Results

Both the experimental and calculated values are shown in Fig. 3. The measured values of $\Delta$ for the iodide concentration are shown as symbols, and the calculated results are shown as curves. The estimated uncertainties in the experimental values are approximately the size of the symbols, and are not shown as error bars for reasons of visual clarity. The values of $\Delta$ for the 1.0 mol/L KCl system were divided by ten so that they could be included on the same plot.

Figure 3: Concentration difference $\Delta$ across each sample as a function of time. The experimental values are shown as filled symbols, the calculated values are shown as solid curves. The measurement uncertainties would appear as the same size as the symbols, so are omitted for visual clarity. The value of $\Delta$ for the 1.0 mol/L KCl system is divided by ten in order to appear on the same scale as the other data.

The KCl and the NaCl systems behaved similarly. The data for the KCl systems are nearly collinear, demonstrating nearly ideal diffusive behavior that can be accurately characterized by Fick's law. There is virtually no concentration dependence in the results for the KCl/Kl systems because the self diffusion coefficient D$D_\infty$ for K⁺, Cl^-, and I^- are nearly equal to one another. The experimental data for the NaCl/Kl system was also practically linear, also indicating nearly ideal diffusive behavior. The estimated diffusion potential for both of the KCl systems was less than 1 mV, and was less than 5 mV for the NaCl system.

The KOH system showed a noticeable difference in behavior. The curvature in the experimental data indicates behavior that cannot be characterized by Fick's law of diffusion. The calculated values, based on the electro-diffusion equation (Eqn. 6), also exhibit the same curvature as was observed in the experimental data. One reason for this is that the self diffusion coefficient D$D_\infty$ for OH^- is significantly greater than that of the other ions present, resulting in a calculated diffusion potential of approximately 16 mV.

Table 3:The values for the slope (ADb/L) of the experimental data shown in Fig. 3. Also shown is the ratio D$D_\infty /D_b$/Db, using D$D_\infty$ for iodide. The uncertainties shown for the slopes are the estimated standard deviation reported by the statistical software, and also characterize the uncertainty in the ratio D$D_\infty /D_b$/Db reported.
System	ADb/L (cm3·h-1)	D$D_\infty /D_b$Db
KCl - 0.1 mol/L	0.2004±0.0032	11.1
NaCl - 0.1 mol/L	0.2118±0.0032	10.3
KOH - 0.1 mol/L	0.2381±0.0047	9.3
KCl - 1.0 mol/L	0.2079±0.0050	10.7

The measured slopes of the experimental data, on semi-logarithmic axes, are shown in Table 3. The estimated standard deviations shown are typically less than 3 % of the slope value, suggesting that the CGA is reasonable approximation for these systems. The systems reached a constant gradient state at a relatively early age. Output from the computer program suggested that a nearly constant gradient is achieved in less than 12 h.

Also shown in Table 3 are the values for the ratio D$D_\infty /D_b$/D_b, using the iodide value for D$D_\infty$ (2.045 x 10^-5 cm² s^-1. [5]). This ratio represents an incorrect application of using the formation factor to determine the apparent bulk diffusion coefficient. Since the quantity D$D_\infty$ in this ratio is a constant, the ratios are simply proportional to the apparent bulk diffusivity D_b. This ratio, however, does not reflect the actual formation factor, because the iodide self diffusion coefficient within the pore solutions is not equal to D$D_\infty$. Also, arbitrary changes in the pore solution will lead to changes in the apparent bulk diffusion coefficient of iodide in these systems, while the formation factor is nearly equal for all systems.

The correct values of the formation factor ${\cal F}$ were determined from the experimental data using the aforementioned computer program, and are shown in Table 4, labelled ${\cal F}_{sim}$_sim. Also shown in the table are the values of the formation factor calculated from the impedance spectroscopy measurements, labelled ${\cal F}_{IS}$_IS. The uncertainty in ${\cal F}_{IS}$_IS reflects the variation in the dc resistance measurement as mentioned previously. The values of ${\cal F}_{sim}$_sim shown in Table 4 were used to calculate values for $\Delta$, and these values of $\Delta$ are plotted in Fig. 3, denoted by the curves. The experimental data and the calculated values for the KCl and the NaCl systems were all nearly linear. The values of $\Delta$ for the KOH system are easily distinguished from the other systems.

The calculated values of ${\cal F}_{sim}$_sim shown in Table 4 were consistent with the measured values ${\cal F}_{IS}$_IS. The values of ${\cal F}_{sim}$_sim varied by approximately 7 %, compared to the 18 % variation in the values of D$D_\infty /D_b$/D_b. The differences between the values of ${\cal F}_{sim}$_sim and ${\cal F}_{IS}$_IS were less than 3 % for the KCl and the NaCl system, and less than 8 % for the KOH system.

The calcuated values ${\cal F}_{sim}$_sim were generally greater than the measured ${\cal F}_{IS}$_IS values. A partial explanation for this can be found in the data in Table 2. In that table, the estimated diffusion coefficients were consistently greater than the handbook values. This suggests that results from the computer program yield a bulk diffusion coefficient that is larger than it is in reality. Therefore, for the computer program to agree with experimental data, the formation factor ${\cal F}_{sim}$_sim must be made greater than its true value, which is consistent with the data in Table 4.

Sighting along the KOH data reveals that these data have some curvature. It is interesting to note that the computed output (solid curve) also exhibits this nonlinear behavior. This suggests that this nonlinear behavior is due to effects of the pore solution chemistry since output from the computer program indicates that the iodide concentration profile across the sample is stable within 12 h. The calculated concentration profile of iodide, however, is not linear due to the diffusion potential.

Table 4:Measured and calculated formation factors from impedance spectroscopy (${\cal F}_{IS}$IS), computer simulation ( ${\cal F}_{sim}$sim), and apparent diffusivity (D$D_\infty /D_b$/Db) using D$D_\infty$ for iodide.
System	${\cal F}_{IS}$IS	${\cal F}_{sim}$sim	D Db
KCl - 0.1 mol/L	10.7±0.2	10.9	11.1
NaCl - 0.1 mol/L	10.9±0.2	11.2	10.5
KOH - 0.1 mol/L	10.6±0.2	11.4	9.3
KCl - 1.0 mol/L	10.7±0.2	10.6	10.7

This electro-chemical effect of the KOH system is revealed in Table 4. The apparent diffusion coefficient of iodide in this system is considerably greater than that for the other systems, even though the computer calculation reveals that the calculated formation factors ${\cal F}_{sim}$_sim are all nearly identical to the electrical values F_IS. This fact demonstrates the effect of using the apparent diffusion coefficient D_b to characterize a microstructure. For that particular test solution, the apparent diffusion coefficient describes how the iodide ion behaves in the presence of KOH, but does not characterize its behavior in the presence of other test solutions. Similarly, it does not necessarily characterize how other ions behave in the same, or similar, microstructure.

Since the pore solution of cementitious systems is typically alkaline, the results for the KOH system have direct relevance to the prediction of ion transport in portland cement systems. The pore solution ionic strength in cementitious systems can be nearly ten times greater than the 0.1 mol/L KOH system studied here. Further, there will be number of additional ions present, with a corresponding number of different self diffusion coefficients. This raises the question of the correct method for characterizing the microstructure of these systems. Either predicting the formation factor from diffusion data or predicting the diffusion coefficient from a formation factor measurement will require a knowledge of the pore solution chemistry. For a formation factor measurement that implements pore solution extraction, a chemical analysis of the extracted pore fluid would be a logical extension of the measurement procedure. Typical diffusion experiments do include these measurements, and so additional analysis may be required for experimental programs that use diffusion measurements to characterize the microstructure of pore cementitious systems.