Two-network method

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 The present paper is a discussion of recently published and unpublished work. 

Part 2. Two-Network Method. Different Types of Strain. 

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.

The two-network method has several advantages, especially when the free energy is expressed in terms of moduli as shown in eq. 5. The following information need not be known ... 

The new method (27) has a number of advantages in comparison to the original two-network method (10). Sample dimensions and the... 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

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. 

The challenge is therefore to develop an experiment which allows an experimental separation of the contributions from chain entangling and cross-links. The Two-Network method developed by Ferry and coworkers (17,18) is such a method. Cross-linking of a linear polymer in the strained state creates a composite network in which the original network from chain entangling and the network created by cross-linking in the strained state have different reference states. We have simplified the Two-Network method by using such conditions that no molecular theory is needed (1,21). 

The Simplified Two-Network Method and Contour Lenqth Relaxation. 

An interpenetrating polymer network (IPN) is defined as a material comprising two or more networks which are at least partly interlaced on a molecular scale, hut not covalently bonded to each other. These networks caimot he separated unless chemical bonds are broken. Two possible methods exist for preparing them, as follows ... 

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. 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

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. 

Example 15.3 The Flow Analysis Network Method Clearly Eq. E15.2-22 is identical to Eq. E15.2-21. This is the basis for the flow analysis network (FAN) method developed by Tadmor et al. (30) to solve two-dimensional steady or quasi-steady state flow problems in injection molds and extrusion dies. In two-dimensional flows the pressure distribution is obtained by dividing the flow region into an equal-sized mesh of square elements... 

This paper presents a thermodynamic availability analysis of an important process design problem, namely, the synthesis of networks of exchangers, heaters and/or coolers to transfer the excess energy from a set of hot streams to streams which require heating (cold streams). Emphasis is placed on the discussion of thermodynamic and economic (i.e., thermoeconomic) aspects of two recent methods for the evolutionary synthesis of energy-optimum and minimum-cost networks. These methods include the... 

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