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Pore Space Percolation

Effects of w/c and degree of hydration

The connectivity of the capillary porosity has been evaluated for various hydration conditions. For C₃S pastes containing no mineral admixtures, simulations were run for w/c ratios of 0.35, 0.45, 0.5, 0.6, and 0.7. Results are presented in Fig. 2a, which plots the fraction of the total porosity that is connected, as a function of degree of hydration.

Figure 2a: Percolation plot of porosity in neat C₃S paste showing fraction connected porosity vs. degree of hydration for w/c = 0.35, 0.45, 0.5, 0.6, and 0.7.

Initially, the porosity is entirely connected. At a given w/c ratio, this fractional connectivity decreases monotonically as hydration proceeds and parts of the capillary pore space become isolated. In Fig. 2a, it is apparent that the w/c ratio has a large effect on pore connectivity. For lower w/c ratios (les than 0.5), as the hydration proceeds, a point is reached at which pore connectivity drops rapidly, ultimately leading to discontinuity in the capillary pore structure. The value of α at which this occurs is denoted by α_c, the critical degree of hydration for percolation. As the w/c ratio decreases, α_c decreases as well, indicating that less hydration is required to close off the capillary pore space. For the higher w/c ratios (greater than 0.6), the pore space also becomes less connected as hydration progresses, but the sharply descending portion of the curves is absent as the high initial porosity prevents closing off the water-filled pore space with hydration products. These results are in agreement with experimental data for cement paste given in [16], where Powers defined capillary pore discontinuity as the point at which the measured fluid permeability, as a function of , showed a marked change in slope.

In Fig. 2b, the fractional connected porosity has been replotted with total capillary porosity, rather than α, as the independent variable. Now, the curves for the different w/c ratios all overlap onto a single curve, suggesting a degree of self-similarity in the hydration process.

Figure 2b: Percolation plot of porosity in neat C₃S paste showing fraction connected porosity vs. total porosity for w/c = 0.35, 0.45, 0.5, 0.6, and 0.7.

Thus, the pore space percolation characteristics of a low w/c ratio cement with a few days of hydration may be equivalent to those of a higher w/c ratio cement with several weeks hydration. The large change in pore connectivity occurs over the total porosity range of 0.2 to 0.4. The range suggested by the model is in agreement with results obtained by (i) Parrott et al [17], who measured diffusion parameters at various ages for C₃S paste and noted a large drop in diffusion parameter occurring as the porosity decreased from 40 to 20%, and by (ii) Powers [18], who observed a large increase in permeability of various cement pastes as capillary porosity increased from 20 to 40%.

The percolation threshold of approximately 18% porosity is in reasonable agreement with the 16% "universal" value of Scher and Zallen [11]. This means that the three-dimensional porosity is still highly connected at a total porosity of 45%, which is Scher and Zallen's "universal" value for the percolation threshold in two dimensions [11]. Thus, Scher and Zallen's results support the general observation that in cements the pore space appears discontinuous in two dimensional micrographs much earlier than it actually becomes discontinuous in three dimensions. Running the microstructural model in a two-dimensional form changes the porosity percolation threshold to about 45%, thus lending support to the conclusion that the agreement with Scher and Zallen's result in three dimensions is not just a numerical coincidence. Knowledge of the 3-d percolation threshold can be utilized by researchers viewing two-dimensional micrographs of a hydrated cement paste. For example, if the capillary porosity quantified by image analysis is found to be 25%, it is likely that the capillary porosity is still connected in three dimensions.

Effect of Inert Mineral Admixture

Next, the effects of mineral admixtures on the "universal" percolation curve were investigated. The presence of an inert mineral admixture was considered first. Fig. 3a provides a comparison of filled and neat C₃S pastes at w/s and w/c ratios of 0.45 and 0.5, with fractional connected porosity being plotted against degree of hydration.

Figure 3a: Percolation plot of porosity in neat C₃S paste and paste containing an inert mineral admixture showing fraction connected porosity vs. degree of hydration for w/c = 0.45 and 0.5.

More hydration is required for the filled pastes to achieve a given value of fractional connected porosity than in the neat pastes. While this may seem counterintuitive, it must be remembered that 13.9% of the cement volume has been replaced with a non-reactive filler. Thus, there is less cement to undergo the expansive hydration reaction, so that pore space is filled less efficiently than when the initial solids are 100% reactive cement particles. These results imply that substitution of inert mineral admixtures, no matter how fine, for cement will generally not improve the percolation properties of concrete at a constant degree of hydration. Enhancement of percolation properties may be realized if such a filler is added to concrete at a constant w/c ratio, mainly due to a reduction in the w/s ratio because of the volume occupied by the mineral admixture. Figure 3b shows that the universal curve for fractional connected porosity vs. total capillary porosity is maintained even when the inert mineral admixture, at 10% weight fraction of the total solids, is present in the paste.

Figure 3b: Percolation plot of porosity in neat C₃S paste and paste containing an inert mineral admixture showing fraction connected porosity vs. total porosity for w/c = 0.45 and 0.5.

Effect of a Reactive Mineral Admixture (Silica Fume)

Next, a reactive mineral admixture, in this case condensed silica fume, is considered. As outlined in the model description, the CH diffusing species are allowed to react with the filler particles to produce C-S-H. Figure 4a plots fractional connected porosity vs. degree of hydration for three (0.35, 0.5, and 0.7) w/c and w/s ratios for the neat and silica fume-filled pastes.

Figure 4a: Percolation plot of porosity in neat C₃S paste and paste containing a reactive mineral admixture showing fraction connected porosity vs. degree of hydration for w/c = 0.35, 0.5, and 0.7.

For the lower w/s ratios, adding silica fume shifts the curves markedly to the left, reducing the value of α_c. This reduction is caused by the extra volume of hydration products produced by the expansive pozzolanic reaction. As seen in Fig. 4b, however, the universal curve still holds when a pozzolanic mineral admixture is present.

Figure 4b: Percolation plot of porosity in neat C₃S paste and paste containing a reactive mineral admixture showing fraction connected porosity vs. total porosity for w/c= 0.35, 0.5, and 0.7.

Feldman and Cheng-Yi have measured the porosity characteristics of portland cement- silica fume blends at w/s ratios of 0.25 and 0.45 [19]. They found relatively discontinuous pores formed in the pastes containing silica fume after seven days of curing. From Fig. 4a, this observation is supported by the simulation results where the 0.35 w/s ratio paste containing silica fume achieves a discontinuous pore structure at less than 50% hydration, the value typically quoted for seven days of curing. Similar trends have been observed in measurements of the permeability of silica fume-containing pastes [20] and strength of silica fume-containing concrete [21]. Furthermore, in studying subcritical crack growth in the cement paste-steel transition zone [22], Detwiler found the following ranking in terms of cement-steel bond at an age of 7 days: cement paste containing silica fume > neat paste > cement paste containing an inert filler (carbon black). These rankings are consistent with the model predicting that, at a constant degree of hydration, the pore connectivity of paste is in the order of paste containing silica fume < neat paste < paste containing an inert filler. Detwiler directly related the bond measurement results to the pore structure of the pastes, concluding that the inferior performance of the paste containing carbon black may be due to its coarser pore structure.

The so-called "seven-day delay" in improved performance properties for cement-based materials containing silica fume has been sometimes attributed to a delayed pozzolanic reaction between the CH and silica fume [21]. However, Fig. 4a suggests another possibility. In the simulation, there is no delay in the starting of the pozzolanic reaction. Note that comparing two curves from Fig. 4a, at the same w/s ratio, one containing silica fume, the other no mineral admixture, the curves diverge farther from each other as the hydration proceeds. Thus, even if the pozzolanic reaction is ongoing from time zero, initially, it will little affect pore connectivity compared to the large effect at later ages. This early hydration corresponds to the flat portion of the curve in Fig. 4b (porosity > 0.40) where substantial changes in capillary porosity have only minor effects on pore connectivity in three dimensions. This is not to suggest that pore percolation is the only critical parameter in terms of permeability and strength, but rather that the initial stages of the "pozzolanic" hydration could have only minor effects on properties in comparison to later stages of the hydration where percolation characteristics change rapidly.

Calculation of Hydration Required to Achieve Pore Discontinuity

The universal curve obtained for fractional connected porosity vs. total porosity, allows the degree of hydration required to achieve pore discontinuity at any given w/c ratio to be calculated. From this curve, a total porosity of 18% or a solids fraction of 82% is required for discontinuity. On a volume basis, for neat cement pastes, the hydration products occupy a volume equivalent to β times the volume occupied by the reactants, where for C₃S, β = 1.7 + 0.61 = 2.31. Thus, at any point in the hydration,

Using equation (1), we obtain, in terms of w/c ratio and degree of hydration,

where the right-hand side of eq. (8) has been evaluated for β = 2.31. Using equation (8), with 0.82, or any other suitable value, substituted for V_solids, the value of α_c can be calculated for any given w/c ratio or vice versa. Table I summarizes the results obtained for a broad range of w/c and α_c values. Clearly, as w/c is decreased, less hydration is required to achieve pore discontinuity.

With an inert mineral admixture in the paste, assuming a 10% weight replacement and a specific gravity of 2.2 for the filler, equations (7) and (8) become

where f ^/ is the initial volume fraction of total solids. Combining equations (4) and (9) results in

Because 10% C₃S replacement with silica fume is inadequate to react with all of the CH produced by the C₃S hydration, for pastes containing a pozzolanic mineral admixture, two sets of equations must be developed. Ten percent, by weight, of silica fume is sufficient to react with 55% of the initial C₃S (in terms of the CH produced) so for α 0.55,

where β ^' = 3.048 is the volume expansion factor for C₃S hydration when CH reacts pozzolanically with silica fume and 0.293 is the volume of silica fume required to react with the CH produced by 1 volume unit of C₃S. Combining equations (4) and (11) and substituting for ^' results in

For > 0.55, the pozzolanic reaction will in theory have been completed (all of the silica fume will have been consumed) and subsequent hydration will produce CH. Thus,

and

It should be noted that equations (11)- (14) are approximations, as they assume that all CH produced for < 0.55 reacts pozzolanically to form C-S-H. In the model execution, a small amount of "stable" CH (CH which nucleates before it encounters a silica fume particle) does form during this early period of the hydration. Referring to Table I, in the presence of a pozzolanic mineral admixture, at a given w/s ratio, less hydration is required to achieve pore discontinuity than that required for neat C₃S paste, as observed in Fig. 4a.

Generalization to Variable Silica Fume Concentrations

The above calculation can be extended to determine the degree of hydration required to achieve pore discontinuity as a function of both w/c ratio and silica fume concentration, x. Once both of these parameters are specified, w/s can be determined and equation (4) can be used to relate w/s ratio to f ^/, the total solids fraction. Then for a given concentration of silica fume, the maximum degree of hydration for which CH can react with the silica fume is determined. Finally, the value of required to first achieve pore discontinuity is determined by formulating and solving equations similar either to (12) or (14) depending on the relationship of to the maximum degree of (pozzolanic) hydration.

Results of this analysis are summarized in Table II, where degree of hydration is tabulated vs. w/c ratio and percentage of silica fume by mass of cement. In this table, w/c ratio and percent silica fume based on mass of cement have been chosen as the independent parameters, as opposed to w/s ratio and percent silica fume based on mass of total solids, because the former are what are typically quoted in the cement and concrete literature. The data indicates that, as w/c is increased, either more hydration or more silica fume is required to achieve discontinuity of the capillary porosity.

For a fixed degree of hydration, the critical amount of silica fume required to achieve capillary pore discontinuity varies with w/c. Consider, for example, = 0.6. At w/c ratios of 0.375, 0.4, 0.425, 0.45, and 0.475, the amounts of silica fume required to achieve discontinuity are 2.5, 5, 7.5, 10, and 12.5% by weight, respectively. Thus, the higher w/c ratios require a higher concentration of silica fume. Similar results, a portion of which are shown in Table III, have been obtained experimentally by Berke and Roberts [23], who measured the chloride permeability of concrete samples containing several concentrations of silica fume. The data in Table III shows that as w/c ratio is increased from 0.38 to 0.48, the critical silica fume content shifts from 3.75 to 7.5%. The critical silica fume content is defined for the data in Table III as that amount needed to make the permeability reading (in coulombs) drop from thousands to hundreds. Large proportional decreases in transport properties are expected as the pore space percolation threshold is approached [5]. Assuming that this is the reason for the behavior of the data in Table III, then this trend is in qualitative agreement with the model results in Table II, further supporting the validity of the model and the influence of percolation on properties of cement and concrete.

Generalization of C₃S Model to Cement Paste

The rationale for comparing model results for C₃S paste to experimental results for cement pastes (containing C ₃S, C₂S, C₃A, and C₄AF) is as follows. Beyond the closeness of the agreement observed between model and experiment, justification for doing this relates to a generalization of the model. Although the hydration products have been specifically considered to be C-S-H and CH (for neat pastes), the model could be generalized to consider a surface product (C-S-H) and a pore product (CH) [24]. For C₃S hydration, each volume unit of reactant produces 1.7 volume units of surface product and 0.61 volume units of pore product. For C₂S and C₃A hydration [6,25], assuming C₃A hydrates to form a C₃AH₆ pore product, one volume element of C₂S produces 2.39 volume elements of surface product and 0.191 volume elements of pore product while hydration of one volume unit of C₃A produces 1.69 volume units of pore product. Based on these values, the amounts of surface product and pore product produced for various cement blends (ignoring C₄AF hydration) can be computed. Table IV summarizes the results for C₃S pastes and typical cements of Type I and II, which indicate that cement pastes of Type I and II are quite similar to C₃S paste in terms of the amounts of surface and pore products produced. Thus, it is not so surprising that porosity percolation results obtained using this model of C₃S hydration compare favorably to experimental measurements made on portland cement pastes.

This generalization of the model raises an interesting question concerning the percolation of the pore space. That is, is the percolation threshold of cement paste controlled mainly by the randomly-formed pore product or by the more abundant, more localized surface product? This question is of interest, because in contrast with sedimentary rocks, another randomly-formed, granular material, a percolation threshold of 18% porosity is quite high. The percolation thresholds of sedimentary rocks are on the order of one percent or less [26]. To investigate this question, the model was executed in a mode in which only surface product (C-S-H) was allowed to form. To keep the volume of product generated constant, 2.31 volume units of diffusing C-S-H species were created for each volume unit of C₃S which dissolved. Fig. 5 provides a comparison of results (with and without CH generation) for a w/c ratio of 0.35. As can be seen in the figure, the absence of a pore product results in only a small shift in the percolation curve, indicating that surface product formation is largely responsible for achieving discontinuity of the capillary pore space, although the random pore product does have a small effect on the absolute location of the percolation threshold.

Figure 5: Percolation plot of fraction connected porosity vs. total porosity for C₃S paste with and without calcium hydroxide formation.

The "pozzolanic" C-S-H which forms in the presence of a reactive mineral admixture should be considered as a pore product, as the fine filler particles are virtually the equivalent of the initially-formed CH nuclei. This helps account for the finding that the universal curve for fractional connected porosity vs. total porosity holds even in the case of a reactive mineral admixture where the quantity of solid CH produced during hydration is drastically reduced.