Most models for cement hydration kinetics start at the particle level and derive rate equations for an individual particle as a function of its radius, for example [1, 5-7]. This derivation is then often extended to considering a complete realistic PSD. Here, a different approach is taken conceptually. Since all hydration products must form in the available water-filled porosity, it is first assumed that the hydration rate is simply proportional to the volume fraction of this water-filled porosity (first order "physical" kinetics):
where k1 is analogous to a first-order rate constant and depends on the specific cement composition, PSD, curing temperature, etc. In this first model, the hydration rate is dependent only on the space available for the deposition of hydration products and not even on the amount of cement present in the hydrating system. It is equivalent to assuming that the local probability of dissolution (or precipitation) of a cement compound is directly proportional to the local (and global) water-filled porosity. Substituting equation (1) into equation (4), the result is easily integrated and solved with the boundary condition that α(0)=0 to yield:
The minimum function has been added to equation (5) to assure that the degree of hydration of the cement does not exceed 1, a physical impossibility. Although derived from a different perspective, equation (5) is similar in form to kinetics equations often derived considering nucleation and growth kinetics for cement hydration (so called Avrami behavior) [8]. In the "Avrami" form, the t (time) term in equation (5) is often raised to a power n and an induction time ti is often subtracted from the t term. Substituting the various parameter values into equation (5), the dependence of hydration rates on w/c can be easily computed. Equation (5) can be applied to both saturated and sealed curing conditions, using values for CS of 0.0 and 0.07, respectively.
At the next level of complexity, one can make an analogy to a bimolecular-type reaction for cement hydration, where the hydration rate has a first order dependence on both the available water-filled porosity and the available unhydrated cement volume. Based on an analogy to the sol-gel process in producing ceramics, Wojcik et al. have presented such an equation where the hydration rate of the binder components is a function of both the binder component and water concentrations [9]. Here, it is assumed that the local probability of reaction of the cement is linearly proportional to both the available water-filled porosity and the available volume of unhydrated cement. In terms of these phase fractions, one thus has:
Substituting equations (1) and (3) into equation (6), the result can still be solved analytically as [10]:
with p=ρcem(w/c)/(ƒexp+ ρcemCS) and R = k2(ƒexp + ρcemCS) / [1+ρcem(w/c)] 2. For the special case where p=1, application of L´Hopital´s rule gives the solution as [10]: α(t) = Rt/(Rt + 1).
Equation (7) should be directly applicable to saturated systems, with CS=0. For sealed curing conditions, an additional complication may be that the creation of empty porosity may effectively "inactivate" a portion of the unhydrated cement (surfaces) [11], as well as reducing the amount of water-filled capillary porosity (e.g., CS=0.07). To a first approximation, the fraction of active unhydrated cement could be estimated as the total fraction of unhydrated cement multiplied by the ratio of water-filled to total porosity. In this case, equation (6) would be extended to:
When equations (1), (2), and (3) are substituted into equation (8), upon reduction one obtains an equation of the form:
While equation (9) is solvable in an analytical fashion only for very limited cases such as C=1, it can be solved numerically using a graphical symbolic manipulation package.
The models developed above (equations (5), (7), and (9)) will be evaluated in comparison to data sets available from the literature and measured recently at NIST. Two types of comparisons will be presented, a direct comparison of the α(t) vs. t behavior for a set of two w/c values for a specific cement and a comparison of the predicted dependence of degree of hydration on w/c to the experimentally observed trends for a variety of cements. These comparisons will be conducted for both saturated and sealed curing conditions.