Transport properties of Ar–Kr binary mixture in nanochannel Poiseuille flow

International Journal of Heat and Mass Transfer 55 (2012) 1732–1740

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Transport properties of Ar–Kr binary mixture in nanochannel Poiseuille flow Chengzhen Sun a,b, Wen-Qiang Lu b,⇑, Bofeng Bai a, Jie Liu b a b

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China College of Physical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 20 July 2011 Received in revised form 26 October 2011 Accepted 16 November 2011 Available online 9 December 2011 Keywords: Transport properties Confined mixture Nanochannel Poiseuille flow Molecular dynamic simulation

a b s t r a c t Transport properties, including thermal conductivity and shear viscosity, of the Ar–Kr binary mixture confined in a nanochannel under Poiseuille flow are calculated by equilibrium molecular dynamics (EMD) simulation through Green–Kubo formula. An external force is applied in the x-direction to drive the Poiseuille flow. Thermal conductivity of the confined mixture in the x- and y-direction is obviously higher than that in macroscale, as a result of the strong interacting potential between the fluid atoms and the wall atoms. Thermal conductivity of the flowing binary mixture is obviously anisotropic. With increasing the external driving force, in the x-direction the thermal conductivity increases, whereas in the y-direction it keeps constant. The xz- and yz-component of the shear viscosity of the confined mixture are enhanced comparing with the xy-component owing to the collisions between the fluid atoms and the wall atoms in the z-direction. They are higher than the results in macroscale and decrease with the external driving force increasing. For the binary mixture, thermal conductivity and shear viscosity vary with the mole fraction of the Kr atoms. The interactions between the fluid atoms and the wall atoms play a key role in the transport properties of the binary mixture confined in the nanochannel. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Study of fluids confined in nanoscale geometry has been drawing interest of researchers from the fields of physics, chemistry, biology, medicine and engineering. The study categorized as nanofluidics [1,2] has experienced a rapid development over the past few years. It finds applications in areas such as micro-electromechanical-systems (MEMS), lab-on-chip, e-paper, micro-reaction devices, photonic devices, material sciences, biotechnology, separation technology and process engineering. In nanoscale geometry, a number of forces that can be neglected in macroscale play an important role, such as van der Waals forces, electrostatic forces and hydrophobic forces. These forces govern the behaviors of the fluids in nanoscale geometry. Fluids confined in nanoscale geometry exhibit behaviors not observed in larger geometry, e.g. vastly increased viscosity near the wall, changes in thermodynamic properties and chemical reactivity of species at the fluid–solid interface. Fluid transport properties are important parameters needed in engineering application, especially in numerical analysis of fluid flow and heat transfer. Transport properties of fluids were studied widely, even if in nanochannels/nanotubes. Xu and Zhou [3] studied the Poiseuille flow of liquid argon in a nanochannel by molecular dynamics (MD) simulation and found that the liquid viscosity at the wall surfaces increased with increasing the applied external ⇑ Corresponding author. Tel./fax: +86 10 81717831. E-mail address: [email protected] (W.-Q. Lu). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.11.028

force. Bitsanis et al. [4] found that the shear viscosity of the fluid under nanochannel Poiseuille flow was distinguishable from the homogenous fluid as the channel width decreased to 4r (r is potential length parameter). Sofos et al. [5] calculated the transport properties of liquid argon under Poiseuille nanoflow between krypton walls and concluded that all the properties approached the macroscale values as the channel width increased and the layers close to the walls presented distinctly different behaviors owing to the interactions with the wall atoms. They found that in the nanochannel the thermal conductivity was lower whereas the shear viscosity was higher compared with the macroscale values of their own. Liu et al. [6] calculated the thermal conductivity and shear viscosity of water confined in single-walled carbon nanotubes (CNTs) and found that the axial thermal conductivity and shear viscosity in CNTs were obviously larger than the macroscale values and those in the radial direction and they increased sharply as the diameters of CNTs decreased. Liu et al. [7] found that the thermal conductivity of liquid argon ultra thin films confined between two plates depended on the distance between the two plates, the existence of solid-like liquid layering at the liquid–solid interface and the average migration frequency of all the liquid molecules. In short, the previous results on the transport properties of fluids in nanochannels/nanotubes are controversial and even opposite. Knowledge of transport properties of confined multicomponent fluids is of particular importance in many industrial applications. It can provide better understanding of the phenomena relevant to biological system, separation devices, micro-reaction devices etc.

C. Sun et al. / International Journal of Heat and Mass Transfer 55 (2012) 1732–1740

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Nomenclature rij rcut N V T E kB Jq Jp t Dt ~ I v ha ma Nstep M m, n Hn (zij) z JN JM h

distance between atom i and atom j, m cutoff distance, m total number of system atoms system volume, m3 system temperature, K system total energy, J Boltzman constant, 1.38  1023 J/K heat current vector, N m2/s off diagonal elements of the stress tensor, J time, s time step length, s unit tensor velocity of atom, m/s mean enthalpy, J mass of a particle, kg total time step in the simulation time step needed for the calculation of thermal conductivity and shear viscosity time step function in statistical velocity profile and number density z-component of coordinate, m start time step in the statistics end time step in the statistics channel width, m

[8–10]. For example, transportation of ions and organic molecules across nanochannels in biological system, separation of mixture performed in nanofluidic channels on silicon or glass chips, and chemical reactions of diverse substances in micro-reaction devices. For the multicomponent fluids, one may expect additional effects due to the interactions of the different components and change their properties by increasing/decreasing the concentration of one of the components. When the mixture fluids are confined in nanochannels, nanoscale effect and interesting phenomenon appears. Presently, little attention was devoted to the transport properties of mixture fluids in nanochannels. Kaushal et al. [11] evaluated the shear viscosity of argon–krypton (Ar–Kr) binary mixture fluids confined in a nanochannel at different mass ratios and concentrations of one of the fluids and found that mixing fluid with lighter mass reduced an enhancement in the shear viscosity and the enhancement decreased as increased the concentration of the fluid. However, the shear viscosity enhancement mechanisms and the nanoscale effect on the shear viscosity of the confined mixture fluids were not illustrated clearly. One possible route to better predict the transport properties of mixture fluids is to use MD simulation. MD simulation is a computational method that solves Newton’s equation of motion for a system of particles interacting with a given potential. As MD simulation directly calculates the movement of particles at the atomic level, it can describe the flow and transport phenomenon accurately, especially at atomic scales that are difficult to perform experimentally. MD simulation was used extensively to determine the transport properties of simple binary mixture fluids [12–15]. Meanwhile, transport properties of complex mixtures were also well predicted [16–17]. Two fundamentally different methods for calculating transport properties by MD simulation are available, i.e. equilibrium molecular dynamics (EMD) using the Green–Kubo formula and nonequilibrium molecular dynamics (NEMD) analyzing the system’s response to an externally applied perturbation. Both methods have different advantages and disadvantages [18] and are used according to one’s preference for comparable efficiency.

xKr F Lw Lf TW

mole fraction of Kr atoms external driving force, N dimensions of krypton walls, m dimensions of fluidic zone, m wall temperature, K

Greek symbols / LJ potential, J e energy parameter, J r length parameter, m k thermal conductivity, W/mK gs shear viscosity, Pas q number density, 1/m3 wG ; w0G Gaussian-distributed random numbers w random number in (0, 1) Superscript ⁄ nondimensional quantities Subscripts 1,2 component 1 and component 2 x,y,z three directions of the coordinate j,k number of atoms a,b kind of atoms

Among the transport properties, thermal conductivity and shear viscosity are of particular interest. Thermal conductivity and shear viscosity represent heat transfer performance and momentum transfer performance, respectively. Therefore, this paper aims to calculate the thermal conductivity and shear viscosity of Ar–Kr binary mixture confined in a nanochannel with Poiseuille flow by EMD method using the Green–Kubo formula. The nanoscale effect on the transport properties of the confined binary mixture will be investigated in detail. The paper is organized as follows. In Section 2, MD simulation model are illustrated, including the atomic interaction potential and the Green–Kubo formula for the calculation of thermal conductivity and shear viscosity. At the same time, some MD details are made clear. In Section 3, density distribution, equilibrium system configuration and velocity profile are obtained, and then thermal conductivity and shear viscosity of the Ar–Kr binary mixture are calculated. Finally, Section 4 provides a summary of the obtained results. 2. Simulation model 2.1. Lennard–Jones potential The simulation is carried out with the system where Ar–Kr (denoted 1–2) binary mixture is confined between two parallel infinite walls. The geometric model of the simulation system is shown in Fig. 1. The two physically infinite walls are considered by limited computational lengths through periodic boundary conditions in the x- and y-direction. The distance between the two walls (h) is of nanometers during the simulation. The two walls are stationary and consisted of Kr atoms to reduce the difficulty in the calculation, as the consideration of Sofos et al. [5]. The widely accepted Lennard–Jones (LJ) potential matches experimental data well for argon atoms and other inert atoms. Meanwhile, LJ potential requires reasonable computation time. In this system, the atomic interactions are modeled by the wellknown LJ 12-6 potential [19,20].

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For a two-component system, the heat current is expressed as the constitution of the kinetic part, the virial part, and the potential part. An extended form is used to calculate the heat current vector [5,21]

J xq ¼

Nb Na Na X 2 X 2 X 2 X X 1 1X ma v 2ja v xja  2 2 a¼1 b¼1 j¼1 k¼1 a¼1 j¼1 k–j



 Na 2 X X ! @/ðr jakb Þ  rjakb  /ðr jakb Þ I v xja  ha v xja @r jakb a¼1 j¼1

Fig. 1. Geometric model of the simulation system where the Ar–Kr mixture is confined between two krypton parallel walls. The krypton wall has the size of Lwx = Lwy = 2.99 nm, Lwz = 1.20 nm. The lengths of the fluidic zone in the x- and ydirection Lfx = Lfy = 2.32 nm. h represents the channel width, F denotes the external driving force.

h i 4e ðrr Þ12  ðrr Þ6 ðr ij < rcut Þ

( /ðr ij Þ ¼

ij

ij

ð1Þ

ðr ij P r cut Þ

0

where rij is the distance between particles i and j, and e and r are the energy parameter which governs the strength of the interaction and the length scale, respectively. To improve the computational efficiency, only the neighboring particles within a certain cutoff radius (rcut) are included in the force calculation since the distant particles have a negligible contribution. In this calculation, the cutoff distance rcut is set to be 2.5r. For argon the LJ parameters r11 and e11 are equal to 0.3405 nm and 1.670  1021 J, respectively [5]. For krypton the LJ parameters r22 and e22 are equal to 0.3633 nm and 2.328  1021 J, respectively [5]. To determine the parameters between argon and krypton, the common Berthelot–Lorentz mixing rule [19] is used:

r12 ¼

r11 þ r22 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi

e12 ¼ e11 e22

Z

VkB T

0

1

< J xq ðtÞ  J xq ð0Þ > dt

1 VkB T

Z

1

0

xy < J xy p ðtÞ  J p ð0Þ > dt

ð5Þ

where gs is the shear viscosity, Jp the off diagonal elements of the stress tensor. For a two-component system, the off diagonal element of the stress tensor is consisted of kinetic part and virial part [5,22]

J xy p ¼

Na 2 X X

ma v xja v yja 

a¼1 j¼1

Nb Na X 2 X 2 X @/ðr jakb Þ 1X rx 2 a¼1 b¼1 j¼1 k¼1 jakb @r yjakb

ð6Þ

k–j

where subscripts j and k are the number of particles, and a and b denote two different kinds of particles. Na and Nb are the number of particles of a and b, respectively. ma is the mass of particles of a, v ja is the velocity of a particle j of a. The xz- and yz-component of shear viscosity can be obtained by corresponding formula. Since the simulation is performed for discrete time steps, Eq. (3) for the calculation of thermal conductivity is written as N step M  Xm  x Dt X 1 J q ðm þ nÞ  J xq ðnÞ 2 N  m VkB T m¼1 step n¼1

ð2bÞ

where tM is given by MDt, Jxq ðm þ nÞ is the heat current vector at MD time step m + n, Nstep is the total time step in the simulation and M is the time step needed for the calculation of thermal conductivity. Eq. (5) for the calculation of shear viscosity is written as

gsxy ðtM Þ ¼

EMD can simulate transport properties, such as self-diffusion, thermal conductivity and shear viscosity, based on the linear response theory. The thermal conductivity in the x-direction is obtained by integrating the equilibrium heat current autocorrelation function (HCACF) through the Green–Kubo formula [19] 2

gsxy ¼

kx ðt M Þ ¼

2.2. Green–Kubo formula

1

where subscripts j and k are the number of particles, and a and b denote two different kinds of particles. Na and Nb are the number of particles of a and b, respectively. ma is the mass of particles of a, v ja is the velocity of a particle j of a, ha stands for the mean en! thalpy per particle of a, and I is the unit tensor. The mean enthalpy is calculated as the sum of the average kinetic energy, potential energy and virial terms per particle of each species [19]. For a singlecomponent system, the last term of Eq. (4) equals zero. The thermal conductivity in the y- and z-direction can be obtained by corresponding formula. The xy-component of shear viscosity is obtained by integrating the time autocorrelation function of the off diagonal elements of the stress tensor through the Green–Kubo formula [22]

ð2aÞ

Therefore, r12 and e12 between argon and krypton are equal to 0.3519 nm and 1.971  1021 J, respectively.

kx ¼

ð4Þ

N step M Xm xy Dt X 1 ðJ ðm þ nÞ  J xy p ðnÞÞ VkB T m¼1 Nstep  m n¼1 p

ð7Þ

ð8Þ

where tM is given by MDt, J xy p ðm þ nÞ is the off diagonal elements of the stress tensor at MD time step m + n, Nstep is the total time step in the simulation and M is the time step needed for the calculation of shear viscosity. 2.3. MD details

ð3Þ

where k is the thermal conductivity, V the system volume, T the system temperature, kB the Boltzman constant, Jq the heat current vector, and the angular brackets denote the ensemble average or the average over time.

The atoms, including wall atoms, Ar atoms and Kr atoms, in the system are originally arranged in regular face-centred cubic (FCC) lattices. The Kr atoms are distributed uniformly in the fluidic zone. The walls consist of two layers of Kr atoms in the z-direction such that the distances between any atom of the fluid and the wall

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atoms out of consideration are beyond the cutoff distance. All the wall atoms are assumed to be stationary. The wall surface is treated as smooth in the calculation and the effect of the wall roughness on the transport properties is not considered. The original system configuration for the Ar–Kr binary mixture with mole fraction of Kr atoms xKr = 0.25 is shown in Fig. 2. The constant channel width h = 5.22 nm is connected with the number of Ar atoms by the average number density q⁄ = 0.8074 (in units of r3). Periodic boundary conditions are imposed in the x- and y-direction of the simulation system. The motion tracks of the atoms are obtained by integrating the motion equation with an effective Leapfrog–Verlet algorithm [23]. Thermal wall model is applied at the interfaces of the walls and the fluid to ensure that all the fluid atoms are confined between the two parallel walls. In this model, when a fluid atom strikes the wall surface, it is assumed to undergo a series of collisions with the wall surface atoms, and rebound in a randomized velocity which is not correlated with the atom’s initial velocity, but determined by the wall temperature and chosen by random sampling from the Maxwell distribution. The three velocity components after the liquid atoms striking the wall surfaces are [24]:

vz

qffiffiffiffiffiffiffiffi

kB T W wG m qffiffiffiffiffiffiffiffi v y ¼ kBmT W w0G qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼   2kBmT W ln w

vx ¼

The Poiseuille flow is driven by applying an external force (F) in the x-direction to the atoms of the fluid, as shown in Fig. 1. The nondimensional simulation time step is chosen from 0.001 to 0.005 (all nondimensional quantities are reduced from the parameters of the component 1, i.e. e11, r11 and m1). A typical simulation requires 600,000–800,000 MD time steps. The initial 100,000 time steps are used to allow the system to reach the temperature equilibrium in NVT ensemble without applying external forces. In the NVT ensemble, no external forces are applied and the temperature is calculated from the total kinetic energy. Subsequently, an external force is applied and NVE ensemble is used. The velocity profile corresponding to the external force reaches stable in 500,000 time steps. Then, the primary data are collected to calculate the thermal conductivity and shear viscosity according to the Green–Kubo formula. The magnitudes of the external driving forces selected in the calculation of the thermal conductivity and shear viscosity are small enough to avoid non-linear variations in the fluid temperature [5]. The Ar–Kr mixture is in liquid state (temperature T = 94.4 K) [14]. The number density distribution and velocity profile are obtained by dividing the fluidic space between the two walls into a series of parallel layers and averaging the fluid atoms within the layers. The nondimensional number density distribution is obtained by

ð9Þ

q r3 ¼

r3

JM X N X

DVðJM  JN þ 1Þ

j¼J N i¼1

  Hn zi;j

ð10Þ

The velocity profile is given by where TW is the temperature of the boundary walls, wG , w0G are Gaussian-distributed random numbers with zero mean and unit variance, w is an uniformly distributed random number in (0, 1).

v x ðzi Þ ¼ PJ

M

j¼JN

1 PN

JM X N X

 i¼1 H n ðzi;j Þ j¼J N i¼1

Hn ðzi;j Þv xi;j

ð11Þ

where the function Hn(zij) is defined as

Hn ðzij Þ ¼



1 ðn  1ÞDz < zi < nDz 0 otherwise

ð12Þ

and vx⁄ij is the nondimensional x-component velocity of atom i at j time step, JN is the statistical start time step, JM is the end time step, z⁄ is the nondimensional z-component of the coordinate, Dz and DV are the height and the volume of one layer. 3. Results and discussion 3.1. Verification of the simulation model Due to lack of the reliable studies relevant to the transport properties of fluids in nanochannels, thermal conductivity and shear viscosity of Ar–Kr mixture with xKr = 0.33 in macroscale (out of confinement) are calculated as Min et al. [14] to partially validate our simulation model. The system temperature T = 94.4 K and the number density q⁄ = 0.8074 (in units of r3). The calculated thermal conductivity and shear viscosity are compared with the results by Min et al. [14] and good agreements are found in Table 1. For a further validation, the velocity profile of the liquid argon Poiseuille flow between platinum parallel walls is obtained. The number of the fluid atoms N = 1372 and the nondimensional external driving force F⁄ = 1.0. The obtained velocity profile is compared with that by Xu and Zhou [3] in Fig. 3 and excellent agreement appears. Hence, our computer code and simulation model is validated indirectly. 3.2. Density distribution Fig. 2. Original system configuration for the Ar–Kr binary mixture with xKr = 0.25 confined in the nanochannel of channel width h = 5.22 nm. The atoms are originally arranged in regular FCC lattices.

Fig. 4 shows the nondimensional number density distribution for the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0.1, 0.2 and

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3.3. Equilibrium system configuration

Table 1 Comparison between our results and those by Min et al. [14] in macroscale. Our results

Results by Min et al. [14]

2

k (W/mK)

2

6.561  10 5.074  104

gs (Pas)

6.413  10 5.122  104

6.0 5.5 5.0

* *

vx (z )

4.5 4.0

3.4. Velocity profile

3.5

Xu and Zhou [3] our results

3.0 2.5 2.0 1.5 0

2

4

6

8

10

12

*

z

Fig. 3. Velocity profiles of liquid argon between platinum parallel walls. The number of the fluid atoms N = 1372 and the nondimensional external driving force F⁄ = 1.0.

1.4 *

F = 0.1 * F = 0.2 * F = 0.3

1.2

3

1.0

ρ*σ

Fig. 5 shows the equilibrium system configurations for the Ar– Kr binary mixture with xKr = 0.25 as F⁄ = 0.1, 0.2 and 0.3, respectively. In the case of F⁄ = 0.1 and F⁄ = 0.2, the fluid atoms appear to be influenced by the attraction and repulsion forces of the neighboring atoms and the wall atoms, causing frequent collisions and well-distribution along the channel height. However, in the case of F⁄ = 0.3, the centrally-located fluid atoms distribute orderly and move straightly. With the increase of the external driving force, the fluid atoms move along the x-direction more regularly and receive marginal interferes. The phenomenon is related to the weakening roles of the wall atoms and the neighboring atoms as the external driving force increases, especially for the centrallylocated fluid atoms.

0.8

0.6

0.4 -2

0

2

4

6

*

z

8

10

12

14

16

Fig. 4. Number density distributions for the Ar–Kr binary mixture with xKr = 0.25 at different external driving forces.

0.3. As shown in the figure, the density of the mixture has a uniform and symmetric distribution, except large oscillations near the two walls. From the oscillation distribution of the number density near the walls, it can be concluded that the interactions between the wall atoms and the fluid atoms near the walls are very strong. As the external driving force increases, the number density in the center of the channel increases slightly. For the centrally-located atoms, the roles of the interacting forces with the walls weaken relatively with increasing the external driving force. The atoms near the walls have the trend of moving to the center of the channel and accordingly the number density of the centrallylocated atoms increases.

Fig. 6 shows the average velocity profiles of the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0.1, 0.2 and 0.3. It can be seen that the velocity profile has a higher distribution with an increase in the external driving force. While the velocity profile as F⁄ = 0.3 is not parabolic distribution, but appears a flat part in the center of the channel. The distinct velocity profile as F⁄ = 0.3 also can be predicted through the system configuration in Fig. 5, where the centrally-located fluid atoms tend to move straightly. Therefore, the flat velocity distribution appears in the center of the channel since the wall atoms have insignificant attractions to the centrally-located fluid atoms. Absolutely, the Poiseuille flow as F⁄ = 0.1, F⁄ = 0.2 and F⁄ = 0.3 are in the ‘‘coupling domain region’’ elucidated by Xu and Zhou [3] where the flow is governed by not only the applied external force but also the interatomic force due to the LJ potential. The larger external driving forces in Fig. 6 will not be involved in the calculation of thermal conductivity and shear viscosity in Sections 3.5 and 3.6. Furthermore, the velocity profiles of the Ar–Kr binary mixture with different mole fractions of Kr atoms as F⁄ = 0.1 are obtained, as shown in Fig. 7. As the figure shows, great changes in the velocity profile occur with a slight variation in the mole fraction of Kr atoms. Lower distribution and smaller maximum value of the velocity profile appear as the mole fraction of Kr atoms increases owing to the heavier mass of Kr atoms (m1 = 6.633  1026 kg, m2 = 13.914  1026 kg [5]). The non-linear variations in the fluid temperature are related to the degree of system non-equilibrium and the velocity profile. Hence, it can be concluded that the external driving force needed to cause non-linear variations in the fluid temperature for the mixture with higher mole fraction of Kr atoms is greater than that with lower mole fraction of Kr atoms. 3.5. Thermal conductivity Thermal conductivity of the confined Ar–Kr binary mixture with xKr = 0.25 in the x-, y- and z-direction is calculated by integrating the corresponding heat current autocorrelation function (HCACF) at time step according to the Green-Kubo formula, respectively. As shown in Fig. 8, the thermal conductivity in the z-direction is remarkably lower than that in the x- and y-direction as F⁄ = 0. This is caused by the inhibited fluid mobility in the z-direction owing to the presence of the boundary walls, because thermal conductivity is related to the velocities of fluid atoms (Eqs. (3) and (4)). Therefore, the thermal conductivity in the z-direction is limited and will not be discussed below. The limited thermal conductivity in the zdirection is similar to the lower radial thermal conductivity of water confined in single-walled carbon nanotubes (CNTs) concluded by Liu et al. [6].

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Fig. 5. Equilibrium system configurations for the Ar–Kr binary mixture with xKr = 0.25 in the x–z plane at different external driving forces. Solid walls are replaced with lines, Ar atoms are in green and Kr atoms in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0.5

1.2

λx λy

λ(W/m·K)

0.8

* *

vx (z )

λz

0.4

1.0

0.6

*

F = 0.1 * F = 0.2 * F = 0.3

0.4

0.3

0.2

0.1

0.2 0.0 0

0.0 2

4

6

*

8

10

12

14

z

Fig. 6. Velocity profiles of the Ar–Kr binary mixture with xKr = 0.25 at different external driving forces.

0.5

0.4

0.3

xKr = 0.12

* *

5000

7500

10000

12500

15000

17500

20000

m 0

vx (z )

2500

xKr = 0.16

0.2

xKr = 0.20 0.1

0.0 0

2

4

6

8

10

12

14

*

z

Fig. 7. Velocity profiles of the Ar–Kr binary mixture with different mole fractions of Kr atoms as F⁄ = 0.1.

Fig. 8. Thermal conductivity in x-, y- and z-direction of the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0.

As F⁄ = 0, the thermal conductivity in the x- and y-direction of the mixture with different mole fractions of Kr atoms is compared with the results in macroscale obtained by Min et al. [14] in Fig. 9. There is slight difference between the thermal conductivity in the x- and ydirection resulted by the limited size of the simulation system. As shown in the figure, the thermal conductivity in the x- and y-direction is obviously higher than that in macroscale. The enhanced longitudinal (x–y plane) thermal conductivity in the nanochannel is resulted by the strong interacting potential (potential part in Eq. (4)) between the fluid atoms and the wall atoms, which has been illustrated in detail in our early work [25]. Meanwhile, the thermal conductivity in the x- and y-direction increases with the increase of the mole fraction of Kr atoms, because of the strengthening interactions between the fluid atoms and the wall atoms. Fig. 10 shows the thermal conductivity in the x- and y-direction with m (in Eq. (7)) of the mixture with xKr = 0.25 as F⁄ = 0.005 is applied. As the figure shows, the thermal conductivity of the flowing binary mixture is obviously anisotropic, i.e. there is a difference between the thermal conductivity in the x-direction and that in the ydirection. The fluid atoms are more orderly and receive marginal interferes from the neighboring atoms in the x-direction under the action of the external driving force. Hence, the thermal conductivity in the x-direction is further enhanced in comparison with that in the y-direction. In the case of xKr = 0.25 and xKr = 0.50, the thermal conductivity in the x- and y-direction is further calculated at different external

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C. Sun et al. / International Journal of Heat and Mass Transfer 55 (2012) 1732–1740

(a) 1.0

1.2

λx

0.9

λy

1.0

xKr = 0.25

λ(W/m·K)

0.8

MD results in nanochannel 0.6

0.7

λx λy

0.6 0.5

0.4

0.4

0.2

-0.001

results in macroscale by Min et al. [14]

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

F*

0.0

(b) 1.2 0.0

0.2

0.4

xKr

0.6

0.8

1.0

1.1

xKr = 0.50

1.0

Fig. 9. Thermal conductivity in x- and y-direction of the Ar-Kr binary mixture with different mole fractions of Kr atoms as F⁄ = 0 compared with the results in macroscale.

0.9

λ(W/m·K)

λ(W/m·K)

0.8

λx λy

0.8 0.7 0.6 0.5

0.9

0.4

λx

0.8

-0.003

0.003

0.006

0.009

0.012

0.015

0.018

0.021

*

F

Fig. 11. Thermal conductivity in x- and y-direction of the Ar–Kr binary mixture with (a) xKr = 0.25 and (b) xKr = 0.50 varies with external driving forces.

0.6

λ(W/m·K)

0.000

λy

0.7

0.5 0.4

10

0.3

ηsxy

8

ηsxz

0.1

7

0.0 2500

5000

7500

10000

m

12500

15000

17500

20000

ηsyz

6 5

4

0

10 xηs(Pa·s)

9

0.2

Fig. 10. Thermal conductivity in x- and y-direction of the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0.005.

driving forces, as shown in Fig. 11. The range of the external driving forces for xKr = 0.25 is smaller than that for xKr = 0.50 to avoid non-linear variations in the fluid temperature, which has been indicated in Section 3.4. The thermal conductivity in the xdirection increases with an increase in the external driving force, while that in the y-direction keeps constant. The regularity of the fluid atoms in the x-direction strengthens with the external driving force increasing and accordingly the thermal conductivity in the x-direction increases. When the external driving force equals zero, the thermal conductivity in the x-direction approaches to that in the y-direction. 3.6. Shear viscosity Fig. 12 shows the xy-, xz- and yz-component of the shear viscosity with m (in Eq. (8)) of the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0. It can be found that the xz- and yz-component of the shear viscosity are enhanced compared with the xy-component. Because the shear viscosity is related to the off diagonal elements of the stress tensor (consisted of kinetic part and virial part in Eq. (6)), the enhanced xz- and yz-component of the shear viscosity in the nanochannel are caused by the collisions (virial part) between

4 3 2 1 0 0

2500

5000

7500

10000

12500

15000

17500

20000

m Fig. 12. xy-, xz- and yz-component of shear viscosity of the Ar–Kr binary mixture with xKr = 0.25 as F⁄ = 0.

the fluid atoms and the wall atoms in the z-direction. This is similar to the enhanced longitudinal thermal conductivity resulted by the strong interacting potential between the fluid atoms and the wall atoms. Furthermore, the xz- and yz-component of the shear viscosity of the mixture with different mole fractions of Kr atoms as F⁄ = 0 are compared with the shear viscosity in macroscale obtained by Min et al. [14], as shown in Fig. 13. The xz- and yz-component of the shear viscosity in the nanochannel are obviously higher than the shear viscosity in macroscale and increase with the increase of the mole fraction of Kr atoms. The phenomenon is related to the strengthening collisions between the fluid atoms and the wall atoms as increases the mole fraction of Kr atoms.

C. Sun et al. / International Journal of Heat and Mass Transfer 55 (2012) 1732–1740 24 22

decrease with increasing the external driving force owing to the weakening collisions between the fluid atoms and the wall atoms.

ηsxz

20

ηsyz

4. Conclusion

18 16

4 10 xηs(Pa·s)

14

MD results in nanochannel

12 10 8 6 4

results in macroscale by Min et al. [14]

2 0

0.0

0.2

0.4

0.6

0.8

1.0

xKr Fig. 13. xz- and yz-component of shear viscosity of the Ar–Kr binary mixture with different mole fractions of Kr atoms as F⁄ = 0 compared with the results in macroscale.

(a)

10

ηsxz ηsyz

xKr = 0.25

9

104xηs(Pa·s)

8 7 6 5 4 -0.003

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0.021

*

F

(b)

15

ηsxz ηsyz

xKr = 0.50

14

4 10 xηs(Pa·s)

13 12 11 10 9 8 -0.01

0.00

1739

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

*

F

Fig. 14. xz- and yz-component of shear viscosity of the Ar–Kr binary mixture with (a) xKr = 0.25 and (b) xKr = 0.50 vary with external driving forces.

In this paper, transport properties, including thermal conductivity and shear viscosity, of the Ar–Kr binary mixture confined in a nanochannel under Poiseuille flow are calculated by EMD simulation using the Green–Kubo formula. An external force is applied in the x-direction to drive the Poiseuille flow. The main conclusions are as follows. For the Ar–Kr binary mixture, as increases the external driving force the fluid atoms move along the x-direction regularly and receive marginal interferes as a result of the weakening roles of the wall atoms and the neighboring atoms. Meanwhile, the number density in the center of the channel increases slightly and the velocity profile has a higher distribution with the external driving force increasing. The longitudinal thermal conductivity of the confined mixture is higher than that in macroscale resulted by the strong interacting potential between the fluid atoms and the wall atoms. Meanwhile, the thermal conductivity increases with the mole fraction of Kr atoms increasing. As an external driving force is applied, the thermal conductivity of the flowing mixture is obviously anisotropic. With increasing the external driving force, in the x-direction the thermal conductivity increases, whereas in the y-direction it keeps constant. The xz- and yz-component of the shear viscosity of the confined mixture are enhanced compared with the xy-component owing to the collisions between the fluid atoms and wall atoms in the zdirection. Meanwhile, the xz- and yz-component of the shear viscosity are higher than the results in macroscale and increase as the mole fraction of Kr atoms increases. As applying an external driving force, the collisions between the fluid atoms and wall atoms weaken and accordingly the xz- and yz-component of the shear viscosity decrease. Transport properties of the Ar–Kr binary mixture confined in the nanochannel exhibit a significantly different behavior compared with those in macroscale geometry. In nanochannels, the thermal conductivity and shear viscosity are obviously enhanced compared with the results in macroscale. For the binary mixture, transport properties vary distinctly with the concentration of one of the components. The different behaviors are attributed to the interactions between the fluid atoms and the wall atoms, which are very strong in the nanochannel and vary with the concentrations of the components. In the nanochannel, a majority of fluid atoms are affected by the walls and, as a result, the fluid attains non-macroscale behaviors. These results should be taken into consideration in the design of nanofluidic devices. Further researches are needed to investigate the effects of wall materials, surface roughness, etc. on the transport properties of fluids in nanochannels. Acknowledgments

Fig. 14 shows that the xz- and yz-component of the shear viscosity of the mixture with xKr = 0.25 and xKr = 0.50 vary with the external driving forces. Comparing to the calculation of thermal conductivity, the range of the external driving forces in the calculation of shear viscosity is larger for the reason of numerical convergence. As shown in the figure, the xz- and yz-component of the shear viscosity decrease with an increase in the external driving force. With the appearance of an external driving force in the x-direction, the flow of the mixture speeds up and the collisions between the fluid atoms and the wall atoms weaken [26]. Accordingly, the xz- and yz-component of the shear viscosity

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