Maxwell Relationship Research Articles

Effects of Pore Morphology and Grain Size on the Dielectric Properties and Tetragonal–Cubic Phase Transition of High‐Purity Barium Titanate

The effects of pore morphology and grain size on the dielectric behavior of high‐purity stoichiometric BaTiO3 have been intensively investigated. It was found that the dielectric constant was influenced not only by grain size but also by pore morphology. Dielectric constants below the Curie temperature could be evaluated by the Maxwell relationship for specimens with fractional density >90%ρt and be estimated by the modified Niesel's equation, but depolarization might be involved for specimens with fractional density <90%ρt. Dielectric Behavior above the Curie temperature followed the Curie–Weiss low. The Curie constants could be separated into two regions depending on the pore morphology, decreasing linearly with increasing porosity at different rates. The results suggest that the tetragonal–cubic phase transition temperature of specimens with fractional density <90%ρt is affected by depolarization due to the presence of continous channel pores. The dissipation factor was increased with increasing porosity due to the adsorption of water. In this study, a high‐density (<99%ρt), uniform, and fine‐grained (∼1.2 μm) microstructure of high‐purity stoichimetric barium titanate has been produced by using wet processing ad pressureless sintering, in which a high dielectric constant (>6100 at 25°C and 1 kHz) and a low dissipation factor (<0.025) could be achieved.

Shear, volume, enthalpy and structural relaxation in silicate melts

The measured relaxation times for shear, volume and enthalpy in Na2Si2O5 vary from 104 to 10−9 s over a temperature range of 414–1176°C and a viscosity range of 13.8–1.3 log10 Pa s. At each temperature of measurement, the observed relaxation timescales for these properties are equivalent. Over this range of viscosities, the measured relaxation times for shear, volume and enthalpy are also in agreement with the relaxation times calculated from the Maxwell relationship. The timescale of oxygen exchange between bridging and non-bridging sites in Na2Si2O5 determined using NMR techniques is within error of the relaxation time calculated from the Maxwell relationship. This is consistent with the observed relaxation in shear and volume deformation and enthalpy being due to the exchange of Si-O bonds in the melt.

Non‐Newtonian rheology of igneous melts at high stresses and strain rates: Experimental results for rhyolite, andesite, basalt, and nephelinite

The stress‐strain rate relationships of four silicate melt compositions (high‐silica rhyolite, andesite, tholeiitic basalt, and nephelinite) have been studied using the fiber elongation method. Measurements were conducted in a stress range of 10–400 MPa and a strain rate range of 10−6 to 10−3 s−1. The stress‐strain rate relationships for all the melts exhibit Newtonian behavior at low strain rates, but non‐Newtonian (nonlinear stress‐strain rate) behavior at higher strain rates, with strain rate increasing faster than the applied stress. The decrease in calculated shear viscosity with increasing strain rate precedes brittle failure of the fiber as the applied stress approaches the tensile strength of the melt. The decrease in viscosity observed at the high strain rates of the present study ranges from 0.25 to 2.54 log10 Pa s. The shear relaxation times τ of these melts have been estimated from the low strain rate, Newtonian, shear viscosity, using the Maxwell relationship τ = ηs/G∞. Non‐Newtonian shear viscosity is observed at strain rates ( ) equivalent to time scales that lie 3 log10 units of time above the calculated relaxation time. Brittle failure of the fibers occurs 2 log10 units of time above the relaxation time. This study illustrates that the occurrence of non‐Newtonian viscous flow in geological melts can be predicted to within a log10 unit of strain rate. High‐silica rhyolite melts involved in ash flow eruptions are expected to undergo a non‐Newtonian phase of deformation immediately prior to brittle failure.

The onset of non-Newtonian rheology of silicate melts

The viscoelastic behavior of silicate melts has been measured for a range of compositions (NaAlSi3O8, NaCaAlSi2O7, CaMgSi2O6, Li2Si4O9, Na2Si4O9, K2Si4O9, Na2Si3O7, K2Si3O7 and Na2Si2O5) using the fiber elongation method. A1l compositions exhibit Newtonian behavior at low strain-rates, but non-Newtonian behavior at higher strain-rates, with strain-rate increasing faster than the applied stress. The decrease in shear viscosity observed at the high strain-rates ranges from 0.3 to 1.6 log10 units (Pa s). The relaxation strain-rates, έrelax, of these melts have been estimated from the low strain-rate, Newtonian, shear viscosity, using the Maxwell relationship; έrelax=τ −1=(ηs/G∞)−1. For all compositions investigated, the onset of non-Newtonian rheology is observed at strain-rates 2.5+0.5 orders of magnitude less than the calculated relaxation strain-rate. This difference between the non-Newtonian onset and the relaxation strain-rate is larger than that predicted by the single relaxation time Maxwell model. Normalization of the experimental strain-rates to the relaxation strain-rate predicted from the Maxwell relation, eliminates the composition. and temperature-dependence of the onset of non-Newtonian behavior. The distribution of relaxation in the viscoelastic region appears to be unrelated to melt chemistry. This conclusion is consistent with the torsional, frequency domain study of Mills (1974) which illustrated a composition-invariance of the distribution of the imaginary component of the shear modulus in melts on the Na2O-SiO2 join. The present, time domain study of viscoelasticity contrasts with frequency domain studies in terms of the absolute strains employed. The present study employs relatively large total strains (up to 2). This compares with typical strains of 10−8 in ultrasonic (frequency domain) studies. The stresses used to achieve the strain-rates required to observe viscoelastic behavior in this study approach the tensile strength of the fibers with the result that some of our experiments were terminated by fiber breakage. Although the breakage is unrelated to the observation of non-Newtonian viscosity, their close proximity in this and earlier studies suggests that brittle failure of igneous melts, may, in general, be preceded by a period of non-Newtonian rheology.

Structural relaxation in silicate melts and non-Newtonian melt rheology in geologic processes

The timescale of structural relaxation in a silicate melt defines the transition from liquid (relaxed) to glassy (unrelaxed) behavior. Structural relaxation in silicate melts can be described by a relaxation time, τ, consistent with the observation that the timescales of both volume and shear relaxation are of the same order of magnitude. The onset of significantly unrelaxed behavior occurs 2 log10 units of time above τ. In the case of shear relaxation, the relaxation time can be quantified using the Maxwell relationship for a viscoelastic material; τS = ηS/G∞ (where τS is the shear relaxation time, G∞ is the shear modulus at infinite frequency and ηS is the zero frequency shear viscosity). The value of G∞ known for SiO2 and several other silicate glasses. The shear modulus, G∞, and the bulk modulus, K∞, are similar in magnitude for every glass, with both moduli being relatively insensitive to changes in temperature and composition. In contrast, the shear viscosity of silicate melts ranges over at least ten orders of magnitude, with composition at fixed temperature, and with temperature at fixed composition. Therefore, relative to ηS, G∞ may be considered a constant (independent of composition and temperature) and the value of ηS, the relaxation time, may be estimated directly for the large number of silicate melts for which the shear viscosity is known.

Temperature Dependence of the Dielectric Constant of Diamond

The temperature dependence of the dielectric constant of diamond has been measured over the temperature range 50-2OOc. The value of E-ldc dT over this range is + 1 x 10-j. Details of the method of measuring the temperature coefficient of dielectric constant are also given. The magnitude and sign of c-ldc, dT for diamond has been theoretically calculated using Maxwell's relationship and Kramers-Heisenberg theory. The agreement between theoretical and experimental values is extremely good.