Random copolymers melting equation

The beads comprise an uncrosshnked propylene random copolymer having a melting point of at least 140C, as a base resin. The time required to attenuate an air pressure within the foamed beads applied by a pressurising treatment with air from 1.2 to 0.8 kgf/sq.cm.(G) under atmospheric pressure at 23C is at least 80 minutes and the CNl value of the foamed beads, which is defined by a given equation, is smaller than 3.80. 

Thermodynamic equations are formulated for the isomorphic behavior of A-B type random copolymer systems, in which both A and B comonomer units are allowed to cocrystallize in the common lattices analogous to, or just the same as, those of the corresponding homopolymers poly(A) or poly(B). It is assumed that, in the lattice of poly (A), the B units require free energy relative to the A units and vice versa. On the basis of the derived thermodyn-amie equations, phase diagrams are proposed for the A-B random copolymers with cocrystallization. The melting point versus comonomer composition curve predicted by this diagram is very consistent with that experimentally observed for the P(3HB-co-3HV) copolymers, as shown in Fig. 21.1. It is suggested that the minor comonomer unit with a less bulky structure cocrystallize thermodynamically simpler than that with a more bulky structure. 

According to Flory (10) the dependence of the equilibrium melting point on the concentration of crystallizable units in a random copolymer should be given by the equation... 

The DSC results for HyPPS03Na materials demonstrate the existence of low-melting crystals that form when these polymers are maintained at room temperature. The volume fraction of such crystals increases as the degree of sulfonation is increased, but the melting temperature is unaffected by the level of sulfonation. The higher-melting crystallites qualitatively obey the Flory equation for melting-point depression in random copolymers. 

The principles of polymer fractionation by solubility or crystallization in solution have been extensively reviewed on the basis of Hory-Huggins statistical thermodynamic treatment [58,59], which accounts for melting point depression by the presence of solvents. For random copolymers the classical Flory equation [60] applies ... 

Completely transreacted blends showed only single T s. It Is a few degrees lower than the values calculated (dotted Hto) by the Fox equation (Figure 4). These new random copolymers are also amorphous, showing no indications by thermal analysis to crystallize or exhibit melting. 

SgConf jjig eonformational entropy per molar struetural unit at Tg h, is also given by rewriting the modified Flory s equation, which expresses the melting point depression as a function of the mole fraction of major component, X, for binary random copolymers " ... 

If the steric structures of both comonomer imits in random copolymers are similar, the melting temperature depression equation will be the same as Eq. 1, with the interaction parameter calculated with Eq. 4. For a given copolymer, the crystallizabilities of copolymer chains in dilute solution strongly depend on the chain composition. From thermodynanuc considerations, this can be explained from the fact that changes in copolymer composition alter the value of the interaction parameter de ed by Eq. 4. For copolymers with two chemically similar comonomers, xia will be very dose to xiB, ind Xab will approach zero, hi this system, one can simply use Eq. 1 with Xl = XiA Xib-... 

Equation (5.33) differs from that for a random copolymer (most probable or binomial distribution) with a pure crystalline phase, by the last term in the argument of the logarithm. The result embodied in Eq. (5.33) is a perturbation on the melting point equation pertinent to a pure crystalline phase. When e is very large the change in free energy that is involved becomes excessive. The B units will then not enter the lattice and Eq. (5.33) becomes... 

It can be expected that for kinetic reasons crystallites smaller than predicted from equilibrium theory will usually develop. The appropriate melting temperature relation can be formulated in a straightforward manner by invoking the Gibbs-Thomson equation. The result for an ideal random copolymer is (16)... 

The effect of small crystallite thickness on the observed melting temperature-composition relation of random copolymers of vinylidene chloride and methyl methacrylate was analyzed by utilizing the Gibbs-Thomson equation.(16) However, to adapt this procedure to copolymers the dependence of both the crystallite thickness and the interfacial free energy aec on copolymer composition needs to be specified. It was possible to explain the observed melting temperature-composition relation for this copolymer by assuming the dependence of these two quantities on composition. 

Tlie basic equation that can be used to describe the melting of random copolymers was derived by Flory (1951,1953). Originally, this equation showed how much the melting point of a polymer decreases in a solvent ... 

According to Flory, the effect of the comonomer units on the melting of a polymer is similar to the effect of a diluent, so in the last equation N2 simply means the mole fraction of noncrystalhzable units, while AHt is the molar heat of fusion of the crystalUzable component. Therefore, the slope of a plot of l/Tn -l/Tm versus the mole fraction of noncrystalhzable units will be RIAH. When AHt is determined from such plots, it is somewhat smaller than the expected value. The reason for this discrepancy is that the lamellar thickness is ignored by this equation moreover, this equation does not account for the presence of the repeating units of the second component in the crystals. In addition, no blockiness is allowed by this equation it describes only ideally random copolymers. 

The unit distribution was estimated by IR and melting-point methods. IR data for the 4-methylpentene-l unite are based on the relative intensity of the 997 cm band in the copolymer spectra (Fig. 10) and show that these copolymers have a significant tendency to blodc formation ( i f2 3—5). This conclusion was supported by melting-point measurements 114,167). The copolymer melting points are evidently lower than those for homopolymer mixtures, thus demonstrating that a real copolymerization takes place (167), but they have an upward deviation from the theoretical curve calculated for the random copolymer model by means of the Flory equation (Section III.E) with d ff = 4710 kal/ mol (171). 

Figure 11.1 Final melting temperatures of ethylene copolymers as a function of branch content ethylene copolymers containing methyl (open circles), ethyl (open square), and n-propyl (solid triangles) branches hydrogenated polybutadiene (open triangles) ethylene-vinyl acetate (solid circles). Dashed line represents Flory s equilibrium theory for random copolymers (p = Xa), as given in Equation (11.2). Reprinted with permission from Reference [12]. Copyright 1984, American Chemical Society.

The most significant consequence, or prediction, of equation (45) is that the melting temperature of a copolymer, in which only one type unit is crystallizable, depends only on the sequence propagation probability p and not directly on composition. This is a rather unusual result and is unique to long chain molecules. Considering the major categories of copolymer structure we find that for a random copolymer p = Xp for a block copolymer p X, and for an alternating copolymer Many real systems will not fit these conditions exactly but will fall between the... 

The component which brings about the depression in Tm may be a constituent of the polymer itself. In a copolymer consisting of A units which crystallize and B units which do not, with the two units occurring in random sequence along the chain, it is easy to show that the latter should depress the melting point of the former according to the equation... 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

The melting point depression of a random binary AB copolymer, where the comonomer B does not crystallize, is described by the equation 15... 

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