James and Guth models

The second approach, pioneered by Arruda and Boyce [12], involves the generalization of the non-Gaussian model of James and Guth [8], The James and Guth model was essentially one of three mutually perpendicular molecular chains,... 

There are many different expressions proposed to model the elastic part They find their origins either in some statistical mechanical model like the Gaussian chains, the Flory [32] or the James and Guth models [33], or in some phenomenological laws tending to reproduce experimental observations [34]. In the... 

For the James and Guth model with arbitrary functionality the modulus is then... 

An early model based on crosslinked rubbers put forward by Flory and Rehner (1943) assumed that chain segments deform independently and in the same manner as the whole sample (affine deformation) where crosslinks were fixed in space. James and Guth (1943) then described a phantom-network model that allowed free motion of crosslinks about the average affine deformation. The stress (cr) described from these theories can be described in the following equations ... 

A numerical calculation needs knowledge of the solvent activity of the corresponding homopolymer solution at the same equilibrium concentration 92 (here characterized by the value of the Flory-Huggins x-fimction) and the assumption of a deformation model that provides values of the factors A and B. There is an extensive literature for statistical thermodynamic models which provide, for example, Flory A = 1 and B = 0.5 Hermans " A = 1 and B = 1 James and Guth or Edwards and Freed A = 0.5 and B = 0. A detailed explanation was given recently by Heinrich et al. ... 

The classical affinity model assumes that the doss-links are immobile with respect to the whole network. The fluctuations of the positions of cross-links induced by thermal motion are taken into account in the phantom model proposed by James and Guth. It suggests that the fluctuations of a given CTOss-link proceed independently of the presence of subchains linked to it, and during such fluctuations the subchains can pass freely through each other like phantoms. The classic phantom theory predicts the shear modulus G as ... 

For example, the phantom network model of James and Guth (1,2) gave a recipe for predicting the deformation of a polymer network by an applied stress, and allowed predictions of the change in chain dimensions as a function of network expansion or distortion. In an effort to make the phantom model more realistic, and to fit the model to a variety of experimental results, P.J. Flory and collaborators (3,4,5) proposed that the fluctuation of crosslink junction points calculated by the James-Guth method should be very much restricted by chain entanglements. 

James and Guth dispensed with the premise of an affine displacement of all network junctions conceived of as fixed in space. Only those Junctions which are located on the boundary surfaces are specified as fixed, and all other Junctions are allowed complete statistical freedom, subject only to the restrictions imposed by their interconnectedness. This theory was later called the phantom network model because the chains are devoid of material characteristics. Their only action is to exert forces on the Junctions to which they are attached, but they can move freely through one another. This also leads to a stress-strain relation of the form of Eq. (7) with Sg(X) given by Eq. (8), but with ah equilibrium modulus equal to... 

One such approach is the three-chain model of James and Guth (1943) in which it is assumed that the network may be replaced by... 

In this equation the final term on the right-hand side remains non-zero even when the rubber is incompressible. As with eqn (3.45) this expression is still only valid under those conditions where Gaussian statistics are valid, viz when rchain model to take into account the fact that A, 17, and A1A2A3 may all have values that are not equal to unity. 

In the model of James and Guth each chain of the network imparts a force between the junctions so connected that is proportional to the distance between them (see eg.3) but the chain is devoid of all other material properties. The chains may pass through one another freely and parts of two or more of them may occupy the same space. Being free of constraints by neighboring chains, the junctions of this "phantom network" may undergo displacements that are Independent of their immediate surroundings. 

A key assumption of the single molecular theory is that the junction points in the network move affinely with the macroscopic deformation that is, they remain fixed in the macroscopic body. It was soon proposed by James and Guth [9] that this assumption is unnecessarily restrictive. It was considered adequate to assume that the network junction points fluctuate around their most probable positions [9,10] and the chains are portrayed as being able to transect each other. This has been termed the phantom network model. The vector r joining the two junction points is considered as the sum of a time average mean r and the instantaneous fluctuation Ar from the mean so that... 

The early molecular-based statistical mechanics theory was developed by Wall (1942) and Flory and Rehner (1943), with the simple assumption that chain segments of the network deform independently and on a microscopic scale in the same way as the whole sample (affine deformation). The crosslinks are assumed to be fixed in space at positions exactly defined by the specimen deformation ratio. James and Guth (1943) allowed in their phantom network model a certain free motion (fluctuation) of the crosslinks about their average affine deformation positions. These two theories are in a sense limiting cases, with the affine network model giving an upper... 

The phantom network model of James and Guth predicts a different free energy-deformation expression (cf. eq. (3.32)) ... 

It is interesting to compare eq. (3.54) with the expressions obtained from the statistical theories (Fig. 3.20). According to both the affine network model and the phantom network model of James and Guth, the reduced stress remains constant and independent of strain, which is not the case for the Mooney-Rivlin equation. 

The above picture of the network structure of vulcanized rubber is supported by -the success of the kinetic theory of rubberlike elasticity (see part 4, page 14) calculations based on this model agree well with experimental measurements of stress-strain curves and other properties (James and Guth, 1943 Flory, 1944). Excellent evidence that the swollen gel contains the same network as the unswollen rubber has been presented by Flory (1944, 1946), based on studies of butyl rubber. Using the network model, the number of cross-links in the structure can be calculated in three ways (o) from measurements of the proportions of insoluble (network) and soluble (unattached) material in samples of different initial molecular lengths (b) from the elastic modulus of the unswollen rubber (c) from the maximum amount of liquid imbibed by the gel when swollen in equilibrium with pure solvent. The results of these three calculations for butyl rubber samples were in good agreement. 

The Phantom Network Model. The theory of James and Guth, which has subsequently been termed the phantom network theory, was first outlined in two papers (186,187), followed by a mathematically more rigorous treatment (188-190). More recent work has been carried out by Duiser and Staverman (191), Eichinger (192), Graessley (193,194), Flory (82), Pearson (195), and Kloczkowski and co-workers (196,197). The most important physical feature is the occurrence of junction fluctuations, which occur asymmetrically in an elongated network in such a manner that the network chains sense less of a deformation than that imposed macroscopically. As a result, the modulus predicted in this theory is substantially less than that predicted in the affine theory. 

In discussing more realistic models we consider first the modulus of the constrained fluctuation theory of Flory. Flory s assumption, that entanglements only restrict the fluctuations of the crosslinks, gives at once the result that the modulus is between the extremes — affine and James and Guth. The constraint parameter k interpolates between both models. This is revealed by the following expression " ... 

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