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Thermal Actuator — Parameterized
Introduction
This example model consists of a two-hot-arm thermal actuator made of polysilicon. The actuator is activated through thermal expansion. The temperature increase required to deform the two hot arms, and thus displace the actuator, is obtained through Joule heating (resistive heating). The greater expansion of the hot arms, compared to the cold arm, causes a bending of the actuator. In an actual device, a wide range of electric resistance values is possible through doping of the polysilicon material.
The actuator’s operation thus involves three coupled physics phenomena: electric current conduction, heat conduction with heat generation, and structural stresses and strains due to thermal expansion.
Model Definition
Figure 1 shows the actuator’s parts and dimensions as well as its position on top of a substrate surface.
Figure 1: The thermal microactuator.
Boundary Conditions and Constraints
An electric potential is applied between the bases of the hot arms’ anchors. The cold arm anchor and all other surfaces are electrically insulated.
Figure 2: Electrical boundary conditions.
The temperature of the base of the three anchors and the three dimples is fixed to that of the substrate’s constant temperature. Because the structure is sandwiched, all other boundaries interact thermally with the surroundings by conduction through thin layers of air. This can be implemented as thermal contact conditions or as a convective heat-flux condition, where the heat-flux coefficient represents one over the thermal resistance. In this model we choose to use a heat flux condition. The heat-transfer coefficient is given by the thermal conductivity of air divided by the distance to the surrounding surfaces for the system. This exercise uses different heat-transfer coefficients for the actuator’s upper and other surfaces.
Figure 3: Heat-transfer boundary conditions.
All three arms are mechanically fixed at the base of the three anchors. The dimples can move freely in the plane of the substrate (the xy-plane in the figure) but do not move in the direction perpendicular to the substrate (the z direction).
Figure 4: Structural boundary conditions and constraints.
Results
Figure 5 shows the surface temperature distribution for the actuator. Figure 6 illustrates the displacement field through color and deformation plot.
Figure 5: Temperature plot.
Figure 6: Displacement in the thermal actuator (here exaggerated by a factor of about 6.6).
Reference
1. D.M. Burns and V.M. Bright, “Design and performance of a double hot arm polysilicon thermal actuator,” Proc. SPIE 3224, Micromachined Devices and Components III, 1997; doi: 10.1117/12.284528.
Application Library path: MEMS_Module/Actuators/thermal_actuator_tem_parameterized
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics > Thermal–Structure Interaction > Joule Heating and Thermal Expansion.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Thermal Actuator
1
In the Model Builder window, right-click Component 1 (comp1) and choose Rename.
2
In the Rename Component dialog, type Thermal Actuator in the New label text field.
3
Click OK.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
d
3[um]
3E-6 m
Height of the hot arm
dw
15[um]
1.5E-5 m
Height of the cold arm
gap
3[um]
3E-6 m
Gap between arms
wb
10[um]
1E-5 m
Width of the base
wv
25[um]
2.5E-5 m
Difference in length between hot arms
L
240[um]
2.4E-4 m
Actuator length
L1
L-wb
2.3E-4 m
Length of the longest hot arm
L2
L-wb-wv
2.05E-4 m
Length of the shortest hot arm
L3
L-2*wb-wv-L/48-L/6
1.5E-4 m
Length of the cold arm, thick part
L4
L/6
4E-5 m
Length of the cold arm, thin part
htc_s
0.04[W/(m*K)]/2[um]
20000 W/(m²·K)
Heat transfer coefficient
htc_us
0.04[W/(m*K)]/100[um]
400 W/(m²·K)
Heat transfer coefficient, upper surface
DV
5[V]
5 V
Applied voltage
Geometry 1
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, click  Build Selected.
3
Click  Go to Plane Geometry.
Work Plane 1 (wp1) > Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L3.
4
In the Height text field, type dw.
5
Locate the Position section. In the xw text field, type L-L3.
6
Click  Build Selected.
Work Plane 1 (wp1) > Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L4.
4
In the Height text field, type d.
5
Locate the Position section. In the xw text field, type L-L3-L4.
6
In the yw text field, type dw-d.
7
Click  Build Selected.
8
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Rectangle 3 (r3)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb.
4
In the Height text field, type dw.
5
Locate the Position section. In the xw text field, type L-L3-L4-wb.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Rectangle 4 (r4)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L2.
4
In the Height text field, type d.
5
Locate the Position section. In the xw text field, type L-L2.
6
In the yw text field, type dw+gap.
7
Click  Build Selected.
Work Plane 1 (wp1) > Rectangle 5 (r5)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb.
4
In the Height text field, type dw+gap+d.
5
Locate the Position section. In the xw text field, type L-L2-wb.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Rectangle 6 (r6)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L1.
4
In the Height text field, type d.
5
Locate the Position section. In the xw text field, type L-L1.
6
In the yw text field, type dw+d+2*gap.
7
Click  Build Selected.
8
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Rectangle 7 (r7)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb.
4
In the Height text field, type dw+gap+d.
5
Locate the Position section. In the yw text field, type dw+d+2*gap.
6
Click  Build Selected.
7
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1) > Rectangle 8 (r8)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type d.
4
In the Height text field, type gap.
5
Locate the Position section. In the xw text field, type L-d.
6
In the yw text field, type dw+gap+d.
7
Click  Build Selected.
Work Plane 1 (wp1) > Rectangle 9 (r9)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type d.
4
In the Height text field, type gap.
5
Locate the Position section. In the xw text field, type L-d.
6
In the yw text field, type dw.
7
Click  Build Selected.
Work Plane 1 (wp1) > Union 1 (uni1)
1
In the Work Plane toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
5
Click  Build Selected.
Work Plane 1 (wp1) > Fillet 1 (fil1)
1
In the Work Plane toolbar, click  Fillet.
2
In the Settings window for Fillet, locate the Radius section.
3
In the Radius text field, type d/3.
4
On the object uni1, select Points 1, 2, 4–9, 11–14, 16, 17, 19–23, and 28 only.
It might be easier to select the points by using the Selection List window. To open this window, in the Home toolbar click Windows and choose Selection List. (If you are running the cross-platform desktop, you find Windows in the main menu.)
5
Click  Build Selected.
Extrude 1 (ext1)
1
In the Model Builder window, under Thermal Actuator (comp1) > Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
2e-6
4
Click  Build Selected.
5
Click the  Go to Default View button in the Graphics toolbar.
Work Plane 2 (wp2)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, click  Build Selected.
3
Click  Go to Plane Geometry.
Work Plane 2 (wp2) > Plane Geometry
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 2 (wp2) > Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb-2*d.
4
In the Height text field, type 2.5*(wb-2*d).
5
Locate the Position section. In the xw text field, type d.
6
In the yw text field, type (dw+d+2*gap)+(dw+gap+d)-2.5*(wb-2*d)-d.
7
Click  Build Selected.
Work Plane 2 (wp2) > Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb-2*d.
4
In the Height text field, type 2.5*(wb-2*d).
5
Locate the Position section. In the xw text field, type L-L2-wb+d.
6
In the yw text field, type d.
7
Click  Build Selected.
Work Plane 2 (wp2) > Rectangle 3 (r3)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type wb-2*d.
4
In the Height text field, type 2.5*(wb-2*d).
5
Locate the Position section. In the xw text field, type L-L3-L4-wb+d.
6
In the yw text field, type d.
7
Click  Build Selected.
Work Plane 2 (wp2) > Fillet 1 (fil1)
1
In the Work Plane toolbar, click  Fillet.
2
In the Settings window for Fillet, locate the Radius section.
3
In the Radius text field, type d/3.
4
Select all four vertices for all three rectangles as follows:
5
Click the  Select Box button in the Graphics toolbar.
6
In the Graphics window, draw a box encompassing the three rectangles you just created, then right-click to confirm the selection.
7
Click  Build Selected.
Work Plane 2 (wp2) > Circle 1 (c1)
1
In the Work Plane toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type d/2.
4
Locate the Position section. In the xw text field, type L-L3/4.
5
In the yw text field, type dw/2.
6
Click  Build Selected.
Work Plane 2 (wp2) > Circle 2 (c2)
1
In the Work Plane toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type d/2.
4
Locate the Position section. In the xw text field, type L-L3/2.
5
In the yw text field, type dw/2.
6
Click  Build Selected.
Work Plane 2 (wp2) > Circle 3 (c3)
1
In the Work Plane toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type d/2.
4
Locate the Position section. In the xw text field, type L-3*L3/4.
5
In the yw text field, type dw/2.
6
Click  Build Selected.
Extrude 2 (ext2)
1
In the Model Builder window, under Thermal Actuator (comp1) > Geometry 1 right-click Work Plane 2 (wp2) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
2e-6
4
Select the Reverse direction checkbox.
5
Click  Build Selected.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click the  Zoom Extents button in the Graphics toolbar.
3
Click in the Graphics window and then press Ctrl+A to select both objects.
4
In the Settings window for Union, click  Build All Objects.
Definitions
substrate contact
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 10, 30, 50, 70, 76, and 82 only.
5
Click the  Go to XY View button in the Graphics toolbar three times to view the geometry from below.
6
In the Model Builder window, click Explicit 1.
7
In the Label text field, type substrate contact.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select MEMS > Semiconductors > Si - Polycrystalline silicon.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Si - Polycrystalline silicon (mat1)
By default, the first material you add applies on all domains so you can keep the Geometric Entity Selection settings.
1
In the Settings window for Material, locate the Material Contents section.
2
In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Electric conductivity
sigma_iso ; sigmaii = sigma_iso, sigmaij = 0
5e4
S/m
Basic
Solid Mechanics (solid)
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundaries 10, 30, and 50 only.
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundaries 70, 76, and 82 only.
Heat Transfer in Solids (ht)
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
This boundary condition applies to all boundaries except the top-surface boundary and those in contact with the substrate. A Temperature condition on the substrate contact boundaries will override this Heat Flux condition so you do not explicitly need to exclude those boundaries. In contrast, because the Heat Flux boundary condition is additive, you must explicitly exclude the top-surface boundary from the selection. Implement this selection as follows:
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
In the Graphics window, click on the top surface and then right-click to remove it from the selection.
A convective heat flux is used to model the heat flux through a thin air layer. The heat transfer coefficient, htc_s is defined as the ratio of the air thermal conductivity to the gap thickness.
5
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
6
In the h text field, type htc_s.
Heat Flux 2
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
Select Boundary 4 only.
A convective heat flux is used to model the heat flux through a thin air layer. The heat transfer coefficient, htc_us is defined as the ratio of the air thermal conductivity to the gap thickness.
3
In the Settings window for Heat Flux, locate the Heat Flux section.
4
From the Flux type list, choose Convective heat flux.
5
In the h text field, type htc_us.
Temperature 1
1
In the Physics toolbar, click  Boundaries and choose Temperature.
2
In the Settings window for Temperature, locate the Boundary Selection section.
3
From the Selection list, choose substrate contact.
Electric Currents (ec)
Ground 1
1
In the Model Builder window, expand the Thermal Actuator (comp1) > Electric Currents (ec) node.
2
Right-click Electric Currents (ec) and choose Ground.
3
Select Boundary 10 only.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundary 30 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type DV.
Mesh 1
1
In the Model Builder window, under Thermal Actuator (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Extra fine.
4
Click  Build All.
Study 1
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, locate the Study Settings section.
3
Select the Include geometric nonlinearity checkbox.
The default solver sequence is not optimal for this problem. The only deviation from linearity comes from the geometric nonlinearity. Both the Electric Currents and Temperature groups in the segregated solver contain damping intended to make solution of nonlinear problems more stable, but slower. Removing that damping gives a significant speed up. Also, if iterations are performed for the Solid Mechanics group where the actual nonlinearity is present, there will be fewer iterations over the whole set of segregated groups.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 > Segregated 1 node, then click Electric Currents.
4
In the Settings window for Segregated Step, click to expand the Method and Termination section.
5
In the Damping factor text field, type 1.
6
In the Model Builder window, under Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 > Segregated 1 click Temperature.
7
In the Settings window for Segregated Step, locate the Method and Termination section.
8
In the Damping factor text field, type 1.
9
In the Model Builder window, under Study 1 > Solver Configurations > Solution 1 (sol1) > Stationary Solver 1 > Segregated 1 click Solid Mechanics.
10
In the Settings window for Segregated Step, locate the Method and Termination section.
11
From the Termination technique list, choose Tolerance.
12
Locate the General section. From the Linear solver list, choose Suggested Iterative Solver solid (te1).
13
In the Study toolbar, click  Compute.
Results
Stress (solid)
The first default plot show the von Mises stress.
Volume 1
1
In the Model Builder window, expand the Stress (solid) node, then click Volume 1.
2
In the Settings window for Volume, locate the Expression section.
3
From the Unit list, choose MPa.
4
In the Stress (solid) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Temperature (ht)
1
Click the  Go to Default View button in the Graphics toolbar.
The second default plot shows the temperature field.
Create a new plot for displacement.
Displacement
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Displacement in the Label text field.
Surface 1
1
Right-click Displacement and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
From the Unit list, choose µm.
4
Locate the Coloring and Style section. From the Color table list, choose SpectrumLight.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Displacement toolbar, click  Plot.
Note that the deformation is exaggerated by the Scale factor in order to be clearly visible.