 Okay, good morning. Thank you for coming. I will talk about computer simulation of the initial stage of condensation with fragmentation of charge drops. We take this walk with collaboration with Zmievsky-Galina from Kielderch Institute of Applied Mathematics from Moscow. So in this presentation, I want to first describe the model of non-equilibrium stage of a first order phase transitions in the process of condensation in this walk. We will do this on the example of silicon-carabit melt vapor. Then I talk a little about the theory beyond this model, Fokker-Planck or Fokker-Planck-Kolmogorov equations and corresponding to these equations, Itho Stratanovich. And in the last I'll show the benefits of our model and the results of simulations. So why are we interested in this problem? Interest is in creation of amorphous metallic nanostructures, the structures which have high corrosion resistance and strength. And the nanostructures that you can obtain could be used in different fields, like for preparation, five divided powders, or for sputtering. Let's go to problem formulation. First of all, I want to talk that all the model is carried with assumption of open physical system. And this is the point. So the process of germ formation can be described as consecutive reactions. So we have some initial size of a cluster. And then we just have a diffusion manner model in which we have step-by-step growth or decreasing the size of a model. Also the necessary condition, the necessary condition of creation with amorphous nanostructures is that we want to get nano-sized clusters, nano-sized drops. And so we can get this by charging our cluster, our drop, and we can, there is a walks which show that the high charged drops of metal can emit the small nano-sized drops. And the effect of this limitation of droplets, of high charged droplets, corresponds to relay instability threshold. The relay criterion for the charged drops can be written in this form. There is a constant charge and the radius of the cluster. Let's talk a little about the model of charging the drop. It shows that it consists of three parts. First, we should put our drop in a charged plasma, for example, so we need a constant charge. Then the drop will become the form of an ellipse. And then in the third part, the drop will emit small nano-sized particles. And now I should say about why do we need the constant charge, because it's very important. If we just charge in the initial stage, then in the third part, we have only several nano-sized drops emitting by the material drop, and it will, these small drops will, for example, take away like 25, for example, percent of the charge, but it will only have several percent of a myosofol system. So the material drop can be able to emit another nano-sized drops because the relay criterion can be done because the charge is fewer. So in the whole system, we will observe only a little part of nano-sized particles. So in this case, we need constant charge of the whole time of our simulation. Now let's see the kinetic model, the thumb theory. So non-equilibrium processes in the gas and plasma could be described with Fokker Planck, also known as Fokker Planck-Lmogorov equations. But mostly Fokker Planck equations worked with density function in the phase coordinates. And here we will work with density function or in the phase space of cluster sizes. So G is the amount of monomers in the cluster. Also we have the diffusion coefficient and thermodynamic potential. Also we have the boundary, yes. So let's see the form of the free Gibbs energy and the diffusion coefficient. In general case, the free Gibbs energy consists of three parts and ideal part of interaction potential. The parts that correspond to surface tension at the cluster and the part that corresponds to the charging model. In previous works, there was established the form of these functional coefficients. So the diffusion coefficient is proportional to the G in power two-thirds with dimensionless coefficient and the free Gibbs energy or thermodynamic potential is also consists of three parts which correspond that I talked about. So let's see the plot of Gibbs free energy this is the plot in the case of uncharged drop and here we can see the hump on the plot and if we take partial of the Gibbs free energy by G by the size of the cluster way can find the critical point and we assume that if we take the initial conditions of cluster size greater than criteria point we will observe the growth of the cluster growth of the drop and on the other hand, if we take the initial condition smaller than critical point we will see the getting the drop smaller but what we can see that we have due to the stochastic term in the stochastic model we have the boundaries of instability near the critical point. So we can, for example, take the initial conditions little smaller than critical point but in some cases we will see that the drop is growing. This is the Gibbs free energy in the case of charged drop and here we can see why do we need the charging of the drop because here we won't see the disappearing of the drop because of this well. So we will get the size of the clusters as we needed and also if we, also due to the relay criterion we, if the clusters will grow very high, very big it will emit small parts and getting smaller. So we didn't get even now such a very big drops and also we didn't find the drops to be disappeared. So the Fokker Planck equation that we see above will corresponds to the either stochastic differential equation mainly the walks use just either stochastic differential equation but here we can see the stochastic term which parts is known as term and it's, we use it because of, using this term we can, instead of in either case we can use, can calculate not stochastic integral but we can go to Lebesgue still yes integral with this term. Also we have the parts corresponding to stochastic model which is described in by Wiener process. So for numerical solution of this stochastic equation, differential equation we use generalized method of the Rosenberg type. We use, this is a Rosenberg part and we use it because of, due to the Rosenberg part we can, we can get sustainability of the equation of the answer. So here this is the part response to stochastic part. This is Q is intensivity of the, white noise and this stochastic part is from the normal distribution and this equation we calculate to for 10 power six independent trajectories so to obtain the statistics. Okay so here we get some results. These plots showed the evolution of cluster size. This is a phrase of equal values. Here we take for initial size G like 400. If we see, if you look at the here, so we take initial size greater than the critical point so to look, if we take the initial conditions lesser than critical point we can't see, you can't observe the early criterion. So we took it's like 400 and in uncharged case we see that it have a small growth but it not what we wanted. So in in charge case we see that because of the relay criterion big drops are just breaking up and the distribution of size of the cluster is getting smaller. So in these plots we can see just histograms of G by by trajectories. It's like a slice of the previous graphs on the time scale and here we also can see the B-model distribution of size clusters because here we green corresponds to the earlier time and black for the final stage and we want to get the particles smaller than we have in the initial conditions. So we get what we wanted. Here on the other hand we have the initial conditions of the parts smaller than the critical point and but here we can see that it's not in the amount of monomers but it's on extremes and here we also can see that the distribution is B-model but we have only a small amount of trajectories with very small drops and the main amount is all the sizes we wanted to see. So I want to conclude first the algorithm shows that the algorithm allows simulate the phase transition of the first kind at the initial stage with a fixed charge, with a constant charge. The numerical experiments show the fluctuation in stability just near the critical point and because of it we can see the B-model distribution of size clusters and this approach make possible to see the role of relay instability or lay criterion in production of metal and powders. So thank you. Thank you. Congratulations, it's the last question. Yes, it's the last question. Yeah, for different temperature, for different pressure, it will be different, yes. This is for a particular temperature, for a particular pressure, I just leave it outside of my presentation but yeah, it's for silicon carbate here. Yeah, melt vapor. Melt vapor. Ah, yeah, yeah. Change, you can't do the composition of the droplet. Wait, oh, I'm sorry. Yeah, you're talking about this? Yeah, just because, no, no, it's due to the sort of material, we will get peak of the, on this plot in different places. Percentage, yeah. Percentage from this? Yeah, sure, but here we can see. One silicon, one carbon. What? One silicon, one carbon. Yeah, yeah. No, it's just if we get, if we take the other percentage of the... Change chemical composition. It also wasn't the real question. Sorry? Chemical composition, changing from chemical composition. No, in these equations, no, there isn't. It's hard to understand this plot. It is not a result of calculation. I guess there's a result from just different calculation of different initial samples. Yeah, sure, yeah, sure. You can change the plot of the simulation. Different of the... Do you mean the simulation of different materials? Yeah, it's not changing in the middle simulation. Not represent different simulations. So you can make composition of the materials. So you can combine, does the silicon work? Yeah, the percentage may vary just in the sense... Yeah, sure, in the initial stage. In the initial stage. Yeah, just we'll gain, we'll get the peak in the, in other place. Poor silicon has larger... Yeah, yeah. Yeah, sure. More questions? Sorry? Can we make a simulation of pure... No. So it is spherical, but it is liquid, and it is ferroid, this type, yeah? Relay? Yeah, yeah, we have... No, in the second part of the... Firstly, the job is charging by the relaying stability, then it gets to the form of ellipse. Yeah. Yeah, yeah. It's getting larger on one axis. Yes, one axis is larger. Yeah. In the session, so let's take home the speakers.