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<details>
<summary>x1.png Details</summary>
### Visual Description
\n
## Image: Blank White Canvas
### Overview
The image presents a completely white canvas with no discernible features, text, or graphical elements. It is essentially a blank slate.
### Components/Axes
There are no components, axes, or legends present in the image. The image lacks any visual structure.
### Detailed Analysis or Content Details
There is no data or content to analyze. The image is uniformly white.
### Key Observations
The most notable observation is the complete absence of information. This suggests the image may be a placeholder, an error, or intentionally devoid of content.
### Interpretation
The image, in its emptiness, signifies a lack of data or information. It could represent a starting point, a void, or a deliberate absence of visual communication. Without any context, it's impossible to determine the intended meaning. The image does not provide any facts or data to interpret. It is a purely visual statement of nothingness.
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<details>
<summary>x2.png Details</summary>
### Visual Description
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## Chart: Global Mean Land-Sea Temperature Anomaly
### Overview
The image presents a line chart depicting the global mean land-sea temperature anomaly from 1880 to approximately 2020. The chart illustrates the temperature deviation from an average baseline, showing a clear warming trend over the period.
### Components/Axes
* **X-axis:** Years, ranging from 1880 to 2020 (approximately). The axis is labeled "Year".
* **Y-axis:** Temperature Anomaly (°C), ranging from -0.6 to +1.2. The axis is labeled "Global Mean Temperature Anomaly (°C)".
* **Data Series:** Three lines are plotted:
* **Black Line:** Represents the annual mean.
* **Red Line:** Represents the 5-year mean.
* **Blue Line:** Represents the land-only mean.
* **Legend:** Located in the top-right corner, identifying each line by color and description.
### Detailed Analysis
The black line (annual mean) shows significant year-to-year fluctuations, but exhibits a clear upward trend overall. The red line (5-year mean) smooths out these fluctuations, revealing a more pronounced and consistent warming trend. The blue line (land-only mean) also shows an upward trend, but appears to be more volatile than the other two lines, particularly in the earlier years.
Here's a breakdown of approximate data points, noting the inherent uncertainty in reading values from the image:
* **1880:**
* Black Line: Approximately -0.2 °C
* Red Line: Approximately -0.2 °C
* Blue Line: Approximately -0.3 °C
* **1900:**
* Black Line: Approximately -0.3 °C
* Red Line: Approximately -0.3 °C
* Blue Line: Approximately -0.4 °C
* **1920:**
* Black Line: Approximately -0.1 °C
* Red Line: Approximately -0.1 °C
* Blue Line: Approximately -0.2 °C
* **1940:**
* Black Line: Approximately +0.0 °C
* Red Line: Approximately +0.0 °C
* Blue Line: Approximately +0.1 °C
* **1960:**
* Black Line: Approximately -0.0 °C
* Red Line: Approximately -0.0 °C
* Blue Line: Approximately -0.1 °C
* **1980:**
* Black Line: Approximately +0.1 °C
* Red Line: Approximately +0.1 °C
* Blue Line: Approximately +0.2 °C
* **2000:**
* Black Line: Approximately +0.4 °C
* Red Line: Approximately +0.4 °C
* Blue Line: Approximately +0.5 °C
* **2020:**
* Black Line: Approximately +1.0 °C
* Red Line: Approximately +1.1 °C
* Blue Line: Approximately +1.2 °C
The trend for all three lines is clearly upward, with the rate of warming accelerating in the latter half of the 20th century and continuing into the 21st. The 5-year mean (red line) consistently shows a higher temperature anomaly than the annual mean (black line) in the later years, indicating a sustained warming trend.
### Key Observations
* The most significant warming occurs after 1970.
* The land-only temperature anomaly (blue line) is generally higher than the global mean (black line), suggesting that land areas are warming faster than oceans.
* The 5-year mean (red line) provides a clearer picture of the long-term warming trend by smoothing out short-term fluctuations.
* The temperature anomaly has increased from approximately -0.2°C in 1880 to approximately +1.0°C in 2020.
### Interpretation
The chart demonstrates a clear and consistent warming trend in global temperatures over the period from 1880 to 2020. This warming trend is evident in all three data series (annual, 5-year, and land-only means). The accelerating rate of warming in recent decades suggests a significant impact from anthropogenic factors, such as greenhouse gas emissions. The higher temperature anomaly observed in land areas compared to the global mean indicates that land surfaces are more sensitive to climate change. The smoothing effect of the 5-year mean highlights the long-term warming trend, minimizing the influence of short-term climate variability. This data provides strong evidence for the ongoing phenomenon of global warming and its potential consequences for the planet. The chart is a visual representation of the scientific consensus on climate change.
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<details>
<summary>x3.png Details</summary>
### Visual Description
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## Text Block: Article Type Label
### Overview
The image contains a single block of text on a light blue background. It appears to be a label or heading.
### Content Details
The text reads: "ARTICLE TYPE". The text is white and appears to be in a sans-serif font.
### Key Observations
The text is centered horizontally within the image. There is no other information present.
### Interpretation
This image simply presents a label indicating a category or field related to "ARTICLE TYPE". It doesn't contain any data or complex relationships, but rather serves as a descriptor. It suggests that further information or content related to article types would follow.
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|
<details>
<summary>x4.png Details</summary>
### Visual Description
Icon/Small Image (327x28)
</details>
| Curvature induces and enhances transport of spinning colloids through narrow channels † |
| --- | --- |
| Eric Cereceda-López, a,b Marco De Corato, c Ignacio Pagonabarraga, a,d Fanlong Meng, e,f,g Pietro Tierno, ∗,a,b,d and Antonio Ortiz-Ambriz, ∗,h | |
|
<details>
<summary>x5.png Details</summary>
### Visual Description
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## Text Block: Publication Metadata
### Overview
The image contains a block of text representing metadata likely associated with a publication (e.g., a research paper). It includes fields for the received date, accepted date, and a Digital Object Identifier (DOI).
### Components/Axes
The text consists of three labels and their corresponding values:
* **Received Date:** (No value provided)
* **Accepted Date:** (No value provided)
* **DOI:** 00.0000/xxxxxxxxxx
### Detailed Analysis or Content Details
The DOI is partially obscured, represented as "00.0000/xxxxxxxxxx". The 'x' characters indicate missing or redacted information. The DOI format is standard, consisting of a prefix (00.0000) and a suffix (xxxxxxxxxx).
### Key Observations
The received and accepted dates are missing. The DOI is present but incomplete. This suggests the metadata is either preliminary or intentionally partially hidden.
### Interpretation
This text block provides basic publication tracking information. The absence of dates suggests the publication process is ongoing or the information is not yet finalized. The redacted DOI prevents identification of the specific publication. The format suggests this is a formal publication record, likely from an academic or professional source.
</details>
| The effect of curvature and how it induces and enhances the transport of colloidal particles driven through narrow channels represent an unexplored research avenue. Here we combine experiments and simulations to investigate the dynamics of magnetically driven colloidal particles confined through a narrow, circular channel. We use an external precessing magnetic field to induce a net torque and spin the particles at a defined angular velocity. Due to the spinning, the particle propulsion emerges from the different hydrodynamic coupling with the inner and outer walls and strongly depends on the curvature. The experimental findings are combined with finite element numerical simulations that predict a positive rotation translation coupling in the mobility matrix. Further, we explore the collective transport of many particles across the curved geometry, making an experimental realization of a driven single file system. With our finding, we elucidate the effect of curvature on the transport of microscopic particles which could be important to understand the complex, yet rich, dynamics of particle systems driven through curved microfluidic channels. |
footnotetext: a Departament de Física de la Matèria Condensada, Universitat de Barcelona, 08028, Spain. E-mail: ptierno@ub.edu footnotetext: b Institut de Nanociència i Nanotecnologia, Universitat de Barcelona (IN2UB), 08028, Barcelona, Spain. footnotetext: c Aragon Institute of Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain. footnotetext: d University of Barcelona Institute of Complex Systems (UBICS), 08028, Barcelona, Spain footnotetext: e Chinese Academy of Sciences, CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Beijing 100190, China footnotetext: f University of Chinese Academy of Sciences, School of Physical Sciences, Beijing 100049, China footnotetext: g University of Chinese Academy of Sciences, Wenzhou Institute, Wenzhou 325000, China footnotetext: h Tecnologico de Monterrey, School of Science and Engineering, 64849, Monterrey, Mexico. E-mail: aortiza@tec.mx footnotetext: † Electronic Supplementary Information (ESI) available: Two videos illustrating the single and collective particle dynamics within the ring. See DOI: 00.0000/00000000.
1 Introduction
In viscous fluids the Reynold ( $Re$ ) number (the ratio between inertial and viscous forces) is relatively low and the Navier-Stokes equations become effectively time-reversible. 1 Under such conditions, inducing the transport of microscale objects may become problematic since any reciprocal motion, namely a series of body/shape deformations that are identical when reversed, do not produce a net movement. 2 However, many artificial prototypes are often powered by external fields, such as magnetic ones, which indeed induce periodic deformation or translations. Thus, at low $Re$ these systems require a subtle strategy to overcome the time-reversibility and propel, such as the presence of flexibility, 3, 4, 5, 6 non linearity of the dispersing medium, 7, 8 or the interaction between different parts. 9, 10, 11
The problem of propulsion at low $Re$ becomes even more complex when the active object, either a microswimmer or a synthetic self-propelled particle, is confined within a narrow channel. 12 The presence of confining walls can affect the microswimmer dynamics in many ways. In particular, planar surface can attract swimming bacteria due to hydrodynamic interactions, 13, 14 induce circular trajectories, 15, 16 or curved posts can be used as reflecting barriers or traps. 17 Moreover, it has been recently found that the curvature of the fallopian tubes guides the locomotion of sperm cells to promote fertilization, 18 and similar situations are not limited to bacteria or sperm cells, but extend to other types of situations. 19, 20, 21, 22
On the other hand, confinement can be used to rectify the random motion of microswimmers using patterned channels and other ratcheting methods. 23, 24, 25, 26, 27, 28, 29, 30, 31 At low $Re$ , the close proximity of a flat wall can break the spatial symmetry of the flow field and induce propulsion in simpler artificial systems, such as rotating magnetic particles, 32, 33, 34, 35 doublets 36 or elongated structures. 37, 38
Here we show that the geometry of the channel can be used to induce translation of a driven particle by breaking the rotational symmetry of the hydrodynamic interactions. We confine spinning magnetic colloids within narrow circular channels characterized by a width close to the particle diameter. We find that curvature can rectify the particle rotational motion inducing a net translational one. Moreover, these surface rotors are able to translate along the ring at a velocity that depends on the curvature of the channel. We explain this experimental observation using numerical simulations which elucidate the mechanism of motion based on the difference between the hydrodynamic coupling with the two confining walls, where the largest coupling is with the wall characterized by the largest curvature. Further, we investigate the collective propulsion of many driven rotors, and show that within the circular ring the magnetic particles behave as a driven single-file, with an initial diffusive regime followed by a ballistic one at a large time scale.
<details>
<summary>x6.png Details</summary>
### Visual Description
\n
## Diagram: Microfluidic Vortex and Particle Trapping
### Overview
The image presents two panels (a and b) illustrating a microfluidic vortex generated within a circular chamber and the trapping of a particle within it. Panel (a) shows a top-down view of the vortex flow pattern, while panel (b) provides a 3D representation of the vortex and a cross-sectional profile of the chamber's height.
### Components/Axes
**Panel a:**
* **Coordinate System:** A Cartesian coordinate system is overlaid on the image with the x and y axes labeled. An additional radial coordinate, Φ (phi), is also indicated.
* **Flow Visualization:** Red and blue arrows depict the direction of fluid flow, forming a vortex pattern.
* **Scale Bar:** A black scale bar is present in the bottom-left corner, indicating a length scale (though the exact length is not specified).
**Panel b:**
* **Coordinate System:** A Cartesian coordinate system (x, y) and a z-axis are shown.
* **Vortex Representation:** A gray, conical shape represents the vortex.
* **Magnetic Field:** A blue arrow labeled "H" indicates the direction of a magnetic field.
* **Rotation Vectors:** A red arrow labeled "Ω" represents the angular velocity of the vortex, and another red arrow labeled "ω" indicates the rotation of the trapped particle.
* **Height Profile:** A graph is embedded in the bottom-left corner, plotting height (h) in micrometers (µm) against radial distance (ρ) in micrometers (µm). The axes are labeled "h [µm]" and "ρ [µm]".
### Detailed Analysis or Content Details
**Panel a:**
The vortex flow pattern is clearly visible as concentric rings. The red arrows indicate a clockwise rotation near the outer edge of the vortex, while the blue arrows indicate a counter-clockwise rotation closer to the center. The flow appears to converge towards the center of the circular chamber.
**Panel b:**
* **Height Profile:** The embedded graph shows a cross-section of the chamber's height. The height is approximately 0 µm at ρ = 3.5 µm and dips to approximately -4 µm at ρ = 6.5 µm, indicating a concave shape at the bottom of the chamber. The profile shows a rapid increase in height around ρ = 6.5 µm.
* **Vortex Orientation:** The vortex is oriented with its apex pointing upwards along the z-axis.
* **Magnetic Field and Rotation:** The magnetic field (H) is aligned with the z-axis. The angular velocity of the vortex (Ω) and the particle (ω) are both oriented in the same direction.
### Key Observations
* The vortex is generated within a circular microfluidic chamber.
* A particle is trapped near the center of the vortex.
* The chamber has a non-flat bottom, as evidenced by the height profile.
* A magnetic field is present and may be involved in the particle trapping mechanism.
### Interpretation
The diagram illustrates a microfluidic system designed to trap particles using a vortex flow. The vortex is likely generated by applying a tangential force to the fluid within the chamber. The particle is drawn into the center of the vortex and held there by the balancing forces of the fluid flow and potentially a magnetic field. The concave shape of the chamber bottom, as shown in the height profile, may contribute to the stability of the vortex and the particle trapping. The magnetic field (H) could be used to manipulate the particle's position or orientation within the vortex. The coordinate systems provide a framework for quantifying the flow dynamics and particle behavior. The diagram suggests a potential application in microfluidic particle sorting, concentration, or analysis.
</details>
Fig. 1: (a) Optical microscope image of two circular channels in a PDMS mold each with a spinning paramagnetic colloidal particle. The red (blue) arrow in the image indicates the direction of the particle rotation (trajectory). Scale bar is $5\rm{\mu m}$ , see also corresponding Movie S1†in the Supporting Information. (b) Schematic of a particle within the circular channel and of the applied precessing magnetic field $\bm{H}$ . The inset shows the transverse profile of the channel measured using confocal profilometry.
2 Methods
2.1 Experimental protocol
The circular microchannels are realized in Polydimethylsiloxane (PDMS) with soft lithography. We first use commercial software (AUTOCAD) to design a sequence of lithographic channels with a width $w=3\,\rm{\mu m}$ , see Fig. 1 (a). The channels are placed concentrically so that they are always separated by a distance of $10\rm{\mu m}$ . To sample the channels in 5 $\rm{\mu m}$ intervals, the rings are separated in two sets, one starting at $5\,\rm{\mu m}$ and the other at $10\,\rm{\mu m}$ . The microchannels present a depth of $\sim 4\rm{\mu m}$ thus, larger than the particle width, such that the particles are completely surrounded by solid walls, as shown by the confocal profile in Fig. 1 (b).
We use the following procedure to realize the structures in PDMS. First, we fabricate reliefs in SU-8, an epoxy-based negative photoresist, on top of a silicon (Si) wafer. The latter was prepared by dehydrating it for $10$ minutes at $200^{\circ}$ C on a hot plate and then plasma cleaned it for $1$ minute at a pressure of $1$ Torr. After that, a layer of SU-8 $3005$ is spin-coated on top of the Si wafer for $10$ s at $500$ revolution per minute (rpm), followed by spinning at $4000$ rpm for $120$ s, and later baked on a hot plate heated at a temperature of $95^{\circ}$ C for $2$ minutes. The SU-8 is then polymerized by exposure to UV light through a Cromium (Cr) mask for $5.6$ s with a light intensity of $100\,\rm{mJ\,cm^{-2}}$ (Mask aligner, i-line configuration with wavelength $\lambda$ =365nm). Then the exposed film is baked again for $1$ min at $65^{\circ}$ C and for another $1$ min at $95^{\circ}$ C, and developed for $30$ s in propylene glycol methyl ether acetate. We then transfer the obtained master to PDMS by first cleaning the SU-8 structure in a plasma cleaner, operating at a pressure of $1$ Torr for $1$ min, and then silanizing it in a vacuum chamber with a drop of silane (SiH 4) for one hour. After that, the PDMS is spin coated onto the Si wafer at $4000$ rpm during $1$ min and baked for $30$ min above a hot plate at a temperature of $95^{\circ}$ C. The PDMS is then transferred to a cover glass by plasma cleaning both surfaces ( $1$ min at $1$ Torr), then joining the PDMS and the glass and finally peeling very carefully with a sharp blade the resulting PDMS film.
Once the PDMS channels are fabricated, we realize a closed fluidic cell by first painting the PDMS patterned coverglass with two strips of silicon-based vacuum grease. After that, we deposit a drop containing the colloidal particles dispersed in water on top of it, we close it with another clean coverglass and then we seal the two open sides with more grease. The colloidal suspension is prepared by dispersing $1.5\,\rm{\mu l}$ from the stock solution of paramagnetic colloidal particles (Dynabeads M-270, $2.8\rm{\mu m}$ in diameter) in $1\,\rm{ml}$ of ultrapure water (MilliQ system, Millipore).
The fluidic cell is placed on a custom-made inverted optical microscope equipped with optical tweezers that are controlled using two Acousto-Optic Deflectors (AA Optoelectronics DTSXY-400-1064) driven by a radio frequency wave generator (DDSPA2X-D431b-34). The optical tweezers ensure that the channels are filled with the desired number of particles. Since it is often necessary to push the deposited particles out of the channels, the tweezers have been set up to trap from below by driving the laser radiation (Manlight ML5-CW-P/TKS-OTS) through a high numerical aperture objective (Nikon $100×$ , Numerical aperture $1.3$ ). The trapping objective is also used for observation and a custom-made TV lens is mounted in the microscope setup to observe a wide enough field of view, $\sim 100\,\rm{\mu m^{2}}$ .
The external magnetic fields used to spin the particles are applied using a set of five coils, all arranged around the sample. In particular, two coils have the main axis aligned along the $\hat{\bm{x}}$ direction, two for the $\hat{\bm{y}}$ direction and one for the $\hat{\bm{z}}$ direction which is perpendicular to the particle plane ( $\hat{\bm{x}},\hat{\bm{y}}$ ). The sample is placed directly above the coil oriented along the $\hat{\bm{z}}$ axis. The magnetic coils are connected to three power amplifiers (KEPCO BOP 20-10) which are driven by a digital-analog card (cDAQ NI-9269) and controlled with a custom-made LabVIEW program.
2.2 Numerical simulations
To understand the mechanism of propulsion of the rotating particles in the channel, we perform finite element simulations. We consider a sphere suspended in a Newtonian liquid with viscosity $\eta$ and confined in a channel obtained from two concentric cylinders of height $h$ , as schematically depicted in Fig. 3 (a). The Reynolds number, estimated from the characteristic particle velocity in the ring, is vanishingly small and thus we neglect inertial effects. We consider a Cartesian reference frame with origin at the center of the channel (Fig. 1). We assume that the distance between the particle and the bottom wall, $z_{p}$ , is fixed due to the balance between gravitational forces and electrostatic repulsion. The ratio of the lateral size of the channel $w$ and the particle radius is fixed to ${w}/{a}=2.2$ which for our $a=1.4\rm{\mu m}$ particles produces a channel $3.1\rm{\mu m}$ wide, similar to that shown in the inset of 1 b). To model an open channel in the experimental system, we chose $h$ much larger than $w$ . The particle is placed near the bottom of the domain, and horizontally at the center, as depicted in Fig. 3 (a). The rotating magnetic field generates a constant torque $\bm{\tau}_{m}$ along the $\hat{\bm{z}}$ axis, which leads to a constant angular velocity $\omega_{p}$ along the same direction. As a result of the rotation, we expect the particle to translate along the direction that is tangential to the channel centerline, $\hat{\bm{\phi}}$ , due to the hydrodynamic coupling with the channel walls (see Fig. 1 ).
To investigate how the motion of the particle along the channel depends on the radius of curvature, we compute the mobility matrix of the particle. The mobility matrix, $\bm{M}$ , relates the translational ( $\bm{v}_{p}$ ) and angular velocity ( $\bm{\omega}_{p}$ ) of the particle to external forces ( $\bm{F}$ ) and torques ( $\bm{\tau}$ ):
$$
\displaystyle\begin{bmatrix}\bm{v}_{p}\\
\bm{\omega}_{p}\end{bmatrix}=\bm{M}\begin{bmatrix}\bm{F}\\
\bm{\tau}\end{bmatrix}\,\,\,\,. \tag{1}
$$
The mobility matrix, $\bm{M}$ , contains all the information about the hydrodynamic interactions between the particle and the surrounding boundaries. In this particular case, $\bm{M}$ depends on the radius of the channel $R$ . Here, we are interested in the coefficient that couples the magnetic torque, acting along the $\hat{\bm{z}}$ direction, to the translational velocity along the $\hat{\bm{\phi}}$ direction.
To obtain $\bm{M}$ , we first find the resistance matrix $\bm{R}$ , and then compute $\bm{M}$ as $\bm{M}=\bm{R}^{-1}$ . To do so, we solve the Stokes equation by imposing a single component of the velocity vector and then computing the resulting force and torque acting on the particle:
$$
\displaystyle\nabla^{2}\bm{v}-\nabla p=0 \displaystyle\nabla\cdot\bm{v}=0 \tag{2}
$$
where $\bm{v}$ is the velocity of the fluid and $\bm{p}$ is the pressure. We consider no-slip boundary condition on the walls of the domain $\bm{v}|_{\mathcal{S}}=0$ , being $\mathcal{S}$ the surface of the particle. On $\mathcal{S}$ we fix the velocity or the angular velocity. The total forces and torques acting on the particle are computed using:
$$
\displaystyle\int_{\mathcal{S}}\bm{T}\cdot\bm{n}\ \mathrm{d}S=\bm{F} \displaystyle\int_{\mathcal{S}}\left(\bm{r}-\bm{r_{p}}\right)\times\left(\bm{T%
}\cdot\bm{n}\right)\ \mathrm{d}S=\bm{\tau} \tag{4}
$$
where $\bm{r}_{p}=(x_{p},y_{p},z_{p})$ is the position vector of the particle center, $\bm{r}$ is the position of a point on the surface of the particle and $\bm{T}$ is the stress tensor defined as: $\bm{T}=\eta(∇ v+∇ v^{T})-p\bm{I}$ , with $p$ the pressure. By repeating the simulation for all the six components of the velocity vector, we can construct the resistance matrix $\bm{R}$ which is the inverse of the mobility matrix, $\bm{M}=\bm{R}^{-1}$ .
<details>
<summary>x7.png Details</summary>
### Visual Description
\n
## Chart/Diagram Type: Scatter Plot with Inset Scatter Plot & Schematic Diagrams
### Overview
The image presents a scatter plot illustrating the relationship between particle radius (R) and tangential velocity (vφ) with error bars. An inset scatter plot shows the tangential velocity (vφ) as a function of angle (Φ). Above the main plot are three schematic diagrams depicting a particle within a fluid, with arrows indicating forces and movement.
### Components/Axes
* **Main Plot:**
* X-axis: R [µm] (Particle Radius) - Scale ranges from approximately 5 µm to 40 µm.
* Y-axis: vφ [µm/s] (Tangential Velocity) - Scale ranges from approximately -0.05 µm/s to 0.15 µm/s.
* Data Points: Blue circles with error bars.
* Trend Line: Black solid line fitted to the data points.
* **Inset Plot:**
* X-axis: Φ [rad] (Angle) - Scale ranges from -π to π.
* Y-axis: vφ [µm/s] (Tangential Velocity) - Scale ranges from approximately 0.00 µm/s to 0.25 µm/s.
* Data Points: Dark grey scatter plot.
* Horizontal Line: Black solid line at approximately vφ = 0.08 µm/s.
* **Schematic Diagrams (Top Row):**
* Three diagrams showing a particle (red circle) in a fluid (light blue background).
* Arrows indicate the direction of movement and forces.
* Labels: x, y, Φ (angle).
### Detailed Analysis or Content Details
* **Main Plot Data Points:**
* R ≈ 8 µm, vφ ≈ 0.11 µm/s (Error bar extends from approximately 0.08 µm/s to 0.14 µm/s)
* R ≈ 12 µm, vφ ≈ 0.05 µm/s (Error bar extends from approximately 0.02 µm/s to 0.08 µm/s)
* R ≈ 18 µm, vφ ≈ 0.01 µm/s (Error bar extends from approximately -0.02 µm/s to 0.04 µm/s)
* R ≈ 25 µm, vφ ≈ -0.01 µm/s (Error bar extends from approximately -0.04 µm/s to 0.02 µm/s)
* R ≈ 35 µm, vφ ≈ 0.00 µm/s (Error bar extends from approximately -0.02 µm/s to 0.02 µm/s)
* **Trend Line (Main Plot):** The black line shows a steep decrease in tangential velocity as particle radius increases from approximately 5 µm to 20 µm, then plateaus around 0 µm/s for radii greater than 20 µm.
* **Inset Plot Data Points:** The inset plot shows a relatively constant tangential velocity across all angles, with some scatter. The data points are clustered around approximately 0.08 µm/s.
### Key Observations
* The tangential velocity decreases significantly with increasing particle radius up to a certain point.
* The error bars on the main plot indicate considerable uncertainty in the measurements, particularly at smaller radii.
* The inset plot suggests that the tangential velocity is largely independent of the angle.
* The schematic diagrams illustrate the coordinate system and the angle used in the analysis.
### Interpretation
The data suggests that the tangential velocity of a particle in a fluid is strongly dependent on its size. Smaller particles experience a higher tangential velocity, while larger particles exhibit a negligible tangential velocity. This could be due to increased drag forces on smaller particles or a transition in the flow regime as the particle size increases. The inset plot indicates that the tangential velocity is not significantly affected by the particle's angular position, suggesting a symmetrical flow pattern. The error bars indicate that the measurements are noisy, which could be due to experimental limitations or inherent variability in the system. The diagrams provide a visual representation of the physical setup and the parameters being measured. The overall data suggests a relationship between particle size, tangential velocity, and fluid dynamics.
</details>
Fig. 2: (a) Schematic showing a spinning particle close to a single, flat wall (left), confined between two flat walls (center) and between two curved walls (right). All three images show the particles from the top view, i.e. the particle rotates in the ( $\hat{\bm{x}},\hat{\bm{y}}$ ) plane. The bottom wall, placed at $z=0$ , is not shown for clarity. (b) Tangential velocity $v_{\phi}$ of the rotating magnetic particle inside the channels for different values of the central radius $R$ . The filled points are experimental measurements, and the continuous line is calculated from the numerical simulations using a magnetic torque equal to $\tau_{m}=0.55\,\rm{pN\,\mu m}$ . The inset shows the mean velocity at each value of the angle for a ring with radius $R=5\,\rm{\mu m}$ . The angle $\phi$ is measured with respect to the $\hat{\bm{x}}$ axis.
<details>
<summary>x8.png Details</summary>
### Visual Description
\n
## Diagram: Cell Migration and Traction Forces
### Overview
This diagram illustrates a cell interacting with a substrate, and the distribution of traction forces generated by the cell. It consists of a schematic representation of the cell and its environment (a), and three visualizations of traction force distributions (b, c, d). The visualizations appear to be heatmaps representing the magnitude and direction of traction forces.
### Components/Axes
* **(a) Schematic:** Shows a cell (light blue) interacting with a substrate. Labels indicate:
* `R`: Horizontal distance, approximately 10-15 units.
* `w`: Vertical distance, approximately 5-7 units.
* `h`: Vertical distance, approximately 10-12 units.
* `in`: Arrow indicating fluid/material entering the cell.
* `out`: Arrow indicating fluid/material exiting the cell.
* **(b) Outer hemisphere:** Heatmap labeled "Outer hemisphere". Axes:
* `ẑ`: Vertical axis.
* `φ`: Horizontal axis.
* Color scale: Ranges from -5 to 5, labeled `Nφ`.
* **(c) Inner hemisphere:** Heatmap labeled "Inner hemisphere". Axes:
* `ẑ`: Vertical axis.
* `φ`: Horizontal axis.
* Color scale: Ranges from -5 to 5, labeled `Nφ`.
* **(d) Total traction:** Heatmap labeled "Total traction". Axes:
* `ẑ`: Vertical axis.
* `φ`: Horizontal axis.
* Color scale: Ranges from -2.5 to 2.5, labeled `Nφ`.
### Detailed Analysis or Content Details
* **(a) Schematic:** The cell is depicted as a roughly rectangular shape with a curved underside. The dimensions `R`, `w`, and `h` provide a scale for the cell's size. The "in" and "out" arrows suggest a flow of material or fluid across the cell membrane.
* **(b) Outer hemisphere:** The heatmap shows a strong red region (positive `Nφ` values) at the center, surrounded by blue regions (negative `Nφ` values). The intensity of the red is highest at the center, decreasing radially outwards. The distribution appears roughly circular.
* **(c) Inner hemisphere:** The heatmap shows a more diffuse distribution of traction forces. There is a central region of moderate intensity, with less pronounced red and blue areas compared to the outer hemisphere. The distribution is also roughly circular.
* **(d) Total traction:** This heatmap displays two distinct, elongated regions of positive traction force (red) flanking a central region of negative traction force (blue). The red regions are more intense than the blue region. The overall shape is bilaterally symmetric.
### Key Observations
* The traction forces are not uniformly distributed.
* The outer hemisphere exhibits a more concentrated traction force distribution than the inner hemisphere.
* The total traction force distribution shows a clear pattern of opposing forces, suggesting a mechanism for cell movement or deformation.
* The color scales indicate that the magnitude of traction forces in the outer and inner hemispheres is generally higher than the total traction.
### Interpretation
This diagram likely represents a model or experimental data related to cell migration and the forces a cell exerts on its surrounding environment. The schematic (a) provides context for the traction force visualizations. The heatmaps (b, c, d) show the spatial distribution of these forces.
The outer hemisphere (b) likely represents the traction forces at the leading edge of the cell, where the cell is actively adhering to and pulling on the substrate. The inner hemisphere (c) may represent the traction forces within the cell body, which are less concentrated. The total traction (d) shows the net effect of these forces, resulting in a pattern consistent with cell polarization and directional movement. The opposing forces suggest that the cell is generating tension to propel itself forward.
The difference in color scale ranges between the hemisphere maps and the total traction map suggests that the total traction is a summation of the forces, and therefore has a lower overall magnitude. The bilaterally symmetric pattern in the total traction map indicates that the cell is exerting force in two opposing directions, which is consistent with the process of cell elongation and migration.
</details>
Fig. 3: (a) Schematic showing the simulation domain considered, which consists of the space between two concentric cylinders separated by a spacing $w$ , with height $h$ , and a central radius $R$ . (b,c) Surface traction in the outer (b) and inner (c) hemispheres of the particle, as defined in panel (a). (d) Total traction given by the sum of the functions shown in panels (b) and (c), which is positive, indicating a resulting force that pushes forward the particle.
3 Single particle motion
We start by illustrating the dynamics of individual spinning particles dispersed within circular channels characterized by different radius $R$ . Fig. 1 (a) shows a microscope image of two channels with radii $R=5,15\,\rm{\mu m}$ and filled with one paramagnetic colloidal particle each, while the corresponding schematic of the experimental system is shown in Fig. 1 (b). The spinning is induced by the application of an external magnetic field that performs conical precession at a frequency $\omega$ and an angle $\theta$ around the $\hat{\bm{z}}$ axis:
$$
\bm{H}\equiv H_{0}[\cos{\theta}\hat{\bm{z}}+\sin{\theta}(\cos{(\omega t)}\hat{%
\bm{x}}+\sin{(\omega t)}\hat{\bm{y}})]\,\,\,, \tag{6}
$$
being $H_{0}$ the field amplitude. The precessing field has an in-plane component $\bm{H_{xy}}\equiv\sin{\theta}(\cos{(\omega t)}\hat{\bm{x}}+\sin{(\omega t)}%
\hat{\bm{y}})$ which is used to spin the particles, and a static one perpendicular to the particle plane, $\bm{H}_{z}=H_{0}\cos{\theta}\hat{\bm{z}}$ which is used to control the dipolar interactions when many particles are present. In particular, we keep constant the magnetic field values to $H_{0}=900\,\rm{A\,m^{-1}}$ , the frequency $f=50$ Hz such that $\omega=2\pi f=314.2\,\rm{rad\,s^{-1}}$ and the cone angle at $\theta=44.5^{\circ}$ .
For the single particle case, the precessing field at relative high frequency, is able to induce an average magnetic torque, $\bm{\tau}_{m}=\langle\bm{m}×\bm{H}\rangle$ , which sets the particles in rotational motion at an angular speed $\omega_{p}$ close to the plane. This torque arises due to the finite relaxation time $t_{r}$ of the particle magnetization, as demonstrated in previous works. 39, 40 Note that here, for simplicity, we assume that the particles are characterized by a single relaxation time. For a paramagnetic colloid with radius $a=1.4\,\rm{\mu m}$ and magnetic volume susceptibility $\chi\sim 0.4$ , the torque can be calculated as 41, 42,
$$
\bm{\tau}_{m}=\frac{4a^{3}\pi\mu_{0}\chi H_{0}^{2}\sin^{2}{(\theta)}\omega t_{%
r}}{3(1+t_{r}^{2}\omega^{2})}\hat{\bm{z}} \tag{7}
$$
being $\mu_{0}=4\pi 10^{-7}\,\rm{H\,m}$ the permeability of vacuum, similar to that of water. Assuming $t_{r}\sim 10^{-3}$ s we obtain $\tau_{m}=0.32\,\rm{pN\,\mu m}$ .
In the overdamped limit, the magnetic torque is balanced by the viscous torque $\bm{\tau}_{v}$ arising from the rotation in water, $\bm{\tau}_{m}+\bm{\tau}_{v}=\bm{0}$ . Close to a planar wall, $\bm{\tau}_{v}$ can be written as, $\bm{\tau}_{v}=-8\pi\eta\bm{\omega}_{p}a^{3}f_{r}(a/z_{p})$ , where $\eta=10^{-3}\,\rm{Pa· s}$ is the viscosity of water, and $f_{r}(a/z_{p})$ is a correction term that accounts for the proximity of the bottom wall at a distance $z_{p}$ . 43 In particular, assuming the space between the particle and the wall $\sim 100$ nm, $f_{r}(a/h)=1.45$ . In the overdamped regime, the balance between both torques allows estimating the particle mean angular velocity: $\langle\omega_{p}\rangle=\mu_{0}H_{0}^{2}\sin^{2}{\theta}\chi\tau_{r}\omega/[6%
\eta(1+\tau_{r}^{2}\omega^{2})]$ , which gives $\omega_{p}=3.27\rm{rad\,s^{-1}}$ . Clearly, this estimate does not consider the presence of a top wall and how it reduces the particle angular rotation, as we will see later.
Now let’s consider the general situation of a rotating free particle, far away from any surface. The rotational motion is reciprocal, and unable to induce any drift. However, if close to a solid wall, as shown in the left schematic in Fig. 2 (a), the rotational motion can be rectified into a net drift velocity due to the hydrodynamic interaction with the close substrate, 43 which breaks the spatial symmetry of the flow. The situation changes when the particle is squeezed between two flat channels, as in the middle of Fig. 2 (a). Now the asymmetric flow close to the bottom surface is balanced by the one at the top, and the spatial symmetry is restored. As a consequence, the particle is unable to propel unless small thermal fluctuations can displace the colloid from the central position making it closer to one of the two surfaces. However, even in this case, the rectification will be balanced on average. Here we show that, in contrast to previous cases, when the particle is confined between two curved walls, the effect of curvature breaks again the spatial symmetry, and induce a net particle motion, as shown in the last schematic in Fig. 2 (a).
<details>
<summary>x9.png Details</summary>
### Visual Description
## Chart/Diagram Type: Scientific Data Visualization - Particle Tracking & Diffusion
### Overview
The image presents three panels (a, b, and c) depicting data related to particle tracking and diffusion. Panel (a) shows a microscopic image of concentric rings, likely representing the experimental setup. Panel (b) is a log-log plot of Mean Squared Displacement (MSD) versus time difference (δt), colored by a parameter ρL. Panel (c) shows a scatter plot of <r²> versus ρL, with an inset time series plot of <r²> versus time (t) for different ρL values.
### Components/Axes
**Panel a:**
* Image: Concentric rings, likely a microscopic view of the experimental setup. No explicit labels.
**Panel b:**
* X-axis: δt [s] (Time difference in seconds), logarithmic scale from 10⁻¹ to 10³ seconds.
* Y-axis: <Δs²> [µm²] (Mean Squared Displacement in square micrometers), logarithmic scale from 10⁻² to 10⁴ µm².
* Colorbar: ρL (parameter value), ranging from 0.01 to 0.30, with color gradient from blue to red.
* Lines: Multiple lines, each representing a different ρL value. Two dashed lines are labeled as ~t and ~t².
**Panel c:**
* X-axis: ρL (parameter value), ranging from 0.10 to 0.30.
* Y-axis: <r²> (Mean Squared Radius), ranging from 0.0 to 1.0.
* Markers: Red circles and blue squares, representing data points.
* Inset: Time series plot of <r²> versus t [s] (time in seconds), ranging from 0 to 1000 seconds. Different colored lines represent different ρL values (0.28, 0.24, 0.20, 0.14).
### Detailed Analysis or Content Details
**Panel b:**
* The dashed line labeled ~t has a slope of approximately 1 on the log-log plot, indicating a linear relationship between <Δs²> and δt.
* The dashed line labeled ~t² has a slope of approximately 2 on the log-log plot, indicating a quadratic relationship between <Δs²> and δt.
* The colored lines representing different ρL values generally follow a trend between the ~t and ~t² lines.
* For short δt (around 10⁻¹ s), the lines are closer to the ~t line.
* For longer δt (around 10³ s), the lines are closer to the ~t² line.
* As ρL increases (from blue to red), the lines tend to shift upwards, indicating a larger MSD for a given δt.
* Approximate data points (reading from the lines, with uncertainty):
* ρL = 0.01: At δt = 1 s, <Δs²> ≈ 1 µm²
* ρL = 0.12: At δt = 1 s, <Δs²> ≈ 10 µm²
* ρL = 0.24: At δt = 1 s, <Δs²> ≈ 100 µm²
* ρL = 0.30: At δt = 1 s, <Δs²> ≈ 300 µm²
**Panel c:**
* The red circles generally show a decreasing trend, with <r²> decreasing as ρL increases.
* The blue squares show an increasing trend, with <r²> increasing as ρL increases.
* Approximate data points:
* ρL = 0.10: <r²> ≈ 0.2
* ρL = 0.15: <r²> ≈ 0.4
* ρL = 0.20: <r²> ≈ 0.6
* ρL = 0.25: <r²> ≈ 0.8
* ρL = 0.30: <r²> ≈ 0.9
* Inset: The time series plot shows that for higher ρL values (e.g., 0.28), <r²> remains relatively constant over time. For lower ρL values (e.g., 0.14), <r²> increases over time.
### Key Observations
* The MSD (<Δs²>) increases with both time difference (δt) and the parameter ρL.
* The relationship between MSD and δt transitions from being approximately linear to quadratic as δt increases.
* <r²> exhibits an inverse relationship with ρL for the red circles and a direct relationship for the blue squares.
* The inset suggests that the dynamics of <r²> are dependent on ρL, with higher ρL values leading to more stable radial positions.
### Interpretation
The data suggests that the particles are undergoing diffusion, with the MSD increasing with time as expected. The parameter ρL appears to influence the diffusion behavior. The log-log plot in panel (b) indicates a transition from ballistic motion (linear relationship between MSD and time) to diffusive motion (quadratic relationship). The different colored lines likely represent different experimental conditions or particle types, each characterized by a specific ρL value.
Panel (c) reveals a complex relationship between ρL and <r²>. The opposing trends for the red circles and blue squares suggest that ρL may be influencing the radial distribution of the particles. The inset shows that higher ρL values lead to more stable radial positions, potentially indicating that the particles are being confined or attracted to the center of the rings.
The concentric ring structure in panel (a) likely represents a confining potential or a patterned substrate that influences the particle motion. The parameter ρL could be related to the strength of this confinement or the properties of the substrate. The data suggests that by tuning ρL, one can control the diffusion behavior and radial distribution of the particles. The opposing trends in panel (c) could be due to different particle populations or different mechanisms governing their motion at different ρL values. Further investigation would be needed to fully understand the underlying physics.
</details>
Fig. 4: (a) Optical microscope image of paramagnetic colloidal particles spinning inside a circular channel with radius $R=35\,\rm{\mu m}$ . The scale bar is 10 $\mu m$ , see also Movie S2†in the Supporting Information. (b) Mean squared displacement of the circumference arc $\langle\Delta s^{2}\rangle$ as a function of the time intervals $\delta t$ for different densities $\rho_{L}$ . Left (right) plot shows the experimental data for a channel with radius $R=15\mu m$ ( $R=35\mu m$ respectively). The dashed-dotted line indicate the normal diffusion ( $\alpha=1$ ) while the dotted line shows the ballistic regime ( $\alpha=2$ ). Colorbar, legend, and axes are common in both plots. (c) Average Kuramoto order parameter $\langle r^{2}\rangle$ (Eq. (10)) versus particle density $\rho_{L}$ . Blue squares (red disks) are experimental data for a channel with radius $R=15\,\rm{\mu m}$ ( $R=35\,\rm{\mu m}$ respectively). Inset shows the temporal evolution of $\langle r^{2}\rangle$ for different particle densities, identified by the color bar. The color code of the color bar is the same as in the main plot.
To demonstrate this point, we perform a series of experiments by varying the channel radius $R∈[5,40]\,\rm{\mu m}$ and measuring the tangential velocity of the particle $v_{\phi}$ , Fig. 2 (b). We find that while for large rings ( $R>20\,\rm{\mu m}$ ) the particle does not display any net movement, $v_{\phi}\sim 0$ , below $R\sim 10\rm{\mu m}$ the effect of curvature becomes important and it can rectify the rotational motion of the particle. Indeed, for $R=5\rm{\mu m}$ the particle circulates along the channel at a non-negligible speed of $v_{\phi}\simeq 0.14\rm{\mu m\,s^{-1}}$ . The rectification process decreases by increasing $R$ , while the sense of rotation is equal to that of the precessing field, sign that the spinning particle is more strongly coupled to the bottom wall of the curved channel.
This can be also appreciated from the Video S1†in the Supporting Information, where a driven particle in the small, $R=5\,\rm{\mu m}$ ring moves clockwise, even if the velocity is not uniform. Regarding this last point, we note that due to the resolution limit of the lithographic technique, our small channels may present imperfections such as deformations and non-uniform edges. These defects may pin a rotating particle during its motion for a longer time. This effect can be observed in the inset to Fig. 2, where we have discretized space and calculated the mean tangential velocity $v_{\phi}$ for each slice of the circular trajectory. Here the time spent by the particle near $\phi=0$ is larger, and at this point, the colloidal particle is trapped by a defect, taking a long time to be released by thermal motion. To avoid this problem, we have calculated the mean velocity directly from the binned series shown in the inset, giving equal weights to all the discrete slices.
We can use the equation
$$
\displaystyle v_{\phi}=M_{v_{\phi},\tau_{z}}\tau_{z} \tag{8}
$$
to calculate the tangential velocity, where the entry of the mobility matrix, $M_{v_{\phi},\tau_{z}}$ , is calculated from finite element simulations. To match the experimental value we fit the value of $\tau_{z}$ , which is constant for all values of the curvature $R$ . We have performed a non-linear regression of the experimental data and fit the velocity profile using the torque as a single, free parameter. The calculated off-axis component of the mobility matrix $\bm{M}$ is multiplied by a constant factor, which gives the continuous line shown in Fig. 2 (b). The data shows excellent agreement with the simulation for a torque of $\tau$ = 0.55 pN $\mu$ m, which is slightly lower than the calculated value of the magnetic torque, $\tau_{m}=0.73\,\rm{pN\,\mu m}$ . We also performed simulations with the sphere center placed away from the channel centerline and closer to the inner or the outer wall. However, these simulations did not agree well with the experimental data. The small difference can be attributed to the different approximations used to calculate $\tau_{m}$ , such as the presence of a single relaxation time or the value of the magnetic volume susceptibility of the particle which is a factor that can change from one batch to the other since it depends on the chemical synthesis used to produce these particles.
To understand the origin of the propulsion, we have used the simulation to calculate the surface traction along a sphere rotating at a constant angular velocity. The traction is computed in each point on the particle surface from the stress tensor as $\bm{T}·\bm{n}$ , where $\bm{n}$ is the unit vector normal to the surface of the sphere. In Fig. 3 we are showing only the $\hat{\bm{\phi}}-$ component of the traction vector, which points in the direction of the particle’s translational motion. In Figs. 3 (b) and (c), where we show the normalized traction force $N_{\phi}$ calculated in the outer and inner hemispheres of the particle respectively, we find that the coupling of the particle surface to the outer wall pushes the particle back towards the negative $\hat{\bm{\phi}}$ -direction, corresponding to a negative angular velocity. In contrast, the interaction between the inner hemisphere of the particle, and the inner wall pushes the particle along the forward direction. The last panel, Fig. 3 (d) shows the sum of both interactions, whose integral yields the total hydrodynamic force acting along the $\hat{\bm{\phi}}$ direction. Even though there is a negative component in the center of the circle, which indicates a higher coupling to the outer wall, it is clear that the strongest contribution comes from the two positive lobes where the inner hemisphere coupling is stronger, and the total force along the $\hat{\bm{\phi}}$ axis is positive. Thus, our finite element simulations confirm that the rectification process of a single rotating particle within a curved channel results only from the hydrodynamic interactions between the particle and the surfaces.
4 Driven magnetic single file
We next explore the collective dynamics of an ensemble of interacting spinning colloids that are confined along the circular channel. Since the lateral width is $w\sim 2a$ the colloidal particles cannot pass each other, thus effectively forming a "single file" system. 44, 45. While the particle dynamics in a "passive" single file, driven only by thermal fluctuations, have been a matter of much interest in the past, 46, 47, 48 more scarce are realizations of "driven" single files 49, 50, where the particles undergo ballistic transport due to external force and collective interactions. 51, 52, 53, 54 In our case, a typical situation is shown in Fig. 4 (a), where $N=23$ particles move within a channel of radius $R=35\rm{\mu m}$ (linear density $\rho_{L}=Na/(2\pi R)=0.15$ ), see also Video S2†in the Supporting Information. The applied precessing magnetic field apart from inducing the spinning motion within the particle is also used to reduce the magnetic dipolar interaction between them. In particular, in the point dipole approximation, we can consider that the interaction potential $U_{dd}$ between two equal moments $\bm{m}_{i,j}$ at relative position $r=|\bm{r}_{i}-\bm{r}_{j}|$ is given by: $U_{dd}=\mu_{0}/4\pi\{(\bm{m}_{i}·\bm{m}_{j}/r^{3})-[3(\bm{m}_{i}·\bm{r%
})(\bm{m}_{j}·\bm{r})/r^{5}]\}$ . Here the induced moment is given by, $m=V\chi H_{0}$ , being $V=(4\pi a^{3})/3$ the particle volume. If we assume that the induced moments follow the precessing field with a cone angle $\theta$ , these interactions can be time-averaged, and become:
$$
\langle U_{dd}\rangle=\frac{\mu_{0}m^{2}}{4\pi r^{3}}P_{2}(\cos{\theta})\,\,\,, \tag{9}
$$
where $P_{2}(\cos{\theta})=(3\cos^{2}{\theta}-1)/2$ is the second order Legendre polynomial. For the field parameters used, the dipolar interaction is small since $\theta=44.5^{\circ}$ which is close to the "magic angle" ( $\theta=45.7^{\circ}$ ) where $P_{2}(\cos{\theta})\sim 0$ . From these values, we estimate that the pair repulsion is of the order of $U_{dd}=5k_{B}T$ being $T=293\,\rm{K}$ the thermodynamic temperature and $k_{B}$ the Boltzmann constant. Thus, the magnetic interactions are slightly repulsive to avoid the particles attracting each other at high density. Indeed, for a purely rotating magnetic field ( $H_{z}=0$ , $\theta=90^{\circ}$ ) the spinning particles aggregate forming compact chains that induce clogging of the channel. In contrast, to obtain a strong repulsion we would require a smaller in-plane field component that would induce a smaller magnetic torque.
Thus, we perform different experiments by varying both the particle linear density within the ring, $\rho_{L}$ , and the channel radius $R$ , while keeping fixed the applied field parameters, which are the same as in the single-particle case. From the particle positions, we measure the mean squared displacement (MSD) along the circumference arc $\Delta s$ connecting a pair of particles within the ring. Here the MSD is calculated as $\langle\Delta s^{2}(\Delta t)\rangle\equiv\langle(s(t)-s(t+\delta t))^{2}\rangle$ , being $\delta t$ the lag time and $\langle…\rangle$ a time average. We use the MSD to distinguish the dynamic regimes at the steady state. Here the MSD behaves as a power law $\langle\Delta s^{2}(\Delta t)\rangle\sim\tau^{\alpha}$ , with an exponent $\alpha=1$ for standard diffusion and $\alpha=2$ for ballistic transport, i.e. a single file of particles with constant speed. As shown in the left plot of Fig. 4 (b) for a small channel width, $R=15\,\rm{\mu m}$ , all MSDs behave similarly, showing an initial diffusive regime followed by a ballistic one at a longer time, and the transition between both dynamic regimes occurs at $\sim 10-20$ s. Increasing the particle density has a small effect in the ballistic regime, where the velocity is higher at a lower density. This effect indicates that the inter-particle interactions reduce the average speed of the single file. Since magnetic dipolar interactions are minimized by spinning the field at the "magic angle", this effect may arise from hydrodynamics. Indeed it was recently observed that these interactions may affect the circular motion of driven systems. 55, 56 In contrast, the situation changes for a larger channel, $R=35\,\rm{\mu m}$ and right plot in Fig. 4 (b). There, at low density $\rho_{L}$ the particle tangential speed vanishes, $v_{\phi}=0$ (as shown in Fig. 2 (b)), and thus we recover a diffusive regime also at long time until the slope fluctuates due to lack of statistical averages. Increasing $\rho_{L}$ we recover the transition from the diffusive to the ballistic one, but occurring at a relatively longer time, $\sim 100$ s. The emergence of a ballistic regime results from the cooperative movement of the particles which generate hydrodynamic flow fields near the surface. Indeed, while single particles display negligible translational velocity in channel of radii $R=15\rm{\mu m}$ and $35\rm{\mu m}$ (Fig. 2(a)), we observed a ballistic regime in both cases at high density. As previously observed for a linear, one-dimensional chain of rotors 42, the spinning particles generate a cooperative flow close to a bounding wall which act as a “conveyor belt effect”. Here, the situation is similar, however the larger distance between the particles due to their repulsive interactions and the presence of two walls which screen the flow reduce the average speed of the single file and increase the time required for this cooperative effect to set in.
We can estimate the transition time between both regimes for the single particle. We start by measuring the diffusion coefficient of the single particle within the channel from the MSD in the diffusive regime, $D=0.0379± 0.0002\rm{\mu m^{2}\,s^{-1}}$ . Thus, the motion of an individual magnetic particle will turn from the Brownian motion with $\langle\Delta s^{2}\rangle=2Dt$ to a ballistic one with $\langle\Delta s^{2}\rangle=(R\omega_{p}t)^{2}$ after a characteristic time $\tau_{c}=D/(R\omega_{p})^{2}$ . From the experimental data in Fig. 2 (b) we have that $\omega_{p}=0.0073\,\rm{rad\,s^{-1}}$ for $R=15\,\rm{\mu m}$ which gives $\tau_{c}=3.1$ s, similar to the small channel in the left of Fig. 4 (b).
We further analyze the relative particle displacement, and the presence of synchronized movement by measuring a Kuramoto-like 57, 58 order parameter, defined as:
$$
\langle r^{2}(t)\rangle=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\cos{(\Delta\varphi_{ij%
}(t))} \tag{10}
$$
where $N$ is the number of particles, $\Delta\varphi_{ij}=\varphi_{i}-\varphi_{j}$ the difference between the angular position of two nearest particles $i,j$ and $\varphi_{i,j}$ is measured with respect to one axis ( $\hat{\bm{x}}$ ) located in the particle plane. This order parameter is normalized such that $\langle r^{2}\rangle∈[0,1]$ , where $1$ corresponds to the fully coordinated relative movement of all spinning particles. Fig. 4 (c) show the average value of $r^{2}$ at different density $\rho_{L}$ . In particular, we find that $r^{2}$ increases with $\rho_{L}$ since, for a large number of particles the repulsive particles start to interact with each other and their translation becomes more coordinated. On the other hand, we find that $\langle r^{2}\rangle$ decreases with the radius of curvature (data from red from blue) meaning that the curvature reduces the cooperative translation between the spinning particles. The cooperative movement revealed from the Kuramoto order parameter results from the repulsive magnetic dipolar interactions. Indeed, this parameter increases with density i.e. when the particles are closer to each other and experience stronger repulsive forces. However, the emergence of self-propulsion in our system is a pure hydrodynamic effect induced by the wall curvature. Indeed, the precessing field induces reciprocal magnetic dipolar interactions which could not give rise to any net motion of the spinning particles.
Conclusions
We have demonstrated that a circular channel can be used to induce and enhance the propulsion of spinning particles while the effect becomes absent for a flat one. The propulsion appears from the different hydrodynamic coupling to each of the two curved walls; even if the no-slip condition of each wall pushes the particle along an opposite direction, the two interactions are not symmetric due to the channel curvature. The dominating interaction is the one from the inner wall, which produces a rotation along the channel along the same direction as the angular velocity. We confirm this hypothesis both via experiments and finite element simulations.
We note here that the realization of field-driven magnetic prototypes able to propel in viscous fluids is important for both fundamental and technological reasons. In the first case, the collection of self-propelled can be used as a model system to investigate fundamental aspects of non-equilibrium statistical physics. 59, 60, 61, 62, 63 On the other hand, it is expected that these prototypes can be used as controllable inclusions in microfluidic systems, 64, 65, 66, 67, 68 either to control or regulate the flow 69, 70, 71 or to transport microscopic biological or chemical cargoes. 38, 72, 73, 74, 75, 76, 77 The propulsion mechanism unveiled in this work could be used to move particles along a microfluidic device, either by designing it with a circular path, or by coupling many semicircles in a zig-zag to form a straight path.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
We thank Carolina Rodriguez-Gallo who shared the protocol for the sample preparation, and the MicroFabSpace and Microscopy Characterization Facility, Unit 7 of ICTS “NANBIOSIS” from CIBER-BBN at IBEC. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 811234). M.D.C. acknowledges funding from the Ramon y Cajal fellowship RYC2021-030948-I and the grant PID2022-139803NB-I00 funded by the MICIU/AEI/10.13039/501100011033 and by the EU under the NextGeneration EU/PRTR & FEDER programs. I. P. acknowledges support from MICINN and DURSI for financial support under Projects No. PID2021-126570NB-100 AEI/FEDER-EU and No. 2021SGR-673, respectively, and from Generalitat de Catalunya (ICREA Académia). P. T. acknowledges support from the Ministerio de Ciencia, Innovación y Universidades (grant no. PID2022- 137713NB-C21 AEI/FEDER-EU) and the Agència de Gestió d’Ajuts Universitaris i de Recerca (project 2021 SGR 00450) and the Generalitat de Catalunya (ICREA Académia).
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