Flory distribution

The FTS mechanism could be considered a simple polymerization reaction, the monomer being a Ci species derived from carbon monoxide. This polymerization follows an Anderson-Schulz-Flory distribution of molecular weights. This distribution gives a linear plot of the logarithm of yield of product (in moles) versus carbon number. Under the assumptions of this model, the entire product distribution is determined by one parameter, a, the probability of the addition of a carbon atom to a chain (Figure 4-7). ... 

This is the famous Flory distribution. Here, it is expressed in terms of the conversion, but Equation (13.21) can be used to replace X with The result is... 

Use Equation (13.26) to find the moments /xq through /X2 for the Flory distribution. Use your results to validate Equation (13.27). 

Find the standard deviation of the Flory distribution as given by Equation (13.26) and relate it to the polydispersity. Extend the calculations in Problem 13.5 to /X3. Find the kurtosis of the distribution in the limit of high conversion. 

The Flory distribution gives a polydispersity of 2 in the limit of high conversion. Yet, a thought experiment suggests that a small batch of self-condensing molecules would eventually condense to form a single, cyclic molecule. Reconcile this apparent inconsistency. 

Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

Figure 1 Poisson and Schulz-Flory distribution with identical (N)n = 50. The arrows indicate (,N)w = 51 (Poisson) and (N)w = 100 (Schulz-Flory).

The above phenomenon is due to the pronounced polydispersity of these products in their chemical size l described by the Flory exponential distribution. Because the composition of each macromolecule of the sample under investigation is unambiguously related to its degree of polymerization l, the Flory distribution for l in a polymer sample is responsible for its significant composition inhomogeneity. 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

Polymers prepared by a polycondensation reaction have the Flory distribution for the chain length, which can be approximated by the function... 

For the same values of ea g, r0 and u0 as for the monodisperse distribution, Figs. 45 and 46 show the results for the Flory distribution of chain lengths. The curves start at degrees of polymerisation determined by zn=[(2a)-1+l]. A comparison of Fig. 45 with Fig. 42 shows that, for a diameter equal to the chain di-... 

Fig. 44 Flory distribution, half-Gauss and uniform chain length distribution for an average DP of 100 monomeric units (m.u.)...

Fig. 45 Ultimate strength of PpPTA fibres versus the degree of polymerisation applying the Flory distribution of chain lengths for various values of the diameter 2 r calculated with Eq. 93...

As an example of a distribution with a smaller number of long chains than the Flory distribution, a half-Gauss distribution is chosen,... 

All computations so far have been performed with the full width of the chain length distributions, i.e. chains with aspect ratios Lld< 10 have also been included. However, the effects of stress transfer across chain ends and the stress concentrations may become important below this aspect ratio. In the theory by Yoon these effects are neglected. In particular, for the Flory distribution containing a relatively large proportion of very short chains, the effects may be considerable. Therefore, calculations are performed in which the negative effect of the very short rods is approximated by assuming that for an aspect ratio L/d<4 the contribution of these rods to the strength is set to zero. 

Fig. 50 The ultimate strength for a Flory distribution in which rods with an aspect ratio L/d<4 do not contribute to the strength... 

Schrock alkylidyne catalysts, 26 948-949 Schrodinger s equation, 16 734-735 Schultz-Flory distribution, 20 156 Schultz-Flory equation, 17 714 Schulze—Hardy rule, 7 289, 10 121 Schweizer, M. E., 11 248 Schwenzfeier process, 3 641 Science... 

Figure 9.4. Typical Schulz-Flory distribution fory=0.85...

Figure 9.5. Comparison of Poisson and Schulz-Flory distribution, see text...

---