- Email: [email protected]
Effects of the magnetodipolar interactions in the alternating magnetic fields
Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
E!ects of the magnetodipolar interactions in the alternating magnetic "elds G. Bossis , A. Cebers * LPMC CNRS U.M.R. 6622, Universite de Nice-Sophia Antinopolis, Parc Valrose 06108 Nice, Cedex 2, France Institute of Physics, University of Latvia, Salaspils -1, LV-2169, Latvia Received 27 May 1998; received in revised form 30 July 1998
Abstract A macroscopic theory for the description of the structure formation in a system of magnetic dipoles under the action of a high-frequency rotating "eld is presented. Continuum equations for the e!ective magnetic "eld strength describing particle interactions are derived. It is shown that, contrary to the case of a constant magnetic "eld, where the demagnetizing "eld arising from a concentration #uctuation stabilizes the system with respect to the phase separation, the same concentration #uctuation will be ampli"ed in the case of a rotating magnetic "eld, leading to the formation of a layered structure. The extensions of the model necessary for the description of the formation of the periodic structures in the rotating "eld are considered and the relevance of the present model for the explanation of phase separation in shear #ows of magnetorheological suspensions is indicated. 1999 Elsevier Science B.V. All rights reserved. Keywords: Rotating "eld; Magnetodipolar interactions; Phase separation; Stripes
1. Introduction Peculiarities of the interaction of the magnetic #uids and magnetorheological suspensions with alternating "elds have rised a lot of interest in the past. For instance, we can point out complicated dynamics of the magnetic #uid droplets in the alternating magnetic "elds [1,2], excitation of parametric free surface oscillations [3], `negative viscositya e!ect [4] and others. Among issues not understood completely at present moment we can point out the
* Corresponding author. Fax: #371-2-2250-39. E-mail address: [email protected] (A. Cebers)
peculiarities of the structure formation by magnetic particles in the alternating magnetic "elds. Besides that similar problems can arise when describing the rheological behaviour of the magnetic suspensions in external magnetic "eld [5]. Here, we want to demonstrate that if an homogeneous suspension of magnetic particles is placed in a rotating "eld then, for strong enough magnetodipolar forces, we shall observe the formation of a layered structure. This illustrates that, contrary to the common belief that long-range magnetodipolar interactions are repulsive and thus in competition with short-range attractive forces for the structure formation [6], long-range magnetodipolar forces can be also attractive and initiate a phase separation.
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 5 3 - 0
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
2. Average potential energy of magnetic interaction Let us consider the magnetic dipole m aligned with y-axis of a coordinate system at position characterized by radius-vector r. We want to calculate its interaction energy with a layer of dipoles of thickness dz perpendicular to the z-axis. The repartition of the dipoles is homogeneous with a number density n and all the dipoles are aligned in the y direction. The interaction energy can be written as follows: ;"!mH"!m
m 3(r!r )(m(r!r ) ! "r!r " "r!r "
;n(r ) dS dz .
(1)
Since the distribution of the particles is homogeneous in x, y plane the function under the sign of the integral can be averaged with respect to the directions of the projection q!qY of the radius-vector r!r on this plane. So we have
;" dS dz
m 3m(q!q ) ! n(r ). "r!r " 2"r!r "
(3)
where m (3 cos a!1) , H"! "r!r" a is the angle of the radius-vector r!r with the z-axis and m "m/(2. Equivalence of the interaction between two rotating dipoles and average interaction of the "xed dipole with dipoles in the plane follows from the rotational invariance of the magnetodipolar energy. Formally, it can be illustrated as follows. Since due to the rotational invariance *; *; d;" [du;r]# [du;m]"0, *r *m
rotation of the dipoles in the plane is equivalent in the sense of the change of interaction energy to the rotation of the radius-vector on the same angle "du" but in the opposite direction. That means that averaging with respect to the directions of the dipole moment in the plane of the rotating "eld is equivalent to the averaging with respect to the directions of the projection of the radius-vector on the same plane at "xed dipole moments. Expression (3) for the interaction energy of two dipoles rotating in the phase in the plane has been derived in Ref. [7] and much earlier was considered by Bibik [8]. Note that this is the physical situation which happens when the rotation period is much smaller than the characteristic time of the relative motion of the dipoles. Relation (3) shows that the interaction of the dipoles are attractive in the plane and repulsive in the direction normal to it. This is the reason why the particles will gather into layers. In the following, we develop a model, based on the interaction energy given by Eq. (3), which predicts a formation of a layered structure.
(2)
Relation (2) shows, for example, that average interaction energy between the dipole and the dipoles of the layer on the circle with radius "q!q " in this case can be expressed as follows: m(3 cos a!1) ;"!m H" , 2"r!r "
219
3. Macroscopic e4ective 5eld equations We introduce the `"ctitiousa magnetic "eld strength which corresponds to Eq. (3): h(r!r ) HI "m r "r!r " with the interaction energy given by } ;"!m hHI , where h is the unit vector perpendicular to the plane of the rotating "eld. `Fictitiousa magnetic "eld strength for a nonhomogeneous distribution of particles will write
HI "m r
h(r!r )n(r ) d< , "r!r "
(4)
Taking the divergence of Eq. (4) and applying *
1 "!4pd(r!r ) "r!r "
we have div HI "4p div(m hn).
(5)
220
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
The magnetic energy in the presence of this "eld is given by
1 E"! nm hHI d<, 2
(6)
where the factor 2 avoids counting twice the dipolar energy of a pair of dipoles.Since rot HI "0, and introducing HI " t the last relation can be transformed in the following way:
1 1 E"! nm h t d<" (nm h)t d< 2 2 1 HI d<. "! 8p
4. Critical conditions for phase separation (7)
Thus, it is energetically advantageous for the magnetic particles to gather in layers normal to the symmetry axis which is the axis of the rotation of the magnetic "eld. That follows from the fact that energy of the &&"ctitious'' magnetic "eld (7) is negative. Since nonzero &&"ctitious'' "eld strength according to (5) appears when the particle distribution is nonhomogeneous along h direction then development of a structure leads to the diminution of the energy and consequently to a phase separation. Thus, in this condition arises an unusual situation where the long-range part of magnetodipolar interactions is causing the phase separation. To emphasize this point more clearly let us consider the local Lorentz "eld associated with the &&"ctitious'' magnetic "eld strength HI . Indeed, since in the case of homogeneous particle distribution HI "0 the "eld HI created by particles outside the Lorenz sphere < can be expressed as follows: * n(r )(r!r )h HI "!HI *"!m r d< 4\4* 4 "r!r " * 4 n(r )h* dS 4p "!m
"! nm h. "r!r " 3 1* The sign is opposite to the usual one and so is the interaction energy with the local "eld
2p E " nm d<. * 3
case of the phase separation under the action of constant "eld, the Lorentz "eld and the demagnetizing "eld have interchanged roles. In the constant "eld case the demagnetizing "eld arising due to a concentration #uctuation is stabilizing the system and its repulsive character determines the period of the pattern [9,10] whereas in the case of a rotating "eld this is demagnetizing "eld which is causing the phase separation, and the Lorentz "eld is playing stabilizing role.
(8)
Relation (8) shows that contrary to the usual case the Lorentz "eld plays a stabilizing role. In other words, it is possible to say that, compared to the
In order to calculate the phase diagram let us start with an homogeneous system which is perturbed by a concentration #uctuation with a gradient in the h direction. According to the Eq. (5) the HI "eld perturbation will be expressed by dHI "4pm hdn. Accounting for the conservation of the particle number and the e!ect of the Lorentz "eld the second variation of the free energy of the system will be
1 *u 2p dF" (dn) d<# m (dn) d< 3 2 *n
1 ! (dHI ) d< 8p
1 *u 4n " (dn) d
(9)
Formula (9) shows the increase of the critical value of magnetodipolar interaction parameter at uP0 and uP0.25. The last value corresponds to the limit of dense packing in the frame of van der Waals approach when large magnetic interactions are ne-
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
cessary to compress the almost dense-packed system. In this case concrete value of u (0.25) for dense packing limit has only illustrative purpose and can be made more realistic by considering more complex equations of state for hard sphere system [5]. The period of the layered structure arising from the phase separation can, in principle, be determined by accounting for the repulsive part of the magnetodipolar interactions (due to "nite extension of the layers in x, y plane) and surface energy on the phase boundaries.
Quite simple expressions for the repulsive part of the magnetodipolar energy can be obtained in the case of plane layers. The surface energy coming from the phase boundaries is introduced by the Cahn}Hilliard term b( n) d<. In the case of periodic concentration perturbation along z direction we have (2h the thickness of the layer).
1 E # b ( dn) d< K 2 "¸ 2h V
5. Period of the structure
m * (z!z ) n(r ) d< H(r)" 2 *z "r!r " m*(r )n(r ) # r ! dS , (10) "r!r " 1 where * the normal to the boundary of the sample which is supposed to be perpendicular to z direction. Di!erentiating the "rst term in Eq. (10) and keeping in mind that the concentration depends only on z we obtain
(z!z ) * nm d< "r!r " *z 2
which, after di!erentiating with respect to z, gives
*1H2 * nm "4p . *z *z 2
(11)
The magnetic energy of the system and its second variation due to the concentration perturbation can be expressed as (H"1H2#H ) K 1 E"! nmH d< 2
dk"dn(k)"
m(1!exp(!2kh)) bk # 4p 2kh
.
The function
In the case where dimensions of the system in x, y plane are "nite the magnetic "eld strength, after averaging along the directions in x, y plane, can be expressed as
1H2"
221
1 1 dE"! dnmH d
m(1!exp(!2kh)) bk # 2kh 4n has a minimum for "nite value of the wave number k and thus allows to describe the formation of the periodic system of particle sheaths. Such a layering has been observed experimentally both in a rotating "eld and in a shear #ow of a magnetic suspension [11].
References [1] J.-C. Bacri, A. Cebers, S. Lacis, R. Perzynski, J. Magn. Magn. Mater. 149 (1995) 143. [2] S. Lacis, J.-C. Bacri, A. Cebers, R. Perzynski, Phys. Rev. E 55 (1997) 2640. [3] J.-C. Bacri, A. Cebers, J.-C. Dabadie, R. Perzynski, Phys. Rev. E 50 (1994) 2712. [4] J.-C. Bacri, R. Perzynski, M.I. Shliomis, G.I. Burde, Phys. Rev. Lett. 75 (1995) 2128. [5] S. Cutillas, G. Bossis, A. Cebers, Phys. Rev. E 57 (1998) 804. [6] M. Seul, D. Andelman, Science 267 (1995) 476. [7] T.C. Halsey, R.A. Anderson, J.E. Martin, Proceedings of Fifth International Conference on Electrorheological Fluids, Magnetorheological Suspensions and Associated Technology, 1995, 192. [8] E.E. Bibik, Magn. Gidrodin. (3)(1973) 25 (in Russian). [9] A. Cebers, Magn. Gidrodin. (2)(1982) 42 (in Russian). [10] A. Cebers. Magn. Gidrodin. (4) (1986) 132 (in Russian). [11] O. Volkova, S. Cutillas, P. Carletto, G. Bossis, A. Meunier, A. Cebers, J. Magn. Magn. Mater. 201 (1999), these Proceedings.
E!ects of the magnetodipolar interactions in the alternating magnetic "elds G. Bossis , A. Cebers * LPMC CNRS U.M.R. 6622, Universite de Nice-Sophia Antinopolis, Parc Valrose 06108 Nice, Cedex 2, France Institute of Physics, University of Latvia, Salaspils -1, LV-2169, Latvia Received 27 May 1998; received in revised form 30 July 1998
Abstract A macroscopic theory for the description of the structure formation in a system of magnetic dipoles under the action of a high-frequency rotating "eld is presented. Continuum equations for the e!ective magnetic "eld strength describing particle interactions are derived. It is shown that, contrary to the case of a constant magnetic "eld, where the demagnetizing "eld arising from a concentration #uctuation stabilizes the system with respect to the phase separation, the same concentration #uctuation will be ampli"ed in the case of a rotating magnetic "eld, leading to the formation of a layered structure. The extensions of the model necessary for the description of the formation of the periodic structures in the rotating "eld are considered and the relevance of the present model for the explanation of phase separation in shear #ows of magnetorheological suspensions is indicated. 1999 Elsevier Science B.V. All rights reserved. Keywords: Rotating "eld; Magnetodipolar interactions; Phase separation; Stripes
1. Introduction Peculiarities of the interaction of the magnetic #uids and magnetorheological suspensions with alternating "elds have rised a lot of interest in the past. For instance, we can point out complicated dynamics of the magnetic #uid droplets in the alternating magnetic "elds [1,2], excitation of parametric free surface oscillations [3], `negative viscositya e!ect [4] and others. Among issues not understood completely at present moment we can point out the
* Corresponding author. Fax: #371-2-2250-39. E-mail address: [email protected] (A. Cebers)
peculiarities of the structure formation by magnetic particles in the alternating magnetic "elds. Besides that similar problems can arise when describing the rheological behaviour of the magnetic suspensions in external magnetic "eld [5]. Here, we want to demonstrate that if an homogeneous suspension of magnetic particles is placed in a rotating "eld then, for strong enough magnetodipolar forces, we shall observe the formation of a layered structure. This illustrates that, contrary to the common belief that long-range magnetodipolar interactions are repulsive and thus in competition with short-range attractive forces for the structure formation [6], long-range magnetodipolar forces can be also attractive and initiate a phase separation.
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 5 3 - 0
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
2. Average potential energy of magnetic interaction Let us consider the magnetic dipole m aligned with y-axis of a coordinate system at position characterized by radius-vector r. We want to calculate its interaction energy with a layer of dipoles of thickness dz perpendicular to the z-axis. The repartition of the dipoles is homogeneous with a number density n and all the dipoles are aligned in the y direction. The interaction energy can be written as follows: ;"!mH"!m
m 3(r!r )(m(r!r ) ! "r!r " "r!r "
;n(r ) dS dz .
(1)
Since the distribution of the particles is homogeneous in x, y plane the function under the sign of the integral can be averaged with respect to the directions of the projection q!qY of the radius-vector r!r on this plane. So we have
;" dS dz
m 3m(q!q ) ! n(r ). "r!r " 2"r!r "
(3)
where m (3 cos a!1) , H"! "r!r" a is the angle of the radius-vector r!r with the z-axis and m "m/(2. Equivalence of the interaction between two rotating dipoles and average interaction of the "xed dipole with dipoles in the plane follows from the rotational invariance of the magnetodipolar energy. Formally, it can be illustrated as follows. Since due to the rotational invariance *; *; d;" [du;r]# [du;m]"0, *r *m
rotation of the dipoles in the plane is equivalent in the sense of the change of interaction energy to the rotation of the radius-vector on the same angle "du" but in the opposite direction. That means that averaging with respect to the directions of the dipole moment in the plane of the rotating "eld is equivalent to the averaging with respect to the directions of the projection of the radius-vector on the same plane at "xed dipole moments. Expression (3) for the interaction energy of two dipoles rotating in the phase in the plane has been derived in Ref. [7] and much earlier was considered by Bibik [8]. Note that this is the physical situation which happens when the rotation period is much smaller than the characteristic time of the relative motion of the dipoles. Relation (3) shows that the interaction of the dipoles are attractive in the plane and repulsive in the direction normal to it. This is the reason why the particles will gather into layers. In the following, we develop a model, based on the interaction energy given by Eq. (3), which predicts a formation of a layered structure.
(2)
Relation (2) shows, for example, that average interaction energy between the dipole and the dipoles of the layer on the circle with radius "q!q " in this case can be expressed as follows: m(3 cos a!1) ;"!m H" , 2"r!r "
219
3. Macroscopic e4ective 5eld equations We introduce the `"ctitiousa magnetic "eld strength which corresponds to Eq. (3): h(r!r ) HI "m r "r!r " with the interaction energy given by } ;"!m hHI , where h is the unit vector perpendicular to the plane of the rotating "eld. `Fictitiousa magnetic "eld strength for a nonhomogeneous distribution of particles will write
HI "m r
h(r!r )n(r ) d< , "r!r "
(4)
Taking the divergence of Eq. (4) and applying *
1 "!4pd(r!r ) "r!r "
we have div HI "4p div(m hn).
(5)
220
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
The magnetic energy in the presence of this "eld is given by
1 E"! nm hHI d<, 2
(6)
where the factor 2 avoids counting twice the dipolar energy of a pair of dipoles.Since rot HI "0, and introducing HI " t the last relation can be transformed in the following way:
1 1 E"! nm h t d<" (nm h)t d< 2 2 1 HI d<. "! 8p
4. Critical conditions for phase separation (7)
Thus, it is energetically advantageous for the magnetic particles to gather in layers normal to the symmetry axis which is the axis of the rotation of the magnetic "eld. That follows from the fact that energy of the &&"ctitious'' magnetic "eld (7) is negative. Since nonzero &&"ctitious'' "eld strength according to (5) appears when the particle distribution is nonhomogeneous along h direction then development of a structure leads to the diminution of the energy and consequently to a phase separation. Thus, in this condition arises an unusual situation where the long-range part of magnetodipolar interactions is causing the phase separation. To emphasize this point more clearly let us consider the local Lorentz "eld associated with the &&"ctitious'' magnetic "eld strength HI . Indeed, since in the case of homogeneous particle distribution HI "0 the "eld HI created by particles outside the Lorenz sphere < can be expressed as follows: * n(r )(r!r )h HI "!HI *"!m r d< 4\4* 4 "r!r " * 4 n(r )h* dS 4p "!m
"! nm h. "r!r " 3 1* The sign is opposite to the usual one and so is the interaction energy with the local "eld
2p E " nm d<. * 3
case of the phase separation under the action of constant "eld, the Lorentz "eld and the demagnetizing "eld have interchanged roles. In the constant "eld case the demagnetizing "eld arising due to a concentration #uctuation is stabilizing the system and its repulsive character determines the period of the pattern [9,10] whereas in the case of a rotating "eld this is demagnetizing "eld which is causing the phase separation, and the Lorentz "eld is playing stabilizing role.
(8)
Relation (8) shows that contrary to the usual case the Lorentz "eld plays a stabilizing role. In other words, it is possible to say that, compared to the
In order to calculate the phase diagram let us start with an homogeneous system which is perturbed by a concentration #uctuation with a gradient in the h direction. According to the Eq. (5) the HI "eld perturbation will be expressed by dHI "4pm hdn. Accounting for the conservation of the particle number and the e!ect of the Lorentz "eld the second variation of the free energy of the system will be
1 *u 2p dF" (dn) d<# m (dn) d< 3 2 *n
1 ! (dHI ) d< 8p
1 *u 4n " (dn) d
(9)
Formula (9) shows the increase of the critical value of magnetodipolar interaction parameter at uP0 and uP0.25. The last value corresponds to the limit of dense packing in the frame of van der Waals approach when large magnetic interactions are ne-
G. Bossis, A. Cebers / Journal of Magnetism and Magnetic Materials 201 (1999) 218}221
cessary to compress the almost dense-packed system. In this case concrete value of u (0.25) for dense packing limit has only illustrative purpose and can be made more realistic by considering more complex equations of state for hard sphere system [5]. The period of the layered structure arising from the phase separation can, in principle, be determined by accounting for the repulsive part of the magnetodipolar interactions (due to "nite extension of the layers in x, y plane) and surface energy on the phase boundaries.
Quite simple expressions for the repulsive part of the magnetodipolar energy can be obtained in the case of plane layers. The surface energy coming from the phase boundaries is introduced by the Cahn}Hilliard term b( n) d<. In the case of periodic concentration perturbation along z direction we have (2h the thickness of the layer).
1 E # b ( dn) d< K 2 "¸ 2h V
5. Period of the structure
m * (z!z ) n(r ) d< H(r)" 2 *z "r!r " m*(r )n(r ) # r ! dS , (10) "r!r " 1 where * the normal to the boundary of the sample which is supposed to be perpendicular to z direction. Di!erentiating the "rst term in Eq. (10) and keeping in mind that the concentration depends only on z we obtain
(z!z ) * nm d< "r!r " *z 2
which, after di!erentiating with respect to z, gives
*1H2 * nm "4p . *z *z 2
(11)
The magnetic energy of the system and its second variation due to the concentration perturbation can be expressed as (H"1H2#H ) K 1 E"! nmH d< 2
dk"dn(k)"
m(1!exp(!2kh)) bk # 4p 2kh
.
The function
In the case where dimensions of the system in x, y plane are "nite the magnetic "eld strength, after averaging along the directions in x, y plane, can be expressed as
1H2"
221
1 1 dE"! dnmH d
m(1!exp(!2kh)) bk # 2kh 4n has a minimum for "nite value of the wave number k and thus allows to describe the formation of the periodic system of particle sheaths. Such a layering has been observed experimentally both in a rotating "eld and in a shear #ow of a magnetic suspension [11].
References [1] J.-C. Bacri, A. Cebers, S. Lacis, R. Perzynski, J. Magn. Magn. Mater. 149 (1995) 143. [2] S. Lacis, J.-C. Bacri, A. Cebers, R. Perzynski, Phys. Rev. E 55 (1997) 2640. [3] J.-C. Bacri, A. Cebers, J.-C. Dabadie, R. Perzynski, Phys. Rev. E 50 (1994) 2712. [4] J.-C. Bacri, R. Perzynski, M.I. Shliomis, G.I. Burde, Phys. Rev. Lett. 75 (1995) 2128. [5] S. Cutillas, G. Bossis, A. Cebers, Phys. Rev. E 57 (1998) 804. [6] M. Seul, D. Andelman, Science 267 (1995) 476. [7] T.C. Halsey, R.A. Anderson, J.E. Martin, Proceedings of Fifth International Conference on Electrorheological Fluids, Magnetorheological Suspensions and Associated Technology, 1995, 192. [8] E.E. Bibik, Magn. Gidrodin. (3)(1973) 25 (in Russian). [9] A. Cebers, Magn. Gidrodin. (2)(1982) 42 (in Russian). [10] A. Cebers. Magn. Gidrodin. (4) (1986) 132 (in Russian). [11] O. Volkova, S. Cutillas, P. Carletto, G. Bossis, A. Meunier, A. Cebers, J. Magn. Magn. Mater. 201 (1999), these Proceedings.