5.1 Plain Cement Paste Results
We have simulated the relative diffusivity D/Do for 0.4, 0.45, 0.5, and 0.6 w/c ratio cement pastes, as a function of degree of hydration . The results, obtained using 1003 pixel unit cells, are plotted in Figs. 4, 5, 6, and 7, along with reported experimental data [3,22,32]. All simulations were run using 32-bit precision real numbers, with no significant difference between 32 and 64- bit precision runs.
In Figs. 4, 5,6, and 7, the open squares are simulation results, and the filled circles are the experimental results. One initial cement particle-packing was used at each w/c ratio to generate the simulation results. The fairly small (at most 10-20%) variation between different initial cement particle-packings is less than the expected error in the experimental results, so that it was not worth averaging over several configurations. There is reasonably good agreement between simulation and experiment.
In Figs. 6 and 7, however, there is one experimental data point that is significantly different (by a factor of 2 or 3) from the simulation results. The experimental data points did not have a measured degree of hydration, as only their ages were recorded. Consequently the degree of hydration was estimated as follows. The 28-day old samples measured in Ref.  were assigned = 0.7, the 60-day old samples measured in Ref.  were assigned = 0.8, and the 180-day old samples measured in Ref.  were assigned = 0.9. These values of were not picked to give the best agreement with simulation, but rather were based on data presented in Ref. . This correlation between and age was assumed to be independent of w/c ratio. However, it is known that higher w/c ratio cement pastes can hydrate faster than lower w/c ratio pastes , so that the apparent disagreement in Figs. 6 and 7 could just be due to an incorrectly assigned degree of hydration. In all fairness, it should be stated that this caveat also applies to the points that agreed well with the simulation results.
Figure 4: Showing the logarithm (base 10) of the relative diffusivity, D/Do vs. degree of hydration , for both simulation and experiment for a 0.4 w/c cement paste.
Figure 5: Showing the logarithm (base 10) of the relative diffusivity, D/Do vs. degree of hydration , for both simulation and experiment for a 0.45 w/c cement paste.
Figure 6: Showing the logarithm (base 10) of the relative diffusivity, D/Do vs. degree of hydration , for both simulation and experiment for a 0.5 w/c cement paste.
Figure 7: Showing the logarithm (base 10) of the relative diffusivity, D/Do vs. degree of hydration , for both simulation and experiment for a 0.6 w/c cement paste.
5.2 Silica Fume Results
Condensed silica fume, a small (0.2-0.4 µm), highly- reactive, almost pure amorphous silica material, is being increasingly used as a mineral admixture in concrete where a low chloride diffusivity is desired . Its effect on transport properties in concrete or mortar is probably due partly to modification of the sand-cement paste interfacial zone [10,35], and partly to modification of bulk cement paste microstructure . Ref.  showed how the incorporation of silica fume into the cement paste could be simulated. The silica fume reacts with the CH phase to produce more (pozzolanic or secondary) C-S-H, which has a larger volume than the original CH and silica fume combined. Therefore, using silica fume tends to further reduce the capillary porosity of a cement paste relative to that of a plain paste.
The incorporation of silica fume has been simulated at two different water to solid (w/s) ratios, 0.6 and 0.4, where w/s ratio is defined similarly to w/c ratio in eq. (1), but with the weight of cement replaced by weight of cement plus silica fume. Ten and 20 percent of the cement, by weight, has been replaced by silica fume, which keeps the w/s ratio constant, permitting a fair comparison with plain cement paste at an equivalent w/c ratio . Silica fume has a lower specific gravity than cement, so that the fraction of solid volume initially taken up by the silica fume is greater than its fraction by weight.
Fig. 8 shows the simulated diffusivity results for 0, 10, and 20% replacement of the cement by silica fume, for w/s = 0.60.
Figure 8: Showing the logarithm (base 10) of the relative diffusivity, D/Do vs. degree of hydration , for an 0.6 w/s (water to solid) cement paste, in which 10 and 20% of the cement, by weight, has been replaced by condensed silica fume.
The 0% data is the same as that shown in Fig. 7, connected with straight line segments to facilitate comparison with the silica fume results, which do not lie at exactly the same degrees of hydration. Both the 10 and 20% results lie below the 0% diffusivities, which is as expected, since the reaction of silica fume with CH reduces the capillary porosity. However, it is a little surprising that the 10% silica fume pastes have a lower diffusivity at all degrees of hydration past 0.2 than the 20% silica fume results. This result can be easily explained by studying Fig. 9.
In Fig. 9 capillary porosity vs. weight percent silica fume, taken as a percent of total original solid weight, is plotted for various degrees of hydration, using relationships developed in Ref. , for w/s = 0.60. The = 0.2 curve shows that the
Figure 9: Showing capillary porosity vs. percent of total original solids, by weight, of silica fume mixed with the original cement, for various degrees of hydration, at w/s = 0.6.
capillary porosity is almost identical for 10 and 20% silica fume replacement, which is the reason that the two diffusivities in Fig. 8 at = 0.12 are almost identical. For < 0.2, though, the capillary porosity is always smaller for the 10% silica fume pastes than for the 20% silica fume pastes. Physically, this is because there is a tradeoff when replacing cement with silica fume. Less cement means that there will be less CH produced to react with the silica fume. At small silica fume fractions, there is more than enough CH to react with all the silica fume, but as cement content decreases and silica fume content increases, there comes a point when there is too much silica fume to react with the CH produced, and part of the silica fume begins to act as an inert filler, which cannot fill pore space as effectively as reactive cement . This is the physical explanation for the minima in the plots shown in Fig. 9. It should also be noted that the greatest differences in diffusivity between the 10 and 20 % results are at intermediate degrees of hydration, 0.4 < < 0.6, where the greatest differences in capillary porosity are also found in Fig. 9.
This explanation suffices for the w/s = 0.60 data, as the capillary pore space always remains continuous, so that the relative diffusivity is always dominated by the capillary pore space. Differences in capillary porosity are then directly and easily related to differences in relative diffusivity. However, the w/s = 0.40 results are somewhat different, because the capillary porosity falls below c = 0.18.
Figure 10: Showing the logarithm (base 10) of the relative
diffusivity, D/Do vs. degree of hydration ,
for an 0.4 w/s (water to solid) cement paste, in which 10 and 20% of the cement, by weight, has been replaced by condensed silica fume.
Fig. 10 shows the diffusivity results for a w/s ratio of 0.40. Again, the 10% silica fume pastes have about the same diffusivity as the 20% silica fume pastes at the lowest degrees of hydration. The 0% silica fume data is from Fig. 4. For intermediate degrees of hydration, the 10% results are systematically lower. Fig. 11 shows the capillary porosity plotted as a function of silica fume weight fraction for the 0.4 w/s ratio pastes, which explains the intermediate degree of hydration results of Fig. 10 in the same way as Fig. 9 explained the results of Fig. 8. For degrees of hydration greater than 0.6, however, the 10% and 20% silica fume results converge to the value D/Do = 0.0025, where log10 (0.0025) = -2.6, and are systematically above the plain cement paste results. This is because if there is enough silica fume to convert nearly all the CH to secondary C-S-H, at high degrees of hydration, where there is very little unreacted cement or capillary pore space left, the cement paste will consist almost entirely of C-S-H. The C-S-H phase has been modelled as having a relative diffusivity of D/Do, so that the bulk value of relative diffusivity for the paste will be the same. For lesser amounts of silica fume replacement, so that not all the CH is reacted, the minimum value of D/Do will vary continuously between 0.001, the minimum for plain cement paste (to be derived in Sec. 6.3) , and 0.0025, the value obtained for complete reaction of cement and CH.
Figure 11: Showing capillary porosity vs. percent of total original solids, by weight, of silica fume mixed with the original cement, for various degrees of hydration, at w/s = 0.4.
The above results imply that silica fume replacement of cement can reduce the diffusivity of bulk cement paste by reducing the capillary porosity, but at low porosities and high degrees of hydration, silica fume replacement can actually increase diffusivity by replacing impervious CH with microporous C-S-H. Silica fume in concrete could also reduce the diffusivity by reducing the porosity of the sand-cement paste interfacial zone, if transport through the concrete were dominated by pathways connecting through the interfacial zone regions.
An additional possibility not considered in this work is that silica fume modifies the microstructure of the C-S-H gel phase, changing its effective diffusivity relative to the C-S-H in plain paste. Reductions in the C/S ratio of the C-S-H, from 1.7 to 1.4, have been observed in cements containing pozzolanic admixtures like silica fume [36,37], as well as an increase in the polymerization of the C-S-H gel phase . These structural changes could mean that a different effective bulk diffusivity should be assigned to the C-S-H phase in cement pastes containing silica fume. An additional difficulty exists when comparing equal age specimens of silica fume-modified cement paste, as is usually done, in that silica fume may accelerate the hydration process , so that specimens of the same age, but with different amounts of silica fume, may have different degrees of hydration. Thus, the presence of silica fume may modify the relationship between age and degree of hydration used above.