Fresenius J Anal Chem (1995) 352:625-632 I:resenius' Journal of
@ Springer-Verlag 1995
On the potential of two- and multi-dimensional separation systems Michel Martin
Ecole Sup6rieure de Physique et Chimie Industrielles, Laboratoire de Physique et M~canique des Milieux H~t6rog~nes (URA CNRS 857), 10, rue Vauquelin. F-75231 Paris Cedex 05, France
Received: 27 December 1994/Revised: 4 February 1995/Accepted: 11 February 1995
Abstract. A probability model of zone overlapping in n-dimensional (n-D) separation systems has been de- veloped. The probability that all sample components are separated is given as a function of the number of compo- nents, m, and of the peak capacity, no, of the n-D system. Application to 1-D separations provides the same expres- sion as that previously obtained with a more rigorous peak overlapping model, in the limit of large m and no and of low saturation of the separation space. The major result is that the probability of total resolution of the sample decreases exponentially with the square of the number of sample components and with the reciprocal of the peak capacity, whatever the dimension of the separ- ation system. In addition, a simple general relationship is obtained between this probability and the probability to separate one or a few components of interest from all other sample components. It is found that, for a given number of components and a given peak capacity, these probabilities slightly depend on the dimension of the separation system, which indicates that the peak capacity is not a fully universal index of characterization of the resolving power. The peak capacity required to separate all sample components at a given probability level in- creases with the square of the number of components. Accordingly, the individual peak capacity per dimension does not increase as fast as m when the dimension of the system exceeds 2.
Tremendous progress has been accomplished during the last three decades in the development of analytical zonal separation methods (gas chromatography, liquid chromatography, capillary electrophoresis, field-flow fractionation). The demands for the performances of these methods have been steadily increasing as they have been applied to samples of increasing complexity. However, in
Dedicated to Professor Dr. h.c. mult. J.F.K. Huber on the occasion of his 70th birthday
spite of the level of performances reached, the separation capability of the individual methods is still insufficient to handle highly complex samples, especially those from environmental or biological origin. The limitations of these individual methods are better understood since a decade or so, owing to the development of various statistical models of the overlapping of peaks in multi- component chromatograms [1 3] which have been re- cently reviewed .
In order to overcome these limitations, various more or less sophisticated procedures of coupling of separ- ation/sample preparation methods have been attempted. They include heartcutting and column switching methods, variations in the mode of coupling chromato- graphic columns , or on-line coupling of gas and liquid chromatographic columns . Planar two-dimensional separations have been performed using different retention mechanisms in liquid chromatography  or gel elec- trophoresis [8, 9]. More recently, methods have been developed to perform two-dimensional separations by microcolumn coupling in a so-called comprehensive mode that mimic planar separations [10 14]. Some of these methods have recently been reviewed  and clas- sified according to their intrinsic characteristics .
The main driving force in developing these two- or multi-dimensional separation methods arises from the fact that their separation power can be significantly in- creased over that of one-dimensional systems. This was quantitatively expressed by Giddings in terms of the peak capacity, no, of the separation system, a convenient index of the overall separation power [17, 18]. Basically, the peak capacity of a two-dimensional system is of the order of the product of the peak capacities of the two individual one-dimensional contributing systems. Therefore, even if one of these two 1-D (for one-dimensional) peak capaci- ties is modest (i.e. of a few units), the resulting 2-D peak capacity will reach a value which would have been im- possible or hard to obtain with an individual 1-D system. Indeed, in order to increase by a factor of, let say, 5 the peak capacity of a column, one must multiply its plate number, i.e. its length at constant plate height, by 25. This will frequently not be possible due to technological or time limitations. Although the 1-D peak capacity of
a given separation system can vary in a wide range de- pending on the operating conditions, it can be useful to have in mind some typical values: 25 in thin-layer chromatography, 50-70 in isocratic liquid chromatogra- phy, 200-500 in gradient elution liquid chromatography, up to 2000-5000 in temperature programming capillary gas chromatography, 500-5000 (?) in capillary elec- trophoresis (in the latter case, this depends on the extent of the relative retention time interval which can be effec- tively used). Although some loss may occur in the peak capacity obtained in one direction when the separation is performed in the second one, these values and the multi- plicative law for nc allow to get rough estimates of the peak capacities which could be reached in multi-dimen- sional separations.
Using a Poisson 2-D zone overlap model, Davis has computed the expected numbers of singlets, doublets and triplets in 2-D separations . This model was later refined and extended to any multiplet  by taking profit of a model developed by Roach . The Roach model was tested by computer simulations in 2-D , then extended to n-D for obtaining the expected numbers of any kind of multiplet as well as their sum which is the expected number of "peaks" in terms of the mean "den- sity" (or saturation factor) of components in the separ- ation space .
Resulting expressions are very useful to interpret or predict the outcome of a n-D separation. In the present work, one uses a different but complementary statistical perspective to estimate the potential of multi-dimensional separations. Namely, one is mainly looking for the prob- ability that all the m components of a complex sample mixture are separated. In the present context, a n-D separation system is understood as a system for which the whole separation space can be occupied by the sample components. In practice, such a multi-dimensional system has, up to now, been realized only for 2-D, for instance in two-dimensional thin-layer chromatography or two-di- mensional electrophoresis, by introducing the sample in a corner of the rectangular plate, developing the separ- ation in one direction, then in the second (orthogonal) direction by means of a different separation mechanism. One can imagine to physically extend this separation scheme to 3-D using a parallelepipedal configuration, but not to more than 3-D. Another possibility to implement a 2-D system is to use a set of two columns operated by different mechanisms or different separation methods, to collect fractions eluting from the first column and inject- ing them sequentially onto the second column, possibly after some concentration step. Let q be the number of fractions so collected. If the q fractions collected from the first column are small enough (in practice of the size of about one unit of peak capacity) and all are reinjected onto the second column, the overall result of the develop- ment of the q separations in the second column will, after some appropriate reconstruction, mimic that which would be obtained in a continuous planar two-dimen- sional separation system. This comprehensive mode of operation [-10-14] can, in principle, be extended to 3-D by collecting, for all q runs, small effluent fractions from the second column and reinjecting them separately onto a third column operating by a separation mechanism or
principle different from those of the two first columns. One easily imagines that this scheme can, in principle, be extended to n-D by using n different columns. In order for this approach to be effective, it is essential that the separ- ation in a given column is not correlated to that in any of the n - 1 other columns, hence that the n separation mechanisms or separation methods involved are mutually independent.
Theory Whatever the dimension of the zonal separation system and physico-chemical nature of the individual (one-di- mensional) separation methods, the representation of the separation is called a diachorismogram, from the greek word 81~Z~plc~ocy for separation [24, 25]. It generally gives the concentration vs. the position in the n-D separ- ation space. It does not matter for the present purpose whether the individual axes of the separation space have dimensions of time, length or of any other appropriate unit suitable for describing the location of the compo- nents (e.g., retention index).
Hypotheses of the model
One is not concerned here with the relative amount of the various components (i.e. by the concentration axis) as one focuses on the position of each component zone in the separation space. However, it is assumed, for the sake of simplicity, but without loosing generality in the trends, that the space occupied by each sample component in n-D hyperspace is a hyperball (of hyperspherical shape) and that the hypervolumes Of all components are equal. Since the model presented applies whatever the dimension of the separation system, the general terms hyperball and hypervolumes are used in the following, remembering that a hyperball is a segment in l-D, a disk in 2-D, a real sphere in 3-D, . . . .
Let ro be the radius of each individual component zone (for a 1-D separation, r0 is half the segment length) and Vo its hypervolume, ro is related to the standard deviation, cy, of the concentration profile along an individual direction. The exact relationship between ro and ~ depends on the level of resolution required for ascertaining that two components do not overlap (for instance, ro = 2cy corresponds to a unit resolution). Since all component zones are assumed to have the same hypervolume, a given component is separated from all others if no other component center is included in an exclusion hyperball of radius 2ro around its center. Let mo be the hypervolume of this exclusion hyper- ball and V the total hypervolume of the separation space. In this model, V is supposed to be large enough compared with Vo for the shape of the separation space to be irrelevant and edge effects negligible. One has:
mo= 2nVo (1)
where n is the dimension of the separation space.
Peak capacity of the separation system
The peak capacity, n .... in n-D is defined, as in l-D, as the maximum number of separated zones which can be ac- commodated in the separation space with the required resolution between two adjacent zones along the line of their centers. In l-D, one has:
V n~, 1 = - - (2 )
In that case, V is the length of the separation space, i.e. the length of the retention interval between the first and last component peaks, Vo the length of a given peak. By extension, the peak capacity in n-D is defined as:
V no,, = d). - - (3)
Although this definition would be simplified by defining n,, as in Eq. (2), as was done by Davis [19, 23], it is found necessary here to introduce a numerical coefficient d?,, smaller than 1, which may vary with the dimension n of the separation space, in order to respect the isotropic (hyperspherical) shape of the component zones. Indeed, for n > 1, there is not a unique way to define no. For instance, in 2-D, if one uses a square grid for accommo- dating the individual components disks, (~2 is equal to re/4 = 0.785 since x/4 is the ratio of the area of the disk to the area of the square within which it is included. If a compact 2-D lattice model is used, then d2a becomes closer to 1 and equal to rex/3/6 = 0.907. In 3-D, for a cubic grid or a compact hexagonal lattice, ~3 is equal to re/6 = 0.524 or rex/2/6 = 0.740, respectively.
Peak capacity in n-D in terms of peak capacities of individual dimensions
Defining the peak capacity by means of the hypercubic grid method evoked above (i.e. square grid in 2-D, cubic grid in 3-D, ... ) has the advantage that the n-D peak capacity, n .... is easily expressed in terms of the individual 1-D peak capacities of the constituting dimensions, no, 1,i, as;
.L nc, n = |1 no, l,i (4)
In this equation, it is implicitly assumed that the separ- ations in the n directions are mutually independent, as discussed above. Indeed, if, for instance, for a 2-D system, the separations in the two directions were totally corre- lated (as might be the case, for instance, for two gas chromatographic columns of similar polarity), the result- ing peak capacity would be, at most, the sum of the individual peak capacities instead of their product. Then, the advantage of such a correlated 2-D system would not be larger than that of an increase of the length of the separation space in a single direction.
In order to obtain the simple result expressed by Eq. (4), qbn must be selected as the ratio of the hyper- volume, Vo, of the hyperball to the hypervolume, (2r0) n, of
the hypercube which includes it, which gives : ~n/2
~n -- 2n/ln C(n/2) (5)
For this reason, the hypercubic grid definition of no, n and (~n is assumed in the following. Corresponding values of d~2 and ~3 have been given above. For larger dimensions, one gets, for instance, 04 = re2/32 = 0.308, qb5 = ~2/60 = 0.164 and d~6 = rc3/384 = 0.081. If the peak capacities of all individual dimensions are equal to nc, l,i,a, Eq. (4) becomes:
nc, n = n~n,l,ind (6)
One notes that the selection of the hypercubic grid basis for defining no, n may not just be a question of convenience but corresponds to the correct definition of the peak capacity in some multi-dimensional strategies using multi-column or so-called "comprehensive" ap- proaches.
Probability that a component is a singlet
One considers a sample mixture containing m compo- nents, distributed within the separation space in such a way that each component hyperball has an equal probability to be located at any position. A given component will be separated from a second one if the center of the zone of this second component does not fall within the exclusion volume determined by the first one. The probability that this happens is 1 - c%/V. The prob- ability, P1, that this first component is a singlet, i.e. that it is separated from all the other m - 1 sample compo- nents, is then:
v>m exp( ,m I,V) ,7, The second equality holds when ~0o/V is much smaller than 1, which is generally the case for multi-dimensional separations. PI is also the ratio of...