# Reverse Coarsening and the Control of Particle Size Distribution through Surfactant
## 2 Results and Discussion
### Simulation of Reverse Coarsening in the Particle Ensemble
Therefore, the surface energy \(\gamma(R)\) decreases with increasing particle radius, making larger particles relatively more stable. With this surface energy, the Gibbs-Thomson concentration is now  
\[n_{GT}(R)=\exp\left[-\frac{\varepsilon}{T}+\frac{2\gamma_{0}}{T\cdot R}+\frac{ 2\gamma_{1}}{T\cdot R^{2}}-\frac{2}{R}\cdot\ln(1+n_{B}E_{1})\right] \tag{7}\]  
Using Equations (4)-(7), we introduced molecular absorption and emission probabilities by the particle (Equations (16)-(17) in [51]) and implemented them in the original Fortran code to simulate the evolution of a \(10^{4}\) particle ensemble. For illustration purposes, we considered two values of \(\gamma_{1}\): 0 and 0.01 eV. The results shown in Figure 2 correspond to some realistic values of known model parameters, chosen for illustrative purposes. We see that both the mean \(\langle R\rangle\) and maximum \(R_{m}\) radii continuously grow with time for \(\varepsilon_{AB}=0.18\) eV, which is characteristic of ordinary coarsening. In contrast, for \(\varepsilon_{AB}>0.197\) eV, the evolution displays reverse coarsening, as the radii \(\langle R\rangle\) and \(R_{m}\) both converge to the same value \(R_{eq}\) predicted by Equation (3), which, for \(\gamma_{1}=0\), gives a value of \(R_{eq}\) that is very close to \(\langle R\rangle\) of the initial PSD. For \(\gamma_{1}=0.01\) eV, the value of \(R_{eq}\) is larger than the average radius \(\langle R\rangle\) of the initial PSD. As discussed, this effective increase in \(\langle R\rangle\) in the early stages is due to the mass transfer from the smaller particles with \(\Gamma>0\) to the bigger particles with \(\Gamma<0\). Clearly, there is a threshold value for \(\varepsilon_{AB}\) above which the system displays reverse coarsening. This threshold is determined by the requirement \(\Gamma=0\) and depends on the surfactant concentration \(n_{B}\), the molecular-surfactant interaction energy \(\varepsilon_{AB}\), the temperature \(T\) and the size dependence of the surface energy determined by \(\gamma_{1}\). In the case of \(\gamma_{1}\approx 0\), the threshold value \(\varepsilon_{AB}^{*}\) is estimated for the values in Figure 1 as  
\[\Gamma=\gamma_{0}-T\cdot\ln(1+n_{B}E_{1})\ \approx\ 0\ \to\ \varepsilon_{AB}^{*}\ \approx\ 0.197\ \text{eV} \tag{8}\]  
Choosing a surfactant with \(\varepsilon_{AB}\approx\varepsilon_{AB}^{*}\) preserves the initial PSD (see curves 4 in Figure 2). The evolution of the PSD is relatively slow (note the log scale for the \(x\)-axis in Figure 2); therefore, replacing the surfactant molecules with those having \(\varepsilon_{AB}\approx\varepsilon_{AB}^{*}\) or \(\varepsilon_{AB}>\varepsilon_{AB}^{*}\) allows it to be stopped or reversed, respectively, providing a great opportunity for thermodynamic control of the PSD for generated nanoparticles. It is worth noting that the values of \(\varepsilon_{AB}\) generating reverse coarsening must also have an upper limit, as at values \(\varepsilon_{AB}>\varepsilon\), the A-molecules would rather form molecular-surfactant complexes in the solution than stay bound together as a particle. Therefore, in our case, the inequality \(\varepsilon_{AB}<\varepsilon\) must hold. The value of \(\varepsilon_{AB}^{*}\) in our simulations is chosen for illustration purposes without reference to any particular material's surface or binding molecule. Determining \(\varepsilon_{AB}^{*}\) in practice is difficult, as it requires measuring the particle's surface energy \(\gamma\) in solution. In our case, it is reasonable to demand that \(\varepsilon_{AB}^{*}\) is low enough for the surfactant molecules not to affect the particle shape during its growth or dissolution [58]. Note that the fine tuning of the coarsening process, which is determined by the value of the effective surface energy \(\Gamma\), can be more easily achieved by varying the bulk concentration of surfactant molecules \(n_{B}\) and the system temperature \(T\) rather than \(\varepsilon_{AB}\) using different surfactant molecules.