Junction model, affine

Convolution of Eq. (7.27) (with = 1) by Eq. (7.29) leads to the conformation of a mesh inside a deformed netwoA for the junction affine model often called Kuhn modelas shown by Ullmann for the uniaxial extension, who gives ... 

The classical rubber elasticity model considers, however, that the crosslink points are particular, such that the cut-off occurs by these points in real space. The corresponding calculations for a chain obliged to pass by several crosslinks are recalled in Ref The calculation for the junction affine model was accomplished by Ull-mann for R and by Bastide for the entire form factor for the case of the phantom network model, this was achieved by Edwards and Warner using the replica method. 

Fig. 14. Comparison of data for long chains in the plateau regime with the calculated form factor for a labeled chain (M, = 2,6 10 ) crosslinked in a rubber of mesh Me = 20000. Solid line below, isotropic above, junction affine model dashed line, phantom network. Data set VI (Table 1) 0> = 10 mn , t " = 40 mn O, isotropic...

However, it remains possible to compare qualitatively S(q) and on the same scale, i.e. for the unit for length b, i.e. q 1/b. For the junction affine model (example 2) for instance, the order of magnitude of S(qper)/Sis (q) at q (1/10 A) and of P2(0) = (3 — l)/2 is comparable. However, we here enter the region where the orientation effect is weaker than the usual accuracy of the SANS experiment. We will now report a recent experiment which compares the two quantities and is related to these problems. 

These networks (99) were extended up to a = 1.6 and characterized by SANS. Owing to large experimental error, no definitive conclusion could be reached, although the data fit the junction affine model better than either the chain affine model or the phantom network model (Section 9.10.5.2). 

From the observed changes in the parallel and pernendicular components of the radius of gyration relative to macroscopic extension ratio, after appropriate correction for the dangling chain contributions, the chain extensive deformation is found to follow a behavior intermediate between the junction affine model and the phantom network model which allows unrestricted fluctuations of network junctions. On the other hand, the chain contractive deformation follows closely the chain affine model, indicating an asymmetry between extensive and contractive chain deformation. In either case, the deformation behavior is found to be the same for the two molecular weights. 

Secondly, the junction affine model, where the junctions deform affinely and a strand between a junction pair remains Gaussian-like. This gives... 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

A treatment of the classic phantom network model is contained in Ref. [6], page 252-256. However, the statement on p. 256 that this model leads to the same stress-strain relation as the affine model is incorrect because it neglects the effect of junction fluctuations on the predicted shear modulus. See Graessley [17],... 

In the affine model of the network it is assumed all junction points are imbedded in the network, and each Cartesian component of the chain end-to-end vector transforms linearly with macroscopic deformation... 

The affine and the phantom models derive the behavior of the network from the statistical properties of the individual molecules (single chain models). In the more advanced constrained junction fluctuation model the properties of these two classical models are bridged and interchain interactions are taken into account. We remark for completeness that other molecular models for rubber networks have been proposed [32,57,75-87], however, these are not nearly as widely used and remain the subject of much debate. Here we briefly summarize the basic concepts of the affine, phantom, constrained junction fluctuation, diffused constraint, tube and slip-tube models. 

The Constrained Junction Fluctuation Model. The affine and phantom models are two limiting cases on the network properties and real network behavior is not perfectly described by them (recall Fig. 29.2). Intermolecular entanglements and other steric constraints on the fluctuations of junctions have been postulated as contributing to the elastic free energy. One widely used model proposed to explain deviations from ideal elastic behavior is that of Ronca and Allegra [34] and Hory [36]. They introduced the assumption of constrained fluctuations and of affine deformation of fluctuation domains. 

Importantly, the model spans the behavior between the phantom and affine models. When k = oo and = 0 we recover the affine network behavior. In this case the junction fluctuations are completely suppressed, i.e., o = 0. When K = 0, i.e., the junctions are free to fluctuate, we recover the phantom network model. 

In summary, the common feature of all constrained chain models is that they impose only limited constraints on chain fluctuations. [101] The constrained-junction fluctuation model restricts fluctuations of junctions and of the center of mass of network chains. The diffused constraint model restricts fluctuations of a single randomly chosen monomer for each network strand. Consequently, all these models can only represent the crossover between the phantom and afflne limits. [101] The phantom limit corresponds to a weak constraining case, while the affine limit corresponds to a very strong constraining potential. 

The earliest and the simplest model of mbber elasticity is the affine model which assumes that the junction points in the network transform affinely with maaoscopic deformation, that is,... 

The second example is the junction affine network model for a chain fixed only... 

Strict adherence to this "affine" model at high strains, and especially at high dilutions, appears implausible. The rearrangements of chains that inevitably take place with strain must relax the confinement of a junction within the domain defined by its neighbors in the reference state. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

C. Fluctuations are partially damped by chain interferences. This leads to a result intermediate between A. and B. To be specific, a model proposed by Flory (16) is adopted here. In this model, the inhibitory influence on junction fluctuations is assumed to be affine in the strain. One finds... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

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