Journal of Molecular Liquids 114 (2004) 89–96
Investigation of the field dependence of magnetic fluids exhibiting aggregation P.C. Fannin*, A.T. Giannitsis Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
Abstract Measurement of the frequency dependent, complex susceptibility, x(v)sx9(v)yix0(v), over the frequency range 100 Hz to 1 MHz, of a colloidal suspension of cobalt ferrite particles, by means of the toroidal technique, indicates the presence of aggregation in the samples: the aggregation being identified by the appearance of a low-frequency loss-peak in the x0(v) component of the susceptibility. By means of the application of a polarising field, H (Aym), the field dependence of the complex susceptibility of the aggregates, x(v, H), the frequency dependent loss tangent, tand and mean power loss, T (Jym3 ), are investigated for polarising fields over the approximate range 0–13.6 kAym. It is shown that the plots of both x(v), H and tand may be fitted in terms of the Cole–Cole parameter, a, and it is demonstrated how the fits can be used to determine data outside the measurement frequency range. A novel method for determining the tand data is also presented. It is also shown how, with increase in polarising field, the magnetic losses of the fluid are reduced. 䊚 2004 Elsevier B.V. All rights reserved. PACS: 75.50.Tt Fine-particle systems; 75.50.Mm Magnetic liquids; 76.60.Es Relaxation effects Keywords: Toroidal technique; Magnetic fluid; Aggregation
1. Introduction Magnetic fluids consist of colloidal suspensions of nanoparticles of ferromagnetic or ferrimagnetic materials dispersed in a carrier liquid and stabilised by a suitable organic surfactant. The surfactant coating creates an entropic repulsion between particles w1x, such that thermal agitation alone is sufficient to maintain them in a stable dispersion. The particles are single-domain and are considered to be in a state of uniform magnetisation with magnetic dipole moment (Wb m), msMsØv
(1)
where Ms is the saturation magnetisation (Wbym2) of the material and v is the magnetic volume of the particle. The preferred orientation of the magnetic moment is along an axis, or axes, of easy magnetisation and this direction depends generally on a combination of shape and magneto-crystalline anisotropy denoted by the symbol K. Also, when in suspension their magnetic proper*Corresponding author. Fax: q353-1-6081860y1580. E-mail address:
[email protected] (P.C. Fannin).
ties can be described by the Langevin function (L(j)), suitably modified to take account of a distribution of particle sizes. The magnetisation M is described by the Langevin expression, MsMswcothjy1yjx.
(2)
jsmHykT, where k is Boltzmann’s constant and H the magnetizing field and Tstemperature. The formation of aggregates w2–4x can arise due to the effects of short range van der Waals attraction or by the effects of magnetic dipolar interactions between particles w5x. Aggregation can also arise due to incomplete coverage of the particles with surfactant during the preparation process or simply weak absorption in which an equilibrium exists between the absorbed surfactant and the surfactant free in solution. One convenient method of determining whether aggregates exist in magnetic fluids is by measurement of the frequency-dependent complex, relative susceptibility, x(v), which may be written in terms of its real and imaginary components, where, xŽv.sx9Žv.yix0Žv.
0167-7322/04/$ - see front matter 䊚 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2004.02.007
(3)
P.C. Fannin, A.T. Giannitsis / Journal of Molecular Liquids 114 (2004) 89–96
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It has been shown that the theory of Debye w6x developed to account for the anomalous dielectric dispersion in dipolar fluids may be used w7–9x to account for the analogous case of magnetic fluids. According to Debye’s theory, x(v) has a frequency dependence given by the equation, x(v)s(x0yx`)y(1qivteff)qx`
(4)
In the case where tN4tB, from Eq. (8) teffs1y2pfmaxstBs4phr3 ykT
(10)
Thus by determining f max, Eq. (10) enables one to obtain the particle or aggregate size for the sample. For a distribution of particle sizes, a distribution of relaxation times, t will exist so that x(v) may also be expressed in terms of a distribution function, F(t) giving
where the static susceptibility 2
x0s(nm y3kTm0)
(5)
x` is the high frequency susceptibility at a frequency below that of resonance, m is the particle magnetic moment, n is the particle number density, teff is the effective relaxation time and m0 is the permeability of free space. The magnetic moment of the particles may relax through either rotational Brownian motion of the particle within the carrier liquid, with relaxation time tB w10x or ´ mechanism with relaxation time tN through the Neel w11x. The Brownian relaxation time tB is given by w10x tBs4pr3hyk
(6)
xŽv.sx`qŽx0yx`.
|
`
FŽt.dtyŽ1qivt.
F(t) may be represented by a range of distribution functions w16x including the Cole–Cole distribution function. The relation between x9(v) and x0(v) and their dependence on frequency, v y2p, can be displayed by means of the magnetic analogue of the Cole–Cole plot w17x. In the Cole–Cole case, the circular arc cuts the x9(v) axis at an angle of ap y2; a 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 xŽv.sx`qŽx0yx`.ywyŽ1qŽivteff.1ya.z~ 0-a-1 x
where r is the hydrodynamic radius of the particle, h is the dynamic viscosity of the carrier liquid. ´ relaxation mechanism, the In the case of the Neel 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. This reversal time may be described approximately in terms of Brown’s w12x expressions for high and low barrier heights, as, tNst0s
y1y2
expŽs., sG2
st0s,
s
