Hydrodynamic collective modes and time-dependent correlation functions of a multicomponent ferromagnetic mixture

O.F.Batsevych, I.M.Mryglod, Yu.K.Rudavskii, M.V.Tokarchuk

     Nowadays magnetic liquids, mixtures of magnetic and nonmagnetic particles in the external fields of mechanical or electromagnetic origin, play significant role in chemical, electronic and other modern technologies. This makes the investigations of the thermodynamical, structural and dynamical properties of liquid magnets very interesting and actual. Most of methods, proposed for the study of magnetic fluids, are based on the phenomenological treatment (see, i.e., [1-3]). Rigorous statistical approach for the consideration of dynamical properties of a Heisenberg model ferrofluid has been developed in [4,5]. The equilibrium properties of this model have been considered in [6].

    The goal of this paper is to investigate the hydrodynamic behaviour of a mixture of magnetic and nonmagnetic particles. Starting from the consideration of a binary magnetic mixture, the scheme, which allows us to find also the excitation spectrum and time correlation functions for more general case of a multicomponent magnetic mixture, is proposed.

     Using the approach based on the Zubarev's method of nonequilibrium statistical operator, the hydrodynamic equations and equations for time correlation functions were derived [8]. For the small deviations from the equilibrium these equations can be written as follows:

               (1)

               (2)

where with denote the deviations of hydrodynamic variables (parameters of abbreviated description) from the equilibrium values , and , are the Laplace transforms of time-dependent and static correlation functions, respectively, is the frequency matrix, is the matrix of memory functions. The parameters of abbreviated description can be chosen as Fourier transforms of microscopic densities of conservative quantities. For the binary mixture one has:

               (3)

where , are the partial densities of nonmagnetic (species 1) and magnetic (species 2) particles, is the density of total momentum, and , are related with the densities of magnetization and energy, respectively. In more general case of a multicomponent mixture, in which some components can posses magnetic momentum, we deal with parameters of abbreviated description, related with the densities of conserved quantities. It is important to note that in the hydrodynamic limit, the matrices and are proportional to and , respectively. Moreover, they have crosslike and anti-crosslike structure, [8].

     The equation for excitation spectrum follows from (2). Solving this equation in general case, we found two sound modes and diffusive modes , describing spin-diffusion, thermodiffusion, etc. [9] with:

               (4)

where and are the sound velocity and damping coefficient. The damping coefficients of all the diffusive modes , , can be found from algebraic equation of -th order [9]. For the case of a binary magnetic mixture , and damping coefficients , , can be expressed via the thermodynamical parameters and transport coefficients.

     Laplace transforms of all time correlation functions can be found from the equation (2). For example, the dynamical structure factors of binary magnetic mixture can be written as follows:

               (5)

where and can be derived from by substitution . Here , . , , are the partial molar volume per molecule, partial concentration and chemical potential of species , respectively. are the coefficients of interdiffusion between particles of -th and -th species, is a longitudinal viscosity, are the coefficients of thermodiffusion. , , are the isothermal compressibility, isobaric thermal expansion coefficients and specific heat in ensemble with constant volume, respectively.

     We note that the matrix equations (1) and (2) are solved in analytical form for a wide class of fluids with parameters of abbreviated description. The collective modes spectrum is derived from the equation (1), and the solutions of equation (2) give the expressions for time correlation functions. It is important to stress that all the input variables of the theory are the thermodynamical parameters and transport coefficients. For the special case of a binary magnetic mixture, equations (4) and (5) give the collective modes spectrum and partial dynamical structure factors.

References

записки