The Four-Parameter Model

FIGURE 13-95 Comparison of the Maxwell and Voigt equations with ones that account for a continuous range of relaxation times. 

This does display the three elements of real behavior, an instantaneous elastic response, primary creep (retarded elastic response) and secondary creep (permanent deformation). However, the fit to real data is not good and again it is because real materials have behavior that is characterized by a spectrum of relaxation times. 

FIGURE 13-96 Schematic diagram of die Maxwell-Wiechert model. 

Instead of solving this differential eqnation, the creep solution is more easily constructed from physical considerations by combining the solutions already available for the linear elastic solid, the Kelvin solid, and the linear viscous fluid. Thus, 

The four unknown parameters to be determined from experiments are E, E2, Tjj, and Tjj. If T 3 is large, 113 — 00 the model reduces to a three-parameter solid. 

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When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

Most screening designs are based on saturated fractional factorial designs. The firactional factorial designs in Section 14.8 are said to be saturated by the first-order factor effects (parameters) in the four-parameter model (Equation 14.27). In other words, the efficiency E = p/f = 4/4 = 1.0. It would be nice if there were 100% efficient fractional factorial designs for any number of factors, but the algebra doesn t work out that way. 

Fig. 11.5 Ca titration of porcine calmodulin and fractional species calculation [25]. (A) Ca-titration for 15 pM of porcine calmodulin in 50 mM HEPES (pH 7.4, T = 21.5 °C, 90% D2O). Error bars were based on the deviation from two sets of Q-TOF data. The solid curve was the best fit for the average data using the four-parameter model. (B) Fractional species as a function of [Ca " ] for CaM interacting with four Ca ". ...

Fig. n.6 Ca titration of 15 aM porcine CaM in three different media (99% D2O) [25]. Line(a) 50 mM HEPES/0.1 M KCI, apparent pH 7.4. Line (b) 50 mM HEPES, apparent pH 7.4. Line (c) 2 mM NH4OAC, apparent pH 7.0. Error bars were based on two sets of LCQ titration data. Solid curves were taken from the four-parameter model and are the best fit for the average data. 

One may question the need for a four parameter enthalpy equation, i.e., whether describing an acid or base by two parameters each is redundant. The following simple matrix algebra shows the conditions whereby a four parameter model reverts to a less redundant two parameter equation. Letting A be the transformation matrix, E and C represent the parameters for the four parameter model, and a represent the acid parameters for the two parameter model, the following equation results ... 

In all of the above there is assumed no product inhibition. Nonlinear regression of the kinetic data revealed the four parameter model, Eq. 1, to be marginally better than the two parameter model of Eq. 2 with both models the maximum error was less than 20%, as shown in Figures 1 and 2. Eq. 2 is quite adequate for the range of variables studied and affords a more direct comparison of deactivation effects, hence it is used in the following. Fresh catalyst parameters are given below. 

Figure 6. Stereoregular structures of polymer chains described by the four-parameter model...

Two quantities—Pi, the concentration of the isotactic triad and ai, the probability that an isotactic triad be followed by another isotactic triad—are sufficient to describe the sequence length of the isotactic units. The concentration of the various triad states can be expressed in terms of the various transition probabilities of the four-parameter model discussed previously (1). It can be shown that in the expressions for the probabilities of the various triads, the four transition probability parameters occur in such a combination that the concentration of the various triad states like Pi, etc. depend on only two independent parameters (Appendix I). 

E3 = — + and E4 = - 7 where the + and - define the relative configuration of two adjacent monomer units in the polymer chain. In accordance with the nomenclature of Bovey (7), we call the sequence (+ +) isotactic, (+ -), or (- +) heterotactic, and (- -) syndiotactic. The transition probabilities between various states were defined for the four-parameter model (1). 

The critical chain length for the four-parameter model can be found in a similar manner to that used above. In this case, as discussed earlier (Equation 1), the probability P+( ) that a given sequence of + s is exactly of length is given by ... 

Four- and five-parameter models have also been proposed. An example of the four-parameter model is the Cross-Williamson model given by (14)... 

A crucial point of the four-parameter model is that T2 is assumed to be independent of cross-link density which is valid only in a first approximation [FU13]. Moreover, the experimental relaxation signal can be modelled at short and intermediate time scales with good agreement without the inclusion of dangling chains in the model the free induction decay of one such cross-link chain can be written as a product of an inhomogeneous and... 

Neither the simple Maxwell nor Voigt model accurately predicts the behavior of real polymeric materials. Various combinations of these two models may more appropriately simulate real material behavior. We start with a discussion of the four-parameter model, which is a series combination of the Maxwell and Voigt models (Figure 14.9). We consider the creep response of this model. 

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. 

The Nitta et al. s equation contains four parameters, Sq, n, b, and u. As a first approximation, we can set u = 0 to reduce the number of parameter by one. This is reasonable in systems where the adsorbate-adsorbate interaction is not as strong as the adsorbate-adsorbent interaction. If the fit of the three parameter model with the data is not acceptable, then the four parameter model is used. In the attempt to... 

By substituting Equations (6.16) and (6.17) back into Equations (6.14) and (6.15), the four-parameter model for describing the temperature dependence of the LC refractive indices is derived, as [24]... 

At aU technically relevant temperatures, polymers deform by creep. To describe the time-dependence of plastic deformation, we again exploit equation (8.3). In contrast to the viscoelastic deformation, there is no restoring force in viscoplasticity. Equation (8.3) is thus used to describe the dashpot element connected in series in the four-parameter model from figure 8.7(b). 

By eliminating various elements in the four-parameter model the response of a Maxwell fluid, Kelvin solid, three-parameter solid (a Kelvin and a spring in series) can be obtained and the model can be used to represent thermoplastic and/or thermoset response as illustrated in Fig. 3.13. For example the creep response of a three-parameter solid is obtained by eliminating the free damper in Eq. 3.44 and gives the creep and creep recovery response shown in Fig. 3.13 for a crosslinked polymer. 

The four-parameter model (Figure 15.If) is a series combination of a Maxwell element with a Voigt-Kelvin element. Its differential equation is... 

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, retarded elastic strain, steady-state viscous flow, instantaneous elastic recovery, retarded elastic recovery, and permanent set. It also describes at least qualitatively 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 mechanisms in polymers, and can, therefore, be used to predict the influences that changes in molecular structure will have on mechanical response. The following analogies may be drawn. 

Example 15.2 Using the four-parameter model as a basis, qualitatively sketch the effects of (a) increasing molecular weight and (b) increasing degrees of crosslinking on the creep response of a linear, amorphous polymer. 

Consider, for example, the creep response of the four-parameter model (Rgure 15.8). For this model, a logical choice for Xc would be the time constant for its Voigt-Kelvin component, r]2/G2- For De1 (t Xc), the Voigt-Kelvin element and dashpot 1 will be essentially immobile, and the response will be due almost entirely to spring 1, that is, almost purely elastic. For De 0 Xc), the instantaneous and retarded elastic... 

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 four-parameter model discussed in the text consists of a series combination of Maxwell and Voigt-Kelvin elements. Here, consider a four-parameter model that consists of a parallel combination of Maxwell (Gi, tji) and Voigt-Kelvin (Gi, rj2) elements. 

The four-parameter model (Figure 15. If) is subjected to the following stress history, in which Xo is a constant stress ... 

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