Separation of nanoparticles by flow past a patterned substrate

Motivated by the problem of efficiently separating nanoparticles of different character held in solution, we investigate trajectory deflection and particle trapping in flows of nanoparticle suspensions past patterned surfaces. We consider rigid atomistic particles suspended in a viscous liquid solvent and driven by a pressure gradient through a channel, one side of which has a pattern of alternating stripes which attract or repel the particles. We first consider van der Waals forces alone, where the wall interaction is obtained by summing over semi-infinite slabs of material having a Lennard-Jones interaction with or without an attractive term, yielding a force field with nontrivial three-dimensional spatial variation. This wall interaction can either trap particles on the attractive stripes or deflect the trajectories of mobile particles away from the direction of mean flow. Using molecular dynamics simulations we determine the motion of particles of different sizes in this potential, and observe distinct but modest deflections of several degrees from the direction of the imposed fluid flow. The effects of electrostatic interactions are considered by decorating the particles and walls with opposite charges, resulting in significantly more trapping and larger deflection angles. We use Langevin simulations to treat the motion of larger particles in the van der Waals case, and again observe particle trapping and deflection, although the numerical details of the results differ from the molecular dynamics simulations. In the Langevin case we are furthermore able to obtain bounds on the deflection angle from an analysis of the associated Fokker-Planck equation. We conclude that patterned surfaces deflect particle trajectories to a degree depending on their size, and may be used as a vector chromatography separation technique.

Figure 1

Illustration of vector chromatography based on surface gradients.

Figure 2

Geometry of the gradient surface method. Particles flow past a solid surface containing alternating regions which do or do not attract them strongly. In response, the particles move at an orientation angle $\theta$ which differs from the orientation ${\theta }_F$ of the underlying fluid flow.

Figure 3

Geometry of the computation box: fluid flows parallel to the $x$-$z$ plane, confined between a uniform and repulsive top wall and a bottom wall divided equally into attractive and repulsive regions. The box repeats periodically in the $x$ and $z$ directions. The LJ coefficients $C$ and $D$ for wall-particle interactions are as indicated, where the parameter $\lambda$ controls the wall attraction strength.

Figure 4

Typical variation of forces near the bottom wall for $c_1=2.0$ and $c_2=0.0$. The solid, dashed, and dash-dotted curves represent $y=1.2$, $y=1.0$, and $y=0.8$, respectively. The $x$ force vanishes at the center of the stripe and is maximum at the stripe boundaries, while the $y$ force takes its maximal values at the center of the stripe.

Figure 5

Competition between attractive and repulsive forces near the bottom wall. Far from the wall, the $x$ force is small and attractive toward the stripe boundaries, but reverses on closer approach. The $y$ force similarly reverses in sign on approach to the wall.

Figure 6

Complete force field at various distances $\Delta$ from the wall, combining discrete wall atom and continuum wall interactions. (a) $x$ force near the wall, (b) $x$ force away from wall, (c) $y$ force near the wall, and (d) $y$ force away from wall.

Figure 7

Point particle model: Final configurations at 7500$\tau$ for various parameter sets viewed end-on down the channel. The attractive (black) and neutral (light gray) wall atoms form stripes at the top and bottom of each frame, and the particles are the red (dark gray) dots between them. From top to bottom: $\phi$ $=$ 0.01, 0.05, and 0.1; from left to right: $\lambda =2.0$, 4.0, 6.0, and 8.0.

Figure 8

Probability distribution of particle's distance from the wall, averaged over various time intervals for $\phi =0.02$ and $\lambda =1.5$. (a) Before and (b) just after turning on the wetting interactions, (c) in the middle of the run, and (d) at equilibrium. Full (red) and dashed (blue) lines are the distributions above the wetting and nonwetting stripes, respectively.

Figure 9

Time dependence of the partition coefficient (left) and the fraction of stuck particles (right) for $\phi =0.02$ (top) and 0.1 (bottom), for various values of the interaction strength.

Figure 10

Effects of varying the attraction strength. Left: asymptotic partition coefficient at different concentrations. Right: width of the gap between particles and the attracting solid wall.

Figure 11

Three examples of typical particles motion for the case $\lambda =1.5$, $r=2.15$, and $g=0.02$ oriented at 45${}^{\circ }$. The trajectories are the irregular lines and the mean flow direction is the dashed line. From left to right we see trapping, restrained motion, and free motion.

Figure 12

Final configurations for the case $\lambda =1.5$, $r=2.0$, and $g=0.02$ and 45${}^{\circ }$ applied flow. (a) Variation of the average particle-substrate gap with time. (b) Side view of the final configuration. (c) Positions at the end of the simulation in the substrate plane; the dashed line is at 455${}^{\circ }$. (d) Trajectory angle vs mean in-plane velocity.

Figure 13

Variation of deflection angle $\Delta \theta$ with imposed pressure gradient angles $\theta$ at various pressure gradient $g$.

Figure 14

Transport angle $\theta$ vs pressure gradient $g$ for particle of different radii. Left to right: $\lambda =1.1$, 1.5, and 2.0.

Figure 15

Scatter plot of velocities in the $x$-$z$ plane against $y$-position for $\lambda =1.1$ and $r=2.64$. Left: $g=0.01$, right: $g=0.10$.

Figure 16

The effect of continuum half-space interactions on particle motion. Final particle positions, (left) with and (right) without, for a simulation with parameters $r=2.15$, $\lambda =1.5$, and $g=0.02$.

Figure 17

Comparison of the final configurations with different particle-solvent interaction coefficients: (a) $C=1.0$, $D=1.0$, (b) $C=0.0$, $D=1.0$, (c) $C=0.5$, $D=0.5$, and (d) $C=1.5$, $D=1.5$.

Figure 18

Illustration of the different charge distribution schemes. Positive (wall) and negative (particle) charges are shown as plus and minus symbols, respectively, and the wetting stripe is shown as a thick line.

Figure 19

MD results for charged system with particle radius $r=2.15$, $\lambda =1.5$, $g=0.02$, and a charge per particle of $6e$. Left to right: schemes 1 to 4. Top row: final positions in the ($x$-$z$) flow plane; middle row: side ($x$-$y$) view of the final configurations; and bottom row: transport angle $\theta$ vs transport distance $\Delta x$.

Figure 20

Comparison of 2D and 3D Ewald sum: transport angle vs transport distance $\Delta x$ for particles of radius $r=2.15$ with charge $6e$. Open symbol: 3D method, solid symbol: 2D method; scheme 1 (triangle), scheme 2 (square), scheme 3 (diamond), and scheme 4 (circle).

Figure 21

Results of MD simulation with water's dielectric constant accounted for: transport angle $\theta$ vs transport distance ${\Delta }_x$ for particles of radius $r=2.15$ with charge $9e$. Water's relative permittivity ${\epsilon }_r=81$. Triangle: scheme 1, square: scheme 2, diamond: scheme 3, and pentagon: scheme 4. Circle: uncharged system.

Figure 22

Mean transport angle $\theta$ as a function of charge density $q$ in mixture system. Particle number: 12 small particles with $r=2.15\sigma$ and charge per particle $q$ and 12 big particles with radius $r=4.3\sigma$ and charge per particle $4q$.

Figure 23

Left: Side view of final configurations for mixture system at $q=3e$. Right: Transport angle $\theta$ vs transport distance ${\Delta }_x$ for individual particle for mixture system at $q=3e$.

Figure 24

Comparison of solutions of the Fokker-Planck (red solid line) and Langevin solutions (blue dashed line). Transport angle $\theta$ vs Péclet number for (a)$A=10.0$ and $r=2.0$ and (b) $A=3.0$ and $r=2.0$. Transport angle $\theta$ vs particle radius $r$ at (c) $A=10.0$ and $u_x=0.03$ and (d) $A=3.0$ and $u_x=0.03$.

Figure 25

Comparison of MD and Langevin simulations at $r=2.15$ and $\lambda =1.5$. Top row: MD, bottom row: Langevin. Left to right: $g=0.005$, 0.01, 0.02, and 0.05.