The effect of anisotropy in the Dimer Model on Ferrofluids in

One-Dimension.

Dima Al-Safadi

Advisor: Dr. Abdalla Obeidat

Co-Advisor: Prof. Nabil Ayoub

Contents

• Introduction

• Theory

• Randomized easy axis

• Fixed easy axis

• Special cases

• Discussion and Conclusion

Introduction Magnetic anisotropy

In the absence of the applied magnetic field, the magnetic moments of the particles tends to align themselves to the direction which makes the magnetoststic energy minimum, this direction is called the easy direction, briefly, magnetic anisotropy shows how the magnetic properties depend on the direction of measuring the magnetization. There are several types of anisotropy and the most common one is the magnetocrystalline anisotropy.

Magnetocrystalline anisotropy (crystal anisotropy)

When single domain fine particles magnetized to saturation, the magnetization has an easy axis along which is prefer to stay. In this case, the total internal energy is minimum. Rotation of the magnetization vector away from an easy axis is possible only by applying an external magnetic field. Thus, the magnetic energy is direction-dependent, and this kind of energy is called magnetic anisotropy energy. It is also called magnetocrystalline anisotropy because it has the same symmetry as the crystal structure of the particle material.

one of the simplest expression of the magnetic anisotropy energy that is uniaxial in symmetry is

2sinVKEa

E

What is the ferrofluids?

Ferrofluid is a stable colloidal suspension consists of single-domain fine magnetic particles (ferromagnetic or ferrimagnetic particles), of diameter (30-150) in a carrier liquid (such as water or kerosene). To prevent the agglomeration and enhance the stability of the suspension, a nonmagnetic surfactant layer (such as polymers) covers particles. Recently, the publications concerned with ferrofluids have increased because these materials have a wide uses in industry, medicine and agriculture

Randomize easy axis on ferrofluid dimer model in one-dimension

We will study the effect of the magnetic anisotropy and the effect of the dipole-dipole interaction on the magnetic properties of a dilute ferrofluid consists of a single domain magnetic fine particles.

The easy axis of each particle ( )directed randomly and the magnetic moment ( ) is freely to rotate in three dimensions.

E

Dimer model

To simplify our calculations, we will use a simple theoretical model called (Dimer model) in this model, the uniaxial single domain magnetic fine particle interacts with only one other adjacent particle thus, our assembly consists of ( ) independent non-interacting systems move in one dimension. Since each system composed of two interacting particles, we call it a dimer.

2N

For one system we can write

E

x

kji ˆsinsinˆcossinˆcos

E

kjiE ˆcosˆsinsinˆcossinˆ

Theoretical back groundThe partition function for one system is

Where

and

deZ kT

ET

aT EEEE 0int

53.int

).)(.(3.

r

rr

rE

HHE

..0

2sinKVEa

Therefore

To calculate This integral we assume the following:1-The interparticle interaction energy is small compared with thermal agitation energy2-The anisotropy energy is small compared with the thermal energy 3-The applied magnetic field is very small.

deZ kT

EEE a )( 0int

For system the total partition function is given by

The magnetization is defiand as

And the initial susceptibility is given by

)2(N

)!2(

2

N

ZZ

N

T

TZkTln

H

MH 0lim

Field parallel to the assembly

The following expression have been calculated for the partition function

Where

2152413

202

211

200 iCxFCxFCiCiCiCZ

)(256 0

24

0 ikT

KVzzeC

20

2

224

1

11256

zze

kTC

i

kTKV

)(3

512 0

24

2 ikT

KVzze

kT

KVC

50

5

2

22

44

3

11

1016

zze

TkC

i

kTKV

And

ikTKV

zzeTk

VKC

0

2

22

224

4 232

20

2

2

22

24

5

11

61024

zze

Tk

KVC

i

kTKV

2

121020

28825664 i

xii

xi xF1

xF2 ]45

224

5

18[ 2

010 iiix

The magnetization M is given by

And the initial susceptibility is calculated to be

45

298)

9

66(

33.63.33

33

4320

5432

10

CCCC

CCCC

CC

xNM

0

1

2

3

3

C

TCT

kN

However, the general form of Curie-Weiss law is:

Therefore our result obeys Curie-Weiss law with

and

0TT

C

k

NC

3

2

22

0

02

0 3 i

i

zz

zz

kT

Magnetic field perpendicular to the assembly

Thank you