ELECTROOSMOTIC FLOW IN A MICROCHANNEL PACKED WITH MICROSPHERES

Transcription

1 ELECTROOSMOTIC FLOW IN A MICROCHANNEL PACKED WITH MICROSPHERES KANG YUEJUN SCHOOL OF MECHANICAL & AEROSPACE ENGINEERING NANYANG TECHNOLOGICAL UNIVERSITY 5

2 Elecoosmoic Flow i a Micochael Packed wih Micosphees Kag Yueju School of Mechaical & Aeospace Egieeig A Thesis Submied o he Nayag Techological Uivesiy I fulfillme of he equieme fo he degee of Doco of Philosophy 5

3 This wok is dedicaed wih affecio o my wife Raa, whose cae, couage, ad paiece have bee a ispiaio o me, ad o my dea gadpaes, paes, ad youge bohe, whose love, suppo, ad ecouageme have pu me whee I am ow. i

4 Absac The elecokieic EK micopump, which employs elecoosmosis ahe ha hydosaic pessue, is a apidly emegig pumpig echique o aspo eages ad elecolyes hough micochael ewoks. Due o may advaages such as o movig pas, low solve ad sample cosumpio, ad pecise cool, i has bee favoed i developig he advaced micofluidic devices which ca pefom oal biochemical aalysis o a sigle fabicaed chip. Use of poous sucues gealy elages he iefacial aea, ad hus sigificaly ehaces he pessue-buildig capaciy of he EK micopump. Alhough elecoosmosis i poous media has bee widely paciced as a kieic pumpig souce i may eleco-sepaaio echiques e.g., Capillay Eleco-Chomaogaphy, he heoeical ivesigaios o he udelyig mechaism sill emai limied. I ligh of his, his disseaio povides a fudameal, sysemaic ad i-deph exploaio o he elecoosmosis i poous micosucues ad is dyamic aue. The heoeical developme i his disseaio maily compises hee aspecs. Fisly, a mahemaical model is developed o descibe he elecoosmoic flow i a micocapillay packed wih micosphees ude elecokieic wall effec. The model is based o he Cama-Kozey heoy, also kow as capillay model. The Dacy velociy of he elecoosmoic flow i he poous media is obaied usig volume aveagig mehod, akig io cosideaio of he poosiy ad ouosiy of he poous packig. Ad a velociy coecio due o he elecokieic wall effec is icluded by aalyically ad umeically solvig he modified Bikma s macoscopic momeum equaio. Secodly, moivaed by poeial applicaio i elecokieic mico-acuaos, his ii

5 sudy also peses a aalysis o AC elecoosmoic flow i boh ope-ed ad closed-ed micocapillaies packed wih micosphees. The oscillaig Dacy velociy i a ope-ed capillay i espose o a AC elecic field is obaied usig Gee s fucio fomulaio. The backpessue associaed wih he coue-flow i a closed-ed capillay is obaied by solvig he modified Bikma s momeum equaio. Thidly, a sysemaic umeical simulaio is caied ou o evaluae he Joule heaig effec o he elecoosmoic flow i a packed micocapillay. Specifically, he o-uifom elecic field esuled fom he empeaue gadie is aalyzed. I is also foud ha a iduced pessue field peses i he capillay due o he velociy vaiaio caused by he empeaue elevaio. I addiio, he dyamic elecoosmoic flows i simple geomeies, such as cylidical ad aula micocapillaies, ude he ime-depede elecic field ae aalyzed. The aalyical soluios ae deived as a basis of model developme fo he elecoosmosis i poous media usig capillay model. As he ohe esseial pa of his disseaio, a expeimeal sudy o elecoosmosis i a packed micocapillay is coduced. Elecokieic micopumps ae fabicaed usig high pessue dive sluy packig echique. Two diffee mehods o chaaceize he fabicaed micopumps ae employed ad compaed ude he iflueces of he capillay size, paicle size, soluio coceaio ad ype of he elecolye soluios, ad he capillay legh. I is foud ha he expeimeal daa ae geeally i easoable ageeme wih he pedicios by he model developed i his sudy. Howeve, deviaio is obseved ude he codiio of high elecic field segh. iii

6 Ackowledgemes The wok peseed i his hesis was coduced bewee Jue ad July 4 i he eseach goup led by D Yag Chu, Chales. As my picipal supeviso, D Yag opeed he doo of he wold of scieific eseach fo me ad augh me how o ejoy he pleasue a evey sage. The fis coac bewee D Yag ad me bega eve oe yea befoe I joied NTU whe I was sill a fial yea udegaduae i USTC. He waed o ake i some gaduae sudes sholy afe he joied NTU faculy. Howeve due o may easos I could o maage o joi his goup igh afe I gaduaed. Alhough I bega o wok as a egiee i a IT compay, my log-em caee emaied misy. I my secod ial of applicaio, D Yag coaced me agai ad gave me a lo of good suggesios ad help. Fially, I was fouae eough o become his fis PhD sude i summe of. Duig he hee yeas a NTU, I have vey clea diecio ude he close supevisio of D Yag. He always gave me ispiig suggesios ad shaed me wih his ich eseach expeiece hough ou feque discussios. Mos impoaly, he gadually helped me build up my cofidece fo a successful academic caee. I also appeciaed vey much he cae exeded o my family by D Yag ad his wife, Ms Xu Li. The compleio of his wok is also owed o my co-supeviso D Huag Xiaoyag. Alhough my ieacio wih D Huag is o so much, i has always bee simulaig ad hough povokig. Especially his vey pacical suggesios o my each peseaio have helped me impove my commuicaio ad peseaio skill o a moe effecive ad efficie way. D Huag also helped me ge some pa-ime eachig jobs, which gave me some elief fom he pessue of livig expeses. iv

7 My sicee gaiude should also be exeded o some of my colleagues ad fieds. Acually pa of his wok would o be fiished wihou hei coibuios. M Ta Say Chog, who was a fial yea udegaduae sude, had spe almos oe yea wokig closely wih me o he expeime. No oe has expeiece o his expeime befoe hus we had o sa fom scach. We had may sleepless ighs whe i was difficul. Bu fially i was doe afe so may ial ad eos, ad we all leaed much duig he eie couse of expeime. M Macos, whose is a vey sma guy doig his mase sudy, had happily coopeaed wih me i some pojecs. His ucodiioal help i my modelig ad expeimes was always gealy appeciaed. My fellow eseach sudes, M. Tag Gogyue, M. Liu Yue, ad M. Che Xuyag gave me a lo of advices o umeical simulaios. I would also hak he echicia office M. Yap Pow Khim Eic i Fluid Dyamics Lab ad M. Yua Kee Hock i Themal ad Fluids Reseach Lab fo hei echical assisace. Seio cleical office Ms Soh Meow Chg i geeal office of he School of MAE geeed me whe I fis joied NTU ad coiued o advice me fo vaied admiisaive issues duig my sudy hee. I am also gaeful fo may ohe people, wihou poducig a log lis of hem, who have helped me i diffee ways ad made my hee yeas a NTU memoable. I would like o hak he diec suppo fo hee yeas fom Nayag Techological Uivesiy hough a eseach scholaship, wihou which I would o have a chace o wok wih so may sma ad billia people. v

8 Via Kag Yueju was bo o Mach 4, 977, i Xiagfa, Hubei, People s Republic of Chia. He obaied Bachelo of Egieeig degee i July afe five yeas udegaduae sudy a Uivesiy of Sciece ad Techology of Chia, Hefei, Ahui, Chia. Afe ha he had woked as a Poduc Specialis fo a yea i Ace Commuicaio & Mulimedia Co. Ld., Suzhou, Chia. I Jue, he joied Nayag Techological Uivesiy as a posgaduae eseach sude fo a PhD degee i Mechaical ad Aeospace Egieeig. Refeeed Jouals: Lis of Publicaios Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Joule Heaig Effecs of Elecoosmoic Flow i Micocapillaies Packed wih Micosphees" Lagmui ude eview, 5 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Aalysis of Elecoosmoic Flow i a Micochael Packed wih Micosphees" Joual of Micofluidics ad Naofluidics, Macos, Yueju Kag, Kim Tiow Ooi, Chu Yag, ad Teck Neg Wog "Fequecy Depede Velociy ad Voiciy Fields of Elecoosmoic Flow i a Closed-Ed Cylidical Micochael". Joual of Micomechaics ad Micoegieeig 5, Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Aalysis of he Elecoosmoic Flow i a Micochael Packed wih Homogeeous Micosphees ude Elecokieic Wall Effec ". Ieaioal Joual of Egieeig Sciece 4, Yueju Kag, Chu Yag, ad Xiaoyag Huag. "AC Elecoosmosis i Micochaels Packed wih a Poous Medium". Joual of Micomechaics ad Micoegieeig 4, vi

9 6 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Modellig of he Capillay Elecochomaogaphy wih Applicaio i BioMEMS". Ieaioal Joual of Compuaioal Egieeig Sciece, Vol. 4, No., Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Dyamic aspecs of elecoosmoic flow i a cylidical micocapillay". Ieaioal Joual of Egieeig Sciece 4, 3-8 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Elecoosmoic Flow i a Capillay Aulus wih High Zea Poeials". Joual of Colloid ad Ieface Sciece 53, Cofeece papes: Chu Yag, Yueju Kag, ad Xiaoyag Huag. "AC Elecokieic Micopump Usig Packed Micocapillaies" 3d Ieaioal Cofeece o Maeials fo Advaced Techologies & IUMRS - Ieaioal Cofeece i Asia 5, 3-8 July, 5, Sigapoe acceped. Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Tasie Joule Heaig ad is Effecs o Elecoosmoic Flow i a Micocapillay Packed wih Micosphees" The 3d Ieaioal Cofeece o Micochaels ad Miichaels, 3-5 Jue, 5, Tooo, Caada acceped. 3 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Aalysis of Elecoosmoic Flow i a Micochael Packed wih Micosphees" The d Ieaioal Cofeece o Micochaels ad Miichaels, 7-9 Jue, 4, Rochese, New Yok, USA. 4 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Fequecy Depede Elecoosmoic Flow i a Capillay Packed wih Micosphees" The s Ieaioal Symposium o Mico & Nao Techology, 4-7 Mach, 4, Hoolulu, Hawaii, USA. 5 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Modellig of he Capillay Elecochomaogaphy wih Applicaio i BioMEMS". d Ieaioal Cofeece o Maeials fo Advaced Techologies & IUMRS Ieaioal Cofeece i Asia 3, 7-4 Decembe, 3, Sigapoe 6 Yueju Kag, Chu Yag, ad Xiaoyag Huag. "Modellig of he Capillay Elecochomaogaphy wih Applicaio i BioMEMS". 5h IFAC Symposium o Modellig ad Cool i Biomedical Sysems, -3 Augus 3, Melboue, Ausalia. vii

10 Table of Coes Chape Ioducio.... Backgoud ad Moivaio. Objecives 3.3 Oulie of he Thesis 5.4 The Elecic Double Laye ad he Elecoosmoic Flow 6.5 Lieaue Review.9.5. Seady-sae elecoosmoic flow Dyamic elecoosmoic flow Elecoosmoic flow hough poous media Themal effecs i elecoosmoic flow sysems Deig ad fabicaio of elecokieic micopump... Chape Dyamics of Elecoosmoic Flow i a Capillay Cylide.3. Ioducio 3. Poblem Fomulaio.4.3 Gee s Fucio Mehod fo Ihomogeeous Diffusio Equaio..7.4 Useady ad Fequecy-depede Elecoosmoic Flow 3.4. Siusoidally aleaig elecic field.3.4. Sep chage elecic field Pulsed elecic field.3.5 Resuls ad Discussio Poeial pofile Tasie velociy pofile AC peubaio Summay 46 Chape 3 Aalyical Soluio of Elecoosmoic Flow i a Capillay Aulus wih High Zea Poeials 47 viii

11 3. Ioducio Dyamics of he Elecoosmoic Flow Geealiy of Aulus Model Resuls ad Discussio Poeial pofile Velociy pofile Pedicio fo he EDL-elaed coecio faco, G Pedicio fo he coecio faco o he Smoluchowski equaio, J Summay...75 Chape 4 DC Elecoosmoic Flow i a Micochael Packed wih Micosphees ude Elecokieic Wall Effec Ioducio Mehod of Volume Aveagig Macoscopic EOF i Homogeeous Chaged Micosphees Iesiial EOF velociy Iesiial EDL poeial field Macoscopic EOF i a Chaged Micocapillay Packed wih Neually Chaged Micosphees EOF velociy field EDL poeial field Aalyical soluio of he modified Bikma momeum equaio Slip velociy appoximaio based soluio o he modified Bikma momeum equaio Oveall Macoscopic EOF Velociy Resuls ad Discussio Summay.. Chape 5 AC Elecoosmoic Flow i a Micochael Packed wih Micosphees.4 5. Ioducio..4 ix

12 5. Flow i a Packed Micochael Coeced o Two Ope Resevois Flow i a Packed Micochael wih Closed Eds. 5.4 Resuls ad Discussio 5.4. Oscillaig flow velociy i a packed micochael coeced o wo esevois Oscillaig backpessue i a packed micochael wih wo closed eds Summay..4 Chape 6 Joule Heaig Effec o he Elecoosmoic Flow i a Micochael Packed wih Micosphees 5 6. Ioducio Poblem Fomulaio Equaios of eegy Equaios of coiuiy ad momeum Coiuiy of elecic cue Numeical Mehod Resuls ad Discussio Tasie developig empeaue field ad is effec o EOF Compaiso wih published woks Effecs of wokig paamees Summay...6 Chape 7 Expeimeal sudies of he Elecoosmoic Flow i Micocapillaies Packed wih Micosphees Ioducio Colum Fabicaio Pepaaio of he maeials Packig pocedue Chaaceizaio ad Flow Measueme Poosiy ad ouosiy..68 x

13 7.3. Tes colum mehod Cue moioig mehod EOF velociy ad zea poeial Resuls ad Eo Aalysis Scaig eleco micoscopy Poosiy ad ouosiy Eo aalysis he cue moioig vs. he es colum mehod Effecs of wokig paamees Summay..89 Chape 8 Coclusio ad Fuue Sudies 9 8. Coibuios Made by This Sudy Recommedaios fo Fuue Sudies..95 Appedices: A. Validaio of Bolzma Disibuio ude Dyamic Elecoosmoic Flow...99 B. Elecic Double Laye Poeial Disibuio i a Capillay Cylide...3 C. Slip Velociy Appoximaio fo he AC Elecoosmosis i a Cylidical Capillay. D. Validaio of Osage Relaioship fo Tasie ad Fequecy-depede EOF i a Capillay wih Closed-eds E. Elecic Double Laye Poeial Disibuio i a Capillay Aulus... F. Fequecy-depede Elecoosmoic Flow Usig Gee s Fucio Fomulaio 3 G. Numeical Scheme fo Solvig he Tempeaue Fields i he Joule Heaig Effec..38 Refeeces.4 xi

14 Lis of Tables 6- Maeial popey i simulaio Resuls fo poosiy measueme Resuls fo ouosiy measueme Zea poeials a he paicles suface fo diffee elecolye coceaios...85 xii

15 Lis of Figues - a Schemaic epeseaio of he ioic disibuio close o a posiively chaged suface i he Gouy-Chapma model Hiemez, b The egio vey close o he solid, showig he Se plae whee he poeial is ψ B ad he shea plae whee he poeial is ζ Pobsei, Elecical poeial a fo he Gouy-Chapma diffuse egio b fo he Se model showig expoeial decay fom he Se plae Pobsei, Compaiso of he esuls fo dimesioless EDL poeial, ψ / ζ vesus dimesioless adius, / a, obaied fom he Debye-Hückel liea appoximaio, he aalyical scheme poposed i his sudy, ad he umeical iegaio of he complee Poisso-Bolzma equaio i a cylidical capillay fo wo cases: i κ a, ad ii κ a 5, wih a fixed dimesioless zea poeial, Ψ s a Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a.35 b Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a...36 c Time devoluio wih fixed he elecokieic diamee, κ a 3.57 ad he zea poeial, Ψ s Seadily oscillaig velociy disibuios alog he dimesioless adius fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he elecokieic diamee, κ a 3.57, he zea poeial, Ψ s 4 ad eigefequecy, f * 5.5 KHz. Sapshos ae peseed a five diffee chaaceisic momes: ω, π/4, π/, 3π/4, π. Compaiso wih he esuls obaied fom slip velociy appoximaio epeseed by dashed cuves. a Low fequecy of he exeal field, f. f *...4 b The eigefequecy of he exeal field, f f *.4 c High fequecy of he exeal field, f f * 4-4 Dimesioless mea velociy vesus ime wih fixed he elecokieic diamee, κ a 3.57 ad he zea poeial, Ψ s 4. a Seadily oscillaig mea velociy fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he sysem eigefequecy of f * 5.5 KHz.44 b Evoluio ad devoluio of he mea velociy whe he exeal field is swiched o fo diffee duaio of ime δ xiii

16 3- Schemaic diagam of a aulus. The adii of he ie ad he oue cylides ae α a ad a, especively No-dimesioal EDL poeial, Ψ vesus o-dimesioal adius, / a of he aulus. The geomey aio of he aula ie adius o oue adius, α.4. The zea poeial aio of he ie cylide o he oue cylide, β ad β -, deoig a symmeicallychaged ad opposiely-chaged aulus, especively. a Compaiso of he esuls obaied fom he Debye-Hückel liea appoximaio, he aalyical scheme poposed i his sudy, ad he umeical iegaio of he complee Poisso-Bolzma equaio i a aulus fo wo cases: i κ a, ad ii κ a 5, wih fixed he o-dimesioal zea poeial of he oue cylide, Ψ s 8 ad he zea poeial aio of he ie cylide o he oue cylide, β.56 b Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s 57 c Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a Dimesioless asie velociy, u / u s vesus dimesioless adius, / a fo he case of adius aio, α.4 ad he zea poeial Ψ s 4. a Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a 5 ad β 6 b Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a 5 ad β c Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a ad β..63 d Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a ad β No-dimesioal elecoosmoic flow velociy, u / u s vesus o-dimesioal adius, / a of he aulus. The geomey aio of he aula ie adius o oue adius, α.4. a Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s ad adius aio β ±.. 65 b Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s ad adius aio β ±..66 c Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a 5 ad adius aio β ± 67 d Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a 5 ad adius aio β ± 68 xiv

17 e Effec of he zea poeial aio, β ζ i / ζ wih fixed he elecokieic diamee, κ a 5 ad he o-dimesioal zea poeial of he oue cylide, Ψ s Vaiaio of he EDL-elaed coecio faco -G wih he elecokieic diamee, κ a fo vaious zea poeials of he oue cylide, Ψ s. The geomey aio of he aula ie adius o oue adius, α.4. The ie ad oue aula walls ae equally chaged, β Coecio faco fo he Smoluchowski Equaio, J vesus he geomeic aio of he aula ie adius o oue adius, α fo vaious values of he zea poeial aio, β ζ i / ζ. Wihou icludig he EDL-elaed coecio faco, G, we ecove Tsao s esuls epeseed by he dashed lies Ciical value combiaios of he adius aio, α ad he zea poeial aio, β fo zeo e flow ae Schemaic illusaio of epeseaive elemeay volume REV: he legh scale of he REV is much lage ha he poe scale, bu cosideably smalle ha he legh scale of he macoscopic flow domai Schemaic illusaio of decomposig he oveall macoscopic EOF velociy i a packed capillay io wo sepaae compoes due o: i he coibuios fom homogeeous isoopic desely packed wih chaged micopaicles, ad ii he coibuios fom he chaged capillay wall wih eual packig Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee values of zea poeial aio, ζ w / ζ p. The esuls ae obaied o he basis of a he umeical soluio of Eq b he aalyical soluio based o Eq c he slip velociy appoximaio based o Eq Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee sizes of packig paicles, d p. The esuls ae obaied o he basis of a he umeical soluio of Eq b he aalyical soluio based o Eq c he slip velociy appoximaio based o Eq Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee elecic field seghs, E Elecoosmoic flow velociy disibuios i a chaged micocapillay packed xv

18 wih chaged micosphees, fo diffee chael sizes, R w. 4-7 Elecoosmoic flow ae vesus applied elecic field fo diffee values of paicle diamee, d p, ad zea poeial aio, ζ w / ζ p Schemaic illusaios of he AC elecoosmoic flow i a poous medium. a flow i a packed micocapillay coeced wih wo esevois. b flow i a packed micocapillay wih wo closed eds. Whee u De is he file Dacy velociy due o elecoosmosis, ad u Dp is he file Dacy velociy due o backpessue Dimesioless ime-peiodic oscillaig Dacy elecoosmoic flow velociy vesus ime. a seadily oscillaig Dacy velociy fo hee diffee exciaio fequecies,.f*, f* ad f* a a fixed poe size, κr poe b seadily oscillaig Dacy velociy fo hee diffee poe sizes,.4, 4.8 ad a a fixed exciaio fequecy.f* 5 c oscillaig Dacy velociy fo hee diffee poe sizes,.4, 4.8 ad a a fixed exciaio fequecy f* Dimesioless maximum Dacy elecoosmoic flow velociy vesus exciaio fequecy fo hee diffee poe sizes,.4, 4.8 ad Time evoluio of he DC elecoosmoic flow velociy fo hee diffee poe sizes,.4, 4.8 ad Dimesioless ime-peiodic oscillaig backpessue dop vesus ime fo hee diffee exciaio fequecies,.f*, f* ad f* a a fixed poe size, κr poe Dimesioless maximum backpessue dop vesus exciaio fequecy fo hee diffee poe sizes,.4, 4.8 ad Time evoluio of he backpessue dop due o DC elecoosmosis i a closed-ed micocapillay fo hee diffee poe sizes,.4, 4.8 ad.3 6- Schemaic illusaio of he wo modelig subsysems of he packed micocapillay Tasie developme of he empeaue field. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a alog he axis 38 b adial disibuio a dowseam Tasie developme of he elecic field segh. Wokig paamees ae ideical o xvi

19 hose i Figue Tasie developme of he elecoosmoic velociy. Wokig paamees ae ideical o hose i Figue 6-. a axial disibuio...4 b adial disibuio a dowseam Tasie developme of he iduced pessue. Wokig paamees ae ideical o hose i Figue Resuls compaiso wih published woks by Keim ad Ladisch 3. Wokig paamees: capillay ie diamee d w 3.8 cm, capillay legh L 38. cm, packig paicle size d p 5 µm, poosiy φ.36, ouosiy τ.5, elecolye coceaio C M, zea poeial a paicle suface ζ p 6 mv, applied elecic field Φ 49 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a expeime daa of asie empeaue ise a he cee of he colum oule.46 b modelig of he adial empeaue pofile a colum oule Effec of applied elecic field. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a axial empeaue disibuio...49 b axial elecoosmoic velociy disibuio Effec of elecolye coceaio ad zea poeial a paicle suface. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a axial empeaue disibuio...5 b axial elecoosmoic velociy disibuio Effec of he covecio hea asfe coefficie a he capillay oue suface. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, ad applied elecic field Φ 3 V. a axial empeaue disibuio...54 b axial elecoosmoic velociy disibuio Effec of he capillay diamee. Wokig paamees: capillay legh L 5 cm, xvii

20 packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a axial empeaue disibuio b axial elecoosmoic velociy disibuio Effec of he paicle size. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. a axial empeaue disibuio...6 b axial elecoosmoic velociy disibuio Schemaic illusaio of he expeimeal seup fo colum packig Schemaic illusaio of he packig pocedue Schemaic illusaio of he expeimeal seup usig he es colum mehod A ypical elaioship of cue-ime usig cue moioig mehod Schemaic illusaio of he expeimeal seup usig he cue moioig mehod Scaig eleco micoscopic image of he eaiig fi. a oveall view of he fi i fused silica capillay of ID 7 µm, OD 85 µm. b a magified poio of he fi Scaig eleco micoscopic image of he middle coss-secio of he micocapillay. a oveall view of he coss-secio i fused silica capillay of ID 7 µm OD 85 µm. b a magified poio of he packed beds wih ODS paicles of 6 µm Elecic cue chage wih applied volage fo packed ad upacked capillaies 7µm i ie diamee, 5cm i legh filled wih sauaed NaCl soluio a 5 o C Compaiso of he aveage elecoosmoic velociy usig he cue moioig mehod ad he es colum mehod. The wokig fluid is -5 M NaCl soluio i capillay of ie diamee 7 µm 8 7- Aveage EOF velociy fo diffee capillay ie diamees usig he cue moioig mehod. Wokig soluio is -4 M NaCl soluio. Capillay legh 5 cm. Packig paicle size 6 µm. Fi legh mm Aveage EOF velociy fo diffee NaCl soluio coceaios usig he cue moioig mehod. Capillay legh 5 cm. Capillay ie diamee 7 µm. Packig paicle size 6 µm. Fi legh mm...85 xviii

21 7- Aveage EOF velociy fo diffee packig paicles. Wokig soluio is -4 M NaCl soluio usig he cue moioig mehod. Capillay legh 5 cm. Capillay ie diamee 7 µm. Fi legh mm Aveage EOF velociy fo diffee elecolye soluios usig he cue moioig mehod. Capillay legh 5 cm. Capillay ie diamee 7 µm. Packig paicle size 6 µm. Fi legh mm Aveage EOF velociy fo diffee capillay leghs usig he cue moioig mehod. Wokig soluio is -4 M NaCl. Capillay ie diamee 7 µm. Packig paicle size 6 µm. Fi legh mm..88 B- Aalyical appoximaio fo he hypebolic sie fucio as defied i Eq. B.3..4 B- Esseial geomey of capillay cylide fo solvig he Poisso-Bolzma equaio wih high zea poeials.5 C- Schemaic epeseaio of he slip velociy appoximaio. E- Esseial geomey of a aulus fo solvig he Poisso-Bolzma equaio wih high zea poeials 3 F- Seadily oscillaig velociy disibuios alog he dimesioless adius fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he elecokieic diamee, κ a 65.4, he zea poeial, Ψ s 4 ad eigefequecy, f * KHz. Sapshos ae peseed a five diffee chaaceisic momes: ω, π/4, π/, 3π/4, π. a Low fequecy of he exeal field, f. f *.. 34 b The eigefequecy of he exeal field, f f *...35 c High fequecy of he exeal field, f f *..36 F- Dimesioless mea velociy vesus ime wih fixed he elecokieic diamee, κ a 65.4 ad he zea poeial, Ψ s 4. Seadily oscillaig mea velociy fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he sysem eigefequecy of f * KHz 37 G- Cool volume used fo he disceizaio of he eegy equaio 38 G- Lie-by-lie applicaio of he TDMA mehod. 4 xix

22 Nomeclaue A coss-secioal aea of he specified geomey [m ] a cylidical capillay adius, o he oue adius of aula capillay [m] C iegaio defied by C ξ J ξ sih[ Ψ ξ ] ξ C p Hea capaciy [J kg - K - ] D i diffusio coefficie of he ype-i io [m s - ] d diamee of he cylidical geomey [m] E exeal applied elecic field [V m - ] E he magiude of he elecic field [V m - ] e expoeial fucio, same as expx e e z F elemeay chage,.6-9 [C] flowig axial coodiae diecio foce [N] f fequecy of he applied siusoidally aleaig elecic field [s - ] f i hydodyamic esisace coefficie [J m - s] G Gee s fucio; o aio of he mea elecosaic poeial acoss he aulus egio o he zea-poeial of he oue chaged wall H Heaviside sep fucio I, I zeo-ode ad fis-ode of he fis kid modified Bessel fucio, especively i ui imagiay umbe J a coecio faco o he Smoulochowski equaio J, J zeo-ode ad fis-ode of he fis kid Bessel fucio, especively. K Dacy pemeabiliy of he poous media [m ] K, K zeo-ode ad fis-ode of he secod kid modified Bessel fucio, especively k hemal coduciviy [W m - K - ] k b Bolzma cosa,.38-3 [J K - ] xx dξ

23 L legh of he micocapillay N zeo-ode Bessel fucio of he secod kid ioic umbe coceaio i he bulk phase [m -3 ] i local umbe coceaio of he ype-i io [m -3 ] P pessue gadie [Pa m - ] p pessue [Pa] Q V dimesioless oal volumeic e flow quaiy q hea poducio pe ui volume [J m -3 ] R dimesioless spaial vaiable defied as R κ i he o-dimesioal P-B equaio chael adius [m] T dimesioless adius dimesioless adius of he zeo-velociy plae absolue empeaue [K] ime [s] dimesioless ime U velociy veco [m s - ] u z, u, u θ velociy compoes [m s - ] u s efeece velociy [m s - ] u dimesioless velociy as defied by Eq u m u dimesioless mea velociy dimesioless asie pa of he oal velociy u u dimesioless seady limiig velociy of he oal velociy u u z aveage velociy of he fluid acoss he aulus coss-secio [m s - ] Q V volumeic flow ae [m 3 s - ] V velociy veco i Navie-Sokes equaio [m s - ] z i valece of he ype-i io xxi

24 Geek Symbols α α m β aio of he ie adius o he oue adius of he aula capillay walls hemal diffusiviy aspec aio of he capillay adius, a o he Sokes peeaio deph, δ s β aio bewee he zea poeial of he ie ad oue adii of he aulus β δ chaaceisic zea poeial aio whe he zeo flow ae occus ime duaio whe he cosa elecic field is swiched o [s] µ δ dimesioless ime duaio, defied as δ δ ρa δ s Sokes peeaio deph [m] ε pemiiviy of vacuum, [C V - m - ] ε Φ φ dielecic cosa of he elecolye elecic field poeial poosiy of he poous media e κ Debye-Hückel paamee, defied as κ [m - ] ε ε k T f elecic coduciviy of he elecolye soluio posiive oos of he zeo-ode Bessel fucio J, o J N α J α N ψ elecical poeial of he EDL field [V] xxii b µ dyamic viscosiy [N s m -3 ] ξ dummy spaial vaiable i he Gee s fucio ρ mass desiy [kg m -3 ] ρ e e chage desiy [C m -3 ] τ dummy ime vaiable i he Gee s fucio; ouosiy of he poous media Ω viscous dissipaio fucio of he fluid i he eegy equaio ω agle fequecy of he siusoidally aleaig elecic field [s - ] eζ Ψ dimesioless elecical poeial of he EDL field, Ψ kt b

25 ζ zea poeial of he specified sufaces [V] Subscips D diff e eff H i L m max o p poe ef ev s V z θ wih efeece o Dacy paamee wih efeece o viscous diffusio wih efeece o elecical chage wih efeece o effecive paamee wih efeece o high EDL poeial egime wih efeece o ype-i io species, i his wok i +, -; value a ie wall of he aulus wih efeece o low EDL poeial value wih efeece o he mea value maximum value value a oue wall of he aulus o he wall of a cylide wih efeece o he packig paicles wih efeece o he poe i he poous media value a adius coodiae diecio efeece value wih efeece o he epeseaive elemeay volume i he poous media wih efeece o Sokes laye; efeece slip velociy; wih efeece o capillay ie suface value of asie aspec wih efeece o volumeic quaiy value a axial coodiae diecio value a agle coodiae diecio value a seady sae Supescips + he value fo obaied i he pevious ieaio xxiii

26 wih efeece o chaaceisic value o eigevalue dimesioless vaiable i figues Abbeviaios AC CEC DC DI EDL EK EOF HPLC LTE ODS REV SEM Aleaig Cue Capillay Eleco-Chomaogaphy Diec Cue Deioised Elecic Double Laye Elecokieic Elecoosmoic Flow High-Pefomace Liquid Chomaogaphy Local Themal Equilibium Ocyldecyl Silica Repeseaive Elemeay Volume Scaig Eleco Micogaph xxiv

27 Chape Ioducio. Backgoud ad Moivaio Miiauizaio ad auomaio have evoluioized he wold of micoelecoics. I ece decades hese cuig-edge egieeig echologies have bee applied o he eeds of he biomedical idusy, givig ise o a bad ew iediscipliay aea Micofluidics. Micofluidic devices hold gea pomise fo biomedical applicaios. They cosume sample maeial ad eages i exemely low volumes. Idividual micofluidic device ca be iexpesive ad disposable. The pocess ime fom samplig o esul eds o be vey sho. Ad he mos advaced chip desigs ca pefom all aalyical fucios, icludig sample maipulaio Mieick e al., 3, sample peeame Peled, 996, sepaaio, diluio, mixig Be ad Chag, ; Takhisov e al., 3, ejecio Thamida ad Chag,, chemical eacios Nohup e al., 995, ad deecio Meiha e al., 998, i a sigle iegaed micofluidic cicui. As a keel compoe i he micofluidic devices, micopump povides he kieic souce o oue he liquid hough micochael ewoks. Micopumps ae caegoized io wo majo ypes Nguye e al., : mechaical ad o-mechaical pumps. The fome ofe ivolves movig pas such as check valves, micoubies, o oscillaig membaes. Whe devices ae miiauized o micoscale, he suface-ovolume aio becomes vey lage ad he suface foces, such as suface esio may domiae i he micosysem. Thus he mechaical micopump ofe cao povide eough powe o ovecome he high fluidic impedace due o he size scalig dow. I

28 addiio, sice he mechaical pumps oly geeae a cosa fluid volume i each pump cycle, i is difficul o accuaely cool a vey fie amou of fluids. I coas, he omechaical micopump ceaes momeum i he fluid by coveig ohe foms of eegy io kieic eegy. Fo isace, he elecokieic EK micopump is solely depede o he ieacio bewee he fluid he applied elecic field. Ad i has may advaages ove ohe ypes of micopumps. Fis, EK micopumps have o solid movig pas ad have much simple desigs. Secodly, EK micopumps ca aspo wokig fluids of a wide age of coduciviy, eve o-elecolye liquid samples Takhisov e al.,, which is esseial fo biological ad medical applicaios. Thidly, pecise amou of liquid ca be maeuveed by easily coollig exeal elecic field. The physical mechaism udelyig he elecokieic pumpig capabiliy is elecoosmosis, which ogehe wih elecophoesis cosiue he basic iefacial elecokieic pheomea. Elecoosmosis was fis ivesigaed by eseaches i geophysical scieces, who foud wae migaio hough poous clay diaphagms ude a applied elecic field Reuss, 89. I is oly i ece decades ha elecoosmosis has foud a vaiey of pacical applicaios i geophysical o eviomeal idusies, such as dewaeig of soils fo cosucio pupose, ad emovig coamias o wase sludges fom soils Hue, 98. Wih he apid developme of micofluidics i he pas few yeas, elecoosmosis has daw wide aeio due o is pessue-buildig abiliy. Exesive heoeical ad expeimeal woks o elecoosmosis have bee epoed o fuhe he physical udesadig ad ealize he mos advaced applicaio of his basic pheomeo. Fo isace, i has bee employed as a basic maipulaio o aspo ad cool liquid samples of aovolumes i micodevices used fo chemical ad biological aalysis ad medical diagosis Bousse e al.,.

29 Afe a boad eview of he cuely available lieaue o elecoosmosis, i ca be cocluded ha mos of he sudies doe so fa i his aea maily focus o he elecoosmosis i simple geomeies, such as paallel slis, cylides, o ecagula chaels. Sice elecoosmosis is a iefacial pheomeo, o maximize iefacial effecs, some complex ad o-coiuous geomeies ae applied i ode o icease he effecive ieface aea, such as he saioay poous sucue used i eleco-sepaaio. Howeve, he epoed sudies o elecoosmosis i poous media ae limied. Ad majoiy of hem ae eihe expeimeal ivesigaio wih simple exesios of he basic heoy, o complicaed saisical model which igoes he coac of he saioay phase. Fuhemoe, he isigh o he dyamic aue of he elecoosmosis ca povide guidace o he desig fo pecisio cool of he elecokieic micopump o ohe micomixig isumes. Theoeically, he woks epoed i he lieaue ae based o umeical mehod, o simplified heoeical aalysis, ad maily focus o he paameic sudies wihou fuhe coceig of he keel causes of he dyamic elecoosmosis. To he bes of he auho s kowledge, hee is sill o sudy epoed o he dyamic elecoosmosis i poous media. I addiio, hee is sill much scope o exploe he elecoosmosis i poous sucues. Thus his sudy seves as a aemp o fulfill he pese void by cayig ou a fudameal, sysemaic ad i-deph exploaio o he elecoosmosis i poous micosucues ad is dyamic aue fom boh hydodyamic ad hemal aspecs.. Objecives Elecoosmosis is a iefacial pheomeo i aue. The momeum acquied by pola fluid oigiaes fom he ieacio bewee he elecic field ad a exemely hi laye of fluid close o he liquid-solid ieface. Theefoe he elecoosmosis i 3

30 poous media ad i upacked micocapillay shae he same fudameal mechaism. I ode o fuhe udesad moe complicaed physics of elecoosmosis i poous media, i is a peequisie ha oe should ivesigae he dyamic aue of elecoosmosis i simple sucues, such as cylidical geomey ha is widely applied i elecosepaaio echology. Use of poous sucues gealy elages he ieface aea. Cosequely he acive egio whee he elecokieic divig foce is pese ges maximized ad he pessue buildig capaciy is sigificaly ehaced. Bu he fluid flow iside poous sucues ae scaeed ad iewied. The fluid domai is o coiuous due o he pesece of saioay phase. Theefoe he aalysis becomes moe difficul. I is expeced ha he heoeical fidigs obaied by simple geomey aalysis should be impoved by accouig fo he geomeical ad elecokieic complexiy i poous media. The ohe impoa objecive of his sudy is o povide he exesive expeimeal daa o he elecoosmosis i poous media as validaio of cue models ad guidace of elecokieic micopump developme. I summay, he objecives of his hesis ae highlighed as followig: Sudy elecoosmoic flow hough cylidical micocapillay, ad a moe geeal mico-sized aula geomey, ad ivesigae he dyamic esposes ude he ime-depede elecic field ad ohe impoa wokig paamees; Chaaceize saic ad dyamic aspecs of he elecoosmoic flow i a micocapillay packed wih micosphees, akig io cosideaio of he bouday wall effec ad he Joule heaig effec; 3 Coduc a expeime o desig, fabicae, ad chaaceize a elecokieic micopump usig packed micocapillaies. Exesive expeimeal daa ude a wide vaiey of hydodyamic ad physicochemical codiios will be accumulaed. 4

31 Alhough majoiy of his sudy is heoeical i aue, he ifomaio ucoveed by he modelig ad expeimeal esuls ca be of gea impoace o he developme ad opimizaio of he elecokieic micopump o micoacuao..3 Oulie of he Thesis The oveall hesis compises five majo pas summaized as followig: Chape seves as a ioducio o he backgoud ad moivaio of his wok. The applicaios ad laes developme of micofluidics ae peseed ad he objecives of his hesis ae oulied. Followig is a bief of he elecic double laye EDL heoy. I he ed he lieaue of exisig sudies o he elecoosmoic flow i micochaels is hooughly eviewed ad emaked. The secod pa icludes chapes ad 3, i which he complee models fo dyamic aspecs of he elecoosmoic flow i a cylidical ad aula capillay wih high zea poeials ae peseed. Aalyical soluios of he Poisso-Bolzma equaio fo he EDL poeial disibuio ad he Navie-Sokes equaio fo he elecoosmoic flow field i micocapillaies ae obaied, ad he esuls ude iflueces of he EDL, exeal elecic field, ad geomeic paamees ae peseed. Specifically, as a majo coibuio of his pa, he dyamic elecoosmoic flow ude AC elecic field is aalyically solved usig Gee s fucio fomulaio. The hid pa compises chapes 4, 5, ad 6, i which a sysemaic sudy of he elecoosmoic flow i a micocapillay packed wih micosphees is coduced based o he capillay model. The elecokieic wall effec is icluded by aalyically solvig he modified Bikma s equaio i chape 4. The dyamics of he AC elecoosmoic flow i a packed capillay is sudied i chape 5. Specifically, he backpessue occuig i a packed capillay wih closed-eds is discussed. The Joule heaig effec associaed wih 5

32 use of high volages, coceaed elecolyes, o lage capillay size is umeically solved i chape 6. The asie developme of he empeaue field iside he packed capillay ad is effec o elecoosmoic velociy is aalyzed i deails. The fouh pa is he expeimeal ivesigaio o he elecoosmoic flow i a micocapillay packed wih micosphees. I chape 7, elecokieic micopumps ae fabicaed usig high pessue dive sluy packig echique. The elecoosmoic velociy iside he packed capillay is measued ude diffee physicochemical codiios. The iflueces of he capillay ad paicle size, he coceaio ad he ype of he elecolye soluios ae peseed. Two diffee mehods o measue he flow velociy ae employed fo diffee volage codiios. The expeimeal esuls ae compaed wih he pedicios by he heoeical model developed i chape 4. Fially, Chape 8 povides majo esuls ad fidigs fom his hesis. The coibuios made by his hesis ae highlighed ad some possible diecios fo fuhe sudies ae biefly oulied..4 The Elecic Double Laye ad he Elecoosmoic Flow Geeally, mos sufaces will acquie a ceai amou of elecic chages whe hey ae bough io coac wih a aqueous pola medium. Some of he chagig mechaisms ae ioizaio, io adsopio, ad io dissoluio. The suface chage, i u, will ifluece he disibuio of eaby ios i he soluio. Ios of opposie chage coue-ios o ha of suface ae aaced owads he suface while ios of like chage co-ios ae epelled fom he suface. This elecosaic ieacio ogehe wih he mixig edecy esuled fom he adom hemal moio of he ios, leads o he fomaio of a elecic double laye EDL: a compac laye ad a diffuse laye. The elecic double laye is a egio close o he chaged suface i which hee is a excessive 6

33 of coue-ios ove co-ios o eualize he suface chage, ad hese ios ae disibued i a diffuse mae. Evidely hee is o chage eualiy wihi he double laye because he umbe of coue-ios is moe ha he umbe of co-ios. Figue - a Schemaic epeseaio of he ioic disibuio close o a posiively chaged suface i he Gouy-Chapma model Hiemez, 986. b The egio vey close o he solid, showig he Se plae whee he poeial is ψ B ad he shea plae whee he poeial is ζ Pobsei, 994. The heoy fo he diffuse double laye was fis developed idepedely by Gouy ad Chapma Hiemez, 986. Accodig o his model, oe laye of chage is assumed o be smeaed ou uifomly ove a plae suface immesed i a elecolye soluio as illusaed i Figue -a. This suface has a elecosaic poeial, ψ. Based o he assumpio ha he compesaig ios ae egaded as poi chages immesed i a coiuous dielecic medium, Gouy-Chapma model idicaes ha he coceaio of he ios i he sheah follows he Bolzma disibuio. Howeve, he ios ae of fiie size ad hus limis he ie bouday of he diffuse pa of he double laye, sice he cee of a io ca oly appoach he suface o wihi is hydaed adius wihou becomig specifically adsobed. To ake his effec io 7

34 accou, he Gouy-Chapma model was lae modified by Se Pobsei, 994 who ioduced a ie pa of he double laye immediaely ex o he chaged suface. The ie laye, he oue bouday of which is appoximaely a hydaed io adius fom he suface usually of seveal Agsoms, coais a laye of coue-ios ha ae sogly aaced o he suface ad ae immobile. This compac laye is also called he Se laye. The plae sepaaig he ie compac laye ad he oue diffuse laye is called he Se plae as show i Figue -b. Ios whose cees ae locaed beyod he Se plae fom he diffuse mobile pa of he double laye. The hickess of he diffuse double geeally ages fom seveal o a few hudeds of aomees, depedig upo he bulk ioic coceaio ad ohe physicochemical popeies of liquid. As show i Figue -, he elecical poeial chages fom he suface poeial, ψ, o he Se plae poeial, ψ B, wihi he Se laye, ad decays o zeo fa away fom he Se plae. The poeial a he Se plae, ψ B is close o he elecokieic poeial o zea poeial, ζ, which is defied as he poeial a he shea suface bewee he chage suface ad elecolye soluio ad is measuable fom expeimes. Figue - Elecical poeial a fo he Gouy-Chapma diffuse egio b fo he Se model showig expoeial decay fom he Se plae. Pobsei, 994 8

35 Elecoosmosis aises whe he mobile poio of he diffuse double laye ieacs wih a exeal applied elecic field i he viscous shea laye ea he chaged suface. Wihi he diffuse laye, due o he pesece of EDL, he coue-ios pedomiae ad he local e chage desiy is o zeo. If a elecic field is applied ageially alog a chaged suface, he he elecic field will exe a coulombic foce o he ios i he diffuse laye. The migaio of he mobile ios will cay he adjace liquid wih hem leadig o a elecoosmoic flow EOF. Due o viscous foces, he flow is caied hough beyod he EDL egio o he es of he liquid i he micochael. This elecokieic pheomeo was fis ivesigaed by Reuss 89, who demosaed ha ude he ifluece of a applied elecic field wae migaed hough poous clay diaphagms owads he cahode. This fac ca be well udesood oday i ha he clay, sad ad ohe mieal paicles usually cay egaive suface chages whe i coac wih wae; he wae omally coais small quaiies of dissociaed sals. As descibed above, elecoosmoic flow is iduced hough he poous medium i he clay, upo applicaio of elecic field..5 Lieaue Review I he lieaue, umeous heoeical ad expeimeal sudies have bee epoed o he elecoosmoic flow i fie capillaies. A deailed eview of he lieaue will be peseed i he followig, focusig o five majo aspecs: seady-sae EOF, dyamic EOF, EOF i poous media, Joule heaig o EOF, ad he desig ad fabicaio of elecokieic micopump..5. Seady-sae elecoosmoic flow The seady sae elecoosmosis has bee well sudied decades befoe. The classical model fo elecoosmosis is aibued o Smoluchowski Hiemez, 986 o he 9

36 cosequeces of applyig pessue ad poeial gadies acoss capillaies filled wih a elecolye. His fomulaio of he poblem is based o he assumpio of lage elecokieic diamee; he coibuio due o he EDL hickess is egleced. Despie ha his classical esuls fo EOF plug velociy ad seamig poeial ae sill fequely employed i ouie expeimeal woks o EOF. Bugee ad Nakache 964 fomulaed a mahemaical model fo he elecoosmoic flow i a ulafie sli. Fom he aalyical soluio obaied hey discussed he possible effec of he elecokieic adius o he elecoosmoic flow hough he micochael. Rice ad Whiehead 965 calculaed aalyically he coecio facos ha mus be applied o he Smoluchowski s esuls whe dealig wih aow capillaies havig abiay elecokieic diamees. Howeve he Rice ad Whiehead heoy iself is subjec o he sevee esicio ha he zea poeial be sufficiely low o pemi he Debye-Hückel appoximaio, effecively limiig he applicaio of hei pedicios i ζ 5 mv fo a - elecolye. Levie e al. 975 exeded he Rice ad Whiehead model o high zeapoeials fo he elecokieic flow i cylidical capillaies. They developed a aalyical appoximaio o solve he Poisso-Bolzma P-B equaio wihi he capillay, i a fashio simila o he mehod used by Philip ad Woodig 97 who solved he P-B equaio ouside a chaged cylidical paicle immesed i a elecolye. Ad i was show by Philip ad Woodig ha he yielded esul diffes oly slighly fom hose by umeical mehod. Keh ad Liu 995 aalyically sudied he seady EOF i a log uifom cicula capillay beaig a solve-pemeable ad io-peeable laye of adsobed polyelecolyes o is iside wall. They obaied he elecical poeial ad space chage

37 desiy disibuio by solvig he lieaized P-B equaio. Thei esuls o he EOF demosaed ha he sucue of he suface chage laye ca esul i a augmeed o dimiished EOF elaive o ha i a capillay wih bea walls, depedig o he chaaceisics of he elecolye soluio, suface chage laye, ad capillay. Koh ad Adeso 975 expeimeally ivesigaed he elecoosmoic flow i chaged micocapillaies of ellipse shape. They compaed he daa wih umeical calculaios ad showed ha adsopio of he poeial deemiig io is depede o elecosaic poeial a he poe wall. Mala e al. 997a ivesigaed he iefacial elecokieic effecs o chaaceisics of liquid flow ad hea asfe bewee wo paallel plaes. Expeimes wee coduced o sudy he effec of EDL o flow chaaceisics wih diffee ioic coceaios ad plae maeials. A mahemaical model was developed fo a seadysae liquid flow wih cosideaio of he EDL effecs. The pediced flow ae agees well wih he measued daa. Recely, due o he elevace o micofluidics fabicaed by micomachiig echology, elecokieic flows i ecagula micochaels have bee sudied by Yag e al. 997, 998, Aulaadam ad Li, ad Yag ad Huag. I hese sudies a wo dimesioal elecokieic model has bee poposed o accou fo he coe effec. Based o he Debye-Hückel appoximaio, a aalyical soluio of he lieaized wo-dimesioal P-B equaio is peseed o descibe he EDL poeial disibuio i he coss-secio of a ecagula chael. By usig he Gee s fucio mehod, hey also obaied a exac soluio fo he EOF velociy. Paaka ad Hu 998 developed a umeical scheme o simulae seady-sae elecoosmoic flows i complicaed geomey. Thei esuls agee wih he expeime o elecoosmoic ijecio a he iesecio of a coss-chael doe by Fa ad Haiso

38 994. The esuls show ha applicaio of elecic field ca be used o cool he shape of ijeced fluid. Michell e al. simulaed a seady-sae eleco-osmoic flow o hee diffee geomeies by usig meshless aalysis based o Fiie Cloud Mehod FCM. Thei esuls show ha liea appoximaio of Poisso-Bolzma equaio fo lage zea poeial ca pedic he plug velociy accuaely bu fails o pedic he velociy vaiaio close o he walls. Biachi e al. sudied elecoosmoic flow a a T-jucio by usig fiie eleme simulaio. Thei esuls idicae ha elaive zea poeial ad chael widhs ae wo pedomia paamees affecig he disibuio of flow a he iesecio. Cummigs e al. examied he codiios fo similiude bewee he fluid velociy ad elecic field i elecoosmoic flow. They showed ha he codiios ecessay ae a seady elecic field, uifom fluid ad elecical popeies, a elaive hi Debye laye compaed o he physical dimesios, ad fluid velociies o all ile ad oule boudaies ha saisfy he Helmholz-Smoluchowski elaio. Tsao sudied he elecoosmosis flow hough a aulus ude he Debye- Hückel liea appoximaio, idicaig ha his wok is valid oly fo he case of low zea poeials e.g., ζ 5mV fo a - elecolye. He ioduced a geomeydepede coecio faco elecoosmoic mobiliy descibed by he Helmholz Smoluchowski equaio. He also foud hee exis e flows eve fo zeo aea-aveaged suface chage desiy due o he cuvaue diffeeces bewee he ie ad oue walls. Ude ceai cicumsaces he flow diecio i a aulus is opposie o ha i a capillay wih he same sig of he e chage. He e al., Re ad Li, ad Gleeso, ivesigaed he chaaceisics of he EOF i a cylidical micochael wih o-uifom zea poeial.

39 Thei heoeical ad umeical esuls show he disoed elecoosmoic velociy pofiles esuled fom he axial vaiaio of he zea poeial. Also, he iflueces of he uequal secio size ad he diecio of he zea poeial chage o he velociy pofile, he iduced pessue disibuio, ad he volumeic flow ae ae discussed. The simulaio esuls evealed possible effecs of bio-adhesio i micochaels o he elecoosmoic flow i biochip devices. Eickso ad Li sudied he effecs of suface elecokieic heeogeeiy o he elecoosmoic flow ad mixig efficiecy of a T-shaped micomixe hough 3D umeical simulaios. While all cases of suface heeogeeiy wee show o ehace species mixig, hey foud ha he geaes impovemes ca be achieved whe he zea poeial of he heeogeeous suface is of opposie sig o ha of he homogeeous suface, esulig i localized ciculaio zoes wihi he bulk flow field. They also showed ha he mixig efficiecy impoved i geeal mehod, such as deceasig applied volage ad he chael size, ca be ehaced by he ioducio of suface heeogeeiy, i some cases esulig i a 7% educio i he equied mixig legh..5. Dyamic elecoosmoic flow Compaed wih he saic aue, he dyamic aspecs of he EOF have eceived elaively less ieio. Howeve, he sudy o he useady EOF o oly ca povide moe isigh io he chaaceisics of he EOF bu also is impoa o he developme ad pecise cool of he EOF based micofluidic device. Lopez-Gacia e al. who have doe a aalysis of he dyamics of EDL i boh ime ad fequecy domai povide some isighs io he asie behavio of poeial, velociy, ad io coceaio pofiles. They aalyzed how poeial ad io coceaio pofiles, paicle o fluid velociy evolve i he aosecod o micosecod 3

40 ime age afe he applicaio of a elecic field. A ewok mehod is poposed o gai ifomaio abou he evoluio wih ime of he poeial, he coue-io, ad co-io peubaios, he paicle velociy, ad he fluid velociy pofile. Yag e al. developed a exac soluio fo he asie elecoosmoic flow i a sli micochael. Exac soluios fo he elecical poeial pofile ad he asie elecoosmoic flow field ae obaied by solvig he complee Poisso Bolzma equaio ad he Navie-Sokes equaio ude a aalyical appoximaio fo he hypebolic sie fucio. Saiago sudied he effecs of fluid ieial ad pessue o he asie velociy ad voiciy fields of elecoosmoic flow i a wo-dimesioal micochael. His wok is based o he classical Debye-Hückel liea appoximaio i solvig he P-B equaio o obai he elecic double laye poeial disibuio ad a validaed slip velociy codiio. I ypical o-chip elecokieics applicaios, he flow field ca be sepaaed io a ie flow egio domiaed by viscous ad elecosaic foces ad a oue flow egio domiaed by ieial ad pessue foces. These wo egios ae sepaaed by a slip velociy codiio deemied by he Helmholz-Smoluchowski equaio. The validiy of his assumpio is ivesigaed by aalyzig he velociy field i a pessue-dive, wo-dimesioal flow chael wih a impulsively saed elecic field. The egime fo which he ie/oue flow model is valid is descibed i ems of o-dimesioal paamees deived fom he example poblem. Wihou assumpio of hi double laye hickess, Keh ad Tseg aalyically sudied he asie espose of elecolye soluios i a aow capillay ube ad sli o a sep chage i he applied elecic field ad pessue gadie by solvig he lieaized P-B equaio. Thei esuls demosaed ha he behavio of he asie elecokieic flow i a capillay ube ad i a capillay sli is simila; howeve, he ae of 4

41 evoluio of he flow i a ube wih ime is fase by a faco of abou ha ha i a sli wih is half hickess equal o he ube adius. By umeically solvig he combiaio of he Poisso, he Nes-Plack, ad he Navie-Sokes equaios, Yag e al. ivesigaed he ime ad space developme fo he ey egio of a elecoosmoic flow hough micochaels bewee wo paallel plaes. They discussed he effecs of he eace egio o he fluid velociy disibuio, chage desiy bouday laye, eace legh, ad shea sess. They foud he eace legh of he elecoosmoic flow is loge ha ha of classical pessue-dive flow. The hickess of he elecical double laye EDL i he ey egio is hie ha ha i he fully developed egio. The chage of velociy pofile is appae i he eace egio, ad he axial velociy pofile is o loge fla acoss he chael heigh whe he Reyolds umbe is lage. Saic elecic fields ae usually used i above eseach wok o he useady elecoosmoic flow. Howeve, i ece yeas, he use of ime depede elecic fields has poved aohe pacical ad useful echique i maeuveig he elecoosmoic flow. Fo isace, Södema ad Jösso 996 developed a heoeical famewok fo he descipio of he ime ad spaial esoluio of elecoosmosis fo boh plaa ad cylidical geomeies ude he effec of pulsed elecic fields. Specifically, siusoidally aleaig AC elecic fields ae moe ofe applied o ivesigae he fequecy depede aue of he fluid velociy o he moio of he ao-paicles, whose mechaism i he lieaue is emed as AC elecoosmosis Ramos e al., 999. Due o poeial applicaios i micomixig ad micoacuaos, AC elecoosmosis has daw much moe eseach ieio. Baagá ad Bauzá pefomed elecoosmosis expeimes hough a caio-exchage membae ad showed ha he pesece of AC peubaio affecs he 5

42 elecoosmoic flow, depedig o he fequecy of AC sigal ad o he soluio siig codiios. I he fequecy age sudied, wo egios have bee obseved whee he elecoosmoic flow eaches a maximum value: oe a low fequecies ~Hz; he ohe a fequecies of he ode of khz. These egios could be elaed o membae elaxaio pheomea. I a ece wok o elecokieic isabiliy micomixig, Oddy e al. developed a elecokieic pocess o apidly si mico-ad ao lie volume soluios fo micofluidic bioaalyical applicaios. They apidly sied mico flow seams by iiiaig a flow isabiliy, which hey obseved i siusoidally oscillaig, elecoosmoic flows. As he effec occus wihi a oscillaig elecoosmoic flow, hey efe o i as a elecokieic isabiliy EKI. They obaied he elecoosmoic velociy pofile i a micomixe by usig slip velociy appoximaio, i which he velociy o he ouskis of he flow field is bouded by he slip velociy. The slip velociy model was fis poposed by Ovebeek 95 who showed ha fo mico-chaels wih elaively hi elecical double layes, he flow field ouside he double laye is a ioaioal flow wih a slip velociy bouday codiio deemied by he well-kow Helmholz- Smoluchowski equaio..5.3 Elecoosmoic flow hough poous media As i above discussio, he elecoosmosis is a iefacial pheomeo. Use of poous sucues ca gealy elage he iefacial aea, ad hus sigificaly ehace he pessue-buildig capaciy of he EK micopump. Theefoe he mechaism of he EOF i poous media is deseved fo fuhe eseach ipu. Rahoe ad Hováh 997 peseed wo models i hei eview pape o he heoies of he EOF i poous media. The fis is based o Smoluchowski s wok as 6

43 adaped ad exeded by Ovebeek 95. I deals wih he EOF hough packed capillaies ude a codiio of low elecic field segh whee he EOF vaies liealy wih he field segh because hee is o polaizaio of he double laye. The secod model poposed by Dukhi 99 pedics he EOF of a leas a ode of magiude highe ha ha expeced by classical heoies. I was cocluded ha hee had o ye bee esablished a solid heoeical udesadig of he EOF i capillay chomaogaphy CEC. They sogly idicaed ha he cosucio ad soluio of he mahemaical model ha could popely descibe he elecokieic pheomea udelyig he EOF i packed beds could have he poeial o develop he CEC io a poweful sepaaio pocess. Liapis ad Gimes cosuced a mahemaical model o descibe quaiaively he pofiles of he elecosaic poeial, pessue, ad velociy of he EOF i chaged cylidical capillaies ad i he CEC sysems. They compaed hei heoeical esuls of he EOF velociy wih he expeimeal values of he EOF velociy obaied fom a fused-silica colum packed wih chaged poous silica paicles; he ageeme bewee he heoeical esuls ad he expeimeal daa is good. Also, he esuls fom model simulaios idicaed he codiios ha pemi high values fo he aveage velociy of he EOF o be obaied, fo a give opeaioally pemissible value of he applied elecic field. As a coiue wok, Gimes e al. employed he poe ewok heoy ad he model cosuced by Liapis ad Gimes ogehe o esimae he magiudes of he iapaicle EOF flow ae, velociy i he poes of he chaged poous silica paicles. Tallaek e al. sudied he macoscopic heeogeeiies i elecoosmoic ad pessue-dive flow hough fixed beds a low colum-o-paicle diamee aio ad 7

44 demosaed ha a sigificaly supeio pefomace, which has bee show fo he EOF hough packed capillaies compaed o pessue-dive flows, ca be obsuced by wall effecs, ad i is difficul ace back emaiig diffeeces i he asympoic dispesio obseved fo hese ypes of fluid flow o a ihee pefomace coceig iapaicle ad film mass asfe o a macoscopic flow heeogeeiy. Blokha ad Joshi 999 ivesigaed he effecs of he mageic field o he elecokieic aspo coefficies of elecolye a diffee poeials, coceaios, ad mageic fields. The pheomeological coefficies chaaceizig he EOF ad he membae chaaceisics ae also esimaed fo he vaious soluios wih he objec of deemiig he efficiecies of elecokieic eegy covesio ad zea poeials. Lee e al. heoeically modeled he EOF of a geeal elecolye soluio hough a fibous medium, akig effec of he EDL polaizaio io accou. I was show ha if he effec of EDL polaizaio is egleced usig he lieaized P-B equaio will udeesimae he elecoosmoic velociy. The deviaio becomes iappeciable if he elecokieic diamee is eihe vey lage o vey small. Coelho e al. 996 developed a geeal heoy of elecoosmoic pheomea i poous media possessig spaially peiodic sucue. Geeal expessios fo he elecic coduciviy, pemeabiliy, ad couplig elecoosmoic eso coefficies ae obaied i ems of soluios of seveal aspo ui cell poblems, posed fo he lieaized elecokieic equaios. Maio e al. exeded he esuls by Coelho e al. 996 o he impoa case of deemiisic siusoidal ad adom self-affied facues i he liea limi fo vaious double laye hickess. Thei umeical esuls showed a defiie ifluece of he suface ampliude o elecoosmoic pocesses. Ad i a moe ece pape, Maio e al. 8

45 fuhe exeded hei wok o iclude he effecs of macoscopic coceaio gadies, i addiio o he ifluece of elecical ad pessue gadies..5.4 Themal effecs i elecoosmoic flow sysems Sice he elecoosmosis is elecic field dive flow, he Joule esisive hea geeaio is ieviable due o he elecic cue icued. This effec becomes sigifica fo lage capillay size due o deceased suface-aea-o-volume aio o high elecolye coceaio due o high elecic coduciviy. I applicaio of elecokieic sepaaio, he empeaue ise due o Joule heaig will i u affec he elecoosmoic flow ad he sepaaio efficiecy. The sigifica empeaue ise may eve damage he hemal labile sample ad cause vapo bubbles. Due o is impoace, he Joule heaig poblem has daw wide aeio of he eseaches. Based o Poisso-Bolzma model fo elecic double laye, Tag e al. 4a, 4b peseed umeical models o evaluae he Joule heaig effec o he elecoosmoic flow ad mass asfe i o-packed micocapillaies fo seady sae ad asie siuaios. Thei umeical simulaios showed he sigifica adial ad axial empeaue ise. Ad hey also compaed he simulaio esuls based o P-B equaio ad Nes-Plack equaio especively Tag e al., 3. They poied ou he wo models gives he same esuls fo he soluio ioic coceaio i he fully-developed flow egio o i case of lage elecokieic diamee. I hei calculaio, hey assumed uifom elecic field segh, which i effec igoes he coiuiy of he elecic cue. Cosideig he empeaue-depede elecic coduciviy, howeve, he elecic field segh should be sogly coupled wih he empeaue field. Xua e al. 4a umeically ivesigaed he hemal ed effecs o he elecoosmoic flow i a o-packed micochael. Thei model icludes he couplig 9

46 elecic field segh which is subjec o local chage due o he o-uifom elecic coduciviy. Ad as a coiual wok, hey use fluoescece-based echiques o measue he liquid empeaue vaiaio ad he elecoosmoic velociy pofile alog a micochael Xua e al., 4b. Thei expeimeal esuls wee foud o agee well wih he pedicios of he umeical model. Kox 988 poied ou ha Joule heaig causes he mai limiaio of pefomace of fou capillay eleco-sepaaio mehods. Oe of hem is capillay elecochomaogaphy i which he capillaies packed wih micopaicles ae used as sepaaio media. The sepaaio efficiecy is gealy affeced because of he peak dispesio aisig fom Joule heaig. He foud he hea is geeaed homogeeously ove he ceal egio ad he empeaue vaiaio acoss he boe of he cylidical capillay is paabolic. The hemal gadie is lages i he ai zoe ad leas i he ube wall. I addiio, he deived simple aalyical expessio of he coss-seam empeaue excess wihi he coe egio ad he capillay wall a seady sae. Bu his model is oo simple because he did o coside he empeaue depedece of he liquid popeies as well as he o-uifom elecic field. Keim ad Ladisch 3 developed a wo-dimesioal asie empeaue model fo elecochomaogaphy. Thei model accous he empeaue depedece of he physical popeies of he saioay ad mobile phase. Thei model also icluded he empeaue depedece of he elecic coduciviy ad hus he o-uifom elecic field segh. The modelig esuls wee compaed wih expeime daa. The pedicio was foud o be wihi 3 o C of he acual empeaue. I addiio hey epoed hee was a asie volage dop a he colum oule due o uifom elecic field. Howeve hey did o fuhe discuss he Joule heaig effecs o he elecoosmoic flow hough he packed capillay.

47 .5.5 Deig ad fabicaio of elecokieic micopump Due o he advaages of he EK micopump ad is poeial of deliveig liquid i high flow ae ad geeaig high pessue, he developme of he acual EK micopump have become a aea of iese eseach duig ece yeas. The fis effo made o desig a elecokieic micopump may aibue o Paul e al. 998 who epoed he abiliy o geeae high pessues usig elecokieic pumpig of liquid hough poous media. Pessues i excess of 8 psi have bee achieved usig silica capillaies packed wih mico-size silica beads. Che e al. developed a pelimiay model of plaa EK micopump o chaaceize he flow ae, pessue capaciy, ad hemodyamic efficiecy of he pump. The pump was fabicaed o a mm glass subsae usig sadad mico lihogaphy ad chemical we echig echiques. The pump is mm log i he flow diecio ad.9 µm 38mm i coss-secio. DI wae was used as wokig fluid. Thei expeimeal esuls showed a liea elaioship bewee flow ae ad coue pessue. Howeve, hei plaa EK pump ca oly geeae flow ae up o 5 µl/mi ad pessue up o.3 am a a applied volage of kv. Zeg e al. fabicaed a EK pump by packig he 3.5 µm diamee opoous silica paicles io 5-7 µm diamee fused-silica capillaies ad usig a silicae fi fabicaio scheme o hold he paicles i place. The pump ca geeae maximum pessues i excess of am o maximum flow aes of 3.6 µl/mi fo a kv applied poeial. Bu hey also addessed he limiaios associaed wih elecolyic gases ad is egaive effec o he log-em pefomace of he device. I desigig a lage flow ae EK micopump, oe of he key issues is he fabicaio of he obus fis. Zeg e al. epoed a ovel fabicaio echique by

48 employig phoopolymeizaio mehod o make fis ad by applyig a side-boe packig mehod o poduce a bed of -3 µm paicles. The phoopolyme offes sigifica desig flexibiliy because i ca be phoopolymeized i vaied sucues. The maximum flow aes ad pessue geeaed by hei pumps ae.8 ml/mi. ad am, a. kv applied volage. Che e al. 3 developed he EK micopump usig sigle ad muliple micocapillaies packed wih silica micosphees. Thei pump is esed wih mehaol, phosphae sodium buffe ad hei mixue, which ca geeae pessue fom. o 5 MPa ad flow aes of es of aolies o seveal micolies pe miue. Specifically, he ieupio of he pumpig by bubble fomaio due o Joule heaig, vapoizaio o decompoudig of he fluid i side he capillaies was obseved, especially fo high volage o lage capillay size. Yao e al. 3a, b demosaed a EK pump pooype by modifyig a commecially available sieed glass fi. The fis hey used wee poous cylides of 4 mm i diamee ad o 5 mm hick ad povide he high weed-suface-o-volume aio. Oe of hei pumps achieved a maximum flow ae of 33 ml/mi ad a maximum pessue capaciy of.3 am a a V applied volage. They also developed a heoeical model o simulae he elecoosmoic flow i he pump ad compaed he esuls wih he expeimeal daa. I is foud he model ca pedic he pefomace ed ad useful fo desig of EK pump. Howeve, lage discepacy fom % o 49% bewee modelig esuls ad expeimeal daa had bee foud. A leas pa of he discepacy, as hey said, may be due o he adom geomey of he acual poous sucue, which is diffee fom he idealized cylidical poe geomey i hei modelig.

49 Chape Dyamics of Elecoosmoic Flow i a Capillay Cylide. Ioducio As discussed i he fis chape, sudy of useady elecoosmoic flow o oly ca povide moe isighs io he chaaceisics of elecoosmoic flow bu also is impoa o he pacical developme ad opeaio of micofluidic devices. The cue sudy peses a heoeical aalysis of he dyamics of elecoosmoic flow i a cylidical micocapillay. We sa wih he ime evoluio of he asie elecoosmoic flow moivaed by a ime depede elecic field, ad discuss he flow field oscillaios excied by a siusoidally aleaig elecic field. Impoaly, he complee Poisso-Bolzma equaio goveig he EDL elecic field is aalyically solved wih abiay zea poeials ude a poposed aalyical scheme. Exac soluios fo asie velociy disibuios ad ime-depede mea velociies ae obaied usig he Gee s fucio mehod. As a compaiso, he soluio ude slip velociy appoximaio is also peseed. I he aalysis of elecoosmosis peseed below we have made he followig assumpios i The empeaue is assumed uifom ove he eie sample. Thus we eglec ay possible effec due o Joule heaig. ii iii iv The mageic field poduced by he movig chages is egleced. The flow of he liquid is lamia i aue. The liquid is assumed o be a icompessible, Newoia, symmeic elecolye 3

50 of cosa viscosiy ad pemiiviy ove he cosideed volume. v vi All he bouday sufaces ae assumed o be of uifom chage desiy. The chagig fequecy of he exeal elecic field is o vey high e.g., less ha MHz, so he Poisso-Bolzma equaio is sill valid fo dyamic elecoosmoic flows.. Poblem Fomulaio Fo he case of a foced, lamia flow hough a cylidical micochael of adius a, he equaio of moio of a icompessible Newoia fluid of desiy, ρ, ad viscosiy, µ, is give by he Navie-Sokes equaio Pobsei, 994. V V P + F + µ V ρ V + ρ. Fo a fully developed flow i he cylidical micochael, he compoes of velociy V saisfy u z u, ad u u i ems of cylidical coodiaes. Thus he ieial em θ V V vaishes. Whe a exeal elecic field, E is applied alog he axis of he capillay, he liquid sas o move as esul of he ieacio bewee he e chage desiy i he elecic double laye EDL ad he applied elecic field. As he flow is o pessue iduced, he em P is o pese. If he gaviy effec is egligible, he body foce F is oly he Coulomb foce caused by he acio of a applied elecic field E o he e chage desiy ρ i he EDL egio. Wih hese cosideaios, Eq.. is educed o Pobsei, 994 e 4

51 u, u, ρ µ E ρ e. whee u, is he asie velociy field. ρ is he local volumeic e chage desiy of he elecolye due o he pesece of he EDL. e Coside a fluid phase coaiig posiive ad egaive ios i coac wih a chaged capillay wall. A EDL field will be esablished. Assume he suface beas a uifom zea poeial, ζ o. We coside a simple fully dissociaed symmeical sal i soluio, whee fa away fom he wall, he coceaio of he posiive ad egaive ae equal, i.e., z + z z ad +. The elecosaic poeial ψ, a ay poi ea he suface, is elaed o he e umbe of elecical chage pe ui volume ρ, i he eighbohood of he poi, which measues he excess of he posiive ios ove egaive ios o vice vesa. Accodigly o he heoy of elecosaics, he elaio bewee ψ ad ρ e is give by Poisso s equaio Hue, 98, which fo a cylidical suface is e d d dψ ρ e d ε ε.3 whee ε is he dielecic cosa of he elecolye ad ε is he pemiiviy of vacuum. Fo ay fluid cosisig of wo kids of ios of equal ad opposie chage z + ad z, he umbe of ios of each ype is give by he Bolzma equaio Hue, 98 ± zeψ exp.4 kbt whee e is he elemeay chage, is he ioic umbe coceaio i he bulk phase i.e., fa fom he chaged sufaces, k b is he Bolzma cosa, T is he absolue 5

52 empeaue, ad ψ is he elecic poeial of he EDL. A poof of he validiy of he Bolzma disibuio i applicaio of ime-depede elecoosmoic flow is peseed i he appedix A. The e chage desiy i a ui volume of he fluid is give by zeψ ρ e + ze ze sih.5 kbt whee z is he valece of he ios. Fo a moovalece elecolye cosideed i his sudy, z. Subsiuig Eq..5 io Eq..3, we obai a o-liea Poisso-Bolzma equaio, which solely deemies he EDL poeial disibuio d dψ ze d d ε ε ze sih kbt ψ.6 Ioducig he followig dimesioless paamees u µ u u s ρ a a e Ψ ψ.7 k T b We ca odimesiolize Eq.. ad Eq..5 especively as u u a µ u s ρ E e.8 ad Hee u s is he efeece velociy defied as ρ e sih[ Ψ ].9 e u s ε ε Eζ o. µ 6

53 The efeece velociy, u s ca also be egaded as he slip velociy o he shea plae of he EDL Ovebeek, 95. Obviously, u s is popoioal o he magiude of he exeal field, E ad he zea poeial, ζ o. Subsiuig Eq..9 io Eq..8, we obai a ihomogeeous diffusio equaio u u κ a Ψ E s sih [ Ψ ] E. whee e o Ψ ζ s ad k T κ e ε ε k T. κ is he Debye-Hückel paamee, ad / κ deoes b b he chaaceisic hickess of he EDL Hue, 98. Eq.. is subjec o he iiial ad bouday codiios u.a u u.b.3 Gee s Fucio Mehod fo Ihomogeeous Diffusio Equaio I Gee s fucio echique, o obai he field caused by a disibued souce o chage hea geeao o whaeve i is ha causes he field we calculae he effecs of each elemeay poio of souce ad add hem all. If G x, ξ, τ is he field a he obseve s poi x a he mome of caused by a ui poi souce a he souce poi ξ a he mome of τ, he he field a x a he mome of caused by a souce disibuio ρ ξ, τ is he iegal of G ρ ove he whole age of space ξ ad ime τ occupied by he souce. The fucio G is called Gee s fucio. 7

54 8 Thus he soluio of ihomogeeous diffusio equaio, Q u D u x.3 is give by Mose ad Feshbach, dsd G u u G d u G d d Q G u,, ;,,, ;,,, ;,, τ τ τ α τ τ τ ξ ξ x ξ ξ ξ x ξ ξ ξ x x.4 whee, ;, τ ξ x G is he Gee s fucio of he specified diffusio equaio. The secod ad hid em o he igh had side epese he effecs of iiial ad bouday codiios, especively. The sough Gee s fucio,, τ ξ G should saisfy Bukov, 968 τ δ ξ δ G G.5 subjec o he iiial ad bouday codiios,, τ ξ G.6a,, τ ξ G.6b,, τ ξ G.6c whee ξ δ ad τ δ ae Diac dela fucios. We ca see ha he souce is a impulse a τ, locaed a ξ. G he gives he descipio of effec of his impulse as i popagaes away fom ξ i he couse of ime. Takig Hakel asfom wih espec o G H H {G} ad usig he bouday

55 codiios Eqs..6b-.6c, we obai dg d H + G ξ J ξ δ τ H.7 whee ae he posiive oos of he zeo-ode Bessel fucio J. Now, we ake he Laplace asfom wih espec o G HL L {G H } ad use he iiial codiio Eq..6a o obai s G H L + G ξ J ξ e H L sτ.8 We solve fo G HL ad ge G H L ξ J ξ e s + sτ.9 akig ivesive Laplace asfom τ G H τ ξ J ξ e. H whee H τ is he Heaviside sep fucio. The ivesio wih espec o Hakel asfom leads o J G, ξ, τ H τ ξ J exp[ ] ξ τ. J Hece we obai he Gee s fucio fo ihomogeeous diffusio equaio i cylidical coodiaes. Usig Gee s fucio mehod ad applyig he iiial ad bouday codiios i Eqs.., we ca show ha he soluio o Eq.. is expessed as κ a u, G, ξ, τ sih[ Ψ ξ ] E τ Ψ dξ dτ. s E τ ξ Subsiuig Eq.. io Eq.. leads o 9

56 3 Ψ ] exp[, s d E E J J C a u τ τ τ τ κ.3 whee [ ] Ψ sih ξ ξ ξ ξ ξ d J C. Iegaig Eq..3 alog he adius of he cylide, we obai he o-dimesioal mea velociy Ψ ] exp[ 4, s m d E E J C a d u u τ τ τ τ κ.4 Sice he elecoosmoic flow velociy is coupled wih he EDL poeial disibuio i Eq..3 by he faco C, he EDL poeial field should be obaied idepedely befoe we solve he flow field. A deailed deivaio is peseed i appedix B..4 Useady ad Fequecy-depede Elecoosmoic Flow.4. Siusoidally aleaig elecic field Coside he applicaio of a siusoidally aleaig elecic field wih a agle fequecy ω i e E E ω.5 Subsiuig Eq..5 io Eq..3, we ca show ex + + Ψ + Ψ 4 4 exp si cos exp exp, s s J J C a i i J J C a REAL u β β β β κ β β κ.6 whee i is he ui imagiay umbe. REAL deoes he eal pa of he soluio. Hee a

57 ew paamee β is defied as Teliois, 98 a β.7 µ / ρω β epeses he aspec aio of he capillay adius a o he Sokes peeaio deph δ s, defied as Teliois, 98 δ s µ µ.8 ρω π ρ f whee ω f is he fequecy of he applied elecic field. π Accodigly, he mea velociy is give by u m 4 κ a Ψ s u, d C J cos β + β si β exp β.9.4. Sep chage elecic field The exeal elecical field is applied ad emais cosa fom he ime i.e., he elecical field follows sep fucio chage whee E E H.3 H δ is he well-kow Heaviside sep fucio. δ < δ.3 Subsiuig Eq..3 io Eq..3 ad Eq..4, we obai 3

58 3 Ψ exp, s J J C a u κ.3 ad Ψ 3 exp 4 s m J C a u κ.33 I fac, we also ca deive Eq..3 ad Eq..33 by seig he agle fequecy of he elecical field ω i Eq..6 ad Eq..9 equal o zeo, ad he β. Hece Eq..6 ad Eq..9 ca educe o Eq.. ad Eq..3, especively..4.3 Pulsed elecic field Fo a moe geeal siuaio whe he elecic field is swiched o a he ime ad emais cosa uil i is swiched off a he ime δ, he elecic field is he descibed by [ ] δ H H E E.34 Subsiuig Eq..34 o Eq..3 ad Eq..4, we ge Ψ exp ] exp[, s H J J C a u δ δ κ.35 ad Ψ 3 exp ] exp[ 4 s m J H C a u δ δ κ.36 The oal volumeic e flow quaiy fom he elecic field is swiched o o he emiaio of he eie flow due o ficioal sess ca be deemied fom

59 V 4 κ a Q δ u d δ C.37 m 3 Ψ s J I is obvious ha he oal e flow quaiy is popoioal o he duaio of ime δ hee µ δ, duig which he elecic field is applied. ρa δ.5 Resuls ad Discussio I calculaios, NaCl is used as he elecolye soluio. A oom empeaue of T 98 K, he elecolye popeies ae dielecic cosa ε 8, viscosiy 3 µ.9 sm kg /, ad desiy ρ kg / m..5. Poeial pofile Figue - shows he EDL poeial pofiles i a cylidical capillay fo wo diffee elecokieic diamees, κ a wih a fixed zea poeial, Ψ s. Fo compaiso, we also iclude he esuls obaied fom he Debye-Hückel liea appoximaio ad complee umeical mehod. To exclude he effec of zea poeial o he magiude of he EDL poeial, we choose he abiay efeece poeial, ζ o. I is show i Figue - ha he Debye-Hückel liea assumpio povides a good appoximaio fo a lage elecokieic diamee. Howeve, he lieaized soluio deviaes fom he complee Poisso-Bolzma equaio fo a smalle elecokieic diamee, i.e., a hicke EDL coespodig o a dilue elecolye o smalle capillay. Fuhemoe, i is oed fom Figue - ha he diffeece bewee he aalyical scheme used i he pese sudy ad he umeical soluio of he Poisso-Bolzma equaio is visually idisiguishable, suggesig a excelle 33

60 appoximaio esimaed by Eq. B.3.. Dimesioless EDL poeial ψ'/ζ κ a κ a 5 Ψ s 8 Debye - Huckel liea Poposed aalyical scheme Complee umeical Dimesioless Radius ' / a Figue - Compaiso of he esuls fo dimesioless EDL poeial, ψ / ζ vesus dimesioless adius, / a, obaied fom he Debye-Hückel liea appoximaio, he aalyical scheme poposed i his sudy, ad he umeical iegaio of he complee Poisso-Bolzma equaio i a cylidical capillay fo wo cases: i κ a, ad ii κ a 5, wih a fixed dimesioless zea poeial, Ψ s Tasie velociy pofile Time evoluio of he elecoosmoic flow field i cylidical capillaies ude a cosa elecic field is show i Figues -a-b. Hee he efeece velociy is chose as u ε ε k T b s E so ha he zea poeial effec ca be show. Accodig o he esuls µ e displayed i Figues -a-b, we ca obseve ha upo he applicaio of a elecic field, he flow is acivaed i a egio adjace o he chael wall ad he velociy iceases apidly fom zeo o he wall o a maximum wihi he EDL whee he divig 34

61 foce is pese. As he ime goes, he flow gadually exeds o he es poio of he chael due o hydodyamic sesses oigiaed fom liquid viscosiy. Whe he flow i eie chael becomes seady, he flow velociy acoss he chael ohe ha he EDL egio emais almos a cosa value, esemblig a plug flow. Fuhemoe, we ca esimae he 8 ' if Dimesioless Velociy u'', ' 6 4 Ψ s 8 Ψ s 4 κ a ' if Dimesioless Radius ' / a Figue -a Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a. The efeece velociy is defied as ε ε kbt us E. µ e chaaceisic ime scale fo he elecoosmoic flow o each is seady sae fom Eq..3 µ ρ a by choosig. The we ca obai ρ a µ, whee. 45 deemied fom J ; his shows ha he chaaceisic ime, is popoioal o he 35

62 squae of he chael adius, a. I addiio, i is oed ha he velociy iceases wih he iceasig zea poeial, idicaig a almos liea elaioship bewee he velociy ad he zea poeial. 8 ' if Dimesioless Velociy u' ', ' 6 4 Ψ s 8 Ψ s 4 κ a ' if Dimesioless Radius ' / a Figue -b Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a. The efeece velociy is he same as i -a. As a compaiso, whe he elecic field is swiched off he ime devoluio of he elecoosmoic flow is show i Figue -c. I ca be see ha he flow decays followig a paabolic ed, ad is simila o he well-kow Sokes fis poblem, i.e., a seamig flow ove a suddely sopped plae, which was discussed i legh by Teliois 98. Howeve, such a similaiy is o coicidece. As poied ou ealie, he elecoosmoic flow is acivaed iside he EDL, which ca be cosideed as he movig plae i he Sokes fis poblem. Oce he divig foce is emoved he movig plae is suddely sopped ad he 36

63 exeal elecic field is ued off, he mechaism goveig fluid flow is he same, i.e., hydodyamic ficioal sesses due o liquid viscosiy. Dimesioless Velociy u'', ' Ψ s 4 κ a ' Dimesioless Radius ' / a Figue -c Time devoluio wih fixed he elecokieic diamee, κ a 3.57 ad he zea poeial, Ψ s 4. The efeece velociy is he same as i -a..5.3 AC peubaio Fo a capillay of a µ m i adius, we ca esimae he chaaceisic ime fo he elecoosmoic flow o each a seady sae is ρa 9.µ s. The coespodig µ eigefequecy of he sysem is f µ 5. KHz a 5 ρ. Hece, hee diffee chaaceisic fequecies of he exeal elecic field,,. f f ad ω / f f π ae chose o sudy he chaaceisic feaues of he oscillaig elecoosmoic flow. 37

64 The seadily oscillaig velociy disibuios ae show i Figues. -3a-c fo five diffee π π 3π chaaceisic momes: ω,,,, π. I is obvious ha he peubed flow egime 4 4 becomes smalle wih a icease i he fequecy of he elecic field. This sceaio ca be logically aicipaed because ude vey high fequecies, he elecic field chages is diecio so fas ha he flow eve ca ge developed acoss he eie chael. Accodig o Eq..8, he fequecy-depede Sokes peeaio deph, δ s, epeseig a ypical legh scale of he oscillaoy lamia viscous flows i espose o a hamoic exeal exciaio Teliois, 98, is ivesely popoioal o he squae oo of he fequecy, f. This specific depedece o fequecy agees well wih ha fo classical Sokes secod poblem, i.e., he oscillaig viscous flow iduced by siusoidally oscillaig ifiie fla plae Teliois, 98. Fo example, a a vey high fequecy, e.g., f f, he peubed aea is esiced oly i a small egio ea he chael wall, while he fluid i he es of he chael emais zeo velociy, as show i Figue -3c. I coas, i Figue -3a, a a vey low fequecy, e.g., f f., he flow is exeded hough he eie capillay. Fuhemoe, i also should be oed ha, i Figues -3a-c he magiude of he maximum velociy deceases as he fequecy iceases. This ca be explaied by fuhe examiaio of he aalyical soluio deived i he pevious secio. Whe he oscillaoy flow becomes seady, i.e.,, he expoeial em i Eq..6 decays, he value of he maximum velociy should be bouded by u max κ a J κ a C Ψ s 4 4 J + Ψ β s C J J.38 Accodig o he defiiio of β give i Eq..7, we ca easoably ife ha he 38

65 maximum velociy u max iceases as he fequecy ω deceases. The maximum velociy eaches is peak value whe he fequecy of he elecic field is zeo, i.e., β. The slip velociy appoximaio used by Södema ad Jösso 996 ad Saiago is ooed fom a aalogy o he Sokes secod poblem. To compae he slip velociy appoximaio wih he aalyical mehod used i his wok, we also pese he esuls fo he elecoosmoic flow field ude he slip velociy appoximaio, epeseed by dashed cuves i Figues -3a-c. Deailed deivaio of he elecoosmoic velociy wih he slip velociy appoximaio is povided i appedix C. I is demosaed i Figues -3a-c ha a he cee egio of he capillay, he wo mehods give ealy he same esuls fo asie velociy disibuios, while adjace o he chaged chael wall he discepacy is obseved. Such a discepacy is due o he assumpio of movig bouday codiio used i he slip velociy appoximaio, i which he bouday velociy is assumed o be popoioal o he oscillaig elecic field efe o Eq. C.b, while he bouday velociy is chose as zeo o-slip codiio i he aalyical scheme used i pese sudy. The slip velociy appoximaio eglecs he velociy pofile i he EDL egio. Thus he aalyical scheme used i his wok ca pese a moe accuae ad complee sceaio fo he oscillaig elecoosmoic flow i he micochael. 39

66 f. f*. ω Dimesioless Velociy u'', Aalyical scheme i his wok ω π/4 ω π/ ω 3π/4 ω π Slip velociy appoximaio Doimesioless Radius ' / a Figue -3a Low fequecy of he exeal field, f. f *. 4

67 f f*. Dimesioless Velociy u'', Aalyical scheme i his wok ω π/4 π/ 3π/4 π Slip velociy appoximaio Dimesioless Radiuis ' / a Figue -3b The eigefequecy of he exeal field, f f *. 4

68 f f*. Dimesioless Velociy u'', Aalyical scheme i his wok 3π/4 ω π/4 π/ π Slip velociy appoximaio Dimesioless Radius ' / a Figue -3c High fequecy of he exeal field, f f *. Figue -3 Seadily oscillaig velociy disibuios alog he dimesioless adius fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he elecokieic diamee, κ a 3.57, he zea poeial, Ψ s 4 ad eigefequecy, f * 5.5 KHz. Sapshos ae peseed a five diffee chaaceisic momes: ω, π/4, π/, 3π/4, π. Refeece velociy is chose as ε ε us Eζ. Compaiso wih he esuls obaied fom o µ slip velociy appoximaio epeseed by dashed cuves. 4

69 I Figue -4a, he ime depede mea velociy is ploed fo wo-cycle peiods of he AC elecic field. The flow ca follow he siusoidally oscillaio of he elecic field, bu hee exiss a phase lag. Fuhemoe, i ca be obseved ha as iceasig fequecy he flow lags behid he exciaio field wih a lage phase lag. Fom Eqs..6-.8, we ca show ha he phase lag of he oscillaig velociy o he exciaio field is expessed as β πρ f θ a a fa a π *, idicaig ha he phase lag θ iceases µ f moooically wih iceasig he exciaio fequecy f. These esuls ae i accod wih he fidigs by Dua ad Beskok i hei sudies o AC elecoosmosis i micochaels. Ad i ca be fuhe ifeed ha whe he exciaio fequecy f appoaches o zeo DC elecic field, he phase lag agle θ appoaches o zeo, i.e., hee is o phase lag. Whe he exciaio fequecy emais a vey low value e.g., f.5 f *, he phase lag is * popoioal o he exciaio fequecy ad ca be simply esimaed as θ π f / f. O he ohe had, whe he exciaio fequecy appoaches o ifiiy he phase lag agle appoaches o 9 degee. The simila ed exiss fo he effec of he capillay size, he phase lag iceases moooically wih he capillay coss-secioal aea, A π a. This is o supisig because lage chael size equies loge ime fo he momeum asfe by diffusio. The magiude of he maximum mea velociy he ampliude of he velociy flucuaio is also depede o he fequecy. As show i Eq..39, he ampliude of velociy flucuaio ca each a asympoe whe he fequecy is below a ciical value. This ciical value ca be deemied by seig 4 β 4 * f π. i Eq..39, oe ca f 43

70 ge f *.5 f. κa κa 4 4 u C C.39 max Ψs J + β / Ψ s J Figue -4b shows depedece of he mea elecoosmoic flow velociy o he ime, duig which he elecic field is poweed o. I is show ha he flow cao each is seady sae uil he ime is lage ha he chaaceisic ime. O he ohe had, he flow decays expoeially afe he elecic field is swiched off.. Mea velociy. Dimesioless Mea Velociy u' m f* f* f* Elecic field Dimesioless Elecic Field E/E π π 3π 4π ω Figue -4a Seadily oscillaig mea velociy fo hee diffee aspec fequecies of he exeal field,. f *, f * ad f *, wih he sysem eigefequecy of f * 5.5 KHz. 44

71 . Mea velociy. Dimesioless Mea Velociy u' m ' δ '. δ '.3 δ '.5 δ '.7 δ '.9 Elecic field Dimesioless Elecic Field E'/E Dimesioless Time ' Figue -4b Evoluio ad devoluio of he mea velociy whe he exeal field is swiched o fo diffee duaio of ime δ. Figue -4 Dimesioless mea velociy vesus ime wih fixed he elecokieic diamee, κ a 3.57 ad he zea poeial, Ψ s 4. Refeece velociy is chose as ε ε us Eζ. o µ 45

72 .6 Summay A aalyical scheme is used o solve he Poisso-Bolzma equaio fo sudy of he dyamics of elecoosmoic flow i a cylidical micocapillay. The esuls show ha he soluio of he Poisso-Bolzma equaio based o he aalyical scheme is viually o diffee fom umeical soluio. I is demosaed ha he geomey ad zea poeial of he chael wall have a sog impac o he elecoosmoic flow. The elecokieic diamee deemies he elaive hickess of he elecic double laye while zea poeial deemies he magiude of he velociy, ad hece he flow ae. The chaaceisic ime fo he flow o each is seady sae is popoioal o he squae of he chael adius. The evoluio of he elecoosmoic flow upo applicaio of a cosa elecic field exhibis a uique flow pofile, which is esuled fom he coibuios due o he elecic body foce ad hydodyamic viscous sess. O he ohe had, he flow devoluio afe he exeal elecic field is swiched off follows a flow pae solely coolled by hydodyamic ficio due o liquid viscosiy. I addiio, i is foud ha he oscillaig elecoosmoic flow is sogly depede o he modulaio fequecy of he applied siusoidally aleaig elecic field, which deemies he hickess of usable Sokes laye, ad hus goves he exe of he hamoic oscillaio ad he velociy disibuios of he oscillaig elecoosmoic flow. 46

73 Chape 3 Aalyical Soluio of Elecoosmoic Flow i a Capillay Aulus wih High Zea Poeials 3. Ioducio The aulus model is moe geeal ha a model like a cylidical poe o wo paallel plaes. Whe he ie adius of he aulus becomes zeo, he aulus acs as a cylidical poe. The effec associaed wih he ie cylide ca be igoed. O he ohe had, as he ie adius appoaches he oue adius, he aulus behaves like wo paallel plaes, ad heefoe boh cylides play a equal impoa ole. I pacice, he elecoosmoic flow i a capillay aulus is elaed o modellig of he elecokieic moio of a paicle i a polymeic poous membae, which is impoa o udesadig of elecophoeic sepaaio of poeis O Coo e al., 996. I addiio, he aulus geomey is employed as a ovel micofluidic mode fo he bledig of chemical ad biological fluids Adeeev e al., 999. Recely, Tsao sudied he seady sae elecoosmosis flow hough a aulus ude he Debye-Hückel liea appoximaio, idicaig ha his wok is valid oly fo he case of low zea poeials e.g., ζ 5mV fo a - elecolye. I pacical siuaios, howeve, he zea-poeials as high as ~ mv ae o ucommoly ecoueed. Theefoe, o cicumve his poblem, i his chape we exed Tsao s model by developig a aalyical appoximaio o obai he geeal soluio of he complee Poisso- Bolzma equaio wih high zea poeials, 47

74 i a fashio simila o he mehod used by Philip ad Woodig 97. As such, aalyical soluios of he Poisso-Bolzma equaio fo he EDL poeial disibuio ad he Sokes equaio fo he asie elecoosmoic flow field i he aulus egio ae obaied, ad hei esuls ude iflueces of he EDL ad geomeic paamees ae peseed. Specifically, we also discuss he siuaios whe wo cylidical walls ae opposiely chaged. As aohe coibuio of his chape, he aalyical soluios fo he fequecy-depede velociy disibuios i a capillay aulus ae obaied usig Gee s fucio mehod. The flow field oscillaios excied by a siusoidally aleaig elecic field is discussed ude diffee modulaio fequecies. The geealiy of he aulus model is also peseed. 3. Dyamics of he Elecoosmoic Flow Coside a elecoosmoic flow i a aula capillay wih ifiie legh bewee wo coaxial cicula cylides of adii, α a ad a, as show i Figue 3- α is he aio of he ie adius o he oue adius. The aulus is filled wih a icompessible, Newoia, symmeic moovalece elecolye of cosa dielecic cosa, ε, viscosiy, µ ad desiy, ρ. The ie ad oue capillay walls ae uifomly chaged wih he zeapoeials ζ i ad ζ o, especively. Whe a exeal elecic field, E is applied alog he axis of he capillay, he liquid sas o move as esul of he ieacio bewee he e chage desiy i he elecic double laye EDL ad he applied elecic field. The moio of liquid hough he cylidical micocapillay is goveed by he Navie-Sokes equaio Pobsei, 994 u, u, ρ µ + E ρ e 3. 48

75 αa a z Figue 3- Schemaic diagam of a aulus. The adii of he ie ad he oue cylides ae α a ad a, especively. Hee u is he velociy field. ρ is he volumeic chage desiy due o he pesece of e he elecic double laye EDL, ad i is give by he Poisso equaio, which akes he fom Hue, 98 d d dψ ρ e d ε ε 3. Subsiuig Eq. 3. io Eq. 3., we obai 49

76 5,,, E u E u u ψ µ ε ε µ ψ ε ε µ ρ 3.3 Ioducig dimesioless paamees u s u u a ρ µ a T k e b ψ Ψ 3. 4 Hee s u is he efeece velociy defied by he well-kow Smoluchowski equaio Hue, 98 o s E u ζ µ ε ε 3.5 Obviously, s u is popoioal o he magiude of he exeal field, E ad he zea poeial, o ζ. Eq. 3.3 akes he dimesioless fom as Ψ Ψ + s u u,, 3.6 whee T k e b o s ζ Ψ. The EDL poeial Ψ ad velociy u saisfy he dimesioless iiial ad bouday codiios u 3.7a α u s Ψ Ψ β 3.7b u s Ψ Ψ 3.7c whee o ζ i ζ β / deoes he aio bewee he zea poeial of he ie ad oue adii.

77 Compaig wih he esuls i he pevious secio, we ca ife he sysem will become seady sae whe he ime goes o ifiie, he soluio should ake he followig odimesioal fom u, u + u, 3.8 The seady sae limiig soluio u ca be obaied fom Eq. 3.6 by seig u /, subjec o he bouday codiios Eqs. 3.7b ad 3.7c. Hece we ca obai he seady sae velociy, which is defied as u Ψ l Ψ lα β s 3.9 Subsiuig Eq. 3.8 io Eq. 3.6, usig he soluio of Eq. 3.9, we ca ge he goveig equaio fo he asie velociy u,, u, u, 3. subjec o he iiial ad bouday codiios u 3.a u α u 3.b u 3.c Usig he classical mehod of sepaaio of vaiables, he soluio of Eq. 3. ca be give by whee u, B [ J N α J α N ] exp 3. 5

78 Ψ l β [ J ] N α J α N d Ψ lα α s B 3.3 d ad ae he posiive oos of α J [ J N α J α N ] N α J α N 3.4 whee J ad N ae he zeo-ode Bessel fucio of he fis ad secod kid, especively. The asie aspec is simila o he EOF i cylidical capillay. So he seady aspec is emphasized hee. Accodig o Eq. 3.9, he seady sae velociy is u ψ l / a us β 3.5 ζ o lα The aveage velociy of he fluid acoss he aulus coss-secio is obaied by u a u d u J G, α, β s a α 3.6 αa z ad he volumeic flow ae is expessed as Q V a π α a u d u π a α J G, α, β 3.7 s whee β α + α lα J G, α, β G lα α ad a G ψ d 3.9 ζ a α αa o J is egaded as a coecio faco o he Smoluchowski equaio Hue 98 ha 5

79 assumes a egligible hickess of he EDL, ad i depeds o he geomey aio, α ad he aio of zea-poeials, β. Clealy, he flow diecio is also deemied by he sig of J. Ad G is he aio of he mea elecosaic poeial acoss he aulus egio o he zeapoeial of he oue chaged wall, epeseig he effec of he fiie EDL hickess. Fo a wide capillay i.e., κ a >>, G is egligibly small, ad hece Eq. 3.8 educes o Tsao, β α + α lα J + 3. lα α By seig J we ca obai he codiio ha he zeo flow ae occus is Tsao, ζ i α lα β α + < ζ o α + α lα 3. Eq. 3. shows ha fo a give α, he liquid flow due o elecoosmosis moves owad oe diecio whe β > β, while he flow goes opposie whe β < β. Sice he elecoosmoic flow velociy is coupled wih he EDL poeial disibuio i Eq. 3.5, he EDL poeial field should be obaied idepedely befoe we solve he flow field. A aalyical soluio of he EDL poeial i a capillay aulus is peseed i appedix E. 3.3 Geealiy of Aulus Model I has bee meioed ha he aulus model is moe geeal ha a model like a cylidical poe o wo paallel plaes. The geealiy of aulus model is peseed hee. The seady sae limiig soluio u has bee obaied i Eq Acually, he cylidical capillay model ad paallel plaes model ae wo limiig cases of he aulus model. 53

80 Limiig Case I: whe he ie adius becomes zeo i.e., α, he capillay aulus behaves as a capillay cylide. Because l α whe α, Eq. 3.9 educes o u Ψ 3. Ψ s The esul i Eq. 3. is cosise wih Rice ad Whiehead s esuls fo elecoosmoic flow i capillay cylide Rice ad Whiehead, 965. Limiig Case II: whe he ie adius appoaches he oue adius i.e., α, he capillay aulus behaves as paallel plaes. Ioducig ew spaial vaiable bewee he wo paallel slis y α, ad because l α + y lim lim α lα α α + y α, Eq. 3.9 educes o + y Ψ y β u y Ψ y s If he wo slis walls ae equally chaged, i.e., β. Eq. 3.3 educes o Ψ y u y 3.4 Ψ s The esul i Eq. 3.4 is cosise wih Yag s esuls fo elecoosmoic flow i paallel plaes Yag e al.,. 3.4 Resuls ad Discussio I he pevious chape, basic equaios have bee deived fo he EDL ad he elecoosmoic velociy disibuios i a aulus egio. Examiaio of hese equaios eveals ha he chaaceisics of he elecoosmoic flow i a aulus ae 54

81 deemied by he followig o-dimesioal paamees: he geomeic adius aio, α, he zea poeial aio, β, he elecokieic diamee, κ a, ad he zea poeial, Ψ s. I calculaios, we coside hese paamees bouded by: α. 4, β 3 ~ 3, κ a 5 ~, ad ~, which ae of mos pacical ieess. Ψ s 3.4. Poeial pofile Figue 3-a shows he compaiso esuls of he o-dimesioal elecical poeial pofiles obaied fom he Debye-Hückel liea appoximaio, he aalyical scheme poposed i his sudy, ad he umeical iegaio of he complee Poisso-Bolzma equaio i a aulus fo wo cases: a κ a, ad b κ a 5. I is oed ha i eihe case, use of he Debye-Hückel liea soluios always gives ise o eos i he elecical poeial. I coas, he diffeece bewee he aalyical scheme poposed i cue sudy ad he complee umeical soluio of he Poisso-Bolzma equaio is visually idisiguishable, idicaig a excelle appoximaio give i Eq. E.4. Figue 3-b peses he pofile of he o-dimesioal EDL poeial fo hee diffee values of he elecokieic diamee, κ a. As illusaed i Figue 3-b, he EDL poeial is symmeically disibued whe he ie ad oue walls ae equally chaged i.e., β ±. I ca also be see ha fo lage elecokieic diamees, he EDL field exiss oly i he egio close o he wall. Howeve, fo small elecokieic diamees, he EDL poeial ca exed o a lage poio of he aulus, cosise wih pevious sudies of he elecoosmoic flow hough micocapillaies of ohe geomeic coss-secioal shapes Bugee ad Nakache 964, Levie e al. 975, Koh ad Adeso 975, Yag ad 55

82 Huag. I is, heefoe, expeced ha he EDL effec is moe sigifica fo smallsized auluses. I Figue 3-c, he EDL poeial vesus he o-dimesioal adius is ploed fo hee diffee zea poeials, Ψ s. Sice he elecokieic diamee, κ a keeps as a cosa, he EDL egime i all hese hee cases emais he same. As such, vaiaio of he zea poeial oly esuls i a chage of he magiude of he EDL poeial. I is oed ha as he zea poeial iceases, he poeial dops off moe shaply wihi he EDL egime..8 ψ ζ o.6.4 κa 5 κa 5 α.4 β Ψ s 8 Debye - Huckel liea Poposed aalyical scheme. Complee umeical / a Figue 3-a Compaiso of he esuls obaied fom he Debye-Hückel liea appoximaio, he aalyical scheme poposed i his sudy, ad he umeical iegaio of he complee Poisso-Bolzma equaio i a aulus fo wo cases: i κ a, ad ii κ a 5, wih fixed he o-dimesioal zea poeial of he oue cylide, Ψ s 8 ad he zea poeial aio of he ie cylide o he oue cylide, β. 56

83 .5.5 Ψ β β -.5 α.4 Ψ s / a,, 3, κa 5 κa 5 κa Figue 3-b Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s. 57

84 6 Ψ β β -6 6 α κa.4 5,, 3, Ψ Ψ Ψ s s s / a Figue 3-c Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a 5. Figue 3- No-dimesioal EDL poeial, Ψ vesus o-dimesioal adius, / a of he aulus. The geomey aio of he aula ie adius o oue adius, α.4. The zea poeial aio of he ie cylide o he oue cylide, β ad β -, deoig a symmeically-chaged ad opposiely-chaged aulus, especively. 58

85 3.4. Velociy pofile Time evoluio of he elecoosmoic flow field i aula capillaies ude a cosa elecic field is show i Figues 3-3a~d. The efeece velociy is chose as u s ε ε E k T b. I is obvious ha he velociy evoluio i aulus is simila o ha i µ e cylidical capillay, which is discussed i deail i chape. Thus oly seady-sae flow chaaceisics of EOF i aulus will be discussed i his secio. The o-dimesioal seady-sae velociy disibuio fo diffee elecokieic diamees, κ a, is show i Figues 3-4a~b. Whe wo cylidical walls ae equally chaged i.e., β, he velociy field exhibis a pofile esemblig a plug flow. I ca be see ha he velociy iceases apidly fom zeo o he wall o a maximum velociy wihi he EDL egio ha exeds abou a few Debye-Hückel leghs fom he chael wall. The he velociy emais as a cosa hough he es of he chael. This uique pofile ca be aibued o he fac ha he flow is dive by he elecical foces esuled fom he ieacio of a exeally imposed elecic field ad he EDL field. Such divig foces exis oly iside he EDL egio whee he elecical e chage is o zeo. The flow i his egio may be viewed as acive. I coas, he flow ouside he EDL egio moves due o ficioal sesses oigiaed fom liquid viscosiy, ad hece i ca be cosideed as passive flow. Fuhemoe, a examiaio of Figue 3-4a eveals ha fo a fixed zea poeial, Ψ s, he magiude of he maximum elecoosmoic velociy is he same, egadless of diffee values of κ a. Takig advaages of he aalyical soluio fo he elecoosmoic flow i he aulus, we ca iepe his sceaio by usig Eq Accodig o Eq. 3.5, he maximum elecoosmoic velociy, i he case of β, is give by he 59

86 appae slip velociy, u s defied i Eq. 3.5 ha is idepede of he elecokieic diamee, κ a. Figue 3-4b shows he effec of asymmeically chaged walls. As explaied ealie, he elecoosmoic flow is he esul of he ieacio bewee he applied elecic field ad he EDL. The zea poeial, Ψ s, heefoe, has a sigifica ifluece o he elecoosmoic flow, which is amply demosaed i Figues 3-4c~e fo boh symmeically-chaged i.e., β ad asymmeically-chaged β auluses. I is oed ha he velociy iceases wih a icease of he zea poeial, idicaig a ealy liea elaioship bewee zea poeial ad velociy. Now we coside a siuaio whe he wo cylidical walls ae opposiely chaged i.e., β <. As illusaed i Figues 3-4a~e, he diecio of he elecoosmoic flow i he aulus is diecly coelaed o he polaiy of he chaged chael wall. I is obseved ha hee is a cylidical plae iside he aulus, o which he fluid velociy is zeo. The liquids divided by such a plae flow i wo opposie diecios, udisubed o each ohe as if hee exiss a wall dividig hem. Ieesigly, a ispecio of Figue 3-4a ad Figue 3-4c idicaes ha give a fixed geomey aio, α, he locaio of he zeo-velociy plae i a equally-chaged aulus emais uchaged despie he diffeece i he zea poeial ad he elecokieic diamee. I fac, he locaio of he zeo-velociy plae ca be pediced fom Eq Give he fac ha fo he case of β < ad lage elecokieic diamees, fo example κ a 5, he elecical poeial, ψ a he zeo-velociy plae is egligible, i.e., ψ. As such, he locaio of he zeo-velociy plae, as show i Figs. 3-4a~e, ca be esimaed as β α, which gives he values of. 63 ad.737 fo α. 4 ad, β ad β, especively. 6

87 u u s α β κ a Ψ s 5 : / a :. :. 3 :.5 4 :. Figue 3-3a Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a 5 ad β. 6

88 3 3 u u s - - α.4 β κ a Ψ s 5 4 :. :. 3: / a Figue 3-3b Time evoluio fo he case of hick elecic double laye wih he elecokieic diamee, κ a 5 ad β -. 6

89 u u s α.4 β κa Ψ 4 s 5 : / a :. :. 3 :.5 4 :. Figue 3-3c Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a ad β. 63

90 4 3 3 u u s α.4 β κa Ψ s 4 :. :. 3: / a Figue 3-3d Time evoluio fo he case of hi elecic double laye wih he elecokieic diamee, κ a ad β. Figue 3-3 Dimesioless asie velociy, u / u s vesus dimesioless adius, / a fo he case of adius aio, α.4 ad he zea poeial Ψ s 4. The efeece velociy is defied as ε ε kbt us E. µ e 64

91 3 u u s , 6 α.4, 5 3, 4 Ψ s β β κa κa 5 κa / a Figue 3-4a Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s ad adius aio β ±. 65

92 4 3 3 u u s α Ψ s / a β β,, 3, κa κa 5 κa 5 Figue 3-4b Effec of he elecokieic diamee, κ a wih a fixed o-dimesioal zea poeial of he oue cylide, Ψ s ad adius aio β ±. 66

93 8 6 4 u u s α.4 κ a β β / a,, 3, Ψ s Ψ Ψ s s 8 4 Figue 3-4c Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a 5 ad adius aio β ±. 67

94 5 5 u u s β β - 6 α κ a / a,, 3, Ψ Ψ Ψ s s s 8 4 Figue 3-4d Effec of he zea poeial, Ψ s wih a fixed elecokieic diamee, κ a 5 ad adius aio β ±. 68

95 5 4 β 3 3 β u β u s - - β - β - α.4 Ψ s κ a / a Figue 3-4e Effec of he zea poeial aio, β ζ i / ζ wih fixed he elecokieic diamee, κ a 5 ad he o-dimesioal zea poeial of he oue cylide, Ψ s. Figue 3-4 No-dimesioal elecoosmoic flow velociy, u / u s vesus o-dimesioal adius, / a of he aulus. The geomey aio of he aula ie adius o oue adius, α.4. Refeece velociy is chose as ε ε kbt us E. µ e 69

96 3.4.3 Pedicio fo he EDL-elaed coecio faco, G The quaiy G is impoa i a sese ha i deoes he effec of he fiie EDL hickess. I ca be cosideed as he EDL-elaed coecio faco o he Smoluchowski equaio. I Figue 3-5, G vesus κ a is ploed fo vaious values of he zea poeial, Ψ s Ψ s. The pedicios display ha whe he elecokieic diamee, κ a is small, G is fa away fom, idicaig ha he impac of he fiie EDL hickess o he elecoosmoic flow becomes elaively soge, ad hece his coecio faco should be ake io accou. Fuhemoe, fo a give value of he elecokieic diamee, κ a, G iceases wih a icease i he wall zea poeial, Ψ s. Howeve, wih he liea appoximaio Rice ad Whiehead, 965, G was epoed o be idepede of Ψ s. O he ohe had, as κ a becomes lage, iespecive of Ψ s all cuves fo G ae appoachig o, suggesig ha while he capillay elecokieic diamee is lage eough, he effec of he fiie EDL hickess, G is egligible. 7

97 ..8 -G Ψ s α.4 β. κa Figue 3-5 Vaiaio of he EDL-elaed coecio faco -G wih he elecokieic diamee, κ a fo vaious zea poeials of he oue cylide, Ψ s. The geomey aio of he aula ie adius o oue adius, α.4. The ie ad oue aula walls ae equally chaged, β. 7

98 3.4.4 Pedicio fo he coecio faco o he Smoluchowski equaio, J.5 β. β J.5. β Ψ s κa 5 wih G his sudy wihou G Tsao's esuls β β α Figue 3-6 Coecio faco fo he Smoluchowski Equaio, J vesus he geomeic aio of he aula ie adius o oue adius, α fo vaious values of he zea poeial aio, β ζ i / ζ. Wihou icludig he EDL-elaed coecio faco, G, we ecove Tsao s esuls epeseed by he dashed lies. Accodig o he defiiio give i Eq. 3.8, he quaiy J epeses he coecio faco o he Smoluchowski equaio, ad i akes io accou of coibuios due o he EDL ad he geomeic aio of he aulus. As poied ou by Tsao, physically J also ca be cosideed as he omalized mobiliy wih espec o ha of a chaged suface wih ζ o see Eq The effecs of he adius aio, α ad he zea poeial aio, β o J ae show i Figue 3-6. Fo a compaiso, he esuls obaied by Tsao who used he soluio of he lieaized Poisso-Bolzma equaio ad 7

99 egleced he EDL-elaed coecio, G ae ploed as he dashed lies i Figue 3-6. As expeced, he mobiliy, J pediced fom his sudy is diffee fom ha obaied by usig Tsao s appoach. Fo isace, Figue 3-6 clealy shows ha fo β > Tsao pediced he ie cylide ehaces he fluid mobiliy ha iceases wih iceasig α. While ou calculaios show a compleely diffee ed of he depedece of J o α. I is aicipaed ha such discepacy would become lage fo he cases of highe zea poeials, Ψ s ad smalle elecokieic diamees, κ a, ude which he coibuios of he o-liea Poisso-Bolzma equaio ad he EDL-elaed coecio, G, ae moe sigifica. J > J J < Figue 3-7 Ciical value combiaios of he adius aio, α ad he zea poeial aio, β fo zeo e flow ae. 73

100 O he ohe had, ou calculaios based o he appoach developed i his wok ecofim he impoa feaues of he elecoosmoic flow i he aulus whe he wo cylidical walls ae opposiely chaged. As meioed ealie, evese of he sig of J fom posiive o egaive implies a chage of he e flow diecio. The e flow diecio chage also implies ha hee exiss some combiaio of α ad β whe he e flow ae is zeo J.Wihou loss of geealiy, he α - β combiaio fo zeo e flow ae i wide capillay is show as he solid cuve i Figue 3-7, accodig o Eq. 3.. Regadless of he value of α, he value of J is foud always posiive fo β >. Howeve, fo β, J deceases fom posiive o egaive as α iceases. The wo egios divided by he solid cuve deoes wo opposie e flow diecios. The e flow ae igh o his cuve is zeo. This α-β elaioship is vey ieesig because i is elaed o he codiio coespods o he elecokieic displaceme of ai bubble i a micochael Takhisov e al.,. The elecokieic flow bewee he bubble suface opposiely chaged compaed o chael wall ad he chael wall ca be modeled as a aula hi film of EOF aoud he bubble. Whe he e flow i his aula film is zeo, he bubble emais saioay while hee is EOF aoud i. Oce e flow exiss i he hi film, he fluid behid he bubble accumulaes ad a pessue gadie is buil up o move he bubble fowad i ode o accommodae he accumulaed fluid. Thus he esuls i Figue 3-7 ca pedic ha whe he bubble would begi o aslae. The α-β elaioship a zeo e flow ae is a impoa heshold codiio fo aspoig bubble o dop i micochaels. 74

101 3.5 Summay Aalyical scheme is poposed o solve he oliea Poisso-Bolzma equaio i a aulus domai. The calculaed esuls show ha he diffeece bewee he poposed aalyical scheme ad he complee umeical soluio of Poisso-Bolzma equaio is visually idisiguishable, idicaig a excelle appoximaio poposed i his sudy. A aalyical expessio fo he elecoosmoic flow filed i a aulus is deived. The calculaios show ha fo a give zea poeial, he magiude of he maximum elecoosmoic velociy is idepede o κ a. I he case ha he wo walls of a aulus ae opposiely chaged, hee exis a zeo-velociy plae, which ca be esimaed by o β α. I addiio, i is foud ha he coecio faco o he Smoluchowski equaio fo aulus eeds o accou fo boh he EDL-elaed ad he geomey-depede coibuios. 75

102 Chape 4 DC Elecoosmoic Flow i a Micochael Packed wih Micosphees ude Elecokieic Wall Effec 4. Ioducio I his chape, we aemp o cosuc a mahemaical model fo he EOF i a chaged micocapillay packed wih chaged micosphees. Ou appoach is based o he capillay model ad he volume aveagig mehod. Sice he packig paicles ae o eual bu chaged wih a zea poeial, ζ p, he local EDL egime will occupy a fiie poio of each void space amog paicles as log as hee is a ieface bewee he paicle ad he fluid. Due o he ieacio of he local EDL ad he applied elecic field, EOF occus eveywhee i he space occupied by fluid phase, icludig i he void space bewee he chaged packig paicles ad he chaged chael wall. We sa wih he aalysis of he EOF i a micocapillay packed wih homogeeously chaged micosphees. I is obvious ha ude he volume aveagig mehod, he macoscopic EOF velociy should be uifom eveywhee i a homogeeous o bouday, uifom paicle size poous medium. Howeve, i his sudy, he poous medium is bouded by a chaged capillay wall. Thus he volume aveagig mehod is employed o obai he macoscopic EOF velociy disibuio, akig io accou ihomogeeiy due o he geomeical vaiaio of packig poosiy. Nex, he modified Bikma momeum equaio is solved o obai he elecokieic wall effec 76

103 which is due o he excessive zea poeials of he capillay ie suface. The effecs of geomeical ad elecokieic paamees o he EOF velociy fields ae discussed. 4. Mehod of Volume Aveagig I a aual poous medium, he disibuio of poes wih espec o shape ad size is iegula. O he poe scale micoscopic scale he flow quaiies such as velociy, pessue, ec., ae also iegula Nield ad Beja, 999. Bu i ypical expeimes he quaiies of iees ae measued ove aeas ha acoss may poes ad such space-aveaged macoscopic quaiies chage i a egula mae wih espec o space ad ime ad hece ae ameable o he heoeical eame. A macoscopic vaiable is defied as a appopiae mea ove a sufficiely lage epeseaive elemeay volume REV. I is assumed ha he esul is idepede of he size of he REV. As illusaed i Figue 4-a, he legh scale of he REV is much lage ha he poe scale, bu cosideably smalle ha he legh scale of he macoscopic flow domai. Repeseaive elemeay volume REV Flow domai a 77

104 E z A poe A ev b φ V f / V A poe / A ev θ E E eff E cosθ L eff L / cosθ E eff θ L L eff c τ / cos θ L eff / L Figue 4- Schemaic illusaio of epeseaive elemeay volume REV: he legh scale of he REV is much lage ha he poe scale, bu cosideably smalle ha he legh scale of he macoscopic flow domai. Ude he volume aveagig mehod, he flow quaiy, such as he velociy will be loweed i magiude i a packed capillay ha i a upacked capillay by some geomeic 78

105 cosai facos Remcho ad Vallao,. Two majo geomeic cosai facos o chaaceize a poous medium ae called he poosiy ad he ouosiy. The poosiy is defied as φ V f / V, whee V f ad V ae he void ad oal volumes of he poous medium, especively as illusaed i Figue 4-b. Thus he poosiy faco is a volume aveagig icludig he space occupied by he solid packig maeial. The ohe faco, as illusaed i Figue 4-c, ouosiy icludes he o-aligme of mos flow chaels wih he elecic field, which is applied alog he capillay axis. The fluid flow i a capillay ube packed wih solid spheical paicles will occu oly wihi he iepaiculae spaces. The idividual flow chaels compise a highly complex, diecioally o-uifom ewok, of which oly a facio will be aliged axially wih he elecic field. Mos flow chaels will be off-axis wih espec o he field esulig i a lowe effecive field segh deceased by a faco of cosθ. I addiio, he fluid flowig i a off-axis chael mus avel a geae disace iceased by a faco of /cosθ fo a give displaceme alog he capillay axis. The ouosiy is defied as τ /cos θ L eff / L, whee L eff is he effecive legh of avel fo flow alog he poe pah ad L is he physical legh of he poous sucue Kaviay, 995. Combiaio of above facos esuls i he educio of he velociy compoe alog he axis by a faco of φ /τ. The ohe wo paamees ae he poe size, d poe, ad he Dacy pemeabiliy, K, which will be defied i ex secio. These sucue-elaed paamees ca be deemied by expeime. I his sudy, we coside a chaged cylidical micocapillay of he adius, chaged spheical micopaicles of uifom diamee, R w packed wih d p. The liquid i he micocapillay is 79

106 assumed o be a icompessible, Newoia, moovalece elecolye of desiy, ρ, ad viscosiy, µ. The zea poeials of he ie wall suface ad he paicle suface ae ζ w ad ζ p, especively. Whe a exeal elecic field, E is applied alog he axis of he micochael, he elecolye i he micochael will move due o he elecoosmosis esuled fom he ieacio of he EDL close o he ieface of he paicles as well as he chael wall ad he applied elecic field. Joule heaig effec due o axially imposed elecic field is igoed i ou model developme. The oveall macoscopic EOF velociy i he packed capillay is decomposed io wo sepaae compoes due o: he coibuio fom homogeeous desely packed chaged micopaicles ad he coibuio fom he chaged capillay wall wih eual packig, as illusaed i Figue chaged paicles chaged bouday chaged paicles o bouday eual paicles chaged bouday Figue 4- Schemaic illusaio of decomposig he oveall macoscopic EOF velociy i a packed capillay io wo sepaae compoes due o: i he coibuios fom homogeeous isoopic desely packed wih chaged micopaicles, ad ii he coibuios fom he chaged capillay wall wih eual packig. As meioed above, he pese model is based o he space-aveagig mehod which is 8

107 commoly applied i he classical pessue-dive aspo pheomea i poous media. This aveagig mehod focuses o he equivale macoscopic physics wihou cosideig he local complexiy i he micoscale. Howeve, he EOF i poous media is oigiaed fom he ieacio of he exeal elecic field ad he EDL a ieal suface of he poous sucue. I ealiy fom micoscopic viewpoi, he divig foce is depede o he local cofiguaio of he ieface ad local pessue gadie may be iduced because of he complicaed geomeic sucue ad ovelapped EDLs, which makes he eal sceaio much moe complex. Sicly speakig, a moe accuae hee dimesioal poe-size level modelig isead of space-aveagig modelig should be developed o iclude such effecs. Howeve due o he eomous difficulies ivolved i umeically simulaig he ue 3-D poblem ad he ime limi fo his pojec, he macoscopic space-aveagig model fo he EOF i he poous media is applied, as a fis-ode appoximaio, houghou his hesis. 4.3 Macoscopic EOF i Homogeeous Chaged Micosphees 4.3. Iesiial EOF velociy The pese wok is based o a semi-heuisic model of flow hough solid maices usig he cocep of hydaulic diamee, which is also kow as he Cama-Kozey heoy Pobsei, 994. The heoy assumes he poous medium o be equivale o a seies of paallel ouous ubules. The chaaceisic diamee of he ubules is ake o be a hydaulic diamee o effecive poe diamee. This diamee is coveioally defied as Kaviay, Void Volume 4φ d poe 4. Suface Aea A φ o 8

108 whee φ is he aveage poosiy of he poous medium. A o is he volumeic o specific aea based o he solid volume, i.e., A fs / Vs, whee fs A is he iefacial aea bewee he fluid ad he solid phase ad V s is he solid volume. Fo spheical paicles cosideed i his wok A o 6/ d p, whee d p is he diamee of spheical paicles. So he hydaulic diamee ca be expessed as φ d poe d p 3 φ 4. We popose he iesiial ubules o be cylidical wih a effecive diamee, i.e., effecive poe size, d R. The ie wall of he cylidical ubules is chaged wih zea poe poe poeial ζ. The iesiial EOF velociy u i i each ubule is goveed by he Sokes p equaio Pobsei, 994. d dui µ + E ρe + Pz Rpoe 4.3 d d whee P z P/L is he global pessue gadie alog he flow diecio. L is he physical legh of he flow domai. I he pese sudy, o global pessue gadie is applied, i.e., P z. As oed i secio 4., fom micoscopic poi of view, a local back pessue gadie may be iduced whee he EOF becomes o-uifom due o he complicaed geomey ad packig codiios. Fo isace, a aow cosicios he local elecic field segh becomes highe if he poous packig has a lowe pemiiviy ha he wokig fluid. As we kow, he EOF velociy is popoioal o he elecic field segh. Thus he local EOF eds o be segheed o icease local velociy. Whe he EOF is o-uifom, local pessue-dive back flow mus be iduced o maiai he local mass cosevaio Tallaek 8

109 e al.,. Howeve, he geomeical o-uifomiy ad he iduced local pessue ae highly adom wihi he bulk packig excep a he ed of he packed capillay he ieface of he eaiig fis, whee he flow pah may subjec o dasic decease i cosssecioal aea, depedig o he qualiy of he fis. I pese sudy he ed fi effec is igoed ad we focus o he elecoosmosis-dive flow hough he poous packig. Ude he pese space aveagig mehod, he oveall effec of he adom local pessue gadie vaishes. Theefoe he poous media is assumed as equivale paallel ubules of uifom coss-secioal aea wihou global pessue gadie. ρ is he local volumeic chage desiy due o he pesece of he EDL iside each iesiial ubule, ad i is give by he Poisso equaio, which akes he fom Hue, 98 e d d dψ i ρe d ε ε R poe 4.4 whee ε is he dielecic cosa of he elecolye ad ε is he pemiiviy of vacuum. Subsiuig Eq. 4.4 io Eq. 4.3, ad usig he followig bouday codiios d ui d dψ i d 4.5a u ψ i ζ p i R poe Rpoe 4.5b we ca obai a aalyical expessio fo he velociy disibuio, which akes he fom as u ε ε ψ i E ζ p µ ζ i p 4.6 Ude he volume aveagig mehod, he macoscopic supeficial velociy also called he Dacy velociy of he fluid ove he epeseaive elemeay volume ca be expessed by akig io accou he ouosiy τ ad he poosiy φ, as 83

110 u Di φ φ Rpoe ε ε φ ui u d E G i p R ζ τ τ µ τ poe 4.7 whee G ζ R p R poe ψ i poe d 4.8 ad φ φ is he local poosiy disibuio of he poous medium. I he lieaue, φ is assumed o ake a fom of Nield ad Beja, 999 Rw φ φ + C exp N R w 4.9 d p whee R w is he disace fom he micocapillay wall. C.4 ad N5 ae empiical cosas Hsu ad Cheg, Iesiial EDL poeial field The iesiial EDL poeial disibuio ψ i i each ubule is goveed by Poisso- Bolzma P-B equaio, which is expessed i he omalized fom as Hue, 98 R d dr dψi R R sih Ψi R dr R κ Rpoe 4. whee e R κ, κ, dimesioless EDL poeials ε ε k T b e i Ψ i ψ ad k T b eζ p Ψ ip, k T b e is he elemeay chage, is he ioic coceaio i he bulk phase i.e., fa fom he chaged ubula wall, k b is he Bolzma cosa, ad T is he absolue empeaue. The bouday codiios give by Eq. 4.5 ca be ewie as 84

111 d dr Ψ i R Ψ κ Ψ 4. i R Rpoe ip A aalyical soluio o Eq. 4. is povided i appedix B. 4.4 Macoscopic EOF i a Chaged Micocapillay Packed wih Neually Chaged Micosphees 4.4. EOF velociy field We coside a chaged cylidical micocapillay of adius R w, wih he ie suface chaged wih zea poeial ζ w. Iside he micochael hee ae desely packed, eually chaged spheical micopaicles of diamee d p. The Bikma momeum equaio, which oigially was deived fo a pessue-dive Dacy flow i poous media, he has bee geealized by Bikma o accou fo he ieial foces, pessue gadie, body foces, ad shea sesses Kaviay, 995. I he pese siuaio, o pessue is exeed ad hece o pessue gadie is pese. Fuhe, due o low Reyolds umbe flow i mico-sized poes, he macoscopic ieial foce is igoed. The body foce is he columbic foce due o he ieacio of he elecic double laye EDL ad he exeal elecic field. Thus he modified Bikma momeum equaio used fo descibig he macoscopic EOF of a icompessible, Newoia elecolye soluio i a packed micocapillay ude a seady sae ca be expessed as d du Dw µφ u Dw Fφ u Dw µ eff + E φρef + ρ Rw 4. d d K K Hee u Dw is he macoscopic EOF velociy Dacy velociy field hough he eual 85

112 poous medium iside he micocapillay. φ φ is he local poosiy disibuio of he poous medium, deemied by Eq The Dacy pemeabiliy K is defied by he Cama-Kozey equaio Pobsei, 994, 3 K φ d p / a φ, ad he ieial coefficie 3/ F b a, wih 5 / φ a ad b. 75 beig he Egu cosas Hsu ad Cheg, 99. µ eff is he effecive fluid viscosiy, a fucio of he fluid viscosiy ad he geomey of he pemeable medium. Fo a isoopic poous medium, µ eff / µ / φτ Nield ad Beja, 999. ρ is he aveage volumeic chage desiy due o he pesece of he EDL, ad i is ef assumed o saisfy he Bolzma disibuio Hue, 98 e ψ w ρ ef e sih Rw 4.3 kbt Eq. 4. is subjec o he bouday codiios du Dw d dψ w d 4.4a u Dw ψ R w ζ w 4.4b w Rw 4.4. EDL poeial field Hue, 98 The EDL poeial disibuio is goveed by he Poisso-Bolzma equaio R d dr dψw R R sih Ψ dr w R R κ R 4.5 w whee R κ, subjec o he bouday codiios 86

113 dψw dr R 4.6a eζ w Ψ w R 4.6b κrw k T b A aalyical soluio of he Eq. 4.5 is povided i appedix B, ad such a aalyical soluio will be also used whe we use he umeical mehod o solve he o-liea modified Bikma s momeum equaio, Eq Aalyical soluio of he modified Bikma momeum equaio, Eq. 4. I geeal, o aalyical soluio is available fo Eq. 4., which ca oly be solved umeically. Howeve, due o he low Reyolds umbe lamia flow i micochaels, we ca safely igoe he high Reyolds umbe em, i.e., ρ Fφ u / K. Ad o emphasize oly he elecokieic wall effec, we ca fuhe assume ha he poosiy is uifom i he poous medium, i.e., φ φ. Ude hese assumpios, we ca exclude he geomeical wall effec, ad hus Eq. 4. ca be simplified o Dw d du aτ φ φ d φ τ E Dw u Dw d d p µ ρ e R 4.7 w Accodig o Poisso s equaio Hue, 98 ρ d dψ w ε ε d d e 4.8 Subsiuig Eq. 4.8 io Eq. 4.7 gives d u d ηψ w β u d Dw Dw

114 whee φ τ ε ε E µ η φ d p aτ φ β 4. Ioducig v ηψ 4. u Dw w Eq. 4.9 becomes d d dv β v β ηψ d w 4. Ude he Debye-Hückel appoximaio, i has bee show ha Rice ad Whiehead, 965 Hece Eq. 4. ca be ewie as I κ ψ w ζ w 4.3 I κ R w d d dv d v β β ηζ w I κ I κ R w 4.4 This is a o-homogeeous zeo-ode modified Bessel equaio, subjec o he bouday codiios dv d 4.5a v Rw ηζ w 4.5b A homogeeous soluio ad a paicula soluio o Eq. 4.4 is give especively by κ I β v ηζ w 4.6a κ β I β R w w β I κ v p ηζ w 4.6b κ β I κ R 88

115 89 The geeal soluio o Eq. 4.4 is w w w w p R I I R I I v v v κ κ ηζ β κ β β β ηζ β κ κ Makig use of Eq. 4. ad Eq. 4.3, we ca obai a aalyical soluio fo he EOF velociy disibuio i a chaged micochael packed wih eually chaged micosphees + w w w w Dw R I I R I I E v u κ κ β β β κ κ τ φ ζ µ ε ε ηψ Slip velociy appoximaio based soluio o he modified Bikma momeum equaio Eq. 4. Cosideig a lage chael size, he momeum equaio ca be fuhe simplified by applyig he slip velociy appoximaio, i which he EDL hickess is igoed. I his siuaio he bouday velociy a he chael wall is deemied by Smoluchowski equaio Pobsei, 994. Thus Eq. 4.9 is educed o Dw Dw u d du d β 4.9 subjec o bouday codiios Dw d du 4.3a w R u Dw E w ζ µ ε ε 4.3b Soluio o Eq. 4.9 ca be eadily obaied as w w Dw R I I E u β β ζ µ ε ε 4.3

116 4.5 Oveall Macoscopic EOF Velociy We e-couple he wo velociy compoes fom he chaged wall ad he chaged packig paicles, ad obai he oveall macoscopic EOF velociy i a chaged cylidical micocapillay wih chaged micosphee packig. The excess zea poeial is ake io accou by such a eame ha he zea poeial of he capillay wall is eplaced by ζ w - ζ p, which is esposible fo he elecokieic wall effec Liapis ad Gimes,. ζ p u D udw + u Di 4.3 ζ w Specifically, i he case whe ζ w ζ p, Eq. 4.3 is educed o u D udi, idicaig o elecokieic wall effec. 4.6 Resuls ad Discussio The mahemaical models goveig he EOF i a micochael packed wih micosphees have bee developed. I his secio we will sudy how he EOF i a packed micocapillay is affeced by he geomeical ad elecokieic paamees. I calculaio, he followig paamees ae bouded as: ioic segh 7.5 µm, fluid viscosiy. -3 kg m - s -, desiy 998 kg m -3, dielecic cosa CV - m -, ad coespodig Debye legh,. µm, D /κ, κ e ε ε k T b, which is he chaaceisic hickess of he EDL egime. Ohe fixed paamees ae: zea poeial mv, poosiy.37, ad ouosiy.5. Fo all of he case sudies, he efeece velociy is u -ε ε E ζ /µ, whee E 3 V/cm ad 9

117 ζ mv. I he followig, he esuls ude hee diffee appoaches: he complee umeical soluio of Eq. 4., he aalyical soluio based o Eq. 4.8, ad he slip velociy appoximaio based o Eq. 4.3 will be compaed. Figue 4-3a-c shows he elecokieic wall effecs fo a give paicle size d p 5 µm ude diffee chage codiios of he capillay wall. The wall zea poeials ae chose o vay fom posiive o egaive, elaive o he polaiy of he paicle zea poeial. Accodig o Eq. 4.3, i ca be eadily udesood ha whe he zea poeial aio, ζ w /ζ p, hee is o elecokieic wall effec. The velociy disibuios exhibi a uifom pofile i.e., a saigh hoizoal lie alog he coss-secioal aea. We also ca fuhe ife ha a such siuaio, his homogeeous velociy is equal o u Di. The pedicio esuls i Figue 4-3a-b show ha whe he capillay wall is moe egaively chaged ha he paicles, i.e., ζ w /ζ p >, he EOF close o he wall is ehaced. I hese cases, he EOF velociy sas o icease fom u Di o he wall o a maximum velociy, ad he deceases o u Di a he locaio aoud.98r w. The local ehaceme is due o he ieacio bewee he applied elecic field ad he excessive EDL poeial, which is limied wihi a egio of seveal Debye leghs fom he wall. I ou calculaio, he poio whee he ehaced EOF occus is limied i a aula egio wih hickess of aoud.75 µm, i.e., abou 7 imes of Debye leghs. Beyod his egime, he fluid emais uifom. This velociy ehaceme by he chael wall cao exed o he es poio of he chael because he pesece of packig paicles causes he facioal foce bewee he fluid ad he suface of packig paicles. This ed was expeimeally veified by Tallaek e al.. They used he pulsed field gadie uclea mageic esoace PFG-NMR ad NMR imagig echiques o sudy he elecoosmoic 9

118 flow i packed capillaies. I was epoed ha highe EOF close o he capillay wall is deeced whe he chael wall is moe egaively chaged ha he paicles. Fuhemoe, i ca be pediced o he basis of he pese model ha whe he capillay wall is less egaively chaged ha he paicles <ζ w /ζ p <, eual ζ w, o posiively chaged ζ w >, he EOF close o he capillay wall becomes weakeed, eve leadig o chage of flow diecio. Dimesioless Dacy Velociy u D / u R w 5 µm d p 5 µm ζ p - mv E 4 V / cm ζ w / ζ p / R w Figue 4-3 a 9

119 .4 Dimesioless Dacy Velociy u D / u.3... R w 5 µm d p 5 µm ζ p - mv E 4 V / cm ζ w / ζ p / R w Figue 4-3 b 93

120 Dimesioless Dacy Velociy u D / u - - R w 5 µm d p 5 µm ζ p - mv E 4 V / cm ζ w / ζ p / R w Figue 4-3 c Figue 4-3 Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee values of zea poeial aio, ζ w /ζ p. The esuls ae obaied o he basis of a he umeical soluio of Eq. 4.; b he aalyical soluio based o Eq. 4.8; c he slip velociy appoximaio based o Eq Refeece velociy ε ε u Eζ, whee E 3 V/cm ad ζ - mv. µ 94

121 Compaiso of Figue 4-3a-c shows ha a he cee egio of he capillay, he hee mehods give ealy he same esuls fo velociy disibuios, while adjace o he chaged chael wall he discepacy is obseved. I is demosaed ha he ehaced egime ude he aalyical soluio i.e., Eq. 4.8 o he slip velociy appoximaio i.e., Eq. 4.3 is wide ha ha ude he complee umeical soluio of Eq. 4.. The discepacy is due o he Debye-Hückel appoximaio used i deivaio of he soluios give by Eq. 4.8 ad Eq I has bee show Kag e al., ha use of he Debye-Hückel appoximaio usually ove-esimaes he EDL egio eaby he chaged wall, ad hus he affeced aea by he excessive zea poeial becomes lage. Fuhe, he diffeece bewee he esul of he slip velociy appoximaio i.e., Eq. 4.3 ad ha obaied fom eihe he umeical mehod Eq. 4. o he aalyical soluio i.e., Eq. 4.8 is esuled fom he assumpio of he movig bouday codiio. I he slip velociy appoximaio, he bouday velociy is assumed o be popoioal o he elecic field efe o Eq. 4.3b, while he zeo o-slip bouday velociies ae used i boh he umeical soluio of Eq. 4. ad he aalyical soluio based o Eq Noeheless, evaluaio of Figue 4-3a-c idicaes ha compaed o he complee umeical soluio of Eq. 4., he aalyical soluio Eq. 4.8 ude-esimaes he elecokieic wall effec while he slip velociy appoximaio Eq. 4.3 ove-esimaes he elecokieic wall effec. Figue 4-4 shows he size effec of he packig paicles o he EOF i a packed capillay. I calculaio, hee diffee sizes of spheical paicles wih diamee of, 5, ad 8 µm ae cosideed. Two ypical codiios ae peseed: wih wall effec ζ w /ζ p.5 ad o wall effec ζ w /ζ p. As discussed ealie, if hee is o wall effec, he velociy exhibis a uifom pofile alog he chael adius. I his siuaio, use of lage packig paicles ca 95

122 geeae a highe uifom velociy u Di. This is because, accodig o Eq. 4., he iapaicle poe size iceases wih iceasig he packig paicle size. Moeove, he value of G i Eq. 4.7, which deoes he aveage EDL poeial iside he iapaicle poe, deceases wih iceasig he poe size. Thus he EOF velociy iside lage packig paicles is highe. I ems of he wall effec, lage packig paicles ca geeae soge ehaceme of local EOF velociy i he viciiy of he capillay wall. This ca be explaied ha he void space bewee he paicles is lage fo lage paicle sizes. Thus he dag foce of he packig paicles should be smalle.. Dimesioless Dacy Velociy u D / u R w 5 µm ζ p - mv E 4 V / cm wih elecokieic wall effec ζ / ζ. 5 o elecokieic wall effec ζ w / ζ p w p d p 8 µm 5 µm µm / R w Figue 4-4 a 96

123 Dimesioless Dacy Velociy u D / u d p 8 µm R w 5 µm wih elecokieic wall effec ζ / ζ. 5 o elecokieic wall effec ζ w / ζ p w p ζ p - mv E 4 V / cm 5 µm µm / R w Figue 4-4 b 97

124 . Dimesioless Dacy Velociy u D / u R w 5 µm ζ p - mv E 4 V / cm d p 8 µm wih elecokieic wall effec ζ / ζ. 5 o elecokieic wall effec ζ w / ζ p 5 µm w µm p / R w Figue 4-4 c Figue 4-4 Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee sizes of packig paicles, d p. The esuls ae obaied o he basis of a he umeical soluio of Eq. 4.; b he aalyical soluio based o Eq. 4.8; c he slip velociy appoximaio based o Eq Refeece velociy is he same as i Figue

125 I Figue 4-5 he EOF velociy i chaged poous media is demosaed ude diffee applied volages. I is oed ha he velociy iceases liealy wih iceasig volage. This is due o he pesece of he local EDL of he chaged paicle suface. I he middle poio of he chael whee o elecokieic wall effec is pese, we ca ife fom Eq. 4.3 ha he local Dacy velociy should be u Di, which is popoioal o he exeal elecic field accodig o Eq Howeve close o he chael wall, boh he umeical soluio of Eq. 4. ad he aalyical soluio Eq. 4.8 show ha he EOF velociy iceases o a maximum ad he deceases o he oigial value a he wall. This is because i he EDL egio close o he wall, he excessive wall zea poeial fuhe ehaces he elecoosmosis..5 Dimesioless Dacy Velociy u D / u umeical soluio of Eq. 4. aalyical soluio based o Eq. 4.8 slip velociy appoximaio based o Eq. 4.3 R w 5 µm d p 5 µm ζ w -5 mv ζ p - mv E 6 V/cm 4 V/cm V/cm / R w Figue 4-5 Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee elecic field seghs, E. Refeece velociy is he same as i Figue

126 Ohewise we ca expec he velociy o be uifom houghou he eie coss-secio of he chael if he zea poeial of he wall is he same as ha of he paicle suface. Fo he same easo discussed above, he slip velociy appoximaio Eq. 4.3 gives diffee esuls because he movig bouday velociy is used..9 Dimesioless Dacy Velociy u D / u d p 5 µm ζ w -5 mv ζ p - mv E 4 V/cm umeical soluio of Eq. 4. aalyical soluio based o Eq. 4.8 slip velociy appoximaio based o Eq. 4.3 R w 4 µm R w 5 µm R w 6 µm / R w Figue 4-6 Elecoosmoic flow velociy disibuios i a chaged micocapillay packed wih chaged micosphees, fo diffee chael sizes, R w. Refeece velociy is he same as i Figue 4-3. The velociy disibuios fo he same packig paicles i diffee chael sizes ae show i Figue 4-6. I is clea ha he chael size dose o affec he velociy magiude i cee egio of he chael. This is expeced because he EOF velociy is oly depede o he poe size a he chael cee whee he packig codiio is homogeeous ad o

127 elecokieic wall effec is pese. Sice he size of he packig paicles is he same, he poe size is also he same, ad hece he homogeeous velociy should keep uchaged. The chael size effec due o he excessive zea poeial oly exiss i he excessive EDL egio close o he wall. I calculaio he Debye legh, D, which measues hickess of he excessive EDL poeial, keeps cosa, ad hus fo a smalle chael, he egio whee ehaced EOF is obseved exeds elaively fuhe away fom he wall. Volumeic Flow Rae Q [µl / mi] d p 8 µm ζ w / ζ p.5 ζ w / ζ p.5 3 ζ w / ζ p -.5 d p 5 µm 4 ζ w / ζ p.5 5 ζ w / ζ p.5 6 ζ w / ζ p -.5 d p µm 7 ζ w / ζ p.5 8 ζ w / ζ p.5 9 ζ w / ζ p { 4, 5, 6 7, 8, 9 R w 5 µm ζ p - mv φ.37 τ Elecic Field Segh E [ V / cm] Figue 4-7 Elecoosmoic flow ae vesus applied elecic field fo diffee values of paicle diamee, d p, ad zea poeial aio, ζ w / ζ p. Of pacical iees fo micopumps, we also sudy he effec of he chael wall ad packig paicles o he oveall elecoosmoic flow ae i a micocapillay. Flow ae vesus applied exeal elecic field, fo diffee packig paicle size ad he zea poeial aio,

128 ζ w / ζ p is ploed i Figue 4-7. I is oed ha he flow ae iceases liealy wih iceasig applied volage. Fo lage packig paicles he flow ae is always highe. This is because of he same easo ha lage packig paicles ca geeae highe velociy as show i Figue 4-4. Fo a give packig sucue, he flow ae slighly iceases whe ζ w / ζ p >, ad deceases whe ζ w / ζ p <, which is due o he elecokieic wall effec. 4.7 Summay A mahemaical model has bee developed o quaiaively descibe he elecoosmoic flow i a chaged micocapillay packed wih chaged micosphees. The model is based o he Cama-Kozey heoy, which assumes he poous medium o be equivale o a seies of paallel ouous ubules. The iesiial ubula velociy is obaied by solvig he Navie-Sokes equaio ad he Poisso-Bolzma equaio. The Bikma macoscopic momeum equaio is modified fo descibig elecoosmoic flow i a poous medium, ad i is solved usig hee diffee mehods: i he umeical mehod, ii he aalyical soluio, ad iii he slip velociy appoximaio o ake io accou he elecokieic wall effec. I is demosaed ha he ehaced egime ude he aalyical soluio i.e., Eq. 4.8 o he slip velociy appoximaio i.e., Eq. 4.3 is wide ha ha ude he complee umeical soluio of Eq. 4.. I is also foud ha compaed o he complee umeical soluio of Eq. 4., he aalyical soluio Eq. 4.8 udeesimaes he elecokieic wall effec while he slip velociy appoximaio Eq. 4.3 ove-esimaes he elecokieic wall effec. I is show ha he EOF i a chaged micocapillay packed wih chaged micopaicles is

129 affeced by seveal paamees: he popey of he wokig fluid, which deemies he hickess of he EDL; he poosiy ad ouosiy of he poous packig; he size of he capillay ad packig paicles; he chage codiio of he paicle ad capillay sufaces. The elecokieic wall effec is soge if he size of packig paicles is compaable o ha of he micocapillay. Fuhemoe, whe he capillay wall is moe egaively chaged ha he paicles ζ / >, he EOF close o he capillay wall ges ehaced; ohewise, w ζ p whe he capillay wall is less egaively chaged ha he paicles ζ / ζ <, eual < w p ζ, o posiively chaged ζ >, he EOF close o he capillay wall becomes w weakeed, eve leadig o a chage of flow diecio. w 3

130 Chape 5 AC Elecoosmoic Flow i a Micochael Packed wih Micosphees 5. Ioducio The iees o he AC elecoosmosis lies i is poeial applicaio i micofluidic devices fo mixig Oddy e al., ad sample maipulaio Mieick e al., 3. Some sudies have bee epoed o he AC elecoosmosis i o-packed micocapillaies. Howeve, o sudy has bee doe o he AC elecoosmosis i a micocapillay packed wih a poous medium. This chape seves as a aemp o ivesigae he dyamic espose of he EOF i a cylidical micocapillay packed wih micosphees o AC elecic field. Two diffee codiios egadig he opeess of chael eds ae cosideed. Whe he chael is ope, i.e., boh eds ae coeced o wo esevois ad subjec o he ambie pessue, he oscillaig elecoosmoic flow is aalyzed usig he capillay flow model as employed i chape 4. Whe he wo capillay eds ae closed, he AC elecoosmoic flow is balaced by he oscillaig coue-flow, esulig i zeo e flow ae. The coue-flow is due o he geeaed backpessue. The dyamic aspec of he iduced backpessue is aalyzed by solvig he modified Bikma momeum equaio. Coside a cylidical micocapillay desely packed wih chaged spheical micopaicles of diamee, d p, as illusaed i Figue 5-. The zea poeial of micopaicles is ζ p. The 4

131 liquid flowig hough he poous sucue is assumed o be a icompessible, Newoia, moovalece elecolye of desiy, ρ, ad viscosiy, µ. I omal applicaios fo elecoosmoic pumpig, he aio of he capillay diamee o he packig paicles diamee is lage ha 5, i.e., d c /d p 5 Zeg e al.,. Thus accodig o Tallaek e al., he eie poous medium ca be assumed o be homogeeous ad he capillay wall effecs ca be safely egleced. The AC elecoosmoic flow i boh ope-ed ad closed-ed micocapillaies will be discussed. a u De d p Flow d p b u Dp u De Figue 5- Schemaic illusaios of he AC elecoosmoic flow i a poous medium. a flow i a packed micocapillay coeced wih wo esevois. b flow i a packed micocapillay wih wo closed eds. Whee u De is he file Dacy velociy due o elecoosmosis, ad u Dp is he file Dacy velociy due o backpessue. 5

132 5. Flow i a Packed Micochael Coeced o Two Ope Resevois I his siuaio, he flow is soely elecoosmosis-dive ad o pessue gadie is pese. We use he so-called capillay model, also kow as he Cama-Kozey heoy Kaviay, 995, i which he poous medium is assumed o be equivale o a seies of paallel ouous ubules wih a effecive poe size d poe φ Rpoe d p, whee φ is 3 φ he poosiy of he poous medium. The moio of liquid hough he cylidical ubules is goveed by he Navie-Sokes equaio Pobsei, 994 u, u, ρ µ E ρ e Rpoe 5. whee u, is he asie velociy field. ρ e is he local volumeic e chage desiy due o he pesece of he EDL, ad is assumed o be goveed by he Bolzma disibuio Hue, 98, e ψ ρ e e sih Rpoe 5. kbt whee e is he elemeay chage, is he ioic coceaio i he bulk liquid phase i.e., fa fom he chaged sufaces, k b is he Bolzma cosa, T is he absolue empeaue, ad ψ is he elecosaic poeial of he EDL. Ioducig he followig dimesioless paamees: u u u s diff R poe e Ψ ψ 5.3 k T b whee he efeece ime ρ / µ is he ime scale of viscous diffusio acoss a diff R poe legh of R poe, we ca odimesiolize Eq. 5. ad Eq. 5. especively as 6

133 u u R µ poe ρe us E 5.4 ad Hee u s is he efeece velociy defied as ρ e sih[ Ψ ] 5.5 e u s ε ε µ Eζ p 5.6 whee ε is he dielecic cosa of he elecolye ad ε is he pemiiviy of vacuum. Subsiuig Eq. 5.5 io Eq. 5.4, we obai u u κ Rpoe Ψ E s sih [ Ψ ] E 5.7 whee eζ p Ψ s ad κ k T b e ε ε k T b. κ is he Debye-Hückel paamee, ad /κ deoes he chaaceisic hickess of he EDL. Eq. 5.7 is subjec o he iiial ad bouday codiios u 5.8a u u 5.8b Usig he Gee s fucio mehod as deailed i chape, we ca obai he soluio o Eq. 5.7, subjec o he homogeeous bouday codiio specified i Eq. 5.8b, as u, τ κ R poe G,, τ sih[ Ψ ξ ] E τ Ψ s E ξ ξ dξ dτ 5.9 whee G x, ; ξ, τ is he specified Gee s fucio which ca be expessed as Kag e al.,, 7

134 8 ] exp[,, J J J H G τ ξ ξ τ τ ξ 5. whee is he posiive oo of he zeo-ode Bessel fucio J, ad τ H is he Heaviside sep fucio. Subsiuig Eq. 5. io Eq. 5.9 gives Ψ ] exp[, s poe d E E J J C R u τ τ τ τ κ 5. whee [ ] Ψ sih ξ ξ ξ ξ ξ d J C. Iegaig Eq. 5. alog he adius of he cylidical ubule, we obai he o-dimesioal mea velociy i each ubule Ψ ] exp[ 4, s poe m d E E J C R d u u τ τ τ τ κ 5. Coside he applicaio of a siusoidally aleaig elecic field wih a agle fequecy ω i e E E ω 5.3 Subsiuig Eq. 5.3 io Eq. 5., we ca show ex + + Ψ + Ψ 4 4 exp si cos exp exp, s poe s poe J J C R i i J J C R REAL u β β β β κ β β κ 5.4 whee i is he ui imagiay umbe. REAL deoes he eal pa of he soluio. Hee a ew paamee β is defied as Kag e al., s R poe δ β 5.5 β epeses he aspec aio of he ubule adius R poe o he Sokes peeaio deph δ s, defied as

135 9 f s π ρ µ ρω µ δ 5.6 whee f ω / π is he fequecy of he applied elecic field. Accodigly, he mea velociy ca be obaied fom Eq. 5.4 as + + Ψ 4 4 exp si cos 4 s poe m J C R u β β β β κ 5.7 Ude he volume aveagig mehod as descibed i chape 4, he macoscopic file velociy he Dacy velociy of he fluid due o elecoosmosis ca be expessed as he mea velociy i each ubule akig io accou of he ouosiy τ ad he poosiy φ, as + + Ψ 4 4 exp si cos 4 s poe m De J C R u u β β β β κ τ φ τ φ 5.8 Combiig Eqs. 5.5, 5.6, ad 5.8, we ca fuhe obai he expessio of he fequecy-depede maximum Dacy velociy, + Ψ 4 4 max 4 s poe De J C R f u β κ τ φ 5.9 Accodig o Eqs. 5.8 ad 5.9, i shows ha he elecoosmoic Dacy velociy is popoioal o he poosiy ad ivesely popoioal o he ouosiy of a poous medium. I case of a applied DC elecic field, i.e., f, hus β, Eq. 5.8 educes o [ ] Ψ 3 exp 4 s poe De J C R u κ τ φ 5. Physically Eq. 5. deoes he ime developig of he asie elecoosmoic flow afe applicaio of a DC elecic field.

136 5.3 Flow i a Packed Micochael wih Closed Eds I a closed-ed micocapillay, he elecoosmoic flow iduces a backpessue gadie. The backpessue gadie i u geeaes a coue-flow o balace he elecoosmoic volume flow, esulig i a zeo flow ae. I poous media, o mae he flow is pessue-dive o elecoosmosis-dive, he fluid velociy is epeseed by a uifom file Dacy velociy u D. Thus o balace he volumeic elecoosmosis-dive flow, he back flow velociy, u Dp, which is dive by he backpessue, mus equal i magiude o he flow velociy, u De dive by elecoosmosis illusaed i Figue 5-b, i.e., udp ude 5. Usig he Bikma s momeum equaio Kaviay, 995 fo geeal pessue-dive flow i poous media, ad eglecig he ieial foces ad he chael wall effec, we ca obai he modified Bikma s momeum equaio fo a elecoosmoic flow ρ dudp p µ udp 5. φ d z K Hee u Dp is he pessue-dive Dacy velociy hough he poous medium packed iside he micocapillay. φ is he poosiy of he poous medium. The Dacy pemeabiliy K is defied i he Cama-Kozey equaio Kaviay, 995, 3 K φ d p / a φ, wih a 5 beig he Egu cosa Hsu ad Cheg, 99. The eaaged fom of Eq. 5. is p µ u z K s u Dp K + φ R poe du Dp d 5.3

137 Subsiue Eqs. 5.8 ad 5. io Eq. 5.3, we obai he ime-peiodic oscillaig pessue gadie Ψ exp si cos exp si cos 4 poe s poe s J C R K J C R u K z p β β β β β φ β β β β κ µ τ φ 5.4 A ime-peiodic seady sae, i.e.,, combiig Eqs , Eq. 5.4 is educed o Ψ 4 4 si cos 4 s poe s K K J C R u K z p β β µφ ρω β β β µφ ρω κ µ τ φ 5.5 Fom Eq. 5.5, we obai fequecy-depede maximum pessue gadie, expessed as + + Ψ Ψ max 4 4 s poe s s poe s K J C R u K K K J C R u K f z p β µφ ρω κ µ τ φ β µφ ρω β β µφ ρω κ µ τ φ 5.6 I case of a applied DC elecic field, i.e., f, hus β, Eq. 5.4 educes o Ψ 3 exp 4 poe s poe s R K J C R u K z p φ κ µ τ φ 5.7 Whe he fluid flow becomes seady sae, i.e.,, he pessue gadie becomes cosa

138 p φ µ u z τ K s 4 κ R Ψ poe C 3 s J 5.8 Sice he elecoosmoic flow velociy is coupled wih he EDL poeial disibuio by he faco C i Eqs , he EDL poeial field should be obaied idepedely befoe we solve he flow field. A aalyical soluio of he EDL poeial is povided i appedix B. 5.4 Resuls ad Discussio I he paameic sudy i his secio, he calculaios ae based o he values epoed by Zeg e al.. The popeies of he wokig soluio wee epoed as: ioic segh 7.5 µm, he Debye legh. µm, fluid viscosiy. -3 kg m - s -, desiy 998 kg m -3, ad dielecic cosa CV - m -. Thee diffee packig paicles of size d p,.35 µm,.7 µm, ad 5.6 µm ae cosideed i calculaios. The coespodig poe sizes, R poe ae.64 µm,.58 µm, ad. µm, especively. Ohe fixed paamees measued by expeimes ae: zea poeial 95 mv, poosiy.37, ad ouosiy.5. Seveal impoa paamees ae also highlighed befoe discussio. The chaaceisic viscous diffusio ime scale is deoed by he efeece ime ρ / µ. The ime diff R poe peiod fo exciaio elecic field is deoed by E / f. Fom he expoeial em i Eq. 5.8, we ca fuhe esimae he chaaceisic ime scale fo a AC elecoosmoic flow o * µ each is ime-peiodic seady sae by choosig. The we ca obai ρ R poe ρ R µ poe, whee.45 deemied fom J. The coespodig chaaceisic

139 µ fequecy of he sysem is defied as f ρ R poe. The faco β pese i all he aalyical expessios, defied by Eq. 5.5, deoes he aspec aio of he ubule adius R poe o he Sokes peeaio deph δ s. Ad we ca fuhe ife ha β also epeses he squae oo of he aspec aio of he viscous diffusio ime scale o he peiod of elecic field, i.e., β π /. diff E 5.4. Oscillaig flow velociy i a packed micochael coeced o wo esevois The seadily ime-peiodic AC elecoosmoic flow velociy fo wo-cycle peiods i a packed micochael wih ope eds ae show i Figue 5-. I is clealy show ha he flow peses a hamoic siusoidal oscillaio oly wih diffee magiudes ad phase lags. Figue 5-a gives a compaiso of he effecs of hee diffee exciaio fequecies fo a fixed poe size. I is obvious ha, ude he same packig codiio, usig a AC field of highe fequecy deceases he EOF velociy. This is aicipaed because ude high fequecy, he elecic field chages is diecio so fas ha he flow ca eve ge fully developed acoss he bulk iapaicle domai. I case of vey high fequecy, coespodig o a lage value of β, he diffusio ime scale is much geae ha he oscillaio ime peiod, i.e., diff > E. Theefoe hee is o sufficie ime fo he flow momeum o diffuse fa io he iapaicle bulk phase. Cosequely he peubed aea is esiced oly wihi a hi laye ea he paicle suface, while he fluid i he bulk phase emais saioay. This ypical legh scale of he oscillaoy lamia viscous flow i espose o a hamoic exeal exciaio is epeseed by he fequecy-depede Sokes 3

140 peeaio deph, δ s, as defied by Eq Fuhemoe, i ca be obseved ha as iceasig fequecy he flow lags behid he exciaio field wih a lage phase lag. Fom Eq. 5.8 ad Eqs , we ca show ha he phase lag of he oscillaig velociy o he exciaio field is expessed as β πρ f θ a a fr a poe π *, µ f idicaig ha he phase lag θ iceases wih iceasig he exciaio fequecy f. These esuls ae i accod wih he fidigs by Dua ad Beskok ad Kag e al. i hei sudies o AC elecoosmosis i o-packed micochaels. Dimesioless Dacy Velociy u' De f* f* f* Dacy velociy Elecic field κ R poe Dimesioless Elecic Field E / E π / ω π / ω 3π / ω 4π / ω Figue 5- a 4

141 Dimesioless Dacy Velociy u' De κ R poe Dacy velociy Elecic field f. f* Dimesioless Elecic Field E / E π / ω π / ω 3π / ω 4π / ω Figue 5- b 5

142 Dimesioless Dacy Velociy u' De κ R poe Dacy velociy Elecic field f f* Dimesioless Elecic Field E / E π / ω π / ω 3π / ω 4π / ω Figue 5- c Figue 5- Dimesioless ime-peiodic oscillaig Dacy elecoosmoic flow velociy vesus ime. a seadily oscillaig Dacy velociy fo hee diffee exciaio fequecies,.f*, f* ad f* a a fixed poe size, κr poe 4.8. b-c seadily oscillaig Dacy velociy fo hee diffee poe sizes,.4, 4.8 ad a a fixed exciaio fequecy, b.f* ad c f*, φ ε ε especively. Refeece velociy is chose as uef Eζ p. τ µ 6

143 The effecs of poous poe sizes ae peseed i Figue 5- b-c fo low ad high exciaio fequecies, especively. I case of low fequecy, e.g., f.f* show i Figue 5-b, diff is smalle ha E, hus he momeum diffusio is fase ha he peiod of oscillaio. The flow momeum has eough ime o diffuse fa o bulk iapaicle domai, ad hus he Dacy velociy i.e., he mea velociy aveaged ove he poe coss-secio is highe fo lage poe sizes. Howeve he phase lag becomes lage fo lage poe sizes. Physically his is because a lage poe size deoes moe ime eeded fo momeum diffusio. I also ca be π ρ iepeed by usig he mahemaical expessio fo he phase lag, θ a f R poe. µ Appaely, he phase lag iceases wih he squae of poe adius coss-secioal aea of poe. I heefoe ca be expeced ha a vey low fequecies he phase lag is popoioal o he poe coss-secioal aea. I coas, fo a high exciaio fequecy, e.g., ff* show i Figue 5-c, hee is isufficie ime fo flow momeum diffusio. Hece he Dacy velociy is lowe fo lage poe sizes. I his siuaio, he fluid ieial effec pedomiaes ad he poe size effec dimiishes. Mahemaically, whe he exciaio fequecy f appoaches o ifiiy, he phase lag θ appoaches o π. The poe size has o effec o he phase lag. Thus he phase lags fo vaious sizes ae almos he same ad close o π, as show i Figue 5-c. To achieve a bee udesadig of he fequecy-depede velociy, i Figue 5-3 we compae he magiude of he oscillaig velociy vesus fequecy fo hee diffee poe sizes. I accod wih above aalysis, fo a fixed poe size, he magiude of maximum velociy deceases wih iceasig exciaio fequecy. Wih iceasig poe size, he 7

144 magiude of he maximum velociy ges highe i case of low fequecy domai lowe ha abou.f*, ad ges lowe i case of high fequecy domai highe ha abou. f*. Dimesioless Maximum Dacy Velociy u' De max f..8 κ Rpoe κ Rpoe κ Rpoe Exciaio Fequecy f / f* Figue 5-3 Dimesioless maximum Dacy elecoosmoic flow velociy vesus exciaio fequecy fo hee diffee poe sizes,.4, 4.8 ad. Refeece velociy is he same as i Figue 5-. I Figue 5-4 we sudy he ime evoluio of he asie elecoosmoic flow fo a special case of a applied DC elecic field fo hee diffee poe sizes. I is appae ha he flow develops apidly afe imposig elecic field. I akes a ime peiod of aoud diffusio ime scale, diff fo he flow o ge fully developed io a seady sae, a which he lage poe size coespods o a lage velociy. 8

145 . Dimesioless Dacy Velociy u' De ' κ R poe κ R poe 4.8 κ R poe.4 Exciaio fequecy f Hz Dimesioless Time ' Figue 5-4 Time evoluio of he DC elecoosmoic flow velociy fo hee diffee poe sizes,.4, 4.8 ad. Refeece velociy is he same as i Figue Oscillaig backpessue i a packed micochael wih wo closed eds Figue 5-5 peses he seadily ime-peiodic oscillaio of he backpessue i a capillay of closed eds. I is show ha he backpessue associaed wih he coue-flow o balace he AC elecoosmoic flow afe applicaio of he aleaig volage also peses a hamoic oscillaio. 9

146 Dimesioless Pessue Dop P' Pessue dop f. f* Elecic field.8 f* κ Rpoe 4.8 f* π / ω π / ω 3π / ω 4π / ω Dimesioless Elecic Field E / E Figue 5-5 Dimesioless ime-peiodic oscillaig backpessue dop vesus ime fo hee diffee exciaio fequecies,.f*, f* ad f* a a fixed poe size, κr poe 4.8. Refeece µ φ ε ε pessue is chose as pef Eζ p. K τ µ κr poe 4.8 The magiude of he maximum pessue dop vesus exciaio fequecy fo hee diffee poe sizes is show i Figue 5-6. The maximum pessue dop geeaed by a small poe size coespodig o a smalle packig paicle size is always geae ha ha geeaed by a lage poe size coespodig o a lage packig paicle size, fo he eie domai of he exciaio fequecy. This agees wih he expeimeal esuls epoed by Paul e al.

147 998 i hei sudy o he geeaio of high pessues usig DC elecoosmosis i poous sucues. They epoed ha he pessue geeaed pe applied vol iceases wih packig beads of smalle diamees. I is also oed i Figue 5-6 ha, fo a fixed poe size, he backpessue deceases wih iceasig exciaio fequecy, uil i eaches is sysem chaaceisic fequecy f*, afe which he backpessue iceases wih iceasig exciaio fequecy. Dimesioless Maximum Pessue Dop P' max f.5 κ Rpoe. κ Rpoe 4.8 κ Rpoe Exciaio Fequecy f / f* Figue 5-6 Dimesioless maximum backpessue dop vesus exciaio fequecy fo hee diffee poe sizes,.4, 4.8 ad. Refeece pessue is he same as i Figue 5-5.

148 I Figue 5-7 we also sudy he special case of ime evoluio of he backpessue afe applicaio of a DC elecic field. Simila o he asie flow velociy i he ope-ed capillay, he backpessue aais a cosa value whe he ime peiod eaches o he viscous diffusio ime scale, E. A he seady sae, use of smalle packig paicles ca geeae a highe backpessue. The esuls also eveal ha hee is a suddely fall of magiude of he backpessue immediaely afe he applicaio of a elecic field. This pheomeo ca be iepeed by examiig Eq Befoe he developme of he coue-flow, he magiude of he flow velociy, u Dp is zeo, howeve, he ime deivaive of he flow velociy, du Dp / d is o zeo. As coue-flow quickly develops, du Dp / d deceases i magiude while velociy iceases. Theefoe he backpessue peses a pofile ha he magiude fis deceases ad he iceases o a cosa.

149 .5 Dimesioless Pessue Dop P' ' κ Rpoe κ Rpoe 4.8 κ Rpoe.4 Exciaio fequecy f Hz Dimesioless Time ' Figue 5-7 Time evoluio of he backpessue dop due o DC elecoosmosis i a closeded micocapillay fo hee diffee poe sizes,.4, 4.8 ad. Refeece pessue is he same as i Figue

150 5.5 Summay A aalyical model has bee developed fo he AC elecoosmoic flow i boh ope-ed ad closed-ed cylidical micocapillaies packed wih uifom spheical paicles. The ime-peiodic oscillaig elecoosmoic flow i a ope-ed capillay is obaied usig he Gee s fucio appoach. The aalysis is based o he Cama-Kozey heoy. Ad he backpessue i a closed-ed capillay is also aalyically solved usig he modified Bikma s momeum equaio. I is foud ha i a capillay wih ope eds coeced o wo esevois ad subjec o he ambie pessue, he oscillaig Dacy velociy pofile depeds o boh he poe size ad he exciaio fequecy; such effecs ae coupled hough a impoa aspec aio of he ubule poe adius o he Sokes peeaio deph. Fo a fixed poe size, he magiude of he AC elecoosmoic flow deceases wih iceasig fequecy. Wih iceasig poe size, howeve, he magiude of he maximum velociy shows wo diffee eds wih espec o he exciaio fequecy: i ges highe i low fequecy domai, ad ges lowe i high fequecy domai. Similaly, he backpessue associaed wih he coue-flow i a closed capillay wih closed eds, also peses a hamoic oscillaio. Geeally fo a fixed exciaio fequecy, use of smalle packig paicles ca geeae highe backpessue. Fo a fixed poe size, hee exis wo diffee eds of he backpessue magiude vesus he exciaio fequecy. Whe he exciaio fequecy is lowe ha he sysem chaaceisic fequecy, he backpessue deceases wih iceasig exciaio fequecy. Wheeas he exciaio fequecy is highe ha he sysem chaaceisic fequecy, he backpessue iceases wih iceasig exciaio fequecy. 4

151 Chape 6 Joule Heaig Effec o he Elecoosmoic Flow i a Micochael Packed wih Micosphees 6. Ioducio Sice he elecoosmoic flow is elecic field iduced, he Joule heaig of he coducive elecolye soluio is ieviable. As poied ou i chape, i is a majo disadvaage i he applicaio of elecoosmoic flow, especially fo lage capillay size o high elecolye coceaio. The Joule heaig effec o he elecoosmoic flow i a opacked micochael has bee sudied i he lieaue Tag, e al., 4a, b. I is also epoed ha magifice empeaue ise occus due o Joule heaig i he applicaio of elecochomaogaphy Keim ad Ladisch, 3, i which he elecoosmoic flow hough poous media is he divig mechaism fo sepaaio pupose. Kox 988 poied ou ha he Joule heaig effec ca cause peak dispesio ad hus gealy educe he sepaaio efficiecy. I addiio, a sigifica empeaue elevaio of he elecolye soluio may chage he chemical popeies o eve lead o he degadaio of he biochemical molecules Xua e al., 4a. Theefoe i is impoa o cool he empeaue wihi he micofluidic sysem. Some popula measues o educe he heaig effec iclude usig maeials wih apid hea coducio ad hemal ieia, ad usig coolig jacke aoud he micofluidic sysem. 5

152 The objecive of his chape is o umeically aalyze he asie Joule heaig effec o he elecoosmoic flow i a micocapillay packed wih micosphees. The iflueces of he elecic field, he ioic coceaio, he capillay size, he packig paicle size, ad he covecio hea asfe coefficie ouside he capillay will be discussed. The empeaue depede viscosiy, dielecic cosa, ad elecic coduciviy of he elecolye soluio ae ake io accou. Specifically, he o-uifom elecic field segh, which is due o vaiaio of empeaue-depede local elecic coduciviy, will be ake io cosideaio. As fa as he auho kows, hee is sill o published lieaue which peses such a sysemaic sudy o he Joule heaig effec of he elecoosmoic flow hough poous media. A complee umeical model will be developed which icludes he coupled equaios of he eegy, momeum, mass ad elecic cue coiuiy. The model is solved umeically usig fiie diffeece mehod. 6. Poblem Fomulaio Coside a cylidical micocapillay desely packed wih chaged micosphees of diamee, d p. The zea poeial of micopaicles is ζ p. The liquid flowig hough he poous sucue is assumed o be a icompessible, Newoia, moovalece elecolye of desiy, ρ f, ad viscosiy, µ f. I omal applicaios fo elecoosmoic pumpig, he diamee aio of he capillay o he packig paicles is lage ha 5, i.e., d c /d p 5 Zeg e al.,. Thus accodig o Tallaek e al., he eie poous medium ca be assumed o be homogeeous ad he capillay wall effecs ca be safely egleced. 6

153 6.. Equaios of eegy I he followig, we will deive he goveig equaio ha expesses he fis law of hemodyamics i elecoosmoic flow i a micocapillay packed wih micosphees. Fo modelig pupose, we divide he packed capillay io wo disic subsysems as show i Figue 6-. The fluid ad saioay phases packig paicles foms he fis subsysem, wih axial boudaies cosisig of he capillay ile ad oule. The oue adial bouday of he fis sysem is he capillay ie wall. The secod subsysem is he capillay wall iself ad is subjec o he fee covecio o he oue suface, ad hea coducio hough he solid wall. Subsysem I: Poous packig ad mobile phase d Subsysem II: Capillay wall ad coaig R w Figue 6- Schemaic illusaio of he wo modelig subsysems of he packed micocapillay. 7

154 Subsysem I homogeeous poous media We sa wih a simple siuaio i which he medium is homogeeous, ad whee adiaive effec ad hemal dispesio effec ae egligible. Vey sholy we shall assume ha hee is local hemal equilibium so ha T T f T, whee T s ad s T f ae he empeaues of he solid ad fluid phases, especively. Whe sysem dimesio L is much loge ha paicles d p L/d p >> ad whe he vaiaio of empeaue acoss d p is egligible compaed o ha acoss L fo boh he solid ad fluid phases, he we ca assume ha wihi a disace d p boh phases ae i local hemal equilibium LTE Kaviay, 995. Hee we also assume ha hea coducio i he solid ad fluid phases ake place i paallel so ha hee is o e hea asfe fom oe phase o he ohe. I a epeseaive elemeay volume REV of he medium we have, fo he solid phase, T s ρ c p s ks Ts + qs 6. ad, fo he fluid phase, T f ρ cp f + U p Tf kf Tf + µ fω+ q f 6. Hee he subscip s ad f efe o he solid ad fluid phases, especively; ρ c ad p s ρ c ae he hea capaciies of he solid ad fluid phases a cosa pessue, especively; p f k is he hemal coduciviy, ad q is he hea poducio pe ui volume. µ f is he fluid viscosiy. The em Ω efes o he viscous dissipaio fucio of he fluid, which is popoioal o he squae of he velociy. Due o he low Reyolds umbe aue of elecoosmosis i poous media, he viscous dissipaio em ca be egleced i cue sudy. U is he local poe velociy veco. p 8

155 Takig aveages ove he REV of he poous medium, we have, fo he solid phase, T + q s 6.3 s φ ρc p s φ ks Ts φ ad, fo he fluid phase, T f φ ρc p f + ρc p f U D T f φ k f T f + φq f 6.4 whee we have used he space-aveaged velociy, i.e., Dacy velociy veco is defied as U D V p V p U p dv p 6.5 whee V p is he void space which is also called poe whee he fluid phase ca pass hough of he elemeay volume, V e is he oal space of he elemeay volume. Fom he defiiio of poosiy, we ca ife V p /V e V f /V φ. Seig T T T as above discussio ad addig Eq. 6.3 ad 6.4 we have s f T ρ c p m + ρc p f U D T km T + q m 6.6 whee we defie he hemal capaciy, hemal coduciviy, ad hea geeaio of he poous medium, especively, as ρ c φ ρc + φ ρc 6.7a p m p s p f k m φ k + φk 6.7b s f q m φ q + φq 6.7c s f wih fuhe aageme ad usig q, ad assumig cosa hea capaciies, we have s T + U σ D T φq f α m T + ρc p m 6.8 whee we ioduce he capaciy aio 9

156 ρc p m σ 6.9 ρc p f ad he hemal diffusiviy of he poous medium k α m m ρc p 6. m The Joule hea geeaio i he fluid phase ca be obaied usig Ohm s law I q f 6. f whee f is he elecic coduciviy of he elecolye which is give by f T + T C+ + T C 6. Hee T ad T +.5 T 98.3 ae ioic T coduciviy of he caio ad aio of he elecolye, especively; C + ad C especively deoe he mole coceaio of caios ad aios of he elecolye. Fo sodium chloide NaCl soluio, m S / mol ad 76.3 m S / mol Tag, e al., + + 4a. I ealiy, he elecic cue desiy I should be he summaio of wo pas. Oe is due o he exeal elecic field applied o he coducive soluio E p, he ohe is due o he moveme of he e chage wihi he double laye u p ρ e, also kow as suface coducace Li,. Due o he highe elecolye coceaio i he double laye, cue desiy goig hough he double laye is sigificaly highe ha ha i Ohmic bulk liquid. Especially whe he double laye hickess exceeds % of he capillay adius, he suface coducio becomes so sog ha is coibuio o he oal cue should be cosideed Li, 4. Howeve, he o-eual double laye iside he ie-paiculae space is sigificaly hi compaed wih he poe size less ha % of he poe size i 3

157 pese sudy, ad he elecoosmoic flow velociy i he poous media is vey low abou. mm/s. The oal cue by bulk coducio is seveal odes of magiude lage ha ha by suface coducio. Thus he elecic cue due o he suface coducio ca be safely egleced. Cosideig he ouosiy effec i he poous media, he local elecic field segh is E / τ Figue 4-. Theefoe, he Joule hea geeaio is expessed as p E z Ep Ez Φ q f 6.3 τ τ z Ez Φ/ z is he axial elecic field segh, whee Φ is he applied elecic field poeial. I is assumed ha T T ad he hemal coduciviy k is cosa, Eq. 6.8 z ca be expaded as T uz T T T φ Φ + αm + + σ z z τ ρcp m z 6.4 Subsysem II capillay wall Geeally, he empeaue bouday codiio a he ie capillay wall is ukow. Sice he hea geeaed by Joule heaig i he elecolye soluio is pimaily dissipaed hough he capillay wall o he suoudig eviome, a cojugae hea asfe poblem has o be solved o simulaeously accou fo hea asfe i boh he soluio ad he capillay wall. The goveig equaio fo hea coducio i he capillay wall, i cylidical coodiaes, is T T T α w z he hemal diffusiviy of he capillay wall is defied as 3

158 k α w w ρc p 6.6 w whee k w is he hemal coduciviy of he capillay wall, ad ρ c p w is he hea capaciy of he capillay wall, all of which ae assumed as cosa. The goveig equaios ae subjec o followig bouday codiios T, z T z T z zl 6.7a T k w T Rw + d h T T 6.7b whee h is he covecio hea asfe coefficie a he oue capillay wall. R w is he ie adius of he capillay ad d is he hickess of he capillay wall. 6.. Equaios of coiuiy ad momeum The equaio of coiuiy of he fluid flow i he poous media is Beja, 995 ρ + ρu D 6.8 Sice he fluid is assumed icompessible, Eq. 6.8 educes o u u z z Sice he elecoosmoic flow is sogly coupled wih empeaue field by he empeauedepede viscosiy ad dielecic cosa, i is ca be ife ha he velociy field is o uifom if empeaue gadie is pese. I his case, a pessue field should be iduced o saisfy he mass cosevaio, i.e., cosa flow ae. Theefoe a pessue-dive flow compoe hough poous media is ioduced i pese model. Accodig o he wellkow Dacy s law which descibes he pessue-dive flow i poous media ad he 3

159 deivaio of he elecoosmoic flow iside poous media i chape 4, he Dacy velociy due o elecoosmosis ad iduced pessue ca be deemied by K U φ ε ε D ζ p GE P τ µ µ whee he elecic field segh E Φ f f ad he eassue gadie P p 6.. K is he pemeabiliy of he poous medium, as defied i chape 4. Neglecig he adial gadie of he elecic poeial field, Φ /, Eq. 6. ca be e-wie as u K µ f p 6.a φ ε ε Φ K p uz ζ p G τ µ z µ z f f 6.b whee ε is he dielecic cosa of he elecolye ad is cosideed as a fucio of empeaue, expessed by T ε T 35.7 exp Tag, e al., 4a. ε is he 9 pemiiviy of vacuum. µ f is he viscosiy of he elecolye soluio ad is vaiaio wih he empeaue is give by 6 73 µ f T.76 exp Tag, e al., 4a. ζ p is he T zea poeial a he suface of he packig paicles. G is defied as G ζ R p R poe ψ i poe d 6. Accodig o he capillay model, ψ is he iesiial EDL field iside he imagiay i ubules, ad i is goveed by Poisso-Bolzma equaio Pobsei, 994 d dψ ze d d ε ε ze i sih kbt i ψ Rpoe

160 subjec o he followig bouday codiios dψ i d ψ ζ i Rpoe p 6.4 whee he poe size is defied as d poe φ Rpoe d p φ 6..3 Coiuiy of elecic cue Sice he capillay wall is o-coducig, he cosevaio of he elecic cue desiy, eglecig he covecio cue as i above discussio, gives f Φ 6.6 Neglecig he adial gadie of he elecic poeial field, Φ/, Eq. 6.6 is educed o Φ f z z 6.7 whee he elecic poeial field Φ is subjec o Φ Φ Φ 6.8 z z L Sice he fluid elecic coduciviy f, dielecic cosa ε, ad viscosiy µ f ae empeaue depede, i is ecessay o solve he coupled eegy equaio , mass coiuiy equaio 6.9, momeum equaio 6., ad he elecic cue coiuiy equaio 6.7 simulaeously. 34

161 6.3 Numeical Mehod The complee se of coupled equaios is solved umeically usig he fiie diffeece mehod. Fo which pupose, a i-house code is developed o solve he diseized equaios. Fo his asie poblem, a each ime sep, he physical paamees ae evaluaed by he cue empeaue field iiially a oom empeaue, o deemie he elecic poeial field, iduced pessue field, ad hus he velociy field. Nex, he kow velociy ad poeial field ae used o solve he ew empeaue field. These seps ae ieaed uil he empeaue vaiaio bewee wo cosecuive ime-seps ae less ha a oleace, i.e., he empeaue eaches a seady sae. I ode o veify he code, we compaed esuls ude he pese model wih wo published woks o he Joule heaig i elecochomaogaphy. Oe is he impoved fomulae deived by Kox 988 o he seady sae empeaue excess acoss boh he poous ad he capillay wall subsysems. The ohe is he umeical ad expeimeal sudy by Keim ad Ladisch 3. I is foud ha he umeical pedicio of his sudy is i good ageeme wih he aalyical soluio by Kox 988 ad he expeimeal daa epoed by Keim ad Ladisch 3. The deails will be show i he followig secio. 6.4 Resuls ad Discussio I his secio, he joule heaig effec o he asie elecoosmoic flow i a micocapillay packed wih micosphees will be peseed. Iflueces of elecic field, elecolye soluio, chael ad packig sizes, ad hea exchage wih ambie amosphee will be discussed. I simulaio, NaCl soluio of coceaio -4 ~ -3 M is chose as he fluid phase; saioay phase is Ocyldecyl silica ODS paicles. The packig colums ae 35

162 fused silica capillaies wih polyimide coaig. Accodig o he poduc specificaios Polymico Techologies, USA, he aio of capillay adius o wall hickess is chose as 3:. Accodigly he aio of he wall hickess o he coaig laye hickess is chose as 3:. The physical popeies of he capillay ad paicles ae lised i Table.. Sice he elecokieic bouday wall effec is egleced due o lage colum-paicle aio, he poosiy ad ouosiy facos of he packed capillay ae fixed as.4 ad.5, especively. Table 6- Maeial popey i simulaio Popeies Desiy, ρ [kg m -3 ] Hea capaciy, C p [J kg - K - ] Themal coduciviy, k [W m - K - ] Mola coduciviy, [m S mol - ] Dyamic viscosiy, µ [kg m - s - ] Dielecic cosa, ε Elecolye soluio NaCl Packig paicles ODS Capillay wall Fused Silica Coaig [+.5T-98] -4.76exp73/T exp-T/ Tasie developig empeaue field ad is effec o EOF The asie ime developme of he empeaue field iduced by Joule heaig is show i Figue 6-. I is demosaed ha hee exiss a hemal eace egio whee he empeaue disibuio vaies o oly alog he capillay adius bu also alog he capillay axis. Figue 6-a is he axial disibuio of he empeaue a he capillay ceelie. The empeaue of he fluid sas o icease fom he oom empeaue 5 o C ad fially 36

163 eaches a seady empeaue a a disace alog he dowseam diecio. The legh of he disace is adiioally called hemal eace legh Beja, 995. The axial empeaue gadie is highes a he ile ad deceases o zeo a he ed of he hemal eace egio. I is oed ha he hemal eace legh fo he elecoosmoic flow i a packed micocapillay exeds abou 5 imes of chael diamee, which is excepioally loge ha he adiioal hemal eace poblem of Newoia fluid i a o-packed chael 5- D h Tag e al., 4a. The adiioal hemal eace poblem has bee well ivesigaed. I is epoed ha he hemal eace legh is popoioal o he Pecle umbe poduc of Reyolds umbe ad Padl umbe, x,.5dre P. Howeve, he hemal eace legh fo he EOF i packed chael has eve bee epoed as fa as he auhos kow. Obviously he hemal eace legh i his case does follow his simple elaioship. Sice he hemal eace legh is sigifica, i is deseved fo a fuhe sudy o iclude he eace effec whe desigig a pacical elecokieic micopump sysem. I will be show i ex secio ha he hemal eace legh will be affeced by diffee wokig paamees. Figue 6-b shows he ime developme of adial empeaue disibuios a he hemal fully-developed egio. I is demosaed ha he adial empeaue gadie iside he poous packig ad he chael wall is isigifica compaed wih he asie empeaue icease wih ime. This pheomeo is due o he low Bio umbe of he whole sysem as fd D a ough esimaio, Bi 3 O. Because he Reyolds umbe of he macoscopic elecoosmoic flow is vey small Re O, he hea geeaed i he packed egio D c is dissipaed maily by adial coducio hough he chael wall ad he by covecio wih he suoudig ai, ahe ha by axial advecio of he fluid flow. If Bi, he 37

164 esisace o hea coducio wihi he capillay sysem is much less ha he esisace o hemal covecio acoss he chael wall-suoudig ai bouday laye Icopea ad DeWi,. Hece viually he empeaue vaiaio is oly he diffeece bewee he oue wall ad he suoudig ai, wheeas he empeaue wihi he whole capillay sysem emais ealy uifom. Howeve, wih a close examiaio of he seady sae adial empeaue disibuio > 6s as i he ise of Figue 6-b, he empeaue disibuio i he poous packig shows a paabolic pofile. The highes empeaue occus a he chael ceelie. This esul cofims he fidigs by Kox 988 i a sudy o he hemal effecs i capillay elecochomaogaphy. Fuhe quaiaive compaiso bewee he cue esuls ad he Kox s fomulae will be show i he followig secio. Figue 6-a 38

165 Figue 6- b Figue 6- Tasie developme of he empeaue field: a alog he axis, b adial disibuio a dowseam. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. 39

166 The mos diec effec of he empeaue ise is ha i causes vaiaio of he fluid elecic coduciviy. The local elecic coduciviy iceases popoioally wih empeaue elevaio. Sice he oal elecic cue desiy alog he capillay is cosa, as deemied by Eq. 6.7, highe local elecic coduciviy leads o lowe local elecic field segh. Theefoe he elecic field segh becomes much highe a he eace egio whee he empeaue is ude developme, ad i becomes lowe ad cosa i he hemally developed egio, as show i Figue 6-3. Figue 6-3 Tasie developme of he elecic field segh. Wokig paamees ae ideical o hose i Figue 6-. 4

167 Accodig o Eq. 6., he EOF velociy field is coupled wih he empeaue field by he empeaue-depede dielecic cosa, ε, viscosiy, µ f, ad elecic field segh, Φ / z. The asie developme of he axial ad adial velociy disibuios is show i Figue 6-4. Alhough boh viscosiy ad he dielecic cosa decease wih iceasig empeaue Table 6-, he effec o viscosiy is much moe sigifica hus esulig i a icease i EOF velociy. Fuhe, as show i Figue 6-4a, he high local elecic field segh gealy iceases he local velociy a he eace. Figue 6-4a 4

168 Figue 6-4b Figue 6-4 Tasie developme of he elecoosmoic velociy: a axial disibuio, b adial disibuio a dowseam. Wokig paamees ae ideical o hose i Figue 6-. I Figue 6-4b, i is oed ha he EOF velociy begis o icease gadually ove he ime uil i eaches a seady sae. The ime fo his velociy developig pocess is he same as ha fo empeaue field. Acually, if hee is o Joule heaig effec, he velociy field will emai uifom i he poous domai. Fom Eq. 5., we ca ife he ime scale fo he EOF velociy i poous media o each is seady sae afe applicaio of elecic field, ρ R poe / µ O s 6, wihou cosideig Joule heaig. This ime scale is vey small 4

169 due o he small ie-paiculae poe size, which is abou mico i his case. Howeve, whe he Joule heaig effec is pese, he ime fo velociy o be fully developed becomes almos eigh imes of ode of magiude highe O s, depedig o he size of he capillay. Also he fully developed velociy iceases o abou 6% highe ha is iiial magiude. Theefoe he Joule heaig effec is sigifica ad should be ake io accou ude he codiios of high ioic coceaio, lage chael size, o high elecic field segh. Fuhemoe, he seady sae velociy disibuio i Figue 6-4b also shows a paabolic pofile wih he highes velociy i chael cee. This is because he highes empeaue ise occus a he cee. Bu he adial empeaue diffeece is so small ha he adial velociy is viually uifom i is asie developme. I is ieesig o oe ha he pese model pedics a iduced pessue field alog he flow diecio, as show i Figue 6-5. As meioed above, he vaiaio of he empeaue alog he flow diecio causes he o-uifom flow velociy. The pesece of he iduced pessue field iside he capillay is o adjus he flow velociy o saisfy he mass cosevaio i.e., a cosa flow ae. Simila o he fidig by Xua e al. 4a o he Joule heaig i o-packed capillaies, he iduced pessue gadie is posiive a he eace egio o decease he flow ae, wheeas i becomes egaive a he fully developed egio o icease he flow ae. Howeve, he velociy pofile a he eace i his sudy is i covex shape, which is coay o he cocave pofile i o-packed capillay epoed by Xua e al. 4a. This icosisecy is due o he diffee aue of he pessue-dive flow i packed ad o-packed capillaies. I o-packed capillaies, he flow pofile is paabolic Poiseuille flow. Howeve i packed capillaies, accodig o Dacy s law, he 43

170 flow pofile is fla plug flow. Theefoe he posiive pessue gadie back pessue a he eace ca geeae a cocave shape i case of a o-packed capillay. Figue 6-5 Tasie developme of he iduced pessue. Wokig paamees ae ideical o hose i Figue Compaiso wih published woks To veify he cue umeical model, case sudies compaed wih Kox s fomulae ad he umeical ad expeimeal wok by Keim ad Ladisch 3 will be show i he followig, especively. Kox 988 peseed simple fomulae o esimae he empeaue diffeece bewee he E CφDc chael axis ad is ie wall, θ coe ad empeaue excess acoss he chael 6k m 44

171 wall, θ wall E CφD c D o l. Wih simple modificaio icludig he ouosiy faco of 8kw Dc he poous packig, Kox s fomulae ca give θ φ E CD c o coe.46 C ad τ 6km θ wall φ E CD c D o l τ 8kw Dc o.57 C. The umeical model i his sudy ise of Figue 6- b gives quaiaively he same pedicios as ha by Kox s fomulae, especively. Keim ad Ladisch 3 developed a umeical model o ivesigae he empeaue pofiles i lage diamee elecochomaogaphy. I addiio hey measued he asie empeaue elevaio a he cee of he colum oule. Thei expeime daa is compaed wih he esul by umeical modelig i his sudy i Figue 6-6a. Whe he elecic field is pese, he empeaue ise i hei expeime is highe ha ha pediced by his model. Wheeas duig he coolig dow pocess elecic field is ued off afe 3 hous ad 35 miues, he empeaue dop i he expeime is fase ha ha pediced by his model. This sceaio is due o he special siuaio applied i hei expeime, i.e., he flow ae was kep cosa duig he couse of hei expeime, alhough hey did o meio how o keep a cosa flow ae. Howeve i his sudy he elecoosmoic flow velociy is sogly coupled wih he empeaue field, i.e., he flow ae may chage ove he ime. As discussed i he pevious secio, duig he heaig up sessio, he high axial empeaue gadie a he eace gealy ehaces he local elecic field segh esulig i a iceased flow field. Theefoe he iceased axial flow covecio has a bee hea dissipaio effec which keeps he empeaue ise a a elaively low level. Duig he coolig sessio, hee is o divig foce due o he absece of elecic field. Thus he flow velociy will decease uil he fluid comes o a full sop. I his siuaio, he hea dissipaio is maily depede o he 45

172 hea coducio i adial diecio. No wode ha he empeaue dop is fase i hei expeime i which he flow ae was o affeced. Figue 6-6a 46

173 Figue 6-6b Figue 6-6 Resuls compaiso wih published woks by Keim ad Ladisch 3: a expeime daa of asie empeaue ise a he cee of he colum oule, b modelig of he adial empeaue pofile a colum oule. Wokig paamees: capillay ie diamee d w 3.8 cm, capillay legh L 38. cm, packig paicle size d p 5 µm, poosiy φ.36, ouosiy τ.5, elecolye coceaio C M, zea poeial a paicle suface ζ p 6 mv, applied elecic field Φ 49 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. Figue 6-6b shows he compaiso of he adial empeaue pofile pediced by Keim ad Ladisch s umeical model ad he model developed i his sudy. I ca be see he wo models give close esuls. The majo discepacy lies a he bouday bewee he poous 47

174 packig subsysem I ad he capillay wall subsysem II whee hei model peseed a o-coiuous empeaue disibuio. This is because a empiical bouday codiio bewee he wo subsysems was used i hei umeical model. Howeve he empeaue a his bouday is ukow. I coas, he cojugaed hea asfe equaios wihi he wo subsysems ae solved simulaeously i his sudy. The ukow bouday poblem is avoided ad he empeaue disibuio is coiuous bewee he wo subsysems Effecs of wokig paamees Elecic field The Joule hea is geeaed maily by he elecic coducio cue i he elecolye soluio, ad i is popoioal o he squae of he elecic field segh Φ / z accodig o Eq Theefoe i is aicipaed ha he magiude of he elecic field will gealy ifluece he Joule heaig. I is show ha, ude he codiio specified i Figue 6-7, he empeaue a he hemally developed egio iceases o above 7 o C fo elecic field of 3 V. This empeaue ise is sigifica ad may cause some disasous poblems i applicaios of elecokieic micopump sysems. Fo isace, he opeaig empeaue i he capillay may easily icease beyod he boilig poi of he liquid phase ude vey high elecic field segh. The vapo bubble will be geeaed, which will block he flow pah of he liquid phase hough he poous packig ad fially sop he eie flow. I addiio, he hemal eace egio exeds much loge as iceasig elecic field, as show i Figue 6-7a. The elecoosmoic velociy is also deemied by he elecic field. Fom Eq. i ca be ifeed ha he EOF velociy is popoioal o he elecic field segh ad uifom iside 48

175 he capillay if hee is o Joule heaig effec. Howeve, fom Figue 6-7b, he pesece of empeaue ise chages he velociy disibuio hough he empeaue-depede viscosiy ad dielecic cosa of he fluid phase, as well as he local elecic field segh as discussed i he pevious secio. Figue 6-7a 49

176 Figue 6-7b Figue 6-7 Effec of applied elecic field: a axial empeaue disibuio, b axial elecoosmoic velociy disibuio. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. Ioic coceaio Ioic coceaio is a iisic faco which has a sog effec o he Joule hea geeaio by ifluecig he elecic coduciviy of he fluid phase. I ca be ifeed ha, fom Eqs. 6.-3, he elecic coduciviy ad hece coespodig Joule hea 5

177 geeaio ae popoioal o he ioic segh C. Fuhemoe, he zea poeial of he solid phase usually deceases wih iceasig he ioic coceaio of he elecolye Hue, 984. The effec of vayig ioic segh ad zea poeial is show i Figue 6-8a. I is appae ha use of elecolye of high ioic segh will geeae sigifica Joule heaig. Fo isace he empeaue ise i coceaed soluio of -3 M is abou 5 degee while he empeaue ise i dilue soluio of -4 M is abou degee oly. Whe he empeaue ise iceases, he hemal eace legh iceases accodigly fo soluio of highe ioic segh. Figue 6-8a 5

178 Figue 6-8b Figue 6-8 Effec of elecolye coceaio ad zea poeial a paicle suface: a axial empeaue disibuio, b axial elecoosmoic velociy disibuio. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. The lage empeaue ise has a gea impac o he elecoosmoic velociy as show i Figue 6-8b. Nomally, he EOF velociy, which is popoioal o he zea poeial ζ p wihou cosideig he Joule heaig effec, should be highe fo lowe elecolye coceaio due o he highe zea poeial. Howeve, high ioic coceaio ca 5

179 geeae magifice empeaue elevaio whe Joule heaig peses. Sice he high empeaue ise educes he fluid viscosiy damaically, he EOF velociy i moe coceaed soluio -3 M is geae ha ha i dilue soluio -4 M, which is i a opposie ed fo he case wihou Joule heaig. Covecio hea asfe coefficie As discussed above, he Joule hea is asfeed o he ambie maily by adial coducio hough he chael wall ad he by covecio wih he suoudig ai. Theefoe i is expeced ha he hea asfe codiio ouside he chael wall will affec he empeaue ad velociy field iside he capillay. Fom he esuls show i Figue 6-9a, he coolig codiio ouside wall chages fom aual covecio h 5 W/m K o foced covecio h 5 W/m K, he ceelie empeaue dops by ove 4 degee. Ad he hemal eace legh is educed fom 4 cm o cm. The velociy field i Figue 6-9b basically follows he chage of empeaue wih deceased magiude fo sog coolig codiio. This coolig effec is impoa o educe he Joule heaig ad impove he pefomace of he elecokieic pump whe usig high elecic field segh o high ioic coceaio. 53

180 Figue 6-9a 54

181 Figue 6-9b Figue 6-9 Effec of he covecio hea asfe coefficie a he capillay oue suface: a axial empeaue disibuio, b axial elecoosmoic velociy disibuio. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, ad applied elecic field Φ 3 V. 55

182 Capillay size Figue 6-a demosaes he empeaue disibuios fo hee diffee capillay sizes. I is see ha he empeaue ise is geae fo lage capillaies. This is due o he fac ha as he capillay size becomes lage he suface-aea-o-volume aio of he sysem becomes smalle, esulig i a lowe aio of he hea dissipaio measued by suface o he Joule hea geeaio measued by volume. This pheomeo is simila o he fidigs o Joule heaig of he EOF i empy chaels epoed by Tag e al. 4a ad Xua e al. 4a. Fuhemoe he hemal eace egio ad he ime eeded fo empeaue o each he seady sae become geae fo lage capillaies. Due o he low Reyolds umbe, he diffusio coducio effec pedomiaes ove he covecio effec i cue hea asfe pocess. Thus he ime scale fo he empeaue o each is seady sae ca be oughly esimaed by leig he poduc of Bio umbe ad Fouie umbe of he sysem equal o, i.e., * hr k Bi Fo k ρc R Icopea ad DeWi,. Hece he p ρc chaaceisic ime is expessed as * p R, which pedics ha he ime scale fo h empeaue developme is popoioal o he chael adius, R. I Figue 6- he ime fo he empeaue o each he seady sae ae abou 4s, 6s, ad 8s fo capillaies of diamees 3 mico, 53 mico, ad 7 mico, especively. I is well kow ha, i he absece of Joule heaig, he EOF velociy is idepede of capillay size accodig o Eq. 6.. Howeve i he pesece of Joule heaig, he fluid empeaue iceases wih iceasig capillay size as discussed above. This leads o a educed liquid viscosiy ad i u esuls i a geae EOF velociy as show i Figue 6- b. 56

183 Figue 6-a 57

184 Figue 6-b Figue 6- Effec of he capillay diamee: a axial empeaue disibuio, b axial elecoosmoic velociy disibuio. Wokig paamees: capillay legh L 5 cm, packig paicle size d p 6 µm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. 58

185 Size of packig paicles Figue 6- peses he empeaue ad velociy fields fo hee diffee packig paicles. I is see ha he ifluece of he packig paicles o he Joule heaig is isigifica. The empeaue diffeece bewee 3 mico ad 8 mico packig sysems ae almos idisiguishable. This is because of ha vaiaio of he paicle size dose o chage ay hemophysical popey of he sysem. I coas he EOF velociy i 8 mico packig sysem is slighly highe ha ha i 3 mico packig sysem. This sceaio is aibued o he fac ha he poe size of he packed capillay R poe iceases popoioally o he paicle diamee d p fom Eq. 6.5 ad he iceased poe size will elage he coecio faco -G i Eq. 6. hus esulig i a highe EOF velociy. The dimesioless faco G, as defied i Eq. 6., physically deoes he aio of he aveage EDL poeial ψ i ove he poe coss-secioal aea o he zea poeial ζ p. Fo he same zea poeial, iceasig he poe size leads o a educed aveage EDL poeial ad hece a lage coecio faco -G. Howeve he chage of he EOF velociy due o he paicle size is small because he ie-paiculae EDL poeial oly occupies vey small facio of he poe coss-secioal aea, e.g., less ha %. This small iceme of velociy will i u slighly ehace he advecio of he flow iside he poous packig, esulig i a educed empeaue fo 8 mico packig sysem as show i Figue 6-a. Bu he Reyolds umbe is so small ha he coolig effec due o he slighly ehaced advecio is idisiguishable fo diffee paicle sizes. 59

186 Figue 6-a 6

187 Figue 6-b Figue 6- Effec of he paicle size: a axial empeaue disibuio, b axial elecoosmoic velociy disibuio. Wokig paamees: capillay ie diamee d w 53 µm, capillay legh L 5 cm, poosiy φ.4, ouosiy τ.5, elecolye coceaio C -3 M, zea poeial a paicle suface ζ p 5 mv, applied elecic field Φ 3 V, ad covecio hea asfe coefficie a he capillay oue suface h 5 W/m K. 6

188 6.5 Summay A umeical model fo evaluaig he Joule heaig effec o he elecoosmoic flow i a capillay packed wih micosphees is developed i his chape. The model icopoaes he momeum equaio fo he elecoosmoic velociy deived i chape 4, he eegy equaio fo Joule heaig iduced empeaue disibuios i boh poous packig ad he capillay wall, ad he mass ad elecic cue coiuiy equaios. The empeauedepede physical popeies of he elecolye soluio, icludig he viscosiy, dielecic cosa, ad he elecic coduciviy, ae ake io cosideaio. The coupled goveig equaios ae umeically solved by he fiie diffeece mehod. The simulaio pedics ha, i he pesece of Joule heaig, hee exiss a sigifica axial empeaue gadie i he hemal eace egio. This high empeaue gadie sogly ehaces he local elecic field a he eace, esulig i a o-uifom disibuio alog he flow diecio. The adial empeaue pofile shows a paabolic disibuio bu he gadie is vey small due o he small sysem Bio umbe. I is foud ha he Joule heaig effec is sigificaly sog fo high elecic field segh, high ioic coceaio of he soluio, lage capillay size, ad weak coolig codiio ouside he capillay. All hose facos have a gea impac o he Joule heaig by deemiig magiude of empeaue ise ad he hemal eace legh. The capillay size ad he hea asfe coefficie ouside he capillay deemie he ime scale fo asie developme of he empeaue field. The o-uifom empeaue disibuio i u gealy affecs he EOF velociy by meas of chagig he local viscosiy ad he dielecic cosa of he fluid phase, ad he local elecic field segh. The esuls by his model ae compaed wih available aalyical ad expeimeal woks i he lieaue. Ad he 6

189 empeaue diffeece bewee he capillay ceelie ad he ie wall as well as he empeaue excess acoss he capillay wall is accuaely i a ageeme wih he esuls pediced by Kox s fomulae. The discepacy of he asie empeaue ise wih Keim ad Ladisch s expeime daa is aalyzed. 63

190 Chape 7 Expeimeal Sudies of he Elecoosmoic Flow i Micocapillaies Packed wih Micosphees 7. Ioducio I his chape, a effo o fabicae ad chaaceize a pooype of he elecokieic pump is made. The eie expeime coais wo mai sages: colum fabicaio ad flow measueme. The fabicaed micopump is chaaceized ude he iflueces of diffee wokig paamees, such as he size of he capillay ad he packig paicles, he wokig elecolye ad is ioic segh, he capillay legh ec. Ad he expeimeal esuls will be compaed wih he heoeical model developed i chape Colum Fabicaio The packed capillaies used i cue expeime ae fabicaed by usig high pessue sluy packig echique Zeg e al., as illusaed i Figue 7-. Sluy pessue packig is he mos widely used mehod o pack capillay colums fo Capillay Eleco-Chomaogaphy CEC Maloey,. I his expeime, he upacked capillay is coeced o a i-house desiged acylic sluy esevoi, ad he sluy esevoi is coeced o a High Pefomace Liquid Chomaogaphy HPLC pump. Duig he packig pocess, deioised DI wae is coiuously pumped ude high pessue io he sluy esevoi o dive he paicle sluy io he silica capillay uil 64

191 he eie capillay is filled wih packig paicles. I he followig evey sep i he packig pocess will be peseed i deails. Sluy esevoi Fused silica capillay HPLC pump Ulasoic bah DI wae Wase fluid Figue 7- Schemaic illusaio of he expeimeal seup fo colum packig. 7.. Pepaaio of he maeials Colum pepaaio Fused silica capillaies Polymico Techologies Ic. of ie diamee 3 µm, 53 µm, 7 µm ae used as empy colum. The capillaies ae cu io pieces of a desied legh e.g., 5 cm log fom he capillay oll caefully ad ised wih aceoe ad DI wae. Afe ha he cleaed capillaies should be died ad kep i a dy place. This is because he bae fused silica maeial wihou polyimide coaig a boh eds will become fagile ad easy o be boke i he mois ai. Sluy composiio The packig maeial Iesil C8 ocyldecyl silica ODS paicles HPLC gade, GL Scieces Ic. ca povide he fucioal silaol goups which poduce a egaive suface chage whe i coac wih elecolye soluio. This suface chage is ecessay o geeae he elecoosmoic flow. The behavio of he packig maeial depeds o he ype of he solve ad he desiy of he packig maeial i he sluy solve. Fo he Iesil ODS paicles used i cue expeime, i is foud ha Aceoe o mehaol alcohols ae suiable solve fo bee mixig. The coceaio of 65

192 he paicle is depede o he desied legh of packig. The loge he packig legh, he highe he paicle coceaio. Fo packig legh of abou 5 cm i cue expeime, 5-8 mg of paicle mixed wih l ml of mehaol is ecommeded. Fo homogeeous mixig, he sluy is pu i a ulasoic bah pio use. Sluy esevoi A sluy esevoi of ie diamee 3.9 mm ad legh 7 cm is fabicaed usig acylic Pespex. The sluy esevoi is subjec o a sigificaly high pessue i he sluy packig pocess, which equies he esevoi maeial be sog. Acylic Pespex is good choice because i ca wihsad high pessue ad also ca be easily machied io desied size ad shape. 7.. Packig pocedue Reaiig fi A eaiig fi mus be fabicaed a oe ed of he empy colum befoe colum packig. The fucio of he eaiig fi is o hold he packig maeial ad esis he high pessue due o packig, flushig, o EOF. So i mus be mechaically sog. Meawhile he fi has o be highly pemeable fo he solve flow; ohewise he EOF geeaed will be educed a he oule. The mos commo appoach is he sieig mehod, i which he fi is cosuced by sieig a plug of silica gel weed wih sodium silicae soluio. Fi fabicaed i his way ca povide he bes mechaical sabiliy. The ODS paicles ae mixed wih sodium silicae soluio 4% NaOH soluio, Sigma-Aldich wih a aio of 3: by weigh. Oly he gel o pase like subsaces i he mixig should be used. A gel plug of mm log is packed io oe ed of he empy colum by appig as i figue 7-a. The he capillaies wih he gel plug ae heaed i a ove a 35 o C fo abou 3 miues. This sieig pocess hades he pase ad allows he silica-sodium silicae mixig o adhee sogly o he capillay wall. Thus he eaiig fi is fomed. 66

193 a Reaiig fi b Paicle packig c Ile fi Figue 7- Schemaic illusaio of he packig pocedue. Sluy packig Oce he eaiig fi is eady, he ohe ed of he capillay was coeced o a pessue-dive sluy packig sysem, which icludes a sluy esevoi ad a high-pessue liquid chomaogaphy HPLC pump PU-8, Jasco. PEEK fiig ad ubig Upchuch Scieific ae used fo all coecios. The paiclemehaol sluy is soicaed fo abou miues befoe beig loaded io he sluy esevoi. DI wae is used as wokig fluid o dive he sluy io he capillay. The HPLC pump povides a cosa pessue of 6 MPa o foce he paicles ad he fluids io he capillay. The paicles will accumulae begiig fom he eaiig fi as illusaed i figue 7-b, while he fluids pass hough he iepaiculae space ad he fi io he wase esevoi. Due o he paabolic pofile of he pessue dive flow, he paicles ed o accumulae a he cee of he capillay ad he paicle close o he capillay wall will be loosely disibued. To educe his effec, he packed capillay is kep i a ulasoic bah duig he eie packig pocess. Aoud 5- miues, he capillay is fully packed wih ODS paicle. DI wae is sill pumped io he packed 67

194 capillay sill i ulasoic bah a 5 MPa fo 3 miues o esue he paicles ae desely ad homogeeously packed. Ile fi Afe he capillay is fimly packed ad ised, a ile fi should be cosuced o hold he ohe ed of he packig. The sieig mehod of he ile fi is diffee fom ha fo he eaiig fi which eeds a log ime i high empeaue. The ile fi does eed o be as mechaically sog as he eaiig fi because i is o subjec o high pessue. Theefoe a local sieig mehod usig a ho coppe wie coeced wih a soldeig io is used o hea he paicles a capillay ile as illusaed i figue 7-c. This mehod ca peve he apid evapoaio of wae iside he packig. Afe sieig fo 5- miues, he ile he fi is cosuced. Till ow he micopump usig packed capillay is fiished, ad i is eady fo chaaceizaio. 7.3 Chaaceizaio ad Flow Measueme 7.3. Poosiy ad ouosiy Poosiy ad ouosiy ae wo impoa geomeic facos chaaceizig of he packed EOF micopump. Poosiy ca be diecly measued while ouosiy faco ca oly be measued idiecly. Poosiy Accodig o he defiiio, he poosiy deoes he aio of he void space o he oal space of he poous packig. Oe mehod o calculae he void space is by measuig he weigh of he fluid iside he fully sauaed poous packig. I cue expeime, he poosiy is calculae by φ V V weigh we weighdy void 7. oal whee weigh we ad weigh dy deoe he we weigh sauaed wih DI wae ad dy weigh of he micopump, especively. ρ is he desiy of he DI wae. A is he ie AL ρ 68

195 coss-secioal aea of he pump ad L is he physical legh of he pump. I should be oed ha he weigh of he pump, we o dy, is coeced by excludig he weigh of he fi weighfi, we weighfied colum, we weigh ufied colum, we 7.a weighfi, dy weighfied colum, dy weighufied colum, dy 7.b Touosiy As discussed i chape 4, he ouosiy faco, defied by τ /cos θ L e / L, deoes he o-aligme of mos flow chaels wih he elecic field, which is applied alog he capillay axis. I was suggesed by Rahoe e al. 999 ha he ouosiy of he poous packig ca be evaluaed by measuig he esisaces of he packed ad upacked capillaies. The effecive coss-secioal aea of he ouous ubules is he as illusaed i Fig. 4-b-c e cosθ φ τ poe 7.3 A A A The esisaces of he empy ad packed capillaies ae ςl/a ad ςl e /A e, especively, whee ς is he esisiviy of he wokig fluid. Thus he aio of he esisaces gives Rempy L/ A R ς φ ς L / A τ 7.4 packed e e Thus usig he poosiy value measued i pevious sep, we ca calculae he value of he ouosiy by measuig he esisaces of he empy ad packed capillaies Tes colum mehod Oe diec mehod o measue he elecoosmoic flow velociy is by obsevig he moveme of he fluid-ai ieface i a es colum coeced a he oule of he micopump. Figue 7-3 demosaes a seup of he es colum mehod. Plaium wie is used as elecode ad coeced wih a high-volage powe supply Safod Reseach Sysems Ic.. To peve he bubble geeaed by elecolysis fom eeig he 69

196 micopump, he elecode coeced o he cahode is made io mesh-like shape, ad kep i a ope soluio esevoi o le he bubble escape o he ambie. The micopump is pimed wih he wokig fluid wih a HPLC pump fo abou 5 miues befoe each u. The es colum should also be pimed wih he wokig fluid ad a ai bubble is specially apped iside he es colum o peve he eea of he upseam fluid-ai ieface due o evapoaio as illusaed i Figue 7-3. The aveage volumeic elecoosmoic flow ae Q av ca be calculaed by measuig he ime spe fo he fluidai ieface o avel a defied disace l iside he es colum of ie diamee D es, ie., Q av l π Des ad he aveage velociy iside he micopump is deemied by U av Q l D A D av es 7.6 whee A is he ie coss-secioal aea of he packed capillay, D is he ie diamee of he packed capillay. _ High volage powe supply + Q av l π D 4 es Tes colum Aode Packed capillay Cahode Ai-fluid ieface Soluio esevoi Figue 7-3 Schemaic illusaio of he expeimeal seup usig he es colum mehod. 7

197 7.3.3 Cue moioig mehod A aleaive mehod is also used o deemie elecoosmoic flow velociy by moioig he elecic cue chage whe a elecolye soluio displaces he same bu slighly moe coceaed soluio o vice vesa iside he packed capillay Aulaadam ad Li, b, ad Sze e al., 3. Decease i he soluio coceaio will icease he elecic esisace of he micopump, esulig i a educed elecic cue. As he soluio of highe coceaio iside he packed capillay is compleely eplaced by ha of lowe coceaio due o he elecoosmoic flow, he cue will sop doppig ad each a cosa value. The ime ake fo he cue o each his cosa value is egaded as he ime fo he elecoosmoic flow o pass a disace of he capillay legh L. Theefoe he aveage elecoosmoic flow velociy ca be calculaed as Uav L 7.7 A ypical esul fo he cue-ime elaioship is show i Figue 7-4. I his case, he iiial cue upo applicaio of a elecic field a V/cm is aoud 8 ma. As he ime goes by, he cue dops uil i eaches a cosa value a 3.5 ma. The ime fo he cue chage is abou 4 secods. Theefoe, he esimaed aveage velociy of he elecoosmoic flow hough a packed capillay of 5 mm log is calculaed as 5/4.8 mm/s. 7

198 Figue 7-4 A ypical elaioship of cue-ime usig cue moioig mehod. The seup fo he cue moioig mehod is show i Figue 7-5. Boh eds of he packed capillay ae coeced wih he elecodes ad kep i wo ope elecolye esevois. The ioic segh of he soluio i he upseam ope esevoi is coolled a 8% of ha of he moe coceaed soluio i he dowseam ope esevoi. Befoe beig coeced wih he esevois, he packed capillay is pimed wih elecolye of highe coceaio. The high-volage powe supply ad he cue daa acquisiio ae coolled by usig Labview sofwae Naioal Isume Ic.. 7

199 _ High volage powe supply + Elecic cue mee A Packed capillay Soluio esevoi highe coceaio Soluio esevoi lowe coceaio Figue 7-5 Schemaic illusaio of he expeimeal seup usig he cue moioig mehod EOF velociy ad zea poeial Based o he discussio i chape 4, if igoig he bouday chael wall effecs, he elecoosmoic flow velociy ca be esimaed, fom Eq. 4.7, as, ε ε φ Uav E ζ p G 7.8 µ τ whee G ζ R p R poe ψ i poe d 7.9 If he Debye-Hückel liea appoximaio is used o solve he Poisso-Bolzma equaio, he EDL poeial field iside he iepaiculae ubules ca be solved, fom Eq. B. as ψ i I κ ζ p 7. I κ Rpoe Subsiuig eq. 7.9 ad Eq. 7. io Eq. 7.8, we ca obai 73

200 U φ εε I κ Rpoe E ζ p τ µ κrpoe I κrpoe av 7. whee he poe size R poe ca be deemied as R poe φ d 3 φ p Chape 4. Hece fom he poosiy, ouosiy, ad aveage EOF velociy measued i his sudy, he zea poeial a he paicle suface ca be evaluaed by Eq Resuls ad Eo Aalysis 7.4. Scaig eleco micoscopy Scaig eleco micogaph SEM of a fabicaed eaiig fi is illusaed i Figue 7-6. The ie diamee of he fused silica capillay is 7 µm i ie diamee. I ca be see fom Figue 7-6a ha he solid maix fomed by died sodium silicae sogly adhees he ie suface of he capillay wall ad he paicles. The micofilames of solid sodium silicae povide void space which is small eough o eap he packig paicles bu lage eough o le he fluid pass hough, as show i Figue 7-6b. I his way, he fi ca hold he poous packig iside he capillay wihou educig he pemeabiliy of he micopump. I fac, DI wae was flushed hough a upacked capillay wih a eaiig fi usig a HPLC pump. The backpessue ecoded is slighly geae less ha.3 MPa ha ha eeded o flush a upacked capillay wihou fi, idicaig ha he fi does o sigificaly icease he flow esisace of he sysem. I addiio, he fi fabicaed usig his mehod ca wihsad high packig pessue up o 5 MPa i his expeime, showig a good mechaical sabiliy. 74

201 a b Figue 7-6 Scaig eleco micoscopic image of he eaiig fi. a oveall view of he fi i fused silica capillay of ID 7 µm, OD 85 µm. b a magified poio of he fi. 75

202 a b Figue 7-7 Scaig eleco micoscopic image of he middle coss-secio of he micocapillay. a oveall view of he coss-secio i fused silica capillay of ID 7 µm OD 85 µm. b a magified poio of he packed beds wih ODS paicles of 6 µm. 76