# Comprehensive Thermal Modeling of ElectroThermoElastic Microstructures

## Abstract

**:**

## 1. Introduction

## 2. Multipurpose Electrothermoelastic Microstructures

**Figure 1.**Examples of E-T end-effectors fabricated on Silicon On Insulator (SOI) material: (

**a**) Micro grasper (combining bent-beam and folded-beam mechanisms) [7]. (

**b**) The end-effector has grasper (obtained from folded beam mechanism), micro-heater (obtained from bent beam structure), and positioning sensor (obtained from resistive Zigzag structure); (

**c**) A packaged end-effector incorporates multifunctional sensing and actuation capabilities.

## 3. Steady-State Thermal Modeling

_{cs}) and h

_{cu}correspond to the thermal convection heat losses off the side wall and the faces of microbeams, respectively. λ is a numeric factor introduced here to describe the heat loss condition. (λ) is equal to one when heat conduction through air gap is considered. However, (λ) is two when microbeams are experiencing convection from all sides with no conduction to other layers, i.e., shape factor (S) is zero. ρ

_{d}and C

_{p}are material density and specific heat of actuator, respectively. The resistivity of microbeam (ρ

_{r}) is linearly dependent on temperature. The thermal conductivity of the actuator material (k

_{p}(T)) will be later assumed constant. (T

_{p}, T

_{a}and T

_{b}) are constant temperatures of the bottom surface of the substrate, ambient and of a black body, respectively. All are assumed here to be equal to some constant temperature (T

_{s}). The nonlinear radiation term in Equation (1) is proportional to Stefan-Boltzmann constant (σ) and micro-surface emissivity (ε

_{e}).

_{T}) is a thermal resistance between the thermal microbeam and the substrate. For microbeam suspended on two sandwiched layers, such as SOI, the thermal resistance is

_{v}) and (t

_{s}) are the air gap and substrate thicknesses, respectively. (k

_{v}) and (k

_{s}) are the thermal conductivity of the air and a substrate, respectively.

_{p}) of silicon is generally dependent on temperature variation. It is generally approximated in a third order polynomial [16]

_{(}

_{i}

_{)}) or (β

_{(i)}) value, and it is obtained from the roots of the characteristic equation of the ODE

#### 3.1. Folded or Bent Beam E-T Actuators

_{h}+ g + l

_{c}+ l

_{f}). The total number of boundary conditions needed to solve the unknown constants is 2n.

**Case 1**. Suspended microstructures with exponential profiles when (I

_{h}, I

_{c}and I

_{f}< I)

_{h}, β

_{c}, β

_{f}> 0); i.e., the input current is greater than the right hand side in Equation (6) or (I

_{h}, I

_{c}and I

_{f}< 1). The line-shape temperature profiles along each beam is

_{∞i}= (T

_{p}+ J

_{i}

^{2}ρ

_{0}/β

_{i}). The temperature profile along the three connected microbeams is continuous. The unknown are solved from the continuity of temperature profile and heat flow. Applying the boundary conditions into the Steady State Heat Conduction Equation (SSHCE) gives the unknown constants (C

_{ij})

**Case 2**. Suspended microstructures with mixed profiles when (I

_{h}, I

_{f}< I and I

_{c}> I)

_{h}, β

_{f}< 0 and β

_{c}> 0), the temperature profiles of hot and flexure arms are given by

**Case 3**. Suspended microstructures with sinusoidal profiles when ((I

_{h}, I

_{c}and I

_{f}) > I)

_{h}, β

_{c}, β

_{f}< 0) becomes true, the solution becomes

_{u}. Where the equations derived in the above three cases can be used for “bent beam” given that w

_{u}= w

_{h}= w

_{c}= w

_{f}. For microbeams of the same width, the critical condition β

_{h}, β

_{c}, β

_{f}= 0 gives constant temperature profile T

_{h}(x) = T

_{c}(x) = T

_{f}(x) = T

_{p}. In this case, the amount of heat generation is equal to the amount of heat lost across a substrate. There are only two cases in “bent beam” microstructures: (i) exponential profile which corresponds to low temperature; or (ii) sinusoidal profile pertaining high temperature response.

#### 3.2. Combined Bent and Folded Beam Actuator

_{s}+ l

_{l}+ l

_{h}+ g + l

_{c}+ l

_{f}). The (g) is the gap between the hot and cold arm. The current density across the hot arm, linkage arm, and bent beam causes high thermal expansion as compared to cold arm. Concurrently, the bent beams push the linkage arm forward. Thus, each tip in the grasper bends toward the cold arm with a greater overall all “opening and closing”.

**Figure 3.**Micro grasper combining folded and bent beam mechanisms: (

**a**) sketch; (

**b**) E-T actuator fabricated from SOI.

**Case 1**. Suspended microstructure with exponential profiles (I

_{f}, I

_{c}, I

_{h}, I

_{l}, I

_{s}< I)

_{f}, β

_{c}, β

_{h}, β

_{l}, β

_{s}> 0) is hold. The temperature profile along microbeams is continuous, start from pad temperatures at a flexure arm and end up with same pad temperature at the end of bent beams. The temperature profiles are

_{s}) and (R

_{h}) are the overall electrical resistance of folded and bent beams in the microstructure, respectively. I is the total current drawn across the pad due to overall resistance, where I = I

^{h}+ I

^{s}and I

^{s}is the current passing in each one of the (q) bent beams

_{a})

**Case 2**. Overhanging microstructure operating on ambient

**Case 3**. Sinusoidal profiles

_{i}= ρξ and T

_{∞i}=T − 1/ξ.

## 4. Simulation and Experimental Results

_{0}, ε > 0), (ω) and (χ) are integers that refer to the time and the space mesh, respectively. (Δx) and (Δt) are space and time grid resolution, respectively. Equation (19) is a general equation which can solve for any serially connected microstructures. The intermediate B.C’s between different microbeams are automatically embedded in the formula. Also, it can handle non-homogenous material and thickness properties across different microbeams. When an m parallel network of microbeam is introduced, a set of m n-serially connected equations must be solved simultaneously for the unknowns.

SOI Layers Parameter | Si Device | Si wafer substrate | Air |
---|---|---|---|

Density, ρ_{d} (Kg/m^{3}) | 2,330 | 2,330 | 0.524 |

Thermal conductivity, (Wm^{−1} °C^{−1}) | 100 | 30 | 3.37 × 10^{−2} |

Thermal Expansion, α (10^{−6} × °C^{−1}) | 3.1 | 3.1 | 1.49 × 10^{3} |

Thermal Capacity, C_{p} (J·Kg^{−1}·°C^{−1}) | 787 | 787 | 1,013 |

Temperature coefficient, ξ (10^{−3} × °C^{−1}) | 1.25 | 1.25 | - |

Electrical resistivity, ρ_{o} (Ω·m) | 1.51 × 10^{−4} | 2.5 × 10^{−2} | 3 × 10^{13} |

Modulus of Elasticity, E (GPa) | 169 | 169 | - |

Poison ratio, ν | 0.22 | 0.22 | - |

_{k}) and temperature profile (T

_{k}(x), l

_{k}≥ x ≥ l

_{k−}

_{1}), can be lumped into

**Figure 9.**Thermal failure experiments: (

**a**) SOI folded beam; (

**b**) failure on hot arm due to melting at (~14.4 V); (

**c**) multi failures in hot and flexure arms due to (~20 V).

## 5. Conclusions

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Mayyas, M. Comprehensive Thermal Modeling of ElectroThermoElastic Microstructures. *Actuators* **2012**, *1*, 21-35.
https://doi.org/10.3390/act1010021

**AMA Style**

Mayyas M. Comprehensive Thermal Modeling of ElectroThermoElastic Microstructures. *Actuators*. 2012; 1(1):21-35.
https://doi.org/10.3390/act1010021

**Chicago/Turabian Style**

Mayyas, Mohammad. 2012. "Comprehensive Thermal Modeling of ElectroThermoElastic Microstructures" *Actuators* 1, no. 1: 21-35.
https://doi.org/10.3390/act1010021