Viscoelastic relaxation transient

Cooperative diffusion of transient polymer networks, heterogeneity mode related to polymer self-diffusion in diblock copolymers, entanglement mode, chain reptation, viscoelastic relaxation, diffusion of clusters Viscoelastic relaxation, a- and 3-relaxation... 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

In the preceding section, it was assumed that a monotonously decreasing viscoelastic function, such as the relaxation modulus, can be expressed in terms of the Laplace transform of an unknown function N s). Owing to the fact that the transient creep eompliance function, J t), is a monotonous increasing function of time, the use of a strategy similar to the one used for the relaxation modulus requires that J t) be related to a decreasing function. By taking into account that for viscoelastic solids J t) = Jg +... 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function... 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

Chapter 9 examines transient and nontransient viscoelastic functions in terms of the retardation times (comphance functions) and relaxation times (relaxation moduli) and compares retardation and relaxation times. Methods to determine retardation and relaxation spectra from compliance functions and relaxation spectra, respectively, are presented. 

The purpose of this paper is to explore various aspects of the rheological behaviour of lyotropic liquid crystalline systems. Lyotropics are often used as model systems for thermotropics because their viscoelastic behaviour seems to be quite similar (1) and solutions are much more easier to handle and can be studied more accurately than melts. The emphasis is on transient data as these are essential for verifying viscoelastic models but are hardly available in the literature. Transient experiments can also provide insight in the development of flow—induced orientation and structure. The reported experiments include relaxation of the shear stress and evolution of... 

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

The four commonly used techniques to extract information on the viscoelastic behavior of suspensions are creep-compliance measurements, stress-relaxation measurement, shear-wave velocity measurements, and sinusoidal oscillatory testing (25-27). In general, transient measurements are aimed at two types of measurements, namely, stress relaxation, which is to measure the time dependence of the shear stress for a constant small strain, and creep measurement, which is to measure the time dependence of the strain for a constant stress. 

The viscoelasticity of the worm-like micelles arises because of the entanglement of very long and flexible worm-like micelles to form a transient network, similar to a solution of flexible polymers. Unlike polymers, however, worm-like micelles break and re-form dynamically. When the network of worm-like micelles is deformed or the equilibrium conditions are suddenly changed, the relaxation occurs within a definite time, and the equilibrium condition is restored again. For a deformation with a hme period shorter than the relaxation time, Tr, the system... 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

DMA measures the viscoelastic properties of a sample using either transient or dynamic oscillatory tests. Transient tests include creep and stress relaxation. In creep, a stress is applied to the sample and held constant, while deformation is measured versus time. After some time, the stress is removed and the recovery measured. In stress relaxation, a deformation is applied to the sample and held constant the degradation of the stress required to maintain the deformation is measured versus time. The most common test is the dynamic oscillatory test, where a sinusoidal stress (or strain) is applied to the material and the resultant sinusoidal strain (or stress) is measured. Also measured is the phase difference, 8, between the two sine waves. The phase lag will be 0° for a purely elastic material and 90° for a purely viscous material. Viscoelastic materials such as polymers will exhibit an intermediate phase difference. Since modulus equals stress divided by strain, the complex modulus, E, can be calculated. From E and 8, the storage modulus, E, the loss modulus, E", and tan 8 can be calculated ... 

Viscoelastic phenomena may be described through three aspects, namely stress relaxation, creep and recovery. Stress relaxation is the decline in stress with time in response to a constant applied strain, at a constant temperature. Creep is the increase in strain with time in response to a constant applied stress, at a constant temperature. Recovery is the tendency of the material to return partially to its previous state upon removal of an applied load. The material is said to have memory as if it remembers where it came from. Because of the memory effect, in transient flows the behavior of viscoelastic fluids wUl be dramatically different from that of Newtonian fluids. Viseoelastie fluids are fiiU of instabilities. Some examples inelude instabilities in Taylor-Couette flow, in eone-and-plate and plate-and-plate flows (Larson 1992). The extrudate distortion, commonly called melt fraeture, is a notorious example of viscoelastic instability in polymer processing. The viseoelastie instability in injection molding can result in specific surface defects such as tiger stripes (Bogaerds et al. 2004). 

Two manifestations of linear viscoelasticity are creep and stress relaxation-, the respective two testing methods are known as transient tests. One can also apply sinusoidal load, an increasingly more used method of study of viscoelasticity by dynamic mechanical analysis (qv) (DMA). We shall now briefly discuss each of these three approaches. 

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