Viscoelastic behavior, regions

FIGURE 8.9 Flow behavior of a viscoelastic system. Regions AB and CD represent viscoelastic behavior region BC represents structural breakdown during steady shear. 

Dynamic mechanical experiments yield both the elastic modulus of the material and its mechanical damping, or energy dissipation, characteristics. These properties can be determined as a function of frequency (time) and temperature. Application of the time-temperature equivalence principle [1-3] yields master curves like those in Fig. 23.2. The five regions described in the curve are typical of polymer viscoelastic behavior. 

The mean times t and tw will be called the number-average and weight-average relaxation times of the terminal region, and tw/t can be regarded as a measure of the breadth of the terminal relaxation time distribution. It should be emphasized that these relationships are merely consequences of linear viscoelastic behavior and depend in no way on assumptions about molecular behavior. The observed relationships between properties such as rj0, J°, and G and molecular parameters provides the primary evidence for judging molecular theories of the long relaxation times in concentrated systems. 

The effects attributed to entangling interactions, e.g., the plateau region in stress relaxation, appear most prominently at high concentrations and in melts. It is important, however, to distinguish this interaction from other types which are present at lower polymer concentrations. To make the separation properly, it is necessary to examine viscoelastic behavior at all levels of concentration, beginning at infinite dilution. 

In 97- it was also shown on the basis of dilatometric data that the free-volume of PMMA in the mixture with polyvinylacetate increases with the increase in FVA concentration. In 98) a large difference was reported in the viscoelastic behavior of block copolymer from that predicted by WLF theory. This theory is believed to be useful only near the Te of each component, not in the broad temperature interval including the transition from glassy to rubberlike state. This anomaly is thought to be connected with certain motions in the interphase regions, which should be looked upon as independent components of the mixture. 

Butter and milk fat exhibit viscoelastic behavior at small stresses (Chwiej, 1969 Pijanowski et al., 1969 Shama and Sherman, 1970 Sherman 1976 Shukla and Rizvi, 1995). To probe this behavior, a very small stress or deformation is applied to a sample and the relationships between stress, strain and time are monitored. Viscoelastic testing is performed in the linear viscoelastic region (LVR) where a linear relationship between stress and strain exists and where the sample remains intact. Depending on the material, this region lies at a strain of less than 1.0% (Mulder and Walstra, 1974) or even less than 0.1% (Rohm and Weidinger, 1993). Figure 7.10 shows the small deformation test results for milk fat at 5°C. 

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

So Figures 13-78 and 13-80 are our idealized representations. This correspondence is obviously interesting and important, but we will defer a discussion of the molecular origin of this until later. First we want to explore this time-temperature correspondence and the various regions of viscoelastic behavior in a little more detail. 

Sketch a plot of the modulus of an amorphous polymer as a function of temperature, labeling the different regions of viscoelastic behavior. Briefly describe the types of relaxation behavior that occur in each region. 

Fig. 4. The five regions of viscoelastic behavior. All polymers exhibit these five regions, but crosslinking, crystallinity, and varying molecular weight alter the appearance of this generalized curve. The loss modulus Tg peak appears just after the storage modulus enters the glass transition region.

Further aspects of the viscoelastic behavior of ESIs which have been reported to date include linear stress relaxation behavior of amorphous ESI [40] and the creep behavior of amorphous ESI in the glass transition region [41]. Chen et al. [42]... 

Figure 4.13 shows the deformation behavior of a viscoelastic material. In response to a stress at time tj, the material deforms exponentially with time. When the stress is removed (or lowered, as is the case when the pad passes from a high region to a low region), the viscoelastic material rebounds with an exponential time dependence. However, the strain in the viscoelastic material does not necessarily completely relax upon unloading. Viscoelastic behavior is described mathematically by ... 

The deformation behavior of two pads are shown in Figure 4.14. Which pad shows greater elastic behavior Which pad shows viscoelastic behavior A CMP process is required to planarize a surface with a maximum step height of 5000 A. If the velocity of the pad is 50 cm/sec, which pad will polish faster inside a 5 pm wide trench A 10 pm wide trench A 15 pm wide trench What is the maximum width of a low region that may be planarized by each pad (Note assume that the pad relaxation and deformation behaviors (curves) are similar and symmetric.)... 

Another aspect of viscoelastic behavior is the influence of temperature and in the slow crack propagation region is has also been well documented. Marshall et al. observed that in PMMA the crack speed curves are shifted to lower Kpvalues with increasing temperature and that also Ki decreases in the temperature range from -60 Cto 80 C. 

The morphological location of the fibrous protein principally responsible for the deformation and viscoelastic behavior is uncertain. Both the cell membrane and intracellular regions are composed of fibrous proteins which differ considerably in amino acid composition. Since the alpha-keratin within the cells shows few orientation properties until high elongations, it has been suggested that the membrane proteins determine the viscoelastic behavior at low deformations (84). 

There have been a number of studies that demonstrate that crystallized AMF and butter exhibit linear (ideal) viscoelastic behavior at low levels of stress or strain (4), where the strain is directly proportional to the applied stress. For most materials, this region occurs when the critical strain (strain where structure breaks down) is less than 1.0%, but for fat networks, the strains typically exceed 0.1% (4, 66). Ideally, within the LVR, mUkfat crystal networks will behave like a Hookean solid where the stress is directly proportional to the strain (i.e., a oc y), as shown in Figure 15 (66, 68). Within the elastic region, stress will increase linearly with strain up to a critical strain. Beyond that critical strain (strain at the limit of linearity), deformation of the network will occur at a point known as the yield point. The elastic limit quickly follows, beyond which permanent deformation and sample fracture occurs. Beyond these points, the structural integrity of the network is compromised and the sample breaks down. 

Although motional averaging might occur in ways other than that envisioned by Cates, temperature-jump experiments have yielded values of Tbr that indicate Tbr < in the region where the relaxation is nearly monoexponential, in agreement with Cates theory. In addition, Cates theory offers distinctive predictions for the concentration-dependencies of the viscoelastic behavior these allow the theory to be tested rather stringently. To obtain these predictions, we note that in the semi-dilute regime, the mean-field reptation time is L 4>, where 0 is the volume fraction of surfactant. Hence, from Eqs. (12-31) and... 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Theoretical calculations for llquld/llquid systems predict that the viscosity goes through a maximum at the spinodal. Depending on the type of system and its regularity, the increase may be quite large for example, Larson and Frederickson ( ) predicted that for block copolymers. These authors concluded that in the spinodal region a three-dimensional network is formed and that the system exhibits non-linear viscoelastic behavior. Experimentally, sharp Increases of n near the phase separation have been reported for low molar mass solutions as well as for oligomeric and polymeric mixtures (21). 

As we have seen above, the transition that separates the glassy state from the viscous state is known as the glass-rubber transition. This transition attains the properties of a second-order transition at very slow rates of heating or cooling. In order to clearly locate the region of this transition and to provide a broader picture of the temperature dependence of polymer properties the principal regions of viscoelastic behavior of polymers will be briefly discussed. 

Figure 2.24 Five regions of viscoelastic behavior for a linear, amorphous polymer I (a to b), II (b to c), III (c to d), IV (d to e), and V ( e to f). Also illustrated are effects of crystallinity (dotted line) and cross-linking (dashed line).

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