Calculation of the surface energy of fcc-metals with the empirical electron surface model

Applied Surface Science 256 (2010) 6899–6907

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Calculation of the surface energy of fcc-metals with the empirical electron surface model Baoqin Fu a , Wei Liu b , Zhilin Li a,∗ a b

College of Materials Science and Engineering, Beijing University of Chemical Technology, Beijing 100029, PR China College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, PR China

a r t i c l e

i n f o

Article history: Received 1 January 2010 Received in revised form 29 April 2010 Accepted 29 April 2010 Available online 6 May 2010 Keywords: Surface energy fcc-metals Empirical electron theory Valence electron structure Dangling bond

a b s t r a c t The empirical electron surface model (EESM) based on the empirical electron theory and the dangling bond analysis method has been used to establish a database of surface energy for low-index surfaces of fcc-metals such as Al, Mn, Co, Ni, Cu, Pd, Ag, Pt, Au, and Pb. A brief introduction of EESM will be presented in this paper. The calculated results are in agreement with experimental and other theoretical values. Comparison of the experimental results and calculation values shows that the average relative error is less than 10% and these values show a strong anisotropy. As we predicted, the surface energy of the close-packed plane (1 1 1) is the lowest one of all index surfaces. For low-index planes, the order of the surface energies is  (1 1 1) <  (1 0 0) <  (1 1 0) <  (2 1 0) . It is also found that the dangling bond electron density and the spatial distribution of covalent bonds have a great influence on surface energy of various index surfaces. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The anisotropic properties of fcc-metal film texture are frequently observed in some experiments. For instance, the elastic modulus of Cu in the [1 1 1] direction is 2.9 times higher than that in the [1 0 0] direction [1]. Reaction rates for oxide [2] and silicide [3] formation on Cu are faster along the (1 0 0) surface than the (1 1 1) surface. Surface energy, especially surface energy anisotropy, determines the stability of surfaces, and the equilibrium shape of fcc-metals. So a detailed knowledge of the structure and property of crystal surfaces is important for the understanding of many surface phenomena such as adsorption, oxidation, corrosion, catalysis, and crystal growth [4,5]. However, the surface energies are difficult to determine experimentally and just few data exist [6–9]. Some of these experiments are performed at high temperature [6,7]. Some of the experimental surface energy data [8,9] stem from surface tension measurements in the liquid phase, which are extrapolated to zero temperature [10]. But these data include uncertainties of unknown magnitude. Especially, they did not yield information, such as the surface energy of one particular surface. Although there were classic measurements on Pb and In [11,12], there are no direct experimental determinations of the anisotropy in the surface energy of other crystals. Recently, Bonzel and Edmundts [13]

∗ Corresponding author. Tel.: +86 10 64421306; fax: +86 10 64437587. E-mail addresses: [email protected] (B. Fu), [email protected], [email protected] (Z. Li). 0169-4332/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2010.04.108

have shown that analyzing the equilibrium shape of crystallites at various temperatures by scanning tunneling microscopy can yield absolute values of the surface energies versus temperature, but this technique has not yet been widely applied. Therefore, to determine the surface energy theoretically is very important. Previous theoretical calculations were based on perturbation theory [14] or non-perturbative variational method [15]. Recently, there have been several methods to calculate the surface energy of metals, including the first-principles calculations [10,16–35] and semi-empirical methods [36–53]. However, the first-principle method is a tough job, especially for a large-scale structure, due to not only the computer time required but also the reliability of the calculated results, which depends strongly on the selection of basis-set, exchange-correlation functional and cutoff energy. And most semi-empirical methods are based on some existing experimental results, do much more approaches, and usually use fitted parameters, focusing on a few systems which have some needed parameters and functions. In the present paper, we will estimate the surface energies of fcc-metals with the empirical electron surface model (EESM), which has been found remarkably successful in predicting the surface energies of bcc-metals [54], hcp-metals [55] and diamond cubic crystals. The starting point of EESM is the valence electron structure (VES) of these fcc-metals calculated by the empirical electron theory (EET) in solid and molecule established by Prof. Yu [56–58]. The reliability of EET has been extensively examined in the fields of metals, alloys, metallic compounds and ceramics [54,55,59–68].

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Table 1 The lattice constants of the fcc-metals. fcc-metals

a = b = c (nm)

˛ = ˇ =  (◦ )

fcc-metals

a = b = c (nm)

˛ = ˇ =  (◦ )

Al Mn Co Ni Cu

0.40495 0.3529 0.3544 0.3524 0.36147

90 90 90 90 90

Pd Ag Pt Au Pb

0.38907 0.40857 0.39239 0.40783 0.49502

90 90 90 90 90

2. Summary of EESM

neglected can be given as the following: DA = 2

Surface energy is usually defined as the additional value of the free energy per unit area of increase on a particular crystal surface, therefore, the surface energy of crystal surface is given by the below expression: E = S

(1)

where E represents the additional value of the free energy with the forming of new surface; S is the added area of the crystal surface. In EET, valence shell electrons are divided into four types based on the spatial distribution and the action characteristics of these electrons in the formation of solids or molecules, such as dumb pair electrons, magnetic electrons, covalent bond electrons (bonding electrons) and lattice electrons. The dumb pair electrons represent either a bonding and an anti-bonding electron with their resultant bonding power mutually cancelled by each other and their spins opposite to each other, or a pair of non-valence electrons of opposite spins deeply sinking down to the atomic orbit. Hybridization states of these fcc-metals, the number and the distribution of these electrons, are listed in Appendix A. The contribution of these four types of electrons to the crystal cohesive energy and the calculation formula can be found in Refs. [58,59]. In EET, the crystal cohesive energy is mainly dependent on the number and the spatial distribution of the four types of electrons in the crystal structure. For fcc-metals, the energy of the covalent bond takes up major constituent of the crystal cohesive energy. Therefore, when all covalent bonds between two nearest crystal planes were broken, the two nearest crystal planes form two new crystal surfaces. Therefore the total energy of all bonds between two nearest crystal planes is just the additional free energy (E): E =



Z˛ E˛

DB = a, DC = (0.5a) + (0.5a) + a2 . In EET, the equivalent bond numbers (I˛ ) of the bonds in the structure unit can be calculated with the below formula [57,58]: I˛ = iM iS iK

(4)

where the bond name ˛ is A, B or C, which represent nearest neighbor, 2nd nearest neighbor and 3rd nearest neighbor interactions, respectively, iM represents the reference atom number in the structure unit, iS represents the equivalent bond number for a reference atom to form ˛ bond, iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are of different kinds. Therefore, the equivalent bond numbers of ˛ bond are: IA = 4 × 12 × 1 = 48, IB = 4 × 6 × 1 = 24, IC = 4 × 24 × 1 = 96. According to the Pauling bond length formula [57,58,69]: D˛ = 2R(1) − ˇlog na

(5)

where D˛ represents the bond length of ˛ bond; R(1) represents the single bond radius of the atom which forms the bond; n˛ represents the number of covalent electron pairs on the bond; and ˇ is a parameter with units of length. Here a modification for Pauling’s original theory is made as the follows: ˇ=

 0.0710 (nm)

0.0600 (nm) 0.0710 − 0.22ε (nm)

M when nM ˛ < 0.25 or n˛ > 0.75 when 0.300 ≤ nM ˛ ≤ 0.700 M when nM ˛ = 0.250 + ε or n˛ = 0.750 − ε

r˛ =

n˛ = 10(DA −D˛ )/ˇ nA

(2)

(3)

3. Calculation of the VES and the bond energies of the fcc-metals These fcc-metals, the crystal cell is shown in Fig. 1 and the lattice constants are shown in Table 1, belong to A1 type crystal structure and the space group of Oh5 − fm3m, and the atom coordinates are listed as follows: (0, 0, 0), (0.5, 0.5, 0), (0.5, 0, 0.5), and (0, 0.5, 0.5). So the experimental bond lengths which cannot be

(6)

where 0 ≤ ε < 0.050, nM ˛ represent the largest n˛ in the structure. Therefore, the following r˛ (˛ = B and C) equation can be obtained:

where ˛ represents bond name; E˛ represents the bond energy of ˛ bond; and Z˛ , obtained by DBAM, is the equivalent dangling bond number of ˛ bond on one particular crystal plane. The analysis of the equivalent dangling bond number is presented in Section 4 in detail. In EET, the bond energy (E˛ ) can be obtained from the value of the bonding capability (f) of covalent electron, the screen factor (b) upon the core electron, the bond length (D) and the number of covalent electron pairs (n): n D

(0.5a)2 + (0.5a)2 ,

2

˛

E = bf



Fig. 1. Unit cell of fcc-metals.

(7)

B. Fu et al. / Applied Surface Science 256 (2010) 6899–6907

By substituting the experimental bond lengths DA , DB and DC into formula (7), the ratio r˛ can be calculated out. A structure unit should be electron neutral, so the covalent electron of all atoms in the structure unit (there are four atoms in the unit cell of fcc-metals) should be distributed on all ˛ covalent bonds in it. In other words, the total covalent electron number of all atoms (nc ) should equal the sum of the electron number on all the covalent bonds (I˛ n˛ ) in the structure unit, i.e.:



nc =



 I˛ n˛ = nA

IA +

 nc 

IA +

I˛ r˛

(8)

˛

˛

nA =





I˛ r˛

=

4nc IA + IB rB + IC rC

(9)

˛

For certain hybrid level  of an atom, the covalent electron number and single bond radius can be found in the hybridization table [57,58]. By selecting a suitable ˇ value and substituting the number of covalent electrons and the single bond radii of the atoms of the corresponding hybrid levels, the number of covalent electron pair nA on the strongest covalent bond can be calculated out. By substituting the obtained nA into the r˛ equations, the numbers of covalent electron pairs on all the other covalent bonds n˛ (˛ = B and C) can be calculated out. The hybrid states of the atoms and the corresponding distributions which are in accordance with the reality have to be determined. In EET, the bond lengths calculated with the calculated n˛ are called theoretical bond lengths. Substituting the obtained n˛ into the bond length formula (5) yields the theoretical bond lengths of all the bonds: D˛ = Ru (1) + Rv (1) − ˇlog n˛ = 2R (1) − ˇlog n˛

(10)

Bond length difference (BLD) D˛ , is the absolute value of the difference between the experimental bond length and the theoretical one, i.e.:





D˛ = D˛ − D˛ 

(11)

In EET, D˛ < 0.005 nm is the criterion to determine whether the given atom state and the corresponding valence electron distribution accord with reality. If D˛ < 0.005 nm, the assumed hybrid levels of the atom are believed being in accordance with the actual state, therefore, the obtained corresponding parameters are the possible VES of the calculated crystal. Through programming the calculation according to the above equations and substituting the parameters of all hybrid levels of the fcc-metal atoms, all the possible existing hybrid levels of the atoms can be obtained. So the VES of these fcc-metals can be calculated, and the calculation results with the smallest bond length difference D˛ are shown in Table 2. Based on the formula (3), the calculation formula for bond energy (E˛ ) of the bond in which the two atoms are the same is: E˛ = bf

n˛ D˛

(12)

where f represents the bonding capability of covalent electron [54,55,58,59]: f =

  √ ˛ + 3ˇ + g 5 

(13)

where the factor g represents the contribution of spin–orbit coupling of d-electrons to the bonding capability. For 4, 5 and 6 periods, g is equal to 1, 1.35 and 1.70, respectively. ˛ , ˇ and   are contents

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of s, p and d electrons in covalent bonds, respectively [54,55,58]: lCh + l   Ct nT mCh + m Ct ˇ = nT nCh + n Ct   = nT

˛ =

(14)

where l, m, n and l , m , n represent the sum of the numbers of s, p, d covalent electron and lattice electron of the h (head) and t (tail) states of  hybrid level, respectively. And the h and t states are two original states of hybridization, one of which is the ground or near excited state, both of which correspond to two stationary states. The terms  and   are parameters for the h and t states, respectively, and equal to 1 when the s electron is covalent electron or 0 when the s electron is lattice electron. The values of these parameters can be taken from Refs. [57,58], and some values of fcc-metal elements are given in Appendix A. nT , Ch␴ , and Ct represent the total numbers of covalent electron and lattice electron, the relative compositions for the h state and for the t state of  hybrid level, respectively. In the formula (12), b is screen factor upon the core electron [54,55,58,59]: b=

31.395 n − 0.36ı

(15)

The factor n and ı in the denominator reflects the total effect of the screen, coulomb, exchange and interrelated interaction of the inner electrons in solids. The values of n, ı and b of some elements can be taken from Refs. [57,58], and the values of fcc-metals are tabulated in detail in Appendix B. So the calculation results are tabulated in detail in the last column of Table 2. 4. Analysis of dangling bond of the related crystal plane in fcc-metals From DBAM, the number and the type of the dangling bonds on one particular crystal surface can be analyzed. Based on the calculated bond energy (see Table 2), the surface energy of one particular crystal plane can be calculated. The formula for the equivalent dangling bond number (Z˛ ) on one particular crystal plane can be deduced out [54,55]: Z˛ = iP iK



iD NP

(16)

NP

where iP represents the reference atom number on particular plane with the area S; and iK is a parameter, which equals 1 when the two atoms that form the bond are of the same kind or 2 when the two atoms are different kinds; iD represents the equivalent dangling bond number for a reference atom to form one type bond with the atoms of the same neighbor crystal plane; NP (NP = 1, 2, . . .) represents the crystal plane number passed through by the bond. As is shown in Fig. 2, there are two atoms on the (1 0 0) plane with the area of a2 nm2 . Every atom with the atoms of the nearest neighbor crystal plane forms four A bonds and eight C bonds, and with the atoms of the second neighbor crystal plane forms one B bond and four C bonds. And based on the formula (16), the equivalent dangling bond numbers on the (1 0 0) plane are ZA = 2 × 1 × 4 × 1 = 8, ZB = 2 × 1 × 1 × 2 = 4, and ZC = 2 × 1 × (8 × 1 + 4 × 2) = 32. Therefore, according to the analysis of Figs. 3–5, the equivalent dangling bond number of the other orientation can be obtained. So the equivalent dangling bond numbers on the (1 1 0) plane are ZA = 2 × 1 × (4 × 1 + 1 × 2) = 12, ZB = 2 × 1 × 2 × 2 = 8, and ZC = 2 × 1 × (4 × 1 + 2 × 2 + 4 × 3) = 40. The equivalent dangling bond number on the (1 1 1) plane can be obtained. They are ZA = 2 × 1 × 3 × 1 = 6, ZB = 2 × 1 × 3 × 1 = 6, and ZC = 2 × 1 × (6 × 1 + 3 × 2) = 24. And the

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Table 2 The VES and bond energies of the fcc-metals. fcc-metal (hybrid type/)

Bond name (˛)

D˛ (nm)

D˛ (nm)

n˛

Dn˛ (nm)

E˛ (kJ mol−1 )

Al (A/4)

A B C

0.28634 0.40495 0.49596

0.28640 0.40501 0.49602

0.20811 0.00444 0.00023

0.00006 0.00006 0.00006

25.50269 0.38510 0.01643

Mn (A1/8)

A B C

0.24954 0.35290 0.43221

0.24946 0.35282 0.43213

0.39162 0.00742 0.00035

0.00008 0.00008 0.00008

51.63322 0.69131 0.02690

Co (D/12)

A B C

0.25060 0.35440 0.43405

0.25138 0.35518 0.43483

0.46447 0.00865 0.00041

0.00078 0.00078 0.00078

52.43866 0.69105 0.02655

Ni (A/13)

A B C

0.24918 0.35240 0.43160

0.24795 0.35117 0.43037

0.52507 0.01000 0.00048

0.00123 0.00123 0.00123

48.16400 0.64762 0.02529

Cu (B/8)

A B C

0.25560 0.36147 0.44271

0.25598 0.36185 0.44309

0.43607 0.00750 0.00033

0.00038 0.00038 0.00038

37.39334 0.45492 0.01644

Pd (A/15)

A B C

0.27511 0.38907 0.47651

0.27474 0.38869 0.47613

0.57425 0.00724 0.00025

0.00038 0.00038 0.00038

49.24018 0.43890 0.01250

Ag (A/8)

A B C

0.28890 0.40857 0.50039

0.28909 0.40875 0.50058

0.36465 0.00369 0.00011

0.00018 0.00018 0.00018

25.88074 0.18539 0.00446

Pt (A/12)

A B C

0.27746 0.39239 0.48058

0.27748 0.39241 0.48060

0.43687 0.00531 0.00018

0.00002 0.00002 0.00002

40.31798 0.34635 0.00959

Au (A/9)

A B C

0.28838 0.40783 0.49949

0.28834 0.40779 0.49944

0.37704 0.00385 0.00011

0.00004 0.00004 0.00004

31.34491 0.22635 0.00548

Pb (A/1)

A B C

0.35003 0.49502 0.60627

0.34140 0.48639 0.59765

0.16583 0.00151 0.00004

0.00863 0.00863 0.00863

9.33852 0.05949 0.00131

equivalent dangling bond numbers on the (2 1 0) plane of are ZA = 1 × 1 × (3 × 1 + 2 × 2 + 1 × 3) = 10, ZB = 1 × 1 × (1 × 2 + 1 × 4) = 6, and ZC = 1 × 1 × (2 × 1 + 4 × 3 + 2 × 4 + 2 × 5) = 32. 5. Calculation of surface energy of the corresponding surface Using the above analysis the surface energy can be calculated for a clean, ideal surface from the following equation:



=

Z˛ E˛

˛

2S

(17)

where ˛ (˛ = A, B and C) is the bond name, and E˛ is the the bond energy of ˛ bond (see Table 2), and S is the area of the related crystal planes in the unit cell. The physical meaning of the number 2 in this formula is that when these bonds between two nearest crystal planes were broken, there will be two crystal surfaces in the region. For these fcc-metals, the area of the (1 0 0) crystal plane in one unit cell is S(1 0 0) = a2 , and that of the (1 1 0) crystal plane, S(1 1 0) = √ √ 2a2 , and that of the (1 1 1) crystal plane, S(1 1 0) = 3a2 /2, and that √ of the (2 1 0) crystal plane, S(1 1 0) = 5a2 /2. Through programming the calculation according to these parameters and the surface energy formula (17), we can calculate the surface energy of the corresponding clean and ideal surface

Fig. 2. The spatial distribution of dangling bond on the fcc-metals (1 0 0) crystal plane.

B. Fu et al. / Applied Surface Science 256 (2010) 6899–6907

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Table 3 The surface energy of fcc-metals. fcc-metal

Plane

Z˛ E˛ (eV)

S (nm2 )

 (eV nm−2 )

10.985b , 15.163c , 15.192e , 9.861f , 9.8j , 10.922k 12.358b , 14.8c , 14.873e , 10.798f , 12.339k , 10.174b , 12.569c , 12.701e , 9.050f , 8.3j , 10.111k 13.045b , 15.535e , 18.787i

15.291l 14.855f 15.763g , 15.938h 15.541i

0.13066 0.18478 0.11316 0.14608

11.95761 12.70479 10.41810 13.38049

8.676b , 13.519c , 10.305e , 7.989f , 7.7j , 11.247m , 5.555n 9.487b , 13.962c , 10.242e , 8.738f , 5.992n 8.051b , 12.183c , 8.794e , 7.302f , 6.4j , 5.243n , 12.108p , 8.0q 9.861b , 10.692e , 13.544i

11.047o 11.172g , 11.391h 12.732i

4.10509 6.16569 3.09246 5.13487

0.15138 0.21408 0.13109 0.16924

13.55928 14.40062 11.79473 15.17012

14.518c , 10.367e , 8.8j , 11.609r , 14.355s , 13.294t 13.887c , 10.429e , 12.296r , 15.604s 11.984c , 8.626e , 7.2j , 10.236q 10.972e

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

2.15506 3.23607 1.62206 2.69538

0.16693 0.23607 0.14457 0.18663

6.45501 6.85393 5.61014 7.22108

7.49c , 7.958e , 5.4j , 7.552r , 8.114s , 7.927u , 8.051v 7.727c , 7.652e , 7.864r , 8.738s , 8.364v 7.315c , 6.816e , 4.4j , 7.552r , 7.240v 7.920e

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

3.36050 5.04713 2.53114 4.20342

0.15397 0.21775 0.13334 0.17214

10.91285 11.58948 9.49118 12.20905

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

2.61016 3.91948 1.96466 3.26459

0.16633 0.23522 0.14404 0.18596

7.84655 8.33153 6.81975 8.77780

10.155c , 6.76e , 5.5j 10.61c , 6.953e 8.008c , 5.53e , 4.3j , 5.0q , 6.491x 7.377e

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

0.77720 1.16693 0.58475 0.97201

0.24504 0.34655 0.21222 0.27397

1.58584 1.68366 1.37773 1.77395

2.356c , 2.646e , 1.9j 2.781c , 2.69e , 3.7y 2.006c , 2.284e , 1.5j , 3.1y , 2.746z 2.815e

4.31874 6.49022 3.26055 5.40336

0.12454 0.17612 0.10785 0.13924

17.33900 18.42514 15.11567 19.40332

Co

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

4.38540 6.59022 3.31054 5.48671

0.12560 0.17762 0.10877 0.14042

17.45789 18.55102 15.21775 19.53618

Ni

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

4.02875 6.05445 3.04170 5.04055

0.12419 0.17563 0.10755 0.13884

Cu

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

3.12478 4.69523 2.35773 3.90932

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

Au

Pb

a b c d e f g h i j k l m n o p q r s t u v w x y z

Experiment (eV nm−2 ) e

16.22066 17.23685 14.14112 18.15186

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

Pt

d

9.644g ,10h

6.51271 6.92205 5.68233 7.28860

Ag

c

19.375c

0.16398 0.23191 0.14201 0.18334

Pd

b

7.115f 7.144g , 7.25h 7.178i

2.13597 3.21057 1.61395 2.67259

Mn

a

6.756 , 5.742 , 8.419 , 5.368 , 5.617 6.813a , 6.366b , 7.944c , 6.678d , 6.067e 5.869a , 5.555b , 7.494c , 4.743d , 3.870e 6.616b , 6.666e

(0 0 1) (1 1 0) (1 1 1) (2 1 0)

Al

Other theoretical results (eV nm−2 )

17.064c , 13.531e , 9.8j 17.595c , 13.307e 14.349c , 10.336e , 7.8j , 8.4q , 12.901w 14.162e

The first-principles calculation [22]. Embedded atom method (EAM) [70]. The full charge density (FCD) method in the generalized gradient approximation (GGA) [10]. Density function theory (DFT) [34]. Modified embedded atom method (MEAM) [50]. The corrected effective-medium method (CEM) [47]. Experiment [8]. Experiment [9] CEM [48]. Johnson EAM (J-EAM) [45,46]. The first-principles calculation [32]. EAM [49]. The modified augmented-plane-wave (MAPW) method [71]. Full potential linear muffin-tin orbitals (LMTO) method [31]. Molecular dynamics simulation [72]. Full potential LMTO method [24]. DFT [30]. Full potential LMTO method [16]. Full potential-linearised augmented-planewave (FP-LAPW) [26]. The first-principles calculation [25]. FP-LAPW [27]. The first-principles calculation [33]. The first-principles calculation [28]. The first-principles calculation [29]. The first-principles calculation [23]. Three-dimensional (3D) equilibrium crystal shapes (ECS) [44].

12.502g , 12.795h

7.777g , 7.802h

15.535g , 15.448h

9.4g , 9.362h

3.706g , 3.75h

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of the fcc-metals. The calculation results are tabulated in detail in Table 3. 6. Results and discussion

Fig. 3. The spatial distribution of dangling bond on the fcc-metals (1 1 0) crystal plane.

Fig. 4. The spatial distribution of dangling bond on the fcc-metals (1 1 1) crystal plane.

The surface energies for 10 fcc-metals Al, Mn, Co, Ni, Cu, Pd, Ag, Pt, Au and Pb, presented in Table 3 in eV nm−2 have been calculated. For comparison, we also present some theoretical results and experimental derived values. These theoretical methods include the first-principles calculation in Refs. [22,23,25,28,29,32,33], the embedded atom method (EAM) in Refs. [45,46,49,50,70], the full charge density (FCD) method in the generalized gradient approximation (GGA) in Ref. [10], the density function theory (DFT) in Refs. [30,34], the corrected effective-medium method (CEM) in Refs. [47,48], the modified augmented-plane-wave (MAPW) method in Ref. [71], the full potential linear muffin-tin orbitals (LMTO) method in Refs. [16,24,31], molecular dynamics simulation in Ref. [72], the full potential-linearised augmented-planewave (FP-LAPW) in Refs. [26,27], and the three-dimensional (3D) equilibrium crystal shapes (ECS) in Ref. [44], and so on. The experimental values in Refs. [8,9] are determined from measurements of the surface tension of liquid metals extrapolating through the liquid–solid phase transition. It must be pointed out that experimental measurements of the surface energy are more commonly found for polycrystalline materials. But for fcc-Co, there is a lack of surface energy data in the literature. In the present calculation, the relaxation and reconstruction of the atomic positions are neglected, which may lead to errors of up to a few percent. The research including the relaxation will be the next step. From Table 3, the surface energy values calculated in this paper are close to the values from the other references under the first-order approximation, and the average relative errors are less than 10% (relative to the experimentally derived values). From Table 3, it is clearly seen that, the calculated surface energy shows a strong anisotropy, the surface energy values of the (0 0 1), (1 1 0), (1 1 1), and (2 1 0) surface are different, and the order is  (1 1 1) <  (1 0 0) <  (1 1 0) <  (2 1 0) . The close-packed (1 1 1) surface energy is the lowest of all these index surfaces as predicted. And according to the analysis of dangling bond, the spatial distribution of broken bond of the (0 0 1), (1 1 0), (1 1 1), and (2 1 0) crystal surface is also different. So the dangling bond electron density and the spatial distribution of covalent bonds have a great influence on surface energy of various index surfaces. The meaning of this paper is not only to extend the EESM to fcc-metals and thereby fill out that gap in the literature, and to calculate surface energy of various index surfaces of fcc-metals, but also to provide comprehensive surface energy results for fcc-metals, bcc-metals [54], hcp-metals [55] and diamond cubic crystals. Such extensive results obtained with the same theoretical model, are useful to both theorists and experimentalists. The EESM can be used to calculate surface energy of the more metals, alloys, ceramics, compounds, etc. 7. Conclusions

Fig. 5. The spatial distribution of dangling bond on the fcc-metals (2 1 0) crystal plane.

With the empirical electron surface model (EESM), we have built a database of the (1 0 0), (1 1 0), (1 1 1), and (2 1 0) surface energy for the fcc-metals Al, Mn, Co, Ni, Cu, Pd, Ag, Pt, Au and Pb, and the spatial distribution of dangling bonds of various index surfaces has been analyzed with dangling bond analysis method (DBAM). The dangling bond electron density and the spatial distribution of covalent bonds have a great influence on surface energy of various index surfaces. Under the first-order approximation, the results are close to the other theoretical values and the experimental values. These surface energy values may be used as a consistent starting point for study of surface science phenomena. The calculated surface energy shows a strong anisotropy. And the close-packed (1 1 1)

B. Fu et al. / Applied Surface Science 256 (2010) 6899–6907

surface energy is the lowest of these index surfaces as predicted. For the low-index planes, the order of the surface energy result is  (1 1 1) <  (1 0 0) <  (1 1 0) <  (2 1 0) . This research model can be further extended to the surface energy estimation of more crystals, which will be the next research objects.

A.5. A type and A1 type of Co

Acknowledgements This work was supported by the Beijing Natural Science Foundation (Grant No. 2072014) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200800100006).

A.6. B type of Co (-Co)

Appendix A. Hybridization states in EET of the elements In EET, dumb pair electrons, magnetic electron, covalent electron, and lattice electron are denoted by ||, ↑, 䊉, and , respectively. When the covalent electron and lattice electron are equivalent electrons, they are denoted by , or s , p , respectively. The hybridization states of the elements are as the follows.

A.7. C type and C1 type of Co

A.1. A type of B and Al

A.8. D type of Co A.2. A type of Mn, Tc, Re and A1 type of Mn

A.9. E type and E1 type of Co

A.3. B type and B1 type of Mn

A.10. A type and A1 type of Ni

A.4. C type of Mn (-Mn)

6905

6906

A.11. A type of Pd, Pt

A.12. B type of Ni, Pd, and Pt

B. Fu et al. / Applied Surface Science 256 (2010) 6899–6907 Table A.1 The values of the factors (n and ı) and the screen factors (b) of fcc-metals. fcc-metal

n

ı

b (kJ nm mol−1 )

fcc-metal

n

ı

b (kJ nm mol−1 )

Al Mn Co Ni Cu

2 3 3 4 4

1 1 0 0 1

19.14329268 11.89204545 10.465 7.84875 8.625

Pd Ag Pt Au Pb

5 5 5 5 2

1 1 1 1 0

6.766163793 6.766163793 6.766163793 6.766163793 15.6975

Appendix B. The values of the factors (n and ı) and the screen factors (b) of fcc-metals The calculation formulae for bond energy (E˛ ) of the bond in which the two atoms are the same have been given in the preceding part of this paper. The needed values of the factors (n and ı) and the screen factors (b) of these metals, which are total effects of shielding, coulombs and exchange-correlation interactions of these atoms, are tabulated in detail in Table A.1. References

A.13. C type of Pd, Pt

A.14. A type of Cu, Ag, and Au

A.15. B type of Cu

A.16. A type of C, Si, Ge, Sn, and Pb

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