Onsager nematics

Camp P J, Mason C P, Allen M P, Khare A A and Kofke D A 1996 The isotropic-nematic transition in uniaxial hard ellipsoid fluids coexistence data and the approach to the Onsager limit J. Chem. Phys. 105 2837-49... 

Otlier possibilities for observing phase transitions are offered by suspensions of non-spherical particles. Such systems can display liquid crystalline phases, in addition to tire isotropic liquid and crystalline phases (see also section C2.2). First, we consider rod-like particles (see [114, 115], and references tlierein). As shown by Onsager [116, 117], sufficiently elongated particles will display a nematic phase, in which tire particles have a tendency to align parallel to... 

Disc-like particles can also undergo an Onsager transition—here tire particles fonn a discotic nematic, where tire short particle axes tend to be oriented parallel to each other. In practice, clay suspensions tend to display sol-gel transitions, witliout a clear tendency towards nematic ordering (for instance, [22]). Using sterically stabilized platelets, an isotropic-nematic transition could be observed [119]. 

Chen s analysis is more accurate than the procedure in which the Onsager trial function is used with a(N) given by Eq. (18), but it is very involved to carry through. On the other hand, the Onsager trial function procedure is simple enough for practical purposes. As shown in Appendix A, it predicts the isotropic-nematic phase boundary concentrations that can be favorably compared with those by Chen s procedure. 

The Gaussian trial function for f(a) used by Odijk [6] is mathematically simpler than Onsager s and allows p to be expressed by the leading term of Eq. (22) and ct(N) to be derived analytically. However, it becomes less accurate as the orientation gets weaker. As shown in Appendix A, its use leads to the isotropic-nematic phase boundary concentrations largely different from those by Chen s method and hence is not always relevant for quantitative discussion. 

A) Onsager s rigid-rod model (1949) was the first correct model of an athermal Isotropic —> Nematic phase transition. It is a relatively simple model that predicts phase transitions. It is based on excluded volume between two rigid rods, which reads... 

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... 

The virial expansion has enjoyed greater appeal, especially as applied to lyotropic systems. Onsager s classic theory rests on analysis of the second virial coefficient for very long rodlike particles. It was the first to show that a solution of hard, asymmetric particles such as long rods should separate into two phases above a threshold concentration that depends on the axial ratio of the particles. One of these phases should be anisotropic (nematic), the other completely isotropic. The former is predicted to be somewhat more concentrated than the latter, but it is the alignment (albeit imperfect) of the solute molecules that is the predominent distinction. The phase separation is a consequence of shape asymmetry alone intervention of intermolecular attractive forces is not required. 

Onsager s potential also applies to oblate particles or disks of high aspect ratio, if the parameter Uq ts redefined as 47rvdisk-shaped particles or molecules can therefore also undergo a transition to a nematic phase, called a discotic nematic. 

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-... 

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

For lyotropic LCPs, there is a hiphasic window of concentrations over which nematic and isotropic phases coexist, t he polymer concentrations in the coexisting isotropic and nematic phases are designated by C (or = Trd Lv /A) and Cj (or (pj = jrd Lv2/A), respectively. There is also a theoretical concentration C, at which the isotropic phase becomes unstable to orientational fluctuations. According to the Onsager theory, 02/0 = 1.047 and 02/0i = 1.27 (see Section 2.2.2). Thermotropic LCPs often have a biphasic window of temperatures over which isotropic and nematic phases coexist. This biphasic window exists in nominally single-component thermotropics because of polydispersity the nematic phase is typically enriched in the longer molecules relative to the coexisting isotropic phase (D Allest et al. 1986). 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

Nearly exact numerical solutions of the Smoluchowski equation show that for the Maier-Saupe potential, A < 1 when S = S2 > 0.524. For the Onsager potential, A < 1 for all values of the order parameter within the nematic range. Values of A for the Onsager potential are plotted in Fig. 11-18. 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

Rod—coil block copolymers have both rigid rod and block copolymer characteristics. The formation of liquid crystalline nematic phase is characteristic of rigid rod, and the formation of various nanosized structures is a block copolymer characteristic. A theory for the nematic ordering of rigid rods in a solution has been initiated by Onsager and Flory,28-29 and the fundamentals of liquid crystals have been reviewed in books.30 31 The theoretical study of coil-coil block copolymer was initiated by Meier,32 and the various geometries of microdomains and micro phase transitions are now fully understood. A phase diagram for a structurally symmetric coil—coil block copolymer has been theoretically predicted as a... 

It should be pointed out that to meet the second virial approximation, molecules must have a large L/D so that at the transition the solution is dilute. For molecules of axial ratio less than 10 the theory does not work well. In addition, the Onsager value of the density difference at the nematic — isotropic transition is greater than the experimental data. 

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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