Eight-chain model

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

We consider instead only an eight-chain model proposed by Arruda and Boyce (1993), in which eight identical chains that are connected at the center of an initial cube radiate out to the eight corners (Fig. 6.6). As with the three-chain model of Wang and Guth (1952), the eight-chain model considers deformation in the principal-axis system of the cube on the basis of the argument that for any other... 

Fig. 6.6 A depiction of the eight-chain model of Arruda and Boyce (a) the unstretched state, in which the vector ro connects the center of the cube to a corner and (b) the stretched state (from Arruda and Boyce (1993) courtesy of Elsevier).

This response was already discussed in Section 6.5.3 comparing the predictions of the eight-chain model with experimental results. 

Then the equilibrium response acting on the network orientation element can be defined following the Arruda-Boyce eight-chain model [40] as... 

Arruda and Boyce [12] showed that the eight-chain model performed much better than three- and four-chain models in predicting the behaviour of vulcanized rubber in uniaxial and biaxial tension and shear, and also performed excellently in modelling uniaxial and plane strain compression of gum and neoprene rubber. 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

K., Synthesis of retinals with eight- and nine-membered rings in the side chain. Models for rhodopsin photobleaching intermediates. J. Org. Chem., 58, 3533, 1993. 

The eight-chain-network model is centered aroimd following the entropy of any one of its identical eight chains as the assembly deforms. Thus, the consideration starts with eqs. (6.43) and (6.44) or (6.45), in which rc now represents the end-to-end distance of the chain. In the unstretched network the initial length of the chain is ro = /nl in the cube with initial edge length oq, giving... 

Using the eight-chain entangled-network model of rubber elasticity (Arruda and Boyce 1993), the back stress is conveniently expressed as... 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

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