A cellular-automata (CA) is basically a computer algorithm that is discrete in space and time and operates on a lattice of sites 40, 41 (i.e., a two-dimensional or three-dimensional array of pixels). Starting from some initial configuration, at each iteration of the algorithm, the state (value) of each site is updated based on its current value and the current value(s) of one or more neighboring sites. The evolution of the system structure (microstructure in our case) over a series of iterations (time) then is monitored in terms of both the visual appearance of the system and any number of global or local properties, such as phase volume fractions or percolation characteristics. The state space of each lattice site is usually limited to a few values; for the cement hydration model, this state space is comprised of individual values (integers) corresponding to each solid phase (including water) and each possible diffusing species. Update rules for dissolution and diffusion/reaction are then implemented as part of the computer algorithm, as illustrated in Fig. C1. Here, for the center pixel in the 3 x 3 grid, a random direction is chosen (± x, ± y), with the CA outcomes shown for each of the four possible choices.
The cement hydration model is not a pure CA because nonlocal information is often used in determining the state space evolution (e.g., using a global concentration of a diffusing species to determine the likelihood of the nucleation of a new crystal), but is largely based on principles utilized in CA. Using a graphics workstation to implement the CA for cement hydration in real time provides the added advantage of visualization of the ongoing hydration process, extremely useful for debugging purposes and for gaining new insights into the effects of the selected rules on the evolving microstructure.
Although often relatively simple in structure, the behavior produced by CA can be quite complex. The field of CA has been developing over the past 10 years 40, 41 or so, with recent applications in materials science (sintering 32, 42and dendritic growth 43, 44) and biological systems. 45
Fig. C1. Illustration of the simple CA algorithms used for dissolution and diffusion/reaction in the cement hydration model. In far left images, central pixel being considered is marked with an X. Images to the right then show the resultant systems for each of the four possible chosen directions for dissolution/diffusion (red is solid materials, green is water-filled porosity, and blue is diffusing species).