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Three-Dimensional Cement Hydration Model

The cement hydration model was originally developed in two dimensions 12 to operate directly on SEM images such as those in Fig. 2. Recently, the model was extended to three dimensions; additions were made to determine model heat of hydration and chemical shrinkage; and the densities/molar volumes of C-S-H, ettringite, and iron hydroxide were adjusted to better model the experimental data generated for actual cements. 14 To begin, one must decide the phases and reactions to consider in the cement hydration model. Table V provides a list of the phases included in the present version of the three-dimensional cement hydration model, with their densities, molar volumes 35 , 36 and heats of formation.37, 38 Figure 5 summarizes the reactions included in the current version of the model, as modified from those provided previously.12 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 V. For C-S-H, the density and molar volume given in Table V were obtained by calibrating measured chemical shrinkage to that predicted by the model,14 because the values given in the literature36 for these properties predict no chemical shrinkage for the hydration of C3S and C2 S, in contrast to the experimentally observed behavior. 39 For both C 3S and C2S hydration, the values given in Table V result in chemical shrinkages of ~6.7 g of H2O/(100 g of cement) at complete hydration, in reasonable agreement with the value of 5.3 g of H2O/(100 g of cement) measured by Powers39 for C3S after 28 d hydration.

The reactions provided in Fig. 5 are implemented as a series of cellular automata-like rules (see Panel C for a brief introduction to cellular-automata) that operate on the original three-dimensional representation of cement particles in water. Rules are provided for the dissolution of solid material, diffusion of the generated diffusing species, and reactions of diffusing species with each other and with solid phases. These rules are summarized in the state transition diagram provided in Fig. 6. Their implementation is as follows.

For dissolution, first, an initial scan is made through all pixels (elements) present in the three-dimensional microstructure, to identify all pixels that are in contact with pore space. Thus, any solid pixels that have one or more immediate (± 1 in the x, y, or z 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 always are soluble during the hydration process. Conversely, some phases, such as ettringite, are initially insoluble but become soluble during the hydration (e.g., when the gypsum is nearly consumed). The calcium hydroxide also is 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.35 In the current model configuration, C3A and C3S are assigned dissolution probabilities 5-8 times greater than those given C4AF and C2S. Because the latter two phases generally account for <30% of the cement, variations in their dissolution probabilities do 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 C4 AF fractions. 46

Silicate Reactions
C3S + 5.3H C1.7SH4 + 1.3CH
1   1.34   1.521  0.61
C2S + 4.3H C1.7SH4 + 0.3CH
1   1.49   2.077   0.191
Aluminate and Ferrite Reactions
C3A + 6H C3AH 6
1   1.21   1.69
C3A + 3CH2 + 26H C6 A 3H 32
0.4   1   2.1   3.3
2C3A + C6A3H 32 + 4H 3C4 AH12
0.2424   1   0.098   1.278
C4AF + 3CH2 + 30H C6A3H 32 + CH + FH3
0.575   1   2.426   3.3   0.15   0.31
2C4AF + C6A3H 32 + 12H 3C4AH12 + 2CH + 2FH3
0.348   1   0.294   1.278   0.09   0.19
C4AF + 10H C3AH6 + CH + FH3
1   1.41   1.17   0.26   0.545

Fig. 5. Cement model reactions (numbers below reactions indicate volume stoichiometry).

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 uniform random number between 0 and 1 to the dissolution probability), the dissolution is allowed, and one or more diffusing species are generated, as indicated in Fig. 6. These diffusing species are not individual ions but, rather, represent a collection of ions that occupy one pixel (~ 1 µm3) unit of volume, consistent with a cellular-automata approach to modeling microstructure development.12 If the dissolution is not allowed, the surface pixel simply remains as its current solid phase, but it may dissolve later in the hydration. The locations of all diffusing species are stored in a linked list data structure that can expand and contract dynamically during execution to optimize memory usage.47 In this way, unlike in previous versions of the NIST hydration model, 12, 13 diffusing species may remain in solution from one dissolution step to the next. Previously, all diffusing species were reacted before a new dissolution step was performed.

Fig. 6. State transition diagram for three-dimensional cement hydration model. Arrow patterns denote the collision of two species to form a hydration product. f ([X]) denotes the nucleation or dissolution probability is a function of concentration or volume fraction of phase x . Asterisk (*) indicates diffusing species. ETTR is ettringite, MONO is monosulfate, pozz is pozzolanic materials (silica fume, etc.), Gyp is gypsum, col is collision, nuc is nucleation, and dis is dissolution.

The generated diffusing species execute random walks in the available pore space, until they react according to the rules provided in Fig. 6. For each diffusing species, the reaction rules included in the present version of the three-dimensional cement hydration model are as follows.

(1) Diffusing C-S-H: When a diffusing C-S-H species collides with either solid C3S or C2 S or previously deposited C-S-H, it is converted into solid C-S-H with a probability of 1.

(2) 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.

(3) 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.

(4) Diffusing gypsum: The diffusing gypsum can react only by collision with some other species in the microstructure. If it collides with solid C-S-H, it can be absorbed if 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. 5.

(5) 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 C4 AF, 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.

(6) DiffusingC3A: If nucleation is probable or the diffusing C3A collides with solid C3AH 6 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 and ettringite is currently soluble, monosulfoaluminate is formed.

For C3AH6 CH, and FH3, the probability of nucleation (Pnuc,) of diffusing species is governed by an equation of the form

Pnuc (Ci ) = A i [ l - exp(- [Ci ] / [ Bi ] )](5)

where Ci, is the current number of diffusing species i, and Ai , and Bi, constants that control the number and formation rate of crystals. 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. 48

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 needlelike structures by evaluating the surface curvature using a pixel-counting algorithm.33, 34 When new ettrin gite is forming, an attempt is made to maximize the number of non-ettringite pixels in contact with the new ettringite pixel. This naturally results in the formation of needlelike ettringite structures.

Prior to each dissolution, the three-dimensional 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 V and the reactions in Fig. 5) 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. For those experiments conducted in this study for which hydration was executed under saturated conditions, simulations were performed assuming that external water always was available to fill the pores emptied by the chemical shrinkage. Thus, all porosity remained water filled during the complete execution of the hydration model. However, the model has been further extended to consider hydration under sealed conditions. 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 three-dimensional microstructure is then scanned to identify the largest water-filled pore regions (using different-sized spherical templates), which then are emptied sequentially from largest to smallest until the correct number of empty pore pixels has been created. In this way, the effects of self-dessication on the evolving hydration process can be simulated to compare to the experimental measurements of degree of hydration versus time.

The heat of hydration can be computed based on the heats of formation given in Table V or the tabulated enthalpy values for each of the four major phases as listed in Table VI. For the model, degree of hydration is calculated as the mass of cementitious material that has reacted divided by the starting mass of cement, with the conversion from a volume basis being performed using the densities of the starting materials given in Table V.

Table V. Physical Properties of Cementitious Materials
Compound nameCompound
(Mg/m3 )
Molar volume
(cm3 /mol)
Heat of formation
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 CH2 2.32 74.2 -2022.6
Calcium silicate hydrate C1.7SH 4 2.12 108 -3283.
Calcium hydroxide CH 2.24 33.1 -986.1
Ettringite C6A3H 32 1.7 735. -17539.
Monosulfate C4AH12 1.99 313. -8778.
Hydrogarnet C3AH6 2.52 150. -5548.
Iron hydroxide FH3 3.0 69.8 -823.9
Water H 1.0 18.0 -285.83

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