The multi-scale modeling approach used here to predict concrete diffusivity has been outlined previously . It is based on the combination of a cellular-automaton-based three-dimensional model for cement paste hydration and microstructure development  and a three-dimensional hard core/soft shell (HCSS) model for concrete . The approach follows these steps:
The cement paste-single aggregate microstructural model was executed at a resolution of 1 µm/pixel with system sizes ranging from 170 pixels x 170 pixels x 170 pixels to 234 pixels x 234 pixels x 234 pixels depending on the specific value of VITZ/V bulk. The system volume for the concrete microstructural model was a cube 3 cm on a side. Both models employed periodic boundary conditions to eliminate artificial edge effects.
The cement used in all of the computer experiments had the following composition on a volume basis: C3 S - 0.665, C2S - 0.1425, C3A - 0.095, C4AF - 0.0475, and gypsum- 0.05. Its PSD corresponded to that of a cement with a measured Blaine fineness of 387 m2/kg and a Rosin-Rammler mean particle diameter of approximately 15 µm . The silica fume (specific gravity of 2.2) is modelled as one-pixel particles, 1 µm in diameter. To simulate "good" curing conditions, the cement hydration model was executed under saturated conditions until the capillary porosity depercolated and under sealed conditions thereafter. Under sealed conditions, the appropriate empty capillary pores are created to account for the chemical shrinkage occuring during cement hydration [12,19]. The aggregate PSD corresponded to the midpoint of the upper and lower limits given in ASTM C33  for coarse (maximum diameter of 19.0 mm) and fine aggregates . This resulted in approximately 500,000 aggregates being present in the 27,000 mm3 concrete computational cell.
For the computer experiment, the following variables and settings were examined in a full factorial experimental design:
The total number of computer experiments was thus 36 (=3 x 3 x 2 x 2). The hydration model was executed for fixed numbers of cycles (times) instead of fixed degrees of hydration because the ultimately obtainable degree of hydration is a function of the w/c ratio. 850 cycles should be fairly typical of 28 days of room temperature curing, while 2000 cycles should correspond to approximately 180 days of curing.
The results were analyzed using ordinary least squares regression analysis . Including only the interaction effects (2nd order cross product terms) that were determined to be significant, the base ten logarithm of the concrete diffusivity, Dconc , was fit to a function of the form:
where CSF is the silica fume addition rate, is the degree of hydration of the cement, and a0 to a9 are the fitting coefficients. For the settings examined in this study, all four independent variables will have values between 0 and 1. Viewing this equation, it can be seen that, specifically, the interaction effects between the volume fraction of aggregates, Vagg , and each of the other three variables were determined to be insignificant based on the regression analysis of the results presented below.