Relaxation and resonance in ferrofluids

Journal of Magnetism and Magnetic Materials 122 (1993) 159-163 North-Holland

Relaxation

and

resonance

in ferrofluids

P.C. F a n n i n a, B.K.P. Scaife a and S.W. C h a r l e s b Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland h Department of Chemistry, Uni~'ersity College of North Wales, Bangor, Gwynedd, LL57 2UW, UK Measurement of the complex, frequency-dependent, relative susceptibility of a number of ferrofluids have been made in the frequency range 10 Hz-500 MHz. These measurements have revealed the presence of Brownian and N6el relaxation and in addition, for the first time, a resonance behaviour characterised by the real part of the susceptibility going negative above a frequency in the range 30-60 MHz.

1. Introduction

the influence of thermal fluctuations as discussed by NEel [2] and Brown [3]. The real part of the complex susceptibility, X'(w), decreases monotonically with increase of w, whilst the imaginary component, X"(co), has a maximum at a frequency fmax = mmax/2"ri" = ( 2 w r ) - I t - 1. In the situation where cr >> 1, the magnetic moment in the particle is 'blocked', i.e. the moment cannot surmount the energy barrier KV to reverse the direction of magnetisation in the time of the measurement. Thus the relaxation time is extremely long with the result that that Xll(W) ~ 0 as ~r ~ oo. However under these conditions, precession of the moment in the anisotropy field,

In 1974, Raikher and Shliomis [1] derived expressions for the complex ac susceptibility, X(w) = X ' ( o ~ ) - i x " ( ~ o ) , of a single domain uniaxial particle parallel Xll(~O), and perpendicular X . (w), to the easy axis of magnetisation. They showed that the frequency dependence of the susceptibility depended on the ratio of the magnetic anisotropy energy to the thermal energy (¢ =. K V / k T ) where K is the effective anisotropy constant (magnetocrystalline and shape) and V is the volume of the particle. For ~r << 1 the expressions for X l ( w ) and Xll(W) are identical, as expected for an isotropic particle for which K--* 0. In this case, the complex susceptibilities are described by the Debye relationship with

occurs with a characteristic angular frequency, w 0, given by

X±(oo) =Xll(W) =X0(1 + i w r ) - '

(1)

co0 = 3,HA ~.~

(2)

and a relaxation time,

=

- ix"(,o)

where

~- = (crw,,) - ' = Ms/2a3,Ktz o

X ' ( w ) =X,,(1 + w Z r 2 ) - l

x"(,o)

HA = 2K/M S

and

+,o27 =) ',

(3)

Xo = nm~/3kTI-to is the static susceptibility, m v is the magnetic moment of the particle and r is the relaxation time of the magnetic moment under

Correspondence to: Dr P.C. Fannin, Department of Microelectronics & Electrical Engineering, Trinity College, Dublin "~ Ireland.

(4)

(5)

(6)

where a is a damping constant [4], M S is the saturation magnetisation per unit volume and 3' is the magnetogyric ratio. It has been shown [1] that in the case of X.(to), when a radio frequency field is applied perpendicular to HA, the motion of the magnetic moment has a typical resonant character with the real and imaginary components of the ac susceptibility having a form as shown in fig. 1. This would apply whether the particles were dispersed in a solid or a liquid as in a magnetic fluid.

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

P. C Fannin et al. / Relaxation and resonance in ferrglluids

160

permeability of free space, the susccptibility is given by

I.oI

x(,o)

i:~ {<))

II

:

~(x,,(~o)+ 2x,,(,o))

(7)

i

I i

where n is the n u m b e r density of the magnetic particles. In the Debye regime, i.e. for ~r << 1, thc static susceptibility is given as before by X, =

C:.C:

nm~,/3kT#o. i

c.ae l 06

(')-2

0,,.

0.0

~.2

CO l
Fig. I. Plot of normalised X_(to) against ~o/toil. (I) o- = 5: (2) ~r =

10; (3)

cr = ~ c .

In this paper we report on measurements, in the frequency range 10 H z - 5 0 0 MHz, of the complex, relative magnetic susceptibility of a n u m b e r of magnetic fluids which exhibit both relaxation and resonance characteristics. 2. E x p e r i m e n t a l

For values of o- = 0.5 (fig. 2) the dispersion of X • (w) has characteristics of both relaxation (Debye) and resonance. At or = 0.1, the dispersion is practically identical to pure Debye behaviour. Calculations by Raikher and Shliomis [1] for the case of a single particle, have shown that the 'transition' between the region of Debye behaviour and that of resonance occurs for cr = 0.7 and that the critical value, o-*, at which the characteristic frequency of precession vanishes is 0.2. For a system of non-interacting particles, with easy axes randomly oriented, in a magnetic fluid, i.e. for m2p/i,,od 3
1.0

o

<

1

3. R e s u l t s a n d d i s c u s s i o n

0.0

q -0.2 &

,-3

, 2

-

f 1

, 0

Three magnetic fluids, called A, B and C respectively, were studied in this investigation. Fluid A contained particles of magnetic iron oxide (nominally y - F % O 3) of median diameter (of the lognormal volume fraction) of 10 nm dispersed in a hydrocarbon Isopar M and stabilised with oleic acid. Fluid B contained similar particles to that of fluid A but the particles were dispersed in water using oleic acid as surfactant and lauric acid as cosurfactant. Fluid C contained cobalt ferrite particles (nominally CoFe204) of median diameter 13 nm dispersed in lsopar M. M e a s u r e m e n t s of the complex ac susceptibility were made using the technique of Fannin et al. [5]. The value of the effective anisotropy constant K = 2 - 5 x 10 4 Jm ~, of the magnetic iron oxide samples were determined from low temperature measurements [6], and a value of K = 2 × 10 5 Jm ~ for the cobalt ferrite was determined by a similar method [7].

1

Ioq (--~) Fig. 2. Plot of normalised ~,, (to) against to/to,,. (1) o" = 0.5: (2) ~r = 1.

For a magnetic fluid the variation of X'(w) and X"(a)) with frequency may d e p e n d on tile relaxation time, r n, of the magnetic m o m e n t within the particlc (Ndel relaxation) or the Debye relaxation time, rB, via rotational diffusion of the particle itself within the carrier liquid (Brownian

P.C. Fannin et al. / Relaxation and resonance in ferrofluids

161

aggregate and r/ is the solvent dynamic viscosity. The N6el relaxation time may also be increased because of an effective increase in K due to the magnetic dipole interactions between particles [9]. i

Fluid A

5

~

4

For fluid A, because of the particle size distribution, the particles would be expected to have 0.3 < KV/kT < 10 corresponding to N6el relaxation times very approximately 10 8 s <~'N < 10 -5 S. These values are shorter than the corresponding Brownian relaxation times for particles in a carrier liquid of = 10 -2 Nsm -2 namely 5 × 1 0 6 s < 7 B < 5 x 1 0 -5 s. Thus for a fluid in which the particles are well dispersed (i.e. aggregation is not significant), which is the case for fluid A, the appearance of X"(w) (fig. 3) should have characteristics of Debye behaviour resulting from the presence of N6el and Brownian relaxation [10]. In addition, for particles for which KV/kT> 0.7, X'(o~) should show characteristics of resonance behaviour as seen experimentally. The X •tt and X •¢ profiles observed are qualitatively similar to those predicted by Raikher and Shliomis [1] with resonance occuring at a frequency of 32 MHz. To obtain a quantitative fit between theory and experiment is complicated by the fact that a particle size distribution exists, the easy axes are randomly aligned and that an uncertainty in the value(s) of K also exists [11]. The values of K quoted here were determined at low temperatures whilst measurements were made at room temperature.

)

log(f),

f in Hz

Fig. 3. Plot of X ' ( w ) and X"(w) against log frequency for fluid A.

relaxation) or both. As magnetic fluids contain a distribution of particle sizes both mechanisms will, in general, contribute to the magnetisation with an effective relaxation time ~'eff [8], where Teff = r N T B / ( r N + TB).

(8)

Thus for the smaller particles in the distribution one would expect %re to be dominated by the Ndel relaxation, whilst for the larger particleg Brownian relaxation would be the dominant mechanism. In addition t"(6o) and X ' ( w ) will also depend on precession of the magnetic moment about the anisotropy field along the easy axis under the appropriate conditions. The situation is further complicated by the possible presence of aggregates of particles which can lead to significantly higher values of r B due to their large size since r B = 3V'rl/kT, where V' is the hydrodynamic volume of the p a r t i c l e / ,22]

.

\\\\\\\\\

i

:o1!

_'zzf5COHz ~____~ I

I i

1

2

3

~ 4

log(f),

5

_ 6

~'w18b41-1z ~ 7

8

f in Hz

Fig. 4. Plot of g'(o~) and X"(w) against log frequency for fluid B.

P.C. Fannin et al. / Relaxation and resonance in fi'rro/luids

162

~.29 l '

\

i

i*.I< i ",,

:" 3kg/~ ' 2 5 b~qH ,,

, (i .

.

.

.

.

. 2

.

. . 3

.

.

.

.

.

.

.;

locJ(f),

, f

> - - ~ ,c,

c)

Ln Hz

Fig. 5. Plot of X'(w) and X"(w) against log frequency for fluid C.

Fluid B In this case, the particles in the fluid are similar to those in composition and size to those used in Fluid A. However, in this case, the carrier is water. Various techniques [12], have been used to determined how well dispersed the particles are, and in all cases aggregation has been found to be prevalent and significant. The aggregate diameter is typically > 50 nm, compared with a median particle diameter of 10 nm. The presence of aggregates results in a low frequency loss peak ( = 500 Hz) for X"(w) as a result of rotational Brownian relaxation [10] (fig. 4). Substituting estimates of r/ ( ~ 10 -2 Nsm -2) and V' ( = 1.5x 10 22 m 3) in the expression for r u produces a comparable value. Ndel relaxation can still occur via isolated particles or particles contained within the aggregates but not strongly magnetically coupled to other particles. There is evidence of resonance behaviour (fig. 4) at a frequency of 40 MHz.

Fluid C In this fluid, the particles are a cobalt ferrite which have a median diameter (13 rim) greater than that for the particles in fluids A and B. In addition, the magnetic anisotropy constant is significantly greater (K ~ 2 × 105 Jm-3). Because of the larger particles, and hence the larger magnetic interaction, aggregation is significant. Further, coupling of the moment to the easy axis is much greater than for the iron oxide particles, so that the dominant relaxation process is by Brown-

ian rotational diffusion. This combined with the larger aggregates present, results in a strong loss peak at a frequency of about 3 kHz in the X"(w) curve (fig. 5). As for the other fluids, resonance behaviour is observed, with X'(w) going negative at a frequency of 53 MHz. Conclusions Measurements of the complex susceptibility of magnetic fluids have revealed the presence of Brownian and Ndel relaxation and in addition resonance behaviour characterised by the precession of the magnitude moment about the easy axis of magnetization in a particle. This resonance behaviour is in qualitative agreement with the predictions of Raikher and Shliomis [1]. Anderson and Donovan [13] in 1959 observed resonance in a colloidal sample of magnetite. However, the resonance frequency reported was significantly higher ( = 250 MHz) than the frequencies of 32 MHz, 40 MHz and 53 MHz observed in this work. The reason for this difference is under further investigation and an attempt is being made to elucidate this phenomenon in the light of the theory of Landau and Lifshtz [14]. We thank the EC for financial support under the B R I T E - E U R A M programme. References [t] Y.L. Raikher and M.1. Shliomis, Soy. Phys. J E T P 40 (1975) 526.

P.C. Fannin et al. / Relaxation and resonance in ferrofluids

[2] L. N~,el, Ann. Geophys. 5 (1949) 99. [3] W.F. Brown, Phys, Rev. 130 (1963) 1677. [4] J.C. Anderson and B. Donovan, Proc. Phys. Soc. 73 (1959) 593. [5] P.C. Fannin, B.K.P. Scaife and S.W. Charles, J. Phys. E Sci. Inst. 19 (1986) 288. [6] M. EI-Hilo, Ph.D. Thesis, UCNW, Bangor (1991). [7] S.W. Charles, R. Chandrasekhar, K. O'Grady, and M. Walker, J. Appl. Phys. 64 (1988) 5840. [8] M.I. Shliomis, Soy. Phys. Usp. 17 (1974) 153.

163

[9] M. EI-Hilo and K. O'Grady, J. Magn. Magn. Mater. 114 (1992) 295. [10] P.C. Fannin and S.W. Charles, J. Phys. D. Appl. Phys. 22 (1989) 187. [11] A. Aharoni, Phys. Rev. 177 (1969) 793. [12] S.W. Charles, Chem. Eng. Comm. 67 (1987) 350. [13] J.C. Anderson and B. Donovan, Proe. Phys. Soc. B 75 (1960) 149. [14] L.C. Landau and E.M. Lifshitz, Phys. Zs. Soviet Union 8 (1935) 153.