Your SlideShare is downloading. ×
COMSOL Multiphysics - MEMS and Microfluidics
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Introducing the official SlideShare app

Stunning, full-screen experience for iPhone and Android

Text the download link to your phone

Standard text messaging rates apply

COMSOL Multiphysics - MEMS and Microfluidics

7,319
views

Published on

How to use MEMS and Microfluidics module for researche

How to use MEMS and Microfluidics module for researche

Published in: Education, Technology

1 Comment
11 Likes
Statistics
Notes
  • can u mail me some of the models?i want to model a micropump and i want study the moving mesh example.thank u so much.my email is:mo.hg.eng20@yahoo.com
       Reply 
    Are you sure you want to  Yes  No
    Your message goes here
No Downloads
Views
Total Views
7,319
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
0
Comments
1
Likes
11
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. M L SO UNIST CO MULTIPHYSIC S MEMS/ Microfluidics 모델링교육 V4
  • 2. 1. MEMS Introduction 2. Cantilever Beam 3. Microfluidics Introduction 4. Electroosmotic Micromixer 5. Electrowetting Lens Tel : 02) 547 - 2344 Fax : 02) 547-2343 E-mail : comsol@altsoft.co.kr
  • 3. MEMS Modeling with COMSOL Multiphysics • MEMS • MEMS • MEMS • MEMS
  • 4. MEMS – Micro Electro Mechanical System • Micro – the devices are extremely tiny, in the order of micrometers or smaller • Electro – some electrical component is involved • Mechanical – the system performs some mechanical motion • Systems – all these and parts are combined in one package
  • 5. MEMS – Micro Electro Mechanical System Electrical Structural MEMS Thin-Film Flow • MEMS • MEMS • MEMS • MEMS Thermal
  • 6. Individual physics interfaces (which include equations and boundary conditions for modeling particular physical phenomena) are accessed from the model wizard (shown below). The MEMS module adds new interfaces and extends those in the base package, as shown below. Electric Circuits interface with SPICE import Enhanced interfaces for Electric Currents and Electrostatics Piezoelectric Devices interface for modeling the direct and inverse Piezoelectric effects Extended structural mechanics capabilities including geometric nonlinearity and viscoelasticity Electromechanics interface for modeling electrostatic actuators Individual physics interfaces (which include equations and boundary conditions for modeling particular physical phenomena) are accessed from the model wizard (shown below). The MEMS module adds new interfaces and extends those in the base package, as shown below. Interfaces for physical models of damping and Fluid Structure interactions. These include: Fluid Structure Interactions, Thin Film Flow (for modeling gas and liquid flow in thin layers via the Reynolds Equations) and Thermoelasticity Interfaces for modeling thermal structure interactions, film stress and Joule heating Piezoresistance interface models fully anisotropic piezoresistivity
  • 7. Electromechanics Electrostatics coupled together with moving mesh and structural mechanics for modeling electrostatic actuators and sensors Electrostatic attraction between objects at different potentials causes structural deformations V=0 Cantilever beam fabricated on silicon wafer V+ Surrounding Air Cantilever with applied bias Ground plane Initial Mesh COMSOL computes electrical forces on structure, both structure and mesh deform accordingly Final result showing deformed structure, axial strain (color in beam) and electric potential (contours) Space dimensions supported: 3D, 2D Plane Strain, 2D Axisymmetric Electromechanics: Structural Features Full range of structural boundary conditions available including: Free, Fixed, Boundary Loads, Follower Loads, Advanced Perioidc Conditions, Thin Elastic Layers, Rollers, Spring foundations, etc. Thin Film Damping feature allows complex damping models to be easily added to the model. y Electrostatic forces and mesh constraints are automatically applied on the surfaces of the structure. Model electrics fields in dielectric layers including isotropic electrostriction. Other material properties (permittivity, elasticity) can be fully anisotropic If the field inside the material is not important, solve only the structural problem (the material is assumed to be conducting). Isotropic, orthotropic, anisotropic and viscoelastic materials can be modelled
  • 8. Electromechanics: Structural Features Arbitrary fictitious forces due to a rotating or accelerating reference frame, or any combination of these can be added using easy to enter expressions for the body force. Add phenomenological damping, thermal stresses and a range of other options to the model. Model anchor losses using perfectly matched layers or low reflecting boundaries Figure from: P.G. Steeneken,1, J.J.M. Ruigrok, S. Kang, J.T.M. van Beek, J. Bontemps and J.J. Koning,, “Parameter Extraction and Support-Loss in MEMS Resonators”, COMSOL Users Conference, Grenoble, 2007 Electromechanics: Electrical Features Moving mesh automatically accounts for contraction of gaps due to structural displacement Full range of electrical boundary conditions available including Floating Potentials, Ground, Thin Capacitors, Thin Low Permittivity Gaps, Dielectric shielding, Zero Charge (symmetry) etc. Infinite Elements can be used to account for electric fields far away from the device, enabling highly accurate capacitance calculations
  • 9. Electromechanics: Electrical Features Advanced perturbation machinery: at different DC bias points, the solution for the AC voltage signal is computed at the correct linearization point. This unique functionality enables straightforward and accurate modeling of effects such as spring-softening and stress stiffening in the frequency domain. Terminal boundary conditions make it straightforward to couple devices to a circuit and automatically define lumped parameters such as capacitance to ground. For high frequency applications terminals can be used to compute the S, Y or Z parameters. p Displacement VAC VDC Voltage Electromechanics: Study Types Prestressed Analysis, Eigenfrequency: First solve a stationary problem with the applied DC bias. Then compute the resonant modes of the structure, accounting for all the stresses induced by the DC bias. Advanced solver automatically includes coupling between modes (as in gyroscopes) and incorporates spring softening etc. Time dependent study type for computing the transient response to arbitrary excitations. Prestressed Analysis, Frequency Domain: First solve a stationary problem with the applied DC bias. Then account for the effect of a (smaller) AC bias (or perturbation) taking account of the changes in the system stiffness as a result of the DC bias (spring softening /stiffening). Resonant Frequency (Hz) Stationary study type for computing static displacements and pullin voltages Spring softening DC Bias (V)
  • 10. Electromechanics: Applications Applications include: • • • • • • • • • Capacitive pressure sensors Accelerometers Gyroscopes Combo sensors Resonant sensors MEMS oscillators / timing Comb Drives Parallel Plate Actuators General capacitive sensors Structural Mechanics All the advanced features discussed in the slides entitled ‘Electromechanics: Structural Features’ can be used in structural analysis. Additionally the following extra features are available (which are not currently integrated into Electromechanics). Space Dimensions Supported:3D, 2D plane stress, 2D plane strain, 2D axisymmetric. Compute effective mass and spring constants for the structure using a ‘Modal Reduced Order Model’ study Support for frequency domain analysis without prestresses. Modal study types allow a reduced set of vibrational modes to be used in a Frequency Domain or Transient study, dramatically increasing the efficiency of the problem Structural contact Structural contact and friction problems (without electrostatic fields operating in the t t l t t f gap) can also be solved, as can linear buckling problems.
  • 11. Electrostatics and Electric Currents All the advanced features discussed in the slides entitled ‘Electromechanics: Electrical Features’ can be used in standalone electrical analyses. Additionally the following extra features are available (which are not currently integrated into Electromechanics). • Electric Currents – – – – • 3D, in plane and axisymmetric Metals and other conductive solids Conductive Fluids Enhanced boundary conditions such as: distributed impedance, floating potentials, contact im pedance, terminals etc. Lumped Parameters – – – Terminal features provide lumped parameters (e.g. capacitance) automatically S, Y and Z parameter computation p parameters in p p g Conversion between lumped p post-processing Electric Circuits Specify circuit using built in components Easily couple circuits to domain models with the terminal boundary condition Import SPICE netlists
  • 12. Piezoelectric Devices • • • • • • • • • Similar range of boundary conditions and advanced structu ral/electrical features to electromechanics Fully coupled piezoelectric equations for linear piezoelectri city Direct and inverse effects Stress-charge and strain-charge formulations Material properties and orientation specified in a convenien t manner – support for both stress and strain charge materi al properties Mixed material modeling: combine piezoelectric materials with decoupled isotropic and anisotropic structural and elec trical materials Pre-defined coupling with acoustics Static, modal, frequency-response and time-dependent an alysis Perturbations: ability to model in the frequency domain with a DC signal bias. Setting up Piezoelectric Materials Easily specify material orientation using Euler angles Strain charge and stress charge forms of material data included Select co-ordinate system / material properties in physics settings
  • 13. Piezoresistance • • • • Similar range of boundary conditions and advanc ed structural/electrical features to electromechani cs Piezoresistance(stress) or Elastoresistance(strain ) equation forms Fully anisotropic equation formulation 3 interfaces allow flexibility in modeling conductin g and structural layers – Treat piezoresistors and conductors as thin layers ( surface features) and model structure in 3D – Model both structure and piezoresistors in 3D, to c apture effects such as non-uniform doping profiles – Use the shell interface for the structure and treat th e piezoresistors and conductors as thin layers withi n the shell (requires the Structural Mechanics Mod ule) • Static, modal, frequency-response and time-depe ndent analysis Setting up Piezoresisitive Materials Select Material from Library Simple definition of material co-ordinate axes using Euler angles (similar to Piezoelectric Materials) Specify a constant, or even a spatially varying dopant density
  • 14. Thermal MEMS • Thermal stresses in structures can be comput ed at a fixed temperature using the thermal e xpansion feature – – • Thermal stress interface to compute the stres ses in structures at non-uniform temperatures – • Available in other interfaces such as Electromechanic s and Piezoresistive Devices Can be used to compute packaging/fabrication stress es in devices held at a fixed temperature Can be used to compute packaging/fabrication stress es in devices with temperature gradients. Joule heating and thermal expansion interfac e to compute thermal stresses, generated by electrical heating – Typical applications include thermal actuators Thermoelasticity • Dedicated interface to compute the damping induced by thermoelastic effects in flexural re sonators. • Includes the entropy driven coupling between structural mechanics and heat transfer autom atically along with the structural and thermal equations. • Compute thermal and structural eigenvalues together with quality factors. Can also be use d for frequency domain, transient and freque ncy-transient studies. Temperature variation in a bending micro-beam
  • 15. Thin Film Flow • • • • • Interfaces solve for fluid flow in thin layers. Two formulations of the Reynolds equation available to model flow in thin-films of gases or liquids. 2D planar, axisymmetric and 3D shell formul ations available. Features to model low pressure (rarefied) g as in the gap, including accurate force comp utation. Film damping available as a feature in the st ructural interfaces with automatically implem ented two way coupling. Includes all the fun ctionality of the stand alone Thin-film flow int erfaces. Fluid Structure Interactions • • Based on the ALE moving mesh technolo gy in COMSOL Fully bidirectional coupling between fluid a nd structure – – • • • • Viscous, pressure and inertial fluid forces o n the structure Momentum transfer back to fluid Small and large deformations Highly accurate fluid load computation (we ak constraints). Predefined physics interface makes it eas y to build models rapidly – no manual inter facing between different solvers Automatic segregated solver settings for t he stationary and parametric analysis of la rge 3D models
  • 16. MEMS Acoustic Devices / Microphones u T Damped vibrations of a shell above a thin air gap. Deformation plot and two cross sections depicting the acoustic particle velocity and the acoustic temperature field. The Acoustics Module includes the thermoacou stic interface for MEMS acoustics applications: • Acoustics in small devices: mobile devices, t ransducers, microphones, hearing aids, perf orates, headsets, couplers etc. • Includes thermal conduction and viscous los ses explicitly • Linearization of Navier-Stokes and energy e quations • Solves for acoustic variations in (p,u,T). • Predefined interfaces to couple acoustics wi th solid mechanics / shell interfaces (shell in terfaces require the Structural Mechanics M odule) RF MEMS T following application areas can be simulated with th The e MEMS Module: • Silicon oscillators • Variable capacitors • Tunable filters W With the RF Module more advanced RF MEMS problems c can be solved: • Cavity resonators / tunable cavity resonators • Antennas / reconfigurable antennas
  • 17. Microfluidics COMSOL offers a dedicated Microfluidics Module which in cludes: • Two Phase Flow • Electrokinetic Effects • Mixing and Diffusion • Porous Media flow eCAD import (requires eCAD import module) Import eCAD data in GDS/NETEX-G or ODB++ format Layers can be imported in 2D or automatically extruded for a full 3D import
  • 18. • MEMS • MEMS • MEMS • MEMS Capacitive Pressure Sensor • • Pressure sensor example designed to provide an introduction to modeling techniques used in MEMS. Shows how to set up a coupled structural/electrical model using important features of the MEMS Module: – – – • Electromechanics Thermal Stress Terminal Boundary Conditions Shows how to compute the sensor performance and the effects of thermally induced packaging stresses on the sensor response
  • 19. Biased Resonator In this sequence of models, an electrostatically actuated MEMS resonator is simulated. The device is biased with a DC voltage. And then driven by a smaller AC voltage. A series of models shows how to compute: 1. The biased displacement 2. The pull-in voltage 3. The biased resonant frequencies (shown below) 4. The frequency domain response 5. Transient response to an applied step function Resonant mode shape Resonant frequency vs applied bias (showing spring softening) DC displacement of biased structure showing electric potential contours Electrostatically Actuated Cantilever Mesh and structural displacement The elastic cantilever beam is one of the elementary structures used in MEMS designs. This model shows the bending of a cantilever beam under an applied electrostatic load. The model solves the deformation of the beam under an applied voltage. C-V curve Electric field and electric potential
  • 20. Electrostrictive Disc Settings for linear elastic dielectric Electric potential and structural displacement Axial strain vs electric field Microresistor Beam This model treats the movement of a beam by conducting a current through it to generate a temperature increase. This leads to a displacement through thermal expansion. The coupled thermal, electrical, and structural analysis makes it possible to estimate the current and temperature increase needed to displace the beam. This model shows how to compute the electrostrictive strains and stresses in an isotropic disc , using the electromechanics interface.
  • 21. Thermal Stresses in a Layered Plate A plate consisting of two layers (a coating and a substrate layer) is stress and strain free at 800 C. The temperature of the plate is reduced to 150 C, which induces thermal stresses. A third layer, the carrier layer, is then added. The thermal stresses in the two plates are included as an initial stress, and the temperature is finally reduced to 20 C. Prestressed Micromirror This model shows the fundamentals of how to set up and solve lift-off of a prestressed plated micromirror. A parametric study reveals how much variations in the prestress affect the displacements.
  • 22. Fluid-Structure Interaction This model exemplifies how to model fluidstructure interactions (FSI) using the MEMS Module. Viscous forces and the system's pressure impose forces to the surface of a structure. The deformation in the soft structure is not small and the fluid regime will therefore change. This means that changes in the structure and the fluid dynamics are coupled. Composite Piezoelectric Transducer A quartz oscillator, operated in the thickness shear mode, is simulated. The model shows how to set up the co-ordinate system correctly for AT cut quartz and to model the response of a device driven at resonance. The resonant frequency of the oscillator is altered by the changing the capacitance of a shunt capacitor. Displacement Electric Potential Mechanical response with different shunt capacitors
  • 23. Composite Piezoelectric Transducer A composite piezoelectric ultrasonic transducer is analyzed. An eigenfrequency analysis is followed by a frequency response analysis to calculate the input admittance as a function of the excitation frequency. Piezoceramic Tube Displacement along radial direction This model performs a static 2D axisymmetric analysis of a piezoelectric actuator. A radially polarized piezoelectric tube is simulated, with two sets of boundary conditions. The first case illustrates the direct piezoelectric effect (see image), and the second case shows the inverse piezoelectric effect. Induced voltage along radial direction Induced voltage Displacement
  • 24. Surface Acoustic Wave Gas Sensor This model analyzes the eigenfrequencies of a surface acoustic wave (SAW) gas sensor. In particular, the model studies how the additional mass load from an adsorbed gas lowers the resonance frequency. Piezoresistive Pressure Sensor Electric current (arrows) and electric potenial (contours) A piezoresistive pressure sensor is simulated. This model shows how to set compute the stress induced potential difference produced by a four terminal piezoresistor when the membrane in which it is embedded is deformed by an applied pressure. Two versions of the model are available - the surface currents model shown here, and a model that uses the shell interface which requires the Structural Mechanics Module. Shear stress along diaphragm edge in local co-ordinates Diaphragm displacement
  • 25. Gecko Foot This model contains the nano/micro hierarchy of synthetic gecko foot hair, where cantilever beams on different scales describe the hairs. The analysis shows the stresses and deflection of the gecko foot caused by contact and friction forces. A Tunable MEMS Capacitor* This is a model of an electrostatically tunable parallel plate capacitor. The distance between the plates is tuned via a spring, to which one of the plates is attached. For a given voltage difference between the plates, the distance of the two plates can be computed, if the characteristics of the spring are known. * This Model is Part of the AC/DC Module Model Library, but the model can also be set up and solved with the MEMS Module.
  • 26. Uniform Layer Waveguide* In this model the thermoacoustic wave field in a shallow uniform waveguide is modeled and compared to an analytical solution. Because of the small waveguide height (1 mm) the thermal and viscous boundary layers are significant. Detailed air or fluid damping for vibrating MEMS devices may require accounting for thermoacoustic damping effects. This can be seen as a generalization of more conventional gas-film damping. * This model and analysis requires the Acoustics Module. Nonlinear Magnetostrictive Transducer* The magnetic field and displacement as functions of the applied current are computed for a magnetostrictive transducer where the BH curve is nonlinear. This model considers the case when the material is sufficiently prestressed so as to obtain the maximum magnetostriction. * This model is made with 2D axisymmetric magnetics available in the COMSOL Multiphysics base package. For 3D magnetics modeling, the AC/DC Module is required. This model is available in the Structural Mechanics Module Model Library but also runs with the MEMS Module.
  • 27. Thermal Expansion in a MEMS Device* This model analyzes the thermal expansion in a MEMS device, such as a microgyroscope, where the thermal expansion should be as small as possible. The model uses temperature-dependent material properties from the Material Library. * The Material Library is required to build the model. Thermoelastic Damping Settings for Linear Thermoelastic Material Temperature Axial stress This model shows how to compute the thermoelastic damping in a vibrating beam.
  • 28. Squeeze Film Disc Pressure on the disc Non-linear response This benchmark model computes the forces on a vibrating disc in for small (using the frequency domain) and large (using a transient study) amplitudes. Piezoelectric Shear-Actuated Beam This model performs a static analysis of a composite cantilever beam equipped with a piezoceramic actuator. An electric field is applied perpendicular to the poling direction, thereby introducing a transverse deflection of the beam.
  • 29. Piezoelectric Microgripper This model shows the fundamentals of how to set up a piezoelectric model with mechanical contact. The microgripper contains a stacked piezoactuator, which operates in the longitudinal mode. Simultaneous contraction in the transversal direction and elongation in the longitudinal direction closes the gripper and moves objects. • MEMS • MEMS • MEMS • MEMS
  • 30. COMSOL is a Fully Integrated Software Suite All modeling steps are available from a single unified environment. The model tree (shown on the left) provides quick and easy access to all the settings: Geometry setup / CAD Import User defined and built in material libraries Simple and intuitive Mulitphysics Problem setup Meshing Solving Visualization + Postprocessing Data Import/Export COMSOL is designed from the bottom up for arbitrary combinations of physical equations and easy user customization Problem Definition • Electrostatically biased cantilevered beam above ground Infinite Free Space 300 um V+ 5 um gap not to scale 2 um thick
  • 31. Electrostatics • The electrostatic equation must be solved in and around the beam 0 r V 0 V+ Forces on Beam • Once the voltage field is computed, the Maxwell Stress Tensor is c omputed • These forces deflect the cantilever
  • 32. How do we couple electrostatics and solid m echanics? 1. Solve for electrostatics 2. Calculate Maxwell stresses and add t hem on the solid boundaries 3. Solve for the displacements in solids 4. Repeat the steps until solution conver ges All these steps are automated in COMSOL What if solids undergo large deformation? Linear strain (Cauchy strain) Green-Lagrange strain • COMSOL can account for large deformation in solids • We cannot use the same computational mesh to solve for the electrostatics problem anymore
  • 33. Reference mesh • Mesh is “rigid” in the structural domain • You solve the structural equation here • We are using the Lagrangian (material) coordinate system Moving Mesh • Mesh is allowed to move in the air domain based on the structural displacement • You solve the electrostatics equation here • We are using the Eulerian (spatial) coordinate system
  • 34. Lagrangian vs. Eulerian • Lagrangian – Mesh follows the material deformation – Easier for representing structural mechanics • Eulerian – Mesh is fixed – Continuum moves with respect to this grid – Easier for most physics (fluids, EM, etc.) oth er than structures • Arbitrary Lagrangian Eulerian (ALE) – Best of both worlds – Moving mesh What this means • Solid mechanics – I am sitting on the mesh that represents t he beam and moving along with it • Electrostatics – I am sitting outside and seeing the mesh that represents air get deformed as the b eam moves All this is automatically taken care of in COMSOL
  • 35. Electric Potential Displacement You can also solve this in 3D But…2D is faster, so we will do that first
  • 36. Objectives • Set up an electromechanical interaction problem • How to mesh high aspect ratio 2D structures • Vary the voltage on the beam and compute the capacitance – Tunable capacitor! Cantilever Beam 1
  • 37. Modeling steps • New > 2D • Structural Mechanics > Solid Mechanics ( Add Selected ) • AC/DC > Electrostatics (Add Selected) • Mathematics > Deformed Mesh > Moving Mesh (Add Selected) • Next • Stationary • Finish Geometry – Change Units to Microns • Highlight “Geometry 1” • Change Length unit to “ m”
  • 38. Geometry - Beam • Rt click on “Geom 1” • Choose “Rectangle” Size and Shape • Width: 320 • Height: 2 Position: Corner • x: 0 • y: 5 Geometry - Air • Rt click on “Geom 1” • Choose “Rectangle” Size and Shape • Width: 320 • Height: 25 Layers to the right • Thickness: 20
  • 39. Materials – Set up Air and Polysilicon • • • • Rt Click on “Materials” Open Material Browser Built-in > Rt Click on “Air” Add to Model • • • • • Rt Click on “Materials” Open Material Browser Built-in > Rt Click on “Polysilicon” Add to Model Assign the beam (domain #2) to Polysilicon Solid Mechanics • Pick “Solid Mechanics” • Domain Selection > “Polysilicon” • Thickness: 150[um] • Rt Click on Solid Mechanics > Fixed Constraint • Choose Left End of Beam (#3)
  • 40. Structural – Electrostatic Load • • • • • • Rt Click on “Solid Mechanics” Choose “Boundary Load” Select Bottom Boundary (#4, 6, 10) Enter: es.nTx_Fes es.nTy_Fes Set up Parameter for Applied Voltage • Rt Click on “Global Definitions” • Choose “Parameters” • Enter Name as “Vdc” • Enter Value as “20[V]”
  • 41. Specify the out-of-plane thickness • Click “Electrostatics” • In the “Thickness” section, set d as 150[um] • This is used to calculate the capacit ance Electrical Boundary Condition – Ground • Rt Click on “Electrostatics” • Choose “Ground” • Pick the bottom two boundarie s (#2, 9)
  • 42. Electrical Boundary Condition – Terminal • Rt Click on “Electrostatics” • Choose “Terminal” • Pick the far end, top and bottom of the beam (#4, 6, 10) • Terminal Type “Voltage” • Enter Voltage as “Vdc” Electrostatics Forces on Beam • • • • Rt Click on “Electrostatics” Choose “Force Calculation” Select Domain: “Polysilicon” Name the force “Fes”
  • 43. Moving Mesh – Specify Domain Motions • Rt Click on “Moving Mesh” • Add “Free Deformation” • Choose Domain: “All Domains” • • • • • Rt Click on “Moving Mesh” Add “Prescribed Deformation” Pick Domain: “Polysilicon” Set dx to “u” Set dy to “v” Free Deformation Boundary Conditions • • • • Rt Click on “Moving Mesh” Choose “Prescribed Mesh Displacement” Choose Boundaries: “4, 6, 10” Set to “u” and “v”
  • 44. Mapped Mesh • Rt Click on “Mesh 1” • Choose “Mapped” • Highlight “Size” • Change “Predefined” to “Extra Fine” We are using 2nd order Lagrange elements…so one element across the thickness of the beam is ok Solve • Rt Click on “Study 1” • Select “Parametric Sweep” • Set up the parametric sweep to vary Vdc from 0 to 20 V at a step of 2 V • Rt Click on “Study 1” • Hit Compute Set up the model once but solve it for multiple inputs
  • 45. Results: Electric Potential Actual aspect ratio Results: Displacement Not to scale…but better visualization for high aspect ratio structures
  • 46. Plot Capacitance vs. Vdc Capacitance is automatically calculated when you use the Terminal boundary condition Cantilever Beam 2
  • 47. Modeling steps • New > 2D • Structural Mechanics > Electromechanics • Next • Stationary • Finish Geometry – Change Units to Microns • Highlight “Geometry 1” • Change Length unit to “ m”
  • 48. Geometry - Beam • Rt click on “Geom 1” • Choose “Rectangle” Size and Shape • Width: 320 • Height: 2 Position: Corner • x: 0 • y: 5 Geometry - Air • Rt click on “Geom 1” • Choose “Rectangle” Size and Shape • Width: 320 • Height: 25 Layers to the right • Thickness: 20
  • 49. Materials – Set up Air and Polysilicon • • • • Rt Click on “Materials” Open Material Browser Built-in > Rt Click on “Air” Add to Model • Rt Click on “Materials” • Open Material Browser • Built-in > Rt Click on “Polysilicon” • Add to Model • Assign the beam (domain #2) to Polysilicon Specify the out-of-plane thickness • Click “Electromechanics” • In the “Thickness” section, set d as 150[um] • This is used to calculate the capacit ance
  • 50. Define What is Solid • Expand “Electromechanics” • Highlight “Linear Elastic Dielectric” • Pick the Polysilicon Beam (#2) Structural Boundary Conditions – Fixed • Rt Click on “Electromechanics” • Choose “Structural” > Fixed Constraint” • Pick the End of the beam “
  • 51. Set up Parameter for Applied Voltage • Rt Click on “Global Definitions” • Choose “Parameters” • Enter Name as “Vdc” • Enter Value as “20[V]” Electrical Boundary Condition – Ground • Rt Click on “Electromechanics” • Choose “Electrical” > “Ground” • Pick the bottom two boundarie s (#2, 9)
  • 52. Electrical Boundary Condition – Terminal • Rt Click on “Electromechanics” • Choose “Electrical” > “Terminal” • Pick the far end, top and bottom of the beam (#4, 6, 10) • Terminal Type “Voltage” • Enter Voltage as “Vdc” Mapped Mesh • Rt Click on “Mesh 1” • Choose “Mapped” • Highlight “Size” • Change “Predefined” to “Extra Fine” We are using 2nd order Lagrange elements…so one element across the thickness of the beam is ok
  • 53. Solve • Rt Click on “Study 1” • Select “Parameteric Swe ep” • Set up the parametric sw eep to vary Vdc from 0 to 20 V at a step of 2 V • Rt Click on “Study 1” • Hit Compute Set up the model once but solve it for multiple inputs Results: Electric Potential Actual aspect ratio
  • 54. Results: Displacement Not to scale…but better visualization for high aspect ratio structures Plot Capacitance vs. Vdc Capacitance is automatically calculated when you use the Terminal boundary condition
  • 55. AND MORE… EigenFrequencies
  • 56. Modeling steps • Start from the last model • Rt click on the root node • Select Add Study • Select Prestressed Analysis, Eigenfrequency • Finish This solves a static problem to evaluate the solution at the DC bias point and then performs an eigenfrequency analysis where it uses the solution of the static analysis to create the “shift” Solve • • • Rt Click on “Study 2” Select “Parameteric Sweep” Set up the parametric sweep to v ary Vdc from 0 to 20 V at a step of 4 V • Click on Study 2 > Step 2: Eige nfrequency Set Desired number of eigenfr equencies to 1 • • • • Click on “Study 2” and uncheck Generate default plots Rt Click on “Study 2” Hit Compute We will only calculate the first or “fundamental” resonance
  • 57. Results: Resonant frequency vs. DC bias voltage • By using a DC bias voltage you can tune: – Capacitance – Resonant frequency • Now we have a tunable resonator! Frequency Domain
  • 58. Modeling steps • Start from the last model • Rt click on the root node • Select Add Study • Select Prestressed Analysis, Frequency Domain • Finish This solves a static problem to evaluate the solution at the DC bias The frequency domain problem is then solved as a linearized small perturbation around this bias point Set up additional parameters • Rt Click on “Global Definitions” • Choose “Parameters” • Enter Name as “Vac” • Enter Value as “0.2[V]” • Enter Name as “eta_s” • Enter Value as “0.01”
  • 59. Add damping • • • • Rt click “Electromechanics > Linear Elastic Dielectric 1” Select “Damping” Choose “Damping type” as “Isotropic loss factor” Set it as a User defined value of eta_s Add the AC input • Rt click “Electromechanics > Terminal 1” • Select “Harmonic perturbation” • Set the Electric potential as Vac • This is the magnitude of the AC signal • This information will only be used for the frequency-domain pertu rbation part of the study
  • 60. Solve • • • Rt Click on “Study 3” Select “Parameteric Sweep” Set up the parametric sweep to a lter Vdc to 2 V and 6 V • Click on “Study 3 > Frequency-D omain Perturbation” Set up the frequency to vary fro m 25 kHz to 35 kHz at a step of 0.2 kHz • Tweaking the solver settings • Rt Click on “Study 3” • Select “Show Default Solver” • Click on “Study 3 > Solver Conf igurations > Solver 1 > Stationa ry Solver 2 > Direct” • Expand the section on “Error” • Set the Factor in error estimat e as 50 • Rt Click on “Study 3” • Hit Compute Setting this factor as a small number is helpful when we are solving a linearized model but the geometry has high aspect ratio and we are using less mesh elements to save memory and time
  • 61. Results: Displacement vs. Frequency at different DC bias voltages • By using a DC bias voltage you can tune: – Capacitance – Resonant frequency – Dynamic response • Now we have a tunable resonator!
  • 62. Microfluidics Modeling with COMSOL Multiphysics • Microfludics • Microfludics • Microfludics • Microfludics
  • 63. Microfluidic Processes – Dimensionless Numbers • Pressure driven flow (Re): laminar flow or creeping flow, slip/no slip wa ll, moving and leaking wall • Species transport with reaction (PeMT): dispersion, mixing and separ ation, filtration, surface reactions, chaotic advection, electrokinetics • Interface tension and multiphase flow (Ca, We): free surface deform ation, jet and droplet, drop dynamics, moving contact • Heat transfer (PeHT, Mg): heat transfer, heat of reaction, phase change , evaporation, condensation, Marangoni effects Reynolds number (Re): measure of the ratio of inertial forces to viscous forces. Peclet number (Pe): the ratio of the rate of convection/advection to the rate of diffusion. Capillary number (Ca): the relative effect of viscous forces versus surface tension. Weber number (We): fluid's inertia compared to its surface tension. Marangoni number (Mg): the ratio thermal surface tension forces to viscous forces. Re U 0 L0 Pe MT U 0 L0 D Ca We U0 U 02 L Pe HT c pU 0 L0 Mg d L T dT
  • 64. Key Application Areas: Lab on a Chip Microfluidic platforms / Lab on a Chip Systems Capillary Driven Droplet Based Droplet production in a T-junction Electroosmotic micropump Pressure Driven Electrokinetic Magnetokinetic Centrifugal Pressure Driven Micromixer Key Application Areas Inkjet printing Drug delivery Electrowetting optical / display devices Fuel cells
  • 65. • Microfludics • Microfludics • Microfludics • Microfludics Single Phase Flow • • • • • • Laminar Flow (Re 1000) Reynolds number: U L 0 0 Re Creeping Flow (Re«1) Incompressible flow Compressible flow (Ma<0.3) Shallow channel approximation Non-Newtonian Flow: – Power law: – Carreau model: – User defined: • look-up table or mathematical expression • dependent on any physics variable: e.g. shear rate variable, Temperature
  • 66. Multi-Phase Flow Three complimentary methods available: Moving Interface, Fixed Mesh Level Set Phase field Moving Mesh Multi-Phase Flow: Moving Interface, Fixed Mesh Level Set and Phase Field Interfaces • • • • Problem solved on a fixed mesh. Level set/Phase Field function switch between a low value (LS: 0 PF:-1) in one phase and high value in the second phase (LS/ PF:1) – density, viscosity etc., is scaled accordingly. Interface is diffuse and centered on the center value of these fu nctions (LS:0.5 PF:0). Level Set – – • Level set function is solved for in addition to the Navier-Stokes equations Level set method usually represents surface tension more accurately than p hase field method Phase field – – Both phase field and phase field helper functions are solved in addition to th e Navier-Stokes equations Phase field method is physically motivated, and is generally more numericall y stable than the level set method. It can be extended to more phases (diffic ult!) and is compatible with fluid structure interactions.
  • 67. Multi-Phase Flow: Moving Mesh • • Problem solved on a moving mesh, using the ALE method. Interface is sharp and corresponds to a boundary in the initial geometry. – – • Since physical interfaces are usually much thinner than practical mesh res olutions this technique offers the most accurate representation of the interf ace. The sharp interface means that different physics can be solved in the dom ains either side of the interface – for example chemical migration can be a dded to the model on one side of the boundary only. The mesh must deform continuously, which means problems involving topological changes cannot be solved. Porous Media Flow • Darcy’s Law – Viscosity dominated flow. – Incompressible flow only • Brinkman Equations – Fluid inertia included (can be neglected b y using Stokes-Brinkman option) – Compressible/Incompressible flow – Body forces can be added • Free and porous media flow – Pre-defined coupling between Brinkman equations and the laminar flow interface. Note: Porous media flow in COMSOL has been applied to model intracellular transport
  • 68. Fluid Structure Interactions (with MEMS or Structural Mechanics Module) • • Based on the ALE moving mesh technolo gy in COMSOL Fully bidirectional coupling between fluid a nd structure – – • • • • Viscous, pressure and inertial fluid forces o n the structure Momentum transfer back to fluid Small and large deformations Highly accurate fluid load computation (we ak constraints). Predefined physics interface makes it eas y to build models rapidly – no manual inter facing between different solvers Automatic segregated solver settings for t he stationary and parametric analysis of la rge 3D models Species Transport • Enhanced interface for transport of diluted spec ies – • Diffusion, Convection & Migration of multiple species Together with Chemical Reaction Engineering Module: – – – Transport of concentrated species Chemical reactions: Bulk / Surface Reaction engineering interface
  • 69. Electrokinetics/Magnetohydrodynamics • • Electrowetting Electroosmosis – – • • • • Explicit modeling of electric double layers on small lengt h scales On larger scales, the electroosmotic mobility can be spe cified as part of a slip boundary condition. Electrophoresis Dielectrophoresis Magnetophoresis (with AC/DC Module) Electrothermal flows Thermal Flows • • • • • • • Non-isothermal flows Natural and forced convection Conduction Radiation (with Heat Transfer Module) Joule Heating (extended with AC/DC Module) Thermophoresis Marangoni flows
  • 70. Rarefied Flows Continuum o n Continuum Flow l Flow Degree of rarefaction characterized by the Knudsen number, Kn: Slip Flow Slip Slip Flow Flow Transitional Flow Transitional s Transitional Flow w Flow Molecular c Molecular Flow Molecular o Flow Flow Continuum Flow Kn L Rarefied Flow Interfaces: 0.01<Kn<0.1: Slip Flow 0.1<Kn<10: Transitional Flow Kn>10: Molecular Flow Transitional Flow and Molecular Flo w interfaces are available in the Mol ecular Flow Module Slip Flow Requires Microfluidics Module • • • • Navier Stokes equations still apply except f or a small region near the boundaries (the Knudsen Layer). COMSOL’s slip flow interface can be used t o model laminar, isothermal or thermal flow s in the slip flow regime. Slip and temperature jump boundary conditi ons are available to model the Knudsen lay er. Specifying the slip can be done in two ways : 1. Through Maxwell’s model with arbitrary accommod ation coefficients. 2. By specifying directly the viscous and thermal slip coefficients along with the temperature jump coeffi cient.
  • 71. Transitional Flow • • The transitional flow interface can handle rarefied flows from the Navier Stokes to the Free Molecular Flow limits. It employs the discrete velocity method: • A finite set of velocities are chosen to represent all the potential velocities of the molecules (the more velocities that are chosen – the better a rarefied flow will be represented). • Atoms are assigned into these velocity bins and the interface solves for the number density in each bin. A convection equation is solved in the domains, along with a scattering term that moves the molecules between the bins. • Total accommodation is assumed on the boundaries The transitional flow interface uses a discrete set of velocities, similar to the set shown, to model the flow. Molecular Flow • • Molecules collide only with the chamber walls – no collisions between molecule s Angular Coefficient Method – – • • • • Molecules collide only with the chamber walls – collisions between molecules are negligible. Diffuse reflection only through Knudsen’s cosi ne law (emitted flux intensity cos( )) No thermal fluctuations. Faster than DSMC for low velocity flows – easier to perform parametric studies e tc. Stationary flows only. Computation of pressure and number d ensity is performed using correct results from Kinetic Theory – More accurate than previously employed ‘radi ation analogy’ modeling.
  • 72. • Microfludics • Microfludics • Microfludics • Microfludics Controlled Diffusion Micromixer • • Micromixer example designed to provi de an introduction to modeling techniq ues used in microfluidics. Shows how to set up a simple fluid flo w problem using some of the core feat ures of the Microfluidics Module: – – – • Creeping flow Species transport Laminar inflow boundary conditions Demonstrates how to use a range of p ost-processing features including: – – – 3D, 2D and 1D plot groups Creating derived data sets such as a cut plan e from the solution data sets generated auto matically by COMSOL Evaluating the numerical values of derived q uantities such as the mean outlet concentrati on.
  • 73. Other Micromixer Models – Enhanced Mixing • Key features: – Import the geometry from industr y-standard CAD – any level of co mplexity is possible – Moving mesh, rotating frame – Moving/rotating parts – Concentrated or diluted species c onvection and diffusion – Mixing index calculation – Nonlinear effects and more physi cs can be added: thermal, chemic c al, electric or magnetic field … Inkjet Printing • • • • Microfluidic technology Propelling droplet onto paper Pressure oscillation generated by piezo or heater Key Features: – Interface tracked exactly; Interface normal, curvature and physical qua ntities can be computed – Good mass conservation with reinit ialization – Mapped mesh – Multiphysics coupling: electrostatic s, heat transfer…
  • 74. Electroosmotic Pump • • • • Essential components in microfluidic l ab-on-a-chip devices No moving parts Easy to integrate in microfluidic circui ts Key Features: – Flow field: momentum balance (S tokes or Navier-Stokes) – Electric field: current balance – Using built-in boundary condition – Parametric study on pressure find s the strength of the pump Transport in an Electrokinetic Valve • Key Features: – Flow field: momentum balance (Stokes or Navier-Stokes) – Electric field: current balance – Mass transport: diluted species, NernstPlanck – Fully integrated – Solved the model in sequence accordin g to the actual device – Yon can specify: charge number, mobili ty, diffusivity (isotropic or anisotropic), n onlinear material properties
  • 75. Capillary Filling • Key Features: – Capillary driven two phase flow – Surface tension and wall adhesive forc es – Built-in moving contact: interface movin g along the wall, hydrophobic or hydrop hilic – Specify friction using wall slip length an d contact angle – Wettability gradients: thermal gradient ( built-in), electrowetting, optowetting – Ready to couple with other physics Electrophoresis • • • • • Coupling of – Fluid flow – Electric current – Moving mesh Walls and particle surfaces are rigid a nd insulating Walls and particle have uniform nega a tive zeta potential Electric double layers not resolved User’s model presented at COMSOL 2006 Conference Model courtesy of Davison and Sharp, COMSOL 2006 Conference
  • 76. Dielectrophoresis • how to model dielectrophoresis of particles in an alternating currents (AC) field • Two groups of particles are considered: one group with positive DEP (pDEP : red) and one group with negative DEP (nDEP : blue). • with Particle Tracing High field region V fCM AC Electrokinetically Enhanced Surface Re actions • Key features: – Coupling of electrostatics, heat transf er with joule heating, species transfer with surface reaction and laminar flow – Electrothermal force generated by te mperature gradient due to inhomogen eous joule heating – Swirling patterns in the fluid enhance t he transport of the analyte to the react t ion surface Model courtesy of Gaurav Soni, Marin Sigurdson, and Carl Meinhart of the Department of Mechanical and Environmental Engineering, University of California, Santa Barbara.
  • 77. Surface Reactions in a Biosensor • A flow cell in a biosensor contains an array of micropillars coated with an active • Micropillars introduces passive mixing • Adsorbtion of species produce a signal • Surface Reactions coupled with mass transport in a fluid stream Drug Delivery System • Drug delivery system that supplies a variable concentration of a water soluble drug. • A droplet with a fixed volume of water travels down a capillary tube at a constant velocity. • Part of the capillary wall consists of a permeable membrane , as the drop passes by the membrane, its contact angle changes and drug dissolves into the water. • Modeling using Transport of Diluted Species, Laminar Two-Phase Flow, and Moving Mesh
  • 78. Split-Recombine Mixer Benchmark • Fluid flow benchmark that computes the lamination pattern in a complex split and recombine mixer. • Lamination patterns are computed in the absence of physical diffusion. • COMSOL compares well with the results in the reference below. T. Glatzel et al, Computational fluid dynamics (CFD) software tools for microfluidic applications - A case study, Computers & Fluids, vol. 37, pp. 218235, 2008. . Rotating Channel Benchmark Pressure (Pa) • Fluid flow benchmark that computes pressure along the axis of a rotating channel. • Example showing how to implement centrifugal and centripetal forces – a good starting point for a centrifugal microfluidics model. • COMSOL compares well with the results in the reference below. Distance along axis (mm) T. Glatzel et al, Computational fluid dynamics (CFD) software tools for microfluidic applications - A case study, Computers & Fluids, vol. 37, pp. 218235, 2008.
  • 79. Tesla Microvalve Optimization Requires Optimization Module Fluid Velocity (m/s), Forward Flow • Geometrical optimization of a Tesla microvale, which inhibits backwards flow on a fixed geometry by utilizing friction forces instead of moving parts. • Model is optimized to minimize the ratio of frictional losses in forward flow to that of frictional losses in reverse flow. • Pressure drop across the valve for forward flow is approximately 200 Pa, for reverse flow it is approximately 380 Pa. • Shows how to perform shape optimization on a fluid flow geometry. Fluid Velocity (m/s), Reverse Flow Ion Implanter Number density along beam path for different angles of the wafer to the beam This model computes the number density of outgassed hydrogen along the ion beam path in an ion implanter. When the ion beam strikes part of the wafer covered in photoresist, the resist breaks down, outgassing undesirable species (including hydrogen) into the system. If the ions strike these outgassed species along the beam path they can become ionized and can be accelerated into the wafer. The implantation of these secondary species degrades the process. • A parametric study is performed to asse ss the effect of altering the angle betwee n the ion beam and the wafer. • The number density along a line interior to the flow domain is computed using th e number density reconstruction feature.
  • 80. Evaporator In an evaporator a metallic sample is heated beyond its melting point. The liquid metal then evaporates into the chamber, at a rate controlled by the temperature of the sample. This model shows how to compute the evaporated film thickness on a wafer segment and on the walls of the system. • • The deposition boundary condition is used to compute the film thickness on the chamber w alls as a function of time. The evaporation feature is used to specify the outgoing flux of evaporated metal from the so urce. The surface temperature feature is also used to specify an elevated temperature for th e source. Water Adsorption/Desorption This model shows how to model the time dependent adsorption and desorption of water in a vacuum system at low pressures. The water is introduced into the system when a gate valve to a load lock is opened and the subsequent migration and pumping of the water is modeled. • • Illustrates how to model chamber pumpdown pro blems. Gives simple examples for the equations required to specify sticking coefficients, desorption rates et c. when modeling adsorption and desorption.
  • 81. Differential Pumping Differentially pumped vacuum systems use a small orifice or tube to connect two parts of a vacuum system at very different pressures. Such systems are necessary when processes run at higher pressures and are monitored by detectors which require UHV for operation. In this model the gas flow through a narrow tube into a high vacuum chamber is approximated using an analytic expression for the flow rate down the tube. • Shows how to couple molecular flow to (isotherm al) transitional flow down a small tube, using anal ytic expressions for the flow rate through the tube . Analysis assesses the effect of molecular beamin g in the tube in the molecular flow regime. Model can be adapted to couple transitional flow t hrough a narrow tube to any geometry. • • Outgassing Pipes Monte-Carlo 1D simulation COMSOL This benchmark model computes the pressure in a system of outgassing pipes with a high aspect ratio. The results are compared with a 1D simulation and a MonteCarlo simulation of the same system from the literature.
  • 82. RF Coupler (Requires particle tracing module) This model computes the transmission probability through an RF coupler using both the angular coefficient method available in the Free Molecular Flow interface and a Monte Carlo method using the Mathematical Particle Tracing interface. The computed transmission probability by the two methods is in excellent agreement, less than 1% difference Trajectories of molecules computed by Particle Tracing interface Molecular flux computed by Molecular Flow interface Rotating Plate Number density on the surface of a rotating plate in a molecular beam computed by different methods, as a function of the angle of the plate to the beam Radiation method COMSOL: Total number density COMSOL: Outgoing number density COMSOL: Incident number density This model computes the particle flux, number density and pressure on the surface of a plate that rotates in a highly directional molecular flow. The results obtained are compared with those obtained from the so-called ‘radiation method’, previously employed for this type of computation. • The schematic (lower left) shows t hat when a plate is perpendicular t o a molecular beam the incident fl ux goes to zero but the number de nsity does not. • The radiation method incorrectly p redicts that the number density go es to zero when the plate is perpe ndicular to the flow.
  • 83. S-Bend Benchmark Particle trajectories (Particle Tracing) Molecular flux (Free Molecular Flow) Number density (Free Molecular Flow) (requires the Particle Tracing Module) This model computes the transmission probability through an s-bend geometry using both the angular coefficient method available in the Free Molecular Flow interface and a Monte Carlo method using the Mathematical Particle Tracing interface. The computed transmission probability by the two methods is in excellent agreement, less than 1% difference. • 2D example which shows how to co mpute transmission probabilities with the particle tracing and free molecul ar flow interfaces • Model also shows how to compute t he number density on a domain.. Vacuum Capillary Molecular flow down a cylinder was one of the first problems in the field to be treated analytically. In this benchmark model the transmission probability is computed for molecular flow down a capillary tube of variable length and the results are compared with the analytic solution.
  • 84. • Microfludics • Microfludics • Microfludics • Microfludics Electroosmotic micromixer
  • 85. Electrowetting Lens AND MORE…
  • 86. Molecular Flow Interface: Modeling Process The COMSOL user interface presents simulation information in a compact, easy to understand format. Graphics Area: Used to select geometric features and to display results Active Node Model Tree: A hierarchical list of all the model properties Settings Area: displays the detailed settings for a node in the model tree Molecular Flow Interface: Modeling Process 1. Create the geometry. Define the geometry Or use your preferred CAD Directly in COMSOL tool and import the file or dynamically link to COMSOL using a LiveLink Product
  • 87. Molecular Flow Interface: Modeling Process 2. Set up materials and physics COMSOL supports a range of boundary conditions to conveniently specify the flow. Industry standard units (e.g. torr, mBar, litres/s, sccm) are supported (SI units are used by default but others can be entered in square brackets) 1000 [l/s] Molecular Flow Interface: Modeling Process 3. Mesh the geometry. A range of options are available for the mesh. It is possible to mesh just the surfaces of the model or the entire domain.
  • 88. Molecular Flow Interface: Modeling Process 4. Solve A range of studies are available including: • Stationary • Time dependent (to study secondary effects such as pump down, getters etc.) • Parametric (parameterize geometry, settings etc.) • Optimization COMSOL automatically suggests the best settings for the study, but advanced users have full flexibility to change all the solver settings. Molecular Flow Interface: Modeling Process 5. Post-Processing COMSOL’s powerful post-processing tools are available to analyze the results.
  • 89. Electroosmotic Micromixer
  • 90. Solved with COMSOL Multiphysics 4.3a Electroosmotic Micromixer 1 Introduction Microlaboratories for biochemical applications often require rapid mixing of different fluid streams. At the microscale, flow is usually highly ordered laminar flow, and the lack of turbulence makes diffusion the primary mechanism for mixing. While diffusional mixing of small molecules (and therefore of rapidly diffusing species) can occur in a matter of seconds over distances of tens of micrometers, mixing of larger molecules such as peptides, proteins, and high molecular-weight nucleic acids can require equilibration times from minutes to hours over comparable distances. Such delays are impractically long for many chemical analyses. These problems have led to an intense search for more efficient mixers for microfluidic systems. Most microscale mixing devices are either passive mixers that use geometrical stirring, or active mixers that use moving parts or external forces, such as pressure or electric field. In a passive mixer, one way of increasing the mixing is by “shredding” two or several fluids into very thin alternating layers, which decreases the average diffusion length for the molecules between the different fluids. However, these mixers often require very long mixing channels because the different fluids often run in parallel. Another way of improving mixing efficiency is to use active mixers with moving parts that stir the fluids. At the microscale level moving parts in an active mixer are very fragile. One alternative is to use electroosmotic effects to achieve a mixing effect that is perpendicular to the main direction of the flow. This model takes advantage of electroosmosis to mix fluids. The system applies a time-dependent electric field, and the resulting electroosmosis perturbs the parallel streamlines in the otherwise highly ordered laminar flow. Model Definition This example of a rather simple micromixer geometry (Figure 1) combines two fluids entering from different inlets into a single 10 μm wide channel. The fluids then enter a ring-shaped mixing chamber that has four microelectrodes placed on the outer wall at angular positions of 45, 135, −45, and −135 degrees, respectively. Assume that the 1. This model is courtesy of H. Chen, Y. T. Zhang, I. Mezic, C. D. Meinhart, and L. Petzold of the University of California, Santa Barbara (Ref. 1 and Ref. 2). ©2012 COMSOL 1 | ELECTROOSMOTIC MICROMIXER
  • 91. Solved with COMSOL Multiphysics 4.3a aspect ratio (channel depth to width) is large enough that you can model the mixer using a 2D cross-sectional geometry. The material parameters relevant for the model are given in Table 1. −V0sin(ωt) V0sin(ωt) 1 2 10 μm −V0sin(ωt) 4 3 V0sin(ωt) Figure 1: Geometry of the micromixer with four symmetric electrodes on the wall of the mixing chamber. This example does not model the two inlet channels. Here you assume a parabolic inflow at the beginning of the computational domain (the gray area). The Navier-Stokes equations for incompressible flow describe the flow in the channels: ρ T ∂u – ∇ ⋅ η ( ∇u + ( ∇u ) ) + ρu ⋅ ∇ u + ∇p = 0 ∂t ∇⋅u = 0 Here η denotes the dynamic viscosity (kg/(m·s)), u is the velocity (m/s), ρ equals the fluid density (kg/m3), and p refers to the pressure (Pa). Because you do not model the two inlet channels, assume that the entrance channel starts at a position where the flow has a fully developed laminar profile. The mixed fluid flows freely out of the right end boundary, where you specify vanishing total stress components normal to the boundary: T n ⋅ [ – pI + η ( ∇u + ( ∇u ) ) ] = 0 When brought into contact with an electrolyte, most solid surfaces acquire a surface charge. In response to the spontaneously formed surface charge, a charged solution forms close to the liquid-solid interface. Known as an electric double layer, it forms because of the charged groups located on the surface that faces the solution. When the operator applies an electric field, the electric field generating the electroosmotic flow displaces the charged liquid in the electric double layer. This scheme imposes a force on the positively charged solution close to the wall surface, and the fluid starts to flow 2 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 92. Solved with COMSOL Multiphysics 4.3a in the direction of the electric field. The velocity gradients perpendicular to the wall give rise to viscous transport in this direction. In the absence of other forces, the velocity profile eventually becomes almost uniform in the cross section perpendicular to the wall. This model replaces the thin electric double layer with the Helmholtz-Smoluchowski relation between the electroosmotic velocity and the tangential component of the applied electric field: εw ζ0 u = ----------- ∇TV η In this equation, εw = ε0εr denotes the fluid’s electric permittivity (F/m), ζ0 represents the zeta potential at the channel wall (V), and V equals the potential (V). This equation applies on all boundaries except for the entrance and the outlet. Assuming that there are no concentration gradients in the ions that carry the current, you can express the current balance in the channel with Ohm’s law and the balance equation for current density ∇ ⋅ ( – σ∇V ) = 0 where σ denotes conductivity (S/m) and the expression within parentheses represents the current density (A/m2). The electric potentials on the four electrodes are sinusoidal in time with the same maximum value (V0 = 0.1 V) and the same frequency (8 Hz), but they alternate in polarity. The potentials on electrodes 1 and 3 are V0sin(2πft), whereas those on electrodes 2 and 4 are −V0sin(2πft) (see Figure 1). Assume all other boundaries are insulated. The insulation boundary condition – σ∇ V ⋅ n = 0 sets the normal component of the electric field to zero. At the upper half of the inlet (see Figure 1) the solute has a given concentration, c0; at the lower half the concentration is zero. Thus, assume that the concentration changes abruptly from zero to c0 at the middle of the inlet boundary. The mixed solution flows out from the right outlet by convection, and all other boundaries are assumed insulated. Inside the mixer, the following convection-diffusion equation describes the concentration of the dissolved substances in the fluid: ©2012 COMSOL 3 | ELECTROOSMOTIC MICROMIXER
  • 93. Solved with COMSOL Multiphysics 4.3a ∂c ----- + ∇ ⋅ ( – D∇c ) = R – u ⋅ ∇c ∂t (1) Here c is the concentration, D represents the diffusion coefficient, R denotes the reaction rate, and u equals the flow velocity. In this model R = 0 because the concentration is not affected by any reactions. TABLE 1: MODEL INPUT DATA PARAMETER VALUE ρ η 1000 kg/m3 10-3 Pa·s DESCRIPTION Dynamic viscosity of the fluid U0 0.1 mm/s Average velocity through the inlet εr 80.2 Relative electric permittivity of the fluid ζ -0.1 V Zeta potential on the wall-fluid boundary σ 0.11845 S/m Conductivity of the ionic solution D 10-11 m2/s Diffusion coefficient c0 1 mol/m3 Initial concentration Density of the fluid Results and Discussion Figure 2 shows a typical instantaneous streamline pattern. It reveals that electroosmotic recirculation of the fluid vigorously stirs the flow, typically in the form of two rotating vortices near the electrodes. The fundamental processes of effective mixing involve a combination of repeated stretching and folding of fluid elements in combination with diffusion at small scales. As the system applies the AC field (Figure 3), the resulting electroosmotic flow perturbs the laminar pressure-driven flow such that it pushes the combined stream pattern up and down at the beginning of the mixing chamber, causing extensive folding and stretching of material lines. 4 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 94. Solved with COMSOL Multiphysics 4.3a Figure 2: Fluid streamlines in an electroosmotic micromixer at t = 0.0375 s. Figure 3: Electric potential lines for an electroosmotic micromixer. The contour lines show the shape when the device uses maximal potentials (±V0). ©2012 COMSOL 5 | ELECTROOSMOTIC MICROMIXER
  • 95. Solved with COMSOL Multiphysics 4.3a The following plots further exemplify how the mixer operates. Figure 4 shows the concentration at steady state when the electric field is not applied. The flow is laminar and the diffusion coefficient is very small, so the two fluids are well separated also at the outlet. When the alternating electric field is applied, the mixing increases considerably owing to the alternating swirls in the flow. Figure 5 depicts the system at the instant when the electric field and the electroosmotic velocity have their largest magnitudes during the cycle (that is, when |sin ωt| = 1). From the plot you can estimate that the concentration at the output fluctuates with the same frequency as the electric field. Thus, this mixer should be further improved to get a steadier output. Figure 4: Steady-state solution in the absence of an electric field. 6 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 96. Solved with COMSOL Multiphysics 4.3a Figure 5: Time-dependent solution at the time when the alternating electric field has its largest magnitude. This example demonstrates a rather simple and effective use of electrokinetic forces for mixing. The scheme is easy to implement, and you can easily control both the amplitude and the frequency. At low Reynold numbers the inertial forces are small, which makes it possible to calculate stationary streamlines patterns using the parametric solver to control amplitude. Notes About the COMSOL Implementation Cummings and others (Ref. 3) have shown that in order to use the Helmholtz-Smoluchowski equation at the fluid-solid boundaries, the electric field must be at least quasi-static to neglect transient effects. In other words, the time scale of the unsteady electric field must be much larger than that of the transient flow. Y. T. Zhang and others (Ref. 1) estimated that the time scale of the transient effect in the modeled micromixer (with a channel width of 10 microns) is roughly 0.0127 s. In this simulation the frequency of the applied electric potential is 8 Hz, which corresponds to a time scale of the electric field 10 times larger than that of the flow. ©2012 COMSOL 7 | ELECTROOSMOTIC MICROMIXER
  • 97. Solved with COMSOL Multiphysics 4.3a Because you can model the time-dependent electric field as a product of a stationary electric field and a time-dependent phase factor (sinωt), it is possible to reduce the simulation time and memory requirements by dividing the solution into two stages. In the first, calculate the amplitude of the electric potential field and the initial state for the time-dependent flow model using a stationary solver. In the second stage, you deactivate the Electric Currents interface and calculate the transient solution for the Laminar Flow and the Transport of Diluted Species interfaces. You obtain the tangential electric field components used in the electroosmotic velocity boundary condition by multiplying the stationary DC solution by sin(ωt). This approach is permissible because there is only a one-way coupling between the electric field and the fluid fields. References 1. H. Chen, Y.T. Zhang, I. Mezic, C.D. Meinhart, and L. Petzold, “Numerical Simulation of an Electroosmotic Micromixer,” Proc Microfluidics 2003 (ASME IMECE), 2003. 2. Y.T. Zhang, H. Chen, I. Mezic, C.D. Meinhart, L. Petzold, and N.C. MacDonald, “SOI Processing of a Ring Electrokinetic Chaotic Micromixer,” Proc NSTI Nanotechnology Conference and Trade Show (Nanotech 2004), vol. 1, pp. 292–295, 2004. 3. E. Cummings, S. Griffiths, R. Nilson, and P. Paul, “Conditions for Similitude Between the Fluid Velocity and the Electric Field in Electroosmotic Flow,”, Anal. Chem., vol. 72, pp. 2526–2532, 2000. Model Library path: Microfluidics_Module/Micromixers/ electroosmotic_mixer Modeling Instructions MODEL WIZARD 1 Go to the Model Wizard window. 2 Click the 2D button. 3 Click Next. 8 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 98. Solved with COMSOL Multiphysics 4.3a 4 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf). 5 Click Add Selected. 6 In the Add physics tree, select AC/DC>Electric Currents (ec). 7 Click Add Selected. 8 In the Add physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 9 Click Add Selected. 1 Click Next. 0 1 Find the Studies subsection. In the tree, select Preset Studies for Selected 1 Physics>Stationary. 1 Click Finish. 2 GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Description U0 0.1[mm/s] Mean inflow velocity sigma_w 0.11845[S/m] Conductivity of the ionic solution eps_r 80.2 Relative permittivity of the fluid zeta -0.1[V] Zeta potential V0 0.1[V] Maximum value of the AC potential omega 2*pi[rad]*8[Hz] Angular frequency of the AC potential t 0[s] Start time D 1e-11[m^2/s] Diffusion coefficient of the solution c0 1[mol/m^3] Initial concentration You need the constant t (used in the scalar expressions below) when first solving the model using a stationary solver. In the time-dependent simulation, the internal time variable, t, overwrites this constant (the red color is just a warning signaling that t is an internal variable). Now define a smoothed step function that you will later use to impose a step in the concentration in the middle of the channel entrance. ©2012 COMSOL 9 | ELECTROOSMOTIC MICROMIXER
  • 99. Solved with COMSOL Multiphysics 4.3a Step 1 1 In the Model Builder window, right click Global Definitions and choose Functions>Step 1. 2 In the Step settings window, locate the Smoothing section. 3 In the Size of transition zone edit field, type 0.1e-6. GEOMETRY 1 1 In the Model Builder window, under Model 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose µm. Rectangle 1 1 Right-click Model 1>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 80. 4 In the Height edit field, type 10. 5 Locate the Position section. From the Base list, choose Center. 6 Click the Build Selected button. Circle 1 1 In the Model Builder window, right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type 15. 4 Click the Build Selected button. Circle 2 1 Right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type 5. 4 Click the Build Selected button. Compose 1 1 Right-click Geometry 1 and choose Boolean Operations>Compose. 2 Click in the Graphics window, press Ctrl+A to highlight all objects, and then right-click to confirm the selection. 3 In the Compose settings window, locate the Compose section. 10 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 100. Solved with COMSOL Multiphysics 4.3a 4 In the Set formula edit field, type (r1+c1)-c2. 5 Clear the Keep interior boundaries check box. 6 Click the Build Selected button. To add vertices for the electrode endpoints on the outer boundary, first add a square whose boundaries intersect the outer circle at the desired locations. Square 1 1 Right-click Geometry 1 and choose Square. 2 In the Square settings window, locate the Size section. 3 In the Side length edit field, type 22.27. 4 Locate the Position section. From the Base list, choose Center. 5 Click the Build Selected button. Next, remove the parts of the square not contained inside the mixer geometry. Compose 2 1 Right-click Geometry 1 and choose Boolean Operations>Compose. 2 Select both objects (co1 and sq1). 3 In the Compose settings window, locate the Compose section. 4 In the Set formula edit field, type co1*sq1+co1. Here, co1*sq1 is the intersection of co1 and sq1. ©2012 COMSOL 11 | ELECTROOSMOTIC MICROMIXER
  • 101. Solved with COMSOL Multiphysics 4.3a 5 Clear the Keep interior boundaries check box. 6 Click the Build Selected button. To see the effects of these operations, switch to point selection mode. 7 Click the Select Points button on the Graphics toolbar. The model geometry is now essentially complete. However, before proceeding to the Materials branch, add an auxiliary vertex midway along the inlet boundary for use when creating the mesh. Point 1 1 Right-click Geometry 1 and choose Point. 2 In the Point settings window, locate the Point section. 3 In the x edit field, type -40. 4 Click the Build Selected button. 12 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 102. Solved with COMSOL Multiphysics 4.3a Form Union 1 In the Model Builder window, under Model 1>Geometry 1 right-click Form Union and choose Build Selected. MATERIALS Material 1 1 In the Model Builder window, under Model 1 right-click Materials and choose Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Density rho 1e3[kg/m^3] Dynamic viscosity mu 1e-3[Pa*s] Electric conductivity sigma sigma_w Relative permittivity epsilonr eps_r LAMINAR FLOW 1 In the Model Builder window’s toolbar, click the Show button and select Discretization in the menu. 2 In the Model Builder window, click Laminar Flow. ©2012 COMSOL 13 | ELECTROOSMOTIC MICROMIXER
  • 103. Solved with COMSOL Multiphysics 4.3a 3 In the Laminar Flow settings window, click to expand the Discretization section. 4 From the Discretization of fluids list, choose P2 + P1. Using higher-order elements can improve the accuracy of the solution significantly for low Reynolds number flows such as those in this model. Inlet 1 1 Right-click Laminar Flow and choose Inlet. 2 Select Boundaries 1 and 3 only. To do this, you can use Ctrl-click to highlight the two segments of the inlet boundary and then right-click to confirm the selection. Alternatively, click the Paste Selection button in the Boundary Selection section and enter the boundary numbers in the dialog box that appears. A third possibility is to copy the text '1 and 3' from this document, click in the Selection box, and then press Ctrl+V. 3 In the Inlet settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Laminar inflow. 5 Locate the Laminar Inflow section. In the Uav edit field, type U0. 6 Select the Constrain endpoints to zero check box. Outlet 1 1 Right-click Laminar Flow and choose Outlet. 2 Select Boundary 7 only. 3 In the Outlet settings window, locate the Boundary Condition section. 4 From the Boundary condition list, choose Normal stress. Wall 1 1 In the Model Builder window, under Model 1>Laminar Flow click Wall 1. 2 In the Wall settings window, locate the Boundary Condition section. 3 From the Boundary condition list, choose Electroosmotic velocity. 4 In the E table, enter the following settings: ec.Ex*sin(omega*t) x ec.Ey*sin(omega*t) y 5 From the Electroosmotic mobility list, choose Built-in expression. 6 In the ζ edit field, type zeta. 7 In the εr edit field, type eps_r. 14 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 104. Solved with COMSOL Multiphysics 4.3a ELECTRIC CURRENTS Electric Potential 1 1 In the Model Builder window, under Model 1 right-click Electric Currents and choose Electric Potential. 2 Select Boundaries 10 and 21 only. 3 In the Electric Potential settings window, locate the Electric Potential section. 4 In the V0 edit field, type -V0. Electric Potential 2 1 In the Model Builder window, right-click Electric Currents and choose Electric Potential. ©2012 COMSOL 15 | ELECTROOSMOTIC MICROMIXER
  • 105. Solved with COMSOL Multiphysics 4.3a 2 Select Boundaries 11 and 20 only. 3 In the Electric Potential settings window, locate the Electric Potential section. 4 In the V0 edit field, type V0. TR A N S P O R T O F D I L U T E D S P E C I E S Raise the element order to match that of the Laminar Flow interface. 1 In the Model Builder window, under Model 1 click Transport of Diluted Species. 2 In the Transport of Diluted Species settings window, click to expand the Discretization section. 3 From the Concentration list, choose Quadratic. Convection and Diffusion 1 1 In the Model Builder window, expand the Transport of Diluted Species node, then click Convection and Diffusion 1. 2 In the Convection and Diffusion settings window, locate the Diffusion section. 3 In the Dc edit field, type D. 4 Locate the Model Inputs section. From the u list, choose Velocity field (spf/fp1). 16 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 106. Solved with COMSOL Multiphysics 4.3a Concentration 1 1 In the Model Builder window, right-click Transport of Diluted Species and choose Concentration. 2 Select Boundaries 1 and 3 only. 3 In the Concentration settings window, locate the Concentration section. 4 Select the Species c check box. 5 In the c0,c edit field, type c0*step1(y[1/m]). The concentration condition on Boundaries 1 and 3 gives a sharp but smooth concentration gradient in the middle of the channel entrance. Outflow 1 1 Right-click Transport of Diluted Species and choose Outflow. 2 Select Boundary 7 only. MESH 1 In the Model Builder window, under Model 1 right-click Mesh 1 and choose Free Triangular. Size 1 In the Model Builder window, under Model 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extra fine. Size 1 1 In the Model Builder window, under Model 1>Mesh 1 right-click Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 10, 11, 20, and 21 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type 0.2. 8 Select the Maximum element growth rate check box. 9 In the associated edit field, type 1.1. ©2012 COMSOL 17 | ELECTROOSMOTIC MICROMIXER
  • 107. Solved with COMSOL Multiphysics 4.3a Size 2 1 Right-click Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Point. 4 Select Point 2 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type 0.1. 8 Select the Maximum element growth rate check box. 9 In the associated edit field, type 1.1. 10 Click the Build All button. STUDY 1 Set up the study to start by computing the stationary solution for velocity, pressure, concentration, and electric potential. Then, add a transient simulation stage that solves only for the variables of the Laminar Flow and Transport of Diluted Species interfaces. Begin by adding a study step for the transient part. 18 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 108. Solved with COMSOL Multiphysics 4.3a Step 2: Time Dependent 1 In the Model Builder window, right-click Study 1 and choose Study Steps>Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,0.125/60,0.5). 4 Locate the Physics and Variables Selection section. In the table, enter the following settings: Physics Solve for Electric Currents (ec) × 5 In the Model Builder window, click Study 1. 6 In the Study settings window, locate the Study Settings section. 7 Clear the Generate default plots check box. This is convenient if you want to create specialized plots while keeping the number of plot groups down. 8 Click the Compute button. RESULTS The following instructions show how to reproduce the plots in the Results and Discussion section. 2D Plot Group 1 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Time list, choose 0.0375. 4 Right-click Results>2D Plot Group 1 and choose Streamline. 5 In the Streamline settings window, locate the Streamline Positioning section. 6 From the Positioning list, choose Uniform density. 7 In the Separating distance edit field, type 0.01. 8 Click the Plot button. 9 Click the Zoom Extents button on the Graphics toolbar. 1 Click the Zoom In button on the Graphics toolbar. 0 Compare the result with Figure 2. ©2012 COMSOL 19 | ELECTROOSMOTIC MICROMIXER
  • 109. Solved with COMSOL Multiphysics 4.3a 2D Plot Group 2 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Time list, choose 0.0375. 4 Right-click Results>2D Plot Group 2 and choose Contour. 5 In the Contour settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electric Currents>Electric>Electric potential (V). 6 Click the Plot button. Compare the result with Figure 3. 2D Plot Group 3 1 In the Model Builder window, right-click Results and choose 2D Plot Group. 2 Right-click 2D Plot Group 3 and choose Surface. 3 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Transport of Diluted Species>Species c>Concentration (c). 4 In the Model Builder window, right-click 2D Plot Group 3 and choose Streamline. 5 In the Streamline settings window, locate the Streamline Positioning section. 6 From the Positioning list, choose Uniform density. 7 In the Separating distance edit field, type 0.01. 8 In the Model Builder window, click 2D Plot Group 3. 9 In the 2D Plot Group settings window, locate the Data section. 10 From the Time list, choose 0. 11 Click the Plot button. 12 Click the Zoom Extents button on the Graphics toolbar. 13 Click the Zoom In button on the Graphics toolbar. Compare the result with Figure 4. 14 In the 2D Plot Group settings window, locate the Data section. 15 From the Time list, choose 0.46875. 16 Click the Plot button. Compare the result with Figure 5. 20 | ELECTROOSMOTIC MICROMIXER ©2012 COMSOL
  • 110. Solved with COMSOL Multiphysics 4.3a Electrowetting Lens Introduction The contact angle of a two-fluid interface with a solid surface is determined by the balance of the forces at the contact point. The equilibrium contact angle, θ0, is given by Young’s equation: γ s1 + σ 12 cos θ 0 = γ s2 Here γs1 is the surface energy per unit area between fluid 1 and the solid surface, γs2 is the surface energy per unit area between fluid 2 and the solid surface, and σ12 is the surface tension at the interface between the two fluids. In electrowetting the balance of forces at the contact point is modified by the application of a voltage between a conducting fluid and the solid surface. In many applications the solid surface consists of a thin dielectric deposited onto a conducting layer; this is often referred to as “Electrowetting on Dielectric” (EWOD). In this case the capacitance of the dielectric layer dominates over the double layer capacitance at the solid-liquid interface (Ref. 1). The energy stored in the capacitor formed between the conducting liquid and the conducting layer in the solid reduces the effective surface energy of the liquid to which the voltage is applied. For the case when a voltage difference occurs between fluid 1 and the conductor beyond the dielectric Young’s equation is modified as follows: 2 εV γ s1 – --------- + σ cos θ ew = γ s2 2d f 12 Here ε is the permittivity of the dielectric, V is the potential difference applied, and df is the dielectric thickness. This equation can be re-written as 2 εV cos θ ew = cos θ 0 + ----------------2σ 12 d f (1) Electrowetting can therefore be used to modify the contact angle dynamically by changing the voltage applied to the conducting liquid. In this example, the meniscus between two immiscible liquids is used as an optical lens. A change in curvature of the meniscus caused by the electrowetting effect is used to change the focal length of the lens over a large range. This model is based on the work ©2012 COMSOL 1 | ELECTROWETTING LENS
  • 111. Solved with COMSOL Multiphysics 4.3a of the Philips FluidFocus team (Ref. 2). The principle of the device is illustrated in Figure 1 and the miniature, variable focus camera developed around the technology is shown in Figure 2. Figure 1: (A) Schematic cross section of the Philips FluidFocus lens. (B) When a voltage is applied, the electrowetting effect alters the contact angle and hence the focal distance of the lens. (C) to (E) Shapes of a 6-mm diameter lens taken at 0 V, 100 V and 120 V respectively. Diagrams and photos: Philips. Figure 2: The miniature variable focus lens and the camera that was developed to contain it. Photo: Philips. 2 | ELECTROWETTING LENS ©2012 COMSOL
  • 112. Solved with COMSOL Multiphysics 4.3a Model Definition The model consists of a sealed chamber with radius 1.5 mm filled with two immiscible liquids. Because the geometry is cylindrical, the axisymmetric geometry shown in Figure 3 can be used. The lower fluid in Figure 3 is a conducting solution of lithium chloride, with a density of 1120 kg/m3 and a viscosity of 1.5 mPa·s. The upper fluid is insulating, with a matching density and a viscosity that is altered by varying its composition to optimize the camera performance. The surface tension at the interface between the two fluids is 50 N/m. The walls of the cylinder are coated with 3 μm of paylene N (relative dielectric constant, 2.65). Because this layer is thin it is not modeled explicitly in COMSOL and Equation 1 is used for the contact angle. The contact angle of the fluid in the absence of applied voltage is 140°. In this model the response of the fluid surface is modeled as a function of time after the voltage is switched from 100 V to 120 V. It is desired to optimize the viscosity of the insulating fluid to achieve a fast response time for the switching of the lens, so the time dependent switching of the system is studied. Viscosities of 10 mPa·s, 30 mPa·s, and 50 mPa·s are investigated. Figure 3: Axisymmetric model geometry. ©2012 COMSOL 3 | ELECTROWETTING LENS
  • 113. Solved with COMSOL Multiphysics 4.3a Results and Discussion When the voltage is switched the contact angle of the fluid changes abruptly but the system takes some time to respond to the change in the force at the contact point. The resonant modes of the interface are excited by this disturbance and, depending on the system damping, the oscillations of the interface take some time to decay. The higher order modes are damped out more rapidly than the fundamental mode, but are apparent in the plots shown in Figure 4 and Figure 5, which show the fluid velocity and pressure respectively 2 ms after the voltage is switched. Figure 4: Fluid velocity magnitude (color) and direction (arrows) for a lens 2ms after the voltage is switched from 100 V to 120 V. The viscosity of the insulating fluid is 10 mPa·s. 4 | ELECTROWETTING LENS ©2012 COMSOL
  • 114. Solved with COMSOL Multiphysics 4.3a Figure 5: Pressure in the fluid (color) and velocity of the boundary (arrows) for a lens 2 ms after the voltage is switched from 100 V to 120 V. The viscosity of the insulating fluid is 10 mPa·s. Clearly for optimum performance of the lens, the oscillation of the meniscus should be damped out as rapidly as possible; the system should therefore be critically damped. Because the viscosity of the insulating fluid can be altered by changing its composition, it is possible to adjust the damping and hence to produce a lens with the fastest possible response time. Figure 6 shows the response of the system for three different values of the viscosity of the insulating fluid. From this plot it is clear that a viscosity of 50 mPa·s will produce a system that is close to being critically damped. ©2012 COMSOL 5 | ELECTROWETTING LENS
  • 115. Solved with COMSOL Multiphysics 4.3a Figure 6: Location of the center of the meniscus as a function of time for different values of the viscosity of the insulating fluid. References 1. F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys. Condens. Matter, vol. 17, pp. R705–R774, 2005. 2. S. Kuiper and B.W. Hendriks, “Variable focus lens for miniature cameras,” Appl. Phys. Lett., vol. 85(7), pp. 1128–1130, 2004. See also: http://www.research.philips.com/technologies/fluidfocus.html Model Library path: Microfluidics_Module/Two-Phase_Flow/ electrowetting_lens 6 | ELECTROWETTING LENS ©2012 COMSOL
  • 116. Solved with COMSOL Multiphysics 4.3a Modeling Instructions MODEL WIZARD 1 Go to the Model Wizard window. 2 Click the 2D axisymmetric button. 3 Click Next. 4 In the Add physics tree, select Fluid Flow>Multiphase Flow>Two-Phase Flow, Moving Mesh>Laminar Two-Phase Flow, Moving Mesh (tpfmm). 5 Click Next. 6 Find the Studies subsection. In the tree, select Preset Studies>Time Dependent. 7 Click Finish. GEOMETRY 1 Define the model geometry. 1 In the Model Builder window, click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose mm. Rectangle 1 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 1.5. Bézier Polygon 1 1 Right-click Geometry 1 and choose Bézier Polygon. 2 In the Bézier Polygon settings window, locate the Polygon Segments section. 3 Find the Added segments subsection. Click the Add Linear button. 4 Find the Control points subsection. In row 1, set z to 0.55. 5 In row 2, set r to 1.5. 6 In row 2, set z to 0.55. 7 Click the Build All button. 8 Click the Zoom Extents button on the Graphics toolbar. Define parameters for the material properties and for the constants in Equation 1. ©2012 COMSOL 7 | ELECTROWETTING LENS
  • 117. Solved with COMSOL Multiphysics 4.3a GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Description theta0 140[deg] Zero voltage contact angle gamma 0.05[N/m] Surface tension muoil 8e-3[Pa*s] Insulating fluid viscosity epsr 2.65 Relative dielectric constant d_f 3[um] Dielectric thickness Vapp 120[V] Applied voltage Define the contact angle according to Equation 1. DEFINITIONS Variables 1 1 In the Model Builder window, under Model 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Description theta acos(cos(theta0)+Vapp^2*epsr* epsilon0_const/(2*gamma*d_f)) Contact angle MATERIALS Set up material properties. Material 1 1 In the Model Builder window, under Model 1 right-click Materials and choose Material. 2 In the Material settings window, locate the Geometric Entity Selection section. 3 Click Clear Selection. 4 Select Domain 2 only. 5 In the Material settings window, locate the Material Contents section. 8 | ELECTROWETTING LENS ©2012 COMSOL
  • 118. Solved with COMSOL Multiphysics 4.3a 6 In the table, enter the following settings: Property Name Value Density rho 1000 Dynamic viscosity mu muoil Material 2 1 In the Model Builder window, right-click Materials and choose Material. 2 Select Domain 1 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Density rho 1000 Dynamic viscosity mu 1.5e-3 L A M I N A R TW O - P H A S E F L O W, M O V I N G M E S H Define the physics settings for the problem. Boundary conditions must be applied for both the moving mesh and the fluid flow. Select the Fluid-Fluid Interface boundary condition for the two-phase boundary. Fluid-Fluid Interface 1 1 In the Model Builder window, under Model 1 right-click Laminar Two-Phase Flow, Moving Mesh and choose Fluid-Fluid Interface. 2 Select Boundary 4 only. 3 In the Fluid-Fluid Interface settings window, locate the Surface Tension section. 4 In the σ edit field, type gamma. Define the contact angle settings at the wall fluid interface. Wall-Fluid Interface 1 1 Right-click Model 1>Laminar Two-Phase Flow, Moving Mesh>Fluid-Fluid Interface 1 and choose Wall-Fluid Interface. 2 In the Wall-Fluid Interface settings window, locate the Wall-Fluid Interface section. 3 In the θw edit field, type theta. The Navier Slip boundary condition must be used in the moving mesh interface for a boundary on which a contact point moves. Use this condition for the wall on which the electrowetting effect occurs. ©2012 COMSOL 9 | ELECTROWETTING LENS
  • 119. Solved with COMSOL Multiphysics 4.3a Navier Slip 1 1 In the Model Builder window, right-click Laminar Two-Phase Flow, Moving Mesh and choose Navier Slip. 2 Select Boundaries 6 and 7 only. Set up the mesh deformation boundary conditions. The mesh is fixed on the upper and lower boundaries. Prescribed Mesh Displacement 2 1 Right-click Laminar Two-Phase Flow, Moving Mesh and choose the boundary condition Moving Mesh>Prescribed Mesh Displacement. 2 Select Boundaries 2 and 5 only. The mesh is allowed to move vertically on the wall with the three phase contact. Prescribed Mesh Displacement 1 1 In the Model Builder window, under Model 1>Laminar Two-Phase Flow, Moving Mesh click Prescribed Mesh Displacement 1. 2 In the Prescribed Mesh Displacement settings window, locate the Prescribed Mesh Displacement section. 3 Clear the Prescribed z displacement check box. Apply a pressure point constraint so that the pressure is constrained. Pressure Point Constraint 1 1 In the Model Builder window, right-click Laminar Two-Phase Flow, Moving Mesh and choose Points>Pressure Point Constraint. 2 Select Point 6 only. Quadrilateral elements are used for the mesh as they are typically stiffer and hence less susceptible to inverted elements than triangular elements. The mesh is also scaled at the contact point to improve the accuracy of the simulation. MESH 1 Scale 1 1 In the Model Builder window, under Model 1 right-click Mesh 1 and choose Scale. 2 In the Scale settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Point. 4 Select Point 5 only. 5 In the Scale settings window, locate the Scale section. 10 | ELECTROWETTING LENS ©2012 COMSOL
  • 120. Solved with COMSOL Multiphysics 4.3a 6 In the Element size scale edit field, type 0.2. Size 1 In the Model Builder window, under Model 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 From the Calibrate for list, choose Fluid dynamics. Free Quad 1 1 In the Model Builder window, right-click Mesh 1 and choose Free Quad. 2 In the Settings window, click Build All. Define an integration operator that can be used to compute the height of the center of the meniscus above the base of the lens. DEFINITIONS Integration 1 1 In the Model Builder window, under Model 1 right-click Definitions and choose Model Couplings>Integration. ©2012 COMSOL 11 | ELECTROWETTING LENS
  • 121. Solved with COMSOL Multiphysics 4.3a 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundary 1 only. Add a parametric sweep on the viscosity of the insulating fluid. STUDY 1 In the Model Builder window, expand the Study 1 node. Parametric Sweep 1 Right-click Study 1 and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names Parameter value list muoil 10e-3 30e-3 50e-3 Solve the problem over an appropriate time interval. Step 1: Time Dependent 1 In the Model Builder window, under Study 1 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type range(0,1e-3,5e-2). 4 In the Model Builder window, right-click Study 1 and choose Compute. RESULTS Reproduce the plot shown in Figure 4. Note that to display the deformed geometry the plot should be viewed in the spatial reference frame rather than in the default material frame. 2D Plot Group 1 1 In the 2D Plot Group settings window, locate the Plot Settings section. 2 From the Frame list, choose Spatial (r, phi, z). 3 Locate the Data section. From the Time list, choose 0.002. 4 From the Parameter value (muoil) list, choose 0.01. 5 Locate the Plot Settings section. Select the x-axis label check box. 6 Select the y-axis label check box. 12 | ELECTROWETTING LENS ©2012 COMSOL
  • 122. Solved with COMSOL Multiphysics 4.3a 7 In the x-axis label edit field, type r (mm). 8 In the y-axis label edit field, type z (mm). 9 Right-click Results>2D Plot Group 1 and choose Arrow Surface. 1 In the Arrow Surface settings window, locate the Coloring and Style section. 0 1 From the Color list, choose White. 1 1 Click the Plot button. 2 Reproduce the plot shown in Figure 5. 2D Plot Group 2 1 In the Model Builder window, under Results click 2D Plot Group 2. 2 In the 2D Plot Group settings window, locate the Plot Settings section. 3 From the Frame list, choose Spatial (r, phi, z). 4 Locate the Data section. From the Time list, choose 0.002. 5 From the Parameter value (muoil) list, choose 0.01. 6 Locate the Plot Settings section. Select the x-axis label check box. 7 Select the y-axis label check box. 8 In the x-axis label edit field, type r (mm). 9 In the y-axis label edit field, type z (mm). 1 In the Model Builder window, expand the 2D Plot Group 2 node, then click Surface 1. 0 1 In the Surface settings window, click to expand the Range section. 1 1 Select the Manual color range check box. 2 1 Set the Minimum value to 1. 3 1 Set the Maximum value to 5. 4 1 In the Model Builder window, right-click 2D Plot Group 2 and choose Arrow Line. 5 1 In the Arrow Line settings window, locate the Coloring and Style section. 6 1 From the Color list, choose White. 7 1 Click the Plot button. 8 3D Plot Group 4 Reproduce the plot shown in Figure 6. The integration operator intop1(1) is used to integrate unity along the center line of the lens, to compute the height of the meniscus as a function of time. ©2012 COMSOL 13 | ELECTROWETTING LENS
  • 123. Solved with COMSOL Multiphysics 4.3a 1D Plot Group 5 1 In the Model Builder window, right-click Results and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 Locate the Plot Settings section. Select the y-axis label check box. 5 In the associated edit field, type Displacement (mm). 6 Right-click Results>1D Plot Group 5 and choose Global. 7 In the Global settings window, locate the y-Axis Data section. 8 In the table, enter the following settings: Expression Unit Description intop1(1) 9 In the Global settings window, locate the x-Axis Data section. 10 From the Axis source data list, choose Inner solutions. 11 Click the Plot button. 14 | ELECTROWETTING LENS ©2012 COMSOL
  • 124. Bring your ideas to life with COMSOL Multiphysics (주)알트소프트 TEL: 02-547-2344 FAX: 02-547-2343 E-mail: comsol@altsoft.co.kr Homepage: www.altsoft.co.kr