Landau-Peierls

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]... 

One characteristic feature of the behavior of Xs(T) for organic metals is illustrated in Fig. 4. In contrast to ordinary metals, Xs(X) increases quite substantially with temperature from (say) 60 to 300 K. This increase is strongest for the most one-dimensional compound, TTF-TCNQ [53], and becomes progressively weaker for (TMTSF)2C104 [54], (3-(BEDT-TTF)2I3 (a genuine two-dimensional compound) [25,26], and the more three-dimensional compound (TSeT)2Cl [18] (also, unpublished results of M. Mil-jak and B. Hilti). For HMTSF-TCNQ [33] such a discussion is complicated by the presence of Landau-Peierls diamagnetism from small pockets of electrons and holes, although estimates of Xs(T) have been made by Soda... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

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. 

Landau-Ginzbuig coefficients 282 Landau-Ginzburg Hamiltonian, critical 303 Landau-Khalatnikov mechanism 564 Landau-Lifshitz theory, phase transitions 366 Landau-Peierls instabilities 285, 647 Langmuir-Blodgett film, atomistic simulations 85 Laplace equation 445... 

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... 

In the nematic phase, one can induce macroscopic orientation by controlling the surface boundary conditions. However, because nematic ordering is the result of a spontaneously broken symmetry, fluctuations of the director n are a soft mode. Indeed a macroscopically oriented nematic phase is much more turbid than a macro-scopically oriented smectic-A phase because of light scattering from orientational fluctuation domains. The layered structure of the smectic-A phase suppresses these orientational fluctuations, and it is this coupling that affects the character of the transition (see [5] for a broad survey of such phase transitions). However, smectic phases exhibit one-dimensional orientational order, characterized by the Landau-Peierls fluctuation of the layer spacing [6, 7]. As a result, the essential features needed to capture the NA transition are ... 

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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