15
Part 29 Micro- and Nano-Fluidics
Part 29
Micro- and Nano-Fluidics
1
Massively Parallel Mesoscopic Simulations of Gas Permeability of Thin Films Composed of Carbon Nanotubes
Alexey N. Volkov and Leonid V. Zhigilei
Abstract: A mesoscopic computational model for simulation of gas flow through carbon nanotube (CNT) films is developed. The model is implemented in a parallel computational code enabling massively parallel dynamic simulations of CNT materials at length scales relevant to experimental studies. Self-diffusivity of Ar within CNT films with 9% volume fraction of the nanotubes and the effective diffusivity of Ar through the films are calculated for two different structures of the films: a continuous network of CNT bundles and a layered arrangement of dispersed individual CNTs. The results of the simulations suggest a moderate structural sensitivity of the gas diffusivities, with about 3-4.5 times lower values of self-diffusivities predicted for films with dispersed CNTs, and a smaller difference in the values of the effective diffusivities that are found to be on the order of 10-6 m2s-1 for both film structures.
1 Introduction
Unique mechanical, thermal, and electrical properties of carbon nanotubes (CNTs) provide opportunities for their application as a basic component in novel multifunctional materials. The porous structure of CNT films or “buckypaper” also suggests a potential use of these materials in gas separation and storage. The design and optimization of new CNT-based materials can be facilitated by computational modeling of their mechanical, transport and gas permeability properties. Computational studies of the effective properties of CNT-based materials, however, have been limited by the absence of models capable of describing large systems of interacting CNTs. Recent development of a mesoscopic dynamic model for CNT materials [11,13,16] opens up new opportunities for investigation of the structural dependence of various properties of this promising class of materials. In particular, first application of the mesoscopic model for analysis of thermal properties of buckypaper has revealed a strong effect of self-organization of CNTs into a network of bundles on the effective thermal conductivity of the material [14]. In this paper we report an extension of the mesoscopic model to investigation of gas flow through thin CNT films and analysis of the connections between the film structure and gas permeability.
2 Mesoscopic model for dynamic simulations and structural characterization of CNT materials
In the mesoscopic model, every individual CNT is represented as a sequence of stretchable cylindrical segments [16]. Forces applied to the ends of the segments are described by a mesoscopic force field (MFF). The internal part of the MFF accounts
Alexey N. Volkov University of Virginia, Materials Science and Engineering Department, 3 9 5 M c C o rm i c k R o a d , Charlottesville, Virginia 22904-4745, USA, e-mail: [email protected] Leonid V. Zhigilei University of Virginia, Materials Science and Engineering Department, 3 9 5 M c C o rm i c k R o a d , Charlottesville, Virginia 22904-4745, USA, e-mail: [email protected]
2
for stretching and bending of CNTs [16]. Interactions between CNTs in MFF are described by the tubular potential method [11] based on the approximate inter-tube (tubular) potential [13]. The model is implemented in a parallel computational code designed to ensure a good scalability for massively parallel simulations.
Fig. 1 Distributions of individual nanotubes in the initial Sample I composed of dispersed straight CNTs arranged into layers (a) and in Sample II obtained after 6 ns of dynamic mesoscopic simulation that resulted in the formation of a steady-state continuous network of bundles (b). The samples are composed of 7829 (10,10) single-walled CNTs with radius RT = 0.6785 nm. The length of each CNT is 200 nm, the dimensions of the systems are 500 nm × 500 nm × 100 nm, density of the material is 0.2 g cm-3 and porosity is 91 %. Periodic boundary conditions are applied in x- and y-directions.
In buckypaper and CNT films, individual nanotubes are arranged into continuous networks of interconnected bundles [10,15]. These structures can be produced in mesoscopic dynamic simulations, in which initial samples with straight CNTs dispersed within layers stacked on top of each other, Fig. 1(a), quickly (within several ns) evolve into steady-state random networks of interconnected CNT bundles, Fig. 1(b) [12,13]. In order to study the effect of the structural arrangement of CNTs on the gas permeability of the CNT films, we perform simulations for Sample I with layered dispersed CNTs shown in Fig. 1(a) and Sample II with continuous network of CNT bundles shown in Fig. 1(b). Both samples are anisotropic films with preferred orientation of CNTs within the plane of the film and have the same density and porosity. The porous structures of the samples, however, are quite different, Fig. 2, with much smaller sizes of the pores present in Sample I. The most probable pore diameter of 23 nm in Sample II is in a good agreement with experimental value of 21 nm measured for SWCNT buckypaper [5]. Other structural characteristics of Sample II, e.g., averaged bundle diameter, also agree with experimental observations [10,15].
3 Mesoscopic model for gas-CNT systems
In the mesoscopic model used in the simulations of gas flow through a CNT material, motion of individual gas atoms is described by equations of motion similar to those of conventional molecular dynamics simulations [2]. Assuming that CNTs are not movable and form a rigid porous network, the forces experienced by the gas atoms are defined by the following potential:
∑∑∑ ∑= == +=
+ϕ=N
i
M
kikCNT
N
i
N
ijijgg UrU
1 1)(
1 1
)( , (1)
3
where φgg(rij) is the Lennard-Jones potential describing the interaction between gas atoms i and j separated by a distance of rij, UCNT(ik) is a mesoscopic potential for interaction between gas atom i and CNT k, N and M are the total numbers of gas atoms and CNTs, respectively. The potential φgg(rij) is parameterized for Argon atoms [1,8] and a cubic cutoff function [13] is used to smoothly bring the potential to zero at rij = 11.9 Å.
Fig. 2 Probability density function of opening diameters calculated in granulometric analysis of digitized CNT samples I (square symbols, Fig. 1(a)) and II (triangular symbols, Fig. 1(b)) with a voxel size of 0.5 nm. Distributions are not calculated for opening diameters larger than 30 nm due to the limited thickness of the CNT films.
Openings diameter (nm)
Pro
babi
lity
dens
ityfu
nctio
n(
nm-1)
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
Sample I
Sample II
Sr
x
TR
Infinitely long nanotube
Gas atom
φ
(a)
O
A
Segment 1 Segment 2
D
B
CSpherical junctionbetween segments
Semisphericalcap at the tube end
Node
Node
Node
TR
Sr
Sr
Sr
1Sr 2Sr
1Dl 2Dl
(b)
2D1D
Fig. 3 Schematic drawings illustrating the interaction between a gas atom and a straight infinitely long CNT of radius RT (a) and between a gas atom and a part of a CNT consisting of a semispherical cap on the nanotube end, two segments, and a spherical junction between them (b). In panel b, letters A, B, C, and D indicate various possible positions of gas atoms interacting with the nanotube.
For a gas atom – CNT pair, the corresponding interaction potential UCNT(ik) is calculated based on potential UCNT∞ describing the interaction between an atom and a straight infinitely long nanotube, Fig. 3(a). This potential depends only on the shortest distance rS from the atom to the surface of the nanotube and can be calculated as follows
∫ ∫+∞π
σ∞ φ
φ++φ−+ϕ=
0 0
2222 sin))cos1((4)( dxdRrRxRnrU TTgcTSCNT , (2)
where φgc(r) is the interaction potential between a gas atom and a carbon atom on the surface of the CNT separated by interatomic distance r, RT is the radius of the CNT, n
σ is the surface density of carbon atoms on the surface of the CNT, and the
distance x and angle φ are defined as shown in Fig. 3(a). The potential φgc(r) is adopted in the form of the Lennard-Jones potential with a cubic spline cutoff function [13] and parameterized for interaction between C and Ar atoms [1,8]. The values of UCNT∞ and its derivative are calculated numerically and used in a tabulated form in the simulations.
The description of the interaction between a gas atom and a curved CNT is based on representation of the nanotube by a chain of cylindrical segments with
4
spherical junctions between them and semispherical caps at the nanotube ends, Fig. 3(b). For each of the gas atoms, a list of the nanotube elements (segments, junctions, or caps) with surfaces that are within the potential cutoff distance rc from the atom is formed. Segments are included in this list only if the points on their axes closest to the atom are within the segment, e.g., point D1 for atom D and segment 1 in Fig. 3(b). The length of the segments is chosen to be larger than rc and, in cases shown in Fig. 3(b) for atoms A, B, and C, this list contains no more than one segment of the nanotube. In these cases, UCNT(ik) = UCNT∞ (rmin), where rmin is the closest distance to the surfaces of the nanotube elements from the list. If the list contains several segments, then UCNT(ik) is calculated as a weighted sum of potentials for all segments in the list. For example, for atom D in Fig. 3(b), the potential UCNT(ik) is calculated as UCNT(ik) = w1UCNT∞ (rS1) + w2UCNT∞ (rS2), where wm = lDm/(lD1+lD2) and lDm is the distance between the point Dm on the axis of the segment m (m=1,2) that is the closest point to the gas atom and the node joining segments 1 and 2. This approach can easily be adopted for interactions of gas atoms with more than two segments of a highly curved CNT.
4 Gas self-diffusivity within CNT films
The computational setup for simulations of self-diffusivity of argon atoms within CNT films is shown in Fig. 4(a). The films, generated without periodic boundary condition in z (out of plane) direction, are immersed into a gas reservoir with small gaps of thickness δ between the film surfaces and reservoir boundaries. The periodic boundary conditions are then imposed in all three directions and dynamic simulations of Ar gas equilibrated at a temperature of 300 K and different values of pressure are performed. The Ar self-diffusivities in the in-plane, Dx and Dy, and out-of-plane, Dz, directions are calculated with the Einstein relation [2],
∑=∞→α α−α=N
iiit
tNt
D1
2|)0()(|1
2
1lim , (3)
where α = x, y, z and αi(t) is the true (corrected to eliminate the effect of the periodic boundary conditions) α coordinate of Ar atom i at time t.
Hfi
lm
x
z
Periodic boundary
conditionsNanotube filmof thickness Hfilm
(a)
Gas atom
δδ
High-pressure gas reservoir with initial pressure p0
Hfi
lm
x
zPeriodic boundary
conditions
Nanotube filmof thickness Hfilm
(b)
Gas atom
Hre
s
Vacuum
Fig. 4 Computational cell used in the simulations of self-diffusivity of gas atoms within a CNT film (a) and gas permeability through a CNT film from a high-pressure reservoir to vacuum (b). Periodic boundary conditions are imposed in all directions in (a) and in the lateral (parallel to the film surface) directions in (b).
The self-diffusivity D = (Dx + Dy + Dz)/3 in pure Ar gas (without the presence of a CNT film) at a temperature of 273 K and pressure of 101325 Pa is found to be equal to 1.56×10-5 m2s-1, which is in a good agreement with experimental value of 1.57×10-5 m2s-1 [4]. The inverse proportionality of D to pressure (dashed lines in Fig. 5) also agrees with predictions of the kinetic theory of gases [4].
5
The mean free path of argon atoms at 0.1 bar is about 500 nm, much larger than the most probable diameter of openings revealed in the granulometic analysis of CNT samples (Fig. 2). Therefore, the Knudsen regime of gas flow through the porous CNT film is realized in simulations at pressure values below ~0.1 bar. Under these conditions, the collisions between the gas atoms and CNTs play the dominant role in defining the values of self-diffusivity, which can be expected to be independent on the gas pressure. The results of the simulations shown in Fig. 5 are consistent with this analysis. Moreover, in the range of partial gas pressure, p, up to 3 bars, the self-diffusivities of Ar exhibit only a weak tendency to decrease with increasing p. The presence of nanotubes reduces the self-diffusivity of argon in the out-of-plane direction more appreciably than in the in-plane direction (Fig. 5). The self-diffusivities exhibit moderate sensitivity to the film structure, with the values predicted for Sample II (network of bundles) being ~3-4.5 times larger than in Sample I (layered dispersed CNTs). For Sample I, which thickness, Hfilm, can be easily changed in simulations, no noticeable changes in the results are observed with increase of Hfilm.
Partial gas pressure (Pa)
Sel
f-di
ffus
ivity
Dx
(m2 s-1
)
103 104 105 10610-7
10-6
10-5
10-4
10-3
10-2
Pure Ar gasSample I,Hfilm = 20 nmSample I,Hfilm = 100 nmSample I,Hfilm = 200 nmSample II,Hfilm = 100 nm
(a)
Partial gas pressure (Pa)
Se
lf-d
iffu
sivi
tyD
z(m
2 s-1)
103 104 105 10610-7
10-6
10-5
10-4
10-3
10-2
Pure Ar gasSample I,Hfilm = 20 nmSample I,Hfilm = 100 nmSample I,Hfilm = 200 nmSample II,Hfilm = 100 nm
(b)
Fig. 5 The values of self-diffusivities of Ar in x- and z-directions, Dx (a) and Dz (b), predicted in simulations of Ar gas diffusion in Samples I and II shown in Fig. 1. The simulations are performed at temperature 300 K and various values of the partial gas pressure. The partial gas pressure is calculated based on the ideal gas equation p = nkT, where n = N / V and T are the gas number density and temperature, k is the Boltzmann constant, N is the total number of Ar atoms, and V is the sample volume. Dashed lines correspond to the self-diffusivity obtained in simulations of pure Ar gas.
5 Gas permeability of CNT films
The computational setup for simulations of gas permeability of CNT films shown in Fig. 4(b) mimics the experimental setups used for measurement of gas permeability of porous materials [6,7,9]. A CNT film is placed in between a high-pressure reservoir with initial gas pressure p0 and a vacuum reservoir, where the pressure is zero. The simulations are performed at a constant temperature of 300 K maintained by the Berendsen thermostat method [3]. Gas atoms gradually permeate through the film and the pressure in the high-pressure reservoir, p(t), drops. Assuming that the size of the reservoir is sufficiently large and the normal diffusion (where the effective diffusivity is independent on pressure) is realized [6], the pressure decay, after completion of the initial filling of the film by the gas atoms (Fig. 6(a)), should be exponential, i.e. p(t)=p(t0)exp(-(t-t0)/τ) [6,9]. Once p(t) is measured, one can use it for calculation of the time constant τ and, in turn, the effective gas diffusivity through the film, Deff = VresHfilm / (Afilmτ) = HresHfilm / τ (Fig. 6(b)), where Vres = HresAfilm and Hres are the volume and thickness of the gas reservoir, and Afilm
6
and Hfilm are the surface area and thickness of the film. Fast convergence or Deff with increasing Hres enables computationally efficient
evaluation of the gas permeability of thin nanoporous films in the mesoscopic simulations. In particular, Hres = 400 nm is found to ensure a sufficiently accurate calculation of Deff for samples I and II, with errors being within 5% with respect to further increase in Hres. The time t0, needed for the initial filling of the film depends on the effective diffusivity of the film and, hence, is larger for Sample I than for Sample II. With proper choice of t0, the time dependence of Deff exhibits only small variations around its averaged value caused by the finite thickness of the film (Fig. 6(b)). This observation suggests that the gas permeation occurs in the normal diffusion regime. The values of Deff are found to be within 25% from the corresponding values of self-diffusivity Dz. The value of Deff determined for sample I is ~3 times smaller than that for Sample II.
Time (ns)
Pre
ssu
re(P
a)
1000
2000
3000
4000
50006000700080009000
10000
Sample II(network of bundles)
210 3 4
(a)
Normal diffusion
Initialfilling ofthe film
765
Averaged partialgas pressure
inside the film
Pressure in the reservoir
Time (ns)
Eff
ectiv
edi
ffusi
vity
(m2 s-1
)
4E-07
6E-07
8E-07
1E-06
1.2E-06
1.4E-06
1.6E-06
1.8E-06
2E-06
Self-diffusivityDz in sample I
10
Effectivediffusivity,sample II
2 53
Self-diffusivityDz in sample II
Effectivediffusivity,sample I
(b)
8764
Fig. 6 Pressure in the high pressure reservoir and averaged partial gas pressure inside the film vs. time (a) and the effective diffusivity Deff vs. time (b) in the gas permeation simulations. In panel a, solid curves correspond to the pressure in the reservoir and inside the film obtained for Sample II. In panel b, solid curves are obtained for samples I and II, dashed and dash-dotted lines correspond to the self-diffusivity Dz in samples I and II at a pressure of 0.1 bar. Simulations are performed with Hres = 400 nm, p0 = 104 Pa, and temperature of 300 K. Time t0 is chosen to be 5 ns for Sample I and 2.5 ns for Sample II.
In Ref. [9], the effective diffusivity of a few common gases through films composed of dispersed multi-walled CNT was measured and values of Deff for N2 (kinetic diameter of N2 gas at 300 K is close to that of Ar) and sufficiently thick samples were found in the range of (7.2-8.2)·10-6 m2s-1.* In order to compare the results of the mesoscopic simulations with these experiments, an additional Sample III was generated with basic properties that were similar to those in experimental samples. This sample of size 500 nm × 500 nm × 200 nm was composed of dispersed CNTs with external radius of 9.73 nm and length of 200 nm. The CNTs are arranged in parallel layers similar to those in Sample I. The density of Sample III, 0.31 g cm-3, was calculated assuming that every CNT has 23 walls with the radius of the internal wall equal to 2.25 nm. The volume fraction of CNTs was equal to 15%. The simulation of Ar permeation through Sample III, performed at 300 K with Hres = 1000 nm and p0 = 104 Pa, predicts Deff = 5.9·10-6 m2s-1, which is within 20-30% from the experimental values. Based on the comparison of the results obtained for samples I and II, one can expect that the film composed of the
* As clarified through communication with Dr. Akos Kukovecz, the values of the effective
diffusivity reported in Table 4 of Ref. [9] are given in units of mol s kg-1 and not in units of m2 s-1 as erroneously stated in the paper. The values should be multiplied by a factor of RT (R is the universal gas constant, T = 298 K) to be converted to units of m2 s-1.
7
multi-walled CNTs would have a somewhat higher effective diffusivity if the layered system of dispersed straight CNTs would be allowed to evolve in a dynamic mesoscopic simulation into a more realistic structure of interconnected network of CNT bundles with larger pore sizes as compared to layered systems.
6 Conclusion
Self-diffusivity of Ar gas within CNT films and the effective diffusivity of Ar gas through the films are studied in mesoscopic simulations. In permeation simulations, the pressure in the high-pressure reservoir follows the exponential dependence characteristic of the normal diffusion. The effective diffusivity is found to be of the same order of magnitude as the gas self-diffusivity in the direction perpendicular to the surface of the film. Simulations predict the effective diffusivities on the order of 10-6 m2s-1 for films with 9% volume fraction of (10,10) CNTs and a moderate sensitivity of the diffusivity on the film structure.
Acknowledgements
The financial support is provided by NASA (Grant No NNX07AC41A) and NSF (Grant No CBET-1033919). Computational support is provided by NCCS at ORNL (project MAT009). The authors would like to thank Dr. Akos Kukovecz of the University of Szeged, Hungary for helpful communication at the final stage of the paper preparation.
References
1. Ackerman, D.M., Skoulidas, A.I. Sholl, D.S., Johnson, J.K.: Diffusivities of Ar and Ne in carbon nanotubes. Molecular Simulation 29, 677–684 (2003)
2. Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Clarendon Press, Oxford (1987)
3. Berendsen, H.J.C., Postma, J.P.M., van Gunsteren, W.F., DiNola, A., Haak, J.R.J.: Molecular dynamics with coupling to an external bath. Chem. Phys. 81, 3684–3690 (1984)
4. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge (1970)
5. Cinke, M., Li, J., Chen, B., Cassell, A., Delzeit, L., Han, J., Meyyappan, M.: Pore structure of raw and purified HiPco single-walled carbon nanotubes. Chem. Phys. Lett. 365, 69-74 (2002)
6. Cooper, S.M., Chuang, H.F., Cinke, M., Cruden, B.A., Meyyappan, M.: Gas permeability of a buckypaper membrane. Nano Lett. 3, 189–192 (2003)
7. Dullien, F.A.L.: Porous Media. Fluid Transport and Porous Structure. Academic Press, New York (1979)
8. Skoulidas, A.I., Sholl, D.S.: Transport diffusivities of CH4, CF4, He, Ne, Ar, Xe, and SF6 in silicalite from atomistic simulations. J. Phys. Chem. B 106, 5058–5067 (2002)
9. Smajda, R., Kukovecz, Á., Kónya, Z., Kiricsi, I.: Structure and gas permeability of multi-wall carbon nanotube buckypapers, Carbon 45, 1176–1184 (2007)
10. Thess, A., Lee, R., Nikolaev, P., Dai, H., Petit, P., Robert, J., Xu, C., Lee, Y.H., Kim, S. G., Rinzler, A.G., Colbert, D.T., Scuseria, G.E., Tománek, D., Fischer, J.E., Smalley, R.E.: Crystalline ropes of metallic carbon nanotubes. Science 273, 483–487 (1996)
11. Volkov, A.N., Simov, K.R., Zhigilei, L.V.: Mesoscopic model for simulation of CNT-based materials. In: Proceedings of the ASME International Mechanical Engineering Congress and Exposition, ASME paper IMECE2008-68021 (2008)
12. Volkov, A.N., Simov, K.R., Zhigilei, L.V.: Mesoscopic simulation of self-assembly of carbon nanotubes into a network of bundles. In: Proceedings of the 47th AIAA Aerospace Sciences Meeting, AIAA paper 2009–1544 (2009)
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13. Volkov, A.N., Zhigilei, L.V.: Mesoscopic interaction potential for carbon nanotubes of arbitrary length and orientation. J. Phys. Chem. C 114, 5513–5531 (2010)
14. Volkov A.N., Zhigilei, L.V.: Scaling laws and mesoscopic modeling of thermal conductivity in carbon nanotube materials. Phys. Rev. Lett. 104, 215902 (2010)
15. Wang, S., Liang, Z., Wang, B., Zhang, C.: High-strength and multifunctional macroscopic fabric of single-walled carbon nanotubes. Adv. Mater. 19, 1257–1261 (2007)
16. Zhigilei, L.V., Wei, C., Srivastava, D.: Mesoscopic model for dynamic simulations of carbon nanotubes, Phys. Rev. B 71, 165417 (2005)
Predicting Three-Dimensional Inertial ThinFilm Flow over Micro-Scale Topography
Sergii Veremieiev, Philip H. Gaskell, Yeaw Chu Lee and Harvey M. Thompson
Abstract Alternative lubrication and depth-averaged formulationsof the unsteadyNavier-Stokes equations are used to explore the problem of gravity-driven inertialthin film flow on substrates containing topography that is either fully submergedor protrudes through the film. The resulting discrete form ofthe equation sets aresolved using a multigrid strategy incorporating automaticadaptive time-stepping,enabling accurate mesh independent solutions to be generated very efficiently. Twobenchmark test problems are solved revealing the extent of the free surface distur-bance that ensues, together with the effect of inertia on thesame.
1 Introduction
The deposition and flow of continuous thin liquid films over man-made or naturallyoccurring functional substrates containing regions of micro-scale topography is ofgreat importance in numerous engineering and biologicallyrelated fields. For ex-ample, in the context of engineering processes thin film flowsplay a key role inphoto-lithography and the laying down of precision coatings [1], while in biological
Sergii VeremieievSchool of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail:[email protected]
Philip H. GaskellSchool of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail:[email protected]
Yeaw Chu LeeSchool of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail:[email protected]
Harvey M. ThompsonSchool of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail:[email protected]
1
2 Sergii Veremieiev, Philip H. Gaskell, Yeaw Chu Lee and Harvey M. Thompson
systems they impact on areas as diverse as tissue engineering [6] and plant diseasecontrol [10].
s0
lt
lp
wp
wt
è
x
yz
(a)(b)
wp
wt
èlp
lt
x
yz
Fig. 1 Schematic of gravity-driven film flow: (a) over a square submerged peak topography and(b) past a square occlusion.
At present, the modelling and solution of three-dimensional free-surface filmflow on substrates containing topography is still at an earlystage of developmentwith most existing analyses based on the long-wave approximation (LWA). The lat-ter enables the reduction of the governing time-dependent Navier-Stokes equationsto a more tractable coupled system of partial differential equations which reducesthe dimensionality of the problem by one. Two reduction scenarios are consideredand solved for: (i) a lubrication formulation (LUBF) [5], where the dependent vari-ables are the film height and pressure; (ii) a depth-averagedform (DAF) [9], whichinvolves two velocity components and the film height, and is able to account forinertia.
2 Lubrication and Depth-Averaged Formulations
The problem of interest, shown schematically in Figure 1, isof gravity-driven lam-inar film flow down a planar surface inclined at angleθ to the horizontal and con-taining either a square submerged topography (Figure 1a) orocclusion (Figure 1b).The liquid is assumed to be Newtonian and incompressible, with constant viscosityand surface tension. The chosen coordinate system is as indicted and the parametersconcerned written in non-dimensional form following [5] and [9]. The solution do-main is bounded from below by the inclined surfaces(x,y) and from above by thefree-surfacef (x,y,t) at timet; therefore the film thickness is given byh = f − s.
Predicting Three-Dimensional Inertial Thin Film Flow overMicro-Scale Topography 3
The variablesf andh are apriori unknown and their prediction forms the main taskat hand.
Following several recent investigations [5, 8, 9], the LWA [7] is employed; themain assumption being that the ratioε = H0/L0 ≪ 1, whereH0 is the asymptoticfilm thickness andL0 = H0/(6Ca)1/3 is the capillary length.
The LUBF and DAF share two common equations:
∂h∂ t
+∂∂x
(hu)+∂∂y
(hv) = 0, (1)
p = −ε3
Ca∇2 (h + s)+2ε (h + s)cotθ , (2)
wherep(x,y,t), u(x,y,t) and v(x,y,t) are dimensionless pressure, averaged alongthe film streamwise and spanwise components of the velocity respectively.
The LUBF employs the following explicit expressions for ¯u andv, obtained fromthe x- and y-momentum equations using the LWA and neglecting the convectiveterms:
u = −h2
3
(
∂ p∂x
−2
)
, v = −h2
3∂ p∂y
, (3)
which are then substituted into equation (1). In the case of the DAF,u andv are alsoobtained from thex- andy-momentum equations using LWA but without neglect-ing the convective terms; together with the assumption of self-similarity of velocityprofiles, they result in the following implicit forms for ¯u andv:
εRe
[
∂ u∂ t
−u5h
∂h∂ t
+65
(
u∂ u∂x
+ v∂ u∂y
)]
= −∂ p∂x
−3uh2 +2, (4)
εRe
[
∂ v∂ t
−v5h
∂h∂ t
+65
(
u∂ v∂x
+ v∂ v∂y
)]
= −∂ p∂y
−3vh2 , (5)
where Re is the Reynolds number. Clearly for Re= 0 the DAF equations (4) and(5) reduce to the LUBF equations (3) and both models are equivalent. The bound-ary conditions applied assume fully developed flow both upstream and downstream[5], the prescription of a static contact angle,θS, condition [8] at the surface of anocclusion, and in the case of the latter, either zero-flux in the case of the LUBF [8],or no-slip in the case of the DAF there.
3 Method of Solution
The corresponding methods of solution appropriate to the two formulations are de-scribed in detail in [5] and [9], hence only a very brief outline is provided here.Space discretisation is on a rectangular computational domain and involves a collo-cated arrangement of the unknowns,h andp, when using the LUBF and a staggered
4 Sergii Veremieiev, Philip H. Gaskell, Yeaw Chu Lee and Harvey M. Thompson
arrangement of unknownsh, u andv in the case of the DAF. The convection termsin equations (4) and (5) are discretized using a second-order accurate total variationdiminishing (TVD) scheme, see [2].
For both approaches the associated time discretisation includes the use of an ex-plicit and second-order accurate in time predictor and a semi-implicit β -method [2]solution stage. The automatic adaptive time-stepping procedure employs an estimateof the local truncation error (LTE) to minimise computational waste. The discretisedequations are solved using a multigrid strategy with a combined Full ApproximationStorage (FAS) and full multigridding (FMG). For the LUBF therelaxation method-ology employs a red-black smoothing Gauss-Seidel scheme, while for DAF, dueto the staggered nature of the discretization involved, therelaxation methodologyadopted employs a lexicographic box smoothing Gauss-Seidel scheme.
4 Results
Problems involving thin film flow over two-dimensional (localised) topographicalfeatures are explored and results obtained with both approaches compared againsteach other for the caseθ = π
6 , θS = π2 , ε = 0.1, Ca= 0.000167 and Re = 15, 30,
50. Following related work [5], a submerged topography is specified via arctangentfunctions with steepnessδ = 0.001. The two-dimensional flow domain is definedas having lengthlp = 80 and widthwp = 80; the topography has lengthlt = 1 andwidth wt = 1 and is centred at(xt ,yt) = (30,40). The submerged topography hasheights0 = 0.5 with the coordinate system(x∗,y∗) = (x− xt,y− yt) defined with itsorigin at the centre of the topography.
−20 −10 0 10 20 30 40
−30−20
−100
1020
300.95
1
1.05
1.1
1.15
x*y*
f
(a)
LUBF
−20 −10 0 10 20 30 40
−30−20
−100
1020
300.95
1
1.05
1.1
1.15
x*y*
f
(b)
DAF Re = 50
Fig. 2 Comparison of (a) LUBF and (b) DAF (Re = 50) predictions for flow over a localised squaresubmerged peak topography. Three-dimensional free-surface plots.
Figures 2 and 3 compare LUBF with DAF predictions for flow overa three-dimensional localised square peak topography. Figure 2 shows the resulting three-dimensional free-surface disturbance generated, while Figure 3 shows the corre-
Predicting Three-Dimensional Inertial Thin Film Flow overMicro-Scale Topography 5
sponding streamwise (top) and spanwise (bottom) free-surface profiles through thecentre of the peak topography.
−15 −10 −5 0 5 10 150.95
1
1.05
1.1
1.15
x*
f
(a)
DAF Re = 50DAF Re = 30DAF Re = 15LUBF
DAF Re = 30DAF Re = 50
LUBFDAF Re = 15
−15 −10 −5 0 5 10 150.95
1
1.05
1.1
1.15
y*
f
(b)
LUBFDAF Re = 15
DAF Re = 50DAF Re = 30
Fig. 3 LUBF vs. DAF (Re = 15, 30 and 50) predictions for flow over a localised square submergedpeak topography: (a) streamwise and (b) spanwise free-surface profiles.
Increasing Re from 15 to 50 leads to an enhancement and widening of the free-surface disturbance, where the characteristic capillary ridge and downstream surgereported for flow over a trench topography [9] have been replaced by two small free-surface depressions. For Re = 50 the DAF predicts a widening and amplification ofthe characteristic capillary depressions, which is not captured by the LUBF. Themaximum and minimum free surface disturbance away from the asymptotic filmthickness predicted by the LUBF and the DAF (Re = 50) arefmax= 1.05, fmin=0.99and fmax= 1.09, fmin=0.98, respectively.
Finally, Figure 4 considers the effect of replacing the submerged square peaktopography by a corresponding square occlusion that is muchtaller than the charac-teristic film thickness and therefore penetrates through the free surface.
−15 −10 −5 0 5 10 150.9
0.95
1
1.05
1.1
1.15
1.2
1.25
x*
h
(a)
DAF Re = 50DAF Re = 30DAF Re = 15LUBF
DAF Re = 30DAF Re = 50
LUBFDAF Re = 15
−15 −10 −5 0 5 10 150.9
0.95
1
1.05
1.1
1.15
1.2
1.25
y*
h
(b)
LUBFDAF Re = 15
DAF Re = 50DAF Re = 30
Fig. 4 LUBF vs. DAF (Re = 15, 30 and 50) predictions for flow around a localised square occlu-sion: (a) streamwise and (b) spanwise free-surface profiles.
The DAF predicts a doubling of the free-surface disturbanceupstream of theocclusion, when Re= 50, with the result that the local film thickness is increased
6 Sergii Veremieiev, Philip H. Gaskell, Yeaw Chu Lee and Harvey M. Thompson
from hmax = 1.08 as predicted by the LUBF tohmax = 1.20. Inertia also has an impacton the degree of film thinning downstream of the occlusion: comparing the LUBFto the DAF (Re= 50) results, reveals the degree of film thinning to differ from avalue of approximatelyhmin = 0.98 tohmin = 0.93. As noted in [1], knowledge ofsuch localised film thickness variation is very important inrelation, for example, tothin film cooling applications since they can have a major impact on achievable heattransfer rates.
5 Conclusions
It has been shown, using two reduced equation sets of the Navier-Stokes equations,that the flow of thin viscous films on surfaces containing topography can be mod-elled satisfactorily and used to predict the associated free surface disturbance. Forsmall Re values the results obtained using both approaches are practically indistin-guishable; however, for high Reynolds numbers the DAF better captures the ampli-fication and enhancement of the free-surface disturbances that result from flow overand around surface topography. The ability to predict and better understand suchfeatures is beneficial in coating applications where the goal is often that of ensur-ing predictable product properties by accurate control of film thickness variationswithin coated films.
Acknowledgements S. Veremieiev gratefully acknowledges the financial support of the EuropeanUnion Marie Curie Action, contract MEST-CT-2005-020599.
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