Phase separation with conserved order parameters has gained renewed interest due to recent experimental progress in the field of bio-molecular condensates, however an analytical solution for phase boundaries is intractable due to the transcendental nature of the equilibrium equations, except in the simplest 1-D case [1]. This is to be contrasted with experimental systems where the plentitude of solute species results in a high-dimensional phase space, and these extra dimensions can have strong impacts on the phase space [2-3], rendering a simplified 1-D description inappropriate. There is thus a pressing need for an analytical framework that both treats the high-dimensional system and requires relatively simple input from experiments. We achieve this by not solving for, but perturbing, the equilibrium conditions. This gives an analytical form of the bulk dilute phase response function in terms of the phase boundary surface normal and tie line, that connects experimental observables to the underlying stabilisation free energy [4]. Consequently, this framework can be applied to study modulations of phase separation in a rigorous manner.

[1]: Qian, D., Michaels, T. C. T. & Knowles, T. P. J. Analytical Solution to the Flory-Huggins Model. J. Phys. Chem. Lett. 13, 7853–7860 (2022).

[2]: Arter, W. E. et al. Biomolecular condensate phase diagrams with a combinatorial microdroplet platform. Nat. Commun. 13, 7845 (2022).

[3]: Qian, D. et al. Tie-Line Analysis Reveals Interactions Driving Heteromolecular Condensate Formation. Phys. Rev. X 12, 041038 (2022).