Rubber phantom model

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 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 classical models of rubber elasticity reduce the elastic properties to a study of entropic springs. In the phantom model the EV interaction only gives the volume conservation, while in the affine deformation model it also is responsible for the affine position transformation of the crosslinks. Otherwise the entropic forces and the excluded volume interactions and consequently the entanglements, as they are a result of the EV interaction, completely decouple from the chain elasticity. The elasticity is entirely determined by the strand entropy. It is obvious that this is a... 

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. 

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 ... 

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. 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

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 recent years, important advances in the theory of rubber elasticity have been made. These include the introduction of the so-called phantom networks by Flory (I) and a two-network model for crosslinks and trapped entanglements by Ferry and coworkers (2,3). 

A point worth noting here is that several of the molecular models that will be described in the subsequent sections are Neo-Hookean in form. Normally, dry rubbers do not exhibit Neo-Hookean behavior. As for the Mooney-Rivlin form of strain energy density function, rubbers may follow such behavior in extension, yet they do not behave as Mooney-Rivlin materials in compression. In Fig. 29.2, we depict typical experimental data for a polydimethylsiloxane network [39] and compare the response to Mooney-Rivlin and Neo-Hookean behaviors. The horizontal lines represent the affine and the phantom limits (see Network Models in Section 29.2.2). The straight line in the range A <1 shows the fit of the Mooney-Rivlin equation to the experimental data points. 

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

In recent years, important advances in the theory of rubber elasticity have been made. These include the introduction of the so-called phantom networks by Flory and a two-network model for crosslinks and trapped entanglements by Ferry and co-workers, The latter builds on work by Flory and others on networks crosslinked twice, once in the relaxed state, and then again in the strained state. In other studies, Kramer and Graessley distinguished among the three kinds of physical entanglements as crosslink sites the Bueche-Mullins trap, the Ferry trap, and the Langley trap. 

Birefringence of Phantom Networks. This theory is the basis for all theories that deal with birefringence of elastomeric polymer networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network to consist of phantom (ie, non-interacting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network the birefringence induced... 

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...

The reader will note that in this theory of rubber elasticity we disregard the architecture of the C-C-C chain, and the van der Waals forces between chains. It is remarkable that a quantitative description of rubber elasticity can be obtained by assuming that the molecules are infinitely supple chains whose conformation is described by the equation of Gauss they can be looked upon as phantom chains in the sense that, as they diffuse to and fro through space, they pass through one another in a way real chains could not do. This assertion of the model, that real chains from a mechanical viewpoint can be... 

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). 

More recent network models of rubber elasticity by Ronca and Allegra [11] and Hory and Erman [12] are based on the phantom network model but assume that the junction point fluctuations are restricted due to the presence of entanglements. The strength of the constraints is defined by a parameter... 

Small-angle neutron scattering (SANS) of labelled (deuterated) amorphous samples and rubber samples detects the size of the coiled molecules and the response of individual molecules to macroscopic deformation and swelling. It has been shown that uncrosslinked bulk amorphous polymers consist of molecules with dimensions similar to those of theta solvents in accordance with the Flory theorem (Chapter 2). Fernandez et al (1984) showed that chemical crosslinking does not appreciably change the dimensions of the molecules. Data on various deformed network polymers indicate that the individual chain segments deform much less than the affine network model predicts and that most of the data are in accordance with the phantom network model. However, defmite SANS data that will tell which of the affine network model and the phantom network model is correct are still not available. 

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