2409.07661v1

# \makeFNbottom <details> <summary>x1.png Details</summary>  ### Visual Description ## Chart/Diagram Type: None (Solid Gray Square) ### Overview The image is a completely blank, uniform gray square with no discernible textual, graphical, or structural elements. There are no visible labels, axes, legends, data points, or diagrams. ### Components/Axes - **No axes, labels, or legends** are present. - **No data series, categories, or sub-categories** are identifiable. - **No textual content** (e.g., titles, annotations, or descriptions) is visible. ### Detailed Analysis - The image lacks any numerical values, trends, or distributions. - No components (e.g., lines, bars, heatmap cells) exist to analyze. - No spatial relationships or flow directions can be inferred. ### Key Observations - The absence of content suggests the image may be a placeholder, corrupted file, or intentionally minimalist design. - No actionable data or interpretive insights can be derived from the image. ### Interpretation This image provides no factual or analytical information. It does not represent a chart, diagram, or textual document. The uniformity of the gray square implies either a technical failure (e.g., missing content) or a deliberate omission of visual elements. Without additional context, no meaningful conclusions can be drawn. </details> <details> <summary>x2.png Details</summary>  ### Visual Description ## Chart/Diagram Type: None (Black Image) ### Overview The image is entirely black with no discernible elements, text, or graphical components. No axes, legends, data points, or labels are visible. ### Components/Axes - **No axes, titles, or scales** are present. - **No legends, labels, or categorical markers** can be identified. - **No data series, lines, or markers** are visible. ### Detailed Analysis - **No textual content** (e.g., axis labels, legends, annotations) is detectable. - **No numerical values, trends, or distributions** are present. - **No spatial grounding** (e.g., legend placement, axis alignment) is applicable. ### Key Observations - The image contains **zero extractable information** based on the provided criteria. - No elements align with the requirements for charts, diagrams, or data tables. ### Interpretation The absence of visible content suggests either: 1. A technical failure (e.g., corrupted file, blank placeholder). 2. Intentional omission of data for privacy, placeholder purposes, or illustrative use. 3. A test case to validate extraction protocols for empty inputs. No actionable insights or conclusions can be derived from this image. Further investigation into the source or context of the image is recommended. </details> <details> <summary>x3.png Details</summary>  ### Visual Description ## Text Graphic: "ARTICLE TYPE" Label ### Overview The image is a minimalist horizontal graphic featuring a single text element. It consists of a solid blue background with white uppercase text reading "ARTICLE TYPE" positioned centrally on the left side. No additional elements, annotations, or contextual details are present. ### Components/Axes - **Text Element**: - Content: "ARTICLE TYPE" (all uppercase, sans-serif font) - Positioning: Centered horizontally on the left third of the image - Color: White (#FFFFFF) - Background: Solid blue (#0000FF or similar) - **No axes, legends, or numerical scales are present.** ### Detailed Analysis - The text "ARTICLE TYPE" is the sole focus of the image. Its placement suggests it may serve as a header, label, or categorical identifier for a section or data grouping in a larger document or interface. - The absence of additional text, graphics, or interactive elements implies this is either a standalone placeholder or part of a modular design system. ### Key Observations - The text is unambiguous and lacks contextual modifiers (e.g., no subtext, icons, or formatting variations). - The blue background provides high contrast with the white text, ensuring readability. - No spatial relationships or alignments with other elements exist due to the image's simplicity. ### Interpretation This graphic likely functions as a categorical label or section header in a technical or informational context. Its simplicity suggests it could be part of a standardized template for organizing content, such as in a knowledge base, documentation portal, or data visualization dashboard. The lack of additional details prevents further analysis of trends, relationships, or anomalies. The phrase "ARTICLE TYPE" may imply categorization of content (e.g., blog posts, research papers, news articles) but requires external context to confirm its specific application. </details> | <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 ## Screenshot: Form Submission Interface ### Overview The image depicts a minimalistic form interface with three labeled fields: "Received Date," "Accepted Date," and "DOI." The layout is text-based, with no graphical elements, charts, or diagrams. The fields are arranged vertically, with labels aligned to the left and input areas below them. ### Components/Axes - **Labels**: - "Received Date" (top-left position) - "Accepted Date" (directly below "Received Date") - "DOI:" (below "Accepted Date") - **Input Fields**: - "Received Date" and "Accepted Date" fields are empty, indicating placeholder positions for date entries. - "DOI:" field contains a placeholder value: `00.0000/xxxxxxxxxx`. ### Content Details - **DOI Placeholder**: - The DOI format follows the standard structure: `prefix/suffix`. - Prefix: `00.0000` (likely a placeholder for an institutional or publisher identifier). - Suffix: `xxxxxxxxxx` (masked with "X" characters, suggesting sensitive or redacted information). - **Date Fields**: - No dates are populated, implying the form is either a template or awaiting user input. ### Key Observations 1. The form is designed for tracking submission and acceptance dates of a document or work. 2. The DOI field is pre-formatted but contains no valid identifier, only placeholders. 3. The absence of dates suggests the form is not yet completed. ### Interpretation This interface likely belongs to an academic or institutional submission system where authors or reviewers input metadata for a document. The DOI placeholder indicates the system may auto-generate or validate DOIs upon submission. The lack of populated dates implies the form is either a draft, a template, or part of a workflow where dates are populated later (e.g., upon receipt or acceptance). The use of "xxxxxxxxxx" for the DOI suffix suggests privacy measures or redaction for demonstration purposes. No numerical data, trends, or anomalies are present in this image. The focus is on textual structure and placeholder content. </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 ## Diagram: Rotational Dynamics and Magnetic Field Interaction ### Overview The image consists of two parts: - **a**: A circular interference pattern with labeled axes (x, y) and angular displacement (φ). Red and blue arrows indicate rotational directions. - **b**: A 3D schematic of a conical structure with vectors (Ω, H, ω) and an inset graph showing height (h) vs. radial distance (ρ). ### Components/Axes #### Part a: - **Axes**: - x-axis (horizontal) and y-axis (vertical) labeled with standard Cartesian coordinates. - Angular displacement φ (phi) marked near the origin. - **Arrows**: - Red arrow: Clockwise rotation (↔). - Blue arrow: Counter-clockwise rotation (↔). - **Scale bar**: 10 µm (bottom-right corner). #### Part b: - **3D Diagram**: - Conical structure with apex labeled Ω (angular velocity, red vector). - Blue vector H (magnetic field) intersecting the cone. - Gray vector ω (angular displacement) on the right. - **Inset Graph**: - Axes: - Radial distance ρ (µm) on the x-axis (3.5–6.5 µm). - Height h (µm) on the y-axis (-4 to 0 µm). - Blue line: Represents a measured profile with a dip at ρ ≈ 5 µm. - **Legend**: - Red: Ω (angular velocity). - Blue: H (magnetic field). - Gray: ω (angular displacement). ### Detailed Analysis #### Part a: - The circular pattern suggests wave interference or rotational motion. - Red and blue arrows indicate opposing rotational directions (clockwise vs. counter-clockwise). - φ (angular displacement) is centered at the origin, with axes x and y defining the plane. #### Part b: - **3D Diagram**: - Ω (red) points upward along the z-axis, indicating rotational motion. - H (blue) is perpendicular to Ω, suggesting a magnetic field aligned with the y-axis. - ω (gray) represents angular displacement, oriented tangentially. - **Inset Graph**: - The blue line shows a parabolic trend with a minimum at ρ ≈ 5 µm (h ≈ -3 µm). - Scale bar: 10 µm (bottom-right corner). ### Key Observations 1. **Part a**: - Opposing rotational directions (red vs. blue arrows) may represent competing forces or wave polarizations. - The scale bar (10 µm) provides spatial context for the interference pattern. 2. **Part b**: - The inset graph’s dip at ρ ≈ 5 µm suggests a localized minimum in height, possibly due to resonance or field interaction. - The 3D vectors (Ω, H, ω) imply a relationship between rotational motion, magnetic fields, and angular displacement. ### Interpretation - **Part a**: The interference pattern and rotational arrows likely model a system with competing rotational dynamics, such as vortices or polarized waves. - **Part b**: The 3D diagram and inset graph suggest a physical system where angular velocity (Ω) and magnetic fields (H) interact to produce a measurable height profile (h) as a function of radial distance (ρ). The dip at ρ ≈ 5 µm could indicate a critical interaction point (e.g., magnetic trapping or resonance). - **Cross-Referencing**: The legend in part b confirms color coding for vectors, ensuring alignment with the 3D model and inset graph. ### Uncertainties - Approximate values (e.g., ρ ≈ 5 µm, h ≈ -3 µm) are inferred from the inset graph’s scale and trend. - The exact nature of the rotational motion (e.g., fluid dynamics vs. electromagnetic) is not explicitly stated. </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 ## Diagram: Sphere Motion and Velocity Analysis ### Overview The image consists of two primary components: 1. **Three schematic diagrams** (top section) illustrating a sphere's motion and rotation in different orientations. 2. **A graph** (bottom section) showing the relationship between velocity (\(v_\phi\)) and radius (\(R\)), with an inset graph detailing angular dependence. --- ### Components/Axes #### Top Diagrams (a) - **Diagram 1 (Top-Left)**: - Sphere with arrows indicating motion in the \(x\)-direction (rightward green arrow) and rotation about the \(y\)-axis (clockwise). - Labels: \(x\), \(y\), and a green arrow labeled \(\rightarrow\). - **Diagram 2 (Top-Middle)**: - Sphere with arrows in \(x\)- and \(y\)-directions (horizontal and vertical black arrows). - Green arrow labeled \(\leftarrow\) (leftward). - **Diagram 3 (Top-Right)**: - Sphere rotating about a vertical axis (green arrow labeled \(\phi\), counterclockwise). - Labels: \(\phi\) (angle in radians) and \(R = 5\ \mu m\). #### Bottom Graph (b) - **Main Graph**: - **X-axis**: Radius \(R\) (in \(\mu m\)), ranging from 10 to 40. - **Y-axis**: Velocity \(v_\phi\) (in \(\mu m\ s^{-1}\)), ranging from 0 to 0.15. - **Data**: Blue line with error bars showing a decreasing trend. - **Legend**: Blue line labeled \(v_\phi\). - **Inset Graph**: - **X-axis**: Angle \(\phi\) (in radians), ranging from \(-\pi\) to \(\pi\). - **Y-axis**: Velocity \(v_\phi\) (in \(\mu m\ s^{-1}\)). - **Data**: Blue points with error bars, showing minimal variation except near \(\phi = \pm \pi/2\). - **Text**: \(R = 5\ \mu m\) explicitly labeled. --- ### Detailed Analysis #### Main Graph Trends - The blue line decreases monotonically as \(R\) increases, with \(v_\phi\) dropping from ~0.15 \(\mu m\ s^{-1}\) at \(R = 10\ \mu m\) to ~0.00 \(\mu m\ s^{-1}\) at \(R = 40\ \mu m\). - Error bars are smallest at \(R = 10\ \mu m\) and largest at \(R = 40\ \mu m\), suggesting increasing uncertainty with larger radii. #### Inset Graph Trends - \(v_\phi\) remains nearly constant (~0.00–0.05 \(\mu m\ s^{-1}\)) across \(\phi \in [-\pi, \pi]\), except for a slight dip near \(\phi = \pm \pi/2\). - Error bars are consistent across \(\phi\), indicating stable measurements. --- ### Key Observations 1. **Radius Dependence**: \(v_\phi\) inversely correlates with \(R\), decreasing sharply for smaller radii and plateauing at larger radii. 2. **Angular Independence**: \(v_\phi\) is largely unaffected by \(\phi\), except near \(\phi = \pm \pi/2\), where minor deviations occur. 3. **Diagram Consistency**: The green arrows in the top diagrams align with the velocity direction (\(v_\phi\)) in the graph, confirming directional relationships. --- ### Interpretation - The data suggests that the sphere's rotational velocity (\(v_\phi\)) is governed by its radius \(R\), with smaller radii enabling higher velocities. This could reflect viscous drag or geometric constraints in the system. - The angular independence of \(v_\phi\) implies that the motion is not sensitive to orientation, except at extreme angles (\(\phi = \pm \pi/2\)), where external factors (e.g., boundary interactions) may play a role. - The inset graph’s flat trend reinforces that \(v_\phi\) is primarily a function of \(R\), not \(\phi\). - The error bars in the main graph highlight measurement limitations at larger radii, potentially due to experimental resolution or environmental variability. --- **Note**: All textual labels, axis titles, and numerical values are extracted as shown. No non-English text is present. </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 ## Diagram: Stress Distribution in Hemispherical Geometry ### Overview The image presents a technical diagram of a hemispherical geometry with a ball interacting with its surface, accompanied by three heatmaps visualizing stress distribution. Diagram (a) illustrates the physical setup, while (b)-(d) show quantitative stress maps with color-coded scales. ### Components/Axes **Diagram (a):** - **Geometry**: Cylindrical container with: - Radius **R** (horizontal dimension) - Height **h** (vertical dimension) - Width **w** (depth dimension) - **Key elements**: - Ball labeled **in** (incoming) and **out** (outgoing) - Coordinate system: - **ẑ** (vertical axis) - **φ** (azimuthal angle) **Heatmaps (b)-(d):** - **Axes**: - **φ** (horizontal axis, 0° to 360°) - **ẑ** (vertical axis, -h/2 to h/2) - **Color scales**: - **b (Outer hemisphere)**: -5 to +5 (N_φ) - **c (Inner hemisphere)**: -2.5 to +2.5 (N_φ) - **d (Total traction)**: -2.5 to +2.5 (N_φ) ### Detailed Analysis **Diagram (a):** - Ball positioned at the bottom-right corner of the container - Dimensions labeled with approximate values (R, w, h) but no numerical scale provided **Heatmap (b) - Outer Hemisphere:** - **Color distribution**: - Dark red center (φ=0°, ẑ=0) - Gradual transition to blue at φ=180° - Symmetric pattern around ẑ=0 - **Key values**: - Maximum stress: ~+5 N_φ (center) - Minimum stress: ~-5 N_φ (φ=180°) **Heatmap (c) - Inner Hemisphere:** - **Color distribution**: - Dark blue center (φ=0°, ẑ=0) - Gradual transition to light blue at φ=180° - Symmetric pattern around ẑ=0 - **Key values**: - Maximum stress: ~-2.5 N_φ (center) - Minimum stress: ~+2.5 N_φ (φ=180°) **Heatmap (d) - Total Traction:** - **Color distribution**: - Central orange stripe (φ=0°) - Blue regions at φ=90° and φ=270° - Symmetric pattern around ẑ=0 - **Key values**: - Maximum positive stress: ~+2.5 N_φ (φ=0°) - Maximum negative stress: ~-2.5 N_φ (φ=180°) ### Key Observations 1. **Stress Cancellation**: Total traction (d) shows opposing stress regions canceling each other, creating a neutral zone at φ=90°/270° 2. **Hemispherical Symmetry**: All maps exhibit perfect φ-symmetry 3. **Scale Consistency**: Color scales use identical N_φ units across all heatmaps 4. **Stress Magnitude**: Outer hemisphere experiences twice the stress magnitude of inner hemisphere ### Interpretation The data suggests a mechanical interaction between the ball and hemispherical surfaces, likely modeling: - **Contact mechanics**: Stress concentration at the point of contact (φ=0°) - **Force distribution**: Outer hemisphere bears compressive stresses (+N_φ), while inner hemisphere experiences tensile stresses (-N_φ) - **Equilibrium state**: Total traction map (d) indicates balanced forces with minimal net stress at φ=90°/270° The hemispherical geometry's curvature creates stress concentration at the contact point, with the inner surface experiencing opposite stress polarity due to material deformation. The total traction map reveals how these opposing stresses partially cancel, leaving residual stress patterns that could inform material failure analysis or contact mechanics studies. </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 ## Image Analysis: Microscopic and Temporal Dynamics of a System ### Overview The image consists of three distinct sections (a, b, c) depicting experimental data related to a physical system. Section **a** shows a grayscale microscopic image of concentric rings with a central bright spot and peripheral dots. Sections **b** and **c** present time-dependent and parameter-dependent analyses of the system's behavior, including mean squared displacements (⟨Δs²⟩, ⟨r²⟩) and their relationships to time (δt) and a parameter (ρ_L). --- ### Components/Axes #### **a** (Microscopic Image) - **Structure**: Concentric rings with a central bright spot and smaller peripheral dots. - **Scale Bar**: 10 µm (bottom right corner). - **Color**: Grayscale (no explicit color legend). #### **b** (Time-Dependent Analysis) - **Left Graph**: - **Y-axis**: ⟨Δs²⟩ (mean squared displacement) in [μm²], log scale (10⁻⁴ to 10³). - **X-axis**: δt (time interval) in [s], log scale (10⁻¹ to 10³). - **Lines**: - Green dashed: ~t (linear time dependence). - Black dotted: ~t² (quadratic time dependence). - Colored lines: Correspond to different ρ_L values (see color bar). - **Color Bar**: ρ_L (0.01 to 0.30), with a gradient from blue (low) to red (high). - **Right Graph**: - **Y-axis**: ⟨r²⟩ (mean squared displacement) in [μm²], log scale (10⁻² to 10³). - **X-axis**: δt (time interval) in [s], log scale (10⁻¹ to 10³). - **Lines**: Same as left graph but with different color coding for ρ_L. #### **c** (Parameter-Dependent Analysis) - **Main Plot**: - **Y-axis**: ⟨r²⟩ (mean squared displacement) in [μm²], linear scale (0.0 to 1.0). - **X-axis**: ρ_L (parameter), linear scale (0.10 to 0.30). - **Data Points**: - Red circles: Higher ρ_L values (e.g., 0.24, 0.28). - Blue squares: Lower ρ_L values (e.g., 0.18, 0.20). - **Inset**: - **Y-axis**: ⟨r²⟩ (mean squared displacement) in [μm²], linear scale (0.0 to 1.0). - **X-axis**: t (time) in [s], linear scale (0 to 1000). - **Lines**: Different ρ_L values (e.g., 0.18, 0.20, 0.24, 0.28), with colors matching the main plot. --- ### Detailed Analysis #### **a** (Microscopic Image) - The concentric rings suggest a radial symmetry, possibly indicating a vortex or rotational structure. The central bright spot may represent a core region, while the peripheral dots could be particles or features in the outer layers. The scale bar (10 µm) provides spatial context. #### **b** (Time-Dependent Analysis) - **Left Graph**: - ⟨Δs²⟩ increases with δt, following both linear (~t) and quadratic (~t²) trends. The colored lines (for different ρ_L) show varying slopes, indicating ρ_L modulates the diffusion behavior. - **Key Observation**: At short δt, ⟨Δs²⟩ aligns with ~t² (diffusive behavior), but at longer δt, deviations suggest anomalous diffusion or saturation. - **Right Graph**: - ⟨r²⟩ also increases with δt, but the colored lines (ρ_L) show distinct trends. For example, higher ρ_L values (red lines) exhibit slower growth compared to lower ρ_L (blue lines). - **Key Observation**: The divergence between ⟨Δs²⟩ and ⟨r²⟩ suggests different spatial scales or mechanisms governing the system's dynamics. #### **c** (Parameter-Dependent Analysis) - **Main Plot**: - ⟨r²⟩ decreases with increasing ρ_L, indicating that higher ρ_L suppresses the system's spatial spread. The red circles (high ρ_L) and blue squares (low ρ_L) confirm this inverse relationship. - **Inset**: - ⟨r²⟩ approaches a plateau over time for all ρ_L values, suggesting the system reaches a steady state. The inset lines (e.g., 0.18, 0.28) show that higher ρ_L values stabilize faster. --- ### Key Observations 1. **Time Scaling**: - ⟨Δs²⟩ and ⟨r²⟩ exhibit diffusive behavior (∼t²) at short times but deviate at longer times, possibly due to boundary effects or non-linear interactions. 2. **Parameter Dependence**: - ρ_L inversely correlates with ⟨r²⟩, implying it acts as a damping or confinement parameter. 3. **Steady-State Behavior**: - The inset in **c** shows ⟨r²⟩ stabilizing over time, indicating the system transitions from transient to equilibrium dynamics. --- ### Interpretation - **Physical Significance**: - The concentric rings in **a** may represent a vortex or rotational flow, with the central spot as a core and peripheral dots as tracer particles. The time-dependent analysis (**b**) reveals how these particles diffuse, with ρ_L modulating the diffusion coefficient. - The inverse relationship between ρ_L and ⟨r²⟩ (**c**) suggests ρ_L could represent a viscosity, confinement strength, or interaction parameter. Higher ρ_L restricts particle motion, reducing ⟨r²⟩. - **Anomalies**: - Deviations from ∼t² in **b** may indicate non-Gaussian diffusion or external forces. The inset in **c** shows that ⟨r²⟩ plateaus, which could reflect a balance between diffusion and confinement. - **Implications**: - The data highlights the interplay between time, spatial scale, and system parameters. Understanding ρ_L's role could inform control strategies for similar systems (e.g., microfluidic devices, biological processes). --- ### Final Notes - All labels, axis titles, and legends were extracted with spatial grounding (e.g., color bar position, legend placement). - Trends were verified by cross-referencing line colors with legends and comparing numerical data to visual slopes. - The image provides empirical evidence of time-dependent and parameter-dependent dynamics, critical for modeling and optimizing the system. </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. 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