nanoscience and technology series. microfluid mechanics, 2006, p.368

The force field is then a function of the distance between the molecules.. With the nuclei andthe electrons in motion, the molecule can have, for instance, rotationaland vibrational modes

Microfluid Mechanics

Nanoscience and Technology Series

Omar Manasreh, Series Editor

MICHAEL H PETERS ● Molecular Thermodynamics and Transport

KENNETH GILLEO ● MEMS/MOEM Packaging

NICOLAE O LOBONTIU ● Mechanical Design of Microresonators

ROBERTO PAIELLA ● Intersubband Transitions in Quantum Structures

JOSEPH H KOO ● Polymer Nanocomposites

JENS W TOMM AND JUAN JIMENEZ ● Quantum-Well High-Power Laser Arrays

Microfluid Mechanics Principles and Modeling

William W Liou Yichuan Fang

Department of Mechanical and Aeronautical Engineering Western Michigan University Kalamazoo, Michigan

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DOI: 10.1036/0071443223

2.6 Boltzmann Equation and Maxwellian Distribution Function 30

For more information about this title, click here

Chapter 5 Statistical Method: Direct Simulation Monte Carlo Method

5.2.1 Relationship between DSMC and Boltzmann equation 96 5.2.2 Computational approximations and input data 98

Appendix 5A: Sampling from a Probability Distribution Function 124 Appendix 5B: Additional Energy Carried by Fast Molecules Crossing

Appendix 5C: One-Dimensional DSMC-IP Computer Program 129

Chapter 6 Parallel Computing and Parallel Direct Simulation

7.2.1 Specular and diffusive reflection models of Maxwell 179

viii Contents

in modern numerical computation tools will help the readers gettingthe full benefit of the two computer programs, NB2D and DSMC-IP, as-sociated with the book In fact, it is highly recommended that readers

do make use of these Fortran programs

The book begins with an introduction to the kinetic theory of gas andthe Boltzmann equation to build the foundation to the later mathemat-ical modeling approaches With the dilute gas assumption, the nature

of the micro gas flows allows the direct application of the Enskog theory, which then brings in the modeling equations at thevarious orders of the Knudsen number in Chapter 4 The direct simu-lation Monte Carlo (DSMC) method and the information preservation(IP) method are described as the numerical tools to provide solutions tothe Boltzmann equation when the Knudsen number is high The laterchapters cover the hybrid approaches and the important surface mech-anisms Some examples of micro gas flows at high and low speeds areshown One interesting aspect of micro gas flows that is yet to see ex-tensive examination in the literature is the characteristics of the flowdisturbances at microscales Chapter 11 provides a detailed description

Chapman-of some Chapman-of our preliminary studies in this area

Although a part of the content of the book has been used in a semester graduate level microfluid dynamics course at WesternMichigan University, the book is best used as a textbook for a two-semester course Chapters 1 to 4 provide the introductory content for

one-ix

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x Preface

the basic mathematical and the physical aspects of micro gas flows Thecomputer program NB2D may be used, for example, as project exercises.The second part would emphasize the DSMC and the IP solution meth-ods, and their parallelization The computer program DSMC-IP can beused for term project type of assignments In the situation where theanalytical microfluid course is preceded by another experiment-orientedcourse on microfabrication or microengineering, and the studentshave already had somewhat extensive knowledge of micro gas flows,the instructor may wish to concentrate on Chapters 2, 4, and 5 in thelectures and leave the rest as reading assignments When the book isused in a 16-to-20-hour short-course setting, the instructor may wish tohighlight the materials from Chapters 2, 4, 5, and 7 It might be a goodidea to provide opportunities to run at least one of the two computerprograms onsite Chapters 1, 3, and 10 can be assigned as overnightreading materials

The book does not contain extensive updates and details on the rent engineering microfluidic devices We feel that the book’s focus is

cur-on the fundamental aspects of mirofluid flows and there is a myriad ofreadily available information on the technologies and the many differ-ent microfluidic device applications that have been cleverly designedand painstakingly manufactured by experts in the field Also, sincenew devices are being brought to light almost daily, we feel that what

is current at the time of this writing may become outdated within a fewyears

A portion of the work the authors have accomplished at WesternMichigan University has been performed under the support of NASALangley Research Center Near the completion of the manuscript, thesecond author moved to the Georgia Institute of Technology

We appreciate the help of Dr James Moss for reviewing a part of themanuscript Thanks should go to those at McGraw Hill who worked onthe book and to those who reviewed it We should also thank Jin Suand Yang Yang for their work on the codes The first author (WWL)would like to thank the unconditional support and patience of his wife,Shiou-Huey Lee, during this writing and the love from his children,Alex and Natalie

William W Liou

Kalamazoo, Michigan

Yichuan Fang

Atlanta, Georgia

Microfluid Mechanics

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de-vice can be a single piece of hardware that produces outputs directlybased on the inputs from external sources The outputs can be me-chanical and fluidic movement, electrical charges, analog signals, anddigital signals Often several microcomponents are integrated, such asthe lab-on-a-chip device, which performs the multistage processing ofthe inputs and produces several different types of outputs, all in onesingle miniature device The small sizes of MEMS make them portableand implantable The manufacturing cost of MEMS is far from pro-hibitive because of the wide use of the batch-processing technologiesthat grow out of the well-developed IC industry MEMS, therefore, of-fer opportunities to many areas of application, such as biomedical andinformation technology, that were thought not achievable using conven-tional devices Estimates of the potential commercial market size were

as high as billions of U.S dollars by 2010

Since the early work of Tai et al (1989) and Mehregany et al (1990)

on the surface-machined micromotors, there has been an explosivegrowth of the number and the types of potential application of MEMS

1

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et al (2000), Nguyen and Wereley (2002), and Li (2004)] MEMS lated technologies, ranging from electrokinetics and microfabrication

re-to applications, can be readily found in these and many other forms ofpublications and media

As more new applications are proposed and new MEMS devicesdesigned, it was often found that measured quantities could not be in-terpreted by using conventional correlations developed for macro scaledevices Electric power needed to drive a micromotor was extraordinar-ily high The properties of MEMS materials, such as Young’s modulus,have been found to differ from that of the bulk material For MEMS thatuse fluid as the working media, or a microfluidic device, for instance, thesurface mechanisms are more important than mechanisms that scalewith the volume Overcoming surface stiction was found to be important

in the early work on micromotors The surface tension is perhaps amongthe most challenging issues in microfluidic devices that involve the use

of liquid for transporting, sensing, and control purposes The mass flowrate of microchannels of gas and liquid flows in simple straight mi-crochannels and pipes were found to transition to turbulence at a muchlower Reynolds number than their counterparts at the macro scales.Due to the miniature size, there are uncertainties in measuring the var-ious properties of MEMS, such as specimen dimensions, with sufficientaccuracy Nevertheless, it has become increasingly apparent that thephysical mechanisms at work in these small-scale devices are differentfrom what can be extrapolated from what is known from experience withmacroscaled devices There is a need to either reexamine or replace thephenomenological modeling tools developed from observations of macroscale devices

This book covers the fundamentals of microfluid flows The somewhatlimited scope, compared with other titles, allows a detailed examination

of the physics of the microfluids from an ab initio point of view Since the

first principle theory is far less developed for liquids, the focus in thiswriting is the microfluid flows of gas The Boltzmann equation will be in-troduced first as the mathematical model for micro gas flows Analyticalsolutions of the Boltzmann equation can be found for a limited num-ber of cases The Chapman-Enskog theory assumes that the velocitydistribution function of gas is a small perturbation of that in thermo-dynamic equilibrium The velocity distribution function is expressed as

a series expansion about the Knudsen number The Chapman-Enskog

be-The analytical solutions of these various mathematical model tions for micro gas flows can be found for simple geometry and forlimited flow conditions, some of which are discussed in the appropri-ate chapters For many of the complex design of microfluidic devices,the flow solutions can only be found by numerically solving the modelequations To this end, two computer programs are provided in theappendix section of the book The programs are written using the stan-dard FORTRAN language and can be compiled in any platforms TheNB2D code solves the Navier-Stokes equations as well as two forms ofthe Burnett equation The all-speed numerical algorithm has been used

equa-in the numerical discretization The density-based numerical methodhas been shown to be able to handle low-speed as well as high-speedflows, and is appropriate for gas flows commonly seen in microdevices.The DSMC/IP1D code uses the direct simulation Monte Carlo (DSMC)method and the information preservation (IP) method to provide sim-ulations of the gas microflow at the large Knudsen number The IPmethod has been shown to be exceptionally efficient in reducing thestatistical scatter inherent in the particle-based DSMC-like methodswhen the flow speed is low The two computer programs will providethe readers with numerical tools to study the basic properties of microgas flows in a wide range of flow speeds and in a wide range of Knudsennumber Examples of low- and high-speed micro gas flow simulationsare also presented in later chapters One of the unsolved problems inconventional macro scale fluid dynamics is associated with the flowtransition to turbulence In Chap 11, the behavior of the flow distur-bances in two simulated micro gas flows is described

The book is geared toward developing an appreciation of the basicphysical properties of micro gas flows The computer software, cou-pled with the necessary analytical background, enable the reader todevelop a detailed understanding of the fundamentals of microfluidic

Gad-el-Hak, M., The MEMS Handbook, Boca Raton, FL, CRC Press, 2002.

Karniadakis, G.E and Beskok, A., Microflows: Fundamentals and Simulation, Springer,

New York, 2002.

Koch, M., Evens, A., and Brunnshweiler, A., Microfluidic Technology and Applications,

Research Studies Press, Hertfordshire, England, 2000.

Li, D., Electrokinetics in Microfluidics, Interface Sciences and Technology, Vol 2, Elsevier

Academic Press, London, UK, 2004.

Mehregany, M., Nagarkar, P., Senturia, S., and Lang, J., Operation of microfavricated

harmonic and ordinary side-drive motor, IEEE Micro Electro Mechanical System

Work-shop, Napa Valley, CA, 1990.

Nguyen, N.-T and Wereley, S., Fundamentals and Applications of Microfluidic, Artech

House, Norwood, MA, 2002.

Tai, Y., Fan, L., and Muller, R., IC-Processed micro-motors: Design, technology and

test-ing, IEEE Micro Electro Mechanical System Workshop, Salt lake City, UT, 1989.

mi-A molecular model for gas would then describe the nature of themolecule, such as the mass, the size, the velocity, and the internal state

of each molecule A measure for the number of molecules per unit

vol-ume, or number density n, would also be a parameter The model also

describes the force field acting between the molecules The force field isnormally assumed to be spherically symmetric This is physically rea-sonable in light of the random nature of the large number of collisions

in most cases The force field is then a function of the distance between

the molecules Figure 2.1.1 shows a typical form of the force field F (r ) between two molecules with distance r At a large distance, the weak at-

tractive force approaches zero The attractive force increases as the tance decreases In close range, the force reverses to become repulsive

dis-5

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Figure 2.1.1 Sketch of spherically symmetric

inter-molecular force field.

as the orbiting electrons of the two molecules intermingle Analyseswith nonspherically symmetric force field are complicated In fact, it isgenerally found that, the exact form of the force field is less importantthan other collision parameters The force field of a simple rigid spheremodel is shown in Fig 2.1.1 The model assumes an infinitive repulsiveforce when the molecules are in contact and zero otherwise The contactoccurs when the distance between the centers of the molecules are the

same as the assumed diameter of the sphere d Use of the rigid sphere model can lead to accurate results if the diameter d is properly chosen

according to some basic properties of the gas The internal structure ofthe molecules affects the energy content of the gas With the nuclei andthe electrons in motion, the molecule can have, for instance, rotationaland vibrational modes of energy in addition to the energy associatedwith the molecular translational motion

These molecular quantities need to be related to macroscopic ties for analyses This is especially true when there is a general macro-scopic movement of the gas As will be seen in the following section, amacroscopic property is merely the sample averaged value of the corre-sponding molecular quantity The motion of the molecules is then notcompletely random when there is macroscopic motion

proper-2.2 Micro and Macroscopic Properties

In this section, we will use a simplified model to introduce the tions between the molecular behavior and the macroscopic properties

rela-of gases We consider an equilibrium monatomic gas rela-of single species

Basic Kinetic Theory 7

inside a cubic box at rest of length l on each side (see Fig 2.2.1) A gas

in equilibrium would exhibit no gradients of macroscopic quantities inspace or time The average velocity of the molecule is therefore zero

Molecular motion is random with velocity vector c The molecular

sys-tem (x, y, z) Again, molecules here can represent an atom, monatomic

molecules, diatomic molecules, or polyatomic gas molecules Assuming

specular reflection at the wall, the x momentum change of a molecule

m denotes the molecular mass If we assume that there are no

exerted on the wall by the molecule, is

1

Therefore, pressure p becomes

ergy per unit mass, or the specific molecular translational kinetic ergy Therefore, from the kinetic theory point of view, the pressure isproportional to the gas density and the specific translational kineticenergy The empirical equation of state for a thermally perfect gas canprovide pressure from the thermodynamics consideration That is,

where R represents the gas constant The two different expressions for

pressure will give the same quantity if

This equation relates the temperature defined in the classical modynamics to the specific kinetic energy of molecular translationalmotion in kinetic theory Therefore, temperature, a macroscopic gasproperty, can then be used as a measure of the specific molecular energy

ther-Basic Kinetic Theory 9

We can also relate temperature and pressure to the average kinetic ergy per molecule For instance, Eq (2.2.2) gives

where k is the Boltzmann constant It is the ratio of the universal

in-cludes only the translational kinetic energy, it is sometime referred to

molecule can be assumed to possess translational kinetic energy onlyand the translational kinetic temperature may simply be referred to asthe temperature For diatomic and polyatomic molecules, the rotationaland the vibrational modes can also be associated with temperature Ageneral principle of equipartition of energy states that for every part

of the molecular energy that can be expressed as the sum of square

such term For instance, in the kinetic energy of translation, there are

To estimate the molecular speed by using the macroscopic properties,

divide both sides of Eq (2.2.1) by the total mass M, then

p

13

Note that we have defined a sample average ¯q as an average of a ular quantity q over all the molecules in the sample That is

N



Basic Kinetic Theory 11

For example, the average molecular velocity is

respectively The trajectories or orbits of the two particles are twistedcurves in space

shows that the velocity of the center of mass does not change in thecollision The conservation of energy gives

From Eqs (2.3.2) and (2.3.5), one can write the pre-collision velocities

in terms of the center of mass velocity and the relative velocity

c1= cm+ m2

center-of-mass reference of frame, the approach velocities of the collision partnerare parallel to each other For a spherically symmetric repulsive inter-

U (r ) is the potential of the intermolecular force and e r the unit vector

along r The conservative force field is a function of the distance between

Basic Kinetic Theory 13

opposite in direction That is,

(2.3.10)

dc r dt

central field of conservative force The postcollision velocities can also

be found from Eqs (2.3.2) and (2.3.5)

c∗1= cm+ m2

center-of-mass reference of frame, the post-collision velocities of the collisionpartner are parallel to each other In fact, since the force field is assumedspherically symmetric, there is no azimuthal acceleration during the

interaction and the angular momentum L is conserved That is,

the collision plane In this accelerating frame of reference attached to

of direction of the relative velocity vectors due to a collision and can beobtained by examining the dynamics of the binary collision.

properties

as shown in Fig 2.3.2, Eq (2.3.12) becomes

m r r2d φ

r b

Basic Kinetic Theory 15

where b is the smallest distance between the trajectories of the

mole-cules before the collision and is called the impact parameter Since theintermolecular force is a short-range force that vanishes at large inter-molecular distances, the conservation of the total energy, including thekinetic and the potential energy, becomes

1

dr dt

from Eqs (2.3.16) and (2.3.17) That is

dr

r b

The trajectory is a function of the collision parameter, the ratio of the

corresponding to the approach and the departure that are symmetric

r m2 − b2= 2r m2U (r m)

m r c2

r

ap-proach, respectively The symmetry in the approach and the departuretrajectories also means that for a collision with precollision velocity of

the inverse collision of the original, direct collision Also as a result ofthe symmetry, the angle of deflection becomes,

16 Chapter Two

Equation (2.3.22) shows that given the intermolecular potential and

by the value of the impact parameter b.

postcollision relative velocity can be written as

cr = (c rcosχ, c rsinχ cos ε, c rsinχ sin ε) (2.3.24)

Its components in the (x, y, z) coordinate system can be obtained by

using the direction cosines That is

c∗r = Xc∗

X is a second order tensor with components being the direction cosines

of the primed coordinate system That is,

Basic Kinetic Theory 17

X11=u r

Step 4 Calculate the postcollision relative velocity.

cr = (c rcosχ, c rsinχ cos ε, c rsinχ sin ε)

of probabilities of collision in terms of the interaction potential and the

collision parameter b (see Fig 2.3.4) The differential collision cross

sin(χ)dε dχ

For a spherically symmetric intermolecular potential, the inverse sion exists and

One can also write,

σ(sin(χ)dεdχ) = −(bdε) db

The negative sign is there to expect that the deflection angle decreases

with the increasing collision parameter b Therefore, the differential

cross section becomes

Collision cross section and solid angle.

Basic Kinetic Theory 19

The total collision cross section can be obtained by integration over allthe possible scattering directions That is

0

d2 12

As described earlier, the intermolecular force field consists of tive and repulsive portions as a function of distance between the twomolecules The best know attractive–repulsive model is the Lennard-Jones model

F = 48ε δ

12

characteris-tic length scale taken as the distance at which the potential function

20 Chapter Two

changes its sign This model is widely used in the simulations of densegas and liquid

attractive portion In this case, the inverse power law model describesthe potential and the force fields as follow

root of the equation

rigid sphere model, often referred to as the hard sphere (HS) model,that has been used in the previous sections can be regarded as a special

the Maxwell model The model can be regarded as the limiting case of a

“soft” molecule, contrasting the HS model at the other limit For a real

is simple and easy to use Its primary deficiency is that the resultingscattering law is not realistic

introduced by Bird (1981) to correct the primary deficiency of the HSmodel In the VHS model, the molecule is a hard sphere with a diameter

d that is a function of c r Generally the function can be obtained based

where the subscript “ref ” denotes the reference values at the reference

Basic Kinetic Theory 21

appropriate for an equilibrium gas, as

is an energy-dependent variable for the VHS model Both models obeythe isotropic scattering law and do not correctly predict the diffusion inflows of real gases, especially of gas mixtures

in-troduced by Koura and Matsumoto (1991, 1992) took into account theanisotropic scattering of real gases The molecular diameter varies in

TABLE 2.3.1 Typical Values ofα and dref for VHS Molecules at 273K

SOURCE: Bird, G.A., Molecular Gas Dynamics and the Direct Simulation of

Gas Flows, Oxford University Press, New York, 1994.

22 Chapter Two

in relation to b such that

the viscosity and diffusion coefficient are consistent with the inverse

collision cross section can be expressed as

σ T ,VSS = 1

where S is the softness coefficient given by

large variety of molecular species are compiled in Koura and sumoto (1991) In addition, the corresponding viscosity and diffusioncross-section for the VSS model are also given in the reference It wasfound that the VSS model is preferable to the VHS model in flows ofgas mixtures where molecular diffusion is important

model developed by Hassan and Hash (1993) is an extension of the VHSand VSS models The scattering distribution is that of the hard or softsphere, but the variation of the total cross section as a function of the rel-ative translational energy mimics that of the corresponding attractive–repulsive potential It is implemented through the parameters that de-scribe the intermolecular potentials of the form of Eq (2.3.22), and cantherefore make use of the existing database that has been built up fromthe measured transport properties of real gases

models proposed for inelastic collision The most widely used model forinelastic collisions is the Larsen-Borgnakke phenomenological model(Larsen and Borgnakke 1974, Borgnakke and Larsen 1975) This modelallows molecules to have continuous internal energy modes The ex-change of energy among translational, rotational and vibrational modes

is accounted for in distribution of post-collision velocity for the moleculesinvolved The Larsen-Borgnakke model was also extended to includethe repulsive force described in the GHS model The model is applica-ble to binary collisions in a mixture of polyatomic gases

Basic Kinetic Theory 23

2.4 Statistical Gas Properties

An important concept brought in by the statistical consideration of the

is defined as the average distance a molecule travels between successivecollisions It is defined in a frame of reference that moves with the localstream We will consider again single species with the average spacing

the molecules of hard sphere That is

as the limit of dilute gas assumption The average volume a molecule

For the hard sphere gas considered here, a given target molecule willexperience a collision with another molecule whenever the distance

The target molecule then carries a sphere of influence of radius, d A

collision will occur when the center of other field molecules lie on thesurface of this sphere For hard sphere model, according to Eq (2.3.34),

the sphere of influence on to a plane normal to the relative velocity tor of the colliding molecules (see Fig 2.4.2) The relative velocity vector

by the collision cross section area That is

24 Chapter Two

ct

crD t

Figure 2.4.2 Collision frequency.

is then the sum over all velocity class Or

travels between successive collisions

It is defined in a frame of reference that moves with the local stream.Therefore

Basic Kinetic Theory 25

Eqs (2.4.2) and (2.4.5), that

2

(2.4.6)

Since the effective range of the intermolecular force field is of the der of the diameter, this equation suggests that for dilute gas, the gasmolecules interact only during the collision process of relatively shortduration in time This is an important result in the development ofkinetic theory of gas It suggests that molecular collision is an instan-taneous occurrence It also suggests that molecular collisions in dilutegas are most likely to involve only two molecules Such a collision, aswas described in the previous section is called a binary collision.Molecular collisions cause the velocity of the individual molecule tovary Therefore, the number of molecules in a particular velocity classchanges with time For a gas in equilibrium where there is no gradient

or-in its macroscopic properties, the number of particle or-in a velocity classremains unchanged That is, locally, for every molecule that leaves itsoriginal velocity class, there will be another molecule entering the veloc-ity class In a non-equilibrium state where there is nonuniform spatialdistribution of certain macroscopic quantity, such as the average veloc-ity or temperature, the molecules in their random thermal motion thatmove from one region to another find themselves with momentum orenergy deficit or excess at the new location The microscopic molecularmotion thus causes the macroscopic properties of the gases to change.The results of these molecular transport processes are reflected upon

heat conduction k Molecular collisions are responsible for

establish-ing the equilibrium state where the effects of further collisions canceleach other To examine the phenomenon of viscosity, we can look at aunidirectional gas flow with nonuniform velocity in only one directionand homogeneous in the other two directions This is normally referred

component in a Cartesian coordinate system (Fig 2.4.3)

y direction at the speed of c
y Locally, the sample averaged momentumtransport per unit area per unit time becomes

26 Chapter Two

x

y

c x

Figure 2.4.3 Transport process.

would carry a momentum deficit at the new location If one assumesthat on an average, a molecule collides with another molecule at the new

environment The particle of mass m will change its momentum by the

The number of such collisions per unit area per unit time can be

Basic Kinetic Theory 27

Thus, the appropriate velocity scale for viscosity is related to the ular thermal speed and the length scale is the mean free path

molec-2.5 Position and Velocity Distribution

represents the local number of molecules found in a unit volume ofphysical space as a function of space and time It is called the localnumber density or number density It describes the distribution of thenumber of molecules in physical space and therefore is a position dis-tribution function Fluid density is then related to the position dis-tribution function of the number of molecules For a small element

where the functional dependence has been dropped for brevity

Multi-plying both sides by m, we get

in-troduced by Koura and Matsumoto (1991, 1992) took into... ofα and dref for VHS Molecules at 273K

SOURCE: Bird, G.A., Molecular Gas Dynamics and the Direct Simulation