Chain length distribution, CLD

A useful alternate to the Mayo plot for determining Cs (as well as other transfer constants) is the chain length distribution (CLD) method based on the molecular weight distributions obtained from size exclusion chromatography (SEC) [Christie and Gilbert, 1996, 1997 Heuts et al., 1998, 1999 Moad and Moad, 1996]. The limiting slope AHIGH of a... 

A simple alternative method was proposed by Gilbert et al. [296, 297] to determine the chain transfer constants based on the chain length distribution (CLD). If the dominant chain termination mechanism is chain transfer to monomer, the instantaneous numerical MWD (the number fraction distribution) is given by ... 

In living polymerization processes it is in fact possible to adjust the final degree of polymerization by simply tuning the initiator amount while keeping narrow the chain length distribution (CLD), i.e. with polydispersity values lower than 1.3 (this quantity reflects the broadness of the final CLD typical minimum values in FRP range from 1.5 to 2). Also, by suitable selection of chemistry and structure... 

It should be emphasized that the discrete variable used here is In some textbooks and research papers, the moments of the MWD are defined in terms of the chain length distribution (CLD) (with chain length, x, being the discrete variable), which means that, in that case, the corresponding averages of the MWD defined in equations analogous to Equations (1.13), (1.16), and (1.17) need to be multiplied by Mq. 

To apply the method of moments, the definitions of the /fth moments of the chain length distributions (CLDs) for live and dead polymer, respectively, are invoked ... 

Notice that no new concepts have been introduced here simply the conventional definitions of DP and DP have been used and reformulated as a ratio of moments of the chain length distribution (CLD). Now, our task is to find the equation for the zeroth, first and second moments of the CLDs for living and dead polymer chains. Since balances for these polymer populations have been developed already, the final expressions for the method of moments are just a few steps away. 

As the reaction proceeds to a fractional conversion of p for either functional group A or B, the reaction mixture contains polymer molecules with a variety of chain lengths. Thus, the modeling objective here is to quantify the development of polymer chain length distribution (CLD) at different end-group conversions. 

One peculiarity of polymerisation reactions is the fact that the product consists of molecules which differ in chain length. Product properties are strongly related to the chain length distribution (CLD) of a polymer. An explicit consideration of all possible chain lengths would be both elaborate and mathematically redundant. Hence, there exist various approaches for reducing the complexity of the reaction system. The most popular approach is the method of moments [35], in which the... 

It is also useful to define the moments of the chain length distribution (CLD). The jth moment is defined in eqn [40] ... 

The methods used for evaluating transfer coeffidents are the same as for conventional chain transfer and involve evaluating the molecular weight (as in Mayo method ) or the chain length distribution (CLD) (as in the Log CLD method ) in low-conversion polymerizations or relative rate of consumption of transfer agent and monomer. Transfer coeffidents for addition-fragmentation chain transfer agents from the literature are summarized in the sections that follow. 

The Cs can also be measured by chain length distribution (CLD), also described as molecular weight distribution (MWD), which can be obtained directly by The... 

It has been shown that equivalent information can be obtained by analysis of logfnumbcr chain length distribution) plots (the log CLD method)."4" 7 For the case where termination is wholly by disproportionation or chain transfer, it is possible to show that (eq. 6) applies ... 

While it is, in principle, desirable to take the limiting slope of the log CLD plot, in practice the limiting slopes are very susceptible to experimental noise and baseline choice issues. Moad and Moad24 have shown that very little error is introduced by systematically taking the slope over the top 10% or the top 20% of the chain length distribution. The values for the slopes will overestimate ln(0). However, because the discrepancy is systematic, the Mayo" analysis still provides a good estimate for C ( 6% error for the example in Figure 6.1). 

Principle of Laplace transformations (or probability generating function (pgf) method) to reconstruct the CLD based on the moment equations, k, vector with rate coefficients [A], vector with concentrations of nonmacromolecules At, time step for integration CLD, chain length distribution ... 

Polyethylene microstructure is defined by its distributions of chain length (CLD) or molecular weight (MWD), chemical composition (CCD), comonomer sequence length (CSLD), and LCB. In addition, polypropylene microstructure is further characterized by its distribution of regio- and stereoregularity [53, 54]. 

In the simplest case, the mass CLD of a heterogeneous polymerization, which is defined as the mass fraction of polymer molecules with a given chain length i, is represented by a weighted sum over different Flory distributions (Flory, 1953), each corresponding to one type of catalyst site ... 

We will illustrate this procedure on a simple linear step-polymerization (/ = 2) to obtain the CLD and on single metallocene ethylene polymerization to find the 2D distribution of chain lengths and numbers of branch points. 

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