Effect of scaling on material selection of electro-thermal initially retracting V-Beam microactuators

Karthik Rao

National Institute of Technology Karnataka, Surathkal 575025

G.K Ananthasuresh

Indian Institute of Science, Bangalore 560012

Abstract

Initially Retracting V-beam microactuators have already been proposed in the microscale and their dimensions, as well as resulting displacement, are in the order of microns. An effort has been made in this study to examine the scaling of Initially Retracting V-Beam microactuators to the meso-scale. To do this, the electro-thermal and thermo-elastic equations were first solved to determine the heat transfer and resulting displacement in the mesoscale. Unlike the microscale, in the mesoscale heat transfer happens through convection and conduction. The value of the convective heat transfer coefficient was then determined and was plugged into the equation. After this, a material selection study was undertaken, using Ashby plots to choose the appropriate materials which can be used to fabricate the actuator. It was found out that no material can produce appreciable actuation that can be detected by the naked eye, for feasible values of current. The results indicate that fabrication of an Initially Retracting V-Bean Microactuator cannot be done in the mesoscale.

Keywords: initially retracting V-Beam actuators, convective heat transfer coefficient, Ashby plots

List of symbols

INTRODUCTION

Initially Retracting Electro Thermal V-Beam Actuators

An Initially-Retracting V-Beam micro-actuator makes use of two sets of opposing V-Beams to achieve actuation. The two beams are of different lengths and are connected to the same voltage terminal. With the application of voltage across the terminal, Joule heating of the actuator takes place. This, in turn, results in non-uniform temperature distribution giving rise to thermally induced strains which produce deformations. Since the two sets of beams are of different lengths, the shorter beam gets heated up faster, undergoes deformation and pushes the longer set of beams. This results in a "retraction". Subsequent heating will make the longer beams more hotter than the shorter ones which makes them push the shorter beams and attain a steady-state, resulting in a forward actuation​​.​^[1]​​

The competition between different modes of heat transfer depends on the scale. In the microscale, conduction dominates over convection. However, in the mesoscale heat transfer happens by both conduction and convection.

Fig 1 Working Principle of Initially retracting V-Beam microactuator

Free Convection Over a Flat,Vertical Plate

In order to study the effect of convection on heat transfer, the electrothermal V-beam actuator was modelled as a flat, vertical plate. Using this model, the heat transfer equation considering both conduction and free or natural convection was derived. In free convection, the motion of fluid over the body is not externally imposed, i.e the fluid is not driven over by the body with some velocity, by an external force. Instead, it occurs due to two factors: density gradient and body force. The effect of both these forces is a net buoyant force that drives the free convection.

In the convective flow over a vertical flat plate, the density gradient is imposed due to the temperature gradient and the body force is due to the gravitational force.

Fig 2 Formation of the convective boundary layer over a flat vertical plate

In ​​Fig 2​​, the plate has a temperataure T_s, which is greater than the temperature of the surroundings T_∞. Therefore the fluid which is in contact with the hot plate will have a density less than that in the bulk of the fluid. The hot fluid rises vertically upwards, entraining the fluid from the quiescent region due to the buoyancy effect.​​^[2]​​

The average Nusselt number for free flow over a flat vertical plate is given by the equation:

Also,

$\overline{Nu_L}=\frac{\overline hL}k$<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>N</mi><msub><mi>u</mi><mi>L</mi></msub></mrow><mo>&#xAF;</mo></mover><mo>=</mo><mfrac><mrow><mover><mi>h</mi><mo>&#xAF;</mo></mover><mi>L</mi></mrow><mi>k</mi></mfrac></math>\overline{Nu_L}=\frac{\overline hL}k

Since the Nusselt number is a function of the convective heat transfer coefficient, the latter can be found out by using ​ Equation 1​​.

Material Selection Based on Design Index

Material selection for a given design can be done with the help of Ashby plots. In Ashby plots, material properties or a combination of material properties are plotted along the log-log axes. Log-log axes are chosen to accommodate the material properties which vary over a wide range. The design objective (f), of any element, can be defined as a function of geometrical parameters (G), functional requirements (F) and material properties (M).

$f=f(F,G,M)$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo><mi>M</mi><mo>)</mo></math>f=f(F,G,M)

When the objective function f can be expressed as a separable function of the functional requirements (f_f(F)), geometrical properties (f_g(G)) and material properties (f_m(M)), i.e,

$f=f_f(F)\times f_g(G)\times f_m(M)$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><msub><mi>f</mi><mi>f</mi></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>&#xD7;</mo><msub><mi>f</mi><mi>g</mi></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>&#xD7;</mo><msub><mi>f</mi><mi>m</mi></msub><mo>(</mo><mi>M</mi><mo>)</mo></math>f=f_f(F)\times f_g(G)\times f_m(M)

Then f_m(M) is called as the material index. This material index helps us choose materials from the Ashby plot for given values of f_g(G) and f_f(m). ​​​^[3]​

Generally, there are two methods of designing any structural element. Structural engineers either assume a material before optimising their design or choose a suitable material for an already existing geometry (which is done with the help of material indices). However, either of these approaches does not guarantee an optimal combination of geometry and material. ​​

Instead of Ashby's form, the objective function can also be expressed as,

Where D, which depends entirely on geometry for given functional requirements, is called the Design Index. ​^[4]​

Fig 3 Material selection chart

In ​Fig 3​, seven materials are taken into consideration. The major and minor axes of the ellipses representing the materials, give us the range of values over which the two material indices vary in each of these materials. The figure shows contours of constant f values for a given value of D(f,g). Fixing the value of D(f,g) means that the functional and geometrical properties are fixed. From the plot, it is obvious that material M₆, minimizes the objective function f, for a value of f₃. Since f is constant along a curve, all materials that intersect with a curve are equally good. Therefore, for a given design, it is possible to have more than one suitable material.

The plots can be drawn with the help of Cambridge Engineering Selector (CES) software which has a database of a wide variety of materials.

Objectives of the Research

• To derive an equation for heat transfer in a mesoscale electro-thermal actuator.
• To find the value of the convective heat transfer coefficient for the heat transfer in the actuator.
• To obtain the expression for displacement of the meso-scaled actuator, using the thermo-elastic model.
• To obtain the objective function of current as a function of material and design indices.
• To find a suitable material for the electro-thermal actuator using Ashby Plots.

Overall objective

To find a suitable material for the fabrication of mesoscale electro-thermal actuator using Ashby Plots.

Scope

Applications of micro-scaled Initially retracting V-Beam actuator are already present in the literature. A two-terminal electro thermally actuated RF bistable switch has already been presented^[5]. The fabrication of the electrothermal actuator in the mesoscale, if at all possible, will have numerous biological applications since they act as large scaled agonist-antagonist pair.

LITERATURE REVIEW

The main motivation for this project is obtained by the paper on Initially Retracting V-Beam actuator ​^[1]​. This paper provides a complete treatment on analytical and finite element analysis of the said actuators. It provides a strong platform for extending the work into the mesoscale.

The next step in this project was the study of convective heat transfer. The textbook of Heat and Mass Transfer by Incorpera​^[2]​​ provides a detailed study on convection, along with elaborate proofs and results. Separate chapters for each type of convection also helped in understanding the concept in a systematic way. The paper by​^[6]​ provided good insights on thermal modelling and scaling of electrothermal microactuators.

Moving on to the materials selection part, the textbook by Ashby​​^[3]​​ explained in great detail the method of selecting a material for a given design. It also had several case studies which were interesting and useful in understanding the methodology of deriving the expression for the objective function in terms of material indices. The concept of design index was understood by reading​​^[4]​​, in which the said concept was explored in detail for designing a statically determinate truss.

Finally, the Ashby plots were created using the Cambridge Engineering Selector (CES) Software by Granta Design, Cambridge, U.K.

METHODOLOGY

There are three domains, namely electrical, thermal and elastic that have to be taken into account to analyze the meso-scaled electro-thermal actuator. In the electrical domain, we find the current distribution for the applied voltage. This current will cause joule heating of the actuator material. Coming to the thermal domain, joule heating is the internal power generated in the electrothermal actuator. Heat transfer due to conduction and convection have to be considered. Thus, the temperature distribution is obtained. Non-uniform temperature distribution will result in thermally induced strains. This leads us to the elastic domain that has to be analyzed. The output of analysis of the elastic domain is the resulting displacement of the actuator.

Next comes the process of material selection. For this, the objective function of current is first obtained as a function of design index and material indices. Then the required Ashby plot is drawn with the two material indices forming the two log-log axes. Contours of constant objective function are drawn in this plot and the optimized region is obtained. From this region, the suitable material is chosen.

Electro Thermal Model

When voltage is applied, current starts flowing through the beams of the actuator. From ​​Fig 1​​ it is clear that since all the beams share the same terminals, there will not be any current flow between the beams as the potential difference is zero. Also, since they share the same terminals, the temperature at the two ends of the beam will be equal to the ambient temperature T_∞. Since we consider only slender V Beams, 1D transient heat models are used. Taking only conduction and convection into account, the required expression is derived.

In the case of conduction, from Fourier Law we have,

$Q=-k_{th}\times\nabla T$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mo>-</mo><msub><mi>k</mi><mrow><mi>t</mi><mi>h</mi></mrow></msub><mo>&#xD7;</mo><mo>&#x2207;</mo><mi>T</mi></math>Q=-k_{th}\times\nabla T

Where T is the temperature, Q is the thermal heat flux and k_th is the thermal conductivity of the actuator.

Similarly, for convection, the heat flux can be given as,

$Q=h_{conv}\times(T-T_o)$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><msub><mi>h</mi><mrow><mi>c</mi><mi>o</mi><mi>n</mi><mi>v</mi></mrow></msub><mo>&#xD7;</mo><mo>(</mo><mi>T</mi><mo>-</mo><msub><mi>T</mi><mi>o</mi></msub><mo>)</mo></math>Q=h_{conv}\times(T-T_o)

Where h_conv is the convective heat transfer coefficient.

Therefore the combined heat transfer equation in the transient state, resulting due to Joule heating and taking both the convective and conductive terms to account for heat transfer will be

$T(0,t)=T_o$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>T</mi><mi>o</mi></msub></math>T(0,t)=T_o

$T(L,t)=T_o$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>(</mo><mi>L</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>T</mi><mi>o</mi></msub></math>T(L,t)=T_o

$T(x,0)=T_o$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mi>T</mi><mi>o</mi></msub></math>T(x,0)=T_o

The extra term that appears in the meso scaled electrothermal modelling, compared to micro scaled modelling is $-\frac{\operatorname dh}{\operatorname dT}(T-T_\infty)(\frac{\partial T}{\partial x})$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mo>(</mo><mi>T</mi><mo>-</mo><msub><mi>T</mi><mo>&#x221E;</mo></msub><mo>)</mo><mo>(</mo><mfrac><mrow><mo>&#x2202;</mo><mi>T</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac><mo>)</mo></math>-\frac{\operatorname dh}{\operatorname dT}(T-T_\infty)(\frac{\partial T}{\partial x}).

Now, Nusselt number, N_u is given by the expression,

Where, k is the thermal conductivity of air, L is the out of plane thickness of the actuator and h is the convective heat transfer coefficient of the flow.

Using ​​​Equation 4​ and ​Equation 1​, the value of the convective heat transfer coefficient h can be found out:

Before proceeding further, it is important to see whether the extra term present in the mesoscale heat transfer equation is significant or not. To do this, we differentiate h from ​Equation 5​ with respect to temperature T and see if it is a significantly large number. From literature, we find that the value of the Prandtl number of air, present in ​Equation 5​ does not vary a lot over a wide range of temperature. So, the denominator of ​Equation 5​ can be found out. The Rayleigh number, for free convection over a flat vertical plate, present in the numerator of ​Equation 5​ is given by the expression:

Where g is acceleration due to gravity, L is the out of plane thickness of the actuator, α is the thermal diffusivity of air, ν is the kinematic viscosity of air and β is the thermal expansion coefficient of air. It can be seen from the property charts that the value of ν and α do not change appreciably over the allowable working range of temperature.

The values of ν and α of air are then plugged into ​​Equation 6​​. For an ideal gases $\beta={\textstyle\frac1T}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3B2;</mi><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mi>T</mi></mfrac></mstyle></math>\beta={\textstyle\frac1T}. Taking the out of plane thickness of the plate, L to be 1mm, the simplified form of ​​​Equation 6​​ substituted in ​Equation 5​ yields the final result:

Differentiating ​Equation 7​ with respect to temperature T gives us,

To see whether h'(T) is significant in magnitude, we plot a graph of h'(T) over the allowable working range of temperature [298K, 1200K].

Fig 4 Variation of h'(T) over the working range of temperature

From ​Fig 4​, it can be seen that the term h'(T) in ​​Equation 3​​ is nearly equal to zero over the allowable working range of the temperature. So the term $-\frac{\operatorname dh}{\operatorname dT}\left(T-T_\infty\right)\frac{\partial T}{\partial x}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mfenced><mrow><mi>T</mi><mo>-</mo><msub><mi>T</mi><mo>&#x221E;</mo></msub></mrow></mfenced><mfrac><mrow><mo>&#x2202;</mo><mi>T</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></math>-\frac{\operatorname dh}{\operatorname dT}\left(T-T_\infty\right)\frac{\partial T}{\partial x} in ​​Equation 3​​ can be neglected.

Finally, ​​Equation 3​​ can be solved with the help of the initial and boundary conditions only if the value of the convective heat transfer coefficient h is known since it contains a term $-h\frac{\partial T}{\partial x}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>h</mi><mfrac><mrow><mo>&#x2202;</mo><mi>T</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></math>-h\frac{\partial T}{\partial x}. The value of h is taken to be the average value of the convective heat transfer coefficient found over the allowable working range of temperature using ​​Equation 7​.

Solving ​​​Equation 3​​ using Finite Fourier Transfor​m gives us,

Where,

$\zeta_n=\frac{2V^2}{cpn\mathrm{πρL}^2}\left[1-\cos\left(n\mathrm\pi\right)\right]$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x3B6;</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>V</mi><mn>2</mn></msup></mrow><mrow><mi>c</mi><mi>p</mi><mi>n</mi><msup><mi>&#x3C0;&#x3C1;L</mi><mn>2</mn></msup></mrow></mfrac><mfenced open="[" close="]"><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfenced><mrow><mi>n</mi><mi mathvariant="normal">&#x3C0;</mi></mrow></mfenced></mrow></mfenced></math>\zeta_n=\frac{2V^2}{cpn\mathrm{πρL}^2}\left[1-\cos\left(n\mathrm\pi\right)\right]

$\gamma_n=\left(\frac{n\mathrm\pi}L\right)^2\frac{k_s}{cp}$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x3B3;</mi><mi>n</mi></msub><mo>=</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mi mathvariant="normal">&#x3C0;</mi></mrow><mi>L</mi></mfrac></mfenced><mn>2</mn></msup><mfrac><msub><mi>k</mi><mi>s</mi></msub><mrow><mi>c</mi><mi>p</mi></mrow></mfrac></math>\gamma_n=\left(\frac{n\mathrm\pi}L\right)^2\frac{k_s}{cp}

​$\beta=\frac h{bcp}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x3B2;</mi><mo>=</mo><mfrac><mi>h</mi><mrow><mi>b</mi><mi>c</mi><mi>p</mi></mrow></mfrac></math>\beta=\frac h{bcp}

Thermo Elastic Model

Due to the slenderness of the V-Beams, Euler's beam theory is applicable. This can be used to analyze the elastic deflection under thermal loads. The deflection δ can be found using Maizels Theorem as,

Where σ is the axial normal stress,α is the thermal expansion coefficient, V is the volume of the body and T is the temperature field obtained in ​Equation 9​.

Since the beams are slender, we can use 1D approximation. So ${\widehat\sigma}_y\;and\;{\widehat\sigma}_z$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>&#x3C3;</mi><mo>^</mo></mover><mi>y</mi></msub><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><msub><mover><mi>&#x3C3;</mi><mo>^</mo></mover><mi>z</mi></msub></math>{\widehat\sigma}_y\;and\;{\widehat\sigma}_zare equal to 0. Due to the fixed-fixed boundary conditions, there is an in-plane bending caused by the thermal stress developed. The axial stiffness, k is given by ​^[1]​,

Finally, the displacement of the tip as a function of time is given by

Where $\int_0^L(T-T_o)\operatorname dx$<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>&#x222B;</mo><mn>0</mn><mi>L</mi></msubsup><mo>(</mo><mi>T</mi><mo>-</mo><msub><mi>T</mi><mi>o</mi></msub><mo>)</mo><mo>d</mo><mi>x</mi></math>\int_0^L(T-T_o)\operatorname dxis obtained from ​​Equation 9​.

Calculation of Design Index

Let us consider the length L of the electrothermal actuator to be a free variable, i.e its value can vary with different designs. Now we need to select a material for the fabrication of the actuator such that the current consumed by the actuator is minimized.

Fig 5 Differential element of V Beam with the known dimensional parameters

​Fig 5​ is the pictorial representation of the dimensional parameters that are taken for consideration. Note that the out of plane thickness is 1mm, satisfying the value of L which has been substituted in ​Equation 6​. Using the values of the known parameters from ​Table 1​ in ​​Equation 12​​, we get

​$\triangle=\frac{-0.2026\times\left(e^{{}^{-\left(\frac{9.869k_sh+39.9081L^2}{L^2cph}\right)t}}-1\right)L^2bd\alpha V^2}{(d^2+w^2)\rho(9.868k_sb+39.9081L^2)}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2206;</mo><mo>=</mo><mfrac><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>2026</mn><mo>&#xD7;</mo><mfenced><mrow><msup><mi>e</mi><msup><mrow/><mrow><mo>-</mo><mfenced><mfrac><mrow><mn>9</mn><mo>.</mo><mn>869</mn><msub><mi>k</mi><mi>s</mi></msub><mi>h</mi><mo>+</mo><mn>39</mn><mo>.</mo><mn>9081</mn><msup><mi>L</mi><mn>2</mn></msup></mrow><mrow><msup><mi>L</mi><mn>2</mn></msup><mi>c</mi><mi>p</mi><mi>h</mi></mrow></mfrac></mfenced><mi>t</mi></mrow></msup></msup><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mi>L</mi><mn>2</mn></msup><mi>b</mi><mi>d</mi><mi>&#x3B1;</mi><msup><mi>V</mi><mn>2</mn></msup></mrow><mrow><mo>(</mo><msup><mi>d</mi><mn>2</mn></msup><mo>+</mo><msup><mi>w</mi><mn>2</mn></msup><mo>)</mo><mi>&#x3C1;</mi><mo>(</mo><mn>9</mn><mo>.</mo><mn>868</mn><msub><mi>k</mi><mi>s</mi></msub><mi>b</mi><mo>+</mo><mn>39</mn><mo>.</mo><mn>9081</mn><msup><mi>L</mi><mn>2</mn></msup><mo>)</mo></mrow></mfrac></math>\triangle=\frac{-0.2026\times\left(e^{{}^{-\left(\frac{9.869k_sh+39.9081L^2}{L^2cph}\right)t}}-1\right)L^2bd\alpha V^2}{(d^2+w^2)\rho(9.868k_sb+39.9081L^2)}​

where 't' is the time elapsed after applying the required voltage. However, we are interested only in the final displacement at steady state, for which the value of time 't' is very large. In such a case, the entire exponential term becomes very small and can be neglected. This was verified numerically and the exponential term gave a value of 10^-53. The known geometrical parameters in the above expression can be clubbed together to form constant terms k₁,k₂,k₃ and so on for the sake of simplification. Doing this yields us the equation,

Where,

​$k_1=0.2026\times b\times d\times V^2$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>2026</mn><mo>&#xD7;</mo><mi>b</mi><mo>&#xD7;</mo><mi>d</mi><mo>&#xD7;</mo><msup><mi>V</mi><mn>2</mn></msup></math>k_1=0.2026\times b\times d\times V^2​

$k_2=(d^2+w^2)\times39.9081$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><msup><mi>d</mi><mn>2</mn></msup><mo>+</mo><msup><mi>w</mi><mn>2</mn></msup><mo>)</mo><mo>&#xD7;</mo><mn>39</mn><mo>.</mo><mn>9081</mn></math>k_2=(d^2+w^2)\times39.9081

$k_3=\frac{9.869}{40.61}\times b$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mn>3</mn></msub><mo>=</mo><mfrac><mrow><mn>9</mn><mo>.</mo><mn>869</mn></mrow><mrow><mn>40</mn><mo>.</mo><mn>61</mn></mrow></mfrac><mo>&#xD7;</mo><mi>b</mi></math>k_3=\frac{9.869}{40.61}\times b

The required displacement Δ, must be expressed relatively with respect to the length L. So, let $\frac{∆}{L}=\psi$<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>&#x2206;</mo><mi>L</mi></mfrac><mo>=</mo><mi>&#x3C8;</mi></math>\frac\triangle L=\psi , where Ψ is the ratio of the displacement achieved with respect to the free length. Using this in ​Equation 13​, and solving for L, we get,

The current consumed by the electro-thermal actuator is given by the expression,

Since L is the free variable, it must be eliminated. Therefore, using ​Equation 14​ in ​Equation 15​ to eliminate L, we get,

Where,

$k_4=VbwN\times 2\psi \times k_2$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mn>4</mn></msub><mo>=</mo><mi>V</mi><mi>b</mi><mi>w</mi><mi>N</mi><mo>&#xD7;</mo><mn>2</mn><mi>&#x3C8;</mi><mo>&#xD7;</mo><msub><mi>k</mi><mn>2</mn></msub></math>k_4=VbwN\times2\psi\times k_2

$k_5=\frac{4{\psi }^2{k_2}^2{k}_3}{{k_1}^2}$<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mn>5</mn></msub><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>&#x3C8;</mi><mn>2</mn></msup><msup><msub><mi>k</mi><mn>2</mn></msub><mn>2</mn></msup><msub><mi>k</mi><mn>3</mn></msub></mrow><msup><msub><mi>k</mi><mn>1</mn></msub><mn>2</mn></msup></mfrac></math>k_5=\frac{4\psi^2k_2^2k_3}{k_1^2}

​Equation 16​ on simplification, gives us,

​​Equation 17​ is of the form $P=Dm_1+m_2$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>D</mi><msub><mi>m</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub></math>P=Dm_1+m_2 where,

• P is the objective function
• D is the design index which is a function of geometrical and functional properties.
• m₁ and m₂ are the material indices.

Therefore,​ ​​Equation 17​​ can be used to draw contours of constant P in the Ashby plot obtained by using the two material indices m₁ and m₂.​

Note that minimizing the current means, maximising the term $\frac{k_4}{2I}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>k</mi><mn>4</mn></msub><mrow><mn>2</mn><mi>I</mi></mrow></mfrac></math>\frac{k_4}{2I} in ​Equation 17​.

From ​Fig 3​, it is clear that Ashby Plots for objective functions which are of the form of ​​Equation 17​ are drawn by plotting $\log\left(m_2\right)$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><msub><mi>m</mi><mn>2</mn></msub></mfenced></math>\log\left(m_2\right)along the y-axis and $\log\left(m_1\right)$<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><msub><mi>m</mi><mn>1</mn></msub></mfenced></math>\log\left(m_1\right)along the x-axis. To do this, ​​Equation 2​​ must be treated as follows:

Using ​Equation 18​ and ​Equation 19​ in ​​Equation 2​​, we get,

Where,

$\tilde{I} = \frac{k_4}{2I}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>I</mi><mo>~</mo></mover><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfrac><msub><mi>k</mi><mn>4</mn></msub><mrow><mn>2</mn><mi>I</mi></mrow></mfrac></math>\widetilde I\;=\;\frac{k_4}{2I}

Solving ​Equation 20​ for y gives,

Drawing Contours of the Objective Function

The design index D is given by the term $\frac{-k_5}{4}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><msub><mi>k</mi><mn>5</mn></msub></mrow><mn>4</mn></mfrac></math>\frac{-k_5}4, i.e,

The values of d, w, b and V on which k₂, k₃ and k₁ depend on are already known from ​​Table 1​​. Therefore, the value of design index D can be found for different values of the required ratio of $\frac{∆}{L}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>&#x2206;</mo><mi>L</mi></mfrac></math>\frac\triangle L which is given by Ψ.

Once the value of the design index is found out for different values of Ψ, some small, feasible values of current can be considered. Both these values (of current and D) can then be plugged into ​Equation 21​ to obtain the equation for the contours, which are then drawn in the Ashby plot.

Drawing Ashby Plots for Material Selection

The material selection chart was generated using the CES (Cambridge Engineering Selector) software by Granta Design, Cambridge. The database of metals was selected and therefore the analysis was done for 1832 materials. Once the contours are drawn, the required search region for optimizing or reducing the value of current is identified. Then, the process of material selection starts. The Ashby plot obtained is shown below.

Fig 6 Ashby's Plot for material selection

As can be seen from the plot, the highest ordinate is 10^-4. The process of finding the optimized region is as follows:

If current I has to be minimized, the left hand term in ​Equation 17​ must be maximized. ​Equation 17​ is of the form Dm₁+m₂ where D is negative. Therefore, to minimize the value of current by maximizing the term $\frac{k_4}{2I}$<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>k</mi><mn>4</mn></msub><mrow><mn>2</mn><mi>I</mi></mrow></mfrac></math>\frac{k_4}{2I}, m₁ should be minimized and m₂ should be maximized. This gives us the optimized region, as shown in ​Fig 7​.

Fig 7 Plot showin the optimized region.

The drawback of CES software is that it doesn't allow us to adjust the scale and draw contours. As a result, ​Fig 7​ could not be zoomed out more. Neither does this software allow us to export the data into MATLAB or Maple. The maximum ordinate of the plot as shown by the CES software is 0.1, thus it is not favourable to draw the "feasible contours", all of whose y-intercepts are greater than 0.1, here. So, the contours were drawn in Maple as shown below:

Fig 8 Plot for 2A current and 10% displacement

Fig 9 Plot for 2A current and 20% displacement

Fig 10 Plot for 5A current and 10% displacement

Fig 11 Plot for 5A current and 20% displacement

Note that current greater than 5A was not taken into consideration as it is not a feasible value.

RESULTS AND DISCUSSION

Interpretation of Ashby plots

Consider the contour plots. The table below gives the y intercept of these plots:

From ​Table 2​, it is clear that if we increase the expected percentage displacement or decrease the input current, we move deeper into the optimized region, indicated in ​Fig 7​. From ​Fig 7​, it is also clear that there are no materials in the optimized region.

The contours that we draw are curves of constant current I for a given value of D, which is a function of the percentage displacement of the actuator. Since I is constant along a curve, all the materials that overlap with the curve are equally good. However, by looking at the range of the plots in ​Fig 8​, ​Fig 9​, ​Fig 10​ and ​Fig 11​, it is obvious that none of these contours will pass through any material which is plotted in ​Fig 6​, since the highest ordinate of the latter is only around 10^-4. Since no material passes through the contour, no material will give us the exact value of current and percentage displacement, as desired by the contour.

Fig 12 Figure to interpret Ashby Plot (Not to scale)

From ​​Fig 12​​ it can be said that the materials lie very far, out of the optimized region, even for the contour of 2A and 10% displacement since the latter's y-intercept from ​​Table 2​​ is 1.2575, and the highest ordinate for the Ashby plots is only 10^-4 (From ​​Fig 6​​). Since the materials lie very far out of the optimized region for this contour, it can be said that these materials can be actuated only for very high values of current, that are not feasible. Even then the resulting percentage displacement will be very small (See ​​Fig 12​​).

CONCLUSION

• The effect of convection in heat transfer while modelling meso scaled electrothermal actuators is quite significant and cannot be neglected.
• There are no materials that give us appreciable displacement for feasible current since no material passes through the contours drawn, or lies inside the optimized region. ​​
• Thus there are no materials that can be used for building meso scaled actuators.

SHORTCOMINGS

The CES software that was used to draw the Ashby plot, did not allow us to zoom out beyond some extent, i.e beyond the ordinate 0.1. Therefore, the contours of feasible current and high displacement, all of whose y-intercepts were greater than 0.1, could not be drawn in the same Ashby plot and had to be drawn in Maple. Thus even for explaination, a rough plot had to be drawn (​Fig 12​). Also, there was no provision to export the data to MATLAB and then draw the plots.

ACKNOWLEDGEMENTS

I would like to extend my heartiest gratitude to Prof. G.K Ananthasuresh, Department of Mechanical Engineering, IISc for guiding me in every step of the project. It was inspiring, having him as my guide. Despite being busy, he always found time to clear my doubts when I approached him. It has been a great learning experience, interning under him.

Next, I would like to thank Mr Dhananjay Yadav, for mentoring me in the lab. He set up a good road map to follow so that I could complete my study in time. He was always ready to answer even the most trivial doubts of mine. His suggestions regarding some aspects of my project helped me get out of some tight spots.

I am also grateful to all other lab mates and interns from M2D2 lab who helped me in this project. I was exposed to a lot of wonderful, new ideas and learnt plenty of new things.

We were provided with an amazing platform to write well-designed, neat and professional-looking reports by Author Cafe. They were also kind enough to conduct a lecture-demonstration in our residency to familiarize us with their product. Their chat support was very handy in solving any technical difficulties faced while using the software.

Finally, I would like to thank the Indian Academy of Sciences for providing me with an opportunity to intern at one of the premier institutes of our country. The lecture session that they conducted for the interns working in Bangalore was informative and interesting. Overall, they ensured a great learning experience for all the interns.