- Email: [email protected]
Monte-Carlo simulation of mixed multilayer adsorption
Thin Solid Films 304 (1997) 344-352
ELSEVIER
Monte-Carlo simulation of mixed multilayer adsorption K. Grabowski, A. Patrykiejew *, S. Sokoiowski Faculty of Chemisto', MCS Universio', 20031 Lubtin, Poland
Received 2 January 1996; accepted 10 April 1997
Abstract
The formation of two-component multilayer films on solid surfaces is discussed using the mean-field theory and Monte-Carlo simulation. Phase diagrams for an adsorption system exhibiting complete mixing of both components are determined. It is shown that the sequence of layering transitions present in the system depends strongly on the properties of both adsorbates and the temperature and composition of the adsorbing mixture. © 1997 Elsevier Science S.A. Keb~,ords: Multilayer; Mean-field; Monte-Carlo simulation
1. Introduction
The formation of one-component multilayer films on solid surfaces has been intensively studied both experimentally [la, lb,2-10] and theoretically [11-21]. A detailed discussion of various possible scenarios for film growth has been presented by Pandit et al. [13] on the basis of mean-field results for a lattice gas model. Also, numerous computer simulation studies have contributed greatly to our understanding of such systems. It is now commonly accepted that the behaviour of any particular adsorption system is essentially determined by the relative strength of the gas-solid and gas-gas interactions. In particular, three main different regimes for film development are distinguished. The first, called the strong substrate regime, corresponds to the situation in which the adsorbate wets the surface at any temperature down to zero and adsorption occurs in a layer-by-layer mode. This leads to the appearance of a sequence of the first-order layering transitions when the temperature is sufficiently low. When the strength of the gas-solid interaction becomes weaker (with respect to the gas-gas interaction) the intermediate substrate regime is met. In this case the adsorption system exhibits nonzero wetting temperature. Finally, when the gas-solid interaction becomes sufficiently weak, the wetting is incomplete at any temperature and this situation corresponds to the weak substrate regime. In each of these
* Corresponding author. 0040-6090/97/$17.00 © t997 Elsevier Science S.A. All rights reserved. PII S0040-6090(97)00214-9
regimes, several subregions can be defined and the properties of any given adsorption system appear to be very sensitive to the details of the operating interaction potentials [13,17,18,20]. Despite this complexity, the formation of one-component multilayer films is now quite well understood. On the other hand, our knowledge about the multilayers formed by mixed adsorbates is very limited. This results mostly from the serious difficulties appearing in experimental determination of surface coverage for different components [22]. Only when both adsorbates exhibit very different volatility and when one component is practically nonvolatile under experimental conditions, does the preadsorbed layer technique give precise results [23-30]. In the previous paper [31], we presented some preliminary results of the Monte-Carlo simulation for multilayer adsorption of binary mixtures with similar volatility of both components. The primary aims of our study was to extend the results obtained by Hommeril and Mutaftschiev [32] for monolayer films and to determine the mechanism of displacemem of preadsorbed particles due to the adsorption of the second component. It has been shown that the structure and composition of mixed multilayer films depend on the relative magnitudes of the gas-solid and the gas-gas interaction energies for different species. The results presented in Ref. [31] have also demonstrated that, for sufficiently strong gas-solid interaction of both components at sufficiently low temperatures, the adsorbed films exhibit a sequence of layering transitions. This is quite the same as those observed for one-component films
345
K. Grabowski et a l . / T h i n Solid Films 304 (1997) 344-352
[13,17,18]. In general, the behaviour of binary multilayer films may be very complex and depends on several factors. For example, when one component wets the surface completely while the other does not exhibit complete wetting, the system may behave differently depending on the relative interaction between different pairs of like and unlike particles, and on the composition of the adsorbed layer. Also, when the adsorbates exhibit partial mixing, one can expect that the concentration gradients appearing in the film may induce demixing in some layers. Even in the case of simple systems exhibiting complete mixing of both adsorbates, one expects to observe important changes of their properties with respect to one-component systems. In this paper, we present the results of Monte-Carlo and mean-field calculations of the properties of binary multilayer films for the systems exhibiting a sequence of layering transitions and complete mixing of both components. The primary aim of our study has been to determine how the phase diagrams for the mixed films change with the composition of the adsorbed mixture and to demonstrate some qualitative differences in the behaviour of mixed films with respect to one-component systems. The paper is arranged as follows. The model of the adsorption system and calculational methods are described in Section 2. In Section 3, we present the ground-state properties of the considered systems. The results of our calculations, their discussion and some final remarks are given in Section 4.
2. The model and methods The model of the adsorption system is the same as described in Ref. [31]. Thus, we consider a lattice gas model in which the particles of both components, A and B, may occupy the sites of a simple cubic lattice and are in contact with the solid surface, also represented by a square lattice of adsorption sites. The interaction between the adsorbates and the surface is represented by the potential V K ( Z ) , (K---A, B), being a function of the distance from the surface z, and defined as [18,31]
in tile surface layer and in the bulk phase. Thus, the lattice gas Hamiltonian for our model reads /max l=i
z) = 0,
i~l
+ E vB(0 E l=l
- "AAE ,4n9
i~l
- ,,BB E n f 4 - "AB E
(ij)
-
Xnf
E i
-
E i
(2)
where n~ represent the occupation variable that is equal to 1 (0) if the ith site is occupied by an atom of the component K (empty), UKL is the measure of the nearest neighbour interaction energy for a pair of particles K and L (K, L = A, B) and /.tK is the chemical potential of the Kth component in the system. Next, we assume that the energy of interaction between a pair of atoms of component A, UAA, is the unit of energy and define the following reduced quantities: blKL = R K L / / l l A A ,
K, L = A, B
(3)
V~* = V[/UAA, K = A , B
(4)
tzk = tZK/UAA, K = A, B
(5)
and the reduced temperature T* = kT/UA A
(6)
The Monte-Carlo simulation method applied here has been described in Ref. [31]. We used a standard Metropolis sampling method in the grand canonical ensemble with the fixed value of the chemical potential of one component (B). In all cases, the simulation runs were performed for the systems of the size L~ = Ly = L, = 30 with the periodic boundary conditions in x and y directions. In order to obtain better statistics, the solid surface was located at both sides of the system, at l = 0 and at l = L z, and the preferential sampling of surfacial region [15,18,33] was applied. We worked at temperatures well below the bulk critical temperature. Moreover, the chemical potentials of both components never reached the corresponding bulk coexistence points, located at
,~;,~ = Z<--Zmox
lmax
z--- E vA(l) E
-- 3t,;:,~,
(7)
(1)
Z ~ Zmax
where the amplitude V{ detern'fines the strength of the gas-solid interaction for the K th component ( K = A , B) and Zm~ is the cut-off distance. In the lattice gas model, the distance from the surface assumes only discrete values (given by the layer number l). Moreover, throughout this paper, as in Ref. [31], we assume that Zm~x= / m a x = 5 for both components. The interactions between particles in the system are restricted to the first nearest neighbours and are the same
and the thickness of adsorbed film never exceeded 5 - 6 layers. This ensures statistical independence of both surfacial regions. In each simulation run, we recorded the densities of both components in each layer (l), P~:.z= ~i~ lnif/(L.,Ly), K = A , B, as well as the corresponding layer energies e l . These quantities were later used to calculate the surface excess densities (the surface coverage) for both components
1 f~ E ( PK:z--P~), K = A , B
O,,.(K) =-~
l=l
(8)
K. Grabowski et al. / Thin Solid Fihns 304 (1997) 344-352
346
the total surface coverage, O~,(A and the surface excess energy 1
e~x =
+ B)= O~.~(A) + O~.~(B),
L:
~ E ( e , - eh)
(9)
l=1
In the above, pbx represents the bulk density of the K th component while e b represents the corresponding bulk energy (per lattice site). The butk properties were obtained by averaging the results obtained for layers from 11 to 20. The number of Monte-Carlo steps (per lattice site) used to calculate averages has been different for different temperatures, and ranged from 2500 to 15 000 in the surfacial region and from 500 to 3000 in the bulk phase. The number of Monte-Carlo steps used in particular runs have been adjusted to produce sufficient statistics. Also, considering that autocorrelation times change with temperature (become longer as the temperature approaches the critical temperature), the actual number of performed Monte-Carlo steps was considerably greater since measurements have been done using configurations spaced by a certain number of Monte-Carlo steps (ranging from 5 to 20). At low temperatures, the simulations produced well-pronounced hysteresis loops at the adsorption-desorption curves, connected with the appearance of metastable states. In order to locate the transition points, we calculated the system surface excess free energies and applied the thermodynamic integration method [34,35]. This can be done by performing Monte-Carlo simulations along different paths. First, one can obtain the free energy differences for different states (corresponding to the different number of occupied layers) with respect to the known ground state energies (see Section 3) for the fixed values of /*A and /x~ as functions of temperature. Then, the changes of the free energies with respect to/x a (for any given temperature) are readily obtained by integrating the corresponding adsorption and desorption isotherms. At temperatures below the critical point, the location of the first-order transition is found at the condition stating that the thermodynamically stable state corresponds to the minimum of the free energy.
relative strengths of the gas-solid and the gas-gas interactions for both components is practically the same; they differ only by the temperature range in which both systems exhibit first-order layering transitions. The system with the weaker gas-solid potential (Va°" = 1.0) shows somewhat different behaviour. The first two layering transitions [18], occurring in layers 1 and 2, merge into one transition corresponding to a simultaneous condensation of the adsorbate in the first two layers. This transition is then followed by a series of layering transitions corresponding to the condensation in subsequent layers 3, 4, etc. Finally, we assumed that the interaction energy between unlike particles, UAB is equal to 0.5(UAA + U~9)= 0.78. This assumption corresponds to ideal complete mixing of both components. All the calculations have been carried out assuming the constancy of the chemical potential of component B. In most cases we have taken the values of /.t~ lower than those corresponding to the first layering transition of pure component B at T* = 0. The ground-state properties of the one-component lattice gas systems considered here are well known [13,t8]. In particular, the system exhibits the sequence of layering transitions: 0, 1, 2, ..., ~ when uxK < Vx(1 ) - V~(2)
and these transitions are located at the chemical potential values given by b~x,,r(0 --+ 1) = --2uxqv + Vx(1 )
(tl)
and
t~x,,r(l--+l+
1) = - 3 u K ~ +
VK(I+
1)
(12)
where K = A or B. For the weaker surface potential, a different sequence of layering transitions may appear. The adsorbate may simultaneously condense in the first n layers adjacent to the solid surface. Then, this transition is followed by the condensation occurring in subsequent layers. Thus, the sequence of the layering transitions reads: 0, n, n + 1. . . . . (n > 1), with n determined by the condition
3. The systems and their ground state properties Here, we assume that the properties of pure component B are fixed and determined by the parameters u~B = 0.56 and VB°* = 1.67. With this choice of interaction parameters, the system containing only particles of component B belongs to the strong substrate regime. Thus, the adsorbate completely wets the surface at any temperature down to T* = 0 and the film develops in a layer-by-layer mode. In particular, at sufficiently low temperatures, a sequence of the first-order layering transitions corresponding to the condensation in layers 1, 2, etc. is observed. For component A, we have considered two different values of the parameter VA°* , 3.0 and 1.0. For the first choice, the
(10)
v (i)
1) >,,K,,.
i=1
> ~ VK(i ) --nVK(n )
(13)
i=J The first transition corresponding to the simultaneous condensation of the adsorbate in the first n layers is located at >K,,,(0--+ n) = -
1[
-(3n-
H
Vx(i )
1),q,.x+
]
(14)
k=l
and the subsequent transitions between m and m + t (m > n) occupied layers are located at the chemicat potential given by Eq. (12).
e t ak / T h i n Solid Films 304 ( 1 9 9 7 ) 3 4 4 - 3 5 2
K. G r a b o w s M
~) AAA
AAB
q
AA
-3
the minimum of the system surface excess free energy (per lattice site) given by
AAAB
AAAA
/--
347
--AABB
fe~,~(/z~,/x~ ) = rn)n {e;x ( k)}
--ABB
(16)
AB --ABBB
A
/
The transition between different states, a and /3, occurs when
--BBB
-4,5 --BBBB
f**<~( IZ~, ~; ) =f~*~,p ( t~ ,tz; )
-5 -5,5
-a
-2,8
-2.6
-2,4
-2.2
AAAJ
-3.05
-1.8
(b)
AAA -3
-2
./
AA
-3,1 • -3.I5 "
--BBBB BB
0
BBB
-3,2 '
i i
(17)
Fig. 1 presents the ground-state phase diagrams for the tWO systems considered in this work. One should note that only the states with a maximum of four occupied layers are included here. At finite temperatures, the assumption of complete ideal mixing of both adsorbates excludes any possibility of sharp transition between the states differing only by the composition of the adsorbed layer, but of the same thickness. An example of such process is the transition between monolayers filled with A or B particles only.
-3.25 i" -3.;1 -3
I -2,8
t
J
i
r
i
-2.6
-2.4
-2.2
-2
-I,8
4. Results and discussion
Fig. I. The ground-state phase diagrams for the systems with V~* = 3.0 (part a) and VA° * = 1.0 (part b).
One can readily check that pure component A with Va°* = 3.0, as well as pure component B both exhibit the sequence of layering transitions: 0, 1, 2 . . . . . % while pure component A with V~* = 1.0 undergoes the transitions: 0, 2 , 3 . . . . . ~. In our model of mixed adsorbate, the only possible stable ordered states at T * = 0 correspond to the films with a different number of completely filled layers. Since we consider only systems with the attractive nearest neighbour interactions between adsorbate particles and assume ideal complete mixing, no other ordered state can appear. Only when the energy of interaction between unlike particles (UAB) is greater than the arithmetic mean of UAA and uas, can other (mixed) ordered states within a single layer appear (see Refs. [36-38]). The surface excess energy for the film with k filled layers is given by
Fig. 2 shows examptes of adsorption-desorption isotherms obtained for the system with Va°* = 3.0 at different temperatures and for the chemical potential of component B equal to - 3.45. It is clear that, at the considered temperatures, the development of multilayer film occurs via a sequence of the first-order layering transitions. In all cases, we find that condensation in each layer is accompanied by rather large changes in its composition. In particular, the increase in the film thickness leads to an increase in the concentration of component A in the adsorbed film. Fig. 3 presents the adsorption isotherm for the system with
3.0 ~ooeeoeomooooooooeooeeooooooooaJo••°°°~
¢~(4 + B)
T " = 0,56
2.0
k
e:~(k)
2 -= E [ - 2 PA,i i=l
, PB,i 2 2 UBB
4U~B Pn,i
o east•
PB,i
T* = O.48
F,,
1.0
- Pa,i P A , i - 1 -- I~;B PB,i P B , i - 1 -- llAB
"-
T* = 0.45
X( f)A,i PB,i-! "{- PB,i PA,i-1) "q- VA* ( i) pA,i + VB* (i) PS,i - IZA PA,i
-- t'@ PB,i]
(15)
where PA,0= PB,O= 0, &,i + PB,i = 1 and PA,i Ps,i = 0 for each occupied layer i. For any given value of the chemical potentials /x~ and /x~, the stable state ( a ) corresponds to
-5.0
-41.6
-4..21
-31.8
I
-3.4
Fig. 2. The adsorption isotherms for the system with I~°* = 3.0 and /z~ = - 3.45 at T* = 0.45, 0.48, 0.52 and 0.56. Note that the results for different temperatures are shifted upward by 0.5 for a better display.
348
K.
(a)
7.0
G.(A
+
Grabowski et al. / Thin Solid Films 304 (1997) 344-352
O0•
B)
~CA
hi'leo ) ,Be•
6.0
0,~
0.9
o
0.8
1.5
-- O~(A)
• --
O=(B)
• - - O=(A + B)
5.0 0.7
~
°
J
D
o
mo I
4.0 1.0
0.6 3.0
QOID
•
oooOO o°
0.5
2.0
0.5 1.0
• o• oe• e ~-...-..o ~t eeo•
-10
eel° • o o o • o o • o e o e • 0 ~ i ~ m ~
-1.5
I
-4.0
0.4 >oO0 ~
-i.5
-4.6
-4.5
-4'.3 -1.2 -4'.1
-4.4
Fig. 3, The total adsorption isotherm and the mole fraction of c o m p o n e n t A in the adsorbed film for the s y s t e m with V ° * = 3.0 a n d ,u,~ = - 3.45 at T * = 0.55.
/x~ = - 3 . 4 5 at T* = 0.55 together with the changes of the mole fraction of component A in adsorbed film (XA), where
G(A) XA= O,x(a + B)
(18)
We observe that in the region of the first plateau, corresponding to one adsorbed layer, the increase of the chemical potential of component A leads to the gradual removal of component B from the film. This is a direct consequence of the stronger interaction between the particles of component A than between any other pair of particles (A and B, or B and B) and of the low chemical potential of component B. The mechanism of displacement of preadsorbed particles B is different when the adsorbed layer is initially rich in component B. For example, when /z~ is higher than the value corresponding to the first layering transition of pure component B (equal to - 2 . 7 9 at T * = 0), the addition of component A leads to the transfer of particles B from the first to the second adsorbed layer. This situation is illustrated in Fig. 4(a) which shows the adsorption isotherm of component A on the preadsorbed monolayer of component B at /x~ = - 2 . 0 and T* = 0.30. From the results given in Fig. 4(a) and from the behaviour of densities of both components in the first two layers (shown in Fig. 4(b)), we find that this system shows a strong effect of enhanced adsorption (studied by Rikvold et al. [36], Rikvold and Deakin [37] and Rikvold [38]) in monolayer films. From the obtained adsorption isotherms and density profiles, we constructed phase diagrams corresponding to different values of the chemical potential of component B in the region of the first three layering transitions. The example corresponding to /x~ = - 3 . 4 5 is presented in Fig. 5 which gives the total surface coverage of coexisting phases versus the temperature for the first three layers. The most striking difference between the obtained phase diagrams and similar results for one-component systems [18]
0.8 - ° o O O
e~
p~(O 0.6
0.4
0.2
~-o--a-,I : ~. . . . . . . . 1. . . . . . . . . ~----m2~'~"]"~
-4.6
-4,5
-4.4
-&3
F
P
-4.2
-4.1
Fig. 4. The adsorption isotherms O~(A + B), Oex(A) a n d O,~(B) for the system with Va° * = 3.0 at T * = 0.30 and /x~ = - 2.0 (part a) and the corresponding densities o f both c o m p o n e n t s in the first t w o adsorbed layers (part b).
is a pronounced increase in the critical point temperature of the layering transitions occurring in subsequent layers 1, 2 and 3. These changes of T,?(I) can be attributed to the
oo~(A+ B) 2.0
1,0
•
•
•
•
nil
el
o ~
•
•
:11|•
°~
.J
---4:i
2:d.3 . . .0.4 .
Fig. 5, The g l o b a l - p h a s e d i a g r a m Oe'(A with Va° * = 3.0 and ~ = - 3.45.
0k.5 T*
+ B) versus T " for the system
K. Grabowski
el a[./Thin
Table 1 Critical temperatures and critical compositions for layering transitions in systems with VA °* = 3.0, for different chemical potentials of component B I
/zB " = - 3.35
/xB * = - 3.45 Tc
1 2 3
Tc *(l) 0.466 0.545 0.552
xt,c(l) 0.59 0.87 0.905
/.z~ = - 3.52
xt,~(l) 0.63 0.87 0.89
"(l)
0.47 0.541 0.544
Tc *(I) 0.495 0.545 0.548
XA,c(I) 0.70 0.90 0.93
Solid Films 304 (1997) 3 4 4 - 3 5 2
349
result is rather obvious and is a direct consequence of the negligence of statistical fluctuations by the mean-field theory and hence a considerable underestimation of •ntropic effects. The second difference is connected with the mean-field prediction concerning the changes of the layering critical temperature with the layer number. It is known [ 11,18,20] that lim T~MF(l) = T f f F ( 3 D )
(19)
l -.., ~¢
differences in the composition of each layer near the corresponding critical point. In Table 1, we sun'unarize the values of T / ( l ) for three different choices of #~. We have also determined the layering critical temperatures using the mean-field theory described in Ref. [31]. The results for the first three layering transitions are shown in Fig. 6(a). Two important differences between the results of the Monte-Carlo simulation and the mean-field predictions should be emphasized. First, we observe that the mean-field theory gives considerably higher values of the layering critical temperatures than the Monte-Carlo simulation. This
(~)
z2M~(1) 1.0
lol---
.
I=2 ........
,
0.9 /
/
// ../
0.8 0.7 /..2
0.6 -~,~
0'.2
0'.4
O..6
0'.8
where T~MF(3D) is the bulk mean-field value of the critical temperature of the adsorbate. In the lattice gas model considered here, the mean-field bulk critical temperature of the pure component K ( K = A , B) is given by T~MF(3D) = 1.5 UKK, while the critical point for the first layering transition T~m'(1) is equal to UKK. In the binary system exhibiting complete mixing, the locations of the layering critical points depend, not only on the layer number, but also on the composition of each adsorbed layer. Also, the bulk critical temperature changes with the bulk phase composition. For the systems considered here, exhibiting complete mixing of both components and with the parameter UAB given by the arithmetic mean of u~, A and UBB, the critical temperature in any given layer should change monotonically with the composition of this layer, between the values corresponding to the critical points for pure components A and B, and with only rather small deviations from linearity. In particular, for the first adsorbed layer, the exact value of critical temperature is known and equal to T ~ ( 1 ) = 0.5674U~K, where K = A or B. We also know that, in the one-component version of the lattice gas model considered here, the roughening temperature is only about 5% higher than the two-dimensional critical temperature [12]. Thus, the observed changes in the layering critical temperature with the layer number can be attributed to the changes in the composition of subsequent layers.
.;CA
(a) .... /
¢'--l=2
r;(o ] A _
t
=3
<9'
0.5
3
-3.05 •
o
•
-3.10 0
c~.~'
%0
I
-3.15
(b)
3
0.4 -3~05
2
#,4,tr
•
-3.10
0.4
0.6
0.8
Fig. 6. The dependencies between the layering critical temperatures and the composition of each layer at the critical point obtained from the mean-field theory (part a) and from Monte-Carlo simulation for the systems with VA ° * = 3.0 (Table 1) and VA °" ~ 1.0 (Table 2) (part b).
~*' • °
~" " ' " *'• • • • ~ 0
0.2
•
•
i
eeoc
-3.15 0'.i
O'.2
O'.3
o'.4
o'.5 T* Fig. 7. The phase diagrams /.z~,cr vs. T * for the systems with VA ° * = 1.0 and two choices of #~ equal to -3.45 (part a) and -3.30 (part b).
350
K. Grabowski et aL / Thin Solid Films 304 (1997) 344-352
Table 2 Locations of triple points and for systems with V[" = 1.0 and different values of the chemical potential of component B IzB *
2.0
,
. . . .
.~::: e
1.0 o
'
0'.1
~2 ~ 63
•
41
~
4,
~e
• • IQb I
o , ( A + B)
(bt 2.0
.
.
.
.
~,.,. N,, ; :,~ I I I
l [ [
1.o
~...~.. • .~' I l l l I
0'.1
~.2 : ~.3
tza., *
l
Oex(A)
Oex(B)
Oex(A + B)
xa
0.053 0.786 1.816 0.017 0.778 1.824 0.005 0.758 1.828 0.00047 0.744 1.856
0.038 0.187 0.168 0.022 0.209 0.170 0.013 0.236 0.170 0.004 0.255 0.144
0.091 0.973 1.984 0.039 0.987 1.994 0,018 0.994 1.998 0,00447 0.999 2.000
0.582 0.808 0.915 0.436 0.788 0.915 0.278 0.763 0.915 0.105 0.745 0.928
0.5 T*
0.4
&
Tt *
-3.60 0.475 -3.0988 0 1 2 -3.45 0.415 -3.0967 0 1 2 -3.30 0.350 -3.0920 0 1 2 -3.15 0.262 -3.0830 0 1 2
0.4
or.5 T* Fig. 8. The phase diagrams O,,(A + B) vs. T* for the systems with VA °" = 1.0 and two choices of ,a~ equal to -3.45 (part a) and --3.30 (part b).
This is not true for the mean-field results, however. In this case, Eq. (19) holds and one expects rather large changes in the layering critical temperatures with the layer number. Within any given layer, the changes of Tc ~ M F ( l ) with the layer composition should be similar. The results presented in Fig. 6 confirm the above predictions very welt. The values of To*(1) deduced from Monte-Carlo simulation change nearly linearly with x A for all transitions considered here. The corresponding values of T~.*(2) a n d To* (3) exhibit only small positive deviations. On the other hand, the values of To* g F ( l ) show much larger changes with the layer number I, for which Tc*MF(l) changes smoothly with the adsorbate composition (see Fig. 6(a)). In the case of the adsorption system with VA°* = 1.0, the addition of the second component, B, leads to much more serious changes in the behaviour of adsorbed films at finite temperatures. Figs. 7 and 8 present the phase dia-
grams (/ZA,tr, T* ) and (0~x(A + B), T* ) obtained for two different values of /z~, - 3 . 4 5 and - 3 . 3 0 . We know that in the system containing pure component A, the first layering transition involves simultaneous condensation in the first and second layer [18]. In the mixed systems, this picture is preserved only at sufficiently low temperatures, lower than a certain triple-point temperature, Tt*. Above Tr", the presence of component B leads to the splitting of this transition into two first-order layering transitions occurring separately in the first and the second layer. The location of the triple point depends strongly on the value o f / x ~ , i.e. on the composition of the adsorbate. In Table 2, we give the triple-point temperatures and the compositions of the coexisting phases for systems characterized by different values of the chemical potential of component B. Table 3 contaius the critical temperatures and the critical compositions of the layering transitions for three selected systems. Fig. 9 gives the plots of Tr* and T~*(1) against the chemical potential of component B. It is clear that the triple-point temperature meets the critical point for the first layer at a certain value of the chemical potential of component B. Here we estimate that this occurs for/x~ = - 3 . 7 . On the other hand, the temperature of the triple point goes to zero when the chemical potential of component B
Table 3 Layering critical temperatures and compositions for systems with Va°* = 1,0 and different values of /zB * ~z~ *
t
T~ *(l)
Xc.A(I)
- 3.30
1 2 3 1 2 3 1 2 3
0.46 0.545 0.552 0.482 0.552 0.555 0.502 0.553 0.556
0.58 0.87 0.905 0.67 0.9 0.92 0.75 0.91 0.93
- 3.45 - 3.60
K. Grabowski et aL / Thin Solid Films 304 (1997) 344-352
o - r;(1) ,,=T[ 0.5
*. r-.e." '"'o ...... O , ,
T*
%
"",-¢.,
0.4 "e
0.3
0.2
0.I
-3.6
-3.4
-3.2
-3~.0
-218
~5 Fig. 9. The dependencies of the triple-point temperatures and the first layering transition critical point temperatures on the chemical potential of component B for the systems with VA°* = 1.0. * denotes the estimated temperature where the triple point and the critical point temperatures meet, while ~ denotes the location of the first layering transition of pure component B at T* = 0.
approaches the value corresponding to the first layering transition of pure component B, equal to - 2.79 (at T* = 0). Again, the critical temperatures for the first layering transition are quite consistent with the dependence on the adsorbate composition found for VA° * = 3.0, while the critical points for higher layering transitions exhibit small positive deviations from this dependence (compare Fig. 6(b)). In conclusion, we would like to stress that even in very simple systems, exhibiting ideal mixing of both components, their behaviour may be quite complex and very sensitive to the interactions operating in the system and to the composition of the adsorbate. Some of the presented results can be readily anticipated from the behaviour of simple one-component systems, but we also find some qualitatively new results. The most interesting finding of this paper is the demonstration of triple points in mixed films. In our earlier study of one-component films [18], we have shown that the triple points exist only within a narrow interval of the relative strength of the adsorbateadsorbate and adsorbate-solid interactions. We have demonstrated that this regime can also be met by addition of the second component, characterized by the different relative strength of the adsorbate-adsorbate and the adsorbate-solid interactions. In binary systems exhibiting complete mixing, we can determine the effective strength of the adsorbate-adsorbate and the adsorbate-solid interactions by defining the parameter ,,,,,,x I + 2,t,,~ < , ( 1 - x,,) + t , ~ ( 1 ~,,(~o)
=
and hence the value of y ( x A) changes with temperature as it is demonstrated in Fig. 10. We know that in one-component systems, the triple points appear for the ratio uxx/V~ ( K = A or B)just slightly below the value delimiting the reNmes leading to the sequences of layering transitions: 0, 1, 2, 3, etc. and 0, 2, 3, etc. at the ground state. The same situation appears in mixed systems. Pure component A with VA°* = t.0 gives the sequence of layering transitions: 0, 2, 3, etc. at any temperature below the corresponding critical point; pure component B gives the sequence: 0, t, 2, 3, etc. The addition of component B modifies the effective interactions and shifts y ( x a) towards lower values. For a sufficiently high concentration of B, we reach the region in which the splitting of the layering transition, 0 ~ 2, into two separate transitions, 0 ~ 1 and 1---* 2, occurs at finite temperature, leading to the appearance of a triple point. In the light of the above discussion, it is clear that the triple-point temperature should decrease as the concentration of component B increases. It should be emphasized that the results presented in this paper are only preliminary. In particular, our choice of the parameters describing adsorbate-adsorbate interactions exclude any possibility of the formation of mixed ordered states in a single adsorbed layer [36-38], the existence of azeotropic states, and phase separation phenomena [39-42]. We have not attempted any finite-size scaling analysis. However, this does not in any way affect the qualitative picture emerging from our study. Of course, our estimations of the layering critical temperatures are inevitably subject to systematic errors due to the smallness of the computational cell. Considering, however, that our aim here was to present qualitative effects resulting from the changes in the composition of the adsorbate and not any direct quantitative comparison with real systems, we believe that the negligence of finite-size effects analysis is justified. The use of larger systems would require much more computing time and would not give any deeper insight into the physics of the problem. °°°Oo%
(20) One should note that even for the fixed chemical potentials of both components, the composition of adsorbed film
0.99
ee
G(A + B)
0.9
0.98
,"
,'
0.97
° o%
, '
0.8 '
0.8
%e %o
0.96 %e
0'.I
v;):~ + W(1 -x~)
***~**** eo
.r A
01,8
- xA) ~
351
0.9 2: A 0.2
0.3
0r.4 T*
Fig. 10. Temperature changes of the total surface coverage ( O ~ ( A + B)) and the mole fraction of component A(x a) for the system with Va° * = 1.0, VB° * = 1.67 at /~,~ = - - 3 . I 1 and /z~ = - - 3 . 4 5 . The insert shows the changes of the parameter y ( x A) with the concentration of component A.
352
K. Grabowski et aL / Thit, Solid Fibns 304 (1997) 344-352
A n o t h e r open question i n v o l v e s wetting p h e n o m e n a in m i x e d systems consisting o f c o m p o n e n t s such that one wets the surface while the other does not. It is very unfortunate that the experimental study o f multilayer adsorption o f binary mixtures meets so m a n y difficulties and it is hard to find suitable data that can be c o m p a r e d with theoretical predictions.
References [la] A. Thorny, X. Duval, J, Chim. Phys. 66 (1969) 1966. [lb] A. Thorny, X. Duval, J. Chim. Phys. 67 (286) (1970) 110t. [2] J. Menaucourt, A. Thorny, X. Dural, J. Physiol. (Paris) 38 (1977) C4-195. [3] A. Thomy, X. Dural, R. Regnier, Surf. Sci. Rep. 1 (1981) 1. [4] L.J. Sequin, J. Suzanne, M. Bienfait, LG. Dash, J+A. Venables, Phys. Rev. Lett. 51 (1983) 122. [5] M. Sutton, S.GJ+ +Mochrie, PCJ. B~rgenau, D.E. Moncton, P.M. Horn, Phys. Rev. B 30 (1984) 263. [6] Y. Larher, F. Angerand, Y. Manrice, J. Chem. Soc., Faraday Trans. 1 83 (1987) 3355. [7] H.S. Nham, G.B. Hess, Phys. Rev. B 38 (1988) 5166. [8] P+ Pfeifer, Y.J. Wu, M.W. Cole, J. Krim, Phys. Rev. Lett. 62 (1989) 1997. [9] H. Tillborg, A. Nilsson, B. Hernniis, N. M~rtensson, R.E. Palmer, Surf. Sci. 295 (1993) 1. [10] H.S. Youn, X.F. Meng, G.B. Hess, Phys. Rev. B 48 (1993) 14556. [11] M.J. de Oliveira, R.B. Griffiths, Surf. Sci. 71 (i978) 687. [12] I.M. Kim, D.P. Landau, Surf. Sci. 110 (1981) 415. [13] R. Pandit, M. Schick, M. Wortis, Phys. Rev. B 26 (i982) 5112. [14] C. Ebner, Phys. Rev. B 28 (1983) 2890. [15] K. Binder, D.P+ Landau, Phys. Rev. B 37 (1988) t745. [16] W.F. Saam, Surf. Sci. 125 (1983) 253.
[17] C. Ebner, Phys. Rev. A 22 (i980) 2776. [18] A. Patrykiejew, D+P. Landau, K. Binder, Surf. Sci. 238 (1990) 317. [19] D.E. Sullivan, M.M. Telo de Gamma, Fluid Interfacial Phenomena, in: C.A. Croxton (Ed.), Wiley, Chichester, 1986. [20] M. Kruk, A. Patryldejew, S. Sokolowski, Thin Solid Films 238 (1994) 302. [2I] W. Gac, M. Kruk, A. Patrykiejew, S. Sokotowski, Langmnir 12 (1996) 159. [22] M.A. Hamdi Alaoui, J.P. Coulomb, N. Dupont-Pavlovsky, I. Mirebeau, B. Mutaftschiev, Surf. Sci. 295 (1993) L1031. [23] J.H. Singleton, G.D. Halsey Jr., J. Phys. Chem. 58 (i954) 330. [24] M. Abdelmoula, T. Ceva, B. Croset, N. Dupont-Pavlovsky, Surf. Sci. 272 (1992) 167. [25] H. Asada, S. Doi, H. Kawano, Surface Sci. Lett. 265 (1992) L233. [26] H. Asada, S. Doi, H. Kawano, Surface Sci. Lett. 273 (1992) IA03. [27] D.A. Slater, P. Hollins, M.A. Chesters, Surf. Sci. 306 (1994) 155. [28] A. Patrykiejew, S. Sokotowski, T. Zientarski, H. Asada, Surf. Sci. 314 (1994) 129. [29] N. Dupont-Pavlovsky, M+ Abdelmoula, S. Rakotozafy, J.P. Coulomb, B. Croset, E. Ressouche, Surf. Sci. 317 (1994) 388. [30] J. Menacourt, C. Bockel, J. Physiot. (Paris) 5I (1990) 1987. [31] M. Kruk, A. Patrykiejew, S. Sokotowski, Surf, Sci. 340 (1995) 179. [32] F. Hommeril, B. Mutaftschiev, Phys. Rev. B 40 (1989) 296. [33] K. Binder, D.P. Landau, J. Appl. Phys. 57 (i985) 3306. [34] W. Helbing, B. Diinweg, K. Binder, D.P. Landau, Z. Phys. B 80 (1990) 401. [35] K. Binder, Monte-Carlo Methods in Statistical Physics, in: K+ Binder (Ed.) Springer, Berlin, I986, p. 1. [36] P+A. Rikvold, J.B. Collins, G.D. Hansen, J.D. Gunton, Surf. Sci. 203 (1988) 500. [37] P.A. Rikvold, M.R. Deakin, Surf+ Sci. 0~49(1991) 180. [38] P.A+ Rikvold, Electrochlm. Acta 36 (1991) 1689. [39] A. Patryldejew, Thin Solid Films 147 (1987) 333. [40] M. Kaburagi, T. Tonegawa, K. Ebina, Surf. Sci+ 242 (1991) 107. [41] J. Sivardiere, Phys. Rev. A 11 (1975) 2090. [42] N.K. Mahale, M.W. Cole, Surf. Sci. 176 (1986) 319.
ELSEVIER
Monte-Carlo simulation of mixed multilayer adsorption K. Grabowski, A. Patrykiejew *, S. Sokoiowski Faculty of Chemisto', MCS Universio', 20031 Lubtin, Poland
Received 2 January 1996; accepted 10 April 1997
Abstract
The formation of two-component multilayer films on solid surfaces is discussed using the mean-field theory and Monte-Carlo simulation. Phase diagrams for an adsorption system exhibiting complete mixing of both components are determined. It is shown that the sequence of layering transitions present in the system depends strongly on the properties of both adsorbates and the temperature and composition of the adsorbing mixture. © 1997 Elsevier Science S.A. Keb~,ords: Multilayer; Mean-field; Monte-Carlo simulation
1. Introduction
The formation of one-component multilayer films on solid surfaces has been intensively studied both experimentally [la, lb,2-10] and theoretically [11-21]. A detailed discussion of various possible scenarios for film growth has been presented by Pandit et al. [13] on the basis of mean-field results for a lattice gas model. Also, numerous computer simulation studies have contributed greatly to our understanding of such systems. It is now commonly accepted that the behaviour of any particular adsorption system is essentially determined by the relative strength of the gas-solid and gas-gas interactions. In particular, three main different regimes for film development are distinguished. The first, called the strong substrate regime, corresponds to the situation in which the adsorbate wets the surface at any temperature down to zero and adsorption occurs in a layer-by-layer mode. This leads to the appearance of a sequence of the first-order layering transitions when the temperature is sufficiently low. When the strength of the gas-solid interaction becomes weaker (with respect to the gas-gas interaction) the intermediate substrate regime is met. In this case the adsorption system exhibits nonzero wetting temperature. Finally, when the gas-solid interaction becomes sufficiently weak, the wetting is incomplete at any temperature and this situation corresponds to the weak substrate regime. In each of these
* Corresponding author. 0040-6090/97/$17.00 © t997 Elsevier Science S.A. All rights reserved. PII S0040-6090(97)00214-9
regimes, several subregions can be defined and the properties of any given adsorption system appear to be very sensitive to the details of the operating interaction potentials [13,17,18,20]. Despite this complexity, the formation of one-component multilayer films is now quite well understood. On the other hand, our knowledge about the multilayers formed by mixed adsorbates is very limited. This results mostly from the serious difficulties appearing in experimental determination of surface coverage for different components [22]. Only when both adsorbates exhibit very different volatility and when one component is practically nonvolatile under experimental conditions, does the preadsorbed layer technique give precise results [23-30]. In the previous paper [31], we presented some preliminary results of the Monte-Carlo simulation for multilayer adsorption of binary mixtures with similar volatility of both components. The primary aims of our study was to extend the results obtained by Hommeril and Mutaftschiev [32] for monolayer films and to determine the mechanism of displacemem of preadsorbed particles due to the adsorption of the second component. It has been shown that the structure and composition of mixed multilayer films depend on the relative magnitudes of the gas-solid and the gas-gas interaction energies for different species. The results presented in Ref. [31] have also demonstrated that, for sufficiently strong gas-solid interaction of both components at sufficiently low temperatures, the adsorbed films exhibit a sequence of layering transitions. This is quite the same as those observed for one-component films
345
K. Grabowski et a l . / T h i n Solid Films 304 (1997) 344-352
[13,17,18]. In general, the behaviour of binary multilayer films may be very complex and depends on several factors. For example, when one component wets the surface completely while the other does not exhibit complete wetting, the system may behave differently depending on the relative interaction between different pairs of like and unlike particles, and on the composition of the adsorbed layer. Also, when the adsorbates exhibit partial mixing, one can expect that the concentration gradients appearing in the film may induce demixing in some layers. Even in the case of simple systems exhibiting complete mixing of both adsorbates, one expects to observe important changes of their properties with respect to one-component systems. In this paper, we present the results of Monte-Carlo and mean-field calculations of the properties of binary multilayer films for the systems exhibiting a sequence of layering transitions and complete mixing of both components. The primary aim of our study has been to determine how the phase diagrams for the mixed films change with the composition of the adsorbed mixture and to demonstrate some qualitative differences in the behaviour of mixed films with respect to one-component systems. The paper is arranged as follows. The model of the adsorption system and calculational methods are described in Section 2. In Section 3, we present the ground-state properties of the considered systems. The results of our calculations, their discussion and some final remarks are given in Section 4.
2. The model and methods The model of the adsorption system is the same as described in Ref. [31]. Thus, we consider a lattice gas model in which the particles of both components, A and B, may occupy the sites of a simple cubic lattice and are in contact with the solid surface, also represented by a square lattice of adsorption sites. The interaction between the adsorbates and the surface is represented by the potential V K ( Z ) , (K---A, B), being a function of the distance from the surface z, and defined as [18,31]
in tile surface layer and in the bulk phase. Thus, the lattice gas Hamiltonian for our model reads /max l=i
z) = 0,
i~l
+ E vB(0 E l=l
- "AAE ,4n9
i~l
- ,,BB E n f 4 - "AB E
(ij)
-
Xnf
E i
-
E i
(2)
where n~ represent the occupation variable that is equal to 1 (0) if the ith site is occupied by an atom of the component K (empty), UKL is the measure of the nearest neighbour interaction energy for a pair of particles K and L (K, L = A, B) and /.tK is the chemical potential of the Kth component in the system. Next, we assume that the energy of interaction between a pair of atoms of component A, UAA, is the unit of energy and define the following reduced quantities: blKL = R K L / / l l A A ,
K, L = A, B
(3)
V~* = V[/UAA, K = A , B
(4)
tzk = tZK/UAA, K = A, B
(5)
and the reduced temperature T* = kT/UA A
(6)
The Monte-Carlo simulation method applied here has been described in Ref. [31]. We used a standard Metropolis sampling method in the grand canonical ensemble with the fixed value of the chemical potential of one component (B). In all cases, the simulation runs were performed for the systems of the size L~ = Ly = L, = 30 with the periodic boundary conditions in x and y directions. In order to obtain better statistics, the solid surface was located at both sides of the system, at l = 0 and at l = L z, and the preferential sampling of surfacial region [15,18,33] was applied. We worked at temperatures well below the bulk critical temperature. Moreover, the chemical potentials of both components never reached the corresponding bulk coexistence points, located at
,~;,~ = Z<--Zmox
lmax
z--- E vA(l) E
-- 3t,;:,~,
(7)
(1)
Z ~ Zmax
where the amplitude V{ detern'fines the strength of the gas-solid interaction for the K th component ( K = A , B) and Zm~ is the cut-off distance. In the lattice gas model, the distance from the surface assumes only discrete values (given by the layer number l). Moreover, throughout this paper, as in Ref. [31], we assume that Zm~x= / m a x = 5 for both components. The interactions between particles in the system are restricted to the first nearest neighbours and are the same
and the thickness of adsorbed film never exceeded 5 - 6 layers. This ensures statistical independence of both surfacial regions. In each simulation run, we recorded the densities of both components in each layer (l), P~:.z= ~i~ lnif/(L.,Ly), K = A , B, as well as the corresponding layer energies e l . These quantities were later used to calculate the surface excess densities (the surface coverage) for both components
1 f~ E ( PK:z--P~), K = A , B
O,,.(K) =-~
l=l
(8)
K. Grabowski et al. / Thin Solid Fihns 304 (1997) 344-352
346
the total surface coverage, O~,(A and the surface excess energy 1
e~x =
+ B)= O~.~(A) + O~.~(B),
L:
~ E ( e , - eh)
(9)
l=1
In the above, pbx represents the bulk density of the K th component while e b represents the corresponding bulk energy (per lattice site). The butk properties were obtained by averaging the results obtained for layers from 11 to 20. The number of Monte-Carlo steps (per lattice site) used to calculate averages has been different for different temperatures, and ranged from 2500 to 15 000 in the surfacial region and from 500 to 3000 in the bulk phase. The number of Monte-Carlo steps used in particular runs have been adjusted to produce sufficient statistics. Also, considering that autocorrelation times change with temperature (become longer as the temperature approaches the critical temperature), the actual number of performed Monte-Carlo steps was considerably greater since measurements have been done using configurations spaced by a certain number of Monte-Carlo steps (ranging from 5 to 20). At low temperatures, the simulations produced well-pronounced hysteresis loops at the adsorption-desorption curves, connected with the appearance of metastable states. In order to locate the transition points, we calculated the system surface excess free energies and applied the thermodynamic integration method [34,35]. This can be done by performing Monte-Carlo simulations along different paths. First, one can obtain the free energy differences for different states (corresponding to the different number of occupied layers) with respect to the known ground state energies (see Section 3) for the fixed values of /*A and /x~ as functions of temperature. Then, the changes of the free energies with respect to/x a (for any given temperature) are readily obtained by integrating the corresponding adsorption and desorption isotherms. At temperatures below the critical point, the location of the first-order transition is found at the condition stating that the thermodynamically stable state corresponds to the minimum of the free energy.
relative strengths of the gas-solid and the gas-gas interactions for both components is practically the same; they differ only by the temperature range in which both systems exhibit first-order layering transitions. The system with the weaker gas-solid potential (Va°" = 1.0) shows somewhat different behaviour. The first two layering transitions [18], occurring in layers 1 and 2, merge into one transition corresponding to a simultaneous condensation of the adsorbate in the first two layers. This transition is then followed by a series of layering transitions corresponding to the condensation in subsequent layers 3, 4, etc. Finally, we assumed that the interaction energy between unlike particles, UAB is equal to 0.5(UAA + U~9)= 0.78. This assumption corresponds to ideal complete mixing of both components. All the calculations have been carried out assuming the constancy of the chemical potential of component B. In most cases we have taken the values of /.t~ lower than those corresponding to the first layering transition of pure component B at T* = 0. The ground-state properties of the one-component lattice gas systems considered here are well known [13,t8]. In particular, the system exhibits the sequence of layering transitions: 0, 1, 2, ..., ~ when uxK < Vx(1 ) - V~(2)
and these transitions are located at the chemical potential values given by b~x,,r(0 --+ 1) = --2uxqv + Vx(1 )
(tl)
and
t~x,,r(l--+l+
1) = - 3 u K ~ +
VK(I+
1)
(12)
where K = A or B. For the weaker surface potential, a different sequence of layering transitions may appear. The adsorbate may simultaneously condense in the first n layers adjacent to the solid surface. Then, this transition is followed by the condensation occurring in subsequent layers. Thus, the sequence of the layering transitions reads: 0, n, n + 1. . . . . (n > 1), with n determined by the condition
3. The systems and their ground state properties Here, we assume that the properties of pure component B are fixed and determined by the parameters u~B = 0.56 and VB°* = 1.67. With this choice of interaction parameters, the system containing only particles of component B belongs to the strong substrate regime. Thus, the adsorbate completely wets the surface at any temperature down to T* = 0 and the film develops in a layer-by-layer mode. In particular, at sufficiently low temperatures, a sequence of the first-order layering transitions corresponding to the condensation in layers 1, 2, etc. is observed. For component A, we have considered two different values of the parameter VA°* , 3.0 and 1.0. For the first choice, the
(10)
v (i)
1) >,,K,,.
i=1
> ~ VK(i ) --nVK(n )
(13)
i=J The first transition corresponding to the simultaneous condensation of the adsorbate in the first n layers is located at >K,,,(0--+ n) = -
1[
-(3n-
H
Vx(i )
1),q,.x+
]
(14)
k=l
and the subsequent transitions between m and m + t (m > n) occupied layers are located at the chemicat potential given by Eq. (12).
e t ak / T h i n Solid Films 304 ( 1 9 9 7 ) 3 4 4 - 3 5 2
K. G r a b o w s M
~) AAA
AAB
q
AA
-3
the minimum of the system surface excess free energy (per lattice site) given by
AAAB
AAAA
/--
347
--AABB
fe~,~(/z~,/x~ ) = rn)n {e;x ( k)}
--ABB
(16)
AB --ABBB
A
/
The transition between different states, a and /3, occurs when
--BBB
-4,5 --BBBB
f**<~( IZ~, ~; ) =f~*~,p ( t~ ,tz; )
-5 -5,5
-a
-2,8
-2.6
-2,4
-2.2
AAAJ
-3.05
-1.8
(b)
AAA -3
-2
./
AA
-3,1 • -3.I5 "
--BBBB BB
0
BBB
-3,2 '
i i
(17)
Fig. 1 presents the ground-state phase diagrams for the tWO systems considered in this work. One should note that only the states with a maximum of four occupied layers are included here. At finite temperatures, the assumption of complete ideal mixing of both adsorbates excludes any possibility of sharp transition between the states differing only by the composition of the adsorbed layer, but of the same thickness. An example of such process is the transition between monolayers filled with A or B particles only.
-3.25 i" -3.;1 -3
I -2,8
t
J
i
r
i
-2.6
-2.4
-2.2
-2
-I,8
4. Results and discussion
Fig. I. The ground-state phase diagrams for the systems with V~* = 3.0 (part a) and VA° * = 1.0 (part b).
One can readily check that pure component A with Va°* = 3.0, as well as pure component B both exhibit the sequence of layering transitions: 0, 1, 2 . . . . . % while pure component A with V~* = 1.0 undergoes the transitions: 0, 2 , 3 . . . . . ~. In our model of mixed adsorbate, the only possible stable ordered states at T * = 0 correspond to the films with a different number of completely filled layers. Since we consider only systems with the attractive nearest neighbour interactions between adsorbate particles and assume ideal complete mixing, no other ordered state can appear. Only when the energy of interaction between unlike particles (UAB) is greater than the arithmetic mean of UAA and uas, can other (mixed) ordered states within a single layer appear (see Refs. [36-38]). The surface excess energy for the film with k filled layers is given by
Fig. 2 shows examptes of adsorption-desorption isotherms obtained for the system with Va°* = 3.0 at different temperatures and for the chemical potential of component B equal to - 3.45. It is clear that, at the considered temperatures, the development of multilayer film occurs via a sequence of the first-order layering transitions. In all cases, we find that condensation in each layer is accompanied by rather large changes in its composition. In particular, the increase in the film thickness leads to an increase in the concentration of component A in the adsorbed film. Fig. 3 presents the adsorption isotherm for the system with
3.0 ~ooeeoeomooooooooeooeeooooooooaJo••°°°~
¢~(4 + B)
T " = 0,56
2.0
k
e:~(k)
2 -= E [ - 2 PA,i i=l
, PB,i 2 2 UBB
4U~B Pn,i
o east•
PB,i
T* = O.48
F,,
1.0
- Pa,i P A , i - 1 -- I~;B PB,i P B , i - 1 -- llAB
"-
T* = 0.45
X( f)A,i PB,i-! "{- PB,i PA,i-1) "q- VA* ( i) pA,i + VB* (i) PS,i - IZA PA,i
-- t'@ PB,i]
(15)
where PA,0= PB,O= 0, &,i + PB,i = 1 and PA,i Ps,i = 0 for each occupied layer i. For any given value of the chemical potentials /x~ and /x~, the stable state ( a ) corresponds to
-5.0
-41.6
-4..21
-31.8
I
-3.4
Fig. 2. The adsorption isotherms for the system with I~°* = 3.0 and /z~ = - 3.45 at T* = 0.45, 0.48, 0.52 and 0.56. Note that the results for different temperatures are shifted upward by 0.5 for a better display.
348
K.
(a)
7.0
G.(A
+
Grabowski et al. / Thin Solid Films 304 (1997) 344-352
O0•
B)
~CA
hi'leo ) ,Be•
6.0
0,~
0.9
o
0.8
1.5
-- O~(A)
• --
O=(B)
• - - O=(A + B)
5.0 0.7
~
°
J
D
o
mo I
4.0 1.0
0.6 3.0
QOID
•
oooOO o°
0.5
2.0
0.5 1.0
• o• oe• e ~-...-..o ~t eeo•
-10
eel° • o o o • o o • o e o e • 0 ~ i ~ m ~
-1.5
I
-4.0
0.4 >oO0 ~
-i.5
-4.6
-4.5
-4'.3 -1.2 -4'.1
-4.4
Fig. 3, The total adsorption isotherm and the mole fraction of c o m p o n e n t A in the adsorbed film for the s y s t e m with V ° * = 3.0 a n d ,u,~ = - 3.45 at T * = 0.55.
/x~ = - 3 . 4 5 at T* = 0.55 together with the changes of the mole fraction of component A in adsorbed film (XA), where
G(A) XA= O,x(a + B)
(18)
We observe that in the region of the first plateau, corresponding to one adsorbed layer, the increase of the chemical potential of component A leads to the gradual removal of component B from the film. This is a direct consequence of the stronger interaction between the particles of component A than between any other pair of particles (A and B, or B and B) and of the low chemical potential of component B. The mechanism of displacement of preadsorbed particles B is different when the adsorbed layer is initially rich in component B. For example, when /z~ is higher than the value corresponding to the first layering transition of pure component B (equal to - 2 . 7 9 at T * = 0), the addition of component A leads to the transfer of particles B from the first to the second adsorbed layer. This situation is illustrated in Fig. 4(a) which shows the adsorption isotherm of component A on the preadsorbed monolayer of component B at /x~ = - 2 . 0 and T* = 0.30. From the results given in Fig. 4(a) and from the behaviour of densities of both components in the first two layers (shown in Fig. 4(b)), we find that this system shows a strong effect of enhanced adsorption (studied by Rikvold et al. [36], Rikvold and Deakin [37] and Rikvold [38]) in monolayer films. From the obtained adsorption isotherms and density profiles, we constructed phase diagrams corresponding to different values of the chemical potential of component B in the region of the first three layering transitions. The example corresponding to /x~ = - 3 . 4 5 is presented in Fig. 5 which gives the total surface coverage of coexisting phases versus the temperature for the first three layers. The most striking difference between the obtained phase diagrams and similar results for one-component systems [18]
0.8 - ° o O O
e~
p~(O 0.6
0.4
0.2
~-o--a-,I : ~. . . . . . . . 1. . . . . . . . . ~----m2~'~"]"~
-4.6
-4,5
-4.4
-&3
F
P
-4.2
-4.1
Fig. 4. The adsorption isotherms O~(A + B), Oex(A) a n d O,~(B) for the system with Va° * = 3.0 at T * = 0.30 and /x~ = - 2.0 (part a) and the corresponding densities o f both c o m p o n e n t s in the first t w o adsorbed layers (part b).
is a pronounced increase in the critical point temperature of the layering transitions occurring in subsequent layers 1, 2 and 3. These changes of T,?(I) can be attributed to the
oo~(A+ B) 2.0
1,0
•
•
•
•
nil
el
o ~
•
•
:11|•
°~
.J
---4:i
2:d.3 . . .0.4 .
Fig. 5, The g l o b a l - p h a s e d i a g r a m Oe'(A with Va° * = 3.0 and ~ = - 3.45.
0k.5 T*
+ B) versus T " for the system
K. Grabowski
el a[./Thin
Table 1 Critical temperatures and critical compositions for layering transitions in systems with VA °* = 3.0, for different chemical potentials of component B I
/zB " = - 3.35
/xB * = - 3.45 Tc
1 2 3
Tc *(l) 0.466 0.545 0.552
xt,c(l) 0.59 0.87 0.905
/.z~ = - 3.52
xt,~(l) 0.63 0.87 0.89
"(l)
0.47 0.541 0.544
Tc *(I) 0.495 0.545 0.548
XA,c(I) 0.70 0.90 0.93
Solid Films 304 (1997) 3 4 4 - 3 5 2
349
result is rather obvious and is a direct consequence of the negligence of statistical fluctuations by the mean-field theory and hence a considerable underestimation of •ntropic effects. The second difference is connected with the mean-field prediction concerning the changes of the layering critical temperature with the layer number. It is known [ 11,18,20] that lim T~MF(l) = T f f F ( 3 D )
(19)
l -.., ~¢
differences in the composition of each layer near the corresponding critical point. In Table 1, we sun'unarize the values of T / ( l ) for three different choices of #~. We have also determined the layering critical temperatures using the mean-field theory described in Ref. [31]. The results for the first three layering transitions are shown in Fig. 6(a). Two important differences between the results of the Monte-Carlo simulation and the mean-field predictions should be emphasized. First, we observe that the mean-field theory gives considerably higher values of the layering critical temperatures than the Monte-Carlo simulation. This
(~)
z2M~(1) 1.0
lol---
.
I=2 ........
,
0.9 /
/
// ../
0.8 0.7 /..2
0.6 -~,~
0'.2
0'.4
O..6
0'.8
where T~MF(3D) is the bulk mean-field value of the critical temperature of the adsorbate. In the lattice gas model considered here, the mean-field bulk critical temperature of the pure component K ( K = A , B) is given by T~MF(3D) = 1.5 UKK, while the critical point for the first layering transition T~m'(1) is equal to UKK. In the binary system exhibiting complete mixing, the locations of the layering critical points depend, not only on the layer number, but also on the composition of each adsorbed layer. Also, the bulk critical temperature changes with the bulk phase composition. For the systems considered here, exhibiting complete mixing of both components and with the parameter UAB given by the arithmetic mean of u~, A and UBB, the critical temperature in any given layer should change monotonically with the composition of this layer, between the values corresponding to the critical points for pure components A and B, and with only rather small deviations from linearity. In particular, for the first adsorbed layer, the exact value of critical temperature is known and equal to T ~ ( 1 ) = 0.5674U~K, where K = A or B. We also know that, in the one-component version of the lattice gas model considered here, the roughening temperature is only about 5% higher than the two-dimensional critical temperature [12]. Thus, the observed changes in the layering critical temperature with the layer number can be attributed to the changes in the composition of subsequent layers.
.;CA
(a) .... /
¢'--l=2
r;(o ] A _
t
=3
<9'
0.5
3
-3.05 •
o
•
-3.10 0
c~.~'
%0
I
-3.15
(b)
3
0.4 -3~05
2
#,4,tr
•
-3.10
0.4
0.6
0.8
Fig. 6. The dependencies between the layering critical temperatures and the composition of each layer at the critical point obtained from the mean-field theory (part a) and from Monte-Carlo simulation for the systems with VA ° * = 3.0 (Table 1) and VA °" ~ 1.0 (Table 2) (part b).
~*' • °
~" " ' " *'• • • • ~ 0
0.2
•
•
i
eeoc
-3.15 0'.i
O'.2
O'.3
o'.4
o'.5 T* Fig. 7. The phase diagrams /.z~,cr vs. T * for the systems with VA ° * = 1.0 and two choices of #~ equal to -3.45 (part a) and -3.30 (part b).
350
K. Grabowski et aL / Thin Solid Films 304 (1997) 344-352
Table 2 Locations of triple points and for systems with V[" = 1.0 and different values of the chemical potential of component B IzB *
2.0
,
. . . .
.~::: e
1.0 o
'
0'.1
~2 ~ 63
•
41
~
4,
~e
• • IQb I
o , ( A + B)
(bt 2.0
.
.
.
.
~,.,. N,, ; :,~ I I I
l [ [
1.o
~...~.. • .~' I l l l I
0'.1
~.2 : ~.3
tza., *
l
Oex(A)
Oex(B)
Oex(A + B)
xa
0.053 0.786 1.816 0.017 0.778 1.824 0.005 0.758 1.828 0.00047 0.744 1.856
0.038 0.187 0.168 0.022 0.209 0.170 0.013 0.236 0.170 0.004 0.255 0.144
0.091 0.973 1.984 0.039 0.987 1.994 0,018 0.994 1.998 0,00447 0.999 2.000
0.582 0.808 0.915 0.436 0.788 0.915 0.278 0.763 0.915 0.105 0.745 0.928
0.5 T*
0.4
&
Tt *
-3.60 0.475 -3.0988 0 1 2 -3.45 0.415 -3.0967 0 1 2 -3.30 0.350 -3.0920 0 1 2 -3.15 0.262 -3.0830 0 1 2
0.4
or.5 T* Fig. 8. The phase diagrams O,,(A + B) vs. T* for the systems with VA °" = 1.0 and two choices of ,a~ equal to -3.45 (part a) and --3.30 (part b).
This is not true for the mean-field results, however. In this case, Eq. (19) holds and one expects rather large changes in the layering critical temperatures with the layer number. Within any given layer, the changes of Tc ~ M F ( l ) with the layer composition should be similar. The results presented in Fig. 6 confirm the above predictions very welt. The values of To*(1) deduced from Monte-Carlo simulation change nearly linearly with x A for all transitions considered here. The corresponding values of T~.*(2) a n d To* (3) exhibit only small positive deviations. On the other hand, the values of To* g F ( l ) show much larger changes with the layer number I, for which Tc*MF(l) changes smoothly with the adsorbate composition (see Fig. 6(a)). In the case of the adsorption system with VA°* = 1.0, the addition of the second component, B, leads to much more serious changes in the behaviour of adsorbed films at finite temperatures. Figs. 7 and 8 present the phase dia-
grams (/ZA,tr, T* ) and (0~x(A + B), T* ) obtained for two different values of /z~, - 3 . 4 5 and - 3 . 3 0 . We know that in the system containing pure component A, the first layering transition involves simultaneous condensation in the first and second layer [18]. In the mixed systems, this picture is preserved only at sufficiently low temperatures, lower than a certain triple-point temperature, Tt*. Above Tr", the presence of component B leads to the splitting of this transition into two first-order layering transitions occurring separately in the first and the second layer. The location of the triple point depends strongly on the value o f / x ~ , i.e. on the composition of the adsorbate. In Table 2, we give the triple-point temperatures and the compositions of the coexisting phases for systems characterized by different values of the chemical potential of component B. Table 3 contaius the critical temperatures and the critical compositions of the layering transitions for three selected systems. Fig. 9 gives the plots of Tr* and T~*(1) against the chemical potential of component B. It is clear that the triple-point temperature meets the critical point for the first layer at a certain value of the chemical potential of component B. Here we estimate that this occurs for/x~ = - 3 . 7 . On the other hand, the temperature of the triple point goes to zero when the chemical potential of component B
Table 3 Layering critical temperatures and compositions for systems with Va°* = 1,0 and different values of /zB * ~z~ *
t
T~ *(l)
Xc.A(I)
- 3.30
1 2 3 1 2 3 1 2 3
0.46 0.545 0.552 0.482 0.552 0.555 0.502 0.553 0.556
0.58 0.87 0.905 0.67 0.9 0.92 0.75 0.91 0.93
- 3.45 - 3.60
K. Grabowski et aL / Thin Solid Films 304 (1997) 344-352
o - r;(1) ,,=T[ 0.5
*. r-.e." '"'o ...... O , ,
T*
%
"",-¢.,
0.4 "e
0.3
0.2
0.I
-3.6
-3.4
-3.2
-3~.0
-218
~5 Fig. 9. The dependencies of the triple-point temperatures and the first layering transition critical point temperatures on the chemical potential of component B for the systems with VA°* = 1.0. * denotes the estimated temperature where the triple point and the critical point temperatures meet, while ~ denotes the location of the first layering transition of pure component B at T* = 0.
approaches the value corresponding to the first layering transition of pure component B, equal to - 2.79 (at T* = 0). Again, the critical temperatures for the first layering transition are quite consistent with the dependence on the adsorbate composition found for VA° * = 3.0, while the critical points for higher layering transitions exhibit small positive deviations from this dependence (compare Fig. 6(b)). In conclusion, we would like to stress that even in very simple systems, exhibiting ideal mixing of both components, their behaviour may be quite complex and very sensitive to the interactions operating in the system and to the composition of the adsorbate. Some of the presented results can be readily anticipated from the behaviour of simple one-component systems, but we also find some qualitatively new results. The most interesting finding of this paper is the demonstration of triple points in mixed films. In our earlier study of one-component films [18], we have shown that the triple points exist only within a narrow interval of the relative strength of the adsorbateadsorbate and adsorbate-solid interactions. We have demonstrated that this regime can also be met by addition of the second component, characterized by the different relative strength of the adsorbate-adsorbate and the adsorbate-solid interactions. In binary systems exhibiting complete mixing, we can determine the effective strength of the adsorbate-adsorbate and the adsorbate-solid interactions by defining the parameter ,,,,,,x I + 2,t,,~ < , ( 1 - x,,) + t , ~ ( 1 ~,,(~o)
=
and hence the value of y ( x A) changes with temperature as it is demonstrated in Fig. 10. We know that in one-component systems, the triple points appear for the ratio uxx/V~ ( K = A or B)just slightly below the value delimiting the reNmes leading to the sequences of layering transitions: 0, 1, 2, 3, etc. and 0, 2, 3, etc. at the ground state. The same situation appears in mixed systems. Pure component A with VA°* = t.0 gives the sequence of layering transitions: 0, 2, 3, etc. at any temperature below the corresponding critical point; pure component B gives the sequence: 0, t, 2, 3, etc. The addition of component B modifies the effective interactions and shifts y ( x a) towards lower values. For a sufficiently high concentration of B, we reach the region in which the splitting of the layering transition, 0 ~ 2, into two separate transitions, 0 ~ 1 and 1---* 2, occurs at finite temperature, leading to the appearance of a triple point. In the light of the above discussion, it is clear that the triple-point temperature should decrease as the concentration of component B increases. It should be emphasized that the results presented in this paper are only preliminary. In particular, our choice of the parameters describing adsorbate-adsorbate interactions exclude any possibility of the formation of mixed ordered states in a single adsorbed layer [36-38], the existence of azeotropic states, and phase separation phenomena [39-42]. We have not attempted any finite-size scaling analysis. However, this does not in any way affect the qualitative picture emerging from our study. Of course, our estimations of the layering critical temperatures are inevitably subject to systematic errors due to the smallness of the computational cell. Considering, however, that our aim here was to present qualitative effects resulting from the changes in the composition of the adsorbate and not any direct quantitative comparison with real systems, we believe that the negligence of finite-size effects analysis is justified. The use of larger systems would require much more computing time and would not give any deeper insight into the physics of the problem. °°°Oo%
(20) One should note that even for the fixed chemical potentials of both components, the composition of adsorbed film
0.99
ee
G(A + B)
0.9
0.98
,"
,'
0.97
° o%
, '
0.8 '
0.8
%e %o
0.96 %e
0'.I
v;):~ + W(1 -x~)
***~**** eo
.r A
01,8
- xA) ~
351
0.9 2: A 0.2
0.3
0r.4 T*
Fig. 10. Temperature changes of the total surface coverage ( O ~ ( A + B)) and the mole fraction of component A(x a) for the system with Va° * = 1.0, VB° * = 1.67 at /~,~ = - - 3 . I 1 and /z~ = - - 3 . 4 5 . The insert shows the changes of the parameter y ( x A) with the concentration of component A.
352
K. Grabowski et aL / Thit, Solid Fibns 304 (1997) 344-352
A n o t h e r open question i n v o l v e s wetting p h e n o m e n a in m i x e d systems consisting o f c o m p o n e n t s such that one wets the surface while the other does not. It is very unfortunate that the experimental study o f multilayer adsorption o f binary mixtures meets so m a n y difficulties and it is hard to find suitable data that can be c o m p a r e d with theoretical predictions.
References [la] A. Thorny, X. Duval, J, Chim. Phys. 66 (1969) 1966. [lb] A. Thorny, X. Duval, J. Chim. Phys. 67 (286) (1970) 110t. [2] J. Menaucourt, A. Thorny, X. Dural, J. Physiol. (Paris) 38 (1977) C4-195. [3] A. Thomy, X. Dural, R. Regnier, Surf. Sci. Rep. 1 (1981) 1. [4] L.J. Sequin, J. Suzanne, M. Bienfait, LG. Dash, J+A. Venables, Phys. Rev. Lett. 51 (1983) 122. [5] M. Sutton, S.GJ+ +Mochrie, PCJ. B~rgenau, D.E. Moncton, P.M. Horn, Phys. Rev. B 30 (1984) 263. [6] Y. Larher, F. Angerand, Y. Manrice, J. Chem. Soc., Faraday Trans. 1 83 (1987) 3355. [7] H.S. Nham, G.B. Hess, Phys. Rev. B 38 (1988) 5166. [8] P+ Pfeifer, Y.J. Wu, M.W. Cole, J. Krim, Phys. Rev. Lett. 62 (1989) 1997. [9] H. Tillborg, A. Nilsson, B. Hernniis, N. M~rtensson, R.E. Palmer, Surf. Sci. 295 (1993) 1. [10] H.S. Youn, X.F. Meng, G.B. Hess, Phys. Rev. B 48 (1993) 14556. [11] M.J. de Oliveira, R.B. Griffiths, Surf. Sci. 71 (i978) 687. [12] I.M. Kim, D.P. Landau, Surf. Sci. 110 (1981) 415. [13] R. Pandit, M. Schick, M. Wortis, Phys. Rev. B 26 (i982) 5112. [14] C. Ebner, Phys. Rev. B 28 (1983) 2890. [15] K. Binder, D.P+ Landau, Phys. Rev. B 37 (1988) t745. [16] W.F. Saam, Surf. Sci. 125 (1983) 253.
[17] C. Ebner, Phys. Rev. A 22 (i980) 2776. [18] A. Patrykiejew, D+P. Landau, K. Binder, Surf. Sci. 238 (1990) 317. [19] D.E. Sullivan, M.M. Telo de Gamma, Fluid Interfacial Phenomena, in: C.A. Croxton (Ed.), Wiley, Chichester, 1986. [20] M. Kruk, A. Patryldejew, S. Sokolowski, Thin Solid Films 238 (1994) 302. [2I] W. Gac, M. Kruk, A. Patrykiejew, S. Sokotowski, Langmnir 12 (1996) 159. [22] M.A. Hamdi Alaoui, J.P. Coulomb, N. Dupont-Pavlovsky, I. Mirebeau, B. Mutaftschiev, Surf. Sci. 295 (1993) L1031. [23] J.H. Singleton, G.D. Halsey Jr., J. Phys. Chem. 58 (i954) 330. [24] M. Abdelmoula, T. Ceva, B. Croset, N. Dupont-Pavlovsky, Surf. Sci. 272 (1992) 167. [25] H. Asada, S. Doi, H. Kawano, Surface Sci. Lett. 265 (1992) L233. [26] H. Asada, S. Doi, H. Kawano, Surface Sci. Lett. 273 (1992) IA03. [27] D.A. Slater, P. Hollins, M.A. Chesters, Surf. Sci. 306 (1994) 155. [28] A. Patrykiejew, S. Sokotowski, T. Zientarski, H. Asada, Surf. Sci. 314 (1994) 129. [29] N. Dupont-Pavlovsky, M+ Abdelmoula, S. Rakotozafy, J.P. Coulomb, B. Croset, E. Ressouche, Surf. Sci. 317 (1994) 388. [30] J. Menacourt, C. Bockel, J. Physiot. (Paris) 5I (1990) 1987. [31] M. Kruk, A. Patrykiejew, S. Sokotowski, Surf, Sci. 340 (1995) 179. [32] F. Hommeril, B. Mutaftschiev, Phys. Rev. B 40 (1989) 296. [33] K. Binder, D.P. Landau, J. Appl. Phys. 57 (i985) 3306. [34] W. Helbing, B. Diinweg, K. Binder, D.P. Landau, Z. Phys. B 80 (1990) 401. [35] K. Binder, Monte-Carlo Methods in Statistical Physics, in: K+ Binder (Ed.) Springer, Berlin, I986, p. 1. [36] P+A. Rikvold, J.B. Collins, G.D. Hansen, J.D. Gunton, Surf. Sci. 203 (1988) 500. [37] P.A. Rikvold, M.R. Deakin, Surf+ Sci. 0~49(1991) 180. [38] P.A+ Rikvold, Electrochlm. Acta 36 (1991) 1689. [39] A. Patryldejew, Thin Solid Films 147 (1987) 333. [40] M. Kaburagi, T. Tonegawa, K. Ebina, Surf. Sci+ 242 (1991) 107. [41] J. Sivardiere, Phys. Rev. A 11 (1975) 2090. [42] N.K. Mahale, M.W. Cole, Surf. Sci. 176 (1986) 319.