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It has been shown that two fairly simple analytical formulas can give a quite accurate description of numerical results on a wide range of concrete mixtures. These formulas must be evaluated numerically, using numerical quadrature techniques like Gaussian integration, but these kind of integration techniques are easily available. The numerial integrations can easily be done on a workstation or a fast personal computer (486 processor or higher, with math coprocessor chip). Therefore the supercomputer simulations inherent in two out the three key steps in the multi-scale model of concrete diffusivity can be replaced with a simple analytical formulation. It is also encouraging to note that the second key step, that of numerically hydrating cement grains around an aggregate, is the easiest step computationally, since it uses only integers. The system sizes used in this paper and in Ref.  were of the order 30 · 106 pixels. For the hydration code necessary for this model, only one byte per pixel is needed to keep track of the phases before, during, and after hydration, so that approximately 30 Mbytes of memory are needed for this part of the multi-scale model. This much memory is even becoming common for modern personal computers, and has long been common for low-end engineering workstations, so that even without an analytical replacement, this part of the model should be accessible to many concrete designers, at least in principle. Of course, run times on a personal computer will be slow, on the order of probably tens of hours, but quite faster on low-end workstations.
The largest difference between this multi-scale model and previous work [7,23], as was mentioned in the Introduction, is the ability to take into account, even in an approximate way, the redistribution of cement due to the presence of the aggregates. This gives more accurate values of DITZ and Dbulk. This step appears to be crucial, especially at the high volume fractions of aggregate common in concrete, as this redistribution of cement plays an important role in determining the bulk properties of the concrete. The procedure used in the multi-scale model to determine this ratio, step (2) as given above, is still approximate, since the gradient of properties in the interfacial zone is treated as being equivalent to a fixed width, fixed property interfacial zone surrounded by fixed property bulk cement paste. This matching of gradient to interfacial zone can be done more accurately, and has been the subject of further research . It turns out that the end results are not significantly affected .
One consequence of this cement redistribution is a sort of "negative feedback" loop, in the following sense. Suppose the interfacial transition zone is made wider, by using a coarser cement, perhaps. This would tend to drive up the value of DITZ, and lower the value of Dbulk, so that the ratio of the two is larger, implying a larger value of D/Dbulk . However, the actual value of D will not be as much higher as one would think, since the higher value of D/Dbulk must be multiplied by the lower value of Dbulk to get the overall concrete diffusivity. So just increasing the diffusivity of the interfacial zone by thickening it will not increase the overall concrete diffusivity as much as one would think . Increasing the surface area of the aggregate by reducing the average aggregate diameter results in similar behavior. Other interplays between the variables of the problem are discussed in Ref. .
The proper experimental validation of the multi-scale model remains to be done. In order to be able to use the model to compare with experimental results, one cannot just prepare concretes at various aggregate volume fractions, including the zero volume fraction of aggregate cement paste matrix, and then simply normalize the concrete measurements by cement paste measurements taken at equal times. The redistribution of cement in the concrete makes the value of Dbulk in the concrete not the same as the plain cement paste sample, even at equal degrees of hydration.
What must be done experimentally is the following. The degree of hydration of the concrete must be determined, along with the volume fraction and particle size distribution of the aggregates, and the particle size distribution of the cement, or at least its median particle size. The diffusivity or electrical conductivity of the concrete can then be measured. If the experimental measurements are taken in this manner, then the model can be quantitatively tested and improved, making it a useful tool for designing the ionic diffusivity of concrete at the mix design stage. This procedure is now being carried out for mortars with various amounts of sand, and will be reported on in the near future .