Model-based Interpretation of the Influence of Silica Fume on Diffusivity

Previously, Garboczi and Bentz [17] have shown that the relative diffusivity of cement paste should be mainly a function of capillary porosity. At higher porosities (> 20 %), transport is dominated by the percolated pathways through the capillary porosity phase. At lower porosities (< 20 %), the capillary porosity becomes discontinuous and transport is controlled by the properties of the nanoporous C-S-H phase. Plotting relative diffusivity vs. total capillary porosity resulted in a single universal curve for a variety of w/c ratios and degrees of hydration [17], albeit with abrupt changes in slope at the percolation threshold porosity, for model tricalcium silicate cement pastes.

Accepting that the NIST cement hydration and microstructure development model is producing reasonable microstructures and chloride ion diffusivities for systems containing silica fume, one can proceed to investigate the effects of w/c ratio, silica fume addition, and degree of hydration on relative diffusivity. For this study, approximately 70 different microstructures were simulated with w/c ratios ranging from 0.2 to 0.7 and silica fume additions from 0 % to 20 %. The results are summarized in Fig. 2 which provides a plot of the model relative diffusivities vs. total capillary porosity. It should be noted that the general trends of these curves are in good agreement with the experimental data for chloride ion diffusivity under leaching conditions presented by Jensen et al. [8], and the relative electrical conductivity data presented by Christensen [28] for w/c=0.4 cement pastes with (20 %) and without silica fume.

   Figure 2: Model results for relative diffusivity of cement pastes with silica fume vs. total capillary porosity. Top dashed line indicates a previously developed equation [17]. Lower four solid and dashed lines from top to bottom indicate fits for 0 %, 3 %, 6 %, and 10 % silica fume additions, respectively. Fitting coefficients are provided in Table 5.

At high porosities, the data collapse onto a single curve as the capillary porosity and its connectivity regulates the ease of transport. As the capillary porosity de-percolates at about 20 % porosity, however, the curves diverge into individual curves for the different silica fume contents. Once the capillary pore space is disconnected, transport will be regulated by the percolation of the conventional C-S-H and the pozzolanic C-S-H. The percolation properties of each of these three phases, assessed using a pixel-based burning algorithm [29], are provided in Figs. 3, 4 and 5. Each phase is observed to exhibit a percolation threshold somewhere between 20 % and 30 % volume fraction, the connectivity of the capillary porosity strictly decreasing and that of the pozzolanic C-S-H strictly increasing with advancing hydration (when sufficient pozzolan is present). The progress of the percolation of the conventional C-S-H with advancing hydration is more complicated, as discussed below.

  

Figure 3: Model percolation results for capillary porosity phase.

  

Figure 4: Model percolation results for the conventional C-S-H phase. Note that for the 6 % and 10 % CSF additions, none of the data points lie to the right of the 0.3 volume fraction line.

  

Figure 5: Model percolation results for the pozzolanic C-S-H phase. Sufficient volume of the pozzolanic C-S-H to achieve percolation is produced only for the CSF addition rates of 6 % and 10 %.

In the systems with low (0 % and 3 %) silica fume additions, from Fig. 4, the conventional C-S-H is seen to be highly percolated for C-S-H volume fractions greater than 0.30 and its relative diffusivity (0.0025) will control the overall diffusivity once the capillary porosity becomes discontinuous. For the systems with higher (6 % and 10 %) silica fume additions, the conventional C-S-H begins to percolate somewhat, but never exceeds a volume fraction of 0.3 due to the "slow" ongoing conversion of conventional C-S-H into pozzolanic C-S-H via the second chemical reaction presented earlier. This behavior can be seen more clearly in Fig. 6, which plots the conventional C-S-H volume fraction vs. the capillary porosity fraction for various w/c ratios and CSF addition rates. The plot is conveniently divided into four quadrants indicated by the two solid lines parallel to the x and y axes. In the lower right quadrant, the capillary porosity is percolated and the conventional C-S-H is discontinuous. In the upper right quadrant, both phases are percolated. In the upper left quadrant, the capillary porosity is discontinuous while the conventional C-S-H is percolated. Finally, in the lower left quadrant, both the capillary porosity and the conventional C-S-H are discontinuous and transport would be controlled by the remaining porous and percolated phase, the pozzolanic C-S-H. In Fig. 6, in every case, as the hydration proceeds, the capillary porosity is strictly decreasing, so that each curve in Fig. 6 proceeds from right to left as time advances. For the 0 % and 3 % silica fume additions, the conventional C-S-H fraction is basically linearly increasing with a decrease in porosity. Thus, as hydration proceeds, the capillary porosity disconnects, while the C-S-H becomes highly percolated. However, for the 6 % and 10 % additions, the curves are seen to first increase, but plateau around 15 % to 25 % conventional C-S-H volume fraction and end up in the lower left quadrant of the plot, due to the conversion of the conventional C-S-H to pozzolanic C-S-H. Thus for these systems, it is the pozzolanic C-S-H that becomes highly connected at long hydration times (Fig. 5) and will dominate the overall transport rates.

  

Figure 6: Conventional C-S-H volume fraction vs. capillary porosity for various w/c ratios and CSF additions rates. As time increases, capillary porosity strictly decreases, so time is advancing as each curve proceeds from right to left in the graph. Solid lines parallel to y and x axes indicate the approximate percolation thresholds (50 % connected) for the capillary porosity and the conventional C-S-H, respectively.

Previously, Garboczi and Bentz [17] developed an equation to relate relative diffusivity, D/D₀, to capillary porosity, $\phi$, of the form:

$\frac{D}{D_0}(\phi)= K_1 + K_2 \cdot \phi^2 + K_3 \cdot (\phi-\phi_c)^2 H(\phi-\phi_c)~~~~~~~~~~~$ (1)

where K₁, K₂, and K₃ are fitting coefficients, $\phi_c$_c represents the capillary porosity percolation threshold, and H is the Heaviside function (H(x) = 1 when x>0 and 0 otherwise). This equation was fitted to the data available for each silica fume addition and the resulting coefficients are provided in Table 5. In agreement with the above discussion, one finds that K₃ is independent of silica fume addition, as this coefficient represents the contribution of the percolated capillary porosity to the overall diffusivity. K₁ and K₂ are seen to vary with silica fume content, but in a similar fashion. Thus, one may write:

$\frac{D}{D_0}(\phi,CSF)= \frac{K_1'}{\beta(CSF)} + \frac{K_2'}{\beta(CSF)} \cdot \phi^2 + K_3' \cdot (\phi-0.17)^2 H(\phi-0.17)~~~~~~~~~~~$ (2)

where K₁'=0.0004, K₂'=0.03, K₃'=1.7, and $\beta$ is a function of silica fume addition, according to the values provided in Table 5.

 

Table 5: Coefficients for fitting equation 1 to model data with $\phi _c=0.17$ = 0.17.

% CSF addition	K1	K2	K3	$\beta$
0	0.0004	0.03	1.7	1.0
3	0.0003	0.0225	1.7	1.33
6	0.0001	0.0075	1.7	4.0
10	0.00005	0.00375	1.7	8.0

The value for $\beta$ for a 10 % silica fume addition in Table 5 predicts an eight-fold reduction in chloride ion diffusivity for mature specimens hydrated to equal porosities. However, the actual experimental and model data provided in Table 4 indicate a 17-28 fold reduction in diffusivity. The apparent difference in these two reduction factors is due to the fact that the data values in Table 4 were obtained at equal hydration times and not at equivalent capillary porosities. Because the pozzolanic reactions contribute to a more efficient reduction of capillary porosity than cement hydration by itself (as can be witnessed by the additional water consumed in the presented pozzolanic reactions), at equal ages, the reduction in diffusivity in samples containing silica fume additions is due both to a reduction in capillary porosity and to a reduction in the inherent diffusivity of the C-S-H. The reduction in capillary porosity at equal hydration times (cycles) can also be observed by comparing the leftmost data points for each curve in Fig. 6. For the w/c=0.3 systems, for example, the "final" (2000 cycles of hydration) capillary porosity fractions are approximately 0.044 and 0.014 for the 0 % and 10 % CSF additions, respectively. For the corresponding w/c=0.5 systems, the reduction after 2400 cycles of hydration is from 0.235 to 0.156.

Silica fume is also known to accelerate the early hydration reactions, but this is likely not a major factor for mature specimens of the lower w/c ratios generally characterizing high performance concretes, as the ultimate degree of hydration of the cement will be less in the systems containing silica fume due to space (capillary porosity) limitations. A further consideration in concrete is the improvement in the microstructure of the interfacial transition zone provided by the addition of silica fume [30]. In part II of this paper, a suite of multi-scale models incorporating all of these various influences will be applied to developing an equation to predict the chloride ion diffusivity of a concrete as a function of w/c ratio, silica fume addition, degree of hydration, and volume fraction of aggregates. But, suffice it to say that the addition of silica fume may (when cracking due to self-desiccation and thermal gradients is avoided or minimized) dramatically increase the service life of a steel-reinforced concrete exposed to chloride ions, resulting in substantial cost savings when a life cycle costing approach [31] is applied to concrete construction.