The Phantom Network Model

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 

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These measurements for the first time allowed experimental access to the microscopic extent of cross-link fluctuations. The observed range of fluctuation is smaller than predicted by the phantom network model, for which... 

Whereas k = 1.3 is derived from the above-presented NSE data, k = 2.75 is expected for a four-functional PDMS network of Ms = 5500 g/mol on the basis of Eq. (67). Similar discrepancies were observed for a PDMS network under uniaxial deformation [88]. Elowever, in reality this discrepancy may be smaller, since Eq. (67) provides the upper limit for k, calculated under the assumption that the network is not swollen during the cross-linking process due to unreacted, extractable material. Regardless of this uncertainty, the NSE data indicate that the experimentally observed fluctuation range of the cross-links is underestimated by the junction constraint and overestimated by the phantom network model [89],... 

The expressions given in this section, which are explained in more detail in Erman and Mark [34], are general expressions. In the next section, we introduce two network models that have been used in the elementary theories of elasticity to relate the microscopic deformation to the macroscopic deformation the affine and the phantom network models. 

According to the phantom network model, the fluctuations Ar are independent of deformation and the mean f deform affinely with macroscopic strain. Squaring both sides of Equation (23) and averaging overall chains gives... 

These two relations result from the phantom network model, as shown in derivations given elsewhere [4,25]. 

The true stress for the phantom network model is obtained by substituting Equation 27 into Equation 15 ... 

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). 

The subscripts L and V denote that differentiation is performed at constant length and volume. To carry out the differentiation indicated in Equation (59), an expression for the total tensile force / is needed. One may use the expression given by Equation (28) for the phantom network model. Applying the right-hand side of Equation (59) to Equation (28) leads to... 

The phantom network model contains a crucial deficiency, well known to its originators, but necessary for simplifying the mathematical analysis. The model takes no direct account of the impenetrability of polymer chains, not is the impossibility of two polymer segments occupying a common volume provided for in this model. Different views have been presented to remedy these deficiencies, no consensus has been reached on models which are both physically realistic and mathematically tractable. 

Fluctuations are larger in networks of low functionality and they are unaffected by sample deformation. The mean squared chain dimensions in the principal directions are less anisotropic than in the macroscopic sample. This is the phantom network model. 

Figures 2, 3 and 4 show S(x) versus for the phantom network model and for the fixed junction case. The largest changes with angle are if the junctions are fixed, the smallest changes are with the phantom network of lowest functionality.

Table II contains measured Rg/Rg from the experiment and calculated Rg/Rg from the phantom network model. Since, the functionality of these networks is not well known but is probably high, calculated data at several different functionalities are shown.

In network BI in which chain expansion was the greatest, the measured results show more chain swelling than a network with f=3 but less than a network with f=4. Chain swelling was less than that of the phantom network model for the other two networks, and in one case, the chains coiled to a size slightly less than that of the unperturbed molecule. 

The SANS experiments of Clough et al. (21) on radiation crosslinked polystyrene are presented in Figure 9, and appear to fit the phantom network model well. However, these networks were prepared by random crosslinking, and the calculations given are for end-linked networks, which are not truly applicable. 

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. 

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

For the affine network model, is the actual strand molar mass (Mx = Ms) whereas the phantom network model requires a longer combined strand length 2) [Eq. (7.40)]. 

In both the affine and phantom network models, chains are only aware that they are strands of a network because their ends are constrained by crosslinks. Strand ends are either fixed in space, as in the affine network model, or allowed to fluctuate by a certain amplitude around some fixed position in space, as in the phantom network model. Monomers other than chain ends do not feel any constraining potential in these simple network models. 

The phantom network model assumes there are no interactions between network strands other than their connectivity at the junction points. It has long been recognized that this is an oversimplification. Chains surrounding a given strand restrict its fluctuations, raising the network modulus. This is a very complicated effect involving interactions of many polymer chains, and hence, is most easily accounted for using a mean-field theory. In the... 

The theoretical approach for determining the deformation behaviour of a network due to swelling or due to a mechanical force (stress-strain measurements, compression experiment) is based on a hypothetical phantom network. A phantom network is, by definition, a network with the fictitious property that chains and junctions can move freely through one another without destroying the cormectivity of the network. Usually, the network chains behave as Gaussian chains. Within the phantom network model, three network types can be distinguished ... 

If N is replaced by a front factor F whose value depends on the model used, the general form for AG derived is essentially the same for both theories and is given by Equation 14.19. Experimental work has shown that the behavior of a swollen network is best described by the phantom network model and further equations are derived on that basis. 

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. 

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