Discussion

The data for each system were analyzed using Eqn. 9. The porosity was fixed at 0.26 and the formation factor was the only remaining free parameter. The value of the formation factor F required to best approximate the experimental data is shown in Table 2 for each system. The coefficient of variation for these values was approximately 10 %. The corresponding results are shown as solid curves in Fig. 3(b). The curves for the other systems are not shown for clarity. Moreover, remaining systems had only minor curvature by comparison.

The CaCl₂ and the K₂CO₃ data are shown in Fig. 3(b) to delineate the envelope over which the concentrations could vary over long times. Note that the slope of each curve is not a constant. While the change in slope for the 2CO3/KI system is gradual, the change in slope for the 2/KI system is dramatic. Therefore, characterizing the iodide transport in the 2/KI system using Fick's law and a constant apparent diffusion coefficient is unwarranted. Further, the apparent iodide diffusion coefficient decreases in the 2CO3/KI system and increases in the 2/KI system. Any "adjustments" to an apparent diffusion coefficient to account for one system would certainly not be applicable to another. This is analogous to determining the apparent diffusion coefficient under one set of experimental conditions and using that to predict the behavior of the concrete in the field under a different species exposure.

Figure 4:The ratio of the apparent diffusion coefficient D^a to the microstructural diffusion coefficient D^µ (Eqn. 7) of iodide for the various counter diffusing electrolytes, as a function of time t.

The electro-diffusion equation can be used to quantify the magnitude of the nonideal behavior within each system. The apparent diffusion coefficient can be calculated at any time by using the slope of the curve calculated from the electro-diffusion equation. The calculated time-dependent apparent diffusion coefficient D^a can then be compared to the constant microstructural diffusion coefficient D^µ. The ratio of these two coefficients are shown in Fig. 4 for all of the systems.

The curves shown in Fig. 4 express quantitatively the deficiency in characterizing multi-species diffusion with a single apparent diffusion coefficient. Accurately characterizing the diffusive transport of iodide in these nonreactive systems using Fick's law requires an apparent diffusion coefficient that must vary in time. For the 2/KI system, the apparent diffusion coefficient increases by a factor of 5 at 60 d.

Figure 5: Concentration difference $\Delta$ of each diffusing species in the (a) 2/KI, and the (b) 2CO3/KI systems, each as a function of time t. The error bars represent an estimated standard deviation.

The behavior of the 2/KI system at very long time was very interesting. The curve for the 2/KI system in Fig. 3(b) appears to be diverging at long times. Analyzing the experiment using the electro-diffusion equation, one expects that the concentration difference between the vessels will eventually become negative. The measured concentration differences $\Delta$ for the 2/KI system beyond 60 d are shown in Fig. 5(a) on a linear scale. After approximately 70 d, the observed concentration difference was, in fact, negative. As a means of comparison, the corresponding data for the 2CO3/KI system are shown in Fig. 5(b).

A negative concentration difference $\Delta$ is significant. Based on Eqn. 1, when the concentration difference $\Delta$ becomes negative, the concentration gradient $\nabla c$ changes sign. Since the experimentally measured flux is still pointing in the same direction, Eqn. 1 is satisfied only by having a negative apparent diffusion coefficient D^a. Similarly, based on Eqn. 1, the only way to diffuse a species from regions of low concentration to regions of higher concentration is through a negative diffusion coefficient. This unphysical apparent behavior is due to the macroscopic diffusion potential that arises from the electrostatic interactions that are not accounted for in the Fick equation (Eqn. 1), but are accounted for in the electro-diffusion equation (Eqn. 9).

Also shown in Figs. 5 are the relative concentration differences for all the species present, as calculated by Eqn. 9. For the 2/KI system, the "slow" species is the Ca²⁺ ion. For the 2CO3 system, it is the 3 ion. The effect of these two ions is to greatly control the macroscopic diffusion potential across the specimen. The relatively slow moving Ca²⁺ ions contribute to a diffusion potential in the CaCl₂ vessel that is positive with respect to the KI vessel. Correspondingly, the 3 ion contributes to a negative potential with respect to the KI vessel. The result is a macroscopic diffusion potential that attracts the iodide ion to the 2 vessel, and repels it from the 2CO3 vessel. In the 2/KI system, the macroscopic diffusion potential is sufficient to drive iodide from the KI vessel to the 2 vessel, even though the iodide concentration in the KI vessel is less than that of the 2 vessel. In time, the diffusion potential decreases and the concentration difference $\Delta$ approaches zero again.