Multiscale Multilevel Approach to Solution of Nanotechnology Problems

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1 Mathematica Modeing and Computationa Physics 2017 Mutiscae Mutieve Approach to Soution of Nanotechnoogy Probems Sergey Poyakov 1,2, and Viktoriia Podryga 1,3, 1 Kedysh Institute of Appied Mathematics of RAS, 4 Miusskaya square, , Moscow, Russia 2 Nationa Research Nucear University MEPhI (Moscow Engineering Physics Institute), 31, Kashirskoe shosse, , Moscow, Russia 3 Nationa Research Center "Kurchatov Institute", 1, Akademika Kurchatova p., , Moscow, Russia Abstract. The paper is devoted to a mutiscae mutieve approach for the soution of nanotechnoogy probems on supercomputer systems. The approach uses the combination of continuum mechanics modes and the Newton dynamics for individua partices. This combination incudes three scae eves: macroscopic, mesoscopic and microscopic. For gas meta technica systems the foowing modes are used. The quasihydrodynamic system of equations is used as a mathematica mode at the macroeve for gas and soid states. The system of Newton equations is used as a mathematica mode at the mesoand microeves; it is written for nanopartices of the medium and arger partices moving in the medium. The numerica impementation of the approach is based on the method of spitting into physica processes. The quasihydrodynamic equations are soved by the finite voume method on grids of different types. The Newton equations of motion are soved by Veret integration in each ce of the grid independenty or in groups of connected ces. In the framework of the genera methodoogy, four casses of agorithms and methods of their paraeization are provided. The paraeization uses the principes of geometric paraeism and the efficient partitioning of the computationa domain. A specia dynamic agorithm is used for oad baancing the sovers. The testing of the deveoped approach was made by the exampe of the nitrogen outfow from a baoon with high pressure to a vacuum chamber through a micronozze and a microchanne. The obtained resuts confirm the high efficiency of the deveoped methodoogy. Introduction The modern computing aows modeing very arge and compex systems and processes. Computer modeing has become one of the most effective toos in many branches of science and production. The present work is devoted to the deveopment and appication of computer methods in the fied of studying the compex gasdynamic processes in technica micro- and nanosystems. A feature of mathematica probems in this fied is the simutaneous study of processes at many scaes, incuding microand nanoscae. One of the modern and activey deveoping approaches to soving such probems is the mutiscae approach which combines the methods of the continuum mechanics (MCM) and the Newton dynamics for individua partices. This aows efficient computer simuation of expensive and difficut to impement physica experiments. e-mai: poyakov@imamod.ru e-mai: pvictoria@ist.ru The Authors, pubished by EDP Sciences. This is an open access artice distributed under the terms of the Creative Commons Attribution License 4.0 (

2 Mathematica Modeing and Computationa Physics 2017 In the present paper, an aspect of modeing technica microsystems is considered, which is reated to computing the parameters of gas microfows under technica vacuum conditions. For a correct description of such processes, it is necessary to know in detai the properties of the rea gases and reproduce them in a numerica experiment. Many properties of the gases are we studied experimentay in certain ranges of temperatures and pressures and are described in the iterature. However, there are properties which can ony be predicted theoreticay on the basis of the kinetic theory of gases [1]. In this case, the obtained theoretica data correspond to a very imited range of temperatures and pressures and can differ substantiay from the rea properties of the gas. One of the ways to obtain the missing information about the properties of the gas medium in a given range of temperatures and pressures is the moecuar dynamic simuation [2, 3]. A mutiscae two-eve approach to cacuating the rea gas fows in the microchannes of technica systems, appied in a wide range of Knudsen numbers, was presented in [4 6]. At the moecuar eve it aows to determine the parameters of the equation of state of a rea gas [7], to cacuate the kinetic properties of the gas [8], to determine the type of the boundary conditions at the was of the microchanne [5, 6]. This approach is based on a combination of MCM and Newton dynamics for individua partices and incudes two eves of modeing: macroscopic and microscopic. As a mode at the macroeve, the system of quasigasdynamic (QGD) [9] equations is used, as a mode at the microeve, the method of moecuar dynamics (MD) [2, 3] is used. The impementation of this approach is based on spitting the probem into physica processes. At the macroeve of detai the description of muticomponent gas media fows takes pace. At the microeve interactions are cacuated for: 1) gas moecues between themseves (forming the equation of state of the mixture, determining the transport coefficients and reaizing the mixing of the components); 2) gas moecues and atoms of soid surfaces (describing phenomena in boundary ayers). In what foows, a mutiscae mutieve approach to soving nanotechnoogy probems with the hep of supercomputer computing systems is described in some detai. This unification covers three scae eves: macroscopic, mesoscopic and microscopic. To reaize it at the mesoscopic eve the cacuation of the motions of arge partices in the gas fow is added, and at the microeve the interactions of the gas moecues with the partice surface are cacuated. The consideration of the mesoscopic eve aows to track the detaied behavior of the individua arge partices, for exampe, during their deposition on a substrate in order to create a given spatia nanostructure. 1 Mathematica formuation The mathematica mode incudes three components corresponding to the three scae eves. At the macroscopic eve the QGD equations are used. In the case of a gas mixture, the system of QGD equations is written for each gas separatey [9]. These equations are written beow in the three-dimensiona case for a mixture of gases in invariant form with respect to the coordinate system together with the equations of constraints and the equations of state: t ρ t + div W (ρ) = 0, (1) ( ρ u,k ) + div W (ρu k ) = S (ρu k), (2) E t + div W (E) = S (E), (3) E = ρ ( 1 2 u 2 + ε ), H = E + p ρ, p = Z ρ R T, ε = c V, T, γ = c p, c V,, Pr = µ c p, χ, Sc = µ ρ D, Ma = u a, Re = ρ u λ µ. (4) 2

3 Mathematica Modeing and Computationa Physics 2017 Here a variabes with the index reate to a gas of the type, each component has its own concentration n, mass density ρ = m n (m is the mass of a moecue of the gas ). Each gas is aso characterized by its temperature T and macroscopic veocity u. Other parameters of the mixture components: p are partia pressures of gases in the mixture; E, H and ε are tota energy densities, enthapies, and interna energies of the mixture components; µ = µ (T ), D = D (T ) and χ = χ (T ) are kinetic coefficients of mixture components, namey, coefficients of dynamic viscosity, diffusion and therma conductivity. Variabes Z = Z (T,ρ ), γ = γ (T,ρ ), c V, = c V, (T ), c p, = c p, (T ) and R = k B /m are compressibiity coefficients, adiabatic indices, specific heat capacities and individua gas constants of the mixture components (k B is the Botzmann constant); Pr, Sc, Ma and Re denote Prandt, Schmidt, Mach, and Reynods numbers for the mixture components; λ are mean free paths; vectors W (ρ), W (ρu k) and W (E) coincide, up to a sign, with the fuxes of mass density, the momentum density of the corresponding components, and energy density. The exchange terms S (ρu k) and S (E) take into account the redistribution of momentum and energy between the components of the mixture. The system of equations (1) (4) is cosed by the corresponding initia and boundary conditions. The initia conditions are taken in accordance with the equiibrium state of the gas medium in the absence of interaction with externa factors. The boundary conditions on soid surfaces are given in the form of mass density, momentum density and energy density fuxes across the boundary, cacuated from empirica formuas or cacuated using MD methods [4 6]. On the free surfaces of the computationa domain, the so-caed soft boundary conditions [9] are enforced. At the microeve, the evoution of the system under investigation is described by Newton s equations for a system of partices. The system of the equations of motion for partices of type, which can be a gas, or a meta, has the foowing form m dv,i dt = F,i, v,i = dr,i dt, i = 1,...,N, (5) where i is partice index, is partice type, N is tota number of partices of type. A partice of type with index i has its own mass m, position vector r,i = ( ) r x,,i, r y,,i, r z,,i, veocity vector v,i = ( ) vx,,i,v y,,i,v z,,i and tota force F,i = ( ) F x,,i, F y,,i, F z,,i acting on it. The forces acting on the i-th partice are cacuated as a sum of its interactions with the surrounding partices, depending on the potentia energy and the component responsibe for the externa action. The potentia energy of the system is represented as a sum of partia energies, the cacuation of which takes pace according to the formua of the chosen interaction potentia. F,i = U ( r,1,...,r,n,... ) + F ext,i, U = U, i = 1,...,N, (6) r,i where U is tota potentia energy, U is interaction potentia for partices of type with partices of type, F ext,i is the force of interaction with the externa environment. The choice of the interaction potentia is based on comparing the mechanica properties of the computer potentia mode and the rea materia. The initia conditions at the microeve are determined by the equiibrium or quasiequiibrium thermodynamic state of the partice system at a given temperature, pressure, and average momentum. The boundary conditions at the moecuar eve depend on the simuated situation. At the mesoscopic eve, in the presence of finey dispersed soid impurities in the gas mixture of the same chemica composition, a diffusion approximation is used in which the principa unknown quantity is the partice concentration C. For it, the foowing noninear convection-diffusion equation is used: C = div (D C C µ C F C C) + (u, C). (7) t 3

4 Mathematica Modeing and Computationa Physics 2017 Here D C is the interdiffusion coefficient of the gas suspension that can be cacuated in various ways [10, 11]. The coefficient µ C describes the mobiity of the impurity partices; in the genera case it is determined in a compex manner and depends on the shape of the mesopartices, the temperature and the viscosity of the medium. In this paper the we-known Stokes formua was used to determine the mobiity coefficient, which is vaid for spherica partices. The term F C is the tota force acting on the partices and cacuated at the moecuar eve, u is average veocity of the gas mixture fow. The initia conditions for (7) consist in assigning a constant and, as a rue, a very sma concentration of partices in the entire free space. The boundary conditions at soid was are determined by the chosen interaction mode (adhesion, refection, siding, etc.). The boundary conditions at the entrance to the medium consist in the assignment of the partice fux. The soft boundary conditions are used at the output. If necessary, to trace the movements of the individua arge partices (substantiay exceeding the dimensions of the moecues of the gaseous medium), equations simiar to (5), and (6) are used together with the equation (7). 2 Numerica approach The numerica approach to the soution is based on the probem spitting into physica processes. In this case, the equations of the quasihydrodynamics (1) (3) and the equation (7) are soved by the method of finite voumes on grids of various types. Newton dynamics equations are soved according to the Veret integration [2] either in each grid ce independenty or in groups of couped ces. The cacuation of the macroparameters according to the QGD equations (1) (3) and the equation (7) is carried out over a tempora grid using an expicit numerica agorithm which is based on the finite voume method on grids of arbitrary type. For the convenience of soving the probem in areas of compex geometry, hybrid bock meshes consisting of ces of various shapes and sizes can be used. In the two-dimensiona case it is proposed to use quadranges and trianges, in the three-dimensiona case poyhedrons with a number of vertices from four to eight. A parameters of the gas components (density, pressure, temperature, veocity vector components, etc.) refer to the mass centers of the ces. Fux variabes refer to the centers of the faces of the ces. Spatia approximations of the basic terms of the QGD equations are performed by standard methods. The chosen computing scheme on time is expicit and two-stage (predictor corrector). The system of MD equations is used in additiona cacuations independenty, or as a subgrid agorithm appied within each contro voume. At the microscopic eve, at every step of the simuation, a system of ordinary differentia equations is soved, corresponding to the Newton second aw and describing the motion of partices of the moecuar dynamic system. To integrate the equations of motion, the Veret integration [2] is used. To carry out a correct cacuation of the QGD, the mode is suppemented by rea gas equations of state, transport coefficients and other accompanying parameters (enthapies, average mean free paths, etc.), as we as rea boundary conditions. In the case of a mixture of gases, it is necessary to add the corresponding exchange terms to the equations for the momentum and energy of each component. The cacuation of these dependencies, coefficients, and conditions is performed using the MD methods. The stabiity condition of the spitting method by physica processes incudes standard conditions for the stabiity of an expicit finite-difference scheme ( t 1 h 2, where t 1 is time step, h is space step) at the macroeve and the conditions for the stabiity of the Veret schemes ( t 2 1/ max F 2, t 3 1/ max F 3, where t 2, t 3 are time steps, max F 2, max F 3 are maximum absoute vaues of the forces of interaction between the partices at a considered step in time) at the meso- and microeves. 4

5 Mathematica Modeing and Computationa Physics 2017 To verify the proposed numerica methodoogy, a verification procedure was used, based on performing cacuations with grinding the mesh at the macroeve, and on performing cacuations for systems with different number of partices and increasing the number of partices at the microeve. 3 Agorithms and program impementation The probem modeing on the basis of the mutiscae approach under consideration with three eves of detai is carried out with the hep of specia agorithms that in genera, depending on the degree of microeve use, are divided into four casses. Agorithms of cass 1 are of interest for the study of the properties of gas media and the properties of soid surfaces with which the gas medium contacts in technica appications. As a numerica impementation of the approach in this case, the Veocity Veret scheme acts. With the hep of 1 cass agorithms, a database of moecuar dynamic cacuations (DMDC) is accumuated for the properties of gases and soid materias, which can be used in the framework of other agorithms. Agorithms of cass 2 assume the soution of probems ony at macro- and mesoeves based on QGD equations (1) (3) and the convection-diffusion equation (7). In this case, the properties of the gas mixture components (the equations of state by pressure and energy, the kinetic coefficients viscosity and therma conductivity, the exchange terms in the equations for momentum and energy, the parameters of the boundary conditions) and the kinetic coefficients for equation (7) are determined from the above-mentioned DMDC accumuated in advance for the desired temperature and pressure range. Agorithms of cass 3 impy the simutaneous use of equations (1) (3), equations (7) and Newton equations of mechanics (5), (6) in the cacuations of QGD. Agorithms of cass 3 are reaized in the framework of the method of spitting into physica processes. It is assumed here that in the gaseous medium and at its boundaries it is possibe to confine ourseves to a oca consideration of the processes of interacting the gases of the mixture with each other, with the surfaces of arge partices and with soid was. Agorithms of cass 4 aso assume the simutaneous use in the cacuations of equations (1) (3), (7) and (5), (6). The difference of this case from the previous one is that in some areas of the environment (usuay at soid surfaces and in zones of a strong drop of the gas parameters), moecuar dynamic cacuations are carried out continuousy without going to the macroeve. In the same areas, the principe of ocaity of moecuar interactions is not used, that is, in the genera case, the agorithms of cass 4 are non-oca at the moecuar eve. The paraeization is impemented based on the principes of geometric paraeism and rationa partitioning of the computationa domain. To baance the task oading, a dynamic agorithm is used. The code reaization is made using MPI and OpenMP programming. 4 Cacuation resuts The code vaidation is carried out on the exampe of the probem of the nitrogen outfow from a baoon with high pressure to a vacuum chamber through a micronozze and a microchanne. The computationa geometry is shown in Fig. 1. A symmetric domain is seected, but the expected soution may not be symmetric. The cyindrica micronozze has a cross section of rectanguar shape, the diameter D µm and ength L 0 = 6D µm. The computationa domain sizes of the baoon equate D 1 = 6D 0 and L 1 = 10L 0. The diameter and ength of the microchanne equate D 2 = 6D 0 and L 2 = 50L 0. The nozze and the microchanne connect the baoon with nitrogen and the open space of the vacuum 5

6 Mathematica Modeing and Computationa Physics 2017 Figure 1. Computationa domain chamber. We assumed that a soid surfaces of the microsystem are covered with a ayer of nicke. Copper custers of a reguar cubic form with a inear dimension of nm ( eementary ces with the edge ength of the FCC attice nm) were considered as arge partices. At the initia moment the gas is at rest: u 1 = u 2 = 0. In this case, in the baoon the gas is under standard norma conditions: T 1 = K, p 1 = Pa; in the nozze, the microchanne and the vacuum chamber, the gas is at the same temperature, but at a ower pressure: T 2 = T 1, p 2 = 10 5 p 1. The nozze is bocked on the eft by a partition which opens instanty at the beginning of the cacuation. The inner surface of the nozze is considered to be perfecty smooth and thermay insuated. The cacuations were performed using an agorithm of cass 2 and were aimed at studying the fow parameters at the initia stage of fow acceeration. They showed that the interna energy of the gas in the baoon is, for a certain period of time, graduay converted into kinetic energy of the gas in the nozze and in the channe. At the same time, a therma shock wave passes through the gas medium. Subsequenty, the gas temperature returns to the norma and the profies of a gas macroparameters are aigned. As an iustration of the above the Fig. 2 shows the distributions of the density, concentration of moecues, temperature, and oca Mach number in the gas aong the symmetry axis of the computationa domain for a number of instants of time. An abbreviation a.u. means arbitrary units, here the spatia coordinates were normaized to the diameter of the micronozze and the gas macroparameters were normaized to their vaues under norma conditions. Two-dimensiona distributions of the concentration of gas moecues and the veocity moduus are shown in Fig. 3. They show the genera picture of the fow near the nozze and the entrance to the microchanne at the time t = µs, when the gas acceeration has aready stopped, and a reguar fow has been estabished. In Fig. 4 the trajectories of copper custers cacuated in the case of triggering of singe partices are shown. Custers were paced in the midde of the cacuated area of the baoon at zero speed. The fow of gas carried through the nozze into the chamber acceerated them to a maximum speed equa to the average fow veocity. The cacuated trajectories as a whoe correspond to theoretica estimates. However, gravity was not taken into account. An even more accurate picture in the future wi be obtained when this force is taken into account as we as the surface structure of the custers and was of the nozze and microchanne. Concusion A mutiscae mutieve approach to the soution of nanotechnoogy probems by supercomputer systems is presented. The soution combines continuum mechanics modes and Newton dynamics for individua partices at three scae eves: macroscopic, mesoscopic and microscopic. The description of the gas-meta technica systems as a mathematica mode at the macroeve is done using the quasihydrodynamic equations for gas and soid states. As a mathematica mode at the meso- and 6

7 Mathematica Modeing and Computationa Physics 2017 Figure 2. Distributions of density, concentration, temperature and oca Mach number aong the symmetry axis (x coordinate) of the computationa domain. The curves abeed 1 to 9 correspond to the time moments t = 0, , , , 1.474, 2.947, 4.421, 11.79, µs Figure 3. Two-dimensiona distributions of gas concentration and veocity moduus near nozze area at the time moment µs 7

8 Mathematica Modeing and Computationa Physics 2017 Figure 4. Iustration of coarse partices moving: streamines of partices microeves, the systems of Newton equations written for nanopartices of the medium and arge partices moving in the medium are used. The numerica impementation of the soution invoves the probem spitting into physica processes, the method of grids at the macroeve and the Veret integration at the meso- and microeves. Four casses of agorithms and their parae impementation are discussed. The impementation uses the principes of geometric paraeism, rationa partitioning the computationa domain, oad baancing of the tasks. A specia dynamic agorithm is used for oad baancing the sovers. The agorithm vaidation is done on the exampe of the probem of the nitrogen outfow containing custers of copper into vacuum. The obtained preiminary resuts correspond to theoretica estimates and confirm the high efficiency of the deveoped methodoogy. Acknowedgments This work was supported by the Russian Science Foundation (project No ). References [1] J.O. Hirschfeder, C.F. Curtis, and R.B. Bird, Moecuar Theory of Gases and Liquids (Wiey, New York, 1954) [2] J.M. Haie, Moecuar Dynamics Simuations. Eementary Methods (John Wiey & Sons, Inc., New York, 1992) [3] D.C. Rapaport, The Art of Moecuar Dynamics Simuation (Second edition, Cambridge University Press, Cambridge, 2004) [4] V.O. Podryga, Dokady Mathematics 94 (1), (2016) [5] V.O. Podryga, IOP Conference Series: Materias Science and Engineering 158, (2016) [6] V.O. Podryga, Yu.N. Karamzin, T.A. Kudryashova, and S.V. Poyakov, Proceedings of the VII European Congress on Computationa Methods in Appied Sciences and Engineering (ECCO- MAS Congress 2016) 2 (ECCOMAS Congress 2016, Crete Isand, Greece, June 5 10, 2016) [7] V. Podryga and S. Poyakov, Proceedings of the Fourth Internationa Conference on Partice- Based Methods-Fundamentas and Appications (PARTICLES 2015) (CIMNE, Barceona, Spain, September 28 30, 2015) [8] V.O. Podryga, Lecture Notes in Computer Science 10187, (2017) [9] T.G. Eizarova, Quasi-Gas Dynamic Equations (Springer-Verag, Berin, 2009) [10] V.Ya. Rudyak, A.A. Bekin, and S.L. Krasnoutskii, Thermophysics and Aeromechanic 12(4), (2005) [11] V.Ya. Rudyak and S.L. Krasnoutskii, Technica Physics. The Russian Journa of Appied Physics 47(7), (2002) 8