Network junction theory

G.B. Ouyang, N. Tokita, M.J. Wang. Hysteresis mechanisms for carbon black filled vulcanizates - a network junction theory for carbon black reinforcement. ACS, Rubb. Div. Mtg, Cleveland, OH, Oct. 17-20, 1995. Paper 108. 

A5.1.3 Strain Amplification Factor from the Network Junction Theory... 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

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... 

These observations can be qualitatively explained in terms of the constrained-junction theory. If a network is cross-linked in solution and the solvent then removed, the chains collapse in such a way that there is reduced overlap in their configurational domains. It is primarily in this regard, namely reduced chain-junction entangling, that solution-cross-linked samples have simpler topologies, and these diminished constraints give correspondingly simpler elastomeric behavior. 

The transient net work model is an adaptation of the network theory of rubber elasticity. In concentrated polymer solutions and polymer melts, the network junctions are temporary and not permanent as in chemically crosslinked rubber, so that existing junctions can be destroyed to form new junctions. It can predict many of the linear viscoelastic phenomena and to predict shear-thinning behavior, the rates of creation and loss of segments can be considered to be functions of shear rate. 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

S.6.2.3 Finite Dimensional Approximations The complexity of the IDEAS formulation resides in the number of possible junctions (the Z, s) that may interact with other junctions throughout the network. In theory, infinitely many possible outputs may be formulated from the network, leading to an infinitely large number of potential variables to be solved. 

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]. 

As shown in more detail elsewhere, the rubbery-state modulus Er showed a trend of increasing with crosslink density, with measured values lying close to or between the predictions of the affine and phantom chain theories of rubber elasticity [63]. However, we also observe an influence from the chemical composition, with the actual values between these two limits reflecting the intrinsic stiffnesses of the three diisocyanates and hence the molecular mobility at the network junctions [63]. 

Fig. 4.12 Phantom network theory. Network junctions are classified into

These calculations reproduce, at least qualitatively, the experimental observations HEUR/SDS reported by Annableeffl/. [18] (Figure 10.5(a)), apart from that in the theory all curves cross each other at a certain surfactant concentration, whereas the experimental data reveal the same tendency only for relatively higher polymer concentrations. The maximum in the modulus appears as a result of the existence of a forbidden gap (from = 2 to 0 - 1) in the multiplicity of the network junctions. 

This article presents uniaxial extension measurements on cis-1,4-polybutadiene networks of known junction functionality. The observed values of the reduced force from uniaxial extension measurements conform to the constrained junction theory of Flory. The reduced force intercept at 1/cX = 0 is fully comprehensible in terms of the cycle rank of the network, and can be calculated from chemical considerations. This holds even though the polybutadiene melt has a high plateau modulus. Therefore, discrete topological entanglements do not contribute perceptibly to the equilibrium modulus of polybutadiene networks. 

The early versions of the statistical theory of rubber elasticity assumed an affine displacement of the average positions of the network junctiOQs with the macroscopic strain This is tantamount to the assertion that the network Junctions are firmly embedded in the medium of which they are part. The elastic equation of state derived on this basis for simple extension at constant volume takes the familiar neo-Hookean form, i.e. Eq. (7), with... 

James and Guth dispensed with the premise of an affine displacement of all network junctions conceived of as fixed in space. Only those Junctions which are located on the boundary surfaces are specified as fixed, and all other Junctions are allowed complete statistical freedom, subject only to the restrictions imposed by their interconnectedness. This theory was later called the phantom network model because the chains are devoid of material characteristics. Their only action is to exert forces on the Junctions to which they are attached, but they can move freely through one another. This also leads to a stress-strain relation of the form of Eq. (7) with Sg(X) given by Eq. (8), but with ah equilibrium modulus equal to... 

Ronca and Allegra, and independently Flory, advanced the hypothesis that real rubber networks show departures from these theoretical equations as a result of a transition between the two extreme cases of behaviour. In subsequent papers Floryl >l and Flory and Ermanl derived a theory based on this concept. At small deformations the fluctuations of the network junctions are constrained by the extensive interpenetration of neighbouring, but topologically remote chains. The severity of these constraints is characterized by the value of the parameter k (k - 0 corresponds to the phantom network, k = to the affine network). With increasing deformation these constraints become less restrictive in the direction of the principal extension. The parameter t describes the departures from affine transformation of the shape of the domains of constraints. The resulting stress-strain relation also takes the form of Eq. (7) with... 

Experimental results indicate that the response to deformation of a network generally falls between the affine and phantom limits [31-34]. At low deformations, chain-junction entangling suppresses the fluctuations of the junctions and the deformation is relatively close to the affine limit. This is illustrated in Fig. 1.8, which shows schematically some of the results of the constrained-junction theory based on this qualitative idea [32-34]. In the case of the two limits, the affine deformation and the non-affine deformation in the phantom-network limit, the reduced stress should be independent of a. Because of junction fluctuations, the value for the... 

Experimental results on networks of natural rubber in shear deformation [134] are not well accounted for by the simple molecular theory of rubber-like elasticity. The constrained-junction theory, however, was found to give excellent agreement with experiment. Shear measurements have also been reported for some unimodal and... 

Prediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions (EANJs) and chains (EANCs), respectively and [260, 261]. EANJs are the intersection of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13). 

A key assumption of the single molecular theory is that the junction points in the network move affinely with the macroscopic deformation that is, they remain fixed in the macroscopic body. It was soon proposed by James and Guth [9] that this assumption is unnecessarily restrictive. It was considered adequate to assume that the network junction points fluctuate around their most probable positions [9,10] and the chains are portrayed as being able to transect each other. This has been termed the phantom network model. The vector r joining the two junction points is considered as the sum of a time average mean r and the instantaneous fluctuation Ar from the mean so that... 

In the 1930 s several scientists presented evidence showing that rubber elasticity is essentially an entropy phenomenon, related to the change in randomness of location of the rubber segments when the material is extended. On this basis they derived relationships between the initial elastic modulus, the average chain length between network junctions, etc. The basic idea-appealed to me. It was a natural extension of my 1922 theory of conformations in simple molecules. 

Fig. 3. Reduced stress as a fiinction of reciprocal extension ratio. The upper and lower horizontal lines represent results from affine and phantom network models, respectively. Circles show representative data from experiments, and the curves are from the constrained junction theory.

The constrained-junction theory successfully describes most of the features of numerous investigations that have been made on stress-strain relationships involving a variety of t5rpes of deformations (1-3,13,220,249-252). Specifically, the decrease in modulus f ] with the increase in elongation is viewed as the deformations becoming more nonaffine as the stretching of the network chains... 

For undiluted polymers and concentrated solutions, there are two types of theories (a) the singlechain theories or reptation theories , in which one focuses on the motions of one polymer molecule in the fluid as it moves in some kind of mean field provided by the surrounding polymer molecules and (b) the network theories , in which one visualizes the fluid as a loosely joined network in which the network junctions have a distribution of lifetimes. The chain theories are similar in structure to the dilute solution theories, and one has to make some kinds of assumptions about how the surrounding molecules affect the hydrodynamic drag and the Brownian motion. The network theories are similar in structure to the kinetic theory of rubber elasticity, and one has to make some kinds of assumptions about the junction kinetics. 

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