Onsager Equation Research Articles

Dr. Virkar's Self-Reflections on Research Career

I have been helped by many mentors and mentees. My greatest mentor has been my father, who received PhD in organic chemistry in 1942, more than two years before I was born. That made both my brother (who has a PhD in mathematics) and myself pursue higher education. I have been immensely helped by many mentors. I will briefly talk about them.At Northwestern University, Prof. Morris E. Fine, who was not my advisor, but had a profound influence on me. His dedication to work is unparalleled. He retired at age 65 (as was the case then) but continued as Professor Emeritus until his death at the age of 97. It was amazing for me to see that he went to work at least 3 days a week well into his 90’s. One week, he did not come, so people from the MSE department at Northwestern went to his house, only to find that he had passed away in his sleep.I came to Utah in September 1973, as a post-doctoral fellow, working for Prof. Ronald S. Gordon, on sodium-sulfur batteries. I was hired to measure strength of Na-beta”-alumina ceramics. He had been my mentor for over 35 years, and passed away in 2008. He gave me many opportunities, and was instrumental in my getting a faculty position at Utah. He also founded Ceramatec, which provided jobs to hundreds over the years. He was one of the closest friends I have had in my life. During my early career, Prof. Richard H. Boyd, Chair of MSE, helped me enormously. He gave me various courses to teach; from the basic MSE course to statistical thermodynamics. I can say without hesitation that it helped me develop various areas of research; basically, my teaching drove my research, rather than the other way around, which is often the case.I have been extremely fortunate to have had excellent students working in my group over the years; they include Bill Rafaniello, L. Viswanathan, Tom Yuan, Gajavalli Srinivasan, Peter Kuo, Jong Chen, Jow-Lay Huang, Quing Tian, Cameron Tanner, Srikanth Gopalan, S-J.Hong, Pomin Su, Jan-Fong Jue, R. Ganeshananthan, Lei Zhang, Alex Szendrei, Kuan-Zong Fung, Seong J. Kim, Hyung D. Baek, Hyung-Tae Lim to name a few, and post-doctoral fellow Liangzhu Zhu, among others. The list is by no means complete.On the technical part, I will talk about what significant changes occur to yttria-stabilized zirconia during electrolysis and the need to conduct property measurements after electrolysis. I have worked on many research topics over the years. The most recent efforts have been on the application of Onsager equations to solid electrolytes.

Power-optimization of multistage non-isothermal chemical engine system via Onsager equations, Hamilton-Jacobi-Bellman theory and dynamic programming

Power-optimization of multistage non-isothermal chemical engine system via Onsager equations, Hamilton-Jacobi-Bellman theory and dynamic programming

Maximum work configuration of finite potential source endoreversible non-isothermal chemical engines

Abstract Chemical engine is an abstract model of some devices, such as solid state, photochemical, and electrochemical devices, photovoltaic cell, and mass exchangers. Finite chemical-potential source is one of its features. Finite time thermodynamics provides effective theoretical tool for determining performance limits for given thermal systems, and determining optimal process paths of thermal systems for given performance objectives. Endoreversible model is its basic model. A model of endoreversible non-isothermal chemical engines operating between a finite chemical-potential source and an infinite chemical-potential sink with mass resistance and heat resistance is established. Mass transfer processes between chemical potential reservoir and working fluid of the model are assumed to obey Onsager equations in linear irreversible thermodynamics. With a fixed cycle period, optimal cycle configuration for the maximum work output of the model is derived by applying optimal control theory. The results obtained include optimal performance and optimal path results in many previous literatures, and can provide some theoretical guidelines for optimal designs of practical chemical plants.

Electrical conductivity and thermodynamic studies on Sodium Dimethyldithiocarbamate in non aqueous solvents Dimethylformamide (DMF), at different temperatures

This paper threw some light on the behavior of Sodium N,N-Dimethyldithiocarbamate as an electrolyte. The effect of solvents on the conductance of this salt would be discussed via measurements of Λo, ao and KA, since it can be assumed that the different solvents have a little chance to impose great variations on the solvation processes. The conductance method was chosen as a tool to illustrate the electrolyte-solvent interactions. Fuoss–Onsager equation would be tested using Sodium Dimethyldithiocarbamate in presence of dimethylformamide solvent at different temperatures. The conductance of dilute solutions of Sodium N,N-Dimethyldithiocarbamate is measured in Dimethylformamide, at different temperatures (25, 30, 35 and 40 °C). Accurate values of Λo were obtained by applying the (Fuoss–Kraus–Shedlovsky) equation. Finally, the (Fuoss–Onsager) equation was solved to give the correct values of the constants Λo, J, KA and a° (the closest distance of approach) for Sodium N,N-Dimethyldithiocarbamate salt in Dimethylformamide solvent. Λo and a° (solvation) increase with increasing temperatures. Thermodynamic parameters (∆G°, ∆H°, ∆S° and ∆Es) of Sodium N,N-Dimethyldithiocarbamate in Dimethylformamide were calculated from conductance measurements, the activation energy (∆Es), the enthalpy change (heat of association) (∆H°) and the entropy change (∆S°) are positive, however The free energy change (∆G°) values was negative for Sodium N,N-Dimethyl dithiocarbamate in DMF systems studied with increasing the temperature.

Maximizing power output of endoreversible non-isothermal chemical engine via linear irreversible thermodynamics

Maximum power performance of an endoreversible non-isothermal chemical engine (NICE) with simultaneous heat and mass transfer is investigated in this paper. The heat and mass transfer processes are assumed to obey Onsager equations in linear irreversible thermodynamics. The power output of the endoreversible NICE as well as the corresponding vector efficiency [See Eq. (13) in this paper for its definition] are obtained analytically. Special cases for an endoreversible Carnot heat engine with the linear phenomenological heat transfer law [q∝Δ(T−1)] and an endoreversible isothermal chemical engine with the linear mass transfer law [g∝Δ(μ)] are further derived based on the general optimization results. Some numerical examples are provided, and the effects of changes of absorbed energy flux rate, mass flux rate, and heat and mass transfer phenomenological coefficients on the optimization results are analyzed. The results show that there are optimal absorbed energy flux rate and optimal mass flux rate for the power output of the endoreversible NICE to approach to its maximum; the relationship between the power output of the endoreversible NICE and its vector efficiency is paraboloid.

Nonequilibrium thermodynamic model of diffusion processes in the steel - carbon thin film tribological system

A nonequilibrium thermodynamic model of dissipative processes in the tribological system steel – carbon thin film was developed. Cross coefficients and driving forces in the Onsager equation for the diffusion of carbon onto the friction surface of hypereutectoid steel are calculated. The thermodynamic conditions for the implementation of the selective transfer process in the tribological system under consideration are determined. Simple differential equations are obtained for the tribological system of carbon steel - graphite, which describe the effect of stresses on the diffusion of carbon, production based on nonequilibrium thermodynamics. It is shown, that the implementation of selective transfer requires the development of diffusion processes without the formation of excess phases. Therefore, carbon steel should be alloyed mainly with elements of the first group that increase the thermodynamic activity of carbon in steel - silicon, nickel, copper, cobalt and phosphorus. Recommendations are formulated on metallic materials choosing for minimal contact wear in contacted pairs of machines parts.

The absolute thermoelectric power of chromium, molybdenum, and tungsten

Abstract In pure metals or homogeneous alloys, the number of electrons per atom as well as the pressure are constant in a temperature gradient, and the electrochemical potential of the electrons is a unique function of the temperature. That is why it is possible to obtain the thermopower of these materials not only in the conventional evaluation of the Onsager equations, which involves the calculation of two transport coefficients, but also with an alternative approach, which involves the calculation of the temperature dependence of the electrochemical potential. The method is illustrated with the calculation of the thermopower of Cr, Mo, and W from the density of states of these metals.

Optimization of a conical cascade using invasive weed optimization (IWO) algorithm for multi-component systems: Investigation of the effect of feed flow on the separation factor of centrifuge machines in the cascade by DSMC method

One of the desirable properties that can be mentioned in the design of a cascade is to maximize the amount of separation work for the minimum number of machines so that the input feed does not exceed its maximum amount. For this purpose, the dilute gas area and part of the continuums area inside gas centrifuge are considered as the computational area for calculating the mass source. The mass source is calculated by direct simulation Monte-Carlo (DSMC) method. Based on the separation parameters obtained from the proposed flow function, the optimum cascade design was performed using the invasive weed optimization (IWO) algorithm. The results show that by considering the exact effect of feed by DSMC method in the Onsager equation, with increasing the separation parameters in a centrifuge machine, for a certain amount of separation in the product and waste streams in an optimal cascade, fewer centrifuge machines were obtained.

Electronic and heat transport phenomena in the nanogranular thiospinelFe3S4

The iron sulphide ${\mathrm{Fe}}_{3}{\mathrm{S}}_{4}$ (greigite) is similar to its oxide counterpart ${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$ (magnetite) as to the crystal structure and ferrimagnetic order, but differs in electronic states. The ab initio calculations have evidenced $p$-type carriers at three spin-minority and three spin-majority Fermi surfaces, which is in contrast to the half-metallic character of magnetite with $n$-type carriers at three spin-minority Fermi surfaces. The transport properties including Hall and Nernst effects have been studied over the range 2--300 K by using nanogranular ceramics prepared by cold isostatic pressing of ${\mathrm{Fe}}_{3}{\mathrm{S}}_{4}$ particles with the mean crystallite size of ${d}_{\mathrm{XRD}}\ensuremath{\approx}80$ and 30 nm, respectively. The samples show metalliclike electrical resistivity with large residual value at $T$ \ensuremath{\rightarrow} 0 K. The $p$-type character of the charge carriers is reflected by the positive sign of both the thermopower and Hall effect. Temperature dependencies of the electrical resistivity, thermal conductivity, and thermopower are analyzed by considering processes of the grain boundary and defect intragrain scattering; simultaneously the role of magnons and their dynamics in electronic and heat transport is revealed. The Nernst and Hall effects show dominant contributions of anomalous type (ANE and AHE) with signs exactly opposite to those of ${\mathrm{Fe}}_{3}{\mathrm{O}}_{4}$, i.e., positive AHE and negative ANE in ${\mathrm{Fe}}_{3}{\mathrm{S}}_{4}$. The results are interpreted by evoking the original Callen treatment of thermoelectric and thermomagnetic phenomena using Onsager equations. Scaling between the longitudinal and transverse components of the electrical resistivity and thermoelectric conductivity tensors is checked. The analysis of the temperature dependent AHE using the relation between transverse and longitudinal resistivity ${\ensuremath{\rho}}_{yx}^{A}(T)\ensuremath{\sim}{\ensuremath{\rho}}_{xx}{(T)}^{n}$ gives the characteristic exponent $n=1.15$, which is close to the $n=1$ predicted by the skew-scattering mechanism.

Non-equilibrium thermodynamics model for calculating diffusion fluxes under phase transformations in alloy steels

The phase transformations in alloyed iron-carbon alloys is largely related to diffusion of components, foremost to the carbon. For the analysis of diffusive processes in alloy steels, it is possible to use the mathematical methods of non-equilibrium thermodynamics. The equation for the diffusive fluxes of the system contains unknown in general case of coefficients activity of elements and vacancies, and their derivatives for to the concentrations, that extraordinarily makes it difficult being of values of cross coefficients. In the article a non-equilibrium thermodynamics methodology of calculation of diffusive fluxes at presence of two phases in alloy steels is described. It allows one to calculate both direct- and cross coefficients in the Onsager equations. Formulas for calculation of thermodynamics forces in the alloy steel – for iron, alloying element of substitution – chrome, of element of introduction – carbon and vacancies, are presented. Common expressions are suggested for calculation of cross-factors, motive forces and fluxes in the Onsager’s equations for a multicomponent thermodynamic system. The example of using the developed model to find changes in concentrations and diffusion fluxes over time is given. For the model system used, it was established that at the stage of predominant diffusion of carbon in the alloy steel, cementite inclusions with a size of about 18 nm are formed rather quickly (within ~ 200 s). The technique developed in the article allows one to perform diffusion kinetics calculations in multicomponent thermodynamic systems, which are also iron-carbon alloys and to control the size of the phases formed, for example, of carbide nanoparticles.

Transport and optical properties of 1-octyl 3-methylimidazolium bromide ionic liquid in water + Ethanol/1-propanol mixtures at T = (298.2, 308.2 and 318.2) K

In this work, viscosities (η), conductivities (Ʌ) and refractive index (nD) for the ternary systems of 1-octyl-3-methylimidazolium bromide [OMIm]Br ionic liquid in the Ethanol/1-propanol+water mixtures were measured in different mass fractions of alcohol in water (w/w% = wAlcohol/wmixture = 0, 10, 20 and 30%) at T = (298.2, 308.2 and 318.2) K. The viscosity data was analyzed using jones-dole equation to derive the viscosity B-coefficient which indicates the solute-solvent interactions. The temperature dependence of the viscosity was described by fluidity equation. Ion association constants (Ka) and limiting molar conductivities (Ʌ0) of [OMIm]Br were obtained by using Fuoss–Onsager equation based on conductivity measurements. In addition, refractive index (nD) of [OMIm]Br were measured for the binary and ternary water+[OMIm]Br + Ethanol/1-propanol mixtures. The obtained results were compared to the predicted values using the Heller equation. Excess refractive index (nDE) for binary and ternary mixtures and regression parameters of the Redlich–Kister and Cibulka equation were obtained at different temperatures.

Molecular free energy profiles from force spectroscopy experiments by inversion of observed committors.

In single-molecule force spectroscopy experiments, a biomolecule is attached to a force probe via polymer linkers and the total extension of the molecule plus apparatus is monitored as a function of time. In a typical unfolding experiment at constant force, the total extension jumps between two values that correspond to the folded and unfolded states of the molecule. For several biomolecular systems, the committor, which is the probability to fold starting from a given extension, has been used to extract the molecular activation barrier (a technique known as "committor inversion"). In this work, we study the influence of the force probe, which is much larger than the molecule being measured, on the activation barrier obtained by committor inversion. We use a two-dimensional framework in which the diffusion coefficient of the molecule and of the pulling device can differ. We systematically study the free energy profile along the total extension obtained from the committor by numerically solving the Onsager equation and using Brownian dynamics simulations. We analyze the dependence of the extracted barrier on the linker stiffness, molecular barrier height, and diffusion anisotropy and, thus, establish the range of validity of committor inversion. Along the way, we showcase the committor of 2-dimensional diffusive models and illustrate how it is affected by barrier asymmetry and diffusion anisotropy.

Method for the Calculation of the Hamaker Constants of Organic Materials by the Lifshitz Macroscopic Approach with Density Functional Theory

The Hamaker constants, which are coefficients providing quantitative information on intermolecular forces, were calculated for a number of different materials according to the Lifshitz theory via simple DFT calculations without any experimental measurements being performed. The physical properties (polarizability, dipole moment, molecular volume, and vibrational frequency) of organic molecules were calculated using the B3LYP density functional and the aug-cc-pVDZ basis set. Values for the Hamaker constants were obtained using the approximation of the Lorentz-Lorenz equation and Onsager's equation with these properties. It was found that, in the case of "nonassociative" materials, such as hydrocarbons, ethers, ketones, aldehydes, carboxylic acids, esters, nitriles, and hydrosilanes, and halides, the calculated Hamaker constants were similar in value to their experimentally determined counterparts. Moreover, with this calculation method, it is easy to create the molecular model and the CPU time can be shortened.

A finite element solution to the generalized Onsager model of fluid flow in a countercurrent gas centrifuge

ABSTRACT Based on the Onsager Equation, a linearized sixth-order partial differential equation describing the flow in the volume of the rotor of a gas centrifuge is solved using a finite element algorithm employed by the CurvSOL code. The results are compared to those from Pancake, an existing code that uses eigenfunction expansion to solve the Onsager equation. Excellent agreement is demonstrated between the solutions of the wall temperature gradient and scoop drive mechanisms for both the Rome and Iguaçu centrifuge designs. Results suggest rotor speed and radius play key roles in the influence of wall curvature on the flow solution.

Non-equilibrium thermodynamics as gauge fixing

By considering a thermodynamic force as gauge field, we extend constitutive equations of Onsager's non-equilibrium thermodynamics to non-linear equations. In Onsager's non-equilibrium thermodynamics, the thermodynamic force corresponds to a pure gauge, for which the constitutive equations are obtained by gauge fixing. If we extend the thermodynamic force from pure gauge to physical one, we obtain the non-linear constitutive equations of non-equilibrium thermodynamics.

The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

We study the small Deborah number limit of the Doi–Onsager equation in the case when hydrodynamic effects are neglected. This is a Smoluchowski-type equation that describes the dynamics of nematic liquid crystals at a molecular level, by characterizing the evolution of a number density function, depending upon both particle position x∈Rd(d=2,3) and orientation vector m∈S2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S2. This flow is a special case of the gradient flow of the well known Oseen–Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions. The proof relies on the strong compactness of the corresponding Q-tensor (namely the second moment), a detailed analysis of the linearized operator near the limit local equilibrium distribution, and an energy dissipation estimate.

The Role of Solvation in the Kinetics and the Mechanism of Hydroperoxide Radicals Addition to π‐Bonds of 1,2‐Diphenylethylene and 1,4‐Diphenylbutadiene‐1,3

ABSTRACTA combination of microcalorimetry, the rotating sector method, and ESR at 323 K in the environment of 10 solvents of different polarities was used to measure rate constants of addition of hydroperoxide radicals () to π bonds of trans‐1,2‐diphenylethylene and trans,trans‐1,4‐diphenylbutadiene‐1,3 (k2) and disproportionation rate constants of these radicals (k3). With increasing dielectric constant of the medium, k2 values increase from 69 to 410 M−1 · s−1, and k3 values almost do not change and are in the range of (1.0 ± 0.2) × 108 M−1 · s−1. A linear dependence of logarithm values of rate constants from the dielectric constant of the medium in the coordinates of the Kirkwood–Onsager equation was found that allows to make a conclusion about the effect of nonspecific solvation in the studied systems. The quantum‐chemical analysis (NWChem, DFT B3LYP/6‐311G**) of the detailed mechanism for addition shows that the influence of the medium polarity reflects the superposition of the effects of nonspecific and specific solvation. The scale of the polar effect will depend on how different solvation energies of the transition and the initial reaction complexes. If a value of the solvation energy of the transition complex is larger than the solvation energy of the initial reaction complex, then the reaction rate should increase with an increase of the solvent's polarity and decrease otherwise.

The effect of solvation on the reactivity of 1,1-substituted ethylenes in hydroperoxyl radical addition reactions

The effect of solvation on the rate constants of hydroperoxyl radical addition to the double bonds of 1,1-substituted ethylene vinyl compounds was studied in the medium of different polarity solvents. The values of rate constants of this process were determined in a series of experiments on the co-oxidation of diphenylethylene (was used as the source of hydroperoxyl radicals) with the studied vinyl compounds at 323 K using the method of microvolumetry in combination with ESR spectroscopy and computer kinetic modeling. The observed linear relationship between the values of rate constant logarithms and the dielectric permittivity of reaction medium in coordinates of the Kirkwood–Onsager equation allows for the conclusion that the effect of nonspecific solvation due to the electrostatic forces does take place in the studied systems.

Conductometric and refractometric study of 1-Ethyl-3-methylimidazolium Bromide ionic liquid in water + ethanol/1-propanol mixtures at T = (298.2, 308.2 and 318.2) K

In this work, thermophysical properties such as conductivity (Λ) and refractive index (nD) of 1-Ethyl-3-methylimidazolium Bromide, [EMIm]Br, in ternary mixtures of [EMIm]Br+ethanol/1-propanol+water have been measured at T=(298.2, 308.2 and 318.2)K and P=0.1MPa. The conductometric measurements were performed for [EMIm]Br ionic liquid from 0.0001 to 0.2molkg−1 in different mass fractions of 1-propanol and ethanol in water+ethanol/1-propanol mixtures ((w/w)%=walcohol/wmixture=10%, 20% and 30%) at T=(298.2, 308.2 and 318.2)K. Ion association constants (Ka) and limiting molar conductivities (Λ0) of [EMIm]Br were obtained by using Fuoss Onsager equation. The critical aggregation concentration (cac) and the degree of ionization (α) were determined by conductivity measurements. The obtained parameters were used to calculate the standard Gibbs free energy of aggregation (ΔG0). In addition, refractive indexes (nD) of [EMIm]Br were measured for the binary and ternary water+[EMIm]Br+ethanol/1-propanol mixtures at T=(298.2, 308.2 and 318.2)K. The obtained results were compared to the predicted values using the Lorentz-Lorenz equation. Excess refractive index (nDE) for binary and ternary mixtures and regression parameters of the Redlich–Kister and Cibulka equations were obtained at different temperatures.

Differential Solvation.

Solvation effects influence reaction equilibria by preferentially stabilizing reactants or products (differential solvation). We propose using simple electrostatic concepts to qualitatively interpret and understand these effects, applying the Born and Kirkwood-Onsager equations. Scenarios include, for charged species, redox potentials, different total absolute charges between reactants and products, and size differences between reactants and products. In addition, for neutral species, they are differences in total dipole moment and size differences. These scenarios are illustrated with several examples from different areas of chemistry.