John T. Bendler

Time‐Scale Invariance in Transport and Relaxation

An early theme in probability was calculating the fair ante for various games of chance. Nicolas Bernoulli introduced a seemingly innocent game, first published in 1713, that yielded a paradoxical result. The result has become known as the St. Petersburg paradox, because of an analysis written later by Daniel Bernoulli in the Commentary of the St. Petersburg Academy.

Generalized Vogel law for glass-forming liquids

A model for non-Arrhenius structural and dielectric relaxation in glass-forming materials is based on defect clustering in supercooled liquids. Relaxation in the cold liquid is highly hindered, and assumed to require the presence of a mobile defect to loosen the structure near it. A mild distribution of free-energy barriers impeding defect hopping can generate a wide distribution of waiting times between relaxation events. When the mean waiting time is longer than the time of an experiment, no characteristic time scale exists. This case directly yields the Kohlrausch-Williams-Watts (KWW) relaxation law. A free-energy mismatch between defect and nondefect regions produces a defect-defect attraction, which can lead to aggregation. This may occur in defect-rich “fragile” liquids which also exhibit Vogel kinetics. Defect aggregation and correlation in the “high-temperature” region above the critical consolute temperatureTc is described using the Ornstein-Zernike theory of critical fluctuations. For a defect correlation length divergence (T-Tc)-γ/2, a generalized Vogel law for the structural relaxation time τ results: τ=τ0exp[B./(T-Tc)1.5γ] In the mean-field limit (γ=1) this provides as good an account of dielectric and structural relaxation in glycerol,n-propanol, andi-butyl bromide as does the original Vogel law, and for the mixed salt KNO3−Ca(NO3)2 and B2O2 it also describes kinetics over their entire temperature ranges. A breakdown of the Vogel law in the immediate vicinity ofTg is avoided, and the need to invoke extra low-temperature mechanisms to explain an apparent “return to Arrhenius behavior” is removed.

Defect-diffusion models of relaxation

Abstract A model of molecular relaxation in glassy materials is presented to describe configurational fluctuations which decay via the mediation of mobile defects. A mild distribution of free-energy barriers impeding the defect motion can generate a wide distribution of waiting times between defect displacements. When the mean waiting time is longer than the time of an experiment, no characteristic time scale exists. This case directly yields the Kohlrausch- Williams-Watts (KWW) stretched exponential relaxation law. Correlations between defects described by the Ornstein-Zernicke theory of critical fluctuations cause aggregation of defects and produce a Vogel-Fulcher (VF) type law for the disappearance of singlet defects, and therefore a VF law for the dipole relaxation time. The model is also applied to predict the molecular weight dependence of mechanical relaxation times (physical aging) in glassy polymeric systems.

Levy (stable) probability densities and mechanical relaxation in solid polymers

Early investigations by Weber, R. and F. Kohlrausch, Maxwell, and Boltzmann of relaxation in viscoelastic solids are reviewed. A two-state model stress-tensor describing strain coupling to internal conformations of a polymer chain is used to derive a linear response version of the Boltzmann superposition principle for shear stress relaxation. The relaxation function of Kohlrauschφ(t)= exp[−(t/τα)] is identical to the Williams-Watts empirical dielectric relaxation function and in the model corresponds to the autocorrelation function of a segments differential shape anisotropy tensor. By analogy with the dielectric problem, exp[−(t/τα)] is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distributionAt−α for defect motion. The exponent a is the fractal dimension of a hierarchical scaling set of defect hopping times. Integral transforms ofφ(t) needed for data analysis are evaluated; the cosine and inverse-Laplace transforms are stable probability densities. The reciprocal kernel for short-time compliance is discussed.

The Need to Reconsider Traditional Free Volume Theory for Polymer Electrolytes

Abstract Pressure–temperature–volume ( PVT ) data have been obtained for poly(propylene glycol) of molecular weight 1025 containing LiCF 3 SO 3 in the mole ratio 20:1. The PVT data were used to calculate the specific volumes, V / V P =0, T =296 K , associated with the pressures and temperatures for previously published variable temperature, high-pressure electrical conductivity data. It is found that the electrical conductivity depends strongly on temperature at a constant volume. Consequently, traditional free volume theory is not consistent with the data. Finally, it is shown that the features of the electrical conductivity data can be accounted for by a recently developed generalized Vogel theory.

Effect of high pressure on the electrical conductivity of ion conducting polymers

Complex impedance and differential scanning calorimetry (DSC) studies have been carried out on poly(propylene glycol) (PPG) with an average molecular weight of 1025 and poly(ethylene glycol mono-methyl-ether) (PEG) with an average molecular weight of 350, both containing NaCF3SO3 in an approximately 20:1 ratio of polymer to salt. The impedance studies were carried out over a range of frequencies, temperatures and pressures. As expected, PEG:NaCF3SO3 exhibits the tendency to crystallize while PPG:NaCF3SO3 is a glass-forming liquid. The fit to the zero pressure data for PPG:NaCF3SO3 using a recently developed generalized Vogel equation (based on a defect diffusion model) is better than that for the standard VTF equation while for PEG:NaCF3SO3 the two expressions give about the same level of fit to the data. In the theory, the effect of pressure is due to a pressure dependent critical temperature, Tc, and a defect–defect separation that follows the dimensions of the material. It is found empirically that the pressure dependence of Tc is similar to the pressure dependence of the glass transition temperature, Tg, for structurally related polymers containing no salt. However, the details of the relationship between Tc and Tg remain to be determined.

Strain, birefringence, and volume relaxation and recovery in polymer glasses

Abstract Stress and volume relaxation, creep, strain recovery and thermally activated strain recovery measurements have been made on bisphenol-A polycarbonate, a polyetherimide, and poly(2,6-dimethyl-1,4-phenylene oxide) as functions of thermal history, room temperature aging, strain, hold time and temperature. It was observed that as the temperature was increased, the amount of persistent strain not only increased, but also affected the stress relaxation behavior, and was observed to be partially recoverable at the test temperature. These data are interpreted in terms of a network model wherein it is proposed that the motion of defects between and past entanglements, their loss at chain ends, and the disentanglement of chain ends from other segments serve as the primary mechanisms of relaxation in the glassy state.

Diffusion bonding between BPA polycarbonate and poly(butylene terephthalate)

Abstract The diffusion bonding of poly(butylene terephthalate) (PBT) to BPA polycarbonate (BPAPC) has been studied over the temperature range 100–230°C. The studies were carried out on laminates of the two polymers prepared by a novel spray application procedure. Variations in adhesive bond strength with temperature at fixed times and with time at fixed temperatures were determined by means of peel tests. Concurrently, TEM studies of interfacial cross sections were used to monitor the extent of diffusion. The results suggest that adhesion is strongly coupled to the level of solubility between the two resins and is independent of the rate or extent of diffusion once an equilibrium number of interfacial chains has been established. A phenomenological model is presented.

Analysis of dielectric loss data using the Williams–Watts function

There are many polymeric materials whose dielectric properties can be derived from the Williams–Watts relaxation function φα(t)=exp [−(t/τ)α] . In this paper we propose a method for estimating the parameters α, τ, and e(0)−e(∞), from dielectric loss data. In contrast to earlier techniques the parameters can be estimated one at a time. Simple approximations to the required functions are given accurate to better than 1%.

Physical Basis of Fragility

Fragility of glass-forming liquids in the supercooled region is considered in the context of a defect diffusion theory. It is shown that a necessary condition that a liquid be “fragile” is that there is an attractive interaction between the mobile defects, i.e., that the defects cluster with falling temperature. The relationship between the model parameters and a widely used fragility index is described. Each of the model parameters provides a contribution to and insight into the fragility value. The behavior of exceptional cases, such as orientationally disordered crystals and aliphatic monohydric alcohols, is also naturally accounted for in terms of the model.