Microsyst Technol
DOI 10.1007/s00542-015-2700-7

TECHNICAL PAPER

Analytical modeling of a silicon‑polymer electrothermal
microactuator
Huu Phu Phan1 · Minh Ngoc Nguyen1 · Ngoc Viet Nguyen1 · Duc Trinh Chu1 

Received: 13 July 2015 / Accepted: 30 September 2015
© Springer-Verlag Berlin Heidelberg 2015

Abstract  This paper illustrates both thermal and mechanical analysis methods for displacement and contact force
calculating of a novel sensing silicon-polymer microgripper when heat sources are applied by an electric current
via its actuators. Thermal analysis is used to obtain temperature profile by figuring out a heat conductions and convections model. Temperature profile is then applied into
the mechanical structure of the gripper’s actuators to form
the final equation of displacement and contact force of the
jaws. Finally, the comparison among the calculation, simulation and actual measurement concludes that materialization methods are appropriate. Achieving the final equation
of gripper’s jaws displacement and contact force is a major
step to optimize or reform this novel structure for different
sizes to meet specific applications.

1 Introduction
In recent years, microelectromechanical systems (MEMS)
have been widely applied in diverse science and engineering domains (Cheng et al. 2008). MEMS-based microgrippers provide advantages in terms of their compact size
and low cost, and hence play an important role in microassembly and micromanipulation fields for manipulating
micromechanical elements, biological cells (Cheng et al.
2008; Zhang et al. 2013). During the past two decades,
microactuators based on different actuation principles
such as shape-memory alloys, electrostatic, electrothermal,


It is known that force sensing is necessary for a delicate micromanipulation task. Nonetheless, before the
force feedback sensor is applied, the optical method has
been widely studied (Miao et al. 2004; Rembe et al. 2001).
Recently, researchers showed a great interest in the sensors
with high resolution and sensitivity. In order to enhance the
reliability and safety of the manipulation, the integrated
position and force sensors such as piezoelectric sensor,
piezoresistive sensor and capacitive sensor were designed

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to provide the real-time position and force information
(Chu Duc et al. 2006; Menciassi et al. 2003; Chronis and
Lee 2005). Thanks to the advances in the technologies, the
sensitivity and resolution of the sensor have been improved
substantially.
A novel design of polymer-silicon electrothermal integrating force sensor microgripper is presented and characterized (Chu Duc et al. 2007b, c; 2008). The device consists
of laterally stacked structures based on a three-element
composite: the metal heating layer, heating conducting silicon structures and a polymer. The heat is highly efficient
transferred from the heater to the polymer by employing
the high heat conduction rate of the deep silicon serpentine
structures that have a large interface with the surrounding
polymer. The proposed device is based on the SU8-2002
polymer with a large thermal expansion coefficient. This
design overcomes the weakness of the other designs and it
boats a large lateral jaw movement with low coupled vertical motion and fast response time. Another advantage is
that the device is made of regular silicon wafers which are

from assumed heat inputs to movements, while the gripper works base on a voltage source. Therefore, completely
model (with electrical input parameters) and careful analysis are needed to improve the accuracy of the simulated and
calculated values and physical properties of the gripper.
The heat transfer and mechanical calculation of the
microgripper basing on thermal, mechanical and thermal–
mechanical combination analysis are presented in this
paper. Firstly, the operation principle of the sensing microactuator based on silicon-polymer electrothermal actuator
and piezoresistive force sensing cantilever is thoroughly
understood using thermal and mechanical analysis. Following these steps, calculation results are compared with 3-D
simulation and the fabricated sample characterized parameters for verification of gripper’s mathematical equations.
Finally, a method for structure optimization is proposed
basing on combination of changing equations’ factors and
the simulation.

2 Design and operation of silicon‑polymer
electrothermal microactuator
The microgripper is designed for the normal opened operating mode with two actuators on opposite sides. Each actuator has a silicon comb finger structure with the aluminum
metal heater on top (Chu Duc et al. 2007d). A thin layer
of silicon nitride is employed as the electrical isolation
between the aluminum structure and the silicon substrate.
Each actuator consists of silicon comb fingers with SU8
polymer layers in between. When the heater is activated,
the generated heat is efficiently transferred to the surrounding polymer through the deep silicon comb finger structure
that has a large interface area with the polymer layer. The
polymer layers expand along lateral direction which leads
to bending displacement of the actuator arms.
The design of the actuator is shown on Figs. 1 and 2,
which is the right arm of the sensing microgripper system. Ideally, both arms of the gripper are similar geometry
and characteristic. Therefore, calculations and simulations
of the gripper are took place on one arm. The structure is

Fig. 3  SEMS pictures of a fully sensing electrothermal silicon-polymer microactuator; b removed silicon cantilever configuration

HAl

L comp

HSU8

Wbone
Wgap

Wcan

L jaw
Silicon

SU- 8 polymer

(see Fig. 3). The configuration without silicon cantilever
removes the heat conduction through the sensing cantilever
for analyzing the mechanism operation of the electrothermal actuator (Fig. 3b). The actuator displacement is then
calculated by using a traditional mechanical method.

Aluminum

3.1 Thermal analysis
Fig. 2  Front-side view of the silicon-polymer electrothermal microactuator with geometry symbols and parameters

beam. Besides that, the contact force between the microactuator jaws and clamped object is then determined, relying
on displacement and stiffness of microactuator arms (Chu

Temperature can be assumed to be uniform throughout the
thickness because it is very thin; therefore the actuator is
regarded as a one-dimensional case. Eventually, calculations and analysis are conducted in x-axis while y-axis is
ignored.
In the steady state, the heat is stored in volume unit
between x and x + ∆x given by (Stephen 2001):
x + ∆x

QG =

qG .y.dx

(1)

x

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Microsyst Technol

The heat loss from left and right side of polymer-silicon
stack is given by:

∂T (x + ∆x) ∂T (x)
−
∂x
∂x

t

(5)

This is the quadratic differential equation which has the
root given by:

T (x) = C1 .e

2α
tx

−

+ C2 .e

2α
tx

+ C3

(6)

=0
Applying boundary conditions: T(0) = T0; dT (x=L)
dx
The coefficients C1, C2 and C3 are given by:
2α
tL


2α
tL

+e

−

2α
tL

qG
t

(8)

(9)

For the fabricated microgripper, the resistor of aluminum layer is about 149.018 Ω. When it is applied a voltage of 4 V, the heat power is calculated about 0.107 W.
Thus, qG ∼
= 6.941e6 W/m2 (it is the power dissipation over
the aluminum filament area).
The value of α is in the range from 2 to 25 W/m2K
(Howell and Robert Siegel), thus, the highest value of
2αTair is 15 × 103 W/m2. Comparing to the value of
qG = (6.941) × 106, the convection is neglected, therefore:
′′

T (x) = −

qG

Actuator/cantilever length

L

390

Actuator/cantilever thickness

T

30

µm

Silicon finger width

HSi

6

µm

SU-8 layer width

HSU8

3

µm


Cantilever width

Wcan

12

µm

µm

Microactuator jaw length

Ljaw

100

µm

Aluminum thickness

TAl

0.6

µm

The heat capacity of actuator

C


J

distribution rises dramatically along the actuator in form of
half parabola. The maximum temperature of actuator peaks
nearly 270 °C at the tip when 4 V between two terminals of
the aluminum heater is applied.
3.2 Mechanical analysis
Considering that the microgripper is a bimorph cantilever
that consists of two different materials: the silicon-polymer
stack layer and silicon layer. It can be supposed as single
material bars because these parts are calculated to obtain
apparent parameters. Therefore, this simplified model is


Microsyst Technol

(a)

(b)

Fig. 7  Cross-side and front-side view of the silicon-polymer electrothermal microactuator for thermal analysis

Fig. 5  Calculated temperature distribution on the microgripper

The displacement d of the bimorph cantilever is:

d=

2
kcur Lact


probably appropriate for the structure. When the bimorph
cantilever is heated, causing the different expansion of two
materials, the cantilever is bent as shown in Fig. 6 (Chu
Duc et al. 2007c).
It is assumed that the average temperature increases ∆T,
and the bending displacement of microgripper is d. Thus,
the curvature of cantilever can be calculated as follows:

1
Tx =
x

(−

qG 2 qG L
qG 2 q G L
x +
x + T0 )dx = −
x +
x + T0
2 t
t
6 t
2 t

0

(15)
The average temperature increase ∆T:

stack; n = EEstack
, m= W
Wc ; ESi is the Young’s modulus of
silicon; Estack is the Young’s modulus of silicon-polymer
stack.

kcur =
=

1
ρ
6(αstack − αSi )(1 + m)2
(Wc + Wb )(3(1 + m)2 + (1 + mn) m2 +
−

qG 2 qG L
x +
x
6 t
2 t

1
mn

)

(17)

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Inserting the Eq. (23) into (24), we obtain:
2
6(αstack − αSi )(1 + m)2 Lact

(Wc + Wb )(3(1 + m)2 + (1 + mn) m2 +

qG
2α

C3 = T0 +
1
mn

)

x4
qG
(− + Lx 3 )
4 t
3

(25)

Applying boundary conditions:
T(0) = T0
Thus,

(18)

4 Microgripper based on silicon polymer

2α
e
t

2α
tL

(19)

qcond

C1 = −

QC = .t.y.

2α
tL

e

(20)

The heat loss due to convection is expressed (Arfken
1985; Trodden 1999; Snieder 1994):

2α
tL

=−


−

(30)
2α
tL

qG
2α

(31)

Temperature profile on the silicon cantilever is also calculated by:

Tcan (x) =

qcond

x + T0

(32)

At x = L, T (L) = Tcan (L), so that:

Applying the conservation law:

QG + QC + Qconv = 0

(22)

The equation obtains:


Si

Si

Qconv = −2α(T (x) − T0 ).y.∆x

qcond

2α
tL

qG −
2α .e
−

qcond

e

∂T (x + ∆x) ∂T (x)
−
∂x
∂x

+

2α
t


2α L
−
t +e

e

−

e
e

2α L
t −e
2α L
−
t +e

2α L
t
2α L
t
2α L
t

− 1)
(33)

−L

The temperature distribution in the actuator is given by

C2 = −
2α

2
( eτ +e
−τ −1)
e−τ −eτ
eτ +e−τ

eτ

−τ

+ eτ

(35)

+ e−τ

Thus,
τ

τ

T (x) = C1 .e L x + C2 .e− L x + T0 +

qG
2α

(36)

h
h2

F

EEF, IEF

G
L

Ljaw

Fig. 9  Frame structure to analyse the sensing microactuator

A

∆Τ

B

C

∆Τ

D

4.2 Thermal–mechanical analysis
A simplified structure which is used to analyze the sensing
microactuator under the change of temperature is shown in
Fig. 9. It can be seen from the figure that lines AB, CD and

4.3 Sensing microactuator displacement analysis
Figure 10 shows the deformation of the structure under the
change of temperature in beams AB and CD. As shown in
the graph, Z1 and Z2 denote the unknown rotation and vertical deflection of the rigid beam BDF. Here, it is assumed
that the axial expansion of elements EF is neglected.
In order to calculate the displacement and the output
force at the jaw tip of the sensing micro gripper, the direct

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Microsyst Technol
2 E AB I AB

4 E AB I AB

L
4 ECD I CD

2 ECD I CD

∆Τ

B

C

∆Τ


h3

Manipulating object

2 EEF I EF

4 EEF I EF

L

F

L

Fig. 13  Structure for solving the output force

Fig. 11  Diagram of the bending moment due to Z1 = 1

6 E AB I AB

6 ECD I CD

Note that axial forces in elements AB and CD due to
Z2 = 1 are zero. Based on conditions for static balance of
the structure, stiffness coefficients are given by:

L2

r11 =

6EAB IAB
6EEF IEF
+ L2 + L2
r21 = −
L2

+

h22 ECD ACD
,
L

Components R1(T) and R2(T) of the force vector are calculated basing on axial forces on elements AB and CD due
to the change of temperature ΔT. We have

R1 (T ) = α1 ∆TEAB AAB h + α2 ∆TECD ACD h2 ,
R2 (T ) = 0
Fig. 12  Diagram of the bending moment due to Z2 = 1

(37)

Z1 (T ) =

r12 R1 (T )
r22 R1 (T )
; Z2 (T ) = −
det K
det K

r11 r12

h2 ECD ACD
L

(39)

(42)

where det K = r11r22 − (r12)2.
Vertical displacement y(T) of the microactuator jaw tip
under the change of temperature is therefore given by

where
K=

(41)

Solving the Eq. (1), we yield

displacement method is used. Under the change of temperature ΔT, the governing equation for the system is given
by:

KZ(T ) = R(T )

(40)

y(T ) = Z1 (T )LJaw + Z2 (T ) =

R1 (T )
r22 Ljaw − r12
det K


R1 (F) = −FLjaw ; R2 (F) = −F
Solving the Eq. (45), we yield

Z1 (F) = det1 K {r22 R1 (F) − r12 R2 (F)},
Z2 (F) = − det1 K {r12 R1 (F) − r11 R2 (F)}

(46)

The vertical displacement of the jaw tip due to the reaction F is given by:
y(F) = Z1 (F)LJaw + Z2 (F)
= det1 K r22 R1 (F) − r12 R2 (F) Ljaw + R2 (F)r11 − R1 (F)r12

(47)
Taking the above result into Eq. (44), we obtain the
value of gripping force:

F=

R1 (T ) r22 Ljaw − r12 − h3 det K
r11 − 2r12 Ljaw + r22 (Ljaw )2

(48)

5 Measurement, calculation, simulation results
and discussions
The design, fabrication and initially characterization of
the proposed sensing microgripper is reported in the ref
(Chu Duc et al. 2007b; Chu Duc et al.) and calculation
results were mention in this paper. In addition, a 3-Dimention computer model of this device which comprises two

simulation. Due to the limitation of the measurement
method (Chu Duc et al.), temperature on each position
of actuator and cantilever could not be gathered precisely. Thus, the results measured in comparison with

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these of other methods are ignored. Obviously, there are
striking similarities in the results of calculation and simulation, and therefore, not only the mathematics methodology but also simulation model of the microgripper are
confirmed.
As is shown in the results comparison among methods,
one methodology has confirmed the accuracy of others and
vice versa. Although there are some errors and tolerances
of each method itself, the model for simulations and calculation scheme is appropriate. Consequently, it is an important factor to improve or adapt the gripper’s structure to
specific application. Moreover, it can be used to optimize
the structure in a particular aspect. For example, a new
microgripper which performs the same range of displacement with the original one, but the operating temperature
below 100 °C can be designed. Firstly, determine size of
the gripper (the number of polymer stacks or the length of
actuator) by using the final equation. After that, conducting the simulation with model which has obtained parameters from the first steps, and therefore, the design via those
results is affirmed.

6 Conclusions
The design of sensing polymer-silicon electrothermal
microgripper was proposed, characterized and simulated.
This device has many advantages in comparison with other
actuators, such as large movement, fast response time,
low driving voltage and CMOS compatible. However, it

Monolithically fabricated microgripper with integrated force
sensor for manipulating microobjects and biological cells aligned
in an ultrasonic field. J Microelectromech Syst 16(1):7–15
Beyeler F, Neild A, Oberti S, Bell DJ, Sun Y, Dual J et al (2007b)
Mono-lithically fabricated microgripper with integrated force
sensor for manipulating microobjects and biological cells aligned
in an ultra-sonic field. J Microelectromech Syst 16(1):7–15
Butefisch S, Seidemann V, Buttgenbach S (2002) Novel micro-pneumatic actuator for MEMS. Sens Actuator A-Phys 97(8):638–645
Carrozza MC, Dario P, Jay LPS (2003) Micromechanics in surgery.
Trans Inst Meas Control 25(4):309–327
Chan EK, Dutton RW (2000) Electrostatic micromechanical actuator with extended range of travel. J Microelectromech Syst
9(3):321–328
Chen L, Liu B, Chen T, Shao B (2009) Design of hybrid-type MEMS
microgripper. In: Proceedings of the International Conference on
Manufacturing Automation. HongKong
Cheng CH, Chan CK, Cheng TC, Hsu CW, Lai GJ (2008) Modeling,
fabrication and performance test of an electro-thermal microactuator. Sens Actuator A-Phys 143:360–369
Chronis N, Lee LP (2005) Electrothermally activated SU-8 microactuator for single cell manipulation in solution. J. MEMS
14(4):857–863
Chu Duc T, Creemer JF, Sarro PM (2006) Piezoresistive cantilever beam for force sensing in two dimensions. IEEE J Sens
7(1):96–104
Chu Duc T, Lau GK, Sarro PM (2007a) Polymeric thermal microactuator with embedded silicon skeleton: part II—fabrication,
characterization and application for 2D Microactuator, JMEMS
Chu Duc T, Lau GK, Creemer JF, Sarro PM (2007b) Electrothermal
microactuator with large jaw dispalcement and intergrated force
sensors. In: Proceeding 21th IEEE Conference on MEMS, Tucson, USA, 13–17 January 2008, pp 519–522
Chu Duc T, Lau GK, Wei J, Sarro PM (2007c) 2D electro-thermal
microactuators with large clamping and rotation motion at
low driving voltage. In:Proceeding 20th IEEE Conference on
MEMS, Kobe, Japan, 21–25 January 2007, pp 687–690

University, Utrecht
Stephen NG (2001) Discussion:shear coefficients for Timoshenko
beam theory. J Appl Mech 68(11):959–960
Trodden M (1999) Methods of mathematical physics. Lecture Notes,
Department of Physics, Case Western Reserve University,
Cleveland
Zhang R, Chu JK, Wang HX, Chen ZP (2013) A multipurpose electrothermal microgripper for biological micromanipulation.
Microsyst Technol 19(1):89–97

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