Phase-field model for the Brownian motion of droplets*

Brownian motion (BM) plays an important role in natural science for the stochastic

motion of particles. Despite of its importance, BM was originally applied for a single

solid-particle, the modelling of BM for droplets coupling with hydrodynamics remains

a knotty issue. We present a multicomponent phase-field model coupled with the

full Navier-Stokes equation to simulate BM of droplets which is perturbed by the

thermal composition noise amid a liquid matrix. The model validation is proceeded for

different mesh finenesses, various droplet radii, diverse interfacial tensions, and distinct

viscosities of the droplet-matrix system. In addition, the structure factor of the disturbed

droplet interface is compared with the capillary wave theory (CWT). We find that

the composition noise induced BM shows distinct behaviors from the viscosity related

random body force in the Langevin mechanics. Similar to the quintessential scaling

law D∼k_BT/(6πηr) in Einstein’s theory for BM, we demonstrate the scaling laws

also depending on the surface tension, and droplet viscosity. Our finding paves an

alternative way for comprehending the microstructural formation consisting of droplets

in soft matters.