Flory point

The i gTinfinite dilution expression for the osmotic pressure, and the second term can be referred to as the second virial term with a second virial coefficient B2[= (1/2 - Xi)< / i]- The second virial coefficient becomes 0 when xi = 1/2. This point is called the Flory point or the... 

The excluded volume problem of polymer chains was taken up early in 1943 by Flory [6]. His arguments based on the chemical thermodynamics brought the conclusions (i) the existence of the Flory point ( point) where two body interactions apparently vanish, and (ii) that in non-solvent state chains behave ideally-... 

As noted above, the first study of the problem of partial chain flexibility has been done by Flory (1) - one more problem in polymer science which he was the first to tackle. Flory has assumed the existence of a favorable arrangement of a number of consecutive base units. The configurational free energy of this arrangement differs by an amount e from other possible sequences. Apparently, these other arrangements do not have to be all identical thus e represents an average value. Flory points out that the stiffness of the chain is involved. He places the chains and solvent molecules on a lattice, a convenient although not a necessary step. 

They observed that the effective exponent v(w) decreases steadily when w increases. For w = 1, i.e. in the absence of attractive interaction v(w) 0.60. When w increases, v(w) decreases and for w = wh it takes the value v = 1/2 which defines the Flory point. The corresponding values of w are given by... 

The excluded volume parameter depends on temperature and, in the vicinity of the Flory point, can be approximated by a linear function of 1 / T... 

We want to calculate the swelling of a chain and the osmotic pressure of a set of chains in the vicinity of the Flory point. For this purpose, we can calculate, on one side + (k, — H S) to first-order in k2, on the other side the quantities (iVxS). 

Thus, to calculate the swelling of an isolated chain and the osmotic pressure of a set of chains in the vicinity of the Flory point, a determination of the partition functions 3 (E, — H S) and 3 N x S) by dimensional renormalization is sufficient. The calculations will be performed to first orders in x and y for a space dimension d = 3 — e (0 < s < 1), and we note that the purely repulsive terms have been already calculated in Chapter 10. The partition functions will be represented by series in terms of x and y. Finally, in order to study the behaviour of long polymers, we shall treat these series by using the direct renormalization method. 

In 1951, Flory reported the condensation reaction of diacid chlorides, e.g. with potassium salts of imides, e.g. the condensation of sebacyl chloride with potassium phthalimide. In this way, A -acyl diimides are formed. Flory pointed out the possibility of forming polymers, when components with higher functionality are used. Poly(imide)s (PI)s from pyromeUitic acid were reported in 1955 by Edwards and Maxwell at DuPont. Tbe diamines used were of aliphatic nature. Later, in addition, aromatic diamines were used. The two major types of Pis are ... 

The real Flory point 0 is defined by the vanishing of v and is thus shifted significantly. 

When our coil is exactly at the Flory point, we say that it has a quasiideal behavior. The prefix quasi is used to recall that some interactions are still present because the three-body term has some residual effects (apart from the renormalization of v). Thus, some subtle correlations remain at T=Q. Since they are probably too small to be observed, we shall not insist very much on their properties. Mathematically, however, they are associated with unusual logarithmic factors. The origin of such factors can be understood from the following crude argument. 

Flory pointed out that volume exclusion must cause (R ) to increase. The amount of increase can be expressed by the expansion factor a , defined as... 

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... 

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. 

These researchers have pointed out that the basic principles of polymer physics apply to tackified elastomers, such as natural rubber. Thus the Tg relationship provided by the Fox-Flory equation applies ... 

As pointed out by Flory [16], the principle of equal reactivity, according to which the opportunity for reaction (fusion or scission) is independent of the size of the participating polymers, implies an exponential decay of the number of polymers of size / as a function of /. Indeed, at the level of mean-field approximation in the absence of closed rings, one can write the free energy for a system of linear chains [11] as... 

Melting of ECC involving transition into the isotropic melt was shown by Flory to be a first-order process. It can be seen in Fig. 18 b that there occurs a transition from a complete order to a fully random chain arrangement in the isotropic melt (Fig. 16, point 4). 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

Both extrapolated lines meet on the A axis at the same point, and this corresponds to 1/M. Other solution properties of the polymer may also be determined once the Zimm plot has been prepared. Along the line of 0 = 0, A = 1/M (1 + 2T2 c +. ..). Hence the slope of this line is 2T2/M, from which r of the Flory equation may be evaluated. 

Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

As early as 1952, Flory [5, 6] pointed out that the polycondensation of AB -type monomers will result in soluble highly branched polymers and he calculated the molecular weight distribution (MWD) and its averages using a statistical derivation. Ill-defined branched polycondensates were reported even earlier [7,8]. In 1972, Baker et al. reported the polycondensation of polyhydrox-ymonocarboxylic acids, (OH)nR-COOH, where n is an integer from two to six [ 9]. In 1982, Kricheldorf et al. [ 10] pubhshed the cocondensation of AB and AB2 monomers to form branched polyesters. However, only after Kim and Webster published the synthesis of pure hyperbranched polyarylenes from an AB2 monomer in 1988 [11-13], this class of polymers became a topic of intensive research by many groups. A multitude of hyperbranched polymers synthesized via polycondensation of AB2 monomers have been reported, and many reviews have been published [1,2,14-16]. 

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