Journal of Crystal Growth 603 (2023) 126978

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Journal of Crystal Growth

journal homepage: www.elsevier.com/locate/jcrysgro

The transformation of amorphous calcium carbonate to calcite and classical

nucleation theory
C.L. Freeman *, J.H. Harding
Department of Materials Science and Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom

A R T I C L E I N F O A B S T R A C T

Communicated by James J. De Yoreo The interfaces of calcite with water, dehydrated and hydrated amorphous calcium carbonate are studied with
molecular dynamics simulations. The interfacial energies demonstrate that the calcite interface is most stable
Keywords: when in contact with water or low concentration solutions rather than amorphous calcium carbonate. These
A1 Interfaces values are used to test the interplay between supersaturation and the interfacial energy for calcite. They
A1 Computer simulation
demonstrate that a dissolution-reprecipitation process should always be energetically preferred to a solid state
A1 Supersaturated solutions
transformation of amorphous calcium carbonate to calcite.
A2 Growth from solutions
B1 Minerals
B1 Calcium compounds

1. Introduction phase may proceed by various mechanisms. For CaCO3 the majority of
authors have reported that the conversion process of the ACC phase to
There is a large (and still growing) literature on the possible mech­ calcite (or other polymorphs) proceeds by either a dissolution, re-
anisms by which calcium carbonate crystals are produced from a su­ precipitation processes [10–13] or nucleation on the surface of precur­
persaturated solution. Depending on the authors and the experimental sor phases [14]. The assumption is that the supersaturation reduces
conditions, prenucleation (nanometre-sized) clusters [1,2], dense liquid during growth of the ACC phase and that this ultimately leads to
phases [3] solid amorphous clusters of various sizes and compositions instability of the ACC phase and its dissolution providing the opportu­
[1], crystalline particles [4] and spinodal decompositions [5] have been nity for formation of a more stable crystalline phase. The majority of
invoked either singly or in combination to give a multi-step pathway. reports of a solid state transformation presume direct dehydration of
These varied mechanisms may coexist; the dominating mechanism bulk ACC [15–18] but there are also cases involving aggregation of
depending on conditions. Studies [6] on the solution speciation at mild nanoparticles [19,20] and hetero-nucleation at surfaces [21]. These all
supersaturation suggest that the solution contains free cations and an­ suggest that the ACC phase may also interconvert without dissolution.
ions (CO2−3 and/or HCO3 depending on the conditions) and a limited
−
This raises the question - under what conditions is a dissolution re-
population of larger ion clusters that are formed by density fluctuations precipitation process preferred to a solid state transition?
in the solution, consistent with classical nucleation theory [7]. On the Unless the polymorph formed is ikaite or calcium carbonate mono­
other hand, it has been claimed that these larger clusters are thermo­ hydrate a solid state transformation requires the expulsion of water. This
dynamically stable and, with increasing supersaturation, can aggregate process is generally thought to involve a high energy barrier although
to form an amorphous phase [8] which is usually hydrated. This can the debate continues as to whether it is thermodynamic or kinetic in
then convert to one of the calcium carbonate polymorphs (usually origin [17,22]. Albéric et al [22] argue strongly for the importance of
calcite or vaterite; aragonite can also be formed in the presence of water in the stabilisation of ACC. Although the transformation enthalpy
suitable additives [1]). The formation of other less stable phases before from ACC to calcite has a linear correlation with the water content, the
formation of the final stable solid phase (Ostwald’s law of stages) is free energy of transformation is independent of this from the anhydrous
common in solution-based nucleation [9]. The formation of these phases case to a composition of CaCO3⋅1.3H2O (a mole fraction of 57% water).
means that we are altering the supersaturation of our system by This is ascribed to a countervailing entropy term. Water undoubtedly
depleting ions. The conversion of these early phases to our final stable has a strong effect on the ion mobility. Both advanced experimental

* Corresponding author.
E-mail address: c.l.freeman@sheffield.ac.uk (C.L. Freeman).

https://doi.org/10.1016/j.jcrysgro.2022.126978
Received 14 June 2022; Received in revised form 21 October 2022; Accepted 4 November 2022
Available online 8 November 2022
0022-0248/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
C.L. Freeman and J.H. Harding Journal of Crystal Growth 603 (2023) 126978

characterisation techniques [1,18,23] and molecular dynamics simula­ where continued growth becomes spontaneous.
tions [24] show that there are two states of water in ACC, “mobile” and Akin depends on the mechanism whereby the structural unit goes
“rigid” and the ratio between them controls both the degree of order in from the bulk solution to attachment to the growing nucleus. The kinetic
ACC and its stability against crystallisation [25]. The synthesis condi­ factor is given by [30–32]
tions are likely to control this ratio. √̅̅̅̅̅̅( )
γ kT
Whatever pathway the formation of calcite follows, at some point the Akin = exp( − ΔGA /kT) (5)
new, separate phase must form an interface with the surrounding me­ kT hR2
dium. This process can be understood from the viewpoint of classical where h is Planck’s constant, R the radius of the critical nucleus and
nucleation theory. There are many extended reviews of classical ΔGA the activation free energy of the process of transferring a structural
nucleation theory (CNT) e.g. [9,26–28], so it is not necessary to give unit of the nucleus from the solution to the nucleus. If Akin is dominated
more than a brief outline here. The earliest forms of CNT confined by the process of diffusion of the mobile species in solution, it can be
themselves to the formation of a condensed phase from its vapour. It was approximated by Akin ≈ D/Ω5/3 where D is the diffusion coefficient of the
assumed that the condensed phase was favoured over the vapour phase species in water. Akin is of the order of 1039 m− 3s− 1
under the conditions of the experiments, giving a chemical potential In this paper we attempt to understand the interface between calcite
driving force arising from the difference between the bulk free energy of and ACC (or a highly saturated water precursor). Assuming the forma­
the condensed phase and that of the vapour. This driving force has to tion of calcite will occur from a solution or an ACC phase, then by
overcome the free energy penalty associated with the formation of an examining this interface and determining the interfacial energy we can
interface between the condensed and vapour phases and depends on the comment on calcite nucleation. Our study allows us to examine the role
nature of the interface formed. The nucleus is treated as a macroscopic of supersaturation variation at the nucleating crystal interface through a
object, with a size-independent chemical potential. Similarly, the classical nucleation theory interpretation.
interface is presumed to be a single interface and any effects of nucleus
size or faceting are ignored. When this approach was extended to the 2. Methods
precipitation of a solid from solution, the partial pressure (strictly
speaking the fugacity) of the vapour was replaced by the activity of the All molecular dynamics simulations were performed using DLPOLY
solute in solution. The change in chemical potential between a species in Classic [33] with a timestep of 0.5 fs. Forcefield terms were taken from
a homogeneous solution of a given activity and the chemical potential of Raiteri et al [34]. Three-dimensional simulation supercells with periodic
the condensed phase (assumed to be size-independent) is the supersat­ boundary conditions containing an interface between calcite and hy­
uration of the species in solution and represents the driving force of drated amorphous calcium carbonate (h-ACC) were constructed for
phase separation of the solute from the solvent. This model introduces different compositions of h-ACC using the following procedure. Calcite
further complications. In solutions, except in the ideal case of infinite slabs exposing the (10.4) surface of 576 formula units were built with an
dilution, the activity is not always well approximated by the concen­ approximate surface area of 2650 Å2 for each surface. Simulation cells of
tration. Also, when an interface is present the local solute concentrations h-ACC were then made using the PACKMOL package [35] with
close to it may differ from the bulk [29]. This may produce a different approximately 1200 H2O + CaCO3 units for mole fractions of CaCO3 of
interfacial energy to that expected at the concentration and also one 0.005, 0.01, 0.03, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4 0.5, 0.6, 0.7, 0.8 0.9 and
which will could vary during the nucleation process as the ion concen­ 1.0. Table S1 in the appendix lists the specific atom numbers for each
tration varies. simulation. Following a similar procedure to Malini et al, the ACC cells
In the basic version of CNT, the nucleation rate, J, (i.e. the number of were heated to 3000 K in steps of 300 K (i.e. 300 K, 600 K, 900 K etc) for
thermodynamically stable nuclei formed per unit volume of solution) is 0.5 ns under a NVT ensemble. These were then maintained at 3000 K for
given by 2 ns. The cells were then quenched back down to 300 K in steps of 300 K
J = Akin exp( − ΔGc /kT) (1) for 250 ps. These cells were then relaxed using a NPT Nosé-Hoover
ensemble (0.01 ps thermostat and 0.05 ps barostat relaxation times
where Akin is a kinetic prefactor and ΔGc is the free energy barrier to respectively) at a pressure of 0 atm and a temperature of 300 K. After
the formation of a nucleus that will grow without limit to form a relaxation each cell was joined to a (10.4) calcite slab to make the final
macroscopic crystal. For a spherical nucleus we can write a general simulation supercell (see Fig. 1). The supercells were then equilibrated
expression for the free energy at 0 atm. and 300 K until convergence was achieved in energy and
4 3 volume (typically this required 2–3 ns). Three different configurations
ΔG = − π r Δμ + 4πr2 γ (2) were run for each solution composition and the configuration of lowest
3
energy was selected for further analysis. The variation in interfacial
where r is the radius of the nucleus, Δμ is the change in chemical
energy between the lowest and highest interfaces is listed in Table S1.
potential between solution and bulk solid and can be written as
The interfacial energy, γ, is the energy of the calcite h-ACC interface with
(∏ )
respect to bulk pure calcite and bulk h-ACC (i.e. not a cleavage energy).
Δμ = kTln ai /Ksp (3)
It was therefore calculated using
where ai are the activities of dissolved components of the molecular Ecalcite∨h− − bEcalcite − Eh−
(6)
ACC ACC
formula unit of the crystal and Ksp is the product of the activities of those γ=
2A
components when the solution is in equilibrium with the bulk crystal. γ
where Ecalcite\/h-ACC is the configurational energy of the simulation
is the interfacial energy. Differentiation of equation (1) gives the ther­
supercell with the calcite h-ACC interface. Ecalcite is the configurational
modynamic barrier to nucleation, ΔGc,
energy of the same number of formula units (b) of calcite as present in
ΔGc = f γ 3 Ω2 /(Δμ)2 ≡ f γ 3 Ω2 /k2 T 2 σ2 (4) the simulation supercell and Eh-ACC is the configurational energy of the
(∏ ) relaxed cell of hydrated ACC. A is the area of the interface between
where the supersaturation, σ = ln ai /Ksp . f is a shape factor which calcite and h-ACC.
for a spherical nucleus is equal to 16π /3, Ω is the molar volume of the We use our calculated values of γ combined with literature values [4]
structural unit added to the cluster. From equation (4) it is clear that in equation (4) to calculate the barrier to nucleation at several super­
CNT suggests that the change in free energy of growing nuclei is saturation values. Alternatively, when considering a solid state trans­
dominated by interfacial energies at small sizes. Nuclei with relatively formation and nucleation rather than solution based nucleation, the
low interfacial energies are more likely to grow beyond a critical size, supersaturation, σ, can be replaced with the free energy difference

2
C.L. Freeman and J.H. Harding Journal of Crystal Growth 603 (2023) 126978

Fig. 2. Interfacial energy of calcite (10.4) surface with h-ACC for

0 < xCaCO3 < 1. Error bar shown covers the range between the minimum value
calculated for the three configurations (purple circle) and the maximum value
for the three configurations (top of the error bar). (For interpretation of the
references to colour in this figure legend, the reader is referred to the web
version of this article.)

affect the interfacial energy significantly. However it will still play an

Fig. 1. Simulation cell for interfacial calculations showing z-axis definition for important role as the driving force for nucleation and altering the den­
the other Figures. Oxygen (red), carbon (blue), calcium (blue), hydrogen (grey). sity of ions in the solution volume. Comparison between the three
(For interpretation of the references to colour in this figure legend, the reader is different configurations for each composition is listed in supplementary
referred to the web version of this article.) Table S1 and the effect is shown in Fig. 2. These show the average
variation between the three configurations is 0.03 Jm− 2. The variation is
between the phases where k2BT2σ2 is replaced with ΔG2. The energy larger for the higher CaCO3 mole fraction systems which can be un­
change for ACC - calcite is taken from a calculation of the mixing derstood by the slower dynamics of these systems which means the final
configurational energy of calcite and ACC, energy will be more dependent on the starting configurations. The
largest variation is 0.1 Jm− 2 for xCaCO3 = 0.9. The trends remain the
ΔE = nEcalcite + mEwater − EACC(nCaCO3 +mH2 O) (7) same even with the variation and the low CaCO3 mole fraction systems
where Ecalcite is the energy per formula unit of bulk calcite, Ewater is have consistently lower interfacial energies than the medium to high
the energy of a single water molecule taken from a bulk water simulation CaCO3 mole fraction systems.
and EACC(n,m) is the energy of a hydrated ACC with a molecular The interfacial energy calculated in this study is only a configura­
composition of n CaCO3: m H2O. The entropic component of the free tional energy and does not include the entropic component. This
energy is assumed to be dominated by the change in the configurational component will be primarily associated with the loss of entropy from the
and librational entropy of the water molecules as they transition from water being organised at the calcite surface and therefore we would
the solid state ACC to a liquid phase, ΔGwater. There have been several expect the interfacial free energy to be larger than the values we present
attempts to calculate this value and we used that produced by Rateri et al here. Calculating this value is a complex process and beyond the remit of
[34] of 42.3 Jmol− 1 K− 1 per water molecule. Note this is an estimated this study but previous attempts on water-calcium sulphate interfaces
value based around a 50:50 mix of CaCO3:H2O and would not be ex­ have produced values up to ~ 0.1 Jm− 2 [36]. Given the same Ca ions are
pected to be constant across the full mixing range as the water envi­ present we would expect similar values and therefore though there could
ronments will vary. The final free energy change is: be some variation in this entropic component with the concentration of
the solution as well but we would not expect the component to be so
ΔG ≈ ΔE + mΔGwater (8) large as to change the effective order of the interfacial energies
calculated.
3. Results and Discussion Cleaving the calcite surface creates under-coordinated ions in the
surface plane. This under-coordination is potentially reduced by inter­
Fig. 2 shows the variation of the energy of the calcite h-ACC interface action with another condensed phase at the interface helping to stabilise
as a function of the mole fraction of CaCO3, xCaCO3 , in the h-ACC. These the surface and lower the interfacial energy. This effect can be observed
clearly show that the interfacial energy is lowest for very low mole in Fig. 3(a) which shows the Z-density profile of the oxygen atoms in the
fractions of CaCO3 (xCaCO3 ≤ 0.05) with values less than 0.03 Jm− 2. As water molecules, Owater, in the h-ACC with respect to the interface.
the mole fraction of CaCO3 increases the interfacial energy increases up Previously, it has been shown that the water molecules align with the
to 0.4 Jm− 2 at xCaCO3 = 0.5 - close to a CaCO3:H2O ratio of 1:1). Beyond surface, helping to satisfy the under-coordination of the ions on the
this point the interfacial energy fluctuates but remains higher than for calcite side [37,38]. This alignment deteriorates as xCaCO3 increases. At
the low mole fraction region (even fully dehydrated h-ACC (xCaCO3 = 1) xCaCO3 = 0.005 the first peak is sharp and there is a second peak. For
gives an interfacial energy of 0.22 Jm− 2). This suggests that there might xCaCO3 = 0.2 the first peak is significantly broader and shorter and the
be little variation in the interfacial energy of a growing crystal for su­ second peak has almost vanished. The CO3 ions in the h-ACC system are
persaturations that might be expected within a typical experimental able to bind at this under-coordinated interface as the mole fraction of
solution. Hence the difference between the local supersaturation (i.e. CaCO3 increases within the h-ACC to mitigate some of this effect but the
accounting for the fluctuations caused by surfaces, non-perfect mixing structuring of the CO3 ions is not as well organised and breaks up the
and natural variation) and the average bulk supersaturation may not inter-water interactions that further stabilise the surface water layers.

3
₃ ₃ ⁻
 C.L. Freeman and J.H. Harding Journal of Crystal Growth 603 (2023) 126978

Fig. 3. Z-density profiles of species in h-ACC. The Z-direction (ordinate in the graphs) is the normal to the calcite h-ACC interfacial plane (the origin is in the plane
of the calcite slab closest to the interface) for (a) Owater in h-ACC. And (b) Ca ions in h-ACC.. Colours refer to h-ACC compositions: xCaCO3 = 0.005 (purple); xCaCO3 =
0.05 (green); xCaCO3 = 0.1 (blue); xCaCO3 = 0.2 (orange). Lines are vertically offset to make them visually clearer. (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)

In Fig. 3(b) the Z-density profile for the Ca2+ ions in h-ACC is plotted. destabilisation of the nucleus which could further inhibit the process.
At low mole fraction (xCaCO3 = 0.005) there are no Ca2+ ions near the Although it is possible that different nuclei and other species may
interface due to their low concentration. At xCaCO3 ≥ 0.05 the Ca2+ ions exist before the formation of calcite in the system, if we are to form a
sit approximately 3.0–3.5 Å from the surface which places them either calcite crystal then at some point we must form a calcite surface and this
between the two water layers or within the second one. These ions surface will be in contact with water or a ACC based phase. We are
generate strong solvent shells that structure the surrounding water essentially assuming that the nucleus will adopt the equilibrium
molecules, preventing them from organising at the calcite surface. This morphology of the calcite crystal and the surfaces will have an interfa­
leads to the disruption of the water structure observed for xCaCO3 = 0.2 cial energy comparable to the infinite surface. Therefore the interfacial
and xCaCO3 = 0.3. At higher mole fractions they create an amorphous energy of that calcite surface will be a barrier on the pathway and will
system of low mobility (due to the strong interactions) preventing the constrain the rate of formation. Since the (10.4) surface has a much
water molecules from reorganising with respect to the surface. This lower energy than any other surface, it is likely to be the preferred
explains why the energy of the calcite:h-ACC interface increases as surface. The differences may appear quite small in Fig. 2 but these are
xCaCO3 increases as the water is no longer able to satisfy the under- very significant since the nucleation rate depends exponentially on the
coordination of the ions that are part of the calcite structure. cube of the interfacial energy. In Fig. 4(a) we have used equation (4) to
Our simulation results suggest that the penalty to nucleus formation calculate ΔGc at a range of compositions and supersaturations in terms of
arising from the interface energy (γ) will be smallest for solutions with a kBT (values not calculated in this paper were taken from Hu et al [4]).
low value of xCaCO3 (which corresponds to the smallest supersaturation The values imply that the barrier to the formation of a calcite nucleus
and driving force towards nucleation). Conversely a high supersatura­ will be much larger from a highly concentrated solution than from a
tion will correspond to a large interfacial energy, increasing ΔGc. dilute solution.
Therefore we have a potentially interesting case where the two main Alternatively to using arbitrary supersaturations, we can calculate
factors are working against each other. Our observations also suggest the approximate supersaturation that relates to the actual solution
that the interfacial energy will lower as ions are depleted from the so­ concentration of the interface represented by our hydrated ACC phase
lution surrounding the nuclei. In an extreme case where local super­ and use this value (Fig. 4(b) – purple squares). These will be very large
saturation decreases substantially this could stabilise a growing nucleus supersaturations as the molality of the solution (see supplementary
of calcite but in a system with the opposite trend (i.e. increasing inter­ Table S1) are very large and could provide the extra driving force
facial energy with decreasing concentration) we could see a needed to overcome the large energy barriers associated with the

​ ​

CS
ACC vs.
BS

Fig. 4. Critical nucleation barrier, ΔGc, to the formation of calcite with an interface of h-ACC with differing mole fractions of CaCO3 for (a) Supersaturation 3 (green
circles), 4 (purple squares) and (b) supersaturation directly approximated to the mole fraction (purple squares) or using a driving force calculated from the free
energy difference between the ACC phase and calcite (green circles). (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)

4
C.L. Freeman and J.H. Harding Journal of Crystal Growth 603 (2023) 126978

interfacial energy. These larger supersaturations do lower all the bar­ can see that many of the different mechanistic cases reported for the
riers significantly so values become ~ 1 kT for 0 < xCaCO3 < 0.1, rising formation of calcite with an ACC precursor can be entirely commensu­
to ~ 10 kT at 0.1 < xCaCO3 < 0.2 and then larger still. Although the rate with each other and a simple CNT interpretation and understood
barriers are reduced by the very high supersaturations at the high CaCO3 from a viewpoint of interfacial energies.
mole fractions these are still an order or orders of magnitude larger for
xCaCO3 > 0.1. CRediT authorship contribution statement
At the high mole fractions of CaCO3 it can be argued that the system
is not a solution and we should consider this as a solid state phase change C.L. Freeman: Conceptualization, Data curation, Formal analysis,
instead. Calculation of the diffusion coefficients for the water molecules Funding acquisition, Investigation, Methodology, Project administra­
and ions suggests this transition occurs around . xCaCO3 > 0.3 (see sup­ tion, Resources, Software, Supervision, Validation, Visualization,
plementary table S2). For this purpose we can calculate the free energy Writing – original draft, Writing – review & editing. J.H. Harding:
differences between the ACC and calcite phases and use this as our Conceptualization, Formal analysis, Funding acquisition, Project
driving force (in a similar appraoach to that used in calculations around administration, Resources, Validation, Writing – original draft, Writing –
metal phase transitions e.g. [39,40]). Calculating this free energy is not a review & editing.
trivial set of simulations and we approximate the value by taking the
bulk energetic (configurational) differences between the ACC phase and
calcite and water, the entropic contribution is assumed to be most Declaration of Competing Interest
dependent on the liberation of water from the ACC phase and we use the
value proposed by Rateri et al [34] of 42.3 Jmol− 1K− 1. Further details The authors declare that they have no known competing financial
are listed in the methods section. The green circles of Fig. 4(b) shows ΔGc interests or personal relationships that could have appeared to influence
calculated using these values. As can be seen most of the values are far the work reported in this paper.
higher to those taken using supersaturation and therefore sit off the plot.
The full range of values (using a natural log plot) can be seen in sup­ Data availability
plementary material; Figure S1.
Using three different values to the driving force for classical nucle­ Data will be made available on request.
ation we observe in all cases that the large interfacial energies associated
with higher CaCO3 content dominate and make nucleation with these Acknowledgements
interfaces present very unlikely. This suggests that a calcite nucleus will
struggle to form if surrounded by an ACC phase. Conversely the for­ This work was supported by the “Crystallisation in the Real World”
mation of a calcite nucleus in water will have to overcome a much programme grant (EPSRC Grant number EP/R018820/1).
smaller barrier as the water molecules stabilise the surfaces.
These results agree well with the commonly reported dissolution of Appendix A. Supplementary data
ACC followed by re-precipitation of calcite. Hydrated ACC is often
observed to be very stable but dehydrated ACC has also been reported Supplementary data to this article can be found online at https://doi.
with long lifetimes when kept in low humidity environments. Ihil et al org/10.1016/j.jcrysgro.2022.126978.
[15] showed that ACC trapped within a lipid bilayer was stable for very
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