Geometrically frustrated interactions drive structural complexity in amorphous calcium carbonate

Amorphous calcium carbonate (ACC), with a composition of CaCO₃·xH₂O (x ≈ 1), is a fascinating material because of its prevalence in biomineralization processes and its surprising metastability. Unlike many simple inorganic salts, calcium carbonate readily precipitates in this hydrated, amorphous form, persisting for extended periods. This metastability can be further enhanced by incorporating dopants such as Mg²⁺ or PO₄³⁻. Conversely, varying pH or temperature can direct ACC to crystallize into various polymorphs. Organisms utilize this complex phase behavior to control the formation of shells and skeletal structures, leveraging the amorphous nature for transformations into different crystalline forms and for constructing hierarchical morphologies essential in biomineral architectures. Understanding the structural basis of ACC’s complex phase behavior, stemming from such a seemingly simple chemical system, is crucial for developing bio-inspired approaches in crystal engineering to control phase and morphology selection in synthetic systems. The theoretical understanding of phase complexity has been extensively studied in soft-matter systems governed by multi-well pair potentials. Simple single-well potentials (e.g., Lennard-Jones) typically lead to simple crystalline structures. However, the addition of multiple energy minima, particularly when the distances of these minima are incommensurate, can drive significant structural complexity due to geometric frustration. The double-well Lennard-Jones-Gauss (LJG) potential provides a compelling example, capable of stabilizing complex crystals with large unit cells or even frustrating crystallization entirely, depending on parameter values. Previous observations in synthetic ACC have hinted at competing length scales influencing its structure, manifested as two preferred Ca…Ca distances in the medium-range order. These distances are associated with different carbonate ion bridging modes: direct Ca-O-Ca pathways and indirect Ca-O-C-O-Ca pathways. This paper investigates the hypothesis that ACC's structure is governed by effective Ca²⁺ interactions reflecting these two competing length scales, leading to the observed structural complexity due to their competition.

Numerous studies have explored the structure and properties of ACC. Gebauer et al. (2008) demonstrated the existence of stable prenucleation calcium carbonate clusters. Aizenberg et al. (2002) and Raz et al. (2003) investigated the role of dopants and magnesium ions in stabilizing ACC. Aizenberg (2004) highlighted the bio-inspired approach to crystallization patterns. Addadi et al. (2003) and Weiner et al. (2005) emphasized the advantages of disorder in biomineralization. Beniash et al. (1997) showed the transformation of amorphous calcium carbonate into calcite during sea urchin larval spicule growth. Previous work by Michel et al. (2008) and Goodwin et al. (2010) characterized the structure of synthetic ACC, revealing a nanoporous structure and medium-range order. These studies, along with others employing techniques such as neutron and X-ray diffraction, solid-state NMR, and molecular dynamics simulations, provided valuable insights into ACC's structural characteristics, but a complete understanding of the underlying driving forces remained elusive. The theoretical framework of multi-well potentials and geometric frustration, as explored by Rechtsman et al. (2006), Engel and Trebin (2007, 2008), Jain et al. (2014), and Dshemuchadse et al. (2021), provides a useful lens to interpret the complexities observed in ACC.

This research employed a hybrid reverse Monte Carlo (HRMC) approach to generate high-quality atomistic models of ACC consistent with experimental data and energetically stable. The HRMC method simultaneously optimizes the goodness-of-fit to experimental X-ray total scattering data and the cohesive energy using state-of-the-art interatomic potentials. This approach addresses limitations of previous reverse Monte Carlo (RMC) studies that neglected energetic considerations, leading to physically unrealistic models. The study used experimental X-ray total scattering data from synthetic ACC, previously shown to be comparable to data from biogenic sources. The HRMC refinements involved 12,960 atoms (1,620 CaCO₃·H₂O units) and simulation cells of approximately 5 nm. Atomic moves were proposed and accepted or rejected based on a Metropolis Monte Carlo criterion and a cost function combining fit to the experimental data and energetic stability. The custom Python code utilized PyLammpsMPI, Numba, and ASE packages for simulations and data analysis. The effective Ca²⁺-ion pair potential, *u*Ca(r), was extracted from the HRMC models using a recently developed inversion algorithm, equating measured and calculated pair distribution functions via a test-particle insertion approach. The resulting potential was fitted to a modified Lennard-Jones-Gauss (LJG) model incorporating an additional repulsive Gaussian term. Monte Carlo (MC) and molecular dynamics (MD) simulations were conducted using the parameterized LJG potential to assess the validity of the derived potential and to create coarse-grained representations of ACC. Orientational correlation functions were calculated to verify the validity of using an isotropic effective pair potential. The analysis included examination of coordination environments (Ca²⁺ and CO₃²⁻ coordination numbers, bond lengths, and bond angles), partial pair distribution functions, and comparison of the coarse-grained LJG models to the full atomistic HRMC models.

The HRMC refinements yielded a high-quality structural model of ACC, characterized by a 'blue cheese' structure with CaCO₃-rich regions separated by filamentary water networks. This contrasts with previous RMC models that showed unrealistic charge separation. The model revealed average Ca²⁺ coordination numbers of around 7, consistent with prior computational and experimental studies. Importantly, the analysis of Ca-pair correlations in the HRMC model revealed two distinct maxima in the gCa(r) function, corresponding to Ca…Ca distances associated with Ca-O-Ca and Ca-O-C-O-Ca carbonate bridging pathways. The inversion of the gCa(r) function provided the effective Ca…Ca interaction potential, *u*Ca(r), which displayed two distinct minima, corresponding to the two preferred Ca…Ca separations. This double-well potential was well-represented by a modified LJG model, parameterized by ε (well depth), r₀ (Gaussian well position), and σ (Gaussian well width). The obtained LJG parameters fell within a region of the LJG phase diagram known to promote structural complexity and inhibit crystallization. Coarse-grained MC and MD simulations driven by the parameterized LJG potential reproduced key structural features of the HRMC model, including the inhomogeneous Ca distribution. The similarity extended beyond simple pair correlations to encompass higher-order correlations, ring statistics, and Voronoi volume distributions. The findings suggest that the inherent structural complexity and metastability of ACC originate from the geometric frustration arising from the competition between the two favored Ca…Ca separations mediated by the carbonate ions.

The successful mapping of the effective Ca…Ca interaction in ACC onto the LJG model offers a compelling explanation for its complex structure and metastability. The competition between the two favored Ca…Ca distances, combined with the density of ACC, makes it difficult to satisfy both simultaneously within a crystalline structure. The LJG model parameters, especially r₀ which is sensitive to the ratio of the Ca…Ca distances, lie in a region of the phase diagram that is known to exhibit high structural complexity and a suppression of crystallization. The incorporation of dopants like Mg²⁺ or PO₄³⁻ would introduce further disorder in the effective potential, favoring the amorphous state akin to the effect of magnetic exchange disorder in spin glasses. The results emphasize that focusing on effective interactions, rather than solely on the individual chemical components, is crucial for understanding the structure and behavior of complex materials. This study suggests that exploring the intermediate length scale of nanoscale materials might be a more productive avenue to find other real-world examples governed by isotropic multi-well potentials. The varying coordination modes of molecular ions often lead to multiple preferred counterion separations, and when geometrically frustrated, a multi-well effective potential is expected.

This study demonstrates that the complex structure and metastability of amorphous calcium carbonate are driven by geometrically frustrated effective interactions between Ca²⁺ ions, well-described by a Lennard-Jones-Gauss potential. The findings provide a fundamental understanding of ACC's structure and offer a new framework for the design of complex inorganic materials through 'interaction engineering'. Future work could focus on extending this approach to other poorly ordered inorganic solids, such as amorphous calcium phosphate and calcium-silicate-hydrates, and explore ways to control the effective potential through varying parameters like cation radius, hydration level, and dopant concentration.

The study's use of an isotropic effective pair potential assumes that local anisotropy is short-ranged. While orientational correlation functions indicate this is largely true, the effect of stronger anisotropy at higher densities, closer to crystallization, remains to be fully investigated. The influence of temperature and density on the LJG phase behavior is also not fully characterized, thus limiting the precise quantitative extrapolation from the LJG model to real-world conditions. Further work is needed to fully explore the phase space of the LJG potential and to extend the approach to more complex systems with anisotropic interactions.