# Reverse Coarsening and the Control of Particle Size Distribution through Surfactant
## 2 Results and Discussion
### Isolated Particle in a Binary Solution  
In our thermodynamic analysis, we considered a conservative system consisting of a dilute binary A-B solution and a precipitated spherical A-particle of radius \(R\) measured in the units of typical intermolecular distance and containing \(N_{P}=4\pi R^{3}/3\) molecules. The system contained \(M_{\rm A}\) molecules of material A, \(M_{\rm B}\) molecules of material B and \(M_{\rm L}\) molecules of a solvent. For simplicity, we assumed that a molecule A can bind only one molecule B and vice versa with characteristic energy \(\varepsilon_{AB}\). This binding resulted in the formation of \(N_{C}\) number of A-B complexes (dimers) in the solution, and in concentration \(n_{S}\) of B-molecules on particle surfaces. The free energy of the system is  
\[ \begin{array}{l}G=-\varepsilon N_{P}+\gamma S_{P}(1-n_{S})-\varepsilon_{AB} n_{S}S_{P}+TS_{P}\cdot[n_{S}\ln(n_{S})+(1-n_{S})\ln(1-n_{S})]\\ -\varepsilon_{AB}N_{C}+T\cdot(N_{C}\ln(N_{C})+N_{A}\ln(N_{A})+N_{B}\ln(N_{B})+ N_{L}\ln(N_{L})-M_{\Sigma}\ln(M_{\Sigma})),\end{array}  \tag{1}\]  
where \(\varepsilon\ >\ 0\) and \(\gamma\ >\ 0\) are the cohesive and surface energies per A-molecule in the particle, respectively, \(T\) is the temperature in energy units (eV), \(S_{P}=4\pi R^{2}\) is the number of molecules on the particle surface, \(N_{A}=M_{A}-N_{P}-N_{C}\), \(N_{B}=M_{B}-S_{P}n_{S}-N_{C}\), \(R\gg 1\) and \(M_{\Sigma}=N_{A}+N_{B}+N_{C}+M_{L}\). In Equation (1), we neglected possible interactions between B-molecules (see Appendix A for details) and assumed a simple form for the entropy, which provides a lower limit estimate [52]. The minimization of the free energy \(G\) with respect to the independent variables \(N_{P},\ N_{C}\) and \(n_{S}\) in the limit of a dilute solution with \(M_{L}>>N_{A}+N_{B}+N_{C}\) yields (see Appendix B)  
\[ \begin{array}{c}T\ln(n_{A})=-\varepsilon+\frac{2}{R}\cdot\Gamma,\\ n_{C}=E_{0}\cdot n_{A}\cdot n_{B},\\ n_{S}=E_{1}\cdot n_{B}\cdot(1+E_{1}\cdot n_{B})^{-1}\end{array}  \tag{2}\]  
where \(E_{0}\ =\ \exp(\varepsilon_{AB}/T)\), \(E_{1}=\exp((\gamma+\varepsilon_{AB})/T)\) and \(\Gamma=\gamma-T\cdot\ln(1+n_{B}E_{1})\) is the renormalized surface energy of the A-particle, and \(n_{A(B)}=N_{A(B)}/M_{\Sigma}\) and \(n_{C}=N_{C}/M_{\Sigma}\) are the number concentrations of corresponding species in the solution. The first equation in Equation (2) is a statement of the equality of chemical potentials for A-molecules in the solution \(\mu_{A}(sol)=T\ln(n_{A})\) and in the particle \(\mu_{A}(R)=-\varepsilon+2\Gamma/R\) of radius \(R\). The second and third equations of Equation (2) give the equilibrium concentrations of A-B complexes in the solution and of B-molecules on the particle surfaces, respectively. Note that the surfactant B-molecules always reduce the surface energy, as \(\Gamma<\gamma\) due to \(\ln(1+n_{B}E)>0\). For high enough values of \(\varepsilon_{AB}\), the effective surface energy \(\Gamma\) can become negative (see Figure 1) resulting in a narrowing of the size distribution for an ensemble of particles in solution. The same figure also shows the surface concentration of surfactant \(n_{S}\) as a function of \(\varepsilon_{AB}\) illustrating an increase in surface concentration \(n_{S}\) and a decrease in \(\Gamma\) with increasing bulk concentration \(n_{B}\) of the surfactant molecules. The effect of surface energy renormalization by A-B bonding can be obtained by considering the molecular chemical potential \(\mu_{A}(R)=-\varepsilon+2\Gamma/R\) for a particle of radius \(R\) in solution containing other A-particles. The case of positive \(\Gamma\) corresponds to ordinary coarsening when A-molecules are transferred from smaller particles to bigger ones until a single big particle remains. If \(\Gamma=0\) the molecular chemical potential according to the first line in Equation (2) becomes independent of \(R\) hence the PSD should also be time-independent.