Instantaneous Average Chain Lengths

If the growing chains terminate exclusively by disproportionation, they undergo no change in length in the process but if combination is the exclusive mode of termination, the growing chains, on the average, double in length upon termination. Therefore, 

The average chain length may be expressed more generally in terms of a quantity the average number of dead chains produced per termination reaction, which equals the rate of dead chain formation/rate of termination reactions. Since each disproportionation reaction produces two dead chains and each combination reaction one. 

Rate of dead chain formation = (2ktj + fc,c) P V Rate of termination reactions = [k,c -I- fe,rf)[F ]  

The instantaneous number-average chain length is the rate of addition of 

When k,i k,c, = 2 and when k,e kt, = 1, duplicating the previous result, but (10.27) can also handle various degrees of mixed termination. 

FIGURE 9.2 Conversion versus time for an isothermal, iiee-radical batch polymerization (data of Example 9.2) Equation 9.15, [1] = constant Equation 9.19, s— 0.136 Equation 9.20, = 0. 

As in step-growth polymerization, a distribution of chain lengths is always obtained in a free-radical addition polymerization because of the inherently random nature of the termination reaction with regard to chain length. Expressions for the number-average 

FREE-RADICAL ADDITION (CHAIN-GROWTH) POLYMERIZATION 

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In the presence of CTAs, chain transfer to CTA is the main chain-termination event and the instantaneous MWD has a number-average chain length given by ... 

When the on-line measurement of the MWD or average molecular weight is not possible or not feasible, state estimationbecomes an alternative. The simplest state estimation scheme is the one that can be applied for linear polymers when CTAs are employed to control the MWD of the polymer. In these systems, the ratio of unreacted monomer to CTA provides an estimate of the instantaneous number chain length, DP = Rp/Rt = fcp[M]/fc [CTA]. Therefore, if the ratio can be measured on-line by one of the techniques described in the previous section, a good estimate of DP can be achieved. This feature has been used to monitor and control the MWD of linear homopolymer and copolymers produced in emulsion polymerization systems [104—107]. 

The calculation of the condition to produce a latex with a given MWD is based on the fact that for linear polymers produced by free-radical polymerization, the polymer chains do not suffer any modification once they are formed. This opens the possibility of decomposing the desired final MWD in a series of instantaneous MWDs to be produced at different stages of the reaction [130]. When chain transfer to a CTA is the main termination event, each of those MWDs can be characterized by the number-average chain length, according to Eq. (76). 

Therefore the problem reduces to calculating the sequence of the values of X that provide the desired final MWD. In order to calculate the X values that should be produced at each value of Xj, the final MWD is discretized as in Eq. (77), where Xj-f is the final overall conversion X , is the instantaneous number-average chain length produced in the conversion increment j, Wj n) is the instantaneous MWD... 

In Eq. (15), w r) is the weight distribution of chains of length r and r is the number-average chain length of the polymer population. Therefore, from a polymer microstructure point of view, a coordination catalyst is considered to have only one site type if it produces polymer with a chain length distribution that follows Eq. (15) instantaneously. 

For polymers made with multiple-site catalysts, the instantaneous number- and weight-average chain lengths are given by Eqs. (25) and (26). 

The molar mass can be addressed using the kinetic chain length concept for free radical polymerization the instantaneous number average chain length, is given in the QSSA for the aforementioned conditions... 

The number- and weight-average chain lengths, as well as the polydispersity, of the instantaneous polymer population can be derived from eqn [61]... 

The instantaneous number-average chain length is the rate of addition of monomer units to all chains (r ) over the rate of dead chain formation ... 

FIGURE 9.5 Instantaneous and cumulative number-average chain lengths versus conversion for an isothermal, free-radical, batch polymerization (data of Example 9.10). 

RELATIONS BETWEEN INSTANTANEOUS AND CUMULATIVE AVERAGE CHAIN LENGTHS... 

Given any instantaneous or cumulative average chain length as a function of conversion in a batch reactor, the others may be calculated as follows. Recall that conversion is defined as... 

Here, GH is the catalyst and G is the gegen ion. The ionization step is assumed to be essentially instantaneous, and always at equilibrium, with K being the equilibrium constant. Obtain expressions for the rate of polymerization and the number-average chain length according to this mechanism. 

The quantity Fi, the instantaneous copolymer composition, is analogous to x , the instantaneous number-average chain length in free-radical addition polymerization. Like x , it depends on the conditions in the reactor at a particular instant. It, too, is really an average, since not all the copolymer formed at a particular instant has exactly the same composition. However, the instantaneous distribution of compositions is normally much narrower than the instantaneous distribution of chain lengths, and because the fact that it is an average is not normally of great practical importance and cannot be controlled anyhow, the overbar is left off of Fj. 

In addition to the instantaneous copolymer composition another quantity of interest is (F/), the cumulative composition of the copolymer that has been formed up to a particular conversion. Fj) is exactly analogous to (x ), the cumulative number-average chain length from Chapter 9. For a batch reactor, it is obtained through a material balance ... 

Relations Between Instantaneous and Cumulative Average Chain Lengths 161... 

Many conformations were sampled by the usual MC procedure. The result is of course that there is no preferred orientation of the molecule. Each conformation can, however, be characterised by an instantaneous main axis this is the average direction of the chain. Then this axis is defined as a director . This director is used to subsequently determine the orientational order parameter along the chain. The order is obviously low at the chain ends, and relatively high in the middle of the chain. It was found that the order profile going from the centre of the molecules towards the tails fell off very similarly to corresponding chains (with half the chain length) in the bilayer membrane. As an example, we reproduce here the results for saturated acyl chains, in Figure 10. The conclusion is that the order of the chains found for acyl tails in the bilayer is dominated by intramolecular interactions. The intermolecular interactions due to the presence of other chains that are densely packed around such a chain,... 

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