Phantom chain behaviour

Here T serves as a dimensionless measure of the strength of the topological constraints. The free-fluctuation phantom result follows from Eq. (42) in two ways. Firstly, in the limit F = 0 (d, -> x), that is, for negligible topological constraints, this limit has to be viewed as the phantom result in the true sense. Secondly, in the case of deformation-independent tube dimension d = d, the free energy differences also simulate phantom chain behaviour. Using the relation... 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

To evaluate Pj. g from modulus measurements, a value of A has to be assumed and used together with the experimental value of Mg/AM. Accordingly, Figures 4 and 5 show pj- g plotted versus Pr.c affine (A=l) and phantom (A = 1 - 2/t) chain behaviour using the results in Figure 1. One condition that should be obeyed in the plots is that Pr e Pr c Pr e includes pre-gel and post-gel reaction. 

There are two assumptions in the Wall-Flory theory which have to be criticized. Why should the crosslinks be fixed and why should entanglements and intermolecular forces be ignored The latter will be discussed in detail below. The problem of the behaviour of the crosslinks for phantom chains is well understood. The basic theory was provided by James and Guth. We do not present the... 

This model assumes that the positions of the junctions are fixed and transformed affinely when the network is strained. The chains, only compelled to join the junctions, are free to rearrange between themselves. This model can be considered as a limiting case of the phantom network behaviour, which is discussed below. 

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to 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 ... 

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. 

Ronca and Allegra, and independently Flory, advanced the hypothesis that real rubber networks show departures from these theoretical equations as a result of a transition between the two extreme cases of behaviour. In subsequent papers Floryl >l and Flory and Ermanl derived a theory based on this concept. At small deformations the fluctuations of the network junctions are constrained by the extensive interpenetration of neighbouring, but topologically remote chains. The severity of these constraints is characterized by the value of the parameter k (k - 0 corresponds to the phantom network, k = to the affine network). With increasing deformation these constraints become less restrictive in the direction of the principal extension. The parameter t describes the departures from affine transformation of the shape of the domains of constraints. The resulting stress-strain relation also takes the form of Eq. (7) with... 

The result (37) has been obtained without making any assumptions about the cross-linking process. In case the network should be able, like a free chain in bulk polymer, to reach the 0-state given by Eq. (29) at every temperature, then the elastic force of the network should be strictly proportional to the absolute temperature, since is a constant of a network of given structure. The well-established experimental observation " that the elastic force is not proportional to T, but reflects the temperature dependence of iq and of ie, indicates that the elastic behaviour of real networks is not adequately described by a phantom network that attains the state of minimum free energy at every temperature. 

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