Characterisation of magnetic fluids

Journal of Alloys and Compounds 369 (2004) 43–51

Characterisation of magnetic fluids P.C. Fannin∗ Department of Electronic and Electrical Engineering, University of Dublin, Trinity College, Dublin 2, Ireland

Abstract One method suitable for the accurate characterisation of magnetic fluids is by the measurement of the frequency-dependent, complex susceptibility, χ(ω) = χ (ω) − iχ (ω). Such measurements enable data on, relaxation mechanisms, ferromagnetic resonance, stochastic resonance, non-linear properties, magnetic losses and the signal-to-noise ratio (SNR) of these colloidal nano-particle systems to be readily and successfully determined. Here, a brief review is given of the above-mentioned topics and examples of results obtained, presented. © 2003 Elsevier B.V. All rights reserved. Keywords: Magnetic fluids; Magnetic measurements; Nanostructures; Anisotropy

ated with the rotational diffusion is the Brownian relaxation time τ B [1] where

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 = Ms v, where Ms denotes saturation magnetisation and v is the magnetic volume of the particle. The magnetisation M is described by the Langevin expression, M = Ms [coth ξ − 1/ξ].

(1)

ξ = mH0 /kT, where k is Boltzmanns constant and H0 the magnetizing field. 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 associ∗

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τB =

3Vη kT

(2)

V is the hydrodynamic volume of the particle and η 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 exp(σ) where σ is the ratio of anisotropy energy to thermal energy (Kv/kT). This reversal time is characterised by a time τ N , which is referred to as the Néel relaxation time [2], and given by the expression, τN = τ0 exp(σ)

(3)

τ 0 is a decay time, often quoted as having an approximate value of 10−8 to 10−10 s. According to Brown [3], for high and low barrier heights, τN = τ0 σ −1/2 exp(σ), σ ≥ 2 = τ0 σ, σ1

(4)

Other workers [4,5] have subsequently derived a single expression to cater for a continuous range of σ, however because of the difficulty in characterising small magnetic particle systems it is perfectly adequate to use the expressions of Eq. (4).

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P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

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 τ eff [6], where, for a particular particle, τN τB (5) τeff = = τ (τN + τB ) the mechanism with the shortest relaxation time being dominant. For the particle sizes used in this study, Néel relaxation would be observed in the MHz–GHz region [7]; thus in the frequency range measured here Néel relaxation is considered to be dominant.

2. Susceptibility The frequency dependent complex susceptibility, χ(ω), may be written in terms of its real and imaginary components, where χ(ω) = χ (ω) − iχ (ω).

(6)

The theory developed by Debye [8] to account for the anomalous dielectric dispersion in dipolar fluids has been successfully used [9] to account for the analogous case of magnetic fluids. According to Debye’ s theory the complex susceptibility, χ(ω), has a frequency dependence given by the equation, (χ0 − χ∞ ) χ(ω) − χ∞ = (7) (1 + iωτeff ) where the static susceptibility or low frequency susceptibility, χ0 , is defined as χ0 =

nm2 . 3kTµ0

and where 1 1 = , τeff = ωmax 2πfmax

(8)

(9)

fmax is the frequency at which χ (ω) is a maximum, n the particle number density and χ∞ 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 χ(ω) may be expressed in terms of a distribution function, f(τ), giving
∞ f(τ)dτ χ(ω) = χ∞ + (χ0 − χ∞ ) (10) 0 (1 + iωτ) The relation between χ (ω) and χ (ω) and their dependence on frequency, ω/2π, can be displayed by means of the magnetic analogue of the Cole–Cole plot [10] where the data fits a depressed circular arc. In the Cole–Cole case the circular arc cuts the χ (ω) axis at an angle of αc π/2; αc 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 χ(ω) = χ∞ + (χ0 − χ∞ )/(1 + (iωτeff )1−αc ), 0 < α < 1 (11) which for αc = 0, reduces to that of Eq. (7). A simple method of measuring the complex susceptibility of a magnetic fluid sample essentially consists of inserting the sample into the alternating magnetic field of a coil, of inductance L and resistance R, and observing the changes in it’s inductance, L, and resistance, R, as the frequency is varied. The ratio of L/L is proportional to χ (ω), whilst χ (ω) is proportional to R/␻L. From Debey’s equations, at low frequencies χ ≈ χ ≈ χ0 ≈

L . L

(12)

Whilst this technique is suitable for single frequency measurement [11] it has many limiting factor for use over a broad range of frequencies; in such situations the toroidal technique [12] is far superior. The Cole–Cole plot of an ideal Debye profile is a semi circle and an example of how close to this ideal one gets in reality is shown by the susceptibility plots of Fig. 1(a), obtained by means of the toroidal technique [12], for a suspension of magnetite in water (sample 1); 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 χ (ω) occurring at 1.25 kHz; from Eq. (2) and using a viscosity of 10−4 Nsm−2 , a corresponding approximate hydrodynamic radius of 75 nm is obtained. This value of hydrodynamic radius is far greater than the value of magnetic radius plus surfactant thickness (say 2–4 nm), and is thus indicative of the existence of aggregation. Fig. 1(a) also shows the fit obtained by fitting Eq. (11) with αc = 0.45, to the data, whilst Fig. 1(b) shows the corresponding Cole–Cole plots of the data and the fit. In both case the fits prove to be a good approximation to the data. By application of a polarising winding, the toroidal technique is ideally suited for the study of dependence of the low-frequency ac susceptibility χ(ω, H0 ). Assuming the Langevin function for the magnetization of the fluid, an expression for the field can be written as follows, χ(ω, H) = with [13],

χ0 (1 + f(H)) − χ∞ + χ∞ 1 + iωτeff 

(1 + f(H)) = 3 1 +



kT mH

(12 ) 

2 − coth

2

mH kT

 (13)

and [14] τeff =

τ(H=0) [ξ − 2L(ξ) − ξL(ξ)2] L(ξ)

(14)

P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

1.0

45

Debye fit with α=0.45

Fit

(a) χ’(ω )

0.8

0.6

Data 0.4

fmax=1250 Hz χ’’(ω)

0.2

0.0 0

1

10

2

10

3

10

4

10

5

10

6

10

10

f (Hz)

(a) 0.4

Data

(b)

0.3

0.2

Fit χ ’’(ω)

0.1

0.0

φ=απ /2

-0.1

-0.2

-0.3 0.0

(b)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

χ’(ω)

Fig. 1. (a) Plot of χ (ω) and χ (ω) for sample 1 and fit against f (Hz) and (b) Cole–Cole plot of sample 1 and fit.

The above equations predict: (i) a reduction in both χ (ω) and χ (ω) with increasing biasing field and; (ii) a corresponding shift in fmax to higher frequencies. These points are clearly confirmed by the results obtained for polarising fields over the range 0–11,250 A/m for sample 1, where, in Fig. 2(a and b), a reduction in both χ (ω) and χ (ω) is apparent whilst in Fig. 2(a), fmax is shown to shift from 1.25 to 8.9 kHz corresponding to an approximate variation in effective particle radius from 75 to 39 nm. Fig. 3 shows a comparison between the variation in τ eff as a function of polarising field determined by Eq. (9) and that obtained by means of the theoretical Eq. (14) for τ .

It can be seen that both plots are almost identical thereby confirming the accuracy of the data and the usefulness of the technique in investigating the parallel relaxation time, τ . 3. Non-linear susceptibility In direct analogy with the dielectric case, the non-linear magnetic increment (χ(ω)) [15–17] is defined as the difference between the susceptibility value measured by a small ac probing field in the presence (χ(ω)H ) and absence (χ(ω)H=0 ) of a dc field, H, of much higher magnitude, giving, χ(ω) = χH (ω) − χH=0 (ω)

(15)

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P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

1.0 H=0

χ′ ( ω) 0.5

H=11,250 Am

-1

0.0 10

1

2

10

3

10

10

4

10

5

10

6

f(Hz)

(a)

0.5

0.4

0.3 H=0

χ ′′(ω )

fmax=1.25 kHz. 0.2

fmax=8.9 kHz.

0.1

H=11,250 Am

0.0 10

1

10

2

10

3

10

4

-1

10

5

10

6

f(Hz)

(b)

Fig. 2. (a) Plot of χ (ω, H) against f (Hz), for sample 1. (b) Plot of χ (ω, H) against f (Hz), for sample 1.

which can be represented in the complex form, χ(ω) = χ (ω) − iχ (ω)

4. Magnetic losses (16)

and again the toroidal technique is ideally suited for the investigation of this parameter. Fig. 4(a and b) shown the non-linear results obtained in the case of sample 1 and the profiles are similar to those obtained in non-linear dielectric studies. This result raises the future possibility of fitting this data to the existing non-linear dielectric models in order to determine one suitable for the magnetic case.

With the advent of magnetoelectronics, the magnetic losses of magnetic fluids are of particular current interest and such losses may be expressed in terms of the loss tangent, tan δ [18–21], which is also known as the dissipation factor. This property can be investigated via the permeability of the magnetic fluid, µ(ω) = µ (ω)−iµ (ω), which is a complex quantity and 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

P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51 2.0x10

-4

1.5x10

-4

τ (s) 1.0x10

-4

5.0x10

-5

47

Data +++++ Theoretical ------

τ ||

0.0 0

2000

4000

6000

8000

10000

12000

H (A/m)

Fig. 3. Plot ofτ , against polarising field, H for sample 1.

field, H, by a phase angle δ, and in this context an important relations is the loss tangent, tan δ, which is also known as the dissipation factor. The dissipated energy per cm3 of the ferrofluid sample is directly proportional to both tan δ [19], where in direct analogy with the dielectric case, [18] tan δ =

µ (ω) µ (ω)

(17)

Now, as µ (ω) = χ (ω), and µ (ω) = χ (ω) + 1, the corresponding equations for the loss factor and power factor in terms of χ (ω) and χ (ω), are tan δ =

χ (ω) (1 + χ (ω))

(18)

convenient to describe χ(ω) in terms of its parallel (relaxational) χ (ω) and perpendicular (resonant) χ⊥ (ω), components, with, χ(ω) = 13 (χ (ω) + 2χ⊥ (ω))

(20)

with corresponding relaxation times τ and τ ⊥ , respectively. χ⊥ (ω) is associated with resonance which is indicated by a change in sign of the value of χ (ω) at an angular frequency, ωres ; where for a small polar angle, ωres = γHA =

γ2K . Ms

(21)

The χ (ω) component enables, a measure of the mean heat per unit volume, T J/m3 , [21] of the sample to be easily determined, as

The transverse or resonant component of the susceptibility, χ⊥ (ω), may be described by equations derived in [22,23] with,

T = πχ µ0 h2 J/m3

χ⊥ (ω) = χ⊥ (0)

(19)

where h is the ac probing field. Fig. 5 presents a plot of tan δ as a function of frequency for different values of polarising field, H, for sample 1. Whilst the frequency at which tan δ is a maximum increases with increase in H, the amplitude at this frequency decreases and is indicative of the fluid becoming less-lossy with increase in H. In the context of the application of magnetic fluids in the area of low-loss devices, this result is of significance as it indicates a possible technique for controlling such losses.

5. Resonance In the GHz frequency range the character of the dispersion changes from relaxation to one of resonance and it is

1 + iωτ2 +  (1 + iωτ2 )(1 + iωτ⊥ ) + 

(22)

where χ⊥ (0), τ ⊥ ,  and τ 2 are as given in [23]. When combined with Eq. (7), the overall frequencydependent susceptibility is approximately given as, χ(ω) =

  χ (0) 1 1 + iωτ2 +  + 2χ⊥ (0) 3 1 + iωτ (1 + iωτ2 )(1 + iωτ⊥eff ) + 

= χ (ω) − iχ (ω)

(23)

Fig. 6 shows high-frequency susceptibility measurements obtained for a 600 G suspension of magnetite in isopar M (sample 2) measured by means of the transmission line technique. From the figure it is seen that for the unpolarised case, resonance occurs at a frequency fres = 1.67 GHz whilst

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P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

0.4 0.3 0.2 0.1 0.0 -0.1

∆χ′

-0.2 -0.3 increasing polarising field

-0.4 -0.5 -0.6 -0.7 -0.8 1

2

10

3

10

4

10

10

5

6

10

10

f(Hz)

(a)

0.100 0.075 0.050 0.025 0.000 -0.025

∆χ′′

-0.050 -0.075 increasing polarising field of

-0.100 -0.125 -0.150 -0.175 -0.200 10

1

10

2

10

(b)

3

10

4

5

10

10

6

f(Hz)

Fig. 4. Plots of (a) χ and (b) χ as a function of frequency for different values of polarising field, H, for sample 1.

the maximum of the χ (ω) loss-peak is shown to occur at a frequency of fmax = 1.1 GHz. Variation of the polarising field, H, over the range 0–100 kAm−1 , results in fres and fmax increasing up to a frequencies of 5.4 and 5.2 GHz, respectively, thereby indicating that the value of fmax approaches the value of fres as resonance becomes the dominant process. A plot of fres against H (ωres = 2πfres = γ(H + H A )), as shown in Fig. 7, enables the value of H A , a mean value

of the anisotropy field to be determined and is found to be equal to 48 kAm−1 . This corresponds to a mean value of anisotropy constant, K, at room temperature and bulk Ms of 0.4 T, of 9.6 × 103 J/m3 . A value of gyromagnetic constant, γ = 2.25 × 105 s−1 A−1 m, is obtained from the slope of Fig. 7, whilst an estimate of the damping parameter, α, may be obtained by fitting [24] the original susceptibility data to theoretical susceptibility profiles generated by Eqs. (7) and (22),

P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

49

0.200 0.175

2.2 kHz

0.150 0.125

tanδ

0.100 0.075 0.050 0.025

9.7 kHz

0.000 1

2

10

3

10

4

10

10

5

10

6

10

f(Hz) Fig. 5. Plot of tan δ as a function of frequency for different values of polarising field, H, for sample 1.

1.2

χ ′(ω)

χ′ ( ω) 1.0

fmax=1.1 MHz

0.8 0.6

χ′′(ω)

χ′′(ω) 0.4 0.2 0.0

fres=1.67 MHz

-0.2 1E8

1

E9

f(Hz Fig. 6. Plot of χ (ω) and χ (ω) against f (Hz) for sample 2.

suitably modified to cater for a distribution of particle size, r, and anisotropy constant, K. This data then enables the exponential prefactor (τ 0 ) of Néels expression for τ N , where τ0 = Ms /2αγK, and the magnetic viscosity, where ηm = Ms /6αγ, to be determined.

6. Stochastic resonance With the possible use of nanoparticles as devices, such as switches, one is interested in the signal-to-noise (SNR) [25,26] ratio of such systems. Here again the usefulness of the availability of complex susceptibility data comes to the

50

P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

6

5

4

fres(GHz)

3

2

γ = 2.25 10

5

1

0 -40

-20

0

20

40

60

80

100

H(kA/m)

HA=48kA/m

Fig. 7. Plot of fres against H for sample 2.

fore since the SNR for a field driven magnetisation of a nano-particle can be determined in terms of χ (ω) and χ (ω). As has been mentioned previously, in the absence of an external field, the magnetic moment of a uniaxial particle flips spontaneously between two anti-parallel equilibrium positions ensuing customary superparamagnetism. On imposition of an external ac field, H1 , it, together with random noise, influences the magnetic switching and the SNR can

Thus, being relevant for low-frequency processes, SNR, however, essentially contains (by way of ηm ) the parameters, which characterize the ferromagnetic resonance and high-frequency relaxation, namely, the magnetomechanical ratio γ and the spin-lattice damping parameter α. In [26], the following data in Table 1 was obtained for a 400 G ferrofluid sample, the SNR being = 3.6 × 104 Hz−1 . Further examples are to be found in [27].

Saturated magnetism G

Mean diameter 10−9 m

¯A H (kAm−1 )

¯ 104 K (Jm−3 )

γ (s−1 A−1 m)

α

ηm 10−6 (Nsm−2 )

τ0 (10–10 s)

τN (10−9 s)

SNR (Hz−1 )

400

7.8

126

6.3

2.02 × 105

0.1

8.0

4.0

9.0

3.6 × 104

be found in the framework of micromagnetic theory. The main material parameters essentially involved in the resulting expression are, the Néel relaxation time, τ N , the precessional decay time, τ 0 , and also the magnetic viscosity coefficient, ηm , however, these are just the parameters obtained in the previous section on ferromagnetic resonance. The description of the SNR of a field-driven magnetization of a fine ferroparticle may be developed by means of the linear response theory. On doing this, one finds that the SNR at the frequency of excitation may be written in the form [25,26], SNR =

πµ0 ωH12 |χ(ω)|2 4kTχ (ω)

In this paper an attempt has been made to demonstrate the usefulness of complex susceptibility measurements, χ(ω) = χ (ω) − iχ (ω), which, when measured over the a frequency range from Hz to GHz, enables the convenient determination of the macroscopic and microscopic properties of the fluids. By means of such data, relaxation mechanisms, ferromagnetic resonance, stochastic resonance, non-linear properties, magnetic losses and the signal-to-noise ratio (SNR) of such colloidal nano-particle systems can be readily and successfully investigated.

(24)

Separating the real and imaginary parts and then arranging them in the expression for the modulus |χ(ω)|, one finds that πMS2 vH21 πµ0 H12 χ(0) SNR = = . 4kTτD 36ηm kT

7. Conclusion

(25)

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.

P.C. Fannin / Journal of Alloys and Compounds 369 (2004) 43–51

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