IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 20 (2008) 204147 (5pp)

doi:10.1088/0953-8984/20/20/204147

Study of audio speakers containing
ferrofluid
R E Rosensweig1 , Y Hirota2 , S Tsuda2 and K Raj3
1

34 Gloucester Road, Summit, NJ 07901, USA
Ferrotec, 1-4-14 Kyobashi, chuo-Ku, Tokyo 104-0031, Japan
3
Ferrotec, 33 Constitution Drive, Bedford, NH 03110, USA2

Received 5 April 2008
Published 1 May 2008
Online at stacks.iop.org/JPhysCM/20/204147
Abstract
This work validates a method for increasing the radial restoring force on the voice coil in audio
speakers containing ferrofluid. In addition, a study is made of factors influencing splash loss ofthe ferrofluid due to shock. Ferrohydrodynamic analysis is employed throughout to model
behavior, and predictions are compared to experimental data.

1. Introduction
The use of ferrofluid in audio speakers and the benefits
achieved including voice coil cooling and centering are wellknown [1, 2]. This work examines a proposal for enhancing
the centering force by filling just the one gap between the voicecoil and the inner pole piece. Experimental limits on the drop
height at which fluid loss onsets by splashing is also studied todetermine means for increasing ruggedness. Increased shock
resistance holds interest also for the use of ferrofluids in seals,
instruments, motors, and other devices.

2. Analysis of voice coil centering
SI units are employed throughout unless otherwise noted.
The unperturbed geometry of the pole/voice-coil/pole
structure of an audio speaker is modeled as three concentric
cylinders with the magnetic field radiating symmetrically fromthe inner pole to the outer one; see figure 1. From Gauss’s law
the distribution of the induction field magnitude B is given by
B R = B0 R0 where R is the radial distance from the center,R0 the radius of the inner pole piece, and B0 the induction at
R = R0 . The defining equation is B = μ0 H + M expressed ina modified form such that M carries units of teslas. In the usualhigh field of an audio gap the magnetization is approximatedas saturated, M = Ms . Thus, magnetic field in the ferrofluid is
distributed according to μ0 H = B0 R0 /R − Ms .
The net magnitude of force Fi on the inner surface of thevoice coil having wetted length L when its center undergoes asmall displacement is given by

Fi =

 2π

p(θ )cos(θ )Ri L dθ.

Figure 1. Geometry for deriving radial distance R(θ) from the centerof the inner pole piece having radius R0 to a point at a position on thevoice coil having inner radius Ri .

Pressure p(θ ) in a stationary volume of ferrofluid is related
to the magnetic field via the ferrohydrodynamic Bernoulli
equation [3] with velocity absent and gravity force negligible:p(θ ) − p0 =

M dH

(2)

H0

p0 is pressure at the inner pole surface where the field
For the highly saturated ferrofluid,magnitude is H0.
equations (2) integrates approximately to p(θ ) − p0 =Ms [H (θ ) − H0]. Substituting for p(θ ) in equation (1) andmaking the approximation that displacement of the voice coil

(1)00953-8984/08/204147+05$30.00

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J. Phys.: Condens. Matter 20 (2008) 204147

R E Rosensweig et al

Figure 2. Left: comparative displacements of the voice coil in response to a radial force with the fill fraction of 1. The dashed line is the initialslope for a filled inner gap predicted from equation (4). Right: calculated location of ferrofluid interfaces for a fill fraction of 1 in the gaps.

has a small effect on the distribution of the field yields

B0 Ms R0 Ri L 2π cosθ
Fi =
dθ .
μ0
R (θ )
0

3. Measurements of voice coil centering
An apparatus was devised to measure radial force versus
displacement; the results are shown in figure 2. The apparatus
in these tests contained a central magnet inside a U-shaped
permeable piece. Parameter values are (APG 836 ferrofluid):
Ms = 0.022 T, B0 = 0.9 T, Ri = 12.75 mm, R0 =12.25 mm, L = 0.0022 m. From equation (4) the predicted
spring constant with only the inner gap filled is K  = 1.16 ×
10−2 g μm−1 . Figure 2 with fill ratio f = 1 shows theprediction for the initial slope compared to experiment. The
voice coil thickness was t = 0.3 mm, so from equation (7)the ratio K i /K = 408. The measured ratio is much smaller
but significant. The variance is believed to be due to nonideal
distribution of the field.
Tests in which only the outer gap was filled with ferrofluid
resulted in spontaneous anti-centering of the voice coil,bringing it into contact with the outer wall in agreement withtheory.

(3)

From geometry, the distance R(θ ) is given by R(θ ) =Ri [(δ/Ri )2 + 2(δ/Ri )cosθ + 1]1/2 . For small displacements,
R(θ ) expands to give R(θ ) ≈ Ri [1 − (δ/Ri ) cos θ ]−1 .
Substituting this expression into equation (3) and integratingyields the final result for the spring constant K i = Fi /δ :Ki = −

π B0 Ms L
R0
μ0 R i(4)

K i is the contribution to the spring constant due to the ferrofluidpressure force exerted on the inner surface of the displaced
voice coil. The minus sign indicates that this contribution to
the spring force is restorative.
If only the outer gap is filled with ferrofluid a similar
treatment yields the spring constant contribution K o due toforces exerted on the outer surface of a thin voice coil having
thickness t :π B0 Ms L
R0 .
Ko =
(5)
μ0 (Ri + t)
The sign in this case is positive, as the pressure forces at any
location are oppositely directed to those acting on the innersurface. The net spring constant when both gaps are filledwith ferrofluid is given by K = K i + K o . Substituting fromequations (4), and (5) and assuming t  Ri , yields
K =−π Ms H0 L
t.
Ri

4. Description of the splashing phenomenon
A shock test apparatus is used in these studies in which a
fixture is dropped from a known height, slides down verticalrails and impacts face down on a hard surface. The fixture
contains a magnetic circuit having an annular gap that is filledwith ferrofluid. The acceleration at impact is measured andthe free surface of the ferrofluid photographed with a highspeed video camera (9000 frames s−1 ). Acceleration peaks are
as high as 1000 g at the highest experimental drop heights.
The duration of the acceleration interval is experimentallyinvariant of drop height, consistent with a simple magnetic
mass–magnetic spring dynamic analysis and equals 1 ms.
From the videos it is seen that prior to impact the ferrofluid
is confined to the gap space. Following impact the ferrofluid
continues to move downward tending to escape from the fixture
as a plug. The fluid is subsequently drawn back into thegap by the gradient magnetic fringe field. Modeled as adamped spring–mass system, the impact process establishes a(6)

It can be seen that K provides a net restoring force. Asidefrom a small difference in the constant factor this relationshiprecovers the result of Bottenberg [2] and the ratio of spring
constants is given by
R0
Ki
=
.
(7)
K
t
Always R0  t , so it is seen that the restoring force withferrofluid present only in the inner gap is predicted to be greatly
enhanced compared to that for filling both the inner and theouter gap [4].
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J. Phys.: Condens. Matter 20 (2008) 204147

R E Rosensweig et al

1500

1500

1000

1000

500

500

0

0

-500

-500

-1000
0

50
100
Wave number, k (cm-1)

150

0

50

100

150

Wave number, k (cm-1)

a)

b)

140

120

0.80

120

100

0.75

80

0.70

60

0.65

40

0.60

20

0.55

Height, H (cm)Height, L (cm)

100
80
60
40
20
0
0

100

200

300

400

500

0
0.6

600

Viscosity, µ(m Pas)

c)

0.8

1.0
1.2
1.4
Gap width, w (m m)

1.6

Magnetic field, H (Tesla)

-1000

0.50
1.8

d)

Figure 3. Theoretical and experimental dependences of the shock instability on the ferrofluid viscosity and gap width. H = 50 cm in theRayleigh–Taylor computations. See the text for other reference parameter values. (a) Predicted viscosity influence. (b) Predicted gap width
influence. (c) Experimental height versus viscosity. (d) Experimental height versus viscosity. In (c) and (d) circles denote no splash; squares
denote splash.

with the growth factor γ defined by
z = ẑ 0 Re exp(γ t) cos(ks)positive acceleration directed from the less dense medium (air)towards the more dense medium (ferrofluid), the requirementfor onset of usual Rayleigh–Taylor instability [5]. Peak adverse
acceleration tends to occur at the moment when the ferrofluid
is maximally displaced and begins to reverse its velocity. The
unstable ferrofluid interface may subsequently pinch off into
droplets as seen in experiments [6].

(9)The analysis can define conditions for the initial growthrate of surface perturbations. The dispersion relationshipfor dependence of the growth factor γ on perturbation
wavenumber k is formulated aswhere Re denotes real part, ẑ 0 is the amplitude of theinterfacial perturbation, t time, s distance along the mean
gap perimeter, σ interfacial tension, a acceleration, L layer
thickness, μ coefficient of viscosity, ρ mass density, k0 =(ρa/σ )1/2 the Taylor wavenumber, σ the interfacial tension,
N = (M/ρa) d H /dz ≈ M Hm /ρaw the ratio of magnetic
to accelerative body force, and Hm the maximum field in the
gap of width w. The viscous term of equation (8) is thatof Mikaelian [7]4 , while the gradient field term of N wasproposed by Zelazo and Melcher [8]. A positive value of γ
represents instability, i.e., deviations from a flat interface growwith time.



ak
k 2μ
k2
γ−
γ +2
1− 2 −N =0ρcoth(k L)
k0

4 Note that the second line above equation (24) omits the factor 2 of theterm X 2 Y ; the corrected equation (24) results in much closer agreement of
the approximate result with the analytic result in figure 2.

5. Rayleigh–Taylor instability analysis

2

(8)
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J. Phys.: Condens. Matter 20 (2008) 204147

R E Rosensweig et al

Figure 4. Splash patterns illustrating aspects of the Rayleigh–Taylor instability. (a) Peak acceleration 640 g ; time on shelf: four days at fill
fraction 1.2; wave crests are evident. (b) Peak acceleration 704 g ; tested at time of filling with fill fraction 1.0; only tiny drops are seen.

Ferrofluid splash is collected on a flat sheet of paper
located 6.5 mm below the test fixture at impact; see figure 4.
When splash occurs, the ferrofluid emanates from all aroundthe gap, at times with a characteristic spacing between radial
wave crests; see figure 4(a). This test was made four days after
the ferrofluid was injected. By definition, the wavelength λ of
the Rayleigh–Taylor instability is related to the wavenumberby λ = 2π/k . The wavelength of maximum growth for thereference case is 0.32 cm, corresponding to k = kpeak =19.5 radian cm−1 . The splash pattern has ∼48 peaks aroundthe periphery of radius R0 = 12.25 cm, yielding λ ∼ 0.16 cm.
Thus, the predicted and observed wavelengths are in the same
league. In figure 4(b) it appears that the initial instability
producing wave crests as in figure 4(a) subsequently underwent
secondary instability resulting in the production of a spray.

Reference physical data corresponding to ferrofluid APG810
Viscosity coefficient: 100 m Pa s (100 cP)
Mass density: 940 kg m−3 (0.940 g cm−3 )Surface tension: 0.032 N m−1 (32 dyn cm−1 )Saturation magnetization: 0.011 T (ferric induction, B − H =110 G).
Reference system parameters
Drop height: 0.50 m (50 cm)
Magnetic field in gap: 477 500 A m−1 (6000 Oe)
Gap width: 0.0016 m (0.16 cm)
Gap length: 0.0022 m (0.22 cm)
Gap outer circumference: 0.099 m (9.9 cm)
Fill fraction: 1.
Figure 2 on the right illustrates computed locations offerrofluid boundaries with f = 1. It can be seen that end
effects are significant for the short wetted length of the voicecoil and hence may affect the accuracy of the predictions.
Figures 3(a) and (b) illustrate the influence of viscosityand gap width on the stability of the ferrofluid interface.
The vertical axis denotes the value of the growth factor γ ,and the horizontal axis denotes the value of the perturbation
wavenumber. The crossover value of stability corresponds toγ = 0, with k0 the corresponding wavenumber. In the unstable
range of wavenumbers (k < k0 ) a peak value of γ corresponds
to the wavenumber km of fastest growth.
In figure 3(a) it can be seen that although k0 is invariant
with viscosity, increasing the viscosity decreases the growthrate. This is beneficial as a sufficiently small growth rateover a limited amount of time can prevent a splash loss ofthe ferrofluid. From figure 3(b), it is seen that a modest
reduction of the gap width to w = 0.11 cm achieves incipientstability over the total range of wavenumbers. This results fromgenerating a larger magnetic field gradient at the interface.

7. Conclusion
A method for achieving enhanced voice coil centering
forces was validated experimentally and theoretically withcomparisons made of filling both gaps, filling only the innergap, and filling only the outer gap. It was also concluded
that Rayleigh–Taylor instability in the gradient magnetic fieldis the mechanism of ferrofluid splash loss when it occurs.
Although the experimental spacing of the ejected droplets
can at times be predicted from the theory, the relationshipfor onset of splashing to drop height is elusive, as themagnitude of initial perturbations on the interface is unknown.
However, the instability theory provides clear directional
guidance for the alleviation or prevention of splashing under
given circumstances. It can be said that the onset of instability
predicted by the theory provides a lower bound on maximum
safe drop height.

6. Experimental splash data
The trends indicated in figures 3(a) and (b) are confirmed by
the data of figures 3(c) and (d), respectively. These figures plot
experimental values of the drop height for the onset of splash.
In figure 3(b) the acceleration is assumed constant throughout
at a = 500 cm s−2 .

Acknowledgment
The authors thank Professor Seiichi Sudo of Akito Prefectural
University for making the high speed camera videos.
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R E Rosensweig et al

References[4] Tsuda S and Rosensweig R E, Ferrofluid centered voice coilspeaker, patent applied[5] Taylor G I 1950 The instability of liquid surfaces when
accelerated in a direction perpendicular to their planes Proc.
R. Soc. A 201 192–6[6] See images at http://www.ame.arizona.edu/research/fluidlab//rayleigh.html[7] Mikaelian K O 1996 Rayleigh–Taylor instability infinite-thickness fluids with viscosity and surface tension
Phys. Rev. E 54 3676
[8] Zelazo R E and Melcher J R 1969 Dynamics and stability offerrofluids: surface interactions J. Fluid Mech. 39 1

[1] Raj K 1996 Magnetic fluids and devices: a commercial survey
Magnetic Fluids and Applications Handbook
ed B Berkovsky and V Bashtovoi (New York: Begell House)
chapter 5, pp 657–751[2] Bottenberg W, Melillo L and Raj K 1980 The dependence of
loudspeaker design parameters on the properties of magnetic
fluids J. Audio Eng. Soc. 28 17[3] Rosensweig R E 1985 Ferrohydrodynamics (New York:
Cambridge University Press) (1997) (New York: Dover)(reprinted with minor corrections and updated)

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