Viscoelastic behavior quantities

Viscoelastic Measurement. A number of methods measure the various quantities that describe viscoelastic behavior. Some requite expensive commercial rheometers, others depend on custom-made research instmments, and a few requite only simple devices. Even quaHtative observations can be useful in the case of polymer melts, paints, and resins, where elasticity may indicate an inferior batch or unusable formulation. Eor example, the extmsion sweU of a material from a syringe can be observed with a microscope. The Weissenberg effect is seen in the separation of a cone and plate during viscosity measurements or the climbing of a resin up the stirrer shaft during polymerization or mixing. 

Real world materials are not simple liquids or solids but are complex systems that can exhibit both liquid-like and solid-like behavior. This mixed response is known as viscoelasticity. Often the apparent dominance of elasticity or viscosity in a sample will be affected by the temperature or the time period of testing. Flow tests can derive viscosity values for complex fluids, but they shed light upon an elastic response only if a measure is made of normal stresses generated during shear. Creep tests can derive the contribution of elasticity in a sample response, and such tests are used in conjunction with dynamic testing to quantity viscoelastic behavior. 

Before concluding this discussion of cell walls, we note that the case of elasticity or reversible deformability is only one extreme of stress-strain behavior. At the opposite extreme is plastic (irreversible) extension. If the amount of strain is directly proportional to the time that a certain stress is applied, and if the strain persists when the stress is removed, we have viscous flow. The cell wall exhibits intermediate properties and is said to be viscoelastic. When a stress is applied to a viscoelastic material, the resulting strain is approximately proportional to the logarithm of time. Such extension is partly elastic (reversible) and partly plastic (irreversible). Underlying the viscoelastic behavior of the cell wall are the crosslinks between the various polymers. For example, if a bond from one cellulose microfibril to another is broken while the cell wall is under tension, a new bond may form in a less strained configuration, leading to an irreversible or plastic extension of the cell wall. The quantity responsible for the tension in the cell wall — which in turn leads to such viscoelastic extension — is the hydrostatic pressure within the cell. 

The viscoelastic response of polymeric materials is a subject which has undergone extensive development over the past twenty years and still accounts for a major portion of the research effort expended. It is not difficult to understand the reason for this emphasis in view of the vast quantities of polymeric substances which find applications as engineering plastics and the still greater volume which are utilized as elastomers. The central importance of the time and temperature dependence of the mechanical properties of polymers lies in the large magnitudes of these dependencies when compared to other structural materials such as metals. Thus an understanding of viscoelastic behavior is fundamental for the proper utilization of polymers. 

Linear viscoelasticity is the simplest type of viscoelastic behavior observed in polymeric liquids and solids. This behavior is observed when the deformation is very small or at the initial stage of a large deformation. The relationship between stress and strain may be defined in terms of the relaxation modulus, a scalar quantity. This is defined in Equation 22.7 for a sudden shear deformation ... 

We have used the generalized phenomenological Maxwell model or Boltzmann s superposition principle to obtain the basic equation (Eq. (4.22) or (4.23)) for describing linear viscoelastic behavior. For the kind of polymeric liquid studied in this book, this basic equation has been well tested by experimental measurements of viscoelastic responses to different rate-of-strain histories in the linear region. There are several types of rate-of-strain functions A(t) which have often been used to evaluate the viscoelastic properties of the polymer. These different viscoelastic quantities, obtained from different kinds of measurements, are related through the relaxation modulus G t). In the following sections, we shall show how these different viscoelastic quantities are expressed in terms of G(t) by using Eq. (4.22). 

It should be repeated that H and L are not essential to the treatment of lineaf viscoelastic behavior. The predictions of molecular theories can, for example, be expressed in the forfti of directly measurable quantities such as G t), and... 

It is found empirically that the concentration dependence of 7 can be expressed by proportionality of n r) oo/Vs) to c, and the proportionality constant can be shown to be the high-frequency intrinsic viscosity [rf ]oo- An example is shown in Fig. 9-27 for five linear polystyrenes with widely different molecular weights and three branched samples.The high-frequency intrinsic viscosity is independent of branching as would be expected for a quantity which reflects a very local motion within the molecule it is also independent of molecular weight for M > 19,800. At M = 19,800, it is almost equal to the ordinary (steady-flow) intrinsic viscosity. However, W]oo does depend on detailed chemical structure as shown in Fig. 9-28, where data for several polymers are similarly plotted. Thus the high-frequency behavior is in a sense just the opposite of the low-frequency viscoelastic behavior... 

Other features of the viscoelastic behavior in the terminal zone are represented by the constants rjo and Rearrangement of equations 7 and 9, together in the equation 51 of Chapter 9 with omission of r)s as appropriate for undiluted polymer, gives for these quantities... 

If a constant shear rate 7 is imposed on a viscoelastic liquid, the stress rises as described by cr(t) or the time-dependent viscosity function Tj+Ct) = o (t)/7, and these quantities approach their steady-state values as steady flow is reached. If the steady-state flow is terminated, the subsequent stress relaxation is described by or the time-dependent viscosity function rj t)y, discussed in Sections C2 and C3 of Chapter 1. If the viscoelastic behavior is linear, these functions are monotonic and related to the relaxation modulus G(t) by equations 13 and 14 of Chapter 1. The functions and are independent of 7 and their sum is simply jjo- At large strain rates, however, both and T (t) become strongly dependent on 7. 

In oscillatory shear flow, a sinusoidal strain is imposed on the fluid under test. If the viscoelastic behavior of the fluid is linear, the resulting stress will also vary sinusoidally, but it will be out of phase with the strain, as schematically shown in Figure 5.5. Since the sinusoidal motion can be represented in the complex domain, the following complex quantities may be defined ... 

They implicitly very well recognize the molecular origin of the viscoelastic behavior of a material — but explicitly they do not refer to noncontinuous quantities such as the discrete size, structure, and arrangement of molecules, the anisotropy of molecular properties, or the distribution of molecular stresses or stored strain energy. If individual molecular events or discontinuities of stress or strain are either not discernible or not of particular importance, a representation of a solid as a continuum is quite adequate. 

The quantity is a friction coefficient and Un is a strain rate. Figure 1.6, showing a spring and a dashpot, is a pictorial representation of Eqs.(1.29) and (1.30). Figure 1.7 shows three simple combinations (a, b, c) of the two elements depicted in Fig. 1.6. These combinations may be translated into differential equations and serve as simple models for so called viscoelastic behavior (Wrana 2009). Important viscoelastic materials are the tread compounds in automobile tires. In the following we merely focus on sketch (a). Its translation is... 

The significance of G G tan 5, Tj, and Tj is that they can be determined experimentally and used to characterize real materials. These parameters depend on frequency and temperature, and this dependence can be used to define behavior. For example, viscoelastic fluids are often characterized by log—log plots of one or more of these quantities vs the angular frequency CO, as shown in Figure 21, which illustrates the behavior of a polymer melt (149). 

Suppose one wanted to compare the behavior of two polymers and their blends. Let us define the signal as the difference between the logarithims of the viscoelastic quantities and the noise as the error calculated for a specific set of viscoelastic properties associated with a specific composition. The signal to noise ratio would have the appearance of the three curves shown in Figure 2 for a PMMA/Hytrel blend >3/1. Selection of the optimum conditions for comparison is apparent in that figure. Emphasis should be placed at those temperatures with high signal/noise ratios. 

It is customary in dealing with linear (or linearized) behavior of dynamical systems to describe the properties of viscoelastic materials as complex quantities (2). For example, we write Young s modulus... 

An exception to the generally observed drag reduction in turbulent channel flow of aqueous polymer solutions occurs in the case of aqueous solutions of polyacrylic acid (Carbopol, from B.F. Goodrich Co.). Rheological measurements taken on an oscillatory viscometer clearly demonstrate that such solutions are viscoelastic. This is also supported by the laminar flow behavior shown in Fig. 10.20. Nevertheless, the pressure drop and heat transfer behavior of neutralized aqueous Carbopol solutions in turbulent pipe flow reveals little reduction in either of these quantities. Rather, these solutions behave like clay slurries and they have been often identified as purely viscous nonnewtonian fluids. The measured dimensionless friction factors for the turbulent channel flow of aqueous Carbopol solutions are in agreement with the values found for clay slurries and may be correlated by Eq. 10.65 or 10.66. The turbulent flow heat transfer behavior of Carbopol solutions is also found to be in good agreement with the results found for clay slurries and may be calculated from Eq. 10.67 or 10.68. 

As noted in Subsection 24.1.2, viscoelasticity of polymers represents a combination of elastic and viscous flow material responses. Dynamic mechanical analysis (DMA, also called dynamic mechanical thermal analysis, DMTA) enables simultaneous study of both elastic (symbol ) and viscous flow (symbol ") types of behavior. One determines the response of a specimen to periodic deformations or stresses. Normally, the specimen is loaded in a sinusoidal fashion in shear, tension, flexion, or torsion. If, say, the experiment is performed in tension, one determines the elastic tensile modulus E called storage modulus and the corresponding viscous flow quantity E" called the loss modulus. 

Analysis of the dynamical viscoelastic quantities shows that the relaxation spectrum H r) of the two-dimensional network goes as H(t) 1/r [65,68-70]. Hence 2-D networks do indeed show dynamical behavior intermediate between that of linear chains and that of 3-D networks. Moreover, in a fractal picture, square networks may be viewed as being fractals and as having a spectral dimension of 2. Now H(r) 1/t leads to an -behavior for the storage modulus G (a>), see Eig. 4, and to G(f) 1/t. 

The relaxation spectriun (as well as the viscoelastic quantities derived from it) obeys features typical of two dimensions while one finds the standard Rouse chain behavior, H(r) at rather short times (on small... 

For example, in Chapter 6, to begin with three parameters, p (shear stress), e (shear strain), and E (modulus or rigidity), are introduced to define viscosity and viscoelasticity. With respect to viscosity, after the definition of Newtonian viscosity is given, a detailed description of the capillary viscometer to measure the quantity t follows. Theories that interpret viscosity behavior are then presented in three different categories. The first category is concerned with the treatment of experimental data. This includes the Mark-Houwink equation, which is used to calculate the molecular weight, the Flory-Fox equation, which is used to estimate thermodynamic quantities, and the Stockmayer-Fixman equation, which is used to... 

The practical significance of the terminal relaxation time r in several qualitative aspects of behavior has already been mentioned in connection with equation 7 for polymers of low molecular weight. The same considerations apply to polymers of high molecular weight, where t (or Ta, in the framework of the tube model) and the other two viscoelastic constants ijo and Jg which characterize the terminal zone are even more important in the processing and use of polymeric materials. Rough estimates of these quantities can sometimes be made from the equations in Section C3 above for practical purposes. 

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