International Journal of Heat and Mass Transfer 55 (2012) 5087–5092

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International Journal of Heat and Mass Transfer

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Unified description of size effects of transport properties of liquids flowingin nanochannels

A.E. Giannakopoulos, F. Sofos, T.E. Karakasidis ⇑, A. LiakopoulosSchool of Engineering, University of Thessaly, 38834 Pedion Areos, Volos, Greece

a r t i c l e i n f o

Article history:Available online 8 June 2012

Keywords:NanofluidicsDiffusion coefficientThermal conductivityShear viscosityMolecular dynamics

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05

⇑ Corresponding author. Tel.: +30 2421074163; faxE-mail address: thkarak@uth.gr (T.E. Karakasidis).

a b s t r a c t

The present work connects transport properties of simple fluids flowing in nanochannels: the diffusioncoefficient, the shear viscosity and the thermal conductivity. We used non-equilibrium molecular dynam-ics (NEMD) simulations of liquid argon flowing in a nanochannel formed by krypton walls, macroscopi-cally equivalent to the planar Poiseuille flow, at constant temperature. All properties approach bulkvalues as the channel width increases. As the channel width decreases, the transport properties changedramatically: the diffusion coefficient decreases with the channel width, the shear viscosity increaseswith the channel width and the thermal conductivity increases with the channel width. The novel resultof this work is that, if the size effect of one of the transport properties (say the diffusion coefficient) isknown, then all the others can be estimated from results that apply to classic fluid mechanics.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The third generation of Nanotechnology products are ‘‘Nano-structures by Design’’ and includes systems of nanosystems withmulti-scale architectures. The scientific and engineering challengesto develop ‘‘Nanostructures by Design’’ are enormous. The presentwork attempts to bridge models of different scale and aims to in-crease our ability to employ scientific principles in creating nano-structures with specific functionalities, providing design strategiesfor their synthesis and assembly. In this work we developed struc-ture (channel)-fluid property relationships and their dependenceon the macroscale (bulk) behavior of fluids. We believe that linkingmolecular simulations to constitutive laws (scaling laws) would en-able rapid estimates for response parameters, such as diffusion, shearviscosity modulus and thermal conductivity, that will help under-standing the functionality of atomically precise nanochannels andtheir assembly into nanochannel systems. Potential applications ofour work include biosensors, fuel cells, artificial organs, drug deliverysystems, smart membranes for clean water technology, targetednutrition proliferation through the stomach wall, microheatexchangers and catalytic microreactors (see for example [1–3]).

A liquid molecule can be viewed as a Brownian particle that expe-riences a hydrodynamic resistance to its motion, corrections arenevertheless needed. Einstein [4] showed that the diffusivity of sucha molecule in an unbounded liquid equals the thermal fluctuationenergy divided by the drag coefficient. Hindered transport theorystarted in the early 1950s, motivated by the desire to understand

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the effects of molecular size on microvascular permeability [5,6].The molecular radius r and the parallel-plate channel width h,define the key parameter of the relative molecular size �k ¼ r=h. Thisparameter introduces a size effect, not only on the diffusion coeffi-cient D, but on the shear viscosity l and the thermal conductivityk. In this work, the channel width h is taken as the distance betweenthe first solid layer of atoms at each side of the channel.

It is commonly accepted [7] that diffusion at nanoscale increaseswith channel width and reaches its bulk value at h � 12r. However,liquid/solid slippage, playing an increasing role at the nanoscale,may change this picture (see for example [8,9]). Shear viscosity in-creases as the channel width decreases [10]. Thermal conductivityincreases as temperature rises [11].

There are also studies concerning fluids in confined mediawhich are macroscopically immobile such as water inside CNTs,where transport properties have been calculated [12]. The effectof nanoparticles added to the fluid on the thermal conductivityof Ar confined in a nanochannel was examined in [13].

In this work we study the interrelation of the average transportproperties ðD;l; kÞ of liquid, as derived by non-equilibrium molecu-lar dynamics (NEMD) simulation of Poiseuille flow between two‘‘infinite’’ walls. Particular examples of liquid argon flowing betweenatomically smooth krypton walls are obtained from the work of Sofoset al. [14]. Possible liquid/solid slippage was not addressed further inthis work, since for the particular examples of atomically smooth so-lid surfaces that were examined it was not found to be important.

The work is organized as follows: In Section 2 the molecular sys-tem is described. Section 3 presents the calculation methodology ofthe three transport properties ðD;l; kÞ. The results are interrelatedin Section 4. The conclusions are presented in Section 5.

Nomenclature

C concentrationCV specific heat at constant volumeD (self-) diffusion coefficientd number of system dimensionsEi dispersion coefficients, , i = x, y, zFext external driving force (magnitude)h channel widthI unitary tensorJp microscopic stress tensorJq microscopic heat flow tensorkB Boltzman constantKw wall spring constantLx length of the computational domain in the x-directionLy length of the computational domain in the y-directionLz length of the computational domain in the z-directionm atomic massMSD mean square displacementN number of atomsp pressureri position vector of atom irij distance vector between ith and jth atom

req position of a wall atom on fcc lattice siteR universal gas constantT temperatureu(rij) LJ potential of atom i with atom jV volume of the computational domainv kinematic viscosity

Greek symbolsap coefficient of thermal expansionr length parameter in the LJ potentiale energy parameter in the LJ potentialn average displacement of the walls due to the pressureq⁄ reduced densitygs shear viscosityjT isothermal compressibilityk thermal conductivity�k molecular sizeti velocity vector of atom iti ith component of atomic velocity, i = 1, 2, 3tm molecular volume

Fig. 1. The atomic system under examination.

5088 A.E. Giannakopoulos et al. / International Journal of Heat and Mass Transfer 55 (2012) 5087–5092

2. System modeling

The NEMD technique is used to simulate planar Poiseuille flowof liquid argon. The geometry of the system under examination ispresented schematically in Fig. 1. The walls consist of two layers ofkrypton atoms arranged on fcc lattice. Atomic interactions are de-scribed by a Lennard–Jones 12-6 potential:

uLJðrijÞ ¼ 4eððr=rijÞ12 � ðr=rijÞ6Þ ð1Þ

where the parameters of the Lennard–Jones potential are: rAr�Ar ¼0:3405 nm;rKr�Kr ¼ 0:3633 nm;rAr�Kr ¼ 0:3519 nm; eAr�Ar=kB ¼119:8K; eKr�Kr=kB ¼ 167:0 K; eAr�Kr=kB ¼ 141:4 K, the cut-offradius is rc = 2.5r, the atomic mass of argon is mAr = 39.95 a.uand the atomic mass of krypton is mKr = 83.8 a.u. (kB is theBoltzmann’s constant). The formulation is typical for the, socalled, simple fluids.

Flows in several nanochannels of variable width (h) have beensimulated (see Table 1). The average fluid density q⁄ = 0.642 (inunits of r�3) has been kept constant. The streamwise and spanwisedimensions Lx and Ly as well as the number of wall atoms are keptconstant throughout this work (Lx = Ly = 36.15 Å, NKr = 288). Peri-odic boundary conditions are assumed in the x- and y-direction.

Wall atoms are kept around their original fcc lattice positionsdue to an elastic spring potential uwallðjrðtÞ � reqjÞ ¼12 KwðjrðtÞ � reqjÞ2, where r(t) is the position of an atom at time t,req is its initial lattice position and Kwis the spring constant. As ex-plained by Liem et al. [15], if we compute the second derivative ofthis potential and the second derivative of the LJ potential (Eq. (1)),evaluated both derivatives at r = r0 = 21/6r, and equate the results,we get Kw = 57.15 (e/r2). We remind the reader that r0 = 21/6r is thedistance where the LJ potential reaches its minimum. The value ofKw = 57.15 (e/r2) was used in all our simulations.

An external driving force Fext is applied along the x-direction toevery fluid particle during the simulation. The magnitude of theexternal applied force is selected in order to avoid non-linear vari-ations in fluid temperature induced by the flow [16], and is in therange 0.018–0.18pN. The choice of the force employed in the re-sults was based on the fact that previous studies has shown thatfor such values the obtained transport properties did not seem todepend on the value of the force employed [17].

The wall atoms absorb the increase in kinetic energy of the fluidatoms and Nosé–Hoover thermostats at the thermal walls are em-ployed in order to keep the system’s temperature constant. A de-tailed description of the use of Nosé–Hoover thermostats isprovided in Evans & Holian [18] and Holian & Voter [19]. We usetwo independent thermostats one for the upper wall and anotherfor the lower wall in order to achieve better thermalisation ofthe wall atoms.

The simulation step is 10�2 ps. In the beginning of a simulationrun fluid atoms are located on fcc sites. Atoms are assigned appro-priate random initial velocities in order to reach the desired tem-perature. The system reaches equilibrium state after a run of 105

timesteps (units:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimr2=e

p). Then, NEMD simulations are per-

formed with duration of 105 timesteps. The temperature profilesinside the channel were found to be almost uniform [14].

3. Theoretical relations for the molecular dynamic estimates ofthe transport properties

The diffusion coefficient can be obtained using either the Ein-stein’s relation

D ¼ limt!1

12dNt

XN

j¼1

½rjðtÞ � rjð0Þ�2* +

ð2Þ

or Green–Kubo’s relation

Table 1Dimensions of the systems under examination.

Case Lz (Å) h (Å) h/r Fluid atoms

1 24.100 9.04 2.65 1922 30.125 15.06 4.42 3203 42.175 27.11 7.9 5764 78.325 63.26 18.58 13445 180.8 171.1 50.1 3648

1 An estimate of the time interval s can be given by dimensional analysis, using thebulk modulus of the shear viscosity lo: min s = los3/(kBT). For T = 100K, 120K and150K, min s = 7.12ps, 4.75ps and 2.37ps, respectively. These results are close to ourNEMD calculations for the steady state estimates of the transport properties.

A.E. Giannakopoulos et al. / International Journal of Heat and Mass Transfer 55 (2012) 5087–5092 5089

D ¼ 1dN

Z 1

0

XN

j¼1

tjð0Þ � tjðtÞ* +

dt ð3Þ

where rj is the position vector of the jth atom and d is the dimen-sionality of the system (d = 1 for diffusivity calculation in one direc-tion, d = 2 in two directions and d = 3 in three directions) and tj isthe velocity vector of the jth atom. The brackets indicate time aver-age and N is the number of the fluid atoms (Ar) in the channel.

The two relations (2) and (3) are equivalent and provide thesame results [7]. They are derived for systems in equilibrium, butthey can be used for non-equilibrium systems as well, providedone excludes the drift contribution from the flow (see [14] for re-lated details). In the present work we have used Einstein’s relationfor the calculation of the diffusion coefficient. The computation iscarried out in two steps. In the first step the Mean Square Displace-ment (MSD) is obtained from the definition

MSDðtÞ ¼ 1N

XN

j¼1

½rjðtÞ � rjð0Þ�2* +

ð4Þ

and subsequently D is evaluated based on Eq. (2) which can be writ-ten as

D ¼ limt!1

12dt

MSDðtÞ ð5Þ

Shear viscosity and thermal conductivity for systems in equilibriumcan be calculated using the Green–Kubo formalism, as described inFernandez et al. [20]. These relations can be used without modifica-tion in NEMD as long as we stay in the linear regime close to equi-librium [21]. When the system is not in equilibrium, due to theexistence of an external force, the magnitude of the external forceshould be small enough for linearization to hold [16]. Our systemis in the linear regime for all channel widths studied for the rangeof forces we employed as discussed in Section 2.

Shear viscosity l for a pure fluid is computed by the relation

l ¼ 1VkBT

Z 1

0h Jxy

p ðtÞ � Jxyp ð0Þidt ð6Þ

where Jxyp is the off-diagonal component of the microscopic stress

tensor

Jxyp ¼

XN

i¼1

mitxi t

yi �

XN

i¼1

XN

j>1

rxij@uðrijÞ@ry

ij

ð7Þ

u(rij) is the LJ potential of atom i interacting with atom j, rij is thedistance vector between atoms i and j, and tj

i is the j-component(j = x, y or z) of the velocity of atom i. V is the computational domainfluid volume (V = Lx � Ly � h).

Pressure has been calculated by averaging the sum of the diag-onal elements of the stress tensor (equals to the force applied onone wall divided by the wall area).

p ¼ 13ðJxx

p þ Jyyp þ Jzz

p Þ ð8Þ

On the other hand, thermal conductivity k can be calculated by theintegration of the time-autocorrelation function of the microscopicheat flow Jx

q, i.e.,

k ¼ 1VkBT2

Z 1

0hJx

qðtÞ � Jxqð0Þidt ð9Þ

where the microscopic heat flow Jxq is given by

Jxq ¼

12

XN

i¼1

miðtiÞ2tx

i �XN

i¼1

XN

j>1

rxij :@uðrijÞ@rx

ij

� I � uðrijÞ" #

� txi ð10Þ

where ti is the speed velocity magnitude of atom i and I is the uni-tary matrix.

For all computed transport properties presented in this paper,error bars were calculated based on five sets of simulations, eachone with duration of 105 timesteps.

4. Analysis of the relations between transport properties

4.1. The anisotropy of the diffusion coefficient

We now turn to a classic assessment of the molecular dynamicsresults, linking then to existing results from continuum fluidmechanics. We expect the velocity ti and the concentration C tofluctuate around the time average values Vi and C, respectively(see for example [20]). Then

ti ¼ �ti þ t�iC ¼ �C þ C�

ð10Þ

The dispersion coefficients Ei are defined, if the time average of t�i C�

is expressed as

1s

Z s

0t�i C�dt ¼ �Ei

@�C@xi

ðno sum over iÞ ð11Þ

where s is the time interval during which we are observing thephenomenon.1

In the slit case we have calculated that Ez � 0, Ex � Ey � E. Wedenote by D the self-diffusion coefficient [22]. Then, ‘‘anisotropic’’diffusion in the slit can be considered to be due to the dispersioncoefficient E

Dz � D; Dx � Dþ E; Dy � Dþ E ð12Þ

The molecular dynamics computations indicate that E depends onthe Schmidt number ðvDÞ ¼

lqD, following the relation [14]

E � blqD

ð13Þ

where q is the fluid density and l is the shear viscosity of the fluid.For the particular Ar density and h/r > 4, our molecular dynamicscalculations estimate b � 0.38 � 10�10(m2/sec). Note that E is nota Taylor–Aris type of dispersion (does not depend on hV2

i ), nor a tur-bulent type of dispersion (does not depend on hVi).

4.2. The average pressure and the relation with the macroscopicconstitutive equation

The influence of the wall deformation is that the channel widthwill be increased due to the pressure inside the channel. If �p is theaverage pressure in the computational domain V, the averagedisplacement n of the walls due to the pressure is

n � LxLy�pKw

ð14Þ

50 100 150 200

−40

−20

0

20

40

60

80

Temperature / (K)

Pres

sure

/ (M

Pa)

h=2.65σ

h=4.42σ

h=7.9σ

h=18.58σ

h=50.1σ

Ptp, Ttp

Pc, Tc

Fig. 2. State diagram, temperature versus pressure. (constant specific volumet � 0.037m3/kmol).

2 Another useful hindrance factor estimator is due to Oseen [26–28]

D=Do ¼ 1��k

ð1� �kÞf ð�kÞ98� 81

128�k

� �tan�1 1� �k

f ð�kÞ

� �

with f ð�kÞ ¼ 1� 81256

�k2� �1=2

5090 A.E. Giannakopoulos et al. / International Journal of Heat and Mass Transfer 55 (2012) 5087–5092

For a very stiff wall, Kw ?1 and so n ? 0. The above approximationholds for n < < h, as the NEMD simulations showed. Let max|sxz| bethe maximum shear stresses that develops at the channel walls.From classic Poiseuille flow analysis:

max jsxzj ¼ l 3�tx

h=2ð15Þ

where �tx is the average flow velocity (2/3 of the maximum fluidflow velocity tmax) and l is the shear viscosity of the fluid. Equilib-rium (Body force = Friction force) along the flow direction gives

NFext ¼ 2 max jsxzjLxLy ) �tx �NFextðhþ 2nÞ

18LxLylð16Þ

The estimate of Eq. (16) proved a very good estimate of the averageflow velocity from the NEMD simulations (see also [23]). The exter-nal force Fext was kept low, in order to avoid non-linear terms of themacroscopic constitutive law.

The principle of corresponding states for a stationary fluid con-nects the absolute temperature T, the pressure p and the molecularvolume tm in a state equation [24]

pD ¼RT

tm � be�a=tmRT ð17Þ

where R is the universal gas constant (R = 8.3144 J/molK) and a, bare constants. For Ar, the constants are [25] a ¼ 0:1746 MPa m6

kmol2and

b ¼ 0:0348 m3

kmol. The above constitutive equation works reasonablewell around the critical point given by tC = 2b, pc ¼ a

4e2b2, Tc ¼ a4Rb.

Note that in our case the molecular volume is

t ¼ hLxLyNA

Nð18Þ

where NA ¼ 6:023� 1023 atomsmol is the Avogadro’s number. The critical

point of Ar (T > Tc) saturated gas, T < Tc) saturated fluid) is [25]:Tc = 150.8 K, pc = 4.87 MPa, tC = 0.0749 m3/kmol. The triple pointof Ar (coexistence of gas, liquid and solid) is at Ttp = 83.81 K andptp = 68.9 kPa. All our calculations are in the region Ttp < T < Tc,p < ptp and t < tC.

In Fig. 2 we present the NEMD calculations of the average pres-sure in the channel as a function of the temperature for a molecularvolume tm = 0.037 m3/kmol. The NEMD results suggest that theflow is that of a compressible linear Neutonean fluid with constantviscosity [14]. For high channel widths (h/r > 10), we note that theMaxwell relation @p

@T

��V¼ ap

jT� 1 MPa

K holds (ap is the coefficient of

thermal expansion and jT is the isothermal compressibility). Con-tinuum fluid mechanics (within the context of a Newtonean con-stitutive law) can predict either a pressure gradient or a bodyforce along the flow.

Using the estimate (Eq. (16)) and neglecting the wall deforma-tion (n << h), we obtain a body force of the type

bx � �1

27Nh

Fext

LxLyð19Þ

Equilibrium equation becomes

bx ¼ l @2tx

@y2 ¼ �2l tmax

ðh=2Þ2¼ �12l

�tx

h2 ð20Þ

where for the Poiseuille flow the maximum flow velocity is relatedto the average flow velocity as tmax ¼ 3=2�tx.

The exact value of the pressure due to the steady state flow can-not be given by the continuum analysis, unless the pressure isdetermined at a certain point along the flow direction (this is notthe case in the present analysis). Therefore, we can only give anupper and a lower bound for the average pressure �p in the compu-tational domain.

�p ¼ pD �12

bxLx ð21Þ

Note that in the absence of fluid motion ð�tx � 0Þ, we have�p � pD, which is the Dieterici estimate given by Eq. (17). For thepresent calculations, �p ¼ pD � 35:4 MPa and the results are shownin Fig. 2. All the NEMD results fall within the bounds of Eq. (21).Equation (17) is also plotted for comparison.

4.3. The size effect on the diffusion coefficient and its influence on theshear viscosity

Below the critical point temperature, T < Tc, the hindrance factorD/Do for diffusion in slits can be used to capture the size effect (Ddecreases as the channel width decreases) [5]2

DDo¼ ð1� �kÞ�1

"1þ 9

16�k ln �k� 1:19358�kþ 0:4285�k3

� 0:3192�k4 þ 0:08428�k5

#ð22Þ

where �k ¼ r=h is the relative size of the atoms and Do is the bulkself-diffusion coefficient (h ?1). Fig. 3 compiles our MD resultsand the expressions of the hindrance factor as proposed by Dech-adilock & Deen [5] and by Oseen [26]. The hidrance factor by Oseenis summarized in Dufresne et al. [27] and Happel & Brenner [28].The range of validity of Dechadilock & Deen is h/r > 18, while therange of validity of Oseen is h/r < 7r. Next we examine the wellknown Stokes–Einstein (lower bound) and the Sutherland (upperbound) equations [3,29–30]:

D ¼ kBT6plr

ðStokes—EinsteinÞ ð23Þ

¼ kBT4plr

ðSutherlandÞ ð24Þ

We have compared the above predictions with our NEMD calcula-tions in Fig. 4. The Sutherland estimate [29] assumes that the atomsare slipping one past the other, so it is a good estimate for h > 20r.

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Channel width [σ]

Dz/D

0

Dechodilok and DeenOseenMD Diffusion

Fig. 3. Hindrance factor versus channel width, where we observe that MD diffusionis similar to Ossen Diffusion for h < 7 � 8r and similar to Dechodilok and Deendiffusion for h > 18 � 20r.

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Channel width [σ]

102

×λ

[W /

m−1

K−1

]

Approximation Eq(25)MD calculations

Fig. 5. Thermal conductivity, extracted from proposed relation and through MDcalculations versus channel width at 120K.

0 10 20 30 40 500

2

4

6

8

10

12

Channel width [σ]

109

× D

[m2 s

−1]

Stokes−EinsteinSutherlandMD

Fig. 4. Self diffusion from Stokes-Einstein relation, Sutherland and MD results at100K. Stokes-Einstein is a good estimate for h < 18 � 20r and Sutherland for h > 20r.

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

Channel width [σ]

Nor

mal

ized

tran

spor

t pro

pert

ies

D/D∞

μ/μ∞

λ/λ∞

Fig. 6. Transport properties are shown (in normalized values, divided by theirrespective bulk value (D1;l1; k1) versus channel width.

A.E. Giannakopoulos et al. / International Journal of Heat and Mass Transfer 55 (2012) 5087–5092 5091

The Stokes–Einstein estimate [30] assumes that the atoms are stick-ing when they come in contact, so it is a good estimate for h < 20r.The important conclusion is that the size effect of D is directly influ-enced by the size effect of the shear viscosity l: as D is decreasingwith the channel width, l is increasing. For T > Tc, the above esti-mates of the hindrance factor D/Do break down.

4.4. The size effect on the thermal conductivity

The laminar flow along a flat plate is a solid–fluid interfaceproblem (essentially a boundary layer problem). A thermal ana-logue can be invoked between the heat transfer coefficient H be-tween the channel wall and the fluid and the mass transfercoefficient k [31]. In this analogy, the Schmidt number l/(qD),for the mass transfer is replaced by the Prandtl number lCV=k ,where CV is the specific heat at constant volume and k is the ther-mal conductivity [32]. The Sherwood number, kLx/D, for the masstransfer is replaced by the Nusselt number, HLx=k. The moleculardynamic calculations show that

klCv

¼ aqDl

� 1=2

) k ¼ aCvðqDlÞ1=2 ð25Þ

For argon at 120 K, Cv ¼ 0:523 kJkg K and q ¼ 1078 kg

m3 (molecularweight M ¼ 39:948 kg=kmol) and q ¼ M=t. Taking the constanta � 5 � 106 (by fitting Eq. (24) with the k’s as computed by theNEMD calculations), the agreement of approximation (24) withour NEMD calculations is excellent, as shown in Fig. 5. Observe thatk is independent of l, since Dl kBT/r. We conclude that k has avery small dependence on the relative channel size h/r. k can beconsidered to depend point-wise on D. Indeed, the spacial variationof D along the channel width is the same with that of k (lower closeto the channel walls).

Regarding the influence of temperature (Ttp < T < Tc), our NEMDcalculations show that the diffusion coefficient increases linearlywith temperature and the shear viscosity modulus decreases inver-sely with temperature. The thermal conductivity is not strongly tem-perature dependent. These results are similar with the experimentalobservations for the bulk properties of argon [33]. Our limitingNEMD results (h/r > 20) approach the experimental bulk results.

5092 A.E. Giannakopoulos et al. / International Journal of Heat and Mass Transfer 55 (2012) 5087–5092

The slight reduction of thermal conductivity for small nano-channels comes from the fact the respective density profiles pres-ent a layered structure, while, as the channel width increases, wehave a nearly uniform density profile approaching that of the mac-roscopic theory, with only the region very close to the wall to bealways of low density [23,34–36]. In this layered structure of thefluid, the transfer of energy through momentum exchange is moredifficult to occur than in a larger channel and this is why the ther-mal conductivity is reduced. On the contrary, this ordered struc-ture presents increased shear viscosity since the motion of onelayer over the other is more difficult than when layers are more li-quid like.

5. Conclusions

In the present work we presented a unifying approach that con-nects the important transport properties of simple fluids flowing innanochannels at constant temperature: the diffusion coefficient D,the shear viscosity l and the thermal conductivity k. Results haveshown that there is a considerable size effect on the transportproperties for small channel sizes (the channel width h in ourcase). However, we managed to relate the numerical estimates ofnon-equilibrium molecular dynamics (NEMD) simulations withmacroscopic relations like Eqs. (22)–(24), that are used in the anal-ysis of fluids flowing in bulk channels. The key result is that, if thesize effect on one of the transport property is known (sayD = D(h,T)), then all the others (l = l(h,T), k ¼ kðh; TÞ) can be foundfrom the forementioned relations that apply to bulk flow mechan-ics. Fig. 6 epitomizes our results and indicates that the diffusioncoefficient reduces with the channel width h, the shear viscosityincreases with the channel width and the thermal conductivity isalmost constant with the channel width. All properties approachtheir respective bulk values ðDo;lo; koÞ as the channel width in-creases (e.g. for the argon case h > 20r). We expect that these re-sults will be very useful in the design of micro and nano channelsystems, providing the step for faster development of complexdrug delivery systems, biosensors and catalytic micro reactors.The present work suggests that the knowledge of the D or l trans-port property, either from simulations (as in the present case) orfrom direct experiments allows for the estimation of the other two.

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