Measuring the deformation of a ferrogel sphere in a homogeneous magnetic field

0.64 MB·English

by  Christian Gollwitzer

APS/123-QED
Measuring the deformation of a ferrogel sphere in a homogeneous magnetic field
Christian Gollwitzer,1 Alexander Turanov,1 Marina Krekhova,2
Gu¨nter Lattermann,2 Ingo Rehberg,1 and Reinhard Richter1
1Experimentalphysik V, Universit¨at Bayreuth, 95440 Bayreuth, Germany
2Makromolekulare Chemie I, Universit¨at Bayreuth, 95440 Bayreuth, Germany
(Dated: February 2, 2008)
A sphere of a ferrogel is exposed to a homogeneous magnetic field. In accordance to theoretical
predictions, it gets elongated along thefield lines. The time-dependenceof the elastic shear modu-
luscausestheelongation toincreasewithtimeanalogously tomechaniccreepexperiments,andthe
rapid excitation causes thesphereto vibrate. Both phenomenacan bewell described bya damped
8
harmonic oscillator model. By comparing the elongation along the field with the contraction per-
0
pendicular to it, we can calculate Poisson’s ratio of the gel. The magnitude of the elongation is
0
compared with thetheoretical predictions for elastic spheresin homogeneous fields.
2

n PACSnumbers: 41.20.Gz, 47.65.Cb,46.35.+z
a Keywords: Ferrogel,softmatter
J

8
I. INTRODUCTION Helmholtz Coils

]
t PC
f A sphere is the mostperfect, most symmetric geomet- Hall Probe
o
s ricalobject. ThePythagoreansbelievedthatthesunand Power Supply
. the earth are perfect and thus spherical. Of course the
t
a rotationoftheEarthisbreakingthesphericalsymmetry.
m
WhetherthischangestheshapeoftheEarthtoaprolate FIG. 1: Schemeof theexperimental setup.
- or oblate rotational ellipsoid has been the subject of a
d
longlastingquarrelbetweenNewtonandCassini[1]. The
n
breakingofthesphericalsymmetryisfoundtobeimpor- magnetization. Recently,theelongationhasbeenrecom-
o
c tantinvariousfields ofphysics. The efficiencyofnuclear puted by Raikher [17] without constraining the shape to
[ fission depends on the shape of the nucleus [2]. Partic- an ellipsoid. In this case, the elongation is expected to

ularly convenient is a breaking of the symmetry by an be 30% larger. This effect has not yet been observed,
1
external electric or magnetic field. A drop of a magnetic a possible reason being that it is rather small for the
v
4 liquid, a colloidal dispersion of magnetic nanoparticles large values of G, characteristic for most of the cova-
4 [3], elongates along the direction of the applied homoge- lently cross linked polymer gels [9, 19]. In contrast, the
2 neous magnetic field [4, 5, 6]. Very recently an amount elasticity of the new class of thermoreversible ferrogels
1 of “quantum ferrofluid”,consisting of dipolar Cr-Atoms, [20] can reversibly be tuned via their temperature. In
.
1 hasbeenfoundaswelltoelongateinamagneticfield[7]. the present paper, we cast thermoreversible ferrogels in
0 All examples presented above have in common that sphericalsamplesandexposethemtoauniformmagnetic
8 they deal with fluid bodies, without any elasticity. In field, to test the above predictions.
0
contrast smart nanocomposite polymer gels [8] are kept
:
v together by their elastic polymer matrix. Especially fer-
i rogels[9,10]areapromisingclassformanyapplications,
X II. EXPERIMENT
like soft actuators, magnetic valves, magnetoelastic mo-
r
bile robots [11, 12], artificial muscles [13], or magnetic
a
A. Setup
controlled drug delivery [14]. All these applications are
basedonthe magneticdeformationeffect, acouplingbe-
tween the mechanic and magnetic degrees of freedom. OurexperimentalsetupissketchedinFig.1. Aferrogel
The magnetic deformation effect is studied in its sim- ballisimmersedinarectangularcontainer,positionedon
plest form for a spherical gel body subjected to a homo- the common axis midway between two Helmholtz coils.
geneousmagneticfield. Anapproximationfortheresult- For the empty Helmholtz pair of coils, the spatialhomo-
ing deformation has been given in 1960 by Landau for geneity is better than ±1%. This grade is valid within
the case of a dielectric elastic sphere [15] and can read- a cylinder of 1cm in diameter and 14cm in height ori-
ily be transferred to ferrogels [16, 17, 18]. The relative ented symmetrically around the center of the coils. The
elongation ε in this case was calculated as coils are powered by a current amplifier (fug electronic
GmbH), which is controlled by a computer. The mag-
κµ0 2
ε= M (1) netic system cannot follow the control signal immedi-
G
ately. For a maximal jump height ∆B = 36mT the
where κ = 1/15, G is the shear modulus and M is the field is reachedafter tB =30ms, as recordedby the Hall
2
10
H
) 9
m
/ 8
A
k 7
(
M 6
n 5
o
ati 4
z
ti 3
e
gn 2
a
M 1
0
0 5 10 15 20 25
InternalmagneticfieldH(kA/m)
FIG. 2: Image of the symmetrical (left) and distorted
ball (right). The dotted line displays a fit with a circle (left)
FIG. 3: Magnetization curveof theball.
and an ellipse (right).
C. Sample preparation
probe(Group3-LPT-231)connectedtoadigitalteslame-
ter (DTM 141).
The temporal evolution of the ball shape is recorded Above its softening temperature at 45 C the mate-
°
using a highspeed camera, capable of taking 400 frames rial becomes a magnetic liquid. We produce the ferrogel
per second with a resolution of 768×768 pixels. Fig- sphere by casting the liquid into an aluminum mould at
ure2showstheoriginalanddistortedshape. Thedotted 55 C. The mould consists of two parts with a spherical
°
curvesstemfromafitofacircletotheedgesintheimage. cavity, tightly screwed together, which is connected by
The circle is located utilizing a normalized correlation a thin channel to a hopper mounted on top of the up-
technique [21], where the correlation between the gradi- per part. To avoid any air bubbles in the final sphere,
entofthe imageanda circleis maximized. The gradient the mould is first filled with the liquefied material un-
is estimated by the Sobel operator [22], and the circle is der vacuum (≈ 1mPa). Then atmospheric pressure is
rasterized with anti-aliasing provided by a Gaussian fil- applied, which compresses any low-pressure air bubbles.
ter [23]. This method is able to extract the radius and Byrepeatingtheprocessofvaryingtheambientpressure
the coordinates of the center with subpixel resolution. andunderthe influence ofgravity,eventuallyallbubbles
leave the mould via the hopper. Next we cool down the
mould and separate the two parts of it.
The ball is then immersed into water with a temper-
ature of 24 C, where all experiments have been carried
B. Material
°
out. The water contains an amount of salt, so that the
density of the liquid is only slightly less than that of the
The ball was prepared with a magnetic, thermo- ball. Thisisnecessarytoreducetheinfluenceofthegrav-
plastic elastomer gel or in other words with a ther- ity on the shape of the sample; the ball is so soft that,
moreversible ferrogel [20]. Therefore, an ABA-type without buoyancy,it is deformed into an oblate ellipsoid
poly(styrene-b-(ethylene-co-butylene)-b-styrene)(SEBS) under gravity. As an additional benefit we can easily
triblock copolymer (Kraton G-1650) was used as a gela- measure the density of the gel by examining the density
tor. As measured with size exclusion chromatography of the fluid. The density determined by this method is
(SEC), Kraton G-1650 exhibits a molar mass of 99000. ρ=1.085±0.005g/cm3.
The styrene content is 29%(manufacturer information).
Thegelatorconcentrationis3.5w%perparaffinoilused.
Theferrofluidwaspreparedintheclassicalway[20]with
24.5w% magnetite per paraffin oil Finavestan A 50B
(Total). The shear viscosity at 20 C of 11mPas and
D. Magnetization
°
the molar mass of 280 (manufacturer information) are
lower than that of the earlier used paraffin oil Finaves-
tanA80B [20]. Using A 80B,the resultingferrogelsare Next, we measure the magnetization of the sphere
toostifftoberemarkablydeformedbythemagneticfield. for various fields, utilizing a fluxmetric magnetometer
Ontheotherhand,usingthegelatorKratonG-1652with (Lakeshore, Model 480). The resulting magnetization
a lower molar mass of 79000 (SEC), the prepared ferro- curve is plotted in Fig. 3. The solid line displays an ap-
gelsarestillsofter,i.e.theyareevenlessshape-retaining. proximation with the model presented in Ref. [24]. The
Furthermore, they sweat out ferrofluid, slowly on stand- sample is superparamagnetic with an initial susceptibil-
ing, faster in the magnetic field. ity χ0 =0.81.
3
1600 1.5
)
Pa 1200 1
( )
%
G
(
s ε 0.5
u n
ul 800 o
d ti
mo ga 0
n
ar 400 Elo
he -0.5
S
0 -1
0.001 0.01 0.1 1 10 100 1000
(a)
Elapsedtime (s) z)
H 22
(
f
FIG. 4: The force response to a jump in the deformation. y 21
c
The solid line displays a fit to Eq. (2) with the parameters n
e
Grheo =1480Pa, t0=127 s and β=0.218. qu 20
e
Fr
10 20 30 40 50 60 70 80 90
2.5
MagnetizationM2(kA/m)
(b)
2
)
% FIG. 6: a) The static elongation as a function of M2. The
(
ε 1.5 circles (triangles) denote the elongation (contraction) par-
n
o allel (perpendicular) to the applied field, respectively. The
ati 1 solid(dashed)lineisthebestlinearfit. b)Thefrequenciesof
g
n theinitial vibrations obtained from Eq. (3).
o
El
0.5
III. RESULTS AND DISCUSSION
0
0.001 0.01 0.1 1
Time(s) We performed time-resolved measurements of the rel-
ative elongation ε of the ball after applying a magnetic
FIG. 5: Elastic response of theball, when a magnetic field is induction B in a jumplike fashion for ten different val-
suddenly applied. The solid line represents a fit to Eq.(3). ues of B. Figure 5 shows ε(t) for the case B = 36mT.
The elongation ε = (d−d0)/d0 measures the scaled dif-
ference of the diameters in the direction of the field (d)
and without a field (d0). Due to the sudden increase of
E. Rheology
the magnetic field and the inertia of the ball, the latter
performs uniaxial damped vibrations with a frequency
For the characterization of the elastic properties we f =22.4Hz. When the oscillations cease, the elongation
measure the shear modulus G. with a rheometer continues to grow due to the softening under load. Be-
(MCR301,AntonPaar)inconeandplategeometry. The cause the timescale of the softening is much larger than
conehasadiameterof50mmandabaseangleof1 . We that of the oscillation, the experiment may be approxi-
°
perform a stress relaxation experiment on the sample: matedbyaharmonicoscillatorpulledbyaconstantforce,
from the equilibrium position we shear the sample by where the spring constant relaxes according to Eq. (2)
a deformation of γ = 1% and measure the stress τ as
a function of time. Figure 4 displays the results. The ε¨+δε˙+ω02exp −(t/t0)β ε=F. (3)
restoring force decays by 50% during one second, which
Here δ is the damping const(cid:0)ant, F th(cid:1)e pulling force (i.e.
meansthattheshearmodulusG=τ/γcannotbetreated
as constant. This decay can well be approximated by a relatedtothemagneticinduction)andω0thenaturalfre-
quency. Afit with thatequationis displayedin Fig.5 as
stretched exponential (the solid line in Fig. 4)
thesolidline. TheratioF/ω2 correspondstotherelative
0
elongationinthe equilibriumstate if the springconstant
G(t)=Grheoexp −(t/t0)β . (2) would not change. Therefore, this ratio compares to the
elongation predicted by the static theories.
(cid:0) (cid:1)
Thematerialthereforesoftensunderload. Toaccountfor The dependence of the elongation on the magnetiza-
that time dependence, the magnetic experiment cannot tion is shown in Fig. 6a. For ten different values of the
be performed in a static manner. magnetic induction we have recorded and evaluated the
4
elongationε of the ball. Figure 6a presents the outcome 1600
for the elongation parallel (ε ) and perpendicular (ε )
z x 1400
to the magnetic field. The data have been plotted ver- a)
sus M2. In agreement with Eq. (1) we find a linear re- (P 1200
lationship ε = c M2 with c = 13.4 · 10−5(m)2 and G 1000
cx = −6.1·i10−5(ikmA)2. Figurze 6b shows the oksAcillation dulus 800
frequency f =ω0/(2π) versus M2. mo 600
Nextwecomparetheratiooftheslopescz/cx =κz/κx ar 400
with the theoretical predictions. For a deformation re- he
S 200
stricted to an ellipsoidal shape and the assumption of a
uniform strain field, the expressionsgiven in [15, 17] can 0
5 10 15 20 25 30 35 40
be rewritten in terms of Poisson’s ratio σ
MagneticinductionB(mT)
3−2σ
κ = (4a)
z
20σ+20 FIG. 7: The shear modulus measured by different methods:
1−4σ Commercialrheometer(dashed),elongationtheory[17](short
κ = . (4b)
x 20σ+20 dashed),elongationtheory[15](dash-dotted),vibrations(solid
circles).
1
Foranincompressiblegel(σ =1/2)thisleadstoκ = .
z 15
For arbitrary σ one obtains for the ratio
1−4σ 1.2
κ /κ = . (5)
x z 3−2σ
1
We exploit Eq. (5) to determine σ, and by substituting γ
n 0.8
σ in Eqs. (4b,1) one finally arrives at G. This yields o
ti
σ =0.48±0.01 G=0.65kPa. (6) ma 0.6
r
o
Forthe moregeneralcaseofanon-uniformstrainfield Def 0.4
andashapenotrestrictedtoanellipsoid[17],oneobtains
0.2
6σ2+σ−7
κ =− (7a) 0
z 20σ2+48σ+28 -1000 0 1000200030004000500060007000
σ2+2σ Elapsedtime (s)
κ =− (7b)
x 10σ2+24σ+14
FIG. 8: The shear deformation as a function of time, after
which yields applyinga constant deformation of 1.2% for 800s.
2σ2+4σ
κ /κ = (8)
x z 6σ2+σ−7
Since the radius of the sphere r and the density ρ are
and finally gives
known,wecancomputetheshearmodulusfromthemea-
σ =0.47±0.01 G=0.87kPa. (9) sured vibration frequency f. Figure 7 shows a compar-
ison of the value of the shear modulus obtained from f
Both approaches yield a Poisson’s ratio σ very close with the values from the elongation and the mechanical
to the limit of incompressibility σ = 1/2, which is char- measurement. Theaverageshearmodulusdeterminedby
acteristic for rubberlike materials [25]. As reported be- this method is Gvib = 0.1kPa, which differs by a factor
fore[17],thevaluesderivedfortheshearmodulusGdiffer of15fromGrheo. Thislargedeviationmayarise,because
by ≈ 30%. But the value obtained from the rheometer, the model for the vibrations does not take into account
Grheo = 1.48kPa, exceeds the old and new predictions the surrounding water. This needs to oscillate together
by a factor of 2.3 and 1.7, respectively. So none of them withthe sphere,leadingtoanincreasedeffectivemassof
is corroboratedby the experiment. the oscillator, and thus a reduced frequency.
The frequency of the vibrations after the sudden in- Now we reconsider the deviation between the shear
crease of the magnetic field offers another possibility to modulusGasdeterminedfromtheelongationsandGrheo
measure the shear modulus. For an incompressible elas- determined via rheology. They differ by factor of about
ticspherethatperformsspheroidalvibrations,wherethe two. This difference cannot be explained solely by the
spheregetsalternatelydeformedintoaprolateandoblate inaccuracy of the measurement of Grheo, which is typi-
ellipsoid of revolution, the frequency is given [26] by cally around 10% for a commercial rheometer. Rather,
the deviationis likely to stem fromthe factthat the ide-
G alistic models [15, 17] consider only Hookean elasticity,
f =0.848 . (10)
s4ρr2 which is insufficient for our material. In fact, the defor-
5
mation of a gel put under load can be made up of three gation parallel and perpendicular to the field, we calcu-
contributions,namelyinstantaneouselasticdeformation, late Poisson’s ratio, which is close to the value σ = 0.5
retardedanelasticdeformationandviscousflow[27]. Our expected for an incompressible material. More impor-
material clearly shows anelasticity, as demonstrated in tantly,the absolutevalue ofthe elongationis 70%larger
Fig. 4. Moreover, our sample shows the phenomenon of than the one calculated from the models. This is pre-
viscous flow, as illustrated in Fig. 8. In this measure- sumable caused by the neglection of the anelasticity and
ment we first apply a constant strain for 800s, and then viscous flow of the ferrogel. So further theoretical inves-
record the relaxation of the strain for zero stress. The tigation, including the full viscoelastic properties of the
remaining deformationat t=7000sis about 15%of the gel, is needed to explain the experiment quantitatively.
straininitially applied. This valuecanbe regardedasan With such a model one would also be able to compute
upper boundforthe plastic contributiontothe deforma- the dynamic response of the gel under a sudden change
tion (viscous flow). While the existence of anelasticity of the external magnetic field. This is not only of funda-
and viscous flow gives no straight-forward explanation mentalinterest,butalsoimportantforpossibletechnical
of the 70% deviation between experiment and theory, it applications of these smart materials.
is obvious that the full viscoelastic behaviour of the gel
should be included in the computations from the very
beginning. V. ACKNOWLEDGMENTS
The authors thank Helmut R. Brand and Yurij
IV. CONCLUSION L. Raikher for fruitful discussions. Financial support
by the Deutsche Forschungsgemeinschaftvia FOR608is
Wehavemeasuredthedeformationofaferrogelsphere gratefully acknowledged. A.T. is grateful for an INTAS
in response to a uniform magnetic field by direct optical YoungScientistsFellowship(05-109-4521). WethankTo-
means. We compare the results for the first time with tal (Finavestan A 50B) and Kraton Polymers (Kraton
the models in Refs. [15, 17]. From the ratio of the elon- G-1650) for providing samples.
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