Modeling Optical Absorption for Heat-Induced Actuation of 

Single-Use Microgrippers

by Christopher J. Morris

ARL-TR-6063 July 2012

Approved for public release; distribution unlimited.
NOTICES

Disclaimers

The findings in this report are not to be construed as an official Department of the Army position 

unless so designated by other authorized documents.

Citation of manufacturer’s or trade names does not constitute an official endorsement or 

approval of the use thereof.

Destroy this report when it is no longer needed. Do not return it to the originator.
Army Research Laboratory 

Adelphi, MD 20783-1197 

ARL-TR-6063 July 2012 

Modeling Optical Absorption for Heat-Induced Actuation of 

Single-Use Microgrippers 

Christopher J. Morris 

Sensors and Electron Devices Directorate, ARL 

Approved for public release; distribution unlimited. 
ii

REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid 

OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

July 2012

2. REPORT TYPE

Final

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Modeling Optical Absorption for Heat-Induced Actuation of Single-Use 

Microgrippers

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Christopher J. Morris

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

U.S. Army Research Laboratory

ATTN: RDRL-SER-L

2800 Powder Mill Road

Adelphi MD 20783-1197

8. PERFORMING ORGANIZATION

 REPORT NUMBER

ARL-TR-6063

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT

Approved for public release; distribution unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Optical power represents a viable way to wirelessly actuate microsystems and minimize the need for conventional onboard 

power storage. One approach is to use mechanical stored energy in residually stressed thin films and trigger the release of this 

energy by mechanically softening a polymer through thermal or chemical stimuli. Building on past reports of circular 

arrangements of actuators based on this approach to form free-standing microgrippers, this report presents analytical and 

numerical thermal models that help interpret experimental measurements of actuation behavior as a function of applied optical 

power. The models showed that steady-state microgripper surface temperature rise ΔT was directly proportional to irradiance 

and microgripper size, which for the example of a 1.4-mm-diameter microgripper leads to a lower limit of 343 mW/cm2to 

reach ΔT = 20 °C, at which actuation can be expected. The transient relationship between the time to the reach actuation 

temperature and irradiance was found to range between a 

 

and a power law, as the distance to a neighboring 

substrate ranged from zero to infinity. These results yielded reasonable predictions of microgripper closing temperatures in 

different experimental conditions, and successfully explained different the power law regimes observed.

15. SUBJECT TERMS

MEMS, microgrippers, origami, thermal conduction, ANSYS, finite element analysis

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION

 OF 

 ABSTRACT

UU

18. NUMBER

 OF 

 PAGES

32

19a. NAME OF RESPONSIBLE PERSON

Christopher J. Morris

a. REPORT

Unclassified

b. ABSTRACT

Unclassified

c. THIS PAGE

Unclassified

19b. TELEPHONE NUMBER (Include area code)

(301) 394-0950

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
iii

Contents 

List of Figures iv

List of Tables v

1. Introduction and Background 1

2. Theory 4

3. Model Methods 6

4. Results and Discussion 9

5. Conclusion 19

6. References 20

Distribution List 23
iv

List of Figures 

Figure 1. Schematic showing principle of microgripper operation, and past work on radio 

frequency identification (RFID) integration (1). .......................................................................1

Figure 2. Experimental data from reference 5 showing an inverse square power law between 

actuation time and irradiance. Insets show three high-speed video frames capturing 

microgripper closure. .................................................................................................................3

Figure 3. Actuation time vs. laser intensity from reference 25, showing an inverse 

relationship between actuation time and laser irradiance or intensity. ......................................4

Figure 4. The 2-D Cartesian coordinate FEA model domain, grid, and boundary conditions 

for a 1-D Cartesian problem. .....................................................................................................7

Figure 5. The 2-D axisymmetric FEA model domain, grid, and boundary conditions for an 

axisymmetric microgripper arm approximating a 3-D spherical coordinate solution at 

locations far from the microgripper arm. ...................................................................................8

Figure 6. Calculated time vs. irradiance for the 1-D FEA model. ...............................................10

Figure 7. Calculated time vs. polymer thickness for solutions obtained with I(z) applied to 

five slices and with the application of equation 19 to the entire polymer region. The 

absorption coefficient was 1, and the applied irradiance was 160 mW/cm2. .......................10

Figure 8. Percent difference between the application of equation 19 and I(z) applied to five 

slices as a function of the product of polymer thickness and absorption coefficient . The 

applied irradiance was 160 mW/cm2. ......................................................................................11

Figure 9. Two dimensional finite element analysis showing temperature distribution at 

different times (shown in inset of each image). .......................................................................12

Figure 10. FEA calculated microgripper surface temperature change vs. time for several 

cases. Base parameters were those described in the model methods section. The other 

cases shown were for 50% thicker Au frames, 50% thicker polymer, and 50% higher air 

thermal conductivity. ..............................................................................................................12

Figure 11. FEA-calculated microgripper surface temperature change vs. time for different 

microgripper radii. Base parameters were those described in the model methods section. 

Solid lines show exponential fits according to equation 20 with values for τ shown in the 

legend. ......................................................................................................................................14

Figure 12. FEA-calculated steady state surface temperature rise vs. product of microgripper 

radius and irradiance for several cases. ...................................................................................14

Figure 13. Two-dimensional finite element analysis showing temperature distribution at 

0.01 s and at 0.1 s for a microgripper and substrate separated by a 50 μm gap. .....................15

Figure 14. FEA-calculated microgripper surface temperature change vs. time for a 

microgripper arm situated at different gap distances from a substrate, compared to the 

case with no substrate. .............................................................................................................16
v

Figure 15. FEA calculated times to reach a temperature change of 20 °C vs. irradiance, for 

different substrate gaps. Solid lines show curve fits according to equations shown on 

plot, with power law exponents ranging from –1 for a large gap to –2 for a small gap. .........17

List of Tables 

Table 1. Material properties used in FEA model. ...........................................................................9
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction and Background

The development of passive microactuators powered by stored energy in residually stressed thin 

films has lead to a new type of single use, battery-free actuator. Each actuator contains an 

initially rigid polymer, which softens upon heating above 43 °C, allowing the pre-stressed 

structures to bend. The circular arrangement of series segments of these actuators results in a 

microgripper, as shown in figure 1. These microgrippers have potential use as microscale 

surgeons (2, 3) or as a method of attachment of microtransponders to fibers, textiles, and insects 

(1). 

Figure 1. Schematic showing principle of microgripper operation, and past 

work on radio frequency identification (RFID) integration (1).

We have previously used heat and chemical inputs as stimuli for actuation (4) and have more 

recently developed the ability to trigger these actuators with focused light from handheld optical 

lasers (5). In contrast to electrical (6, 7), pneumatic (8, 9), or fluidic actuation (10), optical 

actuation circumvents the need for wires, pneumatic lines, or other tethers to the system, as well 

as the need for integrating a large, bulky power source into the system. Other examples of 

optical actuation include polymer/carbon nanotube (CNT) composites, which exploit the 

absorption characteristics of CNTs to cause a conformal change and significant strain in the 

CNTs (11), the polymer matrix (12), or a CNT/polymer biomorph beam as a result of a thermal 

expansion coefficient mismatch (13). Polarized light can also interact with liquid crystal 

elastomeric films, which exhibit localized cross-linking or other phase changes leading to 

mechanical shrinkage and which can be reversed by irradiating with light of a different 

wavelength (14–16). Non-polarized light can also cause localized shrinkage of shape memory 

polymers such as prestressed polystyrene through patterning of an optically absorbing material 
2

(17). Selective laser ablation of indium tin oxide (ITO) films leads to interesting folded 

structures controlled by laser positioning and fluence (18), while the same high powered laser 

system can be used to create localized heat-induced strain effects in other materials (19). Each 

of these examples of optical actuation may be characterized as requiring either high irradiance 

levels (>1 W/cm2); long actuation times (10’s to 100’s of seconds); bulk films or materials that 

are not readily amenable to lithographic patterning and other microfabrication techniques; or a 

combination of these characteristics. By integrating optical actuation materials and methods into 

a microfabricated device or system, actuators could be applied more generally to microsystems,

which are arbitrarily small in size. 

Alternative microscale wireless energy transfer approaches include photovoltaic cells for opticalto-electrical conversion. Typical conversion efficiencies are around 10% (20, 21), up to 22% for 

specific wavelengths (20), and surpass 40% for cooled, multi-junction, commercial cells (22). In 

order to generate the high voltage levels required by typical microelectromechanical system 

(MEMS) actuators, arrays of series-connected photovoltaic cells lead to larger power modules,

which are difficult to integrate with microscale devices (23, 24). The use of antennas for 

harvesting microwave energy also requires centimeter-scale or larger areas for reasonably 

efficient conversion (24). The lower efficiency of these methods inevitably leads to larger 

optically active areas and/or higher necessary irradiance levels, but as previously stated, lower 

irradiance and smaller scale integration could lead to optical actuation being applied more 

generally to arbitrarily small microsystems. 

The optical actuation of residually stressed, thin-film bilayer actuators in fully released, 

microfabricated microgrippers can meet all of these desired requirements. We have 

demonstrated actuation with 532-nm laser irradiance levels as low as 0.6 W/cm2(25), with 15% 

conversion efficiency due to high gold (Au) reflectivity (26), suggesting that much lower 

irradiance levels are possible with higher material absorption properties. We have also 

demonstrated typical actuation times of less than 1 s (5, 25). In reference 5, certain experimental 

conditions led to a dependence of actuation time on irradiance according to a 

 

power law,

as shown in figure 2. In that report, we postulated that actuation time or time to close was the 

time 

for a microgripper to reach a certain temperature rise , and that followed the 

differential equation for transient thermal conduction into a semi-infinite air domain, bounded on 

one side by the microgripper surface. The heat flux boundary condition at the microgripper 

surface was equal to the optical power per unit area absorbed at the microgripper surface. The 

solution to this differential equation for a semi-infinite Cartesian domain is given in reference 27, 

which is rearranged below to show the expected 

 

power law at the boundary: 

 

 

 

 

 

 

 

 

(1)

Here, is thermal diffusivity, and k is thermal conductivity. 
3

Figure 2. Experimental data from reference 5 showing an inverse square power law 

between actuation time and irradiance. Insets show three high-speed video 

frames capturing microgripper closure.

However, several observations suggest that equation 1 does not represent the best model. First, 

the fitting of equation 1 to experimental values obtained in reference 5 using thermal properties 

for air and an assumed absorption of 15% for Au (26) results in ridiculous values of 800 °C

and higher. Furthermore, equation 1 predicts that the surface temperature will increase 

indefinitely as long as the source of irradiation remains in place, which is counter to 

experimental observations. Finally, in reference 25, we found that other experimental conditions 

result in power laws approaching rather than 

 

(figure 3). Therefore, a better model is 

needed to understand the system and how the required optical intensity level for actuation might 

be reduced.
4

Figure 3. Actuation time vs. laser intensity from reference 25, showing an inverse 

relationship between actuation time and laser irradiance or intensity.

This report details my derivation of analytical solutions for microgripper surface temperatures in 

spherical coordinates, as well as the development of axisymmetric finite element analysis (FEA) 

models with and without a neighboring substrate. These models together explain the various 

experimental trends observed in reference 5 and 25, and result in calculated theoretical minimum 

optical power requirements for this form of actuation. 

2. Theory

To start, the equation for three-dimensional transient heat conduction in a coordinate system with 

spherical symmetry accurately describes a heat flux source near and radial heat conduction 

in all directions away from that source, with the final solution being a function of only one 

geometric variable, r. This equation in spherical coordinates is

 

 

 

 

 

 

 

 

 

 

 

 

(2)

which may be simplified by considering only the steady-state solution with the time derivative 

set to zero,

 

 

 

 

 

(3)

Integrating equation 3 once 

 

 

 

 

(4)
5

or 

 

 

 

 

 

(5)

where is a constant. Although a boundary condition at is not readily tractable, a 

boundary condition at may be approximated by considering a sphere of radius with 

surface area equivalent to the area over which heat is lost from the microgripper, . This 

approximation should be valid as long as the length scale of steady-state thermal gradients in the 

surrounding air are much larger than the length scale of . For a microgripper of radius R, 

 

 

 

(6)

where is a ratio of heat loss area to the area of a disk with radius R. The factor of two 

accounts for heat loss out both sides of the disk, or microgripper surface. Equation 6 leads to 

 (7)

The power generated by optical absorption within this sphere must be equal to the power leaving 

the sphere, which is the power leaving the microgripper. The power generated is equal to the 

irradiance absorbed times the area of absorption , and the power leaving the microgripper 

is equal to Fourier’s law of thermal conduction times the area over which that heat is lost, , 

 

 

 

 

(8)

Rearranging, 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(9)

where is a factor describing the ratio of to the area of a disk of radius R, and where again 

R is the microgripper radius. Applying equations 7 and 9 to equation 5, 

 

 

 

 

 

 

 

 

 

 

 

(10)

Solving for ,

 

 

 

 

 

 

 

(11)

Next we substitute equation 11 into 5, 

 

 

 

 

 

 

 

 

 

 

 

(12)

and check that the assumption about steady-state thermal gradient length scales being much 

larger than the length scale of was valid. For a typical 1-mm-diameter microgripper, with 

 as a typical area fill ratio for a six-point folding star, . The 
6

distance for the temperature to change by or 0.36 at is , which for 

k = 0.257 mW/cm-K is 0.185 cm, which is roughly 10 times larger than . Thus, the 

assumption about steady-state thermal gradient length scales was valid. 

Integrating equation 12 a second time, 

 

 

 

 

 

 

 

 

 

 

(13)

The second integration constant is determined from the semi-infinite boundary condition that 

T approach the initial temperature as . Thus, 

 

(14)

Finally, the solution to equation 3 with the aforementioned boundary conditions is 

 

 

 

 

 

 

 

 

 

(15)

The steady-state temperature change at the surface, , is found by substituting equation 

(7)

 

 

 

 

 

(16)

Thus, the final temperature is proportional to , the gripper radius, inversely proportional to the 

surrounding medium thermal conductivity, and somewhat dependent on area fill factors. It is 

surprisingly not dependent on much else. 

3. Model Methods 

I used ANSYS Workbench v13 to model transient thermal conduction in a one-dimensional 

(1-D) Cartesian coordinate system as described by equation 1, and in a two-dimensional (2-D) 

axisymmetric model. The axisymmetric model approximated the 1-D spherical coordinate 

solution in equations 15 and 16 and allowed investigation of more detailed microgripper arm 

features. Also, I developed a 2-D axisymmetric model to investigate substrate effects. 

The 1-D FEA model was actually a 2-D Cartesian model, with a single element division in the y

direction and zero flux boundary conditions on both y direction boundaries, as shown in figure 4. 

There were 200 element divisions along the x-direction, with elements near biased to be 

1000 times smaller than those at the rightmost boundary of mm. The rightmost 

boundary approximated . I used an initial time step of s, a maximum time step 

of s, and automatic time stepping under a program controlled solver. 
7

Figure 4. The 2-D Cartesian coordinate FEA model domain, grid, 

and boundary conditions for a 1-D Cartesian problem.

A modified version of the 1-D model helped determine the correct way to model internal heat

generation, as opposed to the heat flux boundary condition used in the other 1-D models. 

Generally, the absorption of optical energy at a particular depth z follows the Beer-Lambert law:

 (17)

where is the incident irradiance and is the absorption coefficient of the material. The basis 

for deriving this law is that the differential power per unit area absorbed in a slice is equal to the 

power per unit area incident on that slice multiplied by its thickness and ,

 (18)

Based on equation 18, there are at least two ways to derive an internal heat generation per unit 

volume in a film of thickness . One is to integrate across the thickness and divide it by the 

film thickness,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  (19)

A second way is to actually create several slices in the model, and apply to each of 

them. For the first case, I replaced the heat flux boundary condition in figure 4 by a single slice 

at x=0 with internal heat generation applied. For the second case, I divided this slice into five 

separate slices and applied to each of them.

The 2-D axisymmetric FEA model domain is shown in figure 5, with the axis of symmetry at 

r=0. The four boundary conditions are also shown, with approximating the infinite 

domain boundary conditions at y = ±10,000 cm and at r = 7,500 cm. With sharper temperature 

gradients in the radial direction, I found that the distance approximating infinity could be 
8

shortened in that direction. Again, elements were 300–500 times smaller at the microgripper 

than near the infinite boundaries, where temperature changes were negligible. In order to use 

mapped meshing on the majority of the air domain, I used a region of air around the 

microgripper arm extending 40 μm from the microgripper surface to allow free meshing and 

matched elements with the more complicated features of the microgripper arm. Although I 

originally used a large number of elements in this region, I found that the solution did not depend 

significantly on this mesh spacing, and I therefore relaxed it to speed up computation. The total 

number of elements varied depending on exact geometry, but was generally in the range of 5,400 

elements and 16,000 nodes. Solver settings were the same as for the 1-D Cartesian model. 

The insets of figure 5 show the various features of a microgripper arm cross section, with 

material properties applied as shown in table 1. Internal heat generation was applied to the 

polymer regions according to equation 19. The chromium (Cr) layer was 77 nm thick, the 

evaporated Au layer was 300 nm thick, and the electroplated Au layer was 3 μm thick. The 

optically absorbing polymer layer was 3 μm thick with an optical absorption coefficient of 

1.5 μm–1, thereby modeling the case of 99%, or near 100%, absorption. The microgripper hinge 

length was 50 μm, and the radius R was 794 μm.

Figure 5. The 2-D axisymmetric FEA model domain, grid, and boundary conditions for an 

axisymmetric microgripper arm approximating a 3-D spherical coordinate solution at 

locations far from the microgripper arm.
9

Table 1. Material properties used in FEA model. 

Material Thermal 

conductivity 

(mW/cm-K)

Density (g/cm3) Specific Heat 

(J/g-K)

Air 0.257 1.205e-3 1.005

Gold (Au) 3180 19.3 0.13

Chromium (Cr) 939 7.14 0.45

Optically absorbing polymer 5.2 1.4 1.15

Substrate 2.5 1.3 1.34

I also developed an axisymmetric FEA model with a substrate, with the same boundary 

conditions, similar geometry, and similar element spacing used in the 2-D axisymmetric model. 

However, I replaced most of the lower portion of the air domain with elements having the 

thermal properties given in table 1 for a typical plastic substrate. 

4. Results and Discussion 

I first used the 1-D FEA model to compare results with the analytical solution given by equation 

1 to determine appropriate boundaries for the approximation, appropriate grid spacing to 

accurately capture thermal gradients near , and appropriate initial time steps and other 

solution parameters to accurately reproduce the analytical solution. At 20 mW/cm2, using 

thermal properties of air, and a of 20 °C, the time calculated from equation 1 was 244 ms. 

The 1-D FEA model geometry, grid spacing, and solution parameters given in the methods 

section yielded a calculated time of 246 ms, which differed from the analytical solution by less 

than 1%. Figure 6 shows the FEA calculated trend of time to reach a temperature change of 

20 °C vs. irradiance I0, resulting in a I0

–1.95 power law. The –1.95 exponent was close to the I0–2

power law predicted by equation 1. Generally, fewer elements, shorter distances to approximate 

infinite boundaries, and larger time steps all shortened computation time, but at the expense of a 

few percent in accuracy. With the parameters used, the time to calculate temperatures at each of 

the 100–200 time steps for this 200 element model was about 10 s on a 2.53-GHz, quad core 

CPU with 12 GB of memory. 

Figure 7 shows that the use of equation 19 for internal heat generation is valid for thin polymer 

layers, which makes sense because the derivation of equation 19 assumes that absorption is 

occurring in a thin slice. But as the polymer thickness increased, the higher absorption occurring 

near the incident surface was more accurately captured by applying I(z) to each slice. Figure 8 

depicts these results more generally for a variety of polymer thickness and absorption coefficient 

values, and shows the difference between the two methods of modeling heat generation from 

optical absorption. If the product of film thickness and absorption coefficient was less than 4.5, 

the percent difference was only 3.5%. Therefore, the more accurate but much more laborious 
10

process of creating and applying different heat generation quantities to separate slices was not 

necessary, and the application of equation 19 was sufficient.

Figure 6. Calculated time vs. irradiance for 

the 1-D FEA model.

Figure 7. Calculated time vs. polymer thickness for solutions obtained with I(z) 

applied to five slices and with the application of equation 19 to the entire 

polymer region. The absorption coefficient was 1, and the applied 

irradiance was 160 mW/cm2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20

time to reach ∆T=20C (s)

polymer thickness (μm)

I(z)*alpha on 5 slices

I(int)/z0
11

Figure 8. Percent difference between the application of equation 19 and I(z) applied to 

five slices as a function of the product of polymer thickness and absorption 

coefficient . The applied irradiance was 160 mW/cm2.

The 2-D axisymmetric model allowed investigation into finer details more representative of an 

actual microgripper arm, but at the cost of computation time. Figure 9 shows representative 

results at various times, highlighting the optical absorption of the polymer regions at early times 

and the more locally uniform temperature distribution at later times as the temperature front 

spread into the surrounding air domain. The circular isotherms spreading out from the 

microgripper arm suggest that the spherical domain assumed in the derivation of equation 16 was 

appropriate. 

Figure 10 shows the temperature versus time profile for several different cases at an irradiance 

level of 300 mW/cm2, equivalent to 15% absorption of a 2-W/cm2beam. Any microgripper 

geometry change other than overall radius had a minimal effect on the results. Changes to 

microgripper material thermal properties had equally minimal effects on the final calculated 

temperatures. But a 50% increase in air thermal conductivity resulted in a 33% decrease in final 

temperature. This result closely followed the inverse relationship expected between and in 

equation 16, which predicted a 35% decrease. 

0

5

10

15

20

25

0 5 10 15

% difference from I(z)*alpha on slices

polymer thickness * α

alpha = 0.25

alpha = 0.5

alpha = 1

alpha = 2
12

Figure 9. Two dimensional finite element analysis showing temperature distribution at 

different times (shown in inset of each image).

Figure 10. FEA calculated microgripper surface temperature change 

vs. time for several cases. Base parameters were those described 

in the model methods section. The other cases shown were for 50% 

thicker Au frames, 50% thicker polymer, and 50% higher air thermal 

conductivity.

Figure 11 shows further evidence of the validity of equation 16, and that a 50% increase or 50% 

decrease in microgripper radius resulted in a 46% increase and 49% decrease in final temperature 

change, respectively. These results again followed equation 16 according to a linear relationship 

0

5

10

15

20

25

30

35

40

0 0.5 1 1.5 2

Temperature Change ( C)

time (s)

base parameters

50% thicker Au

50% thicker polymer

50% higher k_air
13

between and microgripper radius. The base parameters were the same as in figure 10, but at 

an irradiance level of 1.5 W/cm2, equivalent to 15% absorption of a 10 W/cm2

beam. Also 

shown in figure 11 is the effect that radius had on the shape of each transient curve, with solid 

lines depicting exponential fits according to 

 

 

 

 

(20)

where was a time constant for each case. Transient behavior approximating exponential decay 

is common to many heat transfer problems given the first order partial differential term involving 

time in equation 1. Although I have not derived an analytical solution for the full transient 

behavior from equation 1, it is interesting to note that the time constant resulting from each fit 

followed closely (but not precisely) a linear relationship with radius R: 51.4% less for a 50% 

decrease in R, and 40% greater for a 50% increase in R. The dependence of both and on R

helps explain the empirical observation that at very high irradiance levels, the final temperature 

of a folded microgripper does not seem to be as high as equation 16 predicts. Once a 

microgripper reaches its folding temperature and it folds, its effective radius decreases by a 

factor of three or more, suggesting a rapid adjustment to a final which may three or more 

times lower. 

To more generally validate equation 16, figure 12 shows a plot of final temperature versus the 

product of irradiance and gripper radius. Using fill factors and the thermal 

conductivity of air from table 1, equation 16 predicts a proportionality constant between and 

 of 1.38. The average slope for all cases in figure 12 was 1.43, which was only 3.6% greater 

than the slope calculated in figure 12. Therefore, was proportional to both R and as 

expected. 
14

Figure 11. FEA-calculated microgripper surface temperature change 

vs. time for different microgripper radii. Base parameters 

were those described in the model methods section. Solid 

lines show exponential fits according to equation 20 with 

values for τ shown in the legend.

Figure 12. FEA-calculated steady state surface temperature rise 

vs. product of microgripper radius and irradiance for 

several cases.

While the temperature rise was proportional to irradiance and radius, the time to reach a 

particular temperature depended on the transient response of . Following the observation that 

calculated curves in figures 10 and 11 follow exponential behavior according to equation 20, an 

expression for the time for the microgripper to reach a temperature of closure is

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2

Temperature Change ( C)

time (s)

50% larger; τ=0.297

base parameters; τ=0.212

50% smaller; τ=0.109
15

 

 

 

 

(21)

Expressing the natural logarithm as a Taylor series expansion,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(22)

Assuming , 

 

 

 

(23)

Substituting equation 16 for , 

 

 

 

(24)

Therefore, the time to reach the closing temperature is approximately proportional to 1/ . 

This proportionality is in contrast to the 1-D Cartesian problem modeled by equation 1, where 

the time to reach a particular is proportional to 

 

. Past experimental data have exhibited 

both (25) and 

 

trends (5). 

To reconcile these results, I modeled the interaction between the axisymmetric microgripper arm 

and a substrate. Figure 13 shows the evolution of temperature over time with a substrate gap of 

50 μm and shows that the substrate not only changed the shape of the isotherms, but also 

significantly slowed the spatial propagation of isotherms in the direction of the substrate. 

Figure 14 shows the calculations of microgripper surface temperatures versus time for a variety 

of substrate gaps at an irradiance level of 1.5 mW/cm2. As the gap decreased, the substrate had a 

greater effect on both the transient response and the final temperature distribution. At a gap of 

300 μm, or nearly half the microgripper arm radius, the final temperature change was only about 

half of what it would be without considering the substrate. Besides these effects introduced by 

the substrate, the relationships between , , and were generally the same as for the cases 

without a substrate. 

Figure 13. Two-dimensional finite element analysis showing temperature distribution at 0.01 s and at 0.1 s for a 

microgripper and substrate separated by a 50 μm gap.
16

Figure 14. FEA-calculated microgripper surface temperature change vs. time for a 

microgripper arm situated at different gap distances from a substrate, compared 

to the case with no substrate.

Figure 15 shows the effect of the substrate on the transient behavior of temperature, along with 

power law fits according to the labeled equations. Interestingly, a 

 

power law fit well when 

the substrate gap was small, an exponent between 1 and 2 fit at intermediate substrate gaps, and 

the results at a gap of 300 μm followed closely a power law—despite the fact that there was 

a significant effect on the magnitude of temperature change as shown in figure 14. Figure 15 

therefore shows that the different power law regimes may be simply a result of the interaction 

between the microgripper and substrate. 

0

20

40

60

80

100

120

140

160

180

0 0.5 1 1.5 2

Temperature Change ( C)

time (s)

no substrate

300 μm

50 μm

1 μm
17

Figure 15. FEA calculated times to reach a temperature change of 20 °C 

vs. irradiance, for different substrate gaps. Solid lines show 

curve fits according to equations shown on plot, with power 

law exponents ranging from –1 for a large gap to –2 for a small 

gap.

To revisit the first of two experimental cases exhibiting different power laws in reference 5 and 

in figure 2, I note that the microgrippers were resting on the black, plastic surface of a staticdissipative wafer holder. It was likely that nearly 100% of the optical energy irradiating the 

substrate around the microgripper was absorbed by the substrate. Furthermore, as evident by the 

images in figure 2, two of the six arms in that particular batch of microgrippers consistently 

failed to close, due to a mask problem which was discovered some time later. These 

experimental details insured that the transient response of the microgrippers closely followed that 

of the substrate, or the 

 

power law of equation 1. The rewriting of equation 1 for as a 

function of , and plotting the data from figure 2 in this manner yielded an average slope of 

2837.5 . This slope corresponded to an average of 48.5 °C using the thermal 

properties for plastic from table 1 and an assumed 100% absorption. This value for 

corresponded to 71.5 °C with an initial room temperature of 23 °C, which was consistent with 

past observations of microgripper closure between 43 and 95 °C (3), or temperature changes 

between 19 and 72 °C. The use of a black substrate in the experiments of figure 2 likely resulted 

in such close correlation between microgripper and substrate temperatures that the substrate’s 

higher thermal conductivity and optical absorption dominated. These observations also explain 

why in reference 5, approximately ten times higher irradiance levels were required to actuate 

microgrippers sitting on nanoporous energetic silicon. The thermal conductivity of nanoporous 

silicon is 10 times larger at 25 mW/cm-K (28), suggesting that a 10 times higher irradiance level 

was required to reach an equivalent or .
18

In contrast, the experiment depicted in figure 3 and covered in reference 25 was performed on a 

piece of white paper. The porous structure of this substrate meant that thermal conduction 

properties could be approximated by those of air. Therefore, the transient response and time to 

close followed the relationship in equation 24. Using the empirical power law fit for the 

532 nm source in figure 3, the assumption that was precisely 15% of the applied irradiance, a 

numerically calculated value for of 0.186, and equation 24, I calculated an average of 

46 °C. This result corresponded to an average closing temperature of 69 °C for an initial room 

temperature of 23 °C, which was also consistent with both past observations and the calculations 

above for the 

 

case. Although more accurate calculations would require knowledge of the 

actual relationship between and laser irradiance, and validation with actual measured values 

for , the approximations of equations 1 and 24 are useful to predict how microgrippers might 

perform in different environments. 

One useful prediction is the lower limit of operation, or the lowest intensity at which actuation 

can be expected. For a six-point folding star in air, , and from equation 16, 

 - . For 20 °C and a microgripper diameter of 1.4 mm, 

the minimum irradiance predicted is 343 mW/cm2

. The lowest irradiance for which closure 

was actually observed in reference 25 for this microgripper size was 628 mW/cm2, which given 

the assumption that , was actually much lower than expected. If the experiment were 

to be repeated with an alternative material on microgrippers such that , microgripper 

closure could be expected at 95 mW/cm2, which is about the intensity of direct sunlight across 

the entire spectrum. In a separate measurement of reflectivity with a 200-nm evaporated Au film 

on a silicon substrate, I measured 30.7% absorption at 532 nm. Although this absorption value 

differed significantly from the reference value of 15%, if in fact and if the 

experiment in reference 25 were to be repeated with , microgripper closure might be 

expected at 193 mW/cm2. Therefore, the lower limit of operation is most likely between 95 and 

343 mW/cm2

for a 1.4-mm-diameter microgripper, which would increase or decrease according 

to the inverse proportionality with microgripper size. 

Besides microgripper size, the minimum actuation irradiance level could also be affected by area 

fill ratios and the temperature change needed for closure, . Higher values for and could 

be achieved through a larger central area or through arms with folding joints along multiple axes, 

thereby enabling an unfolded structure that more closely represents a disk. For , 

equation 16 predicts - , resulting in 40% lower minimum irradiance 

levels. The minimum actuation level might also be influenced by a desire for higher , which 

would result in more rapid closure. Alternative actuation materials may necessitate a higher , 

such as the pre-strained polystyrene used in reference 17, which requires an 80 °C temperature 

rise, or other shape memory alloys such as titanium-nickel (7, 29, 30), which require a 60 °C 

temperature elevation. The use of shape memory materials would also have the advantage of 

reversible actuation, but at the disadvantage of higher transition temperatures and therefore 
19

higher irradiance levels. In any case, the analysis presented here may guide the use of this 

actuation method in present and in future configurations. 

5. Conclusion 

I have derived analytical solutions and developed finite element models predicting microgripper 

temperatures with and without substrate interaction. Interestingly, the final, steady-state 

temperature rise of a microgripper was directly proportional to irradiance and microgripper size. 

The transient response of microgripper temperature was such that the time to reach a constant 

temperature of closure was inversely proportional to irradiance, . However, the 

introduction of a substrate caused the trend to approach 

 

as the gap to the substrate 

decreased due to the heat sinking properties of the substrate. Regardless of whether a substrate 

was present or not, the relationship between , , and was generally the same. These 

different power law regimes helped explain different power law regimes observed 

experimentally, and the models yielded reasonable predictions of microgripper closing 

temperatures in the different experimental conditions. 

Using the analytical model, I calculated the lowest intensity at which actuation can 

be expected for a 1.4-mm-diameter microgripper to be 343 mW/cm2

. Smaller or larger 

microgrippers reduce or increase this minimum optical actuation power level according to the 

inverse proportionality between and R. Interestingly, for experiments on a microgripper 

design that only absorbed between 15% and 30.7% of the applied optical power, the measured 

minimum power level at which actuation was observed was between 1.7 and 3.6 times lower 

than the value calculated in this report. The reason for this range was uncertainty in actual 

optical power absorption, but the fact that the calculations over-predicted the measurement is 

encouraging. I expect to verify the actual minimum optical actuation intensity in the near future, 

using a new microgripper designed to absorb nearly 100% of applied optical power. 

In the meantime, model predictions presented in this report may be used to conservatively guide 

the development of this actuation method in present and future configurations. Future 

configurations may involve the optical actuation of shape memory alloys or polymers, which 

would give the added advantage of reversible actuation, while maintaining the advantages of 

wirelessly controlling mechanical actuation in fully miniaturized microsystems. 
20

6. References 

1. Laflin, K.; Morris, C. J.; Bassik, N.; Jamal, M.; Gracias, D. H. Tetherless Microgrippers 

with Transponder Tags. J. Microelectromech. Sys. 2011, 20 (2), 505–511. 

2. Bassik, N.; Brafman, A.; Zarafshar, A. M.; Jamal, M.; Luvsanjav, D.; Selaru, F. M.; Gracias, 

D. H. Enzymatically Triggered Actuation of Miniaturized Tools. J. Am. Chem. Soc. 2010, 

132, 46, 16314–16317. 

3. Leong, T. G.; Randall, C. L.; Benson, B. R.; Bassik, N.; Stern, G. M.; Gracias, D. H. 

Tetherless Thermobiochemically Actuated Microgrippers. Proc. Natl. Acad. Sci. 2009, 106

(3), 703–708. 

4. Randhawa, J. S.; Leong, T. G.; Bassik, N.; Benson, B. R.; Jochmans, M. T.; Gracias, D. H. 

Pick-and-place Using Chemically Actuated Microgrippers. J. Am. Chem. Soc. 2008, 130

(51), 17238–17239. 

5. Morris, C. J.; Laflin, K. E.; Churaman, W. A.; Becker, C. R.; Currano, L. J.; Gracias, D. H. 

Initiation of Nanoporous Energetic Silicon by Optically-triggered, Residual Stress Powered 

Microactuators. in Proc. 25th IEEE International Conference on MEMS, (Paris, France), 

pp. 1245–1248, Jan. 30–Feb. 2 2012. 

6. Bell, D. J.; Lu, T. J.; Fleck, N. A.; Spearing, S. M. MEMS Actuators and Sensors: 

Observations on Their Performance and Selection for Purpose. J. Micromech. Microeng 

2005, 15 (7), S153. 

7. Hawkes, E.; An, B.; Benbernou, N. M.; Tanaka, H.; Kim, S.; Demaine, E. D.; Rus, D.; 

Wood, R. J. Programmable Matter by Folding. Proc. Natl. Acad. Sci. 2010, 107 (28), 

12441–12445.

8. Bütefisch, S.; Seidemann, V.; Büttgenbach, S. Novel Micro-pneumatic Actuator for MEMS. 

Sens. Actuators A 2002, 97–98, 638–645.

9. Hubschman, J. P.; Bourges, J. L.; Choi, W.; Mozayan, A.; Tsirbas, A.; Kim, C. J.; Schwartz, 

S. D. The Microhand: A New Concept of Micro-forceps for Ocular Robotic Surgery. Eye 

2009, 24, 364.

10. Ninomiya, T.; Okayama, Y.; Matsumoto, Y.; Arouette, X.; Osawa, K.; Miki, N. MEMSbased Hydraulic Displacement Amplification Mechanism with Completely Encapsulated 

Liquid. Sens. Actuators A 2011, 166 (2), 277–282. 

11. Ahir, S. V.; Squires, A. M.; Tajbakhsh, A. R.; Terentjev, E. M. Infrared Actuation in 

Aligned Polymer-nanotube Composites. Phys. Rev. B Feb 2006, 73, 085420. 
21

12. Yang, L.; Setyowati, K.; Li, A.; Gong, S.; Chen, J. Reversible Infrared Actuation of Carbon 

Nanotube-liquid Crystalline Elastomer Nanocomposites. Adv. Mater. 2008, 20 (12), 2271–

2275. 

13. Lu, S.; Panchapakesan, B. Nanotube Micro-optomechanical Actuators. Appl. Phys. Lett 

2006, 88 (25), 253107. 

14. Ikeda, T.; Nakano, M.; Yu, Y.; Tsutsumi, O.; Kanazawa, A. Anisotropic Bending and 

Unbending Behavior of Azobenzene Liquid-crystalline Gels by Light Exposure. Advanced 

Materials 2003, 15 (3), 201–205.

15. Yu, Y.; Nakano, M.; Ikeda, T. Directed Bending of a Polymer Film by Light. Nature 2003, 

425, 145.

16. Ikeda, T.; Mamiya, J.-i.; Yu, Y. Photomechanics of Liquid-crystalline Elastomers and Other 

Polymers. Angewandte Chemie International Edition 2007, 46 (4), 506–528. 

17. Liu, Y.; Boyles, J. K.; Genzer, J.; Dickey, M. D. Self-folding of Polymer Sheets Using 

Local Light Absorption. Soft Matter 2012, 8, 1764–1769.

18. Birnbaum, A. J.; Piqué, A. Laser Induced Extraplanar Propulsion for Three-dimensional 

Microfabrication. Appl. Phys. Lett. 2011, 98 (13), 134101. 

19. Piqué, A.; Matthews, S.; Birnbaum, A.; Charipar, N. Microfabricating 3D Structures by 

Laser Origami. SPIE Newsroom Nov 2011.

20. Bermejo, S.; Ortega, P.; Jimenez, J. J.; Castañer, L. Response of c-Si PV Arrays Under 

Monochromatic Light for MEMS Power Supply. J. Micromech. Microeng. 2005, 15 (8), 

1446. 

21. Warneke, B.; Scott, M.; Leibowitz, B.; Zhou, L.; Bellew, C.; Chediak, J.; Kahn, J.; Boser, 

B.; Pister, K. An Autonomous 16 mm3 Solar-powered Node for Distributed Wireless Sensor 

Networks. in Proc. IEEE Sensors, vol. 2, pp. 1510–1515, 2002.

22. King, R. R.; Law, D. C.; Edmondson, K. M.; Fetzer, C. M.; Kinsey, G. S.; Yoon, H.; Sherif, 

R. A.; Karam, N. H. 40% Efficient Metamorphic GaInP/GaInAs/Ge Multijunction Solar 

Cells. Appl. Phys. Lett. 2007, 90 (18), 183516. 

23. Lee, J.; Chen, Z.; Allen, M.; Rohatgi, A.; Arya, R. A Miniaturized High-voltage Solar Cell 

Array as an Electrostatic MEMS Power Supply. J. Microelectromech. Sys. 1995, 4 (3), 102–

108. 

24. Sakakibara, T.; Izua, H.; Shibata, T.; Tarui, H.; Shibata, K.; Kiyama, S.; Kawahara, N.

Multi-source Power Supply System Using Micro-photovoltaic Devices Combined with 

Microwave Antenna. Sens. Actuators A 2002, 95 (2), 208–211. 
22

25. Laflin, K. E.; Morris, C. J.; Muqeem, T.; Gracias, D. H. Laser Triggered Sequential Folding 

of Microstructures. Appl. Phys. Lett. 2012, (submitted). 

26. Bass, M.; Stryland, E.W.V. eds., Handbook of Optics, vol. 2. McGraw-Hill, 2nd ed., 1994. 

27. Incropera, F. P.; Dewitt, D. P. Introduction to Heat Transfer; John Wiley & Sons, 3rd ed., 

1996. 

28. Currano, L.; Churaman, W. Energetic Nanoporous Silicon Devices. J. Microelectromech. 

Sys. 2009, 18 (4), 799–807. 

29. Fu, Y.; Du, H.; Huang, W.; Zhang, S.; Hu, M. TiNi-based Thin Films in MEMS 

Applications: a Review. Sens. Actuators A 2004, 112 (2–3), 395. 

30. Gill, J.; Ho, K.; Carman, G. Three-dimensional Thin-film Shape Memory Alloy 

Microactuator with Two-way Effect. J. Microelectromech. Sys. Feb 2002, 11, 68–77.
23

NO. OF 

COPIES ORGANIZATION 

 1 ADMNSTR 

 DEFNS TECHL INFO CTR 

 ATTN DTIC OCP (ELECTRONIC COPY) 

 8725 JOHN J KINGMAN RD STE 0944 

 FT BELVOIR VA 22060-6218 

 1 US ARMY RSRCH DEV AND ENGRG CMND 

 ARMAMENT RSRCH DEV & ENGRG CTR 

 ARMAMENT ENGRG & TECHNLGY CTR 

 ATTN AMSRD AAR AEF T J MATTS 

 BLDG 305 

 ABERDEEN PROVING GROUND MD 21005-5001 

 1 US ARMY INFO SYS ENGRG CMND 

 ATTN AMSEL IE TD A RIVERA 

 FT HUACHUCA AZ 85613-5300

 1 US GOVERNMENT PRINT OFF 

 DEPOSITORY RECEIVING SECTION 

 ATTN MAIL STOP IDAD J TATE 

 732 NORTH CAPITOL ST NW

 WASHINGTON DC 20402 

 8 US ARMY RSRCH LAB 

 ATTN IMNE ALC HRR MAIL & RECORDS MGMT 

 ATTN RDRL CIO LL TECHL LIB 

 ATTN RDRL CIO LT TECHL PUB 

 ATTN RDRL SEG S RAGER 

 ATTN RDRL SER L C MORRIS (2 HCS) 

 ATTN RDRL SES J EICKE 

 ATTN RDRL SES P G FISCHER 

 ADELPHI MD 20783-1197 
24

INTENTIONALLY LEFT BLANK.