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Model Description

Cement and Mineral Admixture Particles

The microstructural model used to simulate the interfacial zone has been described in more detail elsewhere [7,17], and will only briefly be presented here. The key to the model is the representation of space (volume) as a series of discrete elements, called pixels, arranged on a simple cubic lattice. For the simulations presented in this paper, this three-dimensional lattice is 200 x 200 x 200, so that a total of eight million pixels are present. Each pixel represents a volume element occupied by a single phase of the concrete microstructure. For these simulations, relevant phases are porosity, C₃S, C-S-H, CH, pozzolanic C-S-H (C-S-H produced from the pozzolanic reaction of CH with a mineral admixture), mineral admixture, and aggregate. In the model, portland cement is considered to be composed entirely of C₃S. This simplification has not limited the model's applicability, however, as it has been applied successfully to i) computing the continuity of the capillary porosity of portland cement as a function of the degree of hydration [17], ii) characterizing the cement paste-aggregate interface in ordinary portland-cement concrete [7], and iii) calculating diffusion coefficients for cement pastes as a function of w/c ratio and [18]. The hydration chemistry of portland cement is certainly more complicated than that of C₃S. However, the spatial (geometrical and topological) arrangement of solid and pore phases is what actually determines most performance properties of cement paste, such as strength, diffusivity, or permeability. There is good reason to believe that this physical microstructural model accurately represents these spatial properties of portland cement pastes as well as C₃S pastes, for the following reasons.

Although the hydration products in the model are specifically considered to be C-S-H and CH (for plain pastes), one can think of the hydration products in a more general way. They can be classified as surface product (C-S-H), which forms on cement particle surfaces, and as pore product (CH), which forms in the capillary pore space [19]. For C₃ hydration, each volume unit of C₃S produces 1.7 volume units of surface product and 0.61 volume units of pore product. For C₂S and C₃A hydration [20,21], assuming C₃A hydrates to form a C₃AH₆ pore product (although in the presence of gypsum, calcium sulfoaluminates will form [20]), 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 the hydration of the minor constituent C₄AF can be computed. Table I summarizes the results for pastes of C₃S and typical cements of ASTM Type I and II, which indicate that cement pastes of Type I and II are quite similar to C₃S paste in terms of the separate amounts of surface and pore products produced, as well as the total volume of hydration products produced. This is a more general physical way of looking at the hydration process in cement paste with its reactive growth, and lends support to the application of the model to portland-cement concretes, in spite of the fact that its growth rules are based on C₃S hydration chemistry.

Cement	C3 (%)	C2S (%)	C3A (%)	Surface Product	Pore Product	Total Product
C3S	100	--	--	1.70	0.61	2.31
Type I	57.5	28.7	13.8	1.66	0.63	2.29
Type II	54.9	36.6	8.5	1.81	0.55	2.36

Table 1: Volume Units of Products Produced per Unit Volume of Cement

To begin a simulation, a starting microstructure is created. For simulation of interfacial zones, a 100 x 100 x 100 pixel cubic aggregate is placed in the center of the 200 x 200 x 200 pixel model. In these studies, the aggregate is considered to be inert. Thus, neither C-S-H nor CH may precipitate on the aggregate surface. Next, a water-to-solids (w/s) ratio, where solids includes the cement and mineral admixture but not the aggregate, and mineral admixture volume concentration are specified and the needed number of C₃S and admixture particles are placed inside the 200³ pixel hydration volume. C₃S particles are modelled as digitized spheres, with two different diameters, 21 and 11 pixels, being utilized for this study, so that the cement particle size distribution is bimodal. Thus, on the basis of scale, the aggregate must be considered as a sand particle, as it is five to ten times larger than the cement particles. Similar results would be obtained with larger aggregate particles, as it is the size of the cement particles that controls the interfacial zone properties. The cement particles are placed using periodic boundary conditions. That means if a cement particle is placed so that part of it projects outside the model box, that piece is "wrapped around" to the other side of the box. This reduces any artificial edge effects that arise from having a fairly small model in terms of number of cement particles (about 2000). To investigate the effects of mineral admixture particle size on microstructure, two scenarios are investigated. In one case, the admixture particles are modelled as fine one-pixel elements, where the scale of the model is such that one pixel is roughly equivalent to a one micrometer cube. This case corresponds approximately to a typical well- dispersed silica fume, though with somewhat larger particles. In the second case, the admixture particles are assigned the same size distribution as the C₃S particles. This case corresponds more closely to a fly ash-type mineral admixture. Admixture concentrations are assigned to be either 10% or 20% by weight of total solids. Assuming specific gravities of 3.2 for C₃S and 2.2 for any mineral admixture, these mass concentrations correspond to 13.9 and 26.7% on a volume basis of total solids.

Reactive Growth Processes

Within the microstructural model, hydration is represented as a three step cyclic process consisting of dissolution, diffusion, and reaction, illustrated schematically in two dimensions in Fig. 1. Basically, material randomly dissolves from the C₃S surfaces in contact with water-filled pore space, diffuses within the capillary pore network, and reacts to form either C-S-H or CH. Once again, to reduce artificial edge effects, periodic boundaries are used during the "diffusion" process such that a species may exit one face (side) of the 3-D microstructure and enter the opposite face. After all material dissolved in one cycle has reacted, the next cycle is begun with a new dissolution. Correct volume stoichiometry is maintained explicitly by creating the correct number of "diffusing" C-S-H and CH species for each species of C₃S that dissolves, based on data from Young and Hansen [21]. Solid C-S-H gel is allowed to form only on the surfaces of C₃S particles or on solid C-S-H formed earlier in the hydration. Conversely, CH forms by a nucleation and growth mechanism within the available pore space [7].

Figure 1: Schematic of the rules of the microstructural model.

Pozzolanic reactions can be included in the model by allowing the diffusing CH species to react at the admixture particle surfaces to produce pozzolanic C-S-H. For this model, the following pozzolanic reaction is assumed [21]:

S + 1.7CH + 2.3H → C_1.7SH_4.0

On a volume basis, each unit volume of silica is capable of reacting with 2.08 volume units of CH to produce 4.6 volume units of pozzolanic C-S-H. Although reductions in the C/S ratio, from 1.7 to 1.4, have been observed in cements containing pozzolanic admixtures [22,23], to keep the model tractable, the constant C/S ratio of 1.7 has been used throughout for both primary C-S-H and pozzolanic C-S-H. Reaction stoichiometry is maintained volumetrically by counting the number of CH diffusing species that have reacted with the admixture particles, and then terminating the pozzolanic reaction when this number reaches 2.08 times the initial number of admixture volume units (pixels). Since the pozzolanic reaction is expansive (in terms of solids), volume stoichiometry is maintained by specifying a probability that two volume units of pozzolanic C-S-H are produced instead of one when a diffusing CH species encounters a reactive (pozzolanic) surface. Since 2.08 volume units of CH should produce 3.6 volume units of pozzolanic C-S-H, in addition to the volume unit originally occupied by the mineral admixture, this probability of expansion is (3.6 - 2.08) / 2.08 = 0.73.

Mineral admixtures of various pozzolanic activities were included in this study to assess the influence of pozzolanic activity on final microstructure. In the model, a reactivity factor is defined by a number specifying how many volume units of CH can react with a unit volume of pozzolanic mineral admixture. The following reactivities were studied: a) zero, corresponding to a totally inert mineral admixture such as carbon black, b) 2.08, corresponding to a mineral admixture composed of pure reactive silica, approximately realized in silica fume, which is typically 95% SiO₂ [14], c) 0.47, corresponding approximately to fly ash, where the value of 0.47 was obtained from the long term reduction in CH observed in portland cement containing 25% by volume basis of fly ash [24], and d) 1.04, corresponding to a theoretical maximum pozzolanic reactivity of fly ash, which is typically 50% SiO₂ [22, 25] ( (0.5)(2.08) = 1.04). The reactivity in d) might be achieved if the fly ash were ground sufficiently fine so as to expose all silica surfaces.

Although fly ash may also contain CaO and Al₂O₃, only the pozzolanic reaction of the SiO₂ is considered in this study. When admixtures less reactive than silica fume are used in the model, the pozzolanic reaction is terminated whenever the number of CH diffusing species that have reacted pozzolanically equals the reactivity factor times the initial number of mineral admixture volume units present in the concrete.

To enable a meaningful comparison to be made between model concretes containing different mineral admixtures, the microstructural model was in all cases executed at a constant w/s ratio of 0.45, and hydrated to a constant degree of hydration, α = 0.77. This degree of hydration would normally correspond to a few months curing time, which is more than long enough to see any effects of the mineral admixtures. After this degree of hydration is reached, the final microstructure is quantitatively analyzed by measuring the volume fractions of all cementitious phases present as a function of distance from the aggregate surface. For a more easily visualized qualitative assessment, images of two-dimensional slices through the microstructure are obtained. Images of slices taken through the middle of the aggregate (middle slices) and just in front of the aggregate surface (surface slices) were created and photographed to enable a qualitative evaluation of microstructure to be made. These images were produced for both the starting microstructures and the microstructures formed after 77% hydration.