grow. As a result, the average size of the nanoparticles in the solution will grow, and the dispersion of sizes will decrease. Therefore, if a solution is left for a long time, in the extreme case of , its particles would evolve until they would finally form a single huge spherical particle to minimize the total surface area. The history of research progress in quantitatively modeling Ostwald ripening is long, with many derivations. In 1958, Lifshitz and Slyozov performed a mathematical investigation of Ostwald ripening in the case where diffusion of material is the slowest process. They began by stating how a single particle grows in a solution. This equation describes where the boundary is between small, shrinking particles and large, growing particles. They finally conclude that the average radius of the particles ⟨R⟩, grows as follows: where Note that the quantity is different from , and only the latter one can be used to calculate average volume, and that the statement that ⟨R⟩ goes as relies on being zero; but because nucleation is a separate process from growth, this places outside the bounds of validity of the equation. In contexts where the actual value of is irrelevant, an approach that respects the meanings of all terms is to take the time derivative of the equation to eliminate and . Another such approach is to change the to with the initial time having a positive value. Also contained in the Lifshitz and Slyozov derivation is an equation for the size distribution function of particles. For convenience, the radius of particles is divided by the average radius to form a new variable, ρ = . Three years after that Lifshitz and Slyozov published their findings (in Russian, 1958), Carl Wagner performed his own mathematical investigation of Ostwald ripening, examining both systems where diffusion was slow and also where attachment and detachment at the particle surface was slow. Although his calculations and approach were different, Wagner came to the same conclusions as Lifshitz and Slyozov for slow-diffusion systems. This duplicate derivation went unnoticed for years because the two scientific papers were published on opposite sides of the Iron Curtain in 1961. It was not until 1975 that Kahlweit addressed the fact that the theories were identical and combined them into the Lifshitz-Slyozov-Wagner or LSW theory of Ostwald ripening. Many experiments and simulations have shown LSW theory to be robust and accurate. Even