Cement hydration is modelled as a three part process: 1) material dissolves from the original cement particle surfaces, 2) diffuses within the available pore space, and 3) ultimately reacts with water and other dissolved or solid species to form hydration products through aggregation. Therefore, in order to simulate the microstructure development of hydrating cement, the physical processes of dissolution, diffusion, and reaction must be simulated. Each of these processes may be conveniently simulated using cellular-automaton-type rules.
The overall structure of the CA program is shown in Figure 3. The program uses a double nested loop to first apply rules which effect dissolution and then repeatedly applies rules which effect reaction and diffusion until no more reaction or diffusion can occur, then the whole process repeats. The way rules are used to simulate the three parts of the process is outlined in a general way below. Following this general description, the specific rules used in different variants of our model are then illustrated in state transition diagrams. These diagrams show how the dissolution, reaction and diffusion of the different chemical species interrelate.
Figure 3. CA program flowchart.
Dissolution. A pixel belonging to a soluble phase may dissolve by attempting a one-step random walk to a neighbouring pixel. If the neighbouring pixel is empty (i.e. representing pore space), the dissolution is allowed and an appropriate diffusing species is created. Otherwise, the pixel remains in its current solid state. This simple rule has the desired effect surfaces with high curvature will dissolve at a faster rate than low curvature surfaces, and thus smaller particles will dissolve faster than larger ones. To control relative dissolution rates, each soluble phase can also be assigned a dissolution probability between zero and unity. Thus, these probabilities represent relative mass transfer rates at the particle surfaces. In this manner, the hydration can be tuned such that C3A reacts faster than C3S, which reacts faster than C2S and C4AF, in agreement with experimental observations.
Dissolution can be globally prohibited via the assignment of a dissolution flag for each phase. Only those phases with a dissolution flag value of unity are soluble in any given time step (update or execution) of the cellular automaton. For example, ettringite's dissolution flag might be originally assigned a value of zero but updated to a value of unity when the majority of the gypsum (say 90%) has been consumed by hydration reactions. Alternatively, dissolution may be locally inhibited by checking for the presence of a particle of some type in the local vicinity (such as a 5 x 5 x 5 cubic neighbourhood) of the pixel undergoing dissolution.
Since the hydration products generally occupy a larger volume than the solid reactants, a volume correction step is often necessary to maintain the appropriate reaction stoichiometry. For example, when a pixel of C3S dissolves, 1.7 diffusing C-S-H species and 0.61 diffusing CH species should be created according to the reaction given in Figure 1. Extra diffusing species can be added using a random probability based on the required volume stoichiometry. When averaged over all the reactions in a time step, the desired non-integral volume correction is produced. The extra diffusing species may be added near the dissolution site to simulate a low species mobility or may be randomly scattered throughout the solution volume to represent an effectively infinite species mobility. Specifically, in this work extra C-S-H diffusing species are added in the local vicinity of the dissolution site due to low mobility of the silicate species while extra diffusing CH species are added at random pore locations to simulate the higher mobility of the calcium and hydroxide ions.
Diffusion. Possible diffusing species created by a dissolution process include diffusing CH,diffusing C-S-H, diffusing C3A, diffusing gypsum, and diffusing FH3. To implement diffusion, each diffusing species present in the microstructure is allowed to execute a one-step random walk to a neighbouring pixel. If the neighbouring pixel is pore space, the diffusing species is transferred to this new location; otherwise, it remains at its original location. The relative rate of diffusion of a particular species can be enhanced by increasing the number of spacings in the lattice to which it may move in one time step or decreased by decreasing the probability that the species is allowed to move at all in a given time step. The diffusing species continue to undergo this random Brownian motion until reaction occurs according to the rules outlined below.
Reactions. To model cement hydration, several types of reactions must be considered. Several of the hydration products (CH, FH3, and C3AH6) form by nucleation and growth processes. To simulate the nucleation processes, a diffusing species is assigned a certain probability of changing to its solid form without the presence of a surface onto which to aggregate. This probability (p) is made to be exponentially dependent on the number of diffusing species in solution (c):
p = p0 (1 - e- c / c m )
where p0 and cm are constants. This functional form gives a nucleation probability that will decay rapidly as c becomes much less than cm so that few new crystals will form late in the hydration. The nucleation rate of one diffusing species could even be made dependent on the concentration of an entirely different diffusing species to maintain the correct solution thermodynamics (involving solubility products, etc.). Although we do not consider the possibility in the current model, nucleation can also be spatially inhibited; for instance, CH nucleation could be inhibited in the vicinity of solid C-S-H to force the CH to form 'away from' the C-S-H material.
Other reactions require a collision of two species in order for a reaction to occur. Examples are the deposition of C-S-H on silicate surfaces and the reaction of diffusing gypsum and aluminate species to form ettringite. Here again, when a collision does occur, a probability can be specified as to whether a chemical reaction also takes place so that 'bimolecular' reaction rates can themselves be adjusted within the model. Depending on the precise reaction, volume correction may not be possible during dissolution and may have to be executed when the reaction occurs. To do this, neighbouring pore pixels are converted into the appropriate solid hydration product to satisfy the needed reaction volume stoichiometry. For example, when diffusing gypsum reacts at an aluminate surface, on average, 3.21 pixels of ettringite must be created. The fractional portion (0.21) of extra pixels of ettringite can be realised by using the appropriate probability to form three units of solid ettringite 79% of the time and four units of solid ettringite 21% of the time when the reaction occurs. This volume correction cannot be handled a priori during dissolution since several subsequent reactions are possible for diffusing C3A species.
The above discussion has indicated that CA rules can be used to model the physical processes of dissolution, diffusion and reaction. In a conventional reaction engineering approach, the system of interest is modelled using a set of differential equations describing the mass transfer and kinetics of reactions (Billingham and Coveney 1993). The parameters described in the previous discussion relate directly to the parameters used in this conventional approach. For instance, the dissolution probability for a phase controls the mass transfer rate for the phase entering solution. Further, although not implemented in the present version of the model, by keeping track of how many of each species are present in the solution phase, absolute solubilities could be maintained. Diffusion transport rates are directly related to the probability with which a given species makes a random step in a given time step. Reaction rates are controlled by the probabilities for nucleation and reaction during a bimolecular collision. If a time basis were specified for the model (e.g., each time step is equivalent to some small fraction of a second), the relative probabilities of diffusion, dissolution and reaction could be adjusted to represent the actual rate values proposed for these processes in a cement system. This approach has not been entirely implemented here because, based on current knowledge of cement hydration, these hypothesized values would be guesses at best. In the end, however, the processes that are being simulated at the pixel level must be representative of those actually occurring at the molecular level. In creating this computer-based microstructural model, an attempt has been made to capture the relevant physics of the hydration process without adhering strictly to every detail at the atomic and molecular level. However, the CA framework is flexible enough to incorporate such information as it becomes available.
Figure 4: Diagram of transitions between species
The processes included in the microstructure development model can be conveniently summarized in a state transition diagram such as the one shown in Figure 4. Here, arrows indicate a change of state with dissolution, nucleation, and collision (leading to bimolecular chemical reaction) being the possible transition processes. Diffusing species are indicated with an asterisk and the dependence of a process on a variable is indicated by function notation. For example f([CH*]) indicates that a process depends on the diffusing CH species concentration. Periodic boundary conditions were used in all our models -- no special conditions were needed in our CA rules to deal with edge effects.
The computer-generated microstructure that develops when the model is executed to various degrees of hydration can subsequently be used to compute physical parameters such as the diffusion rates of ions through the pore structure, thereby providing insights into the long term behaviour of concrete (Bentz and Garboczi 1991a, Garboczi and Bentz 1992). It should be noted that CAtype rules can just as easily be developed to model the later degradation of cement paste microstructure (Bentz and Garboczi 1992) as they can be used to model its original formation during hydration, which is the primary focus of the present paper.