The Four-Parameter Model and Molecular Response

The four-parameter model (Fig. 18.1/) is a series combination of a Maxwell element with a Voigt-Kelvin element Its difierential equation is 

Its creep response is the sum of the creep responses of the Maxwell and Voigt-Kelvin elements. 

The four-parameter model provides at least a qualitative representation of all the phenomena generally observed in the creep of viscoelastic materials instantaneous elastic strain, retarted elastic strain, equilibrium viscous flow, instantaneous elastic recovery, retarded elastic recovery, and permaneait set It also describes at least qualitiatively the behavior of viscoelastic materials in other types of deformation. Of equal importance is the fact that the model parameters can be identified with the various molecular response mechanians in polymers, and can therefore be used to predict the influences that changes in molecular 

Dashpot 1 represents molecular slip— the translational motion of molecules. This slip of polymer molecules past one another is responsible for flow. The value of /i alone (molecular friction in slip) governs the equilibrium flow of the material. 

Example 2. Using the four-parameter model as a basis, sketch qualitatively the efifects of (a) increasing molecular weight and ( ) increasing degrees of crosslinking on the creep response of a linear, amorphous polymer. 

FIGURE 15.8 Creep response of a four-parameter model for an input shear stress. Note that the model also works for tensile stress, t, with its corresponding tensile strain, s. 

FIGURE 15.9 The effects of molecular weight and crosslinking on the creep response to application of a shear stress on an amorphous polymer. A similar response would he observed for 

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The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

Although the four-parameter model is useful from a conceptual standpoint, it does not often provide an accurate fit of experimental data and therefore cannot be used to make quantitative predictions of material response. To do so, and to infer some detailed information about molecular response, more general models have been developed. The generalized Maxwell model (Figure 15.10) is used to describe stress-relaxation experiments. The stress relaxation of an individual Maxwell element is given by... 

The viscosity of some polymers at constant temperature is essentially Newtonian over a wide shear rate range. At low enough shear rates all polymers approach a Newtonian response that is, the shear stress is essentially proportional to the shear rate, and the linear slope is the viscosity. Generally, the deviation of the viscosity response to a pseudoplastic is a function of molecular weight, molecular weight distribution, polymer structure, and temperature. A model was developed by Adams and Campbell [18] that predicts the non-Newtonian shear viscosity behavior for linear polymers using four parameters. The Adams-Campbell model is as follows ... 

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