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.

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.

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.

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.

THE DEGRADATION OF POLYSTYRENE: II. THE SYNTHESIS AND DEGRADATION OF POLYSTYRENE POSSESSING THE KUHN–SCHULZ CHAIN LENGTH DISTRIBUTION

1950 ◽  
Vol 28b (7) ◽  
pp. 416-428
Keyword(s):  
Molecular Weight ◽  
Benzoyl Peroxide ◽  
Chain Length ◽  
Average Molecular Weight ◽  
Length Distribution ◽  
Chain Scission ◽  
Limited Range ◽  
Number Average Molecular Weight ◽  
Chain Length Distribution ◽  
Synthesis And Degradation

By polymerizing styrene in emulsion it was possible to synthesize polystyrenes of known number average molecular weight, the chain length distributions of which corresponded closely to the Kuhn–Schulz relation Ny = N0py−1(1 − p)2. This enabled a relation between intrinsic viscosity and number average molecular weight to be established for polystyrenes with chain length distributions of this functional form. Assuming this form of chain length distribution to remain unaltered on degradation, it was possible to estimate the average number of scission points per structural unit from viscosimetric measurements. The extent of thermal scission of polystyrene after one week at 144 °C. was shown to be negligible while benzoyl peroxide caused appreciable chain scission at 80 °C. and 100 °C. The number of scission points estimated from viscosimetric measurements was directly proportional to the mass of benzoyl peroxide added to the system, and the scission appeared to be essentially random over the limited range investigated.

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.

High-polymer solutions

1952 ◽  
Vol 212 (1110) ◽  
pp. 339-361 ◽  
Keyword(s):  
Molecular Weight ◽  
Statistical Method ◽  
Osmotic Pressure ◽  
Chain Length ◽  
Natural Extension ◽  
Length Distribution ◽  
Published Data ◽  
Chain Length Distribution ◽  
Chain Molecules ◽  
Random Flight

In describing the configurations of a polymer molecule in terms of the ‘equivalent chain’ of N elements, each of length l , it has been usual to simplify the problem by assuming the equivalent chain to have position only and zero volume. The weights of the various configurations of such a ‘random flight’ chain are different from those of a real chain in which there exists an interaction potential between any pair of chain elements. These differences are particularly important in the theory of solutions of chain molecules, since they are responsible for the deviation of the osmotic pressure from van’t Hoff’s law. In this paper the average dimensions of a chain with interactions are calculated by a statistical method. For < s 2 >, the average square distance of the elements from the centre of gravity, the result is < s 2 > = ( Nl 2 /6) [1 - 0⋅857( β 1 / l 3 ) N -½ ], (i) where β 1 is the ‘excluded volume’ integral for free chain elements. For large N this reduces to the well-known result for a random flight chain. Similar results are obtained for other average dimensions. The possibility of checking (i) from experimental determinations of <s 2 > for chain molecules using the light-scattering technique is examined, and it is shown that a very accurate knowledge of the chain-length distribution in the fractions used will be required if the influence of the second term in (i) is to be detected in this way. A natural extension of the statistical method is used to calculate the pair distribution function F 2 ( X 12 ) governing the probability of occurrence of the centres of gravity of two chains in equal volume elements separated by the distance X 12 . This function is needed to calculate the second coefficient A 2 in the osmotic pressure expansion π = RT [ M -1 c + A 2 c 2 +...]. Here M is the molecular weight of the solute and c the concentration. For random flight chains F 2 is unity for all values of X 12 ; A 2 is zero and the osmotic pressure follows van’t Hoff’s law. Values of F 2 different from unity, and hence finite values of A 2 are only obtained if there are interactions between chain elements. The first approximation to F 2 is F 2 ( X 12 ) = exp {(9/2 π ) 3/2 ( β 1 / l 3 ) N ½ exp (-9 X 2 12 /2 Nl 2 )}. The theory predicts a rather complicated dependence of A 2 on z , the degree of polymerization and the log-log plot of A 2 against z is curved. Over a limited molecular weight range A 2 may be approximated by a formula of the form A 2 = Cz - ε , (ii) where C is constant for a given polymer-solvent system, ε depends upon z and lies between — ∞ and ½. If A 2 is positive, ε goes from 0 to ½ as z goes from 0 to ∞ and A 2 decreases slowly with z . For systems in which A 2 is negative, ε goes from 0 to — ∞ as z goes from 0 to ∞ and | A 2 | increases extremely rapidly with z . There are complications if the solutions are not homogeneous with respect to chain length, but it is shown that, with well-fractionated samples, little difficulty should arise if z is replaced by the number average < z > n . The theory is illustrated by applying it to some recently published data on the systems: polystyrene-butanone, polystyrene-toluene, and poly iso butylene - cyclo hexane.

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.

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