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4. Cellular Automation Simulations

Figure 8. Reactions used in current version of cement hydration model.

The reactions provided in Figure 8 are implemented as a series of cellular automata-like rules (Click here for a tutorial on cellular automata) which operate on the original three-dimensional representation of cement particles in water. Rules are provided for the dissolution of solid material, the diffusion of the generated diffusing species, and the reactions of diffusing species with each other and with solid phases. These rules are summarized in the state transition diagram provided in Figure 9 and illustrated in the movie hydrating tricalcium silicate paste . Their implementation is as follows.

Figure 9. State transition diagram for cement hydration model.

For dissolution, first, an initial scan is made through all pixels (elements) present in the 3-D microstructure, to identify all pixels which are in contact with pore space as illustrated for a 2-D image in Figure 10. Thus, any solid pixels which have one or more immediate (+/-1 in the x, y, or z directions) neighbors which are classified as porosity are eligible for dissolution. In addition, each solid phase is characterized by two dissolution parameters, a solubility flag and a dissolution probability. The solubility flag indicates if a given phase is currently soluble during the hydration process, with a value of 1 indicating that the phase is soluble. The initial cement phases are always soluble during the hydration process. Conversely, some phases, like ettringite, are initially insoluble but become soluble during the hydration (e.g., when the gypsum is nearly consumed). The calcium hydroxide is made to be soluble to allow Ostwald ripening of the smaller calcium hydroxide crystals into larger ones.

Figure 10. Illustration of various steps in the digital-image-based cement hydration model showing, from left to right, initial cement particles in water (black), highlighting (white) of all cement particle surfaces in contact with water, generation of one-pixel diffusing species, and hydrated images at ~32% and 76% hydration, respectively (C₃S is red, C₂S is blue, C₃A is bright green, C₄AF is orange, gypsum is pale green, C-S-H is yellow, CH is dark blue, and aluminate hydration products (ettringite, monosulfoaluminate, and C₃AH₆) are green).

The second parameter indicates the relative probability of a phase dissolving when a pixel containing that phase "steps" into pore space. This is included in the model to allow the cement minerals to react at different rates as has been observed experimentally [23]. In the current model configuration, the C₃A and C₃S are assigned relatively high dissolution probabilities (> 0.8) while the C₄AF and C₂S are given relatively low ones (< 0.2). Since the latter two phases generally account for less than 30% of the cement, variations in their dissolution probabilities will not have a major effect on the results of the hydration model, although recent research has shown that enhancing the dissolution of C₄AF can significantly influence the properties of cements with substantial C₄AF fractions [27].

In a second pass through the microstructure, all identified surface pixels are allowed to take a one step random walk. If the step lands the pixel in porosity, the phase comprising the pixel is currently soluble, and dissolution is determined to be probable (by comparing a U[0,1) random number to the dissolution probablility), the dissolution is allowed and one or more diffusing species (as identified in Figure 9) are generated as illustrated in Figure 10. The diffusing species do not represent individual ions, but rather a collection of ions in the appropriate stoichiometric proportions to correspond to one pixel in volume of one of the reactant or product phases. If the dissolution is not allowed, the surface pixel simply remains as its current solid phase, but may dissolve later in the hydration. The locations of all diffusing species are stored in a linked list data structure which can expand and contract dynamically during execution to optimize memory usage. In this way, unlike in previous versions of the NIST model [8, 9], diffusing species may remain in solution from one dissolution phase to the next. Previously, all diffusing species were reacted before a new dissolution step was performed.

The generated diffusing species execute random walks in the available pore space, until they react according to the rules provided in Figure 9. For each diffusing species, the reaction rules included in the present version of the 3-D cement hydration model are as follows:
diffusing CSH: when a diffusing CSH species collides with either solid C₃S or C₂S or previously deposited CSH, it is converted into solid CSH with a probability of 1.
diffusing CH: for each diffusion step, a random number is generated to determine if nucleation of a new CH crystal is probable; if so, the diffusing CH is converted into solid CH at its present location. In addition, if a diffusing CH collides with solid CH, it is converted into solid CH with a probability of 1.
diffusing FH₃ : for each diffusion step, a random number is generated to determine if nucleation of a new FH₃ crystal is probable; if so, the diffusing FH₃ is converted into solid FH₃ at its present location. In addition, if a diffusing FH₃ collides with solid FH₃, it is converted into solid FH₃ with a probability of 1.
diffusing gypsum: the diffusing gypsum can only react by collision with some other species in the microstructure. If it collides with solid CSH, it can be absorbed as long as the previously absorbed gypsum is less than some constant (e.g., 0.01) multiplied by the number of solid CSH pixels currently present in the system. If it collides with either solid or diffusing C₃A, ettringite is formed. If it collides with solid C₄AF, ettringite, CH, and FH₃ are formed to maintain the appropriate volume stoichiometry as shown in Figure 8.
diffusing ettringite: when diffusing ettringite is created, it also reacts only by collision with other species. If it collides with solid or diffusing C₃A, monosulfoaluminate is formed. If it collides with solid C₄AF, monosulfoaluminate, CH, and FH₃ are formed. Finally, if it collides with solid ettringite, there is a small probability that it is converted back into solid ettringite. This latter rule is provided to avoid the possibility of a large buildup of diffusing ettringite in the microstructure.
diffusing C₃A: If nucleation is probable or the diffusing C₃A collides with solid C₃AH₆ and precipitation is probable, solid C₃AH₆ is formed. If it collides with diffusing gypsum, ettringite is formed. If it collides with diffusing or solid ettringite, monosulfoaluminate is formed.

For C₃AH₆, CH, and FH₃, the probability of nucleation, P(nuc), of diffusing species is a function of C(i), the current number of diffusing species, and is governed by an equation of the form:

In general, the hydration reaction products are allowed to grow with a completely random morphology. An exception to this is ettringite, where an attempt is made to grow the solid ettringite as needle-like structures by evaluating the surface curvature using a pixel counting algorithm [21, 22]. When new ettringite is forming, an attempt is made to maximize the number of non-ettringite pixels in contact with the new ettringite pixel. This will naturally result in the formation of maximum surface area (or needle-like) ettringite structures.

Prior to each dissolution, the 3-D microstructure is scanned to determine the number of pixels of each phase currently present in the system. From these volumes, chemical shrinkage and heat of hydration can be calculated. The chemical shrinkage is calculated by determining the amount of water consumed by reaction (based on the values in Table 1) in comparison to the volume of capillary porosity remaining in the microstructure. For low w/c ratio systems, all of the water may be consumed while some capillary porosity remains. When external water is available, simulations can be performed assuming that all pores remain water-filled during the full course of the hydration. However, the model can also consider hydration under sealed conditions [29]. To do this, prior to each new dissolution cycle, the volume of remaining porosity is compared to the volume of remaining water. The difference in these two values is converted into a number of porosity pixels to be converted into empty porosity. In an attempt to simulate the actual physical process of pore emptying, the 3-D microstructure is then scanned to identify the largest pore regions (using different size spherical templates), which are then emptied sequentially from largest to smallest until the correct number of empty pore pixels have been created. In this way, the effects of self-desiccation on the evolving hydration process can be simulated to compare to the experimental measurements of degree of hydration vs. time for equivalent real systems.

The heat of hydration can be based on the heats of formation given in Table 1, or the tabulated enthalpy values for each of the four major phases as listed in Table 2. For the model, degree of hydration is calculated as the mass of cementitious material which has reacted divided by the starting mass of cement.

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