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A realistic molecular model of cement hydrates

Edited by Zdeněk P. Bažant, Northwestern University, Evanston, IL, and approved July 21, 2009 (received for review February 27, 2009)
Abstract
Despite decades of studies of calciumsilicatehydrate (CSH), the structurally complex binder phase of concrete, the interplay between chemical composition and density remains essentially unexplored. Together these characteristics of CSH define and modulate the physical and mechanical properties of this “liquid stone” gel phase. With the recent determination of the calcium/silicon (C/S = 1.7) ratio and the density of the CSH particle (2.6 g/cm^{3}) by neutron scattering measurements, there is new urgency to the challenge of explaining these essential properties. Here we propose a molecular model of CSH based on a bottomup atomistic simulation approach that considers only the chemical specificity of the system as the overriding constraint. By allowing for short silica chains distributed as monomers, dimers, and pentamers, this CSH archetype of a molecular description of interacting CaO, SiO_{2}, and H_{2}O units provides not only realistic values of the C/S ratio and the density computed by grand canonical Monte Carlo simulation of water adsorption at 300 K. The model, with a chemical composition of (CaO)_{1.65}(SiO_{2})(H_{2}O)_{1.75}, also predicts other essential structural features and fundamental physical properties amenable to experimental validation, which suggest that the CSH gel structure includes both glasslike shortrange order and crystalline features of the mineral tobermorite. Additionally, we probe the mechanical stiffness, strength, and hydrolytic shear response of our molecular model, as compared to experimentally measured properties of CSH. The latter results illustrate the prospect of treating cement on equal footing with metals and ceramics in the current application of mechanismbased models and multiscale simulations to study inelastic deformation and cracking.
By mixing water and cement, a complex hydrated oxide called calciumsilicatehydrate (C–S–H) precipitates as nanoscale clusters of particles (1). Much of our knowledge of CSH has been obtained from structural comparisons with crystalline calcium silicate hydrates, based on HFW Taylor's postulate that real CSH was a structurally imperfect layered hybrid of two natural mineral analogs (2): tobermorite of 14Å interlayer spacing [Ca_{5}Si_{6}O_{16}(OH)_{2}.7H_{2}0, (3)] and jennite [Ca_{9}(Si_{6}O_{18})(OH)_{6}.8H_{2}O (4)]. While this suggestion is plausible in morphological terms, this model is incompatible with two basic characteristics of real CSH; specifically the calciumtosilicon ratio (C/S) and the density. Recently, smallangle neutron scattering measurements have fixed the C/S ratio at 1.7 and the density at 2.6 g/cm^{3} (1), values that clearly cannot be obtained from either tobermorite (C/S = 0.83, 2.18 g/cm^{3}) or jennite (C/S = 1.5 and 2.27 g/cm^{3}). From the standpoint of constructing a molecular model of CSH, this means that these crystalline minerals are not strict structural analogs. Here we adopt the perspective that the chemical composition of CSH is the most essential property in formulating a realistic molecular description. We show that once the C/S ratio is described correctly, a number of characteristic structural features and physical properties follow naturally in atomistic simulations. We view the present model and its subsequent refinements as enabling a bottomup perspective on the broad science of cementitious materials and the innovative engineering of concrete. Manipulation of such a testable model should ultimately allow the establishment of the critical links between nanoscale microstructure and macroscale behavior, which requires fundamental understanding of the effects of confined water in the context of creep and durability.
Model Construction.
One of the key issues in designing a realistic CSH molecular model is the calciumtosilicon ratio (C/S). Indeed, confirming earlier measurements by Groves et al. (6) and Richardson and Groves (7), energy dispersive Xray analyses of CSH 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 watertocement (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 twofold classification to clarify CSH chemistry (5). This classification references socalled tobermorite/jennite (T/J) models on one hand and tobermoritecalcium 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 CSH as an assembly of tobermorite regions followed by jennite domains. While the T/CH class was found to be relevant for hydrated KOHactivated 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 CSH 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 ^{29}Si 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), Q_{1} the fraction of tetrahedra at the end of a chain (a silicate dimer would have two Q_{1}), Q_{2} the fraction of tetrahedra in the middle of a chain (a silicate pentamer would have three Q_{2}).
To construct a molecular model of CSH that has a C/S ratio consistent with smallangle 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 Ca_{6}Si_{6}O_{16}. 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 SiO_{2} (neutral) groups in silica tetrahedra guided by the NMR results, Q_{0} ∼ 10%, Q_{1} ∼ 67%, and Q_{2} ∼ 23% (10), obtaining a defective CSH structure that has a distribution of Q_{0} = 13%, Q_{1} = 67%, and Q_{2} = 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 coreshell potential model at 0 K to find a density of 2.12 g/cm^{3}, 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/cm^{3}, which is close to the experimental value provided by neutron scattering of 2.6 g/cm^{3} (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 CSH 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/cm^{3}; molecular dynamics simulations under constant pressure and temperature (NPTMD) 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 14Å 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/cm^{3}, with the adsorbed water molecules being in an ultrahigh 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 CSH is thus found to be (CaO)_{1.65}(SiO_{2})(H_{2}0)_{1.75}, which is in reasonable agreement with the neutron scattering experiments (CaO)_{1.7}(SiO_{2})(H_{2}0)_{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).
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 Xray absorption fine structure (EXAFS) spectroscopy signals measuring shortrange order around Ca atoms (Fig. 2A), longer range correlations revealed in Xray 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 shortrange 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 interatomic distances with same relative intensities that allow discriminating CSH against all other calciosilicate crystalline solids (12). The fact that the first peak in the experimental EXAFS signal is broader than that obtained in simulation suggests that real CSH may exhibit an even larger volume fraction of shortrange structural disorder. The xray diffractogram of our CSH model (Fig. 2B) clearly indicates the reduced degree of crystallinity as compared with tobermorite. The suggestion that our CSH 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 CSH model with that of a true nonporous calciosilicate glass at room temperature with the same C/S ratio and a density of 2.34 g/cm^{3} (Fig. 2 C and D): the structure of the second peaks in the g(r) for SiO and CaO pairs in tobermorite show characteristic structural features that are absent for both the calciumsilicate glass and our CSH model. For the sake of consistency, the Caglass and crystalline tobermorite simulations were carried out with the same empirical potential model (see Methods), with the Caglass potential obtained by following the method given in (13). We may interpret this comparison to indicate that CSH 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.
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 CSH 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 CO_{3}^{2−} 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 CSH. Since our model only represents the CSH, 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 SiO_{4}^{−} tetrahedra. The band in the range 660–670 cm^{−1} is due to SiOSi bending while that at 810 and 970 cm^{−1} is attributed to SiO 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 HOH bending, while that at 3,300 cm^{−1} is attributed to OH stretching. Interestingly, these band positions are lower than that of bulk liquid (bulk liquid water is also present in the mesopores of CSH) and are characteristic of a strong confining environment, as is also suggested from neutron quasielastic experiments (16).
Additionally, we consider mechanical properties of the model CSH, 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 CSH 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 micromechanicsbased 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 CSH particle indentation modulus, m_{s} = E_{s}/(1 − v^{2}_{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 CSH model, using for the elasticity constants the ReussVoigtHill average (m_{s} = 65 GPa, Table 1) calculated from the full elasticity tensor (Table 2) to compare with the elasticity properties of randomly oriented CSH 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 planewave GGADFT calculations (22). This comparison underscores the importance of considering a realistic CSH structure for the prediction of elasticity and strength properties of cementbased materials. Moreover, combining the elastic properties determined from our CSH 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 CSH particles. Fig. 2F compares the prediction of two micromechanics models along with nanoindentation results; one is a porous bicontinuous matrix approach captured by the socalled MoriTanaka scheme (20), and the other a granular approach captured by the selfconsistent scheme (19). From this comparison, we observe first that the granular approach better describes the experimental data over the entire domain of CSH particle packing fractions. Second, both approaches give acceptable predictions at larger packing fractions. That is, at the micrometerscale, MoriTanaka 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.
StrengthControlling Shear Localization.
Probing atomiclevel 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 CSH presents an opportunity to initiate similar investigations of cementitious materials, thereby opening up a class of microstructures with unique chemistryrich 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 strainrate dependence. We have simulated the stressstrain behavior of our CSH 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 stressstrain curves of the CSH 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 sawtooth behavior has been observed in deformation simulation of metallic glasses (25), glassy polymers (26) as well as nanoindentationinduced 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 CSH 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 stressstrain response in Fig. 3A we can conclude that the shear response of the CSH 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 CSH 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 fivefold reduction in the compressive strength of αquartz (28). Simulations of watersilica interactions have identified three distinct competing mechanisms in the water attack on the siloxane bridging bond, SiOSi (29). Since we do not allow dissociation in our interatomic potential description, further considerations of this mechanism will be facilitated by firstprinciples MD studies of this CSH model.
In summary, our study provides an atomisticlevel structural model for CSH, developed from a bottomup 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. CSH 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 CSH 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 mechanismbased 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 CSH 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.
Methods
From Energy Minimization to Elastic Properties Using Empirical Interatomic Potentials.
All calculations were carried out with the GULP code (28–30). These calculations were performed using a set of empirical but transferable interatomic potentials calibrated on quartz and CaO compounds. Anions (here oxygens in tobermorite layers and in water molecules) were modeled as polarizable species using the coreshell model. These transferable empirical interatomic potentials based on the use of the formal electric charge for each interacting species, have successfully reproduced the structure and properties of many oxides (33) including silicates (34–36) and phyllosilicates (37, 38) [see reference (39) for liquid water]. The set of potentials used include twobody and threebody analytical functions that allow calculating the energy between pairs and triplets of atoms. They depend on the choice of some parameters that can be advantageously calibrated using ab initio calculations in some simple cases. Note that the calculation of electrostatic interactions between pairs of ions is carried out using the Ewald summation scheme. The advantage of such an approach compared to ab initio quantum mechanical methods is that one can compute for large systems with low symmetry, not only structural but also thermodynamic and elastic properties [from the elastic tensor using both Voigt and Reuss equations for bulk, shear, Young's, plane stress (or indentation) modulus and Poisson's ratio]. Periodic boundary conditions were used for all directions of space. Reference (40) gives all details and equations on such calculations in the case of lizardite, a layered magnesium silicate of central importance in tectonophysics. Energy minimization for finding an equilibrium structure consists in tracking stationary points that correspond to a minimum energy gradient with positive energy curvature (i.e., finding a set of atomic positions that minimizes system energy and give a Hessian operator with positive eigenvalues only). A phonon spectrum calculation at the center of the Brillouin zone is then used as a final validation from which one obtains the list of lattice vibration frequencies that should be all positive except the first three that should be zero (unit cell translational invariance). Such minimization procedure gives a zero temperature solution. In this work, all degrees of freedom were considered including atomic positions, unit cell dimensions and angles. All potential mathematical form and parameters are given in the SI. The same approach but at fixed volume was used to perform deformation and shear calculations: all atomic degrees of freedom were allowed to relax for a given cell volume modifying one cell parameter at a time. The derivative of the system energy versus the incremental modified cell parameter (the interlayer distance) is used to calculate the cohesive pressure that can be compared with experimental nanoindentation data (17, 18). Numerical shear experiments were carried out with the same strategy by modifying at each step a cell angle parameter, then followed by molecular dynamics simulation in the Canonical Ensemble (i.e., at constant volume, see below).
The Grand Canonical Monte Carlo Technique for Water Adsorption.
In this work, we first produced an anhydrous version of the CSH substrate and subsequently calculated the maximum amount of water that can be accommodated in its pore voids. For this purpose, we used the Grand Canonical Monte Carlo (GCMC) simulation technique that is wellsuited to study adsorption/desorption processes. GCMC simulations involve the determination of the properties of a system at constant volume V (the pore with the adsorbed phase) in equilibrium with a fictitious infinite reservoir of particles imposing its chemical potential μ and its temperature T (41, 42). For different values of μ, the absolute adsorption isotherm can be determined as an ensemble average of the adsorbed atom numbers in the system versus the pressure of the gas reservoir P (the latter can be obtained from the chemical potential according to the equation of state for the bulk gas). The adsorption and desorption processes can be respectively simulated by increasing or decreasing the chemical potential of the reservoir; the final configuration of a simulation is the initial state for the next point. Periodic boundary condition was used in all directions of space as for the energy minimization procedure. An equal number of attempts for translation, rotation, creation or destruction of molecules was chosen. The isotherm was calculated for 300 K. Acknowledging the very restricted available space in between tobermoritic layers, one should not expect capillary condensation to occur in contrast to larger pore systems such as vycor (43). In our case, the adsorption/desorption process is expected to be close to that observed for microporous zeolite as far as water adsorption is concerned (44). We did not calculate the entire water adsorption/desorption isotherm, but performed a single GCMC simulation with the water chemical potential fixed to a value that corresponds to the bulk liquid phase with a density of 1 g/cm^{3} at room temperature (μ = 0 eV for the used water potential model; note that the wateroxygen atomic shells were switchedoff and water molecules and substrate were treated as rigid bodies during the GCMC procedure). A complete study of the state of water confined in our realistic model of CSH including thermodynamics and dynamics properties will be presented in a separate publication.
NVT and NPT Molecular Dynamics.
Room temperature relaxation and deformation were carried out using molecular dynamics simulation in the NPT and NVT statistical ensembles integrating motion equations with the leapfrog Verlet algorithm with NoséHoover thermostat and pressiostat (with the default GULP parameters for each of these constraints). When relaxing in NPT conditions (with zero external pressure) the resulting system from energy minimization, the simulation considers finite temperature entropic effects.
XRay Diffraction.
Xray diffraction patterns were calculated with the CRYSTALDIFFRACT code as part of the CRYSTALMAKER package www.crystalmaker.com/crystaldiffract/ at a wave length of 1.54 Å and an apparatus aperture broadening of 0.4 Å^{−1}.
Reproducibility of Results.
Details for the calculation including the interatomic potential functions and related parameters, as well as details on the numerical strategy and simulation techniques used, and the final atomistic model (and relevant coordinates) are provided in the SI to this manuscript. A file containing all of the atomic coordinates is available upon request to RJM Pellenq (pellenq{at}mit.edu) or FJ Ulm (ulm{at}mit.edu).
Acknowledgments
We thank Cimpor Corporation, Portugal, for support of the “Liquid Stone” project, enabled through the MITPortugal program. We thank Dr. P. Ganster and Prof A. Baronnet (CINaM, CNRSMarseille Université, France) for making available to us respectively the calciosilicate glass atomistic model and the transmission electron microscopic image of CSH.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: ulm{at}mit.edu

Author contributions: R.J.M.P., K.J.V.V., S.Y., and F.J.U. designed research; R.J.M.P. and R.S. performed research; R.J.M.P., A.K., and M.J.B. contributed new reagents/analytic tools; R.J.M.P., A.K., R.S., S.Y., and F.J.U. analyzed data; and R.J.M.P., K.J.V.V., M.J.B., S.Y., and F.J.U. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0902180106/DCSupplemental.

Freely available online through the PNAS open access option.
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