# Reverse Coarsening and the Control of Particle Size Distribution through Surfactant
## 2 Results and Discussion
### Simulation of Reverse Coarsening in the Particle Ensemble  
The value of \(R_{eq}\) given by Equation (3) is a good estimate if the PSD before its collapse is close to the initial PSD. This estimate implicitly assumes that the surface energy \(\gamma\) does not depend on the particle radii or shape. In reality, the shape of small particles may deviate from spherical, which, in the context of our theory, means that \(\gamma\) would increase with \(R\). In turn, this dependence on the radius implies that \(\Gamma>0\) for small enough particles (depending on \(\varepsilon_{AB}\) and \(T\)), which will dissolve and transfer their mass to bigger particles with \(\Gamma<0\). Such mass transfer modifies the initial PSD and shifts the mean radius, hence \(R_{eq}\), towards higher values. To confirm this effect, we conducted numerical simulations using the Kinetic Monte Carlo technique [53]. We started with a distribution of spherical particles nucleated and grown so that their evolution is only possible by an exchange of molecules between particles. This initial configuration is chosen so that the vast majority of A-molecules are bound in particles. The particle surfaces are covered with B-molecules, which significantly slows down an exchange of A-molecules between particles and solution. Therefore, it is reasonable to assume that the molecular diffusion in solution is the fastest process and that the coarsening process is limited by the attachment-detachment events with effective reaction rate constant \(K\). Then, the evolution of the \(i\)-th particle's radius \(R_{i}\) (measured in the units of inter-atomic distance) within this [PERSON] approximation [15] can be described by  
\[\frac{dR_{i}}{dt}=K\cdot(n-n_{GT}(R_{i})) \tag{4}\]  
where \(n_{GT}(R_{i})\) is the Gibbs-Thomson concentration (equilibrium concentration) for the \(i\)-th particle, and \(n\) is the mean field concentration of free A-molecules in solution. The latter is determined using mass conservation for all particles (valid because of our earlier assumption about the dominant fraction of A-molecules contained in particles),  
\[\frac{d}{dt}\sum\limits_{i=1}^{N}\frac{4}{3}\pi R_{i}^{3}=\sum\limits_{i=1}^{N }4\pi R_{i}^{2}\frac{dR_{i}}{dt}=\sum\limits_{i=1}^{N}4\pi KR_{i}^{2}\cdot(n- n_{GT}(R_{i}))=0\to n=\frac{\sum\limits_{i=1}^{N}R_{i}^{2}\cdot n_{GT}(R_{i})}{ \sum\limits_{i=1}^{N}R_{i}^{2}} \tag{5}\]  
The Gibbs-Thomson concentration is given by the equality of molecular chemical potentials in the particle and solution \(T\ln(n_{GT})=-\varepsilon+2\Gamma/R\) (see Equation (2)) taking a more realistic expression for surface energy \(\gamma\). This expression is obtained considering the generic expression for particle energy \(\varepsilon(N)\) as a function of the number \(N\) of atoms/molecules it contains [54; 55; 56; 57]. \[-\varepsilon(N)=-\varepsilon+\varepsilon_{S}\cdot N^{-1/3}+\varepsilon_{C} \cdot N^{-2/3}+\varepsilon_{E}\cdot N^{-1} \tag{6}\]  
where \(\varepsilon>0\) is the bulk contribution to energy (cohesive energy), \(\varepsilon_{S}>0\) is the contribution of facets' surfaces, \(\varepsilon_{C}>0\) is the contribution of edges, and \(\varepsilon_{E}\) defines the energy origin (reference point). Adopting this expression for the generic case of spherical particles with radius \(R\) and neglectingthe small energy origin \(\varepsilon_{E}\), we use \(N\sim R^{3}\) in Equation (6) so that the energy per molecule in a particle (or molecular chemical potential) takes the form \(-\varepsilon(R)=-\varepsilon+2\gamma(R)/R\) with \(\gamma(R)=\gamma_{0}+\frac{\gamma_{1}}{R}\). Both parameters \(\gamma_{0}\) and \(\gamma_{1}\) are positive, as the former corresponds to true surface energy for \(R\gg 1\), while the latter takes into account non-sphericity effects according to Equation (6). With such a generic form, this equation can also effectively account for the contributions to the surface energy coming from the possible roughness of particle surfaces.