Elastic constants critical divergence

An important source of error in these calculations is the neglect of short-range order. In particular, the theory fails for the bend and twist elastic constants when smectic-like short-range order is present in the nematic liquid crystal. Under such circumstances these two constants exhibit a critical divergence as the temperature approaches the smectic-nematic point and the light scattering also shows a marked temperature dependence. The present treatment is then inadequate and more elaborate models have been proposed. The phenomenological theory of this aspect of the problem will be discussed in chapter 5. 

Since the coherence lengths diverge rapidly near the A-N transition, the elastic constants for twist and bend should also show critical behaviour. In the mean field approximation... 

Where po is the zero field pitch, A (m, the diamagnetic anisotropy of the liquid crystal, K22, the twist elastic constant. As the field increases the pitch is predicted to oncrease slowly at first and then diverge logarithmically as the critical field is ap>proached. The theory has been verified for lyotropic liquid crystals of PBG in a number of different solvents (Chandrasekhar, 1977, lizuka, 1973, DuPre Duke, 1974,1975, DuPre et al., 1976,1977, Patel DuPre, 1979). Molecules of liquid crystals orient themselves in the magnetic field so that their long chains are oriented parallel to the magnetic field lines (Miller, 1978). This orientation is associated with the molecular anisotropy of macromolecules rather than the existence of permanent magnetic moments. 

P. E. Cladis, Phys Lett. 1974, 48A, 179 and unpublished. This is a qualitative result. While it may not be possible to make elastic constant measurements for and deep enough into the N-SmA critical regime to establish quantitative certainty about critical exponents, there should be no problem for Xi which does not diverge at this transition. 

It is also possible to use turbidity measurements to evaluate elastic constants (Fig. 5, though elastic constants measured in this way for alkylcyanobiphenyls (nCBs) [25] and mixtures [26] are consistently higher than those determined by Frederiks transition studies. Hakemi [27] measured j, 22 and 33 for 8CB (octylcyano biphenyl) close to the nematic-smectic A phase transition and deduced a value of the critical exponent from the divergence of 22 and 33. 

The measurement of elastic constants is of particular interest near the nematic-smee-tic A (Tn-sitia) transition. The bend and twist elastic constants contain terms that diverge at TiM.smA according to a power law in reduced temperature [329] KjjCritical exponents have been measured for the twist [77-79, 81-87, 171] and bend [77-79,81,82,85,88-91,93-97,168,171, 172] constants. X-ray diffraction measurements made by Bradshaw et al. [98] indicate a correlation between short-range smecticlike order and the TTjg/A i, ratio, even for non-smectogenic substances. 

The variety of data on this homologuous series allows for a comparison of the reliability of different methods. Moreover, the octyl homologue (8-CB) [165-180] exhibits a nematic phase followed by a SmA phase and is therefore particularly interesting for the study of the critical divergence of the Kjj and K22 elastic coefficients on approaching the transition from the nematic to the SmA phase. Elastic coefficient measurements show that short-range pretransitional smectic-Uke order is found more or less in all n-CB homologues [98]. In Fig. 2a-c the elastic constants of the cyanobiphenyl series, as determined by Karat and Madhusu-... 

A number of publications deal explicitly with the theory of elastic constants at phase transitions. The behaviour of nematic elastic constants in the vicinity of the nematic-smectic transition has been analysed by means of the Ginzburg-Landau Hamiltonian [295]. At the transition to the smectic A phase, a critical divergence of K22 and A 33 is predicted. 

Yet another technique, measuring ultrasonic velocity anisotropies in the vicinity of the NA transition, however, also find anisotropies consistent with crossover behaviour. Sonntag et al [52] studied the divergence of three elastic constants, the bulk compression constant A, the layer compression constant B and the bulk-layer coupling constant C. B and C have critical exponents that are unequal and are in between that of the 3DXY values and anisotropic scaling values. 

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