Strain gauges based on polyimide carrier foils and piezoresistive
granular thin films are highly sensitive to strain. Unlike
conventional metal foil, granular film strain gauges also have
a pronounced sensitivity to strain acting in the transverse
direction. A novel method that allows for the modification of the strain transfer is
proposed and proven experimentally. The method is based on the
creation of stand-alone polyimide paths, on top of which the
piezoresistive thin film is located. In this way, the granular film
hardly receives any transverse strain; hence, the transverse
sensitivity is drastically reduced. A picosecond laser system can be
used for both patterning of the thin film and for controlled ablation
of polyimide in order to generate well-defined high path
structures. The working principle of the method is demonstrated by
simulation, followed by an experimental verification using measurements
of the transverse gauge factor. Furthermore, the output signal of
force transducers may be increased using granular thin film strain
gauges of reduced transverse sensitivity.
Introduction
Foil strain gauges are used for sensor applications, such as load cells or
torque transducers, and also for the stress analysis of mechanical
components. The strain gauges (SGs) are usually mechanically attached
to a component or spring element by bonding. Surface strain due to
a force acting on the component or spring element is directly
transferred to the strain gauge and results in a change in its
electrical resistance. The constant of proportionality between the
relative resistance change and strain is called the gauge factor
(k). SGs are not only sensitive to strain in the direction of the
grid design but also if the strain is acting in the perpendicular or
transverse direction. Therefore, it is necessary to differentiate
between the longitudinal gauge factor (kL) and the
transverse gauge factor (kT).
Commonly used metal foil strain gauges have longitudinal gauge factors
of 2 to 4 depending on the grid material utilized. The most common
grid materials are constantan and NiCr alloys
. The transverse gauge factor of metal foil
SGs is typically around ±1% of the longitudinal gauge
factor; hence, it is virtually negligible. Since the kT of metal
SGs is caused by elastical deformations of the measuring grid and
reverse loops, it can be influenced by the layout of the SGs. Another
way of influencing the transverse gauge factor without changing the
measuring grid is the introduction of additional grid lines (of
a different material) without electrical connections between the gauge
grid lines. These additional grid lines increase the mechanical
stiffness of the gaps and change the transfer of transverse strain
.
A novel type of highly sensitive foil strain gauges consists of
a polyimide foil as the carrier and a piezoresistive granular thin film,
such as the nearly isotropically responding films of Au nanoparticles
or granular metal–carbon films. Metal–carbon
films with thicknesses of approx. 150 nm are based on granular
Ni–C or NiCr–C
. The highly sensitive foil SGs
show different properties compared with conventional metal foil SGs,
especially concerning the gauge factor values
. Highly sensitive SGs can reach
longitudinal gauge factors up to 30 ,
allowing for the measurement of very small strains, which can be
advantageous for stress analysis and sensor applications. For load
cells or pressure sensors, the higher gauge factor also allows the use
of stiffer spring elements with increased overload capability. The
kT/kL ratio of granular thin films based on
NiCr–C is typically around 50 %. The transverse strain
sensitivity is the result of local, transverse current paths caused by
the particle arrangement of the granular thin film. A straight, global
current path in a granular thin film is comprised of many
particle-to-particle currents with different directions. This
intrinsic property of the thin film causes its transverse gauge
factor. .
Our approach to reduce the transverse gauge factor of highly sensitive
SGs is based on an invention concerning the polyimide substrate with
the objective of decreasing the transverse strain transfer
. The usage of a picosecond laser system
allows for the production of a stand-alone polyimide path with the granular
film on top. By creating such a polyimide path with a certain width
(w) and height (h), as shown in
Fig. , the transfer of transverse strain from
the metal component to the active granular thin film may be
reduced. Therefore, a strain perpendicular to the current direction and parallel
to the surface plane of the thin film only results in
a comparatively small resistance change in the SG. This principle was
first examined by means of finite element analysis (FEA), and
it was verified via measurements of the transverse gauge factor. Using
these measurements, the influence of the path geometry and different SG
cover materials was evaluated. The resultant low kT of
the strain gauges was then utilized for tension/compression force
transducers with the benefit of a higher sensitivity.
Model of an SG that allows for the strain transfer from the metal component to the active thin film to be influenced; the (a) granular thin film, (b) polyimide carrier, (c) transversally strained metal component, (d) solder pad, (e) polyimide path with height h and width w, and (f) active granular thin film element are shown.
The FEA model (according to Fig. ) is composed of an aluminum plate and a polyimide SG with a stand-alone path (h=30µm and w=100µm). The boundary conditions (roller, fixed and displacement) are marked accordingly. (a) The resulting strain ϵxx of the model. (b) The deformation of the circled path section that occurred (with a scale factor of 500).
Simulation of the strain transfer by FEA
The operating principle of SGs with a stand-alone polyimide path
according to Fig. was investigated by means of
FEA with COMSOL Multiphysics. The geometry used, which is shown in
Fig. , consists of an aluminum plate (6.25mm×1.25mm×0.05mm) and a polyimide SG
(6mm×1mm×0.05mm) with
a stand-alone path of h=30µm and
w=100µm. The length of the path is 4 mm. The
geometry is simulated with respect to the transfer of transverse strain from
the plate to the top of the path using a linear elastic material
model. The metal plate is strained by 1 ‰ in the transverse
direction by applying a displacement on its right lateral
surface. A fixed constraint is placed in the bottom left corner of the
aluminum plate, while roller constraints are placed on its bottom,
front and left lateral surfaces. A roller is a boundary condition
that constrains displacement in the normal direction of the selected
surface. The resulting transverse strain ϵxx is
recognizable by means of the scale in Fig. a. While the
surface strain on the aluminum plate equals the applied strain of
1 ‰, the strain on top of the path and on the edges of the
polyimide carrier is slightly above or below 0 ‰. The
deformation, shown in Fig. b, illustrates that material
at the bottom of the polyimide path is strained by 2 ‰
(red areas), whereas the path itself is only strained slightly. The
excessive strain at the bottom pulls the material in the center of the
path downwards, creating a concave surface on top. The strain on top
of the path at a certain displacement of the metal component is highly
dependent on the width and height of the path.
To identify the influences of the width and height of the path,
a parametric sweep was performed. In order to investigate the method's
dependence on the amount of tensile and compressive strain, the
applied strain was varied between -1, -0.5, 0.5 and
1 ‰. The sweep was executed for
w=50µm and w=100µm, while
h was varied in 5 µm steps from 5 to 45 µm. The resulting transverse strain was averaged over the top surface
of the path and is illustrated in
Fig. . Noticeably, all curves intersect at
a certain path height at which the resulting strain is 0 ‰,
leading to a transverse gauge factor of zero. This is the case for
both path widths; however, the slope of the curves is different. For
w=50µm and h=15µm the
transmitted strain is only -1 % of the applied strain,
whereas for w=100µm and h=15µm the
transmitted strain is 29 %. For path heights beyond the
intersection point an overshoot of the curves can be observed. In
practice, this would lead to a negative transverse gauge factor. In
further simulations, the metal component was not only strained in the
transverse direction, but it was additionally strained in the longitudinal direction. This
leads to the transverse contraction of the path and, consequently, results
in a reduced longitudinal gauge factor. The extent of the reduction
depends on the path geometry and the Poisson's ratio ν of
the polyimide. The maximally reduced longitudinal gauge factor
kL, red can be approximated by means of
Eq. (). Assuming a Poisson's ratio of 0.3, kL,
red equals 85 % of kL.
kL, red=kL⋅(1-0.5⋅ν)
The results of this parametric sweep reveal that the
transverse gauge factor of granular strain gauges with a certain path
width can be reduced to almost zero if the height of the path is
chosen correctly. Broader paths need to be cut deeper in order to
eliminate the transverse strain sensitivity. For thinner paths, small
changes of the path height have a stronger impact on the transverse
gauge factor compared with broader paths. The simulation also showed
that the reduction of the transverse gauge factor accompanies
a reduced longitudinal gauge factor caused by the transverse
contraction of the polyimide.
The graphs show the resulting transverse strain (xx component) averaged over the top surface of the path and plotted vs. the path height for different applied strains. Panel (a) shows the results for a path width of 50 µm, and panel (b) shows the results for a path width of 100 µm.
Experimental realization
Thin film strain gauges were produced according to
. The piezoresistive, NiCr–C-based thin film (thickness of 150 nm) was deposited onto
a polyimide carrier (thickness of 50 µm) via reactive sputter
deposition. Subsequently, the thin film and the polyimide foil were
structured using a Nd:YVO4 laser system (3D-Micromac)
with a wavelength of 355 nm and a pulse
duration of less than 15 ps. Moreover, the mark speed was set to
100 mms-1 with a repetition rate of 50 kHz. The
laser type used allows for a local ablation of the thin film with only
minimal damage to the polyimide carrier while maintaining its
electrical insulation . The usage of ultrashort
UV laser pulses also allows for the systematic, high quality removal of polyimide ; thus, the
production of fine path structures is possible. Consequently, the
structuring of the thin film and the polyimide foil was executed in
one single process step. For both the structuring of the thin film
and the ablation of the polyimide, the laser parameters for polyimide
ablation were used, whereby the laser scan path had to be repeated
several times. The first scan repetition structures the thin film and
ablates some polyimide, while the following repetitions continuously
remove the polyimide. The minimal line width that can be removed is
dependent on the diameter of the laser spot – in our case 20 µm. In order to ensure a sufficient trench width, the scan path used
consisted of 10 lines with an overlap of 50 %. Prior to
application, the SGs produced were cleaned with isopropanol, annealed
and provided with solder pads that consisted of a solderable,
sputter-deposited thin film. Figure a presents
a microscopic image of an SG with bone structure that was produced in this
manner. This simple bone structure consists of a path
(w=50µm, h=18µm) and two solder
pads. The topography of the SGs produced was characterized by means of
a chromatic white-light profilometer (CWL, FRT), as illustrated
in Fig. b. The line scan in Fig. c
displays the path height of approximately 20 µm. Due to the
steep edges of the path, the light is not reflected in some spots which
causes missing points in the plot.
(a) Microscopic image of an SG produced with bone structure. (b) Topography of a strain gauge with bone structure measured by means of a chromatic white-light sensor. (c) The height of the path that is recognizable in the line scan is around 20 µm.
ResultsLaser ablation of polyimide structures
The height of the polyimide path depending on the number of scan
repetitions was measured using a tactile profilometer (Veeco
Dektak 150). For each number of scan repetitions (1, 5, 10 and 15
repetitions) four samples were produced, cleaned in an ultrasonic bath
and measured. Figure shows the arithmetic means of the
measured path heights vs. the scan repetitions; the SD are also stated. The path height (h) increases
almost linearly with the scan repetitions, as does the SD. showed that the ablation rate for
polymers starts to decrease above a certain number of laser pulses,
ultimately resulting in a spontaneous ablation stop. In our case, this
leads to a reduced path height and probably greater deviations above
a certain number of scan repetitions. The results show that it is
possible to create reproducible polyimide path structures via laser
ablation. The path height can be controlled by the number of scan
repetitions.
The figure depicts the measured path height vs. the number of scan repetitions. The height of four samples for each number of scan repetitions was measured using a tactile profilometer. The arithmetic means of the measured heights as well as the SD (represented using “s” in the figure) are shown. The dotted line represents a linear fit of points one to three.
Characterization of the transverse gauge factor
The transverse gauge factor of SGs with a bone structure was
determined using a device according to
guideline 2635. The device, illustrated in
Fig. b, consists of a wide metal beam with
thickened ends and metal plates on both sides. A force is applied
between the plates, creating a bending moment of the beam. The
resulting ratio of transverse to longitudinal strain
ϵT/ϵL in the middle of the
device is almost zero . To measure the
transverse gauge factor of the strain gauges, they were bonded in the
middle of the beam: two SGs in the longitudinal direction and two in the transverse
direction. As a reference, a 90∘ T-rosette metal foil SG
(designation: FAET-A6259L-35-S13E; Vishay Precision Group) was also
bonded onto the beam. Three load and four unload cycles with a strain
of 1 ‰ and 0 ‰, respectively, were executed, resulting in 10
measurement values per cycle and strain gauge. The electrical
resistances of the SGs were measured by four-wire sensing using an
Agilent 34970A data acquisition/switch unit. The longitudinal and
transverse gauge factors (kL and kT, respectively) were
calculated by means of the measured resistance changes and the strain
ϵL according to
Eq. (). ϵL is the longitudinal strain
on the device measured using the reference SG. The resistance of the
longitudinally or transversely applied SGs is designated as
RL or RT, respectively.
kL=ΔRLRL⋅ϵLandkT=ΔRTRT⋅ϵL
Figure a illustrates the relative
resistance change for two strain gauges with the same path width
(w=50µm) but different path heights that are both bonded in
transverse direction. The SG with the higher path
(h=18µm) showed a drastically reduced
transverse sensitivity compared with the SG with the lower path
(h=2.5µm). Thus, the transverse gauge factor is
reduced from 4.0 to 0.5, while the longitudinal gauge factor is
also slightly reduced from 10.3 to 9.5, as predicted by the
FEA simulation. The kT/kL ratio is
38.5 % for the SG with the lower path and only 5.6 %
for the SG with the higher path.
(a) Relative resistance change in transversely applied SGs (w=50µm) depending on the path height (h=2.5 and 18 µm). Three load cycles with a strain ϵL of 1 ‰ were performed. (b) The device, according to VDI/VDE standard 2635, used to characterize the transverse sensitivity of SGs.
In order to verify the greater path height of the SGs as the reason for the reduced transverse gauge factor, the lasered trenches were filled with soft silicone rubber or with hard UV varnish based on epoxy acrylate. Depending on
the elastic modulus of the cover material used a more or less
pronounced increase in the transverse gauge factor should be
observed. The transverse gauge factor of SGs with a path width of
50 µm and a path height of 18 µm was first
measured without and then with a cover material. The
kT/kL ratio increased from 5.1 % to
24.7 % for SGs covered with UV varnish. On the contrary,
kT/kL was 5.6 % for SGs without as
well as with silicone rubber cover. Since the cover material creates
a new mechanical connection from the polyimide path to the polyimide
carrier material, the strain transfer to the thin film increases
depending on the cover material's elastic modulus. Cover materials
like silicone rubber with a low modulus around 1 MPa do not increase the transverse gauge
factor, whereas materials with a higher elastic modulus of
approximately 1 GPa, like UV
varnish, lead to a (partial) return to the initial level of strain
transfer.
To further examine the influence of the path width on the transverse
sensitivity, SGs with a path height of 18 µm and path
widths of 50, 40, 30 and 20 µm were
measured. Figure shows that the
kT/kL ratio of the strain gauges increases with
the path width. In contrast to the simulation results, no negative
transverse gauge factors were measured. Therefore, it can be assumed that the
transverse gauge factor declines asymptotically towards zero for
increasing path heights or decreasing path widths,
respectively. kT/kL could be reduced to
0.93 % for w=20µm and
h=18µm, which is a value comparable with conventional
strain gauges.
The graph illustrates the influence of the path width on the kT/kL ratio of highly sensitive strain gauges. The path height was fixed at 18 µm.
Application of strain gauges with reduced transverse gauge factor
A low transverse sensitivity of strain gauges can be advantageous for
applications on biaxial strain fields occurring on tension/compression
force transducers as shown in Fig. . An applied tensile force
generates a deformation of the transducer's membrane. Therefore, tensile
strain occurs in the force direction, whereas compressive strain is
perpendicular to the force direction. The resulting output signal of
a Wheatstone bridge, consisting of two strained and two compressed
SGs, is highly dependent on the SG's longitudinal and transverse gauge
factors. The relation between the nominal SG resistance R0, the
resistance changes of the SGs ΔRi with i=1,2,3,4, the
bridge output voltage VM and the bridge supply voltage
VB under the assumption that ΔRi≪R0 is
given in Eq. () .
3VMVB=14ΔR1R0-ΔR2R0+ΔR3R0-ΔR4R04ΔRiR0=kL⋅ϵL,i+kT⋅ϵT,i
Considering the definition of the gauge factor in Eq. () and
the present strain field with
ϵL,1=ϵL,3=ϵT,2=ϵT,4=ϵ+
and
ϵL,2=ϵL,4=ϵT,1=ϵT,3=ϵ-,
the sensitivity of the bridge can be transformed to
VMVB=14((2kL-2kT)ϵ++(2kT-2kL)ϵ-)
(a) Model of a tension/compression force transducer
with the applied force and resulting strains on the membrane surface. (b) A sectional view of the transducer shows the position and thickness of the membrane.
(a) Signal of four tension/compression force transducers vs. the applied mass. The full bridges of two transducers (red squares/circles) consist of SGs with w=100µm and h=2.5µm, resulting in kT/kL=48.6%. For the full bridges of two additional transducers (black squares/circles), SGs with w=50µm and h=18µm were used, leading to kT/kL=5.5%. Panel (b) shows the positioning of the SGs on the transducer's membrane.
Since ϵ+>0 and ϵ-<0, a large transverse gauge
factor will result in a reduced signal
(VMVB). To verify this concept,
force transducers of a tension/compression type made of aluminum were
characterized using SGs with different path geometries and, accordingly,
different transverse gauge factors. The SG layout for the
characterization is linear, consisting of two paths with a length of
2.1 mm, two solder pads and a reverse loop with a length of
0.7 mm. On the one hand, SGs with a path width of 100 µm, a path height of 2.5 µm and a longitudinal gauge factor
of 10.9 were used. The transverse gauge factor was 5.3 for these SGs
(kT/kL=48.6%). On the other hand,
SGs with a path width of 50 µm, a path height of
18 µm, a longitudinal gauge factor of 9.1 and a transverse
gauge factor of 0.5 were applied (kT/kL=5.5%). Per transducer, four SGs were bonded onto one side of
the membrane, according to Fig. b, and connected
to a Wheatstone bridge. The force transducers were loaded in
20 kg steps up to a mass of 100 kg, which corresponds
to a tensile strain ϵ+ of 0.45 ‰ and
a compressive strain ϵ- of -0.4 ‰. The
graph in Fig. a shows the measured signals
vs. the applied mass for two SG transducers. The
transducer sensitivity is nearly doubled due to the 90 % reduction in the
transverse gauge factor, despite the slightly
lower longitudinal gauge factor. The higher sensitivity allows for
a thicker membrane for this transducer type, increasing the overload
capability.
Conclusions and outlook
The transverse gauge factor of highly sensitive strain gauges based
on a piezoresistive, granular thin film and a polyimide carrier can be
dramatically reduced by modifying the strain transfer. This can be
achieved by means of a picosecond laser which allows for the reproducible
production of strain gauges consisting of a stand-alone polyimide path
with the thin film on top. FEA showed that the width and height of
such a polyimide path has a massive impact on the transfer of
transverse strain. The thinner and higher the path, the less strain
will be transferred to the granular film; as a result, the
transverse gauge factor will be reduced to almost zero. This was
proven by measurements of the transverse gauge factor for different SG
path geometries. The ratio of the transverse to the longitudinal gauge factor
was reduced to about 1 % for a path width of 20 µm
and a path height of 18 µm. The reduced transverse gauge
factor will increase again if the previously created laser trenches
are filled with a hard material-like UV varnish that has a high
elastic modulus compared with polyimide. On the contrary, soft materials
like silicone rubber do not affect the reduced transverse gauge
factor. A Wheatstone bridge consisting of highly sensitive SGs with
a greatly decreased kT shows significantly higher
sensitivity than a bridge of SGs with a regular kT when
applied on a biaxial strain field of tension/compression force
transducers. Therefore, this new method to reduce the transverse
sensitivity of granular thin film SGs allows for the use of stiffer force
transducers with an even more increased overload
capability. Furthermore, the low transverse sensitivity might be
advantageous for uniaxial stress analysis. In this case, the direction
of stress is known, and the amount of transverse contraction, for
example, needs to be measured. In further experiments, the impact of
stand-alone polyimide paths on creep error, temperature-related errors
and moisture sensitivity will be investigated.
Data availability
The underlying data are available from 10.6084/m9.figshare.11996250.v1.
Author contributions
MM was responsible for the investigation, methodology, visualization and writing the original draft of the paper. DV and ML contributed to the conceptualization, methodology, and reviewing and editing the paper. DG was responsible for the validation. AL carried out project administration. GS was responsible for supervision and review and editing the paper.
Competing interests
The authors declare that they have no conflict of interest.
Financial support
This research has been supported by the German Federal Ministry for Economic Affairs and Energy (grant no. 49VF170017, INNO-KOM).
Review statement
This paper was edited by Michael Kraft and reviewed by two anonymous referees.
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