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The law of corresponding states for chemisorbed layers with attractive lateral interactions
333
Surface Science 151 (1985) 333-350 North-Holland, Amsterdam
THE LAW OF CORRESPONDING STATES FOR CHEMISORBED LAYERS WITH ATTRACTIVE LATERAL INTERACTIONS J. KOLACZKIEWICZ
* and E. BAUER
Physikalisches Insriiul, Technische Unruersiiiit ClausthaI, D - 3392 Clausthal- ZeNerfeld, Fed. Rep. of Germany and Sonderforschungsbereich 126, Giittingen - ClausthaI, Fed. Rep. of Germany Received
16 September
1984; accepted
for publication
19 November
1984
Chemisorbed layers which form two-dimensional islands at low temperature consist in general of two phases: condensed phase and a gaseous phase. The coexistence region of the two phases is derived from work function change data of Ni, Cu, Pd and Ag on W(110). The experimental coexistence lines are compared with those obtained from various equations of state with the result that a law of corresponding states is found in the form of the reduced Van der Waals equation or of the quasichemical approximation.
1. Introduction The theory of phase transitions in two dimensions as well as the experimental study of such transitions in physisorbed layers has reached a high level of sophistication in recent years [l]. Rather detailed information including critical exponents has been obtained in several cases, mainly noble gases on graphite. Much less is known about phase transitions in chemisorbed layers [2]. Practically all work has been concerned with adsorbates with repulsive lateral interactions which usually produce several ordered structures with order-order and order-disorder transitions. Many metal adsorbates with large dipole-dipole repulsion such as the alkalis, alkaline earths and rare earths on metal surfaces with large work functions fall into these category [3]. Very little is known about the phase state of metal adsorbates with attractive lateral interactions. Thus, on finds often in the literature the statement that the adsorbate is disordered if the geometry of the diffraction pattern of the substrate does not change during adsorption.
* Permanent address: Institute of Experimental 36, PL-50-205 Wroclaw, Poland.
Physics,
University
0039-6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
of WrocIaw,
B.V.
UI. Cybulskiego
334
J. Kofaczkiewicz,
E. Bauer / Law of corresponding states
It is the purpose of the present paper to correct this situation by a detailed study of the phase state of metal adsorption systems which are known or expected to be governed by attractive lateral interactions. The study makes use of a recently developed method [4] which is based on the Smoluchowski effect [5]: isolated atoms on a flat surface have a large dipole moment while atoms incorporated into a two-dimensional island on such a surface have a comparatively small dipole moment. Thus a study of the temperature dependence of the work function of a surface covered with such an adsorbate gives information on the phase state of the adsorbate. Preceding thermal desorption spectroscopy studies [6-81 have shown that the desorption energies of Cu, Ag, Au and Pd from W(110) surfaces increase with coverage 8, i.e. that the lateral interactions are attractive, and the same is expected by analogy also for Ni. The structural properties of these layers are also well known [7,9,10]: the lateral periodicity of the adsorbate is either identical (Cu, Pd and (for 8 =<0.7) Ni) or very close to that of the substrate (Ag, Au). Thus a lattice gas model seems appropriate to describe .the adsorbate. On the other hand, the W(110) surface is rather smooth so that a two-dimensional gas model appears equally reasonable. The study of the phase state of a chemisorption system differs in general fundamentally from that of a physisorption system. In physisorption studies the adsorbate is usually in (quasi) equilibrium with the surrounding gas phase, in chemisorption studies the adsorbate is bound so tightly to the substrate that the gas from which it was produced can be pumped off and the adsorbate/substrate system may be considered as a closed system. Before a meaningful analysis of the phase state data of a chemisorption system can be made it is, therefore, first necessary to analyze critically the consequences of the experimental differences between physisorption and chemisorption studies. This is the subject of the next section. In section 3 the experimental procedures will be described briefly, section 4 reports the experimental results and their analysis. The concluding discussion is the subject of section 5.
2. Theory In physisorption systems the phase state of the adsorbate is most directly obtained by measuring adsorption isotherms which are determined by the equality of the chemical potentials of gaseous and adsorbed phase, ps = pLa.ps is determined by the pressure p and temperature T of the gas phase (assumed to be perfect), ps = kT ln( p/pa), pa by adsorbate coverage 8, surface temperature T, = T, lateral interaction energy E and the specific model of the adsorption layer. Thus a characteristic adsorption isotherm p = p( 6) is obtained for each model. In a chemisorption system in which the desorption rate is negligible in the (T, 6) range considered p (0) cannot be measured. However,
J. Koiaczkiewicr,
335
E. Bauer / Law of corresponding states
the system may be considered as a closed system just like a condensable gas in three dimensions with negligible diffusion through the walls of the container in which it is enclosed. Just like the three-dimensional system, the two-dimensional system can now be described by an equation of state which expresses the relationship between two-dimensional pressure (“spreading” pressure) 7~, B and T. Again the relationship r = a(&J, T) depends on the lateral interactions and the specific adsorbate model. 7~ is linked with pa by the Gibbs adsorption isotherm dp = a dn = (a,,/B)dm where a is the area available for each adsorbed atom (molecule) and a,, the area occupied by each adsorbed atom (molecule, “co-area”). In table 1 the equations of state +rr= ~(8, T) are compared with the corresponding adsorption isotherms p(8, T) for the specific adsorbate models used later in the comparison with experiment. For the derivations the reader is referred to elementary introductions [ll] and standard text books [12]. Two of the models, Van der Waals’ and Dieterici’s (ref. [12a], p. 273), assume nonlocalized adsorption, the other two, the Bragg-Williams and the quasichemical approximation (ref. [12a], p. 1006), assume localized adsorption. The energy parameter of the various models is hidden in the critical temperature T,, the critical coverage 0, is l/2 except for the first model for which 8, = l/3. 8 = a,/a = Na,/A or 8 = N/N, for non-localized and localized adsorption, respectively. PO*=po
exd-E&T) j,(T)
with X = (2mkT)‘/‘/h,
Table 1 Comparison
= kT.!g(T)exp(_Eo,kT)
of adsorption
Model
Adsorption
A3j,(T) j,( j,)
isotherms
being
the partition
and equations
isotherm
function
of state of various Equation
for the internal
adsorption
layer models
of state
Van der Waals
Dieterici
Not expressible
with elemen-
tary functions Bragg- Williams
$=fln c
Quasichemical
& c
i
1
-2e2
336
J. Kolaczkiewicz, E. Bauer / Law of corresponding states
degrees of freedom of the molecule in the gaseous (adsorbed) state and E, being the desorption energy in the zero coverage limit. z is the number of nearest neighbour sites: z = 6,4, 2 on surfaces with 6-, 4- and 2-fold symmetry. Two-fold symmetry is appropriate, e.g. for fcc(ll0) and bcc(211) metal surfaces. For the 2-fold symmetric bcc(ll0) metal surface, which may be considered as a
t 6-
E 1.4 k
.2
0
0
.2
.4
.6
6-
.8
1 .I3
fs.4
c
G
R
.
.2
0 0
.4
.2
-2
.4
9-----
.6
.8
0t 0
l\\\ .2
I 4
\
\a 6
8
8-
Fig. 1. Isotherms v( 0, T) of the two-dimensional Van der Waals equation (a), Dieterici equation (b), Brad-Williams approximation (c) and quasiche~~ appro~mation (d). i”, ranges from OST, to T, in steps of O.O5T,. Also shown is the coexistence region of two-dimensional gas and condensate according to Maxwell’s construction.
J. Kolaczkiewicz,
E. Bauer / Law of corresponding states
331
somewhat distorted hexagonal surface, z = 6 will be used. Fig. 1 shows the isotherms ~(8, q) of the equations of state of table 1 for T/T, = 0.5, 0.55,. . .) 0.95, 1.0. There is apparently little difference between the two localized adsorption models (c) and (d), while the two non-localized adsorption models (a) and (b) differ strongly from each other and from the localized adsorption models. The differences change significantly if the boundary lines between coexistence and single-phase regions as obtained from Maxwell’s construction (horizontal lines in fig. 1) are plotted as a function of the reduced coverage 8/e,. This is done in fig. 2 for the gas/gas + condensate boundary (e/8,, < 1) which is of main interest in this work. It is evident that the quasichemical and Van der Waals approximation have a very similar gas/gas + condensate boundary line - called coexistence line in the following - while the Bragg-Williams coexistence line differs significantly from both and the Dieterici coexistence line even more. Also shown is the empirical Guggenheim coexistence line [13] B/B, = 1 + ;(l
- T/T,)
- $(l - T/q)1’3
which gives a much better description of many real gases in three dimensions than the Van der Waals, Dieterici and other semi-empirical equations of state.
G
0
0.2
GUGGENHEIM
0.4 e,~, 0.6
0.8
1.0
Fig. 2. Boundary he between coexistence region (gas + condensate) and gas for T/r, > 0.5. D denotes Dieterici, W Van der Waals, B Bragg-Williams and Q quasichemical approximation for z = 6. G is the empirical Guggenheim curve.
338
J. Koiaczkiewicz,
E. Bauer / Law of corresponding states
The Guggenheim equation is a good approximation even close to the critical point: the critical exponent /3 = l/3 of the order parameter is close to the value /3 = 0.312 of the three-dimensional Ising model. The aim of this work is to measure the gas/gas + condensate coexistence lines of condensable two-dimensional gases and to compare them with those of the equations of state which were just discussed in order to determine which one of them - if any - provides an approximate description of experiment. In this comparison several things must be kept in mind. (i) Like in three dimensions it cannot be expected that the equations of state of fig. 1 will agree with experiment using the critical parameters predicted for the various models. However, the equations v/~T, =f(T/T,, e/0,) and T/T, = g( d/l?,) should be the same for all similar materials, expressing the law of corresponding states, with characteristic rC, T, and 19,values for each material. (ii) The critical region is in some cases not accessible to the experimental technique used in this work or the accuracy of the measurements is too low in this region as to allow a reliable determination of the coexistence curve close to T,. The comparison between experiment and theory has, therefore, to rely on the agreement over a wide (0, T) range.
3. Experimental procedure The which A+. If of the dipole
experimental technique measures a quantity as a function of 0 and T is related to the phase state of the adsorbate, the work function change the adsorbate consists of a mixture of monomers, dimers, trimers, . . . and condensed phase with n, , n 2, n *, . . . and no atoms per unit area and the moments per atom pi, pz, p3,. _. and pO, respectively, then A+ is given by
A+=
-4ne
c
nip,.
i=O
Extensive measurements of the B and T dependence of A$ [8] show that the dipole moments pi decrease rapidly with i (i > 0) so that the best defined A+ changes upon condensation occur at low coverages and high temperatures at which the gas consists only of monomers and dimers. The gas/gas + condensate boundary can therefore be studied particularily well. However, even at high coverages (0 > 0.6) distinct changes of A$ with 8 and T occur which can be assigned to the gas + condensate/condensate boundary line on the basis of their r3 and T dependence. This boundary has been studied in much less detail and will be discussed only briefly here. The experimental arrangement for the A+(T) measurements is extremely simple. It is a retarding field electron beam system which is operated in the constant current mode. This mode is achieved with a standard constant voltage source and a 2 GO resistor in series. The voltage change necessary to maintain
J. Kolaczkiewicz,
E. Bauer / LAW of corresponding states
339
a constant current is equal to A+/e. It is measured with a battery-operated electrometer whose output is displayed on the y axis of an xy recorder versus thermocouple output on the x axis. Typical data of this type are shown in fig. 3 for Cu in the T range from 400 to 900 K over a wide coverage range. All curves were taken during cooling and terminated at about 400 K in order to avoid excessive cooling times. It is evident that in all cases except at the highest coverage (19 > 2) the A+ change at 400 K is still significantly larger than that expected for clean metal surfaces. For example d+/dT = 8 x 1O-6 eV/K for clean W (curve 0) or d+/dT < 5 X lop6 eV/K for 2 monolayers (ML) of Cu (last curve) which consist of two slightly distorted Cu(ll1) planes. This does not mean, however, that there is still a significant number of atoms condensing with further decreasing temperature, because even the complete monolayer has an unusually large d+/dT -4 10 X lop5 eV/K. With 8 decreasing from 8 = 1 d$/dT at 400 K decreases so that the d+/dT seen at that temperature may be attributed mainly to the d+/dT of the islands of the condensed phase. In the very low coverage range (6 = 0.1-0.3, depending upon adsorbate) the temperature dependence of A$ is more complicated than that shown in fig. 3, as illustrated for Pd in fig. 4 and for Au elsewhere [4]: the rapid decrease with increasing T of the low coverage curves seen already in fig. 3 terminates in a slow linear decrease with further increase of T. The slope of the linear decrease
I I 800 700 TCMDCD*TIIDC ILIIrLnl+l"nL IYI,1\, I
400
500
600
TEMPERATURE
700 IKI-
800
I
4(IO 500L0600
900
Fig. 3. Work function change with temperature of a W(110) surface with various Cu coverages in units of 1.2 X 1014 atoms/cm*.
340
J. Kolacrkiewicz, E. Bauer / Law
of corresponding states
is constant up to a certain coverage 8, which depends on the adsorbate - for Pd 6’, = 0.06 - and increases with increasing 0 above 8,. On the basis of a detailed A+(8, T) study [8] the coverage-independent d+/dT value may be assigned to monomers, the increasing d+/dT value to a mixture of monomers and dimers. In any case, the linear part of A+(T) is typical for the single phase (gas) region while the curved part corresponds to the two phase region in which with increasing T an increasing number of atoms vaporize into the two-dimensional gas until all condensate has vaporized. The intersections of the extrapolations of the linear and curved parts of A+(T) thus determine the boundary between gas and gas + condensate region, i.e. the coexistence curve. In this manner the data presented in section 4.1 are obtained. Before presenting these data some experimental details concerning sample preparation and characterization should be mentioned. All experiments were performed at about 6 x lo- ” Torr on W(110) surfaces which were oriented within 0.05” of the (110) plane. The sample was cleaned in the usual way heating in lo-’ Torr 0, and flashing in UHV to 2300 IS - until carbon contamination was less than 10m3 ML as determined by AES. The metals were evaporated from W wire loops heated by well stabilized current supplies which produced very constant deposition rates over long periods. The monolayer coverages were determined from the breaks in the Auger amplitude versus deposition time curves as described in several papers (e.g. refs. [9,10]), combined with LEED. Submonolayer coverages were obtained in various ways:
PARAMETER:
I
TEMPERATURE(K)
-
Fig. 4. Work function change with temperature of a W(110) surface with various Pd coverages. The Pd coverages and corresponding room temperature A+ values (+(O, 300 K)- $J(0, 300 K)) are given as curve parameters. All curves have been shifted to coincide at 430 K, the lowest temperature reached during cooling.
J. Kolaczkiewicz,
E. Bauer / LAW of corresponding stales
341
from the deposition time, from the Auger amplitudes or from the work function change after calibration by AES, depending upon what method was most accurate for the metal studied. Thus a relative coverage accuracy of several thousandths of a monolayer could be obtained at small coverages with a series of small identical deposition doses while the absolute 8 accuracy over a wider coverage range was 0.01-0.02 ML. The crystal was heated by a well isolated floating power supply and the temperature was measured with a W5%Re-W26%Re thermocouple to *5 K. The maximum temperature was chosen such that no thermal desorption occurred which was verified by A+(T) reproducibility tests and coverage measurements before heating and after cooling. 4. Results and discussion 4.1. The low coverage range Fig. 5 shows the boundary lines between gas and gas + condensate for Cu, Ni, Ag and Pd on W(110) as obtained from the kind of data shown in fig. 4 in the manner described in section 3. The points are from several series of experiments and scatter so much not because of reproducibility difficulties but mainly because it is difficult to locate the intersection between the extrapolated curved and straight parts of A+(T) with better accuracy. Nevertheless, the scatter is small enough for a meaningful comparison with the various theoretical coexistence curves of fig. 2. Although only the best fit Van der Waals curve is shown in fig. 5, it was verified that the quasichemical approximation curve did not fit quite as well - the curvature is somewhat too large - that the Bragg-Williams approximation was worse and the Dieterici equation out of question. Surprisingly, the Guggenheim equation which was so successful for many gases in three dimensions does not fit at all. The T, and 6, values obtained by the fit with the Van der Waals equation are shown in table 2 together with the T, values derived from thermal desorption spectroscopy (cTDS) [8]. Th e a g reement between the T, values is within the present limits of error of the two techniques. Except for Ag, the 0, values are significantly smaller than the ti, values of the Van der Waals equation, 0, = l/3, in particular for Ni and Cu. The difference between the /3, values obtained by fitting from that of the model are even larger for the quasichemical approximation (0, = l/2) because they are only a few percent larger than those shown in table 2. If only the quality of the fit over a wide temperature range is used as a criterion, the Van der Waals equation and quasichemical approximation give a comparably good law of corresponding states over the (0, T) range which could be studied. If, however, also the 0, difference is taken into account, the Van der Waals equation is significantly better.
J. Koiaezkiewicz, E. Buuer / Law of corresponding states
342
0
01
. a2
0.3
0.4
0.9
‘1
I
1200 -
: x
I I
7
llOO-
1
1r s.
z
I- toooE r;i
VAN CIERWAALS
~6
4 5 900t-
c
Ni/W (110) Tc =t&OOK e, 0.205 VAN DER WAALS
;2-
q
-a6 EOO0
0.2
0.L
0.6
I 1 0.06 0.02 O.Oh COVERAGE 8 IMONOLAYERSI-
I a08
0.8
VAN DER WAALS
- 05 0 C@4ERAGE0ihONOLA&---
0.3
0
CWERAGE 8 CM!hLAYERU-a*
Fig. 5. Coexistence line (boundary between single and two-phase region) of Cu, Ni, Ag and Pd on W(110) as obtained from A+(T) data as shown in fig. 4. The lines are the best fits of the Van der Waals equation of state which were obtained with the indicated critical parameters T,, 0,.
343
J. Kofaczkiewicz, E. Bauer / Law of corresponding states
It may be argued that in view of the deviation of the corresponding states 8, values (table 2) from the model 0, values (l/3, l/2) it is meaningless to draw further conclusions both from the 8, and c values. This may be true for the absolute values of the quantities derived from 6, and T, but trends in the series of atoms studied should not be obscured by this problem. If to a first approximation three-body forces are neglected and the simple Lennard-Jones pairwise interaction potential with the hard core diameter (I and the depth E* of the potential well is assumed, then these potential parameters may be expressed by T, and 8, as follows: c2 = (45/16)kT, and u = (+aN18,)-“2. The T,, 8,, N, values and the cZ and u values derived from them are shown for Cu, Ag, Au [4], Ni and Pd in Table 2. The table also contains the corresponding c! and ub values derived from bulk properties, the ratios e,/ct and u/ub, the to the maximum number maximum desorption energies Edmax- corresponding of lateral bonds - the zero coverage limit of the desorption energy Ed0 corresponding to no lateral bonds - both obtained from thermal desorption spectroscopy [7,8], the difference A Ed = Edmax- Ei, the bond energy ~oc of the quasichemical approximation which is obtained from T, via the relation cot = 2kT, ln[(z/z - 2)] assuming z = 6, and the ratio A Ed/cw. It is interesting to note that for all adsorbates e,/ct = 2/3 but that u/ub Table 2 Characteristic parameters of metal adsorbates on W(110); the symbols are explained in the text
cu
A8
T, (K)
1170
980
0: TTDS (K) N, (lo-l4 cm-‘) - z2 (ev) a (A) - G (ev) 0b (A) c,/G a/oh E.$‘= (eV) Ed0(eV) A Ed (eV) -cw(eV) - A&i/coc cS (ev)
1150 0.215 14.1 b, 0.284 2.646 0.409 2.338 0.69 1.13 3.85 o 3.2 n 0.65 0.082 7.9 0.72
985 0.35 13.6 =) 0.238 2.111 0.345 2.644 0.69 0.80 3.55 o 2.8 0 0.75 0.069 10.9 0.71
a) b, ‘) d, e, o
Ref. Ref. Ref. Ref. Ref. Ref.
[4]. 191. [15]. [IO]. 17). [8].
AU 1130 a) 111800.26 =) 13.4 =) 0.274 2.468 0.441 2.637 0.62 0.94 4.1 I-I 3.35 o 0.75 0.079 9.5 0.82
Ni
Pd
1400
1170 _
1375 0.205 17.8 (14.1) d, 0.339 2.411 (2.709) 0.520 2.282 0.65 1.06 (1.19) 4.95 o 4.4 0 0.45 0.098 4.6 0.97
0.28 14.1 e’ 0.284 2.318 0.427 2.520 0.66 0.92 4.4 e’ 3.6 e’ 0.8 0.082 9.8 0.90
344
J. Kolaczkiewicz, E. Bauer / Law of corresponding slates
changes strongly from adsorbate to adsorbate. That the e2 values for the lateral interactions are smaller than in the bulk is easy to understand. The adsorbate atoms are bound much stronger to the substrate atoms than to atoms of their own kind so that fewer electrons are available for lateral bonding which reduces 6*. Quantitatively it is surprising that these reductions are so similar in spite of the differences in the electron configuration and in adsorbate-substrate bond strength as expressed in the desorption energies [8]. Surprising is also the result that the hard core diameter (I of the adsorbed atom may be larger than its bulk value. One would expect that the polarization normal to the surface resulting from the adsorbate-substrate bond would reduce the lateral dimensions of the adatom as is the case for Ag, Au and Pd. For Cu and Ni, however (I > ub, in particular for Ni (a/ah = 1.19 if iVi = 14.1 x 1014 cm-*, the density of the condensed phase at low coverage is used instead of the monolayer density. This is more appropriate than the monolayer value N, = 17.8 X 1014 cm-* because the u value has been deduced from low coverage data.) Ni and Cu differ from Ag, Au and Pd in that their atomic diameter is significantly smaller than that of the substrate atoms. Thus, pictorially speaking, it appears that atoms which are smaller than the substrate atoms are flattened by strong interaction with the substrate while larger ones became more oblong. Another energy characteristic for the lateral bonding can be obtained from the quasichemical approximation which gives the energy eoc = 2kT, ln[(z/(z - 2)] per lateral bond [12]. For z = 6 which probably is appropriate for the distorted hexagonal atomic distribution of the bcc(ll0) plane eoc = 0.8109 kT, which is only 0.288 of the Lennard-Jones potential well depth c2 = (45/16)kT,. This gives bond energies ~oc which are typically only about 0.1 - Ni excepted - of the experimental energy difference AE,, between single atom and atom with 6 neighbours. Keeping in mind that the bonding charge redistributes with increasing number of neighbours from the substrate to the neighbours which decreases the bonding to the substrate, the AEd values are actually a lower bound for the energy which is to be assigned to the lateral bonds. Thus, the bond energies derived from the quasichemical approximation are much to small compared to A Ed. Ni has an abnormally small A Ed because it forms a coincidence lattice at coverages above about 0.7 ML [lo] in which many Ni atoms sit in energetically unfavorable positions which reduces Edmax. The remark made at the beginning of this section that the absolute values of the quantities derived from experimental 0, and T, have to be taken with caution applies in particular to the eoc data not only because of the large 0, difference but also because of the inaccuracy of the c--T, relationship, even within the pairwise interaction approximation. Thus, within the quasichemical 4 - 1.386kT, while the exact value of the approximation for a square lattice cot interacIsing model is cIsi,,s = 1.762kT, = 1.27~6,. In addition, non-pairwise tions increase the c/T, ratio further. Milchev [16] has taken them effectively
J. Kofaczkiewicz,
E. Batter / Law of corresponding states
345
into account in the quasichemical approximation by a linear decrease of the pairwise interaction energy E, with the number z of neighbours. For E/E4 = 1.5 he obtains E = 1.970kT,, for E,/E4 = 2, z = 2.475kT, = 1.79~:~. For z = 6 the increase is much smaller with E = l.l53kT, = 1.42e& for E/E6 = 2. Nevertheless, it is evident that going beyond the quasichemical approximation either within the pairwise interaction model or by taking non-pairwise interactions into account, invariably increases the E/T, ratio so that the eqc values in table 2 are extreme lower bounds. On the other hand, when compared to the AE, values the Lennard-Jones bonding energies ez appear to high. However, neglecting all neighbours but the first nearest neighbours, leaves for the bond strengths to the three substrate neighbours of each adsorbate atom which is surrounded by 6 neighbours the amount c, = +( Edmax- 6]e2]) shown in table 2. These bond energies are still much larger than the lateral bond energies, expressing the stronger affinity of the adsorbate atoms to the substrate than to atoms of the same kind. As this rough estimate shows, the Lennard-Jones modelling of the Van der Waals equation of state seems to be a reasonable first approximation. Of course, it is not more than that because there is little doubt that three-body forces play an important role as indicated by Monte Carlo simulations of the equation of state of Ag on W(110) [17]. In these simulations attractive three-body interactions of -0.037 eV had to be added to an attractive nearest neighbour interaction of -0.074 eV and repulsive second and third nearest neighbour interactions of +0.037 eV in order to obtain agreement with experiment. 4.2. The high coverage range The high coverage data illustrated by fig. 3 do not show the distinct features seen in the low coverage data (fig. 4). This is not surprising because the vaporization of the condensed phase leads to a very dense gas with a high degree of association and correspondingly small work function changes. If one wants to see phase change related work function changes then inspection of the isotherms in fig. 1 suggests not to plot A+(T) for fixed B - which are the original data shown in fig. 3 - but rather A+(B) at fixed T. At the gas + condensate/condensate boundary a noticeable $ change should occur because of the sudden change of the isotherm. In fig. 6 such a plot is made for Au from data similar to those of fig. 3 for several temperatures. The low coverage range is shown only schematically. Several distinct features are seen which have to be studied in more detail yet in the future such as the sharp break of A+ at 8 = 0.45 independent of coverage. Here only the sharp change of the slope of A+(8) at the coverage 8, which decreases with increasing temperature - as expected for the coexistence line - is of interest. Similar but less pronounced changes are seen for the other adsorbates. In fig. 7 the high coverage coexistence line obtained from an average over several experiments has been added
J. Kolaczkiewicz,
346
E. Bauer / Law of corresponding states
to the low coverage coexistence line of fig. 2 of ref. [4] in order to complete the boundary of the coexistence region. It is seen that is far from being symmetric with respect to 8 = 8, = l/2 as required by the quasiche~cal approximation but it also deviates strongly from the high coverage Van der Waals boundary line. This shows that the reduced Van der Waals equation gives a good law of corresponding states for these adsorbate systems only in the low coverage range.
5. Summary and conclusions The results presented in this paper clearly show that - at least at the present level of sophistication of the experiment - the behaviour of chemisorbed layers with attractive lateral interactions on smooth surfaces can at low coverages be described quantitatively by a law of corresponding states in the form of the
\
A”
_-
17, /
R
PARAMETER:T/T,
I
COVERAGE
0.5 0 (MONOLAYERS)
Fig. 6. Work function change of a W(ll0) surface temperatures. Plotted is A$+(@, T)- A+(@, 430 K).
I 1.0
as a function
J of Au coverage
for various
J. Koiaczkiewicz,
E. Bauer / Luw of corresponding states
347
two-dimensional Van der Waals equation. The quasichemical approximation fits the boundary between gas and gas/condensate coexistence region also quite well but the critical coverages needed for the fit are much further below the theoretical value (13, = l/2) than in the case of the Van der Waals equation (t9, = l/3). Other adsorbate layer models do not even reproduce the low coverage coexistence line. The disagreement at the higher coverages indicates that the reduced Van der Waals equation has also only limited applicability. At present there are not enough experimental data at high coverages to decide whether or not a law of corresponding states is valid at the high coverages as it is at low coverages in form of the reduced Van der Waals equation. The experimental results of this work do not give any reliable information about the region around the critical point. In Cu, Ni and Pd this region is outside of the experimentally accessible range, in Ag and Au the accuracy of the data is not sufficient to determine critical exponents. Although within the present scatter of the Ag and Au data the reduced Van der Waals equation describes the coexistence curve up to the critical temperature rather well it is quite possible that close to T, a term a(1 - 7’/T,)B (p K 1) has to be added if the scatter could be reduced. By analogy to three dimensions in which jI = l/3 according to the Guggenheim equation [13] is close to the three-dimensional Ising model exponent /3 = 0.31 one would expect in the present case the
0
02
04 e [ML)
06
0.8
10
Fig. 7. Phase diagram of an adsorption layer with attractive lateral interactions. Solid line: experimental phase boundary for Au on W(110); dashed line: Van der Waals equation phase boundary, dotted line: quasichemical approximation phase boundary.
348
.I. Kolaczkiewicz,
E. Bauer
/ Law of corresponding
states
two-dimensional Ising model exponent /3 = l/8. In fact, the value p = 0.127 _t 0.020 has been recently reported for CH, adsorbed on graphite [18] and a value close to l/8, /3 = 0.16 earlier for Ar adsorbed on CdCl, [19]. It must be pointed out, however, that /3 may be non-universal (see, e.g. ref. [20]) and actually be much smaller. Thus p values as low as 0.096 + 0.005 have been obtained for order-disorder transitions of adsorbates with repulsive first nearest neighbour interactions on the surface studied here [21]. Future work with improved experimental procedures and with adsorbates which are more suitable from this point of view - i.e. T, is in a temperature range in which no desorption occurs - will hopefully also allow determination of the critical exponents of chemisorbed layers with attractive lateral interactions. Another subject of future work will have to be to determine to what extent a recent generalization of the quasichemical approximation [16] can improve the agreement between theory and experiment. In this generalization one of the flaws of the quasichemical approximation, the additivity of the lateral interactions, is eliminated, with the result that the phase diagram becomes asymmetric and that 8, decreases with increasing non-additivity. This is in tendencial agreement with experiment, in particular for Cu and Ni for which thermal desorption spectroscopy shows a strong tendency to dimer formation and for which the present data show particularily low f3, values.
Acknowledgement The authors wish to thank Dr. H. Steffen connection with the phase diagrams.
Appendix. Comments
for the numerical
calculations
in
on the retarding field electron beam method
This appendix is in response to some questions rised by one of the referees which may occur also to other readers who are similarly unfamiliar with the technique. (1) Correct application of the method requires a highly constant temperature source instead of a constant current source. While the first part of this statement is obvious the second part is a misunderstanding of the expression constant current mode. The current which is kept constant is the current collected at the crystal whose work function change is measured. This is achieved by automatic adjustment of the voltage between emitter and collector at a selected fraction of the saturation current of the clean surface. The total emission current is not influenced by this adjustment because it is determined by the anode voltage which is kept constant and by the temperature and work function of the cathode. The temperature is constant because heating is done
J. Kofaczkiewicz,
E. Bauer / Law
of corresponding states
349
with a constant current source and the environment temperature is stable. The work function is constant because of the high cathode temperature and the low pressure which ensure negligible adsorption. (2) The method does not supply an arithmetical average of the work function of the various domains of the sample. While this statement is true in principle as discussed in Herring and Nichols’ classic paper [22] it is not true in practice, at least sufficiently far below the saturation current as shown a long time ago by Heil [23] and reviewed more recently by Knapp [24]. Heil has shown that independent of the size, distribution and work function differences of the patches (domains) only about 6% of the electrons which can surmount the barrier due to the average work function are reflected. This is due to the deflection of the electrons approaching the high + regions into the low + regions by the patch fields. Heil’s calculations were for high and low (p patches of equal size. In the low coverage range of the present study the high (p regions (condensate) cover only a small fraction of the surface so that complete collection of electrons surmounting the average work function potential barrier should be approached even more. Thus the method measures the arithmetical average of the work function similar to the vibrating capacitor method, at least when used close to zero electron energy, e.g. at l/10 of the saturation current. For details the reader is referred to the reviews by Knapp 1241 and Haas and Thomas [25].
References [l] Recent reviews: (a) J.G. Dash and J. Ruvalds, Eds., Phase Transitions in Surface Films (Plenum, New York, 1980); (b) T. Riste, Ed., Ordering in Strongly Fluctuating Condensed Matter Systems (Plenum, New York, 1980); (c) K. Sinha, Ed., Ordering in Two Dimensions (North-Holland, New York, 1980); (d) M. Bienfait, in: Current Topics in Materials Science, Vol. 4, Ed. E. Kaldis (North-Holland, Amsterdam, 1980) p. 361; (e) A. Thorny, X. Duval and J. Regnier, Surface Sci. Rept. 1 (1981) 1. [2] Recent reviews: E. Bauer, in ref. [la], p. 267; E. Bauer, in: Interfacial Aspects of Phase Transformations, Ed. B. Mutaftschiev (Reidel, Dordrecht, 1982) p. 411; L.D. Roelofs and P.J. Estrup, Surface Sci. 125 (1983) 51, and references therein. [3] Recent reviews: L.A. Bolshov, A.P. Napartovich, A.G. Naumovets and A.G. Fedorus, Soviet Phys.-Usp. 20 (1977) 432; E. Bauer, in: Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Vol. 3B, Eds. D.A. King and D.P. Woodruff (Elsevier, Amsterdam, 1984) p. 1. [4] J. Kdaczkiewin and E. Batter, Phys. Rev. Letters 53 (1984) 485. [5] R. Smoluchowski, Phys. Rev. 60 (1941) 661.
350 [6] [7] [8] [9] [lo] [ll] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
J. Kotaczkiewicz,
E. Bauer / LAW of corresponding states
E. Bauer, F. Bonczek, H. Poppa and G. Todd, Surface Sci. 53 (1975) 87. E. Schlenk and E. Bauer, Surface Sci. 93 (1980) 9. J. Kdaczkiewicz and E. Bauer, to be published. E. Bauer, H. Poppa, G. Todd and F. Bonczek, J. Appl. Phys. 45 (1974) 5164, and references therein. J. Kdaczkiewicz and E. Bauer, Surface Sci. 144 (1984) 495. R. Aveyard and D.A. Haydon, An Introduction to the Principles of Surface Chemistry (Cambridge University Press, Cambridge, 1973). (a) R. Fowler and E.A. Guggenheim, Statistical Thermodynamics, (Cambridge University Press, Cambridge, 1960); (b) T.L. Hill, Statistical Thermodynamics (Addison-Wesley, Reading, MA, 1960). E.A. Guggenheim, J. Chem. Phys. 13 (1945) 253. T. Halicioglu and G.M. Pound, Phys. Status Solidi (a) 30 (1975) 619. E. Bauer, H. Poppa, G. Todd and P.R. Davis, J. Appl. Phys. 48 (1977) 3773. A. Milchev, J. Chem. Phys. 78 (1983) 1994. L.C.A. Stoop, Thin Solid Films 103 (1983) 375. H.K. Kim and M.H.W. Chan, Phys. Rev. Letters 53 (1984) 170. Y. Larher, Mol. Phys. 38 (1979) 789. M. Schick, Progr. Surface Sci. 11 (1981) 245. W. Witt, PhD Thesis ClausthaI (1984); W. Witt and E. Batter, in preparation. C. Herring and M.H. Nichols, Rev. Mod. Phys. 21 (1949) 185. H. Heil, Phys. Rev. 164 (1967) 887. A.G. Knapp, Surface Sci. 34 (1973) 289. G.A. Haas and R.E. Thomas, in: Techniques of Metals Research, Vol. VI, Part 1, Ed. E. Passaglia (Interscience, New York, 1972) p. 91.
Surface Science 151 (1985) 333-350 North-Holland, Amsterdam
THE LAW OF CORRESPONDING STATES FOR CHEMISORBED LAYERS WITH ATTRACTIVE LATERAL INTERACTIONS J. KOLACZKIEWICZ
* and E. BAUER
Physikalisches Insriiul, Technische Unruersiiiit ClausthaI, D - 3392 Clausthal- ZeNerfeld, Fed. Rep. of Germany and Sonderforschungsbereich 126, Giittingen - ClausthaI, Fed. Rep. of Germany Received
16 September
1984; accepted
for publication
19 November
1984
Chemisorbed layers which form two-dimensional islands at low temperature consist in general of two phases: condensed phase and a gaseous phase. The coexistence region of the two phases is derived from work function change data of Ni, Cu, Pd and Ag on W(110). The experimental coexistence lines are compared with those obtained from various equations of state with the result that a law of corresponding states is found in the form of the reduced Van der Waals equation or of the quasichemical approximation.
1. Introduction The theory of phase transitions in two dimensions as well as the experimental study of such transitions in physisorbed layers has reached a high level of sophistication in recent years [l]. Rather detailed information including critical exponents has been obtained in several cases, mainly noble gases on graphite. Much less is known about phase transitions in chemisorbed layers [2]. Practically all work has been concerned with adsorbates with repulsive lateral interactions which usually produce several ordered structures with order-order and order-disorder transitions. Many metal adsorbates with large dipole-dipole repulsion such as the alkalis, alkaline earths and rare earths on metal surfaces with large work functions fall into these category [3]. Very little is known about the phase state of metal adsorbates with attractive lateral interactions. Thus, on finds often in the literature the statement that the adsorbate is disordered if the geometry of the diffraction pattern of the substrate does not change during adsorption.
* Permanent address: Institute of Experimental 36, PL-50-205 Wroclaw, Poland.
Physics,
University
0039-6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
of WrocIaw,
B.V.
UI. Cybulskiego
334
J. Kofaczkiewicz,
E. Bauer / Law of corresponding states
It is the purpose of the present paper to correct this situation by a detailed study of the phase state of metal adsorption systems which are known or expected to be governed by attractive lateral interactions. The study makes use of a recently developed method [4] which is based on the Smoluchowski effect [5]: isolated atoms on a flat surface have a large dipole moment while atoms incorporated into a two-dimensional island on such a surface have a comparatively small dipole moment. Thus a study of the temperature dependence of the work function of a surface covered with such an adsorbate gives information on the phase state of the adsorbate. Preceding thermal desorption spectroscopy studies [6-81 have shown that the desorption energies of Cu, Ag, Au and Pd from W(110) surfaces increase with coverage 8, i.e. that the lateral interactions are attractive, and the same is expected by analogy also for Ni. The structural properties of these layers are also well known [7,9,10]: the lateral periodicity of the adsorbate is either identical (Cu, Pd and (for 8 =<0.7) Ni) or very close to that of the substrate (Ag, Au). Thus a lattice gas model seems appropriate to describe .the adsorbate. On the other hand, the W(110) surface is rather smooth so that a two-dimensional gas model appears equally reasonable. The study of the phase state of a chemisorption system differs in general fundamentally from that of a physisorption system. In physisorption studies the adsorbate is usually in (quasi) equilibrium with the surrounding gas phase, in chemisorption studies the adsorbate is bound so tightly to the substrate that the gas from which it was produced can be pumped off and the adsorbate/substrate system may be considered as a closed system. Before a meaningful analysis of the phase state data of a chemisorption system can be made it is, therefore, first necessary to analyze critically the consequences of the experimental differences between physisorption and chemisorption studies. This is the subject of the next section. In section 3 the experimental procedures will be described briefly, section 4 reports the experimental results and their analysis. The concluding discussion is the subject of section 5.
2. Theory In physisorption systems the phase state of the adsorbate is most directly obtained by measuring adsorption isotherms which are determined by the equality of the chemical potentials of gaseous and adsorbed phase, ps = pLa.ps is determined by the pressure p and temperature T of the gas phase (assumed to be perfect), ps = kT ln( p/pa), pa by adsorbate coverage 8, surface temperature T, = T, lateral interaction energy E and the specific model of the adsorption layer. Thus a characteristic adsorption isotherm p = p( 6) is obtained for each model. In a chemisorption system in which the desorption rate is negligible in the (T, 6) range considered p (0) cannot be measured. However,
J. Koiaczkiewicr,
335
E. Bauer / Law of corresponding states
the system may be considered as a closed system just like a condensable gas in three dimensions with negligible diffusion through the walls of the container in which it is enclosed. Just like the three-dimensional system, the two-dimensional system can now be described by an equation of state which expresses the relationship between two-dimensional pressure (“spreading” pressure) 7~, B and T. Again the relationship r = a(&J, T) depends on the lateral interactions and the specific adsorbate model. 7~ is linked with pa by the Gibbs adsorption isotherm dp = a dn = (a,,/B)dm where a is the area available for each adsorbed atom (molecule) and a,, the area occupied by each adsorbed atom (molecule, “co-area”). In table 1 the equations of state +rr= ~(8, T) are compared with the corresponding adsorption isotherms p(8, T) for the specific adsorbate models used later in the comparison with experiment. For the derivations the reader is referred to elementary introductions [ll] and standard text books [12]. Two of the models, Van der Waals’ and Dieterici’s (ref. [12a], p. 273), assume nonlocalized adsorption, the other two, the Bragg-Williams and the quasichemical approximation (ref. [12a], p. 1006), assume localized adsorption. The energy parameter of the various models is hidden in the critical temperature T,, the critical coverage 0, is l/2 except for the first model for which 8, = l/3. 8 = a,/a = Na,/A or 8 = N/N, for non-localized and localized adsorption, respectively. PO*=po
exd-E&T) j,(T)
with X = (2mkT)‘/‘/h,
Table 1 Comparison
= kT.!g(T)exp(_Eo,kT)
of adsorption
Model
Adsorption
A3j,(T) j,( j,)
isotherms
being
the partition
and equations
isotherm
function
of state of various Equation
for the internal
adsorption
layer models
of state
Van der Waals
Dieterici
Not expressible
with elemen-
tary functions Bragg- Williams
$=fln c
Quasichemical
& c
i
1
-2e2
336
J. Kolaczkiewicz, E. Bauer / Law of corresponding states
degrees of freedom of the molecule in the gaseous (adsorbed) state and E, being the desorption energy in the zero coverage limit. z is the number of nearest neighbour sites: z = 6,4, 2 on surfaces with 6-, 4- and 2-fold symmetry. Two-fold symmetry is appropriate, e.g. for fcc(ll0) and bcc(211) metal surfaces. For the 2-fold symmetric bcc(ll0) metal surface, which may be considered as a
t 6-
E 1.4 k
.2
0
0
.2
.4
.6
6-
.8
1 .I3
fs.4
c
G
R
.
.2
0 0
.4
.2
-2
.4
9-----
.6
.8
0t 0
l\\\ .2
I 4
\
\a 6
8
8-
Fig. 1. Isotherms v( 0, T) of the two-dimensional Van der Waals equation (a), Dieterici equation (b), Brad-Williams approximation (c) and quasiche~~ appro~mation (d). i”, ranges from OST, to T, in steps of O.O5T,. Also shown is the coexistence region of two-dimensional gas and condensate according to Maxwell’s construction.
J. Kolaczkiewicz,
E. Bauer / Law of corresponding states
331
somewhat distorted hexagonal surface, z = 6 will be used. Fig. 1 shows the isotherms ~(8, q) of the equations of state of table 1 for T/T, = 0.5, 0.55,. . .) 0.95, 1.0. There is apparently little difference between the two localized adsorption models (c) and (d), while the two non-localized adsorption models (a) and (b) differ strongly from each other and from the localized adsorption models. The differences change significantly if the boundary lines between coexistence and single-phase regions as obtained from Maxwell’s construction (horizontal lines in fig. 1) are plotted as a function of the reduced coverage 8/e,. This is done in fig. 2 for the gas/gas + condensate boundary (e/8,, < 1) which is of main interest in this work. It is evident that the quasichemical and Van der Waals approximation have a very similar gas/gas + condensate boundary line - called coexistence line in the following - while the Bragg-Williams coexistence line differs significantly from both and the Dieterici coexistence line even more. Also shown is the empirical Guggenheim coexistence line [13] B/B, = 1 + ;(l
- T/T,)
- $(l - T/q)1’3
which gives a much better description of many real gases in three dimensions than the Van der Waals, Dieterici and other semi-empirical equations of state.
G
0
0.2
GUGGENHEIM
0.4 e,~, 0.6
0.8
1.0
Fig. 2. Boundary he between coexistence region (gas + condensate) and gas for T/r, > 0.5. D denotes Dieterici, W Van der Waals, B Bragg-Williams and Q quasichemical approximation for z = 6. G is the empirical Guggenheim curve.
338
J. Koiaczkiewicz,
E. Bauer / Law of corresponding states
The Guggenheim equation is a good approximation even close to the critical point: the critical exponent /3 = l/3 of the order parameter is close to the value /3 = 0.312 of the three-dimensional Ising model. The aim of this work is to measure the gas/gas + condensate coexistence lines of condensable two-dimensional gases and to compare them with those of the equations of state which were just discussed in order to determine which one of them - if any - provides an approximate description of experiment. In this comparison several things must be kept in mind. (i) Like in three dimensions it cannot be expected that the equations of state of fig. 1 will agree with experiment using the critical parameters predicted for the various models. However, the equations v/~T, =f(T/T,, e/0,) and T/T, = g( d/l?,) should be the same for all similar materials, expressing the law of corresponding states, with characteristic rC, T, and 19,values for each material. (ii) The critical region is in some cases not accessible to the experimental technique used in this work or the accuracy of the measurements is too low in this region as to allow a reliable determination of the coexistence curve close to T,. The comparison between experiment and theory has, therefore, to rely on the agreement over a wide (0, T) range.
3. Experimental procedure The which A+. If of the dipole
experimental technique measures a quantity as a function of 0 and T is related to the phase state of the adsorbate, the work function change the adsorbate consists of a mixture of monomers, dimers, trimers, . . . and condensed phase with n, , n 2, n *, . . . and no atoms per unit area and the moments per atom pi, pz, p3,. _. and pO, respectively, then A+ is given by
A+=
-4ne
c
nip,.
i=O
Extensive measurements of the B and T dependence of A$ [8] show that the dipole moments pi decrease rapidly with i (i > 0) so that the best defined A+ changes upon condensation occur at low coverages and high temperatures at which the gas consists only of monomers and dimers. The gas/gas + condensate boundary can therefore be studied particularily well. However, even at high coverages (0 > 0.6) distinct changes of A$ with 8 and T occur which can be assigned to the gas + condensate/condensate boundary line on the basis of their r3 and T dependence. This boundary has been studied in much less detail and will be discussed only briefly here. The experimental arrangement for the A+(T) measurements is extremely simple. It is a retarding field electron beam system which is operated in the constant current mode. This mode is achieved with a standard constant voltage source and a 2 GO resistor in series. The voltage change necessary to maintain
J. Kolaczkiewicz,
E. Bauer / LAW of corresponding states
339
a constant current is equal to A+/e. It is measured with a battery-operated electrometer whose output is displayed on the y axis of an xy recorder versus thermocouple output on the x axis. Typical data of this type are shown in fig. 3 for Cu in the T range from 400 to 900 K over a wide coverage range. All curves were taken during cooling and terminated at about 400 K in order to avoid excessive cooling times. It is evident that in all cases except at the highest coverage (19 > 2) the A+ change at 400 K is still significantly larger than that expected for clean metal surfaces. For example d+/dT = 8 x 1O-6 eV/K for clean W (curve 0) or d+/dT < 5 X lop6 eV/K for 2 monolayers (ML) of Cu (last curve) which consist of two slightly distorted Cu(ll1) planes. This does not mean, however, that there is still a significant number of atoms condensing with further decreasing temperature, because even the complete monolayer has an unusually large d+/dT -4 10 X lop5 eV/K. With 8 decreasing from 8 = 1 d$/dT at 400 K decreases so that the d+/dT seen at that temperature may be attributed mainly to the d+/dT of the islands of the condensed phase. In the very low coverage range (6 = 0.1-0.3, depending upon adsorbate) the temperature dependence of A$ is more complicated than that shown in fig. 3, as illustrated for Pd in fig. 4 and for Au elsewhere [4]: the rapid decrease with increasing T of the low coverage curves seen already in fig. 3 terminates in a slow linear decrease with further increase of T. The slope of the linear decrease
I I 800 700 TCMDCD*TIIDC ILIIrLnl+l"nL IYI,1\, I
400
500
600
TEMPERATURE
700 IKI-
800
I
4(IO 500L0600
900
Fig. 3. Work function change with temperature of a W(110) surface with various Cu coverages in units of 1.2 X 1014 atoms/cm*.
340
J. Kolacrkiewicz, E. Bauer / Law
of corresponding states
is constant up to a certain coverage 8, which depends on the adsorbate - for Pd 6’, = 0.06 - and increases with increasing 0 above 8,. On the basis of a detailed A+(8, T) study [8] the coverage-independent d+/dT value may be assigned to monomers, the increasing d+/dT value to a mixture of monomers and dimers. In any case, the linear part of A+(T) is typical for the single phase (gas) region while the curved part corresponds to the two phase region in which with increasing T an increasing number of atoms vaporize into the two-dimensional gas until all condensate has vaporized. The intersections of the extrapolations of the linear and curved parts of A+(T) thus determine the boundary between gas and gas + condensate region, i.e. the coexistence curve. In this manner the data presented in section 4.1 are obtained. Before presenting these data some experimental details concerning sample preparation and characterization should be mentioned. All experiments were performed at about 6 x lo- ” Torr on W(110) surfaces which were oriented within 0.05” of the (110) plane. The sample was cleaned in the usual way heating in lo-’ Torr 0, and flashing in UHV to 2300 IS - until carbon contamination was less than 10m3 ML as determined by AES. The metals were evaporated from W wire loops heated by well stabilized current supplies which produced very constant deposition rates over long periods. The monolayer coverages were determined from the breaks in the Auger amplitude versus deposition time curves as described in several papers (e.g. refs. [9,10]), combined with LEED. Submonolayer coverages were obtained in various ways:
PARAMETER:
I
TEMPERATURE(K)
-
Fig. 4. Work function change with temperature of a W(110) surface with various Pd coverages. The Pd coverages and corresponding room temperature A+ values (+(O, 300 K)- $J(0, 300 K)) are given as curve parameters. All curves have been shifted to coincide at 430 K, the lowest temperature reached during cooling.
J. Kolaczkiewicz,
E. Bauer / LAW of corresponding stales
341
from the deposition time, from the Auger amplitudes or from the work function change after calibration by AES, depending upon what method was most accurate for the metal studied. Thus a relative coverage accuracy of several thousandths of a monolayer could be obtained at small coverages with a series of small identical deposition doses while the absolute 8 accuracy over a wider coverage range was 0.01-0.02 ML. The crystal was heated by a well isolated floating power supply and the temperature was measured with a W5%Re-W26%Re thermocouple to *5 K. The maximum temperature was chosen such that no thermal desorption occurred which was verified by A+(T) reproducibility tests and coverage measurements before heating and after cooling. 4. Results and discussion 4.1. The low coverage range Fig. 5 shows the boundary lines between gas and gas + condensate for Cu, Ni, Ag and Pd on W(110) as obtained from the kind of data shown in fig. 4 in the manner described in section 3. The points are from several series of experiments and scatter so much not because of reproducibility difficulties but mainly because it is difficult to locate the intersection between the extrapolated curved and straight parts of A+(T) with better accuracy. Nevertheless, the scatter is small enough for a meaningful comparison with the various theoretical coexistence curves of fig. 2. Although only the best fit Van der Waals curve is shown in fig. 5, it was verified that the quasichemical approximation curve did not fit quite as well - the curvature is somewhat too large - that the Bragg-Williams approximation was worse and the Dieterici equation out of question. Surprisingly, the Guggenheim equation which was so successful for many gases in three dimensions does not fit at all. The T, and 6, values obtained by the fit with the Van der Waals equation are shown in table 2 together with the T, values derived from thermal desorption spectroscopy (cTDS) [8]. Th e a g reement between the T, values is within the present limits of error of the two techniques. Except for Ag, the 0, values are significantly smaller than the ti, values of the Van der Waals equation, 0, = l/3, in particular for Ni and Cu. The difference between the /3, values obtained by fitting from that of the model are even larger for the quasichemical approximation (0, = l/2) because they are only a few percent larger than those shown in table 2. If only the quality of the fit over a wide temperature range is used as a criterion, the Van der Waals equation and quasichemical approximation give a comparably good law of corresponding states over the (0, T) range which could be studied. If, however, also the 0, difference is taken into account, the Van der Waals equation is significantly better.
J. Koiaezkiewicz, E. Buuer / Law of corresponding states
342
0
01
. a2
0.3
0.4
0.9
‘1
I
1200 -
: x
I I
7
llOO-
1
1r s.
z
I- toooE r;i
VAN CIERWAALS
~6
4 5 900t-
c
Ni/W (110) Tc =t&OOK e, 0.205 VAN DER WAALS
;2-
q
-a6 EOO0
0.2
0.L
0.6
I 1 0.06 0.02 O.Oh COVERAGE 8 IMONOLAYERSI-
I a08
0.8
VAN DER WAALS
- 05 0 C@4ERAGE0ihONOLA&---
0.3
0
CWERAGE 8 CM!hLAYERU-a*
Fig. 5. Coexistence line (boundary between single and two-phase region) of Cu, Ni, Ag and Pd on W(110) as obtained from A+(T) data as shown in fig. 4. The lines are the best fits of the Van der Waals equation of state which were obtained with the indicated critical parameters T,, 0,.
343
J. Kofaczkiewicz, E. Bauer / Law of corresponding states
It may be argued that in view of the deviation of the corresponding states 8, values (table 2) from the model 0, values (l/3, l/2) it is meaningless to draw further conclusions both from the 8, and c values. This may be true for the absolute values of the quantities derived from 6, and T, but trends in the series of atoms studied should not be obscured by this problem. If to a first approximation three-body forces are neglected and the simple Lennard-Jones pairwise interaction potential with the hard core diameter (I and the depth E* of the potential well is assumed, then these potential parameters may be expressed by T, and 8, as follows: c2 = (45/16)kT, and u = (+aN18,)-“2. The T,, 8,, N, values and the cZ and u values derived from them are shown for Cu, Ag, Au [4], Ni and Pd in Table 2. The table also contains the corresponding c! and ub values derived from bulk properties, the ratios e,/ct and u/ub, the to the maximum number maximum desorption energies Edmax- corresponding of lateral bonds - the zero coverage limit of the desorption energy Ed0 corresponding to no lateral bonds - both obtained from thermal desorption spectroscopy [7,8], the difference A Ed = Edmax- Ei, the bond energy ~oc of the quasichemical approximation which is obtained from T, via the relation cot = 2kT, ln[(z/z - 2)] assuming z = 6, and the ratio A Ed/cw. It is interesting to note that for all adsorbates e,/ct = 2/3 but that u/ub Table 2 Characteristic parameters of metal adsorbates on W(110); the symbols are explained in the text
cu
A8
T, (K)
1170
980
0: TTDS (K) N, (lo-l4 cm-‘) - z2 (ev) a (A) - G (ev) 0b (A) c,/G a/oh E.$‘= (eV) Ed0(eV) A Ed (eV) -cw(eV) - A&i/coc cS (ev)
1150 0.215 14.1 b, 0.284 2.646 0.409 2.338 0.69 1.13 3.85 o 3.2 n 0.65 0.082 7.9 0.72
985 0.35 13.6 =) 0.238 2.111 0.345 2.644 0.69 0.80 3.55 o 2.8 0 0.75 0.069 10.9 0.71
a) b, ‘) d, e, o
Ref. Ref. Ref. Ref. Ref. Ref.
[4]. 191. [15]. [IO]. 17). [8].
AU 1130 a) 111800.26 =) 13.4 =) 0.274 2.468 0.441 2.637 0.62 0.94 4.1 I-I 3.35 o 0.75 0.079 9.5 0.82
Ni
Pd
1400
1170 _
1375 0.205 17.8 (14.1) d, 0.339 2.411 (2.709) 0.520 2.282 0.65 1.06 (1.19) 4.95 o 4.4 0 0.45 0.098 4.6 0.97
0.28 14.1 e’ 0.284 2.318 0.427 2.520 0.66 0.92 4.4 e’ 3.6 e’ 0.8 0.082 9.8 0.90
344
J. Kolaczkiewicz, E. Bauer / Law of corresponding slates
changes strongly from adsorbate to adsorbate. That the e2 values for the lateral interactions are smaller than in the bulk is easy to understand. The adsorbate atoms are bound much stronger to the substrate atoms than to atoms of their own kind so that fewer electrons are available for lateral bonding which reduces 6*. Quantitatively it is surprising that these reductions are so similar in spite of the differences in the electron configuration and in adsorbate-substrate bond strength as expressed in the desorption energies [8]. Surprising is also the result that the hard core diameter (I of the adsorbed atom may be larger than its bulk value. One would expect that the polarization normal to the surface resulting from the adsorbate-substrate bond would reduce the lateral dimensions of the adatom as is the case for Ag, Au and Pd. For Cu and Ni, however (I > ub, in particular for Ni (a/ah = 1.19 if iVi = 14.1 x 1014 cm-*, the density of the condensed phase at low coverage is used instead of the monolayer density. This is more appropriate than the monolayer value N, = 17.8 X 1014 cm-* because the u value has been deduced from low coverage data.) Ni and Cu differ from Ag, Au and Pd in that their atomic diameter is significantly smaller than that of the substrate atoms. Thus, pictorially speaking, it appears that atoms which are smaller than the substrate atoms are flattened by strong interaction with the substrate while larger ones became more oblong. Another energy characteristic for the lateral bonding can be obtained from the quasichemical approximation which gives the energy eoc = 2kT, ln[(z/(z - 2)] per lateral bond [12]. For z = 6 which probably is appropriate for the distorted hexagonal atomic distribution of the bcc(ll0) plane eoc = 0.8109 kT, which is only 0.288 of the Lennard-Jones potential well depth c2 = (45/16)kT,. This gives bond energies ~oc which are typically only about 0.1 - Ni excepted - of the experimental energy difference AE,, between single atom and atom with 6 neighbours. Keeping in mind that the bonding charge redistributes with increasing number of neighbours from the substrate to the neighbours which decreases the bonding to the substrate, the AEd values are actually a lower bound for the energy which is to be assigned to the lateral bonds. Thus, the bond energies derived from the quasichemical approximation are much to small compared to A Ed. Ni has an abnormally small A Ed because it forms a coincidence lattice at coverages above about 0.7 ML [lo] in which many Ni atoms sit in energetically unfavorable positions which reduces Edmax. The remark made at the beginning of this section that the absolute values of the quantities derived from experimental 0, and T, have to be taken with caution applies in particular to the eoc data not only because of the large 0, difference but also because of the inaccuracy of the c--T, relationship, even within the pairwise interaction approximation. Thus, within the quasichemical 4 - 1.386kT, while the exact value of the approximation for a square lattice cot interacIsing model is cIsi,,s = 1.762kT, = 1.27~6,. In addition, non-pairwise tions increase the c/T, ratio further. Milchev [16] has taken them effectively
J. Kofaczkiewicz,
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345
into account in the quasichemical approximation by a linear decrease of the pairwise interaction energy E, with the number z of neighbours. For E/E4 = 1.5 he obtains E = 1.970kT,, for E,/E4 = 2, z = 2.475kT, = 1.79~:~. For z = 6 the increase is much smaller with E = l.l53kT, = 1.42e& for E/E6 = 2. Nevertheless, it is evident that going beyond the quasichemical approximation either within the pairwise interaction model or by taking non-pairwise interactions into account, invariably increases the E/T, ratio so that the eqc values in table 2 are extreme lower bounds. On the other hand, when compared to the AE, values the Lennard-Jones bonding energies ez appear to high. However, neglecting all neighbours but the first nearest neighbours, leaves for the bond strengths to the three substrate neighbours of each adsorbate atom which is surrounded by 6 neighbours the amount c, = +( Edmax- 6]e2]) shown in table 2. These bond energies are still much larger than the lateral bond energies, expressing the stronger affinity of the adsorbate atoms to the substrate than to atoms of the same kind. As this rough estimate shows, the Lennard-Jones modelling of the Van der Waals equation of state seems to be a reasonable first approximation. Of course, it is not more than that because there is little doubt that three-body forces play an important role as indicated by Monte Carlo simulations of the equation of state of Ag on W(110) [17]. In these simulations attractive three-body interactions of -0.037 eV had to be added to an attractive nearest neighbour interaction of -0.074 eV and repulsive second and third nearest neighbour interactions of +0.037 eV in order to obtain agreement with experiment. 4.2. The high coverage range The high coverage data illustrated by fig. 3 do not show the distinct features seen in the low coverage data (fig. 4). This is not surprising because the vaporization of the condensed phase leads to a very dense gas with a high degree of association and correspondingly small work function changes. If one wants to see phase change related work function changes then inspection of the isotherms in fig. 1 suggests not to plot A+(T) for fixed B - which are the original data shown in fig. 3 - but rather A+(B) at fixed T. At the gas + condensate/condensate boundary a noticeable $ change should occur because of the sudden change of the isotherm. In fig. 6 such a plot is made for Au from data similar to those of fig. 3 for several temperatures. The low coverage range is shown only schematically. Several distinct features are seen which have to be studied in more detail yet in the future such as the sharp break of A+ at 8 = 0.45 independent of coverage. Here only the sharp change of the slope of A+(8) at the coverage 8, which decreases with increasing temperature - as expected for the coexistence line - is of interest. Similar but less pronounced changes are seen for the other adsorbates. In fig. 7 the high coverage coexistence line obtained from an average over several experiments has been added
J. Kolaczkiewicz,
346
E. Bauer / Law of corresponding states
to the low coverage coexistence line of fig. 2 of ref. [4] in order to complete the boundary of the coexistence region. It is seen that is far from being symmetric with respect to 8 = 8, = l/2 as required by the quasiche~cal approximation but it also deviates strongly from the high coverage Van der Waals boundary line. This shows that the reduced Van der Waals equation gives a good law of corresponding states for these adsorbate systems only in the low coverage range.
5. Summary and conclusions The results presented in this paper clearly show that - at least at the present level of sophistication of the experiment - the behaviour of chemisorbed layers with attractive lateral interactions on smooth surfaces can at low coverages be described quantitatively by a law of corresponding states in the form of the
\
A”
_-
17, /
R
PARAMETER:T/T,
I
COVERAGE
0.5 0 (MONOLAYERS)
Fig. 6. Work function change of a W(ll0) surface temperatures. Plotted is A$+(@, T)- A+(@, 430 K).
I 1.0
as a function
J of Au coverage
for various
J. Koiaczkiewicz,
E. Bauer / Luw of corresponding states
347
two-dimensional Van der Waals equation. The quasichemical approximation fits the boundary between gas and gas/condensate coexistence region also quite well but the critical coverages needed for the fit are much further below the theoretical value (13, = l/2) than in the case of the Van der Waals equation (t9, = l/3). Other adsorbate layer models do not even reproduce the low coverage coexistence line. The disagreement at the higher coverages indicates that the reduced Van der Waals equation has also only limited applicability. At present there are not enough experimental data at high coverages to decide whether or not a law of corresponding states is valid at the high coverages as it is at low coverages in form of the reduced Van der Waals equation. The experimental results of this work do not give any reliable information about the region around the critical point. In Cu, Ni and Pd this region is outside of the experimentally accessible range, in Ag and Au the accuracy of the data is not sufficient to determine critical exponents. Although within the present scatter of the Ag and Au data the reduced Van der Waals equation describes the coexistence curve up to the critical temperature rather well it is quite possible that close to T, a term a(1 - 7’/T,)B (p K 1) has to be added if the scatter could be reduced. By analogy to three dimensions in which jI = l/3 according to the Guggenheim equation [13] is close to the three-dimensional Ising model exponent /3 = 0.31 one would expect in the present case the
0
02
04 e [ML)
06
0.8
10
Fig. 7. Phase diagram of an adsorption layer with attractive lateral interactions. Solid line: experimental phase boundary for Au on W(110); dashed line: Van der Waals equation phase boundary, dotted line: quasichemical approximation phase boundary.
348
.I. Kolaczkiewicz,
E. Bauer
/ Law of corresponding
states
two-dimensional Ising model exponent /3 = l/8. In fact, the value p = 0.127 _t 0.020 has been recently reported for CH, adsorbed on graphite [18] and a value close to l/8, /3 = 0.16 earlier for Ar adsorbed on CdCl, [19]. It must be pointed out, however, that /3 may be non-universal (see, e.g. ref. [20]) and actually be much smaller. Thus p values as low as 0.096 + 0.005 have been obtained for order-disorder transitions of adsorbates with repulsive first nearest neighbour interactions on the surface studied here [21]. Future work with improved experimental procedures and with adsorbates which are more suitable from this point of view - i.e. T, is in a temperature range in which no desorption occurs - will hopefully also allow determination of the critical exponents of chemisorbed layers with attractive lateral interactions. Another subject of future work will have to be to determine to what extent a recent generalization of the quasichemical approximation [16] can improve the agreement between theory and experiment. In this generalization one of the flaws of the quasichemical approximation, the additivity of the lateral interactions, is eliminated, with the result that the phase diagram becomes asymmetric and that 8, decreases with increasing non-additivity. This is in tendencial agreement with experiment, in particular for Cu and Ni for which thermal desorption spectroscopy shows a strong tendency to dimer formation and for which the present data show particularily low f3, values.
Acknowledgement The authors wish to thank Dr. H. Steffen connection with the phase diagrams.
Appendix. Comments
for the numerical
calculations
in
on the retarding field electron beam method
This appendix is in response to some questions rised by one of the referees which may occur also to other readers who are similarly unfamiliar with the technique. (1) Correct application of the method requires a highly constant temperature source instead of a constant current source. While the first part of this statement is obvious the second part is a misunderstanding of the expression constant current mode. The current which is kept constant is the current collected at the crystal whose work function change is measured. This is achieved by automatic adjustment of the voltage between emitter and collector at a selected fraction of the saturation current of the clean surface. The total emission current is not influenced by this adjustment because it is determined by the anode voltage which is kept constant and by the temperature and work function of the cathode. The temperature is constant because heating is done
J. Kofaczkiewicz,
E. Bauer / Law
of corresponding states
349
with a constant current source and the environment temperature is stable. The work function is constant because of the high cathode temperature and the low pressure which ensure negligible adsorption. (2) The method does not supply an arithmetical average of the work function of the various domains of the sample. While this statement is true in principle as discussed in Herring and Nichols’ classic paper [22] it is not true in practice, at least sufficiently far below the saturation current as shown a long time ago by Heil [23] and reviewed more recently by Knapp [24]. Heil has shown that independent of the size, distribution and work function differences of the patches (domains) only about 6% of the electrons which can surmount the barrier due to the average work function are reflected. This is due to the deflection of the electrons approaching the high + regions into the low + regions by the patch fields. Heil’s calculations were for high and low (p patches of equal size. In the low coverage range of the present study the high (p regions (condensate) cover only a small fraction of the surface so that complete collection of electrons surmounting the average work function potential barrier should be approached even more. Thus the method measures the arithmetical average of the work function similar to the vibrating capacitor method, at least when used close to zero electron energy, e.g. at l/10 of the saturation current. For details the reader is referred to the reviews by Knapp 1241 and Haas and Thomas [25].
References [l] Recent reviews: (a) J.G. Dash and J. Ruvalds, Eds., Phase Transitions in Surface Films (Plenum, New York, 1980); (b) T. Riste, Ed., Ordering in Strongly Fluctuating Condensed Matter Systems (Plenum, New York, 1980); (c) K. Sinha, Ed., Ordering in Two Dimensions (North-Holland, New York, 1980); (d) M. Bienfait, in: Current Topics in Materials Science, Vol. 4, Ed. E. Kaldis (North-Holland, Amsterdam, 1980) p. 361; (e) A. Thorny, X. Duval and J. Regnier, Surface Sci. Rept. 1 (1981) 1. [2] Recent reviews: E. Bauer, in ref. [la], p. 267; E. Bauer, in: Interfacial Aspects of Phase Transformations, Ed. B. Mutaftschiev (Reidel, Dordrecht, 1982) p. 411; L.D. Roelofs and P.J. Estrup, Surface Sci. 125 (1983) 51, and references therein. [3] Recent reviews: L.A. Bolshov, A.P. Napartovich, A.G. Naumovets and A.G. Fedorus, Soviet Phys.-Usp. 20 (1977) 432; E. Bauer, in: Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Vol. 3B, Eds. D.A. King and D.P. Woodruff (Elsevier, Amsterdam, 1984) p. 1. [4] J. Kdaczkiewin and E. Batter, Phys. Rev. Letters 53 (1984) 485. [5] R. Smoluchowski, Phys. Rev. 60 (1941) 661.
350 [6] [7] [8] [9] [lo] [ll] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
J. Kotaczkiewicz,
E. Bauer / LAW of corresponding states
E. Bauer, F. Bonczek, H. Poppa and G. Todd, Surface Sci. 53 (1975) 87. E. Schlenk and E. Bauer, Surface Sci. 93 (1980) 9. J. Kdaczkiewicz and E. Bauer, to be published. E. Bauer, H. Poppa, G. Todd and F. Bonczek, J. Appl. Phys. 45 (1974) 5164, and references therein. J. Kdaczkiewicz and E. Bauer, Surface Sci. 144 (1984) 495. R. Aveyard and D.A. Haydon, An Introduction to the Principles of Surface Chemistry (Cambridge University Press, Cambridge, 1973). (a) R. Fowler and E.A. Guggenheim, Statistical Thermodynamics, (Cambridge University Press, Cambridge, 1960); (b) T.L. Hill, Statistical Thermodynamics (Addison-Wesley, Reading, MA, 1960). E.A. Guggenheim, J. Chem. Phys. 13 (1945) 253. T. Halicioglu and G.M. Pound, Phys. Status Solidi (a) 30 (1975) 619. E. Bauer, H. Poppa, G. Todd and P.R. Davis, J. Appl. Phys. 48 (1977) 3773. A. Milchev, J. Chem. Phys. 78 (1983) 1994. L.C.A. Stoop, Thin Solid Films 103 (1983) 375. H.K. Kim and M.H.W. Chan, Phys. Rev. Letters 53 (1984) 170. Y. Larher, Mol. Phys. 38 (1979) 789. M. Schick, Progr. Surface Sci. 11 (1981) 245. W. Witt, PhD Thesis ClausthaI (1984); W. Witt and E. Batter, in preparation. C. Herring and M.H. Nichols, Rev. Mod. Phys. 21 (1949) 185. H. Heil, Phys. Rev. 164 (1967) 887. A.G. Knapp, Surface Sci. 34 (1973) 289. G.A. Haas and R.E. Thomas, in: Techniques of Metals Research, Vol. VI, Part 1, Ed. E. Passaglia (Interscience, New York, 1972) p. 91.