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A theoretical study of melting
. . . . . . .. . . .. .. . . ..m.
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Surface Science 281(1993) 317-322 North-Holland
A theoretical
...... _
:.:.:.:,,:;
‘.“‘::.:p$
surface science ~~~3~%+:;~......,_. :“.,:.: ...(,,,.i, : ,..: :+:.:.:.:.:.:::j::. ::,.... :::r.:::::<: :,:,:,.. ““‘:‘:‘.‘.:.:.:.:..: ‘(,‘,‘:‘...-.‘........... :.:.:.: .,:, ,.., ,,, ‘yL:::i,:(., .......,..._ .,,,.,, .........,.,,,y ..._,,,
study of melting
Y. Teraoka Department of Physics, University of Osaka Prefecture, Sakai 593, Japan
Received 19 August 1992; accepted for publication 19 October 1992
In order microscopic a first-order temperature. quasi-liquid
to understand bulk melting phenomena, a semi-infinite system with a surface is theoretically investigated from a point of view. Our simple theory based on the lattice gas model predicts that an infinite system without surfaces reveals melting transition, while a semi-infinite system with a surface undergoes a second-order melting transition at the same With approaching the melting point T,, the number of layers which undergo transitions from the solid state to the one increases. The result is consistent with the experimental results observed on Pb(ll0) and Al(110).
1. Introduction
Melting is well-known as one of the typical phase transition phenomena. From a microscopic point of view, many authors have attempted to obtain an understanding of melting phenomena. In theoretical treatments, some physical situations have been taken into account as an origin of melting, such as lattice instability [l], vanishing of shear moduli [2-S], spontaneous generation of crystal lattice imperfection [6-81. However, experiments have failed to find such evidences [8-111. The idea that surfaces may play a key role in the melting of a solid has been proposed by Tammann [12,13]. This is consistent with Lindemann’s empirical criterion [6], because, on the surface, the criterion is satisfied at much lower temperatures than the melting point due to a reduced coordination number of a surface atom. Recent observations on the surface melting of Al(110) and Pb(ll0) by ion scattering spectroscopy (ISS) have confirmed the idea that the bulk melting is induced by the surface [14,15]. The surface melting starts at a somewhat lower temperature than the bulk melting point T,, and more layers from the surface start to melt with approaching T,. The liquid-like layers in the surface region are not perfectly disordered, but 0039-6028/93/$06.00
slightly keep ordered structures, reflecting the structure of solid layers inside. Recent theoretical attempts [16-201 and computer simulations [21-271 also attribute a key role of surface melting to the bulk melting. In particular, Lipowsky et al. [16,17] have found, in a phenomenological theory, that an infinite system without surfaces undergoes a first-order transition, while a semi-infinite system with a surface reveals a second-order transition at the same temperature. The result has been obtained by an appropriate choice of the Landau expansion coefficients of a free energy with short-range interactions. However, a microscopic theory on the role of surface melting in bulk melting phenomena has not yet been obtained. The purpose of the present paper is to investigate the role of surface melting in bulk melting phenomena, from a microscopic point of view, by use of our simple theory treating the gas, liquid and solid phases on the same ground [28,29]. This theory is based on the lattice gas model. Application of the theory to multilayer adsorption systems has given a good explanation in a qualitative way. The difference between the solid and liquid phases has been described by a long-range order parameter reflecting the periodicity of the solid. There is no finite value of the long-range order
0 1993 - Elsevier Science Publishers B.V. All rights reserved
I’. Teraoka /A
318
theoretical study of melting
parameter in the bulk liquid phase. In section 2, our theory is developed for an infinite system without surfaces (3D bulk system) and a semi-infinite system with a surface. In section 3, the phase diagram of the bulk system is discussed. In section 4, the relation between the surface and bulk meltings is discussed in detail. Section 5 is devoted to a summary.
and X,(r), respectively. These quantities are described by use of the concentration of atoms in the ith cell, Xi, and the long-range order parameter Y. as follows: Xi(~) = (Xi + 2x)/3, and xi(P)
2. Theory First we explain our basic idea on the difference between the liquid (L) and solid (S) phases in a bulk crystal (see fig. 1). The space is divided into equivalent cells. Each cell can contain no more than one atom. An atom in a cell can occupy one position among three ones denoted as (Y, /3 and y sites. If only one kind of sites is occupied, the system is a perfect solid. If sites of all kinds are equally occupied, it is a perfect liquid. A long-range order parameter is defined by the difference between occupations on sites of one kind and others. Now we denote the concentrations of atoms on (Y, p and y sites in the ith cell by Xi(a), X,(p)
,I 0
a
0B
0
=xi(Y)
=(xi-yi)/3,
(I)
where it was assumed that p and y sites in the ith cell are equivalent to each other. We assume that there is an interaction energy between two atoms, only when the two atoms occupy the nearest-neighbor cells. The interaction energies are assumed to be V, for sites of the same kind and Vi for sites of different kinds. For example, the interaction energy is V, for CY-_(y, p-/3 and y-y sites, and Vi for a-+?, P-r and y-cu sites. V, and Vi are assumed to be negative, because two atoms in the nearestneighbor cells have a lower interaction energy than in other cases. We consider an infinite system without surfaces (3D bulk system). Assuming that Xi(~), X,(p) and X,(-y> do not depend on i, the free energy of the system with the bulk concentration of atoms X is evaluated within the framework of the Bragg-Williams approximation: ~=Z((V,/2)[X(~)2+2X(P)2]
7
+v,[2X(a)X(P) +knT[(l +X(a)
-X) log X(a)
+X(P)*]} log(1 -X) +2X(P)
log
where k, is the Boltzmann constant temperature. Z is the number of neighbor cells. X(a) and X(/3> are concentrations on (Yand p sites and X(a) Fig. 1. Liquid (L) and solid (S) phases for X = 1 are schematically shown. The space is divided into equivalent cells. Each cell has three sites a, p and y sites. If all atoms occupy sites of the same kind, the system is in S, and if all sites are occupied equivalently, it is in L.
= (X+
2Y)/3,
=X(y)
= (X-
X(P)13 (2) and T is the the nearestthe average defined by
and X(P)
Y)/3,
(3)
where Y is the bulk long-range order parameter.
Y. Teraoka /A
theoretical study
by minimizing F with respect to
Y is determined y: Z(T/,-V,)Y+k,T
log[x(cu)/x(p)]
=o.
(4)
Next we consider a semi-infinite system with a surface. Taking into account experimental conditions, we neglect the gas phase. The surface layer is denoted by n = 1. The concentrations of atoms at (Y, p and y sites per cell on the nth layer, X,(a), X,(P) and X,(y) (=X,(P)>, are given by x,(a)
= (X, + 2y,)/3,
and X,(P)
=x(Y)
= (Xn
-
(5)
U/3,
where X, is the average concentration per cell on the nth layer and Y, is the long-range order parameter on the nth layer. X, and Y, are determined by minimizing the free energy of the system with respect to X, and Y,: (V0 + 2VJ(ZclXI + k,~[(1/3) -(l
+-&X,)/3 log X,(a)
-X1)
log(1 -X1)]
(V0+2~,)(ZIX,-,
+Z,X,
+ kn~[(113)
log X,(a)
+ (213) log X,(P) -p
= 0,
(6)
+ (213) log X,(P)
We show a phase diagram of an infinite system without surfaces in fig. 2, where Z = 12, V0 < 0 and VI/V, = 0.5. L + S means a mixed phase of L and S. TCL, LS) and T(LS, S) are the temperatures at which the bulk system undergoes phase transitions from L to L + S and from L + S to S, respectively, as temperature decreases. For example, we consider the case of X=X, = 0.95261. With increasing T, the system undergoes a firstorder transition from S to L + S at T,,, = TCLS, S) = 2.04 I V, I /kB, which is the melting point. The concentrations of atoms in L and S are X, = 0.93329 and X, =X0 at T,, respectively. There coexist two phases L and S between T(LS, S> and T(L, LS) = 2.074711 V, 1k,. X, (X,) increases from X, = 0.93329 (X,) to X, (Xs = 0.96470) with increasing T from T(LS, S) to T(L, LS). The ratio of L (S) increases (decreases) from 0 (1) to 1 (0). The transition at T(L, LS) is of first-order, too. Above T(L, LS), there exists only L (Y = 0). The temperature dependence of the long-range order parameter is shown in fig. 3. Y decreases with increasing T, and decreases more slowly
I
forn22,
-tZIY,)
log[X,(+X,(P)]
3. Bulk melting
2.2
(7)
+ k,T
319
+ZIX*+1)/3
-(l-X,)log(l-X,)1-p=O,
(V0 - Vr)(ZJl
of melting
= 0,
I
(8)
and (V,-
VI)(ZIY,-I
+ k,T
+Z&
+Z,Y,+r)
log[ X,( a)/XJ
p)] = 0,
for 122 2, (9)
where Z, and Z, are the numbers of the nearest-neighbor cells on the same layer and on the adjacent one (Z = Z, + 22,). The effective chemical potential F is given by /-L= (Z/3)(V, +(2/3)
+ 2V,)X+ log X(p)
k,T[(1/3)
- (1 -X)
log X(a) log(1 -X)]. (10)
Fig. 2. Phase diagram of an infinite system without surfaces (3D bulk system) for Va < 0, VI / Vu = 0.5 and Z = 12. T(LS, S) a is the melting point T,, at which the system undergoes first-order transition from S to L+S with increasing T. T(L, LS) is the first-order transition temperature from L + S to L. For X = 1, T,,, = (LS, S) = T(L, LS). The transition is also of first-order. The dashed line means X = Xa = 0.95261.
320
Y. Teraoka /A
theoretical study of melting
0.901~
a.,,-
keT/iVd
2.0
! ! +
Fig. 3. Temperature dependence of the long-range order parameter for X = Xe in the 3D bulk system. At r, = T&S, S) < T(L, LS), there coexist L and S, and Xs increases from X0 to 0.964’70.At T(L, LS), Y deceases discontinuously from 0.51417 to 0.
between T(LS, S> and TCL, LSI, and then decreases discontinuously from 0.51417 to 0 at T(L, LS). For X = 1, T, = T&S, S> is equal to T(L, LS) = 2.16404 1V, I/k,, at which Y decreases discontinuously from Y = 0.5 to 0.0. The transition is of first-order. In this case, we cannot find a mixture of L and S.
4. Surface meiting Numericat calculations are carried out by assuming 2, = 4, Z, = 4 and X=X,. We determine self-consistently X, and Y, by use of eqs. (5)-(10). The depth profiles of X, for various temperatures lower than T, = 2.04 I V, 1/k, are shown in fig. 4. The concentration of atoms on the surface Iayer X, is much smaller than X = X0 (X, = 0.751, and so they are not shown there. X, increases with increasing T. X, is smaller than X,, as far as T is finite. At a given T, X, approaches X0 with an increase of n. With increasing T toward T,, there appears a region with X, = 0.9333, which is nearly equal to X, at T,,,. With approaching T, more closely, such a region expands deep inside. At T(L, IS) > T > T,, all layers except for the first few layers have concentrations equal to X,. Of course X, in-
.n
15 20 25 5 10 Fig. 4. Depth profiles of X, for various temperatures (n 2 2). The X, are - 0.75, and so they are not shown. X, and X, are the concentrations of atoms in L and S at 7’,, respectively. Cases (a)-(h) are for T = 2.0, 2.03, 2.039, 2.0399, 2.03992, 2.03994, 2.03996, and 2.03998 1V, I/k,, respectively. With approaching T, = 2.04 / V. I /kB, (a)-+(h), there appears a region with X, = X,, and it expands deep inside.
creases from 0.9333 at T, to X0 at TIL, LS). Above T(L, LS), X, =X0. As is shown in fig. 5, where the depth profiles of Y, are shown, Y, is very small in the region with X, =X,. With increasing T, Y decreases and Y, also decrease. The region with Y, = 0 expands with approaching T,. As a result we can say as follows: with approaching T,, there appears a region with X;, =X, and Y, = 0, and it expands deep inside the bulk crystal. In other words, the so-called quasi-liquid layers, in which Y, is not zero but very small, accumulates in the surface region. At T,, even the layer with n = 00 has X, =X, and Y, = 0. This is seen from the
Fig. 5. Depth profiles of Y,. With an increase of n, Y approaches 0.52530, which is indicated by the dashed line. In the region with X,, ax,_, the Y, are very small. With approaching T,, (a>+(h), the region expands deep inside. The cases (a)-(h) are the same as in fig. 4.
Y Teraoka /A
theoretical study of melting
ks(T.-Tl/IVol 10-l . \ \ lo+
lo+
10-4-
'\ '? '\ \ \ '\\
'*
.. \
\\
\ ‘\ 10
5
nc
Fig. 6. Temperature dependence of n,. We can find the logarithmic divergence of n, with approaching T,. This means the semi-infinite system with a surface undergoes a secondorder melting transition at T,,,.
fact that n, is linear with log[(T, - T)/ I V, II (see fig. 6), where n, is given by n at which Y, = 0.3. We can say that layers with n L 12, are in S and ones with n 2 n, are in L. This means that, at T, - O,, the liquid phase is induced near the surface and the solid phase remains deep inside the bulk crystal. This is consistent with the observation on Al(110) and Pb(ll0) [14,15]. As was above-mentioned, an infinite system without surfaces reveals a first-order melting transition. On the other hand, the semi-infinite system with a surface undergoes a second-order melting transition at the melting point. This origin is considered to be in the segregation of vacancies at the
321
surface: as is suggested by fig. 5, a layer with a smaller concentration of atoms has a smaller long-range order parameter, and so the surface layer has a smaller order parameter than other layers. In other words, it becomes a quasi-liquid layer at a temperature lower than T,. This induces successive reduction of the long-range order parameters of other layers. Next we consider the temperature dependence of Y,. For example, as is seen from fig. 7, Yi approaches zero with increasing T toward T,. Y, is proportinal to (T, - T)“.51779. We can say that the surface layer reveals a continuous transition between the liquid and solid states at T,. Moreover, we can find that other Y, decrease with power laws of (T, - T) with approaching T,. All order parameters undergo continuous transitions between liquid and solid states at T,.
5. Summary
We have investigated the surface melting on the basis of a simple theory within the framework of the Bragg-Williams approximation. We have found that an infinite system without surfaces reveals a first-order melting transition, while the semi-infinite system with a surface reveals a second-order melting transition at the bulk melting point. All layers reveals a continuous transition from the solid state to the liquid state at T,,,. This result qualitatively reproduces the experimental findings.
References
111F.A. Lindemann, Z. Phys. 14 (1910) 609. 121P. Sutherland, Philos. Mag. 32 (1891) 31. [31 P. Sutherland, Philos. Mag. 32 (1891) 215. [41 P. Sutherland, Philos. Mag. 32 (1891) 524. 151 M. Born, J. Chem. Phys. 7 (1939) 591. [61 J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955). ke(T.-T)/IVol 1o-2
IO'
Fig. 7. Temperature dependence of Yr. We can find a relation: Yr a (T, - T)“.51779. Y, decreases coninuously to zero with approaching T,.
171 J. Lennard-Jones
and A.F. Devonshire, Proc. Roy. Sot. (London) A 170 (1939) 464. 181 R.M.J. Cotterill, E.J. Jensen and W.D. Kristensen, Anharmonic Lattices, Structural Transitions and Melting, Ed. T. Riste (Noordhoff, Leiden, 1974). [91 Y.P. Varshni, Phys. Rev. B 2 (1970) 3952.
322
Y Teraoka /A
theoretical study of melting
[lo] J.K. Kristensen and R.M.J. Cotterill, Philos. Mag. 36 (1977) 437. [ll] D.P. Woodruff, The Solid-Liquid Interface (Cambridge University Press, London, 1973). 1121 Tammann, Z. Phys. Chem. 68 (1910) 205. [13] Tammann, Z. Phys. 11 (1910) 609. [14] B. Pluis, A.W. Denier van der Gon, J.W.M. Frenken and J.F. van der Veen, Phys. Rev. Lett. 59 (1987) 2678. [15] J.W.M. Frenken, P.M.J. Maree and J.F. van der Veen, Phys. Rev. B 34 (1986) 7506. [16] R. Lipowsky, Phys. Rev. Lett. 49 (1982) 1575. [17] R. Lipowsky and R. Speth, Phys. Rev. B 28 (1983) 3983. [18] L. Pietronelo and E. Tosatti, Solid State Commun. 32 (1979) 255. [19] C.S. Jayanthi, E. Tosatti and A. Fasolino, Surf. Sci. 152/153 (1985) 155.
[20] C.S. Jayanthi, E. Tosatti and L. Pietronelo, Phys. Rev. B 31 (1985) 3456. [21] J.Q. Broughton and G.H. Gilmer, J. Chem. Phys. 79 (1983) 5105. [22] J.Q. Broughton and G.H. Gilmer, J. Chem. Phys. 79 (1983) 5119. [23] F.F. Abraham, Phys. Rev. B 23 (1981) 6145. [24] J.Q. Broughton and G.H. Gilmer, Acta Metall. 31 (1983) 845. [25] W. Schommers and P. von Blackenhagen, Vacuum 33 (1983) 733. [26] W. Schommers and P. von Blackenhagen, Surf. Sci. 162 (1985) 144. [27] R.M.J. Cotterill, Philos. Mag. 32 (1975) 1283. [28] Y. Teraoka and T. Seto, Surf. Sci. 278 (1992) 202. [29] Y. Teraoka and T. Seto, Surf. Sci., in press.
......
:.~.~‘:‘,:.:,:,z ~,:, ,.,.,,,,(, ,,,,.,
. ..‘“...:: ........A :.:.: . . . . . . . .. . . . . .. . :.:.:.: . . . . . . .. . . . . :.:.,:.:~:: “‘.‘.“.:.:.:....‘.‘.U.:i .......n.. >:.:.:.: . . . . . . . .. . . . :,:.:..>
i.z+:.:.:..v.:.:.:.:.:~ ,:,:.:.::::::::: .... ,.,.............. ,:.:.:...
Surface Science 281(1993) 317-322 North-Holland
A theoretical
...... _
:.:.:.:,,:;
‘.“‘::.:p$
surface science ~~~3~%+:;~......,_. :“.,:.: ...(,,,.i, : ,..: :+:.:.:.:.:.:::j::. ::,.... :::r.:::::<: :,:,:,.. ““‘:‘:‘.‘.:.:.:.:..: ‘(,‘,‘:‘...-.‘........... :.:.:.: .,:, ,.., ,,, ‘yL:::i,:(., .......,..._ .,,,.,, .........,.,,,y ..._,,,
study of melting
Y. Teraoka Department of Physics, University of Osaka Prefecture, Sakai 593, Japan
Received 19 August 1992; accepted for publication 19 October 1992
In order microscopic a first-order temperature. quasi-liquid
to understand bulk melting phenomena, a semi-infinite system with a surface is theoretically investigated from a point of view. Our simple theory based on the lattice gas model predicts that an infinite system without surfaces reveals melting transition, while a semi-infinite system with a surface undergoes a second-order melting transition at the same With approaching the melting point T,, the number of layers which undergo transitions from the solid state to the one increases. The result is consistent with the experimental results observed on Pb(ll0) and Al(110).
1. Introduction
Melting is well-known as one of the typical phase transition phenomena. From a microscopic point of view, many authors have attempted to obtain an understanding of melting phenomena. In theoretical treatments, some physical situations have been taken into account as an origin of melting, such as lattice instability [l], vanishing of shear moduli [2-S], spontaneous generation of crystal lattice imperfection [6-81. However, experiments have failed to find such evidences [8-111. The idea that surfaces may play a key role in the melting of a solid has been proposed by Tammann [12,13]. This is consistent with Lindemann’s empirical criterion [6], because, on the surface, the criterion is satisfied at much lower temperatures than the melting point due to a reduced coordination number of a surface atom. Recent observations on the surface melting of Al(110) and Pb(ll0) by ion scattering spectroscopy (ISS) have confirmed the idea that the bulk melting is induced by the surface [14,15]. The surface melting starts at a somewhat lower temperature than the bulk melting point T,, and more layers from the surface start to melt with approaching T,. The liquid-like layers in the surface region are not perfectly disordered, but 0039-6028/93/$06.00
slightly keep ordered structures, reflecting the structure of solid layers inside. Recent theoretical attempts [16-201 and computer simulations [21-271 also attribute a key role of surface melting to the bulk melting. In particular, Lipowsky et al. [16,17] have found, in a phenomenological theory, that an infinite system without surfaces undergoes a first-order transition, while a semi-infinite system with a surface reveals a second-order transition at the same temperature. The result has been obtained by an appropriate choice of the Landau expansion coefficients of a free energy with short-range interactions. However, a microscopic theory on the role of surface melting in bulk melting phenomena has not yet been obtained. The purpose of the present paper is to investigate the role of surface melting in bulk melting phenomena, from a microscopic point of view, by use of our simple theory treating the gas, liquid and solid phases on the same ground [28,29]. This theory is based on the lattice gas model. Application of the theory to multilayer adsorption systems has given a good explanation in a qualitative way. The difference between the solid and liquid phases has been described by a long-range order parameter reflecting the periodicity of the solid. There is no finite value of the long-range order
0 1993 - Elsevier Science Publishers B.V. All rights reserved
I’. Teraoka /A
318
theoretical study of melting
parameter in the bulk liquid phase. In section 2, our theory is developed for an infinite system without surfaces (3D bulk system) and a semi-infinite system with a surface. In section 3, the phase diagram of the bulk system is discussed. In section 4, the relation between the surface and bulk meltings is discussed in detail. Section 5 is devoted to a summary.
and X,(r), respectively. These quantities are described by use of the concentration of atoms in the ith cell, Xi, and the long-range order parameter Y. as follows: Xi(~) = (Xi + 2x)/3, and xi(P)
2. Theory First we explain our basic idea on the difference between the liquid (L) and solid (S) phases in a bulk crystal (see fig. 1). The space is divided into equivalent cells. Each cell can contain no more than one atom. An atom in a cell can occupy one position among three ones denoted as (Y, /3 and y sites. If only one kind of sites is occupied, the system is a perfect solid. If sites of all kinds are equally occupied, it is a perfect liquid. A long-range order parameter is defined by the difference between occupations on sites of one kind and others. Now we denote the concentrations of atoms on (Y, p and y sites in the ith cell by Xi(a), X,(p)
,I 0
a
0B
0
=xi(Y)
=(xi-yi)/3,
(I)
where it was assumed that p and y sites in the ith cell are equivalent to each other. We assume that there is an interaction energy between two atoms, only when the two atoms occupy the nearest-neighbor cells. The interaction energies are assumed to be V, for sites of the same kind and Vi for sites of different kinds. For example, the interaction energy is V, for CY-_(y, p-/3 and y-y sites, and Vi for a-+?, P-r and y-cu sites. V, and Vi are assumed to be negative, because two atoms in the nearestneighbor cells have a lower interaction energy than in other cases. We consider an infinite system without surfaces (3D bulk system). Assuming that Xi(~), X,(p) and X,(-y> do not depend on i, the free energy of the system with the bulk concentration of atoms X is evaluated within the framework of the Bragg-Williams approximation: ~=Z((V,/2)[X(~)2+2X(P)2]
7
+v,[2X(a)X(P) +knT[(l +X(a)
-X) log X(a)
+X(P)*]} log(1 -X) +2X(P)
log
where k, is the Boltzmann constant temperature. Z is the number of neighbor cells. X(a) and X(/3> are concentrations on (Yand p sites and X(a) Fig. 1. Liquid (L) and solid (S) phases for X = 1 are schematically shown. The space is divided into equivalent cells. Each cell has three sites a, p and y sites. If all atoms occupy sites of the same kind, the system is in S, and if all sites are occupied equivalently, it is in L.
= (X+
2Y)/3,
=X(y)
= (X-
X(P)13 (2) and T is the the nearestthe average defined by
and X(P)
Y)/3,
(3)
where Y is the bulk long-range order parameter.
Y. Teraoka /A
theoretical study
by minimizing F with respect to
Y is determined y: Z(T/,-V,)Y+k,T
log[x(cu)/x(p)]
=o.
(4)
Next we consider a semi-infinite system with a surface. Taking into account experimental conditions, we neglect the gas phase. The surface layer is denoted by n = 1. The concentrations of atoms at (Y, p and y sites per cell on the nth layer, X,(a), X,(P) and X,(y) (=X,(P)>, are given by x,(a)
= (X, + 2y,)/3,
and X,(P)
=x(Y)
= (Xn
-
(5)
U/3,
where X, is the average concentration per cell on the nth layer and Y, is the long-range order parameter on the nth layer. X, and Y, are determined by minimizing the free energy of the system with respect to X, and Y,: (V0 + 2VJ(ZclXI + k,~[(1/3) -(l
+-&X,)/3 log X,(a)
-X1)
log(1 -X1)]
(V0+2~,)(ZIX,-,
+Z,X,
+ kn~[(113)
log X,(a)
+ (213) log X,(P) -p
= 0,
(6)
+ (213) log X,(P)
We show a phase diagram of an infinite system without surfaces in fig. 2, where Z = 12, V0 < 0 and VI/V, = 0.5. L + S means a mixed phase of L and S. TCL, LS) and T(LS, S) are the temperatures at which the bulk system undergoes phase transitions from L to L + S and from L + S to S, respectively, as temperature decreases. For example, we consider the case of X=X, = 0.95261. With increasing T, the system undergoes a firstorder transition from S to L + S at T,,, = TCLS, S) = 2.04 I V, I /kB, which is the melting point. The concentrations of atoms in L and S are X, = 0.93329 and X, =X0 at T,, respectively. There coexist two phases L and S between T(LS, S> and T(L, LS) = 2.074711 V, 1k,. X, (X,) increases from X, = 0.93329 (X,) to X, (Xs = 0.96470) with increasing T from T(LS, S) to T(L, LS). The ratio of L (S) increases (decreases) from 0 (1) to 1 (0). The transition at T(L, LS) is of first-order, too. Above T(L, LS), there exists only L (Y = 0). The temperature dependence of the long-range order parameter is shown in fig. 3. Y decreases with increasing T, and decreases more slowly
I
forn22,
-tZIY,)
log[X,(+X,(P)]
3. Bulk melting
2.2
(7)
+ k,T
319
+ZIX*+1)/3
-(l-X,)log(l-X,)1-p=O,
(V0 - Vr)(ZJl
of melting
= 0,
I
(8)
and (V,-
VI)(ZIY,-I
+ k,T
+Z&
+Z,Y,+r)
log[ X,( a)/XJ
p)] = 0,
for 122 2, (9)
where Z, and Z, are the numbers of the nearest-neighbor cells on the same layer and on the adjacent one (Z = Z, + 22,). The effective chemical potential F is given by /-L= (Z/3)(V, +(2/3)
+ 2V,)X+ log X(p)
k,T[(1/3)
- (1 -X)
log X(a) log(1 -X)]. (10)
Fig. 2. Phase diagram of an infinite system without surfaces (3D bulk system) for Va < 0, VI / Vu = 0.5 and Z = 12. T(LS, S) a is the melting point T,, at which the system undergoes first-order transition from S to L+S with increasing T. T(L, LS) is the first-order transition temperature from L + S to L. For X = 1, T,,, = (LS, S) = T(L, LS). The transition is also of first-order. The dashed line means X = Xa = 0.95261.
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0.901~
a.,,-
keT/iVd
2.0
! ! +
Fig. 3. Temperature dependence of the long-range order parameter for X = Xe in the 3D bulk system. At r, = T&S, S) < T(L, LS), there coexist L and S, and Xs increases from X0 to 0.964’70.At T(L, LS), Y deceases discontinuously from 0.51417 to 0.
between T(LS, S> and TCL, LSI, and then decreases discontinuously from 0.51417 to 0 at T(L, LS). For X = 1, T, = T&S, S> is equal to T(L, LS) = 2.16404 1V, I/k,, at which Y decreases discontinuously from Y = 0.5 to 0.0. The transition is of first-order. In this case, we cannot find a mixture of L and S.
4. Surface meiting Numericat calculations are carried out by assuming 2, = 4, Z, = 4 and X=X,. We determine self-consistently X, and Y, by use of eqs. (5)-(10). The depth profiles of X, for various temperatures lower than T, = 2.04 I V, 1/k, are shown in fig. 4. The concentration of atoms on the surface Iayer X, is much smaller than X = X0 (X, = 0.751, and so they are not shown there. X, increases with increasing T. X, is smaller than X,, as far as T is finite. At a given T, X, approaches X0 with an increase of n. With increasing T toward T,, there appears a region with X, = 0.9333, which is nearly equal to X, at T,,,. With approaching T, more closely, such a region expands deep inside. At T(L, IS) > T > T,, all layers except for the first few layers have concentrations equal to X,. Of course X, in-
.n
15 20 25 5 10 Fig. 4. Depth profiles of X, for various temperatures (n 2 2). The X, are - 0.75, and so they are not shown. X, and X, are the concentrations of atoms in L and S at 7’,, respectively. Cases (a)-(h) are for T = 2.0, 2.03, 2.039, 2.0399, 2.03992, 2.03994, 2.03996, and 2.03998 1V, I/k,, respectively. With approaching T, = 2.04 / V. I /kB, (a)-+(h), there appears a region with X, = X,, and it expands deep inside.
creases from 0.9333 at T, to X0 at TIL, LS). Above T(L, LS), X, =X0. As is shown in fig. 5, where the depth profiles of Y, are shown, Y, is very small in the region with X, =X,. With increasing T, Y decreases and Y, also decrease. The region with Y, = 0 expands with approaching T,. As a result we can say as follows: with approaching T,, there appears a region with X;, =X, and Y, = 0, and it expands deep inside the bulk crystal. In other words, the so-called quasi-liquid layers, in which Y, is not zero but very small, accumulates in the surface region. At T,, even the layer with n = 00 has X, =X, and Y, = 0. This is seen from the
Fig. 5. Depth profiles of Y,. With an increase of n, Y approaches 0.52530, which is indicated by the dashed line. In the region with X,, ax,_, the Y, are very small. With approaching T,, (a>+(h), the region expands deep inside. The cases (a)-(h) are the same as in fig. 4.
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ks(T.-Tl/IVol 10-l . \ \ lo+
lo+
10-4-
'\ '? '\ \ \ '\\
'*
.. \
\\
\ ‘\ 10
5
nc
Fig. 6. Temperature dependence of n,. We can find the logarithmic divergence of n, with approaching T,. This means the semi-infinite system with a surface undergoes a secondorder melting transition at T,,,.
fact that n, is linear with log[(T, - T)/ I V, II (see fig. 6), where n, is given by n at which Y, = 0.3. We can say that layers with n L 12, are in S and ones with n 2 n, are in L. This means that, at T, - O,, the liquid phase is induced near the surface and the solid phase remains deep inside the bulk crystal. This is consistent with the observation on Al(110) and Pb(ll0) [14,15]. As was above-mentioned, an infinite system without surfaces reveals a first-order melting transition. On the other hand, the semi-infinite system with a surface undergoes a second-order melting transition at the melting point. This origin is considered to be in the segregation of vacancies at the
321
surface: as is suggested by fig. 5, a layer with a smaller concentration of atoms has a smaller long-range order parameter, and so the surface layer has a smaller order parameter than other layers. In other words, it becomes a quasi-liquid layer at a temperature lower than T,. This induces successive reduction of the long-range order parameters of other layers. Next we consider the temperature dependence of Y,. For example, as is seen from fig. 7, Yi approaches zero with increasing T toward T,. Y, is proportinal to (T, - T)“.51779. We can say that the surface layer reveals a continuous transition between the liquid and solid states at T,. Moreover, we can find that other Y, decrease with power laws of (T, - T) with approaching T,. All order parameters undergo continuous transitions between liquid and solid states at T,.
5. Summary
We have investigated the surface melting on the basis of a simple theory within the framework of the Bragg-Williams approximation. We have found that an infinite system without surfaces reveals a first-order melting transition, while the semi-infinite system with a surface reveals a second-order melting transition at the bulk melting point. All layers reveals a continuous transition from the solid state to the liquid state at T,,,. This result qualitatively reproduces the experimental findings.
References
111F.A. Lindemann, Z. Phys. 14 (1910) 609. 121P. Sutherland, Philos. Mag. 32 (1891) 31. [31 P. Sutherland, Philos. Mag. 32 (1891) 215. [41 P. Sutherland, Philos. Mag. 32 (1891) 524. 151 M. Born, J. Chem. Phys. 7 (1939) 591. [61 J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955). ke(T.-T)/IVol 1o-2
IO'
Fig. 7. Temperature dependence of Yr. We can find a relation: Yr a (T, - T)“.51779. Y, decreases coninuously to zero with approaching T,.
171 J. Lennard-Jones
and A.F. Devonshire, Proc. Roy. Sot. (London) A 170 (1939) 464. 181 R.M.J. Cotterill, E.J. Jensen and W.D. Kristensen, Anharmonic Lattices, Structural Transitions and Melting, Ed. T. Riste (Noordhoff, Leiden, 1974). [91 Y.P. Varshni, Phys. Rev. B 2 (1970) 3952.
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