Compressible lattice theory

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

Huggins theory but differs in one important respect in that it allows the lattice to have some vacant sites and to be compressible. Thus the compressible lattice theory is capable of describing volume changes on mixing as well as LCST and UCST behaviours. As with the theory of Flory and his co-workers, (which is proportional to the change in energy that accompanies the formation of a 1-2 contact from a 1-1 and a 2-2 contact) is obtainable from experimental values of heats of mixing. 

A generalized density gradient theory of interfaces has been combined with a compressible lattice theory of polymers. This yields a unified theory of bulk and surface thermodynamic properties. A unique feature of this theory is that it is parameterless. The only parameters required to calculate a surface tension are obtained from pure component thermodynamic properties. Since the theory is a mean field theory, it is only applicable to non-polar and slightly polar liquids. For such systems, surface tensions can be accurately calculated. 

For a molten, immiscible, binary blend of polymers A and B, Helfand et al developed a quantitative lattice theory of the interphase [4-6]. The model assumed that interactions between statistical segments of pol)nners A and B are determined by x 12 the isothermal compressibility is negligible, and there is no volume change upon blending. The segmental density profile, p/ where i = 1 or 2, was solved for infinitely long... 

Several other equation-of-state models have been proposed The lattice-fluid theory of Sanchez and Lacombe (1978), the gas-lattice model proposed by Koningsveld (1987), the strong interaction model proposed by Walker and Vause (1982), and the group contribution theory proposed by Holten-Anderson (1992), etc. These theories are reviewed by Miles and Rostami (1992) and Boyd and Phillips (1993). The lattice-fluid theory of Sanchez and Lacombe has similarities with the Flory-Huggins theory. It deals with a lattice, but with the difference from the Flory—Huggins model in that it allows vacancies in the lattice. The lattice is compressible. This theory is capable of describing both UCST and LCST behaviour. 

In fact, as a student, 1 had been very concerned about the relationship of the Flory theory of polymer solutions and the Onsager theory. Whereas the Onsager theory is about compressible liquids, the Flory theory is about incompressible liquids, and thus the so-called lattice theory. Eater, it has been shown that the extension of the Onsager theory under incompressible assumptions results in the Flory theory of polymer solutions [41]. 

The dependence of the elastic pressure on the density can be expressed approximately by a power function p = Bpn, usually called polytropic. It could alternatively be considered that the force centers are repelled according to the relationship F = a/(3n-2) as assumed in the Bohr theory of crystal lattices. The thermal motion, at this degree of compression, consists of small oscillations. To each vibrational degree of freedom there corresponds an energy RT (per mole). The total oscillatory energy equals cvT, where cv is independent of the volume in this approximation... 

Structures of the f.c.c. parent and b.c.t. martensite phases are shown in Fig. 24.3. The f.c.c. parent structure contains an incipient b.c.t. structure with a c/a ratio which is higher than that of the final transformed b.c.t. martensite. The final b.c.t. structure can be formed in a very simple way if the incipient b.c.t. cell in Fig. 24.3a is extended by factors of rji = 772 = 1.12 along x[ and x 2 and compressed by 773 = 0.80 along x 3 to produce the martensite cell in Fig. 24.36. This deformation, which converts the parent phase homogeneously into the martensite phase, is known in the crystallographic theory as the lattice deformation-1 Unfortunately,... 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

Basically, the process of tablet compression starts with the rearrangement of particles within the die cavity and initial elimination of voids. As tablet formulation is a multicomponent system, its ability to form a good compact is dictated by the compressibility and compactibility characteristics of each component. Compressibility of a powder is defined as its ability to decrease in volume under pressure, and compactibility is the ability of the powdered material to be compressed into a tablet of specific tensile strength [1,2], One emerging approach to understand the mechanism of powder consolidation and compression is known as percolation theory. In a simple way, the process of compaction can be considered a combination of site and bond percolation phenomena [5]. Percolation theory is based on the formation of clusters and the existence of a site or bond percolation phenomenon. It is possible to apply percolation theory if a system can be sufficiently well described by a lattice in which the spaces are occupied at random or all sites are already occupied and bonds between neighboring sites are formed at random. 

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