Numerical study of dc-biased ac-electrokinetic flow over symmetrical electrodes

Nume ical s udy o dc-biased ac-elec okine ic low o e symme ical elec odes Wee Yang Ng, 1,2,a) An onio Ramos, 3,b) Yee Cheong Lam, 2 and Isabel Rod iguez 1 1 Ins i u e o Ma e ials Resea ch and Enginee ing, Agency o Science, Technology and Resea ch (A*STAR), 3 Resea ch Link, Singapo e 117602, Singapo e 2 School o Mechanical and Ae ospace Enginee ing, Nanyang Technological Uni e si y, Nanyang A enue 50, Singapo e 639798, Singapo e 3 Depa amen o de Elec o´nica y Elec omagne ismo, Facul ad de Fı´sica, Uni e sidad de Se illa, A enida Reina Me cedes s/n, 41012 Se illa, Spain (Recei ed 6 Augus 2011; accep ed 20 No embe 2011; published online 15 Ma ch 2012) This pape p esen s a nume ical s udy o DC-biased AC-elec okine ic (DC-biased ACEK) low o e a pai o symme ical elec odes. The low mechanism is based on a ans e se conduc i i y g adien c ea ed h ough incipien Fa adaic eac ions occu ing a he elec odes when a DC-bias is applied. The DC biased AC elec ic ield ac ing on his g adien gene a es a luid low in he o m o o exes. To unde s and mo e in dep h he DC-biased ACEK low mechanism, a phenomenological model is de eloped o s udy he e ec s o ol age, conduc i i y a io, channel wid h, dep h, and aspec a io on he induced low cha ac e is ics. I was ound ha low eloci y on he o de o mm/s can be p oduced a highe ol - age and conduc i i y a io. Such apid low eloci y is one o he highes epo ed in mic osys ems echnology using elec okine ics. V C2012 Ame ican Ins i u e o Physics. [doi:10.1063/1.3668262] I. INTRODUCTION Dynamic manipula ion o luids in mic ochannel is c ucial o he de elopmen o lab-on-a- chip (LOC) de ices. 1,2 One key equi emen o he ac ua ion sys em in LOC de ices is o a oid mechanical mo ing pa s which can po en ially complica e he ab ica ion and inc ease he size o he sys em. 3,4 Elec okine ics has been demons a ed as a iable app oach o e ec- i ely manipula e luids in mic ochannels h ough he use o an elec ical d i ing o ce. 5,6 One key ad an age lies in i s ease o implemen a ion which makes elec oosmosis (i.e., he use o DC elec ic ield o d i e luid low, due o he Coulomb o ce on he di use double laye ) one o he mos widely used pumping app oaches in mic o luidics in addi ion o p essu e-d i en me hods. 7,8 AC-elec okine ic (ACEK) makes use o AC elec ic ields ac ing on AC cha ge densi y o induce ec i ied low; wo impo an examples o ACEK a e AC-elec oosmo ic (ACEO) and AC-elec o he mal (ACET) lows. 9–16 ACEO low is based on he Coulomb o ce ac ing on he induced cha ge in he double laye in he p esence o a angen ial AC elec ic ield. 10–13 ACET low is based on he in e ac ion o AC elec ic ields wi h conduc i i y g adien s in he luid induced by a he mal g adien . 14–16 Mo e ecen ly, a DC-bias AC ol age (V applied ¼V DC þV AC cos x ) has been employed e ec i ely o concen a e pa icles/cells 17–19 and o pumping/mixing 20–23 applica ions in mic o- luidics. The e has been se e al hypo hesis on he d i ing mechanism o he low induced by DC-bias AC ol age. 6,18,19 Recen ly, we conduc ed expe imen al in es iga ion o elucida e he d i ing mechanism o he DC-biased ACEK low phenomenon. 21,24 In he publica ions, we a) Elec onic mail: [email p o ec ed]. b) Elec onic mail: [email p o ec ed]. 1932-1058/2012/6(1)/012817/10/$30.00 V C2012 Ame ican Ins i u e o Physics6, 012817-1 BIOMICROFLUIDICS 6, 012817 (2012) demons a ed ha he applied DC-bias ol age gi es ise o incipien Fa adaic eac ions a he elec odes’ su ace. 21,24 Indeed, i was es ed ha hese eac ions cause changes on he local pH due o he gene a ion o co-ions abo e he elec odes. As a esul , he egion abo e he anode became acidic due o he inc ease o H þ ions, and he egion abo e he ca hode was basic due o he inc ease o OH – ions. In addi ion, i was measu ed ha he basic egion is 11%–12% mo e conduc i e han he acidic egion. 24 Hi he o, he e is no exis ing heo e ical model o desc ibe he DC-biased ACEK low. In his pape , we p esen a phenomenological model based on he ac ual conduc i i y di e ence be ween he anode and he ca hode o gi e a desc ip ion o he luid low. I should be no ed ha he e ec o DC-bias could no be included by simply adding a DC-bias o he AC ol age in es ablished ACEO model. 12,13 This is because we do no know wi h ce ain y all he elec o- chemical eac ions aking place a he elec odes’ su ace. The e o e, we ake he alue o he inc emen o conduc i i y om expe imen s. 24 Mo eo e , since he DC ol age is jus enough o gene a e he species, he applied DC ol age is d opped ac oss he double laye and no in he bulk. A con ec ion-di usion equa ion is used he e o desc ibe he dis ibu ion o conduc i - i y. The in e ac ion o he elec ic ield wi h he conduc i i y g adien leads o luid low. This pape will demons a e ha he de eloped model could p edic well he expe imen al obse a- ions 23,24 namely he single and dominan low o ex and he di ec ion o o ex o a ion. I is expe imen ally di icul o cha ac e ize he c oss-sec ional luid eloci y p o iles; he e o e a nu- me ical simula ion se es as a ool o be e unde s and he a ious ac o s a ec ing he induced low eloci y. Pa ame ic ac o s ( ol age, conduc i i y a io) and geome ical ac o s (wid h, dep h, and aspec a io) we e a ied in his s udy o unde s and hei e ec s on he induced DC-biased ACEK low eloci y. II. NUMERICAL ANALYSIS DC-biased ACEK low phenomenon was modeled using COMSOL MULTIPHYSICS V R . A phenom- enological model was p esen ed he e o gi e a desc ip ion o he DC-biased AC-elec okine ic luid low. The nume ical model is based on he measu ed conduc i i y di e ence be ween he anode and he ca hode and was used as he basis o he bounda y condi ions o he conduc i - i y equa ion. I was de e mined ha he liquid abo e he ca hode is mo e conduc i e han he one on he anode side. AC impedance analysis was adop ed o measu e he conduc i i y le el abo e he ca hode and he anode. I was ound ha wi h a DC bias o 2 V DC , he ca hode is mo e conduc i e han he anode by 11%–12%. 24 We assume ha he applied DC ol age is jus abo e he h eshold o gene a e Fa adaic eac ions a he elec odes. In his way, V DC is d opped a he elec ode/elec oly e in e ace o d i e he eac ions and negligible DC ol age is d opped ac oss he bulk elec oly e. No ice ha he minimum ol age o p oduce elec olysis o wa e is abou 1.23 V. The e o e, we can neglec V DC in he medium and sol e only o he AC po en ial (V applied ¼V AC cos x ). Mo e- o e , since he equency o he AC signal is much g ea e han he ecip ocal RC ime o cha ging he double laye , 10 he AC ol age is d opped mainly ac oss he bulk elec oly e and negligible AC ol age is d opped a he elec ode/elec oly e double laye s. In addi ion, he e ec o joule hea ing is igno ed since he conduc i i y used is low. In his s udy, a 2D analysis was pe o med o a pai o coplana and symme ical elec o- des enclosed by a mic ochannel as shown in Fig. 1. To limi he scope in his s udy, he elec- ode wid h and gap we e ixed a 40 lm and 20 lm, espec i ely, bu he mic ochannel wid h (W) and dep h (D) we e a ied om 100 o 400 lm and 20 o 200 lm, espec i ely. A. Elec ical equa ions Fo an elec oly e, he elec ic po en ial (/) dis ibu ion a any poin in he solu ion can be ela ed o he elec ic cha ge densi y (qe) which is desc ibed by he Poisson equa ion ðe~ EÞ¼qe;(1) 012817-2 Ng e al. Biomic o luidics 6, 012817 (2012) whe e eis he pe mi i i y and ~ E¼ /is he elec ic ield. The mo emen o he ions in he solu ion will cons i u e o an elec ical cu en ~ J¼Xqinili~ EqiDi niþqini~ V  ;(2) whe e qi(¼zie, wi h zi he ionic alence and e he elemen a y cha ge) is he ion cha ge, niis he ion numbe densi y, li(¼zieD i=kT) is he mobili y, Diis he di usion coe icien , and ~ Vis he eloci y. The i s , second, and hi d e ms ep esen , espec i ely, elec omig a ion (i.e., conduc ion), di usion, and con ec ion. Typically o elec oly es, he con ec i e cu en qini~ V is usually smalle han he conduc ion cu en qiniliEwhich is go e ned by he elec ical Reyn- olds numbe . 25 In mic osys ems, he elec ical Reynolds numbe is e y small o elec oly e. Fo semi-insula ing liquids, a conduc i i y smalle han 10 9 S/m would be equi ed in o de o ha e a con ec i e cu en o he same o de han he conduc ion cu en . In addi ion, he a io o di usi e cu en o elec omig a ion Di ni jj=jnili~ Ejis smalle han uni y in he liquid bulk. 25 Typically, o ol age d op in he bulk much g ea e han 0.025 V, di usion is negligible as compa ed o elec omig a ion. The a io be ween hese wo cu en s is on he o de o 0.025/V. In ou expe imen s, we apply a ound 10 V 0.025 V. In his si ua ion, we can w i e ~ J¼ ~ E, whe e ¼Pqiniliis he elec oly e conduc i i y. Wi h hese app oxima ions, he cha ge conse a ion equa ion o he AC po en ial becomes 25–27 ðð þiexÞ UÞ¼0;(3) whe e Uis he phaso o he elec ical po en ial. Conside a bina y elec oly e (ni¼1), assum- ing elec oneu ali y, 16 he solu ion conduc i i y can be ep esen ed in he model wi h a con ec ion–di usion equa ion 27–29 @ @ D 2 þð~ V Þ ¼0;(4) whe e he second and hi d e ms ep esen , espec i ely, di usion and con ec ion. B. Mechanical equa ions In a mic oscale sys em, Reynolds numbe is usually small (<1) and low is lamina in na- u e. The luid eloci y is go e ned by he Na ie -S okes equa ions, o s eady-s a e solu ions, we ha e q ð~ V Þ~ V¼ Pþg 2~ Vþ~ Fe;(5) FIG. 1. Geome ical con igu a ion and bounda y condi ions o he simula ed DC-biased AC-elec okine ics p oblem. Bounda y condi ions o igh elec ode (conduc i i y: Righ , elec ical: V GND , low: no slip), le elec ode (conduc i i y: Le , elec ical: V AC , low: no slip), and channel walls (conduc i i y: insula ion=symme y, elec ical: elec ical insula ion low: no slip). 012817-3 dc-biased ac-elec okine ic low Biomic o luidics 6, 012817 (2012) ~ V¼0;(6) whe e q is he luid densi y, Pis he p essu e, gis he dynamic iscosi y, and ~ Feis he elec i- cal body o ce. The elec ical body o ce on he luid is gi en by he ime-a e age Coulomb o ce 25–27 ~ Fe¼qe~ E  ¼1 2Re½qe~ E;(7) whe e qeis he cha ge densi y phaso , Re[...] ep esen s he eal pa , ~ Eis he complex conju- ga e o he elec ic ield phaso . In his case, he ime-a e age Coulomb o ce is used in he cal- cula ion. This is because he ac signals ha e pe iods much sho e han he ypical mechanical ime o a ia ion o he sys em. Despi e he elec oly e being quasi-elec oneu al, he esidual cha ge can lead o a signi ican o ce. The cha ge densi y (qe) in he bulk can be ob ained om Eqs. (1) and (3) leading o a cha ge densi y phaso 26,27 qe¼e þiex  U:(8) C. Nume ical me hod and se ings To sol e o he DC-biased ACEK low, “con ec ion and di usion,” “conduc i e media DC,” and “incomp essible Na ie -S okes” modules in he COMSOL so wa e we e employed. COMSOL so wa e employs he ini e elemen me hod (FEM) in sol ing he pa ial di e en ial equa ions (PDEs). The nume ical scheme disc e izes he de ined spa ial domain in o iangula ini e elemen s o app oxima e he solu ions o he PDEs. Typical compu a ion consis s o app oxima ely 1300 iangula elemen s and he minimum elemen quali y is 0.86. Fu he in es iga ion on he nume ical con e gence was conduc ed. Wi h mesh e inemen up o 5000 o e en 20 000 iangula elemen s, he compu ed eloci y does no change app eciably (less han 1%). In he module o “con ec ion and di usion,” cons an conduc i i ies ( 1and 2) we e se on he wo elec odes and insula ion/symme y on he channel walls. In he module o “conduc i e media DC,” a ious po en ials we e se on he wo elec odes and elec ical insula- ion was applied on he channel walls. In he module o “incomp essible Na ie -S okes,” no slip bounda y condi ions we e se o he elec odes and he channel walls. In he sol ing p ocess, an i e a i e app oach was adop ed whe e he p oblem was sol ed in h ee s eps: Fi s , he con ec ion-di usion equa ion was sol ed o he conduc i i y ield, Eq. (4). Nex , he compu ed conduc i i y solu ion was used o sol e o he elec ical po en ial ield, Eq. (3). A pa ame ic sol e was adop ed o se he ol age whe e he ol age was inc eased om ze o o he se ol age le el a a ol age s ep o 0.01 V. Las ly, he compu ed po en ial solu ion was used o sol e o he elec okine ic low eloci y ield, Eqs. (5) and (6). The sol ing p ocess was epea ed un il we ob ained a con e ged solu ion whe e he ela i e ol- e ance is se a 10 3 . The pa ame e s used in he simula ion a e lis ed in Table I. III. RESULTS AND DISCUSSIONS A. DC-biased AC-elec okine ic low Figu e 2shows he simula ed esul s o he eloci y dis ibu ion in he 2D c oss-sec ion model (W ¼100 lm, D ¼50 lm, elec ode wid h ¼40 lm, and elec ode gap ¼20 lm) o bo h nega i e and posi i e DC-biased AC-elec okine ic low. The ol age ampli ude used was 10 V and he conduc i i y le el a he wo elec odes was se wi h a 10% di e ence. In bo h cases, a single and dominan low o ex was ob ained in he di ec ion om highe conduc i i y (basic, ca hode) o lowe conduc i i y (acidic, anode). As explained, due o Fa adaic eac ions, a con- duc i i y g adien is gene a ed om he basic (highe conduc i i y, ne nega i e elec ode) o 012817-4 Ng e al. Biomic o luidics 6, 012817 (2012) he acidic (lowe conduc i i y, ne posi i e elec ode) domains. When an AC signal is applied, a ho izon al o ce componen is gene a ed close o he elec ode su ace whe e he ield is s onges . As such, he luid is ac ua ed om he highe conduc i i y o he lowe conduc i i y domains close o he su ace o he elec odes; his de e mines he di ec ion o o a ion and e en ually esul s in a single and dominan o ex o e he elec odes pai . 24 Fo a nega i e DC-bias AC ol age case (see Fig. 2(a)), he luid lows om igh o le , i.e., om highe conduc i i y OH – ions ich ca hode o lowe conduc i i y H þ ions ich anode, a egion nea he elec odes’ su ace. This low p oduces a esul an clockwise (CW) o ex o a ion. Fo a posi i e DC-bias AC ol age case (see Fig. 2(b)), he luid lows om le o igh , i.e., i is s ill lowing om highe conduc i i y OH – ions ich ca hode o lowe conduc i - i y H þ ions ich anode, a egion nea he elec odes’ su aces. This low p oduces a esul an an i-clockwise (an i-CW) o ex o a ion. As such, he p edic ed esul s on he o ex di ec ion co obo a e he epo ed expe imen al obse a ions. 23,24 TABLE I. Simula ion pa ame e s. P ope y Value Fluid densi y (q)110 3 kg=m 3 Fluid dynamic iscosi y (g)110 3 kg=ms Rela i e pe mi i i y o KCl (e )80 Pe mi i i y o ee space (e0) 8.85 10 12 F=m Coe icien o di usion (D)110 9 m 2 =s Vol age ampli ude (V) 1–20 V AC equency ( ) 100 kHz Conduc i i y a elec ode 1 ( 1) 1–10 mS=m Conduc i i y a elec ode 2 ( 2) 1–10 mS=m FIG. 2. Simula ed eloci y dis ibu ions. (a) Nega i e DC-bias, 1< 2and (b) posi i e DC-bias, 1> 2. Condi ions: ol - age ampli ude ¼10 V, conduc i i y di e ence ¼10%, W ¼100 lm, D ¼50 lm, elec ode wid h ¼40 lm, and elec ode gap ¼20 lm. Fo nega i e DC-bias o (a), plo s o X- eloci y (ho izon al) ac oss (c) channel dep h a a ious loca ions along he le elec ode, and (d) channel wid h a di e en channel heigh s. 012817-5 dc-biased ac-elec okine ic low Biomic o luidics 6, 012817 (2012) In gene al, he nega i e and he posi i e DC-bias simula ed esul s a e simila and symme - ical. The x- eloci ies (Vx) o he simula ed esul s o he nega i e DC-biased case (Fig. 2(a)) we e ex ac ed and plo ed. The ho izon al eloci ies we e plo ed ac oss he wid h and ac oss he dep h a di e en loca ions on he channel as shown in Figs. 2(c) and 2(d), espec i ely. F om Figs. 2(c) and 2(d), i can be obse ed ha Vxa e as e on he egion nea e o he elec ode su ace. This is expec ed as he eloci y is d i en by he g adien s o conduc i i y and he elec ical po en ial, whe e he conduc i i y and he elec ic ield a e he s onges nea he su ace a he elec ode gap. Taking he poin whe e Vx¼0 along he cen e o he channel (y ¼0) as he cen e o he o ex; he cen e o o ex is a a heigh o 20 lm. This co esponds o 40% (i.e., 20 lm/ 50 lm) o he o al channel heigh . De ining H as he pe cen age o he o ex heigh o he channel dep h (D), he H alue compu ed is hus a he high a 40%. Simila ly, we ha e p e iously epo ed an obse ed high H alue (>30%) by expe imen al imaging he induced DC-biased ACEK low o ex. 24 This high cen e o o ex implies ha he low mechanism is bulk d i en. In he li e a u e, ACET low which is bulk induced gene ally has a o ex cen e be ween 30% and 40% o he channel heigh . 14–16 B. E ec o ol age and conduc i i y a io The applied ol age and he conduc i i y a io a ec he eloci ies o he low induced in he sys em. He e, he conduc i i y a io (c) is de ined as he a io o he highe conduc i i y ( 2) o he lowe conduc i i y ( 1). To quan i y he eloci ies o he low a a ious condi ions ( ol age, conduc i i y a io), he a e age eloci y ampli ude in he c oss-sec ion domain was compu ed and de ined as 27 Va e ¼1 AððVdxdy;(9) whe e V¼jVðx;yÞj is he eloci y ampli ude a ðx;yÞand A(m 2 ) is he a ea o he c oss- sec ion domain. Simila ly, he adop ed geome y o be s udy we e W ¼100 lm, D ¼50 lm, elec ode wid h ¼40 lm, and elec ode gap ¼20 lm. The a e age eloci y ampli udes o nega- i e DC-bias a di e en pa ame e s ( ol age, conduc i i y a io) we e calcula ed and p esen ed in Fig. 3. Figu e 3(a) shows ha he a e age eloci y ampli udes inc ease wi h he applied ol age a a ious conduc i i y a ios. The highes eloci ies ob ained o c¼1.1, 1.3, and 1.5 a 20 V we e 2.78 mm/s, 7.64 mm/s, and 11.78 mm/s, espec i ely. Nex , by ixing he ol age ampli ude a 10 V, he e ec o conduc i i y a io on eloci y was in es iga ed. Figu e 3depic s esul s o a ious conduc i i y a ios. I indica es a clea end ha a highe applied ol age and/o highe conduc i i y a io will esul in a as e low eloci y. F om Fig. 3(a), he simula ed eloci y a 10 V and c¼1.1 is 695 lm/s, which compa es a o ably wi h he expe imen al measu ed eloci y o 700 lm/s. 24 I should be no ed ha he conduc i i y g adien / a io gene a ed by he DC-bias Fa adaic cha ging has a limi a ion. This is because elec olysis and elec odes deg ada ion will occu a oo high ol age. The e o e, he conduc i i y a io ia his me hod is limi ed by he le el o DC-bias (i.e., <3V DC ). Ne e heless, o acili a e an unde s anding o mechanism, simula ion is conduc ed o conduc i i y a io up o 10 which is he ypical a io adop ed in elec okine ic ins abili y (EKI) low phenomenon. 30–32 F om Fig. 3(d), he achie ed eloci y a 20 V and c¼10 was ex emely as , 14.44 mm/s. As a compa ison wi h a mo e complex nume ical model, 33 i was epo ed ha he achie ed eloci y was up o ens o mm/s (i.e., a 0.5 10 5 V/m and c¼10). Thei p edic ed eloci y is o he same o de o magni ude as ob ained by he cu en model. In addi ion, his esul implies ha apid induced low can be achie ed by inco po a ing a mic oelec ode (wi h applied AC ol age) wi h manually induced conduc i i y s eams. Typically, he EKI con igu a ion in oduces he high AC ol age h ough 012817-6 Ng e al. Biomic o luidics 6, 012817 (2012) bulk elec odes om he ese oi s. In eg a ed mic oelec odes can indeed supply elec ic ield o he same o de o magni ude (10 4 –10 5 V/m) due o he educed elec odes gap. C. E ec o channel wid h, dep h, and aspec a io Channel dimensions can a ec he e ec i eness o luid manipula ion in he mic ochannel. As such, he e ec s o he channel wid h (W), dep h (D), and aspec a io (i.e., AR ¼D/W) on he a e age o ex low eloci y (Va e) we e in es iga ed. Simila ly, he elec ode wid h and gap we e ixed a 40 lm and 20 lm, espec i ely. The ol age ampli ude used was 10 V and conduc i i y a io was a c¼1.1. Figu e 4(a) shows he a e age eloci y ampli udes (Va e)as he wid h o he channel (W) a ies om 100 lm o400lm, a ixed dep h o ei he 50 lmo 100 lm. I can be obse ed ha he Va e gene ally dec eases as he W inc eases. This is because he e is a limi whe e he induced DC-biased ACEK low can d ag he su ounding luids in o mo ion. 34 The e o e, he e is an op imum ange whe e he luids can be e ec i ely ac ua ed, de e mined by he d i ing eloci y (i.e., eloci y nea he su ace o he elec odes). As W inc eases (see Fig. 4(a)), he eloci ies a e only con ined wi hin he cen al egion o he mic o- channel, and hus, he ac ua ion e icien gene ally dec eases. These indings ha e implica ion on he design o pump and mixe de ices, as i would be p e e ed o loca e he elec odes in close p oximi y o he pumping/mixing egion. The inse in Fig. 4(a) shows he co esponding H pe cen age o he simula ed da a poin s. The end indica es ha he o ex cen e heigh s ays ela i ely cons an (28%–40%) o di e en channel wid h (100–400 lm) in es iga ed. This is because o he ixed channel dep hs (50 lm, 100 lm) adop ed. Figu e 4(b) shows he a e age eloci y ampli udes (Va e) as he dep h o he channel (D) a ies om 20 lm o200lm, a a ixed wid h o ei he 100 lm o 200 lm. The a e age eloc- i y inc eases o a maximum and hen dec eases wi h inc easing channel dep h. I can be obse ed ha he e is an op imal dep h whe e a maximum a e age eloci y is ob ained. The maximum eloci ies we e 748 lm/s (a W ¼100 lm, D ¼100 lm, AR ¼1) and 698 lm/s (a FIG. 3. A e age eloci y ampli ude a di e en (a) applied ol age, (b) conduc i i y a io, up o 1.1, (c) conduc i i y a io, up o 2, and (d) conduc i i y a io, up o 10. Con igu a ions: W ¼100 lm, D ¼50 lm, elec ode wid h ¼40 lm, and elec- ode gap ¼20 lm. 012817-7 dc-biased ac-elec okine ic low Biomic o luidics 6, 012817 (2012) W¼200 lm, D ¼150 lm, AR ¼3/4) o he ixed wid hs in es iga ed. No e ha when he c oss-sec ion is la ge, he luid eloci y does no ex end e y a om he gap be ween elec o- des. This means ha he a e age eloci y should dec ease, since egions a om he gap has e y small eloci y. I he c oss-sec ion is small (sho heigh ), he eloci y is educed because o iscous ic ion wi h he walls. The e o e, he e is a maximum in be ween he wo limi s. In addi ion, he maximum a e age eloci y gene ally occu s a channel AR close o 1, his gi es us clue on designing he geome y o he pump wi h be e e iciency. The esul indica es ha o la ge channel c oss-sec ion (AR 1) he induced low would no ex end a om he elec- ode su ace, hence will no be e icien . Fo small channel c oss-sec ion (AR 1), he induced low eloci y is dec eased om expec ed due o iscous ic ion. Nex , he inse in Fig. 4(a) shows he co esponding H pe cen age o he simula ed da a poin s. The H pe cen age gene ally dec eases wi h inc easing dep h. Simila ly, his is because he induced DC-biased ACEK low also has a limi in se ing he luids in mo ion in he y- di ec ion (dep h-wise). In Fig. 4(b), i should be no ed ha o he ixed wid h o 200 lm, he channel AR <1, and he ob ained H pe cen age is s ill conside ably high (>23%). Fo he ixed wid h o 100 lm, as he geome ical con igu a ion changes om a la mic ochannel (AR <1) o a all mic ochannel (AR >1), a d ama ic d op in he H pe cen age is obse ed. This also sugges s ha he design o mic ochannel geome y should adop a squa e (AR ¼1) o la (AR <1) ype o channel c oss-sec ion o elec okine ic low ac ua ion o be mo e e ec i e. FIG. 4. A e age eloci y ampli ude wi h a ying channel (a) wid h, o a ixed dep h o 50 lm o 100 lm, and (b) dep h, o a ixed wid h o 100 lm o 200 lm. Inse shows H alue o each da a poin . Condi ions: ol age ampli ude ¼10 V, con- duc i i y di e ence ¼10% (c¼1.1), elec ode wid h ¼40 lm, and elec ode gap ¼20 lm. 012817-8 Ng e al. Biomic o luidics 6, 012817 (2012) As obse ed in Fig. 4(b), he channel AR can a ec he manipula ion o luid inside a mic ochannel. He e, we in es iga e he eloci y o he induced low in a channel o a ious ARs (1/2, 1/3, 1/4, and 1/5), bu we es ic ed ou s udy o AR <1 ( la mic ochannel), he p e- e ed channel con igu a ion. Figu e 5shows he a e age eloci y ampli udes in di e en chan- nel wid hs o a ious cons an channel ARs. Figs. 5(a)–5(d) show he plo s o AR o 1/2, 1/3, 1/4, and 1/5, espec i ely. Wi h a dec ease in AR om 1/2 o 1/5, he Va e gene ally dec eases. The AR measu es he la ness o he mic ochannel, hus a lowe AR indica es ha W is much la ge han he D. No e in Fig. 4(a) wi h ixed D, he compu ed a e age eloci y is also ound o dec ease wi h inc easing W. The e o e, he end in Figs. 4(a) and 5a e he same depic ing a dec easing AR wi h ixed D and inc easing W. Typically in such con igu a ion, he induced DC-biased ACEK low has a limi ed ac ua ion ange as discussed p e iously o Fig. 4(a). I should be no ed ha he H pe cen age emains ela i ely high a 27%–39%, 30%–42%, 33%–44%, and 35%–45% o AR o 1/2, 1/3, 1/4, and 1/5. IV. CONCLUSIONS In his pape , a nume ical model desc ibing DC-biased AC-elec okine ic low has been p esen ed. “Con ec ion and di usion,” “conduc i e media DC,” and “incomp essible Na ie - S okes” modules we e adop ed in he COMSOL so wa e. The phenomenological model is based on expe imen ally measu ed conduc i i y g adien gene a ed upon he applica ion o a DC-bias o one o he elec ode pai s. The simula ed esul s indica e good co ela ion wi h epo ed expe imen s in e ms o low induced o ex cha ac e is ics, magni ude, and o a ing di ec ion. I was con i med ha he low di ec ion was om he highe conduc i i y o he lowe conduc- i i y egions nea he su aces o he elec odes and hus o ming a single and uni ied o ex ac oss he wid h o he mic ochannel. Pa ame ic s udies we e conduc ed by a ying he ol - age, conduc i i y a io, channel wid h, dep h and aspec a io. I was ound ha he low eloc- i y induced in DC-ACEK can each alues in he ange mm/s by selec ing an app op ia ed highe ol age and conduc i i y a io. In addi ion, i was ound ha mic o channels wi h squa e dimensions (AR ¼1) a e mo e e icien o op imum low eloci y. FIG. 5. A e age eloci y ampli ude a di e en channel aspec a ios (a) AR ¼1=2, (b) AR ¼1=3, (c) AR ¼1=4, and (d) AR ¼1=5. Inse shows H alue o each da a poin . Condi ions: ol age ampli ude ¼10 V, conduc i i y di e ence ¼10% (c¼1.1), elec ode wid h ¼40 lm, and elec ode gap ¼20 lm. 012817-9 dc-biased ac-elec okine ic low Biomic o luidics 6, 012817 (2012)