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Electric field-induced deformation of polydimethylsiloxane polymers
T. Ioppolo, J. Stubblefield, and M. V. Ötügen
Citation: Journal of Applied Physics 112, 044906 (2012); doi: 10.1063/1.4747832
View online: http://dx.doi.org/10.1063/1.4747832
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JOURNAL OF APPLIED PHYSICS 112, 044906 (2012)
Electric field-induced deformation of polydimethylsiloxane polymers
€ u
€gen
T. Ioppolo,a) J. Stubblefield, and M. V. Ot
Southern Methodist University, Dallas, Texas 75205, USA
(Received 31 May 2012; accepted 6 July 2012; published online 30 August 2012)
The deformation of polydimethylsiloxane (PDMS) spheres under uniform external electric field
was studied experimentally and analytically. In the experiments, 1 mm diameter PDMS spheres
with base-to-curing-agent mixing ratios of 10:1 and 60:1 were exposed to uniform external electric
field with varying magnitudes and poling durations. The spheres elongate along the electric field
direction. For a given electric field strength, the sphere deformation is initially a time function but
reaches a terminal strain value over a certain time period. This terminal strain value is larger for
stronger external electric fields and larger PDMS mixing ratio spheres. At this state, the material is
no longer poled and the surface charge distribution remains constant. In the analysis, an expression
for the sphere deformation is obtained by modeling the PDMS as a linear elastic solid and solving
the Navier equation along with Maxwell’s equations for boundary conditions. The analysis takes
C 2012
into account the surface charge distribution and predicts well the experimental trends. V
American Institute of Physics. [http://dx.doi.org/10.1063/1.4747832]
I. INTRODUCTION
In recent years, there have been several studies investigating the deformation of solids under the effect of an external electric field. Materials that deform under external
electric field have potential for applications in actuation and
sensing.1–3 For example, polymeric materials with embedded
nano- or micro-dielectric particles, commonly referred to as
electrorheological (ER) elastomers, have been investigated
due to the wide range of potential applications.4–7 In the
presence of an electric field, the embedded particles become
polarized exerting force upon one another due to dipolar
interactions. These forces lead to changes in the mechanical
and electric properties of the ER elastomer. Dielectric materials that are not doped with embedded particles can also experience deformation under an external electric field.
However, the amount of deformation is typically smaller
compared to the ER elastomers. The deformation of a medium under the effect of an external electric field is commonly referred to as “electrostriction.” A medium exposed
to an external electric field experiences electrostriction body
forces due to gradients in both the dielectric permittivity and
the electric field magnitude. Also, a discontinuity in the
dielectric permittivity at the material interface results in surface forces at the boundary of the medium. Therefore, even
for solid object with uniform dielectric permeability will
deform under electric field due to this surface electrostatic
pressure.8,9
In Ref. 9, we demonstrated that the electrostriction can
be used as a means to tune polymeric optical resonators. In
the study, we observed that under a fixed external electric
field the elastic deformation of polydimethylsiloxane
(PDMS) 60:1 (60 parts base silicon elastomer to 1 part polymer curing agent by volume) spheres increases when
exposed to an electric field over a period of time, with longer
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-8979/2012/112(4)/044906/7/$30.00
durations under electric field leading to larger deformations.
We attributed this effect to the poling of the material over
time under the influence of the external field.9
In the present, we carry out an analysis and systematic
experiments to study the effect of poling duration on the
extent of the electrostriction effect on dielectric spheres. In
the analysis, we model the sphere as a linear elastic solid and
obtain an expression for the deformation of the sphere
induced by the external electric field by solving the Navier
equation in closed form with the appropriate boundary conditions. In the experiments, we use polymeric spheres made
of PDMS with (10:1) and (60:1) mixture ratios. The deformation of the sphere is monitored by tracking the shifts of
the whispering gallery optical modes (WGM). The optical
modes of the sphere are excited by coupling light from a tunable diode laser using a single mode optical fiber.10
II. EXPERIMENTAL ARRANGEMENT
A. Optical measurement technique
The measurement technique is based on the
morphology-induced shift of WGM of dielectric spheres. In
recent years, several applications of the whispering gallery
modes have been proposed. Some of these include those in
spectroscopy11 and micro-cavity laser technology.12 Sensors
exploiting the WGM shifts of microspheres have also been
proposed for biological applications,13 trace gas detection,14
impurity detection in liquids15 as well as mechanical sensing
including force,10 pressure,16 temperature,17 and wall shear
stress for aerodynamic applications.18 In the present application, the electric field imposes a force on the dielectric
microsphere (commonly referred to as electrostriction effect)
leading to the deformation of the sphere. The magnitude of
the deformation is measured by tracking the WGM shifts of
the sphere.
Geometric optics affords a simple but useful description
of the WGM phenomenon. This description is valid when
112, 044906-1
C 2012 American Institute of Physics
V
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J. Appl. Phys. 112, 044906 (2012)
the wavelength of the light used to excite the optical modes
is much smaller than the size of the optical cavity. In this
geometric view, light coupled at a grazing angle into a
microsphere whose refractive index is larger than that of the
surrounding medium circles the interior surface of the sphere
through total internal reflection (as shown in Fig. 1) and
returns in phase. The condition for optical resonance is
2p Rn ¼ lk, where k is the vacuum wavelength of the light
(supplied by a laser), l is an integer, R is the sphere radius,
and n is the sphere’s refractive index. When the sphere is
subjected to the external electric field, its morphology
changes (elastic deformation) due to the electrostriction
effect. This in turn causes a perturbation of both the radius
(DR) and refractive index (Dn) leading to a shift in the optical resonance (WGM) as follows:
DR Dn Dk
þ
¼
:
R
n
k
(1)
Therefore, any change in the index of refraction and the
radius of the microsphere induced by the external effect electric field can be measured by monitoring the shift in the resonance (WGM) of the microsphere.
FIG. 2. Experimental setup.
B. Experimental setup
The opto-electronic setup is similar to that reported in Ref.
10 in which a WGM-based force sensor was demonstrated. A
schematic of the overall setup is given in Fig. 2(a). Briefly, the
output of a distributed feedback (DFB) laser, whose nominal
wavelength is 1312 nm, is coupled into a single mode optical
fiber. A section of the optical fiber is tapered to facilitate light
coupling between the microsphere and the fiber. Two brass
plates (1 cm 1 cm 0.1 cm) connected to a DC power supply
provide the external electric field as shown in Fig. 2(b).
III. EXPERIMENTAL RESULTS
A. Time evolution of sphere deformation
Typical transmission spectra and the WGM shift due to
external electric field are shown in Fig. 3 for a 60:1 PDMS
FIG. 1. Microsphere coupled with optical fiber.
sphere of 900 lm diameter. When the electric field of 50 kV/m
is turned on, the WGM optical mode, seen as a dip in the
transmission spectrum, experiences a blue-shift of Dk
1.9 pm indicating that the sphere is elongated along the field
direction. Note that the optical path length inside the sphere is
on the equatorial plane normal to the electric field (Fig. 2(b)).
The WGM shift is related to relative deformation, DR/R,
through Eq. (1).
Figure 4 shows the evolution of the electric fieldinduced deformation, DR/R, of a 10:1 PDMS microsphere
with radius R ¼ 1 mm over a period of 2 h. The measurements are repeated for three different external electric field
strengths, E0. The elastic deformation increases with increasing external electric field strength. The figure also confirms
our earlier observation in Ref. 9 that for a fixed electric field
strength, the sphere deformation increases with time, but
reaches an asymptotic value after a certain amount of time
(1 h). The corresponding initial (short time) response of the
FIG. 3. Transmission spectra trough the sphere-coupled fiber.
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FIG. 4. The effect of poling duration on sphere deformation; 10:1 PDMS
microsphere with a diameter of 1 mm.
sphere to the imposed electric field is shown in Fig. 5. The
electric field is turned on at t ¼ 0. Even during the first two
minutes, the rate of sphere deformation becomes smaller
with time. Figures 6 and 7 show the corresponding results
for a 60:1 sphere (R ¼ 700 mm). These results are similar to
those for the 10:1 PDMS sphere; the rate of deformation of
the sphere becomes smaller with increasing time reaching a
saturation value, in this case, in about six hours. It is also
observed that the relative deformation of the 60:1 PDMS
sphere is three orders of magnitude larger than that for the
10:1 PDMS sphere under a similar electric field magnitude.
A major contribution to this increase in sphere deformation
is due to the smaller elastic modulus of the 60:1 PDMS; the
modulus for 60:1 PDMS is Y 3 kPa whereas for 10:1
PDMS, Y 1 MPa.
B. Surface charge effect on microspheres deformation
In the next set of experiments, the effect of switching
the electric field direction is investigated. After exposing the
spheres to an electric field for a time period, the field is
turned off and the polarity of the electrodes is reversed to
change field direction 180 . Figure 8 shows three separate
experiments for a 10:1 PDMS sphere. In two cases, the
sphere is initially exposed to a 1 MV/m electric field (in one
J. Appl. Phys. 112, 044906 (2012)
FIG. 6. The effect of poling duration on sphere deformation; 60:1 PDMS
microsphere with a diameter of 700 lm.
experiment for tp ¼ 5 min and in the other, tp ¼ 3 min) and
then the field is turned off. After quickly switching the electrode polarity to reverse the electric field direction, the field
is again turned on and its magnitude is ramped up as shown
in the figure. The third experiment represents a base line in
which case the sphere was not initially exposed to an electric
field (tp ¼ 0). Note that total duration of these experiments is
20 s. For the base line case, the sphere is elongated in the
E-field direction as the field strength is increased. However,
when the spheres are initially subjected to the electric field
in one direction and then the polarity is reversed, they initially experience a compressive deformation that is nearly
linear with the applied electric field (up to about E0 ¼ 0.6
MV/m). As the field strength is further increased, a maximum deformation is reached around E0 ¼ 0.7 MV/m beyond
which the deformation starts decreasing with increasing field
strength. Corresponding results for a 60:1 PDMS sphere are
shown in Fig. 9. The sphere that is exposed to an initial
external field of 1 MV/m for two hours before switching the
field direction exhibits a behavior similar to that for the 10:1
PDMS. In this case, however, the peak compressive deformation is obtained at a smaller field magnitude of E0 0.25
MV/m.
FIG. 5. Short term poling effect on sphere deformation;
10:1 PDMS microsphere with a diameter of 1 mm.
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J. Appl. Phys. 112, 044906 (2012)
FIG. 9. Polarity reversal test for a 60:1 PDMS microsphere at varying poling
durations, tp.
FIG. 7. Short term poling effect on sphere deformation; 60:1 PDMS microsphere with a diameter of 700 lm.
The results shown in Figs. 8 and 9 indicate that two competing forces are acting on the spheres when the electric field
is reversed. When a dielectric microsphere is subjected to an
external electric field, over time, a net charge distribution
develops at the surface of the material causing a timedependent sphere deformation observed in Figs. 4–7. The surface charge essentially creates a piezo-electric effect and tends
to enhance the electrostriction force and further elongate the
sphere under the same electric field strength (as depicted in
Fig. 10(a)). After a certain period of time (1 h and 6 h for
the 10:1 and 60:1 PDMS spheres, respectively), a steady state
is reached for the surface charge distribution and hence, a constant DR/R value is reached. This surface charge-inducted
force is complementary to the electrostriction force discussed
earlier in Ref. 9. However, if the electric field direction is
reversed abruptly, opposite charges will repel leading to a surface charge-induced compressive piezo-electric force that is
linearly proportional to the field strength. At lower field
strengths, this compression force dominates over electrostricition (as depicted in Fig. 10(b)) which has a quadratic dependence on electric field magnitude.8,9
IV. ANALYSIS
We consider an isotropic, uniform solid dielectric sphere of
radius R and dielectric constant e1, embedded in an inviscid
fluid with dielectric constant, e2. The sphere is subjected to a
uniform external electric field, E0, in the direction of z as shown
in Fig. 1. The elastic deformation of the dielectric sphere is governed by Navier equation which, in steady state, is as follows:
uþ
r2~
~
1
f
rðr ~
u Þ þ ¼ 0:
1 2v
G
(2)
FIG. 8. Polarity reversal test for a 10:1 PDMS microsphere at varying poling durations, tp.
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J. Appl. Phys. 112, 044906 (2012)
FIG. 10. (a) and (b) The effect of electric field direction reversal on sphere morphology.
Here, ~
u is the displacement vector of a given point
within the sphere and ~
f is volumetric (body) force. G and v
are the shear modulus and Poisson ratio of the dielectric material, respectively. Electrostrictive body force induced by an
external electric field is given by8
~ is the electric field
where A is the electric potential and E
vector. In order to obtain a unique solution to these equations, the following boundary conditions have to be satisfied:
U1 ðRÞ ¼ U2 ðRÞ
(7a)
1 ~2
1
~
~2 ;
re1 ða1 þ a2 ÞrE
f ¼ E
2
4
@U2 @U1 e2
e1
þr ¼ 0:
@r r¼R
@r r¼R
(7b)
(3)
~ is the electric field within the sphere, e1 is the dielecwhere, E
tric constant, and a1 and a2 are coefficients that describe the
electro-elastic properties of the sphere material. The parameters a1 and a2 represent the dependence of dielectric constant
e1 on mechanical strain in the directions parallel and normal
to the electric field direction, respectively. For a homogeneous
and isotropic solid sphere, the first term on the right hand side
of Eq. (3) is zero. Since the electric field inside the sphere is
uniform, the second term is also zero. Therefore, the electrostrictive body force acting inside the sphere is zero.
At the sphere-fluid interface, the electric field exerts a
surface force. Per unit surface area, this force is given by8
~E
~ ~
~ ¼ ½aEð
~E
~ ~
n Þ1 ½bE2~
P
n Þ2 þ ½aEð
n 2 ½bE2~
n 1 ; (4)
where, n is the unit surface normal vector. The subscripts 1
and 2 represent the sphere and the surrounding medium sides
at the interface. The constants a and b are given as8
a2 a1
;
a¼eþ
2
e þ a2
b¼
:
2
Furthermore, for a homogeneous sphere under uniform
external electric field, the surface charge density can be
expressed in the form r ¼ r0 cos (h). Solving Eq. (6) together
with the boundary conditions (Eq. (7a)), we obtain the radial
and tangential components of the electric field inside the
sphere as
3e2 E0
r0
cos ðhÞ
Er1 ¼
2e2 þ e1 2e2 þ e1
(8)
3e2 E0
r0
sin ðhÞ:
Eh1 ¼
þ
2e2 þ e1 2e2 þ e1
The electric field at the outside surface of the sphere can
be calculated by inserting Eq. (8) back into Eq. (7b) as follows:
Er2 ¼
(5)
"
Pr ¼
~ ¼ rU ;
E
(6)
9½ðb1 b2 Þe22 þ ðða2 b2 Þe21 þ ða1 þ b2 Þe22 Þ cos2 ðhÞ
"
E0 þ
ðe1 þ 2e2 Þ2
r20 ½b1 b2 ða1 4a2 þ 3b2 Þcos2 ðhÞ
ðe1 þ 2e2 Þ2
#
:
(9)
Eh2 ¼ Eh1 :
The electric field distributions both within the sphere and in
the surrounding medium are governed by Maxwell’s equations
r2 U ¼ 0
2r0 þ 3E0 e1
cos ðhÞ
ðe1 þ 2e2 Þ
The radial component of the force (per unit area) in
Eq. (4) is then expressed as
#
"
E20 þ
#
6r0 ½ðb1 þ b2 Þe2 þ ð2a2 e1 þ a1 e2 b2 ð2e1 þ e2 ÞÞcos2 ðhÞ
ðe1 þ 2e2 Þ2
ð10Þ
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044906-6
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The first term on the right hand side of Eq. (10) represents the electrosctriciton effect and is quadratic with E0.8,9
This force works to elongate the sphere in the electric field
direction. The second term represents the piezoelectric effect
due to the surface charge and is linear with E0. For a given
charge distribution, its direction depends on the external
electrical field direction. The third term represents force due
to the surface charge distribution. The separation of charges
at the opposite sides of the sphere exerts a compressive force
even in the absence of an external electric field.
The general solution of Eq. (2) for a uniform sphere is in
the form19
P
½An ðn þ 1Þðn 2 þ 4vÞr nþ1 þ Bn nr n1 Pn ðcos #Þg
P
Srr ¼ 2G ½An ðn þ 1Þðn2 n 2 2vÞrn þ Bn nðn 1Þrn2 Pn ðcos #Þ
ur ¼
Sr# ¼ 2G
P
½An ðn2 þ 2n 1 þ 2vÞrn þ Bn ðn 1Þrn2 (11)
@Pn ðcos #Þ
:
@#
A unique solution to Eq. (11) can be obtained by satisfying the appropriate boundary conditions for the mechanical stress
components Srr and Srh at the sphere surface. If we assume that the fluid medium surrounding the sphere is inviscid, the tangential component of the surface force induced by the external electric field is Ph ¼ 0. Then, the appropriate boundary conditions
are
Srr ðRÞ ¼ Pr ; Sr# ðRÞ ¼ 0:
(12)
Following the same approach used in Ref. 9, the relative change in the sphere radius at h ¼ p/2 (equatorial plane parallel
to the electrodes in Fig. 2) is obtained as
3
2
2
2
2
2
18a
e
vð2
þ
vÞ
þ
18a
e
vð2
þ
vÞ
9b
e
ð7
þ
9v
þ
10v
Þ
þ
b
vð2
þ
vÞ
þ e22 ð7 þ 9v þ 10v2 Þ
2e
2 1
1 2
1 2
2
1
DR 4
5 E2
¼
0
R
2ðe1 þ 2e2 Þ2 Gð1 þ vÞð7 þ 5vÞ
"
#
6ð2a2 e1 þ a1 e2 Þv ð2 þ vÞ þ 3b1 e2 ð7 9v þ 10v2 Þ þ 3b2 ð4e1 vð2 þ vÞ þ e2 ð7 þ 5v þ 8v2 Þ
r0 E0
2ðe1 þ 2e2 Þ2 Gð1 þ vÞð7 þ 5vÞ
"
#
r20 ½2ða1 4a2 Þvð2 þ vÞ þ b1 ð7 vð9 þ 10vÞÞ þ b2 ð7 þ 21v þ 16v2 Þ
:
2ðe1 þ 2e2 Þ2 Gð1 þ vÞð7 þ 5vÞ
2
(13)
The three terms in Eq. (13) represent the effect of the
three distinct types of force discussed above. Figure 11
shows the relative change in sphere radius calculated from
Eq. (13) for two values of shearing modulus, G, and a surface charge density of r0 ¼ 3.3 106 C/m2. For comparison, the calculated deformation without surface charge
(r0 ¼ 0) is also shown in the figure. The values for the shearing modulus represent the upper and lower bounds of G for
10:1 PDMS reported in the literature. The Poisson ratio is
taken to be v ¼ 0.49. The values for r0 correspond to poling
electric fields of E0 ¼ 1 MV/m and E0 ¼ 0 MV/m and are
obtained from (Ref. 8)
FIG. 11. Relative deformation of a 10:1 PDMS sphere calculated from
Eq. (13).
r0 ¼ ðe1 e2 Þ
3e2 E0
:
2e2 þ e1
(14)
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For electric properties, the following values are used:
e1 ¼ 2.8e0, e2 ¼ e0 (Ref. 20), a1 ¼ (e2 þ 1.8e0), a2 ¼ (e2
e0).21 Figure 11 shows that Eq. (13) captures well the
trends obtained in the experiments (Fig. 8). Further, there is
a reasonable quantitative agreement between the calculations
and measurements for un-poled case (r0 ¼ 0).
V. CONCLUSION
The experimental and analytical results show that
PDMS spheres elongate along the direction of external electric field. The deformation is initially a time function. The
time constant for deformation is shorter for the 10:1 PDMS
(1 h) compared to the 60:1 PDMS (6 h). The mathematical model provides a good qualitative prediction of the
PDMS sphere deformation in uniform external electric field.
Poling of the material results in an electric charge distribution at the surface of the material. This charge distribution in
return results in a piezoelectric force in addition to electrostriction in the presence of external electric field. The piezoelectric force is linear with external electric field magnitude
whereas electrostriction has quadratic dependence on the
field. For small external electric field strengths, the piezoelectric effect dominates while for large values of the field,
the dominant force is electrostriction.
ACKNOWLEDGMENTS
This research is sponsored by the US Defense Advanced
Research Projects Agency under Centers in Integrated Photonics Engineering Research (CIPhER) program with Dr. J.
Scott Rodgers as project manager. The information provided
in this report does not necessarily reflect the position or the
policy of the US Government and no official endorsement
should be inferred.
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