The Statistical Mechanics of Solution-Phase Nucleation: CaCO$_3$ Revisited

Abstract

Providing a molecular level description of amorphous CaCO$_3$ nucleation in aqueous solutions necessitates the use of theory and simulation in addition to a direct connection to experiments. Furthermore, the ability to simulate experimental conditions relevant to nucleation requires a systematic reduction in the complexity of a molecular-based model that can be compared to experimental observations and tested for self-consistency. In this review, we further expand on the theoretical details described in (Henzler et al. Sci Adv 4:eaa06283, (2018)), where a molecular-based solution model was constructed based on a solvent-mediated quantum mechanical potential of mean force for Ca$^{2+}$ and CO$_3^{2-}$ ion pairing. The statistical mechanics and simulation approaches for solution-phase nucleation are presented and connected to the various CaCO$_3$ polymorph chemical potentials and interfacial surface energies. The resulting theory presented herein is general and is shown to accurately describe the initial stages of amorphous calcium carbonate nucleation.

5 Appendix

The following development is a generalization of the thermodynamics of clusters to the case of two components. The one component case was established by Reiss and Bowles [44]. This serves to define our thermodynamic references and make a formal connection between molecular statistical mechanics and thermodynamics. We write expressions without internal molecular degrees of freedom for simplicity. Further generalization is straightforward. The $(i_{1},i_{2})$ component partition function is defined as $q_{(i_{1},i_{2})}=\frac{1}{i_{1}!i_{2}!}\gamma _{1}^{i_{1}}\gamma _{2}^{i_{2}}\int dr_{1}^{\left( i_{1}\right) }dr_{2}^{\left( i_{2}\right) }e^{-\beta U}\Theta$, with the ideal gas partition function, $q_{(i_{1},i_{2})}^{\mathrm {IG}}=\frac{1}{i_{1}!i_{2}!}\gamma _{1}^{i_{1}}\gamma _{2}^{i_{2}}\int dr_{1}^{\left( i_{1}\right) }dr_{2}^{\left( i_{2}\right) }\Theta$. Here, $\gamma _{k}=\left( \frac{m_{k}k_{\mathrm {B}}T}{2\pi \hbar }\right) ^{3/2}$, where $m_{k}$ is the mass of the $k'th$ component, $k_{\mathrm {B}}$ is the Boltzmann constant, $\hbar$ is the Plank constant, and T is the temperature. U is the potential energy of the system. $\Theta$ serves to partition configuration space defining a cluster of size $(i_{1},i_{2})$. For the case of initial studies of DNT, the center of the mass of the cluster is defined by $R=\frac{1}{i_{1}+i_{2}}\sum _{k=1}^{2}\sum _{j=1}^{i_{k}}r_{k,j}$, and the definition of the cluster depends on the radius cutoff parameter, $r_{\mathrm {cut}}$. The spherical volume of the cluster is $v=\frac{4\pi }{3}r_{\mathrm {cut}}^{3}$. We then write, $\Theta =\prod _{k=1}^{2}\prod _{j=1}^{i_{k}}\left( r_{\mathrm {cut}}-\left| r_{k,j}-R\right| \right)$. With, $\bar{i}=i_{1}+i_{2}$ , and $\bar{r}^{\left( \bar{i}\right) }=\left( r_{1}^{\left( i_{1}\right) },r_{2}^{\left( i_{2}\right) }\right)$, $q_{(i_{1},i_{2})}^{\mathrm {IG}}=\frac{1}{i_{1}!i_{2}!}\gamma _{1}^{i_{1}}\gamma _{2}^{i_{2}}\bar{i}^{3/2}Vv^{\left( \bar{i}-1\right) }a_{\bar{i}}$, where $a_{\bar{i}}$ constants that appear in the single component case. We define monomer partition functions as $q_{(1,0)}=\gamma _{1}V$ and $q_{(0,1)}=\gamma _{2}V$. Here, V is the volume of the system. The partition function for the full, non-interacting cluster system is Q, where $\ln Q=\sum _{i_{1}}\sum _{i_{2}}\left\{ N_{(i_{1},i_{2})}\ln \left[ q_{(i_{1},i_{2})}\right] -N_{(i_{1},i_{2})}\ln N_{(i_{1},i_{2})}+N_{(i_{1},i_{2})}\right\}$. Following Reiss and Bowles [44], we introduce particle and volume conservation constraints through Lagrange multipliers and consider the extended partition function, $\Phi$, to apply the stationary phase analysis:

$$\begin{aligned} \begin{aligned} \ln \Phi =\ln Q+\beta \mu _{1}\sum _{i_{1}}i_{1}\sum _{i_{2}}N_{(i_{1},i_{2})}+\beta \mu _{2}\sum _{i_{2}}i_{2}\sum _{i_{1}}N_{(i_{1},i_{2})} \\ -\beta p\left[ V+\sum _{i_{1}}\sum _{i_{2}}v_{(i_{1},i_{2})}N_{(i_{1},i_{2})}\right] . \end{aligned} \end{aligned}$$

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Here, $v_{(i_{1},i_{2})}$ is a volume of a cluster, $\mu _{k}$ is the chemical potential of monomer k, p is the pressure of the system.

We then have the conditions, $\frac{d\ln \Phi }{dN_{\left( 1,0\right) }}=\ln \left[ \gamma _{1}V\right] -\ln N_{(1,0)}+\beta \mu _{1}=0$, $\frac{d\ln \Phi }{dN_{\left( 0,1\right) }}=\ln \left[ \gamma _{2}V\right] -\ln N_{(0,1)}+\beta \mu _{2}=0$, $\frac{d\ln \Phi }{dV}=\frac{N_{\left( 1,0\right) }+N_{\left( 0,1\right) }}{V}-\beta p=0$, and $\frac{d\ln \Phi }{dN_{(i_{1},i_{2})}}=\ln \left[ q_{(i_{1},i_{2})}\right] -\ln N_{(i_{1},i_{2})}+\beta \mu _{1}i_{1}+\beta \mu _{2}i_{2}-\beta pv_{(i_{1},i_{2})}=0$. Further simplification gives $\beta \mu _{1}=\ln \left[ \frac{N_{(1,0)}}{\gamma _{1}V}\right]$, $\beta \mu _{2}=\ln \left[ \frac{N_{(0,1)}}{\gamma _{2}V}\right]$, and $\beta pV=N_{\left( 1,0\right) }+N_{\left( 0,1\right) }$. The equilibrium population resulting from the stationary phase analysis, $N_{(i_{1},i_{2})}^{\mathrm {EQ}}$, is $\ln N_{(i_{1},i_{2})}^{\mathrm {EQ}}=\ln \left[ q_{(i_{1},i_{2})}\right] +\beta \mu _{1}i_{1}+\beta \mu _{2}i_{2}-\beta pv_{(i_{1},i_{2})}$. It is natural to define the total Helmholtz free energy, $A_{(i_{1},i_{2})}^{\mathrm {TOT}}$, by $-\beta A_{(i_{1},i_{2})}^{\mathrm {TOT}}=\ln \left[ q_{(i_{1},i_{2})}\right]$. Collecting terms, we have $\ln N_{(i_{1},i_{2})}^{\mathrm {EQ}}=-\beta \left[ A_{(i_{1},i_{2})}^{\mathrm {TOT}}+pv_{(i_{1},i_{2})}-\mu _{1}i_{1}-\mu _{2}i_{2}\right]$. It is useful to separate out the translational component of the free energy, $A_{(i_{1},i_{2})}^{\mathrm {trans}}=-k_{\mathrm {B}}T\ln \left( \gamma _{\mathrm {TOT}}V\right)$, where $\gamma _{\mathrm {TOT}}=\left( \frac{\left( m_{1}i_{1}+m_{2}i_{2}\right) k_{\mathrm {B}}T}{2\pi \hbar }\right) ^{3/2}$ and $A_{(i_{1},i_{2})}^{\mathrm {TOT}}=A_{(i_{1},i_{2})}+A_{(i_{1},i_{2})}^{\mathrm {trans}}$. The natural definition of the Gibbs free energy follows, $G_{(i_{1},i_{2})}=A_{(i_{1},i_{2})}+pv_{(i_{1},i_{2})}$. We then have $N_{(i_{1},i_{2})}^{\mathrm {EQ}}=e^{-\beta \left[ G_{(i_{1},i_{2})}+A_{(i_{1},i_{2})}^{\mathrm {trans}}-\mu _{1}i_{1}-\mu _{2}i_{2}\right] }$. This may be recast to recover our main result that connects microscopic statistical mechanics (cluster populations) and macroscopic thermodynamics (Gibbs free energies):

$$\begin{aligned} N_{(i_{1},i_{2})}^{\mathrm {EQ}}=e^{-\beta G_{(i_{1},i_{2})}}\left( \gamma _{\mathrm {TOT}}V\right) \left( \frac{N_{(1,0)}}{\gamma _{1}V}\right) ^{i_{1}}\left( \frac{N_{(0,1)}}{\gamma _{2}V}\right) ^{i_{2}}. \end{aligned}$$

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