Long-chain branch formation

The weight distribution of chain length for polymer populations containing i LCBs per chain, w(r, i) is given by the following equation [45]  

In Equation 2.115, the parameter r has a shghtly different definition from that used 

In the absence of LCB formation, r = t = 1/DP , and Equation 2.115 with i = 0 becomes Flory s most probable distribution for linear chains. 

An analytical solution for the instantaneous CLD for the whole polymer produced in a CSTR has also been derived [47]  

The function Ii is the modified Bessel function of the first kind of order 1, defined as  

The polymerization mechanism described by Eqs. (1)-(14) for homopolymerization needs to be augmented by only one additional equation, Eq. (29), to include long-chain branch formation. 

the subscripts i and j indicate the number of LCBs per chain, while the subscripts r and s are their respective chain lengths. Therefore, when a linear growing chain i = 0) reacts with a linear macromonomer (j = 0), a chain with one LCB is formed (i -I- j -I-1 = 1). Similar equations are easily developed for copolymerization. 

An analytical solution for the instantaneous chain length distribution for the whole polymer produced in a CSTR is also available [Eq. (32)] [15]. The function Ii is the modified Bessel function of the first kind of order 1, given by Eq. (33). Bessel functions are easily found in mathematical tables and are readily available is most scientific software applications [43, 44]. 

The parameter a is defined by Eq. (34), where /= is the molar fraction of macromonomer in the reactor, measured with respect to the total concentration of polymer, s is the reciprocal of the average reactor residence time kicB is the rate constant for LCB formation and Y is the concentration of growing polymer chains in the reactor. 

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Copolymerization of macromonomers formed by backbiting and fragmentation is a second mechanism for long chain branch formation during acrylate polymerization (Section 4.4.3.3). The extents of long and short chain branching in acrylate polymers in emulsion polymerization as a function of conditions have been quantified.20 ... 

Fig. 6. Long-chain branch formation through reincorporation of vinyl end groups.

In treatments of polymerization reactions that concentrated on a single feature, the effect of molecular weight upon the termination rate constant has been deduced, the relative rates of initiation of two monomers in a copolymerization have been assessed, constants for chain transfer to monomer have been obtained in an emulsion copolymerization, the relative amounts of chain termination by combination and disproportionation have been discovered from a molecular weight distribution, and the rate constant for long-chain branch formation in the free-radical polymerization of ethylene has been found by fitting a probalistic model. ... 

In addition to the copolymerization ability, it is of special interest to understand the structure of end groups when defining a catalyst s potential to promote long-chain branch formation or to facilitate end-group functionalization. Moreover, analysis of... 

Scheme 2 Long-chain branch formation through incorporation of macromonomer in ethylene polymerization. The catalyst first produces another vinyl-terminated polyethylene chain (macromonomer) (1) and then copolymerizes it into another growing chain (2)...

To summarize long-chain branch formation with metallocenes, as discussed above ... 

Scheme 4.9. Long-chain branch formation by chain transfer to polymer.

No bimolecular termination reactions - termination by combination or disproportionation - as observed in free-radical polymerization take place with coordination catalysts. Some catalysts, under certain polymerization conditions, may polymerize dead polymer chains containing terminal vinyl unsaturations, leading to the formation of chains with long-chain branches. We will discuss the mechanism of long-chain branch formation with coordination catalysts in Section 8.3.4. 

Fig. 8.27. Mechanism of long-chain branch formation with coordination catalysts.

A suitable cocatalyst is tris(pentaiiuorophenyl) borane. There is no evidence in the literature that methylaluminoxane cocatalysts are suitable for the synthesis of polyolefins containing long chain branches. It can be speculated that the presence of methylaluminoxane will promote transfer to aluminum and therefore produce dead polymer chains with saturated chain-ends which are unavailable for long chain branch formation. 

Although there is not much information available in the literature concerning optimal solvents for long chain branch formation, it appears that paraffinic solvents are preferable to aromatics. The use of aliphatic solvents for long chain branch synthesis has been recommended. This choice has also been supported for the catalytic system bis(cyclo-pentadienyl) zirconium dimethyl/perfluorotriphenyl boron. It was found that this catalyst system gave... 

There are at least two separate phenomena which contribute to the extrusion problem of compounds containing a high amount of silica one is poor dispersion or a poor wettability of silica with NR. The other is a gel or long-chain-branch formation involving the coupling agent. 

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