A binary alloy A-B forms a regular solution with free energy of mixing per site…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Point defects, dislocations and alloys

Problem

A binary alloy A-B forms a regular solution with free energy of mixing per site ΔF(x,T)=Ωx(1−x)+kBT[xln⁡x+(1−x)ln⁡(1−x)],\Delta F(x,T)=\Omega x(1-x)+k_BT\big[x\ln x+(1-x)\ln(1-x)\big], where xx is the mole fraction of B and Ω>0\Omega>0 favours phase separation. ((a)) Derive the spinodal condition ∂2ΔF/∂x2=0\partial^2\Delta F/\partial x^2=0 and show it can be written Tsp(x)=2ΩkBx(1−x)T_{\rm sp}(x)=\dfrac{2\Omega}{k_B}x(1-x). ((b)) Show that the maximum of Tsp(x)T_{\rm sp}(x) over xx occurs at x=12x=\tfrac12 and equals the critical temperature Tc=Ω/2kBT_c=\Omega/2k_B. ((c)) A particular A-B system is well described by Ω=0.20 eV\Omega=0.20\,\mathrm{eV}. Find TcT_c, and find the two spinodal compositions at T=900 KT=900\,\mathrm{K}, that is the range of xx inside which a homogeneous alloy of that composition is unstable to any small composition fluctuation.

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