Fluctuations of junctions

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... 

If restricted junction fluctuations are taken into account, the chain deformation is increased, and is more anisotropic. The effect of increasing k is much more evident in networks of low functionality, since fluctuations of junction points are of minor importance in networks of high functionality. 

Concluding, we can state that the absolute values of the small-strain moduli, which are greater for networks having comblike crosslinks, than for those with tetrafunctional junctions, are understandable, if we assume that the fluctuations of junctions are restricted by the very short chains. The strain dependent measurements do not agree quantitatively with the recent theory, although the trends are in accordance. An exact correspondence... 

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 ... 

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... 

The constraining potential represented by virtual chains must be set up so that the fluctuations of junction points are restricted, but the virtual chains must not store any stress. If the number of monomers in each virtual chain is independent of network deformation, these virtual chains would act as real chains and would store elastic energy when the network is deformed. A principal assumption of the constrained-junction model is that the constraining potential acting on junction points changes with network deformation. In the virtual chain representation of this con-... 

Small-angle neutron scattering has also been applied to the analysis of networks that were relaxing after a suddenly applied constant uniaxial deformation (Boue et al., 1991). Results of dynamic neutron scattering measurements of Allen et al. (1972, 1973, 1971) indicate that segments of network chains diffuse around in a network, and the activation energies of these motions are smaller than those obtained for the center of mass motion of the whole chains. Measurements by Ewen and Richter (1987) and Oeser et al. (1988) on PDMS networks with labeled junctions show that the fluctuations of junctions are substantial and equate approximately to those of a phantom network model. Their results also indicated that the motions of the junctions are diffusive and... 

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... 

For example for the infinitely large network with the symmetrical tree-like topology (such as shown in Fig. 5.3) the mean-square fluctuations of junctions ((AR) ) and... 

The extent to which entanglements contribute to network elasticity is not yet fully resolved. In the model of Langley [45], Dossin and Graessley [46-49] a contribution to the equilibrium modulus is associated with the plateau modulus of viscoelasticity. On the other hand, Flory [36] and Erman [38 0] assume that interpenetration of chains is solely reflected by suppression of the fluctuations of junctions. 

The Constrained Junction Fluctuation Model. The affine and phantom models are two limiting cases on the network properties and real network behavior is not perfectly described by them (recall Fig. 29.2). Intermolecular entanglements and other steric constraints on the fluctuations of junctions have been postulated as contributing to the elastic free energy. One widely used model proposed to explain deviations from ideal elastic behavior is that of Ronca and Allegra [34] and Hory [36]. They introduced the assumption of constrained fluctuations and of affine deformation of fluctuation domains. 

In Eq. (29.23) W 0) is the distribution of constraints among different points along the network chain and 0 = i/n is the position of the /th segment of the chain as a fraction of the contour length between two crosslinks. If the distribution is uniform, then W 0) = 1 inside the integrand of Eq. (29.23). In the case when constraints are assumed to affect only fluctuations of junctions (as in the constrained-junction theory), 0 is limited to 0 = 0 or 0 = 1 only. [95] It is important to note that this theory does not reduce identically to the constrained-chain theory, because the latter characterizes the deformation-dependent fluctuations of the centers of mass of the chains and not the deformation-independent fluctuations of the midpoints [95]. 

In summary, the common feature of all constrained chain models is that they impose only limited constraints on chain fluctuations. [101] The constrained-junction fluctuation model restricts fluctuations of junctions and of the center of mass of network chains. The diffused constraint model restricts fluctuations of a single randomly chosen monomer for each network strand. Consequently, all these models can only represent the crossover between the phantom and afflne limits. [101] The phantom limit corresponds to a weak constraining case, while the affine limit corresponds to a very strong constraining potential. 

It is worth noting that > G, which is a coni pKnce of the that the fluctuations of junctions in a phantom n wOTk are unaffected by deformation. Flory has recently pointed out that the klmtification of v, with the numb of effective chains v is valid only for perfect networks (see Sect. 6.1). For an imperfect tetra-functional network, Flory 1] shown that... 

Flory has derived the elastic free energy of dilation of a network with account of restrictions of fluctuations of junctions. Quantitative agreement has been reported for vapor sorption measurements. Particularly impressive is reproduction of the observation that the product of the linear expansion ratiok and the elastic contribution (pi — p.i)e, to the chemkal potential of the dilumt in a swollen network exhibits a maximum with increase in k, which is contrary to previous theory It is convenient to compare the phantom modulus obtained by stress-str measurements to that obtained from swelling equilibrium studies... 

In an affine network, the fluctuations of junctions are completely suppressed. According to the Flory theory, the behavior of a real network is between the affine and the phantom limits, closer to affine near the isotropic state, and tending to the phantom one with increasing deformation. It is interesting to known the nature of the Isotropic starting state. This is done with the structure factor at 2jero deformation which is defined as the ratio between the actual modulus of the network at L = 1 and the theoretical affine modulus [Eq. (10)]... 

The Ronca-Allegra theory (177), and Flory-Erman theory (3,178,182) are both based on the idea that effects of constraints are local and decrease with increasing strain and swelling. The basic difference between the two theories is that in the Ronca-Allegra theory the fluctuations of junctions become exactly affine as the undeformed state is approached, whereas in the Flory theory they are close to but below those of the affine state. 

Because the phantom network is a system with 31 elastic degrees of freedom regardless of the number and functionality of the junctions, an expression for the free energy in I only and not containing N or v or Np explicitly is more satisfactory than an expression discriminating between junctions of different functionality. However, some discrimination can arise firom the fact that the free fluctuations of junctions of high functionality are less extensive than those of junctions of low functionality. 

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