Diffusivity as a Function of Concentration

Each step function evolves according to an error-function solution of the type given by Eq. 4.31, and their superposition is 

When c is assigned units of particles per unit, length. nd corresponds to the total number of particles in the source, and Eq. 4.40 describes the one-dimensional diffusion from a point source as in Fig. 4.5a. Also, when c has units of particles per unit area, nd has units of particles per unit length and Eq. 4.40 describes the one-dimensional diffusion in a plane in two dimensions from a line source initially containing nd particles per unit length as in Fig. 4.55. Finally, when c has units of particles per unit volume, nd has units of particles per unit area, and Eq. 4.40 describes the one-dimensional diffusion from a planar source in three dimensions initially containing nd particles per unit area as in Fig. 4.5c. These results are summarized in Table 5.1. 

This differential equation is generally nonlinear [depending upon the form of D(c), and solutions therefore can be obtained analytically only in certain special cases which are not discussed here [4]. 

When Fick s law applies, the concentration profile generally contains information about the concentration dependence of the diffusivity. For constant D, step-function initial conditions have the error function (Eq. 4.31) as a solution to dc/dt = Dd2c/dx2. When the diffusivity is a function of concentration, 

For identical initial conditions, the difference between a measured profile and the error-function solution is related to the last (nonlinear) term in Eq. 4.43. When diffusivity is a function of local concentration, the concentration profile tends to be relatively flat at a concentration where D(c) is large and relatively steep where D(c) is small (this is demonstrated in Exercise 4.2). Asymmetry of the diffusion profile in a diffusion couple is an indicator of a concentration-dependent diffusivity. 

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If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

Equation 4.51 is an integral equation that can be used to determine D(c ) by a graphical construction or numerical solution. The derivative required in Eq. 4.51 is provided by the measured concentration profile at time t and the integration is performed on the inverse of c x) [6]. However, this historically important method is only moderately accurate, and it would be preferable to obtain diffusion profiles for various assumed diffusivities as a function of concentration by computation. D(c) could be deduced by fitting calculated results for a parametric representation of D(c) to an experimentally determined diffusion profile. 

Anderson, J., F. Rauh, and A. Morales, Particle diffusion as a function of concentration and ionic strength. Journal of Physical Chemistry, 1978, 82, 608-616. 

The physical properties of surface active agents differ from those of smaller or non-amphipathic molecules in one major aspect, namely the abrupt changes in their properties above a critical concentration [1], Figure 2.1 illustrates with plots of several physical properties (osmotic pressure, turbidity, solubilisation, magnetic resonance, surface tension, equivalent conductivity and self-diffusion) as a function of concentration for an ionic surfactant [1]. 

Equation (2.33) now defines the double layer in the final model of the structure of the electrolyte near the electrode specifically adsorbed ions and solvent in the IHP, solvated ions forming a plane parallel to the electrode in the OHP and a dilfuse layer of ions having an excess of ions charged opposite to that on the electrode. The excess charge density in the latter region decays exponentially with distance away from the OHP. In addition, the Stern model allows some prediction of the relative importance of the diffuse vs. Helmholtz layers as a function of concentration. Table 2.1 shows... 

These experiments suggest that as the long time self-diffusion coefficient approaches zero the relaxation time becomes infinite, suggesting an elastic structure. In an important study of the diffusion coefficients for a wide range of concentrations, Ottewill and Williams14 showed that it does indeed reduce toward zero as the hard sphere transition is approached. This is shown in Figure 5.6, where the ratio of the long time diffusion coefficient to the diffusion coefficient in the dilute limit is plotted as a function of concentration. 

In contrast, diffusion of MeOH measured via permeabilify measurements (assuming a partition coefficient of 1) was lower (1.3 x 10 and 6.4 x 10 cm s for Nafion 117 and BPSH 40, respectively) and showed no concentration dependence. The differences observed between the two techniques are related to the length scale over which diffusion is monitored and the partition coefficient, or solubility, of MeOH in the membranes as a function of concentration. For the permeability measurements, this length is equal to the thickness of the membrane (178 and 132 pm for Nafion 117 and BPSH 40, respectively), whereas the NMR method observes diffusion over a lengfh of approximately 4-8 pm. 

Larry Duda and Jim Vrentas were the first to systematically study the diffusion of small molecules in molten polymers, formulate a free volume-based theoretical model, and elucidate the sharp dependence of the diffusion coefficient on temperature and concentration.2 Figure 8.8 shows diffusivities of toluene in polystyrene as a function of concentration and temperature. The values were computed using the Vrentas and Duda (17) free volume model and, as shown, coincide well with available data. 

Early calculations by Borghi et al (59) determined the contributions of kinetics and external diffusion as a function of bed temperature, particle size, and oxygen concentrations. These show that for particles of 1 mm diameter 1116 K, conditions typical of commercial practice, the diffusion and kinetic resistances are of equal importance. Experimental support for these predictions is provided by Ross and Davidson (60, 61). 

When the diffusivity changes as a function of concentration (SOC), the governing equation is... 

The lifetime of the ionic bicyclobutane is a parameter of prime importance. An attempt was made to estimate this parameter for the thio-derivative using the clocking method . The latter involves competition between two nucleophiles e.g., PhS" and the solvent MeOH. Assuming that the PhS " reacts with carbocations at a diffusion-controlled rate and given the product ratio as a function of concentration, the lifetime of 118 in methanol was calculated to be of the order of 10" seconds . ... 

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