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Simulations were executed in which 10% by weight of the cement was replaced by a mineral admixture, either inert or reactive (pozzolanic). During these studies, the filler was substituted for cement, as opposed to adding filler to cement, to enable a direct comparison to be made between neat and filled pastes. The mineral admixture has been assigned a specific gravity of 2.2, which is approximately that of condensed silica fume [6]. The reactive mineral admixture in this study represents condensed silica fume. Due to the difference in specific gravity between C3S and the mineral admixtures, a new relationship must be derived for w/s or water-to-solids ratio. Water-to-solids ratio for a filled cement is defined as:
Now, consider a unit volume of total solids, composed of cement and filler. Based on a 10% weight fraction of filler, this unit volume is comprised of 86.1% by volume of C3S, and 13.9% by volume of filler. Based on this information, the water-to-solids ratio is defined as:
where f / is the initial volume fraction of total solids and x is the volume fraction of the total solids
which is filler. The right-hand side of eq. (4) has been evaluated for the specific value of x =
0.139. Eq. (2) will still hold for
keeping in mind that f is the volume fraction of
C3S, so that f / = f + volume
fraction of filler.
It is assumed that the initial solids (C3S and mineral admixture) do not absorb a substantial amount of water, and that no overall volumetric expansion or shrinkage occurs as a result of the ongoing hydration process. Based on eqs. (1) and (4), a constant w/s ratio can be maintained to compare pastes with and without mineral admixtures.
Mineral admixture particles are modelled as fine one-pixel particles. For comparison, C3S particles are typically 3 to 20 pixels in diameter, with the weight average in these simulations being about 11 pixels. Thus the filler particles are eleven times smaller than the average cement particle. For real materials, the average cement particle is typically 15 micrometers, while the average condensed silica fume particle, for example, is on the order of 0.2-0.4 micrometers [8], or 40-80 times smaller, which is the same order-of-magnitude as used in the model study. Inert mineral admixtures are incorporated into the model by being randomly placed into the pore space of the three-dimensional box after the C3S particles have been placed. Neither the CH nor the C-S-H diffusing species react with these inert filler particles.
Pozzolanic or reactive mineral admixtures, on the other hand, require additional rules to be included in the model, as they can react with the diffusing CH species. For this study, these rules are based on a silica-fume-type reactive filler. The reactive mineral admixture is initially dispersed throughout the paste volume in the same way as was the inert mineral admixture. However, the diffusing CH species are allowed to react when they encounter a filler surface during their random walk diffusion process. Here again, the volume stoichiometry is based on that provided by Young and Hansen [6]. A slightly modified pozzolanic reaction is assumed:
Assuming specific volumes of 27 cm3/mole for S, 33.1 cm3/mole for CH, and 124 cm3mole for C1.7SH4.0 and considering the mineral admixture to be pure SiO2 (S), on a volume basis, this reaction is equivalent to each unit volume of mineral admixture being capable of reacting with 2.08 volume units of CH to form 4.6 total volume units of C-S-H [6]. To maintain the correct reaction stoichiometry, the model monitors the amount of "pozzolanic" (or secondary) C-S-H formed as the hydration progresses. When the number of CH diffusing species that have reacted with mineral admixture particles reaches 2.08 times the initial number of these one-pixel filler particles, the pozzolanic reaction is terminated. It is also necessary to maintain correct volume stoichiometry for this pozzolanic reaction, since the products occupy a larger volume than the solid (non-water) reactants. When a CH diffusing species reacts with a reactive mineral admixture, a probability is specified that two volume elements of product, instead of one, are formed. Accounting for the one volume element occupied by the reactive mineral admixture particle, each 2.08 CH volume elements should form 3.6 extra volume elements of product. Thus, the probability of expansion is given by
Because of the digitized nature of the model, these probability-type rules are easily incorporated into the hydration algorithm.