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Distribution of adsorption energy and the effect of surface heterogeneity on the vibrational motion of adsorption
Surface Science 218 (1989) L461-L466 North-Holland. Amsterdam
SURFACE
SCIENCE
L461
LETTERS
DISTXIBUTION OF ADSORPTION ENERGY AND THE EFFECT OF SURFACE HETEROGENEITY ON THE VIBRATIONAL MOTION OF ADSORPTION Joaquin CORTES Fact&ad de Ciencias Fisicas y Matemciticas,
Uniuersidad de Chile, Casilla 2777, Santiago,
Chile
Received 30 January 1989; accepted for publication 6 April 1989
The calculation of the energy distribution for argon adsorbed on the 100 face of sodium chloride is made from the surface potential energy of adsorption. By a similar procedure the distribution of vibrational frequencies normal to the surface is determined to find the effect of the heterogeneity on the vibrational energy and entropy of adsorption.
1. Introduction One of the most interesting challenges of the physical chemistry of surfaces, which has been the source of a great number of publications during the last 30 years, is the study of adsorption on heterogeneous surfaces. In recent publications [1,2] we have proposed a method for calculating the distribution of the adsorption energies of a gas on the surface of a heterogeneous solid. By a similar procedure it is possible to determine the distribution of vibrational frequencies normal to the surface, and thereby find the effect of the heterogeneity on the vibrational energy and entropy of adsorption. This determination bears a formal analogy to the well known problem of the distribution of the characteristic vibrations of a crystal.
2. Surface heterogeneity The homotattic patch approximation [3] states that the overall adsorption isotherm on a heterogeneous surface in terms of adsorbent surface coverage B(P, T) is given by the equation
e(P, T) = jE24(p, T, E,)f(E,) dE,, El
0)
where 0,( P, T, E,) is the local adsorption isotherm, f( E,) d E, is the fraction of the surface energy of the solid between E, and E, + d E,, and E, and E, are the respective limits of E,. 0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Within the restrictions imposed by the model, the dist~buti~n function f( E,) is, therefore, a measure of the surface heterogeneity. A number of a priori expressions for f( E,) have been proposed in the literature, such as Gaussian [3-51 or M~well-Boltzmann functions 161, log-normal distribution [7], beta [S] or gamma [9] functions, etc., the most interesting of which has undoubtedly been the first kind. In a recent paper [l] we have proposed a method for calculating f(.&) in the case of intrinsic heterogeneity [TO], that is a natural consequence of the periodicity of the adsorption potential energy field affecting the substrate, and it can alsobe used in some cases of residual heterogeneity due to factors that include surface impurities and structural defects such as different exposed crystalline faces. The procedure consists in using a semiempirical numerical method to calculate the adsorption potential as a sum of interaction between the ad-
(aI
1.8
2.4
3.2
E (Kcailmole)
Fig. 1. (a) Adsorption energy distribution function for argon. Solid line: sodium chloride, dashed line: potassium chloride from ref. [2]. (b) Distribution of the frequencies normal to the surface For argon. Solid line: sodium chloride, dashed line: potassium chloride system.
J. Comes / Enerp distriburion landeffect ofsurface heterogeneity
L&3
sorbate molecule and each atom of the solid. If this is done for every point (x, u) of the surface, one obtains a matrix of discrete equilibrium values of E,(x, y) that correspond to the minimum energy in the z direction perpendicular to the surface at each point. The distribution curve is then obtained by counting the number of superficial sites having energies that fall within each interval AE. As an illustration of this method, fig. la shows the distribution of surface energies for the adsorption of argon on sodium chloride. The energy E, (x, y, z ) of site i was calculated by the additivity-by-pairs approximation over the j atoms of the solid by means of the expression [1,2] i
i
I
where the constants C!‘) and C$,?) can be expressed as functions of the polarizability and susce&biiity of atoms i and j by the equations of Kirkwood and Muller [11,12]. The repulsion constant Bij was calculated from the condition of ~~rnurn adsorption energy at the equilibrium distance, which is assumed to be equal to the sum of the van der Waals radii of atoms i and j, The potential’s electrostatic component, E,, which in this type of system is less than 5% of the total, was determined in the usual way [13]. The summations were carried out for 256 atoms of the solid in each case, giving a sufficient approximation for the purposes of the problem. The value of E,(x, y, z) was determined for ldgl superficial sites located uniformly in a unit cell, taking AZ intervals of 0.05 w for the deter~nation of E,(x, u) while for the distribution curve the AE intervals were 0.05 kcal/mol,
3. Heterogeneity
of the vibrational motion
Fig. lb shows the distribution of the frequencies, P=, normal to the surface for argon over NaCl and KC1 systems, whose surface energy distributions appear in fig. la. The vibrational energy and entropy distributions for the same systems are shown in fig. 2. These determinations are similar to the method given in the previous section. The frequency, vzI is calculated for each site (x, y) assuming a parabolic approximation about the minimum E,(x, y) for which the following equation is thought to be valid: Ei(X,
..)‘I Z) - Ea(x> Y) = $f(Z - r,)**
The determination
(3)
of the constant f allows vz to be calculated from
V*=t&7Z where m is the mass of the adsorbate unidimensianal simple harmonic motion.
(4) molecule,
as an approximation
of
J. Cartes / Energy distribution and effect of surfaceheterogeneity
L464
161
E, (coilmole
o.l30.1 2 g 0.07zr L 0.04O.Ol2.2
2.6
(u.e.1 for argon. Solid line: sodium S,
Fig. 2. (a) Vibrational energy distribution potassium
chloride
system. (b) Vibrational entropy distribution chloride, dashed line: potassium chloride
chloride, for argon. Solid system.
dashed line: line: sodium
In the case of mobile adsorption, the vibrational energy, E,, and entropy, S,, can be calculated from V, by the well known relations [14] E=
s=
Nh v, eJ4kT
(e
- 1
+ +Nhv,,
Nhv, -R hv,/kr - 1) T
In(1 - e-h”=/kr),
where h is Planck’s constant, k is Boltzmamr’s constant, N is Avogadro’s number and I” is the temperature. Eqs. (4)-(6) allow vr, E, and S, to be calculated for each site. The d~te~nation of the corresponding dist~butions is done in a manner similar to that used in the previous section for the adsorption energy, counting the number of superficial sites having values of E,
J. Cartes / Energy distribution
and effect of surface heterogeneity
L465
and S, that fall within the intervals E, and S, required by the determination of the distributions.
4. Conclusions It is curious that the intrinsic heterogeneity is normally described as being caused by the periodicity of the adsorption potential energy field, but the energy distribution curve is not deduced from it as a natural consequence. That was the purpose of the method we proposed recently, of which a new example is given here in fig. la. Taking into account that, in the literature, the distribution function has generally been postulated a priori and arbitrarily, these first direct determinations of distributions in the case of specific heterogeneity are interesting, particularly in those systems that have a low residual heterogeneity [lo]. The basic of the method also makes it possible to study the effect of surface heterogeneity on the vibrational adsorption motion, which in this paper has been illustrated for cases of mobile physisorption. Some analogy between the distribution of vz and the characteristic vibrations of a crystal [15-171 should be noted. The extreme case of a homogeneous surface, that corresponds to the usual approximation of considering uz constant, can be related to Einstein’s model [18] of a solid crystal, while Debye’s model [19] of an approximation to an elastic continuum reminds us of the existence of a maximum value of vz on the solid’s surface. Based on these classical formulations, a large number of publications have dealt with various methods related to the frequency distribution in crystals, of which we may mention the two-peak graphs corresponding to our solids KC1 [20] and NaCl [21], but naturally for a frequency range greater than those of fig. lb, since the interionic energies of these crystals have nothing to do with the weak interactions involved in the physisorption of argon on the surfaces of these solids. However, the analogy may suggest the application of these studies in the analysis of surface heterogeneity. In the examples given in this paper, however, it should be noted that the heterogeneous effect on the vibrational motion of adsorption is small, with a greater relative incidence on the entropy than on the vibrational energy. Among other things, this provides a criterion for the approximation of vz as a constant in statistical mechanics calculations of adsorption.
Acknowledgment The author would like to thank FONDECYT Chile) for the financial support of this work.
and DTI (Universidad
de
IA66
J. Cartes / Energy distribution and effect of surface heterogeneity
References [l] (21 [3] [4] [5] [6]
J. Cortts, J. Chem. Phys. 88 (1988) 8011. J. CortCs, J. Chem. Phys., in press. S. Ross and J.P. Ohvier, J. Phys. Chem. 65 (1961) 608. W.A. Steele, Surface Sci. 36 (1973) 317. J. Hove and J.A. Krumhansl, Phys. Rev. 92 (1953) 569. B. Kind], R.A. Pachovsky, B.A. Spencer and B.W. Wojciechowski, J. Chem. Sot. Faraday Trans. I, 69 (1973) 1162. [7] S.E. Hoory and J.M. Prausnitz, Surface Sci. 6 (1967) 377. [S] A.L. Myers and D.Y. Ou, Annual Meeting Am. Inst. Chem. Eng., USA, 1981. [9] S. Sircar, J. Chem. Sot. Faraday Trans. I, 80 (1984) 1101. [lo] W.A. House and M.J. Jaycock, Proc. Roy. Sot. (London) A 348 (1976) 317. [ll] J.G. Kirkwood, Phys. 2. 33 (1932) 57. 1121 H.R. Muller, Proc. Roy. Sot. (London) A 154 (1936) 624. [13] W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). 1141 S. Ross and J.P. Olivier, On Physical Adsorption (Interscience, New York, 1964). [15] M. Blackman, Proc. Roy. Sot. (London) A 159 (1937) 416. [16] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954). (17) R.F. Wallis, Ed., Lattice Dynamics (Pergamon Press, New York, 1965). [18] A. Einstein, Ann. Phys. (Leipzig) 22 (1907) 180. [19] P. Debye, Ann. Phys. (Leipzig) 39 (1912) 789. [20] M. Iona, Jr, Phys. Rev. 60 (1941) 822. [21] R.H. Lyddane and K.F. Herzfeld, Phys. Rev. 54 (1938) 846.
SURFACE
SCIENCE
L461
LETTERS
DISTXIBUTION OF ADSORPTION ENERGY AND THE EFFECT OF SURFACE HETEROGENEITY ON THE VIBRATIONAL MOTION OF ADSORPTION Joaquin CORTES Fact&ad de Ciencias Fisicas y Matemciticas,
Uniuersidad de Chile, Casilla 2777, Santiago,
Chile
Received 30 January 1989; accepted for publication 6 April 1989
The calculation of the energy distribution for argon adsorbed on the 100 face of sodium chloride is made from the surface potential energy of adsorption. By a similar procedure the distribution of vibrational frequencies normal to the surface is determined to find the effect of the heterogeneity on the vibrational energy and entropy of adsorption.
1. Introduction One of the most interesting challenges of the physical chemistry of surfaces, which has been the source of a great number of publications during the last 30 years, is the study of adsorption on heterogeneous surfaces. In recent publications [1,2] we have proposed a method for calculating the distribution of the adsorption energies of a gas on the surface of a heterogeneous solid. By a similar procedure it is possible to determine the distribution of vibrational frequencies normal to the surface, and thereby find the effect of the heterogeneity on the vibrational energy and entropy of adsorption. This determination bears a formal analogy to the well known problem of the distribution of the characteristic vibrations of a crystal.
2. Surface heterogeneity The homotattic patch approximation [3] states that the overall adsorption isotherm on a heterogeneous surface in terms of adsorbent surface coverage B(P, T) is given by the equation
e(P, T) = jE24(p, T, E,)f(E,) dE,, El
0)
where 0,( P, T, E,) is the local adsorption isotherm, f( E,) d E, is the fraction of the surface energy of the solid between E, and E, + d E,, and E, and E, are the respective limits of E,. 0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Within the restrictions imposed by the model, the dist~buti~n function f( E,) is, therefore, a measure of the surface heterogeneity. A number of a priori expressions for f( E,) have been proposed in the literature, such as Gaussian [3-51 or M~well-Boltzmann functions 161, log-normal distribution [7], beta [S] or gamma [9] functions, etc., the most interesting of which has undoubtedly been the first kind. In a recent paper [l] we have proposed a method for calculating f(.&) in the case of intrinsic heterogeneity [TO], that is a natural consequence of the periodicity of the adsorption potential energy field affecting the substrate, and it can alsobe used in some cases of residual heterogeneity due to factors that include surface impurities and structural defects such as different exposed crystalline faces. The procedure consists in using a semiempirical numerical method to calculate the adsorption potential as a sum of interaction between the ad-
(aI
1.8
2.4
3.2
E (Kcailmole)
Fig. 1. (a) Adsorption energy distribution function for argon. Solid line: sodium chloride, dashed line: potassium chloride from ref. [2]. (b) Distribution of the frequencies normal to the surface For argon. Solid line: sodium chloride, dashed line: potassium chloride system.
J. Comes / Enerp distriburion landeffect ofsurface heterogeneity
L&3
sorbate molecule and each atom of the solid. If this is done for every point (x, u) of the surface, one obtains a matrix of discrete equilibrium values of E,(x, y) that correspond to the minimum energy in the z direction perpendicular to the surface at each point. The distribution curve is then obtained by counting the number of superficial sites having energies that fall within each interval AE. As an illustration of this method, fig. la shows the distribution of surface energies for the adsorption of argon on sodium chloride. The energy E, (x, y, z ) of site i was calculated by the additivity-by-pairs approximation over the j atoms of the solid by means of the expression [1,2] i
i
I
where the constants C!‘) and C$,?) can be expressed as functions of the polarizability and susce&biiity of atoms i and j by the equations of Kirkwood and Muller [11,12]. The repulsion constant Bij was calculated from the condition of ~~rnurn adsorption energy at the equilibrium distance, which is assumed to be equal to the sum of the van der Waals radii of atoms i and j, The potential’s electrostatic component, E,, which in this type of system is less than 5% of the total, was determined in the usual way [13]. The summations were carried out for 256 atoms of the solid in each case, giving a sufficient approximation for the purposes of the problem. The value of E,(x, y, z) was determined for ldgl superficial sites located uniformly in a unit cell, taking AZ intervals of 0.05 w for the deter~nation of E,(x, u) while for the distribution curve the AE intervals were 0.05 kcal/mol,
3. Heterogeneity
of the vibrational motion
Fig. lb shows the distribution of the frequencies, P=, normal to the surface for argon over NaCl and KC1 systems, whose surface energy distributions appear in fig. la. The vibrational energy and entropy distributions for the same systems are shown in fig. 2. These determinations are similar to the method given in the previous section. The frequency, vzI is calculated for each site (x, y) assuming a parabolic approximation about the minimum E,(x, y) for which the following equation is thought to be valid: Ei(X,
..)‘I Z) - Ea(x> Y) = $f(Z - r,)**
The determination
(3)
of the constant f allows vz to be calculated from
V*=t&7Z where m is the mass of the adsorbate unidimensianal simple harmonic motion.
(4) molecule,
as an approximation
of
J. Cartes / Energy distribution and effect of surfaceheterogeneity
L464
161
E, (coilmole
o.l30.1 2 g 0.07zr L 0.04O.Ol2.2
2.6
(u.e.1 for argon. Solid line: sodium S,
Fig. 2. (a) Vibrational energy distribution potassium
chloride
system. (b) Vibrational entropy distribution chloride, dashed line: potassium chloride
chloride, for argon. Solid system.
dashed line: line: sodium
In the case of mobile adsorption, the vibrational energy, E,, and entropy, S,, can be calculated from V, by the well known relations [14] E=
s=
Nh v, eJ4kT
(e
- 1
+ +Nhv,,
Nhv, -R hv,/kr - 1) T
In(1 - e-h”=/kr),
where h is Planck’s constant, k is Boltzmamr’s constant, N is Avogadro’s number and I” is the temperature. Eqs. (4)-(6) allow vr, E, and S, to be calculated for each site. The d~te~nation of the corresponding dist~butions is done in a manner similar to that used in the previous section for the adsorption energy, counting the number of superficial sites having values of E,
J. Cartes / Energy distribution
and effect of surface heterogeneity
L465
and S, that fall within the intervals E, and S, required by the determination of the distributions.
4. Conclusions It is curious that the intrinsic heterogeneity is normally described as being caused by the periodicity of the adsorption potential energy field, but the energy distribution curve is not deduced from it as a natural consequence. That was the purpose of the method we proposed recently, of which a new example is given here in fig. la. Taking into account that, in the literature, the distribution function has generally been postulated a priori and arbitrarily, these first direct determinations of distributions in the case of specific heterogeneity are interesting, particularly in those systems that have a low residual heterogeneity [lo]. The basic of the method also makes it possible to study the effect of surface heterogeneity on the vibrational adsorption motion, which in this paper has been illustrated for cases of mobile physisorption. Some analogy between the distribution of vz and the characteristic vibrations of a crystal [15-171 should be noted. The extreme case of a homogeneous surface, that corresponds to the usual approximation of considering uz constant, can be related to Einstein’s model [18] of a solid crystal, while Debye’s model [19] of an approximation to an elastic continuum reminds us of the existence of a maximum value of vz on the solid’s surface. Based on these classical formulations, a large number of publications have dealt with various methods related to the frequency distribution in crystals, of which we may mention the two-peak graphs corresponding to our solids KC1 [20] and NaCl [21], but naturally for a frequency range greater than those of fig. lb, since the interionic energies of these crystals have nothing to do with the weak interactions involved in the physisorption of argon on the surfaces of these solids. However, the analogy may suggest the application of these studies in the analysis of surface heterogeneity. In the examples given in this paper, however, it should be noted that the heterogeneous effect on the vibrational motion of adsorption is small, with a greater relative incidence on the entropy than on the vibrational energy. Among other things, this provides a criterion for the approximation of vz as a constant in statistical mechanics calculations of adsorption.
Acknowledgment The author would like to thank FONDECYT Chile) for the financial support of this work.
and DTI (Universidad
de
IA66
J. Cartes / Energy distribution and effect of surface heterogeneity
References [l] (21 [3] [4] [5] [6]
J. Cortts, J. Chem. Phys. 88 (1988) 8011. J. CortCs, J. Chem. Phys., in press. S. Ross and J.P. Ohvier, J. Phys. Chem. 65 (1961) 608. W.A. Steele, Surface Sci. 36 (1973) 317. J. Hove and J.A. Krumhansl, Phys. Rev. 92 (1953) 569. B. Kind], R.A. Pachovsky, B.A. Spencer and B.W. Wojciechowski, J. Chem. Sot. Faraday Trans. I, 69 (1973) 1162. [7] S.E. Hoory and J.M. Prausnitz, Surface Sci. 6 (1967) 377. [S] A.L. Myers and D.Y. Ou, Annual Meeting Am. Inst. Chem. Eng., USA, 1981. [9] S. Sircar, J. Chem. Sot. Faraday Trans. I, 80 (1984) 1101. [lo] W.A. House and M.J. Jaycock, Proc. Roy. Sot. (London) A 348 (1976) 317. [ll] J.G. Kirkwood, Phys. 2. 33 (1932) 57. 1121 H.R. Muller, Proc. Roy. Sot. (London) A 154 (1936) 624. [13] W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). 1141 S. Ross and J.P. Olivier, On Physical Adsorption (Interscience, New York, 1964). [15] M. Blackman, Proc. Roy. Sot. (London) A 159 (1937) 416. [16] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, 1954). (17) R.F. Wallis, Ed., Lattice Dynamics (Pergamon Press, New York, 1965). [18] A. Einstein, Ann. Phys. (Leipzig) 22 (1907) 180. [19] P. Debye, Ann. Phys. (Leipzig) 39 (1912) 789. [20] M. Iona, Jr, Phys. Rev. 60 (1941) 822. [21] R.H. Lyddane and K.F. Herzfeld, Phys. Rev. 54 (1938) 846.