  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.
