THE MAGNETIC PROPERTIES OF A FERROFLUID WITH DIPOLAR INTERACTIONS

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THE MAGNETIC PROPERTIES OF A FERROFLUID WITH DIPOLAR INTERACTIONS

A. Bradbury, S. Menear, R. Chantrell, K. O’Grady

To cite this version:

A. Bradbury, S. Menear, R. Chantrell, K. O’Grady. THE MAGNETIC PROPERTIES OF A FER-

ROFLUID WITH DIPOLAR INTERACTIONS. Journal de Physique Colloques, 1985, 46 (C6), pp.C6-

283-C6-286. �10.1051/jphyscol:1985649�. �jpa-00224904�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°9, Tome 46, septembre 1985 page C6-283

THE MAGNETIC PROPERTIES OF A FERROFLUID WITH DIPOLAR INTERACTIONS

A. B r a d b u r y , S. Menear, R.W. C h a n t r e l l and K. O'Grady

Department of Physics., University College of North Wales, Bangor, Gwynedd LLS7 2UW, U.K.

+

School of Physios and Astronomy, Lancashire Polyteohnic, Preston, Lanes. PR1 2TQ, U.K.

++

Department of Physics, University of Technology, Loughborough, Leies LE11 2TU, U.K.

Résumé - Un modèle décrivant les propriétés d'un ferrofluide contenant des particules interagissant faiblement est présenté. Le modèle est fondé sur la méthode de Monté-Carlo et il est utilisé pour étudier l'effet des inter- actions sur les courbes d'aimantation. Cet effet est de renforcer l'aimanta- tion à un degré qui dépend du champ magnétique.

Abstract - A model of the properties of a ferrofluid containing weakly interacting particles is presented. The model is based on the Monte-Carlo method, and is used to study the effects of interactions on the magnetisation curve. It is shown that the effect of interactions is to enhance the magnetisation curve to a degree dependent on the magnetic field.

I - INTRODUCTION

The magnetic properties of dispersions of magnetic particles are affected to some extent by the dipolar interactions which exist between the particles. Stable ferrofluids, however, can generally be considered to be weakly interacting systems since strong interactions tend to produce large agglomerates which are likely to result in colloidal instability. We have previously investigated /1,2/ the low field properties of weakly interacting systems and shown that interactions result in a Curie-Weiss like variation of initial susceptibility with temperature, of the form Xi a ( T - T

)

where the ordering temperature T

is dependent only upon the strength of the interactions between particles.

In this paper, we investigate the variation of magnetisation for an interacting system with applied field over the whole range of field values. Comparison with the properties of a non-interacting system shows the extent to which the magnetic behaviour is modified by the effects of dipolar interactions.

II - MONTE-CARTJO MODEL OF INTERARTICT.E INTERACTIONS

We have carried out a computer simulation of an interacting ferrofluid using a method based on the Monte-Carlo method of Metropolis et a\ /3/. This is based on a representative cell of N(=1000) particles. Each particle in turn is given a random displacement in its coordinates and the energy difference involved in this Monte-Carlo 'move' calculated. If AE<0 the move is allowed, otherwise the move is allowed with a probability exp (-AE/kT). After many such moves of particle, the system evolves into a thermal equilibrium configuration with energies characterised by a Boltzmann distribution.

Once thermal equilibrium has been established, it is possible to calculate useful thermal averages, for example, the magnetisation, which is given by

(1)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985649

JOURNAL DE PHYSIQUE

where

is the a n g l e between the magnetic f i e l d

and the magnetic moment ^p of the particle at the end of the t t h Monte-Carlo move. I n o r d e r t o extend the s i z e o f the cell, p e r i o d i c boundary c o n d i t i o n s are used, ie,

a p a r t i c l e l e a v e s the c e l l , it is assumed t o re-enter from the o p p o s i t e s i d e . In a l l the simulations, c a r e was t a k e n t o e n s u r e that e q u i l j b r i u m had been reached b e f o r e the summation o f equation ( 1 ) was c a r r i e d o u t . I n o r d e r t o q u a n t i f y the e x t e n t o f the spatial o r d e r i n the system, pair c o r r e l a t i o n f u n c t i o n s (PCF) were c a l c u l a t e d . The PCF is d e f i n e d as g ( r )

n ( r ) / k where n ( r ) is t h e number d e n s i t y a t a d i s t a n c e r from a c e n t r a l particle and R is the average number d e n s i t y for t h e c o l l o i d . For s i m p l i c i t y , a two dimensional s i m u l a t i o n was used. I n a d d i t i o n t o c o n t r i b u t i o n s t o the t o t a l energy from the d i p o l a r i n t e r a c t i o n s and the a p p l i e d f i e l d , a r e p u l s i v e term was added i n o r d e r t o s i m u l a t e the effects o f the s u r f a c t a n t molecules w i t h which the particles are coated. The fona used w a s t h a t o f the e n t r o p i c r e p u l s i o n f o r c e given

Rosenweig et al.

1

Et -

ANArrYSIS

W e have carried o u t s i m u l a t i o n s for a ferrof 1.ui.d c o n t a i n i n g monodisperse c o b a l t particles coated wifh s u r f a c t a n t molecules o f 1.ength 20

m g n e t i . s a t t o n c u r v e s were c a l c u l a t e d for v a r i o u s v a l u e s o f p a r t i c l e diameter. The average p a . r t i c l e s e p a r a t i o n was k e p t c o n s t a n t

making the c e l l s i z e a

The c e l l corresponds t o a s e c t i o n through a f e r r o f 1ui.d o f volumetric packtng fracti.on

E =

0.05. I n general, the f e r r o f l u i d s s t u d i e d showed short-range o r d e r only. This is demonstrated i n F i g

which shows a three

dimensional p l o t o f g ( r ) f o r

c o b a l t p a r t i c l e s i n z e r o a p p l i e d fie1.d at rorna tf?mperature. The short range o r d e r is r e f l e c t e d i n the n e a r e s t neighbour peak.

a p p a r e n t s t r u c t u r e wi.thi.n

peak is a consequence of the s u r f a c e f i t t i n g routi.ne and is non p h y s i c a l . S t u d i e s were

t n the particle s i z e range 5 0 < 0 < 8 0 A . These systems Fig.

- P l o t o f t h e s p a t i a l d i s t r i b u t i o n were _weakly i n t e r a c t i n g f u n c t i o n g ( r ) f o r

cobalt p a r t i c l e s i n where long range o r d e r was z e r o a p p l i e d f i e l d a t room temperature. n o t observed f o r particle d i a m e t e r s smaSler than

1.0

0.8

C 0

.-

- _.g

+ 0.6

m

0.4

P

0.0

0 100 200 300 400 500

c o n f i g u r a t i o n s whjch l e a d t o a

Magnetic Field (Oel

reduced magnetisati.on f o r a

given f i e3 d

- ^{P i g}

^shows

F i g .

- Magnetisation c u r v e s for 75A c o b a l t the

between t h e p a r t i c l e s : open circles w i t h m a g n e t o s t a t i c I n t e r a c t i ng and non-jnter- i , n t e r a c t i o n s and s o l i d circles without. ~ = 3 0 0 K . a c t i n g c u r v e s a s a f u n c t i o n

I I I I I

h-----0-

o

~ , . - - -

* - - -

oP'rp

- /cop -

,

! I

F i g

shows the magnetisation curve f o r 75

c o b a l t particles. Also gjven i n F i g

i s t h e magnetisation curve f o r a non-interacting system.

I t

can be seen t h a t t h e effect

o f i n t e r a c t i o n s is t o enhance

t h e magnetjsation o f the

f l u i d . T h j s is c o n s i s t e n t

w i t h t h e e x i s t e n c e o f short

range order. Long range o r d e r

r e s u l t s I n c l o s e d l o o p

TV

- RXPERWENTAL MFSSURRWWTS AND

CON

0

12 o f t h e applied- f i e l d K. The

increase i n

with f i e l d ia

C

0

- o

presumably a r e f l e c t i o n o f f i e l d induced aggregatton, an effect which has been observed on a l a r g e r scale in more strongly i n t e r a c t i n g

This e f f e c t

due t o

We

have measured magnetisation curves a t various temperatures f o r a f e r r o f l u i d c o n s i s t i n g o f c o b a l t p a r t j c l e i n toluene. Thls was prepared by t h e c o b a l t carbonyl decomposition method o f H e s s and Parker /E/. By e l e c t r o n microscopy, t h e mean p a r t i c l e s i z e o f t h e sample was found t o be 5 7

which is i n the weak i n t e r a c t i o n regiuie. It was found t h a t magnetisation curves measured a t d i f f e r e n t t e m p r a t u r e a d i d not superimpose when p l o t t e d a s a function o f E/T. i n agreement with t h e p r e d i c t i o n s of the Monte+rlo modeJ outlined previously.

m m

L 0

0 -001-O I

the increased altgignment o f t h e p a r t i c l e moments as the field increases, r e s u l t i n g i n an increase i n t h e i n t e r a c t i o n strengiA at la-e

0 10 0 200 300 400 500

f i n a l l y decreases as the

Magnetlc Fleld toe1

system approaches magnetic

s a t u r a t i o n . Using the Monte- Fig. 3 - Increase i n t h e magnetisation due Carlo model, we have obtained

t o magnetostatlc i n t e r a c t i o n s p l o t t e d a s a magnetisation curves f o r a function o f applied f l e l d . T=300K. v a r i e t y o f p a r t i c l e sizes.

The m a u l t s a r e shown i n f i g

I I I I l 4,

which gives t h e reduced

08 - magnetisation a s a function o f

t h e parameter a

wlth p

C

the magnetic moment o f t h e

p a r t i c l e . I n t h e absence o f

- i n t e r a c t i o n s , the magnet- i s a t i o n i n 2-D is given by /7/

- -

1 - cose exp(acoee)ae

E exp( acos9)de

-

The non-interacting curve was

I

_ calculated using this

0 0.4 0 8 1 2 16 20 expression. Pig 4 also gives

r H l k T reduced magnetisation curves

f o r diameters of

Fig.

- Reduced magnetisation p l o t t e d as a

and

~t can be function of The lowest curve is f o r seen t h a t t h e r e ^is an t h e non i n t e r a c t i n g case. increasing deviation from t h e non-interacting curve as t h e p a r t i c J e diameter increases.

I n f a c t , t h e deviation from the non-interacting curve depends on the parameter

l r 2 / ~ 3 k ~ (with R the average d i s t a n c e between p a r t i c l e s ) which represents the

z e l a t i v e s t r e n g t h of the i n t e r p a r t i c l e i n t e r a c t i o n s . Thus there

an implicit

temperature dependence i n the curves of fxg

Hence t h e model p r e d i c t s t h a t f o r an

tnteract'ing 8ystem. magnetisation curves a t a x f f e r e n t temperatures should n o t

superimpose when p l o t t e d as a function of H/T, as is usually expect& f o r a

superparamagnetic 8ystm.

C6-286 JOURNAL DE PHYSIQUE

The low field portions of the experimental magnetisation curves are shown in Pig 5, where f is plotted as a

function of H/T. The non- superposition of the magnefe isation curves is in agreement with our Monte-Carlo model, and shows that H/T super- position cannot be used as a test for superparamagneti.sm.

A detailed comparison between theory and experiment is not possible at present, due to the two dimensional nature of the Monte-Carlo model.

However theory and experiment are broadly in agreement regarding the enhancement of the magnetisation and consequent non H/T superposition. Further comp- utations are presently being carried out using a 3-D mode.!

in order to effect a comparison between theory and experiment.

Thus interactions can be seen to affect the properties of even weakly interacting systems. It is possible that the predicted H/T nonsuperposition (since it depends only on interactions) could be used to investigate the effects of interactions. This is particularly interesting since it allows the effects of magnetic fields on the interactions to be studied.

Fig. 5 - Magnetisation curves plotted as a function of H/T to show non superposition at low field values, squares are at 200K, open circles at 230K and solid circles at 293K.

REFERENCES

/ V O'Grady, K., Bradbury, A., Charles, S.W., Menear, S., Popplewell J. and Chantrell, R.W., J. Magn. Mag. Mater., 31-34 (198?.) 958

/Z/ Menear, S., Bradbury, A. and Chantrell, R.W., J. Magn. Magn. Mater., 39 (1983) 17

/3/ Metropolis, N., Rosehbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, F...

J. Chem. Phye.. 21 (1.953) 1087

/ 4 / Rosenweig, R.K., Nestor, J.W. and Timmlns, R.S., A. t. Che. E. t. Chem, K., Symposium series No 5 (1965) 104

/5/ Chantrell, R.W., Bradbury, A., Popplewell, J. and Charles, S.W., J. Appl. Phys.

53 (1982) 2742

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