\documentstyle[preprint,aps]{revtex}
\input epsf
\input psfig.sty
\input amssym.def
\input amssym.tex

\begin{document}
\draft

\title{Elongation of confined ferrofluid droplets under applied fields\\}
\author{S. Banerjee$^1$, M. Fasnacht$^1$, S. Garoff$^{1,2}$ and M. Widom$^1$}
\address{$^1$Department of Physics \\ 
$^2$ Colloids, Polymers, and Surfaces Program \\
Carnegie Mellon University, Pittsburgh, Pa. 15213}

\date{\today}
\maketitle
\begin{abstract}
Ferrofluids are strongly paramagnetic liquids. We study the behavior of 
ferrofluid droplets confined between two parallel plates with a weak applied 
field parallel to the plates. The droplets elongate under the applied field 
to reduce their demagnetizing energy and reach an equilibrium shape where the 
magnetic forces balance against the surface tension. This elongation 
varies logarithmically with aspect ratio of droplet thickness to its original 
radius, in contrast to the behavior of unconfined  droplets. 
Experimental studies of a ferrofluid/water/surfactant emulsion confirm this 
prediction.

\end{abstract}

\pacs{}

\newpage

\section{introduction}
\label{intro}

Ferrofluids~\cite{rr} are oil- or water-based colloidal suspensions of
permanently magnetized particles. In an applied magnetic field, the particles 
align creating a strong paramagnetic response in the ferrofluid. Because they 
are fluids, these suspensions can flow in response to forces. For example, 
ferrofluid droplets elongate parallel to applied 
fields~\cite{d3droplets1,d3droplets2,d3droplets3,ellipsoid,bashtovoi,d2droplets} 
and undergo tip-sharpening
transitions~\cite{sharpen1,sharpen2}. When a ferrofluid droplet is confined
between two plates in a ``thin film'' geometry, surrounded by an 
immiscible fluid, and a field is applied perpendicular to the plates, 
it undergoes field induced bifurcations~\cite{goldstein} leading to 
intricate labyrinthine patterns~\cite{pattern}. Ferrofluid 
emulsions~\cite{bibette} undergo structural 
transitions under an applied field from a randomly dispersed structure of 
the emulsion droplets to droplet chains, columns and worm like 
structures~\cite{liu1,liu2} depending on volume fraction, sample geometry
and the rate of field application.


We are interested in the elongation of ferrofluid droplets under applied
magnetic fields fields. While the elongation of freely suspended, 
3-dimensional droplets has been well analyzed and 
observed~\cite{d3droplets1,d3droplets2,d3droplets3,ellipsoid},
the elongation of droplets in the thin film geometry still remains relatively
unstudied. In this geometry droplets are surrounded by an immiscible fluid and
are confined between parallel plates (see figure 1) such that the thickness  
between the plates is much smaller than its undeformed diameter. The applied 
field is parallel to the  plates. We calculate the elongation of the thin film 
droplets as a function of various parameters of the problem. Prior 
studies of ferrofluid droplets in this geometry~\cite{d2droplets,bashtovoi} have 
considered droplet elongation under applied field and its use as a tool for 
measuring surface tension between the ferrofluid and the surrounding immiscible 
fluid. 

A droplet of ferrofluid elongates under an applied field because of 
the demagnetizing fields of magnetic
poles on the surface of the droplet. Surface poles arise wherever the
droplet magnetization has a component perpendicular 
to the surface. The demagnetizing field that they create opposes the 
magnetization,
creating a demagnetizing energy that depends on the shape of the
droplet. The droplet elongates to reduce its demagnetizing field and
energy. Because elongation increases the surface energy of the
system, an equilibrium shape is reached when the magnetic forces  balance
against the surface tension forces.


In the case of freely suspended 3-dimensional droplets the elongation can be
calculated assuming that the droplets are ellipsoidal in shape for small 
elongation. The demagnetizing field is thus uniform 
and the elongation (major axis minus minor axis divided by minor axis) is 
found to be proportional to the undeformed 
droplet radius. The case of droplets confined in the thin film geometry
however is more complex as it involves two length scales, droplet thickness 
and its undeformed diameter. In the limit of small aspect ratio 
(droplet thickness divided by its undeformed diameter) the demagnetizing 
fields are non-uniform. They are stronger near the edge of the droplet 
and fall off as $1/r$ away from the edge into the bulk of the droplet. Due to 
this $1/r$ fall-off of the field we find that the elongation divided by droplet 
thickness is 
proportional to the logarithm of the aspect ratio. This improves
on the existing theory~\cite{d2droplets} which assumes that the demagnetizing 
field is uniform. We also perform an experiment which supports our predicted 
logarithmic behavior.

Section~\ref{theory} of this paper presents our theoretical study of the 
elongation of a ferrofluid droplet confined within a thin film. Our principal
result is a predicted logarithmic dependence of elongation on droplet
aspect ratio. We contrast this result with the corresponding elongation of
unconfined droplets. Section~\ref{expt} describes an experiment done with
ferrofluid emulsions that tests our theory. The experiment is in
qualitative agreement with our theoretical prediction, but differs
quantitatively in at least one respect. In section~\ref{discussion} we discuss
possible explanations of the discrepancy.


\section{theory}
\label{theory}

Consider a paramagnetic liquid  droplet confined in a thin film between two 
parallel plates with a gap, $\Delta$, in the ${\hat z}$ direction 
(see figure 1). An immiscible liquid surrounds the droplet. Let the thickness,
$\Delta$, be much smaller than the radius of the undeformed droplet, $r_0$. 
This small aspect ratio
\begin{equation}
p={\Delta \over 2r_0}
\end{equation}
provides the pseudo-two-dimensional character of the problem. If a uniform,
weak, field ${\bf H}_0$ is applied parallel to the plate, the droplet 
magnetizes. The magnetization creates an opposing demagnetizing 
field whose strength depends on the droplet shape. The droplet elongates 
to decrease its magnetic energy, reaching equilibrium when the magnetic 
forces balance against the restoring forces due to surface tension.
In this section we define the elongation of the droplet and calculate
the surface energy, $E_S$, and the magnetic energy, $E_M$, of the droplet as 
a function of its elongation. By minimizing the total energy with respect 
to the elongation we obtain the elongation as a function of ${\bf H}_0$, 
$r_0$, and $\Delta$.

For simplicity assume the elongated droplet has a uniform cross 
section, $\cal C$, independent of $z$. This corresponds to a contact 
angle of $90^\circ$ between the paramagnetic liquid, the surrounding fluid
and the glass plates, and a plate spacing much less than the capillary length
of the two liquids. Thus the droplet has straight edges if viewed from the 
side (see figure 1). The role of 
contact angle will be discussed later in section~\ref{discussion}. 
We write the equation for $\cal C$ in polar coordinates as a generic 
smooth perturbation to a circle,
\begin{equation}
\label{curve}
r=\alpha_1+\alpha_2 \cos 2 \theta.
\end{equation}
We only include a single harmonic, since we expect coefficients for
the higher harmonics to be much smaller than $\alpha_2$ for small
perturbations. The cross section $\cal C$ has semi-major axis $a$, and
semi-minor axis $b$ (see figure 1b), with $\alpha_1=(a+b)/2$ and
$\alpha_2=(a-b)/2$. We define the elongation of the droplet 
\begin{equation}
\label{definition}
\epsilon \equiv {a \over b}-1.
\end{equation}
We assume that the elongation, $\epsilon$, is much less than $1$.
Imposing the constraint that the volume of the droplet ($\Delta$ times
cross-sectional area) remains constant we calculate 
\begin{equation}
\label{alpha}
\alpha_1= {r_0 \over (1+k^2/2)^{1/2}},~~~~\alpha_2={r_0 k
\over (1+k^2/2)^{1/2}},
\end{equation}
where $k=\epsilon / (2+\epsilon)$.

The surface energy is the sum of interfacial areas times surface tensions
between all pairs of the three phases (solid glass, ferrofluid droplet and
immiscible fluid). For the case of uniform cross-section ($90^\circ$ 
contact angle) droplets, the glass-ferrofluid and glass-immiscible fluid
interfacial areas are independent of the shape of ${\cal C}$ due to the
fixed volume constraint. Hence we concern ourselves with the 
droplet-surfactant solution interface, the area of which is $\Delta$ times
the perimeter. The perimeter of cross section $\cal C$ can be calculated 
as a power series in $\epsilon$,
\begin{equation}
\label{circum}
S= 2 \pi r_0 (1+{3 \over 16} \epsilon^2+O(\epsilon^3)).
\end{equation}
As expected, the leading correction to $S$ is second order in $\epsilon$
since the perimeter should increase regardless of the sign of $\epsilon$.
The relevant surface energy of the droplet is 
\begin{equation}
\label{surface-nrg}
E_S=\sigma_{FI} S \Delta 
\end{equation}
where $\sigma_{FI}$ is the surface tension of the
ferrofluid-immiscible fluid interface.

The total magnetic energy of any paramagnetic body under applied field
is~\cite{wfb}
\begin{equation}
\label{magnetic-nrg}
E_M= -{1 \over 2}\int_V d^3{\bf r}~{\bf H}_0 \cdot {\bf M}({\bf r}).
\end{equation}
The magnetization ${\bf M}({\bf r})$ is determined by the self consistent 
equation
\begin{equation}
\label{M}
{\bf M}({\bf r})=\chi({\bf H}_0+{\bf H}_D({\bf r}))
\end{equation}
for linear susceptibility $\chi$, where
\begin{equation}
\label{demag-field}
{\bf H}_D({\bf r})= \int_{S} d^2 {\bf r'}~ 
({\bf M}({\bf r'}) \cdot  {\bf \hat{n}}({\bf r'}))
{ {\bf r-r'} \over |{\bf r-r'}|^3}
+\int_{V} d^3 {\bf r'}~
(\nabla \cdot{\bf M}({\bf r'}))
{{\bf r-r'} \over |{\bf r-r'}|^3}
\end{equation}
is the demagnetizing field due to the magnetization ${\bf M}({\bf
r})$, with ${\bf {\hat n}}({\bf r'})$ being the outward normal at any
point on the surface. The surface integral gives the demagnetizing
field due to the surface poles which appear wherever the magnetization
has a component normal to the surface. The volume integral gives the 
contribution to the demagnetizing field due to volume charges which 
appear at points where the magnetization has non-zero divergence.

To calculate the magnetic energy we expand ${\bf M}$ and ${\bf H}_D$ 
in power series in the susceptibility $\chi$,
\begin{equation}
\label{mag}
{\bf M}({\bf r})={\bf M}^{(1)}({\bf r})+{\bf M}^{(2)}({\bf r})
+{\bf M}^{(3)}({\bf r})+...
\end{equation}
\begin{equation}
\label{demag}
{\bf H}_D({\bf r})={\bf H}_D^{(1)}({\bf r})+{\bf H}_D^{(2)}({\bf r})+
{\bf H}_D^{(3)}({\bf r})+... ,
\end{equation}
where ${\bf M}^{(n)}({\bf r})$ and ${\bf H}_D^{(n)}({\bf r})$ are proportional 
to $\chi^n$. Equating terms in~(\ref{M}) of equal order in $\chi$ we get
\begin{equation}
\label{first-order}
{\bf M}^{(1)}({\bf r})=\chi {\bf H}_0
\end{equation}
and
\begin{equation}
\label{higher-order}
{\bf M}^{(n+1)}({\bf r})=\chi {\bf H}_D^{(n)}({\bf r}).
\end{equation}
Note that ${\bf M}^{(1)}({\bf r})$ is independent of ${\bf r}$ because
the applied field is uniform whereas ${\bf M}^{(n)}({\bf r})$ may depend on
$({\bf r})$ for $n>1$ because ${\bf H}_D({\bf r})$ may be non-uniform. To 
second order in $\chi$ we write the magnetic energy of the droplet 
in~(\ref{magnetic-nrg}) as
\begin{equation}
\label{mag-nrg2}
E_M=-{1 \over 2}\chi{H}_0^2 V -{1 \over 2}\int_V d^3{\bf r}
~{\bf M}^{(1)}\cdot {\bf H}_D^{(1)}.
\end{equation}

The first term in equation~(\ref{mag-nrg2}) for the magnetic energy is
independent of the shape of the droplet and hence unimportant for our
consideration. The second term in the energy is the demagnetizing
energy $E_D$ due to a uniform magnetization ${\bf M}^{(1)}=\chi{\bf
H}_0$. Because ${\bf M}^{(1)}$ is uniform there are no volume charges,
and the surface poles appear only along the droplet-immiscible
fluid interface, to first order in $\chi$.  Rewrite the second term
in~(\ref{mag-nrg2}) as an energy due to the induced surface charges
along the curved surface of the droplet 
\begin{equation}
\label{demag-nrg}
E_D={1 \over 2} \chi^2  \int_0^\Delta dz \int_0^\Delta dz'
\oint ds \oint ds' {({\hat {\bf n}} \cdot {\bf H}_0)
({\hat {\bf n'}} \cdot {\bf H}_0) \over |{\bf r}-{\bf r'}|}.
\end{equation}
Here $ds$ and  $ds'$ are infinitesimal arc-lengths along the contour of the 
droplet ${\cal C}$, and ${\hat {\bf n}}$ and ${\hat {\bf n'}}$ are the outward 
normals to the curved surface of the droplet at points $(s,z)$ and $(s',z')$ 
respectively.

Write $|{\bf r}-{\bf r'}|= {\sqrt {R^2+(z-z')^2}}$, where 
$R$ is the in-plane distance between points at 
positions $s$ and $s'$ on $\cal C$. Integrating over $z$ and $z'$
in~(\ref{demag-nrg})  yields~\cite{goldstein}
\begin{equation}
\label{demag-nrg2}
E_D=  \chi^2 \Delta \oint ds \oint ds' ({\hat {\bf n}} \cdot {\bf H}_0)
({\hat {\bf n}}' \cdot {\bf H}_0) \Phi(R/ \Delta)
\end{equation}
where
\begin{equation}
\Phi (R/\Delta)= R/\Delta - {\sqrt {1+(R/\Delta)^2}} + 
\ln {\Big [}(R/\Delta)/({\sqrt {1+(R/\Delta)^2}}-1){\Big ]}.
\end{equation}
Using equation~(\ref{curve}) for $\cal C$ we calculate the
demagnetizing energy in~(\ref{demag-nrg2}) as a series expansion in
$\epsilon$ and the aspect ratio $p=\Delta/2r_0$
\begin{equation}
\label{expansion}
E_D=  \chi^2 H_0^2 V {\Big\{}{2p} \ln {B \over p}- 3\epsilon p  
\ln { C \over p}+ \cdot \cdot \cdot {\Big \}}
\end{equation}
where $V=\pi r_0^2 \Delta$ is the volume of the droplet, and $B=4
e^{-1/2}=2.43$ and $C=4 e^{-5/6}=1.74$ are geometrical constants. The term in
the brackets can be identified as $2 \pi$ times the demagnetizing 
factor~\cite{wfb}
of the droplet along the direction of applied field. Additional terms
in the series in equation~(\ref{expansion}) are of higher order in
$\epsilon$ or in $p$. For small elongation and small aspect ratio
we may neglect these higher order terms.

Minimizing the total energy $E=E_S+E_M$ with respect to $\epsilon$ gives
\begin{equation}
\label{elongation}
\epsilon= {\chi^2 H_0^2 \Delta \over \sigma_{FI}}\ln {C \over p}.
\end{equation}
Corrections to this result are higher order in aspect ratio $p$ or higher 
order in $\epsilon$ itself. Interestingly, the elongation depends only 
logarithmically on the undeformed radius $r_0$, and has a much stronger 
dependence on the thickness, $\Delta$, of the droplet.  This result differs
from an earlier theory~\cite{d2droplets} which omits the logarithm because
it assumes that the demagnetizing field is uniform inside the droplet. 

In the case of unconfined, nearly ellipsoidal 
droplets~\cite{d3droplets3,ellipsoid},
the demagnetizing field is quite uniform inside the droplet. The demagnetizing
energy is therefore proportional to the volume ($(4/3)\pi r_0^3$) of the 
droplet according to equation~(\ref{mag-nrg2}). The surface energy is 
proportional to the area ($4 \pi r_0^2$) and the elongation is thus 
proportional to $r_0$. In the case of thin film geometry, however, the 
demagnetizing field is very non-uniform. For distances much less than $\Delta$ 
near the droplet edge, the component of the demagnetizing field is of order 
$M$, since the edge acts like an infinite sheet of charge in the first 
approximation. For distances much greater than $\Delta$ the demagnetizing 
field  is of order $M \Delta/r$, with $r$ the distance away from the edge, 
since the edge acts as a line charge in 
this case. The contribution to the integral for the 
demagnetizing energy in equation~(\ref{mag-nrg2})  mainly comes from the bulk
of the droplet and goes like $r_0 \Delta^2 \ln (r_0/\Delta)$. The 
surface energy is proportional to $2 \pi r_0 \Delta$ and the 
elongation is therefore proportional to $\Delta \ln (r_0/\Delta)$. The 
logarithmic variation of elongation with the aspect ratio is thus a 
signature of the non-uniform nature of the demagnetizing field inside the 
droplet. 


\section{experiment}
\label{expt}

\subsection{Setup}
\paragraph{Sample Preparation and Structure}

Our sample consisted of a ferrofluid/aqueous solution emulsion confined 
between two glass plates. The oil-based ferrofluid used was EMG 905  made 
by Ferrofluidics with susceptibility $\chi=1.9$ and saturation magnetization 
of $400$ Gauss~\cite{ferrofluidics}. To reduce the surface tension between 
the ferrofluid and
the immiscible aqueous external  phase, we incorporated surfactants in 
the aqueous phase. A solution of a commercial detergent made the best 
emulsions while solutions
with other pure anionic surfactants either showed hardly any elongation 
of the ferrofluid droplets under an applied field or produced droplets 
without sharp boundaries with the aqueous phase. In contrast, our stable, well 
behaved emulsions allowed us to probe and confirm the fundamental aspects 
of our model.

To prepare the emulsions,  a single drop of ferrofluid ($\sim 0.1$ ml) 
was added to $10$ ml of surfactant solution which was a $12$ times dilution
of the commercial detergent. The liquid was shaken (by hand) to prepare the 
emulsion, creating ferrofluid droplets with diameters varying from 
$\sim 5-200~\mu$m. A small amount of this emulsion was then put between 
two glass plates which were circular, about $2$ cm  in diameter and  $4$ mm 
in thickness. These plates were cleaned using soap and alcohol and then rinsed 
with ROPure water. We also tried acid cleaning of the glass plates, however it 
did not result in any noticeable change in the quality of the sample.

We used a rectangular spacer made of mylar foil to separate the plates and
prevent the emulsion from leaking out from the edges of the plates. The mylar 
foil extended to the edges of the glass plates and had a rectangular
 hole in the center into
which the emulsion was inserted. The thickness of a single mylar spacer was 
measured to be $6.54\pm 0.06~\mu$m. The experiment was performed  with one 
and two spacers to ensure small aspect ratio.

For the cell assembly, the mylar spacers were placed on the first plate 
and a drop of the emulsion was put in the center of the plate. The second 
plate was placed on top and the two plates were  clamped together using 
a pair of brass rings. The rings were tightened by a set of $4$ equally 
spaced screws. We measured the thickness variation across the sample by 
making a ``dry'' sample (without the emulsion) and counting resulting 
white light interference fringes. Although the thickness of mylar spacers 
was measured to an accuracy of $1$ percent, the thickness variation across 
the sample was found to be $10 \%$ resulting from  the stresses 
due to clamping and possible entrapment of dust in the cell. 
 


\paragraph{Apparatus}
A schematic diagram of the experimental setup is shown in figure 2. 
We put the sample at the center of a pair of Helmholtz coils 
to insure a homogeneous magnetic field. The field measured 
close to the sample using a Hall probe showed a variation of less than 
$4\%$ across the sample. The sample was set up horizontally to prevent 
gravitational settling of the ferrofluid droplets. Horizontal alignment
was achieved using a bubble level.

The sample was illuminated from below using a diffused light source and
observed from above using a tele-microscope. The 
tele-microscope was connected to a CCD camera  and the image from it was 
fed into a video recorder and recorded on video tape. Images 
from the recording were later processed using {\it NIH Image}. We calibrated 
the optical system using a measuring reticule aligned along the two 
orthogonal directions of the CCD array. Figure 3 
shows a low magnification view of a typical sample. The ferrofluid droplets 
appear much darker in the image than the surfactant solution around them.

\paragraph{Experimental Procedure and Image Analysis}
During the experiment the applied field was incremented every few seconds.
We found the response of the droplets to the field to be  nearly 
instantaneous and the shape of the droplets remained constant at constant 
field. Experiments with decreasing field strength showed no hysteresis in 
droplet shape. While droplet elongations were observed to be small we 
incremented the field in steps of about $1$ Gauss ($10^{-4}$ Tesla), and 
increased the increments
up to about $5$ Gauss as the elongation increased. Droplet elongations appeared
to vary smoothly with applied fields over the entire range from $0$ to $50$
Gauss.

During each experiment the droplets were observed on a video monitor and 
recorded on tape. Figure 4 shows a droplet with $r_0 = 132.5~\mu$m 
and $\Delta=13.1~\mu$m elongating under applied field at $50$ Gauss, typical 
of the highest field used in our experiment. 
After grabbing images of distorted droplets, we used a 
cut-off in pixel gray scale level to identify the droplet edge. The 
semi-major axis  $(a)$ and the semi-minor axis $(b)$ were directly read off 
the image using {\it NIH Image}. At zero field measured elongations were small 
(RMS magnitude around $0.003$) and in random directions. These minor 
perturbations from a circular shape were likely due to microscopic distortion
of the contact line pinned on weak surface heterogeneities. The ``observed 
radius'' $r_0$ was calculated as the average of the two semi-axes at zero 
field and the elongation at each field value was calculated using data 
analysis software.



\paragraph{Results}
For each of the $48$ droplets studied  we plotted 
elongation  $\epsilon$, versus the square of the applied field, ${\bf H}_0$. 
Figure $5$ shows typical plots. The elongation is  proportional to the 
square of the applied field for small applied fields as predicted. 
Saturation effects, although small, can be seen at higher values of the field. 
To compare the experimental data to our theoretical 
prediction~(\ref{elongation}), the plot of elongation, for each droplet was  
fitted to
\begin{equation}
\label{fit}
{\epsilon \over \Delta}=k_0+k_1 H_0^2 + k_2 H_0^4.
\end{equation}
We included terms only up to order $H_0^4$ because the saturation effects
were observed to be small. We include $k_0$ to allow for the observed small 
elongations at zero field. 

The coefficients $k_1$ of each droplet were then plotted versus the inverse
of the aspect ratio $1/p=2 r_0 /\Delta$  
on a semi-log plot (see figure 6). The theory predicts a slope of $\chi^2 /
\sigma_{FI}$ and an intercept of $1/C$ on the horizontal axis with $C=1.74$.
The data points in figure 6(a) fall on a straight line as predicted 
by the theory. Also, as predicted by the theory, the data points for
two different droplet thicknesses overlay each other.
There is substantial scatter in the data, but the deviations from a 
straight line are random and consistent with the error bars.  The chief 
source of uncertainty was the $10 \%$ uncertainty in thickness due to the 
variation observed across the sample.  Figure 6(b) displays the deviation 
of $k_1$ from the best fit normalized by the uncertainty. The uncertainties  
in measuring $\epsilon, r_0$ and ${\bf H}_0$ were found to be negligible in 
comparison. 

Dividing the susceptibility $\chi=1.9$ for the ferrofluid 
used by the slope 
= $0.119\pm0.004$~cm/dyne obtained from the fitted line we get 
$\sigma_{FI}=30.4 \pm 1.1$ dynes/cm,
typical of oil-water surface tensions. 
From the fitted line we also get $C=0.35\pm 0.08$, differing substantially 
from our theoretically predicted value of $1.74$. This discrepancy can be
explained by considering the deviations from our two main assumptions of uniform
magnetization and uniform cross-section of the droplet. In the discussion 
section below we explore the validity of our assumptions and the qualitative
effect of any deviations from them to our result.

In figure 7(a) we plot $k_1$ (the coefficient of $H_0^2$ in $\epsilon /\Delta$) 
versus $1/p=2r_0/\Delta$ on a linear
scale. If the demagnetizing field inside the droplet was uniform as in 
the case of unconfined droplets,
the plot would be a straight line. However, the plot is clearly not a straight
line and the deviations from the best fitted straight line are systematic (see
figure 7(b)).
This further supports our theoretical result that the demagnetizing field
inside a confined droplet is non-uniform and the elongation divided by 
thickness is proportional to the logarithm of the aspect ratio.




\section{Discussion}
\label{discussion}

The results discussed in section~\ref{expt} agree with our theoretical
prediction~(\ref{elongation}) of logarithmic variation of $\epsilon
/\Delta$ as a function of $1/p=2r_0/\Delta$. The predicted coefficient
of the logarithm, $\chi^2 H_0^2/\sigma_{FI}$, is consistent with
experimental observations.  However, our theoretical value for $C$ is
$4e^{-5/6}=1.74$, whereas the experimentally measured value for $C$ is
$0.35 \pm 0.08$. To explain this discrepancy we consider the validity
of two main assumptions, uniform magnetization and contact angle of
$90^{\circ}$. We explore, qualitatively, consequences of deviations
from these assumptions.

In section~\ref{theory} we employ a perturbation expansion for small
susceptibility. To lowest order, the magnetization ${\bf M} = {\bf
M}^{(1)} = {\bf H}_0$ which is constant. For our experiments the
susceptibility of the ferrofluid used was large, causing strong
demagnetizing fields near the edge of the droplet which, in turn,
reduce the in-plane component of the magnetization and create an
out-of-plane component of the magnetization near the edges.
Consequently, the magnetic surface poles near the droplet edge spread
out in a manner that reduces the magnetic energy $E_M$. We discuss
this effect further in section~\ref{magnetic-energy}.

We treated the case of contact angle $\beta=90^\circ$ between the
glass plate and liquid droplet. The experiment, however, was performed
with an oil-based ferrofluid in a surfactant solution for which the
oil-glass contact angle $\beta < 90^\circ$ (see figure 8). A contact
angle of other than $90^\circ$ redistributes the magnetic surface
charges and changes interfacial areas. We consider these two effects
in sections~\ref{magnetic-energy} and~\ref{surface-energy}
respectively. First, however, we address an ambiguity in the
definition of aspect ratio and elongation which results from the
non-uniformity of droplet cross-section.

Our experiment observes the profile of the largest cross-section of
the droplet. For a circular droplet with $\beta<90^\circ$ this is the
radius $r_1$ defined as the radius at mid-gap as shown in figure 8.
For an elongated droplet we measure the semi-major and -minor axes
$a_1$ and $b_1$ and, through equation~(\ref{definition}), the
elongation $\epsilon_1$. We also define $r_2$, $a_2$, $b_2$ and
$\epsilon_2$ associated with the ferrofluid-immiscible fluid-glass
plate contact line. Since $\Delta$ is much less than the capillary
length of the ferrofluid/immiscible fluid, to a good
approximation~\cite{contact} the profile of the droplet will be an arc
of a circle, so the difference between $r_1$ and $r_2$ is of order
$\Delta$, and likewise for the semi-major and -minor axes. The
difference $\epsilon_1-\epsilon_2$ is of order $\Delta/r_1$ relative
to the elongation.  Recall that our result~(\ref{elongation}) for the
elongation is only the lowest order term in a series expansion in the
aspect ratio. Thus the distinction between $r_1$ and $r_2$, and
between $\epsilon_1$ and $\epsilon_2$, does not alter our result at
the lowest order in aspect ratio.



\subsection{Corrections to magnetic energy}
\label{magnetic-energy}
Our assumption of uniform magnetization is valid for small $\chi$ but
for large $\chi$ strong demagnetizing fields reduce the in-plane
component of the magnetization $M_x$ and create an out-of-plane
component $M_z$.  This out-of-plane magnetization induces a charge
density on the top and bottom surfaces of the droplet while the
reduction of $M_x$ removes some charge from the droplet
edge.  This charge redistribution necessarily reduces the energy
$E_M$.  Deviations of the contact angle from $90^\circ$ affect the 
magnetic energy in a similar fashion because the inward or
outward bulging spreads a given amount of surface charge over a
slightly larger area. Charge redistribution reduces the values of the 
constants $B$ and $C$ in equation~(\ref{expansion}) but does not alter 
the coefficient of the logarithmic terms.  

The coefficient of the logarithmic terms remains unchanged because the 
total charge within a distance of order $\Delta$ from a given point on the 
edge remains unchanged under redistribution. Gauss's law relates the charge 
near the edge of the droplet
to the magnetization in the bulk. Although the demagnetizing fields are 
strong near the edge they fall-off as $1/r$ beyond a distance of order 
$\Delta$ from the edge. The magnetization in the bulk therefore  remains
uniform and equal to ${\bf M}^{(1)}$. Hence the total charge within a region 
of order $\Delta$ remains unchanged although it gets redistributed.


Now consider the origin of the order $V p \ln p$ terms in 
equation~(\ref{expansion}). These terms arise because ${\bf H}_D$
falls off like $1/r$ far from the edge of the droplet. The $1/r$
fall-off in ${\bf H}_D$ causes a $1/r$ variation in the deviation of
${\bf M(r)}$ from ${\bf M}^{(1)}$. After integrating over space as
indicated in equation~(\ref{magnetic-nrg}) this $1/r$ variation leads
to a logarithm of the aspect ratio. Charge redistribution leaves
the coefficient of the $1/r$ fall-off unchanged because of the
conservation of charge described above. Viewed from afar, the droplet
edge remains effectively a line charge with a fixed linear charge
density. Therefore the coefficient of the $V p \ln p$ term is
independent of the charge distribution.

In contrast, the order $V p$ terms in equation~(\ref{expansion})
depend upon details of the region close to the droplet edge. This is
evident by considering the total volume within a distance $\Delta$ of
the edge, estimated as an area $\Delta^2$ times the circumference
$2\pi r_0$. We consider, for the moment, the case $\epsilon=0$. Since
the droplet volume $V=\pi r_0^2 \Delta$, and $p=\Delta/2r_0$, we see
that the volume near the edge is of order $V p$. Surface magnetic
poles create an energy density in this region of space,
leading to an order $V p$ contribution to the demagnetizing energy
that determines the value of the constant $B$. Since charge
redistribution lowers the magnetic energy, the value of $B$ is
reduced.

A similar effect is observed in the self energies of a
current-carrying ribbon compared with a wire of circular cross-section
carrying the same current. The wire has a smaller self energy because
its current is distributed over a volume whereas the current is
concentrated on a surface for the ribbon. The functional form of these
self energies are related to the demagnetization energy in
equation~(\ref{expansion}). In particular, there are logarithmic terms
whose coefficients depend on the total current but not on the current
distribution, while the quantity equivalent to $B$ is smaller for the
wire than for the ribbon.

The demagnetization energy is always positive and decreases with
elongation.  A smaller demagnetization energy (due to smaller value of
$B$) should exhibit weaker variation with respect to the elongation.
Thus, we expect the value of $C$ for a bulging droplet with
non-uniform magnetization to be smaller than that for a straight-edged
droplet with uniform magnetization. This effect may explain why the
experimentally determined value of $C$ is lower than the theoretically
calculated value.



\subsection{Corrections to surface energy}
\label{surface-energy}
When the contact angle differs from $90^\circ$, the cross section of
the droplet depends on $z$. Consequently, the contact areas of the
glass plates with the droplet and with the surfactant solution may
vary as the droplet elongates. All the three interfacial areas must be
taken into account to calculate the surface energy. The total surface
energy is
\begin{equation}
E_S= \sigma_{FI} A_{\cal C}+2\sigma_{FG}A_G+2\sigma_{IG}(A-A_G),
\end{equation}
where the three surface tensions between ferrofluid-immiscible fluid, 
ferrofluid-glass, and surfactant solution-glass, are denoted by 
$\sigma_{FI},\sigma_{FG}$, and $\sigma_{IG}$ respectively, $A_C$ and $A_G$
are defined below and the total area
of the sample is denoted by $A$. The factors of $2$ in the second and
third terms of the surface energy account for the two glass surfaces. 

The area of the droplet-surfactant solution  interface $A_{\cal C}$ is 
given approximately by the circumference of $\cal C$ multiplied by the 
arc length of the bulge
\begin{equation}
\label{AC}
A_C=2 \pi r_1 (1+{3\over 16}\epsilon^2)
\Delta {(\pi/2-\beta) \over {\cos \beta}}.
\end{equation}
We use $r_1$ here to calculate the circumference of the droplet because it is
the radius observed during the experiment. To first order in the aspect ratio,
using $r_1$ or $r_2$ in equation~(\ref{AC}) yields the same result.


The droplet's contact area with the glass plates must be adjusted
to maintain a constant total volume of ferrofluid as the droplet elongates. 
We approximate the volume  of the bulging region by the circumference of 
$\cal C$ multiplied by the projected area of the bulge. The contact area
$A_G$ must be adjusted so that $A_G \Delta$ changes by the negative of the
change in volume of the bulge. Thus we write
\begin{equation}
A_G=2\pi r_2^2 {\bigg [}1- {3 \over 32} {\bigg \{}{(\pi/2-\beta)\over 
{\cos^2 \beta}}-{\tan \beta}{\bigg \}} {\Delta \over r_2} 
\epsilon^2{\bigg ]}.
\end{equation}
Using $r_2$ instead of $r_1$ makes the above result exact for zero elongation.
The area of ferrofluid in contact with the glass plates decreases with 
elongation for an acute contact angle because the volume of the fluid 
contained in the outward bulge of the droplet increases and therefore the 
fluid contained in the bulk of the droplet decreases. For obtuse contact 
angles exactly the opposite happens for similar reasons.


Finally, consider how these corrections to magnetic and surface
energies affect the elongation calculated in equation~(\ref{elongation})
for the case $\beta=90^{\circ}$ with uniform magnetization. The functional
form of the magnetic energy~(\ref{expansion}) remains the same but the values
of constants $B$ and $C$ are smaller. The $\epsilon$ dependence of the
surface energy remains quadratic, but the coefficient now depends upon
a linear combination of the three surface tensions $\sigma_{FI}$,
$\sigma_{FG}$ and $\sigma_{IG}$. In our result for the elongation~(\ref{elongation})
the constant $C$ will now have a value smaller than $1.74$ and $\sigma_{FI}$ will
be replaced by a linear combination of the three surface tensions. For 
$\beta \ne 90^\circ$ the 
experiment cannot be used to determine $\sigma_{FS}$ unless $\sigma_{FG}$ and 
$\sigma_{IG}$ are known. Since $\beta$ in general is not  
$90^\circ$, it is only possible to  measure the effective surface 
tension during elongation, and not  $\sigma_{FI}$ itself.

\section{Conclusions}
We study the elongation ferrofluid droplets, confined in thin film
geometry, under weak applied field. Our theoretical calculations
predict the elongation of a droplet depends logarithmically on aspect
ratio. This behavior contrasts with the case of unconfined
3-dimensional droplets where elongation is directly proportional to
undeformed droplet radius. We measured the elongation of ferrofluid
droplets in an experiment performed on ferrofluid droplets in a
ferrofluid/water/surfactant emulsion. The results of our experiment
agree with the functional form our theoretical prediction, however the
experimentally measured value of $C$ differs from the predicted value.
We suggest corrections due to high susceptibility and the droplet
contact angle with the confining plates as a source of this
discrepancy.

\acknowledgements
We acknowledge partial support for this research under NSF grant DMR-9732567.

\begin{references}

\bibitem{rr} R.E. Rosensweig, {\it Ferrohydrodynamics} (Cambridge, 1985).

\bibitem{d3droplets1} V.I. Arkhipenko, Y.D. Barkov and V.G. Bashtovoi, 
Magnetohydrodynamics {\bf 14}, 373 (1978). 

\bibitem{d3droplets2} V.I. Drozdova, T.V. Skrobotova and  V.V. Chekanov,
Magnetohydrodynamics {\bf 15}, 16 (1979).  

\bibitem{d3droplets3} J.C. Bacri and D. Salin, J. Physique. Lett. {\bf 43},
L-649 (1982).

\bibitem{ellipsoid}  V.G. Bashtovoi, S.G. Pogirinitsakaya and A.G. Reks,
Magnetohydrodynamics {\bf 23}, 23 (1987). 

\bibitem{d2droplets} C. Flament, S. Lacis, J.C. Bacri, A. Cebers, S. Neveu
and R. Perzynski, Phys. Rev. E {\bf 53}, 4801 (1996).

\bibitem{bashtovoi} V.G. Bashtovoi, {\it et al.} (unpublished, cited in
ref. 6).
 
\bibitem{sharpen1} J. C. Bacri and D. Salin, J. Physique. Lett. {\bf 44},
415 (1983).

\bibitem{sharpen2} H. Li, T.C. Halsey and A. Lobkovsky, Europhys. Lett. 
{\bf 27}, 575 (1994).

\bibitem{goldstein} S.A. Langer, R.E. Goldstein and D.P. Jackson,
Phys. Rev. A {\bf 46}, 4894 (1992).

\bibitem{pattern} R.E. Rosensweig, M. Zahn and R. Shumovich, J. Magn. Magn. 
Mater. {\bf 39}, 127-32 (1983).

\bibitem{bibette} J. Bibette, D. Roux and F. Nallet, Phys. Rev. Lett. {\bf 65},
2470-3 (1990).

\bibitem{liu1} J. Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier,
J. Bibette and J. Richard  Phys. Rev. Lett. {\bf 74}, 2828-31 (1995).

\bibitem{liu2} G.A. Flores,  M.L. Ivey, J. Liu,  M. Mohebi and  N. Jamasbi, 
{\it et al.}, Int. J. Mod. Phys. B. {\bf 10}, 3283-92 (1996).


\bibitem{wfb} W.F. Brown, ``Magnetostatic Principles in Ferromagnetism''
(North-Holland, Amsterdam, 1962).

\bibitem{ferrofluidics} Private communication, Ferrofluidics Inc. (1998).

\bibitem{contact} A.W. Adamson, ``Physical Chemistry of Surfaces'' 
(Interscience, 1967).

\bibitem{landau} L.D. Landau and E.M. Lifshitz, ``Electrodynamics of Continous 
Media'' (Addison Wesley, 1960).
\end{references}

\begin{figure}[tb]
\epsfxsize=6in \epsfbox{parallel-plates.eps}
\caption{(a) A side view of a ferrofluid droplet confined between 
two glass plates.
(b) A top view of a ferrofluid droplet elongating under applied field. The 
dashed line shows the undeformed droplet. N and S indicate the north and 
south magnetic poles. ${\bf H}_D$ is the demagnetizing field.}
\label{f1}
\end{figure}


\begin{figure}[tb]
\epsfxsize=3in \epsfbox{figure1.eps}
\vspace{0.5cm}
\caption{A schematic diagram of the experimental setup.  LS=Light Source, 
HC=Helmholtz Coil, S=Sample, TM=Tele-Microscope, CCD=CCD Camera, VR=Video 
Recorder, C=Computer}
\label{f2}
\end{figure}


\begin{figure}[tb]
\centerline{\psfig{figure=ferro.ps,angle=-90,width=3in}}
\caption{A low magnification view of the sample showing ferrofluid droplets
in emulsion.}
\label{f3}
\end{figure}

\begin{figure}[tb]
\vspace{0.5cm}
\centerline{\psfig{figure=droplet.ps,angle=-90,width=3.5in}}
\vspace{0.5cm}
\caption{A ferrofluid droplet with $r_0=132.5~\mu$m and $\Delta=13.1~\mu m$ 
elongating under an applied magnetic field of $50$ Gauss. The observed 
elongation is $0.56$.}
\label{f4}
\end{figure}

\begin{figure}[tb]
\psfig{figure=new_e_vs_H2.ps,angle=-90,height=4.0in,width=6.0in}
\vspace{0.5cm}
\caption{The plot of elongation vs $H^2$ for droplets with different radii
 and two different thicknesses. The error bars are smaller 
than the size of the symbols on the plot. The symbols and radii for droplets 
with thickness spacing of $\Delta=6.5~\mu m$ are $(\times,~96.5~\mu m)$, 
$({\Huge \bullet},~112.0~\mu m)$ 
and  $(+,~216.5~\mu m)$. The symbols and radii for 
droplets with  thickness spacing of $\Delta=13.1~\mu m$ are 
$(\boxdot,~50.0~\mu m)$, $(\blacksquare,~98.0~\mu m)$ and $(\odot,~177.0~\mu m)$.}
\label{f5}
\end{figure}


\begin{figure}[tb]
\psfig{figure=logfit.ps,angle=-90,height=4.0in,width=6.0in}
\psfig{figure=log_div.ps,angle=-90,height=4.0in,width=5.45in}
\vspace{0.5cm}
\caption{(a) The plot of $k_1$ vs $1/p= (2r_0/\Delta)$ on a log scale. The
dashed line is the best fit of the data to a straight line. (b) The deviation 
plot of the data from the best fit normalized by uncertainty of each data 
point.}
\label{f6}
\end{figure}

\begin{figure}[tb]
\psfig{figure=linfit.ps,angle=-90,height=4.0in,width=6.0in}
\psfig{figure=lin_div.ps,angle=-90,height=4.0in,width=5.45in}
\vspace{0.3cm}
\caption{(a) The plot of $k_1$ vs $1/p=2r_0/\Delta$ and the best fit straight
line. (b) The deviation plot of the data from the best linear fit.}
\label{f7}
\end{figure}

\begin{figure}[tb]
\epsfxsize=6in \epsfbox{contactangle.eps}
\caption{A ferrofluid droplet making an acute contact angle with the glass 
plates.}
\label{f8}
\end{figure}
\end{document}