Birth conversion

Using Equation 5 and 6, it is possible to calculate the crosslinking density distribution as a function of the birth conversion (0). Figure 1 shows one of the calculation results. Though it is quite often assumed that the crosslinking density is the same for all polymer molecules, this assumption is not valid for free radical polymerization. Generally, this distribution becomes significant when the conditions deviate from the idealized Flory s conditions, namely, 1) the reactivities of all types of... 

Qualitatively, at a given conversion if the ratio E (9,4>) does not deviate much from unity for any birth conversion 0 (0<4>) /... 

Generalization of Flory s Theory for Vinyl/Divinyl Copolvmerization Using the Crosslinkinq Density Distribution. Flory s theory of network formation (1,11) consists of the consideration of the most probable combination of the chains, namely, it assumes an equilibrium system. For kinetically controlled systems such as free radical polymerization, modifications to Flory s theory are necessary in order for it to apply to a real system. Using the crosslinking density distribution as a function of the birth conversion of the primary molecule, it is possible to generalize Flory s theory for free radical polymerization. 

The accumulated sol fraction which is measured in experiments is given by the integration over all birth conversion. 

In a first step, the so-called birth conversion x is sampled at which an initial zeroth-order building block (label 0) is generated taking into account the maximum conversion, x ax, reached. In a second step, the chain length i of this block is obtained by random selection, typically following the Flory mass CLD (Tobita, 1993) ... 

In a fourth step, the birth conversions as well as the lengths have to be selected for these branches. To obey physical boundaries, the conversion interval [x, x axl has to be considered... 

Fig. 9.14. Monte Carlo sampling of primary polymers and their connectivity for radical polymerization with transfer to polymer in a batch reactor. Conversion y/, first pp sampled at birth conversion x = d.

The latter equality follows from the quasi-steady-state-assumption. Note that Pi, in a batch reactor is a function of conversion. If other transfer mechanisms are present, the denominator is extended with the corresponding contributions to the initiation process. Whether or not the pp is attached to another pp indeed follows by selecting a random number between 0 and 1 and determining whether the inequality rand(l) < Py is false or true. If connected [true) then the birth conversion of the earlier pp simply follows from the conditional distribution (given that the first sampled pp is created atx = d and grows from an earlier one) ... 

Ptdb (6) this determines the probability of the randomly chosen pp at x = 0 becoming connected to a later pp. We should consider what happens to pps with a TDB after their creation at x = 6. A fraction of them is incorporated, but as the rate at which this happens depends on their concentration, this rate will decrease. In fact, the fractional decrease of these pps exactly follows the decrease in the fraction of TDBs of pps created at x = 6. The TDB mole-fraction, FrM( >w), starts at Cm = km/kj, for all birth conversions, so also atx=9, while it decreases according to the balance describing the TDB consumption starting from 6 ... 

Whether or not it is connected follows by the checking of the inequality rand l) < Pb,TDB P> ) Now, the birth conversion u at which incorporation takes place has to be determined. This is realized by using the conditional probability distribution in CFa, tob(u 0), over the interval 0 to y/, namely girm that the pp has reacted... 

Sampling proceeds in the same manner as discussed in the cases with transfer to polymer [see Eq. (139)]. Connectivity in generation 0 can also occur, when the pp sampled first itself incorporates pp chains with a TDB during its growth 3i x = 9. Obviously, such chains should have been created at birth conversions z before 9 hence 0 < z < 9. The probability of receiving branch points in this way in fact equals the instantaneous branching density which is given by the ratio of... 

Recombination termination is implemented in the same way in the batch reactor and in the CSTR. First to note is that termination through recombination happens to two pps growing simultaneously, which implies that birth conversions or residence times are identical for the two. The algorithm starts with a check on which of the new pps created in a certain generation by some mechanism is connected to another one by recombination. This probability is obtained from the relative reaction rates for example, in the case of transfer to polymer only ... 

When recombination is at hand rand(l) < Ptc), determination of the birth conversion or residence time is performed in no other way, as before. The total length of the two pps connected can be found by sampling once from w and once from n , and addition. The correctness of this can be understood by realizing that the second pp can be connected to the first one only by one chain end. 

Fig. 9.18. Example of the construction of a linear pp chain undergoing scission and recombination in radical polymerization. The square on the LHS marks the (non-scission) initiation point of this chain at birth conversion 02.

Figure 9.18 shows the situation where Seg-0 has grown at a scission point on Seg-2 after scission of the latter at birth conversion 02 < 0o, to be selected as a random number between 0 and Go- The growth direction of Seg-2 has then to be selected. There is a probability of that it is in the direction indicated, to the LHS. Its length is determined by comparing lengths with Eqs. (139) and (171). In this case no sets-... 

S branching coefficient empirical exponent C parameter in free volume 0 birth conversion... 

Cydization processes can be complex as shown in the Scheme 7, 49o present conversion 0, the aoss-link density p(ft0) of primary chains of birth conversion 0 (0< ) is the sum of instantaneous aoss-link density Pi(0) and additional cross-link density p (0,0) ... 

Equations [259[-[265[ can be solved numerically for the aoss-link and cyclization density distributions p 0, ) and 0) at the present conversion 0 as a function of the birth conversion 6. The average cross-link density is... 

Figure 17 Cross-link density distribution of primary polymer chains with respect to their birth conversion. ...

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