Entanglements in networks

The importance of entanglements in network elasticity is proven beyond any doubt by three experimental observations. 

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

The paper is organized in the following way In Section 2, the principles of quasi-elastic neutron scattering are introduced, and the method of NSE is shortly outlined. Section 3 deals with the polymer dynamics in dense environments, addressing in particular the influence and origin of entanglements. In Section 4, polymer networks are treated. Section 5 reports on the dynamics of linear homo- and block copolymers, of cyclic and star-shaped polymers in dilute and semi-dilute solutions, respectively. Finally, Section 6 summarizes the conclusions and gives an outlook. 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

Recent Two-Network Results on the Effect of Chain Entangling in Cross-linked Elastomers... 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

In spite of these important results, the two-network method has had little impact on the discussion of the role of chain entangling in cross-linked elastomers. It was therefore decided to make a more detailed examination of the method and to try to develop a simpler method which would require fewer assumptions. The present paper is a discussion of recently published and unpublished work. 

Figure 2. The principle of the two-network method for cross-linking in a state of simple extension. First network with modulus Gy is entirely due to chain entangling. Second network with modulus Gx is formed by cross-linking in the strained state. Both Gy and Gx can be calculated from the two-network theory.

Unfortunately, the method is based on a fairly large nunber of assumptions. If we want to relate GN to the pseudo-equilibrium rubber plateau modulus, G , and to the effect of chain entangling in ordinary networks produced by cross-linking in the unstrained state, the following assumptions are required ... 

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

It is clearly shown that chain entangling plays a major role in networks of 1,2-polybutadiene produced by cross-linking of long linear chains. The two-network method should provide critical tests for new molecular theories of rubber elasticity which take chain entangling into account. 

Table 42.5 Enantioselective hydrogenation of methyl-2-aceta-midoacrylate with [Rh(MeDuphos)(COD)][CF3S03] entangled in a polymer network.

Dalmas F, Dendievel R, Chazeau L, Cavaille JY, Gauthier C (2006) Carbon nanotube-filled polymer of electrical conductivity in composites. Numerical simulation three-dimensional entangled fibrous networks. Acta Materialia 54 2923-2931. 

If we accept an elastic contribution from chain entangling in cross-linked networks, the problem is to find the relative magnitudes of the contributions from chain entangling and from cross-links. 

This is a theoretical study on the structure and modulus of a composite polymeric network formed by two intermeshing co-continuous networks of different chemistry, which interact on a molecular level. The rigidity of this elastomer is assumed to increase with the number density of chemical crosslinks and trapped entanglements in the system. The latter quantity is estimated from the relative concentration of the individual components and their ability to entangle in the unmixed state. The equilibrium elasticity modulus is then calculated for both the cases of a simultaneous and sequential interpenetrating polymer network. 

Note 5 Physical entanglements between network chains can lead to an increase in the concentration of elastically active network chains and, hence, increases in the shear modulus and the Young s modulus above the values expected for a perfect network structure. 

Figure 3.2 A mechanical force causes a loss of an entanglement in a covalent polymeric network by one of two major mechanisms process A, chain slippage via reptation process...

It is clear that the application of Langley s method in other polymer systems is essential to settle questions about Me and g in networks satisfactorily. The Ferry composite network method (223, 296) appears to be broadly applicable as well, although requiring special care to minimize slippage prior to introduction of the permanent crosslinks. (One is also still faced with the difficult question of whether g is the same for entanglements in crosslinked networks and in the plateau region of dynamic response.) Based on the limited results of these two methods in unswelled systems, Me values deduced by equilibrium and dynamic response appear to be practically the same. 

Chompff,A.J., Duiser,J.A. Viscoelasticity in networks consisting of crosslinked or entangled macromolecules. I. Normal modes and mechanical spectra. J. Chem. Phys. 45,1505-1514(1966). 

As is obvious from the above discussion, a very detailed understanding of entangling in elastomeric networks is required for interpretation of the elastic modulus, in particular its dependence on deformation and swelling. 

The theoretical treatment of line widths and intramolecular dipole-dipole interactions leads to an exponential dependence of Av on x, with the exponent 0.25 (fourth root) if the length of the crosslink is less than the average chain length between crosslinks 138). In poly(dimethyl siloxane) gels the predicted exponent is found to be 0.75, but experimental results give slightly smaller values (about 0.67), probably due to the effects of chain entanglements in these particular networks u9). 

---