A modified Sips distribution for use in adsorption isotherms and in fractal kinetic studies

Abstract

The classical energy distribution introduced by Sips to account for the Langmuir–Freundlich adsorption isotherm is extended to the case when the exponent α of the isotherm is higher than 1. The mean free energy of desorption E₀ is estimated from the apparent dissociation constant K by the classical formula E₀ = −RT ln K. The dispersion of the energy values, which reflects the heterogeneity of the adsorbing sites, is estimated from the exponent α. The dispersion decreases when the exponent increases. Comparisons are made with the Gaussian approximation and the condensation approximation. The Sips distribution is also applied to activation energies, resulting in a Mittag–Leffler kinetic equation, as used in ‘fractal kinetic’ studies. The adsorption of the dye Basic Yellow 28 onto a surfactant-modified aluminium-pillared clay is studied as an example.

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1 Introduction

It has been known for a long time that the heterogeneity of an adsorbing material, i.e. the presence of many adsorption sites having different affinities for the adsorbed molecules, can be represented by a statistical distribution of binding energies. Moreover, it has been shown that this distribution, or ‘energy profile’, can be computed by taking the inverse Stieltjes transform of the observed binding isotherm.

Sips^1,2 has introduced a statistical distribution of energies which, combined with the Langmuir local isotherm, gives a global isotherm in the form of the Langmuir–Freundlich equation, also known in the biochemical domain as the Hill equation. This distribution has been extended to the kinetic domain by Glöckle and Nonnenmacher.³

The Langmuir–Freundlich equation involves a dimensionless exponent α which is related to the ‘sharpness’ of the energy distribution. Sips considered only the case α < 1. However, values of α > 1 are often found experimentally: in these cases, the observed isotherm has an inflection point and a sigmoidal shape. So, the purpose of the present work is to modify the Sips distribution to account for such values.

2 Theory

2.1 Distribution of binding constants

We start with the classical integral equation for adsorption:where: P is the experimental pressure (or concentration) divided by the corresponding standard state value, i.e. P = p/p⁰ or P = C/C⁰, Θ(P) is the global (observed) isotherm, θ(b, P) is the local isotherm, related to a particular site with binding constant b, w_b(b) is a weighting function.

So, the global isotherm is the weighted mean of the many local isotherms.

In the special case when the local isotherm is of the Langmuir type, the integral equation becomes:

Sips^1,2 applied this method to the case where the global isotherm is of the Langmuir–Freundlich type:

where K is an apparent dissociation constant and α a positive real exponent.

The integral in eqn (2) can be written as a Stieltjes transform, from which the expression of w_b(b) can be found (see Appendix 1). We obtain:

with ϕ = α(2κ + 1)π, κ being an integer such that sin ϕ > 0.

From this equation we deduce the integral (Appendix 2):

where ϕ* is such that ϕ = ϕ* + 2κ*π, κ* integer and ϕ* ∈ ]0, π[. Hence, sin ϕ = sin ϕ* and cos ϕ = cos ϕ*. In practice we can take:

ϕ* = π·frac(α) (6)

where frac denotes the fractional part.

The probability density function (p.d.f.) of b is therefore:

2.2 Distribution of energies

Let E = RT ln b be the free energy of desorption and E₀ = −RT ln K. We take as new variable [scr E, script letter E] = (E − E₀)/RT = ln bK, so that (Kb)^α = e^{α[scr E, script letter E]}. The p.d.f. of [scr E, script letter E] is f_{[scr E, script letter E]}([scr E, script letter E]) such that:

So:

Since ϕ* = π·frac(α) the probability density depends only on α. Fig. 1 shows some curves plotted for several α values; the curve becomes sharper when α increases.

Fig. 1 Energy distributions. The curves correspond to increasing values of the exponent α of the Langmuir–Freundlich equation. E is the standard free energy of desorption and E₀ is its mean value. From bottom to top, α increases from 0.5 to 4.5 by step of 1.

The function has some interesting properties (Appendix 3):

(1) f_{[scr E, script letter E]}([scr E, script letter E]) is maximal when [scr E, script letter E] = 0, or E = E₀ = −RT ln K.

(2) The curve f_{[scr E, script letter E]} ([scr E, script letter E]) is symmetric with respect to the Oy axis; so, the function is even and the mean energy is equal to E₀.

(3) The cumulative distribution function (c.d.f.) is given by:

from which we obtain the energy [scr E, script letter E]_{[scr P, script letter P]} such that Prob([scr E, script letter E] ≤ [scr E, script letter E]_{[scr P, script letter P]}) = F([scr E, script letter E]_{[scr P, script letter P]}) = [scr P, script letter P]:

The function F⁻¹ such that [scr E, script letter E]_{[scr P, script letter P]} = F⁻¹([scr P, script letter P]) is the quantile function. It gives the interquartile range:

Δ[scr E, script letter E] = [scr E, script letter E]_0.75 − [scr E, script letter E]_0.25 (12)

This parameter defines an interval centered on the mean and containing 50% of the energy values; it can be used to characterize the energy dispersion since we cannot compute a standard deviation (the integral of [scr E, script letter E]²f_{[scr E, script letter E]}([scr E, script letter E]) cannot be evaluated).

Eqn (12) shows that the energy dispersion, which characterizes the heterogeneity of the binding sites, depends only on the exponent α. A multi-site sorption model is outside of the purpose of the present study. We should however note that the case of heterogeneous ‘patchwise’ binding of a solute at the solid sorbent surface^4–6 is important both from theoretical and experimental points of view.

Fig. 2 shows the variation of Δ[scr E, script letter E] with α. As expected from the probability density curves (Fig. 1), the dispersion decreases when α increases. When α is integer (α = n) there is a discontinuity since Δ[scr E, script letter E] → 0 when α → n⁻ and Δ[scr E, script letter E] > 0 when α → n⁺. Of course, the nonzero value must be chosen to characterize the dispersion.

Fig. 2 Dispersion of the energy values, expressed by the interquartile range Δ[scr E, script letter E], vs. the exponent α of the Langmuir–Freundlich equation. Solid curve: modified Sips distribution. Dotted curve: condensation approximation (logistic distribution).

2.3 Approximate distributions

Two distributions have been used in the past to approximate the Sips distribution.

2.3.1 The normal (Gaussian) distribution. This distribution has a mean of zero and a standard variation σ = Δ[scr E, script letter E]/1.35, since for the normal distribution the interquartile range is equal to 1.35σ. So, the p.d.f. is:

2.3.2 The condensation approximation. This approximation^4,5,7 assumes that the c.d.f. of the binding constants, F̂_b(b), has the same form than the global isotherm. In our case, this leads to a log-logistic distribution:

The p.d.f. is therefore given by:

Multiplying the numerator and denominator by (Kb)^2α we get:

From this equation and the methods described in paragraph 2.2 we can compute the p.d.f. of the energy distribution:

This is a logistic distribution, and also the limit of the Sips distribution (eqn (9)) when frac(α) → 0, sin(ϕ*)/ϕ* → 1 and cos(ϕ*) → 1.

Since this function is still maximal for [scr E, script letter E] = 0, the condensation approximation gives the same mean energy than the Sips distribution. The c.d.f. of this distribution is:

From which we compute the quantile function:

and the interquartile range:

This approximation model is applied on Fig. 2 (dotted line) where we can observe that it constitutes a global envelope upper to the Sips distribution (full curve).

2.4 Application to kinetics

When dealing with sorption kinetics, several empirical models can be applied to experimental data;⁸ they were also established theoretically from the corresponding heterogeneous sorption isotherm equations.^9,10

Assuming that each site reacts by pseudo-first order kinetics with rate constant k, the fraction of reactant remaining at time t will be:

So, y(t) is the Laplace transform of w_k(k).

The rate constant k is related to the free energy of activation E by:

where k_B denotes Boltzmann's constant and h denotes Planck's constant.

The weighting function w_k(k) is obtained from eqn (4) by replacing the equilibrium constant k with a characteristic time τ:

The integral (21) can then be computed (Appendix 4):

where E_α is the Mittag–Leffler function, defined by its series expansion:

This function has been widely used in ‘fractal kinetic’ studies, for instance in pharmacokinetics¹¹ or in the analysis of fluorescence decay curves.¹² For instance, a drug can diffuse into the various tissues of the organism by a set of pseudo-first order reactions with different rate constants.

3 Material and methods

3.1 Experimental data

As an example we started from the data of Cheknane et al.¹³ The adsorption of two basic dyes: Basic Yellow 28 (BY28) and Basic Green 4, was studied at 25 °C and two pH values (3 and 6). The adsorbent was a surfactant-modified aluminium-pillared clay available in three physical forms: powder, granules 300–400 μm, granules 700–800 μm. These conditions resulted in 12 adsorption isotherms which were then fitted by the Langmuir and the Freundlich equations, the first one giving the best fit. In their original paper, the authors did not try the Langmuir–Freundlich equation. Supplementary kinetic data for BY28 at pH 3 were obtained, using the methods described in the original study.

3.2 Statistical methods

3.2.1 Adsorption isotherms and kinetics. The isotherms were analysed by the Langmuir–Freundlich equation, written in the following form:where q denotes the amount of dye adsorbed at equilibrium (mg g⁻¹), Q_max its maximal value and C the concentration of dye remaining in solution (mg L⁻¹). The Langmuir equation (α = 1) was also fitted to the data.

The kinetic curves were analysed by the pseudo-first order rate equation:⁸

q = Q_max(1 − e^−kt) (27)

where q denotes the amount of dye adsorbed at time t, Q_max its maximal value and k the rate constant.

For each equation, the parameters (Q_max, K, α, k) were determined by minimizing the sum of squared residuals:

where n denotes the number of experiments, q_i the observed value for the i-th experiment and q̂_i the calculated value. The minimization was performed by Marquardt's algorithm, using our computer program WinReg for Windows XP.¹⁴ Comparison between the Langmuir and Langmuir–Freundlich equations was performed by the F ratio (explained variance/residual variance).

The mean desorption energy E₀ and the energy distribution were determined from the fitted values of K and α. A computer program to evaluate the relevant distribution functions is given in Appendix 5. Although the mathematical derivations may look difficult, the resulting formulas are very easy to use, since in practice we need only eqn (6), (9)–(11).

3.2.2 Comparison with other distributions. Comparison of the generalized Sips distribution with either the normal distribution or the condensation approximation was performed by evaluating the total variation distance:¹⁵where f_app([scr E, script letter E]) denotes the p.d.f. of the approximating distribution, i.e. g ([scr E, script letter E]) from eqn (13) or f̂_{[scr E, script letter E]}([scr E, script letter E]) from eqn (17).

The integral was computed for several α values by the trapezoidal rule, using 2000 points in the range [−10σ, 10σ], where σ denotes the standard deviation of the normal distribution.

4 Results

4.1 Adsorption isotherms

All curves could be fitted by the Langmuir equation, thus confirming the original results of Cheknane et al.¹³ However, an improvement was noticed with the Langmuir–Freundlich equation for the dye BY28 at pH 3. In this case, the preliminary results suggested that the three curves corresponding to the different physical forms of the adsorbent differed mainly by their Q_max values, while the values of K and α were not statistically different. So, a second fit was performed by grouping the three curves (after dividing each q value by the corresponding Q_max in order to get Θ = q/Q_max) and fitting the Langmuir–Freundlich equation in the form:with only two parameters: K and α.

These procedures gave the following parameter estimates:

K = 3.50 ± 0.16 × 10⁻⁵ M

E₀ = 25.42 ± 0.11 kJ mol⁻¹

α = 2.19 ± 0.28

Q_max(1) = 340 ± 41 mg g⁻¹

Q_max(2) = 246.0 ± 9.5 mg g⁻¹

Q_max(3) = 132.2 ± 4.2 mg g⁻¹

For the Q_max values, the indices 1, 2 and 3 refer to the three forms of the adsorbent, respectively: powder, granules 300–400 μm, granules 700–800 μm.

Fig. 3 shows that the points corresponding to the three forms are regularly spaced around the fitted curve. This curve displays an inflection point, which is characteristic of an exponent α > 1.

Fig. 3 Adsorption isotherm of the dye Basic Yellow 28 at pH 3 onto surfactant-modified aluminium-pillared clay present as: powder (full circles), granules 300–400 μm (full triangles), or granules 700–800 μm (full squares). The curve corresponds to the Langmuir–Freundlich eqn (30) with adjusted parameters given in the text.

From the fitted value of α we computed a Δ[scr E, script letter E] value of 0.98 RT units (about 2.4 kJ mol⁻¹), from which the Sips distribution could be plotted, as well as its Gaussian approximation (Fig. 4). In this case the two curves match very well, but the Gaussian decreases slightly faster. This result could be expected since the normal distribution decreases as exp(−[scr E, script letter E]²) while the Sips distribution decreases as exp(−[scr E, script letter E]).

Fig. 4 Energy distribution for the adsorption of Basic Yellow 28 at pH 3 onto surfactant-modified aluminium-pillared clay. The solid curve corresponds to the modified Sips distribution (eqn (9)). The dotted curve corresponds to the normal Gaussian distribution having the same interquartile range (eqn (13)). The mean energy E₀ is equal to 25.4 kJ mol⁻¹.

The energy distribution extends on ±3 RT units around the mean energy E₀. This corresponds to a range of 18 to 33 kJ mol⁻¹ for the free energy of desorption. This is a relatively wide range which can be ascribed to non-specific binding such as hydrophobic interactions. Indeed, Cheknane et al. found that the binding of BY28 at pH 3 corresponds to the weakest affinities. Since this is precisely the system leading to an energy distribution, as shown by the better fit of the Langmuir–Freundlich equation, we can suggest that the highest affinities are related to the most specific interactions, where all the binding sites have similar affinities.

Since the binding energies of BY28 at pH 3 were found to be distributed, it was interesting to investigate the corresponding activation energies. From the kinetic data we determined the following rate constants:

k(1) = 0.0258 ± 0.0042 min⁻¹

k(2) = 0.0234 ± 0.0024 min⁻¹

k(3) = 0.0325 ± 0.0013 min⁻¹

The curves were very well described by the pseudo-first order (exponential) kinetics (Fig. 5), in agreement with the previous results of Cheknane et al.¹³ for the results at pH 6. Since the exponential function is equivalent to the Mittag–Leffler function with exponent α = 1, the activation energies do not appear to be distributed, unlike the desorption energies.

Fig. 5 Adsorption kinetics of the dye Basic Yellow 28 at pH 3 onto surfactant-modified aluminium-pillared clay present as: powder (full circles), granules 300–400 μm (full triangles), or granules 700–800 μm (full squares). The curves correspond to the pseudo-first order rate eqn (27) with adjusted parameters given in the text.

4.2 Comparison with other distributions

In order to evaluate the error done by approximating the modified Sips distribution with either the normal distribution or the condensation approximation, the total variation distance was computed as a function of the exponent α. It was found that the distance depends only on the fractional part of the exponent (Fig. 6). The following interpolating functions were fitted by nonlinear regression:

Fig. 6 Approximation of the modified Sips distribution by the normal distribution (solid circles) or the condensation approximation (open circles). The curves correspond to eqn (31) and (32) resp.

• For the Gaussian approximation:

δ = 0.00698 exp[3.13·frac(α)] + 0.0238 (31)

• For the condensation approximation:

δ = 0.0242 exp[3.30·frac(α)] − 0.0335 (32)

For frac(α) < 0.35 the condensation approximation works better; but beyond this point the Gaussian approximation should be preferred. In any case, the quality of the approximations is poor for the highest values of frac(α).

5 Discussion

5.1 Modified Sips distribution

Sips has introduced a statistical distribution of adsorption energies, suitable when the adsorption isotherm follows the Langmuir–Freundlich equation. This equation involves two parameters: an apparent equilibrium (dissociation) constant K and a dimensionless exponent α. The mean energy is given by the classical thermodynamic relationship, E₀ = −RT ln K while the energy dispersion is related to the exponent. This statistical analysis provides a physico-chemical interpretation of the parameters K and α, which are often viewed as purely empirical. The Sips distribution was further extended to activation energies by Glöckle and Nonnenmacher. However, these early studies considered only the case α < 1. We propose here a modification of the Sips distribution to account for exponents higher than 1. This case can be encountered in adsorption studies, as shown by the example presented in the Results; it is also widely found in biochemical and pharmacological studies, where the equation is known as the Hill equation. So, the modified distribution has a wide range of applications.

5.2 Comparison with the normal distribution

Sips noticed a similarity between his distribution and the classical normal (Gaussian) distribution, and provided an example with α = 0.5 (Fig. 1 in ref. 1). However our results show that, although the similarity holds for frac(α) values up to about 0.5, the discrepancy between the two distributions increases exponentially for higher values. So, care should be exercized when deciding to replace the Sips distribution with the Gaussian.

It is of course possible to postulate a normal distribution of energies and to compute the resulting isotherm from eqn (2), as described by Davis and DiToro.¹⁶ However, in this case the integral must be computed numerically.

5.3 The condensation approximation

The present work has shown that the equations involved in the application of the method of Stieltjes transform are easily handled, once the necessary demonstrations have been carried out. In spite of this, the method does not seem to have been widely used in experimental adsorption studies. Instead, many researchers have applied an alternative method known as the ‘condensation approximation’.⁷ This method assumes that the statistical distribution of the binding constant b has the same form than the global isotherm. It has been shown that the various distributions derived from most of the many published isotherms are special cases of a general statistical distribution known as Burr's XII-distribution.¹⁷ This is true for the log-logistic distribution derived from the Langmuir–Freundlich isotherm. In this case, however our results show that the approximation may not be acceptable for the highest values of the fractional part of the exponent.

5.4 Kinetic application

We have shown that the modified Sips distribution also works with activation energies, leading to a kinetic equation having the form of a Mittag–Leffler function. A similar result (for α < 1) was obtained by Glöckle and Nonnenmacher,³ using the formalism of fractional order differential equations, in which the exponent α becomes the order of the equation. Here too the statistical interpretation helps to give a physical meaning to a rather abstract formalism.

Of course, the activation energy is the difference between the energy of the transition state and the energy of the reactants. Since both species are linked to the sorbent, we can expect a high degree of similarity between the distributions of these two energies, so that their differences may appear to be non-distributed. This means that the exponent α of the kinetic curve may be quite different from the exponent of the isotherm. Especially, if the two distributions are very similar, we can observe a classical pseudo-first order kinetic (α = 1) even if the adsorption isotherm follows the Langmuir–Freundlich equation. This is exactly the case encountered in our example.

5.5 The Freundlich equation

This equation, which corresponds to a power function, can be viewed as an approximation of the Langmuir–Freundlich equation when P ≪ K, so that Θ(P) ≈ (P/K)^α. In this case, the energy distribution f_{[scr E, script letter E]}([scr E, script letter E]) would be a decreasing exponential, but its integral would be infinite, unless a lower bound E_min is placed on the energy. For instance, if E_min = 0, the lower bound of the integral (2) becomes equal to 1, and the integral is no more a Stieltjes transform. However, if an exponential distribution is used for the energies, the integral can be expressed as a hypergeometric function, of which the Freundlich equation is still an approximation^18,19 (see the Appendix in the ESI section of ref. 19).

6 Conclusions

The Langmuir–Freundlich equation can be used advantageously in adsorption studies, especially when the observed isotherm is sigmoidal. The modified Sips distribution proposed in this work provides a physico-chemical interpretation of the fitted parameters. This interpretation extends to the ‘fractal’ kinetic domain, when the observed kinetics is described by a Mittag–Leffler function. Other applications can be found in the study of molecularly imprinted polymers.²⁰

Appendix 1

Computation of the weighting function

With the change of variable x = 1/P, eqn (2) and (3) become:

The integral is a Stieltjes transform; so, b·w_b(b) is the inverse Stieltjes transform of Φ(x) and can be computed by the classical inversion formula:^21,22

or:

where κ is an integer while R and θ denote respectively the modulus and argument of the complex number −b + εi (with b > 0).

Taking into account the Euler relationships:

e^iθ + e^−iθ = 2 cos θ; e^iθ − e^−iθ = 2i sin θ (36)

We get:

with ϕ = α(2κ + 1)π.

Appendix 2

Integral of the weighting function

This integral is:

With the two changes of variables: z = (Kb)^α, then t = (z + cos ϕ)/sin ϕ, we obtain:

This equation can be simplified if we notice that the function arctan(tan) is defined on with values on ]−π/2, π/2[ such that for any ϕ* ∈ ] −π/2, π/2[ and any κ ∈ [Doublestruck Z], we have

arctan[tan(ϕ* + κπ)] = arctan(tan ϕ*) = ϕ* (41)

where [Doublestruck Z] is the set of the relative integer numbers.

From the condition sin ϕ > 0, we must choose an integer κ verifying For such angle ϕ, it exists an angle ϕ* ∈ ]0, π[ and an integer κ* such that ϕ = ϕ* + 2κ*π.

We have two cases:

(1) ϕ* ∈ ]0, π/2[

We deduce that:

arctan[tan(ϕ* + 2κ*π)] = arctan(tan ϕ*) = ϕ* (42)

It also wellknown that

From the previous equations, and tan x > 0 if x ∈ ]0, π/2[, we get:

which implies

(2) ϕ* ∈ ]π/2, π[

If we put , we have and of which we deduce:

It also wellknown that

From the previous equations, and tan x < 0 if x ∈ ] −π/2, 0[, we get:

which implies

Appendix 3

Properties of the energy distribution

Computation of the maximum. The derivative of the p.d.f. is:f_{[scr E, script letter E]}([scr E, script letter E]) is maximal when f′_{[scr E, script letter E]}([scr E, script letter E]) = 0, so that e^{2α[scr E, script letter E]} = 1, hence [scr E, script letter E] = 0 and E = E₀ = −RT ln K.

Symmetry

The curve f_{[scr E, script letter E]}([scr E, script letter E]) is symmetric with respect to the Oy axis, since:

Cumulative distribution function

This function is given by the integral:

With the two changes of variables: z = e^αu, then t = (z + cos ϕ*)/sin ϕ*, we obtain eqn (10)

Quantile function

We search [scr E, script letter E]_{[scr P, script letter P]} such that F([scr E, script letter E]_{[scr P, script letter P]}) = [scr P, script letter P]. From eqn (10) we get successively:from which we deduce eqn (11)

Appendix 4

Application to kinetics

According to eqn (21), the function y(t) is the Laplace transform of w_k(k). Let us define two consecutive Laplace transforms of w_k(k) as follows:

w_k(k) → y(t) → Y(s) (57)

From the properties of the Laplace transform, and the condition y(0) = 1, we obtain the similar sequence for k·w_k(k):

k·w_k(k) → −y′(t) → 1 − s·Y(s) (58)

Following Glöckle and Nonnenmacher,³ we use the fact that the Stieltjes transform is equivalent to two consecutive Laplace transforms. We know that the Stieltjes transform of k·w_k(k) is Φ(s) such that:

From which we get:

The inverse Laplace transform of this function can be computed by making the change of variable x = (τs)^−α followed by a series expansion:

The inverse Laplace transform of s^−α is t^α−1/Γ(α) for α > 0, so:

which is the Mittag–Leffler function, eqn (25).

Appendix 5

Computer program

This program written in Pascal computes the energy distribution from the exponent α of the Langmuir–Freundlich equation. The functions pdf, cdf and quantile correspond respectively to eqn (9)–(11).

Acknowledgements

We are indebted to John Schwartz and Georgi Smirnov for helpful advices concerning the Stieltjes transform.

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