Random polycondensation

In a seminal and seemingly forgotten paper, Burchard et al. " discussed the analysis of various polymer architectures based on integrated light scattering (LS) and quasielastic light scattering (QELS). They considered mono- and polydisperse linear and star-branched polymers with/number of arms ( rays ), and random polycondensates of Af or ABC type (identical or different... 

The chemical constraint reduces the number of possible reactions considerably, and consequently it leads to a much narrower molar mass distribution. Furthermore, the extent of reaction a of the A-group can cover all values from zero to unity, but the extent of reaction P of the equally reactive 5-groups cannot become larger than P=a/(f-l). One important consequence of this strict constraint is that gelation can never occur [1,13]. A much higher branching density than by random polycondensation can be achieved. For this reason one nowadays speaks of hyperbranching. 

Random crosslinking is equivalent to random polycondensation of RAt with jR A2 monomers, if the distribution of primary chains is the so-called "most probable (55). 

In the following, we will briefly outline the use of the link p.g.f. (l.p.g.f.) for the calculation of the gel point in /-functional polycondensation without and with cyclization including the f.s.s.e. In Chapter II, section 2.2 we will consider an application in connection with the number of elastically active network chains in random polycondensates or in a collection of randomly crosslinked chains. 

Theory of ring chain equilibria in branched non-random polycondensation systems, with application to POCls/P,Os. Proc. Roy. Soc. (London) A 292, 380 (1966). 

The number of units in the n-th generation can now in principle be found in a similar manner as outlined for the random polycondensates with equal reactive groups. We first notice that the result depends on how many of the units in the n-th generation are linked with their A, or B or C groups to a unit in the preceding generation. Therefore, it will be useful to introduce a population vector N (n)... 

Comparing Eq. (C.42) with Eq. (C.28), we recognize a remarkable formal equivalence106 The only difference consists of the fact that the scalar transition probability a (f - 1) for the random polycondensates of equal reactivity has to be replaced by the transition matrix P, and the population number in the 1-st generation has to be replaced by a vector (N(l)) which contains the population of the root linked units in the first generation. Furthermore, the general Eq. (C.42) reduces exactly to the case of the random trifunctional polycondensation when all link probabilities are the same107 Or in... 

In other words, the random polycondensates have a critical exponent of yc = 1 while the ABC type under restriction has an exponent of yc = 2. Percolation theory yields Yc — 1.843,44). It has often been stated that the FS theory is characterized by a mean-field critical exponent of yc = 1, but now we see that the FS theory is more flexible and shows critical exponents between 1 and 2. 

In other words, each generation is weighted after differentiation with a function that depends on the path length. One easily verifies Eq. (C.84) for the random polycondensates when a - Gaussian statistics) is assumed. 

Fig. 21. Zimm-plot of an A/B2 random polycondensate. A3 benzene 1.3.5.-triacetic acid B2 decamethylene glycol. Measurement have been made in benzene at 20 °C, with the two wave lengths of Xa = 546 nm ( ) and A0 = 436 nm (O). h = (4 jiIX) sin 0/2138)...

It is of interest to compare the average number of units in the n-th generation (N (n))x for these fractions of isomers with that of the polydisperse random polycondensate. This number was first calculated by Kurata and Fukatsu158) and can again be obtained by Lagrange expansion83) to give... 

Fig. 22. Average number of units in the n-th generation for a three-functional random polycondensate, A3, the three-functional restricted polycondensate, AB2, and a monodisperse fraction from A3. X = 1000 in all cases...

The exponents are also given in Table 3. In most cases one obtains 2 v = 1 (no excluded volume effect is taken into account). Only in the two cases of the ABC polycondensates with the stringent constraint aB + etc = a a, and for the fractions of the random polycondensates, is the much lower exponent of 2v = 1/2 obtained90,104 (see Fig. 31). 

No restriction is made to the same molecular weight distribution. Instead of this, the natural distributions for f > 2 and f = 2 are taken. For star-molecules, f = 2 corresponds to the monodisperse linear chain or to a linear chain that obeys the most probable distribution, and in the case of random polycondensates, f > 2 corresponds to the branched non-fractionated sample, and f = 2 to the linear polycondensate. The g and h-factors so defined no longer have the appearance of shrinking factors in all cases, as may be recognized from Figs. 43 and 44. For star-molecules, both factors decrease as... 

Fig. 43. Dependence of the g-factor, defined by Eq. (D.34), on the functionality f for random polycondensates, and on the ray number f for star-molecules102 ...

The principles of the calculation by means of the cascade theory is sketched in Fig. 58 and compared with the random polycondensation. Instead of selecting a single monomeric unit as root of a tree, a whole primary chain is placed on the zero-th generation, and the same is done for all the other primary chains from the cross-linked polymer. 

The formalism of construction a path-weight generating function appears at first sight the same as for the random polycondensates, with the essential difference, however, that s and U in the equations for a polycondensate with functionality y for the monomers... 

Example Weight distribution of linear random polycondensate... 

Gordon, M., and Scantlebury, G. R., Non-random polycondensation, statistical theory of the substitution effect. Trans. Faraday Soc. 60, 604-621 (1964). 

The bottleneck in the origin of life is the formation of the functional biopolymers— enzymes and nucleic acids. The answer cannot be the random polycondensation from a chaotic mixture of the monomers, as this process would afford an astronomic number of different chains—ca. 10 for chains with a polymerization degree of 60. Given that, the probability that the same chain is produced more than once by a random polymerization process is in first approximation equal to zero the single active individual macromolecule, even if formed, would decompose before it could be made again by another chance event. How then can active macromolecules be formed ... 

M. Gordon and G. R. Scantlebury, Trans. Faraday Soc., 60, 605, (1964). Theory of Ring-chain Equilibrium in Branched Non-Random Polycondensation Systems. 

In the traditional gelation theory formulated by Flory (1) and Stockmayer (2), it is assumed that like functional groups are equally reactive and all reactions occur intenaolecularly before the gel point. Subsequently, beyond gelation, finite species formed in the sol portion are limited to acyclic trees. This is not correct because intramolecular reaction leading to the formation of ring structures in a random polycondensation must occur. The effect of cyclization has been treated by various approaches, e.g. JacObson-Stockmayer ring-chain factors (3), cascade theory (4), and rate theory... 

Here p can be calculated for various model polymers, and it depends on molecular structure as well as molecular weight distribution ( ). Thus p provides a useful information concerning molecular structure and polydispersity, when the structure is known. For example, polydispersity causes an increase and branching a decrease of p. Excluded volume increases this value. Further examples are discussed in detail in ( ). Here we mention in particular the special case of the f-functional random polycondensates where according to theory the decrease of p is exactly balanced by the increase as the result of the very pronounced polydispersity the p-parameter remains constant in the whole pre-gel region up to the gelpoint. 

The dominant contribution of simultaneous chain growth from all the chain ends, controlled by the deprotonation level of the initiator, leads to relatively narrow MWD B 1.1-1.4) while maintaining branching levels typical of random polycondensation reactions (degree of branching a 0.5, Eq. 30.7) [70]. The SCROP technique has also been used to synthesize hyperbranched polyglycerols [70], polyethers [71], and polyamines [72]. 

The numerical results obtained via an MD/percolation simulation are also shown to be in excellent agreement with those obtained via the corresponding gelation simulation of the random polycondensation of tetrafunctional units with intramolecular reaction allowed [10]. 

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