Polymerization Flory statistics

When the mass fraction of the long-chain hydrocarbon products of the F-T synthesis (W) is plotted against the carbon number (TSf) it is found that W decreases approximately monotonically with molecular size. Thus the major product is the Ci, methane, followed by the C2 hydrocarbons (ethylene and ethane), the C3 hydrocarbons, and so forth, as shown in Figure 15. This distribution follows Schultz-Flory statistics for a polymerization involving the sequential addition of Ci units to a chain, given by the dotted line in Figure 15. Further and more detailed consideration of the mechanisms is in Annex 1. 

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... 

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

Self-avoiding random walks (SARW) statistics has been proposed [1] for single that is for non-interacting between themselves ideal polymeric chains (free-articulated Kuhn s chains [2]) into ideal solvents, in which the all-possible configurations of the polymeric chain are energetically equal. From this statistics follows, that under the absence of external forces the conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar radius Rf of which is described by known expression [3, 4]... 

Polymeric chains in the concentrated solutions and melts at molar-volumetric concentration c of the chains more than critical one c = (NaR/) ] are intertwined. As a result, from the author s point of view [3] the chains are squeezed decreasing their conformational volume. Accordingly to the Flory theorem [4] polymeric chains in the melts behave as the single ones with the size R = aN112, which is the root-main quadratic radius in the random walks (RW) Gaussian statistics. 

Two main approaches for osmotic pressure of polymeric solutions theoretical description can be distinguished. First is Flory-Huggins method [1, 2], which afterwards has been determined as method of self-consistent field. In the initial variant the main attention has been paid into pair-wise interaction in the system gaped monomeric links - molecules of solvent . Flory-Huggins parameter % was a measure of above-said pair-wise interaction and this limited application of presented method by field of concentrated solutions. In subsequent variants such method was extended on individual macromolecules into diluted solutions with taken into account the tie-up of chain links by Gaussian statistics [1]. 

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... 

Experimental values are presented of the molar Kerr constants /x and dipole moments squared, lx, for the copolymers poly(styrene-co-p-bromostyrene), where x is the degree of polymerization. Some results are also presented for poly(styrene-co-p-chlorostyrene) and related polymers. The RIS model of Yoon etal. (Yoon, D. Y. Sundararajan, P. R. Flory, P. J. Macromolecules 1975, 8, 776) is used to calculate mK/x and /x values as a function of tacticity and composition. The statistical weight matrices are identical with those used by Saiz etal. (Saiz, E. Mark, J. E. Flory, P. J. Macromolecules 1977, 10, 967), with the following parameters h = 0.8 exp 397/RT), co = o = 1.3 exp - 1987/RT) and m,= 1.B exp -(2186/RT), where T = 298 K is the temperature. 

A single polymer particle contains a statistically large number of polymers, of the order of 1013. The corresponding MWD can be described with the Flory-Schulz equation only if the single-site type is guaranteed, and under constant reaction conditions, i.e., constant temperature, pressure, and concentrations near the active sites. However, in an industrial polymerization process, a particle often encounters different conditions over its lifetime. In... 

In GPC, the product [77] M, (or the hydrodynamic radius Re) has been widely accepted as a universal calibration parameter. In the Ptitsyn-Eizner modification of the Flory-Fox equation the quantity 4>, which relates the dimensional parameters to the above product, is taken as a variable. The value of < depends upon molecular expansion in solution as represented by a function f(e). Because of this dependence polymeric species having the same [77] M value cannot have the same statistical dimensions (radius of gyration or end-to-end distance) unless they have the same e value. Thus, if [77] M is a universal calibration parameter, the statistical parameters cannot be used as such. A method is presented for obtaining the Mw/Mn ratio from GPC data even though universal calibration is used. 

The Flory model is the version where the equivalence between kinetics and statistical descriptions is extended to the post-gel stage of polymerization. Consequently, the functional groups are assumed to continue to react at random with no distinction on whether they belong to sol molecules or to gel. To analyze this version one can use the explicit form of function H. As usual, the moments are available through successive derivatives of H (Eq. 76) with respect to x calculated at x=l. We may rewrite Eq. (77) in the form... 

Batch Polymerization. Batch polymerization with this mechanism was first treated by Flory (19) using a statistical development. The same results were obtained by Biesenberger (8) using a kinetic analysis with an analytical solution. This was also one of the cases treated by Kilkson (35) using Z-transforms. In the simple cases, his result reduces to the Flory, or random, MWD with the dispersion index of 2. In more complex cases, he solves directly for the moments of the distribution. The Z-transform is probably the most powerful tool for solving condensation MWD problems the convolution theorem allows the nonlinear product terms in the kinetic equation to be handled conveniently. 

If this reaction is indeed third-order, and if Flory s assumption that the intrinsic reactivity of a functional group is independent of chain length is correct, then a plot of 1/c2 versus t should be linear. Because it provides a direct link to the statistics of polymerization, however, it is useful to first follow Flory and define a new parameter, p, the extent of reaction Equation 4-15. 

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 most useful and most commonly employed simplified approach dates back to Flory [27,28] and is based on the premise of equal reactivity of functional groups and statistical growth. The most important application is to polymerization of bifunctional monomers and can be sketched as follows (Flory s derivation is more elaborate). 

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. 

Cross-linking Effect on Polymerization. The general cross-linking reaction that occurs during polymerization and involves components with a functionality greater than one (two or more double bonds, divinyl monomer) has been studied extensively. The statistical analysis of molecular distributions in such reactions is due to Flory (36). 

---