Flory expression

Attempts have been made to obtain a closed expression for a, or at least a form amenable to numerical computation. The Flory expression, Eq. (8), and the Fixman (9S) differential equation,... 

A somewhat similar equation has been given by Ptitsyn (219) on the basis of related arguments, and the deviations from the Flory relation are in the same direction. Thus there seems to be general agreement that some modification in the Flory expression for the long-range interaction effects on chain dimensions is needed. 

The problem of the second virial coeffident of flexible chain polymers has three phases. In the first place, there is the problem of interchain interactions in the second, the problem of the inteochain interactions and finally, the problem of the coupling of the inter- and intra-chain interactions. Different approximations are possible for each phase of the problem, various combinations of such approximations are also possible, and consequently we are confronted by a wide variety of possible theories. For example, the present development of a new excluded-volume equation for the intrachain problem almost doubles the variety of Az-theories, since most existing theories have been developed on the basis of the Flory expression for the excluded volume effect and the new theory can be combined, in prindple, with all of these theories. On the other hand, we have as yet no computational data for A2 and furthermore, the experimental accuracy of A2 measurements is still rather poor, as illustrated in Fig. 2. It is therefore unlikely that one particular combination of several approximations can be properly chosen from among the... 

Huggins independently and almost simultaneously obtained a more rigorous equation for the free energy, but it is this simpler Flory expression that is most often used and which is known as the Floiy-Huggins equation. 

Langmuir (1926) was probably the first to suggest that van der Waals forces should cause a hydrocarbon chain molecule in a dilute gas to roll up into spheres with a density comparable to that of the liquid phase. Stockmayer (1960) extended this concept to polymer molecules in worse than 0-solvents. He recognized that the classical Flory expression for the excluded volume parameter... 

For the random combinatorial term, earlier theories nsed the Flory expression [2], resulting in the LF theory [3,4], or the Guggenheim expression [1], resnlting in the PV model [51] and QCHB model [7]. 

As discussed above it may be dubious to use the Flory expression in the discussion of mixtures of substances which are relatively similar in size, A method based on experimental m. andx-- was sugges-... 

The average degree of branching per monomer unit, Qm = Ntr/Mo c, is given by the well-known Flory expression [27] that is valid for all addition polymerization processes independent of their mechanisms ... 

The most familiar presentation is given in Flory s 1953 book, but in the 1970 Speirs lecture Flory expressed the combinatorial result in a particularly illuminating form... 

This problem can be treated by using the Flory expression for the free energy of fusion, AGf, of a crystallite units long and p units broad selected from A chains each having x repeating units. As has been shown (1)... 

In polymer solutions or blends, one of the most important thennodynamic parameters that can be calculated from the (neutron) scattering data is the enthalpic interaction parameter x between the components. Based on the Flory-Huggins theory [4T, 42], the scattering intensity from a polymer in a solution can be expressed as... 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

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 ... 

Since the Flory-Huggins theory provides us with an analytical expression for AG , in Eq. (8.44), it is not difficult to carry out the differentiations indicated above to consider the critical point for miscibility in terms of the Flory-Huggins model. While not difficult, the mathematical manipulations do take up too much space to include them in detail. Accordingly, we indicate only some intermediate points in the derivation. We begin by recalling that (bAGj Ibn ) j -A/ii, so by differentiating Eq. (8.44) with respect to either Ni or N2, we obtain... 

To arrive at an expression for AS, we follow a series of steps which parallel-for a different model-the development of the Flory-Huggins model for AS,... 

To use the Flory-Huggins theory as a source for understanding the second virial coefficient, we return to Eq. (8.53), which gives an expression for jui -jui°. Combining this result with Eq. (8.79) gives... 

The objective of the Flory-Krigbaum theory is to find a quantitative expression for the placement probability n(d) of the two coils as a function of their separation d. There are three stages to the derivation ... 

The full Flory-Krigbaum theory results in the following expression for the excluded volume ... 

The quantity in parentheses on the right-hand side is reminiscent of the expression AH - T AS, with the quantity 1/2R a contribution from the configurational entropy of the Flory-Huggins theory. Since our objective is to incorporate a solvation entropy into the discussion, we add the latter -in units of R for convenience-to 1/2R ... 

Dispersive Interactions. For pairs of nonpolar polymers, the intermolecular forces are primarily of the dispersive type, and in such cases the energy of interaction between unlike segments is expected to be closely approximated by the geometric mean of the energies of interaction between the two like pairs (98). In this case, the Flory-Huggins interaction energy between this polymer pair can be expressed in terms of the solubiUty parameters 5 of the pure components. 

Real solutions are rarely completely athermal, even when there is considerable similarity between the nature of the molecules. For cases in which some energy effects must be taken into account, Flory introduced an additional term into the expression for excess Gibbs free energy. Adapting the format of the Scatchard-Hildebrand equation, the additional contribution to the excess Gibbs free energy is assumed to be of the form ... 

According to Flory [121], Mandelkern [122], and Price [123], the free energy change in the formation of a bundle-like nucleus shown in Fig. 14 can be expressed as ... 

Considering an (incompressible) polymer mixture with volume fraction (j)A = (j) of A-monomers and volume fraction (j)B = 1 - (j) of B monomers, the mean-field expression for the excess-free energy of mixing is given by the well-known Flory-Huggins expression " ... 

Various modifications of the Flory theory [4] are usually applied to describe the uncharged gels. Their crosslinking density can be simply calculated from the swelling degree using Eqs. (3.1) and (3.2) or analytical expressions for the Mc value (see, for example, Ref. [124]). 

Two schemes may be used to obtain explicit expressions for F. One is a so-called Flory approximation where the terms of Eq. 1 are estimated for ideal uncorrelated chains. The corresponding free energy per chain, in units of the thermal energy kT, is ... 

The Flory-Huggins activity expression for solvent in a solvent(l)/polymer(2) solution is... 

Equivalent expressions for equations (3) and (6) exist for the polymer (7). The Flory-Huggins expressions can also be extended to multicomponent systems (2) ... 

From the outset, Flory (6) and Huggins (4,5 ) recognized that their expressions for polymer solution thermodynamics had certain shortcomings (2). Among these were the fact that the Flory-Huggins expressions do not predict the existence of the LCST (see Figure 2) and that in practice the x parameter must be composition dependent in order to fit phase equilibrium data for many polymer solutions 3,8). 

---