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Critical CH + Capillary Pore Space Phase Fraction

The data in Figs. 5 and 6 indicate that diffusivity increases much more rapidly with an increase in φ, once the capillary pore space reconnects (percolates). Thus, to minimize the effects of leaching of the CH phase on transport properties, it is necessary to assure that the capillary pore space does not reconnect during the leaching process. Assuming the worst case, complete removal of the CH phase, percolation of the capillary pore space will never be obtained after leaching only if the total original volume fraction of CH plus capillary pore space is such that the two phases considered as one composite phase do not form a spanning cluster.

To quantify this effect, it is necessary to determine the percolation characteristics of the combined CH and capillary pore space phase. Figure 7 shows a plot of fraction connected vs. total (CH + capillary pore space) phase fraction for various w/s ratios and silica fume contents. A single curve, with some spread, is obtained, similar to the percolation curves for the capillary pore space alone [10]. "Fraction connected" means the volume fraction of the spanning cluster, normalized by the total volume fraction of the phase of interest, in this case CH plus capillary pore space. From the plot, the percolation threshold of the two phases in combination appears to be about 18%. This value should logically fall somewhere between 15 and 18% since the percolation thresholds of the CH and capillary pore space phases separately are 15 and 18%, respectively. Based on this threshold value, given a w/s ratio and silica fume content, it is possible to determine the degree of hydration required to produce a paste where the total CH + capillary pore space phase fraction is below 18%. Leaching will have only a small effect on the diffusivity of such a paste, as it will be impossible for the capillary pore space to repercolate during the leaching process.

Figure 7: Fraction connected of CH + capillary pore space vs. total CH + capillary pore space phase fraction for a variety of w/s ratios and silica fume contents.

The calculation is performed as follows. For a given silica fume content, there is a specific degree of hydration, α^/, corresponding to complete reaction of all the silica fume with CH given by

where x is the silica fume volume fraction of the total solids volume, 2.08 is the volume of CH consumed by reaction with one volume unit of silica fume, 0.61 is the amount of CH generated by one volume unit of C₃S [9], and m is the weight fraction equivalent to x. The variables x and m are related by the following pair of equations:

where 3.2 and 2.2 are the specific gravities of cement and silica fume respectively.

If the degree of hydration expected is less than the cutoff value α^/, then no CH will be produced. As long as the capillary porosity is below 18%, then the capillary pore space will always be disconnected, as observed in Fig. 4b on the hydration curve. In this case, the volume of solids is given by the sum of the volumes of the reaction products, any unreacted cement, and any unreacted silica fume or

where β^/ = 3.048 is the volumetric expansion factor for conversion of C₃ to all C-S-H (in the presence of silica fume), f^/ is the volume fraction of total solids, and 0.293 is the volume of silica fume required to react with the CH produced by one volume unit of C₃S [10]. This expression reduces to

Furthermore, f^/ relates to the w/s ratio by the expression

Finally, by using eq. (8) to substitute for f^/ in eq. (7) and using the numeric values for all known parameters, the capillary porosity φ = (1 − total solids volume fraction) can be obtained:

Given a w/s ratio and silica fume volume fraction, eq. (9) may be used to determine the value of α required for the porosity to be below the critical value of 18% [10], which will insure that the capillary pore space is discontinuous. This is done by setting eq. (9) equal to φ_c = 0.18, and then solving for α ≡ α₂ as a function of x and w/s ratio. In terms of m, the weight fraction of silica fume replacement, α₂(m) is given by

If α exceeds α^/, then some CH will be produced and the criterion becomes that the sum of the CH volume fraction and capillary porosity does not exceed 18%. For this case, the volume fraction of CH is given by

where once again, 0.61 is the amount of CH generated by one volume unit of C₃S and 2.08 is the volume of CH consumed by reaction with one volume unit of silica fume. In analogy to eq. (9), the capillary porosity is given by

Using eqs. (11) and (12), for given values of w/s ratio and x, a value of α can be determined so that the sum of V_CH and φ is below 18%. This is done by setting the sum of eqs. (11) and (12) equal to 0.18, the critical volume fraction for the combined CH and capillary pore phase, and then solving for α = α₁ as a function of x and w/s ratio. In terms of m, the weight fraction of silica fume replacement, the result is

The point at which α₁ and α₂ meet is by definition at a value of m with α₁ = α₂ = α^/ . This value of m, denoted m_int, is given by

Also, there is a value of m, denoted m_min, below which the α₁(m) curve becomes unphysical, since it predicts a negative capillary porosity. As a function of w/s ratio, m_min is given by

Results of these calculations are shown in Fig. 8, which plots, for several w/s ratios, the degree of hydration needed to produce a paste with (V_CH + φ < 0.18), as a function of silica fume weight percent m. The dashed curve is a graph of eq. (4), the equation for α^/ vs. weight percent of silica fume for a silica fume-modified paste. If α > α^/ , then there is not enough silica fume to consume all the CH produced. If α < α^/, then there is more than enough silica fume to consume all the CH produced. The curves on the left of the α^/ curve use eq. (13) for α₁(m), and begin at m = m_min, while the curves on the right of the dashed curve use eq. (10) for α₂(m). The intersection point with the α = α^/ curve is at m = m_int. Several points are worth noting in this figure.

Second, for a given w/s ratio, there exists an optimum silica fume content that minimizes the hydration required for producing a paste with (V_CH + φ) < 0.18. As the w/s ratio is increased, this optimum silica fume content also increases. The physical reason for the minima in the curves in Fig. 8 at m = m_int is because beyond a certain level of silica fume addition, the additional silica fume acts only as an inert filler, since there is no CH remaining with which the silica fume can react. Thus, the cement present must hydrate even further to produce the desired paste since a portion of the solids (the additional silica fume) is not contributing any solid reaction products to help fill in the capillary pore space.

Finally, when the w/s ratio is high enough, no amount of silica fume will be sufficient to achieve the critical CH plus capillary pore space volume fraction. For example, if α = 0.8 were the maximum degree of hydration to be expected before the beginning of leaching, pastes with w/s ratios of 0.35 and 0.45 would require about 4 and 9% silica fume additions by weight replacement of cement, respectively, to meet this condition, while a paste with a w/s ratio of 0.55 cannot achieve the critical volume fraction of CH plus capillary pore space, regardless of silica fume content. For this latter system, even when sufficient silica fume is added to remove all of the CH, the porosity of the hydrated system will still be greater than 18% at α=0.8, so that the diffusivity will be dominated by the connected capillary pore space before leaching, and will remain so throughout the leaching process.