584769
research-article2015
JLAXXX10.1177/2211068215584769Journal of Laboratory AutomationLiu et al.
Original Report
Numerical Simulation on the Response Characteristics of a Pneumatic Microactuator for Microfluidic Chips
Journal of Laboratory Automation 1–11 © 2015 Society for Laboratory Automation and Screening DOI: 10.1177/2211068215584769 jala.sagepub.com
Xuling Liu1, Songjing Li1, and Gang Bao1
Abstract This article presents a multiphysical system modeling and simulation of a pneumatic microactuator, which significantly influences the performance of a particular pneumatic microfluidic device. First, the multiphysical system modeling is performed by developing a physical model for each of its three integrated components: microchannel with a microvalve, a gas chamber, and an elastomer membrane. This is done for each step of operation for the whole system. The whole system is then considered a throttle blind capacitor model, and it is used to predict the response time of the pneumatic microactuator by correlating its characteristics such as gas pressurizing, hydraulic resistance, and membrane deformation. For this microactuator, when the maximum membrane deformation is 100 µm, the required actuated air pressure is 80 kPa, and the response time is 1.67 ms when the valve-opening degree is 0.8. The response time is 1.61 ms under fully open conditions. These simulated results are validated by the experimental results of the current and previous work. A correlation between the simulated and experimental results confirms that the multiphysical modeling presented in this work is applicable in developing a proper design of a pneumatic microactuator. Finally, the influencing factors of the response time are discussed and analyzed. Keywords pneumatic microactuator, PDMS membrane, TBC model, response time, influencing factors
Introduction Microfluidic large-scale integration (mLSI)1 refers to the fabrication and operation of hundreds or thousands of active microfluidic components on a single chip, to achieve specific chemical and biological processes. The pneumatic microfluidic chip allows for dense integration. It is feasible for mass production and commercialization because its control system and pressure generation are usually off chip.2 Using a soft lithography process involving thick-film micromachining and replica molding of elastomeric materials, pneumatic microactuators are most widely used and they offer a number of advantages.3 They are easy to manufacture at high density, which makes the mLSI (thousands of valves per cm2) or microfluidic very-large-scale integration (approaching 1 million valves per cm2)4 of microfluidic devices possible, which is important for the integration of a microfluidic system. In practical applications, it is preferable to apply the pneumatic actuator to elastic materials, such as polydimethylsiloxane (PDMS)–based systems. This is due to the key merits of PDMS material, such as transparency, mechanical properties, simple structure bonding process, and low production cost. PDMS has a much lower Young’s modulus
compared with other hard materials, such as silicon and silicon nitride,5 enabling it to be used as the deformable membrane of microfluidic devices, which is important for the pneumatic microactuator. PDMS membranes employed in pneumatic actuation are active in microvalves, micropumps, and micromixers. They have become popular and promising in these applications because of rapid and reliable fabrication.6,7 In recent years, while focusing on the performance of deformable membranes under different applied pressures, researchers have studied modified geometries of the pneumatic microactuator. The Quake research group5 studied the effects of variation of membrane-related parameters (such as membrane width, membrane thickness, and actuation 1
Department of Fluid Control and Automation, Harbin Institute of Technology, Harbin, China Received Dec 21, 2014. Corresponding Author: Songjing Li, Department of Fluid Control and Automation, Harbin Institute of Technology, Box 3040, Science Park, No. 2, Yikuang Street Nangang District, Harbin, 150001 P.R China. Email: [email protected]
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channel width) on membrane deflection of a pneumatic microactuator using an ANSYS computational model and then compared the model with a series of experimental results. It was found that the required closed air pressure, in the air chamber, for a push-down pneumatic actuator (microvalve) was 137.9 kPa and only 16.55 kPa for a pushup pneumatic actuator (microvalve). Zhou et al.8 developed the influence of parameters on leakage ratio of the pneumatic micropump, which composed three pneumatic microactuators. These parameters include the aspect ratio of the air chamber, applied force to the membrane, and so on. The authors presented the two-dimensional contact model established with ANSYS finite element analysis. Also, a finite element model of PDMS pneumatic actuator membrane deflection and two sets of experiments for validating the model were studied in refs. 9, 10, and 11. In addition, Mohan et al.12 analyzed the geometry of three different normally closed valves and found that the pneumatic actuators of the normally closed valves, with a sharp corner feature (V-shaped), actuated at lower driving pressures compared with straight-shaped valves and determined that membrane thickness does not significantly influence the actuation pressures by observing a series of relevant experiments. The understanding and management of deformable membrane performance have also been supported by many numerical models of pneumatic microactuator-related researches. The authors of refs. 13 and 14 had studied the coupling models of thermo-pneumatic PDMS microactuators. The pneumatic microactuator was used with a heating electrode, which can be coupled to a flexible membrane for generating actuation. The relationship between the applied voltage, on the heating electrode, and the membrane deflection can be obtained through the coupling model using COMSOL multiphysics. Lee and coworkers15 designed and investigated a finite element model of the pneumatic actuator components to calculate the displacement distribution of membrane deflection under pressures. The mathematical models, another method for evaluation of membrane deflection displacements versus different applied pressures, were introduced. The research of Yang et al.16 proved the effectiveness of the mathematical model through a series of experiments. The mathematical models, the COMSOL simulation models, and the corresponding experiments about membrane deflection of a PDMS chamber were proposed by Hardy et al.17 Cui et al.18 had analyzed and understood the different factors that influenced the performance of the pneumatic microactuator, according to the theory of membrane distortion, and created the computational model about sealing surface area versus membrane deflection. Kartalov et al.19 presented and analyzed three basic linear models (thick beam, thin spring, and thick spring) for the PDMS membrane of the pneumatic microactuator and proposed that the thin spring model was closest to reality and finally gave a superposed systemic model to predict the membrane
property. The results were proved by experimentation, using push-down microactuators. Lau et al.20 determined that the dynamics of the pneumatic actuators (microvalves) on chip were indeed influenced not only by the configurations, such as membrane thickness and driving pressure, but also by the levels of design complexity and positions in the devices. Previous studies, both experiments and simulations about pneumatic actuator, have focused on the influence of various membranes, air chamber geometries, or pressure on the deformation of thin membranes. The influence, however, on the whole performance of the pneumatic actuator, such as response time and air transient pressure changes in the air chamber versus time, have not yet been studied. In addition, the studies of air pressure changes during the response period, the response time, and its factors related to the pneumatic microactuator are far from being understood, and further research is required. Therefore, the motivation of this work is to obtain the suitable actuated air pressures and response times when optimizing the design of a system and to provide a promising simulation for pneumatic microfluidic chips; therefore, the response time of the pneumatic microactuator and its influencing factors are studied in this work, and a methodology to investigate optimal response time is provided to improve the whole efficiency of microchips. By using a multiphysics system model, we numerically analyze the performance of a standard pneumatic actuator to reduce the size of the driving pressure in real applications and the duration of response time to improve its efficiency. Each individual physical model is built separately. All of the models are then interfaced with a throttle blind capacitor (TBC) systemic model for predicting the global operation and actuation performance.
Architecture and Working Principle A length of gas microchannel with a flow control microvalve, an air chamber connecting the microchannel, and a PDMS thin membrane, which locates the upper air chamber, constitute the pneumatic microactuator.21,22 The elastomer thin membrane is the most important and only active part. Its deformation exerts a tremendous influence on the response time of the pneumatic microactuator. The structure of the pneumatic microactuator with a normally closed microvalve is shown in Figure 1. The geometrical parameters taken for the simulations and experiments are described in Table 1. When the actuation is required, a voltage is supplied on the normally closed microvalve, which opens it, and the gas is supplied to the microactuator. The pressure in the air chamber increases and deforms the membrane. PDMS deforms under applied pressures, which can be used in robust manipulation schemes that pinch-off, drive, and mix working fluids in the microfluidic systems.
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Figure 1. Schematic diagram of the pneumatic microactuator with a normally closed microvalve (a) and cut view (b–d; not to scale). Table 1. Pneumatic Microactuator Geometrical Parameters. Geometrical Parameter
la, lm
wa, wm
ha, h0
tm
l0
w0
Value (µm)
500
200
100
40
3000
100
Mathematical Models The comprehensive model formulated in this study was developed by using several different physics describing the different parts of the pneumatic operation: the valve-opening degree of the normally closed microvalve decides the local hydraulic resistance; the length, width, and height of the microchannel dominate the frictional hydraulic resistance; the geometry of the gas chamber decides the volume of the air capacity; and the deflection level of the elastomer membrane related to the length, width, and thickness settles the size of the variable volume of the air capacity and also has a tremendous influence on the response time of the pneumatic microactuator.
Gas Flow Theory Pneumatic systems use gas to apply force. The perfect gas law is the equation of state of a hypothetical ideal gas, which is an approximation to the behavior of many different gases under different conditions. Thus, the gas used in our simulation is considered a perfect gas. In a miniaturized gas system, the flow velocity is very low (nL/min ~ L/min), and the hydraulic diameter of a microchannel is very small
(µm ~ mm); therefore, the Reynolds number23 is always much smaller than 2300 (10–25~1) and the gas is always in the condition of laminar flow. In this article, N2 (nitrogen) is the medium in which the gas is supplied to the membrane deformation through the air capacity.
Hydraulic Resistance Hydraulic resistance refers to the friction that prevents the gas from going forward. For the pneumatic microactuator, which is integrated with microvalves, micropumps, or micromixers, the hydraulic resistance, which relates the small size PDMS microchannel, should be considered in the microfluidic networks. The total hydraulic resistance (Rw) of the TBC includes two parts,24 as shown in eq 1: the constant resistance (Ri) created by a microchannel and the variable resistance (Rv) created by the valve-opening degree of the normally closed microvalve. Rw = Ri + Rv
(1)
The constant resistance (Ri) can be estimated using the following empirical formula for a rectangular microchannel with a large width (w0 > h0)25:
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Journal of Laboratory Automation Maximum displacement of deflection is a function of the pressure over the membrane, which can be calculated by eq (4).21 1
δ max = wm (3∆Pwm )3 4 Etm
Figure 2. The schematic diagram of the throttle blind capacitor. Qv, volume flow rate through the microvalve; Rw, total hydraulic resistance.
Ri =
12µl0 w0h03
∞ 192 h0 1− 5 w 0 n,odd π
∑
1 tan h (nπ w0 ) 0 5 2h0 n
(2)
Rv = ε
ρ Qv 2 Av2
where δmax is the maximum deformation of the membrane; ΔP = Pa – Pl is the applied pressure to the membrane; Pa is the pneumatic pressure in the gas microchannel; Pl are the other forces to the membrane, which are in the opposite direction to the air pressure; E = E / (1 −ν )2 is the plane-strain modulus; E is the Young’s modulus; and v is the Poisson ratio of the material. The volume (V) under the deflection membrane can be estimated by eq 5:
where µ is the viscosity of the fluid in the gas microchannel and l0, w0, and h0 are the length, width, and height of the gas microchannel, respectively. It is shown from eq 2 that the value of hydraulic resistance (Ri) is controlled by the geometry of the microchannel and fluid viscosity, which are assumed to be independent of the flow rate inside the microchannel. To the authors’ knowledge, Rv is a variable value that is caused by microvalve pressure loss26: (3)
Per eq 3, Qv is the volume flow rate through the microvalve, ε is the pressure loss coefficient, p is the density of fluid through the microvalve, and Av is the effective flow area of the microvalve port.
Membrane Deformation Theory The PDMS elastomer is modeled as a near-incompressible, Neo-Hookean material. This constitutive model describes the behavior of rubberlike materials undergoing large deformations.5 The large deformation membrane behavior theory is also called plane-strain theory. Using this theory, for a rectangular shape, three assumptions must be satisfied: (1) the length of the membrane (lm) is at least two times larger than the width (wm), (2) the width (wm) of the membrane is 10 times the membrane thickness (tm), and (3) the deflection of the membrane is more than the membrane thickness.21 If all the assumptions are satisfied, then the variation in deflection along the length can be ignored and the centerline stretch of the membrane dominates bending. We resort to the finite element method (FEM) to represent the geometry of the rectangle elastic thin membrane and to solve the equations governing its deformation.
(4)
V = 2lm wmδ max 3
(5)
After the pneumatic microactuator reflects, the deflection of the membrane is parabolic.
Response Time of Pneumatic Microactuator The pneumatic microactuator always needs an external microvalve to control the pressure of the gas chamber and to further control the degree of the thin membrane deformation. The microvalve and the pneumatic microactuator form a whole pneumatic system, also named TBC,27 shown in Figure 2, which is a function of the variable hydraulic resistance and the variable air capacitor. An electromagnetic microvalve acts as the variable hydraulic resistance. The pneumatic chamber acts as the variable air capacitor because the volume of the chamber is always variable according to the air pressure. In Figure 2, total hydraulic resistance Rw is defined as the ratio of differential pressure across the system and air flow of the system. C is the variable air capacity of the gas chamber.27 Given the previous index analysis, Rw and C are two variable values. They are independently controlled by the valve-opening degree of the microvalve and the volume change of the variable air capacity. When P1 > P2, the TBC is in the inflation process. The TBC system time constant is T (T = Rw · C), and it is a variable value because of variable hydraulic resistance (Rv) and variable volume of air capacity (C). According to the authors’ knowledge, the transfer function of the TBC is a first-order inertia unit, and its storage energy element is the variable air capacity. The period from the closed to the open moment of the microvalve gave the system an input signal—the unit step signal, which is one of the fundamental signals. Then, −t T P2 = P1 (1 − e ) t P = Pe− T 2 1
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P1 > P2 P1 < P2
(6)
5
Liu et al. Using eq 6 in the inflation course of the TBC, its transient pressure in the air chamber increased exponentially. When P2 > P1, the TBC is in the deflation process. Hydraulic resistance was introduced in the Hydraulic Resistance section. The volume (V) under the deflecting membrane is defined in eq 5, which is a function of the maximum deflection δmax, and δmax is a function of the pressure over the membrane: C (∆P) =
dV (∆P) d 1 2 3wm ∆P 13 = ( lw ( ) ) d ∆P d ∆P 6 a m Etm l w2 3w 1 − 2 = a m ( m ) 3 ∆P 3 18 Etm −2 3
(8)
la wm2 3wm 13 ( ) 18 Etm
(9)
C (∆P) = C0 ∆P
C0 =
(7)
The response time (t) of the RwC circuit is about 3.5-fold of the RwC time constant (T), t = 3.5T. When pressure P2 is approximately equal to the air supply pressure P1, the inflation course is finished. The time constant (T) is equal to the required time, which is needed for P2 to reach 63.2% of P1, when a step signal is given.
Model Implementation
Figure 3. Meshing of the polydimethylsiloxane elastic membrane. Table 2. Parameters of Polydimethylsiloxane Membrane for Simulation and Experiment. Prepolymer solution ratio: (base/curing agent) Cured temperature (°C) Cured time (h) Young’s modulus (MPa) Possion ratio Density (kg/m3) Specific heat (Pj/kg K) Thermal conductivity (W/m K)
15:1 65 3 0.75 0.499 0.97 1.46 0.15
In this study, the gas permeability of PDMS and pressure change due to channel deflections were neglected for simplified pressure analysis.
The residual stress (σ0) equaled zero at steady state; it also equaled zero when the applied pressure was 100 kPa, and the boundary condition of the membrane was held rigid on all sides of the model.
Membrane Deformation
Total Hydraulic Resistance
The FEM was used to represent the three-dimensional geometry of the PDMS membrane and to simulate its deformation. The membrane deformation model is implemented using ANSYS software. Because of the twofold symmetry of the problem, the computational model is restricted to one-fourth of the membrane, with symmetry boundary conditions applied on all symmetry planes. The elements employed in the calculations are 10-node quadratic tetrahedral. The meshing for the PDMS membrane contains 58,042 tetrahedral elements. The mesh density in the membrane region was increased to obtain better accuracy on the elastic membrane’s deformation area, shown in Figure 3. The mesh density on the deformation area was quadrupled in reference to the other parts. A uniform pressure was applied over the surface of the PDMS elastic membrane. Table 2 provides the parameters of the PDMS membrane used in the model and the experiments.
The value of the fixed resistance (Ri) is determined by the length, width, and height of the microchannel, which is located between the gas chamber and the gas supply, as well as the viscosity of the fluid in the gas microchannel. The viscosity of N2 is 17.544 µPa•s when the room temperature is 25 °C. According to eq 2, Ri = 1.57*1012 Pa/(m3/s). The value of variable hydraulic resistance (Rv), which is related to the microvalve pressure loss, was determined by the valve-opening degree. In this article, the characteristics of flow rate – differential pressure were obtained by performing a numerical simulation under different valve-opening degrees using computational fluid dynamics software Fluent (in ANSYS 12.0) and the user-defined function, which can be programmed according to the user’s demand and dynamically linked to the solver. The local hydraulic resistance of the microvalve is calculated by eq 3. Under a standard atmospheric pressure, the density of N2 is 1.160 kg/m3.
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Figure 4. Operational steps and model of the multiphysics system for the microactuator.
Variable Air Capacitance For an elastic material device, there are two main parameters to reflect the characteristics of the device: elastic compliance constant and capacitance. The former is a measure of flexible for elastomer, and the latter is an ability to store air quantity in specific applications. What is a feature about capacitance for an elastic device is not only that it stores or releases air, as a container, but also that it can buffer or steady air pressure. In this study, capacitance of the air capacity decides the charging time to a large extent. On the basis of eq 8, C as a function of differential pressure (ΔP) can be obtained by solving the power function using MATLAB (R2009a; MathWorks Company, Natick, MA) compiler.
Multiphysics System Models The goals of the final system model are to predict the response time of the pneumatic microactuator and to further discuss its two main influencing factors. Figure 4 describes each operational step as well as the model for the microactuator system. According to eq 4, the maximal deformation of the membrane is a function of the differential air pressure on the membrane and differential pressure, as well as a function of inflation/deflation time, which depends on the area of valve port and variable air capacitance, per eq 6. That is, δmax = f(ΔP) and ΔP = f(t, A, C). The relationship t = 3.5T reflects that t = f(A, C). Transforming eq 6 and substituting eqs 2, 3, and 5 obtains the general expression
T = ( Ri + Rv )C = ( Ri + ε
−2 ρ Qv )C0 (∆P) 3 ) 2 2 Av
(10)
t = 3.5 × ( Ri + ε
−2 ρ Qv 3 ) C ( ∆ P ) 0 2 Av2
(11)
Result and Discussion The hydraulic resistance of the microchannel created by using a PDMS material, the deflection of PDMS thin film under different pressures, and the response time of the pneumatic microactuator are extracted from the experiments of refs. 13, 28, and 29. In the simulation, the value of the supplied gas (N2) pressure was varied from 5 kPa to 200 kPa to investigate the evolution of the response performance of the pneumatic microactuator under various supplied pressures and different valve-opening degrees, to predict the response time, and to evaluate the effect of different valve-opening degrees and different supplied pressures on operating conditions of the pneumatic microactuator.
Membrane Deformation The simulated results for the elastic membrane deformation simulation are illustrated in Figure 5a. The results are for a PDMS membrane. The parameters are described in Table 2, and the supplied pressure was 100 kPa. The deformation of the membrane under different pressures was simulated using the ANSYS parametric design language (APDL) and the program module circulating function of the ANSYS software (12.0; ANSYS Company, Canonsburg, PA). Figure 5b compares the theoretical, simulated, and experimental results for a maximum membrane deformation as a function of the applied pressure. It can be seen that the simulated results obtained from the APDL program module are similar to the results of Gustavo and coworkers,13 from the laboratory LAAS-CNRS in Toulouse, France.
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Figure 5. Membrane deformation for (a) a 40 µm thick, 500 × 20 µm membrane and (b) comparison between theoretical, simulated, and experimental results for the deformation of a 500 × 200 × 40 µm membrane.
According to their experimental results, the authors13 reported that the maximum membrane deformation was a function of the air pressure inside the gas chamber by “Bulge test,” which is a well-known simple method for measuring thin-film elastic mechanical properties. The experimental values of the maximum deformation, of the elastomeric membrane, can be expressed by eq 12:
δ max = 0.0001*(∆P)3 − 0.0148 *(∆P)2 + 2.1505 * ∆P + 0.5352
(12)
Figure 5b also explains the correlation between simulated and experimental results for maximum membrane deformation of the pneumatic microactuator. According to the simulation results, the membrane deformation – differential pressure are fitted in the following linear relation: δ max =1.284 * ∆P . When the range of the differential pressure is between 100 kPa and 110 kPa, or close to 0 (initial stage of the curves in Figure 5b), the simulated maximum membrane matches well with the experimental results. The maximum error between the simulated and experimental results is 23% when the differential pressure is equal to about 50 kPa. This comparison shows a difference in the nonlinearity of the curves. The simulation curves obtained from APDL are linear because of their assumptions, such as neglecting the compressibility of gas and the slip boundary condition of air in the microchannel. The theoretical curves have a larger mean curvature because an empirical formula has certain unreliability. The differences in the nonlinearity of the curves are similar with the comparison curves, and a similar cause analysis has previously been found for the inconsistent curves.13
Hydraulic Resistance The local resistance of the valve port for various valveopening degrees is shown in Figure 6a. It can be confirmed that different valve-opening degrees have a great impact on local hydraulic resistance. A valve-opening degree of 0% means that the microvalve is fully closed, whereas a valveopening degree of 100% means that it is fully open. It can be seen in Figure 6a that the hydraulic resistance drops sharply at smaller valve-opening degrees with constant differential pressure. The local hydraulic resistance becomes constant when the valve-opening degree is higher than 20%. This phenomenon agrees with the installed flow characteristic of a flow control valve.29 It is important to note that the data of hydraulic resistance under the 0% valve-opening degrees are obtained from the experiments (the apparatus is described in Figure 5a). The size of the local resistance by microvalve depends not only on the valve-opening degree but also on the differential pressure across the microvalve, as shown in Figure 6b. The results prove that when the valve-opening degree decreases, the local hydraulic resistance, introduced by the microvalve, increases with constant differential pressure. In addition, when the differential pressure is increased, the hydraulic resistance becomes greater under the condition of a fixed valve-opening degree. To evaluate the hydraulic resistance of the microvalve under different valve-opening degrees, the pressure-flow characteristics must first be measured, and then the hydraulic resistance of the microvalve must be calculated according to eq 3. As illustrated in Figure 7a and b, the microvalve was connected to a screw at the bottom of the electromagnetic actuator; this enables the valve-opening degree to be
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Figure 6. Hydraulic resistance under different valve-opening degrees (a), hydraulic resistance under differential pressures (b), and comparison of simulated results of hydraulic resistance with experimental data (c).
Figure 7. (a) Picture of the manual measuring hydraulic resistance setup of a microvalve. (b) Schematic diagram of a measuring setup with an external manipulator consisting of a micrometer screw, which positions the valve-opening degree (not to scale).
adjustable.30 The exact valve head position could be determined via the micrometer M3*30 screw (GB818-85). The inlet of the microvalve was connected to a gas supply, the outlet tube was connected to a port of the flow sensor (ASF1430, SENSIRION Company, Switzerland), and another port of the flow sensor was connected to the atmosphere. The differential pressure across the microvalve and displacement of valve head were measured by a miniature pressure sensor (Xcl-080; Kulite, Leonia, NJ) and a micro displacement sensor (LK-G5000; Keyence, Osaka, Japan). The simulated and experimental results of the local hydraulic resistance under different valve-opening degrees are shown in Figure 6c. The simulation and experimental data of the hydraulic resistance are a combination of frictional hydraulic resistance and local hydraulic resistance. The graph explains the characteristics of the hydraulic resistance, of the TBC system, as a function of the valve-opening degree
in which the total hydraulic resistance decreases with increasing increments of the valve-opening degree. The modeling error was evaluated from the comparison of simulated and experimental results of the total hydraulic resistance of the TBC system at different valve openings. The maximum error of the total hydraulic resistance was up to 37.5% when the valve opening was 0.2. The difference between the modeling results and the experimental measurements could be due to the uncertainty in the calculation of an invariable resistance using an empirical formula as well as the assumption of the material surface roughness in the simulation model and the imperfection of the ideal gas theory. All enhance the simulation error, but the trend of simulated results is consistent with what was observed in the experiment. Further experiments are required to determine what caused the errors in both the test fictional hydraulic resistance and local hydraulic resistance.
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Figure 8. (a) Air capacitance. (b) Twelve-response curve of the throttle blind capacitor (TBC) with certain hydraulic resistance (RwC0 = 0.6). (c) Response time of TBC with different hydraulic resistance. (d) Response time of TBC with different valve-opening degree.
Variable Capacitance The air variable capacitance has nonlinear pressure-displacement characteristics enclosed in PDMS-material enclosures. Figure 8a represents the variable C as the function of the differential pressure between chamber pressures P1 and P2. According to eq 8, E = 0.75/(1 – 0.5)2 = 3*103 kPa, C0 = 0.19*10–13.
Response Time of the Pneumatic Microactuator The simulated response time of the pneumatic microactuator is shown in Figure 8b. The system output (maximum membrane deformation) was 100 µm, and the simulation results showed that it needed a differential pressure of 77.9 kPa to reach its maximum deformation. To have a certain margin to guarantee adequate output, the pressure was increased to 80 kPa. Thus, 63% of P1 was equal to 50.4 kPa. In Figure 8b, when the valve-opening degree is 0.8 (RwC0 = 0.6), the response time (t) of the pneumatic microactuator was t = 3.5 × 0.477 = 1.67 ms. The response time is dependent on the hydraulic resistance of the system (Fig. 8c). A linear relation between RwC0 and response time of the pneumatic microactuator is illustrated, which agrees with its mathematic model (T = RwC and t = 3.5T). When the valve-opening degree is 1, the total hydraulic resistance is 2.87 × 1013 Pa/(m3/s) (Fig. 6c), and RwC0 = 0.55 and the response time is 1.61 ms, as shown in Figure 8c. This is the same for the recovery time of the micro pneumatic, when the deformation time of the PDMS membrane does not have a recovery force. When the actuated pressure was 100 kPa and membrane deformation was 10 µm, the open response time was 3.63 ms and the close response time was 2.15 ms.28 The pneumatic microactuator can be reset
back to its initial position only by its elastic recovery power. The close response time was a tiny bit smaller than the open response time. Comparing the simulated results with experimental data, the former were smaller than the latter because the simulated volume of the gas chamber (500 × 200 × 100 µm) was larger as was the active membrane area (200 × 500 µm). In addition, the conditions for the simulation were completely idealized. This is the reason for the deviation between the simulated results and the experimental data. In a real application, however, the microactuator membranes collapse and rise in a few cases because the elastomeric nature of PDMS may cause shrinking or sagging. If this happens, both of the open and close response time will be much longer than the simulated results in this work. Equation 11 displays the calculated results for the response time of TBC, and the results are shown in Figure 8d. It should be noted that the simulations are performed under ideal conditions in which, to simplify the model, the pressure across the microvalve was assumed to be constant. In an actual microfluidic system, however, the pressure drop could change and couple with the hydraulic resistance in the inflation and deflation processes of the TBC.
Influencing Factors to Response Time The main factors influencing the response time of a pneumatic microactuator are its local hydraulic resistance and capacitance. The local hydraulic resistance is determined by the valve-opening degree of a microvalve, which is used to control the gas flow. The capacitance is determined by the volume of the gas chamber. In this work, the volume of the gas chamber was variable and similar to the pneumatic bellow. Its size varies depending on membrane curvature, which was affected by the air pressure in the gas chamber. Thus, the valve-opening degree, air pressure in the gas
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chamber, initial volume of the gas chamber, and variable volume of the gas chamber are a coupling system. Their combined effects influence the response time of the pneumatic microactuator.
Future Work In this article, we present a systematic TBC response process that is analogous to a pneumatic microactuator using multiphysics theoretical models. The simulation method was validated by comparing simulated membrane deformation, hydraulic resistance, and response time with corresponding experimental results. According to the simulated results in this work, a pressure of 80 kPa should be applied for 1.67 ms to the TBC in order to obtain a maximal membrane deformation of 100 µm under the valve-opening degree of 0.8 (RwC0 = 0.6). The response time was 1.61 ms when the microvalve was fully open. Two main influencing mechanisms of the response time, valve-opening degree of the microvalve and air capacitance of the gas chamber, were determined by using a finite element model and then comparing the experimental data to the theoretical data, which offered a useful quantitative prediction for the response time as a function of its dimensions and valve-opening degree. The main application of the present model was to design more accurate pneumatic actuators, which would decide the efficiency of the whole pneumatic microfluidic chips. In particular, it was shown that the model could be used to investigate the mechanical performance of a pneumatic actuator before planning expensive, time-consuming, and difficult experimental programs. Further work needs to be done to consider the permeability of air to the PDMS membranes and elastic channel deformation, which can lead to a significant deviation from the true phenomenon inside the chip. As we mentioned, the effect of ideal air models and roughness can play an important role on the effect; thus, we take into account that future research can include atomistic models, such as molecular dynamics models or dissipative particle dynamics or couple them to larger-scale models in the frame of multiscale simulations.
Nomenclature Rw: total hydraulic resistance (Pa/(m3/s)) Ri: invariable resistance that relates to the microchannel (Pa/(m3/s)) Rv: variable resistance that relates to the microvalve (Pa/(m3/s)) µ: viscosity of the fluid in the gas microchannel (Pa•s) l0, w0, h0: the length, width, and height of the gas microchannel (µm) Qv: fluid flow through the microvalve (m3/s)
ε: pressure loss coefficient of the microvalve p: density (kg/m3) Av: effective flow area of the microvalve port (m2) lm,, wm,, tm: length, width, and thickness of the thin PDMS membrane (µm) δmax: maximum displacement of the membrane (µm) ΔP: pressure difference across the membrane (kPa) Pa: hydrostatic pressure in the gas microchannel (kPa) Pl: constant pressure in the liquid microchannel (kPa) E: Young’s modulus (MPa) v: Poisson ratio : plane-strain modulus (MPa) E σ0: residual stress (Pa) V: volume under the deflection membrane (µm3) t: response time (ms) P1: air pressure outside the air capacity (kPa) P2: air pressure inside the air capacity (kPa) T: time constant of the variable air capacity C: variable air capacity of the gas chamber (m3/Pa) C0: coefficient la, wa, ha: length, width, and height of the gas chamber (µm) Acknowledgments The authors would like to acknowledge Nicole Louise Drey, a PhD student in the Department of Engineering Science and Mechanics at Pennsylvania State University, for valuable discussion and help.
Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is financially supported by the National Natural Science Foundation of China (51175101) and Abroad Short-Term Visiting Program for PhD candidates of HIT.
References 1. Hong, J. W.; Quake, S. R. Integrated Nanoliter Systems. Nat. Biotechnol. 2003, 21, 1179–1183. 2. Thorsen, T.; Maerkl, S. J.; Quake, S. R. Microfluidic LargeScale Integration. Science 2002, 298, 580–584. 3. Zheng, Y.; Dai, W.; Wu, H. A Screw-Actuated Pneumatic Valve for Portable, Disposable Microfluidics. Lab Chip 2009, 9, 469–472.
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Liu et al. 4. Araci, I. E.; Quake, S. R. Microfluidic Very Large Scale Integration (mVLSI) with Integrated Micromechanical Valves. Lab Chip 2012, 12, 2803–2806. 5. Studer, V.; Hang, G.; Pandolfi, A.; et al. Scaling Properties of a Low-Actuation Pressure Microfluidic Valve. J. Appl. Phys. 2004, 95, 393–398. 6. McDonald, J. C.; Whitesides, G. M. Poly(dimethylsiloxane) as a Material for Fabricating Microfluidic Devices. Accounts Chem. Res. 2002, 35, 491–499. 7. Sia, S. K.; Whitesides, G. M. Microfluidic Devices Fabricated in Poly(dimethylsiloxane) for Biological Studies. Electrophoresis 2003, 24, 3563–3576. 8. Du, M.; Ye, X.; Wu, K.; et al. Simulation and Experiments for Optimization of Circular Peristaltic Micropumps. Nano. Precision Eng. 2010, 8, 321–329. 9. Pandolfi, A.; Michael, O. Numerical Analysis of Elastomeric Fluidic Microvalves. Sens. Lett. 2008, 6, 43–48. 10. Kim, T. K.; Jeong, O. C. Effects of Mechanical Strain on Intracellular Calcium Expression in Cell. 15th International Conference on Miniaturized Systems for Chemistry and Life Sciences, MicroTAS, 2011, 335–337. 11. Guan, Y.; Liu, B.; Xu, Z. Finite Element Analysis of PDMS Pneumatic Peristaltic Micropump. Instr. Tech. Sens. 2007, 4, 68–75. 12. Mohan, R.; Schudel, B. R.; Desai, A. V.; et al. Design Considerations for Elastomeric Normally Closed Microfluidic Valves. Sens. Actuat. B 2011, 160, 1216–1223. 13. Gustavo, A. R.; Carloe, R.; Kaili, Z. Multi-Physics System Modeling of a Pneumatic Micro Actuator. Sens. Actuat. A 2008, 141, 489–498. 14. Mongpraneet, D.; Wisitsora-at, A.; Kamnerdtong, T.; et al. Simulation and Experiment of PDMS Based Thermopneumatic Microvalve in Microfluidic Chip. 6th International Conference on Electrical Engineering/ Electronics, Computer, Telecommunications and Information Technology, ECTI-CON. 2009, 458–461. 15. Lee, S. W.; Kim, D. J.; Ahn, Y.; et al. Simple Structured Polydimethylsiloxane Microvalve Actuated by External Air Pressure. Mech. Eng. Sci. 2006, 220, 1283–1288. 16. Yang, Q.; Kobrin, P.; Seabury, C.; et al. Mechanical Modeling of Fluid-Driven Polymer Lenses. Appl. Optics 2008, 47, 3658–3668.
17. Hardy, B. S.; Uechi, K.; Zhen, J.; et al. The Deformation of Flexible PDMS Microchannels under a Pressure Driven Flow. Lab Chip 2009, 9, 935–938. 18. Cui, H.; Li, Z.; Jin, G. Preparation and Performance Analysis of a PDMS-Membrane Microvalve. Microfab. Tech. 2004, 3, 70–75. 19. Kartalov, E. P.; Scherer, A.; Quake, S. R.; et al. Experimentally Validated Quantitative Linear Model for the Device Physics of Elastomeric Microfluidic Valves. J. Appl. Phys. 2007, 101, 1–4. 20. Lau, A. T. H.; Yip, H. M.; Ng, K. C. C.; et al. Dynamics of Microvalve Operations in Integrated Microfluidics. Micromachines 2014, 5, 50–65. 21. Yu, B. Microfluidic Large Scale Integration and Its Application in Image Based Microflow Cytometry. M.Phil. Thesis, University of Waterloo, Canada, 2010. 22. Anjewierden, D. An Enectrostatic Microvalve for Pneumatic Control of Microfluidic Systems. M.Phil. Thesis, University of Utah, 2011. 23. Ni, J. Research and Application on Planar PDMS Microvalve and Micropump for Microfluidic System. Ph.D. Thesis, Donghua University, China, 2011. 24. Wu, Z. Pneumatic Transmission and Control. Harbin Institute of Technology Press: Harbin, China, 2003. 25. Huang, S. B.; Wu, M. H.; Cui, Z.; et al. A Membrane-Based Serpentine-Shape Pneumatic Micropump with Pumping Performance Modulated by Fluidic Resistance. J. Micromech. Microeng. 2008, 18, 1–12. 26. Li, S.; Liu, X.; Jia, W. Simulation on Characteristics of a Pneumatic Electromagnetic Microvalve. Chin. Hydraul. Pneum. 2013, 7, 6–8. 27. Wu, Q. Instrumentation and Control Devices. Chemical Industry Press: Peking, 2002. 28. Unger, M. A.; Chou, H.; Thorsen, T.; et al. Monolithic Microfabricated Valves and Pumps by Multilayer Soft Lithography. Science 2000, 288, 113–116. 29. Zhang, Y. Experimental Research on Flux Characteristic of Valves. J. Meas. Control Tech. Suppl. 2008, 27, 256–259. 30. Liu, X.; Li, S. An Electromagnetic Microvalve for Pneumatic Control of Microfluidic Systems. J. Lab. Autom. 2014, 4, 1–10.
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research-article2015
JLAXXX10.1177/2211068215584769Journal of Laboratory AutomationLiu et al.
Original Report
Numerical Simulation on the Response Characteristics of a Pneumatic Microactuator for Microfluidic Chips
Journal of Laboratory Automation 1–11 © 2015 Society for Laboratory Automation and Screening DOI: 10.1177/2211068215584769 jala.sagepub.com
Xuling Liu1, Songjing Li1, and Gang Bao1
Abstract This article presents a multiphysical system modeling and simulation of a pneumatic microactuator, which significantly influences the performance of a particular pneumatic microfluidic device. First, the multiphysical system modeling is performed by developing a physical model for each of its three integrated components: microchannel with a microvalve, a gas chamber, and an elastomer membrane. This is done for each step of operation for the whole system. The whole system is then considered a throttle blind capacitor model, and it is used to predict the response time of the pneumatic microactuator by correlating its characteristics such as gas pressurizing, hydraulic resistance, and membrane deformation. For this microactuator, when the maximum membrane deformation is 100 µm, the required actuated air pressure is 80 kPa, and the response time is 1.67 ms when the valve-opening degree is 0.8. The response time is 1.61 ms under fully open conditions. These simulated results are validated by the experimental results of the current and previous work. A correlation between the simulated and experimental results confirms that the multiphysical modeling presented in this work is applicable in developing a proper design of a pneumatic microactuator. Finally, the influencing factors of the response time are discussed and analyzed. Keywords pneumatic microactuator, PDMS membrane, TBC model, response time, influencing factors
Introduction Microfluidic large-scale integration (mLSI)1 refers to the fabrication and operation of hundreds or thousands of active microfluidic components on a single chip, to achieve specific chemical and biological processes. The pneumatic microfluidic chip allows for dense integration. It is feasible for mass production and commercialization because its control system and pressure generation are usually off chip.2 Using a soft lithography process involving thick-film micromachining and replica molding of elastomeric materials, pneumatic microactuators are most widely used and they offer a number of advantages.3 They are easy to manufacture at high density, which makes the mLSI (thousands of valves per cm2) or microfluidic very-large-scale integration (approaching 1 million valves per cm2)4 of microfluidic devices possible, which is important for the integration of a microfluidic system. In practical applications, it is preferable to apply the pneumatic actuator to elastic materials, such as polydimethylsiloxane (PDMS)–based systems. This is due to the key merits of PDMS material, such as transparency, mechanical properties, simple structure bonding process, and low production cost. PDMS has a much lower Young’s modulus
compared with other hard materials, such as silicon and silicon nitride,5 enabling it to be used as the deformable membrane of microfluidic devices, which is important for the pneumatic microactuator. PDMS membranes employed in pneumatic actuation are active in microvalves, micropumps, and micromixers. They have become popular and promising in these applications because of rapid and reliable fabrication.6,7 In recent years, while focusing on the performance of deformable membranes under different applied pressures, researchers have studied modified geometries of the pneumatic microactuator. The Quake research group5 studied the effects of variation of membrane-related parameters (such as membrane width, membrane thickness, and actuation 1
Department of Fluid Control and Automation, Harbin Institute of Technology, Harbin, China Received Dec 21, 2014. Corresponding Author: Songjing Li, Department of Fluid Control and Automation, Harbin Institute of Technology, Box 3040, Science Park, No. 2, Yikuang Street Nangang District, Harbin, 150001 P.R China. Email: [email protected]
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channel width) on membrane deflection of a pneumatic microactuator using an ANSYS computational model and then compared the model with a series of experimental results. It was found that the required closed air pressure, in the air chamber, for a push-down pneumatic actuator (microvalve) was 137.9 kPa and only 16.55 kPa for a pushup pneumatic actuator (microvalve). Zhou et al.8 developed the influence of parameters on leakage ratio of the pneumatic micropump, which composed three pneumatic microactuators. These parameters include the aspect ratio of the air chamber, applied force to the membrane, and so on. The authors presented the two-dimensional contact model established with ANSYS finite element analysis. Also, a finite element model of PDMS pneumatic actuator membrane deflection and two sets of experiments for validating the model were studied in refs. 9, 10, and 11. In addition, Mohan et al.12 analyzed the geometry of three different normally closed valves and found that the pneumatic actuators of the normally closed valves, with a sharp corner feature (V-shaped), actuated at lower driving pressures compared with straight-shaped valves and determined that membrane thickness does not significantly influence the actuation pressures by observing a series of relevant experiments. The understanding and management of deformable membrane performance have also been supported by many numerical models of pneumatic microactuator-related researches. The authors of refs. 13 and 14 had studied the coupling models of thermo-pneumatic PDMS microactuators. The pneumatic microactuator was used with a heating electrode, which can be coupled to a flexible membrane for generating actuation. The relationship between the applied voltage, on the heating electrode, and the membrane deflection can be obtained through the coupling model using COMSOL multiphysics. Lee and coworkers15 designed and investigated a finite element model of the pneumatic actuator components to calculate the displacement distribution of membrane deflection under pressures. The mathematical models, another method for evaluation of membrane deflection displacements versus different applied pressures, were introduced. The research of Yang et al.16 proved the effectiveness of the mathematical model through a series of experiments. The mathematical models, the COMSOL simulation models, and the corresponding experiments about membrane deflection of a PDMS chamber were proposed by Hardy et al.17 Cui et al.18 had analyzed and understood the different factors that influenced the performance of the pneumatic microactuator, according to the theory of membrane distortion, and created the computational model about sealing surface area versus membrane deflection. Kartalov et al.19 presented and analyzed three basic linear models (thick beam, thin spring, and thick spring) for the PDMS membrane of the pneumatic microactuator and proposed that the thin spring model was closest to reality and finally gave a superposed systemic model to predict the membrane
property. The results were proved by experimentation, using push-down microactuators. Lau et al.20 determined that the dynamics of the pneumatic actuators (microvalves) on chip were indeed influenced not only by the configurations, such as membrane thickness and driving pressure, but also by the levels of design complexity and positions in the devices. Previous studies, both experiments and simulations about pneumatic actuator, have focused on the influence of various membranes, air chamber geometries, or pressure on the deformation of thin membranes. The influence, however, on the whole performance of the pneumatic actuator, such as response time and air transient pressure changes in the air chamber versus time, have not yet been studied. In addition, the studies of air pressure changes during the response period, the response time, and its factors related to the pneumatic microactuator are far from being understood, and further research is required. Therefore, the motivation of this work is to obtain the suitable actuated air pressures and response times when optimizing the design of a system and to provide a promising simulation for pneumatic microfluidic chips; therefore, the response time of the pneumatic microactuator and its influencing factors are studied in this work, and a methodology to investigate optimal response time is provided to improve the whole efficiency of microchips. By using a multiphysics system model, we numerically analyze the performance of a standard pneumatic actuator to reduce the size of the driving pressure in real applications and the duration of response time to improve its efficiency. Each individual physical model is built separately. All of the models are then interfaced with a throttle blind capacitor (TBC) systemic model for predicting the global operation and actuation performance.
Architecture and Working Principle A length of gas microchannel with a flow control microvalve, an air chamber connecting the microchannel, and a PDMS thin membrane, which locates the upper air chamber, constitute the pneumatic microactuator.21,22 The elastomer thin membrane is the most important and only active part. Its deformation exerts a tremendous influence on the response time of the pneumatic microactuator. The structure of the pneumatic microactuator with a normally closed microvalve is shown in Figure 1. The geometrical parameters taken for the simulations and experiments are described in Table 1. When the actuation is required, a voltage is supplied on the normally closed microvalve, which opens it, and the gas is supplied to the microactuator. The pressure in the air chamber increases and deforms the membrane. PDMS deforms under applied pressures, which can be used in robust manipulation schemes that pinch-off, drive, and mix working fluids in the microfluidic systems.
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Figure 1. Schematic diagram of the pneumatic microactuator with a normally closed microvalve (a) and cut view (b–d; not to scale). Table 1. Pneumatic Microactuator Geometrical Parameters. Geometrical Parameter
la, lm
wa, wm
ha, h0
tm
l0
w0
Value (µm)
500
200
100
40
3000
100
Mathematical Models The comprehensive model formulated in this study was developed by using several different physics describing the different parts of the pneumatic operation: the valve-opening degree of the normally closed microvalve decides the local hydraulic resistance; the length, width, and height of the microchannel dominate the frictional hydraulic resistance; the geometry of the gas chamber decides the volume of the air capacity; and the deflection level of the elastomer membrane related to the length, width, and thickness settles the size of the variable volume of the air capacity and also has a tremendous influence on the response time of the pneumatic microactuator.
Gas Flow Theory Pneumatic systems use gas to apply force. The perfect gas law is the equation of state of a hypothetical ideal gas, which is an approximation to the behavior of many different gases under different conditions. Thus, the gas used in our simulation is considered a perfect gas. In a miniaturized gas system, the flow velocity is very low (nL/min ~ L/min), and the hydraulic diameter of a microchannel is very small
(µm ~ mm); therefore, the Reynolds number23 is always much smaller than 2300 (10–25~1) and the gas is always in the condition of laminar flow. In this article, N2 (nitrogen) is the medium in which the gas is supplied to the membrane deformation through the air capacity.
Hydraulic Resistance Hydraulic resistance refers to the friction that prevents the gas from going forward. For the pneumatic microactuator, which is integrated with microvalves, micropumps, or micromixers, the hydraulic resistance, which relates the small size PDMS microchannel, should be considered in the microfluidic networks. The total hydraulic resistance (Rw) of the TBC includes two parts,24 as shown in eq 1: the constant resistance (Ri) created by a microchannel and the variable resistance (Rv) created by the valve-opening degree of the normally closed microvalve. Rw = Ri + Rv
(1)
The constant resistance (Ri) can be estimated using the following empirical formula for a rectangular microchannel with a large width (w0 > h0)25:
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Journal of Laboratory Automation Maximum displacement of deflection is a function of the pressure over the membrane, which can be calculated by eq (4).21 1
δ max = wm (3∆Pwm )3 4 Etm
Figure 2. The schematic diagram of the throttle blind capacitor. Qv, volume flow rate through the microvalve; Rw, total hydraulic resistance.
Ri =
12µl0 w0h03
∞ 192 h0 1− 5 w 0 n,odd π
∑
1 tan h (nπ w0 ) 0 5 2h0 n
(2)
Rv = ε
ρ Qv 2 Av2
where δmax is the maximum deformation of the membrane; ΔP = Pa – Pl is the applied pressure to the membrane; Pa is the pneumatic pressure in the gas microchannel; Pl are the other forces to the membrane, which are in the opposite direction to the air pressure; E = E / (1 −ν )2 is the plane-strain modulus; E is the Young’s modulus; and v is the Poisson ratio of the material. The volume (V) under the deflection membrane can be estimated by eq 5:
where µ is the viscosity of the fluid in the gas microchannel and l0, w0, and h0 are the length, width, and height of the gas microchannel, respectively. It is shown from eq 2 that the value of hydraulic resistance (Ri) is controlled by the geometry of the microchannel and fluid viscosity, which are assumed to be independent of the flow rate inside the microchannel. To the authors’ knowledge, Rv is a variable value that is caused by microvalve pressure loss26: (3)
Per eq 3, Qv is the volume flow rate through the microvalve, ε is the pressure loss coefficient, p is the density of fluid through the microvalve, and Av is the effective flow area of the microvalve port.
Membrane Deformation Theory The PDMS elastomer is modeled as a near-incompressible, Neo-Hookean material. This constitutive model describes the behavior of rubberlike materials undergoing large deformations.5 The large deformation membrane behavior theory is also called plane-strain theory. Using this theory, for a rectangular shape, three assumptions must be satisfied: (1) the length of the membrane (lm) is at least two times larger than the width (wm), (2) the width (wm) of the membrane is 10 times the membrane thickness (tm), and (3) the deflection of the membrane is more than the membrane thickness.21 If all the assumptions are satisfied, then the variation in deflection along the length can be ignored and the centerline stretch of the membrane dominates bending. We resort to the finite element method (FEM) to represent the geometry of the rectangle elastic thin membrane and to solve the equations governing its deformation.
(4)
V = 2lm wmδ max 3
(5)
After the pneumatic microactuator reflects, the deflection of the membrane is parabolic.
Response Time of Pneumatic Microactuator The pneumatic microactuator always needs an external microvalve to control the pressure of the gas chamber and to further control the degree of the thin membrane deformation. The microvalve and the pneumatic microactuator form a whole pneumatic system, also named TBC,27 shown in Figure 2, which is a function of the variable hydraulic resistance and the variable air capacitor. An electromagnetic microvalve acts as the variable hydraulic resistance. The pneumatic chamber acts as the variable air capacitor because the volume of the chamber is always variable according to the air pressure. In Figure 2, total hydraulic resistance Rw is defined as the ratio of differential pressure across the system and air flow of the system. C is the variable air capacity of the gas chamber.27 Given the previous index analysis, Rw and C are two variable values. They are independently controlled by the valve-opening degree of the microvalve and the volume change of the variable air capacity. When P1 > P2, the TBC is in the inflation process. The TBC system time constant is T (T = Rw · C), and it is a variable value because of variable hydraulic resistance (Rv) and variable volume of air capacity (C). According to the authors’ knowledge, the transfer function of the TBC is a first-order inertia unit, and its storage energy element is the variable air capacity. The period from the closed to the open moment of the microvalve gave the system an input signal—the unit step signal, which is one of the fundamental signals. Then, −t T P2 = P1 (1 − e ) t P = Pe− T 2 1
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P1 > P2 P1 < P2
(6)
5
Liu et al. Using eq 6 in the inflation course of the TBC, its transient pressure in the air chamber increased exponentially. When P2 > P1, the TBC is in the deflation process. Hydraulic resistance was introduced in the Hydraulic Resistance section. The volume (V) under the deflecting membrane is defined in eq 5, which is a function of the maximum deflection δmax, and δmax is a function of the pressure over the membrane: C (∆P) =
dV (∆P) d 1 2 3wm ∆P 13 = ( lw ( ) ) d ∆P d ∆P 6 a m Etm l w2 3w 1 − 2 = a m ( m ) 3 ∆P 3 18 Etm −2 3
(8)
la wm2 3wm 13 ( ) 18 Etm
(9)
C (∆P) = C0 ∆P
C0 =
(7)
The response time (t) of the RwC circuit is about 3.5-fold of the RwC time constant (T), t = 3.5T. When pressure P2 is approximately equal to the air supply pressure P1, the inflation course is finished. The time constant (T) is equal to the required time, which is needed for P2 to reach 63.2% of P1, when a step signal is given.
Model Implementation
Figure 3. Meshing of the polydimethylsiloxane elastic membrane. Table 2. Parameters of Polydimethylsiloxane Membrane for Simulation and Experiment. Prepolymer solution ratio: (base/curing agent) Cured temperature (°C) Cured time (h) Young’s modulus (MPa) Possion ratio Density (kg/m3) Specific heat (Pj/kg K) Thermal conductivity (W/m K)
15:1 65 3 0.75 0.499 0.97 1.46 0.15
In this study, the gas permeability of PDMS and pressure change due to channel deflections were neglected for simplified pressure analysis.
The residual stress (σ0) equaled zero at steady state; it also equaled zero when the applied pressure was 100 kPa, and the boundary condition of the membrane was held rigid on all sides of the model.
Membrane Deformation
Total Hydraulic Resistance
The FEM was used to represent the three-dimensional geometry of the PDMS membrane and to simulate its deformation. The membrane deformation model is implemented using ANSYS software. Because of the twofold symmetry of the problem, the computational model is restricted to one-fourth of the membrane, with symmetry boundary conditions applied on all symmetry planes. The elements employed in the calculations are 10-node quadratic tetrahedral. The meshing for the PDMS membrane contains 58,042 tetrahedral elements. The mesh density in the membrane region was increased to obtain better accuracy on the elastic membrane’s deformation area, shown in Figure 3. The mesh density on the deformation area was quadrupled in reference to the other parts. A uniform pressure was applied over the surface of the PDMS elastic membrane. Table 2 provides the parameters of the PDMS membrane used in the model and the experiments.
The value of the fixed resistance (Ri) is determined by the length, width, and height of the microchannel, which is located between the gas chamber and the gas supply, as well as the viscosity of the fluid in the gas microchannel. The viscosity of N2 is 17.544 µPa•s when the room temperature is 25 °C. According to eq 2, Ri = 1.57*1012 Pa/(m3/s). The value of variable hydraulic resistance (Rv), which is related to the microvalve pressure loss, was determined by the valve-opening degree. In this article, the characteristics of flow rate – differential pressure were obtained by performing a numerical simulation under different valve-opening degrees using computational fluid dynamics software Fluent (in ANSYS 12.0) and the user-defined function, which can be programmed according to the user’s demand and dynamically linked to the solver. The local hydraulic resistance of the microvalve is calculated by eq 3. Under a standard atmospheric pressure, the density of N2 is 1.160 kg/m3.
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Figure 4. Operational steps and model of the multiphysics system for the microactuator.
Variable Air Capacitance For an elastic material device, there are two main parameters to reflect the characteristics of the device: elastic compliance constant and capacitance. The former is a measure of flexible for elastomer, and the latter is an ability to store air quantity in specific applications. What is a feature about capacitance for an elastic device is not only that it stores or releases air, as a container, but also that it can buffer or steady air pressure. In this study, capacitance of the air capacity decides the charging time to a large extent. On the basis of eq 8, C as a function of differential pressure (ΔP) can be obtained by solving the power function using MATLAB (R2009a; MathWorks Company, Natick, MA) compiler.
Multiphysics System Models The goals of the final system model are to predict the response time of the pneumatic microactuator and to further discuss its two main influencing factors. Figure 4 describes each operational step as well as the model for the microactuator system. According to eq 4, the maximal deformation of the membrane is a function of the differential air pressure on the membrane and differential pressure, as well as a function of inflation/deflation time, which depends on the area of valve port and variable air capacitance, per eq 6. That is, δmax = f(ΔP) and ΔP = f(t, A, C). The relationship t = 3.5T reflects that t = f(A, C). Transforming eq 6 and substituting eqs 2, 3, and 5 obtains the general expression
T = ( Ri + Rv )C = ( Ri + ε
−2 ρ Qv )C0 (∆P) 3 ) 2 2 Av
(10)
t = 3.5 × ( Ri + ε
−2 ρ Qv 3 ) C ( ∆ P ) 0 2 Av2
(11)
Result and Discussion The hydraulic resistance of the microchannel created by using a PDMS material, the deflection of PDMS thin film under different pressures, and the response time of the pneumatic microactuator are extracted from the experiments of refs. 13, 28, and 29. In the simulation, the value of the supplied gas (N2) pressure was varied from 5 kPa to 200 kPa to investigate the evolution of the response performance of the pneumatic microactuator under various supplied pressures and different valve-opening degrees, to predict the response time, and to evaluate the effect of different valve-opening degrees and different supplied pressures on operating conditions of the pneumatic microactuator.
Membrane Deformation The simulated results for the elastic membrane deformation simulation are illustrated in Figure 5a. The results are for a PDMS membrane. The parameters are described in Table 2, and the supplied pressure was 100 kPa. The deformation of the membrane under different pressures was simulated using the ANSYS parametric design language (APDL) and the program module circulating function of the ANSYS software (12.0; ANSYS Company, Canonsburg, PA). Figure 5b compares the theoretical, simulated, and experimental results for a maximum membrane deformation as a function of the applied pressure. It can be seen that the simulated results obtained from the APDL program module are similar to the results of Gustavo and coworkers,13 from the laboratory LAAS-CNRS in Toulouse, France.
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Figure 5. Membrane deformation for (a) a 40 µm thick, 500 × 20 µm membrane and (b) comparison between theoretical, simulated, and experimental results for the deformation of a 500 × 200 × 40 µm membrane.
According to their experimental results, the authors13 reported that the maximum membrane deformation was a function of the air pressure inside the gas chamber by “Bulge test,” which is a well-known simple method for measuring thin-film elastic mechanical properties. The experimental values of the maximum deformation, of the elastomeric membrane, can be expressed by eq 12:
δ max = 0.0001*(∆P)3 − 0.0148 *(∆P)2 + 2.1505 * ∆P + 0.5352
(12)
Figure 5b also explains the correlation between simulated and experimental results for maximum membrane deformation of the pneumatic microactuator. According to the simulation results, the membrane deformation – differential pressure are fitted in the following linear relation: δ max =1.284 * ∆P . When the range of the differential pressure is between 100 kPa and 110 kPa, or close to 0 (initial stage of the curves in Figure 5b), the simulated maximum membrane matches well with the experimental results. The maximum error between the simulated and experimental results is 23% when the differential pressure is equal to about 50 kPa. This comparison shows a difference in the nonlinearity of the curves. The simulation curves obtained from APDL are linear because of their assumptions, such as neglecting the compressibility of gas and the slip boundary condition of air in the microchannel. The theoretical curves have a larger mean curvature because an empirical formula has certain unreliability. The differences in the nonlinearity of the curves are similar with the comparison curves, and a similar cause analysis has previously been found for the inconsistent curves.13
Hydraulic Resistance The local resistance of the valve port for various valveopening degrees is shown in Figure 6a. It can be confirmed that different valve-opening degrees have a great impact on local hydraulic resistance. A valve-opening degree of 0% means that the microvalve is fully closed, whereas a valveopening degree of 100% means that it is fully open. It can be seen in Figure 6a that the hydraulic resistance drops sharply at smaller valve-opening degrees with constant differential pressure. The local hydraulic resistance becomes constant when the valve-opening degree is higher than 20%. This phenomenon agrees with the installed flow characteristic of a flow control valve.29 It is important to note that the data of hydraulic resistance under the 0% valve-opening degrees are obtained from the experiments (the apparatus is described in Figure 5a). The size of the local resistance by microvalve depends not only on the valve-opening degree but also on the differential pressure across the microvalve, as shown in Figure 6b. The results prove that when the valve-opening degree decreases, the local hydraulic resistance, introduced by the microvalve, increases with constant differential pressure. In addition, when the differential pressure is increased, the hydraulic resistance becomes greater under the condition of a fixed valve-opening degree. To evaluate the hydraulic resistance of the microvalve under different valve-opening degrees, the pressure-flow characteristics must first be measured, and then the hydraulic resistance of the microvalve must be calculated according to eq 3. As illustrated in Figure 7a and b, the microvalve was connected to a screw at the bottom of the electromagnetic actuator; this enables the valve-opening degree to be
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Figure 6. Hydraulic resistance under different valve-opening degrees (a), hydraulic resistance under differential pressures (b), and comparison of simulated results of hydraulic resistance with experimental data (c).
Figure 7. (a) Picture of the manual measuring hydraulic resistance setup of a microvalve. (b) Schematic diagram of a measuring setup with an external manipulator consisting of a micrometer screw, which positions the valve-opening degree (not to scale).
adjustable.30 The exact valve head position could be determined via the micrometer M3*30 screw (GB818-85). The inlet of the microvalve was connected to a gas supply, the outlet tube was connected to a port of the flow sensor (ASF1430, SENSIRION Company, Switzerland), and another port of the flow sensor was connected to the atmosphere. The differential pressure across the microvalve and displacement of valve head were measured by a miniature pressure sensor (Xcl-080; Kulite, Leonia, NJ) and a micro displacement sensor (LK-G5000; Keyence, Osaka, Japan). The simulated and experimental results of the local hydraulic resistance under different valve-opening degrees are shown in Figure 6c. The simulation and experimental data of the hydraulic resistance are a combination of frictional hydraulic resistance and local hydraulic resistance. The graph explains the characteristics of the hydraulic resistance, of the TBC system, as a function of the valve-opening degree
in which the total hydraulic resistance decreases with increasing increments of the valve-opening degree. The modeling error was evaluated from the comparison of simulated and experimental results of the total hydraulic resistance of the TBC system at different valve openings. The maximum error of the total hydraulic resistance was up to 37.5% when the valve opening was 0.2. The difference between the modeling results and the experimental measurements could be due to the uncertainty in the calculation of an invariable resistance using an empirical formula as well as the assumption of the material surface roughness in the simulation model and the imperfection of the ideal gas theory. All enhance the simulation error, but the trend of simulated results is consistent with what was observed in the experiment. Further experiments are required to determine what caused the errors in both the test fictional hydraulic resistance and local hydraulic resistance.
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Figure 8. (a) Air capacitance. (b) Twelve-response curve of the throttle blind capacitor (TBC) with certain hydraulic resistance (RwC0 = 0.6). (c) Response time of TBC with different hydraulic resistance. (d) Response time of TBC with different valve-opening degree.
Variable Capacitance The air variable capacitance has nonlinear pressure-displacement characteristics enclosed in PDMS-material enclosures. Figure 8a represents the variable C as the function of the differential pressure between chamber pressures P1 and P2. According to eq 8, E = 0.75/(1 – 0.5)2 = 3*103 kPa, C0 = 0.19*10–13.
Response Time of the Pneumatic Microactuator The simulated response time of the pneumatic microactuator is shown in Figure 8b. The system output (maximum membrane deformation) was 100 µm, and the simulation results showed that it needed a differential pressure of 77.9 kPa to reach its maximum deformation. To have a certain margin to guarantee adequate output, the pressure was increased to 80 kPa. Thus, 63% of P1 was equal to 50.4 kPa. In Figure 8b, when the valve-opening degree is 0.8 (RwC0 = 0.6), the response time (t) of the pneumatic microactuator was t = 3.5 × 0.477 = 1.67 ms. The response time is dependent on the hydraulic resistance of the system (Fig. 8c). A linear relation between RwC0 and response time of the pneumatic microactuator is illustrated, which agrees with its mathematic model (T = RwC and t = 3.5T). When the valve-opening degree is 1, the total hydraulic resistance is 2.87 × 1013 Pa/(m3/s) (Fig. 6c), and RwC0 = 0.55 and the response time is 1.61 ms, as shown in Figure 8c. This is the same for the recovery time of the micro pneumatic, when the deformation time of the PDMS membrane does not have a recovery force. When the actuated pressure was 100 kPa and membrane deformation was 10 µm, the open response time was 3.63 ms and the close response time was 2.15 ms.28 The pneumatic microactuator can be reset
back to its initial position only by its elastic recovery power. The close response time was a tiny bit smaller than the open response time. Comparing the simulated results with experimental data, the former were smaller than the latter because the simulated volume of the gas chamber (500 × 200 × 100 µm) was larger as was the active membrane area (200 × 500 µm). In addition, the conditions for the simulation were completely idealized. This is the reason for the deviation between the simulated results and the experimental data. In a real application, however, the microactuator membranes collapse and rise in a few cases because the elastomeric nature of PDMS may cause shrinking or sagging. If this happens, both of the open and close response time will be much longer than the simulated results in this work. Equation 11 displays the calculated results for the response time of TBC, and the results are shown in Figure 8d. It should be noted that the simulations are performed under ideal conditions in which, to simplify the model, the pressure across the microvalve was assumed to be constant. In an actual microfluidic system, however, the pressure drop could change and couple with the hydraulic resistance in the inflation and deflation processes of the TBC.
Influencing Factors to Response Time The main factors influencing the response time of a pneumatic microactuator are its local hydraulic resistance and capacitance. The local hydraulic resistance is determined by the valve-opening degree of a microvalve, which is used to control the gas flow. The capacitance is determined by the volume of the gas chamber. In this work, the volume of the gas chamber was variable and similar to the pneumatic bellow. Its size varies depending on membrane curvature, which was affected by the air pressure in the gas chamber. Thus, the valve-opening degree, air pressure in the gas
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chamber, initial volume of the gas chamber, and variable volume of the gas chamber are a coupling system. Their combined effects influence the response time of the pneumatic microactuator.
Future Work In this article, we present a systematic TBC response process that is analogous to a pneumatic microactuator using multiphysics theoretical models. The simulation method was validated by comparing simulated membrane deformation, hydraulic resistance, and response time with corresponding experimental results. According to the simulated results in this work, a pressure of 80 kPa should be applied for 1.67 ms to the TBC in order to obtain a maximal membrane deformation of 100 µm under the valve-opening degree of 0.8 (RwC0 = 0.6). The response time was 1.61 ms when the microvalve was fully open. Two main influencing mechanisms of the response time, valve-opening degree of the microvalve and air capacitance of the gas chamber, were determined by using a finite element model and then comparing the experimental data to the theoretical data, which offered a useful quantitative prediction for the response time as a function of its dimensions and valve-opening degree. The main application of the present model was to design more accurate pneumatic actuators, which would decide the efficiency of the whole pneumatic microfluidic chips. In particular, it was shown that the model could be used to investigate the mechanical performance of a pneumatic actuator before planning expensive, time-consuming, and difficult experimental programs. Further work needs to be done to consider the permeability of air to the PDMS membranes and elastic channel deformation, which can lead to a significant deviation from the true phenomenon inside the chip. As we mentioned, the effect of ideal air models and roughness can play an important role on the effect; thus, we take into account that future research can include atomistic models, such as molecular dynamics models or dissipative particle dynamics or couple them to larger-scale models in the frame of multiscale simulations.
Nomenclature Rw: total hydraulic resistance (Pa/(m3/s)) Ri: invariable resistance that relates to the microchannel (Pa/(m3/s)) Rv: variable resistance that relates to the microvalve (Pa/(m3/s)) µ: viscosity of the fluid in the gas microchannel (Pa•s) l0, w0, h0: the length, width, and height of the gas microchannel (µm) Qv: fluid flow through the microvalve (m3/s)
ε: pressure loss coefficient of the microvalve p: density (kg/m3) Av: effective flow area of the microvalve port (m2) lm,, wm,, tm: length, width, and thickness of the thin PDMS membrane (µm) δmax: maximum displacement of the membrane (µm) ΔP: pressure difference across the membrane (kPa) Pa: hydrostatic pressure in the gas microchannel (kPa) Pl: constant pressure in the liquid microchannel (kPa) E: Young’s modulus (MPa) v: Poisson ratio : plane-strain modulus (MPa) E σ0: residual stress (Pa) V: volume under the deflection membrane (µm3) t: response time (ms) P1: air pressure outside the air capacity (kPa) P2: air pressure inside the air capacity (kPa) T: time constant of the variable air capacity C: variable air capacity of the gas chamber (m3/Pa) C0: coefficient la, wa, ha: length, width, and height of the gas chamber (µm) Acknowledgments The authors would like to acknowledge Nicole Louise Drey, a PhD student in the Department of Engineering Science and Mechanics at Pennsylvania State University, for valuable discussion and help.
Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article is financially supported by the National Natural Science Foundation of China (51175101) and Abroad Short-Term Visiting Program for PhD candidates of HIT.
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