Chain-Length Distribution Functions of Polymers after Random Degradation and Cross-Linking, with Particular Reference to Elastomers

1954 ◽  
Vol 27 (3) ◽  
pp. 629-633
Author(s):  
W. F. Watson
Keyword(s):  
Chain Length ◽  
Statistical Treatment ◽  
Length Distribution ◽  
Distribution Functions ◽  
Average Chain Length ◽  
Cross Linking ◽  
Chain Length Distribution ◽  
Polymer Formation ◽  
Chain Lengths

Abstract The distribution of chain lengths of polymers on formation, random degradation and random cross-linking, have been derived by a simple statistical treatment. Chain-length distribution functions for all cases are represented by special forms of the expression : Nx/N=(α+β+γ)exp[−(α+β+γ)x] where β is the reciprocal of the average chain length on polymer formation, α is the degree of random degradation, and γ is the degree of cross-linking.

Chain-Length Distribution Functions during Polymerization

1954 ◽  
Vol 27 (3) ◽  
pp. 622-628 ◽  
Author(s):  
W. F. Watson
Keyword(s):  
Reaction Mechanism ◽  
Chain Length ◽  
Length Distribution ◽  
Kinetic Scheme ◽  
Distribution Functions ◽  
Average Chain Length ◽  
Chain Length Distribution ◽  
Molecular Weights ◽  
Measurable Rate ◽  
Chain Lengths

Abstract Functions for the distribution of chain lengths of a polymer formed during polymerization have been evaluated in terms of the directly measurable rate and rate of initiation, or the single equivalent measurement of number-average chain length. No details of the reaction mechanism are required, except for the occurrence of termination by combination of polymer radicals. This is in contrast to the usual derivation of distribution functions from the postulated kinetic scheme. The three types of termination are considered, (1) combination absent, (2) combination predominant, and (3) a mixture of combination with other modes of termination. The application to copolymerization is outlined. Relationships between the various average molecular weights are considered.

Networks Having Multimodal Chain-Length Distributions

1997 ◽  
Author(s):  
Burak Erman ◽  
James E. Mark
Keyword(s):  
Free Radical ◽  
Chain Length ◽  
Length Distribution ◽  
Bimodal Distribution ◽  
High Energy ◽  
Chain Length Distribution ◽  
Network Chain ◽  
Ultimate Properties ◽  
Non Gaussian ◽  
Chain Lengths

As was mentioned in chapter 10, end-linking reactions can be used to make networks of known structures, including those having unusual chain-length distributions. One of the uses of networks having a bimodal distribution is to clarify the dependence of ultimate properties on non-Gaussian effects arising from limited-chain extensibility, as was already pointed out. The following chapter provides more detail on this application, and others. In fact, the effect of network chain-length distribution, is one aspect of rubberlike elasticity that has not been studied very much until recently, because of two primary reasons. On the experimental side, the cross-linking techniques traditionally used to prepare the network structures required for rubberlike elasticity have been random, uncontrolled processes, as was mentioned in chapter 10. Examples are vulcanization (addition of sulfur), peroxide thermolysis (free-radical couplings), and high-energy radiation (free-radical and ionic reactions). All of these techniques are random in the sense that the number of cross-links thus introduced is not known directly, and two units close together in space are joined irrespective of their locations along the chain trajectories. The resulting network chain-length distribution is unimodal and probably very broad. On the theoretical side, it has turned out to be convenient, and even necessary, to assume a distribution of chain lengths that is not only unimodal, but monodisperse! There are a number of reasons for developing techniques to determine or, even better, control network chain-length distributions. One is to check the “weakest link” theory for elastomer rupture, which states that a typical elastomeric network consists of chains with a broad distribution of lengths, and that the shortest of these chains are the “culprits” in causing rupture. This is attributed to the very limited extensibility associated with their shortness that is thought to cause them to break at relatively small deformations and then act as rupture nuclei. Another reason is to determine whether control of chain-length distribution can be used to maximize the ultimate properties of an elastomer. As was described in chapter 10, a variety of model networks can be prepared using the new synthetic techniques that closely control the placements of crosslinks in a network structure.

Network Chain Distribution and Strength of Vulcanizates

1969 ◽  
Vol 42 (3) ◽  
pp. 659-665 ◽  
Author(s):  
S. D. Gehman
Keyword(s):  
Physical Properties ◽  
Chain Length ◽  
Random Distribution ◽  
Length Distribution ◽  
Chain Scission ◽  
Point Of View ◽  
Average Chain Length ◽  
Chain Length Distribution ◽  
Network Chain ◽  
Chain Molecules

Abstract Physical characteristics of rubber network structures usually enumerated and discussed are network chain density, crosslink functionality, average chain length between crosslinks, entanglements which act somewhat like crosslinks, and free chain ends which are network defects. Chemical factors include structure of the chain molecules, type of crosslinks, whether monosulfide, disulfide or polysulfide, or direct carbon-to-carbon bonds. Side effects of vulcanization reactions such as chain scission or combination of minor quantities of chemical fragments from the vulcanizing system are also recognized. One might think that these variables would be adequate to account for physical properties of elastomers but explanations of strength aspects of vulcanizates are still unsatisfactory. Something is missing in these considerations, that is, the distribution of crosslinks along a main chain or the length sequences of monomer units in network chains. Usually a random distribution is implicitly assumed. If the distribution is always random and nothing can be done about it and it cannot be measured anyway, there may seem to be little point in writing about it. However, an ideally random distribution for all crosslinking systems and polymers seems very improbable. The importance of network chain length distribution for physical properties has been, of course, well recognized in theory. Bueche's calculations showed that viscoelastic resistance to deformation increased markedly with increased crosslink functionality, that is, as more chains are involved in the displacement of a crosslink. His molecular theory of tensile strength was based on the concept of short, overloaded network chains which snapped and transferred their loads to neighboring chains. An alternate point of view is that short chains are detrimental because they do not stress orient as well as longer chains.

Chain length distribution and kinetic characteristics of an enzymatically produced polymer

e-Polymers ◽  
2013 ◽  
Vol 13 (1) ◽  
Author(s):  
K.J.M. Mulders ◽  
H.H. Beeftink
Keyword(s):  
Chain Length ◽  
Rate Constant ◽  
Rate Constants ◽  
Reaction Order ◽  
Length Distribution ◽  
Monomer Concentration ◽  
Order Rate Constant ◽  
Reaction Rate Constant ◽  
Chain Length Distribution ◽  
Chain Lengths

Abstract Non-processive enzymatic polymerization leads to a distribution of polymer chain lengths. A polymerization model was developed to investigate the relation between the extent of this distribution on one hand, and the polymerization start conditions and reaction kinetics on the other hand. The model describes changes in concentration of chains of length n as the result of two elongation reactions: elongation by monomer addition to length n-1 and elongation by monomer addition to length n. Polymerization reactions were assumed to be zero order in monomer concentration and to obey Michaelis-Menten kinetics with respect to polymer concentrations. In addition, the amount of enzyme available for each individual reaction (n → n+1) is assumed to be proportional to the concentration of polymer substrate of length n. The development of the shape of the chain length distribution was found to be independent of the value of the overall reaction rate constant; only the rate at which these shapes developed was influenced by the 1st-order rate constant. The value of the Michaelis parameter did affect the form of the chain length distribution curve since it affects the reaction order. An increase in reaction order was found to promote widening of the chain length distribution. Differences in kinetic parameters between the subsequent polymerization reactions, if any, were also found to have a large effect on the development of the chain length distribution. An increase in rate constants with chain length entailed a wider distribution; a more narrow distribution would require a decrease in rate constants with chain length.

scholarly journals Going Beyond the Carothers, Flory and Stockmayer Equation by Including Cyclization Reactions and Mobility Constraints

Polymers ◽  
2021 ◽  
Vol 13 (15) ◽  
pp. 2410
Author(s):  
Lies De Keer ◽  
Paul H. M. Van Steenberge ◽  
Marie-Françoise Reyniers ◽  
Dagmar R. D’hooge
Keyword(s):  
Chain Length ◽  
Kinetic Monte Carlo ◽  
Polymer Network ◽  
Length Distribution ◽  
Average Chain Length ◽  
Network Synthesis ◽  
Dimensionless Numbers ◽  
Chain Length Distribution ◽  
Intramolecular Reactions ◽  
Joint Consideration

A challenge in the field of polymer network synthesis by a step-growth mechanism is the quantification of the relative importance of inter- vs. intramolecular reactions. Here we use a matrix-based kinetic Monte Carlo (kMC) framework to demonstrate that the variation of the chain length distribution and its averages (e.g., number average chain length xn), are largely affected by intramolecular reactions, as mostly ignored in theoretical studies. We showcase that a conventional approach based on equations derived by Carothers, Flory and Stockmayer, assuming constant reactivities and ignoring intramolecular reactions, is very approximate, and the use of asymptotic limits is biased. Intramolecular reactions stretch the functional group (FG) conversion range and reduce the average chain lengths. In the likely case of restricted mobilities due to diffusional limitations because of a viscosity increase during polymerization, a complex xn profile with possible plateau formation may arise. The joint consideration of stoichiometric and non-stoichiometric conditions allows the validation of hypotheses for both the intrinsic and apparent reactivities of inter- and intramolecular reactions. The kMC framework is also utilized for reverse engineering purposes, aiming at the identification of advanced (pseudo-)analytical equations, dimensionless numbers and mechanistic insights. We highlight that assuming average molecules by equally distributing A and B FGs is unsuited, and the number of AB intramolecular combinations is affected by the number of monomer units in the molecules, specifically at high FG conversions. In the absence of mobility constraints, dimensionless numbers can be considered to map the time variation of the fraction of intramolecular reactions, but still, a complex solution results, making a kMC approach overall most elegant.

Recent Advances in Controlled Grafting of Elastomers

1984 ◽  
Vol 57 (3) ◽  
pp. 557-582 ◽  
Author(s):  
Roderic P. Quirk
Keyword(s):  
Chain Length ◽  
Graft Copolymers ◽  
Length Distribution ◽  
Careful Examination ◽  
Average Chain Length ◽  
Chain Length Distribution ◽  
Structure Morphology ◽  
Graft Polymers ◽  
Exciting Time ◽  
Grafting From

Abstract This review describes recent results for preparing graft copolymers with controlled structures. In general, the macromonomer approach appears to be the most promising method for preparation of graft polymers with well defined structures. Not only can macromonomers be prepared using radical, cationic, or anionic polymerization procedures, but the resultant macromonomers can be polymerized using these same methods. Obviously, the macromonomer functionality must be high and well defined. In addition, the average chain length and chain length distribution should be well characterized. In principle, the macromonomer approach can provide comb-type graft copolymers with a random distribution of well characterized graft branches. However, the actual copolymerization of the macromonomers with a variety of comonomers requires careful examination under a variety of reaction conditions before that expectation can be realized. These macromonomers provide an excellent opportunity to examine the effects of chain length on the problem of heterogeneity (i.e., phase-separation) which is expected and found in many grafting systems. “Grafting-from” reactions also should provide graft polymers whose structures are more amenable to prediction and analysis. Like the macromonomer method, it should be possible to eliminate the presence of unwanted homopolymer using this method. However, much more research is required before the generality and value of this method can be evaluated. In conclusion, this era appears to be an exciting time for graft polymerization research. The potential exists for the preparation of model graft polymers with precise structural definition using the methods described herein. This development will, at last, provide graft polymers which can be used to define the relationships between the structure, morphology, and properties of these materials.

Direct Observation of Chain Lengths and Conformations in Oligofluorene Distributions from Controlled Polymerization by Double Electron-Electron Resonance

2019 ◽  
Author(s):  
Dennis Bücker ◽  
Annika Sickinger ◽  
Julian D. Ruiz Perez ◽  
Manuel Oestringer ◽  
Stefan Mecking ◽  
...  
Keyword(s):  
Chain Length ◽  
Length Distribution ◽  
Catalyst System ◽  
Chain Conformation ◽  
Controlled Polymerization ◽  
Chain Length Distribution ◽  
Electron Resonance ◽  
Double Electron ◽  
The Individual ◽  
Double Electron Electron Resonance

Synthetic polymers are mixtures of different length chains, and their chain length and chain conformation is often experimentally characterized by ensemble averages. We demonstrate that Double-Electron-Electron-Resonance (DEER) spectroscopy can reveal the chain length distribution, and chain conformation and flexibility of the individual n-mers in oligo-(9,9-dioctylfluorene) from controlled Suzuki-Miyaura Coupling Polymerization (cSMCP). The required spin-labeled chain ends were introduced efficiently via a TEMPO-substituted initiator and chain terminating agent, respectively, with an in situ catalyst system. Individual precise chain length oligomers as reference materials were obtained by a stepwise approach. Chain length distribution, chain conformation and flexibility can also be accessed within poly(fluorene) nanoparticles.

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