Isotropic-nematic phase transition Onsager theory

The concept potential of mean force was used by Onsager [3] in his theory for the isotropic-nematic phase transition in suspensions of rod-like particles. Since the 1980s the field of phase transitions in colloidal suspensions has shown a tremendous development. The fact that the potential of mean force can be varied both in range and depth has given rise to new and fascinating phase behaviour in colloidal suspensions [4]. In particular, stcricaUy stabilized colloidal spheres with interactions close to those between hard spheres [5] have received ample attention. 

The Onsager theory, or second virial expansion, is very successful in predicting the qualitative behavior of the isotropic-nematic phase transition of hard rods. H owever, it is an exact theory in the limit ofI/D—>oo. Straley has estimated that, for L/D < 20, the contribution of the third virial coefficient to the free energy in the nematic phase should be at least comparable to that of the second virial coefficient [22]. For more concentrated solutions, an alternative approach such as the Flory lattice model [3, 5] is required. The full phase diagram of rod-like colloidal systems has been obtained in computer simulations [23, 24]. The effects of polydispersity of rods [19, 25] and charged rods [10] are also important in the phase transitions. The comparison between Onsager theory and experimental results has been summarized by Vroege and Lekkerkerker [26]. 

B) Flory s lattice theory (1956) has received most attention. It is an extension of Onsager s model to higher concentrations. It is ideally suited for lyotropic LCPs consisting of solvent and rigid rods. It predicts that at a phase transition an isotropic and a nematic phase coexist with respective volume fractions... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

The nematic phase being the liquid crystal of highest symmetry, its condensation from the isotropic liquid should be the simplest to describe. Indeed, molecular theories convincingly explain the natural onset of nematic ordering in a population of anisotropic molecules with excluded volume interaction (Onsager) or in mean field theory (Maier-Saupe). Regarding the effect of symmetry on the isotropic to nematic (I-N) phase transition, the phenomenological approach is useful too. 

Linear viruses are known to form liquid crystalline structures since tobacco mosaic virus (TMV) was foimd to form nemafic liquid crystalline phase. This inspired Onsager to write his seminal paper about the isotropic-nematic transition in hard rods. According to his theory (see Chapter 3), the isotropic phase becomes unstable with respect to the nematic arrangement when. 

The Onsager and Flory theories are both statistical theories on rigid rod liquid crystalline polymers, but the former is a virial approximation while the latter is a lattice model. The first is more applicable to dilute solutions while the latter works especially well for high concentrations and a highly ordered phase. These theories with experiments, especially critical volume fractions 4>i and critical order parameter Sc at nematic-isotropic transition are made below. 

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