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Scaling behavior of the surface energy in face-centered cubic metals
Computational Materials Science 92 (2014) 166–171
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Scaling behavior of the surface energy in face-centered cubic metals Minho Jo a, Y.W. Choi a, Y.M. Koo a,b, S.K. Kwon a,⇑ a b
Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
a r t i c l e
i n f o
Article history: Received 3 February 2014 Received in revised form 17 May 2014 Accepted 20 May 2014
Keywords: Surface energy scaling Universal binding energy curve Molecular dynamics simulation
a b s t r a c t The binding energy of metals shows a universal feature which can be described by an equation of state. We explore the scaling behavior of the surface energy in face-centered cubic metals and propose the concept of equivalent structures. The surface energies were calculated on various orientations using the modified embedded-atom method. A strong linear correlation was observed between the surface energies of different metals. Based on the results, we established a scaled surface energy-to-element relationship. This scalability suggests an efficient scheme to estimate the orientation dependency of the surface energy by two characteristic parameters. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The surface energy, defined by the excess energy of surface atoms compared to bulk, is a fundamental physical quantity to understand surface-related phenomena in materials science. For examples, surface morphology, surface reconstruction, and the shape of nano-sized particles have been described in terms of the surface energy [1–7]. Nonetheless, experimental determination of the surface energy is still difficult and inaccurate. The surface energy in experiments is usually derived by extrapolating the surface tension of liquid phase to low temperature, which involves unavoidable errors [8]. Meanwhile, extensive theoretical studies have been performed on the surface energy of various metals by using ab initio method [9–14] and semi-empirical method [15,16]. Although the surface energy of low indexed surfaces can be obtained with accuracy, determination of the surface energy of high indexed surfaces is time-consuming with ab initio method and semi-empirical values are suspicious in reliability. Therefore, in order to identify the orientation dependency of the surface energy, it is essential to understand a general behavior of the surface energy. It is well known that the binding energy of metals exhibits a universal behavior regardless of the crystalline structure [17–19]. In other words, the binding energy curves of different metals can be merged into a single curve through a scaling with the ground state quantities: cohesive energy, equilibrium lattice constant, and bulk modulus. Considering that the surface energy of metals
⇑ Corresponding author. Tel.: +82 54 2799024. E-mail address: [email protected] (S.K. Kwon). http://dx.doi.org/10.1016/j.commatsci.2014.05.042 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
can be roughly estimated by counting the number of broken bonds of the nearest-neighbor [20,21], one may imagine a possibility of an element-independent description of the surface energy, similarly to the binding energy. In this work, we examine the scalability of the surface energy in face-centered cubic (fcc) metals. Surface energies of various orientations were obtained by using the modified embedded-atom method (MEAM) [22,23]. We show a linear relationship among the surface energies of different metals. Based on the result, the variation of the surface energy on orientation is successfully scaled to a single energy curve. We discuss the scalability and the binding energy in a general view of the structure–element–energy correlation. 2. Computational method We employed the second nearest-neighbor MEAM potential for eight fcc metals to obtain the total energy by molecular dynamics method: Al, Ni, Cu, Pd, Ag, Pt, Au, and Pb [23]. The universality of the binding energy of the fcc metals was checked with the total energy calculation by changing the lattice constant. Surfaces were modeled by the double-sided slab geometry using the bulk equilibrium lattice parameter. For each metal, we considered 16 (12) 0ð½1 0 0Þ direction different surface orientations normal to the ½1 1 as listed in Table 1. The slab thickness was taken to be more than 20 atomic layers depending on the surface orientation, which guarantees that the surface energy in the calculations was converged within the numerical error of less than 0.01 eV/surface atom. The surface energies of both unrelaxed and relaxed structures were considered to inspect how the atomic relaxation modifies the structure–energy correlation.
M. Jo et al. / Computational Materials Science 92 (2014) 166–171 Table 1 The surface orientations and angles for calculation. The angles are measured relative to the (1 1 0) and (0 0 1) surface orientation. Orientation
Angle (°)
Normal to 0 ½1 1
(1 1 0) (4 4 1) (3 3 1) (2 2 1) (3 3 2) (5 5 4) (1 1 1) (3 3 4) (2 2 3) (1 1 2) (2 2 5) (1 1 3) (1 1 4) (1 1 6) (1 1 8) (0 0 1)
0.0 10.0 13.3 19.5 25.2 29.5 35.3 43.3 46.7 54.7 60.5 64.8 70.5 76.7 80.0 90.0
Normal to [1 0 0]
(0 0 1) (0 1 8) (0 1 5) (0 1 4) (0 1 3) (0 2 5) (0 1 2) (0 3 5) (0 2 3) (0 3 4) (0 4 5) (0 1 1)
0.0 7.1 11.3 14.0 18.4 21.8 26.6 31.0 33.7 36.9 38.7 45.0
3. Results and discussion It was shown by Rose et al. [17] that the binding energy of metals can be scaled into a single energy curve through three steps; (i) the total energy is divided by the cohesive energy, (ii) the lattice parameter is shifted and scaled with the equilibrium lattice constant, and (iii) the width of the total energy curve is adjusted with the bulk modulus. We followed an equivalent procedure as above except the last step. For a simplification, the width of the total energy curve was scaled by the difference of the lattice constant between at the equilibrium and half of the cohesive energy. We show the results of the fcc metals in Fig. 1(a). It is clear that all scaled binding energy curves are well fitted into a single curve
167
throughout the interatomic distance. This indicates that the employed interatomic potentials provide reliable cohesive properties of considering materials. The total energy of a crystalline material generally depends on two factors, lattice structure and constituent elements. It is an interesting question to what extend the two factors can be decomposed to be dealt with independently. When it is the case, the total energy can be described solely with the element-dependent factor after confining the structural factor. Definitely, even for a given structure the system energy relies on the composition of constituent elements and it is not an easy task to estimate the total energy. However, there are cases in which two system energies of different compositions or elements can be matched by a relationship in the structural parameter space and a universal class can be identified. In that circumstance, we define the pair of structures as an equivalent structure. In other words, all pair of equivalent structures between the structural group A and A0 having a different composition a and b, respectively, will satisfy a relation Ea; A ¼ f ðEb; A0 Þ where f is a correlation function. Note that the equivalent structures need not be the same structure. The binding energy is an example of the equivalent structure. As demonstrated above, the binding energy behavior of metals is universal with the scaled interatomic distance. Although we figured out only several discrete points in Fig. 1(a), the curve is continuous and every point on it can be reached by all the elements. When such a scaled energy curve is available for a class of material systems, we will define the group of material structures which belongs to the same point of the scaled curve as an equivalent structure. Thus, there is an infinite number of equivalent structure sets in a scaled energy curve and the equilibrium structures of metals, as a trivial example, are equivalent to each other. Note that the structures in an equivalent structure set do not have the same lattice parameter and total energy; the equivalent structure imposes a capability to occupy the same point in the scaled energy curve for the structures of different materials. The equivalent structure implies a very remarkable feature. In Fig. 1(b), we denoted eleven equivalent structures of seven fcc metals, which are matched with those of Al. Each of seven points located along the vertical axis are equivalent and able to be overlapped by the scaling of the cohesive energy. Reminding the prescribed scaling steps, it can be understood that the scaling step (ii) and (iii) identify the group elements of the equivalent structure and the step (i) equalize the energy values in the same equivalent structure. From the fact, we give an energy relationship
Fig. 1. (a) The scaled binding energy of fcc metals as a function of the scaled interatomic distance. (b) The energy–energy correlation of the binding energies with Al as a basis. Inset is the scaled binding energy between the equivalent structures. The equivalent structure is defined as a group of structures which occupies the same points in the scaled energy curve.
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0 (upper panels) and [1 0 0] direction (lower panels). The Fig. 2. Unrelaxed (left) and relaxed (right) surface energies of fcc metals of various orientations normal to the ½1 1 0 direction is defined by the angle from the (1 1 0) plane and that normal to the [1 0 0] direction by the angle from the (0 0 1) plane. surface orientation normal to the ½1 1
Ea ¼ cEb ;
ð1Þ
where Ea and Eb are the total energies of element a and b in an equivalent structure, respectively, and c is the ratio of the cohesive energies. It is most important to note that once a set of the equivalent structure can be scaled by the relation, other sets of the equivalent structure should be also scalable by the same parameters. In other words, the binding energy curve of a given metal can be transformed into that of other metals with the universal relationship. Therefore, the equivalent structure can be regarded as the structures having the same structure-dependent factor of the total energy so that the energy of an element in the set correlates with those of other elements following the specific relationship which reflects the effect of the element-dependent factor. The idea of the equivalent structure can be applied to other cases. Here, we test it to the surface energy (c). Fig. 2 shows the calculated surface energies of Al, Ni, Cu, Pd, Ag, Pt, Au, and Pb in the fcc structure with and without lattice relaxation. We consid 0 ered two classes of the surface orientation normal to the ½1 1 and [1 0 0] direction. All the surface energy curves show the minimum for the (1 1 1) surface which is the most closed-packed plane of the fcc structure. Pt (Pb) has the highest (lowest) surface energy and other metals are located between them, regardless of the surface orientation and relaxation. This fact can be understood by taking into account the melting temperatures of metals. The melting temperature of Pb is Tm = 600.7, which is significantly lower than those of other metals, and Pt has the highest melting temperature Tm = 2045 K [24].
We would like to emphasis that the surface energy values in Fig. 2 are Pb < Al < Ag < Au < Cu < Pd < Ni < Pt in order, irrespective of the surface orientation. This observation suggests a possibility of the surface energy scaling. It is straightforward to guess that the surface energies of the same orientation specify the equivalent structure and they might be connected by a universal relationship. At first, we try a single parameter scaling of the surface energy. Fig. 3 is the dimensionless surface energies which were scaled with c111 and c100 for the surfaces normal to the ½1 1 0 and [1 0 0] direction, respectively. It is evident that the surface energy scaling to reach a universality cannot be accomplished with a single parameter. Both c/c111 and c/c100 behaviors are highly dependent upon metals and the scaled surface energies do not overlap each other except c111 and c100 points. For example, c110/c111 is about 1.20 and 1.46 for Cu and Al, respectively, for the relaxed surface. Reminding that the equivalent structure was defined as a correlation of the total energy between different materials, we start a scaling procedure by searching for an energy–energy correlation between the same oriented surfaces of different metals. Fig. 4 shows the correlation of the surface energies for the pairs of fcc metals in a cyclic order: Al ? Ni ? Cu ? Pd ? Ag ? Pt ? Au ? Pb ? Al and Al ? Ag. Red and blue points represent unrelaxed and relaxed surface energies, respectively. We allowed both inplain and out-of-plane surface relaxation. Obviously, the surface energies of all the pairs of considering fcc metals show a strong linear correlation, which implies that the surfaces of the same orientation are in an equivalent structure. This fact can be also checked with previous surface energy data [9–13], though it was
M. Jo et al. / Computational Materials Science 92 (2014) 166–171
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0 (upper panels) and Fig. 3. Scaled surface energies c/c111 and c/c100 for unrelaxed (left) and relaxed (right) surfaces of fcc metals of various orientations normal to the ½1 1 [1 0 0] direction (lower panels).
Fig. 4. The energy–energy correlation of the surface energies between the equivalent surfaces of fcc metals. Red and blue dots represent unrelaxed and relaxed surface energy, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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M. Jo et al. / Computational Materials Science 92 (2014) 166–171
not recognized due to the limited available data. Relatively large deviations are observed in the relaxed surface energies of Ag–Pt, Pt–Au, and Pb–Al pairs, which come from the rearrangement of surface atoms. That is, the surface relaxation can lead to a mismatching of the equivalent structure. However, the correlation between the surface energies largely retains its linearity when the surface relaxation is not severe. In a similar way to Eq. (1), we write down a relationship between the surface energies of two metals as:
ca ¼ acb þ b;
ð2Þ
where ca and cb are the surface energies of element a and b, respectively, with the same orientation (equivalent surface) and a and b are correlation parameters which are independent upon the surface orientation. By choosing the surface energy of Al (cAl) as a reference, we determined the correlation parameters for other metals by
fitting the surface energy data. The numerical values of the parameters and the cohesive energy are listed in Table 2. For the considering metals, we have 3d (Ni, Cu), 4d (Pd, Ag), and 5d (Pt, Au) elements. The cohesive energy increases from the left and right side within a given period of the periodic table. We found that a and b follow the same trend as that of the cohesive energy except b of Au in the relaxed case. Therefore, Eq. (2) has a similar electronic origin to the universality of the binding energy curve. With the obtained parameters, a and b, the surface energies are scaled and plotted in Fig. 5. It is manifest that the variation of the surface energy on orientation can be described in a single curve, especially for the unrelaxed surface. As mentioned above, the surface relaxation gives rise to a slight modification of the surface energy correlation. The two correlation parameters, a and b, are evidently element-dependent factors from the definition of the equivalent structure. Since the parameters are measurable quantities, they might be able to be used for a characterization of fcc metals.
Table 2 The scaling parameters of the surface energy, a and b in Eq. (2). a is dimensionless and b has the same dimension as the surface energy. The parameters were determined by choosing the surface energy of Al as a reference. Al
Ni
Cu
Pd
Ag
Pt
Au
Pb
Unrelaxed
a b
1 0
1.447 0.721
0.850 0.672
1.089 0.788
0.531 0.528
2.018 0.475
0.813 0.435
0.202 0.257
Relaxed
a b
1 0
1.485 0.702
0.823 0.683
1.046 0.813
0.485 0.554
2.102 0.311
0.569 0.566
0.164 0.271
3.36
4.45
3.54
3.91
2.85
5.77
3.93
2.04
Cohesive energies (eV/atom)
0 (upper panels) and [1 0 0] direction (lower Fig. 5. The scaled surface energy of fcc metals for unrelaxed (left panels) and relaxed (right panels) surfaces normal to the ½1 1 panels).
M. Jo et al. / Computational Materials Science 92 (2014) 166–171
The number of required scaling parameters is one for the binding energy and two for the surface energy. While the equivalent structure of the binding energy can be assigned by a single parameter, i.e., the cohesive energy, that of the surface energy needs two parameters, i.e., a and b. In this sense, the element–energy correlation of the binding energy is 1D and that of the surface energy is 2D. While the surfaces of the same orientation were intuitively guessed to be equivalent, it was not straightforward how to describe the equivalent structure. Defining the equivalent structure indeed corresponds to the identification of the structuredependent parameters. In case of the binding energy, the lattice constant and bulk modulus are the structure-dependent parameters. Although the grouping scheme is not yet clear for other structures, the generalization of method would be a key to reveal the full relation among element, structure, and energy. Recently, a single-parameter scalability of the grain boundary energy was also proposed [25]. Material structures in different dimensions might require a different number of element–energy correlation parameters. Therefore, the dimensionality of the element–energy correlation can be regarded as a feature of the material structure. 4. Conclusion We have investigated the universal behavior of the surface energy in fcc metals. Considering the structure–element–energy correlation, it was shown that the factors of structure and element can be separately treated for the surface energy. Two parameters were introduced to scale the surface energy. While the scaling parameters for the binding energy are related to basic measurable quantities, the physical meaning of the correlation parameters is not yet fully identified for the surface energy. Further investigations are left to clarify this point which will extend our view of the element–energy correlation. Since it is not realistic to measure and/or calculate all necessary energies of structures and elements, revealing the underlying physics of the structure–element–energy
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correlation will be helpful for materials development and application. Acknowledgements This research was supported by the Steel Innovation Program of POSCO, the Technology Innovation Program funded by the Ministry of Knowledge Economy (10041187), and the World Class University program through the National Research Foundation of Korea (R32-1014). References [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]
R.C. Cammarata, Prog. Surf. Sci. 46 (1994) 1–38. H. Ibach, Surf. Sci. Rep. 29 (1997) 195–263. N. Moll, M. Scheffler, E. Pehlke, Phys. Rev. B 58 (1998) 4566–4571. V.A. Shchukin, D. Bimberg, Rev. Mod. Phys. 71 (1999) 1125–1171. D.-H. Wang, G.-F. Wang, Appl. Phys. Lett. 98 (2011) 083112. T. Goudarzi, R. Avazmohammadi, R. Naghdabadi, Mech. Mater. 42 (2010) 852– 862. P. Müller, A. Saúl, Surf. Sci. Rep. 54 (2004) 157–258. W.R. Tyson, W.A. Miller, Surf. Sci. 62 (1977) 267–276. M. Methfessel, D. Hennig, M. Scheffler, Phys. Rev. B 46 (1992) 4816–4829. H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157–7168. L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollár, Surf. Sci. 411 (1998) 186–202. S.K. Kwon, Z. Nabi, K. Kádas, L. Vitos, J. Kollár, B. Johansson, R. Ahuja, Phys. Rev. B 72 (2005) 235423. N.E. Singh-Miller, N. Marzari, Phys. Rev. B 80 (2009) 235407. B. Medasani, I. Vasiliev, Surf. Sci. 603 (2009) 2042–2046. P. Stoltze, J. Phys.: Condens. Matter 6 (1994) 9495–9517. A.M. Rodríguez, G. Bozzolo, J. Ferrante, Surf. Sci. 289 (1993) 100–126. J.H. Rose, J.R. Smith, J. Ferrante, Phys. Rev. B 28 (1983) 1835–1845. A.P. Sutton, Electronic Structure of Materials, Oxford University Press, Oxford, 1993. A. Banerjea, J.R. Smith, Phys. Rev. B 37 (1988) 6632–6645. I. Galanakis, N. Papanikolaou, P.H. Dederichs, Surf. Sci. 511 (2002) 1–12. Q. Jiang, H.M. Lu, M. Zhao, J. Phys.: Condens. Matter 16 (2004) 521–530. M.I. Baskes, Phys. Rev. B 46 (1992) 2727–2742. B.-J. Lee, J.-H. Shim, M.I. Baskes, Phys. Rev. B 68 (2003) 1441121. C. Kittel, Introduction to Solid State Physics, eighth ed., John Wiley & Sons Inc, New York, 2005. E.A. Holm, D.L. Olmsted, S.M. Foiles, Scr. Mater. 63 (2010) 905–908.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Scaling behavior of the surface energy in face-centered cubic metals Minho Jo a, Y.W. Choi a, Y.M. Koo a,b, S.K. Kwon a,⇑ a b
Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
a r t i c l e
i n f o
Article history: Received 3 February 2014 Received in revised form 17 May 2014 Accepted 20 May 2014
Keywords: Surface energy scaling Universal binding energy curve Molecular dynamics simulation
a b s t r a c t The binding energy of metals shows a universal feature which can be described by an equation of state. We explore the scaling behavior of the surface energy in face-centered cubic metals and propose the concept of equivalent structures. The surface energies were calculated on various orientations using the modified embedded-atom method. A strong linear correlation was observed between the surface energies of different metals. Based on the results, we established a scaled surface energy-to-element relationship. This scalability suggests an efficient scheme to estimate the orientation dependency of the surface energy by two characteristic parameters. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The surface energy, defined by the excess energy of surface atoms compared to bulk, is a fundamental physical quantity to understand surface-related phenomena in materials science. For examples, surface morphology, surface reconstruction, and the shape of nano-sized particles have been described in terms of the surface energy [1–7]. Nonetheless, experimental determination of the surface energy is still difficult and inaccurate. The surface energy in experiments is usually derived by extrapolating the surface tension of liquid phase to low temperature, which involves unavoidable errors [8]. Meanwhile, extensive theoretical studies have been performed on the surface energy of various metals by using ab initio method [9–14] and semi-empirical method [15,16]. Although the surface energy of low indexed surfaces can be obtained with accuracy, determination of the surface energy of high indexed surfaces is time-consuming with ab initio method and semi-empirical values are suspicious in reliability. Therefore, in order to identify the orientation dependency of the surface energy, it is essential to understand a general behavior of the surface energy. It is well known that the binding energy of metals exhibits a universal behavior regardless of the crystalline structure [17–19]. In other words, the binding energy curves of different metals can be merged into a single curve through a scaling with the ground state quantities: cohesive energy, equilibrium lattice constant, and bulk modulus. Considering that the surface energy of metals
⇑ Corresponding author. Tel.: +82 54 2799024. E-mail address: [email protected] (S.K. Kwon). http://dx.doi.org/10.1016/j.commatsci.2014.05.042 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
can be roughly estimated by counting the number of broken bonds of the nearest-neighbor [20,21], one may imagine a possibility of an element-independent description of the surface energy, similarly to the binding energy. In this work, we examine the scalability of the surface energy in face-centered cubic (fcc) metals. Surface energies of various orientations were obtained by using the modified embedded-atom method (MEAM) [22,23]. We show a linear relationship among the surface energies of different metals. Based on the result, the variation of the surface energy on orientation is successfully scaled to a single energy curve. We discuss the scalability and the binding energy in a general view of the structure–element–energy correlation. 2. Computational method We employed the second nearest-neighbor MEAM potential for eight fcc metals to obtain the total energy by molecular dynamics method: Al, Ni, Cu, Pd, Ag, Pt, Au, and Pb [23]. The universality of the binding energy of the fcc metals was checked with the total energy calculation by changing the lattice constant. Surfaces were modeled by the double-sided slab geometry using the bulk equilibrium lattice parameter. For each metal, we considered 16 (12) 0ð½1 0 0Þ direction different surface orientations normal to the ½1 1 as listed in Table 1. The slab thickness was taken to be more than 20 atomic layers depending on the surface orientation, which guarantees that the surface energy in the calculations was converged within the numerical error of less than 0.01 eV/surface atom. The surface energies of both unrelaxed and relaxed structures were considered to inspect how the atomic relaxation modifies the structure–energy correlation.
M. Jo et al. / Computational Materials Science 92 (2014) 166–171 Table 1 The surface orientations and angles for calculation. The angles are measured relative to the (1 1 0) and (0 0 1) surface orientation. Orientation
Angle (°)
Normal to 0 ½1 1
(1 1 0) (4 4 1) (3 3 1) (2 2 1) (3 3 2) (5 5 4) (1 1 1) (3 3 4) (2 2 3) (1 1 2) (2 2 5) (1 1 3) (1 1 4) (1 1 6) (1 1 8) (0 0 1)
0.0 10.0 13.3 19.5 25.2 29.5 35.3 43.3 46.7 54.7 60.5 64.8 70.5 76.7 80.0 90.0
Normal to [1 0 0]
(0 0 1) (0 1 8) (0 1 5) (0 1 4) (0 1 3) (0 2 5) (0 1 2) (0 3 5) (0 2 3) (0 3 4) (0 4 5) (0 1 1)
0.0 7.1 11.3 14.0 18.4 21.8 26.6 31.0 33.7 36.9 38.7 45.0
3. Results and discussion It was shown by Rose et al. [17] that the binding energy of metals can be scaled into a single energy curve through three steps; (i) the total energy is divided by the cohesive energy, (ii) the lattice parameter is shifted and scaled with the equilibrium lattice constant, and (iii) the width of the total energy curve is adjusted with the bulk modulus. We followed an equivalent procedure as above except the last step. For a simplification, the width of the total energy curve was scaled by the difference of the lattice constant between at the equilibrium and half of the cohesive energy. We show the results of the fcc metals in Fig. 1(a). It is clear that all scaled binding energy curves are well fitted into a single curve
167
throughout the interatomic distance. This indicates that the employed interatomic potentials provide reliable cohesive properties of considering materials. The total energy of a crystalline material generally depends on two factors, lattice structure and constituent elements. It is an interesting question to what extend the two factors can be decomposed to be dealt with independently. When it is the case, the total energy can be described solely with the element-dependent factor after confining the structural factor. Definitely, even for a given structure the system energy relies on the composition of constituent elements and it is not an easy task to estimate the total energy. However, there are cases in which two system energies of different compositions or elements can be matched by a relationship in the structural parameter space and a universal class can be identified. In that circumstance, we define the pair of structures as an equivalent structure. In other words, all pair of equivalent structures between the structural group A and A0 having a different composition a and b, respectively, will satisfy a relation Ea; A ¼ f ðEb; A0 Þ where f is a correlation function. Note that the equivalent structures need not be the same structure. The binding energy is an example of the equivalent structure. As demonstrated above, the binding energy behavior of metals is universal with the scaled interatomic distance. Although we figured out only several discrete points in Fig. 1(a), the curve is continuous and every point on it can be reached by all the elements. When such a scaled energy curve is available for a class of material systems, we will define the group of material structures which belongs to the same point of the scaled curve as an equivalent structure. Thus, there is an infinite number of equivalent structure sets in a scaled energy curve and the equilibrium structures of metals, as a trivial example, are equivalent to each other. Note that the structures in an equivalent structure set do not have the same lattice parameter and total energy; the equivalent structure imposes a capability to occupy the same point in the scaled energy curve for the structures of different materials. The equivalent structure implies a very remarkable feature. In Fig. 1(b), we denoted eleven equivalent structures of seven fcc metals, which are matched with those of Al. Each of seven points located along the vertical axis are equivalent and able to be overlapped by the scaling of the cohesive energy. Reminding the prescribed scaling steps, it can be understood that the scaling step (ii) and (iii) identify the group elements of the equivalent structure and the step (i) equalize the energy values in the same equivalent structure. From the fact, we give an energy relationship
Fig. 1. (a) The scaled binding energy of fcc metals as a function of the scaled interatomic distance. (b) The energy–energy correlation of the binding energies with Al as a basis. Inset is the scaled binding energy between the equivalent structures. The equivalent structure is defined as a group of structures which occupies the same points in the scaled energy curve.
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M. Jo et al. / Computational Materials Science 92 (2014) 166–171
0 (upper panels) and [1 0 0] direction (lower panels). The Fig. 2. Unrelaxed (left) and relaxed (right) surface energies of fcc metals of various orientations normal to the ½1 1 0 direction is defined by the angle from the (1 1 0) plane and that normal to the [1 0 0] direction by the angle from the (0 0 1) plane. surface orientation normal to the ½1 1
Ea ¼ cEb ;
ð1Þ
where Ea and Eb are the total energies of element a and b in an equivalent structure, respectively, and c is the ratio of the cohesive energies. It is most important to note that once a set of the equivalent structure can be scaled by the relation, other sets of the equivalent structure should be also scalable by the same parameters. In other words, the binding energy curve of a given metal can be transformed into that of other metals with the universal relationship. Therefore, the equivalent structure can be regarded as the structures having the same structure-dependent factor of the total energy so that the energy of an element in the set correlates with those of other elements following the specific relationship which reflects the effect of the element-dependent factor. The idea of the equivalent structure can be applied to other cases. Here, we test it to the surface energy (c). Fig. 2 shows the calculated surface energies of Al, Ni, Cu, Pd, Ag, Pt, Au, and Pb in the fcc structure with and without lattice relaxation. We consid 0 ered two classes of the surface orientation normal to the ½1 1 and [1 0 0] direction. All the surface energy curves show the minimum for the (1 1 1) surface which is the most closed-packed plane of the fcc structure. Pt (Pb) has the highest (lowest) surface energy and other metals are located between them, regardless of the surface orientation and relaxation. This fact can be understood by taking into account the melting temperatures of metals. The melting temperature of Pb is Tm = 600.7, which is significantly lower than those of other metals, and Pt has the highest melting temperature Tm = 2045 K [24].
We would like to emphasis that the surface energy values in Fig. 2 are Pb < Al < Ag < Au < Cu < Pd < Ni < Pt in order, irrespective of the surface orientation. This observation suggests a possibility of the surface energy scaling. It is straightforward to guess that the surface energies of the same orientation specify the equivalent structure and they might be connected by a universal relationship. At first, we try a single parameter scaling of the surface energy. Fig. 3 is the dimensionless surface energies which were scaled with c111 and c100 for the surfaces normal to the ½1 1 0 and [1 0 0] direction, respectively. It is evident that the surface energy scaling to reach a universality cannot be accomplished with a single parameter. Both c/c111 and c/c100 behaviors are highly dependent upon metals and the scaled surface energies do not overlap each other except c111 and c100 points. For example, c110/c111 is about 1.20 and 1.46 for Cu and Al, respectively, for the relaxed surface. Reminding that the equivalent structure was defined as a correlation of the total energy between different materials, we start a scaling procedure by searching for an energy–energy correlation between the same oriented surfaces of different metals. Fig. 4 shows the correlation of the surface energies for the pairs of fcc metals in a cyclic order: Al ? Ni ? Cu ? Pd ? Ag ? Pt ? Au ? Pb ? Al and Al ? Ag. Red and blue points represent unrelaxed and relaxed surface energies, respectively. We allowed both inplain and out-of-plane surface relaxation. Obviously, the surface energies of all the pairs of considering fcc metals show a strong linear correlation, which implies that the surfaces of the same orientation are in an equivalent structure. This fact can be also checked with previous surface energy data [9–13], though it was
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0 (upper panels) and Fig. 3. Scaled surface energies c/c111 and c/c100 for unrelaxed (left) and relaxed (right) surfaces of fcc metals of various orientations normal to the ½1 1 [1 0 0] direction (lower panels).
Fig. 4. The energy–energy correlation of the surface energies between the equivalent surfaces of fcc metals. Red and blue dots represent unrelaxed and relaxed surface energy, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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not recognized due to the limited available data. Relatively large deviations are observed in the relaxed surface energies of Ag–Pt, Pt–Au, and Pb–Al pairs, which come from the rearrangement of surface atoms. That is, the surface relaxation can lead to a mismatching of the equivalent structure. However, the correlation between the surface energies largely retains its linearity when the surface relaxation is not severe. In a similar way to Eq. (1), we write down a relationship between the surface energies of two metals as:
ca ¼ acb þ b;
ð2Þ
where ca and cb are the surface energies of element a and b, respectively, with the same orientation (equivalent surface) and a and b are correlation parameters which are independent upon the surface orientation. By choosing the surface energy of Al (cAl) as a reference, we determined the correlation parameters for other metals by
fitting the surface energy data. The numerical values of the parameters and the cohesive energy are listed in Table 2. For the considering metals, we have 3d (Ni, Cu), 4d (Pd, Ag), and 5d (Pt, Au) elements. The cohesive energy increases from the left and right side within a given period of the periodic table. We found that a and b follow the same trend as that of the cohesive energy except b of Au in the relaxed case. Therefore, Eq. (2) has a similar electronic origin to the universality of the binding energy curve. With the obtained parameters, a and b, the surface energies are scaled and plotted in Fig. 5. It is manifest that the variation of the surface energy on orientation can be described in a single curve, especially for the unrelaxed surface. As mentioned above, the surface relaxation gives rise to a slight modification of the surface energy correlation. The two correlation parameters, a and b, are evidently element-dependent factors from the definition of the equivalent structure. Since the parameters are measurable quantities, they might be able to be used for a characterization of fcc metals.
Table 2 The scaling parameters of the surface energy, a and b in Eq. (2). a is dimensionless and b has the same dimension as the surface energy. The parameters were determined by choosing the surface energy of Al as a reference. Al
Ni
Cu
Pd
Ag
Pt
Au
Pb
Unrelaxed
a b
1 0
1.447 0.721
0.850 0.672
1.089 0.788
0.531 0.528
2.018 0.475
0.813 0.435
0.202 0.257
Relaxed
a b
1 0
1.485 0.702
0.823 0.683
1.046 0.813
0.485 0.554
2.102 0.311
0.569 0.566
0.164 0.271
3.36
4.45
3.54
3.91
2.85
5.77
3.93
2.04
Cohesive energies (eV/atom)
0 (upper panels) and [1 0 0] direction (lower Fig. 5. The scaled surface energy of fcc metals for unrelaxed (left panels) and relaxed (right panels) surfaces normal to the ½1 1 panels).
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The number of required scaling parameters is one for the binding energy and two for the surface energy. While the equivalent structure of the binding energy can be assigned by a single parameter, i.e., the cohesive energy, that of the surface energy needs two parameters, i.e., a and b. In this sense, the element–energy correlation of the binding energy is 1D and that of the surface energy is 2D. While the surfaces of the same orientation were intuitively guessed to be equivalent, it was not straightforward how to describe the equivalent structure. Defining the equivalent structure indeed corresponds to the identification of the structuredependent parameters. In case of the binding energy, the lattice constant and bulk modulus are the structure-dependent parameters. Although the grouping scheme is not yet clear for other structures, the generalization of method would be a key to reveal the full relation among element, structure, and energy. Recently, a single-parameter scalability of the grain boundary energy was also proposed [25]. Material structures in different dimensions might require a different number of element–energy correlation parameters. Therefore, the dimensionality of the element–energy correlation can be regarded as a feature of the material structure. 4. Conclusion We have investigated the universal behavior of the surface energy in fcc metals. Considering the structure–element–energy correlation, it was shown that the factors of structure and element can be separately treated for the surface energy. Two parameters were introduced to scale the surface energy. While the scaling parameters for the binding energy are related to basic measurable quantities, the physical meaning of the correlation parameters is not yet fully identified for the surface energy. Further investigations are left to clarify this point which will extend our view of the element–energy correlation. Since it is not realistic to measure and/or calculate all necessary energies of structures and elements, revealing the underlying physics of the structure–element–energy
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correlation will be helpful for materials development and application. Acknowledgements This research was supported by the Steel Innovation Program of POSCO, the Technology Innovation Program funded by the Ministry of Knowledge Economy (10041187), and the World Class University program through the National Research Foundation of Korea (R32-1014). References [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]
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