There exist two competing NP transport processes at the liquid–vapor interface of a Leidenfrost droplet, whose joint effect determines the number of NPs to be present at the interface. On the one hand, NPs accumulate at the interface through self-organization as a result of the fast evaporation of the liquid. It has been observed by several researchers recently [28,29] that self-organized NP superlattices can nucleate and grow at the liquid–air interface when a droplet of a NP colloidal solution evaporates, controlled by the evaporation kinetics and particle interactions at the phase-change interface. The rate of accumulation, *v*_{a}, is related to the evaporation rate through $va=\kappa v(Ts-Tb)Ac0/hL\rho l$, where *c*_{0} is the bulk NP concentration of the droplet. On the other hand, NPs depart from the liquid–vapor interface driven by the surface tension gradient surrounding the droplet (Marangoni convection) and the concentration gradient between the interface and the bulk droplet (mass diffusion). The rate of departure can be calculated as *v*_{d} = *H*_{m}*A*(*c*_{i} − c_{0}), where *H*_{m} is the mass transfer coefficient, *c*_{i} is the NP concentration at the liquid–vapor interface. Similar to heat transfer, *H*_{m} is related to the dimensionless parameter, Sherwood number, in the form of Sh = *H*_{m}*R*/*D*, where *R* is the radius of the fuel droplet and *D* is the diffusivity of the NPs in the base fuel. At steady-state before ignition, the accumulation–departure processes reach equilibrium, which means *v*_{d} = *v*_{a}, i.e.,Display Formula

(9)$Hm(ci-c0)=\kappa v(Ts-Tb)c0/hL\rho l$