- Email: [email protected]
Investigating magnetic fluids by means of complex susceptibility measurements
Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
Investigating magnetic fluids by means of complex susceptibility measurements P.C. Fannin* Department of Electronic and Electrical Engineering, University of Dublin, Trinity College, Dublin 2, Ireland
Abstract There is an ever-increasing interest in the study and application of magnetic fluids and thus it is important to be able to accurately characterise such fluids. One method is by the measurement of the frequency dependent, complex susceptibility, wðoÞ ¼ w0 ðoÞ iw00 ðoÞ; which, when measured over a frequency range of Hz to GHz, enables the convenient determination of the macroscopic and microscopic properties of the fluids. Ferromagnetic resonance, identified by the w0 ðoÞ component passing from a positive to a negative value at a frequency f res, together with relaxation mechanisms, both Brownian and N!eel, can be readily investigated. The latter components can be determined from maxima occurring in a plot of w00 ðoÞ against frequency and enable the effective particle radius of the suspension to be estimated. From measurements of wðoÞ one can also investigate magnetic losses arising in the fluids. An alternative method for determining the effective particle size is the principal or differential susceptibility technique. This fixed frequency technique consists of applying an external DC polarizing magnetic field, H0 ; to the ferrofluid sample contained within a cylindrical coil and measuring the parallel, w8 ðH0 Þ; and perpendicular, w> ðH0 Þ; incremental susceptibility components. In this paper, a brief review is given of the above-mentioned topics and examples of results obtained, presented. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Complex susceptibility; Differential susceptibility; Relaxation; Resonance; Debye; Cole–Cole
The magnetisation M is described by the Langevin expression,
1. Introduction A ferrofluid is a colloidal suspension of single domain ferromagnetic particles dispersed in a liquid carrier and stabilised by means of a suitable organic surfactant. The particles have radii ranging from approximately 2– 10 nm and when they are in suspension their magnetic properties can be described by the Paramagnetism theory of Langevin, suitably modified to cater for a distribution of particle sizes. The particles are considered to be in a state of uniform magnetisation with a magnetic moment, m; given by m ¼ Ms v;
ð1Þ
where Ms denotes saturation magnetisation and v is the magnetic volume of the particle. *Fax: +353-1-6081860/1580. E-mail address: [email protected] (P.C. Fannin).
M ¼ Ms ½cothx 1=x:
ð2Þ
x ¼ mH0 =kT , where k is Boltzmanns constant and H0 the magnetizing field.
2. Relaxation There are two distinct mechanisms by which the magnetisation of ferrofluids may relax after an applied field has been removed: either rotational Brownian motion of the particle within the carrier liquid, with its magnetic moment, m; locked in an axis of easy magnetisation, or by rotation of the magnetic moment within the particle. The time associated with the rotational diffusion is the Brownian relaxation time tB
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P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
[1] where
wðoÞ wN ¼ ðw0 wN Þð1=ð1 þ o2 t2eff Þ
tB ¼ 3V Z=kT:
ð3Þ
V is the hydrodynamic volume of the particle and Z is the dynamic viscosity of the carrier liquid. In the case of the second relaxation mechanism, the magnetic moment may reverse direction within the particle by overcoming an energy barrier, which for uniaxial anisotropy, is given by Kv; where K is the anisotropy constant of the particle. The probability of such a transition is expðsÞ where s is the ratio of anisotropy energy to thermal energy (Kv=kT). This reversal time is characterised by a time tN ; which is referred to as the N!eel relaxation time [2] and given by the expression, tN ¼ t0 expðsÞ
ð4Þ
t0 is a decay time, often quoted as having an approximate value of 108–1010 s. According to Brown [3], for high and low barrier heights, tN ¼ t0 s1=2 expðsÞ; ¼ t0 s;
447
sX2;
s51:
ð5Þ
Other workers [4,5] have subsequently derived a single expression to cater for a continuous range of s; however, because of the difficulty in characterising small magnetic particle systems it is perfectly adequate to use the expressions of Eq. (5). 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 teff [6], where, for a particular particle, teff ¼ tN tB =ðtN þ tB Þ ¼ t8 :
ð6Þ
The mechanism with the shortest relaxation time is being dominant.
3. Susceptibility The frequency-dependent complex susceptibility, wðoÞ may be written in terms of its real and imaginary components, where wðoÞ w0 ðoÞ iw00 ðoÞ:
ð7Þ
The theory developed by Debye [7] to account for the anomalous dielectric dispersion in dipolar fluids has been successfully used [8,9] to account for the analogous case of magnetic fluids. According to Debye’s theory the complex susceptibility, wðoÞ; has a frequency dependence given by the equation wðoÞ wN ¼ ðw0 wN Þ=ð1 þ ioteff Þ
ð8Þ
iot=ð1 þ o2 t2eff ÞÞ;
ð9Þ
where the static or low frequency susceptibility, w0 ; is defined as w0 ¼ nm2 =3kTm0
ð10Þ
and where teff ¼ 1=omax ¼ 1=2 pfmax :
ð11Þ
fmax is the frequency at which w00 ðoÞ is a maximum, n is the particle number density and wN indicates the susceptibility value at very high frequencies. For the case of Brownian relaxation, a distribution of particle volumes corresponds to a distribution of relaxation times where wðoÞ may be expressed in terms of a distribution function, f ðtÞ; giving Z N wðoÞ ¼ wN þ ðw0 wN Þ f ðtÞ dt=ð1 þ iotÞ ð12Þ 0
f ðtÞ may be represented by a normal, log-normal, or a Cole–Cole distribution function. In the Cole–Cole [11] case where the complex susceptibility data fits a depressed circular arc, the relation between w0 ðoÞ and w00 ðoÞ and their dependence on frequency, o=2p; can be displayed by means of the magnetic analogue of the Cole–Cole plot [2]. In the Cole–Cole case, the circular arc cuts the w0 ðoÞ axis at an angle of ac p=2; ac is referred to as the Cole–Cole parameter and is a measure of the particle-size distribution. The magnetic analogue of the Cole–Cole circular arc is described by the equation wðoÞ ¼ wN þ ðw0 wN Þ=½ð1 þ ðiot0 Þ1ac Þ; 0oao1
ð13Þ
which for ac ¼ 0; reduces to that of Eq. (8). Fig. 1(a) presents a plot of the Debye Eq. (9) which clearly indicates that the maximum possible value of the absorption component, w00 ðoÞ; is equal to half that of w0 ðoÞ component. Fig. 1(b) shows the corresponding ideal Cole–Cole plot with ac ¼ 0: An example of how close to this ideal one gets in reality is shown by the susceptibility plots of Fig. 2(a) for a sample (Sample 1), obtained by means of the toroidal technique [10], for a suspension of magnetite in isopar M; the corresponding mean particle radius as determined by electron microscopy was 5 nm. It is apparent that the plots have a Debye-type profile with a maximum in w00 ðoÞ occurring at 1.6 kHz; from Eq. (1) and using a viscosity of 104 N sm2, a corresponding hydrodynamic radius of 69 nm is obtained. This value is far in excess of 5 nm and is indicative of the presence of aggregation. Fig. 2(b) shows the Cole–Cole plot together with a fit obtained by Eq. (13), realised with ac ¼ 0:25 and wN ¼ 0:17: Fig. 2a also shows the fit to the
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
448
susceptibility components obtained with the same parameters and Eq. (13).
χ'(ω)
1.0
3.1. Differential susceptibility
0.8 ω τ=1 0.6
χ''(ω)
0.4
0.2
0.0 10−3
10−2
10−1
100 ωτ
(a)
101
102
103
0.6 Cole-Cole
0.5
χ''
0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
(b)
0.6
χ'
0.8
1.0
and thus
Fig. 1. (a) Debye plot of w0 ðoÞ and w00 ðoÞ against ot; (b) corresponding Cole–Cole plot.
χ''
χ'
1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1
χ0=1 χ00=1.17 α =0.25
101
102
103
χ''
104
105
106
data fit
απ/2 0.1
0.2
0.3
0.4
0.5 χ'
ð16Þ
d½w> ðx0 Þ wjj ðx0 Þ=dx0 ¼ 0;
χ0=1 χ00=1.17 α =0.25
0.0
n 2 ½w> ðH0 Þ wjj ðH0 Þ ¼ bm 1 þ coth2 ðbmH0 Þ: m0 2 cothðbmH0 Þ ; þ bmH0 ðbmH0 Þ2
where b ¼ 1=kT and n is the particle number density. Thus Eqs. (14)–(16) enable the behaviour of w> ðx0 Þ; wjj ðx0 Þ and w> ðx0 Þ wjj ðx0 Þ for values of H0 to be determined (x0 ¼ mH0 =kT), as illustrated in Fig. 3a. Furthermore, the maximum of Eq. (16), say x1 ; is obtained by differentiating and equating to zero, i.e.,
f(Hz)
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05
(b)
data fit
1600Hz
(a)
Before passing on to the frequency region where resonance can be identified, it is worthwhile to consider a simple, single frequency technique, for the determination of the particle size. The measurement of the principal or differential susceptibilities [12,13] (w8 ðH0 Þ) and (w> ðH0 Þ), provides for such a technique and consists of applying an external DC polarizing magnetic field, H0 ; to the ferrofluid sample contained within a cylindrical coil and measuring the parallel, w8 ðH0 Þ; and perpendicular w> ðH0 Þ incremental susceptibility components. The incremental susceptibility is defined as the rate of change of the magnetic moment in a particular direction with respect to a small measuring magnetising force parallel to H0 : In Refs. [12,13] it is shown that n 1 ; ð14Þ wjj ðH0 Þ ¼ bm2 1 coth2 ðbmH0 Þ þ m0 ðbmH0 Þ2 n cothðbmH0 Þ 1 w> ðH0 Þ ¼ bm2 ð15Þ m0 bmH0 ðbmH0 Þ2
0.6
0.7
0.8
0.9
1.0
Fig. 2. (a). Normalised plot of w0 ðoÞ and w00 ðoÞ and fit against f (Hz) for sample 1, (b) corresponding Cole–Cole plot and fit.
ð17Þ 1 4 1 d½w> ðx0 Þ wjj ðx0 Þ=dx0 ¼ þ 3 þ coth z 2 z z z z 1 2 ð18Þ coth z þ coth z ; z
where z ¼ bmH0 : Thus by plotting Eq. (18) in the manner shown in Fig. 3(b), the value, x1 ; at which zero occurs can be readily obtained and since x1 ¼ mH1 =kT; by determining H1 experimentally, the effective magnetic moment, m, and hence the corresponding effective particle radius, r; can be evaluated. The technique essentially consists of inserting the magnetic fluid sample into the alternating magnetic field of a coil, of inductance L and resistance R; and observing the changes in its inductance, DL; and resistance, DR; as the frequency is varied.
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
at an angular frequency, ores ; where for a small polar angle
1.0
χ⊥
0.6
χ⊥−χ||
χ||
Theoretical plot 0.8
χ⊥
ores ¼ gHA ¼ g2K=Ms :
χ||
χ⊥− χ|| 0.2
w> ðoÞ ¼ w> ð0Þ
ξ1=3.72 0.1
10
1
100
1000
ξ=mH/kT
(a) 1.0
Theoretical plot 0.8
1 þ iot2 þ D ; ð1 þ iot2 Þð1 þ iot> Þ þ D
ð22Þ
where w> ð0Þ; t> ; D and t2 are as given in Ref. [15]. When combined with Eq. (8), the overall frequencydependent susceptibility is approximately given as 1 wjj ð0Þ 1 þ iot2 þ D þ2w> ð0Þ wðoÞ ¼ 3 1 þ iotjj ð1 þ iot2 Þð1 þ iot>eff Þ þ D
0.6 d(χ'⊥−χ'||)/dξ
ð21Þ
The transverse or resonant component of the susceptibility, w> ðoÞ; may be described by equations derived in Refs. [14,15] with
0.4
0.0 0.01
449
¼ w0 ðoÞ iw00 ðoÞ:
ð23Þ
0.4 0.2 0.0 ξ1= 3.72
-0.2 0.01
0.1
(b)
1
10
100
1000
ξ=mH/kT
Fig. 3. (a). Theoretical plots of w> ðH0 Þ; wjj ðH0 Þ and w> ðx0 Þ wjj ðx0 Þ against x: A peak in w> ðx0 Þ wjj ðx0 Þ is shown to occur at x1 ¼ 3:72; (b) theoretical plot of d½w> ðH0 Þ wjj ðH0 Þ=dx against x; zero is shown to occur at x1 ¼ 3:72:
The ratio of DL=L is proportional to w0 ðoÞ; whilst w ðoÞ is proportional to DR=oL: From Debey’s equations, at low frequencies 00
wEw0 Ew0 EDL=L:
ð19Þ
This measurement is performed for both orientations of the polarising field, in Ref. [13] examples of this technique are presented.
In Eq. (23) t> has been replaced by t>eff ¼ t> tB = ðt> þ tB Þ in order to take account of the effects of Brownian relaxation; in Ref. [16], it is shown that this latter effect can often be negligible. Fig 4 shows high-frequency susceptibility measurements obtained for a 450 G suspension of manganese ferrite (MnFe2O4) in dibutylphthalat (DBH) (Sample 2) measured by means of the transmission line technique [17] and subjected to a polarising field over the range 0–100 kA m1. From the figure it is seen that for the unpolarised case, resonance occurs at a frequency fres ¼ 1:2 GHz whilst the maximum of the w00 ðoÞ losspeak is shown to occur at a frequency of fmax ¼ 0:8 GHz. Variation of the polarising field, H; over the stated range results in fres and fmax increasing up to a frequencies of 8.2 and 8.4 GHz, respectively. The application of H to the sample effectively results in an increase in the barrier Kv (and hence s ¼ Kv=kT) which the magnetic moment of the particles must overcome. This increase in H
χ'(ω)
0.3
In the GHz frequency range the character of the dispersion changes from relaxation to one of resonance and it is convenient to describe wðoÞ in terms of its parallel (relaxational) w8 ðoÞ and perpendicular (resonant) w> ðoÞ; components with wðo ¼ 1=3ðw8 ðoÞ þ 2w> ðoÞÞ
χ'(ω)
0 kAm−1
0.2
0.8 GHz 8.2 GHz
0.1 χ''(ω)
χ''(ω)
4. Resonance
0.0
100 kAm−1
1.2 GHz 8.4 GHz
ð20Þ 1E8
with corresponding relaxation times t|| and t>, respectively. w> ðoÞ is associated with resonance which is indicated by a change in sign of the value of w0 ðoÞ
1E9
1E10
f (Hz) 0
00
Fig. 4. Plot of w ðoÞ and w ðoÞ over the frequency range 20 MHz–18 GHz for 17 values of polarising field over the range 0–100 kA m-1.
450
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
results in a reduction in spontaneous flipping of the magnetic moments (N!eel relaxation) leading to an increasing dominance from w> ðoÞ to wðoÞ: The effect is to shift fmax and fres as indicated. The result of this is that the value of fmax approaches the value of fres as resonance becomes the dominant process. A plot of fres against Hðores ¼ 2pfres ¼ gðH þ H% A ÞÞ; enables the value of H% A ; a mean value of the anisotropy field to be determined and is found to be equal to 18 kA m1 [17]. This corresponds to a mean value of % at room temperature and bulk anisotropy constant, K; Ms of 0.4 T, of 2.9 103 J/m3. This compares favourably % with a value of K=4
103 J/m3 obtained by means of the decay of remanence technique. The gyromagnetic constant, g; can be obtained from the slope of fres against H whilst an estimate of the damping parameter, a; may be obtained by fitting [18] the original susceptibility data to theoretical susceptibility profiles generated by Eq. (23), suitably modified to cater for a distribution of particle size, r; and anisotropy constant, K: This data then enables the exponential prefactor (t0 ) of N!eels expression for tN ; where t0 ¼ Ms =2agK; to be determined.
5. Magnetic losses The permeability of a magnetic fluid, mðoÞ ¼ m0 ðoÞ im ðoÞ; is a complex quantity which expresses the loss of energy which occurs as the magnetisation alternates. The loss mechanisms cause the flux density, B; to lag behind the applied alternating field, H; by a phase angle d; and in this context two important relations are the loss tangent, tan d; which is also known as the dissipation factor and sin d; which is referred to as the power factor. The dissipated energy per cm3 of the ferrofluid sample is directly proportional to both tan d and sin d [19], where in direct analogy with the dielectric case 00
tanðdÞ ¼ m00 ðoÞ=m0 ðoÞ and sin d ¼ m00 ðoÞ=jmðoÞj: Now, as m00 ðoÞ ¼ w00 ðoÞ; and m0 ðoÞ ¼ w0 ðoÞ þ 1; the corresponding equations for the loss factor and power factor in terms of m00 ðoÞ and w00 ðoÞ; are tan d ¼ w00 ðoÞ=ð1 þ w0 ðoÞÞ
ð24Þ
and sin d ¼ w00 ðoÞ=ðð1 þ w0 ðoÞÞ2 þ w00 ðoÞ2 Þ1=2 :
ð25Þ
Ref. [19] shows the high-frequency dependence of tan d and sin d for four fluids of different magnetisation. It is demonstrated that, with reducing fluid magnetisation, the profiles of both tan d and sin d gradually become closer to that of the w00 ðoÞ profile, until, in the case of a
75 G fluid, the frequency-dependent profiles become almost identical. These results enable one to conclude that in the case of the most dilute fluid, the highfrequency-dependent loss tangent and power factor can be modelled by the imaginary component of Eq. (23). It is of interest to note that in Ref. [20] losses resulting from power dissipation at low frequencies is investigated as a function of particle size and composition. Finally, with the advent of magnetoelectronics and the possible use of nanoparticles as devices, such as switches, one is interested in the signal-to-noise (SNR) [21,22] ratio of such systems. Here again the usefulness of the availability of complex susceptibility data comes to the fore since the SNR for a field-driven magnetisation of a nano-particle has the form SNR ¼ ðp=12t0 Þ½6agZK 1 2 H12 s2 expðsÞ;
ð26Þ
where H1 is the amplitude of the exciting field and where the other parameters are the macroscopic and microscopic components discussed previously.
6. Conclusion In this paper an attempt has been made to demonstrate the usefulness of complex susceptibility measurements, wðoÞ ¼ w0 ðoÞ iw00 ðoÞ; which, when measured over the frequency range from Hz to GHz, enables the convenient determination of the macroscopic and microscopic properties of the fluids. Relaxation mechanisms, ferromagnetic resonance, magnetic losses and the SNR of such colloidal nano-particle systems can be readily determined from such data. The differential susceptibility technique has also been suggested as an alternative method for determining the effective particle size of magnetic fluids.
Acknowledgements Acknowledgement is due to B.K.P. Scaife for useful discussions and to the Irish Higher Education Authority and Prodex for funding this work.
References [1] [2] [3] [4]
W.F. Brown, J. Appl. Phys. 34 (1963) 1319. L. N!eel, Ann. Geophys. 5 (1949) 99. W.F. Brown, Phys. Rev. 130 (1963) 1677. L. Bessais, L.Ben. Jaffel, J.L. Dormmann, Phys. Rev. B 45 (14) (1992) 7805. [5] A. Aharoni, Phys. Rev. B 46 (9) (1992) 5434. [6] M.I. Shliomis, Sov. Phys.-Usp. 17 (1974) 53. [7] P. Debye, Polar Molecules, The Chemical Catalog Company, New York, 1929.
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451 [8] P.C. Fannin, A.T. Giannitsis, S.W. Charles, Eur. Phys. J. 12 (2000) 93. [9] M.M. Mairov, Magnetohydrodynamics (Trans. Magnitnaia Hidrodinamika) 2 (21) (1979) 135. [10] P.C. Fannin, B.K.P. Scaife, S.W. Charles, J. Phys. E 19 (1986) 238. [11] K.S. Cole, R.H. Cole, J. Chem. Phys. 9 (1941) 341. [12] E. Blums, A. Cebers, M. MM aiorov, Magnetic Fluids, Walter de Gruyter, Berlin, 1997. [13] P.C. Fannin, B.K.P. Scaife, A.T. Giannitsis, S.W. Charles, J. Phys. D 35 (2002) 1305. [14] Y.L. Raikher, M.I. Shliomis, Sov. Phys. JETP 40 (1975) 526. [15] W.T. Coffey, Yu.P. Kalmykov, E.S. Massawe, Adv. Chem. Phys. 85 (2) (1993) 667.
451
[16] W.T. Coffey, P.C. Fannin, J. Phys.: Condens. Matter 34 (2002) 3677. [17] P.C. Fannin, D. Vincent, G. Massart, P. Perov, S. Neveu, Eur. Phys. J. 8 (1999) 247. [18] P.C. Fannin, T. Relihan, S.W. Charles, Phys. Rev. B 55 (21) (1997) 14423. [19] P.C. Fannin, S.W. Charles, J. Magn. Magn. Mater. 226 (2001) 1887. [20] R.E. Rosensweig, J. Magn. Magn. Mater. 252 (2002) 370. [21] P.C. Fannin, Y.L. Raikher, J. Phys. D 34 (2001) 612. [22] P.C. Fannin, Yu.L. Raikher, A.T. Giannitsis, S.W. Charles, J. Magn. Magn. Mater. 252 (2002) 114.
Investigating magnetic fluids by means of complex susceptibility measurements P.C. Fannin* Department of Electronic and Electrical Engineering, University of Dublin, Trinity College, Dublin 2, Ireland
Abstract There is an ever-increasing interest in the study and application of magnetic fluids and thus it is important to be able to accurately characterise such fluids. One method is by the measurement of the frequency dependent, complex susceptibility, wðoÞ ¼ w0 ðoÞ iw00 ðoÞ; which, when measured over a frequency range of Hz to GHz, enables the convenient determination of the macroscopic and microscopic properties of the fluids. Ferromagnetic resonance, identified by the w0 ðoÞ component passing from a positive to a negative value at a frequency f res, together with relaxation mechanisms, both Brownian and N!eel, can be readily investigated. The latter components can be determined from maxima occurring in a plot of w00 ðoÞ against frequency and enable the effective particle radius of the suspension to be estimated. From measurements of wðoÞ one can also investigate magnetic losses arising in the fluids. An alternative method for determining the effective particle size is the principal or differential susceptibility technique. This fixed frequency technique consists of applying an external DC polarizing magnetic field, H0 ; to the ferrofluid sample contained within a cylindrical coil and measuring the parallel, w8 ðH0 Þ; and perpendicular, w> ðH0 Þ; incremental susceptibility components. In this paper, a brief review is given of the above-mentioned topics and examples of results obtained, presented. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Complex susceptibility; Differential susceptibility; Relaxation; Resonance; Debye; Cole–Cole
The magnetisation M is described by the Langevin expression,
1. Introduction A ferrofluid is a colloidal suspension of single domain ferromagnetic particles dispersed in a liquid carrier and stabilised by means of a suitable organic surfactant. The particles have radii ranging from approximately 2– 10 nm and when they are in suspension their magnetic properties can be described by the Paramagnetism theory of Langevin, suitably modified to cater for a distribution of particle sizes. The particles are considered to be in a state of uniform magnetisation with a magnetic moment, m; given by m ¼ Ms v;
ð1Þ
where Ms denotes saturation magnetisation and v is the magnetic volume of the particle. *Fax: +353-1-6081860/1580. E-mail address: [email protected] (P.C. Fannin).
M ¼ Ms ½cothx 1=x:
ð2Þ
x ¼ mH0 =kT , where k is Boltzmanns constant and H0 the magnetizing field.
2. Relaxation There are two distinct mechanisms by which the magnetisation of ferrofluids may relax after an applied field has been removed: either rotational Brownian motion of the particle within the carrier liquid, with its magnetic moment, m; locked in an axis of easy magnetisation, or by rotation of the magnetic moment within the particle. The time associated with the rotational diffusion is the Brownian relaxation time tB
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 1 0 9 3 - 4
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
[1] where
wðoÞ wN ¼ ðw0 wN Þð1=ð1 þ o2 t2eff Þ
tB ¼ 3V Z=kT:
ð3Þ
V is the hydrodynamic volume of the particle and Z is the dynamic viscosity of the carrier liquid. In the case of the second relaxation mechanism, the magnetic moment may reverse direction within the particle by overcoming an energy barrier, which for uniaxial anisotropy, is given by Kv; where K is the anisotropy constant of the particle. The probability of such a transition is expðsÞ where s is the ratio of anisotropy energy to thermal energy (Kv=kT). This reversal time is characterised by a time tN ; which is referred to as the N!eel relaxation time [2] and given by the expression, tN ¼ t0 expðsÞ
ð4Þ
t0 is a decay time, often quoted as having an approximate value of 108–1010 s. According to Brown [3], for high and low barrier heights, tN ¼ t0 s1=2 expðsÞ; ¼ t0 s;
447
sX2;
s51:
ð5Þ
Other workers [4,5] have subsequently derived a single expression to cater for a continuous range of s; however, because of the difficulty in characterising small magnetic particle systems it is perfectly adequate to use the expressions of Eq. (5). 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 teff [6], where, for a particular particle, teff ¼ tN tB =ðtN þ tB Þ ¼ t8 :
ð6Þ
The mechanism with the shortest relaxation time is being dominant.
3. Susceptibility The frequency-dependent complex susceptibility, wðoÞ may be written in terms of its real and imaginary components, where wðoÞ w0 ðoÞ iw00 ðoÞ:
ð7Þ
The theory developed by Debye [7] to account for the anomalous dielectric dispersion in dipolar fluids has been successfully used [8,9] to account for the analogous case of magnetic fluids. According to Debye’s theory the complex susceptibility, wðoÞ; has a frequency dependence given by the equation wðoÞ wN ¼ ðw0 wN Þ=ð1 þ ioteff Þ
ð8Þ
iot=ð1 þ o2 t2eff ÞÞ;
ð9Þ
where the static or low frequency susceptibility, w0 ; is defined as w0 ¼ nm2 =3kTm0
ð10Þ
and where teff ¼ 1=omax ¼ 1=2 pfmax :
ð11Þ
fmax is the frequency at which w00 ðoÞ is a maximum, n is the particle number density and wN indicates the susceptibility value at very high frequencies. For the case of Brownian relaxation, a distribution of particle volumes corresponds to a distribution of relaxation times where wðoÞ may be expressed in terms of a distribution function, f ðtÞ; giving Z N wðoÞ ¼ wN þ ðw0 wN Þ f ðtÞ dt=ð1 þ iotÞ ð12Þ 0
f ðtÞ may be represented by a normal, log-normal, or a Cole–Cole distribution function. In the Cole–Cole [11] case where the complex susceptibility data fits a depressed circular arc, the relation between w0 ðoÞ and w00 ðoÞ and their dependence on frequency, o=2p; can be displayed by means of the magnetic analogue of the Cole–Cole plot [2]. In the Cole–Cole case, the circular arc cuts the w0 ðoÞ axis at an angle of ac p=2; ac is referred to as the Cole–Cole parameter and is a measure of the particle-size distribution. The magnetic analogue of the Cole–Cole circular arc is described by the equation wðoÞ ¼ wN þ ðw0 wN Þ=½ð1 þ ðiot0 Þ1ac Þ; 0oao1
ð13Þ
which for ac ¼ 0; reduces to that of Eq. (8). Fig. 1(a) presents a plot of the Debye Eq. (9) which clearly indicates that the maximum possible value of the absorption component, w00 ðoÞ; is equal to half that of w0 ðoÞ component. Fig. 1(b) shows the corresponding ideal Cole–Cole plot with ac ¼ 0: An example of how close to this ideal one gets in reality is shown by the susceptibility plots of Fig. 2(a) for a sample (Sample 1), obtained by means of the toroidal technique [10], for a suspension of magnetite in isopar M; the corresponding mean particle radius as determined by electron microscopy was 5 nm. It is apparent that the plots have a Debye-type profile with a maximum in w00 ðoÞ occurring at 1.6 kHz; from Eq. (1) and using a viscosity of 104 N sm2, a corresponding hydrodynamic radius of 69 nm is obtained. This value is far in excess of 5 nm and is indicative of the presence of aggregation. Fig. 2(b) shows the Cole–Cole plot together with a fit obtained by Eq. (13), realised with ac ¼ 0:25 and wN ¼ 0:17: Fig. 2a also shows the fit to the
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
448
susceptibility components obtained with the same parameters and Eq. (13).
χ'(ω)
1.0
3.1. Differential susceptibility
0.8 ω τ=1 0.6
χ''(ω)
0.4
0.2
0.0 10−3
10−2
10−1
100 ωτ
(a)
101
102
103
0.6 Cole-Cole
0.5
χ''
0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
(b)
0.6
χ'
0.8
1.0
and thus
Fig. 1. (a) Debye plot of w0 ðoÞ and w00 ðoÞ against ot; (b) corresponding Cole–Cole plot.
χ''
χ'
1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1
χ0=1 χ00=1.17 α =0.25
101
102
103
χ''
104
105
106
data fit
απ/2 0.1
0.2
0.3
0.4
0.5 χ'
ð16Þ
d½w> ðx0 Þ wjj ðx0 Þ=dx0 ¼ 0;
χ0=1 χ00=1.17 α =0.25
0.0
n 2 ½w> ðH0 Þ wjj ðH0 Þ ¼ bm 1 þ coth2 ðbmH0 Þ: m0 2 cothðbmH0 Þ ; þ bmH0 ðbmH0 Þ2
where b ¼ 1=kT and n is the particle number density. Thus Eqs. (14)–(16) enable the behaviour of w> ðx0 Þ; wjj ðx0 Þ and w> ðx0 Þ wjj ðx0 Þ for values of H0 to be determined (x0 ¼ mH0 =kT), as illustrated in Fig. 3a. Furthermore, the maximum of Eq. (16), say x1 ; is obtained by differentiating and equating to zero, i.e.,
f(Hz)
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05
(b)
data fit
1600Hz
(a)
Before passing on to the frequency region where resonance can be identified, it is worthwhile to consider a simple, single frequency technique, for the determination of the particle size. The measurement of the principal or differential susceptibilities [12,13] (w8 ðH0 Þ) and (w> ðH0 Þ), provides for such a technique and consists of applying an external DC polarizing magnetic field, H0 ; to the ferrofluid sample contained within a cylindrical coil and measuring the parallel, w8 ðH0 Þ; and perpendicular w> ðH0 Þ incremental susceptibility components. The incremental susceptibility is defined as the rate of change of the magnetic moment in a particular direction with respect to a small measuring magnetising force parallel to H0 : In Refs. [12,13] it is shown that n 1 ; ð14Þ wjj ðH0 Þ ¼ bm2 1 coth2 ðbmH0 Þ þ m0 ðbmH0 Þ2 n cothðbmH0 Þ 1 w> ðH0 Þ ¼ bm2 ð15Þ m0 bmH0 ðbmH0 Þ2
0.6
0.7
0.8
0.9
1.0
Fig. 2. (a). Normalised plot of w0 ðoÞ and w00 ðoÞ and fit against f (Hz) for sample 1, (b) corresponding Cole–Cole plot and fit.
ð17Þ 1 4 1 d½w> ðx0 Þ wjj ðx0 Þ=dx0 ¼ þ 3 þ coth z 2 z z z z 1 2 ð18Þ coth z þ coth z ; z
where z ¼ bmH0 : Thus by plotting Eq. (18) in the manner shown in Fig. 3(b), the value, x1 ; at which zero occurs can be readily obtained and since x1 ¼ mH1 =kT; by determining H1 experimentally, the effective magnetic moment, m, and hence the corresponding effective particle radius, r; can be evaluated. The technique essentially consists of inserting the magnetic fluid sample into the alternating magnetic field of a coil, of inductance L and resistance R; and observing the changes in its inductance, DL; and resistance, DR; as the frequency is varied.
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
at an angular frequency, ores ; where for a small polar angle
1.0
χ⊥
0.6
χ⊥−χ||
χ||
Theoretical plot 0.8
χ⊥
ores ¼ gHA ¼ g2K=Ms :
χ||
χ⊥− χ|| 0.2
w> ðoÞ ¼ w> ð0Þ
ξ1=3.72 0.1
10
1
100
1000
ξ=mH/kT
(a) 1.0
Theoretical plot 0.8
1 þ iot2 þ D ; ð1 þ iot2 Þð1 þ iot> Þ þ D
ð22Þ
where w> ð0Þ; t> ; D and t2 are as given in Ref. [15]. When combined with Eq. (8), the overall frequencydependent susceptibility is approximately given as 1 wjj ð0Þ 1 þ iot2 þ D þ2w> ð0Þ wðoÞ ¼ 3 1 þ iotjj ð1 þ iot2 Þð1 þ iot>eff Þ þ D
0.6 d(χ'⊥−χ'||)/dξ
ð21Þ
The transverse or resonant component of the susceptibility, w> ðoÞ; may be described by equations derived in Refs. [14,15] with
0.4
0.0 0.01
449
¼ w0 ðoÞ iw00 ðoÞ:
ð23Þ
0.4 0.2 0.0 ξ1= 3.72
-0.2 0.01
0.1
(b)
1
10
100
1000
ξ=mH/kT
Fig. 3. (a). Theoretical plots of w> ðH0 Þ; wjj ðH0 Þ and w> ðx0 Þ wjj ðx0 Þ against x: A peak in w> ðx0 Þ wjj ðx0 Þ is shown to occur at x1 ¼ 3:72; (b) theoretical plot of d½w> ðH0 Þ wjj ðH0 Þ=dx against x; zero is shown to occur at x1 ¼ 3:72:
The ratio of DL=L is proportional to w0 ðoÞ; whilst w ðoÞ is proportional to DR=oL: From Debey’s equations, at low frequencies 00
wEw0 Ew0 EDL=L:
ð19Þ
This measurement is performed for both orientations of the polarising field, in Ref. [13] examples of this technique are presented.
In Eq. (23) t> has been replaced by t>eff ¼ t> tB = ðt> þ tB Þ in order to take account of the effects of Brownian relaxation; in Ref. [16], it is shown that this latter effect can often be negligible. Fig 4 shows high-frequency susceptibility measurements obtained for a 450 G suspension of manganese ferrite (MnFe2O4) in dibutylphthalat (DBH) (Sample 2) measured by means of the transmission line technique [17] and subjected to a polarising field over the range 0–100 kA m1. From the figure it is seen that for the unpolarised case, resonance occurs at a frequency fres ¼ 1:2 GHz whilst the maximum of the w00 ðoÞ losspeak is shown to occur at a frequency of fmax ¼ 0:8 GHz. Variation of the polarising field, H; over the stated range results in fres and fmax increasing up to a frequencies of 8.2 and 8.4 GHz, respectively. The application of H to the sample effectively results in an increase in the barrier Kv (and hence s ¼ Kv=kT) which the magnetic moment of the particles must overcome. This increase in H
χ'(ω)
0.3
In the GHz frequency range the character of the dispersion changes from relaxation to one of resonance and it is convenient to describe wðoÞ in terms of its parallel (relaxational) w8 ðoÞ and perpendicular (resonant) w> ðoÞ; components with wðo ¼ 1=3ðw8 ðoÞ þ 2w> ðoÞÞ
χ'(ω)
0 kAm−1
0.2
0.8 GHz 8.2 GHz
0.1 χ''(ω)
χ''(ω)
4. Resonance
0.0
100 kAm−1
1.2 GHz 8.4 GHz
ð20Þ 1E8
with corresponding relaxation times t|| and t>, respectively. w> ðoÞ is associated with resonance which is indicated by a change in sign of the value of w0 ðoÞ
1E9
1E10
f (Hz) 0
00
Fig. 4. Plot of w ðoÞ and w ðoÞ over the frequency range 20 MHz–18 GHz for 17 values of polarising field over the range 0–100 kA m-1.
450
P.C. Fannin / Journal of Magnetism and Magnetic Materials 258–259 (2003) 446–451
results in a reduction in spontaneous flipping of the magnetic moments (N!eel relaxation) leading to an increasing dominance from w> ðoÞ to wðoÞ: The effect is to shift fmax and fres as indicated. The result of this is that the value of fmax approaches the value of fres as resonance becomes the dominant process. A plot of fres against Hðores ¼ 2pfres ¼ gðH þ H% A ÞÞ; enables the value of H% A ; a mean value of the anisotropy field to be determined and is found to be equal to 18 kA m1 [17]. This corresponds to a mean value of % at room temperature and bulk anisotropy constant, K; Ms of 0.4 T, of 2.9 103 J/m3. This compares favourably % with a value of K=4
103 J/m3 obtained by means of the decay of remanence technique. The gyromagnetic constant, g; can be obtained from the slope of fres against H whilst an estimate of the damping parameter, a; may be obtained by fitting [18] the original susceptibility data to theoretical susceptibility profiles generated by Eq. (23), suitably modified to cater for a distribution of particle size, r; and anisotropy constant, K: This data then enables the exponential prefactor (t0 ) of N!eels expression for tN ; where t0 ¼ Ms =2agK; to be determined.
5. Magnetic losses The permeability of a magnetic fluid, mðoÞ ¼ m0 ðoÞ im ðoÞ; is a complex quantity which expresses the loss of energy which occurs as the magnetisation alternates. The loss mechanisms cause the flux density, B; to lag behind the applied alternating field, H; by a phase angle d; and in this context two important relations are the loss tangent, tan d; which is also known as the dissipation factor and sin d; which is referred to as the power factor. The dissipated energy per cm3 of the ferrofluid sample is directly proportional to both tan d and sin d [19], where in direct analogy with the dielectric case 00
tanðdÞ ¼ m00 ðoÞ=m0 ðoÞ and sin d ¼ m00 ðoÞ=jmðoÞj: Now, as m00 ðoÞ ¼ w00 ðoÞ; and m0 ðoÞ ¼ w0 ðoÞ þ 1; the corresponding equations for the loss factor and power factor in terms of m00 ðoÞ and w00 ðoÞ; are tan d ¼ w00 ðoÞ=ð1 þ w0 ðoÞÞ
ð24Þ
and sin d ¼ w00 ðoÞ=ðð1 þ w0 ðoÞÞ2 þ w00 ðoÞ2 Þ1=2 :
ð25Þ
Ref. [19] shows the high-frequency dependence of tan d and sin d for four fluids of different magnetisation. It is demonstrated that, with reducing fluid magnetisation, the profiles of both tan d and sin d gradually become closer to that of the w00 ðoÞ profile, until, in the case of a
75 G fluid, the frequency-dependent profiles become almost identical. These results enable one to conclude that in the case of the most dilute fluid, the highfrequency-dependent loss tangent and power factor can be modelled by the imaginary component of Eq. (23). It is of interest to note that in Ref. [20] losses resulting from power dissipation at low frequencies is investigated as a function of particle size and composition. Finally, with the advent of magnetoelectronics and the possible use of nanoparticles as devices, such as switches, one is interested in the signal-to-noise (SNR) [21,22] ratio of such systems. Here again the usefulness of the availability of complex susceptibility data comes to the fore since the SNR for a field-driven magnetisation of a nano-particle has the form SNR ¼ ðp=12t0 Þ½6agZK 1 2 H12 s2 expðsÞ;
ð26Þ
where H1 is the amplitude of the exciting field and where the other parameters are the macroscopic and microscopic components discussed previously.
6. Conclusion In this paper an attempt has been made to demonstrate the usefulness of complex susceptibility measurements, wðoÞ ¼ w0 ðoÞ iw00 ðoÞ; which, when measured over the frequency range from Hz to GHz, enables the convenient determination of the macroscopic and microscopic properties of the fluids. Relaxation mechanisms, ferromagnetic resonance, magnetic losses and the SNR of such colloidal nano-particle systems can be readily determined from such data. The differential susceptibility technique has also been suggested as an alternative method for determining the effective particle size of magnetic fluids.
Acknowledgements Acknowledgement is due to B.K.P. Scaife for useful discussions and to the Irish Higher Education Authority and Prodex for funding this work.
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