The Temporary Network Model

Let 1/r be the probability, per unit time, that a strand breaks free of a junction point and relaxes. Then the probability, that the strand docs nnf break free any time during the time interval t to time t obeys the differential equation 

When the material is deformed, each strand is stretched affinely until it breaks free from its junction. After it breaks free, it relaxes to a configuration typical of equilibrium. As often as a strand breaks free, another relaxed strand becomes entangled. The probability that a strand breaks free and becomes re-entangled in an interval of time between t and t - - dt is dt fx. The probability that it survives without breaking from time t to time t is Pt t. The contribution da to the stress from those strands that meet both of these conditions is. according to Eq. ( 3-211.  

However, Eq. (3-24) can be extended to allow multiple relaxation modes simply by setting mtf — t equal to a sum of exponentials (Lodge 1956.1968 Yamamoto 1956.1957. 

To solve Eq. (3-24) for a particular flow history, one must compute the tensor B and carry out the integration. An example of how this is done for start-up of steady uniaxial extension is given in Worked Example 3.1 at the end of this chapter. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model 

---

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

An example of the success of the temporary network model for a practical application is shown in Fig. 3-11. Here, the predictions of Eq. (3-24) are compared to experimental force-deflection data for impact tests in which a heavy flat-bottomed object is dropped onto a flat circular pad of dissipative Sorbothane rubber at various velocities and two different temperatures. Since the material is nearly incompressible under these conditions, the impact... 

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

While empirically useful, the temporary network model gives no indiction of the relationship between the relaxation spectrum and the molecular relaxation processes. In the next sections, this deficiency is addressed by returning to the bead-spring models of Fig. 3-5. 

Worked Examples 3.1 and 3.2 (at the end of this chapter) show how calculations of stress in simple flows are carried out using the temporary network model and the elastic dumbbell model. 

Although the expression for the stress tensor in the Doi-Edwards model is the same as that of the temporary network model, except for the coefficient [see Eq. (3-13)],... 

This latter approximation shows that the strain dependence of the Doi-Edwards equation is softer than that of the temporary network model roughly by the factor 1 -p (7i — 3)/5, There is also a differential approximation to the Doi-Edwards equation (Marmcci 1984 Larson 1984b) ... 

Miiller-Plathe F (2002) Coarse-graining in polymer simulation From the atomistic to the mesoscopic scale and back. J Chem Phys Phys Chem 3 754—769 Muller R, Picot C, Zang YH, Froelich D (1990) Polymer chain conformation in the melt during steady elongational flow as measured by SANS. Temporary network model. Macromolecules 23(9) 2577—2582... 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

Both these models find their basis in network theories. The stress, as a response to flow, is assiimed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. 

Conformational relaxation of polymers at temperatures below their glass transition temperature is retarded by lack of segmental motions. The conformation and free volume at the glass transition temperature continues at lower temperatures since equilibrium cannot be attained over typical experimental times. Cooperative relaxation towards conformational equilibrium depends upon temperatures, relaxation time spectrum, and the disparity between the actual and equilibrium states. The approach of the vitrified polymer to equilibrium is called thermal aging. Aging is both non-linear and nonexponential and several descriptions and models have been proposed. One model is based on a concept of temporary networks where the viscoelastic... 

Our study focuses on selecting location of temporary warehouses (local distribution centers) at post-disaster for facilitating an efficient and effective response operation and determining amount of relief supplies that will be delivered through the relief network in order to minimize cost as well as maximize customer satisfaction under uncertainly environment. The proposed relief network encompasses three tiers multi-collection centers, candidates of local distribution centers (LDCs), and demand points. It should be noted that the majority of papers address to historical information to select the distribution center locations. Our model is applied for designing relief plaiming based on the future disaster predicted by scientist and agreed by disaster stockholders. 

Validity of the linear stress-optical rule points at the dominant role of the network forces in pol3mier melts. The Lodge equation of state can be interpreted on this basis. We introduced the equation empirically, as an ap>-propriate combination of properties of rubbers with those of viscous liquids. It is possible to associate the equation with a microscopic model. Since the entanglement network, although temporary in its microscopic structure, leads under steady state conditions to stationary viscoelastic properties, we have to assume a continuous destruction and creation of stress-bearing chain sequences. This implies that at any time the network will consist of sequences of different ages. As long as a sequence exists, it can follow all imposed deformations. 

Here three constants appear Go is the equilibrium modulus of elasticity 0p is the characteristic relaxation time, and AG is the relaxation part of elastic modulus. There are six measured quantities (components of the dynamic modulus for three frequencies) for any curing time. It is essential that the relaxation characteristics are related to actual physical mechanisms the Go value reflects the existence of a three-dimensional network of permanent (chemical) bonds 0p and AG are related to the relaxation process due to the segmental flexibility of the polymer chains. According to the model, in-termolecular interactions are modelled by assuming the existence of a network of temporary bonds, which are sometimes interpreted as physical (or geometrical) long-chain entanglements. 

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. 

---