Phantom network functionalities

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],... 

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. 

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.

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 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.

Eichinger,B.E. Elasticity theory. I. Distribution functions for perfect phantom networks. Macromolecules 5,496-505 (1972). 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

If entanglements acted like ordinary crosslinks (vN/2) per unit volume) the stored energy function would be given by the usual expression for tetrafunctional phantom networks with the spatial fluctuations of junctions suppressed ... 

For any functionality /, the phantom network modulus is lower than the affine network modulus [Eq. (7.31)] because allowing the crosslinks to fluctuate in space makes the network softer. The phantom network has the... 

What would be the size of a phantom network, made from strands with N monomers of size h and with functionality / of crosslinks, if it were not... 

Consider a circular cylindrical gel extended along its axis so that the circular symmetry is maintained. Let ho and h be the initial (reference) and deformed length of the cylinder. Then, the principal deformation ratio along the axis z is kz, kx and ky are deformation ratios for the axes perpendicular to the axis z. The strain energy density function for and ideal phantom network, Wgi, can be expressed as follows [3,4,84] ... 

Equation (22) holds for phantom networks of any functionality, irrespective of their structural imperfections. In case b), fluctuations of junctions are assumed to be suppressed fully. The junctions themselves are considered to be firmly embedded in the medium and their position is transformed affinely with the macroscopic strain. This leads to the free energy expression for an f-functional network possibly containing free chain ends... 

The shortcomings of the phantom network concepts have stimulated a number of attempts to find theoretically a more satisfactory elastic free energy function to describe the properties of elastomers at different states of deformation. The efforts to explain the real network behaviour by a special non-topological mechanism can be divided into four types. The first group considered intra- and intermolecular effects los-ni) wjtii these assumptions it is hard to explain values of the Mooney-Rivlin parameter which are of the order of the corresponding Cj parameter. 

For uniaxially stretched networks, the molecular deformations are characterised by the radii of gyration R u an respectively parallel and perpendicular to the stretching direction. From the small-angle scattering function, the molecular deformation of a stretched elastic phantom chain has been calculated for three cases 1) Free-fluctuation phantom network... 

The configurational partition function of the phantom network is the product of the configurational partition functions of its individual chains, junctions i and j ... 

FIGURE 29.4. Effect of constraints on the fluctuations of network junctions, (a) Phantom model and (b) constrained junction fluctuation model. Note that the domain boundaries (circles in the figures) are diffuse rather than rigid. The action of domain constraint is assumed to be a Gaussian function of the distance of the junction from B similar to the action of the phantom network being a Gaussian function of AR from the mean position A. 

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] ... 

For a perfect phantom network of functionality/, the front factor contains the term (f - 2)1 f, leading to equation (9.71), which considers chemical crosslinks of arbitrary functionality, but no physical cross-links. 

In Figure 3, the ratio of the orientation function of a network (with X =8 and =0.l) to that of a phantom network is plotted as a function of OC for various values of the empirical constant e. Calculations show... 

Figure 3. Segmental orientation in a uniaxially deformed network. denotes the ratio of the orientation function for a real network to that of a phantom network. Values of the ratio are expressed in terms of the reciprocal extension ratio o( Calculations are made for a network with and J =0.1 for various values of e, where e represents the strength of coupling of a segment to its environment.

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