Rubber junction model

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. 

The elastic modulus of a rubber according to the phantom network theory is much lower than the modulus of the same network with all junction fluctuations suppressed. If the fluctuations are partially suppressed, the calculated modulus lies between these limits. In fact, in many cases, the measured modulus is many times greater than predicted by fixed junction models (8,9). 

The model so set up is the simplest one to treat and known as the fixed junction model with reference to the last point. Let us refer again to a prismatic piece of rubber, with edges Lx, Ly and Lz, and consider a homogeneous orthogonal deformation, which changes the lengths of the edges as follows... 

Whether to use the first or the second form of Finger s constitutive equa tion is just a matter of convenience, depending on the expression obtained for the free energy density in terms of the one or the other set of invariants. For the system under discussion, a body of rubbery material, the choice is clear The free energy density of an ideal rubber is most simply expressed when using the invariants of the Finger strain tensor. Equation (7.22), giving the result of the statistical mechanical treatment of the fixed junction model, exactly corresponds to... 

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

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... 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

The transient net work model is an adaptation of the network theory of rubber elasticity. In concentrated polymer solutions and polymer melts, the network junctions are temporary and not permanent as in chemically crosslinked rubber, so that existing junctions can be destroyed to form new junctions. It can predict many of the linear viscoelastic phenomena and to predict shear-thinning behavior, the rates of creation and loss of segments can be considered to be functions of shear rate. 

The simplest model of rubber-like behaviour is the phantom network model. The term phantom is used to emphasize that the configurations available to each strand are assumed to depend on the positions of the junctions only. Consequently, the configurations of one chain are independent of the configurations of neighbouring strands. For many purposes, the strands can be treated as Gaussian random coils. Even in this simplest case, an exact solution is not a trivial task as will be outlined in Sect. 3. 

As outlined in Sect. 1, a phantom network is defined as a network with the fictitious property that chains and junctions can move freely through one another without destroying the connectivity of the network. Usually, models of rubber elastic networks are built up from Gaussian chains and the topology of connectivity is completely described by the reduced Kirchhoff matrix of Eq. (6). However, Staverman pointed out that for a network with a given Kirchhoff matrix, the model has to be completed by additional assumptions. 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

In accordance with the tube model, Eq. (64) assumes a length scale d to be available for junctions to fluctuate. For example, in the case of natural rubber the value of A according to Eq. (64) changes from A 0.55 to A 0.85 when the molecular weight of network strands increases from M =4x10 g/mol to M = 40 x 10 g/mol. 

One of the best-known of such schemes is the AFFINE deformation model for rubbers. The rubber is considered to be a network of flexible chains, and the macroscopic strain is imagined to be transmitted to the network such that lines joining the network junction points rotate and translate exactly as lines joining corresponding points marked on the bulk material. If we assume that the flexible chains consist of rotatable segments called random links , and that some statistical model can describe the configurational situation, it is then possible to obtain explicit expressions which relate the segmental orientation to the macroscopic deformation. 

In vulcanized rubber the molecular chains are cross linked by chemical bonds which inhibit macro-flow. That is why vulcanized stocks are more perfectly elastic than raw rubber (cf. the models having piston and spring in series.) The chemical cross links, randomly distributed over the mass of rubber, act as permanent junction points. A netlike structure is obviously present in vulcanized rubber. 

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