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Journal ArticleDOI

On the Interplay of Magnetic and Molecular Forces in Curie-Weiss Ferrofluid Models

01 Oct 1998-Journal of Statistical Physics (Kluwer Academic Publishers-Plenum Publishers)-Vol. 93, Iss: 1, pp 79-107

Abstract: We consider a mean-field continuum model of classical particles in Rd with Ising or Heisenberg spins. The interaction has two ingredients, a ferromagnetic spin coupling and a spin-independent molecular force. We show that a feedback between these forces gives rise to a first-order phase transition with simultaneous jumps of particle density and magnetization per particle, either at the threshold of ferromagnetic order or within the ferromagnetic region. If the direct particle interaction alone already implies a phase transition, then the additional spin coupling leads to an even richer phase diagram containing triple (or higher order) points.
Topics: Phase transition (60%), Quantum phase transition (58%), Ising model (58%), Curie–Weiss law (54%), Magnetization (53%)

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Journal
of
Statistical Physics, Vol.
93,
Nos. 1/2, 1998
On the
Interplay
of
Magnetic
and
Molecular
Forces
in
Curie-Weiss
Ferrofluid
Models
Hans-Otto
Georgii1
and
Valentin
Zagrebnov2
Received February
8,
1998
We
consider
a
mean-field
continuum model
of
classical particles
in R
d
with
Ising
or
Heisenberg spins.
The
interaction
has two
ingredients,
a
ferromagnetic spin
coupling
and a
spin-independent molecular
force.
We
show that
a
feedback
between these
forces
gives
rise
to a first-order
phase transition
with
simul-
taneous jumps
of
particle density
and
magnetization
per
particle, either
at the
threshold
of
ferromagnetic
order
or
within
the
ferromagnetic region.
If the
direct
particle interaction alone already implies
a
phase transition, then
the
additional
spin
coupling leads
to an
even richer phase diagram containing triple
( or
higher
order)
points.
KEY
WORDS: Classical continuous system;
first-order
phase transition;
mean-field;
tricritical; large deviations; maximum entropy principle.
1.
INTRODUCTION
Classical systems
of
particles located
in R
d
and
having some internal
degrees
of
freedom
are a
natural object
of
physical study. Examples
of
such
systems
are
ferromagnetic
fluids,
where each particle
has an
Ising
or
Heisenberg
spin;
liquid crystals consisting
of
long molecules with
a
dipole-dipole
interaction;
Coulomb gases
of
charged particles; and, more generally,
1
Mathematisches
Institut
der
Universitat
Munchen,
D-80333
Munich,
Germany;
e-mail:
georgii(a rz.mathematik.uni-muenchen.de.
2
Departement
de
Physique, Universite
de la
Mediterranee, C.P.T., CNRS-Luminy-Case 907,
F-13288 Marseille, Cedex
09,
France;
e-mail: zagrebnov(a:cpt.univ-mrs.fr.
79
0022-4715/98/1000-0079$15.00/0 © 1998 Plenum Publishing Corporation

80
Georgii
and
Zagrebnov
multitype particle systems—the internal degrees
of
freedom
then
correspond
to the
types
of the
particles.
The
last class includes
the
Widom-Rowlinson model
of two
types
of
par-
ticles
with
a
hard-core interspecies repulsion,
the first
continuum model
for
which
a
phase transition
was
established rigorously.
(29,22)
In a
closely
related
ferrofluid
model
of the first
class, spontaneous magnetization
was
established
by
Gruber
and
Griffiths.
(12)
A
common generalization
of
these
models
is the
continuum
Potts
model,
for
which
the
existence
of a
phase
transition
was
recently
proved;
(9)
see
also
the
references therein
for
related
work.
In
all
these examples,
the
phase transition originates
from
an
interac-
tion
between
the
internal degrees
of
freedom,
e.g.,
the
spin orientations,
and
it
manifests
itself
as an
orientational
order
resembling
the
familiar situation
in
lattice spin models.
(8,23)
For
continuum models, however,
one is
primarily
interested
in a
different
kind
of
critical phenomenon, namely posi-
tional
order, which corresponds
to a
liquid-vapor transition
and
involves
only
the
positions
of the
particles rather
than
their orientations
or
types.
In
fact,
positional order
has
been established
for
some models without inter-
nal
degrees
of
freedom—in
one
dimension,
(14)
in the
Kac-van
der
Waals
limit,
(15,17)
and
recently
for
certain long-range interactions
by
perturbation
about this limit.
(16)
In
this paper
we ask
whether
a
direct interplay
of
positional
and
orien-
tational order
can be
observed
in
specific situations.
3
A
physical picture
illustrating
an
interplay between positions
and
orientations
is the
following.
Consider
a
ferrofluid
with
a
ferromagnetic spin interaction which decreases
with
the
distance
of the
particles.
A
ferromagnetic ordering then induces
an
effective
increase
of the
indirect attractive
forces
between
the
particles.
This
effect
should increase
the
particle density.
By the
monotonicity
of the
spin
coupling,
the
resulting lowering
of the
average particle distance implies
an
increase
of the
effective
spin couplings,
and
thereby
a
strengthening
of the
ferromagnetic
order. This
in
turn increases
the
particle density again,
and
so
on. In
thermodynamic terms, this means
that
certain values
of the
particle density
and of the
magnetization
are
impossible,
so
that these
quantities must exhibit
a
jump.
In
other words,
one
expects that
a
direct
feedback
between
the
positional
and the
orientational structure
of a
system
can
change
the
nature
of a
phase transition
from
second order
to first
order.
3
This question,
of
course, does
no
refer
to the
trivial
fact
that positional order
in one
system
can be the
consequence
of
orientational order
in
another system. This
is
well-known
to be
the
case
for the
single-type Widom-Rowlinson model,
which
is the
one-type marginal
of the
two-types
Widom-Rowlinson model.
(29)

Curie-Weiss
Ferrofluid
Models
81
It
is the aim of the
present
paper
to
justify
the
above heuristics
in a
specific
model. Since more realistic systems seem
to be out of
reach
presently,
we
consider
a toy
model
of
mean-field
type. Namely,
we
consider
a
system
of
classical particles with Ising
or
Heisenberg spins which
are
coupled
by a
ferromagnetic Curie-Weiss interaction.
The
point
is
that
the
exchange rate
is
inversely proportional
to the
volume rather than
the
par-
ticle
number
(with
factor J>0),
so
that
the
effective
field
acting
on
each
spin
is
proportional
to the
magnetization
per
volume rather than
per
par-
ticle. This allows
for a
feedback between
the
ferromagnetic
and
positional
features
of the
model.
The
spin-independent interaction
will
be
modelled
by
a
suitable
"phenomenological"
function
g of the
particle density.
The
shape
of
g and its
relation
to J
determine
the
interplay
of
molecular
and
ferro-
magnetic
forces.
In
addition
to
these constituents
of the
model
we
have,
of
course,
the
standard parameters
B > 0, the
inverse temperature,
and z > 0,
the
activity
or "a
priori particle density."
We
will
show that,
for
suitable choices
of
these quantities,
the
model
exhibits
a first-order
phase transition
with
simultaneous jumps
of
particle
density
and
magnetization. This holds even
if the
direct particle interaction
g
alone does
not
induce
a
phase transition.
On the
other hand,
a
density-
independent
spin-coupling would only lead
to a
second-order transition.
It
is
thus clear that
the first-order
phase transition
is
indeed
a
consequence
of
some feedback mechanism.
If the
molecular interaction
g
already gives rise
to a
liquid-vapor transition,
the
interplay
of
positional
and
orientational
order
will
create
an
even richer phase diagram containing
a
tricritical point.
We
will
now
describe
our
results
in
some more detail.
For
simplicity
we
assume here that
the
particles have Ising spins with values
+ 1. The
ferromagnetic
coupling constant
J>0 is
kept
fixed, and we
look
for the
phase diagram
in the (z,
B)-quadrant.
The first
basic
fact
is the
existence
of
a
continuous curve
z =
z
m
(B)
which
separates
the
nonmagnetic
and the
ferromagnetic
parameter regions:
The
limiting phases
of our
system
are
nonmagnetic when
z<z
m
(B)
and
ferromagnetic when
z>z
m
(B).
Whether
or not the
phase transition
at
z
m
(B)
is of first
order
(with
jumps
of
both
density
and
magnetization) depends
on the
specific
features
of g. We
there-
fore
sketch three scenarios which
exemplify
the
variety
of
possibilities.
( For
stability
reasons
we
always assume that
g
grows
faster
than quadratically.)
Scenario
A.
While
the
ferromagnetic phase transition
at z =
z
m
(B)
is
only
of
second order
for
small
B, it is of first
order when
B is
sufficiently
large.
This scenario occurs
if g" is
increasing with
0 <
2g"(0)
< J; cf.
Proposi-
tion
3.2 and
Theorem
3.3
below.
The
simplest example
is
g(p)
= cp
3
with
c> 0; see
Fig.
1.

82
Georgii
and
Zagrebnov
Fig.
1. The
case
J= 1,
g(p)
=
p
3
/30.
(a)
Phase diagram.
The
ferromagnetic transition curve
z
=
z
m
(B)
splits
into
two
parts:
a
bold part indicating
a first-order
phase transition between
nonmagnetic
vapor (NV)
and
ferromagnetic
liquid
(FL),
and a
broken part corresponding
to
a
second-order transition
at
high temperatures.
The
point
A is the
liquid-vapor
critical point;
the
associated inverse temperature
B
A
is
determined
by the
equation
2g"(1/B
A
J)
= J. (b)
Den-
sky
(p) and
magnetization
per
particle
(m) for B =
0.5.
Scenario
B. The
ferromagnetic phase transition
at z =
z
m
(B)
is
still
of
second order
for
small
B and of first
order
for
intermediate values
of B.
For
large
B,
however,
the
phase transition
at the
magnetization threshold
z
m
(B)
is
only
of
second
order,
while
a first-order
transition with simul-
taneous jumps
of
density
and
magnetization occurs
at
some
z>z
m
(B),
i.e.,
in
the
interior
of the
ferromagnetic parameter region.
This holds,
for
example,
if g" is
convex with minimum
0 at
some
p
mm
>0,
and
2g"(0)>J;
see
Corollary
3.6 and
Theorem 3.4(c).
A
typical
example
is
g(p)
= c(p 1 )
4
with
c >
J/24;
cf.
Fig.
2.
Scenario
C. For
some critical inverse temperature
B
c
and z
c
=
z
m
(B
c
),
there
exist
phases with three
different
densities p_<p
#
<p
+
. The
phases with density
p
+
are
ferromagnetic (with positive
or
negative orien-
tation).
The
triple point
(z
c
,B
c
)
is the
endpoint
of a first-order
(liquid-
vapor) transition line
in the
(nonmagnetic) region {z<z
m
(B),
B<B
c
};
see
Fig.
3.
Following this line towards
(z
c
,B
c
)
one
obtains
the
limiting den-
sities
p _ and p
#
. On the
other hand,
in a
neighborhood
of the
triple point
the
ferromagnetic transition
at the
line
z =
z
m
(B)
is
also
of first
order.
Approaching (z
c
,
B
c
)
along this line
from
below
(B
<B
c
)
one
arrives
at the
limiting
densities
p
#
and p
+
, and
coming
from
above (B>B
c
)
one
ends
up
with
p _ and p
+
.In
other words,
for B < B
c
there
are two
density jumps
at

Curie-Weiss Ferrofluid Models
83
Fig.
2. The
case
J = 1,
g(p)
= (p - 1 )
4
. (a)
Phase
diagram.
The
bold
and
broken curves
and
the
critical point
A
have
the
same meaning
as in
Fig.
1. In
addition
to the two
phases
NV and
FL,
there
is an
intermediate phase
FV
(ferromagnetic vapor);
the
solid line separating
FV
from
FL
indicates
a first-order
phase transition. Since
the
transition
from
NV to FV is
second
order,
the
critical point
B is not a
triple point,
(b)
Density
and
magnetization
for B = 2.
different
activities,
one due to the
molecular forces
and one at the
incipience
of
ferromagnetic
order.
These
jumps
add up at B = B
c
, and an
enhanced jump persists
for B > B
c
.
This
scenario occurs,
for
instance,
if g" is
convex with
min g" < 0, and
J
is
suitably
chosen;
see
Theorem
3.8 and the
example
in
Fig.
3.
Fig.
3. The
case
J =
2.5, g(p)
= p
4
4p
2
.
(a)
Phase
diagram. Bold
and
broken lines
are as
in
Fig.
1. The
solid
line
indicates
a first-order
vapor-liquid transition
within
the
nonmagnetic
region.
The
point
C is a
triple point
of
coexistence
of
three phases: nonmagnetic vapor (NV),
nonmagnetic liquid (NL),
and a
ferromagnetic high-density
phase
which should probably also
be
interpreted
as a
liquid
(FL').
B is the V-L
critical point,
and A the
L-L' critical point,
(b)
Density
and
magnetization
for B =
0.48.

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01 Feb 1977

5,933 citations


Journal ArticleDOI
Abstract: We study a two-dimensional ferrofluid of hard-core particles with internal degrees of freedom (plane rotators) and O(2)-invariant ferromagnetic spin interaction. By reducing the continuous system to an approximating reference lattice system, a lower bound for the two-spin correlation function is obtained. This bound, together with the Frohlich–Spencer result about the Berezinskii–Kosterlitz–Thouless transition in the two-dimension lattice system of plane rotators, shows that our model also exhibits the same kind of ordering. Namely for a short-range ferromagnetic interaction the two-spin correlation function does not decay faster than some power of the inverse distance between particles, for small temperatures and high densities of the ferrofluid. For a long-range ferromagnetic interaction the model manifests a non-zero order parameter (magnetization) in this domain, whereas for high temperatures spin correlations decay exponentially.