Flory’s theory

The effect of different types of comonomers on varies. VDC—MA copolymers mote closely obey Flory s melting-point depression theory than do copolymers with VC or AN. Studies have shown that, for the copolymers of VDC with MA, Flory s theory needs modification to include both lamella thickness and surface free energy (69). The VDC—VC and VDC—AN copolymers typically have severe composition drift, therefore most of the comonomer units do not belong to crystallizing chains. Hence, they neither enter the crystal as defects nor cause lamellar thickness to decrease, so the depression of the melting temperature is less than expected. 

Hence, Flory s theory offers an objective criterion for chain flexibility and makes possible to divide all the variety of macromolecules into flexible-chain (f > 0.63) and rigid-chain (f < 0.63) ones. In the absence of kinetic hindrance, all rigid-chain polymers must form a thermodynamically stable organized nematic phase at some polymer concentration in solution which increases with f. At f > 0.63, the macromolecules cannot spontaneously adopt a state of parallel order under any conditions. 

Generalization of Flory s Theory for Vinyl/Divinyl Copolvmerization Using the Crosslinkinq Density Distribution. Flory s theory of network formation (1,11) consists of the consideration of the most probable combination of the chains, namely, it assumes an equilibrium system. For kinetically controlled systems such as free radical polymerization, modifications to Flory s theory are necessary in order for it to apply to a real system. Using the crosslinking density distribution as a function of the birth conversion of the primary molecule, it is possible to generalize Flory s theory for free radical polymerization. 

According to Flory s theory [20], the osmotic pressure due to ionic distribution n is given by the following equation,... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

Flory ( 9) has treated the interesting case of subsequent removal of the first stage cross-links without chain scission. Even after complete removal, i.e. =0, there is still a certain memory of the structure of the first network since the composite network strands were physically part of both networks. According to Flory s theory, the resulting network may be treated as if a certain fraction, 0, of the strands of the second network were effectively converted into strands of the first network, >ie. 

Since the time of Flory, only a few papers have appeared in the hterature in which the kinetics of A2B condensation reactions are treated. A purely theoretical paper was recently pubhshed by Moller et al. where Flory s theory of AjjB polycondensations was expanded to describe the distribution of molecules containing arbitrary numbers of branching units [43]. In another paper, Hult and Malmstrom studied the kinetics of a reacting system based on 2,2-bis(hy-droxymethyljpropionic acid [44]. 

G. Application of Flory s theory to explain starch gelatinization... 

When Flory s theory (1953) of melting point depression is applied to starch gelatinization (or phase transition) in the presence of water, the situation can be described as follows. Af equilibrium state, the chemical potentials between amorphous (pu) and crystalline repeating units (p of fwo phases are equal ... 

The coagulation process can now be considered in perspective of a ternary polymer-solvent-nonsolvent system, A schematic ternary phase diagram, at constant temperature, is shown in Figure 8. The boundaries of the isotropic and narrow biphasic (isotropic-nematic) regions are based on an extension of Flory s theory ( ) to a polymer-solvent-nonsolvent system, due to Russo and Miller (7). These boundaries are calculated for a polymer having an axial ratio of 100, and the following... 

Interactions between different distant parts of the molecule tend to expand it, so that in the absence of other effects a would be greater than unity, but in solution in poor solvents interactions with the solvent tend to contract it. According to Flory s theory (18) these two tendencies will just balance so that a — 1 at a particular temperature T—0 (the theta temperature ), and at this temperature A2 =0 and further this temperature is the limit as Mn- go of the upper critical solution temperature for the polymer-solvent system in question. Quantities relating to T=0 will be denoted by subscript 0. Flory s theory implies that ... 

In Flory s theory (/< ), a polymer-solvent system is characterized by a temperature 0 at which (i) excluded-volume effects are just balanced by polymer-solvent interactions, so that os=l, (ii) the second virial coefficient is zero, irrespective of the MW of the polymer, and (iii) the polymer, of infinite molecular weight, is just completely miscible with the solvent The fundamental definition of the temperature is a macroscopic one, namely that for T near 0 the excess chemical potential of the solvent in a solution of polymer volume fraction v2 is of the form (18) ... 

The unperturbed mean-square radius is calculable for polymers of known structure, on the assumption of random-flight chains (Section 3). It has usually been assumed that random-flight conformations are adopted at the temperature Al at which A2 is zero, according to Flory s theory (18). Light... 

Solvent power parameter entering Flory s theory of dilute solutions, degree of neutralization in polyelectrolyte solutions, free-volume parameter entering Vrentas-Duda theory subscript (1,24) denotes molecular species in solution. 

In their investigation of polydimethylsiloxane and polyethylene oxide) in solution with various solvents, Tanner, Liu, and Anderson40 extrapolated the observed polymer diffusion coefficients to zero polymer concentration c. They applied Flory s theory of dilute solutions 45) to the case of diffusion ... 

Let me close with two more familiar examples. We all know that Flory s theory of polymeric solutions is not in agreement with experiment. However, the insight on polymer compatibility gained from the theory was enormous. The same thing may be said about the van der Waals equation of state. We all know that this equation is very approximate, but something remains in our intuition that is extremely helpful. 

In this chapter we will mostly focus on the application of molecular dynamics simulation technique to understand solvation process in polymers. The organization of this chapter is as follow. In the first few sections the thermodynamics and statistical mechanics of solvation are introduced. In this regards, Flory s theory of polymer solutions has been compared with the classical solution methods for interpretation of experimental data. Very dilute solution of gases in polymers and the methods of calculation of chemical potentials, and hence calculation of Henry s law constants and sorption isotherms of gases in polymers are discussed in Section 11.6.1. The solution of polymers in solvents, solvent effect on equilibrium and dynamics of polymer-size change in solutions, and the solvation structures are described, with the main emphasis on molecular dynamics simulation method to obtain understanding of solvation of nonpolar polymers in nonpolar solvents and that of polar polymers in polar solvents, in Section 11.6.2. Finally, the dynamics of solvation with a short review of the experimental, theoretical, and simulation methods are explained in Section 11.7. 

Rudin combined Flory s theory of the dimension of polymer coils with Zimm s expression (Zimm, 1946) for the second virial coefficient in a dilute suspension of uniform spheres. He further assumed that the swelling factor (e) of the polymer coil (identical to a3 in Flory s formalism) reduces to 1 at a certain critical concentration cx, Ewhereas it tends to a value e0 (= [//] (/[//]w ) at infinite dilution. The functional relation between e and c, would therefore be... 

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