Restricted junction fluctuations

The parameters of neutron scattering theory of polymer networks are A, the macroscopic stretching of the sample, or linear degree of swelling, f, the network functionality, K. which accounts for restricted junction fluctuations and a, a measure of the degree to which chain extension parallels the macroscopic sample deformation. The functionality is known from knowledge of the chemistry of network formation, and A is measured. Both K and a must be extracted from experiments. 

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. 

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. 

To focus on the mean field approach (see Sect. 2), the statistical weight of any configuration of the network may be expressed as a product of the contributions of the (restricted) network chains. Starting with (in our opinion) more natural models for restricted chains, a model with restricted junction fluctuations can be derived by integrating over the configuration space of the network chains. The constraints acting on the network chains are transformed in this way into a mean force potential acting on the junctions. This mean force potential is approximated within the model of restricted junction fluctuation by the domain accessible to each junction. 

Here, x and are material parameters x is proposed to be proportional to the degree of interpenetration of chains and junctions, and it defines the strength of restrictions on junction fluctuations. The value x = 0 corresponds to the free-fluctuation limit and for X - 00 the affine network is obtained. The parameter (< 1) characterizes departures of the shapes of domains from the affine transformation assumption and reflects the effect of structural inhomogeneities on the network structure Although its presence is not as critical as that of x, comparison of experiment with the theory of restricted junction fluctuations shows that it is necessary... 

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... 

Queslel combined the Ball model with the Flory model of restricted junction fluctuations to explain the origin of junction and trapped entanglement contributions to the total modulus exhibited by networks. There, the total contribution to the reduced force is considered to be the sum of the full Flory term (see Sect. 4.2) and the entanglement contribution of the Ball term. Such an expression has the practical advantage that it is valid over the entire range of deformation, but it seems to be an artificial construction. 

Clearly, the extrapolated front factor Ci — in principle — offers no upper limit and would be greater than 1 for (of course umealistic) very large Gn/Gc. In the case of usual values Gn/Gc 1, the difference between Cj and the free-fluctuation phantom limit result Cj = 0.5, if the functionality f is 4, plays only a minor role. On the other hand, the theory of restricted junction fluctuations limits Ci to values smaller (or equal to) 1, and the upper bound corresponds to the affine-deformation phantom limit where complete suppression of junction fluctuation has been assum-mj4i-44) Furthermore, Cj passes through a maximum for values of Cj in the range between 0.5 and 1, and the sum of the reduced parameters Cj and Cj, the reduced shear modulus g = G/v kT c, -t- 03, is limited to the range 0.5 g g 1 The different predictions of the tube model with deformation-dependent constraints and of the model of restricted junction fluctuations concerning the c, — C2. relation are shown in Fig. 9. 

In order to remove the main differences and confusions mentioned in Sect. 1, the relations between the tube model and other approaches which have been used successfully, such as the model of restricted junction fluctuations and the concept of trapped entanglements, have been discussed in detail and a number of new insights have been achieved. 

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. 

The phantom network model assumes there are no interactions between network strands other than their connectivity at the junction points. It has long been recognized that this is an oversimplification. Chains surrounding a given strand restrict its fluctuations, raising the network modulus. This is a very complicated effect involving interactions of many polymer chains, and hence, is most easily accounted for using a mean-field theory. In the... 

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. 

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... 

The expression of the orientation function derived on the basis of the real network model differs markedly from those obtained in previous treatments. The strain function includes the non-affine molecular deformation consecutive to restriction on junction fluctuations and distorsions of constraint domains. The configurational factor for the real chain accounts also for local intermolecular orientational correlations. 

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... 

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