Abstract
Amorphous calcium carbonate is an important precursor for biomineralization in marine organisms. Key outstanding problems include understanding the structure of amorphous calcium carbonate and rationalizing its metastability as an amorphous phase. Here we report highquality atomistic models of amorphous calcium carbonate generated using stateoftheart interatomic potentials to help guide fits to Xray total scattering data. Exploiting a recently developed inversion approach, we extract from these models the effective Ca⋯Ca interaction potential governing the structure. This potential contains minima at two competing distances, corresponding to the two different ways that carbonate ions bridge Ca^{2+}ion pairs. We reveal an unexpected mapping to the LennardJones–Gauss model normally studied in the context of computational soft matter. The empirical model parameters for amorphous calcium carbonate take values known to promote structural complexity. We thus show that both the complex structure and its resilience to crystallization are actually encoded in the geometrically frustrated effective interactions between Ca^{2+} ions.
Main
Calcium carbonate is relatively unusual among simple inorganic salts in that it precipitates from aqueous solution in a metastable hydrated, amorphous form^{1}. Amorphous calcium carbonate (ACC)—with a nominal composition of CaCO_{3}⋅xH_{2}O (x ≈ 1)—can be stabilized for weeks by incorporating dopants such as Mg^{2+} or PO_{4}^{3−} (refs. ^{2,3}), or alternatively directed to crystallize into a number of different polymorphs by varying the pH or temperature^{4}. Nature exploits this complex phase behaviour in a variety of biomineralization processes to control the development of shells and other skeletal structures^{5,6}. Not only does the amorphous nature of biogenic ACC allow transformation to different crystalline CaCO_{3} polymorphs, but it helps organisms fashion largerscale hierarchical morphologies that are important in biomineral architectures^{7}. In seeking to develop bioinspired crystalengineering approaches for synthetic control over phase and morphology selection, there is an obvious need to understand why such a chemically simple system can exhibit such complex phase behaviour.
One domain in which a similar kind of phase complexity has been studied deeply from a theoretical perspective is that of softmatter systems governed by multiwell pair potentials. Whereas isotropic particles that interact via a simple singlewell potential (for example, LennardJones, LJ) selfassemble into structurally simple crystalline phases (for example, facecentred cubic), the inclusion of one or more additional energy minima to the interaction potential can drive remarkable complexity if the distances at which these minima occur lead to geometric frustration^{8,9,10,11}. An elegant example in two dimensions is that of quasicrystal selfassembly for specific parameters of the doublewell LennardJones–Gauss (LJG) potential^{8,12}. In three dimensions, the same interaction model can be tuned to stabilize complex crystals with enormous unit cells^{11}, and some combinations of the LJ and Gauss well positions even appear to frustrate crystallization altogether^{9}.
That competing length scales might be relevant to ACC is a point hinted at by the observation of two preferred Ca⋯Ca distances dominating mediumrange order in synthetic ACC^{13,14}. These two distances are directly evident in the experimental Xray pair distribution function (PDF) of ACC and are attributed to different bridging modes of the carbonate ion, which can connect a pair of Ca^{2+} ions either directly through one of the oxygen atoms (Ca–O–Ca pathway) or by inserting the carbonate ion fully between the cations (Ca–O–C–O–Ca pathway).
In this Article we explore the possibility that the structure of ACC is governed by effective interactions between Ca^{2+} ions that also reflect these two length scales, and that its wellknown complexity emerges because the distances involved are in competition with one another. Our approach begins by obtaining a highquality measure of the Capair correlation function in ACC. We do this by applying a hybrid reverse Monte Carlo (HRMC) approach^{15} to generate the first structural model of ACC that is simultaneously consistent with experiment and stable with respect to stateoftheart potentials. The Capair correlation function that emerges is then inverted using a recently developed algorithm^{16} to reveal the effective, carbonate/watermediated Ca⋯Ca interaction potential. We find that this potential is closely related to the LJG formalism, with empirical parameters that are known to frustrate crystallization. Monte Carlo (MC) simulations driven by our LJG model yield coarsegrained representations of ACC that capture key aspects of our fully atomistic HRMC models. In this way, we explain the structural complexity of ACC in terms of geometric frustration of two competing energetically favoured Ca⋯Ca separations. Mapping the problem of ACC structure onto the phase behaviour of multiwell potentials is important not only because it establishes the first experimental system for which these potentials are relevant, but also because it suggests how structural complexity in inorganic phases might be controllably targeted through suitable tuning of effective interactions.
Results
Structure of ACC
Our HRMC refinements made use of atomistic configurations containing 12,960 atoms (1,620 CaCO_{3}⋅H_{2}O formula units), with simulation cell sizes of ~5 nm. During refinement, atomic moves were proposed and then rejected or accepted according to a Metropolis Monte Carlo criterion, with a cost function that measured the quality of fit to Xray total scattering data^{17} and also the energetic stability evaluated using the stateoftheart interatomic potential of ref. ^{18}. We favoured an HRMC approach over empirical potential structure refinement (EPSR), because calcium carbonate potentials are notoriously delicately balanced^{18}, such that EPSRderived potentials are unlikely to be physically meaningful for this system. To allow for direct comparison with previous studies, we used a weighted variant of the experimental Xray total scattering function (that is, QF_{X}(Q)) of synthetic ACC reported in ref. ^{13}, which has been shown to be indistinguishable from that obtained for biogenic ACC samples^{19}. For consistency, our HRMC configuration size and refinement constraints were also identical to those used in the reverse Monte Carlo (RMC) investigation of synthetic ACC in ref. ^{14}. The absence of key energetic considerations in that RMC study allowed the development of physically unreasonable charge separation to give a model of cationic Ca^{2+}rich domains separated by channels of carbonate anions and water molecules. Our expectation was that the explicit consideration of electrostatics in our HRMC implementation would guide refinement towards a solution that was equally consistent with experiments, but also energetically sensible.
HRMC does indeed find a suitable compromise between experiment and theory. We show in Fig. 1a the QF_{X}(Q) function calculated from a representative configuration, and compare it to the equivalent functions predicted using pure RMC refinement, on the one hand, and unconstrained molecular dynamics (MD) simulations with the potentials of ref. ^{18}, on the other hand. Both RMC and HRMC give very similar highquality fits to experiment—unsurprising, of course, as they have been refined against these data. The MD simulation, however, misses some aspects of the QF_{X}(Q) function. By contrast, the HRMC and MD models give similar cohesive energies, whereas the RMC model is less stable than both by more than 800 kJ mol^{−1} per formula unit. These results are summarized in Fig. 1b, which captures the motivation for our use of HRMC as a suitable balancing act: it has, for the first time, allowed access to an atomistic representation of ACC structure that is consistent with experiment and gives sensible energies using established hydrated calcium carbonate potentials. We also checked whether the HRMC model could reproduce the neutron total scattering data of refs. ^{20,21,22} (it can—see Supplementary Fig. 1 and Supplementary Discussion 1) and whether it is stable in MD simulations driven by the potentials of ref. ^{18} (it is—see Extended Data Fig. 1).
The HRMC model itself is illustrated in Fig. 1c. We observe that water is not homogeneously distributed throughout the configuration, but rather that the ACC structure consists of CaCO_{3}rich regions separated by a filamentary network of water: we term this a ‘blue cheese’ model. Qualitatively similar descriptions were obtained in MD simulations^{23}, in EPSR refinements of combined neutron/Xray total scattering data^{22}, and also inferred from solidstate NMR measurements^{24}. In all cases, the interpretation is that nearly all H_{2}O molecules are bound to Ca^{2+} (as implied by ^{1}H NMR; ref. ^{25}), but neighbouring water molecules are sufficiently close to form a network that percolates the ACC structure. As anticipated, the charge separation that developed in the RMC model of ref. ^{14} has now vanished: the waterrich filaments we observe do not contain any free carbonate and are substantially narrower than the nanopore channels reported previously. The fact that RMC and HRMC models give similar QF_{X}(Q) fits despite their very different descriptions of the structure reflects the difficulty of discriminating between C and O atoms using Xray scattering methods alone^{26}.
Locally, our HRMC model shows coordination environments that are consistent with the consensus of recent computational and experimental studies. For completeness, we show in Fig. 2a,b the distribution of calcium and carbonate coordination numbers. The average Ca^{2+} coordination number of 7.0—defined for a cutoff distance of 2.8 Å—is similar to that reported in refs. ^{22,27,28}. Whereas most Ca^{2+} bind either five or six distinct carbonate anions, the carbonates tend to bind one fewer Ca^{2+} ion each, reflecting a bindingmode distribution of ~80:20 monodentate:bidentate. Representative modal coordination geometries for Ca^{2+} and CO_{3}^{2−} are shown in Fig. 2c,d. As anticipated^{14}, we find that carbonate anions bridge calciumion pairs in two ways: either a pair of Ca^{2+} ions share a common carbonate oxygen neighbour, or they bind distinct oxygens and so are connected formally by a Ca–O–C–O–Ca pathway. A full analysis of coordination environments, including bondlength and bondangle distributions, is provided in Extended Data Figs. 1–4 and Supplementary Discussions 2 and 3.
Coarse graining
In seeking to improve our understanding of the structure of ACC, we focused on the Capair correlation function g_{Ca}(r)—after all, this is the contribution to the pair distribution function that exhibits the strongest persistent welldefined oscillations. In Fig. 2e, we show this function as extracted from our HRMC configurations compared against that reported in the RMC study of ref. ^{14}. We also include the normalized Fourier transform of the experimental F_{X}(Q) function; this transform is an approximate (total) pair distribution function that emphasizes contributions from Ca–Ca pairs as a consequence of the larger Xray scattering crosssection for Ca relative to C, O and H. Although all three realspace correlation functions show maxima at r ≈ 4 and 6 Å, it is the HRMC result that resolves these features most clearly. By counting Ca–Ca pairs separated by Ca–O–Ca and Ca–O–C–O–Ca pathways in our HRMC configuration, we confirmed that the two maxima in g_{Ca}(r) occur at the preferred Ca⋯Ca distances associated with these two different carbonate bridging motifs (Extended Data Fig. 3). Indeed these same preferred separations recur among crystalline calcium carbonates, as noted in ref. ^{14}.
Access to a smoothly varying measure of g_{Ca}(r), together with the HRMC configurations from which it is calculated, allowed us to exploit a recently developed approach for the direct measurement of effective pair potentials from particlecoordinate data^{16}. The method works by equating the pair distribution functions measured directly, on the one hand, and calculated using a testparticle insertion approach^{29,30}, on the other^{31}. Applying this methodology to the atomic coordinates in our HRMCrefined model, we extracted the effective Ca^{2+}ion pair potential u_{Ca}(r) shown in Fig. 3a. This function has two distinct potential wells, the minima of which are centred near distances for which g_{Ca}(r) has its maxima. The physical meaning of u_{Ca}(r) is that it captures the effective twobody interactions between Ca–Ca pairs, mediated by carbonate and water, as required to account for the observed g_{Ca}(r). That this interaction energy is minimized when Ca–Ca pairs are separated by distances corresponding to either Ca–O–Ca or Ca–O–C–O–Ca pathways makes intuitive sense (of course); the longer (6 Å) separation appears energetically more favourable, probably because it minimizes the electrostatic repulsion between calcium ions bound to a common carbonate.
One assumption in using an isotropic effective pair potential to capture interactions mediated by anisotropic molecular species, such as H_{2}O and CO_{3}^{2−}, is that any local anisotropy is sufficiently shortranged. We checked this point by calculating from our original (allatom) HRMC configurations the orientational correlation functions:
where P_{2}(x) is the secondorder Legendre polynomial, and S are vectors parallel to a suitable local axis (for example, C_{3}, C_{2}) of molecules separated by distance r (ref. ^{32}). We find that ϕ(r) is essentially featureless—for both carbonate and water—for all but the very shortest intermolecular approaches, which occur for distances smaller than the nearestneighbour Ca⋯Ca separation (Fig. 3a, inset). Accordingly, the anisotropy of individual molecules is relevant only at a length scale smaller than that over which the effective interactions between Ca^{2+}ion pairs operate. The doublewell form of u_{Ca}(r) is important for a number of reasons. First, it explains why simple hardsphere or LJ potentials fail to reproduce the experimental g_{Ca}(r): if tuned to capture the 4 Å nearestneighbour correlation, such potentials predict a second maximum just below 8 Å that is not observed in experiments. Second, it allows us to rationalize the inability of earlier EPSR studies to determine the correct Ca distribution function; for example, the study of ref. ^{20} included an LJ parameterization of Ca⋯Ca interactions that then resulted in a g_{Ca}(r) function with maxima at positions clearly inconsistent with the inverse Fourier transform of the experimental F_{X}(Q). Third, multiwell potentials are well known to support very complex phase behaviour^{9,11,12,33,34,35}, suggesting a qualitative explanation of the complexity of ACC.
LJG parameterization
To test this hypothesis of a link between effective potential and structural complexity, we sought to map u_{Ca}(r) onto a suitable multiwell potential for which the corresponding theory is already well established. We found that a doublewell LJG interaction^{8} described the observed functional form surprisingly well, as long as we included an additional broad, repulsive Gaussian term that helps capture the local maxima between and beyond the two minima of the LJG function. The behaviour of the LJG potential is characterized by three parameters—ε, r_{0} and σ—which describe, respectively, the depth, position and width of the Gaussian well relative to the LJ component^{9,11}. A leastsquares fit to u_{Ca}(r) gave the values ε = 4.1, r_{0} = 1.4 and σ = 0.14 (Fig. 3a); note that r_{0} is particularly well defined because it is closely related to the ratio of the Ca⋯Ca separations resulting from the two different carbonate bridging motifs (~6 Å and 4 Å). We will return to the importance of these empirical parameters in due course. We note, in passing, related work on the description of coarsegrained molecular systems with doublewell potentials, where one minimum is explicitly assigned to enthalpic and one to entropic terms^{36}, and on the ‘learning’ of effective pair potentials from simulation data by backpropagation^{37}.
As a simple check that our combination of g(r) inversion and potential parameter fitting does indeed result in a meaningful effective pair potential, we carried out direct MC simulations driven by our parameterized LJG potential. These simulations were performed using the experimentally determined Ca particle density in ACC, and although we focus here on MC simulations (for fairest comparison with HRMC), the same results emerge if we use MD instead (Methods and Extended Data Fig. 5). The corresponding pair correlation function closely matches our HRMC g_{Ca}(r), as expected (Fig. 2e), and the resulting structure resembles the Ca distribution in the fully atomistic model (Fig. 3b; cf. Fig. 1c). But the similarity between coarsegrained LJGdriven and HRMC Ca distributions turns out to extend beyond pair correlations, and we explore this more extensive similarity in various forms (for example, ring statistics, Voronoi volume distributions and higherorder correlation functions) in Supplementary Figs. 2–4 and Supplementary Discussion 4. A particular point of interest is that the MC configurations contain Carich/poor regions that are qualitatively similar to those observed in the original atomistic model (Fig. 3b)—the clear implication being that, at the density of ACC, an inhomogeneous Ca distribution is encoded within the effective Ca⋯Ca interaction itself.
Discussion
From the perspective of softmatter theory, much of the interest in the LJG potential lies in the complexity of phase behaviour that it supports^{9,11,12,38}. This complexity arises because of competition between the structuredirecting effects of LJ and Gaussian components, which operate at different length scales. When the positions of the LJ and Gaussian minima are related by a factor 1.2 ≤ r_{0} ≤ 1.6, there is particularly strong geometric frustration that results in a general resistance to crystallization^{9} and the emergence of many competing lowsymmetry groundstate structures^{11}. This behaviour extends across a wide variety of relative well depths ε ≥ 1 for the corresponding relative Gaussian width σ = 0.14 (ref. ^{11}). Hence the empirical parameters we have determined to be relevant to effective Ca⋯Ca interactions in ACC—that is, r_{0} = 1.4, ε = 4.1 and σ = 0.14—locate the system in one of the most complicated parts of the LJG phase diagram (Fig. 3c)^{11}. In developing this mapping onto the LJG model, we assume that the additional broad Gaussian term we have included in our parameterization does not strongly influence the phase behaviour (note that the value of r_{0} does not vary substantially if it is omitted from our fit). Similarly, we cannot be certain of the effects of finite temperature and fixed density on LJG phase behaviour, as these have not yet been studied from theory. It is nevertheless a general phenomenon in frustrated systems that regions of strong geometric frustration are characterized by the existence of many competing ground states with suppressed ordering temperatures, above which the system is disordered but far from random^{39,40}.
In the context of ACC, the implications are twofold. First, the different favoured Ca⋯Ca separations for carbonatebridged Ca pairs—that is, 4 and 6 Å—are configurationally difficult to satisfy in a threedimensionally periodic (that is, crystalline) structure with the density of ACC and in the absence of anisotropic interactions. This rationalizes, in general terms, why an amorphous form of hydrated CaCO_{3} is so readily accessible. Second, the sensitivity of the LJG potential around r_{0} = 1.4 hints—again in general terms—at why ACC might be directed to crystallize into a number of different polymorphs. Of course, as the water content of ACC varies during ageing of the material, both the effective Ca⋯Ca interactions and the bulk density must change, which makes extrapolation from the LJG effective potential of ACC difficult. Increased material density also strengthens the orientational correlations in molecular components, and so one expects the isotropic effective pair potential description to break down as crystallization is approached. Nevertheless, at the density of ACC and in the absence of orientational order, the occurrence of an amorphous state is now more easily rationalized. Furthermore, the incorporation of Mg^{2+} or PO_{4}^{3−} ions into ACC will introduce statistical variations into the effective potential governing cation arrangements, favouring further the amorphous state—much as magnetic exchange disorder can stabilize spin glasses^{41}.
Whereas the (hitherto unsuccessful) search for ‘realworld’ materials governed by isotropic multiwell potentials has focused on the atomic (alloys) and mesoscopic (colloids) length scales^{11,12}, our study of ACC suggests that the intermediate domain of nanoscale materials may in fact be a more fertile source of relevant examples. By their very nature, molecular ions tend to exhibit a range of coordination modes, which must give rise to multiple specific preferred distances between the counterions they coordinate. Whenever these distances are geometrically frustrated, as in ACC, one expects that the corresponding effective potential must contain multiple wells so as to stabilize both separations at once. Through suitable choice of chemical components, one might hope to navigate the complicated phase space associated with such unconventional potentials. Doing so will provide an important test of theory, on the one hand, and also allow the targeted design of complex structures through selfassembly, on the other.
Our coarsegrained approach to understanding ACC structure may be applicable to other poorly ordered inorganic solids of particular scientific importance. Amorphous calcium phosphate (the precursor to bone) and calcium–silicate–hydrates (that is, Portland cements) are obvious evercontentious examples^{42,43} where one might expect the different bridging motifs—now of the phosphate or (poly)silicate anions—to again moderate an effective interaction potential with multiple minima. Better understanding the structures of longstudied materials is certainly one fruitful avenue for future research, but our work also suggests an alternative pathway towards complex materials design. By varying the inorganic cation radius for a given molecular counterion, or by changing the degree of hydration, one might hope to control the ratio of the distances at which minima in the effective potential occur. What emerges is a strategy of ‘interaction engineering’ to target a particular phase of interest, or to stabilize or destabilize an amorphous form of matter.
Methods
HRMC refinements
Our approach to carrying out HRMC refinements was modelled closely on the previous RMC refinements of ref. ^{14}, which were carried out using the RMCProfile code^{44}. For the present study we used a related custom code, written in Python. The Xray scattering calculations carried out by this code were exactly those employed by RMCProfile, but the code was also able to interface with potentialenergy evaluations carried out using the LAMMPS software^{45}. The custom Python code imports functionality from the following packages: PyLammpsMPI^{46}, Numba^{47} and ASE^{48}.
Starting configurations were generated in the following manner. We began by placing 1,620 Ca atoms randomly in a box of dimensions 52.8 × 54.8 × 45.2 Å^{3}—this being the same box size as used in ref. ^{14}. An initial energy minimization was carried out with a repulsive potential between Ca atoms (an LJ functional form, truncated at the zerocrossing distance), so as to prevent any Ca–Ca distances of <3.0 Å. To this homogeneous distribution of Ca atoms we added a stoichiometric number of rigid carbonate and water molecules with idealized geometries. Carbonate molecules were fixed to adopt a C–O bond length of r_{CO} = 1.284 Å—as observed in calcite and as defined for the relevant CaCO_{3} potential^{18}. Water molecules were treated using the foursite rigid model, TIP4PEw^{49}, which uses a fixed geometry of the water molecule (O–H bond length r_{OH} = 0.9572 Å, H–O–H bond angle θ_{HOH} = 104.52°). Shortrange repulsive terms (as described above) were applied between the rigid bodies to prevent close contacts during a subsequent energy minimization and a short NVTensemble MD run, aiming to achieve a homogeneous distribution of particles (cf. the method of ref. ^{50}).
The HRMC cost function, to be optimized during refinement, was defined as
where
represents the mismatch between calculated and experimental Qweighted Xray total scattering functions QF(Q) (ref. ^{51}), and E is the configurational energy as obtained from the empirical potential. The experimental QF(Q) data were those reported in ref. ^{13} and ref. ^{14}, which span the range of momentum transfers 1.2 ≤ Q ≤ 25.0 Å^{−1}. The corresponding HRMC QF(Q) function was computed by weighted Fourier transform of the partial pair correlation functions, g_{ij}(r) (refs. ^{44,51}), themselves obtained as histograms with a bin width of Δr = 0.02 Å. Totalscattering calculations were parallelized using the package Numba, a justintime compiler for Python^{47}. The value of E was determined using the rigidbody force field for aqueous calcium carbonate of ref. ^{18}, with the corresponding calculation delegated to the LAMMPS software^{45} via the PyLammpsMPI parallel LAMMPS–Python interface^{46}. The simulation temperature was set to T = 300 K.
The parameter σ in equation (3) controls the relative weights of fittodata and potential energies in the overall HRMC cost function, and its value must be determined empirically. Through systematic tests of the variation in \({\chi }_{QF(Q)}^{2}\) and E with σ, we determined the value σ = 0.057 to provide the most appropriate balance in this case.
Individual moves were proposed and accepted or rejected according to the usual Metropolis–Hastings criterion. These moves involved a combination of atomic displacements (maximum value 0.1 Å) and rigidbody translations or rotations (maximum values of 0.1 Å and 1°, respectively) of carbonate and water molecules. In contrast to the RMC study of ref. ^{14}, no closestapproach constraints were employed. The incorporation of interatomic potentials within the HRMC cost function ensured that such constraints were unnecessary.
HRMC refinements were continued until the values of χ^{2} and E were deemed to have converged (Extended Data Fig. 1).
MD simulations
MD simulations were performed using LAMMPS^{45} in the NVT ensemble at 300 K, with a Nosé–Hoover thermostat controlling the temperature. The time step was 1 fs with a relaxation time of 0.1 ps for temperature control (cf. the ACC MD simulations in refs. ^{27,52}). We tested the stability of our HRMC ACC structural model by carrying out a 1ns MD simulation (one million time steps) starting from the final HRMC configuration. The simulation was stable with respect to NVT MD (Extended Data Fig. 1), by which we mean that no substantial structural reorganization occurred.
As anticipated, the HRMC configuration is indeed a compromise between fittodata and energy minimization. During the MD simulation, it quickly decreased in energy by ~0.3 eV per Ca before reaching thermal equilibrium (Extended Data Fig. 1). The negative pressure (Extended Data Fig. 1) indicates that the system is driven towards a more compact atomic arrangement than implied by the experimental density. An MD simulation performed in the NPT ensemble led to an increase in density from 2.43 to 2.91 g cm^{−3}.
Pair distribution function inversion
The testparticle pair distribution function inversion method was carried out using the code described in ref. ^{30}, modified for application to a threedimensional system. The Ca coordinates and the averaged g_{Ca}(r) (up to a cutoff of 12 Å with a bin size of Δr = 0.2 Å) from the final 12 frames of the HRMC trajectory were used as input for the inversion algorithm. Approximately 10,000 trial insertions were used per configuration, so as to satisfy convergence criteria.
Coarsegrained simulations
MC and NVT MD simulations were performed using a custom Python program with potentialenergy evaluations and MD protocols carried out using the LAMMPS software^{45}.
Twelve starting configurations for each method (24 in total) were generated by randomly placing 1,620 Ca atoms in a box with identical cell dimensions as the HRMC configuration (52.8 × 54.8 × 45.2 Å^{3}). The simulation temperature was set to T = 300 K (cf. HRMC simulations). For MC simulations, the maximum atomic displacement was 0.1 Å. MC simulations were terminated after four million moves were accepted, at which point convergence with respect to the ensemble energy was met (Extended Data Fig. 5). An initial energy minimization (using the the conjugategradient algorithm for 100 steps) was performed for the starting MD configurations to prevent close contacts rendering the simulation unstable. This gave an average minimum energy of −0.03028 ± 0.00090 eV per Ca. The MD simulations were then run in the NVT ensemble for 1 ns, with a 0.5fs time step and 50fs temperature damping constant.
Data availability
Data supporting the findings of this study are available within the paper and its Supplementary Information files, or at https://doi.org/10.5281/zenodo.8238547. Source data are provided with this paper.
Code availability
All custom code used in this study was developed using widely available algorithms. A copy of the Python scripts used to carry out the HRMC refinements is included as Supplementary Code 1.
References
Gebauer, D., Völkel, A. & Cölfen, H. Stable prenucleation calcium carbonate clusters. Science 322, 1819–1822 (2008).
Aizenberg, J., Lambert, G., Weiner, S. & Addadi, L. Factors involved in the formation of amorphous and crystalline calcium carbonate: a study of an ascidian skeleton. J. Am. Chem. Soc. 124, 32–39 (2002).
Raz, S., Hamilton, P. C., Wilt, F. H., Weiner, S. & Addadi, L. The transient phase of amorphous calcium carbonate in sea urchin larval spicules: the involvement of proteins and magnesium ions in its formation and stabilization. Adv. Funct. Mater. 13, 480–486 (2003).
Aizenberg, J. Crystallization in patterns: a bioinspired approach. Adv. Mater. 16, 1295–1302 (2004).
Addadi, L., Raz, S. & Weiner, S. Taking advantage of disorder: amorphous calcium carbonate and its roles in biomineralization. Adv. Mater. 15, 959–970 (2003).
Weiner, S., Sagi, I. & Addadi, L. Structural biology. Choosing the crystallization path less traveled. Science 309, 1027–1028 (2005).
Beniash, E., Aizenberg, J., Addadi, L. & Weiner, S. Amorphous calcium carbonate transforms into calcite during sea urchin larval spicule growth. Proc. R. Soc. Lond. B 264, 461–465 (1997).
Rechtsman, M., Stillinger, F. & Torquato, S. Designed interaction potentials via inverse methods for selfassembly. Phys. Rev. E 73, 011406 (2006).
Engel, M. & Trebin, H.R. Structural complexity in monodisperse systems of isotropic particles. Z. Kristallogr. 223, 721–725 (2008).
Jain, A., Errington, J. R. & Truskett, T. M. Dimensionality and design of isotropic interactions that stabilize honeycomb, square, simple cubic and diamond lattices. Phys. Rev. X 4, 031049 (2014).
Dshemuchadse, J., Damasceno, P. F., Phillips, C. L., Engel, M. & Glotzer, S. C. Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials. Proc. Natl Acad. Sci. USA 118, e2024034118 (2021).
Engel, M. & Trebin, H.R. Selfassembly of monatomic complex crystals and quasicrystals with a doublewell interaction potential. Phys. Rev. Lett. 98, 225505 (2007).
Michel, F. M. et al. Structural characteristics of synthetic amorphous calcium carbonate. Chem. Mater. 20, 4720–4728 (2008).
Goodwin, A. L. et al. Nanoporous structure and mediumrange order in synthetic amorphous calcium carbonate. Chem. Mater. 22, 3197–3205 (2010).
Opletal, G. et al. Hybrid approach for generating realistic amorphous carbon structure using metropolis and reverse Monte Carlo. Mol. Simul. 28, 927–938 (2002).
Stones, A. E., Dullens, R. P. A. & Aarts, D. G. A. L. Modelfree measurement of the pair potential in colloidal fluids using optical microscopy. Phys. Rev. Lett. 123, 098002 (2019).
McGreevy, R. L. & Pusztai, L. Reverse Monte Carlo simulation: a new technique for the determination of disordered structures. Mol. Simul. 1, 359–367 (1988).
Raiteri, P., Gale, J. D., Quigley, D. & Rodger, P. M. Derivation of an accurate forcefield for simulating the growth of calcium carbonate from aqueous solution: a new model for the calcite–water interface. J. Phys. Chem. C 114, 5997–6010 (2010).
Reeder, R. J. et al. Characterization of structure in biogenic amorphous calcium carbonate: pair distribution function and nuclear magnetic resonance studies of lobster gastrolith. Cryst. Growth Des. 13, 1905–1914 (2013).
Cobourne, G. et al. Neutron and Xray diffraction and empirical structure refinement modelling of magnesium stabilised amorphous calcium carbonate. J. NonCryst. Sol. 401, 154–158 (2014).
Wang, H.W. et al. Synthesis and structure of synthetically pure and deuterated amorphous (basic) calcium carbonates. Chem. Commun. 53, 2942–2945 (2017).
Jensen, A. C. S. et al. Hydrogen bonding in amorphous calcium carbonate and molecular reorientation induced by dehydration. J. Phys. Chem. C 122, 3591–3598 (2018).
Demichelis, R., Raiteri, P., Gale, J. D., Quigley, D. & Gebauer, D. Stable prenucleation mineral clusters are liquidlike ionic polymers. Nat. Commun. 2, 590 (2011).
Ihli, J. et al. Dehydration and crystallization of amorphous calcium carbonate in solution and in air. Nat. Commun. 5, 3169 (2014).
Nebel, H., Neumann, M., Mayer, C. & Epple, M. On the structure of amorphous calcium carbonate—a detailed study by solidstate NMR spectroscopy. Inorg. Chem. 47, 7874–7879 (2008).
Goodwin, A. L. Opportunities and challenges in understanding complex functional materials. Nat. Commun. 10, 4461 (2019).
Singer, J. W., Yazaydin, A. Ö., Kirkpatrick, R. J. & Bowers, G. M. Structure and transformation of amorphous calcium carbonate: a solidstate ^{43}Ca NMR and computational molecular dynamics investigation. Chem. Mater. 24, 1828–1836 (2012).
Clark, S. M. et al. The nano and mesoscale structure of amorphous calcium carbonate. Sci. Rep. 12, 6870 (2022).
Henderson, J. R. On the test particle approach to the statistical mechanics of fluids. Hard sphere fluid. Mol. Phys. 48, 389–400 (1983).
Stones, A. E., Dullens, R. P. A. & Aarts, D. G. A. L. Contact values of pair distribution functions in colloidal hard disks by testparticle insertion. J. Chem. Phys. 148, 241102 (2018).
Widom, B. Some topics in the theory of fluids. J. Chem. Phys. 39, 2808–2812 (1963).
Cinacchi, G. & Torquato, S. Hard convex lensshaped particles: characterization of dense disordered packings. Phys. Rev. E 100, 062902 (2019).
Elenius, M., Zetterling, F. H. M., Dzugutov, M., Fredrickson, D. C. & Lidin, S. Structural model for octagonal quasicrystals derived from octagonal symmetry elements arising in βMn crystallization of a simple monatomic liquid. Phys. Rev. B 79, 144201 (2009).
Mihalkovič, M. & Henley, C. L. Empirical oscillating potentials for alloys from ab initio fits and the prediction of quasicrystalrelated structures in the AlCuSc system. Phys. Rev. B 85, 092102 (2012).
Zhou, P., Proctor, J. C., van Anders, G. & Glotzer, S. C. Alchemical molecular dynamics for inverse design. Mol. Phys. 117, 3968–3980 (2019).
Pretti, E. & Shell, M. S. A microcanonical approach to temperaturetransferable coarsegrained models using the relative entropy. J. Chem. Phys. 155, 094102 (2021).
Wang, W., Wu, Z., Dietschreit, J. C. B. & GómezBombarelli, R. Learning pair potentials using differentiable simulations. J. Chem. Phys. 158, 044113 (2023).
Suematsu, A., Yoshimori, A., Saiki, M., Matsui, J. & Odagaki, T. Control of solidphase stability by interaction potential with two minima. J. Mol. Liq. 200, 12–15 (2014).
Ziman, J. M. Models of Disorder. The Theoretical Physics of Homogeneously Disordered Systems (Cambridge Univ. Press, 1979).
Moessner, R. & Ramirez, A. P. Geometrical frustration. Phys. Today 59, 24–29 (2006).
Saunders, T. E. & Chalker, J. T. Spin freezing in geometrically frustrated antiferromagnets with weak disorder. Phys. Rev. Lett. 98, 157201 (2007).
Lowenstam, H. A. & Weiner, S. Transformation of amorphous calcium phosphate to crystalline dahllite in the radular teeth of chitons. Science 227, 51–53 (1985).
Richardson, I. G. Model structures for C(A)SH(I). Acta Crystallogr. B Struct. Sci. Cryst. Eng. Mater 70, 903–923 (2014).
Tucker, M. G., Keen, D. A., Dove, M. T., Goodwin, A. L. & Hui, Q. RMCProfile: reverse Monte Carlo for polycrystalline materials. J. Phys. Condens. Matter 19, 335218 (2007).
Thompson, A. P. et al. LAMMPS—a flexible simulation tool for particlebased materials modeling at the atomic, meso and continuum scales. Comput. Phys. Commun. 271, 108171 (2022).
Janssen, J. PyLammpsMPI (version 0.0.8). GitHub https://github.com/pyiron/pylammpsmpi (2022).
Lam, S. K., Pitrou, A. & Seibert, S. Numba: a LLVMbased Python JIT compiler. In Proc. Second Workshop on the LLVM Compiler Infrastructure in HPC, LLVM ‘15, Vol. 7, 1–6 (Association for Computing Machinery, 2015).
Larsen, A. H. et al. The Atomic Simulation Environment—a Python library for working with atoms. J. Phys. Condens. Matter 29, 273002 (2017).
Horn, H. W. et al. Development of an improved foursite water model for biomolecular simulations: TIP4PEw. J. Chem. Phys. 120, 9665–9678 (2004).
Bushuev, Y. G., Finney, A. R. & Rodger, P. M. Stability and structure of hydrated amorphous calcium carbonate. Cryst. Growth Des. 15, 5269–5279 (2015).
Keen, D. A. A comparison of various commonly used correlation functions for describing total scattering. J. Appl. Crystallogr. 34, 172–177 (2001).
Saharay, M. & Kirkpatrick, R. J. Water dynamics in hydrated amorphous materials: a molecular dynamics study of the effects of dehydration in amorphous calcium carbonate. Phys. Chem. Chem. Phys. 19, 29594–29600 (2017).
Acknowledgements
T.C.N. was supported through an Engineering and Physical Sciences Research Council DTP award (grant no. EP/T517811/1). A.E.S. acknowledges financial support from the University of Oxford Clarendon Fund. A.L.G. acknowledges useful discussions with M. T. Dove (QMUL) and financial support from the ERC (grant no. 788144). F.M.M. acknowledges financial support provided by the National Science Foundation through CAREER1652237 and the Virginia Tech National Center for Earth and Environmental Nanotechnology Infrastructure (‘NanoEarth’, NSF Cooperative Agreement 1542100). This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DEAC0206CH11357. We acknowledge use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work (https://doi.org/10.5281/zenodo.22558).
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T.C.N. carried out the hybrid reverse Monte Carlo refinements and molecular dynamics simulations, under the supervision of V.L.D. and A.L.G., building on a preliminary investigation performed with A.P. These refinements made use of Xray total scattering data collected and normalized by F.M.M. and R.J.R. T.C.N. performed the effective pairpotential determination, using an algorithm developed by A.E.S. and D.G.A.L.A., and coded by A.E.S. T.C.N., V.L.D. and A.L.G. analysed the data and developed the main conclusions regarding interpretation of the structure of amorphous calcium carbonate. All authors contributed to discussions. T.C.N., V.L.D. and A.L.G. drafted the paper, and all authors contributed to its final version.
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Extended data
Extended Data Fig. 1 Evolution of system properties during HRMC refinement and MD simulation.
a The cost function relating to the Xray scattering data during HRMC refinement. b The change in potential energy of the simulated structural model during HRMC refinement (black) and subsequent MD simulation (red). Energies are reported relative to that obtained in equivalent NVT MD simulations of the crystalline monohydrocalcite polymorph of CaCO_{3}.H_{2}O. The c temperature, d pressure, and e meansquared displacement (MSD) of atoms during the HRMC MD simulation, respectively.
Extended Data Fig. 2 Partial pair distribution functions extracted from the HRMC ACC model.
Each partial pair distribution function (PDF) is calculated with a bin width of Δr = 0.02 Å, averaged over 12 HRMC trajectory configurations exported in intervals of 100,000 proposed moves. Sharp features arise from rigidbody correlations (light grey). For the relevant partial PDFs we omit rigidbody terms contributions from the data shown in black.
Extended Data Fig. 3 Key bondlength and bondangle distributions in the HRMC ACC model.
a The bondlength distributions for CaO divided into CaO_{C} and CaO_{W} contributions. The average CaO_{C} bond length (2.5(1) Å) is shifted to larger r values relative to the average CaO_{W} bond length (2.3(1) Å). b The interatomic distance between calcium atoms bound to the same carbonate can be divided into those which share a common oxygen atom and those which are connected via the carbonate molecule. These two coordination interactions give rise to the characteristic geometric frustration observed for ACC. c Bondangle distributions for O_{C}CaO_{C} triplets, decomposed into mono and bidentate coordination modes. The bidentate coordination mode gives rise to a sharp peak centred at 50^{∘}. d O_{W}CaO bondangle distributions. The lower relative intensity of O_{W}CaO_{W} correlations reflects the low likelihood of finding multiple water molecules bound to the same calcium atom. For each panel, the plots at the top show the corresponding cumulative distribution functions.
Extended Data Fig. 4 Characteristics of water distribution in the HRMC ACC model.
a Radial distribution functions for neighbours of carbonate oxygen atoms in the structure. b Radial distribution functions for neighbours of oxygen atoms in water molecules (‘O_{W}’) only. Intramolecular correlations are omitted for clarity. c The distribution of cluster sizes for those water molecules that do not take part in the percolating water network. A cluster size of 1 means an isolated water molecule that is not within reach of any others according to the defined cutoff distances (see Supplementary Table 1).
Extended Data Fig. 5 Evolution of coarsegrained simulation ensemble energies.
The energy of each ensemble, relative to the minimum configuration energy across all trajectories, as a function of a accepted MC moves, and b MD simulation time. Each of the 12 trajectories is given a different colour. This initial sharp decrease and subsequent increase in E_{rel} for the MD configurations arises from the initially randomised particle positions, followed by a short energy minimisation, before beginning the NVT MD run.
Supplementary information
Supplementary Information
Supplementary Figs. 1–4, Table 1, Discussions 1–4 and references.
Supplementary Data 1
Atomic coordinates in crystallographic information file format for a representative hybrid reverse Monte Carlo model of amorphous calcium carbonate.
Supplementary Data 2
Atomic coordinates in crystallographic information file format for a representative reverse Monte Carlo model of amorphous calcium carbonate.
Supplementary Data 3
Atomic coordinates in crystallographic information file format for a representative Molecular Dynamics model of amorphous calcium carbonate.
Supplementary Data 4
Atomic coordinates in crystallographic information file format for a representative coarsegrained LennardJones–Gauss model of calcium arrangements in amorphous calcium carbonate.
Supplementary Code 1
Python scripts used to carry out HRMC refinements as discussed in the main text.
Source data
Source Data Fig. 1
Statistical source data.
Source Data Fig. 2
Statistical source data.
Source Data Fig. 3
Statistical source data.
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Nicholas, T.C., Stones, A.E., Patel, A. et al. Geometrically frustrated interactions drive structural complexity in amorphous calcium carbonate. Nat. Chem. 16, 36–41 (2024). https://doi.org/10.1038/s41557023013392
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DOI: https://doi.org/10.1038/s41557023013392
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