Heating magnetic fluid with alternating magnetic field
Introduction
It appears that heating of colloidal magnetic fluid (ferrofluid) due to time-varying magnetic induction has not been significantly studied. Heating ferrofluid to achieve hyperthermia in medical treatments is an emerging area of importance [1], [2]. Heating effects in ferrofluids may also be important in loudspeakers where temperature rise adversely affects performance [3]. Other applications exploit the ability to heat the fluid magnetically, as in the thermal actuation of a polymer gel in contact with a ferrofluid.
This work develops dissipation relationships based on rotational relaxation of single domain magnetic particles dispersed in a liquid matrix. Eddy current heating is assumed negligible due to the small size of the particles (<15 nm). The SI system of units is employed.
Section snippets
Power dissipation
From the first law of thermodynamics for a constant density system of unit volume dU=δQ+δW, where U is the internal energy, Q the heat added and W the magnetic work done on the system. For an adiabatic process δQ=0 with the differential magnetic work, given in general [4] by , the result iswhere H (A m−1) is the magnetic field intensity and B (T) the induction, both in the sample. Because the fields are colinear the relationship reduces to where H and B are magnitudes. B=μ0(
Relationship to material parameters
The relaxation equation of Shliomis [5] reduces to the following form for motionless fluid in an oscillatory field:where τ is the relaxation time, is the equilibrium magnetization in the applied field whose value is given in Eq. (3), and χ0 is the equilibrium susceptibility. Substituting the complex representations of M0 and M(t) into Eq. (7) yieldswhich gives the dependence of complex susceptibility on frequency and from which the
Time constants
With the Brownian mechanism of relaxation the magnetic moment is locked to the crystal axis and when the magnetic moment aligns with the field, the particle rotates as well. A second mechanism exists (Néel relaxation) in which the magnetic moment rotates within the crystal. To achieve high heating rates the Néel relaxation must not be allowed to dominate.
The Brownian time constant is given by the following relationship [9]:where η is the viscosity coefficient of the matrix fluid, k
Polydispersions
It is found that the log normal particle size distribution g(R) provides a reasonably good fit to the measured distribution for ferrofluids [11]. g(R) is obtained by substituting ln(R) for R in a Gaussian distribution function, thereby circumventing the unphysical aspect of contributions from negative particle sizes associated with the Gaussian:where is the median and σ the standard deviation of . The most probable radius is given by
Calculated results
The integrand of Eq. (17) is a huge expression when the expressions for g(R),P, χ and τ are substituted in it, and analytical integration is not desirable. Instead, numerical values of temperature rise are calculated as examples. These are based on physical property values listed in Table 1. A value of τ0=10−9 s is employed throughout.
Fig. 3a illustrates the heating rate for monodisperse magnetite samples in a hydrocarbon carrier as a function of particle radius and field intensity (B0=μ0H0).
Conclusion
This study utilizes a collection of well-known principles in magnetic fluid science to formulate and compute heating rate in samples subjected to an alternating magnetic field. The methodology should be of interest to investigators in numerous fields of technology.
Acknowledgements
The author thanks M. Zahn for helpful comments on this work.
References (11)
- et al.
Int. J. Hyperthermia
(1993) - R. Hiergeist, W. Andra, N. Buske, R. Hergt, I. Hilger, U. Richter, W. Kaiser. J. Magn. Magn. Mater. 201 (1999) 420...
- K. Raj, in: B. Berkovski (Ed.), Magnetic Fluids and Applications Handbook, Begell House Inc., New York, 1996, pp....
- J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, p....
Sov. Phys. Uspekhi (Engl. transl.)
(1974)