The Peierls-Landau instability

Theoretical developments in the early 1980s showed that the nonlinear interaction of thermally excited layer undulations, which as we have seen have large amplitudes because of the Peierls-Landau instability, leads to interesting new effects in the hydrodynamics of smectic A at small wavevector and frequency. We present below a very brief outline of the physical arguments involved. ... 

As in the S phase, Sg lacks true long-range translational order because of the Peierls-Landau instability. Theoretically, the existence of molecular tilt implies that there must be a certain degree of bond-orientational order in the Sg phase. This has been verified experimentally by a high resolution synchrotron X-ray study of the transition from Sg to S, in a monodomain freely suspended film. It will be recalled that S, is a tilted hexatic phase. The bond-orientational order parameter Cg (as defined in (5.7.1), but appropriately modified to allow for the fact that the molecules are tilted) is plotted as a function of temperature in fig. 5.8.2. The weak bond-orientational order in evolves continuously into S, showing... 

The smectic A phase is a liquid in two dimensions, i.e. in tire layer planes, but behaves elastically as a solid in the remaining direction. However, tme long-range order in tliis one-dimensional solid is suppressed by logaritlimic growth of tliennal layer fluctuations, an effect known as tire Landau-Peierls instability [H, 12 and 13]... 

The weak logarithmic divergence with the sample size L is known as the Landau-Peierls instability. As a result, for sufficiently large L the fluctuations become of the order of the layer spacing, which means that the layer structure would be wiped out. However, for samples in the miUimolar range and typical values of the elastic moduli K 10 N and B 10 N/m, the layer displacement amplitude a = (ifi) does not exceed 0.5-0.7 run. For a typical smectic period d ra 3 nm this gives relative displacements ajd k, 0.2 the smectic layers are still well defined. Nevertheless, the displacements are large compared to those of a typical 3D crystal for which ... 

The classical physics of SmA phases is recovered by expanding Eq. (72) in powers of the gradients of u (r). The quadratic terms in (V m) and (Vj u) are responsible for the Landau-Peierls instability [1,6]. The nonharmonic terms in (u) (u) and (V l u) lead to a break-down of conventional elasticity B (q) and (q) respectively vanishes and diverges as powers of In( ) at small wave vectors q [102]. Other anhar-monic terms are irrelevant in the renormalization group sense in the smectic phase. 

It is seen that for smectics the fluctuation logarithmically diverges with L.+ This was first shown by Peierls and Landau and is called Landau-Peierls instability. This is behavior is very different from the fluctuations in crystal and columnar liquid crystals (3D and 2D elasticity, respectively) where the amplitude of the fluctuations remain finite even for infinite samples, as u )cryM = (1/Z-1/L), and (u ) columnar I Inserting typical... 

With one-dimensional periodicity, the smectic phase cannot exhibit true long-range order due to the Landau-Peierls instability [6, 7]. An anisotropic scaling analysis [18] (see [3], page 521 for a summary) predicts the divergence of the layer compression modulus B oc thus a divergence with exponent <]) = vy -2i j. ... 

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