Relaxation Time Τα Research Articles

High-precision computer simulations of entangled polymer chains: 1. Determination of entanglement parameters of bond-fluctuation model

High-precision computer simulations of bond-fluctuation (BF) model (volume fraction φ=0.5) are performed in the transition region between the non-entanglement and the entanglement regime. Defects of the model called X-traps are newly found which result in serious errors in the long-time behavior of the system. For samples free from X-traps, diffusion coefficient D of the center of mass, relaxation times τα for Rouse coordinate Rα (α=1,2) and τL for the end-to-end vector L are determined for chains of length N=16–180 within few percent of statistical errors. Their N-dependence changes around N=100 at which entanglement coupling is supposed to begin. By comparing D obtained by the simulations with experimental data of polystyrene melts and solutions, the average chain length per entanglement Ne was estimated to be 89, which is much larger than 30–42 reported by Paul et al. (J Phys II 1991;1:37). To see the origin of the discrepancy, statistical errors and system size effects are studied in detail and it was found that, to determine D within few percent of error, the number of independent chain samples Nsample should be larger than 10 000 and the size of simulation cells ℓcell should be larger than 4Rg; these conditions are not satisfied in the previous simulations. Critical chain length Ncη for entanglement of BF model is estimated to be Ncη=170 using the empirical relationship Ncη/Ne=1.92 for polystyrene melts. It is argued that Ncη=170 is a universal parameter of entanglement but Ne=89 is a specific value for polystyrenes and it may change with the materials with which comparisons are made.

LARGE SCALE LONG-LIVED HETEROGENEITY IN THE DYNAMICS OF SUPERCOOLED LIQUIDS

The local mobility of particles in highly supercooled liquids is demonstrated to be spatially heterogeneous on time scales comparable to the structural relaxation time τα. The particle motions in the active regions dominantly contribute to the mean square displacement, giving rise to a diffusion constant systematically larger than the Stokes–Einstein value. The diffusion process eventually becomes homogeneous on time scales longer than the life time of the heterogeneity structure (~ 3τα). The heterogeneity structure in the local mobility is very analogous to the critical fluctuation in Ising spin systems with their structure factor being excellently fitted to the Ornstein–Zernike form.

Relation between some secondary relaxations and the α relaxations in glass-forming materials according to the coupling model

Some secondary or β relaxations in glass-forming materials involve molecular motions that bear strong resemblance to the primitive α relaxations of the coupling model, although the two are not identical. For these β relaxations, at the glass transition temperature Tg the relaxation time τβ(Tg) is expected to be shorter than but not too different in order of magnitude from τ0(Tg), the primitive α-relaxation time at Tg. The latter can be calculated by the coupling model from the relaxation time τα(Tg), the exponent (1−n) of the Kohlrausch–Williams–Watts (KWW) correlation function exp[−(t/τα)1−n], and the experimental crossover time, tc≈2 ps, of the α relaxation. From experimental data of β and α relaxations in a variety of glass-forming materials, it is found that τβ(Tg) and τ0(Tg) are close to each other in order of magnitude as anticipated. The results indicate these β relaxations indeed bear some close relation to the corresponding primitive α relaxation, although they are not the same process. Since the relaxation times of the majority of these β relaxations have the Arrhenius temperature dependence, τβ(T)=τβ∞ exp(Eβ/RT), where τβ∞ is of the order of 10−13–10−16 s, knowing, approximately, the value of τβ(T) at one temperature Tg means the location of the β relaxation in the relaxation map can be roughly determined from the α relaxation. The findings can be restated as the empirical result: there exists a strong correlation between the value of log[τβ(Tg)] and the KWW exponent (1−n) of the α relaxation in many glass-formers. A smaller KWW exponent of the α relaxation corresponds to shorter τβ(Tg) or smaller log[τβ(Tg)]. This remarkable cross correlation between the α relaxation and the β relaxation should be of interest for any model or theory of molecular dynamics of glass formers.

Temperature-domain analysis of primary and secondary dielectric relaxation phenomena in a nonlinear optical side-chain polymer

The investigation of dipole relaxation processes in nonlinear optical (NLO) polymers containing chromophore dipoles with large hyperpolarizabilities is important for optimizing the poling process and for predicting the long-term stability of the poling-induced order. The primary or α relaxation is difficult to assess by dielectric spectroscopy in polymers with high glass transition temperature due to thermally induced chromophore degradation. A fast experimental procedure is developed for the investigation of dielectric relaxation processes in NLO polymers, without severely inducing chromophore degradation. The procedure is based on the measurement of the dielectric function ε̃(T)=ε′(T)−iε″(T) at a few frequencies from 30 Hz to 30 kHz, while heating the polymer at a constant rate. The complex plane representation of the temperature-dependent dielectric function is used to determine the distribution of relaxation times, while the temperature-dependent mean relaxation time τα(t) is numerically determined from the dielectric loss ε″(T). With only three decades in frequency, information on five decades in time or 90 K in temperature is gained above the glass transition temperature. The strong α and the weak β relaxation below the glass transition are separately investigated by thermally stimulated depolarization after a suitable two-step poling procedure. The method has been applied to a typical polyimidelike side-chain nonlinear optical polymer with modified Disperse Red 1 chromophores. Even below room temperature, a γ-relaxation process is observed, demonstrating significant mobility of the chromophores much below the glass transition. From the results of thermally stimulated depolarization it is concluded that the initial fast decay of the electro-optical response to a temporally stable value is related to the partial depolarization caused by the β relaxation.