The venerable "no slip" boundary condition of fluid mechanics is violated on a regular basis. Examples include the flow of polymeric liquids and, in the example discussed here, when rarefied gases flow through nanopores. Here, fluid "slips" at a wall, having a velocity vs at the wall.
Introduction
Continuum hydrodynamics accurately describes the flow of fluids over a wide range of systems. However, it is now accepted that continuum models are unable to adequately describe the flow of fluid under extreme confinement. In particular, the no-slip boundary condition invoked in continuum flow calculations is violated for low density gases flowing in nanoporous materials. So what is the boundary condition? This is the question we have addressed in this research.
Slip in Nanopores
The diffusion of gases at low density is an old problem. Maxwell was the
first to study this problem in 1860. His work led to the definition of wall
slip, as characterized by a slip length, the proportionality constant relating
the velocity at the wall to its gradient 
This model, details of which are described in our publications listed below, relates slip length to the tangential momentum accomodation coefficient, f, through the following relationship
where lambda is the mean free path of the gas. The problem is, what is f?
The tangential momentum accomodation coefficient, f, is defined as the fraction of molecules that are diffusely reflected by the surface. When f=1 there is no slip, and f=0 means total slip. Typically, f is assumed to be 1, or else it is determined experimentally by assuming the Maxwell model is valid and fitting experimental results to the model. This approach has several shortcomings. First, it treats f as an adjustable parameter, and gives no way of predicting how f should vary with the type of gas or nature of the wall. It is also not clear if the main assumption of the Maxwell model, namely that reflections are either completely diffuse of completely specular, is justified.
Model
We tested the Maxwell model and sought to relate f to the nature of the diffusing molecules and wall through use of molecular simulations. We used two approaches. In the first, we performed direct simulation of pressure driven flow through a pore. A schematic of the way in which this was done is shown in Figure 1.
Fig 1: Pressure driven flow through an atomically rough slit pore was simulated by imposing an external force on gas molecules in one direction and recording the flow profile. Here, all atoms interact through a Lennard-Jones potential. The lattice is assumed to be rigid. Similar simulations with a vibrating lattice yielded results that were indistinguishable from the rigid lattice model.
Flow profiles calculated from the simulations are shown on the right as a function of gas density. Slip is clearly seen for low density gases near the edges of the cell (10 nm in width). As density increases, the degree of slip decreases.
For very low density gases (in the high Knudsen number regime), intermolecular interactions among gas molecules become unimportant. For this case, the entire flow beavior can be quantified by knowledge of the interactions with the pore walls. Thus a simpler simulation scheme can be used to examine this system. A schematic of this simulation method is shown below.
Fig 2: Computed tangential momentum accomodation coefficient (f) as a function of reduced interaction energy and surface roughness for a Lennard-Jones molecule interacting with a Lennard-Jones fcc lattice. Recall that f=1 corresponds to the no-slip boundary condition, while f=0 is perfect slip. As expected, increasing surface roughness and energy interactions lead to greater momentum accomodation than for smooth, non-interacting walls.
These results are consistent with a recent simulation study of light gases in carbon nanotubes ("Rapid transport of gases in carbon nanotubes", Skoulidas AI, Ackerman DM, Johnson JK, Sholl DS, Phys. Rev. Lett.89 (18): art. no. 185901, 2002). In this study, molecules adsorbed within smooth-wall carbon nanotubes diffused several orders of magnitude faster than when they were adsorbed in comparably sized but rough zeolite pores.
If the fraction of wall collisions that are diffuse is given by f, it is straightforward to show that the self-diffusivity is given by the so-called Smoluchowski model
where Dk is the classic Knudsen diffusivity. This result is turns out NOT to agree with molecular dynamics simulations. The reason stems from the fact that wall collisions are neither purely diffuse nor purely specular, as assumed in the theory. Our contribution was to show that by relaxing this assumption, one can get an analytic expression that does agree with MD.
We assume that each collision has a specular and diffuse component, such that the reflected velocity of a collision is given by the following expression
Figure 4: Definition of incident and refelcted velocities.
where vinc is the incident velocity and vG is consistent with a Gaussian velocity distribution. Since the incident and reflected velocities must be consistent with a Maxwell-Boltzmann distribution, k is given by
which then yields the value of the self-diffusivity
Epsilon is a cutoff distance that can be taken as the length of the pore. We refer to this solution as an extended Smoluchowski model, which can be compared to the tradional Smoluchowski model and MD simulations. The figure below shows the results of a series of MD simulations for pores of varying width and relative roughness. The self-diffusivity is normalized by the Knudsen diffusivity, which assumes f=1. Values of the accommodation coefficient f were determined from scattering simulations as a function of reduced sorbate diameter (see inset). It can be seen that the traditional Smoluchowski result (dashed line) significantly overpredicts the diffusivity, while the new extended Smoluchowski model (solid line) agrees perfectly with the simulations.
Figure 5: Results of a range of MD simulations (symbols) plotted in terms of the self-diffusivity reduced by the Knudsen diffusivity. The dashed line is the Smoluchowski model, which significantly overpredicts the diffusivity, while the solid line (our model) agrees perfectly with the simulations.
The implications of this result is that smooth walls (small f) will lead to very high self-diffusivities. This result has been observed in the simulations of Sholl and co-workers who computed extraordinarily high self-diffusivities for sorbates in smooth-wall carbon nanotubes.
Support for this work was provided by the Notre Dame Center for Applied Mathematics, and the Petroleum Research Fund (administered by the American Chemcial Society).Publications
G. Arya H. -C. Chang and E. J. Maginn, "Molecular Simulation of Knudsen Wall Slip: Effect of Wall Morphology", Molecular Simulation, 29, 687-709 (2003).
G. Arya, H. -C. Chang and E. J. Maginn,"Knudsen Diffusivity of a Hard Sphere in a Rough Sli Pore", Physical Review Letters, 91, 026102 (2003).
Funding for this work has been provided by the Department of Energy and the donors to the Petroleum Research Fund, administered by the American Chemical Society.
