6. Cement Paste Diffusivity Dependence on Pore Structure

6.1 Capillary Porosity Contribution

Since the microstructural model gives a detailed quantitative picture of the cement paste pore structure at any degree of hydration, it can be used to determine the dependence of diffusivity on pore structure in a fundamental way.

Of the two phases that contribute to the diffusivity, the capillary pore space and the C-S-H gel phase, the capillary pore space is first considered. The capillary pore space percolation threshold occurs when the capillary porosity is 18%, expressed as a percent of total volume. For porosities well above this threshold, the diffusivity should be dominated by the capillary pore space, since its conductivity is so much higher (by a factor of 400) than the C-S-H, although the C-S-H contribution is still not totally negligible. To separate the contributions of the two phases, the diffusivity was computed with _C-S-H = 0 [8]. The diffusivity of course then tends towards 0 as the capillary porosity approaches the percolation threshold of 18%. In Fig. 12, the logarithm of the diffusivity is plotted against the logarithm of the quantity ( - 0.18), where is the capillary porosity. From percolation theory, it is expected that such a plot will result in a straight line with a slope of about t = 2 [30] as approaches _c = 0.18 [20]. This scaling is expected from the concept of the universality of critical transport exponents [20]. A very good straight line is indeed found, with the correct slope of about 2.0. The complete equation of the line is

6.2 C-S-H Contribution

When the capillary porosity falls below 18%, then the diffusivity will be controlled by transport through C-S-H gel pores. However, there is still some capillary pore space left, in the form of isolated clusters. The physical picture of the dominant diffusive flow pathways in this regime consists of isolated capillary pore clusters linked together by C-S-H gel pore pathways. Although the C-S-H phase is itself continuous, pathways that also include the much more conductive capillary pores should be more important to the total diffusivity. This physical picture is similar to that proposed by Atkinson [3]. Capillary porosity will still be an appropriate variable in this regime, with the diffusivity continuing to decrease as the capillary porosity decreases. For < 0.18, then, the diffusivity is fitted with an Archie's law [40] form, (a ^m), with a and m constants, but modified by having a cutoff value R_min, where R_min is the value of the relative diffusivity when the capillary porosity is zero. For sandstone rocks, for which Archie's law was defined, the critical value of the porosity is approximately zero, so that the transport properties and approach zero simultaneously [40,41]. However, a zero capillary porosity cement paste would be composed of C-S-H, CH, and unreacted cement, which will have a non-zero relative diffusivity R_min because of the connected gel pores of the C-S-H phase. The value of R_min is not a fitting parameter, but can be calculated using composite theory and the known value of _C-S-H, as will be described in section 6.3.

If we consider the pure capillary pore space diffusivity, above = 0.18, and the C-S-H/capillary pore space pathways for all values of to be roughly in parallel, then a reasonable functional form for the relative diffusivity as a function of capillary porosity, which is well-justified physically, is

6.3 Zero Porosity Diffusivity Derivation

The cutoff value of 0.001 of the plain cement paste relative diffusivity at = 0 is justified by the use of a recent equation that gives a percolation theory-based description of the effective conductivity of a two-component composite [42]. For plain cement paste, with no silica fume, at = 0, the two components are C-S-H, with D/D_o = 0.0025, and the combination of CH and any unreacted cement, with D/D_o assumed to be zero for this phase. The equation to be solved for the effective relative diffusivity, D_m / D_o, for the composite is

Using eqs. (8) and (9) and _S = 1.7 and _P = 0.61, we find that D_m / D_o = 0.0012 for a w/c ratio 0.4, 0.00098 for w/c=0.35, and 0.0008 for w/c=0.3, thus justifying the choice of 0.001 as a reasonable approximation for any w/c ratio less than 0.41, when no silica fume is present. With silica fume present, a more reasonable value of the cutoff value is 0.001 < R_min < 0.0025, depending on the amount of silica fume replacement, as was discussed in section 5.2.