Fixed phantom network

Fixed phantom network Some junctions are fixed in space. Such constraints do not occur in reality. 

Limit of full suppression of junction fluctuation (fixed phantom network, affine... 

Three types of phantom networks can be distinguished free phantom networks, fixed phantom networks, and localised phantom networks The first type is without any constraints and will consequently collapse. The second type is a phantom network with some junctions fixed in space. As a result, it is subjected to contraints that do not really exist. The most natural phantom network model is the last variant, in which the equilibrium positions of all segments are determined by suitable boundary conditions without any need for segments or junctions being fixed. 

Equation (27) shows the main Umitation of the theory of restricted junction fluctuation. The topological contribution to the modulus is limited by the value of the fixed phantom network. As an example, the relation > C, is impossible in the case f = 4. On the other hand, experimental studies on networks produced by endlinking of chains have been interpreted successfully within the theory of restricted junction fluctuation As already discussed in Sect. 2, this may be explained as... 

The tube model presented here yields values of the front factor of the crosslink contribution close to the front factor of the free-fluctuating phantom network. It is felt that the stronger constraints acting on the crosslinks have to be simulated by tube dimensions that depend on the distance from the crosslinks. In this way, the crossover from the free-fluctuating to the fixed phantom network value of the front factor, characteristic for the model of restricted junction fluctuation, can also be reproduced by tube models. 

However, for a network with given Kirchhoff matrix three types of phantom networks can be defined free phantom networks, fixed phantom networks and localised phantom networks. 

The properties of such a localised expanded network can be derived from James theory in Flory s version. Consider a large phantom network with variable Kirchhoff matrix T. The configuration function of the free phantom network is Zfree = exp[- J T J ] where R represents the set of N vectors (3N components) R, Rj of the junctions i, j. James and Flory have shown that for a fixed phantom network with a junctions fixed and r junctions free, the configuration function can be written... 

Clearly, this assumption implies that, once the Kirchhoff matrix and the size of the fluctuation domains ate known, also the reai configuration of the real network is known. It also implies that the ),eai configuration is not in equilibrium with the network forces. Therefore, the theory of localised or fo fixed phantom networks is not applicable. 

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.

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. 

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 ... 

Localised phantom network The equilibrium position of all segments of the net chains is determined by suitable boundary conditions without any need for segments or junctions to be fixed. 

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. 

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. 

Birefringence of Phantom Networks. This theory is the basis for all theories that deal with birefringence of elastomeric polymer networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network to consist of phantom (ie, non-interacting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network the birefringence induced... 

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). 

An important consequence of the memory-lattice model is that high moduli can be accommodated provided that the memory term in the potential function does not vanish. In terms of the model, a memory effect is present if is not equal to unity. This is evident in Eq(14d), low fluctuations corresponding to small Af lead to large values ofa A, the free energy of deformation, and corresponding large values of the retractive force. The coupling of the modulus to junction fluctuations does not appear in either the phantom network or fixed junction models. It arises here only because the minimum in the total potential is not exactly centered on the lattice. 

The theory is consistent with both low and high moduli, including cases where the moduli are higher than predicted by either phantom network or fixed junction models. 

James and Guth dispensed with the premise of an affine displacement of all network junctions conceived of as fixed in space. Only those Junctions which are located on the boundary surfaces are specified as fixed, and all other Junctions are allowed complete statistical freedom, subject only to the restrictions imposed by their interconnectedness. This theory was later called the phantom network model because the chains are devoid of material characteristics. Their only action is to exert forces on the Junctions to which they are attached, but they can move freely through one another. This also leads to a stress-strain relation of the form of Eq. (7) with Sg(X) given by Eq. (8), but with ah equilibrium modulus equal to... 

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... 

It can be seen that the first term in Equation (4.34) is the free energy of a phantom network, equivalent to the first term in Equation (4.32) of the constrained junction model. The second term can be thought of as equivalent to the second term in Equation (4.32), but because there is no fixed limit to the ratio of slip links Ns to permanent cross-links Nc, the maximum value of the free energy can be greater than that of the affine network that would replace V2 Nc in the first term by Nc. 

The early molecular-based statistical mechanics theory was developed by Wall (1942) and Flory and Rehner (1943), with the simple assumption that chain segments of the network deform independently and on a microscopic scale in the same way as the whole sample (affine deformation). The crosslinks are assumed to be fixed in space at positions exactly defined by the specimen deformation ratio. James and Guth (1943) allowed in their phantom network model a certain free motion (fluctuation) of the crosslinks about their average affine deformation positions. These two theories are in a sense limiting cases, with the affine network model giving an upper... 

Figure 3.10 shows schematically the difference between the affine network model and the phantom network model. The affine deformation model assumes that the junction points (i.e. the crosslinks) have a specified fixed position defined by the specimen deformation ratio L/Lq, where L is the length of the specimen after loading and Lq is the length of the unstressed specimen). The chains between the junction points are, however, free to take any of the great many possible conformations. The junction points of the phantom network are allowed to fluctuate about their mean values (shown in Fig. 3.10 by the points marked with an A) and the chains between the crosslinks to take any of the great many possible conformations. 

As discussed briefly in the introduction the elastic and relaxational properties of polymer networks are also expected to be influenced significantly by the presence of entanglements. The classical theories, the phantom network modeP and the affine deformation model, describe the two extreme points of view. In the first, at least in its original form, the network strands and the crosslinks are not subject to any constraint besides connectivity and functionality. The other extreme considers the crosslinks to be fixed in space and deform affinely under deformation. A number of modifications of these theories have been proposed in which the junction fluctuations are partially suppressed. All of these models however consider the network strands as entropic springs. The entropic force, as... 

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

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