Flory’s method

The idea behind the replacement of the excluded volume interactions by a self-consistent potential was put forward by Edwards6 and subsequently developed by various other physicists.5 7 Concerning the critical exponent v, this method gives the same result as Flory s method (v = 3/([Pg.298]

The AHfis determined by differential scanning calorimetry (DSC), and is determined by differential thermal analysis (DTA) and from heat of fusion tables such as values obtained by Flory s method. 

In Section 1.4.2, we derived Rp = for an excluded-volume chain using Flory s method. Here, we use a similar method to derive Rp = for theta chains. The difference in the free energy A<,h of the chain is in the second term of Eq. 1.63. For the theta chains, binary interaction is effectively absent (A2 = 0) and therefore the leading term in the polymer-polymer interaction is b R (N/R y = b N /R, which is due to the ternary interactions. Then, Ajh is given as... 

Problem 2.17 We apply Flory s method (Section 1.4) to find how much a small deviation from the theta condition changes the end-to-end distance. For this purpose, we express the fiee energy per chain that has an end-to-end distance R by... 

Problem 4.7 We can apply Flory s method to evalnate the chain contraction in the semidilnte solntion. In Flory s expression for the free energy of a polymer chain in Eq. 1.63, the binary monomer-monomer interaction was limited to a pair on the same polymer chain. In the semidilnte solntion of monomer density p, the partner of the interaction is mostly the monomers on the other chains. We can approximate this change by introdncing the probability (N/R )/(N/R + p). A monomer on a given polymer chain interacts with other monomers on the same chain with this probability. Then, Eq. 1.63 is changed to... 

Notice that in the derivation of equation (59), the only input is the assumption of equation (54) and the obvious results of equation (55) and Vi = 1, and everything else is simple changes of variables. This is to be contrasted with Flory s method of optimizing the chain free energy to obtain v. ... 

However, Szabo-Rethy293) showed that Flory s assumption leads to wrong values of rate constants and that the error can be as high as 15 to 35%. This author gave a method of calculation which takes into account the elimination of water. Similar relations have been used recently by Lin and Hsieh270) but in our opinion erroneously. 

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

That is why logically to assume the possibility of specific interactions also during the swelling process, since the values of parameters of the coal solubility S2, which are determining accordingly to the Flory Renner s equation are differed. It depends on fact if the data for all solvents are taking into account in calculations or such calculations are performed with the exclusion of results for solvents able to be as acceptors of hydrogen bonds (amines, ketones). Different results have been obtained also under application of other methods for calculations, especially of the Van-Krevelen s method [14],... 

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. 

Cycllzatlon of polyoxyethylene to form the macrocycle 3x-crown-x is studied for even x from 4 to 20 units using Monte-Carlo methods. POE is assumed to behave in accord with the model developed by Mark and Flory (S 137, S 138). The fraction of acyclic chains satisfying criteria for cyclozation reaches a maximum of 1.2 x 10-3 for x = 6 and falls of to 0.1 x 10-3 when x Is 20. The implication that 18-crown-6 is the most easily formed macrocycle is in harmony with experiment. Conditional probabilities and a priori probabilities are evaluated at 298 K conditional probabilities, in their matrix representations, are ... 

The proton NMR shifts of diad, triad, and tetrad protons in an isolated chain of PP are calculated, taking into account its configurations and conformations using Pople s approximation. The agreement between the observed and calculated results, however, is insufficient with respect to the large difference between syn and anti protons and the order of tetrad protons. Here, a priori probabilities of specified conformations necessary to the estimation of the chemical shifts are calculated using Flory s matrix method. 

Rubber Elasticity. This method is based on the hypothesis that Flory s relationship can be applied to thermosets, which is only a rough approximation (Chapter 10) ... 

The second method is based on Flory s principle of the independence of the reactivity of the bonding sites is independent of their position in the long chain or a low-molecular weight analogue with a correct selection of the model reaction components (a central particle in the form of a metal ion). For a single-site bonding the K value is determined as in the case of low-molecular weight compounds... 

Several methods have been proposed in the literature for this purpose. The first method, which may be called the Flory-Fox-Schaefgen method, is based on the combination of Flory and Fox s viscosity equation (4) and Flory s excluded volume equation (8) [see, Flory and Fox (103)]. The substitution of Eqs. (5) and (6) into Eq. (8) yields... 

In fact, no explanation of this method is given in Flory s book (5). 

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. 

Flory s approach (10) has often been used to study phase separation, but this method does not always predict the exact conditions of separation. The equation of Huggins was used to draw the phase diagram of Figure 1 since values could be calculated over the whole range of concentrations. Also the ft values agreed more closely with the parameter values. This equation is (II) ... 

The modern theory, grounded on perturbation calculations, predicts the validity of the relation ZN ZaNy 1 p. However, if this formula is right, it will not be possible to prove it without using very sophisticated methods (see the criticism of Flory s theory in Chapter 8). 

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