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Table 2 lists the numerical point counting and random walker results for the interfacial zone volume fraction and D/Dbulk, respectively, for each of the 23 systems studied previously . The parameters listed are enough, along with the sieve data in Table 1, to allow the analytical equations listed previously to be used to calculate the interfacial zone volume, and the value of D/Dbulk.
center>Table 2: Parameters of model systems used, including point-counting result for VITZ and myopic random walk results for D/Dbulk.
Figure 2 shows the ITZ volume fraction, calculated from eq. (2), plotted against the exact numerical ITZ volume fraction as given by point counting. Equation (2) is evaluated for r = tITZ , the interfacial zone width. Clearly there is excellent agreement, within a few percent, over a large range of ITZ volume fractions. At the low volume fractions, there is very little overlap of the interfacial zones, so that an accurate prediction of the total ITZ volume can be made simply by adding up the interfacial zone volume of each aggregate particle and assuming there is no overlap. Equation (2) does correctly reduce to this limit and so still works well. In the higher volume fractions in Fig. 2, there is substantial ITZ overlap, and so the close agreement of eq. (2) with the numerical results is even more impressive. Further investigations of this equation applied to concrete models can be found elsewhere .
Figure 2: Showing the interfacial transition zone (ITZ) volume fraction, as computed by eq. (2), plotted against the exact (numerical) results of point counting, for each of the 23 concretes listed in Table 2 . The abscissa values are given in Table 2. The solid line is the line of equality.
Figure 3 shows the D-EMT result for D/Dbulk plotted against the myopic ant numerical result. The D-EMT result takes a weighted average of the two possible choices for solving the D-EMT equation, as discussed in the previous section. If choice number one is that of fixing the value of DITZ, resulting in the value D1/Dbulk, and choice number two is fixing the value of DITZ/Dbulk, resulting in the value D2/Dbulk, then the average used is
The dashed lines in Fig. 3 correspond to the line of equality and the ± 20% error boundaries. All the D-EMT results, using eq. (11), stay within the ± 20% lines, and most stay well within. This choice of weighting of these two extremes agrees well with the results of direct simulation for a wide range of concretes, and is expected to be quite robust and apply to other concretes as well. Choosing a weighted average is equivalent to saying that in a real concrete, the interfacial zone diffusivity becomes partially blended in with the rest of the matrix phase due to the action of nearby aggregates.
Figure 3: Showing the D-EMT value of D/Dbulk, plotted against the exact (numerical) myopic
listed in Table 2. The dashed lines consist of the line of equality, bracketed by lines of ± 20% error.
We note that, a priori, there is no reason to expect an EMT to work well for a particular problem. The dilute limit used in the development of an EMT is exact, but the approximation used to produce predictions for larger volume fractions of aggregate are essentially uncontrolled. Having numerically exact diffusivity calculations available makes it possible for us to quantitatively evaluate the D-EMT predictions and optimize them for practical use. The D-EMT results can now be used to replace the random walker numerical results, but the validity of this replacement could only have been determined by having the random walker techniques available.