^{1,a)}, Boris Khusid

^{2}, Joel Koplik

^{3}and Andreas Acrivos

^{4}

### Abstract

Molecular dynamics simulations are used to study the behavior of closely fitting spherical and ellipsoidal particles moving through a fluid-filled cylinder at nanometer scales. The particle, the cylinder wall, and the fluid solvent are all treated as atomic systems, and special attention is given to the effects of varying the wetting properties of the fluid. Although the modification of the solid-fluid interaction leads to significant changes in the microstructure of the fluid, its transport properties are found to be the same as in bulk. Independently of the shape and the relative size of the particle, we find two distinct regimes as a function of the degree of wetting, with a sharp transition between them. In the case of a highly wetting suspending fluid, the particle moves through the cylinder with an average axial velocity in agreement with that obtained from the solution of the continuum Stokes equations. In contrast, in the case of less-wetting fluids, only the early time motion of the particle is consistent with continuum dynamics. At later times, the particle is eventually adsorbed onto the wall and subsequently executes an intermittent stick-slip motion. We show that van der Waals forces are the dominant contribution to the particle adsorption phenomenon and that depletion forces are weak enough to allow, in the highly wetting situation, an initially adsorbed particle to spontaneously desorb.

The authors would like to thank H. Brenner and P. M. Adler for comments on their work on the continuum description of the motion of spheres in cylindrical tubes, J. J. L. Higdon for calling their attention to his numerical work on the resistivity of spherical particles in such systems, and V. A. Kirsch for discussing with them relevant work on the van der Waals interactions present in the system. G.D. thanks J. Halverson for carefully reading the manuscript and for helpful comments. This work was supported by the Engineering Research Program, Office of Basic Energy Sciences, U.S. Department of Energy under Grant No. DE-FG02-03ER46068.

I. INTRODUCTION

II. MOLECULAR DYNAMICS SIMULATIONS

III. FLOW OF A SINGLE FLUID THROUGH A CYLINDRICAL NANOCHANNEL

IV. MOTION OF A SPHERE THROUGH A CYLINDRICAL NANOCHANNEL: SHORT TIMES

A. Comparison with continuum hydrodynamics

V. MOTION OF A SPHERE THROUGH A CYLINDRICAL NANOCHANNEL: LONG TIMES

A. Adsorption phenomena in poorly wetting fluids

B. Particle-wall interactions: van der Waals forces

C. Particle-wall interactions: Depletion forces

D. Dynamics after adsorption: Stick-slip motion

VI. MOTION OF A PROLATE SPHEROID THROUGH A CYLINDRICAL CHANNEL

VII. CRITERION FOR THE ADSORPTION TRANSITION

VIII. SUMMARY AND CONCLUSIONS

### Key Topics

- Wetting
- 52.0
- Adsorption
- 38.0
- Viscosity
- 20.0
- Molecular dynamics
- 13.0
- Hydrodynamics
- 9.0

## Figures

Schematic and explicit views of a colloidal spherical particle moving in a nanochannel.

Schematic and explicit views of a colloidal spherical particle moving in a nanochannel.

Pure fluid profiles for channel radius ; open and solid circles correspond to and , respectively. The data points are obtained as histograms by dividing the interior of the tube into cylindrical shells of thickness . (a) Density vs ; the curves are drawn through the data points to guide the eye. (b) Axial velocity vs ; the curves are best fits to a parabolic profile.

Pure fluid profiles for channel radius ; open and solid circles correspond to and , respectively. The data points are obtained as histograms by dividing the interior of the tube into cylindrical shells of thickness . (a) Density vs ; the curves are drawn through the data points to guide the eye. (b) Axial velocity vs ; the curves are best fits to a parabolic profile.

Shear viscosity for the Lennard-Jones fluid as a function of the centerline density . The solid circles, triangles, and squares are the simulation results listed in Table I for , 10.26, and 20.52, respectively. The open circles are published numerical results for the shear viscosity of LJ bulk systems (Ref. 54), and the solid line is an empirical fit to the experimental shear-viscosity data for argon (Ref. 55). , where is the dilute-gas limit value of the viscosity at (Ref. 56).

Shear viscosity for the Lennard-Jones fluid as a function of the centerline density . The solid circles, triangles, and squares are the simulation results listed in Table I for , 10.26, and 20.52, respectively. The open circles are published numerical results for the shear viscosity of LJ bulk systems (Ref. 54), and the solid line is an empirical fit to the experimental shear-viscosity data for argon (Ref. 55). , where is the dilute-gas limit value of the viscosity at (Ref. 56).

Slip length for the Poiseuille flow in a narrow tube as a function of the wetting parameter . The solid (open) squares, circles, and triangles correspond to simulation results for the slip length in tubes of radius , 10.26, and 20.52, respectively.

Slip length for the Poiseuille flow in a narrow tube as a function of the wetting parameter . The solid (open) squares, circles, and triangles correspond to simulation results for the slip length in tubes of radius , 10.26, and 20.52, respectively.

Radial position of the spherical particle, , as a function of time for tube radii (circles), (triangles), and (squares). Time corresponds to the end of the force ramp. In all cases there is perfect wetting, .

Radial position of the spherical particle, , as a function of time for tube radii (circles), (triangles), and (squares). Time corresponds to the end of the force ramp. In all cases there is perfect wetting, .

Particle average axial velocity, normalized by the Stokes velocity (; ), as a function of the relative size of the tube radius: (closed circles), (open triangles), and (closed squares). The solid curve is the theoretical prediction of Ref. 42, as discussed in the text.

Particle average axial velocity, normalized by the Stokes velocity (; ), as a function of the relative size of the tube radius: (closed circles), (open triangles), and (closed squares). The solid curve is the theoretical prediction of Ref. 42, as discussed in the text.

Particle average axial velocity as a function of the wetting properties of the system. (The tube radius is and the particle radius is .) (a) The velocity is given in MD units. (b) The velocity is normalized by a Stokes velocity that accounts for the variations in the viscosity, , with being the viscosity values listed in Table I.

Particle average axial velocity as a function of the wetting properties of the system. (The tube radius is and the particle radius is .) (a) The velocity is given in MD units. (b) The velocity is normalized by a Stokes velocity that accounts for the variations in the viscosity, , with being the viscosity values listed in Table I.

Radial position of the sphere vs time for four single realizations with : 0.1, 0.4, 0.7, and 1.0. The sudden jump to a radial position for the three lower values of corresponds to adsorption at the tube wall. (The tube radius is and the particle radius is . The dashed line corresponds to a radial position equal to , the nominal maximum radius.)

Radial position of the sphere vs time for four single realizations with : 0.1, 0.4, 0.7, and 1.0. The sudden jump to a radial position for the three lower values of corresponds to adsorption at the tube wall. (The tube radius is and the particle radius is . The dashed line corresponds to a radial position equal to , the nominal maximum radius.)

Cross-section snapshots of the particles at the end of the simulations in Fig. 8. For visual clarity, we use circles of different sizes to represent the different atoms, with radii in the ratio 1.0 to 0.8 to 0.4 for the particle, wall, and fluid atoms, respectively. (The tube radius is and the particle radius is .)

Cross-section snapshots of the particles at the end of the simulations in Fig. 8. For visual clarity, we use circles of different sizes to represent the different atoms, with radii in the ratio 1.0 to 0.8 to 0.4 for the particle, wall, and fluid atoms, respectively. (The tube radius is and the particle radius is .)

Radial position of the sphere as a function of time for six single realizations with , 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. The time axes are shifted so as to be able to observe the jump into contact in all cases. (The tube radius is and the particle radius is .)

Radial position of the sphere as a function of time for six single realizations with , 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7. The time axes are shifted so as to be able to observe the jump into contact in all cases. (The tube radius is and the particle radius is .)

Radial position of the sphere as a function of time for two independent realizations of the MD simulations. In both cases the wetting parameter is set, initially, equal to , leading to the adsorption of the particles. At the wetting parameter is changed to , and a spontaneous desorption of the spheres can be observed. After changing, at , the wetting parameter back to the spheres are readsorbed. Finally, at , changing the wetting parameter to leads, once again, to the immediate desorption of the spheres. (The tube radius is and the particle radius is .)

Radial position of the sphere as a function of time for two independent realizations of the MD simulations. In both cases the wetting parameter is set, initially, equal to , leading to the adsorption of the particles. At the wetting parameter is changed to , and a spontaneous desorption of the spheres can be observed. After changing, at , the wetting parameter back to the spheres are readsorbed. Finally, at , changing the wetting parameter to leads, once again, to the immediate desorption of the spheres. (The tube radius is and the particle radius is .)

(a) Position of the particle along the tube as a function of time. (b) Radial position of the sphere as a function of time. The results correspond to a disordered sphere moving through a disordered tube for . (The tube radius is and the particle radius is .)

(a) Position of the particle along the tube as a function of time. (b) Radial position of the sphere as a function of time. The results correspond to a disordered sphere moving through a disordered tube for . (The tube radius is and the particle radius is .)

Angle between the spheroid major axis and the tube axis as a function of time for four realizations with , 0.4, 0.7, and 1.0 (for each value of one independent MD simulation is presented). The abrupt locking of the polar angle observed at , 0.4, and 0.7 corresponds to the adsorption of the spheroid on the tube wall. The tube radius is . The spheroids have minor axis and aspect ratio , and both the spheroids and the tube are ordered.

Angle between the spheroid major axis and the tube axis as a function of time for four realizations with , 0.4, 0.7, and 1.0 (for each value of one independent MD simulation is presented). The abrupt locking of the polar angle observed at , 0.4, and 0.7 corresponds to the adsorption of the spheroid on the tube wall. The tube radius is . The spheroids have minor axis and aspect ratio , and both the spheroids and the tube are ordered.

Probability distribution function for the orientation angles of a spheroid after adsorption, averaged over 50 or more realizations with wetting properties ranging from to 0.7. The left (right) hand plots are the probability distribution functions for the polar angle (azimuthal angle ), and the top (bottom) plots correspond to an ordered (disordered) spheroid and wall. The tube radius is . The spheroids have minor axis and aspect ratio .

Probability distribution function for the orientation angles of a spheroid after adsorption, averaged over 50 or more realizations with wetting properties ranging from to 0.7. The left (right) hand plots are the probability distribution functions for the polar angle (azimuthal angle ), and the top (bottom) plots correspond to an ordered (disordered) spheroid and wall. The tube radius is . The spheroids have minor axis and aspect ratio .

Three cases of spheroid adsorption in three distinct realizations for , with side views on the left and cross-sectional views on the right. The tube radius is . The spheroids have minor axis and aspect ratio .

Three cases of spheroid adsorption in three distinct realizations for , with side views on the left and cross-sectional views on the right. The tube radius is . The spheroids have minor axis and aspect ratio .

(a) Position of the spheroid along the tube as a function of time, and (b) polar angle of the spheroid as a function of time, for a disordered spheroid moving through a disordered tube, for . The tube radius is . The spheroids have minor axis and aspect ratio .

(a) Position of the spheroid along the tube as a function of time, and (b) polar angle of the spheroid as a function of time, for a disordered spheroid moving through a disordered tube, for . The tube radius is . The spheroids have minor axis and aspect ratio .

Long-time average particle velocity along the tube. The velocity is normalized by the Stokes velocity that a sphere having the same radius would have had in an unbounded fluid, for the same applied force and no-slip boundary conditions. The time span of the simulations, , was always longer than the diffusive time required for the particle to reach the wall, . The average velocity of particles that were adsorbed during the simulation is computed for times after adsorption had occurred, whereas, for particles that are not adsorbed the velocity is averaged over the entire simulation. The dispersion in the velocities of adsorbed particles is a consequence of the stick-slip motion after adsorption that has already been described.

Long-time average particle velocity along the tube. The velocity is normalized by the Stokes velocity that a sphere having the same radius would have had in an unbounded fluid, for the same applied force and no-slip boundary conditions. The time span of the simulations, , was always longer than the diffusive time required for the particle to reach the wall, . The average velocity of particles that were adsorbed during the simulation is computed for times after adsorption had occurred, whereas, for particles that are not adsorbed the velocity is averaged over the entire simulation. The dispersion in the velocities of adsorbed particles is a consequence of the stick-slip motion after adsorption that has already been described.

## Tables

Numerical results for pure fluid flow. Average centerline density and Poiseuille fitting results: viscosity and slip lengths.

Numerical results for pure fluid flow. Average centerline density and Poiseuille fitting results: viscosity and slip lengths.

Adsorption times in MD simulations. Note that an estimate of the diffusive time required to reach the tube wall by a single sphere yields . : Particle type, ◻ ordered; ⧄ disordered. : Tube wall type, ◻ ordered; ⧄ disordered. : Time at which the particle is adsorbed in units of ; − means that there is no adsorption; { } list of adsorption times corresponding to multiple realizations with identical parameters and total duration. : Duration of the simulation in units of .

Adsorption times in MD simulations. Note that an estimate of the diffusive time required to reach the tube wall by a single sphere yields . : Particle type, ◻ ordered; ⧄ disordered. : Tube wall type, ◻ ordered; ⧄ disordered. : Time at which the particle is adsorbed in units of ; − means that there is no adsorption; { } list of adsorption times corresponding to multiple realizations with identical parameters and total duration. : Duration of the simulation in units of .

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