2. Computer Modeling

The multi-scale modeling approach used here to predict concrete diffusivity has been outlined previously [9]. It is based on the combination of a cellular-automaton-based three-dimensional model for cement paste hydration and microstructure development [12] and a three-dimensional hard core/soft shell (HCSS) model for concrete [13]. The approach follows these steps:

• 1) the "median" cement particle diameter is used to establish the thickness of the interfacial transition zone (t _ITZ) surrounding each aggregate;
• 2) the HCSS model is executed for the volume fraction and particle size distribution (PSD) of aggregate and t_ITZ of interest to determine the volume fractions of bulk (V_bulk ) and ITZ (V_ITZ) cement paste (alternately, the analytical equations of Lu and Torquato [14,15] can be used for this determination);
• 3) these paste fractions are duplicated in the cement paste microstructural model in a system containing a single flat plate aggregate with the desired cement particle size distribution, overall w/c ratio, and silica fume addition;
• 4) hydration of the cement paste microstructure is simulated for a fixed number of cycles or until a degree of hydration of interest;
• 5) the hydrated microstructure is analyzed to determine the capillary porosity present in the cement paste as a function of distance from the aggegate surface;
• 6) these local porosities and the initial local silica fume addition are substituted into a previously developed equation relating relative diffusivity of cement paste to capillary porosity and silica fume addition [11];
• 7) the local relative diffusivities are averaged into two subsets- those within the ITZ region and those within the bulk cement paste;
• 8) the ratio of "average" ITZ diffusivity (D _ITZ) to average bulk cement paste diffusivity (D_bulk) is computed and used as input back into the HCSS model to determine the effective diffusivity of the concrete composite (D_eff ) relative to the average value for the bulk cement paste (using random walker techniques [16], typically with 10,000 random walkers each taking on the order of 1,000,000 random steps); and
• 9) the overall diffusivity of the concrete is computed by multiplying the result of step 8 by the average bulk cement paste relative diffusivity determined in step 7 and by the diffusivity of chloride ions in bulk water (taken to be 1.81 x 10^-9 m²/s at 20 ºC [17] for this study).

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 V_ITZ/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 [18]. 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 [20] for coarse (maximum diameter of 19.0 mm) and fine aggregates [9]. 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:

w/c:      0.3      0.4      0.5
• CSF addition (mass fraction of cement basis):      0.00      0.05      0.10
• V_agg (volume fraction of aggregates):       0.62           0.70
• Cycles of hydration:    850      2000

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 [21]. Including only the interaction effects (2nd order cross product terms) that were determined to be significant, the base ten logarithm of the concrete diffusivity, D_conc , was fit to a function of the form:

$$log_{10} (D_{conc})= a_0~+~a_1 (\frac{w}{c})~+~a_2 ({\frac{w}{c}})^2~+~a_3 CSF~+~a_4 (CSF)^2~+~a_5 (\frac{w}{c})CSF~$$ (1)

$~~~~~+a_6 \alpha~+~a_7 (\frac{w}{c}) \alpha ~+~a_8 (CSF) \alpha ~+~a_9 V_{agg}$

where CSF is the silica fume addition rate, $\alpha$ 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, V_agg , and each of the other three variables were determined to be insignificant based on the regression analysis of the results presented below.