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Adsorption and surface energy (I): The effect of adsorption on the γ-plot
ADSORPTION
AND
SURFACE
ADSORPTION N.
A.
ENERGY ON
THE
(I):
THE
EFFECT
OF
y-PLOT*
GJOSTEINt
Based on the idea that an adsorbing gas atom can interact with any of three different types of surface sites-surface, single ledge, and double ledge sites-each being characterized by discretely different adsorption free energies-a phenomenological theory is developed, governing the shape alterations of the y-plot due to adsorption from the gas phase. When single and double ledges are preferred sites for adsorption, the theory predicts that an appreciable increase in the curvature of y-plot can occur as the pressure of the system approaches the characteristic pressures of these sites. At some higher pressure, depending on the relative adsorption energies of single and double ledge sites, a marked decrease in the slope of the y-plot at the origin of the cusp will occur. When the surface sites are preferred for adsorption the curvature of the y-plot decreases, and the slope at the origin of the cusp increases as the system pressure approaches the characteristic pressure of the surface site. In addition to, and simultaneously with these changes, each point of the y-plot will shrink towards the origin of the y-plot. ADSORPTION
ET ENERGIE L’ADSORPTION
DE SURFACE (1): 1’INFLUENCE SUR LE DIAGRAMME y
DE
Elle est basee sur l’idee qu’un atome gazeux peut s’adsorber en trois genres d’emplacements, caract&i&s par des energies libres d’adsorption differentes, placement en surface proprement dit, contre un gradin peu marque (“single ledge”) au creux dun gradin accentue(“double ledge”). Les auteurs developpent une theorie qui regit les modifications de la forme du diagramme de l’energie de surface y, dues 21 I’adsorption dune phase gazeuse. Quand l’adsorption a lieu preferentiellement contre des gradins, cette theorie prevoit une augmentation sensible de la courbure du diagramme y quand la pression du systeme s’approche des pressions Pour une certaine pression plus elevee, en relation avec les caracteristiques de ces emplacements; energies relatives dadsorption, se produit une diminution marquee de la pente du diagramme y vers son origine .
Quand l’adsorption a plut6t lieu en surface proprement dite la courbure du diagramme y diminute, et la pente a l’origine augmente quand la pression du systeme d’approche de la pression caracteristique du site d’adsorption. En m&me temps, les points du diagramme y se rassemblent aux environs de l’origine. ADSORPTION
UND OBERFLACHENENERGIE (I): EINFLUSS ADSORPTION AUF DIE 1/-FLACHE
DER
Ein adsorbiertes Gasatom kann mit drei verschiedenen Oberlliichenpliitzen wechselwirken: mit Pliitzen auf der F&he, an einer einfachen und an einer doppelten Stufe; die Freie Energie der Adsorption ist fur die drei Pliitze verschieden. Diese Vorstellung wird einer phanomenologischen Theorie zugrunde gelegt, die die Anderungen der y-F&he infolge Adsorption aus der Gasphase behandelt. Wenn einfache und doppelte Kanten bei der Adsorption bevorzugt werden, kann nach der Theorie eine betrachtliche Zunahme der Krtimmung der y-Fliiche auftreten, sobald der Druck des Systems sich den charakteristisohen Drlicken dieser Oberfliichenpliitze ntihert. Bei einem hijheren Druck, der von den relativen Adsorptionsenergien der einfachen und doppelten Stufen abhangt, tritt eine ausgepriigte Abnahme der Steigung der y-Flache an der Spitze auf. Wenn die bevorzugten Pliitze der Adsorption die auf der Fl5che sind, nimmt die Krtimmung der y-Fliiche ab, und Steigung an der Spitze nimmt jedoch zu, wenn der Druck des Systems sich dem oharakteristischen Druck des Flachenplatzes nahert. Gleichzeitig mit diesen Veranderungen tritt eine zusiitzliche Schrumpfung der gesamten y-Fliiche auf.
INTRODUCTION
Recently, determining
cusps at the (100) and (111) orientations.
methodso-4)
have
been
developed
the variation
of the surface energy
In nickel
for
some minor
(y)
notably (llO), (210) and (410), but the results here are not as well established.(i)
of metals with crystallographic orientation of the surface. When interpreted in terms of the polar y-plot,(5*6) these results indicate that the shape of
In carrying a strong
cusps may occur
out experiments
likelihood
that
at other orientations,
of this type there is
impurity
atoms,
adsorbed
the y-plot for nickel and copper is similar to that predicted by Herring(b) several years ago. For these
from the gas phase or from the crystal interior, will be
metals, curved
understand the alterations in shape of the y-plot which will occur as a result of gaseous adsorption. Lacmann and Stranski”) have given a treatment for the change in surface energy of low index planes with
the y-plot appears surface except for
to be a continuously major inward-pointing
* Received July 19, 1962. t Metallurgy Department, Scientific Motor Company, Dearborn, Michigan. ACTA
METALLURGICA,
VOL.
Laboratory,
11, AUGUST
1963
Ford
present
on the surface,
gaseous adsorption, 957
and thus it is important
to
but there has been no treatment
ACTA
958
METALLURGICA,
of the effect of adsorption on the shape of the y-plot in the vicinity of a cusp. The present work is directed towards solving this latter problem. Utilizing a model of surface struoture, which has evolved from the surface energy work described above, a phenomenologi~al theory is developed, which predicts the change in shape of a cusp in the y-plot with the extent of adsorption. From the results of this analysis, an estimate will be made of the extent to which the present determinations of y-plot have been influenced by adsorption. In an accompanying paper,(*) it will be shown that dependence of surface-energy-induced thermal faceting on gaseous impurity adsorption oan be explained on the basis of this analysis. SURFACE
STRUCTURE
Low index planes Burton, Cabrera and Frank(s) and &Iullins(lo) have treated the problem of atomic roughening of an atomi~a~y smooth low index plane that occurs with increasing temperatures. At elevated temperatures, a small concentration (approximately 5 X IOP5, at temperatures near the melting point in copper) of surface adatoms and vacancies are stable due to the lowering of free energy of the surface that occurs from the configurational entropy of the roughened surface. Both treatments conclude, however, that for most metals no large soale disordering (surface melting) of low index planes will occur even at temperatures near their melting points. For the purpose of this treatment, they will be considered as atomically smooth. Vicinal prunes In discussing the origin of inward-dinting cusp in the y-plot, Herring(5@) suggested a structure for the vieinal planes of a low index plane. According to this picture, the vicinal planes should differ in structure from the low index plane only by the presence of certain density of monatomic ledges of height S and average separation
&G
Wu3
where M is the small angle of deviation from the low index plane. The surface energy y of a vicinal plane for small tc is given by
where y0 = surface energy of low index plane (ergslcmz) yI = excess free energy of surface per unit length of ledge (ergs/cm).
VOL.
II,
1963
(III)
PLANE
FIG. 1. Torque terms for (111) cusp of copper, 0 -0FHC e-high purity copper in copper in graphite boat; graphite boat; a-OFHC copper with no graphite in system (3). Taken from Robertson and Shewmon, to be published in Trans. Amer. Inst. Min. (MetaEZ.) Engrs. 224 (1962).
Equation (1) predicts a cusp wit8ha parabolic shape, having
a constant torque
term,
ay 75 acr = x -
yOa.
Experimental determinations of torque terms for nickel(l) and coppeG show that the torque term decreases approximately linearly* with angular deviation from the low index plane, Fig. 1. The magnitude of the decrease, however, is generally much smaller than would be expected from the coefficient yO. This indicates that another second order term in u should appear in equation (1). Robertson and Shewmon(3) have shown how this term can arise, if the steps are randomly distributed, from the probability of forming double ledges-that is, ledges of height 2s.t Their equation may be written:
y=yo+
Fu- [~ + I&]u2
(2)
* In the data of Robertson and Shewmon,‘3) the scatter is too large to conclude that there is any deviation from linearity. Mykura”) indicates some curvature in his plots, but this will be ignored in the present treatment. t Other models also can give rise to a second order term. For example, if ya varies inversely with the single ledge spacing, to a first approx~tion, a relationship similar to equation (2) can be derived. Alternatively, it can be postulated that the ledges are in thermal equilibrium and that two single ledges combine with the formation energy yrbe, where _5? is a length per atom along a ledge. In this case, the COefficient of cxzin equation (2) will be yr/S*M exp [+y&‘/kT]. The treatment given in this paper wa8 carried out only for the statistical model of Robertson and Shewmon because it appears to be the simplest model consistent with the experimental facts. It would not be difficult to modify the present treatment, to t&e into account these other ideas.‘aO’ 3 In equation (2), the density of single ledges a/S has not been modified to take into account the presence of double ledges because the density of double ledges is small in comparison to x/X, especially for small a.
GJOSTEIN:
ADSORPTION
AND
SURFACE
treatment
where is YI = 2YL - YD (positive or negative)
the
interaction
energy
between ledges, yD is the
energy per unit length of double ledge, and M
ENERGY
shows
evident
two
quantities
mentally,
which
to define for later use
may
be
determined
experi-
account
the
&_L
(a2r1=
yoO 8x2
2y,
to this model
for believing
that
+k
(4)
S2MYCW
where yoOrefers to y0 for perfectly
the
heterogeneous
there appears a linear
to be no
extrapolation
curvature
of since
any higher order terms in equation
would give rise to appreciable
(2),
of the torque
term plot at large angles rather than small. For both nickel and copper, however, there are no data at small
angles
information
to verify
a minimum
of a facet.
except
for some
from the equations
These equations
will give
value for the torque terms at the origin
of the cusp.
Torque
these equations taken
this point,
that may be obtained
of equilibrium
from
terms for nickel obtained
agree well with extrapolated
the twin
boundary
method,
from value
according
to Mykura,(l)
but this does not seem to be the case
for
Gjostein(l2)
37 1
copper. 1
MODEL
discussion
it is obvious
estimated
value for -aa to be about Y00 [ 0
structure-that
the
minimum
0.4 for the
(100)
character
is, the defect
of the
sites,
an adsorption
simplicity, such
the
defects
impurity
atoms
considered different double
site
kinks,
and sites arising
sites.
so too, Instead,
adsorption is
shown
although
AG,“,
three
surface,
AG,‘,
in
Fig.
point
3,
with
evident
of these
quantities
bond
be in a state
facets on which measurements
it is necessary
further experimental
is needed.
evidence
an adsorption An
> AG,“,
AG,o,
The relative magnibe determined
from
knowledge
of the nature
of the
in which AG, to
a phenomeno-
can
that an adsorbing
Shewmon. The former value was based on only a few observations due to the difficulty of obtaining Thus,
and
in Fig. 2.
and AGF may be
atom
theory of surface bonding
can be made.
ledge
> AG,’
From
AG,’
lated
a serious discrepancy,
single
respectively.
AG,’
later.
of view, both
either a theoretical
AG,O,
will be
energetically
this order is not essential to the treatment
either greater or less than AG,‘. tude
of
bulk
energy diagram, which depicts this model,
as will become logical
AG,,
of
dislocations
the gas atom
the ledge or double ledge, or from
before this can be considered
adatoms,
ledge sites, as shown schematically
energy,
sake
energy
will be eliminated
with
of sites:
for
from
they
a
that
could serve
surface
To each type of site will be attributed free
In
characterized
Since,
to
vacancies,
as interacting types
being
free energy.
as
as adsorption
each
contribution
have been ignored,
electron
This is much larger than the extrapoof 0.067 obtained by Robertson and
surface.
all the defects that have been mentioned
plane in copper. value
that
must take into
general sense, it would be expected
as adsorption by
clean surface.
term curve to cc = 0 is not valid,
presumably
between
of a surface.
completely
the torque
will
The purpose of this section is to derive a set of adsorption isotherms which reflect (3)
reason
will be made
of gaseous adsorption surface
distance
As will become
sites in this treatment.
the previous
any treatment nature
namely
According
interkink
ADSORPTION
to the mean ledge
959
distances.
later no distinction
kinked and non-kinked
From
direction. At this point, it is convenient
the
be about five interatomic
is the number of possible ledge sites per cm of length along a line normal
that
(I)
with The
presently does not seem to the relative
can be predicted, rely
makes
experiment. magnitude
on experimental
meagre as it may be, regarding
evidence,
this point.
Ledge structure At elevated temperatures, will have a constant
the ledges at equilibrium
mean direction,
but they will
not be atomically straight.@) In analogy to the structure of a low index plane, a ledge will contain a certain density of kinks at temperatures above absolute zero. In contrast to the structure of a low index plane, however, the defect concentration is much larger in the case of the ledge. For example, for copper near it#s melting point, the Burton
et uZ.(~)
of
and therefore
FIG. 2. Schematic representation of interaction of an atom in gas phase with the surface (s), single ledge (I), and double ledge (d) sites on the surface.
ACTA
960
METALLURGICA,
VOL.
11,
1963
different sites on the surface, and are, of course, dependent on the first three reactions. Due to similarity of the first three reactions, the same formalism applies to all of them. It is necessary, therefore, only to derive the adsorption isotherm for the ith site. Consider the Gibbs free energy change SG whieh occurs as a result of transferring an infinitesimal number of atoms, Etnp,from the gas phase to occupy a certain number, 6n,, of the ith site. This free energy change can be written in terms of the chemical potentials, ,u, of the three species participating in reaction (5a), and is given by:
Gas Phoss
‘P FIG. 3. Adsorption energy dirtgram.
That atoms are more strongly adsorbed at ledges than on the surface receives support from the flashfilament-desorption studies of Hiekmott and Ehrlichol) and from low energy eleetron diffr~otion studies,(12) but these investigations do not provide information about the relative magnitudes of AC,“, AC,’ and AC,“. As will be demonstrated later,(*) an analysis of faceting phenomena can give some idea as to the magnitude of AG,‘. The adso~tion isotherm will depend on the mechanisms of adsorption and desorption as well as the type of adsorption layer that is formed, i.e. whether it is mobile or immobile. Since the main features of the present analysis can be illustrated quite well by considering the simplest kind of adsorption process, namely, the localized (immob~e) adsorption of single atoms, this will be considered first, while the complexity arising from dissociation of diatomic molecules will be treated later. It is necessary to develop this latter case also, since there is a strong likelihood that diatomic moleoules do dissociate upon adsorption, although this has been demonstra~d convincingly only for nitrogen on tungsten.ol) As will become evident, the type of adsorption isotherm does not alter the kind of changes that adsorption produces in the y-plot, but only their magnitude. The model just described gives rise to the following equilibria: Gi-X?r?:Ge G+LZG, G+DtiG= Gs*GL
(5a) (5b) (50) (5d)
G,ZG, GLZZGD
OW @f)
G symbolizes the gas atom; S, L, and D represent the surface, single ledge and double ledge sites; the last three reactions represent the equilibria among
where N is Avogadros number. Since the adsorption of a single gas atom destroys one of the ith sites and creates one g-i complex, it follows that
tin,
lingi = 6n, =
At equilibrium, 6G = 0, and equation (6) becomes IQ@6 = PI + I%
(7)
Making use of the thermodynamic relationship pi= & + RT1 n ai,in which pi0 is the chemical potential of the ith species in the standard state and ai is the activity of the ith species, it is possible to rewrite equation (7) in the form a 25
= exp
a@ai
1Pi0+;;-
aq _exp[-;;q
(*)
where ,QO - ,ugO- &’ z AGiQ is the adsorption free energy for the ith site, when the species are in their standard states. The activities aj have the following definitions in terms of surface concentration Ni (atoms/cm2 or atoms/cm), the saturation surface concentration Ni, (atomslom2 or atoms/em), the activity coefficients fsi, fi and f,, the pressure in the gas phase P and the pressure in the standard state PO.
(W
@2=f,
P p
() 0
(90)
GJOSTEIN:
ADSORPTION
AND
From these definitions, equation (8) can be placed in the form*
which defines a characteristic pressure Pi for the ith site given by (II)
SURFACE
ENERGY
expression for y0 is ye - yw =
-kTN,,
’ - p’ps s 0 (1 + P/P,,
=
-kTN,,
In (1 _t P/P,)
dlnp
yzo = --kTfi,,ln
Ideal behavior
(1 + P/PJ + kTflN,9, ln (I + P/P,)
The quantity f,&‘Ji the relation
may be considered, through (12)
as & measure of the lateral interaction energy AGir for this particular site. In general, AGil is a complex quantity and will be some function of the occupation probability pi = NJN,,. The most complete treatment can be worked out for adsorbing atoms which behave ideally. Ideal behavior will be defined as AG,r = 0 for 0 5 2
zs
yI -
I 1. This casewill be treated
for both dissociative and non-di~o~iative adsorption. An attempt will be made, later, to comment on the perturbing influence that AGir # 0 will have on ideal behavior.
+ kTSN,, In (1 + P/P,)
4% -_=
d 1nP
-kTI’,
(dT = 0)
where yk may be yO, yE or yr (from equation (Z)), and where the corresponding surface excess quantities rk are defined by l?,=N,-NN,rNs
(I3a)
rc = N, - XN,
(13b)
I’r = 2I’, -
I’& = 2N, -
equations
(lo),
N& - XN,
(13~)
(12) and (13a), the
* If the ratio P/P, is replaced by the concentmtion ratio C/C,, equation (10) applies equally well to solute adsorption from the crystal interior.
(14~)
where yO,, ylo and yr,, refer to surface, ledge and ledge in~r&ction energies at P = 0, that is, for a perfectly clean surface. Making use of the fact that the total number of all types of sites N, available for adsorption is a constant given by
the final expression can be arrived at by combining equations (2), (14) and (15) y -
y,,,,= -kTN, + i
In (1 + P/P,) yzo -
kTN,,
+ kT(N,, + IIaving obtained equations, giving the degree of occupancy of each type of site as a function of the variables of the system (P, AGiO, T), it is now necessary to relate the occupation probabi~ty to the change in surfaoe free energy, y. This can be done through the use of the Gibbs-Adsorption Isotherm, which may be stated in the form:
(14b)
yro = --BkTN,, In (1 + P/PJ + kTN, In (1 + P/P,)
AG: = - RT In (f,#JJ
(144
remembering that for ideal behavior P, is not a function of N,/N,,. Similarly the expressions for yI and yr are found to be yt -
Combining
961
(I)
-
fiN,J ln (1 +
f’/PJ
PIPJ]
-YOOa2 2 a2
- SrX -
In (1 +
[
‘yr, + kTNd, ln (1 + PIP,)
2kTN,, In (1 + P/P,)
+ kT[XiWN,,
-NT, .,
-
flJf)I In (1 + P/P,)
1
+ -t?-
-
SMln (I + P/P,)]
(16)
-
N,, + Sfl,(l
XSM2
[ETN,,
Some simplification of equation (16) is possible. The third and fourth order terms are present only when P > 0. They have an appreciable magnitude only when ccis large and when there is preferential adsorption at surface sites, P, < fPd, P&. When ot is large, the model used here should be least applicable, particularly without the inclusion of higher order terms in the y vs. M:relationship for P = 0. Accordingly, they will be ignored in deriving expressions for the slope
ACTA
962
METALLURGICA,
and intercept of the torque term plot from equations (3) and (4). In the equations that follow, N,, = N,,.
(17) 2kTN,, +----S2MY,0 f12MY,
23=1+-
X Pn (1 + P./PA + e
[(SM -
i
00
2 ln (1 + P/P,)]
l)(iVz, - SN,)] lnfl -
VOL.
11,
1963
can be viewed as an interaction energy for ith sit,e.
A comparison of equations (10) and (21), for the case where P, = P,,, it shows that at small degrees of occupation equation (21) gives much higher values of N~lNi~. may be derived following Expressions for ylc - ylEO the integration procedure used previously, but for sake of brevity they will not be given here. Moreover, the analog of equation (16), which is very similar to it in appearance will be omitted. If the final result is simplified in the manner discussed in connection with equation (lci), the parameters A and 3 that characterize the y-plot are given by
PIP,) (18)
Adsorption
with dissociation
Allowing the ith site to be represented by Si, a prototype equilibrium for dissociative adsorption can be written as G, + 2X,*
2(G - SJ
+ rg
so that equation (8) becomes --.aoi
2 =
exp
[(~/GO
+
~~2’
-
‘g2’i2
2,Uu,i”)/RT]
= exp (-_G,,‘/RT)
(20)
AG,,O is the adso~tion site for the dissociative case free energy of the itsh when the species are in their standard states. If use is made of the definitions given in equations (9), the adsorption isotherm for the dissociative case can be expressed in the form where 2,~’
-
2pio -
,ugzo =
ENJOY,, -
2 In [l +
SN,) In [I + (PIJ’sa)1’21 (23)
+ 4kTN,, m
oo
(In [l + (fYPdf21
tPI'P,,P21) -tg
00 x [(EM - l)(N,, + SN,)] In [l + (P/f’,,)l~z]
The free energy change (SC result from the transfer of Sng2 diatomic molecules from the gas phase to an, sites on the surface is the same as given in equation (6), except in this case 6?z,z = 2&b, = 2&Z.,,
2~1,
B,=1$-
(19)
(N,, 00
(24) DISCUSSION
According to the above t~atment, there are three major effects that an adsorbing gas will have on the shape of the y-plot in the vicinity of an energy cusp. Each effect comes into prominence as the ambient pressure P approaches the characteristic pressure of the site. In the analysis that follows it will be necessary to treat limiting cases, i.e., cases where one (or two) of the ~harac~risti~ pressures is much larger than the others. Such limiting cases are not without experimental interest. For example, if the AG,O’s are separated by 10 kcal/mole-a figure that is not unreasonable in view of the fact that heats of adsorption
may be as large as 150 kcal/mole-the
characteristic pressures will differ by approximately two orders of magnitude at 1000°K. Consider first the ease where there is enhanced binding at the ledge
where the characteristic pressure P,, of the ith site for the dissociative case can be written as P
id
= L!@- Pli2 exp (AG.ad“/RT) f$ a
Analogously to the previous case, RT
soi_ lnfTi = -
(221
and double ledge sites. Case I:
P, > (Pd, PI) or P,, > (P,,, P,,) As the ambient pressure P is increased from zero the first major change in the y-plot occurs in its as reflected by changes in B. In equations (18) and (24) the terms depending on P, or P,, will
curvature
AG:”
be neglected,
and for convenience
the following
new
GJOSTEIN:
ADSORPTION
AND
SURFACE
ENERGY
963
(I)
variables are defined. 2.0
x=PIP,
Xd = PIP,, 1.5
R=
Pa/P1
With these definitions it is possible to write equations (18) and (24) in the form
t
1.0
a? 2
0.5
0 al" 0 I 0 " -0.5 -1.0
B, = l3,, + 2I3, In
-1.5
1 + x2’2
I
(25b)
II 3 (fi&Pli
The parameters K and K, are a measure of the relative binding energies of the single and double ledge sites. As (Ii, K,) -+ 0 double ledges tend to have the dominant binding energy, which means that the concentration of adsorbed atoms in single ledge sites can be neglected with respect to that in double ledge sites. Figs. 4 and 5 give plots of (B - B,,) /3,and (Bd - 3,,)/2B,,vs. x and xd, for various values of K and K,. For the non-dissociative case, when 0 < K < 0.5, the curves in Fig. 4 show a maximum, which is located at x, = (1 - 2K)lK. For x > x, = (1 - 2K)jK2, (B - B&B, -=c 0. When K > 0.5, (B - &J/B, is never positive and decreases monotonically with increasing x due to the dominance of the single ledge term. In the dissociative case, a maximum at xdm = (1 - 2K,1'2)2/K, appears in the curves in Fig. 5, when 0 < K, < 0.25, and ( Bd - B,,)/2B, < 0 for all
0
5
IO
I5
20 _ 25
30
35
40
45
50
x,j’YPdd FIG. 5. Effect of reduced pressure (X, = P/P& on yz-plot curvature parameter (Rd) for dissociative adsorption. Kd( =P,/P,,) is a measure of the relative adsorption energies of single and double ledge sites. x& when Kd > 0.25. If a comparison is made between dissociative and non-d~sociativ~ cases, by setting x = x~, it can be seen that for a given value of (x, zd), the increase (or decrease, depending on K) in curvature of the y-plot is somewhat greater for non-dissociative adsorption, and that for a given K = Kd value, the maximum occurs earlier for dissociative adsorption. The second major change in the shape of the y-plot occurs in the slope of the energy cusp at its origin, (A, Ad), as is evident from equations (17) and (23). To describe this effect, use is made of the definitions
so that equations (17) and (23) can be rewritten as A=&,-&ln(l
+Kx)
(264
A, = A,,- 2A, In [ 1 + (X,Z#~]
0
I
I
I
,
I
I
I
,
I
IO
20
30
40
50
60
70
60
90
Xf
IO0
P%
pl-3SSUre {x = P/P,) On yz-plot curvature parameter (B), for non-dissociative adsorption. K(= Pa/P,) is a measure of the relative adsorption energies of single and double ledge sites.
FIG.
4.
Effect
Of
lY?dUC%d
(26b) Fig. 6 gives plots of (A - A,,)/A, and (A, - A,)/2A, vs. it: and xcl for various values of K and K,. From equation (26) it is clear that at some value of (x, x& (A, Ad) will become zero and finally negative. This will occur when the logarithmic functions attain a value of A,,/A, (or A,,/2A,). Generally it can be said that when (K,KJ < 1, the effect of adsorption on curvature of the plot. y-plot occurs at much smaller values of (x, z8) than does the effect on the slope at the origin, but that when K - 0.5(orK, -+ 0.25) both effects tend to be present simultaneously. It was mentioned above that it is possible for (A,Ad) to become negative*. When this is the case has discussed surface energy.
* Herring’5)
negative
the
possibility
of having
a
ACTA
964
‘CA,
~ETALLURG
: B
- 1.0
I I
11,
1963
conditions, ledges can bunch together, resulting in a phenomenon known as thermal faceting. The criterion for this to occur will be discussed in an accompsnying paper.(8) A summary of the adsorption effects for Case I is shown schematically in Fig. 8.
0
t 8
VOL.
----_-_
s
N
-2.0 P”0
b
::
a* I 4
4
P -
-3.0
P-
- 4.0 t
1 0
Pd
P,
I 11
20
1
$1 40
x.Por
pd
I 60
x d
11
I
1 80
IO0
-p-m pdd
?&a.
6. Effeot of reduced pressure (X, X,) an the slope of y2-piot at the vertex of the cusp (A, A?). (K, K,) is a mewmro of the relative adsorption energies of single and double ledge sites. The dashed curves represent dissociative adsorption and the solid curves represent non-dissociative adsorption.
an outward pointing cusp will appear in the y-plot. Fig. 7 schematic&~y shows this tradition. There is no ex~riment&l evidence indicting whether or not this effect actually occurs. At first glance, the concept of a negative ledge energy might be rejected, since apparently the ledges could increase in length spont&neously with it decrease in free energy of the system. The total surface energy is not negative, however, and any change in length of the ledges must be subject to the criterion that total surface energy of system be B minimum. TTndercertain thermodynamic
FIU. 7. y,-plots and corresponding Gibbs-Wulff shapes, depicting the trrsnsition from an inward to an outwardpointing cusp.
0 -Ct-
FIG. 8, Schematic ~umm5ry of major effects of adsorption on shape of y vs. cc plots.
Case II: P, < (Pd, PJ or Ptd < (P,,, P,,) A third major change in the y-plot occurs when the ambient pressure P approaches P, (or Pad). This effect which would not be important in Case I, results when there is stronger bonding of the adsorbed atoms on surface sites relative to the ledge sites. This is a complex effect, involving alterations in B, A snd yO. Examination of equations (18) and (24) shows that the important term controlling the magnitude of the curvature effect is, (XiW - l)(Nls + SN,),, which takes on values of -0.086 N,, or -0.17 N,, for the (100) and (111) planes, respectively. Thus, the curvature (B, B,) should decrease with increasing (x, Q) in a manner shown by the curves for (K, KS) = 0 in Figs. 4 and 5, when they are reflected about the zero baseline. Equations (17) and (23) show that the effect on (A, Ad) depends on the term, N,, + SN,, which takes on values of 1.71 N,, and 2.41 N,, for the (100) and (111) plane, respectively. Thus, in contrast to Case I, (A, A,) should increase with increasing (X, X,) for both the (100) and (111) cusp. The form the (A, Ad) vs. (X, X,) relationship can be seen by reflecting the curves in Fig. 6 about the zero baseline. These effects are essentially the reverse of Case I. They can be illustrated by viewing the sequence of curves, from bottom to top, in Fig. 8, as representing increasing pressure.
ADSORPTION
GJOSTEIN:
AND
SURFACE
ENERGY
TABLE 1. Adsorption of oxygen on copper at 1000°C
Heat of adsorption
- 110,000 ealfmole 0,
965
(I)
__I__ Dissociated coverage -_I
Dew point
PO,
Pebar&eristie
Undissociated coverage
- 36°C
3.5 x IO-“2
5 x 10-13
7 X IO-10
2.7 x 10-z
+23*c
3.5 x lo-‘8
5 x 10-13
7 x LO-6
2.7 x 10-a
The effect on y0 can be determined from equation (16). As is evident, y0 will decrease with increasing pressure by an amount kTN, In (1 + P/P,)* for non-dissociative adsorption or by an amount kTN, In [l + (P/PS)1i2] for dissociative adsorption. For the (160) plane in copper ~TN~ = 270 ergs/cm2 at lOOO”C, which means that by, = (0.1, 0.2)~~ when (P/P,, P/P,,) = 1. Since y0 decreases with increasing P, all the vicinal planes will be reduced by the same amount, and thus each point of the y-plot will shrink towards the origin.
The y-plot has been explored for two metals, copper’3) and nickel,(1t2) under entirely different ambient atmospheres. The nickel specimens were annealed at 1000°C in a vacuum of about low5 mm, while the copper specimens were annealed in a hydrogen atmosphere with a dew point of either -36°C or +23”C at temperatures in range 950-1020°C. The torque terms repotid by Robertson and Shewmon(3) for copper annealed under hydrogen are somewhat smaller than those for nickel. This might be viewed as an impurity effect, but as will be shown, there is a strong possibility that this is not the case, and that probably they are intrinsically lower. Dell et aZ.(lQ found the heat of adsorption of oxygen on copper at low coverages to be about Oxygen appears to be the - 110 kcaljmole 0,. most likely adsorbent on copper, since hydrogen has a much smaller, although uncertain, heat of adsorption (it is reported in the range -9.0 to -35.0 kcal./mole H,.) (17~18)Assuming a reasonable value for the entropy of adsorption of hydrogen, it should have a very small free energy of adsorption at elevated ~m~eratu~s (l~O°C~, and thus should have a negligible coverage, unless there is a change in bonding mechanism at these temperatures, which is not reflected in the heat measured at lower temperatures. Table 1 gives the degree of occupancy to be expected for a heat of adsorption of - 110 kcal/mole 0, and * These values c&r&ted by assuming that every lattice
site on the substrate can serve as sn adsorption site. Ledges were assumed to lie along close-packed directions in an f.c.c. lattice. Lscmann et aZ.(‘) were concerned soley with this term; they assumed P and N, to vary in a specific manner with the crystallographic plane under consideration.
entropy of adsorption of -30 e.u.7, under various conditions. It can be seen that the worst possible case is for a dissociated: molecule when the dew point is +23”C, where the degree of occupancy would be 2.7 x 10m3. Since there is no information available which would allow the charac~ristic pressure given in Table 1 to be associated with any particular site, the most unfavorable assumption will be made. This means the characteristic pressure should be equated to Pa,,. If this is true, the change in Bd will be, Bd - B,, = 5.4 x 1O-3 fz,, due to the terms containing P,,. For the (100) cusp in copper, B, is found to be at lOOO*C B, -
2kT Nzs _ 2kT 4 y0 PA!l y0 uf g(1.38 x lo-16)(1.273 x 103) = o 6.
=
(1.7 x 103)(3.61 x 10-8)2
’
For the (ill) cusp of copper, B, is a factor 0.43 smaller than this value. Thus it appears that Robertson and Shewmon essentially de~rmined B,,,. It could be argued, however, that I(AH,,O, AH,,O)I > j AH,“j, and that this larger heat was not detected calorimetrically. If this were the case there should be appreciable contribution to B, from terms containing (P,,, P,,). However, at the lower dew point this contribution should decrease considerably. This means they should have found B to vary with the partial pressure of oxygen whereas no appreciable effect on B was observed when the partial pressure of oxygen was decreased by four orders of magnitude. Similar reasoning can be applied to A,, and again no effect of adsorption seems likely. It must be concluded, then, that adsorbed oxygen did not influence the results of Robertson and Shewmon, and that it seems doubtful that adsorbed hydrogen exerted any influence either, although this latter point must be left open until further information is availab1e.g t The entropy of adsorption was assumed to be the same as entropy of formation Cu,O, namely -30 Cal/mole 0, - “K, since an atom loses essentially the same number of degrees of freedom in both the oxidation and adsorption processes. f It has not been established conclusively whether or not oxygen exists as dissociated atoms on metal surfaces. 3 If it is assumed that B,, W&Rmeasured, it is interesting to note that B, < 1, and therefore it follows that yro, i.e. there is a rep&non between single ledges. The significance of this f&zt is not understood as yet.
ACTA
966
Mykura
found
larger
the
parameters
A and B for nickel than those reported
for copper;
TABLE 2. Comparison
values
METALLURGICA,
of
A and B for nickel and
of parameters copper
11, 1963
VOL.
Deviations from ideality In any real system such as those discussed deviations
from ideal behavior
is independent
of degree of occupancy
and therefore,
of the pressure of the adsorbing
Very little is known theoretically
Nickel
copper
* Actually
so
that it is no longer possible to assume that Pi (or Pid)
A
(111) (100)
above,
can be expected,
gas.
about the deviations
0.17 0.30
0.24t 1.13
that might
not be treated within the scope of this paper.
Experi-
0.11 0.067
0.22 0.14
mentally,
heat
the slope of the experiment
be expected,
of given site,
it is known
adsorption
torque vs. GLplot is
which,
decreases
increasing
coverage
at any rate, could
generally (becomes
that
the
less negative)
of the surface.
The manner
which it decreases is not always simple.
rather than These two quantities are not much and nickel values, where yO/v is nearly unity and A is small. B’ is not constant for the (111) cusp in nickel. This value is for a = 0; B’ increases to 0.8 at cc = 0.3.
viewed
Pi increase
as having
occupancy
of the ifh site.
with
would
retain
the
degree
In this case,
essentially
the
in
This can be
changes in shape, which were described case,
of
with
of
the basic
for the ideal
same
character,
but will occur at higher pressures
(or lower temper-
that the values of A for copper are for a clean surface,
atures).
Given enough information
about the changes
it appears that either Case II-type
on
y-plot
a comparison
place
on nickel,
higher yet,
is given in Table
single
ledge
or that nickel
ledge
energies
principles
energy,
shape of small nickel
by Mykura.
This indicates
have
influenced
been
pressures Without
surface
pressure
it is not
cleanliness
is about
this
of the Gibbs-
by Sundquistc4)
adsorption. systems
The
possible to
the partial pressure of oxygen fraction
of
electron
diffraction
total
pressure
and
from ideal behavior
The phenomenological in the y-plot, has been
developed,
structure
theory
of shape alterations
due to adsorption
based
utilizing
from the gas phase, a model
on three types
that the slope of the y-plot
pressure in a degree
(15).
bulk In
of the adsorbing
decrease
there is enhanced
ledge sites.
binding
pressure
temperature)
of the adsorbing
when
atom at
ledge sites relative to the surface sites (Case I).
The
at
slope of the cusp origin can decrease to a point where
order
to
it becomes negative,
even
the surface
causing a cusp inversion.
sites have
the stronger
of nickel,
(Case II),
the slope of the y-plot
must be a very small
increasing
pressure of the adsorbing
of
sites: Theory
in the vicinity
with increasing
gas (decreasing
surface
NiO
(or probably oxygen?)
The
of a cusp should
of
of adsorption
predicts
1O-5 mm.
it
for a given system.
used by Mykura
set of data.
pressure
temperature,
should be possible to determine the extent of deviation
single ledge and double
to form
with oxidation
with
surface,
to assign
either
required
the
total
the same,
with chemisorbed
the
first
that the results for nickel
8 x 1O-s mm
problems
more critically
from
to decide
check on the partial
of oxygen
eliminate
As
terms than those found
were essentially
a further
system,
1000°C
crystals
by
in the vacuum
and Sundquist
copper.
determinations
much larger torque
takes
has an intrinsically
be calculated
Wulff
of
adsorption
ylO, than
cannot
Some preliminary
each
Since it appears
and thus it is not possible
issue.
indicate
2.
10e5 mm.
The
binding
will increase gas.
forces with
For Case I
the curvature
increases
worthog) show that carbon is also a possible adsorbent
goes through The maximum
a maximum and finally decreases. tends to vanish as the difference in
on nickel, in which case it is to be expected
binding
energy
becomes
large.
adsorption
studies
of
Schlier
and
Farnsthat CO
may be another problem.
There are no other data on the shape of the y-plot for metals,
except
for information
that can be ob-
tained from thermal faceting. This subject cussed in detail in an accompanying paper.(s)
is dis-
t This parenthetical comment was added because normally it would be expected that chemisorbed oxygen would be bonded more strongly than bulk oxide. However, Paravano et aZ.‘l*’ reported a heat of adsorption of - 111 kcal/mole, which is very close to the heat of formation of bulk oxide, - 116 kcal/mole.
with pressure
When
between When
single
critical value, the curvature with increasing curvature involved.
with
and
this difference
pressure. pressure
at first, then
double
ledge
drops below
a
decreases monotonically
Case II gives a decreasing increase;
no minimum
is
References 1. H. MYKURA, Acta Met. 9, 570 (1961). 2. J. M. BLARELY and H. MYKURA, Acta Met. 9. 595 (1961). 3. W. M. ROBERTSON and P. G. SHEWMCIN, Trans. Amer. Inst. Min. (M&U.) Engrs. 224, 804 (1962).
GJOSTEIN:
ADSORPTION
AND
4. B. E. SUNDQUIST. Private communication. 5. C. HERRING, Structure and Properties of Solid Surfaces. (Edited by R. GOMER and C. S. SMITH), p. 5. University of Chicago Press (1953). 6. C. HERRING, Phys. Rev. 82, 87 (1951). 7. R. LACMANN and I. N. STRANSKI, Growth and Perfection of Crystals. (Edited by R. H. DOREMUS, B. W. ROBERTS and D. TURNBULL), p. 427. Wiley, New York (1958). 8. N. A. GJOSTEIN. Acta Met., this issue p. 969. 9. W. K. BURTON, N. CABRERA and F. C. FRANK, Phil. Trans. Roy. Sot. Lond. 243A, 299 (1951). 10. W. W. MULLINS, Acta Met. 7, 746 (1959). 11. T. W. HICKMOTT and G. EHRLICR, J. Phys. Chem. Solids 5, 47 (1958). 12. E. BAUER, Phys. Rev. 123, 1206 (1961). 13. N. A. GJOSTEIN, Acta Met. 7, 812 (1959).
SURFACE
ENERGY
967
(I)
14. 0. D. GONZALEZ and G. PARRAVANO, J. Amer. Chem. Sot. 78, 4533 (1956). 15. F. D. RICHARDSON and J. H. E. JEFFES, J. Iron St. Inst. 160, 261 (1948). 16. R. M. DELL, F. S. STONE and P. F. TILEY, Trans. Fu”aTaday Sot. 49, 201 (1953). 17. T. TAKEUCHI and
M.
SAKAGUCHI,
Japan 39, pp. 177-182 (1957). 18. T. KWAN, Advances in Catalysis 6, 67.
Bull.
Chem. Sot..
Academic
Press,
New York (1954). 19. R. E. SCHILIER and H.
E. FARNSWORTH, Advances in Catalysis 9, 434. Academic Press, New York (1957). 20. The author is indebted to Professor G. M. POUND for discussions on this problem, and to Drs. H. MYKURA and J. HIRTR for communications, which help crystallize some of these ideas.
AND
SURFACE
ADSORPTION N.
A.
ENERGY ON
THE
(I):
THE
EFFECT
OF
y-PLOT*
GJOSTEINt
Based on the idea that an adsorbing gas atom can interact with any of three different types of surface sites-surface, single ledge, and double ledge sites-each being characterized by discretely different adsorption free energies-a phenomenological theory is developed, governing the shape alterations of the y-plot due to adsorption from the gas phase. When single and double ledges are preferred sites for adsorption, the theory predicts that an appreciable increase in the curvature of y-plot can occur as the pressure of the system approaches the characteristic pressures of these sites. At some higher pressure, depending on the relative adsorption energies of single and double ledge sites, a marked decrease in the slope of the y-plot at the origin of the cusp will occur. When the surface sites are preferred for adsorption the curvature of the y-plot decreases, and the slope at the origin of the cusp increases as the system pressure approaches the characteristic pressure of the surface site. In addition to, and simultaneously with these changes, each point of the y-plot will shrink towards the origin of the y-plot. ADSORPTION
ET ENERGIE L’ADSORPTION
DE SURFACE (1): 1’INFLUENCE SUR LE DIAGRAMME y
DE
Elle est basee sur l’idee qu’un atome gazeux peut s’adsorber en trois genres d’emplacements, caract&i&s par des energies libres d’adsorption differentes, placement en surface proprement dit, contre un gradin peu marque (“single ledge”) au creux dun gradin accentue(“double ledge”). Les auteurs developpent une theorie qui regit les modifications de la forme du diagramme de l’energie de surface y, dues 21 I’adsorption dune phase gazeuse. Quand l’adsorption a lieu preferentiellement contre des gradins, cette theorie prevoit une augmentation sensible de la courbure du diagramme y quand la pression du systeme s’approche des pressions Pour une certaine pression plus elevee, en relation avec les caracteristiques de ces emplacements; energies relatives dadsorption, se produit une diminution marquee de la pente du diagramme y vers son origine .
Quand l’adsorption a plut6t lieu en surface proprement dite la courbure du diagramme y diminute, et la pente a l’origine augmente quand la pression du systeme d’approche de la pression caracteristique du site d’adsorption. En m&me temps, les points du diagramme y se rassemblent aux environs de l’origine. ADSORPTION
UND OBERFLACHENENERGIE (I): EINFLUSS ADSORPTION AUF DIE 1/-FLACHE
DER
Ein adsorbiertes Gasatom kann mit drei verschiedenen Oberlliichenpliitzen wechselwirken: mit Pliitzen auf der F&he, an einer einfachen und an einer doppelten Stufe; die Freie Energie der Adsorption ist fur die drei Pliitze verschieden. Diese Vorstellung wird einer phanomenologischen Theorie zugrunde gelegt, die die Anderungen der y-F&he infolge Adsorption aus der Gasphase behandelt. Wenn einfache und doppelte Kanten bei der Adsorption bevorzugt werden, kann nach der Theorie eine betrachtliche Zunahme der Krtimmung der y-Fliiche auftreten, sobald der Druck des Systems sich den charakteristisohen Drlicken dieser Oberfliichenpliitze ntihert. Bei einem hijheren Druck, der von den relativen Adsorptionsenergien der einfachen und doppelten Stufen abhangt, tritt eine ausgepriigte Abnahme der Steigung der y-Flache an der Spitze auf. Wenn die bevorzugten Pliitze der Adsorption die auf der Fl5che sind, nimmt die Krtimmung der y-Fliiche ab, und Steigung an der Spitze nimmt jedoch zu, wenn der Druck des Systems sich dem oharakteristischen Druck des Flachenplatzes nahert. Gleichzeitig mit diesen Veranderungen tritt eine zusiitzliche Schrumpfung der gesamten y-Fliiche auf.
INTRODUCTION
Recently, determining
cusps at the (100) and (111) orientations.
methodso-4)
have
been
developed
the variation
of the surface energy
In nickel
for
some minor
(y)
notably (llO), (210) and (410), but the results here are not as well established.(i)
of metals with crystallographic orientation of the surface. When interpreted in terms of the polar y-plot,(5*6) these results indicate that the shape of
In carrying a strong
cusps may occur
out experiments
likelihood
that
at other orientations,
of this type there is
impurity
atoms,
adsorbed
the y-plot for nickel and copper is similar to that predicted by Herring(b) several years ago. For these
from the gas phase or from the crystal interior, will be
metals, curved
understand the alterations in shape of the y-plot which will occur as a result of gaseous adsorption. Lacmann and Stranski”) have given a treatment for the change in surface energy of low index planes with
the y-plot appears surface except for
to be a continuously major inward-pointing
* Received July 19, 1962. t Metallurgy Department, Scientific Motor Company, Dearborn, Michigan. ACTA
METALLURGICA,
VOL.
Laboratory,
11, AUGUST
1963
Ford
present
on the surface,
gaseous adsorption, 957
and thus it is important
to
but there has been no treatment
ACTA
958
METALLURGICA,
of the effect of adsorption on the shape of the y-plot in the vicinity of a cusp. The present work is directed towards solving this latter problem. Utilizing a model of surface struoture, which has evolved from the surface energy work described above, a phenomenologi~al theory is developed, which predicts the change in shape of a cusp in the y-plot with the extent of adsorption. From the results of this analysis, an estimate will be made of the extent to which the present determinations of y-plot have been influenced by adsorption. In an accompanying paper,(*) it will be shown that dependence of surface-energy-induced thermal faceting on gaseous impurity adsorption oan be explained on the basis of this analysis. SURFACE
STRUCTURE
Low index planes Burton, Cabrera and Frank(s) and &Iullins(lo) have treated the problem of atomic roughening of an atomi~a~y smooth low index plane that occurs with increasing temperatures. At elevated temperatures, a small concentration (approximately 5 X IOP5, at temperatures near the melting point in copper) of surface adatoms and vacancies are stable due to the lowering of free energy of the surface that occurs from the configurational entropy of the roughened surface. Both treatments conclude, however, that for most metals no large soale disordering (surface melting) of low index planes will occur even at temperatures near their melting points. For the purpose of this treatment, they will be considered as atomically smooth. Vicinal prunes In discussing the origin of inward-dinting cusp in the y-plot, Herring(5@) suggested a structure for the vieinal planes of a low index plane. According to this picture, the vicinal planes should differ in structure from the low index plane only by the presence of certain density of monatomic ledges of height S and average separation
&G
Wu3
where M is the small angle of deviation from the low index plane. The surface energy y of a vicinal plane for small tc is given by
where y0 = surface energy of low index plane (ergslcmz) yI = excess free energy of surface per unit length of ledge (ergs/cm).
VOL.
II,
1963
(III)
PLANE
FIG. 1. Torque terms for (111) cusp of copper, 0 -0FHC e-high purity copper in copper in graphite boat; graphite boat; a-OFHC copper with no graphite in system (3). Taken from Robertson and Shewmon, to be published in Trans. Amer. Inst. Min. (MetaEZ.) Engrs. 224 (1962).
Equation (1) predicts a cusp wit8ha parabolic shape, having
a constant torque
term,
ay 75 acr = x -
yOa.
Experimental determinations of torque terms for nickel(l) and coppeG show that the torque term decreases approximately linearly* with angular deviation from the low index plane, Fig. 1. The magnitude of the decrease, however, is generally much smaller than would be expected from the coefficient yO. This indicates that another second order term in u should appear in equation (1). Robertson and Shewmon(3) have shown how this term can arise, if the steps are randomly distributed, from the probability of forming double ledges-that is, ledges of height 2s.t Their equation may be written:
y=yo+
Fu- [~ + I&]u2
(2)
* In the data of Robertson and Shewmon,‘3) the scatter is too large to conclude that there is any deviation from linearity. Mykura”) indicates some curvature in his plots, but this will be ignored in the present treatment. t Other models also can give rise to a second order term. For example, if ya varies inversely with the single ledge spacing, to a first approx~tion, a relationship similar to equation (2) can be derived. Alternatively, it can be postulated that the ledges are in thermal equilibrium and that two single ledges combine with the formation energy yrbe, where _5? is a length per atom along a ledge. In this case, the COefficient of cxzin equation (2) will be yr/S*M exp [+y&‘/kT]. The treatment given in this paper wa8 carried out only for the statistical model of Robertson and Shewmon because it appears to be the simplest model consistent with the experimental facts. It would not be difficult to modify the present treatment, to t&e into account these other ideas.‘aO’ 3 In equation (2), the density of single ledges a/S has not been modified to take into account the presence of double ledges because the density of double ledges is small in comparison to x/X, especially for small a.
GJOSTEIN:
ADSORPTION
AND
SURFACE
treatment
where is YI = 2YL - YD (positive or negative)
the
interaction
energy
between ledges, yD is the
energy per unit length of double ledge, and M
ENERGY
shows
evident
two
quantities
mentally,
which
to define for later use
may
be
determined
experi-
account
the
&_L
(a2r1=
yoO 8x2
2y,
to this model
for believing
that
+k
(4)
S2MYCW
where yoOrefers to y0 for perfectly
the
heterogeneous
there appears a linear
to be no
extrapolation
curvature
of since
any higher order terms in equation
would give rise to appreciable
(2),
of the torque
term plot at large angles rather than small. For both nickel and copper, however, there are no data at small
angles
information
to verify
a minimum
of a facet.
except
for some
from the equations
These equations
will give
value for the torque terms at the origin
of the cusp.
Torque
these equations taken
this point,
that may be obtained
of equilibrium
from
terms for nickel obtained
agree well with extrapolated
the twin
boundary
method,
from value
according
to Mykura,(l)
but this does not seem to be the case
for
Gjostein(l2)
37 1
copper. 1
MODEL
discussion
it is obvious
estimated
value for -aa to be about Y00 [ 0
structure-that
the
minimum
0.4 for the
(100)
character
is, the defect
of the
sites,
an adsorption
simplicity, such
the
defects
impurity
atoms
considered different double
site
kinks,
and sites arising
sites.
so too, Instead,
adsorption is
shown
although
AG,“,
three
surface,
AG,‘,
in
Fig.
point
3,
with
evident
of these
quantities
bond
be in a state
facets on which measurements
it is necessary
further experimental
is needed.
evidence
an adsorption An
> AG,“,
AG,o,
The relative magnibe determined
from
knowledge
of the nature
of the
in which AG, to
a phenomeno-
can
that an adsorbing
Shewmon. The former value was based on only a few observations due to the difficulty of obtaining Thus,
and
in Fig. 2.
and AGF may be
atom
theory of surface bonding
can be made.
ledge
> AG,’
From
AG,’
lated
a serious discrepancy,
single
respectively.
AG,’
later.
of view, both
either a theoretical
AG,O,
will be
energetically
this order is not essential to the treatment
either greater or less than AG,‘. tude
of
bulk
energy diagram, which depicts this model,
as will become logical
AG,,
of
dislocations
the gas atom
the ledge or double ledge, or from
before this can be considered
adatoms,
ledge sites, as shown schematically
energy,
sake
energy
will be eliminated
with
of sites:
for
from
they
a
that
could serve
surface
To each type of site will be attributed free
In
characterized
Since,
to
vacancies,
as interacting types
being
free energy.
as
as adsorption
each
contribution
have been ignored,
electron
This is much larger than the extrapoof 0.067 obtained by Robertson and
surface.
all the defects that have been mentioned
plane in copper. value
that
must take into
general sense, it would be expected
as adsorption by
clean surface.
term curve to cc = 0 is not valid,
presumably
between
of a surface.
completely
the torque
will
The purpose of this section is to derive a set of adsorption isotherms which reflect (3)
reason
will be made
of gaseous adsorption surface
distance
As will become
sites in this treatment.
the previous
any treatment nature
namely
According
interkink
ADSORPTION
to the mean ledge
959
distances.
later no distinction
kinked and non-kinked
From
direction. At this point, it is convenient
the
be about five interatomic
is the number of possible ledge sites per cm of length along a line normal
that
(I)
with The
presently does not seem to the relative
can be predicted, rely
makes
experiment. magnitude
on experimental
meagre as it may be, regarding
evidence,
this point.
Ledge structure At elevated temperatures, will have a constant
the ledges at equilibrium
mean direction,
but they will
not be atomically straight.@) In analogy to the structure of a low index plane, a ledge will contain a certain density of kinks at temperatures above absolute zero. In contrast to the structure of a low index plane, however, the defect concentration is much larger in the case of the ledge. For example, for copper near it#s melting point, the Burton
et uZ.(~)
of
and therefore
FIG. 2. Schematic representation of interaction of an atom in gas phase with the surface (s), single ledge (I), and double ledge (d) sites on the surface.
ACTA
960
METALLURGICA,
VOL.
11,
1963
different sites on the surface, and are, of course, dependent on the first three reactions. Due to similarity of the first three reactions, the same formalism applies to all of them. It is necessary, therefore, only to derive the adsorption isotherm for the ith site. Consider the Gibbs free energy change SG whieh occurs as a result of transferring an infinitesimal number of atoms, Etnp,from the gas phase to occupy a certain number, 6n,, of the ith site. This free energy change can be written in terms of the chemical potentials, ,u, of the three species participating in reaction (5a), and is given by:
Gas Phoss
‘P FIG. 3. Adsorption energy dirtgram.
That atoms are more strongly adsorbed at ledges than on the surface receives support from the flashfilament-desorption studies of Hiekmott and Ehrlichol) and from low energy eleetron diffr~otion studies,(12) but these investigations do not provide information about the relative magnitudes of AC,“, AC,’ and AC,“. As will be demonstrated later,(*) an analysis of faceting phenomena can give some idea as to the magnitude of AG,‘. The adso~tion isotherm will depend on the mechanisms of adsorption and desorption as well as the type of adsorption layer that is formed, i.e. whether it is mobile or immobile. Since the main features of the present analysis can be illustrated quite well by considering the simplest kind of adsorption process, namely, the localized (immob~e) adsorption of single atoms, this will be considered first, while the complexity arising from dissociation of diatomic molecules will be treated later. It is necessary to develop this latter case also, since there is a strong likelihood that diatomic moleoules do dissociate upon adsorption, although this has been demonstra~d convincingly only for nitrogen on tungsten.ol) As will become evident, the type of adsorption isotherm does not alter the kind of changes that adsorption produces in the y-plot, but only their magnitude. The model just described gives rise to the following equilibria: Gi-X?r?:Ge G+LZG, G+DtiG= Gs*GL
(5a) (5b) (50) (5d)
G,ZG, GLZZGD
OW @f)
G symbolizes the gas atom; S, L, and D represent the surface, single ledge and double ledge sites; the last three reactions represent the equilibria among
where N is Avogadros number. Since the adsorption of a single gas atom destroys one of the ith sites and creates one g-i complex, it follows that
tin,
lingi = 6n, =
At equilibrium, 6G = 0, and equation (6) becomes IQ@6 = PI + I%
(7)
Making use of the thermodynamic relationship pi= & + RT1 n ai,in which pi0 is the chemical potential of the ith species in the standard state and ai is the activity of the ith species, it is possible to rewrite equation (7) in the form a 25
= exp
a@ai
1Pi0+;;-
aq _exp[-;;q
(*)
where ,QO - ,ugO- &’ z AGiQ is the adsorption free energy for the ith site, when the species are in their standard states. The activities aj have the following definitions in terms of surface concentration Ni (atoms/cm2 or atoms/cm), the saturation surface concentration Ni, (atomslom2 or atoms/em), the activity coefficients fsi, fi and f,, the pressure in the gas phase P and the pressure in the standard state PO.
(W
@2=f,
P p
() 0
(90)
GJOSTEIN:
ADSORPTION
AND
From these definitions, equation (8) can be placed in the form*
which defines a characteristic pressure Pi for the ith site given by (II)
SURFACE
ENERGY
expression for y0 is ye - yw =
-kTN,,
’ - p’ps s 0 (1 + P/P,,
=
-kTN,,
In (1 _t P/P,)
dlnp
yzo = --kTfi,,ln
Ideal behavior
(1 + P/PJ + kTflN,9, ln (I + P/P,)
The quantity f,&‘Ji the relation
may be considered, through (12)
as & measure of the lateral interaction energy AGir for this particular site. In general, AGil is a complex quantity and will be some function of the occupation probability pi = NJN,,. The most complete treatment can be worked out for adsorbing atoms which behave ideally. Ideal behavior will be defined as AG,r = 0 for 0 5 2
zs
yI -
I 1. This casewill be treated
for both dissociative and non-di~o~iative adsorption. An attempt will be made, later, to comment on the perturbing influence that AGir # 0 will have on ideal behavior.
+ kTSN,, In (1 + P/P,)
4% -_=
d 1nP
-kTI’,
(dT = 0)
where yk may be yO, yE or yr (from equation (Z)), and where the corresponding surface excess quantities rk are defined by l?,=N,-NN,rNs
(I3a)
rc = N, - XN,
(13b)
I’r = 2I’, -
I’& = 2N, -
equations
(lo),
N& - XN,
(13~)
(12) and (13a), the
* If the ratio P/P, is replaced by the concentmtion ratio C/C,, equation (10) applies equally well to solute adsorption from the crystal interior.
(14~)
where yO,, ylo and yr,, refer to surface, ledge and ledge in~r&ction energies at P = 0, that is, for a perfectly clean surface. Making use of the fact that the total number of all types of sites N, available for adsorption is a constant given by
the final expression can be arrived at by combining equations (2), (14) and (15) y -
y,,,,= -kTN, + i
In (1 + P/P,) yzo -
kTN,,
+ kT(N,, + IIaving obtained equations, giving the degree of occupancy of each type of site as a function of the variables of the system (P, AGiO, T), it is now necessary to relate the occupation probabi~ty to the change in surfaoe free energy, y. This can be done through the use of the Gibbs-Adsorption Isotherm, which may be stated in the form:
(14b)
yro = --BkTN,, In (1 + P/PJ + kTN, In (1 + P/P,)
AG: = - RT In (f,#JJ
(144
remembering that for ideal behavior P, is not a function of N,/N,,. Similarly the expressions for yI and yr are found to be yt -
Combining
961
(I)
-
fiN,J ln (1 +
f’/PJ
PIPJ]
-YOOa2 2 a2
- SrX -
In (1 +
[
‘yr, + kTNd, ln (1 + PIP,)
2kTN,, In (1 + P/P,)
+ kT[XiWN,,
-NT, .,
-
flJf)I In (1 + P/P,)
1
+ -t?-
-
SMln (I + P/P,)]
(16)
-
N,, + Sfl,(l
XSM2
[ETN,,
Some simplification of equation (16) is possible. The third and fourth order terms are present only when P > 0. They have an appreciable magnitude only when ccis large and when there is preferential adsorption at surface sites, P, < fPd, P&. When ot is large, the model used here should be least applicable, particularly without the inclusion of higher order terms in the y vs. M:relationship for P = 0. Accordingly, they will be ignored in deriving expressions for the slope
ACTA
962
METALLURGICA,
and intercept of the torque term plot from equations (3) and (4). In the equations that follow, N,, = N,,.
(17) 2kTN,, +----S2MY,0 f12MY,
23=1+-
X Pn (1 + P./PA + e
[(SM -
i
00
2 ln (1 + P/P,)]
l)(iVz, - SN,)] lnfl -
VOL.
11,
1963
can be viewed as an interaction energy for ith sit,e.
A comparison of equations (10) and (21), for the case where P, = P,,, it shows that at small degrees of occupation equation (21) gives much higher values of N~lNi~. may be derived following Expressions for ylc - ylEO the integration procedure used previously, but for sake of brevity they will not be given here. Moreover, the analog of equation (16), which is very similar to it in appearance will be omitted. If the final result is simplified in the manner discussed in connection with equation (lci), the parameters A and 3 that characterize the y-plot are given by
PIP,) (18)
Adsorption
with dissociation
Allowing the ith site to be represented by Si, a prototype equilibrium for dissociative adsorption can be written as G, + 2X,*
2(G - SJ
+ rg
so that equation (8) becomes --.aoi
2 =
exp
[(~/GO
+
~~2’
-
‘g2’i2
2,Uu,i”)/RT]
= exp (-_G,,‘/RT)
(20)
AG,,O is the adso~tion site for the dissociative case free energy of the itsh when the species are in their standard states. If use is made of the definitions given in equations (9), the adsorption isotherm for the dissociative case can be expressed in the form where 2,~’
-
2pio -
,ugzo =
ENJOY,, -
2 In [l +
SN,) In [I + (PIJ’sa)1’21 (23)
+ 4kTN,, m
oo
(In [l + (fYPdf21
tPI'P,,P21) -tg
00 x [(EM - l)(N,, + SN,)] In [l + (P/f’,,)l~z]
The free energy change (SC result from the transfer of Sng2 diatomic molecules from the gas phase to an, sites on the surface is the same as given in equation (6), except in this case 6?z,z = 2&b, = 2&Z.,,
2~1,
B,=1$-
(19)
(N,, 00
(24) DISCUSSION
According to the above t~atment, there are three major effects that an adsorbing gas will have on the shape of the y-plot in the vicinity of an energy cusp. Each effect comes into prominence as the ambient pressure P approaches the characteristic pressure of the site. In the analysis that follows it will be necessary to treat limiting cases, i.e., cases where one (or two) of the ~harac~risti~ pressures is much larger than the others. Such limiting cases are not without experimental interest. For example, if the AG,O’s are separated by 10 kcal/mole-a figure that is not unreasonable in view of the fact that heats of adsorption
may be as large as 150 kcal/mole-the
characteristic pressures will differ by approximately two orders of magnitude at 1000°K. Consider first the ease where there is enhanced binding at the ledge
where the characteristic pressure P,, of the ith site for the dissociative case can be written as P
id
= L!@- Pli2 exp (AG.ad“/RT) f$ a
Analogously to the previous case, RT
soi_ lnfTi = -
(221
and double ledge sites. Case I:
P, > (Pd, PI) or P,, > (P,,, P,,) As the ambient pressure P is increased from zero the first major change in the y-plot occurs in its as reflected by changes in B. In equations (18) and (24) the terms depending on P, or P,, will
curvature
AG:”
be neglected,
and for convenience
the following
new
GJOSTEIN:
ADSORPTION
AND
SURFACE
ENERGY
963
(I)
variables are defined. 2.0
x=PIP,
Xd = PIP,, 1.5
R=
Pa/P1
With these definitions it is possible to write equations (18) and (24) in the form
t
1.0
a? 2
0.5
0 al" 0 I 0 " -0.5 -1.0
B, = l3,, + 2I3, In
-1.5
1 + x2’2
I
(25b)
II 3 (fi&Pli
The parameters K and K, are a measure of the relative binding energies of the single and double ledge sites. As (Ii, K,) -+ 0 double ledges tend to have the dominant binding energy, which means that the concentration of adsorbed atoms in single ledge sites can be neglected with respect to that in double ledge sites. Figs. 4 and 5 give plots of (B - B,,) /3,and (Bd - 3,,)/2B,,vs. x and xd, for various values of K and K,. For the non-dissociative case, when 0 < K < 0.5, the curves in Fig. 4 show a maximum, which is located at x, = (1 - 2K)lK. For x > x, = (1 - 2K)jK2, (B - B&B, -=c 0. When K > 0.5, (B - &J/B, is never positive and decreases monotonically with increasing x due to the dominance of the single ledge term. In the dissociative case, a maximum at xdm = (1 - 2K,1'2)2/K, appears in the curves in Fig. 5, when 0 < K, < 0.25, and ( Bd - B,,)/2B, < 0 for all
0
5
IO
I5
20 _ 25
30
35
40
45
50
x,j’YPdd FIG. 5. Effect of reduced pressure (X, = P/P& on yz-plot curvature parameter (Rd) for dissociative adsorption. Kd( =P,/P,,) is a measure of the relative adsorption energies of single and double ledge sites. x& when Kd > 0.25. If a comparison is made between dissociative and non-d~sociativ~ cases, by setting x = x~, it can be seen that for a given value of (x, zd), the increase (or decrease, depending on K) in curvature of the y-plot is somewhat greater for non-dissociative adsorption, and that for a given K = Kd value, the maximum occurs earlier for dissociative adsorption. The second major change in the shape of the y-plot occurs in the slope of the energy cusp at its origin, (A, Ad), as is evident from equations (17) and (23). To describe this effect, use is made of the definitions
so that equations (17) and (23) can be rewritten as A=&,-&ln(l
+Kx)
(264
A, = A,,- 2A, In [ 1 + (X,Z#~]
0
I
I
I
,
I
I
I
,
I
IO
20
30
40
50
60
70
60
90
Xf
IO0
P%
pl-3SSUre {x = P/P,) On yz-plot curvature parameter (B), for non-dissociative adsorption. K(= Pa/P,) is a measure of the relative adsorption energies of single and double ledge sites.
FIG.
4.
Effect
Of
lY?dUC%d
(26b) Fig. 6 gives plots of (A - A,,)/A, and (A, - A,)/2A, vs. it: and xcl for various values of K and K,. From equation (26) it is clear that at some value of (x, x& (A, Ad) will become zero and finally negative. This will occur when the logarithmic functions attain a value of A,,/A, (or A,,/2A,). Generally it can be said that when (K,KJ < 1, the effect of adsorption on curvature of the plot. y-plot occurs at much smaller values of (x, z8) than does the effect on the slope at the origin, but that when K - 0.5(orK, -+ 0.25) both effects tend to be present simultaneously. It was mentioned above that it is possible for (A,Ad) to become negative*. When this is the case has discussed surface energy.
* Herring’5)
negative
the
possibility
of having
a
ACTA
964
‘CA,
~ETALLURG
: B
- 1.0
I I
11,
1963
conditions, ledges can bunch together, resulting in a phenomenon known as thermal faceting. The criterion for this to occur will be discussed in an accompsnying paper.(8) A summary of the adsorption effects for Case I is shown schematically in Fig. 8.
0
t 8
VOL.
----_-_
s
N
-2.0 P”0
b
::
a* I 4
4
P -
-3.0
P-
- 4.0 t
1 0
Pd
P,
I 11
20
1
$1 40
x.Por
pd
I 60
x d
11
I
1 80
IO0
-p-m pdd
?&a.
6. Effeot of reduced pressure (X, X,) an the slope of y2-piot at the vertex of the cusp (A, A?). (K, K,) is a mewmro of the relative adsorption energies of single and double ledge sites. The dashed curves represent dissociative adsorption and the solid curves represent non-dissociative adsorption.
an outward pointing cusp will appear in the y-plot. Fig. 7 schematic&~y shows this tradition. There is no ex~riment&l evidence indicting whether or not this effect actually occurs. At first glance, the concept of a negative ledge energy might be rejected, since apparently the ledges could increase in length spont&neously with it decrease in free energy of the system. The total surface energy is not negative, however, and any change in length of the ledges must be subject to the criterion that total surface energy of system be B minimum. TTndercertain thermodynamic
FIU. 7. y,-plots and corresponding Gibbs-Wulff shapes, depicting the trrsnsition from an inward to an outwardpointing cusp.
0 -Ct-
FIG. 8, Schematic ~umm5ry of major effects of adsorption on shape of y vs. cc plots.
Case II: P, < (Pd, PJ or Ptd < (P,,, P,,) A third major change in the y-plot occurs when the ambient pressure P approaches P, (or Pad). This effect which would not be important in Case I, results when there is stronger bonding of the adsorbed atoms on surface sites relative to the ledge sites. This is a complex effect, involving alterations in B, A snd yO. Examination of equations (18) and (24) shows that the important term controlling the magnitude of the curvature effect is, (XiW - l)(Nls + SN,),, which takes on values of -0.086 N,, or -0.17 N,, for the (100) and (111) planes, respectively. Thus, the curvature (B, B,) should decrease with increasing (x, Q) in a manner shown by the curves for (K, KS) = 0 in Figs. 4 and 5, when they are reflected about the zero baseline. Equations (17) and (23) show that the effect on (A, Ad) depends on the term, N,, + SN,, which takes on values of 1.71 N,, and 2.41 N,, for the (100) and (111) plane, respectively. Thus, in contrast to Case I, (A, A,) should increase with increasing (X, X,) for both the (100) and (111) cusp. The form the (A, Ad) vs. (X, X,) relationship can be seen by reflecting the curves in Fig. 6 about the zero baseline. These effects are essentially the reverse of Case I. They can be illustrated by viewing the sequence of curves, from bottom to top, in Fig. 8, as representing increasing pressure.
ADSORPTION
GJOSTEIN:
AND
SURFACE
ENERGY
TABLE 1. Adsorption of oxygen on copper at 1000°C
Heat of adsorption
- 110,000 ealfmole 0,
965
(I)
__I__ Dissociated coverage -_I
Dew point
PO,
Pebar&eristie
Undissociated coverage
- 36°C
3.5 x IO-“2
5 x 10-13
7 X IO-10
2.7 x 10-z
+23*c
3.5 x lo-‘8
5 x 10-13
7 x LO-6
2.7 x 10-a
The effect on y0 can be determined from equation (16). As is evident, y0 will decrease with increasing pressure by an amount kTN, In (1 + P/P,)* for non-dissociative adsorption or by an amount kTN, In [l + (P/PS)1i2] for dissociative adsorption. For the (160) plane in copper ~TN~ = 270 ergs/cm2 at lOOO”C, which means that by, = (0.1, 0.2)~~ when (P/P,, P/P,,) = 1. Since y0 decreases with increasing P, all the vicinal planes will be reduced by the same amount, and thus each point of the y-plot will shrink towards the origin.
The y-plot has been explored for two metals, copper’3) and nickel,(1t2) under entirely different ambient atmospheres. The nickel specimens were annealed at 1000°C in a vacuum of about low5 mm, while the copper specimens were annealed in a hydrogen atmosphere with a dew point of either -36°C or +23”C at temperatures in range 950-1020°C. The torque terms repotid by Robertson and Shewmon(3) for copper annealed under hydrogen are somewhat smaller than those for nickel. This might be viewed as an impurity effect, but as will be shown, there is a strong possibility that this is not the case, and that probably they are intrinsically lower. Dell et aZ.(lQ found the heat of adsorption of oxygen on copper at low coverages to be about Oxygen appears to be the - 110 kcaljmole 0,. most likely adsorbent on copper, since hydrogen has a much smaller, although uncertain, heat of adsorption (it is reported in the range -9.0 to -35.0 kcal./mole H,.) (17~18)Assuming a reasonable value for the entropy of adsorption of hydrogen, it should have a very small free energy of adsorption at elevated ~m~eratu~s (l~O°C~, and thus should have a negligible coverage, unless there is a change in bonding mechanism at these temperatures, which is not reflected in the heat measured at lower temperatures. Table 1 gives the degree of occupancy to be expected for a heat of adsorption of - 110 kcal/mole 0, and * These values c&r&ted by assuming that every lattice
site on the substrate can serve as sn adsorption site. Ledges were assumed to lie along close-packed directions in an f.c.c. lattice. Lscmann et aZ.(‘) were concerned soley with this term; they assumed P and N, to vary in a specific manner with the crystallographic plane under consideration.
entropy of adsorption of -30 e.u.7, under various conditions. It can be seen that the worst possible case is for a dissociated: molecule when the dew point is +23”C, where the degree of occupancy would be 2.7 x 10m3. Since there is no information available which would allow the charac~ristic pressure given in Table 1 to be associated with any particular site, the most unfavorable assumption will be made. This means the characteristic pressure should be equated to Pa,,. If this is true, the change in Bd will be, Bd - B,, = 5.4 x 1O-3 fz,, due to the terms containing P,,. For the (100) cusp in copper, B, is found to be at lOOO*C B, -
2kT Nzs _ 2kT 4 y0 PA!l y0 uf g(1.38 x lo-16)(1.273 x 103) = o 6.
=
(1.7 x 103)(3.61 x 10-8)2
’
For the (ill) cusp of copper, B, is a factor 0.43 smaller than this value. Thus it appears that Robertson and Shewmon essentially de~rmined B,,,. It could be argued, however, that I(AH,,O, AH,,O)I > j AH,“j, and that this larger heat was not detected calorimetrically. If this were the case there should be appreciable contribution to B, from terms containing (P,,, P,,). However, at the lower dew point this contribution should decrease considerably. This means they should have found B to vary with the partial pressure of oxygen whereas no appreciable effect on B was observed when the partial pressure of oxygen was decreased by four orders of magnitude. Similar reasoning can be applied to A,, and again no effect of adsorption seems likely. It must be concluded, then, that adsorbed oxygen did not influence the results of Robertson and Shewmon, and that it seems doubtful that adsorbed hydrogen exerted any influence either, although this latter point must be left open until further information is availab1e.g t The entropy of adsorption was assumed to be the same as entropy of formation Cu,O, namely -30 Cal/mole 0, - “K, since an atom loses essentially the same number of degrees of freedom in both the oxidation and adsorption processes. f It has not been established conclusively whether or not oxygen exists as dissociated atoms on metal surfaces. 3 If it is assumed that B,, W&Rmeasured, it is interesting to note that B, < 1, and therefore it follows that yro, i.e. there is a rep&non between single ledges. The significance of this f&zt is not understood as yet.
ACTA
966
Mykura
found
larger
the
parameters
A and B for nickel than those reported
for copper;
TABLE 2. Comparison
values
METALLURGICA,
of
A and B for nickel and
of parameters copper
11, 1963
VOL.
Deviations from ideality In any real system such as those discussed deviations
from ideal behavior
is independent
of degree of occupancy
and therefore,
of the pressure of the adsorbing
Very little is known theoretically
Nickel
copper
* Actually
so
that it is no longer possible to assume that Pi (or Pid)
A
(111) (100)
above,
can be expected,
gas.
about the deviations
0.17 0.30
0.24t 1.13
that might
not be treated within the scope of this paper.
Experi-
0.11 0.067
0.22 0.14
mentally,
heat
the slope of the experiment
be expected,
of given site,
it is known
adsorption
torque vs. GLplot is
which,
decreases
increasing
coverage
at any rate, could
generally (becomes
that
the
less negative)
of the surface.
The manner
which it decreases is not always simple.
rather than These two quantities are not much and nickel values, where yO/v is nearly unity and A is small. B’ is not constant for the (111) cusp in nickel. This value is for a = 0; B’ increases to 0.8 at cc = 0.3.
viewed
Pi increase
as having
occupancy
of the ifh site.
with
would
retain
the
degree
In this case,
essentially
the
in
This can be
changes in shape, which were described case,
of
with
of
the basic
for the ideal
same
character,
but will occur at higher pressures
(or lower temper-
that the values of A for copper are for a clean surface,
atures).
Given enough information
about the changes
it appears that either Case II-type
on
y-plot
a comparison
place
on nickel,
higher yet,
is given in Table
single
ledge
or that nickel
ledge
energies
principles
energy,
shape of small nickel
by Mykura.
This indicates
have
influenced
been
pressures Without
surface
pressure
it is not
cleanliness
is about
this
of the Gibbs-
by Sundquistc4)
adsorption. systems
The
possible to
the partial pressure of oxygen fraction
of
electron
diffraction
total
pressure
and
from ideal behavior
The phenomenological in the y-plot, has been
developed,
structure
theory
of shape alterations
due to adsorption
based
utilizing
from the gas phase, a model
on three types
that the slope of the y-plot
pressure in a degree
(15).
bulk In
of the adsorbing
decrease
there is enhanced
ledge sites.
binding
pressure
temperature)
of the adsorbing
when
atom at
ledge sites relative to the surface sites (Case I).
The
at
slope of the cusp origin can decrease to a point where
order
to
it becomes negative,
even
the surface
causing a cusp inversion.
sites have
the stronger
of nickel,
(Case II),
the slope of the y-plot
must be a very small
increasing
pressure of the adsorbing
of
sites: Theory
in the vicinity
with increasing
gas (decreasing
surface
NiO
(or probably oxygen?)
The
of a cusp should
of
of adsorption
predicts
1O-5 mm.
it
for a given system.
used by Mykura
set of data.
pressure
temperature,
should be possible to determine the extent of deviation
single ledge and double
to form
with oxidation
with
surface,
to assign
either
required
the
total
the same,
with chemisorbed
the
first
that the results for nickel
8 x 1O-s mm
problems
more critically
from
to decide
check on the partial
of oxygen
eliminate
As
terms than those found
were essentially
a further
system,
1000°C
crystals
by
in the vacuum
and Sundquist
copper.
determinations
much larger torque
takes
has an intrinsically
be calculated
Wulff
of
adsorption
ylO, than
cannot
Some preliminary
each
Since it appears
and thus it is not possible
issue.
indicate
2.
10e5 mm.
The
binding
will increase gas.
forces with
For Case I
the curvature
increases
worthog) show that carbon is also a possible adsorbent
goes through The maximum
a maximum and finally decreases. tends to vanish as the difference in
on nickel, in which case it is to be expected
binding
energy
becomes
large.
adsorption
studies
of
Schlier
and
Farnsthat CO
may be another problem.
There are no other data on the shape of the y-plot for metals,
except
for information
that can be ob-
tained from thermal faceting. This subject cussed in detail in an accompanying paper.(s)
is dis-
t This parenthetical comment was added because normally it would be expected that chemisorbed oxygen would be bonded more strongly than bulk oxide. However, Paravano et aZ.‘l*’ reported a heat of adsorption of - 111 kcal/mole, which is very close to the heat of formation of bulk oxide, - 116 kcal/mole.
with pressure
When
between When
single
critical value, the curvature with increasing curvature involved.
with
and
this difference
pressure. pressure
at first, then
double
ledge
drops below
a
decreases monotonically
Case II gives a decreasing increase;
no minimum
is
References 1. H. MYKURA, Acta Met. 9, 570 (1961). 2. J. M. BLARELY and H. MYKURA, Acta Met. 9. 595 (1961). 3. W. M. ROBERTSON and P. G. SHEWMCIN, Trans. Amer. Inst. Min. (M&U.) Engrs. 224, 804 (1962).
GJOSTEIN:
ADSORPTION
AND
4. B. E. SUNDQUIST. Private communication. 5. C. HERRING, Structure and Properties of Solid Surfaces. (Edited by R. GOMER and C. S. SMITH), p. 5. University of Chicago Press (1953). 6. C. HERRING, Phys. Rev. 82, 87 (1951). 7. R. LACMANN and I. N. STRANSKI, Growth and Perfection of Crystals. (Edited by R. H. DOREMUS, B. W. ROBERTS and D. TURNBULL), p. 427. Wiley, New York (1958). 8. N. A. GJOSTEIN. Acta Met., this issue p. 969. 9. W. K. BURTON, N. CABRERA and F. C. FRANK, Phil. Trans. Roy. Sot. Lond. 243A, 299 (1951). 10. W. W. MULLINS, Acta Met. 7, 746 (1959). 11. T. W. HICKMOTT and G. EHRLICR, J. Phys. Chem. Solids 5, 47 (1958). 12. E. BAUER, Phys. Rev. 123, 1206 (1961). 13. N. A. GJOSTEIN, Acta Met. 7, 812 (1959).
SURFACE
ENERGY
967
(I)
14. 0. D. GONZALEZ and G. PARRAVANO, J. Amer. Chem. Sot. 78, 4533 (1956). 15. F. D. RICHARDSON and J. H. E. JEFFES, J. Iron St. Inst. 160, 261 (1948). 16. R. M. DELL, F. S. STONE and P. F. TILEY, Trans. Fu”aTaday Sot. 49, 201 (1953). 17. T. TAKEUCHI and
M.
SAKAGUCHI,
Japan 39, pp. 177-182 (1957). 18. T. KWAN, Advances in Catalysis 6, 67.
Bull.
Chem. Sot..
Academic
Press,
New York (1954). 19. R. E. SCHILIER and H.
E. FARNSWORTH, Advances in Catalysis 9, 434. Academic Press, New York (1957). 20. The author is indebted to Professor G. M. POUND for discussions on this problem, and to Drs. H. MYKURA and J. HIRTR for communications, which help crystallize some of these ideas.