- Email: [email protected]
Potentials of mean force for adatoms on surfaces
Surface Science 119 (1982) 71-78 North-Holland Publishing Company
POTENTIALS
71
OF MEAN FORCE FOR ADATOMS
ON SURFACES
Steven M. VALONE, Jimmie D. DOLL and H. Keith MCDOWELL* Los Alamos Scientific Laboratory, University of California, L.os Alamos, New Mexico 87545, Received
USA
1 February 1982
Differences between the adatom-adatom potential of mean force and the pair potential in the limit of low adatom density are illustrated through the Rirkwood equation for a two-component solid. The major difference which remains at all adatom densities arises from the adatom-substrate contribution to the potential of mean force. A method designed to extract an estimate of the adatom-adatom pair potential from field ion microscopy data is suggested.
1. Introduction The thrust of this article is to highlight the more decisive differences between the Kirkwood equations for single component fluids and for multicomponent solids [l]. An importance for these differences is assumed in the case of pairs of adatoms adsorbed on a solid surface. The two-particle (or pair) distribution function for the adatoms may be observed experimentally for certain types of system through field ion microscopy (FIM) [2]. The information about the system which may be derived from the pair distribution is the subject of concern. It had been thought by some researchers that a pair potential, a direct pair interaction between adatoms independent of the substrate, could be derived from the experimental pair distributions [3]. In other words, it was assumed that the potential of mean force (PMF), the difference between the interaction energy of the adatoms and the substrate and the adatom-substrate interaction with the adatoms infinitely far apart but still adsorbed on the surface with the substrate coordinates averaged out, approximates the pair potential, since the density of the adatoms is very low for these experiments. However, this conclusion, derived through a Kirkwood equation, holds only in the case of single component fluids. The systems under investigation in the FIM studies are neither single component nor fluid. Consequently, the relation between the potential of mean force and the pair potential is much more complicated. * Permanent address: Department South Carolina 2963 1, USA.
of Chemistry
and Geology,
Clemson
0039-6028/82/0000-ClOOO/$O2.75 0 1982 North-Holland
University,
Clemson,
12
S.M. Valone et al. / Potentials of mean force for adatoms on surfaces
The relationship between the PMF and the adatom-adatom pair potential is specified in the next section (eqs. (6), (7) and (23)), by means of a Kirkwood equation. A method for deriving the pair potential from FIM data is suggested in the third section (eqs. (25) and (26)).
2. A Kirkwood equation for a two-component
solid
As suggested by the title to this section, the actual derivation will be for only a two-component solid. Our intention however is that the initial framework for the derivation be sufficiently general so as to make the derivation of more general Kirkwood equations obvious. In deriving Kirkwood equations for multi-component systems, the only change from the derivation for a single component systems is a notational one. An appropriate notation is already established in section 40 of I [l]. We adopt this system with a few modifications. A classical canonical ensemble of N-particle systems of volume 5l and temperature T will form the backdrop for the discussion. The distribution functions for the system will be averages over the Boltzmann factor of the nuclear potential energy in the nuclear adiabatic approximation. The potential energy will be approximated by a sum of pair interactions. Such an approximation is germane to the concept of an adatom-adatom pair potential. After establishing some notation, the definitions of the multi-component distribution functions are given. The Kirkwood equation of interest is derived directly from the definition of the two-particle distribution. The low adatom density limit of this equation demonstrates the desired PMF pair potential relation. 2. I. Definitions
and notation
Notation is first established for a system of C components with a total of N particles with N, particles in each component, s = 1,. . ., C. The bold-faced letters N and n represent C-dimensional vectors: N = (N,, . . . , Nc) and n = (n , , . . . , n c). The position vectors of the system are grouped similarly. Vectors 1, vectors rN, + , , . . . , r,, + N reprerl...., r,,,, represent the particles of component sent the particles of component 2, . . . , and vectors rN_-Nc+, , . . . , r,..,repreient the particles of component C. For shorthand, {rJ,=
6
{G+lY.‘Jo+n,}>
(1)
s=l
where (I = ES,;‘, Nk and where it is understood that if n, = 0, then there are no elements of component s contributing. Similarly, dN=
fi dN,, s=l
(2)
S. M. Valone et al. / Potentials of mean force for adatoms on surfaces
73
where dN,=
$
dr,,,,
(3)
i=l
and CJis as in eq. (1). Further,
(4
=,p, d(N,-n,),
W-n) where d(N,--n,)=
;
drO+i.
i:=n,+
(5)
I
This notation will greatly simplify the expressions required to define the statistical quantities of interest. The form of the potential energy U in the pair additivity approximation is WM
= i i v,,({‘L s=l ,=s
WA
(6)
(1 - (1 -~)s,,s,i)Vsr(r,+;,,+i)t
(7)
where W’L
W,)
;
= ; i=i
j=l+iS,,
is as in eq. (l), r = ZiL’, Nk and rp4 = 1rp - rq I. The parameter 5 represents the strength of interaction of any one of the N, component 1 particles which happens to be at r, with all the other particles of the system. Eq. (7) indicates that we are making the further approximation that the interactions between any pair of particles for any two specified components is the same function. The derivative of U with respect to the coupling parameter is also required:
u
g
= f: t--l
._$ hh+,)*
(8)
J--1+6,,
This mathematical quantity is required in the derivation of the Kirkwood equation. Using the expression for U given in eq. (6), multi-component distribution functions may be defined as follows. The probability P(“)<(r),,) of finding particles n at {r},, averaged over the positions of the remaining N - n particles is P’“‘({r},)
J d(N--n)
exp(-W({rI,))
Ln
ZhJ
7
(9)
where Z,=
Ju
dNexp(-/?U)
(10)
14
SM.
Valone et al. / Potentials of mean force for adatoms on surfaces
is the classical configuration integral, p = l/k,T with k, as the Boltzmann constant and T is the constant temperature of the ensemble. The probability p(“)({r},) of finding any of the N, particles of component 1 at r,, any of the remaining N, - 1 particles at r,, . . . . any of the remaining N, - PZ,particles at r etc., averaged over all configurations of the remaining N - II particles, just n%tiplies P(“) by a constant accounting for the ways of choosing n, particles from N,, etc.:
(11) When necessary for clarity, the dependences on N or $ are expressed as PP({r),;5)9
P%)({r),A)>
G({rLv;S).
One also needs to define conditional probabilities. If n = (n,, . . . , nc) particles are fixed at {r},,, the probability Pt”*“]({r},,;[r],,,) of finding m= Cm ,,..., m,) particles at [r], = {r},+, \ {r }” averaged over all configurations of the remaining N - n - m particles is
P[“~ml({r},;
[r]J
-
/ d(N--n-m)exp(--PU({r},)) n /51d(N-4
exp(-Bu({rM)
P(“+mY{4”+m) =
02)
P’“‘({r},)
.
(13)
The backslash notation “1” in the definition of [t-l,,, represents set substraction. By analogy one may define
(14) Correlation functions are then defined by P’“‘({&J ~t~+l((r)~;
A,
G, PYY7+,)
P’({4”]~
IrIm) =,@, j?, P(r,+,,+,) Pml({rI,; IrIm).
(15)
(16)
We are now in a position to derive the Kirkwood equations for any of the distribution or correlation functions in the pair additivity approximation. The procedure is the same in each case. The technique used below is from section 32 of I.
S.M. Valone et al. / Potentials of mean force for adatoms on surfaces
2.2. The Kirkwood
75
equation
At this point it is appropriate to specialize to the case of interest. Namely the essential constants have the values: C = 2, N= (N,, N,) where Ni and N, are arbitrary, N = N, + N2 and n = (2,O). Ni will represent the adatoms and N2 the substrate. The analogous macroscopic densities will be denoted as p1 = N,/V and pz = N,/V, respectively. N, n and C will have these special values unless stated otherwise. From eqs. (9)-(1 l), the two-particle distribution is given by J dW- n> exp(-W(E)) - 1) o / dN exp( -W(l))
p(“)(r,,r2;5)=N,(N,
(17) *
n
Differentiating In p(“’ with respect to the coupling parameter 6 and scaling the result by - l/p produces
-$
$
lnp(“)(<)=
J’ d(N--n)g d(N-n)
Jh2
exp(-/W)
1 dN5 -
exp(-j3U)
n
exp(-/W) ZN
(18) Substituting for Xl/i35 from eq. (8), using the identity of the interactions between components and using the scaling relations between the probability distributions and the density distributions (eqs. (11) and (14)), one obtaines -i
T$ln p(“)(t) = vl,(r12)+ i r=1
- / dr, o
/ dr,
n
(V,,(r,,) p[“*“‘](r,,r2;r,;t)
p((‘*“)+m)( i, , r,; 0 N,
(19)
where 7 = 3 + (N, - 2)6,, and m = (a,,, a,,). Integrating with respect to E from 0 to 1 and changing to correlation functions yields -- 1 ln p$(r,) P
_
Jn
p%Vrz) g$Yr,,rz;l) &G-M
+‘Yr,;t)
dr
1
4
g((‘~“)+m)(r,,rT;t) v,,(r,,)
The quantity p,p(&)_l(rz) = pI;))(r,, r,;O) as may be verified from the definition
16
SM.
I/alone et al. / Potentials of mean force for adatoms on surfaces
of p(Nn,and the pair additivity approximation expressed to emphasize that at [ = 0 the system function V,,(r,,) is the pair potential of interest. At this time it should be noted that slightly required if either there are two types of adatom on the surface [3]. 2.3. Potentials
different pair distributions are or one of the adatoms is fixed
of mean force and the low adatom density limit
As described in section associated with the particles exp(-/?w’“‘({r},) where PMF PMF’s of the
to U. The N - 1 and N are contains one less particle. The
31 of I, a standard choice for the PMF represented by n is defined by
=WP’“‘({r},),
w(“,
(21)
n = 2:= ,n,. The quantity which earlier work sought to measure is the associated with particles n relative to the sum of the single particle of the particles of n. This relative PMF IV(“) may be defined in terms particle distribution functions as
exp( -j?W’“‘(
{r},))
E
pC:,)(‘r’n’ II, ;n, wr~+i) c
As may be seen from the above correlation functions (eq. (16)). In (22) is also g$‘, (r,, r2; 1) in eq. (20) Expressing the left hand side of W’“‘(r,,r,)
= V,,(r,z)
r=l
J
- W#,(T,) - w$(r*)
+ w#L,(r*)
j’dEJ,d 0
+ i
-
definition, IV(“) is closely related to the the present case, the right hand side of eq. times the normalization factor N,/( N, - 1). eq. (20) in terms of PMF’s yields
d,.
cl
I
Pha N,
I/
(r,
It7
) g((l.O)+m)
(rlrr,;5) .
(23)
1
By analogy with the case of fluids [ 11, it may be argued that, in the low p, (adatom) limit, the t = 1 term of the sum is negligible, since p(,,(r,) is small on the average while the factor in the square brackets is relatively independent of the adatom density for sufficiently low p,. One may also argue that
w$‘,(r*) - w$)(r2) is negligible in the face of strong adatom-substrate interactions and at low p,. Strictly speaking, it is probably necessary to decide on the aptness of these
S. M. Vdlone et al. / Potentials of mean force for adatoms on surfaces
77
arguments on a case-by-case basis. Regardless of the above, the w(‘) (r,) PMF cannot be neglected for solids. More importantly, it is incorrect, even for fluids, to neglect the t = 2 (adatom-substrate) term, since neither the substrate density pz nor the adatom-substrate interactions Vi, are necessarily small. The conclusion to be drawn is that in the limit of low adatom density, the PMF for the adatoms is not identical to the pair potential. Differences arise from the contribution of the substrate to the PMF of the adatoms. The magnitude of these differences depends on the strength of the adsorbate-substrate interaction and the substrate density. In the next section, a suggestion is tendered for obtaining an estimate of the pair potential V, ,(r,*) from a combination of FIM data and a theoretically derived quantity: the pair distribution with the adatom-adatom interaction switched off.
3. A possible method for deriving the pair potential In spite of the negative result of section 2, it is still of interest to try to recover the adatom-adatom interaction from FIM data. Benignly, a slightly different manipulation of the data does yield the desired quantity. Instead of calculating the pair correlation function from the FIM-derived pair distribution [3], the pair distribution should be divided by the pair distribution for the system with the adatom-adatom interaction switched off. For this result one takes C = 2, n = (2,O) and N, = 2. The potential energy is written so that the coupling parameter .$is only between the adatom-adatom interaction, U,, = V, , : w
=Ev,,h)
+ uash
Ws)
+ v,h
{&)
+ U,SWS)~
(24)
where {r}, represents the set of substrate position vectors, U, is the adatomsubstrate interaction and US,is the substrate-substrate interaction. Note that some form of additivity approximation is still necessary. With p(“)( r,, rz; 5) defined as (eq.( 17))
p(yr,,r2;[)
=2
J d& exp(-Bu(t)) *
-%(I)
(25)
’
the logarithm of the ratio of p(“) at 6 = 1 and at I = 0 gives U,, up to a constant defining the zero of energy: -$
In
PYl=
p’“‘(
1) =
5 = 0)
V,,(k)
-L
P
In
z&=
z,(t=1)’
0)
(26)
The ratio of configuration integrals represents a free energy difference between the two systems [1,4]. A Kirkwood-like derivation of eq. (26) yields the
78
SM.
Valone et al. / Potentials of mean force for adatoms on surfaces
identification (27) Thus if derived function derived adatoms
one computes p(“)(t = 0) by some theoretical technique, the FIM pair distribution in conjunction with the calculated quantity yields a which differs from the pair potential by a constant. Pair potentials through eq. (26) could help clarify the roles of the substrate and the in the formation of clusters and thin film growth.
4. Conclusions We have shown (section 2) that the potential of mean force for adatoms in the limit of low adatom density does not equal the adatom-adatom pair potential (eq. (23)), as defined in the pair additivity approximation for the potential (eq. (6)). The adatom-adatom pair potential may be estimated by combining the experimental pair distribution with a theoretical pair distribution generated from a system in which the adatom-adatom pair potential has been switched off. Using FIM data in this way could lead to a better understanding of the role of the adatom-adatom interaction in the formation of clusters on solid surfaces and in thin film growth.
[ 1) T.L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956). especially sections 29, 31, 32 and 40. Hereafter referred to as I. For a lucid discussion of the Kirkwood equations for multi-component fluids, see: T.L. Hill, An Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, MA, 1960) pp. 331-334, section 18-2. [2] E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Principles and Applications (Elsevier, Amsterdam, 1969); E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Field Ionization, and Field Evaporation, in: Progress in Surface Science, Vol. 4, Part 1, Ed. S.G. Davidson (Pergamon, Oxford, 1973); G. Ehrlich, Surface Sci. 63 (1977) 422; G.L. Kellogg, T.T. Tsong and P. Cowan, Surface Sci. 70 (1978) 485. (31 G. Ehrlich, Phys. Today (June 1981) 51; H.-W. Fink, K. Faulian and E. Bauer, Phys. Rev. Letters 44 (1980) 1008; R. Casanova and T.T. Tsong, Phys. Rev. B22 (1980) 5590. [4] D.R. Squire and W.G. Hoover, J. Chem. Phys. 50 (1969) 701; C.H. Bennett and B.J. Alder, J. Chem. Phys. 54 (1971) 4796; C.H. Bennett, Exact Defect Calculations in Model Substances, in: Diffusion in Solids, Eds. A.S. Nowick and J.J. Burton (Academic Press, New York, 1975).
POTENTIALS
71
OF MEAN FORCE FOR ADATOMS
ON SURFACES
Steven M. VALONE, Jimmie D. DOLL and H. Keith MCDOWELL* Los Alamos Scientific Laboratory, University of California, L.os Alamos, New Mexico 87545, Received
USA
1 February 1982
Differences between the adatom-adatom potential of mean force and the pair potential in the limit of low adatom density are illustrated through the Rirkwood equation for a two-component solid. The major difference which remains at all adatom densities arises from the adatom-substrate contribution to the potential of mean force. A method designed to extract an estimate of the adatom-adatom pair potential from field ion microscopy data is suggested.
1. Introduction The thrust of this article is to highlight the more decisive differences between the Kirkwood equations for single component fluids and for multicomponent solids [l]. An importance for these differences is assumed in the case of pairs of adatoms adsorbed on a solid surface. The two-particle (or pair) distribution function for the adatoms may be observed experimentally for certain types of system through field ion microscopy (FIM) [2]. The information about the system which may be derived from the pair distribution is the subject of concern. It had been thought by some researchers that a pair potential, a direct pair interaction between adatoms independent of the substrate, could be derived from the experimental pair distributions [3]. In other words, it was assumed that the potential of mean force (PMF), the difference between the interaction energy of the adatoms and the substrate and the adatom-substrate interaction with the adatoms infinitely far apart but still adsorbed on the surface with the substrate coordinates averaged out, approximates the pair potential, since the density of the adatoms is very low for these experiments. However, this conclusion, derived through a Kirkwood equation, holds only in the case of single component fluids. The systems under investigation in the FIM studies are neither single component nor fluid. Consequently, the relation between the potential of mean force and the pair potential is much more complicated. * Permanent address: Department South Carolina 2963 1, USA.
of Chemistry
and Geology,
Clemson
0039-6028/82/0000-ClOOO/$O2.75 0 1982 North-Holland
University,
Clemson,
12
S.M. Valone et al. / Potentials of mean force for adatoms on surfaces
The relationship between the PMF and the adatom-adatom pair potential is specified in the next section (eqs. (6), (7) and (23)), by means of a Kirkwood equation. A method for deriving the pair potential from FIM data is suggested in the third section (eqs. (25) and (26)).
2. A Kirkwood equation for a two-component
solid
As suggested by the title to this section, the actual derivation will be for only a two-component solid. Our intention however is that the initial framework for the derivation be sufficiently general so as to make the derivation of more general Kirkwood equations obvious. In deriving Kirkwood equations for multi-component systems, the only change from the derivation for a single component systems is a notational one. An appropriate notation is already established in section 40 of I [l]. We adopt this system with a few modifications. A classical canonical ensemble of N-particle systems of volume 5l and temperature T will form the backdrop for the discussion. The distribution functions for the system will be averages over the Boltzmann factor of the nuclear potential energy in the nuclear adiabatic approximation. The potential energy will be approximated by a sum of pair interactions. Such an approximation is germane to the concept of an adatom-adatom pair potential. After establishing some notation, the definitions of the multi-component distribution functions are given. The Kirkwood equation of interest is derived directly from the definition of the two-particle distribution. The low adatom density limit of this equation demonstrates the desired PMF pair potential relation. 2. I. Definitions
and notation
Notation is first established for a system of C components with a total of N particles with N, particles in each component, s = 1,. . ., C. The bold-faced letters N and n represent C-dimensional vectors: N = (N,, . . . , Nc) and n = (n , , . . . , n c). The position vectors of the system are grouped similarly. Vectors 1, vectors rN, + , , . . . , r,, + N reprerl...., r,,,, represent the particles of component sent the particles of component 2, . . . , and vectors rN_-Nc+, , . . . , r,..,repreient the particles of component C. For shorthand, {rJ,=
6
{G+lY.‘Jo+n,}>
(1)
s=l
where (I = ES,;‘, Nk and where it is understood that if n, = 0, then there are no elements of component s contributing. Similarly, dN=
fi dN,, s=l
(2)
S. M. Valone et al. / Potentials of mean force for adatoms on surfaces
73
where dN,=
$
dr,,,,
(3)
i=l
and CJis as in eq. (1). Further,
(4
=,p, d(N,-n,),
W-n) where d(N,--n,)=
;
drO+i.
i:=n,+
(5)
I
This notation will greatly simplify the expressions required to define the statistical quantities of interest. The form of the potential energy U in the pair additivity approximation is WM
= i i v,,({‘L s=l ,=s
WA
(6)
(1 - (1 -~)s,,s,i)Vsr(r,+;,,+i)t
(7)
where W’L
W,)
;
= ; i=i
j=l+iS,,
is as in eq. (l), r = ZiL’, Nk and rp4 = 1rp - rq I. The parameter 5 represents the strength of interaction of any one of the N, component 1 particles which happens to be at r, with all the other particles of the system. Eq. (7) indicates that we are making the further approximation that the interactions between any pair of particles for any two specified components is the same function. The derivative of U with respect to the coupling parameter is also required:
u
g
= f: t--l
._$ hh+,)*
(8)
J--1+6,,
This mathematical quantity is required in the derivation of the Kirkwood equation. Using the expression for U given in eq. (6), multi-component distribution functions may be defined as follows. The probability P(“)<(r),,) of finding particles n at {r},, averaged over the positions of the remaining N - n particles is P’“‘({r},)
J d(N--n)
exp(-W({rI,))
Ln
ZhJ
7
(9)
where Z,=
Ju
dNexp(-/?U)
(10)
14
SM.
Valone et al. / Potentials of mean force for adatoms on surfaces
is the classical configuration integral, p = l/k,T with k, as the Boltzmann constant and T is the constant temperature of the ensemble. The probability p(“)({r},) of finding any of the N, particles of component 1 at r,, any of the remaining N, - 1 particles at r,, . . . . any of the remaining N, - PZ,particles at r etc., averaged over all configurations of the remaining N - II particles, just n%tiplies P(“) by a constant accounting for the ways of choosing n, particles from N,, etc.:
(11) When necessary for clarity, the dependences on N or $ are expressed as PP({r),;5)9
P%)({r),A)>
G({rLv;S).
One also needs to define conditional probabilities. If n = (n,, . . . , nc) particles are fixed at {r},,, the probability Pt”*“]({r},,;[r],,,) of finding m= Cm ,,..., m,) particles at [r], = {r},+, \ {r }” averaged over all configurations of the remaining N - n - m particles is
P[“~ml({r},;
[r]J
-
/ d(N--n-m)exp(--PU({r},)) n /51d(N-4
exp(-Bu({rM)
P(“+mY{4”+m) =
02)
P’“‘({r},)
.
(13)
The backslash notation “1” in the definition of [t-l,,, represents set substraction. By analogy one may define
(14) Correlation functions are then defined by P’“‘({&J ~t~+l((r)~;
A,
G, PYY7+,)
P’({4”]~
IrIm) =,@, j?, P(r,+,,+,) Pml({rI,; IrIm).
(15)
(16)
We are now in a position to derive the Kirkwood equations for any of the distribution or correlation functions in the pair additivity approximation. The procedure is the same in each case. The technique used below is from section 32 of I.
S.M. Valone et al. / Potentials of mean force for adatoms on surfaces
2.2. The Kirkwood
75
equation
At this point it is appropriate to specialize to the case of interest. Namely the essential constants have the values: C = 2, N= (N,, N,) where Ni and N, are arbitrary, N = N, + N2 and n = (2,O). Ni will represent the adatoms and N2 the substrate. The analogous macroscopic densities will be denoted as p1 = N,/V and pz = N,/V, respectively. N, n and C will have these special values unless stated otherwise. From eqs. (9)-(1 l), the two-particle distribution is given by J dW- n> exp(-W(E)) - 1) o / dN exp( -W(l))
p(“)(r,,r2;5)=N,(N,
(17) *
n
Differentiating In p(“’ with respect to the coupling parameter 6 and scaling the result by - l/p produces
-$
$
lnp(“)(<)=
J’ d(N--n)g d(N-n)
Jh2
exp(-/W)
1 dN5 -
exp(-j3U)
n
exp(-/W) ZN
(18) Substituting for Xl/i35 from eq. (8), using the identity of the interactions between components and using the scaling relations between the probability distributions and the density distributions (eqs. (11) and (14)), one obtaines -i
T$ln p(“)(t) = vl,(r12)+ i r=1
- / dr, o
/ dr,
n
(V,,(r,,) p[“*“‘](r,,r2;r,;t)
p((‘*“)+m)( i, , r,; 0 N,
(19)
where 7 = 3 + (N, - 2)6,, and m = (a,,, a,,). Integrating with respect to E from 0 to 1 and changing to correlation functions yields -- 1 ln p$(r,) P
_
Jn
p%Vrz) g$Yr,,rz;l) &G-M
+‘Yr,;t)
dr
1
4
g((‘~“)+m)(r,,rT;t) v,,(r,,)
The quantity p,p(&)_l(rz) = pI;))(r,, r,;O) as may be verified from the definition
16
SM.
I/alone et al. / Potentials of mean force for adatoms on surfaces
of p(Nn,and the pair additivity approximation expressed to emphasize that at [ = 0 the system function V,,(r,,) is the pair potential of interest. At this time it should be noted that slightly required if either there are two types of adatom on the surface [3]. 2.3. Potentials
different pair distributions are or one of the adatoms is fixed
of mean force and the low adatom density limit
As described in section associated with the particles exp(-/?w’“‘({r},) where PMF PMF’s of the
to U. The N - 1 and N are contains one less particle. The
31 of I, a standard choice for the PMF represented by n is defined by
=WP’“‘({r},),
w(“,
(21)
n = 2:= ,n,. The quantity which earlier work sought to measure is the associated with particles n relative to the sum of the single particle of the particles of n. This relative PMF IV(“) may be defined in terms particle distribution functions as
exp( -j?W’“‘(
{r},))
E
pC:,)(‘r’n’ II, ;n, wr~+i) c
As may be seen from the above correlation functions (eq. (16)). In (22) is also g$‘, (r,, r2; 1) in eq. (20) Expressing the left hand side of W’“‘(r,,r,)
= V,,(r,z)
r=l
J
- W#,(T,) - w$(r*)
+ w#L,(r*)
j’dEJ,d 0
+ i
-
definition, IV(“) is closely related to the the present case, the right hand side of eq. times the normalization factor N,/( N, - 1). eq. (20) in terms of PMF’s yields
d,.
cl
I
Pha N,
I/
(r,
It7
) g((l.O)+m)
(rlrr,;5) .
(23)
1
By analogy with the case of fluids [ 11, it may be argued that, in the low p, (adatom) limit, the t = 1 term of the sum is negligible, since p(,,(r,) is small on the average while the factor in the square brackets is relatively independent of the adatom density for sufficiently low p,. One may also argue that
w$‘,(r*) - w$)(r2) is negligible in the face of strong adatom-substrate interactions and at low p,. Strictly speaking, it is probably necessary to decide on the aptness of these
S. M. Vdlone et al. / Potentials of mean force for adatoms on surfaces
77
arguments on a case-by-case basis. Regardless of the above, the w(‘) (r,) PMF cannot be neglected for solids. More importantly, it is incorrect, even for fluids, to neglect the t = 2 (adatom-substrate) term, since neither the substrate density pz nor the adatom-substrate interactions Vi, are necessarily small. The conclusion to be drawn is that in the limit of low adatom density, the PMF for the adatoms is not identical to the pair potential. Differences arise from the contribution of the substrate to the PMF of the adatoms. The magnitude of these differences depends on the strength of the adsorbate-substrate interaction and the substrate density. In the next section, a suggestion is tendered for obtaining an estimate of the pair potential V, ,(r,*) from a combination of FIM data and a theoretically derived quantity: the pair distribution with the adatom-adatom interaction switched off.
3. A possible method for deriving the pair potential In spite of the negative result of section 2, it is still of interest to try to recover the adatom-adatom interaction from FIM data. Benignly, a slightly different manipulation of the data does yield the desired quantity. Instead of calculating the pair correlation function from the FIM-derived pair distribution [3], the pair distribution should be divided by the pair distribution for the system with the adatom-adatom interaction switched off. For this result one takes C = 2, n = (2,O) and N, = 2. The potential energy is written so that the coupling parameter .$is only between the adatom-adatom interaction, U,, = V, , : w
=Ev,,h)
+ uash
Ws)
+ v,h
{&)
+ U,SWS)~
(24)
where {r}, represents the set of substrate position vectors, U, is the adatomsubstrate interaction and US,is the substrate-substrate interaction. Note that some form of additivity approximation is still necessary. With p(“)( r,, rz; 5) defined as (eq.( 17))
p(yr,,r2;[)
=2
J d& exp(-Bu(t)) *
-%(I)
(25)
’
the logarithm of the ratio of p(“) at 6 = 1 and at I = 0 gives U,, up to a constant defining the zero of energy: -$
In
PYl=
p’“‘(
1) =
5 = 0)
V,,(k)
-L
P
In
z&=
z,(t=1)’
0)
(26)
The ratio of configuration integrals represents a free energy difference between the two systems [1,4]. A Kirkwood-like derivation of eq. (26) yields the
78
SM.
Valone et al. / Potentials of mean force for adatoms on surfaces
identification (27) Thus if derived function derived adatoms
one computes p(“)(t = 0) by some theoretical technique, the FIM pair distribution in conjunction with the calculated quantity yields a which differs from the pair potential by a constant. Pair potentials through eq. (26) could help clarify the roles of the substrate and the in the formation of clusters and thin film growth.
4. Conclusions We have shown (section 2) that the potential of mean force for adatoms in the limit of low adatom density does not equal the adatom-adatom pair potential (eq. (23)), as defined in the pair additivity approximation for the potential (eq. (6)). The adatom-adatom pair potential may be estimated by combining the experimental pair distribution with a theoretical pair distribution generated from a system in which the adatom-adatom pair potential has been switched off. Using FIM data in this way could lead to a better understanding of the role of the adatom-adatom interaction in the formation of clusters on solid surfaces and in thin film growth.
[ 1) T.L. Hill, Statistical Mechanics (McGraw-Hill, New York, 1956). especially sections 29, 31, 32 and 40. Hereafter referred to as I. For a lucid discussion of the Kirkwood equations for multi-component fluids, see: T.L. Hill, An Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, MA, 1960) pp. 331-334, section 18-2. [2] E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Principles and Applications (Elsevier, Amsterdam, 1969); E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Field Ionization, and Field Evaporation, in: Progress in Surface Science, Vol. 4, Part 1, Ed. S.G. Davidson (Pergamon, Oxford, 1973); G. Ehrlich, Surface Sci. 63 (1977) 422; G.L. Kellogg, T.T. Tsong and P. Cowan, Surface Sci. 70 (1978) 485. (31 G. Ehrlich, Phys. Today (June 1981) 51; H.-W. Fink, K. Faulian and E. Bauer, Phys. Rev. Letters 44 (1980) 1008; R. Casanova and T.T. Tsong, Phys. Rev. B22 (1980) 5590. [4] D.R. Squire and W.G. Hoover, J. Chem. Phys. 50 (1969) 701; C.H. Bennett and B.J. Alder, J. Chem. Phys. 54 (1971) 4796; C.H. Bennett, Exact Defect Calculations in Model Substances, in: Diffusion in Solids, Eds. A.S. Nowick and J.J. Burton (Academic Press, New York, 1975).