Minimum Relaxation Time Research Articles

Exact solution of the AC hopping conductivity problem at low site densities

A sum rule is derived for the hopping conductivity ( sigma ( omega )) in the rate equation formalism. It is shown that the pair approximation to ( sigma ( omega )) satisfies the sum rule exactly in the limit of low site densities. Detailed comparison of the exact high-frequency expansion of ( sigma ( omega )) with that derived from the pair approximation shows that the pair approximation becomes exact for all omega in the limit of low site densities. The real and imaginary parts of ( sigma ( omega )) are calculated numerically in this limit on the basis of model assumptions about the site statistics and the intersite transition rates which are appropriate to doped crystalline semiconductors. The magnitude of the ratio of the imaginary to the real part of ( sigma ( omega )) is shown to be 0.293 lg ( omega tau 0)-1 in this case where tau 0 is the minimum pair relaxation time.

Kinetics of Stress Relaxation Properties of Oat Coleoptile Cell Wall after Geotropic Stimulation

This study describes the stress relaxation of the cell wall of oat (Avena sativa) coleoptiles after different periods of geotropic stimulation. The upper and lower tissues (with respect to gravity) of geotrophically stimulated coleoptiles exhibit different wall properties. The lower tissues are less resistant to deformation than the upper. The ratio of stress to strain is significatly less in the lower than in the upper tissue. Similarly, the relaxation time and the minimum relaxation time, derived from the Maxwell model which describes the physical characteristics of polymers, are also shorter in the lower tissue. However, the maximum relaxation time shows no difference between the upper and lower tissues of a geotropically stimulated coleoptile. The differences between the tissues begin at about 8 minutes after the commencement of stimulation, similar to the time for the initiation of dictyosome redistribution, and precede the onset of geotropism. The above responses of the cell wall of the lower tissue are similar to those induced by indoleacetic acid. The parameters of wall properties of the coleoptiles of both the control and the geostimulated fluctuate rhythmically with time. The periodic changes in wall properties of the coleoptile are compared to other cyclic physiological phenomena.

Spin-lattice relaxation in the system epoxy resin-thermoplastic polymer

In our earlier papers [1–4] we reported our findings regarding change in the molecular mobility of oligomer molecules on solid surfaces based on dielectric relaxation and NMR investigations. We found that for linear oligomers there was a single relaxation process associated with the motion of molecular segments and of the molecular chains themselves. Moreover it was found that on solid surfaces interaction between the latter and oligomer molecules (adsorption) leads to displacement of the minimum spin-lattice relaxation time T 1 towards higher temperatures. This was attributed to adsorption and to limitation of the number of possible molecular conformations near the surface with consequent stiffening of the oligomer molecules. In all the previous investigations we dealt with processes taking place at the interface with a solid substance, and only the effect of the latter on the behaviour of the boundary layer of polymer were considered. In view of the growing interest being shown in mixtures of polymers it appeared necessary to investigate the behaviour of the boundary layers in polymer-polymer or oligomer-polymer systems, the assumption being that the properties of the substrate (the latter being the more rigid polymer) may also vary as a result of interaction with the boundary layer of the other more flexible polymer or oligomer. In this investigation our aim was to clarify changes in molecular mobility in the system oligomer-thermoplastic polymer the latter having the role of filler or substrate. A study was made of relaxation processes (based on change in T 1), in the surface leyers ofED-5 and ED-6 epoxy resins coated on polymeric substrates consisting of the stren (St+-methyl methacrylate (MMA) copolymer investigated by us in reference [5]. The behaviour of a mixture of both the epoxy resins was also investigated.

Analysis of Flow of Shear Thinning Colloidal and Polymeric Systems Which Exhibit Elastic Recovery or Rigidity

Abstract The hydrodynamic-structural theory of viscosity is extended to take into account the possibility of a distribution of relaxation times. A new equation is presented which is easily applied to experimental data. The effect of a distribution of lifetimes is to extend the range of shear rate over which shear thinning occurs. When the ratio of the largest relaxation time to the smallest relaxation time is greater than 1 but less than 10 the new equation gives results in agreement with the Willamson equation. When this ratio of maximum relaxation time to minimum relaxation time is 300 ± 100 the new equation agrees with an equation recently suggested by Cross. The utility of the new equation is discussed and is illustrated by using it to determine the relaxation time spectra in polystyrene melts.

Analysis of the flow of shear thinning colloidal and polymeric systems which exhibit elastic recovery or rigidity

The hydrodynamic-structural theory of viscosity is extended to take into account the possibility of a distribution of relaxation times. A new equation is presented which is easily applied to experimental data. The effect of a distribution of lifetimes is to extend the range of shear rate over which shear thinning occurs. When the ratio of the largest relaxation time to the smallest relaxation time is greater than 1 but less than 10 the new equation gives results in agreement with the Williamson equation. When this ratio of maximum relaxation time to minimum relaxation time is 300 ± 100 the new equation agrees with an equation recently suggested by Cross. The utility of the new equation is discussed and is illustrated by using it to determine the relaxation time spectra in polystyrene melts.

Viscoelasticity of Moderately Concentrated Polymer Solution

Investigation has been made on viscoelastic properties of moderately concentrated high-polymer solutions of the order of small percentage. Tschoegl, Ferry et al. have developed a general theory on the viscoelasticity of random coiled chain polymers, adopting into it the so-called hydrodynamic interaction effect and the excluded volume effect between the segments from the well-known Rouse-Zimm theory, Making a comparison between their theoretical complex modulus and experimental ones, they have pointed out that the hydrodynamic interaction seems to decrease with increasing concentration of solutions and increasing molecular weight of polymers. With increase of interpolymer interactions the polymers will behave with increased free-draining coils.This is rather a strange thing from the theoretical view-point. It even betrays some discrepancy between the theory and the experimental results in some cases. Though the Tschoegl's calculation shows the relation G"-ωηs>G' excepting very high frequency region corresponding to the minimum relaxation time, there are some systems that give the inverse relation G"-ωηs<G' at moderately high frequency region.In the case of concentrated solutions or melts of polymers, there exists the flat, so-called“box” region of relaxation spectrum due to the entanglement of polymer chains in the longer relaxation time region than the“wedge”region corresponding to the mechanism described by the Rouse-Zimm-Tschoegl's theory. Even in our moderately concentrated solution, the relaxation spectrum may have the box region to some extent, though the entanglement is not so remarkable. This effect may decrease the average gradient of spectrum as a whole as the reduction of the hydrodynamical interaction.In this paper, we assume the relaxation spectrumH(lnτ)={τ-α τα<τ<τc=11 1=τc<τ<τM0 otherwiseα=0.63 for non-draining (Zimm's) caseα=0.50 for free-draining (Rouse's) caseand calculate the complex modulus G' and G" for some values of maximum relaxation time τM. Our complex modulus is qualitatively in good agreements with the experimental results carried out by Ferry et al. It is also pointed out that the value of τM in this case, ≈20, is well explained by Hayashi's theory of weakly coupled rubber-like network structure.

Maximum relaxation time on polymers in the vicinity of critical molecular weight

AbstractIt is a well‐known fact that the relaxation spectra in the rubbery region of linear amorphous polymers show “box‐type” functions which are characterized by three elements, τe τm and E0, the minimum relaxation time, maximum relaxation time, and height of the curve, respectively. The equation (τm − τe)/log (τm/τe) = 2.303 αM/Er(τe) can be derived by a simple treatment described in this paper. Then log τe = −log E0/α(1 −1/10K) + β log M̄w can be obtained from the above equation for the polymers synthesized by us. According to Fujita's results, however, τm is independent of molecular weight and τm becomes τ3 when M̄w conforms to Mc. For such a case the above equation is transformed into log (τm/τe − 1) − log log τm/τm = β log M̄w/Mc. On the other hand, log τm experimentally obtained was plotted against log M̄w for the same polymers and it was found that the results were raher on the straight line indicated by the former equation than on the curve shown by the latter. Also log (τm + 0.234) = —5.87 + 1.65 log M̄w was found to be suitable for the polymers in the vicinity of Mc.

Maximum Relaxation Times of PMMA

It is widely known that the“box”type curve is depicted in the rubbery region of the relaxation spectrum of a linear amorphous polymer. Three parameters, τl, τm and E0 are defined as the minimum relaxation time, the maximum relaxation time, and the height in the “box”curve function respectively.One of these characterizing factors, τm was shown to be the function of molecular weight and temperature in a quantitative form for polymers by Tobolsky and Andrews, and later by Tobolsky and Murakami in a more refined form. The method of obtaining the value of τm suggested previously by Tobolsky and Andrews, is less natural than those proposed by Tobolsky and Murakami, which is so-called Procedure X. In the previous papers, the possibility of measuring the molecular weights by Procedure X was briefly described for some amorphous polymers such as polyisobutylene, polyvinyl acetate, and polystyrene.This paper deals with such a new procedure for polymethyl methacrylate with more details.The relation among τm, a maximun relaxation time, nw, weight-average chain length, T temperature and Tg, glass transition temperature, or Ts, WLF temperature, or Td, Tobolsky's temperature is indicated as follows:logτm(sec.)=logAi-C1T-Ti/C2+T-Ti+3.4lognwThe value of i is g, s or d, changing the values of the constants C1 and C2 in each case.The samples of polymethyl methacrylate whose monomers were purified by means of vacuum distillation under the nitrogen currents were prepared in the presence of varying concentration of the initiator 2-azobis isobutyronitrile at 60°C, using lower conversion around 10%. The radical-initiated polymers thus obtained were found to have a heterogeneity index of 1. 9.If the values of logτm are plotted against that of logPn, the lines consisting of two straight lines are obtained, in which the slope has a value of 3.4 in the ranges of more than logPn=3.4, or about Pn=2500, and has a value of approximately unity in the ranges of less than logPn=3.4.It is interesting to consider that the knick point by two straight lines may have a close relation with the critical molecular weight Mc, though this point looks located at the place somewhat higher than Mc of polymethyl methacrylate.If the value of {logτm+17.44T-Tg/51.6+T-Tg} is plotted against that of log nw beautiful straight lines are obtained for the data including polyisobutylene, polyvinyl acetate, and polystyrene.Finally, the dependency of τm on temperatures is studied for polymethyl methacrylate whose slope is unity. For this polymer whose slope is unity, the following equation is obtained.logτm/nw·As'+C1=C1C2/C2+T-TsIt is found that the experimental relation between τm and temperature can be on the theoretical line shown by the above equation almost exactly.