Analysis

The analysis of the data can be performed using any one of many possible methods. In this experiment, the concentration of iodide on both sides of the specimen is changing with time. Rather than try to "fit" the time-dependent concentration data on both sides of the cell, the analysis was based upon a linear idealization of Fick's law so that behavior that cannot be sufficiently modeled using Fick's law will be apparent.

Assuming ideal diffusive behavior, for a sufficiently low diffusivity sample and sufficiently large vessels, the concentration profile across the specimen should become practically linear after some initial induction period. At this point, the flux of iodide would be constant across the sample, and the corresponding concentration gradient would also be constant. This behavior is referred to here as the constant gradient approximation (CGA), and has been used elsewhere to analyze diffusion data [13].

Figure 2: Schematic of the constant gradient approximation (CGA) for a sample with thickness L. Vessels 1 and 2 contain an ionic species with concentrations c₁ and c₂, and have volumes v₁ and v₂, respectively.

The schematic shown in Fig. 2 depicts a system in the constant gradient state. The thin line depicts the concentration of iodide throughout the system; constant in each vessel and a straight line with constant slope across the sample. The specimen apparent bulk diffusivity is D_b, the thickness is L, the area is A, and the volume of each vessel is v₁ and v₂, each with iodide concentration c₁ and c₂, respectively. Under the CGA, the flux is constant and the rate of change in iodide concentration in each vessel is also a constant:

$$\frac{\partial c_1}{\partial t} = \frac{AD_b}{v_1}\, \frac{c... ...}{L} = \frac{-v_2}{\,\,v_1}\,\frac{\partial c_2}{\partial t}$$ (12)

Upon making the following substitution for the concentration difference, Δ = c₂ − c₁, the time dependent behavior for − can be expressed as an exponential [13]:

$$\Delta = \Delta_\circ \, \exp\left[\frac{-AD_b}{L} \left(\frac{1}{v_1}+\frac{1}{v_2}\right)t\right]$$ (13)

The quantity $\Delta_\circ$ is the concentration difference at the onset of the constant gradient. Based on Eqn. 13, a semi-logarithmic plot of Δ versus time data would be a straight line, with a slope that is proportional to the apparent bulk diffusion coefficient D_b.

The formation factor is estimated from the diffusion data through the use of a computer program that simulates the diffusion experiment by implementing the electro-diffusion equation (Eqn. 6). A similar computer program has been described previously [14]. The computer program performs an electro-diffusion transport calculation using Eqn. 6 and knowledge of the sample porosity and the pore solution chemistry. The microstructural diffusion coefficient D_µ is calculated from the formation factor using Eqn. 11. The formation factor is varied (porosity is held constant) until the slope of the calculated values of Δ equals the slope of the experimental values.

The computer program calculated the solution to Eqn. 6 using a finite difference scheme. The system was represented by a one-dimensional mesh composed of 21 nodes. The differencing algorithm was fully explicit, but the stability criterion was satisfied by a factor of five. The computer program calculated the activity coefficients using an implementation of the Pitzer equations [15] that was based on the PHRQPITZ [16] computer program. The diffusion potential was calculated using the local electro-neutral (zero current) hypothesis [6]. For the species flux j_i, as given in Eqn. 6, the total current I_T is the sum over the individual fluxes, each proportional to the species charge z_i:

$${\bf I}_T = \sum_i z_i\ {\bf j}_i = 0$$ (14)

The diffusion potential gradient is chosen so that this relation is satisfied at the boundary of each computational element, assuring both local and global charge neutrality.

Table 2: Comparison of diffusion coefficients D from the computer program (CP) and handbook (HB) values for some 1:1 valence salts. The handbook values are from the CRC Handbook of Chemistry and Physics.
Salt	conc. mol·L-1	DHB 10-9             m2·s-1		DCP 10-9             m2·s-1
KCl	0.01	1.917		1.902
	0.10	1.844		1.807
	1.00	1.892		1.801
NaCl	0.01	1.545		1.539
	0.10	1.483		1.476
	1.00	1.484		1.571
Kl	0.10	1.865		1.829
	1.00	2.065		1.911

As a test of the computer program, the diffusion coefficient of 1:1 valence salts in bulk liquid are calculated by the computer program (CP) and the values are compared to values reported in a chemistry handbook (HB) [17]; the computer program was executed with both the formation factor and the porosity fixed at a value of one. The calculations are performed over a range of salt concentrations and the results are shown in Table 2. Generally, the computed results agree quite favorably with reported values, with the worst case being a difference of approximately 10 %. Having thus demonstrated the ability and accuracy of the computer program, one can made direct comparisons between its prediction of a particular scenario and the corresponding experimental data.

Based on Eqn. 13, a semi-logarithmic plot of Δ versus time data would be a straight line. The slope of this line is determined first for the experimental data. To determine the formation factor ${\cal F}$ from these data, values for ${\cal F}$ are input to the computer program and are varied until a linear least squares regression of the calculated values of Δ versus time, on a semi-logarithmic plot, gives the same slope as for the corresponding experimental data. Deviations from linearity would indicate behavior that cannot be characterized using Fick's law.