A binary mixture below its critical point has a local free energy density (in units…

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 binary mixture below its critical point has a local free energy density f(ϕ)f(\phi) (in units of kBTk_BT per unit volume) and a square-gradient free energy F[ϕ]=∫dV [f(ϕ)+κg2∣∇ϕ∣2],F[\phi] = \int \mathrm{d}V\,\left[f(\phi) + \frac{\kappa_g}{2}|\nabla\phi|^2\right], where ϕ(r,t)\phi(\mathbf r,t) is the local composition and κg>0\kappa_g>0 is a constant gradient coefficient. The order parameter ϕ\phi is conserved (no chemical reaction creates or destroys it) and moves down gradients of the exchange chemical potential μ(r,t)=δF/δϕ\mu(\mathbf r,t) = \delta F/\delta\phi with a constant mobility MM, so the local flux is J=−M∇μ\mathbf J = -M\nabla\mu. Composition varies smoothly enough that ϕ→\phi\to constant (or ϕ\phi is periodic) at the boundary of the region considered, so all boundary terms in integrations by parts vanish. ((a)) Using the calculus of variations, show that μ(r,t)=f′(ϕ)−κg∇2ϕ\mu(\mathbf r,t) = f'(\phi) - \kappa_g\nabla^2\phi. ((b)) Combine this with conservation of ϕ\phi to obtain the Cahn-Hilliard equation ∂ϕ/∂t=M∇2[f′(ϕ)−κg∇2ϕ]\partial\phi/\partial t = M\nabla^2\big[f'(\phi)-\kappa_g\nabla^2\phi\big]. ((c)) Write ϕ(r,t)=ϕˉ+δϕ(r,t)\phi(\mathbf r,t) = \bar\phi + \delta\phi(\mathbf r,t) about a uniform background ϕˉ\bar\phi, with δϕ=A eiq⋅r+R(q)t\delta\phi = A\,e^{i\mathbf q\cdot\mathbf r + R(q)t} and ∣δϕ∣≪1|\delta\phi|\ll1, and show that the growth rate obeys R(q)=−Mq2[f′′(ϕˉ)+κgq2].R(q) = -Mq^2\left[f''(\bar\phi) + \kappa_g q^2\right].

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