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Computation of Diffusivities

All of the computer runs executed in this study and their resultant predicted chloride diffusivity values are provided in Table 2. The diffusivities are seen to range well over two orders of magnitude, from 2.1 x 10^-13 to 8.18 x 10^-11 m²/s. For one system, three runs using different random number seeds were conducted to examine the variability in D due to the randomness of the microstructures. The determined coefficient of variation, less than 1.5%, justified the execution of only one computer run to obtain representative results for each of the experimental conditions. Figure 3 summarizes the effects of the seven variables considered in this study on the computed concrete diffusivity. In this figure, the variables are identified by the IDs given in the first column of Table 1. For each variable, the results obtained for the low variable setting lie to the left of the ID, while those computed for the high variable setting lie to the right of the ID. The horizontal line in each figure indicates the global mean value for the 16 runs. The scatter plot illustrates all of the data values obtained in the first 16 runs, while the mean plot shows only the mean values obtained for the eight runs with the low variable setting contrasted against the mean for the eight runs with the high setting. From the mean plot, one can observe that w/c ratio, degree of hydration, and volume fraction of aggregate are the most significant variables. The thickness of the interfacial transition zone and air content are less significant and the coarse and fine aggregate particle size distributions have almost no effect on the computed diffusivities.

Some insight into the effects of these variables can be gained by considering two components of the overall concrete diffusivity, the inherent diffusivity of the bulk paste portion, D_bulk and its attenuation/diminution by the presence of aggregates (and air voids) and their accompanying interfacial transition zones, as indicated by the value of
D_conc / D_bulk. This second parameter is strongly influenced by the volume of interfacial zone paste and the ratio of D_ITZ to D_bulk [17, 18]. The effects of each variable on these two parameters can be summarized as follows:

w/c ratio: Numerous experimental [21,22,23,24] and computer model [10] results have indicated that a reduction in w/c ratio will lead to a significant reduction in diffusivity. This large reduction is only partially offset by the fact that a decrease in w/c ratio does actually lead to a slight increase in the ratio of D_conc to D_bulk, as it increases the ratio D_ITZ / D_bulk, particularly at the lower degrees of hydration.

degree of hydration: Increasing the degree of hydration (age) is observed to both decrease the diffusivity of the bulk cement paste and also decrease the value of D_conc / D_bulk. The latter reduction is due to the fact that the porosities in the interfacial transition zone and bulk regions become closer to one another as hydration proceeds [4] and to the reduced slope of Eqn. 1 at low porosities.

volume fraction of aggregate: Increasing the volume fraction of the aggregates creates a greater volume of ITZ cement paste, which in turn leads to a larger reduction in the w/c ratio of the "bulk" paste [4]. Thus, an increased volume fraction of aggregate will result in a reduction in the value of D_bulk. The effect of the volume fraction of aggregate on the value of D_conc / D_bulk is a bit more complex as discussed in Ref. [17]. Because in these studies, due mainly to the smaller aggregate surface area of a concrete relative to that of a mortar, the ratio of D_ITZ to D_bulk is generally less than 5, adding additional aggregates will, on the whole, result in a decrease in the value of D_conc / D_bulk [17].

coarse aggregate particle size distribution: Changing the coarse aggregate particle size distribution from "fine" to "coarse" lowers the volume of ITZ cement paste present in a given concrete due to the reduction in aggregate surface area. This in turn increases the w/c ratio of the bulk paste and thus increases D_bulk, but only slightly for the distributions examined in this study. As the coarse aggregate PSD varies from fine to coarse, the ratio of D_ITZ / D_bulk decreases very slightly and, because there is also less ITZ cement paste, the ratio D_conc / D_bulk decreases as well. These two effects tend to nullify each other so that the overall influence of coarse aggregate PSD on concrete diffusivity is negligible within the range examined in this study.

fine aggregate particle size distribution: The arguments introduced above for the coarse aggregate PSD also apply for the fine aggregate PSD, but even more so due to the increased surface area per unit volume of the fine aggregates. Once again, the overall effect is observed to be negligible.

interfacial transition zone thickness: Increasing the value of T_ITZ naturally increases the volume of ITZ cement paste. This in turn once again decreases the effective "bulk" w/c ratio, so that D_bulk is decreased. However, an increase in t_ITZ significantly increases both V_ITZ and D_ITZ / D_bulk, and thus increases D_conc / D_bulk. These two effects are in opposition to one another, with the latter dominant, so that the overall concrete diffusivity is only slightly increased due to the larger value of t_ITZ.

air content: Air voids should tend to mimic the effects described above for an increase in aggregate volume fraction, the main difference being their much smaller size and higher surface area per unit volume. Thus, while their addition does slightly decrease the value of D_bulk, due to the large amount of additional ITZ cement paste, the value of D_conc / D_bulk increases. The two effects are again competing and result in a small decrease in concrete diffusivity as the air content is increased from 0% to 10%.

Considering only the three most significant variables (w/c ratio, degree of hydration, and volume fraction of aggregate), the following equation for computing chloride diffusivity of concrete has been estimated by ordinary least squares regression:

The diffusion coefficients calculated according to this equation are compared to the computer simulation actual results in Figure 4. In general, the predicted values are within a factor of 1.5 of the actual values as indicated by the upper and lower solid lines in Figure 4, with a ratio of 2 being the worst case.

Figure 4: Comparison of predicted results based on Eqn. 4 with those from the computer experiment.

Figure 5 provides a contour plot for the log of the concrete diffusivity vs. degree of hydration and w/c ratio at a fixed volume fraction of aggregates of 0.675. Because the slope of the isolevel lines is greater than 45º, one can infer that w/c ratio has a stronger influence on diffusivity than degree of hydration, except for very high values of chloride diffusivity. Two other points are worth noting on the figure. First, because for low w/c ratio concretes, there is a maximal achievable degree of hydration which is less than one (as the capillary porosity goes to a value of zero) [25], the uppermost isolevel line indicating a chloride diffusivity of 10^-12.5 (3.2 x 10^-13) m²/s can basically be considered as a lower bound for the chloride ion diffusivity of conventional concretes, according to the computer simulation results. For low w/c ratio cement pastes, the maximal achievable degree of hydration is typically given by (w/c)/0.4 or (w/c)/0.42, so that for a w/c ratio of 0.3, the maximum value would be on the order of 0.73, which corresponds quite closely to the uppermost isolevel diffusivity line. Thus, there is definitely a lower limit on the diffusivities achievable using conventional concrete mixture proportioning and curing practices. Second, for higher w/c ratios, the effects of a change in w/c ratio on concrete diffusivity are much less marked, as indicated by the greater distance between the -10.5 and -10 isolevel lines. Thus, only minimal hydration (less than 50%) is needed to obtain a chloride diffusivity on the order of 10^-10 m²/s. Of course, it should be kept in mind that this value is only a factor of 20 below that of chloride ions in bulk water.

Figure 5: Contour plot for the logarithm (base 10) of concrete diffusivity vs. w/c ratio and degree of hydration for a fixed volume fraction of aggregate of 0.675.

Equation 4 can be compared to that established by Walton et al. [26] based on data compiled for chloride ion diffusion in cement paste by Atkinson et al. [27]:

Figure 6 compares the equation of Walton et al. to Eqn. 4 with four sets of values for degree of hydration and volume fraction of aggregate. The curves for the lower volume fraction of aggregate fit well with Eqn. 5 for w/c ratios between 0.3 and 0.6, with significant deviations at higher w/c ratios. Thus, the equation derived here appears to be reasonable in terms of one previously determined with w/c ratio as the only variable. However, Eqn. 4 has the advantage of accounting for variability in degree of hydration and mixture proportions which should allow for a more accurate prediction of chloride diffusivity in concrete.

Figure 6: Comparison of current results to those of Walton et al. for diffusivity vs. w/c ratio.