⇦Chinese Physics Letters, 2017, Vol. 34, No. 2, Article code 026401 Up-Hill Diffusion of Phase-Separated FeCu Melt by Molecular Dynamics Simulation * Wen-Chao Cui(崔文超), Chuan-Xiao Peng(彭传校), Yun Cheng(程昀)**, Kai-Kai Song(宋凯凯), Xue-Lian Li(李雪莲), Zhen-Ting Zhang(张振庭), Sheng-Zhong Yuan(袁胜忠), Li Wang(王丽)** Affiliations School of Mechanical & Electrical and Information Engineering, Shandong University, Weihai 264209 Received 6 October 2016 *Supported by the National Natural Science Foundation of China under Grant No 51371108, 51501104 and 51501103.
**Corresponding author. Email: wanglihxf@sdu.edu.cn; Chy81@wh.sdu.edu.cn Citation Text: Cui W C, Peng C X, Cheng Y, Song K K and Li X L et al 2017 Chin. Phys. Lett. 34 026401 Abstract Molecular dynamics simulation is performed to characterize the concentration fluctuation of FeCu melts during the liquid–liquid phase separation process, which undergoes the following stages: the formation of interconnected structure and its coarsening, migration and coagulation of droplets driven by the decreasing of potential energy. The up-hill diffusion happens at the early relaxation period in which Cu atoms in Fe-rich region are forced to move toward Cu-rich region by spinodal decomposition with 90% Cu content in Cu-rich region and 95% Fe content in Fe-rich region at temperature of 1500 K. The higher diffusion rate of homogeneous atom can be observed at lower temperature, which is attributed to the larger potential energy difference between Cu-rich region and Fe-rich region. It also exhibits energy heterogeneity in the separated liquid. The domain size decreases sharply during the aggregation and coarsening of droplets, after that it keeps unchanged until the coagulation of droplets begins. The studies characterize concentration and energy heterogeneity of phase-separated liquid on the atomic scale. DOI:10.1088/0256-307X/34/2/026401 PACS:64.75.Gh, 65.20.-w, 66.10.cd © 2017 Chinese Physics Society Article Text The bulk metallic glass has attracted extensive attention due to its excellent mechanical properties. However, its fragility at room temperature limits its applications as structure materials. Extensive efforts have been made over recent years to solve the problem of low plasticity.[1-5] Among them, nano-sized phase separated phase is introduced to the glass by controlling the solidification process to fabricate phase separation bulk glass with high strength and toughness.[6-11] It is due to the fact that the uniformly distributed atomic-scale heterogeneous structure obtained by phase separation can successfully enhance the overall plastic deformability through the formation of multiple branched shear band.[7,10,11] However, if miscibility gap of the liquid is relatively larger, the droplet is liable to migrate and coagulate due to the variance of density, surface tensor etc., leading to the larger size of the second phase glass after rapid solidification, therefore, its plasticity at room temperature cannot be improved due to the different elastic module of the two separated phases. The liquid–liquid phase separation (LLPS) is a process of migration and coagulation of homogeneous atom pairs. The size and shape as well as distribution of droplets have great influence on the final properties of glass after rapid solidification. Therefore, it is of significance to control the formation of droplets. However, the characterization of LLPS and its formation mechanism have not been understood fully and verified. The peculiar types of LLPS are classified into an interconnected structure or a droplet-like structure by the mechanisms[12-15] of spinodal decomposition and nucleation growth, respectively. At the primary separation period during spinodal decomposition process, the liquid phase is unstable with respect to such small concentration fluctuation, which results in the decrease of the free energy, therefore a spontaneous phase separation will happen in the liquid, the concentration difference between the two phases will be increased due to the up-hill diffusion. After that, the liquid phase with small volume fraction will be transformed to droplets with globe shape. During the nucleation and growth reaction, the new liquid phase with the globe shape is generated in the bulk liquid. It grows continuously into droplet-like structure with time relaxation due to the absence of the anisotropies between liquids.[16] The LLPS is the result of the concurrent action of the nucleation and diffusional growth of droplets, the collisions and coagulations and the spatial phase separation due to the Stokes and Marangoni motions of droplets.[17,18] Thus it is difficult to observe clearly the LLPS behavior, especially for the liquid metal. One of the main features of the phase diagram of the Fe-Cu binary system is that the metastable miscibility gap of the liquid phase exists at higher temperatures[19] due to its positive mixing enthalpy, while others think that Fe-Cu alloys are completely miscible in the stable liquid state, and the system shows a wide metastable miscibility gap at rather low undercooling values.[14] Thus the Fe-Cu system provides a model system to study LLPS. However, LLPS of the Fe-Cu system is very difficult to observe because of its high temperature and opaque characteristics. For the liquid phase of high-temperature metallic alloys, such data are scarce. Little information of the phase separation of metallic liquid has been discussed in literature. In our earlier work, the LLPS of Fe-Cu melts above liquidus has been studied by molecular dynamics (MD) simulation.[20] The pair correlation function (PCF), structure factor, enthalpy release and diffusion coefficient serve as a function of concentration or temperature, and suggest the typical phase separation characteristics of Fe-Cu melts. The partial PDFs and coordination numbers of Fe$_{60}$Cu$_{40}$ undercooled liquid have been calculated in the literature,[21] showing that Cu and Fe atoms are not apt to come together on the atomic scale at low temperatures, which leads to large fluctuations and phase separation in liquid Fe-Cu alloy. LLPS of Fe$_{75}$Cu$_{25}$ undercooled melt has also been explored by MD simulation. The Cu-rich liquid droplet is found to be surrounded by the Fe-rich matrix controlled by nucleation growth mechanism.[22] For the spinodal decomposition, the research on the domain size during the early period and the later period of phase separation indicates that the driving force of phase separation is transformed from atom diffusion to interfacial minimization. MD calculation in combination with a reasonable potential model has been applied as a powerful tool to study the separation properties of liquid alloy. The objective of the present work is to characterize concentration fluctuation of FeCu melt during LLPS by computing the structure and thermodynamics parameters. The domain size of droplets, concentration fluctuation and potential energy as a function of relaxation time during the phase separation process are discussed. In this study, calculations are performed using the Fe-Cu potential as proposed by Bonny et al. based on the EAM.[23] The simulations are performed at zero external pressure. Systems of $N=25600$ atoms are put in a cubic simulation box under periodic boundary conditions. To observe the structural change clearly, the box sizes along $x$, $y$ and $z$ directions are set to 40$L$, 40$L$ and 4$L$ ($L$ is the crystal lattice of the Fe), respectively. The number ratio of Fe and Cu is 1:1. MD simulations in NPT ensemble are performed, Newton's equations of motion are integrated with the velocity form of the Verlet algorithm using time step of 1 fs. The masses of Fe and Cu atoms are set to $m_{\rm Fe}=55.845$ a.u. and $m_{\rm Cu}=63.546$ a.u. Firstly, the system is fully equilibrated at 3500 K by performing runs over time steps in the NPT ensemble. To keep the temperature constant, we apply the Nose–Hoover thermostat after every 100 cycles. We have chosen new velocities from a Maxwell–Boltzmann distribution in accordance with the temperature of the system. Next the configuration is used as the initial configuration for further calculation of dynamical properties in undercooled state by running another time step in the NPT ensemble. The temperatures considered are from 2000 K to 1400 K with 100 K interval.
cpl-34-2-026401-fig1.png
Fig. 1. The snapshot of atom at temperature of 1500 K: (a)–(h) correspond to the time of 0 ns, 0.3 ns, 1.6 ns, 3 ns, 5.2 ns, 13.3 ns, 13.5 ns and 22.3 ns.
Figure 1 shows the LLPS process of FeCu melt at the temperature of 1500 K. The initial configuration at 0 ns seems to be homogeneous, after that, the homogeneous atom pairs begin to aggregate because of the positive mixing enthalpy, and the Fe-rich phase precipitates in the Cu-rich matrix due to the relatively larger volume fraction of Cu atom than that of the Fe atom. The interconnected-type structure by the mechanism of spinodal decomposition can be found in the melts at the beginning of relaxation time, as shown in Figs. 1(a) and 1(b). It coarsens gradually and then disappears as the relaxation time is about 3–5 ns, as shown in Figs. 1(c) and 1(d); after that, the droplet tries to become round by the motion of the Fe atom at the edge of droplets due to the minimization of interfacial area, as shown in Fig. 1(e). There are two droplets in the liquid, one is large, and the other is small. The shape of the large droplet almost keeps unchanged even if the time reaches 6 ns, while the small one shows an irregular shape, and always changes with time. Gradually, the two droplets move towards each other due to the fact that Fe atoms at the edge of the small droplet show strong interaction with the Fe atom in the Cu-rich region, although the amount of the Fe atom around the Cu atom is quite small. Fe atoms at the center of the droplet also move toward the Cu-rich region attracted by the Fe atoms at the edge. Although the distance between the two droplets is quite short, it takes them a long time to come together. After that, the coagulation happens at the edge of the small and large droplets. The small one is swallowed up by the large one. The whole coagulation process is finished within 1 ns, the droplet becomes round after coagulation caused by the interfacial minimization of the droplet. Generally speaking, the LLPS of undercooled FeCu melts is controlled by spinodal decomposition mechanism. It undergoes the formation and coarsening of interconnected-type structure, the formation of droplets, and its migration and coagulation.
cpl-34-2-026401-fig2.png
Fig. 2. Concentration fluctuation as a function of the relaxation time at the temperature of 1500 K.
Fe-rich and Cu-rich regions can be observed clearly in Fig. 1. However, the concentration fluctuation of the two regions during the relaxation process is still unknown. We distinguish the Fe (Cu)-rich region by calculating the number of atoms around its first-nearest-shell neighbor, whose radius refers to the valley distance of the first peak in partial correlation function $g_{\alpha \beta}(r)$. If the number of Fe atoms is larger than that of the Cu atom in the neighbor, it is called Fe-rich region, otherwise, it is called Cu-rich region. The atomic percentages of Fe and Cu in the Fe (Cu)-rich region are shown in Fig. 2. We can find that Fe content in the Fe-rich region increases rapidly at the beginning, and it is on the slow rise at about 3–5 ns, and then keeps almost unchanged as time goes on, with 95% Fe content in the Fe-rich region. The growth of concentration fluctuation is realized by the uphill diffusion: the Fe (Cu) atom moves from Cu (Fe)-rich region to Fe (Cu)-rich region; as the LLPS proceeds, and the amount of Fe (Cu) atoms in Cu (Fe)-rich region should be decreased by spinodal decomposition with 5% Cu in Fe-rich region and 10% Fe in Cu-rich region at temperature of 1500 K. The concentration fluctuation reaches the equilibrium state after up-hill diffusion at first 5 ns, while a longer time is needed for migration of the droplets. In any reaction occurring spontaneously, there is a reduction of the Gibbs free energy due to the increasing concentration difference, and the changes of Cu content in the Cu-rich region shows the same trend with 90% Cu in the Cu-rich region. The two separated phases Fe with 10% Cu content and liquid Cu with 90% content exist at temperature of 1500 K in Fe-Cu equilibrium phase diagram, about 5% Cu content lower than that in our simulated system because of the non-equilibrium undercooled liquid. Both the Marangoni migration and the Stokes motion can be used to describe the motion of droplets.[17,18] However, the former emphasizes the temperature gradient induce the migration from lower temperature to higher temperature to reduce the interfacial energy. The latter stresses the density difference between two separated liquids, which may lead to the Stokes motion of the droplets under gravity field. In our simulated system, the temperature in the liquid system is controlled by the Nose/Hoover thermostat under the period boundary condition, which shows uniform temperature distribution in the liquid. The gravity field is also absent and the motion of atom is driven by the interatomic potential function. The formation of droplets, and their migration and coagulation certainly happen, which make us consider the changes of potential energy during the LLPS process.
cpl-34-2-026401-fig3.png
Fig. 3. The variation of potential energy and domain size with relaxation time at the temperature of 1500 K.
Figure 3 shows the changes of average potential energy per atom as a function of relaxation time. The changes of domain size, defined as the boundary length along Fe-rich liquid phase in the simulated system, shown in Figs. 1(g) and 1(h), are also presented in Fig. 3. The domain size of Fe-rich region can be used to evaluate the changes of interfacial area of a cylinder since the simulated box is a cube with the size of $40L\times40L\times4L$, where the Fe-rich region becomes cylindrical along the $z$-direction. The larger the domain size is, the greater the interfacial area of cylinder is. It is obvious that potential energy and domain size show a similar trend during the relaxation process. The potential energy decreases sharply at the early 2–4 ns, and then keeps unchanged at about 4–12 ns. After that, slightly decreasing energy is observed at about 12–15 ns, which corresponds to the migration and coagulation process of the two droplets. The sharp decrease of potential energy drives the up-hill diffusion of the atom and promotes the formation of interconnected structure and its coarsening. The domain size of Fe-rich liquid phase also drops sharply during this period, as shown in Figs. 1(b)–1(e). A minor decrease of potential at 12–15 ns leads to the migration of the two droplets and its coagulation becomes a large one, as a result of decreasing in domain size. Compared with the time spent on the coagulation, longer time is needed for the migration of droplets. The growth of droplets can be achieved easily by the minor driving force. There could exist a link between potential energy $E$ and domain size $D$. As $D$ decreases, $E$ is decreasing. As illustrated in the inset in Fig. 3, a direct linear relationship between $E$ and $D$ is deduced, $$ E=E_{0}+\xi D,~~ \tag {1} $$ where $E$ is the potential energy as the domain size is zero, and the fitting parameter $E=3.442$ eV. If the liquid is homogeneous, no LLPS happens, the lowest energy is 3.442 eV, and the slope of the linear $\xi=0.02$ eV/Å. It indicates that the decrease of domain size is driven by the minimization of potential energy. The average potential energy of per atom at the early relaxation is $-$3.421 eV, higher than the final energy of $-$3.438 eV, the decrease of energy promotes the LLPS process. However, we still wonder which types of atoms in Fe-rich region or in Cu-rich region contribute to the decreasing potential energy, Fe atoms or Cu atoms. The potential energies of Fe-rich region and Cu-rich regions combined with the per-atom energy of Fe and Cu are plotted in Fig. 4. Obviously, Cu atoms show higher energy than Fe atoms in the liquid and the potential energy of Fe atoms decreases with relaxation time from $-$3.562 eV to $-$3.671 eV; in contrast, Cu atoms increase from $-$3.239 eV to $-$3.201 eV. The energy of Fe atoms in Fe-rich region shows a decreasing trend from $-$3.604 eV to $-$3.708 eV, and that of Cu atoms in Fe-rich region changes slightly decreasing from $-$3.276 eV to $-$3.287 eV. However, the energy of Fe atoms shows increasing trends from $-$3.384 eV to $-$3.291 eV in Cu-rich region at the early relaxation time of 0–5 ns, the energy of Cu atoms in Cu-rich region increases slightly from $-$3.231 eV to $-$3.197 eV. Compared with that of Fe atoms in both Fe-rich region or Cu-rich region, the energy of Cu atoms changes only slightly. We can make a speculation from the fact that Fe atoms seem to be more active in the LLPS process due to the decreasing energy, while Cu atoms seem to be more unzealous, and the aggregation of Fe atoms due to stronger interaction of homogeneous atom pairs force Cu atoms to move away from the surrounding area of Fe atoms, leading to formation of Cu-rich region. It also shows energy heterogeneity in the separated liquid. It is stressed that LLPS is driven by the decreasing energy, which is attributed to the decreasing potential energy of Fe atoms.
cpl-34-2-026401-fig4.png
Fig. 4. Relaxation time dependence of potential energy of the two separated phases at the temperature of 1500 K.
cpl-34-2-026401-fig5.png
Fig. 5. Fe atomic percentage in Fe-rich region (a) and potential energy difference of Cu-rich region and Fe-rich region of per atom (b) at various temperatures.
The driving force of up-hill diffusion originates from the decreasing potential energy, while the energy increases with time in Cu-rich region, and decreases in Fe-rich region. The Cu atoms are forced to move from Fe-rich region to Cu-rich region. The concentration fluctuation of Fe-rich region at various temperatures is shown in Fig. 5(a). The demixing behavior of FeCu melts is enhanced with decreasing temperature: the Cu content is 20% in Fe-rich region at temperature of 2000 K, only 4% at temperature of 1400 K. LLPS does not happen if temperature is higher than 1900 K, although the concentration heterogeneity still exists in the liquid. The slope of the curve can be considered as the diffusion rate, it increases with the decreasing temperature, since high temperature is in favor of the diffusion of atoms, while the diffusion rate shows the lowest value at 1900 K. Furthermore, longer time is needed to reach the concentration balance at higher temperature. As shown in Fig. 4, LLPS is driven by the decreasing potential energy, which makes us consider the energy changes in Fe-rich region and Cu-rich region at various temperatures. The potential energy difference between Cu-rich region and Fe-rich region is as follows: $$\begin{align} \Delta E=E_{\rm Cu-rich} -E_{\rm Fe-rich}.~~ \tag {2} \end{align} $$ Figure 5(b) shows the time dependence of $\Delta E$. The shape of the curve is quite similar to that in Fig. 5(a). Since Cu atoms show higher energy than that of Fe atoms, $\Delta E$ shows a positive value. At early relaxation period, a large number of Fe atoms in the Cu-rich region move to the Fe-rich region to decrease the energy, leading to the motion of Cu atoms in the Fe-rich region toward the Cu-rich region. The energy difference $\Delta E$ becomes larger at lower temperature than that at higher temperature, indicating larger driving force of Fe atoms moving toward the Fe-rich region at lower temperature, which also accounts for the larger diffusion rate of Fe atoms. Consequently, larger Fe content at lower temperature can be observed. Once the droplet is formed in the liquid, if the cooling rate is high enough to suppress the coagulation of the two droplets, the droplet with smaller size could be kept in the glass, which is in favor of the improvement of plasticity of the glass. FeCu melts are characterized by the miscibility gap below the liquidus. It separates into Fe-rich and Cu-rich liquid upon relaxation through the gap controlled by the spinodal decomposition mechanism. The LLPS undergoes the formation of interconnected structure, the coarsening of interconnected structure and the formation of droplet with irregular shape, the rounding of droplet, migration and coagulation by potential energy minimization. During the LLPS process, the Fe (Cu) content in Fe (Cu)-rich region increases quickly, and the lower temperature implies the higher diffusion rate because of the larger potential energy difference between Cu-rich region and Fe-rich region. Fe atoms are more active to move to the Fe-rich region due to energy minimization, therefore Cu atoms around Fe atoms are pushed out to the Cu-rich region. The concentration almost does not vary after up-hill diffusion. Compared with the up-hill diffusion, longer time will be spent on migration of droplets. The concentration difference in Fe-rich region and Cu-rich region becomes larger and larger with the decreasing temperature. The driving force of LLPS comes from the decreasing potential energy, although it is inhomogeneous in the separated liquid. We acknowledge the supercomputing system in the supercomputing center of Shandong University for the numerical calculations. 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