A reaction model for cement solidification: Evolving the C–S–H packing density at the micrometer-scale
Introduction
Powers and Brownyard, two renowned cement scientists, described cement paste as “an amorphous mass enclosing microcrystalline Ca(OH)2 and unhydrated residues of the original cement grains” (Powers and Brownyard, 1946). Since then, significant progress has been made to understand the structure of this “amorphous mass”—the calcium–silicate–hydrates (C–S–H)—that lends the gel its mechanical performance. At its most elementary scale, C–S–H has been found to precipitate as nanometer-sized units (Jennings, 2000, Pellenq, Kushima, Shahsavari, Van Vliet, Buehler, Yip, Ulm, 2009) with a molecular structure that exhibits both glass-like short-range order and regions of crystallinity (Plassard et al., 2005). At larger length scales, the treatment of cement paste as a colloidal dispersion has elucidated the hierachical organization of these sub-units and the impact of their assembly on the material’s elasticity (Ioannidou, Krakowiak, Bauchy, Hoover, Masoero, Yip, Ulm, Levitz, Pellenq, Del Gado, 2016b, Masoero, Del Gado, Pellenq, Ulm, Yip, 2012). For instance, the statistical frequency distribution of the local modulus, which was constructed by nanoindentation measurements, showed that C–S–H particles organize themselves into low-density (LD) and high-density (HD) assemblies reminiscent of loose-packed and close-packed granular morphologies (Constantinides, Ulm, 2007, Jennings, Dalgleish, Pratt, 1981, Tennis, Jennings, 2000). Later, molecular dynamics simulations of coarse-grained C–S–H nanoparticles recreated the mesoscale structure of cement paste at 100s of nanometers, deciphering the role of the physicochemical environment on the gelation dynamics of the hydrates (Bonnaud, Labbez, Miura, Suzuki, Miyamoto, Hatakeyama, Miyamoto, Van Vliet, 2016, Ioannidou, Kanduč, Li, Frenkel, Dobnikar, Del Gado, 2016a, Ioannidou, Krakowiak, Bauchy, Hoover, Masoero, Yip, Ulm, Levitz, Pellenq, Del Gado, 2016b). Here, it was found that the C–S–H nanograins precipitate into an out-of-equilibrium structure, whose solid packing fraction is the principal determinant of the local elastic modulus (Masoero et al., 2012). Nonetheless, modeling via such potential-of-mean-force approaches is not optimal when the number of discrete units needed to describe a heterogeneous structure becomes computationally cumbersome. But to make a prediction of bulk behavior, the long-range distributions of the solid packing fraction, anhydrous clinker residuals, and macro-porosity need to be resolved. This begs the question: How do we connect the evolution of the continuum phases of hydrating cement paste to the physics that underlie its particle-based nature?
Revolutionary to materials science, Allen–Cahn reaction kinetics have successfully described a plethora of phase transformation phenomena: dendritic growth in undercooled metals, ion intercalation in Lithium battery materials, and hydrate formation on methane bubbles, to name a few (Allen, Cahn, 1979, Chen, 2002, Singh, Ceder, Bazant, 2008, Svandal, Kvamme, Grànàsy, Pusztai, Buanes, Hove, 2006, Warren, Boettinger, 1995). It is thus surprising that this generalized phase-field approach has eluded application to cement solidification, which exhibits nucleation and growth kinetics similar to many solute-solvent systems. Moreover, because the Allen–Cahn reaction model (ACR) relies principally on the formulation of a system’s free energy (for cement hydrates this includes, e.g., bulk chemical free energy, interfacial energy, and/or electrostatic energy), it is an excellent vehicle to upscale information gleaned from out-of-equilibrium particle simulations. Hence, the objective of the current study is to present an ACR model that reproduces the cement paste hydration kinetics and microstructure as function of the degree of hydration. Though several simplifying assumptions are made about the chemistry, namely
- •
the product phase does not distinguish between C–S–H and CH (or other secondary hydration products),
- •
and the diffusion of all aqueous species is represented by a single ionic concentration,
To further elaborate on the usefulness of phase-field approaches to model cement solidification, we briefly introduce its mathematical construction: a free energy G is defined by a set of field variables and their gradients such that (Chen, 2002)Herein, V is the system’s volume, gh is the bulk free energy density with local energy minima that represent stable phases, and the gradient-dependent terms introduce energy penalties that force the field variables to vary smoothly in space. Using the expression for G, ACR models use variational calculus to evolve the local field variables and push the system toward its energy minimum. Take for instance the C–S–H packing density ϕ, which evolves heterogeneously in the porespace of the cement paste. By defining a free energy density with minima for the liquid phase and the solid phase (see Fig. 1), the gradient energy term κϕ|∇ϕ|2/2 produces interfaces that interpolate between regions of densely packed C–S–H (high ϕ) and the dilute solution (sparse in nanoparticles; low ϕ). Because can be made a function of the local chemical environment, where the energy minimum for the solid phase is shifted according to observations from experiments, we can reproduce the measured distributions of ϕ. As will be shown, HD-C–S–H regions are predicted in the phenograins of the clinker phase, where a hydration shell increases the local ionic concentration, and LD-C–S–H regions develop in the spaces between the source particles. As a result, this continuum approach offers a vehicle to upscale the elasticity of cement paste by adopting relations between the local packing density and the local stiffness found from nanoindentation experiments and particle-based simulations.
Although a multitude of cement hydration models exist (e.g., the vector-based μic model Bishnoi and Scrivener, 2009, and cellular automata models CEMHYD3D Bentz, 1997 and HydratiCA Bullard et al., 2010), most focus on how the chemical environment influences the formation of different product phases and achieving agreement with observed hydration kinetics. The present work will demonstrate the ability of phase-field approaches to capture the features of cement paste solidification and the microstructural information salient to its mechanical characterization. By exploring the parameter space we predict the influence of temperature, clinker coarseness and the water-to-cement ratio on the early-age hydration kinetics, morphology of the microstructure, and the homogenized elasticity. Thus, our manuscript is organized in the following manner: Section 2 constructs the landscape of our local energy density by connecting expressions from solution chemistry to the nanoparticulate behavior observed in experiments and modeling the source phase as a moving boundary that dissolves in function of the ion concentration of the pore solution. Next, Section 3 relates the free energy variation due to changes in the local field variables to the reaction kinetics of the dissolution and nucleation and growth processes. The results of our simulations are presented in Section 4.1, where the evolution of the system’s free energy is vetted against typical hydration heat curves, and the structural evolution of the cement paste is matched to observations. Lastly, we link the local packing density to an experimentally validated relation for the local elastic modulus and upscale the cement paste’s mechanical properties. Here, the HD-, and LD-C–S–H signatures of the cement paste are recovered and it is found that the space filling procedure of the precipitation reaction plays a critical role in the development of its Young’s modulus. The paper concludes with a discussion of the results and an outlook of studies to come.
Section snippets
Energy landscape for cement hydration
In this section, we define the energy landscape used to model the precipitation of C–S–H product and dissolution of the source phase. We begin by revisiting Fig. 1(a) and (b) and rewriting Eq. (1) aswhere we postulate that the local energy density depends on three fields:
- 1.
The ionic concentration of the pore solution c,
- 2.
the packing density of C–S–H nanoparticles ϕ,
- 3.
and an order parameter ψ that identifies the source phase and measures the space available to
Kinetic equations for cement solidification
In this section, we connect Gibb’s energy expression for our representative elementary volume (REV; Eq. (1)) to the kinetic equations that evolve the fields according to laws of non-equilibrium thermodynamics. Before detailing the form of the reaction rates, we stipulate the mass conversion rates of our chemical system and the relation between the set of differential equations that advance our field variables.
Simulations and results
In the succeeding paragraphs, we present the numerical results of several two-dimensional simulations of cement hydration. Running trials at different temperatures and source particle coarseness, we compare the rate of change in Gibb’s free energy to experimental hydration heat curves, calculate the pore and solid chord-length distributions at intermediate degrees of hydration, and predict the evolution of the elasticity field.
Conclusions
This paper presented a reactive phase-field model for the precipitation dynamics of C–S–H in cement-based systems. Using only a few parameters that define Gibb’s free energy densities of the C–S–H nanoparticles and the source phase, and rate constants that define the reaction and diffusion timescales, we provide a tool to analyze the hydration kinetics of cement paste and a vehicle to upscale its mechanics. Though the model makes no distinction between the various product phases (C–S–H, CH, or
Acknowledgments
Thomas Petersen would like to thank Prof. Martin Bazant for valuable discussions on chemical kinetics and non-equilibrium thermodynamic processes. The numerical simulations were facilitated by the Finite Volume PDE Solver using Python (FiPy) developed at the National Institute of Standards and Technology. Financial support was generously provided by the National Science Foundation Graduate Research Fellowship, and the Concrete Sustainability Hub at the Massachusetts Institute of Technology
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