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solid adsorption isotherms by energy distribution functions
Journal of Colloid and Interface Science 331 (2009) 329–334
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Design of liquid/solid adsorption isotherms by energy distribution functions Grit Kalies ∗ , Peter Bräuer, Michael v. Szombathely Institute of Experimental Physics I, University of Leipzig, Linnéstrasse 5, D-04103 Leipzig, Germany
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 October 2008 Accepted 27 November 2008 Prof. Dr. Mietek Jaroniec on the occasion of his 60th birthday Keywords: Adsorption excess isotherm Binary liquid mixture Adsorption energy distribution function Adsorption integral equation
Adsorption excess isotherms of binary liquid mixtures have been calculated from synthetic adsorption energy distribution functions characterizing energetic heterogeneity at the liquid/solid interface. In order to see consequences for the adsorption isotherms, the distribution functions were varied systematically. In this way, the sensitivity or the lack of sensitivity of liquid-phase adsorption isotherms over the whole concentration range for changes in energy distribution functions became evident. The question of to what extent it makes sense to use liquid adsorption measurements to obtain relevant information on a solid’s energetic heterogeneity is answered. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Two kinds of heterogeneity of porous solids can be distinguished: geometric and energetic heterogeneity [1,2]. While pore size distribution functions describe the geometric heterogeneity, adsorption energy distribution functions describe the energetic heterogeneity. In determining the engineering characteristics of solids and understanding various processes inside pores, such as immersion, adsorption, storage, diffusion, and catalysis, the knowledge of both distribution functions is indispensable. In recent years, significant progress has been achieved in determining porosity, pore size distributions, and other textural parameters of porous solids from adsorption isotherms [3]. The calculation of relevant energetic information from gas or liquid adsorption isotherms on porous solids and the design of surface chemistry are a challenging research field. Adsorption energy distributions can be obtained from measured adsorption isotherms (so-called total isotherms) by different mathematical methods such as regularization [4–7] and Stieltjes transformation [8,9]. The regularization method is a purely numerical method taking into consideration the so-called ill-posed problem of the adsorption integral equation. The Stieltjes transformation provides an analytical solution of the adsorption integral equation. In the case of low-temperature pure gas adsorption, many attempts have been made to calculate relevant energy distributions, e.g., [7,10,11], and commercial software tools are even available [12]. Compared with gas adsorption, liquid adsorption measure-
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0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.11.072
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2008 Elsevier Inc. All rights reserved.
ments are rather seldom used to characterize porous solids. Since from the liquid phase only competitive adsorption can be studied, energy distribution functions at the liquid/solid interface are quite complex. Only the differences F (U 21 ) = F (U 2 − U 1 ) of the adsorption energies of components 2 and 1 in a binary liquid mixture can be obtained. However, liquid adsorption can provide additional or confirming energetic information. A number of works have investigated (n) the calculation of F (U 21 ) distributions from liquid-phase Γ2 (x2 ) adsorption measurements [13–17]. In this paper, we will go the inverse way. A set of synthetically created Gaussian adsorption energy distribution functions are used (n) to design Γ2 (x2 ) adsorption excess isotherms. By systematic variation of (1) the position of peaks in the F (U 21 ) energy distribution function, (2) the shape of peaks in the F (U 21 ) energy distribution function, (3) and the interaction model of the surface and the bulk phases, a spectrum of adsorption isotherms is generated. It forms the basis for a deeper understanding of the liquid-phase isotherms’ dependence on the assumed energetic heterogeneity of an adsorbent. The relevance of energy distributions calculated from measured liquid adsorption isotherms can then be assessed. 2. Calculations 2.1. F (U 21 ) energy distribution functions in the interval −∞ < U 21 < ∞ Gaussian peak distribution functions for positive and negative U 21 values were calculated from
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G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
F (U 21 ) =
m
f jm exp −
j =1
1
U jm − U 21
2Δ2jm
2
;
−∞ < U 21 < ∞; m = 1, 2, 3,
(1)
where Δ2jm is the peak’s variance, U jm is the position of the peak maximum, f jm is the magnitude of the Gaussian peak, and m is the number of Gaussian peaks. The normalization condition for the adsorption energy distribution,
∞ F (U 21 ) dU 21 = 1,
(2)
−∞
yields after a few mathematical manipulations
√
2π
m
f jm Δ jm = 1;
j = 1, 2, . . . , m,
(3)
j =1
which can be used to estimate f jm in Eq. (1), if Δ jm j = 1, 2, . . . , m is given. 2.2. F (U 21 ) energy distribution functions in the interval 0 U 21 < ∞ Gaussian peak distribution functions for positive U 21 values alone were calculated by F (U 21 ) =
m
f jm exp −
j =1
1 2Δ2jm
U jm − U 21
2
;
0 U 21 < ∞; m = 1, 2, 3.
(4)
The normalization condition
∞ F (U 21 ) dU 21 = 1
(5)
0
yields
√
m 2π
2
f jm Δ jm
j =1
U jm 1+Φ √ 2Δ jm
= 1;
j = 1, 2, . . . , m
(6)
(n)
tributions by the well-tried equation of Ostwald and Izaguirre [19],
mA
s x2t (x2 ) − x2 ,
(7)
(n)
where Γ2 (x2 ) is the reduced adsorption excess of component 2 as a function of the equilibrium mole fraction x2 of component 2 in the bulk phase; ns is the total mole number in the surface phase; s (x2 ) is the total mole fraction of m A is the mass of adsorbent; x2t component 2 in the surface phase, and the fundamental integral equation for the physical adsorption of binary nonelectrolytic liquid mixtures on energetically heterogeneous solid surfaces [1,13, 20,21], U 21,max s (x2 ) x2t
x2s (x2 ) F (U 21 ) dU 21 ,
= U 21,min
(9)
where x21 = x2 /(1 − x2 ), and C is the interaction function, T the absolute temperature, and R the universal gas constant. The interaction function C takes into account all molecular interactions between molecules in the surface and bulk phases and hence depends on the model of the surface and bulk phases and the topography of adsorption sites on the solid surface [1,13]. (n) In this paper, the Γ2 (x2 ) adsorption isotherms were obtained by assuming ideality as well as nonideality in the bulk and the surface phases. For assumed ideality, C = 1. For assumed reality, the molecular interactions (the number of nearest neighbors, the interaction energies between neighbors) were described by the lattice theory, applying the Bragg–Williams approximation (BWA) and the quasi-chemical approximation (QCA). The topography of adsorption sites on the solid surface was described by the random distribution approximation (RDA). The equations for the calculation of the interaction function C have been taken from [13]. In the following, the parameters applied are listed:
• U 21,min = −6 kJ/mol, U 21,max = 6 kJ/mol for F (U 21 ) in the interval −∞ < U 21 < ∞, • U 21,min = 0 kJ/mol, U 21,max = 6 kJ/mol for F (U 21 ) in the interval 0 U 21 < ∞, • ns /m A = 7 mmol/g, • the numbers of nearest neighbors in the bulk and sorption phases: c b = 12, c s = 6 (Fig. 8), • the Lennard–Jones parameter for ideal mixtures (C = 1):
• the Lennard–Jones parameter for real mixtures (Fig. 8):
b s ε21 /k = ε21 /k = −110 K,
Γ2 (x2 ) adsorption excess isotherms of binary liquid mixtures were calculated from the synthetic continuous F (U 21 ) energy dis-
(n)
,
b s ε22 /k = ε22 /k = −100 K,
(n)
Γ2 (x2 ) =
C exp(U 21 / R T )x21 1 + C exp(U 21 / R T )x21
b s ε11 /k = ε11 /k = −150 K,
2.3. Γ2 (x2 ) adsorption excess isotherms
x2s (x2 ) =
b s b s b s ε11 /k = ε11 /k = ε22 /k = ε22 /k = ε21 /k = ε21 /k,
with the known definite error integral Φ( z) [18]. Equation (6) is used to estimate the f jm values in Eq. (4).
ns
s where x2t (x2 ) is the so-called total isotherm; x2s (x2 ) is the local mole fraction of component 2 in the surface phase, the so-called local isotherm; U 21 is the difference U 2 − U 1 of the adsorption energies of the molecules 2 and 1 on identical surface sites; and F (U 21 ) dU 21 is the fraction of surface sites with differences in adsorption energies between U 21 and U 21 + dU 21 . The x2s (x2 ) local isotherm is given by [13]
(8)
• R = 8.31441 × 10−3 kJ/mol/K, • T = 298.15 K. The molecular partition functions for the internal degrees of freedom are assumed to be equal in the bulk and the sorption phases for each component; i.e., q1s = qb1 , q2s = qb2 . 3. Results and discussion The discussion of the results will be based on combined presentations of synthetic F (U 21 ) energy distributions and the cor(n) responding calculated Γ2 (x2 ) adsorption excess isotherms. The parameters of the F (U 21 ) distributions are listed in Table 1. On the left-hand side of Fig. 1, seven different F (U 21 ) energy distributions calculated by means of Eq. (1) in the interval −6 U 21 6 are presented:
• 1 narrow and high one-Gaussian peak energy distribution function at 4 kJ/mol (distribution 1);
G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
331
Table 1 Parameters of the Gaussian F (U 21 ) energy distributions F (U 21 ).a Figure
Fig. 1
Number of different F (U 21 )
Name of F (U 21 )
m
7
1 2
1 2
3
2
4
2
5
2
6
2
7 1 2 3 1 2
1 1 1 1 1 2
3
2
4
2
5
2
6
2
7 1 2 3 1 2 3 1 2
1 1 1 1 1 1 1 1 2
3
3
1
3
2
3
1
2
Fig. 2
3
Fig. 3
7
Fig. 4
3
Fig. 5
3
Fig. 6
3
Figs. 7, 8
Fig. 9
2
1
j
1 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 3 1 2 3 1 2 3 1 2
f jm (mol/kJ) 0.797900 0.050000 0.747900 0.170000 0.627900 0.250000 0.547900 0.398942 0.398942 0.547900 0.250000 0.332452 1.329808 0.664904 0.332452 1.595800 0.100000 1.495800 0.340000 1.255800 0.500000 1.095800 0.797884 0.797884 1.095800 0.500000 0.664904 1.330370 1.330370 1.329810 2.659616 1.329808 0.664904 0.664904 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.443269 0.221635 0.295513 0.398942 0.398942
U jm
Δ jm
(kJ/mol)
(kJ/mol)
4
−4 4
−4 4
−4 4
−4 4
−4 4 0 0 0 0 5 1 5 1 5 1 5 1 5 1 5 3 1 3 5 3 3 3 3 1 5 1 3 5 1 3 5 1 3 5 −4 4
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 1.200 0.300 0.600 1.200 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.600 0.300 0.300 0.300 0.150 0.300 0.600 0.600 0.250 0.250 0.167 0.167 0.167 0.167 0.167 0.167 0.300 0.600 0.450 0.500 0.500
See Eqs. (1) and (6). m = the number of Gaussian peaks, j = the name of the jth Gaussian peak, Δ2jm = the peak’s variance, U jm = the position of the peak’s maximum, and f jm = the magnitude of the isolated or nonisolated Gaussian peaks.
Fig. 1. Seven different Gaussian peak distributions F (U 21 ) in the interval −6 (n)
U 21 6 and inverted U- and S-shaped Γ2
isotherms for C = 1.
Fig. 2. Three different Gaussian peak and one delta function distributions F (U 21 ) in (n)
the interval −6 U 21 6 and S-shaped Γ2
isotherms for C = 1.
• the position of the adsorption azeotropes is shifted to smaller mole fractions.
a
• 5 two-Gaussian peak energy distribution functions (the shape of the distribution is varied systematically; the peaks at 4 kJ/mol decrease, whereas the peaks at −4 kJ/mol increase from distribution 2 to 6); • 1 wide one-Gaussian peak energy distribution function at 0 kJ/mol (distribution 7). (n)
The corresponding seven ideal Γ2 adsorption isotherms (C = 1) are given on the right-hand side of Fig. 1. Isotherm 1 (calculated from distribution 1) is the highest and corresponds to type II of the classification of Schay and Nagy [22,23]. In the order from isotherm 1 to isotherm 6 we find that
It can be seen that the isotherms’ dependence on changes in the shape and symmetry of the F (U 21 ) distributions becomes evident. As distribution 5 is symmetric, we find a symmetric S-shaped Type IV isotherm 5. The distributions 4 and 6 are laterally reversed. Hence, the isotherms 4 and 6 are mirrored through the X axis. From distribution 7 with a peak maximum at 0 kJ/mol follows a very small Type IV isotherm 7 with a maximum at x2 = 0.25 of (n) Γ2 ≈ 0.073 mmol/g. Fig. 2 shows four F (U 21 ) energy distributions with peak maxima at 0 kJ/mol:
• 1 delta function, • 3 one-Gaussian peak functions of different magnitudes (distributions 1–3; distribution 3 is equivalent to distribution 7 of Fig. 2).
(n)
• the magnitude of the Γ2 adsorption isotherms decreases, • the maximum is shifted to smaller mole fractions, • the type of isotherm is changed from type II/III (isotherms 1 and 2) to type IV/V (isotherms 3–6) with adsorption azeotropes,
(n)
The corresponding S-shaped Γ2 isotherms are presented on the right-hand side of Fig. 2. For the delta function at 0 kJ/mol, Γ2(n) has to be zero, since components 1 and 2 do not differ in their adsorption energies.
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G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
Fig. 3. Seven different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 (n)
6 and inverted U-shaped Γ2
isotherms for C = 1.
Fig. 5. Three different Gaussian peak and one delta function distributions F (U 21 ) in (n)
the interval 0 U 21 6 and inverted U-shaped Γ2
isotherms for C = 1.
As the ideal isotherms calculated from Gaussian peaks and delta functions at the same position overlap nearly completely, only (n) three Γ2 isotherms can be distinguished with the naked eye. (n)
The magnitude of the Γ2 adsorption isotherms increases noticeably with increasing adsorption energy difference U 21 = U 2 − U 1 (n) from 1 to 5 kJ/mol. Additionally, the maximum of Γ2 is shifted in the direction of smaller mole fractions with increasing U 21 energy differences. As component 2, with higher adsorption energy, is preferentially adsorbed over the whole concentration range, the (n) Γ2 isotherms correspond to type I or II of the classification of Schay and Nagy. Let us summarize the results of Figs. 1–4. In Figs. 1–3 we changed the shape of peaks in the F (U 21 ) energy distribution functions systematically, in Fig. 4 the position of peaks in the interval (n) 0 U 21 6. We found that Γ2 isotherms respond sensitively to both the position and the shape of the peaks in F (U 21 ); i.e., both (n) factors influence the size and shape of Γ2 systematically. On the (n)
Fig. 4. Three different Gaussian peak and three delta function distributions F (U 21 ) (n)
in the interval 0 U 21 6 and inverted U-shaped Γ2
isotherms for C = 1.
In Fig. 3, the energy distributions of Fig. 1 are shifted to the interval 0 U 21 6 and the peak’s variance is decreased (see Eq. (4) (n) and Table 1). For the corresponding seven inverted U-shaped Γ2 adsorption isotherms (C = 1), we find in the order of isotherms 1 to 6 that
• the magnitude of the Γ2(n) adsorption isotherms decreases, and • the isotherms are all of type II, but the maximum is slightly shifted in the direction of higher mole fraction (decreased selectivity). The isotherms’ dependence on changes in the shape and symmetry of F (U 21 ) again becomes evident. Isotherms 5 and 7 are quite similar, as they are both calculated from distributions with the symmetric axis at 3 kJ/mol. On the left-hand side of Fig. 4, six different F (U 21 ) energy distributions calculated by means of Eq. (4) in the interval 0 U 21 6 are shown:
• 3 one-Gaussian peak energy distribution functions with average peak values at 1, 3, and 5 kJ/mol (the position of the peaks is varied), and • 3 delta functions at 1 kJ/mol, 3 kJ/mol and 5 kJ/mol.
other hand, Γ2 isotherms are not very sensitive to changes in F (U 21 ) if the symmetric axis is maintained (cf. Fig. 2 or the distributions 5 and 7 in Fig. 3). In Fig. 4, delta and Gaussian functions even provide indistinguishable inverted U-shape isotherms. In the following, we will consider the sensitivity of inverted (n) U-shaped Γ2 adsorption isotherms to differences in F (U 21 ) by keeping the symmetric axis at 3 kJ/mol. First, Fig. 5 shows four F (U 21 ) distributions with peak maxima at 3 kJ/mol:
• 1 delta function, • 3 one-Gaussian peak functions of different magnitudes (distributions 1–3; distribution 3 is equivalent to distribution 7 of Fig. 3). (n)
The corresponding Γ2
isotherms (C = 1) are very similar, and
(n)
the maximal ΔΓ2 difference of about 0.015 mmol/g is smaller than the maximal difference between isotherms 1 and 3 in Fig. 2 (n) (ΔΓ2 ≈ 0.07 mmol/g). Taking into account the experimental er(n)
ror of Γ2 isotherms of ±0.03 mmol/g [24,25], it will be difficult to extract clear information from such adsorption isotherms. Second, Fig. 6 shows three F (U 21 ) distributions with different peak numbers:
• 1 one-Gaussian peak distribution at 3 kJ/mol, • 1 two-Gaussian peak distribution at 1 and 5 kJ/mol, • 1 three-Gaussian peak distribution at 1, 3, and 5 kJ/mol.
G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
Fig. 6. Two different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 6 (n)
and inverted U-shaped Γ2
isotherms for C = 1.
333
Fig. 8. Two different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 6 (n)
and inverted U-shaped Γ2
isotherms for BWA and QCA.
Fig. 9. One Gaussian peak distribution F (U 21 ) in the interval −6 U 21 6 and Fig. 7. Three different Gaussian peak distributions F (U 21 ) in the interval 0 (n)
U 21 6 and inverted U-shaped Γ2
(n)
S-shaped Γ2
isotherms for BWA as functions of the values of c B and c s .
isotherms for C = 1.
(n)
As can be seen, the Γ2 isotherms (C = 1) differ from each other, and the magnitude decreases with increasing distance of peak area in the order isotherm 1, isotherm 3, isotherm 2. How(n) ever, the maximal ΔΓ2 difference between isotherm 2 (calculated from two peaks) and isotherm 3 (calculated from three peaks) is only 0.04 mmol/g; i.e., it lies again in the region of the (n) experimental error of Γ2 . Third, Figs. 7 and 8 present the results for two different threeGaussian peak distributions:
• a three-Gaussian peak distribution with unconnected narrow and high peaks at 1, 3, and 5 kJ/mol,
• a three-Gaussian peak distribution with connected peaks at 1, 3, and 5 kJ/mol. (n)
On the right-hand side of Fig. 7, the ideal Γ2 isotherms (C = 1) are shown; on the right-hand side of Fig. 8, the real ones calculated by means of QCA and BWA. In all three cases, the two isotherms are very similar. Much more pronounced quantitative differences are found between ideal and real isotherms, indicat(n) ing that the model assumptions influence the Γ2 isotherms more strongly than differences in the F (U 21 ) distribution functions. The QCA and BWA isotherms differ more from each other than in the case of gas phase calculations [26], whereas the type of isotherm
is not changed by varying the statistical thermodynamic model for molecular interactions. Fig. 9 illustrates the well-known influence of another model assumption: the number of nearest neighbors in the bulk and sorption phases. The two-Gaussian peak distribution F (U 21 ) on the left-hand side of Fig. 9 corresponds to distribution 5 in Fig. 1. (n) By applying BWA, we get significantly different Γ2 isotherms as functions of a varied number of nearest neighbors in the sorption phase. Figs. 5–8 confront us with the limitations of F (U 21 ) calculations (n) from Γ2 liquid adsorption isotherms. Especially in the case of inverted U-shaped isotherms, considerable differences in the F (U 21 ) (n) distributions yield only weak differences in the Γ2 isotherms. We have to realize that this fact makes it difficult to obtain trustworthy (n) energy information from Γ2 . Let us summarize the facts influencing the reliability of F (U 21 ) energy distributions from liquid-phase adsorption measurements: (n)
1. Liquid adsorption isotherms are fixed at x2 = 0, Γ2 (n)
= 0 and
x2 = 1, Γ2 = 0; i.e., they possess less freedom than gas adsorption isotherms to react to differences in energy distribu(n) tions. Thus, Γ2 isotherms are a priori less sensitive than gas adsorption isotherms. (n) 2. It is true that Γ2 isotherms respond sensitively to changes of the position or the shape of peaks in F (U 21 ), but a shift of the symmetric axis is required. Different distributions, by
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G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
keeping the same symmetric axis, yield very similar isotherms (see Figs. 5–8); i.e., the isotherms do not “feel” differences in peak width of peak connectivity and even reflect changes in the peak number inadequately (cf. Fig. 6). At this point, the isotherms are under-determined and insensitive to changes in F (U 21 ). 3. Inverted U-shaped isotherms are less sensitive to changes in F (U 21 ) than S-shaped isotherms; i.e., the calculation of welldefined F (U 21 ) distributions from inverted U-shape isotherms is complicated. (n) 4. The experimental error of Γ2 adsorption isotherms is about
(n)
Our study indicates that experimental errors of Γ2 , model assumptions, uncertain geometric effects, etc. decisively influence the F (U 21 ) distributions, even if the numerical method for solving the ill-posed problem in the adsorption integral equation is advanced. The limitations of the evaluation of heterogeneity effects from liquid adsorption isotherms are evident. Regarding the various facts influencing the reliability of F (U 21 ), we have to recognize that there is a risk of overrating characterization results from liquidphase adsorption. In interpreting F (U 21 ) distributions, we have to act with caution and should use other independent methods to confirm information on energetic heterogeneity of solids.
(n)
±0.03 mmol/g; i.e., the changes in Γ2 caused by differences in F (U 21 ) are often smaller than the error of the isotherms. (n) 5. Model assumptions can influence Γ2 more strongly than differences in F (U 21 ) (cf. Figs. 8, 9); i.e., our ideas from reality determine the calculation results, not the given facts. 6. In liquid-phase adsorption, the distinction between geometric and energetic information is more complicated than in gas-phase adsorption. While the low-pressure region in gas isotherms is connected with energetic information and the (n) high-pressure region with geometric information, Γ2 isotherms do not have distinct regions that can be assigned to geometric or energetic effects. As we know that especially in micro- and nanoporous adsorbents, the adsorption effects caused by geometric or energetic heterogeneity strongly interfere with each other, the extraction of purely energetic information is complicated. 7. In view of the above-mentioned points, the well-known problems in the solution of the adsorption integral equation (Eq. (8)) as an ill-posed problem, i.e., the required a priori assumption of the local isotherm and the fact that small changes s in x2t caused by experimental errors, can significantly influence the energy distribution play an increased role in the (n) calculation of F (U 21 ) from Γ2 . 4. Summary Synthetic F (U 21 ) adsorption energy distribution functions and (n) their corresponding Γ2 adsorption excess isotherms for binary liquid mixtures over the whole concentration range have been presented. By systematic variation of the parameters of F (U 21 ), its (n) influence on the size and shape of Γ2 has been studied. It is shown that both the position and the shape of en(n) ergy peaks in F (U 21 ) influence the Γ2 isotherms systemati(n)
cally. On the other hand, the sensitivity of Γ2 to changes of F (U 21 ) keeping the symmetric axis is quite limited. Inverted Ushaped isotherms only slightly reflect differences in peak width, peak connectivity, and peak number, which makes it much more complicated to extract trustworthy energy information from such isotherms.
Acknowledgment The financial support for this project by the Deutsche Forschungsgemeinschaft (DFG, Ka 1560/3-1, 4-1) is gratefully acknowledged. References [1] M. Jaroniec, R. Madey, Physical Adsorption on Heterogeneous Solids, Elsevier, Amsterdam, 1988. ´ [2] W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, London, 1992. [3] P. Llewellyn, J. Rouquérol, N. Seaton (Eds.), Characterization of Porous Solids VII: Proceedings of the 7th International Symposium on the Characterization of Porous Solids (Cops-VII), Elsevier, Amsterdam, 2006. [4] A.N. Tichonov, Dokl. Akad. Nauk SSSR 39 (1943) 195. [5] A.N. Tichonov, Dokl. Akad. Nauk SSSR 153 (1963) 49. [6] A.N. Tichonov, Sov. Math. 4 (1963) 1624. [7] M. v. Szombathely, P. Bräuer, M. Jaroniec, J. Comput. Chem. 13 (1992) 17. [8] P. Bräuer, M. Fassler, M. Jaroniec, Thin Solid Films 123 (1985) 245. [9] P. Bräuer, M. Fassler, M. Jaroniec, Chem. Phys. Lett. 125 (1986) 241. [10] M. Heuchel, M. Jaroniec, R.K. Gilpin, P. Bräuer, M. v. Szombathely, Langmuir 9 (1993) 2537. [11] D.D. Do, Adsorption Analysis: Equilibrium and Kinetics, Series on Chemical Engineering, vol. 2, Imperial College Press, London, 1998. [12] J.P. Olivier, W.B. Conklin, M. v. Szombathely, DFT-Users Guide, Micromeritics Instrument Corporation, 1994. [13] M. Heuchel, P. Bräuer, M. v. Szombathely, U. Messow, W.D. Einicke, M. Jaroniec, Langmuir 9 (1993) 2547. [14] M. Heuchel, M. Jaroniec, Langmuir 11 (1995) 1297. [15] P. Podko´scielny, A. Dabrowski, ˛ M. Bülow, Appl. Surf. Sci. 196 (2002) 312. [16] A. Dabrowski, ˛ P. Podko´scielny, M. Bülow, Colloids Surf. A Physicochem. Eng. Aspects 212 (2003) 109. [17] P. Podko´scielny, K. Nieszporek, P. Szabelski, Colloids Surf. A Physicochem. Eng. Aspects 277 (2006) 52. [18] G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers, second ed., Dover, New York, 2000. [19] Wo. Ostwald, R. de Izaguirre, Koll. Z. 30 (1922) 279. ´ [20] W. Rudzinski, J. O´scik, A. Dabrowski, ¸ Chem. Phys. Lett. 20 (1973) 5. [21] M. Jaroniec, P. Bräuer, Surf. Sci. Rep. 6 (1986) 65. [22] G. Schay, L. Nagy, T. Szekrenyesy, Period. Polytechnika 4 (1960) 95. [23] L. Nagy, G. Schay, Acta Chim. Hung. 93 (1963) 365. [24] G. Kalies, P. Bräuer, U. Messow, J. Colloid Interface Sci. 275 (2004) 410. [25] R. Rockmann, G. Kalies, O. Klepel, Adsorption 13 (2007) 515. [26] P. Bräuer, M. Jaroniec, J. Colloid Interface Sci. 108 (1985) 50.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Design of liquid/solid adsorption isotherms by energy distribution functions Grit Kalies ∗ , Peter Bräuer, Michael v. Szombathely Institute of Experimental Physics I, University of Leipzig, Linnéstrasse 5, D-04103 Leipzig, Germany
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 21 October 2008 Accepted 27 November 2008 Prof. Dr. Mietek Jaroniec on the occasion of his 60th birthday Keywords: Adsorption excess isotherm Binary liquid mixture Adsorption energy distribution function Adsorption integral equation
Adsorption excess isotherms of binary liquid mixtures have been calculated from synthetic adsorption energy distribution functions characterizing energetic heterogeneity at the liquid/solid interface. In order to see consequences for the adsorption isotherms, the distribution functions were varied systematically. In this way, the sensitivity or the lack of sensitivity of liquid-phase adsorption isotherms over the whole concentration range for changes in energy distribution functions became evident. The question of to what extent it makes sense to use liquid adsorption measurements to obtain relevant information on a solid’s energetic heterogeneity is answered. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Two kinds of heterogeneity of porous solids can be distinguished: geometric and energetic heterogeneity [1,2]. While pore size distribution functions describe the geometric heterogeneity, adsorption energy distribution functions describe the energetic heterogeneity. In determining the engineering characteristics of solids and understanding various processes inside pores, such as immersion, adsorption, storage, diffusion, and catalysis, the knowledge of both distribution functions is indispensable. In recent years, significant progress has been achieved in determining porosity, pore size distributions, and other textural parameters of porous solids from adsorption isotherms [3]. The calculation of relevant energetic information from gas or liquid adsorption isotherms on porous solids and the design of surface chemistry are a challenging research field. Adsorption energy distributions can be obtained from measured adsorption isotherms (so-called total isotherms) by different mathematical methods such as regularization [4–7] and Stieltjes transformation [8,9]. The regularization method is a purely numerical method taking into consideration the so-called ill-posed problem of the adsorption integral equation. The Stieltjes transformation provides an analytical solution of the adsorption integral equation. In the case of low-temperature pure gas adsorption, many attempts have been made to calculate relevant energy distributions, e.g., [7,10,11], and commercial software tools are even available [12]. Compared with gas adsorption, liquid adsorption measure-
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0021-9797/$ – see front matter doi:10.1016/j.jcis.2008.11.072
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ments are rather seldom used to characterize porous solids. Since from the liquid phase only competitive adsorption can be studied, energy distribution functions at the liquid/solid interface are quite complex. Only the differences F (U 21 ) = F (U 2 − U 1 ) of the adsorption energies of components 2 and 1 in a binary liquid mixture can be obtained. However, liquid adsorption can provide additional or confirming energetic information. A number of works have investigated (n) the calculation of F (U 21 ) distributions from liquid-phase Γ2 (x2 ) adsorption measurements [13–17]. In this paper, we will go the inverse way. A set of synthetically created Gaussian adsorption energy distribution functions are used (n) to design Γ2 (x2 ) adsorption excess isotherms. By systematic variation of (1) the position of peaks in the F (U 21 ) energy distribution function, (2) the shape of peaks in the F (U 21 ) energy distribution function, (3) and the interaction model of the surface and the bulk phases, a spectrum of adsorption isotherms is generated. It forms the basis for a deeper understanding of the liquid-phase isotherms’ dependence on the assumed energetic heterogeneity of an adsorbent. The relevance of energy distributions calculated from measured liquid adsorption isotherms can then be assessed. 2. Calculations 2.1. F (U 21 ) energy distribution functions in the interval −∞ < U 21 < ∞ Gaussian peak distribution functions for positive and negative U 21 values were calculated from
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F (U 21 ) =
m
f jm exp −
j =1
1
U jm − U 21
2Δ2jm
2
;
−∞ < U 21 < ∞; m = 1, 2, 3,
(1)
where Δ2jm is the peak’s variance, U jm is the position of the peak maximum, f jm is the magnitude of the Gaussian peak, and m is the number of Gaussian peaks. The normalization condition for the adsorption energy distribution,
∞ F (U 21 ) dU 21 = 1,
(2)
−∞
yields after a few mathematical manipulations
√
2π
m
f jm Δ jm = 1;
j = 1, 2, . . . , m,
(3)
j =1
which can be used to estimate f jm in Eq. (1), if Δ jm j = 1, 2, . . . , m is given. 2.2. F (U 21 ) energy distribution functions in the interval 0 U 21 < ∞ Gaussian peak distribution functions for positive U 21 values alone were calculated by F (U 21 ) =
m
f jm exp −
j =1
1 2Δ2jm
U jm − U 21
2
;
0 U 21 < ∞; m = 1, 2, 3.
(4)
The normalization condition
∞ F (U 21 ) dU 21 = 1
(5)
0
yields
√
m 2π
2
f jm Δ jm
j =1
U jm 1+Φ √ 2Δ jm
= 1;
j = 1, 2, . . . , m
(6)
(n)
tributions by the well-tried equation of Ostwald and Izaguirre [19],
mA
s x2t (x2 ) − x2 ,
(7)
(n)
where Γ2 (x2 ) is the reduced adsorption excess of component 2 as a function of the equilibrium mole fraction x2 of component 2 in the bulk phase; ns is the total mole number in the surface phase; s (x2 ) is the total mole fraction of m A is the mass of adsorbent; x2t component 2 in the surface phase, and the fundamental integral equation for the physical adsorption of binary nonelectrolytic liquid mixtures on energetically heterogeneous solid surfaces [1,13, 20,21], U 21,max s (x2 ) x2t
x2s (x2 ) F (U 21 ) dU 21 ,
= U 21,min
(9)
where x21 = x2 /(1 − x2 ), and C is the interaction function, T the absolute temperature, and R the universal gas constant. The interaction function C takes into account all molecular interactions between molecules in the surface and bulk phases and hence depends on the model of the surface and bulk phases and the topography of adsorption sites on the solid surface [1,13]. (n) In this paper, the Γ2 (x2 ) adsorption isotherms were obtained by assuming ideality as well as nonideality in the bulk and the surface phases. For assumed ideality, C = 1. For assumed reality, the molecular interactions (the number of nearest neighbors, the interaction energies between neighbors) were described by the lattice theory, applying the Bragg–Williams approximation (BWA) and the quasi-chemical approximation (QCA). The topography of adsorption sites on the solid surface was described by the random distribution approximation (RDA). The equations for the calculation of the interaction function C have been taken from [13]. In the following, the parameters applied are listed:
• U 21,min = −6 kJ/mol, U 21,max = 6 kJ/mol for F (U 21 ) in the interval −∞ < U 21 < ∞, • U 21,min = 0 kJ/mol, U 21,max = 6 kJ/mol for F (U 21 ) in the interval 0 U 21 < ∞, • ns /m A = 7 mmol/g, • the numbers of nearest neighbors in the bulk and sorption phases: c b = 12, c s = 6 (Fig. 8), • the Lennard–Jones parameter for ideal mixtures (C = 1):
• the Lennard–Jones parameter for real mixtures (Fig. 8):
b s ε21 /k = ε21 /k = −110 K,
Γ2 (x2 ) adsorption excess isotherms of binary liquid mixtures were calculated from the synthetic continuous F (U 21 ) energy dis-
(n)
,
b s ε22 /k = ε22 /k = −100 K,
(n)
Γ2 (x2 ) =
C exp(U 21 / R T )x21 1 + C exp(U 21 / R T )x21
b s ε11 /k = ε11 /k = −150 K,
2.3. Γ2 (x2 ) adsorption excess isotherms
x2s (x2 ) =
b s b s b s ε11 /k = ε11 /k = ε22 /k = ε22 /k = ε21 /k = ε21 /k,
with the known definite error integral Φ( z) [18]. Equation (6) is used to estimate the f jm values in Eq. (4).
ns
s where x2t (x2 ) is the so-called total isotherm; x2s (x2 ) is the local mole fraction of component 2 in the surface phase, the so-called local isotherm; U 21 is the difference U 2 − U 1 of the adsorption energies of the molecules 2 and 1 on identical surface sites; and F (U 21 ) dU 21 is the fraction of surface sites with differences in adsorption energies between U 21 and U 21 + dU 21 . The x2s (x2 ) local isotherm is given by [13]
(8)
• R = 8.31441 × 10−3 kJ/mol/K, • T = 298.15 K. The molecular partition functions for the internal degrees of freedom are assumed to be equal in the bulk and the sorption phases for each component; i.e., q1s = qb1 , q2s = qb2 . 3. Results and discussion The discussion of the results will be based on combined presentations of synthetic F (U 21 ) energy distributions and the cor(n) responding calculated Γ2 (x2 ) adsorption excess isotherms. The parameters of the F (U 21 ) distributions are listed in Table 1. On the left-hand side of Fig. 1, seven different F (U 21 ) energy distributions calculated by means of Eq. (1) in the interval −6 U 21 6 are presented:
• 1 narrow and high one-Gaussian peak energy distribution function at 4 kJ/mol (distribution 1);
G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
331
Table 1 Parameters of the Gaussian F (U 21 ) energy distributions F (U 21 ).a Figure
Fig. 1
Number of different F (U 21 )
Name of F (U 21 )
m
7
1 2
1 2
3
2
4
2
5
2
6
2
7 1 2 3 1 2
1 1 1 1 1 2
3
2
4
2
5
2
6
2
7 1 2 3 1 2 3 1 2
1 1 1 1 1 1 1 1 2
3
3
1
3
2
3
1
2
Fig. 2
3
Fig. 3
7
Fig. 4
3
Fig. 5
3
Fig. 6
3
Figs. 7, 8
Fig. 9
2
1
j
1 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 2 3 1 2 3 1 2 3 1 2
f jm (mol/kJ) 0.797900 0.050000 0.747900 0.170000 0.627900 0.250000 0.547900 0.398942 0.398942 0.547900 0.250000 0.332452 1.329808 0.664904 0.332452 1.595800 0.100000 1.495800 0.340000 1.255800 0.500000 1.095800 0.797884 0.797884 1.095800 0.500000 0.664904 1.330370 1.330370 1.329810 2.659616 1.329808 0.664904 0.664904 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.797852 0.443269 0.221635 0.295513 0.398942 0.398942
U jm
Δ jm
(kJ/mol)
(kJ/mol)
4
−4 4
−4 4
−4 4
−4 4
−4 4 0 0 0 0 5 1 5 1 5 1 5 1 5 1 5 3 1 3 5 3 3 3 3 1 5 1 3 5 1 3 5 1 3 5 −4 4
0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 1.200 0.300 0.600 1.200 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.600 0.300 0.300 0.300 0.150 0.300 0.600 0.600 0.250 0.250 0.167 0.167 0.167 0.167 0.167 0.167 0.300 0.600 0.450 0.500 0.500
See Eqs. (1) and (6). m = the number of Gaussian peaks, j = the name of the jth Gaussian peak, Δ2jm = the peak’s variance, U jm = the position of the peak’s maximum, and f jm = the magnitude of the isolated or nonisolated Gaussian peaks.
Fig. 1. Seven different Gaussian peak distributions F (U 21 ) in the interval −6 (n)
U 21 6 and inverted U- and S-shaped Γ2
isotherms for C = 1.
Fig. 2. Three different Gaussian peak and one delta function distributions F (U 21 ) in (n)
the interval −6 U 21 6 and S-shaped Γ2
isotherms for C = 1.
• the position of the adsorption azeotropes is shifted to smaller mole fractions.
a
• 5 two-Gaussian peak energy distribution functions (the shape of the distribution is varied systematically; the peaks at 4 kJ/mol decrease, whereas the peaks at −4 kJ/mol increase from distribution 2 to 6); • 1 wide one-Gaussian peak energy distribution function at 0 kJ/mol (distribution 7). (n)
The corresponding seven ideal Γ2 adsorption isotherms (C = 1) are given on the right-hand side of Fig. 1. Isotherm 1 (calculated from distribution 1) is the highest and corresponds to type II of the classification of Schay and Nagy [22,23]. In the order from isotherm 1 to isotherm 6 we find that
It can be seen that the isotherms’ dependence on changes in the shape and symmetry of the F (U 21 ) distributions becomes evident. As distribution 5 is symmetric, we find a symmetric S-shaped Type IV isotherm 5. The distributions 4 and 6 are laterally reversed. Hence, the isotherms 4 and 6 are mirrored through the X axis. From distribution 7 with a peak maximum at 0 kJ/mol follows a very small Type IV isotherm 7 with a maximum at x2 = 0.25 of (n) Γ2 ≈ 0.073 mmol/g. Fig. 2 shows four F (U 21 ) energy distributions with peak maxima at 0 kJ/mol:
• 1 delta function, • 3 one-Gaussian peak functions of different magnitudes (distributions 1–3; distribution 3 is equivalent to distribution 7 of Fig. 2).
(n)
• the magnitude of the Γ2 adsorption isotherms decreases, • the maximum is shifted to smaller mole fractions, • the type of isotherm is changed from type II/III (isotherms 1 and 2) to type IV/V (isotherms 3–6) with adsorption azeotropes,
(n)
The corresponding S-shaped Γ2 isotherms are presented on the right-hand side of Fig. 2. For the delta function at 0 kJ/mol, Γ2(n) has to be zero, since components 1 and 2 do not differ in their adsorption energies.
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Fig. 3. Seven different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 (n)
6 and inverted U-shaped Γ2
isotherms for C = 1.
Fig. 5. Three different Gaussian peak and one delta function distributions F (U 21 ) in (n)
the interval 0 U 21 6 and inverted U-shaped Γ2
isotherms for C = 1.
As the ideal isotherms calculated from Gaussian peaks and delta functions at the same position overlap nearly completely, only (n) three Γ2 isotherms can be distinguished with the naked eye. (n)
The magnitude of the Γ2 adsorption isotherms increases noticeably with increasing adsorption energy difference U 21 = U 2 − U 1 (n) from 1 to 5 kJ/mol. Additionally, the maximum of Γ2 is shifted in the direction of smaller mole fractions with increasing U 21 energy differences. As component 2, with higher adsorption energy, is preferentially adsorbed over the whole concentration range, the (n) Γ2 isotherms correspond to type I or II of the classification of Schay and Nagy. Let us summarize the results of Figs. 1–4. In Figs. 1–3 we changed the shape of peaks in the F (U 21 ) energy distribution functions systematically, in Fig. 4 the position of peaks in the interval (n) 0 U 21 6. We found that Γ2 isotherms respond sensitively to both the position and the shape of the peaks in F (U 21 ); i.e., both (n) factors influence the size and shape of Γ2 systematically. On the (n)
Fig. 4. Three different Gaussian peak and three delta function distributions F (U 21 ) (n)
in the interval 0 U 21 6 and inverted U-shaped Γ2
isotherms for C = 1.
In Fig. 3, the energy distributions of Fig. 1 are shifted to the interval 0 U 21 6 and the peak’s variance is decreased (see Eq. (4) (n) and Table 1). For the corresponding seven inverted U-shaped Γ2 adsorption isotherms (C = 1), we find in the order of isotherms 1 to 6 that
• the magnitude of the Γ2(n) adsorption isotherms decreases, and • the isotherms are all of type II, but the maximum is slightly shifted in the direction of higher mole fraction (decreased selectivity). The isotherms’ dependence on changes in the shape and symmetry of F (U 21 ) again becomes evident. Isotherms 5 and 7 are quite similar, as they are both calculated from distributions with the symmetric axis at 3 kJ/mol. On the left-hand side of Fig. 4, six different F (U 21 ) energy distributions calculated by means of Eq. (4) in the interval 0 U 21 6 are shown:
• 3 one-Gaussian peak energy distribution functions with average peak values at 1, 3, and 5 kJ/mol (the position of the peaks is varied), and • 3 delta functions at 1 kJ/mol, 3 kJ/mol and 5 kJ/mol.
other hand, Γ2 isotherms are not very sensitive to changes in F (U 21 ) if the symmetric axis is maintained (cf. Fig. 2 or the distributions 5 and 7 in Fig. 3). In Fig. 4, delta and Gaussian functions even provide indistinguishable inverted U-shape isotherms. In the following, we will consider the sensitivity of inverted (n) U-shaped Γ2 adsorption isotherms to differences in F (U 21 ) by keeping the symmetric axis at 3 kJ/mol. First, Fig. 5 shows four F (U 21 ) distributions with peak maxima at 3 kJ/mol:
• 1 delta function, • 3 one-Gaussian peak functions of different magnitudes (distributions 1–3; distribution 3 is equivalent to distribution 7 of Fig. 3). (n)
The corresponding Γ2
isotherms (C = 1) are very similar, and
(n)
the maximal ΔΓ2 difference of about 0.015 mmol/g is smaller than the maximal difference between isotherms 1 and 3 in Fig. 2 (n) (ΔΓ2 ≈ 0.07 mmol/g). Taking into account the experimental er(n)
ror of Γ2 isotherms of ±0.03 mmol/g [24,25], it will be difficult to extract clear information from such adsorption isotherms. Second, Fig. 6 shows three F (U 21 ) distributions with different peak numbers:
• 1 one-Gaussian peak distribution at 3 kJ/mol, • 1 two-Gaussian peak distribution at 1 and 5 kJ/mol, • 1 three-Gaussian peak distribution at 1, 3, and 5 kJ/mol.
G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
Fig. 6. Two different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 6 (n)
and inverted U-shaped Γ2
isotherms for C = 1.
333
Fig. 8. Two different Gaussian peak distributions F (U 21 ) in the interval 0 U 21 6 (n)
and inverted U-shaped Γ2
isotherms for BWA and QCA.
Fig. 9. One Gaussian peak distribution F (U 21 ) in the interval −6 U 21 6 and Fig. 7. Three different Gaussian peak distributions F (U 21 ) in the interval 0 (n)
U 21 6 and inverted U-shaped Γ2
(n)
S-shaped Γ2
isotherms for BWA as functions of the values of c B and c s .
isotherms for C = 1.
(n)
As can be seen, the Γ2 isotherms (C = 1) differ from each other, and the magnitude decreases with increasing distance of peak area in the order isotherm 1, isotherm 3, isotherm 2. How(n) ever, the maximal ΔΓ2 difference between isotherm 2 (calculated from two peaks) and isotherm 3 (calculated from three peaks) is only 0.04 mmol/g; i.e., it lies again in the region of the (n) experimental error of Γ2 . Third, Figs. 7 and 8 present the results for two different threeGaussian peak distributions:
• a three-Gaussian peak distribution with unconnected narrow and high peaks at 1, 3, and 5 kJ/mol,
• a three-Gaussian peak distribution with connected peaks at 1, 3, and 5 kJ/mol. (n)
On the right-hand side of Fig. 7, the ideal Γ2 isotherms (C = 1) are shown; on the right-hand side of Fig. 8, the real ones calculated by means of QCA and BWA. In all three cases, the two isotherms are very similar. Much more pronounced quantitative differences are found between ideal and real isotherms, indicat(n) ing that the model assumptions influence the Γ2 isotherms more strongly than differences in the F (U 21 ) distribution functions. The QCA and BWA isotherms differ more from each other than in the case of gas phase calculations [26], whereas the type of isotherm
is not changed by varying the statistical thermodynamic model for molecular interactions. Fig. 9 illustrates the well-known influence of another model assumption: the number of nearest neighbors in the bulk and sorption phases. The two-Gaussian peak distribution F (U 21 ) on the left-hand side of Fig. 9 corresponds to distribution 5 in Fig. 1. (n) By applying BWA, we get significantly different Γ2 isotherms as functions of a varied number of nearest neighbors in the sorption phase. Figs. 5–8 confront us with the limitations of F (U 21 ) calculations (n) from Γ2 liquid adsorption isotherms. Especially in the case of inverted U-shaped isotherms, considerable differences in the F (U 21 ) (n) distributions yield only weak differences in the Γ2 isotherms. We have to realize that this fact makes it difficult to obtain trustworthy (n) energy information from Γ2 . Let us summarize the facts influencing the reliability of F (U 21 ) energy distributions from liquid-phase adsorption measurements: (n)
1. Liquid adsorption isotherms are fixed at x2 = 0, Γ2 (n)
= 0 and
x2 = 1, Γ2 = 0; i.e., they possess less freedom than gas adsorption isotherms to react to differences in energy distribu(n) tions. Thus, Γ2 isotherms are a priori less sensitive than gas adsorption isotherms. (n) 2. It is true that Γ2 isotherms respond sensitively to changes of the position or the shape of peaks in F (U 21 ), but a shift of the symmetric axis is required. Different distributions, by
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G. Kalies et al. / Journal of Colloid and Interface Science 331 (2009) 329–334
keeping the same symmetric axis, yield very similar isotherms (see Figs. 5–8); i.e., the isotherms do not “feel” differences in peak width of peak connectivity and even reflect changes in the peak number inadequately (cf. Fig. 6). At this point, the isotherms are under-determined and insensitive to changes in F (U 21 ). 3. Inverted U-shaped isotherms are less sensitive to changes in F (U 21 ) than S-shaped isotherms; i.e., the calculation of welldefined F (U 21 ) distributions from inverted U-shape isotherms is complicated. (n) 4. The experimental error of Γ2 adsorption isotherms is about
(n)
Our study indicates that experimental errors of Γ2 , model assumptions, uncertain geometric effects, etc. decisively influence the F (U 21 ) distributions, even if the numerical method for solving the ill-posed problem in the adsorption integral equation is advanced. The limitations of the evaluation of heterogeneity effects from liquid adsorption isotherms are evident. Regarding the various facts influencing the reliability of F (U 21 ), we have to recognize that there is a risk of overrating characterization results from liquidphase adsorption. In interpreting F (U 21 ) distributions, we have to act with caution and should use other independent methods to confirm information on energetic heterogeneity of solids.
(n)
±0.03 mmol/g; i.e., the changes in Γ2 caused by differences in F (U 21 ) are often smaller than the error of the isotherms. (n) 5. Model assumptions can influence Γ2 more strongly than differences in F (U 21 ) (cf. Figs. 8, 9); i.e., our ideas from reality determine the calculation results, not the given facts. 6. In liquid-phase adsorption, the distinction between geometric and energetic information is more complicated than in gas-phase adsorption. While the low-pressure region in gas isotherms is connected with energetic information and the (n) high-pressure region with geometric information, Γ2 isotherms do not have distinct regions that can be assigned to geometric or energetic effects. As we know that especially in micro- and nanoporous adsorbents, the adsorption effects caused by geometric or energetic heterogeneity strongly interfere with each other, the extraction of purely energetic information is complicated. 7. In view of the above-mentioned points, the well-known problems in the solution of the adsorption integral equation (Eq. (8)) as an ill-posed problem, i.e., the required a priori assumption of the local isotherm and the fact that small changes s in x2t caused by experimental errors, can significantly influence the energy distribution play an increased role in the (n) calculation of F (U 21 ) from Γ2 . 4. Summary Synthetic F (U 21 ) adsorption energy distribution functions and (n) their corresponding Γ2 adsorption excess isotherms for binary liquid mixtures over the whole concentration range have been presented. By systematic variation of the parameters of F (U 21 ), its (n) influence on the size and shape of Γ2 has been studied. It is shown that both the position and the shape of en(n) ergy peaks in F (U 21 ) influence the Γ2 isotherms systemati(n)
cally. On the other hand, the sensitivity of Γ2 to changes of F (U 21 ) keeping the symmetric axis is quite limited. Inverted Ushaped isotherms only slightly reflect differences in peak width, peak connectivity, and peak number, which makes it much more complicated to extract trustworthy energy information from such isotherms.
Acknowledgment The financial support for this project by the Deutsche Forschungsgemeinschaft (DFG, Ka 1560/3-1, 4-1) is gratefully acknowledged. References [1] M. Jaroniec, R. Madey, Physical Adsorption on Heterogeneous Solids, Elsevier, Amsterdam, 1988. ´ [2] W. Rudzinski, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, London, 1992. [3] P. Llewellyn, J. Rouquérol, N. Seaton (Eds.), Characterization of Porous Solids VII: Proceedings of the 7th International Symposium on the Characterization of Porous Solids (Cops-VII), Elsevier, Amsterdam, 2006. [4] A.N. Tichonov, Dokl. Akad. Nauk SSSR 39 (1943) 195. [5] A.N. Tichonov, Dokl. Akad. Nauk SSSR 153 (1963) 49. [6] A.N. Tichonov, Sov. Math. 4 (1963) 1624. [7] M. v. Szombathely, P. Bräuer, M. Jaroniec, J. Comput. Chem. 13 (1992) 17. [8] P. Bräuer, M. Fassler, M. Jaroniec, Thin Solid Films 123 (1985) 245. [9] P. Bräuer, M. Fassler, M. Jaroniec, Chem. Phys. Lett. 125 (1986) 241. [10] M. Heuchel, M. Jaroniec, R.K. Gilpin, P. Bräuer, M. v. Szombathely, Langmuir 9 (1993) 2537. [11] D.D. Do, Adsorption Analysis: Equilibrium and Kinetics, Series on Chemical Engineering, vol. 2, Imperial College Press, London, 1998. [12] J.P. Olivier, W.B. Conklin, M. v. Szombathely, DFT-Users Guide, Micromeritics Instrument Corporation, 1994. [13] M. Heuchel, P. Bräuer, M. v. Szombathely, U. Messow, W.D. Einicke, M. Jaroniec, Langmuir 9 (1993) 2547. [14] M. Heuchel, M. Jaroniec, Langmuir 11 (1995) 1297. [15] P. Podko´scielny, A. Dabrowski, ˛ M. Bülow, Appl. Surf. Sci. 196 (2002) 312. [16] A. Dabrowski, ˛ P. Podko´scielny, M. Bülow, Colloids Surf. A Physicochem. Eng. Aspects 212 (2003) 109. [17] P. Podko´scielny, K. Nieszporek, P. Szabelski, Colloids Surf. A Physicochem. Eng. Aspects 277 (2006) 52. [18] G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers, second ed., Dover, New York, 2000. [19] Wo. Ostwald, R. de Izaguirre, Koll. Z. 30 (1922) 279. ´ [20] W. Rudzinski, J. O´scik, A. Dabrowski, ¸ Chem. Phys. Lett. 20 (1973) 5. [21] M. Jaroniec, P. Bräuer, Surf. Sci. Rep. 6 (1986) 65. [22] G. Schay, L. Nagy, T. Szekrenyesy, Period. Polytechnika 4 (1960) 95. [23] L. Nagy, G. Schay, Acta Chim. Hung. 93 (1963) 365. [24] G. Kalies, P. Bräuer, U. Messow, J. Colloid Interface Sci. 275 (2004) 410. [25] R. Rockmann, G. Kalies, O. Klepel, Adsorption 13 (2007) 515. [26] P. Bräuer, M. Jaroniec, J. Colloid Interface Sci. 108 (1985) 50.