1 An Ideal Solid Solution Model for C-S-H 2 3 Jeffrey W. Bullarda and George W. Schererb 4 5 a National Institute of Standards and Technology, Gaithersburg, MD 20878, USA 6 b Princeton University, Eng. Quad. E-319, Princeton, NJ 08544, USA 7 8 Abstract 9 A model for an ideal solid solution, developed by Nourtier-Mazauric et al. [Oil & Gas 10 Sci. Tech. Rev. IFP, 60 [2] (2005) 401], is applied to calcium-silicate-hydrate (C-S-H). Fitting 11 the model to solubility data reported in the literature for C-S-H yields reasonable values for the 12 compositions of the end-members of the solid solution and for their equilibrium constants. This 13 model will be useful in models of hydration kinetics of tricalcium silicate because it is easier to 14 implement than other solid solution models, it clearly identifies the driving force for growth of 15 the most favorable C-S-H composition, and it still allows the model to accurately capture 16 variations in C-S-H composition as the aqueous solution changes significantly at early hydration 17 times. 18 19 Keywords: solubility, cement, equilibrium constant, solid solution 20 21 22 23 24 25 1 26 1. Introduction 27 The purpose of this paper is to present a thermodynamic model that can be used to predict 28 the thermodynamic driving force for growth of calcium silicate hydrate (C-S-H)† during 29 hydration of tricalcium silicate. Extensive study of the solubility of C-S-H, which has been 30 reviewed by several authors1,2,3,4,5, has yielded abundant data that have been interpreted in terms 31 of solid solution models including 2 or 3 types of C-S-H with calcium/silicon atomic ratios 32 ranging from about 0.8 (similar to tobermorite) to 2. Indeed, Richardson6 recently showed that 33 the structure of C-S-H with C/S ratios < 1.5 can be understood as a stacking of sheets with 34 infinite siloxane chains interspersed with sheets having only siloxane dimers. These would be 35 true solid solutions of components with C/S = 2/3 and 3/2. During hydration of cement, the 36 composition of the aqueous solution will change considerably, and the composition of the solid 37 precipitate will change accordingly, reaching C/S ≈ 1.7. Lothenbach and Nonat7 argue that the 38 observed range of C/S ratios can be achieved by removing bridging silicate tetrahedra from a 39 tobermorite-like structure and replacing them with interlayer calcium ions, while preserving 40 structural continuity. This is consistent with the idea that C-S-H is a solid solution containing 41 layers with differing silicate chain lengths, together with interlayer calcium (possibly in the form 42 of layers with the structure of calcium hydroxide). A simple and rigorous model of growth and 43 dissolution rates of an ideal solid solution has been proposed by Nourtier-Mazauric et al.,8 which 44 predicts the driving force for, and composition of, the solid that will grow as a function of the 45 aqueous solution composition. If it could be shown to apply to C-S-H, such a model would be 46 particularly useful in reaction-transport models in cementitious systems because it is † We use conventional cement chemistry notation, in which C = CaO, S = SiO2, H = H2O; the hyphens in C-S-H indicate that it is not a stoichiometric compound, whereas CSH4 represents CaO•SiO2•4H2O. 2 47 considerably more computationally efficient than non-ideal solid solution models,3,5,9 which may 48 even suffer from numerical instabilities.2 Besides this advantage, it is instructive to examine how 49 well an ideal solution model can describe solubility data of C-S-H in comparison to non-ideal 50 solid solution models that require an enthalpy of mixing and therefore more fitting parameters 51 than does the current model. The primary weakness of this simpler ideal solid solution model is 52 that it is not directly tied to the underlying molecular structure of the material, as some more 53 complex non-ideal models have striven to be.3 In particular, the current model does not require 54 any particular composition of the end members, such as one of the tobermorite or jennite 55 minerals that are often used for that purpose.3,5,9 Instead, as will be shown, the ideal solid 56 solution model can be fit to determine the compositions of either two or three end members that 57 provide the best agreement with published C-S-H solubility data. Section 3 shows that quite 58 good fits result from end member compositions that are nearly the same as those proposed using 59 some of the most recent thermodynamic and structural models,4,6 despite the fact that it assumes 60 zero enthalpy of mixing. Therefore, the current model provides a simple yet accurate and useful 61 new tool for predicting C-S-H composition for use in reaction transport models. 62 63 64 2. Theory 65 The theory8 applies to an ideal solid solution, which means that the free energy of the 66 solid solution differs from that of a physical mixture of the components only as a result of the 67 entropy of mixing. The absence of an enthalpy of mixing is obviously an approximation, but is 68 plausible if the components of the solution have similar crystal structures, as in the C-S-H 69 system. An obvious extension of the model would be to allow for a finite enthalpy of mixing, 3 70 while retaining an ideal entropy of mixing (i.e., a regular solution model). However, that adds 71 additional parameters and leads to an implicit equation for the composition of the product. Given 72 that excellent fits to experimental measurements are obtained with the ideal solution model, there 73 is no justification for use of the more complicated version. 74 As outlined in Appendix 1, the theory predicts that the composition of the solid that has 75 the lowest free energy, and is therefore the most favored to grow, is given by 76 βk 77 xk = N , k = 1,…, N (1) ∑ β j j=1 78 79 where xk is the most favored mole fraction of component k in a solid solution of N components, 80 and the saturation is βk = Qk/Kk, where Qk is the ion activity product of the aqueous solution and 81 Kk is the equilibrium constant for component k. Therefore, if the concentrations of the ions in 82 solution and the equilibrium constants for the components are known, then eq. (1) can be used to 83 predict the composition of the solid that precipitates. Appendix 2 shows that the driving force for 84 growth of that compound is the total saturation, βTot, which is the sum of the saturations of the 85 components: N 86 (2) βTot =∑ β j j=1 87 88 This equation has important implications. Suppose that the aqueous solution has a composition 89 identical to the equilibrium activity product for any one component, say k = 1 (so Q1 = K1), then 90 β1 = 1; this means that if the solid solution is in equilibrium with the liquid, in which case βTot = 4 91 1, then eq. (2) requires that the other components are absent (i.e., βk≠1 = 0). That is, the solid 92 solution consists of only one component when the activity product in the solution is identical to 93 the equilibrium product for that component. It is sometimes suggested that C-S-H with a high 94 C/S ratio could contain layers with the structure of CH, which would mean that CH could be 95 regarded as a component in the solid solution. This possibility is excluded by eq. (2), because it 96 indicates that C-S-H could not exist in a solution in equilibrium with CH; however, those phases 97 certainly can coexist. This raises two possibilities. Either C-S-H is actually an ideal solid solution 98 that does not contain CH as a component, or C-S-H is a non-ideal solid solution in which CH can 99 be inserted at the price of a positive enthalpy of mixing (see Appendix 1). In the following, we 100 will explore the application of the ideal model, so CH is excluded as a component of the solid 101 solution. Non-ideal solution models of C-S-H have been considered in the past5,9,10, but they are 102 considerably more complicated and not more accurate, at least in terms of mere fitting of 103 published solubility data, than the one presented here. 104 Suppose that C-S-H is an ideal solid solution of three components with C/S ratios of α1, 105 α2, and α3. The dissolution of each component can be written as 106 2+ − − 107 (CaO) ⋅(SiO2 )⋅(H2O) ! α k Ca + H3SiO4 +(2α k −1)OH +(n −α k −1)H2O , k=1,2,3 (3) α k n 108 109 so the activity product for component k is 110 α 2α −1 α 2α −1 2+ k − − k n−α k −1 2+ k − − k 111 Qk ={Ca } {H3SiO4 }{OH } {H2O} ≈ {Ca } {H3SiO4 }{OH } (4) 112 5 113 where the braces indicate activities; square brackets will be used for molar concentrations. In the 114 absence of significant alkali concentrations like those found in portland cement solutions, the 115 aqueous solutions of calcium and silicate are so dilute that the activity of water is very nearly 116 unity. Even in most portland cement solutions, the mole fraction of water exceeds 0.99, so 117 Raoult’s law will apply for water activity and dapproximating it as unity is still valid.