A cut-off criterion for calculating more reliable adsorption energy distributions from gas adsorption isotherms

Abstract

Adsorption energy distributions from experimental gas adsorption isotherms are capable to characterize the energetic heterogeneity of a solid surface. Unfortunately, they can only be computed by the adsorption integral equation, which represents an ill-posed problem, i.e., the solution is highly sensitive to errors in the input data. Ill-posed problems are usually solved by means of regularization, but general regularization schemata do not provide useful criteria to estimate the approximation quality. In this paper, a former presented solution strategy tailor-made for the Langmuir kernel of the adsorption integral equation is extended to Fourier transform. This yields a simple and effective cut-off criterion for the Fourier cosine transform of the adsorption energy distribution. The cut-off criterion is applied to calculate adsorption energy distributions from synthetic and experimental adsorption isotherms.

Introduction

Many models describing the adsorption of gases on solid surfaces are based on the assumption of energetically equivalent adsorption sites. In order to describe the adsorption on heterogeneous solid surfaces, these models can be applied to each type of adsorption site (Rudzinski and Everett, 1992). This leads to the well-known adsorption integral equation (Rudzinski and Everett, 1992; Jaroniec and Madey, 1988)

$${\Theta }_t(p,T)={\displaystyle {\int }_{u_{\text{min}}}^{u_{\text{max}}}{\Theta }_l}(p,T,u)·F(u) \text{d}u$$

(1)

In equation (1), ${\Theta }_t(p,T)$ is the total coverage of the surface at pressure p and constant temperature T. ${\Theta }_l(p,T,u)$ represents the local coverage on adsorption sites of the same molar energy u released by the adsorption. Hence, equation (1) states that the total coverage of the surface is a weighted average over all local coverages where the molar energy u varies between the minimal and the maximal values $u_{\min }$ and $u_{\max }$ (Rudzinski and Everett, 1992).

Usually, the total adsorption isotherm ${\Theta }_t={\Theta }_t(p,T)$ is measured, whereas the local adsorption isotherm ${\Theta }_l={\Theta }_l(p,T,u)$ is given by an adsorption theory. Hence, the function of the weight factors $F=F(u)$ , the so-called adsorption energy distribution (AED), can be computed by an inversion of equation (1). In general, inverse problems related to the Fredholm integral equations of the first kind, such as equation (1), are called unstable and ill-posed, respectively. That means they are very sensitive to the input data. For erroneous data, a straightforward inversion is therefore unusuable (Kress, 1989).

Over the past 60 years, many approaches to calculate AEDs were developed. Initially, exact methods such as the Stieltjes transformation have been widely used (Prasad and Doraiswamy, 1984; Sips, 1948, 1950). Since the total adsorption isotherm has to be of a special analytical form, such methods do not provide an inversion formula of practical use. Other methods such as the so-called condensation approximation aim for simplification of the true integral kernel, i.e., the local adsorption isotherm (Harris, 1968), and were developed further in iterative programs like HILDA (House and Jaycock, 1978). Since the mentioned methods do not consider the instability of the problem, regularization is up to now often the method of choice. General regularization methods, e.g. by Tikhonov (1943) (Honerkamp and Weese, 1990), are independent of the integral kernel, but do not provide useful criteria to estimate the approximation quality. To overcome this disadvantage, Arnrich et al. (2015) presented a regularization scheme by means of differentiation and Fourier series, which is tailor-made for the Langmuir kernel of the adsorption integral equation. Unlike the mentioned methods, this solution strategy is able to give quantitative statements about the error amplification.

In the current paper, the next step is made by passing over from Fourier series to Fourier transform. The aim is to present a simple cut-off criterion for a more effective calculation of AEDs from gas adsorption isotherms. After an introduction into ill-posed problems and their consequences for calculating AEDs by means of equation (1), we shortly explain the general method of regularization. Then a cut-off criterion for the so-called spectral density is presented and illustrated by calculations on the basis of a synthetic AED $F_{\text{synth}}$ shown in Figure 1. In the end, the cut-off criterion is verified by calculating AEDs from synthetic and experimental adsorption isotherms.

Figure 1.

AED $F_{\text{synth}}$ synthetically generated by a sum of normal distributions.

We have to emphasize that our method only belongs to the Langmuir model with all its restrictions, i.e., we suppose that there are no interactions between the different adsorption sites. Therefore, the method can only be applied to adsorption isotherms where these restrictions are fulfilled in a sufficient manner.

Ill-posed problems

By an abstract point of view, equation (1) belongs to the following class of equations

$$\psi =A\phi$$

(2)where A is a mapping (operator) between two sets X and Y.

Hadamard postulated three properties of physical problems described by equation (2) (Arnrich et al., 2015; Hadamard, 1923; Kress, 1989):

For every $\psi \in Y$ , there is a solution $\phi \in X$ .

There is at most one solution (uniqueness of the solution).

The solution $\phi$ depends continuously on the data ψ.

If all three requirements are satisfied, then the problem (2) is called well-posed, otherwise ill-posed. Well-posedness means that the inverse operator $A^{-1}:Y\to X$ does exist and is continuous.

If only property 3 is violated, the problem is called unstable. The instability of equation (1) can be illustrated by a straightforward discretization. In general, a discretization is necessary due to the finite number of measured values of the total adsorption isotherm and is performed by a suitable quadrature with the corresponding quadrature weight w_j ¹ (Kress, 1989)

$${\Theta }_t(\hspace{0.17em}{p}_i,T\hspace{0.17em})\approx {\displaystyle \sum _{j=1}^n{w}_j}·{\Theta }_l(p_i,T,u_j)·F(u_j),i=1,\mathrm{\dots },m$$

(3)

For a total isotherm Θ _t with m measurement points, one obtains a linear system of equations. By means of

$$\Theta :=\left(\begin{array}{c}{\Theta }_t(p_1,T)\\ {\Theta }_t(p_2,T)\\ ⋮\\ {\Theta }_t(p_m,T)\end{array}\right),w:=\left(\begin{array}{cccc}{w}_1& 0& \dots & 0\\ 0& w_2& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \dots & 0& w_n\end{array}\right),F:=\left(\begin{array}{c}F(u_1)\\ F(u_2)\\ ⋮\\ F(u_n)\end{array}\right)$$

equation (3) can be written as a matrix equation

$$\Theta =w·K·F=A·F$$

(4)with

$$K:=\left(\begin{array}{ccc}{\Theta }_l(p_1,T,u_1)& \dots & {\Theta }_l(p_1,T,u_n)\\ ⋮& \ddots & ⋮\\ {\Theta }_l(p_m,T,u_1)& \dots & {\Theta }_l(p_m,T,u_n)\end{array}\right)$$

Assuming that the coefficient matrix $A$ is square (m = n) and that the inverse matrix $A^{-1}$ does exist, then equation (4) has the unique solution

$$F=A^{-1}·\Theta$$

(5)

If $\Theta$ is erroneous with the error $\Delta \Theta$ , then the error of $F$ is given by

$$\Delta F=A^{-1}·\Delta \Theta$$

(6)

For unstable integral equations, the amplification of $\Delta \Theta$ by $A^{-1}$ grows with the accuracy of the approximation of equation (3). At first glance, this seems to be paradox, but reflects the discontinuity of the inverse operator $A^{-1}$ . Thus the recalculated F becomes less and less reliable (Kress, 1989). This fact is illustrated in Figure 2 by using the Langmuir isotherm as local adsorption isotherm in equation (1).

Figure 2.

(a) Total isotherm ${\Theta }_t={\Theta }_t(p,T),T=$ 77 K, corresponding to the AED $F_{\text{synth}}$ in Figure 1. (b) Two noisy AEDs recalculated by inversion of equation (4) for two different discretizations. The error that causes this result is purely numerical.

Regularization

As can be seen in Figure 2, even small errors in the magnitude of round-off errors in the measured total isotherm can cause enormous errors in the AED. In order to obtain stable approximate solutions of ill-posed problems, Tikhonov (1943) introduced a solution strategy called regularization (Kress, 1989; Richter, 2015; Arnrich et al., 2015).

We shortly illustrate this method by means of equation (1), where we suppose that there is a unique solution F for every error-free total adsorption isotherm Θ _t . In this case the inversion operator $A^{-1}$ is defined on the set of all these Θ _t . In the unstable case, $A^{-1}$ is not continuous.

The idea is to approximate $A^{-1}$ by continuous operators $B_{\lambda },\lambda >0$ , which are defined also for erroneous total isotherms ${\Theta }_t^{\epsilon }$ . ${\Theta }_t^{\epsilon }$ can be understood as deviation of Θ _t with the error level ε. Approximation means that there holds

$$\underset{\lambda \to 0}{\lim }{B}_{\lambda }{\Theta }_t=F\text{ for }\text{all}{\Theta }_t$$

(7)

The parameter λ is called regularization parameter, and the family of all $B_{\lambda }$ is called a regularization scheme. For ${\Theta }_t^{\epsilon }$ , every $B_{\lambda }{\Theta }_t^{\epsilon }$ can be considered as an approximation of F. Because of the continuity of $B_{\lambda }$ , the approximation is stable.

The strategy is now to choose the regularization parameter λ in dependence on the error level ε and ${\Theta }_t^{\epsilon }$ in such a way that holds

$$\underset{\epsilon \to 0}{\lim }{B}_{\lambda (\epsilon ,{\Theta }_t^{\epsilon })}{\Theta }_t^{\epsilon }=F$$

(8)

The function $\lambda =\lambda \left(\epsilon ,{\Theta }_t^{\epsilon }\right)$ is called choice of parameter. A regularization scheme together with a choice of parameter is called a regularization. In practice, one strives to find an optimal regularization parameter which yields the best results for a given error level.

A regularization scheme by means of Fourier transform

Due to the fact that general regularizations do not provide useful criteria to estimate the approximation quality, Arnrich et al. (2015) presented an inversion formula tailor-made for the Langmuir kernel of the adsorption integral equation

$${\Theta }_t(p,T\hspace{0.17em})={\displaystyle {\int }_{u_{\min }}^{u_{\max }}\frac{K_L·p}{1+K_L·p}}·F(u) \text{d}u$$

(9)

The dependence of the Langmuir constant K_L on the adsorption energy u is given by the following equation (Arnrich et al., 2015; Jaroniec and Madey, 1988)

$$K_L=K_0(T)·e^{\frac{u}{R·T}}$$

(10)

The temperature-dependent constant $K_0(T)$ is connected with the ratio of the internal partition functions for a molecule in the adsorbed and gas phase, respectively, and has the value $8.9708 \frac{1}{\text{Torr}}$ at $T=77 \text{K}$ (Tóth, 2002). R denotes the ideal gas constant.

In order to obtain practicable inversion formulas, Arnrich et al. (2015) applied the change of variables

$$\xi :=-R·T·\text{ln\hspace{0.17em}}(K_0(T)·p),{\varphi }_t(\xi ,T):={\Theta }_t(p,T)$$

(11)and hence transformed equation (9) into

$${\varphi }_t(\xi ,T)={\displaystyle {\int }_{u_{\min }}^{u_{\max }}\frac{F(u)}{1+e^{\frac{\xi -u}{R·T}}}} \text{d}u$$

(12)

${\varphi }_t={\varphi }_t(\xi ,T)$ is called the transformed total (adsorption) isotherm. Note that the transformed variable ξ has the dimension of a molar energy. In that sense, equation (11) transforms low pressures into large molar energies and high pressures into small ones, respectively, as can be seen in Figure 3.

Figure 3.

Transformed total isotherm ${\varphi }_t={\varphi }_t(\xi ,T),T=$ 77 K, corresponding to the total adsorption isotherm ${\Theta }_t={\Theta }_t(p,T)$ in Figure 2(a).

One of the main results of Arnrich et al. (2015) is the integral formula

$${\displaystyle {\int }_{u_{\min }}^{u_{\max }}F}(u)·g(u)\text{d}u={\displaystyle {\int }_{-\infty }^{+\infty }{\varphi }^t\prime }(\xi ,T)·\frac{G(\xi -i\pi RT)-G(\xi +i\pi RT)}{2i\pi RT}\text{d}\xi$$

(13)which applies for analytical functions g with at most polynomial growth. G is an antiderivative of g, i.e., it holds $G\prime =g$ , and i is the imaginary unit.

Since sine and cosine are analytical and bounded functions, equation (13) allows to compute the Fourier coefficients of F and hence its Fourier series. Thus the Fourier sums of F yield a regularization scheme. The reciprocal of N – the number of Fourier coefficients used – serves as the regularization parameter. This scheme works very well for synthetic isotherms and has the advantage that it includes explicit terms of error amplification in the Fourier coefficients. However, beside the numerical disadvantages connected with using large sums, it has been shown that the optimal N also depends on the energy range of u, which is not exactly known in applications.

In order to overcome difficulties like that, in the current work, we pass from Fourier series to Fourier transform. More precisely, we try to recalculate F by the real part of its Fourier transform which we call the real spectral density or simply spectral density of F

$$R(\omega ):={\displaystyle {\int }_{-\infty }^{+\infty }F}(u)·\text{cos}(\omega u) \text{d}u$$

(14)

Since F(u) = 0 for u < 0, it holds (Brigola, 2013) ²

$$F(u)=\frac{2}{\pi }{\displaystyle {\int }_0^{+\infty }R}(\omega )·\text{cos}(\omega u) \text{d}\omega$$

(15)

Applying equation (13) to the right-hand side of equation (14) by using $G(\xi \pm iRT)=\frac{1}{\omega }\sin (\omega (\xi -i\pi RT)$ yields (cf. Arnrich et al., 2015)

$$R(\omega )=\frac{\sinh (\pi RT\omega )}{\pi RT\omega }{\displaystyle {\int }_{-\infty }^{+\infty }-}{\varphi }_t^{\prime }(\xi ,T)·\text{cos}(\omega \xi ) \text{d}\xi$$

(16)

Thus we can recalculate the AED F from its transformed total isotherm ${\varphi }_t$ by means of equations (16) and (15).

Spectral density

The spectral density $R=R(\omega )$ contains any information on the AED F. A rapidly decaying spectral density $R$ indicates a smooth AED (Trefethen, 2000). In order to illustrate this behavior, we show in Figure 4 two spectral densities with very different decay behavior and their corresponding AEDs.

Figure 4.

(a) Two spectral densities $R$ with different decay behavior. (b) Corresponding AEDs. A rapidly decaying $R$ (red) correlates with a smooth AED, while a slowly decaying $R$ (blue) is related to a non-smooth AED.

In applications, due to numerical and measurement errors, $R$ does not converge to zero for large frequencies. Since $\sinh (\pi RT\omega )$ grows exponentially, the error of ${\varphi }_t^{\prime }$ is exponentially amplified. Therefore, $R$ itself grows exponentially starting at a certain frequency ω (cf. Figure 5), and it becomes necessary to cut off the spectral density at a suitable frequency ${\omega }_{\max }$ right before the influence of the error amplification prevails (Arnrich et al., 2015).

Figure 5.

Spectral density $R_{\text{synth}}$ of the AED $F_{\text{synth}}$ shown in Figure 1 for different ${\omega }_{\max }$ . (a) The error is negligible. (b) Drastic increase of the error (error explosion).

As it can be seen in Figure 5, the spectral density of the AED $F_{\text{synth}}$ presented in Figure 1 decays rapidly and remains at a low level until ω = 15 $\frac{\text{mol}}{\text{kJ}}$ . Afterwards, the amplification of numerical errors increases and causes significant deviations from the correct AED. The best approximation of $F_{\text{synth}}$ is hence obtained when $R_{\text{synth}}$ is cut off at ${\omega }_{\max }$  = 15 $\frac{\text{mol}}{\text{kJ}}$ . This result is confirmed in Figure 6.

Figure 6.

(a) Recalculations for $F_{\text{synth}}$ . (b) Corresponding transformed total isotherms ${\varphi }_{synth}$ for different cut-offs ${\omega }_{\max }$ of the spectral density $R_{\text{synth}}$ .

Obviously, the accuracy of the recalculation increases with rising cut-off frequency ${\omega }_{\max }$ . For ${\omega }_{\max }$  = 15 $\frac{\text{mol}}{\text{kJ}}$ , the recalculation overlays $F_{\text{synth}}$ perfectly. In addition, it can be seen that at a certain accuracy (in this case for ${\omega }_{\max }>0.8 \frac{\text{mol}}{\text{kJ}}$ ), the transformed total isotherms ${\varphi }_t$ do not deviate significantly.

Based on the latest findings, the following cut-off operators are used as a regularization scheme for the adsorption integral equation (9) by means of equation (7)

$$(B_{\lambda }{\varphi }_t)(u):=\frac{2}{\pi }{\displaystyle {\int }_0^{\frac{1}{\lambda }}R}(\omega )·\text{cos}(\omega u) \text{d}\omega$$

(17)where $R$ is defined by equation (16). Note that the cut-off frequency ${\omega }_{\max }$ in equation (17) is given by $\frac{1}{\lambda }={\omega }_{\max }$ .

Now, the right choice of the regularization parameter λ for erroneous ${\varphi }_t$ remains (cf. equation (8)). Therefore, we have to analyze the influence of experimental errors on the spectral density $R$ and on the quality of the approximation.

Influence of experimental errors on the spectral density ³

For illustrating the influence of measurement errors, the following deviation is used for the synthetic transformed isotherm ${\varphi }_{\text{synth}}$ shown in Figure 3

$${\varphi }_{\text{synth}}^{\epsilon }(\xi ):=(1+\epsilon (\xi ))·{\varphi }_{\text{synth}}(\xi )$$

(18)

$$\epsilon (\xi ):=\frac{{\epsilon }_{\max }}{2}·\left(1+\text{tanh}\frac{\xi }{8}\right)·\text{sin}(\xi )$$

(19)with the maximal errors ${\epsilon }_{\max }=0.001, 0.02 \text{ and } 0.05$ . The chosen error function $\epsilon =\epsilon (\xi )$ in equation (19) is based on the assumption that the influence of the measurement error is negligible for $p\to \infty$ .

In order to compute the spectral density $R_{\text{synth}}^{\epsilon }$ by equation (16), we need the derivatives of ${\varphi }_{\text{synth}}^{\epsilon }$ . Equation (18) yields

$${({\varphi }_{\text{synth}}^{\epsilon })}^{\prime }(\xi )={\epsilon }^{\prime }(\xi )·{\varphi }_{\text{synth}}(\xi )+(1+\epsilon (\xi ))·{\varphi }_{\text{synth}}^{\prime }(\xi )$$

(20)

In Figure 7, the negative derivatives of ${\varphi }_{\text{synth}}^{\epsilon }$ for the three chosen values of ${\epsilon }_{\max }$ are shown.

Figure 7.

Negative derivatives of ${\varphi }_{\text{synth}}^{\epsilon }$ defined in equations (18) and (19) for ${\epsilon }_{\max }=0.001, 0.02 \text{ and } 0.05$ . $-{({\varphi }_{\text{synth}}^{\epsilon })}^{\prime }$ approximates $F_{\text{synth}}$ in first order.

Because of equation (20), the spectral density $R_{\text{synth}}^{\epsilon }$ consists of two parts

$$R_{\text{synth}}^{\epsilon }(\omega ):=r_{\text{synth}}^{\epsilon }(\omega )+{\rho }_{\text{synth}}^{\epsilon }(\omega )$$

(21)with

$$r_{\text{synth}}^{\epsilon }(\omega ):=-\frac{\text{sinh}(\pi RT\omega )}{\pi RT\omega }{\displaystyle {\int }_{-\infty }^{+\infty }(1+\epsilon (}\xi )){\varphi }^{\prime }(\xi )\text{cos}(\omega \xi ) \text{d}\xi$$

(22)

$${\rho }_{\text{synth}}^{\epsilon }(\omega ):=-\frac{\text{sinh}(\pi RT\omega )}{\pi RT\omega }{\displaystyle {\int }_{-\infty }^{+\infty }\epsilon }\prime (\xi )\varphi (\xi )\text{cos}(\omega \xi ) \text{d}\xi$$

(23)

While $r_{\text{synth}}^{\epsilon }$ differs from the true spectral density $R_{\text{synth}}$ at most by ${\epsilon }_{\max },{\rho }_{\text{synth}}^{\epsilon }$ is mainly responsible for the deviation of the recalculated AED $F_{\text{synth}}^{\epsilon }$ from the true AED $F_{\text{synth}}$ . The two parts of the spectral density $R_{\text{synth}}^{\epsilon }$ are illustrated in Figure 8.

Figure 8.

(a) $r_{\text{synth}}^{\epsilon }$ according to equation (22) for different ${\epsilon }_{\max }$ in comparison with $R_{\text{synth}}$ . (b) ${\rho }_{\text{synth}}^{\epsilon }$ according to equation (23) for the same values of ${\epsilon }_{\max }$ .

$r_{\text{synth}}^{\epsilon }$ decreases with increasing frequency ω and contains the needed information to recalculate the AED. While $r_{\text{synth}}^{\epsilon }$ is almost resistant to any error in the input data (cf. Figure 8(a)), ${\rho }_{\text{synth}}^{\epsilon }$ tends to an oscillatory behavior with exponential growth of the amplitudes (cf. Figure 8(b)).

Cut-off criterion

In order to find the best approximation of the AED, it is essential to cut off the spectral density $R$

as late as possible to ensure that sufficient information is available for the calculation of the distribution function, and

as early as necessary, due to the influence of the error amplification.

It is evident that the attenuation of ${\rho }_{\text{synth}}^{\epsilon }$ leads to significantly better calculations of AEDs.

Based on the graphical representation of the spectral density $R$ , the following simple and intuitive cut-off criterion can be formulated: Choose λ in equation (17) such that the spectral density passes over from fading out to a strong oscillating behavior with fast growing amplitudes at $R(\omega )=R(\frac{1}{\lambda })$ as illustrated in Figure 9.

Figure 9.

Graphical illustration of the choice of a suitable cut-off.

In Figure 9, the spectral density $R$ is fading out until $\omega \approx 7.6 \frac{\text{mol}}{\text{kJ}}$ . At higher frequencies, $R$ shows a strongly oscillating behavior. Hence, the cut-off should be made at $\omega \approx 7.6 \frac{\text{mol}}{\text{kJ}}$ . It becomes obvious that the suggested criterion is simple and effective, but rather subjective. ⁴

Application

In the following, we will apply equations (16) and (15) to calculate AEDs from erroneous isotherm data: firstly from ${\varphi }_{\text{synth}}^{\epsilon }$ (cf. equation (18)), and secondly from three experimental adsorption isotherms.

Application to erroneous synthetic isotherms

The results of the recalculation of AEDs from ${\varphi }_{\text{synth}}^{\epsilon }$ by equations (16) and (15) are shown in Figure 10.

Figure 10.

(a) Spectral densities $R$ corresponding to ${\varphi }_{\text{synth}}^{\epsilon }$ . (b) AEDs $F_{\text{synth}}^{\epsilon }$ corresponding to ${\varphi }_{\text{synth}}^{\epsilon }$ for a suitable cut-off.

It can be seen that for the very small relative error of 0.1%, the approximation (red line) is almost perfect. The location as well as the size of peaks correspond very well to the original AED $F_{\text{synth}}$ . The quality of this approximation is confirmed by the corresponding spectral density $R$ (cf. Figure 10(a)) indicating a low error amplification. With increasing error, the approximation deviates stronger from the original AED $F_{\text{synth}}$ , which is again suggested by the behavior of $R$ , since the error amplification starts at very low ω (cf. the blue and green lines at small frequencies ω between 0 $\frac{\text{mol}}{\text{kJ}}$ and 2 $\frac{\text{mol}}{\text{kJ}}$ ).

As shown, the presented regularization is quite stable to small relative errors in the adsorption isotherm, and the impact of rising errors to the calculated AEDs is appropriate. It becomes obvious that the spectral density $R$ represents a suitable tool for estimating the quality of the corresponding AED.

Application to experimental adsorption isotherms

In order to enlighten the calculation procedure and to check the plausibility of calculation results, the presented Langmuir tailor-made regularization method with cut-off criterion is now applied to experimental adsorption isotherms.

In Figure 11(a) and (b), three different measured nitrogen adsorption isotherms at 77 K are shown in linear and semi-logarithmic presentations:

Figure 11.

Comparison of the N₂ adsorption isotherms of Carboxen 563 and NaY samples. (a): linear presentation, (b): semi-logarithmic presentation, (c): transformed isotherms according to equation (11), cf. Figure 3.

1. the N₂ isotherm of the Carboxen 563 carbon molecular sieve (CMS) of Supelco (sample α),

2. the N₂ isotherm of the Carboxen 563 carbon molecular sieve (CMS) of Supelco (sample β), and

3. the N₂ isotherm of a NaY zeolite (Faujasite structure).

All measurements were performed with conventional volumetric (manometric) ASAP devices (micromeritics). The α sample was measured with an ASAP 2000 device within a relative pressure range from ${10}^{-6}$ to 1, the β and NaY samples with an ASAP 2010 device in the range ${10}^{-7}<p/p_0<1$ . Before the measurements, the α and β samples were outgassed at $180\hspace{0.17em}°\text{C}$ and the NaY sample at $350\hspace{0.17em}°\text{C}$ for 20 h in the degas port. At low relative pressures, an incremental dosing routine was employed.

In the following, not the quality of the experimental adsorption isotherms presented in Figure 11 is of importance but the effects of the differences in isotherms, especially in the initial pressure range, on the AED calculation results.

Looking at the linear presentation in Figure 11(a), all isotherms exhibit Langmuir shape (type-I-isotherm according to the IUPAC classification of physisorption isotherms (Sing et al., 1985)). Type-I-isotherms are typical for NaY zeolite as a purely microporous adsorbent and also for the micro- and mesoporous Carboxen 563 material in the considered pressure range up to about 200 Torr. Note that in Figure 11, relative pressures up to $p/p_0<0.30$ are presented being still very far from saturation pressure. Capillary condensation in the mesopores of Carboxen 563 is found not before $p/p_0>0.80$ , i.e., in the pore size distributions, micro- and mesopores are well separated.

In Table 1, the approximate starting pressures p and corresponding relative pressures $p/p_0$ of the isotherms in Figure 11 are listed.

Table 1.

Starting pressures of experimental N₂ adsorption isotherms.

Adsorbent	p [Torr]	$p/p_0$
Carboxen 563 (α)	$2.35·{10}^{-3}$	$3.07·{10}^{-6}$
Carboxen 563 (β)	$4.44·{10}^{-4}$	$5.80·{10}^{-7}$
NaY zeolite	$4.30·{10}^{-4}$	$5.66·{10}^{-7}$

From Table 1 and Figure 11(b), it becomes obvious that the starting pressure of the α isotherm (measured with the older ASAP 2000 device) is nearly an order of magnitude higher than the starting pressures of the β and NaY isotherms. Comparing the initial pressure region of the α and β isotherms, a similar shape is found – however, the whole α isotherm is shifted to higher pressures.

This behavior is reflected in the transformed isotherms ${\varphi }_t$ presented in Figure 11(c). Lower pressures are transformed into larger molar adsorption energies – hence, the transformed isotherm ${\varphi }_t$ of sample α already ends at ξ-values of about $14 \frac{\text{kJ}}{\text{mol}},$ whereas the transformed isotherms of β and NaY end later. It can be seen that all inflections of the measured (total) adsorption isotherms in Figure 11(a) and (b) are reproduced in the transformed isotherms ${\varphi }_t$ on another level. Last but not least, the transformed isotherms ${\varphi }_t$ specify the energy range allowing reliable statements about the energetic heterogeneity.

Before presenting the calculated AEDs based on the three isotherms, we add some preliminary considerations. In the calorimetric experiment at low temperatures, a nearly constant isosteric heat $-q_{st}$ is measured for the adsorption of N₂ on Carboxen 563 at 77 K (cf. Figure 12) up to a loading degree ${\Theta }_t\approx 0.6$ . From an energetic point of view, this suggests a relatively homogeneous carbon surface.

Figure 12.

Experimental isosteric heat for the adsorption of N₂ on Carboxen 563 at 77 K.

Therefore, it is expected that Carboxen 563 provides a more or less simple distribution function with one main peak reflecting the predominant and strong N₂-carbon interaction, and possibly any smaller secondary peaks for the interaction between N₂ and the functional groups existing on the carbon surface.

Figure 13 shows the calculated spectral densities $R$ and AEDs based on the two experimental Carboxen 563 isotherms.

Figure 13.

Calculated spectral densities $R$ : (a) of the α sample, (b) of the β sample. Calculated AEDs: (c) of the α sample, (d) of the β sample.

As can be seen in Figure 13(a) and (b), the spectral densities $R$ decay rather slowly. This result indicates AEDs with small distinct peaks (cf. Figure 4(a) and (Trefethen, 2000)). The cut is made as late as possible, but as early as necessary in order to include as much information as possible.

The obtained AEDs (cf. Figure 13(c) and (d)) are qualitatively comparable. As expected, a relative simple distribution function with only one main peak at interaction energies u larger than $13 \frac{\text{kJ}}{\text{mol}}$ is found pointing out the energetic homogeneity of the Carboxen 563 surface and reflecting the predominant N₂-carbon interaction. Additionally, in both AEDs, a smaller distinct peak appears at energies u of about $7 \frac{\text{kJ}}{\text{mol}}$ which can be attributed according to Bräuer et al. (1982) to the condensation of N₂ on the surface of the adsorbent. This peak disappears if less data points at higher pressures are included in the calculation – an additional indication that our approximated AEDs are physically meaningful.

The positions of the main peak in the AEDs differ significantly. For sample α, the main peak lies around $u\approx 13.75 \frac{\text{kJ}}{\text{mol}}$ , and for sample β around $u\approx 15 \frac{\text{kJ}}{\text{mol}}$ . This result shows the high sensitivity of the calculated AEDs to the underlying adsorption isotherms, i.e., even changed measurement conditions, sample heterogeneities etc. can significantly influence the shape of the AED. However, the quantitative difference of the AEDs is acceptable.

For NaY zeolite, a different picture results. The calculated spectral density $R$ and AED are presented in Figure 14. Again, the spectral density $R$ decays rather slowly, i.e., small distinct peaks are expected.

Figure 14.

(a) Calculated AED of the NaY zeolite. (b) Corresponding spectral densities $R$ .

If the first peak at $u\hspace{0.17em}\approx \hspace{0.17em}7 \frac{\text{kJ}}{\text{mol}}$ can be again attributed to the condensation of nitrogen on the surface of the adsorbent (Bräuer et al., 1982), three distinct peaks appear in the energy range of $12 \frac{\text{kJ}}{\text{mol}}<u<16 \frac{\text{kJ}}{\text{mol}}$ , i.e., we find a more heterogeneous AED than in Figure 13. This is physically intuitive because in contrast to the Carboxen 563 carbon material, the NaY zeolite as an alumosilicate provides an energetically heterogeneous surface.

Conclusion

In this paper, the regularization scheme presented by Arnrich et al. (2015) has been further developed by means of Fourier transform. As the main result of this work, a simple cut-off criterion has been established based on the graphical analysis of the so-called spectral density $R$ . For error-free adsorption isotherms, excellent recalculation of AEDs can be obtained.

If the regularization is applied to erroneous adsorption isotherms, e.g. measured ones, the spectral density $R$ provides qualitative information about the error influence and the separation of peaks in AEDs. If $R$ exhibits a strong oscillating behavior with exponentially growing amplitudes at low frequencies, the measurement error is too large for making meaningful statements about the AEDs, i.e., the measurement has to be rejected. The cut-off of the spectral density $R$ is found by simple visual selection. The application of our new method to experimental gas adsorption isotherms indicates that the regularization is able to provide physically meaningful results.

The presented regularization for the Langmuir kernel suggests that:

The calculated AEDs strongly depend on the reliable pressure range of the experimental adsorption isotherms. Especially in the initial pressure range (Henry region) correlated with higher molar adsorption energies, the experimental errors have a strong impact. Not rarely, the AED cannot be calculated due to fluctuations in the low pressure range or due to the lack of measurement points. Therefore, many points with small fluctuations in the initial pressure range are required for calculating reliable AEDs.

The number of values of the recalculated AED depends on the number of measured values of the adsorption isotherm, which is usually too low. In order to increase the number of measured values, it is often necessary to construct a smoothed polygon chain, which describes the measured points of the adsorption isotherm accurately.

The best approximation of the real AED is given by using all isotherm points corresponding to the Langmuir model (because of the non-locality of Fourier transform), but not by using only points in the initial pressure range.

The presented method can be understood as a further step in the right direction; however, it is not the last word on the subject. At the moment, the cut-off criterion by simple visual selection is still subjective. Hence, there is a need of an objective mathematical criterion to assure that the algorithm really yields the best solution for the given data. Furthermore, extended methods have to be developed, e.g. based on damping out the error amplification.