Reduced Temperature Tr Research Articles

Superposition principle and nonlinear response in spin glasses

The extended principle of superposition has been a touchstone of spin-glass dynamics for almost 30 years. The Uppsala group has demonstrated its validity for the metallic spin glass, CuMn, for magnetic fields H up to 10 Oe at the reduced temperature Tr=T/Tg=0.95, where Tg is the spin-glass condensation temperature. For H>10 Oe, they observe a departure from linear response which they ascribe to the development of nonlinear dynamics. The thrust of this paper is to develop a microscopic origin for this behavior by focusing on the time development of the spin-glass correlation length, ξ(t,tw;H). Here, t is the time after H changes, and tw is the time from the quench for T>Tg to the working temperature T until H changes. We connect the growth of ξ(t,tw;H) to the barrier heights Δ(tw) that set the dynamics. The effect of H on the magnitude of Δ(tw) is responsible for affecting differently the two dynamical protocols associated with turning H off (TRM, or thermoremanent magnetization) or on (ZFC, or zero-field-cooled magnetization). This difference is a consequence of nonlinearity based on the effect of H on Δ(tw). Superposition is preserved if Δ(tw) is linear in the Hamming distance Hd (proportional to the difference between the self-overlap qEA and the overlap q[Δ(tw)]). However, superposition is violated if Δ(tw) increases faster than linear in Hd. We have previously shown, through experiment and simulation, that the barriers Δ(tw) do increase more rapidly than linearly with Hd through the observation that the growth of ξ(t,tw;H) slows down as ξ(t,tw;H) increases. In this paper, we display the difference between the zero-field-cooled ξZFC(t,tw;H) and the thermoremanent magnetization ξTRM(t,tw;H) correlation lengths as H increases, both experimentally and through numerical simulations, corresponding to the violation of the extended principle of superposition in line with the finding of the Uppsala Group.11 MoreReceived 22 July 2022Revised 18 May 2023Accepted 31 May 2023DOI:https://doi.org/10.1103/PhysRevB.107.214436©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMagnetismPhysical SystemsMetallic glassesSpin glassesTechniquesCritical phenomenaCondensed Matter, Materials & Applied Physics

Thermodynamic Properties of van der Waals Dimers and Trimers of Nonpolar Gases and Their Correlation with Lennard-Jones Potential Well Depths.

A method of unifying the equilibrium thermodynamic properties ΔHo and ΔGo relating to the van der Waals dimers and trimers of 15 nonpolar gases (He-Xe, H2, D2, CH4, CF4, ethene, ethane, CO2, SF6, propane, and neopentane) is described. Values of ΔHo and ΔGo, obtained at the reduced temperature Tr = 0.7, show good correlation with the respective Lennard-Jones pair potential well depth, calculated from the monomer critical temperature and the acentric factor. Such relationships present the opportunity to estimate the van der Waals dimer and trimer thermodynamic properties of other nonpolar molecules, and examples of seven such applications are given. It is found that the enthalpies of dimerization and trimerization of the 15 gases are about 21 and 29% of the respective condensation enthalpies, providing information about the thermodynamics of small clusters in relation to liquefaction.

An improvement on Martin–Hou equation of state for more precise prediction in the liquid region

AbstractA modified Martin–Hou (M–H) equation of state (EOS) for the accurate representation of thermodynamic property data of some pure substances in liquid phase is presented in this work. The improvement is achieved by substituting a new parameter η into the original M–H EOS. Eighteen pure substances are chosen to construct the dependency relationship of η on reduced temperature Tr and acentric factor ω. Furthermore, to verify the effectiveness and improvement of the modified M–H EOS, the calculated values of the saturated vapor and liquid molar volumes are compared with the literature data in the reduced temperature ranging from approximately 0.6 to 1.0. The results show that the modified M–H EOS perform a good prediction in gaseous phase for pure substances with the average absolute deviation (AAD) generally less than 1%, and more precise liquid molar volume values can be obtained with the modified equation. In liquid phase, the AAD of the modified M–H EOS is 3.71%, which is much lower than 6.16% of the original M–H EOS.

Saturation loadings on 13X (faujasite) zeolite above and below the critical conditions. Part III: Inorganic monatomic and diatomic species data evaluation and modeling

The saturation loadings for subcritical adsorption of inorganic monatomic and diatomic species on 13X zeolite are modeled using the modified Rackett model of Spencer and Danner (J Chem Eng Data 17:236–240, 1972) for the saturated liquid densities combined with crystallographic data for the 13X zeolite. A similar equation is derived for supercritical adsorption involving supercritical adsorbate densities and crystallographic data for the 13X zeolite. Adsorption data from the literature are first critically evaluated and then compared to the model. Log–log plots are used to determine whether each isotherm is near saturation; isotherms that exhibit a $${{\left( {\partial \ln q} \right)} \mathord{\left/ {\vphantom {{\left( {\partial \ln q} \right)} {\left( {\partial \ln p} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\partial \ln p} \right)}}$$ slope of zero at the maximum pressure point are assumed to be saturated. The highest loading is used from each isotherm that approaches saturation. Unsaturated isotherms are not considered further. The theoretical equation satisfactorily models the available experimental data for the data that are subcritical. However, a steric factor is required in the model for iodine. 80% of the data for helium, nitrogen and oxygen are in the supercritical region; this data reveals a linear decreasing trend with increasing reduced temperature Tr. For this data a new supercritical model is developed using the reduced critical adsorbate temperature, TCAR, and the slope of the decreasing linear plot against Tr. The physical phenomena causing this effect is considered to be increasing molecular vibration in the cavity reducing the total molecular loading with temperature rise.

An improved PR equation of state for CO2-containing gas compressibility factor calculation

Peng-Robinson equation of state (PR EoS) is a reliable and widely used cubic EoS to predict natural gases thermodynamic properties. However, large deviations between the calculated compressibility factor values and the realistic values are observed when it is applied to the high volume CO2-content natural gas. In order to enhance its applicability, a new correction parameter associated with three-parameters (the reduced temperature Tr, reduced pressure Pr and CO2 volume content) is introduced into the attractive term of PR EoS by fitting the deviations between 267 groups of experimental compressibility factors and corresponding calculation values. The results reveal that the average absolute deviation between experimental data and predicted compressibility factors from the improved PR EoS is 2.708% when the pressures range from 3 MPa to 15 MPa, temperatures range from 263.15 K to 313.15 K and CO2 volume contents range from 9.84% to 100%. The deviation is reduced by 9.61% when compared to that of the original PR EoS. The results also demonstrate that this improved PR EoS has better performance than PRSV, PRTwu BWRS EoSs, and is essentially comparable in predictability to PC-SAFT EoS.

Significant difference in the attractive energies of ethane and ethanol in supercritical CO2

Vibrational Raman spectra of the CC stretching modes for ethane and ethanol were measured in supercritical carbon dioxide (scCO2) at the isotherm of reduced temperature Tr=T/Tc=1.02 (Tc is the critical temperature) as a function of fluid density, ρr=ρ/ρc, in the range of 0.01–1.5. The peak frequencies of the Raman spectra were analyzed using the perturbed hard-sphere theory, and the attractive and repulsive energy components between solute and solvent were quantified as a function of the fluid density. For ethane, the magnitude of the attractive energy was almost equal to that of the repulsive energy. In contrast, the magnitude of the attractive energy for ethanol was four times greater than that of the repulsive energy. The significant difference in the attractive energies of ethane and ethanol is attributed to the solvation of the CO2 molecules at the oxygen atom of ethanol, according to analyses of local density and quantum chemical calculations. These observations provide useful information to understand the entrainer effect in scCO2.

Liquid Viscosity and Surface Tension of R1234yf and R1234ze Under Saturation Conditions by Surface Light Scattering

In the present study, surface light scattering (SLS) was used for the simultaneous determination of liquid kinematic viscosity and surface tension of 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) and of trans-1,3,3,3-tetrafluoroprop-1-ene (R1234ze) under saturation conditions. A new SLS apparatus was built up and checked with 1,1,1,2-tetrafluoroethane (R134a), and a good agreement of our data from SLS with literature could be found. With the new apparatus, R1234yf and R1234ze were investigated in the temperature ranges between (293 and 365) K and between (295 and 373) K. For determination of the liquid kinematic viscosity the expanded uncertainties on a confidence level of more than 95 % (k = 2) are estimated to be ± 2 % for reduced temperatures Tr (= T/Tc) < 0.95, and ± 6 % for Tr close to 0.99. For the surface tension, the expanded uncertainties are less than ± 1.5 % (k = 2) in the whole temperature range.

Investigation of attractive and repulsive interactions associated with ketones in supercritical CO2, based on Raman spectroscopy and theoretical calculations

Carbonyl compounds are solutes that are highly soluble in supercritical CO2 (scCO2). Their solubility governs the efficiency of chemical reactions, and is significantly increased by changing a chromophore. To effectively use scCO2 as solvent, it is crucial to understand the high solubility of carbonyl compounds, the solvation structure, and the solute-solvent intermolecular interactions. We report Raman spectroscopic data, for three prototypical ketones dissolved in scCO2, and four theoretical analyses. The vibrational Raman spectra of the C=O stretching modes of ketones (acetone, acetophenone, and benzophenone) were measured in scCO2 along the reduced temperature Tr = T∕Tc = 1.02 isotherm as a function of the reduced density ρr = ρ∕ρc in the range 0.05-1.5. The peak frequencies of the C=O stretching modes shifted toward lower energies as the fluid density increased. The density dependence was analyzed by using perturbed hard-sphere theory, and the shift was decomposed into attractive and repulsive energy components. The attractive energy between the ketones and CO2 was up to nine times higher than the repulsive energy, and its magnitude increased in the following order: acetone < acetophenone < benzophenone. The Mulliken charges of the three solutes and CO2 molecules obtained by using quantum chemistry calculations described the order of the magnitude of the attractive energy and optimized the relative configuration between each solute and CO2. According to theoretical calculations for the dispersion energy, the dipole-induced-dipole interaction energy, and the frequency shift due to their interactions, the experimentally determined attractive energy differences in the three solutes were attributed to the dispersion energies that depended on a chromophore attached to the carbonyl groups. It was found that the major intermolecular interaction with the attractive shift varied from dipole-induced dipole to dispersion depending on the chromophore in the ketones in scCO2. As the common conclusion for the Raman spectral measurements and the four theoretical calculations, solute polarizability, modified by the chromophore, was at the core of the solute-solvent interactions of the ketones in scCO2.

Intrinsic adsorption properties of CO2 on 5A and 13X zeolite

Data for the adsorption of CO2 on 5A (CaA) and 13X (NaX) zeolite are critically evaluated. In addition, fresh data for the adsorption of CO2 on 13X zeolite is reported. Three intrinsic properties are examined: q max , the saturation loading, K H , the Henry constant, and (−ΔH) q , the isosteric heat of sorption. Below a reduced temperature T r , of 0.9, the q max values for both 5A and 13X zeolites are similar to theoretical values that may be derived using zeolitic crystallographic properties and the sorbate density calculated using the Rackett equation. For the region 0.9 ≤ Tr ≤ 1.0, the calculated q max values exceed the theoretical values similarly calculated, indicating that the molecules have a smaller molar volume than in a similar liquid phase. This is a similar result to that observed in ionic liquids. Linear regressed equations are derived for q max for the region 0.9 ≤ Tr ≤ 1.25. The Henry constant values for 5A are remarkably consistent for the five studies examined, with a correlation coefficient, R, of 0.999 for the van’t Hoff equation, but for the seven studies examined in 13X the data is more disperse as indicated by a correlation coefficient R of 0.899 for the van’t Hoff equation. The values of (−ΔH) q , the isosteric heat of sorption are in agreement with the literature. An explanation is advanced for the discrepancy between the higher heats of sorption values obtained calorimetrically from those obtained from isosteric adsorption studies. Finally, the fresh data is observed to fit the Toth model with regression coefficients of 0.999. However, the parameters obtained for the Toth equation by regression are significantly different from the intrinsic properties derived earlier, indicating the difficulty of deriving intrinsic parameters from isotherm fits.

Static inhomogeneity of supercritical ethylene studied by small-angle X-ray scattering

Abstract Small-angle X-ray scattering (SAXS) experiments for supercritical C2H4 were carried out, to study its density fluctuation and correlation length, which are quantitative descriptions of the inhomogeneity of molecular distribution. Except for thermodynamic states near the critical point, the density fluctuation from SAXS experiments is in very good agreement with the one calculated from a new state equation. For investigation on the substance dependence of the inhomogeneity in supercritical states, the results were compared with our previous ones of CO 2 , CF 3 H and H2O. The density fluctuation and correlation length for C2H4 show fairly similar behaviors to those of supercritical CO 2 , CF 3 H as a function of reduced density ρr=ρ/ρc and reduced temperature Tr=T/Tc. On the other hand, characteristics for inhomogeneity of supercritical H2O are different from those of others. Good correspondence of static inhomogeneity and dynamic one in supercritical C2H4 is revealed, by comparing the fluctuation from SAXS with its correlation time from dynamic light scattering experiments.

Correlation time of density fluctuation for supercritical ethylene studied by dynamic light scattering

Time dependence of density fluctuation for neat supercritical ethylene (C2H4) is investigated by dynamic light scattering at various densities in isothermal conditions of reduced temperatures Tr=T/Tc=1.02, 1.04, and 1.06. Time correlations of the density fluctuation for all thermodynamic states decay as single exponential functions with the time constant of submicrosecond. Critical slowing down of diffusive motions is observed in the time domain, and the correlation time of the density fluctuation becomes maximum at the extension of the gas–liquid coexistence curve on the P–T phase diagram. It is revealed that the time dependence of the density fluctuation just corresponds to the magnitude of static density fluctuation obtained by small-angle x-ray scattering measurements. By measuring correlation times as a function of scattering angle, the critical slowing down is thermodynamically discussed. It is elucidated experimentally that the critical slowing down of diffusive motions considerably correlates to the increase of specific heat capacity, the decrease of the thermal diffusivity, and rotational relaxation time in the supercritical state of the neat fluid.

Thermodynamic properties of pure fluids using the GEOS3C equation of state

Thermodynamic properties were predicted for two groups of pure fluids, along the saturation curve and in the single-phase region: IUPAC pure fluids (propylene, chlorine, oxygen and nitrogen) and halogenated hydrocarbons (R11, R12, R21, R22, R152a). A new form of equation GEOS, named GEOS3C, was used. A wide comparison with literature recommended data, as well as with results of other six cubic equations of state (Soave-Redlich-Kwong, Peng-Robinson, Stryjek-Vera, Schmidt-Wenzel, Freze et al. and Salim and Trebble) was made. The GEOS3C equation has three parameters estimated by matching three points on the saturation curve (vapor pressures and the corresponding liquid volumes). The three fixed temperatures are the triple point, the boiling point and the reduced temperature Tr=0.7. The GEOS3C equation leads to the best results in predicting vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties in the saturation range, but this do not lead necessarily to similar best predictions of all properties in the single-phase region.

X-Ray Diffraction Studies of the Recrystallization of C60-Based Polymers

A time-dependent x-ray scattering study is presented of the kinetics of the C60-based star polymer containing 6 urethane-connected polyether and the star model polymers containing 4, 3, and 2 urethane-connected polyether arms. By varying the temperature from 40°C to -40°C, the order-disorder phase transition (recrystallization) of the polymers has been observed. Two crystalline domains with the periodicities of 4.5±0.6 Å and 3.7±0.4 Å were detected. The domain sizes ranging from 70 Å to 250 Å, the transition temperatures from -15°C to 25°C and the maximum percentage of recrystallization from 13% to 34% increase as the number of polyether arms increases. The x-ray scattering peaks of the order phase grow up as I1 +I2(l - e−t/t0), where I1 and I2 are the initial intensity and the proportional intensity modulus, respectively. The relaxation time t0 of the order phase is varying as a power law with respect to the reduced temperature TR(=(T - TC)/TC): t0 = (35±10)·|TR|−1.42±0.07 seconds for the C60-based star polymer containing 6 urethane-connected polyether; t0 = (315 ± 64)·|TR|−1.68±0.08 seconds for the star model polymer containing 4 urethane-connected polyether arms.

Characterization of solvent clusters in a supercritical Lennard-Jones fluid

We have developed a new methodology for characterizing the solvent cluster structures which occur in a pure supercritical fluid in its compressible regime. This methodology takes advantage of the time scale separation which exists between collective-cluster and individual-solvent–atom motions in order to classify atoms according to their ‘‘instantaneous’’ local environments. The resultant picture is of a fluid having density inhomogeneities on a mesoscopic length scale—i.e., clusters and cavities. Calculation of partial radial distribution functions shows that atoms residing in different density domains have very different equilibrium structural properties, information which is not available from the usual total radial distribution function. For example, for a 2-dimensional Lennard-Jones fluid at a reduced temperature Tr=1.06 the nearest neighbor coordination number in a high density domain is 4.2, whereas in a low density domain it is only 1.0. We have also found that, for such clustering fluids that in a finite volume system there is an ensemble independent [within terms of 𝒪(1/N)], nonstructural long-range correlation which arises from an excluded volume effect. This long range correlation enables us to determine the average domain size, volume fraction and density for both the high and low density domains.

Second Virial Coefficient of the 2cLJ, 3cLJ and 4cLJ Molecules

The second virial coefficient was evaluated for the two-centre, three-centre and four-centre Lennard Jones molecules with the site site distance l ∈ (0,1) at the reduced temperatures Tr = 0.6 - 3.0. The obtained data are correlated by an expression derived originally for the Kihara non-spherical molecules; the same value of the σ-parameter is considered for the both pair potentials whereas εKihara/εncLJ and lKihara / lncLJ vary with the increasing values of lncLJ. Values of the virial coefficient of the individual ncLJ molecules agree within error bars with experimental values in the whole temperature range studied; with only slightly higher deviations also data for the single 2cLJ, 3cLJ and 4cLJ molecules for all the lncLJ values can be correlated.

Fluids in pores: experimental and computer simulation studies of multilayer adsorption, pore condensation and critical-point shifts

Abstract

On the representation of pure‐compound saturated (ZV − ZL) values

AbstractAn equation for determining the difference between the compressibility factors of saturated vapor and liquid (ZV ‐ ZL) through cubic equations of state has been developed as a function of reduced vapor pressure Pr, reduced temperature Tr and the critical cohesion parameter Ωac of the chosen equation. The regressed equation has been presented in a polynomial expression of (Pr/Tr) and Ωac for simple non‐iterative calculations.

The equivalence of temperature and velocity in determining the non-linear positive ionic motion in normal liquid3He

The non-linear behaviour of the positive ionic motion in liquid 3He has been studied down to 3 mK. It was found that the influence of temperature and velocity is symmetrical. At velocities comparable to kBT/pF the behaviour may be described by the introduction of a reduced temperature TR where TR32/=T32/+(0.31pFv/kB)32/.

Orientational order and tricritical behavior in the trans-p-n-alkoxy-α-methyl-p′-cyanophenyl cinnamate (n-OMCPC) series of liquid crystals

The temperature dependence of the orientational order parameter S has been evaluated from their infrared spectra for six homologues of the n-OMCPC (trans-p-n-alkoxy-α-methyl-p′-cyanophenyl cinnamate) series of liquid crystals with n = 7–12, where n is the number of carbon atoms in the end alkyl chain. The even–odd effect of the S values in the nematic phase is rather weak for homologues 7–10. In homologues 10–12, the changes of S at the transitions have been correlated with their thermodynamic behavior. The smectic A–nematic (A–N) and nematic–isotropic (N–I) transitions in several binary mixtures of 11 with 8, 10, and 12 have been studied by differential scanning calorimetry. With decreasing average alkyl chain length in the mixtures, the nematic range becomes extended and the enthalpy of the A–N transition is found to decrease. We deduce that the tricritical point in the A–N transition occurs when the reduced temperature TR=TAN/TNI=0.94±0.005. The triple point between A, N, and I phases has also been observed in an 11–12 mixture at the mole percent ratio of ∼27:73.

Four-Parameter Generalized Compressibility Charts for Nonpolar Fluids

A series of five generalized compressibility charts for nonpolar and slightly polar substances in the gaseous and liquid states and in the two-phase region is presented. The charts are plots of the compressibility factor Z versus the reduced pressure Pr with the reduced temperature Tr as a parameter. Each chart is for a group of substances with similar values of the critical compressibility factor Zc and the slope of the reduced vapor-pressure curve at the critical point, αc. The use of four independent parameters (Pr, Tr, Zc, and αc) results in an increase in accuracy over existing methods of presenting compressibility charts.