3-D Cement Hydration Model

The cement hydration model was originally developed in two dimensions [23] to operate directly on SEM images such as those in Figs. 2 and 3. Here, the model has been extended to three dimensions, additions made to determine model heat of hydration and chemical shrinkage, and several coefficients adjusted to better model the experimental data generated for actual cements. To begin, one must decide the phases and reactions to consider in the cement hydration model. Table 6 provides a list of the phases included in the present version of the three-dimensional cement hydration model, along with their densities, molar volumes, [24, 25] and heats of formation [26,27]. Figure 6 summarizes the reactions included in the current version of the model, as modified from those provided in [23]. The volume stoichiometries indicated below each reaction have been calculated based on the molar stoichiometries of the reactions and the compound molar volumes tabulated in Table 6.

Table 6: Physical Properties of Cementitious Materials

Compound Name	Compound Formula	Density (Mg / m 3 )	Molar volume (cm 3 / mole)	Heat of formation ( k J / mole)
Tricalcium silicate	C3S	3.21	71.	-2927.82
Dicalcium silicate	C2S	3.28	52.	-2311.6
Tricalcium aluminate	C3A	3.03	89.1	-3587.8
Tetracalcium aluminoferrite	C4AF	3.73	128	-5090.3
Gypsum	$C\bar{S}H_2$	2.32	74.2	-2022.6
Calcium silicate hydrate, C-S-H	C1.7SH4	2.12	108	-3283.
Calcium hydroxide	CH	2.24	33.1	-986.1
Ettringite	$C_6A\bar{S_3}H_{32}$	1.7	735.	-17539.
Monosulfate	$C_4A\bar{S}H_{12}$	1.99	313.	-8778.
Hydrogarnet	C3AH6	2.52	150.	-5548.
Iron hydroxide	FH3	3.0	69.8	-823.9

  

Figure 6: Cement model reactions - numbers below reactions indicate volume stoichiometries.

The reactions provided in Fig. 6 are implemented as a series of cellular automata-like rules 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 Fig. 7. Their implementation is as follows.

  

Figure 7: State transition diagram for 3-D Cement Hydration Model. Arrow patterns denote the collision of two species to form a hydration product. f([X]) denotes that nucleation or dissolution probability is a function of concentration or volume fraction of phase X.

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. Thus, any solid pixels which have one or more (+ 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 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. 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 [24]. 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 [28].

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 are generated as indicated in Fig. 7. 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 [3, 23], 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 Fig. 7. For each diffusing species, the reaction rules included in the present version of the 3-D cement hydration model are as follows:

diffusing C-S-H: when a diffusing C-S-H species collides with either solid C₃S or C₂S or previously deposited C-S-H, it is converted into solid C-S-H 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 C-S-H, 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 C-S-H 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 Fig. 6.

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 governed by an equation of the form:

$$P_{nuc}(C_i) = A_i*(1-e^{\frac{-[C_i]}{[B_i]}})$$ (5)

where C_i is the current number of diffusing species i and A_i and B_i are constants which control the number and rate at which crystals are formed in the microstructure. This results in the effect that few new crystals are formed late in the hydration when the "concentrations" of diffusing species are reduced relative to their initial values, in agreement with experimental observations [29].

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 6 ) 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. However, in the execution of the model, it is assumed that saturation is always maintained to simulate the experimental measurements performed in this study, where additional water is always present on the top surface of the hydrating cement paste. Research to extend the model to hydration under sealed conditions, with the creation of internal voids due to chemical shrinkage, is ongoing. The heat of hydration can be based on the heats of formation given in Table 6 , or the tabulated enthalpy values for each of the four major phases as listed in Table 7. For the model, degree of hydration is calculated as the mass of cementitious material which has reacted divided by the starting mass of cement.

Table 7: Enthalpy of Complete Hydration for Major Phases of Cement

Phase	Enthalpy (kJ/kg phase)	Source
C3S	517	[8]a
C2S	262	[8]
C3A	1144	[8]
C4AF	725	[26]b

^a w/c =  0.4 and T = 21ºC

^b w/c =  0.5 and T = 20ºC

Comments on Assumptions of