1 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 Density of Liquid Lead as Function of Temperature and Pressure Based on the Molecular Dynamics Method Muhammad Abdul Bashar Imanullah 1 , Artoto Arkundato 1,a and Endhah Purwandari 1 1 Physics Department, Faculty of Mathematical and Natural Sciences, Jember, Indonesia a a.arkundato@unej.ac.id Abstract. Simulation research has been carried out to obtain the formula for mass density of liquid lead as a function of temperature and pressure. The simulation method used is the molecular dynamics method. The potential energy used in the simulation is the Morse potential. From the simulation, it is found that the relationship between the mass density of liquid lead and temperature and pressure can be expressed in the equation for pressure 1 – 5 atm and for pressure 7 atm in units kg/m 3 . Keywords: Liquid Lead, Mass Density, Temperature, Pressure, Molecular Dynamics. Introduction Currently, nuclear power plants have become one of the alternative energy sources which are becoming an interesting choice of many countries. If developed countries have long used nuclear energy to support their heavy industries, developing countries think about the use of this energy in a recent time. The nuclear reactor basically produces heat energy from the process of nuclear fission. The subsequent use in the form of conversion to electricity in large quantities is one of the attractions for the development of this nuclear energy, besides there may concerns for utilization in the field of weaponry. The development of nuclear reactors for welfare today generally still relies on the design of thermal reactors. There are still many thermal reactors operating in this world. Unfortunately, many thermal reactors are still legacy of old designs. Since the Chernobyl nuclear reactor accident in Ukraine in the past, a safer new concept reactor design is constantly being considered. For this reason currently developing IV generation nuclear reactors. Some of the advantages of this generation IV reactor design are the inherent safety concept. Liquid metal-cooled fast nuclear reactors, for example liquid lead, are reactor concepts and designs that apply the safety concept inherent in the design so that if there is a potentially catastrophic anomaly in the reactor, the reactor system can automatically shut down the operation of the nuclear reactor without the need for any process of manual shutdown. More specifically, the liquid lead metal cooled fast nuclear reactor is one of the promising reactor concepts apart from applying the inherent safety concept it is also capable of being made in a modular form and producing high power energy [1]. 2 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 Liquid lead metal and its alloys with other metals such as bismuth are currently promising candidate materials for cooling fast nuclear reactors [2]. There are many advantages compared to cooling water in a slow neutron nuclear reactor. One of them is that it has a high boiling point and has a large thermal conductivity, making it very suitable for cooling in fast reactor designs. For this reason, in order for the design of a fast nuclear reactor to be made properly, complete nuclear data, for example, information on mass density, is required. Because in the reactor two factors that are very important to note are temperature and pressure, this mass density needs to be known as a function of temperature and pressure. In this study, therefore, we want to know the density formula as a function of temperature and pressure. To obtain this, the molecular dynamics method will be used so that a lot of data can be obtained for various temperatures and pressures. Theory Molecular dynamics methods in general can be used to predict the physical properties of materials such as melting point, heat conductivity, enthalpy, diffusion coefficient, and so on, including predicting density as a function of temperature. What matters is whether we have sufficient potential energy data to describe the interactions between the atoms of the material system. Arkundato et al has used the molecular dynamics methods to investigate liquid lead coolant [3-4]. In this research, we will use the Morse potential. In the molecular dynamics method, the Newton motion equation is solved to get the trajectories of all the atoms that make up the material. The relationship between Newton's equations of motion and potential is as follows F = m d 2 r/dt 2 (1) F = - dV/dr (2) where F is the interacting force of particles, r position vector of a particle, V is potential energy of interacting particles. The Morse potential for this simulation has a form of [5]. ( ) e - - ( - ) (3) where re is the equilibrium bond distance from the atom, De is the bonding energy balance of an atom and a is a parameter, √ (4) Girifalco dan Weizer (1958) has made table of Morse potential parameters as below [6]. 3 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 Table 1. Morse potential parameters of some cubic metals [5] Metal ( ) D(ev) Pb 2.921 83.02 7.073 1.1836 3.733 0.23480 Ag 2.788 71.17 10.012 1.3690 3.115 0.33230 Ni 2.500 51.78 12.667 1.4199 2.780 0.42050 Cu 2.450 2.450 49.11 10.330 1.3588 0.34290 Al 2.347 4417 8.144 1.1646 3.253` 0.27030 Ca 2.238 39.63 4.888 0.80535 4.569 0.16230 Sr 2.238 39.63 4.557 0.73776 4.988 0.15130 W 2.225 72.19 29.843 1.4116 3.032 0.9906 Cr 2.260 75.92 13.297 1.5721 2.754 0.4414 Fe 1.988 51.97 12.573 1.3885 2.845 0.4174 In our work we will use Morse potential parameter from Girifalco data. Furthermore, based on the trajectory of the atoms of the material as a solution of Newton's equations of motion then using statistical mechanics concepts and theories it can be predicted any physical quantities that we want to know. This physical quantities can be calculated easily when we use the Lammps molecular dynamics software [6] that also we used in this research (https://lammps.sandia.gov/). Method The purpose of this study was to find the mass density relation of liquid lead as a function of temperature and pressure. Physical variables mass density, pressure, temperature were simulated with Lammps software. The procedure for obtaining the relationship between the three variables is carried out according to the following steps: 1. Create a script file that contains data on the position of lead atoms, mass, number of atoms, pressure, temperature, number of integration steps, compute command, etc. Table 2 is a summary of input parameters. Table 3. Input parameters for Lammps simulation Variable Value Mass of Pb 207.2 Lattice constant 4.950 Temperatures 323 , 423 , 523 , 623 , 723 , 823 , 923 , 1023 D 0.2648 ev 3.520 1.3036 - pressure 1 atm, 5 atm, 7 atm 2. Do simulation for different temperature and pressure 3. Make analysis and conclusion 4. Determine mass density as a function of temperature and pressure 4 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 Results From lead simulation results for different temperatures and pressures then we can calculated the density as Table 4. Table 4. Density of liquid lead at various temperatures and pressures Temperature (K) Mass density (kg/m 3 ) pressure1 atm pressure 5 atm pressure 7 atm 323 10918.59013 10918.71963 10918.78437 423 10839.89903 10840.03266 10840.09959 523 10758.77752 10758.90916 10758.97953 623 10673.75396 10674.02894 10674.10180 723 10582.58427 10582.73939 10582.81616 823 10479.97194 10480.13611 10481.66200 923 10356.97857 10357.16451 10357.58981 1023 10189.90371 10190.05928 10190.47230 From Table 4 we can determine the mass density as a function of temperature and pressure using linear regression method, For pressure 1 atm: [1 atm] [5 atm] (5) Equation (5) also applies for pressure 5 atm. For pressure 7 atm there is a little different, i.e., [7 atm] (6) Figure 1 shows the mass density of liquid lead at 1 atm as a result of molecular dynamics simulation. Figure 1. Mass density of liquid lead at 1 atm ρ = 11233-0.9217 x T R² = 0.9937 10300 10400 10500 10600 10700 10800 10900 11000 0 200 400 600 800 1000 ρ (k g /m 3 ) T (K) 5 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 Let’s we compa e ou simulation esult and a reference [7]. Sobelov states that the formulation value for the density of liquid lead at a pressure of 1 atm is: [1 atm] (7) We can check the discrepancy between simulation and reference as shown in Table 5. Table 5. Comparison the mass density by simulation and reference Temperature (K) Mass density (kg/m 3 ) (simulation) Mass density (kg/m 3 ) (Sobelov, 2011) discrepancy (%) 323 10935,2909 10997,7215 0,57% 423 10843,1209 10869,7715 0,25% 523 10750,9509 10741,8215 0,08% 623 10658,7809 10613,8715 0,42% 723 10566,6109 10485,9215 0,77% 823 10474,4409 10357,9715 1,12% 923 10382,2709 10230,0215 1,49% From Table 5 we can conclude that our simulation results are pretty good when compared to references. In our simulation we have used 500,000 atoms of Pb created by 50 x 50 x 50 boxes of unit cells in FCC crystal structure. Then this also describes that the Morse potential and its parameters is good to formulate the interaction among metallic atoms of lead. Conclusions The simulation results are pretty good when compared to references. The simulation method used is the molecular dynamics method. The potential energy used in the simulation is the Morse potential. In our simulation we have used 500,000 atoms of Pb created by 50 x 50 x 50 boxes of unit cells in FCC crystal structure. Then this also describes that the Morse potential and its parameters is good to formulate the interaction among metallic atoms of lead. From the simulation, it is found that the relationship between the mass density of liquid lead and temperature and pressure can be expressed in the equation for pressure 1 – 5 atm and for pressure 7 atm in units kg/m 3 . References [1] T Sofu, 2015, A review of inherent safety characteristics of metal alloy sodium-cooled fast reactor fuel against postulated accidents, Nuclear Engineering and Technology, volume 47, issue 3, pages 227-239. [2] J Zhang, N Li, 2008, Review of the Studies on Fundamental Issues in LBE, Journal Nuclear Mater, volume 373, page 351-377. [3] kundato. Z Su’ud, Abdullah and Widayani, 2013, Molecular Dynamic Simulation on Iron Corrosion Reduction in High Temperature Molten Lead-Bismuth Eutectic, Turk. J. Phys, volume 37, page 132-144. https://www.sciencedirect.com/science/article/pii/S1738573315000753#! https://www.sciencedirect.com/science/journal/17385733 https://www.sciencedirect.com/science/journal/17385733/47/3 https://www.sciencedirect.com/science/journal/17385733/47/3 6 Computational and Experimental Research in Materials and Renewable Energy (CERiMRE) Volume 1, Issue 1, page 1-6 Submitted : August 20, 2018 Accepted : October 10, 2018 Online : November 10, 2018 doi : 10.19184/cerimre.v1i1.19541 [4] kundato, Z Su’ud, M Abdullah, W Sutrisno and M Celino, 2013, Inhibition of iron corrosion in high temperature stagnant liquid lead: A molecular dynamics study, Annals of Nuclear Energy, volume 62, page 298-306. [5] D D Abajingin, 2012, Solution Of Morse Potential for Face Centre Cube Using Embedded Atom Method, Advance in Physics Theories and Applications, volume 8, page 36-44. [6] L A Girifalco, V G Weizer, 1959, Application Of The Morse Potential Function To Cubic Metals, Physical Review, volume 114(3), page 687-690. [7] S Plimpton, 1995, Fast Parallel Algorithms for Short-Range Molecular Dynamics, J Comp Phys, volume 117, page 1-19. [8] V Sobelov, 2011, Database of Thermophysical Properties of Liquid of Metal Coolats for GEN-IV, Belgium, SKC.CEN.