Two-order parameter model of liquid

H. Tanaka, Two-order-parameter description of liquids. I. A general model of glass transition covering its strong to fragile limit. J. Chem. Phys. Ill, 3163-3174 (1999). 

Here we consider a simple two state model of liquid, which corresponds to the weak-coupling limit of our two order-parameter model [25,37]. We first estimate how the average fraction of locally favored structures, S, increases with a decrease... 

A variant of the two-state model of liquid water was used in a recent work by Patey and co-workers that shows that if we categorize a water molecule in the liquid by using the tetrahedral order parameter into liquid-like (low ih) and ice-like (high h) molecules, and then treat the liquid as a binary mixture, such a binary-mixture model can indeed reproduce most of the anomalies of liquid water [13]. Thus, there does not seem to be any need to invoke the existence of the LDL state. 

Overall, the order parameter model provides both a simple physical interpretation of thermodynamic changes at Tg and a semiquantitative estimate of their magnitude. It does not, however, explain why segmental motion freezes in and in the absence of knowledge of the two-state parameters 8s and 8, it does not lead to predictions of Tg and therefore cannot explain how Tg will vary with molecular weight, composition, and chemical structure. The free-volume theory and the GM configurational entropy theory are the two most important attempts to explain why molecular motions eventually stop in a supercooled liquid and hence why the glass transition takes place. 

Recent models rationalize these crucial features of vitrification with the presence of medium-range ordered regions coexisting with the isotropic liquid. In this framework, the Two Order Parameter (TOP) model, proposed by Tanaka, offers a promising advance in the field towards a self-consistent theory of glass transition. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

In order, for the two liquids to separate into two phases, they must be very weakly soluble in each other. When exposed to each other by mixing or shaking in a separatory funnel, they may not interpenetrate each other s realm to any extent. At the molecular level, we infer that the two species of molecules have no significant affinity for each other, rather they are predominantly attracted to other molecules with the same structure. To model this aversion, the joining and breaking rules must encode this behavior. The cells of liquids X and Y must respond to rules typified by those shown in the parameter setup tables below. With these rules we anticipate that liquid X will favor associating with other X molecules, while molecule Y will be found predominantly among other Y molecules. 

About the same time Beutier and Renon (11) also proposed a similar model for the representation of the equilibria in aqueous solutions of weak electrolytes. The vapor was assumed to be an ideal gas and < >a was set equal to unity. Pitzer s method was used for the estimation of the activity coefficients, but, in contrast to Edwards et al. (j)), two ternary parameters in the activity coefficient expression were employed. These were obtained from data on the two-solute systems It was found that the equilibria in the systems NH3+ H2S+H20, NH3+C02+H20 and NH3+S02+H20 could be represented very well up to high concentrations of the ionic species. However, the model was unreliable at high concentrations of undissociated ammonia. Edwards et al. (1 2) have recently proposed a new expression for the representation of the activity coefficients in the NH3+H20 system, over the complete concentration range from pure water to pure NH3. it appears that this area will assume increasing importance and that one must be able to represent activity coefficients in the region of high concentrations of molecular species as well as in dilute solutions. Cruz and Renon (13) have proposed an expression which combines the equations for electrolytes with the non-random two-liquid (NRTL) model for non-electrolytes in order to represent the complete composition range. In a later publication, Cruz and Renon (J4J, this model was applied to the acetic acid-water system. 

We note that earlier research focused on the similarities of defect interaction and their motion in block copolymers and thermotropic nematics or smectics [181, 182], Thermotropic liquid crystals, however, are one-component homogeneous systems and are characterized by a non-conserved orientational order parameter. In contrast, in block copolymers the local concentration difference between two components is essentially conserved. In this respect, the microphase-separated structures in block copolymers are anticipated to have close similarities to lyotropic systems, which are composed of a polar medium (water) and a non-polar medium (surfactant structure). The phases of the lyotropic systems (such as lamella, cylinder, or micellar phases) are determined by the surfactant concentration. Similarly to lyotropic phases, the morphology in block copolymers is ascertained by the volume fraction of the components and their interaction. Therefore, in lyotropic systems and in block copolymers, the dynamics and annihilation of structural defects require a change in the local concentration difference between components as well as a change in the orientational order. Consequently, if single defect transformations could be monitored in real time and space, block copolymers could be considered as suitable model systems for studying transport mechanisms and phase transitions in 2D fluid materials such as membranes [183], lyotropic liquid crystals [184], and microemulsions [185],... 

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