† Corresponding author. E-mail:
Molecular dynamics simulations are performed to investigate the liquid–liquid phase transition (LLPT) and the spatial heterogeneity in Al–Pb monotectic alloys. The results reveal that homogeneous liquid Al–Pb alloy undergoes an LLPT, separating into Al-rich and Pb-rich domains, which is quite different from the isocompositional liquid water with a transition between low-density liquid (LDL) and high-density liquid (HDL). With spatial heterogeneity becoming large, LLPT takes place correspondingly. The relationship between the cooling rate, relaxation temperature and percentage of Al and the spatial heterogeneity is also reported. This study may throw light on the relationship between the structure heterogeneity and LLPT, which provides novel strategies to control the microstructures in the fabrication of the material with high performance.
A possible first-order LLPT between two forms of the liquid phase in some supercooled pure substances has received considerable interest in recent years,[1–8] because LLPT offers an avenue for interesting applications that exploits different properties of distinct liquid phases.[9] Understanding LLPT and its dynamic properties is of great importance to researchers in the field of physics, chemistry and materials science. However, despite intense scrutiny over the years, scientists are still far from reaching a coherent understanding of unusual structural behavior in the process of LLPT.[10–12] The existence of two coexisting liquids with the same composition but different structure[13, 14] has been a controversial issue.
Among the substances having two coexisting liquids in supercooled states are water,[5–8] silicon,[2] germanium,[15] carbon, and hydrogen, etc. There is much evidence from simulation and experiment results that is suggestive of the existence of LLPT for a variety of liquids. For example, the LLPT of water was first put forward[5] to account for its peculiar thermodynamic and dynamic anomalies in supercooled liquid states, which was followed by various theories[16–20] to support it. Recent studies provided sufficient evidence for a liquid–liquid transition in ST2 model[21–26] and TIP4P/2005 model[27–29] which gave the best agreement with the experiment among the TIPnP family of water.[30] Remarkably similar to water, The LLPT in Si was first predicted by Aptekar,[15] and subsequently supported by MD simulations combined with experiment data obtained by x-ray diffraction and Raman spectroscopy.[31–34] In addition, the coexistence of two forms of phosphorus liquids was directly observed by in situ x-ray diffraction experiment.[35–37] Besides the isocompositional LLPTs, supercooled liquid to liquid transition, especially in sulfur, has also been comprehensively studied.[38–40] However, to our limited knowledge, fewer efforts have been focused on the type of binary liquids in the perspective of LLPT, such as liquid Al–Pb binary alloy.
Considerable efforts have been devoted to studying the local structure in LLPT because the local structure leads to its heterogeneity which has close relation with the structural origins of LLPT and associated phenomena. For example, Debenedetti pointed out that the difference between two water liquids lies in their local structures: in the high density liquid, the local tetrahedrally coordinated hydrogen-bond structure is not fully developed, whereas in the low density liquid, locally “ice-like“ hydrogen-bond network is fully realized.[5,9] The low density liquid had a more open network compared to the high density liquid. The two types of liquid phases separated by a line of first-order LLPT were identified by their density and local atomic structure. Tanaka also argued that liquid is not homogeneous and the heterogeneity may be responsible for the LLPT in the molecular liquid.[41] Subsequently, by defining parameters presenting the local favored heterogeneous structure, Tanaka gave a good explanation of LLPT in water.[42–44] By analogy with different structures of two forms of liquid in water, LLPT of liquid Si also involves structure transition from a metallic dense-packed structure at high temperature to a semiconducting open network at low temperature.[45] Furthermore, Treacy and Borisenko[46] believed that the continuous random network (CRN) model represents the structural topology of amorphous silicon which has been used successfully to liquid silicon to find the inhomogeneous paracrystalline structures containing local structure at the 10 to 20 angstrom length scale, indicating the structural heterogeneity. However, the relationship between structure heterogeneity and LLPT process remains uncertain. In addition, how to quantify structure heterogeneity remains unsolved.
Al–Pb alloy is essential for a number of industrially important materials, such as wheel bearings. However, its potential applications[47] are seriously limited because of a major drawback caused by two liquid phases within the miscibility gap having different densities, resulting in phase separation at the macro scale.[48,49] The existence of LLPT in alloy has important implications for understanding the low-temperature liquid state in general.[50,51] It may provide an opportunity for comprehensive and satisfactory understanding of the structures and properties of immiscible liquid alloy. Till now, although there has been much progress in the understanding of the behavior of phase separation, some important questions on the LLPT of binary alloy have remained unanswered.[52–55] In this paper, we report molecular dynamic simulations on the LLPT of Al–Pb alloy, highlighting its relationship with structure heterogeneity.
In this study, MD simulations are used to investigate the structure of liquid Al–Pb alloy containing 10000 atoms placed between two isolated and smooth walls, initially arranged into 50 × 50 × 1 face-centered cubic (FCC) lattice. The interaction among metal atoms is described with embedded-atom method (EAM) which is particularly appropriate for metallic systems. The directions x and y are imposed periodic boundary conditions, while the z direction is bounded with walls. In this work, we use MD package LAMMPS to perform the simulations. All simulations are carried out at a constant lateral pressure, using the constant number, lateral pressure, temperature (NPT) ensemble. The Nose–Hoover thermostat and barostat are used to control the temperature and pressure, respectively, and the velocity–verlet algorithm is used to integrate the equations of motion with a time step of 1 fs. The liquid is equilibrated at 2000 K first. Subsequently, the well-equilibrated liquid alloy is cooled from 2000 K to the target temperature at different quenching rates. The final structures are obtained after a relaxation process at the target temperature.
Figures
![]() |
Figure
It was found in liquid carbon film[57] that the position of the first peak gradually increased as the temperature decreased, which was opposite to the scenario of the normal substance. The author ascribed the anomaly to the LLPT, during which carbon transformed from the twofold coordinated liquid to the threefold coordinated liquid. Remarkably similar to carbon, the partial Al–Pb pair correlation function (PCF) during cooling process also exhibits the anomaly that the position of the first peak increases with the reduced temperature, as shown in Fig.
It is suggested that if the distance between two atoms is less than a given cutoff distance, which is equal to the position of the first minimum in the corresponding pair correlation function, then the pair of atoms is considered to form a bond. Based on this principle, Al–Pb bond fraction at different cooling processes is calculated and displayed in Fig.
To well describe spatial heterogeneity in the LLPT process, micro-structure factor (MSF) is put forward and defined as follows:
![]() |
The definition of MSF is similar to the variance in mathematics, which is the expectation of the squared deviation of a random variable from its mean, and informally measures how far a set of random numbers are spread out from their mean. Similarly, the MSF is used to measure how far the concentration of each atom deviates from the average concentration. If the system tends to be homogeneous, then the concentration of each atom deviates little from the average, and MSF value is small. However, when the system displays a great degree of heterogeneity, the concentration of each atom deviates far from the average, and MSF value is large. Therefore, the MSF is effective to measure the extent of heterogeneity of liquid alloy.
Figure
The parameter to judge the start of the LLPT in water was the density as reported by Wang et al.[58] By detecting the average density of water with warming and cooling scans, the author recognized the position where the maximum density difference between the cooling and warming scans emerged as the starting position of LLPT. Similarly, there is also a parameter MSF to determine the start of the LLPT of Al–Pb alloys. It is understandable that there is a striking close relation between the LLPT and the spatial heterogeneity. Note that MSF curve has a period of fluctuation without apparent rise or drop at the initial stage of the cooling process. The starting temperature where MSF rises can be considered apparently as the starting temperature point of LLPT. It can be seen from Fig.
Next, we compare Al–Pb bond fraction and MSF curves at different cooling rates. When the cooling rate is 100 K/ps, both Al–Pb bond fraction and MSF values exhibit no dramatic change, suggesting low extent of phase separation and spatial heterogeneity; whereas when the cooling rate is 0.1 K/ps, both Al–Pb bond fraction and MSF values show dramatic change, indicating high extent of phase separation and spatial heterogeneity. To directly investigate how cooling rate affects the LLPT process and spatial heterogeneity, we plot Al–Pb bond fraction and MSF curves as a function of cooling rate as displayed in Figs.
To explore how relaxation temperature affects the spatial heterogeneity, the liquid alloys are quenched to the target temperature at 100 K/ps and then relaxed at the target temperature. Since MSF is effective to estimate the spatial heterogeneity, figure
![]() | Fig. 4. (color online) (a) MSF versus temperature at 1000 ps in relaxation process for alloys with different atomic Al percentages. (b) MSF of Al50Pb50 alloy during relaxation process. |
Another important issue is the reaction rate of the LLPT at different temperatures. Contrary to the conventional wisdom that the reaction rate is higher at higher temperature, our results reveal an opposite trend in terms of LLPT process. Figure
Through the example of liquid Al50Pb50 alloy, we explain the reason why MSF curves behave like this. Figure
Finally we report the effect of Al percentage on the spatial heterogeneity during the LLPT process. Figure
![]() | Fig. 6. (color online) (a) MSF as a function of component at different relaxation time. (b) The number of atoms distributed in each component interval when the system is at relaxation equilibrium state. (c) The phase diagram for Al–Pb monotectic liquid alloy. Previous study data are taken from Ref. [59]. |
Wang et al.[58] plotted a phase diagram of the LLPT of the confined heavy water by finding critical points under different pressures and temperatures. By the same token, phase diagram of the LLPT in the confined liquid SiC was also plotted.[56] Here, we calculate the MSF of all alloys during the cooling process when the cooling rate is 1 K/ps. By determining the starting point where the MSF values start to rise conspicuously, we plot the phase diagram of the LLPT of the Al–Pb alloy compared with that obtained by previous study[59] which is shown in Fig.
In summary, molecular dynamics simulations are carried out to investigate the liquid–liquid phase transition (LLPT) and the spatial heterogeneity in Al–Pb monotectic alloys. It is shown that the homogeneous one single liquid phase transforms into two liquid phases with different components and densities, which is considered as a type of LLPT. LLPT process is highly correlated with the spatial heterogeneity. With increasing spatial heterogeneity, LLPT takes place correspondingly. Higher cooling rate can delay the LLPT and prevent the increase of spatial heterogeneity. Spatial heterogeneity would increase with decreasing temperature when the alloys are liquid. As the temperature further reduces, spatial heterogeneity would decrease due to the crystallization of Al-rich domains. Moreover, high temperature can lead to low LLPT reaction rate. Spatial heterogeneity and LLPT reaction rate are strongly component-dependent. Of all alloys, Al50Pb50 alloy has the largest spatial heterogeneity and reaction rate. Specifically, the phase diagram of the LLPT of the Al–Pb alloy is plotted.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] | |
[42] | |
[43] | |
[44] | |
[45] | |
[46] | |
[47] | |
[48] | |
[49] | |
[50] | |
[51] | |
[52] | |
[53] | |
[54] | |
[55] | |
[56] | |
[57] | |
[58] | |
[59] |