Model Construction.

One of the key issues in designing a realistic C-S-H molecular model is the calcium-to-silicon ratio (C/S). Indeed, confirming earlier measurements by Groves et al. (6) and Richardson and Groves (7), energy dispersive X-ray analyses of C-S-H in hardened Portland cement pastes aged 1 day to 3.5 years reveal a composition variation spanning C/S from approximately 1.2 to 2.3 with a mean value of 1.7; this variation also depends on the water-to-cement (W/C) mass ratio at which cement is hydrated (5). Given the shortfalls of the natural analogs, tobermorite and jennite, to meet this compositional constraint, Richardson proposed a two-fold classification to clarify C-S-H chemistry (5). This classification references so-called tobermorite/jennite (T/J) models on one hand and tobermorite-calcium hydroxyle (T/CH) models on the other hand. The T/CH class considers models that are solid solutions of tobermorite layers sandwiching calcium hydroxide, hence providing a means to achieve a higher C/S ratio than the one of tobermorite. The T/J class considers C-S-H as an assembly of tobermorite regions followed by jennite domains. While the T/CH class was found to be relevant for hydrated KOH-activated metakaolin Portland cement, more common water activated Portland cement pastes can be only partly described by the T/J or the T/CH approaches. A realistic model for C-S-H that predicts a realistic C/S ratio thus remains a center piece of any model construction. Furthermore, quantitative information on the fractions of Si present in silicate tetrahedra with different connectivities is provided by ²⁹Si nuclear magnetic resonance (NMR) (8, 9). Such studies have established that the dimer is the most predominant of all silicate species, with the linear pentamer as the second most abundant. Tetrahedral coordination measured by NMR is expressed in terms of the Q_n factor, denoting the fractional chemical shift of a silicon atom bound to n bridging oxygens. Thus Q_o is the fraction of a single tetrahedron (a silicate monomer), Q₁ the fraction of tetrahedra at the end of a chain (a silicate dimer would have two Q₁), Q₂ the fraction of tetrahedra in the middle of a chain (a silicate pentamer would have three Q₂).

To construct a molecular model of C-S-H that has a C/S ratio consistent with small-angle neutron scattering measurements, we begin with a monoclinic periodic computational cell of dry tobermorite of interlayer spacing of approximately 11 Å, with 4, 2, and 1 units along axes a, b, and c with a unit cell chemical formula of Ca₆Si₆O₁₆. The cell contains two CaO layers lying in the ab plane, and eight silicate chains (2 chains each on top and bottom side of each layer). In this initial configuration the C/S ratio is 1.0; note that with this value of C/S, the layers are not electroneutral and there are interlayer calcium ions to maintain electroneutrality. We then remove SiO₂ (neutral) groups in silica tetrahedra guided by the NMR results, Q₀ ∼ 10%, Q₁ ∼ 67%, and Q₂ ∼ 23% (10), obtaining a defective C-S-H structure that has a distribution of Q₀ = 13%, Q₁ = 67%, and Q₂ = 20%, with a C/S ratio of 1.65. There are likely other ways to produce defected silicate chains that are consistent with the NMR constraints; we are currently investigating a combinatorial approach. Our procedure was carried out without the presence of any OH groups, so that a reasonable C/S ratio could be obtained under the electroneutrality constraint (11). After the silicate chain modification, we relax the dry cell using the core-shell potential model at 0 K to find a density of 2.12 g/cm³, the interlayer distance having shifted slightly to 11.3 Å (see Methods). At this stage, one can observe significant distortion of the layer structure. We next performed Grand Canonical Monte Carlo simulation of water adsorption in the above defected unit cell, coupling the system to an external reservoir at a chemical potential corresponding to liquid water at 300 K (see Methods). At equilibrium, the adsorbed water increases the density to 2.56 g/cm³, which is close to the experimental value provided by neutron scattering of 2.6 g/cm³ (1). We regard this agreement, which was an outcome rather than an input or constraint to this model, to be a significant consistency check on our model development procedure. Further relaxation at 0 K of this hydrated C-S-H model yields a slight increase of the interlayer spacing from 11.3 Å to 11.9 Å, reducing the density by 4% to 2.45 g/cm³; molecular dynamics simulations under constant pressure and temperature (NPT-MD) give the same result (see Methods). In the final model, one can observe water molecules adsorbed in cavities inside the calcium oxide layers as a result of relaxation. While the amount of water is similar to that present in the interlayer region of 14-Å tobermorite, the water in our model is adsorbed not only in the interlayer regions, but also in the distorted intralayer regions around the silica monomers. As a consequence of water adsorption the density increases to 2.56 g/cm³, with the adsorbed water molecules being in an ultra-high confining environment. This water may be regarded as part of the structure, reminiscent of structural water or bound water in cement chemistry terminology. The overall chemical composition of the computational model of the hydrated C-S-H is thus found to be (CaO)_1.65(SiO₂)(H₂0)_1.75, which is in reasonable agreement with the neutron scattering experiments (CaO)_1.7(SiO₂)(H₂0)_1.8 (1). The molecular configuration of this model is shown in Fig. 1; and all cell parameters and atomic positions of our model are provided in the SI (see also Methods).

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Fig. 1.

(A) TEM image of clusters of C-S-H (courtesy of A. Baronnet, CINaM, CNRS and Marseille Université, France), the inset is a TEM image of tobermorite 14 Å from (45). (B) the molecular model of C-S-H: the blue and white spheres are oxygen and hydrogen atoms of water molecules, respectively; the green and gray spheres are inter and intra-layer calcium ions, respectively; yellow and red sticks are silicon and oxygen atoms in silica tetrahedra.

Model Validation Against Experiments.

We validated the structure of our model by calculating several experimentally accessible properties. The results, summarized in Fig. 2, consist of extended X-ray absorption fine structure (EXAFS) spectroscopy signals measuring short-range order around Ca atoms (Fig. 2A), longer range correlations revealed in X-ray diffraction intensity (Fig. 2B), vibrational density of states measured by infrared spectroscopy (Fig. 2E), and nanoindentation measurements of elastic properties (Fig. 2F). The various tests provide strong evidence of a short-range structural disorder, the hallmark of a glassy phase. For instance, the simulated and experimental Ca total pair distribution functions as measured in EXASF (Fig. 2A), agree well, showing peaks at the same inter-atomic distances with same relative intensities that allow discriminating C-S-H against all other calcio-silicate crystalline solids (12). The fact that the first peak in the experimental EXAFS signal is broader than that obtained in simulation suggests that real C-S-H may exhibit an even larger volume fraction of short-range structural disorder. The x-ray diffractogram of our C-S-H model (Fig. 2B) clearly indicates the reduced degree of crystallinity as compared with tobermorite. The suggestion that our C-S-H model can be seen as a glassy phase at short length scales is confirmed by a comparison of the partial pair distribution functions g(r) of our C-S-H model with that of a true non-porous calcio-silicate glass at room temperature with the same C/S ratio and a density of 2.34 g/cm³ (Fig. 2 C and D): the structure of the second peaks in the g(r) for Si-O and Ca-O pairs in tobermorite show characteristic structural features that are absent for both the calcium-silicate glass and our C-S-H model. For the sake of consistency, the Ca-glass and crystalline tobermorite simulations were carried out with the same empirical potential model (see Methods), with the Ca-glass potential obtained by following the method given in (13). We may interpret this comparison to indicate that C-S-H should be considered as a glass on the short range of distances associated with the distorted intralayer structure, while retaining some layered crystal features at longer range of distances associated with the interlayer spacing.

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Fig. 2.

Characterization and validation of molecular model of C-S-H. (A) EXAFS Ca-radial distribution function, exp (12). (phase shift of + 0.3 Å, background subtracted); (B) XRD data, exp (46). for C-S-H and (47) for tobermorite 14 Å; (C) SiO radial distribution function, comparison with that for a non porous calcio-silicate with Ca/Si = 1.6 and with that for tobermorite 14 Å; (D) idem for the CaO pair; (E) infrared data, exp (15); (F) nanoindentation data, exp (18), see text.

The comparison between simulation and experimental infrared spectra (Fig. 2E), allows further characterization of our model. Note that calculated infrared intensities were obtained from the relaxed model C-S-H structure by performing a numerical integration over the sampled phonon modes (14). All of the experimental bands are present in the calculated spectrum but the ones in the range 1,200–1,500 cm⁻¹; the absence of these bands is not surprising, as these correspond to experimental carbonation effects (Q3 silicate stretching and vibrational mode of CO₃²⁻ ions) that can be avoided in a computational model. The first low frequency band in the range 200–350 cm⁻¹ corresponds to the vibration of Ca polyhedra including those of other hydration products, namely Ca(OH)₂ grains, that nucleate in the mesopores of the real cementitious material, in addition to C-S-H. Since our model only represents the C-S-H, this Ca polyhedra band is present in the model, but not of as high intensity as in experiment, more closely corresponding to that measured for tobermorite (15). The band in the domain 440–450 cm⁻¹ can be attributed to deformations of SiO₄⁻ tetrahedra. The band in the range 660–670 cm⁻¹ is due to Si-O-Si bending while that at 810 and 970 cm⁻¹ is attributed to Si-O stretching in silica tetrahedra (Q1 and Q2 environments, respectively). Finally, infrared analysis provides some information on the nature of the water molecules: the band at approximately 1,600 cm⁻¹ is characteristic of water H-O-H bending, while that at 3,300 cm⁻¹ is attributed to O-H stretching. Interestingly, these band positions are lower than that of bulk liquid (bulk liquid water is also present in the mesopores of C-S-H) and are characteristic of a strong confining environment, as is also suggested from neutron quasi-elastic experiments (16).

Additionally, we consider mechanical properties of the model C-S-H, computed by stretching the cell dimensions to calculate elastic constants (Tables 1 and 2) as well as the rupture strength. For a quantitative comparison, we use nanoindentation measurements that probe the stiffness and hardness of nanoscale clusters of randomly oriented C-S-H particles at the micrometer scale (Fig. 1), which have been characterized by isotropic stiffness and strength particle properties and particle packing density (17, 18). Then, using micromechanics-based scaling relations pertaining to granular (19) and porous materials (20), of the indentation elastic modulus, M = m_s Π_M (η, v), and indentation hardness, H = h_s Π_H (η, α) (15, 16), we correct for the effect of interparticle porosity, via particle packing density, η, and determine the C-S-H particle indentation modulus, m_s = E_s/(1 − v²_s), and the particle hardness, h_s, (where E_s is the Young's elastic modulus, v_s is the Poisson's ratio, α is the friction coefficient). The experimental values are in excellent agreement with the ones obtained from our computational C-S-H model, using for the elasticity constants the Reuss-Voigt-Hill average (m_s = 65 GPa, Table 1) calculated from the full elasticity tensor (Table 2) to compare with the elasticity properties of randomly oriented C-S-H particles; and for hardness the maximum negative isotropic pressure (h_s = 3 GPa, Table 1) that precedes rupture of the simulation cell perpendicular to the layer plane. These values are somewhat higher than those for 14 Å tobermorite and jennite, for which m_s = 56 GPa obtained from classical (21) and ab initio plane-wave GGA-DFT calculations (22). This comparison underscores the importance of considering a realistic C-S-H structure for the prediction of elasticity and strength properties of cement-based materials. Moreover, combining the elastic properties determined from our C-S-H model with some micromechanics models (19, 20) with no adjustable parameters, we can also probe the texture and extent of anisotropic structures within cement paste at micrometer length scales of randomly oriented C-S-H particles. Fig. 2F compares the prediction of two micromechanics models along with nano-indentation results; one is a porous bicontinuous matrix approach captured by the so-called Mori-Tanaka scheme (20), and the other a granular approach captured by the self-consistent scheme (19). From this comparison, we observe first that the granular approach better describes the experimental data over the entire domain of C-S-H particle packing fractions. Second, both approaches give acceptable predictions at larger packing fractions. That is, at the micrometer-scale, Mori-Tanaka and self consistent micromechanics approaches, parameterized only with nanoscale derived elasticity constants, indicate that cement paste can be conceptualized as a cohesive granular material rather than a porous bicontinuous matrix.

Strength-Controlling Shear Localization.

Probing atomic-level mechanisms to gain insights into structural deformation and failure at larger length scales is currently a central issue in the development of nanomechanics of hard crystalline materials (23, 24). The formulation of a molecular model of C-S-H presents an opportunity to initiate similar investigations of cementitious materials, thereby opening up a class of microstructures with unique chemistry-rich and spatially heterogeneous characteristics. A fundamental question common to all systems is the nucleation and evolution of a “unit process” in the constitutive response to tensile and shear loading, and the effects of specimen size, temperature, and strain-rate dependence. We have simulated the stress-strain behavior of our C-S-H model in affine shear deformation (strain controlled) after first relaxing the computational cell using MD at 300 K, under constant NVT ensemble conditions. A series of shear strains in increments of 0.005 is imposed; after each increment the atomic configuration is relaxed and the shear stress determined from the virial expression. Fig. 3 shows the shear stress-strain curves of the C-S-H model, as well as a “dry” version of this model in which all water molecules have been removed. The responses in both cases are a sequence of elastic loading under incremental strain, interspersed with discrete stress drops reminiscent of strain localization events. This type of intermittent or saw-tooth behavior has been observed in deformation simulation of metallic glasses (25), glassy polymers (26) as well as nanoindentation-induced dislocation nucleation (27). Further investigations of the mechanisms governing stress relaxation in this model are ongoing; here, we will discuss only the first two stress drops in each response curve. These occur at stresses between 2.5 to 3 GPa in the dry sample and approximately 1 GPa in the hydrated sample; these values are lower than the ideal shear strength, or about 10% of the shear elastic modulus, due to the defected microstructure of this C-S-H phase. Moreover, it is not surprising that the presence of water lowers the strength. On the other hand, inspection of Fig. 3A indicates a significant difference between the two response curves. If we take the elastic loading portion of the response after the first drop and extrapolate back to zero stress, we find a significant “residual” strain of approximately 0.1 in the dry sample, an indication of irreversible deformation associated with the first drop. In contrast, in the hydrated sample unloading after the first drop indicates essentially no residual strain, which suggests the deformation to be largely elastic. To observe the atomic displacements that correspond to these stress drops, we display in Fig. 3B the largest individual displacements associated with the four stress drops in Fig. 3A. It is clear that in the dry sample local strains are distributed across the cell, with a slight degree of strain concentration within the layers rather than in the interlayer region, especially at stage (d). It is also quite clear that in the hydrated sample the strains are localized entirely in the interlayer region and are mostly associated with displacements of water molecules. Combining these observations with the characteristics of the stress-strain response in Fig. 3A we can conclude that the shear response of the C-S-H model is strain localization in the interlayer region; this localization occurs in the form of sliding, ostensibly facilitated by the lubricating action of the water molecules. In the absence of water, strain localization appears to manifest as individual events of irreversible deformation. The present results demonstrate the potential to gain insights into the effects of water on the deformation behavior of the C-S-H particle. This problem bears some analogy to the phenomenon of “hydrolytic weakening” in other crystalline and glassy silicates, where it is believed that hydration causes more than a five-fold reduction in the compressive strength of α-quartz (28). Simulations of water-silica interactions have identified three distinct competing mechanisms in the water attack on the siloxane bridging bond, Si-O-Si (29). Since we do not allow dissociation in our interatomic potential description, further considerations of this mechanism will be facilitated by first-principles MD studies of this C-S-H model.

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Fig. 3.

(A) Relationships between the shear stress τ_zx and the strain γ_zx for the C-S-H model with (solid line) and without (dashed line) water molecules. (B) Atom displacements at the stress drops (1)–(4) in panel (A). Only atoms with displacement larger than 0.5 Å are shown, and the arrows indicate the displacements. Dashed lines correspond to the C-S-H layers. (C) Magnified view of regions (A) and (B) that are marked as red boxes in panel (B).

In summary, our study provides an atomistic-level structural model for C-S-H, developed from a bottom-up perspective and validated against several experimental analyses of structure and properties. This model could enable many opportunities for future developments focused on understanding fundamental deformation mechanisms, diffusive properties, electrical properties and many other characteristic material parameters. C-S-H is the primary hydration product and binding phase of concrete, the synthetic material currently produced in volumes larger than any other material on Earth. The insight gained by deformation simulations based on our C-S-H model (Fig. 3) illustrates the prospect of treating cementitious composites on equal footing with less structurally complex materials. The knowledge of unit deformation mechanisms (in analogy to dislocations, shear bands, etc.) in concrete could enable the development of mechanism-based models and multiscale simulation methods to study inelastic deformation, flow, and fracture to complement and improve current empirical strength models of this complex yet ubiquitous material. The existence of an atomistic level model of the C-S-H nanostructure is crucial to enable advances in our understanding of how specific structural arrangements at the nanoscale relate to resulting material properties. In a broader context, the approach illustrated here to carry out an experimentally validated structural prediction of a complex heteronanocomposite could be applied to many other systems such as colloids, hydrated polymer or protein gels, as well as polymer nanocomposites.