 We'll bring a lot of different topics that are all related to energy, structure, dissipation, and surfaces from different areas. And we actually start with a talk given by Jin Zhao, who joins us from University of Constance, is talking about Moray pattern and collective motion, I guess. I'm Jin Zhao from University of Constance in Germany. And thanks a lot for Ariel Tosati and Ryo Vanossi and the other organizers for inviting and organizing such a great conference. It really feels great to have an in-person conference again after more than two years of pandemic. My presentation today is Moray pattern evolution couples rotational and translational friction at crystalline interfaces. In daily life when you try to push a heavy object you would intuitively do it in such a way that you rotate and translate it at the same time. Because by doing so the force required to deepen the object would be significantly smaller. Such rotation, translation, friction, coupling. Can be understood by considering these touching asperities at the interfaces. Subjected when subjected to external force or torque these touching interfaces is touching asperities will undergo in systematic rearrangement which finally leads to the rotation, translation, friction coupling. At microscopic scales the contacting interfaces are often automatically flat surfaces. These contacts exist in many nanomanipulation experiments and are also very important in the operation of nanomechanical devices. However they follow fundamentally different rules compared with the friction in our daily life. And many distinct friction phenomena has been observed such as topological kinks and anti-kinks mediated sliding. Friction duality and friction anisotropy. Superlubric sliding as well as a re-transition from a superlubric state to high friction state. In addition to rotational motion in addition to translational motion nano-objects can also or are often rotating on crystalline surfaces. This kind of rotational motion induces a rotational friction which resist the rotation of the clusters. Such rotational friction can be relevant for for example nanomechanical motors rotating on surfaces for nanomanipulations on surfaces with angular control for the creation of a trist-angle hetero structures and for surface-based catalyst. However such rotational friction has attracted much less attention compared with the translational counterpart not only due to the difficulties in applying the precise talk at the microscopic scale but also due to the difficulties in measuring the static friction, in measuring the friction torque at such small length scales. So here we implement a colloid model to study the rotational motion and the dynamics of crystalline clusters moving on top of periodic surfaces. Specifically we create colloid crystals of several tens to several thousand of particles of close packet monolayer crystals and putting these crystalline clusters on top of periodic structures prepared by photolithography. This actually mimics a finite-sized two-dimensional Franka-Kondorovar model and due to the strong particle particle interaction this model has elastic constant k which is much greater than the potential energy of the particles with the surface. The lattice spacing ratio in our experiments is a few microns such a length scale is large enough that we can easily observe and manipulate the motions of the colloidal crystals. On the other hand it's also small enough that a classic friction loss does not apply. In addition we can vary the lattice-spacian ratio by changing the substrate lattice spacing constant. These two movies here shows the dynamics of the colloidal crystals on periodic surfaces and the first one shows roughly give a rough idea of how the clusters is rotating on the periodic surface by a constant torque and the second one give a rough idea of how the cluster is moving under a simultaneous force and a torque. Now before going to show the results of rotational friction I would like to present some of my previous results first. In the previous works I apply a constant force to a colloidal crystal and let the crystal to slide across periodic surfaces. So this movie here shows the sliding motion of the colloidal crystal across the periodic surface. The force is applied on the horizontal direction. We see that the direction of motion of the crystal is in a different direction so this is called directional locking. We here plotted the orientation which is defined by this angle here and the direction of motion to that D as a function of time. We see that both angles are lock it to 19.1 degrees and for most of the time and only very occasionally they will deviate from this angle and whenever they deviate they always deviate together. So this means directional locking is lost once the cluster rotates away from a stable angle. Similar directional locking has been observed when crystalline clusters are sliding across periodic surfaces of different lattice spacings and even different symmetries. So for this cluster here we changed the substrate lattice spacing from 5.8 in previous movies to 5.4 microns here. So the direction of motion and the orientation has been changed to 13.9 degrees instead of 19.1 degree. Here when we change the substrate lattice spacing to 6.2 micrometers the orientation becomes 33 degrees while the direction of motion is 13.9 degrees. Here we changed to a square surface with our lattice spacing 5 micrometers. The orientation of the cluster is minus 3.4 degree and the direction of motion is 26.6 degrees. Here again for a square surface of 5.4 micrometer both orientation and direction is locked to 45 degrees. We see that directional locking is a robust and quite a general phenomenon for many different lattices in contact. So directional locking actually appears whenever there is a roughly partially commensurate contact which is shown by this equation here. M1A1 plus M2A2 approximately equal to N1B1 plus N2B2. The A1A2 are the primitive vectors of the colloid cluster and B1B2 are the primitive vectors of the periodic surface. M1, M2, N1 and 2 are small integers. This such an equation requires an angular alignment between the two lattices which leads to orientation locking. So this and then finally determines the orientation of the cluster with this equation here. With such an equation in the real space similarly we can construct a similar equation in reciprocal space. Here the capital A and B are primitive vectors in the reciprocal space and this relation in reciprocal space is related to the wave vectors in the potential energy landscape and finally determines the direction of motion. Here we use a triangular cluster on the triangular surface as an illustration but in fact we have generated such a theory to arbitrary lattices in contact in in this work. A direct result of the directional locking is friction and isotropy. Taking this case as an example the cluster is orientationally locked to minus 3.4 degree and the direction of motion is locked to 26.6 degrees. So here we calculated its potential energy u as a function of the cluster center of mass position xc and yc here by fixing by fixing the cluster's orientation at the minus 3.4 degrees. The dark region is the low energy region and the brighter region are the high energy regions. So we clearly identify low energy corridors along the 26.6 degree directions. It is this low energy corridor that guided the directional locking of the of the colloidal cluster. Obviously if the cluster is sliding across the low energy corridor its potential its friction will be very small and if it's sliding perpendicular to the corridor the friction will be very large. So this on the right hand side we measured the we measured the static friction of the colloidal clusters as a function of the cluster's size both in the perpendicular direction and in the parallel direction. In the perpendicular direction we see that the static friction remains more or less constant. The static friction per particle remains more or less constant while in a parallel direction the static friction per particle decreases dramatically when the cluster's size increases. This indicates a directional superlubricity at a very large cluster size. Such directional lubricity can provide a possibility of stable superlubricity in sliding and more details of such superlubricity can be found in the poster section already as already shown on Monday by our collaborators Andrea Silva. Now come back to the topic of rotational friction. The way that we realize the rotational motion is to apply a constant torque. The torque is achieved by a very fast rotating magnetic field in a sample plane with the help of two pairs of magnetic coils which generates magnetic field in the X and the Y direction with a phase shift of 90 degrees. When this magnetic field is applied to the colloidal crystals rotating on the surface without corrugation the clusters will rotate very smoothly with a constant velocity as shown in this plot here. When the magnetic field is turned off the rotational velocity is zero and when it's turned on it remains a constant value and when it's turned off again it goes immediately to zero. The mechanism of the torque is because of a phase lag between the magnetization and the rotating magnetic field. The magnetization tries to catch up with the rotating magnetic field but we'll never be able to do so. So there's a phase lag and the cross product leads to a torque which is proportioned to the square of the magnetic field because the magnetization itself is proportioned to the magnetic field. Taking a closer look at this movie here the cluster is rotating by a torque of about 3.96 piconewton micrometers. We from the movie we see that when the cluster is rotating is not aligned with the substrate lattice the rotation is relatively smooth while it's aligned with the substrate lattice spacing the rotation is stopped for a while. Such kind of intermittent rotation is clearly seen from this plot here where we plotted the orientation of the cluster as a function of time. When we apply a much larger torque of course the rotation becomes much more smoother and continuously the cluster is continuously rotating. We also systematically changed the external torque and the results is shown here. The slope of these curves gives the average rotational velocity of the cluster which is plotted as a function of the applied torque. So from the plot here we clearly see that there is a critical torque or a static friction torque below which the cluster is no longer able to rotate on the periodic surface. To understand the intermittent rotational motion here in this movie we color coded for a very large experimental cluster. We color coded all the particles in the cluster by the interaction energy with the periodic surface. Again the dark region corresponds to the low energy regions and the brighter regions corresponds to high energy regions. From the movie we clearly see that see an evolution of the moire pattern during the rotational motion. Here we plotted the average potential energy of the particles in the cluster as a function of its angle theta. We see that around zero degrees the potential energy is very low. This corresponds to this configuration here where most of the particles has very low energy and the system the cluster contains a single very large moire, a low energy moire spot. When the cluster rotates 1.6 degrees we see that the moire, the low energy moire spot is is significantly reduced in size. So this means that the potential energy increases and at this point at 1.6 degrees there are some other low energy moire spots just arrived at the outer edge of the cluster. You see these shallowly dark regions and this outside the moire spots they are just about to enter into the cluster through the edge and this is the moment when the potential energy reaches maximum and when these regions finally went into the region the potential energy becomes smaller again because they have more low energy moire spots. So this potential energy when they just arrived inside the inner edge of the cluster the potential energy reaches minimum at 2.46 degrees. Similarly when the cluster rotates 4.39 degrees a second layer of moire spots went inside the cluster and this leads to a second low energy minimum. So from this picture we see that the rotational energy profile of this cluster will strongly depends on the cluster size and its shape and its boundary is how the moire pattern and moire spots enters the edges during rotational motion and this will of course strongly influence the measured static friction torque. So here we measure the static friction torque Tau C as a function of the cluster size for three different lattice mismatchings. Unfortunately in experiments we are not able to control the shape of the cluster so that's why the data looks very noisy but we are nevertheless be able to see some clear trends. First of all the static friction will increase when the lattice become better matched and the second the static friction seems to increase with cluster size. This is particular true when the cluster size is small. For this blue for the blue data points corresponds to a very good very well matched situation with mismatch 1.1 the the the static fishing increases up to when the cluster is about 1,000 particles in size. For this red data points with the worst lattice mismatch the static friction torque increases up to about 200 particles and then it seems that it will start to decrease a little bit. For this case with lattice spacing equal to 5.3 the static friction increase up to 100 particle and then becomes almost independent of the cluster size. In experiments the maximum size we are able to achieve is around one or two thousand particles so to to know the trend at larger size we have to turn to numeric simulations. This is done by our collaborators in here by Andrea Silva. So this is the results numerical results for hexagon shape for hexagon shaped clusters with different sizes. The results agrees very well with experimental data at a smaller cluster size regime. From the results we see that at larger cluster sizes for all cases the static friction torque becomes almost constant, fluctuating a little bit but is more or less constant. So we also found similar trends for square shaped clusters and also for triangular shaped clusters. We see that the shape of clusters indeed influence the results a bit and making the data very noisy as observed in experiments where we have random shaped clusters. Interestingly when we go to circular shaped clusters we found something different. The small cluster regime agrees relatively well with the previous results but at a larger cluster size the static friction torque seems to become decreasing with the cluster size again. So this is a bit unexpected results so we decide to construct a theory to understand to confirm whether this is true or not and to understand this. To construct a theory we want to calculate the potential energy at the contact. This can be done by sum over the potential energy of all particles in the cluster but with pure numerical computing power but this is not what we want to do. Instead we want, instead of sum over all the particles from the previous animation we know that the potential energy is actually it's all about the more low energy more response. So we can instead of sum over all particles we can sum over the potential energy of every more response that falls in the cluster. So basically these are these more response to do so. So we assume that HMORA sports has a Gaussian energy density profile as shown by this equation here. The width of the Gaussian profile is proportioned to the largest spacing of the MORA spots which is determined by geometric relations with the cluster's orientation and then the energy of the MORA sports can be easily calculated by an integral of the Gaussian function. And then by some proper dealing with the boundary conditions of the cluster's edge then we can sum over all the MORA patterns inside the cluster and arrive at such a formula for the contact energy as a function of the cluster's orientation theta and its center mass position. This is quite useful because with this equation we are now able to calculate the static friction torque analytically. This plot here shows the results of the previous analytical formulation and indeed it confirms the numerically calculated results for the circular shaped clusters which shows a clear trend where the static friction torque decreases with the cluster's size with a component with a power law like this. Because we have now an analytical formulation for this tau c so we can in principle also try to overlap all these curves into a single master equation so this is how we do it. So we see that a large cluster in better matched configuration is actually equivalent to a small cluster in not very well matched contact. So now with this formula of the contact energy we are now finally able to study the rotation translation coupling. What we do is we construct the hason matrix and we calculate the determinant of the hason matrix as a function of the orientation theta and the translation x and this is the results. So this green region is where the determinant is greater than zero and this pink region is where the determinant is smaller than zero and is unstable. And in between there is a boundary which is the boundary between the stable and unstable region. From this determinant matrix we can construct the dependent boundary via these first derivatives of this generalized entropy. So the results shows that at a small torque and a force the system is pinned and cannot be moved. When the torque and the force exceeds a certain boundary the system will start to move. The theoretical curve, this orange curve, agree well with the experimental data and the simulation results and this dashed line here corresponds to the rotation translation coupling of macroscopic objects. We see that the difference is not so large but actually the mechanisms are fundamentally different. In the microscopic case when the cluster is pinned all the particles there is a large moray pattern centered in the in the cluster and when applying when you try to deping the cluster this large moray spots in the center will will have to move out of the cluster from the cluster's edge. When a torque is applied such this moray low-energy moray spawns becomes significantly reduced in size therefore the reduced size when the size of the spot become reduced it will be much easier to pull it out of the cluster's edge and therefore the the static force is reduced. So this is also the translation coupling for the case where the two surfaces are not have different large spacings and the results between the microscopic model and the macroscopic model become becomes larger. In the end I would like to thanks all my collaborators and those in Constance and in Italy who helped the project and thanks all for your attention. For sharing these beautiful results and the meaningful analysis we have almost used up our time but there's time for one or two questions and Renato. Very nice results indeed I was asking whether in the in the case of a random island so in the random cluster when you rotate you also allow for a lateral translation because in that case the moray is not symmetric with respect to the shape of the island and so the energetics might might change in that case. Well random shape of the clusters when sliding rotating. You show the energetics as a function of the rotation so I was asking if in that case you let the translation during or you force the center of mass of the island fixed during this rotation. No it's not fixed it's it can it will be able to move according to the energetics. Okay it's free. It's free yes. One more question let me ask myself you you have very precise information about the shape of your random cluster so is it not easy to define some quantitative measure for the ruggedness of the of the edge and and just see how that. Yes we have defined something to quantify the shape of the cluster that's will be related to quantifying the applied torque. Okay but is your scatter explained by the roughness? The roughness. No the ruggedness like how how how many low coordinated edges you have along this the edge of your island that could maybe explain the scatter in your in your data. That is related yes but the relation would not be quite trivial because the more a pattern is for a random shape the cluster at the edge is a bit complicated. Yeah okay thank you very much again. I would like to draw everyone