Order-disorder phase transition critical points

One consequence of Eq. (11.4.5) follows immediately. In the neighborhood of certain points of instability such as the gas-liquid critical point or order disorder phase transitions, the susceptibilities corresponding to the fluctuations in the order parameters become very large. Thus if A does not increase as rapidly as/, the corresponding relaxation rates Twill become small. This phenomenon is called critical slowing of the fluctuations. There has been much recent work on this phenomena (Swinney, 1974). 

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

Fig. 35 Phase diagrams AB miktoarm-star copolymers for n = 2, n = 3, n = 4 and n = 5. mean-field critical point through which system can transition from disordered state to Lam phase via continuous, second-order phase transition. All other phase transitions are first-order. From [112]. Copyright 2004 American Chemical Society...

Fig. 6.36 Phase diagram calculated using SCFT for a blend of a symmetric diblock with a homopolymer with fl = 1 (see Fig. 6.32 for a blend with a diblock with / = 0.45) as a function of the copolymer volume fraction

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

If a critical point solid-liquid exists, both phases would have the same equation of condition, analogous to Van der Waals equation for liquids and gases. From considerations based on the molecular theory, Tammann concludes that this is not the case. A liquid can change continuously into an amorphous glass-like sohd, but never into an anisotropic crystal. It is difficult to conceive of continuous transition from a state of disordered motion, such as we must assume in the liquid and gaseous states, to the ordered arrangement of the atoms and molecules in a crystal. 

Figure 5.8 Theoretical phase diagram representing the order to disorder transition in the LiMn2O4 electrode. The dotted line represents the isotherm at T= 298 K. (1 - 6)tr(i) and (1 — 5)tr(2) indicate the disorder-to-order and the order-to-disorder transition points, respectively. The disordered phase is stable over the entire range oflithium content above the critical temperature T. (Reproduced with permission from Ref. [16].)...

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