A two-step nucleation model based on diffuse interface theory (DIT) to explain the non-classical view of calcium carbonate polymorph formation

A two-step nucleation model based on the diffuse interface theory

(DIT) to explain the non-classical view of crystal formation

Bruno M.

1 _{Dipartimento di Scienze della Terra, Università degli Studi di Torino, Via Valperga Caluso 35,}

10125 Torino (TO), Italy

2 _{SpectraLab s.r.l., Spin-off accademico dell’Università degli Studi di Torino, Via G. Quarello 15/a,}

10135 Torino (TO), Italy *Corresponding author

marco.bruno@unito.it

Abstract

We developed a phenomenological two-step nucleation model based on the Diffuse Interface Theory (DIT) for explaining the non-classical pathway of crystallization: (i) initially, formation of an amorphous phase; (i) later, formation of a crystalline phase within the amorphous one. Our model uses an arbitrary function describing the evolution of the Gibbs free-energy of the system over time. We show that the amorphous cluster can nucleate overcoming a low activation energy and successively can grow until to reach a maximum size (step I); the cluster cannot grow further, otherwise the Gibbs free energy of the system increases indefinitely. To reduce further the Gibbs free energy of the system, the crystalline phase nucleates within the amorphous cluster or at its interface (step II). We believe that the amorphous cluster can operate as crystallization chamber, a nanometric environment suitable to the crystal formation, where it is less expensive to form a crystalline phase than continuing to grow the ACC cluster.

1. Introduction

By the term nucleation we mean all processes leading to the formation of a new phase from a supersaturated system. The most employed quantitative theory used to explain and predict the nucleation processes is the Classical Nucleation Theory (CNT).1,2 _{According to CNT, ions}

aggregate to form clusters that are initially unstable due to the existence of a positive crystal/solution interface free energy (i.e., the energy to spend for increasing the crystal/solution interface of an unit area). By increasing the size of the particle, the contribution of the interface free energy decreases with respect to that of the bulk lattice energy and, consequently, the system passes through a free energy maximum and then increases in stability with growing diameter. In the CNT it is supposed that the interface free energy does not vary with the size of the particle (i.e., capillary assumption): the interface free energy is always equal to that of an infinite crystal. It is also supposed that the width of the crystal/solution interface is zero: the volume of the interface is nil.

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 1

Finally, it is supposed that the unstable nucleus has the same structure of the infinite crystal. All these assumptions make the CNT unable to reproduce many experimental observations: e.g., the frequency of nucleation.

A more realistic phenomenological theory was developed independently by Spaepen3 _and

Gránásy,4-6 _{which is physically appealing and can be readily fit to nucleation data using parameters}

that are similar to those used in CNT, as described in a successive paragraph. Such theory (Diffuse Interface Theory; DIT) is based on the description of a crystal/solution interface having a non-nil volume. Indeed, computer simulation,7-10 _{density-functional calculations,}11-14 _{and discrete studies}15,16

showed that the width of the interface between the original phase and clusters of the new phase can be a significant fraction of the cluster radius. As we will see in the following, in this theory the concept of free energy interface acquires a new meaning.

Both theories, CNT and DIT, were developed to describe the formation of a new crystalline phase from a supersaturated system, but DIT can be extended to explain recent experimental observations where at least a two-steps nucleation process is involved (i.e., non-classical view on nucleation): (i) initially, nucleation and growth of an amorphous phase (prenucleation clusters) and (ii) successively, nucleation of a crystalline phase from the amorphous one. Indeed, there is increasing evidence that small polymeric species and stable clusters play a dominant role in the prenucleation stage in many systems; see the interesting review paper by Gebauer et al.17 _{for more}

details, where a deeper discussion on the different nucleation theories is developed.

One of the most studied systems concerns the crystallization of calcium carbonate (CaCO3)

polymorphs (calcite, aragonite and vaterite), which seem to be preceded, in many cases, by the formation of an amorphous phase (Amorphous Calcium Carbonate; ACC). According to experimental observations, it was proposed18,19 _{that the system initially forms stable prenucleation}

nanoparticles of amorphous material of size < 4 nm. Then, after several minutes, such prenucleation particles seem to aggregate or grow to generate greater ACC nanoparticles with a size of  30 nm.19

With the proceeding of the reaction, larger amorphous particles develop (> 70 nm) and, contemporary, polycrystalline material appear.19 _{ACC is believed to be composed mainly by CaCO}

3

and H2O molecules with a variable water content,20 from nominally dry calcium carbonate to an

approximate composition of CaCO3H2O, although 1.38 water molecules were also detected.21

Starting from these observations, we can reasonably hypothesize that when the ACC particle reaches a maximum size (currently unknown) the crystallization (nucleation?) of a calcium carbonate polymorph (e.g., calcite) begins inside the ACC particle and/or at the ACC particle/solution interface. This mechanism is highly reasonable, since the ACC particles cannot grow indefinitely, otherwise calcite crystals will never form from a supersaturated solution. In this work, we suppose that the nucleation of spherical ACC particles from a supersaturated solution occurs overcoming a very low activation free energy. Then, by making some reasonable assumptions and applying the DIT for a closed system a constant temperature (T) and pressure (P), we will show that these ACC particles cannot grow indefinitely, but they reach a maximum size in correspondence of a local minimum of the Gibbs free energy of the system. Successively, the only way to reduce the Gibbs free energy of the system is the nucleation of calcite (or another CaCO3

polymorph) within the ACC particles that act like crystallization chambers, in which a very high calcite supersaturation is reached due to the elevated CaCO3/H2O molecular ratio. The model we

propose is analogous to the nucleation mechanism already suggested by Raiteri and Gale,20 _{but we}

will show that this can occurs for thermodynamic reasons when specific conditions realise. Raiteri

2 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 3

and Gale20 _{performed molecular dynamic simulations at room temperature on anhydrous and}

hydrated ACC particles confirming the experimental results of Gebauer et al.:18 _{in particular, (i) it is}

not necessary to invoke impurities (e.g., Mg2+_{) to stabilize precritical clusters of ACC, but they are}

already stable in pure solutions, and (ii) the addition of ion pairs of calcium carbonate to ACC is thermodynamically favoured, for this reason the amorphous phase is able to grow rapidly in comparison to the ordered phases.

This work wants to be a contribution to the interesting debate concerning the non-classical view of nucleation, by providing a possible thermodynamic explanation to some experimental observations, to validate in a next future by means of more sophisticated computational methods (e.g., molecular dynamic simulations). Our path does develop as follows: (i) a brief description of the DIT and its relationship with the CNT; (ii) application and discussion of the DIT to describe our two-steps nucleation model used to interpret the experimental observations concerning the non-classical view of crystal formation; (iii) conclusions and future developments.

2. Diffuse Interface Theory (DIT) and Classical Nucleation Theory

(CNT)

In order to describe the main features of the DIT we use the same notations reported in ref. 22. Let us to imagine the case of a spherical cluster of radius r (crystalline or amorphous) growing in an isotropic amorphous medium (e.g., solution). Such case is illustrated schematically in Figure 1, where a broad interfacial region () develops between the growing cluster and mother solution. This diffuse interface is in conflict with the central assumption of CNT and has significant consequences for the calculation of the work of cluster formation. This broad interface is due to structural and compositional variations of the system when passing from the cluster surface (at rs, the cluster

radius) to the interior of the solution (at r > rs + ) . We express these modifications by means of the

radial variation of the free-energy density (free-energy per unit volume, e.g., J m-3_{), g(r), where r is}

measured from the center of the cluster. As depicted in Figure 1, g(r) increases, above that in the liquid, gl, and the solid, gs, phases due to the interface (the interfacial free energy is positive, equal

to the excess energy per unit area from the introduction of an interface). By assuming constant pressure, the reversible work of formation for a cluster of the new phase having a radius of rs and an

interfacial width  is,

∆ G(r_s)=4 π

_∫

0 ∞

[

_]

where g(r) = gs for r < rs; g(r) = gl for r > rs + . As for CNT, the critical cluster radius for

nucleation rs

¿

is determined by the condition

(

d rs

)

¿

=0 (2)

Given a functional form for g(r), the problem can be solved. Eq 1 can be rewritten in the following way:

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∆ G

(

)

_∫

[

_]

where ∆ g_sl=g_s−g_{l < 0. Now, we can relate the quantities} ∆ G_{V and} ∆ G_{S in eq 3 with}

their analogous in CNT, ∆ GVCNT and ∆ GSCNT , always for a spherical cluster:

∆ G

₍

₎

where ∆ μ=μ_l−μ_s > 0 is the supersaturation of the system: μ_l and μ_s are the chemical potentials of the formula unit in solution and cluster, respectively;  is the molecular volume of the formula unit in the cluster;  is the cluster/solution interface free energy.

By equating eqs 3 and 4, one obtains:

∆ μ=−∆ gsl (5)

γ=1

rs2

∫

rs+δ

[

_]

From eq 6, it is evident that  is a function of g(r) and . In a closed system, during the growth of a cluster, g(r) is continuously modified, since the supersaturation of the mother solution decreases for the reduction of the amount of solute that moves from the solution to the cluster. Starting from this last consideration, we developed a modified version of the model described in the following paragraph.

Fig. 1 Free energy density profile (J m-3_{), g(r), for a spherical cluster growing from a solution. r}

s is

the radius of the cluster;  is the interfacial width; gs and gl are free-energy densities of the cluster

and solution, respectively.

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3. Two-step nucleation model

At variance with the original description of DIT above reported, we decided to represent the free-energy density (J m-3_{) profile for an amorphous cluster (e.g., ACC) forming from a supersaturated}

solution by means of the function g(r,t), which depends on both the radial distance, r (nm), and time, t (s):

g (r ,t )=

{

{

[

]

}

A (t )lnD (t),∧r ≥ C(t) (7)

where A(t), B(t), C(t) and D(t) are functions of time. When r < C(t), g(r,t) is described with a Gaussian: A(t) is the height of the curve’s peak, C(t) is the position of the center of the peak, and

B(t) controls the width of the bell. When r  C(t), g(r,t) becomes a logarithmic function. It was necessary to split the function g(r,t) in this way to reproduce different constant values of gs and gl

inside cluster and solution, respectively.

It is important to highlight that the function g(r,t) above reported does not derive by any chemical-physics model, it is only an attempt of description of a possible free-energy profile across an interface. At the best of our knowledge, there are not experimental observations or theoretical calculations of free-energy profiles for crystal growing in supersaturated mediums. Thus, we supposed a g(r,t) function with increasing values at the cluster/solution interface, in such a way to describe both the favourable term to the cluster formation (GV < 0) and the unfavourable one (GS

> 0). Since g(r,t) is only a hypothetical function in our model, the values of the parameters we used to describe it in the following are hypothetical as well. The aim of this model is to show, from a thermodynamic point of view, how a diffuse interface and its evolution over time are able to explain why a new amorphous phase forms and grows until to reach a size limit. We will show in the next paragraphs that what determines the evolution of the system is exclusively the trend of the function

g(r,t), not its absolute value. Therefore, all the numerical values given in the discussion of the

model (e.g., time, cluster radius, parameters of g(r,t), G values) must be considered unrealistic from a chemical-physics point of view, but essential to understand its evolution. Nevertheless, the proposed model is qualitatively reasonable and able to explain and unify many experimental and theoretical aspects described in previous works.

We performed all numerical calculations by using a simple homemade program written in

Octave language,23 _{which is freely available at the webpage http://mabruno.weebly.com/download.}

We considered a closed system a constant T and P: e.g., an aqueous solution supersaturated with respect to calcite. Moreover, we supposed a function g(r,t) that evolves with time, according to the relations: A (t)=A₀−A₁t (8) C (t )=C₀+C₁t (9) 5 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 9

where A0 = 0.3, A1 = 3.93×10-3, C0 = 1.0 and C1 = 7.86×10-2. The functions B and D are kept

constants during the calculations: 8.0 and 1.5, respectively. The variation of the functions A and C with time wants (i) to reproduce the increase of the radius cluster (i.e., raising C) and, simultaneously, (ii) to lower the free-energy density in the mother solution and in correspondence of the peak of function g(r,t) (i.e., reducing A). This is clearly depicted in Figures 2a and b, where the function g(r,t) is represented at two different times (t = 0 and 4200 s) of the simulation detailed in the following. We considered a reduction of the free-energy density in the mother solution, since with the nucleation and growth of the clusters, there must be at the same time a decrease of the solute concentration in the aqueous solution. In addition, the thickness of the diffuse interface is maintained constant during the evolution of the system,   2 nm (Figures 2a and b).

Then, we determined how the Gibbs free energy variation of the system changes with time, G(t), by solving numerically the following integral at different t:

∆ G(t )=4 π

_∫

r=0 r=rl

[

_]

where rl is the maximum radial distance (10 nm) considered in our calculations, measured from the

center of the spherical cluster; g(rl,t) represents the constant value gl in the mother solution (Figure

1). We performed the integration at constant t, every 60 s, for an overall time interval of tl = 4200 s.

In the time interval spanned by our calculations, the radius of the cluster goes from rs = 0 (at t = 0)

to rs  5.5 nm (at t = 4200 s). In Figures 2a and b, the functions g(r,0) and g(r, tl) are drawn,

whereas in Figure 2c we reported G(t) normalized with respect to kT at room temperature (T = 300K). Finally, in Figure 2d, the derivative of G(t) with respect to t is reported.

By analyzing the behavior of G(t) (Figure 2c), the following features are evident: 1. in the time interval 0  t < 960 s, G(t) is greater than zero and grows monotonously. 2. At t = 960 s, G(t) reaches the greatest value, ∆ GS

¿

 2 kT, corresponding to the activation energy of the cluster nucleation. The critical radius corresponding to ∆ GS

¿

is

r_S¿ _{ 1.2 nm.}

3. In the time interval 960 < t < 3700 s, G(t) decreases monotonously; at t = 2030 s, G(t) = 0.

4. At t = 3700 s, G(t) reaches the lowest value, ∆ GS max

. The cluster radius corresponding is

rS

max _{=  4.5 nm.}

5. At t > 3700 s, G(t) comes back to increase in a monotonic way.

According to our simulation, the amorphous cluster nucleates overcoming a low activation energy and successively grows until to reach a maximum size (step I). The cluster cannot grow further, otherwise the Gibbs free energy of the system increases indefinitely. Our model justifies with the thermodynamics the existence of a size limit for the amorphous cluster.

It is fundamental to highlight that the achievement of the cluster size limit is due to the consideration of a closed system, which evolves in time lowering the g(r,t) function at the interface and in the mother solution. Indeed, when the g(r,t) is not lowered by the clusters formation (e.g., an open system where diffusion of nutrient at the cluster/solution interface is very fast), G(t) never reaches a minimum, but it continues to decrease indefinitely.

6 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 11

This result is independent of the law used to describe A and B. By way of example, they could be a function of t1/2_{, but the trend described above would always be observed. The only constraint}

required is that the function g(r,t) is reduced with time. Of course, considering different laws leads to obtaining different activation energies, as well as different critical and maximum cluster radii.

At this point, to reduce further the Gibbs free energy of the closed system represented by the supersaturated aqueous solution, it is necessary to nucleate a crystalline phase (step II): in our example, calcite. The nucleation should take place within the cluster of ACC or at its interface, where the high CaCO3/H2O molecular ratio generates an extremely high supersaturation and,

consequently, very low activation energy and critical nuclei radius; again, this nucleation event could be explained by means of the DIT. We believe that ACC cluster can operate as crystallization

chamber, a nanometric environment suitable to the calcite formation, where it is less expensive to

form a crystal than continuing to grow the ACC cluster. It is also probable that at the beginning the calcite crystals grow rapidly, being abundant the growth units (CaCO3 molecules) inside the ACC

clusters.

Fig. 2 Function g(r,t) is depicted at two different times: (a) t = 0 and (b) t = 4200 s. (c) G(t)

normalized with respect to kT at room temperature (T = 300K). (d) Derivative of G(t)/kT with respect to t.

4. Conclusions

We proposed a two-step nucleation model based on the Diffuse Interface Theory (DIT) to explain the formation of an amorphous precursor before the nucleation of the crystalline phase. Others already invoked a two-step nucleation theory24-27 _{to explain protein crystal formation: (i) first,}

formation of a metastable dense liquid phase; (ii) then, nucleation of the crystal inside this intermediate in which an increased supersaturation level is reached. Our model puts together this two-step nucleation theory and DIT, in an attempt to explain on a thermodynamic basis the non-classical crystallization pathway. Specifically, we used a phenomenological model to show that: (i) it is possible to nucleate an amorphous phase by overcoming a low activation energy; (ii) the amorphous cluster cannot grow indefinitely. This is the step I of our model, which is schematically exemplified in Figure 3a. Later, the system reduces its Gibbs free energy by nucleating a crystalline phase (e.g., calcite) within or at the interface of the amorphous cluster; step II of our model (Figure 3b).

We have confirmed the Raiteri and Gale20 _{prediction for ACC: they wrote “As the nanoparticles} increase in size, the driving force for growth will decrease; the free energy for addition of an ion pair (i.e., Ca2+ _{and CO}

32-) is more exothermic for a small cluster than for a larger one”. In fact, we

have seen that the free energy of the system decreases until the cluster reaches a size limit. Beyond this size the free energy of the system can only increase. Then, the lower the ACC cluster size, the higher the driving force for growth.

Fig. 3 A schematic picture of the two-step nucleation model proposed in this work. In step I, the nucleation of ACC cluster occurs, for which a very low activation energy, ∆ GS

¿

, is necessary. Successively, the ACC cluster grows until to reach a maximum size, rSmax , over which the Gibbs

energy of the system can only increase. Then, the only way to decrease the Gibbs energy of a supersaturated system is to nucleate calcite (step II). This should require a very low activation energy, ∆ Gcal

¿

, since the calcite supersaturation inside the ACC cluster is extremely high and the calcite/ACC interface energy is probably very low.

Many criticisms can be made about this work. The main one is definitely the use of a completely arbitrary function to describe the free energy density of a supersaturated aqueous solution, g(r,t). This function and its evolution over time is completely unknown for this system. At the best of our knowledge, free energy density profiles were only determined for crystal surfaces28-30

and crystalline phases in epitaxial contact.31 _{Actually, there are density-functional theories (DFT),}

8 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 15

based on an order-parameter description of the phase transition, allowing a more formal treatment of nucleation, which are intermediate between the macroscopic thermodynamic description of CNT and the atomistic computer simulation methods.22 _{Such DFT theories make use of the concept of}

functional and allow of calculating g(r) around the nucleating cluster by minimizing the free energy of the system. Unfortunately, DFT theories are quite complicated from a mathematical point of view and currently do not allow the treatment of complex systems. Moreover, we were not interested in obtaining the exact behavior of g(r,t) because, according to our model, this aspect is not significant to explain qualitatively the nucleation of an amorphous cluster and its growth until to reach a maximum size. What is important is the decrease of g(r,t) in the mother solution and at the crystal/solution interface, which must occur in a closed system that nucleates a new phase. It is precisely this behavior that generates the evolution of G(t) over time represented in Figure 2c, not the shape of g(r,t).

In this regard, the parallelism with the work of Bienfait and Kern32 _{is remarkable, in which the}

authors determined analytically the evolution, at constant T, of the Gibbs free energy of a closed system consisting of a supersaturated monoatomic ideal gas. As expected, they observed32.33 _the

existence of two critical points in the function G: a maximum corresponding to the activation energy of the nucleation process and a minimum corresponding to the largest size of the crystal. In their model, the condensation of a crystal from a supersaturated vapor is studied: the pressure, then the supersaturation, is continuously reduced for the nucleation and successive growth of the crystal. As in our model, the behavior of G is due to the consideration of a closed system.

It is important to specify that our model should be validated by means of more sophisticated computational methods. In particular, it should be interesting (if possible) to perform molecular dynamic calculations to simulate the growth of ACC clusters in aqueous solution with decreasing supersaturation, in order to verify our findings.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

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