Slip length of confined liquid with small roughness of solid-liquid interfaces

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by  Li Wan

Slip length of confined liquid with small roughness of solid-liquid interfaces
Li Wan∗ and Yunmi Huang
Department of Physics, Wenzhou University, Wenzhou 325035, P. R. China
(Dated: January 4, 2017)
We havestudied theslip length of confined liquid with small roughness of solid-liquid interfaces.
DyadicGreenfunctionandperturbationexpansionhavebeenappliedtogetthesliplengthquantita-
tively. Inthesliplength,botheffectsoftheroughnessoftheinterfacesandthechemicalinteraction
between the liquid and the solid surface are involved. For the numerical calculation, Monte Carlo
method has been used to simulate the rough interfaces and the physical quantities are obtained
7 statistically over the interfaces. Results show that the total slip length of the system is linearly
1 proportional to the slip length contributed from the chemical interaction. And the roughness of
0 the interfaces plays its role as the proportionality factor. For the roughness, the variance of the
2 roughness decreases the total slip length while thecorrelation length of theroughness can enhance
theslip length dramatically to a saturation value.
n
a PACS numbers: 47.10.ad, 47.15.-x, 47.61.-k, 87.85.Rs,47.63.mf
J
Keywords: Slip length, confined liquid, rough interface, correlation length

3

] I. INTRODUCTION And for the interfaces with small roughness, the rough
n
interfaces can be treated as flat ones with the roughness
y
d The dynamics of liquid confined in nano structures involved in the slip length of the flat interfaces. So, for
- have attracted interests due to their potential appli- the latter, a question arises. What is the slip length
u
cations in nanofluidics [1–6]. It has been considered b of the interfaces with the roughness involved? In
l
f that the Navier-Stokes(NS) equation is still valid for this study, we will focus on the interfaces with small
.
s the description of the fluid hydrodynamics even if the roughness and address the problem to get the b relating
c
i liquid is scaleddownto a few molecularlayers[7–14]. In to the roughness in a quantitative way.
s order to solve the NS equation for the confined liquid,
y
h a proper boundary conditions(BC) should be applied The slip length b is defined as the ratio of the shear
p at the solid-liquid interfaces. A commonly used BC viscosity η of liquid to the friction coefficient κ of the
[ is the no-slip BC, which requires that the fluid has solid surface by b = η/κ. It is known that the η is

zero velocity at the interfaces. Such BC increases the position dependent in the liquid [7, 11, 16, 37, 38]. Es-
1
v hydrodynamic resistance to the fluid when the size of pecially, the viscosity of liquid in the region close to the
9 the confined liquid is decreased, which brings difficulties interfaces may be enhanced compared to the viscosity
6 for the applications of the nanofluidics [9]. In order in the region away from the interfaces. The discrepancy
5 to overcome the difficulties, the slippage of liquid at of the viscosities in the two regions leads to the shift of
0 the solid surfaces has been introduced to decrease the the exactlocationofthe BC appliedawayfromthe solid
0
hydrodynamic resistance [2, 9]. The BC with the surfaces by a few molecular layers [16, 39]. However,
.
1 slippage of the liquid at the interfaces is known as the in this study, we consider that the fluid is Newtonian
0 Navier boundary condition (NBC). The NBC requires and the η is constant in the whole computational
7 that the velocity components of the liquid does not domain for simplicity, since we only focus on the ef-
1
vanishattheinterfacesinthe movingdirection. Inorder fectoftheroughnessoftheinterfacesonthesliplengthb.
:
v to quantify the slippage of liquid, a slip length b has
i been introduced in the NBC [15–24]. Generally, the
X The κ is used to show the resistance of the solid
no-slip BC can be recovered from the NBC by setting
r surfaces to the liquid. Generally, two mechanisms
a b = 0. And, larger the slip length b, more slippery the
are contributed to the κ. One mechanism is of the
liquid at the interfaces. Since the slip length b used in
chemical interaction between the liquid molecules and
the NBC determines the solutions to the NS equation
the solid molecules at the solid-liquid interfaces, like
for the confined liquid, many authors have proposed
that the attractive or repulsive force to aqueous liquids
various structures for the interfaces to enhance the slip
depends on whether the solid surfaces are hydrophobic
length b [25–36]. But for the various structures, the
or hydrophilic respectively. The other mechanism is
fundamental structure is the roughness, which is the
due to the fine structures of the interfaces, such as the
nature of the interfaces. For the interfaces with large
roughness of the interfaces. The collusions between
roughness, one has to solve the NS equation by using
the liquid molecules and the rough interfaces bring
mathematical tools such as the finite element method.
the friction force to the liquid. We denote the friction
coefficient contributed from the chemical interaction by
κ and the contribution of roughness of the interfaces
w
∗ [email protected]
by κ in the following. For an ideally flat interface, the
r
2
chemical interaction contributes totally to the κ with A. Fluid Velocity
κ = κ and κ = 0. However, for a real interface, the
w r
roughness is inevitable and κr is nonzero. In such a ThecompleteNavier-Stokes(NS)equationforthefluid
real case, the friction forces respecting the two mecha- in the channel has the form of
nisms then are obtained by f = κ v and f = κ v
w w s r r s d~v 4η
with v the fluid velocity at the interfaces in the η∇×∇×~v+ρ =−∇p+ρ~g−ρ~v·∇~v+ ∇(∇·~v). (1)
s
dt 3
moving direction. Due to the additivity of the friction
forces,wegetthetotalfrictioncoefficientasκ=κ +κ . Here,~v isthe fluidvelocityandρthe liquiddensity. The
w r
ρandthe η havebeenassumedtobe uniformforthe liq-
uidin the channel, since we focus onthe effects fromthe
The determination of friction force of the solid sur- solid surfaces only. In the right hand side of the eq.(1),
faces, hence the b, for the confined liquid has been a the first two terms represent the force of the gradient
long-standing question and has been heavily studied by of pressure and the gravitational force respectively. The
manyauthors[7,11,18,21,25–36,40–43]. Experimental third term is for the convectand the final term is due to
and theoretical results, as well as the Molecular Dynam- the compressibilityofthe fluid. Inthe eq.(1), the equal-
ics Simulations, have been reported. In those studies, ity of ∇2~v =∇(∇·~v)−∇×∇×~v has been applied. We
the roughness of the solid surfaces causing the no-slip think the roughness of the interfaces is not too large to
BC has been pointed out and later confirmed by the bringtheconvecteffectinthechannelandthefluidisin-
Molecular Dynamics Simulations [18, 35, 36]. However, compressible. Thus,wedropoffthelasttwotermsonthe
theroleoftheinterfaceroughnessintheslipBCactually righthandsideofeq.(1). Andforsimplicity,weusef~to
is still lack. A very interesting result basing on the represent the two force terms by f~= −∇p+ρ~g. Based
linear response theory has been obtained, in which the on the above considerations, the complete NS equation
structure factor of the liquid in the region close to the is reduced to be
interfaces is involved in the κ [9, 16]. However, the
ρd~v 1
structure factor does not have an explicit relation to ∇×∇×~v+ = f~. (2)
η dt η
the roughness of the interfaces in that study. In order
to complete the understanding of the flow boundary Now we apply the time Fourier transformations of
conditions for confined liquid, it is necessary to bridge U~(ω) = +∞~veiωtdt and F~(ω) = +∞f~eiωtdt on the
−∞ −∞
the slip length b to the roughness of the solid surfaces, both sideRs of the eq.(2) and transfoRrm the equation in
which is our goal in this study. the frequency domain to be
ρ 1
∇×∇×U~ −i ωU~ = F~. (3)
η η
↔
We introduce a dyadic Green function G(~r,~r′,ω), satis-
fying the following equation
↔ ρ ↔ ↔
∇×∇×G−i ωG= Iδ(~r−~r′), (4)
η
II. THEORY
↔
with I the dyadic unit and ~r(~r′) the position vector in
the channel. We consider the equations of one fixed fre-
We consider a model that simple liquid is confined by quency and drop off the notation of ω for convenience.
two parallel solid plates. We set the coordinate with the ↔
Taking the dot product of G on both sides of the eq.(3),
z axis normal to the plates and the x axis parallel to and taking the dot product of U~ on both sides of eq.(4),
the moving direction of the liquid. The two plates both
finally subtracting the two equations after the dot prod-
are infinitely long to get rid off the ending effects of the
ucts, we obtain
structure. The two solid-liquid interfaces are located at
z = +h and z = −h for the upper and lower plates [∇×∇×U~]·↔G−U~·[∇×∇×↔G]= 1F~·↔G−U~·↔Iδ(~r−~r′). (5)
respectively. The friction force applied on the fluid is η
considered to be opposite to the moving direction of the
Integrating both sides of the eq.(5) over the computa-
xaxis,whilethecomponentofthefrictionforcealongthe tional domain, we get the solution of U~ for the liquid
ydirectionisaveragedtobezero. Forsimplicity,wethink
with the form of
the interfacesareflatalongthe y direction,andthenthe
↔
roughness of the interfaces is the function of x only. We U~(r′)=− [(nˆ×U~(r))·(∇×G(r,r′))
use the function ζ (x) to describe the roughness of the ZZ
u
upper interface anduse ζd(x) forthe lowerone. The two +(nˆ×∇×U~(r))·↔G(r,r′)]ds
roughness functions both are averaged to be zero over
1 ↔
the interfaces at their locations of z = +h and z = −h + F~(r)·G(r,r′)dV. (6)
respectively. ZZZ η
3
On the right hand side of the eq.(6), the first integral of non-ideal interfaces into the friction coefficient κ of the
ds is the integral over the total surface of the com- ideally flat interfaces. The total friction coefficient κ
RpRutational domain, which is obtained by the applying of the flat interfaces after the average then is the sum
of the Green theorem on the volume integral of the left of κ and κ as mentioned in the introduction. The
w r
hand side of the eq.(5). The Green theorem has been total slip length b of the flat interfaces after the average
shown in the Appendix A. The nˆ is the unit vector can be written as b = b b /(b +b ) with b = η/κ
w r w r w w
outward normal to the interfaces. Note that the direc- and b = η/κ by using the equalities of κ = κ +κ
r r w r
tionofthenˆvariesontheinterfacesduetotheroughness. and b = η/κ. It can be seen that for flat interfaces
with b → ∞, the contribution of chemical interaction
r
Forsimplicity,weassumetheinletofthechannelalong dominates the friction force with b = b . On the other
w
thexdirectionisperiodictotheoutlet. Thustheintegral side, when b →∞, the friction force then is dominated
w
over the inlet and outlet cross-section surfaces would be by the roughness of the interfaces with b = b . So, after
r
cancelled. Similar cancellation is hold for the integral getting the total slip length b of the confined liquid,
over the cross-section surfaces normal to the y direction the rough solid-liquid interfaces can be replaced by flat
due to the uniformity along the direction. In this way, ones in an effective way, which simplifies the problems
the term of surface integral in the eq.(6) remains as the of the fluid dynamics in real systems with the roughness
integral over the solid-liquid interfaces only. Without having been involved in the b. In the following study,
losing the generality, we will focus on only the upper we will set the b as a priori since the b depends on
w w
interface to figure out the contribution of the roughness the natural interaction between the solid surfaces and
to the slip length, since the similar treatment can be fluid molecules, and focus on the informations of the
applied to the lower interface. For convenience, we use roughness.
K to represent the integrand of the surface integral by
We start from the upper rough interface. For conve-
↔ ↔
K =−(nˆ×∇×U~)·G−(nˆ×U~)·(∇×G). (7) nience,wedenoteτˆtheunitvectorontheinterfacealong
the tangential direction in the x,z plane and perpendic-
ular to the nˆ. The τˆ can be expressed by τˆ = nˆ×U~×nˆ.
|U~×nˆ|
B. Boundary Condition
Thelocalforcebalanceontheroughinterfaces,hencethe
NBC, can be found generally to be
The essential of the NBC is that the friction force ap-
plied on the fluid per unit length should be balanced by U~ ·nˆ =0, (8)
the shear stressof the fluid at the interface. The friction ηnˆ·(∇U~)·τˆ=−κ U~ ·τˆ, (9)
force and the shear stress have the same magnitude but w
the opposite directions. We denote the velocity compo- nˆ·(∇U~)·(τˆ×nˆ)=0. (10)
nent of the fluid parallel to the interfaces by v . Then
||
In the general NBC, eq.(8) means the component of
in the case that the solid-liquid interfaces are ideally
flat, the shear stress is along the x direction and has U~ normal to the interfaces vanishes, showing that the
been well defined as the product of the liquid viscosity liquid can not penetrate the solid surface. In the eq.(9),
η and the normal gradient of the v to the interface, the term on the left hand side is the shear stress which
||
while the friction force is along the −x direction and is is along the τˆ direction and is on the plan normal to
the product of the friction coefficient κ and the v at nˆ. The term on the right hand side of the eq.(9) is
||
the interface. Thus, for the ideal interfaces, the force the friction force. The final condition eq.(10) means no
balance can be expressed by η∂v|| = −κ v valid at stress along the y direction according to our model.
∂z w ||
the interfaces, with the minus meaning the friction force
It is emphasized here that the slip length used in the
resists the fluid flow. This is exactly the NBC. Here,
eq.(9) is the b = η/κ and no κ is involved. This is
only the κ appears in the NBC, since no any effect of w w r
w
because the force balance considered now is hold locally.
roughness of the interfaces contributes to the friction
Theinvolvementofκ inthesliplengthbwillbeobtained
coefficient in such ideal case. The slip length then is r
aftertheaveragingoftheeq.(6)overtheroughinterfaces
defined as b =η/κ .
w w
by using the general NBC. For the application of the
generalNBC,weneedtomodifytheeq.(8,9,10)tohave
In the non-ideal case that the solid-liquid interfaces
the following forms
are not ideally flat but with roughness, the NBC can be
expressed locally since the force balance is still hold lo-
U~ ·nˆ =0, (11)
callyonthe interfaces. We willaveragethe fluidvelocity
1
eq.(6) over the rough interfaces to get an effective fluid U~ ·[(nˆ·∇)U~]=− U~ ·U~, (12)
velocity. From an alternative view, the effective fluid bw
velocitycanbesolvedfromtheNSequationbyimposing [(nˆ·∇)U~]·[nˆ×U~]=0. (13)
an effective NBC(ENBC) on ideally flat interfaces.
The ENBC has absorbed the effect of roughness of the TheabovemodificationcanbefoundintheAppendixB.
4
C. Resolution of K D. Perturbation Expansion
WeneedtoresolutetheK definedineq.(7)intoseveral Wehaveusedthefunctionζ (x)todescribetherough-
u
terms to figure out the effect of the roughness. Before nessoftheupperinterface. Forconvenience,weomitthe
the resolution, we decompose the vector ∇×U~ into two subscripttodenotetheroughnessbyζ(x). Thelocalunit
vectors with one vector along the y axis and the other vectors of the interface can be expressed in the term of
vector in the x,z plan, having ζ(x), reading
[(∇×U~)·(nˆ×U~)](nˆ×U~) 1 ∂ζ
∇×U~ = nˆ = (− , 0, 1),
(nˆ×U~)·(nˆ×U~) N ∂x
1 ∂ζ
(nˆ×U~)×(∇×U~)×(nˆ×U~) τˆ= (1, 0, ), (21)
+ . (14) N ∂x
(nˆ×U~)·(nˆ×U~)
with
By using the eq.(11, 12, 13,14), the vector n×∇×U in
∂ζ
the K can be split into three terms, reading N = ( )2+1. (22)
r ∂x
nˆ×∇×U~ =T~1+T~2+T~3, (15)
Now we need to average the fluid velocity U~ over the
with rough interface to get an effective velocity W~ =< U~ >.
−U~ ·U~(nˆ×nˆ×U~) Alternatively, the fluid vector W~ can be solved from the
T~1 = , NS equation by imposing the ENBC on a flat interface.
b (nˆ×U~)·(nˆ×U~)
w The ENBC for such flat interface should have the form
−nˆ·((U~ ·∇)U~)(nˆ×nˆ×U~) of
T~2 = ,
(nˆ×U~)·(nˆ×U~) ∂W W
x x
W =W =0, =− . (23)
(nˆ×U~)[nˆ·((nˆ×U~)·∇)U~] z y ∂z b
T~3 = .
(nˆ×U~)·(nˆ×U~) Here, the total slip length b has been used to comprise
the two effects of the interface, the chemical interaction
The details for the above derivation can be found in the
and the roughness. The dyadic Green function for the
AppendixC.WewritetheU~ intheformofU~ =Mτˆwith ↔
effective system with the flat interface is denoted by g.
M the magnitude of U~. Then, we can further simplify ↔
The boundary conditions for the g are the same as the
the terms of T~1,T~2,T~3 to be eq.(23)onlybyreplacingtheW withtheg whereiand
i ij
Mτˆ j represent the x, y and z coordinates. Note that g is
T~1 = , (16) ↔ ij
b the component of g, meaning that the influence of the
w
T~2 =Mτˆ(nˆ·(τˆ·∇)τˆ), (17) source of i component is↔on the object of j component.
We expand the U~ and G of the rough interface in the
T~3 =M(nˆ×τˆ)(nˆ·((nˆ×τˆ)·∇)τˆ). (18) termsofW~ and↔g oftheflatinterfacerespectively. Since
In the above simplification, the equality of nˆ·τˆ= 0 has the friction force consideredis parallelto the x direction
been applied. The similar treatment can be applied to and the y component of the friction force is negligible,
the vector nˆ×U~ in the K, leading to weonlyconsiderthexcomponentofU~ fortheexpansion
andneglecttheyandzcomponents. Thelattertwocom-
nˆ×U~ =T~4 =Mnˆ×τˆ. (19) ponentsareaveragedto be zeroovertheroughinterface.
↔
Thus, the integrandK defined in the eq.(7) can be writ- In this way, for the G, we only consider the components
ten as ofGxi. Wehavenotedthattheroughnessoftheinterface
should not be too large, or the roughness will bring the
↔ ↔
K =−(T~1+T~2+T~3)·G−T~4·(∇×G). (20) convection and turbulence effects to the system. Under
the condition of |ζ/b| << 1, the perturbation expansion
It can be understood that for an ideally flat interface
of the U~ ·xˆ and the G read
where the τˆ is nota variablebut a constantvectoralong xi
the x direction on the interface, the two terms of T~2 and ∂Wx ζ
Mτˆ·xˆ=W +ζ =W (1− ),
T~3 bothdisappearbecausetheoperator∇appliedonthe x ∂z x b
constantvectorτˆcauseszerointheeq.(17,18). Andthen ∂g ζ
xi
only the terms of T~1·↔G and T~4·(∇×↔G) remain in the Gxi =gxi+ζ ∂z =gxi(1− b). (24)
K for the ideally flat interface. Therefor, the terms of
In above derivation, the ENBC of eq.(23) have been
↔
(T~2+T~3)·G in the K are originatedfrom the roughness applied.
of the interface.
5
In the following, we expand the K in the terms of W the correlation function is Gaussian, which reads
and g. For the first term T~1·↔G of the K, we get < ζ(x′)ζ(x′ +x) >= σ2e−x2/l2. Here, σ is the variance
of the roughness. The x is the distance between two
T~1·Gxixˆˆi= Wx(1− ζ)2gxiˆi. (25) positions on the interface for the correlation. And the l
bw b is the correlationlength, meaning that the two positions
with their distance larger than l lose their correlation.
By substituting the eq.(21,24) into the second term of
Theinterfacewithalongerl hasasmoothertopography.
the K, we get
In the expression eq.(29) of b, the information of l has
T~2·Gxixˆˆi=Wx(1− ζ)2(nˆ·(τˆ·∇)τˆ)gxiˆi been reflected by the factor of ∂∂x2ζ2 in the A. We have
b used Mont Carlo method to simulate various rough in-
ζ ∂2ζ terfacesbyvaryingσ andl. The latticeparameterofthe
=W (1− )2 N−3g ˆi (26)
x b ∂x2 xi solid surface is used for the dimensionless. The physical
quantities in this work are evaluated statistically over
We observe that the vectors T3 and T4 both are along the rough interfaces. Fig. 1 shows that the < N > is
they direction,whichhavenegligiblecontributiontothe
functional of the correlation length l of the interface
effect of the roughness as mentioned in this work. Thus,
by varying the roughness variance σ. In the figure, we
we will figure out the roughness information only from
the two terms of eq.(25) and eq.(26).

E. Slip Length 100
Thesurfaceintegraloftheeq.(25)andeq.(26)obtained
=4
from the eq.(6) can be effectively equivalent to
=32
W >10 =64
< [T~1·Gxixˆˆi+T~2·Gxixˆˆi]ds>= xgxiˆidxdy. N
Z Z Z Z b <
(27)
The right hand side of the above equation is the sur-
face integral over the flat interface after the average and 1
canbe understoodfromthe eq.(16)with the eq.(17) and
eq.(18) vanishing at the flat interface. Here, b is used to 0 60 120 180 240 300
comprisethe botheffects ofthe chemicalinteractionand l
theroughnessfortheflatinterface. Therefore,wegetthe
FIG. 1: Statistical average of N over interfaces, noted
equation for the total slip length b, reading
by <N >, as a function of the correlation length l with
N ζ ∂2ζ ζ 1 various roughness variances σ. The N has been defined
< (1− )2+ N−2(1− )2 >= (28)
b b ∂x2 b b in the text by eq.(22). And the <N > is plotted in the
w
logarithmic scale.
by using the ds = Ndxdy. The factor of
W g ˆidxdy onRbRoth sides RofRthe equation are canceled,
x xi
find that the < N > deviates from the number of one
meaning that the b is a character of the confined liquid,
in the range of l close to zero. Larger the σ, stronger
andisindependent onthe velocityoffluidandtime. For
simplicity, we denote A = 1 ∂2ζ + N . After averaging the deviation of the < N >. With the increasing of
N2∂x2 bw l, the < N > decreases and approaches to the unit,
both sides of the eq.(28) statistically over the interface,
indicating that the interface is becoming smooth with
we solve the slip length b as ∂ζ going to zero and τˆ deviating from the unit vector
∂x
2<ζA>+1+ (2<ζA>+1)2−4<A><ζ2A> xˆ slightly. A rougher interface with a larger σ need
b= . a larger l to smooth the interface and to recover the
p 2<A>
< N > to the unit. In the fig.1, the deviation of the
(29)
< N > actually means that the l and the σ both play
This is the main result of this work. their roles in determining the slip length. And the l
gradually weakens its effect in the determination with
the increasing of l. Finally, only the σ remains to de-
III. RESULTS AND DISCUSSIONS terminethesliplengthifthelgoestothelimitofinfinity.
The topography of the rough interface is described Fig. 2 shows the slip lengths as functional of l with
by the ζ(x) together with the correlation func- various σ. It should be noted that in the eq.(24) the
tion of the roughness. For simplicity, we set that |ζ/b| << 1 is used for the perturbation expansion
6
instead of |ζ| <<1. Thus, we use σ/b to represent the
w
variance of the roughness in the plot instead of σ itself,
1.2
(a) considering that the b is equivalent to b in the extreme
w
1.0 case of σ =0. We apply σ/b ≤0.4 in the plot to verify
w
the perturbation expansion. Fig.2(a) and (b) are for
0.8
the total slip length b basing on the eq.(29), while the
w
b
b / 0.6 bw=10 =0.1 fig.2(c) and (d) are for the slip length br contributed
=0.2 from the roughness only. The b is obtained from
0.4 = /bw =0.3 1 = 1 − 1 . The slip lengths are rrenormalized by the
=0.4 br b bw
0.2 b in the figures for clarity. The b has been indicated
w w
in each figure. In the fig.2(a) and (b), it can be found
0.0
0 50 100 150 200 250 300 that the b increases dramatically with the increasing of
l thel andthengoestoasaturationvalue. Thesaturation
value of the b is independent on the l, but decreases

1.2 with the increasing of σ as expected. The increasing of
(b)
b with the l emphasizes that in order to increase the
1.0
b it is essential not only by choosing proper material
0.8 for the channel fabrication, but also by smoothing the
w solid surfaces in the fabrication. The b in the fig.2(c)
b / b 0.6 and (d) are calculated from the data ofrfig.2(a) and (b)
=0.1
0.4 respectively, showing the similar behaviors of b in the
bw=80 =0.2
=0.3 fig.2(a) and (b). In the fig.2(d), we have appended the
0.2 = /bw =0.4 scale of l to 1000 because the b /b does not reach the
r w
0.0 saturation at the l = 300. The fig. 2(c) and (d) show
0 50 100 150 200 250 300
l that the slip length contributed from the roughness is a
function of l and σ in the range of l close to zero, and
finally is a function of σ only when the l goes to the
90
(c) limit of infinity. We need to note that in the fig.2 the
scaling behaviors of the slip lengths are dependent on
the ratio of σ to b rather than the σ itself. And for the
60 w
w =0.1 interfaces with large roughness where the perturbation
b / br bw==1/b0w ==00..23 expansion is invalid, say σ/bw > 0.4, rigorous numerical
=0.4 calculations such as the finite element method should
30
be applied to solve the NS equation, which is out of the
scope of this work.
0
0 50 100 150 200 250 300 Before we figure out the relation between the the
l
b(b ) and the b , we firstly make a simple analysis. We
r w
approximate the following equalities of < A >= 1/b ,
w
90 < ζA >=< ζ ∂2ζ > and < ζ2A >=< ζ2 > /b and
(d) N2∂x2 w
find that b can be expressed by b ∼ b < f(ζ, ∂2ζ) >.
w ∂x2
That means b is linearly proportional to b and the
60 w
w roughness of the interface plays its role as the propor-
b / br bw=80 ==00..12 tionality factor. Now we check the proportion in fig.3
=0.3 with various σ. Here, the correlation length is fixed
30 = /bw =0.4
at l = 200 in the plot and σ is used instead of the
σ/b . Fig.3(a) shows that the ratio of the b to the b
w w
goes up to a constant with the increasing of b . The
0 w
0 200 400 600 800 1000 constant ratio confirms the linear proportion of the b to
l the b and the roughness plays as the proportionality
w
FIG. 2: Slip lengths as functions of the correlation factor. Larger the σ, smaller the ratio, meaning that
length l with various roughness variances σ. (a) and (b) the roughness weakens the effect of bw. Fig.3(b) is for
are for the total slip length b of the system while the (c) the br calculated from the data of fig.3(a), also showing
and (d) are for the slip length b contributed from the the linear behavior as the b. The relationship between
r
roughness of interfaces. In the plot, the slip lengths and the br and the bw indicates that the roughness of the
σ are all renormalizedby the slip length b contributed interface can not play the role itself in determining the
w
from the chemical interaction of the interfaces. b, but need to be with the bw together. That means if
b is infinite, b must be infinite too and then is the b.
w r
7
It has been revealed that some biological nanopores
havepeculiar features in the transportof water. Such as
IV. CONCLUSION intheaquaporinchannelsasthekeycomponentofmany
biological processes, the permeability of water is larger
In this work, we have studied the slip length of the than what is expected in classicalnanofluidic framework
by three orders of magnitude [43]. It has been suggested
confinedliquidwithsmallroughnessofsolid-liquidinter-
faces. The quantitative expression of the slip length has that the hydrophobic surface of the aquaporin channels
is the potential mechanism to the large permeability of
been obtained, in which the roughness of the interfaces
and the chemical interaction between the liquid and the water in the channels. To supplement the suggestion,
we think that the large correlation length of the chan-
solid are involved. In the study, perturbation expansion
of the interface roughness is required. The results we nels may contribute to the water transport as have been
studied in this work, even though how the positions are
have obtained are threefold. First, the slip length b of
the confined liquid is linearly proportional to the slip correlated in the aquaporin channels is not clear. What
length b contributed from the chemical interaction. is more, the biological nanopores are active matters, in
w
Second, the roughness of the interface plays its role which the correlation between the two positions on the
as a proportionality factor for the b and the b . And nanopore interfaces may behave in a complicated way
w
only with the chemical interaction together does the ratherthanwhatdoesin the artificialnanofluidic frame-
roughnesshaveits effect onthe b. Finally,largevariance works. Theconsiderationofthecorrelationlengthinthe
of the roughness decreases the b and the increasing of biological nanopores is the extension of this work and
the correlation length of the roughness increases the under the research.
slip length in a dramatical way to a saturation value.
Those results propose that we need to improve our
techniques to smooth the solid surfaces by enlarging
the correlation length of the roughness as well as Appendix A
decreasing the variance of the roughness for the channel
fabricationbesidestofindpropermaterialswithlargeb .
w The second kind Green theorem is
↔ ↔
[(∇×∇×U~)·G−U~ ·(∇×∇×G)]dV
ZZZ
1.0
(a) = [(nˆ×U~)·(∇×↔G)+(nˆ×∇×U~)·↔G]ds. (A1)
ZZ
0.9
w0.8
b
b / l=200
0.7 Appendix B
=10
=20
0.6
=40
We need to modify the NBC eq.(8, 9, 10)for the appli-
0.5 cationasthefirststepinsplittingthe K. The eq.(8)can
0 200 400 600 800 1000
bw be written in the form of

U n =0, (B1)
80 a a
(b)
70
60 l=200 with the Einstein’s notation. Here, the subscript a
represents the coordinates x,y,z.
w 50 =10
b
b / r 40 ==2400 Then, we substitute the τ = nˆ×U~×nˆ into the eq.(9),
30 |U~×nˆ|
20 and get rid of the common denominators on both sides
10 of the equation. In this way, we get the modified eq.(9)
reading
0
0 200 400 600 800 1000
bw 1
nˆ·(∇U~)·[nˆ×U~ ×nˆ]=− U~ ·[nˆ×U~ ×nˆ]. (B2)
FIG. 3: (a)Proportion of b to bw. (b)Proportion of br to bw
b . In the plot, the b and b have been renormalized by
w r
the b for clarity.
w The left hand side of the eq.(B2) can be simplified by
8
using the Einstein’s notation Thus, for the eq.(C1), we have
nˆ·(∇U~)·[nˆ×U~ ×nˆ]
∂U
=(na~ia)·(~ib∂xc~ic)·(ǫefg~ienfǫghkUhnk) nˆ×[(∇×U~)·(nˆ×U~)](nˆ×U~)
b
∂U 1
=ǫ ǫ n en U n =[− U~ ·U~ −nˆ·[(U~ ·∇)U~]](nˆ×nˆ×U~). (C2)
efg ghk b∂x f h k b
b w
∂U And
e
=(δ δ −δ δ )n n U n
eh fk ek fh b f h k
∂x
b
∂U
=n U e =U~ ·[(n·∇)U~]. (B3)
b e
∂x
b (nˆ×U~)×(∇×U~)×(nˆ×U~)
Here, the eq.(B1) has been applied. Similarly, the right ∂U ∂U
c b
hand side of the eq.(B2) can be simplified as − 1 U~ ·U~. =ǫbefianeUf(ncUa∂x +naUc∂x
bw b c
So the eq.(B2) can be transformed as ∂U ∂U
c b
−n U −n U )
a c c a
∂x ∂x
1 b c
U~ ·[(n·∇)U~]=− U~ ·U~. (B4) =U~nˆ·[(nˆ×U~)·∇]U~ +nˆ(nˆ×U~)·[(U~ ·∇)U~]
b
w
−nˆU~ ·[(nˆ×U~)·∇]U~ −U~[(nˆ·∇)U~ ·(nˆ×U~)].
This equation will be used for the splitting of the K.
(C3)
The eq.(10) then can be transformed to be
∂U
nˆ·(∇U~)·(U~ ×nˆ)=ǫbefnaneUf b =0 The last term of the above equation is dropped off by
∂x
a
using the eq.(B8). For the eq.(C3), we have
⇒[(n·∇)U~]·[n×U~]=0. (B5)
For clarity, we collect the modified NBC here
U~ ·nˆ =0, (B6) nˆ×(nˆ×U~)×(∇×U~)×(nˆ×U~)
1 =(nˆ×U~)[nˆ·[(nˆ×U~)·∇]U~], (C4)
U~ ·[(nˆ·∇)U~]=− U~ ·U~, (B7)
b
w
[(nˆ·∇)U~]·[nˆ×U~]=0. (B8)
By using the equality of nˆ × nˆ = 0. In this way, the
Appendix C nˆ×∇×U~ can be split into three terms by
Before we resolve the K, we need to decompose the
∇×U by two vectors. One vector is along the nˆ ×τˆ.
The other vector is in the y,z plan. The decomposition −U~ ·U~(nˆ×nˆ×U~)
T1 = ,
has been shown in the eq.(14). Now we simplify the nu- b (nˆ×U~)·(nˆ×U~)
w
merators of the equation. The factors of the terms are
−nˆ·((U~ ·∇)U~)(nˆ×nˆ×U~)
simplified as the following. T2 = ,
(nˆ×U~)·(nˆ×U~)
∂U ∂U
(∇×U~)·(nˆ×U~)=n U k −n U k (nˆ×U~)[nˆ·((nˆ×U~)·∇)U~]
j k∂xj k j ∂xj T3 = (nˆ×U~)·(nˆ×U~) .
=U~ ·[(nˆ·∇)U~]−nˆ·[(U~ ·∇)U~]
1
=− U~ ·U~ −nˆ·[(U~ ·∇)U~]. (C1)
b
w
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