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Applied Surface Science 265 (2013) 375–378

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Applied Surface

Science

j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /a p s u s

The calculation of surface free energy based on embedded atom method for solid nickel

Wenhua Luo a ,?,Wangyu Hu b ,Kalin Su a ,Fusheng Liu c

College of Physics and Electronics,Hunan Institute of Science and Technology,Yueyang 414000,PR China b

Department of Applied Physics,Hunan University,Changsha 410082,PR China c

College of Metallurgical Engineering,Hunan University Technology,Zhuzhou 412007,PR China

a r t i c l e

i n f o

Article history:

Received 20September 2012Accepted 5November 2012

Available online 12 November 2012

Keywords:

Surfaces free energy Nanoparticles

Thermal properties

Embedded atom method

a b s t r a c t

Accurate prediction of surface free energy of crystalline metals is a challenging task.The theory calcu-lations based on embedded atom method potentials often underestimate surface free energy of metals.With an analytical charge density correction to the argument of the embedding energy of embedded atom method,an approach to improve the prediction for surface free energy is presented.This approach is applied to calculate the temperature dependent anisotropic surface energy of bulk nickel and sur-face energies of nickel nanoparticles,and the obtained results are in good agreement with available experimental data.

1.Introduction

The knowledge of surface free energies of crystalline metals is of utmost importance for the understanding of a large number of basic and applied phenomena,such as equilibrium surface morphologies,surface reconstructions,faceting and nucleation [1,2].Experimen-tally these quantities are notoriously dif?cult to measure due to such problems as surface defects and/or impurities [3–5],and results from different researchers often being in very poor agree-ment [5,6].Furthermore experimental determination of surface free energy anisotropy is well known to be almost impossible [7],and most resulting measurements of surface free energies repre-sent averages over many stable crystal face orientations.Finally,very little is known about the temperature dependence of the sur-face free energies of solids since most experimental data were obtained at temperature near the melting point [5].

Computational techniques such as molecular dynamics (MD)or Monte Carlo methods can complement experiment with the abil-ity to simulate different surface morphologies of various systems obtaining important structural,dynamics,and thermodynamic properties.Unfortunately for some systems direct comparison with experiment was often hampered by the overly simplistic repre-sentation of bonding inherent in using empirical pair potentials [8].Developments such as the embedded atom method (EAM)type potentials [9,10]provided a more realistic representation of

?Corresponding author.

E-mail address:wenhualuo@https://www.wendangku.net/doc/d57090216.html, (W.Luo).

metallic bonding.However,most EAM potential sets have been developed using only experimental data from bulk materials or,if experimental data for surface were considered,they were not highly weighted in ?tting.This is one reason why most EAM parametrizations do not perform well when used to quantitatively predict material surface properties where charge densities are sig-ni?cantly different from the central bulk region,and some EAM potentials underestimate surface energies of low index metallic faces by up to 50%[10–12].

The nonuniformity of charge density in lattice defects may be a dominant factor to determine the prediction accuracy of EAM.Webb and Grest [12]modi?ed the electron density with the square of the charge density gradient ( i ).In terms of their approach,the predictions of bulk material properties are almost unchanged since the charge distribution in an isotropic environment is uni-form,and inhomogeneity corrections are negligible,while in the

case of surface,| i |/=

0,the prediction results for liquid/vapor surface tension can be improved by adding this correction to the argument of the embedding energy related to the nonuniformity in the charge density.This approach is valid,however,it is dif?cult to calculate the gradient of charge density,and the determination of the parameters with a trial and error approach is computationally expensive.

In the present paper,by using the available data of surface ener-gies and vacancy formation energy,a new solution for correcting the electron density and an analytical method for determining the related parameters are proposed.This approach is applied to calculate the temperature dependent anisotropic surface free ener-gies of bulk nickel material and surface free energies of nickel

376W.Luo et al./Applied Surface Science 265 (2013) 375–378

nanoparticles at 0K,and the results show that correspondence between theoretical prediction and experimental results is found.

2.Theory and method

The surface free energy per area is calculated as [13,14]:

area

=G slab

?Ng bulk A l

(1)

where G slab is the total Gibbs free energy of a slab,N is the total number of atoms in the slab,g bulk is the Gibbs free energy per bulk atom,and A l is the surface area of the slab.The Gibbs free energy at an arbitrary temperature can be obtained according to the Gibbs-Duhem equation [14–17]:

g (T )=T [g (T 0)

T 0

h ( )

d ](2)

where T 0is a predetermined reference temperature,g (T 0)is the Gibbs free energy at T 0,which is calculated by means of the coupling parameter formalism [14,17],h ( )is the enthalpy which is ?rst computed as a function of temperature T using constant-pressure MD simulations and it is ?tted with a second-order polynomial in T ,which allows an analytic integration in Eq.(2).

The interatomic potential of nickel is modeled by using the mod-i?ed analytic embedded atom method (MAEAM)[18,19]in which the energy of an N -atom system is

E =

N i =1

?F ( i )+

j /=i

(r ij )+M (P i )?

(3)

where F ( i )is the embedding energy to embed atom i in an electron density i [20], (r ij )is the pair potential as a function of the atomic distance r ij [18,19],and M (P i )is the modi?ed term [18,21,22],which describes the energy change caused by the nonspherical distribu-tion of atoms and deviation from the linear superposition of atomic density.The functions of F ( i ), (r ij ),and M (P i )for fcc crystal con-?guration take the following forms:

F ( i )=?F 0

1?n ln (

i e

) ·

(4)

(r ij )=k ?1e (r 1/r ij ?1)+

4 i =0

k i e i (1?r ij /r 1)

(5)

M (P i )=?

1?exp

?10

P e

(6) i =

j /=i

f (r ij )

(7)

P i =

f 2

(r ij )

(8)

f (r ij )=

r 1

r ij

4.7 r ce ?r ij r ce ?r 1

2(9)

where f (r ij )is the electron density distribution function of an atom, i is the electron density induced at site i by all others in the sys-tem,and P i is the second order item of electron density. e and P e correspond to their equilibrium values at T =0K,respectively.The pair potential function is truncated at a speci?c cutoff distance r c =r 3+0.75(r 4?r 3),and f (r )is truncated at r ce =r 4+0.75(r 5?r 4).r i is i th nearest neighbor distance.The model parameters are deter-mined analytically by ?tting the physical properties of nickel [18].

In order to improve the prediction effect of the MAEAM on the surface free energy,according to Eq.(1),the total energy of a slab

should increase by means of new MAEAM.Since surface charge densities are different from the bulk,the embedding energy and modi?cation term which are both related to charge densities is con-sidered to enlarge by adding corrections to the argument of them.Due to the contribution from the modi?cation term being trivial,it is reasonable to neglect its effect.Instead of the square of gradient of charge density [12],( b ? i )2is used to correct the argument of the embedding energy:

F ( i )→F [ i +ˇ( b ? i )2]

(10)

where b is the bulk value of electron density,ˇis an adjustable parameter.The surface free energy can be further improved by tak-ing into account the fact that EAM is invariant under the following transformations:

F c ( i )=F [ i +ˇ( b ? i )2]+c [ i +ˇ( b ? i )2]/ e ,

c (r ij )= (r ij )?2cf (r ij )/ e ,

(11)

where c is an arbitrary constant.

3.Results and discussion

The parameters ˇand c can be adjusted based on the difference between calculated surface energy and the ?rst-principles value.For example,the calculation result for Ni (100)surface energy based on the original MAEAM is 2.00(J/m 2),and the ?rst-principles value for same system is 2.42(J/m 2)[23].The deviation between the calculated and the ?rst-principles value is taken as the objec-tive quantity for the surface energy adjustment,which is used to build the equation to determine parameters ˇand c according to the embedding energy difference between after and before the cor-rection for a surface atom,

F [ i +ˇ( e ? i )2]+c [ i +ˇ( e ? i )2]/ e ?[F ( i )+c i / e ]

≈0.42(J/m 2

(12)

It should be pointed out that as the calculation of parameters ˇand c is carried out at T =0K, b equals e (12.7642).In order to avoid the vacancy formation energy becoming signi?cant due to introduction of parameters ˇand c ,we restrict the unrelaxed

vacancy formation energy E f

1v to be less than or equal 1.84eV (the experimental value is 1.6eV [24])while adjusting parameters ˇand c ,that is

E f

1v (ˇ,c )=E t (ˇ,c )?E t +E coh ≈1.84eV

(13)

where E t

(ˇ,c )is the total energy of the lattice in which there is a vacancy,E t is the total energy of perfect crystal con?gurations and E coh is the cohesive energy of a nickel atom.By solving Eqs.(12)and (13)numerically,the adjustable parameters ˇand c can be obtained.Considering the form of the embedding energy func-tion,positive charge density corrections (ˇ>0)are expected to be preferred to avoid producing the negative argument of the embed-ding energy F ( )in extreme cases.Finally,we obtain c =8eV,and ˇ=0.017,which merely leads to a small relative change of 3.5%in the charge densities of surface site atoms,where the unrelaxed electron density i of the ?rst layer surface site atoms is 8.5486.Fig.1shows the behavior of the embedding energy F c ( i )as a func-tion of relative change densities i / e .Due to large rate of change of embedding energy with relative change densities in the case of c =8eV,slight change in the electron density will cause apparent increase in embedding energy.

Using MD simulation with the MAEAM potentials based on Eq.(11)and the results for ˇand c ,the relaxed surface energy for Ni (100)surface and the vacancy formation energy are calculated to be 2.33(J/m 2)and 1.68eV respectively,which are somewhat smaller than the corresponding objective value of 2.42(J/m 2)and

W.Luo et al./Applied Surface Science 265 (2013) 375–378

377

Fig.1.The embedding energy F c ( i )as a function of relative change densities i / e

and the adjustable parameter c .

1.84eV.The differences are attributed to the effect of lattice relax-ation.

By performing MD simulations and thermodynamic integration,and combining Eqs.(1)and (2),the surface free energy l of a crystal face l at an ?nite temperature can be obtained as [14]

l (T )= l (T 0)

T 0

b 2(T ?T 0)+b 1ln T T 0

b 0

T +b 0

T 0

(14)

where b kl (k =0,1,2)is the coef?cient for the surface free energy

calculation for crystal face l ,and l (T 0)is surface free energy at the reference temperature T 0.Each crystal has its own surface free energy and a crystal can be bounded by an in?nite number of face types.Thus,we only consider three low index surfaces (111),(100)and (110)because of their low surface energy,and the results are shown in Fig.2,and the related coef?cients are listed in Table 1.It is found that the free energies of three surfaces are ordered precisely as expected from packing of the atoms in the layers.The close-packed (111)face has the lowest free energy and loosely packed (110)the largest.Note that the value for Ni in Ref.[23]is doubtful where the surface energy of Ni (110)is smaller than that of

Fig.2.Solid surface free energies vs.temperature for the (111),(100),and (110)faces obtained using thermodynamic integration technique and the average value of surface free energy over the three faces.Also shown are the experimental values [5]and semi-theoretical values [28]at 0K and the melting point.

Table 1

Surface free energy parameters for the (111),(100),and (110)faces at T 0=300K.All entries are in units of J/m 2.

Surface s (T 0)

b 2(×10?7)

b 1(×10?4)

b 0(111) 2.0779 1.7683?1.8839 2.1479(100) 2.3069 4.2600?5.3608 2.4768(110)

2.4452

6.4626

?9.0345

2.7312

(100),which is physically unacceptable.In the temperature range of up to about 1200K,there is less change in free energies of three surfaces,especially in that of Ni (110).This might be the reason why some researchers neglected the temperature dependence of the surface free energy in the case where surface free energy is involved [25,26].However,as the temperature is close to melting one,surface free energies decrease rapidly with temperature.

According to Zhang et al.[27],the average surface free energy of a crystal,weighted by the surface area,is

= n

l =1A l l

l =1A l

=n n l =11

(15)

where n is the number of crystal faces under consideration.The calculated result,experimental value [5]and Tyson and Miller’s semi-theoretical estimates [28]for surface free energy are also shown in Fig.2.It is obvious that the solid surface free energy is a nonlinear function of temperature,and the calculated results at 0K and melting temperature are in good agreement with the semi-theoretical estimates and experimental value,which yields the average gradient of the surface free energy curves of 1.73×10?4(J/m 2-K).This result can be compared to the experimental value of 2.1×10?4(J/m 2-K)[29]for the liquid.

In order to examine our model further,the surface energy of nanoparticles at 0K is calculated according to Eq.(1).To obtain the surface area,we de?ne the radius of the nanoparticles as [30]

R c =R g

+R Ni

(16)

where R Ni is half the atomic distance in bulk system,and the radius of gyration R g is given by

R 2g =

(R i ?R cm )2

(17)

where (R i ?R cm )is the distance of atom from i to the center of mass of nanoparticle.The calculation results for the surface energy of nanoparticles with correction MAEAM are shown along with exper-imental result in Fig.3.It is obvious that extrapolating the results from ?nite nanoparticles to D =∞,leads to a bulk surface energy of 2.465(J/m 2)which is in good agreement with the experimental value of 2.45(J/m 2)in reference [31],and somewhat higher than the average surface energy of three low index surfaces in Fig.2.This means that the surface of nanoparticles include high index surfaces besides low ones.In addition,Fig.3also indicates that surface ener-gies of nanoparticles are size-dependent,which is in agreement with Lu and Jiang’s [32]theoretical predictions.

Because of the form of Eq.(11),there is little change for any envi-ronment which,on average,is isotropic.This is because ˇis very small,the time average of the correction is zero in an isotropic envi-ronment,and the MAEAM is invariant to transformations such as those described by Eq.(11).In order to further show that the modi-?ed MAEAM has no effect on the bulk thermodynamic predictions,the variations of the internal enthalpy and thermal expansion for bulk material with temperature,with and without the correction,are shown together with experimental values [33]in Fig.4.In gen-eral,two thermodynamic quantities with the correction are slightly higher than the original result;however,the agreement between

378W.Luo et al./Applied Surface Science 265 (2013) 375–

378

Fig.3.Surface energies of nanoparticles as a function of inverse cluster diameter 1/D

Fig.4.(a)The enthalpy and (b)the linear expansion of Ni as a function of the tem-perature.The asterisks represent the experimental values [33],and the solid line denotes the data with correction,and the dashdot lines represent the original data.

experiment and theories for both cases is reasonable good.The only properties that can change due to the correction are those associ-ated with the nonuniformity in the charge density.As has been demonstrated,these property changes can be con?ned to a small range by optimizing ˇand c .The calculations of vacancy forma-tion energy support the approach.After relaxation,the deviation from experimental vacancy formation energy is 1.7%for the original MAEAM potentials and 5%for the correction approach.The changes are small enough relative to excellent improvement of surface free energy.

4.Conclusion

To summarize,an approach to improve the prediction effect of the embedded atom method on surface free energy is presented.

By adopting a new solution for correcting the argument of the embedding energy F i ,the parameters ˇand c can be obtained analytically.Further,the solution is applied to calculate the tem-perature dependent anisotropic surface free energy and surface energies of nanoparticles,and the result in agreement with exper-iment is obtained.Meanwhile,the calculated results show that surface energies of nanoparticles are size-dependent.In isotropic environment,the approach does not change most predictions of bulk material properties,and in nonuniformity cases,the deviation from experimental results can be con?ned to a small range.Many more functions exist for the standard EAM so it is signi?cant to investigate the effect of the approach.

Acknowledgements

This work is ?nancially supported by the National Natu-ral Science Foundation of China (NNSFC)(No.51171064),and the Key Scienti?c Research Fund of Hunan Provincial Education Department (No.11A044),and Hunan Provincial Natural Science Foundation of China (No.10JJ3052).

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