Flow around Surface-Attached Carbon Nanotubes - ACS Publications

Jan 27, 2006 - Simulations using no-slip and full-slip boundary conditions on the carbon ... velocity of the surfacesthis is the “no-slip” boundar...
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Ind. Eng. Chem. Res. 2006, 45, 1797-1804

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GENERAL RESEARCH Flow around Surface-Attached Carbon Nanotubes Ashlee N. Ford† and Dimitrios V. Papavassiliou*,†,‡ School of Chemical, Biological, and Materials Engineering, The UniVersity of Oklahoma and Sarkeys Energy Center, The UniVersity of Oklahoma, 100 East Boyd Street, SEC T335, Norman, Oklahoma 73019

Flow around surface-attached multiwall carbon nanotubes was studied using computational fluid dynamics. Two patterns of attachment were examined: nanotubes attached in a straight line and nanotubes attached in a forest formation. Simulations using no-slip and full-slip boundary conditions on the carbon nanotubewater interface were performed in order to compare the two limiting cases of no-slip and complete-slip. Appropriate length and velocity scales were found that can describe the features of the flow. These scales were different than those adopted conventionally for flow around infinite cylinders, and they were used to define a Reynolds number and to develop empirical expressions for the drag coefficient as a function of this Reynolds number. 1. Introduction The hydrodynamic behavior of flow around carbon nanotubes and around nanowires is of interest for possible applications, such as developing microfluidic devices, reducing drag on surfaces, and even for improving carbon nanotube synthesis processes. Often nanoscale roughness on a surface results in ultra-hydrophobic behaviorsthe well-known “lotus leaf effect”1,2s that has been shown to reduce drag for laminar flows.3 These drag reduction effects appear to be due to the formation of a thin air layer between the nano-indentations, limiting, thus, the effective contact area between the flowing water and the surface.3 The present work explores the case when more water is present, to the point that it can completely wet the nanoindentations (this is called a transition to a Wenzel state2). The hydrophobic character of the carbon nanotubes is maintained even in the Wenzel state because the graphite-water interfaces that comprise the surface of carbon nanotubes (CN) in water are hydrophobic.4,5,6 Furthermore, CNs coated with polymers can have hydrophobic behavior when water flows between the coated CNs.7 The effect of hydrophobicity on the description of flow past hydrophobic surfaces can be expressed using the concept of slip. (In conventional hydrodynamics, it is assumed that the velocity of the fluid next to a solid surface is equal to the velocity of the surfacesthis is the “no-slip” boundary condition for flow.) The flow of water past hydrophobic surfaces can, therefore, be characterized by slip boundary conditions, i.e., the velocity at the CN-water interface is not zero as would be the case when the standard no-slip velocity boundary condition is applied.4,8,9 The slip condition is associated with hydrophobicity because the hydrophobic repulsive interactions at the interface reduce the shear forces exerted on the fluid at the hydrophobic surface, which leads to a nonzero velocity at the interface. The method of choice for the simulation of the behavior of * To whom correspondence should be addressed. Phone: (405) 3255811. Fax: (405) 325-5813. E-mail: [email protected]. † School of Chemical, Biological, and Materials Engineering. ‡ Sarkeys Energy Center.

fluids in the vicinity of CNs is molecular dynamics (MD). Such studies have been conducted recently using nonequilibrium MD for the case of flow around infinite CNs.4,10 A surprising and yet significant finding from these simulations is that the MD results for the drag on the nanotubes were in very good agreement with predicted behavior from macroscopic StokesOseen equations.4,11 This conclusion suggests that simulations based on macroscopic equations that employ the continuum approximation can provide reliable results for drag and for flow around CNs (within the range of Reynolds numbers employed in the MD simulations). In fact, excellent agreement between simulations with macroscopic equations and MD simulations has also been obtained when the molecular diameter of the fluid is less than 10 times the dominant length scale of the geometry for flow past a micropatterned surface.12 On the basis of the findings described above, the present study is conducted using conventional computational fluid dynamics (CFD) and continuum Navier-Stokes equations to simulate the microscopic flow field around nanotubes. The dominant length scale of the problem geometry is the diameter of the nanotubes. The use of conventional CFD allows the simulation of flow around not only one but multiple nanotubes. The nanotubes can be attached on a surface, and they can exhibit different patterns instead of being infinitely long. The use of CFD offers the additional advantage of small computational time and userfriendly software13 (relative to, usually, custom-written MD codes). In the present study, the simulated tubes are finite, attached to a surface, and arranged in a diamond pitch forest or in a straight line. 2. Background The Stokes-Oseen equation is a semiempirical relationship that can predict drag coefficients as functions of volume fraction and Reynolds number4 for infinitely long, rigid cylinders that are in the vicinity of other identical cylinders. The equation applies at low Reynolds numbers, Re. However, a similar correlation is not proven to work for cylinders with very small diameters or for cylinders of a finite height attached to a surface.

10.1021/ie050932h CCC: $33.50 © 2006 American Chemical Society Published on Web 01/27/2006

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In general, the drag coefficient, CD, and the force, F, that act on an object immersed in a moving fluid are related by

CD )

F 1 2 Fu A 2

υ) (1)

where F is the fluid density, u is a characteristic velocity of the flow (usually the free stream velocity of the fluid), and A is the area on which the force is exerted. One of the contributions of the present study is to present empirical correlations for determining the drag coefficient on nanotubes as a function of the flow and of the CN dimensions. The specific simulated physical problem was the flow of water with a low velocity around a number of identical vertically oriented nanotubes attached to a flat surface. The nanotubes were arranged in two basic patterns: (a) a linear array of nanotubes placed one next to the other on a line perpendicular to the direction of flow and (b) a forest of nanotubes that were placed according to a diamond pitch. The tubes were modeled as smooth, rigid circular cylinders. The simulations were threedimensional and were conducted for various tube dimensions, water velocities, and tube spacings. In addition to simulations with no-slip boundary conditions on the nanotube-water interface, simulations with full-slip conditions were performed in order to compare the two limiting cases of no-slip and complete-slip. Full- or complete-slip refers to the case where the wall sheer stress at the nanotube surface is zero. The criterion for the rigorous applicability of no-slip NavierStokes continuum mechanics14 is that the Knudsen number, Kn, must be less than or equal to 10-3. For 10-3 e Kn e 10-1, the flow can be described using the Navier-Stokes equations even though partial slip at the boundary is possible. The primary concern for flows at the nanoscale is that the length scale approaches the molecular mean free paths. The Kn in the vicinity of nanotubes is given by

Kn )

λ L

(2)

where λ is a characteristic mean free path length for fluid molecules (e.g., for liquid water this length can be the intermolecular distance or the diffusion length) and L is a characteristic length for the macroscopic geometry of the problem, which in this case is equal to the nanotube diameter. Assuming an intermolecular distance of 0.3 nm for liquid water, it is calculated that the Kn is 0.006, 0.003, and 0.0012 for nanotubes with diameters equal to 50, 100, and 250 nm, respectively. It is acceptable, therefore, to apply the NavierStokes equations in this case. The Knudsen number can also be calculated based on a diffusion length, which is determined from the Brownian motion of the water molecules and molecular diffusivity. The molecular dispersion length, λ, for liquid is equal to the standard deviation of the displacement due to Brownian motion15

λ)

2ν xυPr

(3)

where ν is the kinematic viscosity, Pr is the Prandtl number, and υ is the frequency of molecular jumps. An expression for the molecular jump frequency is found by combining the viscosity of a Newtonian fluid,16

µ)

N ˜p exp(∆G ˜+ 0 /RGT) V ˜

and the rate equation for molecular jumps,

(4)

κT exp(- ∆G ˜+ 0 /RGT) p

(5)

where µ is the viscosity, κ is the Boltzmann constant, p is the Planck constant, T is the temperature, RG is the ideal gas ˜ is constant, ∆G ˜+ 0 is the standard Gibbs energy of activation, N Avogadro’s number, and V ˜ is the molar volume of the fluid. Combining eqs 4 and 5 gives

υ)

κN ˜T V ˜µ

(6)

which has a value of 1.35 × 1011 s-1 for water at 293 K. The dispersion length using eq 3 for water with a Pr of 6.96 and a ν of 0.995 × 10-6 m2 s-1 is 1.46 nm. Then, the Kn using the dispersion length has values of 0.02914, 0.0147, and 0.00583 for 50, 100, and 250 nm, respectively. These values are also within the acceptable range for the Navier-Stokes equations to be valid, but it is possible that there is slip on the CN-fluid interface for the length scale of interest. All three tube sizes are simulated with no-slip and with full-slip boundary conditions for comparison between the two limiting cases. Note that if the Kn is too large, simulations based on the continuum approach will fail to predict features of the flow, such as those described in section 3.5. 3. Computational Method 3.1. Geometry and Computational Grid. The simulations involved water flowing transverse to vertically oriented tubes. In this discussion, the streamwise direction is referred to as the x-direction, the spanwise, as the z-direction, and the vertical, as the y-direction. As mentioned already, the first of the two simulated geometries was an array of tubes arranged linearly in the z-direction. The computational model for this case consisted of a single tube surrounded by a rectangular computational box with the following boundary conditions: periodic on xy planes (on the sides of the box), specified bulk velocity at the inlet yz plane, outflow at the outlet yz plane, and no-slip at the bottom wall (xz plane). To model tubes of finite height, the computational domain extended in the vertical dimension to a height equal to twice the tube height. This choice was made somewhat arbitrarily, based on observations of the flow streak lines above the top of the nanotubes (we wanted to ensure that enough free space was available above the nanotube for the development of the flow in the vertical direction without boundary effects). The boundary condition on the top xz plane of the computational domain was set as symmetric, which means that the flux of all quantities was assumed to be zero across the symmetry boundary. This boundary condition generates flow in an infinitely long and wide channel that has a half-height equal to twice the nanotube height. This type of flow ensured that the fluid velocity would develop a parabolic profile. Figure 1a shows the bottom half of a channel with this geometry. The second simulated geometry was a forest of tubes arranged with diamond pitch. The symmetry of this geometry allowed for the reduction in the size of the computational domain by using four half-tubes placed with their centers at the midpoints of the sides of a square (see Figure 2). The computational domain for this case in the x-direction was equal to the domain size in the z-direction. For this nanotube configuration, the boundary conditions were given as symmetric for the xy planes (on the sides of the box) and periodic for the yz planes at the flow inlet and outlet. Figure 1b shows the bottom half of a

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Figure 1. Schematic of the physical problem modeled with the simulations described herein: (a) a linear array of nanotubes oriented perpendicular to the flow, which is along the x-direction, as designated by the dark arrows, (The actual computational domain used in the simulations included only one of the nanotubes, as indicated by the dashed lines.); (b) a forest of nanotubes that are arranged on a diamond pitch, with the flow along the x-direction.

Figure 2. Depiction of the four half-nanotubes used in the computational domain for the nanotube forest simulation. The domain is periodic in the x-direction, which is the flow direction, and symmetric in the z-direction. In the y-direction, the computational domain extends to twice the height of the nanotubes, at which point there is symmetry. Since the nanotubes are attached on a surface at y ) 0, symmetry in the y-direction implies that there is a similar nanotube arrangement attached at the top wall of an infinite channel.

channel with tubes arranged in this geometry. In this case, as was described in the previous paragraph, an empty rectangular volume with height equal to the tube height was created above

the tubes and the boundary condition at the top xz plane of the computational domain was set as symmetric. The numerical grid for both cases was created with the software GAMBIT 2.2.3017 and consisted of tetrahedral grid cells. Due to software limitations and Kn restrictions, the smallest CN diameter simulated was 50 nm, a dimension that corresponds to multiwall CNs.18 Nanotubes with diameter values of 100 and 250 nm were also studied for the case of CN arrays. The diameter of 100 nm is within the range for multiwall CNs,19 but the 250 nm case is at the very high end of multiwall CN diameters20 and corresponds realistically to cases in which the CNs are coated or “sheathed” with polymers or Teflon.7 The height of the nanotubes was 5 µm for most linear array simulations, but nanotubes with heights of 2 and 8 µm were also simulated (see Table 1). Identical nanocylinders with a height of 5 µm and a diameter of 250 nm were used in the forest simulations (see also Table 2). 3.2. Simulation Procedure. The default double precision segregated three-dimensional solver available in the finite volume-based program FLUENT 6.1.22 was used to simulate the laminar flow of isothermal (25 °C) water through the computational domain. The inlet velocity in the x-direction was specified in each run for the array of nanotubes, to simulate flow around only one row of nanotubes, and the inlet mass flow rate was specified in each run for the forest of nanotubes, to simulate the flow in a periodic domain (see Figure 2). The range of flow rates was determined to correspond to Reynolds numbers in the range of those studied by Walther et al.,4 for which the macroscopic equations were shown to agree with molecular dynamics. Numerical iterations were carried out until the residuals of the continuity equation and the x-, y-, and zmomentum equations were less than 10-5. No-slip stationary wall boundary conditions were set on the nanotube-water interface for no-slip cases, and a shear stress equal to zero in all directions was used on the CN-water interface for the slip cases. 3.3. Domain Independence. For the linear CN array, the size of the domain in front of the tubes in the x-direction was found to influence the results of the simulation. To reduce these entrance effects and to allow the velocity profile to fully develop in front of the tube array, the domain in front of the tubes was increased until the numerical results for the wall shear stress along the front of the tubes did not change. Figure 3 shows the values of wall shear stress along the front of a 250 nm tube in the linear array. The results converged for domains larger than 12.25 µm in front of the tube. The domain used for all the subsequent linear array simulations is 19.25 µm long in front of the nanotubes (roughly 4 times the height of the nanotubes used for the domain independence determination). The total dimensions of the computational domain were 21µm in the x-direction and 1.75 µm in the z-direction. For the nanotube forest, entrance effects did not influence the results because of the periodic boundary condition at the flow inlet. However, the effect of varying the spacing between the tubes was considered as a factor that affected the drag coefficient. The dependence on tube spacing is discussed later. 3.4. Grid Resolution Independence. The number of grid cells was initially chosen so that there were 20 grid points on the circumference of the nanotubes and 20 grid points along the height of each nanotube. The number of grid cells was then increased by refining the grid around the nanotubes, until the results of the simulation were grid-independent. To modify the number of grid cells, a region that surrounded each tube was selected. This region had a square base with each side equal to

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Table 1. Simulation Parameters for the Case of Nanotubes Arranged as a Linear Arraya CN diameter (nm)

CN height (µm)

avg velocity seen by the CNs using eq 9 (m s-1)

avg velocity across the domain inlet (m s-1)

Re as defined in eq 7

no. of grid cells in computational domain

250 100 50 250 250 250 250

5 5 5 2 8 5 5

0.0791 0.1978 0.3957 0.0791 0.0791 0.2725 0.3957

0.2532 0.6330 1.2661 0.2532 0.2532 0.8720 1.2661

0.088 0.139 0.197 0.056 0.111 0.303 0.440

819 960 875 163 704 406 330 252 921 910 819 960 819 960

a All cases were repeated with no-slip and slip boundary conditions. The spanwise dimension of the computational box was 1.75 µm, the streamwise dimension was 21 µm in all cases, and the distance from the top of the nanotubes to the top of the computational domain was equal to the height of the nanotubes in each case.

Table 2. Simulation Parameters for the Case of Nanotubes Arranged as a Foresta CN volume fraction

center to center distance (µm)

no. of grid cells in computational domain

0.01603 0.03206 0.06411

3.500 1.750 0.875

1 023 285 1 044 214 1 049 818

a The nanotube forests consisted of identical cylinders with diameters of 250 nm and heights of 5 µm arranged on a diamond pitch. All cases listed here were repeated with no-slip and slip boundary conditions with the following mass flow rates: 0.004423, 0.00729, 0.011058, 0.0150, 0.0180, and 0.02212 µg s-1.

Figure 3. Wall shear stress profile along the nanotube for a 250 nm diameter, 5 µm tall CN. Different symbols correspond to different sizes of the computational domain in front of the nanotube.

here was used for all the simulations. The largest number of grid cells corresponded to 80 grid points along the circumference of the nanotubes and 80 grid points along the height of the nanotubes. Each simulation result reported here is independent of the number of grid cells. 3.5. Validation. The computational methodology used for the simulation of flow around surface-attached nanotubes was first used to simulate flow around infinite nanotubes. All of the boundary conditions of the computational domain were set to periodic except those at the surface of the tubes, which were set as no-slip walls. Comparison between the Reynolds number required for the formation of wakes behind the tubes was used as a validation of the ability of this software to predict the features of flow around a nanocylinder. It is well-known21-23 that separation takes place and that two wakes form behind infinitely long circular cylinders at Reynolds number of approximately 10 (the characteristic length in the definition of this Reynolds number is the cylinder diameter, and the characteristic velocity is the velocity of the fluid at an infinite distance from the cylinders). A series of simulations were conducted by progressively increasing the fluid velocity. For Reynolds number of approximately 12, double wakes appeared behind the tubes, suggesting that the simulations were able to predict flow characteristics even at this scale. Figure 5 illustrates the recirculation of velocity vectors behind a tube. 4. Results and Discussion

Figure 4. Streamwise component of the total force on the CN as a function of the number of grid cells.

five nanotube diameters and with the center of the nanotubes located in the center of the square base. The height of this region extended 1 µm farther than the height of the nanotubes. The grid refinement region was defined as above for every case of nanotube arrangement and nanotube height. Each grid cell in the region was split into four for each refinement level. The change in the x-component of the force on the tubes as the number of grid cells was increased was monitored, as well as the velocity field. Figure 4 shows the plot of force vs number of grid cells for the linear array and forest used to determine grid independence. There was little change as the number of grid cells increased, so the largest numbers of grid cells shown

4.1. Linear Array of Carbon Nanotubes. For the tube array, the diameter, height, and velocity were varied independently in order to determine the influence of those parameters on the drag coefficient. Table 1 lists the parameters employed in the simulations. Interestingly, the simulations for the linear array show that defining a Reynolds number based on the CN diameter, as has been the conventional practice for flow around infinite cylinders, did not result in unique drag coefficient valuessthe drag coefficient was a function of the tube diameter, but it was also a function of the tube height. This is attributed to the finite length of the tubes. Because of this joint dependence on tube height and diameter, a new length scale had to be introduced. The geometric mean of the CN diameter, d, and height, h, was chosen in order to simultaneously account for variations in both tube dimensions. This length scale was used as the characteristic length for calculating the Reynolds number, Re, in Figure 6. In the present case, therefore, the Reynolds number is defined as

Re )

Fuxdh µ

(7)

where F is density, u is the characteristic fluid velocity of the system, and µ is the fluid viscosity. It is understood that this

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Figure 5. Velocity vectors colored by velocity magnitude (m s-1) for a 250 nm diameter, infinite tube with free stream x-velocity of 50.8 m s-1 and a Re of 12.7.

where umax is the maximum fluid velocity at the center of the channel (y ) 0). In that case, the average velocity, uavg, between the surface (y ) H) and a distance from the surface equal to the nanotube height (y ) h ) H/2) can be calculated to be

uavg )

Figure 6. Drag coefficient as a function of the Reynolds number for the case of a linear array of nanotubes and flow across them. The Reynolds number is defined based on eq 7. The solid line presents a fit for the noslip case (R2 ) 0.941), and the dashed line presents a fit for the slip case (R2 ) 0.893). The drag coefficient is smaller for the case of slip on the nanotubes-water interface.

Reynolds number definition does not approach the traditional Reynolds number as the nanotube height becomes infinitely large. However, it is suggested that this definition is suitable for describing small scale flow. The limit of its applicability needs to be investigated further. The Stokes-Oseen equation has been developed for 2D flow and is per unit length of a cylinder. The use of a Re like the one defined in eq 7 is speculated to be appropriate for tubes that are short enough that the average velocity of the fluid around the CN depends on h. Note here that the average velocity of the fluid along the height of the tubes was used as the characteristic velocity for calculating drag coefficients and for calculating Re, instead of a “free stream” velocity that is conventionally used for infinite cylinders. In our specific configuration, this characteristic velocity was calculated by integrating the parabolic velocity profile known to develop in an infinite channel. The velocity profile for laminar flow between two flat stationary plates separated by distance 2H is given by

( )

u(y) ) umax 1 -

y2 H2

(8)

5 u 24 max

(9)

This result is independent of the tube height for our configuration of the flow domain, so it was used for all the simulations for the linear array of nanotubes. The drag coefficients for the linear array simulations as a functions of the Reynolds number is shown in Figure 6 for both the slip and no-slip cases. The dependence of the drag coefficient on the Re is found to be

CD ∼

1 Re0.8

(10)

for both slip and no-slip cases, which is different than the Re-1 dependence that has been the usual approximate22 correlation for infinite cylinders at low Reynolds numbers. The drag coefficients for the cases of infinite slip are lower than that with no-slip boundary conditions, as expected. Actual flow around nanotubes should exhibit partial slip, meaning that the drag coefficient should be between that determined for the no-slip and infinite-slip cases. 4.2. Forest of Carbon Nanotubes. For the nanotube forest simulations, the velocity uavg was found by integrating the velocity profile that resulted from the simulations from the surface to the top of the nanotubes. The spacing between tubes in the forest was expected to influence the drag on each tube. Three different volume fractions, φ, were analyzed with simulations of various velocities. The volume fraction was defined as the ratio of the tube volume to the volume of the computational domain from the bottom wall to the top of the tube. The center to center distances for the three volume fractions were 14, 7, and 3.5 tube diameters. Figure 7 is a typical contour plot of the fluid velocity at different heights around the cylinders in the forest geometry. Table 2 lists the parameters

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nanotube-water interface, and the values of the coefficient in eq 11 are a ) 24.67, 21.95, and 19.33 for φ ) 0.064, 0.032, and 0.016, respectively. The drag reduction (i.e., reduction in the value of a) is 36.4% for φ ) 0.064, 30.8% for φ ) 0.032, and 26.7% for φ ) 0.016. Therefore, the denser the nanotube forest, the more important the hydrophobic effects are for the reduction of drag. For the case of a linear array of CNs, it can be seen (Figure 6) that the drag reduction for complete slip is on the order of 14.8%; this is almost half of the drag reduction effect seen for the forest of nanotubes. If one includes the volume fraction in a simple expression for the drag coefficient of the form

CD ) Figure 7. Contours of fluid velocity around a forest of surface-attached nanotubes that are arranged on a diamond pitch. The nanotube spacing is 1.75 µm, the nanotube diameter is 250 nm, and the nanotube height is 5 µm. The scale of the color bar is in m s-1.

a b φ Re

(12)

then the values of a ) 86.7 and b ) 0.287 and a ) 39.9 and b ) 0.175 can be obtained for the no-slip and the slip boundary conditions, respectively. Even though the regression produces values of R2 ) 0.99, we should be careful in generalizing eq 12, since only three different volume fractions have been simulated. With no experimental drag coefficient data available for the tube sizes and velocities studied here, the results of the simulations for the forest can be compared to the drag coefficients calculated using the Stokes-Oseen equation corrected for the case of multiple cylinders that occupy a volume fraction, φ,4,10

CD )

(

8π ReS-O ln(7.4/ReS-O)

)(

)

3 + 2φ5/3 9 9 3 - φ1/3 + φ5/3 - 3φ2 2 2

(13)

where ReS-O is the Reynolds number used by Stokes-Oseen, which is defined as

ReS-O )

Figure 8. Drag coefficient as a function of the Reynolds number for the case of a forest of nanotubes and different nanotube void fractions. (a) The no-slip boundary condition at the nanotube-fluid interface is applied. (b) The full-slip boundary condition applied at the nanotube-fluid interface. In both a and b, the solid, dashed, and dotted lines present fits for the case of nanotube volume fractions equal to 0.064, 0.032, and 0.016, respectively, with R2 values equal to 0.999 for all cases.

employed for the forest simulations. The computational domain in the xz plane was square, and the center to center distances between neighboring nanotubes were equal in the cross-flow and in the flow direction. The drag coefficient results are shown as functions of Re in Figure 8a and b for the no-slip and slip cases, respectively. The drag coefficients were found to decrease inversely with the Re

CD )

a Re

(11)

with each volume fraction exhibiting a different coefficient a. For the no-slip case, the coefficient a has the values of 38.77, 31.74, and 26.30 for φ ) 0.064, 0.032, and 0.016, respectively. The drag coefficient is smaller for the case of slip on the

Fud µ

(14)

with the characteristic length equal to the tube diameter. When the Reynolds number is defined in a conventional way (using the average fluid velocity in the infinite channel and the tube diameter), the error can be over 1000%, and it increases with φ. If the Re is defined using the average velocity, as proposed in the present study, then the values calculated with eq 13 are within 25% from the simulation results for the no-slip boundary condition cases. This is not a large discrepancy and can be attributed to the fact that the Stokes-Oseen equation has been developed for infinitely long cylinders. It was previously mentioned that the drag coefficients on the finite tubes do depend on the tube height, contradicting the main assumption of the Stokes-Oseen equation. In any case, eq 12 appears to be a simpler form of correlation between the drag coefficient, the Reynolds number, and the nanotube volume fraction. 4.3. Nanotube Deflection. An estimate of the deflection of the nanotubes due to the flow of water can be calculated, to assess our assumption of viewing CNs as rigid cylinders. The tubes can be assumed to behave as cantilever beams with distributed loads along their height. The loads were found from the simulation results for pressure exerted along the front of the tubes. The pressure as a function of tube height is shown in Figure 9 for the Re ) 0.09 case for the forest and the linear arraysthis is the case used as an example for the calculation of the CN deflection. Using fifth-order polynomials fits, the pressure as a function of height, P(y), can be described in the form of a5y5 + a4y4 + a3y3 + a2y2 + a1y + a0. The pressure on the tubes in the forest

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equal to the total load given by the fifth-order polynomial equations. The deflection calculations for nanotubes can then be simplified. If one knows the Reynolds number for a particular case, then Figure 6 or eq 12 can be used to determine the drag coefficient and, from that, the total force on the nanotube. Assuming uniform loading and using the maximum deflection equation for a cantilever beam under uniform loading, xmax ) -wdh4/(8EI), one can estimate the nanotube deflection with good accuracy. 5. Conclusions Figure 9. Pressure along the front of the tube as a function of tube height for a 5 µm tube with a Re ) 0.09.

was found to be P(y) ) 3.383 × 1031y5 - 3.462 × 1026y4 + 1.298 × 1021y3 - 2.178 × 1015y2 + 1.684 × 109y + 1.014 × 105 Pa, with y in meters, and the pressure on the tubes in the array was P(y) ) 1.821 × 1029y5 - 2.449 × 1024y4 + 3.879 × 1018y3 + 2.947 × 1013y2 - 1.281 × 108y + 1.014 × 105 Pa. A fourth-order linear differential equation governs the bending of an elastic beam and can be used to determine the deflection,24

d4x P(y)d ) -EI 4 dy

(15)

where d is the nanotube diameter so that P(y)d is the load on the nanotube, E is the Young’s modulus of the nanotube, I is the moment of inertia, x is the direction of the deflection, and y is the direction of the tube height. Equation 15 can be integrated using the following boundary conditions to determine the constants of integration: the shear force on the free end of the nanotube has to be zero (i.e., d3x/dy3 ) 0 at y ) h), the bending moment on the free end of the nanotube has to be zero (i.e., d2x/dy2 ) 0 at y ) h), the slope of the elastic curve at the fixed end of the nanotube has to be zero (i.e., dx/dy ) 0 at y ) 0), and the deflection at the fixed end of the nanotube has to be zero (i.e., x ) 0 at y ) 0). The resulting expression for the beam deflection is

x)

(

9 a4y8 a3y7 a2y6 a1y5 a0y4 -d a5y + + + + + + EI 3024 1680 840 360 120 24 3

C1

)

(16)

where

(

C1 ) - a5

h6 h5 h4 h3 h2 + a4 + a3 + a2 + a1 + a0h 6 5 4 3 2

)

Acknowledgment This work was partially supported by the Office of Naval Research through Grant N00014-03-1-0684. The donors of the Petroleum Research Fund, administered by the American Chemical Society, should also be acknowledged for support of this research through Grant PRF# 39455-AC9. FLUENT Inc. is acknowledged for allowing the use of their software with an educational license. Nomenclature

2

y y + C2 6 2

To conclude, empirical correlations for the drag coefficients on a linear array and a forest of nanotubes as functions of Re have been presented for slip and no-slip CN surfaces. Hydrophobic effects are more important as the nanotube volume fraction increases. The characteristic length and the characteristic fluid velocity that are appropriate and convenient for the definition of the Reynolds number in the case of a flow field around surface-attached nanotubes have been determined to be the geometric mean of the tube diameter and height and the average fluid velocity along the tubes, respectively. With the appropriate definition of the Reynolds number, the results obtained by CFD and the computational methods described here can be used to model applications that involve water flowing around nanotubes, as well as other fluids. They can also improve our understanding of microfluidics with surface nano-indentations.

(17a)

and

h7 h6 h5 h4 h3 h2 + a4 + a3 + a2 + a1 + a0 (17b) 7 6 5 4 3 2

C2 ) a5

The moment of inertia for a circular cross-sectional area is πr4/4 where r is the radius of the nanotube. Assuming a modulus value of 1.2 TPa for multiwalled CNs,18 the deflection at the end of the tubes in the forest is -8.74 nm and -8.59 nm for the linear array. These values correspond to approximately 3.4% of the tube diameter, so our assumption that the nanotubes are rigid cylinders is reasonable. For the particular example above, the deflection would have values of -8.75 nm for the forest and -8.59 nm for the linear array of nanotubes if the calculations of the deflection were done based on the assumption of a uniform load, wd, which is

a ) empirical constant [eqs 11 and 12] ai ) ith order polynomial coefficient b ) empirical constant [eq 12] A ) surface area on which drag force is exerted (m2) C1, C2 ) constants of integration [(Equations (17a,b)] CD ) drag coefficient d ) nanotube diameter (m) E ) Young’s modulus (Pa) F ) force on tube surface (N) ∆G ˜+ 0 ) standard Gibbs energy of activation (J) h ) nanotube height (m) H ) half-channel height (m) p ) Planck constant (6.626 × 10-34 J s) I ) moment of inertia (m4) Kn ) Knudsen number L ) characteristic large scale length for Kn definition (m) M ) bending moment (N m) N ˜ ) Avogadro’s number (6.0221 × 1023 molecules g-mol-1) Pr ) Prandtl number r ) nanotube radius (m) R2 ) coefficient of determination, defined as R2 ) 1 - χ2/ (∑iσi(yi - yj)2), where yi is the actual data point value, i is the index for each data point, yj is the mean of the actual values, σi is the standard deviation of the actual values, χ2 )

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∑i((yi - f(xi))/σi)2, and f(xi) is the calculated value at point xi from the curve fit of equation f. RG ) ideal gas constant (8.3145 J g-mol-1 K-1) Re ) Reynolds number T ) temperature (K) u ) fluid velocity (m/s) V ˜ ) molar volume of fluid (m3 g-mol-1) w ) uniform load on nanotube (∫h0 P(y) dy/h, Pa) x ) nanotube deflection (m) y ) vertical distance along tube height (m) Greek Characters κ ) Boltzmann constant (1.38066 × 10-23 J molecule-1 K-1) λ ) characteristic length for fluid molecules (m) µ ) dynamic viscosity (kg m-1 s-1) ν ) kinematic viscosity (m2 s-1) υ ) frequency of molecular jumps (s-1) π ) trigonometric constant pi F ) density (kg m-3) φ ) volume fraction of tubes in forest Subscripts avg ) average max ) maximum S-O ) Stokes-Oseen Literature Cited (1) Barthlott, W.; Neinhuis, C. Purity of the sacred lotus, or escape from contamination in biological surfaces. Planta 1997, 202 (1), 1-8. (2) Carbone, G.; Mangialardi, L. Hydrophobic properties of a wavy rough substrate. Eur. Phys. J. E 2005, 16 (1), 67-76. (3) Ou, J.; Perot, B.; Rothstein, J. P. Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 2004, 16 (12), 4635-4643. (4) Walther, J. H.; Werder, T.; Jaffe, R. L.; Koumoutsakos, P. Hydrodynamic properties of carbon nanotubes. Phys. ReV. E 2004, 69, 062201. (5) Lau, K. K. S.; Bico, J.; Teo, K. B. K.; Chhowalla, M.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Superhydrophobic carbon nanotube forests. Nano Lett. 2003, 3 (12), 1701-1705. (6) Jaffe, R. L.; Gonnet, P.; Werder, T.; Walther, J. H.; Koumoutsakos, P. Water-carbon interactionss2: Calibration of potentials using contact angle data for different interaction models. Mol. Simul. 2004, 30 (4), 205216.

(7) Gao, M.; Huang, S.; Dai, L.; Wallace, G.; Gao, R.; Wang, Z. Aligned coaxial nanowires of carbon nanotubes sheathed with conducting polymers. Angew. Chem., Int. Ed. 2000, 39 (20), 3664-3667. (8) Baudry, J.; Charlaix, E.; Tonck, A.; Mazuyer, D. Experimental evidence for a large slip effect at a nonwetting fluid-solid interface. Langmuir 2001, 17, 5232-5236. (9) Tretheway, D. C.; Meinhart, C. D. Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluids 2002, 14 (3), L9-L12. (10) Kotsalis, E. M.; Walther, J. H.; Koumoutsakos, P. Multiphase water flow inside carbon nanotubes. Int. J. Multiphase Flow 2004, 30 (7-8), 9951010. (11) Walther, J. H.; Jaffe, R. L.; Werder, T.; Halicioglu, T.; Koumoutsakos, P. On the boundary condition for water at a hydrophobic, dense surface. In Proceedings of the 2002 Summer Program, Center for Turbulence Research: Stanford, CA, 2002; 317-329. (12) Priezjev, N. V.; Darhuber, A. A.; Troian, S. M. Slip behavior in liquid films on surfaces of patterned wettability: Comparison between continuum and molecular dynamics simulations. Phys. ReV. E 2005, 71 (4), 041608. (13) Fluent 6.1 User’s Guide; Fluent Inc.: Lebanon, NH, 2005. (14) Roy, S.; Raju, R.; Chuang, H. F.; Cruden, B. A.; Meyyappan, M. Modeling gas flow through microchannels and nanopores, J. App. Phys. 2003, 93 (8), 4870-4879. (15) Einstein, A. Uber die von der molekular-kinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen. Ann. Phys. 1905, 17, 549. (16) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley and Sons: New York, 1960; p 28. (17) Gambit 2.2 User’s Guide; Fluent Inc.: Lebanon, NH, 2005. (18) Meyyappan, M., Ed. Carbon nanotubes: science and applications; CRC Press: Boca Raton, FL, 2005; pp 15, 68, 110. (19) Baughman, R. H.; Zakhidov, A. A.; de Heer, W. A. Carbon nanotubessthe route toward applications. Science 2002, 297, 787-792. (20) Shelley, S. Carbon nanotubes: A small-scale wonder. Chem. Eng. 2003, 110 (1), 27-29. (21) Van Dyke, M. An Album of Fluid Motion; Braun-Brumfield, Inc.: Stanford, CA, 1982; p 28. (22) Tritton. D. J. Physical Fluid Dynamics, 2nd ed.; Oxford Science Publications: New York, NY, 1988; pp 23, 113. (23) Wen, C. Y.; Yeh, C. L.; Wang, M. J.; Lin, C. Y. On the drag of two-dimensional flow about a circular cylinder. Phys. Fluids 2004, 16 (10), 3828-3831. (24) Beer, F. P.; Johnston, E. R.; DeWolf, J. T. Mechanics of Materials, 3rd ed.; McGraw-Hill: New York, 2002; pp 533-539.

ReceiVed for reView August 11, 2005 ReVised manuscript receiVed December 5, 2005 Accepted January 4, 2006 IE050932H