The Possibility of Second-Order Transitions

The question of whether the normally first-order smectic-A to nematic phase transition can be second order in some materials is somewhat controversial. All of the published mean-field theories of the smectic-A phase do exhibit second-order phase changes for certain values of the potential parameters. In both McMillan s theory and that of Lee et al, the second-order transition is predicted to occur at that end of homologous series having short chain lengths. More specifically, these models predict the second-order changes to occur when the ratio of transition temperatures Tan/Tni (or Tac/Tci) is at or below about 0.88 (see Fig. 5). 

The situation with respect to Landau-type phenomenological theories is also contradictory. Drawing an analogy between the smec-tic-A phase of liquid crystals and the superconducting phase of metals, de Gennes22 23 constructed a phenomenological theory from which he concludes that the smectic-A to nematic phase transition can be second order. Halperin and Lubensky, on the other hand, have improved the analogy with superconductors and conclude that the transition will always be at least weakly first order. 

My own conjecture in this matter is that the smectic-A to nematic phase transition is always first order. The basis of my argument is the analogy between this transition and the melting of a crystalline solid. When a crystal melts, three dimensional long-range translational 

Kirkwood has pointed out that the density distribution function of a crystalline solid (the translational molecular distribution function) can be expanded in a three-dimensional Fourier series. The coefficients in this series are then identified as the order parameters of the crystalline phase. All these order parameters vanish discontin-uously at the first-order melting point. Empirically, there are no second-order melting transitions, nor do there seem to be any solid-liquid critical points. Though not a proven fact (as far as I am aware), it seems reasonable that crystal melting is always first order because all of the order parameters cannot vanish simultaneously and continuously before the free energy of the solid phase exceeds that of the liquid phase. 

In the smectic-A phase, the single-molecule distribution function, Eqs. [7] and [10] can likewise be represented as a (one-dimensional) Fourier series in which all the coefficients may be considered order parameters. The disappearance of smectic-A order requires the simultaneous vanishing of all the order parameters. That they all can vanish simultaneously and continuously before the free energy of the smectic-A phase exceeds that of the nematic phase seems just as unlikely here as in the (empirically verified) case of the crystalline solids. It seems clear to me that the reason the various theoretical treatments mentioned above can exhibit second-order phase changes is that an insufficient number of order parameters is included. In all the treatments, either the potential, the potential of mean force, or the distribution function are expressed in terms of highly truncated Fourier series. Such truncation automatically limits the number of order parameters. Small numbers of order parameters can then vanish simultaneously and continuously under certain conditions providing the spurious second-order phase transitions. 

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