Rosensweig instability in ferrofluids
We propose a simple model to analyze stability of the free surface of horizontally unbound ferrofluid in a vertical magnetic field. With respect to the well known Rosensweig instability (see e.g., R.E.Rosensweig, Ferro hydrodynamics, Cambridge University Press, Cambridge (1993) and references therei...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Rosensweig instability in ferrofluids / E.I. Kats // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 1019–1021. — Бібліогр.: 6 назв. — англ. |
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Kats, E.I. 2017-05-31T07:40:13Z 2017-05-31T07:40:13Z 2011 Rosensweig instability in ferrofluids / E.I. Kats // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 1019–1021. — Бібліогр.: 6 назв. — англ. 0132-6414 PACS: 47.20.Mа, 75.50.Mm https://nasplib.isofts.kiev.ua/handle/123456789/118779 We propose a simple model to analyze stability of the free surface of horizontally unbound ferrofluid in a vertical magnetic field. With respect to the well known Rosensweig instability (see e.g., R.E.Rosensweig, Ferro hydrodynamics, Cambridge University Press, Cambridge (1993) and references therein) we go one step further to include into consideration coupling of surface displacements to non-magnetic degree of freedoms. We show that the coupling can lead to a considerable reduction of the critical magnetic field and as well yields to nontrivial depletion layering near the surface. The author acknowledge support from Russian Federal grant “FTP Kadry”. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Теория электронных свойств Rosensweig instability in ferrofluids Article published earlier |
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Rosensweig instability in ferrofluids Kats, E.I. Теория электронных свойств |
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Rosensweig instability in ferrofluids |
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Rosensweig instability in ferrofluids |
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Rosensweig instability in ferrofluids |
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rosensweig instability in ferrofluids |
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Kats, E.I. |
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Kats, E.I. |
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Теория электронных свойств |
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We propose a simple model to analyze stability of the free surface of horizontally unbound ferrofluid in a vertical magnetic field. With respect to the well known Rosensweig instability (see e.g., R.E.Rosensweig, Ferro hydrodynamics, Cambridge University Press, Cambridge (1993) and references therein) we go one step further to include into consideration coupling of surface displacements to non-magnetic degree of freedoms. We show that the coupling can lead to a considerable reduction of the critical magnetic field and as well yields to nontrivial depletion layering near the surface.
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Rosensweig instability in ferrofluids / E.I. Kats // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 1019–1021. — Бібліогр.: 6 назв. — англ. |
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AT katsei rosensweiginstabilityinferrofluids |
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© E.I. Kats, 2011
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10, p. 1019–1021
Rosensweig instability in ferrofluids
E.I. Kats
Landau Institute for Theoretical Physics, RAS
Kosygina, 2, Moscow 119334 ,Russia
E-mail: kats@ill.fr; kats@landau.ac.ru
Laue-Langevin Institute, Grenoble F-38042, France
Received December 1, 2010
We propose a simple model to analyze stability of the free surface of horizontally unbound ferrofluid in a
vertical magnetic field. With respect to the well known Rosensweig instability (see e.g., R.E.Rosensweig, Ferro
hydrodynamics, Cambridge University Press, Cambridge (1993) and references therein) we go one step further
to include into consideration coupling of surface displacements to non-magnetic degree of freedoms. We show
that the coupling can lead to a considerable reduction of the critical magnetic field and as well yields to non-
trivial depletion layering near the surface.
PACS: 47.20.Mа Interfacial instabilities;
75.50.Mm Magnetic liquids.
Keywords: ferrofluids, Rosensweig instability.
Fluids with ferromagnetic properties (termed tradition-
ally as ferrofluids) are formed by a colloidal suspension of
solid magnetic particles in a parent fluid. When a layer of
such a liquid is subjected to a vertically oriented and uni-
form magnetic field, above a critical value of the field
strength a pattern of periodic peaks appears on the surface
of the liquid. This is the classical Rosensweig instability
observed long ago by Cowley and Rosensweig [1]. Physics
behind the Rosensweig instability is related to a feed back
from the ferrofluid to the applied magnetic field. This feed
back modifies the magnetization drastically and establishes
a new equilibrium state of the fluid. This is a typical sym-
metry breaking phenomenon omnipresent in the realm of
phase transitions. Therefore the theoretical tools developed
for thermodynamic phase transitions can be utilized to de-
scribe the Rosensweig instability. Note in passing that it is
not the case for some other instabilities also known in liq-
uids. For instance Rayleigh–Taylor or Kelvin–Helmholtz
instabilities occurring as results of acceleration or shearing
of liquid interfaces, are basically dynamic in their nature.
The arrangement of peaks resulting from the Rosens-
weig instability is a particular example of pattern forma-
tion in physical systems [2]. For these phenomena, at least
as the first step, Landau phenomenological theory is an
appropriate theoretical tool. Because we are dealing with a
sort of instability occurring at a finite wave vector, we util-
ize so-called weak crystallization Landau theory [3,4]. The
instability threshold itself can be obtained easily from a
harmonic part of the surface energy. It includes two contri-
butions stabilizing the flat surface, namely the surface ten-
sion term 2( )h∝ σ ∇ and gravitation energy 2g hρ (σ is
surface tension, ρ is ferrofluid mass density, g is the
Earth gravitation acceleration, and h is the vertical dis-
placement with respect to the planar on average surface).
On the contrary the vertical magnetic field B destabilizes
the flat surface contributing to the energy density as
2 | |,B h∝ −χ ∇ where χ is ferrofluid magnetic susceptibi-
lity. Combining all terms together and transforming to
Fourier space (q is the wave vector within the surface
which we assume on average stretched along X Y− axis)
we arrive at the following harmonic energy density (i.e.,
the surface energy per unit area)
( )2 2 2( ) = | ( ) | .a q q g B q h qσ +ρ −χ (1)
From the (1) we find the instability threshold cB
1/4
2
4= ,c
gB
⎛ ⎞σ ρ
⎜ ⎟⎜ ⎟χ⎝ ⎠
(2)
and the optimal wave vector 0q
2
0 = = ,
2
cB gq
χ ρ
σ σ
(3)
surprisingly independent of ferrofluid magnetic characte-
ristics.
The Fourier component 0( )h q plays the role of the
Landau theory order parameter ψ for the Rosensweig
E.I. Kats
1020 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10
instability. If one is interested not only in the threshold cB
but also in various pattern selection, the non-linear terms
have to be added to the Eq. (1). Then the standard Landau
functional for the order parameter ψ , borrowed from the
weak crystallization theory, can be written as
22 2 2 3 4
02
0
1= ( ) .
2 6 248
E dxdy q
q
⎧ ⎫β μ λ⎪ ⎪⎡ ⎤αψ + ∇ + ψ + ψ + ψ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
∫
(4)
The functional gives the energy of the surface bending fluc-
tuations with wave vectors near 0q (it determines the pat-
tern period 02 / qπ ). The first term in the Eq. (4), as usually
within Landau theory, reads as a B B∝ − , where B is
close to the critical magnetic field cB (but does not coincide
with it, since in a general case we have to deal with the first
order phase transition). The second term has its minimum at
0q q= and guarantees that only fluctuations with wave vec-
tors close to 0q are relevant. The last terms in the Eq. (4)
are usual cubic and fourth-order terms in the Landau expan-
sion relevant near the critical field. This energy (4) treats the
Rosensweig ''condensed ripple'' formation as a pure static
(thermodynamic) phase transition.
Since the Rosensweig instability we are investigating, is
an athermal one, and characteristics scales (in real space)
can be very large (up to 1 cm), fluctuations of the order
parameter are practically irrelevant. Therefore we can re-
strict ourselves to the mean field treatment of the energy
(4). In this approximation the phase diagram can be found
easily by the straightforward minimization of the (4) and
the results are well known, see e.g., [4], and the Fig. 1
where for the case = constλ and = constμ the phase
diagram is plotted on the plane /μ λ and /α λ . This is the
phase diagram when the both non-linear (interaction) terms
coefficients μ and λ in the Landau energy (4) are con-
stant, independent of wave vectors. This describes the first
instability with hexagonal pattern formation. However,
experimentally upon further increase of the magnetic field
above the threshold cB gives rise to the transition from the
hexagonal to square lattice of peaks [5]. Weak crystalliza-
tion Landau theory is equally suitable to describe this
second instability. Indeed, as we already mentioned above,
neglecting fluctuations is equivalent to fixing 0| | .q=q In
this case (i.e., in the mean field approximation) μ indeed
has to be considered as a constant. However λ is allowed
to be a function of a single angle θ between the four wave
vectors entering into the fourth order term. These wave
vectors have to satisfy the conditions
4
0
=1
| |= ; = 0,i i
i
q ∑q q (5)
and in the most general form it means that
0= (1 cos(2 )),k
k
kλ λ + λ θ∑ (6)
where kλ describe anisotropy of the fourth order interac-
tion. Keeping only 1 0λ ≠ we can obtain the phase dia-
gram (see Fig. 2) which includes also a tetragonal phase.
However recent experimental observations [6] show
that something is missing in this picture. This unknown
something yields to a considerable reduction of the critical
field, and as well produces non-trivial depletion layering
near the surface. Reduction of the critical field means that
there is another destabilizing factor promoting surface un-
dulations. Pure phenomenologically depletion and non-
uniform colloidal particle distribution can be bluntly
lumped into an effective field φ coupled to the surface
curvature 2 2h∇ ≡ ∇ ψ . In a Fourier space
2
int = ( )( ( )).E q q qγφ ψ − (7)
In the spirit of the Landau theory the energy per unit area
responsible for colloid composition variations, reads as
Fig. 1. The mean field phase diagram on the plane /μ λ and
/α λ . SA stands for the flat surface, SB for the hexagonal
structure, and in the mSA phase one dimensional modulation
takes place.
SB
SA
m
SA
–1.0 –0.5 0
� �/
�
�/
Fig. 2. The phase diagram with anisotropic λ with a tetragonal
structure Te .
Te SAm
SB
SA
–1.5 –1.0 –0.5 0
� �/
�
�/
Rosensweig instability in ferrofluids
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 1021
( )22 4
comp
1= .
2 2 4
c dE dxdy t⎧ ⎫φ + ∇φ + φ⎨ ⎬
⎩ ⎭∫ (8)
Within the mean field approximation minimizing
comp int ( )E E a q+ + over ( )qψ we get
2
2 2( ) = .qq
q
q g B q
γφ
ψ
σ + ρ−χ
(9)
Then we find the effective free energy in terms of φ with
the quadratic over ( )qφ term 2| ( ) |q qΓ φ with
2 4
2
2 2= ,q
qt cq
q g B q
γ
Γ + −
σ + ρ−χ
(10)
with its minimum at 2
0 0q ≠ if 2 2
0 0γ ≥ γ ≠ . Note that this
instability might occur even at = 0B . In this simple model
in a single mode approximation there are three equilibrium
phases:
– flat and symmetric phase ( = 0〈φ〉 and = 0〈ψ〉 );
– flat and asymmetric phase ( 0〈φ〉 ≠ and = 0〈ψ〉 );
– asymmetric and modulated phase ( 0〈φ〉 ≠ , and 0〈ψ〉 ≠ ).
Our assumptions of the “weak crystallization» nature of
the phase transition should be treated as a working hypo-
theses. Comparison of the predictions resulting from this
hypothesis with experimental observations will show
whether and when this hypothesis is justified. While the
picture is still not completely clear we do believe that fur-
ther detailed studies of this transition, both from the expe-
rimental and theoretical sides, will increase our under-
standing of the mechanisms enabling ferrofluids to
accommodate different structures and physical properties.
I am very pleased to dedicate this article to V. Pe-
schanskii on the occasion of his 80-th birthday. He certain-
ly loves “Electron phenomena in conducting systems”, the
topic of this special issue, but as well condensed matter
physics in general, and discussions with him have been an
inspiration for this paper.
The author acknowledge support from Russian Federal
grant “FTP Kadry”.
1. M.D. Cowley and R.E. Rosensweig, J. Fluid Mech. 30, 671
(1967).
2. M.C. Cross and P.H. Hohenberg, Rev. Mod. Phys. 65, 851
(1993).
3. S. Brazovskii, Sov. Phys. JETP 41, 85 (1975).
4. E.I. Kats, V.V. Lebedev, and A.R. Muratov, Phys. Rep. 228,
1 (1993).
5. R. Friedrichs and A. Engel, Phys. Rev. E64, 021406 (2001).
6. A. Vorobiev, G. Gordeev, O. Konovalov, and D. Orlova,
Phys. Rev. E79, 031403 (2009).
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