Flory lattice theory

The rheology of isotropic suspensions and solutions of stiff molecules and particles is covered in Section 6.3. The rheology of lyotropic and thermotropic nematic liquid crystals composed of long, stiff molecules is described in Chapter 11. 

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A ternary system consisting of two polymer species of the same kind having different molecular weights and a solvent is the simplest case of polydisperse polymer solutions. Therefore, it is a prototype for investigating polydispersity effects on polymer solution properties. In 1978, Abe and Flory [74] studied theoretically the phase behavior in ternary solutions of rodlike polymers using the Flory lattice theory [3], Subsequently, ternary phase diagrams have been measured for several stiff-chain polymer solution systems, and work [6,17] has been done to improve the Abe-Flory theory. 

Figure 2.6 Illustration of Flory lattice theory for the thermodynamics of long stiff rods. In (a), the rod of length X measured in units of the lattice spacing is oriented at an angle 6 with respect to the preferred horizontal direction. In (b), the rod is represented on a cubic lattice by subdividing the rod into y segments, each of length X/y. (From Flory 1984, reprinted with permission from Springer Verlag.)...

Figure 5.14 (a) Temperature-volume fraction phase diagram for PBLG Ma, = 310,000) in DMF, where I denotes an isotropic phase, LC denotes a chiral nematic liquid-crystalline phase, and I + LC is a gel that is presumed to be two coexisting phases that are unable to separate macroscopically. (b) The x-volume fraction phase diagram predicted by the Flory lattice theory for rigid rods of axial ratio (length/diameter) = 150. (From Miller et al. 1974, with permission.)... 

The lattice theories are the oldest and most frequently used to interpret and to predict the thermodynamic properties of multicomponent systems containing polymers. The Huggins-Flory lattice theory is the best known. To use the theory one must know the temperature, pressure and concentration dependence of the enthalpic and entropic contributions to the binary interaction parameter, P, ) + P, , ) / T. 

The problem is more difficult when polymers are considered. If they are supposed to be rigid rods, without "soft" interactions,the Flory lattice theory provides a model in agreement with experiments. If the macromolecular chain is semi-flexible, the situation is less clear. Several theories are currently developed their main features are described in this volume. They also describe the transition as first order, but with a strong second order component. 

It was de Gennes (1972) who first introduced the Lagrangian theory to interpret the behavior of polymer solutions. Although des Cloizeaux showed the similarity between his and Flory s interpretations of the osmotic pressure, de Gennes pointed out that des Cloizeaux interpretation is superior to Flory s. To compare the two theories, de Gennes first translated the Flory lattice theory into his language. Let represent the fraction of lattice sites occupied by the monomers. Then cubic centimeters and is the volume of the unit needed in the cubic lattice. Flory s equation of osmotic pressure [Eq. (9.20)] can be expressed in the form... 

The thermodynamics of liquid-crystalline polymers can be relatively adequately described as the Flory lattice theory. This predicts that the stability of the anisotropic phase depends on the aspect ratio of the rigid rods and on the volume fraction of polymer. The theory was originally developed for completely rigid molecules but is also applicable to semiflexible... 

At the limit of O < 1, Eq. (1.37) is reduced to Eq. (1.26), valid for a dilute solution. It should also be noted that this equation is essentially the continuous analog of Eq. (1.17) used in the Flory lattice theory (cf. [60]). Equality (1.37), while retaining all of the advantages of the corresponding relation in die Rory theory, is free of the limitations of this theory caused by the correlation with the previously defined spatial lattice. 

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

By using the liquid lattice approach to treat the random mixing of a disoriented polymer and a solvent, the so-called Flory-Huggins theory is often used to correlate the penetrant activity and the composition of the solution ... 

This results in a value of d = 2.5 for bond percolation on a 3-dimensional lattice. The fractal dimension of the Bethe lattice (Flory-Stockmayer theory) is... 

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

The Flory-Huggins theory begins with a model for the polymer solution that visualizes the solution as a three-dimensional lattice of TV sites of equal volume. Each lattice site is able to accommodate either one solvent molecule or one polymer segment since both of these are assumed to be of equal volume. The polymer chains are assumed to be monodisperse and to consist of n segments each. Thus, if the solution contains TV, solvent molecules and TV2 solute (polymer) molecules, the total number of lattice sites is given by... 

Fint is the free energy of non-Coulomb interactions of monomer units. Finl can be expressed, for example, in terms of the Flory-Huggins lattice theory [21]. In the general case, when network is immersed in solvent which includes 1 different components some of which can be polymeric with the degree of polymerization Pi(Pi 1, i = L 2,... k), Fim in the Flory-Huggins theory has the following form [21-22] ... 

Flory-Huggins /u. The Flory-Huggins n value measures the interaction between polymer and solvent (plasticizer). It derives from the so-called lattice theory, which represents a statistical approach to the behavior of polymer molecules in solution (10, 14, 75, 16, 22). The n value may be experimentally determined for any polvmer-plasticizer system (where the plasticizer can dissolve the polymer) by osmotic pressure measurements according to the relation ... 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

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 ... 

Several other studies have also been made in an attempt to account theoretically for the phase transition in terms different from those of the Flory-Huggins theory. Otake et al. [55] thus proposed a theoretical model that takes hydrophobic interaction into account in explaining the thermally induced discontinuous volume collapse of hydrogels. In addition, Prausnitz et al. [56] proposed a lattice model, an improvement of which was made to explain the swelling curves of gels consisting of /V,/V -methylenebis(acrylamide) (MBA)-crosslinked copolymers of AAm with [(methacrylamide)propyl]trimethyl-ammonium chloride (MAPTAC) [57],... 

The fact that the thickness of the interphase estimated here stays unchanged at 34A in the molecular weight range of 30,000-100,000, while the mass fraction and thickness of amorphous phase change remarkably, is particularly meaningful. Flory et al. [6,7] anticipated in 1984 based on their lattice theory that the methylene chains that emerge from the basal plane of lamellar crys-... 

The state of miscibility of any mixture is governed by the Gibbs free energy of mixing, AG, which may be described by the lattice theory of Flory-Huggins (Flory, 1953), as follows ... 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

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