Amorphous relaxation time

The average polymer melt relaxation times between the processing temperature Tp and the solidifying temperature (the Tg in amorphous polymers and somewhere between Tg and with polycrystalline polymers). 

Fig. 6.10 Mossbauer spectra of amorphous frozen aqueous solutions with the indicated concentrations of [Fe(H20)6]. The spectra were obtained at 80 K. Rough estimates of the relaxation times are given...

Using the time-dependent aspect of state diagrams, Roos (2003) illustrated the effects of temperature, water activity, or water content on relaxation times and relative rates of mechanical changes in amorphous systems (Figure 36). This diagram can be considered as a type of mobility map, where mobility increases (relaxation time decreases) as temperature and/or water content/activity increases. Le Meste et al. (2002) suggested the establishment of mobility maps for food materials showing characteristic relaxation times for different types of molecular motions as a function of temperature and water content. 

LIG. 36 Effects of temperature, water activity, or water content on relaxation times and relative rates of mechanical changes in amorphous materials [reproduced with permission from Roos (2003)]. M. Peleg reference is Peleg (1996). 

The Fourier transform of this quantity, the dynamic structure factor S(q, ffi), is measured directly by experiment. The structural relaxation time, or a-relaxation time, of a liquid is generally defined as the time required for the intermediate coherent scattering function at the momentum transfer of the amorphous halo to decay to about 30% i.e., S( ah,xa) = 0.3. 

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

A method of characterising transport mechanisms in solid ionic conductors has been proposed which involves a comparison of a structural relaxation time, t, and a conductivity relaxation time, t . This differentiates between the amorphous glass electrolyte and the amorphous polymer electrolyte, the latter being a very poor conductor below the 7. A decoupling index has been defined where... 

TD-NMR is performed on NMR spectrometers that are equipped with lower magnetic field strength magnets with relatively low field homogeneity. Thns, relatively short FlDs on the order of a few milliseconds are obtained and FT of this signal yields broad lines from which no chemical detail can be obtained. However, the data is rich in information regarding the relative amonnts of different phases that are present in a sample, snch as water and oil, liqnid and solid, crystalline and amorphous. The data can be approached in two ways - analysis of the FID or analysis of relaxation times and their relative distributions. 

H( P) as a function of the nondimensional relaxation time, 7 = u/x, the ratio of local to global relaxation times, and p. When Equations 3 and 5 are used simultaneously in analyzing experimental data, we have found that p= 1/2 for most amorphous polymers which will also be assumed for lightly crosslinking systems. 

The marked difference in the relaxation times for the kaolinite and silica may be attributed to the nature of the surface. Intuitively, the hydrogen bonding which influences the increased structure at the kaolinite surface would be expected to give shorter values for the relaxation time. However this is not observed in the simulations. Instead, shorter values are seen for the silica surface which is a result of water molecules becoming trapped in the cage-like amorphous silica surface. This reflects experimental results where precipitated silica surfaces are microporous and water inclusion in the surface is common. 

The classic example of a NEAS is a supercooled liquid cooled below its glass transition temperature. The liquid solidifies into an amorphous, slowly relaxing state characterized by huge relaxational times and anomalous low frequency response. Other systems are colloids that can be prepared in a NEAS by the sudden reduction/increase of the volume fraction of the colloidal particles or by putting the system under a strain/stress. 

The sensitivity of the three relaxation times to the molecular dynamics and structure will be discussed in a subsequent section. The general temperature dependence of Tj, T1 and T2 for a typical linear amorphous polymer with one side group attached to a backbone is shown in Fig. 4. 

As a liquid is cooled at a finite rate, the relaxation time spectrum will shift to longer times and a temperature region will eventually be reached where the sample is no longer in volume equilibrium. If the sample continues to be cooled at this rate it will become a glass. A glass is a nonequilibrium, mechanically unstable amorphous solid. If the sample is held at a fixed temperature near Tg the volume will relax towards its equilibrium value. In this section we will restrict our attention to equilibrium liquids at temperatures near... 

It is often found that spin 3 nuclei have very long T, relaxation times (up to several hours in some cases) particularly in amorphous solids. Quadrupolar nuclei, however, generally relax quite fast, which makes them of special interest in the study of the solid state. 

Since the relaxation mechanisms characteristic of the constituent blocks will be associated with separate distributions of relaxation times, the simple time-temperature (or frequency-temperature) superposition applicable to most amorphous homopolymers and random copolymers cannot apply to block copolymers, even if each block separately shows thermorheologically simple behavior. Block copolymers, in contrast to the polymethacrylates studied by Ferry and co-workers, are not singlephase systems. They form, however, felicitous models for studying materials with multiple transitions because their molecular architecture can be shaped with considerable freedom. We report here on a study of time—temperature superposition in a commercially available triblock copolymer rubber determined in tensile relaxation and creep. 

The temperature dependence of the relaxation times introduced into mechanical equations by means of Eq. (2.93) was used to calculate the residual stresses in cooling amorphous polymers. 

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