The food that we consume on a day-to-day basis, particularly carbohydrates, is a powerful source of energy for the human body. For the body to utilize energy from carbohydrates and store glucose for later use, it is crucial that its cells properly absorb the sugar. The key to this process is insulin, a hormone the body signals to the pancreas cells to release into the bloodstream, allowing sugars to enter the cells and be used for energy.
But what happens when the body fails to produce enough insulin or if it doesn’t work in the way that it should? In this case, the glucose fails to be absorbed by the cells and will instead remain in the bloodstream, resulting in rather high blood glucose levels. Referred to as diabetes, this metabolic disease relates to cases where the body produces little or no insulin (Type 1) or does not properly process blood sugar or glucose (Type 2). Note that in the latter type, a lack of insulin can develop as the disease progresses.
A device for injecting insulin. Image by Sarah G. Licensed under CC BY 2.0, via Flickr Creative Commons.
In both Type 1 and Type 2 diabetes, insulin injections serve as a viable treatment option. These injections, however, can cause pain when applied by a heavy single-needle mechanical pump. To minimize patients’ discomfort, researchers have investigated the potential of using a microneedle-based MEMS drug delivery device to administer insulin dosages. Not only would the stackable structure be minimal in size and easy to apply to the skin, but it would also provide a safer and less painful approach to applying injections.
Here’s a look at how a research team from the University of Ontario Institute of Technology used simulation to evaluate such a device…
Let’s begin with the design of the micropump model. The researchers developed a MEMS-based insulin micropump, placing a piezoelectric actuator on top of a diaphragm membrane comprised of silicone, with a viscous Newtonian fluid flowing through it. Note that the design itself is based on the minimum dosage requirement for diabetes patients — this typically ranges from 0.01 to 0.015 units per kg per hour.
Vibrations from the actuator create a positive/negative volume in the main chamber of the pump, which then pulls the fluid from the inlet gate and pushes it toward the outlet gate. Two flapper check valves control the direction of the fluid from the inlet to the outlet leading to the microneedle array, with a distributor connecting the outlet gate to the microneedle substrate. The established discharge pressure then pushes the fluid out of the microneedles to the skin’s outer layer.
The following set of images show the dimensions and cross section of the micropump as well as a more detailed layout of the model setup, respectively.
The MEMS-based piezoelectric micropump design. Image by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
A 2D layout of the micropump model setup. Image by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
To accurately study the performance of the micropump, the researchers utilized three different physics interfaces in COMSOL Multiphysics: the Solid Mechanics, Piezoelectric Devices, and Fluid-Structure Interaction (FSI) interfaces. The fluid flow that occurs from the inlet to the outlet via the action of the flapper check valve is described by the Navier-Stokes equations. Upon a wave signal exciting the piezoelectric actuator, the diaphragm disk and piezoelectric actuator move together, with an FSI moving mesh presenting the deformed solid boundary to the fluid domain as a moving wall boundary condition. Within the solid wall of the pump, this moving mesh follows the structural deformation. The FSI interface also accounts for the fluid force acting on the solid boundary, making the coupling between the fluid and solid domains fully bidirectional.
For their simulation analyses, the research team applied different input voltages and input exciting frequencies to the micropump design, studying various elements of the device’s behavior. The range of the voltages spanned from 10 to 110 V, while the exciting frequencies ranged from 1 to 3 Hz.
Let’s look at the results for an input voltage of 110 V and an input exciting frequency of 1 Hz. The plot on the left depicts the inflow and outflow rates, showing very little leakage for both. The plot on the right shows the established discharge and suction pressures at the inlet and outlet. At the inlet gate, a negative pressure denotes suction pressure, while a negative pressure at the outlet gate represents discharge pressure.
Left: Inflow and outflow rates. Right: Discharge and suction pressures. Images by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
As part of their analyses, the researchers measured the stress and deflection of the flapper check valves as well as the velocity field of the fluid. You can see their results in the following set of plots.
Left: Von Mises stress in the flapper check valves and velocity field of the fluid. Right: Deflection of the flapper check valves. Images by F. Meshkinfam and G. Rizvi and taken from their COMSOL Conference 2015 Boston paper.
The studies shown here, as well as those conducted with alternative inputs, indicate that the micropump design performs properly from the minimum to maximum spectrum of pressure and flow rates. Such a configuration can therefore serve as a viable alternative for applying insulin injections, providing a safer and less painful method of treatment for diabetes patients. The researchers hope to use their simulation findings as a foundation for creating more durable and dynamic insulin micropump designs in the future.
In any simulation study, there are complex theories and physics that must be considered in order to obtain accurate, realistic results. What is unique about simulation apps is that they are able to incorporate these elements into the underlying model while hiding such complexities behind a user-friendly interface. Such capabilities give those without simulation expertise the ability to set up their own numerical tests, all while enhancing their understanding of various scientific methods along the way.
Within the university setting, simulation apps are evolving as a powerful tool for introducing students to challenging concepts and advancing their modeling skills. What’s more is that as students continue with their studies, professors can take advantage of the Application Builder’s flexibility and incorporate greater complexity into an app’s design to further advance their learning.
The Tubular Reactor app, available in our Application Gallery, offers just one example of how teachers can simplify the process of teaching mathematical modeling concepts to students.
So how, you might wonder, are professors incorporating apps into their curriculum? Let’s take a look at two cases.
In 2015, Mumbai University in India introduced MEMS technology as a new course within their curriculum, combining theory with hands-on experience in the lab. Like many educators around the world, professors at this university recognize the importance of ensuring that students understand the basic principles of scientific methods at an early stage. The students, however, are often not yet ready to set up numerical problems on their own right away. They need to first invest a lot of time in learning the simulation software and mastering its use before they can carry out their own tests. To address this challenge, the professors turned to the Application Builder in COMSOL Multiphysics, having a set of apps developed that could better illustrate complex ideas.
With the easy-to-use apps — all of which are designed to be self-explanatory — students have been able to more quickly grasp the theory behind various scientific concepts. Each app includes notes that provide a clearer overview of its underlying theory as well as access to numerous design parameters that depict how device performance changes when parameters are modified.
“Since the students do not need to learn the software to use apps, this approach offers a quick way to understand the underlying principles,” notes Darshana Sankhe, a professor at the university’s DJ Sanghvi College of Engineering. “It frees up time for students to focus on other aspects of the topic at hand.”
The apps themselves cover a range of MEMS technology. Take the pressure sensor moisture absorption app, for instance, which shows students how MEMS devices react to a moist environment. The thermal expansion app, meanwhile, offers an overview of MEMS materials and their properties, while demonstrating the working principles behind a thermal actuator. The screenshots below highlight some of the additional apps created for the course.
MEMS-based apps utilized by professors at Mumbai University. Left: Piezoelectric shear-actuated beam app. Right: Piezoresistive pressure sensor app.
The benefits of incorporating such apps into the course’s layout extend not just to the students, but to the professors as well. Chetana Sangar, a professor and head of the Department of Electronics and Telecommunication at the Bharat College of Engineering, offers insight: “Not only have these apps helped students, but they have also helped us professors prepare for this new course. The apps that we have developed are very useful and I have recommended them to my colleagues in other colleges as well.”
At the Eindhoven University of Technology, professors also recognize the importance of teaching students the principles of scientific methods as early on as possible. The challenge here is that students don’t yet possess the knowledge and skills that are required to solve realistic heat transfer problems on their own. Recognizing this factor, professors at the university sought to simplify the learning process — and offer hands-on experience — in a first-year scientific methods course on building physics. Their answer came in the form of building an app.
The app itself is centered around a 3D heating experiment that involves analyzing three different methods for determining the thermal conductivity of polymethylmethacrylat (PMMA), with the goal of finding out which one is most accurate. The simulation study focuses on a box composed of expanded polystyrene, with a PMMA sheet attached at the top. For the case of Fourier’s law of conduction, one of the studied methods, a heat flux sensor is installed to determine the sheet’s thermal conductivity.
By using a simulation app, students are able to perform data analysis at an early stage, simulating output sensor data, from temperatures to heat fluxes. Such data is then used to calculate the sample’s heat conduction coefficient (the objective parameter), information that can then be compared to the input value. In this way, students successfully learn how to verify both their simulation and data analysis methods. Further, they are able to bring the skills and knowledge that they obtain through virtual testing, including how to improve the expected responses of the sensor, and apply them when conducting the actual experiments.
The 3D heating simulation app designed by professors at the Eindhoven University of Technology.
Along with the advantage of helping students understand key concepts faster, Jos van Schijndel, an assistant professor at the university, notes how the hands-on approach is also generating greater student engagement and interest: “The students are much more enthusiastic to start with the simulation because they can interact very intuitively with the simulation results from the beginning. This stimulates their curiosity to find out more about what’s behind the underlying model.”
Simulation apps, by design, hide complex theories and physics behind a user-friendly, customizable interface. These same advantages that are helping to optimize design workflows across a range of industries are also allowing professors to more efficiently teach students. When developing an app, professors can tailor the design in a way that best suits the needs of their specific course, from the included physics to the overall degree of complexity. Such flexibility enables the creation of an interactive learning tool that correlates with the needs of a particular set of students and can evolve with them as their skill set grows.
When using either the AC/DC Module, the MEMS Module, or the Plasma Module, the Terminal condition can be applied to the boundaries of any domains through which conduction or displacement currents can flow. With this boundary condition, it is possible to apply a Current, Voltage, or Power excitation as well as a connection to an externally defined Circuit or a Terminated connection with known impedance.
Regardless of the type of excitation or the physics interface being used, the Terminal condition always specifies the voltage, but optionally adds more equations to the model. For example, when using a Terminal condition with a specified current, the software automatically adds an Integration Component Coupling feature to integrate the total current through the specified boundary. The software also adds a Global Equation that introduces one additional degree of freedom to the model for the terminal voltage, such that the current through the terminal equals the user-specified current.
This combination of Global Equations with an Integration Component Coupling is quite flexible and you may already be familiar with its usage for structural mechanics and heat transfer modeling. Let’s now see how we can easily switch between different terminal types.
A schematic of a block of material with a ground and a Terminal condition on opposite sides. The Terminal boundary condition will be switched between a voltage or a current source.
We will look at a very simple electric current model involving just a block of material with a grounded boundary on one side and a current-type Terminal boundary condition on the other. We will start by considering the steady-state case and address how to apply a current or voltage excitation by adding a Global Equation. The Global Equation itself is added to the Electric Currents interface (To add Global Equations to a physics interface, make sure to toggle on Advanced Physics Options underneath the Show menu in the Model Builder.)
First, we take a look at the Terminal settings. As we can see from the screenshot below, the Terminal type is Current, and the applied current is the variable Current
, which will be solved for via the Global Equation.
The current-type Terminal condition with an applied current, which will be controlled by the Global Equation.
The Global Equations settings control the applied current for the Terminal condition.
The settings for the Global Equation are shown in the screenshot above. There is a single equation for the variable Current
, and the equation that must be satisfied is
(Current-1[A])/1[A]
Since this equation, by definition, must equal zero, the applied current equals 1 Amp. This is a straightforward equation; it does not include any feedback from the model, but rather sets the value of Current
. The Global Equation itself is nondimensionalized, since we will also want to satisfy an equation for voltage. Switching to a voltage excitation can be done by simply changing this equation to
(ec.V0_1-3[V])/3[V]
where the variable ec.V0_1
is automatically defined by the Terminal boundary condition.
We are thus applying a current such that the terminal voltage is equal to 3 Volts. This equation does introduce a feedback from the model, but the model is still linear. It will still solve in a single iteration, but it does require using a direct solver. If you try this out yourself, you can see that you can now switch between a voltage and a current excitation simply by changing the Global Equation for a stationary problem. Next, we will look at how you can dynamically switch between these excitations during a transient simulation.
Let’s suppose that we have a power source driving a system that exhibits variable resistance. For example, resistance changes with temperature due to Joule heating and induction heating. Let’s also suppose that, as the resistance changes, our power source can supply a constant current up to some peak voltage, or a constant voltage up to some peak current.
To model this type of switch, we will use the Events interface. We have previously written about the Events interface for implementing a thermostat for a thermal problem and we recommend reviewing that blog post for technical details and relevant solver settings.
The Events interface contains four features: a Discrete States feature, an Indicator States feature, and two Implicit Events features. First, the Discrete States feature defines a single-state variable, CC
, which acts as a flag indicating if the power supply is in constant-current mode, CC=1
, or constant-voltage mode, CC=0
. Initially, our power supply will be in constant-current mode. Next, there is an Indicator States feature defining two indicator-state variables, PeakV
and PeakI
, which should vary smoothly over time. Lastly, there are two Implicit Events features, which will track these two indicator-state variables and change the discrete-state variable, CC
, to zero or one if the logical conditions are met. These settings are all shown in the screenshots below.
The Discrete States feature defines a flag that signals the Terminal state.
The Indicator States feature defines two different possible events indicators.
The Implicit Events toggle the Discrete States variable.
There is only one task left to do: modify the global equation for the variable Current
to be
CC*((Current-1[A])/1[A])+(1-CC)*(ec.V0_1-3[V])/3[V]
You can see that this is a sum of the two expressions developed earlier for either current control or voltage control using the CC
flag to switch between them. Once this is done, it is only a matter of solving in the time domain with the study settings described in our earlier blog post and using a direct solver for the Electric Currents voltage field, the Terminal voltage, and the Global Equation for the current, as shown in the screenshot below. The Events variables can be solved in a separate segregated step.
Solver settings showing how the variables are segregated and that a direct solver is being used.
With these features, we have now implemented the following power supply behavior:
To generate some representative results, we will explicitly make the total resistance of our domain vary over time, as shown in the plot below. We can see from the subsequent plots of current and voltage that the supplied current is initially constant but the voltage rises due to increased resistance. The power supply then switches over to constant-voltage mode and the current varies. As the resistance goes back down, the current rises to its peak and then the power supply switches back to fixed-current mode.
As the device resistance varies over time, the source switches between constant current and constant voltage to ensure that the maximum current and voltage are never exceeded.
We have demonstrated a control scheme that enforces both a current and voltage maximum, implemented with Global Equations and the Terminal boundary condition. The functionality of the Terminal boundary condition used here is not limited to the Electric Currents interfaces. The Terminal condition is also available within the Magnetic and Electric Fields interface as well as the Magnetic Fields interface, where it is called the Boundary Feed or Gap Feed condition. The Magnetic Fields interface also includes a Multi-Turn Coil domain feature that can be used equivalently.
It is also possible to use this type of control scheme during a Frequency-Transient study, which often occurs in Joule or induction heating, wherein the electromagnetic problem is solved in the frequency domain and the thermal problem, which solves for the temperature variations that lead to impedance changes, is solved in the time domain. This can be particularly useful in the modeling of, for example, RF heating and ablation.
The Batteries & Fuel Cells Module, Corrosion Module, Electrochemistry Module, and Electrodeposition Module all contain the Electrode Current and Electrolyte Current boundary conditions that can be used equivalently to the Terminal condition demonstrated here. In particular, the Capacity Fade of a Lithium-Ion Battery tutorial model shows how to model the charging and discharging of a battery.
We hope you can see that with a little bit of imagination, it is possible to implement some quite complex control schemes with the Terminal boundary condition and the Events interface.
If you have a particular application that you’d like to model with COMSOL Multiphysics, please don’t hesitate to contact us.
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Before we start discussing piezoelectric physics, let’s first review a couple of mathematical conventions. Piezoelectric materials have properties that allow strain to produce electric polarization and vice versa. The former is known as the direct piezoelectric effect, while the latter is referred to as the inverse piezoelectric effect. In mathematics, two forms of a set of matrix equations are conventionally used to describe these effects: the strain-charge form (also known as the d-form) and the stress-charge form (or the e-form). The two forms can be derived from each other by a transformation.
Note: The two forms are convertible, thus you can choose whichever option is preferred. In COMSOL Multiphysics, the strain-charge form is converted to the stress-charge form internally.
Here, we’ll choose the d-form, in which the relations are written as
where the field quantities \varepsilon, D, S, and \bf{E} are the strain, electric displacement, stress, and electric field, respectively.
The material parameters s_E, d, and \varepsilon_{r} are the compliance, coupling matrix, and relative permittivity, respectively. The superscript T stands for transpose. The first equation expresses the relation that the stress field and electric field produce strain. The second equation, meanwhile, shows the electric polarization generated by the stress field and electric field. The second term of the first equation accounts for the electric contribution to the strain.
In solid mechanics, we typically consider an imaginary, small differential volume to understand the force that is exerted on a volume. Let’s consider a volume where the surfaces are aligned to each axis of the global Cartesian coordinate system (xyz-axes).
There are six distinct quantities that describe the forces per unit area on the surface of the imaginary volume. Three of them are normal to each surface and are usually denoted by \sigma_{xx}, \sigma_{yy}, and \sigma_{zz}. The other three are parallel to each surface and are denoted by \tau_{yz}, \tau_{xz}, and \tau_{xy}. (\tau_{zy}, \tau_{zx}, and \tau_{yx} are omitted because of symmetry.)
The first index represents the normal vector of the surface under consideration. The second index indicates the direction of the force. For example, \sigma_{xx} is a force along the x-axis that is exerted on a yz-plane (a plane perpendicular to the x-axis); \tau_{yz} is a force along the z-axis on an xz-plane; and so on.
The force components in a small differential cubic volume.
These six quantities are often expressed as a column vector. In COMSOL Multiphysics, the order of the quantities has two conventions: one is the standard notation and the other is the Voigt notation. In COMSOL Multiphysics, it is important to make sure that the Voigt notation is used in the Piezoelectric Devices interface. (The standard notation is used by default in the Solid Mechanics interface.)
In the Voigt notation, the stress is written as a 6 x 1 column vector (special attention should be paid to the ordering of yz-, xz-, and xy- components):
Similarly, the strain can be written as
In COMSOL Multiphysics, the coupling matrix d is defined by
Here, the notation d_{xyz} reads as the coupling coefficient for the strain component \varepsilon_{yz} caused by the electric field component E_x. Now, in the Voigt notation, all of the subscripts are numeric with the rule:
As such, the coupling coefficient matrix is
Thus, the electric contribution can be rewritten as
In the view of the material property settings in COMSOL Multiphysics, the matrix d is flattened into a 1 x 18 row vector to be concise. It then looks like the following:
We always have the option to view the expression in the matrix form. We can do so by clicking on the Edit button in the “Output properties” section, as shown below:
A screenshot of the COMSOL Desktop shows the d coefficients in a matrix form.
Now that we’ve reviewed some of the basics of piezoelectric theory, let’s turn our attention to performing simulations. Here, we’ll enter a nonzero number only in d_{33}. This means that the piezoelectric material makes a normal force along the z-axis by the z-component of the electric field E_z. We also assume that the bottom surface is mechanically fixed.
The results shown below are obtained, with the surface color representing the total displacement and the arrows indicating the electric field. The contribution of the electric field to the strain is given by E_3 d_{33}. The black outline indicates the initially undeformed shape. As indicated by the plot, the volume is stretched in the z-axis direction.
An example of the d coefficients with the only nonzero in d_{33}. Under an electric field along the z-axis, the volume stretches along the z-axis.
Keeping the same electric field, we’ll now enter a negative coefficient only in d_{31}. With the yz-plane mechanically fixed, the volume shrinks in the x-axis direction.
An example of the d coefficients with the only nonzero negative in d_{31}. Under an electric field along the z-axis, the volume shrinks in the x-axis.
The last preliminary example shows a shear force, with a nonzero number entered only in d_{24}. The xz-plane is mechanically fixed and an electric field is applied along the y-axis.
An example of the d coefficients with the only nonzero in d_{24}. Under an electric field along the y-axis, the volume experiences a shear along the yz-plane.
Building on these basic elements, we’ll now introduce you to a helpful trick for designing piezoelectric actuators — a piezoelec”trick”, if you will.
On its own, a piece of piezoelectric material is not a MEMS device. To become such a device, it must be attached to other elastic materials. Cantilevers are one of the simplest MEMS examples, and there are two different piezoelectric types. The first is the unimorph type, which is typically a sheet made by attaching a single layer of piezoelectric material to an elastic material. The other is the bimorph type, which is composed of two piezoelectric layers and other elastic materials. The materials used for attachment can be almost anything, as long as it is possible to put an electrode layer on their surface. Here, for simplicity, we will consider a unimorph type of MEMS device.
It is worth mentioning that all of the piezoelectric material properties provided by COMSOL Multiphysics are assumed to be poled in the z-axis of the local coordinate system. If the material is poled along another direction, you need to define a coordinate system so that its third direction is aligned with the poling direction.
COMSOL Multiphysics provides convenient functionalities to set up local coordinate systems. To learn more, you can refer to page 106 in the Structural Mechanics Module Users Guide. You can also consult the Piezoelectric Shear-Actuated Beam and Thickness Shear Mode Quartz Oscillator models, found in the Application Libraries under the Piezoelectric Devices interface in the MEMS Module and also on our website.
Under the assumption referenced above, we model a unimorph piezoelectric cantilever that is fixed at one end. From the Material Libraries, we select barium titanate (BaTiO3) as a piezoelectric material and silica (SiO2) as the substrate.
The principle of the MEMS device is simple. An electric field is applied in the z-direction, causing shrinkage in the x-direction due to d_{31} (=-7.8e-11 [C/N]). Note that the elastic properties of the electrode are not modeled here. To achieve the z-directional electric fields, an electric potential is applied on one side of the barium titanate, while the other side is grounded. With the left end mechanically fixed, the MEMS cantilever moves about 9 um at the free end. As the top material (BaTiO3) shrinks, it pulls the bottom material (SiO2), causing the entire device to bend in the upward direction.
A unimorph cantilever with a fixed end shows the total displacement.
But what if we have to fix both ends of the device for some reason — perhaps mechanical ruggedness? The result is obvious: There is almost no displacement, as the top material cannot shrink along the x-axis at all. More accurately, the only displacement is due to the d_{32} effect, which is not intended.
A unimorph cantilever with both ends fixed shows almost no displacement.
It is, however, possible to overturn such a situation. We can do so by modifying the piezoelectric device with a little “trick”. As the resulting plot below shows, the displacement is now back to the same order of magnitude. If you look at the electric field directions, what we have done to obtain this result is apparent. The electric field is reversed in the center part of the beam. You can perform the trick by patterning the electrode into three parts: a center and two sides, grounding them alternately and applying a voltage alternately in the opposite order as in the first of the next two figures.
Material and voltage configurations in a unimorph cantilever with both ends fixed under alternate electric fields.
A unimorph cantilever with both ends fixed. Applying alternate electric fields improves the total displacement.
Let’s look at another example. It may seem a bit mysterious at first sight, as the entire electric field is aligned in the same direction. Still, we have the same performance as the previous result without alternating the electric field. We instead alternate the piezoelectric material, splitting it into three parts (again, a center and two sides) and flipping the center part.
With regards to simulation, this can be done either by using different local coordinate systems or, in a simple case like this, by changing the sign of the coupling coefficient, d_{31}. In practice, we can really split the piezoelectric material up and put the flipped part in the center. An easier and more practical approach is to pattern the electrode in the same manner as the previous example, but pole the center part in the opposite direction. There is no need to split the material in such a situation.
For the unimorph cantilever with both ends fixed, alternate material and voltage configurations.
For the unimorph cantilever with both ends fixed, alternating the poling direction also improves the total displacement.
Our last example is particularly interesting. Without the use of a trick, we would really have no displacement, since the entire perimeter is fixed and, as such, there is no room for any d_{3*} component to play a role. Applying the same trick from the previous examples, the entire-perimeter-fixed membrane can have a tremendous displacement. The simulation shown here can be used as a MEMS device that compensates for, to name an example, optical aberration (particularly spherical and astigmatism).
Alternate material and voltage configurations in a membrane MEMS device with the entire perimeter fixed.
A membrane MEMS device with the entire perimeter fixed. Applying an alternate electric field or poling direction improves the total displacement.
We have reviewed some of the fundamental mathematics and conventions that are necessary when simulating piezoelectric materials. Further, we have walked you through the steps of performing basic simulations, then diving into more interesting applications using our piezoelec”trick”.
When it comes to the trick, the secret to significantly improving the displacement is the “inflection zone”, which is created by alternating the electric field or the material property. (Refer to US Patent 7,369,482 for more details and other examples.) The local coordinate system functionalities of COMSOL Multiphysics make it very easy to set up a simulation for such systems with alternating material orientations.
If you enjoyed today’s blog post, be sure to check out some other related blog posts that might spark your interest:
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To set up the orientation of a crystal within COMSOL Multiphysics, it is necessary to specify the orientation of the crystallographic axes with respect to the global coordinate axes used to define the geometry. This is different than the manner in which the standards define the crystal orientation. Thus, some care is needed when defining the orientation of the geometry. For example, the orientation of the crystal axes will change if the orientation of the plate is changed. Here, we will show how to set up an AT cut quartz plate in different orientations in the physical geometry.
In a previous blog post, we discussed in detail the system that is used in both the IEEE 1978 standard and the IRE 1949 standard. Due to differences in the orientation of the crystallographic axes specified by each standard, the definition of the AT cut differs between them. The table below shows both definitions of the AT cut:
Standard | AT Cut Definition |
---|---|
IRE 1949 | (YXl) 35.25° |
IEEE 1978 | (YXl) -35.25° |
The difference between the standards can be understood by recalling that the plate cut from the crystal has an orientation defined by the l-w-t axes set (l-w-t stands for length, width, and thickness). The first two letters given in parentheses in the cut definition — Y and X — define the crystal axes with which the l and t axes are originally aligned. A rotation of 35.25° is then performed about the l-axis. The sense of the rotation differs between the standards, since the material properties are defined with respect to different sets of axes within the standards. This is illustrated in the figure below, which shows that the rotation about the l-axis is in a positive sense for the 1978 standard, but a negative sense for the 1949 standard.
The AT cut of quartz (mauve cuboid) shown together with a right-handed quartz crystal. The axes sets adopted by the IRE 1949 standard and the IEEE 1978 standard are shown, as well as the orientations of the l-w-t axes set in the plate.
There is another subtle difference between the two standards. As the AT cut is defined in the two standards, the thickness and length directions are reversed between them (shown in the figure above). From the figure, it is clear that to obtain exactly the same plate orientation as the 1949 standard, the 1978 standard would require an additional rotation of 180° about the w-direction. In this case, the AT cut in the 1978 standard would be defined as: (YXlw) -35.25° 180°. We need to carefully account for these differences between the standards when setting up a model in COMSOL Multiphysics.
One way to set up a model is to keep the global coordinate system aligned with the crystal axes and simply rotate the plate to correspond with the first figure. As we will see, this method is perfectly valid, although it results in a rather inconvenient specification of the geometry.
Instead, we will consider how to define the material orientation for an AT cut quartz disc. In this COMSOL Multiphysics model, the crystal orientation is determined by the coordinate system selection in the Piezoelectric Material settings window. The crystal orientation is specified via a user-defined axis system that is selected in the coordinate system combo box, depicted below. This example is based on a simplified version of the Thickness Shear Mode Quartz Oscillator tutorial, available in our Application Gallery.
Changing the coordinate system for a piezoelectric material in COMSOL Multiphysics.
In the example above, the left-handed quartz defined in the 1978 standard is used for the material. If we wish to use the global coordinate system for the crystal orientation, then the quartz disc must be orientated in the manner shown in the first figure, with the axes set up for the 1978 standard. This can be achieved by rotating the cylinder about the x-axis.
A rotation operation is applied to the quartz cylinder.
The images below show the response of the device when it is set up in the selected orientation. The crystal is vibrating in the thickness shear mode. To obtain this response, use Study 1 in the COMSOL Multiphysics Application Gallery file and solve for a single frequency of 5.095 MHz.
IRE 1949 Standard | IEEE 1978 Standard |
---|---|
Thickness shear mode of an AT cut crystal for the same plate set up with the IRE 1949 (left) and the IEEE 1978 (right) standards. The driving frequency is 5.095 MHz. In both cases, the global coordinate axes in COMSOL Multiphysics correspond to the crystal axes.
Setting up the model within the IRE 1949 standard is straightforward, as COMSOL Multiphysics includes the material properties for both left- and right-handed quartz in each standard. To use the alternative standard, simply add the Quartz LH (1949) material to the model and select the quartz disc. This will override the previously added material. Then, change the rotation angle of the disc to -54.75º to orientate the disc equivalent to the plate shown in the first figure. The figure above shows that when these steps are followed, the 1949 standard gives the same result as the 1978 standard. Although the two figures appear identical, the global axes have been rotated so that they correspond to the two axes sets in the first figure.
As this example shows, it is possible to use the global coordinate system for the crystal axes. However, for a cut such as the AT cut, this results in an unusual orientation of the plate within the geometry. In a real world application, one might have several piezoelectric elements in different orientations and then this approach could not be used for all of the crystals. Therefore, it is often more convenient to specify the crystal orientation by means of a rotated coordinate system.
In the COMSOL Multiphysics environment, the most convenient way to specify a rotated coordinate system is through a set of Euler angles. The Euler angles required for a given crystal cut will vary for different orientations of the plate with respect to the model global coordinates. Now we will consider how to specify the Euler angles for two different plate orientations in both of the available standards.
The best way to determine the Euler angles required within a given standard is to carefully draw a diagram that specifies the orientation of the l-w-t axes with respect to the crystal axes. Note that in some of the figures for the 1978 standard, l, w, and t are labeled as dimensions of the plate rather than as a set of right-handed axes. It is best to ensure that they are drawn as a set of right-handed axes to avoid potential confusion when determining the Euler angles for a plate in a COMSOL Multiphysics model. The Euler angles determine the orientation of the crystallographic axes (X_{cr}-Y_{cr}-Z_{cr}) with respect to the global coordinate system (X_{g}-Y_{g}-Z_{g}). Consequently, both the orientation of the plate with respect to the global system and the crystal cut determine the Euler angles.
As an example, we will consider the case where the global axes X_{g}-Y_{g}-Z_{g} align with the l-w-t axes (corresponding to the plate, with its thickness in the Zg direction). This is often the most convenient way to orientate the plate within a larger geometry. The figure below shows what happens when we take the first figure and rotate the plate such that the l, w, and t axes correspond to the global axes X_{g}-Y_{g}-Z_{g} within the two standards. For ease of comparison with the initial figure, the global axes are not in the same orientation for the two standards.
Rotated versions such that the l, w, and t axes correspond to the global axes X_{g}-Y_{g}-Z_{g} within the 1949 standard (left) and the 1978 standard (right). The Y and Z axes lie in a single plane.
The next figure shows the unrotated and rotated axes as seen from a side view of the first figure. This diagram represents an easier “paper and pencil” approach for determining the Euler angles.
IRE 1949 Standard | IEEE 1978 Standard | |
---|---|---|
Orientation in unrotated axes | ||
Orientation in rotated axes |
End on views of the axes orientation when cutting the crystal (top) and when the plate axes are oriented parallel to the global axes (bottom).
The following figure shows how the Euler angles are specified for a rotated system within COMSOL Multiphysics. An arbitrary rotated system can be specified by rotating first about the Z-axis, then about the rotated X-axis (marked as N in the figure below), and finally once again about the rotated Z-axis. This is known as a Z-X-Z scheme.
It is important to note that for cuts specified by means of multiple rotations, the rotations usually need to be applied in reverse when specifying the Euler angles. This is because COMSOL Multiphysics software specifies the orientation of the crystal with respect to the plate, whilst the standards used for cutting the plates from a crystal specify the orientation of the plate with respect to the crystal. It is straightforward to obtain equivalent Euler angles from the figure above.
Z | X | Z | |
---|---|---|---|
IRE 1949 Standard | 0° | 54.75° | 0° |
IEEE 1978 Standard | 0° | 125.25° | 0° |
Euler angles for the AT cut within the two standards. Both angles are positive for a right-handed rotation about the Z-axis.
Specifying a coordinate system using Euler angles through the rotated system feature.
If we use the Euler angles specified in the table above to set up the thickness shear mode for a quartz disc, then we obtain the results shown below for two plates with identical excitation and orientation. What went wrong? The problem is that the thickness direction for the AT cut is defined in opposite directions within the two standards. To obtain identical results from a model using the two standards, we could either switch the polarity of the driving electrodes or try using the alternative 1979 AT cut definition proposed above: (YXlw) -35.25° 180°. As a final exercise, let’s consider how to set up the Euler angles for this doubly rotated cut.
IRE 1949 Standard: (YXl) 35.25° | IEEE 1978 Standard: (YXl) -35.25° |
---|---|
Thickness shear mode of an AT cut crystal for the same plate set up with the IRE 1949 and the IEEE 1978 standards with a driving frequency of 5.095 MHz. In each image, the global axis orientation is shown on the left and the crystal axis orientation is shown on the right. The top images are aligned with the global coordinates and the lower images are shown with the crystal coordinates in the same orientation as in the first figure.
Below, we have the sequence of rotations involved in defining the cut (YXlw) -35.25° 180° and the sequence of Z-X-Z Euler rotations required to rotate the global axes onto the crystal axes. The corresponding Euler angles are provided in the table below. Note that the order of the rotations for the Euler angles is the reverse of that specified in the cut definition.
IEEE 1978 Standard: (YXlw) -35.25° 180° | |
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1.Orientate the thickness direction (Z_{g}) along the Y-axis of the crystal (Y_{cr}) and the width direction (X_{g}) along the X-axis of the crystal (X_{cr}). | |
2. Rotate the cut by 35.35° about the l- (X_{g}) axis. | |
3. Rotate the cut by 180° about the w- (Y_{g}) axis. | |
4. Reorientate the above figure so that the global axes are in a convenient orientation. |
Sequence of rotations that correspond to the cut (YXlw) -35.25° 180° in the IEEE 1978 standard.
Equivalent Z-X-Z Euler Angles | |
---|---|
1. Start with the crystal and the global axes aligned. | |
2. Rotate the crystal axes 180° about their Z-axis (Z_{cr}). | |
3. Rotate the crystal axes -54.75° about the new crystal X-axis (X_{cr}). |
Corresponding rotations that determine the Euler angles of the crystal axes with respect to the global axes.
X | Z | X | |
---|---|---|---|
IEEE 1978 Standard: (YXlw) -35.25° 180° | 180° | -54.75° | 0° |
Euler angles for the cut (YXlw) -35.25° 180° in the IEEE 1978 standard. This cut corresponds to exactly the same orientation of the plate in the IRE 1949 standard AT cut definition.
Finally, the figure below shows the frequency-domain response of the cut (YXlw) -35.25° 180° in comparison to the IRE 1949 standard AT cut. As expected, the responses of the two devices are now identical.
IRE 1949 Standard: (YXl) 35.25° | IEEE 1978 Standard: (YXlw) -35.25° 180° |
---|---|
Thickness shear mode of an AT cut crystal set up with the IRE 1949 standard compared to the cut (YXlw) -35.25° 180° in the IEEE 1978 standard with a driving frequency of 5.095 MHz. In each image, the global axis orientation is shown on the left and the crystal axis orientation is shown on the right. The top images are aligned with the global coordinates and the bottom images are shown with the crystal coordinates in the same orientation as in the first figure.
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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