Relaxation function

Joe T and Albrecht A C 1993 Femtosecond time-resolved coherent anti-Stokes Raman spectroscopy of liquid benzene a Kubo relaxation function analysis J. Chem. Phys. 99 3244-51... 

Show that the relaxation function F can be obtained from elastic precursor measurements of D,u and u according to... 

Figure 5.18 This figure shows how the properties of a glass polyalkenoate cement change as it ages. S is the compressive strength, E the modulus, a a stress-relaxation function, and c a strain-conversion function from elastic to plastic strain (Paddon Wilson, 1976).

Fig. 5. The characteristic frequencies QR and time exponents (3 in the stretched exponential relaxation function obtained for the randomly labelled PDMS melt at 100 °C. (Reprinted with permission from [44]. Copyright 1989 Steinkopff Verlag, Darmstadt)...

The transition strongly affects the molecular mobility, which leads to large changes in rheology. For a direct observation of the relaxation pattern, one may, for instance, impose a small step shear strain y0 on samples near LST while measuring the shear stress response T12(t) as a function of time. The result is the shear stress relaxation function G(t) = T12(t)/ < >, also called relaxation modulus. Since the concept of a relaxation modulus applies to liquids as well as to solids, it is well suited for describing the LST. 

With increasing distance from the gel point, the simplicity of the critical state will be lost gradually. However, there is a region near the gel point in which the spectrum still is very closely related to the spectrum at the gel point itself, H(A,pc). The most important difference is the finite longest relaxation time which cuts off the spectrum. Specific cut-off functions have been proposed by Martin et al. [13] for the spectrum and by Martin et al. [13], Friedrich et al. [14], and Adolf and Martin [15] for the relaxation function G(t,pc). Sufficiently close to the gel point, p — pc <4 1, the specific cut-off function of the spectrum is of minor importance. The problem becomes interesting further away from the gel point. More experimental data are needed for testing these relations. 

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

The first integral denotes the rest period, — oo < t < 0, where the strain rate is zero. The second integral contains a relaxation function which we chose very broad, including relaxation times much larger than the period In/iD. Integration and quantitative analysis clearly showed (without presenting the detailed figures here) that the effect of the start-up from rest is already very small after one cycle... 

Time-crosslink density superposition. Work of Plazek (6) and Chasset and Thirion (3, 4) on cured rubbers suggests that there is one universal relaxation function in the terminal region, independent of the crosslink density. Their results indicate that the molar mass between crosslinks might be considered as a reducing variable. However, these findings were obtained from compliance measurements on natural rubber vulcanizates,... 

The following Eqns. are important for the mathematical analysis of the elements of the transient responses around the wavefront (see (5, 6 ). In these equations y. stays for the state variables, such as concentration in the ambient fluid, or on the catalyst surface, temperature etc., f is the relaxation function (e.g. 

For the special case of linear relaxation functions f. or when the integrand in Eq. (3) is invariant on the wavefront, we have the simplified Equation for the time relaxation of the propagating disturbance ... 

We can see that the different positions along the chain show distinct temperature-dependent relaxation curves. To further analyze these relaxation functions, we must Fourier transform them to determine their spectral density, which is best done employing an analytic representation of the data that... 

Now if we divide the shear stress in Equation (4.13) by the applied strain we obtain an expression in the form of a shear modulus. This term G(t) is described as the relaxation function ... 

At very short experimental times compared with tm the exponential term tends to 1. Under these circumstances the relaxation function tends to the value of the modulus of the spring. The response is simply that of the spring so that the initial stress divided by the strain gives the modulus of the spring. 

Suppose the multiple Maxwell model which describes the material we are interested in is composed of m processes each with an elasticity Gj, a viscous process with a viscosity rjj and a corresponding relaxation time ty. We can form the relaxation function by adding all these models together ... 

These two mathematical Equations (4.59) and (4.60) illustrate an important feature about linear viscoelastic measurements, i.e. the central role played by the relaxation function and the compliance. These terms can be used to describe the response of a material to any deformation history. If these can be modelled in terms of the chemistry of the system the complete linear rheological response of our material can be obtained. 

Finally it is worth noting an alternate form for the stress dependence of a series of strains. Some microstructural models utilise the memory function m t). This is the rate of change of the stress relaxation function ... 

This is the stress relaxation function, so the slope plotted as a function of time provides us with G(t). Now in the limit of short times we find the exponential tends to unity ... 

For a viscoelastic liquid (7(0) = 0. These expressions transform the stress relaxation function to the storage and loss moduli. Being Fourier trans-... 

The mathematics underlying transformation of the data from different experiments can be applied to simple models. In the case of the relationship between G (a>) and G(t) it is straightforward. To give an example, consider a Maxwell model. It has an exponentially decaying modulus with time. We have indicated that the relationship between the complex modulus and the relaxation function is given by Equation (4.117). So if we substitute the relaxation function into this expression we get... 

This expression is the transform of the relaxation function. However, it is not in a readily recognisable form. If we multiply top and bottom of the quotient by rr we get... 

This is an extended exponential. It operates within the remit of linear viscoelastic theory. So for example for a simple exponential we can show that the integral under the relaxation function gives the low shear viscosity ... 

This represents the relaxation of that stress in terms of the Young s relaxation function. Now we can express this as a viscosity by multiplying both sides by time t and integrating to get... 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

The relaxation function has been calculated and is compared with experimental data in Figure 5.16. The agreement between the model and the data is reasonable. The storage and loss moduli for a polystyrene latex have also been measured and compared to the model for the relaxation spectra. The data was gathered for a dispersion in 10 2M sodium chloride at a volume fraction of 0.35 is shown in Figure 5.20. 

The relaxation function has been used to predict the moduli and... 

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