A symmetric polymer blend has segments per chain and is quenched to (critical value

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Non-equilibrium statistical mechanics and stochastic thermodynamics

Soft Matter at the Thermal Scale, Ch Spinodal decomposition and coarsening

Problem

A symmetric polymer blend has N=250N = 250 segments per chain and is quenched to χ=0.012\chi = 0.012 (critical value χc=2/N=0.008\chi_c = 2/N = 0.008, so the quench is inside the spinodal). At some instant a small composition ripple ϕ(x)=0.5+0.01cos⁡(qx)\phi(x) = 0.5 + 0.01\cos(qx) is present. For this estimate, gradient-energy corrections to the local response can be ignored, so the flux obeys J=−M∂μ∂x,μ=f′(ϕ),∂tϕ=Mf′′∇2ϕ.J = -M\frac{\partial \mu}{\partial x}, \qquad \mu = f'(\phi), \qquad \partial_t\phi = Mf''\nabla^2\phi. Someone argues: "diffusion always moves material from a region of high concentration to a region of low concentration, so the flux must be directed down the concentration gradient here too; the ripple's amplitude therefore decreases with time." ((a)) Compute f′′(12)f''(\tfrac12) for this quench (in units of kBTk_BT per site). ((b)) State whether the quoted reasoning is correct, and say whether the ripple's amplitude grows or decays.

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