Moving cooling source-induced phase separation in binary liquids: An interplay of competing velocities

We investigate phase separation dynamics in a binary mixture driven by a moving cooling source from which cold thermal fronts propagate radially into the surrounding mixture. The motion of the source introduces two distinct velocity scales: v_{s} associated with the translation of the source and v related to the propagation of the cooling thermal fronts. Using a modified Cahn-Hilliard-Cook model that explicitly couples the time-dependent temperature and concentration fields, we demonstrate that the competition between these two velocities fundamentally controls the morphology and kinetics of phase separation. Our numerical simulation results reveal a rich variety of nonequilibrium patterns whose characteristics depend not only on the ratio v_{s}/v but also on the absolute values of v_{s} and v. Same value of v_{s}/v yields distinctly different patterns for different v. The spatial extent and shape of the quenched region, determined by the temperature profile delineating spatial regions with local temperatures above and below the demixing temperature dictates the domain morphology. Furthermore, unlike conventional isothermal phase separation, the system does not exhibit dynamical scaling of the equal-time two-point correlation function with a single growing length scale, reflecting the intrinsically nonstationary nature of phase separation driven by a moving quench source. Our results establish moving thermal sources as an effective means of controlling phase-separation pathways and provide a strategy for engineering desired pattern structures by tuning the two velocities.