Relaxations retardation

Ferry showed that superposition required that there be no change in the relaxation/retardation mechanism with temperature and that the T values for all mechanisms must change identically with temperature. Defining the ratio of any relaxation time Tat some temperature Fto that at reference temperature T0 as aT,... 

The reduced expressions of Table 4 form a set of universal viscoelastic functions. Given the polymer molecular weight, material constants [Jg, Je, etc.), and one extremal relaxation/retardation time, one should be able to predict, roughly, the nature of the system response (within the framework of the linear models) from Eqs. (T 1)—(T 6) and Fig. 2—5. 

Isoelectric points require electrokinetic experiments as a function of pAg. pH, etc. which will be discussed in sec. 4.4. Theoretical problems are all but absent because phenomena like surface conduction and relaxation retardation vanish as 0. However, experimental problems may arise because the systems become... 

Other simplifications included the disregarding of surface conduction (i.e. only the case Du = 0 was considered) and the limitation to very simple geometries (spheres, capillaries, etc.) without double layer overlap. Inclusion of all these features is physically and mathematically extremely dlfiicult and as yet only rigorously solved under limiting conditions. In order to identify the various problems we shall, in the present section, retain the restrictions to simple geometries (emphasizing electrophoresis of spheres) and absence of double layer overlap but do automatically consider double layer polarization (i.e. the relaxation retardation, force in sec. 4.3a(i)) and always take surface... 

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

Main signs symptoms A faded, white facial color, a faint, lethargic voice, diminished appetite, lack of strength of the four extremities, loose stools, a pale tongue, and a fine, relaxed/retarded pulse... 

Main signs symptoms Abnormal vaginal discharge which is colored white or pale yellow and is clear, thin, and without odor, an ashen, white facial color, fatigue, loose stools, a pale tongue with white coating, and a relaxed/retarded or soggy, weak pulse... 

When studying a polymer on a large frequency/time scale, the response of a given material under a dynamic stimulus usually exhibits several relaxations. Moreover, the peaks are usually broad and sometimes and are associated with superposed processes. The relaxation rate, shape of the loss peak, and relaxation strength depend on the motion associated with a given relaxation process [41]. In general, the same relaxation/retardation processes are responsible for the mechanical and dielectric dispersion observed in polar materials [40]. In materials with low polarity, the dielectric relaxations are very weak and cannot be easily detected. The main relaxation processes detected in polymeric systems are analyzed next. 

Relaxations in the double layers between two interacting particles can retard aggregation rates and cause them to be independent of particle size [101-103]. Discrepancies between theoretical predictions and experimental observations of heterocoagulation between polymer latices, silica particles, and ceria particles [104] have promptetl Mati-jevic and co-workers to propose that the charge on these particles may not be uniformly distributed over the surface [105, 106]. Similar behavior has been seen in the heterocoagulation of cationic and anionic polymer latices [107]. 

The Dehye-Hbckel theory of electrolytes based on the electric field surrounding each ion forms the basis for modern concepts of electrolyte behavior (16,17). The two components of the theory are the relaxation and the electrophoretic effect. Each ion has an ion atmosphere of equal opposite charge surrounding it. During movement the ion may not be exacdy in the center of its ion atmosphere, thereby producing a retarding electrical force on the ion. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

This is different for the star core. Figure 57 provides a comparison of the spectra at two Q-values with those from an equivalent full star (sample 3). Over short periods of time, both sets of spectra nearly coincide. However, over longer periods of time, the relaxation of the star core is strongly retarded and seems to reach a plateau level. This effect may be explained by the occurrence of interarm entanglements as recently proposed by scaling arguments [135]. 

Fig. 57. Relaxation spectra of the fully labelled star ( , + ) and the star core (, ) at two different Q-values. The solid lines represent the result of a fit for the Zimm dynamic structure factor to the initial relaxation of the fully labelled star. The dashed lines are visual aids showing the retardation of the relaxation for the star core. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)...

We assume that the above solution is valid in about the same time range as the self-similar relaxation time spectrum, Eq. 1-5. The retardation time spectrum is also self-similar. It is characterized by its positive exponent n which takes on the same value as in the relaxation time spectrum. 

In Section II, models were discussed that had only a single relaxation or retardation time. Actual polymers have a large number of relaxation or retardation times distributed over many decades of time. E(t) is then the sum of individual contributions, so equation (5) becomes... 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

---