Micromixers. Fundamentals, Design and Fabrication

356 Pages·2011·12.32 MB·English

by  Nam-Trung Nguyen (Auth.)

CHAPTER
1
Introduction
CHAPTER OUTLINE
1.1 MicromixersandMixinginMicroscale...............................................................................................1
1.2 MicromixersasMicroreactors...........................................................................................................5
1.3 OrganizationoftheBook....................................................................................................................7
References..............................................................................................................................................8
1.1 MICROMIXERS AND MIXING IN MICROSCALE
Thisbookdiscussesthedesign,fabrication,andcharacterizationofmicromixers,whicharedefinedas
miniaturizedmixingdevicesforatleasttwodifferentphases,whichcouldbeliquids,solids,orgases.
The structures of a micromixer are fabricated partially or in whole, using microtechnology or
precisionengineering.Thecharacteristicchannelsizeofmicromixersisinthesub-millimeterrange.
Common channel widths are on the order of 100–500mm, while channel length could be a few
millimetersormore.Thechannelheightisontheorderofthechannelwidthorissmaller.Theoverall
volumedefinedbyamicromixerisfrommicroliterstomilliliters.Comparedtomolecularsizescale,
the length scale and volume scale of micromixers are very large. This fact leads to two key char-
acteristics of micromixers. First, designing the micromixers relies on manipulating the flow using
channel geometry or external disturbances. Second, while micromixers bring advantages and new
featuresintochemicalengineering,molecular-levelprocessessuchasreactionkineticsremainalmost
unchanged.
Mixing is a transport process for species, temperature, and phases to reduce inhomogeneity.
Mixing leads to secondary effects such as reaction and change in properties. In conventional
macroscale mixing techniques, there are three established terminologies for mixing: macromixing,
mesomixing, and micromixing [1]. Macromixing refers to mixing governed by the largest scale of
fluid motion. For instance, the scale of macromixing corresponds to the diameter of the mixing
tank. Micromixing is mixing at the smallest scale of fluid motion and molecular motion. In
conventional macroscale mixing, the smallest scale of fluid motion is the size of turbulent eddies,
also called the Kolmogorov scale. Mesomixing is in the scale between macromixing and micro-
scale. Although micromixers may have dimensions on the order of micrometers, transport process
in micromixers may still be classified as mesomixing. Since structures in micromixers may have
a size approaching the Kolmogorov scale, this book avoids the use of micromixing for describing
mixing processes.
Therearemanydifferentwaystoprovidemixinginmacroscalesuchasmoleculardiffusion,eddy
diffusion,advection,andTaylordispersion.Eddydiffusionisthetransportoflargegroupsofspecies
1
Nam-TrungNguyen:Micromixers,Secondedition.DOI:10.1016/B978-1-4377-3520-8.00001-2
Copyright(cid:1)2012ElsevierInc.Allrightsreserved.
2 CHAPTER 1 Introduction
andrequiresaturbulentflow.Becauseofthedominantviscouseffectinmicroscale,turbulenceisnot
possible in micromixers. Mixing based on eddy diffusion is therefore not relevant for micromixers.
Thus, the main transport phenomena in micromixers include molecular diffusion, advection, and
Taylor dispersion. Molecular diffusion is caused by the random motion of molecules. This transport
mechanism is characterized by the molecular diffusion coefficient. Advection is the transport
phenomenacausedbyfluidmotion.AsimpleEulerianvelocitycanleadtoachaoticdistributionofthe
mixedspecies.Astableandlaminarflowcanalsoleadtochaoticadvection.Thus,chaoticadvection
wouldbeidealforthelaminarflowconditioninmicromixers.Taylordispersionisadvectioncausedby
a velocity gradient. Axial dispersion occurs due to advection and inter-diffusion of fluid layers with
different velocities.Due tothiseffect, mixingbased onTaylordispersion can betwoorthree orders
faster than that based on pure molecular diffusion.
Designingmicromixersisacompletelynewengineeringdiscipline,becauseexistingdesignsin
macroscale cannot simply scale down for microscale applications. One of the main challenges
related to miniaturization is the dominance of surface effects over volume effects. Actuation
conceptsbasedonvolumeforcesworkingwellin macroscalemayhaveproblemsinmicroscale.A
magnetic stirrer is a typical example of the ratio between surface forces and volume forces. It
consists of a magnet bar and a rotating magnet or stationary electromagnets creating a rotating
magnetic field. The driving magnetic force is proportional to the volume of the magnet bar, while
thefrictionforceisproportionaltoitssurface.Scalingdownthestirrerfollowstheso-calledcube-
squarelaw.Thismeansshrinkingdownthestirbar10timeswouldroughlydecreaseitsvolumeby
1000timesanditssurfaceonlyby100times.Withitsoriginalsize,theexternalmagneticfieldcan
generateaforceofthesameorderofthefrictionforceandcausethestirbartomove.Scalingdown
thesize10timesinthesamemagneticfieldwouldcreateasmalldrivingforce,whichisonly1/10th
of the friction force. As a consequence, the stir bar cannot move. A surface force-based actuation
conceptwouldallowscalingdownbecausetheratiobetweendrivingforceandfrictionforcewould
remain unchanged.
The dominant surface phenomena in microscale also affect mixing processes with immiscible
interfaces. For a solid–liquid system, mixing starts with suspension of the solid particles. The dis-
solving process follows suspension. The largesurface-to-volumeratio inmicroscale isan advantage
for the dissolving process, making it easily achievable. Thus, the main challenge is the suspension
process.Becauseoftheirrelativelylargesizesandthecorrespondinglysmalldiffusioncoefficient,the
particlecanonlybesuspendedinmicroscalewiththehelpofchaoticadvection.Therefore,thequality
ofsolid–liquidmixing inmicroscale isdetermined by the suspension process.
Inasystemofimmiscibleliquids,additionalenergyisneededtoovercomeinterfacialtension.On
theonehand,dispersing theimmisciblephasesisadifficulttask.Ontheotherhand,surfacetension
breaksthestretchedfluidintosegmentsandformsmicrodroplets.Theadvantageofmicroscaleisthat
theformationprocesscanbecontrolleddowntoeachindividualdroplet.Therefore,anemulsionwith
homogenous droplet size can beachievedin micromixers.
Gas–liquid systems are other systems affected by the dominant surface phenomena. Some appli-
cationssuchashydrogenation,oxidation,carbonation,andchlorinationrequiregas–liquiddispersion.
Unlike liquid–liquid emulsion, gas molecules can be absorbed into the liquid phase. The gas–liquid
mixing process consists of two processes: dispersion of the gas bubble and absorption of gas mole-
cules.Whileabsorptionispromotedduetothelargeravailableinterfacialarea,dispersionoftinygas
bubblesisthe main challenge indesigning micromixersfor a gas–liquidsystem.
1.1 Micromixers and mixing in microscale 3
Besides surface phenomena, the laminar flow condition is another challenge for designing
micromixers.Theproblemsinmicromixersaresimilartothoseinmacroscalelaminarmixers.Laminar
mixersexistinmanyprocessesoffood,biotechnological,andpharmaceuticalindustriesbecauseofthe
high viscosity and slow flow velocity involved. For many applications, the flow velocity in micro-
mixerscannotbetoohigh.Thesmallsizeofmicromixersleadstoanextremelylargeshearstressin
mixingdevices,evenatrelativelyslowflowvelocities.Thisshearstressmaydamagecellsandother
sensitive bioparticles. In complex fluids with large molecules and cells, the fluid properties become
non-Newtonianathighshearstress.Ontheonehand,thehighshearcompromisesboththemetabolic
andphysicalintegrityofcells.Ontheotherhand,viscoelasticeffectsunderthisconditionmayleadto
flowinstability, which can bewell utilized for improvingmixing.
In this book, micromixers are categorized as passive micromixers and active micromixers [2]
(Fig. 1.1). Except the kinetic energy of the flow itself, passive micromixers do not require external
energyfor disturbance. Themixingprocessrelies entirely on diffusion orchaotic advection.Passive
mixersarefurthercategorizedaccordingtothewayinwhichtheinterfacebetweenthemixedphasesis
arranged:parallellamination,seriallamination,segmentation,chaoticadvection,andmultiphaseflow.
In active micromixers, disturbances are induced by an external field. Thus, active mixers can be
categorized according to the physical phenomenon of the disturbance as a pressure-driven flow,
electrohydrodynamics, dielectrophoretics, electrokinetics, magnetohydrodynamics, acoustics, and
heat.Thedesignsofactivemicromixersareoftencomplexbecauseofadditionalcomponents.External
power sources are needed for the operation of active micromixers. Thus, the integration of active
mixersinamicrofluidicsystemisbothchallengingandexpensive.Incontrast,passivemicromixersdo
notrequireexternalactuatorsexceptthoseforfluiddelivery.Passivemicromixersarerobust,stablein
operation, and easily integrated in a more complex system. Figure 1.1 depicts an overview on the
different typesof micromixers discussedin thisbook.
The time scale of mixing processes changes with miniaturization. Most micromixers are used as
areactionplatformforanalysisorsynthesis.Mixingandchemicalreactionsareinterrelated[3].While
reaction kinetics and reaction time do not change with miniaturization, mixing time can be signifi-
cantlyaffectedbythemixerdesignaswellasbythemixertype.Thisfactleadstotwoimportantissues
relatedtochemicalreaction:measurementofrealreactionkineticsandcontroloverreactionproducts.
Inmacroscale,mixingtimeisusuallymuchlargerthanreactiontime.Thereactionrateistherefore
mostlydeterminedbythemixingtime.Inmicroscale,mixingtimecanbereducedtothesameorderor
evenlessthanthereactiontime.Measurementofrealreactionkineticsisthereforepossibleinmicroscale.
Mixingtimeand,consequently,thereactionproductscanbepossiblycontrolledinmicroscale.If
thereactionresultsinonlyoneproduct,mixingtimecanonlyaffectthereactionrate.Iftherearemore
than one product, mixing time determines the product composition and distribution. The following
example shows the impact of mixing type on reaction results. Assuming a reaction between the
substrate S and reagent R:
SþR/P (1.1)
1
whereP isthedesiredreactionproduct.However,P canreactwithRtoformanundesiredproductP :
1 1 2
P þR/P (1.2)
1 2
If mixing relies on the relatively slow process of molecular diffusion, as in the case of a parallel
laminationmicromixer,P hasenoughtimetoreactwithR.Therefore,themainproductofthereaction
1
4 CHAPTER 1 Introduction
FIGURE1.1
Passiveandactivemicromixers.
1.2 Micromixers as microreactors 5
FIGURE1.2
Effectofmicromixertypeonachemicalreactionwithmorethanoneproduct:(a)fastmixingwithchaotic
advectionand(b)slowmixingwithmoleculardiffusion.(S:substrate,R:reagent,P1:product1,P2:product2).
processisP .Ifmixingoccursquickly,forinstance,throughchaoticadvection,allmoleculesofRare
2
utilizedinthefirstreactiontoformP ,notmanyRmoleculesareleftforthesecondaryreaction.Thus,
1
the main productofthe reaction isP . Figure 1.2 illustrates this problem.
1
1.2 MICROMIXERS AS MICROREACTORS
Since2000,wehavewitnessedincreasingactivitiesintheuseofmicrofluidictechnologyinanalytical
chemistry and chemical production. Mixing is the central process of most microfluidic devices for
medicaldiagnostics,geneticsequencing,chemistryproduction,drugdiscovery,andproteomics.The
impactofmicromixersonmicrofluidicsystemsforchemicalanalysisandsynthesisissimilartothatof
transistorsinintegratedcircuits.Althoughmicromixersforanalysisandsynthesisaredifferent,some
applications require both classes. For instance, in combinatorial chemistry and screening micro-
devices,micromixersare analyticaltoolsforinformation gatheringandsynthetictoolsfor providing
minute quantities ofproducts.
In micromixers for analysis, information gained from this product is the purpose of the mixing
process and the reaction. The amount of the reaction product only needs to fulfill the delectability
requirements.Incontrast,reactionproductsinsynthesisapplicationsareusedtomakematerialswith
improvedpropertiesatfavorableconditionsgivenbymicromixers.Alargeamountoftheproductmay
6 CHAPTER 1 Introduction
beneeded.Thus,thedesignofmicromixersforsynthesisshouldbereadyfornumberingupinthecase
ofa large-scaleproduction [4].
Micromixers as microreactors will potentially have a large impact on chemical technology.
Because of their small size, micromixers allow control over a number of parameters of production
processes in chemistry and the pharmaceutical industry. Reaction conditions that are unusual in
macroscalearetechnicallypossibleinmicromixers.Theadvantagesofreactioninmicromixersare
small thermal inertia, uniform temperature, high gradient of the different physical fields, short
residence time, and a high surface-to-volume ratio. The small thermal inertia allows fast and
precise temperature control in micromixers. Miniaturization leads to higher rates of heat and mass
transfer. Compared to their macroscale counterparts, micromixers can offer more aggressive
reaction conditions. The large surface-to-volume ratio allows for an effective suppression of
homogenoussidereactionsinheterogeneouslycatalyzedgas-phasereactions.Thesmallsizemakes
reaction in micromixers safe because of the suppression of flames and explosions. Explosions can
be suppressed by using mixing channels with hydraulic diameter less than the quenching distance
(cid:2)
[5]. For instance, the fluorination of toulene with fluorine can be carried out at –10 C in micro-
(cid:2)
mixers.Conventionalreactorswouldrequireatemperatureof–70 Cduetotheexplosivenatureof
thereaction[5].Inthecaseofaccidents,thesmallamountsofhazardousreactionproductsareeasy
to contain.
Micromixers as microreactors enable a faster transfer of research results into production. Since
scalingupthemixerdesignisnotpossible,labsetupcanimmediatelybetransferredintolarge-scale
productionbynumberingup.Sincenumberingupistheonlyoptionformicromixers,scalinglawleads
to high ratio between device material and reaction volume. This means, fixed production costs will
increase with miniaturization because of the higher costs of materials and infrastructure. If micro-
reactors deliver a similar performance to their conventional macroscale counterparts, the higher
production costs will make micromixers unprofitable for chemical production. However, for some
particular products, the smaller production capacity may save costs through other factors such as
replacingabatchprocessbyacontinuousprocess.Forinstance,duetoslowmassandheattransferin
macroscalereactors,reactiontimeforfinechemicalsisdeterminedbymixingandismuchlongerthan
needed for reaction kinetics. Replacing a batch-based macroscale reactor by a continuous-flow
microreactor can significantly reduce the reaction time. The reactor volume is smaller, but the total
throughputper unittimeishigher.Asaresult,forthe same amount ofproductsthe reaction process
wouldbe carriedout faster inmicroreactors.
In addition, as illustrated in Fig. 1.2, selectivity of the reaction may increase with micromixers.
Productionyields of microreactors could exceed those of batch-basedmacroscale reactors. The next
cost-saving factor of micromixers for chemical production is the intensification process. The larger
surface-to-volumeratioprovidesmoresurfaceforcatalystincorporation.Comparedtoitsmacroscale
counterpart,theamountofcatalystneededinamicroreactorcanbedecreasedbyafactorof1000.If
thecostofthecatalystsissignificantintheoverallproduction,savingcatalystscancompensateforthe
largeamountof constructionmaterials needed for numbering-up microreactors.
Micromixershaveanindirectimpactonnationalsecurityduetothepossibilityofon-sideportable
detection systemsfor chemical weaponsand explosive.However, due to its portability, micromixers
couldbemisusedbycriminalsandterrorists[5].Aminiaturechemicalplantfittedintoasuitcasecould
be misused for the production of drugs and hazardous gases. Raw chemicals may not be detectable
prior to reactions in these miniature plants. Lethal nerve gases could be formed by two primary
1.3 Organization of the book 7
less-toxiccompoundsinmicromixers.Detectionfacilitiesshouldbeextendedtothesepre-compounds
to counter this potential misuse.
1.3 ORGANIZATION OF THE BOOK
This book offers a wide spectrum for the study of the mixing processes in microscale, from funda-
mentaltransporteffectstoavarietyofdesignstospecificapplicationsinchemistryandlifesciences.
AftertheintroductioninChapter1,Chapters2and3discussthebasicterminologyandfundamental
physicsoftransporteffectsthatwillbeusedfordesigningmicromixers.Chapter2discussesindetail
thethreekeymasstransporteffectsoftenusedinmicromixers:moleculardiffusion,Taylordispersion,
andchaoticadvection.Thechallengesandadvantagesofminiaturizationinmixingarehighlightedin
this chapter with the help of scaling laws. The scaling laws are discussed based on nondimensional
numbers which represent relationships between different transport effects. Chapter 3 discusses the
fundamentals of the different numerical schemes for modeling the transport phenomena in
micromixers.
Chapter 4 gives an overview on available microtechnologies for making micromixers. Basic
techniques of conventional silicon-based microtechnologies are covered. Since polymers are chemi-
cally and biologically compatible, polymeric micromachining is the focus of this chapter. Technol-
ogiesforbondingandsealingarenecessaryformakingamicromixer.Thischapteralsodiscussesthe
design and fabrication of fluidic interconnects that are needed for interfacing micromixer to larger-
scale devices and equipments.
Different concepts and designs for micromixers are discussed in Chapters 5 to 7. Although all
mixing concepts involve molecular diffusion, Chapter 5 only discusses concepts where molecular
diffusionistheprimarymasstransferprocess.Basedonthearrangementofthemixedphases,thefour
mixertypesdiscussedinthischapterareparallelmixer,serialmixer,sequentialmixer,andinjection
mixer.
Chapter6isdedicatedtomicromixersbasedonchaoticadvection.Incontrasttothemicromixers
discussedinChapter5,thisclassofmicromixersreliesonbulkmasstransportformixing.Thegeneral
conceptsforgeneratingchaoticadvectionarestretchingandfoldingoffluidstreams.Thesestretching
andfoldingactionscanbeimplementedinaplanardesignorinacomplexthree-dimensionalchannel
structure. A special case of chaotic advection is mixing in microdroplets. Manipulation of the flow
fieldinsideadropletcanleadtothesamestretchingandfoldingeffectsasachievedinacontinuous-
flowplatform.
Chapter 7 discusses activemixers, where mixing is achievedwith energyinduced by an external
source.Activemixersaresimilartoconventionalmacroscalemixerswherefluidmotionisdrivenbyan
impeller.However,asdiscussedinSection1.1,miniaturizationoftheimpellerconceptwouldnotwork
because of the dominant viscous force in microscale. This chapter discusses different concepts for
inducing a disturbance into the flow field. The use of electrohydrodynamic, dielectrophoretic, elec-
trokinetic, magnetohydrodynamic, acoustic,and thermal effects inmicromixersis discussed here.
Chapter8summarizeskeydiagnosticstechniquesforcharacterizationofmicromixers.Sinceboth
velocityfieldandconcentrationfieldareimportantforgoodmixing,diagnosticstechniquesforthese
fields are the focus of this chapter. The quantification of the extent of mixing is important for the
evaluation ofperformance as well asthe design optimizationof micromixers.
8 CHAPTER 1 Introduction
Chapter9discussesthecurrentapplicationsofmicromixers.Differentapplicationsneeddifferent
designrequirements. Thechapterdiscussesapplications fromthe two major areas: lab-on-a-chip for
chemicalandbiochemicalanalysisandchemicalproduction.Thischapteralsorecommendsmaterials
andmixertypes for each application area.
References
[1] E.L.Paul, V.A.Atiemo-Oberg,S.M.Kresta, Handbookof IndustrialMixing, Wiley,New York,2004.
[2] N.T.Nguyen,Z.G.Wu,“Micromixers–areview”,JournalofMicromechanicsandMicroengineeringvol.
15 (2005), R1–R16.
[3] J.RBourne,“MixingandSelectivityofChemical Reactions”,OrganicProcessResearch &Development,
vol. 7 (2003),471–508.
[4] K.F.Jensen,TheImpactofMEMSonthechemicalandpharmaceuticalindustries,in:TechnicalDigestof
theIEEESolidStateSensorandActuatorWorkshop,HiltonHeadIsland,SC,4–8June,2000,pp.105–110.
[5] H. Lo¨we, V. Hessel, A. Muller, “Microreactors. Prospects already achieved and possible misuse”, Pure
AppliedChemistryvol.74(2002),2271–2276.
CHAPTER
2
Fundamentals of mass transport
in the microscale
CHAPTER OUTLINE
2.1 TransportPhenomena.....................................................................................................................10
2.1.1 Molecularlevel............................................................................................................10
2.1.2 Continuumlevel..........................................................................................................14
2.1.2.1 Conservationofmass...............................................................................................14
2.1.2.2 Conservationofmomentum.....................................................................................15
2.1.2.3 Conservationofenergy.............................................................................................21
2.1.2.4 Conservationofspecies...........................................................................................22
2.2 MolecularDiffusion........................................................................................................................22
2.2.1 RandomwalkandBrownianmotion...............................................................................22
2.2.2 Stokes–Einsteinmodelofdiffusion...............................................................................23
2.2.3 Diffusioncoefficient.....................................................................................................24
2.2.3.1 Diffusioncoefficientingases....................................................................................24
2.2.3.2 Diffusioncoefficientinliquids..................................................................................25
2.2.3.3 Diffusioncoefficientofelectrolytes...........................................................................26
2.3 TaylorDispersion...........................................................................................................................28
2.3.1 Two-dimensionalanalysis.............................................................................................28
2.3.2 Three-dimensionalanalysis...........................................................................................34
2.4 ChaoticAdvection..........................................................................................................................36
2.4.1 Basicterminologies.....................................................................................................36
2.4.2 Examplesofchaoticadvection......................................................................................40
2.4.2.1 Lorentz’sconvectionflow.........................................................................................40
2.4.2.2 Deanflowincurvedpipes........................................................................................40
2.4.2.3 Flowinhelicalpipes................................................................................................46
2.4.2.4 Flowintwistedpipes...............................................................................................47
2.4.2.5 Flowinadroplet......................................................................................................50
2.5 Viscoelasticeffects........................................................................................................................53
2.6 ElectrokineticEffects.....................................................................................................................55
2.6.1 Electroosmosis............................................................................................................55
2.6.1.1 TheDebyelayer......................................................................................................55
2.6.1.2 Electroosmotictransporteffect.................................................................................57
2.6.1.3 Electrokineticflowbetweentwoparallelplates..........................................................58
2.6.1.4 Electrokineticflowinacylindricalcapillary...............................................................60
2.6.1.5 Electrokineticflowinarectangularmicrochannel......................................................61
9
Nam-TrungNguyen:Micromixers,Secondedition.DOI:10.1016/B978-1-4377-3520-8.00002-4
Copyright(cid:1)2012ElsevierInc.Allrightsreserved.
10 CHAPTER 2 Fundamentals of mass transport in the microscale
2.6.1.6 Ohmicmodelforelectrolytesolutions.......................................................................63
2.6.2 Electrophoresis............................................................................................................64
2.6.3 Dielectrophoresis.........................................................................................................65
2.7 MagneticandElectromagneticEffects.............................................................................................66
2.7.1 Magneticeffects..........................................................................................................66
2.7.1.1 Electromagneticeffects............................................................................................68
2.8 ScalingLawandFluidFlowinMicroscale.......................................................................................68
References...........................................................................................................................................71
2.1 TRANSPORT PHENOMENA
Transport phenomena in micromixers can be described theoretically at two basic levels: molecular
levelandcontinuumlevel.Thetwodifferentlevelsofdescriptioncorrespondtothetypicallengthscale
involved. Continuum model can describe most transport phenomena in micromixers with a length
scalerangingfrommicrometerstocentimeters.Mostmicromixersforpracticalapplicationsareinthis
rangeoflengthscale.Molecularmodelsinvolvetransportphenomenaintherangeofonenanometerto
one micrometer. Mixers with length scale in this range should be called “nanomixer.” The term
“micromixer” in thisbook will cover devices with submillimeter length dimension.
At the continuum level, the fluid is considered as a continuum. Fluid properties are defined
continuously throughout the space. At this level, fluid properties, such as viscosity, density, and
conductivity,areconsideredasmaterialproperties.Transportphenomenacanbedescribedbyasetof
conservation equations for mass, momentum, and energy. These equations of change are partial
differentialequations,whichcanbesolvedforphysicalfieldsinamicromixer,suchasconcentration,
velocity, andtemperature.
Miniaturizationtechnologieshavepushedthelengthscaleofmicrodevicesfurther.Intheeventof
nanotechnology, scientistsandengineerswillencountermorephenomena at molecularlevel. Atthis
level, transport phenomena can be described through molecular structure and intermolecular forces.
Because many micromixers are used as microreactors, a fundamental understanding of molecular
processesisimportantfordesigningdeviceswithlengthscaleinthemicrometertocentimeterrange.
2.1.1 Molecular level
Atmolecularlevel,thesimplestdescriptionoftransportphenomenaisbasedonthekinetictheoryof
dilutedmonatomicgases,whichisalsocalledtheChapman–Enskogtheory.Theinteractionbetween
nonpolar molecules isrepresented by the Lennard–Jones potential, which hasan empirical form of:
(cid:1) (cid:5)
(cid:3) (cid:4) (cid:3) (cid:4)
s 12 s 6
f ðrÞ ¼ 43 c (cid:2)d ; (2.1)
ij ij r ij r
wheresisthecharacteristicdiameterofthemolecule,risthedistancebetweenthetwomolecules,and
3 is the characteristic energy, which is the maximum energy of attraction between the molecules. In
(2.1),theterm(s/r)12representstherepulsionpotential,whiletheterm(s/r)6representstheattraction
potentialbetweenthepairofmolecules.Thecoefficientsc andd aredeterminedbymoleculetypes
ij ij