Molecular dynamic simulation of the melting and solidification processes of argon

Transcription

1 Journal of Mechanical Science and Technology 23 (200) 1563~1570 Journal of Mechanical Science and Technology DOI /s Molecular dynaic siulation of the elting and solidification processes of argon Jae Dong Chung Departent of Mechanical Engineering, Sejong University, Seoul, , Korea (Manuscript Received June 24, 2008; Revised February 25, 200; Accepted April, 200) Abstract Molecular Dynaic (MD) siulations have been conducted to look at the elting and solidification of the Lennard- Jones argon (100) interface with sall aounts (up to 6.0K) of undercooling and superheating. By cobining the fully equilibrated bulk phases of liquid and solid in one siulation box and counting the nuber of solid like particles, the interface velocities, i.e. the growth rate or elting rate, were obtained as a function of teperature. The elting teperature, where no growth or elting of crystal particle is expected, is T =0.668 which is close to that of the Gibbs free energy calculation. A linear dependence of growth or elting rate on teperature was found except for high superheating, T > 6K. The high superheating is believed as the ain source of slope discontinuity in the rate, not the isuse of initial regie as discussed in the earlier works. Keywords: Phase transition; Freezing rate; Molecular dynaic siulations; Melting teperature Introduction The study of the growth rates of crystals fro the elt or fro supersaturated solutions is of iportance for the prediction of their acroscopic growth orphologies. Although uch is known about the therodynaics of the solid liquid phase transition, the kinetics of this transition are still poorly understood. Since probing the interface between two dense phases is largely inaccessible with experiental techniques [1], the atoistic details fro coputer siulations have provided an excellent alternative for understanding the icroscopic processes involved in crystallization or elting. One of the open questions on phase transitions concerns the rates of growth and elting. There has been uch debate on whether there exists a slope discontinuity in the rates when crossing the elting point. This paper was recoended for publication in revised for by Associate Editor Dongsik Ki Corresponding author. Tel.: , Fax.: E-ail address: jdchung@sejong.ac.kr KSME & Springer 200 Uhlann et al. [2] ascertained that an abrupt change in the kinetic coefficient would iply a violation of icroscopic reversibility. However, the asyetry of freezing and elting kinetics was shown experientally for crystalline silicon by Tsao et al. [3]. Their results were supported by several researchers [4-6]. Even the sae research group, however, reported contradictory results [7, 8]. They explained this in ters of lattice iperfections. In this study, olecular dynaic (MD) siulations were used to look at the elting and solidification of argon at sall aounts (6.0K) of undercooling and superheating. Special care was taken at the elting teperature and for the rates of growth and elting at the Lennard-Jones (100) interface. 2. Syste and siulation ethod The argon FCC (100) interface was chosen as the siulation syste, which is coonly described by a Lennard-Jones potential. Choosing a siple syste allows for the elucidation of results that ay be difficult to resolve in ore coplex aterials, where

2 1564 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~1570 ulti-ato unit cells can generate additional effects. The Lennard-Jones 6-12 potential describing the twobody interaction between atos i and j separated by distance r ij, is φ ( rij ) 12 6 σ σ = 4ε r ij r ij (1) where σ and ε define characteristic length and energy scales, respectively. The potential φ is 0 at 1/6 rij = σ and its iniu is ε at rij = 2 σ. Argon has values σ = A and ε / kb = 11.8 K. The paraeters ε and σ are used as units of energy and length, respectively. Siulation results have been reported in reduced units such as N = Nσ 3 / V, T = kbt/ ε, E = E/ ε N, t = t/ ( σ 2 / ε) 1/2, P = Pσ 3 / ε. Baidakov et al. [] pointed out that the cut-off radius of the interaction potential in two-phase systes is extreely iportant and should be chosen to be larger (6.78σ in their study) than that for the onephase systes to provide an adequate representation of properties. But in contrast to their liquid-gas syste, the density change between liquid-solid is not as large as that between vapor-liquid, so the cut-off radius is not expected to be as sensitive as liquid-gas syste. Fro the experience in one-phase systes and 2.5σ used in Tepper and Briels [8], the atoic interactions were truncated at a cutoff radius of 3.5σ, which is nearly one-half of the length of the siulation box. Periodic boundary conditions were iposed in all directions. The classical trajectories of the atos were deterined by using the Verlet leap-frog integration algorith [10] with a tie step of 2 fs. One should carefully choose the tie step so that all the tiescales of interest in the syste can be resolved. In siulations of LJ argon where the highest frequency is 2 THz, a tie step of 2 fs is sufficiently sall copared to typical values of 5~10 fs [11]. The effect of the tie step was investigated copared with 1 fs tie step and with the sae heating/cooling rate, no appreciable difference was found. The size effect on elting and crystallization was explored over the range of 500~1372 atos. This showed negligible differences for typical heating and cooling rates, so a siulation box coposed of FCC unit cells (500 atos) was adopted. All the MD siulations presented here were carried Fig. 1. Nuber density variations in the heating and cooling process at K/s. out, except where noted, in the isobaric-isotheral (NPT) enseble. The pressure was kept constant at atospheric pressure, P = by a Nosé- Hoover ethod [12] and Langevin dynaics [13]. The therostat relaxation tie and barostat relaxation tie were chosen as 200 fs and 100 fs, respectively. The elting was achieved by a step-wise increase of the syste teperature T = 0.5K, initially FCC crystal well below its estiated elting teperature. Between increents, the teperature was held constant and the siulation was run for 30,000 steps; thus the heating rate is K/s. The heating rate was varied over the range K/s to K/s (=0.1K/120ps) to exaine its effect on the elting teperature. The sae was done when cooling the elt. All teperature dependent intensive properties have been averaged after one-third of total siulation tie steps at each teperature which is enough for equilibration. 3. Melting teperature The nuber densities of the solid and liquid phases are plotted as a function of teperature in Fig. 1. Upon increental heating, the nuber densities of the syste increases steadily until there appears a catastrophic volue increase at T + = The excellent agreeent between the value obtained here and other reported values [8, 11, 14-16], validates the ethod and siulation code eployed. Deviations fro other research are expected because of the siulation box

3 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~ size, cutoff radius and potential shift. Another validation was done by coparing the zero teperature unit cell paraeter. When relaxed to zero teperature, the MD FCC crystal unit cell paraeter is A, which corresponds to a cutoff radius of 3.5 σ. McGaughey and Kaviany [11] have found values of A for a cutoff radius of 3.1σ. The experiental value is A [17]. Previously, the discontinuity in the potential energy or volue vs. teperature curve was inappropriately assued as an indicator of elting. However, we should distinguish the therodynaic elting T which occurs heterogeneously, and elting due to echanical instability T + [18], which is a hoogeneous process. Since, due to superheating, T + can be considerably higher than T, any failure to distinguish between these two quantities will result in confusion in interpreting the elting characteristics of the odel syste. In contrast to an experient in which superheating is extreely difficult even in the ost favorable cases, nuerical siulations of a perfectcrystal cell allow substantial etastable overheating without suffering fro heterogeneous nucleation sites. Several easureents deonstrate that when the surface conditions are odified, the elting point can be depressed [1] or the solid can be substantially superheated [20]. This iplies that elting is basically a heterogeneous process and the teperature obtained fro the MD results of the abrupt volue change is a echanical elting teperature, not a therodynaic one. Note that even after long runs 6 (up to 1 10 tie steps) above T such as T =0.78, no signal for elting can be detected, indicating the existence of substantial etastable superheating. Since the equilibriu state is the state of iniu Gibbs free energy, free energy calculations should be perfored to deterine the therodynaic elting point T, which is defined as the teperature at which the free energies of the crystalline and liquid phases are equal. Coputation of both free energy differences and absolute free energies has propted a great deal of attention and techniques such as free energy perturbation [21] and therodynaic integration [22] have been well-established. However, the coputational ipleentation poses foridable barriers to its practical use, and the proper convergence of the free energy calculations and quantitative interpretation of the results require extreely long siulations. Thus, the densities of each phase were used to siply evaluate the free energies of the solid and liquid fro the equations of state. The expressions were adopted for the liquid fro Johnson et al. [15] and for the solid fro Hoef [16]; these are known to be ost accurate expressions for each phase. The nuber density of the bulk FCC crystal as a function of teperature in the range fro 75K to 87K (i.e., < T < ) at P = can be fitted fro Fig. 1 with a second-order polynoial: Ns = T T T (2) 2 with a coefficient of deterination R = For the bulk liquid, the nuber density for the sae teperature range is given by Nl = T T T (3) with a coefficient of deterination R 2 = Applying these densities to the equations of state for each phase gives the elting teperature 82.4K ( T =0.6878). The therodynaic elting point is 82.28K fro a siulation [23] and 83.8K fro experient [24]. The therodynaic elting point deterined here by the Gibbs free energy is very close, but not exactly the sae as the experiental value because the potential function does not describe Ar perfectly and the cutoff is used. In the heating process, the solid phase is aintained above the therodynaic elting teperature, until the echanical elting teperature is reached where collapse of the crystal lattice occurs due to the elastic instability associated with an unstable phonon ode. Also, the liquid phase is aintained below the therodynaic elting teperature in the cooling process, giving clear hysteresis as shown in Fig. 1. Luo et al. [25] suggested the hysteresis ethod where T can be deduced given the teperature for the axiu degree of superheating ( T + ) and undercooling T ) as ( T = T T T + T (4) + + They ascertained that the deviation of their hysteresis ethod was about 2% or less. However, we cannot find any physical explanation for this ethod and the elting teperature evaluated fro the data in

4 1566 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~1570 Fig. 1, T =0.652 is far fro the experiental observation. The effect of heating rate and cooling rate was scrutinized for four different rates, 0.5K/60ps, 0.25K/60ps, 0.25K/120ps, and 0.1K/120ps and Fig. 2 shows the energy variations. No significant differences were found except for the cooling process at K/s. We find that at high cooling rate, this solid phase is not crystalline but aorphous as shown in Fig. 1 where solidification occurs but nuber density differs fro crystalline. Nosé and Yonezawa [26] showed that there is a critical cooling rate which separates crystal-foring and glass-foring. McGaughey and Kaviany [11] also found an aorphous phase by 11 quenching at a high cooling rate, K/s, shown in Fig. 1 as open circle sybols. However, they argued that above 20K the theral equilibriu fluctuations in the syste are large enough to return the atos to the FCC crystal structure. In the present work, the aorphous phase is aintained up to 45K. The heat of elting ay also be obtained fro the difference in the configurational energy between the crystalline and liquid phases at the elting point. In Fig. 2, the calculated value of the latent heat is 1.12 kj/ol, which is copatible with the experiental value of 1.18 kj/ol. 4. Growth and elting rate Growth and elting rate are also exained using cobined siulation boxes of the solid and liquid phases at the expected elting teperature. If the given teperature is higher than the elting teperature, the interface between solid and liquid phases will ove towards the solid phase; if the given teperature is lower than the elting teperature, the direction of the interface oveent will be opposite. By counting the nuber of solid like particles, we can estiate the interface velocity, i.e., the growth rate or elting rate, as a function of teperature. In cobining the fully equilibrated bulk phases of the liquid and solid in one siulation box, it is reported that the proper preparation of the twophase syste plays a crucial role in deterining the resulting growth and elting rates [8]. In the present work, an NPT siulation was initially carried out for solid phase at the expected (i.e., experiental) elting teperature T =0.65 and a tieaveraged volue was obtained. Then the bulk NVT siulation at this volue was run for 100,000 tie Fig. 2. Energy variations in the heating and cooling process at various heating/cooling rates. steps of equilibration and was written once every 1,000 tie steps for 15,000 further tie steps. For the liquid phase, the sae procedure was conducted by fixing the cross-section area in the x-y plane to be equal to the tie-averaged value of the solid phase. Thus, only the z-directional volue change can be peritted in the NPT siulation. After equilibration in NVT, the results were written once every 1,000 tie steps. To ake two-phase boxes, one liquid configuration and one solid configuration were taken, and both copied twice in the z direction in the order liquid-solid-solid-liquid. The resulting syste contained two solid-liquid interfaces and consisted initially of 1,000 solid particles ( unit cells) and 1000 liquid phase particles. To release excessive potential energy due to particle overlap, 10,000 tie steps with the NVT siulation were conducted on condition that the solid phase atos reained frozen at their positions. After this, the cobined two-phase systes were quenched to the desired teperature and NPT runs were carried out to study growth and elting rates. The volue relaxation in each direction took place independently. Due to volue fluctuations in the isotheral-isobaric enseble, the siulation was repeated for 15 runs fro different starting configurations and subsequently averaged to obtain good statistical accuracy. Since the fluctuation of the solid-like particle nuber N s in one single run was uch less than that of Tepper and Briels [8] as shown in Fig. 5, only 15 runs of different starting configurations were regarded as

5 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~ necessary. Note that Teppers and Briels used 50 runs. They pointed out that their approach of equilibration included the solid phase oveent together with liquid phase oveent and this had the side effect of reoving iperfections that were present in the bulk crystal phase, which resulted in an asyetry between growth and elting. In their other work, equilibration with a fixed solid phase was ipleented but a uch longer equilibration tie was used. They ephasized that when equilibration was not carried out to its full extent, erroneous conclusions could be drawn about the teperature dependence of growth and elting rates. The solid-like particle recognition criterion is based on the total instantaneous volue of the syste. Since the volues per particle for each phase are known fro the bulk siulations, the proportion of particles between the solid and liquid phases can be calculated at every instance. The nuber of solid particles is estiated by N s () t total l total 1/ Nl 1/ Ns () N / N V t = (5) where N total, V total, N l, and N s are the total nuber of particles, total volue noralized by σ, and 3 particle nuber densities of liquid and solid, respectively. Fig. 3 shows the growth and elting curves showing the nuber of solid-like particles vs. tie along with curve-fitted lines for each teperature T = , , 0.644, , , , , 0.612, 0.65, A change of slope is discernible for the teperature apart fro the elting teperature as arked by the circle in Fig. 3, but is not obvious for teperatures close to equilibriu. Because of this change of slope, Tepper and Briels [8] suggested the presence of two regies. They ascertained that the initial regie was associated with interface relaxation and the second regie was associated with the acroscopic liit of growth and elting and true growth and elting rates should be obtained fro the second regie. With the data of the second regie, they showed that there was no discontinuity in the elting and growth rates. However, they did not show any clear reason why the slope change of N s occurs. Also, the distinction between the two regies ay be abiguous since, as they pointed out, the second regie was not found close to equilibriu. Fig. 3. Growth and elting curves showing the nuber of solidlike particles vs. tie for teperatures T = , , 0.644, , , , , 0.612, 0.65, In addition, the second regie existed only for a very short tie when far fro equilibriu. The teperature dependence of the rates as deterined fro the slopes of the average elting curves is presented in Fig. 4. The present work shows a linear dependency on teperature. It is hard to say if the two fitted lines above and below the elting teperature in Fig. 4 have different slopes considering the uncertainties involved in the MD siulations. Note that the siulation box with 2,000 atos is uch saller than those of Tepper and Briels [8] which had 4048 and 806 atos. They ascertained that the second regie could only be easured accurately for sufficiently large systes and a non-linear elting rate ight appear in a sall siulation box. However, if the data furthest fro the elting teperature T =6.0K is excluded, their results for the initial regie also show a clear linear dependency with a coefficient of deterination R 2 = This is clearly in conflict with their conclusion. Thus, a reassessent will be needed on the slope discontinuity to deterine whether it stes fro the initial siulation regie or high superheating. In Fig. 4, the initial regie has different phase change rate fro that of long-tie regie. Which rate is true phase change rate is another proble, but we can see no slope discontinuity in the growth and elting rates in the long-tie regie but also in the initial regie for the all results of the present and Tepper and Briels [8]. The reason for the appearance of two

6 1568 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~1570 Fig. 4. Melting rates as a function of teperature, deterined fro the slopes of the average elting curves. regies can be understood by exaining the cross sectional area change together with the change in the nuber of solid-like particles as shown in Fig. 5. Although independent volue relaxations in the x, y, z directions were allowed, the area of the x y plane reains constant for a tie because the z direction is uch ore flexible than the x or y directions, which are restricted by the solid phase. After soe tie, the flexibility of the x, y, z di- rections becoes coparable, and each directional contribution to the volue increase becoes siilar. The distinction between the two regies suggested by Tepper and Briels [8] coincides with the abrupt area change at tie indicated by arrow A in Fig. 5 where the area of the x y plane starts to increase. Care should be taken when evaluating the growth rate in the second regie since a variable cross sectional area also affects the growth rate, R = 1/ ( 2 ρsa) dns/ dt, as well as the evolution of the nuber of solid particles. This was not considered by Tepper and Briels [8]. The elting teperature, where no growth or elting of the crystal particles in Fig. 4 is expected, is T = This value is close to that fro the Gibbs free energy calculation. At this teperature, a snapshot of the atoic configuration and sectional nuber density variation is shown in Fig. 6. The sectional nuber density distribution is obtained by averaging 80 tie fraes in each slice along the z direction, with its width is chosen to be 1A. A broadening of the tie-averaged interface profile is observed. Since Fig. 5. Melting curves showing the nuber of solid-like particles and cross sectional area vs. tie at T =0.65. Fig. 6. Snapshot of the atoic configuration and sectional nuber density variation at the elting teperature T = the Lennard-Jones FCC (100) crystal-elt interface is very diffuse [27], the interface extends over several interlayer spacings.

7 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~ Conclusions Molecular dynaic (MD) siulations have been carried out to look at the elting teperature and rates of growth and elting of the Lennard-Jones Ar (100) interface for sall aounts (6.0K) of undercooling and superheating. Clear hysteresis behavior was found. The heating rate was varied over the range fro K/s to K/s, and no significant differences were found for the echanical elting point, but for the highest cooling rate ( K/s), an aorphous phase was ob- served. By cobining the fully equilibrated bulk phases of liquid and solid in one siulation box and counting the nuber of solid like particles, the interface velocity, i.e. growth rate or elting rate, was obtained as a function of teperature. A linear dependency on teperature was found except for high superheating, T > 6K. The high superheating is believed to be the ain source of the slope discontinuity in the rate of growth or elting, not the isuse of initial regie as discussed in the earlier works. Initially the area of the x y plane reains constant due to the higher flexibility in the z direction; the distinction between the two regies found in earlier work coincides with the point where the area of the x y plane starts to increase. Acknowledgent This work was supported by the Korea Research Foundation Grant funded by the Korea Governent (MOEHRD) (KRF D00018) Noenclature A : Area E : Total energy k B : Boltzann constant, J/K : Lennard-Jones ass scale N : Nuber of atos P : Pressure r : Particle position, interparticle separation R : Growth rate t : Tie T : Teperature v : Volue per particle V : Volue x, yz, : Coordinate Greek Sybols ε : Lennard-Jones energy scale φ : Potential energy ρ : Nuber density σ : Lennard-Jones length scale Subscripts i, j : Particle label l : Liquid : Melting s : Solid Superscripts : Reduced unit + : Superheat : Undercooling References [1] B. B. Laird and A. D. J. Hayet, The crystal/liquid interface: Structure and properties fro coputer siulation, Che. Rev., 2 (12) [2] D. R. Uhlann, J. F. Hays and D. Turnbull, The effect of high pressure on B2O3: Crystallization, desification, and the crystallization anoaly, Phys. Che. Glasses, 8 (167) [3] J. Y. Tsao, M. J. Aziz, M. O. Thopson and P. S. Peercy, Asyetric elting and freezing kinetics in silicon, Phys. Rev. Lett., 56 (186) [4] M. D. Kluge and J. R. Ray, Velocity versus teperature relation for solidification and elting of silicon: A olecular-dynaics study, Phys. Rev. B., 3 (18) [5] M. Iwaatsu and K. Horii, Interface kinetics of freezing and elting of Si and Na, Phys. Lett. A., 214 (16) [6] C. J. Tyczak and J. R. Ray, Asyetric crystallization and elting kinetics in sodiu: A oleculardynaics study, Phys. Rev. Lett., 64 (10) [7] H. L. Tepper and W. J. Briels, Siulation of crystallization and elting of the FCC (100) interface: the crucial role of lattice iperfections, J. Crystal Growth, 230 (2001) [8] H. L. Tepper and W. J. Briels, Crystallization and elting in the Lennard-Jones syste: Equilibration, relaxation, and long-tie dynaics of the oving interface, J. Che. Phys., 115 (2001) [] V. G.. Baidakov, G. G. Chernykh and S. P. Protsenko, Effect of the cut-off radius of the interolecular potential on phase equilibriu and surface tension in

8 1570 J. D. Chung / Journal of Mechanical Science and Technology 23 (200) 1563~1570 Lennard-Jones systes, Che. Phys. Lett., 321 (2000) [10] M. P. Allen and D. J. Tildesley, Coputer siulation of liquid, Oxford, 187. [11] A. J. H. McGaughey and M. Kaviany, Theral conductivity decoposition and analysis using olecular dynaics siulations. Part I. Lennard-Jones argon, Int. J. Heat Mass Transfer, 47 (2004) [12] G. J. Martyna, D. J. Tobias and M. L. Klein, Constant pressure olecular dynaics algoriths, J. Che. Phys., 101 (14) [13] S. E. Feller, Y. Zhang, R. W. Pastor and B. R. Brooks, Constant pressure olecular dynaics siulation: The Langevin piston ethod, J. Che. Phys., 103 (15) [14] J. Q. Broughton and G. H. Giler, Molecular dynaics investigation of the crystal-fluid interface. I. Bulk properties, J. Che. Phys., 7 (183) [15] J. K. Johnson, J. A. Zollweg and K. E. Gubbins, The Lennard-Jones equation of state revisited, Mol. Phys., 78 (13) [16] M. A. van der Hoef, Free energy of the Lennard- Jones solid, J. Che. Phys., 113 (2000) [17] O. G. Peterson, D. N. Batchelder and R. O. Sions, Measureents of X-ray lattice constant, theral expansivity, and isotheral copressibility of argon crystals, Phys. Rev., 150 (166) [18] M. Born and K. Huang, Dynaical Theory of Crystal Lattices, Oxford, 162. [1] Ph Buffat and J. P. Borel, Size effect on the elting teperature of gold particles, Phys. Rev. A., 13 (176) [20] J. Daeges, H. Gleiter and J. H. Perepezko, Superheating of etal crystals, Phys. Lett. A., 11 (186) [21] J. G. Kirkwood, Statistical echanics of fluid ixtures, J. Che. Phys., 3 (135) [22] R. W. Zwanzig, High-teperature equation of state by a perturbation ethod. I. Nonpolar gases, J. Che. Phys., 22 (154) [23] R. Agrawal and D. A. Kofke, Therodynaic and structural properties of odel syste at solid-fluid coexistence, Mol. Phys., 85 (15) [24] [25] S. N. Luo, A. Strachan and D. C. Swift, Nonequilibriu elting and crystallization of a odel Lennard-Jones syste, J. Che. Phys., 120 (2004) [26] S. Nosé and F. Yonezawa, Isotheral-isobaric coputer siulations of elting and crystallization of a Lennard-Jones syste, J. Che. Phys., 84 (186) [27] H. E. A. Huitea, M. J. Vlot and J. P. van der Eerden, Siulations of crystal growth fro Lennard-Jones elt: Detailed easureents of the interface structure, J. Che. Phys., 111 (1) Jae Dong Chung received his B.S. degree in Mechanical Engineering fro Seoul National University, Korea, in 10. He then received his M.S. and Ph.D. degrees fro Seoul National University in 12 and 16, respectively. Dr. Chung is currently a Professor at the Mechanical Engineering at Sejong University in Seoul, Korea. He serves as a Director of General Affairs of the SAREK and the theral division of KSME. Dr. Chung s research interests include nano-scale heat transfer, phase change, aterial processing and HVAC&R.