Theory phantom network

The assumption of Gaussian chains in the affine network theory can be removed by using nonlinear chains, such as the RF model (Langevin chain), stiff chain model (KP chain), etc. These models show enhanced stress in the high-stretching region. The effect of nonlinear stretching will be detailed in Section 4.6. 

In contrast, the assumption of affine deformation is difficult to remove. The affine network theory assumes that each subchain deforms in proportion to the macroscopic deformation tensor. However, because the external force neither directly works on the chain nor on the cross-links it bridges, the assumption lacks physical justification. In fact, the junctions change their positions by thermal motion around the average position. It is natural to assume that the nature of such thermal fluctuations remains unchanged while the average position is displaced under the effect of strain. 

James and Guth developed a theory of rubber elasticity without the assumption of affine deformation [18,19,20]. They introduced the macroscopic deformation as the boundary conditions applied to the surface of the samples. Junctions are assumed to move freely under such fixed boundary conditions. The network chains (assumed to be Gaussian) act only to deliver forces at the junctions they attach to. They are allowed to pass through one another freely, and they are not subject to the volume exclusion requirements of real molecular systems. Therefore, the theory is called the phantom network theory. 

JGl The mean positions of the junctions deform affinely to the strain, while their instantaneous positions are not affine to the strain. 

JG2 (fluctuation theorem) The fluctuations Ar around the mean positions are Gaussian with a mean square value that is independent of the strain, and is given by 

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The remaining question is, how the deviations from phantom network theory at high branching densities can be explained. 

Figure 7. Ratio of experimentally observed and theoretically calculated modulus, using phantom network theory with f2 and v2, versus branching density z-...

A subsequent theory [6] allowed for movement of the crosslink junctions through rearrangement of the chains and also accounted for the presence of terminal chains in the network structure. Terminal chains are those that are bound at one end by a crosslink but the other end is free. These terminal chains will not contribute to the elastic recovery of the network. This phantom network theory describes the shear modulus as... 

Both limits correspond to the classical approaches of phantom network theory. 

A very interesting way to explain deviations from the phantom network theory is the approach proposed by Ball et al. who modelled the topological constraints by sliding links which make contacts between the network chains. Formally, this approach is based on the Deam-Edwards concept. Assuming that the chemical junctions are free to fluctuate about their mean positions the calculation of the elastic free energy change leads to the expression... 

Vapour-sorption experiments on different polymer plus solvent systems have shown that the elastic component of the solvent chemical potential exhibits a maximum, contrary to the phantom network theories or the Mooney-Rivlin equation. Furthermore, evidence has been found that the localisation and height of the maximum is dependent upon the nature of the diluent. 

JG3 The elastic free energy of the phantom network theory is... 

Fig. 4.12 Phantom network theory. Network junctions are classified into

Therefore, the phantom network theory gives smaller elastic free energy due to the free fluctuations of the junctions. 

The main idea of the phantom network theory (Figure 4.12) is summarized as follows [1, 5]. It first classifies the junctions into two categories cr-junction and r-junction. The a-junctions are those fixed on the surface of the sample. They deform affinely to the strain i. 

This is the free energy (4.37) of the phantom network theory. 

The elastic free energy (4.46) of the phantom network theory can be derived directly from the fluctuation theorem for the junctions [4]. According to (4.18), the free energy change by deformation is equivalent to... 

The number of elastically effective chains = v(l — 2/(p) in phantom network theory is smaller than its affine value v. In an affine network, all junctions are assumed to displace under the strict constraint of the strain, while in a phantom network they are assumed to move freely around the mean positions. In real networks of rubbers, the displacement of the junctions lies somewhere between these two extremes. To examine the microscopic chain deformation and displacement of the junctions, let us consider deformation of rubbers accompanied by the sweiiing processes in the solvent (Figure4.14) [1,5,14,25]. 

Figure 4.15 plots the reduced tension of a natural rubber against the reciprocal degree of elongation As predicted by the theories, the experimental data lie in-between the upper limit of the affine network theory and the lower limit of the phantom network theory [5]. 

Rg. 4.15 Reduced stresses in the swollen (upper line) and unswollen (lower line) states. Experimental data lie in-between the limits predicted by the affine and phantom network theories. The tension reduces on swelling. (Reprinted with permission from Ref. [5], Chap. 8.)... 

Entanglements between chains also serve as a type of crosslink. In a linear or branched polymer, entanglements can slip or move, and so are very impermanent. However, chemical (or physical) crosslinking limits their motion, and increases their effect on bulk properties. At this time, the phantom network theory is calling into question the reality of entanglements. While a monograph such as this cannot of itself resolve the controversy, some of the properties of interpenetrating polymer networks described in later chapters bear on the problem. 

Theories of rubber elasticity [119], such as the affine network theory [120] or the phantom network theory [121], provide expressions for the network pressure, depending on cross-link functionality and network topology. For a perfect tetrafunctiOTial network without trapped entanglements, the elastic network pressure is given by [120] ... 

In a series of theoretical papers, T Jllman (103-105) reexamined the phantom network theory of rubber elasticity, especially in the light of the new SANS experiments. He developed a semiempirical equation for expressing the lower than expected chain deformation on extension ... 

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). 

Finally, the phantom network theory where Junctions only deform affinely on the average but with strain independent and unconstrained fluctuations For a tri-functional network, this gives... 

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 order to allow the application of the phantom network theory to real networks, die equilibrium positions in the real network should coincide with those of the corresponding phantom network... 

In fact, the first term in Eq. (57) leading to Eq. (63) marks the region of i.x in which not only the fluctuation domain but also the fluctuations themselves are deformed affinely with A. It is the region in which erf is proportional to This region is limited for (= xA) = 0.15 the deviation is 0.08% and for = 0.3 it is 3%. Thus, in the context of the phantom network theory, affine deformation of fluctuations occurs in a limited range of values of xA only. 

However, although the experimental values of A are not in exact agreement with those calculated by Flory, the agreement is satifactory. A strong point in favour of Flory s theory is the fact that A Ai = 1 in exp. II, found in tests using the same number of chains, Vp, and different functionality. Within the framework of the phantom network theory where no discrimination between junctions is made, the value A3/A4 = I3/I4 = 0.67 is expected. In exp. I, a value of 0.79 is found. It is further satisfactory that A3/A4 = la/ 4 = 0.67 in exp. I although this result is not confirmed by exp. II. 

The maximum in the plot of XAfi versus A is more difficult to explain within the framework of the phantom network theory. Of the effects operating in swollen networks and listed in Sect. 5.4 only the fourth one, the reduction of fluctuation domains with increasing internal pressure, can give rise to a decrease in AA, at high elongation. 

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