Molecular weight Flory statistics

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

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]. 

Krigbaum, W. R. and Flory, P. J., Statistical mechanics of dilute polymer solutions. IV. Variation of the osmotic second coefficient with molecular weight, /. Am. Chem. Soc., 75, 1775, 1953. 

In agreement with Flory s predictions, hyperbranched polymers based on A,jB monomers reported in the literature exhibit a broad molecular weight distribution (typically 2-5 or more). The polydispersity of a hyperbranched polymer is due to the statistical growth process. A strategy to overcome this disadvantage is to add a By-functional core molecule, or a chain terminator, which Hmits the polydispersity and also provides a tool to control the molecular weight of the final polymer. The concept of copolymerizing an A2B monomer with a B3 functional core molecule was first introduced by Hult et al. [62] and more recently also utilized by Feast and Stainton [63] and Moore and Bharathi [64]. 

The product of a polymerization is a mixture of polymer molecules of different molecular weights. For theoretical and practical reasons it is of interest to discuss the distribution of molecular weights in a polymerization. The molecular weight distribution (MWD) has been derived by Flory by a statistical approach based on the concept of equal reactivity of functional groups [Flory, 1953 Howard, 1961 Peebles, 1971]. The derivation that follows is essentially that of Flory and applies equally to A—B and stoichiometric A—A plus B—B types of step polymerizations. 

The molecular weight distribution in this type of nonlinear polymerization will be much narrower than for a linear polymerization. Molecules of sizes very much different from the average are less likely than in linear polymerization, since this would require having the statistically determined / branches making up a molecule all very long or all very short. The distribution functions for this polymerization have been derived statistically [Peebles, 1971 Schaefgen and Flory, 1948], and the results are given as... 

A second and distinct era in the development of branched macromolecular architecture encompasses the time between 1940 to 1978, or approximately the next four decades. Kuhn 151 published the first report of the use of statistical methods for analysis of a polymer problem in 1930. Equations were derived for molecular weight distributions of degraded cellulose. Thereafter, mathematical analyses of polymer properties and interactions flourished. Perhaps no single person has affected linear and non-linear polymer chemistry as profoundly as P. J. Flory. His contributions were rewarded by receipt of the Nobel Prize for Chemistry in 1974. 

Although the major interest in experimental and theoretical studies of network formation has been devoted to elastomer networks, the epoxy resins keep apparently first place among typical thermosets. Almost exclusively, the statistical theory based on the tree-like model has been used. The problem of curing was first attacked by Japanese authors (Yamabe and Fukui, Kakurai and Noguchi, Tanaka and Kakiuchi) who used the combinatorial approach of Flory and Stockmayer. Their work has been reviewed in Chapter IV of May s and Tanaka s monograph Their experimental studies included molecular weights and gel points. However, their conclusions were somewhat invalidated by the fact that the assumed reaction schemes were too simplified or even incorrect. It is to be stressed, however, that Yamabe and Fukui were the first who took into account the initiated mechanism of polymerization of epoxy groups (polyetherification). They used, however, the statistical treatment which is incorrect as was shown in Section 3.3. 

The gel point t is one of the most important kinetic characteristics of curing, since it describes the attainment of a certain critical conversion responsible for the transition from the first to the second stage of the process. In the classical statistical theory of gelation developed by Flory [8], the 1 point is characterized by the appearance in a reactive system of a macromolecule with an infinitely large molecular weight, -+ oo. Viscosity becomes infinite which corresponds to the above condition. 

Flory Statistics of the Molecular Weight Distribution. The solution to the complete set (j - I to j = 100,000) of coupled-nonlinear ordinary differential equations needed to calculate the distribution is an enormous undertaking even with the fastest computers. However, we can use probability theory to estimate the distribution. This theory was developed by Nobel laureate Paul Floty. We have shown that for step ipolymeiization and for free radical polymerization in which termination is by disproportionation the mole fraction of polymer -with chain length j is... 

Polymerization in the melt is widely used commercially for the production of polyesters, polyamides, polycarbonates and other products. The reactions are controlled by the chemical kinetics, rather than by diffusion. Molecular weights and molecular weight distributions follow closely the statistical calculations indicated in the preceding section, at least for the three types of polymers mentioned above. There has been much speculation as to the effect of increasing viscosity on the rates of the reactions, without completely satisfactory explanations or experimental demonstrations yet available. Flory [7] showed that the rate of reaction between certain dicarboxylic acids and glycols was independent of viscosity for those materials, in the range studied. The viscosity range had a maximum of 0.3 poise, however, far below the hundreds of thousands of poises encountered in some polycondensations. 

Random Prepolymers. These prepolymers are built up from polyfunctional monomers reacting statistically according to the theories of Flory. Reaction is stopped at a desired prepolymer molecular weight, usually by cooling. Final polymerization is achieved by heating therefore the term thermoset is used for them. 

CDP7-Jb polymerization. Plot disU ibution of molecular weight using Flory statistics. [3rd Ed. P7-15]... 

CDP7-P, Use Flory statistics for molecular weight distribution. [3rd Ed. P7-2IJ... 

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