Stability and Structure of Hydrated Amorphous Calcium Carbonate

Article pubs.acs.org/crystal

Stability and Structure of Hydrated Amorphous Calcium Carbonate Yuriy G. Bushuev,* Aaron R. Finney, and P. Mark Rodger Department of Chemistry, and Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, United Kingdom S Supporting Information *

ABSTRACT: The results of molecular dynamics simulations of hydrated amorphous calcium carbonate (CaCO3·nH2O: ACC) are presented. ACC properties were investigated on atomistic, supramolecular, and thermodynamic levels. The clustering of water occluded in the ionic ACC framework was found to be well described by percolation theory, and with a percolation transition for water through ACC at a hydration level, n, of ca. 0.8. Percolation in ACC systems is quantitatively similar to site percolation on a simple cubic lattice where the percolation threshold is observed at pc = 0.312. Predominantly fourfold tetrahedral molecular coordination of water molecules in the bulk liquid state is changed to sixfold connectivity in ACC. Kinetic stability of ACC is enhanced by dehydration and reaches maximal values when the water content is below the percolation threshold. The computed free energy shows a region of thermodynamic stability of hydrated ACC (1 < n < 6) with respect to calcite and pure water. This region is bounded by two crystallohydrates, monohydrocalcite (n = 1) and ikaite (n = 6), that have lower free energies than ACC. During dehydration at n < 1 the thermodynamic stability of ACC decreases, which favors the processes of nucleation and crystallization. On the other hand, water mobility within ACC also decreases during dehydration, thus making dehydration more difficult. So, the stability of hydrated ACC is controlled by a balance of two opposing factors: kinetics and thermodynamics.

1. INTRODUCTION Calcium carbonate is a ubiquitous material in nature, playing a significant role in areas such as global climate regulation and geology, as well as the building industry, agriculture, and other spheres of human activity. A special scientific interest is focused on the role of CaCO3 in biological materials. Biological systems exhibit a degree of control over the form and structure of inorganic minerals during biomineralization that cannot yet be reproduced in the laboratory.1−3 There are three anhydrous crystalline polymorphs of CaCO3 which may be ordered by increasing thermodynamic stability under standard conditions: vaterite, aragonite, and calcite. Two minerals, ikaite (CaCO3· 6H2O) and monohydrocalcite (CaCO3·1H2O: MHC), are hydrated forms of calcium carbonate. Ikaite is unstable at normal pressure and temperature,4,5 and depending on external conditions it decomposes into MHC or calcite and water. The coexistence of aragonite and calcite in adjacent microscale domains in mollusk shells,1,2,6 the assembly of highly specific and unusual submicron single crystal morphologies in coccolithophores,7 and the control of morphology on larger length scales in sea urchin spicules8,9 all play a functional role in imbuing essential material properties into the resultant biominerals. Much of the current research on carbonate biominerals is focused on the nature and role of amorphous calcium carbonate (ACC). Experimental studies8,10 have shown that ACC is a precursor phase during the growth of anhydrous crystalline polymorphs in biogenic and inorganic conditions. Pure ACC is a metastable material with a lifetime measured in minutes or © XXXX American Chemical Society

hours. Some organic and inorganic impurities in ACC particles, or additives in the parent solution, can promote ACC stability, leading to much longer lifetimes. Factors like pH are also important, affecting the structure and properties of the material formed through the carbonate/bicarbonate equilibrium and the presence of hydroxyl groups.11 In many situations, ACC initially appears in a hydrated form and subsequently transforms into anhydrous crystalline polymorphs. One of the key unsolved questions in this respect is the mechanism for the expulsion of water from hydrated ACC. While there is considerable debate surrounding the mechanism of crystallization from ACC, evidence has been provided which suggests crystal nucleation can take place within ACC, via solid state transformation.12−14 On the other hand, reprecipitation of ACC to feed the growth of crystal nuclei has also been observed,15,16 and so it is likely that both solid state and reprecipitation mechanisms are involved in the growth of crystalline CaCO3. These processes will not readily occur within hydrated ACC, however, since the process of expelling water would also form a barrier to further growth;17 thus it is perhaps not surprising that biogenic ACC has been found to dehydrate prior to the onset of crystallization.18 Elucidating the hydrated ACC → anhydrous ACC transition is therefore an important step in understanding, and hence in mimicking, biomineralization. Received: June 4, 2015 Revised: September 25, 2015

A

DOI: 10.1021/acs.cgd.5b00771 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

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MD simulations was DL_POLY 4.03.4.38 The Edvinsson algorithm39 was used to calculate water cluster statistics and properties. One difficulty in simulating amorphous solids at a molecular scale is that structural relaxation times are inevitably much longer than those accessible to molecular simulations, and while one can generate very long-lived metastable states, it is likely that they are at best a subset of the ensemble of amorphous microstructures found in the bulk material. In order to test the reproducibility of our results, therefore, we have generated model amorphous materials using two substantially different procedures; differences in properties between these two models can then be used to place confidence limits on calculated properties. A more extensive assessment of the inherent variability of ACC microstructures will be reported elsewhere. A set of CaCO3·nH2O configurations was prepared with a “random” particle distribution. For this purpose 2880 CaCO3 formula units (f.u.) and 2880 × n (n = 3 and 6) water molecules were placed in a simulation cell. A 3 ns NVT molecular dynamics simulation was performed at 750 K using a force field in which all electrostatic interactions were switched off and a Ca−Ca Lennard-Jones (LJ) interaction, which is absent in the Demichelis force field, added with parameters ε = 0.00674 eV, σ = 5.17 Å to prevent close contact of atoms or the (temporarily) uncharged Ca atoms. To prepare initial configurations for other n some randomly selected water molecules were removed from the simulated systems. The published intermolecular potential, i.e., including electrostatic interactions and excluding the Ca−Ca LJ term, was then reinstated and the systems relaxed in the NpT ensemble at 300 K, and 0.1 MPa. All simulations were carried out in cubic cells. To generate the second set of initial configurations at n ≤ 1, a sample of the crystalline monohydrocalcite (MHC) supercell containing 2592 CaCO3 f.u. was used. The crystal was melted at 700 K and pressure of 0.1 MPa during a 3 ns NpT simulation and the resulting structure was then relaxed at 300 K. In this case, simulations were conducted using parallelepiped periodic boundary conditions. Initial configurations for n < 1 were obtained by removing from the cell some randomly selected water molecules In all cases after preparation, NpT simulations were performed to allow for equilibration of the initial configurations. The degree of relaxation was monitored during these simulations using the time dependence of the system energy and volume; relaxation took 7−15 ns, depending on system composition. Production simulations were then performed with durations of ca. 10 ns.

Unfortunately, little is known about the thermodynamic and kinetic factors that might affect this transition. The enthalpy of anhydrous ACC crystallization has been well studied, and shown to depend strongly on the conditions of ACC synthesis (pH and chemical composition of solution), as well as the texture and a size of the final calcite particles;11,19−22 as a result, measurements of the enthalpy of calcite crystallization from ACC vary from −4 to −31 kJ/mol,10,21 and questions about the relationship between hydrated and anhydrous ACC enthalpies remains unsolved. The situation becomes even more complicated for the free energy, where there is essentially no quantitative information about the decrease in the entropy of water when it is incorporated within ACC23,24 or the entropy of any consequent changes to the amorphous CaCO3 framework induced by the water. With respect to kinetic factors, the mobility of water is known to be hindered by the properties of the CaCO3 framework,5,25 and potentially trapped in voids within the ACC. Both mobile and static states for water in synthetic ACC are observed in NMR,26,27 though the mobile water is not always seen in biogenic ACC.28 We are unaware of any experimental measurements of water diffusion coefficients within bulk ACC. There have been a number of previous simulation studies of ACC, but most have focused on small isolated ACC nanoparticles23,29−34 for which surface effects will bias the structure and dynamics of the water. Of the few studies that have considered bulk ACC,29,35,36 none have focused on the behavior of water within the material, and how this is affected by the hydration level. As already noted, such a characterization is crucial for understanding the hydrated → amorphous transition in ACC. This paper addresses precisely this issue, reporting the results of an extensive molecular dynamics (MD) study of the structural, thermodynamic, and dynamic properties of bulk ACC across a wide range of water content: CaCO3·nH2O for 0 ≤ n ≤ 6. The remainder of the paper is structured as follows. A comparison with experimental XRD results is given first, to demonstrate that the simulation model is consistent with known structural properties. This is followed by an analysis of the clustering of water within ACC, using percolation theory, and calculation of both water and ion diffusion coefficients. Finally, the simulations are used to calculate a number of thermodynamic properties for hydrated ACC and to estimate the free energy of ACC relative to calcite across the full range of hydration states considered.

3. RESULTS AND DISCUSSION The structure and properties of complex multicomponent systems may be characterized on several levels. Thermodynamic functions are averaged over the whole molecular ensemble of a system. They determine the stability of the system, but relaxation times may be very large in natural or synthetic systems, and metastable states can even persist on geological time scales. Kinetic stability is governed by supramolecular structure and the mobility of particles, by energetic barriers between free energy basins in configurational space, and by the gradient of chemical potentials. On an atomistic level, the quality of theoretical models may be tested by comparison with results obtained from experiments such as XRD. 3.1. Atomistic Structure. Pair Distribution Functions (PDFs) are one of the main characteristics of atomistic structure, and are often used to show the dependence of structure on thermodynamic state; in the current context, it is the dependence on water content that is of particular interest. The total distribution function (TDF), which is a weighted sum of PDFs, has been determined experimentally40 and may be compared with results from computer simulations. TDFs for two ACC hydration levels are presented in Figure 1, together with the experimental XRD data of Radha et al.40

2. METHODOLOGY Classical molecular dynamics (MD) simulations have been employed to investigate CaCO3−water mixtures. The force field of Demichelis et al.,30 which predicts accurately the structure and energetics of crystalline CaCO3 polymorphs, was selected. Within this force field the energy of water−water interactions was calculated with the flexible simple point charge (SPC/Fw) water potential,37 and water−mineral interactions were fitted to results from quantum calculations as well as solution data. All parameters of the force field were previously published,30 and are reproduced in the Supporting Information (SI) for completeness. For all systems, production simulations were performed in the NpT ensemble at T = 300 K and p = 0.1 MPa, employing a Nosé−Hoover thermostat and barostat with relaxation times of 0.1 and 1 ps, respectively. Equations of motion were integrated with the velocity Verlet algorithm using a time step of 1 fs. Periodic boundary conditions were used throughout. The cutoff for short-range interactions was set to 9.0 Å, as specified by the chosen force-field.30 Electrostatic interactions were treated using the smooth particle mesh Ewald (SPME) method. The version of the package used to run the B



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In contrast, the structure of the water included in ACC is very different from that of liquid water. From the PDFs presented in Figure 2a,b it is clear that water molecules form H-

Figure 1. Experimental (red curve)40 and calculated total distribution functions of amorphous CaCO3·nH2O, at n = 0.5 (blue curve) and n = 1 (black dash curve).

Calculations based on the two ACC initialization protocols showed no significant difference, and so only one calculated curve is reproduced in Figure 1. The TDFs were calculated according to the standard equation41 D(r ) = 4πρ0 rg x (r ) n

= 4πρ0 r

∑ cicj i,j=1

ZiZj n (∑k = 1 ckZk)2

[gij(r ) − 1]

(1)

where Zi and ci are the atomic number and the proportion of atoms of type i in the material, respectively, gij(r) are the PDFs for atoms of type i and j, and ρ0 = N/V is the average number density of the material. Comparison of the functions shows generally good agreement between experimental and calculated data. Across a wide range of composition, the calculated function depends only weakly on ACC hydration level. Some discrepancy between experimental and calculated functions is observed for 3.25 < r < 4 Å. Possible differences in the composition of the simulated and experimental systems should be noted: hydroxyl ions and calcium hydroxide may be present in the experimental sample, albeit at very low concentrations within the bulk, but were not included in the simulations. The force field has previously been shown to reproduce a number of CaCO3 properties well,23,30 and this additional comparison with experimental XRD data reinforces the conclusion that the presented ACC models do give a reasonable representation of the real materials. Since the TDF depends only weakly on the hydration level, it is reasonable to expect that many of the corresponding PDFs, gij, demonstrate the same behavior. PDFs at different hydration levels are presented in Figures S1 and S2 (SI) and demonstrate that the C-Oc, Ca-Oc, C-Ow, Ca-Ow, and C-C distributions show only a minimal variation of ACC structure across a wide range of hydration. Some variation is seen, however, in gCa−C and gCa−Ca, particularly at r = 5−7 Å. The largest change is observed in gOw‑Ow. The PDFs show close agreement across the two protocols for generating ACC (Figures S3 and S4). We conclude that the ionic coordination environment in ACC is preserved across a wide range of hydration levels, and that only modest variations in the midrange structure of the ionic network are required to accommodate the water.

Figure 2. Partial distribution functions (PDFs) gij for water in hydrated ACC, CaCO3 nH2O: (a) PDFs involving the water hydrogen atom for n = 1; (b) water oxygen PDF for n = 1; (c) water oxygen PDF for several hydration levels. The hydration levels displayed in (c) are most relevant for the discussion of percolation (see text); a greater range is displayed in Figures S1 and S2. PDFs for pure water are given as dashed lines in (a) and (b) for comparison.

bonds primarily with CO3 oxygen atoms instead of with other water molecules. A strong Ow−H correlation at 1.8 Å, observed in liquid water, is almost completely absent in ACC (Figure 2a), but is observed in the Oc−H curve; a second peak at 3.4 Å in the Oc−H distribution is also strongly reminiscent of the O− H distributions for pure water. Carbonate oxygens have 20% larger negative charge than water oxygen, and so stronger water−carbonate H-bonds are expected. The presence of two peaks for Oc−H near the major intermolecular peak seen for C



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Ow−H in pure liquid water means that predominantly linear O2C−O···H−OH bonds are formed, although the smaller depth of the first minimum indicates that some bifurcated O− CO2···H−OH geometries are present. Additional evidence of the absence of typical liquid water is seen in the gOO curves (Figure 2b). In the case of ACC, a broadening of the first peak of gOwOw compared with that seen in pure liquid water is observed. The peak is lower, wider, and has an asymmetric shape. Its maximum is shifted toward a larger interatomic distance, from 2.75 to3.0 Å. The first minimum of the function is reached at 3.8 Å (cf., 3.3 Å in liquid water); the distance exceeds the diameter of a water molecule (σ = 3.165 Å). The second peak, which characterizes longer range order, is at approximately the same distance as for liquid water (4.4 Å), but additional peaks at 5.2 and 6.2 Å are observed in the case of water in ACC. The positions of the four peaks point to a cubic ordering of water molecules instead of the tetrahedral coordination seen in bulk liquid water: in a simple cubic lattice the distances between the first, second, third, and fourth neighbor nodes are in the ratio 1.00:1.41:1.73:2.00, while the peak positions listed above for water in ACC (3.0 Å, 4.45 Å, 5.2 Å, 6.2 Å) give a ratio of 1.00:1.47:1.73:2.07. A potential criticism of any simulation of amorphous materials is that the time- and/or length-scales of the simulation are too small to sample the diverse range of nanostructures present in the amorphous solid. For this reason, we have used two very different methods for preparing the initial configurations for ACC (see Methodology). A detailed analysis of the PDFs (Figures S3 and S4) shows that the atomistic structure of hydrated ACC materials is the same in both cases. It is reasonable to assume that if ACC is characterized by multiple metastable states with different properties, then these two different preparation protocols would find different metastable states. While not comprehensive, the close agreement with these two protocols does suggest that the prepared CaCO3·nH2O mixtures show quasi-ergodic behavior, representing the same structural basins of configurational space, and so constitute a reliable and reproducible model. A more extensive examination of variability within modeled ACC is in progress, but is beyond the scope of this work. Given the quality of the comparison with experimental XRD data (Figure 1), we also conclude that they provide a good model of the ACC material used in those experiments. 3.2. Supramolecular Structure of Water in Hydrated ACC. We have already noted form the PDFs (Figure 2) that there is very little direct hydrogen bonding between water molecules. There is, however, still evidence of association of water molecules being mediated by carbonate ions. The Ow− Ow PDF shows a peak at 3 Å, which is inside the repulsive branch of the direct Ow−Ow (σ = 3.16 Å) interactions but can be explained by two water molecules hydrogen bonding to the same carbonate (giving rise to the strong Oc−H peak at 1.8 Å in Figure 2a). The Ow−Ow peak is broad, with the first minimum occurring at 3.8 Å; this is larger than would normally be expected for a direct interaction, but can again be explained by hydrogen bonding to the same carbonate. This peak is sensitive to water content (Figure 2c). Thus, in considering the micro- and mesoscale structure of water in ACC, it is useful to define carbonate-mediated water clusters based on a longer than usual distance criterion of about 3.8 Å. Such water clusters are visibly evident in molecular configurations taken from trajectories after relaxation from an initial configuration, in which water was uniformly distributed through ACC (Figure

S5). The shape and distribution of these water clusters suggests a network of water-filled pores and channels that can profitably be analyzed with percolation theory. 3.2.1. Percolation in Water and Aqueous Systems. Percolation theory42 has been used in the past to analyze the experimental properties of water and aqueous solutions43 and found to be in good agreement with computer simulations.44,45 In many studies the theory was tested for lattice models, because in this case large systems may be simulated to give a rigorous check of the theoretical predictions. The idealized lattice model is a simple approximation of a continuous system, and can be fruitful for understanding and highlighting basic properties of the system. The key theoretical parameters of the theory are the topology of the lattice and the probability of either bond formation (the bond percolation) or a lattice site occupation (the site percolation), p.42 All previously investigated systems showed a universal scaling of key properties near a percolation threshold. The cluster properties were found to depend only on the lattice topology and the probability, and were found to scale exponentially toward percolation with universal critical indexes that depend only on the spatial dimension. The spanning cluster at the percolation threshold for a 3D lattice has a fractal dimension of 2.523. In past applications of percolation theory to water, the notional “bonds” that form the clusters were usually identified with H-bonds, and so could be defined in terms of a set of threshold values for geometric quantities such the O−O distance and the O−H···O angle.44 For common threshold values, the liquid water H-bond network shows the same percolation properties as the model of bond percolation on a tetrahedral lattice (diamond-like structure observed in ice Ic).44 3.2.2. Water percolation in ACC. When percolation theory is applied to water in ACC the situation is more complicated. We have already noted from the PDFs that carbonate−water H-bonds are much more prevalent than water−water H-bonds and it is better to think of water−water association as being carbonate-mediated, with consequently longer Ow−Ow distances than one would use for pure water. In this work we present data calculated when connectivity is defined using several different distance thresholds: rcut values of 3.55, 3.65, and 3.8 Å. This range serves to demonstrate the invariance of our conclusions to the choice of rcut. Here the first rcut value corresponds to the position of the van der Waals interaction energy minimum for water molecules and may be attributed to molecular size; meanwhile the last value is the position of the first minimum in gOwOw: 3.8 Å. Both values may be taken as natural cutoff criteria, rcut, to define clustering of water molecules. It is possible to use a large number of criteria to define bonds. To study percolation in liquid water Geiger and Stanley44 used 32 energetic and 1 geometric criteria for bond formation between water molecules. In the present work, a similar approach has been adopted as described above, but in the case of ACC the investigated systems contain different numbers of water molecules. Therefore, chemical composition is an additional parameter and the probability of site occupation, which is the main parameter of the percolation theory, depends on both: a neighbor definition, and water content in ACC, p(rcut, n). The percolation threshold is the lowest concentration of water at which a single cluster spans the available space. At this threshold the percolated cluster has a fractal dimension. Our simulations of ACC were performed with periodic boundary D



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conditions, which introduces system size limitations. In such cases there are two commonly used definitions of when a cluster “spans the available space”: (a) if its length is larger than a size of the periodic cell in any direction;44 (b) if the “tail” of the cluster is connected to the “head” of a periodic image of the same cluster.39 In simulations with periodic boundaries, such clusters are considered to be infinite, while all others are finite. Both of these definitions have been used in the present work, and were found to give consistent results. 3.2.3. Scaling of Cluster Sizes. According to percolation theory, the distribution of molecules belonging to finite clusters has a singular point at the percolation threshold. In the case of spatially limited systems the distribution has a sharp maximum. Average cluster size is the first moment of the distribution, ns:

⟨S⟩ =

∑ sns /∑ ns s

s

Figure 4. Number of finite water clusters, ns, in amorphous CaCO3· nH2O as a function of cluster size, s near the percolation threshold using three values of the cluster “bond” distance, rcut: 3.55 (n = 1), 3.65 (n = 0.9), and 3.8 Å (n = 0.8). All linear fits were constrained to use the theoretical line slope, τ = 2.189. For clarity the plots for n = 1 and 0.8 are shifted vertically to ln(ns) + 1 and ln(ns) − 1, respectively.

(2)

where ns is the number of finite clusters containing s water molecules. Spanning (i.e., infinite) clusters are omitted from the calculation of ⟨S⟩. Average cluster sizes as a function of hydration level are presented in Figure 3 for three values of rcut (3.55, 3.65, and 3.8

follow the exponential size distribution with universal exponent of 2.189; it is also clear that this universal behavior close to percolation is followed irrespective of the value of rcut used to define the clusters. The radius of gyration (Rg) is a fundamental characteristic of finite clusters. It is calculated according to the formula ⎡ s ⎤ R g 2(s) = ⎢∑ (ri − ⟨r ⟩)2 ⎥ /s ⎢⎣ i = 1 ⎥⎦

s

⟨r ⟩ =

∑ ri/s i=1

(3)

where vectors ri define positions of water oxygens in the cluster. Near the percolation threshold Rg should increase with size s as a power function, Rg(s) ∼ s1/df, where df = 2.523 is the fractal dimension of clusters near the percolation threshold in 3-D space and is a universal constant that does not depend on the lattice topology. Plots of Rg(s) calculated from our ACC simulations are depicted in Figure 5. They are clearly in good quantitative agreement with this universal scaling law across 2 orders of magnitude variation in cluster size for compositions near percolation. 3.2.4. Lattice Approximation. The discussion above has shown that the distribution of water in ACC·nH2O is well

Figure 3. Average sizes ⟨S⟩ (eq 2) of finite water clusters in amorphous CaCO3·nH2O as a function of hydration level n. Data is given for three values of the cluster “bond” distance, rcut: 3.55, 3.65, and 3.8 Å.

Å). Below a percolation threshold, pc, all water molecules belong to finite clusters and the average size increases with increased water content in ACC; however for p(rcut, n) > pc most of the water molecules belong to the infinite (spanning) clusters, and so the average size of finite clusters decreases with increasing hydration level. Figure 3 shows that the composition for the percolation threshold (the maximum finite cluster size) does depend on the choice of rcut, with maxima at ACC compositions n = nwat/nfu = 0.8 ± 0.05 (rcut = 3.8 Å), 0.9 ± 0.05 (3.65 Å), and 1.0 ± 0.05 (3.55 Å); however, as shown hereafter, the three analyses still present a consistent picture of percolation and, particularly, universal scaling behavior and the same critical site occupation probability, pc. Percolation and phase transitions have many common features. A percolation threshold, pc, is a point of singularity for many properties, and those properties show a universal scaling behavior around percolation. In particular, the number of finite clusters should decrease with the cluster size according to ns ∼ s−τ, where τ = 2.189(1) is a universal constant.46 Distributions of ns at large s were calculated from our simulations and are depicted in Figure 4. For compositions close to percolation it is clear that the water clusters in ACC

Figure 5. Radius of gyration, Rg, of finite water clusters in amorphous CaCO3·nH2O as a function of cluster size, s, near the percolation threshold and using three values of the cluster “bond” distance, rcut: 3.55 (n = 1), 3.65 (n = 0.9), and 3.8 Å (n = 0.8). All linear fits are constrained to use the theoretical slope 1/df = 0.396. For clarity the plots for n = 1 and 0.8 are shifted vertically to ln(Rg) + 1 and ln(Rg) − 1, respectively. E



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occupancy at the percolation threshold, pc, does depend on the connectivity of the lattice, but not on the distance threshold used. For a simple cubic lattice, the critical probability is pc = 0.3116.46 Since, as noted above, the site occupancy can be calculated from ⟨Nn⟩/k, pc may be evaluated by calculating the average number of neighbors at the percolation threshold indicated by the peaks in Figure 3. Using k = 6, this gives pc = 0.312, 0.311, and 0.314 when using rcut = 3.55, 3.65, and 3.80 Å, respectively, all in good agreement with the required value of 0.3116 for a sixfold simple cubic lattice. In summary, combining the distribution function and percolation analysis, our results indicate that the ACC structure exhibits a sixfold ionic coordination environment that is independent of hydration level; this then imposes a sixfold “connectivity” on water (Figure 6), which is then incorporated into the material in a way that then modulates the intermediate-range structure of Ca2+ CO32− subnetwork. 3.2.5. Typical Clusters. The functions discussed so far characterize the properties of the finite clusters averaged over a statistical ensemble. Individual clusters have a specific shape and topology which depends on ACC composition and cluster size, s. Two examples of finite clusters (s = 50) of water in ACC for conditions near the percolation threshold are shown in Figure 7.

described by a percolation model, with a percolation threshold at n = nwat/nfu ≤ 1; the precise composition for this threshold does depend on the choice of “bonding” threshold, rcut, but we have shown that the ACC follows universal scaling behavior around percolation irrespective of the choice of rcut and n. It is therefore of interest to identify the topology (connectivity) of the underlying effective lattice describing the distribution of water in hydrated ACC. As noted above, a random lattice model is characterized by the topology (connectivity), k, of the lattice, and probability, p, of site occupancy. The proportion of occupied sites with i neighboring sites also occupied (mi) is then given by the binomial distribution: mi =

k! pi (1 − p)k − i i ! (k − i )!

(4)

We have already noted (Figure 2, and discussion thereof) that the position of the first four peaks in gOwOw matches well with those found in a simple cubic array, which suggests that k = 6. More quantitative confirmation that the clustering of water can be described by sixfold connectivity is seen in Figure 6.

Figure 7. Samples of finite clusters containing 50 water molecules. Water oxygen atoms are depicted by spheres, bonds connect oxygen atoms at r < 3.8 Å, and surfaces show the overall cluster shape.

Figure 6. Probability distribution with respect to the number of neighboring water molecules connected to any given water molecule in ACC (CaCO3·0.8H2O): calculated directly from simulation (black), and for random assignment on a simple cubic lattice (red; calculated from eq 4 with p fixed by the average number of neighbors observed in the simulation).

Molecules with 1, 2, or 3 “bonds” prevail in clusters. The molecules with two “bonds” form chains within the cluster, while molecules with three “bonds” enable the formation of branches and rings. All these topological elementschain fragments, chain ends, branch points, and ringsare visible in Figure 7. Such irregularly shaped water clusters are occluded in an ACC ionic framework, which may itself be considered to be built from “bonds” connecting neighboring Ca and C atoms (Figure S5). In most cases molecular surfaces of water clusters correspond to the shape of voids left in the ACC framework if the water is removed (without allowing the framework to relax), as shown in Figure 8. The chain-like fragment of the water cluster is occluded by a Ca-CO3 channel wall. The wall wraps the water cluster and mimics its shape. This figure is typical of all the clusters we have examined, in that the channels and voids have irregular shapes, with cross-sectional areas that change continually along the channels. Transport of water through these channels would be limited by the width of “bottle necks” unless there was a mechanism for the correlated motion of water and ions. In most cases, the CaCO3 framework closes around the finite water clusters, effectively forming cages rather

This depicts the probability distribution of finding water molecules with i neighborscalculated directly from our simulations for a composition near the percolation threshold together with the binomial distribution for k = 6 and using the probability calculated from the average number of water neighbors observed in the simulations (p = ⟨Nn⟩/k, where ⟨Nn⟩ is the average number of neighbor water molecules). The simulated distribution is very well described by the binomial distribution for a lattice with sixfold connectivity, and this quantitative agreement involves no fitted parameters. In general, near percolation, we find about 10% of water molecules are isolated (m0), 28% of molecules form dimers or terminate chain structure (m1), 33% of molecules have two neighboring waters and belong to chain fragments within clusters, and about 20% of water molecules form branching points in the cluster network (m3); of the remaining 10% of water molecules, none were observed to have connectivity higher than 6. As a final confirmation of the appropriateness of an underlying sixfold lattice, we note that the critical site F



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Figure 10. Self-diffusion coefficients for water and ions in hydrated ACC. Lines are included merely as a visual reference.

Figure 8. Fragment of a water cluster in an ACC channel. The surface of the water cluster is given in yellow, and the transparent blue surface represents the ACC channel walls.

10) and so it is reasonable to interpret these values as short time diffusion coefficients. Diffusion coefficients for Ca2+ and CO32− were very similar, and so data is given for an average of the two, Dion. Not surprisingly for an amorphous solid, the diffusion coefficients are small (∼104 times smaller than for liquid water,47 D = 2.3 × 10−9 m2 s−1 at 298 K). Diffusion coefficients for ionic species in calcite have been measured48 in the range 10−23−10−22 m2 s−1. This is much smaller mobility of ions than in ACC (ca. 10−14 m2 s−1). Two distinct regions can be identified in Figure 10. The diffusion coefficients of water and ions show approximately constant values at n < 0.8 and steady, near linear, increase with n when n > 0.9. Intriguingly, the change of behavior coincides with the percolation threshold (n ≈ 0.8−0.9) and is in the range of composition most relevant for biomineralization, which suggests that kinetic stabilization should not be overlooked in characterizing the dehydration of ACC prior to crystallization when biominerals form. Mobility of water within the ACC decreases during dehydration, making dehydration more difficult. 3.4. Thermodynamics. Excess thermodynamic properties of binary solutions characterize mixing effects. For a binary system, excess functions are calculated according to equation

than partially filled channels. This is illustrated in the Supporting Information Figures S5 and S6. In an unlimited system, at the percolation threshold the infinite cluster is a stochastic fractal. This means that the same structural fragments are observed at any scale. For simulated ACC systems, the actual size of clusters is always limited due to computational practicalities. Diversity in shape and topology of “infinite” clusters is observed in molecular configurations representing hydrated ACC. Three such clusters are shown in Figure 9. These clusters represent fragments of an actual fractal.

id ΔB = B12 − B12

(5)

where B12 is an arbitrary thermodynamic state function calculated per mole of mixture. For enthalpy and volume Bid12 = x1B1 + x2B2; for free energy the additional entropy of mixing term should be included. Here Bi is the thermodynamic molar state function of the pure component i, and xi is the mole fraction of component i. In the following, we identify the first component with anhydrous ACC (i = 1) and the second with liquid water (i = 2). 3.4.1. Excess Enthalpy and Volume. The excess volume and energy, calculated from eq 5, are shown in Figure 11. Both are negative at all concentrations investigated. The energy of mixing shows a maximum deviation from ideality of ca. −20 kJ/ mol at x2 = 0.5−0.65 (i.e., n = 1−2), a composition range that is typical of synthetic ACC. Simulations using randomized and melted MHC initiation protocols were in quantitative agreement. The observed volume contraction is 6 cm3/mol and can be understood in terms of the loss of tetrahedral liquid water structure in favor of the sixfold structure embedded within the ACC, as discussed above. Both the excess energy and volume are of greater magnitude than normally found with aqueous binary solutions. For example, water/dimethylacetamide mixtures give maximum deviations from ideality of ΔH = −3

Figure 9. Examples of spanning water clusters contained in ACC near a percolation threshold.

The length of each cluster exceeds the cell size (ca. 60 Å). The first cluster represents a sample where the structure is predominantly chain-like. Both chain-like backbones and more bulky compact fragments are visible in the other two clusters. 3.3. Diffusion. Self-diffusion coefficients for water and ions have been calculated from the linear slope of the mean square displacements (MSD) of molecules over time, and the results are presented in Figure 10. Examples of MSD dependences and statistical uncertainties of fitting are presented in Figure S7. The overall average displacement is small, and it is possible that a long-time diffusion dynamic would emerge once the ions begin to diffuse several ionic radii. Nonetheless, the calculated curves clearly show linear behavior over 10 ns, and statistical uncertainties in determining the slopes are no more than 1.5% (i.e., smaller that the size of the points plotted in Figure G



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Figure 11. Excess enthalpies and volumes of anhydrous ACC−water mixtures, CaCO3·nH2O. The two sets of data were obtained from the randomized and the melted MHC starting configurations. Lines are fits using a Redlich−Kister polynomial as per eq 6 with coefficients: (excess enthalpy/kJ mol−1) a0 = −76.02, a1 = −34.17; (excess volume/ cm3 mol−1) a0 = −20.78, a1 = −13.79, a2 = −9.95.

Figure 12. Excess enthalpy (empty squares, black line) and estimated free energy difference (red dashed line for ΔS1 = 21 J/(mol K); magenta dash-dotted line for ΔS1 = 42 J/(mol K), eqs 8 and 9) of ACC with respect to calcite and liquid water. Experimental enthalpies10 for hydrated ACC are presented as blue circles. Experimental enthalpies10,11,21,22 for anhydrous ACC, synthesized in ethanol and aqueous media at different pH, are presented as dark blue circles.

kJ/mol and ΔV = −1.6 cm3/mol.49,50 The strong ionic interactions and large ACC porosity create conditions for a large water adsorption. The concentration dependence of both sets of data was well fitted by a Redlich−Kister polynomial expansion

synthetic ACC using differential scanning calorimetry at T = 550−650 K and found ΔHcalcite‑ACC = 12.8−15.4 kJ/mol. Different thermodynamic stability is also demonstrated by ACC samples synthesized in water (ΔHcalcite‑ACC = 13.1 kJ/mol) and ethanol (31 kJ/mol).21 According to our simulations, the enthalpy of anhydrous ACC with respect to calcite is about 40 kJ/mol, which is significantly larger than reported experimental values. However, the comparison between calculated and experimental enthalpies is complicated by the phase changes that are seen in experiments. Peaks in the DTA/DSC curves, corresponding to crystallization from ACC, are observed at high temperature, whereas the computer simulations were carried out at 300 K. Slow structural relaxation processes, which occur on time scales measured in minutes or even hours, are activated only at high temperature, and cannot easily be taken into account during the simulations. Further, the enthalpies of calcite and ACC depend on temperature. For calcite there is an increase of ca. 30 kJ/mol on heating from 300 to 600 K.51 For ACC, however, it is difficult to estimate, since substantial and uncharacterized structural changes in the amorphous network must be expected over such a large temperature range. Consequently, the correction needed to estimate the experimental enthalpy of calcite formation from ACC at 300 K from that observed for the phase change at high temperature in the DTA/DSC experiments is unknown. At 300 K the crystallization enthalpy of biogenic ACC (CaCO3·0.25H2O) into biogenic calcite has been measured experimentally (ΔHACC‑calcite = 14.3 kJ/mol),10 and the value does correspond to the enthalpy change measured at 600 K for synthetic ACC samples. However, since the chemical composition, properties, structure, and texture of biogenic and synthetic ACC samples differ substantially this similarity in ΔH values must be considered coincidental. According to our calculated data, the thermodynamic instability of ACC increases rapidly with dehydration (Figure 12), and the crystallization enthalpy (ΔHACC‑calcite) of CaCO3·0.25H2O is −23 kJ/mol, which is not very different from crystallization enthalpy of biogenic ACC, suggesting that uncertainties in the measurement of

n

ΔA = x 2(1 − x 2) ∑ ai(2x 2 − 1)i i=0

(6)

where ai are adjustable parameters and x2 is mole fraction of water. Results of data fitting are presented in Figure 11. Crystalline phases of hydrated CaCO3 are more energetically stable than ACC with the same composition, with ΔΔH = 3 kJ/mol (MHC) and 2.2 kJ/mol (ikaite). Recognizing that the temperature for ikaite decomposition is about 300 K, the difference between ikaite and ACC·6H2O entropies, ΔΔS, may be estimated as ca. 7 J mol−1 K−1 (from ΔΔH = TΔΔS at the two-phase equilibrium). 3.4.2. Comparison with Experiments. Enthalpies of hydrated ACC have been determined experimentally10 at 300 K with respect to calcite, the most stable crystalline polymorph under these conditions. The enthalpy change for the reaction CaCO3 ·nH 2O(ACC) = CaCO3(calcite) + nH 2O

(7)

is reported to be (−17) − (−24) kJ/mol at n = 1.2−1.6 for synthetic ACC. Excess (mixing) enthalpies, calculated per mole of mixture, are presented in Figure 12. Note that these experimental data use calcite, instead of anhydrous ACC, as component 1. Calculated enthalpies for the hydrated ACC are lower than experimental values, but the trend for them to increase on dehydration is clear. Maximal deviations from experimental enthalpies are ∼10 kJ/mol at x2 = 0.5−0.6. It has been shown11,21,22 that the experimental enthalpies depend on factors such as the method of synthesis, size of samples, and lifetime of the unstable ACC. The crystallization temperature and ΔHcalcite‑ACC also increase with increasing pH in aqueous solution, in part because the size and the properties of initial ACC and final calcite particles varies with pH.11 As a result, the enthalpies of crystallization reported in the literature are scattered and vary within a large interval. Radha et al.10 have measured the enthalpies for a set of H



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given the observed stability of these crystallohydrates. Synthetic ACC samples typically have chemical compositions that lie within this estimated zone of stability. For real ACC samples, depending on external conditions, calcite is usually formed only following crystallization of vaterite from ACC,15,16 in agreement with Ostwald’s rule of stages.53 Vaterite has higher free energy than calcite, which means that the nucleation barrier for ACC crystallization into vaterite is lower, and hydrated ACC will have a wider region of stability with respect to vaterite than it does with respect to calcite.

water content within experimental ACC are another source of difficulty in making quantitative comparisons between experiments, and between experiments and simulations. The metastable nature of ACC, and the diversity of shapes, sizes, and textures found in real samples, prevent an exact reproducible measurement of ΔH and, as a result, the experimental ACC crystallization enthalpies vary from −4 to −31 kJ/mol.10,11,21,22 Taking into account the effectively infinite size of the simulated ACC and calcite, and the defectfree calcite structure, we may conclude that the calculated enthalpy of ACC crystallization (−42.1 kJ/mol) does reasonably correspond to the experimental range of values. 3.4.3. Free Energy. According to the results of simulations (Figure 12) a region with negative enthalpy with respect to calcite and liquid water is observed at x2 > 0.55. However, it is the free energy that truly determines the stability of ACC. The free energy of ACC may be expressed as

4. CONCLUSIONS In this paper we present results of computer simulations of amorphous calcium carbonate (ACC), CaCO3·nH2O, for a range of hydration levels 0 ≤ n ≤ 6. Two different protocols were used to generate the ACC configurations, and both were shown to give essentially the same results. Partial and total radial distribution functions calculated from the simulations are in good agreement with experiment, and confirm that the computer models used give a reasonable description of synthetic ACC materials. The PDFs show only a weak dependence on the ACC water content across a wide range of compositions: dehydration of ACC is accompanied by relatively minor reconstruction of the ionic network structure formed by the calcium carbonate. More substantial changes were observed at a supramolecular level with clustering of water within the ACC, and this was found to be well described by percolation theory. A percolation transition was identified at a composition CaCO3·nH2O for n ≈ 0.8, based on a cluster connection distance corresponding to the first minimum in the Ow−Ow radial distribution function (3.8 Å), and the size and radius of the finite water clusters were found to follow universal scaling laws with universal constants for compositions close to percolation. Intriguingly, the percolation analysis suggests that the arrangement of water in ACC is best described by a 6-connected lattice (i.e., equivalent to a simple cubic lattice); the 6-connected lattice was also found to be consistent with the position of the peaks in the Ow−Ow radial distribution function. This contrasts with liquid water and monohydrocalcite, both of which show a predominance of fourfold coordination for water. Our analysis of the hydrated ACC simulations has also enabled us to estimate how the free energy of ACC changes with water content. The computed free energy shows a region of thermodynamic stability/metastability for hydrated ACC with respect to calcite and pure water. This zone of stability is bounded by the two crystallohydrates of calcium carbonate: MHC and ikaite. Both have lower free energies than ACC at the respective hydration level, and their positions in the diagram are close to the borders of the ACC stability region. Calculations show that below n = 1 ACC becomes unstable with respect to calcite and water, with the instability increasing as n approaches zero. Given that biogenic calcite formation is known to involve a hydrous → anhydrous ACC transition, this increasing thermodynamic instability, coupled with the intrinsically slow dynamics for any rearrangement of the amorphous network in ACC and the nature of the carbonate-mediated clustering of water in hydrated ACC, will have substantial implications for the mechanism by which water can be expelled from the ACC during biomineral formation.

ΔG(x 2) = ΔH(x 2) + RT[(1 − x 2) ln(1 − x 2) + x 2 ln(x 2)] − Tf (x 2)

(8)

where the entropy has been subdivided into ideal entropy of mixing, and a correction f(x2) describes the entropy change due to the nonideal mixing of calcite and pure water. This function is unknown but an estimate may be obtained using the equation f (x 2) ≈ ΔS1(1 − x 2) + ΔS2x 2

(9)

In this equation, ΔS1 is the entropy change for the transition of calcite to a disordered ACC. This term is positive and favors ACC formation. ΔS2 is the associated entropy change for water, and is negative because water molecules are much more confined in ACC than in liquid water. According to the experimental data,24 water loses entropy during crystallohydrate formation. For MHC and ikaite the values ΔS2 are approximately the same: − 32 and 33.5 J/(mol K), respectively.24 Water in hydrated ACC is not so strongly ordered as in the crystals, and so the corresponding entropy loss must be smaller; it probably also depends on ACC composition. To estimate ΔS1, we have used a published theoretical correlation51 for entropy of disordering due to high temperature phase transition of calcite: ΔS1 = 0.676 (1260 − T)1/2, which gives a value of 21 J mol−1 K−1 at 300 K. The corresponding enthalpy of the transformation (ΔH1) is 9.8 kJ/ mol.51 These estimates were obtained from high temperature (T = 1240−1260 K) calcite crystal transformations from R3̅c to R3̅m symmetry,51,52 which involve a disordering of the carbonate orientation. We note that ACC is more disordered than the high temperature R3m ̅ phase of calcite, and so both free energy terms (ΔS and ΔH) for this process provide lower bounds for the magnitude of the corresponding terms for a calcite R3̅c → ACC transformation. The Gibbs free energy as a function of ACC composition is also presented in Figure 12. Based on the above discussion, ΔG has been obtained from the enthalpy curve using eqs 8 and 9 with ΔS2 = 33 J mol−1 K−1, and two values for ΔS1: 21 and 42 J mol−1 K−1. The last value is twice the value of the entropy for the calcite high temperature phase transition discussed above, and was used to demonstrate the influence of possible uncertainties in ΔS1. According to these estimates, there is a region of bulk ACC stability covering the range of water mole fraction 0.5 < x2 < 0.85 (1 < n < 6). MHC and ikaite are near the limiting points of this range, and with lower free energy than ACC with the same water contents, as should be expected I



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(11) Koga, N.; Nakagoe, Y.; Tanaka, H. Crystallization of Amorphous Calcium Carbonate. Thermochim. Acta 1998, 318, 239− 244. (12) Pouget, E.; Bomans, P.; Goos, J. The Initial Stages of TemplateControlled CaCO3 Formation Revealed by Cryo-TEM. Science 2009, 323, 1455−1458. (13) Pouget, E. M.; Bomans, P. H. H.; Dey, A.; Frederik, P. M.; de With, G.; Sommerdijk, N. A. J. M. The Development of Morphology and Structure in Hexagonal Vaterite. J. Am. Chem. Soc. 2010, 132, 11560−11565. (14) Gebauer, D.; Völkel, A.; Cölfen, H. Stable Prenucleation Calcium Carbonate Clusters. Science 2008, 322, 1819−1822. (15) Bots, P.; Benning, L. G.; Rodriguez-Blanco, J.-D.; RoncalHerrero, T.; Shaw, S. Mechanistic Insights into the Crystallization of Amorphous Calcium Carbonate (ACC). Cryst. Growth Des. 2012, 12, 3806−3814. (16) Rodriguez-Blanco, J. D.; Shaw, S.; Benning, L. G. The Kinetics and Mechanisms of Amorphous Calcium Carbonate (ACC) Crystallization to Calcite, via Vaterite. Nanoscale 2011, 3, 265−271. (17) Aizenberg, J.; Muller, D. a; Grazul, J. L.; Hamann, D. R. Direct Fabrication of Large Micropatterned Single Crystals. Science 2003, 299, 1205−1208. (18) Gong, Y.; Killian, C.; Olson, I. C.; Appathurai, N. P.; Amasino, A. L.; Martin, M. C.; Holt, L. J.; Wilt, F. H.; Gilbert, P. U. P. A. Phase Transitions in Biogenic Amorphous Calcium Carbonate. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 6088−6093. (19) Kimura, T.; Koga, N. Thermal Dehydration of Monohydrocalcite: Overall Kinetics and Physico-Geometrical Mechanisms. J. Phys. Chem. A 2011, 115, 10491−10501. (20) Kimura, T.; Koga, N. Monohydrocalcite in Comparison with Hydrated Amorphous Calcium Carbonate: Precipitation Condition and Thermal Behavior. Cryst. Growth Des. 2011, 11, 3877−3884. (21) Koga, N.; Yamane, Y. Thermal Behaviors of Amorphous Calcium Carbonates Prepared in Aqueous and Ethanol Media. J. Therm. Anal. Calorim. 2008, 94, 379−387. (22) Koga, N.; Yamane, Y.; Kimura, T. Thermally Induced Transformations of Calcium Carbonate Polymorphs Precipitated Selectively in Ethanol/Water Solutions. Thermochim. Acta 2011, 512, 13−21. (23) Raiteri, P.; Gale, J. D. Water Is the Key to Nonclassical Nucleation of Amorphous Calcium Carbonate. J. Am. Chem. Soc. 2010, 132, 17623−17634. (24) Königsberger, E.; Königsberger, L. C.; Gamsjager, H. LowTemperature Thermodynamic Model for the System Na2CO3MgCO3-CaCO3-H2O. Geochim. Cosmochim. Acta 1999, 63, 3105− 3199. (25) Michel, F. M.; MacDonald, J.; Feng, J.; Phillips, B. L.; Ehm, L.; Tarabrella, C.; Parise, J. B.; Reeder, R. J. Structural Characteristics of Synthetic Amorphous Calcium Carbonate. Chem. Mater. 2008, 20, 4720−4728. (26) Ihli, J.; Wong, W. C.; Noel, E. H.; Kim, Y.-Y.; Kulak, A. N.; Christenson, H. K.; Duer, M. J.; Meldrum, F. C. Dehydration and Crystallization of Amorphous Calcium Carbonate in Solution and in Air. Nat. Commun. 2014, 5, 1−10. (27) Schmidt, M. P.; Ilott, A. J.; Phillips, B. L.; Reeder, R. J. Structural Changes upon Dehydration of Amorphous Calcium Carbonate. Cryst. Growth Des. 2014, 14, 938−951. (28) Reeder, R. J.; Tang, Y.; Schmidt, M. P.; Kubista, L. M.; Cowan, D. F.; Phillips, B. L. Characterization of Structure in Biogenic Amorphous Calcium Carbonate: Pair Distribution Function and Nuclear Magnetic Resonance Studies of Lobster Gastrolith. Cryst. Growth Des. 2013, 13, 1905−1914. (29) Singer, J.; Yazaydin, A. Structure and Transformation of Amorphous Calcium Carbonate: A Solid-State 43Ca NMR and Computational Molecular Dynamics Investigation. Chem. Mater. 2012, 24, 1828−1836. (30) Demichelis, R.; Raiteri, P.; Gale, J. D.; Quigley, D.; Gebauer, D. Stable Prenucleation Mineral Clusters Are Liquid-like Ionic Polymers. Nat. Commun. 2011, 2, 590.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.5b00771. Plots and pictures of molecular configurations (PDF) DL_POLY input files with the used force field (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the EPSRC under grant EP/ I001514/1. Computer time was provided by MidPlus Regional Centre under EPSRC grant EP/K000128/1 and by the Scientific Computing Research Technology Platform at Warwick University. Authors thank A. Fernandez-Martinez for providing the experimental total distribution function of ACC

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REFERENCES

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Stability and Structure of Hydrated Amorphous Calcium Carbonate

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