AN INTEGRAL EQUATION APPROACH TO PHASE TRANSITIONS IN FERROFLUIDS

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AN INTEGRAL EQUATION APPROACH TO PHASE

TRANSITIONS IN FERROFLUIDS

G. Martin, A. Bradbury, R. Chantrell

To cite this version:

(2)

JOURNAL DE PHYSIQUE

Colloque C8, Supplkment au no 12, Tome 49, d6cembre 1988

AN INTEGRAL EQUATION APPROACH TO PHASE TRANSITIONS IN FERROFLUIDS

G. A. R. Martin ( I ) , A. Bradbury (I) and R. W. Chantrell (2)

(I) Physics Department, UCNW, Bangor, Gwynedd, G.B.

(2) Lancashire Polytechnic, Preston, Lancashire, PRI PTQ, G.B.

Abstract. - Recent computational and experimental studies of the magnetic behaviour of ferrofluids have shown the existence of Curie-Weiss type behaviour at high temperatures. There has not, however, been clear evidence of a magnetic phase transition and we present here some results which have beenobtained in order to examine this transition.

1. Introduction

A number of computer simulation studies [I, 21 of ferrofluids using Monte Carlo methods have been per- formed in recent years. A result of particular inter-

est from these calculations is the prediction of Curie- Weiss type behaviour for the magnetisation. This result has been verified experimentally, e.g. O'Grady [3], and so it is of interest to determine whether there are any magnetic phase transitions in a ferrofluid. The Curie-Weiss behaviour immediately suggests some form of ferromagnetic ordering though this seems unlikely since the dipole-dipole interaction used in the models favours both parallel and antiparallel align- ment of the magnetic moments. At low temperatures the suspension tends to agglomerate into clus- ters with low resultant magnetic moment and so some form of antiferromagnetic or spin glass ordering may be present.

Results obtained from Monte Carlo simulation have been rather unclear due to the usual problems of the critical slowing down of the relaxation times near phase transitions. Recently, we have reported [4] an alterna- tive approach to the study of the behaviour of ferrofluids using techniques from the integral equation theory of fluids and in particular, we have used the Refer- ence Hypernetted Chain approximation (RHNC) as a closure approximation to the Omstein-Zernike equation. By looking for any divergences in the expansion of the correlation functions one can infer the existence of phase transitions and in this study we have considered the behaviour of the higher order terms in order to determine the nature of any magnetic transitions in a ferrofluid if they exist.

2. Theory

The Omstein-Zernike (OZ) equation plays an essen- tial role in the integraI equation theories of fluids and is given by:

where h (12) is the pair correlation function, c(12) is

the direct correlation function, p is the particle density and d3 is understood to mean integration over both the translational and orientational coordinates of particle

3. The correlation functions are obtained from equation (1) by an appropriate set of closure relations. In the solution of the mean spherical approximation for systems with anisotropic potentials Blum [5, 61 showed that in Fourier transform space by expanding the correlation functions in terms of rotational invariants the OZ equation can be written as

where

C z X

(k) and

fiEX

(k) are linear combinations of the Hankel transforms of c;I","' (r) and

T l r ~ '

(r)

(= h;I","' (r)

-

c;T;nl (r)) respectively.

For linear molecules (i.e. ,u = v = 0) equation (2)

can be written as

ax

(k) p e X (k)

ex

(k)[l+ (-I)'+' P& (k)]-I 73) where the

fix

(k) and

ex

(k) are matrices whose elel ments ( i , j) are

RZ,,

(k) and

Cg,,

(k) respectively. I is the identity matrix.

The modei we choose for our ferrofluid is that of hard spherical particles with point dipoles embedded in their centres. We have recently [4, 81 solved,$he OZ equation for our model ferrofluid using the reference hypernetted chain approximation (RHNC) as given by Fries and Patey [9] which is an extension of the Linear and Quadratic hypernetted chain approximations.

The existence of phase transitions in the system can be determined by investigating the singularities of the [I -t- ( - ~ ) ~ + l p& (k)] matrix, which we denote by (k)

,

in equation (3). In this study we choose the basis set consisting of the following combinations of mnl for the @;;"on' (a1, 522, F12) occuring in the rota, tional invariant expansion of the correlation functions:

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C8 - 1848 JOURNAL DE PHYSIQUE

(0001, (110), (112), (022), (202), (2201, (2221, (2241, (132), (134), (312), (314), (330), (332), (334), (336). The OZ equation then decouples into four independent matrix equations with

x

ranging from 0 t o 3 (negative values of

x

provide no new information due to sym- metry); the largest matrix (4 x 4) occuring for

x

= 0. We have computed the functions det [,D (f)] for each

value of

x

for the system studied in reference [4] i.e. at a packing fraction of 1 % and a reduced interaction strength

(A)

of 7.2. At this interaction strength difficulties'are exkountered in obtaining a convergent solution during the iterative procedure and it may be expected that this is associated with the onset. of a transition. The results indicate no divergence of the functions except for the possible onset of a divergence a t k = 0 for

x

= 0. Inspection of the o B (k) matrix at Ic = 0 reveals it t o be diagonal with the elements

0 ~ 1 1 (0) = 0.563, oB22 (0) = 1.000, 0 ~ 3 3 (0) = 0.949

and 0 B 4 4 (0) = 0.999.

The computer results indicate the vanishing of the (1, 1) element of the atlove matrix for Ic = 0 which im- plies the divergence of hooO (0). This is associated with the usual liquid-gas transition as can be seen by writ- ing the compressibility equation in terms of hooO (0)

where p is the partial pressure of the suspension. The magnetic response of a ferrofluid to a vanish- ingly small external field Ho is given by

where m is the magnetisation.

The divergence of hl10 (0) would indicate the pres-

ence of ferromagnetic ordering throughout the suspension and would be observed by the vanishing of the

0B22 (0) term, but this is not apparent for the case examined. By inspection of the matrices oczurring in the OZ equation it can be seen that both hl10 (0) and k112 (0) are linear combinations of o d ~ : (0) and

OD:: (0) and thus any long range behaviour in hl10 ( 7 )

will be intimately connected with the corresponding behaviour in h112 ( T )

.

The smallest element after the 0 ~ 1 1 ( 0 ) component is the 0 ~ 3 3 (0) term. This_appears t o be decreasing at

a quicker rate than the 0 D22 (0) term and its vanish-

ing may be associated with higher order orientational transitions. Any ferromagnetic type transition in a ferrofluid will be accompanied by the existence of aligned chains of particles but the dipole interaction will tend

to align the dipoles in adjacent chains antiparallel thus inhibiting this transition. As a consequence of this one may expect that after the liquid-gas transition the orientational transition may be of higher order (e.g. associated with the long range behavior of the hZz0

component) than the ferromagnetic transition.

3. Conclusions

We have considered here the integral equation theories of fluids t o obtain some information on the nature of phase transitions in ferrofluids. By expanding the pair correlation function in terms of rotational invariants one is able to solve the hypernetted chain approximation for angle dependent interactions. This expansion allows us to examine the various orientational structures in the fluid. For exemple, the hllo

term gives information as to the possibility of a ferromagnetic type transition in a ferrofluid. The difficulty in obtaining convergent solutions for the case examined appears t o be associated with the usual liquid gas transition with no corresponding long range behaviour in the angular components.

Further study of the angular components will be nec- essary in the liquid phase (where the ferro%uid particles are no longer dispersed) of the suspension in order t o further investigate the presence of any angular ordering. It is unlikely that any long range ferromagnetic ordering can exist and much more likely seems to be the existence of a spin glass type transition where the particle orientations will "freeze" into long-lived metastable states.

[I] Menear, S., Bradbury, A. and Chantrell, R. W., J. Magn. Magn. Mater. 43 (1984) 166.

[2] Bradbury, A., Menear, S. and Chantrell, R. W.,

J.

Magn. Magn. Mater. 54-7 (1986) 745. [3] O'Grady, K., Bradbury, A., Charles, S. W., Me-

near, S. w . , Popplewell, J., Chantrell, R.

w.,

J.

Magn. Magn. Mater. 31-4 (1983) 958.

[4] Martin, G. A. R., Bradbury, A. and Chantrell, R. W., J. Magn. Magn. Mater. 65 (1987) 177.

[5] Blum, L. and Torruella, A. J., J. Chem. Phys. 56

(1971) 303.

[6] Blum, L., J. Chem. Phys. 5 7 (1972) 1862.

[7] Edmonds, A. R., Angular Momentum in Quan- tum Mechanics (Princeton University, Princeton) 1957.

[8] Martin, G. A. R., Bradbury, A. and Chantrell, R. W., IEEE Trans. Magn. MAG-22 (1986) 1137.

[9] Fries, P. H. and Patey, G. N., J. Chem. Phys. 82