Schulz distribution

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

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 ... 

The polymerisation reaction is a step growth polymerisation similar to a condensation polymerisation of amides or esters. The reaction starts with monomers, which dimerise, trimerise, etc. continuously maintaining a Flory-Schulz distribution. 

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... 

Under same reaction conditions, multi-site catalysts such as Ziegler catalysts provide a MWD that can be considered as a superposition of a number of Flory-Schulz distributions (see ref. 10). Current results show that even metallocene catalysts can be interpreted as "two-site" catalysts [13-15]. Often, it is not clear whether a certain shift of the molecular weight over the reaction time (at constant reaction conditions) can be attributed to small changes of the hydrogen concentration. 

In considering the first case, it is convenient to assume a Schulz distribution W(M) for the molecular weight of the form... 

Note that the dependence on the parent distribution is now only through a. This can be understood from the general discussion in Section III.G The family (77) of density distributions now contains all Schulz distributions with the given a, and the moment free energy is insensitive to which member of this family (specified by p and pf ) is used as the parent. 

There are a number of indications in the literature that might make a reconsideration of their results worthwhile. The authors assume in their model that the intrinsic selectivity and activity of the catalyst does not change over the synthesis period. However, it is shown by Ponec et al. (6), that the catalyst is slowly activated during the synthesis. This activation is accompanied by changes in the selectivity. Ponec et al. attribute these changes in the behaviour of ruthenium catalysts to the deposition of carbon, and not to the low intrinsic propagation activity of the catalyst. Furthermore, Madon (7) showed that the simple Flory-Schulz distribution does not apply to ruthenium catalysts. The applicability of the Schulz-Flory law, however, is an essential part of the treatment of Dautzenberg and coworkers. 

This is the Floty-Schulz distribution. We discuss this distribution further after we discuss chain reactions. 

Peak Shapes. In the case of the Wesslau MWD, the shapes of the peaks from the three detectors are always the same. For the Flory-Schulz distribution, the peak shapes are slightly different and the differences increase with increasing polydispersity. As the polydispersity increases, the LS and viscosity signals become narrower relative to the concentration detector signal and they also become less skewed. Figure 3 shows the peak variance of the viscosity and LS signals relative to the concentration detector peak variance as a function of polydispersity. The concentration detector peak variance increases from 0.25 mL when the polydispersity is 1.1 to 3.65 mL when the polydispersity is 3.3. The LS peak variance increases more slowly. The viscometer variance is in between the two but closer to the LS peak behavior. Figure 4 shows the relative skew of the peaks compared with the refractometer, where the skew is defined as... 

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. 

A similar trend is observed for the polydispersities of the formed PPEs. At the beginning the polydispersity increases fast, to level off after about 24 h at a value of approximately 3—5. These values are higher than for a classic Flory—Schulz distribution, but it may be an effect of increased reactivity of the larger oligomers toward further metathesis. How-... 

Equation 3.6, together with Equation 3.4, describes a random distribution of molecular sizes this distribution is also known as the Flory-Schulz distribution or the most probable distribution [5]. Recently, Wutz and Kricheldorf [6] proposed a model describing the frequency distribution (/ ) and formulated the weight distribution (w,) of linear chains in step-growth polymerizations considering the cyclation reaction, which is one of the most important side reactions in step-growth polymerization. 

Hint 1. If what controls the instantaneous MWD is independent of reaction time or conversion, then the final product will have the most probable (or Flory-Schulz) distribution with a molecular weight dispersity (polydispersity) of 2 (for isothermal operation). 

The other approach is to assume that the molecular-weight distribution is described by the Schulz distribution given by... 

Equation (10.4) indicates that the polydispersity of the Schulz distribution is characterized by the single parameter, Z. In this approach for a certain polymer sample, a series of G t) curves are first calculated by convoluting Eq. (9.19) with Eq. (10.3) at different Z values. Then, the optimum Z value is determined by the best superposition of the measured on the calculated G t) curve. In calculating the Git) curve at a certain Z value, the value used in Eq. (10.3) is first calculated from Eq. (10.4) with the known value of the sample. 

For a series of studied polystyrene samples with a molecular weight less than lOMe, the best Schulz distributions obtained from the closest... 

In Eq. (14.1), /(M) is the molecular-weight distribution of the sample under study, which may be expressed in terms of the Schulz distribution (Eq. (10.3)). The functional forms and relaxation times of the different processes contained in the ERT g.A t), Mx(0i nd gc t) have been... 

For R > 1 and X = 1 the above model approaches random sdssion if R<0.05, the molecules are split practically at the midpoint. Figure (12) shows an example of a model calculation assuming a Schulz distribution with the nonuniformity U = P /P — 1 =0.1 and values of R and X as indicated in the graph. 

Zimm-Schulz distribution (88,89). Using an analysis method similar to that used previously on the poly(Q -MeSty)-6Zoc -poly(4-vinyl pyridine) system, the MMD of both parts of the copolymer were determined. The data analysis method was claimed to verify the random coupling hypotheses. The hypothesis (90) that the polydispersity of individual blocks is higher than the polydispersity of the whole polymer was confirmed (85). That is, block copolymers with narrow MMD have broad complex chemical composition distribution. 

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