Nucleation and growth models for hydration of cement
Introduction
Extensive research has been devoted to the hydration reaction of ordinary Portland cement and its component oxides (particularly, C3S and C2S).2 This work has been discussed in comprehensive books and reviews [e.g., [1], [2], [3], [4]]. It is widely accepted that the early stage of hydration, including the time of initial and final setting of a paste, involves a mechanism of nucleation and growth of hydration product. This idea is supported by the good performance of analytical models based on that idea [e.g., [5], [6], [7], [8], [9]], and the results of simulations based on fundamental reaction models [10], [11]. Recently, Thomas [9] pointed out that the conventional Johnson–Mehl–Avrami–Kolmogorov (JMAK) model [12], [13], [14], which assumes that nuclei are randomly distributed throughout the reaction volume, is not appropriate for describing the hydration reaction, where nucleation occurs primarily on the surface of the cement particles. Instead, he applied a model that was developed by Cahn [15] to describe transformations in polycrystalline materials, where nucleation occurs on grain boundaries. This model (hereafter called BNG = boundary nucleation and growth) was found to give an improved fit to calorimetric data for hydration of C3S [9], under the assumptions that the rates of nucleation (per unit area of exposed clinker surface) and growth are constant throughout the process. However, computer simulations indicate that the evolution of the solution composition is such that a burst of nucleation is expected in the first few minutes of hydration, after which a stable supersaturation is maintained [10], [11]. Under these conditions, growth would be expected to occur from a fixed number of nuclei.
The purpose of this paper is to extend the BNG model to allow for anisotropic growth from a constant number of nuclei or with a constant rate of nucleation. As noted by Taylor [Ref. 1, p. 223], the observed kinetics is the sum of the rates for all the phases undergoing reaction, so we will consider how the model should be applied to Portland cement, where the hydration of several phases is concurrent.
The theoretical development is presented in the next section. In Section 2.1, we show how the volume fraction of product, which is calculated from the nucleation and growth model, is related to the degree of hydration and the chemical shrinkage. In 2.2 BNG model — constant nucleation rate, 2.3 BNG model — constant number of nuclei, we will derive the transformation rate when nucleation occurs at a constant rate or from a constant number of nuclei, respectively. Since nucleation is assumed to occur only on the portion of the surface of the clinker that is not yet covered with hydration products, the rate of production of nuclei drops to zero as the hydrates spread across the surfaces of the cement particles. Therefore, although the rate of nucleation per unit area of exposed surface is constant, the net rate of nucleation drops, and the transformation rates tend to converge for systems having a constant number of nuclei or a constant nucleation rate. Section 2.4 describes the situation where multiple phases are reacting. In Section 3, the model is compared to chemical shrinkage data for Class H cement, and it is demonstrated that the results are slightly better when growth is assumed to occur from a fixed number of nuclei. The implications of these results are discussed in Section 4.
Section snippets
Relating volume fraction, degree of hydration, and chemical shrinkage
Nucleation and growth models predict the volume fraction, X, of the system that has been converted to the product phase; however, the quantity that is directly measured may be the heat released [9], water consumed [6], [7], [8], or chemical shrinkage [16]. Therefore, it is useful to provide an explicit connection between X and the degree of reaction, α, and the chemical shrinkage, s. Suppose that a volume of water, Vw, is mixed with a volume of cement, Vc, to make a paste. The water/cement mass
Comparison to chemical shrinkage data
Measurements of the chemical shrinkage of Class H cement were presented in Ref. [16] and analyzed using the BNG model in the form used by Thomas [9]. In the following, we fit the same data to the model from Section 2.3, where the number of nuclei is fixed. This is more consistent with the predictions of simulations by Bullard [10], which show that the supersaturation is high enough to cause nucleation only during the first few minutes of hydration of C3S. In reality, there should be a
Comparison to calorimetric data
Bishnoi and Scrivener [24] performed a series of calorimetric measurements on the hydration of alite on powders with narrow particle size distributions. Fits to these data should yield similar values of growth rate and nucleation density (allowing for variations in mineralogy and defect concentration in particles of different sizes), since the size fractions were separated from the same source. In Fig. 13, we show the results of simultaneously fitting Eqs. (42), (43) to those data over the time
Discussion
The most striking thing about the fits using the two models is that they are so similar, as illustrated by Fig. 6. Assuming that growth occurs from a constant number of nuclei results in slightly better fits at temperatures from 10 to 40 °C; assuming a constant nucleation rate yields slightly better results at 60 °C, but the data are noisier at that temperature. The temperature dependences of the constants are identical, as demonstrated by Fig. 7. This means that it is impractical to determine
Conclusions
The analysis presented here generalizes the BNG model to allow for anisotropic growth with a constant rate of nucleation or a constant number of nuclei. The latter case is more consistent with the results of detailed simulations [11]. Fitting the model to chemical shrinkage data for Class H cement [16], [37] provides a very good match to the cumulative and derivative curves, and the nucleation density has the same order of magnitude found in simulations [33]. Nevertheless, the application of
Acknowledgment
We are indebted to Jeffrey Bullard of NIST for helpful discussions of the hydration process, and to Shashank Bishnoi for providing calorimetric data (originally obtained by Dr. Mercedes Costoya) for hydration of alite and for enlightening discussions about the μic model.
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Present address: University of Michigan, Civil & Env. Eng., 2314 GG Brown Building, Ann Arbor, MI 48105, USA.