Thermodynamic properties of liquid3He-4He mixtures near the tricritical point. II. Data analysis by the scaling-field method

The scaling theory for tricritical phenomena by Riedel is applied to the analysis of thermodynamic properties of liquid³He-⁴He mixtures near the tricritical point. Within this theory experimental data for the phase diagram, the³He molar concentrationX, and the concentration susceptibility (X/) _T are discussed in terms of two scaling fields that are functions of the temperatureT and the difference = ₃ – ₄ of the chemical potentials of the two helium isotopes. The quantitiesX and (X/) _T in terms of thefields T and as independent variables are obtained for the intervals –0.1<T – T _t<0.53 K and –9< – _t <0.8 J/mole, from vapor pressure and calorimetric data described in a previous paper by Goellner, Behringer, and Meyer. The transformed data are analyzed to yield the tricritical exponents, amplitudes, scaling fields, and scaling functions. The values of the tricritical exponents are found to agree with those predicted by the renormalization-group theory of Riedel and Wegner. (Logarithmic corrections are beyond the precision of the present experiment.) Relations between amplitudes are derived and tested experimentally. The (linear) scaling fields are determined by using their relationship to geometrical features of the phase diagram. The data forX and (X/) _T are found to scale in terms of these generalized scaling variables. The sizes of the tricritical scaling regions in the normal and superfluid phases are estimated; the range of apparent tricritical scaling is found to be appreciably larger in the normal-fluid phase than in the superfluid phase. The tricritical scaling function for the concentration susceptibility is compared with the analogous scaling function for the compressibility of pure³He near thecritical gas—liquid phase transition. Finally, when the critical line near the tricritical point is approached along a path of constant < _t , the experimental data are found to exhibit the onset of the crossover from tricritical to critical behavior in qualitative agreement with crossover scaling.Work supported in part by the National Science Foundation through Grant No. GH-36882 and Grant No. GH-32007, and by a grant of the Army Research Office (Durham).     

Anisotropy of the mechanical α-relaxation in biaxially oriented linear polyethylene

Alternative theories for the origin of the high temperature,-relaxation in oriented linear polyethylene are compared with experiment. The-mechanism is assumed to be simple shear by all theories which differ in the allocation of the shear plane. The theories take the shear plane to be the interlamellar surface (theory I), a crystal plane of type (hk0) (theory IIA), a crystal plane of type (h00) (theory IIB). The experiments comprise measurements of the creep tensile compliance D(t) on oriented specimens at angles 0°, 45° and 90° to the draw axis. The crucial test involves the examination of specimens with approximately the same crystallographic orientation (determined by wide-angle X-ray diffraction) but differing orientation of lamellar normals. The analysis is based on the Reuss, constant stress hypothesis. It is shown that with this assumption, only theory I agrees with experiment.     

A quasi-conforming triangular laminated composite shell element based on a refined first-order theory

A quasi-conforming triangular laminated shell element based on a refined first-order shear deformation theory is presented. The Hu-Washizu variational principle, involving strain and displacement fields as variables, with stresses being considered as Lagrange multipliers, is used to develop the laminate composite shell element. Both strains and displacements are discretized in the element, while displacements alone are discretized at the boundary. The inter-element C ¹ continuity is satisfied a posteriori in a weak form. Due to the importance of rotations and shear deformation in the geometrically non-linear analyses of shells, 7 degrees of freedom per node are chosen, viz. three displacements, two first-derivatives in the in-plane directions of the out-of-plane displacement, and two transverse shear strains at each node. To consider the effect of transverse shear deformation on the global behavior of the laminated composite shell, the Reissner-Mindlin first-order theory, with shear correction factors of Chow and Whitney, is adopted. The transverse shear stresses are obtained through the integration of the 3-D equilibrium equations; and the warping induced by transverse shear is considered in the calculation of the in-plane stresses to improve their accuracy. Numerical examples show that the element has good convergence properties and leads to highly accurate stresses.     

Equation of State for Complex Liquid Mixtures from Surface Tension

The present work shows a successful extension of previous studies to molecular liquids for which the second virial coefficients are not known. Recent advances in the statistical mechanical theory of equilibrium fluids can be used to obtain an equation of state (EOS) for compressed normal liquids and molten alkali metals. Three temperature-dependent quantities are needed to use the EOS: the second virial coefficient, B(T), an effective van der Waals covolume, b(T), and a scaling factor, (T). The second virial coefficients are calculated from a correlation that uses the surface tension, _tr, and the liquid density at the triple point. Calculation of (T) and b(T) follows by scaling. Thus, thermodynamic consistency is achieved by use of two scaling parameters (_tr, _tr). The correlations embrace the temperature range T _tr<T<T _c and can be used in a predictive mode. The remaining constant parameter is best found empirically from _tr data for pure dense liquids. The equation of state is tested on 42 liquid mixtures The results indicate that the liquid density at any pressure and temperature can be predicted within about 5%, over the range from T _tr to T _c.     

Viscosity and diffusivity of a binary liquid mixture of critical composition: Study of the system 2-butoxyethanol/water

Results of measurements of the temperature dependence of the shear viscosity and the mutual diffusion coefficient of a 2-butoxyethanol/water mixture of critical composition are reported. The shear viscosities are measured with a capillary viscometer, and the diffusion coefficients by dynamic light scattering. The viscosity data are used to determine the regular (background) and the singular (critical) part of the viscosity and to analyze the crossover regime.Q ₀ has a value ofQ _o = (1.54 ± 0.60) × 10⁶ cm^–1, which is small for a binary mixture of components of small molar mass. The viscosity is dominated by singular contributions in a narrow temperature range (T _c – T) 1.2 K. The singular contributions can be neglected for temperatures (T_c–T) 11 K. The function =_b expz H, given in the literature, represents the viscosity data in the temperature range 3 mK (T _c – T) 25 K using the asymptotic form of the functionH for (q _d/q _c) 0 (q _d,q _c, parameter of the theory). This limit corresponds to the case in which the background diffusion coefficient Lib can be neglected. The reduced diffusion coefficient D^* calculated from the light-scattering data as a function of the scaling variable x(=q) is represented by the approximation of the dynamic scaling function proposed in the literature.     

Free energy and specific heat of critical films

Based on the field-theoretical renormalization group theory forO(N) symmetric systems in 4 — dimensions the free energy of critical films confined between two parallel plates at distanceL is analyzed. For temperatures aboveT _c,bulk the universal scaling functions are presented and discussed. At bulk criticality the scaling functions reduce to the universal Casimir amplitudes. Using these field-theoretical results quantitative predictions on the specific heat of⁴He films and films of³He-⁴He mixtures close to the onset of superfluidity are made. Wetting of a substrate by⁴He or³He-⁴He mixtures close to the — transition is studied in detail and thus the full scaling function is probed.     

The ISM Equation of State Applied to Refrigerants

In this work, we apply an equation of state based on statistical–mechanical perturbation theory to liquid refrigerants and their mixtures. Three temperature-dependent parameters are needed to use the equation of state: the second virial coefficient, B ₂(T), an effective van der Waals covolume, b(T), and a scaling factor, (T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, H _vap, and the liquid density at the freezing point, _fp. (T) and b(T) can also be calculated from the second virial coefficient by a scaling rule. Based on the theory, these two temperature-dependent parameters depend only on the repulsive branch of the potential function, and therefore, by our procedure, can be found from H _vap and _fp. The theory has considerable predictive power, since it permits the construction of the p–v–T surface from the heat of vaporization plus the triple-point density. The equation of state is tested for pure, two- and three-component liquid refrigerant mixtures.     

Equation of State for Molten Alkali Metals from Surface Tension. Part II

This work presents a new method for predicting the equation of state for molten alkali metals, based on statistical–mechanical perturbation theory from two scaling constants that are available from measurements at ordinary pressures and temperatures. The scaling constants are the surface tension and the liquid density at the boiling temperature (_b, _b). Also, a reference temperature, T _Ref, is presented at which the product (T _Ref T _b ^1/2 ) is an advantageous corresponding temperature for the second virial coefficient, B ₂(T). The virial coefficient of alkali metals cannot be expected to obey a law of corresponding states for normal fluids, because two singlet and triplet potentials are involved. The free parameter of the Ihm–Song–Mason equation of state compensates for the uncertainties in B ₂(T). The vapor pressure of molten alkali metals at low temperatures is very low and the experimental data for B ₂(T) of these metals are scarce. Therefore, an equation of state for alkali metals from the surface tension and liquid density at boiling temperature (_b, _b) is a suitable choice. The results, the density of Li through Cs from the melting point up to several hundred degrees above the boiling temperature, are within 5%.     

Superfluid Transition of 4He for Two-Dimensional Crossover, Heat Capacity, and Finite-Size Scaling

We report heat capacity measurements of confined films of ⁴ He. These studies were undertaken to test predictions of correlation-length scaling. They are the first measurements for completely confined films over a range of confinements, and represent a geometry where criticality changes from 3-dimensions (3D) to 2D. The finite system is realized with a ⁴ He film confined between two, 2 diameter, silicon wafers, which are separated by a small gap. A new technique was developed to bond these wafers at a uniform separation. The gap size, which determines the film thickness, ranges from 0.05 to 0.7 m in the present work, and has better than 1% uniformity. The bonded cells are used to conduct high precision heat capacity measurements using a modified ac technique. This involves oscillating the sample temperature, as in conventional ac calorimetry, but with simultaneous dc regulation of the average temperature. The data are analyzed using a modified Sullivan–Seidel equation, which takes into account in an empirical way the finite conductivity of the cell. This procedure yields heat capacity data with good absolute accuracy and high resolution. Scaling analysis of the data both above and below the bulk transition temperature shows collapse onto universal curves determined only by the ratio of the correlation length to the confinement size. This is true everywhere except near the heat capacity maximum. Here, and into the superfluid side there is lack of scaling which might be associated with 2D crossover. We compare this result with calculations of scaling functions and find that these tend to underestimate the effect of confinement. Comparison with earlier results for cylindrical confinement shows differences which are most striking in the region of the specific heat maximum. The cylindrical and planar confinement data follow similar trends above the superfluid transition of bulk helium. Below the transition, however, the present data show much more structure. Fits of the scaled planar data above the transition to an empirical scaling function yield a correlation length exponent of _eff=0.674±0.001.     

Flow of power-law fluids in cone-plate viscometer

Summary The flow of a power-law fluid in a cone-plate viscometer has been considered. The first order solutions for velocity and pressure have been obtained. Effects of secondary flow on velocity components, pressure, rate of deformation tensors and apparent viscosity have been studied. Both the primary and secondary flows are influenced by flow behavior index n. In shear thinning fluids, the effects on both the primary and secondary flows are of similar nature, whereas in shear thickening fluids, they are of opposite nature. The net result is that for shear thinning fluids the effects of the primary and secondary flows are additive, whereas in shear thickening fluids they almost balance out each other and the resultant effect is insignificant. The other important observation is in Newtonian fluids the secondary flow makes a significant contribution at high value of Reynolds number, which depends on the radius, velocity, density, and viscosity. In power-law fluids, Reynolds number depends on an additional parameter n. With this change, the effect of secondary flows can be important even at low shear rates. A critical value of n is obtained at which flow is independent of shear rates.     

New perspective on Poisson's ratios of elastic solids

Summary A new perspective on Poisson's ratios of elastic solids is presented. We show that, by scaling the Poisson's ratios through the square root of a modulus ratio, thetransformed Poisson's ratios,n ₁,n ₂,n ₃, are bounded in a closed region, which is inside a cube centered at the origin with a range from –1 to 1. The shape of this closed region, depicted in Fig. 1, looks like a Chinese food, Zongzi. With this geometric interpretation, any positive definite compliance of an orthotropic solid can be easily constructed by selecting any point inside the region, together with any three positive Young's moduli and any three positive shear moduli. This provides a new insight to the admissible range of Poisson's ratios. We also provide an example that the inequality proven by Rabinovich [6], i.e.v ₁₂+v ₂₃+v ₃₁3/2, is not generally true.     

Shear testing of mercuric iodide single crystals

A specialized shear testing system has been designed and assembled for the purpose of measuring the response of thin single crystals of mercuric iodide to shear loading. Numerous stress-strain measurements have been made on single crystal samples of thicknesses ranging from 0.3 to 1.0 mm. Shear strains of over 100% are readily achieved. The results are analysed in terms of a two-parameter semiempirical model for yielding that fits the experimental data extremely well. The model employs a normalized Gaussian distribution f(s) for the change f(s)ds in the relative density of mobile dislocations that occurs when the shear stress s is increased from S to s+ds. The Gaussian parameters s ₀ (the shear stress at which the density of mobile dislocations has reached half its steady state value) and (the standard deviation) are determined from a least squares fit of the theoretically computed stress-strain curve to the experimental data. The onset of yielding s _c is defined as s ₀–2, which is the shear stress at which the density of mobile dislocations has reached about 2% of its steady state value. The utility of the model for studying work hardening and time dependent recovery in single crystal HgI₂ is established.This work is based upon parts of the PhD Dissertation of Gary Georgeson, University of California, Santa Barbara, USA.     

The Law of Corresponding States and Surface Tension of Liquid Metals

Empirical relationships for the surface tension of liquid metals (LM) are shown to follow from the principle of corresponding states. In order to relate the surface tension of LM to its bulk properties, a formula is derived by scaling with the melting point T _m (0) at the atmospheric pressure, p = 0 and the atomic volume _m (0) at the melting point as macroscopic parameters for scaling and a characterizing the interatomic potential (r)= *(r/a). Correlation rules, derived for the surface tension and its temperature coefficient, are discussed and compared with experimental data.     

Concentration susceptibility of 3He-4He mixtures near the superfluid transition

Measurements of the concentration susceptibility % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabiqaaiaacaGaaeqabaWaaeaaeaaakeaacuaHhpWygaacai% abg2da9iabgkGi2kaadIfacaGGVaGaeyOaIyRaeuiLdqKaaiykamaa% BaaaleaacaWGubaabeaaaaa!3C99!\[\tilde \chi = \partial X/\partial \Delta )_T \] are reported for ³He-⁴He mixtures at saturated vapor pressure and at constant mole fraction X(³He) as a function of temperature. Here is the isotopic difference in chemical potentials. The mixtures cover the range from X = 0.60 to X = 0.677. Particular emphasis is given to the region near the lambda line for three mixtures and to the region close to the tricritical point. The method makes use of the vertical concentration gradient induced in the mixture by gravity. This difference X is measured via the dielectric constant over a height of 2 mm using a capacitance technique. The predicted peak of the susceptibility at the superfluid transition is observed and as the tricritical point is approached, this peak is progressively broadened by gravity effects. The data are compared with those from other methods and also with estimations based on calorimetric data for more dilute mixtures. The susceptibility data are transformed into results along a path at constant chemical potential . Sufficiently close to T , these can be fitted to an expression with a weakly singular term, which is consistent with the predictions on the grounds of universality. These predictions are based on the calorimetric data of more dilute mixtures. The width of this critical region is found to be consistent with theoretical estimations by Riedel, Meyer, and Behringer. The tricritical scaling scheme by these authors is tested by the new data and the resulting scaling curve is found to be in reasonable agreement with that obtained from earlier data by Goellner, Behringer, and Meyer, except in the region closest to the tricritical point. There the new data appear to be more consistent with measurements from light scattering. In addition, the concentration susceptibility for more dilute mixtures (0.05 < X < 0.4) is calculated both from calorimetric data and from saturated vapor pressure measurements and the results are found to be internally consistent.Supported by grants from the AFOSR and from the NSF.     

Nonequilibrium Molecular Dynamics Simulations of Shear Viscosity: Isoamyl Alcohol, n-Butyl Acetate, and Their Mixtures

Nonequilibrium, NVT, molecular dynamics (NEMD) simulations were used to obtain the shear viscosity, , of isoamyl alcohol, n-butyl acetate, and their binary mixtures at 35°C and 0.1 MPa. The fluids were modeled using rigid bonds, rigid bond angles, appropriate torsional potentials, pairwise-additive Lennard–Jones dispersion interactions between united-atom sites, and partial point charges located at atomic centers. Simulations were performed at different shear rates, , and values obtained at =0 are compared to experimental values. Two methods are commonly used to extrapolate pure-fluid simulated data to zero shear, (0). The applicability of these two methods to mixtures of polar fluids was examined in this study. It was found that linear extrapolation with respect to ^1/2 can lead to ambiguous (0) results for some mixtures because of a curvature in the data that shows no observably distinct change in rheology. On the other hand, a log–log plot of () versus is consistently very linear with a distinct change from shear-thinning to Newtonian rheology at lower shear rates. The latter method is recommended for consistency sake, even though agreement between experiment and (0) values was better with the former method. This agreement was 12 and 21% for the two methods, respectively. A negative bias in the simulated values is attributable to the united-atom model.     

Critical Light Scattering in Pure Liquids

Light-scattering experiments near the critical point (T _c, _c) in fluid systems and, in particular, the central Rayleigh peak in the frequency spectrum are reviewed. Within a nonasymptotic renormalization-group theory, the crossover function is calculated between several regions: (i) from the background to the asymptotic region, (ii) from the hydrodynamic region (wave lengthcorrelation length) to the critical region (wave lengthcorrelation length), and (iii) from critical densities to noncritical densities. Contrary to the mode-coupling expression, the appropriate scaling function is well defined in all limits of its arguments. At T _c the crossover in the wave-vector dependence of the linewidth is also considered. Theoretical results are compared with experiments for pure liquids. Nonuniversal parameters are chosen consistent with the transport coefficients (i.e., the shear viscosity) for the same substance which can be evaluated within the same formalism.     

Surface tension of He II near the λ-point

Measurement of the surface tension of superfluid helium in the neighborhood of the -transition are presented. The data are analyzed to obtain the behavior of the singular part of the surface tension, which is discussed in the light of the calculation by Sobyanin, based on the Ginzburg-Pitaevskii phenomenological theory, and the results from scaling theory as obtained by Hohenberg. It is found that the behavior of the surface tension suggests a divergent and negative specific heat above and below T and that the free surface correlation length is given by =(1.5±0.4)Å|| ^–2/3, in good agreement with the coherence length in the Mamaladze version of the phenomenological theory.Work supported by the National Science Foundation, Grant Nos. 27242 and 31650.     

External field effect on the critical behavior of the interface between fluid phases

The equilibrium structure of the interface between fluid phases ind dimensions in the presence of an external field is investigated. The equilibrium interface is assumed to consist of an intrinsic interface which undergoes capillary-wave fluctuations. It is found that in two dimensions the interfacial thickness is very sensitive to the choice of external field and intrinsic interface. For an intrinsic interface of a thickness proportional to, the bulk correlation length, the exponent co, which describes the divergence of the interfacial thickness as the critical point is approached, depends on the scale of the external field relative to and ranges from=9/32 to = 17/32, in contrast to the prediction=1 of scaling theory. When an intrinsic interface of vanishing thickness is chosen,=9/32 for any external field. This is in strong contrast to the results in three or more dimensions, where is found to be independent of both the external field and the intrinsic interface and satisfies = (d-1), with the critical exponent of the surface tension, in accord with scaling theory.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

A small-angle neutron scattering (SANS) study of worm-like micelles under shear

The structure of a cationic worm-like cylindrical micelle was investigated by SANS (small-angle neutron scattering). Intensities from 0.1% by weight solutions in D₂O, at rest and under shear, were measured on the NIST Cold Neutron Research facility 30-m spectrometer in the wave vector range 0.03Q (nm¹)-2.0. Scattered intensity patterns from the solutions subjected to shears equal to or greater than 40s¹ showed pronounced anisotropy, but such anisotropy could not be detected below this apparent threshold shear. The threshold was characterized by a relaxation time since anisotropy was detected only after several minutes of shearing. In contrast, the anisotropy was apparent immediately the shear was applied at the higher shears. The data were analyzed based on the assumption that the micelles behave as rigid rods. Estimates of the radii and length under shear are given. Polydispersity in rod length is discussed, and we argue that it contributes significantly to the scattering patterns.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

Scaling of the Equilibrium Magnetization in the Mixed State of Type-II Superconductors

No Heading We discuss the analysis of mixed-state magnetization data of type-II superconductors using a recently developed scaling procedure. It is based on the fact that, if the Ginzburg-Landau parameter does not depend on temperature, the magnetic susceptibility (H, T) is a universal function of H/H_c2(T), leading to a simple relation between magnetizations at different temperatures. Although this scaling procedure does not provide absolute values of the upper critical field H_c2(T), its temperature variation can be established rather accurately. This provides an opportunity to validate theoretical models that are usually employed for the evaluation of H_c2(T) from equilibrium magnetization data. In the second part of the paper we apply this scaling procedure for a discussion of the notorious first order phase transition in the mixed state of high-Tc superconductors. Our analysis, based on experimental magnetization data available in the literature, shows that the shift of the magnetization accross the transition may adopt either sign, depending on the particular chosen sample. We argue that this observation is inconsistent with the interpretation that this transition always represents the melting transition of the vortex lattice.PACS numbers: 74.60.-w, 74.60.Ec, 74.60.Ge, 74.72.-h