Pulsed heating of the self-actuated cantilever: a one-dimensional exact solution investigation of non-axial temperature gradients

Abstract

Self-actuated bimorph cantilevers are implemented in a variety of micro-electro-mechanical systems. Their tip deflection relies on the unmatched coefficients of thermal expansion between layers. The thermal bimorph phenomenon is dependent on the temperature rise within the cantilever and, while previous studies have investigated variations in the thermal profile along the cantilever length, these have usually neglected variations in the thermal profile along the cantilever thickness. The current study investigates the thermal distribution across the thickness of the cantilever. The exact closed form solution to the one-dimensional problem of heat conduction in the composite (layered) domain subjected to transient volumetric heating is developed using the appropriate Green’s function. This solution is applied to a one-dimensional case study of a 3-layer cantilever with an Aluminium heater, a silicon dioxide resistive layer, and a silicon base layer. The aluminium heater experiences volumetric heating at a rate of 0.2 mW/μm³ of 5 μs duration at 100 μs intervals (10 kHz with a 1/20 duty cycle). Benchmark solutions of the temperature at select times and positions are provided. It is shown that there are negligible temperature gradients across the cantilever thickness during the heating and the first ~ 5 μs afterward. These short-lived temperature differences are positively biased with the unmatched thermal expansion coefficients between the layers, though their relative influence on bending is not clear. A simple parametric analysis indicates that the relative magnitude of the temperature differences across the cantilever (compared to the overall temperature) decreases substantially with increasing duty cycle.

1 Introduction

Bi-material actuators are a class of thermally self-actuated micro-electro-mechanical systems (MEMs) that are used in microscale sensors and actuator devices [1,2,3]. Applications of bimorph actuators include optical sensors, micro-manipulators, and AFM probes which require precise micro- and nanoscale manipulation of the MEMS [2, 4]. The working principle underlying this class of MEMS is described by the theory of the bending of a bi-material strip subjected to heating [5]. When two materials with different coefficients of thermal expansion are heated, the layer with the higher CTE experiences higher expansion and the uneven strain experienced by the composite will result in a bending whose magnitude is proportional to the temperature rise [6].

In the material choices of many self-actuated cantilevers, the different layers may also have large differences in electrical and thermal conductivities. In many MEMs applications, the cantilever is fabricated by bulk or surface micro-machining of silicon wafers [7, 8]. When an electrical current is used to activate the device, the heated layer must be electrically isolated from the silicon (Si) base layer. Consider, for example, the three-layer cantilever presented in [9, 10] depicted in Fig. 1a. A layer of aluminium (Al) serving as the actuator is electrically isolated from the base layer using a layer of electrically resistive silicon dioxide (SiO₂). As the current passes through the aluminium, electrical energy is converted to thermal energy by resistive (Joule) heating and this heat is conducted through the cantilever resulting in temperature rises within the different layers. Because the aluminium layer has a thermal expansion coefficient that is about one order of magnitude higher than that of the Silicon base layer, the bimorph effect results in the deflection of the cantilever tip from its unheated state (Fig. 1b) to its activated heated state (Fig. 1c). By applying a periodic or pulsed electric signal to the aluminium heater, the cantilever can be operated in a transient oscillatory mode so that the position of the tip oscillates as well.

Fig. 1

a The thermal actuator cantilever with the representative layers; b side view of the cantilever in its unheated state; c side view of the cantilever in its heated state with deflection

The temperatures within the different layers of the cantilever will be influenced by the thermal properties of the layers and this may influence the deflection of the cantilever. In the example depicted in Fig. 1, the SiO₂ layer has a thermal conductivity two orders of magnitude lower than the other layers. This study develops an analytical model to investigate the effect that such inhomogeneity in thermal conductivity has on the temperature distribution of a multilayer cantilever operated in oscillatory mode.

Understanding the deflection of the self-heated micro-cantilever requires an understanding of the thermal profile [6] and published thermal models include both exact mathematical solutions [11,12,13] and numerical simulations [4, 14,15,16] though none investigate the effects of variation in thermal conductivity across the cantilever thickness (across the different layers).

In an early study [14], the authors develop a one-dimensional steady finite difference model that effectively represents the thermal profile of the cantilever as a homogenous extended surface (fin). While the model does neglect thermal resistance across the thickness, the authors are able to provide excellent insight into AFM design noting that future studies should consider adding transience to the model to consider pulsed heating. This is exactly what the authors of [15] do the following year in a one-dimensional transient numerical model of the periodic (sinusoidal) heating of the cantilever. They show that the magnitude of the heater’s temperature oscillations decrease with increasing heating frequency and that the time averaged spatial temperature gradients along the length of the heater decrease with increasing frequency of heating. The study considers only the temperature profile along the length of the cantilever, so it does not capture temperature gradients across the cantilever thickness. Two years later, the authors of [4] develop a comprehensive 3-D transient finite element model of the self-heated cantilever experiencing periodic heating with different activation characteristics including sinusoidal and pulsed heating profiles. The study provides insight into the nature of the thermal profile at the system level noting that about one half of the heater energy leaves the cantilever through the base. While the study does consider multidimensional effects, it does not consider the variations in thermophysical properties across the cantilever thickness.

Exact solutions to the problem of the heated cantilever have not yet described the composite nature of the problem, opting instead to represent the layered cantilever using effective or volume averaged properties. These generally all represent the exact solution using the 1-D fin (extended surface) model [11,12,13]. In [11], the fin approximation is used to model heat transfer along the length of the cantilever, and the problem approximates the heating using a simple boundary condition at one end of the fin. In the study [12] the self-heated cantilever is modelled analytically in a 1-D steady problem that effectively also uses the solution to the problem of heat transfer along extended surface. The temperature solution along the length of the fin is developed using volume averaged effective thermal conductivity values to account for the layered composition of the cantilever. Authors have used the fin approximation in more complicated one-dimensional transient models that estimate tip deflection. In [17], the authors develop the one-dimensional thermal profile along the length of a cantilever that is not uniform in composition along the length and that is subject to periodic heating. The authors couple this thermal profile to a model of mechanical stress in a parametric study. However, again with the fin representation of the cantilever, any inhomogeneity across the cantilever thickness is neglected.

The above studies all assume negligible thermal resistance across the individual layers of the cantilever. This assumption is clearly valid under steady conditions when the dominant thermal resistance in the cantilever is associated with conduction along the cantilever length. However, it is not clear that the assumption holds at the short timescale transience associated with the periodic operation of a cantilever.

The underlying principle of the self-heated cantilever is that when heated, the difference in thermal expansion coefficients result in unmatched expansion and thus, a bending of the cantilever. The coupling between the thermal solution and the cantilever bending was accomplished in a 3-D finite element simulation of steady heating in [10]. The authors consider the different thermophysical parameter values in a three-layered micro-cantilever and couple the steady thermal solution with a mechanical stress model to predict the tip deflection.

The current study seeks to develop an exact transient solution to the thermal profile across the thickness of the layered cantilever undergoing periodic–pulsed heating. Although exact solutions to the transient heat conduction problem in the composite domain are complicated by the unmatched parameter values at the layer interfaces, solutions have been developed previously that primarily use the separation of variables method [18,19,20,21]. Solutions have been developed for the mathematically similar problem of transient mass diffusion in composite biological media [22, 23]. The exact solutions to problems in the composite domain have been presented for volumetric heating in the Pennes’ bioheat model [23,24,25] and for non-uniform convection along the extended surface (segmented fin) [26]. More recently, the exact solution to the transient problem of pulsed surface heating of an axis-symmetric two-layered slab was developed using Green’s functions [27].

The current work uses an exact mathematical solution to investigate the temperature profile across the thickness of a composite multilayer cantilever subject to pulsed heating. In the section that follows, the exact solution to this problem is developed. The appropriate Green’s function is used to develop the integral solution of a cantilever composed of an unspecified number of layers that is exposed to (i) convection on the outer surfaces and (ii) periodic heating within the layers. This solution is applied to a case study of a three-layer cantilever composed of aluminium (Al), silicon dioxide (SiO₂), and silicon (Si) in which the aluminium layer experiences volumetric heating in a square wave pulse train.

2 The Green’s function solution to heat transfer in the composite slab

Descriptions of solution development to the problem of transient diffusion in the composite domain have previously been reported [28, 29] and these discretize the temperature so that each layer, $i$, has its own unique representation of temperature, $T_{i}$. To further simplify the solution presentation, some studies use a spatial coordinate that is unique to each composite layer [25, 26, 30]. With these in mind, the problem in the “M-layered” composite slab subject to volumetric heating may be represented as:

$$ \frac{1}{{\alpha_{i} }}\frac{{\partial T_{i} }}{\partial t} = \frac{{\partial^{2} T_{i} }}{{\partial x_{i}^{2} }} \, + \frac{1}{{k_{i} }}g_{i} \left( {x_{i} ,t} \right) \, 0 < x_{i} < L_{i} \quad t > 0\quad i = 1,2,...,M. $$

(1)

Here $t$ is the time, and the subscript $i$ on the remaining variables and parameters indicate the layer number. The parameters $\alpha_{i}$ and $k_{i}$ are, respectively, the thermal conductivity and thermal diffusivity of layer $i$ and each layer may be subject to volumetric heating, $g_{i}$. The layer spatial coordinate $x_{i}$ is specific to layer $i$ with layer thickness $L_{i}$. In this way, the layer spatial coordinate, $x_{i}$, is related to the global continuum spatial coordinate, $x$, by the relations $\, x_{1} = x$ and for $i > 1: \, x_{i} = x - \sum\nolimits_{n = 1}^{i - 1} {L_{n} }$.

In this paper, Eq. (1) is subject to convection at the outer surfaces of layer 1 and of layer M:

$$ \begin{gathered} k_{1} \left. {\frac{{\partial T_{1} }}{{\partial x_{1} }}} \right|_{{x_{1} = 0}} = h_{1} \cdot T_{1} \left( {0,t} \right) - h_{1} \cdot T_{\infty 1} \hfill \\ \left. {k_{M} \frac{{\partial T_{M} }}{{\partial x_{M} }}} \right|_{{x_{M} = L_{M} }} = - h_{M} \cdot T_{M} \left( {L_{M} ,t} \right) + h_{M} \cdot T_{\infty M} . \hfill \\ \end{gathered} $$

(2)

where $h_{1}$ and $h_{M}$ correspond to the heat transfer coefficients at the outer surface of layers 1 and ${\text{M}}$, respectively, and $T_{\infty 1}$ and $T_{\infty M}$ correspond to the ambient temperatures at those locations.

The system is subject to some general initial condition:

$$ T_{i} \left( {x_{i} ,0} \right) = F_{i} \left( {x_{i} } \right) \, 0 < x_{i} < L_{i} \quad t = 0, $$

(3)

And interface conditions are applied between the layers:

$$ \left. \begin{gathered} \left. {k_{i - 1} \frac{{\partial T_{i - 1} }}{{\partial x_{i - 1} }}} \right|_{{x_{i - 1} = L_{i - 1} }} = k_{i} \left. {\frac{{\partial T_{i} }}{{\partial x_{i} }}} \right|_{{x_{i} = 0}} \hfill \\ T_{i - 1} \left( {L_{i - 1} ,t} \right) = T_{i} \left( {0,t} \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3,...M. $$

(4)

The solution to the problem of Eqs. (1)–(4) may be developed using the appropriate Green’s function (see for example [28, 29, 31, 32]). The Green’s function solution to heat conduction in a composite slab composed of an arbitrary number of $M$ layers is:

$$ \begin{gathered} T_{i} \left( {x_{i} ,t} \right) = \sum\limits_{j = 1}^{M} {\left\{ {\int_{{x^{\prime}_{j} = 0}}^{{L_{j} }} {\left. {G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right)} \right|_{\tau = 0} F_{j} \left( {x^{\prime}_{j} } \right){\text{d}}x^{\prime}_{j} } } \right.} \hfill \\ \quad \quad \quad \quad \left. { + \frac{{\alpha_{j} }}{{k_{j} }}\int_{\tau = 0}^{t} {\int_{{x^{\prime}_{j} = 0}}^{{L_{j} }} {G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right)g_{j} \left( {x^{\prime}_{j} ,\tau } \right){\text{d}}x^{\prime}_{j} } {\text{d}}\tau } } \right\} \hfill \\ \quad \quad \quad \quad + \frac{{\alpha_{1} }}{{k_{1} }}\int_{\tau = 0}^{t} {\left. {h_{1} \cdot T_{\infty 1} \left( {G_{i,1} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{1} ,\tau } \right)} \right)} \right|_{{x^{\prime}_{1} = 0}} {\text{d}}\tau } \hfill \\ \quad \quad \quad \quad - \frac{{\alpha_{M} }}{{k_{M} }}\int_{\tau = 0}^{t} {\left. {h_{M} \cdot T_{\infty M} \left( {G_{i,M} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{M} ,\tau } \right)} \right)} \right|_{{x^{\prime}_{M} = L_{M} }} {\text{d}}\tau } . \hfill \\ \end{gathered} $$

(5)

Here $G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right)$ is the Green’s function of layer $i$ for which $\tau$ is a dummy time variable and $x^{\prime}_{j}$ is the dummy spatial variable of layer $j$. In the cases studied in this paper, the initial temperature and the temperature associated with convection at the boundaries are equal and constant: $F_{i} \left( {x_{i} } \right) = T_{\infty 1} = T_{\infty M} = T_{\infty }$. With this simplification and by defining a variable $\theta_{i} \equiv T_{i} \left( {x_{i} t} \right) - T_{\infty }$, the associated boundary conditions of Eq. (2) and the initial condition (3) may be represented as homogeneous so that expression (5) is reduced and may be represented as:

$$ T_{i} \left( {x_{i} ,t} \right) - T_{\infty } = \sum\limits_{j = 1}^{M} {\frac{{\alpha_{j} }}{{k_{j} }}\int_{\tau = 0}^{t} {\int_{{x^{\prime}_{j} = 0}}^{{L_{j} }} {G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right)g_{j} \left( {x^{\prime}_{j} ,\tau } \right){\text{d}}x^{\prime}_{j} } {\text{d}}\tau } } . $$

(6)

While the Green’s function to the 1-D composite problem has been previously derived in [28, 29], a derivation is provided in Appendix A using notation consistent with Eqs. (1)–(4). For problem (1) subject to (2) and (4), the Green’s function of the $M$-layered slab is represented by:

$$ \begin{aligned} G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right) & = \sum\limits_{n = 1}^{\infty } {\frac{{k_{j} }}{{\alpha_{j} }}\frac{1}{{N_{n} }}\exp \left[ { - \frac{{\mu_{i,n}^{2} \alpha_{i} }}{{L_{i}^{2} }}\left( {t - \tau } \right)} \right]} \hfill \\ & \quad \times \left[ {a_{i,n} \sin \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right) + b_{i,n} \cos \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right)} \right] \hfill \\ & \quad \times \left[ {a_{j,n} \sin \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right) + b_{j,n} \cos \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right)} \right]. \hfill \\ \end{aligned} $$

(7)

For which the constants of (7) may be represented:

$$ \begin{gathered} a_{1,n} = h_{1} L_{1} /k_{1} \quad \hfill \\ b_{1,n} = \mu_{1,n} \hfill \\ \left. \begin{gathered} a_{i,n} = \frac{{L_{i} }}{{L_{i - 1} }}\frac{{k_{i - 1} }}{{k_{i} }}\frac{{\mu_{i - 1,n} }}{{\mu_{i,n} }}\left[ {a_{i - 1,n} \cos \left( {\mu_{i - 1,n} } \right) - b_{i - 1,n} \sin \left( {\mu_{i - 1,n} } \right)} \right] \hfill \\ b_{i,n} = \,a_{i - 1,n} \sin \left( {\mu_{i - 1,n} } \right) + b_{i - 1,n} \cos \left( {\mu_{i - 1,ni} } \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3,..,M. \hfill \\ N_{n} = \frac{1}{2}\sum\limits_{i = 1}^{M} {\frac{{k_{i} L_{i} }}{{\alpha_{i} }}\frac{1}{{\mu_{i,n} }}} \left( \begin{gathered} 2a_{i,n} b_{i,n} \sin^{2} \left( {\mu_{i,n} } \right) + \mu_{i,n} \left( {a_{i,n}^{2} + b_{i,n}^{2} } \right) \hfill \\ - \left( {a_{i,n}^{2} - b_{i,n}^{2} } \right)\sin \left( {\mu_{i,n} } \right)\cos \left( {\mu_{i,n} } \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(8)

The eigenvalues, $\mu_{i,n}$, are determined from:

$$ \mu_{M,n} \left( {a_{M,n} - b_{M,n} \tan \left( {\mu_{M,n} } \right)} \right) + \frac{{h_{M} L_{M} }}{{k_{M} }}\left( {a_{M,n} \tan \left( {\mu_{M,n} } \right) + b_{M,n} } \right) = 0\quad n = 1,2, \ldots, \infty . $$

(9)

Following expression (A20) of Appendix A, the eigenvalues of any two layers $i$ and $j$ are related to one another by the relation:

$$ \mu_{j,n}^{2} \frac{{\alpha_{j} }}{{L_{j}^{2} }} = \mu_{i,n}^{2} \frac{{\alpha_{i} }}{{L_{i}^{2} }}. $$

(10)

Note that for layers $i = 2..M$, the constants of integration $a_{i,n}$ and $b_{i,n}$ are all described using the eigenfunction of the previous layer, $i - 1$. The studies [19, 25, 26] avoid the need for rigorous algebraic manipulation by producing an algorithm that “builds” the solution starting with layer 1 and progressing sequentially through layers $i = 2..M$ and that approach is taken in this study as well.

The transcendental expression used to determine the eigenvalues (9) and the relation (10) may be used together in order to “build” the full expression of the eigenvalues. First one layer is chosen to be a “reference layer” (subscript $R$) and following Eq. (10), the eigenvalues of all other layers are expressed in terms of the reference layer eigenvalues by the relation:

$$ \mu_{i,n} = \mu_{R,n} \frac{{L_{i} }}{{L_{R} }}\sqrt {\frac{{\alpha_{R} }}{{\alpha_{i} }}} . $$

(11)

Traditionally, published solutions of pure diffusion in the composite domain assign the first layer as the reference layer [29, 33] although an alternate reference layer selection has been proposed for heat conduction along the extended surface of the segmented fin [26] and for conduction in layered perfuse tissue represented by the Penne’s bioheat equation [25].

Starting with layer 1 and progressing through layers $i = 2 \ldots M$ , the constants $a_{i,n}$ and $b_{i,n}$ of Eq. (8) are expressed in terms of the reference layer eigenvalues $\mu_{R,n}$. The transcendental Eq. (9) may then be expressed in terms of the reference layer eigenvalues $\mu_{R,n}$ (which need to be determined). Then the left-hand side of Eq. (9) is plotted versus a continuum $\mu_{R}$. From this depiction, the values of $\mu_{R}$ that bound each of the roots are determined and these bounds can be used in a simple method of bisection to evaluate each root (each value of $\mu_{R,n}$). While there exist more elegant methods of root determination that are less prone to human error (see for example the description of the method presented in [20]) with sufficient care, the combined graphical bisection technique is adequate and simple to implement. In the current study, the numerical values of the reference layer eigenvalues are determined using the combined graphical bisection method which has been implemented in similar composite layered problems [18, 30, 34].

2.1 Representation of periodic heating

In the case study presented in the following section, only the layer representing the heater experiences periodic heating. However, in the more general solution developed here, we consider heating in each of the layers and that the heating characteristics may vary between layers. Within any layer “$j$”, the heating is of uniform magnitude, $g_{0,j}$, occurs over a duration $\tau_{P,j}$, and at regular intervals of period $P$ (the same for all layers). The duration of the heating is related to the duty cycle by the relation $\tau_{P,j} = DC_{j} \cdot P$. The heating in layer $j$ begins at an activation time, $t_{0,j}$ that is specific to that layer. In this way, the heating of layer $j$ is restricted to begin within the time interval $0 \le t_{0,j} \le P - \tau_{P,j}$. The volumetric heating term in layer $j$ of the problem in Eq. (1) and of the solution Eq. (6) satisfies:

$$ g_{j} (x_{j} ,t) = \left\{ \begin{gathered} 0{\text{ for }}t \in \left[ {0,t_{0,j} } \right] \, \hfill \\ g_{0,j} {\text{ for }}t \in \left[ {t_{0,j} ,\left( {t_{0,j} + \tau_{P,j} } \right)} \right] \hfill \\ 0{\text{ for }}t \in \left[ {\left( {t_{0,j} + \tau_{P,j} } \right),P} \right] \hfill \\ \end{gathered} \right.. $$

(12)

where the heating is periodic so that $g_{j} \left( {x_{j} ,t + n_{P} P} \right) = g_{j} \left( {x_{j} ,t} \right)$ for any integer value of pulse number $n_{P} > 1$.

The heating of layer $j$ is expressed using its Fourier series representation:

$$ g_{j} \left( {x_{j} ,t} \right) = \sum\limits_{m = - \infty }^{\infty } {c_{j,m} } \exp \left( {{\text{i}}{\kern 1pt} m\omega t} \right). $$

(13)

where ${\text{i}}^{2} = - 1$, ${\kern 1pt} \omega = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } P}} \right. \kern-0pt} P}$, and each of the Fourier coefficients may be evaluated by the integral expression:

$$ c_{j,m} = \frac{1}{P}\int\limits_{0}^{P} {g_{j} \left( {x_{j} ,t} \right)\exp \left( { - {\text{i}}{\kern 1pt} m\omega t} \right){\text{d}}t} . $$

(14)

The evaluation of the integral (14) for a square wave function described by (12) results in:

$$ \begin{gathered} {\kern 1pt} c_{j,m} = g_{0,j} \frac{1}{P}\frac{1}{{{\text{i}}\omega m}}\left( {1 - \exp \left( { - {\text{i}}{\kern 1pt} m\omega \tau_{P} } \right)} \right)\exp \left( { - {\text{i}}{\kern 1pt} m\omega t_{0,j} } \right)\quad m \ne 0 \hfill \\ c_{j,0} = g_{0,j} \frac{{\tau_{P,j} }}{P}\quad m = 0. \hfill \\ \end{gathered} $$

(15)

Substituting (13) and (7) into (6) and rearranging yields:

$$ \begin{aligned} T_{i} \left( {x_{i} ,t} \right) & = T_{\infty } + \sum\limits_{j = 1}^{M} {} \sum\limits_{n = 1}^{\infty } {} \frac{1}{{N_{n} }}\int\limits_{{x^{\prime}_{j} = 0}}^{{L_{j} }} {\left[ {a_{j,n} \sin \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right) + b_{j,n} \cos \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right)} \right]{\text{d}}x^{\prime}_{j} } \hfill \\ &\quad \times \left[ {a_{i,n} \sin \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right) + b_{i,n} \cos \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right)} \right] \hfill \\ &\quad \times \sum\limits_{m = - \infty }^{\infty } {c_{j,m} } \int\limits_{\tau = 0}^{t} {\exp \left[ { - \frac{{\mu_{i,n}^{2} \alpha_{i} }}{{L_{i}^{2} }}\left( {t - \tau } \right)} \right]\exp \left( {{\text{i}}{\kern 1pt} m\omega \tau } \right){\text{d}}\tau } . \hfill \\ \end{aligned} $$

(16)

Evaluating the integrals of Eq. (16):

$$ \begin{aligned} T_{i} \left( {x_{i} ,t} \right) &= T_{\infty } + \sum\limits_{j = 1}^{M} {} \left\{ {\sum\limits_{n = 1}^{\infty } {} } \right.\left\{ {\frac{1}{{N_{n} }}\frac{{L_{j} }}{{\mu_{j,n} }}\left[ {b_{j,n} \sin \left( {\mu_{j,n} } \right) - a_{j,n} \cos \left( {\mu_{j,n} } \right) + a_{j,n} } \right]} \right. \hfill \\ &\quad \times \left[ {a_{i,n} \sin \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right) + b_{i,n} \cos \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right)} \right] \hfill \\ &\quad \times \sum\limits_{m = - \infty }^{\infty } {c_{j,m} } \left. {\left. {\frac{{\exp \left[ {\left( {{\text{i}}m\omega } \right)t} \right] - \exp \left[ { - \left( {{{\mu_{j,n}^{2} \alpha_{j} } \mathord{\left/ {\vphantom {{\mu_{j,n}^{2} \alpha_{j} } {L_{j}^{2} }}} \right. \kern-0pt} {L_{j}^{2} }}} \right)t} \right]}}{{{\text{i}}m\omega + {{\mu_{j,n}^{2} \alpha_{j} } \mathord{\left/ {\vphantom {{\mu_{j,n}^{2} \alpha_{j} } {L_{j}^{2} }}} \right. \kern-0pt} {L_{j}^{2} }}}}} \right\}} \right\}. \hfill \\ \end{aligned} $$

(17)

where relation (10) has been used in the last term.

The Fourier series representation of the periodic heating results in an additional infinite summation term in solution (17). Alternate expressions of the time-dependent conditions could be used instead. For example, in Appendix C of [35], time-dependent conditions are represented as piecewise continuous approximations in the evaluation of the Greens Function solution. However, in this study, the Fourier Series representation is used.

3 Case study

In this section, solution Eq. (17) is applied to a three-layer single heater cantilever arrangement similar to that depicted in Fig. 1 and described in [10]. This 1-D transient solution is representative of the thermal profile on the cantilever at an axial location where the heater is located. Consider the cross section of the cantilever depicted in Fig. 2 whose orientation has been rotated compared to those of Fig. 1. Here, layer 1 represents the aluminium heater with transient heating $g_{1} \left( t \right)$. layer 2 is composed of the thermally resistive SiO₂. layer 3 is the base layer of the cantilever and is composed of Si. The outer boundaries are exposed to convection heat transfer with the ambient air at $T_{\infty }$ and the convection coefficients at these two outer surfaces are equal so that $h_{3} = h_{1} = h$.

Fig. 2

Cross-sectional representation of the 3-layer single heater cantilever. To simplify the presentation of the results, the orientation of the cantilever has been rotated 90° from the depiction in Fig. 1

The resistance heating resulting from the electrical current applied to the aluminium layer is represented by the rate of volumetric generation,$g_{0}$ (as there is only a single heater in this case, the subscript index $j$ appearing in Eqs. (12) and (15) has been dropped). The heating is applied in a periodic square waveform of period $P$ using a duty cycle, $DC$, the duration of the heating for each period is $\tau_{P} = DC \cdot P$. Within Layer 1 $\left( {0 < x_{1} < L_{1} \, } \right)$, the uniform volumetric heating of magnitude $g_{0}$ is activated at the beginning of the period ($t_{0,1} = 0$):

$$ g_{1} (x_{1} ,t) = \left\{ \begin{gathered} g_{0} {\text{ for }}t \in \left[ {0,\tau_{P} } \right] \hfill \\ 0{\text{ for }}t \in \left[ {\tau_{P} ,P} \right] \hfill \\ \end{gathered} \right.. $$

(18)

where the heating is periodic so that $g_{1} \left( {x_{1} ,t + n_{P} P} \right) = g_{1} \left( {x_{1} ,t} \right)$ for any integer value of pulse number $n_{P}$.

Following expressions (13) and (15) the Fourier series representation of the heating is rewritten as:

$$ \begin{gathered} g_{j} \left( {x_{j} ,t} \right) = \sum\limits_{m = - \infty }^{\infty } {c_{j,m} } \exp \left[ {\left( { - {\kern 1pt} {{2\pi {\text{i}}{\kern 1pt} \cdot m} \mathord{\left/ {\vphantom {{2\pi {\text{i}}{\kern 1pt} \cdot m} P}} \right. \kern-0pt} P}} \right)t} \right]\quad \hfill \\ {\kern 1pt} c_{1,m} = g_{0} \frac{1}{{2\pi {\text{i}}{\kern 1pt} \cdot m}}\left( {1 - \exp \left( { - 2\pi {\text{i}}{\kern 1pt} \cdot m{\kern 1pt} {{\tau_{P} } \mathord{\left/ {\vphantom {{\tau_{P} } P}} \right. \kern-0pt} P}} \right)} \right)\quad m \ne 0 \hfill \\ c_{1,0} = g_{0} \frac{{\tau_{P} }}{P}\quad m = 0. \hfill \\ \end{gathered} $$

(19)

There is no volumetric heating in any other layer:

$$ j \ne 1:\quad g_{j} (x_{j} ,t) = 0. $$

(20)

Substituting (19) and (20) into (17) and simplifying results in:

$$ \begin{aligned} & T_{i} \left( {x_{i} ,t} \right) \\ &\quad= T_{\infty } + \sum\limits_{n = 1}^{\infty } {} \left\{ {\frac{1}{{N_{n} }}\frac{{L_{1} }}{{\mu_{1,n} }}\left[ {b_{1,n} \sin \left( {\mu_{1,n} } \right) - a_{1,n} \cos \left( {\mu_{1,n} } \right) + a_{1,n} } \right]} \right. \hfill \\ &\qquad \times \left[ {a_{i,n} \sin \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right) + b_{i,n} \cos \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right)} \right] \hfill \\ &\qquad \left. { \times \sum\limits_{m = - \infty }^{\infty } {c_{1,m} \frac{{\exp \left[ {\left( { - {\kern 1pt} {{2\pi {\text{i}}{\kern 1pt} \cdot m} \mathord{\left/ {\vphantom {{2\pi {\text{i}}{\kern 1pt} \cdot m} P}} \right. \kern-0pt} P}} \right)t} \right] - \exp \left[ { - \left( {{{\mu_{1,n}^{2} \alpha_{1} } \mathord{\left/ {\vphantom {{\mu_{1,n}^{2} \alpha_{1} } {L_{1}^{2} }}} \right. \kern-0pt} {L_{1}^{2} }}} \right)t} \right]}}{{{{2\pi {\text{i}}{\kern 1pt} \cdot m} \mathord{\left/ {\vphantom {{2\pi {\text{i}}{\kern 1pt} \cdot m} P}} \right. \kern-0pt} P} + {{\mu_{1,n}^{2} \alpha_{1} } \mathord{\left/ {\vphantom {{\mu_{1,n}^{2} \alpha_{1} } {L_{1}^{2} }}} \right. \kern-0pt} {L_{1}^{2} }}}}} } \right\}\quad i = 1,2,3.\quad \hfill \\ \end{aligned} $$

(21)

Here, the values of constants, $a_{i,n}$,$b_{i,n}$, and of the Norm, $N_{n}$, are determined using Eq. (8) for $M = 3$, the eigenvalues are determined from Eq. (9), the layer eigenvalues are related to each other through Eq. (10).

The thermophysical parameter values that appear in Eqs. (8)–(10), (19), and (21) have been chosen to reflect those of previous studies and are listed in Table 1. The layer-specific thermophysical and geometrical parameter values, $L_{i}$,$k_{i}$, $\alpha_{i}$, have been adapted from those published in the self-heated micro-cantilever study of [10]. Layer 1 (the Al heater) is heated volumetrically at a rate of $g_{0}$ = 0.2 mW/μm³ for a pulse duration of $\tau_{P}$ = 5 μs in a period $P$ = 100 μs (for a duty cycle $DC$ = 0.05). The transient heating profile resulting from the Fourier series of Eq. (19) using the first 1201 terms (from $m = - 600,...,600$) is depicted in Fig. 3. The pulsing parameters used here are similar to those of [4] who consider heating at a slightly lower rates (0.0625 mW/μm³) but for longer pulse durations (10 μs or 100 μs) over periods in the range between 20 and 200 μs.

Table 1 Thermophysical parameter values used in the case study

Fig. 3

The volumetric heating to the heated layer in a series pulses of duration of 5 μs applied at 100 μs intervals that results using the first 1201 terms of the Fourier series representation Eq. (19)

The heat transfer coefficient of this study has been modified from those of previous work and its value is presented in the context of previous works as follows. In some early studies, for example [11], the heat transfer from the cantilever to the air is not explicitly modelled and the authors instead opt for a prescribed periodic temperature at the base of the cantilever. However empirical observations indicate that the heat transfer from the surface is significant [36]. In the study [13], the authors estimate the effective heat transfer coefficient between an AFM and the surrounding air to be 3400 W/m² K and this lies within the ranges published in other studies [4]. In the finite element study of [4], the authors consider the pulsed heating of the cantilever in air and they seek to estimate the heat transfer coefficient. Their simulations treat air as a still fluid and so the estimates consider only heat conduction to and within the air and the study estimates that the effective heat transfer coefficients lie in the range ~ 2800–4800 W/m² K for pulsed and periodic heating. It is anticipated that when the cantilever is heated periodically, the cantilever will experience deflection at the same frequency of the periodic heating. This flapping should enhance the heat transfer by providing forced convection heat transfer (in addition to the conduction to the air). We anticipate that the heat transfer from a flapping cantilever will enhance the heat transfer to exceed that of the pure conduction to the air described by [4]. To account for this, we use a heat transfer coefficient an order of magnitude larger than in those published works: 5 × 10⁴ W/m² K. The authors of the multidimensional work [4] also note that, at the system scale, their results predict that nearly half of all the thermal energy introduced by the heater is conducted to the base of the cantilever which also supports using a larger heat transfer coefficient in the current 1-D study that cannot explicitly capture multidimensional effects.

While the general solution to transient heat conduction in the composite domain as the solution has been developed previously [28, 29], the solution to the periodic heating of one layer posed in this paper is a new application. In Appendix B, the solution of this study is compared to a numerical simulation in Appendix B with excellent agreement.

3.1 Results and discussions

The parameter values of Table 1 were used to determine the eigenvalues by the combined graphical bisection method described in Sect. 2. The first 30 eigenvalues of layer 1 are provided in Table 4 of Appendix C. In the numerical evaluation of solution (21), the series is truncated at $n = 30$. A simple analysis of the effect of the number of terms is presented in Appendix C.

The temperatures of the outer surfaces during the first 5 pulses are presented in Fig. 4. The temperature at these locations exhibit the characteristic sawtooth temperature profile and rapid approach to a periodic-steady state reported in [4]. Here, periodic-steady state occurs when the cantilever is operated steadily and after the initial start-up transient effects die out such that the temperature distribution at any instant in a cycle is nearly unchanged between periods:$T\left( {x,t} \right) \approx T\left( {x,t - P} \right)$. The largest difference in temperature between periods occurs at the layer 1–layer 2 interface and that at this location, the per cent difference between cycle 4 and 5 is less than 0.2% and between cycles 9 and 10, this difference drops to below $2 \times 10^{-4}$. In the numerical simulation [4], their solution also rapidly approaches a periodic-steady state so that by the 10th period, there is less than 1% difference between maximum cycle temperatures.

Fig. 4

The temperature at the outer surfaces of the cantilever: solid black corresponds to the heated outer surface (x = 0) and dashed red corresponds to the unheated outer surface (x = 3.7 µm)

In Fig. 4, the temperature of the outer surface of the heater (layer 1 at $x_{1} = 0$) is represented by the solid black line and the outer surface of the unheated Si layer 3 ($x_{3} = L_{3}$) is represented by the red dash-dot line). Beginning with the first heating cycle, the temperatures of both outer surfaces experience increases from the initial ambient temperature. The temperature profiles of both surfaces increase during the early phase of the cycle (when the heating is on) and once the heater is turned off (in the cooling phase) the outer surface temperatures decline rapidly. But before the outer surfaces cool to the initial ambient temperature (20 °C), the system is once more in the heating phase. The temperatures of the two surfaces are nearly identical for most of the cooling phase. However, during the heating phase, the temperature of the unheated outer surface of layer 3 does not reach the same maximum temperature as that of the heated surface and its profile is less steep. This is evidence of the high thermal resistance of the SiO₂ layer. During the heating, the differences in temperatures at the outer surfaces indicate large non-uniformity of temperature across the cantilever thickness (which is the focus of the current study and has been treated as negligible by previous studies).

Cross-sectional views of the cantilever at specified times are next presented to illustrate the spatial–temporal behaviour within the cantilever during the first pulse. The temperature profiles during the heating phase of the first pulse (the first 5 μs) are presented in Fig. 5. Here the layer interfaces are indicated by solid black vertical lines. The cantilever is initially at a uniform temperature (at the ambient temperature). The heated section rapidly increases in temperature. Because there is so little thermal resistance in the Al and Si layers, these profiles remain relatively uniform. However, the higher thermal resistance of the SiO₂ layer results in large temperature differences across this layer that exceed 40 °C within the first 1 μs and soon exceed 60 °C. The temperature rises of the unheated Si layer lag those of the heated layer and that is obviously due to the thermal resistance provided by the intermediate layer of SiO₂.

Fig. 5

Spatial temperature profiles at select times during the activation of the heating of the first period. The vertical lines correspond to the interface between layers

The temperature profiles during the first period at selected times of the cooling phase (when the heater is turned off) are presented in Fig. 6. The heated layer immediately cools from its peak temperature as thermal energy leaves by convection to the air and by conduction into the SiO₂ layer 2. Even though the heating is turned off, at early times of this phase the temperature of the Si base layer 3 continues to rise. This is evidence that at these times, the rate at which heat is conducted into the Si layer exceeds the rate of heat removal by convection. Again, the thermal behaviour of layer 3 lags behind that of the Al heater (layer 1) due to the high thermal resistance of the SiO₂ layer. The temperature gradients within the cantilever are short-lived and by 10 μs, the temperature profile within the cantilever is relatively uniform as it continues to cool. The cantilever does not return to its initial temperature (20 °C) by the end of the first period when the next cycle of heating will begin.

Fig. 6

Spatial temperature profiles at select times of the first period when the heater has been deactivated. The vertical lines correspond to the interface between layers

In its practical application, the self-heated micro-cantilever may be operated at periodic-steady state. The exact periodic-steady solution to this problem may be immediately presented by simply noting that the second term in the numerator of the last term of Eq. (21) disappears as $t \to \infty$. The periodic-steady spatial profile at representative times during the 5 μs duration of heating is presented in Fig. 7 where the times referenced correspond to the start of the period, the time at which the heating is activated. At the beginning of the period, the cantilever temperature is nearly uniform at around 53 °C. During the activation, the temperature of the heated layer 1 rapidly increases to its maximum temperature that now exceeds 180 °C. Again, the temperature profiles of layer 1 and layer 3 remain relatively uniform during the heating due to the high conductivity values of the Al and Si. The higher thermal resistance of the SiO₂ results in the noticeable temperature gradient within layer 2 so that the temperature rises of the unheated base layer 3 lag those of the heated layer and only reach about 120 °C by the end of the pulse.

Fig. 7

Periodic-steady solution spatial temperature profiles at select times during the 5 μs heating. The times referenced here correspond to the start of a period. The vertical lines correspond to the interface between layers

For periodic-steady operation temperature profiles at selected times of the cooling phase (when the heater is deactivated) are presented in Fig. 8. Again, the times cited here refer to the time after the start of the period during periodic-steady operation. The heated layer immediately cools from its peak temperature as some thermal energy leaves by convection to the air and some is conducted to the SiO₂ layer. As the heat is conducted into the Si base layer, its temperature again increases during the early stages of the cooling phase when the heater is turned off until it reaches nearly 140 °C. Again the thermal profile of the Si base layer lags behind that of the Al heater layer due to the high thermal resistance of the SiO₂ layer. The temperature gradients within the resistive layer 2 quickly drop so that at around 10 μs the temperature profile within the cantilever is nearly uniform as it continues to cool until it reaches the end of the period at a temperature of around 53 °C.

Fig. 8

Periodic-steady temperature profiles in the cooling phase following an applied pulse. The time referenced here corresponds to the time after the application of a pulse during periodic-steady operation. The vertical lines correspond to the interface between layers

Numerical values of the temperatures within the cantilever at the outer boundaries and at the layer interfaces during the first 10 μs of a single period at periodic-steady operation are presented in Table 2. At all times, the temperature differences across layer 1 are less than 0.5 °C and the temperature differences across layer 3 are always less than 0.6 °C. The descriptions of the periodic-steady temperature profile of this three-layer cantilever can be characterized only by the temperatures of the outer surfaces (as implied by Fig. 4 and quantitatively presented in the last column of Table 2).

Table 2 Temperature values (°C) at the layer boundaries during the first 10 μs of a period during periodic-steady operation with temperature differences between heated and unheated boundary on the right

The principle underlying the bending of the self-heated AFM cantilever is that the unmatched thermal expansion coefficients of the different layers result in unequal expansion between layers and thus bending. A detailed mathematical description of the mechanical model of the thermally actuated bending of the composite micro-cantilever is provided in [17]. While the current study does not model the mechanical dynamics of the cantilever, the thermal solution developed here does provide insight into the anticipated bending dynamics. The material choices of each layer in the current study were informed by those of [10, 17]. The thermal expansion coefficient of the Al in layer 1 at 23 × 10^–6 K⁻¹ is at least an order of magnitude higher than those of the other layers (the expansion coefficient of SiO₂ used in layer 2 is 0.5 × 10^–6 K⁻¹ and that of Si used in layer 3 is 2.6 × 10^–6 K⁻¹.

The temperatures within the cantilever (Figs. 4, 5, 6, 7, 8) coupled with this drastic mismatch in thermal expansion coefficients are anticipated to result in bending and tip deflection. Note that in the deflection study of [10] the authors predict noticeable deflection for this cantilever composition at temperatures below those in the current study (for the pulse parameters of that study they find a temperature rise of nearly 80 °C above ambient. Because the study of [10] is steady, the temperature change resulting in the deflection is with respect to the initial (ambient) temperature. However, in the periodic-steady operation of the pulsed periodic heating of the cantilever, the deflection associated with the temperature rise should be considered relative to the minimum cycle temperature and not relative to the ambient or the initial temperatures. Recalling the results of Fig. 4 and then of Fig. 7, the minimum cycle temperature occurs at the end of the cooling phase and just before the application of the heating of the next cycle. That corresponds to t = 0 in Table 2, for which the temperature within the cantilever is nearly uniform at 53 °C (this is 33 °C above the ambient temperature).

Due to the relatively high thermal resistance of layer 2, and because the heating occurs only in layer 1, during most of the heating phase, the Al heater of layer 1 temperature exceeds that of the Si of layer 3 by more than 50 °C and that temperature difference plateaus at nearly 63 °C (see the last column of Table 2). Because that temperature difference is positively biased towards the layer with the order of magnitude larger thermal expansion coefficient, this short-lived temperature difference across the cantilever layers will act to further enhance bending and flapping.

While this temperature difference is strongly dependent on the thermal properties of the cantilever layers, it is also influenced by the heating characteristics of the actuator. Solution (21) can illustrate how this temperature difference is influenced by the period length, $P$, and the heating duration, $\tau_{P}$, (also represented by frequency and duty cycle). The parameters characterizing the heating only appear in the transient component of the solution and that this component is identical for all layers. At periodic-steady state, this component may be approximated as:

$$ \begin{gathered} \sum\limits_{m = - \infty }^{\infty } {c_{1,m} \frac{{\exp \left[ {\left( { - {\kern 1pt} {{2\pi {\text{i}}{\kern 1pt} \cdot m} \mathord{\left/ {\vphantom {{2\pi {\text{i}}{\kern 1pt} \cdot m} P}} \right. \kern-0pt} P}} \right)t} \right]}}{{{{2\pi {\text{i}}{\kern 1pt} \cdot m} \mathord{\left/ {\vphantom {{2\pi {\text{i}}{\kern 1pt} \cdot m} P}} \right. \kern-0pt} P} + {{\mu_{1,n}^{2} \alpha_{1} } \mathord{\left/ {\vphantom {{\mu_{1,n}^{2} \alpha_{1} } {L_{1}^{2} }}} \right. \kern-0pt} {L_{1}^{2} }}}}} \quad \hfill \\ {\kern 1pt} c_{1,m} = g_{0} \frac{1}{{2\pi {\text{i}}{\kern 1pt} \cdot m}}\left( {1 - \exp \left( { - 2\pi {\text{i}}{\kern 1pt} \cdot m{\kern 1pt} {{\tau_{P} } \mathord{\left/ {\vphantom {{\tau_{P} } P}} \right. \kern-0pt} P}} \right)} \right)\quad m \ne 0 \hfill \\ c_{1,0} = g_{0} \frac{{\tau_{P} }}{P}\quad m = 0. \hfill \\ \end{gathered} $$

(22)

Each term of the summation of Eq. (22) increases with ${{g_{0} \tau_{P} } \mathord{\left/ {\vphantom {{g_{0} \tau_{P} } P}} \right. \kern-0pt} P}$, which means that increasing the heating power, increasing the duration of the heating, or decreasing the period length (increasing the frequency of heating) all act to increase the temperature. The summation of Eq. (22) acts to magnify the spatial solution component equally (regardless of position). Thus, the temperature rises at different locations have a very similar dependence on ${{g_{0} \tau_{P} } \mathord{\left/ {\vphantom {{g_{0} \tau_{P} } P}} \right. \kern-0pt} P}$, and this has implications regarding the temperature gradient across the cantilever. If the temperatures throughout the cantilever behave similarly with changes to ${{g_{0} \tau_{P} } \mathord{\left/ {\vphantom {{g_{0} \tau_{P} } P}} \right. \kern-0pt} P}$, then the resulting temperature differences between locations will not be strongly influenced by these changes. This is illustrated in a pair of very simple parametric analysis of Eq. (21). In the first analysis, the rate of heating and the heating duration are held constant (at the values of the case study) while the period duration is decreased from 100 to 10 μs. The solution to Eq. (21) is evaluated at the outer surfaces at steady-periodic operation and the difference, $\Delta T\left( t \right) = T_{1} \left( {x_{1} = 0,t} \right) - T_{3} \left( {x_{3} = L_{3} ,t} \right)$, is plotted in Fig. 9a. To make direct comparisons, the time has been normalized by the period duration and the time shown begins at the beginning of a period. While the frequency has been increased by an order of magnitude, the magnitude of the temperature difference across the cantilever only decreases slightly. However, the magnitudes of these temperature differences must also be considered in the context of the actual cantilever temperatures. Recall that the temperatures throughout the cantilever are expected to increase with decreasing period duration. The temperature of the outer surface of the heater (at $x = 0$) is evaluated under periodic-steady conditions and plotted in Fig. 9b). While it is unrealistic that in practice the heater would be operated at such high temperatures (the operation is limited by the melting point of the heater), the trends of the analysis still hold. Any effect of these short-lived temperature differences on deflection will be reduced with increasing cantilever temperature, and thus with decreasing period length. Since the bending is proportional to the temperature, the influence of the temperature difference on bending will diminish as the temperature of the cantilever increases. Thus, the temperature differences across the cantilever that this study describes must be considered in the context of the cantilever overall temperature.

Fig. 9

a Temperature difference across the cantilever and b the temperature values at the heated surface (x = 0). The values are at steady-periodic operation at different period durations. All other parameter values are those of Table 1. The time has been normalized by period duration and begins at the start of a period

In a second parametric analysis, the period length is held constant (at 100 µs), but the heating duration and heating rate are both varied such that the total heat addition per cycle is held constant $g_{0} \tau_{P} = 10^{ - 6}$ mJ/μm³. In this case, both the temperature difference across the cantilever (Fig. 10a) and the heater outer surface temperature (Fig. 10b) decrease with increasing duty cycle (and decreased heating rate). In this case it is clear that, at least for these parameters, any influence on bending associated with the temperature difference across the cantilever is reduced with increasing duty cycle. While in practice it would not make sense to operate the heater to result in such low temperature rises that are seen in Fig. 10b, the parametric study does indicate that the temperature differences described in this study must always be considered in the context of operation.

Fig. 10

a Temperature difference across the cantilever and b Temperature values at the heated surface (x = 0). The values are at steady-periodic operation at different heating durations and magnitudes such that the quantity τ_Pg₀ is constant. All other parameter values are those of Table 1. The time begins at the start of a period

Note that the thermal gradients along the cantilever thickness are derived from the one-dimensional solution and this does not directly capture conduction along the cantilever length. In fact, it has been previously suggested that the anchor end of the cantilever acts as a type of heat sink [4]. To account for this simplification, the present study uses a heat transfer coefficient one order of magnitude higher than that described elsewhere. The solution of this study could next be extended to the 2-D domain as has been done in other exact solution studies in the composite domain [37, 38]. Additionally, the solution developed in this study could be used in parametric investigations to determine under which conditions the thermal gradients across the thickness of the cantilever are likely to be influential in the subsequent bending.

4 Conclusions

The exact integral solution to the one-dimensional problem of heat conduction in the composite domain subjected to transient volumetric heating and convection at the outer boundaries is developed using the appropriate Green’s function. The solution is used to investigate the transient temperature profile across the thickness of a self-heated micro-cantilever. A case study is conducted in a 3-layer cantilever using parameter values relevant to those found in both experiment and computational studies. It is shown that the cantilever reaches a periodic-steady state within 5 periods. In the parameters used in this case study, there exist non-negligible temperature differences during the heating phase and the very early stages of the cooling phase. These short-lived temperature differences within the cantilever are positively biased with the unmatched thermal expansion coefficients of the layers. However, the heating parameters of this case study resulted in a small duty cycle. A preliminary parametric investigation on heating characteristics is conducted using the model that indicates that this temperature difference decreases with increasing duty cycle. Future work should include systematic and more comprehensive parametric analysis that includes the heating characteristics as well as the thermophysical parameter values to determine the conditions under which these temperature differences are non-negligible. The one-dimensional exact solution presented in this paper does not capture the thermal profile along the cantilever length. The one-dimensional solution of this study should be extended to two dimensions accounting for conduction along both the length of the cantilever and the cantilever thickness.

Appendices

Appendix A: Green’s function derivation

The determination of the Green’s function of heat conduction in an M-layered composite slab follows the approach described in chapter 4.4 of the book [31] (and more recently in [32]) and in chapter 10.6 of the book [28]. This approach uses the Separation of Variables method to determine the integral solution to the homogenous initial value problem of heat conduction in the composite slab. The Green’s function may then be inferred from the Green’s function solution equation (to the homogenous problem):

$$ T_{i} \left( {x_{i} ,t} \right) = \sum\limits_{j = 1}^{M} {\int_{{x^{\prime}_{j} = 0}}^{L} {\left. {G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right)} \right|_{\tau = 0} F_{j} \left( {x^{\prime}_{j} } \right){\text{d}}x^{\prime}_{j} \, } } 0 < x_{i} < L_{i} ,\quad t > 0\quad i = 1,2...,M. $$

(A1)

The homogenous problem of one-dimensional conduction in the composite slab is represented:

$$ \frac{1}{{\alpha_{i} }}\frac{{\partial T_{i} }}{\partial t} = \frac{{\partial^{2} T_{i} }}{{\partial x_{i}^{2} }}. $$

(A2)

This is subject to homogenous boundary conditions:

$$ \begin{gathered} \left. {\frac{{\partial T_{1} }}{{\partial x_{1} }}} \right|_{{x_{1} = 0}} = \frac{{h_{1} }}{{k_{1} }}\left( {\left. {T_{i} } \right|_{{x_{1} = 0}} } \right) \hfill \\ \left. {\frac{{\partial T_{M} }}{{\partial x_{M} }}} \right|_{{x_{M} = L_{M} }} = - \frac{{h_{M} }}{{k_{M} }}\left( {\left. {T_{M} } \right|_{{x_{M} = L_{M} }} } \right). \hfill \\ \end{gathered} $$

(A3)

The temperature and heat fluxes between layers are related to one another through interface conditions:

$$ \left. \begin{gathered} k_{i} \left. {\frac{{\partial T_{i} }}{{\partial x_{i} }}} \right|_{{x_{i} = 0}} = \left. {k_{i - 1} \frac{{\partial T_{i - 1} }}{{\partial x_{i - 1} }}} \right|_{{x_{i - 1} = L_{i - 1} }} \hfill \\ T_{i} \left( {0,t} \right) = T_{i - 1} \left( {L_{i - 1} ,t} \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3,...,M. $$

(A4)

And the initial condition of each layer is represented by some arbitrary function:

$$ T_{i} \left( {x_{i} ,0} \right) = F_{i} \left( {x_{i} } \right) \, 0 < x_{i} < L_{i} \quad t = 0\quad i = 1,2...,M. $$

(A5)

In order to simplify the development of the solution, the following scaled variables are introduced:

$$ \begin{gathered} \overline{x}_{i} = x_{i} /L_{i} \quad \overline{k}_{i} = k_{i} /L_{i} \quad \overline{\alpha }_{i} = \alpha_{i} /L_{i}^{2} \hfill \\ H_{1} = h_{1} L_{1} /k_{1} \quad H_{M} = h_{M} L_{M} /k_{M} . \hfill \\ \end{gathered} $$

(A6)

The governing equation to the homogenous initial value problem is now:

$$ \frac{1}{{\overline{\alpha }_{i} }}\frac{{\partial T_{i} }}{\partial t} = \frac{{\partial^{2} T_{i} }}{{\partial \overline{x}_{i}^{2} }} \, 0 < \overline{x}_{i} < 1. $$

(A7)

Subject to associated boundary and initial conditions:

$$ \begin{gathered} \left. {\frac{{\partial T_{1} }}{{\partial \overline{x}_{1} }}} \right|_{{\overline{x}_{1} = 0}} = H_{1} T_{1} \left( {0,t} \right) \hfill \\ \left. {\frac{{\partial T_{M} }}{{\partial \overline{x}_{M} }}} \right|_{{\overline{x}_{M} = 1}} = - H_{M} T_{M} \left( {1,t} \right). \hfill \\ \end{gathered} $$

(A8)

and initial condition:

$$ T_{i} \left( {\overline{x}_{i} ,0} \right) = F_{i} \left( {\overline{x}_{i} } \right) \, 0 < \overline{x}_{i} < 1\quad t = 0\quad i = 1,2...,M. $$

(A9)

and interface conditions:

$$ \left. \begin{gathered} \overline{k}_{i} \left. {\frac{{\partial T_{i} }}{{\partial \overline{x}_{i} }}} \right|_{{\overline{x}_{i} = 0}} = \left. {\overline{k}_{i - 1} \frac{{\partial T_{i - 1} }}{{\partial \overline{x}_{i - 1} }}} \right|_{{\overline{x}_{i - 1} = 0}} \hfill \\ T_{i} \left( {0,t} \right) = T_{i - 1} \left( {1,t} \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3, \ldots, M. $$

(A10)

Anticipating the SOV solution, the solution in each layer is represented as the product of a transient term, $\Gamma_{i} \left( t \right)$, and a spatially dependent term, $\overline{X}_{i} \left( {\overline{x}_{i} } \right)$:

$$ T_{i} \left( {\overline{x}_{i} ,t} \right) = \overline{X}_{i} \left( {\overline{x}_{i} } \right)\Gamma_{i} \left( t \right). $$

(A11)

Substituting into the governing equation (A7) yields the expression:

$$ \frac{1}{{\overline{\alpha }_{i} }}\frac{{\Gamma^{\prime}_{i} }}{{\Gamma_{i} }} = \frac{{\overline{X}^{\prime\prime}_{i} }}{{\overline{X}_{i} }} = - \mu_{i}^{2} \, 0 < \overline{x}_{i} < 1\quad t > 0. $$

(A12)

where the prime and double prime denote the first and second ordinary derivatives and $\mu_{i}$ is the eigenvalue of layer $i$ that remains to be determined. This expression has an infinite number of solutions so that for each eigenvalue:

$$ \frac{1}{{\overline{\alpha }_{i} }}\frac{{\Gamma^{\prime}_{i.n} }}{{\Gamma_{i,n} }} = \frac{{\overline{X}^{\prime\prime}_{i,n} }}{{\overline{X}_{i,n} }} = - \mu_{i,n}^{2} \, i = 1,2, \ldots M{ , }n = 1,2, \ldots \infty . $$

(A13)

The solution (A11) may be represented as:

$$ T_{i} \left( {\overline{x}_{i} ,t} \right) = \sum\limits_{n = 1}^{\infty } {\overline{X}_{i,n} \left( {\overline{x}_{i} } \right)\Gamma_{i,n} \left( t \right)} . $$

(A14)

From Eq. (A13), the spatial solution component is satisfied by:

$$ \overline{X}_{i,n} \left( {\overline{x}_{i} } \right){ = }a_{i,n} \sin \left( {\mu_{i,n} \overline{x}_{i} } \right) + b_{i,n} \cos \left( {\mu_{i,n} \overline{x}_{i} } \right), $$

(A15)

and the transient component is satisfied by:

$$ \Gamma_{i,n} = C_{i,n} \exp \left[ { - \left( {\mu_{i,n}^{2} \overline{\alpha }_{i} } \right)t} \right], $$

(A16)

where $a_{i,n}$, $b_{i,n}$, and $C_{i,n}$ are constants that need to be determined.

When the boundary condition (A8) of layer 1 at $\overline{x}_{1} = 0$ is implemented, the constants of layer 1 are related to each other as:

$$ b_{1,n} = \frac{{\mu_{1,n} }}{{H_{1} }}a_{1,n} . $$

(A17)

When (A16) and (A15) are substituted into (A14), both $a_{i,n}$ and $b_{i,n}$ are multiplied by $C_{i,n}$ so that without loss of information, the constants $a_{i,n}$ of layer 1 may all be arbitrarily set. Setting $a_{1,n} = H_{1}$. In Eq. (A17) $b_{1,n} = \mu_{1,n}$, the Eigen function (A15) of layer 1 is:

$$ \overline{X}_{1,n} \left( {\overline{x}_{1} } \right){ = }H_{1} \sin \left( {\mu_{1,n} \overline{x}_{1} } \right) + \mu_{1,n} \cos \left( {\mu_{1,n} \overline{x}_{1} } \right) \, {.} $$

(A18)

Because interface conditions (A10) are satisfied at all times, they will only be satisfied as long as the transient solution component (A16) of all layers are equivalent:

$$ \Gamma_{i,n} = \Gamma_{i - 1,n} \quad t > 0\quad i = 2,3, \ldots ,M, $$

(A19)

which considering (A16) has the two obvious implications that:

$$ \mu_{i,n}^{2} \overline{\alpha }_{i} = \mu_{i - 1,n}^{2} \overline{\alpha }_{i - 1} \quad \quad i = 2,3, \ldots M, $$

(A20)

and

$$ C_{i,n} = C_{i - 1,n} = C_{n} . $$

(A21)

The second result of the interface conditions is that in layers $i = 2 \ldots M$, the constants $a_{i,n}$ and $b_{i,n}$ may be related to those of the previous $\left( {i - 1} \right)$ layer:

$$ \left. \begin{gathered} b_{i,n} = \overline{X}_{i,n} \left( 0 \right) = \overline{X}_{i - 1,n} \left( 1 \right) \hfill \\ a_{i,n} = \frac{{\overline{X}^{\prime}_{i,n} \left( 0 \right)}}{{\mu_{i,n} }} = \frac{{\overline{k}_{i - 1} }}{{\overline{k}_{i} }}\frac{1}{{\mu_{i,n} }}\overline{X}^{\prime}_{i - 1,n} \left( 1 \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3, \ldots ,M. $$

(A22)

The outer boundary condition may then be implemented to determine the eigenvalues:

$$ \overline{X}^{\prime}_{M,n} \left( 1 \right) = - H_{M} \overline{X}_{M,n} \left( 1 \right). $$

(A23)

The temperature distribution in each layer may now be represented:

$$ T_{i} \left( {\overline{x}_{i} ,t} \right) = \sum\limits_{n = 1}^{\infty } {C_{i,n} \overline{X}_{i,n} \left( {\overline{x}_{i} } \right)\exp \left[ { - \left( {\mu_{i,n}^{2} \overline{\alpha }_{i} } \right)t} \right]} \quad i = 1,2 \ldots ,M. $$

(A24)

Substituting (A24) into the initial condition (A9) yields:

$$ \sum\limits_{n = 1}^{\infty } {C_{n} \overline{X}_{j,n} \left( {\overline{x}_{j} } \right)} = F_{j} \left( {\overline{x}_{j} } \right)\quad \quad j = 1,2, \ldots, M. $$

(A25)

, which is evaluated by relying on the orthogonality condition and the Fourier series representation of the initial condition. The value of the constant is determined by multiplying (A25) by ${{\overline{k}_{i} } \mathord{\left/ {\vphantom {{\overline{k}_{i} } {\overline{\alpha }_{i} }}} \right. \kern-0pt} {\overline{\alpha }_{i} }}X_{i,m} \left( {\overline{x}_{i} } \right)$ integrating over the entire domain:

$$ \sum\limits_{j = 1}^{M} {\frac{{\overline{k}_{j} }}{{\overline{\alpha }_{j} }}\left[ {\sum\limits_{n = 0}^{\infty } {C_{n} \int_{{\overline{x}_{j} = 0}}^{1} {\overline{X}_{j,n} \left( {\overline{x}_{j} } \right)\overline{X}_{j,m} \left( {\overline{x}_{j} } \right){\text{d}}\overline{x}_{j} } } } \right]} = \sum\limits_{j = 1}^{M} {\frac{{\overline{k}_{j} }}{{\overline{\alpha }_{j} }}\left[ {\int_{{\overline{x}_{j} = 0}}^{1} {F_{j} \left( {\overline{x}_{j} } \right)\overline{X}_{j,m} \left( {\overline{x}_{j} } \right){\text{d}}\overline{x}_{j} \, } } \right]} . $$

(A26)

The expression on the RHS is the orthogonality condition of the problem of linear diffusion in the composite domain which states that:

$$ \sum\limits_{j = 1}^{M} {\frac{{\overline{k}_{j} }}{{\overline{\alpha }_{j} }}\left[ {\int_{{\overline{x}_{j} = 0}}^{1} {\overline{X}_{j,n} \left( {\overline{x}_{j} } \right)\overline{X}_{j,m} \left( {\overline{x}_{j} } \right){\text{d}}\overline{x}_{j} } } \right]} = \left\{ \begin{gathered} 0\quad n \ne m \hfill \\ N_{n} \quad n = m \hfill \\ \end{gathered} \right., $$

(A27)

where the norm, $N_{n}$, may be represented as:

$$ N_{n} = \sum\limits_{i = 1}^{M} {\frac{{\overline{k}_{i} }}{{\overline{\alpha }_{i} }}\frac{1}{{2\mu_{i,n}^{2} }}} \left( \begin{gathered} \overline{X}^{\prime}_{i,n} \left( 0 \right)\overline{X}_{i,n} \left( 0 \right) - \overline{X}^{\prime}_{i,n} \left( 1 \right)\overline{X}_{i,n} \left( 1 \right) \hfill \\ + \left( {\mu_{i,n} \overline{X}_{i,n} \left( 0 \right)} \right)^{2} + \overline{X^{\prime}}_{i,n} \left( 0 \right)^{2} \hfill \\ \end{gathered} \right). $$

(A28)

Expressions (A27) and (A28) are well known and they have been developed with different notation previously. They are described in detail with the same notation used in this paper in the appendices of [25, 26].

Rearranging (A26) the integration constant is represented in integral form by:

$$ C_{n} = \frac{1}{{N_{n} }}\sum\limits_{j = 1}^{M} {\frac{{\overline{k}_{j} }}{{\overline{\alpha }_{j} }}\left[ {\int_{{\overline{x}_{j} = 0}}^{1} {F_{j} \left( {\overline{x}_{j} } \right)\overline{X}_{j,n} \left( {\overline{x}_{j} } \right){\text{d}}\overline{x}_{j} \, } } \right]} . $$

(A29)

The dimensionless expression of the eigenfunctions of Eq. (A15) that are used in the evaluation of the norm in (A28) and in building the function (A23) used to determine the eigenvalues is very convenient because they are only evaluated at $\overline{x}_{i} = 0,1$. Because this function is seen throughout the solution, it is suggested here to retain it.

Alternately evaluating the function (A15) and its first derivative at $\overline{x}_{i} = 0,1$ yields:

$$ \begin{gathered} \overline{X}_{i,n} \left( {0_{i} } \right){ = }b_{i,n} \, \hfill \\ \overline{X}^{\prime}_{i,n} \left( 0 \right){ = }\mu_{i,n} a_{i,n} \hfill \\ \overline{X}_{i,n} \left( 1 \right){ = }a_{i,n} \sin \left( {\mu_{i,n} } \right) + b_{i,n} \cos \left( {\mu_{i,n} } \right) \, \hfill \\ \overline{X}^{\prime}_{i,n} \left( 1 \right){ = }\mu_{i,n} \left( {a_{i,n} \cos \left( {\mu_{i,ni} } \right) - b_{i,n} \sin \left( {\mu_{i,n} } \right)} \right). \hfill \\ \end{gathered} $$

(A30)

Substituting these into (A28) and rearranging yields:

$$ N_{n} = \sum\limits_{i = 1}^{M} {\frac{{\overline{k}_{i} }}{{\overline{\alpha }_{i} }}\frac{1}{{2\mu_{i,n}^{2} }}} \left( \begin{gathered} 2\mu_{i,n} a_{i,n} b_{i,n} \sin^{2} \left( {\mu_{i,n} } \right) + \mu_{i,n}^{2} \left( {a_{i,n}^{2} + b_{i,n}^{2} } \right) \hfill \\ - \mu_{i,n} \left( {a_{i,n}^{2} - b_{i,n}^{2} } \right)\sin \left( {\mu_{i,n} } \right)\cos \left( {\mu_{i,n} } \right) \hfill \\ \end{gathered} \right). $$

(A31)

Substituting this into (A24), the integral solution to the initial value problem may be represented:

$$ \begin{gathered} T_{i} \left( {\overline{x}_{i} ,t} \right) = \sum\limits_{j = 1}^{M} {\left[ {\int_{{\overline{x^{\prime}}_{j} = 0}}^{1} {\left\{ {\sum\limits_{n = 1}^{\infty } {\frac{{\overline{k}_{j} }}{{\overline{\alpha }_{j} }}\frac{1}{{N_{n} }}\exp \left[ { - \left( {\mu_{i,n}^{2} \overline{\alpha }_{i} } \right)t} \right]\overline{X}_{i,n} \left( {\overline{x}_{i} } \right)\overline{X}_{j,n} \left( {\overline{x^{\prime}}_{j} } \right)} } \right\}F_{j} \left( {\overline{x^{\prime}}_{j} } \right){\text{d}}\overline{x^{\prime}}_{jj} } } \right]} . \hfill \\ \, \hfill \\ \end{gathered} $$

(A32)

Expressing the dimensionless variables and parameters of (A32) in their dimensional form, Eq. (A32) may be rewritten as:

$$ \begin{gathered} T_{i} \left( {x_{i} ,t} \right) = \sum\limits_{j = 1}^{M} {\left[ {\int_{{x^{\prime}_{j} = 0}}^{{L_{j} }} {\left\{ {\sum\limits_{n = 1}^{\infty } {\frac{{k_{j} }}{{\alpha_{j} }}\frac{1}{{N_{n} }}\exp \left[ { - \left( {\mu_{i,n}^{2} \frac{{\alpha_{i} }}{{L_{i}^{2} }}} \right)t} \right]X_{i,n} \left( {x_{i} } \right)X_{j,n} \left( {x^{\prime}_{j} } \right)} } \right\}F_{j} \left( {x^{\prime}_{j} } \right){\text{d}}x^{\prime}_{j} } } \right]} . \hfill \\ \, \hfill \\ \end{gathered} $$

(A33)

For which

$$ X_{i,n} \left( {x_{i} } \right){ = }a_{i,n} \sin \left( {\mu_{i,n} x_{i} } \right) + b_{i,n} \cos \left( {\mu_{i,n} x_{i} } \right) \, {.} $$

(A34)

Comparing the integral solution (A33) with the green’s function solution (A1), it becomes immediately apparent that the green’s function to the composite slab may be represented as:

$$ \begin{gathered} G_{i,j} \left( {\left. {x_{i} ,t} \right|x^{\prime}_{j} ,\tau } \right) = \sum\limits_{n = 0}^{\infty } {\frac{{k_{j} }}{{\alpha_{j} }}\frac{1}{{N_{n} }}\left[ {a_{i,n} \sin \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right) + b_{i,n} \cos \left( {\mu_{i,n} \frac{{x_{i} }}{{L_{i} }}} \right)} \right]} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \times \left[ {a_{j,n} \sin \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right) + b_{j,n} \cos \left( {\mu_{j,n} \frac{{x^{\prime}_{j} }}{{L_{j} }}} \right)} \right] \, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \times \exp \left[ { - \frac{{\mu_{i,n}^{2} \alpha_{i} }}{{L_{i}^{2} }}\left( {t - \tau } \right)} \right]. \hfill \\ \end{gathered} $$

(A35)

For which the constants may be represented:

$$ \begin{gathered} \, \hfill \\ a_{1,n} = h_{1} L_{1} /k_{1} \quad \hfill \\ b_{1,n} = \mu_{1,n} \hfill \\ \left. \begin{gathered} b_{i,n} = \overline{X}_{i - 1,n} \left( 1 \right) \hfill \\ a_{i,n} = \frac{{L_{i} }}{{L_{i - 1} }}\frac{{k_{i - 1} }}{{k_{i} }}\frac{1}{{\mu_{i,n} }}\overline{X^{\prime}}_{i - 1,n} \left( 1 \right) \hfill \\ \end{gathered} \right\}\quad \quad i = 2,3,...,M \hfill \\ N_{n} = \sum\limits_{i = 1}^{M} {\frac{{k_{i} L_{i} }}{{\alpha_{i} }}\frac{1}{{2\mu_{i,n}^{2} }}} \left( \begin{gathered} \overline{X}^{\prime}_{i,n} \left( 0 \right)\overline{X}_{i,n} \left( 0 \right) - \overline{X}^{\prime}_{i,n} \left( 1 \right)\overline{X}_{i,n} \left( 1 \right) \hfill \\ + \left( {\mu_{i,n} \overline{X}_{i,n} \left( 0 \right)} \right)^{2} + \overline{X^{\prime}}_{i,n} \left( 0 \right)^{2} \hfill \\ \end{gathered} \right). \hfill \\ \end{gathered} $$

(A36)

Recalling Eq. (A23), the eigenvalues, $\mu_{i,n}$, are determined from:

$$ \overline{X}^{\prime}_{M,n} \left( 1 \right) = - \overline{X}_{M,n} \left( 1 \right) \cdot h_{M} L_{M} /k_{M} \quad n = 0...\infty . $$

(A37)

When Eq. (A30) is substituted for $\overline{X}_{i,n} \left( {\overline{x}_{i} } \right)$ and $\overline{X}^{\prime}_{i,n} \left( {\overline{x}_{i} } \right)$ evaluated at $\overline{x} = 0,1$, Eq. (8) may be expressed instead as:

$$ \begin{gathered} a_{1,n} = h_{1} L_{1} /k_{1} \quad \hfill \\ b_{1,n} = \mu_{1,n} \hfill \\ \left. \begin{gathered} b_{i,n} = \,a_{i - 1,n} \sin \left( {\mu_{i - 1,n} } \right) + b_{i - 1,n} \cos \left( {\mu_{i - 1,ni} } \right) \hfill \\ a_{i,n} = \frac{{L_{i} }}{{L_{i - 1} }}\frac{{k_{i - 1} }}{{k_{i} }}\frac{{\mu_{i - 1,n} }}{{\mu_{i,n} }}\left[ {a_{i - 1,n} \cos \left( {\mu_{i - 1,n} } \right) - b_{i - 1,n} \sin \left( {\mu_{i - 1,n} } \right)} \right] \hfill \\ \end{gathered} \right\}\quad \quad i = 2, \ldots ,M. \hfill \\ N_{n} = \frac{1}{2}\sum\limits_{i = 1}^{M} {\frac{{k_{i} L_{i} }}{{\alpha_{i} }}\frac{1}{{\mu_{i,n} }}} \left( \begin{gathered} 2a_{i,n} b_{i,n} \sin^{2} \left( {\mu_{i,n} } \right) + \mu_{i,n} \left( {a_{i,n}^{2} + b_{i,n}^{2} } \right) \hfill \\ - \left( {a_{i,n}^{2} - b_{i,n}^{2} } \right)\sin \left( {\mu_{i,n} } \right)\cos \left( {\mu_{i,n} } \right) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(A38)

and (A37) may be rewritten so that the eigenvalues are evaluated from the roots of:

$$ \mu_{M,n} \left( {a_{M,n} \left( {\mu_{M,i} } \right) - b_{M,n} \tan \left( {\mu_{M,n} } \right)} \right) - \frac{{h_{M} L_{M} }}{{k_{M} }}\left( {a_{M,n} \tan \left( {\mu_{M,n} } \right) + b_{M,n} \left( {\mu_{M,n} } \right)} \right)\;{ = }\;{0}{\text{.}} $$

(A39)

Appendix B: Comparison with numerical simulation

It is well understood that the numerical solution is only an approximation of the analytical solution prediction. Even for a relatively straightforward problem presented in the case study, the implementation of exact Solution (21) is complicated and, in its evaluation, there are many opportunities for transcription and typographical error. Perhaps this is one of the reasons that the evaluation of such problems using commercially available software has become popular in recent years. While it is intended to be neither a validation nor a verification, in this section a comparison is made between Solution (21) and a numerical simulation executed on commercially available software.

The governing equation (1), boundary conditions (2), and initial conditions (3) are applied to the three-layer cantilever subject to the periodic pulsed heating and the associated parameter values described in Sect. 3 are evaluated numerically in the commercially available package Comsol 6.1 by the heat transfer in solids physics module. The periodic heat source in layer 1 was represented by a square waveform function with no smoothing. The “extremely fine physics controlled” mesh was used resulting in a grid of 101 nodes of 3.6842E−2 µm spacing within layer 1, and of 3.7037E−2 µm spacing within layers 2 and 3. The implicit backward differentiation formulation with a maximum order of 3 and a minimum order of 1 was used and time stepping was conducted with a strict constant maximum step size of 5E−2 µs.

The major temperature gradient occurs across the SiO₂ layer 2 so that, as explained in Sect. 3.1, the outer surface temperatures (at x = 0 and at 3.7 µm) characterize the thermal profile of the cantilever. The temperatures at these locations are used to compare the exact and numerical solution during the first 5 pulses. The exact solution results at these locations and times have been presented in Fig. 4. The absolute per cent difference between the numerical and the exact solution is presented in Fig. 11. There is excellent agreement between these overall. The magnitude of the difference decreases with pulse number and as the system approaches periodic-steady operation. There are 4 minor instantaneous temperature differences (less than 3% at the heated surface (x = 0) which occur at the start of pulses 2–5. The numerical time stepping is variable so its solution is not evaluated explicitly at these times (t = 0.1, 0.2, 0.3, 0.4 ms) and this difference is indicative of the numerical interpolation from just before and just after these times (hence a slightly larger value in the numerical solution).

Next, a comparison is made at periodic-steady operation. The numerical model is run for the first 25 periods. The numerical solution at the first ten µs of the 25th period (beginning at 24 ms) has been listed at specified locations in Table 3 and ,in parentheses, is the per cent difference from the exact periodic-steady benchmark solution presented in Table 2. There is excellent agreement at most times, though at the instant the heater has been turned on, the numerical solution of the heater layer (layer 1) is 2% above the analytical solution. The software did not provide the numerical solution at this instant and the numerical value at this instant is found by interpolating in time from just before and just after the heater is turned on.

Appendix C: Eigenvalues and the series truncation

Using the parameter values of the case study, the left-hand side of Eq. (9) was used in a combined graphical bisection method to determine each value of the eigenvalues, $\mu_{1,n}$. The first thirty eigenvalues resulting from the parameter values used in the case study are presented in Table 4.

In the case study of this paper, Eq. (21) was truncated at 30 terms. A simple analysis of the behaviour of the solution at different truncations is presented here. Equation (21) was evaluated for different maximum number of terms (nm = 2, 5, 10, 30, and 100) at specified times in the first period. At 0.01 μs, there are oscillations evident in the central layer when the solution is truncated at 10 or fewer terms (Fig. 12, left panel). Of course the effect of the exponential term of Eq. (21) is to dampen such oscillations with increasing time, so that by 0.1 μs these oscillations are no longer obvious at nm = 10 (Fig. 12, right panel) though they persist for nm = 5. The numerical values of the results at $x = 1.2$ μm and at times up to 1 are presented to 6 digits in Table 5. Truncating the series at nm = 30 terms has five-digit agreement with that at nm = 100 terms at this location by 1.0 μs.

Table 3 Numerical Solution Temperature values (°C) at specified locations and at times following the start of the 25th pulse (24 ms)

Table 4 The first 30 Eigenvalues of layer 1 corresponding to the parameter values listed in Table 1

Fig. 11

Per cent difference (absolute) between numerical and exact solutions at the outer surfaces during the first 5 pulses

Fig. 12

Temperature profile within the cantilever in the first period at 0.01 μs (left) and at 0.1 μs when Eq. (21) is truncated at n = nm terms

Table 5 Temperature values (°C) at x = 1.2 μm at specified times resulting from truncation of the series at different numbers of terms (nm)