Network models phantom

The main assumption of the affine network model is that the ends of network strands (the crosslink junctions) are fixed in space and are displaced 

Recall from Problem 2.38 that the fluctuations of a single monomer in an ideal chain with fixed ends are identical to the fluctuations of an end monomer of a single effective chain of K monomers. For the particular case of the center monomer of an ideal chain with 2N monomers, the effective chain has K=Njl monomers. Hence, the constraining effect of the two strands of N monomers is identical to the constraining effect of a single effective chain of K=N/2 monomers. More generally, if there are/chains 

The fluctuations of junction points in a network are quite similar to those of the branch point of an /-arm star polymer. In order to calculate the amplitude of these fluctuations, start with/- 1 strands that are attached at one end to the surface of the network and joined at the other end by a junction point connecting them to a single strand [see the left-most part of Fig. 7.5(a)]. The strands attached to the elastic non-fluctuating network surface are called seniority-zero strands. Each of these /— 1 seniority-zero strands are attached to a single seniority-one strand by a /-functional crosslink [see the left-mostpart of Fig. 7.5(a)]. The seniority of a particular strand is defined by the number of other network strands along the shortest path between it and the network surface. The/- 1 seniority-zero strands 

This single effective chain containing N monomers is connected in series with a single seniority-one N-mer and together with it can be described by an effective chain of K =N + N monomers. In this way, Fig. 7.5(a) sketches how / — 1 zero-seniority strands together with one seniority-one network strand can be replaced by an effective strand with K monomers  

Combining this effective chain with the real seniority-two chain connected to it in series, gives an effective chain representing the combined effect of a tree of strands from seniority-zero through seniority-two [Fig. 7.5(b)]  

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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... 

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]. 

Comparison of the expressions for the elastic free energies for the affine and phantom network models shows that they differ only in the front factor. Expressions for the elastic free energy of more realistic models than the affine and phantom network models are given in the following section. 

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). 

Experimental determinations of the contributions above those predicted by the reference phantom network model have been controversial. Experiments of Rennar and Oppermann [45] on end-linked PDMS networks, indicate that contributions from trapped entanglements are significant for low degrees of endlinking but are not important when the network chains are shorter. Experimental results of Erman et al. [46] on randomly cross-linked poly(ethyl acrylate)... 

Equating the chemical potential to zero gives a relationship between the equilibrium degree of swelling and the molecular weight Mc. The relation for Mc ph is obtained for a tetrafunctional phantom network model as... 

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. 

Segmental orientation in model networks of PDMS in uniaxial tension is measured by infrared dlchroism, Measurements are made for four tetrafunctlonal end-linked networks. Results of experiments are compared with predictions of calculations based in (i) the widely used Kuhn expression and (ii) the RIS formalism. The Kuhn expression is found to considerably overestimate the segmental orientation. The RIS approach leads to values of segmental orientation that fall between predictions of the affine and phantom network models. This indicates that the nematic-like Intermolecular contributions to orientation are not significant. 

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],... 

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)]. 

Both the affine and phantom network models predict the same (classical) dependence of stress on deformation [Eqs (7.32) and (7.33)]. Detailed quantitative comparison of the classical form with experiments indicates two major disagreements (see Fig. 7.8). Experiments demonstrate softening at intermediate deformations and hardening at higher deformations. In... 

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. 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

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... 

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 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 ... 

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