Spectrum of relaxation times

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

The continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

At this point both the spectrum of relaxation times and the way they contribute to the modulus change, so that... 

Note 5 The relaxation spectrum (spectrum of relaxation times) describing stress relaxation in polymers may be considered as arising from a group of Maxwell elements in parallel. 

One of the methods for measurement of chemical reaction constants is the relaxation spectroscopy (Eigen, 1972). Relaxation of a system after an impact gives us a relaxation time or even a spectrum of relaxation times. For catalytic cycle with limitation, the relaxation experiment gives us the second constant whereas the measurement of stationary rate gives the smallest constant, fcrnin. This simple remark may be important for relaxation spectroscopy of open system. 

It is observed that the reduced steady-state compliance is more sensitive to chain branching than the intrinsic viscosity. Whereas the intrinsic viscosity decreases with increasing degree of branching due to the fact that the values of the longest relaxation times are decreasing, the reduced steady-state compliance decreases as a consequence of the fact that the lines of the spectrum of relaxation times come closer together [cf. Ham]. 

Rotation of the core (or its reciprocating rotary vibration) can be even more efficient in processing of high-viscous melts, for example, filled polymers, high- and superhigh-molecular polyethylene (with MM > 10s). We may assume that this is dependent upon two major causes. The introduction of a filler results in a changed spectrum of relaxation time H(9) 41-42-45). Thus, for example, introduction of 10% of chalk (by volume) into polyolefins shifts the spectrum along the axis of coordinates towards... 

These yield a spectrum of relaxation times , according to theories of linear viscoelasticity. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

The quasi-static acquisition and interpretation of the force-distance curves is straightforward for elastic materials. The information is generally incomplete and less reproducible for polymers which demonstrate viscoelastic contact, plastic deformation, and ploughing type friction. Moreover, they exhibit a wide spectrum of relaxation times from 105 to 109 Hz [121]. 

Actually, relaxation processes do not follow a first-order kinetics. In a glass there is a relatively wide variety of situations leading to the existence of a spectrum of relaxation times. A static relaxation kinetics can be approximated by... 

The analysis of the characteristic polynomial (primarily of its roots) is absolutely necessary when studying the non-steady-state behaviour of a complex chemical system. A traditional problem is to study the spectrum of relaxation times r = l/ Re/ [63]. A characteristic polynomial can be written as... 

Let us consider the concept of "relaxation in more detail since no accurate definition for it has been given previously. The term "relaxation is often used for the process by which either an equilibrium or a steady state is achieved in the system, and the relaxation time is treated as the time to achieve complete or partial thermodynamic equilibrium. It is evident that, in this context, the difference between "equilibrium and "steady state is insignificant. The concept of "relaxation time is often used for the time during which a certain function characterizing the deviation from the equilibrium or the steady state diminishes by e ( 2.718) times compared with its initial value. It is evident, however, that this definition is only correct for one-dimensional linear systems. For multi-dimensional linear systems, a spectrum of relaxation times must be used. For non-linear systems, the application of these definitions is correct only in the neighbourhood of a singular point. 

The P zone extend over a large temperature range. This is a characteristic of a secondary process which involve local motions of the lateral groups [155], They are more diversified movements with a large spectrum of relaxation times. Therefore, thermal cleaning of the t.s.c. global spectra is used to study the broad relaxation peaks of the low temperature secondary relaxation [42], This is effective because it allows one to excite only the specific transition of interest [155],... 

The moduli can be expressed in terms of the discrete spectrum of relaxation times given by... 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

The majority of the different chemical and physical properties, as well as the morphology of microemulsions, is determined mostly by the micro-Brownian motions of its components. Such motions cover a very wide spectrum of relaxation times ranging from tens of seconds to a few picoseconds. Given the complexity of the chemical makeup of microemulsions, there are many various kinetic units in the system. Depending on their nature, the dynamic processes in the microemulsions can be classified into three types ... 

The various relaxation processes that occur in entangled chains may be so complex and coupled that only a smooth distribution or spectrum of relaxation times may actually exist. Nevertheless, the concept of modes and even models that are characterized by just one or two relaxation rimes is useful when you start out in this subject, as it gives you a good feel for what is going on. Just keep in mind that the dynamics of entangled polymer chains is a complex subject and one where we presently only have rough theoretical models. 

This parameter is called the relaxation time and what we hope you immediately grasp is that perhaps a model that incorporates a spectrum of relaxation times would provide a better representation of the data. We will discuss such a model later and show how it has the simple form of the Maxwell equation. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

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. 

Which of these models would best give a description of viscoelastic behavior in terms of a spectrum of relaxation times (well, say 2). Again, explain why. 

If the slip parameter a is a non-zero constant, the requirement that the shear stress be a monotonically increasing function of the shear rate in simple shear flow imposes a constraint upon the viscosity ratio. Using a spectrum of relaxation times loosens this constraint and allows for more realistic fitting of the rheological data. 

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