2.4 Transition Rules for Early­Time Behaviour and Simulating Induction

The original program described above is somewhat unphysical in that there is an outer execution loop in which pixels on the surface of a cement grain are dissolved into solution and an inner execution loop in which the dissolved species diffuse by random walk statistics until they have all reacted (Figure 3). The outer loop repeats until no more surfaces of C₃S are exposed to pore space and no more dissolution can take place. The nesting of the diffusion­aggregation loop within the dissolution loop makes the program computationally simple and efficient and is an effective way of reaching the final microstructural state of the system. From the point of view of modelling the temporal evolution of cement hydration, however, the nested loops make the problem of relating model time to real time more difficult. Thus, the program was modified to contain only a single loop so that pixels dissolve, diffuse and aggregate simultaneously in every step. The relative rate of dissolution versus diffusion was controlled by assigning appropriate relative probabilities of dissolution and diffusion to different species.

This program was applied to simulate the induction period found in setting cements by adding a pair of rules, one to create a species which had the effect of blocking the aggregation of diffusing C-S-H onto the cement particles and another to transform the blocking species into a non­blocking one. The rule for C-S-H was also modified to test for the presence of such a blocking species in its neighbourhood; if it is present the diffusing C-S-H does not aggregate. The rate of transformation to the non­blocking form was a sigmoidal function, dependent upon the amounts of other reacting species in solution. This has the effect that as the concentration of the blocking species suddenly decreases, the aggregation restarts quickly; the new rate is faster because of the higher concentration of diffusing C-S-H present in the pore space and the greater surface area of aggregation sites in the form of partially dissolved particles. Though the exact mechanism which slows hydration and precipitation of C-S-H is not known we included these rules to produce the desired overall effect. A candidate for this blocking species is the initial amorphous gel form of ettringite and its subsequent transformation into a crystalline form. Evidence for this mechanism, which comes inter alia from ATR/FTIR spectroscopy, is discussed by Billingham and Coveney (1993). The state transition diagram for these rules is given in Figure 5: diffusing gypsum (GYP*) reacts with diffusing aluminate (C₃A*) forming ettringite (ETR*) which aggregates (ETR(gel)), and then transforms into crystals (ETR(crystalline)). ETR(gel) is the species providing the blocking mechanism. While controversy still surrounds the mechanisms responsible for the induction period, this simple example illustrates how such behaviour can be easily realised in a CA model.

A complementary approach to modelling the induction period that has recently been investigated is a 'chemical clock' model based on an inhibited auto­catalytic nucleation mechanism (Billingham and Coveney 1993). This work draws on a small set of physicochemical transformations involving the conversion of gelatinous ettringite into the crystalline form. Rate equations are developed which describe the kinetics of these processes. One step within the scheme, in particular, involves auto-catalysis: conversion of the gel to the crystalline form is accelerated by the presence of crystals of ettringite. Analysis of the corresponding non­linear differential equations shows that an appropriate selection of rate constants for the reactions can produce non­linear growth in species concentrations, of the kind observed in cement hydration. The CA approach is inherently semi­empirical in that the transition rules and probabilities are chosen in order to create the observed behaviour, while in the chemical clock approach certain aspects of the system can be derived from well established a priori laws of chemical kinetics. However, the simple clock model (Billingham and Coveney 1993) has the disadvantage of requiring the assumption of spatial homogeneity, while the CA incorporates the inhomogenous nature of the evolving microstructure.

Figure 5: Diagram of additional transitions for induction