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Methane Transport through Distorted Nanochannels: Surface Roughness Beats Tortuosity Mariano Martin Ramirez, Marcos Federico Castez, Veronica Muriel Sanchez, and Emilio Andrés Winograd ACS Appl. Nano Mater., Just Accepted Manuscript • DOI: 10.1021/acsanm.8b02190 • Publication Date (Web): 01 Mar 2019 Downloaded from http://pubs.acs.org on March 5, 2019

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ACS Applied Nano Materials

Methane Transport through Distorted Nanochannels: Surface Roughness Beats Tortuosity. M. Martín Ramírez,†,‡ M. F. Castez,∗,†,¶,§ V. M. Sánchez,k,¶,⊥ and E. A. Winograd† †YPF Tecnología S.A., Av. del Petróleo s/n (1923), Berisso, Buenos Aires, Argentina ‡Fac. de Ingeniería, Universidad Nacional de La Plata (UNLP), 1 y 47 (1900), La Plata, Buenos Aires, Argentina ¶Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Godoy Cruz 2290 (1425), Ciudad de Buenos Aires, Argentina §Departamento de Matemáticas, Fac. de Ciencias Exactas, Universidad Nacional de La Plata (UNLP), 115 y 50 (1900), La Plata, Buenos Aires, Argentina kCentro de Simulación Computacional para Aplicaciones Tecnológicas, CSC-CONICET, Godoy Cruz 2390 (1425), Ciudad de Buenos Aires, Argentina ⊥ECyT, UNSAM, 25 de Mayo y Francia (1650), San Martín, Buenos Aires, Argentina E-mail: [email protected]

Abstract

ated with surface roughness have a deeper impact, dominating the overall properties of the flow.

Fluid transport through carbon nanotubes have shown remarkable flow properties, with measured flow rates orders of magnitude larger than the expected from standard continuum flow theories. Related studies have indicated that the observed high flow rates were driven by the extreme smoothness of the cylindrical nanotubes used in the experiments. In this work we consider several types of nanochannels far from the cylindrical geometries. Using a combination of simulation techniques, such as molecular dynamics and the lattice Boltzmann method, we study the flow behavior under tortuous and rough channels, which are of fundamental relevance either for optimizing carbon nanotubes for nanofiltering applications as well as for characterizing nanoporous organic media. We show that, although both features have a detrimental effect on flow rates, when nanochannels have simultaneously both roughness and tortuosity, shorter length-scales associ-

Keywords Nanofluidics, Carbon Nanotubes, tortuosity, surface roughness, Flow Enhancement, Molecular Dynamics

1

Introduction

In the last decade, fluid transport through carbon nanotubes (CNTs) has gained considerable attention due to their promisingly fluid conductivity properties and its many possible applications in diverse fields that require high flow rates nanofilters. 1–4 Comparing to continuum flow models estimations, the flow rates obtained with CNTs were up to four orders of magnitude higher than those expected by applying classical Hagen-Poiseuille flow equation. Experiments

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mains or doing several realizations on statically probable porous media. In this paper, we examine the impact on fluid flow of incorporating geometrical features such as tortuosity, curvature variations and surface roughness, to the description of the nanotubes. At the best of our knowledge, except for roughness, 7,8,13,25,26 there is no related literature on how flow in nanopores is affected by these properties. Throughout this work, we combine atomistic simulations under the Molecular Dynamics (MD) approach with a continuous computational fluid dynamics implementation in the framework of Lattice Boltzmann (LB) methods. Within this two-technique approach, we are able to simulate methane flow for a same geometry of pore-throats using both continuous and discrete representations. Although in this paper we focus on the flow of methane, we expect that our main results remain valid for most light molecular-weight gases. The remaining of this paper is organized as follows: in section 2.1 we describe the atomistic simulations performed using MD, while the continuous modeling with the LB method is introduced in 2.2. Section 2.3 contains a detailed discussion about the geometries employed in this work to mimic complex poral throats. The results of extensive numerical simulations on the implemented system are introduced and discussed in section 3. Finally, in section 4, we summarize our findings and provide some perspectives.

included several fluids (water, inert gases, hydrocarbons, etc.), which made the found behavior to be quite general. Although the precise mechanisms that trigger this phenomenon remain unclear, it was recognized from the beginning that surface effects were the main drivers. 5–7 As the ratio surface/volume increases at the nanoscale, fluidsurface interactions become dominant and continuum or macroscopic flow models start to fail as they usually oversimplify them. The extreme smoothness of the surface and slip flow together with heterogeneous density profiles inside the CNTs, were identified as key-factors to account for the enhancement. 6,8 Most experimental, analytical and numerical work have restricted the studies to cylindrical geometries of the CNTs. Distortions to this shape are important for many reasons. From a fundamental point of view, it is of main interest to understand whether the observed enhancements are related to the ideal cylindrical geometry of smooths CNTs or it is a general feature of transport in confined systems. From the application perspective, altering the geometry of the CNTs may be useful to optimize nanofilter devices, either to increase flow rates, filtering capabilities or to improve their design. From a more general point of view, nanoporous media, as well, exhibits tortuous paths which are far from being cylindrical. Many efforts have been dedicated to upscale the results on CNTs to porous materials in larger scales, especially focusing on the organic matrix of shale rocks. 9–16 To describe this scenario, two main strategies were usually proposed: using pore network models 17–19 or considering complex material descriptions that satisfy the chemical composition or other relevant property of the rock. 20–24 On the former case, pores are connected by pore-throats of simplified morphology (slits or cylinders), where the flow equations are known, and the macroscopic result is obtained after solving the equations for the coupled pores. On the latter, simulations are straight-forward performed over the porous medium. The main disadvantage of this approach is the high computational cost of performing such simulations on macroscopic do-

2 2.1

Methods Molecular dynamics

We use non-equilibrium molecular dynamics to model methane transport through carbon nanotubes by means of the open source simulator LAMMPS. 27 The simulation implementation is analogous to that presented in Ref. 8, whose main features are described in what follows. The system box is composed by three main regions: two reservoirs at different pressures, and a nanochannel that connects them and through which methane molecules can flow driven by


2

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P< being the high and low pressure reservoirs, respectively) is set to ∆P = 100 atm. This relatively high value of ∆P speeds up the simulation and allows to get enough statistical data, without affecting the physical results, as we have already verified the linear dependence of the flow with the pressure drop in a wide range of the parameters space. 8 Methane molecules are initially inserted in each reservoir at a density consistent with their pressure and temperature, estimated through the Peng-Robinson equation of state. 33 A simulation run consists of two stages: in the first one, the nanochannel remains closed and methane molecules equilibrate, independently in each reservoir. At this stage, pistons oscillations are softened by including damping forces. Once reservoir’s volumes and pressures reach equilibrium values, the nanochannel is opened and the second stage (the ‘transport stage’) begins. Methane molecules start to flow towards the low-pressure reservoir and mass density inside the nanochannel increases up to reach stationary values. Simultaneously, flow rates also attain a steady-state. Under this regime, we extract most feedback from the simulation, including flow rates and velocity/density profiles inside the nanochannels. Typically, our simulations include around 0.81.2 million of methane molecules, a number that provides enough statistics. The overall simulated time ranges between 4-12 ns, with a timestep of 4 fs.

the pressure gradient. Pressure is controlled by two independent pistons under imposed external forces, while temperature is maintained using a Nosé-Hoover thermostat. 28,29 Nanochannel ends define our x-axis (pistons are normal to the x-axis and methane particles are confined to the region between them) and periodic boundary conditions are imposed along the yand z- axes. A schematic representation of the full simulation box is shown in Figure 1.

Figure 1: Scheme of the system box: Two reservoirs containing methane molecules (gray) are connected by a tortuous nanochannel (yellow). Pressure in each reservoir is controlled by independent pistons (red). Immobile walls are attached at both ends of the nanochannel (blue). Methane molecules are considered as point particles and molecular interactions methanemethane and methane-walls are simulated through Lennard-Jones potentials. We generically call ‘walls’ to the rigid body particles of the system, namely the pistons, the walls in the reservoirs and the nanochannel surface. Walls are also built from point particles. While nanochannel particles are Carbon atoms, particles that constitute reservoir walls and pistons are unspecific, having the only purpose of confining methane molecules. Except the pistons, that move as a rigid body, wall particles are immobile. We use Lennard-Jones parameters according to the OPLS-UA potentials database 30,31 in which ǫCH4 −CH4 = 0.2941kcal/mol, ǫCH4 −C = 0.1225kcal/mol, σCH4 −CH4 = 0.3730nm and σCH4 −C = 0.3617 nm. 32 All LJ cutoffs are set to 2.5σ. Temperature is fixed at 400 K and the mean pressure is maintained at Pm = 450 atm, reasonable values for shale hydrocarbons reservoirs. At this temperature, methane is a super-critical fluid. Pressure drop ∆P = P> − P< (P> and

2.2

Fluid dynamics

To compare the MD flow rates with those expected from the continuum theory, we perform analogous fluid dynamics simulations with the Lattice Boltzmann method. MD and LB simulations are performed independently over systems in the same length scale, to identify and quantify the differences found in nanofluidic transport processes with respect to the continuum description. It is standard in the literature to define enhancement on nanoscale flows by considering the ratio between the flows on these systems and on the no-slip HP. It is then natural to ex-


3


tend this definition to tortuous geometries by comparing molecular dynamics flows with a noslip fluid dynamics solution, such as that provided by the LB method. However, it should be noticed that the enhancement factor can be computed by other techniques that combine aspects of the molecular and fluid dynamics (see, for instance, Ref. 34 and references therein). We use the Palabos toolkit, 35 which is a flexible open-source implementation of the LB method. Simulations are carried out using the D3q19 lattice, applying non-slip boundary conditions at the nanotube surfaces and a constant pressure at the ends (Neumann boundary conditions). The geometries are generated in the standard stereolithography file format (STL), which can be easily handled by Palabos. We use the regularized BGK method 36 for the collision step and off-lattice boundary conditions. 37 In a typical simulation, the voxel size is equal or below 1 Å3 , where the relative errors due to discretization are found to be below 3%, resulting in meshes with about 1-40 million of voxels. Density (ρb ) and dynamic viscosity (µ) of methane are extracted from the NIST database 38 at mean pressure Pm = 450 atm and T = 400 K, being ρb = 191.94 kg/m3 and µ = 2.5549 × 10−5 P a · s.

2.3

2.3.1

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Tortouosity

We considered two types of tortuous nanochannels: one with smooth and other with sharp curvature variations. For the former, we work with helical shapes which have a constant curvature (Fig. 2a). For the latter, we implement a generatrix curve using spline functions, which exhibit sharp breaks associated with large curvature variations (Fig. 2b). The details of the implementation are presented in the Supporting Information. In Fig. 2c, we plot the curvature (C) as a function of the arc-length (s) for the geometries of Figs. 2a and 2b. The helical channel has a constant curvature, except at the ends, where a straight path is attached to improve its connection with the reservoirs, avoiding undesired effects like particle filtration outside the domain or obstructions at the inlet and outlet. On the contrary, the spline channel has zero-curvature everywhere, except at the specific points where the breaks take place. While more realistic mechanical models over twisted and bent CNTs exhibit changes in crosssectional shapes (see, for instance, Ref. 40 and references therein), in this paper we assume, for simplicity, that cross-sectional changes and tortuosity can be decoupled, so we only consider tortuous nanochannels with a constant circular cross-section. To ease the nomenclature, we use the following acronyms to refer to both classes of tortuous nanotubes: HNT for helical nanotube and SNT for spline-type nanotube. When the tortuosity of these channels equals 1, they become cylinders and we shall use the acronym CiNT (cylinder nanotube). CiNTs are important for the sake of comparison, since most of related work is limited to such geometries. The scheme depicted so far regarding the definition of the surface nanochannel is focused on purely geometric considerations. Once resolved the geometrical properties of nanochannels we use them to generate nanochannels to be inputted in discrete (MD) and continuous (LB) simulations. For the former, the atomistic nanochannels are generated by randomly populating the surface with carbon atoms up to get the desired areal density. This density

Geometries of nanochannels

The main aim of this paper is understanding how geometric features, such as tortuosity and surface roughness, impacts on the flow at the nanoscale. As there is an infinity of possible geometries for such complex channels, we conveniently restrict the analysis to a few geometries that allows us to perform a systematic study over NTs with well-defined characteristics. We only consider tubular channels, in which the surface is centered around a given curve ~r(t) (that we shall call the generatrix curve) in 3D space, and the cross-sections are restricted to circular shapes. The effects of considering CNTs with different cross-sections are discussed in Ref. 39.


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is set to 38 carbon atoms per squared nanometer (graphene’s areal density), and we also ensure that distances between carbon atoms on the nanochannel surface are always above 0.13 nm. For LB simulations, the geometry is directly exported to the STL file format. 2.3.2

Surface roughness

To account for surface roughness, we implement two types of distortions to the nanochannels, one at the atomic scale and other at the nanoscale. For the former, we shift the actual position of carbon atoms in the channel by xr Wa along the radial coordinate, where xr is a random number uniformly distributed in [−0.5, 0.5] and Wa is the parameter that controls the amplitude of the atomic-scale roughness (Wa = 0 corresponds to an atomically smooth nanochannel). Figure 3 shows a cross-sectional view of this type of nanotube. Throughout this work, Wa ranges between 0 − 0.2 nm. We only consider atomic-scale roughness for atomistic MD simulations.

(a)

(b)

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R

6 4 Wa

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Figure 3: Cross-sectional view of an atomicscale rough tunnel. Roughness is quantified by the parameter Wa .

(c)

Figure 2: Implemented tortuous nanochannels. (a) HNT. (b) SNT. (c) Curvature as a function of the arc length parameter for generatrix curves associated to nanochannels in (a) and (b): The continuous line represents the curvature along the HNT, showing a constant curvature except at the ends of the tunnel. The dashed line represents the curvature of the SNT, showing several peaks corresponding to the occurrence of the breaks.

Nanoscale roughness is implemented for both, MD and LB simulations, by introducing a sinusoidal perturbation on the tunnel radius R to the CiNT: ˜ R(x) = R + Wc sin(2πx/λc ) ,

(1)

with x in the range 0 ≤ x ≤ Lx , and where Wc and λc are the amplitude and wavelength of the perturbation, respectively, while Lx is the rectilinear extent of the channel. This model allows us to continuously interpolate the results


5


expected that Jc ≃ 1. As the characteristic system size approaches the atomic dimensions, the continuous theory is no longer applicable and Jc deviates from 1. In this sense, we can use deviations of Jc from unity as a measure of the strength of nanoscale effects.

between the smooth (λc → ∞) and the atomicscale rough (λc ≃Å) nanotubes. Figs. 4(a) and 4(b) represent a single tunnel with nanoscale roughness in its atomistic (MD) and triangulated (LB) versions, respectively.

3.1 3.1.1 (a)

Effect of tortuosity on fluid flow Roughless nanochannels

To identify the influence of the different characteristics of the nanotube geometry, we begin by considering roughless tortuous nanotubes with fixed curvature (HNT), as a natural extension to CiNT. Tortuous channels whose ends are at a distance Lx and with tortuosity τ , have a total length of Ls = τ Lx . It is then expected that jp decreases as tortuosity increases, as the channel has a longer effective length Ls . In Fig. 5 we plot the enhancement Jc obtained for HNTs with two different number of turns (Nt ) of the channel (curvature varies with Nt as shown in the Supporting Information), together with the enhancement obtained for the case of CiNTs as a function of the total length Ls . If the only effect of tortuosity was the associated to the increasing of the effective length Ls , the flow rates for CiNTs and HNTs with a same value of Ls will be the same. However, numerical results show that this is not the case, which implies that tortuosity impacts in a deeper way. Moreover, the curvature of HNTs also affects flow rates: in fact, for a same tortuosity (or, equivalently, Ls in Fig. 5), data associated to HNTs with high curvature are systematically below the ones corresponding to lower curvature values. It is remarkable that both, HNTs and CiNTs data show an increasing enhancement with the nanochannel length. This effect was largely discussed in Ref. 8 and has been attributed to the extreme smoothness of nanochannels, together with the relatively short lengths Ls amenable to MD simulations. The combined action of these effects prevents the emergence of the L−1 s law proper of the classical solutions (as Hagen-

(b)

Figure 4: View of the NTs with nanoscale roughness, whose cross-section radii vary according to Eq. 1. Specific parameters are: Lx = 20 nm, R = 2 nm, λc = 2.5 nm and Wc = 0.2 nm. (a): Geometry for MD simulations. (b): Triangulated version of the surface for LB simulations.

3

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Results & Discussion

In this section we introduce the main results obtained after performing extensive numerical simulations of the transport process across tortuous and rough nanochannels. The main measurable quantity is the particle flow rate QN . Exploiting the linearity of the flow with the pressure drop, we usually analyze jp = QN /∆P . We compare the MD results with the analogous LB simulations to quantify the differences between the nanoscale flow and that expected from the continuum limit. We find jM D useful defining the ratio Jc = jpLB as a measure p of the enhancement of atomistic against continuous models of flow rates at the nanoscale. In fact, for relatively large system sizes, in which the continuous hydrodynamic limit holds, it is


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with the curvature of HNTs with a dependence nearly linear, although this trend seems to accelerate for larger curvatures.

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Figure 6: (a) Flow rates as a function of tortuosity (τ ) for HNTs and SNTs. (b) jp dependence on the number of breaks (Nb ) on SNTs with τ = 1.8. (c) jp as a function of the curvature for HNTs with fixed τ = 4. Simulations in (a) and (b) have the following parameters: R = 1 nm, Lx = 20 nm and Wa = 0 nm. For (c), parameters are: R = 2 nm, Lx = 20 nm and Wa = 0 nm.

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Poiseuille’s law) and, as a consequence, causes the dependence of the enhancement with Ls . Applications of CNTs may profit from this finding. Similar to CiNTs, HNTs also exhibit high flow rates, but provide the opportunity of having a same effective length Ls in a thinner sample (lower Lx ). We have just shown that, when the curvature C remains constant along the NT as in the HNTs, jp depends on the mean value of C. It is then interesting to address what happens in a different kind of tortuous nanotube, like SNTs, that exhibit large localized curvature variations, associated to sudden changes in direction, as discussed in Sec. 2.3.1.

Jc

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jp (ns atm )

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90

jM D

Figure 5: Enhancement Jc = jpLB as a funcp tion of the total length of the tunnel Ls . The enhancement increases with Ls and decreases with increasing curvature and tortuosity. The parameters of the simulations are: R = 2 nm, Lx = 20 nm (Lx equals Ls in the case of CiNTs), Wa = 0 nm.

3.1.2

Effect of roughness

atomic-scale

surface

In addition to tortuosity, we contemplate the effects of surface roughness on flow rates for NTs with different geometries. In Fig. 7, we plot the flow rates for CiNT, HNT and SNT, at fixed Ls and R, varying their surface roughness with the parameter Wa . For the case of roughless nanochannels (Wa = 0), there is remarkably lower jp for the case of the SNT comparing to the CiNT and HNT cases. As Wa increases, jp monotonically decreases for the three types of considered nanotubes. Interestingly, curves coalesce to nearly the same curve when Wa & 0.1 nm, which suggests that, for high enough atomic-scale roughness, tortuosity contribution on flow rates become a less important lower-order effect.

In Fig. 6 we plot jp for both, HNTs and SNTs nanochannels. As shown in Fig. 6(a), the flow rates monotonically decrease with tortuosity, and data associated to both types of NTs exhibit similar characteristics. However, for all the considered values of τ , we find that jp is remarkably lower for SNTs than for HNTs. These results suggest that, regarding the flow rates, high variations of curvature have stronger impact than its proper absolute value. Fig. 6(b) suggests that jp decreases approximately linearly with the number of breaks Nb of the SNTs. As can be seen in Fig. 6(c), jp also decreases


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2

cases, show a monotonic increasing of the flow rate as λc increases. In the asymptotic limit λc → ∞, flow rates tend to the corresponding smooth NT of radius Rmean . When λc decreases and gets closer to R, surface roughness effects become important, and jp diminishes. Two important differences between both cases are worth noting: In the first place, the absolute values of these asymptotic flow rates are nearly five times larger in the discrete case, which is another manifestation of the characteristic enhancement of flows through smooth NTs. In the second place, while in the continuous case flow rates are bounded between the values corresponding to smooth NTs with radius Rmin and Rmean , in the discrete case, flow rates are well below the Rmin threshold for small values of λc . This evidences that surface roughness impacts in nanoflows much more deeply that what it does in their continuous counterpart. To analyze these differences, in Fig. 9 we plot the density (top panel) and the axial component of the velocity (bottom panel) profiles from the MD simulations for the cases without surface roughness for the limiting cases of R = Rmean and R = Rmin (corresponding to the dashed and dotted lines of Fig. 8, respectively) and for a case with surface roughness (with λc = 2.5 nm). At the nanotube center, densities are close to that of bulk methane (ρb ). As the radial coordinate (r) approaches the nanotube wall, the density profile deviates from ρb , and the formation of an adsorbed layer becomes evident, especially for the smooth nanotube. In the case of the rough NT, the adsorbed layer peak is broadened. Despite the presence of the adsorbed layer is a feature of the nanoscale, absent in the continuum limit, the steep decrease in the flow rates of the rough NT can be mainly attributed to the impact on the velocity profile. This can be observed from the bottom panel of Fig. 9, where the velocity profile for the rough NT lies well below those corresponding to the smooth NTs, even for the case of taking the inner radius R = Rmin . The strong reduction of the axial velocity is not constrained to the vicinities of the NT wall, but also to the NT volume. This explains the remarkable difference between flow rates for the smooths and

CiNT HNT SNT

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Figure 7: Flow rates as a function of the atomic-scale roughness parameter Wa for CiNTs, HNTs and SNTs with the same channel radius R = 1 nm and total length Ls . HNTs and SNTs have τ = 1.93. The distance between nanochannels ends is Lx = 20 nm for HNT and SNT cases, while it is Lx = 38.56 nm for the CiNT.

3.2

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Effect of surface roughness length-scale on fluid flow

Considering tortuosity and surface roughness as distortions on the cylindrical shape of the NTs, at the light of the results of Sec. 3.1.2, we can conjecture that distortions on a shorter lengthscale, associated with surface roughness, have a predominant impact on the flow than distortions on longer length-scales, associated with tortuosity. To systematically study this effect, we consider the NTs of Fig. 4 (see Eq. 1), which allow to continuously interpolate between roughless CiNT to atomically-rough NTs, by changing a single parameter (the wavelength λc ). For this case, we consider Wa = 0, and the amplitude of the perturbation to the CiNT is set to Wc = 0.2 nm. In Fig. 8 we plot jp as a function of λc for atomistic MD (top panel) and for continuous LB (bottom panel) simulations. Dashed lines give the corresponding flow rates of roughless CiNTs (Wc = 0) with a radius equal to the mean value Rmean = R, while dotted lines correspond to the minimum value of such radius Rmin = R − Wc . Curves for both, atomistic and continuous


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roughs NTs. 40 Rmean

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Figure 8: Particle flow rate as a function of roughness length-scale λc . Top (bottom) panel corresponds to the atomistic (continuous) system. Dashed lines represent flow rates for a smooth nanotube (Wc = 0), with a radius Rmean , while dotted lines represent flow rates for a smooth nanotube with radius Rmin = Rmean − Wc . While in the continuum scenario the flow slightly varies with λc and it is constrained to the limits of the dashed lines, surface roughness in the molecular case has stronger effects, and flow rates decrease up to about 15%, for these simulation conditions. The parameters for this set of simulations are: R = Rmean = 2 nm, Lx = 20 nm and Wc = 0.2 nm.

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40 20 0

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Figure 9: Particle density (top panel) and axial velocity (bottom panel) radial profiles inside the nanotube. Densities in the center of the nanotube are close to methane bulk density for smooth and rough NTs. Contrarily, the axial velocity profile in the rough NT shows a notorious lowering respect to the smooth case, even in the central region of the NT. Data indicated with circles (squares) correspond to a roughless nanotube with R = 2 nm (R = 1.8 nm) and Lx = 20 nm, while data labeled with rhombus correspond to a rough NT with parameters λc = 2.5 nm, Wc = 0.2 nm, R = 2 nm and Lx = 20 nm.

In Fig. 10 we plot the flow rates as a function of the roughness amplitude Wc , normalized by their roughless counterpart, i. e., JW ≡ jp (Wc ) . Evidently, according to this definijp (Wc =0) tion, JW = 1 for smooth NTs. As expected, JW monotonically decreases as Wc grows. The same behavior is observed at different values of λc , although the decreasing is more pronounced for lower values of λc . In addition to the data referred to nanoscale rough NTs, Fig. 10 includes a data set (values labeled with crosses) associated to atomically rough CiNT. Although Wa and Wc indicate amplitudes for roughness of different nature, the normalized rate dependence with both types of roughness amplitudes behaves similarly, which allows us to generalize our findings on the strong dependence of the flow rates on the small length-scales.


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λ c= 20 nm λ c= 10 nm λ c= 5 nm λ c= 0.8 nm λ c= 2.5 nm

specific nanochannels, lies beyond the scope of the present work.

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Summary and conclusions

We have simulated methane flow through organic nanotubes, including tortuosity and surface roughness at different scales, by means of molecular dynamics. Comparison of atomistic flow with continuous fluid dynamics was achieved using the Lattice Boltzmann technique. The combined use of MD and LB represents a promising approach and, at the best of our knowledge, is novel in this kind of studies. At the light of our MD results, we conclude that tortuosity has a noticeable influence on the flow rate further than the mere increase in the tunnel’s length, although flow enhancements over the macroscopic description, similar to those observed in CiNTs, still remain. Flow rates decrease with increasing curvature and with increasing number of sudden changes in the curvature. However, quantifying these effects, sudden changes are more detrimental to transport rates. Flow rates also showed a major decreasing in the presence of surface roughness. In the case of nanoscale roughness, the decreasing in flow rates was notoriously more pronounced when the characteristic length of roughness λc approached the dimensions of the nanochannel. Comparing to its continuous counterpart, the flow in atomistic MD systems exhibited a much deeper distortion in flows with respect to the roughless case. In fact, while the effects in continuous systems were constrained to intuitive bounds of straight cylinders with different radii, in the atomistic case, flows were entirely modified. Roughness effects were not constrained to the vicinities of the nanochannel walls, but even to its center. In particular, axial velocities were strongly reduced along the whole radial range. Finally, although tortuosity and surface roughness have a measurable effect on flow rates in nanochannels, our extensive numerical study show that, in all considered cases, surface roughness appears being the dominant effect, in the sense that the flow rates show an

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p (Wc ) Figure 10: Normalized flow rate JW ≡ jpj(W c =0) as a function of the roughness amplitude (W ). Flow decreases when W increases in all cases. Data indicated with crosses correspond to atomic-scale rough CiNTs, while remaining data correspond to NTs with roughness in the nanoscale (so the amplitude W in the bottom axis must be understood as Wa for the data set labeled with crosses and as Wc for the remaining data). The remaining parameters of the tunnel are R = 2 nm and Lx = 20 nm.

The fact that high flow rates can also be achieved in tortuous CNT opens the question whether the geometry of the nanotubes can be optimized to improve the filtering properties of potential devices. Beyond the experimental challenges of controlling the geometry of the CNTs, further theoretical work is required to assess this statement. As pointed out by Mattia et al., 34 it is permeability of CNTs rather than enhancement over the HP solution that is more relevant to actual filtration applications. They showed that, under typical conditions for nanofiltrations, the permeability is proportional to WA−1 , where WA is the work of adhesion, measurable by contact angle experiments. Physically, WA is directly related to the affinity between the fluid and the nanochannel walls which incorporate the effect of surface roughness. Thus, our findings regarding the decreasing in flow rates with roughness are qualitatively consistent with their work, although a more detailed study about the relation between WA and the surface roughness for our


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extreme sensitivity to the roughness of NTs. In the presence of both features, it is surface roughness what more severely impacts on the flow properties. Generalizing this result, it is also reasonable to conjecture that distortions on the cylindrical shape associated with short length-scales may have more impact on transport properties than those associated to long length-scales. Flow properties in nanoporous media, that ordinarily exhibit tortuosity, surface roughness and complex connectivity, which have not been deeply studied yet, are expected to be very different to flow properties for roughless and not tortuous nanochannels. It is therefore important to be extremely cautious when extrapolating results regarding flows in CNT to natural nanoporous media.

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Supporting Information Available

(6) Verweij, H.; Schillo, M.; Li, J. Fast Mass Transport Through Carbon Nanotube Membranes. Small 2007, 3, 1996– 2004.

The Supporting Information is available free of charge on the ACS Publications website.

(7) Joseph, S.; Aluru, N. Why are Carbon Nanotubes Fast Transporters of Water? Nano Lett. 2008, 8, 452–458.

• TortuositySI.pdf: Mathematical details on the implementation of tortuous nanochannels.

(8) Castez, M. F.; Winograd, E. A.; Sánchez, V. M. Methane Flow through Organic-Rich Nanopores: The Key Role of Atomic-Scale Roughness. J. Phys. Chem. C 2017, 121, 28527–28536.

Acknowledgement The project was funded by YPF Tecnología S.A.. M. M. R., V. M. S. and M. F. C. received funding from YPF Tecnología S.A. and CONICET (Argentina National Research Council). E. A. W. received funding from YPF Tecnología S.A.. Part of the computational calculations were performed in TUPAC Cluster from CSC-CONICET, Argentina.

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