Theory of Branching Processes and Statistics of Rubber Elasticity

Dobson, Gerard R.; Gordon, M.

doi:10.1063/1.1696794

Cited by 175 publications

(45 citation statements)

References 16 publications

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“…with y as cDP, (24) This formula generalizes and frees from approximations numerous results obtained by lengthy calculations of individual cases in a number of papers6' Three regions of this equation, and of equation (18). are important for studies of elasticity, swelling, etc.…”

Section: Scanlan-case Eancsmentioning

confidence: 99%

The structure and properties of molecular trees and networks

Gordon¹,

Ross‐Murphy²

1975

Pure and Applied Chemistry

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195

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The gel point is a well-understood critical point in polymer science (Flory, Stockmayer). A summary is given of the resolution of those aspects which may seem paradoxical at first sight. The relevant equations for the basic paradigm of f-functional polycondensation are very simple. Special attention is paid to the critically branched state of materials not far from the gel point. The quasiinvariance principle is explained according to which all solution properties appear ultimately to level off as the critical conversion is approached from below. The 'Malthusian' packing paradox is resolved by a proper treatment of the ring-chain competition situation, which also disposes of the spurious divergence of the rate of cyclization predicted by a more naive theory.Network theories not based on extinction probability (Charlesby) are not worth considering. The proper definition of an elastically active network chain (EANC) was based on this concept by Scanlan and by Case in 1960, and it greatly simplifies the graph-like-state theories of network structure (usually called 'network topology'). It allows classical rubber elasticity theory to be applied near the gel point. The point is characterised by a fifth-order Ehrenfest transition due to the contribution of long-range correlations to the configurational free energy. Though directly relevant data on reversible gelation are not available, data on isothermal crosslinking of very diverse systems support this analysis. Except possibly for some highly crosslinked systems, the parameter M (mean chain length between 'crosslinks') is an undesirable ingredient of elasticity or swelling theories. Chain-end corrections are quite undesirable (and usually done incorrectly). Everything is correctly and more simply formulated in terms of Scanlan-Case EANCs.The proper understanding of the gel point as a critical point allows the construction of reduced plots, illustrated with temperature superposition of experimental modulus-conversion plots (by M. Judd) for aqueous gelatin jellies. These fit reasonably to the basic model involving triple-helix junction zones, essentially without adjusting arbitrary parameters. Such reducedvariable treatments for critically branched materials eliminate the otherwise inescapable difficulties of characterization, purity, electrolyte content, etc., from gelatin research.

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Section: Scanlan-case Eancsmentioning

confidence: 99%

The structure and properties of molecular trees and networks

Gordon¹,

Ross‐Murphy²

1975

Pure and Applied Chemistry

Self Cite

195

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“…A mixed gel can be characterized by the composition dependence of C 0 , which is determined by K, f A and f B . Equation (23) shows that C 0 is simply inversely proportional to K, whereas the effect of f A and f B is somewhat complicated. Figure 9 illustrates the effect of f A and f B on the composition dependence of C 0 .…”

Section: Gel Points Predicted By the Cascade Theorymentioning

confidence: 97%

“…The polymer network is then constructed via the branching process described in the cascade theory, 23 which uses the probability generating function for a binomial process of f trials as roots…”

Section: Cascade Theorymentioning

confidence: 99%

Cascade model for coupled two‐component polymer gels

Mao

Chen

2005

J of Applied Polymer Sci

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A cascade model for the network of coupled two-component polymer gels is presented. The network formation is assumed to be primarily due to the interaction between the functional groups of two unlike polymers. The gel modulus G versus polymer concentration C curve is constructed using the cascade formalism. The variation of G versus C curves with respect to the concentration ratio r of the two components is found to be the characteristics of mixed gels. The model parameters can be obtained by fitting G versus C curves using a nonlinear least-squares method (the gel-modulus approach). The approach is applied to the experimental data for galactomannan/xanthan and glucomannan/xanthan mixed gels. Finally we discuss the gel point predicted by the cascade model. The critical concentration C 0 for the gel point is strongly dependent on r and the functionalities for both polymers. The C 0 versus r curve is easily obtained by experiment, and can be used to evaluate the model parameters (the gel-point approach). The experimental data for galactomannan/xanthan mixed gels is tested to demonstrate the validity of the gel-point approach. The advantages and limitations of both approaches are discussed.

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“…They used a laborious approach based on combinatoric considerations of the most probable composition of the system. A much more effective variant of this approach, exploiting theory of branching processes, has been developed later by Dobson and Gordon [14,15]. This approach allows also an analysis of structural details of the system behind the critical point, see, e.g.…”

Section: Combinatoric Approachmentioning

confidence: 99%

Two simple approaches to sol–gel transition

Olemskoĭ

Krakovský

2001

Physica A: Statistical Mechanics and its Applications

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We represent a theory of polymer gelation as an analogue of liquid-glass transition in which elastic fields of stress and strain shear components appear spontaneously as a consequence of the cross-linking of macromolecules. This circumstance is explained on the basis of obvious combinatoric arguments as well as a synergetic Lorenz system, where the strain acts as an order parameter, a conjugate field is reduced to the elastic stress, and the number of cross-links is a control parameter. Both the combinatoric and synergetic approaches show that an anomalous slow dependence of the shear modulus on the number of cross-links is obtained. 05.70Ln, 47.17+e, 61.43Fs PACS:

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Theory of Branching Processes and Statistics of Rubber Elasticity

Cited by 175 publications

References 16 publications

The structure and properties of molecular trees and networks

The structure and properties of molecular trees and networks

Cascade model for coupled two‐component polymer gels

Two simple approaches to sol–gel transition

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