Langmuir 1996, 12, 3969-3975
3969
Simple Analytical Model for the Interaction between a Molecule and a Surface with a Step S. Briquez, A. Marmier, and C. Girardet* Laboratoire de Physique Mole´ culaire, URA CNRS 772, UFR Sciences La Bouloie, Universite´ de Franche-Comte´ , 25030 Besanc¸ on Cedex, France Received January 26, 1996. In Final Form: April 30, 1996X The Fourier transform procedure is used to determine the interaction potential experienced by a molecule adsorbed on a surface with a monoatomic step. The step is modeled as a perfect surface with a reduced set of lines or antilines describing the upper or lower terraces. The model is nearly analytic and, when additional terms issued from asymptotic approximations are considered, it appears to be fairly accurate. The behavior of the potential valleys close to the step is analyzed for ideal situations (Lennard-Jones potential, electric dipole, or quadrupole contributions), and it compares satisfactorily to exact calculations. This analysis allows us to understand the role of each interaction contribution in general situations of physisorption at a surface step.
I. Introduction The interaction between a molecule and a perfectly periodic substrate can be calculated using pairwise potential sums in direct space or a Fourier transform procedure in two-dimensional (2D) reciprocal space.1-4 The second technique presents the twofold advantage of being rapidly convergent when the substrate atom density is sufficiently large and analytic for simple forms of interaction potentials. However, more and more phenomena incriminated in various adsorption processes have their origin in surface defects. Steps on surfaces5,6 or extended defects implying linear atomic vacancies for nonstoichiometric7 surfaces are currently invoked in order to interpret specific adsorption geometries of isolated molecules or the formation of domains during the growth of layer structures.8-11 In that case, the 2D surface periodicity is broken and most of the calculations are limited to sums in direct space. The aim of this paper is to discuss the validity of a Fourier transform procedure which mixes 1D periodicity of the defect and 2D periodicity of the perfect surface. The defect is described by sticking semi-infinite atomic planes which characterize the upper terrace of a step above infinite atomic planes which schematically represent the perfect surface. In an equivalent way, semi-infinite ghost atomic planes can be subtracted from the perfect surface in order to form the lower terrace of the step. While 2D periodicity is clearly broken, the semi-infinite planes can be described as an arrangement of atomic (additional or ghost) lines, which obey the 1D periodicity requirement. Using such a model would be irrelevant for an analytical treatment of the interaction potential if the sum over the * Corresponding author. E-mail:
[email protected]. Fax: (33) 81 66 64 75. X Abstract published in Advance ACS Abstracts, July 1, 1996. (1) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: Oxford, 1974; Chapter 2. (2) Steele, W. A. Chem. Rev. 1993, 93, 2355. (3) Girard, C.; Girardet, C. J. Chem. Phys. 1987, 86, 6531. (4) Lakhlifi, A.; Girardet, C. Surf. Sci. 1991, 241, 400. (5) Pacchioni, G. Surf. Sci. 1993, 281, 207. (6) Nada, R.; Hess, A. C.; Pisani, C. Surf. Sci. 1995, 336, 353. (7) Zschach, P.; Cohen, J. B.; Chung, Y. W. Surf. Sci. 1992, 262, 395. (8) Zeppenfeld, P.; Horch, S.; Comsa, G. Phys. Rev. Lett. 1994, 73, 1259. (9) Bardi, U.; Glachant, A.; Bienfait, M. Surf. Sci. 1980, 97, 137. (10) Picaud, S.; Briquez, S.; Girardet, C. Chem. Phys. Lett. 1995, 242, 212. (11) Heidberg, J.; Meine, D. Ber. Bunsen-Ges Phys. Chem. 1993, 97, 211.
S0743-7463(96)00086-8 CCC: $12.00
line arrangement would be kept discrete. However we show that the influence of the step on the interaction potential between the molecule and the substrate is localized, suggesting that a very restrictive number of lines participate in the potential perturbation. We consider in section II a monoatomic step on an ionic or ionocovalent substrate although the model can be easily generalized to any step geometry. In section III, we briefly present the Fourier transform expressions of dispersionrepulsion and electrostatic contributions. The step model is discussed in section IV, and an application to simple adsorbates (atom, dipolar, or quadrupolar molecule) is done in section V in order to define general statements for the molecular adsorption close to a defect. Note that the present method is limited to the physisorption process, since no charge transfer is considered in the interaction potential. For chemisorption on metals, the dominant local chemical bond tends to relegate the physical interactions to a fraction of the interaction potential. However on ionic or ionocovalent substrates, the electrostatic terms are clearly large and they contribute for the most part to the interaction process at low temperature.6,12 II. System Geometry Figure 1 displays the geometry of an admolecule interacting with a step. The (001) terrace of the ionic substrate (NaCl, MgO, or LiF) exhibits a two-dimensional (2D) square symmetry, and a 1D symmetry is assumed for the step edge along the [100] or [010] direction. The absolute frame (X, Y, Z) is chosen such that the X and Z axes correspond to the step edge and to the surface normal direction, respectively, and its origin is taken at the step. The system is separated in two parts: the planes p e 0 belong to the perfect substrate, with the full 2D symmetry. The step is described by lines (n ) 0, 1, 2, ...) or ghost lines (n ) -1, -2, ...) belonging to semi-infinite planes p > 0 (p ) 0 is thus the first 2D plane of the lower terrace). For the monoatomic step, pmax ) 1. The position of the sth ion (s ) 0, 1, 2, 3) in the lth 2D cell (l ≡ (l1, l2) is a doublet of integer numbers) belonging to the pth substrate plane (p e 0) is written as
r(p,l1,l2,s) ) l1a1 + l2a2 + τsp + phpZ,
(1)
where a1 and a2 are the translation vectors for the 2D (12) Mejı´as, J. A.; Ma´rquez, A. M.; Ferna´ndez Sanz, J.; FernandezGarcia, M.; Ricart, J. M.; Sousa, C.; Illas, F. Surf. Sci. 1995, 327, 59.
© 1996 American Chemical Society
3970 Langmuir, Vol. 12, No. 16, 1996
Briquez et al.
charges qs ) (q at the running point r
(-1)s
∑j |r - r |
Φ(r) ) q
(5)
j
Figure 1. Geometry of the substrate with a monoatomic step; p and n define the plane and line numbers, respectively. The drawn molecule is diatomic with center of mass position r and axis orientation Ω(θ,φ).
lattice (|a1| ) |a2| ) a), hp is the interplanar distance (here hp ) a/2), and τsp locates the sth ion in the (l,p)th cell. Note that the choice of a unit cell which has a surface two times larger than the usual 2D cell and contains four ions instead of two results from the symmetry imposed by the step direction, which is rotated by 45° with respect to the usual primitive translation vectors of the perfect surface (Figure 1). In the same way, the location of the sth ion (s ) 0, 1) in the l1th cell of the nth line belonging to the pth (p > 0) plane is defined as
r(p,n,l1,s) ) l1a1 + τspn + phpZ - nhlY
(2)
where a1 is the unit translation vector for the line, hl is the interline distance (here hl ) a/2), and the vector τspn defines the ion location in the 1D cell. The linear molecule position and orientation are defined by the center of mass location r and by the Euler angles Ω(φ,θ) in an absolute frame tied to the surface step. III. Interaction Energy A. General. The interaction between an ionic substrate and an admolecule is mainly due to pairwise dispersion-repulsion terms and to electrostatic contributions.3,4 We choose a Lennard-Jones form to describe the quantum interactions between a characteristic point r of the molecule and the jth ion of the substrate (j ≡ lps for a terrace ion and j ≡ l1pns for a step ion). This characteristic point can be an atom center of mass, another site in the molecule, or the molecular center of mass itself, depending on the description used for the molecule. This potential is given as
VLJ )
∑j νLJ(rj) ) ∑j R)6,12 ∑ (-1)
CRs R/2
(3)
rRj
while the electrostatic contribution appears as an electric multipolar series expansion3,4
VE )
∑j νE(rj) ) 1 qΦ(r) - µ[∇Φ(r)] + Q[∇∇Φ(r)] + ... (4) 3
Φ(r) is the electrostatic potential created by the substrate
and q, µ, and Q are respectively the electric charges, dipoles, and quadrupoles located at point r in the molecule. The sum over the ions belonging either to the surface or to the step lines is performed in the reciprocal space using a Fourier analysis procedure1,2 for the periodic interaction potential which is adapted to the symmetry of the molecule-terrace-step system. Any interaction contribution V(r) between the probe molecule located at r and a source system assumed to be infinite and exhibiting a symmetry of order N (N e 3) is a periodic function of the r| variable. The vector r is projected onto two subspaces E| and E⊥ with respective dimensions N and 3 - N and is split into r| and r⊥. Since V(r) is expressed as a sum of binary potentials
V(r) )
ν(rs,1 - r) ∑ s,1
(6)
N liai + τs is the general vector defining the where rs,l ) ∑i)1 location of a site in the direct periodic lattice of the source system, it can be expanded as a Fourier series over g N gibi (gi is an integer and vectors defined as g ) 2π/a ∑i)1 the bi’s are the reciprocal vectors in the E| space), as1,2
V(r) )
∑g w˜ g(r⊥)eig‚r
|
(7)
The Fourier coefficient w ˜ g is given by
1
w ˜ g(r⊥) )
detN
∑e-ig‚τ ∫E ν(r)e-ig‚r dr| (a ) s |
s
|
(8)
i
where detN(ai) defines the unit cell volume while the integral is over the space E|. The next section will deal with the cases N ) 1 and N ) 2, which correspond to 1D periodic lines and the 2D periodic substrate. B. Analytical Expressions. 1. Lennard-Jones Interaction. Explicit calculations for the interaction between point r in the molecule and the perfect surface, using the potential defined by eq 3, lead to1-4
VTLJ )
2π a2
4π a
2
∑
s,pe0
[
C6s
-
∑ ∑
C12 s +
4z4p
10z10 p
(-1)R/2CRs
s,pe0 R)6,12
(
1
R 2
]
+ ′
∑g cos g(r| - τ|sp) ×
() )
-1 !
|g|
2zp
(R/2)-1
K(R/2)-1(|g|‚zp) (9)
where Kν(|g|‚zp) is the modified Bessel function of order ν and zp ) z + php is the distance between point r and the pth substrate plane. The prime indicates that the sum is performed over the g values defined by g ) 2π/a (g1b1 + g2b2) with g1 > 0 and g2 or g1 g 0 and g2 < 0. The first term corresponds to the continuum description (g ) 0) whereas oscillating terms g * 0 are successive harmonics which account for the discrete nature of the surface. Due to the decreasing exponential behavior of the Bessel
Interaction of a Molecule and a Surface with a Step
Langmuir, Vol. 12, No. 16, 1996 3971
functions with |g|, the convergence of the expansion is very good for a restricted number of g vectors. For the interaction potential between point r and a set of n lines separated by a/2 and forming a step with p (p > 0) planes, the analytical integration can be easily performed,13 leading to
VSLJ
3π
∑
)
8a n,s,p>0 4π a
[
C6s
21C12 s
-
+ 5 dp,n
∑ ∑
11 32dp,n ′
(-1)R/2CRs
n,s,p>0 R)6,12 (R/2)-1
e-gdp,n
( )
∑
R
(2dp,n)
R-1
2
-1 !
]
+
∑g cos g(x - τs,p,n) ×
(R - k - 2)! (2gdp,n)k
k)0
(
R
k!
2
)
(10)
-k-1 !
where g ) (2π/a)g1, with g1 > 0. dp,n is the distance between point r and the nth line in the pth plane expressed as dp,n ) [(y + n(a/2))2 + (z - p(a/2))2]1/2, and x is the location of point r along the step direction (Figure 1). The first part of eq 10 corresponds to g ) 0 and describes the set of lines as a continuum whereas the second part generates corrugation along these lines. Note also the exponential behavior of the coefficients g * 0, which significantly decrease for reasonably large values of g and dp,n. 2. Electrostatic Interaction. In order to get the electrostatic contributions (eq 4), the electrostatic potential Φ and its successive gradients are calculated in Appendix A. The interaction energy between the dipolar moment µA located at point r and the perfect surface is expressed as
VTD
4πq )-
{
a2
′
∑ ∑(-1) s,p g
s+p -|g|‚zp
e
µAz cos g‚(r| - τ|sp) +
×
g‚µA |g|
sin g‚(r| - τ|sp)
}
(11)
while for the set of lines it becomes
VSD ) -
4q
′
∑ ∑ (-1)s+p+n × s,p,n g
a g[K1(gdp,n)(µA‚er(n)) cos g(x - τspn) + K0(gdp,n)‚µAx sin g(x - τspn)] (12)
In a similar way, the quadrupolar interaction can be expressed, for the perfect surface, as
VTQ )
8π3q 3a
′
∑ ∑(-1)s+p 4 s,p g
e-|g|‚zp QA‚A(g,r|-τ|sp) (13) |g|
Figure 2. Schematic representation of the model for stepmolecule interaction for the two situations: (a) molecule on the lower terrace; (b) molecule on the upper terrace. Dark and crossed circles represent added and removed lines, respectively.
and 2D Fourier expansions to get the general interaction between the admolecule (r,Ω) and the surface with a step along x and located at y ) 0. This interaction energy is thus the sum of terrace and step contributions, as
VMS(r,Ω) ) VT(r,Ω) + VS(r,Ω)
Each term in eq 15 corresponds to sums over planes or lines. Let VP(p) be the interaction between the molecule L and the pth terrace plane and V(n) be the interaction with the nth line of the step; we have
VT(r,Ω) )
P V(p) (r,Ω) ∑ pe0
VS(r,Ω) )
L (r,Ω) ∑n V(n)
VSQ )
case a: VMS(r,Ω) )
P L (r,Ω) + ∑ V(n) (x,dn,Ω) ∑ V(p) ng0
pe0
case b: VMS(r,Ω) )
P L V(p) (r,Ω) - ∑ V(n) (x,dn,Ω) ∑ pe1 n VMS VMS VMS
[
]
6 6 12 12 2π (C+ + C-) (C+ + C- ) + + a2 2z4 5z10
[ [
)
(
( ) ( ) ( ) (
)]
12 5 (C12 2x2π + - C- ) x2π K5 z 5! az a
16πq -(2πx2/a)z e × a2 x2πQ π 2π sin (x + y) cos φ µsin 2θ + a 4 a x2πQ π 2π µsin (x - y) cos φ + sin 2θ + a 4 a x2πQ 2π cos (x + y) µ cos θ × a a π 2π cos2 θ - sin2 θ cos2 φ + cos (x - y) × 4 a x2πQ π µ cos θ cos2 θ - sin2 θ cos2 φ + (19) a 4
) ( ) (
(
(
(
[
(
)[ )[
)[ ( ))]
] ]
(
)
(
P2
( )
(20)
( )
P5
( ) ( ) ( ) ( ) ( ) ( ) ( )
2πd0 2πd0 2πd0 )3+3 + a a a
2
2πd0 2πd0 2πd0 2 ) 252 + 252 + 112 + a a a 2πd0 3 2πd0 4 4 2πd0 +4 + 28 a a 15 a
(18)
(k ) 6, 12) are the potential coefficients for the cation (+)-admolecule pair and the anion (-)-admolecule pair. The electrostatic energy for a point admolecule is similarly written for |g| e (2π/a)x2 as
{ (
( ) ( )]
with
VSa LJ = -
Ck+,-
VTE ) -
]
0 - C6- 2πd0 P2 + a 4d50 12 C12 2πd0 + - CP5 11 a 512d0
C6+
5
Taking advantage of such a convergence, the LennardJones interaction with the full step is thus the sum of the expression given in eq 20 and a contribution due to the other lines (n g 1) viewed as an attractive continuum (g ) 0). Since for reasonably large values of d > a the attractive term is dominant compared to the repulsive contribution, this second term has the asymptotic form
)]
2π 2π 8π cos (x + y) + cos (x - y) × 2 a a a 6 6 (C+ - C-) x2π 2 2x2π K2 z + 2 az a
(
[
0
π 2πx -(2π/a)d0 cos e a a
where δV is the inaccuracy brought by the two kinds of truncation with respect to the exact value. 1. On the Terrace. For dense substrate faces, it has been shown that the first values of g (|g| e 4π/a) are generally sufficient4 to adequately represent the molecule/ substrate interaction. Moreover, because of the fast decrease of interactions through either the modified Bessel function or the exponential behavior, the number of planes which are taken into account for an accuracy better than a few percent is usually restricted to the surface and to the first internal planes. The dispersion-repulsion interaction between site r and the single surface plane can be written for |g| e 2πx2/a as
VTLJ )
[
3π 1 21 12 -(C6+ + C6-) 5 + (C12 + + + C- ) 8a d 32d11
))]}
2. Near the Step. In order to estimate the error due to g and d truncations in the step-molecule interaction, let us define dmin as the equilibrium distance between the molecule and the line; dmin is typically in the [a/2, a] range. Calculations show that, for d > dmin, the g e 2π/a terms are sufficient to produce a good description of the interaction with a relative accuracy |δVS/VS| better than 2%. For d > a, the continuum description is quite good, since the relative error |δVS/VS| is about 2.5% for the system Ar/ MgO at d ) a, which corresponds to |δVS/VMS| ) 5 × 10-3. For g e 2π/a, the Lennard-Jones interaction between point r and the single line (n ) 0, p ) 1) representing the step becomes
12π 6 y (C+ + C6-)ζ 5,2 +1 a a
(
)
(21)
where it has been assumed that z = a/2, and the generalized ζ function of Riemann is defined by13
(
) ()
a y ζ 5,2 +1 ) a 2
5
∞
∑ n)1
(
1
)
a y+n 2
5
(22)
This asymptotic expression, added to the terms g e 2π/a and n ) 0 in eq 20, leads to an interaction expression which is available anywhere, even when the adatom is close to the step, within a relative accuracy of 10-3 for distances to the step edge larger than a/2. According to eqs 12 and 14 the electrostatic energy between a multipolar admolecule and the line is expressed for g1 e 1 as
{
( )[ ( )
2πd0 16πq 2πx sin µK0 cos φ sin θ 2 a a a 2πd0 πQ + F4(R0)K1 2a a 2πd0 2πx µK1 (sin φ sin θ cos R0 + cos a a 2πd0 πQ cos θ sin R0) F1(R0)K0 + 2a a 2πd0 2πd0 aF3(R0) F2(R0)K1 + K1 a πd0 a
VSE ) -
( )[ ( )
( )
( )]
(
( ) ( ))]}
(23)
where F1, F2, F3, and F4 are angular functions defined in Table 1. Note that even values of g1 vanish because of equidistant alternated charges along the line. The convergence study with g1 values show that, for a relative accuracy |δVS/VS| of 3 × 10-3 at d ) a/2 and better than 10-5 for d g a, the first oscillating term (g1 ) 2π/a)
Interaction of a Molecule and a Surface with a Step
Langmuir, Vol. 12, No. 16, 1996 3973
Table 1. Expression of the Fi Functions in Eq 23a F1(Rn) ) 1/3 + [sin φ sin 2θ cos Rn sin Rn + sin2 φ sin2 θ cos2 Rn + cos2 θ sin2 Rn - 2 cos2 φ sin2 θ] F2(Rn) ) -1/3 + [sin φ sin 2θ cos Rn sin Rn + sin2 φ sin2 θ cos2 Rn + cos2 θ sin2 Rn] F3(Rn) ) 1/3 + [sin φ sin 2θ cos Rn sin Rn - sin2 φ sin2 θ sin2 Rn + cos2 θ cos2 Rn] F4(Rn) ) 2[sin 2φ sin2 θ cos Rn + cos φ sin 2θ sin Rn] a
cos Rn ) [y + n(a/2)]/dn; sin Rn ) [z - (a/2)]/dn.
is sufficient to provide a good description of the electrostatic field due to the charges belonging to a line. Moreover, for d > 2a and a relative precision better than 2%, one has14
1 a = e (2πd a ) 2xd
Kν
(24)
-2πd/a
Therefore in close similarity to the Lennard-Jones study, asymptotic expressions for the electrostatic interaction energy can be added to eq 23
VSa E ) -
8πq
x
∑ (-1)n ng1
a
e-2πdn/a × d a n πQ 2πx sin cos φ sin θ F4(Rn) + a 2a 2πx µ(sin φ sin θ cos Rn + cos θ sin Rn) cos a a πQ F′1(Rn) + F3(Rn) (25) 2a πdn 2
{ ( )[ ( )[
]
)]}
(
with F′1 ) F1 + F2 (cf. Table 1). Note furthermore that the alternated series occurring in eq 25 can be limited to the first line thanks to the general result
Kν
(2πd a )
2πd Kν +π a
(
)
> eπ ≈ 23
(26)
V. Numerical Results In this part, we will take advantage of the relatively simple analytical forms of the interaction energy between a step and an admolecule to analyze general features. Therefore we will first consider in a separate way the behavior of the dominant contributions in the potential as a function of the position of the admolecule with respect to the step for various ideal molecules in order to discuss the influence of the step on the diffusion process. This is easily illustrated by considering the two situations (a and b) defining the two potential expressions (eq 17). Indeed depending on the sign in front of VL and on the sign of VL itself, the step will either be attractive or repulsive. For instance, when VL is negative, the step is attractive for a molecule climbing up (case a) and on the contrary repulsive for a molecule descending down (case b). A. Rare Gas Atom. A rare gas atom interacts with the step mainly through the dispersion-repulsion interaction expressed as (eqs 18, 20, and 21)
VMS ) VTLJ ( (VSLJ + VSa LJ)
Figure 3. Minimum potential energy V/ (reduced units) experienced by an adatom along its diffusion valley on MgO. is the average Lennard-Jones parameter which defines the well depth for an Ar adatom/MgO substrate pair. Comparison between exact calculations (upper curve) and the simplified model (lower curve). M is the matching point between cases a and b; see Figure 2.
(27)
where the sign depends on case a or b. Therefore, the rare gas atom will be attracted by the step when it comes from the lower terrace, whereas the step will act as a barrier for the adatom coming from the upper terrace. As (14) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications: New York, 1965.
an example, Figure 3 displays the minimum potential energy experienced by an Ar adatom along its diffusion valley on MgO. This valley exhibits a maximum preventing the adatom from going down at the step edge and a minimum at the counterstep preventing the adatom from climbing the step. This maximum is explained by the fact that an adatom has to pass over the step edge where it has only a few neighbors, i.e. a site where it is less bound to the surface. On the contrary, at the step bottom the number of neighbors increases, leading to a minimum energy. This behavior is well-known as the ‘Schwoebel effect’.15,16 These features have already been observed in computer simulations for Kr adsorbed on heterogeneous surfaces using a continuum description for the interaction potential.17 In Figure 3 the upper curve has been obtained from the minimization in direct space, while the lower curves correspond to reciprocal calculations using the simplified expression (eq 27). Curve a corresponds to the minimum energy experienced by the atom moving toward the step from the lower terrace (case a). The same potential is drawn (case b) for the atom moving toward the step from the upper terrace. Curves a and b match close to the step at point M for which y ) 0.3a and then exhibit divergences, as expected from the model. After the matching, the curves displayed by the discrete and simplified approaches fit in a quite satisfactory way; the potential depth and barrier (15) Schwoebel, R. L.; Shipsey, E. J. J. Appl. Phys. 1966, 37, 3682. (16) Ehrlich, G. Surf. Sci. 1995, 331-333, 865. (17) Bojan, M. J.; Steele, W. A. Surf. Sci. 1988, 199, L395. Bojan, M. J.; Steele, W. A. Langmuir 1989, 5, 625.
3974 Langmuir, Vol. 12, No. 16, 1996
Briquez et al.
Figure 4. Minimum potential energy Va2/qµ (reduced units) experienced by a dipolar molecule along its diffusion valley on MgO. Comparison between exact calculations (upper curve) and the simplified model (lower curve) using cases a and b. The solid part of the lower curve corresponds to a region where curves a and b cannot be distinguished.
Figure 5. Minimum potential energy Va3/qQ (reduced units) experienced by a quadrupolar molecule along its diffusion valley on MgO. Comparison between exact calculations (upper curve) and the simplified model (lower curve) using cases a and b. The solid part of the lower curve corresponds to a region where curves a and b cannot be distinguished.
height are overestimated by only about 5 meV in the simplified approach. B. Polar Molecule. We then discuss the behavior of the electrostatic terms. The quantum interactions have only been considered in order to define the equilibrium distance ze ) f(y) between the admolecule and the substrate along the equilibrium valley. The electrostatic potential energy is written as (eqs 19, 23, and 25)
field due to the step (eq A-6)
VMS ) VTE ( (VSE + VSa E )
(28)
depending on case a or b. Let us first consider a dipolar molecule, whose surface diffusion valley is drawn in Figure 4. We still see a better agreement between the discrete approach and the simplified form, after consideration of the asymptotic corrections. The match is quite good, since we cannot distinguish curves a and b in Figure 4 over a large extent of distances, i.e. 0.2a e y e 0.6a. The potential valley for the dipolar molecule is very different from the shape exhibited by a simple atom, since the barrier/well asymmetry at the step disappears and is replaced by a single well. This valley exhibits furthermore a breaking of the slope between 0 and a/3. To interpret this feature, it may be noted that the study of the dipolar contribution reduces to that of the electrostatic field norm. Indeed, the most stable configuration for every point above the surface corresponds to µ colinear to E. The dipole-surface or dipole-line interaction is therefore minimum when µ is oriented along the field direction, i.e. when the total electric field intensity |E| due to the terrace and the step is maximum. This maximum is found to be in front of an ion, with a perpendicular orientation to the surface plane for the part of the field due to the terrace (eq A-2)
ET(0,0,z) )
32πq -(2x2π/a)z e Z a2
(29)
and to the line representing the step for the part of the
ES(0,d) )
16πq 2πd K1 e a r a2
( )
(30)
The ratio between the two field intensities is calculated for the same dipole-line and dipole-terrace distances equal to l
|ES| 1 2πl (2x2π/a) e ) K1 a |ET| 2
( )
(31)
For typical values of l ) a/2, a, and 3/2a, the electrostatic field is respectively 1.6, 3.6, and 10.5 larger at the line site than at the terrace site. This clearly indicates that the line corresponds to a highly favorable site for a strong dipolar molecule oriented perpendicular to the step which thus acts as a tip. The slope change in the well characterizes the fact that the molecular center of mass climbs up to the step, being at the same distance from the ledge of the step but changing its orientation in order to remain perpendicular to this ledge. A similar agreement is obtained for a quadrupolar molecule between the discrete approach and the simplified expression of the interaction energy (Figure 5). The right potential valley can be successfully fitted using cases a and b without additional matching when the asymptotic contribution is added but on a lesser extent for the stepmolecule distance 0.25a e y e 0.5a. The potential well found close to the step looks like that calculated with the dipolar molecule. Such a behavior can be interpreted within the previous scheme by analyzing the field gradient instead of the electric field, and it still corresponds to a tip effect for the field gradient at the step. The equilibrium configuration corresponds to the molecular axis perpendicular either to the surface or to the line, the line and the molecular axis being coplanar. The line is still more attractive than the plane. This behavior tends to reinforce the dipole effect with the noticeable difference that the ratio |VS|/|VT| is smaller than that for the dipole, since we
Interaction of a Molecule and a Surface with a Step
Langmuir, Vol. 12, No. 16, 1996 3975
get ratios of 1.2, 2.7, and 7.9 for the distances l ) a/2, a, and 3/2a, respectively. C. Discussion. By using a simple model describing the step as a perfect terrace plus or minus a limited number of lines, we obtained information about the step influence on surface diffusion above ionic substrates. This model has the advantage of being nearly analytical and thus easily tractable for further calculations, and it works fairly well for every potential contribution and particularly for multipolar molecules adsorbed on ionic substrates. It shows the well-known feature that a rare gas atom or a nearly spherical molecule (CH4, SF6) diffusing at the surface experiences, a dissymmetric potential valley, which is attractive for an atom coming toward the step from the lower terrace and repulsive from the other side. In contrast, dipolar and quadrupolar molecules are strongly attracted by the step edge, with a perpendicular orientation, and the step appears to have a symmetric influence on the potential. Then, the usual diffusion barrier due to dispersion-repulsion is generally strongly lowered because of electrostatic effects. The global result will depend on the relative importance of electrostatic interactions as compared to quantum interactions. The general situation of a molecule interacting with the surface through all potential contributions must account for the superimposition of the various effects studied here.18 The competition between line-molecule and terrace-molecule interactions tends to complicate the equilibrium configuration because the molecular shape has a significant influence on the equilibrium geometry close to the step. Indeed the dispersion-repulsion interactions tend to align long linear molecules along the surface or the line, whereas electrostatic effects favor perpendicular configurations. The resulting molecular configuration is thus due to a subtle competition between these two antagonistic contributions. As an example the mainly quadrupolar CO2 molecule is found to be flat along the step direction19 while the axis of the strong dipolar NH3 molecule is found to be perpendicular to the step and to the surface.18 The present approach is able to explain most of these features, and it is expected to be well-suited for studying adsorption on stepped surfaces in simulation calculations which require simple expressions of interaction potentials. Appendix A For the surface, the electrostatic interaction potential is written as T
Φ (r) )
′
4πq
∑ ∑(-1) 2 s,p g
s+p -|g|‚zp
e
cos g(r| - τ|sp) |g|
a
(A-1)
The g ) 0 term vanishes because of the surface neutrality. From the successive gradients of ΦT(r), we calculate the electrostatic field ET(r) as
ET(r) )
4πq
′
(-1)s+pe-|g|‚z ∑ ∑ 2 s,p g
p
×
a
{
cos g(r| - τ|sp)‚Z +
g |g|
sin g(r| - τ|sp)
}
(A-2)
(18) Briquez, S.; Girardet, C.; Goniagowski, J.; Noguera, C. J. Chem. Phys., in press. (19) Briquez, S.; Lahklifi, A.; Picaud, S.; Girardet, C. Chem. Phys. 1995, 194, 65.
and the field gradient as
∇∇ΦT(r) )
′
8π3q
e-|g|‚zp s+p | (-1) ) ‚A(g,r|-τsp ∑∑ 4 s,p g |g| a (A-3)
where A is a second rank tensor defined as
[
| A(g,r|-τsp ))
-g21c
-g1g2c
-g1g2c
-g22c
g 1x
g21
+
g22s
g2x
g21
g1xg21 + g22s g2xg21 + g22s
+
g22s
g22(g21
+
g22)c
]
(A-4)
where c ) cos g(r| - τ|sp) and s ) sin g(r| - τ|sp). The same scheme is applied for the line with 1D symmetry. The electrostatic potential is
ΦS(x,z) ) 4q a
′
(-1)s+p+n cos g(x - τspn)K0(gdp,n) ∑ ∑ s,p,n g
(A-5)
where g ) (2π/a)g1 with g1 a strictly positive integer. The electric field is calculated as
ES(x,z) ) 4q a
′
(-1)s+p+ng[K0(gdp,n) sin g(x - τspn)X + ∑ ∑ s,p,n g K1(gdp,n) cos g(x - τspn)er] (A-6)
where er is a radial unit vector with coordinates (0, cos R ) y/d, sin R ) z/d). The double gradient ∇∇ΦS is expressed as
∇∇ΦS(x,z) )
4q a
′
∑ ∑ (-1)s+p+ng2B(g,x-τspn) s,p,n g
(A-7)
with B a second rank tensor defined as
[
B(g,x-τspn) ) -c′K0 s′K1cR s′K1cR s′K1sR
{ {
s′K1sR
} { } {
K1 1 c′ [K0 + K1]c2R - s2R 2 gd K1 1 c′ [K0 + K1] 2 gd
}
K1 1 [K + K1] 2 0 gd K1 1 c′ [K0 + K1]s2R - c2R 2 gd c′
}
]
(A-8)
where c′ ) cos g(x - τspn), s′ ) sin g(x - τspn), sR ) sin R, and cR ) cos R; the modified Bessel functions K0 and K1 depend on the variable gdp,n. The dipole and quadrupole moments for the molecule are defined in the absolute frame as µA ) (µ sin θ cos φ, µ sin θ sin φ, µ cos θ) and QA ) M-1(θ,φ)‚Q‚M(θ,φ). µ and Q ) Qzz ) -2Qxx are the values of the dipolar and the quadrupolar moments expressed in the molecular frame,20 respectively; M(θ,φ) is the usual rotation matrix.21 LA9600863 (20) Buckingham, A. D. In Advances in Chemical Physics; Hirschfelder, J. O., Ed.; Wiley: New York, 1967; Vol. 12, p 107. (21) Rose, M. E. Elementary Theory of Angular Momentum; Wiley: New York, 1967.
