Ordered structure formation from disordered atoms within undercooled liquid rhodium

er Chi s of ring K, t pea bein e str 336 The structure of liquid metals is of fundamental importance in understanding atomic distribution, deter physical properties, performing numerical ing novel materials [1–7]. This subject has interest in the field of materials physics in their f und dynam le liqu ong tim metals are seriously scarce, especially for metastable liquid metals [14,15]. Recently, the structure of liquid metals has been investigated by combining containerless processing, such as electromagnetic levi- tation and electrostatic levitation, with X-ray scattering, synchro- tron radiation, and neutron diffraction methods [15,16]. Kelton et al. [17] removed the solid/liquid interface by an electrostatic levitator, and performed X-ray scattering to verify the formation Ti droplet by synchrotron X-ray diffraction. Holland-Moritz et al. dium (Rh). Because the content of rhodium (Rh) is very rare inside the earth, it cannot be applied everywhere in common life. However, due to its great importance in producing novel alloys, it has been exten- sively applied in various alloys, such as thermocouples, furnace windings, electrodes of aircraft spark plugs, and high temperature crucibles. In addition, Rh has been employed to make jewelries and decorations. When alloying, Rh is always melted into liquid state and then becomes merged with other elements. Accordingly, the structure and properties of liquid Rh is of great importance to produce its alloys, and this results in indispensable requirement ⇑ Corresponding author. Fax: +86 88495926. Chemical Physics Letters 521 (2012) 55–58 Contents lists available at y .e l E-mail address: [email protected] (B. Wei). conditions. Meanwhile, there exists a tendency that a nonlinear change of atom distribution occurs for metastable liquid metals. Liquid metals have an amorphous atomic structure, which are the parent phases for solid metals. The atoms in a liquid metal are ordered in short range and disordered in long distance [12,13]. As compared with the structure of a solid metal, it is diffi- cult to determine the structure of a liquid metal by experiments due to the rapid atom diffusion. This leads to such a situation that the experimental data for the structure and properties of liquid plied in the research on the structure of undercooled liquid metals [21–23]. As compared with the experimental study, the relevant information can be obtained in more details. Jakse et al. [24] per- formed first-principle molecular dynamics simulations to study the evolution of dynamic properties around the melting point of Si. Gheribi [21] simulated the short range order structure of under- cooled liquid Zr by MD method with MEAM potential model. Even so, the research on liquid metals has so far been insufficient, espe- cially for those noble metals with high melting points such as rho- metals can be undercooled below nucleation occurs, i.e. formation o [8,9]. From the viewpoint of thermo metals are in the state of metastab metastable state can be kept for a l 0009-2614/$ - see front matter � 2011 Elsevier B.V. A doi:10.1016/j.cplett.2011.11.009 mining chemical and modeling, and develop- aroused great research recent decades. Liquid melting points and no ercooled liquid metals ics, undercooled liquid id [10,11]. This kind of e under some special [19] also determined the liquid structure of Ti at about 200 K und- ercooling by neutron scattering technique combined with electro- magnetic levitation method. Using synchrotron radiation together with an electromagnetic levitator, Higuchi et al. [20] investigated the atomic structure of molten Si in the undercooled temperature regime. In spite of these experimental reports, there are still many uncertainties about liquid structure owing to the great experimen- tal difficulties. Meanwhile, molecular dynamics (MD) simulation has been ap- [18] measured the structure factor of an electrostatically levitated Ordered structure formation from disord liquid rhodium H.P. Wang, B. Wei ⇑ Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, PR a r t i c l e i n f o Article history: Received 30 September 2011 In final form 1 November 2011 Available online 19 November 2011 a b s t r a c t Here we present the result ular dynamics method. Du temperature drops to 1900 tribution functions. The ap atomic clusters come into are employed to display th 1900 to 2800 K including 1. Introduction Chemical Ph journal homepage: www ll rights reserved. ed atoms within undercooled na ordered structure formation within undercooled liquid rhodium by molec- the cooling of liquid rhodium, the enthalpy decreases linearly before the hen it has a steep decline. This structure change is clarified by the pair dis- rance of peaks beyond the first neighbor distance suggests that ordered g. Meanwhile, the self diffusion coefficient and the atom number density ucture change. This work provides the density data of liquid rhodium from K undercooling. � 2011 Elsevier B.V. All rights reserved. of the icosahedral phase within liquid Ti–Zr–Ni alloy. Lee et al. SciVerse ScienceDirect sics Letters sevier .com/ locate /cplet t for the fundamental knowledge of Rh. However, the melting point of Rh is as high as 2236 K. This leads to the great difficulties of deal- ing with Rh and the scarcity of structure information about liquid Rh. The objective of the present Letter is to study the atomic-scale structure characteristics of liquid Rh when cooling from super- heated state to undercooled temperature regime. The pair distribu- tion function, the atomnumberdensity, the self diffusion coefficient, and the mass density are also applied to explore the structure change. 2. Method formed for calculation at 100 K temperature intervals. At each tem- In order to clarify the above special phenomenon, the states of the simulated cell are examined at each temperature. Here, the pair distribution function (PDF) is employed to reveal the structure characteristics, which can be expressed by gðrÞ ¼ Vhniðr; r þ DrÞi 4pr2DrN ð1Þ where V is the simulated cell volume, ni(r, r + Dr) the atom number around the ith atom in a spherical shell between r and r + Dr, h���i the average symbol, and N the atom number. The temperature range for examining structure is from 2800 to 1000 K, including both a superheating range and a board metastable undercooled regime. Figure 2 presents the PDF results of Rh, in which the typical PDF curves are given at 2800 (the highest temperature in present work, far beyond its melting point), 1900, 1800, 1000 K (the lowest tem- perature in this Letter, far below its melting point). According to Figure 2a, an obvious peak occurs at the first neighbor distance, and the peak at the second neighbor distance is also clear, how- ever, its height is only about 1.3. That is to say, the PDF at 2800 K rapidly begin to fluctuate to 1 once the distance exceeds the first neighbor distance. This characteristic indicates that the atom distribution is ordered in short range and disordered in long distance. This suggests that the simulated Rh cell is in the state of liquid or glass. And it is verified to be in the state of liquid by the diffusion coefficient at the subsequent section. With the decrease of temperature, the liquid structure keeps till to 1900 K, as shown in Figure 2b. It can be seen that the PDF at -14 56 H.P. Wang, B. Wei / Chemical Physics Letters 521 (2012) 55–58 1000 1500 2000 2500 3000 -17 -16 -15 Normal liquid Ordered structure formationΗ , 10 4 ev Undercooled Liquid 2236 K Co olin g Solid perature, 100000 steps are carried out for equilibrium. The last 50000 steps are applied to calculate the final results. During the simulations, the structure of the cell is monitored by the pair dis- tribution function and the mean square displacement versus the simulated time. All of codes run in a Lenovo 1800 Cluster system. 3. Results and discussion 3.1. Structure change In MD simulation of Rh cell, a cooling process is performed from 2800 to 1000 K after achieving an equilibrium liquid state at a high temperature. The enthalpy of the simulated cell is computed as a function of temperature and illustrated in Figure 1. With the de- crease of temperature, the enthalpy drops linearly before the tem- perature falls to 1900 K. When the temperature is lower than 1900 K, the enthalpy has a steep decrease. This change is different from the linear variation in the range of 2800–1900 K, which may have resulted from the change of atom distribution. The potential model is quite important for MD simulation. Here, the modified embedded atom method (MEAM) model proposed by Baskes [25] is selected because it specializes in dealing with the metals with face-centered cubic structure including Rh. In the sim- ulated cell, 32000 Rh atoms are arranged in a cubic box and sub- jected to periodic boundary conditions under constant pressure and constant temperature. The time step is 1 fs and the pressure is set to 105 Pa. The temperature is adjusted every 50 steps. It starts at 4000 K to get an equilibrium liquid state. The initial temperature is kept constant for 200000 steps. In order to obtain much more information of liquid structure and properties, especially for high temperature and highly undercooled states, the maximum temper- ature is selected to be 2800 K, and the minimum temperature is 1000 K. The cooling process with a 1013 K/s cooling rate is per- T, K Figure 1. Simulated enthalpy of Rh versus temperature. 1900 K is very similar to that at 2800 K apart for a little difference in the shapes of the peaks at the second neighbor distance. Figure 2b infers that the simulated Rh cell is still in the state of liquid, although the temperature is 336 K lower than the melting point of 2236 K. In this case, liquid Rh is in metastable undercooled state. 0 2 4 6 8 10 0 2 4 6 (d) 1000 K Solid state 0 2 4 (c) 1800 K Ordered structure formation 0 2 4 0 2 4 PD F PD F PD F (b) 1900 K Undercooled liquid ΔT=336 K PD F (a) 2800 K Normal liquid r, Å Figure 2. Pair distribution function of Rh versus temperature. When the temperature drops to 1800 K, there is a pronounced change for the PDF, as shown in Figure 2c. The height of the first peak remarkably increases when compared with that at 2800 K. Especially, much more peaks appear in the PDF curve at 1800 K, which suggest that a nonlinear change comes into being for the distribution of Rh atoms. These peaks show the appearance of or- dered structure within disordered Rh atoms at this temperature. Here, the simulated cell is not in the normal state of liquid. With the decrease of temperature, more atoms change from disordered distribution to ordered structures. Figure 2d illustrates the PDF at 1000 K, the lowest temperature in the present Letter, where the peaks are much clearer. The atom distribution becomes more or- dered than that at 1800 K. For the sake of clarity, the distributions of Rh atoms at two typ- 6 @t During molecular dynamics simulation, the mean square dis- coefficient descends with the decrease of temperature. The values are 3.53 � 10�9 and 2.84 � 10�10 m2/s at 2800 and 1900 K, i.e. the diffusion coefficient drops for almost one order of magnitude from 2800 to 1900 K. Since ordered structure comes into being when the temperature falls to 1800 K, the self diffusion coefficient begins to rapidly descend once the temperature is lower than 1800 K. At 1000 K, the self diffusion coefficient is extremely small, 7.69 � 10�14 m2/s. Such a low value infers that the simulated cell is in the state of solid. Together with the diffusion coefficient, it is found that solidification of liquid Rh has been finished at 1600 K after examining the atoms distribution versus time. 0.10 0.04 0.06 0.08 0.10 0.04 0.06 0.08 0.10 (c) 1000 K, Solid (b) 1800 K, Ordered structure appearance 0.023 N ρ, Å- 3 N ρ, Å- 3 (a) 1900 K, Undercooled liquid, ΔT=336 K 0.015 1000 1400 1800 2200 2600 10-3 10-1 101 Solid D , 1 0- 10 m 2 s- 1 T, K Undercooled liquid Normal liquid Ordered structure formation H.P. Wang, B. Wei / Chemical Physics Letters 521 (2012) 55–58 57 ical temperatures are given in Figure 3. At 1900 K (Figure 3a), Rh atoms randomly distribute in the cell, and obviously ordered distri- bution occurs at 1800 K (Figure 3b). Figure 2b reflects the distribu- tion characteristics of Figure 3a, i.e. typical liquid structure. Figure 2c displays the structure of Rh atoms in Figure 3b, and ordered structure forms in the system, which can be verified by the peaks in Figure 2c. By monitoring the atom diffusion, the atoms cannot diffuse in long range once ordered structure forms. However, the other atoms can freely move in the remnant space. With the decrease of temperature, more and more Rh atoms change into ordered structures till solidification finishes in the whole system. Furthermore, the atom number density Np is calculated in order to reveal the change of atom distribution. The simulated cell is di- vided into slabs with 1 Å thickness, and the atom number is statis- tically averaged in each slab, then Np can be derived, as shown in Figure 4. In terms of Figure 4a–c, it can be seen that the fluctuation becomes larger and larger with the decrease of temperature. At 1900 K, the Rh atoms randomly distribute in the cell and the fluc- tuation range (FR) is from 0.056 to 0.071. Till to 1800 K, ordered structures form, the fluctuation of Np curve becomes more serious. The FR is from 0.054 to 0.077. At the lowest temperature of 1000 K, the FR is from 0.051 to 0.085. There is a pronounced difference for the Np curve as compared with that of normal liquid structure. The above results were calculated along the x-axis. We also examined the results along the y-axis and the z-axis, and the situations are alike. 3.2. Diffusion coefficient The structure change will influence the atom diffusion. The self diffusion coefficient is related to the mean square displacement (MSD), which can be written as MSDðtÞ ¼ 1 N XN i¼1 ½riðtÞ � rið0Þ�2 * + ð2Þ (a) 1900 K (b) 1800 K ordered Figure 3. Rh atoms distribution at different temperatures (a) 1900 K, disordered structure, (b) 1800 K, ordered structure appearance. placements are computed at different temperatures. On the basis of the results of MSD, the self diffusion coefficient of Rh atoms can be derived by Eq. (3), as shown in Figure 5. The self diffusion where ri(0) is the initial position of the ith particle, and ri(t) the po- sition of the ith particle at some later time t. The self diffusion coef- ficient can be obtained from MSD by D ¼ 1 @ MSDðtÞ ð3Þ 0 20 40 60 80 0.04 0.06 0.08 x, Å 0.034 N ρ, Å- 3 Figure 4. Atom number density of Rh along x-direction versus temperature. Figure 5. Self diffusion coefficient of Rh versus temperature. 3.3. Density The density plays a very fundamental role in most numerical modeling and materials design. In this Letter, we also track the var- iation of the cell volume, and then the density can be obtained. When the temperature attains 1800 K the system is not a normal liquid structure. Accordingly, the density data are valid and valu- able from 1900 to 2800 K. For the data below 1900 K, the density cannot be applied in the reality since the simulated cell includes both ordered structures and disordered atoms. Figure 6 illustrates the simulated density as a function of tem- perature. It can be seen that the density exhibits a linear depen- dence on temperature in the range of 1900–2800 K. Once the 58 H.P. Wang, B. Wei / Chemical Phys temperature drops to 1800 K, the density displays a significant in- crease, which should result from the ordered structure formation at this temperature. That is to say, the density change also reflects the difference of atom distribution. Here, the accuracy of the simulated density will be discussed. From Figure 6, the density q of liquid Rh decreases linearly with the rise of temperature when TP 1900 K as follows: q ¼ 10:32� 1:21� 10�3ðT � TmÞ g cm�3 ð4Þ where the melting point Tm is equal to 2236 K. The calculated tem- perature for the density of liquid Rh is in the range of 1900–2800 K, including 336 K undercooling and 564 K superheating. Such a large temperature span is hard to achieve by experiments. According to the above expression, the density is 10.32 g cm�3 at the melting point, and its temperature coefficient is �1.21 � 10�3 g cm�3 K�1. To evaluate the present results by molecular dynamics method, the density data of liquid Rh obtained by the other researchers are also given in Figure 6. In Smithells Metals Reference Book [26], only the density of liquid Rh at the melting point is listed, which is 10.80 g cm�3, and no relationship to temperature is given. In com- parison with this reported result, a difference of only 0.48 g cm�3 exists between them. In other words, our result is 4.4% smaller than this value. Usually, the experimental error for determining density is about 5%. That is to say, this difference is smaller than the experimental error. Paradis et al. [27] measured the density of liquid Rh in the tem- perature range of 1820–2250 K by an electrostatic levitator, and a linear relationship was obtained as follows: q ¼ 10:82� 7:6� 10�4ðT � TmÞ g cm�3 ð5Þ which is also presented in Figure 6, including a temperature span of 430 K. It can be seen that there is almost no difference between Paradis’s result and the reported value in Smithells Metals Refer- ence Book [26]. This indicates that the present density data also agree well with the results in Ref. [27]. Nevertheless, the present temperature span of 900 K is much larger than that of 430 K in 1000 1500 2000 2500 9.5 10.0 10.5 11.0 11.5 12.0 Solid Ordered structure formation Present work Gale et al [26] P.F. Paradis et al [27] ρ =10 .32 -1 .21x10 -3(T -T m ) ρ, g cm - 3 Undercooled Normal liquid T, K Figure 6. Simulated density of Rh versus temperature. Ref. [27]. It is inferred that the molecular dynamics simulation of Rh not only provides density data at a broader temperature range, but also leads to valuable results of atom distribution change. 4. Concluding remarks The Rh cell with 32000 atoms is cooled from 2800 to 1000 K covering 1800 K span. The characteristics of Rh atom distribution are studied to reveal the structure change. During the cooling of liquid rhodium, the enthalpy decreases linearly before the temperature drops to 1900 K, then it has a steep decline when the temperature is decreased. This peculiar phenom- enon suggests a structure change within metastable liquid Rh. The PDFs show that the liquid structure keeps till 1900 K with the de- crease of temperature. When the temperature drops to 1800 K, there is a pronounced change for the PDF. The height of the first peak remarkably increases when compared with that at 2800 K. Especially, more peaks appear in the PDF curve, which indicate the formation of ordered structure within disordered Rh atoms. Meanwhile, the fluctuation of atom number density becomes more and more serious after ordered structures come into being. With the falling of temperature, the self diffusion coefficient de- creases and thedensity increases. There are two remarkable changes for both of them when the temperature drops to 1900 K. Moreover, this Letter provides the density data of liquid rhodium. The density decreases linearly with the rise of temperature when TP 1900 K including 336 K undercooling. The density is 10.32 g cm�3 at the melting point, and its temperature coefficient is �1.21 � 10�3 g cm�3 K�1. Acknowledgments This work was financially supported by National Natural Sci- ence Foundation of China (Grant Nos. 50971103 and 50971105), the Program for New Century Excellent Talents, the Shaanxi Project for Young New Star in Science and Technology, and NPU Founda- tion for Fundamental Research. The authors are grateful to Mr. S.J. Yang for his helpful discussion. References [1] Y.T. Shen, T.H. Kim, A.K. Gangopadhyay, et al., Phys. Rev. Lett. 102 (2009) 057801. [2] M.N. Schneider, O. Oeckler, Z. Anorg. Allg. Chem. 634 (2008) 2557. [3] N.V. Surovtsev, Chem. Phys. Lett. 477 (2009) 57. [4] G. Nandhini, M.V. Sangaranarayanan, Chem. Phys. Lett. 470 (2009) 224. [5] T. Kordel, D. Holland-Moritz, F. Yang, et al., Phys. Rev. B 83 (2011) 104205. [6] K. Georgarakis, A.R. Yavari, M. Aljerf, et al., J. Appl. Phys. 108 (2010) 023514. [7] N. Jakse, A. Pasturel, Phys. Rev. B 78 (2008) 214204. [8] S.M. Chathoth, B. Damaschke, K. Samwer, et al., Appl. Phys. Lett. 93 (2008) 071902. [9] T.P. 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