Flory theory point

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

Since the pioneering work of Kuhn, the solvent freezing point depression observed in swollen crosslinked rubbers has been the subject of many works. The observed AT can be attributed to two origins. A sizable AT is accounted for by the lowering of the thermodynamic potential of solvent molecules in a polymer solution derived from the Flory theory, and the additional AT observed for crosslinked rubbers has been attributed to confinement effects, fn 1991, Jackson and McKenna [32] studied... 

This fact can be demonstrated as follows. Let us determine the value of the well-known Flory parameter x, which corresponds to the 6 point (i.e. to the point of inversion of the second virial coefficient of the solution of rods) in the Flory theory of Ref.9). This can be done by expanding the chemical potential of the solvent in the isotropic phase (Eq. (16) of Ref.9 ) into powers of the polymer volume fraction in the solution, and by equating the coefficient at the quadratic term of this expansion to zero this procedure gives Xe = 1/2 independently of p. On the other hand, it is well known26,27) that the value of x decreases with increasing p and that X < 1 at p > 1. The contradiction obtained shows that the expressions for the thermodynamic functions used in Ref.9) are not always correct... 

In order to analyze the dependence of the liquid crystalline transition properties on temperature (i.e. on the solvent quality), it is necessary to introduce the attraction of rods parallel to their steric repulsion. This has been done by Rory l. The classical phase diagram of Flory for the solution of rods (see Fig. 2) agrees well with experimental results from the qualitative point of view However, the Flory theory cannot give adequate answers to all the questions connected with the orientational ordering in the system of rigid rods. Indeed ... 

In this section we will calculate the second virial coefficient for the solution of rods interacting as described in Sect. 2.2 and we will find the point of inversion of this coefficient, i.e. the 6 point. As noted above, the Flory theory gives the incorrect value for the 6 temperature. 

The phase diagram of the polypeptide solutions is shown schematically in Fig. 1. According to the Flory theory, the relation between the concentration at the A point (Vja) and the axial ratio (r) is represented as follows ... 

It is worth pointing out that some assumptions of the Flory theory are valid only in a perfectly ordered phase, i.e., the Flory theory is more applicable in the case of high order and of large concentration. 

It is worthwhile to point out that the Maier-Saupe theory has been successful in analyzing the behavior of small molecular mass liquid crystals at transition, such as the temperature change of the order parameter. The jump of the order parameter at transition, Sc = 0.43 is in reasonable agreement with most experiments. The Onsager and Flory theories, which take into account the steric effects predict a higher critical order parameter. 

Family [9] considered the conformations of statistical branched fractals (which simulate branched polymers) formed in equilibrium processes in terms of the Flory theory. Using this approach, he found only three different states of statistical fractals, which were called uncoiled, compensated, and collapsed states. In particular, it was found that in thermally induced phase transitions, clusters occur in the compensated state and have nearly equal fractal dimensions ( 2.5). Recall that the value df = 2.5 in polymers corresponds to the gelation point this allows gelation to be classified as a critical phenomenon. 

The gel melting data were analyzed using the Flory theory for the melting point depression of a polymer by a diluent [181] so that the fundamental thermodynamic parameters referred to earlier could be evaluated. This theory predicts the following dependence of melting point on the volume fraction of the diluent, which in this case is the solvent ... 

The apparent discrepancy between the Flory theory and the entanglement concept of Dossin and Graessley has been addressed by Gottlieb and Macosco [55]. They pointed out that the two parameters h and k, both measuring the severity of constraints are related. For the case of a perfect, incompressible, unswollen network the analytical relationship is given by... 

The Flory theory is not rigorous, but it captures the two most essential features of the coil size variation in good solvents (1) that R oc N with v 0.6 for large N and (2) that the swelling coefficient a depends on the parameter z. More sophisticated theories and computer simulations point to v 0.588 which is very close to the Hoty exponent. 

Clearly, the issue of the conformation of polymer chains in the bulk amorphous state is not yet settled indeed it remains an area of current research. The vast bulk of research to date strongly suggests that the random coil must be at least close to the truth for many polymers of interest. Points such as the extent of local order await further developments. Thus, this book will expound the Mark-Flory theory of the random coil, except where specifically mentioned to the contrary. 

An alternative method which avoids these problems is to apply the Flory theory of melting point depression of polymer solutions. A crystalline polymer may be in equilibrium with a solution of the same polymer at a temperature Ts, lower than the melting point of the pure polymer Tm. At equilibrium the chemical potentials of the polymer in the two phases must be equal, i. e. ( 2)- This can be expressed as... 

Equation (2.40) is identical to the classical Gibbs-Thomson expression for the melting of crystals of finite size. Thus, following the Flory theory (10) nonequilibrium crystallites of high molecular weight chains obey the same melting point relation... 

In the course of analyzing experimental results of melting point depressions, recourse will be made to the different expressions that have been developed. It can be expected, however, that with the many expressions available, and the possibility of adding additional terms to the ideal Flory theory, it will be difficult to differentiate whether or not the crystalline phase is pure based solely on melting temperature-composition relations. Except in a few special cases recourse will have to be made to direct physical measurements to determine the composition of the crystalline phase. 

The failure of the Flory theory, even when extrapolated equihbrium melting temperatures are used, does not necessarily mean that either comonomers or structural defects are entering the lattice.(45) The melting point relation given by Eq. (5.42) is for an ideal melt. Modification of this theory can be legitimately made, while still maintaining equilibrium, without requiring that the co-unit enter the lattice. 

The Flory theory also suffers in that it applies the affine hypothesis to the transformation of the average position of all crosslink points. 

For various elasticity theories, the qualitative phase diagrams have been constructed as functions of the Flory-Huggins interaction parameter (Fig. 13) [93]. Case a describes simultaneously cross-linked IPNs from the point of view of James-Guth theory, case b from the point of view of Flory theory, while case c relates to sequentially cross-linked IPNs using Flory theory. 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

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... 

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