# Reverse Coarsening and the Control of Particle Size Distribution through Surfactant
## 1 Introduction  
The generation of identical nanoparticles [1; 2; 3; 4; 5; 6; 7] and fine nanostructures [8; 9; 10] is a key requirement for self-assembly processes in bottom-up nanotechnology. However, structure coarsening, or Ostwald ripening [11], is a universal phenomenon that stands in the way of forming identical nano-building blocks through quasi-equilibrium methods. Coarsening represents the spontaneous evolution of precipitated particles, during which smaller particles transfer mass to bigger ones, thus broadening the particle size distribution (PSD) [12; 13; 14; 15]. This is a common phenomenon taking place at the final stages of many first-order phase transformations following nucleation and growth, for example, in two-phase mixtures [11], binary alloys [16], clusters on surfaces [17], oil-water emulsions [18], and during epitaxial growth [19; 20] and synthesis of nanoparticles [21]. The main driving force behind coarsening is the minimization of the interfacial area between the two phases, and as such, it depends intimately both on the interface energetics and the system temperature. Much effort has been devoted to finding ways to inhibit or even reverse coarsening in order to achieve a narrower PSD. Known examples of PSD narrowing discussed in the literature include the so-called digestive ripening [1; 2; 3; 22; 23], inverse ripening [24; 25; 26; 27], and size focusing during production-controlled growth [28; 29]. Some of these processes take place in the presence of surfactants, which are widely used in nanoparticle production, mainly to stabilize particles [30; 31; 32] and to tune the growth regimes towards the production of particles with required shapes and sizes [33; 34; 35]. There are also mechanisms of inhibition, or of arresting Ostwald ripening [36; 37; 38], which do not permit real narrowing of the PSD but rather preserveit from broadening. Digestive ripening can be explained by taking into account electric charges on the particle and by including the electrostatic energy in the free energy analysis [22; 23], while inverse ripening has been observed for gold inclusions in an amorphous SiO\({}_{2}\) matrix irradiated with MeV gold ions [25; 27]. The PSD narrowing in the latter case is of transient character. It is affected by mechanical stresses and requires fine-tuning of radiation-annealing cycles. On the other hand, size focusing during production-controlled growth is predicted for effectively open systems where the material deposited on particles is continuously reproduced in the bulk of solution [28; 29]. In summary, there is currently no effective theory describing the phenomenon of PSD narrowing in conservative and charge-neutral systems.  
Besides the quasi-equilibrium fabrication methods when the system is thermodynamically driven, narrow PSDs are also successfully generated using discrete size selection by undercooling [39; 40; 41] and dewetting [42; 43] in a non-equilibrium regime. Here, we study the role of surfactants, or surface binding molecular ligands [44; 45; 46; 47; 48; 49; 50], in controlling PSDs. We present a thermodynamic mechanism for particle size stabilization and narrowing of a PSD by surfactant molecules binding to particle surfaces. This process is illustrated by considering a simple dilute solution of two species (referred to here as _molecules_) in a neutral solvent in the presence of precipitated aggregates (called _particles_) of one of the components. Based on a free energy analysis, we demonstrate that there exists a range of parameters at which reverse coarsening (RC) occurs and, consequently, smaller precipitated particles grow at the expense of bigger ones. This regime is spontaneous: i.e., it is thermodynamically driven and converts any initial distribution of particles into an array of almost identical particles [51]. Notably, the transition into this regime can be obtained by varying the surfactant concentration and/or the system's temperature.