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Hydrodynamic Collective Modes and Time-dependent Correlation Functions of a Multicomponent Ferromagnetic Mixture

O.F.Batsevych^b, I.M.Mryglod^a, Yu.K.Rudavskii^b, M.V.Tokarchuk^a
^aInstitute for Condensed Matter Physics, National Academy of Sciences of Ukraine,
1 Sventsitskii St., UA-290011 Lviv, Ukraine
^bState University ``Lvivska Politekhnika'',12 Bandera St, UA-290013 Lviv, Ukraine

Bibliography

Abstract:

Dynamic properties of multicomponent mixtures of magnetic and nonmagnetic particles were considered. We start from the general case of a multicomponent mixture, some components of which posses magnetic momentum, being described by m+2 parameters of abbreviated description. The equations of generalized hydrodynamics and those for time correlation functions are derived. Then we calculate eigenvalues of the hydrodynamic equations for small wave-number k, which yielded sound velocity and sound damping coefficients for two propagating modes, and damping coefficients for the rest modes which are purely diffusive. The scheme, which allows us to obtain weight coefficients for time correlation functions is proposed. Expressions for time correlation functions are analyzed for paramagnetic case, when the coupling between the spin and fluid degrees of freedom can be neglected.

Keywords: multicomponent mixtures, hydrodynamics, collective mode, time correlation function.

Introduction
Nowadays, due to the very rapid technology development, the theoretical investigation of magnetic fluids and their mixtures are of great interest. Magnetic liquids, mixtures of magnetic and nonmagnetic atoms in the external fields of mechanical or electromagnetic origin, have already taken their significant place in chemical, electronic and other modern technologies. That is why the investigations of the thermodynamical, structural and dynamical properties of liquid magnets are very actual. Considerable number of methods were proposed for studying of thermodynamic, structural and dynamic properties of liquid magnets. Most of them are based on the phenomenological treatment (see, i.e., [1,2,3]). Rigorous statistical approach for the study of dynamical properties of a Heisenberg model ferrofluid has been developed in [4,5]. The equilibrium properties of this model has been considered in [6]. The goal of this paper is to study the hydrodynamic behaviour of the mixtures of magnetic and nonmagnetic particles. Starting from the consideration of the binary magnetic mixture, the scheme, which allows to find the excitation spectrum and time correlation functions for more general case of multicomponent magnetic mixtures was proposed.

Initial Relations
Using the approach developed in [4,5,7], which is based on the Zubarev's method of nonequilibrium statistical operator, the hydrodynamic equations and equations for time correlation functions can be derived [8] and for the small deviations of the system from the equilibrium they can be written as follows:

$\displaystyle \left\{\mbox{\rm i}\omega{\cdot}\tilde 1-\mbox{\rm i}\tilde\Omega(k)+\tilde\Phi(k,\omega)\right\}\langle \Delta \hat Y_i(k)\rangle ^\omega=0.$

(1)

$\displaystyle \left\{z{\cdot}\tilde1-\mbox{\rm i}\tilde\Omega(k)+\tilde\Phi(k,z)\right\}\,\tilde F(k,z)=\tilde F_0(k),<tex2html_comment_mark>25$

(2)

where $\langle \Delta \hat Y_i(k)\rangle ^\omega$ are the deviations of hydrodynamic variables (parameters of abbreviated description) from the equilibrium, $\tilde F(k,z)$ , $\tilde F_0(k)$ are time-dependent and static correlation functions, respectively, $\mbox{\rm i}\tilde\Omega(k)$ is the frequency matrix, $\tilde\Phi(k,z)$ is the matrix of memory functions. The parameters of abbreviated description can be chosen as Fourrier transforms of densities of conservative quantities. For the binary mixture they are following:

$\begin{displaymath} \hat Y(k) = \{ \hat n_1( k), {\hat n_2}(k), \hat p(k), \hat s(k), \hat h(k) \} , \end{displaymath}$

(3)

which are the Fourier transforms of partial densities of particle numbers of nonmagnetic (species 1) and magnetic (species 2) particles, total momentum, projected magnetization and enthalpy, respectively. We will consider more general case of multicomponent mixture, in which some components posses magnetic momentum. In such a case we consider m+2 parameters of abbreviated description, where m+2 gives the number of conserved quantities. As it can be shown, for this system in hydrodynamic limit (wave number $k\rightarrow0$ , frequency $\omega\rightarrow0$ ), the matrices $\mbox{\rm i}\tilde \Omega$ and $\tilde\Phi$ are proportional to $(\mbox{\rm i}k)$ and $(\mbox{\rm i}k)^2$ , respectively:

$\begin{displaymath} \mbox{\rm i}\tilde\Omega=\mbox{\rm i}k\,\tilde\omega,\qquad \tilde\Phi=(\mbox{\rm i}k)^2\,\tilde\varphi \end{displaymath}$

(4)

where matrices $\tilde\omega$ and $\tilde\varphi$ have crosslike and anti-crosslike structure

$\begin{displaymath} \tilde\omega=\left( \begin{array}{ccccccc} &&&\omega_{1,... ...ots&&\tilde\varphi _4&\\ [-1mm] &&&0&&& \end{array}\right). \end{displaymath}$

(5)

In general case the elements of matrix $\tilde\omega$ and $\tilde\varphi$ can be expressed via the k-dependent thermodynamical functions and generalized transport coefficients [8]. In the hydrodynamic limit $k\to0$ the matrix elements of $\tilde\omega$ and $\tilde\varphi$ do not depend on k

Excitation spectrum
The spectrum of collective modes in general case can be found by solving the equation (see (1))

$\begin{displaymath} \det\vert z I - \mbox{\rm i}\tilde\Omega(k)+ \tilde \Phi (k,z)\vert=0. \end{displaymath}$

(6)

From (6) by means of the matrix perturbation theory over the small parameter k we find: a) two complex-conjugated modes $z_s^{\pm}$ , which are responsible for the propagation of sound waves,

$\begin{displaymath} z_s^{\pm} = \pm k v_s + D_* k^2 \end{displaymath}$

(7)

where the sound velocity v_s and damping coefficient D_* are given by:

$\begin{displaymath} v_s^2= {1\over2} \mathop{\mbox{\rm Sp}}\nolimits\,\left(\ti... ...ga^2)\over\mathop{\mbox{\rm Sp}}\nolimits\,(\tilde\omega^2)}, \end{displaymath}$

(8)

b) m purely diffusive modes:

$\begin{displaymath} z_i=-D_i\,k^2,\quad i=1,\dots,m. \end{displaymath}$

(9)

It is shown that the damping coefficients D_i can be found as the solutions of the equation of m-th order:

$\begin{displaymath} \det\left.\left(\tilde\omega+\tilde\varphi -{D}{\cdot}\tild... ...\vert _{ \mbox{{\scriptsize element}}_{\pi,\pi}\equiv0}=0. \end{displaymath}$

(10)

Time Correlation Functions
As it follows from (2), time correlation functions can be given by:

$\begin{displaymath} \tilde F(k,z)=(z-\tilde T_H(k))^{-1}\tilde F_0(k). \end{displaymath}$

(11)

The matrix expression $(z-\tilde T_H(k))^{-1}$ can be written as $\sum\limits _{i=1}^{m+2}{(\tilde g_0^i+\mbox{\rm i}k \tilde g_1^i)/ (z-z_i)}$ , where $\tilde G^i=\tilde g_0^i+\mbox{\rm i}k \tilde g_1^i$ are so-called weight coefficients in the second order with respect to to small parameter k. These coefficients describe the contributions of each mode to time correlation functions. We found explicit expressions for them in the form

$\displaystyle \tilde g_0^\pm=\frac{\tilde\omega(\tilde\omega\pm v_s)}{2v_s^2}, ... ...tilde g_0^\mp)\pm (\tilde P+\frac12\tilde g_0^\mp)\tilde\varphi \tilde g_0^\pm,$

(12)

$\displaystyle \tilde g_0^k=\prod\limits_{i=1(\neq k)}^m\tilde P\frac{D_i-\tilde... ..._0^j+\tilde g_0^j\tilde\varphi \tilde\omega(\tilde\varphi {\cdot}\tilde R_j-1),$

(13)

for sound and dissipative modes, respectively. Here $\tilde P=1- {\tilde\omega^2/v_s^2}$ and $\tilde R_j=\sum\limits _{k=1(\neq j)}^{m}{\tilde g_0^k}/{(D_k-D_j)}$ , $j=1,\dots,m$ .

Discussion
Result for the excitation spectrum and time correlation functions, obtained above, is exact within the hydrodynamic approximation and describe multicomponent mixtures of magnetic and nonmagnetic particles. Applying resulting expressions (7) - (13) to the particular case of binary mixture of magnetic and nonmagnetic particles (3), we find, that the sound velocity is expressed via the adiabatic compressibility (see [5]):

$\begin{displaymath} v_s^2 = -{V^2\over {\bar m}} \left({\partial P\over \partial... ...{NSM} = \left({\partial P\over\partial \rho} \right)_{NSM}, \end{displaymath}$

(14)

where ${\bar m}$ , V, $\rho$ , P are the mass, the volume, the mass density and the pressure, respectively. The spin-spin dynamic structure factor can be written in this case in the well-known form:

$\begin{displaymath} S_{ s, s}(k,\omega)/S_{ s, s}={k^2 D_1 \over \omega^2 + (D_1 k^2)^2}. \end{displaymath}$

(15)

The other time correlation functions do nod include the spin contributions, because the coupling between magnetic and nonmagnetic degrees of freedoms is absent in the paramagnetic case. The partial dynamic structure factors coincide with that for a binary mixture [9]. Comparing with previous works, our consideration allows us to find as well the terms $\tilde g^i_1$ for diffusive modes, which produce non-Lorenzian contributions to time correlation functions.
Acknowledgments. I.M. thanks the Fonds fьr Fцrderung der wissenschaftlichen Forschung for financial support under Project P 12422 TPH.

Bibliography

1: I.A.Akhiezer, I.T.Akhiezer, Sov. Phys. Solid State 29 (1987) 48.
2: J.B.Hubbard, P.J.Stiles, J.Chem. Phys. 84 (1986) 6955.
3: Y.Ido, T.Tanahashi, J. Phys. Soc. Japan 60 (1991) 466.
4: I.M. Mryglod, M.V. Tokarchuk, R. Folk, Physica A 220 (1995) 325.
5: I.M. Mryglod, R. Folk, Physica A 234 (1996) 129.
6: I.A. Vakarchuk, Yu.K. Rudavskii and G.V. Ponedilok, Teor. Matem. Fiz. 58 (1984) 445 [Theor. Math. Phys. 58 (1984) 291].
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9: A.B.Bhatia, D.E.Tornton, N.H.March, Phys. Chem. Phys. 4 (1974) 97.

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записки

		$\displaystyle \left\{\mbox{\rm i}\omega{\cdot}\tilde 1-\mbox{\rm i}\tilde\Omega(k)+\tilde\Phi(k,\omega)\right\}\langle \Delta \hat Y_i(k)\rangle ^\omega=0.$	(1)
		$\displaystyle \left\{z{\cdot}\tilde1-\mbox{\rm i}\tilde\Omega(k)+\tilde\Phi(k,z)\right\}\,\tilde F(k,z)=\tilde F_0(k),<tex2html_comment_mark>25$	(2)

		$\displaystyle \tilde g_0^\pm=\frac{\tilde\omega(\tilde\omega\pm v_s)}{2v_s^2}, ... ...tilde g_0^\mp)\pm (\tilde P+\frac12\tilde g_0^\mp)\tilde\varphi \tilde g_0^\pm,$	(12)
		$\displaystyle \tilde g_0^k=\prod\limits_{i=1(\neq k)}^m\tilde P\frac{D_i-\tilde... ..._0^j+\tilde g_0^j\tilde\varphi \tilde\omega(\tilde\varphi {\cdot}\tilde R_j-1),$	(13)