Affinity distribution functions in multicomponent heterogeneous adsorption . Analytical inversion of isotherms to obtain affinity spectra

An analytical approach for the interpretation of multicomponent heterogeneous adsorption or complexation isotherms in terms of multidimensional affinity spectra is presented. Fourier transform, applied to analyze the corresponding integral equation, leads to an inversion formula which allows the computation of the multicomponent affinity spectrum underlying a given competitive isotherm. Although a different mathematical methodology is used, this procedure can be seen as the extension to multicomponent systems of the classical Sips's work devoted to monocomponent systems. Furthermore, a methodology which yields analytical expressions for the main statistical properties (mean free energies of binding and covariance matrix) of multidimensional affinity spectra is reported. Thus, the level of binding correlation between the different components can be quantified. It has to be highlighted that the reported methodology does not require the knowledge of the affinity spectrum to calculate the means, variances, and covariance of the binding energies of the different components. Nonideal competitive consistent adsorption isotherm, widely used in metal/proton competitive complexation to environmental macromolecules, and Frumkin competitive isotherms are selected to illustrate the application of the reported results. Explicit analytical expressions for the affinity spectrum as well as for the matrix correlation are obtained for the NICCA case.


I. INTRODUCTION
The interpretation of the interactions of small molecules with heterogeneous surfaces and/or macromolecules, both in gas and solution phase, has been the subject of intensive research for the last decades owing to its great importance in several scientific disciplines: environmental speciation studies, [1][2][3] gas/solid adsorption, 4,5 binding of ligands to biochemical receptors, 6 chromatographic studies, 7 polymer science, 8,9 etc.A current strategy to describe monocomponent adsorption consists in expressing the total macromolecular/surface coverage as a weighted superposition of local Langmuirian isotherms 1,2,10 ͑c͒ϭ

͑1͒
where c is the concentration of the adsorbed species and pЈ(k) is the so-called affinity distribution function or affinity spectrum, which expresses the probability density of having sites in the macromolecule/surface with binding stability constant for the adsorbed species lying in between k and k ϩdk.Usually, as indicated in Eq. ͑1͒, the affinity spectrum is represented in the logarithmic space ln k, proportional to the binding free energy.Hence, once a concrete form for the affinity spectrum is proposed, Eq. ͑1͒ produces an analytical isotherm which can further be used to fit an experimental set of coverage versus concentration data.
In other cases, one could be interested in the knowledge of the affinity spectrum underlying an empirically proposed isotherm, which successfully describes the experimental information available.This inverse problem reduces to solving the integral equation ͑1͒ for p(ln k).An inversion formula to obtain p(ln k) for a given analytical isotherm was provided by Sips in a pioneer work. 10The formula has been widely used for a long time in the interpretation of monocomponent isotherms ͑such as Langmuir-Freundlich, Frumkin or Temkin͒ in terms of a underlying affinity distribution function. 10,11On the other hand, several numerical methods have been applied to determine p(ln k) directly from a given set of experimental versus ln(c) data.Among these, the use of regularization methods to solve the integral equation ͑1͒ became the subject of many works devoted to the analysis of heterogeneous monocomponent adsorption, both in gas and in solution phase. 3,4,12he affinity spectrum approach can be straightforwardly extended to multicomponent competitive adsorption, by considering the macromolecular/surface coverage of each component i, i (c 1 ,c 2 ,...,c N ), dependent on the concentrations a͒ Author to whom correspondence should be addressed; Electronic mail: fmas@qf.ub.es of all the N components of the system, as a superposition of competitive Langmuirian isotherms [13][14][15][16] i ͑ c 1 ,c 2 ,...,c N ͒ϭ ͵ Nevertheless, the main obstacle for this generalization lies in the multidimensional nature of the integral equation ͑2͒.This has resulted, in contrast with the equivalent treatment of monocomponents systems, in a lack of simple analytical results allowing an easy interpretation of empirical isotherms in terms of an underlying affinity distribution function.Regularization methods have been successfully applied to the finding of p(ln k 1 ,ln k 2 ,...,ln k N ) for two adsorbing components in solution. 17However, the inverse problem has not been analytically solved for the general case.A first aim of this work is to provide, in Sec.II, a general analytical solution for Eq.͑2͒, based on Fourier analysis, and to apply it, in Sec.III, to the finding of the affinity spectrum which generates nonideal competitive consistent adsorption ͑NICCA͒ and competitive Frumkin isotherms.It is important here to note that the choice of the local competitive isotherm appearing in the kernel of Eq. ͑2͒, in the case of competitive adsorption, could be multiple.In this work we only undertake the problem of Langmuirian local adsorption, but if, for instance, the molecules of one component bind to more that one site at the same time ͑chelate complexation͒, other choices of the local adsorption isotherm would be required. 17nother question of interest in the characterization of the binding to heterogeneous systems is determining the minimum information needed to successfully describe a set of experimental adsorption data.This question arises from the very ill-posed nature of the integral equation ͑1͒, i.e., small changes in the affinity spectrum lead to almost the same binding curve.This means that, from a practical point of view, only a part of the information contained in the affinity spectrum is of interest in the interpretation and prediction of the empirical information for a given range of concentrations.
In the case of monocomponent systems, it has been shown that in a wide range of concentrations corresponding to intermediate coverages, only the mean and the variance 2 of each peak of the affinity spectrum are relevant in the description of the experimental binding curve ͑that is, different isotherms with the same and 2 can reproduce with similar accuracy the same binding curve͒. 18Hence, the knowledge of and 2 is almost enough to characterize the binding properties of the heterogeneous surface.
It is therefore expected that something similar will hold for multicomponent systems, becoming relevant quantities in this case not only the mean and variance of the spectrum of the different components, but also the correlation of the binding among them.The natural mathematical description of such statistical properties is the set of mean affinities ͑one for each component͒ and the elements of the covariance matri-ces.A procedure to obtain means and covariance matrices of multidimensional affinity spectra in terms of the parameters of competitive isotherms is given in Sec.IV, and thereafter applied to NICCA and Frumkin isotherms.

II. FOURIER ANALYSIS OF MULTICOMPONENT BINDING ISOTHERMS: AN INVERSION FORMULA TO OBTAIN THE MULTIDIMENSIONAL AFFINITY SPECTRA
A. Transformation of the integral Eq. "2… in order to apply Fourier analysis Let us assume that the coverage of certain component i, i (͕c j ͖), can be expressed as a superposition of multicom- ponent Langmuirian isotherms, as Eq.͑2͒ indicates.In order to apply Fourier transform techniques, it is convenient to rewrite Eq. ͑2͒ in terms of a new set of variables, where x i is proportional to the binding free energy of component i to the site and i is proportional to the chemical potential of component i.In terms of these new variables, Eq. ͑2͒ rewrites where vectorial notation, ជ ϭ( 1 , 2 ,..., N ) and x ជ ϭ(x 1 ,x 2 ,...,x N ), have been used, and . ͑6͒ Thus, Eq. ͑5͒ takes the form of a convolution equation.
Fourier analysis provides an immediate solution for the Fourier transform ͑FT͒ of the affinity spectrum.Since several definitions of FT are commonly found in the literature, we adhere to the FT definition of an arbitrary function f, as in Ref. 19, where the components of vector ជ ϭ( 1 , 2 ,..., N ) are the variables in the Fourier space.
Bearing in mind that the FT of the convolution of two functions, is the product of their Fourier transforms and recalling that the FT of p(Ϫx ជ ) is the complex conjugate of the FT of p(x ជ ), the Fourier transform of ͑5͒ leads to ˜i͑ ជ ͒ϭ p ˜*͑ ជ ͒g ˜i͑ ជ ͒. ͑8͒ Nevertheless, as functions g i and i do not tend to zero in the limit i →ϱ, we modify Eq. ͑5͒ in such a way that Fourier analysis can be applied with well behaved functions.
From the total macromolecular coverage a new function B can be defined as where and ⌫(i) is the gamma function.
In this way, we have transformed the integral equation ͑2͒ into another one which contain well behaved functions to apply Fourier analysis, since both B and h tend to zero when the respective variables i or i Ϫx i tend to Ϯϱ.

B. Fourier analysis of the transformed equations: Analytical solution of Eq. "2…
Applying Fourier transform to both sides of Eq. ͑13͒ and using the FT convolution property ͑8͒, can be computed analytically by introducing the change of variables u j ϭe x j .Using the property, 20 h ˜( ជ ) can be written as Notice that, regardless of the isotherm, Eq. ͑18͒ relates the macroscopic information contained in B ˜͑which is calculated from the competitive isotherm under study͒ to the microscopic information contained in the affinity spectrum.Isolating p ˜* in ͑18͒ and recalling the inverse FT of p ˜*, p͑Ϫx ជ ͒ϭ 1

͑19͒
we obtain for p(͕ln k i ͖),

͑20͒
Equation ͑20͒ represents the sought inversion formula which provides, for a given competitive isotherm, the affinity spectrum ͑if it exists͒.
For the particular case of Nϭ1, expression ͑20͒ reduces to which can be seen as an alternative to the classical inversion formula deduced by Sips. 10 Although we do not have a direct derivation of Sips inversion formula from the general equation ͑21͒, this expression can be easily tested, for instance, with the Langmuir-Freundlich isotherm, where m is the heterogeneity parameter for monocomponent systems.The Fourier transform of function B() for this isotherm ͑as can easily be deduced from Appendix B, for the particular case of Nϭ1) is Replacing ͑23͒ in ͑21͒ and using Ref. 19, the well-known Sips distribution is obtained,

A. Affinity spectrum underlying NICCA isotherm
One of the most used multicomponent isotherms, especially in the field of environmental chemistry, is the NICCA ͑nonideal competitive consistent adsorption͒ isotherm, which can be written as 21 or, in terms of the variable i ϭln c i , as where x ¯iϭϪln k ¯i and k ¯i , n i , and p are parameters of the isotherm.NICCA has been widely used to discuss the cation binding to humic substances.It accounts for heterogeneous competitive binding, being k ¯i related to the median affinity for ion i, p describing a common distribution of affinities seen by all the ions and n i taking into account an ion specific heterogeneity.The factor n i /n H , where H refers to the proton, was introduced as a more general way to keep thermodynamic consistency from a previous expression of the isotherm 22 which required equal n i values for all the ions in order to satisfy thermodynamic consistency.In the current generalized way, the maximum binding capacity is ion dependent which could reflect some degree of multidentism.As this effect suggests the use of local isotherms other than the competitive Langmuirian one considered in this work, we will restrict ourselves to the case n H ϭn i , ᭙i.
Let us then apply the inversion formula ͑20͒ to a bicomponent system (Nϭ2) described by NICCA isotherm with n 1 ϭn 2 ϭn.Computing the total coverage and performing the derivatives involved in ͑10͒, B ˜corresponding to a bicomponent NICCA isotherm ͓see Appendix B, Eq. ͑B12͔͒ becomes Replacing this expression into Eq.͑20͒ and using the relationship ⌫(1ϩz)ϭz⌫(z), 20 we obtain

͑28͒
Recalling now the Dirac ␦-function, whose integral representation can be written as Eq. ͑28͒ rewrites where and are real-valued functions, defined as

͑32͒
Figure 1 depicts the affinity spectrum corresponding to NICCA isotherm as a function of log(k 1 ) and log(k 2 ) ͑we take decimal logarithms, as it is usually done in the literature͒ with parameters nϭ pϭ0.5 and k ¯1ϭk ¯2ϭ1.The distribution basically corresponds to that obtained numerically in Ref. 13  by using regularization methods, this supporting the suitability of the numerical methods in the obtention of multicomponent affinity spectra.Notice that the spectrum obtained is symmetrical with respect to the line x 1 Ϫx ¯1ϭx 2 Ϫx ¯2 , something that is directly shown by Eq. ͑30͒, which remains invariant under the change x 1 Ϫx ¯1↔x 2 Ϫx ¯2 .The spectrum is clearly elongated along this line, evidencing the fact that NICCA isotherm implies nonzero correlation between the binding energy of both components.Conversely, this elongation is greatly reduced when the p value is equal to 1, see Fig. 2, indicating that this parameter plays an important role in the binding correlation of the two components, as was previously reported by Ref. 13.This result agrees with the classical meaning given to p, 0рpр1, which relates this parameter with a common distribution of affinities seen by all the ions, 21,22 decreasing the width of the distribution as p approaches 1.

B. Affinity spectrum underlying the multicomponent Frumkin isotherm
The inversion formula ͑20͒ can also be used to obtain the spectrum corresponding to other multicomponent isotherms.Among them, we have chosen the competitive Frumkin isotherm, which for a system of two components reads 1 where b i and i j are parameters of the isotherm.
Unfortunately, the FT of function B cannot be obtained analytically, so that we have to compute numerically the FT's appearing in the inversion formula ͑20͒ to obtain the spectrum.The result is shown in Figs.3͑a͒ or 3͑b͒, where parameters b 1 ϭb 2 ϭ1, 11 ϭ 22 ϭ1, and 12 ϭ0.5 have been used.In this case, an ellipsoidal oblate shape is obtained for the resulting spectrum.It can be observed that spurious oscillations appear, especially close to the top of the oblate which we expect to be of a smoothed form with a maximum.It is likely that these oscillations come from the fact that the affinity distribution of the Frumkin isotherm is nonzero only over a finite domain of ln k.This behavior has also been found in the inversion of the monocomponent Frumkin isotherm. 23tice also that the average log k along each axis of the spectrum of Fig. 3 is not zero ͓see Fig. 3͑b͔͒

IV. CENTRAL MOMENTS OF A MULTIDIMENSIONAL AFFINITY SPECTRUM
As it has been commented above, the integral equation ͑2͒ is an ill-posed problem for p(͕ln k i ͖).This means that the coverage predicted by different isotherms can be almost the same over decades of the free metal concentration or, in other words, the description of the binding does not require all the information contained in the affinity spectrum.We can say that there is a great instability in the recovered affinity spectrum from ͑20͒ since the binding data do not contain enough information for a precise recovering of p(͕ln k i ͖).
A robust strategy in these conditions is the determination of the first central moments of the spectrum, which actually contain the relevant information for the binding description.Let us now solve this problem with a method that avoids the recovering of p(͕ln k i ͖) from ͑20͒.
We begin recalling that the mean binding energy of component i is proportional to where the symbol ͗ ͘ indicates average over the distribution p.Likewise, the variance and covariance of the binding energy between two components is quantified by the so-called covariance matrix, C of the probability distribution p, whose components are defined as Obviously, C ii is the variance of the binding energy of component i, while C i j is the covariance between the binding energy of component i and j.The covariance quantifies to which extent an increase of the affinity of a particular site for one component, implies an increase of the affinity of such a site for the others components. 13,21,22Two limiting cases can be considered: when we have full correlation, for any site, there is a fixed shift, ␣, between the binding energy of component 1 and the binding energy of component 2 at the same site, so that the multicomponent affinity spectrum takes the form In this case, the one-dimensional affinity spectrum for both components has the same shape while they may differ in the location along the ln k axis.This situation is also called congruent adsorption.
When the affinity spectrum is fully uncorrelated, C i j ϭ0 for i j; for a given site, there is no relation between the affinity for component 1 and component 2, ln k 1 and ln k 2 are statistically independent variables and the affinity spectrum reads For the most general case, some correlation is present and no factorization of p(ln k 1 ,ln k 2 ) is possible.A dimensionless measure of the correlation between the binding energy of component i and j is the correlation coefficient, R i j , Obviously, for a two-component system, there is only one correlation coefficient which we will label as RϭR 12 ϭR 21 .
On the other hand, for a given probability distribution function, p, the mean and the covariance matrix can be understood as particular cases of the so-called cumulants of the distribution function.The cumulant generating function, K, for a given distribution p can be written as 24

͑40͒
The cumulants, jkl... , of the distribution p(͕ln k i ͖), are de- fined as the coefficients appearing in the series expansion of Although a general relationship between cumulants and central moments is not simple, it is a well-known result of probability theory that i , the first order cumulant, is the average of the distribution function, i ϭ͗ln k i ͘, ͑42͒ while i j , the cumulants of second order, are the elements of the covariance matrix i j ϭC i j ϭ͗ln k i ln k j ͘Ϫ͗ln k i ͗͘ln k j ͘.

͑43͒
A practical fact on the computation of the central moments of the multidimensional distribution functions defined as indicated in Eq. ͑2͒ can be noticed: if we are interested in the first and second order moments, the knowledge of the full affinity spectrum is not necessary.One only needs the spectrum of simpler systems where only one or two components are present.This is easily shown by observing that, for instance, where indicating that the computation of ͗ln k i ln k j ͘ does not re- quire the knowledge of the full spectrum p(͕ln k i ͖), but of only q(ln k i ,ln k j ).Due to the particular form of the kernel of Eq. ͑2͒, q(ln k i ,ln k j ) is the affinity distribution function that is found by taking in the multicomponent local isotherm all the concentrations equal to zero except those of components i and j.In effect i ͑ 0,0,...,0,c i ,0,...,0,c j ,...,0͒

͑46͒
This means that obtaining the mean and covariance matrix for multicomponent systems can be done from the study of all the monocomponent systems and from the study of the bicomponent systems where only the different couples of components are present.

A. Calculation of the mean and covariance matrix of the spectrum associated to a competitive isotherm
We can now take advantage of Eq. ͑18͒ to carry out the calculation of the cumulants of multicomponent affinity spectra.Taking logarithms at both sides of ͑18͒ we obtain

͑47͒
Recalling that ln p ˜*ϭ(ln p ˜)*, the left-hand side of ͑47͒ can be related to the cumulants of p as On the other hand, B has all the properties of a distribution function, as pointed out in Sec.II A. Thus, Eq. ͑47͒ simply expresses the fact that the complex conjugate of the cumulant generating function of p(͕ln k i ͖), is the cumulant generating function of B plus a function independent of the particular isotherm under study.We can now expand both sides of ͑47͒ in terms of the Fourier set of variables, w i .Expansion of the left-hand side ͑lhs͒ of ͑47͒ leads to the cumulants of p(͕ln k i ͖), Eq. ͑48͒, and the expansion of the first term on the right-hand side ͑rhs͒ of ͑47͒ leads to the cumulants of B, where the superscript B recalls that these cumulants refer to the B distribution at the same time that it allows to differentiate both cumulants.The expansion of the second term of the rhs of ͑47͒ can be done using 20 where C is the Catalan constant and (z) is the Riemann's zeta function.Identifying in the resulting equation the terms of the same power on w i , for the first two cumulants of the affinity spectrum ͑mean and covariance matrix͒ we obtain where B (i) ( i ) is the function B corresponding to the monocomponent system obtained by taking all the concentrations equal to zero except that of component i. Proceeding analogously, the covariance matrix is where the elements C i j B are where B (i j) ( i , i ) is the function B corresponding to the two-component system obtained by taking all the concentrations equal to zero except those of components i and j.Equations ͑51͒-͑53͒ give the mean binding energy and the elements of the covariance matrix underlying any competitive isotherm in terms of the isotherm parameters through B, obtained from the derivatives of the isotherm, and without requiring the knowledge of the affinity spectrum.

B. Binding correlation in NICCA and Frumkin isotherms
First we are going to use the results presented above to obtain the mean and the correlation matrix corresponding to NICCA isotherm with n 1 ϭn 2 ϭn.Performing the corresponding integrals we obtain and

͑57͒
the correlation coefficient becoming The diagonal elements of the correlation matrix, C ii give the variance of a Sips distribution with heterogeneity parameter mϭnp, 18 in agreement with the fact that NICCA isotherm reduces to a Langmuir-Freundlich isotherm when only one component is present.This means that mϭnp is the responsible of the heterogeneity of adsorbate i, a heterogeneity that is also present even in absence of the rest of components.
In Fig. 4, the correlation coefficient, given by ͑58͒, is plotted as a function of the parameter p for different values of m.Notice that R is always positive ranging between 0.5 and 1, corresponding these limiting cases to pϭ1 and p ϭm, respectively.For pϭ1, NICCA isotherm reduces to the so-called Langmuir-Freundlich generalized isotherm for which ͑58͒ prescribes Rϭ1/2, independently of the m value.This fact corresponds to the situation of minimum correlation of binding between the different components allowed by this isotherm.For pϭm, i.e., nϭ1, we have R ϭ1, i.e., a situation of fully correlation, in agreement with the original consideration of p as a measure of the heterogeneity of the ligand, in contrast with m, a parameter ion dependent which reflects the deviation of the local binding from the ideal case (nϭ1). 21,22Thus, if nϭ1, the only one source of heterogeneity comes from the ligand and is common for all the ions ͑described by p͒ leading to fully correlation or congruent binding.
The correlation coefficient corresponding to a twocomponent Frumkin isotherm is plotted as a function of the interaction parameter 12 , for different values of 11 and 22 in Fig. 5. Notice that, while in the NICCA isotherm considered above (n 1 ϭn 2 ϭn) we have only two parameters ͑n and p͒ to reproduce three independent statistical properties of the spectrum ͑for instance, the two variances and the correlation coefficient͒, in this case we have three parameters to describe them.In this sense, the present situation is more flexible, which can be seen for instance in the values scanned by the correlation coefficient, depicted in Fig. 5. Since an analytical expression for the correlation coefficient is not available for the two-component Frumkin isotherm, we have calculated it from ͑54͒ and ͑39͒, performing the integrals numerically.Notice that, via a suitable selection of the parameters, any value of the correlation coefficient can be achieved in this case, according to the more flexible situation that we are considering.On the other hand, although we do not have a general condition for the achievement of the maximum level of correlation (Rϭ1), we can see that for 12 ϭ 11 ϭ 22 , Rϭ1.In effect, when all the interaction parameters are equal, all the components of the system behave likely from the point of view of heterogeneity.

V. CONCLUDING REMARKS
We have undertaken the problem of obtaining the affinity spectrum underlying multicomponent adsorption isotherms.The main difficulty lies in the fact that the integral equation to be solved involves the integration over several variables ͑as many as number of components can be adsorb-ed͒.Nevertheless, by rewriting the integral equation in a suitable manner, a general analytical solution can be found.This implies the definition of a new function, B, basically a Nth derivative of the total macromolecular coverage, which has the properties of a probability distribution function.By further applying Fourier transform techniques, we have been able to provide an inversion formula, which is the counterpart to that given by Sips for one-component systems.
The analytical solution obtained has been applied to NICCA isotherm, commonly used in environmental water studies and an analytical expression for the underlying spectrum has been derived.The resulting spectrum agrees with that obtained numerically in the literature by using leastsquare regularization methods.The spectrum associated to Frumkin isotherm is also computed, although in this case numerical FT transform techniques have been used and some spurious oscillations remain.
On the other hand, a method to obtain the main statistical properties of multicomponent affinity spectra underlying competitive isotherms as a function of the isotherm parameters is presented.An expression for the cumulants of the affinity spectrum in terms of the cumulants of the function B has been reported.These results lead to analytical expressions for the mean and covariance matrix of NICCA isotherm, and, by performing the suitable numerical integrations, to those of Frumkin isotherm.These results allow to quantify the level of binding correlation among the different adsorbing molecules.

ACKNOWLEDGMENTS
The authors gratefully acknowledge support of this research by the Spanish Ministry of Education and Science ͑DGICYT: Projects BQU2000-0642 and BQU2003-9698͒, by the European Community under Contract No. EVK1-CT2001-86 and, from the ''Comissionat d'Universitats i Recerca de la Generalitat de Catalunya.''

APPENDIX A: THE FUNCTION B" ᠬ … CAN BE CONSIDERED A PROBABILITY DISTRIBUTION FUNCTION
The function B( ជ ) is defined as where ϭ͚ iϭ1 N i is the total macromolecular coverage and i ϭln c i .If the macromolecular coverage can be expressed by means of certain affinity spectrum p(͕ln k i ͖) we have Performing the derivatives indicated in ͑A1͒, B( ជ ) becomes Since all the terms in the integrals appearing in ͑A3͒ are positive, we have B( ជ )Ͼ0 for all the i values.
On the other hand, the integration of both sides of ͑A3͒ with respect 1 , 2 ,..., N , The Fourier transform of function B for the twocomponent NICCA isotherm can be written as

͑B1͒
where the definition of B, Eq. ͑10͒, the expression of the NICCA isotherm, Eq. ͑26͒ with n H ϭn 1 ϭn 2 , and the definition of FT, Eq. ͑7͒, have been used.Introducing the change of variables y 1 ϭn(  and F ˜( 1 /n, 2 /n) is the FT of F(y 1 ,y 2 ) when i /n label the variables in the Fourier space.
although b 1 ϭb 2 ϭ1 (log b 1 ϭ0,log b 2 ϭ0) in parallel to what happens in the monocomponent Frumkin isotherm, whose mean binding energy does not coincide with the logarithm of the parameter b appearing in the isotherm.

e
ln k i ϩ i ͩ 1ϩ ͚ jϭ1 N e ln k j ϩ j ͪ Nϩ1 ͟