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\date{Juli 2003}

\author{Dissertation\\ 
\vspace{1.2cm}
  zur Erlangung des Grades\\
 % \vspace*{1cm} 
  %\begin{center}
 ``Doktor der 
 \vspace{1.2cm}
  Naturwissenschaften" (Dr. rer. nat.)\\
  vorgelegt am Fachbereich Physik\\ 
  der Universit\"at  Duisburg-Essen\\ 
\vspace{2cm}
Standort Essen\\
%  \vspace{2cm}
   von  
 % \end{center}
  \vspace{2cm}\\
  \textbf{Elena Jarkova}\\
  %\vspace{0.01cm}\\
  geboren in Ryazan, Russ. F\"oderation\\
   \vspace{3cm}
}


\title{\vspace*{-1.5cm}
\textbf{Hydrodynamics of ferronematics and ferrogels\\}
\vspace{2cm}}\maketitle

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{\large
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\begin{tabular}{ll}
Gutachter: &Prof. Dr. H. Pleiner\\ &Prof. Dr. R. Graham\\[0.5cm]
%& \\
%2.\,Gutachter:\ & Prof.\ Dr.\ K.\ Binder\\
%               & Johannes-Gutenberg-Universit\"at, Mainz\\
%& \\
\noalign{\ Tag der m\"undlichen Pr\"ufung:  10 Juli 2003 }
%Tag der m\"undlichen Pr\"ufung:&  10 Juli 2003 
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\section*{Summary of the thesis}
%\addcontentsline{toc}{chapter}{Summary}
\sectionmark{Summary of the thesis}
\vspace*{1cm}
Some 30 years ago the idea was born to mix nano-sized magnetic particles with nematic liquid crystals, in order to get fluids with a large magnetic susceptibility. Quite recently stable and well defined systems have been produced and it became evident that these emulsions are interesting materials not only because of their enhanced magnetic susceptibility. This thesis studies systematically the new, mainly dynamic, features of such ferronematics. It makes a number of theoretical predictions for qualitatively new effects that can be tested experimentally. The method used is that of phenomenological hydrodynamics and its extension to slowly relaxing, non-hydrodynamic variables. This is a well established method to get a reliable set of (partial differential) equations that describe the long wavelength and low frequency behavior of complex fluids and soft matter. It is phenomenological in the sense that parameters (susceptibilities and transport coefficients) are introduced, whose magnitude is not known a priori. Emphasis is laid, thus, on qualitatively new effects.

First, the possible phase structure of these materials has been investigated. Conventional ferrofluids do not show any phase transition and are "super"-paramagnetic at any temperature, particle concentration, or density and no transition. That means, there is a large (saturation) magnetization even for moderate or low external field strengths. No ferromagnetic state with a spontaneous magnetization has been observed. Ordinary nematic liquid crystals show a transition from an isotropic state to the nematic state (with orientational order) at a certain temperature (thermotropic) or concentration (lyotropic) in the case of solutions. The question was, what kind of phase transitions are possible in ferronematics with respect to both, nematic (orientational) and magnetic (ferromagnetic) order. Within a simple Landau free energy description up to fourth order in the appropriate order parameters it could be shown that there are two different ferronematic phases (apart from an isotropic one at high temperature or low concentration):  one with nematic but no magnetic order (superparamagnetic) and one with both types of order. Within this model a phase with magnetic but no nematic order seems to be impossible. The transition from the isotropic to the ferromagnetic phase can be either directly or in two steps via the (super)paramagnetic nematic phase. Those transitions, where the nematic order is built, are always first order.  In the presence of an external magnetic field the transitions are smeared out and at high fields can become second order. In an external magnetic field the behavior of a superparamagnetic and ferromagnetic phase are rather similar. Up to now no clear experimental evidence for a ferromagnetic phase has been found, but experiments at low or vanishing fields are rather scarce. 

In a second step the macroscopic dynamics of (superparamagnetic) ferronematics has been explored. Generally, one has to consider the dynamics of the nematic degree of freedom (reorientation of the preferred direction in space) as well as the magnetic one (rotations and size changes of the magnetization). Since the latter usually is much faster than the former, one can consider the special case that the magnetization has already relaxed to its equilibrium value and direction (set by the external field). Then a ferronematic dynamics has the same structure as that of a conventional nematic liquid crystal. Only the static and dynamic response to external fields is much stronger. In conventional nematics the influence of a magnetic field on the dynamics had been neglected or not considered at all, previously. Here such dynamic effects are studied systematically. They can be linear in the external field, in contrast to the static (superparamagnetic) effect, which has to be quadratic in the field due to time reversal invariance (of the free energy). In the dynamics, currents can be odd or even under time reversal, so describing reversible or irreversible behavior. Thus, the new linear dynamic field effects always toggle between reversible and irreversible, when compared to their non-magnetic counter parts. Among the effects discussed are reversible ("Hall'-like) contributions to heat, concentration, and thermodiffusion currents as well as viscosity and director relaxation, and an irreversible contribution to flow alignment effect. The latter is manifest in a shear flow experiment, with the magnetic field in the shear plane, where in a ferronematic the nematic director orients with a non-vanishing component out of the shear plane, while in conventional nematics the director orients in the shear plane.
Another testable effect of a new linear dynamic effect is the relaxation of the director into a direction set by an external magnetic field. For conventional nematics this is a simple relaxation process, where the director and the magnetic field lie always in the same plane. In ferronematics an oscillation is superimposed on the relaxation connected with a spatial wobble. 

In the more general case the dynamics of the magnetization has to be taken into account. Since the director can be either parallel or perpendicular to the field (and the magnetization) in equilibrium, there is a uniaxial and a biaxial case. The third possible case that the director, the magnetization and the magnetic field are all pointing in different directions, but lie in the same plane, is disregarded here, since it has not been found in experiment yet. The dynamics is rather rich:  In addition to the coupling of the orientational changes of the magnetization and the nematic director reorientations, there are
reversible and dissipative dynamic cross-couplings of
compressional, shear and elongational flow with rotations and changes 
of the absolute value of the magnetization and director
reorientations. For  measuring some combinations of the parameters that
describe these cross-couplings we studied the sound wave spectrum and 
the rheology of shear flow. In particular the sound spectrum is studied and a field-dependent contribution to damping is found. Additionally sound waves (compressional flow) can trigger shear flow (and vice versa), if the wave vector is oblique to the field direction.  Oscillatory shear (without field, or with a field parallel or perpendicular to the wave vector) shows a considerable influence of the magnetic degree of freedom. 
Even without a magnetic field the
apparent viscosity is different from the bare one and the modified nematic 
director diffusion couples to the flow response. In the presence of an external field
the director relaxation is shifted to a finite frequency, which
approximately increases with the third power of the field strength. 

Finally the dynamics of ferrogels is investigated. Here the magnetic particles are put into a matrix of cross-linked polymers. Ferrogels can be isotropic or uniaxial depending whether the cross-linking process is done with or without a magnetic field, although we only deal with the former one.  An external field leads to a static deformation of the gel (magnetostriction).  The dynamic around this strained equilibrium state shows various peculiar features. The dynamic coupling of the magnetic degree of freedom to the elastic one leads to magnetic field dependent effective (longitudinal and transverse) sound velocities. The dispersion relations show characteristic steps at the frequency of the magnetization relaxation. This frequency and field  dependence of the sound spectra, thus provides the opportunity to measure the relevant static and dynamic magneto-elastic parameters that are crucial for the possible applications of ferrogels.
 
   


\tableofcontents

\chaptermark{Contents}
%\addcontentsline{toc}{intro}{entry} 

\chapter{Introduction}
\section{Ferronematics}
\pagenumbering{arabic}
Ferrofluids are suspensions of nano-sized ferromagnetic particles
in some carrier
liquid (for example, water or oil) \cite{Rosen}.
Without an applied external magnetic field the orientations of the
magnetic moments of the particles are random resulting in a vanishing
macroscopic magnetization
(magnetic disorder). An external magnetic field, however, easily orients the
particles' magnetic moments and a large (induced) magnetization is
obtained. This "superparamagnetic"
property is the basis for many applications.

In 1970 the idea emerged \cite{brochard} to dope nematic liquid
crystals with a small amount of ferromagnetic grains (volume fraction of
$10^{-5}-10^{-4}$) in order to  enhance the
magnetic susceptibility.
The expectation was that a strong coupling between the orientation of the
magnetic grains and the nematic ordering
induces
a spontaneous macroscopic magnetization (ferromagnetic state), which allows
easy control of the director alignment by weak external magnetic fields.

Quite rapidly, mixtures of thermotropic calamitic \cite{rault70,chen87} and
discotic \cite{hayes76}, as well as
lyotropic nematics \cite{liebert79} with ferroparticles were produced. However,
these systems were more like dirty liquid crystals, where the magnetic
additives served for a better orientation of the nematics in an external
field. Problems were the stability of these mixtures and the mutual
orientation of the director and the magnetization.  The experimental
situation changed considerably, when it was possible to make stable emulsions,
first as ferrosmectic systems \cite{dabadie,ponsinet,spoliansky}, were the
ferromagnetic nano-particles are embedded in the layers.  These ferrosmectics 
are very dilute
systems, which prevents their use in applications.
Recently, stable
ferronematic systems, where the liquid crystal and the magnetic
aspects are on equal footing, have drawn increasing attention (apart from
other rather exotic phases, like ferrovesicles \cite{bacri2}).
Birefringence
\cite{bacri,raikher,burylov}
behavior in homogeneous electric \cite{potocova}, and
magnetic fields (including the Frederiks transition
\cite{burylov,koneracka,berejnov,zubarev})
in inhomogeneous fields \cite{matuoa}, and
under the influence of bounding surfaces
\cite{fontanini} have been investigated.


In their original work Brochard and de Gennes started from the
so-called ''rigid anchoring'' approximation, implying that the
directions of the director ${\bf n}$ and the local magnetization
${\bf M}$ are perfectly co-aligned. However, with the
synthesis of thermotropic ferronematics \cite{C1} it became
evident that the rigid-anchoring approximation might not be
generally applicable. Within the framework of a microscopic model
of rod-like ferromagnetic grains Burylov and Raikher
\cite{burylov} reconsidered the surface interaction between the
liquid crystalline nematogens and derived an expression for the
free energy of a ferronematic. Thereby the orientations of ${\bf
n}$ and ${\bf M}$ were treated as separate degrees of freedom. As
in Ref.\cite{brochard} the strength of the magnetization was
assumed to be in saturation, even without external fields.
However, the existence of a remanence in ferronematics, being
characteristic for a ferromagnetic ordering (spontaneous
equilibrium-magnetization in the absence of any external magnetic
field) seems to be experimentally unproven yet. Rather the
existing substances all seem to be superparamagnetic like ordinary
ferrofluids.

\section{Ferrogels}
Ferrogels belong to a new class of  magneto-controlled elastic materials, which are 
chemically  cross-linked polymer networks swollen with a ferrofluid.
Coupling the elastic medium with the magnetic properties of the particles
allows to manipulate 
the elastic behavior of ferrogels by external magnetic
fields. This feature offers opportunities for various applications as e.g.
soft actuators, micromanipulators, and artificial muscles \cite {zr0}. Heating
 of these materials in alternating magnetic fields allows for a
promising approach in cancer therapy \cite{babin}. Since the
 magnetic rubber is soft, inexpensive and controlled by the magnetic
field, it can also be used in an apparatus for immunoblotting \cite{uritani}.

The properties of ferrogels  depend on the preparation conditions
(solvent, concentration of cross-linking, concentration of magnetic
particles). Preparing ferrogels in an external magnetic field one can
obtain large columns of magnetic particles, the size of which is much
larger than the mesh size of network. In this case the clusters are
fixed in the network \cite{mit}. As a result the  ferrogel is strongly anisotropic
\cite{bayreuth}. 

Here we consider isotropic ferrogels. The typical size of the magnetic
particles is $\sim$ 10 nm.  The bare particles tend to
coagulate. To prevent this, magnetic grains are 
charged \cite{berejnov} or coated by polymers \cite{chen}. The magnetic gels are
usually only weakly cross-linked, so that the size
of the magnetic particles is much smaller than 
the mesh size of the network. However, still some coagulation
takes place resulting in magnetic clusters comparable in size to that of the mesh
\cite{zrinyi}. Without external field no remnant magnetization is found. 
An external field easily magnetizes the sample (superparamagnetism). Outside
equilibrium the magnetization relaxes to its equilibrium value and orientation
set by the external field. This relaxation is rather slow compared to the
(many) microscopic relaxation processes.  

In inhomogeneous magnetic fields an abrupt shape transition of isotropic
ferrogels was observed \cite{zrinyi}. The force generated by 
such a magnetic gradient field drives the magnetic grains in the
direction of the gradient, thus deforming the network, if there is a 
coupling between the magnetic particles and the network. In these experiments
this seems to be the case.  

\section{Scope of this thesis}
In chapter \ref{1} a Landau-type free energy function is presented to describe the phase
transitions from an isotropic (superparamagnetic) ferrofluid
to a ferromagnetic nematic liquid crystal either directly or via
a superparamagnetic nematic liquid crystal. These
two nematic phases are usually both called 'ferronematic', although they are
distinct phases. Both show nematic ordering, but only the
ferromagnetic phase shows spontaneous magnetic ordering, additionally. The
interplay of nematic with magnetic order is of special interest. A phase
with ferromagnetic order but no nematic order is not possible in the present
model, since the
former always implies the latter (but not vice versa). In the presence of a
strong
external magnetic field these transitions are smeared out and the different
ferronematic phases become more similar to each other.


In  chapter \ref{2} we derive hydrodynamic equations for nematic ferrofluids
  (ferronematics) in the limit that the magnetic degree of freedom has
  relaxed to its equilibrium value. Hydrodynamics describes a system in the long wavelength limit and for long
time scales. The hydrodynamic equations are derived by
means of symmetry and thermodynamic arguments.
The main advantage of the hydrodynamic method lies
in its generality, which allows its application to very different
systems. However, the occurrence of phenomenological parameters in the
static and dynamic expansions are the prize one has to pay for this
generality. Therefore coefficients turning up in the equations below
have to be determined by microscopic models or by experiments.


We concentrate on novel dynamic
  effects linear in  the magnetic field. We show that flow alignment, heat conduction, diffusion,
  thermodiffusion, viscosity and director reorientation are all modified by
  the presence of an external field. In particular, the new effects describe
  reversible (irreversible) couplings, where the conventional effects are
  irreversible (reversible). We discuss, how these effects can be
  measured. In principle, this description is applicable to
  conventional nematics, too, although huge magnetic fields are expected
  to be necessary for detecting the new effects in this case.

In chapter \ref{3} we include the magnetization as an independent
dynamic degree of freedom, thus  deriving the complete set of macroscopic dynamic
equations for ferrofluids in the nematic liquid crystal phase under an
external magnetic field. 
To derive these  equations, the hydrodynamic method has to be generalized.


This an example, where non-hydrodynamic, relaxing processes become so slow
that their dynamics takes place on a macroscopic time scale as well. Then it
is appropriate also to include non-hydrodynamic, but
slowly relaxing variables in the dynamic description of such a system. 
However, there are no first principles to select those slowly relaxing
variables and heuristic arguments have to be used instead. Such a
phenomenological macroscopic description does not have the solid foundation of
a truly hydrodynamic theory. On the other hand, by comparing the
implications of this extended description with experiments, one can learn for
which systems, and under what conditions, the inclusion of those
non-hydrodynamic variables is required for a reasonable description.

Special emphasis is laid on possible static and dynamic
  cross-couplings between magnetization, director reorientations and flow. As
  examples we discuss the field dependence of the sound spectrum and the
  rheology of shear flow.

 In chapter \ref{4} we construct the macroscopic dynamic equations for 
 a ferrogel using the hydrodynamic method described above. We include the
 deformations of a gel and 
the magnetization as  independent dynamical variables. 
 The magneto-mechanical 
cross-couplings between elasticity and the magnetic degree of
freedom are introduced.  
In particular we discuss static elongation, shear and 
the modified sound spectrum in a homogeneous magnetic field by solving
the appropriate generalized hydrodynamic equations.

%\pagebreak

%
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\chapter[Landau description of ferrofluid to ferronematic phase transition]{Landau description of ferrofluid to ferronematic phase transition\boldmath${}^*$\label{1}\footnotetext{${}^*$This chapter is based on
    ref.\cite{PJMB}}}
\renewcommand{\thefootnote}{\arabic{footnote}}
\chaptermark{Landau description}
\section{Introduction}

To study phase transitions in a ferronematic, the Landau mean field theory is
used. The theory is based on a power series expansion of the free
energy in terms of one or more  oder  parameters. It
assumes, that the order parameter is small and uniform near the
transition point  so that only the lowest
order terms allowed by symmetry are kept. 

The starting point of our approach is to determine the order
parameters. In order
to obtain the isotropic-nematic
phase transition,
we make use of the nematic order parameter
originally proposed by de Gennes \cite{PGdG}, a symmetric,
traceless tensor described by $Q_{ij}=\frac{1}{2} S (3 n_{i}  n_{j}-
{\delta}_{ij})$.
The unit vector $ \bm{ n}$ describes the orientation of the
nematic ordering. Since the nematic order does not discriminate between "up"
and "down" along the preferred direction, $ \bm{n}$ and $- \bm{ n}$
are equivalent and $ \bm{ n}$ is called a director. $Q_{ij}$ is the
simplest tensorial quantity compatible with this up-down symmetry.
The quantity S defines the strength of the nematic
ordering (the modulus of the nematic order parameter)
and is zero (one) for complete disorder (order). Thus
in the isotropic phase $ S = 0$ and in the nematic phase $S \neq 0$. 

The magnetic order is described by the magnetization $\bm{M} = M \bm{  m}$
whose modulus $M$ is zero in a (super)paramagnetic and non-zero in
a ferromagnetic state. The unit vector \bm{m} denotes the orientation of the magnetic
ordering. It is an axial vector, which is odd under time reversal symmetry
(like a magnetic field).

We are now in the position to write down the Landau free energy
$\varepsilon$ for a ferronematic, which we study first for the  case
without, and second with a magnetic field. 

\section{Phase transitions without field }
%\sectionmark{Phase transitions}
We first discuss phase transitions in the absence of an external
field, using the nematic and the magnetization
order parameters defined above. 
Keeping terms up to quartic order
the total free energy density near the isotropic/nematic and para-/ferromagnetic
transition can be written as:
%
\begin{eqnarray}
\label{free_1} \!\!\!\! \!\!\!\!\varepsilon\!\!&=&\!\!\!\! \varepsilon_{0}+\frac {A}{2}  Q_{ij}Q_{ij}
-\frac {B}{3} Q_{ij}Q_{jk}Q_{ki}
 +\frac {C_{1}}{4} (Q_{ij}
Q_{ij})^{2}  +
\frac {C_{2}}{4} Q_{ij}Q_{jk}Q_{kl}Q_{li}
\nonumber \\
&+&\!\!\!\!  \frac { \alpha}{2}
\bm{M}^{2}
+\frac {\beta}{4} \bm{M}^{4} +\frac {\gamma}{2}  M_{i} M_{j} Q_{ij} + \frac
{\delta_{1}}{2}
\bm{M}^{2}Q_{ij}Q_{ij} + \frac {\delta_{2}}{2} M_{i} M_{k} Q_{ij} Q_{kj}
\end{eqnarray}
%
where $\varepsilon_0$ is the free energy density of the isotropic (superparamagnetic)
ferrofluid phase. The change of sign of
$A\equiv A_{0}(T-T_{NI}^{*})$ and $\alpha \equiv {\alpha}_{0}(T-
T_{FP}^{*})$ at the critical
temperatures $T_{NI}^{*}$ and $T_{FP}^{*}$ corresponds
 to a  hypothetical second order isotropic to nematic,
and to the para- to ferromagnetic transition, respectively, if the cross
coupling terms ($\sim \gamma$ and $\sim \delta_{1,2}$) are absent. All other
coefficients, as well as $A_{0}$ and ${\alpha}_{0}$,
are assumed to be
constant near the transition point. For
pressure - or (more likely) -
concentration-driven phase transitions one could write $A\equiv
A_{c}(c-c_{NI}^{*})$ and $\alpha \equiv {\alpha}_{c}(c- c_{FP}^{*})$
with a change in sign at some critical concentrations of ferroparticles. For
simplicity we will stick to the temperature related notation in the following.

In addition to the bi-quadratic couplings
between the
nematic and magnetic order governed by the coupling coefficients
$\delta_{1,2}$, there is also a cubic coupling ($\sim \gamma$), which is most
important for these transitions. As will be discussed below, it is responsible
for the fact that magnetic order induces the nematic order.
As we shall see, negative (positive) values of $\delta_{1,2}$ favor (disfavor)
ferromagnetic nematics
over the (super)paramagnetic ferronematics.

We assume $C_1$, $C_2$, $\beta$ and $\beta C_{1,2}-\delta_{1,2}^{2}$
 to be positive to guarantee
the stability of the isotropic phase at high temperatures
and $B > 0$ to get $S> 0 $ in the ferronematic phases, which is suitable for
rod-like nematogens.


Here we consider phases in which the nematic and magnetic order
are spatially invariant, $S=const.$ and $M = const.$, and the ordering
directions $\bm{n}$ and $\bm{ m}$ make an angle $\psi$,
i.e. $\bm{ n}\cdot \bm{ m} = \cos \psi$. In that
case eq.(\ref{free_1}) reads
%
\begin{eqnarray} \label{free2}
\varepsilon-\varepsilon_{0}\! &=&\!\frac{3}{4} AS^{2}-\frac{1}{4}BS^{3}+
\frac{9}{16}CS^{4}
+\frac{1}{2} \alpha {M}^{2}+\frac {1}{4} \beta {M}^{4}
+ \frac{3}{4} \delta_{1} {M}^{2}S^{2}
\nonumber \\ &&
+\,\, \frac{1}{8} \delta_{2}
M^{2} S^{2}(3 \cos^{2}\psi +1) + \frac{1}{4} \gamma M^{2} S \,(3\cos^{2}\psi
- 1)
\end{eqnarray}
%
where $C\equiv C_{1}+ C_{2}/2$.
The presence of the cubic terms ($\sim B$ and $\sim \gamma$) describes
the first order character of the isotropic ferrofluid to ferronematic
transitions. Eq.(\ref{free2}) has a structure quite similar to that of a
Landau free energy for the isotropic to smectic A phase transition in
conventional thermotropic liquid crystals \cite{Muk01}.

Minimization of eq.(\ref{free2}) with respect to
$S$, $M$ and $\psi$ yields the following three phases:
%
\begin{eqnarray}
{\rm Isotropic \,\,\,Ferrofluid~ (I):} &&\!S =0, \quad M=0, \quad \psi={\rm undefined}
\label{iso} \\
{\rm Paramagnetic \,\,\, Nematic~ (N):} &&\!S_{N}=\frac{1}{6C} \left[B+ (B^{2}-
  24AC)^{1/2}\right]>0,
\nonumber \\ && M=0, \quad \psi={\rm undefined} \label{nema} \\
\label{smA}
{\rm Ferromagnetic \,\,\, Nematic~ (F):} &&\!S_{F} >0, \quad M^{2}= -\frac
{1}{\beta}\Bigl(\alpha + \tilde \gamma S_{F} + \tilde \delta S_{F}^{2}
\Bigr), \nonumber
\\  &&{} \psi= 0 \,\,\,{\rm or}\,\,\, \pi/2 \quad
\end{eqnarray}
%
where $S_{F}$ is defined by
%
\begin{equation}
\label{SF}
-\frac{2}{3} \frac{\alpha \tilde \gamma} {\beta} + 2A^{*} S_{F} - B^{*}
S_{F}^{2} + 3 C^{*}
S_{F}^{3} =0
\end{equation}
%
with the abbreviations $A^{*}
= A-(2/3\beta) \tilde \delta \alpha - (1/3\beta){\tilde \gamma}^{2}$, $B^{*} = B 
+(2/\beta) \tilde \delta \tilde \gamma$ and $C^{*} = C-
(4/9\beta) {\tilde \delta}^{2}$. The new coefficients $\tilde \delta$ and
$\tilde \gamma$ depend on the angle $\psi$ and are
%
\begin{equation}
  \label{tilde}
\tilde \delta=
\begin{cases}
{\frac{3}{2}} \delta_{1}+\delta_{2}\\
\frac{3}{2} \delta_{1}+  \frac{1}{4}\delta_{2}
\end{cases}
\quad \quad\quad \tilde \gamma =
\begin{cases}
\gamma
\quad \quad &{\rm for}\quad \quad
\psi=0\\
-\frac{1}{2}\gamma
\quad \quad &{\rm for} \quad \quad \psi=\pi/2
\end{cases}
\end{equation}
%
There is no extremum of (\ref{free_1}) with $S=0$ and $M \neq 0$, i.e. no
ferromagnetic phase without nematic ordering. This can easily be seen directly
from the coupling term $\sim \gamma$ in (\ref{free_1}), where a finite $M^{2}$
acts as an external "field" on the nematic order, thus inducing a non-zero $S$.

Necessary conditions for the different phases to be stable are
%
\begin{eqnarray}
 \label{cond2a}
\frac{{{\partial }^{2}}\varepsilon}{{\partial }S^{2}}>0, \quad\quad
\frac{{{\partial }^{2}}\varepsilon}{{\partial }{M}^{2}}\!\!\!\!&>&\!\!\!\!0, \quad\quad 
\frac{{{\partial }^{2}}\varepsilon}{{\partial } \psi^{2}}>0, \quad\quad
\frac{{{\partial }^{2}}\varepsilon}{{\partial }S^{2}}
\frac{{{\partial }^{2}}\varepsilon}{{\partial }{M}^{2}}-\left(\frac{{{\partial
      }^{2}}\varepsilon}{{\partial }S \,{\partial}{M}}\right)^{2}>0 \quad\quad
\end{eqnarray}
%
The derivatives in (\ref{cond2a}) have to be taken at the
values (\ref{iso}-\ref{smA}) for the appropriate phases.
For the isotropic ferrofluid phase the stability conditions are simply
$\alpha > 0$ and
$A > 0$.
The paramagnetic nematic phase is stable, if
$\alpha +\tilde \gamma  S + \tilde \delta S^{2}>0$
and $24 AC < B^{2}$.

The possible orientation between nematic and magnetic ordering in the
ferromagnetic phase, either parallel
($\psi =0$) or orthogonal ($\psi = \pi/2$), is fixed by the third stability
condition in (\ref{cond2a}), where the former (latter) orientation is obtained for
$\delta_{2} S_{F} + 2 \gamma <0$ ($>0$). Thus, a negative (positive) $\gamma$
favors parallel (orthogonal) orientation.
In the parallel case the phase has uniaxial D$_{\infty h}$ symmetry (as
conventional nematics), while in the orthogonal case the phase is biaxial with
orthorhombic D$_{2h}$ symmetry.
For $\psi$ a value different from
zero or ninety degrees could only be obtained by taking into account terms
higher than forth order in the Landau expansion, a procedure, for which
there is no a
priori reason.
The other stability conditions for the ferromagnetic case read
%
\begin{eqnarray}
 \label{stab1}
A-BS_{F} +\frac{9}{2}CS_{F}^{2}+\frac{2}{3}
\tilde \delta M^{2} &>&0 \\ \label{stab2}
\alpha + \tilde \gamma S_{F}+ \tilde\delta S_{F}^{2}  &<& 0\\
 \label{stab3}
A^{*} -B^{*} S_{F} + \frac{9}{2}C^{*} S_{F}^{2}  &>&0
\end{eqnarray}
%
ensuring $S_{F}$ and  $M^{2}$ to be positive quantities.
Obviously, a negative $\tilde \delta$ enhances the stability of the
ferromagnetic phase.
These stability conditions determine the
existence ranges (in terms of temperature) of the different phases rather
implicitly.
The existence ranges of all three phases generally overlap. That phase
with
the lowest free energy is the stable one. A (first order) transition takes
place, when 2 free energies are identical.
The isotropic ferrofluid to paramagnetic nematic transition, thus, takes
place, when the right hand side of (\ref{free2}) is zero taking for $M$
and $S$ the values (\ref{nema}). This happens for $A=B^{2}/27C$ leading to a
transition temperature $T_{NI}= T_{NI}^{*} + B^{2}/27 C A_{0}$ somewhat larger
than the critical transition temperature of the hypothetical second order
transition.

The transition from the paramagnetic nematic phase (with finite
$S_{N}$) to the ferromagnetic one is then described by (\ref{free2}), which
takes the form
%
\begin{equation}
\label{PF}
\varepsilon - \varepsilon_{N} = \frac{1}{2} \tilde \alpha {M}^{2}+\frac {1}{4} \beta {M}^{4}
\end{equation}
%
where $\varepsilon_{N}$ is the free energy of the paramagnetic nematic and $\tilde
\alpha= \alpha +\tilde \gamma S_{N} +\tilde \delta S_{N}^{2}$. Obviously that
transition is of second order and takes place at the temperature $T=T^{*}_{FP}
- (1/\alpha_{0})(\tilde \gamma S_{N}+ \tilde \delta S_{N}^{2})$, which is
shifted from the critical para- to ferromagnetic transition temperature due to
the finite nematic order. This shift can be negative as well as positive,
i.e.\ the existing nematic order can favor or disfavor an additional
ferromagnetic ordering.
Again, a negative $\tilde \delta$ favors the ferromagnetic ordering.

Of course, there is the competing possibility of a direct transition from the
isotropic ferrofluid to the ferromagnetic nematic phase.
Substituting
 the solution
(\ref{smA}) for the magnetic order parameter
$M \neq 0$ and for $\psi$ into the free energy
(\ref{free2}) we get
the free energy density
for the ferromagnetic phase as a function of $S_{F}$ alone, which
can be written as
%
\begin{equation}
 \label{free*}
\varepsilon-\varepsilon_{0}=
- \frac {\alpha^{2}} {4\beta}
- \frac{\alpha \tilde \gamma}{2\beta} S_{F} +
\frac {3}{4}A^{*}S^{2}_{F}
-\frac{1}{4}B^{*}S^{3}_{F}+\frac {9}{16}C^{*}S^{4}_{F}
\end{equation}
%
where the starred coefficients are defined after (\ref{SF}). Of the
3 possible extrema ($\frac {\partial \varepsilon}{\partial S}=0$ leading to a real
solution for $S_{F}$)
only those are relevant that exist
within the stability range of the ferromagnetic phase.
Although the solutions are involved, one important qualitative
feature can be extracted immediately. It is obvious that a
continuous IF transition ($S_{F}= 0 = M$ at the transition temperature)
is not possible,
except for the
very special case $B=0=\gamma$ (involving $B^{*}=0=\tilde \gamma$) and
$T_{NI}^{*} = T_{FP}^{*}$.
In the general case
there is a jump in $S_{F}$ and $M$ at some temperature $T_{FI}$.
At that temperature the right hand side of
(\ref{free*}) is zero for $S_{F}$
given by eq.(\ref{SF}). Of course, $T_{FI}$ has to be larger than $T_{NI}$ for
the direct transition to happen. Such a case is shown in Fig.2.1  by
choosing an appropriate set of parameters and numerically solving (\ref{SF}).
Fig.2.1 shows that both
order parameters, $S_{F}$ and $M$ jump simultaneously at
the FI transition point. We also see that orientational order
in the ferromagnetic phase is much higher than in the (superparamagnetic)
nematic phase.
%



\begin{figure}\label{p1}
\centerline{\epsfxsize=6.8cm \epsffile{EPL1.eps}}
\caption{$S$ and $M^{2}$ versus temperature. The upper (lower) solid line
represents the orientational order parameter $S_{F}$ ($S_{N}$),
while the dashed
line represents the
(dimensionless)
magnetic order parameter $M^{2}$ in
the ferromagnetic phase.
The values of the parameters in eq.(\ref{free2}) were taken to be
$\alpha_0 = .1 K^{-1}$, $ A_{0} = .1 K^{-1}$,
$T_{NI}^{*}= 301.7 K$, $T_{FP}^{*}= 302 K$,
$B = .8 $, $C = .53 $,
$\beta = 8 $,
$\tilde \delta = -.89 $ (for the FI transition),
$\tilde \delta = .16 $ (for the NI transition), and
$\tilde \gamma=-.05 $ using a dimensionless free energy density $f$.
}
\end{figure}

It is interesting to note that this phase transition is isomorphic to the
direct isotropic to smectic A transition for ordinary liquid crystals, where
the role of the magnetic order is played by the smectic order \cite{Muk01}.


\sectionmark{Phase transitions  }
\section{Phase transitions in an external field \label{withfield}}
%\sectionmark{Phase transitions  }

An external field induces a finite magnetization in the ferrofluid by
orienting the magnetic particles. In addition it also orients the mesogens due
to the diamagnetic anisotropy effect. Thus, any phase has a finite $S$ and
a finite $M$ due to the external field. This is also obvious from the Landau
free energy including the external field $\bm{H}$
%
\begin{eqnarray}
\label{free3}\varepsilon_{H}= \varepsilon - \bm{M\cdot H}- \frac
{\chi_{S}}{2}  H_{i} H_{j} Q_{ij} + \frac{\delta_{3}}{2}
\bm{H}^{2}Q_{ij}Q_{ij} + \frac {\delta_{4}}{2} H_{i} H_{k} Q_{ij} Q_{kj}
\end{eqnarray}
%
with $\varepsilon$ the field-free energy (\ref{free_1}).
The angle between the magnetization and the field is dominated by the
ferromagnetic coupling $-\bm{M\cdot H}$ with $\bm{M}$ parallel to $\bm{H}$
being the ground state. Thus, the angle $\psi$ between $\bm{M}$ and $\bm{ n}$
is also the angle between $\bm{H}$ and $\bm{n}$. As we show later 
there are three possibilities for  the orientation of $\bm{ n}$
and  $\bm{M}$ with respect to the magnetic field. 

The diamagnetic coupling
($\sim \chi_{S}$) and the higher coupling term ($\sim \delta_{4}$) have the
same $\psi$-dependence as the $\gamma$- and $\delta_{2}$ terms in
(\ref{free2}) leading again to the free energy extrema at $\psi = 0\,\,\,
\textrm{or} \,\,\,\pi/2$. The former (latter) is the stable minimum for
$[(\delta_{2} S + 2\gamma)M^{2} + (\delta_{4} S - 2\chi_{S}) H^{2}]S <0$ ($>0$).
The latter case seems to be frequent in
discotic lyotropic \cite{fontanini} and rod-like thermotropic
nematic systems \cite{C1}.


\begin{figure}\label{p2}
\centerline{\epsfxsize=6.1cm \epsffile{Grns.eps} \epsfxsize=6.55cm
\epsffile{Grnm.eps}}
\caption{
$S(T)$ and $M^{2}(T)$ according to
eqs.\ (\ref{cubic1},\ref{cubic2}) for two different values of the
  external magnetic field with $\bm{ n}$ and \bm{M} parallel. The transition to
  the ferromagnetic state ($\tilde
\delta=-.89 $) is discontinuous for (l) lower fields ($H= .28 $) and becomes
smooth for (h) higher fields ($H= .41 $). We have chosen $\chi_{S}=.05$ and
$\delta_{3} + (2/3)\delta_{4}=-.89$; all other parameters are as in
  Fig.2.1.
}
\end{figure}


Minimizing then (\ref{free3}) with respect to $S$
and $M$ leads to the coupled equations for $S(H)$ and $M(H)$
%
\begin{eqnarray}
  \label{cubic1}
0&=& - \tilde \chi_{S} H^{2} + 3 \tilde A S - \frac{3}{2} B S^{2} +
\frac{9}{2} C S^{3} + 2 \tilde \delta
  M^{2} S + \tilde \gamma M^{2}   \\ \label{cubic2}
0&=& -H +\alpha M + \beta M^{3} +\tilde \delta M S^{2} + \tilde \gamma M S
\end{eqnarray}
%
with $\tilde A= A +H^{2}(\delta_{3} +(2/3) \delta_{4})$ and $\tilde \chi_{S}=
\chi_{S}$ for $\psi=0$, and $\tilde A= A +H^{2}(\delta_{3} +(1/6) \delta_{4})$
and $\tilde \chi_{S}=(-1/2)
\chi_{S}$ for $\psi=\pi/2$.
Also there is a third possibility for the orientation of $\bm{ n}$ and
$\bm{M}$ with respect to the external field $\bm{H}$: None is parallel to
another one, but all three lie in a common plane separated by the angles
$\psi$ and $\phi$
(between $\bm{M}$ and $\bm{H}$).
These angles are determined by the
relations $HM\sin\phi = (3/8) SM^{2}(\delta_{2} S + 2 \gamma) \sin 2\psi =
(3/8) S H^{2} (\delta_{4} S - 2 \chi_{S}) \sin 2(\phi - \psi)$. Again, $S(H)$
and $M(H)$ follow from equations of the form (\ref{cubic1},\ref{cubic2}),
where the coefficients, however, are very complicated. Since such a case has
not been found in experiment, we will not discuss it further.

Obviously there is no solution of (\ref{cubic1},~\ref{cubic2}) with $S=0$
and/or $M=0$ for finite magnetic field.
Instead, even the high temperature phase shows a small but finite nematic as
well as magnetic order both induced by the external field.
For the 
transition to the ferromagnetic
nematic phase, where $M$ acquires a spontaneous contribution in addition to the
field-induced one, both jumps (in $M$ and $S$) are
reduced (and shifted to higher temperatures) and eventually, above some
strong external fields, are
replaced by smooth passages from low to high values (Fig.2.2). This field
dependence of $S$ resembles
the isotropic-nematic transition in the presence of an external
{\em electrical} field \cite{lelidis}. In conventional nematics this behavior under
an external magnetic field has never been observed, since the
strong fields necessary
are outside the experimental reach, while in ferronematics the expectation is
that the necessary magnetic fields are much smaller and thus this effect is  
detectable.

The transition to the superparamagnetic nematic state (for positive
coup\-ling coefficients
$\delta_{1,2,3,4}$) is again
characterized by a jump from small $S$ to a larger value of $S$, where the jump
decreases with increasing field and eventually vanishes
leading to a smooth passage from a low $S$ to a high $S$ state. For large
coupling coefficients the
transition temperature is shifted to lower values and at the transition the
field-induced value of $M$ is reduced due to the coupling to $S$, either
discontinuously (low fields) or smoothly (large fields). For smaller coupling
coefficients the transition temperature is almost field-independent
and $M$ is  hardly affected by the change of $S$.


\section{Summary}

Here we discussed  possible phase transitions as a function of temperature (or
pressure or ferrofluid particle concentration)
between an isotropic ferrofluid
phase, the superparamagnetic  and the ferromagnetic  nematic
phase. Using a
Landau-de Gennes expansion of a nematic and a magnetic order parameter up to
fourth order, we found that depending on the signs and magnitude of the
cross-couplings between nematic and magnetic order there is
the possibility to have either a direct
FI transition or a two-step NI and FN transition.
Similarities to the isotropic-nematic-smectic A transitions in
ordinary liquid crystals are discussed. In the
ferromagnetic phase  the magnetic and the nematic orientations are either
parallel or orthogonal. In  an external magnetic field in
addition there is a third possibility: the magnetization, the director and 
the magnetic field orientations are different, but lie in a common plane. 
In the present model there is no possibility to have
a ferromagnetic phase that lacks
nematic ordering. 
In
the last section we studied the phase transitions in an external magnetic 
field. We showed how the
jumps of the order parameters are reduced by the presence of an external
field and eventually are completely smeared out,
if the field is strong enough. Then, the superparamagnetic and the ferromagnetic
nematic phases are identical in their symmetry properties and differ only
quantitatively in the value of $M$. Thus, a clear distinction is best made at
vanishing external field. 

%
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\chapter[Hydrodynamics of ferronematics]{Hydrodynamics of
  ferronematics\boldmath${}^\dagger$\label{2}\footnotetext{${}^\dagger$This chapter is based on
    ref.\cite{jpmfb}}}
\renewcommand{\thefootnote}{\arabic{footnote}}
%\chaptermark{Ferronematics}
\section{Introduction}
Owing to the admixture of magnetic particles
to a nematic
both, the nematic and the magnetic degree of freedom generally  have to be taken into
account. However, we will restrict
ourselves here to situations where the magnetization
has relaxed to its equilibrium value parallel to the applied magnetic
field (the case, when the magnetization is an independent variable is
discussed in the chapter \ref{3}). 
It is then a function of all state
variables (and the external field), but has no independent dynamics,
i.e. magnetic relaxation effects 
are disregarded here. This is
a good approximation for times shorter than typical magnetic relaxation times
($10^{-4}$ to $10^{-6}s$). Since the nematic director dynamics is much slower, it is
appropriate to keep explicitly the latter
(this has nothing to do with the 
''rigid anchoring'' approximation mentioned above, 
since $\bm{n}$ can have any orientation relative to $\bm{M}$). 
This description is, thus, in
principle also valid for ordinary nematics, where no magnetic degree of freedom is
present. However, the new effects described here probably require huge
magnetic fields to be observable in ordinary nematics, while in ferronematics
their detection should be much easier.
 In
ordinary nematics the diamagnetic interaction energy adding up to
the director's molecular field is usually considered to be the
only relevant magnetic contribution. However, in ferronematics
other effects disregarded so far mostly (except in \cite{Lub}) might become
significant. In the
following we will concentrate on those effects, which are {\em
linear} in the magnetic field strength $H$. As this quantity is of
negative parity under time reversal such effects can arise in the dynamics
only and a variety of new Onsager
couplings will appear. These new couplings
change
the character of the contributions in the field free case
from reactive to dissipative or vice versa. The
implications of these new couplings will be illustrated by means
of a series of different examples.

\section{Nematodynamics without magnetic field \label{nofield}}
%\sectionmark{Nematodynamics}
Nematic liquid crystals are characterized by an extended set of hydrodynamic
variables that comprise those of a simple liquid (density $\rho$, momentum
density $\rho \bm{v}$ related to the velocity $\bm{v}$, and entropy density
$\sigma$, or equivalently
energy density $\varepsilon$) and in addition the director
$\bm{n}$, the symmetry variable denoting orientational changes ($\bm{n}^{2}=1$)
of the preferred direction and, in the case of mixtures, the concentration
$c$. In our case the latter is the concentration of the ferromagnetic
particles. The hydrodynamic equations are \cite{MPP,PGdG,PB1}
%
\begin{eqnarray}\label{drho1}
\frac{\partial}{\partial t}\rho
+   div\,\rho \bm{v}&=&0
\\ \label{dv1}
\frac{\partial}{\partial t}g_{i} +\nabla_{j}\Bigl( v_{j}g_{i}+ \delta_{ij}p +
\Phi_{k j}\nabla_i n_k
+ \sigma_{ij}\Bigr)&=&0
\\ \label{dn1}
{\Bigl(\frac{\partial}{\partial
      t}+v_{j}\nabla_{j}\Bigr)}n_{i}+(\bm{n \times \omega})_{i}+Y_{i} &=&0
\\ \label{dc1}
\rho{\Bigl(\frac{\partial}{\partial t}+v_{j}
\nabla_{j}\Bigr)}c + div\,\bm{j}^{c}&=&0
\\ \label{dsigma1}
\frac{\partial}{\partial t}\sigma + div\, \sigma \bm{v}
+ div\bm{j}^{\sigma}&=& \frac{R}{T}
\end{eqnarray}
%
Generally the densities of the currents for heat $\bm{j}^{\sigma}$, concentration
$\bm{j}^{c}$, and momentum, the stress tensor $\sigma_{ij}$
and the quasi-current $Y_{i}$ are the sum of two parts, a reversible and an
irreversible one, where the former (latter) has the same (opposite) time
reversal symmetry as the rest of the equation and leads to zero (positive)
entropy production, i.e. it is reactive (dissipative). Within irreversible
thermodynamics the dissipative parts
can be derived from the
dissipation function $R$ (the source term in
(\ref{dsigma1})) as a potential, while the reversible ones do not follow from
any potential \cite{forster}. The currents
read
%
\begin{eqnarray}\label{csigma1}
\sigma_{ij}&=&-\frac{1}{2}\lambda_{kji}h_{k}-
\nu^D_{ijkl}\nabla_{l}v_{k}
\\ \label{cY1}
Y_{i}&=&-\frac{1}{2}\lambda_{ijk}\nabla_{j}v_{k}+
\frac{1}{\gamma_{0}}\delta_{ij}^{\perp}h^n_{j}
\\ \label{cheat1}
j_{i}^{\sigma}&=&- \kappa_{ij}\nabla_{j}T -D^T_{ij} \nabla_{j} \mu_{c}
\\ \label{cdiff1}
j_{i}^{c}&=& -D_{ij} \nabla_{j} \mu_{c} - D^T_{ij}\nabla_{j}T
\end{eqnarray}
%
with the transverse Kronecker symbol
$\delta_{ij}^{\perp} \equiv \delta_{ij} -n_{i}n_{j}$.
The conjugate quantities temperature $T=T\,(\rho,\sigma,c)$, pressure
$p=p\,(\rho,\sigma,c)$, relative chemical potential
$\mu_{c}\,(\rho,\sigma,c)$, and the 'molecular
fields' $h^n_{i}\,(\rho,\sigma,c)$ and $\Phi_{ij}\,(\rho,\sigma,c)$
follow
from the free energy density $\varepsilon$ (see below (\ref{free})).
This part of the equations, which constitutes the statics of the system, is
derived completely independently
from the dynamics, eqs.(\ref{csigma1}-\ref{cdiff1}).

The heat conduction tensor
$\kappa_{ij}$, the diffusion tensor $D_{ij}$
as well as thermodiffusion tensor $D^T_{ij}$
(related to the Soret/Dufour
effects) are symmetric and have the following form
containing together six coefficients (thermal conductivity, diffusivity and
Soret/Dufour),
%
\begin{equation}\label{heatcond}
\kappa_{ij}=\kappa_{\perp}\delta_{ij}^{\perp}+ \kappa_{\para}n_{i}n_{j}
\end{equation}
%
The tensor $\lambda_{ijk}$, describing  flow alignment in (\ref{cY1}) and back
flow in (\ref{csigma1}) contains one phenomenological
parameter
%
\begin{equation}\label{flowalign}
\lambda_{ijk}=(\lambda-1)\delta_{ij}^{\perp}n_{k}+
(\lambda+1)\delta_{ik}^{\perp}n_{j}
\end{equation}
%
which is reversible, since $h^n_{i}$, $Y_{i}$ and $\sigma_{ij}$ all do not
change sign under time reversal.

The fourth rank viscosity tensor contains five viscosities
%
\begin{eqnarray}\label{viscos}
\nu^D_{ijkl}&=&\nu_{2}(\delta_{jl}\delta_{ik}
+  \delta_{il}\delta_{jk})
\nonumber \\
&&
+2(\nu_{1}+\nu_{2}-2\nu_{3})n_{i}n_{j}n_{k}n_{l}  \nonumber
\\
&&
+(\nu_{3}-\nu_{2})(n_{j}n_{l}\delta_{ik}+n_{j}n_{k}\delta_{il}+
  n_{i}n_{k}\delta_{jl}+n_{i}n_{l}\delta_{jk})  \nonumber     \\
&&
+(\nu_{4}-\nu_{2})\delta_{ij}\delta_{kl} \nonumber \\
&& +(\nu_{5}-\nu_{4}
+ \nu_{2})(\delta_{ij}n_{k}n_{l}+\delta_{kl}n_{i}n_{j})
\end{eqnarray}
%
\section{Static relations}

The statics of a macroscopic system is governed by its free energy. For
nematics the free energy density in harmonic approximation
reads \cite{MPP,PGdG,PB1}
%
\begin{eqnarray}
  \label{free}\nonumber
\varepsilon &=&
\frac{T}{2 C_V} \,(\delta \sigma)^2 + \frac{1}{2 \rho^2 \kappa_s}\,
(\delta \rho)^2 + \frac{\gamma}{2} \,(\delta c)^2 + \frac{1}{\rho \alpha_s}\,
(\delta \sigma)(\delta \rho)
\\ &&+\, \beta_\sigma  (\delta c)(\delta \sigma)
+ \beta_\rho (\delta c)(\delta \rho)
+ \frac{1}{2} \,K_{ijkl} (\nabla_j n_i)(\nabla_l n_k)
\end{eqnarray}
%
where the Frank tensor
%
\begin{eqnarray}
  \label{Ktensor}
  K_{ijkl} = K_1 \delta_{ij}^\perp \delta_{kl}^\perp
+ K_2 n_p \epsilon_{pij} n_q \epsilon_{qkl} +
K_3 n_j n_l \delta_{ik}^\perp \,\,\,\,\,
\end{eqnarray}
%
describes the energy cost for distorting the homogeneous alignment of the
director. The conventional static susceptibilities contained in (\ref{free}) are
the specific heat (at constant density) $C_V$, the isentropic compressibility
$\kappa_s$, the adiabatic volume expansion coefficient $\alpha_s$ and the
appropriate susceptibilities $\gamma$, $\beta_\sigma$ and $\beta_\rho$
related to the concentration instead of the total mass density.

Due to the Gibbs relation (the local manifestation of the first
and second law of thermodynamics)
%
\begin{eqnarray}
  \label{Gibbs}
 d\varepsilon = \mu \,d\rho + T \,d\sigma + \mu_c \,dc +v_i\,d g_i +\Phi_{ij} \,d\nabla_j n_i +
 h_i^{n \prime} \,dn_i   \,\,\,\,
\end{eqnarray}
%
the conjugate quantities follow from the free energy density by partial
differentiation
%
\begin{eqnarray}
  \label{Tconj}
  \delta T &\equiv& \frac{\partial f}{\partial \sigma}
= \frac{T}{C_V} \,\delta \sigma + \frac{1}{\rho \alpha_s} \,\delta \rho +
\beta_\sigma \delta c \\
\delta \mu &\equiv& \frac{\partial f}{\partial \rho} =
\frac{1}{\rho^2 \kappa_s} \,\delta \rho + \frac{1}{\rho \alpha_s} \,\delta \sigma
+ \beta_\rho \delta c \label{muconj}\\
\delta \mu_c &\equiv& \frac{\partial f}{\partial c} =  \gamma \,\delta c
+ \beta_\sigma \delta \sigma + \beta_\rho \delta \rho \label{cconj}\\
\Phi_{ij} &\equiv& \frac{\partial f}{\partial \nabla_j n_i} =
K_{ijkl} \nabla_l n_k  \label{Phiconj} \\
h_{i}^{n \prime} &\equiv& \frac{\partial f}{\partial n_i} =  \delta_{iq}^\perp 
\frac{\partial  K_{pjkl}}{2\,\partial n_q}
\,(\nabla_l n_k)(\nabla_j n_p) \label{hprconj}
\end{eqnarray}
%
The response to static director deformations is given by
%
\begin{equation}
  \label{hconj}
h^n_{i} \equiv \frac{\delta}{\delta n_i}\int f dV =   h_{i}^{n\prime} - \nabla_{j}
\Phi_{ij}
\end{equation}
%
The pressure is related to the other conjugate quantities by the Gibbs-Duhem
relation \cite{PB1}
%
\begin{equation}
  \label{GibbsDuhem}
 \delta p = \rho \delta \mu + \sigma\delta T  - \mu_c \delta c - h_j \delta n_j
\end{equation}
%
neglecting contributions quadratic in the velocity.


\section{Dynamics in a magnetic field \label{dynamics}}
\subsection{Linear vs. quadratic field effects \label{linquad}}

Let's consider nematics in a magnetic field. The effect of an external magnetic field $H$ on the statics is rather
simple. Since all static equations have to be invariant under time reversal
symmetry, and $H$ changes sign under this symmetry, only quadratic
contributions to the free energy are allowed in lowest order
%
\begin{equation}
  \label{fmagn}
  \varepsilon^{(M)} = - \frac{1}{2} \chi_{a}(\bm{H\cdot n})^{2}
\end{equation}
%
which is the diamagnetic anisotropy energy \cite{PGdG} giving rise to the
magnetic molecular field (\ref{diamagH}). Depending on the sign of $\chi_{a}$
the static orientation of $\bm{n}$ is either parallel
or perpendicular to $\bm{H}$.
 An external field breaks the
rotational symmetry externally and $n_{i}$ relaxes according to the
diamagnetic anisotropy to its equilibrium orientation. 
The appropriate molecular field $h^n_{i}$ reads
%
\begin{equation}\label{diamagH}
h_{i}^{(M)}=-\chi_{a}H_{i}(\bm{n\cdot H})
\end{equation}
%
Usually this is the only effect of an external magnetic field that is taken
into account when dealing with ordinary nematic liquid crystals. All other
effects (a few of them are discussed in \cite{Lub}) are assumed to be small and are
neglected. In ferronematics the response to an external magnetic field is
rather enhanced and such hitherto disregarded effects can become important. There
is the (rather trivial) effect that all material coefficients (transport
parameters and susceptibilities) can depend on $H^{2}$. In addition, if
$\bm{n}$ is perpendicular to $\bm{H}$ in equilibrium, the system is biaxial and
the uniaxial tensors
in eqs. (\ref{heatcond}, \ref{flowalign}, \ref{viscos})
are of
the well-known biaxial form.
Furthermore, there are the magnetic forces, which are described by the Maxwell
stress $\sigma_{ij}^{(M)}= - \mu_{eq}(H)H_{i} H_{j}$ with the equilibrium
magnetic susceptibility $\mu_{eq}(H)$ \cite{Rosen} and by a redefinition of
the pressure $p \to p -(1/2)H^{2}$.
However, all these effects (including (\ref{diamagH})) are quadratic in the
external field strength and represent additions to effects already present. In
the following we will discuss additional effects
that are linear in the field and represent new effects thus bearing a good
chance of being observable in
ferronematics.

Since a magnetic field changes sign under time reversal, linear effects
are possible in the dynamics only, since in the statics all relations are
time-reversal symmetric, that is invariant under time reversal.
In the dynamics the currents come in two classes,
either reversible or irreversible, meaning time-reversal symmetric and
antisymmetric, respectively. The introduction of a linear field then toggles
between these two possibilities.


\subsection{Flow alignment \label{flow}}

As a first example we will consider flow alignment in the presence of a
magnetic field. The flow alignment tensor (\ref{flowalign}) can have additions
linear in the field
%
\begin{eqnarray}\label{lambdaprime1}
\lambda_{ijk}^{D}(H)\!\!&=&\!\!\lambda^{D}_{1}\left(\delta_{iq}^{\perp}
\epsilon_{pjq}H_{p}n_{k}+\delta_{iq}^{\perp}\epsilon_{pkq}H_{p}n_{j}\right)
\nonumber \\ &&\!\!+
\lambda^{D}_{2}\left(\delta_{ik}^{\perp}H_{p}\epsilon_{pjq}n_{q}+
\delta_{ij}^{\perp}H_{p}\epsilon_{pkq}n_{q}\right)
\nonumber \\
&&\!\!+\lambda^{D}_{3}\left(H_{j}\epsilon_{ipk}n_{p}
+H_{k}\epsilon_{ipj}n_{p}\right)
\nonumber \\
&&\!\!+\lambda^{D}_{4}
\left(H_{q}n_{q}n_{j}\epsilon_{ipk}n_{p}+H_{q}n_{q}n_{k}
\epsilon_{ipj}n_{p}\right)
\nonumber\\
&&\!\!+{\lambda^{D}_{5}H_{p}\epsilon_{piq}n_{q}n_{j}n_{k}+\lambda^{D}_{6}
H_{p}\epsilon_{piq}n_{q}\delta^{\perp}_{jk}}
\end{eqnarray}
%
Note that these contributions are all dissipative while (\ref{flowalign})
represents only reversible ones. Adding up in eq.(\ref{cY1}) both contributions into
$\lambda^{(Y)}_{ijk}\equiv \lambda_{ijk}+\lambda_{ijk}^{D}(H)$ then the
cross coupling term in (\ref{csigma1})  reads $\lambda^{(\sigma)}_{kji}\equiv
\lambda_{kji}+\lambda_{kji}^{D}(-H)= \lambda_{kji}-\lambda_{kji}^{D}(H)$ due to
Onsager's relation\footnote{
With the currents $C_{\alpha }\equiv\{Y_{i}\,,\,\sigma_{jk}\}$ the forces are
$F_{\beta}\equiv \{h^n_{i}\,,\, -\nabla_{j}
v_{k}\}$, since the entropy production $R \sim - h^n_{i}Y_{i} - v_{k} \nabla_{j}
\sigma_{kj} \sim - h^n_{i}Y_{i} +  \sigma_{kj} \nabla_{j}v_{k}$. Then,  
Onsager's relation
$D_{\alpha \beta}(H) = \epsilon_{\alpha} \epsilon_{\beta} D_{\beta \alpha}
(-H)$, with $C_{\alpha}= D_{\alpha \beta} F_{\beta}$, leads to the symmetries
stated above for the $\lambda$-tensor, since $\epsilon_{\alpha}
\epsilon_{\beta}=-1$ for cross terms connecting variables of different time
reversal symmetry.
}
guaranteeing a positive (zero) entropy production due to
the field-dependent (-independent) parts.

In the stationary case all currents and quasi currents are zero (disregarding
the thermal degree of freedom here)
%
\begin{equation}\label{station1}
\nabla_{i}(p\delta_{ij}+\Phi_{lj}\nabla_{i}n_{l}+\sigma_{ij})=0, \quad \textrm{and} \quad(\bm{n \times \omega})_{i}+ Y_{i}=0.
\end{equation}
%
The first condition is identically
satisfied for constant shear flow and constant field.
The second one
%
\begin{equation}
-\frac{1}{2}\lambda_{ijk}^{(Y)}\nabla_{j}v_{k}+\frac{1}
{\gamma_{0}}\delta_{ik}^{\perp}h_{k}^{(M)}=0
\end{equation}
%
reads explicitly
%
\begin{eqnarray}\label{flowalign2}
n_{k}(\lambda-1) \nabla_{i} v_{k}
+n_{j}\nabla_{j}v_{i}(\lambda+1)
-2\lambda n_{i}n_{j}n_{k}\nabla_{j}v_{k} &&
\\ \nonumber
+ \lambda_{ijk}^{D}(H)\nabla_{j}v_{k}+\chi_{a}'H_{i}n_{j} H_{j}
-\chi_{a}'n_{i}n_{k}H_{k}n_{j}H_{j}&=&0,
\end{eqnarray}
%
where $\chi_{a}^{\prime}= 2 \chi_{a}/{\gamma_{0}}$.

We will solve eq.(\ref{flowalign2}) for a particular case, where the new
linear field terms (\ref{lambdaprime1}) become manifest: The external field
lies in the shear plane
%
\begin{eqnarray}
{n_{x}=\sin\theta}\nonumber\\
{n_{y}=\cos\theta\sin\varphi}\nonumber\\
{n_{z}=\cos\theta\cos\varphi}\nonumber\\
H_{y}=H\sin\psi \nonumber\\  H_{z}=H\cos\psi
\end{eqnarray}
%
with the shear flow $\nabla_{z}v_{y} =S$ and with $\psi$ the angle of the
magnetic field with the shear gradient.

Without the new terms it is known that the director also lies in the shear
plane (i.e.\@ $\theta=0$) making an angle $\varphi$ with the direction of  
the shear gradient,
where $\varphi$ is given by
%
\begin{equation}
  \label{flowold}
  2S\left(\lambda\cos2\varphi+1\right)=\chi^{\prime}_{a}H^{2}\sin
\left(2\left(\varphi-\psi\right)\right)
\end{equation}
%
The new terms (except $\lambda^{D}_{2}$) force the director out
of the shear plane. Taking first $\lambda^{D}_{1}$
as a representative for the other terms
(in order to simplify the formulas) we get
the unchanged condition
(\ref{flowold}) for the in-plane orientation $\varphi$ of the director. There
is, however, now an out-of-plane component of the director given by the
non-zero angle $\theta$
%
\begin{equation}
  \label{theta1}
\tan \theta=\frac{- \lambda^{D}_{1}S H
  \sin\varphi\cos(\varphi+\psi)}{\left(\lambda+1\right)S\cos\varphi
+\chi^{\prime}_{a}H^{2}\sin{\psi}\cos(\psi-\varphi)}
\end{equation}
%
This kinetic expulsion out of the shear plane due to the combined action of shear
flow and (in-plane)
field occurs,
even if the (static) diamagnetic anisotropy would favor
a director parallel to the field ($\chi_{a}>0$), i.e. to be in-plane. Other
contributions in (\ref{lambdaprime1}) will change also the simple expression
(\ref{flowold}) for the in-plane components of $\bm{n}$. E.g.\@ the two last
contributions in (\ref{lambdaprime1}) lead to an out-of-plane orientational angle
%
\begin{equation}
  \label{theta56}
\tan \theta=\frac{ \beta
  \sin^{2}\varphi}{2(\lambda \sin\varphi
+\alpha\cos\varphi)}
\end{equation}
%
with $2\alpha=\chi^{\prime}_{a}H^{2}/S$ and $\beta=(\lambda^{D}_{5} -  
\lambda^{D}_{6}) H$.
Here the in-plane-angle $\varphi$ is not given by (\ref{flowold}) but follows from
%
\begin{eqnarray}
  \label{flow56}
\frac{\beta^{2} \sin^{3}\varphi}{4(\lambda \sin\varphi + \alpha
\cos\varphi)^{2}}= \frac{1+\lambda \cos2\varphi -
\alpha \sin2\varphi}{(\lambda -1)\sin\varphi + 2\alpha \cos\varphi}
\end{eqnarray}
%
In (\ref{theta56}) and (\ref{flow56}) we have simplified the formulas by
assuming $\psi=0$, i.e.\@ the magnetic field parallel to the shear gradient.
For $\lambda_3^D$ and $\lambda_4^D$ the appropriate formulas are even more
bulky and will not be shown here.

%



\subsection{Heat conduction, diffusion and thermodiffusion \label{diffusion}}

As a second example we consider the heat conduction tensor $\kappa_{ij}$.
There are additions linear in the field of the form
%
\begin{eqnarray}
\kappa_{ij}^{R}(H)&=&\kappa_{1}^{R}\epsilon_{ijk}H_{k}
+\kappa_{2}^{R}\epsilon_{ijk}n_{k}n_{p}H_{p}
+
\kappa_{3}^{R}\left(\epsilon_{ipq}H_{p}n_{q}n_{j}-
\epsilon_{jpq}H_{p}n_{q}n_{i}\right)
\label{krev1}
\end{eqnarray}
%
These terms are reversible due to their time reversal behavior, while those of
(\ref{heatcond}) are irreversible. The former are antisymmetric
$\kappa_{ij}^{R}(H)=
\kappa_{ji}^{R}(-H)= - \kappa_{ji}^{R}(H)$
according to the Onsager's relation thus leading to
zero entropy production.

The field-free heat conduction tensor (\ref{heatcond}) leads to a heat current
$\bm{j}^{\sigma}= (\kappa_{\para}-\kappa_{\perp})\,\bm{n}\,
(\bm{n\! \cdot\! \nabla}T) 
\\+\kappa_{\perp}\bm{\nabla}T$ that lies in the plane of the director and the
temperature
gradient. The
field-dependent terms lead to (reversible) contributions to the heat current
that can be perpendicular to both, temperature gradient and field, or to
temperature gradient and director, or to field and director.
For example, if the field $H$ is along the x- and temperature gradient $G$
along the y-direction, there is a reversible component of the heat current
%
\begin{equation}
  \label{csigmaH}
  j^{\sigma}_{z} =
\begin{cases}
(\kappa_{1}^{R} + \kappa_{2}^{R}) H G  \\
(\kappa_{1}^{R} - \kappa_{3}^{R}) H G
\end{cases}
\quad  \textrm{for} \quad \bm{n} \parallel
\begin{cases}
\bm{H} \\
\bm{\nabla} T
\end{cases}
\end{equation}
%
that is orthogonal to the $\bm{n}/\bm{\nabla}T$ plane and perpendicular
to the field.
This effect is quite analogous to the Hall effect related to electric
currents.
There, similarly the diagonal elements (the analogues to
(\ref{heatcond})) are
dissipative, while the antisymmetric parts (the analogues to (\ref{krev1}))
are non-dissipative \cite{LL}.
Instead of a voltage transverse to the dissipative electric current,
eq.(\ref{csigmaH}) leads to a temperature difference transverse to the
dissipative heat current. This effect (especially the part related to
$\kappa_{1}^{R}$) is in principle present in any fluid
(and called Righi-Leduc
effect \cite{Lub}) and
not restricted to ferronematics, but in the latter system chances are much
better that it is observable.


Quite analogously to (\ref{krev1}) one can introduce 3 new reversible
diffusivities $D_{1,2,3}^{R}$ and the discussion between (\ref{krev1}) and
(\ref{csigmaH}) can be taken over replacing ($\bm{j}^{\sigma},\,\,T$) by
($\bm{j}^{c},\,\, \mu_{c}$). Rather similar is also the case of thermal
diffusion. There are three reversible thermal diffusivities $D_{1,2,3}^{T,R}$ of
the form (\ref{krev1}). Since they are antisymmetric w.r.t. interchange of
indices, and linear in the field, they automatically fulfill Onsager's relation
$D_{ij}^{T,R}(H) = D_{ji}^{T,R}(-H)$. Under an external magnetic field
(of strength $H$) orthogonal
to a temperature gradient (of magnitude $G$)
these new contributions give rise to a
concentration current (orthogonal to both) given by the r.h.s. of
(\ref{csigmaH}), if the $\kappa^{R}$'s are replaced by $D^{T,R}$'s.
Thus, Hall-like temperature and concentration gradients are generated,
which are
transverse to the dissipative heat and concentration currents,
respectively.


\subsection{Viscosity \label{viscosity}}

As the third example we consider field dependent generalizations of the
viscosity tensor
%
\begin{eqnarray}\label{viscosH1}
\nu_{ijkl}^{R}(H)&=&\nu^{R}_{1}[\epsilon_{imp}
n_{j}n_{k}n_{l}+\epsilon_{jmp}n_{i}n_{k}n_{l}-\epsilon_{kmp}n_{j}n_{i}n_{l}
\nonumber \\ &&-
 \epsilon_{lmp}n_{j}n_{k}n_{i}]\,n_{p}H_{m} \nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{2}[\epsilon_{jmp}n_{l}\delta_{ik}-\epsilon_{lmp}n_{j}\delta_{ik}+
\epsilon_{jmp}n_{k}\delta_{il}  \nonumber \\
&&\,\,
-\epsilon_{kmp}n_{j}\delta_{il}+\epsilon_{imp}n_{k}\delta_{jl}-
\epsilon_{kmp}n_{i}\delta_{jl}\nonumber \\
&&\,\,
 +\epsilon_{imp}n_{l}\delta_{jk}-
\epsilon_{lmp}n_{i}\delta_{jk}]\,n_{p}H_{m} \nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{3}[\epsilon_{kmp}n_{l}\delta_{ij}
+\epsilon_{lmp}n_{k}\delta_{ij}\nonumber \\
&&\,\,-
\epsilon_{imp}n_{j}\delta_{kl} -\epsilon_{jmp}n_{i}\delta_{kl}]\,n_{p}H_{m}  \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{4}[\epsilon_{ikp}n_{j}n_{l}
+\epsilon_{ilp}n_{j}n_{k} \nonumber \\
&&\,\,+
\epsilon_{jlp}n_{i}n_{k} +
\epsilon_{jkp}n_{i}n_{l}]\, n_{p}n_{m}H_{m}\nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{5}[\epsilon_{ikp}n_{j}n_{l}
+\epsilon_{ilp}n_{j}n_{k}+
\epsilon_{jlp}n_{i}n_{k}+
\epsilon_{jkp}n_{i}n_{l}]
\,H_{p} \nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{6}[\epsilon_{ikp}\delta_{jl}
+\epsilon_{ilp}\delta_{jk}+
\epsilon_{jlp}\delta_{ik}+
\epsilon_{jkp}\delta_{il} ]\,n_{p}n_{m}H_{m}\nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{7}[\epsilon_{ikp}\delta_{jl}
+\epsilon_{ilp}\delta_{jk}+
\epsilon_{jlp}\delta_{ik}+
\epsilon_{jkp}\delta_{il}]\,H_{p} \nonumber \\
+&& \!\!\!\!\!\!\!
\nu^{R}_{8} n_p [\epsilon_{ikp} (H_j n_l + H_l n_j)
+ \epsilon_{ilp} (H_j n_k + H_k n_j)
  \nonumber \\
&&\,\,
+ \epsilon_{jlp} (H_i n_k + H_k n_i)
+ \epsilon_{jkp} (H_i n_l + H_l n_i)] \nonumber
\end{eqnarray}
%
Because of $\nu_{ijkl}^{R}(H)= \nu_{jikl}^{R}(H)$
and $\nu_{ijkl}^{R}(H)= \nu_{ijlk}^{R}(H)$
the stress tensor remains symmetric and (\ref{viscosH1}) does not contain any
coupling to the vorticity. All contributions in (\ref{viscosH1}) are
reversible, where the antisymmetry w.r.t. exchange of the first pair of
indices with the second one, $\nu_{ijkl}^{R}(H)= \nu_{klij}^{R}(-H)=
-\nu_{klij}^{R}(H)$, (according to Onsager's relation)
guarantees zero entropy production.
The major effect of these new terms is that a density wave (sound wave) is not
only connected to $\textrm{div} \bm{v}$,
but also to transverse velocities (and thus to all
other variables). This will be manifest as
(reversible) bulk shear stresses accompanying
the sound wave. For example, for a density wave with
amplitude $\Delta\rho$, frequency $\omega$ and wave vector $k$ along the
x-direction, magnetic field H along the y-direction, this bulk shear
stress will be felt by a tracer particle as a transverse force
%
\begin{equation}
  \label{transforce}
  f_{z} =  \bar \nu^{R} \, \omega\,
   k \, H \, \frac{\Delta \rho}{\rho_{0}}
\end{equation}
%
with $\bar \nu^{R}=2 \nu^{R}_{2} + \nu^{R}_{3} - 2 \nu^{R}_{7}$, $\bar
\nu^{R}= \nu^{R}_{1} + 2 \nu^{R}_{2} - \nu^{R}_{3} -
2 \nu^{R}_{5} - 2 \nu^{R}_{7}$, and
$\bar \nu^{R}=-2 \nu^{R}_{6}  - 2 \nu^{R}_{7}$
if $\bm{n}$ is along the z-, x- and the y-direction, respectively.


\subsection{Director reorientation \label{reoerient}}

In eq.(\ref{cY1}) the rotational viscosity $\gamma_{0}^{-1}
\delta_{ij}^{\perp}\equiv (\gamma^{-1})_{ij}$
acquires linear field-dependent additions
%
\begin{eqnarray}
  \label{rotvisH1}
 (\gamma^{-1})_{ij}^{R}(H) &=& \frac{1}{\gamma_{1}^{R}} \epsilon_{ijk} n_{k} n_{p}
  H_{p}  + \frac{1}{\gamma_{2}^{R}} ( \epsilon_{ijp} +
  \epsilon_{ipk} n_{k}n_{j} -
  \epsilon_{jpk} n_{k} n_{i}) H_{p} 
\end{eqnarray}
%
which are reversible and give zero entropy production, because they are
antisymmetric in $i$ and $j$ according to Onsager's relation. Of course, they
are also transverse to $\bm{n}$
in both indices $n_{i}\,
(\gamma^{-1})_{ij}^{R}(H) = 0 = n_{j}\, (\gamma^{-1})_{ij}^{R}(H)$.

Usually, the so-called rotational viscosity $\gamma_{0}$ is measured by the
homogeneous relaxation of the director towards an
external magnetic field due to the magnetic
anisotropy effect (\ref{diamagH}). In this case $\bm{H}$ and
$\bm{n}(t)$ lie in the same plane all the time with relaxation rate
$(\chi_{a}/\gamma_{0})H^{2}$ \cite{PGdG}.
The new terms in (\ref{rotvisH1}) change this
picture. With (\ref{diamagH},\ref{rotvisH1}) the director relaxation equation
(\ref{dn1}) takes the form
%
\begin{equation}
  \label{nrelax}
  \dot n_{i}= \chi_{a}^{\prime} \delta_{ij}^{\perp} H_{j}(\bm{H \! \cdot \! n})
+ \chi_{a}^{\prime\prime} (\bm{H \! \times \! n})_{i} (\bm{H\! \cdot\! n})^{2}
\end{equation}
%
where $\chi_{a}^{\prime}=\chi_{a}/\gamma_{0}$ and
$\chi_{a}^{\prime\prime}=\chi_{a}(1/\gamma_{1}^{R} + 1/\gamma_{2}^{R})$.
Obviously, $\bm{n}(t)$ does not stay in the (initial) plane given by $\bm{H}$ and
$\bm{n}(0)$, since there is a nonvanishing component $(\bm{H \!\times\!
  n})\bm{\, \cdot \, \dot n}$
in (\ref{nrelax}). Thus, there are two distinct and coupled dynamic
processes involved. For small angles the solutions of (\ref{nrelax}) can be
written
%
\begin{eqnarray}
  \label{relax2}
 \varphi &=& \varphi_{0} \exp(-\chi_{a}^{\prime}H^{2}t)
 \cos(\chi_{a}^{\prime\prime}H^{3}t) \nonumber \\
 \theta &=&  \varphi_{0} \exp(-\chi_{a}^{\prime}H^{2}t)
 \sin(\chi_{a}^{\prime\prime}H^{3}t)
\end{eqnarray}
%
where $\varphi$ is the angle between the field and the projection of $\bm{n}(t)$
onto the initial plane $\bm{H/n}(0)$ and $\theta$ is the angle of $\bm{n}(t)$
with this initial plane. The time dependence of $\varphi$ is not a simple
exponential decay, but shows an oscillation about it. The angle $\theta$
describes spatial oscillations of the director during the reorientation process.
With field reversal $\bm{H}\to \bm{-H}$, also $\theta$ changes sign.
Without the new reversible terms ($\chi_{a}^{\prime\prime}=0$)
a simple relaxation for $\varphi$, which is then the true angle between
$\bm{n}$ and $\bm{H}$ is regained.

\section{Summary \label{concl}}

In this chapter we have derived hydrodynamic equations 
for ferronematics in the limit that the magnetic degree
of freedom has relaxed to its equilibrium value.
When comparing the equations derived here with the usual description of 
ordinary nematic liquid crystals, we find that there is 
no additional contribution to the static behavior
linear in the magnetic field.
This situation changes completely, however, when one investigates 
dynamic coupling terms  between
the various hydrodynamic variables that are linear in the magnetic field.

The new dynamic effects predicted here come in 
four classes.
In most nematics, for temperatures far above a smectic phase,
one observes the phenomenon of flow alignment.
A shear flow applied to a spatially homogeneous
director field leads to a stationary 
configuration in which the director includes an angle 
with its original orientation - the flow alignment
angle. In the case of usual nematics the director
lies in the shear plane.
Here we predict that an additional magnetic field 
\emph{in} the shear plane
applied to a ferronematic  
forces the director \emph{out} 
of the shear plane
due to the dynamic effects given here, which couple 
the director to the shear flow dissipatively.

Applying a temperature gradient to 
a nematic leads to a heat flux that has components parallel to the 
applied temperature gradient and parallel to the director. 
Here we suggest that an additional magnetic field orthogonal to  
the temperature gradient applied to a ferronematic leads to an
additional reversible heat current 
that is perpendicular to both, the applied magnetic field
and the temperature gradient.

In most hydrodynamic systems including ordinary 
nematics only viscous effects couple the various
components of the velocity field.
For a ferronematic there are, in addition, 
several terms coupling the three components of the
velocity field reversibly.
One of the consequences of these new contributions
could be detected experimentally 
studying the effect of a sound wave propagating in
$x-$direction, say, on a tracer particle 
also exposed to a magnetic field 
in $y-$direction.
For this configuration we predict the occurrence 
of a force 
on this tracer particle in $z-$ direction,
that is perpendicular to the plane 
spanned by the two applied fields.

Director reorientation is expected to change as well
when switching from an ordinary nematic to a
ferronematic.
Here we have shown that the director reorientation
picks up reversible contributions 
in addition to the usual director diffusion
associated with the rotational viscosity $\gamma_1$.
The new reversible contributions are predicted to
lead to a relaxation oscillation when the director is
reoriented in an external magnetic field
in contrast to the simple relaxation observed
for usual nematics.
\hfil \break

%\setcounter{equation}{0}
%\renewcommand{\theequation}{A.\arabic{equation}}
%\renewcommand{\thesection}{\!\!}


\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\chapter [Macroscopic dynamics of ferronematics with the
magnetization as an independent variable]{Macroscopic dynamics of ferronematics with the
magnetization as an independent variable\boldmath${}^\ddagger$\label{3}\footnotetext{${}^\ddagger$This chapter is based on ref.\cite{jpmb}}}
\renewcommand{\thefootnote}{\arabic{footnote}}
%\chaptermark{Ferronematics}
\chaptermark{Macroscopic dynamics of ferronematics }
\section{Introduction \label{intro}}
%The ferronematic  exposed to an external magnetic field has the induced
%magnetization, which is  always co-aligned, while the nematic director can
%be parallel
%\cite{matuoa}
%or perpendicular \cite{berejnov} to it. In the former
%case the equilibrium configuration the system is uniaxial.
%This case will be described in the main text below. The biaxial
%case, where the director is perpendicular to the external field,
%is only slightly more complicated and will be discussed briefly
%in Appendix A. %\ref{biax}.


In ferrofluids the magnetization (its orientation as well as its
absolute value) relaxes to the equilibrium value set by the external field. The
appropriate relaxation time is much larger than all microscopic time scales 
and can be relevant for the macroscopic dynamics, particularly in complicated
lyotropic systems \cite{spoliansky}. In this case one should treat the magnetization 
as an additional dynamic variable with its own dynamical (relaxation) equation.
For isotropic ferrofluids this has been
done in \cite{shliomor} and within a hydrodynamic description quite recently
in \cite{ml}.

In ferronematics the orientations of the director and the
magnetization are linked in equilibrium, but the dynamics off equilibrium can
be quite different: an external field reorients the magnetization quite easily
due to the superparamagnetic effect, but is less effective regarding the
director, which only is coupled to the magnetic field by the diamagnetic
anisotropy effect. Vice versa, an electric field strongly orients the director
but not the orientation of the magnetization. Thus, it is easy to create
situations, where the director and the magnetization are quite different and
where they relax independently into their equilibrium orientation. We also
keep the absolute value of the magnetization as a relevant variable, but not
the degree of nematic order, since the latter is not specifically influenced by a
magnetic field or by the magnetic degrees of freedom and, thus, behaves as in
ordinary nematics (and is relevant near the isotropic phase transition, only).



For ferronematics hydrodynamic equations
were derived (chapter \ref{2}) in the limit that the magnetic degree of freedom
has already relaxed to its equilibrium value. Here we generalize the set of hydrodynamic equations for
ferronematics (in a constant external magnetic field) by including the
magnetization as an additional, slowly relaxing variable.
Special emphasis is laid on the static (Sec.\ref{statics}) and dynamic
(Sec.\ref{dynamics1}) cross-couplings between the
nematic and the magnetic degrees of freedom. As examples for their relevance,
and possible ways for measuring such effects, we discuss in more detail sound
propagation and damping (Sec.\ref{sound1}) as well as the rheology during shear
flow (Sec.\ref{rheo}).
 
In Ref. \cite{liu94} an extended description of
electromagnetic effects in nematic
liquid crystals has been given within the framework of generalized
dissipative Maxwell equations. A non-hydrodynamic magnetic degree of freedom
(a generalized dissipative magnetic field) is introduced, the dynamics of which
can be compared to the dissipative part of our magnetization dynamics, while
there is no counterpart in \cite{liu94} to the reversible dynamics of an
independent magnetization degree of freedom (see below). 


\section{Statics and thermodynamics \label{statics}}

The macroscopic description of a system starts with the identification of the
relevant variables. Apart from the quantities that are related to local
conservation laws, like mass density ($\rho$), momentum density
($\bm  g$), energy density ($\varepsilon$) and concentration ($ c$) of the
magnetic particles\footnote{In lyotropic systems one could also take into
  account additionally the concentration of the solvent $c_{S}$ without
  changing the major results, since it has the same transformation behavior as
  $c$ and thus makes the same type of coupling terms.} or that are related to
spontaneously broken
continuous symmetries like reorientations of the director $\bm{n}$, we take,
as discussed in the introduction, the magnetization $\bm{M}$ as slowly
relaxing variable. To take into account Maxwell's equations the
magnetic induction $\bm{B}$ must be considered as well. According to the 
Eulerian
description all variables are local fields, i.e. volume densities that depend
on space and time. The director $\bm{n}$ is a unit vector and due to the 
special
nematic symmetry all equations have to be invariant under the replacement
$\bm{n} \to -\bm{n}$. The magnetization $\bm{M}=M \bm{m}$ is associated with  
rotations
$\delta \bm{m}$ as well as changes in the absolute value $\delta M$.

Assuming local thermodynamic equilibrium, i.e. all other, fast
relaxing quantities are already in equilibrium, the Gibbs relation
%
\begin{eqnarray} \label{energy}
d\varepsilon= T d \sigma +\mu d \rho +\mu_{c} d c+ v_i d g_i +
 H_i d B_i+h_i^{M} d M_i+
h^{n \prime}_{i}d n_{i} +\Phi_{i j} d(\nabla_{j}n_{i})
\end{eqnarray}
%
connects the macroscopic variables to the entropy density $\sigma$.
In eq.(\ref{energy}) the
thermodynamic quantities: chemical potential ($\mu$), temperature ($T$),
relative chemical potential
($\mu_{c}$), nematic molecular fields ($ \Phi_{ij}$, $ h^{n \prime}_{i}$),
velocity ($v_i$),
magnetic Maxwell field ($H_i$) and the magnetic molecular field ($h_{i}^{M}$)
are defined as partial derivatives of the energy density with
the respect to the appropriate variables \cite{PB1}.


In equilibrium the magnetization is generally a nonlinear function of the
external field. This function is known from experiments \cite{pshen} or from
simulations \cite{wang}. The equilibrium
value of the magnetization, $\bm{M}_{0}$ at a given external field, which
may have a field
independent part in case of true ferromagnetism, is an input parameter into
our dynamic theory. In addition, we assume the equilibrium orientation of the
director to be parallel to the field (and the equilibrium magnetization). The 
case of a perpendicular director alignment is discussed in Sec.\ref{biax}.

The statics, i.e. the relation between the conjugate quantities and the
variables, is conveniently set up by providing an energy density function,
usually in harmonic approximation (bilinear in the variables). We will do here
the same for the magnetic part of the energy density and consider
below the  more general description. Thus $\varepsilon =
\varepsilon_{n} + \varepsilon_{M}$ with the magnetic energy density
%
\begin{eqnarray}\label{energyM1}
\varepsilon_{M}(\bm{B,M})&=&\frac{B^2}{2}-\bm{M}\cdot \bm{B}
+ \frac{1}{2} A_1\left(\bm{n}\cdot\bm{M}\right)^2+\frac{1}{2}A_2 \bm{M}^2
\end{eqnarray}
%
and $\varepsilon_{n}$ the energy density of a conventional nematic liquid
crystal \cite{PB1} (without magnetization and external field). The term
$\frac{1}{2}
A_1\left(\bm{n}\cdot\bm{M}\right)^2$
describes the coupling between the nematic director and the magnetic
particles. This strong interaction was observed in a series of
experiments \cite{berejnov, C1, koneracka} studying the Fredericks
transition in ferronematics. For $\bm{n} \para \bm{M}$
being the energy minimum, $A_{1}$ has to be negative.
Generally the coefficients $A_{1,2}$ are functions of the state variables,
like temperature, pressure etc., but also of $\bm{M}_{0}^{2}$ (or the external
field strength). Only for a linearized description, valid for small deviations
from equilibrium, are the coefficients constant.

Using eq.(\ref{energyM1}), the magnetic Maxwell field $H_i$ is defined in
the usual way
%
\begin{eqnarray} \label{H}
H_i=\left(\frac{\partial \varepsilon}{\partial
    B_i}\right)_{\bm{M},\bm{n},\dots}=B_i-M_i,
\end{eqnarray}
%
while the magnetic molecular field $h^M_i$ reads
%
\begin{eqnarray}
\label{molfm}
h_{i}^M =\left(\frac{\partial \varepsilon}{\partial
    M_i}\right)_{\bm{B},\bm{n},\dots}=-B_i+A_1
n_i n_{j} M_{j} + A_2 M_i
\end{eqnarray}
%
Note that because of the definition (\ref{H}), it is not possible to have a
direct coupling between the
external field $\bm{B}$ and the director; the field orientation of the
director is mediated by the magnetization via the term $\sim A_{1}$.

In equilibrium $h_{i}^{M}$ has to be zero and $B_{i}= A_1 n_i n_{j} M_{j} +  
A_2 M_i$
results. With that solution the magnetic energy density (\ref{energyM1}) can  
be expressed
as $\varepsilon_{M}(\bm{B})= \tfrac{1}{2} H_{i}(\bm{B})B_{i}$.
In the regime of linear
magnetism or if linearized for small
deviations from equilibrium, one can use instead the usual magnetic
susceptibility tensors, defined by $B_{i}= \mu_{ij} H_{j}$ and $M_{i} =
\chi_{ij} H_{j}$, which both have the uniaxial form $\mu_{ij}=
\mu_{\perp}\delta_{ij}^{tr} + \mu_{_{\para}}n_{i}n_{j} =
\mu_{\perp}\delta_{ij} + \mu_{a}n_{i}n_{j}$ (where
$\delta_{ij}^{tr} = \delta_{ij}
-n_{i}n_{j}$ is the transverse
Kronecker symbol). Comparison with the equilibrium solution of (\ref{molfm})  
gives $A_{1}=
\mu_{_{\para}}/\chi_{_{\para}} - \mu_{\perp}/ \chi_{\perp}$ and $A_{2}=
\mu_{\perp}/
\chi_{\perp}$.
Written in this way the magnetic energy density
(\ref{energyM1}) obtains the familiar form $\varepsilon_{M}(\bm{B})=
\tfrac{1}{2}\mu_{ij}H_{i}H_{j}$.
In the general case, again, the magnetic
susceptibilities are
not constants, but depend on the magnetization or the external field.

In ordinary nematic liquid crystals instead of $\varepsilon_{M}(\bm{B})$
usually the Legendre-transformed energy
$\varepsilon_{M}(\bm{H})=\varepsilon_{M}(\bm{B}) -\bm{B}\cdot \bm{H}$ is used,
for which we get the well-known form
$2\varepsilon_{M}(\bm{H})= -\chi_{a}(\bm{n} \cdot \bm{H})^{2} -
(\chi_{\perp} -1) \bm{H}^{2}$
where very often the vacuum field energy ($\bm{H}^2$) is omitted \cite{PGdG}.

The molecular field $ h^{n \prime}_i $
%
\begin{equation}
  \label{molf}
 h^{n \prime}_{i} = \delta_{ij}^{tr} \left(\frac{\partial \varepsilon}{\partial
     n_{j}}\right)_{\bm{M},\bm{B},\dots}=
 A_{1} \delta_{ij}^{tr} M_{j} M_{k} n_{k}
\end{equation}
%
where higher order terms are neglected (\ref{hprconj}) and
$\Phi_{ij}$ (\ref{Phiconj})
can be combined to the field 
%
\begin{eqnarray} \label{molfn}
h_{i}^n= \frac{\delta}{\delta n_i}\int \varepsilon dV =   h^{n
\prime}_i-\nabla_j\Phi_{i
   j}.
\end{eqnarray}
%
Since $\bm{n}$ is a unit vector that can only rotate, $n_{i} h^{n
\prime}_{i}=0$, which
is ensured in (\ref{molf}) by the transverse Kronecker symbol. In equilibrium,
$\bm{n}=const.$ and $\delta_{ij}^{tr} M_{j}=0$, hence $h_{i}^{n}=0$.


In the  general case  the equilibrium value of the magnetization $\bm{M^{0}}$
is a nonlinear function of
an external magnetic induction $\bm{B}$,
$M^{0}_i=f_i\left(\bm{B}\right)$. In isotropic ferrofluids simple theoretical
models predict $\bm{f} (\bm{B})= f(B) \bm{B}/B$ with $f(B)$ the Langevin
function and $B=|\bm{B}|$. Actual experiments \cite{pshen} and computer 
simulations
\cite{wang} show a somewhat more complicated form for $f(B)$. We assume the
function $\bm{f}(\bm{B})$ as known and given. To our hydrodynamic theory this
is an input as are any other aspects of the equilibrium structure, e.g. the
existence and the orientation of $\bm{n}$ w.r.t the external field.
Assuming $\bm{f}(\bm{B})$ to be monotonous we can invert it and call it
$\bm{g}(\bm{M})=\bm{f}^{-1}(\bm{B})$. Using the same arguments as in
Sec.\ref{statics} one can construct the magnetic part of the energy density
in the following form
%
\begin{eqnarray}\label{energyM}
\varepsilon\left(\bm{B},\bm{n},\bm{M}\right)&=&\frac{B^2}{2}-\bm{M}\cdot 
\bm{B}+
\int(n_{i}n_{j}g_{j}^{(1)} +
\delta_{ij}^{tr} g_{j}^{(2)}) d M_i
\end{eqnarray}
%
where, in contrast to the isotropic case, $g_{i}$ has been decomposed into a
longitudinal and a transverse part. This leads to
%
\begin{eqnarray}
\label{Hg}
H_i&=&\left(\frac{\partial \varepsilon}{\partial
    B_i}\right)_{\bm{M},\bm{n},\dots}=B_i-M_i \\
\label{molfmg} h_{i}^M&=&\left(\frac{\partial \varepsilon}{\partial
    M_i}\right)_{\bm{n},\bm{B},\dots}=-B_i+ n_{i} n_{j} g_j^{(1)} +
\delta_{ij}^{tr} g_{j}^{(2)}.
\end{eqnarray}
%
In equilibrium the magnetic molecular field $h_i^M$  is zero and
$\bm{B} = \bm{g} \para \bm{n}$ (requiring $n_{i} n_{j} g_{j}=g_{i}^{(1)}$ and
$\delta_{ij}^{tr} g_{j}^{(2)}=0$ in
equilibrium). Outside equilibrium $g_{i}(\bm{M}) \neq B_{i}$ and $\bm{n}$ is
not parallel to $\bm{B}$, generally.


For the nematic molecular field
%
\begin{equation} \label{molfg}
h^{n \prime}_i=  \delta_{ij}^{tr} \left(\frac{\partial \varepsilon}{\partial
    n_j}\right)_{\bm{M},\bm{B},\dots}= \delta_{ij}^{tr} n_{k}\int \Bigl(
 d M_j (g_{k}^{(1)} - g_{k}^{(2)}) +  dM_{k}(g_{j}^{(1)} - g_{j}^{(2)})
\Bigr)
\end{equation}
%
is found. Linearizing about equilibrium the choices
$g_{i}^{(1)}=(A_{1}+A_{2})n_{i}n_{j}M_{j}$  and
$g_{i}^{(2)}=A_{2} \delta_{ij}^{tr}M_{j}$ lead back to
eqs.(\ref{molfm},\ref{molf}).

\section{Dynamics \label{dynamics1}}

The hydrodynamic equations for conserved, broken-symmetry and slowly
relaxing variables are
%
\begin{eqnarray}\label{he}
\frac{\partial}{\partial t}\rho
+   div\,\rho \bm{v}&=&0
\\ \label{dv}
\frac{\partial}{\partial t}\sigma + div\, \sigma \bm{v}
+ div\bm{j^{\sigma}}&=& \frac{R}{T}
\\ \label{ds}
\frac{\partial}{\partial t}g_{i} +\nabla_{j}\Bigl( v_{j}g_{i}+ \delta_{ij}p +
\sigma_{ij}^{th}
+ \sigma_{ij}\Bigr)&=&0
\\ \label{dn}
{\Bigl(\frac{\partial}{\partial
      t}+v_{j}\nabla_{j}\Bigr)}n_{i}+(\bm{n \times \omega})_{i}+Y_{i} &=&0
\\ \label{dc}
\rho{\Bigl(\frac{\partial}{\partial t}+v_{j}
\nabla_{j}\Bigr)}c + div\,\bm{j}^{c}&=&0
\\ \label{dm}
{\Bigl(\frac{\partial}{\partial
      t}+v_{j}\nabla_{j}\Bigr)}M_{i}+\left(\bm{M} \times
\bm{\omega}\right)_i+X_{i}&=&0
\end{eqnarray}
%
with
%
\begin{equation}\label{astress}
\sigma_{ij}^{th}= - B_{j}H_{i} - \frac{1}{2}(M_{j} h_{i}^{M} - M_{i} h_{j}^{M})
- \frac{1}{2}(n_{j} h_{i}^{n} - n_{i} h_{j}^{n}) + \Phi_{kj} \nabla_{i}n_{k}
\end{equation}
%
where $\omega_i=\frac{1}{2}\epsilon_{i j k}\nabla_j v_k$ is the vorticity.
The thermodynamic pressure $p$ is given by
%
\begin{equation}\label{pressure}
p=-\varepsilon + T \sigma +\mu \rho +\mu_c c+ \bm{g \cdot v} + \bm{B \cdot H}.
\end{equation}
%
The parts of the currents shown explicitly in (\ref{he}-\ref{astress}) are
not material dependent, but are given by general symmetry and
thermodynamic principles \cite{PB1}, like transformation behavior under
translations (convective terms) or rotations (e.g. $\ldots \times
\bm{\omega}$) and by the requirement of zero entropy production ($R=0$).

Using the relation \cite{PB1} $\Omega_{ij}( -M_{i}
B_{j} + h_{i}^{M} M_{j} +
h_{i}^{n} n_{j} + \Phi_{ki} \nabla_{j}n_{k} +
\nabla_{k}(n_{j}\Phi_{ik}))=0$, which is valid for any constant
antisymmetric matrix
$\Omega_{ij}= - \Omega_{ji}$ and which follows from the
fact that the energy density (\ref{energy}) has to be invariant under constant
rotation, eq.(\ref{astress}) can be transformed into
%
\begin{equation} \label{stress}
\sigma_{ij}^{th}=
-\frac{1}{2}(B_{j}H_{i}+B_{i}H_{j})  + \frac{1}{2}(\Phi_{ki} \nabla_{j}n_{k}+
\Phi_{kj} \nabla_{i}n_{k})+
\frac{1}{2}\nabla_{k}(n_{j}\Phi_{ik}-n_{i}\Phi_{jk}).
\end{equation}
%
Here the antisymmetric part has the form of
a divergence, which ensures angular momentum conservation. It can be
brought into a manifestly symmetric form by
some redefinitions \cite{MPP}.

The phenomenological parts of the entropy
current $j^{\sigma}_{i}$, the stress tensor $ \sigma_{i j}$, the
concentration current $j^{c}_{i}$ and the quasi-currents $Y_{i}$ and $X_i$
associated with the temporal changes of the director and the
magnetization, respectively, are given below. The source term $R/T$ in
(\ref{dv}) is the entropy production,
which is zero for reversible and
positive for irreversible processes. The phenomenological part of the stress
tensor $\sigma_{ij}$ has to be symmetric guaranteeing angular momentum
conservation.

Since we are not dealing with electromagnetic effects, we can use the static
Maxwell equations to determine $\bm{B}$
%
\begin{eqnarray}\label{MAX}
curl \bm{H} = curl \left(\bm{B}-\bm{M}\right)=0, \qquad div\bm{B}=0.
\end{eqnarray}
%

The dynamic equation for the energy density follows from (\ref{he}-\ref{dm})
via (\ref{energy}) and is not shown here.


The phenomenological currents written down in eqs.(\ref{he}-\ref{dm})  can be 
split into dissipative (superscript $D$) and into
reversible (superscript $R$) parts, depending on whether they give rise to a 
finite amount of dissipation ($R>0$) or to a zero
entropy production ($R=0$). Using general symmetry and invariance arguments and
the fact, that a magnetic field changes sign  under time
reversal,  we obtain
%
\begin{eqnarray}
\label{Q}
j_{i}^{\sigma R}&=&- \kappa_{ij}^R(M)\nabla_{j}T -D^{T R}_{ij}(M)
\nabla_{j} \mu_{c} \\
%
\label{j}
j_{i}^{c R}&=& -D^R_{ij}(M) \nabla_{j} \mu_{c} - D^{T R}_{ij}(M)\nabla_{j}T \\
%
\label{sigma}
\sigma_{ij}^{R}&=& -\frac{1}{2}\lambda_{kji}h_{k}^{n}-c_{kij}^{R}(M)h_{k}^{M}
-\nu_{ijkl}^{R}(M)A_{kl}\\
%
\label{Y}
Y_{i}^R&=& -\frac{1}{2}\lambda_{ijk}A_{jk} +
(\gamma^{-1})_{ij}^{R}(M)h_{j}^n+\chi^R
\left(\bm{n}\times\bm{h}^M\right)_i\\
%
\label{X}
X_i^R&=&b_{ij}^R(M)h_j^M +\chi^R \left(\bm{n}\times\bm{h}^n\right)_i-
c_{ijk}^R(M) A_{jk}
\end{eqnarray}
%
with $A_{ij}=\frac{1}{2}\left(\nabla_i v_j+\nabla_j v_i\right)$.  The
magnetization-dependent tensors $\kappa_{ij}^{R}(M)$,  $D_{ij}^{R}(M)$,
 $D^{T R}_{ij}(M)$, $(\gamma^{-1})_{ij}^{R}(M)$ , $b^{R}_{i j}(M)$,
$\nu^{R}_{i j
  k l}(M)$,  $c_{i j k}^{R}(M)$ are all odd functions of
the magnetization. 

For
the case of the magnetization  already relaxed to its equilibrium value,
similar tensors exist with $M$ replaced by the external
field $H$ (except for $b_{ij}^{R}$ and $c_{ijk}^{R}$). Their physical meaning has
been discussed in chapter \ref{2}.

The
second order tensors  $D_{ij}^{R}(M)$,
 $D^{T R}_{ij}(M)$ and $b^{R}_{i j}(M)$ have the same form as the heat
 conduction tensor $\kappa_{ij}^{R}(M)$ (\ref{krev1}), while the forms
 of  $(\gamma^{-1})_{ij}^{R}(M)$ and $\nu^{R}_{i j
  k l}(M)$ are given in (\ref{rotvisH1}) and (\ref{viscosH1}), respectively.  
The third rank tensor $c_{i j k}^R$ describing a dynamic crosscoupling between
flow and magnetization is symmetric in the two last indices and reads
%
\begin{eqnarray} \label{cR}
c_{ijk}^R(M) &=& c_1^{R} M_i n_j n_k+c_{2}^R(\delta_{ij}M_{k}+\delta_{ik}M_{j})
\nonumber + c_{3}^R
M_i\delta_{jk} +c_4^R n_{i} M_p n_p \delta_{jk}
 \\
&&+ c_5^R(n_i M_j n_k +n_i
M_k n_j)+c_6^{R} n_i M_p n_p n_j n_k
\end{eqnarray}
Such a coupling does not exist in isotropic ferrofluids, nor in conventional  
nematics.

Due to the new degree
of freedom (magnetization) there is an  additional term  in the nematic
quasi-current eq.(\ref{Y}) $\chi^R \left(\bm{n}\times\bm{h}^M \right)_i
$ and a counter term in $X_{i}^{R}$, which describes a dynamic cross-coupling
between magnetization and nematic
director. It does not exist in ordinary nematics nor in isotropic
ferrofluids. Its physical meaning (together with $b_{ij}^{R}$ and
$c_{ijk}^{R}$) will be explored in the
Secs.\ref{sound1} and \ref{rheo}. The flow alignment tensor $\lambda_{ijk}$ has 
the
usual form \cite{MPP} $\lambda_{ijk}= \lambda(\delta_{ij}^{tr}n_{k}+
\delta_{ik}^{tr}n_{j})$.

  For the derivation of the dissipative parts of the phenomenological
 currents one usually expands the
dissipation function $R$
to second order in the
 thermodynamic forces and then obtains  the dissipative currents by
 taking the variational derivatives with
 respect to the forces.
 We find for the dissipation function
%
\begin{eqnarray} \label{R}
R&=&\frac{1}{2} \kappa_{i j}\left(\nabla_{i} T\right)\left( \nabla_{j}
  T\right)+\frac{1}{2} \nu^D_{ijkl} A_{ij}A_{kl} \nonumber
+ \lambda_{ijk}^{D}(M)h_i^n A_{jk}
\\ &+&\frac{1}{2}D_{i j}
\left(
  \nabla_i \mu_c \right)\left(\nabla_j \mu_c \right)
+ \frac{1}{2}b_{i j}^D h_i^M h_j^M  +  c_{ijk}^D  h_i^M A_{jk}
+ \chi_{ij}^D(M) h_j^M h_i^n
\nonumber \\ &+& \frac{1}{2 \gamma_0}h_i^n\delta_{i j}^{tr}h_j^n+D_{i
  j}^{T}\left(\nabla_j T\right) \left(\nabla_i \mu_c \right)
\end{eqnarray}
%
Here $\nu^D_{ijkl}$ eq.(\ref{viscos}) is the uniaxial
viscosity tensor \cite{MPP} and $\kappa_{ij}$ eq.(\ref{heatcond}), $D_{ij}$ and $D^T_{ij}$
describe heat
conduction, diffusion and thermodiffusion, respectively. Director diffusion  
or relaxation
is given by one coefficient $\gamma_{0}$, while magnetization relaxation
$b_{ij}^{D}$ contains a transverse (rotational) and a longitudinal (absolute
value) coefficient $b_{ij}^D=b_{\perp}^{D}\delta_{ij}^{tr}+ b_{\paras}^{D}
n_{i}n_{j}$. All the 2nd rank field-free tensors mentioned above have this form.
The meaning of $\lambda_{ijk}^{D}$ has been
discussed in chapter \ref{2}, while $\chi_{ij}^{D}$ will show up in
Sec.\ref{rheo}. The
3rd rank tensor $c_{ijk}^{D}$ (\ref{cd_bia}) again is specific to nematic ferrofluids and
%contains one coefficient $c_{ijk}^D=c^D(\epsilon_{imk}n_m n_j+\epsilon_{imj}
%n_m n_k)$
describes a dissipative coupling between flow and orientation of
the magnetization.

The range of possible values of the coefficients in eq.(\ref{R}) is
restricted by the positivity of the entropy production.

The dissipative parts of the currents then read
%
\begin{eqnarray} \label{sD}
j_{i}^{\sigma D}&=&- \kappa_{ij}\nabla_{j}T -D^T_{ij} \nabla_{j} \mu_{c}
\\ \label{cdiff}
j_{i}^{c D}&=& -D_{ij} \nabla_{j} \mu_{c} - D^T_{ij}\nabla_{j}T \\
%
\label{sigmaD}
\sigma_{ij}^D&=&-\nu_{ijkl}^D A_{kl}-\lambda_{kij}^D(M) h_k^n-c_{kij}^D h_k^M
\\
%
\label{YD}
Y_i^D&=&\frac{1}{\gamma_{0}}\delta_{ij}^{tr} h_{j}^n+\chi_{ij}^D(M)
h_j^{M} +\lambda^D_{ijk}(M) A_{jk} \\
%
\label{XD}
X_i^D&=&b_{ij}^D h_j^M +\chi_{ji}^D(M) h_j^n +c_{ijk}^D A_{jk}
\end{eqnarray}
%
%
The dissipative second rank tensor $\chi_{i j}^D$ is different from the
reversible ones (determined above), since it has to contain an odd number of 
$\bm{n}$'s
%
\begin{eqnarray}\label{chiD}
\chi_{ij}^D(M)=\chi_1^D\delta_{ik}^{tr}M_k n_j +\chi_2^D\delta_{ij}^{tr} M_k 
n_k
\end{eqnarray}
%
This set of equations is comparable to the dissipative dynamics given in
\cite{liu94} for the magnetic and nematic degrees of freedom. Interpreting
$-H_{i}^{D}$ as our current $X_{i}^{D}$ and $c\, {\rm curl} \bm{E}^{0}$ as our
conjugate $\bm{h}^{M}$, eq.(13) of \cite{liu94} is compatible with
eqs. (\ref{sigmaD}-\ref{XD}) in the uniaxial case ($\bm{n}\parallel \bm{m}$ in
equilibrium). There is however no counterpart in \cite{liu94} to the
reversible dynamics of eqs.(\ref{sigma}-\ref{X}).



\section{Propagation of  sound \label{sound1}}

In this section we derive the longitudinal eigenmodes (sound) of the
system with the nematic director $\bm{n}^{0}$  parallel to
the magnetic field in equilibrium. In ordinary nematics the sound velocity is
isotropic and does not
depend on the external field, since the nematic orientational fluctuations do
not couple to sound. Only sound damping is anisotropic due to the anisotropy
of viscosity and heat conduction. In nematic ferrofluids with the
magnetization relaxed to its equilibrium value $\bm{M}_{0}$, sound is
accompanied by transverse forces (chapter \ref{2}), but the sound velocity is still
isotropic and field-independent. Here we focus on the effect of the new
variable (magnetization) on the sound spectrum. Thus, we can neglect all
diffusional processes connected e.g. with viscosity and
heat conduction as well as their reversible counterparts. Only the relaxations
of the director and the magnetization in
the field are kept. The relevant linearized static fields are then
%
\begin{eqnarray} \label{psis}
h^{n \prime}_i &=& A_1 M_{0}\delta M_i - A_1
{M_{0}}^2 \delta n_i\\
h_{i}^M &=& -\delta B_i+A_1 M_{0}\delta
n_i + A_2\delta M_i
+A_1 n^{0}_i n^{0}_{j} \delta M_{j} \label{hms}
\end{eqnarray}
%
with $n_{i}^{0}$ the unit vector along the field.
Assuming a one-dimensional plane wave the space-time dependence is $\sim \exp
i(-\omega t + \bm{k}\cdot \bm{r})$ for
all deviations from equilibrium and the linearized set of dynamic equations  
becomes an
algebraic one.
Even then the full system of equations is rather complicated in the case of  
ferronematics.
It can be simplified by looking at particular geometries.

Let us consider sound in the case where the external magnetic field,
the equilibrium magnetization, and the director are perpendicular to the wave
vector. Field fluctuations $\delta
B_{i}$ are fixed by the static
Maxwell equations (\ref{MAX}) to $\delta B_{i}= \delta M_{j}(\delta_{ij}-
k_{i}k_{j} k^{-2})$. However, since $A_{2} \gg 1$
(because of $\chi_{\perp} \ll 1$), this contribution can
safely be neglected in (\ref{hms}).
Then there are only three relevant variables left, the
density $\rho$, the longitudinal velocity component ($\bm{v} \para \bm{k}$)
and the (longitudinal) component $\delta \bm{M} \para \bm{B}$, which is
transverse to $\bm{k}$, changing the sound dispersion
relation into
%
\begin{eqnarray} \label{sdisp}
\frac{\omega^2}{k^{2}}- c_{0}^{2}=
 \frac{\mu_{\paras}}{\rho}
(c_{3}^{R}+c_{4}^{R})^{2}
 \chi_{\paras}
    H^2 \frac{i \omega}{i \omega -\tau}
\end{eqnarray}
%
with $c_{0}$ the usual adiabatic sound velocity and
$\tau=b_{\paras}^{D}\mu_{\paras}/\chi_{\paras}$ the relaxation time of the
longitudinal magnetization.
Here $B^{2}$ has been replaced by $(\chi_{\paras}H)^{2}$ and $A_{1}+A_{2}$
by $\mu_{\paras}/\chi_{\paras}$.
The coefficient $c_{3}^{R}$ (\ref{cR}) also
exists in isotropic ferrofluids. Thus, this change in the sound dispersion
relation is possible in the isotropic phase as well \cite{mjl}.

If the sound period is
much bigger than the inverse relaxation time of the magnetization
$k c_{0} \gg 1/\tau$, there is a field dependent addition to the
sound velocity, while in the opposite, realistic case $k c_{0} \ll 1/\tau$,
there is a field-dependent sound damping due to magnetization relaxation
%
\begin{eqnarray} \label{sdamp}
 \mathfrak{Im} (\omega) =-\frac{1}{2} \frac{(c_{3}^{R}+c_{4}^{R})^{2}
  }{\rho b_{\paras}^{D}} \chi_{\paras}^{2} H^2 k^{2}
\end{eqnarray}
%
that comes in addition to the usual magnetic-field-independent
sound damping (not shown here). This
effect can be used to measure the combination $c_{3}^{R}+c_{4}^{R}$ of new
coefficients by varying the external field.
In the case of sound waves along the field direction
($\bm{k}\para \bm{B}$) the situation is quite similar, only a different
combination of transport parameters is involved, i.e. in
(\ref{sdisp},\ref{sdamp}) $c_{3}^{R}+ c_{4}^{R}$ has to be replaced by
$c_{1}^{R}+ 2c_{2}^{R}+ c_{3}^{R}+ c_{4}^{R}+2c_{5}^{R}+c_{6}^{R}$. A
completely different situation arises, when the sound is oblique to the field
$\bm{B}$ (i.e. $\bm{k}$ has an angle different from zero or $\pi$ with
$\bm{B}$). Then the longitudinal variables $div \bm{v}$, $\delta \rho$, and
$\bm{B}\cdot \delta \bm{M}$ are coupled to the transverse ones ($curl \bm{v}$,
$\delta_{ij}^{tr}\delta M_{j}$, $\delta n_{i}$) mainly through the existence of
the tensor $c_{ijk}^{R}(M)$ (\ref{sigma}, \ref{X}, \ref{cR}). This has the
consequence that shear flow is connected to sound and vice versa, and if one
excites one of them the other is excited as well. A similar
phenomena for isotropic ferrofluids is discussed in \cite{mjl}.

The picture is qualitatively similar for the biaxial case ($\bm{n}\perp
\bm{B}$). When
the sound is along the preferred directions (either parallel to $\bm{n}$, to  
$\bm{B}$ or
perpendicular to both) the sound dispersion gets an extra field dependent
damping similar to (\ref{sdamp}), while in the oblique case again sound is
coupled to shear flow.


\section{Rheology \label{rheo}}


To evaluate the influence of the static and dynamic coupling between the
nematic degree of
 freedom and the magnetization we investigate the stress
 induced by applying a shear flow that varies periodically in time.
To make the problem simpler we consider the particular
 geometry, where the equilibrium orientation of the nematic director and the  
magnetization
 are perpendicular to the flow direction and along the gradient direction. The
 linear response of the system that contains
 information on the material properties is described by the frequency
 dependent response function $G(\omega)$, which is defined as the ratio of the
 induced stress (say $\sigma_{xz}$) to the applied "strain" $e_{xz}\equiv
 i A_{xz}/\omega$
%
\begin{equation}
 G(\omega)=\frac{\sigma_{xz}}{e_{xz}}=G{'}(\omega)+i G{''}(\omega),
\end{equation}
%
which has poles at the frequencies of those eigenmodes that couple to shear
flow. The real part
$G'$ (imaginary part $G''$) describes the reversible, in-phase (irreversible,
out-of-phase) response. When calculating $G(\omega)$ the flow is
assumed to show the undisturbed, externally imposed linear profile.

The relaxation process of the magnetization to its equilibrium value is fast
in comparison to the nematic director relaxation/diffusion time. This gives the
possibility to investigate the influence of the nematic-magnetization
couplings on the two modes separately. Since the slow nematic mode seems to be
more convenient for experiments, we will concentrate on 
this mode in the following.
We consider the situation without an external
magnetic field first and then the much more complicated case with a field
(along the gradient direction). This probes the coupled dynamics of the
transverse variables
($curl \bm{v}$,
$\delta_{ij}^{tr}\delta M_{j}$, $\delta n_{i}$).

The response function without magnetic field has the following
form
%
\begin{eqnarray}\label{Gp}
\frac{G{'}}{\omega^2}&=&  \frac{C \Gamma}{\omega^{2} + \Gamma^{2}} \\
\frac{G{''}}{\omega}&=&\nu_{3} - \frac{\xi_{1}}{q_{1}} +
 \frac{C \omega^{2}}
{\omega^2+\Gamma^{2}}, \label{Gpp}
\end{eqnarray}
%
with
%
\begin{eqnarray}
\label{xi1}
 \xi_1 &=&
\frac{1-\lambda^2 }{4} +\frac{(c^D)^2}{b^{D}_{\perp}\gamma_0} -
\frac{c^D\chi^R \,\lambda}{b^{D}_{\perp}} \\
q_{1} &=&
\frac{1}{\gamma_0}+\frac{(\chi^R)^2}{b_{\perp}^{D}} \label{q1}
\\
\label{amplitude}
C&=&\frac{\xi_{1}}{q_{1}} - \frac{(c^{D})^{2}}{b_{\perp}^{D}} =
\frac{1}{q_{1}}\Bigl(\frac{1}{4}  - \bigl[ \frac{\lambda}{2} +
\frac{c^D\chi^R}{b^{D}_{\perp}}\bigr]^{2} \Bigr) \\
 \Gamma &=&  q_{1}  K_{3}k^{2} \label{width}
\end{eqnarray}
%
eqs.(\ref{Gp},~\ref{Gpp}) are, as discussed above, only applicable for low
frequencies with $\omega  \ll A_{2}b_{\perp}^{D}$.
In the loss modulus $G''$ the apparent Newtonian shear viscosity differs from
the bare shear viscosity $\nu_{3}$ by $-\xi_{1}/q_{1}$ where
$1/q_{1}$ (\ref{q1})
is the renormalized nematic orientational viscosity that deviates from
$\gamma_{0}$ due to the reversible dynamic crosscoupling ($\chi^R$) between
director and magnetization (\ref{Y},\ref{X}), and the transverse
magnetization  relaxation
$b_{\perp}^{D}$.
The new contribution $\sim \xi_{1}$
is due to the relaxation of
the magnetization ($b_{\perp}^{D}$) and of the director ($\gamma_{0}$), as 
well as
due to the dissipative crosscoupling between flow and magnetization ($c^{D}$)
and the
reversible crosscouplings of the director with flow ($\lambda$) and
magnetization ($\chi^{R}$). This change in the apparent viscosity
is absent in isotropic ferrofluids, but present in ordinary nematics (for
$\lambda \neq \pm 1$).
However, since ferronematics, ordinary nematics and isotropic ferrofluids are
different phases
with generally different (bare) viscosities, the additional change of
the viscosity due the magnetization degree of freedom is only a quantitative
effect and probably difficult to observe.
There are, however, magnetic oscillations (around $M=0$) induced by the
periodic shear flow (directly via $c^{D}$) or mediated by director
oscillations (via $\chi^{R}$ and $\lambda$). The oscillating magnetization
points out of the shear plane with a peak amplitude that has its
maximum for $\omega \approx \Gamma$. Such magnetic oscillations can be
measured by a Hall probe.

Qualitatively new, compared to isotropic ferrofluids, is the occurrence of a
director diffusional mode in $G'$ and $G''$
with amplitude $C$ (\ref{amplitude}) and width $\Gamma$ (\ref{width}).
The amplitude $C$ shows that, even without a
magnetic field, the magnetization as an independent variable with its own
dynamics influences the relaxation process of the director.
However this mode has its strongest influence at $\omega \approx \Gamma$,
which is very low, if the wave vector $k$ is small, and it may be hard to
detect experimentally.
Therefore we also study the case with an external field, where
not only the relevant frequencies are shifted to higher values, but where
also another, qualitatively new behavior is seen.
Indeed we find
%
\begin{eqnarray} \label{Gpm}
\frac{G{'}}{\omega}&=&\frac{1}{2 M q_2}\left(\frac{\Omega^2
    \xi_{1}-\Omega M q_2 \xi_2+{q_{1}}^2\xi_{1}}{\left(\Omega-
    q_2 M\right)^2+\left(q_1 -
    M^2 q_3 \right)^2} \right. \nonumber\\
&&\quad\quad\quad - \left. \frac{\Omega^2
    \xi_{1}+\Omega M q_2 \xi_2+ {q_{1}^{2}}\xi_{1}}{\left(\Omega+
    q_2 M\right)^2+\left(q_1 -M^2 q_3 \right)^2}\right)\\
\frac{G{''}}{\omega}&=&\nu_{3} - \frac{1}{2 M q_2}\left(\frac{\Omega^3 \xi_{3}
   -\Omega^2 M \xi_{3} q_2 + \Omega {q_{1}}^2\xi_{3} +M q_2 q_{1}\xi_1
}{\left(\Omega-
    q_2 M\right)^2+\left(q_1 -
    M^2 q_3 \right)^2}
\right.\nonumber\\
&&\quad\quad\quad\quad\quad-\left. \frac{\Omega^3 \xi_{3}
   +\Omega^2 M \xi_{3} q_2 + \Omega{q_{1}}^2\xi_{3} -M q_2 q_{1
}\xi_1}{\left(\Omega+
    q_2 M\right)^2+\left(q_1 -
    M^2 q_3 \right)^2}\right) \label{Gppm}
\end{eqnarray}
%
where $\Omega \equiv \omega /(K_3 k^{2}+\chi_{a}H^{2})$ is the scaled 
frequency and
$M=\chi_{\paras}H$.
The abbreviations $q_{2,3}$
%
\begin{eqnarray} \label{q2}
q_2&=&\frac{1}{\gamma_{1}^R} + \frac{1}{\gamma_{2}^R}+ \frac{2
    \chi^D_{2}\chi^R}{b^{D}_{\perp}}-\frac{(\chi^R)^2
   }{(b^{D}_{\perp})^2}(b_{1}^{R}+ b_{2}^{R}) \\
%
q_3&=& \frac{1}{(b_{\perp}^{D})^{3}}\left(
  \chi_2^D b_{\perp}^{D}-\chi^R(b_{1}^R+b_{2}^{R})\right)^2  \label{q3}
\end{eqnarray}
%
are combinations of the various
static susceptibilities and the transport parameters involved, while $\xi_{2} =
\xi_{1} + q_{1} \xi_{3}$, and $
\xi_3=({c^D})^2 / b^{D}_{\perp}$ with $\xi_{1}$ and $q_{1}$ given in
(\ref{xi1},\ref{q1}), respectively. Since we concentrate on the frequency 
dependence here, we have
suppressed some additional $M^{2}$-dependences of $\xi_{1-3}$ and $q_{2}$,
which would render the results (\ref{Gpm},\ref{Gppm}) extremely involved.

%
\begin{figure}\label{p3}\vspace{1cm}
\centering\resizebox*{8.2cm}{6cm}{\includegraphics{Gimag.eps}}
\centering\resizebox*{8.2cm}{6cm}{\includegraphics{Greal.eps}}
\caption{ \label{rheo1} $G''$ (left) and $G'$ (right) as functions of
  the frequency for
  increasing magnetic field strengths: $H=0$ (solid),
$H=320 G$
   (dashed), and $H=350 G$ (dash-dotted); in our rationalized Gaussian unit
  system (cf. eq.(\ref{energyM1})) $1 G^{2} = (1/4\pi) erg\,cm^{-3}$. The 
values
  of the parameters
  were taken to be $\gamma_0=10^{-1}\,g\,s^{-1}cm^{-1}$, $K_3=10^{-6}\,dyn$, 
  $k=100 cm^{-1}$, $\nu_3=10^{-1}\,g\,s^{-1}cm^{-1}$,
  $\chi_{a}=10^{-4}$, $\chi_{\paras}=10^{-4}$, $\chi^{R}=3\,s^{-1}G^{-1}$,
  $\chi^{D}_{2}=3\,cm\,s\,g^{-1}$,
  $b_{\perp}^{D}=1\,s^{-1}$, $b^{R}_{1}+b^{R}_{2}=40\,G^{-1}s^{-1}$,
  $c^{D}=7\times 10^{-3}\,G$, $(1/\gamma^{R}_{1}) + (1/\gamma^{R}_{2})=
  1\,s\,cm\, g^{-1} G^{-1}$, and $\lambda=1$
(to eliminate ordinary nematic effects).
On this scale the zero field effects (\ref{Gp},\ref{Gpp}) 
can hardly be seen.}
\end{figure}
%

In the presence of an external field the relaxation mode is much more
complicated than without a field. The most important, and experimentally most
easily detectable feature is the shift of the nematic pole to finite
frequencies $\Omega = \pm q_{2}M$ (cf. Fig.4.1). The existence of $q_{2}$ is
characteristic for nematic ferrofluids, since neither the reversible coupling
between director and magnetization rotations ($\chi^{R}$), nor the reversible
counterparts to director diffusion/relaxation ($\gamma_{1,2}^{R}$) and
magnetization relaxation ($b_{1,2}^{R}$) exist in ordinary nematics or
isotropic ferrofluids.  For the true frequency $\omega$ the shift increases
$\sim H^{3}$ for fields larger than $\sqrt{K_{3}k^{2}/\chi_{a}}$. In an
oscillating Couette shear experiment, under the approximation of a linear
velocity profile, this behavior can be measured. There is also 
the possibility that
the nematic orientational mode becomes propagating for some intermediate
fields strengths, if $q_2 M > q_1 - M^2 q_3$.
As in the field-free case the oscillating shear flow induces also
magnetization oscillations, but with a field there are oscillating components
not only out of the shear plane, but also along the flow direction.
\section{The biaxial case \label{biax}}

In this section we deal with the case that in equilibrium the nematic
director $\bm{n}^{0}$ is perpendicular to the external field $\bm{B}$ (and the
magnetization ${\bm M}_{0}$). This case is often realized in lyotropic systems
\cite{berejnov}. In the plane perpendicular to the field the direction of
$\bm{n}$ is
not specified, i.e. the rotational in-plane symmetry is spontaneously broken
and the system is biaxial. Thus, there is one rotation of $\bm{n}$ (the
in-plane rotation) that is hydrodynamic (diffusive), while the out-of-plane
rotation is relaxing (non-hydrodynamic) due to the external field. This
difference to the uniaxial case discussed in the main body of this manuscript
(where both rotations of $\bm{n}$ are relaxing) becomes apparent in the
statics, i.e. in the magnetic energy of the system and hence in the molecular
fields $h_{i}^{M}$ and $h^{n \prime}_{i}$.

The static equations (\ref{energyM1}-\ref{molfn}) still apply, however with
$A_{1}>0$ ensuring $\bm{n}^{0} \perp \bm{M}_{0}$. The linearized expressions
(\ref{molfm},\ref{molf}) for $h^{n \prime}_{i}$ and $h_{i}^{M}$ now only
contain the
out-of-plane components of $\bm{n}$ ($M_{k} \delta n_{k}$), while the in-plane
component appears in the gradient part $\Phi_{ij}$ only.

The form of the dynamic equations is not affected. Only the material tensors
have a more complicated form (containing more coefficients generally) due to
the lower orthorhombic symmetry. We will give the biaxial form of the
field-free material tensors, while those that have to contain an odd number of
factors $\bm{M}$ have been given above (for the uniaxial as well as the
biaxial case to linear order in $\bm{M}$).
In the biaxial case we have to discriminate between the
directions of the director and the magnetization, which we denote by its unit
vector $\bm{m}= \bm{M}/M$. The form of the material tensors of usual biaxial
nematics is given in \cite{BP} and is contained below for convenience. The
symmetric 2nd rank tensors, like
the heat conduction tensor $\kappa_{ij}$, the diffusion tensor $D_{ij}$,  the
thermodiffusion tensor $D^{T}_{ij}$, and magnetic relaxation tensor
$b_{ij}^D$, have 3 independent coefficients
%
\begin{eqnarray}
\kappa_{ij}=\kappa_{\paras} n_i n_j +\kappa_{\perp}\delta_{ij}^{3}+\kappa_3  
m_i m_j
\end{eqnarray}
%
where $\delta_{ij}^{3}=\delta_{ij}-n_i n_j-m_i m_j$.

For the third order tensors
%
\begin{eqnarray}\label{cd_bia}
c^D_{ijk}&=&c_1^D\left(\epsilon_{imk} n_m n_j+\epsilon_{imj} n_m
  n_k\right)
+c_2^D \left(\epsilon_{imk} m_m m_j+\epsilon_{imj} m_m m_k\right)\\
%
\lambda_{ijk}&=&\lambda_1 (\delta_{ij}^{3}n_k+ \delta_{ik}^{3}n_j)
+\lambda_2 (m_i m_j n_k+  m_i m_k n_j)
\end{eqnarray}
%
there is one coefficient more than in the uniaxial case, each. The
orientational diffusion and relaxation of the director is governed by 1
coefficient each (cf. eq.(\ref{R}))
%
\begin{eqnarray}
R^{}=\frac{1}{2} (\frac{1}{\gamma_{0}} \delta^{3}_{ij} +
\frac{1}{\gamma_{3}} m_{i}m_{j}) h^{n}_i h_{j}^n + \ldots
\end{eqnarray}
%
and the viscosity tensor has four additional coefficients
%
\begin{eqnarray}\nonumber
\nu_{ijkl}&=&\nu_1 m_i m_j m_k m_l+\nu_2 n_i n_j n_k n_l+\nu_3
\delta_{ij}^3\delta_{kl}^3
+\nu_{4}\left(m_k m_l n_i n_j+m_i m_j n_k n_l\right)\\ \nonumber
&&+\nu_5\left(m_k m_l \delta_{ij}^3+m_i m_j
  \delta_{kl}^3\right)+\nu_6\left(n_k n_l \delta_{ij}^3+n_i n_j
  \delta_{kl}^3\right) \\ \nonumber
&&+\nu_7\left(m_j m_l n_i n_k+m_j m_k n_i n_l+m_i m_k n_j n_l+m_i m_l n_j
n_k\right)\\ \nonumber
&&+\nu_8\left(m_j m_l \delta_{ik}^3+m_j m_k \delta_{il}^3+m_i m_k
  \delta_{jl}^3+m_i m_l \delta_{jk}^3\right) \\
&&+\nu_{9}\left(n_j n_l
  \delta_{ik}^3+n_j n_k \delta_{il}^3+n_i n_k
  \delta_{jl}^3+n_i n_l \delta_{jk}^3\right)
\end{eqnarray}
%

The reversible dynamic coupling between director and magnetization is still
given by only one coefficient $\chi^{R}$ (\ref{Y},~\ref{X}), but now contains
only the out-of-plane components $Y_{i}^{R} m_{i}$ or $h_{i}^{n} m_{i}$ and
$X_{i}^{R} \delta_{ij}^{3}$ or $h_{i}^{M} \delta_{ij}^{3}$.

In the biaxial case some new additional terms cubic in $\bm{M}$ can be
produced in (\ref{lambdaprime1},~ \ref{viscosH1},\\ \ref{cR},~ \ref{chiD}) by replacing any pair $n_{i}n_{j}$ with
$M_{i}M_{j}$, e.g. $\delta_{ij}^{tr}$ by $\delta_{ij}^{3}= \delta_{ij}-
n_{i}n_{j} - M_{i}M_{j}$ and $M_{i}M_{j}$ terms.

\section{Summary \label{conclu}}


In this chapter we derive the complete set of macroscopic dynamic equations for
ferronematics introducing the magnetization as an independent slowly
relaxing variable. We show that orientational changes of the magnetization are
coupled to
nematic director reorientations not only in the statics, but in the dynamics
as well. In addition, there are reversible and dissipative dynamic
cross-couplings between (compressional, shear and elongational) flow,
(rotations and changes of the absolute value of the) magnetization and
director reorientations. Some of these couplings are only possible, when a
finite magnetization is present due to spontaneous magnetic order and/or due
to an external magnetic field. In order to measure some combinations of the
parameters that describe these cross-couplings we study the sound wave spectrum
and the rheology of shear flow. There is an additional field dependent
contribution to sound damping due to such a cross-coupling. For the case of a
sound wave propagating in a direction
oblique to the preferred directions (set by
equilibrium magnetization, nematic
director, or perpendicular to them) compressional flow (and changes of the
absolute value of the magnetization) are coupled to shear flow (and rotations
of the director and the magnetization).
In addition we discussed the linear
response of the system to oscillatory shear flow concentrating on frequencies
below the transverse magnetization relaxation frequency. This allows
to describe the influence of the magnetic dynamic degree of freedom on the
nematic director diffusion/relaxation mode for ferronematic systems and the
influence of the magnetic field on it. Even without a magnetic field the
apparent viscosity is different from the bare one and the modified nematic 
director
diffusion couples to the flow response. In the presence of an external field
the director diffusion/relaxation is shifted to a finite frequency, which
approximately increases with the third
power of the field strength.
Even in the field-free case there are shear-flow-induced oscillations of
the magnetization out of the shear plane.
\\[0.5cm]



\chapter{Macroscopic dynamics of ferrogels \label{4}}
%\chaptermark{ferrogel}
\section{Introduction}

In this chapter we generalize the set of hydrodynamic equations for
ordinary gels to those for ferrogels by including the
magnetization as an additional, slowly relaxing variable. 
The relaxation of the magnetization is rather slow compared to the
(many) microscopic relaxation processes and it is therefore reasonable to keep
the magnetization as a macroscopic, slowly relaxing variable.

We assume in our model that the magnetic particles are "attached"
to the network, although the precise meaning of that is unclear on the
molecular level. On the macroscopic level this leads to a coupling of
rotations of the magnetization as well as
changes of its absolute value with the elastic strains or stresses.     
As an application for these equations we discuss the spectrum of 
longitudinal and transverse sound in the presence of an external 
magnetic field. In the low frequency limit a comparison with static elastic 
measurements is made. A possible way of exciting shear waves by 
oscillating temperature gradients in the presence of a 
gradient field is outlined.

\section{Statics and Thermodynamics \label{statics1}}



The macroscopic description of a system starts with the identification of the
relevant variables. Apart from the quantities that are related to local
conservation laws, like mass density ($\rho$), momentum density
($\bm  g$), energy density ($\varepsilon$) and concentration ($ c$) of the
swelling fluid (and/or that of the magnetic particles), we consider the
elastic strain $u_{ij}$ and the magnetization 
$\bm{M}$ as additional variables. In a crystal the former is related to the broken
translational symmetry due the long range positional order, which gives rise
to the displacement vector $\bm{u}$ as a hydrodynamic symmetry variable. 
Since neither solid
body translations nor rigid rotations give rise to elastic deformations, the
strain tensor is used as variable, which reads in linearized version $u_{i
  j}=\tfrac{1}{2}\left(\nabla_i u_j+\nabla_j u_i\right)$ in. In amorphous
solids, like rubbers, gels etc., linear elasticity is still described by a second
rank, symmetric strain tensor. For a proper description of nonlinear
elasticity cf. \cite{tem}. For the purpose of this work, however, linear
elasticity is sufficient. As discussed in the Introduction the magnetization
$\bm{M}$ is a slowly relaxing variable in the superparamagnetic case.

Assuming local thermodynamic equilibrium, i.e. all microscopic, fast
relaxing quantities are already in equilibrium, we have the Gibbs relation 
%
\begin{eqnarray} \label{energy1}
d\varepsilon= T d \sigma +\mu d \rho +\mu_{c} d c+ v_i d g_i +
 H_i d B_i+h_i^{M} d M_i+\Psi_{i j} d u_{ij}.
\end{eqnarray}
%
relating all macroscopically relevant variables discussed above to the entropy
density $\sigma$. $\bm{B}$ is the magnetic induction field included here in
order to accommodate the static Maxwell equations.
In eq.(\ref{energy1}) the
thermodynamic quantities, chemical potential ($\mu$), temperature ($T$),
relative chemical potential ($\mu_{c}$), velocity ($v_i$), elastic stress ($\Psi_{ij}$),
magnetic field ($H_i$), and the magnetic molecular field ($h_{i}^{M}$),
are defined as partial derivatives of the energy density with
the respect to the appropriate variables \cite{PB1}.

To determine these thermodynamic forces and thus the static
properties of magnetic elastomers one provides an expression of the energy
density in terms of the variables
% 
\begin{eqnarray}\label{efunc}
\varepsilon&=& \varepsilon_{0} + \frac{B^2}{2}-\bm{B}{M}+\frac{\mu_{i j k l}}{2}
u_{ij}u_{kl}-\frac{\gamma_{i j k l}}{2}M_{i} M_{j}
u_{kl}+\frac{\alpha}{2}M_{i}^2+\frac{\beta}{4}\left(M_i^2\right)^2 \nonumber \\ 
&& + u_{ii}\left(\chi^\rho \delta
  \rho+\chi^\sigma\delta\sigma+\chi^c\delta c\right)
\end{eqnarray}
%
where $\varepsilon_{0}$ is the energy density of a fluid binary mixture.
eq.(\ref{efunc}) explicitly contains the elastic and the magnetic energy,  
their crosscoupling (the magneto-elastic 
energy) and bilinear couplings of compression with the scalar variables. 
To discuss large elastic deformations (rubber elasticity) one
should keep terms of higher order of $u_{ij}$, which are neglected here.
The magneto-elastic coupling is cubic \cite{LL} and the $\bm{M}^{4}$ contribution
is kept in order to guarantee the thermodynamic stability. 
The tensors $\mu_{ijkl}$ and $\gamma_{ijkl}$ take the
isotropic form $a_{ijkl}=a_1
\delta_{ij}\delta_{kl} + a_2\left(\delta_{ik}\delta_{jl}+\delta_{il}
\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl}\right)$,
where $\mu_{1}$ is the compressibility and  $\mu_2$ the shear
modulus. The magneto-elastic energy is similar to that for ferromagnetic
materials, where however the compressional magnetostriction is 
neglected ($\gamma_1=0$) \cite{LL}. We will not take this approximation for magnetic
gels and keep $\gamma_{1} \neq 0$, in order to explore effects due to a
non-zero $\gamma_{1}$ and how they can be measured. While $\gamma_{1}$
describes the elastic response to changing the field strength (or vice versa
compression or dilation due to changes in $|\bm{M}|$), $\gamma_{2}$ is
related to elastic shear and to rotations of $\bm{M}$. Depending on how the
"attachment" of the magnetic clusters to the network actually is realized in a
sample, this interaction may be large or small. Thus measuring the $\gamma$'s
by their effects described below, may give some hints on the microscopic structures. 
All static susceptibilities, like the elastic and magneto-elastic moduli as
well as those describing crosscouplings between compression and
the density, entropy density, and concentrations variations ($\chi^\rho$, $\chi^\sigma$ and 
$\chi^c$, respectively) can depend on $\bm{M}^{2}$ and thus on magnetic field strength.. 

Using eqs.(\ref{energy1},\ref{efunc}), the magnetic Maxwell field $H_i$ is defined in
the usual way
%
\begin{eqnarray} \label{H1}
H_i=\left(\frac{\partial \varepsilon}{\partial
    B_i}\right)_{\bm{M},u_{ij},\dots}=B_i-M_i,
\end{eqnarray}
%
while the magnetic molecular field $h^M_i$ reads
%
\begin{eqnarray}
\label{molfm1}
h_{i}^M =\left(\frac{\partial \varepsilon}{\partial
    M_i}\right)_{\bm{B},u_{ij},\dots}=-B_i-\gamma_{ijkl} M_j u_{kl} +
\alpha M_i+\beta M^2 M_i 
\end{eqnarray}
%
Note that because of the definition (\ref{H1}), it is not possible to have a
direct coupling between the
external field $\bm{B}$ and the strain; the deformation of the
network is mediated by the magnetization via the coupling terms $\sim \gamma_{ijkl}$.


The elastic stress $\Psi_{ij}$ has the following form
%
\begin{equation}
  \label{molf1}
\Psi_{ij} =\left(\frac{\partial \varepsilon}{\partial
     u_{ij}}\right)_{\bm{M},\bm{B},\dots}=
 \mu_{ijkl}u_{kl}-\frac{\gamma_{ijkl}}{2}M_k M_l+\delta_{ij}\left(\chi^\rho \delta
  \rho+\chi^\sigma\delta\sigma+\chi^c\delta c\right)
\end{equation}
%
and depends on the magnetization.\\[1cm]


\section{Equilibrium \label{equi}}


In equilibrium, both the elastic strain eq.(\ref{molf1})
and the magnetic molecular field eq.(\ref{molfm1}) have to be zero. Without an
external field or external strain there is no magnetization and no strain in
equilibrium. A finite external field, taken along the $z-$ axis, 
$\bm{B}= B_{0}\bm{e}_{z}$, induces an equilibrium 
magnetization ($\bm{M}^0 = M_0 \bm{e}_z$) and an
non-zero strain ($u_{ij}^0$) due to the magnetostriction effect. 
Neglecting the couplings of density\footnote{A non-zero $\chi^{\rho}$ 
merely renormalizes $\gamma_1$ into $\gamma_1(1-3\kappa_s(\rho \chi^{\rho})^2)$, 
where $\kappa_s$ is the compressibility.}, entropy and concentration to the strain
tensor we have
%
\begin{eqnarray} \label{psis1}
\Psi_{ij} &=&\left(
  \left(\mu_1-\frac{2}{3}\mu_2\right)u_{kk}^0-\left(\frac{\gamma_1}{2}
-\frac{\gamma_2}{3}\right)M_{0}^2\right)\delta_{ij}-\gamma_2 M_i^0 M_j^0
+2\mu_2 u_{ij}^0 =0 \\  
h_{i}^M &=& -B_0 \delta_{iz} -\left(\gamma_1-\frac{2}{3}\gamma_2\right)M_i^0 
u_{k k}^0 -2 \gamma_2 M_j^0 u_{ij}^0+\alpha M_i^0 +\beta M_0^2 M_i^0 =0 \label{hms1}
\end{eqnarray}
%
From $h_{x}^M=0$ , $h_{y}^M=0$ and $\Psi_{xy}=0$ it follows
that $u_{xz}^0=0$, $u_{yz}^0=0$, and $u^0_{xy}=0$ accordingly. The remaining
conditions give
\begin{eqnarray}
u_{xx}^0&=&u_{yy}^0=\frac{\mu_2\gamma_1-\mu_1\gamma_2}{6 \mu_1\mu_2}M_0^2\\
u_{zz}^0&=&\frac{\mu_2\gamma_1+2\mu_1\gamma_2}{6\mu_1 \mu_2}M_0^2
\end{eqnarray}
%
leading to the volume change $U^0 \equiv u^0_{xx}+u^0_{yy}+u^0_{zz}=\frac{\gamma_1}{2\mu_1}M_0^2$. The magnetostrictive
volume change of the ferrogel is determined by the bulk modulus
$\mu_1$ and by the coefficient $\gamma_1$, which couples the trace of the stress tensor
to the magnitude of the magnetization. 

From $h_z^M =0$ we get implicitly
the equilibrium magnetization as a function of the field  
$B_{0}=M_{0}\left(\alpha+\beta M_0^2-\left(3 
    \gamma_1^2\mu_2+4\gamma_2^2 \mu_1\right) M_0^2/\left(6 \mu_1 \mu_2\right)\right)$. 
Writing this relation in the form $\chi B_{0}=(1+\chi) M_{0}$, a
field-dependent magnetic susceptibility $\chi$ results with
%
\begin{equation}
  \label{chiM}
\frac{1+\chi}{\chi}=\alpha+\left(\beta - \frac{3\gamma_1^2\mu_2+4
\gamma^2_2 \mu_1 }{6\mu_1 \mu_2} \right) \left(\frac{\chi B_{0}}{1 + \chi}\right)^{2}. 
\end{equation}
%
The explicit form of eq.(\ref{chiM}) follows from the 
(truncated) expansion (\ref{efunc}) and is suitable for small external fields only.
For high field intensities, when the magnetization reaches its saturation
value, $\chi$ in eq.(\ref{chiM}) has to be replaced by a more complicated
function $\chi_{0}=\chi(B_{0})$, either measured  \cite{zr} or calculated from
reliable microscopic 
models. For the small deviations from equilibrium, which we are dealing with in
the following, the simple form $\chi B_0 = M_0$ is sufficient for
any field strength, when for $\chi$ the appropriate equilibrium value
$\chi_{0}$ is taken. \\[1cm]




\section{Dynamics \label{dynamics2}}

The hydrodynamic equations for conserved and slowly
relaxing variables as well as for those associated with 
spontaneously broken continuous symmetries are
%
\begin{eqnarray}\label{he1}
\frac{\partial}{\partial t}\rho
+   div\,\rho \bm{v}&=&0
\\ \label{dv2}
\frac{\partial}{\partial t}\sigma+ div\, \sigma \bm{v} + div\bm{j^{\sigma}}&=&
\frac{R}{T}
\\ \label{ds1}
\frac{\partial}{\partial t}g_{i} +\nabla_{j}\Bigl( v_{j}g_{i}+ \delta_{ij}[p_0 
+\bm{B \cdot H}]+
\sigma_{ij}^{0} 
+ \sigma_{ij}\Bigr)&=&0
\\ \label{dn2}
{\Bigl(\frac{\partial}{\partial
      t}+v_{k}\nabla_{k}\Bigr)}u_{ij}+Y_{i j} &=&0
\\ \label{dc2}
\rho{\Bigl(\frac{\partial}{\partial t}+v_{j}
\nabla_{j}\Bigr)}c + div\,\bm{j}^{c}&=&0
\\ \label{dm1}
{\Bigl(\frac{\partial}{\partial
      t}+v_{j}\nabla_{j}\Bigr)}M_{i}+\left(\bm{M} \times
\bm{\omega}\right)_i+X_{i}&=&0
\end{eqnarray}
%
where $\omega_i=\frac{1}{2}\epsilon_{i j k}\nabla_j
v_k$ is the vorticity
and
%
\begin{equation}\label{astress1}
\sigma_{ij}^{0}= - B_{j}H_{i} - \frac{1}{2}(M_{j} h_{i}^{M} - M_{i}
h_{j}^{M})+\Psi_{j k}u_{k i}.
\end{equation}
%  
Using the fact that the energy density eq.(\ref{energy1}) has to be
invariant under constant rotation \cite{PB1} eq.(\ref{astress1}) can be simplified as 
\begin{equation}\label{astress2}
\sigma_{ij}^{0}= -\frac{1}{2}\left(B_i H_j+B_j
  H_i\right)+\frac{1}{2}\left(\Psi_{j k}u_{k i}+\Psi_{i k}u_{k j}\right) 
\end{equation}
%
The last term in eq.(\ref{astress2}) is nonlinear, but since there 
is a finite strain in an external field, it will enter linear deviations 
from that constrained equilibrium. 
The thermodynamic pressure $p_0$ is given by
%
\begin{equation}\label{pressure1}
p_0=-\varepsilon + T \sigma +\mu \rho+ \bm{g \cdot v}.
\end{equation}
%
$j^\sigma$ is the entropy current, in eqs.(\ref{dn2})-(\ref{dm}) $Y_{i j} $ 
and $X_i$ are the
quasi-currents of the variables associated with broken translational symmetry 
(network) and slowly relaxing magnetization. To guarantee rotational  
invariance of the dynamical equation for the strain field, one must
require $Y_{i j}=Y_{j i}$. The source term $R/T$ in the dynamic
equation for the entropy density is the entropy production. The second 
law of thermodynamics reads $R\ge 0$ for dissipative and reversible
process respectively. 

Since we are not dealing with electromagnetic effects, we can use the static
Maxwell equations to determine $\bm{B}$
%
\begin{eqnarray}\label{MAX1}
curl \bm{H} = curl \left(\bm{B}-\bm{M}\right)=0, \qquad div\bm{B}=0.
\end{eqnarray}
%

All the currents can be split into dissipative and into reversible
($R=0$ ) and irreversible ($R\ge 0$) parts.
Using general symmetry and invariance arguments and
the fact, that a magnetic field changes sign  under time
reversal,  we obtain the linear currents
%
\begin{eqnarray}
\label{Q1}
j_{i}^{\sigma R}&=&- \kappa^R \varepsilon_{i j k}M_k \nabla_{j}T -D^{T
  R}\varepsilon_{ij k}M_k
\nabla_{j} \mu_{c}+\xi^{TR}\varepsilon_{ijk}M_k\Psi_{j} \\
%
\label{j1}
j_{i}^{c R}&=&-D^R\varepsilon_{ij k}M_k \nabla_{j} \mu_{c} + D^{T
  R}\varepsilon_{ij k}M_k\nabla_{j}T
+\xi^{cR}\varepsilon_{ijk}M_k\Psi_{j} \\ 
%
\label{sigma1}
\sigma_{ij}^{R}&=& -\Psi_{i j}-c_{kij}^{R}(M)h_{k}^{M}
-\nu_{ijkl}^{R}(M)A_{kl}\\
%
\label{Y1}
Y_{i j}^R&=& -A_{i j}+\frac{1}{2}\lambda^M
\left(\nabla_i\left(\nabla\times h^M\right)_j+\nabla_j\left(\nabla\times 
    h^M\right)_i\right)-\\ \nonumber
&-& \frac{1}{2}\left(\nabla_i
  \left(\xi^{R}\varepsilon_{jkl}M_l \Psi_k+\xi^{TR}\varepsilon_{jkl}M_l \nabla_kT+\xi^{cR}\varepsilon_{jkl}M_l \nabla_k
    \mu_c
  \right)+
\left(i\longleftrightarrow j\right)\right)\\
%
\label{X1}
X_i^R&=&b^R\varepsilon_{ijk} M_k h_j^M +\lambda^M \left(\nabla\times\Psi\right)_i-
c_{ijk}^R(M) A_{jk}
\end{eqnarray}
%
with $A_{ij}=\frac{1}{2}\left(\nabla_i v_j+\nabla_j v_i\right)$ and
$\Psi_i=\nabla_j\Psi_{i j}=\nabla_j \Psi_{ji}$. Again, nonlinear elastic
contributions have been neglected. Due to the new degree 
of freedom (magnetization) there is an  additional term  in the stress 
quasi-current eq.(\ref{Y1}) and a counter term in $X_{i}^{R}$, which describes
a dynamic cross-coupling 
between magnetization and the network.
It does not exist in ordinary elastomers nor in isotropic
ferrofluids. Its impact on the sound spectrum will be explored in Sec.\ref{sound}. 
The new coefficient $\lambda^M$ (reversible dynamic
coupling between the magnetization and the strain tensor) gives a small
effect in the dynamics of order $\sim k^4$. The reversible second rank magnetization-dependent material tensors, like the
reversible analogue of heat conduction $\kappa^{R}\varepsilon_{ijk}M_k$,  diffusion
$\sim D^{R}$, thermodiffusion $\sim D^{T R}$, magnetization
relaxation $\sim b^{R}$, and the reversible coupling terms between temperature,
concentration and elasticity $\sim\xi^{T R}$, $\sim \xi^{
  c R}$, $\sim\xi^{R}$  are all of the same form.
The magnetization-dependent
tensors  
\begin{eqnarray}
\nu^{R}_{i j
  k l}(M) &=& \nu^{R}_{7}[\epsilon_{ikp}\delta_{jl}
+\epsilon_{ilp}\delta_{jk}+
\epsilon_{jlp}\delta_{ik}+
\epsilon_{jkp}\delta_{il}]\,M_{p} \\
c_{i j k}^{R}(M)&=&c_{2}^R(\delta_{ij}M_{k}+\delta_{ik}M_{j})
\nonumber + c_{3}^R
M_i\delta_{jk} 
\end{eqnarray}
are  odd
functions of the magnetization. 
  
To derive the dissipative contributions to the currents it is most
convenient to start with the expression for dissipation function
$R$. The dissipative currents are then obtained by taking variational
derivatives with respect to one thermodynamic conjugate while keeping
all others fixed.  Expanding the dissipation function $R$ up
to second order in the thermodynamic forces we obtain
%
\begin{eqnarray} \label{R1}
R&=&\frac{1}{2} \kappa\left(\nabla_{i} T\right)^2+\frac{1}{2}
\nu^D_{ijkl}A_{ij} A_{kl} \nonumber
+ \frac{1}{2}D\left(
  \nabla_i \mu_c \right)^2+ \frac{b}{2}\left(h_i^M\right)^2  
\\ &+&\frac{1}{2} \xi \left(
  \Psi_i\right)^2+ D^{T}\left(\nabla_j T\right)\left(\nabla_j \mu_ 
 c\right)+ \Psi_i \left(\xi^T\nabla_i T+\xi^c
  \nabla_i \mu_c\right)
\end{eqnarray}
%
Here $\nu^D_{ijkl}$ is the
viscosity tensor (and has the same form as the elastic tensor
$\mu_{ijkl}$ defined above)  and $\kappa$, $D$ and $D^T$
describe heat conduction, diffusion and thermodiffusion,
respectively. The quantity $b$ is the inverse magnetization relaxation and
$\xi$ the self diffusion constant of the strain field.
The range of possible values of the coefficients in eq.(\ref{R1}) is
restricted by the positivity of the entropy production. 

We derive the dissipative parts of the currents by taking the variational 
derivative of the dissipation function with respect to the
appropriate thermodynamic force
%
\begin{eqnarray} \label{sD1}
j_{i}^{\sigma D}&=&- \kappa\nabla_{i}T -D^T \nabla_{i}
\mu_{c}-\frac{1}{2}\xi^T\Psi_i
\\ \label{cdiff2}
j_{i}^{c D}&=& -D \nabla_{i} \mu_{c} -
D^T\nabla_{i}T-\frac{1}{2}\xi^c\Psi_i \\
%
\label{sigmaD1}
\sigma_{ij}^D&=&-\nu^D_{ijkl} A_{kl}\\
%
\label{YD1}
Y_{ij}^D&=&-\frac{1}{2}\left(\nabla_i
  \left(\xi \Psi_j+\xi^T \nabla_j T+\xi^c \nabla_j
    \mu_c
  \right)+
\left(i\longleftrightarrow j\right)\right) \\
%
\label{XD1}
X_i^D&=&b h_i^M
\end{eqnarray}


\section{Experiments \label{exp}}
%\addcontentsline{toc}{section}{Experiments}
\subsection{Static elongation and shear \label{ses}}

In preparation for, and for comparison with, the sound spectrum we first 
discuss static elongational and shear deformations. 
We assume an external field (along the $z$ axis) that 
gives a non-zero magnetization as well as a deformation in 
equilibrium. This state is then disturbed by an external 
deformation $\Delta u_{ij}$ by some mechanical device. 
Due to the magnetostriction effect this gives also rise  
to a change in the magnetization. In the static limit 
the magnetic degree of freedom is still in equilibrium and 
the change of the magnetization can be obtained from the 
condition $h_i^M =0$, eq.(\ref{molfm1}). The applied 
deformation gives, directly by Hooke's law, and indirectly 
by the change of the magnetization, an elastic stress. From eq.(\ref{molf1}) we get 
%
\begin{eqnarray} \label{seszz}
\Psi_{zz} &=& ( \mu'' - \chi_0 \gamma''^{\,2} M_0^2) \Delta u_{zz} 
 + (\mu' - \chi_0 \gamma' \gamma'' M_0^2) (\Delta u_{xx} + \Delta u_{yy}) \\ \label{sesxx}
\Psi_{xx} &=& ( \mu'' - \chi_0 \gamma'^2 M_0^2) \Delta u_{xx} + ( \mu' - \chi_0 \gamma'^2 M_0^2)\Delta u_{yy} + 
(\mu' - \chi_0 \gamma' \gamma'' M_0^2) \Delta u_{zz} 
\\ \label{sesxz}
\Psi_{zx} &=& 2( \mu_2 - \chi_0 \gamma_2^2 M_0^2) \Delta u_{zx} \\ \label{sesxy}
\Psi_{xy} &=&  2\mu_2 \Delta u_{xy} 
\end{eqnarray}
%
for the elastic stresses.  Apart from the elastic 
moduli ($\mu'' = \mu_1 + (4/3)\mu_2$ and 
$\mu' = \mu_1 - (2/3) \mu_2$), 
it contains $M_0^2$ corrections due to magnetostriction 
($\gamma' = \gamma_1 -(2/3)\gamma_2$ and 
$\gamma''= \gamma_1 + (4/3) \gamma_2$), except 
for deformations that do not affect the magnetization. 
Due to the compressibility $\kappa_s$, 
defined as $\kappa_s = \rho^{2}( \partial^2 
\varepsilon/ \partial \rho^2)^{-1}$,  
the coefficient $\mu''$ has to be replaced 
by $\mu'=1/(\rho \kappa_s) +\mu''$ for the effective response
to various external deformations in the stress tensor. 
Note that, even if the deformation does conserve 
the volume ($\Delta u_{xx} + \Delta u_{yy} + 
\Delta u_{zz} =0$), the trace of the elastic stress tensor 
is not zero, but given by  $\Psi_{kk} = - 6\chi_0 
\gamma_1 \gamma_2 M_0^2 \Delta u_{zz}$. The formulas 
(\ref{seszz}-\ref{sesxy}) are applicable for 
small strains only, since it is based on Hooke's law, 
while for larger strains deviations  
from this law due to rubber elasticity are to be expected.\\[0.5cm]




\subsection{Propagation of sound \label{sound}}


Due to the presence of the permanent polymer network in ferrogels 
compared to ferrofluids, there are transverse as
well as longitudinal sound eigenmodes. In this section we derive the
longitudinal and the transverse sound of the system with an external
magnetic field parallel to the $z-$ axis. We neglect all diffusional
processes  connected e.g. with viscosity and
heat conduction as well as their reversible counterparts. Terms with
$\lambda^{M}$- and $b^{R}$- coefficients are omitted here as well. The
first one does not contribute to the sound velocity, because it is of fourth
order in the wave vector $\sim k$. Terms with $b^R$ shift the magnetization relaxation
time by $\sim b^R M^2$, and thus give higher order corrections $\sim M^4$  to the
sound spectrum, in which we are not interested here. Only the
relaxation the magnetization in the field is kept.  

Assuming a one-dimensional plane wave with space-time dependence $\sim \exp
i(-\omega t + \bm{k}\cdot \bm{r})$ for all deviations $\delta u_{ij}$, 
$\delta M_i$, $v_i$, $\delta \rho$ from the equilibrium 
values determined in Sec.\ref{equi} 
the linearized set of dynamic equations  becomes an algebraic one.
Let us consider sound in the two cases,  where the external magnetic field and
the equilibrium magnetization are either perpendicular or parallel to the wave
vector. Field fluctuations $\delta B_{i}$ are fixed by the static
Maxwell equations (\ref{MAX1}) to $\delta B_{i}= \delta M_{j}(\delta_{ij}-
k_{i}k_{j} k^{-2})$.

In  the case where an external field is perpendicular to the wave
vector the sound dispersion relations for longitudinal and for
transverse modes, respectively, read, up to order $O(M^2)$  from here on all old $\mu'$ are called $\tilde \mu$
%
\begin{eqnarray}\label{s_p_l}
 \omega^2_l&=&\frac{k^2}{\rho}\left[\mu'-{M_0}^2
\left(\mu'\frac{\mu_2\gamma_1-\mu_1\gamma_2}{6\mu_2\mu_1} +\chi_{0}\left(\gamma'\right)^2
 - \frac{\imath \omega\left(\chi_{0}\gamma'-c_3^R\right)^2}
{\imath \omega \chi_{0}- b}\right) \right]\\    
\label{s_p_t1}
\omega^2_{t1}&=&\frac{k^2}{\rho}
\left[\mu_2 -{M_0}^2 \frac{\mu_2\gamma_1-\mu_1\gamma_2}{6\mu_1}\right]
\\
\label{s_p_t2}
\omega^2_{t2}&=&\frac{k^2}{\rho}\left[\mu_2-{M_0}^2
\left(\frac{2\mu_2\gamma_1+\mu_1\gamma_2}{12\mu_1}+ \chi_{0} \gamma_2^2
+\frac{\gamma_2 }{2} -  \frac{\imath
      \omega\left(\chi_{0} \gamma_2-  c_2^R+\tfrac{1}{2}\right)^2}
{\imath \omega \chi_{0}- b}\right)\right],
\end{eqnarray}
%
where $\mu'=1/(\rho \kappa_s) +\mu_1+\frac{4}{3}\mu_2$ 
and $\gamma'=\gamma_1-\frac{2}{3}\gamma_2$. Generally, 
$\chi_0(B_0)$ brings in an additional dependence on 
$M_0$, which however can be neglected in $O(M^2)$. 


If an external field is parallel to the wave vector we have the following
longitudinal and transverse dispersion relations
%
\begin{eqnarray}\label{s_pa_l}
 \omega^2_l&=&\frac{k^2}{\rho}\left[\mu'-{M_0}^2
\left(\mu'\frac{\mu_2\gamma_1+2\mu_1\gamma_2}
{6\mu_2\mu_1}+\chi_0\left(\gamma''\right)^2 - \frac{\imath \omega\left(\chi_0\gamma''-\left(2 c_2^R+c_3^R\right) 
\right)^2}{\imath \omega\chi_0- b}\right)\right]\\  
\label{s_pa_t2}
\omega^2_t&=&\frac{k^2}{\rho}\left[\mu_2-{M_0}^2
\left(\frac{2\mu_2\gamma_1+\mu_1\gamma_2}{12\mu_1}+\chi_0\gamma_2^2
-\frac{\gamma_2}{2}  -  \frac{\imath
      \omega\left(\chi_0\gamma_2- c_2^R-\tfrac{1}{2}\right)^2}
{\imath \omega\chi_0- b}\right)\right],
\end{eqnarray}
%
where $\gamma''=\gamma_1+\frac{4}{3}\gamma_2$. As the problem is
symmetric in the plane perpendicular to the magnetic field 
the two transverse sounds have identical dispersion relations.

Knowing the dispersion relations for the two geometries one can
study sound velocities. The relaxation time of the magnetization
$\tau_M=\chi_0/b$ is known for ferrofluids and is typically of
order $10^{-4}$. Our approach allows to investigate two frequency
regimes: for $\omega<1/\tau_M$ the low frequency limit and 
for $\omega>1/\tau_M$ the high frequency regime.  
There is no frequency dependence for the transverse sound mode, 
where the magnetic field, the wave vector and the velocity 
are mutually perpendicular, eq.(\ref{s_p_t1}). 
 
Let us first investigate low frequencies  ($\omega<1/\tau_M$). 
The frequency-dependent parts in the square brackets of
the dispersion relations (\ref{s_p_l}-\ref{s_pa_t2}) 
become dissipative and do not contribute to the sound velocities. In  the case of the
external field being perpendicular to the wave vector  the velocities of
the longitudinal $c_l $ and the transverse sounds
$c_{t1}$,  $c_{t2}$ read 
%
\begin{eqnarray} \label{sdisp_p}
c_l^2&=&\frac{\tilde \mu }{\rho}-\left(\tilde \mu \frac{\mu_2\gamma_1-\mu_1\gamma_2}{6\rho 
    \mu_2\mu_1}+\frac{\chi_0\left(\gamma'\right)^2}{\rho}\right)M_0^2
\\ 
\label{tr1}
c_{t1}^2&=&\frac{\mu_2}{\rho}-\left(\frac{\mu_2\gamma_1-\mu_1\gamma_2}{6\rho\mu_1}\right)M_0^2
\\ 
\label{tr2}
c_{t2}^2&=&\frac{\mu_2}{\rho}-\left(\frac{2\mu_2\gamma_1+\mu_1\gamma_2}
{12\rho\mu_1}+\frac{2\chi_0\gamma_2^2+\gamma_2}{2\rho}\right)M_0^2
 \end{eqnarray}
%
while for a parallel field we have
%
\begin{eqnarray} \label{sdisp1}
c_l^2&=&\frac{\mu'}{\rho}-\left(\mu'\frac{\mu_2\gamma_1+2\mu_1\gamma_2}
{6\rho\mu_1\mu_2}+\frac{\chi_0\left(\gamma''\right)^2}{\rho}\right)M_0^2
\\ 
\label{sd}
c_t^2&=& \frac{\mu_2}{\rho}-\left(
\frac{2\mu_2\gamma_1+\mu_1\gamma_2}{12\rho\mu_1}+\frac{2\chi_0\gamma_2^2
-\gamma_2}{2\rho}\right)M_0^2
 \end{eqnarray}
%
The sound speeds at low frequencies and zero field give information about the
compressibility and the elastic moduli (bulk and shear).  
The dependence on $M_0^2$ is due to magnetostrictive effect 
and completely absent for $\gamma_1 = 0 = \gamma_2$. 
Magnetostriction enters the sound speed in different ways.
There is a direct static coupling of magnetization and 
stress in eq.(\ref{molf1}) and the strain dependence of the 
magnetic molecular field (\ref{molfm1}), which give rise to 
terms linear and quadratic in $\gamma_1$ 
and $\gamma_2$. 
A similar term emerges indirectly via the Maxwell stress.
The second route is dynamic, given by the nonlinear elastic 
stress contribution to the stress tensor in 
eq.(\ref{astress2}), which however is effectively linear due 
to the non-zero equilibrium strains. In the sound speeds these 
contributions are of the bilinear $\gamma \mu$ type. In an 
external field the effective moduli measured by sound propagation 
are therefore different from those measured by static elongational 
or shear deformation measurements discussed in the preceding 
subsection. The coincidence of static elasticity and low 
frequency sound speed is restored in the limit of vanishing 
magnetic field only, when in both cases deviations from the 
{\em true} thermodynamic equilibrium are considered.


The sound velocities change with an external magnetic 
field basically with the second power of the field, which is in 
accordance with experiments on longitudinal sound \cite{zr}. 
There is, however, an additional field dependence through $\chi_0$. 
Whether the sound velocities are decreased or increased by 
the field cannot be established by general rules, since the 
signs of $\gamma_{1,2}$ are not fixed and can be material 
dependent.  Measurements of transverse and longitudinal sound 
velocities in the different geometries will provide informations on 
the magnitude and 
sign of the magnetostrictive and elastic moduli. As a first 
approximation the magnetostrictive volume change ($\sim  
\gamma_1 M_0^2 / \mu_1$) can be neglected in those rubbers  and only shape changes remain.

Damping of sound waves generally is rather weak and given by 
the imaginary part of the dispersion relation.
In addition to the usual magnetic-field independent sound damping
due to viscosity and other diffusional processes there is a 
field-dependent sound damping in ferrogels. This is an 
effect of the reversible, dynamic coupling of the magnetization 
to flow, either phenomenological ($c_{ijk}^R(M)$ 
in (\ref{X1})) or kinematic ($\epsilon_{ijk}M_j \omega_k$ in 
(\ref{dm1})) and its counterparts in the Navier-Stokes equation. For
example, when the magnetic field is parallel (perpendicular) to the wave vector a 
field-dependent damping of longitudinal (transverse) 
sound eq.(\ref{s_pa_l}) (eq.(\ref{s_p_t2})) reads, respectively
\begin{eqnarray} \label{Iml}
 \mathfrak{Im}(\omega_l)&=&-\frac{1}{2}\frac{\left(\chi_0\gamma''-\left(2
      c_2^R+c_3^R\right)\right)^2}{\rho b} M_0^2 k^2 \\ \label{Imt2}
 \mathfrak{Im}(\omega_{t2})&=&-
\frac{1}{2}\frac{\left(\chi_{0} \gamma_2-  c_2^R+\tfrac{1}{2}\right)^2}{\rho b} M_0^2 k^2
\end{eqnarray}
the first of which can be related to 
the observed increase of the apparent viscosity 
due to the magnetic field \cite{nikitin}. In all cases 
$\mathfrak{Im}<0$, as it should be according 
to the second law of thermodynamics.

Let us now investigate the high frequency limit for $\omega>1/\tau_M$. In  the
case where an external field is perpendicular to the wave 
vector the velocities of
the longitudinal $c_l $ and the transverse sounds
$c_{t2}$ read ($c_{t1}$ is as in the low frequency limit)
%
\begin{eqnarray} \label{sdisp_p1}
c_l^2&=&\frac{\mu'}{\rho}-\left(\mu'
\frac{\mu_2\gamma_1-\mu_1\gamma_2}
{6\rho\mu_2\mu_1}+\frac{c_3^R\left(2\chi_0\gamma'-c_3^R\right)}
{\chi_0\rho}\right)M_0^2
\\  
\label{str1}
c_{t2}^2&=&\frac{\mu_2}{\rho}-
\left(\frac{2\mu_2\gamma_1+\mu_1\gamma_2}{12\rho\mu_1}+\frac{2 \chi_0\gamma_2
  \left(c_2^R- \tfrac{1}{4}\right)-\left(c_{2}^R-
\tfrac{1}{2}\right)^2}{\chi_0\rho}\right)M_0^2 
 \end{eqnarray}
%
For an external field parallel to the wave vector we have
%
\begin{eqnarray} \label{sdisp2}
c_l^2&=&\frac{\tilde \mu}{\rho}-
\left(\tilde \mu \frac{\mu_2\gamma_1+2\mu_1\gamma_2}{6\mu_2\mu_1
 \rho}+\frac{\left(2
    c_1^R+c_2^R\right)\left(2\chi_0\gamma''-\left(2 c_1^R+c_2^R\right)\right)}
{\chi_0\rho}\right)M_0^2
\\ 
\label{sd1}
c_t^2&=& \frac{\mu_2}{\rho}-
\left( \frac{2\mu_2\gamma_1+\mu_1\gamma_2}{12\mu_1
 \rho }+\frac{2\chi_0\gamma_2
  (c_1^R +\tfrac{1}{4}) -
\left(c_1^R+\tfrac{1}{2}\right)^2}{\chi_0\rho}\right)M_0^2
 \end{eqnarray}
%
The field dependence of the sound velocities contains the magnetostrictive 
contributions already present in the low frequency limit, 
but in addition also the reversible, dynamic crosscouplings 
($c_2^R, c_3^R)$ between magnetization and flow 
(which in the low frequency limit contributed to the sound damping). 
Of course, most interesting experimentally is the transition 
region $\omega \tau_M \approx 1$, where the sound velocities make 
steps and the damping is enhanced.\\[0.5cm]



\subsection{Shear excitation by temperature 
oscillations }


In ferrogels there are several reversible crosscouplings mediated 
by material tensors that are linear in the external 
magnetic field. They belong to the same type of effects 
as the Hall and the Righi-Leduc effect.  
In eq.(\ref{Y1}) there is such a coupling, described by 
$\xi^{TR}$, between a temperature gradient and 
the dynamics of the elastic degree of freedom. From the 
form of $\xi^{TR}$ one can conclude that an applied oscillating temperature $T = T_0 + T_1 exp{[i(kz - \omega t)]}$, 
in the presence of a homogeneous or gradient magnetic field \footnote{ The field component $\bm{\hat e}_z H_1 y$, present due to $curl \bm{H}=0$, does not contribute.} $\bm{H}= \bm{\hat e}_y (H_0 + H_1 z)$, 
leads to an 
oscillating shear in the plane perpendicular to the magnetic field (Fig.\ref{rubber1}),
since $Y_{xz} \sim a \equiv  \tfrac{1}{2} \chi_0 \xi^{TR} T_1 (H_1 + ik H_0)$.
%
\begin{figure} \label{rubber1}
%\begin{minipage}{8cm}
\begin{center}
\includegraphics[width=10cm]{rubber}
\end{center}
%\end{minipage}
\caption{Shear excitation $u_{x z}$ in a ferrogel due to  
applied temperature oscillations along the z-direction, in the presence of a constant or gradient magnetic field in y-direction.
} 
\end{figure}
%
Oscillating shear $u_{xz}$ is part of the transverse sound discussed in the previous subsection. 
After the initial transients have died out the excited shear wave travels with the frequency and wave vector of the applied temperature, $u_{xz} = u_0 \,exp{[i(kz - \omega t)]}$, with the complex amplitude $u_0$ given by
%
\begin{eqnarray}\label{Tshear}
\left( -i \omega - \frac{c_{t1}^2 k^2}{i \omega - \tfrac{\nu}{\rho} k^2} + \xi \mu_2 k^2 \right)u_0 = i k a 
\end{eqnarray}
%
where $c_{t1}$ is the shear wave velocity (\ref{tr1}), $\nu$ the shear viscosity, and $\xi$ the self diffusion of the strain field (\ref{YD1}). 
The former comprises a field dependent part, which is space dependent 
in a gradient field. However,  that part is small 
compared to the constant one ($\mu_2/\rho$) and can be 
approximated by its spatial mean or can be neglected at all here. 
If the ratio $\omega/k$ for the applied temperature is in the range of the transverse sound velocity, we have $\omega \gg (\nu/\rho) k^2$ and $\omega \gg \xi \mu_2 k^2$, and (\ref{Tshear}) gives the amplitude of the excited shear wave in the long time limit in the form of a response of a damped harmonic oscillator
%
\begin{eqnarray}\label{Tshear2}
u_{0}=\frac{ -a k \omega}{\omega_0^2-\omega^2+2 i \omega
  \Gamma}
\end{eqnarray}
%
with eigenfrequency $\omega_0 = c_{t1} k$ and damping $2 \Gamma = k^2 (\xi \mu_2 + \nu /\rho)$.

The real  part of the strain tensor, 
which is the measurable quantity, has the form
\begin{eqnarray}
\mathfrak{Re}\left(u_{x z}\right)=A_{1,2}\cos\left(kz -\omega t+\delta_{1,2}\right)
\end{eqnarray}
where
$A_{1,2}=a_{1,2}\,\,k \omega \left(\left(\omega_0^2-\omega^2\right)^2+4\omega^2\Gamma^2\right)^{-1/2}$ 
  and $\delta_2 = \delta$, $\delta_1 = \delta - \pi/2$ with
$\tan \delta=\left({\omega^2-\omega_0^2}\right)/\left({2\omega\Gamma}\right)$ for the two cases of a constant magnetic field $a_2 = \tfrac{1}{2}\chi_0 \xi^{TR} T_1 k H_0$, and a field gradient $a_1 = \tfrac{1}{2}\chi_0 \xi^{TR} T_1 H_1$, respectively. 
This reduces to 
%
\begin{eqnarray}
u_{x z}=\frac{a_1 k}{2\Gamma}\sin\left(kz -\omega_0 t\right) \quad\quad {\rm or}  \quad\quad u_{x z}=\frac{a_2 k}{2\Gamma}\cos\left(kz -\omega_0 t\right)
\end{eqnarray}
%
in the resonant cases $\omega=\omega_0$. Of course, this coupling of temperature to shear also works the other way round, and an imposed shear wave can excite a temperature wave in the presence of an appropriate magnetic field (gradient).


Another possibility of shear excitation  is the application of  concentration instead of a temperature oscillations (see eq.(\ref{Y1})), although this might be more challenging experimentally. \\[1cm]



\section{Summary \label{conclu1}}

In this chapter we have given the hydrodynamic equations for magnetic 
elastomers including the magnetization as an independent slowly
relaxing variable, which allows to study the system for high
frequencies as well.  Due to presence of  the permanent network the
displacement field $\bf{u} $ is a truly hydrodynamic
variable. The fact, that magnetic grains are "attached" to the network
is expressed by the static coupling of the magnetization and the
strain tensor.  This leads to an additional field dependent
contribution to the
sound spectrum. The contribution to the transverse
sound modes
depends on the relative angle between an external field and the 
wave vector. From the low frequency limit  of the sound spectrum one can obtain information
about the effective, magnetic field dependent elastic moduli.  However, 
these moduli are different from those measured by static
elongations or shear deformations in an external field. The reason is
that due to the finite magnetostriction the linear response theory is
not applicable. Only in the limit of a vanishing field  they are equal and match the true elastic moduli.  In the high frequency
limit one gets a shift in the sound velocities proportional to 
the dynamic coupling
between the flow and the magnetization. This reflects the fact,
that the magnetization is an independent variable. 
Finally, a shear excitation experiment in an oscillating temperature gradient
plus a gradient of the magnetic field has been proposed. 
 




%\chapter*{Bibliography}


\begin{thebibliography}{10}
\chaptermark{Bibliography}
\addcontentsline{toc}{chapter}{Bibliography}

\bibitem{Rosen}
{R.E. Rosensweig}, {\it Ferrohydrodynamics} (Cambridge University Press, New  
York, 1985).


\bibitem{brochard}
{ F. Brochard and P.G. de Gennes}, {\it J. Phys.\@ (France)}
    {\bf 31},  691 (1970).


\bibitem{rault70}
J. Rault, P.E. Cladis, and J.P. Burger, {\it Phys.\@ Lett.
A} {\bf 32,} 199 (1970).

\bibitem{chen87}
S.S. Chen and S.H. Chiang, {\it Mol.\@ Cryst.\@ Liq.\@ Cryst.}
{\bf 144}, 359 (1987).

\bibitem{hayes76}
C.F. Hayes, {\it Mol.\@ Cryst.\@ Liq.\@ Cryst.} {\bf 36},
245 (1976).

\bibitem{liebert79}
L. Li\'ebert and A. Martinet, {\it J. Physique (France)
Lett.} {\bf 40}, 363 (1979).


\bibitem{dabadie}
{ P. Fabre, C. Casagrande, M. Veyssi\'e, V. Cabuil, R. Massart},
  {\it Phys.\@ Rev.\@ Lett.} {\bf 64},  539  (1990).\\
{ J.C. Dabadie, P. Fabre, M. Veyssi\'e, V. Cabuil, R. Massart},
  {\it J.\@ Phys: Condens.\@ Matter} {\bf 2},  SA291 (1990).

\bibitem{ponsinet}
{ V. Ponsinet, P. Fabre, M. Veyssi\'e}, {\it Europhys.\@ Lett.}
  {\bf 30}, 277 (1995).

\bibitem{spoliansky}
{ D. Spoliansky, J. Ferr\'e, J.-P. Jamet, V. Ponsinet},
  {\it J.\@ Magn.\@ Magn.\@ Mater.} {\bf 201},  200 (1999).

\bibitem{bacri2}
{ J.C. Bacri, V. Cabul, A. Cebers, C. Menager, R. Perzynski},
  {\it Europhys.\@ Lett.} {\bf 33}, 235  (1996).

\bibitem{burylov}
{S.~V.~Burylov and Y.~L.~Raikher},
{\it Mol.\@ Cryst.\@ Liq.\@ Cryst.}
{\bf 258}, 107 (1995).

\bibitem{raikher}
{Y.L. Raikher and V.I. Stepanov}, {\it J.\@ Magn.\@ Magn.\@ Mater.}
 {\bf 201}, 182 (1999).

\bibitem{bacri}
J.C. Bacri and A.M.F. Neto, {\it Phys.\@ Rev.} {\bf E50}, 3860 (1994).


\bibitem{potocova}
{I. Poto\v cov\'a, M. Konerack\'a, P. Kop\v cansk\'y,
  M. Timko, L. Tom\v co, J. Jad\.zyn, G.Czechowski},
{\it J.\@ Magn.\@ Magn.\@ Mater.} {\bf 196}, 578 (1999).


\bibitem{koneracka}
{M. Konerack\'a, V. Z\'avi\v sov\'a, P. Kop\v cansk\'y, J. Jad\.zyn,
G. Czechowski, B. \.Zywucki},
{\it J.\@ Magn.\@ Magn.\@ Mater.} {\bf 157/158}, 589 (1996).

\bibitem{zubarev}
{A.Yu. Zubarev and L.Yu. Iskakova},
{\it J.\@ Magn.\@ Magn.\@ Mater.} {\bf 183}, 201 (1998).

\bibitem{berejnov}
{V. Berejnov, J.-C. Bacri, V. Cabuil, R. Perzynski and Y.L. Raikher},
{\it Europhys.\@ Lett.} {\bf 41}, 507 (1998).


\bibitem{matuoa}
{C.Y. Matuo and A.M. Figueiredo Neto},
{\it Phys.\@ Rev.} {\bf E 60}, 1815 (1999).


\bibitem{fontanini}
{S. Fontanini, A.L. Alexe-Ionescu, G. Barbero, A.M. Figueiredo Neto},
{\it J.\@ Chem.\@ Phys.} {\bf 106}, 6187 (1997).


\bibitem{C1}
{S.-H. Chen and N.M. Amer}, {\it Phys.\@ Rev.\@ Lett.} {\bf 51}, 2298 (1983).


\bibitem{zr0}
{M. Zr\'inyi, F. Horkay}, {\it J.\@ of Intelligent Material Systems and
  Structers} {\bf 4}, 190 (1993). 

\bibitem{babin}
{M. Babinc\'ova, D. Leszczynska, P. Sourivong, P. \v Ci\v cmanec,
  P. Babinec}, {\it J.\@ Magn.\@ Magn.\@ Mater.} {\bf 225}, 109 (2001).

\bibitem{uritani}
{M. Uritani and A. Hamada}, {\it Biochem.\@ Education} {\bf 27}, 169 (1999).

\bibitem{mit}
{T. Mitsumata, E. Juliac, K. Furukawa, K. Iwakura, T. Taniguchi,
  K. Koyoma }, {\it Macrom.\@ Rapid\@ Commun.} {\bf 23}, 175 (2002)

\bibitem{bayreuth}
{D. Collin, G. Auernhammer, O. Gavat, Ph. Martinoty and H.R. Brand},
in preparation 

%\bibitem{berejnov}
%{V. Berejnov, J.-C. Bacri, V. Cabuil, R. Perzynski and Y.L. Raikher},
%{\it Europhys. Lett.}   
%{\bf 41}, 507 (1998).

\bibitem{chen}
{S.-H. Chen and N.M. Amer}, {\it Phys.\@ Rev.\@ Lett.} {\bf 51}, 2298 (1983).

\bibitem{zrinyi}
{M. Zr\'inyi, L. Barsi and A. B\"uki}, {\it J.\@ Chem.\@ Phys.} {\bf 104}, 8750 (1996)
and
{\it Polymer Gels and Networks}
{\bf 5}, 415 (1997)


\bibitem{PJMB}
{H. Pleiner, E. Jarkova, H.-W. M\"uller, and H.R. Brand}, {\it
  Magnetohydrodynamics}\@ {\bf 37}, 254 (2001).\\
\vspace{0.01cm}
H. Pleiner, E. Jarkova, H.-W. Müller, and H.R. Brand, {\it 
     J.\@ Magn.\@ Magn.\@ Mater.} {\bf 252}, 147 (2002)


\bibitem{PGdG}
P.G. de Gennes and J. Prost, {\it The Physics of Liquid Crystals}
(Clarendon Press, Oxford, 1993).


\bibitem{Muk01}
{ P.K. Mukherjee, H. Pleiner, and H.R. Brand},
  {\it Eur.\@  Phys.\@  J.}  {\bf E4}, 293 (2001).


\bibitem{lelidis}
{ I. Lelidis and G. Durand},
   {\it Phys.\@ Rev.}  {\bf E48}, 3822  (1993).

\bibitem{jpmfb}
E. Jarkova, H. Pleiner, H.-W. M\"uller, A. Fink, and H.R. Brand,
{\it Eur.\@ Phys.\@  J.} {\bf E5}, 583 (2001).

\bibitem{Lub}
T.C. Lubensky, {\it Mol.\@ Cryst.\@ Liq.\@ Cryst.} {\bf 23}, 99 (1973).

\bibitem{PB1}
H. Pleiner and H.R. Brand,
Hydrodynamics and Electrohydrodynamics of Nematic Liquid Crystals,
in {\it Pattern Formation in Liquid Crystals},
eds.\@ A. Buka and L. Kramer (Springer, New York, 1996) p.\@ 15ff.


\bibitem{MPP}
P. C. Martin, O. Parodi and P.S. Pershan, {\it Phys.\@ Rev.}
{\bf A6}, 2401 (1972).


\bibitem{forster}
D. Forster, {\it Hydrodynamic Fluctuations, Broken Symmetry and Correlation
Functions}, (Benjamin, Reading Mass, 1975)

\bibitem{LL}
L.D. Landau and E.M. Lifshitz,  {\it Electrodynamics of 
Continuous Media}, (Pergamon, Oxford, 1984).

\bibitem{jpmb}
{E. Jarkova, H. Pleiner, H.-W. M\"uller, and H.R. Brand}, {\it
  J.\@ Chem.\@ Phys.} {\bf 118}, 2422 (2003) 

%\bibitem{rosensweig85}
%R.E. Rosensweig, {\it Ferrohydrodynamics}, Cambridge University Press,
%Cambridge (1985).

%\bibitem{berejnov}
%{V. Berejnov, J.-C. Bacri, V. Cabuil, R. Perzynski and Y.L. Raikher},
%{\it Europhys. Lett.} 
%{\bf 41}, 507 (1998).

%\bibitem{Kon}
%{M. Konerack\'a, V. Z\'avi\v sov\'a, P. Kop\v cansk\'y, J. Jad\.zyn,
%  G. Czechowski, and B. \.Zywuski}, {\it J. Magn. Magn. Mater.} 
%{\bf 157/158}, 589 (1996).

\bibitem{shliomor}
M.I. Shliomis, {\it Sov.\@ Phys.\@ Usp.} {\bf 17}, 153 (1974).

\bibitem{ml}
H.-W. M\"uller, M. Liu, 
{\it Phys.\@ Rev.}
{\bf E64}, 061405 (2001).


\bibitem{liu94}
M. Liu, {\it Phys.\@ Rev.\@ Lett.} {\bf 50}, 2925 (1994).

\bibitem{pshen}
A.F. Pshenichnikov, {\it J.\@ Magn.\@ Magn.\@ Mat.} {\bf 145}, 319 (1995).

\bibitem{wang}
{Z.W. Wang, C. Holm, H.-W. M\"uller,}
{\it Phys.\@ Rev.} {\bf E66}, 021405 (2002).


\bibitem{mjl}
H.-W. M\"uller and M. Liu, {\it Phys.\@ Rev.\@ Lett.} {\bf 89}, 067201 (2001);
H.-W. M\"uller, Y. Jiang, and M. Liu, {\it Phys.\@ Rev.} {\bf {E}}, submitted
(2002).

\bibitem{BP}
H.R. Brand and H. Pleiner, {\it Phys.\@ Rev.} {\bf A24}, 2777 (1981).


%\bibitem{yu80}
%L.J. Yu and A. Saupe, {\it Phys. Rev. Lett.} {\bf 45}, 1000
%(1980).

%\bibitem{galerne83}
%Y. Galerne and J.P. Marcerou, {\it Phys. Rev. Lett.}
%{\bf 51}, 2109 (1983).


%\bibitem{fontanini}
%S. Fontanini, A.L. Alexe-Ionescu, G. Barbero, and A.M.F. Neto, {\it
%  J. Chem. Phys.} {\bf 106}, 6187 (1997).

%\bibitem{jamamoto01}
%J. Yamamoto and H. Tanaka, {\it Nature} {\bf 409}, 321
%(2001).

%\bibitem{berejnov01}
%V.V. Berejnov, Y. Raikher, V. Cabuil, 
%{\it J. Colloid Interface Sci.}
%{\bf 199},
%215 (1998).

%\bibitem{burylov95}
%{S.V.~Burylov and Y.L.~Raikher},
%{\it Mol. Cryst. Liq. Cryst.} 
%{\bf 258}, 107 (1995).


%\bibitem{neto}
%C.Y. Matuo and A.M.F. Neto,
%{\it Phys. Rev.} {\bf E60}, 1815 (1999).


%\bibitem{Spo}
%{D. Spoliansky, J. Ferr\'e, J.-P. Jamet, and V. Ponsinet}, {\it
%  J. Magn. Magn. Mater.} {\bf 201}, 200 (1999).


%\bibitem{PB1}
%H. Pleiner and H.R. Brand,
%Hydrodynamics and Electrohydrodynamics of Nematic Liquid Crystals,
%  in {\it Pattern Formation in Liquid Crystals},
%       eds. A. Buka and L. Kramer, Springer, New York, p.\ 15
%       (1996).

\bibitem{tem}
H. Temmen, H. Pleiner, M. Liu and H.R. Brand,  {\it Phys.\@ Rev.\@ Lett.} {\bf 84}, 3228 (2000)

%\bibitem{zr1}
%{D. Szab\'o, G. Szeghy, and
%  M. Zr\'inyi}, {\it Macromolecules}\@ {\bf 31}, 6541 (1998).


\bibitem{zr}
{T. Mitsumata, K. Ikeda, J.P. Gong and Y. Osada, D. Szab\'o and
  M. Zr\'inyi}, {\it J.\@ of Appl.\@ Phys.} {\bf 85}, 8451 (1999).

\bibitem{nikitin}
L.V.Nikitin, L.S. Mironova, G.V. Stepanov and A.N. Samus, {\it Polymer 
  Science, Ser. A}\@
{\bf 43}, 443 (2001).

\end{thebibliography}



\chapter*{Acknowledgement}

\addcontentsline{toc}{chapter}{Acknowledgement}

\begin{itemize}
\item{First of all, I would like to thank my thesis supervisor
 Harald Pleiner for his constant support and encouragement. His
 confident guidance helped me a lot throughout the
 development of this thesis. I also thank Prof.~Robert Graham, who
 agreed to act as the Zweitgutachter for his effort and a valuable discussion during the
 preparation of the thesis. Financial support of the DFG through Schwerpunktpunkt 1104 (Magnetische 
Fluide) and by the International Max Planck Research School
is gratefully acknowledged.
 }
\item{ Second, I would like to thank  all people with whom I have
    collaborated: Helmut R. Brand and Hanns-Walter
    M\"uller  for intensive and fruitful
    collaboration; Thomas A. Vilgis and Namkyung Lee, who
    introduced me to the polymer physics and shared their 
    knowledge on this subject with me; M\'ario Tamashiro, who
    readily devoted his time for valuable discussions; and
    my office mate Carsten Svaneborg, 
    who had always time, when I needed a  scientific or technical
    advice. }
\item{Third, I thank  the director of the Theory group of
    MPIP, Prof.~Kurt  Kremer and all present and former members of the group for their hospitality, in addition to the excellent work
    conditions. Thanks to Doris Kirsch for solving many bureaucratic
    problems and Achim Gelessus for helpful and prompt computer 
    support. Especially, 
    I would like to thank my lunch group for the very nice time I had in
    Mainz.
 Eating with them helped staying alive after Mensa
    food. 
}
\item{Finally, I'm very  thankful to  my family, especially my husband
    Vsevolod Ksenofontov, for their love  and
    support. }
\end{itemize}
%\noindentI enjoyed working with you very much,
% both from a professional as a personal point of view. 
%E.J. wants to thank V. Berejnov for fruitful email discussion. Partial support
%f this work through the Schwerpunktsprogramm
%1104 'Colloidal Magnetic Fluids' of
%he Deutsche Forschungsgemeinschaft is gratefully
%cknowledged.
\end{document}


