Spontaneous liquid-liquid phase separation is commonly understood in terms of phenomenological mean-field theories. These theories correctly predict the structural features of the fluid at sufficiently long time scales and wavelengths. However, these conditions are not met in various examples in biology and materials science where the mixture is slowly destabilised, and phase separation takes place close to the critical point. Using kinetic Monte Carlo and molecular dynamics simulations of a binary surface fluid under these conditions, we show that the characteristic length scale of the emerging structure decreases, in 2D, with the 4/15 dynamic critical exponent of the quench rate rather than the mean-field 1/6th power. Hence, the dynamics of cluster formation governed by thermodynamically undriven Brownian motion is much more sensitive on the rate of destabilisation than expected from mean-field theory. We discuss the expected implications of this finding to 3D systems with ordering liquid crystals, as well as phase-separating passive or active particles.
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