Phantom network networks

Free phantom network Network without any constraints, which consequently collapses,... 

Since, in contrast to experiment, the simulation knows in detail what the connectivity looks hke, how long the strands are, and how the network loops are distributed, one can attribute this behavior to the non-crossability of the chains. Actually, one can even go further by allowing the chains to cross each other but still keep the excluded volume. Such a technical trick, which is only possible in simulations, allows one to isolate the effect of entanglement and non-crossability in such a case. As one would expect, if one allows chains to cross through each other one recovers the so-called phantom network result. 

Staverman, A.J. Properties of Phantom Networks and Real Networks. Vol. 44, pp. 73-102. 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

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... 

A model, which accounts for this effect, is the junction constraint model of Flory [86]. Starting from the phantom network an additional parameter k... 

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 ... 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

Equation (40) shows that the small deformation shear modulus of an affine network increases indefinitely over the phantom network modulus as junction functionality approaches 2. 

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). 

According to the arguments based on the constrained-junction model, the term Gch should equate to the phantom network modulus onto which contributions from entanglements are added. 

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. 

If network unfolding takes place so that distances between junctions connecting the ends of a polymer chain deform less than that of a phantom network, molecular dimensions change less than by any other of the models considered. This is easily seen from the data presented for a not equal to zero. 

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.

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

Figure 2. Scattering intensity versus azimuthal angle for a uniaxially oriented elastomer, X is 3, x is 0.2. Phantom network where , f is 3 A, f is 4 V, f is 10. Crosslink junctions fixed, X.

Figure 5. The ratio S (x)/S (x) plotted as junction of x for the phantom network at different cross-link functionalities, and for a fixed junction network. A is 2. Key Otfis3 n.f s 4 A, f is 6 V, f is 10 O is junctions fixed.

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. 

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