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Wandrei SM, McCarthy DG, Schoen M. Phase Behavior of Magnetic Nanocolloids of Different Sizes Suspended in an Apolar Solvent. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2017; 33:11366-11376. [PMID: 28764322 DOI: 10.1021/acs.langmuir.7b01952] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We employ classical density functional theory (DFT) to investigate the phase behavior and composition of binary mixtures; each compound consists of hard spheres of different sizes with superimposed dispersion attraction. In addition to the dispersion attraction, molecules of one component carry an additional three-dimensional magnetic "spin" where the orientation-dependent spin-spin interaction is accounted for by the Heisenberg model. We are treating the excess free energy using a modified mean-field approximation (second virial coefficient) for the orientation-dependent pair correlation function. Depending on the concentration of the magnetic particles, the strength of the spin-spin coupling, and the size ratio of the particles, the model predicts the formation of ordered (polar) phases in addition to the more conventional gas and isotropic liquid phases. Key features of our model are a particle-size dependent shift of the gas-liquid critical point (critical temperature and density) and a change in the width of the phase diagram. In the near-critical region, the latter can be analyzed quantitativly in terms of an effective critical exponent βeff that may differ from the classical critical exponent [Formula: see text]; the classical value is attained in the immediate vicinity of the critical point as it must. The deviation between βeff and β can be linked to nontrivial composition effects along the phase boundaries.
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Affiliation(s)
- Stefanie M Wandrei
- Stranski-Laboratorium für Physikalische und Theoretische Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische Universität Berlin , Straße des 17. Juni 115, 10623 Berlin, Germany
| | - Dannielle G McCarthy
- Department of Chemistry, Stanford University , 333 Campus Dr., Stanford, California 94305, United States
| | - Martin Schoen
- Stranski-Laboratorium für Physikalische und Theoretische Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische Universität Berlin , Straße des 17. Juni 115, 10623 Berlin, Germany
- Department of Chemical and Biomolecular Engineering, Engineering Building I, Box 7905, North Carolina State University , 911 Partners Way, Raleigh, North Carolina 27695, United States
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Cattes SM, Klapp SHL, Schoen M. Condensation, demixing, and orientational ordering of magnetic colloidal suspensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052127. [PMID: 26066139 DOI: 10.1103/physreve.91.052127] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2015] [Indexed: 06/04/2023]
Abstract
In this work we study the phase behavior of magnetic particles suspended in a simple nonmagnetic solvent. Magnetic particles are modelled as spherical particles carrying a three-dimensional, classical Heisenberg spin, whereas solvent molecules are treated as spherically symmetric Lennard-Jones particles. The binary mixture of magnetic particles and solvent is studied within the framework of classical density functional theory (DFT). Within DFT pair correlations are treated at the modified mean-field level at which they are approximated by orientation dependent Mayer f functions. In the absence of an external magnetic field four generic types of phase diagrams are observed depending on the concentration of magnetic particles. In this case we observe liquid-liquid phase coexistence between an orientationally ordered (polarized) and a disordered phase characterized by slightly different concentrations of magnetic particles. Liquid-liquid phase coexistence is suppressed by an external field and vanishes completely if the strength of the field is sufficiently large.
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Affiliation(s)
- Stefanie M Cattes
- Stranski-Laboratorium für Physikalische und Theoretische Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des 17. Juni 115, 10623 Berlin, Germany
| | - Sabine H L Klapp
- Institut für Theoretische Physik, Fakultät für Mathematik und Naturwissenschaften, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Martin Schoen
- Stranski-Laboratorium für Physikalische und Theoretische Chemie, Fakultät für Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des 17. Juni 115, 10623 Berlin, Germany
- Department of Chemical and Biomolecular Engineering, Engineering Building I, Box 7905, North Carolina State University, 911 Partners Way, Raleigh, North Carolina 27695, USA
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Berim GO, Ruckenstein E. Nanodrop of an Ising magnetic fluid on a solid surface. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2011; 27:8753-60. [PMID: 21671604 DOI: 10.1021/la2011919] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
The density functional theory of inhomogeneous simple fluids is extended to an Ising magnetic fluid in contact with a solid surface, which is subjected to an external uniform or nonuniform magnetic field. The system is described by two coupled integral equations regarding the magnetic moment and fluid density distributions. The dependence of the contact angle that a nanodrop makes with the solid surface on the parameters involved in the magnetic interactions between the molecules of fluid and between the molecules of fluid and an external magnetic field is calculated. For the uniform magnetic field, the contact angle increases with increasing magnetic field, approaching an asymptotic value that depends on the strength of the fluid-fluid magnetic interactions. In the nonuniform field generated by a permanent magnet, the contact angle first increases with increasing magnetic field B(M) and then decreases, with the decrease being almost linear for large values of B(M). The obtained results are in qualitative agreement with the experimental data on the contact angle of magnetic drops on a solid surface available in the literature.
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Affiliation(s)
- Gersh O Berim
- Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260, United States
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Canas-Marin WA, Lira-Galeana C. Calculation of Critical Points in Gas−Condensate Mixtures under the Influence of Magnetic Fields. Ind Eng Chem Res 2010. [DOI: 10.1021/ie900939y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Omelyan IP, Folk R, Kovalenko A, Fenz W, Mryglod IM. Liquid-vapor interfaces in XY -spin fluids: an inhomogeneous anisotropic integral-equation approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:011123. [PMID: 19257017 DOI: 10.1103/physreve.79.011123] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2008] [Indexed: 05/27/2023]
Abstract
An integral-equation approach is developed to study interfacial properties of anisotropic fluids with planar spins in the presence of an external magnetic field. The approach is based on the coupled set of the Lovett-Mou-Buff-Wertheim integro-differential equation for the inhomogeneous anisotropic one-particle density and the Ornstein-Zernike equation for the orientationally dependent two-particle correlation functions. Using the proposed inhomogeneous angle-harmonics expansion formalism we show that these integral equations can be reduced to a much simpler form similar to that inherent for a system of isotropic fluids. The interfacial orientationally dependent direct correlation function can be consistently constructed by means of a nonlinear interpolation via its values obtained in the coexisting anisotropic bulk phases. A soft mean spherical approximation is employed for the closure relation. This has allowed us to solve the complicated integral equations in the situation when both spatial inhomogeneity and orientational anisotropy are present simultaneously. The approach introduced is applied to an XY fluid model with ferromagnetic spin interactions. As a result, the density-orientation and magnetization profiles at the liquid-vapor interfaces are calculated in a wide range of temperatures up to subcritical regions. The influence of the external field on the microscopic structure of the interfaces and the surface tension is also analyzed in detail.
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Affiliation(s)
- I P Omelyan
- Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine
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Almarza NG, Lomba E, Martín C, Gallardo A. Demixing in binary mixtures of apolar and dipolar hard spheres. J Chem Phys 2008; 129:234504. [DOI: 10.1063/1.3039512] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Lado F, Lomba E. Integral equation procedure based on tailored orthogonal functions for the XY spin fluid in an external magnetic field. Phys Rev E 2007; 76:041502. [PMID: 17994989 DOI: 10.1103/physreve.76.041502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2007] [Indexed: 11/07/2022]
Abstract
The classical XY model describes particles in three-dimensional space that carry magnetic moments or spins whose motion is restricted to rotations in a plane. Introduction of an external magnetic field lying in the same plane then generates a system that is anisotropic in the azimuthal angle phi . We use numerical simulations and integral equation techniques to study this system, producing in the latter case a formalism that is identical to that of the simpler isotropic version having no external field. The basis for this simplification is a generalization Em(phi) of the ordinary exponential basis set eimphi that restores orthogonality in the presence of the external field. We display results of sample calculations obtained with two integral equation closures, reference hypernetted-chain and soft mean-spherical approximation, both coupled to the Lovett-Mou-Buff-Wertheim relation, along with results from the numerical simulations for comparison. Construction of the Em(phi) is described in an Appendix.
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Affiliation(s)
- F Lado
- Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202, USA
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Fenz W, Folk R, Mryglod IM, Omelyan IP. Possibility of Fisher renormalization of the critical exponents in an Ising fluid. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:061504. [PMID: 17677266 DOI: 10.1103/physreve.75.061504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2007] [Indexed: 05/16/2023]
Abstract
Using Monte Carlo simulation techniques, we study the ferromagnetic order-disorder phase transition in Ising spin fluids with hard-core Yukawa interaction truncated at various cutoff radii r{c}. We focus our interest on the dependence of critical quantities such as the Binder cumulant and various exponent ratios on the value of r{c}, and on the question whether the Fisher-renormalized exponents expected for such systems can be observed in the simulations. It turns out that the corrections to scaling decaying with a rather small exponent prevent reaching the asymptotic region with the computational power available. Thus, we observe only effective exponents, with different (nonuniversal) values depending on the cutoff radius. The same behavior is also found for the critical Binder cumulant. Nevertheless, an exact investigation of the effective susceptibility exponent gamma{eff} as a function of temperature seems to point towards a Fisher-renormalized value. For two selected cutoff radii, the critical temperature is determined more accurately using, in addition to the cumulant crossing technique, the scanning technique and the shifting technique, taking into account corrections to scaling. Simulations of Ising fluids with constant cutoff radius and varying Yukawa-tail screening lengths lambda also show a nonuniversal dependence of U{c} on lambda. Finally, we have performed simulations of the Ising lattice model with increasing number of couplings which show the expected asymptotic behavior, independent of the range of interactions.
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Affiliation(s)
- W Fenz
- Institute for Theoretical Physics, Linz University, A-4040 Linz, Austria
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Omelyan IP, Folk R, Mryglod IM, Fenz W. Liquid-vapor and liquid-liquid interfaces in Ising fluids: An integral equation approach. J Chem Phys 2007; 126:124702. [PMID: 17411147 DOI: 10.1063/1.2709885] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The microscopic structure and thermodynamic properties of liquid-vapor and liquid-liquid interfaces in Ising fluids are studied using an integral equation approach. The calculations are performed in the absence and presence of an external magnetic field by solving the corresponding set of Lovett-Mou-Buff-Wertheim integrodifferential equations for the one-particle density distribution functions. The two-particle inhomogeneous direct correlation functions are consistently constructed by nonlinear interpolation between the bulk ones. The bulk correlation functions of the coexisting phases are obtained from the Ornstein-Zernike equations with a modified soft mean spherical approximation for the closure relation. As a result, the density and magnetization profiles at liquid-vapor and liquid-liquid interfaces as well as the surface tension and adsorption coefficients are evaluated in a wide temperature range including subcritical regions. The influence of an external magnetic field on the liquid-vapor interfaces is also considered.
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Affiliation(s)
- I P Omelyan
- Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine and Institute for Theoretical Physics, Linz University, A-4040 Linz, Austria
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Köfinger J, Wilding NB, Kahl G. Phase behavior of a symmetrical binary fluid mixture. J Chem Phys 2006; 125:234503. [PMID: 17190563 DOI: 10.1063/1.2393241] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We have investigated the phase behavior of a symmetrical binary fluid mixture for the situation where the chemical potentials mu(1) and mu(2) of the two species differ. Attention is focused on the set of interparticle interaction strengths for which, when mu(1)=mu(2), the phase diagram exhibits both a liquid-vapor critical point and a tricritical point. The corresponding phase behavior for the case mu(1) not equalmu(2) is investigated via integral-equation theory calculations within the mean spherical approximation and grand canonical Monte Carlo (GCMC) simulations. We find that two possible subtypes of phase behavior can occur, these being distinguished by the relationship between the triple lines in the full phase diagram in the space of temperature, density, and concentration. We present the detailed form of the phase diagram for both subtypes and compare with the results from GCMC simulations, finding good overall agreement. The scenario via which one subtype evolves into the other is also studied, revealing interesting features.
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Affiliation(s)
- Jürgen Köfinger
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria
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Lomba E, Martín C, Almarza NG, Lado F. Phase behavior of a hard sphere Maier-Saupe nematogenic system in three dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:021503. [PMID: 17025433 DOI: 10.1103/physreve.74.021503] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2006] [Indexed: 05/12/2023]
Abstract
We present a detailed computer simulation and integral equation study of the phase behavior of a nematogenic system composed of hard spheres with embedded three-dimensional Maier-Saupe spins. For this well-known system, we map the gas-liquid equilibrium, which is coupled to a first-order isotropic-nematic transition. The anisotropic integral equation theory is found to yield excellent agreement with the simulation data within the fluid regime. Additionally, we determine the fluid-solid equilibrium transition by means of computer simulation.
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Affiliation(s)
- E Lomba
- Instituto de Química Física Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain
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Fenz W, Omelyan IP, Folk R. Integral equation study of an ideal Ising mixture. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:056121. [PMID: 16383702 DOI: 10.1103/physreve.72.056121] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2005] [Revised: 09/21/2005] [Indexed: 05/05/2023]
Abstract
We construct an integral equation scheme for magnetic binary mixtures of an ideal soft-core Ising fluid and a soft-sphere fluid by mapping the system onto an equivalent nonmagnetic ternary mixture. We apply the multicomponent Ornstein-Zernike equation together with a closure relation based on the soft mean spherical approximation and a field constraint for the Ising fluid component. Phase coexistence curves are calculated both by directly evaluating the chemical potentials via the bridge function, and by using a Maxwell-like construction which is derived in the text. Our results are compared to Monte Carlo data obtained earlier, and we find that the second method yields a much better agreement with the simulations.
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Affiliation(s)
- W Fenz
- Institute for Theoretical Physics, Linz University, A-4040 Linz, Austria
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Schöll-Paschinger E, Kahl G. Type-IV phase behavior in fluids with an internal degree of freedom. J Chem Phys 2005; 123:134508. [PMID: 16223315 DOI: 10.1063/1.2042447] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We have identified a fourth archetype of phase diagram in binary symmetrical mixtures, which is encountered when the ratio of the interaction between the unlike and the like particles is sufficiently small. This type of phase diagram is characterized by the fact that the lambda line (i.e., the line of the second-order demixing transition) intersects the first-order liquid-vapor curve at densities smaller than the liquid-vapor critical density.
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Omelyan IP, Fenz W, Mryglod IM, Folk R. XY-spin fluids in an external magnetic field: an integral equation approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031506. [PMID: 16241443 DOI: 10.1103/physreve.72.031506] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2005] [Indexed: 05/05/2023]
Abstract
We develop an integral equation approach to study anisotropic fluids with planar spins in the presence of an external field. As a result, the integral equation calculations for these systems appear to be no more difficult than those for ordinary isotropic liquids. The method presented is applied to the investigation of phase coexistence properties of ferromagnetic XY-spin fluids in a magnetic field. The soft mean spherical approximation is used for the closure relation connecting the orientationally dependent two-particle direct and total correlation functions. The Lovett-Mou-Buff-Wertheim and Born-Green-Yvon equations are employed to describe the one-particle orientational distribution. The phase diagrams are obtained in the whole range of varying the external field for a wide class of XY-spin fluid models with various ratios of the strengths of magnetic to nonmagnetic Yukawa-like interactions. The influence of changing the screening radii of the interaction potentials is also considered. Different types of the phase diagram topology are identified. They are characterized by the existence of critical, tricritical, critical end, and triple points related to transitions between gas, liquid, and para- and ferromagnetic states, accompanied by different external field dependencies of critical temperatures and densities corresponding to the gas-liquid and liquid-liquid transitions. As is demonstrated, the integral equation approach leads to accurate predictions of the complicated phase diagram behavior which coincide well with those evaluated by the cumbersome Gibbs ensemble simulation and multiple-histogram reweighting techniques.
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Affiliation(s)
- I P Omelyan
- Institute for Condensed Matter Physics, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine
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Range GM, Klapp SHL. Density-functional study of model bidisperse ferrocolloids in an external magnetic field. J Chem Phys 2005; 122:224902. [PMID: 15974711 DOI: 10.1063/1.1924702] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We present phase diagrams of a model bidisperse ferrocolloid consisting of a binary mixture of dipolar hard spheres (DHSs) under the influence of an external magnetic field. The dipole moments of the particles are chosen proportional to the particle volume to mimic real ferrocolloids, and we focus on dipole-dominated systems where isotropic attractive interactions are absent. Our results are based on density-functional theory in the modified mean-field (MMF) approximation. For one-component DHS fluids in external fields, and for corresponding mixtures dominated by one of the components, MMF theory predicts the tricritical point of the transition between an isotropic gas and a ferromagnetic liquid occurring at zero field to be changed into a critical point separating two magnetically ordered phases of different density. The corresponding critical temperature displays a nonmonotonic dependence on the field strength. Completely different behavior is found for the critical temperature related to the demixing phase transitions appearing in strongly asymmetric mixtures [G. M. Range and S. H. L. Klapp, Phys. Rev. E 70, 061407 (2004)]. For such systems, we find a monotonic decrease of the demixing critical temperature with increasing field. The field strength dependence of the critical temperature can therefore be tuned between nonmonotonic and monotonic behaviors just by changing the composition of the mixture--e.g., by adjusting the chemical potentials. This allows us to efficiently control the influence of external magnetic fields on the phase behavior over a large temperature interval.
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Affiliation(s)
- Gabriel M Range
- Stranski-Laboratorium für Physikalische und Theoretische Chemie, Sekretariat TC 7, Fakultät II für Mathematik und Naturwissenschaften, Technische Universität Berlin, Strasse des 17. Juni 124, D-10623 Berlin, Germany.
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Fenz W, Folk R. Phase behavior of Ising mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:046104. [PMID: 15903723 DOI: 10.1103/physreve.71.046104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2004] [Indexed: 05/02/2023]
Abstract
We present phase diagrams that were calculated both in mean-field theory and via Monte Carlo (MC) simulations for binary mixtures of a ferromagnetic Ising fluid and a nonmagnetic fluid (Ising mixtures) in the absence of an external field. We look at both the simple ideal Ising mixture, consisting of an ideal Ising fluid and a hard-sphere fluid, as well as at the general case with one component being a nonideal Ising fluid and the other a van der Waals fluid. It is shown that the mean-field phase diagram of the ideal Ising mixture in the limit of infinite pressure is identical to that of the Blume-Capel model for 3He-4He mixtures. The MC phase diagrams were obtained using the Gibbs ensemble, the cumulant intersection technique, and the multi-histogram re-weighting method, adapted to the semi-grand ensemble. The results are qualitatively compared with mean-field theory, and both types of tri-critical lines occurring there are verified in the computer simulations.
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Affiliation(s)
- W Fenz
- Institute for Theoretical Physics, Linz University, A-4040 Linz, Austria
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Omelyan IP, Fenz W, Mryglod IM, Folk R. XY spin fluid in an external magnetic field. PHYSICAL REVIEW LETTERS 2005; 94:045701. [PMID: 15783572 DOI: 10.1103/physrevlett.94.045701] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2004] [Indexed: 05/24/2023]
Abstract
A method of integral equations is developed to study anisotropic fluids with planar spins in an external field. As a result, the calculations for these systems appear to be no more difficult than those for ordinary homogeneous liquids. The approach proposed is applied to the ferromagnetic XY spin fluid in a magnetic field using a soft mean spherical closure and the Born-Green-Yvon equation. This provides an accurate reproduction of the complicated phase diagram behavior obtained by cumbersome Gibbs ensemble simulation and multiple histogram reweighting techniques.
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Affiliation(s)
- I P Omelyan
- Institute for Condensed Matter Physics, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine
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