Branching factors

Since our indirect method produces both the linear (b=0) and branched intrinsic viscosities across the chromatogram, it is possible to estimate several LCB parameters as a function of elution volume or number average molecular weight. The branching factor G(V) can be written as... 

In order to estimate the branching factor e for polyvinyl acetate we have analyzed the SEC data obtained on sample PVAc-E4 using the MWBD method with various e values. This sample was synthesized under kinetically controlled conditions (isothermal, T = 60°C, [AIBN] = 10"5 g-mole/1, conversion level of 48.5 percent). The SEC measurements were made at 25°C in tetrahydro-furan. The Mark-Houwink coefficients used for linear polyvinyl acetate are those suggested by Graessley (21), namely K = 5.1 x 10"5 dl/gm and a = 0.791. The whole polymer M, Mj, and B j values obtained are listed in Table II. 

Specifying a as the chain branching factor, then, the previous example was for the condition a = 1. If, however, pure chain branching occurs under exactly the same conditions, then a = 2 and every radical initiating the chain system creates two, which create four, and so on. Then 1019 molecules/cm3 are consumed in the following number of generations (N) ... 

In Chapter 3, the conditions for a chain branching explosion were developed on the basis of a steady-state analysis. It was shown that when the chain branching factor a at a given temperature and pressure was greater than some critical value acrit, the reacting system exploded. Obviously, in that development no induction period or critical chain ignition time rc evolved. 

The effect of branching is to increase the segment density within the molecular coil. Thus a branched molecule occupies a smaller volume and has a lower intrinsic viscosity than a similar linear molecule of the same molecular weight. The degree of branching is often characterized in terms of the branching factor [1] in Eq. (14), where the subscripts B and L, respectively, refer to branched and linear polymers of the same molecular weight ... 

The development of the reaction was followed by measuring pressure change (Ap), light emission (7), reaction rate (dAp/dt), and by chemical analysis. Pressure rise was recorded by a pressure transducer (A.C.B. 504H). Reaction rate (dAp/dt = W) was obtained by using a resistance-capacity circuit of suitable time constant, 6 = RC (76, 78), appropriate to the branching factor of the reaction, < . It was possible to record simultaneously pressure rise vs. time and rate vs. time or rate vs. pressure rise. 

Here P denotes the stable product and a is a branching factor. 

Set up and solve the time-dependent equation for the concentration of [F] and [R ], assuming a branching factor a of 1, 2, and 1.01, respectively. Compare the necessary reaction time for 99% conversion of F for the three cases. 

The growth scheme used in this work is a modification of growth scheme B from Anderson et al. (I) Addition of the carbon atom is permitted on the first two carbon atoms on one end of the growing chain and on substituents on the first three carbon atoms. Results are not very different if branching is allowed on the first three carbon atoms of a chain (8). The chain branching factor is /, as usual, except in the formation of quaternary carbons in which case 0.1 / is used because of the low probability of substitution on tertiary carbon atoms. 

Branching factors were also determined by the ratio of concentrated solution viscosities of branched and linear polymers having the same weight-average molecular weight denoted by g" ... 

Molecular data for the branched polybutadienes are given in Table III. The values for the average branching factor g were calculated from... 

Values for the average branching factors obtained from the concentrated solution viscosity g" were obtained similarly by using the relationship shown in Figure 2. The log-log plot of g and g" is given in Figure 3, and values for g" are given in Table IV. 

Since intrinsic viscosity but not concentrated solution viscosity is known to be sensitive to polydispersity (10, 19), a correction has been applied to g according to the calculations of Berger and Shultz (20). A value of zero was taken for b, exponent a was set equal to 0.75, and Qbi/Qlin was related to Mw/Mn using Equation 21 of Shultz (19). The branching factor corrected for polydispersity, g corr, is given by... 

Table III. Molecular Weight and Conventional Branching Factors for Polybutadienes...

Table IV. Branching Factors f" for Polybutadienes from 10% Solution Viscosity...

Likewise, the samples can be arranged in the order of decreasing coil size and increasing branching, as determined by g COrr and g", again using the catalyst systems to identify the samples. The most linear polymers are the reference butyllithium samples followed by the nickel-based polymer, butyllithium, alfin, cobalt based, titanium based, and emulsion. The correction to the branching factor for polydispersity makes the nickel-based and alfin polybutadienes less branched with respect to the other polymers examined. 

The variation in branching between the two sets of butyllithium polybutadienes is attributable to polymerization impurities such as 1-butyne and vinylacetylene as shown by Adams and his co-workers (23). This factor might be responsible for the small difference between Equations 5 and 6. The agreement of these branching factors with published data... 

Figure 3. Relations between branching factors g and g" (g from Equation 6)...

The sequence of reactions 2,3, and 4 has a chain branching factor of 2, but, the overall chain branching factor for 1, 2, 3, and 4 is 2(1- a) where a is the ratio of the rate constant for reaction 2 to the sum of the rate constants of reactions 1 and 2. Since reaction 1 is faster than 2, a is less than one. This means that for a self sustaining chain reaction to occur another reaction which provides additional chain branching is necessary. It is generally agreed that in the Thermal DeNOx process reaction 5 has that role... 

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