requirements can be translated into higher solubility in nonaqueous media and lower molecular weight. In addition to solubility and diffusion coefficients, the following requirements should also be met by the potential redox additives (1) the formal potential of the redox couple [R]/[0] should be lower than the onset potential for major decom-... 

The fuel cell in Figure 13.9 can be conceptually viewed as a combination of a Nafion film-coated cathode and a Nafion film-coated anode. Hence, the fuel cell is, in essence, a combination of two chemically modified electrodes. This idea is, in fact, more than just a concept, because electrochemical investigations of Nafion film-coated electrodes have been used to obtain fundamental chemical and electrochemical information that is relevant to the operation of such devices [93]. For example, the kinetics of 02 reduction in fuel cells can be investigated at such modified electrodes the solubility and diffusion coefficient for 02 in Nafion and the proton conductivity of this membrane material can also be determined. Chemically modified electrodes have made analogous contributions to battery development. 

Consistency of the model with optical data showing index of refraction as a function of growth temperature (85), data on 02 solubility and diffusion coefficient in Si02, and isotopic tracer data (86). 

Nonlinear, pressure-dependent solubility and permeability in polymers have been observed for over 40 years. Meyer, Gee and their co-workers (5) reported pressure-dependent solubility and diffusion coefficients in rubber-vapor systems. Crank, Park, Long, Barrer, and their co-workers (5) observed pressure-dependent sorption and transport in glassy polymer-vapor systems. Sorption and transport measurements of gases in glassy polymers show that these penetrant-polymer systems do not obey the "ideal sorption and transport eqs. (l)-(5). The observable variables,... 

Pressure-dependent sorption and transport properties in polymers can be attributed to the presence of the penetrant in the polymer. Crank (32) suggested in 1953 that the "non-ideal" behavior of penetrant-polymer systems could arise from structural and dynamic changes of the polymer in response to the penetrant. As the properties of the polymer are dependent on the nature and concentration of the penetrant, the solubility and diffusion coefficient are also concentration-dependent. The concentration-dependent sorption and transport model suggests that "non-ideal" penetrant-polymer systems still obey Henry s and Fick s laws, and differ from the "ideal" systems only by the fact that a and D are concentration dependent,... 

In the dual-mode sorption and transport model the pressure-dependence of a (= C/p), P and 0 in gas-glassy polymer systems arises from the pressure-dependent distribution of the sorbed gas molecules between Langmuir sites and Henry s law dissolution. Although k, Dg and are assumed to be constant, the average or effective solubility and diffusion coefficients of the entire ensemble of gas molecules change with pressure as the ratio of Henry s to Langmuir s population, C /C, changes continuously with pressure [eq. (14)]. 

The basic difference between Mconcentration-dependentM and dual-mode, models is in their assumption about penetrant-polymer interactions. Concentration-dependent sorption and transport models are based on the assumption that the concentration-dependence of the solubility and diffusion coefficients arises... 

In the following chapter we present the matrix model of gas sorption and diffusion in glassy polymers which is based on the observation that gas molecules interact with the polymer, thereby altering the solubility and diffusion coefficients of the polymer matrix. 

The concentration-dependent models attribute the observed pressure dependence of the solubility and diffusion coefficients to the fact that the presence of sorbed gas in a polymer affects the structural and dynamic properties of the polymer, thus affecting the sorption and transport characteristics of the system (3). On the other hand, in the dual-mode model, the pressure-dependent sorption and transport properties arise from a... 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

In the derivation of the simplified expressions for solubility and diffusion coefficients, eqs. (4) and (9), C was assumed to be small. This fact does not limit the usefulness of these expressions for high concentrations. We show below that sorption and transport expressions, eqs. (11) and (14), respectively, derived from the simplified equations retain the proper functional form for describing experimental data without being needlessly cumbersome. Of course, the values of the parameters in eqs. (4) and (9) will differ from the corresponding parameters in eqs. (3) and (8), to compensate for the fact that the truncated power series used in eqs. (4) and (9) poorly represent the exponentials when aC>l or 0C>1. Nevertheless, this does not hinder the use of the simplified equations for making correlation between gas-polymer systems. 

TABLE 8.4 Activation Potentials and Current Sensitivities for the Voltanunetric Oxidation of Dissolved Hydrogen at Activated Platinum Electrodes in Five Solvents (Scan Rate 0.1 V s 1) Solubilities and Diffusion Coefficients for H2 in Several Solvents... 

However, the mathematical formulae of DST satisfactorily present the dependence of the solubility and diffusion coefficients for simple gases and organic vapors on the concentration of the penetrant in the glassy polymer (9,11,13,15,17,33,34,89). 

These results can be rationalized in terms of O2 solubilities and diffusion coefficients. At low pressures, both of these quantities are small and the oxidation is essentially a diffusion controlled process, and since the radical flux is relatively higher than the local concentration of O2 within the films, the rate is independent of I. 

Besides EXAFS and related techniques, electrochemical methods have been used extensively to characterize metals in ILs. Lagunas et al. studied the electrochemistry of the ferricenium/ferrocene couple in Bmim-based ILs [194, 195], In a similar study, Rogers et al. studied ferrocene and cobaltocenium hexafluorophos-phate in a range of common ILs, including phosphonium ILs. The authors report solubilities and diffusion coefficients for a variety of concentrations and suggest that both ferrocene and cobaltocenium hexafluorophosphate are useful and reliable internal standards for use in ILs [196],... 

Sato, Y. Fujiwara, K. Takikawa, T. Sumarno Takishima, S. Masuoka, H. Solubilities and diffusion coefficients of carbon dioxide and nitrogen in polypropylene, high-density polyethylene, and polystyrene under high pressures and temperatures. Fluid Phase Equilib. 1999, 162, 261-276. 

In the dry particles, oxygen uptake occurs on a measurable time scale (minutes). Here the span of the experiment is the particle radius. Such experiments hold the promise of providing sorption rates into the stabilizer phase and into the interphase. We are investigating whether it is in fact possible to obtain information in sufficient detail to measure the magnitude of the interphase in the dry particles and to obtain the solubilities and diffusion coefficients of various sorbents within the various individual phases of the material. 

Tlie use of the term "membrane for these filters is somewhat misleading. Membranes are normally used to separate the components of a gas mixture which have different permeabilities through the membrane material. The permeabilities. In turn, can be related to the solubilities and diffusion coefficients in the membrane which differ for different gases. However, for a membrane filter, the gas passes through the pores of the film by a macroscopic flow process, driven by the pressure gradient. No gas separation takes place. The principal mechani.sms of panicle deposition for both fibrous and membrane filters are the dilTusion and impaction of particles of finite diameter. Settling and electro.siaiic effects may contribute to removal. 

The commonly accepted mechanism for the transport of a solute in LM is solution-diffusion. The solute species dissolve in the liquid membrane and diffuse across the membrane due to an imposed concentration gradient. Different solutes have different solubilities and diffusion coefficients in a LM. The efficiency and selectivity of transport across the LM may be markedly enhanced by the presence of a mobile complexation agent (carrier) in the liquid membrane. Carrier in the membrane phase reacts rapidly and reversibly with the desired solute to form a complex. This process is known as facilitated or carrier-mediated liquid membrane separation. In many cases of LM transport, the facilitated transport is combined with coupling counter- or cotransport of different ions through LM. The coupling effect supplies the energy for uphill transport of the solute. 

The diffusion flux M, g/cm /s) of the species is defined as the amount of matter passing perpendicularly through the unit area during the unit time. Different solutes have different solubilities and diffusion coefficients, in a LM. tn a steady-state permeation experiment, the flux of a species S through a membrane of thickness h is related to the concentration gradient through Fick s first law ... 

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