Version 1; May 15, 1998

Hydrodynamic collective modes and time-dependent correlation functions of a binary ferromagnetic mixture
I.M.Mrygloda, M.V.Tokarchuka, O.F.Batsevychb, Yu.K.Rudavskiib,

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

Contact author:
Dr.I.Mryglod
Institute for Condensed Matter Physics,
National Academy of Sciences of Ukraine,
1 Sventsitskii St., UA-290011 Lviv, Ukraine
Tel/Fax: +380 322 761978
E-mail: mryglod@icmp.lviv.ua

Abstract:

     Using Zubarev nonequilibrium statistical operator method the equations of generalized hydrodynamics for a binary mixture of magnetic and nonmagnetic fluids are derived. In the hydrodynamic limit we find the spectrum of collective excitations. The explicit expressions for the dispersion and damping coefficient of sound modes are derived. It is shown that the sound velocity of the considered model with Heisenberg-like magnetic interaction at constant external magnetic field h is isotropic and can be expressed by the adiabatic compressibility at constant magnetization m and constant concentrations of species $c_\alpha$. The equations for the heat, diffusion and spin-diffusion modes which are purely diffusive are also obtained. The hydrodynamic time correlation functions could be calculated by a matrix perturbation theory. The obtained results are discussed in comparison with the results known in the literature.

          Keywords: Ferrofluid, binary mixture, 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. 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 behavior of more realistic model of ferrofluids which allows to take into account the mutual influence of nonmagnetic solvent on the properties of magnetic subsystem.

Theory. Using the approach developed in [4,5,7], which is based on the Zubarev's method of nonequilibrium statistical operator, we have derived [8] the generalized hydrodynamic equations for a binary model ferrofluid with Heisenberg-like spin interactions in an inhomogeneous external field. These equations, describing the time-dependent behavior of the conserved quantities (averaged densities of particle numbers ${\hat n_1}({\bf k})$ and ${\hat n_2}({\bf k})$, momentum $\hat{\bf p}({\bf k})$, magnetic momentum $\hat{\bf m}({\bf k})$ and energy ${\hat{\varepsilon}}({\bf k})$), can be used for the study both strong and weak nonequilibrium properties of the system considered. In the hydrodynamic region for homogeneous magnetic field h we have to solve the linearized set of equations for the hydrodynamic time correlation functions F(k,t) defined by

\begin{displaymath} F(k,t)= \int\limits_0^1 d\tau \langle \Delta\hat Y(k) \exp\... ...equiv \left( \hat Y(k), e^{-i\hat Lt} \hat Y^+(k)\right), \end{displaymath}

(1)


which for the Laplace transforms $\tilde{F}(k,z)$ of F(k,t) has the following matrix form

\begin{displaymath} \left\{ z I - i \Omega (k) + \tilde \phi (k,z) \right\} \tilde{F}(k,z) = F(k,0), \end{displaymath}

(2)


where the dynamic variables $\hat Y(k)$*xfor the longitudinal fluctuations are

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


and $\hat s(k)$, $\hat h(k)$ denote the magnetization and enthalpy densities (see, i.e., [5]). The elements of frequency matrix $i \Omega(k)$ could be expressed via generalized k-dependent thermodynamic quantities [7,8]. The elements of matrix $\tilde \phi (k,z)$ are directly related to the generalized transport coefficients. In the hydrodynamic limit one find that

\begin{displaymath} i \Omega (k) \simeq i k \, {\hat \omega}, \ \ \tilde \phi (k,z) \simeq k^2\, \hat \varphi. \end{displaymath}

(3)


     The nonzero elements of matrix ${\hat \omega}$ are expressed in the terms of well-known thermodynamic quantities

$\displaystyle \omega_{n_i p} = {c_i \over \bar{m}}, \ \ \omega_{p n_i} = \frac... ...= \frac {\nu_{\scriptscriptstyle P,T}}{ \rho \kappa_{{\scriptscriptstyle T},h}}$

   

$\displaystyle \omega_{p h} =\frac{V}{C_{n,m,c}} \frac{\alpha_{{\scriptscriptsty... ...ac{\nu_{\scriptscriptstyle P,T}^2} {\kappa_{{\scriptscriptstyle T},h}} \right),$

   

(4)


where $\chi_{{\scriptscriptstyle T},n,c}$ and Cn,m,c are the magnetic susceptibility and specific heat at constant volume, and

\begin{displaymath} v_i = \left( \frac{\partial V}{\partial N_i} \right)_{{\s... ...tial V} {\partial h} \right)_{\scriptscriptstyle P, N_i, T}, \end{displaymath}


\begin{displaymath} \kappa_{{\scriptscriptstyle T},h} = -\frac{1}{V} \left(\fra... ...al V}{\partial T} \right)_{{\scriptscriptstyle P, N_i,} m}, \end{displaymath}


are the partial molar volume per molecule of species `i', the coefficient of magnetostriction, the isothermal compressibility at constant magnetization, and the coefficient of isobaric thermal expansion at constant magnetization, respectively (ci=Ni/N and $\rho= M /V = \bar{m} N/V$). Elements of matrix $\hat\varphi$,

\begin{displaymath} \varphi_{ij} = {V \over\beta} \sum\limits_k L_{ik} ({\Delta}{\hat Y},{\Delta}{\hat Y}^+)^{-1}_{kj}, \end{displaymath}

(5)


are expressed via the kinetic coefficients Lik defined by Green-Kubo like formulas

\begin{displaymath} L_{ij} = {\beta\over V}\int\limits_0^\infty \left( f_i, e^{-{{\rm i}\hat L\,} t} f_j \right) dt, \end{displaymath}

(6)


where fi are nonorthogonal parts of fluxes [5].

Results and discussion. In general case the spectrum of collective modes can be found by solving the equation

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

(7)


     Using (3), (4) and (5) by means of the matrix perturbation theory over the small parameter k we find two complex-conjugated sound modes $z_s^{\pm}$, which are responsible for the propagation of sound waves,

\begin{displaymath} z_s^{\pm} = \pm k v_s + D_s k^2 \end{displaymath}

(8)


where vs,

\begin{displaymath} v_s^2 = \frac{1}{2} \ {\rm Sp} ({\hat\omega}^2) = \frac{1}{... ... \over \partial \rho} \right)_{{\scriptscriptstyle N_i,S,}m}, \end{displaymath}

(9)


is a sound velocity. vs is expressed by an adiabatic compressibility $\kappa_{{\scriptscriptstyle T},m}$ at constant magnetization and constant concentrations of species. The damping coefficient Ds can be written in the form

\begin{displaymath} D^s_{1,2} = {1\over 2 v_s^2} {\rm Sp} ({\hat\varphi}{\cdot}{... ...arphi}{\cdot}{\hat\omega}^2)\over{\rm Sp} ({\hat\omega}^2)}, \end{displaymath}

(10)


     Moreover, we find three purely diffusive modes $z_{\alpha}= D_{\alpha} k^2$, $\alpha =\{ h, d, s \}$, which are the heat `h' mode, the diffusion `d' mode, and the spin diffusion `s' mode, respectively. It is worth to note that the first two from these modes are typical for a binary mixture of simple fluids, and the last one describes the time behavior of a magnetic excitation. The expressions for $D_{\alpha}$ can be found by solving the algebraic equation

\begin{displaymath} D_{\alpha}^3 p_3 + D_{\alpha}^2 p_2 + D_{\alpha}^1 p_1 + p_0 = 0, \end{displaymath}

(11)


where coefficients pi are expressed via the elements of matrices ${\hat \omega}$ and ${\hat\varphi}$.

     The solutions for hydrodynamic time correlation functions could be written [9] via eigenvectors $\Vert \hat X_{\alpha} \Vert$ of the hydrodynamic operator $\hat T_{\scriptscriptstyle H} = -ik \hat \omega + k^2 \hat \varphi$, namely,

\begin{displaymath} F_{ij} (k,t) = \sum \limits_{{\gamma}} G_{ij}^{\gamma} e^{-z_{\gamma} t}, \end{displaymath}

(12)


where $\gamma = \{ s^+, s^-, h, d, s \}$ and $ G_{ij}^{\gamma} = \sum \limits_{l} \hat X_{i{\gamma}} \hat X^{-1}_{\gamma l} F_{lj}(k). $ The weight coefficient $G_{ij}^{\gamma}$ describes the contribution from the hydrodynamic collective mode $z_{\gamma}$ to the function Fij (k,t).

     For a paramagnetic case when h=0 it is found that from the formal point of view the expressions for the hydrodynamic time correlation functions, constructed on the variables of `liquid' subsystem, are very similar to the known results for a binary mixture of simple fluids [10]. The time correlation function of ``spin density-spin density'' in this case can be written as follows

\begin{displaymath} F_{ss}(k,t) = \frac{\chi_{{\scriptscriptstyle T},n,c}}{\beta} \ e^{-k^2 \varphi_{ss}t}, \end{displaymath}

(13)


where $\varphi_{ss}$ is a damping coefficient for the spin density fluctuations. For $h \neq 0$ the additional contributions from the sound and heat modes appear in (13).
Acknowledgments. This study is supported in part by the Fonds für Förderung der wissenschaftlichen Forschung under Project P 12422 TPH.

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