Computer studies of ﬂuctuationof conﬁned ﬂuid
Kyoto Univ. Hiroyoshi Nakano
熱場の量子論 2019/9/4
in collaboration with Shin-ichi Sasa

Boundary condition on Navier-Stokes equation
Macroscopic behavior of ﬂuids is described by NS equation.
incompressible ﬂuid

r·v =0@v+ (v · r)v =@t
Boundary condition
Motion of ﬂuid at solid surface⇢ = const1
⌘
rp +v
⇢⇢

Motivation for investigation of boundary condition
Boundary conditions in textbook

no-slip BCv

s=0Surface roughness brings ﬂuid to rest.

Development of nanotechnology from 1990s・control of roughness
・increasing of measurement accuracy(Zhu and Granick, 2001)

breakdown ofno-slip BCNew transport coeﬃcient

no-slip BCv s=0partial slip BCdvv =bdz ss

・Slip length is new transport coeﬃcient that characterizes
the amount of slip. (cf. viscosity, thermal conductivity)
・No-slip BC is a special case of partial slip BC. ( b = 0 )Application of slip phenomena
D. Erickson, D. Li / Analytica Chimica Acta 507 (2004) 11–26

ded PVDF (poly(vinylidenefluoride))
membranes adsorbed
Typically, slip length is about0nm∼30nm
with BSA. Prest et al. [112] describes an integrated single
working electrode PDMS device for the isotachophoretic
separation of metal cations. The electrode was integrated
into the chip by placing it between the two polymer layers
prior to thermal bonding of the two substrates.

Nano- and Microﬂuidics

Control and manipulationof ﬂuids
4. Devices for cell handling, sorting andgeneral analysis
at submicron scale

Here, we review some of the interesting devices w
plications in cell handling and microscale flow cyto
dielectrophoretic sorting, and other general cell atechniques.

L ∼ 4.1. Cell handling and cytometry
100nm Gawad et al. [117] presented a microfluidic cyto

In addition to on-chip DNA analysis and capillary electrophoresis, there has been a large amount of research directed towards the integration of microfluidic technologies
with different aspects of cellular analysis [6,20]. Recentreviews have discussed these directions in the context of
single cell analysis by capillary electrophoresis [113], drug
development [114], tissue engineering [115], sample preparation for molecular diagnostics [11] and biosensors [116].

tool, based on the micro Coulter particle counter prfor cell counting and separation. The device, shown in
consists of a glass–polymer chip with integrated chand electrodes and functions by introducing suspend
ticles into the measurement area, via pressure driven lflow, where the spectral impedance of the cell is meand subsequently used to determine its size. Screeninon the order of 100 samples/s were reported. An ining study on the effectiveness of different electrode aments, based on FEM simulations, and details of tnal conditioning technique were also presented in thi

becomes
newto beuser-controlled.
1022Slip lengthAppl. Phys.
Lett., Vol.a79,No.variable
7, 13 August
2001

Lab-on-a-chip devices

Y. Gogotsi et al., Appl. Rhys. Lett. 79, 1021 (2001)

S. Gawad et al., Lab. Chip 1, 76(2001)

Theme of this presentation
Our goal：to establish relationship between slip length andmolecular parameters(wall structure, ﬂuid structure)
Tools

Molecular dynamical
simulation

Green-Kubo formula

Linearized
ﬂuctuating hydrodynamics

Previous studies

Our proposalMolecular dynamical simulationsquare latticeinteraction between
ﬂuid particle and solidparticles

Interaction between ﬂuid particles

V (r) = 4✏⇣ ⌘12r

⇣ ⌘6r

VFS (r) = 4✏{⇣FSr⌘12cFS⇣FSr⌘6

Microscopic description: Hamiltonʼs equation}

Green-Kubo formulacoarsegraining
Kirkwood (1946), Zwanzig (1960)

GK formula for slip length⌘1=lim limbkB T S t!1 L!1Z t

F̂ (t)

dshF̂t (s)F̂t (0)i0

System size is macroscopic.

microscopic force
acting on wall

Bocquet and Barrat (1994)
Nakano and Sasa (2019)

GK formula × MD simulationis very diﬃcult, mainly because of macroscopic limit.
GK formula for slip length⌘1=
lim limb
kB T S t!1 L!1Z t
0

dshF̂t (s)F̂t (0)i

GK formula × MD simulation
GK formula for slip length⌘1=
lim limb
kB T S t!1 L!1Z t

dshF̂t (s)F̂t (0)i0Calculation method

1, prepare equilibrium state at t = 02, calculate microscopic force F̂ (t) from snapshot
3, calculate force autocorrelation function

1hF̂ (t)F̂ (0)ieq =T
4, calculate time integral

1(t) =kB T SZTdsF̂ (t + s)F̂ (s)0Z t0dshF̂t (s)F̂t (0)iCalculation result
0.25

Parameters

LL = 20.0√−3γ(t)/ mϵσ

0.20(Lx , Ly ) = (48.0 , 48.0 )⇢ = 0.75

3dshF̂t (s)F̂t (0)i

50

100

150

0

0.15
0.10
0.05
0.000

Units
length: diameter of ﬂuid particle
time: microscopic time ⌧micro
vibration

1(t) =
kB T SZ t

0.83⌧microt/τmicro

200

GK formula for slip length⌘= lim lim (t)t!1 L!1bCalculation result
0.25ParametersL
L = 20.0√−3γ(t)/ mϵσ0.20(Lx , Ly ) = (48.0 , 48.0 )⇢ = 0.75

31(t) =kB T SZ tdshF̂t (s)F̂t (0)i

00.15

⌘
L + 2b0.10
0.05

Relaxation time of ﬂuid

0.000
Units
length: diameter of ﬂuid particle
time: microscopic time ⌧micro
vibration

0.83⌧micro

50

100

t/τmicro

150

200

GK formula for slip length⌘= lim lim (t)t!1 L!1bSystem Size Dependence
GK formula for slip length

High computational cost√−3γ(t)/ mϵσ⌘= lim lim (t)t!1 L!1b0.20

red：L = 20.0
blue：L = 30.0

0.15
0.10
0.05
020406080t/τmicro

L100

System Size Dependence
GK formula for slip length

High computational cost√−3γ(t)/ mϵσ⌘= lim lim (t)t!1 L!1b0.20

red：L = 20.0
blue：L = 30.0

0.15
0.10
0.05
020

Where is correct slip length
obtained?

GK formula is not practical.

406080t/τmicro

L100

New strategy
Our goal：to establish relationship between slip length andmolecular parameters(wall structure, ﬂuid structure)
to extract information on slip length from

(t)

Tools
Previous studies

Molecular dynamicalsimulation
Green-Kubo formula

Linearized
ﬂuctuating hydrodynamics
Our proposal

New strategy
Our goal：to establish relationship between slip length andmolecular parameters(wall structure, ﬂuid structure)
to extract information on slip length from

(t)

0.25

√
γ(t)/ mϵσ −30.20
0.15(t) =1kB T S

Z tdshF̂t (s)F̂t (0)i

Linearized
ﬂuctuating hydrodynamics

0in ﬁnite size

0.10

describes this data

0.05
0.000

50100t/τmicro

150200Fluctuating hydrodynamics

Continuum description of ﬂuctuation
Landau and Lifshitz, 1959

Navier-Stokes equation + ﬂuctuationr·v =0@v+ (v · r)v =@t

1⌘
rp +v +f⇢⇢

Continuum description of ﬂuctuation
Landau and Lifshitz, 1959

Navier-Stokes equation + ﬂuctuationr·v =0@v+ (v · r)v =@t
Properties of ﬂuctuation

1⌘
rp +v +f⇢⇢

Fluctuating hydrodynamics

0, Gaussian white noise 1, detailed balance condition
2, conserve law of momentum 3, isotropic properties of bulk
ab@s(r, t)
af (r, t) =@rb2
hs (r1 , t1 )s (r2 , t2 )i = 2kB T ⌘ ac bd + ad bc + ab cd (r13abcdr2 ) (t1t2 )Continuum description of ﬂuctuation
Landau and Lifshitz, 1959

Navier-Stokes equation + ﬂuctuationr·v =0@v+ (v · r)v =@t
Properties of ﬂuctuation1⌘rp +v +f⇢⇢

Fluctuating hydrodynamics

0, Gaussian white noise 1, detailed balance condition
2, conserve law of momentum 3, isotropic properties of bulk
ab@s(r, t)
af (r, t) =@rb2hs (r1 , t1 )s (r2 , t2 )i = 2kB T ⌘ ac bd + ad bc + ab cd (r13abcdr2 ) (t1t2 )Fluctuation of conﬁned ﬂuids Nakano and Sasa (2019)
Linearized ﬂuctuating hydrodynamics

r·v =0@v=@tL

1⌘
rp +v+f⇢⇢

Deterministic partial slip BCdvv =bdz ss

Force acting on wallZ

⇣ @v x⌘Ft =dxdy ⌘+ sxz@zwall
S

Exact calculation
Z t1(t) =dshF̂t (s)F̂t (0)ikB T S 0

Fluctuating hydrodynamics

×MD simulation

Comparison between simulation data andﬂuctuating hydrodynamics

ﬂuctuating
NS equation
Exact solution

0dshF̂t (s)F̂t (0)i1(t) =kB T S
0.25

Z t

dshF̂t (s)F̂t (0)i

5010000.20√
γ(t)/ mϵσ −31(t) =kB T SZ tsimulation

0.15
0.10
0.05
0.000

t/τmicro

150200Comparison between simulation data andﬂuctuating hydrodynamics

ﬂuctuating
NS equation
Exact solution

dshF̂t (s)F̂t (0)i01(t) =kB T S
0.25

Z t

dshF̂t (s)F̂t (0)i

5010000.20Fitting√
γ(t)/ mϵσ −31(t) =kB T SZ tsimulation

0.15
0.100.050.000t/τmicro

150200Fitting parameter(⌘, ⇢)bParameters⌘ ：viscosity in bulk, obtained in advance⇢ ：density in bulk, obtained from simulation datab ：slip lengthﬁtting parameter

Fitting Results
blue：ﬂuctuating hydrodynamics
red：simulation data√−3γ(t)/ mϵσ0.20
0.15
0.10
0.050.000

50100t/τmicro

150200Fluctuating hydrodynamics reproduces simulation data in fulltime region.
Fluctuation of force acting on wall is characterized by only
three parameters (⌘, ⇢, b) .

Fitting Results
blue：ﬂuctuating hydrodynamics
red：simulation data√−3γ(t)/ mϵσ0.20
0.15
0.10
0.05
0.000

50100t/τmicro

150200Fluctuating hydrodynamics reproduces simulation data in fulltime region.
Fluctuation of force acting on wall is characterized by only
three parameters (⌘, ⇢, b) .

Fitting Results
0.225

0.83⌧micro√−3γ(t)/ mϵσ0.200vibration

0.175
0.150

0.3⌧micro

0.125
0.1000246t/τmicro

810

Fluctuating hydrodynamics accurately describes simulation
data even in microscopic time scale.

⟨−0.00002
01

234Comparison with non-equilibrium measurementt/τmicro(a)Slip length is obtained in two diﬀerent way.

FIG. 2. (a) ⟨F (t)F (0)⟩eq and (b) γ(t) for wall (0.6σ, 0Insetofin (a)(t)presents the enlargement of time region [0, τ
1, best-ﬁtting parameterγ(t).

2, observation of velocity ﬁeld in non-equilibrium steady state
cFS
0.8
0.4
0.0

bneq /σ7.5
30.5
118 ± 7.5

beq /σ
7.8
29.2
104

TABLE I. Comparison of slip length calculated fromdiﬀerent methods. Column 1 gives the wall parameters.
and bneq give, respectively, the slip length estimated fromfirst peak of γ(t), calculated by NEMD simulation. bDeviations between
NEMDandEMD
simulations
the slip length measured by new equilibrium measure
are within 10%. method we proposed, specifically, by fitting the simul
data to the linearized fluctuating hydrodynamics. The d
are explained in Sec. V A.

Our goal：to establish relationship between slip length andmolecular parameters(wall structure, ﬂuid structure)

to extract information on slip length from(t)0.2250.83⌧micro√γ(t)/ mϵσ −30.200
We can obtain more information
by analyzing linearized
ﬂuctuating hydrodynamics.

Application

vibration

0.175
0.150 0.3⌧

micro

0.125
0.1000246t/τmicro

810

Green-Kubo-like formula
Exact result of LFH

limt!+0⌘(t) =b0.225⌘b√−3γ(t)/ mϵσ0.200
0.175
0.150 0.3⌧micro
0.125
0.1000246t/τmicro

810

Green-Kubo-like formula
Exact result of LFH

limt!+0

Reasonable estimation⌘peak ⇠b⌘(t) =bL. Bocquet and J.-L. Barrat, 1994

0.225⌘b√−3γ(t)/ mϵσ0.200

peak

0.175
0.150 0.3⌧micro
0.125
0.1000246t/τmicro810

in progress

Application to rough wall

Our goal：to establish relationship between slip length andmolecular parameters(wall structure, ﬂuid structure)
Reasonable estimation⌘peak ⇠bL. Bocquet and J.-L. Barrat, 1994

Rough wall
Our derivation expands the range of application.

Np /S (density of pillars)

When density of pillars are small,Z ⌧01peak =
kB T S0b⇠<latexit sha1_base64="ZYdzrYKWis5vapwbbVwaVdQ5EeU=">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</latexit>

NpdshFt (s)Ft (0)i ' plate +pillarS

1Npbplate + S
1pillar⌘universal relation

Important conclusions
Linearized ﬂuctuatinghydrodynamics
provides description of (t)in ﬁnite systemin full time region

GK formula⌘= lim lim (t)t!1 L!1b
macroscopic limit
inﬁnite time limit
0.225

0.83⌧micro√−3γ(t)/ mϵσ

0.200
vibration

0.175
0.150 0.3⌧

micro

0.125
0.1000246

t/τmicro

810