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The microstructure model used in this study has been described in more detail elsewhere [5,8]. Within the model, a unit volume of cement paste is represented as a three-dimensional cubic array of elements, called pixels. Each pixel is identified as belonging to a particular phase such as tricalcium silicate (C3S) or calcium hydroxide (CH). For the simulations used in this study, in all cases, the unit volume is 100x100x100 or a total of one million elements. In three dimensions, cement particles are represented as digitized spheres, typically 3 to 21 pixels in diameter. To eliminate artificial edge effects, periodic boundaries are employed such that a spherical particle which would extend out one face of the unit volume is completed on the inside of the opposite face. The water:cement (w/c) ratio of a system is related to the volume fraction of solids (cement) by the equation
where f is the solids volume fraction and 3.2 is the specific gravity of cement [9].
Microstructural development due to hydration is simulated as a series of cycles, with each cycle consisting of three steps: dissolution, diffusion, and reaction, as illustrated schematically in Fig. 1.
Figure 1: Schematic diagram of the cement paste microstructural development algorithm.
In the dissolution step, any cement pixels in contact with a water-filled pore space pixel attempt to take a step in a random direction. The pixels whose step lands them in the pore space dissolve, and each such pixel turns into a random diffuser. The pixels whose random step would land them in a solid phase are not allowed to move, and so remain at their original location, undissolved. The number of pixels that dissolve are counted, and the correct number of extra diffusing pixels are added at random locations within the pore system, replacing pore space pixels, to account for the correct amount of surface and pore product formation.
The model considers two types of reaction products: surface products such as C-S-H gel, and pore products such as calcium hydroxide. Pore products can spontaneously nucleate in the pore space. These nucleation sites subsequently grow into larger crystals as more pore product diffusers contact the crystal surfaces. Surface product diffusers precipitate into solid gel on the surfaces of the original cement particles or on previously deposited gel product.
Reaction stoichiometry is explicitly based on the hydration of C3S [9] in that 1.7 volume units of surface product (C-S-H) and 0.61 volume units of pore product (CH) are produced for each unit volume of cement which dissolves. The similarity of these values to those for cements of ASTM types I and II has been noted and discussed [10]. Thus, although its reaction stoichiometry is based on C3S, the microstructure model is considered to be a portland cement paste microstructure model. Favorable comparisons of model systems to real cement systems have been made [5,6,11].
In systems containing silica fume, which is modeled as fine one-pixel particles with a specific gravity of 2.2, pore product (CH) diffusers are allowed to react upon contacting a silica fume or secondary C-S-H gel surface to form pozzolanic or secondary C-S-H. The pozzolanic reaction is terminated when all possible silica fume, according to the proper volume stoichiometry, is consumed [9,10]. This requires, of course, that primary and secondary C-S-H be uniquely identified within the model, so as to be able to keep track of the extent of the pozzolanic reaction [10].
When all random diffusers generated in one cycle have reacted, a new cycle is begun with a new dissolution step. The degree of hydration, α, may be determined after each cycle as the fraction of cement reacted relative to the original amount of cement in the system. The hydration model is ultimately self-limiting, as the remaining cement particles eventually become surrounded by hydration products, so that no further hydration is possible. In the three-dimensional model, however, this typically occurs only after about 90% of the cement has hydrated, which is in agreement with the observed hydration characteristics of real specimens. Since for this work, "fully" hydrated microstructures were desired, in all cases the microstructural model was executed for 200 cycles, which produced degrees of hydration ranging from 0.80 to 0.91 for the various w/s ratio and silica fume content combinations considered in this study.