Electrostatic actuators offer certain advantages, including the use of lithographic techniques for ease of manufacture, as well as the ability to use any conductive material as a resonator.
Let’s consider the schematic of an electrostatic actuator consisting of a parallel plate capacitor connected to a mechanical spring, as shown in the diagram below. The electrostatic force acting on the capacitor plate is given by
where W_e is the total stored electrical energy in the system composed of the capacitor with capacitance C and an ideal battery (not shown) supplying the voltage V to the capacitor. Note that this simplistic force expression only applies for a parallel plate capacitor in which one of the plates is rigidly moving up and down while remaining parallel to the fixed plate. It also ignores any edge effects. We will use this simplification to study the behavior of an actuator as it captures the basic concepts of equilibrium and pull-in phenomenon.
Schematic of an electrostatic actuator.
The capacitance can be calculated as C = \epsilon\frac{A}{d-x}, where \epsilon is the permittivity of the dielectric between the parallel plates and A is the area of the plate. The spring represents the elasticity of the moving electrode.
For this 1D analysis, the restoring force in a linear spring is -k x. However, in a general scenario with a large enough displacement, x, the relationship between restoring force and displacement is no longer linear. This is conceptually referred to as large deformation or geometric nonlinearity. We will therefore define the restoring mechanical force in the nonlinear spring as kx + k_1x^2+k_2x^3. You can learn more about geometric nonlinearity in these previous blog posts: “Modeling Linear Elastic Materials — How Difficult Can It Be?” and “What Is Geometric Nonlinearity?“.
The net force on the capacitor plate is given by
(1)
At equilibrium, the electrostatic force is balanced by the mechanical force so that F = 0. This leads us to the equilibrium equation
(2)
where x_e is the equilibrium displacement.
We can then solve Equation (1) for the equilibrium displacement x_e.
A pull-in point is defined as the equilibrium point beyond which the system becomes unstable. Practically, this means that for values of voltage greater than the voltage corresponding to the pull-in point (pull-in voltage), the plates will snap together.
Estimating this pull-in voltage is an important aspect of designing microresonators. To estimate the pull-in voltage, we will need to analyze the stability of an equilibrium point. For simplicity, we will take the case of a linear spring for estimating the pull-in point. As per the theory of stability of an equilibrium point,
From Equation (2), for the case of a linear spring (small deformation analysis), the voltage at the equilibrium point can be expressed as
(3)
where V_c = \sqrt{\frac{2kd^3}{\epsilon A}}.
Evaluating dF/dx at x = x_e and using Equation (3), we find
Solving the above equation for dF/dx = 0 gives
We note that for x_e > x_p, dF/dx > 0 and for x_e < x_p , dF/dx < 0. This indicates that x_p is our pull-in displacement and for values of x_e > x_p, the system becomes unstable. Using Equation (3), we also find that voltage V has a maxima at x_e = x_p = \frac{d}{3}. The plot below shows the relationship between V and x_e.
The pull-in voltage V_p is given by
In the above derivations for pull-in displacement and pull-in voltage, we assume a linear spring. Inclusion of geometric nonlinearity will result in some changes in these values.
Let’s take a look at the forces on the capacitor plate about an equilibrium point x_e. For a small displacement, x, around the equilibrium point x_e, the total force acting on the parallel plate capacitor is
For a small enough displacement, x, we can perform a series expansion about x_e. Keeping terms to only a second order in x, we get
(4)
Using Equation (2) to cancel out equilibrium terms, we can rewrite the force as
k_s(x) is the stiffness corresponding to the spring force and k_e(x) is the stiffness corresponding to the electrostatic force. k_e(x) acts in a way to reduce the mechanical stiffness. This is known as the spring softening phenomenon. The value of k_s(x), in the case shown here, increases with an increase in x_e. This is known as stress stiffening. Such an effect is a result of geometric nonlinearity and is discussed in this previous blog post.
In the time-harmonic response analysis of an electrostatic actuator, the actuation voltage can be considered a sum of a DC bias voltage and an AC signal, so that V = V_{DC}+v_{AC}, where v_{AC} = v_0 cos(\omega t) and v_0 is the perturbation amplitude. The electrostatic force is given by
where x_e is the equilibrium point given by solving Equation (2) with V = V_{DC}. We will consider a linear frequency response analysis. For a linear harmonic analysis, the displacements will also be of the form x = Xcos(\omega t+ \phi). Expanding the above equation, we get
We can also expand the term \frac{\epsilon A}{\left(d -x_e -x\right)^2} about the equilibrium point x_e to give
Keeping only the terms to a first order in perturbation amplitude and frequency \omega, we get
The electrostatic force is thus a sum of a DC force and a time-harmonic force at the excitation frequency. Note that in this derivation, we are ignoring the small DC component proportional to v_0^2 and a force component at twice the excitation frequency. We can similarly derive the expression for the mechanical force for linear time-harmonic analysis with a DC bias.
We have discussed the theory behind the principles of electrostatic actuation using the case of a spring connected to a parallel plate capacitor to analyze the equilibrium, pull-in phenomenon, and time-harmonic response in an electrostatic actuator. We have also discussed the principles of electrostatic spring softening and stress stiffening that occur in an electrostatic actuator.
While the principles discussed in this blog post are important for understanding the fundamentals of an actuator, the expressions derived above cannot be used for an accurate analysis of an actual 3D microresonator. These are based on simplifications and approximations enabling a lumped analysis. Finite element analysis would give us a more accurate picture of the equilibrium, pull-in, and time-harmonic analysis of a microresonator.
In an upcoming blog post, we will take a look at how we can model microresonators using the dedicated Electromechanics physics interface in COMSOL Multiphysics.
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Wireless sensor networks are used today in a variety of low-power applications, from wearable healthcare to water quality monitoring. These networks include a series of sensors that measure and record physical conditions — often intermittently, over a period of time — at various locations. Through a wireless link, each sensor communicates the information it obtains to other sensors in the network as well as a base location that records the readings from all of the sensors.
One of the biggest challenges in developing wireless sensor networks is balancing energy consumption with efficiency. Battery life in sensors is often limited, making these devices more expensive to deploy, while reducing their number of applications. That’s where energy harvesting comes into play. In this process, energy is gathered from an external source (e.g., solar or thermal power) and converted into usable energy. Energy harvesting is beneficial, as it makes use of energy that would have otherwise been lost, helping to optimize the power of devices while extending their operational lifetime.
As previously referenced, energy can be harvested from the environment in a number of ways. One example is when mechanical strain is converted into electrical energy, also known as piezoelectric energy harvesting. A potential source of mechanical strain is local variations in acceleration. This is the case when a wireless sensor is mounted on a piece of machinery that is vibrating.
Let’s analyze such an energy harvester configuration in COMSOL Multiphysics.
The Piezoelectric Energy Harvester tutorial model is designed to represent a simple “seismic” energy harvester. The device features a piezoelectric biomorph clamped at one end of the vibrating machinery and a proof mass mounted on the other end. A ground electrode is embedded within the biomorph, with two electrodes on the cantilever beam’s exterior surfaces. This design scheme ensures that an equal voltage is induced on the exterior electrodes.
Geometry of a piezoelectric energy harvester.
Our series of simulation analyses begins with addressing the power output as a function of vibration frequency. The plot below shows the input mechanical power and the power harvested, along with the voltage that is induced across the piezoelectric biomorph when acceleration occurs. In this case, the fixed electrical load is 12 kΩ and the acceleration magnitude is 1 g. From the results, we can identify a peak voltage at 76 Hz. This calculation is close to the resonant frequency computed for the cantilever in a separate eigenfrequency analysis (73 Hz).
Power output as a function of vibration frequency.
Let’s now measure the power output as a function of the electrical load resistance. In this scenario, we apply an acceleration of 1 g vibrating at 75.5 Hz. The results, shown in the following graph, indicate that the peak of power harvested correlates to an electrical load of 6 kΩ.
Power output as a function of electrical load resistance.
Lastly, we analyze the voltage and mechanical/electrical power as a function of mechanical acceleration. Here, the acceleration is set at 75.5 Hz, with a load impedance of 12 kΩ. As the plot below illustrates, there is a linear relationship between the voltage and the load, while the harvested energy increases quadratically.
Power output as a function of acceleration.
These results show good qualitative agreement when compared with experimental findings.
Simulation offers a simplified approach to studying and optimizing energy harvesting devices. The ability to easily test different device configurations accelerates the design process, while helping to produce more efficient energy harvesters. As the efficiency of these devices continues to grow, a greater number of technologies will have the chance to benefit from energy harvesting.
Aquatic environments aren’t welcoming spaces for man-made crafts. Lack of light and murky waters make visibility difficult or even nonexistent at times. Underwater vehicles such as submarines need to detect, monitor, and avoid objects in these conditions. If this isn’t hard enough, submarines have a limited energy supply and need to be as energy efficient as possible.
An image of an autonomous submarine. (By the CSIRO ICT Centre. Licensed under Creative Commons Attribution 3.0 Unported, via Wikimedia Commons.)
Many conventional submarines use sonar and optical methods to navigate their surroundings. These methods work, but they have drawbacks. Sonar can harm or kill marine organisms and optical methods don’t function well in low visibility conditions. Additionally, both of these methods utilize active sensing, which uses energy and can be inefficient.
When searching for a more efficient way for aquatic vehicles to monitor their surroundings, a team from the PSG College of Technology, Coimbatore, Tamil Nadu, India looked to a blind cave fish for inspiration.
A blind Mexican cave fish can move quickly and avoid obstacles in its murky and cluttered environment thanks to an array of neuromasts called lateral lines. Specifically, this fish can navigate its surroundings by using superficial neuromasts that respond to flow variations, as well as canal neuromasts that respond to pressure variations.
An image of a blind cave fish with lateral lines shaded. The dots within the lateral lines represent the canal neuromasts. Image by Aarthi E. et al., and taken from their COMSOL Conference 2013 Bangalore paper submission.
Perhaps the most important aspect of this cave fish is that it performs passive sensing to study its environment. This means that the fish can navigate its environment without spending energy to emit waves that can cause harm or reveal its location. Instead, this fish passively senses the flow of water around itself.
By using the COMSOL Multiphysics Laminar Flow interface, the researchers were able to predict the performance of their underwater pressure sensor design based on this energy-efficient passive sensing method.
To build a pressure sensor that can function passively, the researchers mimicked the cave fish’s lateral lines.
When designing the pressure sensor, they arranged ten sensors in an array that is similar to that of the cave fish, with spaces to avoid crosstalk. This design also has a flexible sensing diaphragm mounted over the base of the sensor. The sensing layer is made out of a liquid crystal polymer (LCP), which is flexible, tough, and has good corrosion resistance.
Above this sensing diaphragm are strain gauges, which are used to convert pressure change into resistance change in gold piezoresistors. Finally, a standing structure mimics the fish’s superficial neuromast and measures velocity.
Geometry of the MEMS-based pressure sensor. Image by Aarthi E. et al., and taken from their COMSOL Conference 2013 Bangalore paper submission.
This pressure sensor can detect disruptions in the water around it. For example, let’s imagine that a submarine using this sensor is approaching a sunken ship. The sunken ship causes a change in the water flow and thus creates a pressure difference across the sensor’s membrane. The sensor is able to identify this change because the pressure difference causes the diaphragm to bend, as seen below.
Diaphragm displacement caused by applied pressure. Image by Aarthi E. et al., and taken from their COMSOL Conference 2013 Bangalore paper submission.
This bending causes a change in the resistance value of the piezoresistors, which can be read as a voltage shift. In this way, the pressure sensor is able to passively detect changes in its environment, such as the changes caused by the sunken ship.
This device could allow submarines to easily navigate their surroundings without wasting energy. But first, the researchers had to check how well the pressure sensor works.
The researchers analyzed the pressure sensor’s ability to detect changes in the surrounding environment by computing the velocity and pressure distribution for various levels of boundary stress exerted over the sensor. In their simulations, an increase in boundary stress occurs when an object approaches the sensor. This change in boundary stress should also change the velocity and pressure experienced by the sensor.
Left: Velocity distribution over the sensor. Right: Pressure distribution over the sensor. Images by Aarthi E. et al., and taken from their COMSOL Conference 2013 Bangalore paper submission.
The simulations revealed that the sensor experienced changes in both velocity and pressure when the boundary stress increased. These are the same changes that the blind cave fish detects, but how sensitive is the pressure sensor to these changes?
The researchers defined sensitivity as the change in resistance of the strain gauge per unit stress change. They observed an increase in resistance occurring when the pressure changed and determined that their sensor was sensitive enough to detect a pressure change as small as 5 N/m^{2}.
Overall, this passive pressure sensor is not only sensitive enough to function well, but also energy efficient, safe, and stealthy. This design is a good energy-efficient alternative for underwater pressure sensing applications. The fluid-structure interaction feature, the piezoresistivity physics interfaces, and the hyperelastic material models available in the COMSOL software make it straightforward to simulate this kind of sensor.
Miniature devices have many applications and researchers are constantly finding new uses for them. One such use, which we’ve blogged about before, is a microfluidic device that could let patients conduct immune detection tests by themselves. But to work in the microscale, devices like this one, of course, rely on even smaller components such as micropumps.
Let’s turn to a tutorial model of a valveless micropump mechanism that was created by Veryst Engineering, LLC using COMSOL Multiphysics version 5.1.
The micropump in the tutorial model creates an oscillatory fluid flow by repeating an upstroke and downstroke motion. The fluid flow enters a horizontal channel containing two tilted microflaps, which are located on either side of the micropump. The microflaps passively bend in reaction to the motion of the fluid and help to generate a net flow that moves in one direction. Through this process, the micropump mechanism is able to create fluid flow without the need for valves.
The geometry of the micropump mechanism tutorial.
Please note that the straight lines above the microflaps are there to help the meshing algorithm. Check out the tutorial model document if you’d like to learn how this model was created.
The tutorial calculates the micropump mechanism’s net flow rate over a time period of two seconds — the amount of time it takes for two full pumping cycles. The Reynolds number is set to 16 for this simulation so that we can evaluate the valveless micropump mechanism’s performance at low Reynolds numbers. The Fluid-Structure Interaction interface in COMSOL Multiphysics is instrumental in taking into account the flaps’ effects on the overall flow, as well as making it an easy model to set up.
Left: At a time of 0.26 seconds, the fluid is pushed down and most of it flows to the outlet on the right. Right: At a time of 0.76 seconds, the fluid is pulled up and most of it flows from the inlet on the left.
The simulation starts with the micropump’s downstroke, which is when the micropump pushes fluid down into the horizontal channel. This action causes the microflap on the right to bend down and the microflap on the left to curve up. In this position, the left-side microflap is obstructing the flow to the left and the flow channel on the right is widened. This naturally causes the majority of the fluid to flow to the right, since it is the path of least resistance.
During the following pumping upstroke, fluid is pumped up into the vertical chamber. Here, the flow causes the microflaps to bend in opposite directions from the previous case. This shift doesn’t change the direction of the net flow, because now the majority of the fluid is drawn into the flow channel from the inlet on the left.
Due to the natural deformation of the microflaps caused by the moving fluid, both of these stages created a left-to-right net flow rate. But how well did the micropump mechanism do at maintaining this flow over the entire simulation time period?
The net fluid volume that is pumped from left to right.
During the two-second test, the net volume pumped from left to right was continually increased, with a higher net flow rate during peaks of the stroke speed. This valveless micropump mechanism can function even at a lower Reynolds number.
The valveless micropump mechanism could have many future applications, one of which is to work as a fluid delivery system. In such a scenario, a micropump mechanism could take fluid from a droplet reservoir on its left and move it through a microfluidic channel to an outlet on its right. In this post we have shown just one set of simulation results. By experimenting with the tutorial model set up by Veryst Engineering, you can visualize how a valveless micropump may work in different situations and use this information to discover new uses for micropump mechanisms.
Piezoelectric valves are common in medical and laboratory applications because they offer many advantages, such as energy efficiency, durability, and fast response times. To open and close the valve featured in this tutorial, there is a hyperelastic material with a piezoelectric actuator sitting on top of it. When a voltage is applied to the stacked piezoelectric actuator, it deforms in a way that either pushes the hyperelastic material against the opening of the valve to seal it or moves it away from the valve to open it.
Valve, piezoelectric actuator, and seal.
Stacked piezoelectric actuators consist of two actuators stacked on top of each other. Each of the two actuators is made up of alternating layers of piezoelectric material, PZT, and very thin metal conducting layers between them. Every second metal layer is grounded, while every other layer receives an applied voltage. Similarly, the stacked PZT layers have alternating polarization directions.
Close-ups of the actuator and seal with alternating layers of PZT and metal highlighted. The top images show the PZT layers of alternating polarization directions. The bottom images show the metal substrate with an applied voltage to every other layer and the others set to a ground.
The bimorph actuator under consideration can be thought of as two stacked actuators placed one on top of the other. For a positive applied voltage, the upper and lower actuators are designed to expand laterally and contract laterally, respectively. This results in a bending of the structure (in this case, a disc), such that the center of the disc arches downwards. This forces the hyperelastic seal into contact with the valve seat — closing the valve. In the surface plot below, the stress is indicated by the color scale.
The von Mises stresses in a piezoelectric valve with a bimorph disc actuator.
The Piezoelectric Valve tutorial model, a new addition to the Application Gallery with COMSOL Multiphysics 5.1, demonstrates how to model a stacked piezoelectric bimorph disc actuator in a pneumatic valve. The MEMS Module and Nonlinear Structural Materials Module are used for this simulation.
The valve model consists of a multilayer stacked piezoelectric actuator, which in itself is a complex structure of stacked layers and electrodes. The model also includes a stainless steel substrate and a seal of hyperelastic material over the through hole of the valve.
For the simulation, we apply 50 volts to the layers. The contact pressure is determined here at the two contact pressure points of the seal. We can see that deformation of the disc is greatest at the center, which compresses the hyperelastic seal against the valve’s opening and closes the valve.
Left: The strain at the two contact surfaces of the valve’s seal. Here, we can see that the deformation of the disc is greatest at the center, which closes the valve. Right: The contact pressure at the two surface points of the valve’s seal.
Modeling a piezoelectric valve allows us to analyze the operation of the stacked piezoelectric actuator and evaluate the stress and strain in the seal and the surrounding materials. The analysis could be extended to estimate the performance of the seal with different pressure differentials applied across the valve in the closed state.
MEMS gyroscopes are found in a wide range of consumer products, from phones to game consoles to cars. These devices detect the rotation of an object, often using a coupling between motion in two directions at right angles induced by the Coriolis force.
The figure above shows two degenerate modes of a section through a wine glass. The equation is a mathematical expression of the fact that the two modes are orthogonal, or independent of each other. We could construct other degenerate pairs by taking combinations of these two modes that are also orthogonal. They would look similar in form, except for a rotation about the center of the wine glass.
The principle behind degenerate mode sensors was first discovered by the physicist G.H. Bryan when he struck the edge of a wine glass and listened to its vibrations. He found that when the glass was being rotated about its stem, the tone of vibration undulated, which he recognized as beats between two sound waves with slightly different frequencies.
What Bryan had discovered was the symmetry breaking of degenerate modes — two different modes of vibration that resonate at identical frequencies. In vibrating objects that are symmetric, such as Bryan’s wine glass, two degenerate mode pairs exist for every degree of rotation (natural number). When a symmetric object is rotating, Coriolis forces break the symmetry between sound waves generated by the object, splitting their frequencies in proportion to the rate of rotation. Measuring the difference in frequency of the broken degenerate modes can tell us the rate of rotation.
This concept is the principle behind the design of some MEMS gyroscopes. When the symmetry of the system is broken by an environmental factor of interest (in the case of a MEMS gyroscope — the Coriolis force), this change causes degeneracy breaking in the modes of vibration that can be used as a means to sense the environmental factor. However, instead of using the Coriolis force to cause degeneracy breaking, we could use other factors that change the symmetry of the device. For example, selectively adding a small amount of material to a part of the object can alter its symmetry and cause degeneracy breaking. Thus, sensors other than gyroscopes can utilize this principle.
In the paper “Degeneracy Breaking, Modal Symmetry and MEMS Biosensors” by H.T.D. Grigg, T.H. Hanley, and B.J. Gallacher, researchers at Newcastle University in the UK leveraged simulation to investigate the effects of material and geometric symmetry breaking in a degenerate mode sensor, a device designed using the same principles behind the MEMS gyroscopes that operate through degeneracy breaking.
Using COMSOL Multiphysics software, the team designed a degenerate Rayleigh SAW device with a symmetric geometry — that initially had degenerate modes. The surface of the sensor is treated so that an analyte (a substance or chemical of interest) will bind selectively to parts of the device, leading to a non-symmetric mass loading. This induces a frequency split in the degenerate modes.
The vibrational mode used can be seen in an image from their COMSOL Conference presentation:
Image credit: H.T.D. Grigg, T.H. Hanley, B.J. Gallacher, Newcastle University, Newcastle upon Tyne, United Kingdom.
Since the device is symmetric, the researchers used a quarter of the geometry in order to reduce computation time. The images below show how the device was designed using an anisotropic piezoelectric substrate covered by an isotropic layer. This arrangement of materials creates dispersive, anisotropic Rayleigh waves. The device was modeled with the Piezoelectric Devices interface using Frequency Domain and Time Dependent studies in COMSOL Multiphysics. The image below also shows the Perfectly Matched Layer (PML), which the researchers used to simulate a boundary absorber.
In their simulations, the researchers broke the degeneracy of the modes by changing the boundary conditions and/or the material properties in a non-symmetric manner. In a laboratory experiment, the degeneracy would be broken by asymmetric mass loading of the analyte. Because the sensor is symmetric with respect to other environmental factors (such as changes in temperature, etc.), it is generally robust to environmental disturbances — just like the MEMS gyroscopes that operate on the same principle.
Image credit: H.T.D. Grigg, T.H. Hanley, B.J. Gallacher, Newcastle University, Newcastle upon Tyne, United Kingdom.
To learn more about the MEMS biosensor, download the paper and presentation from the COMSOL Conference 2013.
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When light from an astronomical object, such as a star, passes through the earth’s atmosphere, the atmospheric turbulence causes the blurring and twinkling (scintillation) of star images. A similar effect happens when you look at an object in a pool or over a fire. It limits the resolution of optical instruments and deteriorates the quality of astronomical images. Astronomers fought the effect of this so-called astronomical seeing for centuries. They tried to escape it by building their observatories on mountaintops and launching telescopes into space.
In an effort to overcome these difficulties, the American astronomer Horace W. Babcock presented the groundbreaking idea of adaptive optics in 1953. This concept was first pursued independently for both astronomical and military applications, but it wasn’t until the advent of modern computer technology that adaptive optics became feasible and accessible for broad science and commercial applications.
Nowadays, adaptive optics is used not only in astronomical telescopes, but also in laser communications; laser material processing; meteorology; military and security applications, such as surveillance; biomedical technology, such as ophthalmology and vision science; in consumer devices to improve the quality of images; and even in robotics vision.
The global adaptive optics market has a turnover of about $40 million and is expected to grow to about $40 billion in 2022, according to this report.
Schematic of an adaptive optics system. The wavefront enters the system at the top. The light first hits a tip-tilt (TT) mirror and is then directed to a deformable mirror (DM). The wavefront is corrected and part of the light is tapped off by a beamsplitter (BS). The wavefront is measured by a wavefront sensor (Shack-Hartmann in this case) and the control hardware then sends updated signals to the DM and TT mirror. The two filterwheels (FW1 and FW2) are used during calibration only. Source: Wikimedia Commons. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
The basic principle is simple but powerful. The adaptive optics system measures the incoming wavefront with a wavefront sensor, calculates the correction, and applies it to the optical component to correct the wavefront in real time. The optical component and a key part of the system is mostly a deformable mirror, which consists of an array of actuators connected to an optical surface that deforms according to the movement of the actuators.
Deformable mirrors can be based on different actuation methods, such as magnetic, electrostatic, or piezoelectric methods. Today, micro-electro-mechanical systems (MEMS) deformable mirrors are the most commonly used technology. Recently, some new concepts are being explored, such as micro-optical-electro-mechanical systems (MOEMS) and ferrofluid mirrors.
The image of the star HIC59206 made with the Very Large Telescope as corrected with an adaptive optics system. Source: Wikimedia Commons. This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
The design of adaptive optics systems, especially deformable mirrors with different MEMS actuators, can benefit from the power of multiphysics modeling. The COMSOL Multiphysics simulation software platform is the ideal tool for modeling the key components in such adaptive optics systems.
Not surprisingly, adaptive optics, in particular its application in astronomical systems, has already been a highlighted topic at several of our user conferences. Modern astrophysics is a high-tech science that profits from a strong connection between industry and research to solve various engineering challenges in ambitious scientific projects.
One such scientific and engineering challenge is the development of large advanced telescopes with the mirror diameters ranging from a few meters to as much as 40 meters. For example, the Thirty Meter Telescope, currently under construction at the Mauna Kea observatory in Hawaii, will have an optical performance nearly ten times better than the Hubble Space Telescope, thanks to its innovative adaptive optics system. Another giant telescope, which is being built in Chile, is the 39-meter-long European Extremely Large Telescope — one of the mirrors will involve over 6,000 actuators deforming its shape a thousand times a second!
At the COMSOL Conference 2012 in Milan, a team of scientists from the Arcetri Astrophysical Observatory and the University of Cassino and Southern Latium in Italy presented a paper on using COMSOL Multiphysics to design an adaptive optics actuator called the Variable Reluctance Adaptive mirror Linear Actuator (VRALA), which is based on a magnetic circuit.
Operating an adaptive mirror for a ten-meter-class telescope on visible wavelengths requires the mirror to be more slender and faster than on other wavelengths. VRALA is the ideal candidate for the actuators at those wavelengths. The team used COMSOL simulation software in the design process for electromagnetic, mechanical, and thermal studies.
At the COMSOL Conference 2013 in Rotterdam, a team from the Arcetri Astrophysical Observatory and two Italian companies, ADS International and Microgate, presented their studies on deformable mirrors from the Large Binocular Telescope located in Arizona and the Very Large Telescope in Chile.
The actuator geometries found in their adaptive optics systems are complex. The deformable mirror of the VLT has, for example, 1,170 actuators. The team used COMSOL Multiphysics and LiveLink™ for MATLAB® to calculate the so-called influence functions showing the deformation that any combination of actuators would produce.
At the more recent COMSOL Conference 2014 in Cambridge, a team from the Arcetri Astrophysical observatory presented once again. This time, they showed how they used COMSOL Multiphysics and LiveLink™ for MATLAB® to match simulated influence functions with the measurements, thus validating the simulations. You can find these papers here and here.
At the COMSOL Conference 2014 in Boston, a team of scientists from the NASA Goddard Space Flight Center and two U.S. companies, the Newton Corporation and Iris AO, Inc., showed how they used a finite element model developed with COMSOL software to perform structural mechanics simulations of a MEMS mirror segment. This mirror segment, called the Multiple Mirror Array, will be used as a key component in the instrument called the Visible Nulling Coronagraph, which is intended for the detection of Earth-size exoplanets. The team also used the COMSOL Multiphysics model to predict the dynamic behavior and stresses of the segment when undergoing mechanical shock during a spaceflight.
The MEMS Module provides modeling tools to efficiently simulate different types of MEMS components and devices, including MEMS actuators.
In such devices and at small scales, it is necessary to consider couplings of different physical phenomena, such as electromagnetic-structure, thermal-structure, or fluid-structure interactions. Thus, MEMS devices represent truly multiphysics applications.
In the Model Library and our online Model Gallery, you can find several COMSOL Multiphysics example models relevant to adaptive optics systems.
For instance, the model of the electrostatically actuated cantilever shows the bending of an elastic cantilever beam due to electrostatic forces. The Electromechanics interface found in the MEMS Module allows the calculation of the elastic deformation of a cantilever beam in response to electrostatic forces, which are induced by an applied potential between the cantilever and the substrate beneath it.
As the cantilever bends, the forces are modified, since the shape of the gap between the two surfaces changes. The deformation of the gap region is tracked by the built-in moving mesh functionality (such functionality can also be referred to as ALE or Arbitrary Lagrangian Eulerian) in COMSOL Multiphysics. The COMSOL simulation software computes the electrostatic forces in a self-consistent manner throughout the process.
Displacement in an electrostatically actuated cantilever.
Using the same Electromechanics interface, you can also perform a modal, frequency, and transient analysis of a MEMS cantilever. You can also estimate the critical voltage, or the pull-in voltage, when the system becomes unstable. The sequence of models of the electrostatically actuated MEMS resonator shows how to characterize such microelectromechanical resonators, for example by calculating normal modes and a frequency-dependent response of the system. For tutorial purposes, these models are also available in 2D.
You can also model thermal residual stress in thin-film resonators, which are often a result of the fabrication process. Using the Solid Mechanics interface, the Thermal Expansion feature, and the Eigenfrequency study feature, you can calculate how the thermal stress changes the resonant frequency of the thin-film resonator.
Another example is the model of a prestressed micromirror. Such a mirror could be used as an optical reflection device. To create curved surfaces or a spring-like structure, MEMS device manufacturers sometimes use the plating process to introduce a residual stress in the micromirror. The example demonstrates how to set up such model and include initial stress and strain in the model. You can also see how the deformed structure differs for different materials, such as aluminum and steel.
Micromirror deformation and lift-off for aluminum.
Micromirror deformation and lift-off for steel.
COMSOL Multiphysics also offers a number of other capabilities to perform optical simulations and couple them with mechanical, thermal, or other simulations of MEMS components.
The Wave Optics Module provides dedicated tools for high-frequency electromagnetic wave simulations in optical media by way of the innovative beam envelope method. You may also couple structural mechanics with wave optics simulations, like in this cavity model or in this waveguide simulation.
The new Ray Optics Module can be used to model light propagation in optical media and devices, treating electromagnetic waves as rays. You have a number of ray optics examples at your disposal, including the model of a corner cube retroreflector or light propagation in a Newtonian telescope.
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The piezoelectric modeling interface seeks to:
This will allow you to successfully simulate piezoelectric devices as well as easily extend the simulation by coupling it with any other physics.
You may already be familiar with the three different modules that can be used for simulating piezoelectric materials:
Each of these modules gives you a predefined Piezoelectric Devices interface that you can use for modeling systems that include both piezoelectric and other structural materials. The Acoustics Module offers two predefined interfaces, namely the Acoustic-Piezoelectric Interaction, Frequency Domain interface and the Acoustic-Piezoelectric Interaction, Transient interface. These two allow you to model how piezoelectric acoustic transducers interact with the fluid media surrounding them.
The Piezoelectric Devices interface is available in the list of structural mechanics physics interfaces.
The Acoustic-Piezoelectric Interaction, Frequency Domain and the Acoustic-Piezoelectric Interaction, Transient interfaces are available in the list of acoustics physics interfaces.
These predefined multiphysics interfaces couple the relevant physics governing equations via constitutive laws or boundary conditions. Thus, they offer a good starting point for setting up more complex multiphysics problems involving piezoelectric materials. The new piezoelectric interfaces in COMSOL Multiphysics version 5.0 provide a transparent workflow to visualize the constituent physics interfaces. There is also a separate Multiphysics node that lists how the constituent physics interfaces are connected to each other.
Let’s find out how these multiphysics interfaces are structured.
Upon selecting the Piezoelectric Devices multiphysics interface, you see the constituent physics: Solid Mechanics and Electrostatics. You also see the Piezoelectric Effect branch listed under the Multiphysics node, which controls the connection between Solid Mechanics and Electrostatics.
Part of the model tree showing the physics interfaces and multiphysics couplings that appear upon selecting the Piezoelectric Devices interface.
By default, all modeling domains are assumed to be made of piezoelectric material. If that is not the case, you can deselect the non-piezo structural domains from the branch Solid Mechanics > Piezoelectric Material. These domains then get automatically assigned to the Solid Mechanics > Linear Elastic Material branch. This process ensures that all parts of the geometry are marked as either piezoelectric or non-piezo structural materials and that nothing is accidentally left undefined.
If you are working with other material models that are available with the Nonlinear Structural Materials Module, such as hyperelasticity, you can add that as a branch under Solid Mechanics and assign the relevant parts of your modeling geometry to this branch. The Solid Mechanics node gives us full flexibility to set up a model that involves not only piezoelectric material but also linear and nonlinear structural materials. The best part is that if these materials are geometrically touching each other, the COMSOL software will automatically take care of displacement compatibility across them.
If some parts of the model are not solid at all, like an air gap, you can deselect them in the Solid Mechanics node.
From the Solid Mechanics node, you will also assign any sort of mechanical loads and constraints to the model.
The Electrostatics node allows you to group together all the information related to electrical inputs to the model. This would include, for example, any electrical boundary conditions such as voltage and charge sources. By default, any geometric domain that has been assigned to the Solid Mechanics > Piezoelectric Material branch also gets assigned to the Electrostatics > Charge Conservation, Piezoelectric branch. If you have any other dielectric materials in the model that are not piezoelectric, you could assign them to the Electrostatics > Charge Conservation branch.
The Multiphysics > Piezoelectric Effect branch ensures that the structural and electrostatics equations are solved in a coupled fashion within the domains that are assigned to the Solid Mechanics > Piezoelectric Material (and also the Electrostatics > Charge Conservation, Piezoelectric) branch.
The multiphysics coupling is implemented using the well-known coupled constitutive law for piezoelectric materials. Note that the Electrostatics > Charge Conservation, Piezoelectric branch is mainly used as a placeholder for assigning geometric domains that belong to the piezoelectric material model. This helps the Multiphysics > Piezoelectric Effect branch understand whether a domain assigned to the Electrostatics interface is piezoelectric or not.
Note: For an example of working with the Piezoelectric Devices interface, check out the tutorial on modeling a Piezoelectric Shear Actuated Beam.
It is also possible to add effects of damping or other material losses in dynamic simulations. You can do so by adding one or more of the following subnodes under the Solid Mechanics > Piezoelectric Material branch:
Damping and losses that can be added to a piezoelectric material.
Subnode Name | When to Use the Subnode |
---|---|
Mechanical Damping | Allows you to add purely structural damping. Choose between using Loss Factor (in frequency domain) or Rayleigh damping (for both frequency and time domains) models. |
Coupling Loss | Allows you to add electromechanical coupling loss. Choose between using Loss Factor (for frequency domain) or Rayleigh damping (for both frequency and time domains) models. |
Dielectric Loss | Allows you to add dielectric or polarization loss. Choose between using Loss Factor (for frequency domain) and Dispersion (for both frequency and time domains) models. |
Conduction Loss (Time-Harmonic) | Allows you to add electrical energy dissipation due to electrical resistance in a harmonically vibrating piezoelectric material (for frequency domain only). |
Note: For an example of adding damping to piezoelectric models, check out the tutorial on modeling a Thin Film BAW Composite Resonator.
Additional damping also takes place due to the interaction between a piezoelectric device and its surroundings. This can be modeled in greater details using the Acoustic-Piezoelectric Interaction interfaces.
Upon selecting one of the Acoustic-Piezoelectric Interaction interfaces, you see the constituent physics: Pressure Acoustics, Solid Mechanics and Electrostatics. You also see the Acoustic-Structure Boundary and Piezoelectric Effect branches listed under the Multiphysics node.
Part of the model tree showing the physics interfaces and multiphysics couplings that appear when selecting the Acoustic-Piezoelectric Interaction, Frequency Domain and the Acoustic-Piezoelectric Interaction, Transient interfaces.
By default, all modeling domains are assigned to the Pressure Acoustics interface as well as the Solid Mechanics > Piezoelectric Material and Electrostatics > Charge Conservation, Piezoelectric branches. Note that the Pressure Acoustics interface is designed to simulate acoustic waves propagating in fluid media.
Since COMSOL Multiphysics cannot know a priori which parts of the modeling geometry belong to the fluid domain and which ones are solids, you are expected to provide that information by deselecting the solid domains from the Pressure Acoustics, Frequency Domain (or Pressure Acoustics, Transient) branch and deselecting the fluid domains from the Solid Mechanics and Electrostatics branches.
Once you do that, the boundaries at the interface between the solid and fluid domains are detected and assigned to the Multiphysics > Acoustic-Structure Boundary branch. This branch controls the coupling between the Pressure Acoustics and Solid Mechanics physics interfaces. It does so by considering the acoustic pressure of the fluid to be acting as a mechanical load on the solid surfaces, while the component of the acceleration vector that is normal (perpendicular) to the same surfaces acts as a sound source that produces pressure waves in the fluid.
Note: For an example of Acoustic-Piezoelectric Interaction, check out the tutorial on modeling a Tonpilz Transducer.
The transparency in the workflow as discussed above also paves the way for adding more physics and creating your own multiphysics couplings.
For example, let’s say there is some heat source within your piezoelectric device that produces nonuniform temperature distribution within the device. In order to model this, you can add another physics interface called Heat Transfer in Solids in the model tree and prescribe appropriate heat sources and sinks to find out the temperature profile. You could then add a Thermal Expansion branch under the Multiphysics node to compute additional strains in different parts of the device as a result of the temperature variation.
The Multiphysics > Thermal Expansion branch couples the Heat Transfer in Solids and the Solid Mechanics interfaces. It might also be possible that the piezoelectric material properties have a temperature dependency. You could represent these properties as functions of temperature and let the Multiphysics > Temperature Coupling branch pass on the information related to temperature distribution in the modeling geometry to the Solid Mechanics or even the Electrostatics branches, thereby producing additional multiphysics couplings.
Part of the model tree showing the physics interfaces and multiphysics couplings that you can use to combine piezoelectric modeling with thermal expansion and temperature-dependent material properties.
Similar to adding more physics and multiphysics couplings, it is also possible to disable one or more multiphysics couplings — or even any of the physics interfaces shown in the model tree. This could be immensely helpful for debugging large and complex models.
The model tree on the left shows a scenario where the Piezoelectric Effect multiphysics coupling is disabled. The model tree on the right shows a scenario where the Electrostatics physics interface is disabled.
For example, you can disable the Multiphysics > Piezoelectric Effect branch and solve for the Solid Mechanics and Electrostatics physics interfaces in an uncoupled sense. You could also solve a model by disabling either the Solid Mechanics or the Electrostatics interface.
Running such case studies could help in evaluating how the device would respond to certain inputs if there were no piezoelectric material in place. This approach could also be used to evaluate equivalent structural stiffness or equivalent capacitance of the piezoelectric material.
You could also start by adding only one of the constituent physics, say Solid Mechanics, and after performing some initial structural analysis, go ahead and add the Electrostatics physics interface to the model tree once you are ready to add the effect of a piezoelectric material.
In that case, when you add the Electrostatics physics on top of the existing Solid Mechanics physics in the model tree, the COMSOL software will automatically add the Multiphysics node. From there, you can manually add the Piezoelectric Effect branch. Note that if you take this approach of adding the constituent physics interfaces and multiphysics effect manually, you would also have to manually add the piezoelectric modeling domains to the Solid Mechanics > Piezoelectric Material, the Electrostatics > Charge Conservation, Piezoelectric, and the Multiphysics > Piezoelectric Effect branches.
In a similar fashion, you can continue to add more physics interfaces and multiphysics couplings to your model based on your needs.
To learn more about modeling piezoelectric devices in the COMSOL software environment, you are encouraged to refer to these resources:
Among MEMS manufacturing technologies, surface micromachining is recognized for its ability to merge electronic components and freely moving mechanical parts on the one substrate. The fabrication process begins with the deposition of thin films on top of a silicon substrate. These varying structural layers are then selectively etched to free the moving parts and create the desired structure.
Featuring a low cost and the capability for high-volume production, this form of micromachining has found several application areas within MEMS device manufacture. One example is the surface micromachined accelerometer. This type of accelerometer is particularly noted for its important use in triggering airbags in automobiles.
Now that we’ve acquired some background info on the manufacturing process, let’s dive into the modeling of this device.
The Surface Micromachined Accelerometer model is based on a case study from the book Microsystem Design by Stephen D. Senturia. Using polysilicon as the building material, this model consists of a released proof mass that is supported by anchored springs at both ends, along with sensing and self-test electrodes that extend to the sides of the device.
The geometric building blocks — the proof mass with attached electrodes, the folded spring, and the fixed electrode array — are implemented as Subsequences in the geometry building process in COMSOL Multiphysics, such that the building blocks can be stored in a source model file. By linking to the Subsequences in the source model file, these same geometric building blocks can be re-used in alternate model files. This is enabled by the Linked Subsequences geometry feature, which was first introduced in COMSOL Multiphysics version 5.0.
Subsequences can take on a number of arguments, which result in varying dimensions, orientations, positions, and numbers of features. In the example below, the two electrode arrays are composed with the same Subsequence but have different sets of arguments. This affects the number of electrodes, the dimensions, and how the anchor pads are oriented within each array.
Electrode arrays built from the same Subsequence but featuring varying sets of arguments. This results in different dimensions, anchor pad orientations, and numbers of electrodes.
When acceleration is applied to the device — in this case, by using the Body Load domain feature — the restoring force from the springs causes a displacement in the proof mass that is proportional to the acceleration. This displacement in turn affects the capacitance between the fixed and moving electrodes, a change that can be measured with various standard circuits.
With the Electromechanics interface, we can model the electric field within the deforming gaps between the electrodes. This physics interface also allows us to apply the self-test electrostatic forces to the solids, which produces a corresponding deformation within the structure.
Our initial study begins with analyzing the displacement in polysilicon domains of the model after an applied acceleration of 50 g. A movement of around 0.07 micrometers is observed in the proof mass, including the attached moving electrodes. While very little movement is seen in the anchored spring bases and the fixed electrodes, varying displacement is noted along the length of the folded springs, as predicted.
Displacement resulting from an applied acceleration of 50 g.
Additionally, this study highlights the linear relationship between the applied acceleration and the displacement. The measurement of the displacement is derived from the capacitive coupling between the moving and fixed sense electrodes.
In a real device, an alternating square-wave voltage is applied to the fixed sense electrodes. When the applied acceleration causes the proof mass to move, an alternating voltage that is proportional to the displacement is induced as a result of this capacitive coupling between the moving and fixed electrodes. This set-up enables easier signal processing in the attached circuitry. In our simplified model, only one half of the square wave is modeled as a stationary problem to save computation time without losing generality.
We then turn our attention to the accelerometer’s self-test electrodes. Here, we analyze the displacement that occurs in the polysilicon domains when 0 volt is applied to the fixed electrodes to the left of the moving electrodes that are attached to the proof mass. A bias of 2 volts is applied to those on the right-hand side. The results show a movement in the proof mass of around 0.02 micrometers, a magnitude that is big enough for the self-test purpose.
Displacement following self-test voltage.
When comparing the displacement between applying the self-test bias to the set of fixed electrodes on the left-hand side of the moving electrodes versus the one on the right-hand side, the values of the displacement are found to have the same size with opposite signs. This result shows agreement with the prediction from symmetry.
In this blog post, we have shown you how to model a surface micromachined accelerometer and visualize displacement within the device in response to electric forces. Additionally, we have demonstrated the use of the new Linked Subsequences geometry feature, emphasizing its role in making the geometry building process more efficient.
You’ve probably heard the word “graphene” in the news and here on the blog numerous times, usually with references to its powerful capabilities in advancing technology within various industries. It’s not every day that a material quite as unique and powerful as graphene comes along and it’s safe to say the world has taken notice.
The structure of graphene. Image by K. M. Al-Shurman and H. A. Naseem and is taken from the paper titled “CVD Graphene Growth Mechanism on Nickel Thin Films“.
Graphene has been a relevant topic on the minds of many, ourselves included. In a recent series of blog posts, we highlighted the revolution behind this material, from its exotic properties and production methods to simulating its use in various applications.
Our last post in the series emphasized research on the “wonder material” that led to the accidental discovery of 2D glass. While the discovery in itself is remarkable, the point on which I’d like to focus is how they actually grew the graphene used in the research — through chemical vapor deposition (CVD).
Chemical vapor deposition describes the chemical process designed to create solid materials that perform strongly and are highly pure. In this method, gas molecules are combined in a reaction chamber containing a heated substrate. The interaction between the gases and the heated substrate causes the gases to react and/or decompose on the substrate’s surface, thus producing a material film.
This synthesis method is particularly valued for its ability to produce materials that are rather high in quality. Compared to other coating methods, the resulting materials in chemical vapor deposition tend to possess greater purity, hardness, and resistance to agitation or damage. An additional advantage within this method is the wide range of materials that can be deposited, one of which is graphene.
Among synthesis techniques, chemical vapor deposition has proved promising in the development of high-quality graphene films. The process involves growing graphene films on different kinds of substrate that utilize transition metals. One such example is nickel (Ni). This involves the diffusion of decomposed carbon atoms into nickel at a high temperature, followed by the precipitation of carbon atoms on the surface of the nickel during the cooling process.
Because of the multiplicity of the growth conditions in the CVD method, producing a single-layer graphene and maintaining control over the quality of the graphene film can be very challenging. One research team from the University of Arkansas recognized the need to better understand the growth mechanism as well as optimal conditions for graphene production.
Using COMSOL Multiphysics, the researchers created a graphene synthesis model to analyze the dissolution-precipitation mechanism for CVD graphene growth on nickel. In the study, they analyzed factors affecting the number of graphene layers synthesized, including growth time and temperature, rate of cooling, carbon solubility in nickel, and the nickel’s film thickness.
A schematic showing the mechanism for CVD graphene growth on Ni. Image by K. M. Al-Shurman and H. A. Naseem and is taken from the poster titled “CVD Graphene Growth Mechanism on Nickel Thin Films“.
In analyzing the diffusion of the carbon atoms, the team found that the greater the temperature within the Ni film, the more accelerated the diffusion process was. From their results, they also concluded that additional time was needed for carbon atoms to reach their saturated state in thicker Ni film.
Additionally, the researchers modeled supersaturation by cooling. In the supersaturation process, carbon atoms become segregated on the surface of the Ni thin film. When cooling the film from 900°C to 725°C, 1.7 layers of graphene were obtained on the film’s surface. This resulting number of graphene layers proved reasonable in comparison to experimental data.
Graph highlighting the number of layers produced when the Ni film is cooled from 900°C to 725°C. Image by K. M. Al-Shurman and H. A. Naseem and is taken from the presentation titled “CVD Graphene Growth Mechanism on Nickel Thin Films“.
Piezoelectric materials become electrically polarized when strained. From a microscopic perspective, the displacement of charged atoms within the crystal unit cell (when the solid is deformed) produces a net electric dipole moment within the medium. In certain crystal structures, this combines to give an average macroscopic dipole moment and a corresponding net electric polarization. This effect, known as the direct piezoelectric effect, is always accompanied by the inverse piezoelectric effect, in which the solid becomes strained when placed in an electric field.
Several material properties must be defined in order to fully characterize the piezoelectric effect within a given material. The relationship between the material polarization and its deformation can be defined in two ways: the strain-charge or the stress-charge form. Different sets of material properties are required for each of these equation forms.
To complicate things further, there are two standards used in the literature: the IEEE 1978 Standard and the IRE 1949 standard, and the material properties take different forms within the two standards. IEEE actually revised the 1978 standard in 1987, but this version of the standard contained a number of errors and was subsequently withdrawn. Confused yet? I was when I first started reading the literature!
Today’s blog post describes in detail the different equation forms and standards, with a focus on the particular case of quartz — the material that causes the most confusion. In both academia and industry, the quartz material properties are commonly defined within the older 1949 IRE standard. Meanwhile, other materials are now almost always defined using the 1978 IEEE standard. To make matters worse, it is not common to indicate which standard is being employed when specifying the material properties.
The coupling between the structural and electrical domains can be expressed in the form of a connection between the material stress and its permittivity at constant stress or as a coupling between the material strain and its permittivity at constant strain. The two forms are given below.
The strain-charge form is written as:
where S is the strain, T is the stress, E is the electric field, and D is the electric displacement field. The material parameters s_{E}, d, and ε_{rT} correspond to the material compliance, coupling properties, and relative permittivity at constant stress. ε_{0} is the permittivity of free space. These quantities are tensors of rank 4, 3, and 2, respectively. The tensors, however, are highly symmetric for physical reasons. They can be represented as matrices within an abbreviated subscript notation, which is usually more convenient. In literature, the Voigt notation is almost always used.
Within this notation, the above two equations can be written as:
\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}s_{E11} & s_{E12} &s_{E13} &s_{E14} &s_{E15} &s_{E16}\\
s_{E21} & s_{E22} &s_{E23} &s_{E24} &s_{E25} &s_{E26}\\
s_{E31} & s_{E32} &s_{E33} &s_{E34} &s_{E35} &s_{E36}\\
s_{E41} & s_{E42} &s_{E43} &s_{E44} &s_{E45} &s_{E46}\\s_{E51} & s_{E52} &s_{E53} &s_{E54} &s_{E55} &s_{E56}\\s_{E61} & s_{E62} &s_{E63} &s_{E64} &s_{E65} &s_{E66}\\\end{array}
\right)\left(
\begin{array}{l}T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
d_{11} & d_{21} & d_{31} \\
d_{12} & d_{22} & d_{32} \\
d_{13} & d_{23} & d_{33} \\
d_{14} & d_{24} & d_{34} \\
d_{15} & d_{25} & d_{35} \\
d_{16} & d_{26} & d_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
d_{11} & d_{12} &d_{13} & d_{14} & d_{15} & d_{16}\\
d_{21} & d_{22} &d_{23} & d_{24} & d_{25} & d_{26}\\
d_{31} & d_{32} &d_{33} & d_{34} & d_{35} & d_{36}\\
\end{array}
\right)\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rT11} & \epsilon_{rT12} & \epsilon_{rT13} \\
\epsilon_{rT21} & \epsilon_{rT22} & \epsilon_{rT23} \\
\epsilon_{rT31} & \epsilon_{rT32} & \epsilon_{rT33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}
The stress-charge form is as follows:
The material parameters c_{E}, e, and ε_{rS} correspond to the material stiffness, coupling properties, and relative permittivity at constant strain. ε_{0} is the permittivity of free space. Once again, these quantities are tensors of rank 4, 3, and 2 respectively, but can be represented using the abbreviated subscript notation.
Using the Voigt notation and writing out the components gives:
\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}c_{E11} & c_{E12} &c_{E13} &c_{E14} &c_{E15} &c_{E16}\\
c_{E21} & c_{E22} &c_{E23} &c_{E24} &c_{E25} &c_{E26}\\
c_{E31} & c_{E32} &c_{E33} &c_{E34} &c_{E35} &c_{E36}\\
c_{E41} & c_{E42} &c_{E43} &c_{E44} &c_{E45} &c_{E46}\\c_{E51} & c_{E52} &c_{E53} &c_{E54} &c_{E55} &c_{E56}\\c_{E61} & c_{E62} &c_{E63} &c_{E64} &c_{E65} &c_{E66}\\\end{array}
\right)\left(
\begin{array}{l}S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
e_{11} & e_{21} & e_{31} \\
e_{12} & e_{22} & e_{32} \\
e_{13} & e_{23} & e_{33} \\
e_{14} & e_{24} & e_{34} \\
e_{15} & e_{25} & e_{35} \\
e_{16} & e_{26} & e_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
e_{11} & e_{12} &e_{13} & e_{14} & e_{15} & e_{16}\\
e_{21} & e_{22} &e_{23} & e_{24} & e_{25} & e_{26}\\
e_{31} & e_{32} &e_{33} & e_{34} & e_{35} & e_{36}\\
\end{array}
\right)\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rS11} & \epsilon_{rS12} & \epsilon_{rS13} \\
\epsilon_{rS21} & \epsilon_{rS22} & \epsilon_{rS23} \\
\epsilon_{rS31} & \epsilon_{rS32} & \epsilon_{rS33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}
The matrices defined in the above equations are the key material properties that need to be defined for a piezoelectric material. Note that for many materials, a number of the elements in each of the matrices are zero and several others are related, as a result of the crystal symmetry.
Using the international notation for describing crystal symmetry, the symmetry group of quartz is Trigonal 32. The nonzero matrix elements take different values within different standards, which can result in confusion when specifying the material properties for a simulation, especially for quartz, where two different standards are commonly employed.
Finally, there is another complication in the case of quartz. Quartz crystals do not have symmetry planes parallel to the vertical axis. Correspondingly, they occur in two types: left- or right-handed (this is known as enantiomorphism). Each one of these enantiomorphic forms results in different signs for particular elements in the material property matrices.
The material property matrices appropriate for quartz and other Trigonal 32 materials are shown below. Note that the symmetry relationships between elements in the matrix hold irrespective of the standard used or whether the material is right- or left-handed.
\left(
\begin{array}{cccccc}
c_{E11} & c_{E12} &c_{E13} & c_{E14} & 0 & 0\\
c_{E12} & c_{E11} &c_{E13} & -c_{E14} &0 & 0\\
c_{E13} & c_{E13} &c_{E33} & 0 & 0 & 0\\
c_{E14} & -c_{E14} & 0 & c_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & c_{E44} & c_{E14}\\
0 & 0 & 0 & 0 & c_{E14} & \frac{1}{2}\left(c_{E11}-c_{E12}\right)\\
\end{array}
\right)
&
\left(
\begin{array}{cccccc}
s_{E11} & s_{E12} &s_{E13} & s_{E14} & 0 & 0\\
s_{E12} & s_{E11} &s_{E13} & -s_{E14} &0 & 0\\
s_{E13} & s_{E13} &s_{E33} & 0 & 0 & 0\\
s_{E14} & -s_{E14} & 0 & s_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & s_{E44} & 2 s_{E14}\\
0 & 0 & 0 & 0 & 2 s_{E14} & 2\left(s_{E11}-s_{E12}\right)\\
\end{array}
\right)
\\
\left(
\begin{array}{cccccc}
e_{11} &-e_{11} & 0 & e_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & -e_{14} & -e_{11}\\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)
&
\left(
\begin{array}{cccccc}
d_{11} & -d_{11} & 0 & d_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & -d_{14} & -2d_{11} \\
0 & 0 & 0 & 0 & 0 & 0\\
\end{array}
\right)
\\
\left(
\begin{array}{ccc}
\epsilon_{rS11} & 0 & 0 \\
0 & \epsilon_{rS11} & 0 \\
0 & 0 & \epsilon_{rS33} \\
\end{array}
\right)
&
\left(
\begin{array}{ccc}
\epsilon_{rT11} & 0 & 0 \\
0 & \epsilon_{rT11} & 0 \\
0 & 0 & \epsilon_{rT33} \\
\end{array}
\right)
\\
\end{array}
Having defined a set of material properties in terms of matrices that operate on the different components of the stress or the strain in the x,y,z axes system, all that remains is to define a consistent set of axes to use when writing down the material properties.
Correspondingly, all of the standards define a consistent set of axes for each of the relevant crystal classes. Unfortunately, in the particular case of quartz, subsequent standards have not used the same sets of axes, and the adoption of the most recent standard has not been widespread. Therefore, it is important to understand exactly which standard a given set of material properties is defined in.
The two relevant standards are:
The orientation of the axes set with the crystal can be determined by specifying the orientation with respect to the atoms in the unit cell of the crystal (which is not that helpful in practice) or by specifying the orientation with respect to the crystal forms. A crystal form is a set of crystal faces or planes that are related by symmetry. Particular forms commonly appear in crystal specimens found in rocks and are used to identify different minerals.
The Quartz Page has a series of helpful figures for identifying the common crystal forms, termed m, r, s, x, and z, as well as a further page specifying the Miller indices of the corresponding planes. Since the standards typically use crystal forms to orientate the axes, this approach is adopted in the figure below, which shows the two axes sets that relate to the 1978 and 1949 standards. Note that both left- and right-handed quartz are shown in the figure.
Crystallographic axes defined for quartz within the 1978 IEEE standard (solid lines) and the 1949 standard (dashed lines). Click on the image to view a larger version.
As a result of the different crystal axes, the signs of the material properties for both right- and left-handed quartz can change depending on the particular standard employed. The table below summarizes the different signs that occur for the quartz material properties:
IRE 1949 Standard |
IEEE 1978 Standard |
|||
---|---|---|---|---|
Material Property |
Right-Handed Quartz |
Left-Handed Quartz |
Right-Handed Quartz |
Left-Handed Quartz |
s_{E14} |
+ |
+ |
– |
– |
c_{E14} |
– |
– |
+ |
+ |
d_{11} |
– |
+ |
+ |
– |
d_{14} |
– |
+ |
– |
+ |
e_{11} |
– |
+ |
+ |
– |
e_{14} |
+ |
– |
+ |
– |
Usually, piezoelectrics, such as quartz, are supplied in thin wafers that have been cut at a particular angle, with respect to the crystallographic axes. The orientation of a piezoelectric crystal cut is frequently defined by the system used in both the 1949 and 1978 standards. The orientation of the cut, with respect to the crystal axes, is specified by a series of rotations, using notation that takes the form illustrated below:
Diagram showing how a GT cut plate of quartz is defined in the IEEE 1978 standard. The crystal shown is right-handed quartz.
The first two letters of the notation given in the brackets describe the orientation of the thickness and length of the plate that is being cut from the crystal. From the figure on the left, it is clear that the thickness direction (t) is aligned with the Y-axis and the length direction (l) is aligned with the X-axis. The plate also has a third dimension, its width (w). After the first two letters, a series of rotations are defined about the edges of the plate.
In the example above, the first rotation is about the l-axis, with an angle of -51°. The negative angle means that the rotation takes place in the opposite direction to a right-handed rotation about the axis. Finally, an additional rotation about the resulting t-axis is defined, with an angle (in a right-handed sense) of -45°.
Most practical cuts use one or two rotations, but it is possible to have up to three rotations within the standard, allowing for completely arbitrary plate orientations.
Note that since the crystallographic axes are defined differently in the 1949 and the 1978 standards, the crystal cut definitions differ between the two. A common cut for quartz plates is the AT cut, which is defined in the two standards in the following manner:
Standard |
AT Cut Definition |
---|---|
1949 IRE |
(YXl) 35.25° |
1978 IEEE |
(YXl) -35.25° |
The figure below shows how the two alternative definitions of the AT cut correspond to the two alternative definitions of the axes employed in the standards.
The AT cut of quartz is defined as (YXl) 35.25° in the IRE 1949 standard and (YXl) -35.25° in the IEEE 1978 standard. The figure shows the cut defined in a right-handed crystal of quartz. The reason for the difference between the standards is related to the different conventions for the orientation of the crystallographic axes. In the IRE 1949 standard, the rotation occurs in a positive or right-handed sense about the l-axis (which in this case is aligned with the X-axis). As a result of the different axes set employed in the IEEE 1978 standard, the rotation corresponds to a negative angle in this standard.
We have now seen how the two different standards result in different definitions of the material properties and different definitions of the crystal cuts.
In a follow-up blog post, we will explore how to set up a COMSOL Multiphysics model using the two standards. COMSOL Multiphysics provides material properties for quartz using both of the available standards, so it is possible to set up a model using whichever standard you are most familiar with. Stay tuned for that.
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