Thermodynamics of smectic mixtures

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Abstract
P. G. de Gennes discusses the thermodynamic phase diagram of a mixture of substances generating smectic A and C phases with the aid of the expansion of free enthalpy : F(T, x, ϕ) = F0(T, x) + 1/2a( T, x)ϕ2 + 1/4b(T, x)ϕ 4 + ... where x is mole fraction of second component, ϕ the symmetry-breaking parameter ( tilt angle), and concludes that the phase transition is one of second order, if b is positive where a(T, x) changes sign, which locates the single-line phase boundary x0(T). This is erroneous : the transition so arising is of first order unless in addition ∂2 F0/∂x2 > 1/2b(da/d x)2. and, without satisfaction of this condition there is a coexistence range in the phase diagram delimited by lines on either side of x0(T), which is a spinodal rather than a phase-boundary

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