# Reverse Coarsening and the Control of Particle Size Distribution through Surfactant
## 3 Conclusions  
We have presented a free-energy analysis and simulation studies of a dilute binary molecular solution with precipitated particles of one component and with the other component acting as a surface passivating surfactant. Our analysis predicts that for certain component concentrations and strengths of molecular-surfactant interaction, the system of precipitated particles evolves towards equilibrium in such a way that the PSD spontaneously narrows, leading eventually to a mono-sized array of particles. The narrowing of the size distribution is mainly due to a strong enough surface passivation of particle surfaces by surfactant molecules, which renders the effective surface energy of particles negative. The effect is confirmed by direct simulations of the PSD evolution in a large ensemble of poly-dispersed particles. This reverse coarsening effect is a new paradigm for the thermodynamically controlled solution-processed generation of mono-sized nano-particulate arrays of various materials.  
Conceptualization, V.M.B. and A.G.; Methodology, V.M.B. and A.G.; Software, V.M.B.; Validation, A.G.; Formal Analysis, V.M.B. and A.G.; Writing--Original Draft Preparation, V.M.B.; Writing--Review and Editing, A.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. The above analysis is based on an assumption that the interaction between B-molecules can be neglected. This interaction, if high enough, may significantly affect the coverage of particle surfaces by B-molecules. In particular, a significant B-B interaction \(\varepsilon_{BB}\) could promote the formation of large B-clusters on the particle surface and even cause complete coverage if the B-cluster grows over its critical size. To illustrate the effect, consider the simplified situation of the critical 2D cluster formation of B-molecules on a flat surface of an A-crystal. The condition for the critical cluster formation is given by the equivalence of chemical potentials of B-molecules in the surface gas phase of concentration \(n_{S}\) and in the critical cluster of radius \(r_{C}\), namely, \(T\ln(n_{S})=-\varepsilon_{BB}+\gamma_{B}/r_{C}\), where \(\gamma_{B}\) is the energy per  
Figure 2: (a) Time evolution of the mean \(\langle R\rangle\) and maximum \(R_{m}\) radii calculated for an ensemble of \(10^{4}\) particles with Gaussian initial distribution (\(\langle R\rangle=1.5\) and standard deviation \(\sigma=0.25\)) obtained via numerical solution of Equations (4)–(7) for T = 0.032 eV (100 \({}^{\circ}\)C), \(\gamma_{0}~{}=~{}0.1\) eV, \(\varepsilon~{}=~{}0.4\) eV and \(n_{S}~{}\approx~{}0.002\). The values of \(\varepsilon_{AB}\) and \(\gamma_{1}\), respectively, are (eV units): (1) 0.24 and 0; (2) 0.21 and 0; (3) 0.21 and 0.01; (4) 0.196 and 0; (5) 0.18 and 0. (b) Particle size distribution (PSD) corresponding to the curves (3) at the time moments indicated by arrows in (a).  
B-molecule on the cluster perimeter. For the critical radius, we then obtain \(r_{\rm C}=\gamma_{B}/(\varepsilon_{BB}+T\ln(n_{\rm S}))\). Using this one can easily obtain the critical cluster nucleation barrier \(\Delta_{\rm C}\)  
\[\Delta_{\rm C}=(-\varepsilon_{BB}-T\ln(n_{\rm S}))\cdot\pi r_{\rm C}^{2}+\gamma _{B}2\pi r_{\rm C}=\pi\frac{\gamma_{B}^{2}}{(\varepsilon_{BB}+T\ln(n_{\rm S}))} \tag{10}\]  
Equation (10) shows that for the \(\varepsilon_{BB}\) values that are so low that \(\varepsilon_{BB}+T\ln(n_{\rm S})\leq 0\), no critical cluster can be formed; therefore, the interaction between B-molecules can be disregarded.