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Resolution effects on computed diffusivity and permeability

One of the biggest effects of any change in capillary pore space percolation properties is on the diffusivity of the cement paste. Clearly, if one size system is still percolated when a smaller size is not, at the same overall porosity, then there could be a significant difference between the diffusivities of the two, even though the larger system will be connected through smaller pores.

To test this possible effect on diffusivity, various microstructures, of sizes 1003 to 4003, were saved at several different porosities between 0.12 and 0.32. The C-S-H phase was allowed to have a diffusivity between 0.0 and 0.01, where the pore space had a diffusivity of unity. In previous work, where an equation was developed relating porosity and diffusivity, a value of 0.0025 was used for the C-S-H phase [10]. Since computing the diffusivity for the 4003 systems consumed much CPU time, runs were made at a smaller number of porosities.

The data is summarized in Fig. 13, which shows the computed diffusivity for all systems using only DCSH = 0.0, 0.005. It is clear that as the capillary porosity percolation threshold decreases, the diffusivity at a given porosity increases because the pore space is better connected farther away from its percolation point. As the capillary porosity decreases, the effect of the capillary porosity also decreases, with the overall diffusivity being controlled by the amount and diffusivity of the C-S-H phase. Since at any given capillary porosity, the phase fractions of all the different size systems are the same, the diffusivities should all become similar once the porosity has passed the percolation threshold for all systems. The graphs have almost reached that point in Fig. 13.

Figure 13: Showing the normalized diffusivities for different size systems vs. capillary porosity, for two different values of DCSH , 0.0 and 0.005.

The permeability of each system was computed using a lattice Boltzmann algorithm modified to run on parallel computers [29]. Figure 14 shows the computed fluid permeability curves. This transport property is much more sensitive to pore size than is diffusivity. The extra connectivity of the pore space of the larger models is through smaller pores, so it is interesting to compare permeability and diffusivity, which is not very sensitive to pore size, on the same systems. In Fig. 14, it is clear that the permeability becomes very small for porosities smaller than the percolation threshold for the 1003 system. This is because the biggest pores, which dominate the permeability, are disconnected by the 1003 threshold. The permeability for the larger systems are dominated by much smaller pores for porosities lower than this. This figure broadly agrees with the experimental results of Powers [30] on where the permeability of various water:cement ratio pastes sharply changes, indicating a change in the size of pore that dominates the permeability.

Figure 14: Showing the permeabilities for different size systems (in units of µm2) vs. capillary porosity.

Figure 8 shows the effect of resolution on the capillary pore size very well. Since permeability is so much more sensitive to pore size than is diffusivity [6], digital resolution affects it much more than diffusivity.

Chemical shrinkage results have shown that there is a pronounced flattening in the degree of hydration vs. time curve when the capillary porosity reached values near those predicted for the percolation threshold predicted by 1003 models [19]. This was interpreted as evidence that the capillary porosity was closing off near this porosity, thus restricting the inflow of external water and causing a decrease in hydration. Since the flow of water is controlled by permeability, not diffusivity, the results of Fig. 14 are instructive in this case. This figure shows that, even though the higher resolution models still have some permeability for porosites lower than the 1003 threshold, this permeability is very small. For most purposes, the porosity at the 1003 threshold is the controlling porosity, thus explaining the rough agreement with the experimental evidence.

One conclusion is clear from the above diffusivity work. Extracting an effective conductivity/diffusivity for the C-S-H phase by tuning the C-S-H property so that the bulk property agrees with experiment is not necessarily an accurate process. The interaction between C-S-H diffusivity and digital resolution of the model was clearly seen in Fig. 13.

Now this does not mean that the equation for diffusivity of cement paste as a function of capillary porosity ($\phi$) that was derived in earlier work [10],

D / D f = 0.001 + 0.07 x 2 + 1.7 x H ( - c) ( - c) 2 (1)

is inaccurate. The parameters in this equation are H(x), which equals 0 for negative values of the argument, 1 for positive values of the argument, and $\phi_c$c, which is the percolation threshold of the capillary porosity, as computed several times in earlier parts of this paper. This equation is accurate, as it was fit to experimental measurements, by using a relative diffusivity of 0.0025 for the C-S-H phase, where the value of 1 is used for the capillary pore space. It is just that using this value other than in the context of this equation, or assuming it to be an accurate measurement of a C-S-H property, is not necessarily valid. So while this equation, or an improved version [24], can be used for predicting cement paste transport, one must be cautious about extracting direct C-S-H information from it.

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