2637
Langmuir 1993,9,2537-2646
Energetic Heterogeneity of Reference Carbonaceous Materials M. Heuchel,?M. Jaroniec,*and R. K. Gilpin Department of Chemistry, Kent State University, Kent, Ohio 44242
P. Brauer and M. v. Szombathely Institute of Physical Chemistry, Leipzig University, 0-7010 Leipzig, Germany Received September 28, 1992. In Final Form: December 14, 199F
Systematic studiesof the energetic heterogeneity of graphitized and ungraphitized reference carbonaceous materials have been carried out using several standard adsorption isotherms available in the literature. Adsorption energy distribution functions have been calculated by using a new algorithm based on a regularization method. Analysis of the calculated energy distributionsfor various reference carboneprovides important comparative informationabout their energetic heterogeneity. It has been shown that the surface properties of the reference materials vary depending on differences in the degree of their graphitization.
Introduction Characterization of active carbons and active carbon fibers, which importantly includes assessment of their microporosity, is usually based on a comparison of an adsorption isotherm measured on the material of interest to that of a standard adsorption isotherm measured on a reference nonporous carbon.14 This type of comparison is the principle of well-known procedures such as t-plot, a,-plot, and &plot method^.^^^ However, each of these differs in the manner in which the standard adsorption data are defined. For example, in the t-plot method, the standard adsorption is expressed in terms of the thickness of the surface film formed on the reference adsorbent. The idea behind comparative plots is to utilize differences which exist between the adsorption mechanisms on a flat surface and those that occur in the micropores. Adsorption on a flat surface or on the surface that is contained in large pores (e.g., mesopores and macropores) occurs by a layer-by-layermechanism, whereas adsorption in micropores resembles a volume fiiing mechanism. Since active carbons possess both micropores and large pores, adsorption on these solids occurs via a mixed mechanism. At low pressures the micropores are filled first which is followed by layer-by-layer adsorption on the external surface as well as by capillary condensation in the large pores. A plot of the amount adsorbed on an active carbon against the amount adsorbed on the reference nonporous carbon is linear at high pressures because layer-by-layer adsorption occurs in both ~ases.59~However, at low pressures the adsorption mechanisms are different, resulting in nonlinear behavior. The slope of the linear section of the plot is associatedwith the monolayer capacity
* To whom correspondence should be addressed.
t Permanent addreas: Leipzig University, Department of Chem-
istry, 0-7010 Leipzig, Germany. Abstractpublished in Advance ACS Abstracts, August 15,1993. (1) Sing, K. S. W.; Everett, D. H.; Haul, R.A. W.; Moscou; Pierotti, R. A.; Rouquerol, J.; Siemieniewaka,T. Pure Appl. Chem. 1986,57,603. (2) Gregg,S. J.; Sing, K. S.W. Adsorption, SurfaceArea, and Porosity; @
Academic Press: New York, 1982. (3) Jaroniec, M.; Choma, J. Chem. Phys. Carbon 1989,22,197. (4) Jaroniy, M.; Madey, R.Physical Adsorption on Heterogeneous Solids; Elsenex Amsterdam, 1988. (5) Sing, K. S. W. Ber. Bunaen-Ges. Phys. Chem. 1975, 79,724. (6) Jaroniec,M.;Madey,R.;Choma,J.;McEmney,B.;Mays,T.Carbon 1989, 27, 77.
of the mesopore surface, and ita intercept gives the maximum amount adsorbed in the micropores.6 A key problem in comparative methods is the appropriate selection of the reference adsorbent? For the ideal case ita surface properties, which can be described in terms of the energetic heterogeneity, should be the same as the properties of the mesopore surface of the active carbon being studiede6This is especiallyimportant if the amount adsorbed in the micropores is to be distinguished accurately from the total amount adsorbed. Since the energetic heterogeneity of the reference solid controls the initial part of a comparative plot, accurate knowledge about ita energetic heterogeneity is necessary for selection of an appropriate standard. The aim of the current paper has been to characterize the energetic heterogeneity of reference materials via the distribution function of the adsorption energy e which has been calculated using a regularization method. This numerical regularization has been found to be a good procedure for extracting informationabout an adsorbent’s heterogeneity from experimental adsorption data. The procedure has been evaluated for a standard nitrogen isotherm (i.e., that on Vulcan), in order to illustrate how different factors influence the shape of the energy distribution function F(e). Although the influence of certain factors on F(e) was studied earlier,4l8this was done mainly in the past using approximation methods for calculating F(e). In the current paper a systematic discussion of different aspects of the evaluation of F(e) via numerical regularization is presented. Theory Adsorption Integral Equation for Heterogeneous Solids. In most theoretical treatments of adsorption on heterogeneous solids the adsorbent surface is assumed to have a continuous distribution of adsorption sites with The resulting respect to the adsorption energy distribution function, denoted by F(e), is accepted as a quantitative characteristicof an adsorbent’sheterogeneity, and F(e) de is the fraction of the surface with adsorption energies between e and e + de. The relative surface coverage, e@), for a heterogeneous solid characterized by the energy distribution function F k ) , is expressed by the e.498
(7) Rodriguez-Reiioeo, F.; Martin-Martinez,J. M.; Prado-Burguete, C.; McEnany, B. J. Phys. Chem. 1987,91,516. (8) Rudziiki, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1991.
0743-7463/93/2409-2537$04.00/0 Q 1993 American Chemical Society
Heuchel et al.
2538 Langmuir, Vol. 9, No. 10,1993 well-known integral equation4n8
e,@)
= Je@,e) F(e) de for
T = constant
(1)
A
where et@) can be evaluated experimentally. When measurements are made at a constant temperature T,the overall adsorption isotherm is obtained by assuming a model a priori for the local adsorption isotherm e@,e). The integration region A in eq 1 is over all possible adsorption energies. Since different models can be used to represent local adsorption on sites of the same adsorption energy, i.e., energetically homogeneous, the current treatment s t a r b with the very popular and simplistic model formulated by Langmuir in 1918.9 This model describes localized monolayer adsorption by neglecting lateral attractive interactions in the surface phase, where
additional assumptions be made about the topography of the adsorption sites. Two extreme models have been used to represent the topography of the sites on heterogeneous surfaces: the random distribution approximation (RDA) and the homotattic patch approximation (HPA). For a random distribution of adsorption sites, statistical thermodynamicsgives the following equation for the local isotherm:&8
The zw8, term describes the average force field acting on an adsorbed molecule that arises from molecules located on the nearest-neighbor sites. In the HPA model, the average force field depends on the homotattic patch coverage e(e) and the local isotherm is expressed as follows: Kp exp(zw8) (7) 1 + K p exp(zw8) In contrast to localized models, mobile adsorption models assume that molecules can diffuse freely on the surface. One of the most popular equations used to describe mobile adsorption is that proposed by Hill and de Boer13 =
In eq 2 Kl is Langmuir's constant, which is defined as follows:
Kl = Klo(T)exp(dRT) (3) where Klo(T) is the preexponential factor that contains the partition functions of an isolated molecule in the gas and surface phases with rotational, vibrational, and translational degrees of freedom. A detailed description of this factor is given in a monograph by Clark.lo AdamsongJl has approximatedthe preexponential factor Klo by the following equation: Klo = N , U ~ T ~ / ( ~ T M R T ) ~ / ~ (4) where uo denotes the actual area per molecule, T O is Frenkel's characteristic adsorption time (i.e., which is s in the current study),NOis Avogadro's assumed to be W 3 number, and M is the molecular weight of the adsorbate. Equation 4 has been used to estimate the Kl0 values for the adsorbates studied as discussed in a latter section. Inclusion of multilayer effects in Langmuir's original model leads to the Brunauer-Emmett-Teller (BET) adsorption model, which can be written in the following form? e@,e) = (1
- x)[l
cx
+ (C - 1)x3
(5)
where x = p/pl is the relative pressure, p s is the saturation vapor pressure, and C = Kp,. Although the BET model is a simplification of multilayer adsorption, it provides a relatively good description of the initial stage of this process. Thus, it can be employed to make multilayer corrections of an experimental isotherm in order to evaluate adsorbent heterogeneity (i.e., only the initial submonolayer part is essential in this evaluation). The simplest model which corrects Langmuir's localized monolayer approach for lateral interactions was proposed by Fowler and Guggenheim (FG).I2 Accordingly, lateral interactions are described by the number of nearest neighbors, z, and by the interaction energy parameter, w. As has been shown elsewhere,4la inclusion of lateral interactions into local adsorption models requires that (9) Adamaon,A. W.Physica1 ChemistryofSurfaces;Wiley: NewYork, 1990. (10) Clark, A. The Theory of Adsorption and Catalysis, Academic Prese: New York, 1970. (11) Adamson, A. W.; Ling,1. Adu. Chem. Ser. 1961,33,51. (12) Fowler, R. H.; Guggenheim, E. A. Statiatical Thermodynamics; Cambridge University Press: London, 1949.
-1
pKm= e exp[ e 1-8 1-8 B,RT
(8)
where am and & are two-dimensional van der Waals constants and Km is a constant analogous to Langmuir's constant Kl in eq 3. However, the preexponential factors of the constants KI and Km differ. In the case of localized adsorption, the preexponential factor Kl0 takes into account the vibrations of adsorbing molecules in the x , y, and z directions, whereas Kmo contains only the partition functions for vibration in the z direction and the translational partition function describing the mobility of adsorbing molecules in the x y plane. Equation 8 describes the mobile monolayer adsorption on an energetically homogeneous surface, and thus it is analogous to eq 7, which describes localized adsorption with lateral interactions. Numerical Procedure for Evaluating the Energy Distribution Function. From the mathematical point of view, eq 1 is a linear Fredholm integral equation of the first kind, which can be written in a more general form as follows: g(Y) = JabK(xYy)f ( x ) dx
(9)
The integral kernel K(x,y) represents the local isotherm in eq 1, i.e., the physicochemical model of adsorption on sites of the same energy, g(Y) is the experimentally measurable adsorption isotherm et@), and f ( x ) = F(e) denotes the energy distribution function. Many attempts have been undertaken to invert integral eq 9 with respect to f ( x ) . The so-called "condensation approximation" (CA) method introduced by Roginskij'4 is considered to be the first attempt to solve eq 9. The distribution function determined by this method is used often in other iterative methods as the initial step.lS Many researchers4ps assume a priori the shape of F(c), and integrate eq 1 analytically. Although this approach has been applied to different types of some local isotherms, (13) Ross, 5.;Olivier, J. P. On Physical Adsorption; Wiley-Interscience: New York, 1964. (14) Roginskij, S. S. Dokl. Akad. Nauk SSSR 1994,61, 194. (15) Jaroniec, M. Adv. Colloid Interface Sci. 1983, 18, 149.
Langmuir, Vol. 9, No. 10, 1993 2539
Energetic Heterogeneity of Carbonaceous Materials
Langmuirian types have received the most attention. The resulting expressions for the overall adsorption isotherm contain adjustable parameters, which can be obtained by a numerical least-squares fit of the experimental adsorption data. The disadvantages of this approach are that (1)it is unknown whether the assumed shape of F(e) is correct and (2) often a variety of different analytical functions can be used to describe a given data set with about the same degree of accuracy. Thus, the calculation of F(E)from eq 1is a numerically ill-posed problem; i.e., small changes in et@) caused by experimental errors can distort significantlythe calculated F(e). Additionally, this distortion is caused not only by experimental errors but also by errors generated during the numerical calculations and errors from the quadrature of eq 1. The ill-posedness is mainly a mathematical numerical problem. Another difficulty in evaluating F(e) on the basis of eq 1is associated with the selection of the integral kernel, used to model the local adsorption process. It is necessary to decide whether the assumptions of the local adsorption model are in accordance with the experimental data. Theoretical foundations for solving numerically instable problems were developed by Tich~now,~~J' who introduced the regularization method. This method was first applied to gas adsorption by House18 and Merzl9 and further developed by Papenhuijzen and KoopalZ0and Brown et al.21*22 In the current paper, the regularization method with singular value decomposition programmed by Szombathely23 has been used to study the energetic heterogeneity of several reference carbonaceous solids. Regularization Method. The first step in the regularization method is the discretization of the integral equation by a quadrature. Thus, integral eq 9 needs to be transformed into a system of the following linear equations: g=Af (10) where g and fare one-dimensionalmatrices representing, respectively, the functions g and f and A is a twodimensional matrix representing the integral kernelK(x9). The fundamental idea of numerical regularization is to replace the ill-posed problem of minimizing the function 11Af - gJ12by a well-posed one which smoothes the calculated distribution function and distorts the origin problem insignificantly. This can be done by addition of a second minimizing term to )IAf - ell2: (11) min i llAf - g(I2+ yII W(f$ The regularization parameter y is a measure of the is defined as follows: weighting of both terms and 11 W(f)1I2
(12)
By introducing this latter expression intoeq 11,oscillations of the calculated distribution function may be suppressed. (16) Tichonow, A. N.Dokl. Akad.Nauk SSSR 1943,39,195;1963,153, 49; Sou. Math. (Proutdence) 1963,4,1036,1624. (17) Tichonow, A. N. Areenin, V. Ja. Metody resenija ne-korrektnych radac; Nauka: Moskva, 1979. (18) House, W. A. J. Colloid Interface Sei. 1978, 67, 166. (19) Merz, P. H. J. Comput. Phys. 1980,38,64. (20) Papenhuijzen, J.; Koopal, L. K.In Adsorption from Solutions; ottewill,R. H., Rochester, C. H., Smith, A. L., Eds.; Academic Press: London, 1983; p 211. (21) Brown,L.E.,Travie,B.J. InFundomentals ofAdsorption;Myers, A. L.. Belfort. G.. Eds.:American Instituteof ChemicalEngineers: New York, 1984, p 125. ' (22) Britten, J. A.; Travis, B. J.; Brown, L. F. A1ChE Symp. Ser. 1983, I.,
.
I Y , 1.
(23) Szombathely, M. v. Ph.D. Thesis, University Leipzig, 1988.
The regularization method can be modified further through inclusion of additional restrictions on the function f, e.g., only allowing non-negative solutions (NNLS). Under these latter conditions, the regularization criterion given by eq 11can be written as (13) min = llAf - ell2+ yII W(f)l12 f 2 0 The regularization parameter y is usually chosen through a series of trials by an interactive judgment about the solution. A detailed description of strategies for finding the optimal y value in adsorption applications is given in ref 24. Usually as a starting point, a high regularization parameter is selected, e.g., y = 1,which results in astrongly smoothed distribution function, with a residual, 11Af - gII 2, generally higher than that associated with the experimental errors. Subsequently, y is reduced in an iterative fashion until the experimental accuracy is reached. In the current work, the program INTEG developed by Szombathely2312ahas been used to calculate the distribution f according to the regularization method. It is based on the singular value decomposition (SVD) of the matrix A that represents the discretized integral kernel K(x,y).A combination of regularization and SVD not only has computational advantages, such as the minimization of numerical errors and the fast optimization of the final solution by choosing different regularization parameters, but also provides a means of evaluating the validity of the physicochemical model selected to represent the local adsorption which is contained in A. Application of the Regularization Method. Knowledge of how various factors affect the numericalcalculation of F(e) is important for the practical application of the regularization method to experimental data. The current study was carried out using a typical nitrogen adsorption isotherm (14 data points) that was measured on Vulcan 3. The distribution function F(e) shown in Figure l a was calculated with INTEG using a Langmuir local isotherm with an energy range from 0 to 20 kJ/mol, and a regularization parameter y = 10-3. Subsequently, Figure l a which has two maxima at 8 and 23 kJ/mol with corresponding heights equal to 0.2 and 0.5 mol/kJ was used as the reference distribution for additional comparisons. With other functions of F(e)calculated for the same adsorption system but under different numerical or physical conditions, a physical-chemical interpretation of the distribution function in Figure la is given below. Experimental Points. Since the adsorption sites are covered gradually starting with the high-energetic ones, the low pressure range of the experimental isotherm is essential for accurately evaluatingF(e). This is illustrated by in Figure 1. If the first three points of the reference data set are omitted in the calculations of F(e),the higher energy peak at 12 kJ/mol is considerably smaller (see Figure lb). Excluding two further experimental points gives a one-peak distribution (cf. Figure IC). Preexponential Factor Kl0.Figure 2 illustrates the influence of Klo on the distribution function F(e). Since e = RTln(KdKlO),a relatively large change in Klo results in only a small shift in F(c) on the e axis. For example, if the parameter KO is increased by a factor of 100 over that predicted by eq 4, the F(e) function is shifted about 3 kJ/mol to lower energy. Similarly, the same shift is observed but in the opposite direction when the preexponential factor is made 100times smaller than that given by eq 4. Since the shape of F(e) is insensitive to changes in the Kl0 range studied, a comparison of the F(e) (24) Szombathely,M. v.; Brauer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17.
Heuchel et al.
2540 Langmuir, Vol. 9, No. 10, 1993 0
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7
-
7 -1.0 1-5.-
0
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20
0
a 5
e [W/mole]
10
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e [kl/molel
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e [kJ/molel
0.51 gmw"0.1
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I
Y
0
0
5
10
15
e [W/mole]
Figure 1. Energy distribution functions F(e) for Na on Vulcan 3 (ref 28) calculated by INTEG with the Langmuir model of local adsorptionfor different selectionsof data points. The numerical parameters of regularization were the same for all three parts: (a)pointa 1-14 (the whole data set); (b)data without points 1-3; (c) data without points 1-5.
-0.1
-
0
5
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Figure 2. Dependence of the distribution function F(e) on the preexponential factor Klo in the Langmuir constant for Nz on vulcan 3. distributions for the same adsorbate on different solids can be made for an approximate Klo value, e.g., a value estimated according to eq 4. This feature of F(d is useful for comparing the energetic heterogeneities of different solids because the exact Klo value is not required to obtain information about the shape of F(c). This is especially important for complex molecules since the calculation of Klo is difficult.lg Accuracy of the Experimental Data. Since experimental data are measured at a certain level of uncertainty, it is meaningless to accept a F ( 4 function as solution which describes a given data set with higher accuracy than is experimentally warranted. The regularization method thus takes into account such experimental constraints. The key quantity in this method is the regularization
5
10
15
20
e [kJ/molel
Figure 3. Distribution functions F(e) calculated with INTEG by assuming the different accuracies of the experimental data (see upper right values) for Nz on Vulcan 3. parameter y in eq 11 which weights and combines two strategies for an optimal solution: (1) it minimizes the s u m of the least squares, and (2) it maximizes the smoothing of the solution. For each reasonable local adsorption isotherm when y 0 a nearly perfect representation of the experimental data is obtained; however, the calculated F ( 4 is usually an oscillating function with large positive and negative amplitudes (see Figure 3a). An increase in the y value results in a greater degree of smoothing in F(t),but it also enlarges the error between the calculated and the experimental isotherms. The influence of y on the resulting F ( t ) distribution is illustrated in Figure 3. Strong negative oscillations occur when y = 106(cf. Figure 3a), but this function reproduces the overall isotherm with a 1% accuracy, which is considerably lower than the experimental error. For y = le3, the F ( 4 function reproduces the overall isotherm with an acceptable deviation of 2.8% (Figure 3b). Although further smoothing, e.g., y = 0.01, still results in a two-peak distribution, the peaks are not well separated (cf. Figure 3c) and the overall isotherm is reproduced with only a 5 % accuracy, which is higher than the experimental error. Finally, if y = 0.1, the calculated F(e) distribution is a single peak function (see Figure 3d) with a 20% deviation between the calculated and experimental adsorption data, Shown in Figure 4 are the calculated overall isotherms for two different y values in comparison to the experimental points. The solid line is the calculated isotherm with a 2.8% deviation from the data, whereas the dotted line deviates 20% from the data. For the larger value of y especially poor agreement between the calculated and experimental isotherms was observed at lower pressures, which are essential for obtaining information about the high-energetic sites. Local Adsorption Isotherm. The distribution function F(e) depends on the assumed integral kernel that is
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Energetic Heterogeneity of Carbonaceous Materials
Langmuir, Vol. 9, No. 10, 1993 2541 0.8
0
605
0.5
5
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Pressure, p [Torr] Figure 4. Adsorption isothermsfor N2 on Vulcan 3 reproduced with the distribution functions F(c) shown in Figure 3 in comparison to the experimental points (squares). The solid curve reproduced with 2.8 % accuracy,the dotted curve corresponds to 20% accuracy. used to represent the physicochemical model for adsorption occurring on sites with the same adsorption energy. Each kernel has different smoothing properties and thus represents the experimental data differently. In order to demonstrate the influence of a chosen model on the calculated distribution function, comparative calculations were carried out for different local isotherms using the same numerical conditions. The resulting F(e) functions are shown in Figure 5. The F(e) distribution calculated using a Langmuirian kernel (cf. eq 2) and the BET kernel (cf. eq 5 ) are illustrated, respectively, in parts a and b of Figures 5. In both cases, bimodal curves were obtained; however, the magnitude of the low-energy peak at 8 kJ/ mol is smaller when the BET model is used. The similarity between the F(e)functions obtained for the Langmuir and BET local models is not surprising because in both cases submonolayer adsorption data were used to calculate F ( 4 and thus multilayer effects are relatively small. In the HPA model (eq 7), the data representation is much better. Inclusion of lateral interactions at the surface makes the local isotherm much more flexible than that represented by the Langmuir model (Le., eq 2) and provides a better description of S-shaped isotherms. In comparison to the F(e) function associated with the Langmuir model, the FG eq 7 with HPA gives a sharper F(e) function that is shifted slightly in the direction of lower energies. The higher energy peak is located at 10.5 kJ/mol, and the lowenergy single peak which is centered at 8 kJ/mol shows a small degree of splitting. Also, the negative part of F(e) is nearly absent. A comparison of the F(t) functions obtained for Langmuir and FG local isotherms (cf. Figure 5a,c) shows that when lateral interactions are included in the local adsorption model the negative part of F(e) is eliminated and the reproduction of the overall isotherm is improved. For example, the assumption that y = 103 leads to reproduction of the overall isotherm with a 2.8% accuracy when the Langmuir local model is used, and with a 0.7% accuracy when the FG eq 7 is employed. Equation 7 takes into account the lateral interactions between molecules adsorbed onto homottatic patches (HPA). When the localized monolayer with laterally interacting molecules is formed on a surface with a random distribution of adsorption (RDA) sites (cf. eq 61, the calculatedF(c) function is broader than that for HPA (i.e., as illustrated in parts c and d of Figure 5, respectively) and appears as highly convoluted peaks over a range of
BET
5
e [kJ/molel
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i ; r i BWA HPA
error 10.1%
error=0.7%
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E 0.2
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error =03% 0.6-
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I
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E 0.2-
e [kllmolel
Figure 5. Distribution functions F(e) for N2 on Vulcan 3 calculated for different models of the local adsorption. The functions were calculated with INTEG using the same regularization parameter y = 0.001. The values shown in each part give the mean deviation of the calculated isotherm from the experimental one. energies from 1to 14 kJ/mol. Previously this effect has been observed in terms of the condensation approximation method.s The effect of mobile adsorption on F(e) can be evaluated via use of the Hill-de Boer model for the local isotherm (i.e., eq 8). In thislatter case, theF(e) distribution contains two sharp peaks which are shifted drastically in the direction of lower adsorption energies (cf. Figure 5e). A comparison of the F(e) distributions shown in Figure 5 and obtained for different local adsorption models indicates that the essential features of an adsorbent heterogeneity are reproduced by all. However, when lateral interactions in the surface are included, improvements in the reproduction of the overall adsorption isotherm are obtained as well as reductions in the negative part of F ( d . Also, the calculated distributions show additional features compared to those obtained using a Langmuir local model. It should be noted that the RD approximation generates the F(e) function which differs significantly from that obtained using the Langmuir local
2542 Langmuir, Vol. 9,No. 10,1993 I
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Heuchel et al.
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$ 0.6-
iz
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imental adsorption isotherms. Therefore, application of the non-negative restriction is not needed for calculating F(4. Model Specification. The adsorption energy distribution functions for gases adsorbed on reference carbonous materials have been calculated for all available data points from the lowest pressure value up to pressures which provide monolayer coverage. On the basis of the initial numerical studies discussed above, the Langmuir model was selected to represent the local adsorption because it reproduces well characteristic features of the heterogeneity of the adsorbents studied. Only for highly graphitized carbons does the FG model seem to be better than the Langmuir model. The regularization parameter y should be varied starting with y = 1until the data reproduction reaches the experimental accuracy. Fifty points of F(e) have been calculated using the INTEG program without non-negative restrictions.
Experimental Section A summary of the adsorption systems studied in this paper is given in Tables I and 11. The first table contains information
about the reference carbonaceous materials.All of these materials consist of similar structural units (graphitic planes) containing less than 10-20 aromatic rings, which are stacked more or less parallel by weak van der Waals forces. Most adsorbenta listed in Table I have been used by different research groups aa reference materials for studying microporous active carbons. They differ in the degree of graphitization and are generally divided intotwo groups, graphitizedand nongraphitized 0
10
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[kJ/molel Figure 6. Distribution functions F(E) for N2 on Vulcan 3 calculated by means of INTEG with the restriction F(t) 2 0 (NNLSmethod, see eq 13) for a Langmuir local isotherm without (a) and with (b) additional smoothing (y = 0.001). The values shown in each part give the mean deviation of the calculated isotherm from the experimental one. e
model. Therefore, in this latter case, the surface topography of the adsorption sites is important if F(e) is to be evaluated at submonolayer coverages using high coverage data where the effect of lateral interactions is significant. This problem can be reduced when low-coverage adsorption data are used to calculate F(e). Under these conditions, the Langmuir local adsorption model is thus sufficient for carrying out comparative studies of energetic heterogeneities of different adsorbents. Non-negative Least-Squares (NNLS) Method. A controversial topic in the numerical determination of F(e) is the a priori application of the restriction F(e) 10. From the physical point of view, the F(e) function should be positive. Since the inversion of the integral equation with respect to F(e) is purely a mathematical problem, the positive distribution function obtained by the regularization method without applying a non-negative restriction demonstrates that the assumed models are p h y s i d y realistic. The effect of the NNLS restriction on F(t) is shown in Figure 6 for the reference data using a Langmuir local model. Without regularization (y = 0) it is not possible to find an NNLS solution which represents the experimental data with an accuracy better than 13% (see Figure 6a). Interestingly, the stable NNLS solution with two sharp peaks at 7 and 11kJ/mol was obtained for y = 10-3(cf. Figure 6b). Szombathely et al." have found that the NNLS restriction usually leads to F(e) functions with more than two peaks. Figure 6b demonstrates that additional smoothing of F(e) (e.g., y = 10-3) does not improve the accuracy between the calculated and exper-
Carbons.
Graphitized materials are composed principally of planes of singleindex, e.g., "graphitized" carbon black prepared by heating thermal carbon blacks in a neutral atmosphere up to 3000 K. Thereare many indications- that these materials possess highly homogeneoussurfacesconsistingmainly of basal planes of single graphitic crystals. Their surface is often so homogeneous that adsorption of N2, Ar,or Kr gives stepwise isotherms. In 1961 Isirikyan and Kisele@l combined several adsorption isotherms for Nz at 77 K on graphitized carbon blacks to obtain a standard isotherm for nitrogen adsorption on the basal face of graphite (Graph C). Ita BET specific surface area is 7.6 m2/g. The f i t data points start at a relative pressure p / p , = 3 x 10-5. The relative pressure for monolayer surface coverage is equal to 0.0076 which is much lower than those for other nonporous carboneous materials. In the same paper Isirikyanand Kisele91 also reported standard isotherms for n-hexane and benzene on theaamegraphiticmaterial.The highlygraphitizedcarbon blacks are nonporous and highly homogeneous materialswithout surface groups. Graphitization leads to the enhancement of dispersive interactions and, consequently, to stepwise isotherms in the monolayer region. Typically,the surfaceproperties of carbon blacks are modified in three different ways aa a result of graphitization: (1) The overall surface becomes more uniform, (2) polar groups are removed, and (3)porosity is reduced significantly. The graphitization process leads to enhanced dispersive interactions and (26) Sellea-Perez,M. J.;Martin-Martinez,J. M. J. Chem.Soc., Faraday Trans 1 1991,87,1237. (26) Klinik, J.; Choma, J. Private communication. (27) Sing, K. S. W. Private communication. (28) Femandez-Colinas, J.; Denoyel, R.; Grillet, Y.; Rouquerol, F.; Rouquerol, J. Langmuir 1989,5, 1206. (29) Kaneko, K. Privata communication. (30) Choma, J.; Jaroniec, M. B i d . Wojsk. Akad. Tech. 1990,40, 23. (31) Ieirikyan, A. A,; Kiselev, A. V. J. Phys. Chem. 1991,66, 601. (32) Carrott, P. J. M.; Roberta, R. A.; Sing, K. S. W. Carbon 1987,6, 769. (33) Sella-Perez, M. J.; Martin-Martinez,J. M. Carbon 1992,30,41. (34) Carrott, P. J. M.; Roberta, R. A.; Sing, K. S. W. Langmuir 1988,
4, 740.
(36) Oberlin, A. In Chemistry and Physics of Carbon;Thrower, P. A., Ed.; Marcel Dekker: New York, 1989; Vol. 22, pp 1-143. (36) Rodriguez-Reinoso,F.; Linaree-Solano,A. Chem. Phys. Carbon 1989,21, 1.
Langmuir, Vol. 9, No.10, 1993 2643
Energetic Heterogeneity of Carbonaceous Materials
Table I. Noneorous carbons source remarks A 4.4 active carbons from olive stones 2073 K, Ar,30 min 7.3 active carbons from olive stones 2073 K, Ar,30 min 73.6 acetylene decomposition Elftex 120 37.4 ungraphitized carbon black from Cabot Carbon Ltd. (U.S.A.) vulcan 3 84.5 ungraphitized carbon black from Cabot Carbon Ltd.(U.S.A.) MK 80 carbon black from Mitsubishi Kasei Co. (Japan) 1223 K, Ar, 6 h SA0 107 active furance soot from antracene Graph Ca 7.6 thermal carbon blacks six thermal carbon blacks graphitized at 3273 300 K Ungraph Ca 25-84 carbon-coatedsilica three reference materials (Sooty Silica, Elflex 120,Sterling S) a Hypothetical standard adsorbents obtained by averaging experimental isotherms measured on various carbons. name
SBET (m2/g)
2
*
ref
7 25 26 27 28 29 30 31 32
Table 11. Information About Experimental Systems gas Ar (77 K). .
adsorbent AD
Nz (77K)
A
AP AP
@/P,)m, o.Oo0 1 0.005 0.005 0.005 o.Oo0 065
Elftex 120 Vulcan 3 o.Oo0 3 MK o.Oo0 33 Graph C O.Oo0 03 Ungraph C 0.005 0.005 n-butane (273K) AP 0.005 neopentane (273K) Elftex 120 293 K Graph C o.Oo0 1 O.Oo0 06 benzene (293K) SA0 benzene (293K) Graph C o.Oo0 5 a Error refers to Langmuir's local adsorption isotherm.
hence a sharpening of the Nz isotherm in the monolayer region. The removal of polar groupsfrom the surface leadsto considerable decreases in the specificinteractions. Further, the graphitization lowers the surface area of the carbon blacks. Rodriguez-Reinosoet al.7 have raised objections to the use of graphitized carbon blacks as reference adsorbents for assessment of microporosity of activated carbons because the chemical composition and hence the surface properties of graphite are significantly different from the surface properties of active carbons. The reference carbonaceous material should consist as with activecarbon of twisted graphite layers which contain defeds and which are cross-linked by an extended carbon network, but without micropores. In order to produce such nonmicroporous reference materials, Master and McEnanY7 heated activated carbons above 2000K in an Ar atmosphere. This heat treatment removes the micropores. Such ungraphitized carbons, e.g., A, Ap, Elftex 120, and Vulcan 3, have been used as reference materials. The low-temperature nitrogen isotherms for these reference materials are available in the literature.7s27v28Also, a few isotherms for other adsorbates such as argon, benzene, and neopentane are a ~ a i l a b l e .The ~ ~ isotherms ~ listed in Table I1 have been used to calculate the energy distribution functions for various reference carbonaceous materials. Although no detailed information is available about the structure of carbons AD, MK, and SAO, the latter material was heated to 1223 K, which is comparable to treatments of carbons A and Ap. Under these conditions, SA0 is most likely a nongraphitized carbon with small pore volume and/or which contains active surface groups. The Japanese MK carbon black should be comparable to Vulcan 3. The properties of the Polish AD carbon should be between those of Elftex 120 and Vulcan 3. It is interesting to note a large spectrum of properties for the ungraphitized standard materials. The specific surface areas of carbons A and Ap are around 1 order of magnitude lower than those for the other carbons. (37)Masters, K. J.; McEnaney, B. Carbon 1984,22, 595.
@IPS),
0.05 0.05 0.05
0.08 0.078 0.07 0.04 0.0075
0.075 0.11 0.03 0.06 0.06
error of data descriptiona (% )
points 12 6 6 6 6 14 46 21 11 10 11 17 16 19
5.0 0.5
0.4 1.0 0.3 2.9 2.3 2.8 (BW) 0.6 0.7 0.3 1.3 2.6
ref 26 33 7 25 27 28 29 31 32 33 34 31 30 31
0.6
0.5- Carbon AD
::
ARGON
3 0.4E n
0.3-
4.1'
0
4
5
1
10
I
15
1
e [kJ/molel Figure 7. Distribution functions F(c)for Ar on carbons Ap and AD (see Table I) calculated with the Langmuir model for the local adsorption.
Results and Discussion The adsorption energy distribution functions were calculated from all available data points up to monolayer coverage. Summarized in Table I1 are the number of data points, the lowest and highest value of the relative pressure, and the accuracy of the fit of the calculated isotherm using INTEG with a Langmuir model. The highest pressure value, @/pJ-, corresponds to monolayer coverage or lower. The monolayer coverages were taken from the original references or were determined by the BET method. Argon. Figure 7 shows the adsorption energy distribution functions for Ar on the Ap and AD carbons. The maximum of the F(c) function for these were, respectively, 10 and 12 kJ/mol. These values are in good agreement with the reported experimentaldifferentialheats for argon
2544 Langmuir, Vol. 9, No. 10, 1993
-0.25'
0
I
5
I
I
10
15
Heuchel et al.
1
20
e [kJ/molel Figure 8. Comparison of the distribution functionsF(c) for Ns on carbonaceous reference materials Graph C, Ungraph C, and Ap (see Table I) calculated with the Langmuir model for local adsorption.
0.01
,
i ' ' ' '
, "
-4
Ungrclph.C.
T
'5; 0.4-
-0.25'
0
5
I
to
.
.
5
15
I
20
e [kJ/molel Figure 9. Distribution functionsF(c)for Nz on Graph C for the
local isotherm without (dotted line for Langmuir eq 2) and with lateral interactions (solid line for Fowler-Guggenheim eq 7 with
3 6
6
'
0.6
NITROGEN Elftex120 0.4-
Y
Y
.
'
Figure 10. Calculated adsorption isothermsfor N2 on Graph C with the distribution functions F(c) shown in Figure 9 in comparison with the experimentalpoints (circles): the solid line refers to Fowlel-Guggenheim eq 7 with HPA; the dotted line refers to Langmuir eq 2. NITROGEN
.
'
log(P/P,)
0.6
0.25
'
-3
g'o,2 0.0
0
10
5
15
20
0
5
10
15
20
e IkJ/molel
e IkJ/mole]
0.6
0.6
NllTROQEN
NITMEN
HPA).
adsorption on carbon blacks, e.g., 10.7 and 11.1 kJ/mo12. The energy difference in the maxima for the two samples is consistent with the differences in the specific surface area (Le., the AD carbon specific surface area is 10 times higher than that of the Ap carbon). Carbons with relative large specific surface areas often possess a small amount of fine pores and other active centers, which interact strongly with adsorbate molecules. The small negative part in the F(e) function of AD carbon is likely due to the very sharp peak shape. Nitrogen. The calculated F(e) functions for Na on nongraphitized adsorbents A, Ap, and Ungraph C are very similar. Because the results for carbons A and Ap are similar, only one of these is shown in Figure 8. The F(e) functions for these carbons contain a maximum at 8 kJ/ mol and a small shoulder or second peak a t about 10 kJ/ mol. The maximum of F(e) for N2 on the Ap carbon (cf. Figure 8) is nearly 2 kJ/mol lower than that for the Ar carbon, which is in good agreement with other reported experimental d a h 2 A detailed comparison of the normalized energy distribution functions for Graph C and Ungraph C shows that for graphitized carbon the major peak of F(e) is sharper with a maximum which is located a t higher energy. For this carbon the Langmuir model gives a maximum at 12 kJ/mol (see Figure 9), which is 4 kJ/mol higher than that for ungraphitized carbons. However, the representation of the data with the Langmuir local model is not good (Figure 10). The experimental points form an S-shaped isotherm which does not fit the
0
5
10 E
15
tkllmolel
20
0
5
10 E
15
20
[Wlmolel
Figure 11. Distribution functions F(E)for Nz on carbonaceous reference solids (UngraphC, Elftax 120,MK carbon, and Vulcan 3) calculated with the Langmuir model for local adsorption. Langmuir model. Additionally, it was not possible to eliminate the negative oscillations in the F(E)by changing the regularization parameter y. However, a numerically stable distribution F(e),which has a sharp peak at 11 kJ/ mol (cf. Figure 91,was obtained using an adsorption model which includes lateral interactions, Le., the FowlerGuggenheim eq 6 with the number of nearest neighbors z = 2,the interaction energy w/k = ~ L J= 95 K, and y = 0.01. Although there is a small second peak at 9 kJ/mol, this stable distribution reproduces the S-shaped isotherm in the semilogarithmic coordinates (see Figure 10). The Nz adsorption on ungraphitized carbons MK, Vulcan 3, and Elftex 120 is shown in Figure 11 as well as F(c) for Ungraph C for comparative purposes. The calculated F(e) functions have two peaks for all of the
Langmuir, Vol. 9, No.10, 1993 2545
Energetic Heterogeneity of Carbonaceous Materials 0.15
30
BENZENE
40
50
80
Carbon Ap
70
[kJ/molel Figure 12. Distribution functions F(e) for benzene on Graph C (dotted line) and carbon SA0 (solid l i e ) calculated for the Langmuir local isotherm. e
[kJ/molel Figure 13. Distribution functions F(c) of different gases on carbon Ap: Ns,Ar,and n-butane. e
0.30
reference carbons. The lower-energy peaks are between 6 and 9 kJ/mol, whereas the higher-energy peaks appear between 11 and 12kJ/mol. For MK and Vulcan 3carbons, the second peak is nearly 2 times larger than the first. For Elftex 120 the peaks are nearly identical in magnitude. The presence of two peaks suggesta the existence of two main energetic states for nitrogen molecules on the surface of the reference carbons. A comparison of these peaks with F(c)for Ungraph C indicates that the lower peak can be associated with adsorption on ungraphitized regions of carbon. However,the higher peakmay reflect the energetic heterogeneity of carbon surfaces that are partially graphitized. There is an interesting trend between the peak ratios and the BET surface areas of the samples. For the MK and Vulcan 3carbons which have surface areas nearly twice that of Elftex 120 the high-energy peaks are nearly double the low-energy peaks. Benzene. Benzene adsorption isotherms were available for two reference carbonmaterials. The calculated energy distribution function, F(e),for Graph C is a Gaussian-like peak located in the range 32-48 kJ/mol with a maximum at 43 kJ/mol (see Figure 12). Although experimental isotherms are modeled well with the F(c) function obtained using a Langmuir kernel, it can be improved by taking into account lateral interactions on the surface. Under these latter conditions, the small negative part in F(c) diminishes. The differential heat of adsorption reported in ref 31 is 41 kJ/mol which is in good agreement with the maximum of the F(e)function at 43kJ/mol. The calculated F(c) function for SA0 is much broader (30-53 kJ/mol) and shows two well-separated peaks, at 48 and about 39 kJ/mol. On the basis of these resulta, the SA0 carbon seems to be more heterogeneous than Graph C. A comparison of F(c) with that for Graph C suggesta that the small peak at 48 kJ/mol arises for the energetic heterogeneity associated with active surface groups. Different Gases on the Same Adsorbent. Figures 13-15 compare the calculated adsorption energy distribution functions of different adsorbates on the same carbon. In Figure 13 the F(e) functions for nitrogen and argon have sharp peaks on the Ap carbon, while F(c) for n-butane is a broader distribution with two maxima at 28 and 37 kJ/mol. These values are comparable to experimental values which have been reported by Beebe et al.38 on Spheron (i.e., adsorption energies in the range of 31-65 kJ/mol and the integral heat of adsorption about 43 kJ/ mol). Because nitrogen and butane molecules sample two (38)Beebe, R.A.;Polley,M.H.;Smith, W.R.;Wendell,C.B.J. Am. Chem. SOC.1947,69,2294.
Emex 120
I
4.05
0 1 0 2 0 3 0 4 0 5 0
e
[kJ/molel
Figure 14. Adsorption energy distributions F(e) for Nz and neopentane on Elftes 120.
-
8 8
Y
t
-. NlTROaEN
0.6
......
n-HEXANE
1
O 0.2 'II
-0d 0
'
'
'
'
'
'
1 0 2 0 9 0 4 0 5 0 6 0
e
[kJ/molel
Figure 15. Energy distribution functions F(c) for Ns,benzene, and n-hexane on Graph C.
different kindsof surface heterogeneitieson the Ap surface, the splitting of F(c) into two peaks arises from structural differences between the adsorbate molecules. This observation is supported by the resulta shown in Figure 14 for spherical neopentane on Elftex 120,which gives only a single peak between 25 and 45 kJ/mol. The maximum of this peak is about 34 kJ/mol. Figure 15 shows the distribution functions for nitrogen, benzene, and n-hexane on the graphitized carbon Graph C. Both hydrocarbons have broader distributions than that for nitrogen. This result is in good agreement with theoretical considerations about the adsorption of molecules of different sizes on energetically heterogeneous
2546 Langmuir, Vol. 9, No.10,1993
surfaces.39 The greater value of the maximum of F(r) for n-hexane in comparison to benzene is in agreement with calculated heats of adsorption.31
Conclusions A detailed comparison of the energy distribution functions for various nonporous carbons used as reference adsorbents has shown how much these materials differ. Since adsorption isotherms on such reference materials are employed in complvative characterization methods for microporous carbons, knowledge about the F(c) functions for these solids is important for selecting the proper standard isotherm. For example, the current work demonstrates that the surface heterogeneities of Vulcan and MK carbons differ slightly. Additionally, the surface (39)M “ k i , A. W.; DerylcA4”wka,A.; Jaroniec,M. J. Colloid Interface Sci. 1986,l09,310.
Heuchel et al. properties of these carbons differ from those of the Ap, A, and Ungraph C carbons. Elftex 120 carbon falls between these two groups. Analysis of the energetic distribution functions has shown differences in F(c) for graphitized and nongraphitized carbons. Further studies in this direction should provide answers to the question of how the degree of graphitization changes the F(c)shape. The application of different probe molecules also can provide additional information about the energetic heterogeneity of the reference carbons.
Acknowledgment. The authors wish to thank Drs. J. Choma (Poland), K. Kaneko (Japan), J. Klinik (Poland), and K. S. W. Sing (U.K.) for providing four standard isotherms in tabulated form; information about these isotherms is given in Tables I and 11.
