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Local equilibrium in the dissolution and segregation kinetics of Ag on Cu(1 1 1) surface
Applied Surface Science 297 (2014) 130–133
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Local equilibrium in the dissolution and segregation kinetics of Ag on Cu(1 1 1) surface Min Lin a,b , Xu Chen a,c , XinYi Li a , Chi Huang a , YanXiu Li a , JiangYong Wang a,∗ a b c
Department of Physics, Shantou University, 243 Daxue Road, Shantou, 515063 Guangdong, China Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Department of Mathematics, University of Macau, Av. Padre Tomás Pereira Taipa, Macao, China
a r t i c l e
i n f o
Article history: Received 22 November 2013 Received in revised form 12 January 2014 Accepted 15 January 2014 Available online 25 January 2014 Keywords: Local equilibrium Surface segregation Dissolution Kinetics Modified Darken model
a b s t r a c t The local equilibrium in the kinetic processes of dissolution and segregation of Ag on Cu(1 1 1) surface is addressed. The measured dissolution and segregation kinetic data of Ag on Cu(1 1 1) surface at temperature of 450 ◦ C are well fitted by the modified Darken model. The segregation parameters, i.e. segregation energy, interaction and diffusion parameters, in the Cu(1 1 1)(Ag) binary system are obtained upon fitting. Using the obtained segregation parameters, the discontinuous transition of Ag surface concentration against the bulk concentration of the surface neighboring layers deduced from the local equilibrium model is quantitatively interpreted. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The surface segregation of Ag in the low-index Cu single crystals has been studied extensively because a discontinuous transition of Ag surface concentration accompanying an interesting hysteresis effect in isosteric/isothermal surface segregation was observed in the Cu(1 1 1)(Ag) system [1–8]. To interpret such a hysteresis effect in surface segregation, different opinions exist [1–10]. It was firstly proposed that such an effect is related to the so-called 2D-phase separation [3,4]. Then, it was demonstrated by applying the so-called local equilibrium model [1,2] that the discontinuous transition of Ag surface concentration in isothermal segregation could be deduced from a continuous segregation/dissolution kinetics. By the modified Darken model [6–8], the hysteresis effect in surface segregation is explained quantitatively. In the local equilibrium model, it is assumed an existence of the local equilibrium between the surface and its neighboring layer in the dissolution/segregation kinetics, which is the pre-requirement for applying the local equilibrium model. However, the judgment whether the local equilibrium is indeed established could not be made according to the local equilibrium model itself. In this paper, it will be demonstrated that the modified Darken model can be used to explore the local equilibrium in the dissolution/segregation
kinetics and to reproduce the experimental data of the discontinuous transition of Ag surface concentration in the isothermal dissolution/segregation kinetics deduced based on the local equilibrium model. 2. Theory 2.1. The local equilibrium model The notion of local equilibrium was first suggested in the study of sulphur diffusion in coppers by Pétrino et al. [12] and then formalized by Lagües and Domange [13]. It implies that the existence of a transition area “selvedge” between the surface and the bulk. The solubility and diffusion coefficient of the selvedge are different from the bulk. The dissolution process can be divided into two steps: the diffusion of atoms through the selvedge and the diffusion in the bulk. It is assumed that the diffusion through the selvedge is fast compared with the diffusion in the bulk because of the absence of large potential barrier in the selvedge. In the second step, the local equilibrium is established between the surface layer and its neighboring layer, and the concentration of the layer near the surface depends exclusively on the surface concentration, but weakly on the bulk concentration. The bulk concentration below the selvedge obeys Fick’s diffusion equation [13]:
∗ Corresponding author. Tel.: +86 754 82902225. E-mail address: [email protected] (J. Wang). 0169-4332/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2014.01.099
Cv (x, t) = Cv +
t
J0 (t − ) 0
exp(−x2 /4Dv )d
Dv
(1)
M. Lin et al. / Applied Surface Science 297 (2014) 130–133 1.0
when x = 0, the concentrate near the surface layer is described by: t
J0 (t − )
0
d
(2)
−dCs J0 = , dt
(3)
where Cv (x,t) is the bulk concentration at the distance of x to the surface at time t, Cv is the initial bulk concentration, Cs is the surface concentration, Dv is the volume diffusion coefficient of the solute atom, and J0 is the flux of atoms from the bulk layers into the surface. The dataset {Cv (0,t), Cs } represents the isothermal segregation/dissolution equilibrium. 2.2. The modified Darken model
∂X S DX B1 B1 ,S = RTd2 ∂t ∂X B1 DX B1 DX B2 B2 ,B1 − B1 ,S = RTd2 RTd2 ∂t
∂X (j) DX (j) DX (j+1) (j+1,j) − (j,j−1) = 2 RTd RTd2 ∂t
(4)
.. .
∂X N DX BN BN ,BN−1 , =− RTd2 ∂t where XS is the surface concentration and X(j) is the jth bulk layer’s concentration, R is the gas constant, T is the temperature, d is the interlayer distance, D is the diffusion coefficient (D = D0 exp(−Q/RT), Q is the activation energy and D0 is the frequency factor), (j+1,j) is the difference of chemical potential between two adjacent layers, (j+1) (j) (j+1) (j) i.e. (j+1,j) = 1 − 1 − 2 + 2 . Under the regular solution approximation, the chemical potential is expressed in terms of the standard chemical potential 0 , the interaction parameter ˝ and the concentration X as: (j)
2
0(j)
+ ˝(1 − X (j) ) + RT ln X (j) ,
0(j)
+ ˝(X (j) ) + RT ln(1 − X (j) ),
1 = 1
(5)
and (j)
2 = 2
2
segregation kinetics 2 Cu (111) 4.5at% Ag 0.6
full equilibrium dissolution kinetics
0.4
0.2
0.0 1E- 4
The model based on the Fick’s diffusion equation describes only a down-hill diffusion process, but the surface segregation is an uphill diffusion process. Therefore, any attempts to set up a model based on Fick’s diffusion equation to describe the surface segregation kinetics will not be consistent with this uphill diffusion process. The problem of uphill diffusion was first addressed by Darken [14] and it was assumed that the driving force for the diffusion process is the gradient of chemical potential instead of the concentration gradient as in Fick’s diffusion equation. Followed this assumption, the modified Darken model was proposed for describing the up-hill diffusion process of surface segregation kinetics [15–19]. It is a layer-by-layer model in which the investigated system is regarded as a closed system and is divided into one surface layer in contact with N bulk layers. This modified Darken model has already been successfully applied to describe both equilibrium and kinetic segregation processes [6–8,20–22]. The core of this model is that the kinetic process of surface segregation in a binary system is described by a set of coupled rate equations as follow:
. ..
segregation kinetics 1 Cu (111) 0.45at% Ag
0.8
Dv surface concentration Xs
Cv (0, t) = Cv +
131
(6)
1E-3
0.01
0.1
first bulk layer concentration Xb1 Fig. 1. Two segregation kinetics (dashed and dashed-dotted lines) and one dissolution kinetics (solid line) simulated by the modified Darken model are compared with the full equilibrium segregation curve (dotted line) according to Eq. (7), for G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ and D0 = 24 × 10−6 m2 /s at T = 450 ◦ C. The initial bulk concentration is 0.45% for segregation kinetics 1 (dashed line) and 4.5% for segregation kinetics 2 (dashed-dotted line).
If an equilibrium state is reached, all the rate equations should equal to zero, resulting XB1 = XB2 = · · · = XBN = XB . For the first rate equation, it follows
XB XS G + 2˝(X S − X B ) = exp , B S RT 1−X 1−X
(7)
where G is the so-called segregation energy and is defined as 0S 0B 0S G = 0B 1 − 1 − 2 + 2
(8)
where 0B and 0S are the standard chemical potentials of pure i i element i for the bulk material and surface region material, respectively. In general, the segregation energy G is independent on the bulk/surface concentration and the temperature. Eq. (7) is the well-known Bragg–Williams expression for widely describing the equilibrium surface segregation. Thus, the equilibrium surface segregation is a natural consequence of the rate Eq. (4). In order to distinguish the local equilibrium in segregation kinetics, the equilibrium segregation described by Eq. (7) is referred to the so-called full equilibrium. It has been demonstrated that, according to Eq. (7), a discontinuous transition of surface concentration can be brought about by a continuous change of sample’s bulk concentration [23]. Such a discontinuous transition of surface concentration against the bulk concentration at temperature of 450 ◦ C is shown in Fig. 1 by the dotted line for the segregation energy of 24.2 kJ/mol and the interaction parameter of 14.1 kJ/mol. The two bulk concentration values corresponding to the discontinuous transitions indicated by arrows in Fig. 1 are 0.188 at.% and 0.156 at.%, respectively, upon increasing and decreasing the bulk concentration value. For the segregation parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ and D0 = 24 × 10−6 m2 /s (see Section 3), according to the rate Eq. (4), two isothermal segregation kinetics of surface concentration as a function of first bulk layer concentration are presented in Fig. 1 as dashed and dashed-dotted lines for the initial bulk concentration values of 0.45 at% and 4.50 at%, respectively. Clearly, the segregation kinetic curve follows one of the full equilibrium for the case of the initial bulk concentration of 0.45 at%, indicating that the local equilibrium is indeed established between the surface and the first bulk layers, but not for the case of the initial bulk concentration of 4.50 at%.
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M. Lin et al. / Applied Surface Science 297 (2014) 130–133
0.8 1.0
modified Darken model fitting
dissolution kinetic data
Cu (111) 0.45at% Ag
0.6
surface concentration
0.8
surface concentration
0.7
0.6
0.4
0.2
modified Darken model fitting
0.5
segregation kinetic data 0.4 0.3 0.2 0.1
0.0 0
1000
2000
3000
4000
5000
6000
0.0
time(s)
Besides describing both the equilibrium and kinetic surface segregations, the modified Darken model can also be applied for describing the dissolution kinetics, which is regarded as a reversed process of surface segregation kinetics with respect to the direction of atomic diffusion. As an example, the dissolution kinetics at temperature of 450 ◦ C is simulated and presented in Fig. 1 as solid line for the same segregation parameters as those for the segregation kinetics. It is obvious that the dissolution kinetics follows exactly the same curve as the segregation kinetics for the initial bulk concentration of 0.45 at% (but in the opposite direction). Both the dissolution and segregation kinetics exhibit a vertical S shape and are in between the two transition bulk concentration values of the full equilibrium segregation, implying that the local equilibrium is also established for the case of dissolution kinetics. Actually, for a giving surface concentration value, the bulk concentration value determined by Eq. (7) (corresponding to the full equilibrium segregation) is the lowest value for any segregation/dissolution kinetics. For dissolution kinetics, the initial concentration value in the bulk layer is zero; therefore the dissolution kinetics will follow the full equilibrium segregation as described by Eq. (7), implying that the local equilibrium is always valid for any dissolution kinetics, but only for the segregation kinetics with a low initial bulk concentration value. 3. Result and discussion The experimental details about the segregation and dissolution kinetics of Ag on Cu(1 1 1) surface have been described in Refs. [1,2]. The main points are summarized here. For the dissolution kinetic measurement, a monolayer of Ag was deposited on a single crystal Cu(1 1 1) surface at room-temperature by Joule sublimation of a silver wire. The calibration for silver thickness was based on the Auger line breaks which appeared on signal intensity versus time curve when each monolayer is completed. For the segregation kinetic measurement, the Auger transition intensities for copper (60 eV and 920 eV) and silver (356 eV) were recorded at 450 ◦ C for the sample of Cu(1 1 1)0.45 at%(Ag) (limit of solubility for Ag in Cu at 450 ◦ C: 0.83 at%). The measured dissolution and segregation kinetic data of Ag on Cu(1 1 1) at 450 ◦ C are presented in Figs. 2 and 3 as open points, respectively. Based on the modified Darken model, the best fits to these measured data points using the least square method are
1000
2000
3000
4000
5000
6000
time(s) Fig. 3. Experimental segregation kinetics (open points) of Ag on Cu(1 1 1) are well fitted by the modified Darken model (solid line) with parameters G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
represented by solid lines in Figs. 2 and 3, respectively, for the segregation parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s. These obtained segregation parameter values are very similar to the ones that were used for fitting the discontinuous transition of Ag surface concentration in Cu(1 1 1)0.25 at.%Ag system [8]. Based on the local equilibrium model, the dataset {Cv (0,t), Cs } calculated from the measured dissolution and segregation kinetic data displayed in Figs. 2 and 3 according to Eqs. (2) and (3) are plotted as open points in Figs. 4 and 5, respectively. In the calculation of the dataset, high degree polynomials are used for fitting the experimental data in Figs. 2 and 3 and substituted into Eq. (2). These calculated data points follow almost exactly the same dissolution and segregation kinetic curve (solid line) simulated by the modified Darken model (see above), which exhibits a vertical S shape at the bulk concentration of 0.16 at.%. This vertical S shape indicates a
1.0
surface concentration
Fig. 2. Experimental dissolution kinetics (open points) of Ag on Cu(1 1 1) are well fitted by the modified Darken model (solid line) with parameters G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
0
0.8
deduced from dissolution kinetic data
0.6
modified Darken model full equilibrium curve
0.4
0.2
0.0 1E-4
1E-3
0.01
0.1 B1
first bulk layer concentration X
Fig. 4. The open points represent the discontinuous transition of Ag surface concentration against the first bulk layer concentration and are deduced from the measured dissolution kinetic data presented in Fig. 2 by applying the local equilibrium model. The simulated dissolution kinetics by the modified Darken model and the full equilibrium segregation are presented by solid and dotted lines, respectively, for the parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
M. Lin et al. / Applied Surface Science 297 (2014) 130–133 0.8
surface concentration
0.7
Cu (111 ) 0.45at% Ag
0.6
de duced from segregation kinetic data
0.5 0.4
mod ified Darken mode l
0.3
full equilibrium
133
the segregated Ag atoms immediately segregate into the surface and the surface concentration of Ag increases. The local equilibrium model provides an alternative and easy way to deduce the discontinuous transition of surface concentration from the continuous isothermal segregation/dissolution kinetics. However, the judgment whether the local equilibrium is indeed established can only be made by the modified Darken model instead of the local equilibrium model itself. By combining these two models, the discontinuous transition of surface concentration in isothermal segregation/dissolution kinetics could be thoroughly and quantitatively interpreted.
0.2
4. Conclusion 0.1 0.0 1E-4
1E-3
0.01
first bulk layer concentration X
0.1
B1
Fig. 5. The open points represent the discontinuous transition of Ag surface concentration against the first bulk layer concentration and are deduced from the measured segregation kinetic data presented in Fig. 3 by applying the local equilibrium model. The simulated segregation kinetics by the modified Darken model and the full equilibrium segregation are presented by solid and dotted lines, respectively, for the parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
discontinuous transition of Ag surface concentration against the first bulk layer concentration, namely the first bulk layer concentration stays nearly a constant value of 0.16 at.% while the Ag surface concentration drops from a high value of 88 at.% to a low value of 13 at.% for the dissolution kinetics, or increases from a low value to high value for the segregation kinetics. This fact implies that if the local equilibrium in a kinetic process is indeed established, the discontinuous transition of Ag surface concentration can be deduced from a continuous dissolution/segregation kinetic measurement by applying the local equilibrium model. The dissolution kinetic process can be explained as following: (1) the deposited Ag atoms dissolve into the neighboring layers of the surface; (2) when the first bulk layer (the neighboring layer of surface) concentration reaches the value of 0.16 at.% at the temperature of 450 ◦ C, the local equilibrium is established between the surface and the first bulk layer (the neighboring layer of surface); (3) Ag atoms dissolve from the first bulk layer into the deeper bulk layers and the dissolved Ag atoms are immediately replaced by the atoms from the surface and the surface concentration of Ag decreases. While the segregation kinetic process is opposite and explained as following: (1) the Ag atoms in the first bulk layer (the neighboring layer of surface) segregate into the surface layer; (2) when the first bulk layer concentration reaches the value of 0.16 at.% at the temperature of 450 ◦ C, the local equilibrium is established between the surface and the first bulk layer (the neighboring layer of surface); (3) Ag atoms segregate from deeper bulk layers the into the first bulk layer and
The local equilibrium between the surface layer and its neighboring layers in segregation/dissolution kinetic process is established only if the corresponding kinetics follows the same curve as described by the (full) equilibrium isotherm. The discontinuous transition of Ag surface concentration deduced from the measured continuous Ag dissolution and segregation kinetic data on Cu(1 1 1) surface at temperature of 450 ◦ C is quantitatively interpreted by the modified Darken model. Acknowledgement The work is supported by the National Natural Science Foundation of China (Project No. 11274218). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
J. Eugène, B. Aufray, F. Cabané, Surf. Sci. 241 (1991) 1. J. Eugène, B. Aufray, F. Cabané, Surf. Sci. 273 (1992) 372. Y. Liu, P. Wynblatt, Surf. Sci. 290 (1993) 335. Y. Liu, P. Wynblatt, Surf. Sci. 240 (1990) 245. G. Tréglia, B. Legrand, J. Eugène, B. Aufray, F. Cabané, Phys. Rev. B 44 (1991) 5842. J.Y. Wang, J. du Plessis, J.J. Terblans, G.N. van Wyk, Surf. Sci. 423 (1999) 12. J.Y. Wang, J. du Plessis, J.J. Terblans, G.N. van Wyk, Surf. Interface Anal. 28 (1999) 73. J.Y. Wang, J. du Plessis, J.J. Terblans, G.N. van Wyk, Surf. Sci. 419 (1999) 197. J. du Plessis, Surf. Sci. Lett. 273 (1992) L482. J. du Plessis, Surf. Sci. Lett. 259 (1991) L729. P. Pétrino, F. Moya, F. Cabané-Broutly, J. Solid State Chem. 2 (1970) 439. M. Lagües, J.L. Domange, Surf. Sci. 47 (1975) 77. L.S. Darken, Trans. AIME 180 (1949) 430. J. du Plessis, G.N. van Wyk, J. Phys. Chem. Solids 49 (1988) 1441. J. du Plessis, G.N. van Wyk, J. Phys. Chem. Solids 49 (1988) 1451. J. du Plessis, G.N. van Wyk, J. Phys. Chem. Solids 50 (1989) 237. J. du Plessis, G.N. van Wyk, J. Phys. Chem. Solids 50 (1989) 247. J. du Plessis, G.N. van Wyk, J. Phys. Chem. Solids 50 (1989) 251. E.C. Viljoen, J. du Plessis, H.C. Swart, G.N. Van Wyk, Surf. Sci. 342 (1995) 1. J.K.O. Asante, J.J. Terblans, W.D. Roos, Appl. Surf. Sci. 252 (2005) 1674. J. du Plessis, E.C. Viljoen, Appl. Surf. Sci. 100 (1996) 222. X.Y. Li, M. Lin, X. Chen, C. Huang, Y.X. Li, J.Y. Wang, Adv. Mater. Res. 648 (2013) 35.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Local equilibrium in the dissolution and segregation kinetics of Ag on Cu(1 1 1) surface Min Lin a,b , Xu Chen a,c , XinYi Li a , Chi Huang a , YanXiu Li a , JiangYong Wang a,∗ a b c
Department of Physics, Shantou University, 243 Daxue Road, Shantou, 515063 Guangdong, China Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Department of Mathematics, University of Macau, Av. Padre Tomás Pereira Taipa, Macao, China
a r t i c l e
i n f o
Article history: Received 22 November 2013 Received in revised form 12 January 2014 Accepted 15 January 2014 Available online 25 January 2014 Keywords: Local equilibrium Surface segregation Dissolution Kinetics Modified Darken model
a b s t r a c t The local equilibrium in the kinetic processes of dissolution and segregation of Ag on Cu(1 1 1) surface is addressed. The measured dissolution and segregation kinetic data of Ag on Cu(1 1 1) surface at temperature of 450 ◦ C are well fitted by the modified Darken model. The segregation parameters, i.e. segregation energy, interaction and diffusion parameters, in the Cu(1 1 1)(Ag) binary system are obtained upon fitting. Using the obtained segregation parameters, the discontinuous transition of Ag surface concentration against the bulk concentration of the surface neighboring layers deduced from the local equilibrium model is quantitatively interpreted. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The surface segregation of Ag in the low-index Cu single crystals has been studied extensively because a discontinuous transition of Ag surface concentration accompanying an interesting hysteresis effect in isosteric/isothermal surface segregation was observed in the Cu(1 1 1)(Ag) system [1–8]. To interpret such a hysteresis effect in surface segregation, different opinions exist [1–10]. It was firstly proposed that such an effect is related to the so-called 2D-phase separation [3,4]. Then, it was demonstrated by applying the so-called local equilibrium model [1,2] that the discontinuous transition of Ag surface concentration in isothermal segregation could be deduced from a continuous segregation/dissolution kinetics. By the modified Darken model [6–8], the hysteresis effect in surface segregation is explained quantitatively. In the local equilibrium model, it is assumed an existence of the local equilibrium between the surface and its neighboring layer in the dissolution/segregation kinetics, which is the pre-requirement for applying the local equilibrium model. However, the judgment whether the local equilibrium is indeed established could not be made according to the local equilibrium model itself. In this paper, it will be demonstrated that the modified Darken model can be used to explore the local equilibrium in the dissolution/segregation
kinetics and to reproduce the experimental data of the discontinuous transition of Ag surface concentration in the isothermal dissolution/segregation kinetics deduced based on the local equilibrium model. 2. Theory 2.1. The local equilibrium model The notion of local equilibrium was first suggested in the study of sulphur diffusion in coppers by Pétrino et al. [12] and then formalized by Lagües and Domange [13]. It implies that the existence of a transition area “selvedge” between the surface and the bulk. The solubility and diffusion coefficient of the selvedge are different from the bulk. The dissolution process can be divided into two steps: the diffusion of atoms through the selvedge and the diffusion in the bulk. It is assumed that the diffusion through the selvedge is fast compared with the diffusion in the bulk because of the absence of large potential barrier in the selvedge. In the second step, the local equilibrium is established between the surface layer and its neighboring layer, and the concentration of the layer near the surface depends exclusively on the surface concentration, but weakly on the bulk concentration. The bulk concentration below the selvedge obeys Fick’s diffusion equation [13]:
∗ Corresponding author. Tel.: +86 754 82902225. E-mail address: [email protected] (J. Wang). 0169-4332/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2014.01.099
Cv (x, t) = Cv +
t
J0 (t − ) 0
exp(−x2 /4Dv )d
Dv
(1)
M. Lin et al. / Applied Surface Science 297 (2014) 130–133 1.0
when x = 0, the concentrate near the surface layer is described by: t
J0 (t − )
0
d
(2)
−dCs J0 = , dt
(3)
where Cv (x,t) is the bulk concentration at the distance of x to the surface at time t, Cv is the initial bulk concentration, Cs is the surface concentration, Dv is the volume diffusion coefficient of the solute atom, and J0 is the flux of atoms from the bulk layers into the surface. The dataset {Cv (0,t), Cs } represents the isothermal segregation/dissolution equilibrium. 2.2. The modified Darken model
∂X S DX B1 B1 ,S = RTd2 ∂t ∂X B1 DX B1 DX B2 B2 ,B1 − B1 ,S = RTd2 RTd2 ∂t
∂X (j) DX (j) DX (j+1) (j+1,j) − (j,j−1) = 2 RTd RTd2 ∂t
(4)
.. .
∂X N DX BN BN ,BN−1 , =− RTd2 ∂t where XS is the surface concentration and X(j) is the jth bulk layer’s concentration, R is the gas constant, T is the temperature, d is the interlayer distance, D is the diffusion coefficient (D = D0 exp(−Q/RT), Q is the activation energy and D0 is the frequency factor), (j+1,j) is the difference of chemical potential between two adjacent layers, (j+1) (j) (j+1) (j) i.e. (j+1,j) = 1 − 1 − 2 + 2 . Under the regular solution approximation, the chemical potential is expressed in terms of the standard chemical potential 0 , the interaction parameter ˝ and the concentration X as: (j)
2
0(j)
+ ˝(1 − X (j) ) + RT ln X (j) ,
0(j)
+ ˝(X (j) ) + RT ln(1 − X (j) ),
1 = 1
(5)
and (j)
2 = 2
2
segregation kinetics 2 Cu (111) 4.5at% Ag 0.6
full equilibrium dissolution kinetics
0.4
0.2
0.0 1E- 4
The model based on the Fick’s diffusion equation describes only a down-hill diffusion process, but the surface segregation is an uphill diffusion process. Therefore, any attempts to set up a model based on Fick’s diffusion equation to describe the surface segregation kinetics will not be consistent with this uphill diffusion process. The problem of uphill diffusion was first addressed by Darken [14] and it was assumed that the driving force for the diffusion process is the gradient of chemical potential instead of the concentration gradient as in Fick’s diffusion equation. Followed this assumption, the modified Darken model was proposed for describing the up-hill diffusion process of surface segregation kinetics [15–19]. It is a layer-by-layer model in which the investigated system is regarded as a closed system and is divided into one surface layer in contact with N bulk layers. This modified Darken model has already been successfully applied to describe both equilibrium and kinetic segregation processes [6–8,20–22]. The core of this model is that the kinetic process of surface segregation in a binary system is described by a set of coupled rate equations as follow:
. ..
segregation kinetics 1 Cu (111) 0.45at% Ag
0.8
Dv surface concentration Xs
Cv (0, t) = Cv +
131
(6)
1E-3
0.01
0.1
first bulk layer concentration Xb1 Fig. 1. Two segregation kinetics (dashed and dashed-dotted lines) and one dissolution kinetics (solid line) simulated by the modified Darken model are compared with the full equilibrium segregation curve (dotted line) according to Eq. (7), for G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ and D0 = 24 × 10−6 m2 /s at T = 450 ◦ C. The initial bulk concentration is 0.45% for segregation kinetics 1 (dashed line) and 4.5% for segregation kinetics 2 (dashed-dotted line).
If an equilibrium state is reached, all the rate equations should equal to zero, resulting XB1 = XB2 = · · · = XBN = XB . For the first rate equation, it follows
XB XS G + 2˝(X S − X B ) = exp , B S RT 1−X 1−X
(7)
where G is the so-called segregation energy and is defined as 0S 0B 0S G = 0B 1 − 1 − 2 + 2
(8)
where 0B and 0S are the standard chemical potentials of pure i i element i for the bulk material and surface region material, respectively. In general, the segregation energy G is independent on the bulk/surface concentration and the temperature. Eq. (7) is the well-known Bragg–Williams expression for widely describing the equilibrium surface segregation. Thus, the equilibrium surface segregation is a natural consequence of the rate Eq. (4). In order to distinguish the local equilibrium in segregation kinetics, the equilibrium segregation described by Eq. (7) is referred to the so-called full equilibrium. It has been demonstrated that, according to Eq. (7), a discontinuous transition of surface concentration can be brought about by a continuous change of sample’s bulk concentration [23]. Such a discontinuous transition of surface concentration against the bulk concentration at temperature of 450 ◦ C is shown in Fig. 1 by the dotted line for the segregation energy of 24.2 kJ/mol and the interaction parameter of 14.1 kJ/mol. The two bulk concentration values corresponding to the discontinuous transitions indicated by arrows in Fig. 1 are 0.188 at.% and 0.156 at.%, respectively, upon increasing and decreasing the bulk concentration value. For the segregation parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ and D0 = 24 × 10−6 m2 /s (see Section 3), according to the rate Eq. (4), two isothermal segregation kinetics of surface concentration as a function of first bulk layer concentration are presented in Fig. 1 as dashed and dashed-dotted lines for the initial bulk concentration values of 0.45 at% and 4.50 at%, respectively. Clearly, the segregation kinetic curve follows one of the full equilibrium for the case of the initial bulk concentration of 0.45 at%, indicating that the local equilibrium is indeed established between the surface and the first bulk layers, but not for the case of the initial bulk concentration of 4.50 at%.
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modified Darken model fitting
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segregation kinetic data 0.4 0.3 0.2 0.1
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0.0
time(s)
Besides describing both the equilibrium and kinetic surface segregations, the modified Darken model can also be applied for describing the dissolution kinetics, which is regarded as a reversed process of surface segregation kinetics with respect to the direction of atomic diffusion. As an example, the dissolution kinetics at temperature of 450 ◦ C is simulated and presented in Fig. 1 as solid line for the same segregation parameters as those for the segregation kinetics. It is obvious that the dissolution kinetics follows exactly the same curve as the segregation kinetics for the initial bulk concentration of 0.45 at% (but in the opposite direction). Both the dissolution and segregation kinetics exhibit a vertical S shape and are in between the two transition bulk concentration values of the full equilibrium segregation, implying that the local equilibrium is also established for the case of dissolution kinetics. Actually, for a giving surface concentration value, the bulk concentration value determined by Eq. (7) (corresponding to the full equilibrium segregation) is the lowest value for any segregation/dissolution kinetics. For dissolution kinetics, the initial concentration value in the bulk layer is zero; therefore the dissolution kinetics will follow the full equilibrium segregation as described by Eq. (7), implying that the local equilibrium is always valid for any dissolution kinetics, but only for the segregation kinetics with a low initial bulk concentration value. 3. Result and discussion The experimental details about the segregation and dissolution kinetics of Ag on Cu(1 1 1) surface have been described in Refs. [1,2]. The main points are summarized here. For the dissolution kinetic measurement, a monolayer of Ag was deposited on a single crystal Cu(1 1 1) surface at room-temperature by Joule sublimation of a silver wire. The calibration for silver thickness was based on the Auger line breaks which appeared on signal intensity versus time curve when each monolayer is completed. For the segregation kinetic measurement, the Auger transition intensities for copper (60 eV and 920 eV) and silver (356 eV) were recorded at 450 ◦ C for the sample of Cu(1 1 1)0.45 at%(Ag) (limit of solubility for Ag in Cu at 450 ◦ C: 0.83 at%). The measured dissolution and segregation kinetic data of Ag on Cu(1 1 1) at 450 ◦ C are presented in Figs. 2 and 3 as open points, respectively. Based on the modified Darken model, the best fits to these measured data points using the least square method are
1000
2000
3000
4000
5000
6000
time(s) Fig. 3. Experimental segregation kinetics (open points) of Ag on Cu(1 1 1) are well fitted by the modified Darken model (solid line) with parameters G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
represented by solid lines in Figs. 2 and 3, respectively, for the segregation parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s. These obtained segregation parameter values are very similar to the ones that were used for fitting the discontinuous transition of Ag surface concentration in Cu(1 1 1)0.25 at.%Ag system [8]. Based on the local equilibrium model, the dataset {Cv (0,t), Cs } calculated from the measured dissolution and segregation kinetic data displayed in Figs. 2 and 3 according to Eqs. (2) and (3) are plotted as open points in Figs. 4 and 5, respectively. In the calculation of the dataset, high degree polynomials are used for fitting the experimental data in Figs. 2 and 3 and substituted into Eq. (2). These calculated data points follow almost exactly the same dissolution and segregation kinetic curve (solid line) simulated by the modified Darken model (see above), which exhibits a vertical S shape at the bulk concentration of 0.16 at.%. This vertical S shape indicates a
1.0
surface concentration
Fig. 2. Experimental dissolution kinetics (open points) of Ag on Cu(1 1 1) are well fitted by the modified Darken model (solid line) with parameters G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
0
0.8
deduced from dissolution kinetic data
0.6
modified Darken model full equilibrium curve
0.4
0.2
0.0 1E-4
1E-3
0.01
0.1 B1
first bulk layer concentration X
Fig. 4. The open points represent the discontinuous transition of Ag surface concentration against the first bulk layer concentration and are deduced from the measured dissolution kinetic data presented in Fig. 2 by applying the local equilibrium model. The simulated dissolution kinetics by the modified Darken model and the full equilibrium segregation are presented by solid and dotted lines, respectively, for the parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
M. Lin et al. / Applied Surface Science 297 (2014) 130–133 0.8
surface concentration
0.7
Cu (111 ) 0.45at% Ag
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de duced from segregation kinetic data
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mod ified Darken mode l
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full equilibrium
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the segregated Ag atoms immediately segregate into the surface and the surface concentration of Ag increases. The local equilibrium model provides an alternative and easy way to deduce the discontinuous transition of surface concentration from the continuous isothermal segregation/dissolution kinetics. However, the judgment whether the local equilibrium is indeed established can only be made by the modified Darken model instead of the local equilibrium model itself. By combining these two models, the discontinuous transition of surface concentration in isothermal segregation/dissolution kinetics could be thoroughly and quantitatively interpreted.
0.2
4. Conclusion 0.1 0.0 1E-4
1E-3
0.01
first bulk layer concentration X
0.1
B1
Fig. 5. The open points represent the discontinuous transition of Ag surface concentration against the first bulk layer concentration and are deduced from the measured segregation kinetic data presented in Fig. 3 by applying the local equilibrium model. The simulated segregation kinetics by the modified Darken model and the full equilibrium segregation are presented by solid and dotted lines, respectively, for the parameters of G = 24.2 kJ/mol, ˝ = 14.1 kJ/mol, Q0 = 181 kJ, D0 = 24 × 10−6 m2 /s.
discontinuous transition of Ag surface concentration against the first bulk layer concentration, namely the first bulk layer concentration stays nearly a constant value of 0.16 at.% while the Ag surface concentration drops from a high value of 88 at.% to a low value of 13 at.% for the dissolution kinetics, or increases from a low value to high value for the segregation kinetics. This fact implies that if the local equilibrium in a kinetic process is indeed established, the discontinuous transition of Ag surface concentration can be deduced from a continuous dissolution/segregation kinetic measurement by applying the local equilibrium model. The dissolution kinetic process can be explained as following: (1) the deposited Ag atoms dissolve into the neighboring layers of the surface; (2) when the first bulk layer (the neighboring layer of surface) concentration reaches the value of 0.16 at.% at the temperature of 450 ◦ C, the local equilibrium is established between the surface and the first bulk layer (the neighboring layer of surface); (3) Ag atoms dissolve from the first bulk layer into the deeper bulk layers and the dissolved Ag atoms are immediately replaced by the atoms from the surface and the surface concentration of Ag decreases. While the segregation kinetic process is opposite and explained as following: (1) the Ag atoms in the first bulk layer (the neighboring layer of surface) segregate into the surface layer; (2) when the first bulk layer concentration reaches the value of 0.16 at.% at the temperature of 450 ◦ C, the local equilibrium is established between the surface and the first bulk layer (the neighboring layer of surface); (3) Ag atoms segregate from deeper bulk layers the into the first bulk layer and
The local equilibrium between the surface layer and its neighboring layers in segregation/dissolution kinetic process is established only if the corresponding kinetics follows the same curve as described by the (full) equilibrium isotherm. The discontinuous transition of Ag surface concentration deduced from the measured continuous Ag dissolution and segregation kinetic data on Cu(1 1 1) surface at temperature of 450 ◦ C is quantitatively interpreted by the modified Darken model. Acknowledgement The work is supported by the National Natural Science Foundation of China (Project No. 11274218). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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