 Okay, good. Yeah, so good morning. So you are part of the crowd of the early birds to make it here at nine. So it's a pleasure to welcome you here. Of course, my thanks also to the organizers to provide me with the opportunity to address this nice audience and to meet friends and colleagues here in Dresden at a nice time of the year. So that is really a pleasure. And now early on I also would like to acknowledge the collaborations I enjoyed with a couple of colleagues in the past as well as on-going and the topic of my talk is about capillary forces on collides. So actually it was not our intention to study systems with long-range interactions. So we were interested in two-dimensional structures, and then it turned out in the course of time that the particularly long range of the interactions in these kind of systems fits in the scheme of this workshop. Now as a sort of introduction I would like to say that we are used to the fact that condensed matter and its structures sensitively depend on spatial dimension, and therefore there is a particular interest to study systems which are quasi exactly two-dimensional in character. And this is accomplished by colloidal particles which should deposit at fluid interfaces. So now that's better. Okay, so I have now to juggle with these, so at least as long as they are not three pieces I try to manage. Yeah, so the very fact I want to say is that the binding energy of these collides to the interface is extremely strong. It's typically in the order of 10 to 6 KBT and therefore once such a colloid gets at the interface it stays there basically forever. And these structures have both an interest in basic research as well as in far as applications are concerned. So a famous example is a study of two-dimensional melting. There has been quite a theoretical effort in the past, so it led to the cost of its soulless transition and it was the first time when it was observed absolutely clearly was in terms of colloidal particles at water-air interfaces. So here you see it at low temperatures rather ordered and then you undergo a melting and this has been analyzed so very great detail and universal features have been detected. So that was an early really accomplishment in dealing with two-dimensional systems. And the other interesting aspect is that these colloidal particles can also form clusters, ordered and now you can fix them on a solid substrate by dipping it into the water and pulling it out again and then you have these kind of clusters fixed at the solid substrates and that again is interesting for optical applying applications and scattering and so on. So that's actually the background of it. And now the further advantage of these systems is that basically with your naked eye you are able to monitor what's going on. So you do not have extraordinary experimental efforts to monitor what's going on. You basically see it. Does it now work? It's fantastic. And I have two ten minutes in addition of course. So we make a deal in that direction. Yeah, and so if you now look at them these are examples and here you see also length scales involved and actually I very much like this app upper panel here. So you basically get everything there. You get monomers. So individual ones these are here. You get timers for example here. You get trimmers over here and it just keeps going. You see five, six, seven and so on. So this is I think a particularly nice example of what you see and obviously there are attractive interactions. So there is nothing which presses these collides towards a cluster but it's intrinsic interaction among them. And here you see then a bit on a larger scale how beautiful kind of structures emerge there. Here for example in that panel there we have a mixture of three types of collides between one, three and five micrometer in size and then you see for example depletion areas appearing there and also here and here so you have then an unbelievable wealth of phenomena you can observe in these kind of systems. The upper panel here shows just the accuracy with which you can accomplish these ordered structures. So this is like drilled but it's actually self-organized. And here you have it in even larger scales so that you really can look at them. And as soon as these particles deviate from spherical symmetry become elongated particles then the type of structures is unbelievable rich. So this is what you get here if you have then ellipsoids instead of spheres. And you can imagine that self-assembly with these kind of elements gives you again additional degrees of freedom. And here you have the interaction of rock-like particles with a micro post so that you can also locate them in space at designated areas. Well now the kind of interactions which are acting on these particles are capillary forces and they basically come about by the deformation of this fluid interface. And this is described by a displacement field I called U and what happens is that there is a pressure acting a locally varying pressure acting on this interface and the interface asses by changing its shape. And so what you actually want to know is provided you have given such kind of a lateral pressure what is the kind of structure of the interface which will appear. Now in most of the interesting cases the amplitudes of these deformations are pretty small. So that you have an approximation that the gradient the lateral gradient of this you mention as quantity and it's usually much smaller than one. Which means that then the differential equation which describes the shape of this interface is here given by the Young Laplace equation. It basically says that the system tries to minimize the surface area of this interface. And the surface area is square root of one plus gradient U squared and you expand it and then you end up here with the Young Laplace equation. And this equation contains the external pressure you apply and here what enters is the capillary length which is determined by the surface tension and the mass density contrast between the two coexisting phases. Such that here this acts like a mass in this Laplace equation. The important aspect is that this capillary length is at your experimental disposal because you can modify the strength of the surface tension for example by changing temperature or you can add surfactants to it and then you have a huge variation in the strength of the surface tension. That means this object here is a knob you can play with. This can also be described in terms of lateral forces acting on such kind of a part of the interface and this force is actually given by the curvature integral around here the three-phase contact line and here this is expressed in terms of the displacement field. The capillary length typically is an object which is rather large. So it's typically in the order of a millimeter and larger but typically it's large compared to the size of the coides which should deposit at this interface. Now let's start first of all to describe the system in terms of capillary forces acting on a single co-ride. So we just want to know what happened to a single one and this is shown here on the left-hand side. So this is the Laplace equation. I just mentioned before and here is the description in terms of the force and now what is very striking is that the mathematics of it just tells you that you have a nice analogy between gravitational and electrostatic phenomena. So here on the left-hand side we have our interface problem with the deformation and here this corresponds to the situation of gravitational potential. So this U plays the role of the gravitational potential. The capillary length is like a screening length for the electrostatic. Density of the vertical force pi entering here plays the role of a mass density up to a sign. The vertical force with which you can pull these collides is minus the mass and you encounter here the particle interface contact line and this is a generator for multipolar moments as indicated here. So what we should keep in mind is now that these effective capillary interactions lead to the same problem as a screened gravity, but in 2D. And that is something which to say our friends from cosmology, they cannot modify by hand the gravitational potential. It's there at all scales, but here in our case we are able by changing the capillary length we have a tool to have a screening length there which we can shift force and back. Now once we have understood the single particle, let's now study two collides and so the phenomena is as follows. That here you have a single particle which generates a deformation of this interface. A second particle does that the same and these displacements superimpose and give rise to a mutual attraction. And here I have indicated a situation in which for example by external means for example by an electric field you can pull the colloidal particle. So it's a bit alone pulled out of the water. This is then a capillary monopole corresponding to the vertical force. The capillary force between the two is then the derivative of the potential energy for this configuration. And you can solve this and this is just given in terms of a Bessel function which depends on the distance between these two particles in units of the capillary length. And the amplitude is set by the force with which you pull and in the denominator we have the surface tension. Here it's plotted this capillary interaction and the properties are such that first of all over a certain range this potential decays logarithmically that is very slowly that is the 2D version of gravity and then finally however at a crossover given set by the capillary length this crosses over to an ultimate exponential decay governed by the decay length set by the capillary length. And this is sort to say the key element that now you have a means to decrease or increase the range of the interaction. Now typically we have inter-particle separations of the order of 10 to 100 micrometer the capillary length being about one millimeter. That means that the so-called plasma parameter which counts the number of interacting neighbors is given by this ratio and typically this is much larger than one and that means that you are there really probing the long range interaction. And here I again emphasize that the quantity is lambda the capillary length the external force the surface tension the size of the particle and the nearest neighbor interaction distance is all easily tunable experimental parameters. Now we had almost one particle two particles now let's study several particles this is shown up here. So this pressure I now idealize by having point like particles which are pulled out of the water. And then this displacement field is given by this logarithmic interaction. And here again I have written down the potential which corresponds to this distance and we have then the gravitational potential. What I meant in the introduction was that this trapping energy is very strong but it depends sensitively on the size of the particles. Namely this pulling force is proportional to the cube of the radius of the colloidal particle. This enters in the potential quadratically which means that the potential depends on the six power of the radius of the particle. So that changing the size of the particle has a dramatic effect on these potentials. Now here I introduce you to the fact that you have not only mono poles and then in the next stage dipoles but you have all higher dipole moments which are generated typically by having non-sphere particles at play. And this is displaced here where I have put on an ellipsoid deposited on this interface. And the first observation is that this colloidal particle sinks into the interface such as to accomplish the pre-scribed contact angle of the liquid vapor interface forming with the surface of this colloidal particle. And these contact angles they are just determined in terms of surface tension so they are materials parameters and so if you choose a certain type of material you get a certain contact angle and then you want to like to know what is the shape of the three phase contact line which appears. And that turns out that if you are deviating from a spherical object the three phase contact line no longer lives in a two dimensional plane but it has to escape out of the plane that is the three phase contact line is no longer a planar object. And this is indicated here so you see that the interface here bulges up and this means that you generate higher aura multiples of the interaction as shown here. And of course the theoretical description becomes more complicated and this also influences the range of the interactions and it also then depends on whether these particles are tipped to tail or whether they are parallel to each other and there is now also some experimental evidence which is compatible with the theoretical predictions. Now if you can now spice up the problem by introducing charges to the problem as indicated up here which means then that it is particularly responsive to external electric fields in this sense that the electric field which emanates from these charges gets out of the liquid and puts on a pressure on the liquid interface and deforms it. And this is here indicated as a reference configuration up here you have just a sphere accomplishing here the contact angle and it's an undeformed interface but now if you switch on an external field you press the particle down and you get then here a dipole-dipole interaction. The theoretical prescription is then formulated in terms of the free energy expression the upper one here is the change of surface area multiplied by the surface tension and here we have the restoring force due to the capillary length. The next contribution is that you have to take into account pulling the meniscus this is this piece the pressure times the displacement field and finally pushing the colloid that means external force acting against a change in height of the center here. And finally you have to take into account that the surface free energies of the colloids are different from the liquid vapor interface and here from the colloid vapor interface so you have these different surface tensions and this is taken into account by here. Now once you have a model which tells you how this pressure looks like you just have to minimize this free energy and taking the difference of the free energy for a fixed particle-particle distance relative to the one at large separation gives you an attractive interaction. There is also a direct interaction between these colloids and this comes from the charges here and this is typically a repulsive interaction and then you see already there a competition between an attractive piece coming from the capillary interactions and a repulsive one from direct interactions. And if you work this out then another length case comes into play this is the so-called Debye-Hooker screening length which says about the screening of these coulomb interactions and in the case that the screening length fulfills this inequality with respect to the reference configuration this is the reference radius here and that has to be put into ratio with the Debye-Hooker screening length and then you see here that indeed what you build up here is a minimum in the interaction but that very much depends on these ratios here and so properly you have to take into account also the repulsive interaction and the net result of superimposing them is shown here on the right panel and there indeed you do observe a minimum. Now there are certain conditions that this minimum appears and that means that Debye-Hooker screening length is comparable to R and this is a small perturbation and here the colloidal charge density should reach these kind of values which are rather large. And so a slight disadvantage in this description is that this minimum now happens to occur relatively close to the two particles which means that an asymptotic approximation is difficult to maintain numerically valid. You can now also study what happens if you now replace your liquid by a more complicated complex fluid, a nematic and you can now study what happens if you now deposit these colloids upon a nematic film which means that here you have now elastic interactions in the nematic film due to the presence of your colloidal particles here and this gives an interesting extra contribution to the interaction which is also governed by the topological defects which appear here and here and what you find out here is that then this elastic interaction follows this power law here and this is a quadrupole repulsion whereas then the contribution from the meniscus is this combination so you get here rather interesting long range interactions and they are also competing with each other and the nice aspect which also was experimentally exploited is that these interactions depend also sensitively on the thickness of this nematic film and the kind of configurations which appear at this interface as you can see here they are very sensitive depending on about the thickness of this nematic film so the structure formation you see here can be tuned by changing this nematic film thickness and that would basically deserve a talk of its own the kind of wealth of phenomena which appear here with these nematics now another problem which I would like to address is that so far we have studied the interface which is asymptotically flat and here we now what we do is that we pull the particle with an external force on a sessile droplet so you have a droplet in contact with a substrate and now you would like to know what is the optimal position of this particle on the surface of this drop once it is pulled a little bit out of the water and what turns out is that the answer of this question depends sensitively on whether the contact line is free or whether it is pinned in the case that it is free the minimum of the angle here this is the angle is at zero that is the particle likes to stay on the top of the droplet whereas in the case that the interface three phase contact line is pinned but you have a certain preferred angle so it is here something like 50 degrees so there is a very specific angle these particles do prefer and this is displayed here as the free energy landscape this is what happens if you fix one colloidal particle at a certain position at the surface and now you want to ask if you add a second one where does the second one go and it turns out that they like each other very much and that is the black ones they sit over here there are meta stable states which are on the opposite so there are also configurations where they prefer to sit at the three phase contact line but this is not always so and on the far side of the moon there are no minima at all so the particles want to be here where the first particle has been deposited now so far we have neglected thermal fluctuations of the interface now we want to study what happens to the effective interaction between these particles if you allow for capillary waves here, thermally excited capillary waves they are undercut by the presence of these colloidal particles and this changes the free energy spectrum and this gives rise to additional fluctuation induced force and there are now various ways to do that you can allow the particles to fluctuate too or they are pinned or the contact line is pinned there is a whole zoo of possibilities here are the configurations so here it is a pinned interface, pinned interface and the colloid is pinned in this case you have an extremely long range entropic interaction the log of a log is attractive if you now let the free phase contact line fluctuate on the surface of the colloid instead of having it pinned but the colloids are still fixed you get this kind of attractive interaction and then you just can play it through and you encounter various power laws depending on which kind of fluctuations you unfreeze so far we have been interested in the static properties now let's turn to dynamics and here again I remind you of the basic interactions that here describes the displacement field of the interface in the presence of a certain density field, a two-dimensional density field of the colloids so we have now a whole ensemble of colloids and we now want to describe what happens to such a cluster of these colloidal particles then you have here particle conservation and then dynamics is governed by an overdamped stokes and dynamics which is absolutely suitable for here so here you have no inertial phenomena so you don't have hydrodynamics to do but Stokesian dynamics and this here is the corresponding mirror with self-gravitating fluids you have this to compare with that, that's the same thing provided you kill here the contribution from the capillary length you have particle conservation and here you do have of course inertial Newton dynamics and so in this case what we are studying here on the left column is to say the cosmology in the Petri dish now what is the kind of dynamics? so I provide you here with a flow diagram so you start out with an initial distribution of the colloids this is a certain density rho 0 then you feed this in into the Laplace equation or screen Poisson equation this way you obtain then the displacement field for this kind of initial configuration then you feed this into the dynamic equation which is governed by what you get from dynamic density function theory so density function theory here in terms of classical liquids not electronic degrees of freedom but the classical liquids and this I have no time to get into the details how you derive that but the point is that this dynamic density function theory provides you with a time evolution of this density and then you from feeding in the rho 0 and you you get then rho of t and then you run the whole thing until iteratively until you get a stable solution that's the way you get the time dependence of this clustering now the point I want to make is that you want to know what is the capillary energy in such a cluster in the case that once the cluster is larger or smaller compared to the capillary length and that makes a huge difference so here I have the capillary length in that case being large compared to the size of the cluster and here the opposite and it turns out that the energy density in the lower case is independent of the size of the cluster so this here cancels out with this whereas in the opposite case here it scales like log L times L squared so that means you have a significant difference depending on the size of the cluster compared to the capillary length now this capillary interaction they would like to crush the cluster but then you have opposing to that a two-dimensional pressure which comes about due to thermal motion of these particles for example they are hard disks sitting there and they give rise to a pressure which opposes the contraction governed by the capillary interaction now the question is who wins either you crush them they have collapse or that they stay where they are or if they even evaporate and the statement now is that if you equate the capillary energy density with the one which comes about from the repulsive interactions among them the size of the cluster which fulfills this equality is what is called the jeans length which is to say in the cosmology community well known since more than a hundred years by now and the statement now we find is that once this jeans length is large compared to the capillary length all clusters are stable if however on the opposite the jeans length falls below the capillary length then only clusters which are sufficiently small are stable and all clusters with the size beyond the jeans length they do collapse and this can be understood also in terms of linear stability analysis so typically linear expansion and here you have then the exponential growth and it turns now out that the whole thing is governed by the jeans length and the jeans time which sets the time scale for the growth of the perturbation and what now you can figure out is that if the capillary length divided by the jeans length is less than one all nodes are stable but once this becomes larger than one you see here the exponential collapse occurring from this analysis and now you would like to know whether to say these collapse is experimentally accomplishable and this is indeed the case that conditions are given here by the capillary length much larger than the nearest neighbor distance and here the jeans length which is 1 over K has to be smaller than lambda and larger than the size of the colloid and then this enters here in a dimensionless quantity which here by Q and this tells you now putting in real numbers for various kinds of values of this Q here we have 3, 10 and 30 there is a window of opportunity between lambda times K being 1 reaching up to the jeans length being equal to the size of the particle so in this range here you do see this collapse then you would also like to know whether the time scales are suitable for it and this is shown here and for the 3 examples I showed you the window of opportunity means that you have to wait either minutes or up to days but they are reachable time scales now you would like to know how in case it does collapse what is the kind of dynamics this collapse follows and this can be studied either by Brownian dynamic simulations I show some results in a minute and it means to solve this diffusion equations where the heads are all the dimensionless quantities so that you have here only to deal with non-dimensional objects and the whole thing can be encapsulated in this ratio which plays the role of an effective temperature and now you can first of all now study this collapse by perturbation theory in a cold collapse where this happens to be 0 and then you get this top head distribution they have an over density at a certain constant radius and then as function of time this shrinks and the head grows and this is described here by these time dependencies and you see here that at the finite time the size of the cluster reduces to 0 and at this time this diverges now I show you here now the results of the Brownian simulations and this is shown such that the red particles are those marked which are in close neighborhood with other particles so the red ones are those who form not to say the nucleus of the collapse and this is shown here for the gravitational collapse and I now look at Brownian dynamic simulations for other kind of parameters and what you see there a completely different mechanism namely the shockwave formation here that the shockwave appears now and this contracts different kind of dynamics and then at the very end this shows the appearance of a traveling shockwave you start out with this configuration and then it bulges up here informing a shockwave and the shockwave runs to the center this can be captured by a dynamic phase diagram in which you plot here the effective temperature versus the range and you see here different kinds of mechanism where how this collapse occurs here we have a collective collapse so everything just homogeneously contracts the cold collapse or something which appeared here this transition region to the non-collapse and collapse is formed by shockwaves and out here you observe spinodal decomposition this can be just described by density profiles which is shown here and the position of the panels correspond to the position of the diagram here so the panel up to the left is shown here and the other panels correspond to this arrangement here and then you see quite vividly that here you have the homogeneous gravity collapse here you have the shockwave formation and here where you see these density oscillations here and here this is the hallmark of spinodal decomposition so by varying these parameters you can say switch between the various types of dynamics since I have been just reminded to stop here I think that is the main message I wanted to tell you so think of a system in which you can generate long range interactions with a knob such that you can have a short range, exponential decay or logarithmic long range forces you can play with that and that it gives rise to interesting structures and to a whole variety of different dynamics where you can also by changing the parameters have a crossover between rather distinct different types of dynamics thank you