Moir\'e-pattern evolution couples rotational and translational friction at crystalline interfaces

The sliding motion of objects is typically governed by their friction with the underlying surface. Compared to translational friction, however, rotational friction has received much less attention. Here, we experimentally and theoretically study the rotational depinning and orientational dynamics of two-dimensional colloidal crystalline clusters on periodically corrugated surfaces in the presence of magnetically exerted torques. We demonstrate that the traversing of locally commensurate areas of the moir\'e pattern through the edges of clusters, which is hindered by potential barriers during cluster rotation, controls its rotational depinning. The experimentally measured depinning thresholds as a function of cluster size strikingly collapse onto a universal theoretical curve which predicts the possibility of a superlow-static-torque state for large clusters. We further reveal a cluster-size-independent rotation-translation depinning transition when lattice-matched clusters are driven jointly by a torque and a force. Our work provides guidelines to the design of nanomechanical devices that involve rotational motions on atomic surfaces.


I. INTRODUCTION
To set an object into motion typically requires a finite driving force to overcome the static friction with the surface underneath. Similarly, a finite torque must be applied to initiate a rotation. Although both effects originate from the same mechanisms, i.e. molecular adhesion and surface roughness [1,2], the simultaneous translation and rotation of macroscopic objects demonstrate a nontrivial relation between static friction forces and torques [3]. Compared to macroscopic scales, where the overall tribological behaviour is usually explained in terms of time-honored, yet phenomenological, classical laws, the possible translation-rotation frictional interplay becomes physically much more intriguing, when dealing with atomically smooth crystalline contacts at the micro-and nanoscopic scales.
These contacts appear in many nano-manipulation experiments and are crucial in microand nano-electro-mechanical systems (MEMS, NEMS) [4][5][6]. In such cases, friction strongly depends on the atomic commensurability of the surface lattices in contact [7], which generate a rich tribological behavior including stick-slip motion and superlubric translational sliding [8][9][10][11][12]. Contrary to translational nanofriction which received considerable experimental and theoretical attention during recent years, microscopic rotational friction has remained rather elusive despite being important for the reorientation dynamics and positioning of molecules and nano motors on atomic surfaces [13][14][15][16][17][18][19][20][21][22][23][24][25]. In particular, it is unclear how rotational friction couples to the translational friction at atomic scales and how this depends on the properties of the two lattices in contact. This lack of knowledge is due to the difficulty of applying well-controlled torques at nanoscopic length scales, but also results from the difficulty of the systematic variation of the lattice constant of materials. Such problems can be resolved by using micron-sized colloidal crystals sliding across patterned surfaces since torques and forces in such systems can be applied in a precise manner [26][27][28][29]. In addition, in such colloidal systems, contacts with almost arbitrary interface incommensurability can be created [30,31].
Here, we experimentally and theoretically investigate the complex rotational motion of close-packed two-dimensional (2D) colloidal clusters which interact with a triangular surface lattice in presence of a constant external torque. We observe a non-monotonic contact-size dependence of the critical torque per particle required for rotational depinning of clusters when their lattice spacing differs from that of the substrate. We also discover a size-independent depinning boundary for clusters driven by a combination of external torques and forces. Our results are in excellent agreement with a theoretical model which considers the motion-induced evolution of the moiré pattern at the interface and coarse grains the locally commensurate moiré areas to Gaussian energy-density profiles. In contrast to its linear motion, the evolution of moiré pattern during rotation displays a qualitatively different and rather complex behavior: locally commensurate areas expand or shrink continuously in size, thus crossing the edges of the clusters, which is crucial for the depinning. Interestingly, our theoretical evaluation of the rotational depinning threshold reveals a super low-static-torque state which may find use for the engineering of low-friction nanomechanical gears.

A. Experimental sample preparation and torque realization
Colloidal clusters are made from an aqueous suspension of superparamagnetic colloidal particles (diameter σ = 4.45 µm) where a small amount (0.02% in weight) of polyacrylamide (PAAm) is added. The PAAm causes strong interparticle bonds leading to rigid 2D clusters with the lattice constant a = σ which is fixed in our experiments. Owing to their fabrication process, the clusters have a broad distribution in size and shape.
As illustrated in Fig. 1(a), the clusters are interacting with a periodically corrugated substrate fabricated by photolithography. In our experiments we use four different substrate lattice spacings b = 4.4, 4.6, 4.7, 4.8 µm, producing lattice-spacing mismatches δ = |1 − a/b| = 1.1%, 3.3%, 5.3%, 7.3%, respectively. Application of a torque to the clusters is achieved by two mutually perpendicular pairs of coils [ Fig. 1(b)] which create a magnetic field with components H x = H cos ω H t and H y = H sin ω H t. This leads to rotation of the total H vector in the x − y sample plane. The frequency f H = ω H /(2π) was set to f H = 10 Hz in all measurements. The rotating H vector induces a rotating magnetization M i within each superparamagnetic colloidal particle i of a cluster. Due to a small phase lag in M i , the rotating magnetic field applies a torque Γ = | i M i × H| to the entire cluster [32,33]. This causes the cluster to rotate smoothly on top of a flat surface with an angular velocity ω = τ /[πηa 3 (1 + I)] (see Video 1 and Fig. S1 of the Supplemental Material [34]).
Here τ = Γ/N is the applied torque per particle, N the number of particles, N πηa 3  the viscous torque of the cluster rotating in a liquid with viscosity η, and I = i 3r 2 i /(N σ 2 ) a dimensionless factor which depends on the position of each particle r i relative to the rotation center. For details regarding sample preparation, cluster formation, particle tracking, and the calibration of the torque τ we refer to Appendix A and B. In the following we characterize the cluster's angular velocity ω * = (I + 1)ω = τ /(πηa 3 ), which does not depend on the clusters' size and shape.

B. Orientational cluster motion
The rotational dynamics of clusters is strongly modified in presence of a periodically patterned surface. This is illustrated in Fig. 1(c), which shows the time-dependence of the orientation θ of a cluster consisting of N = 133 particles rotating on a nearly matched surface (δ = 1.1%) under various applied torques τ . As expected, θ(t) displays an increasingly intermittent behavior for decreasing τ , due to the increasing relative influence of the substrate corrugation. The interaction with the substrate leads to plateaus around highsymmetry angles θ = 0 • , 60 • , 120 • where the rotational velocity almost vanishes [inset of Fig. 1(c)]. The intermittent orientational dynamics originates from the rapidly changing cluster-substrate interaction energy s near the high-symmetry angles. This is illustrated in Fig. 1(d). The reason for these energy oscillations are clarified in Fig. 1(e-h), reporting the local energy distribution for four snapshots of a cluster near θ = 0 • . The low-energy spots (i.e. the dark-colored regions) arrange periodically on the cluster, forming the moiré pattern of the two contacting lattices. During rotation, and notably around θ = 0 • , the low-energy moiré spots change drastically in size and spacing [35]. As a consequence, they regularly move in and out of the cluster's edge, as illustrated in Video 2 of the Supplemental Material [34]. The snapshot in Fig. 1(e) corresponds to a situation where the entire cluster is covered by a single, broad moiré spot which determines the absolute potential-energy minimum at θ = 0.06 • in Fig. 1(d). As the cluster rotates, the moiré spot shrinks and the potential energy increases. When the cluster rotates to θ = 1.60 • , the potential energy reaches a maximum in Fig. 1 On the other hand, when the cluster is much larger than the size of the low-energy spots [ Fig. 1(h)], the oscillation becomes smaller. Note that the above picture of the potential energy oscillation is valid for contacts of arbitrary δ. This is verified in Fig. S2 of the Supplemental Material [34], which shows similar potential-energy oscillations when moiré spots are crossing the edges of clusters rotating on surfaces of different δ.

C. Scaling behaviors of static friction torque
The above mentioned energy oscillation leads to a torque τ s = −∂ s /∂θ, which we refer to as the substrate torque since it is acting on the cluster by the substrate.
In combination with the constant external torque τ they determine the angular velocity ω * = (τ + τ s )/(πηa 3 ). This relation is found in good agreement with our experiments which demonstrate an approximate proportionality between ω * and τ s in the presence of a constant torque τ [ Fig. 2(a)]. To allow for a continuous cluster rotation, the applied torque τ must exceed a critical value τ c (i.e. the onset of cluster rotation), to satisfy ω * ∝ τ + τ s > 0 for all θ. To determine τ c we have gradually decreased τ and measured the average rotational velocity ω * t as a function of τ [see Fig. 2(b)]. Note that τ c is smaller for clusters on larger-δ surface. In general, τ c also depends on the cluster size. This is seen in Fig. 2 for δ = 5.3%, τ c becomes nearly independent of the cluster size for N >∼ 50. Note that the scaling behaviour of rotational friction torque observed here is very different from that of the translational friction force [36][37][38], not just because rotation and translation involve different degrees of freedom. Moreover, friction torque and friction force require different ways to measure.

III. THEORETICAL ANALYSIS
A. The analytical model The above experimental findings are well reproduced by numerical simulations of a microscopic model (Appendix C) which explicitly considers all particle-surface interactions as in Fig. 1(e-h) and which can be applied to clusters of arbitrary size and shape. In the following, we will demonstrate that the above results are quantitatively reproduced by a much D1 in Appendix D. Due to the interplay between the contacting lattices, a rotation θ of the cluster results in a rotation ψ = ψ(θ) of the moiré pattern accompanied with a shrinkage or expansion of the lattice spacing L = L(θ); a cluster translation r cm results in a translation t = t(θ, r cm ) of the moiré pattern [35]. The integration yields an analytic expression for the interaction energy per particle s as a function of θ and r cm , i.e.
Note that rotation and translation contribute separately to s (θ, r cm ) through the LJ 1 and the cosine term, respectively. The s calculated with Eq. (1) shows excellent agreement with that obtained from the microscopic model, see Fig. S3-S5 of the Supplemental Material [34].
Differentiating Eq. (1) with respect to θ we obtain the following expression for the mean substrate torque: Here J 2 () is the second-order Bessel function of the first kind. The critical torque for rotational depinning is reported in Fig. 2(c) as a function of cluster size N (solid lines) for δ = 1.1%, 3.3%, 5.3%, where θ c denotes the angle where τ s reaches its minimum for r cm = 0 (i.e. pure rotation).
B. Theoretical understandings of the cluster-size-dependence of the critical torque The above analytical results confirm the experimentally observed N 0.5 scaling at small N . In addition, Eq. (3) predicts a strict N 0.5 scaling at all cluster size N for the δ = 0 contact (see Appendix D). Such scaling results from the coherent summation of all local substrate torques τ s inside the cluster, which applies when the cluster size is smaller than a single moiré spot. This is exemplarily shown in Fig. 3 the (τ, F ) c line (solid) obtained for circular colloidal clusters on perfect crystalline surfaces systematically falls below that obtained for a macroscopic disc in contact with a uniform surface (dashed curve) [3]. The difference originates from a fundamentally distinct depinning mechanism: compared to the spatially uniform depinning of rigid macroscopic contacts [3], the depinning of our colloidal cluster depends on the rotation-and translation-induced moiré-pattern evolution as shown in Fig. 4(c-f)

IV. DISCUSSIONS
The complex depinning of torque-and force-driven colloidal clusters on crystalline surfaces as demonstrated here should be of immediate relevance for nano-manipulation experiments where e.g. atomic-force microscopes often induce not only forces but additional torques which drastically affect the depinning of nanoparticles and their translational friction [40,41]. Similar to macroscopic scales where e.g. circular-shaped clutches or end bearings are used to achieve smooth friction forces, the super-low static rotational friction state found in our work suggests that circular contacts also provide the ideal contact geometry at microscopic scales. This may be useful for the design of atomic actuators and nano-electro-mechanical-devices where low rotational friction is desired. Finally, the complex moiré pattern evolution upon cluster rotation may find use in the area of twistronics where angle-dependent variations of the electronic properties between atomically flat layers are exploited for applications [42]. Afterwards it is exposed to ultraviolet light through a photo mask that contains the cor- During this process they grow in size by collecting more and more colloids (see Fig. S8 of the Supplemental Material [34]). This process yields 2D crystalline clusters with a broad distribution of size (up to N = 1700 particles) and shape. As shown in Fig. S9 of the Supplemental Material [34], these clusters have an extremely small nearest-neighbour bond length fluctuation (0.34%) during their rotation on the periodic surfaces. This leads to a critical size of about N c = (1/0.0034) 2 ≈ 90000 particles, below which the cluster's elasticity effect can be negligible. Like in previous work [30], we obtain the positions of the colloidal particles and those of the substrate wells simultaneously by using computer microscopy as shown in Fig. S10 of the Supplemental Material [34]. This allows us to know the positions of the colloidal particles relative to the substrate wells.
One of the advantages of our colloidal model system is that we can measure the cluster's orientation in a very precise manner. Specifically this is done by measuring the average orientation of all nearest-neighbor bonds in one lattice direction. For a single bond, the error of its orientation is roughly 0.5 • , which is estimated from the uncertainty in the particle position (approximately 40 nm or 1/3 pixel size) divided by the interparticle spacing The viscous torque of a rigid colloid cluster rotating with angular velocity ω in a liquid suspension can be expressed as [32] Γ v = i (τ i + r i F i ). Here τ i = 8πη(a/2) 3 ω is the viscous torque of a single colloidal sphere rotating at angular velocity ω around its center of mass, F i = 6πη(a/2)ωr i the viscous force acting on the colloid when moving at speed ωr i in the suspension, r i the distance of particle i to the axis of rotation (the cluster's center-ofmass position for our 2D clusters), η the solvent's viscosity, and a = 4.45 µm the colloidal diameter. With the above quantities this yields Γ v = 8N πη(a/2) 3 ω + 6πη(a/2)ω i r 2 i . With the shape-dependent factor I = i 3r 2 i /(N a 2 ), this finally leads to The factor I characterizes the cluster shape.
where k depends on the magnetic susceptibility of the colloids as well as the misalignment angle between M bulk and H. The balance between magnetic and viscous torques gives where ω 0 = kH 2 /(πηa 3 ). The Video 1 and To calculate the potential energy s = V well (δr) of a colloidal particle placed at a distance δr from the center of the nearest potential well, we use the formula: sphere on the topographic surfaces as shown in Fig. S14 of the Supplemental Material [34].
Given the Eq. (C1), the potential energy per particle s as calculated in Fig. 1(d) is then the summation of the V well (δr) for all particles in the cluster divided by N . Similarly, the substrate torque τ s of a colloidal particle can be expressed as τ s = (r − r cm ) × ∇V well (δr), where r − r cm is the position of the colloidal particle relative to the center of mass of the colloidal cluster. The τ s reported in Fig. 2(a) is averaged over all the particles in the cluster.
In simulation, we describe a cluster of colloids as a rigid body with particle positions This accounts for the large dispersion of experimental data in Fig. 2(c).
Note that the rigid-contact assumption in our experiments is also well established in various real nanoscale 2D systems over a wide range of sample sizes. For example, an elastic critical length is defined in [43], below which the dislocation induced elasticity will be negligible at the contact interface. Using the experimental data in [44][45][46] and [47], the elastic critical length for MoS 2 /Graphene heterostructure and double-walled carbon nanotube is calculated to be on the order of millimeters and centimeters respectively, which are already far larger than the contact sizes in most of the relevant experiments.
To simulate the cluster depinning in the presence of external torque and force (τ, F) as in Fig. 4(b), we assume an overdamped dynamics of the rigid cluster and integrate the first-order Langevin equations of motion The equations are integrated with a time step dt = 5 · 10 −4 ms. The effective rotational and translational viscous-friction coefficients are defined as γ r = γ i r 2 i and γ t = γN , respectively, with γ = 1 fKg/ms. Since we solve first-order equations, the value of the γ does not affect the dynamic behaviour of the system. η is an uncorrelated Gaussian random variable of unit variance. respectively.
Appendix D: The analytic coarse-grained model.
To obtain an analytical form for the cluster-substrate interaction energy and torque, we resort to a coarse-grained model. Consider a cluster translation r cm = r cm R(θ d )x along direction θ d , wherex = (1, 0) is the unit vector in the x direction. This translation of the cluster yields a translation t = (r cm L/b)R(ψ + θ d )x of the moiré lattice along the direction ψ + θ d [35], where ψ is the orientation of the moiré lattice. Consider also a rotation of the cluster to an orientation θ: this leads not only to the rotation of the moiré lattice according to tan ψ = sin θ/(cos θ − ρ), but also to the shrinkage or expansion of the moiré lattice spacing according to L = bρ/ 1 + ρ 2 − 2ρ cos(θ), where ρ = a/b is the lattice-spacing ratio of the colloidal cluster and the periodic surface. We index each moiré spot by a integer pair n 1 , n 2 . The centers of the moiré spots are expressed by R n 1 ,n 2 = n 1 A 1 + n 2 A 2 + t, where We assume that the energy contribution of each moiré spot amounts to a Gaussian density profile centered at R n 1 ,n 2 , namely S n 1 ,n 2 (θ, r) = M exp − (r − R n 1 ,n 2 ) 2 /(2λ 2 L 2 ) , where  [34]. By integrating over the area of the cluster oriented at θ and translated at r cm , the interaction energy per particle is Here the Heaviside function h() cuts the moiré spots that are inside the cluster and the summation runs over all integer pairs n 1 , n 2 . The sum of the Gaussian contributions S n 1 ,n 2 can be calculated by means of the Fourier transform where III(q) = Q δ(q − Q) is the Dirac comb, Q = m 1 β 1 + m 2 β 2 a reciprocal moiré lattice vector, m 1 , m 2 integers, β 1 , β 2 primitive reciprocal moiré lattice vectors which satisfy β i · A j = 2πδ ij for i, j = 1, 2. δ ij is the Kroneker delta function. This way, Eq. (D1) can be rewritten as: Here Q = |Q| = (Kb/L) m 2 1 + m 2 2 + m 1 m 2 with K = 4π/( √ 3b), r = |r|, the scalar product Since r cm is small compared with t, we take the approximation h(R cl − |r − r cm |) ≈ h(R cl − r). The integral in Eq. (D3) becomes the , we obtain the energy The For δ = 0, the moiré spacing L has a finite maximum value L max . Therefore at cluster size R cl L max , the Bessel function's oscillation amplitude decays as (R cl /L) −0.5 . By substituting these relations into Eq. (2) and using Eq.
On the other hand, for cluster sizes R cl L max , depinning occurs at the angle θ c when the first ring of moiré spots reaches the edge of the cluster, i.e. L ≈ R cl [see Fig. 1(e,f)]. This leads to L ≈ a/[2 sin(θ c /2)], implying L sin θ c ≈ a considering that θ c is generally small (see yields τ c ∝ R cl ∝ N 0.5 . Note that L max → ∞ at δ = 0, and thus the relation τ c ∝ N 0.5 is valid at any cluster size as shown in Fig. 5(a). Notably, this τ c ∝ N 0.5 scaling matches the τ c ∝ A 0.5 law of macroscopic friction between a rotating disc of area A and a flat surface with uniform friction coefficient [48].
For the stability diagram in Fig. 4(a) we focus on the δ = 0 case at small θ. This leads to L ≈ a/θ. For simplicity, we consider only translations alongx, i.e. r cm = (x, 0), the potential energy then reads Fig. S15 of the Supplemental Material [34] reports a comparison between the energy computed in the microscopic model, and the coarse-grained model with Eq. (D5).
In the presence of an external torque τ and driving force F in the x direction, the equilibrium condition satisfies ∂A/∂θ = 0 and ∂A/∂x = 0, where A(θ, x) = s (θ, x) − θτ − xF is the generalised enthalpy. This leads to The sign of the determinant of H provides indications about the mechanical stability of the cluster and is shown by the color pattern reported in Fig. 4(a). The stability boundary is defined by det H = 0. This condition provides the critical (θ, x) c , reported as a solid line in Fig. 4(a). The corresponding (τ, F ) c reported in Fig. 4(b) are obtained by evaluating equations (D6) and (D7) at the (θ, x) c .
Appendix E: Phenomenological scaling law of static translational friction and our extension to static torsional friction.
In a recent work [49], Koren and Duerig (KD) decomposed the static translational friction force F static of a crystalline cluster (or flake) interacting with a periodic surface, as follows: Here F a = F a0 R 2β cl is the area (or bulk) contribution, F e = F e0 R γ cl is the edge (or rim) contribution, R cl ∝ √ N is the radius of the cluster, β and γ are appropriate scaling exponents. According to results of KD, area exponents β = 1 (β = 1/4) are obtained for commensurate (incommensurate) contacts. Edge exponents γ = 1 (γ = 1/2) are obtained for hexagon-shaped (circular-shaped) clusters. To extend KD's scaling law, we introduce a position-dependent scaling relation and assume circular-shaped clusters with radius R cl : f static =f a (r), r < R cl f static =f e (R cl ), r = R cl , where f a (r) = f a0 r 2β−2 is the bulk contribution, f e (R cl ) = f e0 R γ cl is the edge contribution, r < R cl is the distance from the center of the cluster. The integral F static = 2π To evaluate the static torque, and thus rotational friction, we construct the following integral, by multiplying the corresponding position-dependent force in the integrand by the appropriate force arm, namely Γ static = 2π where τ a and τ e are constants. Even though obtained by assuming circular-shaped clusters, Eq. (E3) agrees very well with the observed scaling relation even for clusters of different shapes, with the same exponents as obtained by KD. In a lattice-matched contact (δ = 0) where β = 1, the area contribution dominates Eq. (E3), yielding τ c ∝ N 0.5 regardless of the shape of the cluster. This recovers the small-N scaling we observe in Fig. 2(c).