"Eighth international conference on magnetic fluids",

ICMF 8, June 29-July 3, 1998, Timisoara, Romania, pp. 98-99

STATISTICAL HYDRODYNAMICS OF A HEISENBERG MODEL FERROFLUID

I. Mryglod ¹, R. Folk ², S. Dubyk ³, Yu. Rudavskii ³.

1. Institute for Condensed Matter Physics, National Academy of Scienses of Ukraine, Ukraine, Lviv, UA-290011, Sventsitskii Street, house 1. E – mail: mryglod@icmp.lviv.ua

2. Institut for Theoretische Physik, University Linz, 4040-Linz, Wsterreich

3. State University ''Lvivska Politekhnika'', 12 Bandera St, UA-290013 Lviv, Ukraine

Introduction

Since the invention of magnetic fluid, the characteristics of these fluids have been progressively improved and applications in various fields are growing. From the theoretical point of view, ab initio investigation of static and especially dynamical properties of ferrofluids still remains to be an important problem. In this report the results for hydrodynamic collective mode spectrum, transport coefficients as well as for the hydrodynamic time correlation fuctions of a Heisenberg model ferrofluid obtained within microscopic statistical approach will be presented.

A Heisenberg model ferrofluid

To study ferrofluid, we considered a model in which the particles interact via pair potentials. The Hamiltonian of such system is a sum of two terms. The first described the classical translational degrees of freedom of particles. The second one is the Hamiltonian of «magnetic» subsystem describing spin degrees of freedom (or orientational motions). Contrary to the first term, the Hamiltonian of «magnetic» subsystem can be considered either as classical or quantum mechanical. Such model is rather general and the interactions have to be specified for subsequent calculations. In particular, for the description of translational motions one can choose a Lennard-Jones potential, hard or soft sphere, etc. The spin interaction may be considered either as an isotropic Heisenberg-like or dipolar one.

Results and Discussion

For derivation of the hydrodynamic equations, the Zubarev's method of the non-equilibrium statistical operator [1] has been used. The general procedure for solving the Liouville equation was described in detail in Ref.[2]. The set of hydrodynamic equations for the densities of conserved variables such as mass-, momentum-, energy- and spin-density are obtained. The microscopic expressions for the generalized (k-depending) thermodynamic quantities and generalized (k- and -depending) transport coefficients are derived. The hydrodynamic equations, the structure of frequency matrix and matrix of memory functions are written in a form which allows to consider the limiting cases, where the system reduces to the hydrodynamic description of a pure ``liquid'' or a pure ''magnetic'' system. It is shown the equations transform to the well-known system of molecular hydrodynamic equations [3], the variables of ''magnetic'' subsystem having been formally neglected. This limit is reached in practice when the spin relaxation is much faster than process with typical time scale of the ''liquid'' subsystem. In another limiting case if the relaxation in a ''magnetic'' subsystem is much more slower than typical time scales of a ''liquid'' subsystem, the hydrodynamic equations of ferromagnets (see, e.g., Ref.[4]) are obtained. However, it is important to note the averaging in both limiting cases is more complicated than in case of a simple liquid or for a pure magnetic system. In general the mutual influence of one subsystem on the other takes place. Excluding the variables of ''faster'' subsystem from the hydrodynamic equations, the renormalized transport coefficients are found for the subsystem with ''slower'' dynamical processes. Taking into consideration only magnetic relaxation processes in an external field, the generalized Bloch equation has been derived. This result will be discussed in comparison with other ones obtained within microscopical theories, in particular, Rubi and Miguel [5] and Felderhof and Jones [6]. On this basis the spectrum of hydrodynamic collective modes of a Heisenberg ferrofluid is found [7]. We derive the explicit expressions for the dispersion and dampling coefficient of sound modes depending on the value of external magnetic field. It is shown that the sound velocity is isotropic and can be simply identified with the adiabatic compressibility at constant magnetization. The heat and spin modes are purely diffusive. Explicit expressions for the viscosity, thermal conductivity, spin diffusion and thermomagnetic diffusion coefficients containing the corresponding time correlation functions are also derived. These results are compared with previous ones obtained mainly within phenomenological theories. The hydrodynamic time correlation functions could be calculated by solving eigenvalues problem for the generalized hydrodynamic matrix. The most interesting of them are the ''density-density'' and ''spin-spin'' time correlation functions which can be determined by scattering experiments. In the hydrodynamic limit we obtain [8] the analytical expressions for all the time correlation functions constructed on conserved variables. It is shown that for non-zero value of external field the additional contributions appear in both ''density-density'' and ''spin-spin'' time correlation functions due of coupling both subsystems. For example, the ''spin-spin'' time correlation function has an additional term contributed by sound excitations. Besides, all the parameters in these expressions are functions of external field.

Possibilities for further investigations

It will be also discussed the possibility to use the developed approach for the study of more complicated statistical models in which the shape of particles and many-component structure of a ferrofluid can be taken into account. The obtained results can be also used for interpretation of some experimental data.

Acknowledgment.

This study is supported in part by the Fonds for Forderung der wissenschaftlichen Forschung under Project P 12422 TPH.

Bibliography.

1. D. N. Zubarev, Nonequilibrium statistical thermodynamics, (Plenum Press, 1977).

2. I. M. Mryglod, M. V. Tokarchuk, R.Folk, Physica A 220 (1995) 235.

3. J. P. Hansen, I. R. McDonald, Theory of simple liquids, (Academic Press, 1987).

4. F. Schwabl, K. H. Michel, Phys. Rev. B 2 (1970) 189.

5. J. M. Rubi, M. C. Miguel, Physica A 194 (1993) 209.

6. B. U. Felderhof, R. B. Jones, Phys. Rev. E 48 (1993) 1084; 1142.

7. I. M. Mryglod, R. Folk, Physica A 234 (1996) 129.

8. I. Mryglod, R. Folk, S. Dubyk, Yu. Rudavskii, Physica A (in preparation).