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Review

Chromatography as L´evy Stochastic process Francesco Dondi ∗ , Alberto Cavazzini, Luisa Pasti Department of Chemistry, Via Luigi Borsari, 46, I-44100 Ferrara, Italy Available online 3 July 2006

Abstract The Stochastic Theory of Chromatography has been revised in light of some of the most relevant L´evy’s findings in Theory of Probability, including the so-called L´evy’s distance, the characteristic function and the theory of infinitesimally divisible distributions. These concepts represent the key to exploit and understand, at a molecular basis, phenomena typical of chromatographic separations under linear conditions, such as peak tailing and splitting. In particular, L´evy’s distance has been used to quantify the degree of convergence of real peaks towards an ideal Gaussian shape; the characteristic function properties, introduced by L´evy to deal with the problem of the addition of independent random variables, have been employed to solve a wide variety of chromatographic models (including adsorption on heterogeneous surfaces) and to interpret mobile phase dispersion from a probabilistic point of view. Finally, L´evy’s studies concerning infinitesimally divisible distributions have allowed to introduce in the stochastic description of chromatography, effects associated to dispersion in mobile phase. It has been demonstrated that, according to L´evy’s canonical representation of stochastic processes, the basis of chromatography is a mobile phase Poisson Process. Represented as a L´evy’s process, the microscopic–probabilistic model of chromatography permits the establishment of a connection between single-molecule properties and their statistical fluctuations and shapes of real chromatographic peaks allowing, at the same time, for the constitution of a link between different branches of physical sciences. © 2006 Elsevier B.V. All rights reserved. Keywords: Chromatography fundamentals; L´evy Processes; Stochastic Theory; Peak shape analysis

Contents 1. 2. 3.

4.

5. 6.

∗

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. L´evy’s first period: the limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why are peaks Gaussian? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantify the similarity of a peak to a Gaussian function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. L´evy’s second period: additive processes of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Addition of a discrete number of random variables: the kinetics of chromatography under general conditions handled from a stochastic point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peak tailing at infinite dilution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Stationary stochastic process of addition of random independent increments: the possibility of handling kinetic effects and continuous mobile phase dispersion in chromatography from a stochastic point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The L´evy canonical representation of stochastic processes: the possibility of representing and interpreting the chromatographic process from an unifying stochastic point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to correlate microscopic and thermodynamic–spectroscopic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Corresponding author. Tel.: +39 532 291154 E-mail address: [email protected] (F. Dondi).

0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.06.030

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1. Introduction Separation scientists deal with decoding complex mixtures of different nature to recognize single components therein present. Their work is extremely important for both the general progress of science and society. Georges Guiochon has been a great scientist, whose contributions to separation science span over different domains such as fundamentals of adsorption, theory and modelling of linear and nonlinear chromatography in GC and HPLC, preparative chromatography, only to cite some general categories. F.D. has intensively collaborated with him for more than thirty years in the field of theoretical modelling of chromatography. First discussions on this topic are dated long time ago, ´ when F.D. visited Georges’ laboratories at the Ecole Polytechnique in Paris (before at Rue Descartes and, later, at Palaiseau), a place where the theory of stochastic processes had one of its founders, Paul L´evy (1886–1971, Fig. 1). L´evy became pro´ fessor of Analysis at the Ecole Polytechnique in Paris in 1920, where he remained until he retired in 1959 [1]. As S.J. Taylor wrote:“If there is one person who has inﬂuenced the establishment and growth of probability theory more than any other, that person must be Paul L´evy” [2]. In the glorious context of the ´ Grand Ecole Polytechnique, Georges graduated in 1951 and L´evy was one of his professors of analysis. This review on the Stochastic Theory of Chromatography is dedicated to Georges as a gift for his 75th birthday. The stochastic model of chromatography was firstly proposed by Giddings and Eyring in 1955 [3]. Their model is based on a molecular (microscopic) description of the chromatographic process, in which the progression of a molecule inside the chromatographic column is represented as the sum of a random number n of random quantities [3–6]. n represents the number of adsorption–desorption steps performed by a molecule in the column up to elution. During any adsorption, the molecule spends a random time on the stationary phase, before desorption occurs. This process is graphically represented in Fig. 2a where the space travelled by a molecule inside the column, l, is represented as function of the time elapsed from injection, t. Any of the broken lines shown in Fig. 2a represent possible random paths followed by molecules up to their elution at l = L (being L the column length). These trajectories, called in the language of stochastic processes molecular histories, are composed by segments, parallel to the time axis, during which no progression in length occurs (immobilized adsorption on the

Fig. 1. Paul L´evy.

Fig. 2. (a) Example of random trajectories (t, l) of solute molecules moving forward in chromatographic medium 1, 2 unretained, retained solute; (b) molecule status during migration inside the column, referred to full line case. Reprinted from [11] with permission. Copyright [1982] American Chemical Society.

stationary phase) and parts of constant slope that correspond to residence times in mobile phase where molecules move at the mobile phase velocity u0 = L/t0 . Both the number of the segments forming the broken line and their lengths are randomly distributed. Peak number 1 represents the hold-up time signal (Delta Dirac function): molecules that do not perform any adsorption–desorption step, travel through the system at the same velocity as the solvent does and are all eluted at t0 (in the original Giddings–Eyring description, dispersion phenomena in mobile phase were neglected). On contrary, the chromatographic peak of a retained compound corresponds to the probability density function of times spent by molecules in the chromatographic column (peak number 2). Fig. 2b basically represents the same information, but in a more compact manner, to emphasize the concept of molecular histories as random series of transitions between stationary and mobile phase. The traditional representation of chromatography as stochastic process has been extended, through a series of theoretical works [7–22] and practical interpretation of experimental data [23–26], to account for a significant number of phenomena such as adsorption on heterogeneous surfaces, mobile phase diffusion effects, interconversion phenomena, size exclusion effects, etc. Being substantially limited to infinite dilution conditions (linear part of the adsorption isotherm), however, stochastic models had modest diffusion among the scientific community and limited use in every-day works if compared with models based on differential mass balance equations [27,28] in which nonlinearity effects could be accounted for in a straightforward manner. The extraordinary technological advancements of modern analytical equipments has allowed for the direct observation

F. Dondi et al. / J. Chromatogr. A 1126 (2006) 257–267

of molecular dynamics [30,31]. This opens new frontiers to separation science and gives a new value to microscopic models of chromatography. Thanks to the possibility, offered by these approaches, of taking into account molecular properties and their probabilistic fluctuations and to correlate them with experimental peak shapes recorded at infinite dilution conditions, they constitute a fundamental tool to correlate information proper of specific scientific scopes (single-molecule stationaryphase residence times, adsorption energy distribution functions, adsorptive material heterogeneities, etc.) and chromatography. This paper presents a review of microscopic models of chromatography [11–14,16–22] under the light of L´evy’s studies in Probability Theory [32,33]. Incidentally, these tools have been widely employed even in many other scientific domains [34–39]. Throughout this work, the schematization of L´evy’s scientific work proposed by Lo`eve [40] has been used. According to him, three period of L´evy’s activity can be recognized. The first is that of the study of distribution laws [2,41], the second concerns the study of addition of random variables [32] and, finally, the third period is devoted to investigation of Brownian movement [33]. 1.1. L´evy’s ﬁrst period: the limit laws A fundamental concept in chromatography is that of chromatographic peak. Under linear conditions, chromatographic peaks are often approximated by a Gaussian peak function: 1 1 t − tR 2 f (t) = √ exp − (1) 2 σ σ 2π where t represents the time variable; tR the retention time and σ the standard deviation. f(t) is the probability density function (PDF) of retention times and f(t) dt represents the probability of obtaining a retention time included in the interval [t, t + dt]. The Gaussian distribution can be related to the standard normal distribution, which is a Gaussian distribution with zero mean and standard deviation equal to one: 2 1 c f (c) = √ exp − = z(c) (2) 2 2π where c is the standardized variable: t − tR c= σ

(3)

The advantage of using the normal standard distribution is that it permits a direct comparison of functions differing in position (retention time) and dispersion (peak width). Even if, under linear conditions, peak shapes are expected to be Gaussian or approximately Gaussian, there are experimental cases in which strong asymmetric shapes are encountered. In this paper we will use the microscopic–stochastic description to give an explanation to some basic concepts of chromatography whose effects are often observed by every-day chromatographers such as:

2. How to express the similarity between a real peak and a Gaussian function? 3. Why, even under infinite dilution conditions, chromatographic peaks can be strongly asymmetrical? 4. How to correlate microscopic and thermodynamic— spectroscopic properties derived through a chromatographic output? We will show that these questions can be addressed in a general manner by using some the statistical tools invented by L´evy. 2. Why are peaks Gaussian? As it was previously described, the chromatographic process represented at a molecular level is constituted by a great number (n) of sorption–desorption events characterized by random times spent in stationary phase (τ S ) [4]. The total time spent in stationary phase (tS ) is the sum of these random and independent variables. The Central Limit Theorem guarantees that the PDF of any measurable quantity is Gaussian, provided that a sufficiently large number of statistically independent observations are made [38]. Consequently, the frequency function of total sojourn time of the solute molecule tends to a Gaussian function, provided that n is large enough. Other examples to which the Central Limit Theorem is applied are, for instance, the distribution of random errors or the investigation of free diffusion processes. According to the Central Limit Theorem applied to chromatography, a given experimental peak shape, y(c), not exactly Gaussian also under linear conditions y(c) ≈ z(c)

(4)

will tend to a Gaussian distribution if the number of sorption–desorption events n tends to infinity or, since this number increases with column length, if L tends to infinity: n,L→∞

y(c) −→ z(c)

(5)

Eq. (5) represents a more general relationship known as “probability convergence in law”. In the framework of L´evy representation of stochastic processes (see below), it is convenient to replace PDFs with cumulative distribution functions (CDF). CDFs represent the probability that a random variable c takes a value smaller than or equal to a given value b: b Y (b) = y(c)dc (6) −∞

or, by considering normalized distributions: b z(c)dc Z(b) = −∞

(7)

Accordingly, Eqs. (4) and (5) can be represented as: Y (b) ≈ Z(b)

1. Why, in the majority of cases, are the chromatographic peaks Gaussian-like?

259

n,L→∞

Y (b) −→ Z(b)

(8) (9)

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Eqs. (4) and (5) and (8) and (9) represent qualitatively two fundamental concepts in the theory of chromatography: the difference between an experimental peak and a Gaussian shape (Eqs. (4) and (8)) and the rate of the convergence from the real shape to the Gaussian distribution (Eqs. (5) and (9)). However, they do not quantify how large is this difference and how fast is the convergence. These are not trivial questions. Basically, they refer to the effect of column length on peak evolution inside the column and to the convergence to a Gaussian shape that constitutes an essential condition for accurate quantification in chromatography. L´evy developed the concept of similarity/difference between two CDFs, F(c) and G(c), by defining the concept of distance between F and G, L(G, F). This distance has been called L´evy or L´evy–Kolmogorov distance [41]. Mathematically, it is defined as the infimum of all h, that is the smallest h value, such that for all b the following condition is fulfilled: F (b − h) − h ≤ G(b) ≤ F (b + h) + h

(9) [13]. Other mathematical approaches based on other metrics or on series development have been also employed to investigate these aspects [4]. In conclusion, L´evy’s studies of distribution laws in probability theory bring to a general vision about the essence of peak shape in chromatography and its dependence on column length-number of theoretical plates under linear conditions. Most important – and not yet fully developed – is the study of the dependence of peak shape and its convergence rate towards the Gaussian law in the case of complex sorption kinetics (the so-called Berry Es´een Theorem [11,38]). On contrary, in chromatographic practice, L´evy’s distance can be employed to optimize the column length for a given class of substances or to study kinetic sorption mechanisms by chromatographic techniques.

(10)

An example is reported in Fig. 3a where the CDF of the reference function is the normal distribution P(b) and the specific function G(b), indicated as Fα,N , corresponds to a Gamma Distribution of order N [38]. The L´evy distance, originally defined for CDFs, can also be employed for measurements of distances between frequency functions. In this case, we can refer to it as an improper L´evy distance [13]. In Fig. 3b are reported the PDFs corresponding to the case described in Fig. 3a. Eqs. (5) and (8) can thus be reformulated by the use of the L´evy distance as: Y = P + L(Y, P)

(11)

and y = Z + L(y, Z)

(12)

It can be noted that in Eqs. (11) and (12), the independent variable b (defined in Eqs. (6) and (7)) does not appear. In effect, the L´evy distance refers to the whole distribution domain (or frequency function domain) and not to a specific coordinate value. This distance quantitatively expresses the concept of convergence of the central limit theorem and gives a metric in the space of distribution functions. 3. Quantify the similarity of a peak to a Gaussian function In theoretical chromatography, L´evy’s distance can be used to achieve a quantitative estimation of the approximation degree of a given chromatographic peak to the Gaussian distribution (this concept was qualitatively expressed by Eqs. (5) or (9)) [11–13]. In Fig. 4, this kind of study is applied to two chromatographic models, the Martin-Synge [42] and the Giddings–Eyring model [5]. To perform this investigation a bi-logarithmic plot was used in which, on the y-axis, there is the logarithm of L´evy’s distance and, as the independent variable the average number of sorption–desorption steps or a quantity proportional to it such as the number of theoretical plates, N. Since L is proportional to N, this figure quantifies convergences represented by Eqs. (5) or

Fig. 3. L´evy distance between a theoretical chromatographic model function and a reference function. (a) Distribution function case. Particular function: Martin–Synge model (fα,N or Fα,N , where N is the number of theoretical plates). Reference function: normal function (Z or P). (b) Frequency function case (in this case the distance is referred as “improper L´evy distance”. Reprinted from [13] with permission. Copyright [1982] American Chemical Society.

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261

number of entry, n¯ , is proportional to the time spent in mobile phase tM : n¯ = tM µ =

tM τ¯M

(15)

where µ=

Fig. 4. L´evy distance (L) between reference functions and theoretical models of chromatography, Giddings–Eyring–McQuarrie with ks = 1 or Martin–Synge model, as a function of the number of theoretical plates (N) or of the mean number of adsorption–desorption steps (µ), respectively. Line 1: the reference function in L computation is the Gaussian function, P, reported as function and a reference function. For other details, see Ref. [13]. Reprinted from [13] with permission. Copyright [1982] American Chemical Society.

3.1. L´evy’s second period: additive processes of random variables 3.1.1. Addition of a discrete number of random variables: the kinetics of chromatography under general conditions handled from a stochastic point of view The second period of L´evy’s contributions to Probability Theory was dedicated to the mathematics of addition of random and independent variables and culminated in the publication of a famous monograph published in 1937 [32]. This work can be employed to represent the chromatographic process as stochastic process of addition of random variables. Let us apply these considerations to the fundamental Giddings–Eyring model previously described (Fig. 2). In that case [5], the sorption time has been assumed to be exponentially distributed. An exponential distribution of elementary sorption steps corresponds, in the macroscopic description of the process, to a first order desorption kinetics: 1 τS f (τS ) = exp − (13) τ¯S τ¯S

1 τ¯M

(16)

is the sorption frequency and τ¯M the average time elapsed between two consecutive entry in the stationary phase. A Poisson distribution for the number of adsorption–desorption steps corresponds to an exponential distribution of elementary times between two successive adsorptions (i.e. an exponential distribution for the times spent in mobile phase). Therefore, both desorption and sorption processes obey to first order kinetics. On the basis of the abovementioned hypotheses, the PDF of the solute sojourn time in the column for the Giddings–Eyring model is [5]: n¯ n¯ tS f (tS ) = e[−(tS /¯τS +¯n)] (17) I1 2 tS τ¯S τ¯S where I1 is a Bessel function of first order. Eq. (17) allows for the calculation of the residence time PDF as a function of the microscopic quantities as n and τ S . Eq. (17) holds in the very simplified case of adsorption on a homogeneous surface. In more realistic situations, for instance adsorption on heterogeneous media, the approach proposed by Giddings and Eyring led to extremely complex equations, of limited practical utility. As Giddings himself wrote: “One aspect of the stochastic theory which has been pursued from the beginning is the effect of a nonuniform surface with different kinds of adsorption sites. The mathematics rapidly becomes intractable, however, when we pass from the sheltered simplicity of one-site theory” [4]. L´evy treated, in a very general way, the problem of addition of random variables. By following his method, a general solution of complex chromatographic models can be obtained [11,12,14,18–20]. The mathematical tool introduced by L´evy is the characteristic function (CF) ϕt (ω), which is the Fourier–Stiltjes transform [10,29] of the frequency function f(t): (18) E[eiωt ] = ϕt (ω) = eiωt f (t)dt

(14)

In Eq. (18), ω is an auxiliary variable; i, the imaginary unit; t, the random variable and E[ ], the expected value, i.e. the mean of the quantity into square brackets. Apart from introducing the CF, which is a basic mathematical tool in modern probability theory, L´evy further developed the concept of probability function transformation proposed, for the first time, by H. Poincar´e [2]: (19) E[e−ωt ] = e−ωt f (t)dt

The Poissonian distribution has some important properties and, in particular, it describes so-called memoryless processes. In such processes, the probability to observe a given event is only proportional to the observation length. The average value of the

Due to the uniqueness, inversion and linearity CF properties [38,39], a given probabilistic process can be either described in the frequency domain (CF transformation) besides the original domain of the specific random variable. In the case of

The distribution of the number of visits was described by the Poisson law: Pn =

(µtM )n (−µtM ) e n!

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probability of occurrence, Pn . The Poisson law (Eq. (14)) merely represents one example of Pn distribution. Because of the linearity property of the CF, it is possible to write [38]: ϕtS (ω) = (25) Pn ϕSn n

so that the CF of the resulting distribution is simply a weighted sum of CFs, elevated to the power of n, where the weights are the probabilities of realization of histories characterized exactly by n entries. In Eq. (25), tS is the total time spent in the stationary state: Fig. 5. (a) Adsorption processes of a molecule inside a homogeneous chromatographic column, τ s is the elementary sorption time; (b) Frequency functions, f, convolute, and the characteristic functions, ϕ, multiply (see text).

chromatographic processes, this corresponds to time (see Eq. (17)). The use of CFs instead of distribution functions has some important advantages [12] allowing for a drastic simplification of the mathematics. To demonstrate this, let us again refer to the Giddings–Eyring model of chromatography (Figs. 2 and 5) and, specifically, to the case of molecules that performed two adsorption–desorption steps on two sites (labelled as 1 and 2). The total delay time, τ S,tot , accumulated by these molecules after the two adsorptions will be the stochastic sum of two independent random variables, the time spent in the sorption state 1,τ S,1 , and that in 2, τ S,2 : τS,tot = τS,1 ⊕ τS,2

(20)

where the symbol ⊕ represents the sum of two random variables. The resulting total PDF is the convolution product of the two distributions: (21) fS,tot (t) = fS,1 (t − x)fS.2 (x)dx ≡ fS,1 ∗ fS,2 being fS,tot , fS,1 , and fS,2 the distribution functions of τ S,tot , τ S,1 and τ S,2 , respectively, and the symbol * indicating the convolution. If τ S,1 and τ S,2 are equally distributed random sorption times (adsorption on an homogeneous surface), from Eq. (21) one has (see Fig. 5): fS,tot = fS∗2

(22)

Computation of convolution integrals can be difficult and timeconsuming especially when the number of added random independent variables is elevated. On contrary, the representation of the same process in terms of CF leads to the following expression [12,38]: ϕS,tot (ω) = ϕS,1 (ω) × ϕS,2 (ω)

(23)

where the convolution of PDFs (Eq. (21)) has been replaced by the product of corresponding CFs, ϕ. In the case of adsorption on homogeneous surfaces (see Fig. 5), from Eq. (23), one gets: ϕS,tot (ω) = ϕS (ω)2

(24)

Solute molecules, in their progression inside the column, can follow different trajectories (histories) characterized by different n values. These different histories have different associated

tS = tR − tM

(26)

and tR is the retention time. Eq. (25) can be transformed into a compact, elegant expression by using the so-called log–exp transformation applicable to a mixture of distributions [9,38]. lnϕτS (ω) ϕtS (ω) = ϕn (27) i where ϕtS is the CF of total residence time, tS , and ϕτS and ϕn are the CFs of the time spent in one single adsorption–desorption step and that of the number of entries, respectively. Eq. (27) represents the chromatographic peak in frequency domain. Inverse transformation of it, either through analytical [38] or numerical approaches [43], gives the peak in time domain. The statistical moments of order j, mj are obtained by calculating the j-derivatives of Eq. (27) [12,38]: dj ϕ(ω) j ϕ (0) = = ij mj (28) dω ω=0

From them, fundamental chromatographic quantities, such as the retention factor (k) or the column efficiency (H) can be obtained [11,12]. From the general equation of the chromatographic process (Eq. (27)), it is relatively simple to obtain the expression describing a specific chromatographic model [11,12] once ϕτS and ϕn are specified. For instance, the Giddings–Eyring model finds a simple solution in term of CF [11]:

1 ϕtS (ω) = exp n¯ −1 (29) 1 − iω¯τS More importantly, the solution of the problem of the adsorption on heterogeneous media could be solved in a straightforward manner through the formalism of CFs. In the case of a two-site adsorption surface (BiLangmuir adsorption model), it is simple to demonstrate that the corresponding CF is represented as [14]: ϕt (ω) = exp{¯n[qϕtS,1 (ω) + (1 − q)ϕtS,1 (ω) − 1]}

(30)

where q and (1 − q) are the fraction of sites of type 1 and 2 present on the surface, and ϕtS,1 (ω) and ϕtS,1 (ω) the CFs of the time spent over them, respectively. It can be seen in Eq. (30) that the exponential argument is a weighted average of single site CF, while the weights are their relative abundances. Eq. (30) can be extended to deal with multiple site adsorption surfaces and continuous adsorption energy distribution functions [14].

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4. Peak tailing at inﬁnite dilution Eq. (30) can be used to study the effect of adsorptive material heterogeneity on chromatographic peak shapes. Heterogeneity is responsible for strongly asymmetric peaks even when extremely reduced amounts of material are injected (linear conditions). The stochastic theory allows for a clear explanation of these effects. Fig. 6 shows some peak profiles obtained under the hypotheses of a BiLangmuir adsorption model. As it can be seen from the simulations, peak tailing increases as the difference in adsorption energies of the two sites increases and it is more pronounced if strongly adsorptive sites are present in very reduced amounts. The Stochastic Theory of Chromatography, described in terms of CFs, was further developed to describe different chromatographic processes such as Size Exclusion chromatography. In SEC, separation occurs as a result of molecular partition between mobile phase and a porous stationary phase. The CF describing the size exclusion process is [17]:

1 ϕtp (ω; ρ) = exp n¯ p (ρ) −1 (31) 1 − iω¯τp (ρ) where r ρ= d

(32)

is the so-called size exclusion parameter, r and d being the solute and the pore diameter, respectively, and: n¯ p (ρ) = n¯ p,perm (1 − ρ)mp

(33)

τ¯p (ρ) = τ¯p,perm (1 − ρ)me

(34)

are the average number of entries in the pore structure and the average time spent inside a pore, expressed as function of ρ, respectively. Finally, n¯ p,perm and τ¯p,perm are the average number of visits in the pore structure and the average time spent in pores by a totally permeating species, i.e. a species with r d(ρ ≈ 0).

mp and me are exponents describing the features of the kinetics of entry into the pores from the mobile phase and exit from the pores [19,26]. 4.1. Stationary stochastic process of addition of random independent increments: the possibility of handling kinetic effects and continuous mobile phase dispersion in chromatography from a stochastic point of view The described process of addition of random and independent variable has been noticeably useful in developing stochastic models of chromatography. However, there is a limit in it, connected with the idea of making discrete the theoretical handling of a chromatographic column. Chromatographic media are, instead, continuous and so are many of fundamental chromatographic processes, such as for instance mobile phase diffusion, even if build up by discrete steps. The concept of “stochastic process” exactly concerns the problem of handling a discontinuous process over a continuous variable (time). During the long period dedicated to the study of additive processes, L´evy faced these problems, by developing the concept of “infinitely divisible distribution functions”. This tool, as it will be demonstrated later on, allows to account not only for the kinetics of Chromatography but also for mobile phase dispersion effects. Let us consider how the concept of infinitesimal divisibility is related to chromatography. Fig. 7 compares two different concepts: the process of putting columns in series (top) and that of cutting a column in n equal parts (bottom) as it would be formalized in terms of CF. The reported expressions are straightforward consequence of Eq. (19) [38]. Going to the limit of a “continuous” cutting process (infinitely narrow slices), one has: ϕS,L = (ϕS,L=1 )L

(35)

or ln[ϕS,L ] = L[lnϕS,L=1 ]

Fig. 6. Heterogeneous two-site model. Different peak shapes have been simulated for different values of the number of the adsorption–desorption steps, n = 100–2000; τ 2 /τ 1 = 10 and proportion of the stronger site p2 = 1%, for all the chromatograms. For other details, see Ref. [16]. Reprinted from [16] with permission. Copyright [1999] American Chemical Society.

263

(36)

Fig. 7. Combining the characteristic functions (ϕ). (a) Effect of coupling columns in series; (b) effect of dividing a column in identical parts (see text).

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where ϕS,L−1 indicates the CF of a chromatographic process over a column of L = 1. These two positions presuppose that the CF of the process is “infinitesimally divisible” [38,41,44]. This property allowed to extend the CF handling from discontinuous additions of independent random variables to a continuous addition of independent random increments, i.e. to consider continuous processes running along the continuous time variable. It is not the case to face here the concept of infinitesimal divisibility of a CF. The interested reader can refer to Refs. [38] or [39]. The chromatographic process in general meets these conditions [11]. This is apparent if reference is made to the Eq. (29) where n¯ appears in the exponent. From Eqs. (15) and (16), n¯ can be expressed as: n¯ =

Lµ vM L tM

(38)

By combining Eqs. (29) and (37) and (38), one has: 1 −1 ln[ϕtS (ω)] = tM µ 1 − iω¯τs

(39)

By applying the CF shifting property [38], one obtains from Eq. (38) and (26) a CF expression in terms of tR :

1 − 1 + iω (40) ln[ϕtR (ω)] = tM µ 1 − iω¯τS which shows that the CF of the chromatographic process is infinitesimally divisible since its logarithm is linearly independent on the continuous time parameter tM (see Fig. 8). In analogy with Eq. (36), Eq. (40) can thus be rewritten as: ϕtR = (ϕtR ,tM =1 )tM where

ϕtR ,tM =1 (ω) = exp µ

In Eq. (43), ϕtM is the CF of the continuous function PtM . By solving Eq. (43) for specific models (ϕtR ,t M =1 and ϕtM defined), the so-called of stochastic dispersive models of chromatography can be obtained [12,17].

(37)

where vM is the mobile phase velocity, assumed here constant: vM =

scopic point of view, mobile phase dispersion effects. Fig. 8 enlightens this concept. Due to the mobile phase dispersion – namely, longitudinal diffusion – different statistical replicas of the chromatographic process, with different values of times spent in mobile phase, are possible. The global behaviour will be a weighted average, where the weights, PtM , are function of tM . In terms of CF in analogy with Eq. (27), the general solution is [11]: lnϕtR ,tM =1 (ω) ϕtR (ω) = ϕtM (43) i

(41) 1 − 1 + iω 1 − iω¯τs

(42)

The infinitesimal divisibility property of the CF describing the chromatographic process, allows us to handle, from a micro-

Fig. 8. Chromatographic process linearly dependent on tM . Different mobile phase times are showed and the unit mobile phase time is indicated with the corresponding CF. The CF of the global process is the weighed average of the CF components (see text).

4.2. The L´evy canonical representation of stochastic processes: the possibility of representing and interpreting the chromatographic process from an unifying stochastic point of view One of the highest contributions to the theory of stochastic processes due to L´evy is the general or canonical representation of stochastic processes [32–39]. We will try to give an intuitive idea of it, which hopefully will be useful to chromatographers, as we are, not to mathematicians. Let us consider again Fig. 2 where, as it was said, a simple representation of chromatography as stochastic process is given in terms of random molecular trajectories in the (t, L) space. For the sake of simplification, vM was assumed to be constant. The main concept that we want to single out is that tM must be considered as the true “clock time” of the chromatographic process. In fact, only when tM is running, the chromatographic sorption events are generated (see Fig. 2a). These events determine, in turn, a “time loss” of solute species, and the accumulation of this “retard” results in chromatographic retention. The two states of the molecule, when moving in mobile phase and adsorbed on the stationary phase, are however, conceptually different. The accumulation of retard is, in fact, “directed by” or “subordinated to” the true “clock time”. This becomes clearer if, instead of considering the chromatographic process in the “elution” mode, one considers in it the “development mode”. In this case the “lost time” results in an “accumulated distance” from the solvent front. Additionally, it is not essential for the chromatographic process the time loss in stationary phase to be a random quantity, as in the case of Fig. 5: a chromatographic peak will be equally generated by the random character of the stationary phase entry process (Poissonian distribution, Eq. (14), in the specific case). L´evy rebuilt the entire theory of stochastic processes by using the Poisson Process, the most fundamental subordinating process [32,37]. Under the “direction” of the “time clock”, different “subordinated” processes can run (see Fig. 9). In addition to Poisson mechanism (Fig. 9c), L´evy identified two other fundamental mechanisms: the continuous mechanism that gives rise to a “shift” (Fig. 9a) and the “Brownian” mechanism (Fig. 9b). The shifting increases together with the “clock time” and it is proportional to it (see Fig. 9a). In the case of elution chromatography, the “clock time” is tM and the shift is exactly equal to it.

F. Dondi et al. / J. Chromatogr. A 1126 (2006) 257–267

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Fig. 9. L´evy canonical expression of Stochastic Processes with independent increments over the variable u, as a function of the “clock time” t. v is the shift component, σ is the standard deviation of “Brownian” component and M is the “spectral” function of the Poissonian component, i.e. the distribution of the positive Poissonian increments, as a function of the increment.

In the case of development chromatography, the shift is the solvent front position, which is in this case, proportional to tM . The “Brownian” component of the L´evy representation constitutes the second possible contribution of a stochastic process with independent increments. This contribution is showed in Fig. 9b: it is centred at zero and, in this case, it is of Gaussian type. The third possible component is an always-increasing (or decreasing) random component and it is of “Poissonian” type. This is represented in Fig. 9c, where the x-coordinate increases by constant steps at the random times reported over the y-coordinate. A general process will have several Poissonian increasing (or decreasing) mechanisms, of different lengths, which combine each other. This explains the integral appearing in the third term of the canonical representation reported in Fig. 9. In chromatography, this expresses the fact that the sorption times spent by solute molecules in sorption states are randomly distributed, as it was previously discussed with reference to Fig. 2. The L´evy vision simply collects all of the possible Poisson processes with constant increments and then superimposes all of them with a specific weight. The possibilities offered by L´evy’s representation are fantastic. In essence, at a same time, one can represent and interpret whatever chromatographic process [45] having, as its driving mechanism, the addition of random and independent increments [44]. By looking at the literature on stochastic processes, the L´evy approach has been widely adopted in different field of physical sciences, geology, financing, etc. [34]. The L´evy approach was recently applied also to chromatography [21]. According to this representation the chromatographic process of Fig. 2 is represented as:

∞ iωτS lnϕtR (ω, tM ) = tM iω + (e − 1)dM(τS ) (44)

is the “spectral function” which is related to the distribution of the time spent in stationary phase τ S by dF (τS ) = f (τS )dτS

(47)

f(τ S ) is the frequency distribution of τ S . From Eq. (47), the Giddings–Eyring model can be derived just as an example (see, for the mathematical derivation, Ref. [21]). By referring again to Fig. 9, one can see that in this L´evy representation of the chromatographic process, the “Brownian” component is missing (see Eq. (47)). This is not a simplification: the essence of the “Brownian” component in the L´evy representation does not refer to the mobile phase dispersion. This last effect but must be accounted for by the randomisation procedure expressed by Eqs. (44) and (45) and represented in Fig. 9. It is worthwhile to specify that this effect is referred to, in L´evy’s language, with the significant term of “volatility” effects [46], to indicate that “dispersive” components destroy the basic stochastic mechanism. The “Brownian” component could instead account for stationary surface diffusion. This important aspect has not yet developed but it is here evocated to emphasize, once again, the enormous possibilities offered by L´evy description. The most relevant result achieved through the L´evy description of the chromatographic process [21] is connected with the establishment of a link between single molecule observation [30,31] and chromatographic process. In fact, in the discrete case, Eq. (45) can be rewritten as: lnϕtR (ω, tM ) = tM iω + tM µ

(eiωτS,i − 1)Fi

(48)

0

or as

lnϕtR (ω, tM ) = tM iω + µ

∞

(eiωτS − 1)dF (τS )

(45)

0

where dM(τS ) = µdF (τS )

(46)

From single molecule observation, experimental measurements of sorption time distribution can, in fact, be obtained, which correspond to the quantity Fi (as function of τ S,i ) appearing in Eq. (48). In Ref. [21] one can find a program written under Mathematica, for both the computation of lnϕtR (ω, tM ) and its numerical inversion for obtaining the chromatographic peak shape in time domain.

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5. How to correlate microscopic and thermodynamic–spectroscopic properties The great possibilities offered by the L´evy description of the chromatographic process include the physical interpretation of the microscopic features. In Eq. (41), the sum over the possible Poisson components of given τ S,i is singled out. L´evy’s approach can be applied in an extended manner since the spectral functions M or F (Eqs. (47) and (48)) may be expressed in relation to other characteristics of stationary phases, such as sorption energies, spectroscopic surface quantities, porosity. For example, if one knows the sorption energy distribution, F(ES,i ), ES,i , (i = 1, . . ., k), one can rewrite the Eq. (48) L´evy representation as [21]: lnϕtR (ω, tM ) = tM iω + tM µ (eiωτS (ES,i ) − 1)F (ES,i ) (49) where the function τ S (ES ) gives the link between the sorption energy and the sorption time. For this we can refer, e.g. to the Frenkel–De Boer equation [47] or to experimental approaches such as Attenuated Total Internal Reflection Infrared Spectroscopy [48]. This allows for the establishment of a coherent connection between chromatography and other domains of experimental sciences under an unifying L´evy vision. 6. Conclusion The extraordinary relevance of L´evy’s studies in Probability Theory for chromatography was illustrated by going through the three fundamental periods of his work. Peak shape characterization under linear chromatography was achieved with the tools developed during the first period, connected to the study of probability distribution laws. The second period, centred on the study of addition of random variables, defined three milestones of the modern theory of probability: the CF definition and his employment in the study of random variable addition, the infinitely divisible stochastic processes and, finally, the canonical representation of stochastic processes with independent increments. All of these had a fundamental impact in the stochastic description of the chromatographic process. The CF allowed for a general description of sorption on heterogeneous surfaces, by generalizing the Giddings–Eyring model. Dispersive mobile phase effects were dealt by using the concepts of infinitely divisible distribution functions. Finally, the L´evy canonical representation of stochastic processes was the starting point to reformulate the stochastic description of chromatography and include experimentally observable quantities, such as sorption time distributions. The third fundamental achievement of L´evy was the study of Brownian movement [33], not yet fully exploited in chromatographic domain. Acknowledgements This work was financially supported by University of Ferrara (ex 60%) and the Italian Ministry of University and Scientific Research (Grant 2005037725 002).

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