Lattice Boltzmann simulations in microfuidics : probing the no-slip boundary condition in hydrophobic, rough, and surface nanobubble laden microchannels

REVIEW La ice Bol zmann simula ions in mic o luidics: p obing he no-slip bounda y condi ion in hyd ophobic, ough, and su ace nanobubble laden mic ochannels Jens Ha ing •Ch is ian Kune •Ja i Hy a ¨luoma Recei ed: 14 Augus 2009 / Accep ed: 7 Sep embe 2009 / Published online: 14 Oc obe 2009 ÓThe Au ho (s) 2009. This a icle is published wi h open access a Sp inge link.com Abs ac In his con ibu ion, we e iew ecen e o s on in es iga ions o he e ec o (appa en ) bounda y slip by u ilizing la ice Bol zmann simula ions. We demons a e he applicabili y o he me hod o ea undamen al ques ions in mic o luidics by in es iga ing luid low in hyd ophobic and ough mic ochannels as well as o e su aces co e ed by nano- o mic oscale gas bubbles. Keywo ds Appa en and in insic slip  Rough and hyd ophobic su aces  La ice Bol zmann simula ions 1 In oduc ion Du ing he pas ew decades, he minia u iza ion o ech- nical de ices down o submic ome ic sizes has made conside able p og ess. In pa icula , he so-called mic o- elec o-mechianical sys ems (MEMS) became a ailable o chemical, biological, and echnical applica ions leading o he ise o ‘‘mic o luidics’’ abou 20 yea s ago (Tabeling 2005). A wide a ie y o mic o luidic sys ems including gas ch oma og aphy sys ems, elec opho e ic sepa a ion sys- ems, mic omixe s, DNA ampli ie s, and chemical eac o s we e de eloped. Nex o hose ‘‘p ac ical applica ions,’’ mic o luidics was used o answe undamen al ques ions in physics including he beha io o single molecules o pa - icles in luid low o he alidi y o he no-slip bounda y condi ion (Tabeling 2005; Lauga e al. 2005). The la e is he ocus o he cu en e iew and is in es iga ed in de ail by mesoscopic compu e simula ions. Reynolds numbe s in mic o luidic sys ems a e usually small, i.e., usually below 0.1. In addi ion, due o he small scales o he channels, he su ace- o- olume a io is high causing su ace e ec s such as we abili y o su ace cha ges o be mo e impo an han in mac oscopic sys ems. Also, he mean ee pa h o a luid molecule migh be o he same o de as he cha ac e is ic leng h scale o he sys em. Fo gas lows, his e ec can be cha ac e ized by he so-called Knudsen numbe (Knudsen 1909). While he Knudsen numbe p o- ides a good es ima e o when o expec a e ac ion e ec s in gas lows, o liquids one would nai ely assume ha i s eloci y close o a su ace always co esponds o he ac ual eloci y o he su ace i sel . This assump ion is called he no-slip bounda y condi ion and can be coun ed as one o he gene ally accep ed undamen al concep s o luid mechan- ics. Howe e , his concep was no always well accep ed. Some cen u ies ago, he e we e long deba es abou he eloci y o a New onian liquid close o a su ace, and he accep ance o he no-slip bounda y condi ion was mos ly due o he ac ha no expe imen al iola ions could be ound, i.e., he so-called bounda y slip could no be de ec ed. In ecen yea s, i became possible o pe o m e y well con olled expe imen s ha ha e shown a iola ion o he no-slip bounda y condi ion in submic on-sized geome ies. J. Ha ing (&) Depa men o Applied Physics, TU Eindho en, Den Dolech 2, 5600MB Eindho en, The Ne he lands e-mail: [email p o ec ed] J. Ha ing C. Kune Ins i u e o Compu a ional Physics, Uni e si y o S u ga , P a enwald ing 27, 70569 S u ga , Ge many e-mail: [email p o ec ed] C. Kune e-mail: [email p o ec ed] J. Hy a ¨luoma Depa men o Physics, Uni e si y o Jy a ¨skyla ¨, 40014 Jy a ¨skyla ¨, Finland e-mail: [email p o ec ed] 123 Mic o luid Nano luid (2010) 8:1–10 DOI 10.1007/s10404-009-0506-6 Since hen, mos ly, no only expe imen al (Lauga e al. 2005; C aig e al. 2001; T e heway and Meinha 2004; Cheng and Gio dano 2002; Choi e al. 2003; Baud y and Cha laix 2001; Co in-Bizonne e al. 2002; Vinog ado a and Yakubo 2003), bu also heo e ical s udies (Vinog a- do a 1995; Gennes 2002) as well as compu e simula ions (Succi 2002; Ba a and Bocque 1999; Cieplak e al. 2001; Thompson and T oian 1997; T e heway e al. 2002) ha e been pe o med o imp o e ou unde s anding o bounda y slip. The opic is o undamen al in e es because i has p ac ical consequences in he physical and enginee ing sciences as well as o medical and indus ial applica ions. In e es ingly, also o gas lows, o en a slip leng h much la ge han expec ed om classical heo y can be obse ed. Ex ensi e e iews o he slip phenomenon ha e ecen ly been published by Lauga e al. (2005), Ne o e al. (2005), as well as Bocque and Ba a (2007). The eason o ou unsa is ac o y unde s anding o bounda y slip is ha he beha io o a luid close o a solid in e ace is e y complex and in ol es he in e play o many physical and chemical p ope ies. These include he we abili y o he solid, he shea a e o low eloci y, he bulk p essu e, he su ace cha ge, he su ace oughness, as well as impu i ies and dissol ed gas. Because all hose quan i ies ha e o be de e mined e y p ecisely, i is no su p ising ha ou unde s anding o he phenomenon is s ill e y unsa is ac o y. Owing o he la ge numbe o di e en pa ame e s, a signi ican dispe sion o he esul s can be obse ed o almos simila sys ems (Lauga e al. 2005; Ne o e al. 2005). Fo example, obse ed slip leng hs a y be ween a ew nanome e s (Chu ae e al. 1984) and mic ome e s (T e heway and Meinha 2004) and while some au ho s ind a dependence o he slip on he low eloci y (C aig e al. 2001; Choi e al. 2003; Zhu and G anick 2001), o he s do no (T e heway and Meinha 2004; Cheng and Gio dano 2002). A bounda y slip is ypically quan i ied by he so-called slip leng h b—a concep ha was al eady p oposed by Na ie in 1823. He in oduced a bounda y condi ion whe e he luid eloci y a a su ace is p opo ional o he shea a e a he su ace (Na ie 1823) (a x=x 0 ), i.e., zðx0Þ¼bo zðxÞ ox:ð1Þ In o he wo ds, he slip leng h bcan be de ined as he dis ance om he su ace whe e he ela i e low eloci y anishes. Assuming a ypical Poiseuille se up consis ing o a p essu e-d i en low o an incomp essible liquid be ween wo in ini e planes, he eloci y in low di ec ion ( z )a posi ion xbe ween he planes is gi en by zðxÞ¼ 1 2l oP ozd2x22db  ;ð2Þ whe e 2dis he dis ance be ween he planes, and lis he dynamic iscosi y. qP/qzis he p essu e g adien . In con- as o a no-slip o mula ion, he las e m in Eq. 2linea ly depends on he slip leng h b. Mos ecen compu e simula ions apply molecula dynamics and epo inc easing slip wi h dec easing liquid densi y (Koplik e al. 1989; Thompson and Robbins 1990) o liquid–solid in e ac ions (Cieplak e al. 2001; Nagayama and Cheng 2004), while slip dec eases wi h inc easing p essu e (Ba a and Bocque 1999). These simula ions a e usually limi ed o a ew ens o housand pa icles, leng h scales o a ew nanome e s and ime scales o nanoseconds. Also, shea a es a e usually some o de s o magni ude highe han in any expe imen (Lauga e al. 2005). Owing o he small accessible ime and leng h scales o molecula dynamics simula ions, mesoscopic simula ion me hods, such as he la ice Bol zmann me hod, a e well applicable o he simula ion o mic o luidic expe imen s. The expe imen al in es iga ion o appa en slip can be based on di e en se ups: a luid is pumped h ough a mic ochannel, and he measu ed mass low a e a he end o he channel is compa ed o he heo e ical alue wi h no-slip bounda y condi ions. F om he de ia ion o he wo alues, he magni ude o slip can be compu ed (T e heway and Meinha 2002). Ano he possibili y is o measu e he slip leng h di ec ly using op ical me hods such as pa icle image elocime y (PIV). Ve y popula is he modi ica ion o an a omic o ce mic oscope (AFM) by adding a silicon sphe e o he ip o he can ile e . While mo ing he sphe e owa d he bounda y, he equi ed o ce is measu ed. I is possible o measu e he amoun o slip a he wall by compa ing he o ce needed o mo e he sphe e wi h i s heo e ical alue (Vinog ado a and Yakubo 2003; Vinog ado a 1996). Du ing he pas ew yea s, he subs an ial scien i ic esea ch in es ed in he slip phenomenon has led o a mo e clea pic u e which can be summa ized as ollows: one can a gue ha many su p ising esul s published we e only due o a i ac s o misin e p e a ion o expe imen s. In gene al, he e seems o be an ag eemen wi hin he communi y ha slip leng hs la ge han a ew nanome e s can usually be e e ed o as ‘‘appa en slip’’ and a e o en caused by expe imen al a i ac s. Small slip leng hs a e expe imen- ally e en ha de o de e mine and equi e sophis ica ed se ups such as he modi ied AFMs as desc ibed abo e. He e, small a ia ions o he appa a us such as choosing a di e en shape o he can ile e o modi ying he con ol ci cui o he sample holde can lead o subs an ial a ia- ion o he measu emen s. Also, he heo e ical equa ions co ela ing he measu ed o ce o he slip leng h a e only alid o pe ec su aces and in ini ely slow oscilla ions o he sphe e. The e o e, i is o impo ance o pe o m 2 Mic o luid Nano luid (2010) 8:1–10 123 compu e simula ions which ha e he ad an age ha mos pa ame e s can be changed independen ly wi hou modi- ying any hing else. Thus, he in luence o e e y single modi ica ion can be s udied o p esen es ima es o expec ed slip leng hs. 2 Appa en slip in hyd ophobic mic ochannels The simula ion me hod used o s udy mic o luidic de ices has o be chosen ca e ully. While Na ie –S okes sol e s a e able o co e mos p oblems in luid dynamics, hey lack he possibili y o include he in luence o molecula in e ac ions as needed o model bounda y slip. Molecula dynamics (MD) simula ions a e he bes choice o simula e he luid– wall in e ac ion, bu he compu e powe oday is no su i- cien o simula e leng h and ime scales necessa y o achie e o de s o magni ude which a e ele an o expe imen s. Howe e , bounda y slip wi h a slip leng h bo he o de o many molecula diame e s has been s udied wi h molec- ula dynamics simula ions by a ious au ho s (Baud y and Cha laix 2001; Cieplak e al. 2001; Thompson and T oian 1997; Co in-Bizonne e al. 2004; P iezje e al. 2005). This a icle ocuses on nume ical in es iga ions o he slip phenomenon by means o la ice Bol zmann simula- ions. While an emphasis is pu on e iewing ou own con ibu ions o he ield, he achie emen s o o he g oups a e commonly e e ed o. Howe e , i should be no iced ha while a la ge numbe o g oups u ilizes he la ice Bol zmann echnique o in es iga e mic o luidic p oblems, only a e y small numbe o esea che s a e ac ually applying he me hod o s udying slippage. E en hough in e ac ions ha e o be desc ibed on a mesoscopic scale, his is su p ising since mesoscopic simula ion me hods o e a close ela ion o expe imen ally ele an ime and leng h scales han mic oscopic echniques such as molecula dynamics. In he la ice Bol zmann me hod, one disc e izes he Bol zmann kine ic equa ion o o þ x  gðx; ; Þ¼Xð3Þ on a la ice. The Bol zmann kine ic equa ion desc ibes he e olu ion o he single pa icle p obabili y densi y g(x, , ), whe e xis he posi ion, he eloci y, and he ime. The de i a i es ep esen simple p opaga ion o a single pa icle in eal and eloci y space, whe eas he collision ope a o X akes in o accoun molecula collisions in which a pa - icle changes i s momen um due o a collision wi h ano he pa icle. In o de o ep esen he co ec physics, he col- lision ope a o should conse e mass and momen um, and should be Galilei in a ian . By pe o ming a Chapman Enskog p ocedu e, i can be shown ha such a collision ope a o X ep oduces he Na ie –S okes equa ion (Succi 2001). In he la ice Bol zmann me hod, he ime , he posi ion x, and he eloci y a e disc e ized. A ew g oups ha e applied he la ice Bol zmann me hod o he simula ion o mic o lows and o s udy bounda y slip. A popula app oach is o in oduce slip by gene alizing he no-slip bounce-back bounda y condi ions o allow specula e lec ions wi h a gi en p obabili y (Succi 2002; T e heway e al. 2002; Tang e al. 2005; Sb agaglia and Succi 2005), o o apply di use sca e ing (Ansumali and Ka lin 2002; So onea and Seke ka 2005; Niu e al. 2004). I has been shown by Guo e al. ha hese app oaches a e i ually equi alen (Guo e al. 2007). Ano he possibili y is o modi y he luid’s iscosi y, i.e., he luid iscosi y is modi ied due o local densi y a ia ions o model slip (Nie e al. 2002). In bo h cases, he pa ame e s de e mining he p ope ies a he bounda ies a e ‘‘a i icial’’ pa ame e s, and hey do no ha e any ob ious physical meaning. The e o e, hey a e no easily mappable o expe imen ally a ailable alues. We model he in e ac ion be ween hyd ophobic channel walls and he luid by means o a mul iphase la ice Bol zmann model. Ou app oach o e comes his p oblem by applying a mesoscopic o ce be ween he walls and he luid. A simila app oach is used by Zhu e al. (2005), Benzi e al. (2006a), and Zhang e al. (2004). This o ce applied a he bounda y can be linked o he con ac angle which is commonly used by expe imen alis s o quan i a i ely desc ibe he we abili y o a ma e ial (Benzi e al. 2006b; Huang e al. 2007). The simula ion me hod and ou implemen a ion o bounda y condi ions a e desc ibed as ollows. A mul i- phase la ice Bol zmann sys em can be ep esen ed by a se o equa ions ga iðxþci; þ1Þga iðx; Þ¼Xa i;i¼0;1;...;b;ð4Þ whe e g i a (x, ) is he single-pa icle dis ibu ion unc ion, indica ing he amoun o species awi h eloci y c i , a si e xon a D-dimensional la ice o coo dina ion numbe b(D3Q19 in ou implemen a ion), a ime-s ep . This is a disc e ized e sion o Eq. 3wi hou ex e nal o ces F o a numbe o species a. Fo he collision ope a o X i a we choose he Bha naga –G oss–K ook (BGK) o m (Bha naga e al. 1954) Xa i¼1 saðga iðx; Þgaeq iðuaðx; Þ;gaðx; ÞÞÞ;ð5Þ whe e s a is he mean collision ime o componen aand de e mines he kinema ic iscosi y ma¼2sa1 6:ð6Þ o he luid. The elaxa ion ime s a is kep cons an a 1.0 in his s udy. The sys em elaxes o an equilib ium dis ibu ion Mic o luid Nano luid (2010) 8:1–10 3 123 g i aeq which can be de i ed imposing es ic ions on he mic oscopic p ocesses, such as explici mass and momen um conse a ion o each species. In ou implemen a ion, we choose o he equilib ium dis ibu ion unc ion geq i¼ iga1þciu c2 sþðciuÞ2 2c4 su2 2c2 sþðciuÞ3 6c6 su2ðciuÞ 2c4 s "# ; ð7Þ which is a polynomial expansion o he Maxwell dis ibu ion. c i ’s a e he eloci y ec o s poin ing o neighbo ing la ice si es and i a e he la ice weigh s esul ing om he eloci y space disc e iza ion. cs¼1=ffiffiffi3 p is he speed o sound o he D3Q19 la ice. The mac oscopic alues can be de i ed om he single-pa icle dis ibu ion unc ion g i a (x, ), i.e., he densi y g a (x, ) o he species aa la ice si e xis he sum o e he dis ibu ion unc ions g i a (x, ) o all la ice eloci ies c i , gaðx; ÞX i ga iðx; Þ:ð8Þ u a (x, ) is he mac oscopic eloci y o he luid, de ined as gaðx; Þuaðx; ÞX i ga iðx; Þci:ð9Þ In e ac ions be ween di e en luid species a e in oduced, acco ding o Shan and Chen, as a mean ield body o ce be ween nea es neighbo s (Shan and Chen 1993,1994), Faðx; Þwaðx; ÞX  a ga aX x0 w aðx0; Þðx0xÞ;ð10Þ whe e waðx; Þ¼ð1egaðx; Þ=g0Þis he so-called e ec i e mass wi h g 0 being a e e ence densi y ha is se o 1 in ou case (Shan and Chen 1993). ga ais a o ce coupling cons an , whose magni ude con ols he s eng h o he in e ac ion be ween componen aand  a:The dynamic e ec o he o ce is ealized in he BGK collision ope a o (5) by adding an inc emen du a =s a F a /g a o he eloci y uin he equilib ium dis ibu ion unc ion (7). A epulsi e po en ial be ween su ace and luid can be used o model hyd ophobic luid– su ace in e ac ions. Such a po en ial is ealized by a aching he imagina y luid ‘‘densi y’’ g wall o he i s la ice si e inside he wall. Only he dis ibu ion co esponding o he es eloci y is illed, while he emaining ones a e kep a 0. As a esul , he only di e ence be ween g wall and any o he luid packages on he la ice g ais ha he luid co esponding o g wall is aken in o accoun only o he collision s ep and o he calcula ion o Eq. 10, bu no in he p opaga ion s ep. The e o e, we can adop g wall and he coupling cons an g a,wall o une he luid–wall in e ac ion. g a,wall is kep a 0.08 h oughou his a icle i no men ioned o he wise, and all he alues a e epo ed in la ice uni s. These pa ame e s allow o simula e a wide ange o e ec i e in e ac ions wi hou comp omising on nume ical s abili y. In addi ion, we apply second-o de co ec mid-g id bounce-back bounda y con- di ions be ween he luid and he su ace which assu es anishing eloci ies a solid su aces. He e, a dis ibu ion unc ion ha would be ad ec ed in o a solid node is simply e e sed and ad ec ed in o he opposi e di ec ion (Succi 2001). F om molecula dynamics simula ions, i is known ha he luid–wall in e ac ions causing a slip phenomenon usually ake place wi hin a ew molecula laye s o he liquid along he bounda y su ace (Baud y and Cha laix 2001; Cieplak e al. 2001; Thompson and T oian 1997; Co in-Bizonne e al. 2004). Ou coa se-g ained luid–wall in e ac ion ac s on he leng h scale o one la ice cons an and does no ake he molecula de ails in o accoun . The e o e, coa se-g ained implemen a ions based on he la ice Bol zmann me hod a e only able o ep oduce an a e aged e ec o he in e ac ion and canno ully esol e he co ec low p o ile e y close o he wall and below he esolu ion o a single la ice spacing. Howe e , in o de o unde s and he in luence o he hyd ophobici y on expe i- men ally obse ed appa en slip, i is ully su icien o in es iga e he low beha io on mo e mac oscopic scales as hey a e accessible o expe imen al in es iga ion. Coa se- g ained in e ac ion models could be imp o ed by a di ec mapping o da a ob ained om MD simula ions o he coupling cons an g a,wall allowing a di ec compa ison o he in luence o liquid–wall in e ac ions on he de ec ed slip (Ha ing e al. 2006). Simila app oaches a e known om quan i a i e compa isons o la ice Bol zmann and molec- ula dynamics simula ions in he li e a u e (Ho bach and Succi 2006; Chibba o e al. 2008). The simula ions in his s udy use a se up o wo in ini e planes sepa a ed by he dis ance 2d. We call he di ec ion be ween he wo planes x, and i no s a ed o he wise, 2dis se o 64 la ice si es. In ydi ec ion, we apply pe iodic bounda y condi ions. He e, eigh la ice si es a e su icien o a oid ini e size e ec s since he e is no p opaga ion in his di ec ion. zis he di ec ion o he low wi h ou channels being 512 la ice si es long. A he beginning o he simu- la ion ( =0), he luid is a es . We hen apply a p essu e g adien Pin he z-di ec ion o gene a e a plana Poiseuille low. Assuming Na ie ’s bounda y condi ion, he slip leng h bis measu ed by i ing he heo e ical eloci y p o ile as gi en by Eq. 2in low di ec ion ( z ) a posi ion x, o he simula ed da a ia he slip leng h b. We alida e his app oach by compa ing he measu ed mass low a e Rg ðxÞdx o he heo e ical mass low wi hou bounda y slip and ind a e y good ag eemen . The dynamic iscos- i y las well as he p essu e g adien oP ozneeded o i Eq. 2 a e ob ained om ou simula ion da a. In Ha ing e al. (2006), we show ha his model c ea es a la ge slip bwi h s onge in e ac ion, namely, la ge g a,wall and la ge g wall . The maximum a ailable slip leng h 4 Mic o luid Nano luid (2010) 8:1–10 123 measu ed is 5.0 in la ice uni s. Fo s onge epulsi e po en ials, he densi y g adien a he luid–wall in e ace becomes oo la ge, causing he simula ion o become uns able. A lowe in e ac ions, he me hod is e y s able, and he slip leng h bis independen o he dis ance d be ween he wo pla es and, he e o e, independen o he esolu ion. We also show ha he slip dec eases wi h inc easing p essu e since he ela i e s eng h o he epul- si e po en ial compa ed o he bulk p essu e is weake a high p essu e. The e o e, he p essu e educ ion nea he wall is less in he high p essu e case han in he low p essu e one. Fu he mo e, we demons a e ha bcan be i ed wi h a semianaly ic model based on a wo- iscosi y model. We s udy he dependence o he slip leng h bon he low eloci y o a wide ange o eloci ies o mo e han h ee decades as shown in Fig. 1and in Ha ing e al. (2006). In he igu e, we show da a o di e en luid–wall in e ac ions 0 g wall 2.0 and low eloci ies om 10 -4 10 -1 . Fo simplici y, we es ic ou sel es o g a,wall =0.08 which is a sui able alue ound om pa ame e s udies gi en in Ha ing e al. (2006). Wi hin his egion, we con i m he indings o many s eady-s a e expe imen s (Cheng and Gio dano 2002), namely, he slip leng h is independen o he low eloci y and only depends on he we abili y o he channel walls. Some dynamic expe imen s, howe e , ind a shea a e-dependen slip (Zhu and G anick 2001; Ne o e al. 2003). These expe imen s o en u ilize a modi ied AFM, as desc ibed in he in oduc ion, o de ec bounda y slippage. Since he slip leng h is ound o be cons an in ou simula ions a e su icien ly long simula ion imes, we canno con i m hese esul s. Howe e , i has been p oposed by a ious au ho s ha his eloci y dependence is due o noncon olled e ec s such as impu i ies o su ace nano- bubbles. In simula ions, we can only ind a shea a e dependence i he sys em has no ye eached he s eady s a e o i ime-dependen accele a ions a e p esen (Kune and Ha ing 2008a). Ou mesoscopic app oach is able o each he small low eloci ies o known expe imen s, and ep oduces esul s om expe imen s and o he compu e simula ions, namely, an inc ease o he slip wi h inc easing liquid–solid in e - ac ions, he slip being independen o he low eloci y, and a dec easing slip wi h inc easing bulk p essu e. In addi ion, wi hin ou model, we de elop a semianaly ic app oxima- ion o he dependence o he slip on he bulk p essu e as desc ibed in Ha ing e al. (2006). 3 Roughness induced appa en slip I ypical leng h scales o he expe imen al sys em a e compa able o he scale o su ace oughness, he e ec o oughness canno be neglec ed anymo e. Figu e 2(le ) shows a ypical example o a simula ion se up: Poiseuille low be ween wo ough su aces. The su ace is gene a ed using a andom numbe gene a o o andomly choose he heigh o he obs acles a e e y disc e e su ace posi ion. As can be obse ed in he igu e, he s eam lines o he low a e ge ing dis u bed o apped be ween he obs acles a he su aces. In his sec ion, we show ha an appa en bounda y slip can ha e i s o igin in he misleading assump ion o pe ec ly smoo h bounda ies. The in luence o su ace a ia ions on he slip leng h b has been in es iga ed by nume ous au ho s. I was demon- s a ed by Richa dson ha oughness leads o highe d ag o ces and hus o no-slip on mac oscopic scales. He has shown ha i on a ough su ace e en a ull-slip bounda y condi ion is applied, one ob ains a low speed educ ion nea he bounda y esul ing in a mac oscopic no-slip bounda y condi ion (Richa dson 1973). An expe imen al con i ma- ion was la e p esen ed by by McHale and New on (2004). The MD simula ions o Coue e low be ween sinusoidal walls ha e been p esen ed by Jabba zadeh e al. (2000). They ound ha slip appea s o oughness ampli udes smalle han he molecula leng h scale (Jabba zadeh e al. 2000). Sb agaglia e al. applied he LB me hod o simula e luids in he icini y o mic os uc u ed hyd ophobic su - aces (Sb agaglia e al. 2006), Al-Zoubi e al. demons a ed ha he LB me hod is well applicable o ep oduce known low pa e ns in sinusoidal channels (Al-Zoubi and B enne 2008), and Va nik e al. (Va nik and Raabe 2006; Va nik e al. 2006) ha e shown ha e en in small geome ies, ough channel su aces can cause low o become u bulen . Recen ly, we p esen ed he idea o an e ec i e wall o ough channel su aces (Kune and Ha ing 2007). He e, we in es iga e he in luence o di e en ypes o oughness on he posi ion o he e ec i e bounda y. Fu he , we show how he e ec i e bounda y depends on he dis ibu ion o 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.0×10-4 1.0×10-3 1.0×10-2 slip leng h β eloci y ηwall = 0.0 ηwall = 0.5 ηwall = 1.0 ηwall = 2.0 =50 000 Fig. 1 Slip leng h b e sus bulk eloci y o di e en luid–wall in e ac ions g wall .bis independen o and only depends on g wall (Ha ing e al. 2006). All uni s a e exp essed in la ice uni s h oughou his a icle, i no s a ed o he wise Mic o luid Nano luid (2010) 8:1–10 5 123 he oughness elemen s, and how oughness and hyd o- phobici y in e ac wi h each o he (Kune and Ha ing 2008b). Lecoq e al. (2004) pe o med expe imen s wi h well-de ined oughness, and de eloped a heo y o p edic he posi ion o he e ec i e bounda y. In he expe imen s, hey u ilized a lase in e e ome e o measu e he ajec o y o a colloidal sphe e, and, he eby, de e mined he lub i- ca ion o ce and an e ec i e bounda y posi ion. The used geome y consis s o g oo es wi h a iangula p o ile. Fo a heo e ical desc ip ion, he bounda y is exp essed in a Fou ie se ies ha gi es he bounda y condi ion o he Laplace equa ion. As a esul , an e ec i e bounda y can be de i ed by a as con e ing se ies. In his a icle, we e ise ou p e ious achie emen s and compa e hem wi h he heo e ical and expe imen al esul s o Lecoq e al. (2004). Again, Poiseuille low measu emen s a e u ilized o in es iga e he e ec o in e es . The ough su aces a e cha ac e ized by he highes poin o one plane (h max ), he posi ion o he deepes alley (h min ), and he a i h- me ic a e age o all su ace heigh s gi ing he a e age oughness R a . In he case o symme ical dis ibu ions, we ge R a =h max /2. The posi ion o he e ec i e bounda y h e can be ound by i ing he pa abolic low p o ile ia he dis ance d e . Wi h bse o 0, we ob ain he no-slip case. In o de o ob ain an a e age alue o he e ec i e dis ance be ween he planes d e , a su icien numbe o indi idual p o iles a di e en posi ions za e aken in o accoun . The d e so ound gi es he posi ion o he e ec i e bounda y, and he e ec i e heigh h e o he ough su ace is hen de ined by d max -d e (see Fig. 2, le ). We show ha he posi ion o he e ec i e bounda y heigh is depending on he shape o he oughness elemen s, i.e., o s ong su ace dis o ions, i is be ween 1.69 and 1.90 imes he a e age heigh o he oughness R a =h max /2 (Kune and Ha ing 2007). In Fig. 3, we plo he e ec i e bounda y posi ions o di e en geome ies, i.e., andomly dis ibu ed g oo es wi h a squa e p o ile and g oo es wi h a iangula p o ile. The esul s o he iangula ones ma ch wi h he heo e ical alue o Lecoq e al. (2004) o a simila geome y. By adding an addi ional dis ance be ween oughness elemen s, h e dec eases slowly, so ha he maximum heigh is s ill he leading pa ame e . We a e also able o simula e low o e su aces gene a ed om AFM da a o gold-coa ed glass used in mic o low expe imen s by Vinog ado a and Yakubo (2006). We ind ha he heigh dis ibu ion o such a su ace is Gaussian and ha a an- domly a anged su ace wi h a simila dis ibu ion gi es he same esul o he posi ion o he e ec i e bounda y al hough in his case he heigh s a e no co ela ed (Fig. 4). We can une he wid h o he dis ibu ion and he a e age heigh R a . By scaling wi h R a , we ob ain geo- me ically simila geome ies. This simila i y is impo an because he e ec i e heigh , h e , scales wi h he a e age Ra dmax de max h he hmin x y z Fig. 2 Le A ypical simula ed sys em. Poiseuille low be ween wo ough su aces showing andom su ace a ia ions. S eamlines depic a wo dimensional cu and illus a e he pa abolic eloci y p o ile. This p o ile is dis o ed in he icini y o he ough su aces (Kune and Ha ing 2008b). Righ The e ec i e bounda y heigh h e is ound be ween he deepes alley a h min and he highes peak a h max .I co esponds o an e ec i e channel wid h d e .R a deno es he a e age oughness, and he maximum dis ance be ween he pla es d max is kep cons an (Kune and Ha ing 2008b) 0 2 4 6 8 10 12 14 16 18 0 1 2 3 4 5 6 7 8 9 e ec i e heigh he a e age oughness Ra iangles blocks andom Lecoq e al Fig. 3 Simula ed e ec i e heigh h e e sus R a o di e en su ace geome ies. The iangula shape ma ches he heo e ical esul s o Lecoq e al. (2004) o a simila geome y 6 Mic o luid Nano luid (2010) 8:1–10 123 oughness in he case o geome ical simila i y (Kune and Ha ing 2007). We in es iga e Gaussian-dis ibu ed heigh s wi h di e en wid hs and ind ha he heigh o he e ec i e wall depends linea ly on in he obse ed ange (Kune and Ha ing 2008b). Fu he , we ind ha he slip di e ges as he ampli ude o he oughness inc eases and he low ield ge s mo e es ic ed, which highligh s he impo ance o a p ope ea men o su ace a ia ions in e y con ined geome ies (Kune and Ha ing 2007). 4 S uc u ed su aces wi h en apped mic obubbles A na u al con inua ion o ou p e ious s udies on ough- ness-induced appa en bounda y slip and he collabo a ion men ioned abo e is he analysis o low along supe hy- d ophobic su aces (Hy a ¨luoma and Ha ing 2008). While in ypical expe imen s, slip leng hs o a ew ens o nanome e s can be obse ed, i would be p e e able o echnical applica ions o inc ease he h oughpu o luid in a mic ochannel, i.e., o ob ain subs an ially la ge slip. Supe hyd ophobic su aces a e p omising in his con ex , since i has been ecen ly p edic ed (Co in-Bizonne e al. 2003) and expe imen ally epo ed (Pe o and Ro hs ein 2004) ha he so-called Faki e ec o Cassie s a e con- side ably ampli ies bounda y slippage. Using highly ough hyd ophobic su aces, such a si ua ion can be achie ed. Ins ead o en e ing he a ea be ween he ough su ace elemen s, he liquid emains a he op o he oughness and aps ai in he in e s ices. Thus, a e y small liquid–solid con ac a ea is gene a ed. S einbe ge e al. u ilized su aces pa e ned wi h a squa e a ay o cylind ical holes o demons a e ha gas bubbles p esen in he holes may cause a educed slip (S einbe ge e al. 2007). Nume ically, hey ound e en nega i e slip leng hs o low o e such bubble ma esses, i.e., he e ec i e no-slip plane is inside he channel, and he bubbles inc ease he low esis ance. In his sec ion, we conside nega i e slip leng hs on bubble su aces and also discuss he ques ion o shea - a e dependen slip. In pa - icula , we show ha mic obubbles can gene a e a shea - a e dependence. Ou simula ions u ilize he single componen mul iphase LB model by Shan and Chen (1994), which enables sim- ula ions o liquid– apo sys ems wi h su ace ension. We a e no awa e o u he la ice Bol zmann simula ions o s udy he low o e a bubble ma ess. Howe e , a numbe o au ho s ha e applied a ious LB mul iphase and mul icomponen models o s udy he p ope ies o d ople s on chemically pa e ned and supe hyd ophobic su aces (Kusumaa maja e al. 2006; Kusumaa maja and Yeomans 2007; Pi a e al. 2008; Hy a ¨luoma e al. 2007). The low in ou sys em is con ined be ween wo pa allel walls. One o he walls is pa e ned wi h holes and apo bubbles a e apped o hese holes. The o he wall is smoo h and mo ed wi h eloci y u 0 . S einbe ge e al. (2007) p esen ed ini e- elemen simula ions o low o e igid ‘‘bubbles’’ by applying slip bounda ies a s a ic bubble su aces. The LB me hod allows he bubbles o de o m i he iscous o ces a e high enough compa ed o he su ace ension. We a e also in e es ed in how su ace pa e ning a ec s he slip p ope ies o hese su aces, and how bubbles could be u ilized o de elop su aces wi h special p ope ies o mic o luidic applica ions (Hy a ¨luoma and Ha ing 2008). The dis ance be ween walls is d=1lm (40 la ice nodes) in all he simula ions, and he a ea ac ion o holes is 0.43. A uni cell o he egula a ay is included in a simula ion, and pe iodic bounda y condi ions a e applied a domain bounda ies. The bubbles a e apped in o holes by using di e en we abili ies o bounda ies in con ac wi h he main channel and wi h he hole. The p o usion angle u (see Fig. 5 o de ini ion) is a ied by changing he liquid’s bulk p essu e. The e ec i e slip leng h is b=lu 0 / -d, whe e =ld /dzis he shea s ess ac ing on he uppe wall and l he dynamic iscosi y o he liquid. We in es iga e he e ec o a modi ied p o usion angle and di e en su ace pa e ns by using squa e, ec angula , and hombic bubble a ays. The cylind ical holes ha e a adius a= 500 nm, and he a ea ac ion o he holes is equal in all he cases. The shea a e is such ha he Capilla y numbe Ca =laG s /c=0.16. He e, G s and ca e he shea a e and su ace ension, espec i ely. A snapsho o a simula ion is shown in he le pa o Fig. 5, and he slip leng hs ob ained a e shown in he igh pa . The obse ed beha io is simila o ha epo ed in S einbe ge e al. (2007), whe e a squa e a ay o holes was s udied. In pa icula , we obse e ha when uis la ge enough, b becomes nega i e. Mo eo e , when he p o usion angle Fig. 4 Simula ed e ec i e heigh h e e sus R a o gold-coa ed glass su aces and a andomly gene a ed su ace wi h Gaussian dis ibu ed heigh s. The backg ound image shows he gold coa ed glass su ace on he le and he a i icially gene a ed s uc u e on he igh (Kune and Ha ing 2007) Mic o luid Nano luid (2010) 8:1–10 7 123 equals ze o, he slip leng h is maximized, and he highes possible h oughpu in a mic ochannel is ob ained. The beha io o he slip leng h can be explained by hinking o an inc eased su ace oughness i he p o usion angle is g ea e o less han ze o. Since he a ea ac ion o he bubbles is he same in all he h ee cases, ou esul s clea ly indica e ha slip p ope ies o he su ace can be ailo ed no only by changing he p o usion angle bu also by he a ay geome y. In his s udy, he highes slip leng hs a e ob ained o he hombic uni cell, and i is an ongoing s udy o in es iga e he in luence o he a ay geome y in mo e de ail. Recen ly, ou indings ha e been con i med heo e ically by Da is and Lauga (2009). Nex , he shea - a e dependence o he slip leng h is in es iga ed. As he shea a e and, hus, he iscous s esses g ow, he bubbles a e de o med (see Fig. 6, le ) and he low ield is modi ied. In he cen al pa o Fig. 6, we show he simula ed slip leng h as a unc ion o he Capilla y numbe o h ee di e en p o usion angles. The Capilla y numbe s chosen a e in highe end o he expe imen ally a ailable ange. Ou esul s show shea - a e dependen slip, bu he beha io is opposi e o ha ound in some expe i- men s: in ac , he slip leng hs measu ed by us dec ease wi h inc easing shea due o a de o ma ion o he bubbles. In he expe imen s, su ace o ce appa a uses a e used (see, e.g., Zhu and G anick 2001), whe e a s ong inc ease in he slip is obse ed a e some c i ical shea a e. This shea - a e dependence has been explained, e.g., wi h o ma ion and g ow h o bubbles (Gennes 2002; Lauga and B enne 2004). In ou simula ions, he e is no o ma ion o g ow h o he bubbles as we only simula e a s eady case o gi en bubbles. The expe imen s on he con a y a e dynamic. Howe e , ou esul s indica e ha he changes in he low ield which occu due o he de o ma ion o he bubbles canno be an explana ion o he shea - a e dependence obse ed in some expe imen s. Ou esul s a e consis en wi h Kune and Ha ing (2007) and he p e ious sec ion, whe e i is shown ha smalle oughness leads o smalle alues o a de ec ed slip. In his s udied case, he shea educes he a e age heigh o he bubbles, and hus he a e age scale o he oughness dec eases as well. Finally, we conside a su ace pa e ned wi h g oo es. Cylind ical bubbles p o ude o he low channel om hese holes wi h p o usion angle u=72°, and he a ea ac ion o slo s is 0.53. We apply shea bo h pa allel and pe pen- dicula o he slo s. The slip leng h is s ongly dependen on he low di ec ion (Hy a ¨luoma and Ha ing 2008). Fo pa allel low, he slip leng h is posi i e, bu o he pe - pendicula case, i becomes nega i e. Flow di ec ion a ec s also g ea ly on he shea - a e dependence (c . Fig. 6, igh ). When low is pa allel o he g oo es, no shea - a e depen- dence is obse ed, bu o he pe pendicula case, his dependence is simila o ha seen on hole a ays. These esul s can be unde s ood on he basis o de o ming bubbles. Fo pe pendicula low, he bubbles a e able o de o m, bu o he pa allel case, he bubbles e ain hei shape ega d- less o he shea a e. 5 Conclusion In his a icle, we e iew applica ions o he la ice Bol z- mann me hod o mic o luidic p oblems. The main ocus o his a icle is on ou own esea ch ela ed o he alida ion o he no-slip bounda y condi ion. By in oducing a model o hyd ophobic luid–su ace in e ac ions and s udying p essu e-d i en low in mic ochannels, we show ha an expe imen ally de ec ed slip can ha e i s o igin in hyd o- phobic in e ac ions, bu is cons an wi h a ied shea a es and dec eases wi h inc easing p essu e. Ano he e ec ha was no ully unde s ood so a is he in luence o su aces oughness. We a e able o apply ou simula ions o su ace da a ob ained om AFM measu emen s o expe imen al samples. We show ha igno ing oughness can lead o la ge e o s in a de ec ed slip. In ac , we p opose ha oughness alone could o en be he eason o appa en bounda y slip. Mic oscale bubbles a su aces allow o ailo he slip p ope ies o a su ace. Such a su ace wi h bubbles may yield nega i e slip, i.e., inc eased esis ance o low, i bubbles a e s ongly p o uding o he channel. The la ice Bol zmann simula ions cap u e he de o mabili y o bubbles and hus allow o s udy he in luence o he shea a e on he de o ma ion o he in e ace and i s e ec on he measu ed slip. We ind ha he slip dec eases wi h inc easing shea a e demons a ing ha shea -induced bubble de o ma ion canno explain ecen expe imen al indings whe e slip inc eases wi h inc easing shea a e. In his a icle, we also demons a e he sui abili y o he la ice Bol zmann me hod o modeling mic o luidic applica ions: in con as o molecula dynamics, i is able o 40 20 0 20 40 60 80 100 120 140 Slip leng h [nm] 80 60 40 20 0 20 40 60 80 P o usion angle [°] d Fig. 5 A isualiza ion o he simula ion se up (le ) he lowe su ace is pa e ned wi h holes, while he uppe su ace is mo ed wi h eloci y u 0 .Righ he slip leng h bas a unc ion o p o usion angle u. A uni cell o each a ay is shown in inse s, and co esponding esul s a e gi en by iangles ( hombic a ay), diamonds ( ec angula a ay), and ci cles (squa e a ay). The inse in he op-le co ne shows he de ini ion o u(Hy a ¨luoma and Ha ing 2008) 8 Mic o luid Nano luid (2010) 8:1–10 123 each expe imen ally a ailable ime and leng h scales. This allows one o compa e simula ion esul s o expe imen al da a di ec ly as demons a ed in he case o simula ions o low along su ace da a ob ained om AFM measu emen s o ‘‘ eal’’ samples. Acknowledgmen s This s udy was inanced wi hin he DFG p i- o i y p og am ‘‘nano- and mic o luidics,’’ he collabo a i e esea ch cen e 716, he Ge man academic exchange se ice (DAAD), and by he ‘‘Landess i ung Baden-Wu ¨ embe g.’’ We hank he Neumann Ins i u e o Compu ing, Ju ¨lich and he Scien i ic Supe compu ing Cen e , Ka ls uhe o p o iding he compu ing ime and echnical suppo o his s udy. Jy ki Hokkanen (CSC—Scien i ic Compu ing L d., Espoo, Finland) is g a e ully acknowledged o he bubble isualiza ions. Ch is ian Kune acknowledges he ui ul discussions wi h P. Szymczak. Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion Noncomme cial License which pe - mi s any noncomme cial use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed. Re e ences Al-Zoubi A, B enne G (2008) Simula ing luid low o e sinusoidal su aces using he la ice Bol zmann me hod. 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Phys Re Le 96:224503 Huang H, Tho ne DT, Schaap MG, Sukop MC (2007) P oposed app oxima ion o con ac angles in shan-and-chen- ype mul i- componen mul iphase la ice Bol zmann models. Phys Re E 76:066701 Hy a ¨luoma J, Ha ing J (2008) Slip low o e s uc u ed su aces wi h en apped mic obubbles. Phys Re Le 100:246001 30 20 10 0 10 20 30 Slip leng h [nm] 0.01 0.1 1 Capilla y numbe Ca 1 1 2 2 3 3 4 4 200 100 0 100 200 300 400 500 Slip leng h [nm] 0.01 0.1 1 Capilla y numbe Ca Fig. 6 The le igu e shows a snapsho o a bubble de o med by shea low. In he cen e , he slip leng h as a unc ion o he capilla y numbe o a squa e a ay o bubbles wi h h ee di e en p o usion angles, u=63°,68°, and 71°( om uppe mos o lowe mos ) is shown. The inse shows c oss sec ions o liquid–gas in e aces o ou capilla y numbe s (Hy a ¨luoma and Ha ing 2008). The igh igu e shows he slip leng h as a unc ion o capilla y numbe o a su ace wi h cylind ical bubbles. Ci cles deno e he alues o low pa allel o he bubbles, and diamonds o he pe pendicula di ec ion Mic o luid Nano luid (2010) 8:1–10 9 123