Proceedings TH2002 Supplement (2003) 403 – 418
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/040403-16 $ 1.50+0.20/0
Proceedings TH2002
Condensation Phase Transitions in Ferrofluids
A. Yu Zubarev L. Yu Iskakova
Abstract. We present the results of theoretical study of a new scenario of condensation phase transitions in ensemble of particles in ferrofluids. In this scenario the
phase transition begins with the formation of linear chainlike clusters which collapse into dense globules when number of particles in the chain exceeds a certain
magnitude nc . These globules serve as nuclei of a new dense phase. Final (equilibrium) stage of the phase transition is also considered. Our estimates show that for
actual parameters of ferrofluid without magnetic field nc is about 40 − 70. Calculated magnetic volume concentration of particles in dense phase after separation is
about 0.2; initial susceptibility of the dense phase is 70 - 140. These results are in
agreement with known computer and laboratory experiments.
1 Introduction
Ferrofluids are colloidal suspensions of single-domain ferroparticles in a carrier
liquid medium. Combination of liquid behavior with high response to external
magnetic field, rich set of unique properties, valuable for modern high technologies,
possibility to control their macroscopical behavior and properties using moderate
and weak magnetic fields (of the order of 10 kA/m), attract considerable interest
to these systems (see, for example, [1]-[3]).
Typical ferrofluids consist of magnetite particles with mean diameter about
10nm. To avoid agglomeration of the particles under van der Waalse forces, they
are coated by surfactant layers, screening these molecular interactions. The usual
thickness of the covers is 2 - 3 nm. Simple estimates show that due to these layers the dipole-dipole interaction between magnetite particle with diameter 10 nm
for room temperatures is significantly smaller than kT . However one of peculiarities of real magnetic fluids is that they are polydisperse with relatively broad
distribution over sizes of particles [3]. Tails of these distributions include particles
with diameters about 15-20 nm. Energy of magnetic interaction between these,
relatively big, particles is several times more than kT . Under this interaction the
biggest particles can form various linear or bulk aggregates, observed in many
experiments and numerical simulations (see, for example, [4]-[10] and references
below). Experiments show that these aggregates, in spite of small concentration
of ”big” particles, are able to change macroscopical properties of ferrofluids more
than in order of magnitude [3].
Theoretical investigations of appearance of the dense ”drops” have been done
in refs. [11] - [14]. The common point in these, greatly different models, is that they
treat this phenomenon as van der Waalse gas - liquid phase transition in ensemble
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of single particles. Any heterogeneous clusters (chains, for instance) are ignored in
these models. However various computer experiments [15]-[19] demonstrate that
long linear chain-like clusters occur in magnetic fluids before their separation into
two massive phases. Analytical studies [20],[21] of influence of the chains on the
van der Waalse phase transition demonstrate that their appearance makes the
phase transition impossible - when temperature decreases, chains become longer
and longer in spatially homogeneous ferrofluid. This conclusion coincides with the
results of [15]-[19] where only linear structures have been detected even for very
strong dipole-dipole interactions. It should be noted that in computer experiments
some ”clouds”, consisting of separate chains, have been observed in ref. [19]. But
these clouds look like fluctuations of density rather than massive dense phases.
Equilibrium phase transitions in the system of dipole particles have been observed
in computer experiments [18]. However these transformations are transitions between two very dilute phases and only under external magnetic field. At the same
time numerous laboratory experiments with magnetic liquids demonstrate appearance of very dense drop-like domains in these systems, both under and without
the field.
Thus the conclusion, that the condensation phase transitions and appearance
of dense phases in ferrofluids are impossible, is in principle contradiction with experiments. In our opinion, one can overcome this contradiction considering another,
than the van der Waalse-like, scenario of the phase transition. Indeed,the results
of computer experiments [15]-[19] and analytical investigations [20]-[24] give us a
ground to suppose that the chains appear in ferrofluid before the bulk separation.
Long chains, like polymer macromolecules, must present flexible coil-like structures. If dipole - dipole interaction is strong enough (only in this situation long
chains are expected), these coils can collapse into compact dense globules, which
can serve as bulk nuclei of new dense phase. We can expect that concentration of
particles in these globules is high and close to concentration of dense packing. This
means, in particular, that thermodynamical functions of the globules and dilute
environment must be described on the base of different models, adopted to dense
and relatively dilute phases respectively. As a result, the critical point of phase
transition, presenting in the van der Waalse-like models [11] - [14], must be absent
in this scenario. Thus a gap between branches of binodal of coexistence of dense
and dilute phases is expected. One can expect (and calculations below confirm this
expectation) that, due to this gap, concentration of ferroparticles in dilute phase,
along the corresponding branch of this binodal, must be low and, therefore, probability of appearance of chains in this phase must be low too. Therefore, the chains
can not influence significantly on conditions of equilibrium coexistence between
these phases.
In this work we study initial (collapse of chains into dense globules) and
final, equilibrium, stages of the phase transition. We estimate critical number nc
of particles in the chain, corresponding to its collapse into a dense globule, and
calculate binodals of equilibrium coexistence between dilute and dense phases. The
results of calculations are in agreement with computer and laboratory experiments.
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Condensation Phase Transitions in Ferrofluids
405
2 Chain-globule collapse. Basic model
Consider a linear chain, consisting of n identical ferroparticles. Our aim is to estimate critical number nc of the particles for the chain - globule transformation. For
that we will estimate thermodynamical works, necessary for formation of n-particle
linear as well as globular clusters, and compare them. The stage, corresponding to
minimum work, is more stable.
Minimum work, necessary to form this chain, is equal to the free energy of
the chain:
(1)
Fch = −kT lnZ,
where
Z=

exp (κ0 ·
i
ei ) −

uij  dω
i=j
Here index ch denotes chain, κ0 = µ0 mH0 /kT is the dimensionless magnetic
field in place where the chain is situated, µ0 is the vacuum permeability, uij is
the dimensionless energy of dipole-dipole interaction between i − th and j − th
particles in the chain, ei is the unit vector, aligned along magnetic moment of
i − th particle, ω stands for all ei and radius-vectors of all particles in the chain
under condition that position of one of them (say, the first) is fixed.
As usual in statistical physics, the many-particle integral Z can not be calculated exactly in general form. Decisive simplification can be reached if we take
into account interaction only between particles being nearest neighbors along the
contour of the chain. However even in this approximation Z can be decoupled
into combination of low dimensional integrals only in the cases of zero or infinite
magnetic field. In both limiting cases
lnZ = −(n − 1)εef
(2)
Low estimates for parameters εef have been obtained in [26]. For instance,
under zero field:
3
1
εd
2
f
(Ω)
r
drdΩ
(3)
εef = ln
exp
2πv
2 r3
µ0 m2
4π d3 kT
where d is the hydrodynamical (with surface layer) diameter of the particle, v =
πd3 /6 - its volume, r ≥ d is the distance between centers of two neighboring
particles in chain,
ε=
f (Ω) = 3 cos θ(cos θ cos ω + sin θ sin ω cos(φ − ψ)) − cos ω,
dΩ = d cos θd cos ωdφdψ
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π
].
2
Physical meaning of parameter ε is the characteristic dimensionless energy
of magnetic interaction of two contacting particles. If ε is much more than unity,
the following asymptotic estimate of (3) can be used [23],[26]:
φ, ψ ∈ [0, 2π], ω ∈ [0, π], θ ∈ [0,
εef = ε − ln
3ε3
.
8
(4)
As it was mentioned, for finite κ0 decoupling of Z into low-dimensional integrals is impossible, therefore the free energy Fch can not be calculated using
a regular approximations. To reach physically reasonable estimates, we consider
case of relatively small magnetic fields, when κ0 < ε. Considerations, based on
well-known results of physics of polymer chains, indicate that order of magnitude
of persistence length of chain is εd. This means that for the contour segments with
number of particles smaller than ε, thermal fluctuations of shape are insignificant
and, in the first approximation, this segment can be considered as a straight rod.
In contrast, for parts of chain much longer than εd, the flexibility is of principle importance and correlations between positions and orientations of particles on
opposite edges of this segment are very small.
Thus, to estimate an influence of magnetic field on the free energy of the
chain, we can present this cluster as chain of straight rods with number of particles
n1 ∼ ε and magnetic moment mn1 . We neglect any correlations in orientations
of these rods. This presentation of the chain corresponds to well known model of
0 n1
free-linked polymer chains. Langevine free energy of each rod is −kT ln sinhκ
κ0 n1 ,
therefore the total magnetic energy of the ”free-linked” chain of the rods is
Fmch = −
n
sinhκ0n1
kT ln
.
n1
κ0 n1
Therefore, in the first approximation, we may estimate the free energy of the
chain under moderate magnetic field (κ0 < ε) as:
n sinhκ0 n1
ln
Fch = −kT (n − 1)εef +
,
(5)
n1
κ0 n1
where n1 = ε and εef is estimated in eq. (4).
Let us estimate now minimal work Fg necessary for the formation of the
dense bulk cluster, consisting of n particles. It is important to note that, unlike
polymer macromolecule, which conserves linear structure even in globule state, in
dense phase of ferrofluid the linear contour of chain must be lost. From topological point of view this dense phase presents rather homogeneous fluid of particles
than dense macromolecular coil. Therefore, to estimate Fg we may use statistical
thermodynamical methods, developed for dense homogeneous ferrofluids.
Classical results of theory of nucleation [27] and electrodynamics of continuum media [28] give the following formula for minimal work of formation of the
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Condensation Phase Transitions in Ferrofluids
407
bulk cluster:
Fm
Fg = F0 + Fm + Fs
H0
= −V
Mi dH0 , Fs = σS .
(6)
0
Here F0 is the free energy of this cluster without magnetic field, V and S are
the volume and surface of the globule, Mi is the magnetization inside this cluster,
σ is the surface tension.
Using the model [14], one can present F0 and Mi as follows:
1 2
F0 = kT n fhs (ϕ) − ε ϕ
(7)
3
n L(κi )
dL(κi )
Mi = m
1 + 4εϕ
V κi
dκi
where
nv
πd3
1
mHi
, ϕ=
, κi = µ0
, v=
x
V
kT
6
Here L is the Langevine function, ϕ is the volume concentration of particles
inside the globule, Hi is the magnetic field in this aggregate.
The magnitude fhs in eq. (7) stands for the free energy per one particle of
dense gas of hard spheres with concentration ϕ. Taking into account that, due
to strong magnetodipole interaction, the density of particles inside globule is expected to be high (calculations below confirm this fact), to estimate fhs we use
the equation of state, suggested by Hall in [29]:
L(x) = coth(x) −
phs =
A
kT
ϕ
v ϕm − ϕ
(8)
where ϕm = 0.74 is the density of close packing of spheres, phs is the osmotic pressure of gas of these particles, A is a parameter. Using standard thermodynamical
relations between free energy and pressure, we get:
A
v
ϕ
phs
+C
(9)
fhs =
dϕ =
ln
kT
ϕ2
ϕm ϕm − ϕ
where C is the constant of integration. Following to [29],[30], parameters A and
C can be fixed from condition that gas of hard spheres undergoes entropy driven
phase transition with volume concentrations of particles equaling to 0.494 and
0.545 in relatively dilute and dense phases, respectively. Using relations (8) and
(9) for thermodynamical functions of dense phase, classical Carnagan - Starling
equation for the dilute phase, from condition of equalities of osmotic pressure and
chemical potentials in these phases one can obtain A ≈ 2.2, C ≈ 1.255.
To find the magnetic field Hi inside the globule, we need to determine shape
of this cluster. Exact shape of magnetic ”drop” in non magnetic environment is
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Proceedings TH2002
unknown. In a first approximation one may present the aggregate as ellipsoid of
revolution with long axis aligned along external field H0 . As has been shown in [7],
[8], results of this approximation are in a good agreement with the experimental
evidence in weak and moderate magnetic fields. Below we use this presentation for
shape of the globule. Inside homogeneous ellipsoid dimensionless magnetic field κi
is connected with the external one κ0 as [28]:
κi =
where
χi = 4π
κ0
1 + χi ξ(c)
Mi
L(κi )
= ϕε24
Hi
κi
(10)
dL
1 + 4ϕε
dκi
is the magnetic susceptibility,calculated in model [14], c < 1 is the ratio between
the minor and the major axes of ellipsoid, ξ(c) is the demagnetizing factor, which
is determined from the following relation [28]:
√
c2
1 + 1 − c2
2
√
ξ(c) =
−2 1−c
ln
.
(11)
2(1 − c2 )3/2
1 − 1 − c2
Let us return now to the expression (5). The interfacial tension σ between
dense and dilute phases of ferrofluid has been calculated in [31] and can be estimated as:
1 − 34 ϕ
1
2 −2/3
σ ∼ kT ϕ v
G(ε, κi ) −
(12)
2 ,
2
(1 − ϕ)
1
G = 2εL(κi )2 + ε2 .
3
We will use this estimate without any additional coefficients. The surface S
of the ellipsoidal cluster can be calculated from the well-known formula:
√
2/3
3
1 − c2
2/3
c
1+ √
.
(13)
S = 2π
V
4π
c 1 − c2
Combining eqs. (6)-(13), taking into account that V = nv/ϕ, we come to
expression for Fg as a function of ϕ and c, depending on n, ε and κ0 as parameters.
To determine ϕ, one can use the condition of mechanical equilibrium between the
globule and environment. With the help of approximation of zero concentration of
particles outside the globule, this condition can be written down as:
pi − ps − pM = 0 .
(14)
Here pi is the osmotic pressure inside the globule, ps is the capillary pressure, proportional to σ, pM is Maxwell pressure, proportional to square of normal
component of magnetization. Estimates show that two last terms in left side of
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Condensation Phase Transitions in Ferrofluids
409
(14) are smaller than the first one. Therefore, in a first approximation, they can be
neglected. Using the results of model [14] and expression (8) for osmotic pressure
of dense gas of hard spheres, one can present pi in the form:
kT
A
2
−ϕ G .
(15)
pi =
ϕ
v
ϕm − ϕ
Thus, the condition pi = 0 of mechanical equilibrium between the bulk cluster
and empty environment gives
ϕm + ϕ2m − 4A/G
(16)
ϕ=
2
Formula (16) shows that bulk aggregates, mechanically equilibrium with dilute environment, can appear only when parameter ε of dipole-dipole interaction
between particles is high enough and inequality G > 4A/ϕ2m holds true.
To determine the form-factor c of the ellipsoidal cluster, the condition of
minimum of Fg , with respect to c, must be used, i.e.:
∂Fg
=0.
∂c
(17)
Taking into account eqs. (10) and (16), which present κi and ϕ through
κ0 , ε and n, the solution of eq. (17) gives us c as a function of these magnitudes.
Substituting c(κ0 , ε) into (7), keeping in mind eqs. (8) and (10)-(14), we obtain
Fg as a function of n, ε and κ0 .
Consider now the difference between the works of formation of the linear and
bulk clusters
(18)
δF (n, ε, κ0 ) = Fch (n, ε, κ0 ) − Fg (n, ε, κ0 ) .
For small enough n this difference is negative. This means that linear state
for n-particle cluster is more preferable. If n exceeds a certain critical magnitude
nc , the difference δF becomes positive and, therefore, the chain undergoes collapse
into bulk globule. Therefore, the threshold number nc of particles in the cluster is
determined from equation δFg (nc ) = 0.
3 Chain - globule collapse. Moderate and weak fields
The described procedure from principle point of view is simple, but leads to cumbersome calculations. In this part we suppose that the field is weak enough and
the strong inequality κi << 1 is valid. In this situation the susceptibility χi (κi )
may be taken in its initial form χoi = χi (0).
It is important to note that for large ε, when appearance of linear and bulk
aggregates is expected, and for magnitudes of ϕ, corresponding to dense phase,
the initial susceptibility χoi is much more than unity (see eq. (10)). This equation
shows also that inequality κi << 1 can be valid even for κ0 ≥ 1 if the inequality
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Proceedings TH2002
χoi ξ(c) >> 1 is held. Therefore, we may consider moderate external magnetic field
when κ0 > 1. One needs to check only the strong inequality for κi .
Experiments and calculations of [8],[10] show that the plot of dependence
of the form-factor c on dimensionless external magnetic field κ0 consists of two
stable branches and one unstable branch between them (see Fig. 1). For small or
moderate magnitudes of κ0 , corresponding to the first stable part of plot of c(κ0 ),
this parameter changes slowly and deviates from unity not significantly. Therefore,
on this part of the plot, ξ ≈ ξ(1) = 1/3 and κi << κ0 (see eq. (10)). Then, at a
certain κ0 , parameter c undergoes a stepwise transition to a second stable branch,
where c, ξ << 1. On this branch internal field κi has the same order of magnitude
as the external one κ0 . We will consider only the first stable branch of Fig.1, where
ξ ≈ 1/3 and κi << 1.
Figure 1: (Zubarev, Iskakova “Condensation phase transitions in ferrofluids”) The sketch of
dependence of form-factor of globule c on dimensionless external magnetic field κ0 . Solid lines
correspond to stable state of the aggregate, dashed one - to unstable.
Dependence of concentration ϕ of particles in the globule on parameter ε ,
calculated from eq. (16) for zero internal field, is shown in Fig.2. This plot indicates
that for ε ≥ 9 dense globule (ϕ ≥ 0.55) can be in mechanical equilibrium with
dilute environment. For concentrations ϕ < 0.5 the Hall equation of state (8) is not
precise and must be replaced by a more suitable equation (Carnagan - Starling,
for example). For definiteness, we do not consider here these situations, though it
is not difficult to do, conserving all considerations of this model.
The minimal work of formation of the globular cluster under approximations
Vol. 4, 2003
Condensation Phase Transitions in Ferrofluids
411
Figure 2: Volume concentration ϕ of particles inside globule vs. parameter ε of
magnetic interaction between particles for zero internal magnetic field.
κi << 1, ξ = 1/3 is:
F0 = n
Fg = F0 + Fm + Fs
A
1 2
ϕ
+ C − ε ϕ kT
ln
ϕm ϕm − ϕ
3
(19)
χoi
1 nκ20
kT
48 ϕε 1 + χoi /3
2/3
4π 3 n
Fs =
ε2 ϕ2 kT
6 4π ϕ
4
o
χi = 8ϕε 1 + ϕε
3
Fm = −
A = 2.2, C = 1.255 ϕm = 0.72
The free energy of chain Fch is given in eq. (5), volume concentration ϕ is
estimated in eq. (16). Using eqs. (5),(16) and (19), we can present the free energy
difference δF in the form of explicit function of n, depending on ε and κ0 as
parameters. The results of calculations of the critical number nc , satisfying to
equation δF (nc ) = 0, are shown in Fig.3 for two magnitudes of ε. Using eqs. (7)
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Proceedings TH2002
and (10)-(13) for ϕ ∼ 0, 6 − 0.7, we have checked that relations κi << 1, c ≈ 1
are valid for all values of nc , indicated in Fig.3. Therefore, our calculations are
self-consistent.
Figure 3: Critical number nc of particles in chain, corresponding to its collapse to
globule, as a function of dimensionless external magnetic field κ0 for ε = 9 and 11
(lines 1 and 2 respectively).
For a given magnitudes of ε, below the corresponding curve δF is negative.
Therefore here linear state of the cluster is more preferable. Above the curve
δF > 0 and the globular state is more stable. The obtained results indicate that
for systems with ε ∼ 9 − 10 (for magnetite ferrofluids with thickness of surface
layer on particle about 3nm this magnitude of ε corresponds to particles of 18-20
nm in diameter) the critical number nc for zero field is about 50-70. The number
nc increases with magnetic field and decreases when ε grows. Both of these results are expected: under magnetic field the chains become more rigid; increase of
magnetodipole interaction stimulates condensation of particles into bulk cluster.
We would like to note that the results of calculations of nc are very sensitive to
estimates for the interfacial tension σ, approximately as nc ∼ σ 3 . Therefore they
can be considered only as estimates by the order of magnitude.
Our estimates of nc allow us to explain, at least qualitatively, the results of
computer experiments of refs. [15]- [19], where only linear chains are observed.
From our point of view in these simulations the chains were shorter than it is
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Condensation Phase Transitions in Ferrofluids
413
necessary for their collapse.
The results, shown in Fig.3, are in agreement with computer simulations [33]
of phase transition in Stockmayer fluid with weak central Lennard - Jones and
strong dipole-dipole interaction between particles. These experiments exhibit that
bulk nucleation in polar fluids begins with the formation of chainlike clusters,
which condense into compact drop-like nuclei when number of particles in chain
exceeds a certain magnitude. Taking Lennard - Jones diameter of a soft particle
as a diameter of rigid sphere, for parameters of the Stockmayer fluid, used in [33]
, in our notations we come to ε ∼ 9. The critical number of particles in chain
for its collapse into globule, calculated in [33] for zero field, is 30. Our result for
ε ∼ 9 and zero field is nc ∼ 70. Taking into account that central interaction in the
Stockmayer systems stimulates the condensation of chain into globule and that our
calculations are sensitive to σ (which exact expression for ferrofluid is unknown),
the agreement between our estimates and calculations of [33] can be considered as
reasonable.
One can expect that the sharp increase of nc with magnetic field must produce
a strong magnetoviscous effect in ferrofluids. Indeed, when magnetic field increases,
for many globular clusters n becomes smaller than nc and they are transformed
to linear chains. These chains must be elongated along lines of magnetic field.
It is known [34] that the influence of elongated cluster on effective rheological
properties of suspension is much stronger than influence of compact spherical-like
clusters with approximately the same total volume. Therefore transformations of
globules into chains under magnetic field must lead to a sharp change of viscosity
of ferrofluid. Really, the strong magnetoviscous effect in typical magnetic liquids
under weak and moderate magnetic fields (till 20 kA/m) has been observed in
series of experiments, described in [3].
4 Equilibrium stage of the phase transition
At the beginning we would like to note that all real ferrofluids are polydisperse.
As a rule most part of particles have small sizes (diameter of magnetic core is
about 9 − 10nm for magnetite ferrofluids). Magnetodipole interaction between
them is too weak to provide formation of any heterogeneous structures or, moreover, condensation phase transitions. At the same time relatively small number of
big particles (with diameter of magnetic core about 16 − 20nm), capable to form
the structures and dense phases, presents in many real magnetic fluids. Total hydrodynamical volume concentration ϕ of these particles is about one - two percent
(see diagrams of phase distribution of particles in ref. [3]). Taking into account
that small particles are almost neutral in sense of magnetodipole interaction, for
maximal simplification of analysis we will ignore them and consider only system of
the big ones. This means that we neglect effects of depletion interaction between
small and big particles. Estimates show that since ratio of diameters of big and
small particles is not high, less than two, the depletion effects must not play a
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Proceedings TH2002
principle role in the phase transition.
Let us consider now homogeneous (without any heterogeneous aggregates)
ferrofluid, consisting of N spherical particles. Following [14], we may write down
free energy of the system of particles as:
mH
(20)
kT
where ϕ is again the volume concentration of the particles, fhs is the dimensionless
free energy per one particle of gas of hard spheres, G is determined in eq. (12),
H is the magnetic field inside the fluid. We suppose that this field is the same in
both coexisting phases. This situation corresponds to interface boundary, parallel
to the field.
For the dense phase we take the free energy fhs in the Hall form (9). For dilute
phase we use approximation, following from the well-known Carnagan - Starling
equation of state:
1 − 3ϕ
ϕ
fhs = ln + ϕ
(21)
2
e
(1 − ϕ)
F = kT N [fhs (ϕ) − ϕG(ε, κ)] , κ = µ0
Combining eqs. (12) and (20) for the dense phase and (20),(21) for the dilute
one, we can calculate osmotic pressures p and chemical potentials µ of particles
in both phases. Standard conditions of thermodynamical coexistence between two
phases allow us to calculate volume concentrations ϕ1 and ϕ2 in dilute and dense
phases, therefore, to construct diagrams (binodals)of equilibrium coexistence of
the phases. Some results of calculations of these diagrams are shown in Fig.4 for
zero and infinite fields. These diagrams begin with threshold magnitudes εc of
parameter ε so that for ε > εc the system of equations for p and µ has a branch
of solution with ϕ1 << 1. For all ε this system has a solution with ϕ1 ∼ 0.3 − 0.5,
however the high concentrations of big particles, capable to undergo the structural
transformations, is non realistic for modern ferrofluids and we do not consider
them.
Diagrams in Fig.4 demonstrate, first, that magnetic field stimulates the phase
transition, i.e. decreases εc . Second, unlike models [11]-[14], these diagrams have
no critical point. Concentrations of dilute phases, corresponding to these diagrams,
are very low. Using results of model [26] of ferrofluids with chains, we estimated
ratio x of number of particles, aggregated into chains, to their total number along
left branches of the phase diagrams. In all situations, except regions close to εc ,
calculations give x << 1, i.e. unlike the van der Waalse scenario of the phase
transition, in the suggested scheme chains can not influence significantly on equilibrium conditions of the phase separation and, therefore, they can not make them
impossible. The exception is only narrow region of magnitudes of ε close to εc ,
where influence of chains might be significant. We plan to study this region in
nearest future. At the same time concentration of particles in the dense phase is
high enough, that justifies the use of the Hall approximation for fhs .
For magnetite ferrofluids with thickness 3nm of surface layers on particles,
the magnitudes ε = 7 − 8, under room temperatures, correspond to magnetic
Vol. 4, 2003
Condensation Phase Transitions in Ferrofluids
415
Figure 4: Phase diagrams (binodals) of equilibrium coexistence of dilute and dense
phases. Solid line - magnetic field is infinitely strong, dashed line - the field is zero.
diameter of particles about 18 − 19nm. Particles of this sizes present in many real
ferrofluids. Therefore even under zero field they can condense into dense phases,
observed in experiments. Since εc is decreasing function of magnetic field, increase
of the field induce condensation of particles with diameters smaller than 18−19nm.
Thus, increase of the field must lead to a sharp rise of volume of dense phase. All
known experiments confirm this conclusion.
For particles with magnetic diameter 18 − 20nm, hydrodynamical volume
concentration ϕ = 0.5 − 0.6 corresponds to magnetic volume concentrations ϕm ∼
0.2. Namely this result was obtained in experiments [8]. For this set of parameters
of the system (ε = 7−8, ϕm ≈ 0.2) initial magnetic susceptibility χ = 4π M(H)
H |H=0
calculated from eq. (10), is 70 − 140, which is close to measurements χ ∼ 80 − 130
of [8]. The agreement between our estimates of parameters of dense phase and
measurements of [8], as well as estimates of nc and computer determination of [33]
of this magnitude, show that the suggested model, at least in its principle points,
is robust.
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Conclusion
New scenario, free from difficulties of known models, of condensation phase transitions in ferrofluids, is suggested. In this scenario the phase transition begins with
the formation of linear chains and collapse of longest of them into dense compact
globules. For zero magnetic field and typical magnitudes of parameter of magnetic interaction between particles, calculated critical number nc of particles in
the chain, corresponding to its collapse, is about 40 - 70. This estimate is in agreement with the results of known computer experiments. Evolution of the globules
leads to equilibrium separation of ferrofluid into two phases, with high and very
low concentration of the particles. Because of very small concentration of the particles in dilute phase, probability of existence of chainlike clusters on final stage of
the phase transition is very low and these clusters can not influence significantly
on equilibrium conditions of the phase transition. Results of calculations of volume
concentration of particles and initial susceptibility of the dense phase in final stage
of the phase transition are in agreement with the results of known experiments.
Acknowledgments
This work has been supported by grants of Russian Basic Research Foundation,
projects NN 00-02-17731, 01-02-16072, 01-01-00058, grant CRDF, REC-005 and
grant of BMBF N RUS 00/196
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A. Yu Zubarev, L. Yu Iskakova
e-mail:[email protected]
Department of Mathematical Physics
Ural State University
Lenin Av., 51
620083 Ekaterinburg
Russia