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.
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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
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