^{1}, Adrian Neild

^{1,a)}, Tuck Wah Ng

^{1}and Murat S. Muradoglu

^{1}

### Abstract

We present here a method for sorting nanometer scale Brownian rods by using a switching asymmetric periodic potential. A two stage sorting process is used to isolate particles with specific dimensions, with acceptable sorting times as well as realizable potential barrier lengths. The method was tested using computer simulations. The ability to sort the nanometer scale anisotropic particles, such as goldnanorods, portends important applications in large scale data recording, photothermal surgery, and bioimaging.

I. INTRODUCTION

II. SORTING PRINCIPLE

A. Stage one

B. Stage two

III. RESULTS AND DISCUSSIONS

IV. CONCLUSIONS

### Key Topics

- Diffusion
- 29.0
- Anisotropy
- 6.0
- Hydrodynamics
- 6.0
- Brownian motion
- 5.0
- Photothermal effects
- 5.0

## Figures

(a) Directions of diffusion coefficients superimposed on a typical rod under consideration, with the static laboratory frame in the top left. (b) PDFs for small and large spheres at a given time *t*. The greatest difference in probability between the small and big spheres occurs at the intersection of the PDFs (marked as b). (c) Diagrammatic representation of the position and alignment of rods near the end of the on period from both top and side views.

(a) Directions of diffusion coefficients superimposed on a typical rod under consideration, with the static laboratory frame in the top left. (b) PDFs for small and large spheres at a given time *t*. The greatest difference in probability between the small and big spheres occurs at the intersection of the PDFs (marked as b). (c) Diagrammatic representation of the position and alignment of rods near the end of the on period from both top and side views.

Plot of *D* _{ yy } (upper curves) and *D* _{ xx } (lower curves) vs time for two different rods with dimensions 595.2 × 168 × 168 nm^{ 3 } (black, dashed) and 864 × 72 × 72 nm^{ 3 } (red, solid). Horizontal lines represent and vertical lines represent τ_{θ}.

Plot of *D* _{ yy } (upper curves) and *D* _{ xx } (lower curves) vs time for two different rods with dimensions 595.2 × 168 × 168 nm^{ 3 } (black, dashed) and 864 × 72 × 72 nm^{ 3 } (red, solid). Horizontal lines represent and vertical lines represent τ_{θ}.

Contour plots of diffusion coefficient in the laboratory *x* direction over a range of rod sizes taken at time (a) 30 and (b) 400 μs. The percentage difference in length (*L*) and diameter (*D*) are taken from a 960 × 120 × 120 nm^{ 3 } base rod. The circles on the plot indicate the rod sizes used in simulations, with the blue colored circles indicating the rods that were taken to the second stage of sorting. The red and black circles are rod sizes on adjacent contours at 30 μs, and are overlaid in (b) to emphasize the change in diffusion coefficient over time. The largest diffusion coefficients correspond to the smallest particle sizes in the lower left corner of the plot (red contours) and decrease toward the top right (blue contours). The colors of dots used correspond to the curves in Fig. 5.

Contour plots of diffusion coefficient in the laboratory *x* direction over a range of rod sizes taken at time (a) 30 and (b) 400 μs. The percentage difference in length (*L*) and diameter (*D*) are taken from a 960 × 120 × 120 nm^{ 3 } base rod. The circles on the plot indicate the rod sizes used in simulations, with the blue colored circles indicating the rods that were taken to the second stage of sorting. The red and black circles are rod sizes on adjacent contours at 30 μs, and are overlaid in (b) to emphasize the change in diffusion coefficient over time. The largest diffusion coefficients correspond to the smallest particle sizes in the lower left corner of the plot (red contours) and decrease toward the top right (blue contours). The colors of dots used correspond to the curves in Fig. 5.

Separation time (blue squares) and number of cycles for separation (green squares) vs off time for three different rod sizes taken to stage 2 of sorting. Rod sizes used were 595.2 × 168 × 168, 710.4 × 120 × 120, and 864 × 72 × 72 nm^{ 3 }. Separation time has been defined as number of cycles for separation multiplied by the off time, and is shown as it is a more meaningful performance characteristic than number of cycles required to separate rod populations.

Separation time (blue squares) and number of cycles for separation (green squares) vs off time for three different rod sizes taken to stage 2 of sorting. Rod sizes used were 595.2 × 168 × 168, 710.4 × 120 × 120, and 864 × 72 × 72 nm^{ 3 }. Separation time has been defined as number of cycles for separation multiplied by the off time, and is shown as it is a more meaningful performance characteristic than number of cycles required to separate rod populations.

[(a) and (c)] Plots of 10th and 90th percentiles of expected rod populations vs cycles, and [(b) and (d)] expected number of particles in each potential well at 20 000 cycles using an initial population of 10 000 particles. These were determined for [(a) and (b)] first stage sorting using 30 μs off time with nine different rod dimensions (as shown in Table I) along three contours (overlaid dots in Fig. 3), and [(c) and (d)] second stage sorting using 400 ms off time with the blue population from the first stage (curve colors correspond to circles in Fig. 3).

[(a) and (c)] Plots of 10th and 90th percentiles of expected rod populations vs cycles, and [(b) and (d)] expected number of particles in each potential well at 20 000 cycles using an initial population of 10 000 particles. These were determined for [(a) and (b)] first stage sorting using 30 μs off time with nine different rod dimensions (as shown in Table I) along three contours (overlaid dots in Fig. 3), and [(c) and (d)] second stage sorting using 400 ms off time with the blue population from the first stage (curve colors correspond to circles in Fig. 3).

## Tables

Actual rod dimensions used in simulations, along with percentage differences compared to a 960 × 120 × 120 nm^{ 3 } base rod and diffusion coefficients in the laboratory *x* direction at both early and late off times. Entries are arranged in the same way as in Fig. 3, with the left column corresponding to the red circles, center column to blue circles, and right column to black circles.

Actual rod dimensions used in simulations, along with percentage differences compared to a 960 × 120 × 120 nm^{ 3 } base rod and diffusion coefficients in the laboratory *x* direction at both early and late off times. Entries are arranged in the same way as in Fig. 3, with the left column corresponding to the red circles, center column to blue circles, and right column to black circles.

Scale factors used to adjust the theoretical diffusion coefficients in the three degrees of freedom considered.

Scale factors used to adjust the theoretical diffusion coefficients in the three degrees of freedom considered.

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