Onsager phase transitions

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... 

If Onsager s great achievement with the thermodynamics of irreversible processes met with initial indifference, Onsager s next feat created a sensation ill the scientific world. In a discussion remark in 1942, he disclosed that he had solved exactly the two-dimensional Ismg model, a model of a ferro-magnet, and showed that it had a phase transition with a specific heat that rose to infinity at the transi-... 

As the temperature is decreased, the chains become increasingly rigid zc then approaches 1 if we assume that there is only one fully ordered crystalline structure and Zconf for the liquid becomes smaller than 1. This means that, at this level of approximation, the disordered state becomes less favorable than the crystalline ground state. A first-order disorder-order phase transition is expected to occur under these conditions. Flory interpreted this phase transition as the spontaneous crystallization of bulk semiflexible polymers [12], However, since the intermolecular anisotropic repulsion essential in the Onsager model is not considered in the calculation, only the short-range intramolecular interaction is responsible for this phase transition. 

A very simple model that predicts lyotropic phase transitions is the hard-rod model proposed by Onsager (Friberg, 1976). This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder where the... 

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 changes in the o-Ps lifetimes should be explainable on the basis of eq. (10) and its connection with the free volume. It is interesting to note that in sulfolan, the latter does not change at the liquid/plastic phase transition. The changes in I3 cannot yet be quantified. Changes in the dielectric constant (in the Onsager radius) should be one of the main factors to consider. 

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... 

For example, Bridgman s [43] work shows that Onsager s model can fail for thermodynamic processes F occurring in systems that display hysteresis or that are near a phase transition. We next paraphrase Bridgman s argument for the phase transition case. 

Since then, a large body of work, both theoretical and experimental, has dealt with the Onsager model and the N/I phase transition of hard rods. This subject has been recently reviewed by Vroege and Lekkerkerker [43], A major improvement was brought by Khokhlov and Semenov who considered the effects of rod flexibility [44],... 

Fig. 6.20 Onsager model order parameter dependence on molecular packing factor r for two values of spherocylinder anisotropy ratio x = A dash curve) and X =11 (solid curve). Sc = 0.25 is the amplitude of the order parameter jump at the phase transition...

In the Ising surface model the crystal fluid phase is partitioned at the interface between the solid and fluid, with the crystal interface considered as a gradient in solid and fluids cells, going from complete solid to a complete fluid phase. Using the concepts introduced by Onsager, the dimensionless Ising temperature (dyk) that corresponds to the order-disorder phase transition can be calculated ... 

This roughening transition happens in the interface and thus might at first be identified with Onsager s phase transition in the two-dimensional Ising model at J/IcrT = 0.44069. However, it actually occurs [14] at J/ bTr = 0.40758 and is not described by the two-dimensional Ising model, but by the KosterUtz-Thouless transition of the two-dimensional AF-model which predicts the ratio IF /ln(L) to approach 1/a if the temperature approaches Tr from above. At Tr, no anomaly was seen in the interfadal tension. For more details we refer to Hasenbusch et al. [14]. 

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. 

More convenient calculations of the phase transition can be performed by choosing a trial function for the orientation distribution function / with one or more variational parameters. The free energy as a function of these parameters can then be minimized with respect to these parameters. Onsager [8] chose the following function... 

Disordered systems of rods have also been modeled with MC (276), and the Onsager-Flory transition to an anisotropic phase was observed. Chain configurations and surface energies of individual polymeric nanofibers have been investigated with Mattice s 2nnd lattice method (277). In short, methods exist to treat, say, cord-reinforced tires from atoms up to FEA on the whole tire. 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

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