Distribution function Schulz-Flory

Thus, Equation 27 is in this case a possible distribution function. It is of the type of the Schulz-Flory (25) distribution function. The expressions p and alternating polymerization (chain termination). The validity of the Schulz-Flory distribution function in this example of a polymerization with reversible propagation steps is evident. This type of distribution is always present if the distribution of the chain lengths... 

Thus, a Schulz-Flory distribution function is chosen for the segment number distribution and a Dirac distribution function for chemical polydispersity. In other words, only the first moment of Yb is taken into account. 

Figure 10.1 Differential (a) and integral (b) Schulz Flory distribution function with = lOi) and different k values.

The polymer under consideration can be described with a Schulz-Flory distribution function with = 100 and k = 1. For the thermodynamic modeling, the quantities tm and rz must be calculated. 

For widely distributed polymers such as radically polymerized polyethene the Schulz-Flory distribution function is unable to describe the high degree of asymmetry in the distribution. In this case, the Wesslau distribution (logarithmic normal distribution) is used and given by ... 

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. 

Given the Schulz-Flory distribution, the mole and weight fractions of polymer with j monomer units (based on polymer) as a function of that number j are... 

Krishna and Bell (299) described the results of their steady-state tracing experiments by the model shown in Fig. 31. The scheme is in accord with the Anderson-Schulz-Flory distribution of products, based on chain growth by the successive addition of monomers s to chain fragments C s- is different from Ci,s. ft is assumed that the probability of chain growth a is not a function of n, where... 

The amount of in situ branching produced is a sensitive function of many variables. Catalyst and reaction parameters have a major influence on the following (a) the a-olefin generation relative to polymer formation, (b) 1-hexene generation relative to that of other linear a-olefins, (c) how sharp or flat the Schulz-Flory distribution of linear a-olefins is, and (d) how efficiently the a-olefins are incorporated as branches. 

Solution In radical chain reactions, the overall rate of polymerization, Rp, and the number-average degree of polymerization, X , are functions of the initiator concentration [I], the monomer concentration [M], and also the temperature via the temperature dependence of the individual rate constants. At constant [M] and [I], the Schulz-Flory MWD is produced. However, if [M] and [I] vary with time, a number of Schulz-Flory distributions overlap and thus a broader MWD is produced. In the ideal CSTR [M] and [I] are constant and the temperature is relatively uniform. Consequently, chain polymerizations in CSTR produce the narrowest possible MWD. In the batch reactor, [M] and [I] vary with time (decrease with conversion) while in the tubular reactor [M] and [I] vary with position in the reactor and the temperature increases with tube radius. These variations cause a shift in X with conversion and consequently a broadening of MWD. 

An exponentially decreasing curve is obtained for A = 1 when the mole fraction is plotted against the degree of polymerization (see Figure 8-3). For this reason, and not because an exponential function appears in Equation (8-35), the Schulz-Flory distribution is called an exponential distribution. 

The results above are only valid for tetrafunctional crosslinking of monodisperse polymer. However, in many thermoreversible systems the crosslinks have functionalities that are much larger than four. Moreover, the polymers used are not monodisperse in general. In order to be able to calculate network parameters the present author [39—44] extended the Flory-Stockmayer model for polydisperse polymer which is crosslinked with f-functional crosslinks. It was possible to calculate network parameters for polymers of various molecular weight distributions (monodisperse polymer with D s M, /r3 = 1, a Schulz-Flory distribution with D = 1.5, a Flory distribution with D = 2, a cumulative... 

Marano and Holder have calculated the VLE of the Fischer-Tropseh system. The pseudo-components were defined with the aid of an analytical molar-mass distribution function (Anderson-Schulz-Flory distribution). The properties of a pseudo-component were based on a hypothetical model component in each carbon-number cut. 

In eq 9.44 the distribution function is called generalized Schulz-Flory distribution (or Schulz-Zimm distribution) because of its introduction by Schulz in 1935, and by Flory in 1936. " To evaluate the polydispersity of polymers U = Mw/M - 1 is an important quantity. The quantity k of the Schulz-Flory distribution is given by k= IU. The two parameters f and k of the... 

Figure 6.16.2 Schulz-Flory distribution of ethylene oligomers. Mass and mol fractions are given as a function of the ratio of rate of insertion (ins) and rate ofelimation (eli) (insieli is equal to 1//3).

The normal distribution function, also referred to as the Flory-Schulz distribution, relates the fraction of an x-mer (a polymer molecule consisting of x repeat units) in the entire assembly of molecules to its formation probability. It can be defined either as a number distribution function or as a weight distribution function. The number of moles of an x-mer (Nx) is given by the normal number distribution as follows ... 

Schaefgen and Flory [79] were the first to observe this effect. They prepared star-branched polyamides by co-condensation of A-B types of monomers with central units which carried/-functional A groups. By this technique star molecules were obtained in which the arms are not monodisperse in length. They rather obeyed the Schulz-Flory most probable length distribution with polydis-persity index However, the coupling of f arms onto a star center leads... 

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

Figure 10.3. Schulz-Flory mole-fraction distribution (left) and corresponding molecular-weight distribution (right) at different degrees of fractional conversion of functional groups (adapted from Flory [27]).

Challa [69] found that the monomer content of the polyesters was greater than that predicted by the Flory—Schulz distribution function, the so-called most probable distribution of molecular species. Challa proposed that the monomer molecule — in this case bis(2-hydroxy-ethyl)terephthalate, though the conclusion could be general for all polycondensations — lost more entropy on entering the transition state than did the longer molecules. 

Figure 8-5. Calculated < A/) / ratios as a function of the exponents Orj for a Schulz-Flory (—) or a generalized logarithmic normal (--) molar mass distribution for, in each case, (Mw)l

The polycondensation processes generally produce polyamides that are mixtures of polymer molecules of different molecular weights, the distribution of which usually follows a definite continuous function according to the most probable distribution model by Schulz-Flory [3]. This distribution function may, in principle, be derived from the kinetics of polymerization process, but is more readily derived from statistical considerations. In this case, the extent... 

---