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An experimental observation of the dynamic behaviour of ferrofluids
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Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
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An experimental observation of the dynamic behaviour of ferrofluids P.C. Fannin Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland
Received 5 August 1993; in revised form 26 January 1994
Abstract The frequency dependence of the complex susceptibility of eight ferrofluids have been measured. It reveals the presence of both relaxation mechanisms, Brownian and N6el, and of resonance, indicated by the real part of the susceptibility becoming negative above a critical frequency. The theoretical aspects of these mechanisms are reviewed and compared with the measured dynamical parameters. Plots of the effective relaxation time as a function of the particle radius are presented, and the relaxation time in terms of longitudinal, ~',, and transverse, z ± , components are also considered and compared with measured values for ~'11determined by means of polarised measurements. The resonance data are discussed in light of the work of Raildaer and Shliomis and estimates are made of the damping parameter, a.
1. Introduction A ferrofluid is a colloidal suspension of single-domain ferromagnetic particles dispersed in a carrier liquid and stabilised by a suitable organic surfactant. The surfactant coating creates an entropic repulsion between particles such that thermal agitation alone is sufficient to prevent aggregation. T h e particles have radii in the range approximately 2 - 1 0 nm and when in suspension their magnetic properties can be described by the Langevin theory of paramagnetism, suitably modified to take account of a distribution of particle sizes. Being single domain, the particles are considered to be in a state of uniform magnetisation with magnetic moment, m, given by: m =M~v,
(1)
where M s ( W b / m 2) denotes the saturation mag-
netisation, and v is the volume of the particle. The magnetic moments are fixed in orientation relative to the particles themselves because of magnetic anisotropy K ( j / m 3 ) , which generally arises from a combination of shape and magnetocrystalline anisotropy. The direction of the magnetic m o m e n t is referred to as the axis of easy magnetisation.
2. Characteristic times There are three characteristic times which govern the behaviour of an individual particle in a magnetic field, and two of these are associated with the relaxation of the magnetic m o m e n t of the particle. Equilibrium of the magnetic m o m e n t is attained by the rotation of the magnetic m o m e n t
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00302-8
50
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
along with the particle. The time associated with this 'bulk rotation' is the Brownian relaxation time r B [1], (2)
r B = 3Vrl/kT,
where V is the hydrodynamic volume of the particle and r/ is the dynamic viscosity of the carrier liquid. In the case of the second relaxation mechanism, the magnetic m o m e n t may reverse direction within the particle by overcoming an energy barrier, which for uniaxial anisotropy, is given by Ku. The probability of such a transition is approximately equal to exp(cr), where tr is the ratio of anisotropy energy to thermal energy ( K v / k T ) . This reversal, or switching time, is referred to as the NSel relaxation time, r [2], where NSel estimated the relaxation time r to be ~- = z o e x p ( t r ) ,
(3)
with z 0 having an often quoted approximate value of 10 -9 s [3]. Brown [4] developed N6el's work and arrived at his expressions for high and low barrier heights, described approximately as z N = r o o'-1/2 e x p ( t r ) , -- 70o',
t r > 1,
(4)
o- << 1,
where r 0 is a damping or extinction time, and is in fact the third time component to be considered. U n d e r equilibrium conditions, the magnetic moment, m, and the internal field, H A, of a particle are parallel. Any deviation of the magnetic m o m e n t from its equilibrium position results in the precession of the magnetic m o m e n t about its easy axis with an angular frequency too =/xoYH A,
(5)
where y is the gyromagnetic ratio. The internal field H A has magnitude 2K/M s,
(6)
where M s is the saturation magnetisation per unit volume. In the absence of an external rf field the
precession decays with a decay time ~'o s, where
[5]: (7)
z o =MJ21.%yag;
a is a damping constant that is not accurately known, but is generally approximated to 0.1 or 0.01 [5,61. As an example, for the case where M s = 0.4 T, y = 8.8 X 101° s - i T -1 and K = 2 × 10 4 J m -3, the use of Eq. (7) gives ~'0 = 0.9 x 1 0 - 1 ° / a s. Thus for a = 0.1, r 0 = 9 x 10 -1° s, whilst for a = 0.01, r 0 = 9 × 10 -9 s. Thus it is quite apparent why ~'0 has only an approximate value and also why it plays a significant role in the determination of ~'N of (4). A further impediment to determining an accurate value of z o and indeed of ZN, is the fact that in a ferrofluid the particles not only have a size distribution but also a shape distribution which leads, in turn, to a distribution of values of K, the anisotropy constant [7]. 2.1. E f f e c t i v e r e l a x a t i o n t i m e
A distribution of particle sizes implies the existence of a distribution of relaxation times, with both relaxation mechanisms contributing to the magnetisation. They do so with an effective relaxation time zef f [8], where • o,f
=
+
(8)
the mechanism with the shortest relaxation time being dominant. An insight into the spectrum of relaxation times that may be encountered in a typical ferrofluid sample may be determined from the data of a micrograph particle analysis, as shown in Fig. l(a). A 23-particle fraction analysis of this waterbased sample yields the data in Table 1 on particle radii (in nm). The corresponding values of r N and ~'B as a function of particle radius are shown in Figs. l(b) and (c), and a plot of r~tt is shown in Fig. 2. In calculating ~'a, an allowance of 2 nm was made for the thickness of suffactant surrounding the particles. Values of ~1 = 10 - s N s m -2, K = 4 × 104 J m - s and t o = 10 -9 s at room t e m p e r a t u r e were also assumed.
P.C. Fannin/Journal of Magnetism and MagneticMaterials 136 (1994) 49-58
51
Table 1 1
2
3
4
5
1.44
1.85
2.26
2.67
13 6.36
14 6.77
15 7.18
16 7.79
3.08 17 8.0
6 3.49
7 3.9
8 4.31
9 4.72
10 5.13
11 5.54
18 8.2
19 8.8
20 9.65
21 10.0
22 10.8
23 11.2
With reference to Fig. 2, it is obvious that, over the region A-B, the dominant mechanism is N6el, with a characteristic pronounced exponential dependence, and in the region B-C, the dominant mechanism is Brownian. In the region
|00'
o.J
ra
12 5.94
50
surrounding point B we have ~'N = rB, and here
one can determine the critical particle radius as
d Z
d e f i n e d in Ref. [8]. H e r e r N ~ -
I
2
3
4
5
6
7
8
9
1.56 X 10 - 6 s
q'B =
for an approximate particle size of 5.94 nm so that particles with radii > 6 nm relax by Brownian relaxation whilst the remaining particles in the
1011121314151617181920212223
sample relax through the N 6 e l m e c h a n i s m . T h e relaxation time o f the m a g n e t i c m o m e n t o f the b l o c k e d particles may also be expressed in
Frocfton No. 10"' (b)
terms of its longitudinal, 'rip and transverse, r ± , components [9]. These components are defined in terms of ~B and the Langevin function, L(~) with '~B(s ) ,0'
rll = ( d l n ( L ( ~ : ) ) / d
(9)
ln(~:)) TB,
w h e r e ~ = m h / k T , and h is the a m p l i t u d e o f the applied field. Eq. (9) reduces to 107
-
-
,
-
•
,
"
"
,
"
"
,
"
"
'
"
"
12
0
rll = [ ¢ B ( ~ - 2 L ( ~ )
- ~L2(~))]/L(~),
(10)
R(nm) 10
'* 1o
(C)
] 0 -5
1o tu o
io IO 1o 1o'
10 6,
NIs ) lO'°,o
,-':
lu
. . . . .
i. . . . . . . .
1o 1o 1o °
lo= to1o= iolu =
lo= 1o = 1o"
.
.- -, 2
4
6
8
10
12
Rbm) Fig. 1. (a) Particle distribution plot of a ferrofluid sample. Plots of (a) rn, and (b) CN against particle radius, R (nm) for the 23-fraction sample.
10.9
.A 2
4
6
8
10
12
R (nm) Fig. 2. Plot of q'eff against particle radius, R (nm) for the 23-fraction sample.
52
P.C. Fannin/Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
and for (a) small
~,
1-11=%[1 - 2/15~2],
(11)
(b) large
~,
zll=zB/~:.
(12)
Similarly, z ± = rs(2L(~))/(K
- L(~))
(13) Fig. 3. Toroidal arrangement.
which for (a) small ~,
z±--,B(1-~2/10),
(14)
(b) large ~,
, ± = 2zB/~.
(15)
3. Complex susceptibility The foregoing treatment of relaxation times has been mainly theoretical and it is of obvious interest to try to ascertain what actually happens in practice, in order to test the accuracy of the theory. The key to determining this lies in the measurement of the complex frequency-dependent magnetic susceptibility, X(to), where )((to) =X'(to) - ix"(to).
(16)
Before presenting typical results it is appropriate to describe briefly how one can measure the complex susceptibility and, in particular, the technique employed by the author to obtain the results presented here. The conventional method of determining the frequency dependence of the complex susceptibility of a ferrofluid is to insert the fluid into the alternating magnetic field of a coil and to observe the changes in its inductance and resistance. However, depending on the size of the particles of the sample under investigation, this method may require the use of a large value of inductance. This large inductance will have some self-capacitance, the effect of which is to restrict the frequency range over which the measurements can be made. A more convenient and efficient method of measuring X(to) is to use the 'toroidal technique' [10], which involves placing the fluid in an alternating magnetic field generated within a highpermeability (e.g. Mu-Metal or similar magnetic alloy) toroid. A narrow slit is cut in the toroid (Fig. 3) and the fluid is inserted into the slit. By
measuring the inductance and resistance of the toroid (a) when the slit is empty, and (b) when the slit is full of ferrofluid, the components of X'(to) and X"(to) can be readily determined. Measurements over a wide frequency range can be accommodated by the use a number of toroids. The theory developed by Debye [11] to account for the anomalous dielectric dispersion in dipolar fluids has been used [12,13] to account for the analogous case of magnetic fluids. Debye's theory holds for spherical particles when the magnetic dipole-dipole interaction energy, U, is small compared with the thermal energy k T . According to Debye's theory the complex susceptibility, X(to), has a frequency dependence given by the equation: X(to) =X~ + (X0 - X ~ ) / ( 1 + ito~-eff),
(17)
where %ff is the effective relaxation time given by: reff = 4 ~ r * ? r 3 / k T ,
(18)
and where r is the hydrodynamic radius of the particle, -q is the dynamic viscosity of the carrier liquid, with X0 and X~ indicating values at to = 0 and at very high frequencies. For simplicity, this is often approximated to X(to) = X 0 / ( 1 + ito%ff)
(19)
=Xo/(1 + to2"r2ff) - io)'reffXo/(1 + 002"g2ff), (20) with Xo = n m Z / 3 k T ~ o
(21)
where n is the particle number density. Plots of the ideal Debye curves are shown in Fig 4. They illustrate how X'(to) falls monotoni-
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
/3~
\
53
"1. Z
ooo
~('(~
,,.
......
__
©U='
oO,~)~!~_L_.
X(~I
-6
0 6 In(s) Fig. 4. Plots of Debye profiles; ×'(to) and ×"(to) against In(s).
cally whilst the X"(to) component has a maximum at tOmaxreff = 1. This occurrence is of the utmost significance, since it enables one to determine an effective relaxation time and the corresponding particle size for the ferrofluid sample in a relatively simple manner. An illustration of actual results obtained for four ferrofluid samples (samples 1-4) is given in Fig. 5 for measurements taken up to a frequency of 10 MHz. The median particle radii of samples 1, 2 and 3 were 5 nm, whilst that of sample 4 was 4.5 nm. The samples were colloidal solutions of (1) cobalt ferrite in hexadecane, (2) magnetite in water, (3) cobalt ferrite in toluene and (4) magnetite in Isopar M with corresponding assumed viscosity values of 3.5 x 10 -3, 1 x 10 -3, 0.6 × 10 -3 and 1 x 10 -3 N s m -E, respectively. Fig. 5 presents normalised plots of X'(to) and ×"(to) against l o g ( f ) Hz, and shows four Debye-type profiles with loss peaks at 400 Hz, 2 kHz, 63 kHz and 3.5 MHz, respectively. The corresponding values of hydrodynamic particle radii, as determined by Eq. (18) at room temperature, are 33, 29.7, 10 and 2.4 nm, respectively. Fig. 5 simply illustrates how the loss peak can lie in any part of the frequency spectrum (for f << f0, the Larmor frequency). However, one can comment on the fact that the values of radii obtained for samples 1 and 2 are indicative that aggregation has occurred. This demonstrates that the dominant relaxation mechanism in these samples was Brownian. Using the data of Fig. 2 as a rough guide, the corresponding results for samples 3 and 4 indicate that both N4el and Brown-
0
1
"- - . . . .
I '\
2
3 4 5 6 7 log( F)Hz Fig. 5. Normalised plots of g'(to) and ×"(to) against log(f) (Hz) for samples 1-4.
ian mechanisms contributed to the magnetisation with the dominant mechanism in the former sampie being Brownian and that of the latter sample being N6el. Allowing for a surfactant thickness of 2 nm, the result for sample 3 is a good approximation to the actual mean particle radius. For sample 4, since the dominant relaxation mechanism is N6el, Eq. (4) and not Eq. (18) is applicable, and applying Eq. (4) with a value of K = 4 X 104 J m -3, results in an approximate magnetic radius of 5 nm. These findings are in accord with previously published data [12,13]. In general, one can anticipate a single loss peak in susceptibility measurements taken over the frequency range mentioned; in some cases, however, measurements have revealed the existence of two absorption peaks. In theory, this effect can be demonstrated by considering the case of a ferrofluid sample which has two groups
L~
\1 X"
-
0
12
In(s) Fig. 6. Theoretical plots of )('(to) and ×"(to) against In(s) for two groups of particle fractions with relaxation times 71 and "r 2 .
P.C. Fannin/Journal of Magnetismand MagneticMaterials136 (1994)49-58
54
H(kAm
-I)
U
0.6
"-.
"
0.5
";., ".. "..'. "..'.
t.0 "\
o
0
1
2
3 log(f)
4. He
5
6
Xico)
1.5 .........',..:.
7
3.o
0
'
0
X(OO) = X o ( A / ( l + ioJrl) + B / ( 1 + tour2) ). (22) Corresponding theoretical plots of X(~O) against In(s), (where s = ~or), for values of A = 2B and for % / % = 500, are shown in Fig. 6. The figure displays two absorption peaks occurring at ogr I = 1 and at ~or 2 = 1, and two plateau regions, A - B and C - D . The results of measurements performed on two ferrofluid samples (samples 5 and 6) which closely resemble this profile are shown in Fig. 7. For sample 5 two distinct 'plateau-type' regions A 1 - B 1 and C 1 - D 1 are evident, with two loss peaks occurring at 200 Hz and 40 k Hz, respectively. In the case of sample 6, the plateau regions are not as distinct whilst the loss peaks occur at frequencies of 1 kHz and 2 MHz, respectively.
......: .:'-~:. "-.'.%
Fig. 7. Normalised plots of X'(to) and X"(to) against log(f) (Hz) for samples 5 and 6.
of particle fractions with relaxation times r t and %. This situation can be represented by the expression:
.:.
%",.. '.~..
~
1
;
!
J
'
4
6
7
log( f } I-Iz
Fig. 8. Plot of X'(~o, H) against log(f) for polarising fields of H = 0-3 kA m- I for sample 7. tion of the fluid, an expression for the field dependence of the ac susceptibility, X(oJ, H ) , is X(o~, H ) = g o ( 1 + f ( H ) ) / ( 1
+ iog"reff),
(23)
with
(l +f(n)) = 311 + ( k T / m H ) 2 - c o t h 2 ( m H / k T ) ] .
(24)
Eq. (24) predicts reductions in both X'(oJ) and
X"(~o) with increasing bias, as confirmed by the results shown in Figs. 8 and 9, respectively, for a suspension of cobalt ferrite in hexadecane (sam-
H(kAr'n-1) ....~r700 H z
0.2
o./
I..,.".. "'..'.
•'[ .' ]
"
".'.
'"..?::.
: .. ..,?.:::. 4. Polarised measurements
Susceptibility measurements in the presence of a polarising magnetic field ( H ) [14] can be used to investigate the relaxational behaviour of ~'ll and r _L. The toroidal technique is well suited for this purpose, since a dc biasing field ( H ) can be readily generated by the addition of an auxiliary biasing winding to the arrangement of Fig. 3. Assuming a Langevin function for the magnetisa-
•
. I"
".'.I.
0.5. •-" .".'1i ........-.":!:..,. 1 ~ '.: : /.-..~--::'.';.6.3KHz .u .."..: .')".."'1":"!i'.';'. 1 5 / " ..' .'1/' : "-::~.
g,o
0
1
I ":,2".'.... .
2
3 g log(F) He
5
6
7
Fig. 9. Plot of X"(to) against log(f) for polarising fields of H = 0-3 kA m- 1 for sample 7.
P. C Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
c~
55
5. Relaxation to r e s o n a n c e
o
0
,
,
1
2
,
,
;
,
3 4 6 H(kA m -1 ) ,
,
7
,
,
8
10
Fig. 10. Plots of theoretical ~'~ and ~-± against sc, and experimental "rll against H (kA m-l). pie 7). Fig. 8 shows how, over the biasing range 0 - 3 kA m -1, X0 is reduced to approximately 1 / 3 of its unbiased value. Furthermore, Fig. 9 shows how fmax increases from 700 Hz to 6.3 kHz which, according to (18), is indicative of a decrease in ceff from 1.43 to 0.158 ms. These data enable a comparison to be made of values of ~'ll and z j_ (as a function of H ) as calculated by Eqs. (11) and (14) in order to determine which relaxational component is active. Fig. 10 illustrates this comparison and clearly shows that qualitatively there is excellent agreement between the predicted and measured values of the relaxation component. Furthermore, the data indicate that the active relaxational component is in fact ~'11' which is indeed correct, since in the toroidal system H is applied parallel to the sampie in the slit. Thus Eq. (23) effectively represents Xll(to, H).
The results presented thus far have been obtained from measurements taken at frequencies up to 10 MHz. However, as the frequency is increased, a point is reached where the character of the dispersion changes from relaxation to one of resonance. In a uniaxial particle, this is characterised by the precession of the magnetic moment, m, about its axis of easy magnetisation with an angular frequency too, as previously defined in (5). In the measurement of the complex susceptibility of ferrofluids, resonance is indicated by a transition in the value of X'(to) from a positive to a negative quantity. The frequency to,, at which X"(to) is a maximum, is given by
tor
=
(25)
toO0///0''
where t r - - K v / k T is the ratio of anisotropy energy to the thermal energy, k T , v is the volume of the particle and a is the damping parameter. In the work of Raikher and Shliomis [15] the resonance phenomenon was treated in terms of equations proposed by Landau and Lifshitz [16], modified to include stochastic terms. They derived expressions for X(to) in terms of components Xil(to) and X ± (to), parallel and perpendicular to the easy axis of magnetisation, respectively.
1 O-"
1
~,i {a)
1
-.06
Iog(F) Hz
.75 . . . . . . . .
Fig. 11. Normalisedplots of X'Qo)and X"(~0)against log(f) Hz for o-= 1, 0.75 0.5 and 0.2.
56
P.C. Fannin/Journal of Magnetism and MagneticMaterials 136 (1994) 49-58
For o, << 1, Xll(to) and X±(to) are identical, as expected for an isotropic particle for which K ~ 0 (no energy barrier to surmount), and Xll(oJ) =Xx(OJ) = X 0 / ( 1 + ioJ~'N),
(26)
which has a Debye-type profile. For o, >> 1, the magnetic moment in the particle is 'blocked', i.e. the moment cannot surmount the energy barrier Kv so as to reverse the direction of magnetisation in the time of the measurement. Thus the relaxation time is extremely long, with the result that Xll(o~)---,0 as ~,--* oo and under these conditions, precession of the magnetic moment in the anisotropy field occurs. The equations presented in Ref. [15] do not lend themselves easily to the plotting of susceptibility curves shown in Fig. 11. The latter curves were in fact plotted from more tractable equations which are valid for small ~ [17], where X . (to) is represented as xl(,o) Xa_ o
(1 + ~/5)ioJ + 1/~'N(3 + 16o'/35 + o'2/5a 2) 1/'rN(3 + 160"/35+ 0.2/5a2) -- w2rN + 2i~o(20./35) " (27) An illustration of the changes in the form of the complex components of the susceptibility profiles for changes in tr is shown in Fig. 11 for values of tr of 1, 0.75, 0.5 and 0.2 (in plots 1, 2, 3 and 4), respectively. It should be noted that for tr = 1 the dispersion of X~(to) has a resonance
1"
~'(~ gMHz
0
1
2
3
4
5
6
0.9
7
"-8
9
log(F) Hz Fig. 12. Plots of X'(to) and X"(to) against log(f) Hz for sample 8.
0 Fig. 13. Cole-Cole plot of X"(to) against X'(to) for sample 8.
characteristic, whilst for tr = 0.75 and 0.5 it has the characteristics of both the relaxation and resonance. For tr = 0.2 the resonance effect (negative X'(ta)) has vanished and the dispersion has a Debye-type characteristic. The transition between the region of Debye behaviour and that of resonance is predicted to occur at tr = 0.7 [15]. Resonance observations that are qualitatively similar to those described in Ref. [15] have been reported for a number of magnetic fluids [18]. A similar result is shown in Fig. 12 for a colloidal solution of magnetic iron oxide in Isopar M (sample 8) with particles of median diameter of about 9 nm. The magnitude of K for these particles lies in the range 2 × 104-5 x 104 J / m 3, as determined in Ref. [19], leading to approximate values of tr between 1.8 and 4.6. Here X'(to) is observed to change sign at a frequency of approximately 31 MHz, and X"(to) shows a maximum at a somewhat lower frequency, fr(tOr/2"rr), of 9 MHz. The corresponding Cole-Cole plot [20] of Fig. 13 displays the characteristic resonance condition with the circular arc reaching the origin via the fourth quadrant. It must be emphasised that the theoretical susceptibility curves based on Eq. (27) assume an assembly of identical single-domain particles, which is not the situation in practice as all ferrofluids have a particle size distribution. However, applying the data to Eq. (25) (to r = tOoa/tr) and taking f0 = 109, it is found that for values of tr between 1.8 and 4.6, a lies over the range 0.018-0.046. These values of tr lie within the generally approximated range 0.01-0.1 [5,6].
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58 F o r a system of non-interacting particles, having r a n d o m l y oriented easy axes, as is the situation for m e a s u r e m e n t s p e r f o r m e d at small values of ~: b y m e a n s o f the toroidal technique, the susceptibility can be expressed as [16,21]:
x(o,) = n(xil(¢o) + 2X ± ( ¢ o ) ) / 3 = n ( x l l o / ( 1 + ioJzll ) + 2 X ± 0 / ( 1 + i~o~"± ) ) / 3 ,
(28) where n is the n u m b e r density of the magnetic particles. This expression satisfactorily represents the behaviour of uniaxial particles in a ferrofluid over a wide frequency range and current activity is being directed towards fitting profiles g e n e r a t e d by this equation to m e a s u r e d data.
7. Conclusions M e a s u r e m e n t s o f the dynamic magnetic susceptibility in ferrofluids which reveal the basic response mechanisms o f nanoscale ferrofluid part i d e s to applied m a g n e t i c fields, have b e e n presented. T h e presence o f Brownian and N6el relaxation mechanisms and r e s o n a n c e has b e e n revealed, and B r o w n ' s equations (4) has b e e n used to calculate the N6el relaxation time, eN- It is again emphasised that these equations are only approximations; a major source o f error centres on the value used for the c o m p o n e n t %. T h e often q u o t e d value of 10 -9 s was used here, although the literature values for % range between 10 -8 and 10 -12 s [22-25]. Obviously, since the position o f the m a x i m u m of the loss peak, Wmax, is proportional to 1 / ¢ 0 , any c h a n g e in z 0 is reflected in a c o r r e s p o n d i n g c h a n g e in the position of OJn~~. Finally, as has b e e n explained, the susceptibility resonance discussed here has its origin in the precession of the m a g n e t i c m o m e n t o f the particle and occurs at relatively high frequencies. It is not to be confused with a n o t h e r f o r m o f susceptibility resonance, that due to a tunnelling effect
57
[25-27], which can occur at frequencies below 1 k H z and at t e m p e r a t u r e s a p p r o a c h i n g 0 K.
Acknowledgements Sincere thanks are due to Yu.P. Kalmykov for m a n y helpful discussions; to S.W. Charles w h o provided m a n y of the ferrofluid samples r e p o r t e d on in the text; and to the E C for financial support u n d e r the B R I T E - E U R A M p r o g r a m m e .
References [1] W.F. Brown, J. Appl. Phys. 34 (1963) 1319. [2] L. N~el, Ann. Geophys. 5 (1949) 99. [3] E. Kneller, in: Magnetism, vol. III (Academic Press, New York, 1963) p. 382. [4] W.F. Brown, Phys. Rev. 130 (1963) 1677. [5] M.I. Shliomis and Yu.L. Raikher, 1980, IEEE Trans. Magn. MAG-16 (1980) 237. [6] J.C. Anderson and B. Donovan, Proc. Phys. Soc. B75 (1960) 149. [7] A. Aharoni, Phys. Rev. 177 (1969) 793 [8] M.I. Shliomis, Sov. Phys.-Usp. 17 (1974) 53. [9] W.T. Coffey, Yu.P. Kalmykov and K.P, Quinn, J. Chem. Phys. 96 (1992) 5471. [10] P.C. Fannin, B.K.P. Scaife and S.W, Charles, J. Phys. E: Sci. Instrum. 19 (1986) 238. [11] P. Debye, Polar Molecules (New York, Chemical Catalog Company, 1929). [12] M.M. Maiorov, Magnetohydrodynamics (cover-to-cover translation of Magnitnaia Hidrodinamika) 2 (1979) 21. [13] P.C. Fannin, B.K.P. Scaife and S.W. Charles, J. Magn. Magn. Mater. 72 (1988) 95. [14] P.C. Fannin, B.K.P. Scaife and S.W. Charles, J. Magn. Magn. Mater. 65 (1987) 279. [15] Y.L. Raikher and M.I. Shliomis, Sov. Phys. JETP 40 (1975) 526. [16] L.D. Landau and E.M. Lifshitz, Phys. Z. Soviet Union 8 (1935) 153. [17] W.T. Coffey, Yu.P. Kaimykov and P.J. Cregg. Adv. Chem. Phys, 83 (1992) 264. [18] P.C. Fannin, B.K.P. Scaife and S.W. Charles, Meas. Sci. Technol. 3 (1992) 1014. [19] M. El Hilo, PhD Thesis, UCNW Bangor (1991). [20] K.S. Cole and R.M. Cole, J. Chem. Phys. 9 (1941) 341. [21] M.I. Shliomis and V.I. Stepanov, Adv. Chem. Phys. (1993). [22] D.P.E. Dickson, N.M.K. Reid, C.A. Hunt, H.D. Williams, M. EI-Hilo and K. O'Grady, J. Magn. Magn. Mater. 125 (1993) 345.
58
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
[23] V. Schunemann, H. Winlder, H.M. Ziethen, A. Schiller and A,X. Trautwein in: Magnetic Properties of Fine Particles (Elsevier, Amsterdam, 1992). [24] L. Bessais, L.Ben. Jaffel and J.L. Dormann, Phys. Rev. B 45 (1992) 7805. [25] P.C. Fannin and S.W. Charles, J. Phys. D: Appl. Phys. 27 (1994) 185.
[26] A. Aharoni, Phys. Rev. B 46 (1992) 5434. [27] C.P. Bean and J.D. Livingston, J. Appl. Phys. 30 (1959) 4. [28] L. Gunther, in: Proc. Studies of Magnetic Properties of Fine Particles and their Relevance to Materials Science (North-Holland Delta Series, Amsterdam, 1992).
,•/JNrnlll
mlmusm mambo
ELSEVIER
Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
malerinl8
An experimental observation of the dynamic behaviour of ferrofluids P.C. Fannin Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Ireland
Received 5 August 1993; in revised form 26 January 1994
Abstract The frequency dependence of the complex susceptibility of eight ferrofluids have been measured. It reveals the presence of both relaxation mechanisms, Brownian and N6el, and of resonance, indicated by the real part of the susceptibility becoming negative above a critical frequency. The theoretical aspects of these mechanisms are reviewed and compared with the measured dynamical parameters. Plots of the effective relaxation time as a function of the particle radius are presented, and the relaxation time in terms of longitudinal, ~',, and transverse, z ± , components are also considered and compared with measured values for ~'11determined by means of polarised measurements. The resonance data are discussed in light of the work of Raildaer and Shliomis and estimates are made of the damping parameter, a.
1. Introduction A ferrofluid is a colloidal suspension of single-domain ferromagnetic particles dispersed in a carrier liquid and stabilised by a suitable organic surfactant. The surfactant coating creates an entropic repulsion between particles such that thermal agitation alone is sufficient to prevent aggregation. T h e particles have radii in the range approximately 2 - 1 0 nm and when in suspension their magnetic properties can be described by the Langevin theory of paramagnetism, suitably modified to take account of a distribution of particle sizes. Being single domain, the particles are considered to be in a state of uniform magnetisation with magnetic moment, m, given by: m =M~v,
(1)
where M s ( W b / m 2) denotes the saturation mag-
netisation, and v is the volume of the particle. The magnetic moments are fixed in orientation relative to the particles themselves because of magnetic anisotropy K ( j / m 3 ) , which generally arises from a combination of shape and magnetocrystalline anisotropy. The direction of the magnetic m o m e n t is referred to as the axis of easy magnetisation.
2. Characteristic times There are three characteristic times which govern the behaviour of an individual particle in a magnetic field, and two of these are associated with the relaxation of the magnetic m o m e n t of the particle. Equilibrium of the magnetic m o m e n t is attained by the rotation of the magnetic m o m e n t
0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00302-8
50
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
along with the particle. The time associated with this 'bulk rotation' is the Brownian relaxation time r B [1], (2)
r B = 3Vrl/kT,
where V is the hydrodynamic volume of the particle and r/ is the dynamic viscosity of the carrier liquid. In the case of the second relaxation mechanism, the magnetic m o m e n t may reverse direction within the particle by overcoming an energy barrier, which for uniaxial anisotropy, is given by Ku. The probability of such a transition is approximately equal to exp(cr), where tr is the ratio of anisotropy energy to thermal energy ( K v / k T ) . This reversal, or switching time, is referred to as the NSel relaxation time, r [2], where NSel estimated the relaxation time r to be ~- = z o e x p ( t r ) ,
(3)
with z 0 having an often quoted approximate value of 10 -9 s [3]. Brown [4] developed N6el's work and arrived at his expressions for high and low barrier heights, described approximately as z N = r o o'-1/2 e x p ( t r ) , -- 70o',
t r > 1,
(4)
o- << 1,
where r 0 is a damping or extinction time, and is in fact the third time component to be considered. U n d e r equilibrium conditions, the magnetic moment, m, and the internal field, H A, of a particle are parallel. Any deviation of the magnetic m o m e n t from its equilibrium position results in the precession of the magnetic m o m e n t about its easy axis with an angular frequency too =/xoYH A,
(5)
where y is the gyromagnetic ratio. The internal field H A has magnitude 2K/M s,
(6)
where M s is the saturation magnetisation per unit volume. In the absence of an external rf field the
precession decays with a decay time ~'o s, where
[5]: (7)
z o =MJ21.%yag;
a is a damping constant that is not accurately known, but is generally approximated to 0.1 or 0.01 [5,61. As an example, for the case where M s = 0.4 T, y = 8.8 X 101° s - i T -1 and K = 2 × 10 4 J m -3, the use of Eq. (7) gives ~'0 = 0.9 x 1 0 - 1 ° / a s. Thus for a = 0.1, r 0 = 9 x 10 -1° s, whilst for a = 0.01, r 0 = 9 × 10 -9 s. Thus it is quite apparent why ~'0 has only an approximate value and also why it plays a significant role in the determination of ~'N of (4). A further impediment to determining an accurate value of z o and indeed of ZN, is the fact that in a ferrofluid the particles not only have a size distribution but also a shape distribution which leads, in turn, to a distribution of values of K, the anisotropy constant [7]. 2.1. E f f e c t i v e r e l a x a t i o n t i m e
A distribution of particle sizes implies the existence of a distribution of relaxation times, with both relaxation mechanisms contributing to the magnetisation. They do so with an effective relaxation time zef f [8], where • o,f
=
+
(8)
the mechanism with the shortest relaxation time being dominant. An insight into the spectrum of relaxation times that may be encountered in a typical ferrofluid sample may be determined from the data of a micrograph particle analysis, as shown in Fig. l(a). A 23-particle fraction analysis of this waterbased sample yields the data in Table 1 on particle radii (in nm). The corresponding values of r N and ~'B as a function of particle radius are shown in Figs. l(b) and (c), and a plot of r~tt is shown in Fig. 2. In calculating ~'a, an allowance of 2 nm was made for the thickness of suffactant surrounding the particles. Values of ~1 = 10 - s N s m -2, K = 4 × 104 J m - s and t o = 10 -9 s at room t e m p e r a t u r e were also assumed.
P.C. Fannin/Journal of Magnetism and MagneticMaterials 136 (1994) 49-58
51
Table 1 1
2
3
4
5
1.44
1.85
2.26
2.67
13 6.36
14 6.77
15 7.18
16 7.79
3.08 17 8.0
6 3.49
7 3.9
8 4.31
9 4.72
10 5.13
11 5.54
18 8.2
19 8.8
20 9.65
21 10.0
22 10.8
23 11.2
With reference to Fig. 2, it is obvious that, over the region A-B, the dominant mechanism is N6el, with a characteristic pronounced exponential dependence, and in the region B-C, the dominant mechanism is Brownian. In the region
|00'
o.J
ra
12 5.94
50
surrounding point B we have ~'N = rB, and here
one can determine the critical particle radius as
d Z
d e f i n e d in Ref. [8]. H e r e r N ~ -
I
2
3
4
5
6
7
8
9
1.56 X 10 - 6 s
q'B =
for an approximate particle size of 5.94 nm so that particles with radii > 6 nm relax by Brownian relaxation whilst the remaining particles in the
1011121314151617181920212223
sample relax through the N 6 e l m e c h a n i s m . T h e relaxation time o f the m a g n e t i c m o m e n t o f the b l o c k e d particles may also be expressed in
Frocfton No. 10"' (b)
terms of its longitudinal, 'rip and transverse, r ± , components [9]. These components are defined in terms of ~B and the Langevin function, L(~) with '~B(s ) ,0'
rll = ( d l n ( L ( ~ : ) ) / d
(9)
ln(~:)) TB,
w h e r e ~ = m h / k T , and h is the a m p l i t u d e o f the applied field. Eq. (9) reduces to 107
-
-
,
-
•
,
"
"
,
"
"
,
"
"
'
"
"
12
0
rll = [ ¢ B ( ~ - 2 L ( ~ )
- ~L2(~))]/L(~),
(10)
R(nm) 10
'* 1o
(C)
] 0 -5
1o tu o
io IO 1o 1o'
10 6,
NIs ) lO'°,o
,-':
lu
. . . . .
i. . . . . . . .
1o 1o 1o °
lo= to1o= iolu =
lo= 1o = 1o"
.
.- -, 2
4
6
8
10
12
Rbm) Fig. 1. (a) Particle distribution plot of a ferrofluid sample. Plots of (a) rn, and (b) CN against particle radius, R (nm) for the 23-fraction sample.
10.9
.A 2
4
6
8
10
12
R (nm) Fig. 2. Plot of q'eff against particle radius, R (nm) for the 23-fraction sample.
52
P.C. Fannin/Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
and for (a) small
~,
1-11=%[1 - 2/15~2],
(11)
(b) large
~,
zll=zB/~:.
(12)
Similarly, z ± = rs(2L(~))/(K
- L(~))
(13) Fig. 3. Toroidal arrangement.
which for (a) small ~,
z±--,B(1-~2/10),
(14)
(b) large ~,
, ± = 2zB/~.
(15)
3. Complex susceptibility The foregoing treatment of relaxation times has been mainly theoretical and it is of obvious interest to try to ascertain what actually happens in practice, in order to test the accuracy of the theory. The key to determining this lies in the measurement of the complex frequency-dependent magnetic susceptibility, X(to), where )((to) =X'(to) - ix"(to).
(16)
Before presenting typical results it is appropriate to describe briefly how one can measure the complex susceptibility and, in particular, the technique employed by the author to obtain the results presented here. The conventional method of determining the frequency dependence of the complex susceptibility of a ferrofluid is to insert the fluid into the alternating magnetic field of a coil and to observe the changes in its inductance and resistance. However, depending on the size of the particles of the sample under investigation, this method may require the use of a large value of inductance. This large inductance will have some self-capacitance, the effect of which is to restrict the frequency range over which the measurements can be made. A more convenient and efficient method of measuring X(to) is to use the 'toroidal technique' [10], which involves placing the fluid in an alternating magnetic field generated within a highpermeability (e.g. Mu-Metal or similar magnetic alloy) toroid. A narrow slit is cut in the toroid (Fig. 3) and the fluid is inserted into the slit. By
measuring the inductance and resistance of the toroid (a) when the slit is empty, and (b) when the slit is full of ferrofluid, the components of X'(to) and X"(to) can be readily determined. Measurements over a wide frequency range can be accommodated by the use a number of toroids. The theory developed by Debye [11] to account for the anomalous dielectric dispersion in dipolar fluids has been used [12,13] to account for the analogous case of magnetic fluids. Debye's theory holds for spherical particles when the magnetic dipole-dipole interaction energy, U, is small compared with the thermal energy k T . According to Debye's theory the complex susceptibility, X(to), has a frequency dependence given by the equation: X(to) =X~ + (X0 - X ~ ) / ( 1 + ito~-eff),
(17)
where %ff is the effective relaxation time given by: reff = 4 ~ r * ? r 3 / k T ,
(18)
and where r is the hydrodynamic radius of the particle, -q is the dynamic viscosity of the carrier liquid, with X0 and X~ indicating values at to = 0 and at very high frequencies. For simplicity, this is often approximated to X(to) = X 0 / ( 1 + ito%ff)
(19)
=Xo/(1 + to2"r2ff) - io)'reffXo/(1 + 002"g2ff), (20) with Xo = n m Z / 3 k T ~ o
(21)
where n is the particle number density. Plots of the ideal Debye curves are shown in Fig 4. They illustrate how X'(to) falls monotoni-
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
/3~
\
53
"1. Z
ooo
~('(~
,,.
......
__
©U='
oO,~)~!~_L_.
X(~I
-6
0 6 In(s) Fig. 4. Plots of Debye profiles; ×'(to) and ×"(to) against In(s).
cally whilst the X"(to) component has a maximum at tOmaxreff = 1. This occurrence is of the utmost significance, since it enables one to determine an effective relaxation time and the corresponding particle size for the ferrofluid sample in a relatively simple manner. An illustration of actual results obtained for four ferrofluid samples (samples 1-4) is given in Fig. 5 for measurements taken up to a frequency of 10 MHz. The median particle radii of samples 1, 2 and 3 were 5 nm, whilst that of sample 4 was 4.5 nm. The samples were colloidal solutions of (1) cobalt ferrite in hexadecane, (2) magnetite in water, (3) cobalt ferrite in toluene and (4) magnetite in Isopar M with corresponding assumed viscosity values of 3.5 x 10 -3, 1 x 10 -3, 0.6 × 10 -3 and 1 x 10 -3 N s m -E, respectively. Fig. 5 presents normalised plots of X'(to) and ×"(to) against l o g ( f ) Hz, and shows four Debye-type profiles with loss peaks at 400 Hz, 2 kHz, 63 kHz and 3.5 MHz, respectively. The corresponding values of hydrodynamic particle radii, as determined by Eq. (18) at room temperature, are 33, 29.7, 10 and 2.4 nm, respectively. Fig. 5 simply illustrates how the loss peak can lie in any part of the frequency spectrum (for f << f0, the Larmor frequency). However, one can comment on the fact that the values of radii obtained for samples 1 and 2 are indicative that aggregation has occurred. This demonstrates that the dominant relaxation mechanism in these samples was Brownian. Using the data of Fig. 2 as a rough guide, the corresponding results for samples 3 and 4 indicate that both N4el and Brown-
0
1
"- - . . . .
I '\
2
3 4 5 6 7 log( F)Hz Fig. 5. Normalised plots of g'(to) and ×"(to) against log(f) (Hz) for samples 1-4.
ian mechanisms contributed to the magnetisation with the dominant mechanism in the former sampie being Brownian and that of the latter sample being N6el. Allowing for a surfactant thickness of 2 nm, the result for sample 3 is a good approximation to the actual mean particle radius. For sample 4, since the dominant relaxation mechanism is N6el, Eq. (4) and not Eq. (18) is applicable, and applying Eq. (4) with a value of K = 4 X 104 J m -3, results in an approximate magnetic radius of 5 nm. These findings are in accord with previously published data [12,13]. In general, one can anticipate a single loss peak in susceptibility measurements taken over the frequency range mentioned; in some cases, however, measurements have revealed the existence of two absorption peaks. In theory, this effect can be demonstrated by considering the case of a ferrofluid sample which has two groups
L~
\1 X"
-
0
12
In(s) Fig. 6. Theoretical plots of )('(to) and ×"(to) against In(s) for two groups of particle fractions with relaxation times 71 and "r 2 .
P.C. Fannin/Journal of Magnetismand MagneticMaterials136 (1994)49-58
54
H(kAm
-I)
U
0.6
"-.
"
0.5
";., ".. "..'. "..'.
t.0 "\
o
0
1
2
3 log(f)
4. He
5
6
Xico)
1.5 .........',..:.
7
3.o
0
'
0
X(OO) = X o ( A / ( l + ioJrl) + B / ( 1 + tour2) ). (22) Corresponding theoretical plots of X(~O) against In(s), (where s = ~or), for values of A = 2B and for % / % = 500, are shown in Fig. 6. The figure displays two absorption peaks occurring at ogr I = 1 and at ~or 2 = 1, and two plateau regions, A - B and C - D . The results of measurements performed on two ferrofluid samples (samples 5 and 6) which closely resemble this profile are shown in Fig. 7. For sample 5 two distinct 'plateau-type' regions A 1 - B 1 and C 1 - D 1 are evident, with two loss peaks occurring at 200 Hz and 40 k Hz, respectively. In the case of sample 6, the plateau regions are not as distinct whilst the loss peaks occur at frequencies of 1 kHz and 2 MHz, respectively.
......: .:'-~:. "-.'.%
Fig. 7. Normalised plots of X'(to) and X"(to) against log(f) (Hz) for samples 5 and 6.
of particle fractions with relaxation times r t and %. This situation can be represented by the expression:
.:.
%",.. '.~..
~
1
;
!
J
'
4
6
7
log( f } I-Iz
Fig. 8. Plot of X'(~o, H) against log(f) for polarising fields of H = 0-3 kA m- I for sample 7. tion of the fluid, an expression for the field dependence of the ac susceptibility, X(oJ, H ) , is X(o~, H ) = g o ( 1 + f ( H ) ) / ( 1
+ iog"reff),
(23)
with
(l +f(n)) = 311 + ( k T / m H ) 2 - c o t h 2 ( m H / k T ) ] .
(24)
Eq. (24) predicts reductions in both X'(oJ) and
X"(~o) with increasing bias, as confirmed by the results shown in Figs. 8 and 9, respectively, for a suspension of cobalt ferrite in hexadecane (sam-
H(kAr'n-1) ....~r700 H z
0.2
o./
I..,.".. "'..'.
•'[ .' ]
"
".'.
'"..?::.
: .. ..,?.:::. 4. Polarised measurements
Susceptibility measurements in the presence of a polarising magnetic field ( H ) [14] can be used to investigate the relaxational behaviour of ~'ll and r _L. The toroidal technique is well suited for this purpose, since a dc biasing field ( H ) can be readily generated by the addition of an auxiliary biasing winding to the arrangement of Fig. 3. Assuming a Langevin function for the magnetisa-
•
. I"
".'.I.
0.5. •-" .".'1i ........-.":!:..,. 1 ~ '.: : /.-..~--::'.';.6.3KHz .u .."..: .')".."'1":"!i'.';'. 1 5 / " ..' .'1/' : "-::~.
g,o
0
1
I ":,2".'.... .
2
3 g log(F) He
5
6
7
Fig. 9. Plot of X"(to) against log(f) for polarising fields of H = 0-3 kA m- 1 for sample 7.
P. C Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58
c~
55
5. Relaxation to r e s o n a n c e
o
0
,
,
1
2
,
,
;
,
3 4 6 H(kA m -1 ) ,
,
7
,
,
8
10
Fig. 10. Plots of theoretical ~'~ and ~-± against sc, and experimental "rll against H (kA m-l). pie 7). Fig. 8 shows how, over the biasing range 0 - 3 kA m -1, X0 is reduced to approximately 1 / 3 of its unbiased value. Furthermore, Fig. 9 shows how fmax increases from 700 Hz to 6.3 kHz which, according to (18), is indicative of a decrease in ceff from 1.43 to 0.158 ms. These data enable a comparison to be made of values of ~'ll and z j_ (as a function of H ) as calculated by Eqs. (11) and (14) in order to determine which relaxational component is active. Fig. 10 illustrates this comparison and clearly shows that qualitatively there is excellent agreement between the predicted and measured values of the relaxation component. Furthermore, the data indicate that the active relaxational component is in fact ~'11' which is indeed correct, since in the toroidal system H is applied parallel to the sampie in the slit. Thus Eq. (23) effectively represents Xll(to, H).
The results presented thus far have been obtained from measurements taken at frequencies up to 10 MHz. However, as the frequency is increased, a point is reached where the character of the dispersion changes from relaxation to one of resonance. In a uniaxial particle, this is characterised by the precession of the magnetic moment, m, about its axis of easy magnetisation with an angular frequency too, as previously defined in (5). In the measurement of the complex susceptibility of ferrofluids, resonance is indicated by a transition in the value of X'(to) from a positive to a negative quantity. The frequency to,, at which X"(to) is a maximum, is given by
tor
=
(25)
toO0///0''
where t r - - K v / k T is the ratio of anisotropy energy to the thermal energy, k T , v is the volume of the particle and a is the damping parameter. In the work of Raikher and Shliomis [15] the resonance phenomenon was treated in terms of equations proposed by Landau and Lifshitz [16], modified to include stochastic terms. They derived expressions for X(to) in terms of components Xil(to) and X ± (to), parallel and perpendicular to the easy axis of magnetisation, respectively.
1 O-"
1
~,i {a)
1
-.06
Iog(F) Hz
.75 . . . . . . . .
Fig. 11. Normalisedplots of X'Qo)and X"(~0)against log(f) Hz for o-= 1, 0.75 0.5 and 0.2.
56
P.C. Fannin/Journal of Magnetism and MagneticMaterials 136 (1994) 49-58
For o, << 1, Xll(to) and X±(to) are identical, as expected for an isotropic particle for which K ~ 0 (no energy barrier to surmount), and Xll(oJ) =Xx(OJ) = X 0 / ( 1 + ioJ~'N),
(26)
which has a Debye-type profile. For o, >> 1, the magnetic moment in the particle is 'blocked', i.e. the moment cannot surmount the energy barrier Kv so as to reverse the direction of magnetisation in the time of the measurement. Thus the relaxation time is extremely long, with the result that Xll(o~)---,0 as ~,--* oo and under these conditions, precession of the magnetic moment in the anisotropy field occurs. The equations presented in Ref. [15] do not lend themselves easily to the plotting of susceptibility curves shown in Fig. 11. The latter curves were in fact plotted from more tractable equations which are valid for small ~ [17], where X . (to) is represented as xl(,o) Xa_ o
(1 + ~/5)ioJ + 1/~'N(3 + 16o'/35 + o'2/5a 2) 1/'rN(3 + 160"/35+ 0.2/5a2) -- w2rN + 2i~o(20./35) " (27) An illustration of the changes in the form of the complex components of the susceptibility profiles for changes in tr is shown in Fig. 11 for values of tr of 1, 0.75, 0.5 and 0.2 (in plots 1, 2, 3 and 4), respectively. It should be noted that for tr = 1 the dispersion of X~(to) has a resonance
1"
~'(~ gMHz
0
1
2
3
4
5
6
0.9
7
"-8
9
log(F) Hz Fig. 12. Plots of X'(to) and X"(to) against log(f) Hz for sample 8.
0 Fig. 13. Cole-Cole plot of X"(to) against X'(to) for sample 8.
characteristic, whilst for tr = 0.75 and 0.5 it has the characteristics of both the relaxation and resonance. For tr = 0.2 the resonance effect (negative X'(ta)) has vanished and the dispersion has a Debye-type characteristic. The transition between the region of Debye behaviour and that of resonance is predicted to occur at tr = 0.7 [15]. Resonance observations that are qualitatively similar to those described in Ref. [15] have been reported for a number of magnetic fluids [18]. A similar result is shown in Fig. 12 for a colloidal solution of magnetic iron oxide in Isopar M (sample 8) with particles of median diameter of about 9 nm. The magnitude of K for these particles lies in the range 2 × 104-5 x 104 J / m 3, as determined in Ref. [19], leading to approximate values of tr between 1.8 and 4.6. Here X'(to) is observed to change sign at a frequency of approximately 31 MHz, and X"(to) shows a maximum at a somewhat lower frequency, fr(tOr/2"rr), of 9 MHz. The corresponding Cole-Cole plot [20] of Fig. 13 displays the characteristic resonance condition with the circular arc reaching the origin via the fourth quadrant. It must be emphasised that the theoretical susceptibility curves based on Eq. (27) assume an assembly of identical single-domain particles, which is not the situation in practice as all ferrofluids have a particle size distribution. However, applying the data to Eq. (25) (to r = tOoa/tr) and taking f0 = 109, it is found that for values of tr between 1.8 and 4.6, a lies over the range 0.018-0.046. These values of tr lie within the generally approximated range 0.01-0.1 [5,6].
P.C. Fannin /Journal of Magnetism and Magnetic Materials 136 (1994) 49-58 F o r a system of non-interacting particles, having r a n d o m l y oriented easy axes, as is the situation for m e a s u r e m e n t s p e r f o r m e d at small values of ~: b y m e a n s o f the toroidal technique, the susceptibility can be expressed as [16,21]:
x(o,) = n(xil(¢o) + 2X ± ( ¢ o ) ) / 3 = n ( x l l o / ( 1 + ioJzll ) + 2 X ± 0 / ( 1 + i~o~"± ) ) / 3 ,
(28) where n is the n u m b e r density of the magnetic particles. This expression satisfactorily represents the behaviour of uniaxial particles in a ferrofluid over a wide frequency range and current activity is being directed towards fitting profiles g e n e r a t e d by this equation to m e a s u r e d data.
7. Conclusions M e a s u r e m e n t s o f the dynamic magnetic susceptibility in ferrofluids which reveal the basic response mechanisms o f nanoscale ferrofluid part i d e s to applied m a g n e t i c fields, have b e e n presented. T h e presence o f Brownian and N6el relaxation mechanisms and r e s o n a n c e has b e e n revealed, and B r o w n ' s equations (4) has b e e n used to calculate the N6el relaxation time, eN- It is again emphasised that these equations are only approximations; a major source o f error centres on the value used for the c o m p o n e n t %. T h e often q u o t e d value of 10 -9 s was used here, although the literature values for % range between 10 -8 and 10 -12 s [22-25]. Obviously, since the position o f the m a x i m u m of the loss peak, Wmax, is proportional to 1 / ¢ 0 , any c h a n g e in z 0 is reflected in a c o r r e s p o n d i n g c h a n g e in the position of OJn~~. Finally, as has b e e n explained, the susceptibility resonance discussed here has its origin in the precession of the m a g n e t i c m o m e n t o f the particle and occurs at relatively high frequencies. It is not to be confused with a n o t h e r f o r m o f susceptibility resonance, that due to a tunnelling effect
57
[25-27], which can occur at frequencies below 1 k H z and at t e m p e r a t u r e s a p p r o a c h i n g 0 K.
Acknowledgements Sincere thanks are due to Yu.P. Kalmykov for m a n y helpful discussions; to S.W. Charles w h o provided m a n y of the ferrofluid samples r e p o r t e d on in the text; and to the E C for financial support u n d e r the B R I T E - E U R A M p r o g r a m m e .
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