Bulk acoustic wave piezoelectric micropumps with stationary flow rectifiers: a three-dimensional structural/fluid dynamic investigation

Transcription

1 Microfluid Nanofluid (2015) 18: DOI /s RESEARCH PAPER Bulk acoustic wave piezoelectric micropumps with stationary flow rectifiers: a three-dimensional structural/fluid dynamic investigation Ersin Sayar Bakhtier Farouk Received: 26 January 2014 / Accepted: 7 June 2014 / Published online: 15 June 2014 Ó Springer-Verlag Berlin Heidelberg 2014 Abstract Coupled structural and fluid flow analysis of bulk acoustic wave (BAW) piezoelectric micropumps is carried out for liquid (water) transport applications. The BAW micropump consists of trapezoidal-prism inlet/outlet elements; the pump chamber, a piezoelectric (PZT-5A) actuator and a thin structural layer (Pyrex glass) between the pump chamber and the actuator. Two-way coupling of forces and displacements between the solid and the liquid domains in the system is considered where actuator motion causes fluid flow. Flow contraction and expansion (through the trapezoidal-prism inlet and outlet sections, respectively) generate net fluid flow. The effect of the back-pressure, actuation frequency, inlet/outlet port angles and driving voltage on the structural piezoelectric bi-layer membrane deformation and the resulting flow rate are investigated. For the compressible flow formulation considered, an isothermal equation of state for the working fluid is employed. Three-dimensional governing equations for the flow fields and the structural piezoelectric bi-layer membrane motions are considered. The predicted flow rate increases with actuation frequency up to a critical value. The highest pumping rate is observed at this critical (resonant) actuation frequency due to the combined effects of mechanical, electrical and fluidic capacitances, inductances, and damping. Time-averaged flow rate starts to drop with increase of actuation frequency above the critical value. The present models can be utilized to optimize the design of E. Sayar Department of Mechanical Engineering, Istanbul Technical University, Istanbul, Turkey B. Farouk (&) Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA bfarouk@coe.drexel.edu microelectromechanical system-based micropumps on the basis of fluid flow and structural characteristics. Keywords Micropump Acoustic waves Thin-film bulk acoustic resonator Piezoelectric actuator 1 Introduction There is an increasing demand to develop efficient drug delivery devices to meet medical needs. Microelectromechanical systems (MEMS) technology can be implemented to address the needs. The application of MEMS technology is advantageous for medical needs of site-specific drug delivery, reduced side effects, increased biocompatibility and increased therapeutic effectiveness. A total drug delivery system consists of a drug reservoir, micropumps, valves, microsensors, microchannels and necessary circuitry. Micropumps are essential components in such drug delivery systems. A comprehensive review of micropumps is given in (Laser and Santiago 2004). Numerous micropumps for drug delivery applications have been reported. A review of such micropumps can be found in (Nisar et al. 2008a). These micropumps can be categorized as mechanical (positive displacement) or non-mechanical. The most common actuation methods for mechanical micropumps are electrostatic, piezoelectric, thermo-pneumatic and magnetic. A skin-contact-actuated dispenser/ pump has been studied (Mousoulis et al. 2011). The dispenser consists of stacked PDMS layers mounted on a silicon substrate and operates based on the evaporation and condensation of a low boiling point liquid. A low-cost, disposable and electronic patch-based cancer chemotherapy device was developed and reported (Ochoa et al. 2012). It was designed to be not only be simple, automated

2 434 Microfluid Nanofluid (2015) 18: and unobtrusive, but that also delivers chemotherapy drugs as they exist and that delivers up to three drugs at once. Similar micropumps are also being used for the treatment of hemodynamic dysfunctions in transdermal drug delivery systems (Nisar et al. 2008a). Bulk acoustic wave (BAW) micropumps are similar to diaphragm type-positive displacement micropumps, but they do not use check valves to direct flow. Instead, trapezoidal prism elements are used as flow rectifiers. This study presents the results from a detailed threedimensional time-dependent numerical investigation of the coupled structural and fluid flow analysis of piezoelectrically actuated BAW micropumps for liquid transport applications. The effect of actuation electrical voltage and frequency on pump structural layer deformation and the flow rate are analyzed. Time-averaged velocity fields are computed to predict the flow rates. The previous works (Cui et al. 2007; Fan et al. 2005; Nisar et al. 2008b) explained the dependence of net flow rates to actuation frequency using the instantaneous deformation shapes of the bi-layer membrane without calculating the liquid velocity field. Flow rates are calculated in the present study by integrating the predicted time-averaged velocity fields. Some of the past computational studies on similar problems did not consider coupling of the piezoelectric actuation to the pressure and shear forces generated by the fluid (one-way coupling only) (Cui et al. 2007, 2008; Nguyen and Huang 2000; Tsui and Lu 2008). Two-way fluid solid coupling was considered in several past studies (Ashraf et al. 2010; Bodhale et al. 2010; Fan et al. 2005; Nisar et al. 2008a, b); however, none of them reported the cycle-averaged mass flow rates and time-averaged velocity patterns in the pump chamber. The net cycle-averaged flows generated by such devices need to be evaluated by averaging velocity field over sufficient number of cycles. The present study addresses the prediction of cycle-averaged flows generated by the micropumps. 2 Problem description and geometry Flows generated in the piezoelectric BAW micropump (Fig. 1a c) is modeled using mechanical equations of motion for the solids, the electromagnetic field equations for the actuator and mass/momentum conservation equations for the fluid. The BAW piezoelectric micropump problem geometry considered is similar to the one reported in the experimental study (Olsson et al. 1997) where a double-chamber cylindrical pump geometry was considered. The piezoelectric BAW micropump device configuration adopted in this study has a thin-film bulk acoustic resonator (FBAR) structure. Figure 1a c shows that the BAW micropump consists of trapezoidal prism inlet/outlet Fig. 1 Piezoelectric BAW micropump geometry considered. a Front view, b top view, c side view elements and the pump chamber (bottom layer); one thin structural layer (middle layer) and one piezoelectric element (top layer) as the actuator. Pumps considered in the present work are made of Pyrex 7740 borosilicate glass structural layer and PZT-5A piezoelectric element (see Appendix for material and working fluid properties). The length (L) and the width of the pump chamber are both equal to 6.0 mm. Length and the width of the piezoelectric element are 3.8 mm. The depths of the (un-deformed) pump chamber and the trapezoidal prism inlet/outlet are 80.0 lm. At x = 7.1 mm, the outlet of the micropump is positioned at the far right in Fig. 1a c. Thickness of the structural layer, Pyrex 7740 borosilicate glass is lm and the thickness of the piezoelectric element, PZT-5A is lm. A water layer is considered under the Pyrex glass (0 lm \ y \ 80.0 lm) followed by the Pyrex glass (80.0 lm \ y \ lm) and finally the PZT-5A element (580.0 lm \ y \ lm. The top of the micropump (y = lm) is free to deform. The inlet/outlet port lengths are each 1.1 mm. The width of the narrowest part is 80.0 lm for both the trapezoidal inlet and outlet ports, and three different inlet and outlet taper angles (h in and h out ) are considered, viz., 7.0, 9.8 and The effect of the angles on the generated flow rates is also investigated.

3 Microfluid Nanofluid (2015) 18: Mathematical model Piezoelectric micropump design and analysis require coupled analysis of electrical, mechanical and flow quantities. The governing equations for the solid (piezoelectric and non-piezoelectric) and the fluid domains and the boundary and initial conditions are discussed in the following sections. The coupling relations of the mechanical, electrical and fluid fields are described thereafter. 3.1 Solid (piezoelectric) domain Propagation of acoustic waves in piezoelectric crystals is governed by the mechanical equation of motion and the electromagnetic field equations (Auld 1973). Substituting the equation of motion for a vibrating solid in the piezoelectric constitutive equation yields the first wave equation: o 2 s i q s ot 2 þ o 2 s k o 2 / ce ijkl þ e kij ¼ 0 ð1þ ox j ox l ox k ox j In Eq. (1), s i is the displacement component of the solid in the ith direction (m), u is the electric potential (V), q s is the density of the solid (kg/m 3 ), c E ijkl is the elasticity constant (N/m 2 ), e kij is the piezoelectric stress constants (C/m 2 ), t is the time (s), and x j is the coordinate (m). Piezoelectric materials are insulators; the absence of electric charge within the material can be expressed by the second coupled wave equation as: o 2 s k e ikl e S o 2 / ik ¼ 0 ð2þ ox i ox l ox i ox j where e ikl is the piezoelectric stress constants (C/m 2 ) and s e ik is the dielectric permittivity constant (F/m). The superscripts on the elastic stiffness constants (Eq. 1) and the dielectric permittivity constant (Eq. 2) imply that these are the properties at constant electric field and strain, respectively. In Eqs. (1 2), Einstein summation convention is used. For indices i, j, k, l, the index values are 1, 2 or 3 (representing the three dimensions of physical Euclidean space; x: 1,y: 2 and z: 3). The piezoelectric stress constants (e kij and e ikl ) couple the three displacement equations and voltage equation in Eqs. (1) and (2). The solutions of the displacement and voltage depend on the x-, y- and z- directions. 3.2 Pyrex glass (non-piezoelectric) domain Non-piezoelectric solid domain comprises of structural layer (Pyrex glass) as shown in Fig. 1a c. The piezoelectric stress constants (e ijk and e ikl ) are zero in non-piezoelectric materials. Thus, the mechanical equation of motion, Eq. (1), simplifies to: o 2 s i q s ot 2 þ o 2 s k ce ijkl ¼ 0 ð3þ ox j ox l Similarly, the governing equation for electrical quantities, Eq. (2), simplifies to the Laplace s equation o 2 / ¼ 0 ox i ox j ð4þ Propagation of acoustic waves in non-piezoelectric domains is governed by the Eqs. (3) and (4) which are uncoupled. 3.3 Fluid domains Working fluid (water) entirely fills the fluid domains in the inlet, pump chamber and outlet elements in Fig. 1a c. In the present model, fluid properties of water are used while treating the liquid as compressible and Newtonian. The fundamental equations used to describe the velocity and pressure fields in water are the compressible form of the three-dimensional continuity and momentum equations (Sayar and Farouk 2011). A thermodynamic relation is required to close the set of equations for the fluid medium. For isentropic waves, the pressure is linearly related to the density through the speed of sound in the fluid. The isentropic assumption is often used for acoustic problems. The equation of state is represented as (Weinberg et al. 2003): q f q 0 ¼ p p 0 c 2 ð5þ s where q 0 is nominal fluid density (a constant), p 0 is nominal pressure (atmospheric) and c s is the speed of sound in working fluid. Isentropic assumption is not valid for all frequencies such as very low frequencies when the energy does not radiate. However, for the cases investigated here ( ,000.0 Hz), isentropic equation of state can be applied. The effect of compressibility of water in flexural platewave pumps has been reported (Sayar and Farouk 2011). Compressible form of the equation of state of water was also used by other investigators (Shin et al. 2005; Weinberg et al. 2003; Friend and Yeo 2011) for analyzing acoustically driven water flows in microchannels. The streaming acoustic Reynolds number (Re acoust )is defined as Re acoust ¼ q f u 1 k f ð6þ 2pb When Re acoust is greater than one, fluid motion has to be represented by finite amplitude acoustic wave formulation including the nonlinear and compressible terms in the fluid flow equations (Friend and Yeo 2011). For the present cases studied, fluid motion has to be represented by finite

4 436 Microfluid Nanofluid (2015) 18: amplitude acoustic wave formulation. For the cases studied in this paper (described in Sect. 4 below), the fluid flow Mach numbers are small, and also the variations of the density of the fluid are below 0.1 %. Typically the Mach number criterion is used for gas flows in pressure-driven flows to determine whether a compressible flow model should be used. However, for liquids, the criterion is not strictly applicable (Sayar and Farouk 2011; Shin et al. 2005; Weinberg et al. 2003; Friend and Yeo 2011; Sayar and Farouk 2012a, b). 3.4 Boundary and initial conditions A symmetric domain along the z = 0.0 lm plane is considered. Along the walls of the fluid model, zero slip boundary conditions are used for velocity. We employ the pressure boundary conditions similar to (Nisar et al. 2008b; Cui et al. 2007, 2008; Ullmann 1998) for the open boundaries. Zero pressure is specified along the inlet (left boundary in Fig. 1b), and a back-pressure (P b ) is specified along the outlet (right boundary in Fig. 1b). Back-pressure (pressure at discharge) refers to pressure due to the forces that are operating in a direction opposite to that being considered in a confined place, such as that of a fluid flow. Density along the inlet and outlet boundaries is specified using the equation of state (Eq. 5) and the specified pressure boundary conditions. Due to symmetry, only one-half of the geometry shown in Fig. 1a c is modeled. Three sides and the top surface of the piezoelectric element are traction (stress) free. The symmetry plane is the fourth side. Bottom of the piezoelectric element transfers loads to the structural layer. Two sides (left and right boundaries in Fig. 1b) and the top surface (the exposed regions, shown in blue in Fig. 1b) of the structural layer are traction (stress) free. Back side of the structural layer is considered to be clamped. Bottom of the structural layer is the fluid structure interface. Structural layer is coupled to the liquid layer by the principle of continuity of stresses along the interface. Loads are calculated iteratively and transferred between fluid and solid domains. Side and bottom surfaces of the volume occupied by the water layer are silicon. Silicon-base plate is considered as perfectly rigid. Boundary conditions of electrical variables are as follows: The sides of all three layers are insulated. Two electrodes are patterned on the top and the bottom surfaces of the piezoelectric substrate, and electrodes fully cover the surfaces. Sinusoidal voltage is applied along the bottom of the piezoelectric element according to: / ¼ 0:5V p p sin 2pft þ h ph ð7þ where u is the electric potential (V), V p p is the magnitude of electric potential (peak to peak), f is the cyclic frequency (Hz) and h ph is the phase angle of the input signal. The piezoelectric element is grounded at its top surface. The fluid is initially quiescent, the pressure is atmospheric, and all dependent variables are zero for the pump operation under zero back-pressure. In order to consider a realistic initial condition and to eliminate immediate water hammer effects, the back-pressure is gradually applied on the micropump. 3.5 Coupling of the mechanical, electrical and fluid fields Once the piezoelectric element is powered, the applied electrical energy is converted to mechanical energy according to the piezoelectric constitutive equations and equation of motion. The mechanical waves in the structural layer interact with the liquid at its bottom surface creating displacements in normal and tangential directions. Along the fluid solid interface, the displacements generated by the structural piezoelectric bi-layer membrane generate flow in the fluid according to: u f ¼ u s ¼ os ð8þ ot where s is the displacement vector in structural layer, u s is the velocity vector in structural layer and u f is the velocity vector in fluid. Mathematical representation of the continuity of stresses along the interface is as follows: T s ij ns j þ Tf ij nf j ¼ 0 ð9þ where T ij are the stresses, n s j is the outward normal to the solid at the solid liquid interface, the superscript s represents solid, superscript f represents fluid, so that n s j ¼ n f j [the indices i and j define directions of components according to axes of the applied three-dimensional (3D) coordinate system]. Loads are calculated iteratively and transferred in between fluid and solid domains until the equilibrium condition of stresses, Eq. (9) is satisfied. The shear and normal forces are first calculated (by the solution of pressure and velocities for the fluid) that are to be imposed on the solid domains. Using these forces, governing equations for the solid domains are solved. If the convergence for all variables for flow and structure dynamics is achieved, the calculations proceed to the next time step. Computational grid is reconstructed based on the deformation of the solid model. Pressure and velocities for the fluid are solved again and the process repeated till convergence.

5 Microfluid Nanofluid (2015) 18: Numerical model For the solid domains (piezoelectric and non-piezoelectric Pyrex glass), second-order strain analysis is invoked. A direct linear equations solver PCGLSS 5.0 (ESI-US 2010) is used for the solution of dynamic structural model. The time step was chosen such that the cyclic variation of the piezoelectric actuator is well resolved. The coupled solid fluid domain problem is simulated using 20 time steps per cycle. The same time step is used for the solid and liquid zones. Decreasing the time step by another 50 % did not have any appreciable effect on the time-averaged flow rate results. The time required to reach pseudo-periodic state is about 10.0 cycles. The governing equations for the liquid domain are solved using the SIMPLEc algorithm (Van Doormaal and Raithby 1984). The resultant system of equations is solved by CFD-ACE? 2010 solver (ESI-US 2010). Terms for the discretization of convection terms are formulated according to the second-order accurate upwind scheme for velocity, and second-order accurate central difference scheme for density. For the flows with moving and deforming boundaries, grid generation is considered at every time step by using the transfinite interpolation (TFI) scheme (Eriksson 1985). The velocity and the computational grid cell center coordinates are stored for one cycle (20 time steps). The stored velocity and coordinates are averaged to get time averaged velocity vectors. Hexahedral structured brick elements are used both for the solid and the liquid domains. A grid density of (in x-, y- and z-directions, respectively) is considered for the fluid domain in all cases. For solid domains, we consider a grid density of Computations were carried out on a twelve core 28.0 GB memory Dell Precision T7500. frequency and inlet/outlet port angles are reported, respectively, in Sects. 5.4 and 5.5. The effects of excitation voltage (on a non-resonant frequency value) are discussed in Sect Acoustically driven flows in micropump: quasi-periodic state (for case 1) Instantaneous flow fields in the fluid layer (Fig. 1) are shown in Fig. 2a d (for case 1) at the beginning of tenth cycle, one-third cycle later, two-third cycle later and at the beginning of the eleventh cycle, respectively. Velocity vectors are plotted at the z = 0 mm plane. When estimated velocity differences between subsequent cycles for the calculated points are observed to be less than the specified convergence criteria g ¼ u f ;jþ1 u f ;j =uf ;j 0:01; the flow field is assumed to become quasi-steady. In Fig. 2a, pump is drawing the fluid from the tapered inlet passage (on the left) and delivering it out from the tapered outlet passage. One-third cycle later (Fig. 2b) the pump is drawing water from the inlet with a greater velocity and delivering it out from the outlet. Two-third cycle later (Fig. 2c), the fluid is delivered out of the pump 5 Results and discussion The results for the base case (case 1) are given in Sects. 5.1 and 5.2. Details of the acoustically driven flows in the micropump for case 1 are given in Sect Displacement and stress field predictions of the structural model are presented in Sect. 5.2 for case 1. Parameters for case 1 are as follow: f = Hz (the resonant frequency), V p p = V, P b = 0.0 kpa, h in = 9.8 and h out = 9.8. Here, the resonant frequency refers to the frequency at which micropump yields the maximum flow rate. The resonant frequency is based on the combined effects of mechanical, electrical and fluidic capacitances, inductances, and damping as discussed in Sect The effects of back-pressure on the pump characteristics and validation of the model are given in Sect The effects of actuation Fig. 2 Velocity field in the pump (case 1) at the 10th cycle, z = 0.0 lm with f = 2,000.0 Hz, V p p = V and P b = 0.0 kpa. a At the beginning of tenth cycle, b one-third of a cycle later, c twothirds of a cycle later, d at the beginning of eleventh cycle

6 438 Microfluid Nanofluid (2015) 18: cavity from the inlet and outlet. The channel depth in y- direction is significantly greater than the amplitude of wall motion in y-direction (s y ); therefore, the flow field does not show any asymmetry in the vertical direction. For shallower micropump cavities, velocity profiles deviate from parabolic velocity profile to an asymmetric wall-jet-driven velocity profile (Sayar and Farouk 2011). Figure 2d is almost identical to Fig. 2a (time interval between Fig. 2a, d is one cycle). The instantaneous pressure field (case 1) along the horizontal mid-plane of the water layers (y = 40.0 lm plane) at the end of the tenth cycle in the micropump, t = 5.0 ms, is shown in Fig. 3 below. The deformation of the PZT-5A and glass bi-layer membrane generates suction or discharge of the fluid. The pressure is asymmetric with respect to x = 3.0 mm plane (the vertical mid-plane of the device in x-direction). This asymmetry is the driving force that leads to a net flow in the micropump. Our simulation results indicate that the suction or discharge pressure increases or decreases with increasing distance from the center of the device (x = 3.0 mm plane) depending on the phase that the system goes through in an actuation cycle. Though the pressure field given in Fig. 3 is at a specific time and actuation frequency, it is emphasized here that different actuation frequencies induce different deformation shapes and amplitudes. The variation of design parameters such as applied electric potential, actuation frequency and geometric variables (the thickness of the glass layer and piezoelectric elements, inlet/outlet lengths and taper angles) could facilitate mixing of species which is otherwise molecular diffusion limited. A piezoelectric bimorph actuator composing of three layers (PZT Copper PZT, similar to the present geometry) was experimentally shown by (Zhou et al. 2014) to induce fluid motion in the microchannel for mixing enhancement. A similar device (Liu et al. 2002, 2003; Zhou et al. 2014) showed that fluid mixing is remarkably promoted by the piezoelectricity effect. Varying phase angle of the input electrical signal (h ph ) and operating more than one micropump in series or parallel are other methods to increase pumping rate (Olsson et al. 1995). For flows with acoustic excitations, the energy transmitted by the oscillating wall is attenuated not only by the inertia and viscosity of the fluid, but also through the density variations (accounted for in the present study) in the fluid. Figure 4 shows the axial variation of liquid velocity along the intersection of y = 40.0 lm and z = 1,000.0 lm planes at different time levels (four consecutive phases within a cycle) for case 1 during the tenth cycle. In the BAW micropump, as the boundary (the moving boundary at y = 80.0 lm plane) is oscillating with time, it is expected that the fluid will also oscillate in the y-direction in time with a specific frequency. Individual velocity profiles in Fig. 4 above can be explained by comparing them to the well-established driven cavity problem velocity profiles (Schreiber and Keller 1983). The flow resistances in inlet and outlet of the micropump are not the same in the present study due to the inlet/outlet ports divergence/ convergence angle differences and the differences in deformation shapes (due to the fluid mass load and the pumping system load behavior). As the areas under individual velocity profiles are not identical (Fig. 4), the oscillating bi-layer results in net flow. Same effect can also be seen in Fig. 2a d. The Fig. 2a d is plotted over the Fig. 3 Instantaneous pressure (Pa) field in the flow domain at the end of tenth cycle. y = 40.0 lm, t = 5.0 ms, f = 2,000.0 Hz, V p p = V and P b = 0.0 kpa (case 1) Fig. 4 Instantaneous axial velocity profiles along the x-axis. y = 40.0 lm, z = 1,000.0 lm, f = 2,000.0 Hz, V p p = V and P b = 0.0 kpa (case 1) during the 10th cycle

7 Microfluid Nanofluid (2015) 18: z = 0.0 lm plane, whereas Fig. 4 is plotted over the z = 1,000.0 lm plane. In Fig. 4, the number of local peaks in individual profiles increases with frequency of the driving electrical voltage. 5.2 Displacement and stress field predictions in the solid domains (for case 1) The instantaneous solid domain y-component normal stress and liquid domain pressure variations are plotted in Fig. 5 for case 1 along the y-axis over the z = 25.0 lm plane at t = 5.0 ms for selected constant x lines. The match of pressure and normal stresses at y = 80.0 lm (Pyrex glass/water interface) and at y = lm (PZT-5A/Pyrex glass interface) signifies that the two-way coupling of the model provide satisfactory predictions (see Sect. 3.5). The thickness of the viscous boundary layer (Moroney et al. 1991), d is found to be qffiffiffiffiffiffi lm where d ¼ and x is the angular frequency, 2l q f x l is the dynamic viscosity and q f is the density of the fluid. Slightly lower velocities are observed within the viscous boundary layer (can be seen in Fig. 2a d). In Figs. 6a c, contours of instantaneous displacement in y-direction (s y ) along the interface between solid and fluid layers (y = 80.0 lm plane) are shown for case 1 at the beginning of tenth cycle, one-third cycle later and twothirds cycle later (f = 2,000.0 Hz and V p p = V). The interface is clamped rigidly along its front and back sides. The bi-layer membrane deforms in a complicated Fig. 5 Pressure/stress variations in the liquid and solid domains along the y-axis for various x-locations at t = 5.0 ms, z = 25.0 lm, f = 2,000.0 Hz, V p p = V and back-pressure P b = 0.0 kpa (case 1) shape (solid domains are wavy in Fig. 6a), and two peaks appear at the beginning of tenth cycle. Contours of displacement in y-direction are shown one-third of a cycle later in Fig. 6b. In Fig. 6b, solid domains bend upward (positive y) drawing fluid from the inlet and outlet. The maximum displacement in y-direction is nm at the given instant of time. Two-thirds of a cycle later (Fig. 6c) fluid solid interface are wavy. Interface is bended upward toward the left right sides and bended downward in the center. The maximum amplitude of displacement is observed in the center of the interface in Fig. 6c (308.1 nm). Displacement in y-direction is zero along the rigid sides of the interface plane. Along the left and right sides of the interface plane, solid layer (Pyrex glass) is free to deform. Different actuating frequencies induce different deformation shapes; hence, they affect the pumping rate. When the actuating frequency is low, the membrane bends in one direction and it has only one peak. At higher frequencies, multiple bending peaks can be observed yielding undesirable pressure fields within the pump chamber. The deformation shape and amplitude influence the pumping rate since stroke volume depends on the behavior of the structural piezoelectric membrane motion. 5.3 The effects of back-pressure and validation of the model The time-dependent volumetric flow rates, time-averaged axial velocity and net (time-averaged) flow rates through the outlet of the micropump are shown during the tenth cycle are presented in this section for the cases shown in Table 1. In Fig. 7, the temporal variations of flow rates through the outlet of the micropump are shown during the tenth cycle for various values of the pumping back-pressure for cases 1, 3 and 5 (Table 1). During the supply mode, the instantaneous flow rate is positive (flow is into the pump chamber) and during the pump mode it is negative (flow is out of the pump chamber). Instantaneous velocities are integrated over the outlet area at each time step during one cycle in the present study. The deformation of the pump membrane attenuates yielding to reduced flow rates produced by the micropump at higher values of the back-pressure. The temporal variations of the flow field show that flow rates through the outlet of the micropump decrease linearly with increasing backpressure seen by the pump in Fig. 7. In Fig. 8, the variations of the time-averaged axial component of fluid velocity (z-averaged) along the y- coordinate are shown for various back-pressure values for cases 1 5 at the tenth cycle along the outlet (x = 7.1 mm).

8 440 Microfluid Nanofluid (2015) 18: Fig. 6 Contours of displacement in y-direction along the interface between solid and fluid layers (y = 80.0 lm plane), f = 2,000.0 Hz, V p p = V and P b = 0.0 kpa (case 1). a At the beginning of tenth cycle, b at one-third of a cycle later, c at two-thirds of a cycle

9 Microfluid Nanofluid (2015) 18: Table 1 List of the cases considered: effect of pumping back-pressure f = 2,000.0 Hz, V p p = V, h in = 9.8, h out = 9.8 Case Back-pressure, P b (kpa) Back-pressure (P b ) is varied (from 0.0 to 40.0 kpa) while f = 2,000.0 Hz and V p p = V are kept constant. At high back-pressures, shear and normal forces imposed on the solid membranes increase. The deformation of the pump membrane attenuates yielding to reduced flow rates produced by the micropump at higher values of the back-pressure. At higher values of back-pressure, higher order modes of the deformations are attenuated as well. The velocity profile over the outlet of the pump is parabolic. The velocity profiles slightly deviate from the parabolic, if the back-pressure seen by the pump is sufficiently large (Fig. 8, P b = 17.8 and 40.0 kpa). As it is illustrated in Fig. 8, the magnitude of time-averaged axial component fluid velocity decrease linearly with increasing back-pressure seen by the pump up to P b = 17.8 kpa Validation of the model Fig. 7 The effect of back-pressure on the time-dependent flow rates at the outlet of the pump during the tenth cycle and x = 7.1 mm (at the outlet) P b = kpa, f = 2,000.0 Hz and V p p = V (cases 1, 3 and 5) In Fig. 9, time averaged flow rates through the outlet of the micropump are shown for the present study and (Olsson et al. 1997). Flow rates are reported during the tenth cycle under various back-pressures (Table 1: cases 1 thru 5) in Fig. 9. Net flow rate is estimated by time averaging the instantaneous flow rates over one cycle. Flow rates for piezoelectric double-chamber micropump operated in anti-phase mode are experimentally measured and reported by Olsson et al. Only half of the flow rates reported for the double-chamber micropump (Olsson et al. 1999) are shown in Fig. 9 for comparison. For a singlechamber micropump (present case), the average flow rate is Fig. 8 Effect of back-pressure on the time-averaged axial velocity along the y-axis at the outlet of the pump (x = 7.1 mm), tenth cycle. The time-averaged fluid velocity is averaged over the width (zcoordinate) in Fig. 1a c. P b = kpa, f = 2,000.0 Hz and V p p = V (cases 1 5) Fig. 9 Net (time-averaged) flow rates during the tenth cycle. f = 2,000.0 Hz and V p p = V, x = 7.1 mm (over the outlet), cases 1 5

10 442 Microfluid Nanofluid (2015) 18: predicted as ll/min (case 1, P b = 0.0 kpa). The coupled multifield model results agree well with the normalized results of Olsson et al. (1997) where the difference in flow rate prediction is about 3 % at P b = 0.0 kpa. Deformation amplitude of the structural piezoelectric layers is lm in the present work. Olsson et al. report deformation amplitude as 0.64 lm with the difference to present prediction is 2.6 %. Increase of back-pressure progressively reduces the net flow rate. For high backpressure values, the net output of the pump decreases markedly and the pump suffers in efficiency. 5.4 Effect of actuation frequency The net flow rate developed by the pump varies with the actuation frequency. The actuation frequency (the cyclic frequency of the driving electric signal, f) is varied from to 5,000.0 Hz in cases 6 12 (Table 2), at the excitation voltage V p p = V. Table 2 List of the cases considered: effect of actuation frequency P b = 0.0 kpa, V p p = V, h in = 9.8, h out = 9.8 Case Freq. (Hz) , , , , , ,000 In Fig. 10, time averaged flow rates through the outlet of the micropump are shown during the tenth cycle under various actuation frequencies (from to 5,000.0 Hz). The present fluid solid system shows the highest pumping rate at the actuation frequency of 2,000.0 Hz (case 1) due to the combined effect of mechanical, electrical and fluidic capacitances, inductances, and damping. Elasticity of the piezoelectric element and structural materials is dominant at actuation frequencies below Hz. At frequencies below 2,000 Hz, the predicted flow rate increases with actuation frequency. Time-averaged flow rate starts to drop with increase of actuation frequency above 2,000.0 Hz in Fig. 10. Mechanical system response is more dominant than the fluidic response at frequencies below 2,000.0 Hz; with increasing actuation frequency, the predicted flow rate increases due to the increasing membrane velocity per unit applied potential (Morris and Forster 2003; Sayar and Farouk 2012b). In Fig. 11, the volumetric flow rate variations over a cycle are shown for various values of the actuation frequency (cases 1, 6, 9 and 12) along the outlet. At the resonant frequency (2,000 Hz, case 1), the net flow rate is the highest over a cycle. At higher frequencies ([2,000 Hz), the effect of the high frequency perturbations on the water is weakened due to the reduced cyclic times. This results in a change in the velocity profile whereby, it becomes relatively flat or plug-like (Sayar and Farouk 2012b). As shown in Fig. 11 2,000 Hz, the time-averaged flow rate increases with increasing frequency. Time-averaged flow rate starts to drop with increase of actuation frequency above 2,000 Hz. At high frequency, the working fluid (water) ceases to follow the perturbations due to its inertia and viscosity. At higher frequencies, although the Fig. 10 The effect of actuation frequency on the net flow rates during the tenth cycle and x = 7.1 mm (over the outlet) f = ,000.0 Hz. V p p = V and P b = 0.0 kpa (cases 1, 6 12) Fig. 11 The effect of actuation frequency on the flow rate variations over a cycle during the tenth cycle and x = 7.1 mm (at the outlet) f = ,000.0 Hz. V p p = V and P b = 0.0 kpa (cases 1, 6, 9 and 12)

11 Microfluid Nanofluid (2015) 18: value of instantaneous velocity is higher, the steady component of velocity is lower. 5.5 Effect of inlet/outlet port angles The effect of inlet/outlet port angles was also evaluated by simulating the flow fields with varying port angles. Both inlet (h in ) and outlet (h out ) angles are varied (7.0, 9.8 and 13.0 ) in cases 1, (nine cases) as shown in Table 3. The inlet outlet lengths and the width of the narrowest part of trapezoidal prism ports are kept constant. Here, the walls of the tapered inlet and outlet elements are considered as flat rigid silicon walls. In Fig. 12 below, time-averaged flow rates through the outlet are shown for varying inlet/outlet taper angles, h in and h out (cases 1, 13 20). The flow rate is found to increase with decreasing outlet taper angle and increasing inlet taper angle. Flow separation is encountered at higher values of the outlet angle with reduced pumping performance. The pulsating flow through the inlet port is not found to separate significantly during a cycle. The maximum pumping is found for the case where h in = 9.8 and h out = 7.0. It is also found that the inlet taper angle should be greater than the outlet taper angle. 5.6 Effect of excitation voltage at a non-resonant frequency The excitation voltage effect on pumping was studied at a non-resonant frequency (to avoid the possibility of cavitation in the micropump). In cases (Table 4), the excitation voltage is varied (between 25.0 and V) at frequency f = 3,350.0 Hz. In Fig. 13, time-averaged net flow rates through the outlet of the micropump are shown during the tenth cycle under various excitation voltages. Net flow generated by the micropump increases with increasing excitation voltage. At excitation voltage above V (for an excitation frequency of 3,350.0 Hz), the instantaneous pressure in the pump attained values less than the vapor pressure at the given temperature. A slight nonlinear behavior of net flow rate and excitation voltage relation can be seen in Fig. 13. At lower excitation voltages, the mechanical system response is dominant compared to the fluidic response. On the other hand, at higher excitation voltages, fluidic response is more dominant. Actuation frequency and excitation voltage of the micropump are limited by the possibility of occurrence of cavitation in the fluid systems. Table 3 List of the cases considered: effects of inlet outlet port angles Case h in ( ) h out ( ) Table 4 List of the cases considered: effect of excitation voltage Case Voltage (V) f = 3,350.0 Hz, P b = 0.0 kpa, h in = 9.8, h out = 9.8 f = 2,000.0 Hz, V p p = V, P b = 0.0 kpa Net Flow Rate (µl/min) in = 7.0 in = 9.8 in = Outlet convergence angle, out ( ) Fig. 12 Net flow rates during the tenth cycle. f = 2,000.0 Hz, V p p = V and P b = 0.0 kpa, x = 7.1 mm showing the effect of inlet (h in ) and outlet (h out ) port angles (cases 1, 13 20) Fig. 13 Effect of excitation voltage on net flow rates during the tenth cycle and x = 7.1 mm (over the outlet). V p p = V. f = 3,350.0 Hz, and P b = 0.0 kpa (cases 10, 21 25)

12 444 Microfluid Nanofluid (2015) 18: Summary and conclusions The flow fields generated in a piezoelectric micropump by a thin-film bulk acoustic resonator are simulated by considering the coupled electrical, mechanical and fluidic fields. The two-way coupled fluid solid interactive model considered here is superior to the past studies (Tsui and Lu 2008; Nguyen and Huang 2000) which considered prescribed wall motions. The difference in flow resistance through the inlet and the outlet ports generates net fluid flow as the bi-layer structural piezoelectric membrane vibrates. The compressibility of the liquid is considered with an isothermal equation of state. The coupled model considers the deformation of solid membranes simultaneously with the prediction of fluid flow. The model predictions of the net flow rate generated by the micropump are compared to experimental results (Olsson et al. 1997). Net flow rate results agree well with their results where the difference in flow rate prediction is about 3 %. The solid bi-layer deformation results also agree with the experimental results (Olsson et al. 1997), where the difference is 2.6 %. The present fluid solid system with the chosen geometry shows the highest pumping rate at the actuation frequency of 2,000.0 Hz due to the combined effect of mechanical, electrical and fluidic capacitances, inductances, and damping. At frequencies above and below 2,000.0 Hz, the predicted flow rate increases with actuation frequency. Actuation frequency and excitation voltage of the micropump are limited by the occurrence of the cavitations in the fluid systems and structural integrity of the solid materials used for fabrication. At excitation voltage above V (for the frequency of 3,350.0 Hz), the instantaneous pressure in the pump attained values less than the vapor pressure at the given temperature. These present results quantitatively provide the dependence of net flow rates to the actuation frequency, excitation voltage, back-pressure and pump geometrical parameters (inlet/outlet port angles). The pressure, velocity and flow rate prediction models developed in the present study can be utilized to optimize the design of MEMSbased micropumps that are being used for the treatment of hemodynamic dysfunctions in transdermal drug delivery systems. Appendix: Material and working fluid properties Property data are given below for the piezoelectric material (PZT-5A), pump structural layer (Pyrex 7740 Borosilicate glass) and working fluid (water). The material data are provided following the conventional IEEE standards on piezoelectricity (Meeker 1996). The y- and z-axes in the piezoelectric material crystal frame (as shown on the expressions for piezoelectricity e, electrical permittivity e and elasticity matrix c below) correspond to the z- and y- axes of the global frame (shown in Fig. 1a c). Piezoelectric material PZT-5A (Cui et al. 2007) The density q s, piezoelectric stress constant e, permittivity e and the elasticity matrix c of PZT-5A are given below: Density q s ¼ 7700:0 kg=m 3 Piezoelectric stress constant : : :8 e ¼ 0 12:3 0 C=m : : Permittivity e ¼ 4 0 8: F=m 0 0 7:346 Elasticity constant :1 7:54 7: :54 12:1 7: :52 7:52 11: c ¼ : N=m : :26 Glass (Sollier et al. 2011) Pyrex 7740 borosilicate glass is considered to be isotropic. Material properties of the glass hence can be represented by just two independent quantities, i.e., Young s Modulus and Poisson s ratio. The relations between the elasticity constant, Young s Modulus and Poisson s ratio are summarized in (Fan et al. 2005). Material properties of Pyrex 7740 borosilicate glass: densityq s ¼ 2230:0 kg=m 3, Young s modulus E = Pa and Poisson s ratio t = 0.2. Working fluid (water) (Cui et al. 2007) Properties of the working fluid: density q f ¼ 997:0 kg=m 3, dynamic viscosity l ¼ 0:00104 kg=m s and speed of sound in working fluid c s ¼ 1480:0 m=s. References Ashraf MW, Tayyaba S, Nisar A, Afzulpurkar N, Bodhale DW, Lomas T, Poyai A, Tuantranont A (2010) Design, fabrication and analysis of silicon hollow microneedles for transdermal drug delivery system for treatment of hemodynamic dysfunctions. Cardiovasc Eng 10(3):91 108

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