Scale effect on flow and thermal boundaries in micro-/nano-channel flow using molecular dynamics–continuum hybrid simulation method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 81:207–228Published online 15 July 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2683

Scale effect on flow and thermal boundaries inmicro-/nano-channel flow using molecular dynamics–continuum

hybrid simulation method

Jie Sun, Ya-Ling He∗,† and Wen-Quan Tao

State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy and Power Engineering,Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

SUMMARY

The molecular dynamics (MD)–continuum hybrid simulation method has been developed in two aspectsin the present work: (1) The energy equation has been combined into the coupling method in order to obtainthe hybrid temperature profile and (2) the coupling method has been improved by the local linearization toobtain a smoother parametric profile. The developed method is primarily validated by analytical solutionsand full MD results. Then, it is employed to study the scale effect on the flow and thermal boundariesin micro-/nano-channel flow. The hybrid velocity and temperature profiles are obtained with the channelheight (H) ranging from 60� to 2014� and the solid–liquid coupling (�) ranging from 0.1 to 50. Scaleeffect has shown strong influence on the boundaries. Obvious slip characteristics can be found in theprofiles, i.e. velocity slip and temperature jump, when H is small and � is large. However, the results alsoshow that the profiles can be well predicted to converge to the macroscale non-slip/non-jump analyticalsolutions when H is large enough, where the effect of � can be omitted and the slip characteristicsdisappear. Correlations of relative slip length, relative temperature jump and pressure gradient with H arefitted from the simulation results. Copyright q 2009 John Wiley & Sons, Ltd.

Received 21 January 2009; Accepted 15 May 2009

KEY WORDS: molecular dynamics simulation; finite volume method; hybrid simulation; velocity slip;temperature jump; micro-/nano-fluidics

1. INTRODUCTION

A marked trend of the science and technology in 21st century is the miniaturization featuringthe rapid improvement of micro-/nano-electro-mechanical systems (MEMS/NEMS) that refer to

∗Correspondence to: Ya-Ling He, State Key Laboratory of Multiphase Flow in Power Engineering, School of Energyand Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant numbers: 50736005, 50636050Contract/grant sponsor: National Basic Research Program of China; contract/grant number: 2007CB206902

Copyright q 2009 John Wiley & Sons, Ltd.

208 J. SUN, Y.-L. HE AND W.-Q. TAO

devices that have characteristic length in the order of micrometer/nanometer and that combineelectrical and mechanical components together [1–3]. These devices have shown promising useful-ness in various aspects in the fields of industry and medicine, for example, cooling electroniccircuits with micro-heat-exchangers and micro-pumps, separating biological cells with micro-reactors, controlling high-frequency fluidics with micro-ducts, etc. The majority of the devicesare either directly or indirectly related to the fluid flow and heat transfer such as micro-ducts ormicro-reactors. Early experimental and numerical studies [1–9] have revealed that fluid flow andheat transfer in micro-/nano-channels greatly differ from those in conventional scales because ofthe presence of the microscopic characteristics such as velocity slip, appreciable liquid viscosityvariation, significant viscous dissipation, electrokinetic effects, etc. All that mentioned above hasindicated that the research on the mass and heat transfer in micro-/nano-channels is of greatscientific significance and application values.

As the spatial scale shrinks to being measured with micrometer and nanometer, it often bringsbig problems to experiments because of the difficulty in miniaturized manufacturing processesand the uncertainty in the measurement data [9], which therefore makes the numerical studybecome popular. For liquid flows in micro-/nano-channels, continuum models are still useful inanalysis, but with some necessary considerations for boundary conditions (BCs) [8, 9]. For thesimulations of fluid flow and heat transfer in the nano-systems, the particle-based methods, i.e.molecular dynamics (MD) simulation and direct simulation Monte Carlo (DSMC), are usuallyadopted. The MD simulation is mainly applied for dense gas and liquid [5, 10] and DSMC ismainly applied for dilute gas [11, 12]. Because of the heavy computational work by the particle-based methods, most of the previous MD studies restricted their flow configurations with onlytens of nanometers [5, 10, 13–15]. To simulate flows in channels for larger scale but still in micro-scopic order where the conventional numerical methods based on the continuum model can stillbe applied in the major part of the channels, the so-called hybrid simulation method emerged.In such hybrid simulation, the advantage of conventional computational fluid dynamics (CFD)in continuum description and that of MD and DSMC in discrete description can be appropri-ately combined, which allows the simulation of microscale thermo-fluid phenomena without theprohibitive cost of a fully particle calculation, and has been developed very rapidly in the recentyears [16, 17].

In this paper, the MD-continuum hybrid simulation will be conducted for fluid flow and heattransfer in micro-/nano-channels. In the following, a brief review on the MD-continuum hybridsimulation is presented. In such simulation method, MD is reserved for situations, where thecontinuum approach is inadequate to compute important flow quantities from first principles,and CFD is used for the rest part, where the continuity assumption is still applicable. Gener-ally, the parametric coupling is realized by the inter-region information transfer between particle(P) region and continuum (C) region through an overlap (O) region. The pioneering work wasdone by O’Connell and Thompson [18] by presenting the momentum coupling scheme and itsapplication in an unsteady shear flow. Afterwards, a number of researchers made contributions toits improvements. These include modifying coupling scheme [19, 20], improving the numericalstability [21] and making applications in solving different types of flow problems such as shearflow [8, 19, 21–25], pressure-driven flow [8, 22, 26, 27], lid-driven flow [27–29], oscillatory flow[23, 30], obstructed flow [31] and contact line problem [32, 33]. So far, the hybrid simulationhas been applied in a wide range of fluid flow problems. However, as can be seen from theabove-mentioned references, heat transfer characteristics are seldom simulated in open publica-tions. In order to obtain a better understanding on the microscopic heat transfer phenomena in

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 81:207–228DOI: 10.1002/nme

SCALE EFFECT ON FLOW AND THERMAL BOUNDARIES 209

MEMS/NEMS devices such as micro-heat-exchangers, micro-heat-engines and micro-heat-pumps,fluid flow and heat transfer should be simulated together.

In the present work, the developed hybrid simulation method is introduced and applied to studythe fluid flow and heat transfer in micro-/nano-channels. The effects of the channel height and theliquid–solid interaction on BCs are investigated and analyzed to obtain the correlations for flowand thermal boundaries.

2. SIMULATION METHODS

2.1. Domain decomposition

The overall computational domain is a half planar channel (see Figures 1(a)–(b)) because the flowcondition is symmetric. The geometric parameters of the domain are set to be lx = ly =13.6�.The half channel height lz(=H/2) varies in the range of 30�–1007� in order to investigate thescale effect on BCs. The computational domain is decomposed into three different regions (seeFigure 1(b)). In the main flow area, which is referred to as C region, the flow can still be welldescribed by Navier–Stokes equation; therefore, the conventional finite volume method (FVM) isemployed as a C solver. In the wall-neighboring area, which is referred to as P region, the continuityassumption breaks down and the flow and thermal slippages emerge; therefore, the classical MDsimulation is used as a P solver. In the area where the two regions mentioned above overlaps, acertain type of coupling scheme is needed to ensure the continuity of the state parameters, such asvelocity and temperature, across this overlap region (O region). The coupling scheme is usually

Figure 1. Schematics for the hybrid simulation method: (a) schematic of the flow problem;(b) configuration of the overall domain; (c) configuration of the coupling domain; and

(d) configuration of the solid wall in the P region.



the essence of a hybrid simulation method and is also the major difficulty we need to handle. Inthe following text, each component of the method will be addressed in details.

2.2. MD simulation for particle region

MD simulation is carried out in three dimension. The region is set as lPx = lx , lPy = ly , lPz =23.7�–61.1� (see Figures 1(b)−(c)) and is uniformly occupied by liquid molecules initialized inface-centered cubic (FCC) pattern to ensure the density �=0.81m�−3 (m is the molecular mass).The modified Lennard–Jones (L-J) potential is used to describe the interaction between liquidmolecules

�(r)=4ε

{[(�

r

)12−(�

r

)6]+[6

(�

rc

)12

−3

(�

rc

)6](

r

rc

)2

−[7

(�

rc

)12

−4

(�

rc

)6]}

(1)

where � and ε are the length and energy characteristic parameters, respectively. The cutoff radiusrc is chosen to be 2.5�. The function is recommended by Stoddard and Ford [34] for the reasonthat when r increases to rc, both the potential energy (�) and the potential force (−d�/dr) can gocontinuously to 0. Therefore, no discontinuous forces will appear and the molecules outside therc range can act as free particles. The interaction between the solid and liquid is also described byEquation (1), but the characteristic parameters are changed into �wf=0.91� and εwf=�ε, where� is a factor that adjusts the strength of the liquid–solid coupling. The ratio �wf/� is chosen froman argon–platinum (Ar-Pt) particle pair [13, 14, 35] to model the geometric relation between liquidand solid particles.

Periodic BCs are used in the x- and y-directions. The solid wall is set at the bottom of P region. Inorder to keep the wall temperature constant at a target value Tw, the ‘phantom method’ [13, 14, 35]based on the Langevin equation is chosen because it is less artificial compared with simplevelocity re-scaling. This method is actualized through the following steps: (1) Three layers ofsolid atoms labeled as ‘real’ atoms are set in FCC (111) structure with the neighboring distance�w=0.814� and the spring constant k=3249.1ε�−2. This arrangement leads to a solid density�w=2.63m�−3, which corresponds to the metal of Pt if we take the length characteristic parameterof Ar �=0.34nm; (2) to model the semi-infinite solid wall, another two layers of ‘phantom’ atomsare set right below the ‘real’ ones with the lower layer fixed as a frame; (3) some equivalenttransformations are applied by setting a special spring of 2k in vertical and 0.5k in horizontaldirections for a real-phantom atom pair, and another special spring of 2k in vertical and 3.5k inhorizontal directions for a phantom–phantom pair belonging to different layers. There are 396atoms in a single layer, therefore, 1980 solid atoms altogether in five layers; (4) the phantom atomsin the upper layer are imposed by three types of forces corresponding to the Langevin equation[13, 14, 35, 36]:

dpi (t)dt

=−�pi (t)+fi (t)+F(t) (2)

where pi (t) is the momentum vector of the i th phantom atom, � is the damper constant, whichis assumed to be independent of particle positions and momenta; therefore, the first term onRHS indicates the damper force; fi (t) is the interaction force vector between the i th phantomatom and all its neighbors; F(t) is the random force vector by which the phantom atomsare excited. The random force is sampled from a Gaussian distribution with a mean value�F =0 and standard deviation �F =(2�kBTw/�tP)1/2, where kB is the Boltzmann constant and



�tP=0.005 (=m1/2�ε−1/2) is the time step for P region. For the damper constant, we choose�=168.3−1 according to Reference [13, 14, 35]. It should be emphasized that for differentmolecules, random force components are sampled independently at each time step. With thismethod, the wall temperature can be well controlled at the preset value Tw. The configuration ofthe wall is shown in Figure 1(d) in details and the treatment for the top boundary will be referredto in Section 2.4.

The simulation procedure is now presented. It contains two periods, i.e. equilibrium period andcoupling period. For each run, the initial 500 is used to allow the molecules in P region reachthe thermal equilibrium state at 1.1εk−1

B . After that, the coupling between P and C regions isexecuted for another 5000 to ensure the synchronization in velocity and temperature. P regionis divided equally into 47–120 slices in the z-direction to obtain the specific parametric profiles.The leapfrog method is used for the integration of motion equations. In order to reduce thecomputational consumption, the cell subdivision technique is considered when dealing with themolecular interactions [37].

2.3. FVM for continuum region

At the continuum level, two-dimensional FVM is used as C solver. Although the flow is onedimensional for the present work, the aim of using two-dimensional FVM is to prepare for thefurther work of two-dimensional complex flow. The present problem satisfies the assumption ofincompressible flow; therefore, the fluid has to obey the mass, momentum and energy conservationequations in the incompressible form [38]

∇ ·u= 0 (3)

�u�t

+u·∇u= �

�∇2u− 1

�∇ p (4)

�T�t

+u·∇T =

�cv

∇2T + �

�cv

[2

(�u�x

)2

+2

(�w

�z

)2

+(

�u�z

+ �w

�x

)2]

(5)

where the unsteady term �/�t is retained to account for the time evolution; u and w are thex- and z- components of the velocity vector u; �, and cv are the dynamic viscosity, thermalconductivity and specific heat, respectively. In principle, these thermophysical property coefficientsfor the governing equations should be determined through preliminary MD simulations in orderto ensure the consistence with P region, such as Liu et al. did in their work [24]. However, for thesimple L-J fluid, it is found that the coefficients obtained by MD simulations [24, 39, 40] agreevery well with the experimental data [41] for a large range of temperature, especially for liquidstate. For example, the quantities obtained by Liu et al. with MD simulation at T =1.16εk−1

B and�=0.81m�−3 are 2.08ε�−3, 7.7kB�−1−1 and 2.43kBm−1, while the corresponding experi-mental data of Ar are 2.19ε�−3, 6.91kB�−1−1 and 2.42kBm−1. With the consideration of thisaspect, the coefficients are directly cited from the property database of fluid [41] for the presentwork as 2.18ε�−3, 6.92kB�−1−1 and 2.40kBm−1 corresponding to the mean liquid state ofT =1.25εk−1

B and �=0.81m�−3.The spatial compatibility is satisfied by domain overlapping in O region, whereas the temporal

compatibility is satisfied by setting the time step for C region (�tC) to be �tC=n�tP. The fullyimplicit scheme is employed for the discretized governing equations, which means that the iteration



is unconditionally stable for any �tC theoretically. However, for the hybrid simulation, �tC stillshould not be arbitrarily chosen because the coupling will be strongly disturbed if �tC is notappropriately set. On the one hand, when �tC is excessively large, the promptness for coupling isobviously weaken by temporal lag. On the other hand, when �tC is too small, available informationcan be hardly obtained in P region in a single �tC because of statistical noise. To ensure thepromptness, a comparatively small time step �tC=50�tP=0.25 is chosen based on its practicalperformance.

The BCs for the rectangular computational region (see Figure 1(b)) are:

(1) inlet (left) and outlet (right) boundaries: Periodic BCs;(2) symmetric boundary (top): �u/�z=0,w=0,�T /�z=0;(3) P-C boundary (bottom): u, w and T are assumed to be uniform along x and is provided

from P region.

The computational size for C region is lCx = lCz =23.7�–976.9� (see Figure 1(b)) and is dividedby uniform grids with the grid size �xC=�zC= lPx . The semi-implicit method for pressure linkedequations consistent (SIMPLEC) algorithm is employed to solve the flow and pressure fields [38].The staggered grid is actualized by displacing u- and w-velocities to the faces of the main controlvolumes, i.e. u at the circles and w at the crosses in Figures 1(b)–(c), for avoiding unrealisticchecker-board pressure field.

2.4. Coupling method for overlap region

The overlap region is the significant place where the coupling method is carried out by exchangingthe information between P and C regions. For the present work, O region is composed of thelast three C grids in the middle column (i= L1/2) and the molecules filling these grids. Thereare three layers in the O region from top to bottom, namely C-P layer, relaxation layer whichjust serves as buffer, and P-C layer. Figure 1(c) shows the specific configuration. The couplingparameters include velocity and temperature, and the coupling method in C-P and P-C layers willbe introduced in sequence. In the following text, all the C velocities correspond to the centralnodes unless specially mentioned and only the z-index will be indicated for convenience, i.e. thevelocity uL1/2,4 will be expressed as u4, because the x-index does not change in O region.

In C-P layer, the molecular average velocity should be synchronized with the C velocity u4through the constraint dynamics according to Nie et al. [19]

ri (t)= 1

�tP

[u4(t+�tP)− 1

NC-PNC-P∑j=1

r j (t)

]+

[fi (t)m

− 1

NC-PNC-P∑j=1

f j (t)m

](6)

where fi and f j indicate the interaction force acting on the i th and j th molecule in C-P layer,f=−(�/�r)

∑NC-P �. Therefore, the summation of the second bracket over all the molecules in C-P

layer will definitely equal to 0. It is worth mentioning that the interpolation and extrapolation shouldbe carried out to obtain u4(t+�tP) for each �tP with C solutions at the two nearest time steps.In order to alleviate the relatively high noise-to-ratio and improve the convergent characteristics,Yen et al. [8] introduced a novel treatment to Equation (6) by substituting the transient quantities

1

NC-PNC-P∑j=1

r j (t) and1

NC-PNC-P∑j=1

f j (t)m



with corresponding time-averaged quantities⟨1

NC-PNC-P∑j=1

r j (t)

⟩p�tP

and

⟨1

NC-PNC-P∑j=1

f j (t)m

⟩p�tP

obtained over a period of p�tP. Therefore, Equation (6) can be changed into

ri (t) = 1

(p/2+1)�tP

⎡⎣u4(t+�tP)−

⟨1

NC-PNC-P∑j=1

r j (t)

⟩p�tP

⎤⎦

+⎡⎣ fi (t)

m−

⟨1

NC-PNC-P∑j=1

f j (t)m

⟩p�tP

⎤⎦ (7)

where the factor p is chosen to be equal to n, i.e. p�tP=�tC. By observing Equations (6)–(7),it can be found that all the molecules in C-P layer are adjusted by identical constraint forcecorresponding to the central node (L1/2,4). This will definitely lead to a local uniform velocityin C-P layer and cause deviation in the specific profile. This deviation would not be noticed ifthe width of the P slice is as large as �zC, which is commonly used in the previous literatures[8, 18, 19, 23, 24]. However, it will become obvious if the slice width is much smaller as in thepresent work.

In order to eliminate the deviation, a novel treatment is proposed with the idea of substitutingthe local uniform constraint force with locally linearized constraint forces. Figure 2 shows theschematic for our improvement in details, where � indicates u-velocity. For convenience, thenotation follows that in Reference [38]. The curve abc is the actual solution, whereas the curveab

′c

′is the results obtained by the coupling scheme in Reference [8], in which the deviation

emerges because of the local uniform in C-P layer. The curve ab′′c

′′indicates the result obtained

z

P

(L 1/2, 3) E

(L 1/2, 4)

Relaxation layer C-P layer

W

(L 1/2, 2)

a

c

b

b’

c’

w e

c’’

b’’

ee

Figure 2. Schematic of the improved coupling scheme.



by our improved scheme, in which the deviation has been greatly eliminated by locally linearizingthe constraint force in C-P layer in the form

f (z, t)= fe(t)+k(t)(z−ze)= fe(t)+2[ fE (t)− fe(t)](z−ze)/�zC (8)

where

fe(t)= 1

(p/2+1)�tP

⎡⎣ue(t+�tP)−

⟨1

Ne

Ne∑j=1

r j (t)

⟩p�tP

⎤⎦

is the constraint force at face e

fE (t)= 1

(p/2+1)�tP

⎡⎣uE (t+�tP)−

⟨1

NE

NE∑j=1

r j (t)

⟩p�tP

⎤⎦

is the constraint force at central node E . Note that ue(t+�tP) cannot be obtained directly; therefore,a certain kind of interpolation should be carried out. The idea of the quadratic upstream interpolationfor convective kinetics (QUICK) scheme [38], which is a three-point upstream-weighted quadraticinterpolation of high-order accuracy, is borrowed

ue(t+�tP)= 68uP(t+�tP)+ 3

8uE (t+�tP)− 18uW (t+�tP) (9)

where uP(t+�tP), uE (t+�tP) and uW (t+�tP) are the velocities at the corresponding centralnodes obtained by averaging the values at faces, respectively. Similarly, for w-velocity, wE (t+�tP)is obtained by:

wE (t+�tP)= 68we(t+�tP)+ 3

8wee(t+�tP)− 18ww(t+�tP) (10)

Therefore, z-coordinate of each molecule in C-P layer should be taken into consideration whencalculating the constraint force:

ri (z, t)= f (z, t)+⎡⎣ fi (t)

m−

⟨1

NC-PNC-P∑j=1

f j (t)m

⟩p�tP

⎤⎦ (11)

Finally, Equations (8)–(11) constitute the C-P part of the velocity coupling for our improvedscheme. It can be found that the summations of Equations (6) and (11) are identical, which ensurethe consistence. Besides velocity, the pressure gradient −dp/dx obtained from C solution shouldalso be transferred to P region over all the molecules as an external driving force:

fe=−m

�

(dp

dx

)(12)

The test case for the performance of the improved scheme will be provided in Section 3.1.For temperature coupling, the similar idea of local linearization in C-P layer is employed and

is also illustrated by Figure 2, where � indicates temperature now. The temperature constraintis also actualized through the Langevin method as Equation (2), because it has clear physicalmeaning and contains less artifact compared with simple velocity re-scaling, in which the intrinsic



fluctuations between kinetic and potential energy are removed [24]. The molecules in C-P layerare thermalized only in the y-direction by

yi (z, t)=−�′ yi (z, t)+ fi (t)m

+ F(z, t)

m(13)

where the damper constant �′ is chosen to be 1.0−1 [5, 24]. The target temperature T (z, t+q�tP)is implicitly included in F(z, t) and is linearized as

T (z, t+q�tP)=Te(t+q�tP)+2[TE (t+q�tP)−Te(t+q�tP)](z−ze)/�zC (14)

where the temperature at face e, i.e. Te(t+q�tP) should still be obtained by QUICK interpolation:

Te(t+q�tP)= 68TP(t+q�tP)+ 3

8TE (t+q�tP)− 18TW (t+q�tP) (15)

Note that TP(t+q�tP), TE (t+q�tP) and TW (t+q�tP) are the temperature at central nodes att+q�tP calculated by the BC provided from P-C layer at t+(q−n)�tP. Equations (13)–(15)constitute the C-P part of the temperature coupling for the present work.

In order to avoid the molecules drifting away from P region and to eliminate the unphysicaldensity oscillation, an extra depress force fd is exerted on the molecules in the C-P layer. Thespecific form of this depress force is various in literatures. It can be based on the average pressure[8, 19], an arbitrary weigh function [22] or the repulsive part of the L-J potential [27]. For thepresent work, we choose the first type due to its simplicity in application

fd(z)=−p0�z−ze

1−(z−ze)/�z(16)

where p0 is the average pressure in the P region.In P-C layer, the information transfer is comparatively simpler, which obtaining the velocity

and temperature at central node (L2,1) by a spatial averaging within P-C layer and a temporalaveraging over a period of p�tP for velocity

u1=⟨

1

NP-CNP-C∑j=1

r j

⟩p�tP

(17)

or q�tP for temperature

T1= 2

3kBNP-C

⟨NP-C∑j=1

1

2m(r j −u1)2

⟩q�tP

(18)

and transferring the results to C region as BCs. Equations (17)–(18) constitute the P-C part of ourcoupling method.

In principle, the velocity and temperature couplings should be carried out in sync by settingp=q=n to ensure the promptness, especially for unsteady problems. However, for the presentsteady problem, p and q is unnecessarily restricted to be equal to n. It has been found that p=nsometimes cannot guarantee providing an accurate temperature for C region, but q=n is fine forvelocity. Therefore, we prolong the averaging period for temperature and finally p is optimizedto be 10n by trial and error. Figure 3 illustrates the information transfer and the evolution of ourhybrid method in details.



Figure 3. Evolution of the hybrid method.

3. RESULTS AND DISCUSSION

Unlike the Couette flow where the flow is only driven by shear force and is often characterized byshear rate [5], we have to assign a parameter as the guideline for the channel flow. For the presentwork, we fix the maximum velocity Um=1.0�−1 for all the cases to investigate the effects of thechannel height and the solid–liquid coupling on the BCs.

3.1. Code validation

Two test cases are primarily carried out to verify the practical performance of our coupling methodby comparing with either analytical solution or full MD results. The first one is the typical Poiseuilleflow with non-slip BC, which has the following analytical solutions for velocity:

u(z)= 4zUm

H

(1− z

H

)(19)

and temperature

T (z)=Tw+ �U 2m

3

[1−

(1− 2z

H

)4]

(20)

where Um=−(H2/8�)(dp/dx) is the maximum velocity at the symmetric boundary. Since Um=1.0�−1, the external driving force fe should be about 0.00064ε�−1. The channel height H is setto be 182� and it is found that �=2.8 can well guarantee a non-slip BC in MD simulation underthe present configuration. Figure 4 shows the evolution of uC-P, u4, TC-P, T4 and fe with time,which indicates a well synchronization between the C quantities and the corresponding averagedP quantities, and a good convergence of fe to the target value. It is found that all the quantitiesmentioned above keep steady after 3000; therefore, the profiles shown from then on were averagedwithin 3000–5500. Figures 5(a)–(b) show the comparison between the result of the couplingscheme in Reference [8] and that of the improved scheme. The deviation in C-P layer, which we



0.6

0.7

0.8

0.9

1.0

1.05

1.10

1.15

1.20

1.25

1.30

00.0000

0.0006

0.0012

0.0018

0.06

0.08

uC-P

u4

TC-P

T 4

1000 2000 3000 4000 5000 6000

Figure 4. Transient variations of the coupling parameters and the driving force for the non-slip test case.

have mentioned in Section 2.4, can be clearly seen due to the local uniform. However, it is welleliminated by our improved scheme and the velocity profile is modified indeed. Figure 5(b) alsoshows the comparison between the analytical solution, the full MD result and our hybrid result,from which we can find that both the two simulation results agree quite well with the analyticalsolution with non-slip BC. For temperature profile, the hybrid result should be compared with thefull MD result instead of the analytical solution because the presence of temperature jump (Tj )

at the solid–liquid interface, which is similar to the cases in Reference [24], will cause a wholedeviation from the analytical solution (see Figure 5(c)). Although the agreement of temperaturewith the full MD result is not as good as that of velocity, it is still reasonably satisfying. Besides,the smooth connection between the MD and continuum parts in O region indicates the greatperformance of our improved coupling scheme.

The other test case is a slip flow. The configuration is identical to that of the first case exceptthat � is changed to 1.0 to obtain a slip BC, which will cause a little decrease in the externaldriving force as fe=0.00061ε�−1. The good agreement between the full MD result and the hybridresult in both velocity and temperature profiles (see Figure 6) indicates that the hybrid simulationis quantitatively able to provide accurate profiles, as well as velocity slip (us) and temperaturejump at the boundary, just as a full MD simulation does. In other words, the result equivalent toa full atomistic description can be easily achieved in a more efficient way. Note that the hybridsimulation is not obviously so sensitive to the spatial extension as a full MD simulation is, which



0.0

0.0

0.2

0.4

0.6

0.8

1.0

z /H

Hybrid(MD)

Hybrid(Continuum)

Analytical solution(no-slip)

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Hybrid(MD)

Hybrid(Continuum)

Full MD

Analytical solution (no-slip)

(b)

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

Hybrid(MD)

Hybrid(Continuum)

Full MD

Wall temperature

Analytical solution(no-slip)T

j

(c)

0.1 0.2 0.3 0.4 0.5

0.0

z /H

0.1 0.2 0.3 0.4 0.5

0.0

z /H

0.1 0.2 0.3 0.4 0.5

Figure 5. Hybrid velocity profiles for the non-slip test case obtained by: (a) the coupling schemein Reference [8]; (b) the improved coupling scheme; and (c) hybrid temperature profile obtained

by the improved coupling scheme.



0.0

0.0

0.2

0.4

0.6

0.8

1.0

Hybrid(MD)

Hybrid(Continuum)

Full MD

z /H

z /H

us

(a)

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

Hybrid(MD)

Hybrid(Continuum)

Full MD

Wall temperature

Tj

(b)

0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

Figure 6. Hybrid velocity and temperature profiles for the slip test case.

is one of the most attractive advantages. Therefore, it is a suitable tool for study on the scale effectover a large range.

3.2. Scale effect on flow boundary

In this section, we focus on the effect of the channel height (H =60�–2014�) on flow boundarieswith different solid–liquid couplings (�=0.1–50). Figure 7 shows the hybrid velocity profiles withdifferent � at different H . It can be found that when H is small (see Figure 7(a)), the velocity slipus varies obviously with �, where the slip boundary emerges at small � (��1) and the lockingboundary emerges at large � (��5) with the non-slip boundary existing just in the transitionrange (1<�<5). The differences between slip, non-slip and locking BCs become unconspicuouswith increasing H (see Figures 7(b)–(e)). When H is large enough (H =2014�), the differencesseem undistinguishable and the boundary velocities converge to 0 (see Figure 7(f)). This variationagrees with the conventional understanding that the BC for fluid flow at macroscale can be



0.0

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti. =0.1 =0.5 =1 =5 =10=50

z /H(a)

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti.

=0.1 =0.5 =1 =5 =10=50

(b)

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti. =0.1 =0.5 =1 =5 =10=50

(c)

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti.

=0.1

=0.5

=1

=5

=10

=50

(d)

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti. =0.1 =0.5 =1 =5 =10=50

(e)

0.0

0.2

0.4

0.6

0.8

1.0

MD Conti. =0.1 =0.5 =1 =5 =10=50

(f)

0.1 0.2 0.3 0.4 0.5 0.0

z /H

0.1 0.2 0.3 0.4 0.5

0.0z /H

0.1 0.2 0.3 0.4 0.5 0.0z /H

0.1 0.2 0.3 0.4 0.5

0.0

z /H

0.1 0.2 0.3 0.4 0.5 0.0

z /H

0.1 0.2 0.3 0.4 0.5

Figure 7. Hybrid velocity profiles with different � at: (a) H =60�; (b) H =142�; (c) H =222�;(d) H =408�; (e) H =1018�; and (f) H =2014�.

well described as non-slip. The reason for this variation includes the following aspects: (1) Thesolid–liquid coupling often plays the part of confining the liquid molecules around solid atoms,but the influence area is limited, which can be recognized as the wall-neighboring area. WhenH is very small, the influence area takes larger part of the whole domain and the influence is



predominant. When H is very large, the influence area is hard to be noticed; (2) with fixed Um,the shear rate �(=du/dz) near the boundary and the pressure gradient (−dp/dx) is sensitive to thechannel height. They are supposed to increase rapidly when H is very small. Therefore, extremelylarge shear tensor will definitely emerge at extremely small H , which will play the part of takingthe liquid molecules away from the solid atoms. However, the tensor will also become unnoticeablewhen H is large enough; and (3) Which of the three types of BCs will emerge depends on thecomprehensive effect of the two aspects mentioned above. However, both of them become weakwith increasing H , therefore the differences in BCs are much more obvious in small channels,which are called the scale effect in microscale.

In order to show the effects of � and H on the BCs in a clearer way, the relative slip lengths(Ls/H) are shown in Figure 8, where the slip lengths (Ls) are calculated according to Reference[3, 15, 25] by

Ls=us

/du

dz

∣∣∣∣w

(21)

From Figure 8(a) we can find that the values of the joints of the x-axis and the data, whichcorrespond to non-slip BCs, decrease with increasing H . It can be also found that the � we choosefor the non-slip test case (�=2.8) is just the joint of the x-axis and the data of H =182�, whichdemonstrates that Figure 8(a) can accurately provide the specific values of the � for non-slip BCs.In Figure 8(b), the positive values correspond to slip BCs and the negative values correspond tolocking BCs. It also shows that Ls/H varies in a obvious power form with H , therefore the dataof a certain � are fitted with a power function

Ls

H=a

(H

�

)b

(22)

and the coefficients a and b are fitted again in an exponential function to include �:

a or b= A1 exp

(− �

t1

)+A2 exp

(− �

t2

)+A0 (23)

Finally, all the data in Figure 8(b) can be well fitted by the following correlation:

Ls

H= a

(H

�

)b

a = 3.651exp

(− �

1.221

)+38.706exp

(− �

0.304

)−0.398

b = −0.105exp

(− �

40.732

)−0.441exp

(− �

2.530

)−0.589

(24)

Equation (24) includes the effects of H and � on Ls/H and can provide us a quantitative under-standing about the slip, non-slip and locking BCs. To verify this, the correlation in Reference [8],which can be recognized as a specific case of �=0.6 for Equation (24), is compared with ourcorrelation. Regardless of the small deviation that may be due to the slight difference in �w/�, thegood agreement shown in Figure 9 has directly validated our correlation.

The pressure gradients for different conditions are recorded in Figure 10 and it can be foundthat −dp/dx is very sensitive to scale when channel height is small. The data are fitted in the



0.1-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05 H=60

H=102

H=142

H=182

H=222

H=306

H=408

H=610

H=1018

H=1404

H=2014

ββ =2.8

(a)

0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

L s / H

L s / H

=0.1

=0.5

=1

=5

=10

=50

Full MD data

Fitting curves

(b)

1 10 100

200 400 600 800 1000 1200 1400 1600 1800 2000

2 3 4 5 6

50 100 150 200 250 300 350

Figure 8. Variations of Ls/H with: (a) � and (b) H .

same way as Equation (24) and the similar correlation for −dp/dx is obtained as:

−dp

dx= a

(H

�

)b

a = −5.276exp

(− �

0.908

)−6.854exp

(− �

6.710

)+11.655 (25)



00.00

0.05

0.10

0.15

0.20

0.25

L s / H

Present ( =0.6)

Ref.[8] ( =0.6)

100 200 300 400 500

Figure 9. Comparison between the correlation in the present work and that in Reference [8].

0-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

00.000

0.001

0.002

0.003

0.004

0.005

0.006 =0.1

=0.5

=1

=5

=10

=50

Fitting curves

200 400 600 800 1000 1200 1400 1600 1800 2000

50 100 150 200

Figure 10. Variation of −dp/dx with H .

b = 0.136exp

(− �

6.657

)+0.623exp

(− �

0.421

)−1.862

With Equation (25), the external driving forces for full MD at arbitrary H at Um=1.0�−1 canbe quantitatively predicted. The data of full MD simulations at H =20� are also included inFigure 8(b), which show a good agreement with our relation, especially for the positive data.



3.3. Scale effect on thermal boundary

The scale effect on thermal boundary of liquid flow with different � will be focused on in thissection. The hybrid temperature profiles are shown in Figure 11, from which we can find that theprofiles are remarkably different at small H. Tj is very large with small � because the solid–liquidcoupling is so weak that it cannot confine the liquid molecules around any more to ensure anefficient heat transfer. Since the dissipation heat generated by viscosity in the main flow cannot betransferred across the solid–liquid interface fluently, the average temperature will surely increase.When � increases, the constraint of solid atoms on the liquid molecules is strengthened, thereforethe heat transfer between them is also improved, which will lead to a smaller Tj and a loweraverage temperature. Tj also decreases with increasing H , because the velocity gradient becomessmall and it leads to a decrease in the dissipation heat. When H is large enough, Tj becomes verysmall compared with the temperature difference between the maximum and minimum (�T ) andthe whole profiles converge to the analytical solution at macroscale (see Figure 5(c)) no matterhow � varies. This scale effect on thermal boundary is similar to that on velocity boundary andthey are often coupled together in principle, because the emergence of Tj is generally relative tothe heat transfer caused by the viscous dissipation.

The relative temperature jump (Tj/�T ) of hybrid simulations versus � and H is shown inFigures 12(a)–(b), as well as the data of full MD simulations at H =20�. It is worth mentioningthat when � is very large (�=50), Tj/�T seems not sensitive to H (see Figure 12(a)). This isprobably due to the root of Tj that it is generated from the solid–liquid interaction. If we carefullyexamine Figure 7, it can be found that the ultra thin layer of liquid molecules that are adjacent tothe solid wall are always confined around for �=50 no matter how H varies, which means that thelocal environment for the solid–liquid interaction never changes much. This aspect is unlike theconditions with smaller � and it may cause the special phenomenon mention above. In Figure 12(a),the data of �=50 at small H (H =20�–142�) show quite an opposite variation with the correlation,which suggest that the scale effect on Tj/�T is more complex than that on Ls/H , especially inthe case of a very strong solid–liquid coupling. The data in Figure 12(b) are still fitted in formof Equation (22), but it is more difficult to combine � into the coefficients a and b as a function.Therefore, the coefficients are listed along with the correlation:

Tj

�T= a

(H

�

)b

� = 0.1, a=3.907, b=−0.328, �=0.5, a=6.986, b=−0.505

� = 1, a=6.071, b=−0.589, �=5, a=1.302, b=−0.527

� = 10, a=3.026, b=−0.783, �=50, a=0.412, b=−0.345

(26)

4. CONCLUSIONS

In the present work, the MD-continuum hybrid simulation method has been developed in two mainaspects. One is that the energy equation has been combined into the method in order to obtainthe hybrid temperature profile. The other is that the coupling method has been improved by locallinearization to obtain a better specific parametric profile. The improvements and the validity of the



0.0

1.10

1.14

1.18

1.22

1.26

1.30

1.34

1.38

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

z /H(a)

1.10

1.14

1.18

1.22

1.26

1.30

1.34

(b)

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

(c)

1.10

1.12

1.14

1.16

1.18

1.20

1.22

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

(d)

1.10

1.12

1.14

1.16

1.18

1.20

(e)

1.10

1.12

1.14

1.16

1.18

1.20

(f)

0.1 0.2 0.3 0.4 0.5 0.0

z /H

0.1 0.2 0.3 0.4 0.5

0.0

z /H

0.1 0.2 0.3 0.4 0.5 0.0

z /H

0.1 0.2 0.3 0.4 0.5

0.0

z /H

0.1 0.2 0.3 0.4 0.5 0.0

z /H

0.1 0.2 0.3 0.4 0.5

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

MD Conti. MD Conti. =0.1 =0.5=1 =5 =10 =50

Figure 11. Hybrid temperature profiles with different � at: (a) H =60�; (b) H =142�; (c) H =222�;(d) H =408�; (e) H =1018�; and (f) H =2014�.

developed method are verified by analytical solutions and full MD results. The smoother parametricprofiles indicate that not only the continuity of the state parameters, but also the continuity ofthe fluxes are ensured. Finally, the developed method is employed to study the scale effect onthe flow and thermal boundaries within a large range of the channel height (H =60�–2014�) and



0.1

0.0

0.2

0.4

0.6

0.8

1.0 H =60

H =102

H =142

=182H

H =222

H =306

H =408

H =610

H =1018

H =1404

H =2014

Tj /

ΔT

Tj /

ΔT

(a)

0-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

=0.1

=0.5

=1

=5

=10

=50

Full MD data

Fitting curves

(b)

200 400 600 800 1000 1200 1400 1600 1800 2000

1 10 100

Figure 12. Variations of Tj/�T with: (a) � and (b) H .

a large range of the solid–liquid coupling (�=0.1–50). It has been found that the scale effectshows strong influence on the BCs for micro-/nano- liquid flow. On the one hand, the results showobvious slip in the velocity profiles and jump in the temperature profiles when H is small and� is large. On the other hand, all the profiles can be well predicted to converge to the macroscale



analytical solutions with non-slip/non-jump conditions when H is large enough, where the effectof � can be omitted. The data are fitted in correlations for velocity slip, temperature jump andpressure gradient. The agreement of the data obtained with full MD data at very small channelwith the correlations has further supported our work. Future research needs are required to revealthe special variation rules of the BCs within very small H with very large �.

Our numerical practice once again shows that the hybrid simulation method is a very goodsolution for problems with micro-/nano-geometric scale. Obviously, time-saving is one of thegreatest advantages of the hybrid simulation method. Another outstanding advantage is that themost appropriate approaches for different computational regions can be combined to work together,which can provide the detailed multiscale information that a singular method can never achieve.These distinctive advantages will surely lead the hybrid simulation method to become a verypromising complement to the full MD simulation in more research fields that featuring very strongmultiscale characteristics.

ACKNOWLEDGEMENTS

The present work was supported by the Key Project of National Natural Science Foundation ofChina (No. 50736005, 50636050), and National Basic Research Program of China (973 Program)(No. 2007CB206902).

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Documents

Scale effect on flow and thermal boundaries in micro-/nano-channel flow using molecular dynamics–continuum hybrid simulation method