Langmuir 1999, 15, 5961-5969
5961
Probabilistic Analysis of the Dual Site-Bond Model: The Self-Consistent Case† Alessandra Adrover Dipartimento di Ingegneria Chimica, Universita´ di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy Received September 25, 1998. In Final Form: December 4, 1998 A probabilistic analysis of the generation of dual site-bond models for the energy landscape of adsorbents is presented, focusing attention on the self-consistent case. To generate an uncorrelated site-bond energy landscape with prescribed distribution functions FS/ (E) and FB/ (E) consistent with the two basic laws of the dual site-bond model, the random variables associated with the site and bond energies should possess the local distribution functions FS(E) and FB(E), which are related to FS/ (E) and FB/ (E) through a nonlinear integral equation. In particular, the inverse problem in two dimensions is solved in closed form. The analytical solution for the inverse problem provides a clear description of the intrinsic correlation in the energy landscape induced by the fulfillment of the construction principle.
1. Introduction The analysis of adsorption phenomena on heterogeneous surfaces requires a detailed description of the statistical and correlation properties of the energy distribution. From recent studies,1-3 it is evident that the spatial structure (expressed, e.g., by means of the correlation function) of the energy field plays a fundamental role as regards the value and shape of the resulting macroscopic quantities (e.g., adsorption equilibria), especially in those cases in which the correlation properties of the energy field influence surface-hopping phenomena between nearestneighbor sites (surface diffusion). For this reason, many models describing and characterizing heterogeneous surfaces have been proposed recently to improve the simplified characterization of the correlation properties based on the distinction between random and patchwise topographies.4 Among these, the Dual Site-Bond (DSB) model proposed by Mayagoitia et al.5,6 displays great versatility because it intrinsically contains a description both of the energy-site distribution (associated with the adsorbing centers) and the saddle-point energies between two nearest-neighbor sites (minima between two local maxima of the adsorption energy), which control particle hopping between nearest-neighbor sites. The DSB model has been applied to study porous structures,7 surface diffusion phenomena and adsorption equilibria,2,3 and annihilation kinetics.8 † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998.
(1) Riccardo, J. L.; Pereyra, V.; Rezzano, J. L.; Rodriguez Saa´, D. A.; Zgrablich G. Surf. Sci. 1988, 204, 289. (2) Riccardo, J. L.; Chade, M. A.; Pereyra, V. D.; Zgrablich, G. Langmuir 1992, 8, 1518. (3) Sapag, K.; Bulnes, F.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Langmuir 1993, 9, 2670. (4) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (5) Mayagoitia, V.; Rojas, F.; Riccardo, J. L.; Pereyra, V. D.; Zgrablich, G. Phys. Rev. B 1990, 41, 7150. (6) Mayagoitia, V.; Rojas, F., Pereyra, V. D.; Zgrablich, G. Surf. Sci. 1989, 221, 395. (7) Faccio, R. J.; Vidales, A. M.; Zgrablich, G.; Zhdanov, V. P. Langmuir 1993, 9, 2499. (8) Gonzales, A. P.; Pereyra, V. D., Riccardo, J. L., Zgrablich, G. J. Phys.: Condens. Matter 1994 6, 1.
The generation of a DSB model with prescribed distribution functions for the site and bond energies is subjected to two constitutive laws characterizing the model. These two laws express both globally and locally the fact that the bond energies are the local minima between two energy maxima. They state that (1) the difference between the bond energy distribution and the site energy distribution is always nonnegative and (2) the probability of a site admitting an energy value less than the energies of its bonds is equal to zero. The existence of these two constitutive laws makes the generation of a DSB model slightly more cumbersome, especially for high overlappings of the two distribution functions. For this reason, Riccardo et al.9,10 propose a Monte Carlo method to overcome the intrinsic distortion associated with the iterative assignment of bond and site energies induced by fulfillment of the two constitutive laws of the model. This article presents a probabilistic analysis of the generation of model heterogeneous surfaces according to the DSB description. In particular, attention is focused on the so-called self-consistent (SC) case, which is the method most widely used to generate a DSB energy landscape.9,10 The SC case was introduced by Mayagoitia et al.11,12 in order to achieve the maximum degree of randomness allowed by the construction principle (see section 2 for details). This article is organized as follows. Section 2 briefly reviews the basic definitions of the DSB model and the way of generating two-dimensional energy landscapes fulfilling the SC procedure. Section 3 formulates the basic integral equations for solving the direct problem, i.e., for quantifying the distortion effects in the generation of energy landscapes as a consequence of the fulfillment of the construction principle. Section 4 gives a closed-form solution for the inverse problem, which means finding local site and bond energy distribution functions which yield the desired site and bond energy distributions. Section 5 addresses the important issue of the intrinsic (9) Riccardo, J. L.; Pereyra, V.; Zgrablich, G. Langmuir 1993, 9, 2730. (10) Riccardo, J. L.; Steele, W. A.; Ramirez Cuesta, A. J.; Zgrablich, G. Langmuir 1997, 13, 1064. (11) Mayagoitia, V.; Cruz, J. M.; Rojas, F. J. Chem. Soc. 1 1989, 85, 2071. (12) Cruz, J. M.; Cruz, M. J.; Rojas, F. J. Chem. Soc. 1 1989, 85, 2079.
10.1021/la981321f CCC: $18.00 © 1999 American Chemical Society Published on Web 05/06/1999
5962 Langmuir, Vol. 15, No. 18, 1999
Adrover
correlations induced by the construction principle characterizing the DSB model in the SC case for high overlapping degrees and how these effects can be included in the integral equations for solving the inverse problem. 2. Basic Definitions and Notation Let FS(E) and FB(E) be the distribution functions associated with site and bond energies and pS(E) and pB(E) the corresponding probability density functions FS(E) ) ∫E0 pS() d, FB(E) ) ∫E0 pB() d. Let the intervals S ) [S1, S2) and B ) [B1, B2) be the support of the site and bond probability measures, i.e., the set of values of E for which pS(E) and pB(E) are greater than zero. S2 and B2 may be equal to infinity, and S1 g 0, B1 g 0. Let pSB(ES, EB) denote the joint probability density function of nearest-neighbor site-bond energies. The two basic laws describing the DSB model are
FB(E) - FS(E) g 0
(1)
pSB(ES, EB) ) 0 for ES < EB
(2)
The first law, eq 1, implies that B1 e S1 and B2 e S2. The second law, which is of a local nature, expresses the fact that in the DSB model, bond energies greater than the energies of the adjacent sites are excluded. For this reason, eq 2 can be called the local exclusion principle of DSB models. The fulfillment of the local exclusion principle plays a fundamental role when the intervals S and B overlap, i.e., when S1 < B2. We will concentrate on this case throughout. In the SC case, the joint probability density function pSB(ES, EB) can be expressed as
pSB(ES, EB) ) pS(ES)pB(EB)φ(ES, EB)
(3)
where the correlation function φ(ES, EB) attains the following expression:
φ(ES, EB) )
[
exp -
ES
dFB(x)
B
FB(x) - FS(x)
∫E
]
/[FB(EB) - FS(EB)] )
[g(EB)/pB(EB)]exp [ -
∫EE
S
B
g(x) dx] (4)
E S g EB where g(x) ) [pB(x)]/[FB(x) - FS(x)], and φ(ES, EB) ) 0 for ES < EB. The simulation of a DSB energy landscape consists of a sequential assignment of site and bond energies satisfying eqs 1-3. Let us first consider a one-dimensional model surface, and let EB(k + 1) be the bond energy associated with the bond connecting site k and site k + 1. Given ES(k), the energy EB(k + 1) is generated from the conditional probability density function (pdf)
pB[EB(k + 1)/ES(k)] ) pB[EB(k + 1)]φ[ES(k), EB(k + 1)] (5) and conversely, given EB(k + 1), the site energy ES(k + 1) can be obtained from the conditional pdf:
pS[ES(k + 1)/EB(k + 1)] ) pS[ES(k + 1)]φ[ES(k + 1), EB(k + 1)] (6) In the two-dimensional case on a square lattice topology, an elementary unit should be defined as a site and its right and down bonds as depicted in Figure 1 a. The
Figure 1. (a) Schematic representation of the basic unit cell used in the generation of 2-D DSB energy landscapes on a square lattice. (b) Pictorial representation of a two-dimensional lattice. The solid lines are bonds which are pre-assigned according to the local pdf pB(EB).
generation of the energy landscape follows a sequential procedure which is a slight modification of the method adopted by Riccardo et al.9 in order to diminish boundary effects. First, the bonds of the uppermost row and leftmost column (indicated by solid lines in Figure 1b are assigned according to the pdf pB(EB). Subsequently, proceeding from left to right and from top to bottom in the lattice, each site energy is assigned according to the conditional pdf:
pS(ES/E1B, E2B) ) pS(ES)φ(ES, E1B)φ(ES, E2B)
∫max{E
1 2 B ,EB }
pS()φ(, E1B )φ(, E2B) d
(7)
where E1B and E2B are the energies of the up and left bond of the actual site. Afterward, the two bond energies of the actual unit cell (right and down bonds) are generated according to the conditional pdf:
pB(EB/ES) ) pB(EB)φ(ES, EB)
(8)
In two-dimensional model systems, topological effects (and ultimately eq 7) induce distortion in the generation of a prescribed energy landscape, so that the local pdf’s pB(EB) and pS(ES), used in the assignment procedure, may be different from the effective ones p/B(EB) and p/S(ES). The effective pdf’s p/B(EB) and p/S(ES) are the pdf’s which result as a consequence of the local assignment and are therefore the resulting statistical quantities characterizing the model energy structure. Two basic issues, namely, the direct and the inverse problem, arise naturally in efforts to understand and overcome the distortion effects. The direct problem can be stated as follows: Given the local pdf’s pB and pS, find how they are distorted in the assignment procedure; i.e., predict the effective pdf’s p/B and p/S. In the inverse problem, given the effective pdf’s p/B and p/S, the goal is to find the local ones pB and pS that should be used in order to generate the desired energy landscape with prescribed p/B and p/S. The solution of the inverse problem is of course particularly important in order to generate a prescribed energy landscape properly. For further use, it is advisable to define the overlapping degree I between site and bond energy distributions as
Probabilistic Analysis of the DSB Model
I ) 1/2[
Langmuir, Vol. 15, No. 18, 1999 5963
∫0∞ χB(E) dFS(E) + ∫0∞ χS (E) dFB(E)]
(9)
where χB(E) is the characteristic function of the set B, χB(E) ) 1 if (E) ∈ B and zero otherwise, and χB(E) is the characteristic function of S. For example, in the case of uniform site and bond energy pdf’s pS(ES) ) 1/(S2 - S1) ) 1/∆S and pB(EB) ) 1/(B2 - B1) ) 1/∆B the overlapping degree is I ) (B2 - S1)/(B2 - B1). 3. Solution of the Direct Problem This section analyzes the relationship between local and effective site and bond energy pdf’s. From the local assignment procedure described above and from the expressions for the site and bond conditional pdf’s, eqs 7-8, it readily follows that (1) The effective bond energy pdf p/B(EB) can be expressed as the average of the conditional pdf pB(EB/ES) with respect to the effective site energy distribution, i.e.,
p/B(EB) )
∫SS
pB(EB/)p/S() d )
2
1
pB(EB)
∫ES
2
B
φ(, EB)p/S() d (10)
where the lower bound S1 in the first integral is replaced by EB in accordance with the local exclusion principle, eq 2. From the substitution of the expression for the correlation function eq 4 into eq 10, it follows that
p/B(EB) ) g(EB)
∫ES
2
B
p/S() exp[-
∫E
B
g(η) dη] d ) Ψ ˆ B[p/S]
(11)
(2) The effective site energy pdf p/S(ES) can be expressed as the average of the conditional pdf pS(ES/E1B, E2B), eq 7, with respect to the effective bond energy joint pdf p/B(E1B, E2B), i.e.,
p/S(ES) )
∫BB ∫BB 2
2
1
1
pS(ES/ξ, η)p/B(ξ,η) dξ dη
(12)
We shall employ the approximation of regarding E1B and E2B as uncorrelated to one another, so that p/B(E1B,E2B) is simply given by
p/B(E1B, E2B) ) p/B(E1B)p/B(E2B )
(13)
and the effective site energy pdf p/S(ES) attains the form
p/S(ES) ) 2pS(ES)
∫B
B2 1
dξ
∫B
ξ 1
φ(ES,ξ)φ(ES,η)p/B(ξ)p/B(η)
dη
∫ξS
2
(14)
pS()φ(,η)φ(,ξ) d
From the substitution of the expression for the correlation function eq 4 into eq 14, it follows that
Figure 2. Effective site and bond energy pdf’s p/S(ES) and p/B(EB) as obtained from simulations on a 103 × 103 square lattice (broken lines) for uniform local site-bond energy pdf’s pS ) 1/(S2 - S1) and pB ) 1/(B2 - B1) compared with the theoretical predictions (continuous lines) obtained by solving eqs 11 and 15. ES and EB are expressed in arbitrary units. (A) I ) 0.65. (B) I ) 0.75.
From the substitution of the expression for p/S eq 15 into the integral equation for p/B eq 11 and the use of the explicit expression for the local site and bond energy pdf’s pS and pB and for the correlation function φ(ES, EB), eq 4, the solution of the direct problem implies the solution of the single functional (integral) equation
p/B ) Ψ ˆ B[Ψ ˆ S[p/B]] ) Ψ ˆ [p/B]
which can be solved, for example, by means of a relaxation algorithm such as
ˆ [p/B(n)] p/B(n + 1) ) Rp/B(n) + (1 - R) Ψ
2pS(ES)
∫B
p/B(ξ)F/B(ξ)
ES 1
∫ξS
2
pS()exp[-2
∫E
S
dξ ) g(x) dx] d Ψ ˆ S[p/B] (15)
A discussion of the limits of the validity of this approximation and one possible way to overcome them is given in section 5.
(17)
where R is a real parameter 0 < R < 1 controlling the convergence of the sequence {p/B(n)} toward p/B. To give a numerical example, Figure 2 shows the behavior of the effective site-bond energy pdf’s p/S and p/B as obtained from simulations on a 103 × 103 square lattice, starting from uniform local site-bond energy pdf’s pS ) 1/∆S and pB ) 1/∆B, for two different values of the overlapping degree I, and the comparison with the theoretical prediction obtained by solving eqs 11 and 15. The simulated 103 × 103 DSB energy landscapes were generated by implementing the simple and efficient numerical algorithm described in the Appendix. The algorithm makes use of the auxiliary functions ψ and θ, which are defined in the next section, to solve the inverse problem. The agreement between theory and simulations is good also for high values of the overlapping degree I > 0.5, and this is a consequence of the validity of the approximation, eq 13, over the entire range of I. Indeed, Figure 3 shows the behavior of the bond-bond normalized correlation coefficient CE1 ,E2 between energies of the up and left B B bonds (attached to the site of the elementary cell)
CE1 ,E2 ) B
p/S(ES) )
(16)
B
〈(E1B - 〈EB〉)(E2B - 〈EB〉)〉 〈(E1B - 〈EB〉)2〉
(18)
as numerically evaluated from simulated energy landscapes on a 103 × 103 square lattice starting from uniform site-bond energy pdf’s pS and pB for different values of the overlapping degree. As can be observed, the bondbond correlation coefficient CE1 ,E2 saturates to a constant B B value less than 5 × 10-2 as I increases toward 1. This result bears out the validity of regarding E1B and E2B as uncorrelated and, consequently, the validity of eq 13 in the solution of the direct problem.
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Adrover
To solve eq 23, it is convenient to subdivide the interval [B1, S2] into three subintervals: [B1, S1], [S1, B2], and [B2, S2]. For B1 e E e S1, because p/S(x) ) 0 for x < S1, the lower bound η of the integral appearing at the denominator of eq 23 can be replaced by S1. As a result, the function ψ(E) fulfils in this subinterval the simplified equation
1 ψ(E) ) exp K
[
K) Figure 3. Bond-bond normalized correlation coefficient CEB1,EB2, eq 18, as a function of the overlapping degree I, numerically evaluated from simulated energy landscapes on a 103 × 103 square lattice starting from uniform site-bond energy pdf’s pS(ES) and pB(EB).
4. Closed-Form Solution for the Inverse Problem Solving the inverse problem means finding the local pdf’s pS and pB to be used in the local assignment procedure in order to generate an energy landscape with effective (desired and prescribed) p/S and p/B. Analytically, this means solving eqs 11 and 15 with respect to the unknown functions pS and pB for prescribed p/S and p/B. To solve the inverse problem in closed form, let us introduce two auxiliary integral functions ψ(E, Ea) and θ(E, Ea), defined in E ∈[B1, S2]:
[
ψ(E, Ea) ) exp -
p (x)
∫EE F (x) B- F (x) dx a
B
S
exp[ -
∫E
S2
θ(E, Ea) )
]
)
∫E
E a
2
pS(x)ψ2(x, Ea) dx
(20)
(21)
θ′(η) dη B1 e E e B2 (22) ψ(η)
where A and B are two normalization constants and θ′ ) dθ/dE and ψ′ ) dψ/dE. This problem is addressed in the next two subsections by considering the solutions for ψ and θ separately, because the introduction of these two auxiliary functions makes it possible to decouple the solution of the inverse problem into two independent integral equations for ψ and θ. 4.1. Closed-Form Solution for ψ. Let us start from the integral equation for p/B, eq 11, and apply the definition of ψ(E), eq 19; it is easy to show that ψ(E) satisfies the following integral equation:
[∫∫
ψ(E) ) exp -
p/B(η)ψ(η)
E
S1
S2
η
which can be easily solved to obtain
ψ(E) )
p/S(ξ)ψ(ξ) dξ
dη
]
(23)
(24)
1
p/S(ξ)ψ(ξ) dξ ) const
[
]
F/B(S1) - F/B(E) 1 K ψ(S1)
-1
(25)
When the facts that limEfB1 ψ(E) f ∞ and FB/ (B1) ) 0 are accounted for it follows that K ) FB/ (S1)ψ(S1); henceforth,
ψ(E) ) ψ(S1)[F/B(S1)]/[F/B(E)] B1 e E e S1 (26) where ψ(S1) can be chosen arbitrarily, e.g., ψ(S1) ) 1. The singularity of ψ in B1 is consistent with the fact that the correlation function φ(ES, EB) can be expressed as a function of ψ as follows:
g(x) dx] (19)
pS(E) ) -Aθ′(E)/ψ2(E) S1 e E e S2
∫ES
2
1
φ(ES, EB) )
The function ψ satisfies the group property ψ(E, Ea) ) ψ(E, Eb)ψ(Eb, Ea). As discussed in the Appendix, each choice of the lower bound Ea in the interval (B1, S1] is equivalent in the generation procedure. Henceforth we shall assume Ea ) S1 and make use of the simplified notation ψ(E) and θ(E) instead of ψ(E, S1) and θ(E, S1). Solving the inverse problem in closed form is equivalent to finding an analytic expression for ψ(E) and θ(E) for E ∈ [B1, S2] because, by definition,
ψ′(E) pB(E) ) B 2 ψ (E)
∫SS
∫SE p/B(η)ψ(η) dη]
ψ(ES) 1 FB(EB) ψ(EB)
(27)
Because FB(B1) ) 0, the property limEfB1 ψ(E) f ∞ guarantees that the correlation function φ [and the joint pdf pSB(ES, EB)] is summable in the neighborhood of B1. Let us now consider the overlapping interval S1 e E e B2. By taking the logarithm of eq 23 and the derivative with respect to E, it follows that
d(log ψ(E)) ψ′(E) ) )dE ψ(E)
p/B(E)ψ(E)
(28)
∫ES p/S(ξ)ψ(ξ) dξ 2
A further derivative with respect to E gets rid of the integral at the denominator. In this way, we obtain a second-order nonlinear differential equation to be fulfilled by ψ in the overlapping interval:
-p/Bψψ′′ + [ψ′]2(2p/B - p/S) + [p/B]′ ψψ′ ) 0 (29) Equation 29 reduces to a first-order linear differential equation by means of the transformation ψ(E) ) exp ∫ E dx/y(x)
( )
y′ + 1 -
p/S
p/B
+
p/B′ y)0 p/B
(30)
the solution of which is readily obtained as
ψ(E) ) C exp
[
∫SE 1
p/B(ξ) D-
∫Sξ [p/B(η) - p/S(η)] dη 1
dξ
]
(31)
where C and D are two constants. The constant D can be determined by enforcing the condition that the functional
Probabilistic Analysis of the DSB Model
Langmuir, Vol. 15, No. 18, 1999 5965
expression for ψ, eq 31, should satisfy the integral equation, eq 23. As a result, we obtain D ) -FB/ (S1). The constant C is determined by enforcing the continuity of ψ at S1, i.e., C ) ψ(S1) ) 1, yielding
[∫
ψ(E) ) exp -
p/B(ξ)
E
S1
F/B(ξ) - F/S(ξ)
dξ
]
S1 e E e B2 (32)
It should be observed, by recalling the definition of ψ, eq 19, that in the interval [S1, B2]
[
ψ(E) ) -
p/ (ξ)
∫SE F (ξ) B- F (ξ) dξ 1
B
[∫
exp -
S
)
p/B(ξ)
E
S1
]
FB/ (ξ) - FS/(ξ)
]
dξ ) ψ*(E) (33)
(34)
where ψ(B2) is already known from the expression of ψ(E) in the overlapping range, eq 32, for E ) B2. Through collection of the three expressions eqs 26, 32, and 34, the functional form for ψ over the entire range [B1, S2] is obtained. To give an example, Figure 4 A shows the behavior of ψ(E) in the case of uniform site and bond energy effective pdf’s p/S ) 1/(S2 - S1) ) 1/∆S and p/B ) 1/(B2 - B1) ) 1/∆B with B1 ) 1, B2 ) 3, S1 ) 2, S2 ) 4, i.e., ∆S ) ∆B ) ∆ and I ) 0.5. In this case, FB/ (EB) ) (EB - B1)/∆ and FS/(ES) ) (ES - S1)/∆ and the application of eqs 26, 32, and 34 yields
ψ(E) ) (S1 - B1)/(E - B1)
(36)
S 1 e E e B2 ψ(E) ) const ) exp[- (B2 - S1)/(S2 - B2)] (37)
4.2. Closed-Form Solution for θ. By rewriting eq 15 for p/S in terms of ψ, we obtain
) 2pS(ES)ψ (ES)
∫B
p/B(ξ)FB/ (ξ)
ES 1
∫ξ
S2
2
θ(η)
dξ (38)
pS()ψ () dξ
By rearranging eq 38 and applying the definition of θ, eq 20, we obtain an integral equation for θ:
dξ
(39)
dη
θ(E) ) const B1 e E e S1
(40)
where the constant can be set to an arbitrary value, e.g., θ(E) ) 1. In the overlapping interval [S1, B2], by taking twice the derivatives of eq 39 with respect to E, it is easy to show that θ satisfies the second-order nonlinear differential equation
p/S θθ′′ - p/S′θθ′ - [(FB/ )2]′[θ′]2 ) 0
(41)
which reduces to a linear first-order equation in the auxiliary function y by means of the transformation θ(E) ) exp ∫E dx/y(x):
y′ +
[
(
)
[(FB/ )2]′ p/S′ y + -1 ) 0 p/S p/S
p/ (ξ)
∫SE [F / (ξ)]S2 - F /(ξ) dξ 1
B
S
]
(42)
S1 e E e B2 (43)
In the subinterval B2 e E e S2, because p/B(x) ) 0 for x > B2, the upper bound ξ in the integral appearing at the denominator of eq 39 can be replaced by B2, and therefore
θ(E) )
B2 e E e S 2
2
∫B
p/B(η)FB/ (η)θ(η)
ξ
To solve eq 39, let us subdivide the interval [B1, S2] into the following subintervals [B1, S1], [S1, B2], and [B2, S2], exactly as in the analysis of ψ. For B1 e E e S1, because p/S(x) ) 0 for x < S1, the lower bound E in the main integral appearing in eq 39 can be replaced by S1, and therefore
θ(E) ) exp -
ψ(E) ) exp[-(E - S1)/(S2 - B2)]
p/S(ξ)
S2
which can be solved to obtain
(35)
B 1 e E e S1
p/S(ES)
∫E
1 θ(E) ) 2
1
i.e., the function ψ undergoes no distortion effect due to the fulfillment of the local exclusion principle. Moreover, the function ψ is a smooth monotonically decreasing function of E in [S1, B2], because (FB/ (E) - FS/(E)) > 0 ∀E as a consequence of the first basic law of the DSB model, eq 1. For B2 e E e S2, because p/B(x) ) 0 for x > B2, the upper bound E in the main integral appearing in eq 23 can be replaced by B2, and therefore
ψ(E) ) const ) ψ(B2) B2 e E e S2
Figure 4. (A) ψ(E) and (B) θ(E) in the case of uniform effective site and bond energy pdf’s p/S ) 1/(S2 - S1) ) 1/∆S and p/B ) 1/(B2 - B1) ) 1/∆B with B1 ) 1, B2 ) 3, S1 ) 2, and S2 ) 4, i.e., ∆S ) ∆B ) ∆ and I ) 0.5.
1 2K
1 [1 - FS/(E)] ∫ES p/S(ξ) dξ ) 2K 2
(44)
where the constant K is determined by enforcing the continuity of θ at B2, i.e.,
[1 - FS/(E)] B2 e E e S2 θ(E) ) θ(B2) [1 - FS/(B2)]
(45)
where θ(B2) is known from the expression of θ in the overlapping interval, eq 43, for E ) B2, i.e., θ(B2) ) / / / 2 exp[- ∫B2 S1 [pS(ξ)]/[[FB(ξ)] - FS(ξ)] dξ].
5966 Langmuir, Vol. 15, No. 18, 1999
Adrover
To give an example, Figure 4B shows the behavior of θ(E) in the case of uniform site and bond energy effective pdf’s p/S ) 1/(S2 - S1) ) 1/∆S and p/B ) 1/(B2 - B1) ) 1/∆B with B1 ) 1, B2 ) 3, S1 ) 2, S2 ) 4, i.e., ∆S ) ∆B ) ∆ and I ) 0.5 (the same as those considered for ψ in subsection 4.1). By making use of eqs 40, 43, and 45, we obtain
θ(E) ) const ) 1
(46)
B1 e E e S1 θ(E) ) exp{-[h(E) - h(S1)]}
(47) Figure 5. Local site-bond energy pdf’s pB(EB) and pS(ES) evaluated from the closed-form expression for ψ(E), eqs 35-37 and θ(E), eqs 46-48 to achieve uniform effective site bond energy pdf’s p/B ) 1/∆B and p/S ) 1/∆B with ∆S ) ∆B for an overlapping degree I ) 0.5. The broken-bold lines are the histograms of p/B(EB) and p/S(ES) obtained from simulation on a 103 × 103 square lattice and show an excellent level of agreement with the expected constant behavior p/B ) 1/2 and p/S ) 1/2 (dotted lines).
S1 e E e B 2 h(E) )
[
]
-∆B2 + 2∆S(E - B1) 2∆B2 arctan R R
R ) x∆B2[-∆B2 + 4∆S(S1 - B1)] S2 - E θ(E) ) θ(B2) S2 - B2
(48)
B2 e E e S2 5. Effects of Correlations In this section, we analyze in greater detail the effects of correlations in the direct generation of DSB energy landscapes. Let us consider the expressions for pS(ES) and pB(EB) in terms of the two auxiliary quantities ψ and θ, eqs 21-22. In order that pS(ES) and pB(EB) should always be nonnegative, the functions ψ′(E) and θ′(E) should be nonpositive in the entire range of definition. This condition is always satisfied by ψ in [B1, S2], and by θ(E) in the two subintervals [B1, S1] and [B2, S2] for any prescribed FB/ and FS/. In the overlapping interval [S1, B2], θ(E) is a decreasing function of E if and only if
[FB/ ]2 - FS/ > 0
(49)
This condition places some constraints on the possibility of solving the inverse problem for any desired p/B(EB) and p/S(ES). To give an example, in the case of uniform effective site-bond energy pdf’s p/B ) 1/∆B and p/S ) 1/∆S, eq 49 implies that there exists a critical value of the overlapping degree Ic
Ic ) 1-(1/4)(∆B/∆S)
(50)
such that for I g Ic, it is not possible to obtain the desired uniform p/B(EB) and p/S(ES). This critical condition also becomes evident upon analysis of the denominator (i.e., the functional form for R) appearing in eq 47 for θ(E) in the overlapping range. To give an example of the application of the solution of the inverse problem for I < Ic, Figure 5 shows the behavior of the local site-bond energy pdf’s pB(EB) and pS(ES) as evaluated from the closed-form expressions for ψ(E) and θ(E) in order to obtain the uniform effective site-bond energy pdf’s p/B(EB) and p/S(ES) for an overlapping degree I ) 0.5. The broken-bold lines are the histograms of p/B(EB) and p/S(ES) obtained from simulation on a 103 × 103 square lattice and show an excellent level of agreement with the expected constant behavior p/B ) 1/ and p/ ) 1/ (in the case of ∆S ) ∆B ) 2). 2 2 S
Figure 6. Histograms of p/B(EB) and p/S(ES) as obtained from the simulation on a 103 × 103 square lattice. The simulation results deviate significantly from the expected behavior p/B ) 1/ and p/ ) 1/ (dotted lines). (A) I ) 0.6. (B) I ) 0.7. 2 2 S
As we draw closer to the critical value Ic ) 0.75 (for ∆S ) ∆B), however, the simulation results start to deviate significantly from the expected behavior. This phenomenon is depicted in Figure 6, showing the simulation results for p/B(EB) and p/S(ES) at higher values of the overlapping degree, I ) 0.6 and 0.7. The physical explanation for these deviations is related to the generation of correlations intrinsic to DSB assignment. Indeed, as we draw closer to the critical value Ic, the basic hypothesis that the energies E1B and E2B of the up and left bonds connected to the site in the elementary unit cell (see Figure 1a) can be regarded as uncorrelated variables, eq 13, fails, because the generation process intrinsically starts to induce correlations that should be taken into account in the definition of the joint probability density function p/B(E1B, E2B). A simple way to overcome this difficulty is to introduce correlations in the model, eq 13, which account for generation-induced correlation between the quantities E1B and E2B. Equation 13 should therefore be replaced by the more general expression
p/B(E1B, E1B) ) p/B(E1B) p/B(E2B)[1 + λµ(E1B )µ(E2B)]
(51)
where λ is a real positive parameter and µ(E) a generic function satisfying the following two conditions:
µ(x)µ(y) g -1/λ ∀x,y ∈[B1, B2]
∫BB
2
1
µ(x)p/B(x) dx ) 0
(52) (53)
Probabilistic Analysis of the DSB Model
Langmuir, Vol. 15, No. 18, 1999 5967
The last condition, eq 53, stems directly from the basic / / condition ∫B2 B1 pB(E, x) dx ) pB(E). By the use of eq 51 for the joint probability density function p/B( E1B, E2B), the integral equation for θ(E) (corresponding to eq 39 in the uncorrelated model) attains the form
θ(E) )
∫E
1 2
p/S(ξ)
S2
/ / ξ pB(η)FB(η)
∫B
+ λζ′(η)ζ(η)
θ(η)
1
dξ
(54)
dη
µ(x)p/B(x)
dx and ζ′(E) ) dζ(E)/dE. A closedwhere ζ ) form solution can be derived for θ(E) also in the more general correlated model: E ∫B1
θ(E) ) const ) 1
[∫
θ(E) ) exp -
(55)
B 1 e E e S1 p/S(ξ)
E
S1
[FB/ (ξ)]2 + λζ2(ξ) - FS/(ξ)
Figure 7. Ω(E, λ) vs E ∈[S1, B2] for different values of the bond-bond correlation coefficient λ in the correlated model with µ(E) ) a0 (E - Em) + a1 (E - Em)3 and a0 ) 5/2, and a1 ) 5(1/x3 - a0/3). The overlapping degree is I ) 0.75 (a) λ ) 0, (b) λ ) 0.1, (c) λ ) 0.3, (d) and λ ) 0.4.
dξ
]
(56)
S 1 e E e B2 θ(E) ) θ(B2)
[1 - FS/(E)] [1 - FS/(B2)]
(57) Figure 8. Ic vs λ in the correlated model with µ(E) ) a0 (E Em) + a1 (E - Em)3 (a0 ) 5/2 and a1 ) 5(1/x3 - a0/3)).
B 2 e E e S2 It should be observed that the introduction of correlations does not change the expression of p/B(EB). Consequently, the closed-form solution of ψ(E), valid also in the correlated case, is still given by eqs 26, 32, and 34. This comes as no surprise. Correlations in the generation of DSB energy landscapes are induced by the topology of the lattice, which plays a role (according to the chosen assignment process) exclusively in the generation of the site energies conditional to the values of the bond energies E1B and E2B depicted in Figure 1a. The introduction of correlations in the generation procedure through eq 51 modifies the functional form of θ(E) only in the overlapping interval [S1, B2]. From eq 56, it follows that the condition of monotonicity for θ(E) implies
[FB/ (E)]2
2
+ λζ (E) -
FS/(E)
) Ω(E, λ) > 0
(58)
Because λζ2(E) > 0, the introduction of correlations in the generation procedure moves the critical value of the overlapping degree Ic toward higher values of I (higher with respect to the uncorrelated case). This permits the accurate generation of energy landscapes with desired effective site-bond energy pdf’s for values of the overlapping degree at which the corresponding uncorrelated model would certainly fail. Let us give an application of the correlated model based on eq 51 in the case of uniform p/S(ES) and p/B(EB). A simple way of choosing µ(E) satisfying eq 53 is given by N
µ(E) )
ak(E - Em)2k+1 ∑ k)0
(59)
where Em ) (B1 + B2)/2 and the real coefficients ak can be chosen arbitrarily on the condition that eq 52 is satisfied. For example, it is advisable to choose the set of coefficients {ak} in such a way that the parameter λ would coincide
with the theoretical normalized correlation coefficient CEt 1 ,E2 B
B
CEt 1 ,E2 B B
)
∫BB
2
1
dE1B
∫BB
2
1
dE2B[(E1B - Em)(E2B - Em) p/B(E1B, E2B)]
∫BB (E1B - Em)2 p/B(E1B) dE1B 2
1
(60) thus obtaining the following condition for the set of coefficients {ak}: N
ak
∑ k)0 2k + 3
( ) ∆B 2
2(k+1)
)
( )
1 ∆B
x3
2
(61)
Figure 7 shows the behavior of the quantity Ω(E, λ) defined by eq 58 vs E ∈[B1, S2], for different values of λ in the case of uniform p/S ) 1/∆ and p/B ) 1/∆, for I ) 0.75 by choosing N ) 1, i.e., µ(E) ) a0 (E - Em) + a1 (E - Em)3 with a0 ) 5/2 and Q1 ) (1/x3 - Q0/3). For λ ) 0 there exists a value E of E such that Ω(E, 0) ) 0 because λ ) 0 corresponds to the uncorrelated model eq 13 for which I ) 0.75 is the critical value of the overlapping degree (for ∆S ) ∆B ) ∆). For positive values of λ, Ω(E, λ) is always greater than zero, thus implying that the higher the value of λ is, the higher the value of the critical overlapping degree Ic will be. Indeed, Figure 8 shows that the critical value of the overlapping degree Ic is a linear function of the correlation coefficient λ, which is not surprising because eq 58 is linear in λ. We can summarize the analysis of the correlated model as follows. The introduction of correlations in the generation of DSB energy landscapes through eq 51, i.e., through a function µ(E) satisfying eqs 52 and 53, makes it possible
5968 Langmuir, Vol. 15, No. 18, 1999
Figure 9. (a) λo vs I in the correlated model with µ(E) ) a0 (E - Em) + a1 (E - Em)3 (a0 ) 5/2 and a1 ) 5(1/x3 - a0/3)) in the case of uniform effective site and bond energy pdf’s. (b) Normalized nearest-neighbor site-site correlation coefficient CES1,ES2 vs I in the correlated model.
Adrover
as uncorrelated as possible (as shown in Figure 3). The degeneration of prescribed effective p/B(EB) and p/S(ES) is slightly different, because a fully uncorrelated local assignment, based on eq 13, may still introduce some distortions for high overlapping degrees. To overcome this effect, correlations should be introduced into the local generation procedure, eq 51, as discussed in section 5. The analysis developed in section 5 highlights the delicate balance between the generation of prescribed site and bond energy pdf’s and the intrinsic correlations introduced in the generation method: the prescribed pdf’s p/B(EB) and p/S(ES) can be recovered from the generation of a DSB energy landscape by means of local assignments based on the solution of the inverse problem only if the correlation coefficient λ entering into eq 51 matches the corresponding bond-bond correlation coefficient resulting from the generation of the lattice structure. This effect is intrinsically topological, and the methods developed in this article for 2-d square lattices can be readily extended to other dimensions and topologies. Finally, it should be mentioned that other methods for generating correlated DSB energy landscapes are available, based either on Monte Carlo thermalization10 or the definition of a global correlation function.13 Acknowledgment. The author wishes to thank M. Giona, G. Zgrablich, and J. L. Riccardo for discussions and invaluable suggestions.
Figure 10. Histograms of p/B(EB) and p/S(ES) obtained from simulation on a 103 × 103 square lattice by making use of the correlated model with µ(E) ) a0 (E - Em) + a1 (E - Em)3 (a0 ) 5/ , a ) 5(1/x3 - a /3)) for values of λ depicted in Figure 9. 2 1 0 o (A) I ) 0.65; (B) I ) 0.75.
to find a value λo(I) of the parameter λ for I < Ic(λ) permitting the generation of the desired p/S(ES) and p/B(EB). In particular, by choosing the coefficients {ak} in such a way that λ coincides with the theoretical bondbond correlation coefficient CEt 1 ,E1 (see eq 61), the value B B of λo can be found by matching it with the corresponding value of the bond-bond correlation coefficient resulting from the generation of the lattice. This can be achieved by means of a shooting procedure. Figure 9 shows the behavior of λo as a function of the overlapping degree I in order to obtain uniform effective site-bond energy pdf’s, and Figure 10 shows the good level of agreement between simulation results and the expected uniform behavior for p/B(EB) and p/S(ES) for values of the overlapping degree at which the corresponding uncorrelated model fails (see Figure 6 for the corresponding simulations with the uncorrelated model). It should be observed that the cases considered in Figure 6 correspond to significantly high values of the normalized correlation coefficient between nearest-neighbor sites CES1,ES2 depicted in Figure 9 (curve b). 6. Concluding Remarks A thorough probabilistic analysis of the direct generation of DSB energy landscapes has been developed. Closedform solutions have been found for the direct and the inverse problems (in the case of both uncorrelated and correlated assignments). In the case of the direct problem, the hypothesis of the absence of correlations, eq 13, is fully satisfactory. This is because the general methods (based on local assignments of site and bond energies) develop significant distortion of the effective pdf’s from local pdf’s in order to keep nearest-neighbor bond energies
Appendix This appendix describes a simple and efficient numerical algorithm for generating 2-D DSB energy landscapes. The algorithm is based on the definition of the auxiliary functions ψ(E) and θ(E). Given the local site and bond energy pdf’s pS(ES) and pB(EB) (as in the case of the direct problem), ψ(E) and θ(E) can be directly evaluated from their definitions, eqs 19 and 20. Conversely, if the desired effective site and bond energy pdf’s p/S and p/B are known (inverse problem), ψ(E) and θ(E) are given respectively by eqs 26, 32, and 34 and eqs 40, 43, and 44. It follows from eqs 4, 8, and 19 that the conditional pdf of bonds pB(EB/ES) can be expressed as a function solely of ψ(E), as follows:
pB(EB/ES) ) ψ(ES)ψ′(EB)/[ψ(EB)]2
(62)
Given that limEBfB1 ψ(EB) f ∞, it thus follows that the conditional distribution of bonds (from which bond energies must be sampled in the generation procedure) is given by
FB(EB/ES) )
∫BE
p
1
pB(ξ/ES) dξ )
ψ(ES) ψ(EB)
(63)
The function ψ(E) is monotonically decreasing in the interval [B1, B2] and therefore admits an inverse ψ-1 in this interval. The value of a bond energy in the generation procedure can therefore be simply obtained as follows:
EB ) ψ-1[ψ(ES)/rnd]
(64)
where rnd is an uncorrelated random number with uniform distribution in [0, 1]. Analogously, it is easy to show that the conditional pdf for sites pS( ES/E1B, E2B) is given by (13) Adrover, A.; Giona, M.; Giustiniani, M. Langmuir 1996, 12, 4272.
Probabilistic Analysis of the DSB Model
pS( ES/E1B, E2B ) )
pS(ES)[ψ(ES)]2
∫ES
2
max
pS(η)[ψ(η)]2 dη
Langmuir, Vol. 15, No. 18, 1999 5969
relation
(65)
where Emax ) max{ E1B, E2B, S1 }. The conditional distribution of site energies FS( ES/E1B, E2B) (from which site energies must be sampled in the generation procedure) therefore attains the form
FS( ES/E1B, E2B ) )
∫EE
S
max
pS(η/E1B, E2B ) dη ) 1 - θ(ES)/θ(Emax) (66)
Because θ(E) is monotonically decreasing in the interval [S1, S2], its inverse θ-1 exists, and the value of a site energy in the generation procedure can be computed from the
ES ) θ-1[θ(Emax)(1 - rnd)]
(67)
This algorithm does not require any numerical integration and guarantees great accuracy also in all of those cases in which ψ(E) and θ(E) exhibit very stiff decay. To give a computational benchmark of the efficiency of the implementation of eqs 64 and 67; the generation of an energy landscape on a 103 × 103 square lattice requires less than 3 min, on a 155 MHz Pentium computer. It follows from eqs 63 and 66 that the conditional distribution functions are homogeneous functions of the order zero in ψ and θ. ψ and θ can therefore be defined modulo an arbitrary multiplicative factor. LA981321F
