Geometrically frustrated interactions drive structural complexity in amorphous calcium carbonate

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 high-quality atomistic models of amorphous calcium carbonate generated using state-of-the-art interatomic potentials to help guide fits to X-ray 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 Ca²⁺-ion pairs. We reveal an unexpected mapping to the Lennard-Jones-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 Ca²⁺ ions.

Fig. 1. HRMC yields a balanced model of ACC structure.

a, The experimental Q-weighted X-ray total scattering function QF_X(Q) is well fitted by both RMC (top trace) and HRMC (middle) refinements, but is meaningfully different to that calculated using the interatomic potential of ref. (bottom). Experimental data are shown as black lines, and fits or calculations as red lines. b, Quality criteria for the different structural models. HRMC simultaneously optimizes both goodness-of-fit to experimental data (χ²) and cohesive energy (E_rel), and therefore finds a structure solution that is essentially as consistent with experiment as that obtained using RMC, while also as energetically sensible as that obtained using potentials alone (MD). c, Representation of a converged structure of ACC obtained using HRMC. Water molecules connect to form filamentary strands (aquamarine strings) that separate calcium carbonate-rich domains (Ca atoms are shown as large beige spheres, and carbonate ions as stick representation). Source data

Fig. 2. Coordination environments and Ca-pair distributions in ACC.

a, Histogram of Ca²⁺ coordination environments, decomposed into contributions from carbonate and water oxygen donors (O_c and O_w, respectively). b, Histogram of CO₃²⁻ coordination environments, now decomposed into contributions from Ca²⁺ and water hydrogen donors (H_w). c, Representative Ca²⁺ coordination sphere for the modal coordination environment marked by a star in a. d, Representative CO₃²⁻ coordination sphere for the modal coordination environment marked by a star in b. Note that pairs of Ca²⁺ ions within the same CO₃²⁻ coordination sphere either share a common oxygen donor (for example, red arrow) or are connected by Ca–O–C–O–Ca pathways (for example, blue arrow). Ca, C, O and H atoms are shown in beige, grey, aquamarine and white, respectively. e, Ca-pair correlation functions g_Ca(r) extracted from HRMC, RMC and LJG configurations, compared against the normalized Fourier transform of the experimental X-ray total scattering function (which includes contributions from all atom pairs, for example, the Ca–O peak at 2.4 Å marked with an asterisk). The two principal peaks common to all functions, indicated by red and blue shading, can be assigned to the two types of Ca²⁺-ion pair illustrated in d. Source data

Fig. 3. Effective Ca⋯Ca interactions in ACC.

a, The effective Ca⋯Ca interatomic potential extracted from our HRMC configurations (open circles) and least-squares fit using a modified LJG model (line) as described in the text. The inset shows the orientational correlation functions (equation (1)), which vanish for distances relevant to the Ca⋯Ca separations. In the lower panel, the yellow dashed line corresponds to the additional broad, repulsive Gaussian term. b, Representative MC configuration of Ca atoms (beige spheres) generated by the LJG potential, parameterized by the fit shown in a. The Ca atoms are not uniformly distributed, but cluster to leave Ca-poor voids, shown as aquamarine surfaces. Note the qualitative similarity to the heterogeneous structure of ACC represented in Fig. 1c. c, Approximate location of ACC effective LJG parameters (star) in the LJG phase space as reported in ref. ; shaded regions correspond to the stability fields of different ground-state structures, and solid lines denote the approximate locations of phase boundaries. Source data

Extended Data Fig. 1. Evolution of system properties during HRMC refinement and MD simulation.

a The cost function relating to the X-ray 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 CaCO₃.H₂O. The c temperature, d pressure, and e mean-squared 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 rigid-body correlations (light grey). For the relevant partial PDFs we omit rigid-body terms contributions from the data shown in black.

Extended Data Fig. 3. Key bond-length and bond-angle distributions in the HRMC ACC model.

a The bond-length distributions for Ca-O divided into Ca-O_C and Ca-O_W contributions. The average Ca-O_C bond length (2.5(1) Å) is shifted to larger r values relative to the average Ca-O_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 Bond-angle distributions for O_C-Ca-O_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-Ca-O bond-angle distributions. The lower relative intensity of O_W-Ca-O_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 coarse-grained 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.