ISSN 10628738, Bulletin of the Russian Academy of Sciences: Physics, 2009, Vol. 73, No. 8, pp. 1150–1152. © Allerton Press, Inc., 2009. Original Russian Text © A.Z. Kashezhev, V.K. Kumykov, A.R. Manukyants, I.N. Sergeev, V.A. Sozaev, 2009, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2009, Vol. 73, No. 8, pp. 1214–1215.
Dependence of the Surface Energy of Metals on Pressure A. Z. Kashezheva, V. K. Kumykova, A. R. Manukyantsb, I. N. Sergeeva, and V. A. Sozaevb a
b
KabardinoBalkarian State University, Nalchik, 360004 Russia NorthCaucasian Institute of Mining and Metallurgy, Vladikavkaz, 362021 Russia email:
[email protected]
Abstract—A theoretical model has been developed within the method of electrondensity functional to describe the effect of pressure on the surface energy of pure metals. DOI: 10.3103/S1062873809080395
The dependence of the surface tension (ST) or sur face energy (SE) on pressure makes it possible to esti mate the surface density of a condensed phase (or self adsorption) at the interface with a gas [1]: Γ = – ρ(α)(dσ/dр)T, (1) where Γ is selfadsorption, ρ(α) is the condensed phase density, and р is pressure. Such data are important for developing the theory of surface phenomena. However, there are very little data on the effect of pressure on ST (or SE) of liquid and all the more solid metals in the literature. The thermodynamic analysis within the finite thickness method [2] shows that when a gas is weakly dissolved or weakly adsorbed at the liquid metal–gas interface ST should weakly increase with an increase in pressure. In [3], the effect of inert gases on SE metals was studied within the statistical electronic theory, accord ing to which SE should decrease with an increase in the gas pressure; for example, for copper, the decrease in SE in an argon atmosphere at the melting tempera ture is ∆σ/σ ≈ 0.1%. In [4], the dependence of ST on pressure at the liq uid–gas interface was also theoretically considered. Such calculations require the data on the coefficients of compressibility, which are much lower for metals than for organic liquids; therefore, one would expect a decrease in ST with an increase in pressure. A formula was obtained in [5], where the heat sub limation, specific heat, and coordination number of metals are taken into account. This formula gives a decrease in ST of liquid metals with an increase in pressure. The results of experimental studies of the effect of pressure on ST of organic liquids and water show that ST increases with an increase in pressure [1, 6] while for metals in the pressure range from 10–3 to 103 Pa either the effect of inert gas pressure is not
observed [7, 8] or changes are due to the presence of oxygen and other impurities in the gaseous medium and the related change in the interaction in the gas eous medium (formation of oxides, enhancement of gas dissolution in metals with an increase in pressure). There are no data in the literature on the effect of high pressures on ST in metals. In this context, it is interesting to study the effect of pressure on SE of simple metals within the method of electrondensity functional [9]. The electrondensity distribution n−(z) will be set as a twoparameter exponential trial function, describing the asymmetric electrondensity distribution at the interface:
⎧⎛ ⎞ β n−( z ) = n ⎨⎜1 − exp( α ( z − Z G)) ⎟ θ ( − z ) ⎩⎝ α + β ⎠ ⎫ + α exp( −β ( z − Z G)) θ ( z )⎬ . α+β ⎭
(2)
Here, z is the coordinate counted along the axis directed perpendicularly to the phase boundary and θ(z) is the Heaviside function. The Gibbs interface position ZG is obtained from the condition of electroneutrality of the system: ZG = 1/α – 1/β,
(3)
where α and β are the variational parameters, which are found from the minimum of total SE:
1150
σ = min σ( α , β ).
(4)
α, β
Total SE can be written as σ = σj + σps + δσps + δσcl.
(5)
DEPENDENCE OF THE SURFACE ENERGY OF METALS ON PRESSURE
SE in the jelly approximation, σj, was estimated from the formula ∞
σj = 1 2
+
−∞
∞
+ 0.3(3 π )
2 2/3
∫ [n
−∞ ∞
+ 1 72
()
− 0.75 3 π
− 1/3
− n+ ( z )] dz 5/3
⎧ 1 σ ps = 2 π n 2 ⎨ 1 3 + 5 3 + 2 ( αβ α + β) 6 6 β α ⎩
2
∫
−∞ ∞
∫
5/3 − (z)
∇ n−( z ) dz n−( z )
− (6)
[ n−4/3( z ) − n+4/3( z )] dz
−∞
∞
⎡ ⎤ n−4/3( z ) n+4/3( z ) − 0.056 ⎢ − 1/3 1/3 ⎥ dz ⎣0.079 + n− ( z ) 0.079 + n+ ( z )⎦
∫
−∞
∞
+ C xc( rs)
and taking into account the relaxation of the metal surface structure under pressure. If the pseudopotential cutoff radius rc > d0/2 – δd0, where d0 is the equilibrium interplanar spacing in the absence of pressure,
∫ ϕ( z )[n (z ) − n (z )]dz −
∫
−∞
1151
2
∇ n−( z ) dz. n−4/3( z )
Cxc(rs) = (2.702–0.174rs) × 10–3 and rs = (4π n /3)–1/3 is the Wigner–Seitz cell radius. The electrostatic potential ϕ ( z ) is obtained from the Poisson equation: (7) ∆ϕ( z ) = − 4 π[ n−( z ) − n+( z )] taking into account the boundary conditions and con tinuity conditions for ϕ ( z ) and ϕ '( z ) at the interface z = ZG [9]. The first term in (6) is the contribution of the intrinsic electrostatic energy of interaction of the elec tron gas with the jelly charge and interaction of the electron gas with the positive charge of the adsorption layer, which depends on the position of the Gibbs interface; the second term is the contribution of the kinetic energy of the noninteracting electron gas; the third term is the correction to the kinetic energy for the Weizsacker–Kirzhnits field inhomogeneity; the fourth term is the contribution of the exchange inter action energy; the fifth term is the contribution of the correlation interaction energy; and the sixth term is the nonlocality correction to the exchange–correla tion interaction, taken in the Geldart–Resault approximation. All integrals in expression (6) are easy to take; the corresponding formulas are given in [9]. However, the variational parameters α and β, whose reciprocal val ues characterize the lengths of tails of the electron dis tribution at the interface, depend on the external pres sure in our case. To estimate the other contributions to formula (5), it is necessary to take into account the pressure induced structure relaxation δ. The contribution of the electron–ion interaction σps will be found using the Ashcroft pseudopotential
+
exp(−α Z G) β d0 [cosh[α(d0 − rc)] (α + β )α 2 sinh(α d 0/ 2)
⎤ U 0α rc2 rc2 ⎞ β d0 ⎛ sh[ α ( d 0 − rc)]⎥ + 2 ⎜1 + U 0α 2 2 ⎟⎠ ⎦ (α + β )α ⎝ (8) d0 αd0 ⎡ ⎤ × exp −α rc − + Z G + ⎢⎣ ⎥⎦ ( α + β ) β 2 2
(
(
)
d × exp ⎡−β rc − 0 − Z G ⎣⎢ 2
−
)
(
2 ⎤ ⎛1 + U β rc ⎞ − d 0 r − d 0 c 0 ⎦⎥ ⎜⎝ 2 ⎟⎠ 2 2
(
)
2
)
⎫ d0 ⎛ 1 1 ⎞ d0 d 0rc2U 0 ⎬; ⎜ 2 − 2 ⎟ + 0.5 rc − 2 ⎝α β ⎠ 2 ⎭
⎛ ⎞ d δσ ps = −2π(nd 0)2δ ⎜ rc − 0 (1 − δ) − 1 + 1 ⎟. 2 α β ⎝ ⎠ At rc < d0(1/2 – δ),
(9)
⎧ 1 σ ps = 2πn 2 ⎨ 1 3 + 5 3 + αβ α + β) 2 ( β α 6 6 ⎩ (10) 2 βexp(−αZ G)αd0[cosh(αrc) − rc U 0αsinh(αrc)/2]⎫ − ⎬, sh(αd0 / 2) (α + β)α3 ⎭ where δσps is the dilatation contribution. It is a correc tion to σps, related to the change in the interplanar spacing:
β exp[ −α ( d 0/ 2 (α + β )α 2 + Z G)] ch ( α rc)[1 − exp( αδ d 0)],
δσ ps = 4 π n d 0 (d)
2
(11)
where δ is the dilatation induced by external pressure, which is varied. The Madelung surface energy σcl, caused by the ion–ion interaction, will also be estimated taking into account the dilatation correction: (d) δσ cl = δσ(0) cl + δσ cl .
(12)
Here, δσ(0) cl = α hkl Zn , αhkl is a surface analog of the Madelung constant [10], dependent on the crystal structure and reticular density of the (hkl) face and Z is the number of electrons per Wigner–Seitz cell.
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KASHEZHEV et al.
The dilatation contribution to the Madelung energy can be written as
cosh(2πd0hµ1, µ2)cos(2πx1µ1 − 2πx2µ2) − 1 δσ(cld) = − 1 2 , 4ω µ , µ hµ1, µ2[cosh(2πd0hµ1, µ2) − cos(2πx1µ1 + 2πx2µ2)]2
∑ 1
2
where µ1 and µ2 are integers, which can be taken in the interval [–6, 6] in view of the fast convergence of the sum; hµ1, µ2 is the vector of the surface twodimensional reciprocal lattice; xi is determined by the displacement vector of the Nth layer to (N +1)th layer (0 ≤ xi ≤ 1); and d0 is the interplanar spacing in the absence of pres sure. The prime at the sum means that the terms with µ1 = µ2 = 0 are excluded in summation; ω is the surface area per atom. The pressure and dilatation are related by the expression 2 p / c11 = δ exp{ δd 0*}[1 + 0.15 δd 0* + 0.05( δd 0*) ], (14)
δ is obtained by minimizing σ and is varied from 0 to 0.2.
d 0* = d 0/ l s ,
(13)
(15)
where ls is the scaling parameter, d = d0(1 – δ), n = n /(1 – δ) is the positive charge density of the metal under pressure, n is the positive charge density in the metal bulk, and с11 is the modulus of elasticity. For sodium, с11 = 0.073 × 1011 N/m2 and ls = 1.09. The pre liminary estimations for Na show that at pressures of 100 MPa the relative decrease in SE (∆σ/σ) is ~10%.
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