Transitions and Critical Points Are Universal

In Chapter 25 we noted that the shapes of the coexistence curves for liquid-liquid immiscibility are about the same as for boiling. Both display critical points, and both are described approximately by the same simple model. Advances in theory and experiments since the 1940s have led to a revolution in understanding critical phenomena it goes beyond those models. 

How can you mathematically express the shape of a coexistence curve near the critical point First, you need to define the order parameter of the system. An order parameter is a quantity m on which the free energy depends. The order parameter m is zero above the critical temperature, indicating that the system is disordered or randomly mixed, m is non zero below the critical temperature, indicating that the system is ordered or phase-separated in some way. For liquid-liquid immiscibility, the order parameter could be the difference in phase compositions m = x - x. At the critical temperature, x = x = 0.5, so the quantity m (T) = 2x - 1 is a conveniently normalized order parameter it equals 1 at low- temperature, and 0 at the critical point T = Tc. At low-temperatures, m - 1 indicates that the two coexisting phases are very different. As T Tt, m 0 indicates that the two states of the system become indistinguishable from each other. 

Chapter 26. Cooperativity The Helix-Coil, Ising Landau Models 

C Careri, Order and Disorder in Matter, English language translation by K Jarrett, Benjamin Cummings, Reading, 1984. 

J Als-Nielsen, Phase Transitions and Critical Phenomena, Volume 5a, 

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