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.