Next: Results and discussion
The model operates on a square cell of pixels, typically of edge length equal to 500 pixels, where each pixel is assigned to a single phase, such as pore space or cement. The scale of the model is then approximately one micrometer per pixel length. It should be noted at this point, by way of definition, that cement particles are mixed with water to produce cement paste. Initially, a user- specified number of cement particles are randomly placed in the cell such that no two particles overlap, which simulates the mixing process . This is the well-known "random parking" algorithm . Periodic boundary conditions are used to eliminate any artificial edge effects at the cell walls. In practice, this means that any part of a cement particle that would extend beyond the cell boundary is wrapped around to the other side. The particles may be identical or of different sizes, and are modelled in the work described in this paper as being circular in shape. This was done as a matter of convenience--there are no restrictions on cement particle shape in the model. Any shape that can be represented by pixels can be used. We do not believe that there is any significant effect of cement particle shape on the general conclusions of this paper. For the current application, an aggregate particle, modelled as a 100 x 100 pixel square, is first placed in the center of the cell. Since the largest cement particle used was 21 pixels in diameter, the relative size of the aggregate particle requires it to be thought of as a sand grain. During the placement of the cement, the cement particles are not allowed to overlap the aggregate. Fig. 1a illustrates an initial configuration of monosize cement particles (red) placed around a square aggregate particle (white). The black region is water-filled pore space. A preparation variable commonly used in the cement- based materials literature is the water:cement ratio, defined as the ratio of the weight of water to weight of cement in the original water and cement mixture. A value of 0.47 was used in all the simulations described in this paper.
Figure 1a: Showing original cement particle placement around
aggregate for 21 pixel diameter circular cement particles. The
red circles are the cement particles, the white square is the
aggregate particle, and the black is water-filled pore space.
Figure 1a: Showing original cement particle placement around aggregate for 21 pixel diameter circular cement particles. The red circles are the cement particles, the white square is the aggregate particle, and the black is water-filled pore space.
The model is iterated by cycles, where each cycle consists of three steps: dissolution, diffusion, and reaction.
In the dissolution step, any cement pixels in contact with a water-filled pore space pixel are given a probability to dissolve, so that some fraction of these identified pixels dissolve, and some must wait for the next cycle. This is reasonable, since dissolution is a random process at the molecular level. The pixels which dissolve "step off" into the pore space and become random walkers (diffusers). The number that dissolve are counted, and the correct number of extra diffusing pixels are added at random locations within the pore system to account for calcium hydroxide (CH) formation and the volume expansion which occurs when amorphous calcium silicate hydrate (C-S-H) forms. Other ways of adding the extra material, that would correlate more closely to the actual areas where pixels have dissolved, could be thought of, but this is the simplest and easiest method, and should not introduce any problems for the random cement particle-water system. Standard cement chemistry notation (C=CaO, S=SiO2, and H=H2O) is used to denote the hydration reaction products of tricalcium silicate (C3S), the major component of portland cement. In this paper, the cement particles are taken to be pure C3S. More precisely, if n pixels dissolve from off the cement surfaces, 0.7n extra diffusing C-S-H pixels and 0.61n CH diffusing pixels are added to the system . In this way, the correct stoichiometry and hydration product volumes are achieved.
During the diffusion/reaction steps, the dissolved pixels execute random walks throughout the pore space. Periodic boundary conditions are again used, so that a pixel that diffuses out of the box comes back in on the other side. C-S-H pixels continue to move in this random fashion until they encounter a cement surface, at which point they react and stick to this surface. Once C-S-H product is present, diffusing C-S-H pixels can also react and stick to these surfaces as well. Following known cement hydration chemistry, CH product grows in the pore space in a manner different from the C-S-H material. For any given step taken by a diffusing CH pixel, there is a non-zero probability that it can nucleate at its present location. This probability p is a function of the number of CH diffusers put into solution in the given cycle, and is given by
|P = Po [ 1 - exp(-c/cm)]||(1)|
where p is the probability of nucleation, Po is the maximum probability of nucleation, c is the number of CH diffusing pixels remaining in solution at each time step, and cm is an arbitrary scale factor. Experimental results on the number of CH clusters  indicate that the form of this function must be such that the nucleation probability goes to zero rapidly with decreasing concentration, but reaches a peak at saturation. The form of eq. (1) satisfies these requirements, although it is not unique in doing so. The parameters Po and cm can be adjusted to control the number of CH clusters that are nucleated. Besides the possibility of nucleation, if a CH species encounters a CH cluster, it will react onto this surface, increasing the size of this cluster. However, there is no Ostwald ripening mechanism in the model, where small CH clusters dissolve and the large clusters' sizes are increased. When the aggregate particle is present, its surface is assumed to be non-reactive, in the sense that neither C-S-H nor CH is allowed to react onto it. Reactive aggregates could, however, be easily handled in the model. When all diffusing pixels have reacted or nucleated, the cycle is complete, and the next cycle begins with a new dissolution step.
After a given number of cycles (typically about 200 are needed to achieve maximum hydration) are executed, the resulting microstructure is analyzed to determine the area fraction of each phase as a function of distance from the aggregate edge. The square shape of the aggregate particle was chosen for convenience in carrying out this analysis. Other parameters such as degree of hydration are easily determined from analysis of the final microstructure. Degree of hydration is defined as the
ratio of reacted cement volume to original cement volume, so that starts at zero when the cement particles are first mixed with water, and ends at a value of one when hydration is complete.
Fig. 1b shows the result of 200 hydration cycles on the initial configuration shown in Fig. 1a. The black is again water-filled pore space, red is unhydrated cement particles, the blue is CH, and the yellow is C-S-H. One advantage of implementation of the model on graphics computers that allow animation is that the user can directly view the microstructural development on the display screen as it is taking place. This ability is invaluable in understanding how the growth rules of the model influence the developing microstructure.
Figure 1b: Same as Fig. 1a but after completion of the hydration simulation. White represents the aggregate, red represents the remaining unhydrated cement, CH is blue, C-S-H is yellow, and the remaining water-filled pore space is black.