RELAXATION PHENOMENA IN FERROFLUIDS

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RELAXATION PHENOMENA IN FERROFLUIDS

H. Winkler, H.-J. Heinrich, E. Gerdau

To cite this version:

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JOURNAL DE PHYSIQUE CoUoque C6, s~pplkmetzt uu no 12, Tome 37, Dicembre 1976, page C6-261

RELAXATION

PHENOMENA IN FERROFLUIDS

H. WINKLER, H.-J. HEINRICH and E. GERDAU 11. Institut fur Experimentalphysik der Universitiit

Hamburg, Fed. Rep. of Germany

Rksumk.

-

Un terrofluide h base de diester contenant des particules ultrafines de magnetite a

ete etudik par spectrometrie Mossbauer entre 77 et 300 K. Les processus dynamiques predominants sont les relaxations superparamagnbtiques et la diffusion de translation. Les relaxations superpara- magnktiques sont plus faibles dans la phase liquide que dans le matQiau gelb. Le coefficient de diffusion mesure est en accord satisfaisant avec la thkorie, si on suppose l'existence d'un glissement du fluide ?la surface des particules. i

Abstract. - A ferrofluid on diester base containing ultrafine magnetite particles has been inves-

tigated by means of Mossbauer spectroscopy in the temperature range from 77 K to 300 K. Super- paramagnetic relaxations and translational diffusion have been found to be the predominating dynamic processes. A slowing down of the superparamagnetic relaxations in the liquid phase compared with frozen material points to a local ordering. The measured diffusion coefficient is in satisfactory agreement with the theory, if a slip of the fluid flow at the particle surface is assumed.

1. Introduction.

-

A ferrofluid js a colloidal suspen- sion of ultrafine magnetic particles (about 100

A

diameter) in a carrier fluid giving an homogeneous magnetic material, which has nevertheless retained its fluid characteristics. Under the influence of an external field it gets magnetized nearly instantaneously and loses its magnetization after removal of the field probably within microseconds.

Investigations of ferrofluids by means of "Fe Mossbauer spectroscopy at various temperatures are expected to yield valnable information about dynamic phenomena in liquids. There can be distinguished three principally different mechanisms, which should influence the spectra :

(i) translational motions of the particles (ii) rotations

(iii) superparamagnetic relaxations.

Translational motions lead always to a line broadening irrespective of the underlying mechanism. Rota- tions cause changes of the quantization axis and thus produce also line shifts. They can occur in two different manners, either more diffusionlike or by strong collisions. A diffusionlike behaviour is characterized by a continuous process of reorientation of the particles axes, whereas the term strong collisions specifies changes, which are so abrupt that no correlations exist between initial and final states. The spectra differ in a characteristic manner from each other as will be shown in the following section, where the theoretical treatment of randomly reorienting hyperfine fields is reviewed briefly.

Superparamagnetic relaxations, which can be des-

cribed by a magnetic field flipping up and down, have to be taken into account because of the particle size. It is known from investigations of Roggwiller and Kiin- dig [l] that magnetite particles exhibit superparama- gnetism yet below room temperature. The theoretical spectra for this case have been published for the first time by Wickman et al. [2].

Since the theroy is well established, it should be possible to seperate the processes in the experimental spectra. For this purpose additional information can be obtained by the application of an external magnetic field.

2. Mossbauer spectra in the presence of randomly reorienting hyperfine fields.

-

The starting point for the evaluation of absorption spectra in the presence of stochastic fluctuations of the hyperfine fields is the expression

(1) with

The equation is written in the Liouville formalism.

R^

is the relaxation (super-) operator and

.Ett

the stochastic Liouville operator, which is given by the

(3)

~ ~ 2 6 2 H. WINKLE^, H.J. HEINRICH AND E. GERDAU

hyperfine interactions in the various stochastic states treated in Ref. [ S ] for the case of perturbed angular 13, 41 .X,(ko) is the Hamiltonian for the absorption of ~~rrelations. As has been shown there the

a photon with momentum hk and helicity o by the

nucleus, and p describes the system nucleus plus envi-

I

x(Ff,

JL

;

4 I

ronment in its initial state. representation is Most appropriate to the problem. Randomly reorienting hyperfine fields have been The ,matrix elements of

R^

are given by

DJ(J

+

1) for diffusion with

y(J)

=

(

W(l

-

6j0) for strong collisions.

For the coupling between the nuclei and the extranuclear field the form

is assumed, where the A:' are the components of an irreducible spherical tensor of degree I, which have to be constructed from the nuclear spin operator I, while the B;) ( a ) describe the properties of the reorienting hypefine field. The set of Euler angles o = (a,

b7

B,)

defines the actual orientation with respect to a fixed laboratory frame S. With (3) one gets for the stochastic Liouville operator

For the reduced matrix element (I, I,

L,

(1

11

I, I,,

L,)

the expression (39) of [5] cannot be taken, because in the case of Mossbauer absorption there are involved two nuclear levels with spins I, and I, instead of the

single intermediate level in perturbed angular correlations. As it is shown in the Appendix one gets

Using the basic multipole operators

for the representation of the operator X,(ko) gives [6]

with

L

~ $ $ ~ ( k ) =

C

(- (2k

+

l )

($

L,

k even I - M 2 - N

A model, which is based on randomly reorienting hyperfine fields, is necessarily isotropic in space, so that the initial state p(0) of the system nucleus (n) plus environment (e) is apart from a normalization constant given by the direct product of the unity operators 1, and 1, :

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RELAXATION PHENOMENA IN FERROFLUIDS C6-263

as can be verified from the formulas given in Ref. [5] one gets for the absorption spectrum after averaging over the directions and polarizations of the incident photons

-

1

X (x(L, M,, OL2 ; 0)

I

[PE

-

X:,-

g]

/

x(L1 M,, OL1 ; 0)) 2

=

-

Re

c

(2 L

+

l)-'

I

<

I ,

I1

Ln

II

I ,

>

i2

X L M

X (*(LM, OL; 0)

~e:,

- g]

I

_{x(LM, OL; 0))}

.

₍₁₁₎

Taking into account that the matrix elements of X: and

2

do not depend on the magnetic quantum number f = M it follows finally

If

I

M,) and

(K,

I

are eigenkets and -bras belonging to the eigenvalue i

- L , + i o q of

+ - G + @

h

one gets because of the closure relation

3. Discussion of the theoretical results.

-

For the case of 57Fe one has to deal with a M 1 transition between I, =

3

and I, =

4.

A magnetic hyperfine

interaction requires to diagonalize a 6 X 6 matrix, as

only the matrix elements with

L = 1, J = O , 1 , 2 and L = 2 , J = l , 2 , 3 mix with

The results of the numeric calculations for a transmis- sion spectrum are shown in figure l a and b. The values chosen for

r

and H,, were 0.2 mm/s and 500 kOe.

The two models differ mainly in the way, that the inner lines are broadened stronger than the outer ones by rotational diffusion, while strong collisions have nearly the same effect on all lines of the spectrum. The spectra resulting from the strong collision model are very similar to those given by Dattagupta and Blume [7]. Indeed, formula (12) of the last section can be transformed into (41) of Ref. [7] by applying the operator identity

and

where

R^

has the strong collision form of (2).

Superparamagnetic relaxations can be distinguished unambiguously from the two mechanisms represented by figure la and b, as can be seen from figure 2. The main feature of the field up and down model is the pronounced central line, which appears already at moderate relaxation rates.

4. Experimental results.

-

The experiments have been performed with commercially available material (Ferrofluidics Corp., Burlington, Mass.), which consist- ed of magnetite particles in a diester carrier liquid with a number concentration of about 1017 per cm3. The saturation magnetization was 100 G and the

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C6-264 H. WINKLER, H.4. HEINRICH AND E. GERDAU O = l O lper. D = 20 /per Lhlfvr~on model ~ [ r n m l s ] - l0 - 5 .S . l 0

FIG. 1. - Theoretical Mossbauer spectra for 57Fe in a randomly reorienting magnetic field : a) diffusion, b) strong collision model. The fluctuation rates are measured in units of the excited state Larmor period.

Fig. 6), the geometric mean diameter has been deter- mined from electronmicrographs as d, = 103 (5)

W.

For the volume weighted diameter

d, =

(C

ni d?/C ni)'I3 (16) we got d, = 115 (5)

W.

In figure 3a series of measurements at various temperatures is reproduced. Up to 224 K the spectra exhibit a behaviour, which is characteristic for superparamagnetic systems. The pronounced central line, which appears at elevated temperatures, makes the interpretation unique. The deviations from the theoretical spectra of figure 2 can be explained completely by the size distribution. A comparison with investiga- tions of a frozen ferrofluid on mineral oil base carried out by Roggwiller and Kiindig [l] reveals a slowing down of the relaxation rate for our case. Although our particles are about 15

%

smaller on the average, the spectra of figure 3 are shifted to higher temperatures by more than 100 centigrades. Thus the aniso- tropy constant K, which determines the relaxation time 7, by means of the relation

KV

zR = 70 exp

-

kT

where z, is a time of the order of 1 ns and V the particle volume, turns out a factor of more than 2.7 greater than the 5.1 X 104 erg/cm3 found by Roggwiller and Kiindig.

Although the method of determining absolute values for K bears some arbitrariness, the effect of a decreased relaxation rate seems to be well established. It can be supposed that this is caused by a short range ordering, which takes place in the liquid phase. Thus each individual particle is coupled to some neighbours, which prevent it from flipping into the wrong direction of magnetization by means of a local field. Only if some neighbours flip too, the direction of magnetization can be reversed. This collective process is of course less probable than the flip of an isolated particle. It leads to an enlargement of KV, because the effective volume is greater. In solid material the ordering may be destroyed during the process of freezing. The coupling strength pH of a 100

A

particle (saturation magnetization M, = 480 G) in a field of 200 Oe

is 3.14 X 10q2 eV, which corresponds to a ternpera-

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RELAXATION PHENOMENA IN FERROFLUIDS C6-265

FIG. 2.

-

Field up-down model.

showed that some 100 Oe produce an observable polarization, as can be seen from figure 4. On account of these considerations we can say, that the ferrofluid is in some sense the model of a spin glass, where the spins are represented by the magnetite particles and the coupling by classical dipole-dipole interaction. Above 224 K a considerable line broadening sets in, which cannot be explained by superparamagnetic relaxation. It has to be supposed that this behaviour is caused by linear diffusion. This could be confirmed by measurements with an applied field of l 000 Oe. A field of this magnitude should be strong enough to switch off nearly completely all magnetization flips and eventual particle rotations. Figure 5 shows the resulting spectra, the solid lines are fits assuming pure magnetic hypei4ne interaction. The line widths have been compared with that found from a measurement at 210 K and the broadening is mapped in an Arrhenius plot (Fig. 6 ) . The points can be fitted by a straight line.

Assuming simple diffusion the broadening AT and the diffusion coefficient D are related by [8]

so that also the diffusion coefficient obeys an Arrhenius equation :

L

-1b $ 0 .5 *l0

VELOCITY Immls l

FIG. 3. - Mossbauer spectra of a diester-based ferrofluid (100 cp at 300 K) as a function of temperature. The solid lines are fits with 3 pairs of Lorentzians and an additional central

line.

with E, = 0.61 (4) eV and D,

=

200 cm2/s. Extrapo- lation to room temperature gives :

The most favoured way of explaining diffusion data is the Stokes-Einstein equation

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C6-266 H. WINKLER, H.-J. HEINRICH AND E. GERDAU

VELOCITY lmmlsl

FIG. 4. - Mossbauer spectra of a diester-based ferrofluid at 210 K in various applied fields parallel to the direction of

y emission.

is in satisfactory agreement with the experimental value.

5. Conclusions. - It has been demonstrated, that careful analysis of temperature dependent Mossbauer spectra allows to distinguish rather well between various dynamic processes in liquids.

Determination of the dynamic parameters as a function of macroscopic conditions as concentration or viscosity promises to be a powerful tool for the study of the liquid state, since relaxation times are more or at least in a'different way sensitive to changes of the environment than' hyperfine fields and isomer shifts.

l

-10 - 5 0 r 5

.

?b

V E L O C I T Y l m m l r l

FIG. 5. - Mossbauer spectra of a diester-based ferrofluid in an applied field of 1 000 Oe at various temperatures.

FIG. 6.

-

Line broadening and diffusion coefficient as a function of the inverse temperature. The insert gives the distribution of

the particle sizes.

Acknowledgements.

-

This work has been support- ed by the Bundesministerium fiir Forschung und Technologie.

Appendix. - The applications of the Wigner-Eckart theorem to spherical tensor Liouville operators gives

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RELAXATION PHENOMENA I N FERROFLUIDS

By means of eq. (6) one can go over to the

1

I, m, I, m 3 representation :

(a@, I,, L, N Z )

I

a(Ig I,, L , N I ) ) = (- 1 ) 2 ' g + m 1 + r n z [(2 L1

+

1 ) (2 L,

+

X

mlrnirnzmi

X

(-

' ' g m , N , Ie m:

) (

- I, m, NZ Ie m;

)

.

( I ~ m, Ie m;

I

I g m , I , m ; ) . ( A 2 )

On the other hand one gets from the definition of the Liouville operator

( I , m, I, m', ( A : ) ~

I

I g m , I , m ; )

-

~ 5 , ~ ~ ;

<

I , m,

I

A:")

I

I , m,

>

-

6,,,,

<

I , m;

I

A,'')

I

I , m;

>

so that

(4,

I,, L, N,)

1 I

a(Ig I,, L, N , ) ) = (- 1 ) ~ f i + ' = + ~ 2 - ~ 2 [(2 L ,

+

1) ( 2 L,

+

1 ) ] l J 2

(

= l ) X

- N Z q NI

where the 3 j-symbols have been replaced by the appropriate 6 j-symbol. From comparison with (A 1) one gets for the reduced matrix element the expression (5).

References

[l] ROGGWILLER, P. and K~MDIG, W., Solid State Commun. 12 [6] GEDIKLI, A., WINKLER, H., and GERDAU, E., 2. Phys. 267

(1973) 901. (1974) 61.

L2] H. H.$ KLEIN* M. P. and SHIRLEY* D. Phys. _[7]_DATTAGUPTA,_{S. ~ ~ ~ B L u M E ,}_{M., Phys. Rev. B}₁₀_(1974)4540. Rev. 152 (1966) 345.

[S] GABRIEL, H., BOSSE, J. and RANDER, K., phys. stat. sol. 27 181 S ~ G W I , K. S. and SJ~~LANDER, A., ~ h y s . .Rev. 120 (1960) 1093. (1968) 30. [g] TYRRELL, H. J. V. in Diffusion Processes, ed. J . N . Sherwood,