Ideal Branched Polymers

In Section 3.4 it was explained that polymers having very well defined structures can be prepared by means of anionic polymerization, and this technique has been widely used to prepare samples for rheological study. This has been a particularly fruitful approach to the study of the elfects of various types of long-chain branching structure on rheological behavior. Linear viscoelastic properties are very sensitive to branching. In this section we review what is known about the zero-shear viscosity, steady-state compliance, and storange and loss moduli of model branched polymers. 

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Radius of gyration of an ideal branched polymer (Kramers theorem)... 

The radius of gyration of ideal branched polymers can be calculated using the Kramers theorem [Eq. (2.65)]. 

Recently, McLeish and Larson [95] developed a nonlinear viscoelastic theory for an idealized branched polymer with multiple branches but only two branch points. This molecular structure, called the pom-pom (described in Section 10.9.2), is a generalization of the H polymer in that each of the two branch points of the pom-pom is permitted to have an arbitrary number of branches, q see Fig. 9.4. The pom-pom model contains three basic time constants the backbone reptation time T, the backbone stretch time T, and the arm relaxation time x,. These time constants are given in terms of the molecular parameters of the pom-pom molecule as ... 

Indeed, cumyl carbocations are known to be effective initiators of IB polymerization, while the p-substituted benzyl cation is expected to react effectively with IB (p-methylstyrene and IB form a nearly ideal copolymerization system ). Severe disparity between the reactivities of the vinyl and cumyl ether groups of the inimer would result in either linear polymers or branched polymers with much lower MW than predicted for an in/mcr-mediated living polymerization. Styrene was subsequently blocked from the tert-chloride chain ends of high-MW DIB, activated by excess TiCU (Scheme 7.2). 

Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

A majority of the hyperbranched polymers reported in the literature are synthesized via the one-pot condensation reactions of A B monomers. Such one-step polycondensations result in highly branched polymers even though they are not as idealized as the generation-wise constructed dendrimers. The often very tedious synthetic procedures for dendrimers not only result in expensive polymers but also limit their availability. Hyperbranched polymers, on the other hand, are often easy to synthesize on a large scale and often at a reasonable cost, which makes them very interesting for large-scale industrial applications. 

In ideal random crosslinking polymerization or crosslinking of existing chains, the reactivity of a group is not influenced by the state of other groups all free functionalities, whether attached or unattached to the tree, are assumed to be of the same reactivity. For example, the molecular weight distribution in a branched polymer does not depend on the ratio of rate constants for formation and scission of bonds, but only on the extent of reaction. Combinatorial statistics can be applied in this case, but use of the p.g.f. simplifies the mathematics considerably. 

Whereas the well-characterized, perfect (or nearly so) structures of dendritic macromolecules, constructed in discrete stepwise procedures have been described in the preceding chapters, this Chapter reports on the related, less than perfect, hyperbranched polymers, which are synthesized by means of a direct, one-step polycondensation of A B monomers, where x > 2. Flory s prediction and subsequent demonstration 1,2 that A B monomers generate highly branched polymers heralded advances in the creation of idealized dendritic systems thus the desire for simpler, and in most cases more economical, (one-step) procedures to the hyperbranched relatives became more attractive. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

An example of a branched polymer used as a synthetic elastomer is poly(4-methyl-1-pentene), CAS 25068-26-2 [134]. The idealized structure of poly(4-methyl-1-pentene) and the formulas of a few molecular fragments found in the pyrolysate of this polymer are shown below ... 

An ideal randomly branched polymer is a fractal object with fractal dimension V — 4. In Chapter 6, we will learn how this polymer can fit into three-dimensional space. What is the ratio of molar masses M1/M2 of two ideal randomly branched polymers if the ratio of their sizes is R1/R2 = 3 ... 

Substituting this average [Eq. (2.66)] into the Kramers theorem [Eq. (2.65)] recovers the classical result for the radius of gyration of an ideal linear chain [Eq. (2.54)]. In Section 6.4.6, we apply the Kramers theorem [Eq. (2.65)] to ideal randomly branched polymers. In this case the average is not only over different ways of dividing a molecule into two parts, but also over different branched molecules with the same degree of polymerization N. 

Consider a randomly branched polymer in a dilute solution. Let us assume that the radius of gyration for this polymer in an ideal state (in the absence of... 

Consider a monodisperse melt of randomly branched polymers with N Kuhn monomers of length b. Randomly branched polymers in an ideal state (in the absence of excluded volume interactions) have fractal dimension D = 4. Do these randomly branched polymers overlap in a three-dimensional monodisperse melt ... 

Here we calculate the size of ideal randomly branched polymers, ignoring excluded volume interactions and allowing each molgcule to achieve the state of maximum entropy (recall the discussion of ideal chains in Chapter 2). Since branched molecules have many ends, the mean-square end-to-end distance used to characterize the size of linear chains is not appropriate for them. The simplest quantity describing the size of branched molecules is their mean-square radius of gyration j g [see Eq. (2.44) fonthe definition]. 

As expected, the radius of gyration of an ideal randomly branched polymer is much smaller than that of an ideal linear chain with the same number and size of monomers (see Fig. 6.25). It is important to note that the dependence of the size on the degree of polymerization for randomly branched polymers... 

The fractal dimension of an ideal randomly branched polymer is 27 = 4 (because its degree of polymerization is proportional to its size to the fourth power N 2 ). In spaces with dimension d

Calculate the size of an ideal randomly branched polymer with precursor chains made of Aq Kuhn monomers with Kuhn length b—5A. and total -------number A== 10" monomers. Estimate this size for... 

PAGES 226 227 last sentence on page 226 continuing onto 227 spaces), ideal randomly branched polymers become too dense as reflected in the overlap parameter decreasing with degree of polymerization " increasing is replaced by "decreasing . 

The solution properties of copolymers are much more compHcated. This is due mainly to the fact that the two copolymer components A and B behave differently in different solvents, and only when the two components are soluble in the same solvent will they exhibit similar solution properties. This is the case, for example for a nonpolar copolymer in a nonpolar solvent. It should also be emphasised that the Flory-Huggins theory was developed for ideal Hnear polymers. Indeed, with branched polymers with a high monomer density (e.g. star-branched polymers), the 0-temperature will depend on the length of the arms, and is in general lower than that of a linear polymer with the same molecular weight. 

An important conclusion from this series of investigations is that the MWD of a branched polymer in a reactor with ideal mixing will be broader than in a batch reactor. The same is true for the degree of branching. 

Equation (11.18) has two peculiar features firstly, D depends appreciably on d and, secondly, there exists a critical dimension of Euclidean space d = % for which D = 4 in accordance with the ideal statistical model, i.e., the model withont correlations. At d > 8, the correlations cansed by the effect of excluded volume are no longer significant and Df does not change. The value d = % was found in studies of branched polymers and lattice animals [79]. Calcnlations using formula (11.18) are in good agreement with the known results for lattice animals. ... 

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