Soft Matter at the Thermal Scale, Ch Membranes, fluctuations and soft matter in biology
Problem
Looking at fluorescence images of a protein droplet inside a cell, a colleague argues: "This condensate must be a solid aggregate rather than a liquid droplet, because it appears at an extremely low protein concentration, far below the volume fraction ϕ≈0.5 at which a small solute typically becomes insoluble. Only irreversible aggregation could produce such a low threshold." The protein behaves as a flexible chain of N=400 Kuhn segments, described by Flory-Huggins theory.
((a)) Calculate the composition ϕc of this chain's Flory-Huggins critical point, where demixing into two coexisting phases first appears as χ rises.
((b)) Explain what is wrong with the colleague's argument, referring to how the mixing entropy of a long chain depends on N.