(a) Sketch of aggregation kinetics inside the two compartments I and II. Left: Initially, monomers get enriched on a short diffusive time scales due to the partitioning mediated by the phase separated compartments (Equation 1). Center: Monomers slowly aggregate. More aggregates nucleate and grow in compartment I due to the initial partitioning of monomers. This pronounced, initial aggregation causes a continuous monomer flux into compartment I, further promoting aggregation (positive feedback indicated by arrows). Right: Partitioning of monomers together with the positive feedback can cause a very pronounced accumulation of aggregates relative to compartment II. (b) Aggregate concentration as a function of time obtained from solving numerically and analytically Equation 3 actually confirms that aggregates can enrich by several orders of magnitude. (c) The asymptotic concentrations and inside each of the compartment inversely scale for small compartments, while for large compartment I, aggregate enrichment therein vanishes while depletion inside compartment II is dominated by primary nucleation. The asymptotic concentration in the absence of monomer partitioning, , is denoted as . Dashed line are the scalings given in the the main text. Parameters: . (d) Partitioning factor of aggregates inside compartment I as a function of monomer partitioning can reach very large values. The behavior switches from secondary nucleation dominated increase at small compartment I volumes to primary dominated growth at large volumes. Dashed line are the scalings given in Equation (6). (e) The slope of the partitioning factor as a function of mean volume fraction , equivalently speaking, volume of compartment I, changes its sign when partitioning is dominated by primary () or secondary nucleation (). Parameters: (b,e) consistent with weak interactions.