微流道中稀薄氣流之適當滑移邊界條件與表面粗糙度之相關探討---實驗研究與DSMC計算

行政院國家科學委員會專題研究計畫 成果報告

微流道中稀薄氣流之適當滑移邊界條件與表面粗糙度之相 關探討:實驗研究與 DSMC 計算(第 3 年)

研究成果報告(完整版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 96-2221-E-011-099-MY3

執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學機械工程系

計 畫 主 持 人 ： 蘇裕軒

處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 99 年 11 月 01 日

行政院國家科學委員會專題研究計畫結案報告

微 流道中稀薄氣流之適當滑移邊界條件與表面粗糙度之相關探討 ^: 實驗研究與 ^DSMC 計算

Investigation of proper slip boundary conditions for rarefied gas flow over micromachined channels with controlled roughness:

experiments and DSMC calculations 計畫編號 : NSC 96-2221-E-011-099-MY3 執行期限 _{: 98} 年 ₈ 月 ₁ 日至 ₉₉ 年 ₇ 月 ₃₁ 日 主持人 _: 蘇裕軒 ₍ 國立台灣科技大學機械系 ₎

計畫參與人員 _: 林偉銘 ₍ 國立台灣科技大學機械系 ₎

I. 中文摘要

本計劃之主要課題, 著重於從實驗與數值兩方面,探討固體表面粗糙度與入射氣體分子在固體表面反射之行為之相關性。 最終之目 的,在於將量化固體表面粗糙度所得之參數,當作 「蒙第卡羅直接法」 模擬程式之輸入參數,以期能反映固體表面粗糙度改變滑移邊 界條件效應,進而獲得較真實之模擬結果。

關鍵詞:滑移邊界條件、 蒙第卡羅直接模擬、 動量調節係數。

Abstract

In this research project, our effort is focused on the experimental work of deter- mining the tangential momentum accommodation coefficient of the gas flow through a microchannel due to the very limited budget given. In the past three years, we have developed and constructed the anodic bonding system for bonding pyrex and wafers and ultrasonic machining system for opening the via holes that are necessary for the fabrication of chips containing microchannels. 64 microchips are successfully fabri- cated. Preliminary experimental data agrees well with the theory. Due to the budget limitation, further experiments have to be conducted in the near future. Currently, 2 papers have been submitted for publication.

Keywords: Microchannel; Tangential momentum accommodation coefficient(TMAC);

Anodic bonding; ultrasonic machining.

II. Introduction

Gas flowing through microchannels is a fundamental topic of great importance in the fluid mechanics. The thriving development of various biochips enabled by MEMS technology gives impetus to the researches focused on this interesting topic in the past two decades. Study of gas flow in microchannels is not only of great importance to the microfluidic devices but also may provide the understanding of how the toxic gas flowing through the micro-cracks of chemical vessels.

The transport phenomena of gases flowing through microchannels are quite different from those conventionally observed when the mean free path of the gas is commensurate with the characteristic dimension of its confining structures. Gas rarefaction effect becomes important when Knudsen number goes beyond 0.001. As a first sympton, no- slip boundary conditions may break down due to the insufficient momentum exchange between the gas molecules and its confining structures. Slip flows are even observed when physicists were characterizing the transport properties of gases at a pressure of one atmosphere [1]. The friction factor of the microchannel flow is reduced as slip flow becomes prominant.

The large pressure drop per unit length through a long microchannel leads to the significant variation in gas density along the microchannel. Thus compressibility be- comes the pronounced effect for gas flow through microchannels.

Boundary conditions are particularly important in the prediction of rarefied gaseous flows in microchannels. Although without a compelling argument and any microscopic justification, it is well accepted that no slip boundary conditions do apply macroscopi-

cally for the flow of viscous fluids contacting with solid surface after Stokes’ investiga- tion [4]. A nice overview of the history of slip boundary conditions can be found in [5].

Despite this no slip boundary condition has been remarkably successful in the mod- eling of many flows, there are situations in which this condition results in unrealisitic behaviors such as the liquid spreading on a solid surface [6], [7].

Knudsen number, Kn ≡ _L^λ, determines the degree of rarefaction and the degree of validity of the continuum approximation of a fluid flow. A well accepted categorization for various flow regimes is given below [8]:

Euler equations (neglect molecular diffusion): K_n→0 Navier-Stokes equations with no-slip boundary conditions: K_n ≤10⁻³ Navier-Stokes equations with slip boundary conditions: 10⁻³≤K_n ≤10⁻¹

Transition regime: 10⁻¹≤Kn≤10

Free-molecule flow: 10 < K_n

In general, gas flows with Knudsen number Kn < 10⁻³can be considered as contin- uum, and Navier-Stokes equations with no slip boundary conditions can be used to pre- dict the behaviors of gas flows. As the Knudsen number increases (0.001 < Kn < 0.1), rarefaction effects become significant and no slip boundary conditions are no longer appropriate. Historically, the first slip boundary condition for liquids was proposed by Navier [9]. Navier (1823) argued that the tangential slip velocity v^t at the surface is proportional to the rate of shear strain ∂v_k

∂n at the surface vt = λ0

∂v_k

∂n,

where λ0is the slip length. Slip lengths in the nanometer to micrometer range have been

reported by several experimental studies [10], [11]. Recently, Navier’s slip boundary conditions have received the serious attention of researchers whose major interest is on the physico-chemical nature of solid-liquid interface.

On the other hand, Maxwell (1879) treated the slip boundary conditions for the case of gases rigorously [12]. Based upon the kinetic theory of gas, Maxwell proposed that the tangential velocity vt of the gas on the wall to be calculated as follows:

vt =2 − f

f Kn ∂v

∂y    _wall

,

where f is the tangential momentum accommendation coefficient or the fraction of incident molecules that are reflected diffusively, Kn the local Knudsen number, v the streamwise velocity, y is the normal coordinate directed toward the gas. A slip length of the order of the mean free path of the gas molecules is expected. The case of f = 0 is termed specular reflection, and this means no skin friction exists. The case of f = 1 is called diffuse reflection, and the molecules are reflected with no net tangential velocity macroscopically. As indicated by Sharipov [13], Maxwell assumed that the incident molecules do not interact with the reflected ones in the Knudsen layer. Consequently, the slip velocity calculated may not be accurate.

Though the tangential momentum accommodation coefficient can be determined experimentally [14], [15], the physical nature of this coefficient is still not clear. In particular, slip phenomenon depend crucially on the morphology and structure of the surface which may differ from an experiment to the other drastically. In addition, limitations in measurement accuracy at microscale tend to blur the conclusions that may be drawn from the experiment.

III. Theoretic analysis

The mass flowrate through the channels of high aspect ratio including both the com- pressibility and slip-flow effects can be written as follows [14]:

˙

m= p_o RT

wH³ 12µ

 p_i−p_o L

  P + 1

2 + 6Kn(2 − f ) f

 ,

where pi, p_o are the inlet and outlet pressures respectively, w the channel width, H the height of the channel, L the length of the channel, µ the viscosity of the gas, Kn the Knudsen number of the gas flow, R the specific gas constant, and T is the temperature of the gas.

The mass flowrate can be directly determined by the dual-tank accumulation system as follows:

˙ m= V_f

RT_f d∆p

dt −V_f∆p RT_f²

dT_f dt ,

where Vf, T_f are the volume and temperature of the flow tank respectively, and ∆p is the reading of the differential pressure sensor.

IV. Experiments

In this work, the microchannels with artificaial barriers are formed on a (100) silicon wafer using 2-stage ICP etching. The nominal depth of the channels are 10 µm and the width are nominally 400 µm. This results in a aspect ratio of 40 and a hydraulic diameter Dh = 19.5µm. The lengths of the channels are all 10000 µm. In the first stage, the channels are ICP-etched down first using the first mask. In the second stage, channels with artificial barriers (pillars) are formed using the second mask. The cross

sections of the microchannels are then characterized by a surface profiler (Dektak 6M, Veeco). The averaged depth of the microchannel is 10.45µm. The inset in Figure 1 demonstrated the pillars formed in the channels using white light interferometry.

Figure 1: Chip containing microchannel after anodic bonding. The small inset demonstrated the pillars formed during the second stage ICP etching.

Two via holes (3 mm diameter) functioning as inlet and outlet ports are opened on the Pyrex 7740 glass cover plate by ultrasonic machining developed and fabricated in our laboratory. The silicon dies containing microchannels are then sealed with the cover plates using anodic bonding (400^◦C, 1100V, in air), also developed ans fabricated in our own laboratory. In general, very good bonding (bonding strength > 20 MPa) can be attained for a bonding time of 10 minutes [17]. To ensure perfect hermetic sealing, all anodic bonding are conducted with a bonding time of 1 hour in this work.

Various combinations of electric fields and temperatures are explored. A typical set of bonding conditions is shown in Figure 2. For the best hermetic sealing performance, we decided to use 1100 V for electric field and 400 ^oC for temperature.

Figure 2: The anodic bonding system

In order to interface the microchannels with the external piping system, we proposed the fluidic interconnect system as shown in Figure 3.

The procedures of achieving the interconnect are listed below:

1. Open a connecting hole with a diameter of 2 mm on the pyrex cover plate using laser drilling or ultrasonic machining.

2. Seal the chip containing the microchannels with the machined cover plate via anodic bonding.

3. To prevent the fracture of chips during assemly, a holding jig is used to fix the 1/4” tubes that will be connected with the connecting holes on the pyrex cover plate.

Figure 3: The ultrasonic drilling machine designed and fabricated in our laboratory.

4. Align the 1/4” tubes with the connecting holes on the pyrex cover plate and bond them together with proper chemicals.

5. Test the hermetic sealing of the interconnect by holding pressure for 12 hours.

To measure the minute mass flowrate of the microchannel flows, the dual-tank mass flow measurement system developed in [14], [16] is adopted and modified in this work.

The complete system is shown in Figure 4. Closeup of the dual-tank accumulation system is shown in Figure 5.

For reliability and accuracy, all the test procedures are conducted sequentially by a C program. Consequently, all the valves are pneumatically operated and controlled by the solenoid valves. Pirani type pressure gauges based upon the thermal conductivity of gas are gas-dependent and easily affected by the ambient thermal fluctuation. There- fore, all direct total pressure measurements in this work are conducted with MKS Baratron absolute capacitance pressure gauges. All the three MKS pressure gauges

Figure 4: Dual-Tank Accumulation System for measuring the minute flowrate of gas flow through microchannels.

(ranges: 1 torr., 100 torr., 1000 torr.) are sent back to factory for NIST-traceable calibration before the system is assembled. The differential pressure sensors are then calibrated by the calibration system constructed in our laboratory, see Figure 6. The accuracies of all MKS pressure gauges used in this work are within 0.5% of reading.

After all the experiments are done, the chip is sectioned and cold mounted in epoxy for dimension check. See Figure 7. One critical issue regarding the dual-tank mass flow measurement system is how to calculate the volumes of the space inside the reference

Figure 5: A closeup of the Dual-Tank Accumulation System.

tank and flow tank accurately?

In order to calculate the volumes of reference tank and flow tank very accurately, a simple and reliable procedure was developed in this work. The volume of a tank is first determined by filling distilled water into the tank and weighted on a precision balance. Let the volume be V_t. This tank with know volume is then attached to the system as shown in Figure 8. Pump down the pressure to a pressure pH close to the upper limit (133 Pa) of the differential pressure sensor and isolate the tank with known volume from the rest of the system. Pump down the pressure of the rest of the system to a low pressure pL close to 0. Separate the flow tank and reference tank. After all the pressures reach steady state, open the valve that separates the tank with known volume and flow tank. Take the reading of the differential pressure sensor, ∆P . Then

Figure 6: Differential Pressure Calibration System.

from mass conservation, we have

p_HV_t+ p_LV_f V_t+ Vf

= (∆p + PL).

Or

V_f = p_H −p_L−∆p

∆p V_t.

The volume of the reference tank can be similarly calculated.

The test procedures are listed below: 1. Open all the valves and purge the system clean. 2. Fill the system with gas till the desired outlet test pressure is reached. 3.

Figure 7: Cross section of microchannel under optical microscope.

Close the reference tank valve and let the system settle for 10 minutes. 4. Isolate the two tanks by closing the isolation valve slowly. 5. Fill the system except the dual-tank subsystem with gas till the desired inlet test pressure is reached. 6. Let the system settle for 10 minutes before activating the microchannel flow (preflow measurement).

7. Open the chip valve to activate the microchannel flow till the differential pressure sensor nearly reaches its full capacity (flow measurement). 8. Shut off the chip valve and let the system settle for another 10 minutes (postflow measurement).

Ambient thermal fluctuations can be corrected for with the preflow-flow-postflow signals. It is easy to make the correction for ambient thermal fluctuation with the information of temperature histories. Consequently, this correction is made for all the experimental data.

The low-density viscosities of helium, argon and nitrogen are obtained from [3] and listed in Table 1.

Figure 8: Calculation of the volumes inside the reference tank and flow tank.

Table 1: Viscosities of He, N², and Ar at zero density, η⁰, and at 100 kPa, η¹⁰⁰, when temperature is 298 K. Mean free paths and molecular diameters of He, N², and Ar are calculated using hard sphere model.

Gas η0 (µPa·s) η100 (µPa·s) λ (nm) σ (pm) He 19.833 ± 0.016 19.832 ± 0.016 179 216 N² 17.750 ± 0.016 17.765 ± 0.016

Ar 22.570 ± 0.016 22.587 ± 0.016 64.7 359.7

V. Conclusions

A dual-tank mass flowrate measurement system with a resolution several orders of magnitude higher than what is currently commercially available has been successfully constructed. From the preliminary data, the experimental data for the case close to the Poiseuille flow agrees well with theoretical data (within 10 %). We are currently undertaking measurements of the mass flowrate of microchannels with artificial barriers and the data will be submitted for publication soon.

VI. Self Evaluation

This project is very interesting but involves very delicate measurement in order to reach affirmative conclusion in fundamental science. Under the limited budget, we are able to build the anodic bonding system and ultrasonic machining system in our laboratory.

We don’t have sufficient funding for purchasing the necessary MKS baratron pressure sensors that are critical for this work. Eventually, PI have to acquire these equipments with personal funding. PI does believe this project will definitely run much well if the funding is not too tight. In addition, PI believes very good data will be obtained and published in the near future.

Acknowledgments

This work was supported by the National Science Council of the Republic of China under the grant NSC 96-2221-E-011-099-MY3.

References

[1] D. Seibt, E. Vogel, E. Bich, D. Buttig, and E. Hassel 2006 “Viscosity measurements on nitrogen.”, J. Chem. Eng. Data, 51, 526–533.

[2] G. L. Morini, M. Lorenzini, & M. Spiga 2005 “A criterion for experimental validation of slip-flow models for incompressible rarefied gases through microchan- nels”, Microfluid Nanofluid, 1, 190–196.

[3] E. F. May, R. F. Berg, and M. R. Moldover 2007 “Reference viscosities of H², CH⁴, Ar, and Xe at low densities.”, International Journal of Thermophysics, 28(4), 1085–1110.

[4] G.G. Stokes 1845 “On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids”, Mathematical and Physical Papers by George Gabriel Stokes, Vol.1, pp 75–187, 1966.

[5] C. Neto, D.R. Evans, E. Bonaccurso, H.-J. Butt, & V.S.J. Craig 2005

“Boundary slip in Newtonian liquids: a review of experimental studies”, Reports on Progress in Physics, 68, pp 2859–2897.

[6] L.M. Hocking 1976 “A moving fluid interface on a rough surface”, Journal of Fluid Mechanics, 76, pp 801–817.

[7] E.B. Dussan 1979 “On the spreading of liquids on solid surfaces: static and dy- namic contact lines”, Annual Reviews of Fluid Mechanics, 11, pp 371–400.

[8] M. Gad-el-Hak 1999 “The fluid mechanics of microdevices–The Freeman Scholar Lecture”, Journal of Fluids Engineering, ASME Transactions, 121, pp 5–33.

[9] C.L.M.H. Navier 1823 “M´emoire sur les lois du mouvement des fluids”, M´emoire de l’Acad´emie Royale des Sciences de l’Institut de France, 6, 389–440.

[10] E. Lauga and H.A. Stone 2003 “Effective slip in pressure-driven Stokes flow”, Journal of Fluid Mechanics”, 489, pp. 55–77.

[11] S. Gogte, P. Vorobieff, R. Truesdell, et al. 2005 “Effective slip on textured superhydrophobic surfaces”, Physics of Fluids, 17 (5), pp. 051701.

[12] J.C. Maxwell 1879 “On Stresses in Rarified Gases Arising from Inequalities of Temperature”, Philosophical Transactions of the Royal Society of London, 170 (1879), pp. 231–256.

[13] F. Sharipov and D. Kalempa 2003 “Velocity slip and temperature jump coef- ficients for gaseous mixtures. I. Viscous slip coefficient”, Physics of Fluids, 15 (6), pp. 1800–1806.

[14] E.B. Arkilic, M.A. Schmidt, and K.S. Breuer 1997 “Gaseous slip flow in long microchannels”, Journal of Microelectromechanical systems, 6, pp. 167–178.

[15] E.B. Arkilic, K.S. Breuer, M.A. Schmidt 2001 “Mass flow and tangential momentum accommodation in silicon micromachined channels”, Journal of Fluid Mechanics, 437, pp. 29–43.

[16] E. B. Arkilic, M. A. Schmidt, & K. S. Breuer 1998 “Sub-nanomol per second flow measurement near atmospheric pressure”, Experiments in Fluids, 25, 37–41.

[17] J. A. Dziuban, “Bonding in Microsystem Technology”, Springer, 2006, P. 204.

無衍生研發成果推廣資料

96 年度專題研究計畫研究成果彙整表

計畫主持人：蘇裕軒 計畫編號：96-2221-E-011-099-MY3

計畫名稱：微流道中稀薄氣流之適當滑移邊界條件與表面粗糙度之相關探討:實驗研究與 DSMC 計算 量化

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