Lattice Boltzmann modelling of diverse particle deposition patterns by droplet evaporation

In this work, we propose a lattice Boltzmann model (LBM) to simulate diverse particle deposition patterns induced by isothermal droplet evaporation. The model is composed of two distributions, for the multiphase flow with phase change, and the particle transport with deposition, coupled with a contact angle hysteresis model for the contact line stick-slip dynamics. The model is validated by two benchmarks, and our simulations agree well with the theoretical solutions or experimental results. With the validated LBM, we first reproduced diverse deposition patterns, ranging from the coffee ring, uniform, to mountain-type patterns in single and multiple symmetrical/unsymmetrical forms. Then a parametric study is conducted to investigate how the solvent/particle/substrate properties affect the evaporation dynamics and resultant deposition patterns. Afterwards, we apply the average ratio ( $r_{\phi ,a}$ ) of particles deposited at the droplet periphery and the centre to quantitatively classify the diverse emerging patterns. We show that $r_{\phi ,a}$ is controlled by the competition between the capillary transport and particle diffusion, leading to a linear dependence on the average Péclet number $\textit{Pe}_{a}$ . Finally, we validated the scaling by lattice Boltzmann simulations with the proposed $\textit{Pe}_{a}$ spanning over three orders of magnitude, supplemented by discussions from the aspect of the particle transport equation.