PINSTECH/NPD-124
MOLECULAR DYNAMICS! APPLICATION TO LIQUID SODIUM
Kbawaja Yaldram Abdullah Sadiq Sohail Murad
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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 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°. Apparently, the qualitative features of the short range liquid metal structure depend primarily on the depth and position of the potential minimum and not to any great extent on the potential shape. For our studies we have retained the model potential proposed by JHM. The potentials considered in M.D. simulations are entirely ionic in nature, they do not contain any electronic contribu tions. The results obtained for the specific heat and thermal conductivity will therefore represent only the ionic contribu tion. - 7 - We consider a system of 256 particles, enclosed in a cube of side L, with periodic boundary conditions interacting via the JHM potential. The value of L is adjusted so as to reproduce the density of interest. The interction was assumed R to extend upto a range cut-^ so that a particle interacts with all particles situated within a sphere of that radius. To simplify calculations we choose the reduced units: the lengths are expressed in units of T»n\C v mo* ' The initial staiie in the MD calculations was the F.C.C. lattice. Which was overheated for a rapid destruction of the long-range order. After that the system was cooled to the temperature of interest. The equations of motion were solved through a predictor-corrector algorithm. The time step was chosen as 1.659x10-1 5 sees. This is the average time taken for a particle to travel its own length at temperature of interest. The positions of particles obtained at regular intervals are recorded for use at a later stage. Different quantities like mean-square displacements, specific heat, configurational energy, density distribution function, thermal conductivity etc. are measured at regular intervals. - 8 - 4. RESULTS AND DISCUSSIONS a) Self Diffusion Coefficient The average distance travelled by a particle during a time t may be expressed as: ^ N being the total number of particles and r.(t) the position of particle i at time t. Our results forC*^|are drawn in Fig*J*.. For large time we obtain a linear behaviour. The constant D in equation is the self diffusion coefficient. Prom the slope of the plot of time versus mean square displacement we can get the self diffusion coefficient directly. These have been obtained for five representative temperatures and corresponding densities along the saturation curve and compared with the experimental values (Table-I). Experimentally it is seen that the temperature dependence of the self-diffusion coefficient of liquid sodium can be expressed by an exponential function that contains two constants, a frequency factor D . and the activation energy Q [11,12]. Our values for self diffusion lie within the limits of the errors in the experimental results. - 9 - Specific Heat Certain investigators [13,14] have reported that C is essentially constant over a temperature range from about 100°C whereas others [15,16] have reported a monotonic decline of C with temperature. Still other [17] have reported a steady decline to about 300°C and then essentially constant behaviour. The recent investigation by Stone [18] closely confirms an older data of Ginnings [19]. These data can be expressed by the equation. C (cal/g) * 0.34324-1. 3868xl0~4t + 1.104xl0~7t2, where t is the temperature expressed in °K,which is valid to within 0.4% from the melting point to about 900°C. Experimental data (circles) along with the curve constructed from the above equation is reproduced in Fig.2. Our MD results for the specific heat are in excellent agreement with the experimental values for all ranges of temperatures. c) Viscosity Aothough a MD programme for the direct evaluation of viscosity was available to us, we could not use it due to the nonavailability of a subroutine for matrix inversion. We therefore evaluated the viscosity indirectly through the use of the Navier Stokes theorem. n - JLL L~ mrc>Ak where D is the self diffusion coefficient and A. is the hydraulic radius of sodium atoms. A. for sodium as far as we know is not available in literature, we therefore calculated A. by using the experimental values of D and1). Since there is a significant error in the experimental value of D we obtained Ah to be (2.86 ± 1.43) A0. Our calculated values of'I are in satisfactory agreement with the experimental ones [12]. We hope the agreement would improve significantly once we obtain the viscosity directly from the MD simulation. - 10 - d) Thermal Conductivity The thermal conductivity coefficient is defined with respect to the steady flow of heat when there exists a temperature gradient through the relation. ST QxA wnere Q is the flux of thermal energy. A is expressed in units watt/cm°C. In pure metals most of the heat is carried by electrons. The thermal conductivity is therefore almost totally an electronic phenomena. In our MD calculations we can only measure the lattice or ionic contribution to the thermal conductivity. This contribution as expected turns out to be rather small. Theoretically, the lattice component has been calculated from the relation \= 237/T [23]. Prabhuram and Saksena [24,25] have emphasized the predominant role of structure scattering for the phonons. They have calculated the lattice contribution which seems to agree with the experimental values calculated indirectly as the difference of the total thermal conductivity and the electronic conductivity. However their criteria for choosing the values of some of the constants makes this comparison suspect. In Table-II, we have compared our values with those obtained by the above authors. We find much better agreement with the theoretically predicted results of Cook et. al.[23]. e) Radial Distribution Function Most of the experimental results on liquid metals are obtained either through X-Ray scattering or neutron diffraction studies. The quantity measured in these experiments is the structure factor £(q), which is the ratio of scattered intensity to incident intensity. The radial distribution function g{r) is the Fourier transform - 11 - of the structure factor. ^o g(r) can be measured directly in a M.D. simulation. In Fig.3.we represent g(r) for liquid sodium at 417 °K. The radial distribution function seems to be insensitive to temperature over quite an appreciable temperature range. A compaiison with experimental position and height of the peaks gives a discrepency with our results which we have still not resolved. CONCLUSIONS The use of the JHM potential function reproduces fairly satisfactorily most of the experimental results over a wide temperature range. This gives us confidence that we can use this potential for determining properties of liquid sodium that are not accessible to the experimentalists- We hope to undertake the calculations for viscosity shortly, work is also in progress to study the effect on the thermodynamic properties of Sodium as we pass through the transition from solid to liquid phase. AC KNOWLEDGEMENT S One of the authors (SM) would like to acknowledge the financial assistance of the National Talent Pool, Govt, of Pakistan and the generous hospitality extended to him by PAEC during his stay in Islamabad. The authors would like to place on record their gratitude and sincere thanks to Dr. G.D. Alam, Head Computer Division and has colleagues without whose cooperation and support this project would have been impossible. We would al^o like to than!: Mr. Nisar, Mr. Daniyal, Mr. Gulfam and especially Mr. Dad who helped in their various capacities in making the project a success. - 12 - REFERENCES 1. Y. Khawaja and A. Sadiq: J. Pol. Sc. 1^9, 499(1981). 2. A. Sadiq and Y. Khawaja: Z. Fur. Phys. A2^, 163(1981). 3. A. Sadiq and K. Yaldram: J. Phys. A 2±, L207(1988). 4. A. Sadiq and K. Binder: Surf. Sc. 128, 350(1983). 5. A. Sadiq: Phys. Rev B9, 2299(1974). 6. M.P.Allen and D.J. Tildesiey: "Computer Simulation of Liquids" Clarendon Press-Oxford 1987. 7. "Simulation of Liquids and Solids": editors G. Ciccotti, D. Frenkel and I.R. McDoneld. North Holland (1987). 8. A. Paskin and A. Rehman: Phys. Rev. Lett. 1_6, 300(1966). 9. M.D. Johnson, P.Hutchinson, and N.H. March: Proc. Roy. Soc. (London), Ser. A 282, 283(1964). 10. A. Paskin: Advan. Phys. 16, 223(1967). 11. R.E. Meyer and N.H. Nachtrieb: J.Ch. Phys. 21' 1851 (1955). 12. Sodium-N K Eng. Handbook Vol. I. ed: O.J. Foust. Gordon and Breach. 13. P.Y. Achener and D.L. Fisher: OSAEC Report AGN-8191 (Vol. 6). 14. I.M. Pchelkin: USAEC Report AEC-tr-4511 (1959). 15. R. Hultgren et. al.: "Selected values of thermodynamic properties of metals and alloys" John Wiley and Sons Inc. New York 1963. 16. N.A. Nikolskiy et.al.: Russian Report MCL-714/1,1961 and MCL-714/1,1961. 17. D.L. Martin: Phys. Rev. 1J_4, 571(1967). 18. J.P.Stone et.al.: Report NRL-6241, Naval Research Labs. Dec. 24, 1964. 19. D.C. Ginnings et.al.: J. Res. Nat. Bur. Stand. 45_, 23(1950). 20. C. Kittel: Introduction to Solid State Physics John Wiley and Sons. - 13 - Saksena, Probharam and Desrhora: Ind. J. Pure. Appl. Phys. 22, 620 (1984). G. Fritsch and E.Luscher: J.Phys. Chem. Solids. 3_3_, 2041 (1972). J.G. Gook, M.P. Vander Mcer and M.J. Laufitz: Can. J. Phys. 50, 1386(1972) , Prabhuram and M.P. Saksena: Ind. J. Pure and Appl. Phys. .19, 717(1981). M.P. Sakr-ena, Prabhuram and P. Dashora: Ind. J. Pure and Appl. Phys. 22, 620(1984). - 14 - FIGURE CAPTIONS Fig. 1: Mean square displacement of the particles plotted versus time for three different temperatures along the saturation curve. Both<^r (t)^> and t are in reduced units. Fig. 2: Specific heat (c ) of liquid sodium. Circles represent the experiment values [18], the continuous line is the curve constructed from the equation as presented in the text. Fig. 3: Radial distribution function of liquid sodium at temperature 417 °K. t < 0.18 200 400 600 800 1000 1200 TEMPERATURE CO ^Z. (J)6 TABLE 1 Self-Diffusion Coeff. Viscosity Specific Heat 'UT£ DUO4 cn//Sec (centipoise) C (Cal/g °C) gm/cm Cal.. Exp Cal Exp Cal Exp Cal Exp i i i 0.924 397 * 371.5 0.41 ! 0.4+0.2 0.5640.28 0.705 0.36 !0.33 • 0.89 546 ! 523 0.6 ! 1.06+0.58 0.7+0.35 0.4 0.245J 0.315 « r i • 0.72 * 1.5+0.7 0.853 524 • 623 • 0.67+0.33 0.31 0.34 J 0.305 i 1 0.80 945 J 923 2.18 j 2.9+1.2 0.33+0.17 0.2 0.35 ', 0.30 1 1 1 0.76 1079! 1073 2.36 I 3.5+1.4 0.35+0.18 0.17 0.29 | 0.305 • • 1 i i i i A comparison of calculated and experimental Diffusion Coefficients, Viscosities and Specific Heats.at different temperatures and densities. TABLE 2 id Classical thermal conductivity P T°K "# (Watt/cm °C) ^ - gm/cm" M.D. !2.37[23] Ref.[24] Experimental I4.2T Ref. [25] i T I 0.924 390 10.0035 + 0.0005 |0.006 0.0235 10.0235 I I I 0.924 473 '0.0031+0.0005 '0.005 0.0205 10.021 I I I l 0.0197 0.89 523 ,0.0036+0.0005 |0.0045 0.02 I I 0.853 623 (0.0031+0.0005 (0.0038 0.019 0.0195 I I 0.80 923 10.0021+0.0005 j0.0026 0.013 •0.023 i I | 0.76 1073 |0.0023+0.0005 |0.0022 Calculated and experimental thermal conductivity for different temperatures and densities.