Proceedings of ICONE TH International Conference on Nuclear Engineering Arlington, VA, USA, April -, ICONE-3 COMPUTER SIMULATION OF DIFFUSION OF PB-BI EUTECTIC IN LIQUID SODIUM BY MOLECULAR DYNAMICS METHOD Yingxia Qi Research Laboratory for Nuclear Reactors Tokyo Institute of Technology -- O-okayama, Meguro-ku, Tokyo 5-55, Japan qiyingia@nr.titech.ac.jp Minoru Takahashi Research Laboratory for Nuclear Reactors Tokyo Institute of Technology -- O-okayama, Meguro-ku, Tokyo 5-55, Japan mtakahas@nr.titech.ac.jp ABSTRACT Lead-bismuth eutectic is a potential candidate for coolant of secondary loops of sodium-cooled fast breeder reactors (FBR). The studies on the diffusion of liquid Pb-Bi in liquid Na are carried out corresponding to the case that liquid Pb-Bi leaks to liquid Na by accident. As the diffusion processes are the results of atomic motions, molecular dynamics method has been used to study the diffusion process. The self-diffusion coefficients of pure liquid Pb and Na, and liquid Pb-Bi are calculated and compared with ones by the empirical equations. The discrepancy between them could be eliminated by changing the densities of the liquids. The diffusion of leadbismuth in sodium is simulated based on the changed densities under which the self-diffusion coefficients of individual liquid metals are close to those by the empirical equations. The simulation results show that the diffusion process of liquid Pb- Bi in liquid Na is a heat releasing process and the density of ternary liquid Na-Pb-Bi is higher than the average value of the densities of liquid Na and liquid Pb-Bi. It is also found that the diffusion coefficients of liquid Pb-Bi in liquid Na are much higher than their self-diffusion coefficients, indicating that liquid Pb-Bi are easy and quickly to diffuse in liquid Na. However, the diffusion coefficient of liquid Na is decreased due to the existence of liquid Pb-Bi, implying that liquid Na- Pb-Bi have a higher viscosity than that of pure liquid Na. INTRODUCTION Lead-bismuth eutectic is a potential candidate for coolant of secondary loops of sodium-cooled fast breeder reactors (FBR). On the safety considerations, it is necessary to know what will happen in case that liquid Pb-Bi leaks into liquid Na flowing in the primary loop from the secondary loops. A lump of Pb-Bi may contact with the bulk of Na, fall down and form a pool in the bottom of the intermediate heat exchanger (IHX) with liquid-liquid direct contact. Next, Pb and Bi molecules may diffuse through the liquid-liquid interface and the contaminant may be transported into the reactor core. Remarkable progress has been made over the past years in the development of diffusion theories for liquid metals. A number of equations have been proposed for the self-diffusion coefficient calculation in liquid metals based on the various of theories, e.g. the equation proposed by Swalin (959) based on the fluctuation model; by Cohen and Turnbull (959) the freevolume theory, by Protopapas, Andersen and Parlee (973) the hard-sphere theory, by Breitling and Eyring (97) the significant structure theory etc. and the Stokes-Einstein equation related with the viscosity. Although these equations generally give good results for the magnitude and temperature dependency of the self-diffusion coefficient for liquid metals, the objections to the model theories are that, in general, they make drastic assumptions about the structure of a liquid and the mechanism of diffusion, and further, that they contain one or more parameters which need to be fitted using experimental data. As diffusion is the transport of mass from one region to another on an atomic scale, molecular dynamics theory seems to be a powerful tool, in which the atomic motions are calculated numerically. Currently, with increasing computer power, molecular dynamics method is being applied into more and more the fields to study the macroscopic phenomena from the molecular (atomic) level. For liquid metals, molecular dynamics method have been used to calculate the shear viscosity, diffusion coefficient, and other dynamical and structural properties of liquid sodium [][][3][], the phase transformation phenomena of liquid lead [5], and the properties of the liquid alloys of Na-Pb [] and Na-K [7], and so on. The studies on the diffusion process of liquid lead- Copyright by ASME
bismuth in liquid sodium has not been heard so far. In the present study, computer simulations by molecular dynamics method have been carried out to investigate the atomic mechanism of the diffusion, and to predict the diffusion coefficients of each constitute element in the ternary system along with the thermodynamic phenomena accompanying with the diffusion process, etc. to provide some reference information for design of next generation reactor system. SIMULATION METHOD Molecular dynamics method With the rapid developments of computer power, it is becoming more and more feasible to calculate the macroscopic properties of experimental interest from the microscopic details, such as the masses of the atoms, the interactions between them etc., through molecular dynamics method []. In molecular dynamics method, the model system is composed of a large number of atoms with well-defined interatomic interaction potentials. The positions and velocities of the atoms are traced with time. The atoms are assumed to be the classical ones following the Newton s equations of motion. Assuming a liquid metal system of N atoms, the Newton s equations of motion for atom i is: vi ri F i = mi = mi, ( i =,, N) () t t where F i is the force exerted on atom i of mass m i, v i and r i are the velocity and position vectors, respectively. The force F is determined from the derivative of the interatomic pair potential [9] : φ( r) F =, r φ is the effective pair-wise potential. And, where ( r) F = i j i F = j i φ ( r ) r, ( i, j =,, N) () is the sum of forces F on atom i due to atom j ( j i ) in the system. In principle, it is possible to construct an exact picture of a liquid using the molecular dynamics, provided that the pair potentials are accurately known. The accuracy of the potentials used to determine the forces between the atoms is essential to the success of a computer experiment. At present, the potentials for simple metals, such as alkali metals, are well defined while those for heavy metals such as bismuth are scarcely investigated. Here, Born-Mayer [] type of analytical pair potentials are used because only this type of potentials are available for all the three metals of sodium, lead and bismuth. The form of Born-Mayer potentials is [] Br ( r) = Ae φ, (3) where r is the distance between the pair atoms. The constants in Born-Mayer potentials are shown in Table. Table. The constants in Born-Mayer potentials [9] Sodium Lead Bismuth A (ev) 3. 9 39 B (Å) 3.77759 3.59 3.99 This is a hard-sphere potential. The potentials for sodium, lead and bismuth are shown in Fig. as a function of the distance r between atoms. Potential, φ(r)(ev) Na-Na Pb-Pb Bi-Bi 3 5 Separation distance, r(å) Fig. Born-Mayer potentials [9] The interactions between the atoms of two different species can be described by the geometric mean of the interactions between the atoms of the same species [][7], i.e., Then, ( r) [ φ ( r) φ ( )] φ =, () r F = ( Fφ + Fφ) (5). φ φ Solution of equations of motion The differential equations of motion () were solved by the finite difference method on a step-by-step base with a small time interval δ t. Here, the so-called leap-frog algorithm [] was adopted for integration of the equations of motion. This is a direct solution of the first-order equations of motion. It is demonstrated briefly as the following: Fi ( ) ( ) ( t) vi t + δ t = vi t δt + δt mi, vi ( t) = [ ( t i δt) i ( t v + + v δt) ], r i ( t + δt) = ri ( t) + δtvi ( t + δt), ( i =,, N) (). In practice, the velocity equation is implemented first by assuming that the force F i is constant during the time interval ( t t, t + δt ). Then the position r at time t + δt is δ calculated from the mid-step velocity at t + δt. The current velocity v at time t is necessary to calculate the energy as well as other quantities of the system. The upper limit for δ t is determined by the Debye frequency. By calculation tests, it is found that δt of -5 s is suitable for the present calculations, to maintain the stability of the system and for the system parameters to achieve convergence in short times. Copyright by ASME
Evaluation of the macroscopic properties The temperature of the system can be calculated using the virial theorem [3] as the following equation: T = miv i, (7) k 3N N B ( ) where k B is the Boltzman constant, 3N is the total number of degrees of freedom for the system of N atoms while Nc is the total number of independent internal constraints. In the present calculations, Nc is equal to three as the centre of masses of the system is constrained at three directions. The pressure P was also calculated using the virial theorem [3] as PV = NkB T + W, () where V is the volume of the system, W is the virial item: = w( r ), (9) W 3 w ( r ) c i j > i φ( r ) = r r i (). The self-diffusion coefficient D was calculated using the following equation: D = Nt N i= ( ) r ( ) r, () i t where < >denotes the average on all the atoms and over the whole simulation time period. Model system In order to calculate the properties of liquid metals, a small amount of liquid sample was taken out from a bulk liquid which is a unique and isotropic fluid for liquid sodium, lead and bismuth or lead-bismuth alloy. The sample is confined in a cubic cell which is called a simulation cell. Centered on the simulation cell, the cells are replicated in three directions in space to simulate an infinite system. In this case, the periodic boundary conditions could be imposed on the central simulation cell, that is, a particle which leaves the cell across one face re-enters again across the face opposite in the symmetrical way to keep the number of the particles in the cell. The details of the model systems are described in Table. Table. Model systems used in simulations T= K Density (kg/m 3 ) Edge length of cube (Å) Number of atoms Liquid Na 3. Liquid Na * 33 3. Liquid Pb 3 3. Liquid Pb * 575 39.5 Liquid Bi 975 33. Liquid Pb-Bi 7 3. 37 (5.5%Pb) Liquid Pb-Bi* (5%Pb) 5555 3. 55 *: density is deviated from the experimental data. i Since the interaction potentials decrease exponentially with the distances (see Eq. (3)), a cut off distance r c was applied to save the computer time. That is: φ ( r) =, if r > r c (). The cut off distance was always equal to the half length of the simulation cell in the present simulations. Simulation condition The initial atomic positions in the model system were randomly distributed. In order to reach the equilibrium state, the model systems first underwent a relax process. During the relax process, the extra energies of the systems were extracted by controlling the temperatures at the desired ones (NVT ensemble). Then, the systems will run independently without exchanging energies with externals (NVE ensemble). During this period, the total energy of the system E should be kept constant: E = Ek + E φ, (3) where E k is the kinetic energy and Eφ is the potential energy. The systems become equilibrium at the desired temperatures. These processes are depicted in Fig.. For the first, time steps, the temperature was controlled at 9 K, the rest run independently. The time required for relaxing process was dependent on the system, up to, time steps here. The data for statistical calculations were collected over the equilibrium processes. Energy (KJ/mol) 7 5 NVT Internal energy NVE 3 9 Temperaure x x 3x x 5x x Number of time steps Fig. System variations with simulation time step in NVT and NVE ensembles. Diffusion model For the diffusion process of liquid Pb-Bi alloy in liquid sodium, the physical model is considered as the following. Suppose that a drop of liquid Pb-Bi falls down into a bulk of liquid sodium, and diffuses isotropically. For simplicity, the diffusion on one-dimension was considered here. Therefore, the simulation model system is constructed as shown schematically in Fig. 3. The simulation model system is composed of three cubic cells with same edge length 3. Å, each of which is consist of Na atoms, 5 Pb and 3 Bi atoms, and Na atoms. The cells share the same face on the direction perpendicular to 3 Copyright by ASME
the paper plane between the adjacent cells, and each of them is at equilibrium state individually before diffusion. About the boundary condition, the three cells extend infinitely on the two directions vertical to the diffusion direction. Along the diffusion direction, the cells of liquid sodium extend infinitely towards the two outside directions from the middle cell of liquid Pb-Bi. Na Pb-Bi Na Self-diffusion coefficient ( -5 cm /s) 7 5 3 MD cal. empirical eqn. MD cal.* Sodium Fig. 3 The simulation model system before diffusion. white: Na, smaller black: Bi, larger black: Pb At the first, the system is relaxed due to the overlap of the atoms on the interfaces between the adjacent cells for a short time and equilibrated at K for some time. Then the diffusion process starts under the conditions of constant-nve. RESULTS AND DISCUSSIONS Properties for pure liquid metals The self-diffusion coefficients for pure liquid metals of sodium, lead and bismuth, and lead-bismuth alloy were calculated in order to compare them with experimental ones. It is shown in Fig. that the self-diffusion coefficients at different temperatures converge to the different values with simulation progressing time. Self-diffusion coefficient ( -5 cm /s) 7 5 3 K 5K K 7K 9K K Simulation progressing time (ps) Fig. Convergences of self-diffusion coefficients with simulation time for liquid sodium. It is evident that some long simulation time is necessary for the convergence of the self-diffusion coefficients. The self-diffusion coefficients mentioned later are referred to the conve rged ones. Fig. 5 shows the calculated self-diffusion coefficients for liquid sodium as the function of temperature, compared with [] the empirical equation from fitting eqn () with experimental data: Fig. 5 Variations of self-diffusion coefficient with temperature. *: density is increased by 37%. Empirical equation: by eqn (), valid range: ~ K. H = D D exp () RT where, R is the gas constant (.3 J/K*mol), D and H are constants. For Na and Pb, the constants are shown in Table 3. Table 3. Constants in equation () [5] D ( 5 cm /s) H ( 3 J/mol) Na. Pb 9.5. It can be seen that the calculated self-diffusion coefficients are much higher than that by the empirical equation. However, the temperature dependencies are almost same. The discrepancy between them may stem from the potential models used as the potential models determine solely the atomic motions. By increasing the density of liquid sodium by 37%, the calculated self-diffusion coefficients became close to the curve by the empirical equation, as shown in Fig. 5. The runs with densities changed are marked with symbol *. Self-diffusion coefficient ( -5 cm /s) empirical eqn. for Pb MD. cal. for Pb-Bi* MD. cal. for Pb* 7 9 3 Fig. Variations of self-diffusion coefficient with temperature. *: density is decreased by 5%. Empirical equation: by eqn (), valid range: ~ 9 K. Copyright by ASME
The calculated self-diffusion coefficients for lead and bismuth deviate from ones by the empirical equation, too, being too small. By decreasing the densities by 5 %, the calculated self-diffusion coefficients became close to the curve by the empirical equation as shown in Fig.. Since the constants in the empirical equations for liquid bismuth and liquid Pb-Bi alloy are not available, the curves for their empirical equations are not shown in Fig.. The phenomena happened in the diffusion process of liquid Pb-Bi in liquid sodium are discussed below. Atomic distribution variations The model liquids used in the diffusion process are the liquids with the densities changed as described above in order to make the self-diffusion coefficients close to the practical ones. The positional distributions of lead and bismuth atoms with simulation time are presented schematically in Fig. 7. Fig. 7a Atomic distributions after 7 ps diffusion Concentration (%) Before diffusion t= ps Bi t= ps t= ps t= ps t= ps t= ps t= ps - - - Position (Å) Fig. 9 Variations of the concentration profiles of sodium atoms. The variations of the concentration profiles of the lead and bismuth atoms are shown with simulation time in Fig. and Fig. 9, respectively. It can be seen that the lead atoms and the bismuth atoms diffuse at the same time. But the bismuth atoms diffuse faster than the lead atoms. Thermodynamic properties It is confirmed from Fig. that the total energy of the system E is kept constant.. Energy (KJ/mol).3 Fig. 7b Atomic distribution after ps diffusion white: Na, smaller black: Bi, larger black: Pb Atomic concentration profile variations.. 3.x.x 9.x.x 5.5x 5 Number of steps Fig. Trajectory of the internal energy of the system. Concentration (%) Before diffusion t= ps Pb t= ps t= ps t= ps t= ps t= ps t= ps - - - Position (Å) 9 7 5. 3.x.x 9.x.x 5.5x 5 Number of steps Fig. Variations of the concentration profiles of lead atoms. Fig. Trajectory of the temperature of the system. 5 Copyright by ASME
Pressure (MPa) 3. 3.x.x 9.x.x 5.5x 5 Number of steps Fig. Trajectory of the pressure of the system. The trajectories of the temperature and pressure of the system during the diffusion process are shown in Fig. and Fig., respectively. It can be seen that the temperature is gradually increased from K to K. As the total energy (kinetic energy plus potential energy) of the system is kept constant, it means that the potential energy of the system transforms into the kinetic energy of the system. On the other hand, the pressure of the system is gradually decreased. Obviously, if the pressure is kept constant, the system should be compressed. This indicates that the volume of the ternary liquid Na-Pb-Bi is smaller than that of the sum of the liquid Na and Pb-Bi. In other words, the density of the ternary liquid Na- Pb-Bi is higher than the average density of pure liquid Na and liquid Pb-Bi. Diffusion coefficients The diffusion coefficients for each specie of atoms are shown in Fig. 3. It can be seen that the diffusion coefficients of lead and bismuth in sodium are higher than their selfdiffusion coefficients shown in Fig., indicating that liquid Pb-Bi are easy and quickly to diffuse in liquid Na. However, by contrast with Fig. 5, it is known that the diffusion coefficient of liquid Na is decreased due to the existence of liquid Pb-Bi, and the diffusion coefficient of the ternary liquid Na-Pb-Bi is lower than that of liquid sodium. Self-diffusion coefficient ( -5 cm /s) 3 5 5 5 Na-Pb-Bi Na Pb Bi T= K Simulation progressing time (ps) Fig. 3 Diffusion coefficients for different species of atoms during the diffusion process. CONCLUSION The self-diffusion coefficients of sodium, lead and leadbismuth alloy are calculated using the molecular dynamics method, and compared with ones by the empirical equations. The discrepancy between them may be caused from the potential models used which define solely the atomic motions. By changing the densities of the liquids, the discrepancy between them could be eliminated. The diffusion process of lead-bismuth in sodium is simulated based on the changed densities under which the selfdiffusion coefficients of them are close to that by the empirical equations. The results show that the lead and bismuth atoms diffuse easily in liquid sodium by the interatomic forces. And the diffusion will cause the increase of the temperature and decrease of the volume of the diffusing system. It is implied that the viscosity of the ternary liquid Na- Pb-Bi is higher than that of the pure liquid Na. The future work may should be put on to modify the present potential model to reproduce the self-diffusion coefficients of liquid metals more accurately. ACKNOWLEDGMENTS Financial support from the Japan Society for Promotion of Science (JSPS) is gratefully acknowledged. REFERENCES [] T. koishi et al., J. Non-Cryst. Solids 5-7, 33 (99). [] U. Balucani et al., J. Non-Cryst. Solids 5-7, 99 (99). [3] Anil P. Bhansali et al., HTD-Vol. 3, ASME National Heat Transfer Conference (99). [] Ten-Ming Wu et al., physica A 5, 57 (99). [5] J. Rybicki et al., Computer Physics Communications 97, 5 (99). [] Y. Senda et al., J. Non-Cryst. Solids 5, 5 (999). [7] L E Gonzalez et al., J. Phys.: Condens. Matter, 5 (99). [] Allen, M. P., Tildesley, D. J., Computer Simulation of Liquids, Oxford Science Publications, Oxford, 97. [9] Ian M. Torrens, Interatomic Potentials, Academic press, New York and London, 97. [] A. A. Abrahamson, Phys. Rev. 7, 7 (99). [] F. T. Smith, Phys. Rev. A 5, 7 (97). [] Hockney, R. W., Methods comput. Phys. 9, 3 (97). [3] Munster, A., Statistical Thermodynamics, Academic Press, New York, 99. [] T. Iida, Roderick I. L. Guthrie, The Physical Properties of Liquid Metals, Oxford Science Publication. [5] Meyer, R. E. and Nachtrieb, N. H., J. Chem. Phys. 3, 5 (955). [] T. Nanba et al., J. Non-Cryst. Solids 5, 3 (99). [7] S. M. Foiles, Phys. Rev. B 3, 75 (95). Copyright by ASME