Constrained chains

According to the models, the free energy of deformation and the stress in the network with constrained chains contains an additive contribution to that describing... 

Figure 2. Types of constraint in the molecular theories. In the earliest such constraint theoiy (uppermost portion of the figure) the total effects of the constraints were placed on the cross-links themselves. In the subsequent constrained-chains theory, they were placed at the mass centers of the network chains and, in the diffused-constraints theory, along the entire network chains. The lowermost portion of the figure shows how additional experimental information could suggest a more refined placement of the constraints.

As already described, the upper three portions of Figure 2 summarize the differences in the way the constraints are applied in the constrained-junction theory, constrained-chain theory, and the diffused-constraints theory, respectively [4], Additional comparisons between theory and experiment for a variety of elastomeric properties should be very helpful [20], Also, neutron-scattering measurements conducted on series of networks having different values of the junction functionality , which is the number of chains emanating from a junction (cross-link), would be extremely useful in suggesting how to position the constraints along a chain in refining such models, since should have a pronounced effect on the... 

Figure 10. Sketch of polymer chains in an elastomer constrained in the vicinity of a reinforcing filler particle. The constrained chains are arbitrarily shown by the heavier lines, and are referred to in the elastomer literature as bound rubber [233, 236-238].

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

Figure 4-14 Illustration of Reptation Concept, Credited to de Gennes and Doi and Edwards, of a Polymer Chain in a Concentrated Polymer Solution. The cross sections of the constraining chains and the tube for reptation of the polymer chain are shown.

Samulski, E. T., Constrained chain statistics D NMR of octane in a nematic solvent, Ferro-electrics, 30, 83-93 (1980). 

The strength of the constraints is determined in this model by the prefactors of [R (s) - ft (s)] It is assumed that for the undeformed isotropic system the constraining potential is independent of the direction of the constrained chain. Consequently, the constraining potential has to be diagonal in the main axis system of the deformation tensor in the case of external deformation. 

Fig. 2. Entanglement model. Configuration of the constrained chain Allowed configuration of a constraining chain ------...

The simplest case is that of high crosslink density or small coil interpenetration (Np 1). In this case, the restrictions on the configurations of network chains caused by the crosslinks dominate, and the constraints acting on the constraining chains may be omitted in the course of the calculation of the constraining potential. With the assumption of an affine displacement of the crosslink positions with the deformation of the sample, Eq. (12) was obtained with... 

An analytic expression for d has been derived in the case that effects of the constraints predominate over the crosslink contribution, i.e. in the melt case . The assumption of affine displacement of the tube axis of the constraining chains again gives Eq. (12) with... 

In vinyl acetate (VA) bulk or solution polymerization systems, side reactions (e.g., chain transfer or termination) will inevitably occur to produce highly branched poly(vinyl acetate) (PVA), based on the nonconjugated nature of the propagating radical. However, the polymerization of VA in nanochannels of [Cu2(terephthalate)2ted] effectively suppresses chain branching during the polymerization, and this results in a constrained chain growth in the narrow 1-D nanochannels [26]. 

A tolerance model includes 2D or 3D contours or wires with constrained chains of dimensions and tolerance values. It relates... 

Reptation assumes that the mobility of the matrix polymer plays no role in the relaxation of the tube constraint felt by a test molecule. However, if the matrix chains were very much more mobile than the test chain, additional lateral motion of the chain might be permitted by virtue of the constraining chains themselves moving away. This type of motion is called constraint release or tube renewal , and may be operative if some of the matrix chains are significantly shorter or intrinsically more mobile than the test chain (Green 1991, Composto etal. 1992). 

The models presented in the previous section are of an elementary nature in the sense that they ignore contributions from intermolecular effects (such as entanglements that are permanently trapped on formation of the network). Among the theories that take account of the contribution of entanglements are (1) the treatment of Beam and Edwards [19] in terms of topological invariants, (2) the slip-link model [20, 21], (3) the constrained-]unction and constrained-chain models [22-27], and (4) the trapped entanglement model [11,28]. The slip-link, constrained-junction, and constrained-chain models can be studied under a common format as can be seen from the discussion by Erman and Mark [7]. For illustrative purposes we present the constrained-junction model in some detail here. We then discuss the trapped entanglement models. 

Our analysis will be based on the simple configuration shown in Figure 4.2. This figure depicts a serial-link manipulator chain with no internal closed loops. The joints are arbitrary, and they are modelled using the general joint model of Chapter 2. The base member is fixed to the inertial firame. The spatial f ce vector, f, represents the vector of forces and moments applied by the tip of the chain to the environment. For an open chain, f is identically zero. For a constrained chain, however, f is unknown and in general nonzero. 

Given the recursive dynamic equations for a constrained chain, we will now begin the development of a linear recursive algcxithm fw A the inverse operational space inertia matrix of a single chain. First, we will define a new quantity, (A ) an inertial matrix which relates the spatial acceleration of a link and the propagated spatial contact force exoted at the tip of the same link. We may write a defining equation for this matrix (at link t) as follows ... 

Here k is a parameter which measures the strength of the constraints. For k = 0 we obtain the phantom network limit, and for infinitely strong constraints (k = oo) the affine limit is obtained. Erman and Monnerie [27] developed the constrained chain model, where constraints effect fluctuations of the centers of the mass of chains in the network. Kloczkowski, Mark, and Erman [28] proposed a diffused-constraint theory with continuous placement of constraints along the network chains. 

The constrained junction fluctuation theory was modified by Erman and Monnerie [94]. The fundamental difference between the modified and the original models is the adoption of the assumption that constraints affect the centers of mass of the chains rather than the junction points only. They considered two different cases (1) the fluctuations of aU points along the chains in the phantom network are independent of macroscopic strain (constrained chain scheme, CC) and (2) the fluctuations of the points in the phantom network are dependent on the macroscopic strain, only the junctions are invariant to strain (modified constrained chain... 

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. 

FIGURE 29.5. Mooney-Rivlin reduced stress plot showing comparison of experimental data with modified constrained chain model (MCC) predictions for dry (o) and swollen ( ) natural rubber networks [112,117]. Swelling agent n-Decane. continuous lines are theoretical curves calculated with paremeters /cT/l/o = 0.17MPa and kq =2.0. 

Kloczkowski, Mark, and Erman [95] compared the prediction of the diffused constraint model with the results of the Flory constrained-junction fluctuation theory [36] and the Erman-Monnerie constrained chain theory [94]. They found that the shapes of the [/ ] vs. a curves for all three theories were very similar. Rubinstein and Panyukov [101] reanalyzed the data of Pak and Flory [118] obtained for uniaxially deformed crosslinked PDMS samples. They concluded that the fit of the experimental data by the diffused... 

TABLE 29.5. Network parameters calculated by the constrained chain (CC) and modified constrained chain (MCC) models [116,117]. 

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