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Study of magnetic percolation in heterogeneous materials
Journal of Magnetism and Magnetic Materials 140-144 (1995) 2189-2190
~i
Jeumalof magnetism and magnetic materials
~i ELSEVIER
Study of magnetic percolation in heterogeneous materials J.-L. Mattei *, O. Minot, M. Le Floc'h Laboratoire d'Electronique et des Systkmes de T£l~communications, Universit~de Bretagne Occidentale, 6, av. Le Gorgeu, BP 809, 29285 Brest Cedex, France Abstract Actual heterogeneous magnetic materials are simulated by a macroscopic model, where the coordination number Z of the structure can easily be changed. The experimental values of the effective magnetic permeability are interpreted by an effective medium theory. We show that the magnetic percolation threshold Cp varies as the inverse of the coordination number (ZC o = 1.6).
A macroscopic physical model has been developed for easier and more precise investigations of the properties of the heterogeneous magnetic materials. The actual materials, obtained from magnetic and non-magnetic micron-sized powders mixed together, are simulated by a mixture of steel (magnetic) and glass (non-magnetic) spheres some millimeter in diameter. In the actual materials the random dispersion of the particles always leads to a coordination number nearly equal to 6 [1], and to a percolation threshold equal to 0.27 [2]. Our purpose is to study the magnetic susceptibility of simulated structures with various coordination numbers, with special attention to the percolation threshold. The macroscopic sample is made of balls positioned by hand, in order to form a stacking of 2D hexagonal layers. In each layer, the ball positions can easily be chosen. The final structure is a regular 3D fcc lattice, which is mounted in a toroidal container in order to avoid macroscopic magnetic fields. The theoretical coordination number of an infinite fcc lattice is Z = 12, but because of boundary effects, Z may be smaller than 12. One of the advantages of the physical model developed is the possibility to choose values of the coordination number that are unrealistic in actual materials. This is precisely achieved by using the boundary effects of the sample (variation of either the number or the diameter of the spheres). The magnetic susceptibility of the macroscopic sample is measured by a cell which has been tested previously [2]. The experimental results obtained for a structure with Z = 11 are presented in Fig. 1 (full squares). An effective medium theory (EMT) has been formu-
* Corresponding author. Fax: ( + 33)98.01.63.95.
lated in order to interpret these results, and is developed elsewhere [3]. Only the main steps are reported here. The heterogeneous materials are investigated as a mixture of several groups of particles. Each of them is defined, on the one hand, by its internal field H k and permeability ;zk, and on the other, by the volume concentration Ck, with the condition E Ck = 1. We assume that the effective field acting on the equivalent medium is given by the following expression: H= ECkH~,
(1)
which leads to the final result: F_,[Q(~-~)]/[~+N(n~-~)]
=0,
(2)
where ~ is the permeability of the effective medium. Similar developments have been made previously, but only on dielectric materials [4]. In the simplest case of two components, where only one is magnetic, Eq. (2) gives: (1-N)~(2+[(N-C)xi+I]~(-Cx,=O,
I! ~ l
8
6
(3)
q | y=3.c o.1 ~'/
4 ~ 2
=
CP="0.14
(N-C) ~i" (N-I-----~ 0.5
C
m
Fig. 1. The experimental susceptibility of a macroscopic lattice of heterogeneous material, with Z = 11 (full squares), is very well described by the mean field theory we have developed (continuous curve). The magnetic percolation occurs at Cp = 0.14 (volume concentration).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 0 6 2 4 - 5
2190
J.-L. Mattei et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 2189-2190
where ~ and Xi denote the susceptibilities of the composite material and the magnetic component, respectively. We point out the three major predictions of this EMT (Eq. (3)) based on the experimental results presented in Fig. 1: (i) At low concentrations, the theoretical variation of ~" is linear (~, = C/N). Since the experimental slope gives N = 1/3, this is proof that no interaction exists between the balls. (ii) At the high concentration end, the asymptotic behaviour of ~ is easily visible, and follows the law ~ = x i ( N - C ) / ( N - 1). (iii) An analytical definition of the percolation threshold Cp is obtained, corresponding to the intersection of the asymptote defined above with the concentration axis. Fig. 1 shows that Cp = 0.14 when Z = 11. Two sets of balls of different diameters were used to build three other structures. Since the volume of the container remains unchanged, the mean coordination number of the fcc lattices decreases with the number of balls, or when the ball diameters increase. These structures hold 8000-2400 spheres, depending on the desired value for Z. In each case the predictions of the EMT are verified; in particular, a value for the percolation threshold is always precisely determined (Fig. 2). The main interest of the model lies in its noteworthy ability to describe actual materials in which the particles are randomly dispersed [3]. A new result we get here is the dependence of Cp with the coordination number Z:
ZCp = 1.6.
(4)
This behaviour, specific to heterogeneous magnetic materials, differs from the well known electrical conduction case. Actually, for the latter, the volume concentration below which no electrical conduction takes place is constant, independently of the value of Z, and is equal to 0.15 [5,6]. In conclusion, a new approach for the study of heterogeneous magnetic materials, using a macroscopic physical model, has been successfully tested. The experimental variations of the susceptibilities we obtain on both actual
l!
/ 0~.1
z-I '
~,
Fig. 2. For macroscopic heterogeneous mixtures, the magnetic percolation Cp and the coordination number Z are linked by the relation ZCp = 1.6. For actual materials (full circle) we obtain a similar result (Z = 6, Cp = 0.27). The experimental values obtained with the macroscopic model (full squares) were: (Z= 11, Cp = 0.14), (Z = 10, Cp = 0.16), (Z = 7.8, Cp = 0.21) and (Z = 6.2, Cp = 0.26).
and simulated materials, are very well described by an EMT. The relation between the coordination number and the measured magnetic percolation threshold of the simulated structures is specific to heterogeneous magnetic materials.
References [1] H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon and C.D. Mitescu, J. Phys. C 11 (1978) 1311. [2] J-L. Mattei, A-M. Konn and M. Le Floc'h, IEEE Trans. Instrum. Meas. 42 (1993) 121. [3] M. Le Floc'h, J.-L. Mattei, P. Laurent, O. Minot and A-M. Konn, J. Magn. Magn. Mater. 140-144 (1995) 2191 (these Proceedings). [4] C.G. Grandqvist and O. Hunderi, Phys. Rev. B 18 (1978) 1554. [5] V.A. Vyssotsky, S.B. Gordon, H.L. Frisch and J.M. Hammersley, Phys. Rev. 123 (1961) 1566. [6] H. Scher and R. Zallen, J. Chem. Phys. 53 (1970) 3759.
~i
Jeumalof magnetism and magnetic materials
~i ELSEVIER
Study of magnetic percolation in heterogeneous materials J.-L. Mattei *, O. Minot, M. Le Floc'h Laboratoire d'Electronique et des Systkmes de T£l~communications, Universit~de Bretagne Occidentale, 6, av. Le Gorgeu, BP 809, 29285 Brest Cedex, France Abstract Actual heterogeneous magnetic materials are simulated by a macroscopic model, where the coordination number Z of the structure can easily be changed. The experimental values of the effective magnetic permeability are interpreted by an effective medium theory. We show that the magnetic percolation threshold Cp varies as the inverse of the coordination number (ZC o = 1.6).
A macroscopic physical model has been developed for easier and more precise investigations of the properties of the heterogeneous magnetic materials. The actual materials, obtained from magnetic and non-magnetic micron-sized powders mixed together, are simulated by a mixture of steel (magnetic) and glass (non-magnetic) spheres some millimeter in diameter. In the actual materials the random dispersion of the particles always leads to a coordination number nearly equal to 6 [1], and to a percolation threshold equal to 0.27 [2]. Our purpose is to study the magnetic susceptibility of simulated structures with various coordination numbers, with special attention to the percolation threshold. The macroscopic sample is made of balls positioned by hand, in order to form a stacking of 2D hexagonal layers. In each layer, the ball positions can easily be chosen. The final structure is a regular 3D fcc lattice, which is mounted in a toroidal container in order to avoid macroscopic magnetic fields. The theoretical coordination number of an infinite fcc lattice is Z = 12, but because of boundary effects, Z may be smaller than 12. One of the advantages of the physical model developed is the possibility to choose values of the coordination number that are unrealistic in actual materials. This is precisely achieved by using the boundary effects of the sample (variation of either the number or the diameter of the spheres). The magnetic susceptibility of the macroscopic sample is measured by a cell which has been tested previously [2]. The experimental results obtained for a structure with Z = 11 are presented in Fig. 1 (full squares). An effective medium theory (EMT) has been formu-
* Corresponding author. Fax: ( + 33)98.01.63.95.
lated in order to interpret these results, and is developed elsewhere [3]. Only the main steps are reported here. The heterogeneous materials are investigated as a mixture of several groups of particles. Each of them is defined, on the one hand, by its internal field H k and permeability ;zk, and on the other, by the volume concentration Ck, with the condition E Ck = 1. We assume that the effective field acting on the equivalent medium is given by the following expression: H= ECkH~,
(1)
which leads to the final result: F_,[Q(~-~)]/[~+N(n~-~)]
=0,
(2)
where ~ is the permeability of the effective medium. Similar developments have been made previously, but only on dielectric materials [4]. In the simplest case of two components, where only one is magnetic, Eq. (2) gives: (1-N)~(2+[(N-C)xi+I]~(-Cx,=O,
I! ~ l
8
6
(3)
q | y=3.c o.1 ~'/
4 ~ 2
=
CP="0.14
(N-C) ~i" (N-I-----~ 0.5
C
m
Fig. 1. The experimental susceptibility of a macroscopic lattice of heterogeneous material, with Z = 11 (full squares), is very well described by the mean field theory we have developed (continuous curve). The magnetic percolation occurs at Cp = 0.14 (volume concentration).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 0 6 2 4 - 5
2190
J.-L. Mattei et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 2189-2190
where ~ and Xi denote the susceptibilities of the composite material and the magnetic component, respectively. We point out the three major predictions of this EMT (Eq. (3)) based on the experimental results presented in Fig. 1: (i) At low concentrations, the theoretical variation of ~" is linear (~, = C/N). Since the experimental slope gives N = 1/3, this is proof that no interaction exists between the balls. (ii) At the high concentration end, the asymptotic behaviour of ~ is easily visible, and follows the law ~ = x i ( N - C ) / ( N - 1). (iii) An analytical definition of the percolation threshold Cp is obtained, corresponding to the intersection of the asymptote defined above with the concentration axis. Fig. 1 shows that Cp = 0.14 when Z = 11. Two sets of balls of different diameters were used to build three other structures. Since the volume of the container remains unchanged, the mean coordination number of the fcc lattices decreases with the number of balls, or when the ball diameters increase. These structures hold 8000-2400 spheres, depending on the desired value for Z. In each case the predictions of the EMT are verified; in particular, a value for the percolation threshold is always precisely determined (Fig. 2). The main interest of the model lies in its noteworthy ability to describe actual materials in which the particles are randomly dispersed [3]. A new result we get here is the dependence of Cp with the coordination number Z:
ZCp = 1.6.
(4)
This behaviour, specific to heterogeneous magnetic materials, differs from the well known electrical conduction case. Actually, for the latter, the volume concentration below which no electrical conduction takes place is constant, independently of the value of Z, and is equal to 0.15 [5,6]. In conclusion, a new approach for the study of heterogeneous magnetic materials, using a macroscopic physical model, has been successfully tested. The experimental variations of the susceptibilities we obtain on both actual
l!
/ 0~.1
z-I '
~,
Fig. 2. For macroscopic heterogeneous mixtures, the magnetic percolation Cp and the coordination number Z are linked by the relation ZCp = 1.6. For actual materials (full circle) we obtain a similar result (Z = 6, Cp = 0.27). The experimental values obtained with the macroscopic model (full squares) were: (Z= 11, Cp = 0.14), (Z = 10, Cp = 0.16), (Z = 7.8, Cp = 0.21) and (Z = 6.2, Cp = 0.26).
and simulated materials, are very well described by an EMT. The relation between the coordination number and the measured magnetic percolation threshold of the simulated structures is specific to heterogeneous magnetic materials.
References [1] H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon and C.D. Mitescu, J. Phys. C 11 (1978) 1311. [2] J-L. Mattei, A-M. Konn and M. Le Floc'h, IEEE Trans. Instrum. Meas. 42 (1993) 121. [3] M. Le Floc'h, J.-L. Mattei, P. Laurent, O. Minot and A-M. Konn, J. Magn. Magn. Mater. 140-144 (1995) 2191 (these Proceedings). [4] C.G. Grandqvist and O. Hunderi, Phys. Rev. B 18 (1978) 1554. [5] V.A. Vyssotsky, S.B. Gordon, H.L. Frisch and J.M. Hammersley, Phys. Rev. 123 (1961) 1566. [6] H. Scher and R. Zallen, J. Chem. Phys. 53 (1970) 3759.