Nanofluidic rocking Brownian motors

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Science  30 Mar 2018:
Vol. 359, Issue 6383, pp. 1505-1508
DOI: 10.1126/science.aal3271

Shaking the small from the even smaller

Gently oscillate a can of mixed nuts, and eventually the larger Brazil nuts will rise to the top. Skaug et al. created three-dimensional patterned surfaces to which they applied electric fields. Combined with gentle shaking to induce a rocking Brownian motion, they were able to guide particles smaller than 100 nm along complicated paths. They could also quickly separate particles with different sizes and shapes.

Science, this issue p. 1505


Control and transport of nanoscale objects in fluids is challenging because of the unfavorable scaling of most interaction mechanisms to small length scales. We designed energy landscapes for nanoparticles by accurately shaping the geometry of a nanofluidic slit and exploiting the electrostatic interaction between like-charged particles and walls. Directed transport was performed by combining asymmetric potentials with an oscillating electric field to achieve a rocking Brownian motor. Using gold spheres 60 nanometers in diameter, we investigated the physics of the motor with high spatiotemporal resolution, enabling a parameter-free comparison with theory. We fabricated a sorting device that separates 60- and 100-nanometer particles in opposing directions within seconds. Modeling suggests that the device separates particles with a radial difference of 1 nanometer.

Lab-on-chip devices that can size-selectively transport and collect nanoscale particles are expected to find applications in materials and environmental sciences [e.g., size analysis, filtration, and monodisperse production (1, 2)] as well as in point-of-care diagnostics and biochemistry [e.g., molecular separation and preconcentration (1, 35)]. For example, directed transport may overcome fundamental limits in the detection of dilute species in fluidics (6) by actively transporting and accumulating them at the sensor area. Inspired by molecular motors in biology, Magnasco (7) and Prost et al. (8) proposed that such particle transport could be achieved with artificial Brownian motors (BMs) based on an asymmetric energy landscape and non-equilibrium fluctuations. Previous experiments (914) focused primarily on “flashing ratchet”–type BMs that exploit a periodically generated asymmetric trapping potential and isotropic diffusion to transport micrometer-scale particles. The required potentials were obtained using optical (9, 10) or dielectrophoretic (1113) forces, which scale with the particle volume and therefore are not efficient at the nanoscale. Accordingly, direct charge-charge interactions were required to transport DNA molecules by means of intercalated electrodes (14).

Such flashing BMs were also explored for particle sorting by exploiting the dependence of diffusion on particle size. However, similar to the case of continuous-flow devices (4, 15), it is expected that using external forcing instead of diffusion will result in greater separation precision. Rocking BMs (7, 12, 16) use a zero-mean external force and a static potential landscape to generate directed particle motion. Their transport exhibits a strong nonlinear dependence on particle diffusivity, which is promising for nanoparticle separation (17). However, for nanoscale particles, creating a sufficiently strong static energy landscape remains a challenge.

Electrostatic trapping (18) addresses this challenge by confining like-charged particles between uniformly charged surfaces. A geometrical recess in one of the surfaces lowers the particle-surface interaction energy locally and thus defines a lateral trapping potential. Confinement energies of several multiples of kBT (where kB is the Boltzmann constant and T is absolute temperature) were achieved, resulting in the stable trapping of various types of charged nanoparticles, such as 10- to 100-nm-diameter Au nanospheres (1820), Au nanorods (21), 50-nm-diameter vesicles (18), and even 10- to 60-base DNA oligonucleotides and 10-kDa proteins (22). External forcing by optical and electrical fields was also explored as a means of driving the particles from one stable minimum to another (23).

We expanded the concept of geometry-induced electrostatic trapping to create complex two-dimensional (2D) energy landscapes for nanoparticles by replacing the simple recess geometries with lithographically patterned 3D topographies. The patterns were fabricated using thermal scanning probe lithography (t-SPL) (24), which has a depth accuracy in the nanometer range (25). We implemented nanofluidic rocking BMs for our model particle system of 60-nm Au nanospheres. We created energy landscapes of up to ~10kBT in scale at a lateral resolution of less than 100 nm. Using high spatiotemporal resolution optical microscopy, we determined all relevant physical system parameters in situ, such as the ~10-nm spatially resolved particle interaction potential and the millisecond-resolved particle motion. We find excellent agreement with theory and demonstrate a sorting device.

Gold nanoparticles in electrolyte were confined in a nanofluidic slit of controllable gap distance d (Fig. 1A and fig. S1) (26) using the nanofluidic confinement apparatus described in (27). We used closed-loop t-SPL (25) to pattern the thermally sensitive polymer polyphthalaldehyde with a sawtooth profile along complex-shaped transport tracks having a depth of 30 to 50 nm and a period of L ≈ 550 nm (Fig. 1, B to D). The modulated gap height (Fig. 1A) in the slit results in a static lateral particle energy landscape, which confines the particles to the tracks (Fig. 1E) and provides the static asymmetric potential required for the rocking BM.

Fig. 1 Nanofluidic Brownian motor setup and ratchet topography defined by thermal scanning probe lithography (t-SPL).

(A) Schematic cross section and top view of the nanofluidic slit. Top left: A pillar (height 30 to 50 μm) etched into the cover glass provides close proximity, optical access, and space for electrodes. Top right: The top view (not to scale) depicts the electrodes (yellow, spaced by ~1.2 mm) and the pillar in the center (width ~100 μm). Two voltage signals Ux,y(t), zero mean square–shaped with a phase shift of 90°, are applied in the x and y directions to the electrodes, thus creating a rotating electric field that drives the motor. Bottom: The ratchet topography in polyphthalaldehyde and the 60-nm gold nanoparticle, drawn to scale. The particle experiences a ratchet-shaped energy landscape due to the ion-cloud interactions (orange) between like-charged surfaces. The gap distance is d = 150 ± 1 nm. (B) Topography image of the patterned geometry. (C) Close-up of the small circular ratchet shape of (B). (D) Cross section of the track’s sawtooth geometry measured along the white dashed line in (B). (E) Optical image of particles trapped in the ratchet. The color overlay indicates the average speed measured along the ratchet path in the counterclockwise direction at an applied rotating voltage of 3 V @ 30 Hz.

To elucidate the effect of the topography on the particle’s interaction energy and to demonstrate the operation of the BMs, we resorted to simple linear ratchets (Fig. 2A). We define the direction of the BMs as the direction of the shallow slope profile (Fig. 2B), indicated by the arrows in Fig. 2A. In addition, three control fields, C1, C2, and C3, were written with depths of the maximum, mean, and minimum depth of the ratchets, respectively. The control fields are used for in situ measurement of the mean particle diffusivity and the mean particle speed in the nondriven and driven case, respectively.

Fig. 2 Experimental determination of the average energy landscape.

(A) Topography image of the patterned ratchets R1 to R4 and three control fields, C1 to C3. Arrows indicate the ratchets’ direction. The observed particle positions are overlaid as a heat map. (B) Single-cell averaged cross sections (dashed lines) of the four ratchets including standard deviations (areas, 15 cells per ratchet) along the ratchet direction xr. (C) Average experimental 1D energy profiles V0(xr) of the particles in the ratchet compared to a profile obtained from finite element modeling (FEM; dotted line). The blue line denotes the average interaction energy as estimated from the average occupation probability (inset, 24 profiles) using Boltzmann’s principle. The shaded region marks the standard deviation. (D) V0(xr) for nine gap distances of 127 nm < d < 148 nm (standard deviation of d, ~1 nm). (E) Master curve obtained by linear scaling of the potentials by a factor of α. The inset depicts the exponential dependence of α on d.

We recorded the position of the particles with an effective particle illumination time of ~40 μs and a frame rate of 1000 fps (27) using interferometric scattering detection microscopy (27, 28) (movie S1). The overlaid heat map (2D histogram) in Fig. 2A indicates the density of all recorded particle positions. Owing to the brief illumination time, the particle positions obtained from each frame can be used directly to obtain particle occupation histograms (26) (fig. S2). We collapsed the observed 2D particle positions onto the x axis and calculated an average 1D histogram for a single ratchet cell by cross-correlation averaging. The normalized histograms of the single cells are shown in the inset of Fig. 2C. Identifying the histograms with the average 1D occupation probabilities p(x), we calculated the effective 1D interaction energy V0(x) of the particles using Boltzmann’s principle p(x) ∝ exp[–V0(x)/kBT] (Fig. 2C).

In the nanofluidic confinement apparatus, the gap distance is controlled and measured (27) with nanometer accuracy, which provides a convenient handle to tune the energy scale. Figure 2D shows nine recorded energy profiles for 127 nm < d < 148 nm. Within this range the curves are self-similar, and they collapse on a master curve obtained by linear scaling (Fig. 2E). The scaling coefficient α decreases exponentially: α ∝ exp(–d/2l), where l = 18 ± 3 nm. Finite element modeling simulations (26) (figs. S3 and S4) yield similar potential shapes (Fig. 2C) and indicate that l is slightly greater than the Debye length κ–1 ≈ 12 nm (26) (table S1).

The patterned nanofluidic slit thus defines an asymmetric static potential landscape for the particles—one of the two required ingredients for a rocking BM. The other necessary ingredient is non-equilibrium fluctuations, which we created by applying a 10-Hz electric square-wave potential to gold electrodes (Fig. 1A and fig. S1). The electric field induces both an electrophoretic force on the nanoparticles and an electro-osmotic plug flow of the electrolyte, acting on the particles in opposite directions. Our experiments showed that electro-osmotic effects dominate because the particles are dragged toward the negative electrode. Figure 3A depicts the particle drift velocities measured at time intervals of 1 ms, d = 127 nm, and applied voltage U = 3.5 V @ 10 Hz, resolving each half-cycle of the electric field.

Fig. 3 Brownian motor operation.

(A) Time-resolved average velocities measured in the control fields C1 and C2 and ratchets R1 to R4. (B) Average particle speed in the control fields as a function of applied voltage U and for two gap distances, d = 147 nm and 127 nm. (C) Estimated forward (solid lines) and backward (dashed lines) 1D potential energy V(xr) for the two gap distances and applied voltages of 3 and 5 V. Arrows indicate the magnitude of ΔVB. (D) Average measured particle drift velocity in ratchets R1 to R4. (E) Average particle drift velocities shown separately in the forward (blue and light blue) and backward directions (black and gray) of the ratchets. Lines in (D) and (E) indicate a fit parameter–free comparison with theory. (F) Average overall drift velocity 〈v〉 observed in ratchets R1 to R4 as a function of the driving frequency for three gap distances and their respective maximum voltages. Error bars in (B), (D), (E), and (F) denote SD of measurements obtained for each (ratchet) field.

We used the average drift velocity (fig. S5) in the control fields, 〈vc〉, to quantify the total force as a function of the gap distance (Fig. 3B). Combining Einstein’s relation D0 = kBT/6πηa and Stokes’ equation for particle drag F = 6πηavc〉 (where η is dynamic viscosity and a is particle radius) yields F/kBT = 〈vc/D0. At U = 5 V and d = 127 nm, we observed F = 39 ± 2 kBT/μm for the measured average diffusion coefficient D0 = 4.6 ± 0.2 μm2 s–1 (SD of 18 measurements). Maximum voltages were restricted to 5 V at d = 127 nm and 3 V at d = 147 nm (movies S2 and S3) to prevent particles escaping from the reservoirs.

In each half-cycle of the electric field, the force is approximately constant, which corresponds to a linear potential being added to the static potential V0: V(x) = V0(x) ± xF. Figure 3C shows the tilted potentials for the most- and least-confined experiments (d = 127 nm and d = 147 nm) and the maximally applied voltages of 5 and 3 V, respectively. In the forward direction, all energy barriers disappear and the particles begin to drift. In the backward direction, however, finite energy barriers ΔVB remain, thus hindering drift. Oscillating the force leads to a “rocking” of the asymmetric particle potential, resulting in directed transport due to the rectification of particle drift.

According to Reimann et al., the average drift velocity can be computed for the tilted potentials shown in Fig. 3C using the first passage time model (29):Embedded Image(1)Embedded Image(2)All relevant, nontrivial quantities V0(x), D0, and F in this theoretical description were measured in situ, and a parameter-free comparison with theory is possible. Equation 1 was evaluated numerically and the total drift was calculated as one-half of the drift difference in forward and backward directions (lines in Fig. 3, D and E). The theoretical description assumes slow rocking; that is, the actuation time of each half-cycle is long relative to the particle-drift time across a single ratchet element. Our experiments show excellent agreement with the model at 10 Hz (fig. S6). Moreover, we found a frequency dependence of the ratchet performance (Fig. 3F). Best performance is observed when the actuation time at 30 to 50 Hz is about twice the particle-drift time L/v〉 ≈ 6 ms in the forward direction (movies S4 to S6).

For the curved paths shown in Fig. 1C, a rotating electric field was applied to achieve efficient operation (fig. S7 and movie S7). The electric field direction can also be used to selectively power ratchets written in orthogonal directions—for example, to shuttle particles between several reservoirs (fig. S8 and movie S8).

Finally, we exploited the strong nonlinear gap distance dependence of the energy landscape to separate 60- and 100-nm particles into opposite compartments, LC and RC, of a sorting device (Fig. 4, A and B). The middle area between the compartments contains three deep (local d = 185 to 210 nm) and narrow (width ≈ 140 nm) ratchets pointing to the left and embedded in a wide (8 μm) and shallow (local d = 160 to 190 nm) ratchet pointing to the right. Our experiments show that at a gap distance of d = 150 nm, the large particles are well confined to the deep ratchets (movie S9). Although the small particles experience an energy barrier of ΔV0 ≥ 1.3kBT (Fig. 4C and fig. S9) to move from the deep ratchet into the shallow ratchet, its wide area (4 μm × 2 μm versus 3 nm × 140 nm) leads to a measured occupation probability of 76% in the shallow ratchet. Combined with the measured ratchet potential of ~4.5kBT (red line in Fig. 4C), this leads to a transport of 60-nm particles toward RC.

Fig. 4 Sorting of 60- and 100-nm Au particles.

(A) Topography of the sorting device as determined by t-SPL. Three deep ratchets (white arrows) are embedded in a wide shallow ratchet for transporting the 100- and 60-nm particles into compartments LC and RC, respectively. The deep ratchets comprise wide ends for better entrance of nanoparticles from RC. (B) Close-up of the dotted white box in (A) depicting the central deep ratchet (width 140 nm). (C) Top: Schematics of the sorting device, roughly to scale with experimental conditions. The 60-nm particles (red) in the shallow ratchet and the 100-nm particles (blue) in the deep ratchet are transported in opposite directions (arrows). Bottom: The corresponding measured (solid) and modeled (dashed) static energies. The 60-nm particles experience an energy barrier of ΔV0 ≈ 1.3kBT as they move into the shallow ratchet. (D) Optical microscopy images before sorting, and after 2 s and 6.3 s of applied voltage of 4 V @ 30 Hz AC. Colored circles indicate the 100-nm particle types depicted in (E). (E) Spatial particle distribution (left) and histograms (right) of the median relative brightness of the particle trajectories measured before and after sorting. We observed a third particle population (light blue), which we attribute to platelets according to their relative abundance. Insets: SEM images of the corresponding particles (scale bars, 100 nm). (F) Modeled average particle drift v as a function of particle radius r. The data points indicate the measured speed (mean ± SD) of 15 particles. The radial error corresponds to the measured SD (60 nm, 8%; 100 nm, 5%) of the particle size determined by SEM (26).

The optical images in Fig. 4D demonstrate the efficient sorting of the particle populations in opposite directions. Initially, the particles were distributed randomly across the ratchet field. Applying a square-wave voltage of 4 V @ 30 Hz, we found that the populations were fully separated into the respective compartments after 2 s, except for a single 100-nm particle stuck in the shallow ratchet. After 6.3 s, this particle diffused into the deep channel and was transported into the LC. Figure 4E depicts the median brightness of measured particle trajectories in the left and right halves of the field. The histograms at the right in Fig. 4E indicate that next to the majority particle types of 60-nm (red) and 100-nm (blue) spheres, there is a small population of a third particle type (light blue) in the sample. Inspection by scanning electron microscopy (SEM) revealed that, apart from the spheres, the 100-nm dispersion contained a small fraction of “platelets,” which we assign to the intermediate particle type (26) (figs. S10 and S11). Using the median trajectory contrast for differentiation, we measured a particle potential for this population of ~6kBT in the deep ratchet (Fig. 4A). Note that although the brightness histograms of 60-nm spheres and 100-nm platelets almost overlap, all platelets were actively driven toward LC (movie S9).

The operation and separation potential of the device can be understood using a simple electrostatic model (26, 30) to calculate the particle energy (dashed lines in Fig. 4C), assuming a linear scaling of the applied force with particle radius, and using equation 1 to calculate the particle speed as a function of radius (26). The model yields an average drift speed, which is compared to the measured speeds in Fig. 4F. With increasing particle size, the deep ratchets produce an increasingly dominating bias of a negative drift speed. This is due to a stronger confinement to the deep ratchets and an increasing ratchet potential experienced by the particles in the deep ratchets (from ~2kBT for the 60-nm particles to ~10kBT for the 100-nm particles; Fig. 4C). This leads to a zero crossing of the net drift at a particle diameter of ~82 nm (r = 41 nm). The strong nonlinear character of the curve originates from the intrinsically nonlinear BM transport and the exponentially increasing interaction energy, resulting in a slope of –7 μm s–1 nm–1 at the zero crossing. Thus, two particle species with a radial difference of 1 nm would drift at >3 μm s–1 in opposite directions, which is sufficient to overcome the diffusion current that drives the particles back into the device (26) (figs. S12 and S13).

Our results show that confining nanoparticles in electrolyte between a flat surface and a topographically patterned surface creates an energy landscape defined by the topography. By tuning the gap distance, the potential landscape to first order is simply scaled in magnitude, providing a convenient handle to optimize the system. All relevant physical quantities necessary to model the system are accessible in situ. Agreement between theory and experiment proves the validity of the interpretation and the predictability of the system. The nonlinear character of rocked BM transport (17) and of the electrostatic interaction leads to efficient separation (26) by our device, similar to the case of other concepts based on geometrical constrictions (3, 31).

Moreover, our modeling (26) and the trapping results of Ruggeri et al. (22) indicate that the method should scale to relatively small biomolecules. In contrast to flow-based separation (15), the rocked Brownian motor implementation provides selective transport, precise separation, and accumulation of nanoparticles without a net flow of the electrolyte or sustained thermodynamic gradients. Combined with the small footprint and the low applied voltage, such devices are ideally suited for the precise analysis of small liquid volumes in lab-on-chip technology.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S13

Table S1

Movies S1 to S9

References (3237)

References and Notes

  1. See supplementary materials.
Acknowledgments: We thank U. Drechsler for assistance in fabricating of the glass pillars, U. Duerig and H. Wolf for stimulating discussions, R. Allenspach and W. Riess for support, and C. Bolliger and L.-M. Pavka for proofreading the manuscript. Funding: Supported by European Research Council Starting Grant 307079, European Commission FP7-ICT-2011-8 no. 318804, and Swiss National Science Foundation grant 200020-144464. Author contributions: M.J.S. and A.W.K. jointly conceived the idea and the experimental concept; M.J.S. implemented first working devices and started theoretical analysis; C.S. and S.F. performed the sorting experiment; C.S. performed the analysis of the sorting experiment and the numerical modeling for the BMs and the sorting experiment; S.F. performed the BM experiments and contributed to their analysis; C.D.R. performed the finite element modeling and analysis; A.W.K. performed t-SPL, analyzed the BM function, supervised the work, and wrote the manuscript; C.S., C.D.R., S.F., and A.W.K. wrote the supplementary materials. Competing interests: The authors declare no competing interest. Data and materials availability: All data needed to evaluate our conclusions are present in the paper and/or the supplementary materials. All authors are inventors on U.S. patent application 15/469995 submitted by IBM that covers transport and separation of nanoparticles.

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