Estimation of the Surface Energetic Heterogeneity of a Solid by

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Estimation of the Surface Energetic Heterogeneity of a Solid by Inverse Gas Chromatography† H. Balard‡ Institut de Chimie des Surfaces et Interfaces, CNRS, B.P. 2488, F-68057 Mulhouse, France Received December 11, 1995. In Final Form: December 6, 1996X The determination of the surface heterogeneity of a solid is a difficult challenge. The present work proposes a method for the determination of the adsorption energy distribution functions (DF) directly computed from the chromatographic signal acquired at finite concentration conditions. The calculation of the DF calls on the extended Rudzinski and Jagiełło’s solution method using discrete Fourier transforms to perform the multiple derivation of the distribution function for the condensation approximation. Because of the “ill-posed” character of the fundamental integral equation describing the adsorption on a heterogeneous surface, the good robustness of the method versus the experimental noise and the regularity of the data sampling was evidenced using a simulation program.

I. Introduction Atomic structures present on a solid surface are rigidly connected, and this rigidity is responsible for the formation of surface defects like cracks, corners, steps, and porous structures. Moreover, if we consider the eventual presence of impurities, the surface chemical composition and morphology may strongly change when moving from one surface point to another. Consequently, the surface energy concept that is easily defined and measurable in the case of pure liquids as the reversible work, per area unit, needed to extend its surface, does not hold for a heterogeneous real solid surface. The surface heterogeneity can be seen by looking at the variation, with the coverage degree (θ), of the heats of adsorption ∆Ha of an adsorbate, calculated from two adsorption isotherms determined at close temperatures.1 Generally, ∆Ha decreases most steeply as θ increases indicating the presence of active adsorption sites that are occupied by the first adsorbing molecules. A second proof of surface heterogeneity is also given by the deviation observed, at low partial pressures, between the BET model and the actual isotherm. These observations suggest that information on the surface heterogeneity may be extracted from the analysis of the isotherm shape. Rudzinski and Everett2 and Jaroniec and Madey3 have published extensive reviews on the determination of surface heterogeneity from adsorption measurements. After having reviewed briefly the main methods that are proposed in the literature, we shall describe a method for the estimation of the energetic surface heterogeneity of a solid based, on the one hand, on the acquisition of the experimental data by IGC at finite concentration conditions and, on the other hand, on a method of calculation of the adsorption energy distribution functions combining Rudzinski and Jagiełło’s solution method4 and discrete Fourier transforms to perform the multiple derivation of

the distribution function for the condensation approximation (DFCA). Finally, using a simulation program, we shall test the robustness of the proposed method versus the experimental noise and the regularity of the data sampling that also play a major role on the instability of the solution of the adsorption integral equation. 2. Physical Model and Methods of Resolution of the Adsorption Integral Equation All approaches described in the literature are based on a physical model that supposes that an energetically heterogeneous surface, with a continuous distribution of adsorption energies, may be described, in the simplest way, as a superposition of a series of homogeneous adsorption patches. Hence, the amount of adsorbed molecules (probes) is given by the following integral equation

N(Pm,Tm) ) N0

∫

max

min

θ(,Pm,Tm) χ() d

(1)

† Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. ‡ E-mail: [email protected]. Tel: 33 03 89 60 88 00. Fax: 33 03 89 60 87 99. X Abstract published in Advance ACS Abstracts, February 15, 1997.

where N(Pm,Tm) is the number of molecules adsorbed at the pressure Pm and temperature Tm of measurement, N0 is the number of molecules needed for the formation of a monolayer, θ(,Pm,Tm) is the local isotherm,  the adsorption energy of a site, and χ() is the distribution function (DF) of the sites seen by the probe. The range of adsorption energies is included between minimal (min) and maximal (max) values. From a mathematical point of view, solving eq 1 is not a trivial task because it has no general solution.5 It is an “ill-posed” problem, i.e., a small variation in the experimental data will cause a large variation in the site energy distribution function. Hence the existence, the uniqueness, and the stability of the solution are not generally assured. Testing the robustness of the chosen method of resolution versus experimental noise, using a simulation, is therefore of great importance. Various methods have been proposed and are classified in four main categories. The first category ascribes a given analytical form to the distribution function χ(): a gaussian function,6,7 a gamma function,8 or a combination of two Langmuir

(1) Kiselev, A. V.; Yashin, Y. I. La Chromatographie Gaz-Solide; Masson et Cie: Paris, 1969. (2) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (3) Jaroniec, M.; Madey, R. Physical Adsorption on heterogeneous Solids; Elsevier: Amsterdam, 1988. (4) Rudzinski, W.; Jagiełło, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478.

(5) Lum Wan, J. A.; White, L. R. J. Chem. Soc., Faraday Trans. 1991, 87 (18), 3051. (6) Ross, S. Adsorpt. Technol. 1971, 67, 1. (7) House, W. A.; Jaycock, M. J. J. Colloid Interface Sci. 1974, 47, 50. (8) Gilpin, R. K.; Jaroniec, M.; Martin-Hopkins, M. B. J. Chromatogr. 1990, 513, 1.

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isotherm discrete distributions.9 But, according to these approaches, one assumes a given chemistry and morphology of the solid surface. Therefore, many authors tried to develop solution methods without making any assumption on the shape of the distribution function. The second category supposes a discrete distribution of monoenergetic sites,10 based on a discretization form of eq 1. Considering a number n of measurements, one ends up with a linear system of n equations. Despite the apparent simplicity of this method, the inversion of the characteristic matrix is not a simple task. The instability of the solution increases quickly with increasing values of n and origins partially from border effects due to the data acquisition in a finite window of experimental pressures. To control this natural instability, many authors propose to use a regularization parameter like Szombathely et al.10 and Jagiełło,11 but the calculated distribution function is strongly dependent on the value of this parameter. The choice of a regularization parameter value entails a subjective character to the method. The third category includes the methods of resolution considering local isotherm approximations that were exhaustively reviewed by Nederlof et al.12 The oldest and the simplest approximation of the local isotherm is the condensation approximation.13 The condensation approximation supposes that the sites of adsorption of given energy are unoccupied below a characteristic pressure and entirely occupied above it. The distribution function for the condensation approximation (DFCA) is then directly related to the first derivative of the isotherm corrected for the multilayer adsorption, according to eq 2

χCA() )

(

)

1 P′ ∂N′(P′,T) RT N0 ∂P′

(2)

where N′ and P′ are respectively the amount of adsorbed probe and the pressure of the probe corrected for the multilayer adsorption (see below eq 9), N0 is the amount of adsorbate corresponding to the monolayer, R is the universal gas constant for an ideal gas, and T is the absolute temperature at which the measurement is performed. This approximation is all the better as the temperature of measurement approaches the absolute zero. In the conditions usually retained for IGC measurements, at room temperature and above it, this approximation fails completely and it is necessary to use other approximated forms of the local isotherm. Among them, for Langmuir or FFG local isotherms, the extended Rudzinski-Jagiełło method15 allows the computation of the actual distribution function (DFRJ) from a limited development of the even derivatives of the DFCA, according to eq 3

RT(2j)b2j χ(2j) ∑ CA () j)0 +∞

χ() )

(3)

with b0 ) 1 and b2j ) (-1)j π2j/(2j + 1)! This approach is based on the fact that the product of the first logarithmic derivative, ∂θ/∂ ln(P), of the local (9) Roles, J.; Guiochon, G. J. Phys. Chem. 1991, 95 (10), 4098. (10) Szombathely, M. V.; Brauer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13 (1), 17. (11) Jagiełło, J. Langmuir 1994, 10, 2778. (12) Nederlof, M. N.; Riemsdjik, W. H.; Koopal, K. J. Colloid Interface Sci. 1990, 135 (2), 410. (13) Roginski, S. Z. C. R. Acad. Sci. USSR 1944, 45, 194. (14) Jagiełło, J.; Ligner, G.; Papirer, E. J. Colloids Interface Sci. 1990, 137 (1), 128. (15) Jagiełło, J.; Schwarz, J. A. J. Colloid Interface Sci. 1991, 146 (2), 415.

Figure 1. Simulation of the convolution product of a biGaussian theoretical distribution function by the logarithmic derivative of a Langmuir local isotherm, centered around 37.5 kJ/mol.

isotherm, by the distribution function χ() tends rapidly to zero around the characteristic energy m corresponding to the pressure of measurement Pm, because of the belllike shape of this derivative (Figure 1). The main difficulty of the methods, based on the local isotherms approximation, stems from the necessity to perform multiderivation of the experimental DFCA, without amplifying the experimental noise. On the other hand, its main advantage is to provide a fine control of the calculation process by the comparison of DFRJ of increasing order. To perform the multiple derivations, Jagiełło et al.15 have used a virial expansion in order to fit the experimental isotherms. Some examples of distribution functions calculated according to this method are reported in the literature.2,14,16 The high flexibility of this fitting equation, using a limited number of parameters, brings in the well-known default of polynomial fittings, i.e., adding residual oscillations to the experimental data that are greatly amplified by the multiple derivation procedure leading to artifacts.17 Finally, a fourth category of solution methods of eq 1 proposed by Rudzinski and Wojciechowski18 or, more recently, by Lum Vam and White5 calls on Fourier transforms. According to the Lum Wam and White approach, eq 1 can be considered as a convolution product, a type of equation currently used in the domain of instrumentation relating the output signal (the adsorption isotherm), the input signal (the wanted distribution function), and the transfer function of the instrument (the local isotherm). Then with the application of Plancherel’s theorem, the Fourier transform of the distribution function is directly related to the ratio of the Fourier transforms of the experimental isotherm and the Fourier transform of the local isotherm. As do the preceding methods, this latter method requires, firstly, the first-order derivation of the isotherm and, secondly, the extrapolation of the experimental data, applying Biraud’s method, to prevent undesirable oscillations due to the limited pressure acquisition window (border effects). The authors emphasize also the particular sensitivity of this method to the regularity of the sampling rate of the experimental data in the adsorption energy space. Therefore, the robustness of the chosen method of resolution has to be tested considering this last factor. (16) Papirer, E.; Balard, H.; Jagiełło, J.; Baeza, R.; Clauss, F. In Chemically Modified Surfaces; Mottola, H. A., Steinmetz, J. R., Eds.; Elsevier: 1992. (17) Cortes, J.; Araya, P. An. Quim. 1989, 85, 33. (18) Rudzinski, W.; Wojciechowski, B. W. Colloid Polym. Sci. 1977, 255, 869.

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The first derivative of the isotherm is related to the retention time of the characteristic point by the simplified Conder equation:

(∂N ∂P )

L,tr

)

1 JDtr′ Vn ) RT m m

(4)

One merit of this approach, that seems to have not been extensively applied up to now, it to underline that the calculated distribution function is strongly dependent on the choice of the local isotherm, i.e., on the hypothesis done on the true mechanism of adsorption of the molecule on the solid surface. Finally, whatever the chosen method for the resolution of eq 1, it is very important, firstly, to emphasize that an efficient method has to provide one or more criteria of quality in order to control the calculation process and, secondly, to be validated by simulation. Last but not least, the correct acquisition of experimental data is essential,19 because calculation processes will amplify the random and systematic measurement errors. To perform such an analysis, one has to make some important decisions: firstly, the selection of suitable experimental conditions that authorize the acquisition of the adsorption isotherm in a large pressure domain; secondly, the choice of an adsorption model based on some physical assumptions, leading to an equation relating the distribution function of interest to the experimental isotherm; thirdly, the choice of a suitable mathematical solution method of this equation in order to obtain a stable solution. We shall examine now these different points in the case of an experimental acquisition of the adsorption isotherm using IGC and a method of calculation of the adsorption energy distribution functions based on the Rudzinski and Jagiełło’s method and on a multiple derivation of the DFCA using discrete Fourier transforms. Acquisition of the Adsorption Isotherm by Inverse Gas Chromatography. Volumetric or gravimetric methods are generally employed and are well appropriate for isotherm determination, at low temperature, of adsorbates such as nitrogen, argon, or krypton. For organic adsorbates, having boiling points above room temperature, chromatographic methods are more suitable. Different IGC exploitations have been reviewed by Conder.20 The simplest one “the elution characteristic point method” (ECP) allows the acquisition of the isotherm from a unique chromatographic experiment. Using this method, the first derivative of the adsorption isotherm can be readily calculated starting from the retention times and the signal height of characteristic points taken on the diffuse descending front of the chromatogram, as depicted in Figure 2.

where N is the number of absorbed molecules, P the pressure of the probe at the output of the column, L the column length, tr′ the net retention time of a characteristic point on the rear diffuse profile of the chromatogram, Vn the corresponding net retention volume, J the James and Martin’s coefficient taking into account the compressibility of the gas due to the pressure drop inside the chromatographic column, D the output flow rate, and m the mass of adsorbent. The preceding equation assumes that the rear diffuse front of the chromatographic peaks corresponding to different injected amounts overlay as shown on Figure 2, for three different injected amounts of benzene, the incidence of the injected probe vapor to the gas flow rate across the column is negligible, and the contribution of the width of the injection “band” and longitudinal diffusion processes along the column plays a minor role (ideal chromatographic conditions). Practically, those experimental conditions have to be carefully controlled. If these requirements are not met, one may obain a deformed chromatogram and, consequently, an adsorption isotherm and an energy distribution function that have no physical meaning. When taking into account these precautions, one accedes easily to the first derivative of the isotherm. Choice of the Local Isotherm. In eq 1, only the left termsthe experimental isothermsis known. The local isotherm, θ(,Pm,Tm) has to be selected according to the phyical hypothesis describing the interaction of a molecule with an adsorption site and, eventually, with the neighboring adsorbed molecules. Hence, the calculated distribution function χ() is obviously dependent on the choice of the local isotherm. The common local isotherm equations were reviewed by Lum Wam and White.5 The Langmuir local isotherm supposes localized adsorption sites and no lateral interaction between adsorbed molecules whereas the BET isotherm allows for multilayer adsorption in addition to the preceding assumptions. The Frumkin, Fowler, and Guggenheim’s (FFG) isotherm takes into account lateral interactions between adsorbed molecules and considers motionlesss adsorption. On the contrary, Volmer’s isotherm accounts only for the mobility of the adsorbed molecule and ignores lateral interactions. Finally, Hill and de Boer’s model assumes both mobility and lateral interactions. The required knowledge of the actual value of some parameters, such as the interaction energy between two adsorbed molecules for the FFG local isotherm, constitutes a difficulty for the choice of that local isotherm. For the latter local isotherm, this energy is currently approximated by the heat of vaporization of the probe.22 It is necessarily a rough approximation, because adsorption strongly influences the electronic state of the adsorbed molecule and therefore its capacity to exchange lateral interactions. Moreover, concerning the surface mobility of an adsorbed molecule, only the mobility on a homogeneous domain is assumed by the local isotherm equations whereas, on a real heterogeneous surface, the mobility will be strongly

(19) Stanley, B. J.; Guiochon, G. Langmuir 1994, 10, 4278. (20) Conder, J. R.; Young, C. L. Physicochemical Measurements by Gas Chromatography; Wiley: New York, 1979. (21) Hobson, J. P. Can. J. Phys. 1965, 43, 1941.

(22) House, W. A.; Jaycock, M. J. J. Colloid Polym. Sci. 1978, 256, 52. (23) Gawdzik, J.; Suprynowicz, Z.; Jaroniec, M. J. Chromatogr. 1976, 121, 185.

Figure 2. Typical experimental chromatogram for IGC experiments in finite concentration conditions, corresponding to different injected volumes of a benzene probe, on a column filled with a ground mica.

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Figure 3. (A) Correction for the multilayer adsorption of an experimental isotherm of octane on a ground mica: (a) uncorrected isotherm; (b) corrected isotherm. (B) Correction for multilayer adsorption of the logarithmic derivative of an experimental isotherm of octane on a ground mica: (a) uncorrected isotherm; (b) corrected derivative.

anisotropic depending on the adsorption energies of the neighboring sites. Therefore, we decided to use the simplest model of local isotherms: the one of Langmuir. This choice is all the more reasonable since, whatever the chosen local isotherm, the characteristic energy  of an adsorption site is directly related to the pressure, namely the characteristic pressure of the site Pc, for θ equal to 1/2, by eq 5 c

 ) -RT ln(P /K)

(5)

where K is a constant. For the Langmuir local isotherm, K is related, according to Hobson,21 to the molar mass and the temperature of measurement by

K ) (1.76 × 104)(MT)1/2

(6)

Other equations are proposed in the literature by House and Jaycock,22 Gadwick et al.,21 and Jagiełło et al.14 that nevertheless lead to very close energy scales.2 Finally, because most isotherms belong to type II of BET classification, when using the Langmuir or FFG local isotherms as kernel of eq 1, we have to correct the experimental isotherm for the multilayer adsorption. Correction for the Multilayer Adsorption. Comparing the BET and the Langmuir equation,5 it is apparent that the former (eq 7) can be rewritten as a Langmuir equation (eq 8)

Cx (1 - x)(1 - x + Cx)

(7)

x C 1-x N(1 - x) ) N0 x 1+C 1-x

(8)

N ) N0

replacing the relative pressure x by x′ ) 1/(1 - x) and the adsorbed amount N by N′ ) N(1 - x), we obtained eq 9 for the isotherm corrected for the multilayer adsorption

N′ )

Cx′ 1 + Cx′

(9)

where N′ and x′ are respectively the values of N and x, corrected for the multilayer adsorption relative pressure. This correction becomes effective in the domain of relative pressures higher than 0.05. This correction is purely formal and supposes that the BET equation is valid in this domain of pressure and that no molecular clusters are formed at relative pressures lower than 0.05 at which the surface heterogeneity influences mainly the isotherm shape.

On the other hand, IGC provides directly the first derivative of the isotherm. Integration and subsequent derivation lead to a loss of information. It is therefore of an utmost interest to apply directly this correction for the multilayer adsorption to the first derivative of the isotherm using eq 10

∂N′ ∂N N ) (1 - x)3 - (1 - x)2 ∂P′ ∂P P0

(10)

where N′ and P′ are the values of N and P after multilayer adsorption correction, x being the partial pressure P/P0, P0 the vapor pressure of the probe at the temperature of measurement. Figure 3A depicts respectively the experimental and the multilayer adsorption corrected isotherms whereas Figure 3B shows the first logarithmic derivative of the initial and corrected isotherms, for experimental adsorption data acquired on a grounded mica sample, using the IGC method. As expected, the corrected isotherm tends toward a plateau whereas the first logarithmic derivative goes through a maximum corresponding to the inflection point of the corrected isotherm when plotted versus the logarithm of the pressure. Computation of the Adsorption Energy Distribution Function in the Condensation Approximation. As underlined previously, the DFCA is directly related to the first logarithmic derivative of the experimental isotherm after multilayer adsorption correction (eq 2). Therefore, combining eq 2 and eq 4, the DFCA can be computed knowing the net retention time (tr′) of the probe and the height of the signal (pressure) corresponding to each characteristic point taken on the rear diffuse front of the chromatographic peak. Figure 4A shows an example of DFCA measured by IGC, for a ground mica. Fitting and Multiple Derivation of the DFCA Using Fourier’s Series. Using Rudzinski and Jagiełło’s method for the calculation of the actual distribution function requires an effective method for the multiple derivation of the DFCA. Taking advantage of the shape of the experimental DFCA curve that may be considered as a part of a periodic function, we propose the use of Fourier’s series to perform its fitting (eq 11) n

χCA() ) A0 +

∑ Ak sin(kω) + Bk cos(kω)

(11)

k)1

where k is the harmonic order and ω the frequency corresponding to the period selected on the energy scale. Then the even derivatives of the DFCA can be simply calculated by the derivation of the Fourier series, according

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Figure 4. (A) DFCA of n-octane for a ground mica determined at 30 °C, using the IGC method of acquisition and (B) its extrapolation to zero on low- and high-energy sides.

Figure 5. Unmodified and filtered Fourier moduli spectra of second (A) and fourth (B) derivatives of the DFCA of a ground mica.

to eq 12 n

2j

χCA () )

(kω)2j[Ak sin(kω) + Bk cos(kω)] ∑ k)1

(12)

where 2j is the order of derivation. But the experimental DFCA data correspond only to a part of a period, because points are missing on both sides, particularly on the low-energy side. In order to obtain a periodic function without any discontinuities, one has to extrapolate the DFCA, so as to achieve a complete period (performing no extrapolation leads to a strong discontinuity at the junction between two periods and consequently generates heavy intensive harmonics in the highfrequency range of the spectrum, moreover, the contribution of these high harmonics will be amplified by the multiple derivation). Such a difficulty was also encountered by Lum Van and White,5 that used Biraud’s extrapolation method, to prevent artifactual oscillations. In our case, the experimental DFCA is extrapolated on both sides, using two sections of sinusoidal function, as depicted on Figure 1B, allowing the extrapolated curve to correspond to a complete period. The used computer software assumes, firstly, the continuity of the curve and of its first derivative, at the junction points between the experimental curve and the extrapolated sections and, secondly, the equality of the ordinates for the points located at both extremities of the extrapolated curve, so as to always prevent strong discontinuity at the junction point between two periods. Of course, distribution functions will have a physical meaning only in the window corresponding to the experimental data, and in the following figures displaying the calculated DF, the part corresponding to the extrapolated data will be shown in gray. Moreover, we observed some artifactual peaks at the borders of the experimental data window due to the imperfections occurring at the junction between the experimental and extrapolated data

(see Figure 9, for example) that are amplified by the multiple derivation. From eqs 11 and 12, one calculates the Fourier moduli spectra corresponding to the second and fourth derivatives of the previous DFCA. They are depicted in Figure 5. One notices that these moduli spectra are obviously bimodal. The harmonics corresponding to the lowest frequencies contain mainly the wanted information whereas harmonics having the highest frequencies will origin from both experimental and computational errors. Of course, we observed also that the contribution of the harmonics corresponding to the highest frequencies increases with the order of derivation because the terms of the initial Fourier series are multiplied by the factor (kω)2j. Then, to minimize the noise contribution, we propose to perform a correction of the modulus spectrum, as depicted in Figure 5, by decreasing the relative intensity of the higher harmonics keeping the phase spectrum unchanged, in order to eliminate the high-frequency components in the final filtered derivative, according to a procedure very often used in the signal treatment technique. Extrapolation and correction of the modulus spectra are obviously never performed exactly in the same way. Figure 6 gathers the Fourier spectra of second and fourth derivatives resulting from three different extrapolation operations, starting from the same experimental DFCA. One observes that the high-frequency components of the spectra of the second and fourth derivatives are more or less important depending on the way the extrapolation operation was performed. The influence of both extrapolation and filtering operations on the DF shapes of order 2 (DFRJ2) and 4 (DFRJ4) are depicted respectively in parts A and B of Figure 7. Despite of the unavoidable variability of the Fourier modulus spectra filtration and the extrapolation operation, one notices that the calculated DFRJ2 curves overlay very satisfactory. Of course, for the DFRJ4 curves larger deviations are observed resulting from the higher con-

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Figure 6. Unmodified and filtered Fourier moduli spectra of the second (A) and fourth (B) derivatives of DFCA for different extrapolations.

Figure 7. Calculated DFRJ2 (A) and DFRJ4 (B) of a ground mica corresponding to three different treatments of the same DFCA.

tribution of high-frequency harmonics. The influence of the filtering operations on DFRJ4 shape will also be examined below (see Figure 16). Starting from repeated experimental datasdifferent chromatogramssdeviations may moreover result from the experimental errors, i.e., incertitude on the injected volume or slight variation of the column temperature. In order to decrease the noise/signal ratio, i.e., to overcome the variability originating from the experimental conditions and the DFCA treatments, we propose to accumulate several signals (see Figure 8A) computed from different acquisitions and DFCA treatments. Finally, the

resulting average distribution function is simply smoothed using cubic spline (see Figure 8B) in order to reduce the high-frequency residual noise having a period lower than 1 kJ/mol. As expected, we obtain a noise-free distribution function. Finally, using a simulation program, we shall now test, firstly, the validity of the proposed method and, secondly, its robustness to the experimental noise and to the regularity of the data sampling that may also play a significant role on the instability of the solution of the adsorption integral equation.

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Figure 8. Cumulated (A) and averaged and spline smoothed (B) DFRJ4 for a ground mica.

Figure 9. DFCA, DFRJ2, DFRJ4, and DFRJ6 calculated using FT derivation method from a simulated chromatogram starting from a theoretical bi-Gaussian DF.

Figure 11. Calculated theoretical chromatogram, unnoised and noised (200 µV) starting from the theoretical bi-Gaussian DF (Figure 10).

Figure 10. DFCA, DFRJ2, DFRJ4, and DFRJ6 calculated using the FT derivation method from a simulated chromatogram starting from another theoretical bi-Gaussian DF.

3. Testing of the Method Using a Simulation Program

Figure 12. Calculated DFCA from the theoretical chromatogram, unnoised and noised (100 and 200 µV) starting from the theoretical bi-Gaussian DF (Figure 10).

For testing the efficiency and the robustness of this new approach, we developed a program of simulation that allows the calculation of the theoretical chromatogram from a theoretical distribution function made of a discrete number of local Langmuir isotherms. An example of the results obtained using our method is reported in Figure 9, starting with a bi-Gaussian theoretical distribution function. It appears that a good agreement is observed between the theoretical DF and the RJ approximation of order 6 (sixth derivative of the DFCA) in the case of the bi-

Gaussian distribution, without any supplementary data accumulation and averaging. Another example is depicted in Figure 10 for another theoretical distribution made of two narrow Gaussian peaks. The RJ method again allows confirmation of the bimodality of the initial distribution function, but with a lower resolution consequently to the narrower width of both peaks of the initial theoretical DF. In order to test the robustness versus experimental noise, we have added a random noise having increasing

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Figure 13. Unmodified and filtered Fourier moduli spectra of second (A) and fourth (B) derivatives of the preceding DFCA calculated from the theoretical chromatogram, unnoised and noised (100 and 200 µV) starting from the theoretical bi-Gaussian DF (Figure 10).

Figure 14. Calculated DFRJ2 (A) and DFRJ4 (B) for various random noise from the preceding DFCA (Figure 12).

intensities up to 200 µV to the calculated chromatogram (Figure 11). Calculated DFCA and the Fourier modulus spectra of their corresponding second and fourth derivatives are displayed respectively in Figures 12 and 13. Obviously, the introduction of a random noise increases the contribution of high-frequency harmonics leaving the low-frequency part of the spectra almost unchanged. Applying the preceding filtration procedure, we obtain the DFRJ distribution functions shown in Figure 14. Even for the highest level of added random noise of 200 µV, a value that is much higher than the normal experimental noise (a few microvolts) of a flame ionization detector (FID) detector, the calculated DFRJ2 overlays very correctly the DFRJ2 obtained from the noiseless chromatogram and only a slight change in the shape of

the DFRJ4 was observed located in the high energies domain at the upper limit of the data window. As pointed out above, the main part of the signal is contained in the harmonics having the lowest frequencies. It will be therefore interesting to examine how the filtration procedure will influence the shape of the final DFRJ4. Figure 15 shows different modes of filtration of the Fourier spectra: for parts A and B of Figure 15, a simple cutoff at the harmonics 5 and 12 was done; for Figure 15C a progressive decrease of harmonic intensity in the cutoff region was applied as previously proposed; finally, in Figure 15D, the spectrum was kept unchanged. DFRJ4 calculated from the preceding spectra are depicted in Figure 16. No significant differences can be observed between the DFRJ4 computed from the filtered spectra A and B.

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Figure 15. Different filtration modes for the Fourier moduli spectrum of the 4th derivative of the DFCA corresponding to a noisy theoretical chromatogram (100 µV): A, cutoff at the 5th harmonic; B, cutoff at the 12th harmonic; C, progressive decreasing of the harmonic intensity between the 5th and 7th harmonics; D, unfiltered spectrum.

4. Conclusion

Figure 16. Calculated DFRJ4 from the preceding spectra A, B, C, and D (Figure 15).

Conversely, making a cutoff at the 12th harmonic or keeping the spectrum unchanged induces a lot of artifactual peaks in the DFRJ4 that are more intensive for the unfiltered spectrum. This last observation justifies subsequently the proposed method of filtration proposed for the separation of the signal contribution from the noise contribution and demonstrates the efficiency of the method to overcome the ill-posed character of eq 1. Finally, we will test the robustness of our method versus both noise and irregular sampling. For this purpose, we have simultaneously added a random noise and we deleted randomly a third of the points of the initial DFCA, especially in the lower energies range. The unmodified and modified DFCA curves are shown in Figure 17A whereas Figure 17B displays the corresponding calculated DFRJ. Again, no significant changes are observed between the DFRJ4 calculated from the initial DFCA and that obtained starting from the noisy and irregularly sampled DFCA, demonstrating the remarkable robustness of our method versus both factors: random noise and irregular sampling.

As pointed out previously, the main problem encountered in the calculation of the adsorption energy distribution functions is related to the instability of the solution of the integral equation of Fredholm used to describe the adsorption of molecules on a heterogeneous surface. This instability originates particularly from the experimental noise contained in the experimental data but also from the calculation method and from the precision of the computer language itself that induces calculation errors. The major problem is therefore the right evaluation of the contribution of the signal of interest taking into account experimental and computational noises. To achieve this difficult task, we propose the following: The acquisition of the experimental data using IGC at finite concentration conditions. The principal advantage of this method is its ability to provide directly the first derivative of the isotherm that is richer in information than its corresponding integral form, i.e., the isotherm itself. For this purpose, we developed a method of calculation, including the correction for multilayer adsorption, that allows the direct calculation of the DFCA (distribution function for the condensation). The calculation of DF taking the extended Rudzinski and Jagiełło’s solution method combined with discrete Fourier transforms of DFCA to perform it multiple derivation of second, fourth, and sixth order. The major advantage of this multiple derivation method is the possibility to separate the respective parts of the signal (low-frequency harmonics) and of the experimental noise (high-frequency harmonics) by filtration using the Fourier moduli spectra of the DFCA derivatives. The accumulation and averaging of the distribution functions computed from different chromatograms and/ or coming from different computer treatments of the DFCA. In this way, it is possible to overcome the unavoidable subjective character of the DFCA treatment and to reduce the noise/signal ratio. Before applying our method to real solids, we tested it by computer simulation, especially, looking at the influ-

Surface Heterogeneity of a Solid

Langmuir, Vol. 13, No. 5, 1997 1269

Figure 17. Calculated DFCA (A) and the corresponding DFRJ4 (B) for various regular and irregular reduced sampling (R) and noise levels.

ence of the noise and of the regularity of the data sampling on the stability of the solution. Starting from a theoretical distribution function (DF), we first calculated the corresponding simulated chromatogram. Then we analyzed the latter using our method, without any modification or by adding a given level of random noise or deleting randomly a given number of experimental points. Despite the high added random noise (up to 200 µV, i.e., 2‰ of the full scale) and the degree of irregular data sampling introduced, we were able to find again easily the same DF, demonstrating, in this way, the high robustness of the proposed calculation method versus noise and irregular data sampling. The simulation emphasizes also the importance of the step by step filtration of Fourier spectra of the even derivatives of the DFCA, allowing prevention of the apparition of artifactual peaks in the calculated DF. Thus, we can consider that the main aim of this work is reached: to provide a simple and efficient calculation method for solving the fundamental equation of surface heterogeneity. Finally, considering the “instrumental approach” of Lum Vam and White, namely looking at the Fredholm’s integral equation as a convolution product (a different approach in which they used the Plancherel’s theorem for solving the adsorption integral equation), it becomes obvious that the calculated distribution functions are closely related

to the chosen physical model, here, one molecule/one adsorption site and additivity of the contribution of the different homogeneous patches, to the ideality of the chromatographic process, to the choice of the local isotherm, i.e., the Langmuir isotherm, and to the hypothesis subtending the multilayer adsorption correction, i.e., no formation of molecular clusters at low pressure. Making up a “virtual mathematical instrument” through it, one is looking to an object (the distribution function), starting from the image (the first logarithmic derivative of the adsorption isotherm). The unavoidable imperfections of this instrument will never allow us to reach the actual distribution function. It will only provide fingerprints that will be of a great interest for the sake of comparison as proved by the studies in progress in our laboratory. Indeed, this method was applied to various fillers such as micas and clays (see this issue) or silicas,24 talcs,24 clays,24 and carbon blacks25 using apolar or slightly polar probes such as alkanes and benzene or polar probes like pyridine or propanol-2, as can be seen elsewhere.26 LA951526D (24) Papirer, E.; Balard, H. In Studies in Surface Science and Catalysis; Elsevier: Amsterdam, 1995; Vol. 99, pp 479-502. (25) Papirer, E.; Balard, H. Polym. Mater. Sci. Eng. 1994, 70, 456. (26) Papirer, E.; Saada, A.; Balard, H. Clays Clay Miner., in press.