- Email: [email protected]
Interaction effects in the dynamic response of magnetic liquids
of Magnetism and Magnetic Materials lOl(l991) North-Holland
45-46
Journal
Interaction M. Hanson
effects in the dynamic
response
of magnetic
liquids
and C. Johansson
Physics Department,
Chalmers
University
of Technology, S-412 96 Gijteborg, Sweden
The magnetic susceptibility of magnetic liquids has been measured for frequencies up to 70 MHz. The imaginary part of the susceptibility exhibits a maximum around 10 MHz. This feature is interpreted as due to single-particle Ntel relaxation, where particle
interactions
cause a slight concentration
dependence
1. Introduction An ideal magnetic liquid consists of identical, noninteracting magnetic particles, about 10 nm in diameter, randomly distributed in a carrier liquid. It may be described as a superparamagnet with a single-particle dynamic response x = x0(1 + iw7)-l=
x’ - ix”.
(1)
x0 is the zero-frequency susceptibility, which may be obtained from the Langevin function, w is the angular frequency and 7 the relaxation time for the magnetization reversal of a particle. In a real liquid the behaviour is modified due to the influence of the particle size distribution, the spatial rearrangement of the particles in an applied field and interactions between the particles. We have earlier used a method described in ref. [l] to measure the frequency dependence of the complex magnetic susceptibility of a series of magnetic liquids in the range 5 Hz-13 MHz [2]. The liquids exhibited a broad maximum in x” at frequencies of the order of 10 MHz. This feature was interpreted as mainly due to the NCel relaxation of single particles with a distribution of sizes. The low-frequency behaviour was independent of concentration for particle volume fractions, x, in the range 0.03 < x < 0.2. interaction effects were however discernable in the frequency region approaching the maximum of x”. In order to further investigate the influence of interactions upon the dynamic response we have extended the measurements to frequencies up to 70 MHz. From the position of the maximum of x” we have made a simple estimate of the interaction strength.
in the position
of the maximum.
titles. The corresponding volume fractions are 19% 14%, 10% and 6% respectively. To obtain the susceptibility we used the same sample cell as earlier for the lower frequencies and a HewlettPackard 4191A radio-frequency impedance analyzer. With this instrument the resistance and reactance are determined from measurements of the vector ratio of the incident and the reflected waves in the range 1 MHz-l GHz. A 2-terminal pair configuration is used. The accuracy of the measurements decreases at high frequencies, where the sample cell starts to radiate. This limits the usable range to frequencies below 70 MHz in our case. There is an overlap with the earlier obtained data for frequencies between 1 and 13 MHz. The difference between the calculated susceptibilities from the two methods is less than 2%. 3. Results The frequency dependence of x” at room temperature for three of the investigated liquids is shown in fig. 1. The data for liquid no. 4, which fall between those of no. 1 and no. 2, were omitted in the figure for the sake of legibility. As was discussed in the earlier paper [2] we consider the maximum of x” to be due to single
1: 65 kA/m
2: 34 kA/m 3: 19 kA/m
2. Experimental methods Four magnetic liquids with particles of iron oxide in kerosene, number 1 and 4, or in synthetic ester oil, number 2 and 3, were investigated. The liquids 1, 2 and 3 are the same as investigated earlier. Their properties as well as the preparation method are described in ref. [2]. Liquid 4 was prepared by diluting liquid 1. The saturation magnetizations, M,, of the liquids were 65 (no. l), 48 (no. 4), 34 (no. 2) and 19 kAm-’ (no. 3). it4, is proportional to the concentration of magnetic par0312~8853/91/$03.50
0 1991 - Elsevier Science Publishers
0
10’
1 frequency[Hz]
Fig. 1. The frequency dependence of the imaginary part of the susceptibility for three of the magnetic liquids. The values in the figure give the saturation magnetizations of the liquids.
B.V. All rights reserved
M. Hanson, C. Johansson / Interaction effects in dynamic response of magnetic liquids
46
particles that undergo NCel relaxation. The N&e1 relaxation time for uniaxial, non-interacting particles is given by 7N = T,exp( Kk’/kT) [3]. ra is a characteristic time of the order of 10e9 s, K the crystalline anisotropy constant, V the volume of a particle, k the Boltzmann constant and T the temperature. This relaxation gives a = (271~~))‘. For maximum in x” at a frequency f,, our liquids rr., is of the order of IO-* s. In fig. 1 it can be seen that the frequency spectrum is shifted towards lower frequencies as the particle concentration increases. The single-particle relaxation time may be modified to take weak interactions, kT, << KV, into account; 7 = ra exp( KV/k(TT,)). T, is a measure of the interaction strength which depends on the volume fraction of particles. From this the following relation is obtained: = kT/KV
[ln(f,/f,,,)]-’
- kT,/KV.
(2)
f. = (2~7,)~‘.
pendence quadratic
We have analyzed the concentration deof [ln(fo/fmax)]-t trying a linear as well as a relation. The quadratic dependence gives a
I
I
I
I
somewhat better fit to the experimental data. In fig. 2, [ln(fo/fmax)]-l is plotted versus M,’ for four magnetic liquids. The straight line was obtained by linear regression. Using the values of its slope and intercept, T, and KV were calculated from eq. (2). For the liquids with M, = 19, 34,48 and 65 kAm-’ we obtained T, = 3, 10, 20 and 38 K respectively. The anisotropy KV = 0.07 eV. 4. Discussion The
estimated
values
T,
are small, yielding are made in the hightemperature regime and the effects of the interactions on the susceptibility are small. In order to make a detailed analysis of the concentration dependence of the interaction parameter T,, it would be desirable to cover a wider concentration range. This is not possible, since magnetic liquids with higher concentrations than our 19% would be difficult to stabilize, and the accuracy of the measurements puts a limit at low concentrations of about 5%. Interactions depending on the square of the concentration have, however, been found in other experiments and they have been discussed in a simple model for weak interactions between particles undergoing NCel relaxation [4]. In order to further study the influence of particle interactions we are at present doing measurements of the magnetization at low temperatures, where the interactions lead to magnetic ordering in the frozen liquids. kT,/KV
of
< 0.05. The experiments
This work was supported Technical Development.
by the Swedish
Board of
References I
I
1
2 M,’
[log
I
I
3 (Aim)’
4
]
Fig. 2. [ln(fc/f,,,~)]-’ vs. M,’ for the magnetic liquids. f,, is the frequency where x” has its maximum and f0 the frequency corresponding to the NCel relaxation time. M, is the saturation magnetization of the liquids.
[l] P.C. Fannin, B.K.P. Scaife and SW. Charles, J. Magn. Magn. Mater. 72 (1988) 95. [2] M. Hanson, J. Magn. Magn. Mater. 96 (1991) 105. [3] L. Ndel, Compt. Rend. Acad. Sci. 228 (1949) 664. [4] S. Shtrikman and E.P. Wohlfarth, Phys. Lett. A 85 (1981) 467.
45-46
Journal
Interaction M. Hanson
effects in the dynamic
response
of magnetic
liquids
and C. Johansson
Physics Department,
Chalmers
University
of Technology, S-412 96 Gijteborg, Sweden
The magnetic susceptibility of magnetic liquids has been measured for frequencies up to 70 MHz. The imaginary part of the susceptibility exhibits a maximum around 10 MHz. This feature is interpreted as due to single-particle Ntel relaxation, where particle
interactions
cause a slight concentration
dependence
1. Introduction An ideal magnetic liquid consists of identical, noninteracting magnetic particles, about 10 nm in diameter, randomly distributed in a carrier liquid. It may be described as a superparamagnet with a single-particle dynamic response x = x0(1 + iw7)-l=
x’ - ix”.
(1)
x0 is the zero-frequency susceptibility, which may be obtained from the Langevin function, w is the angular frequency and 7 the relaxation time for the magnetization reversal of a particle. In a real liquid the behaviour is modified due to the influence of the particle size distribution, the spatial rearrangement of the particles in an applied field and interactions between the particles. We have earlier used a method described in ref. [l] to measure the frequency dependence of the complex magnetic susceptibility of a series of magnetic liquids in the range 5 Hz-13 MHz [2]. The liquids exhibited a broad maximum in x” at frequencies of the order of 10 MHz. This feature was interpreted as mainly due to the NCel relaxation of single particles with a distribution of sizes. The low-frequency behaviour was independent of concentration for particle volume fractions, x, in the range 0.03 < x < 0.2. interaction effects were however discernable in the frequency region approaching the maximum of x”. In order to further investigate the influence of interactions upon the dynamic response we have extended the measurements to frequencies up to 70 MHz. From the position of the maximum of x” we have made a simple estimate of the interaction strength.
in the position
of the maximum.
titles. The corresponding volume fractions are 19% 14%, 10% and 6% respectively. To obtain the susceptibility we used the same sample cell as earlier for the lower frequencies and a HewlettPackard 4191A radio-frequency impedance analyzer. With this instrument the resistance and reactance are determined from measurements of the vector ratio of the incident and the reflected waves in the range 1 MHz-l GHz. A 2-terminal pair configuration is used. The accuracy of the measurements decreases at high frequencies, where the sample cell starts to radiate. This limits the usable range to frequencies below 70 MHz in our case. There is an overlap with the earlier obtained data for frequencies between 1 and 13 MHz. The difference between the calculated susceptibilities from the two methods is less than 2%. 3. Results The frequency dependence of x” at room temperature for three of the investigated liquids is shown in fig. 1. The data for liquid no. 4, which fall between those of no. 1 and no. 2, were omitted in the figure for the sake of legibility. As was discussed in the earlier paper [2] we consider the maximum of x” to be due to single
1: 65 kA/m
2: 34 kA/m 3: 19 kA/m
2. Experimental methods Four magnetic liquids with particles of iron oxide in kerosene, number 1 and 4, or in synthetic ester oil, number 2 and 3, were investigated. The liquids 1, 2 and 3 are the same as investigated earlier. Their properties as well as the preparation method are described in ref. [2]. Liquid 4 was prepared by diluting liquid 1. The saturation magnetizations, M,, of the liquids were 65 (no. l), 48 (no. 4), 34 (no. 2) and 19 kAm-’ (no. 3). it4, is proportional to the concentration of magnetic par0312~8853/91/$03.50
0 1991 - Elsevier Science Publishers
0
10’
1 frequency[Hz]
Fig. 1. The frequency dependence of the imaginary part of the susceptibility for three of the magnetic liquids. The values in the figure give the saturation magnetizations of the liquids.
B.V. All rights reserved
M. Hanson, C. Johansson / Interaction effects in dynamic response of magnetic liquids
46
particles that undergo NCel relaxation. The N&e1 relaxation time for uniaxial, non-interacting particles is given by 7N = T,exp( Kk’/kT) [3]. ra is a characteristic time of the order of 10e9 s, K the crystalline anisotropy constant, V the volume of a particle, k the Boltzmann constant and T the temperature. This relaxation gives a = (271~~))‘. For maximum in x” at a frequency f,, our liquids rr., is of the order of IO-* s. In fig. 1 it can be seen that the frequency spectrum is shifted towards lower frequencies as the particle concentration increases. The single-particle relaxation time may be modified to take weak interactions, kT, << KV, into account; 7 = ra exp( KV/k(TT,)). T, is a measure of the interaction strength which depends on the volume fraction of particles. From this the following relation is obtained: = kT/KV
[ln(f,/f,,,)]-’
- kT,/KV.
(2)
f. = (2~7,)~‘.
pendence quadratic
We have analyzed the concentration deof [ln(fo/fmax)]-t trying a linear as well as a relation. The quadratic dependence gives a
I
I
I
I
somewhat better fit to the experimental data. In fig. 2, [ln(fo/fmax)]-l is plotted versus M,’ for four magnetic liquids. The straight line was obtained by linear regression. Using the values of its slope and intercept, T, and KV were calculated from eq. (2). For the liquids with M, = 19, 34,48 and 65 kAm-’ we obtained T, = 3, 10, 20 and 38 K respectively. The anisotropy KV = 0.07 eV. 4. Discussion The
estimated
values
T,
are small, yielding are made in the hightemperature regime and the effects of the interactions on the susceptibility are small. In order to make a detailed analysis of the concentration dependence of the interaction parameter T,, it would be desirable to cover a wider concentration range. This is not possible, since magnetic liquids with higher concentrations than our 19% would be difficult to stabilize, and the accuracy of the measurements puts a limit at low concentrations of about 5%. Interactions depending on the square of the concentration have, however, been found in other experiments and they have been discussed in a simple model for weak interactions between particles undergoing NCel relaxation [4]. In order to further study the influence of particle interactions we are at present doing measurements of the magnetization at low temperatures, where the interactions lead to magnetic ordering in the frozen liquids. kT,/KV
of
< 0.05. The experiments
This work was supported Technical Development.
by the Swedish
Board of
References I
I
1
2 M,’
[log
I
I
3 (Aim)’
4
]
Fig. 2. [ln(fc/f,,,~)]-’ vs. M,’ for the magnetic liquids. f,, is the frequency where x” has its maximum and f0 the frequency corresponding to the NCel relaxation time. M, is the saturation magnetization of the liquids.
[l] P.C. Fannin, B.K.P. Scaife and SW. Charles, J. Magn. Magn. Mater. 72 (1988) 95. [2] M. Hanson, J. Magn. Magn. Mater. 96 (1991) 105. [3] L. Ndel, Compt. Rend. Acad. Sci. 228 (1949) 664. [4] S. Shtrikman and E.P. Wohlfarth, Phys. Lett. A 85 (1981) 467.