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Molecular Dynamics! Application to Liquid Sodium

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Molecular Dynamics! Application to Liquid Sodium PINSTECH/NPD-124 MOLECULAR DYNAMICS! APPLICATION TO LIQUID SODIUM Kbawaja Yaldram Abdullah Sadiq Sohail Murad tk > >\ / \ ! j li ( ! ! ? ii I 1 I ! I ! il: M; NUCLEAR PHYSICS DIVISION Pakistan Institute of Nuclear Science & Technology P. O. Nilore Islamabad. April, 1989 PINSTECH/NPD-124 MOLECULAR DYNAMICS: APPLICATION TO LIQUID SODIUM KHAWAJA YALDRAM ABDULLAH SADIQ +SOHAIL MURAD Pakistan Institute of Nuclear Science and Technology P.O. Nilore, Islamabad +University of Illionis, Chicago, U.S.A. CONTENTS S.No. Description Pages 1. INTRODUCTION 1 2. MOLECULER DYNAMIC SIMULATION {6,71 2 3. SIMULATION OF LIQUID SODIUM 6 4. RESULTS AND DISCUSSIONS 8 5. CONCLUSIONS 11 6. ACKNOWLEDGEMENTS 11 7. REFERENCES 12 8. FIGURE CAPTIONS 14 MOLECULAR DYNAMICS: APPLICATION TO LIQUID SODIUM 1. INTRODUCTION Molecular dynamics (M.D.) and Monte Carlo (M.C.) computer simulation methods have become important tools for the study of equilibrium and transport properties of model condense matter systems. The M.C. method has been extensively used at PINSTECH for the study of different physical systems of interest: spin glass percolation problems, catalytic surface reactions, single chc?in polymers, phase transitions and diffusion of atoms in adsorbed monolayers etc. I1—5]. So far, however, very little wovking experience in M.D. exists either in PINSTECH or elsewhere in the country. In the present study the alternate M.D. technique has been used to study the equilibrium and transport properties of liquid sodium, and in the process to acquire a working knowledge of this powerful simulation technique. In the M.D. method, one investigates the time evolution of position and velocities of atoms and molecules which constitute the system. This is done by constructing a suitable intermolecular potential model, and solving the equations of motion of these particles with a suitable numerical algorithm. The simulations are done on an isolated system so that the number of particles in the system and its total energy E is conserved, as the system moves along its trajectory in its phase space. During the simulations either the temperature and pressure or the temperature and density of the system can be kept constant. The simulation gives a detailed informtion about the velocities and positions of the particles of the system at different tim» s. This information together with the inter-atomic potential can be i?ed to study both its equilibrium as well as some non-equilibrium properties. Such simulations - 2 - can also be used to study dynamical properties of the system such as its velocity, auto-correlation function etc. which cannot be studied with the help of M.C. techniques. Liquid sodium melts at a relatively low temperature (97.8°C) and stays stable upto the boiling point (892°C). This together with its high specific heat, and thermal conductivity makes it a very desirable coolant in nuclear reactors. The Nuclear Materials Division has a programme for the establishment of a small liquid sodium heat transfer loop with an aim to study heat transfer parameters for this medium. The present study in addition to providing physical properties data for the design and experimental calculations in the prevalent working conditions, may also help in studying liquid sodium under conditions not accessible to the experimentalists. 2. MOLECULER DYNAMIC SIMULATION [6,7] a) Periodic Eoundary Conditions Computer simulations are usually performed on a small number of particles, typically lO^N^lOOO, because of limited computer time. Generally the particles are assumed to occupy a cubic box whose dimensions are fixed by the density of the fluid. Because of the limited number of particles, surface effects become very important. This problem is partly overcome by employing periodic boundary conditions in the computer programme by surrounding the cubic box on all sides by periodic replicas of itself. A molecule in the original box can interact with all molecules within its range of interaction including some of the "ghost" molecules in the replica boxes. Whenever, a molecule moves out of the box, it's ghost enters through the opposite side of the box. This ensures that the number of particles in the original box is conserved. - 3 - In such a simulation it is important to ensure that the range of the intermolecular potential is much less than the size of the box. If the potential is long range there is a danger of a significant interaction between a particle and its image in the neighbouring boxes. Moreover the use of periodic boundary conditions inhibits the occurance of long wavelength fluctuations. The properties of systems near the critical point, where fluctuations become very important are, therefore,difficult to study with the help of this technique. b) Spherical Cut Off A particle in principle interacts with all other particles (N) in the box. This would involve N(N-l) terms in the sum: an extremely large number for the computer to handle. In practice most of the contribution to the potential and forces comes from neighbours close to the particle of interest. For short range forces we apply a spherical cut off i.e; the potential for distances greater than a certain critical distance is taken to be zero, this results in a considerable saving of computer time. c) The Heating Up In order to generate the configuration for a particular temperature and density it is convenient (but not necessary) to start from a face centered cubic lattice. The lattice spacing is chosen so as to obtain the appropriate liquid state density. In M.D. the initial velocities of the particles have to be specified. Generally such velocities are chosen from a Gaussian distribution corresponding to a given temperature. The system is heated up to a temperature higher than the one at which it is to be studied to ensure that the system looses all memory of its previous history, and enhance the rate of equili­ bration. The mean - square velocity of the particles is - 4 - related to the temperature of the system by the following equation This equation is used to monitor the temperature. If this temperature is not in the region at which the system is to be studied (T), all velocities are rescaled up or down by the factor Tto^T The system is then allowed to equilibrate to the desired temperature. For subsequent runs we do not have to heat the system, it is sufficient to save the configurations of the system from the last run and use this as the initial configuration of the next run. d) Equations of Motion The force acting upon the i particle in the system at any time t is: where m. is the mass of the particle and V is its potential energy. Computing the particle trajectories, then involves solving 3N second order differential equations. In the M.D. simulation it is done with the help of predictor-corrector method1 . In This method one starts by specifying certain positions and velocities of all the particles at a time t, and guessing their values at a later time t+8t. The equations of motion are solved on a step by step basis, ^"t is typically smaller than the time taken by the particle to traverse its own length. An estimate of the positions, velocities, accelerations etc. at a time t+gt is obtained by the Taylor expansion. - 5 - br(t+st)*feL(0* The superscript P stands for predicted values. From the I* new positions •>• we can calculate the forces at time t+St, and hence the correct accelerations <Mt+£t). These are compared with the predicted accelerations, to estimate the size of the error in the prediction step. Aa (t+ Sfc) - * (t + ^) ~ <? ^ ^) This error is then used to generate the corrected values C t (Uft) »•**(*•«:)• Ca &q (t+£fc) c P + V tt + St) = V (b+ Sb) * S * * 0 **) c a (t^Ct) = a*(U Cfc) t ct &a (t+ St) The choice of C o ...C3, is given in the literature. The process is repeated and variables of interest are calculat­ ed after each such step. e) Choice of Potential The choice of a potential is extremely crucial in determining the correct properties of a system. Usually a potential is chosen so as to give agreement with certain experimental results, the same potential can then be considered good enough for the system under different conditions. For an N particle system the potential energy can be represented ass - 6 - The first term represents the effect of an exte/nal field. The 2 term is the two body interaction (pair interaction). The last term which represents three body interactions is usually not included in the computer simulation. Their effect is extremely small and does not Significantly change the results, on the other hand their inclusion would make the calculations extremely time consuming. 3. SIMULATION OF LIQUID SODIUM Paskin and Rahman 18) investigated the dynamics and structure of liquid sodium using long range oscillatory potentials of the form obtained by Johnson, Hutchinson, and March (JHM) 19]. They calculated the radial distribution function and self diffusion coefficient using a MD simulation. Their model potential had the form. 4^) _. -A C^)^™*!/V. V /*]} - *>*> *xf \?072q - to- m* where \= 3.72A° and B were taken as adjustable parameters. Using the value A=556 °K and p = 0.5954 chosen to fit the JHM potential, Paskin and Rahman were able to recover the radial distribution function from which the potential was obtained. It was subsequently shown by Paskin [10] that a long range oscillatory potential could be substituted by a Lennard - Jones 12-6 potential with parameters 6 = 620 K and (J~ = 3.32 A0 causing only small differences in radial distribution function g(r) in the region 5-7 A°.
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