Weak First-Order Transitions

In reality, the N-A transition is, as a rule, weak first order transition. There are, at least, two ways to understand this in framework of Landau approach. We still use the same smectic order parameter pi but include additional factors, either (a) higher harmonics of the density wave, or (b) consider the influence of the positional order on the orientational order of SmA, the so-called interaction of order parameters. 

We keep equality of nematic orientational order parameters in both phases 5n = Sp, and take only the amplitude of the second harmonic p2 of the density wave as an additional SmA order parameter 

Therefore we have two order parameters (for the same transition) and the free energy density is  

For a typical situation pi p2, it is sufficient to take only one cross-term with coefficient B. Coefficients Oi and 2 re assumed positive and, in addition, we assume T T2 because on cooling, the first Fourier harmonic appears at higher temperature, and afterwards, at a lower temperature, the single harmonic law is violated and p2 appears. The minimization of (6.20) with respect to p2 results in 

Due to smallness of p2, the second and fourth terms are small and we can find p2  

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A Peierls distorsion at zero Kelvin on purely 1 D system is a second order phase transition. The observed transition (Figure 4) might be of weakly first order transition it is interesting to note that the specific heat anomaly increases with the transition temperature in the three considered compounds. We suppose that this critical temperature increases in relation with the 3-D and the disorder effects. 

It is concluded [217] that an interpretation of the ideal herringbone transition within the anisotropic-planar-rotor model (2.5) as a weak first-order transition seems most probable, especially since previous assignments [56, 244] can be rationalized. This phase transition is fluctuation-driven in the sense of the Landau theory because the mean-field theory [141] yields a second-order transition. Assuming that defects of the -v/3 lattice and additional fluctuations due to full rotations and translations in three dimensions are not relevant and only renormalize the nonuniversal quantities, these assignments should be correct for other reasonable models and also for experiment [217]. 

For a weak first order transition, B is small and IFFj aC may be expected to be a very small quantity. 

Fig. 5. Proposed phase diagram of Ar on graphite. S, L, V and F represent, respectively, solid, liquid, vapor and fluid phases. Solid circles near 55 K at submonolayer coverages correspond to positions of heat-capacity anomaly arising from liquid-vapor transition. Other circles are signatures of melting. Dashed hnes are speculative. At submonolayer coverages, Ar solid melts via a weak first-order transition at a triple-point temperature (47.2 K) to a liquid-vapor coexistence region. The liquid phase appears to be orientationally ordered below 54 K. The positions of the broad anomalies centering near 49.5 K, due to the gradual decrease of this order, are not shown. Unless indicated otherwise, the uncertainty in the peak position of the heat-capacity anomaUes is comparable to or smaller than the size of the circles (from Ref 42).

In the low-field range the weak first order transitions are fully reversible, but above 8 teslas we observe the strong hysteresis associated with the formation of a metastable SDW phase. Finally, the direct measurement of the specific heat as a function of the field showed no variation of the density of state at the Fermi level, for the normal metallic state below the FISDW threshold fieldlO... 

For B = 0, the free energy is symmetric with respect to ri and we have a second order transition. For small B 7 0 the transition is called weak first order transition because the discontinuity of the order parameter is small and Tc becomes close to Tc. 

The transition from water to ice at 1 atmosphere pressure is a first-order transition, and the latent heat is about 100 J/g. The isotropic-nematic transition is a weak first-order transition because the order parameter changes discontinuously across the transition but the latent heat is only about 10 J/g. De Gennes extended Landau s theory into isotropic-nematic transition... 

As a rule, the phase transition from the isotropic phase into the nematic phase is a weak first-order transition [6] with a small jump in the order parameter 5 (Fig. 1.3 [7]) and other thermodynamic properties. The so-called clearing point corresponds to this first-order transition temperature Tni. At the same time, in the pretransitional region of the isotropic phase we can observe the temperature divergence in some physical parameters, such as heat capacity, dielectric permittivity, etc., according to the power law (T — T i) where T j is the other, virtual, second-order phase transition point, (Tni — T 0.1 K) and t) is an exponent, depending on the physical property under consideration. 

Are there additional effects, beyond the shift of the transition temperature, on the smectic-d smectic-C phase transition in chiral-racemic systems The transition is, in the vast majority of compounds, of the second-order type, the first examples for a first-order smectic-d-smectic-C transition were found in chiral compounds possessing large Pg values [7], [8]. It has been even observed, that a weakly first-order transition in the chiral enantiomer becomes continuous in the racemate [79], However, it seems that chirality and/or large spontaneous polarization are not the primary reasons for the occurrence of first-order smectic-d-smectic-C transitions, since first-order transitions were also found in racemic or nonchiral compounds [80], [81] (the above-mentioned second-order transition in a racemate results probably from an increased width of the smectic-d temperature range in the racemate compared to the chiral enantiomer). One relevant factor for a first-order... 

Figure 10.10 Prolate-oblate fluctuations of a vesicle [75]. The long axis of the prolate vesicle is located within the focal plane. The oblate vesicle appears quasispherical because its rotational symmetry axis is normal to the focal plane. The elapsed time between snapshots is 6.3 s. A bistable behavior occurs where time sequences of fluctuating prolate and oblate shapes are separated by comparatively short-lived transitional shapes. Such a scenario confirms the theoretical expectations of a weakly first-order transition between prolate and oblate shapes. A statistical analysis of the shape distribution allows identification of the prolate configuration as the meta-stable shape in this example.

Thus, the melting becomes continuous and a commensurate /3 x /3 structure can be foxmd [116] before the bdayer forms, by compression at monolayer completion and at a sufficiently low temperature. Moreover, Heiney et al. s x-ray diffraction results [114] indicate that the melting transitions becomes continuous rather than first order at slightly above the monolayer range. Finally, the thermodynamic data of Gangwar et al. [107] have been interpreted as indicating a weak first order transition from 116 to 147 K, whereas the transition becomes continuous from 147 to 155 K, where the formation of a second layer interferes with the results. 

Figure 1(a) gives the temperature dependence of H near a strongly first-order transition with a large latent A//l at 7,r. The variation of H with T is nearly linear above and below Ttr resulting in almost temperature independent Cp values in the low and high temperature phases. Figure 1(b) represents the case of a weakly first-order transition with only a small latent heat but H... 

The latent heat of this transition is usually small and may even vanish if the width of the nematic temperature range is sufficiently large [23]. Thus, the transition can be either of first or second order. For the second-order transition the discontinuity in the orientational order parameter S and, hence, the dielectric (or diamagnetic) susceptibility, disappears and the field influence on both phases is the same. Thus we do not anticipate any field-induced shift in the N-A transition temperature. For the weak first-order transition there is a small discontinuity in both S and dielectric (and magnetic) susceptibilities, and the shift depends on the competition between two small quantities the difference in susceptibilities for the nematic and smectic A phases on the one hand and transition enthalpy on the other. In particular, the field may induce a change in the phase transition order, from first to the second order, as shown in Fig. 4 [24]. 

Here, 1 will introduce another type of the fluctuation as an example for the isotropic-nematic phase transition, which is a weak first-order transition and large fluctuations of the order parameter appear as a critical phenomenon near... 

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