Surjyasish
Mitra
a,
A.-Reum
Kim
b,
Boxin
Zhao
*b and
Sushanta K.
Mitra
*a
aDepartment of Mechanical & Mechatronics Engineering, Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail: skmitra@uwaterloo.ca
bDepartment of Chemical Engineering, Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail: zhaob@uwaterloo.ca
First published on 11th April 2025
On perfectly vertical surfaces, rolling is conventionally deemed impossible without external torque. While various species like geckos and spiders exhibit vertical locomotion, they cannot achieve rolling; instead, they fall. In this study, we demonstrate the spontaneous rolling of a soft polyacrylamide sphere on a soft polydimethylsiloxane (PDMS) substrate held vertically at a 90° incline, given specific elasticity values for the materials. Our experiments uncover a slow rolling motion induced by a dynamically changing contact diameter and a unique contact asymmetry. The advancing edge behaves like a closing crack, while the receding edge acts as an opening crack. Utilizing adhesion hysteresis theories and crack propagation models, we explain how this contact asymmetry generates the necessary torque and friction to maintain rolling, preventing either pinning or falling. The findings challenge the traditional understanding of vertical surface interactions and open new avenues for exploring soft-on-soft contact systems. This novel phenomenon has potential implications for designing advanced materials and understanding biological locomotion on vertical surfaces.
On the other extreme, a millimetric liquid water droplet will wet a perfectly vertical surface, forming a downward elongated profile and exhibiting finite contact angle hysteresis, which is the difference in contact angles between the advancing and receding edges (see Movie S2, ESI†). The droplet may either remain pinned in its original position or slide down freely while maintaining contact with the surface (see Movie S2, ESI†). Under special circumstances, for instance, on an inclined micro-nano textured superhydrophobic substrate, droplets may display irregular rolling and tumbling motion.4 The question is whether this overarching notion of inability of objects to roll on surfaces when θS = 90° still holds true when both objects and underlying surfaces are made of elastic material (non-rigid) whose elasticity can be tuned at will? At least, when the underlying substrate is soft, a rigid cylinder can sustain spontaneous rolling for an inclination angle of 90°, as first demonstrated by Barquins.5 Further works by Barquins and coworkers have elaborated on the different facets of the observed rolling motion under various normal loads and inclination angles.6,7 In this work, while exploring soft-on-soft contact/wetting systems across a broad range of both top and bottom contacting pair elasticity, we report a scenario where a soft sphere exhibits spontaneous rolling motion on a soft substrate held vertically at a 90° incline without any externally imparted torque. In the following sections, we first present our experimental findings and then unravel the governing principles of this unique phenomenon by invoking a theory based on crack propagation and adhesion hysteresis.
The rolling experiments were performed using shadowgraphy with three different settings. First, moderate magnification experiments were performed with a high-speed camera (FASTCAM AX-200, Photron) operating at acquisition rates of 50–60 frames per second and coupled with a 4× microscope objective lens and an adapter tube providing a spatial resolution of 7–8 μm per pixel (see Movies S3–S6, ESI†). These experiments provide the optimal spatio-temporal resolution and are thus used for extracting key features (advancing/receding edges, velocity, and contact angles) of the rolling dynamics. Second, higher magnification rolling experiments were performed using a similar configuration albeit with a 10× long-working distance objective lens (Optem), providing a spatial resolution of 2 μm per pixel (see Movie S7, ESI†). These recordings provide a close up of the advancing and receding edges and thus a lower field of view. However, these recordings are not used for extracting key features of the rolling dynamics. Lastly, lower magnification (1×) recordings were performed using DSA Kruss' in-built CCD camera at 10 frames per second acquisition rate and a spatial resolution of 22–25 μm/pixel (see Movies S8 and S9, ESI†). These recordings have a larger field of view and exhibits the point of dispensing and extended rolling. Again, these are not used for extracting key features of the rolling dynamics. To enhance observation of the rolling motion, we conducted additional experiments by dispersing 100 μm sized microplastics on the PDMS substrates along the rolling path. These microplastics adhered to the rolling PAAm sphere, acting as markers to help visualize the rolling motion (see Movie S6, ESI†). Note that the microplastics do not create or sustain the rolling motion in any form and the initial PDMS location where the PAAm sphere was placed contained no microplastics (see Movie S6, ESI†). Overall, we repeated the rolling experiments 15–20 times to ensure consistency.
Image processing was conducted using an edge detection algorithm in Python.9,10 Briefly, shadowgraphy rendered the PAAm sphere dark against a bright background. Consequently, we scanned each frame along the line separating the rolling PAAm sphere from the substrate. The first pixel below a threshold intensity value identified the advancing edge position when scanning from the bright background. Similarly, scanning from the opposite direction located the receding edge position. The threshold intensity typically ranged between 70 a.u. and 100 a.u., depending on the image acquisition rate and illumination conditions. Dynamic contact angles at the advancing and receding edges were determined using either a tangent fit or a first-order polynomial fit. To identify the contact asymmetry, a circle was fitted over the PAAm profile to locate its center, and the center's x-coordinate was projected onto the contact interface. Alternatively, the x-coordinate of the outermost advancing point at the central axis can be extracted using edge detection (similar to that performed for advancing/receding edges) and upon subtracting the PAAm sphere's radius, we can identify the center. For rolling experiments without microplastics as markers, a random protrusion or shape irregularity at the periphery of the rolling PAAm sphere is tracked over successive frames to determine angular velocity.
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Fig. 5 (a) Experimental snapshot during rolling of a 1 mm radius PAAm sphere (elasticity, E1 = 169.7 kPa) on a vertical PDMS substrate (elasticity, E2 = 2422 kPa). The asymmetric contact interface is highlighted using the fitted circle (white dotted line) along the center-of-mass xCOM enabling extraction of the two unequal contact radii a1 and a2. Inset shows a close up of the contact interface. Scale bar represents 0.5 mm. (b) Side-view and bottom-view schematic of the contact asymmetry. The blue arrow represents the direction of roll. (c) Static configuration of a similar 1 mm radius PAAm sphere (elasticity, E1 = 169.7 kPa) on a horizontal PDMS substrate (elasticity, E2 = 2422 kPa). The symmetric contact interface is highlighted using equal contact radius a. Scale bar represents 0.5 mm. (d) Evolution of a1 and a2 for the rolling of the PAAm elastic sphere (E1 = 169.7 kPa) on a vertical PDMS substrate (E2 = 2242 kPa), i.e., corresponding to the analysis in Fig. 4. |
As the sphere rolls down, crack propagates through the contact interface, acting like an adhesive joint. Consequently, the crack at the advancing (leading) edge continuously closes whereas the crack at the receding (trailing) edge continuously opens resulting in the observed contact asymmetry. A rough estimate of this crack opening and closing velocity can be obtained from evaluating dxR/dt (receding edge) and dxA/dt (advancing edge) whose magnitudes averaged over the rolling duration are 0.50 mm s−1 and 0.52 mm s−1, respectively (Fig. S6, ESI†). Interestingly, these velocities are approximately the same as the average velocity of the center-of-mass, i.e., vCOM,avg = 0.51 mm s−1. Higher magnification visualization of the rolling event confirms this crack opening/closing phenomenon (see Movie S7, ESI†).
The strain energy release rate associated with the crack propagation process can be expressed as, , where G1 and G2 are the energy release rates at the receding (crack opening) and advancing (crack closing) edges, respectively, and G1−G2 represents the adhesion hysteresis.16 Here, δ is the common indentation depth shared by both contact radii, a1 and a2. A natural consequence of the adhesion hysteresis is a finite torque and as a result, a finite friction force. In other words, rolling can only occur if G1−G2 > 0, i.e., the energy required for opening the crack is higher than that required to close it. Using the contact asymmetry and the differential strain energy release rates, Dominik et al. exhibited how the pressure distribution varies along the contact interface generating a torque about the y-axis (Fig. 5(b)), most pronounced at the periphery of the receding part of the contact interface.15 This torque which essentially counters mgR can be expressed as, τ = C0R(a1G1 − a2G2) (see detailed derivation in ESI† and also ref. 17). Note that in the expression of torque, the prefactor C0 can vary between 0.5 to 2 depending on the geometry of the contacting pairs. For sphere-sphere contacts, C0 = π/2. In the present study, we consider the mean of these limits, i.e., C0 = 1.25. Consequently, we can express the friction force f as f = C0(a1G1 − a2G2). Thus, the force balance reads maCOM = mg − f where aCOM is the linear acceleration of the center-of-mass.
Before proceeding with further analysis, here we make certain considerations relevant to the problem. First, typically the indentation depth δ is significant for static, horizontal contacts.2,3 However, for the present vertical contact configuration and subsequent rolling, it is negligible due to the continuous rolling motion providing negligible time for the indentation to form. Thus, we assume a1,22 ≫ Rδ. Further, so far, we have considered the static Young's modulus for describing the elastic properties of the contacting pairs (see ESI†). However, during rolling, the PAAm sphere induces contact on the underlying PDMS substrate at a finite angular velocity ωavg = 0.94 rad s−1 ≈ 0.15 Hz. Thus, we calculate the effective elasticity E* as E*|ω=0.15Hz = [(1 − ν12)/E1|ω=0.15Hz + (1 − ν22)/E2|ω=0.15Hz]−1. As a result, for the specific combination of PAAm elastic sphere and PDMS substrate which exhibits the rolling motion, we have E1|ω=0.15Hz = 405 kPa and E2|ω=0.15Hz = 2316 kPa providing E*|ω=0.15Hz = 457 kPa (see also Fig. S7, ESI†).
In Fig. 6, we reveal the variation of the experimentally calculated friction force f over the duration of rolling event. Since f is a function of a1 and a2, it also oscillates about a mean value favg = 38.7 μN during the course of rolling. Note that since vCOM ∼ 10−4 m s−1 and aCOM ∼ 10−4 m s−2, maCOM ≪ mg Thus, from the force balance expression, it is expected that f ≈ mg which is reflected in our experiments where favg = 38.7 μN ≈ mg = 39.2 μN. Thus, for rolling on a 90° incline, the average friction force is just enough to balance the weight of the rolling sphere. Conversely, the torque expression reads mgR − fR = Iα and is perfectly balanced since Iα ≪ mgR, fR. Here I = (2/5)mR2 + (1/2)mR2 is the moment of inertia, α is the angular acceleration, and Iα ∼ 10−6 μN m−1 whereas mgR, fR ∼ 10−2μN m−1.
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Fig. 7 Evolution of strain energy release rates, G1 (crack opening, receding edge) and G2 (crack closing, advancing edge) for the rolling of the PAAm elastic sphere (E1 = 169.7 kPa) on a vertical PDMS substrate (E2 = 2242 kPa), i.e., corresponding to the analysis in Fig. 5 and using the relation, ![]() |
Here, we elaborate on the role of slip in the observed rolling. To do so, we first aim to quantify the relevant slip velocity. Consequently, we first draw an analogy with conventional rigid-body rolling with slipping on rigid, horizontal surfaces. For such cases, a finite velocity is observed at the contact point, i.e., the slip velocity, commonly expressed as vslip = vCOM − ωR (Fig. 8a). Consequently the velocity at the apex of the rolling sphere typically is vCOM + ωR. However, for the present soft rolling, instead of a singular contact point, a contact region exists whose average velocity should ideally reflect the slip velocity. Previously, we have highlighted the evolution of the advancing edge, receding edge and the center-of-mass. Here, we also extract the evolution of the apex point of the rolling sphere and calculate its instantaneous translation velocity, i.e., vT = dxT/dt (Fig. 8b). For the sake of simplicity, we consider the average instantaneous velocity of the advancing edge and the receding edge, i.e., (vA + vR)/2 to be representative of the slip velocity. Upon comparison of the different relevant velocities (Fig. 8c), we observe that the average contact region velocity i.e., (vA + vR)/2 mirrors the evolution of the center-of-mass velocity, vCOM, and does not converge to the theoretical value of vCOM,avg − ωavgR (Fig. 8c). Here, note that as previously mentioned, for the experiments shown in Fig. 3, vCOM,avg = 0.51 mm s−1 and ωavgR = 0.15 mm s−1, yielding vCOM,avg − ωavgR = 0.36 mm s−1. At the same time, we observe that velocity at the apex point monotonically decreases and converges to the value of vT = 0.64 mm s−1 = vCOM,avg + ωavgR. Thus, we observe that in contrast to conventional rigid-body rolling with slipping, vCOM,avg − ωavgR does not hold in soft rolling due to a finite contact region even though vCOM,avg + ωavgR still holds due to a singular apex point. Further, it is likely that slippage here occurs intermittently. This fact can be observed from Fig. 8c, where a subtle increase in both vCOM and (vA + vR)/2 occurs at t ≈ 0.25 s indicating the possible onset of slip. Upon comparing with the contact asymmetry (Fig. 5d, t ≈ 0.25 s) and strain energy release rates (Fig. 7, t ≈ 0.25 s), we observe that roughly before the onset of slip, a2 approaches a1 and exceeds it momentarily. Thus, the corresponding G2 approaches and exceeds G1 momentarily. Upon subsequent analysis, we observe that here , where Δac is the critical displacement required to sustain steady rolling.17 Thus, Δa momentarily dropping below the critical value may have induced momentary pinning (at a time scale below the current resolution), followed by the slip phase. Furthermore, since both Δa and ΔG diminishes, friction is also observed to drop significantly at the onset of slip (Fig. 6, t ≈ 0.25 s). A similar analysis can potentially explain other instances of intermittent slippage.
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Fig. 8 (a) Schematic of rigid-body rolling with slipping on horizontal surfaces. (b) Schematic of present soft vertical rolling. Here, xA, xR, xCOM, and xT are the locations of the advancing edge, receding edge, center-of-mass, and apex point, respectively. vA = dxA/dt, vR = dxR/dt, vCOM = dxCOM/dt, and vT = dxT/dt are the respective instantaneous velocities. ω is the angular velocity. Evolution of (vA + vR)/2, vCOM, and vT for rolling of the PAAm elastic sphere (E1 = 169.7 kPa) on a vertical PDMS substrate (E2 = 2242 kPa), i.e., corresponding to the analysis in Fig. 4. |
Lastly, we highlight that multiple experiments revealed that the average center-of-mass (or even the contact edges) velocities varied in a rather small range, i.e., between 0.46 mm s−1 and 2.52 mm s−1, with occasional velocity enhancement due to intermittent slippage. Thus, highlighting the dependence of strain energy release rates and friction (or dissipation) on the relevant velocity scales is beyond the scope of the present work. To perform such an analysis, a broad variation in velocity is crucial which can only be achieved by factors like imparting external forces, varying the inclination angle, or varying the mass (or radius) of the soft sphere.
It is noteworthy to mention that apart from some precedence of rigid cylinders rolling on a perfectly vertical soft surface,5–7 existing theoretical studies19,20 have demonstrated the feasibility of liquid droplets rolling down an inclined plane, supported by experimental evidence.4,20 Given the deformability of liquid droplets and their ability to maintain contact over a finite area, it might be possible for them to roll on a vertical, defect-free, superhydrophobic surface. However, to the best of our knowledge, no experimental evidence has been reported for this phenomenon. Until such evidence is presented, soft-on-soft contact systems may be the most effective method to enable a spherical object to self-roll on a 90° incline. In our previous work, we highlighted how exploiting soft-on-soft contact systems, specifically material elasticity, bridged adhesion and wetting.8 Similarly, the present work indicates that modulating elasticity can bridge sliding and free-fall outcomes, guiding us in formulating a unifying framework to better understand motion on a vertical surface.
Finally, we emphasize that the findings of the present work extends beyond mere scientific curiosity. They offer a promising avenue towards the design of soft micro rovers for navigating unpredictable terrain during space exploration. Additionally, understanding this type of rolling motion is relevant for N-body simulations of dust aggregate compression in protoplanetary discs.21 Such aggregates consist of spherical monomers whose rolling-driven mutual interactions are crucial for understanding compression.21,22
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm01490a |
This journal is © The Royal Society of Chemistry 2025 |