The I-N Transition

It is noteworthy that the strong discrepancy from Eq. (I) occurs on approaching the I-N transition when the correlation length and the lifetime of quasi-nematic heterogeneities in the fluidlike surrounding boost, namely ... 

The change of FDSE exponent visible in Fig. 2 reflects the fact that the MCT eq. (3) fails in the immediate vicinity of the I-N transition whereas such distortion is absent for cr(r) behavior portrayed via eq. (4). This can induce the visible gradual change of the exponent (p. ... 

It is beyond the scope of this review to be exhaustive in the field of supercooled liquids that has drawn intense research activities over several decades. Reviews that are exclusive for this field and deal with specific topics in considerable detail are recommended for supplemental reading [9-11]. In view of the scope this chapter, the next section provides the readers with a brief introduction to the systems of interest and associated nomenclature. Section III sets up the background by reviewing experimental results on the dynamics of thermotropic liquid crystals across the I-N transition, then introducing the central issues in the dynamics of supercooled liquids, and finally comparing the dynamics of the two systems in the light of recent experiments. Section IV presents a summary of some of the well-known theoretical approaches to liquid crystals. Section V provides a detailed account of computational efforts. Finally, we conclude in Section VI with a list of problems for future work. 

We now discuss the orientational dynamics of mesogens. The orientational dynamics in the isotropic phase of thermotropic liquid crystals near the I-N transition have drawn much attention over the years [5-7, 33-40]. The focus was initially on the verification of the Landau-de Gennes (LdG) theory, which predicts a long-time exponential decay with a strongly temperature-dependent... 

The thermodynamics of the I-N phase transition has been extensively investigated for resolving the issue concerning the order of the transition. Following the Ehrenfest scheme, a phase transition is classified into a first-order transition or a second-order one, depending upon the observation of finite discontinuities in the first or the second derivatives of the relevant thermodynamic potential at the transition point. An experimental assessment of the order of the I-N transition has turned out to be not a simple task because of the presence of only small discontinuities in enthalpy and specific volume. It follows from high-resolution measurements that I-N transition is weakly first order in nature [85]. 

The normalization constant G is often conveniently defined by setting Qzz equal to unity in perfectly oriented system. By definition, Q is real, symmetric, and traceless. Q vanishes in the isotropic phase as per the requirement of its suitability for an order parameter to describe the I-N transition. The macroscopic tensor order parameter Q can always be diagonalized ... 

There exist pre-transition effects in the isotropic phase heralding the I-N phase transition. Such pre-transition effects, which are consistent with the weakly first-order nature of the I-N transition, can be attributed to the development of short-range orientational order, which can be characterized by a position-dependent local orientational order parameter Q(r), where all component indices have been omitted [2]. In the Landau approximation, the spatial correlation function < G(0)G(r) > has the Omstein-Zemike form < G(0)G(r) exp(—r/ )/r, where is the coherence length or the second-rank orientational correlation length. The coherence length is temperature-dependent and the Landau-de Gennes theory predicts... 

Li et al. subsequently developed a schematic mode coupling theory description of the short-time as well as long-time dynamics of mesogens in the isotropic phase near the I-N transition [91]. Their treatment started with a very general form [92] for the kinetic equation of the autocorrelation function (j)2(t) of the anisotropy of polarizability... 

The first-rank and second-rank OTCFs are mostly studied because of their relevance to experiments. One of the early computational studies of orientational dynamics in the isotropic phase near the /-A transition is due to Allen and Frenkel [105]. In their molecular dynamics simulation study of a system of N = 144 hard ellipsoids of revolution, the slowdown of orientational dynamics on approaching the I-N transition was captured—in particular, in the time evolution of the... 

The growth of orientational correlations and the slow down of collective orientational dynamics were subsequently investigated using a soft potential [106]. Allen and Warren (AW) studied a system consisting of N = 8000 particles of ellipsoids of revolution, interacting with a version of the Gay-Beme potential, GB (3, 5, 1, 3), originally proposed by Berardi et al. [107]. AW computed the direct correlation function, c( 1, 2), in the isotropic phase near the I-N transition. The direct correlation function is defined through the Omstein-Zemike equation [108]... 

Here the first set of expansion coefficients refers to a space-fixed laboratory frame while the second set refers to a coordinate system based on the intermolecular frame. The two sets of coordinates are interconvertible. A similar expansion holds good for c(l, 2), and it is possible to express the mechanical stability of the isotropic phase relative to the nematic phase in terms of these expansion coefficients and thus to obtain an estimation of T [109, 110]. In spite of the rapid growth of the second-rank orientational correlation length, 2, on approaching the I-N transition from the isotropic side, the simulation results showed the associated component of the direct correlation function to remain short-ranged and showed its spatial integral to approach the mechanical instability limit of the isotropic phase [106]. 

Figure 9. Orientational relaxation in the model liquid crystalline system GB(3, 5, 2, 1) (N = 576) at several densities across the I-N transition along the isotherm at T = 1. (a) Time dependence of the single-particle second-rank orientational time correlation function in a log-log plot. From left to right, the density increases from p = 0.285 to p = 0.315 in steps of 0.005. The continuous line is a fit to the power law regime, (b) Time dependence of the OKE signal, measured by the negative of the time derivative of the collective second-rank orientational time correlation function C t) in a log-log plot at two densities. The dashed line corresponds to p = 0.31 and the continuous line to p = 0.315. (Reproduced from Ref. 112.)...

In the quest for a universal feature in the short-to-intermediate time orientational dynamics of thermotropic liquid crystals across the I-N transition, Chakrabarti et al. [115] investigated a model discotic system as well as a lattice system. As a representative discotic system, a system of oblate ellipsoids of revolution was chosen. These ellipsoids interact with each other via a modified form of the GB pair potential, GBDII, which was suggested for disc-like molecules by Bates and Luckhurst [116]. The parameterization, which was employed for the model discotic system, was k = 0.345, Kf = 0.2, /jl= 1, and v = 2. For the lattice system, the well-known Lebwohl-Lasher (LL) model was chosen [117]. In this model, the particles are assumed to have uniaxial symmetry and represented by three-dimensional spins, located at the sites of a simple cubic lattice, interacting through a pair potential of the form... 

In this section of the review we restrict ourselves to those aspects of dynamics of supercooled liquids that appear to be analogous with dynamical features of thermotropic liquid crystals across the I-N transition. Power law relaxation in short-to-intermediate time scales is well known for supercooled liquids. We first review recent computational efforts that provided insights into power law relaxation in supercooled liquids. We then focus on a model system tailored specifically to study analogous dynamical features of the two seemingly different classes of soft matter systems. 

Figure 19. Typical trajectories of the unit orientation vector for a single ellipsoid revolution in two different systems, (a) Calamitic system GB(3,5,2,1) at four temperatures (i) T = 2.008 (in the isotropic phase), (ii) T = 1.396 (close to the I-N transition), (iii) T = 1.310 (close to the I-N transition), and (ii) T = 1.192 (in the nematic phase), (b) Binary mixture at the highest temperature (left) and the lowest (right) temperature studied. (Reproduced from Ref. 131.)...

E0 and the infinite temperature relaxation time To are independent of temperature, and (ii) in the isotropic phase near the I-N transition, the temperature dependence of ts2(T) shows marked deviation from Arrhenius behavior and can be well-described by the Vogel-Fulcher-Tammann (VFT) equation ts2(T) = TyFrQxp[B/(T — TVFF), where tvff, B, and tvft are constants, independent of temperature. Again these features bear remarkable similarity with... 

The structural features of the inherent structures yielded important information on the interplay between the orientational and translational order in the calamitic mesophases [144]. It follows from Fig. 22b that quenching results in enhanced orientational order for inherent structures than the corresponding pre-quenched configurations. We now examine how the inherent structures evolve as revealed by the pair distribution functions that were obtained by averaging over the quenched configurations. An analysis through the parallel radial distribution function g (V ), which depends only on rj, the pair separation r, parallel to the director n, reveals a remarkable feature as illustrated in Fig. 26a. The onset of growth of the orientational order in the vicinity of the I-N transition induces a... 

In this review, our focus has been largely on the orientational dynamics of calamitic liquid crystals across the I-N transition and their similarity with the dynamics of supercooled liquids. We have reviewed experimental, theoretical,... 

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