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Surface melting and superheating
Surface Science 294 (1993) 273-283 North-Holland
‘surface
science
Surface melting and superheating Y. Teraoka Department of Physics, University of Osaka Prefecture, Sakai 593, Japan
Received 17 February 1993; accepted for publication 2.5 May 1993
In order to obtain microscopic understanding of surface melting, in particular the role in bulk melting transitions, a semi-infinite system with a surface is investigated by use of a simple model describing gas, liquid and solid phases on the same ground. Surface melting appears and the number of quasi-liquid layers increases with increasing temperature towards the melting point Tr,,. At relatively low temperatures, the boundary between quasi-liquid and solid layers continuously moves into the crystal with increasing temperature. While approaching T,, however, it starts to move discontinuously. Such a step-like movement with temperature is found to be due to the existence of many stable states with different numbers of quasi-liquid layers. Each state corresponds to a local minimum of the free energy. Even above T,, many such states can be found. This leads to the appearance of superheating under some experimental conditions. However, superheating cannot be found, if the system is in the thermodynamical equilibrium state.
1. Introduction The melting of a solid is well known to be one of the most popular phase transitions. In order to understand the mechanism on atomic scale, several types of theories have been developed from the point of view that bulk melting is induced by a thermal instability of the crystal lattice [l-8]. However, experimental observations have not suggested the existence of such an instability [8111. Tammann has proposed a very important idea that surfaces play a key role in the bulk melting of a solid [12,13], but, for a long time, it has been difficult to experimentally confirm the idea. Recent development of ultrahigh vacuum technolsuch as medium-energy ion scattering %S), 1ow -energy electron diffraction (LEED), etc., has made it possible to observe atomically clean surfaces, and many interesting aspects of surface melting has been found. In particular, MEIS is very useful to accurately determine the depth over which the crystalline order is lost or reduced. For example, on a Pb(ll0) surface it has been found that the number of quasi-liquid layers, which are layers with reduced order, in0039-6028/93/$06.00
creases logarithmically while approaching the melting point T, (= 600.7 K) and then increases with a power law just below T, [14-161. On an Al(110) surface, a logarithmic increase of the number of quasi-liquid layers with temperature is also found [17]. On the other hand, Pb(lll1 and Al(111) surfaces have not been found to reveal such a temperature dependence [17]. The dependence of surface melting on the crystal surface is understood to be due to the difference in the coordination number of a surface atom [18]. Theoretical studies on surface melting have been carried out from two points of view; one is phenomenological [19,20] and the other is based on computer simulations, such as molecular dynamics [21,22] and Monte Carlo simulation [23]. However, we have not yet had a theory which well describes surface melting on an atomic scale. The reason is that there has not been a microscopic model describing the gas, the liquid and the solid phases in a unified way, and so we have not been able to discuss both the surface and the bulk melting on the same footing. Recently we have proposed a theory in which the gas, the liquid and the solid phases are treated on the same ground 124-261. Our theory has been
0 1993 - Elsevier Science Publishers B.V. All rights
reserved
274
Y Teraoka / Surface melting and superheating
developed on the basis of the lattice gas model. In the solid phase, each atom occupies a regularly arrayed position. On the other hand, in the gas or the liquid phase, each atom occupies a random position. In order to describe such a crystalline order, a new long-range order parameter has been introduced. The theory has successfully been applied to adsorption systems of rare gas atoms onto graphites and has well reproduced the phase diagram of a monolayer adsorption system [2.5]. There are two purposes in the present paper; one is to investigate the role of the surface melting in bulk melting transition phenomena. From a microscopic point of view, based on the abovementioned theory, surface melting is discussed within a framework of the mean-field approximation. The other one is to discuss the possibility of superheating. As is well known, the bulk melting transition is of first-order, and so the existence of undercooling and superheating is expected. Experimentally, superheating has not been found in the bulk crystal, although undercooling has been found. However, in fine Pb particles [27], superheating is found: even at 2 K above T,, fine Pb particles with only (111) surfaces do not melt, and the nonmelting state is maintained for several hours. On the other hand, Pb particles with other surfaces melt just at TM. This observation suggests that the nonmelting state is quasi-stable even above TM, but not stable. In section 2, a theory is developed for an infinite system without surfaces and a semi-infinite system with a surface. In section 3, a bulk phase diagram is discussed. In section 4, surface melting is discussed. The relation between surface and bulk melting is discussed, based on the bulk phase diagram. In section 5, conditions for the appearance of superheating are discussed. Section 6 is devoted to conclusions.
2. Theory The basic idea of our model is discussed [24,26]. The space, in which there are N atoms (or molecules), is divided into cells with equivalent volume. Each cell cannot contain more than one atom. It is assumed that each cell has some sites,
0a
@
07
Fig. 1. The space is divided into cells with equivalent volume. Each cell can contain no more than one atom. Three sites, (Y, p and y, are assumed to be in a cell (see the upper part). In a liquid phase (L), the probabilities of finding an atom at a, p and y sites are equal to one another. On the other hand, in a solid phase (S), the probability of finding an atom at one site is different from those at the other sites. The distributions of atoms in L and S are schematically shown for X = 1.
which are denoted by (Y,/3, y and so on. An atom in a cell can occupy one site among them. Such a degree of freedom represents randomness of positions occupied by an atom. If sites of all kinds are equivalently occupied, the system is in a liquid phase CL) or a gas phase (G). In a solid phase (9, the probabilities of finding an atom at sites differ from one another. If only sites of one kind are occupied, it is in a perfect solid phase (see fig. 1). Long-range order parameters describing the crystalline order are defined by the differences between the probabilities of finding an atom at two sites in a cell. If there is no long-range order parameter, the system is in L or G. The difference between G and L is quantitative. The ratio of the total number of atoms to the total number of cells is small in G and large in L and S. Now, for simplicity, three sites (Y,/3 and y, are assumed in each cell. We denote the probabilities of finding an atom at cy, p and y sites in the ith cell by XJc.u), X,(P) and X,(-y), which are described by use of the probability of finding an atom and the long-range order parameter in the ith cell, Xi and yi, as follows:
Y. Teraoka / Surface melting and superheating
X,(a) =(Xi+2y)/3
and
and
xi(P) =xi( Y) = (xi - x)/3,
(1)
where J3 and y sites in each cell were assumed to be equivalent. There are pair-wise interactions between atoms, only when the two atoms occupy the the first nearest-neighbor cells. The interaction energy is assumed to be VO,if the two atoms occupy sites of the same kind, and VI if they occupy sites of different kinds. For example, in the present case, the interaction energy is V, if two atoms occupy sites CYand (Y(p and p, or y and y), and V, if they occupy sites (Yand p (p and y, or y and (Y). V, is assumed to be negative, because there does not exist a solid if not so. The internal energy of the system per cell, E, is given by E = (l/2)
C
C C
(id P
Q
r/;j(pQ>Xi(p)Xj(Q).
(2)
yj(PQ) is the pairwise interaction energy between an atom at site P in the ith cell and another atom at site Q in the jth cell, where P and Q are (Y or /3 or y. The first summation is taken over all first nearest-neighbor pairs of cells. The second and the third summations are taken over (Y, p and y sites. According to the abovementioned assumption, vj(PQ) = V0 at P = Q and V, at P f Q, if the ith and the jth cells are first neighbors, and 0 otherwise. First we consider an infinite system without surfaces. Within a framework of the mean-field approximation, the free energy per a cell, F, can be written as follows; F= (Z/2)(&[
X(42
+2V[2X(n)X(P) +k,T[(l +X(a)
-X) log X(a)
+ 2X(13)~] +X(B)2])
log[X(cu)/X(p)]
X((Y) and X(p) are defined by
=O.
(5)
The bulk phase diagram is determined by taking into account mixed phases. Next we consider a semi-infinite system with a surface. Taking into account experimental conditions, the system is assumed not to be in thermodynamical equilibrium with G [151. The surface layer is denoted by n = 1. Each layer parallel to the surface has a two-dimensional arrayed cell. The probabilities of finding an atom at site (Y, p and y on the nth layer, X,(a), X,(p) and X,,(y), are defined by X,(a) and
= (1/3)(X,
+ 2Y,)
X,(P) =X,(y) = (1/3)(X, -Y,), (6) where X, and Y, are the probability of finding an atom in a cell and the long-range order parameter on the nth layer, respectively. The total free energy per cell belonging to the surface layer, F, is evaluated in the mean-field approximation. By minimizing F with respect to X,, and Y,, we obtain the following set of equations: (l/3)(1/,+ + k,T[ +(2/3)
21/,)(&X, -log(l -X,) log X,(P)]
log X,(p)]
(V0 - ~I)(-%Y, +
log X(P)].
Z is the number of the first neighbor cells of a cell. k, is the Boltzmann constant and T is the
X((Y) = (1/3)(x+2Y)
Z(V,-I’,)Y+k,T
+ (2/3)
log(1 -X) + 2X(P)
X(P) =X(y) = (1/3)(X-Y), (4) where X is the ratio of the total number of atoms to the total number of cells and Y is the longrange order parameter. X= 1 means that each cell is occupied by an atom. Y is determined by minimizing F with respect to Y,
(1/3)(1/,+2I/,)(zIXVr -/.L + k,T[ -log(l
(3)
temperature.
275
+
kF3T
b+wwx,(P)]
+&X2) + (l/3)
-CL log X,((Y)
= 0, -X,J
(7)
+&Jx, +ZIXn+l) + (l/3) log X,(a)
= 0
for n 2 2,
(8)
ZlY2) =
0
(9)
and (V0 - VI)(Z,Y,-I + -%lr, + ZIY,.,) +k,T log[X,(a)/X,(/?)] =0 for n22. (10)
Y. Teraoka / Surface melting and superheating
276
Z, and Z, are the numbers of the first nearestneighbor cells on the same layer and on the adjacent layer, respectively (Z = Z, + 22,). /L is the effective chemical potential. For example, it is given by /_&=(1/3)(V0+2V,)ZX+knT[-10g(1-X) +(1/3)
log X(a)
+ (213) log
xwl
for S,
(11)
and by /.L=(l/3)(V0+2V&zX+k,T[-log(l-X) + log( X/3)]
for L or G.
(12)
X,,s and Y,s in the present system are determined by use of eqs. (7)-(11).
3. Bulk phase diagram The phase diagram of an infinite system with no surface for X = 1 is shown in fig. 2, where Z = 12, V, < 0 and VI/V, = 0.5 (hereafter these values will be adopted in numerical calculations). The phase diagram over all X, can be found elsewhere [24,25]. T(L, LS) and T(LS, S) mean
the phase transition temperatures from L to a mixture of L and S and from a mixture of L and S to S, respectively, with decreasing T. As an example, we consider a case of X = 1. With increasing T, the system undergoes a phase transition from S with Y = 0.5 to L with Y = 0 at TCLS, S) = T(L, LS), which is the melting point TM = 2.16404257 I V, I/k,. Another example is a case of X = 0.9902211, whose value is denoted by X0. With increasing T, at T(L, LS) = 2.14 I V, l/k,, the system undergoes a phase transition from S with Xs to a mixture of L with X, and S with X,, where X, and X, are the concentrations in L and S, X, = 0.9874874 and X, =X0, respectively. S has also the long-range order parameter Y = 0.5029490 at T(L, LS). L has no finite order parameter. With further increasing T, X, and Xs increase along the curves T(L, LS) and T(LS, S) in the figure, respectively, and then the system undergoes another phase transition from a mixed phase, which consists of L with X, =X0 and Y = 0 and S with X, = 0.9923305 and Y = 0.502244, to L with X, = X, and Y = 0 at T&S, S) = 2.1452289 I V, l/k,. There coexist two phases, L and S, in the range TCLS, S) I T 5 T(L, LS). The melting point TM is given by TKS,
2.2
1 I
1k.T/IVol
9.
The temperature dependence of the bulk long-range order parameter Y in the thermodynamical equilibrium state is shown in fig. 3. Y is
Y 1 .o
T
Fig. 2. Phase diagram of an infinite system with no surfaces for X= 1. Z= 12, V,
o.o* 2.0
T(LS,S)ai k-T(LS,S) =T(L,LS) T(L,LS)j j , I If I, I I 1ksT/IVol 2.1 2.2
Fig. 3. Temperature dependences of the bulk long-range order parameter Y. For X= X,,, two phases coexist, L and S at T&S, S) I T I T(L, LS).
Y. Teraoka / Surface melting and superheating
equal to X at T = 0. Y decreases with increasing T. For X = 1, Y decreases discontinuously from 0.5 to 0 at TM. This can be proved exactly. For X = X0, Y decreases more slowly for TCLS, S) I T I T(L, LS), compared with the variation just below TCLS, S). We find Y = 0 above TCL, LS).
TY"
a
AT=O.O
0 .5-X=1
zo=12 4. Surface melting We consider a semi-infinite system with a surface. The first 59 layers are differently treated than other layers in numerical calculations. For the present system, we can find many solutions satisfying eqs. (7)-(11). One solution corresponds to one state and another solution does to another state. The states have different depth profiles of X,, and Y, from one another. Y, takes various values between 0 and the bulk value Y, and so we cannot say which layer is in the liquid state or in the solid state. Of course we obtain Y = 0 in L for the infinite system with no surface. Now we introduce a simple criterion: the nth layer with Y, < Y/2 (Y, 2 /2) is a liquid (solid) layer. Hereafter a liquid layer with a finite order parameter is called a quasi-liquid layer. The most stable state could be found by comparing the free energies of states with one another. In order to distinguish one state from another one, a parameter n, is defined as follows; the nth layer is in a liquid or a quasi-liquid state if IZI +,, and it is in a solid state if IZ2 IZ~+ 1, where n, is an integer and nonnegative. 2 (Z = Z, + 22,) is fixed to 12 in numerical calculations. In order to investigate the orientation dependence of the surface melting on the crystal surface, the surface melting is discussed as a function of Z,.
o.o-
0..
10
20
30
:7::: _*-- .-
4;
:y,,
_.-:$-_*-
.
F,,“o,4,__.--*--‘AT=0
+__.__.__.__.__.__.__.-.__.__.
4 1O*"--*--.-_.__e_ -.-.__J 0" 7" -.__ .-_* ~ x=1 zo=12 f Fig. 4. Z, = 12 and X = 1. (a) Depth profiles of Y, for various n,,s (n, = 0,. ,201 at AT = 0. AT is defined by AT = k,(T TM)/ I VO I. Y, with with n I n, is equal to 0 and Y, with n 2 na + 1 is equal to Y = 0.5. For example, if no = 5, the first five layers are in the liquid state and the following layers are in the solid state. The so-called quasi-liquid layers cannot be found in this case. (b) ho dependences of the free energy at AT = - 10m5, 0 and lo-‘. The energy scale is arbitrary. The states with na = 0 and m are the most stable at AT < 0 and AT > 0, respectively. Just at AT = 0, all states have the same free energies.
4.1. x= 1 All Y,s are only determined by eqs. (9) and (10). In order to obtain the physical understanding of surface melting, the simplest case is first discussed. It is a case of Z, = 12, where there is no interaction between layers parallel to the surface. An example of the depth profile of Y, for AT = 0 is shown in fig. 4a, where AT is defined by use of the melting point TM as follows; AT =
k,(T - TM)/ I V, I. The solutions are found for all n,s (n, 2 0, n, = 0 means the state in which all layers are in the solid state), as far as T does not exceed 2.18528000 I V,, I /kB, above which there exists only one solution with Y, = 0 for all ns. Of course there are no solutions with a finite Y in the bulk crystal above this temperature. For Z, = 12, there are no quasi-liquid (0 < Y, < Y/2) layers. The system consists of only liquid (Y, = 0)
278
Y. Teraoka / Surface melting and superheating
a
AT=O.O
0.
n
0.
_I
0-5
F’ivoi.. AT=0 *-..__.__.__* _.__.__. 4.. 1o-5
T‘
TYn
1f-jl L__*__ , l
.’ x=1
a
AT=O.U
0.5 _________t *
l ___*_.--e--*--*e -a
,.
with izes that differ by 1 from each other is given by the difference between F, and F,, which are the free energies of one liquid and solid monolayer, respectively. F, - F, is positive for AT < 0, and so the free energy of the system increases linearly with the number of the liquid layers no. Just at AT = 0, the free energy does not depend on IZ@,because FL is equa1 to Fs. For AT > 0, the lowest free energy state is one with n, = m. In this case, the free energy decreases linearly with n,, because F, - F, is negative.
b-----d
t
0.
-*,-.-‘* --0
Zo=l 1 i Fig. 5. 2, = 11 and X = 1. (a) Depth profiles of Y, at AT = 0. a-g correspond to na = 3, 4, 5, 6, 7, 8 and 9, respectively. States with na _<2 cannot be found, and states with n,, > 10 can be found. Y, with n 5 na is finite and smaller than Y/2 = 0.25, and Y, with n 2 n, + 1 is larger than 0.25. For example, in case a, the first three layers are in the quasi-liquid state. (b) n, dependences of the free energy at AT = - 10e5, 0 and 10W5.At AT = - lo-‘, the state with no = 4 is the most stable. Just at AT =O, the free energy approaches the constant value with an increase of n,. At AT = lo-“, F decreases with an increase of n,.
and solid (Y, = Y> layers. There is a clear-cut boundary between the n,th and the (n,, + 0th layers. It is a boundary between the liquid and the solid state regions. The free energy as a function of ~1~is shown in fig. 4b. For AT < 0, a state with no = 0 has the lowest free energy. With an increase of no, the free energy increases Iinearly. The free energy difference between states
‘~no *
4 AT=0 l-_* \\ 1o-7 I. a, -*-*-•__*-_* f .'Zo=4 X=1 b Fig. 6. Za = 4 and X = 1. (a) Depth profiles of Y, at AT = 0. a-g correspond to no = 13, 14, 15, 16, 17, 18 and 19, respectively. States with no I 12 cannot be found, and states with ns 2 20 can be found. fb) na dependences of F at AT = - 1O-6 and 0. At AT = - lo-“, only three states with n, = 12, 13 and 14 can be found. The most stable state is the one with no = 13. Just at AT = 0, F decreases with an increase of no and approaches the constant value. Solutions with n,r I 12 cannot be found. At AT = 10W6,we cannot find states with a finite no.
219
Y. Teraoka / Surface melting and superheating
Next we discuss the cases of 2, = 11 and 4. Examples of depth profiles of Y, at AT = 0 are shown in figs. 5a and 6a. All profiles in each case are very similar to one another. Y, increases with an increase of )2 and then approaches Y = 0.5. Its variation becomes very large in the neighborhood of the boundary between the quasi-liquid and the solid layers. With a decrease of Z,, the variation becomes slower and the width of the boundary region becomes broader. Such a situation comes from the decrease of the coordiation number at the surface. The free energies F for 2, = 11 and 4 as a function of no are shown in figs. 5b and 6b, respectively. In the case of 2, = 11, the state with n,, = 4 gives us the minimum value of F at AT = - 10Pi5. F increases with increasing n, beyond n,, = 4. Just at AT = 0, with increasing +,, F decreases and approaches a constant value. On the other hand, at AT > 0, F decreases with increasing n,, and it does not reach a minimum value. For 2, = 11, we cannot obtain solutions with n, = 0, 1 and 2. In the case of Z, = 4, the state with IZ~= 13 has the lowest energy at AT = - 10e6. The solutions can be found only for eta = 12, 13 and 14. Just at AT = 0, F decreases and approaches a constant value with increasing n,. For no I 12, we cannot find solutions. At AT = 10P6, no solutions with a finite value of it,, can be found. The system in the thermodynamical equilibrium state, i.e. the most stable state, has the lowest free energy. For Z, = 12, the state with no = 0 is the most stable at AT < 0 and one with IZ~= m at AT > 0, and so a transition occurs from
thestatewithn,=Otoonewithn,=mat AT= 0. The depth profiles of Y, in the lowest free energy state for several negative AT, are shown in fig. 7a for Z, = 11 and in fig. 7b for Z, = 4, respectively. With an increase of T towards T, (a + b + c+ d + e +f), a0 increases, in other words, the boundary between the quasi-liquid and the solid regions goes deep inside the crystal. At a sufficiently large n, Y, approaches Y at a given temperature. Y itself decreases with increasing T. With a decrease of Z,, the number of quasi-liquid layers becomes larger. Surface melting more proceeds. Now, in order to investigate the proceeding of surface melting, we define a new quantity n,, at which Yn, is equal to half of Y. ~zc is given by interpolating Y, and Y, + 1, between which there is Y,,. The temperature dependence of n, in the most stable state is shown in fig. 8. n, increases with an increase of T towards TM and is then infinite just at AT = 0. The most striking finding is that step-like increases of n, appear with approaching TM, while the increase is continuous at low temperatures. The behavior depends on Z,. With a decrease of Z,, the region of the continuous increase expands to the high temperature side and the variation of n, becomes larger. Just below TM, nc increases by 1 when -AT reduces by about a factor lo-‘.’ and 1O-o.4 for Z, = 11 and 4, respectively. In the temperature region, nc is a half-integer. Such a step-like behavior has been found in surface segregation of segregating alloys [281. As a result, with increasing T towards TM, layers parallel to the surface undergo succes-
b
a TYn
TYn 0.
Lj..-----
0.0 Fig. 7. Temperature
dependence correspond
--
:$ffjjKy 10
20
30
I 4;
of depth profiles of Y, in the most stable state for X= 1. (a) Z, = 11 and (b) Z,, = 4. a-f to AT = - lo-‘, - 10m3, - 10m4, - 10m5, - 10e6 and - lo-‘, respectively.
I’. Teraoka / Surfnce melting and superheating
280
nc 15i
zo=4 X=1
#
10 Zo=l 1
5
’
j---/ii
-AT
ii-’ 1i3 1 Ii’ 1 o5 1 oa 1o-’
1
Fig. 8. Temperature dependences of nC in the most stable state for 2, = 11 and 4 at X = 1. The abscissa is on a logarithmic scale. n, increases with increasing T towards TM. At low temperatures, the increase of n, is continuous, however, a step-like increase appears with approaching TM. With a decrease of Z,, the region of a continuous increase of no starts to expand to the high temperature side.
sive transitions from a quasi-liquid state to a solid state, in other words, the number of quasi-liquid layers increases. Just below TM, the increase becomes always logarithmically step-like. At T,, an infinite number of liquid layers are induced near the surface and solid layers remain deep inside the bulk crystal. Between liquid and solid layers, there exist a finite number of quasi-liquid layers. The step-like character in the temperature dependence of IZ~ originates from an existence of many stable solutions, which have similar depth profiles of Y,, but have lz,,s that differ from one another by an integer. Other stable solutions do
not exist. In the neighborhood of T,, all profiles of solutions with sufficiently large n,s are similar to one another except for the difference in IZ”. An increase of T makes one solution unstable and makes another solution with no + 1 to be stable, and so on. Therefore an existence of the step-like temperature dependence does not depend on our somewhat artificial definition of the quasi-liquid layer and does not depend on the definition of ~zc, too. 4.2. Xc1 As an example, the surface melting of a semiinfinite system with X=X, is discussed (TM = T(LS, S) = 2.14 I V, I/k,). We find only one solution at relatively low temperatures, but many solutions with different a0s appear with approaching T,. In the case of Z, = 11, we obtain only one solution with n, = 1 at AT = - 10e2, and only one solution with IZ,,= 2 for AT = - 10P3. However, we obtain an infinite number of solutions with n, 2 3 for AT = - 10P4, - lo-” and - 10e6, and an infinite number of solutions with n, 2 4 for AT = - 10m7. With approaching T,, only states with more quasi-liquid layers can been found. The depth profiles of X, and Y, in the thermodynamical equilibrium state are shown in fig. 9, where X, and Xs are the concentrations of L and S in the bulk crystal at T,. With an increase of T towards T, (a + b + c + d --+e -+ f), the region with X,, equal to X, = 0.9875 at T, appears near the surface, and then expands
b
a
o*ggo:~F-” 0.9851' Fig. 9. Temperature
2; 4
10
dependence of depth proflies in the most stable state for X=X,,. (a) X,, and (b) Y,. a-f AT= -lo-‘, -10-3, - 10e4, - 10V5, -10-s and - lo-‘, respectively.
correspond
to
Y Teraoka / Surface melting and superheating
towards a larger n. In such a region, Y, is nearly zero. This means that quasi-liquid layers are induced near the surface and the number increases with approaching TM. At TM, an infinite number of liquid layers are induced and solid layers remain deep inside the bulk crystal.
5. Superheating In order to investigate the possibility of superheating, we discuss physical properties of a semiinfinite system with a surface at positive ATs. An infinite number of states with different n,s can be found at a positive AT. Now we consider the case of X = 1. For Z, = 12, we can find solutions for no 2 0 at T smaller than the critical value, 2.18528 I V,, I/kB (AT I 0.021237431, above which there are no solutions with finite order parameters. The bulk system has not a solution with a finite order parameter, too. Of course, for a positive AT, all layers melt in the most stable state. However, all solutions with n, 2 0 correspond to local minima of the free energy, and so the corresponding states are stable. For Z, smaller than 12, we can also find a similar situation. The corresponding states have very similar profiles of Y, to one another. However, the states with small n,s are excluded, because a decrease of the coordination number at the surface makes strong influence on states with a small no, in particular the position of the boundary between the quasi-liquid and the solid layers. A typical example is the case of Z, = 12, in which case there is no decrease of the coordination number at the surface, and so the profiles have the same forms with one another. With a decrease of Z,, stable solutions can be found only at smaller ATs. For example, we can find solutions at AT = 10e5 and not at AT = low4 for Z, = 11, and we can find solutions at AT = lo-’ and not at AT = lop6 for Z, = 4. Therefore it is considered that a critical temperature, above which solutions with finite order parameters are not found, decreases towards TM with a decrease of Z,, and we suppose that it becomes to be equal to TM for Z, = 0. For a given positive AT, the free energy decreases with an increase of n,, as is seen in figs. 4b, 5b
281
and 6b. For a sufficiently large n,, the free energy decreases linearly. The coefficient is negative and is common for all Zas, because the surface effect disappears for a large ~~ and the coefficients are determined only by F, -F,. With increasing AT, the free energy starts to decrease more rapidly due to a decrease of F, - F,. In the case of X = X0, similar results are obtained. For Z, = 11, we can find solutions with n, 2 3 at AT = 10-4, and no solutions at AT = 10P3. Comparing the case of X=X,, with one of X = 1, we can say that, in the case of X = X,,, solutions are found up to larger ATs. As was discussed in section 4, an increase of ~tc with temperature is continuous at low temperatures. In this temperature region, the surface melting proceeds continuously. However, the increase of n, becomes step-like while approaching TM, so that the proceeding of surface melting just below TM means successively discontinuous transitions from a state with IZ~to one with one more quasi-liquid layer and to one with two more quasi-liquid layers, etc. for all Z,s. n,-., which is a half-integer just below T,, increases discontinuously by 1 in each transition. If the system is kept in the thermodynamical equilibrium state at each temperature, nc is infinite at T,, in other words, bulk melting appears at TM. If not so, nc can be finite at and even above TM, and bulk melting does not appear at T,. Therefore, in the latter case, the system is expected to reveal superheating. From an experimental point of view, this condition for appearance of the superheating is always satisfied. The reason is as follows: nc increases logarithmically step-like with an increase of AT. As far as the raising rate of the temperature is not infinitesimal, the proceeding of surface melting cannot follow the raising of the temperature, and so the system is not able to remain in the thermodynamical equilibrium state and n, remains finite even above TM. This leads to the appearance of superheating. At a given raising rate of temperature, a system with a smaller Z, remains in the thermodynamical equilibrium state up to higher temperatures. Above TM, the decrease of the free energy due to an increase of n, by 1 becomes larger with increasing T, but the decreasing amount is not depen-
282
Y. Teraoka / Surface melting and superheating
dent on Z,, because it is only given by F, - F, at a given temperature. As concerned with an infinite system with no surface, it can be expected that the system always reveals superheating, because there is no surface melting. Unfortunately, such a system cannot be experimentally realized, so that no superheating has been observed so far. A real sample has always a few surfaces with different Z,s from one another. Surface melting proceeds more for surfaces with smaller Z,s and less for ones with larger Z,s with increasing T. At the surface with a small Z,, the first many layers are in quasi-liquid states just at TM. Therefore it seems that bulk melting appears at TM from an experimental point of view. On the other hand, for a large Z,, only the first few layers are in quasi-liquid states and other layers remain in solid states just at TM, and so superheating appears. If a sample which is cut only by surfaces with large Z,s is prepared, superheating can be expected to observe. Such an example is found in fine Pb particles with only (111) surfaces [27]. They remain in a solid phase at 2 K above TM for several hours. Other fine Pb particles do not reveal superheating. Our result can explain this observation. The superheating state is not the most stable state, and the system changes to the thermodynamical equilibrium state, in other words, to the liquid phase, after a sufficiently long time. if a sample size is finite, the relaxation time is also finite. The superheating state exists up to the temperature, below which we can find an infinite number of states with different ltOs from one another. In the case of X= 1, the states exist up to the maximum temperature of an existence of the bulk long-range order parameter, T = 2.18528000 I V. I/k,, for Z, = 12. With a decrease of Z,, the temperature decreases.
minima of the free energy of a semi-infinite system with a surface. The solutions have very similar depth profiles to one another except for the boundary’s position between the quasi-liquid and the solid layers. In the thermodynamic equilibrium state, with increasing T towards T,, surface melting proceeds and the number of quasi-liquid layers increases, and then bulk melting appears at T,. Superheating does not appear. However, due to a logarithmically step-like increase of the number of quasi-liquid layers with the temperature, the proceeding of surface melting cannot follow the raising of the temperature, as far as the raising rate is not infinitesimal. Therefore superheating is expected to appear under general experimental conditions. However, superheating is not experimentally observed. This seems to conflict with the above-expected appearance of superheating. Our opinion on this confliction is as follows; the confirmation of existence of several hundreds or thousands or millions quasi-liquid layers near the surface is experimentally very difficult, so that bulk melting seems to appear at TM.
References [l] [2] [3] [4] [S] [6] [7] [8]
[9] [lo]
6. Conclusion We have investigated the role of surface melting in bulk melting transitions by use of a simple model developed on the basis of the lattice gas model with short-range pairwise interactions. We have found many solutions corresponding to local
[ll] [12] [13] [14] [15]
F.A. Lindemann, Z. Phys. 14 (1910) 609. P. Sutherland, Philos. Mag. 32 (1891) 31. P. Sutherland, Philos. Mag. 32 (1891) 215. P. Sutherland, Philos. Mag. 32 (1891) 524. M. Born, J. Chem. Phys. 7 (1939) 591. J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955). J. Lennard-Jones and A.F. Devonshire, Proc. Roy. Sot. (London) A 170 (1939) 464. R.M.J. Cotterill, E.J. Jensen and W.D. Kristensen, Anharmonic Lattices, Structural Transitions and Melting, Ed. T. Riste (Noordhoff, Leiden, 1974). Y.P. Varshni, Phys. Rev. B 2 (1970) 3952. J.K. Kristensen and R.M.J. Cotterill, Philos. Mag. 36 (1977) 437. D.P. Woodruff, The Solid-Liquid Interface (Cambridge University Press, London, 1973). F. Tammann, Z. Phys. Chem. 68 (1910) 205. F. Tammann, Z. Phys. 11 (1910) 609. J.W.M. Frenken and J.F. van der Veen, Phys. Rev. Lett. 54 (1985) 134. J.W.M. Frenken, P.M.J. Maree and J.F. van der Veen, Phys. Rev. B 34 (1986) 7506.
Y Teraoka / Surface melting and superheating [16] B. Pluis, T.N. Taylor, D. Frenkel and J.F. Van der Veen,
[17]
[18] [19] [20] [21]
Phys. Rev. B 40 (1989) 1353. A.W. Denier van der Gon, R.J. Smith, J.-M. Gay, D.J. O’Connor and J.F. van der Veen, Surf. Sci. 227 (1990) 143. B. Pluis, A.W. Denier van der Gon, J.W.M. Frenken and J.F. van der Veen, Phys. Rev. Lett. 59 (1987) 2678. R. Lipowsky, Ferroelectrics 73 (1987) 69. B. Pluis, D. Frenkel and J.F. van der Veen, to be published. P. Stoltze, J. Chem. Phys., in press.
283
[22] E.T. Chen, R.N. Barnett and U. Landman, Phys. Rev. B 41 (1990) 439. [23] A.W. Denier van der Gon, D. Frenkel, J.W.M. Frenken, R.J. Smith and P. Stoltze, to be published. [24] Y. Teraoka and T. Seto, Surf. Sci. 278 (1992) 202. [25] Y. Teraoka and T. Seto, Surf. Sci. 283 (1993) 371. [26] Y. Teraoka, Surf. Sci. 281 (1993) 317. [27] J.J. Metois and J.C. Heyraud, J. Phys. (Paris) 50 (1989) 3175. [28] Y. Teraoka and T. Seto, Surf. Sci. 255 (1991) 209.
‘surface
science
Surface melting and superheating Y. Teraoka Department of Physics, University of Osaka Prefecture, Sakai 593, Japan
Received 17 February 1993; accepted for publication 2.5 May 1993
In order to obtain microscopic understanding of surface melting, in particular the role in bulk melting transitions, a semi-infinite system with a surface is investigated by use of a simple model describing gas, liquid and solid phases on the same ground. Surface melting appears and the number of quasi-liquid layers increases with increasing temperature towards the melting point Tr,,. At relatively low temperatures, the boundary between quasi-liquid and solid layers continuously moves into the crystal with increasing temperature. While approaching T,, however, it starts to move discontinuously. Such a step-like movement with temperature is found to be due to the existence of many stable states with different numbers of quasi-liquid layers. Each state corresponds to a local minimum of the free energy. Even above T,, many such states can be found. This leads to the appearance of superheating under some experimental conditions. However, superheating cannot be found, if the system is in the thermodynamical equilibrium state.
1. Introduction The melting of a solid is well known to be one of the most popular phase transitions. In order to understand the mechanism on atomic scale, several types of theories have been developed from the point of view that bulk melting is induced by a thermal instability of the crystal lattice [l-8]. However, experimental observations have not suggested the existence of such an instability [8111. Tammann has proposed a very important idea that surfaces play a key role in the bulk melting of a solid [12,13], but, for a long time, it has been difficult to experimentally confirm the idea. Recent development of ultrahigh vacuum technolsuch as medium-energy ion scattering %S), 1ow -energy electron diffraction (LEED), etc., has made it possible to observe atomically clean surfaces, and many interesting aspects of surface melting has been found. In particular, MEIS is very useful to accurately determine the depth over which the crystalline order is lost or reduced. For example, on a Pb(ll0) surface it has been found that the number of quasi-liquid layers, which are layers with reduced order, in0039-6028/93/$06.00
creases logarithmically while approaching the melting point T, (= 600.7 K) and then increases with a power law just below T, [14-161. On an Al(110) surface, a logarithmic increase of the number of quasi-liquid layers with temperature is also found [17]. On the other hand, Pb(lll1 and Al(111) surfaces have not been found to reveal such a temperature dependence [17]. The dependence of surface melting on the crystal surface is understood to be due to the difference in the coordination number of a surface atom [18]. Theoretical studies on surface melting have been carried out from two points of view; one is phenomenological [19,20] and the other is based on computer simulations, such as molecular dynamics [21,22] and Monte Carlo simulation [23]. However, we have not yet had a theory which well describes surface melting on an atomic scale. The reason is that there has not been a microscopic model describing the gas, the liquid and the solid phases in a unified way, and so we have not been able to discuss both the surface and the bulk melting on the same footing. Recently we have proposed a theory in which the gas, the liquid and the solid phases are treated on the same ground 124-261. Our theory has been
0 1993 - Elsevier Science Publishers B.V. All rights
reserved
274
Y Teraoka / Surface melting and superheating
developed on the basis of the lattice gas model. In the solid phase, each atom occupies a regularly arrayed position. On the other hand, in the gas or the liquid phase, each atom occupies a random position. In order to describe such a crystalline order, a new long-range order parameter has been introduced. The theory has successfully been applied to adsorption systems of rare gas atoms onto graphites and has well reproduced the phase diagram of a monolayer adsorption system [2.5]. There are two purposes in the present paper; one is to investigate the role of the surface melting in bulk melting transition phenomena. From a microscopic point of view, based on the abovementioned theory, surface melting is discussed within a framework of the mean-field approximation. The other one is to discuss the possibility of superheating. As is well known, the bulk melting transition is of first-order, and so the existence of undercooling and superheating is expected. Experimentally, superheating has not been found in the bulk crystal, although undercooling has been found. However, in fine Pb particles [27], superheating is found: even at 2 K above T,, fine Pb particles with only (111) surfaces do not melt, and the nonmelting state is maintained for several hours. On the other hand, Pb particles with other surfaces melt just at TM. This observation suggests that the nonmelting state is quasi-stable even above TM, but not stable. In section 2, a theory is developed for an infinite system without surfaces and a semi-infinite system with a surface. In section 3, a bulk phase diagram is discussed. In section 4, surface melting is discussed. The relation between surface and bulk melting is discussed, based on the bulk phase diagram. In section 5, conditions for the appearance of superheating are discussed. Section 6 is devoted to conclusions.
2. Theory The basic idea of our model is discussed [24,26]. The space, in which there are N atoms (or molecules), is divided into cells with equivalent volume. Each cell cannot contain more than one atom. It is assumed that each cell has some sites,
0a
@
07
Fig. 1. The space is divided into cells with equivalent volume. Each cell can contain no more than one atom. Three sites, (Y, p and y, are assumed to be in a cell (see the upper part). In a liquid phase (L), the probabilities of finding an atom at a, p and y sites are equal to one another. On the other hand, in a solid phase (S), the probability of finding an atom at one site is different from those at the other sites. The distributions of atoms in L and S are schematically shown for X = 1.
which are denoted by (Y,/3, y and so on. An atom in a cell can occupy one site among them. Such a degree of freedom represents randomness of positions occupied by an atom. If sites of all kinds are equivalently occupied, the system is in a liquid phase CL) or a gas phase (G). In a solid phase (9, the probabilities of finding an atom at sites differ from one another. If only sites of one kind are occupied, it is in a perfect solid phase (see fig. 1). Long-range order parameters describing the crystalline order are defined by the differences between the probabilities of finding an atom at two sites in a cell. If there is no long-range order parameter, the system is in L or G. The difference between G and L is quantitative. The ratio of the total number of atoms to the total number of cells is small in G and large in L and S. Now, for simplicity, three sites (Y,/3 and y, are assumed in each cell. We denote the probabilities of finding an atom at cy, p and y sites in the ith cell by XJc.u), X,(P) and X,(-y), which are described by use of the probability of finding an atom and the long-range order parameter in the ith cell, Xi and yi, as follows:
Y. Teraoka / Surface melting and superheating
X,(a) =(Xi+2y)/3
and
and
xi(P) =xi( Y) = (xi - x)/3,
(1)
where J3 and y sites in each cell were assumed to be equivalent. There are pair-wise interactions between atoms, only when the two atoms occupy the the first nearest-neighbor cells. The interaction energy is assumed to be VO,if the two atoms occupy sites of the same kind, and VI if they occupy sites of different kinds. For example, in the present case, the interaction energy is V, if two atoms occupy sites CYand (Y(p and p, or y and y), and V, if they occupy sites (Yand p (p and y, or y and (Y). V, is assumed to be negative, because there does not exist a solid if not so. The internal energy of the system per cell, E, is given by E = (l/2)
C
C C
(id P
Q
r/;j(pQ>Xi(p)Xj(Q).
(2)
yj(PQ) is the pairwise interaction energy between an atom at site P in the ith cell and another atom at site Q in the jth cell, where P and Q are (Y or /3 or y. The first summation is taken over all first nearest-neighbor pairs of cells. The second and the third summations are taken over (Y, p and y sites. According to the abovementioned assumption, vj(PQ) = V0 at P = Q and V, at P f Q, if the ith and the jth cells are first neighbors, and 0 otherwise. First we consider an infinite system without surfaces. Within a framework of the mean-field approximation, the free energy per a cell, F, can be written as follows; F= (Z/2)(&[
X(42
+2V[2X(n)X(P) +k,T[(l +X(a)
-X) log X(a)
+ 2X(13)~] +X(B)2])
log[X(cu)/X(p)]
X((Y) and X(p) are defined by
=O.
(5)
The bulk phase diagram is determined by taking into account mixed phases. Next we consider a semi-infinite system with a surface. Taking into account experimental conditions, the system is assumed not to be in thermodynamical equilibrium with G [151. The surface layer is denoted by n = 1. Each layer parallel to the surface has a two-dimensional arrayed cell. The probabilities of finding an atom at site (Y, p and y on the nth layer, X,(a), X,(p) and X,,(y), are defined by X,(a) and
= (1/3)(X,
+ 2Y,)
X,(P) =X,(y) = (1/3)(X, -Y,), (6) where X, and Y, are the probability of finding an atom in a cell and the long-range order parameter on the nth layer, respectively. The total free energy per cell belonging to the surface layer, F, is evaluated in the mean-field approximation. By minimizing F with respect to X,, and Y,, we obtain the following set of equations: (l/3)(1/,+ + k,T[ +(2/3)
21/,)(&X, -log(l -X,) log X,(P)]
log X,(p)]
(V0 - ~I)(-%Y, +
log X(P)].
Z is the number of the first neighbor cells of a cell. k, is the Boltzmann constant and T is the
X((Y) = (1/3)(x+2Y)
Z(V,-I’,)Y+k,T
+ (2/3)
log(1 -X) + 2X(P)
X(P) =X(y) = (1/3)(X-Y), (4) where X is the ratio of the total number of atoms to the total number of cells and Y is the longrange order parameter. X= 1 means that each cell is occupied by an atom. Y is determined by minimizing F with respect to Y,
(1/3)(1/,+2I/,)(zIXVr -/.L + k,T[ -log(l
(3)
temperature.
275
+
kF3T
b+wwx,(P)]
+&X2) + (l/3)
-CL log X,((Y)
= 0, -X,J
(7)
+&Jx, +ZIXn+l) + (l/3) log X,(a)
= 0
for n 2 2,
(8)
ZlY2) =
0
(9)
and (V0 - VI)(Z,Y,-I + -%lr, + ZIY,.,) +k,T log[X,(a)/X,(/?)] =0 for n22. (10)
Y. Teraoka / Surface melting and superheating
276
Z, and Z, are the numbers of the first nearestneighbor cells on the same layer and on the adjacent layer, respectively (Z = Z, + 22,). /L is the effective chemical potential. For example, it is given by /_&=(1/3)(V0+2V,)ZX+knT[-10g(1-X) +(1/3)
log X(a)
+ (213) log
xwl
for S,
(11)
and by /.L=(l/3)(V0+2V&zX+k,T[-log(l-X) + log( X/3)]
for L or G.
(12)
X,,s and Y,s in the present system are determined by use of eqs. (7)-(11).
3. Bulk phase diagram The phase diagram of an infinite system with no surface for X = 1 is shown in fig. 2, where Z = 12, V, < 0 and VI/V, = 0.5 (hereafter these values will be adopted in numerical calculations). The phase diagram over all X, can be found elsewhere [24,25]. T(L, LS) and T(LS, S) mean
the phase transition temperatures from L to a mixture of L and S and from a mixture of L and S to S, respectively, with decreasing T. As an example, we consider a case of X = 1. With increasing T, the system undergoes a phase transition from S with Y = 0.5 to L with Y = 0 at TCLS, S) = T(L, LS), which is the melting point TM = 2.16404257 I V, I/k,. Another example is a case of X = 0.9902211, whose value is denoted by X0. With increasing T, at T(L, LS) = 2.14 I V, l/k,, the system undergoes a phase transition from S with Xs to a mixture of L with X, and S with X,, where X, and X, are the concentrations in L and S, X, = 0.9874874 and X, =X0, respectively. S has also the long-range order parameter Y = 0.5029490 at T(L, LS). L has no finite order parameter. With further increasing T, X, and Xs increase along the curves T(L, LS) and T(LS, S) in the figure, respectively, and then the system undergoes another phase transition from a mixed phase, which consists of L with X, =X0 and Y = 0 and S with X, = 0.9923305 and Y = 0.502244, to L with X, = X, and Y = 0 at T&S, S) = 2.1452289 I V, l/k,. There coexist two phases, L and S, in the range TCLS, S) I T 5 T(L, LS). The melting point TM is given by TKS,
2.2
1 I
1k.T/IVol
9.
The temperature dependence of the bulk long-range order parameter Y in the thermodynamical equilibrium state is shown in fig. 3. Y is
Y 1 .o
T
Fig. 2. Phase diagram of an infinite system with no surfaces for X= 1. Z= 12, V,
o.o* 2.0
T(LS,S)ai k-T(LS,S) =T(L,LS) T(L,LS)j j , I If I, I I 1ksT/IVol 2.1 2.2
Fig. 3. Temperature dependences of the bulk long-range order parameter Y. For X= X,,, two phases coexist, L and S at T&S, S) I T I T(L, LS).
Y. Teraoka / Surface melting and superheating
equal to X at T = 0. Y decreases with increasing T. For X = 1, Y decreases discontinuously from 0.5 to 0 at TM. This can be proved exactly. For X = X0, Y decreases more slowly for TCLS, S) I T I T(L, LS), compared with the variation just below TCLS, S). We find Y = 0 above TCL, LS).
TY"
a
AT=O.O
0 .5-X=1
zo=12 4. Surface melting We consider a semi-infinite system with a surface. The first 59 layers are differently treated than other layers in numerical calculations. For the present system, we can find many solutions satisfying eqs. (7)-(11). One solution corresponds to one state and another solution does to another state. The states have different depth profiles of X,, and Y, from one another. Y, takes various values between 0 and the bulk value Y, and so we cannot say which layer is in the liquid state or in the solid state. Of course we obtain Y = 0 in L for the infinite system with no surface. Now we introduce a simple criterion: the nth layer with Y, < Y/2 (Y, 2 /2) is a liquid (solid) layer. Hereafter a liquid layer with a finite order parameter is called a quasi-liquid layer. The most stable state could be found by comparing the free energies of states with one another. In order to distinguish one state from another one, a parameter n, is defined as follows; the nth layer is in a liquid or a quasi-liquid state if IZI +,, and it is in a solid state if IZ2 IZ~+ 1, where n, is an integer and nonnegative. 2 (Z = Z, + 22,) is fixed to 12 in numerical calculations. In order to investigate the orientation dependence of the surface melting on the crystal surface, the surface melting is discussed as a function of Z,.
o.o-
0..
10
20
30
:7::: _*-- .-
4;
:y,,
_.-:$-_*-
.
F,,“o,4,__.--*--‘AT=0
+__.__.__.__.__.__.__.-.__.__.
4 1O*"--*--.-_.__e_ -.-.__J 0" 7" -.__ .-_* ~ x=1 zo=12 f Fig. 4. Z, = 12 and X = 1. (a) Depth profiles of Y, for various n,,s (n, = 0,. ,201 at AT = 0. AT is defined by AT = k,(T TM)/ I VO I. Y, with with n I n, is equal to 0 and Y, with n 2 na + 1 is equal to Y = 0.5. For example, if no = 5, the first five layers are in the liquid state and the following layers are in the solid state. The so-called quasi-liquid layers cannot be found in this case. (b) ho dependences of the free energy at AT = - 10m5, 0 and lo-‘. The energy scale is arbitrary. The states with na = 0 and m are the most stable at AT < 0 and AT > 0, respectively. Just at AT = 0, all states have the same free energies.
4.1. x= 1 All Y,s are only determined by eqs. (9) and (10). In order to obtain the physical understanding of surface melting, the simplest case is first discussed. It is a case of Z, = 12, where there is no interaction between layers parallel to the surface. An example of the depth profile of Y, for AT = 0 is shown in fig. 4a, where AT is defined by use of the melting point TM as follows; AT =
k,(T - TM)/ I V, I. The solutions are found for all n,s (n, 2 0, n, = 0 means the state in which all layers are in the solid state), as far as T does not exceed 2.18528000 I V,, I /kB, above which there exists only one solution with Y, = 0 for all ns. Of course there are no solutions with a finite Y in the bulk crystal above this temperature. For Z, = 12, there are no quasi-liquid (0 < Y, < Y/2) layers. The system consists of only liquid (Y, = 0)
278
Y. Teraoka / Surface melting and superheating
a
AT=O.O
0.
n
0.
_I
0-5
F’ivoi.. AT=0 *-..__.__.__* _.__.__. 4.. 1o-5
T‘
TYn
1f-jl L__*__ , l
.’ x=1
a
AT=O.U
0.5 _________t *
l ___*_.--e--*--*e -a
,.
with izes that differ by 1 from each other is given by the difference between F, and F,, which are the free energies of one liquid and solid monolayer, respectively. F, - F, is positive for AT < 0, and so the free energy of the system increases linearly with the number of the liquid layers no. Just at AT = 0, the free energy does not depend on IZ@,because FL is equa1 to Fs. For AT > 0, the lowest free energy state is one with n, = m. In this case, the free energy decreases linearly with n,, because F, - F, is negative.
b-----d
t
0.
-*,-.-‘* --0
Zo=l 1 i Fig. 5. 2, = 11 and X = 1. (a) Depth profiles of Y, at AT = 0. a-g correspond to na = 3, 4, 5, 6, 7, 8 and 9, respectively. States with na _<2 cannot be found, and states with n,, > 10 can be found. Y, with n 5 na is finite and smaller than Y/2 = 0.25, and Y, with n 2 n, + 1 is larger than 0.25. For example, in case a, the first three layers are in the quasi-liquid state. (b) n, dependences of the free energy at AT = - 10e5, 0 and 10W5.At AT = - lo-‘, the state with no = 4 is the most stable. Just at AT =O, the free energy approaches the constant value with an increase of n,. At AT = lo-“, F decreases with an increase of n,.
and solid (Y, = Y> layers. There is a clear-cut boundary between the n,th and the (n,, + 0th layers. It is a boundary between the liquid and the solid state regions. The free energy as a function of ~1~is shown in fig. 4b. For AT < 0, a state with no = 0 has the lowest free energy. With an increase of no, the free energy increases Iinearly. The free energy difference between states
‘~no *
4 AT=0 l-_* \\ 1o-7 I. a, -*-*-•__*-_* f .'Zo=4 X=1 b Fig. 6. Za = 4 and X = 1. (a) Depth profiles of Y, at AT = 0. a-g correspond to no = 13, 14, 15, 16, 17, 18 and 19, respectively. States with no I 12 cannot be found, and states with ns 2 20 can be found. fb) na dependences of F at AT = - 1O-6 and 0. At AT = - lo-“, only three states with n, = 12, 13 and 14 can be found. The most stable state is the one with no = 13. Just at AT = 0, F decreases with an increase of no and approaches the constant value. Solutions with n,r I 12 cannot be found. At AT = 10W6,we cannot find states with a finite no.
219
Y. Teraoka / Surface melting and superheating
Next we discuss the cases of 2, = 11 and 4. Examples of depth profiles of Y, at AT = 0 are shown in figs. 5a and 6a. All profiles in each case are very similar to one another. Y, increases with an increase of )2 and then approaches Y = 0.5. Its variation becomes very large in the neighborhood of the boundary between the quasi-liquid and the solid layers. With a decrease of Z,, the variation becomes slower and the width of the boundary region becomes broader. Such a situation comes from the decrease of the coordiation number at the surface. The free energies F for 2, = 11 and 4 as a function of no are shown in figs. 5b and 6b, respectively. In the case of 2, = 11, the state with n,, = 4 gives us the minimum value of F at AT = - 10Pi5. F increases with increasing n, beyond n,, = 4. Just at AT = 0, with increasing +,, F decreases and approaches a constant value. On the other hand, at AT > 0, F decreases with increasing n,, and it does not reach a minimum value. For 2, = 11, we cannot obtain solutions with n, = 0, 1 and 2. In the case of Z, = 4, the state with IZ~= 13 has the lowest energy at AT = - 10e6. The solutions can be found only for eta = 12, 13 and 14. Just at AT = 0, F decreases and approaches a constant value with increasing n,. For no I 12, we cannot find solutions. At AT = 10P6, no solutions with a finite value of it,, can be found. The system in the thermodynamical equilibrium state, i.e. the most stable state, has the lowest free energy. For Z, = 12, the state with no = 0 is the most stable at AT < 0 and one with IZ~= m at AT > 0, and so a transition occurs from
thestatewithn,=Otoonewithn,=mat AT= 0. The depth profiles of Y, in the lowest free energy state for several negative AT, are shown in fig. 7a for Z, = 11 and in fig. 7b for Z, = 4, respectively. With an increase of T towards T, (a + b + c+ d + e +f), a0 increases, in other words, the boundary between the quasi-liquid and the solid regions goes deep inside the crystal. At a sufficiently large n, Y, approaches Y at a given temperature. Y itself decreases with increasing T. With a decrease of Z,, the number of quasi-liquid layers becomes larger. Surface melting more proceeds. Now, in order to investigate the proceeding of surface melting, we define a new quantity n,, at which Yn, is equal to half of Y. ~zc is given by interpolating Y, and Y, + 1, between which there is Y,,. The temperature dependence of n, in the most stable state is shown in fig. 8. n, increases with an increase of T towards TM and is then infinite just at AT = 0. The most striking finding is that step-like increases of n, appear with approaching TM, while the increase is continuous at low temperatures. The behavior depends on Z,. With a decrease of Z,, the region of the continuous increase expands to the high temperature side and the variation of n, becomes larger. Just below TM, nc increases by 1 when -AT reduces by about a factor lo-‘.’ and 1O-o.4 for Z, = 11 and 4, respectively. In the temperature region, nc is a half-integer. Such a step-like behavior has been found in surface segregation of segregating alloys [281. As a result, with increasing T towards TM, layers parallel to the surface undergo succes-
b
a TYn
TYn 0.
Lj..-----
0.0 Fig. 7. Temperature
dependence correspond
--
:$ffjjKy 10
20
30
I 4;
of depth profiles of Y, in the most stable state for X= 1. (a) Z, = 11 and (b) Z,, = 4. a-f to AT = - lo-‘, - 10m3, - 10m4, - 10m5, - 10e6 and - lo-‘, respectively.
I’. Teraoka / Surfnce melting and superheating
280
nc 15i
zo=4 X=1
#
10 Zo=l 1
5
’
j---/ii
-AT
ii-’ 1i3 1 Ii’ 1 o5 1 oa 1o-’
1
Fig. 8. Temperature dependences of nC in the most stable state for 2, = 11 and 4 at X = 1. The abscissa is on a logarithmic scale. n, increases with increasing T towards TM. At low temperatures, the increase of n, is continuous, however, a step-like increase appears with approaching TM. With a decrease of Z,, the region of a continuous increase of no starts to expand to the high temperature side.
sive transitions from a quasi-liquid state to a solid state, in other words, the number of quasi-liquid layers increases. Just below TM, the increase becomes always logarithmically step-like. At T,, an infinite number of liquid layers are induced near the surface and solid layers remain deep inside the bulk crystal. Between liquid and solid layers, there exist a finite number of quasi-liquid layers. The step-like character in the temperature dependence of IZ~ originates from an existence of many stable solutions, which have similar depth profiles of Y,, but have lz,,s that differ from one another by an integer. Other stable solutions do
not exist. In the neighborhood of T,, all profiles of solutions with sufficiently large n,s are similar to one another except for the difference in IZ”. An increase of T makes one solution unstable and makes another solution with no + 1 to be stable, and so on. Therefore an existence of the step-like temperature dependence does not depend on our somewhat artificial definition of the quasi-liquid layer and does not depend on the definition of ~zc, too. 4.2. Xc1 As an example, the surface melting of a semiinfinite system with X=X, is discussed (TM = T(LS, S) = 2.14 I V, I/k,). We find only one solution at relatively low temperatures, but many solutions with different a0s appear with approaching T,. In the case of Z, = 11, we obtain only one solution with n, = 1 at AT = - 10e2, and only one solution with IZ,,= 2 for AT = - 10P3. However, we obtain an infinite number of solutions with n, 2 3 for AT = - 10P4, - lo-” and - 10e6, and an infinite number of solutions with n, 2 4 for AT = - 10m7. With approaching T,, only states with more quasi-liquid layers can been found. The depth profiles of X, and Y, in the thermodynamical equilibrium state are shown in fig. 9, where X, and Xs are the concentrations of L and S in the bulk crystal at T,. With an increase of T towards T, (a + b + c + d --+e -+ f), the region with X,, equal to X, = 0.9875 at T, appears near the surface, and then expands
b
a
o*ggo:~F-” 0.9851' Fig. 9. Temperature
2; 4
10
dependence of depth proflies in the most stable state for X=X,,. (a) X,, and (b) Y,. a-f AT= -lo-‘, -10-3, - 10e4, - 10V5, -10-s and - lo-‘, respectively.
correspond
to
Y Teraoka / Surface melting and superheating
towards a larger n. In such a region, Y, is nearly zero. This means that quasi-liquid layers are induced near the surface and the number increases with approaching TM. At TM, an infinite number of liquid layers are induced and solid layers remain deep inside the bulk crystal.
5. Superheating In order to investigate the possibility of superheating, we discuss physical properties of a semiinfinite system with a surface at positive ATs. An infinite number of states with different n,s can be found at a positive AT. Now we consider the case of X = 1. For Z, = 12, we can find solutions for no 2 0 at T smaller than the critical value, 2.18528 I V,, I/kB (AT I 0.021237431, above which there are no solutions with finite order parameters. The bulk system has not a solution with a finite order parameter, too. Of course, for a positive AT, all layers melt in the most stable state. However, all solutions with n, 2 0 correspond to local minima of the free energy, and so the corresponding states are stable. For Z, smaller than 12, we can also find a similar situation. The corresponding states have very similar profiles of Y, to one another. However, the states with small n,s are excluded, because a decrease of the coordination number at the surface makes strong influence on states with a small no, in particular the position of the boundary between the quasi-liquid and the solid layers. A typical example is the case of Z, = 12, in which case there is no decrease of the coordination number at the surface, and so the profiles have the same forms with one another. With a decrease of Z,, stable solutions can be found only at smaller ATs. For example, we can find solutions at AT = 10e5 and not at AT = low4 for Z, = 11, and we can find solutions at AT = lo-’ and not at AT = lop6 for Z, = 4. Therefore it is considered that a critical temperature, above which solutions with finite order parameters are not found, decreases towards TM with a decrease of Z,, and we suppose that it becomes to be equal to TM for Z, = 0. For a given positive AT, the free energy decreases with an increase of n,, as is seen in figs. 4b, 5b
281
and 6b. For a sufficiently large n,, the free energy decreases linearly. The coefficient is negative and is common for all Zas, because the surface effect disappears for a large ~~ and the coefficients are determined only by F, -F,. With increasing AT, the free energy starts to decrease more rapidly due to a decrease of F, - F,. In the case of X = X0, similar results are obtained. For Z, = 11, we can find solutions with n, 2 3 at AT = 10-4, and no solutions at AT = 10P3. Comparing the case of X=X,, with one of X = 1, we can say that, in the case of X = X,,, solutions are found up to larger ATs. As was discussed in section 4, an increase of ~tc with temperature is continuous at low temperatures. In this temperature region, the surface melting proceeds continuously. However, the increase of n, becomes step-like while approaching TM, so that the proceeding of surface melting just below TM means successively discontinuous transitions from a state with IZ~to one with one more quasi-liquid layer and to one with two more quasi-liquid layers, etc. for all Z,s. n,-., which is a half-integer just below T,, increases discontinuously by 1 in each transition. If the system is kept in the thermodynamical equilibrium state at each temperature, nc is infinite at T,, in other words, bulk melting appears at TM. If not so, nc can be finite at and even above TM, and bulk melting does not appear at T,. Therefore, in the latter case, the system is expected to reveal superheating. From an experimental point of view, this condition for appearance of the superheating is always satisfied. The reason is as follows: nc increases logarithmically step-like with an increase of AT. As far as the raising rate of the temperature is not infinitesimal, the proceeding of surface melting cannot follow the raising of the temperature, and so the system is not able to remain in the thermodynamical equilibrium state and n, remains finite even above TM. This leads to the appearance of superheating. At a given raising rate of temperature, a system with a smaller Z, remains in the thermodynamical equilibrium state up to higher temperatures. Above TM, the decrease of the free energy due to an increase of n, by 1 becomes larger with increasing T, but the decreasing amount is not depen-
282
Y. Teraoka / Surface melting and superheating
dent on Z,, because it is only given by F, - F, at a given temperature. As concerned with an infinite system with no surface, it can be expected that the system always reveals superheating, because there is no surface melting. Unfortunately, such a system cannot be experimentally realized, so that no superheating has been observed so far. A real sample has always a few surfaces with different Z,s from one another. Surface melting proceeds more for surfaces with smaller Z,s and less for ones with larger Z,s with increasing T. At the surface with a small Z,, the first many layers are in quasi-liquid states just at TM. Therefore it seems that bulk melting appears at TM from an experimental point of view. On the other hand, for a large Z,, only the first few layers are in quasi-liquid states and other layers remain in solid states just at TM, and so superheating appears. If a sample which is cut only by surfaces with large Z,s is prepared, superheating can be expected to observe. Such an example is found in fine Pb particles with only (111) surfaces [27]. They remain in a solid phase at 2 K above TM for several hours. Other fine Pb particles do not reveal superheating. Our result can explain this observation. The superheating state is not the most stable state, and the system changes to the thermodynamical equilibrium state, in other words, to the liquid phase, after a sufficiently long time. if a sample size is finite, the relaxation time is also finite. The superheating state exists up to the temperature, below which we can find an infinite number of states with different ltOs from one another. In the case of X= 1, the states exist up to the maximum temperature of an existence of the bulk long-range order parameter, T = 2.18528000 I V. I/k,, for Z, = 12. With a decrease of Z,, the temperature decreases.
minima of the free energy of a semi-infinite system with a surface. The solutions have very similar depth profiles to one another except for the boundary’s position between the quasi-liquid and the solid layers. In the thermodynamic equilibrium state, with increasing T towards T,, surface melting proceeds and the number of quasi-liquid layers increases, and then bulk melting appears at T,. Superheating does not appear. However, due to a logarithmically step-like increase of the number of quasi-liquid layers with the temperature, the proceeding of surface melting cannot follow the raising of the temperature, as far as the raising rate is not infinitesimal. Therefore superheating is expected to appear under general experimental conditions. However, superheating is not experimentally observed. This seems to conflict with the above-expected appearance of superheating. Our opinion on this confliction is as follows; the confirmation of existence of several hundreds or thousands or millions quasi-liquid layers near the surface is experimentally very difficult, so that bulk melting seems to appear at TM.
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[9] [lo]
6. Conclusion We have investigated the role of surface melting in bulk melting transitions by use of a simple model developed on the basis of the lattice gas model with short-range pairwise interactions. We have found many solutions corresponding to local
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