Statistical theory of spot overlap in two-dimensional separations

Anal. Chem. 1881, 63,2141-2152

2141

Statistical Theory of Spot Overlap in Two-Dimensional Separations Joe M.Davis Department of Chemistry and Biochemistry, Southern Illinois University at Carbondale, Carbondale, Illinois 62901

Equatlons are derived for the expected numbers of singlet, doublet, and triplet spots in a two-dlmensionai (2-D) separation bed, in whlch circular component zones are randomly distributed. The basis of these derivations is the selective interpretatlon of the radial distribution functions governlng 2-D Polsson processes. The equations are sufficient to descrlbe overlap In many 2-0 separations and are shown to be adequate in describing the overlap of elliptical zones having aspect ratlos lees than 2. It is demonstrated that, per unit peak capacity, 2-D separatlons are considerably worse than their onedlmenslonal analogues. The equations are verlfied at low saturations by interpretation of several hundred computer simulations of spot dlstrlbutions In rectangular beds. Departures of the equations from the simulations are observed at higher saturations. Caution Is suggested in overinterpreting the quality of 2-D separations.

INTRODUCTION Several theories have been proposed in recent years to quantify peak overlap in high-resolution chromatograms (1-5). The principal conclusions one draws from these theories are that chromatograms of complex multicomponent mixtures contain a large fraction of multiplet peaks and that the separation of such mixtures by a single column is inadequate for the resolution of the mixture components. One of the simplest of these theories, proposed by Giddings and myself ( I ) , is the statistical model of overlap (SMO). Several applications of the SMO to both computer-generated (6-9) and experimental (8,10-14) chromatograms have been reported. In particular, the SMO has been confirmed experimentally by its application to gas chromatograms of synthetic mixtures containing known numbers of detectable components (13,14). The theory also has been reinterpreted (15-17) and extended (18)by others. Other early literature references dealing with problems of overlap by mathematical means (19,20)have been brought to the author’s attention (21). The statistical limitations on separations in one dimension have been interpreted as strong incentives for carrying out separations in two dimensions (22,23),where peak capacities are substantially larger than in one dimension. However, two-dimensional (2-D) separations have statistical limitations as well. This work quantifies some of these limitations, which are surprisingly pronounced. Here, expressions for the expected numbers of singlet, doublet, and triplet spots in a 2-D separation are derived from the radial distribution functions governing 2-D Poisson processes. (The derivation of the number of singlet spots has been addressed by other means by Martin (24).) As is shown below, one’s ability to resolve zones in two dimensions does not increase in direct proportion to the increase in peak capacity. Instead, per unit peak capacity, one actually obtains worse separations in two dimensions than in one. In particular, as will be demonstrated, approximately one-third of all zones are overlapped in a 2-D space having 10 times the peak capacity required to resolve the zones. 0003-2700/91/0363-2141$02.50/0

The 2-D separations considered in this study are of the sequential type (23) and are carried out in a porous (and usually rectangular) bed, through which zones differentially migrate in two separate, discrete stages at different (usually right) angles. Examples of this type of separation include TLC/TLC (W),isoelectronic focusing/sodium dodecyl sulfate (IEF/SDS) electrophoresis (26-28), and a unique form of column liquid chromatography described by Guiochon and co-workers (29,30). Separations based on the coupled column method, such as column switching, strictly are not forms of 2-D separations (except in the limit of very short transfer times) but rather multidimensional separations (22). The latter are not considered here, except to note that the elution profiles developed by certain such separations, e.g., LC/CZE (31)and GC/GC (32),closely approximate 2-D beds and could merit examination in terms of the theory developed here. THEORY Assumptions. In this theory, one will assume that the centers of a larger number m of component zones are randomly distributed over a 2-D bed. More specifically, in accordance with the properties of a 2-D Poisson process, one will assume that the center of any component zone has a constant probability, equal to XdA, of falling within a differential area dA of the bed (33,341. The parameter, X, is the ratio of the mean or expected number rii of component zones in the bed to the bed’s area A X = rii/A

(1)

The exact shape of the bed (i.e., the contour of area A ) is immaterial, as long as many zones are contained (33). The number, rii, is a statistical approximation to the actual number m of components in the bed. The locus describing the boundary of a zone in a 2-D separation is an ellipse (35). To simplify substantially the mathematics required here, one will assume instead that the zone’s boundary is a circle of diameter do. It will be shown later that the results developed for circular zones also apply to elliptical ones of equal area, as long as the aspect ratio (Le., the ratio of the semimajor axis to the semiminor axis of the ellipse) is less than 2 or so. The assumption of a unique zone diameter, equal to do, clearly is simplistic. Detailed examinations of 2-D beds show that variations of zone diameters by a factor of 2 or more are possible. Rigorously, the theory developed here is not applicable to such beds. One hopes that these variations will average out in two dimensions, as they appear to average out in one dimension. In applications of the SMO, for example, the widths of chromatographic peaks at baseline can vary substantially, because of differences in peak amplitudes, and yet reasonable statistical parameters can be calculated by interpreting experiment by theory (36). Further study will be required to determine the validity of this assumption. The criterion by which one assesses the overlap of circular zones is easily stated. As shown in Figure la, the span, do, is the smallest distance by which the centers of two nonoverlapping circular zones of diameter do can be separated. If C 1991 American Chemical Society

2142

ANALYTICAL CHEMISTRY, VOL. 63, NO. 19, OCTOBER 1, 1991

neighbor

Ib)

tours of large multiplets can be quite complex). The total number of such spots is represented here by the variable p . The derivation of spot numbers from radial distribution functions has a major shortcoming, in that one must calculate separately the probabilities of forming singlet and various multiplet spots. In other words, by these theoretical means, one cannot derive a simple expression for the probability of forming a spot containing n zones, as one can for one-dimensional(1-D) separations ( I ) . As will be apparent from the work below, the analytical difficulty of these derivations increases substantially with increasing n. Here, expressions for the expected numbers of singlets (n = I), doublet d (n = 2), and triplet t (n = 3) spots are derived. No reason exists for not deriving expressions for higher order multiplets, except that the required mathematics would become daunting and the article would become prohibitively long. This work consequently should be viewed as a preliminary but not conclusive effort. Fortunately, the expressions derived are sufficient to describe many practical cases of zone overlap in two dimensions. The observable spot number, p , which is the arithmetic sum of the numbers of singlet and all multiplet spots ( I ) , consequently will be approximated here as

p=s+d+t

Flgure 1. (a) Illustration of a "circle of overlap" of radius do and its relationship to neighboring zones. Span do is the distance of closest

approach between nonoverlapping zones of diameter do. (b) F m t l o n of multiplet by inclusion of nearest neighbor in a circle of overlap of radius @d,. (c)Formation of various multiplets by inclusion of various numbers of nearest neighbors in the circle of overlap. (d) Formation of various multiplets by the further overlap, by other zones, of zones lying within the circle of overlap. the central zone in the figure is to be free from overlap, the centers of all other zones must be excluded from the area represented by the dashed circle of radius do,which is centered about the central zone. This circle is designated here as "the circle of overlap". A zone whose circle of overlap contains no zone centers but its own is designated a singlet spot, or simply a singlet. Depending on the objectives of the separation, one possibly could tolerate some degree of overlap between two zones and still consider them to be separated. In this case, the radius of the circle of overlap would be less than do. Here, one will generalize for this case by representing the radius of the circle of overlap by @do,where the parameter, B, lies between 0 and 1. By changing its numerical value, one can adjust the degree of overlap that one finds acceptable for a given purpose, much as one does by adjusting the resolution factor R,* in the SMO (6-8, II). The assignment, @ = 1, implies that zones which are singlets are not overlapped a t all by other zones. In contrast, if one or more neighboring zones lie within a zone's circle of overlap, that zone is overlapped by them (see Figure lb). Such a collection of overlapping zones, whose peripheral members ultimately break away from other zones in the bed (thus forming a zone cluster), will be designated a multiplet spot, or simply a multiplet. In general, various types of multiplets can result from variation of the number of nearest neighbors within the circle of overlap (see Figure IC).Even more complicated multiplets can be formed by the further overlap, by other zones, of zones lying within the circle of overlap (see Figure Id). Clearly, many complex multiplets can be realized in practice. Each singlet and multiplet corresponds to an observable spot in the bed, which may or may not be circular (the con-

(2)

in which higher order multiplets (e.g., quartets and quintets) are neglected, a t least for the moment. Equations for Overlap Probability. To determine expressions for the numbers of singlet, doublet, and triplet spots, the radial distribution functions for the distances between any zone center and the centers of its first, second, and third nearest neighbors must be known. In contrast, in the SMO, one only requires the distribution function for the distance between adjacent neighbors ( I ) . The need of these functions is straightforwardly illustrated. As stated above, a circular zone of diameter dowill be a singlet only if the distance rl between its center and the center of its first nearest neighbor (or, in a less exact but simpler use of language, the distance between it and its first nearest neighbor) is greater than @do,as shown in Figure 2a. For each zone in a doublet, the span rl between it and its first nearest neighbor must be less than pd,, and the span r2 between it and its second nearest neighbor must be greater than @do,as shown in Figure 2b for one of the zones. The second nearest neighbors of the two zones comprising the doublet may be two distinct zones or the same zone; this consideration is immaterial to the analysis. For the central zone in the triplet shown in Figure 2c, the spans r1 and r2 between it and its first two nearest neighbors must be less than pd,, and the span r3 between it and its third nearest neighbor must be greater than @do.Hence, one must consider all three distribution functions a t various stages in the analysis. The distribution function fk(r) for the radial distance r between any zone center and its kth nearest neighbor is (33, 34)

fk(r) = 27rXre-"p(aXr2)k-1/(k - I)!

r20

(3)

The probability of finding the kth nearest neighbor of any zone within an annular ring, which is centered about that zone and defined by ro 5 r 5 roo, is

u = aXr2 where the'substitution, u = air2,has been made to simplify the integral. The integrand on the right-hand side of eq 4 is the gamma function with k degrees of freedom (37).


n, = A / A ,

2143

(6)

then one recognizes that parameter CY is equivalent to its l-D analogue ( I ) , i.e., the ratio of the expected number r?i of components in the bed to the peak capacity ci

n

Figure 2. (a) Formation of a singlet spot by exclusion of the first nearest neighbor by the distance r , , where r 1> @d,. (b) Formation of doublet spot by inclusion of the first nearest neighbor within the distance r , , where r , < @do,and exclusion of the second nearest neighbor by the distance r 2 , where r 2 > @do. Both constraints apply to both zones. (c) Formation of triplet spot by inclusion of first two nearest neighbors within the distances r , and r2, where rl < @doand r 2 < @do,and exclusion of the third nearest neighbor by the distance r 3 , where r 3 > @ d o . Other constraints also must be satisfied.

In particular, the probability p(rk > @do)that the distance rk from any zone center to its kth nearest neighbor is greater than @dois

= rii/nc

Some will undoubtedly object to this definition of nc, which has been more traditionally defined as the number of zones which can be packed into a body-centered cubic structure (22, 23). A simple numerical factor, however, relates the n, so defined to eq 6. For example, the traditional n, for zones packed in a body-centered cubic structure is 7 / 4 % = 79% of the capacity defined by eq 6. The principal reason for defining n, in this manner is to develop a theory of overlap in 2-D separations, which is free of any geometrical factor (e.g., 7/41 that depends on the nature of zone packing and bed shape. The inclusion of such a factor would reduce the theory's resemblance to the theory developed for l-D separations. This reduction would be undesirable because both theories have many features in common. In both theories, peak and spot numbers depend only on f l and CY. In both theories, CY is rigorously defined as the statistical number m of components divided by the ratio of the total space available for separation to the space required for the resolution of one comDonent. In one dimension. this ratio is in accordance with the classical definition of'n,, whereas in two dimensions it is not. To maintain a close correspondence between the theories, it is convenient (but not essential) to think of the 2-D n, as defined by eq 6. Those who object to this definition of n, can simply interpret CY as m~(@cl,)~/4A, about which no ambiguity exists. The probability P(rk I@do)that the distance rk between any zone center and its kth nearest neighbor is less than or equal to pd, is the complement of eq 5

1 - ~(nX(@do)2)ie-*"Bddl/i! (8a) i=O k-1

= 1- C ( 4 ~ ~ ) ' e - ~ ~ / i ! i=O

+

= 1 - (P(0) P(1)

k-1

= C ( 4 c ~ ) ' e - ~ ~ / ci i != ~X(pa1,)~/4

(5b)

is0

= P(0)

+ P(1) + ... + P(k - 1)

(5c)

In eq 5c, P(n) represents the probability that n zone centers are contained in the circle of overlap. In other words, the fact that rk > pd, implies that no more than (k - 1)zone centers lie within this circle (33). For example, a zone's second nearest neighbor (corresponding to k = 2) will lie outside that zone's circle of overlap either if no zones lie within the circle-in which case the first nearest neighbor, and consequently the second, lie outside the circle of overlap-or if one zone lies within the circle. The fact that each term, P(n), corresponds to a unique term in the sum, eq 5b, will be useful later. In eq 5b, the parameter CY equals the product of the mean component density, A, as defined by eq 1,and the scaled area A, = ~ ( @ d , ) ~of/ a4 pure component zone (A, exactly equals the zone area when 0 = 1). If one defines the peak capacity (or, perhaps more accurately, spot capacity) n, of a 2-D bed as the ratio of the total area A of the bed to the area A, required for the resolution of any zone

(7)

+ ,.. + P(k - 1))

(ab) (84

Calculation of Spot Numbers. With the above equations, one can derive expressions for the expected spot numbers s, d , and t. As stated above, these expressions must be derived separately. Calculation of Singlet Number s. The calculation of the expected number s of singlet spots is straightforward. The probability p(rl > @do)that the span between any zone center and its first nearest neighbor is greater than @dois given by eq 5b with k = 1

p ( r l > @do)= P(0) = p1 = e-4a

(9)

where p1represents the probability of forming a singlet spot. Each of f i zones has this probability of being a singlet; therefore, the mean number s of singlets spots is

(10) in agreement with Martin (24). This result is smaller than its l-D analogue, which is me-% (I),by the factor e-%. Hence, a t any saturation, fewer singlets will be found in a 2-D separation than in a l-D separation. By dividing both sides of eq 10 by eq 6 and introducing eq 7, one obtains the number of singlets resolved per unit peak capacity s = fie-4a

s/n, =

(11)

2144


probability that the distance between $ and its second nearest /---3----\ \ \

I

I I

Flgure 3. Zones { and $, which comprise a doublet. The shaded region is common to both clrcles of overlap.

By differentiating eq 11with respect to a and equating the result to zero, one determines that the maximum value of s l n , is e-'/4 = 0.092 and occurs at a = I f 4 . Calculation of Doublet Number d . The calculation of doublet number d is more complicated than that of s. These complications have two principal origins. The f i s t is that only a portion of the probability p ( r z > @do)given by eqs 5a-c is relevant to the calculation of doublet formation. The second is that the two circles of overlap, which are centered about the two zones comprising the doublet, themselves overlap. Figure 3 depicts two zones, $ and l, which overlap. For these zones to comprise a doublet, their second nearest neighbors must lie outside their circles of overlap. For either zone, therefore, one requires that p ( r z > @do)= P(0) + P(1), in accordance with eq 5c. The right-hand side of this equation states that the second nearest neighbor of $ (or {) will lie outside $'s (or r s ) circle of overlap, if either no zones or one zone lies within this circle. However, the contribution P(0) is irrelevant to doublet formation, because a doublet can be formed only when one zone center (which corresponds to P(1)) lies within the circle of overlap. Hence, only the contribution P(1)= 4ae-4u(the latter equality is given by eq 5b) needs to be considered in doublet formation. From this consideration, one might think that the probability of doublet formation is calculated by applying independently the distribution function f i ( r )to both zones $ and 5: The probability so determined would equal the product, P+(l)Pf( l),where the subscripts indicate the zones associated with the probability P(1). This conclusion would be incorrect, however, because P&1) and Pf(l)are not independent or mutually exclusive. They are not independent because a fraction of the area (indicated by the shaded region in Figure 3) from which second nearest neighbors are excluded is counted twice by two independent applications of the distribution function, f i ( r ) . The probability product, P&l)Pf(l), consequently underestimates the likelihood of doublet formation. To account properly for doublet formation, one instead must calculate a probability consistent with answers to the following two questions: First, what is the likelihood that r s second nearest neighbor lies beyond @doand that its first nearest neighbor, which is $, lies within @do?Secondly, given that { overlaps with $, what is the probability that $'s second nearest neighbor lies beyond pd,? The answer to the first question is indeed Pr(l).The answer to the second question, however, must be calculated as a conditional probability, in which one explicitly accounts for the fact that $ is overlapped by {. In general, the conditional probability p(alb) that event a occurs, given that event b occurs, is given by (38)

where p ( a n b )is the intersection of the sample spaces of events a and b, i.e., the set of possible outcomes common to both a and b , and p ( b ) is the probability that event b occurs. Examples of eq 12's application are given in basic texts on probability (38). Here, one will represent the conditional

neighbor is greater than @do,given that {overlaps with $, as P($P>@d,l{ - $). The subscript, 2 > @do,indicates that the second nearest neighbor of $ lies beyond @do,and the notation { - $ indicates that $ is overlapped by {. The intersection of these sample spaces that is consistent with both the exclusion of $'s second nearest neighbor from J/'scircle of overlap a n d the overlap of $ with { is P J l ) . In accordance with eq 12, one must divide this intersection by the probability that { does indeed overlap with $. This probability equals the likelihood that {lies in $'s circle of overlap and is expressed by 1- P$(O),where eq 8b with k = 1has been used. Therefore P.,.(1 The sequence of probabilities required for doublet formation is the product of Pf(l)(which is the answer to the first of the two questions asked above) and P($z>@dol{ - $) - $) (which is the answer to the second of these questions)

P&&)

P{(l)P($2>pdAc- $) = pf(1) -

(14)

However, the probability p z of doublet formation is not equal to eq 14 but to half of eq 14, because the probability sequence described by this equation accounts equally well for $'s overlap with {or r s overlap with $. This results from the symmetry of the zones comprising a doublet spot. Each doublet is counted twice by the above formation; consequently, the probability of doublet formation is

The number d of doublets is given by mpz (y2e-8a

d = 8m- 1 - e-4u

(16)

where eqs 5b,c and 8b,c have been used to express the various probabilities in terms of a. This result differs markedly from its 1-D analogue, which is r??e-2a(1- e-.) (1). By dividing both sides of eq 16 by eq 6 and introducing eq 7 , one obtains (y3e-8a

d / n , = 8- 1 - e-4a

(17)

By differentiating eq 17 with respect to a,equating the result to zero, and solving this equation numerically, one can show that the maximum value of d l n , is about 0.0281 and occurs a t a = 0.312. Calculation of Triplet Number t . The complications encountered above in the calculation of d-the use of only a portion of the probability given by eqs 5a-c and the necessity of calculating conditional probabilities-also complicate the calculation of triplets. Two further complications exist. The f i s t additional complication is that two types of triplets exist, as shown in Figure 4, and each must be addressed separately. One can see that the triplets must be addressed separately, because the multiplet in Figure 4a is formed only if the second nearest neighbor of $ lies outside $'s circle of overlap, whereas the multiplet in Figure 4b is formed only if the second nearest neighbor of $ lies inside J/'s circle of overlap. (Similar arguments can be made for zone [.) The triplet in Figure 4a, in which a central zone overlaps with two zones which, in turn, do not overlap with each other, is called a "straight-chain" triplet and is designated t,. The triplet in Figure 4b, in which each zone overlaps with the other two, is called an "interlocking" triplet and is designated ti. The total number t of triplet spots is the sum, t = t , + ti.


\

/‘

X

X /

\

,f \

I

2145

---A

I

I

I

\

I

\

/ 1

straight-chain triplet ts

Figure 5. Patterns described by the probability, 8a2e-4a. The X represents exclusion of the third nearest neighbor from r s circle of overlap: (a) t,-like pattern; (b) trlike pattern. 140

g 120 n g 100

4b)

E

80 60 40

20

‘

5

1-

0

/

0

0.1

0.2

0.3

0.4

0.5

0.6

interlocking triplet ti Figure 4. Zones t, i , and $, which comprise either (a)a stralghtchain triplet t , or (b) an Interlocking triplet ti. The shaded regions are common to at least two circles of overlap.

The second additional complication, which is more serious, is that one cannot (or more correctly, the author cannot) calculate either t, or ti from a simple interpretation of the radial distribution function, f3(r). Rather, one can only calculate the probability of forming the sum of two spot patterns, which are related to (but not identical with) t , and ti. One does not know a priori what fraction of this sum to assign to the probability of forming t, and what fraction to assign to the probability of forming ti. To obtain independent expressions for t, and ti, additional information is required. This information was obtained here empirically by analysis of several hundred computer simulations of random 2-D spot distributions. Unlike the expressions for s and d , the expressions for t, and ti are consequently not rigorous. Rigorous theory is used whenever possible. In fact, the functional forms of t, and ti are derived rigorously; it is only the coefficients that multiply these expressions that are determined empirically. The difficulties associated with this latter complication are now detailed. As shown in Figure 4, zone t is the only zone common to both t, and ti, from whose circle of overlap one specifically must exclude its third nearest neighbor. The probability p ( r , > pd,) that (‘s third nearest neighbor lies outside r s circle of overlap is given by eq 5b with k = 3

p(r3 > Od,) = P(0) + P(1)+ P(2) (18) Only the contribution P(2) PA2) = We* (the latter equality is given by eq 5b) is relevant to the formation of triplets,

however, because Ps circle of overlap must contain two zones. These two zones may themselves overlap, in which c w e a ti-like pattern is formed, or they may not overlap, in which case a t,-like pattern is formed. It is this interaction between and 5 that one cannot deduce from Pr(2)alone. Rather, Pf(2) accounts for both interactions; i.e., it describes the probability of forming the sum of the t,-like and +like patterns shown in Figure 5. These distributions of zones are designated patterns, and not multiplets, because the interactions of and 6 with any neighbors other than themselves and {have not

+

+

c 0.54

P

’

0.52

t I

0

0.1

0.2

0.3

0.4

0.5

CI

Flgure 6. (a) Plot of t,-like, t,-like, and total pattern numbers vs a. The patterns shown are identical with those in Figure 5. Points represent the average numbers of patterns determined from 500 simulations with m = 500. (b) Plot of fraction of t,-llkepattems vs a. Open clrcles correspond to 500 simulations;filled circles at low a’s,to 5000

simulatlons. been specified (e.g., it is not clear from Figure 5a if $‘s second nearest neighbor is excluded from its circle of overlap). This lack of specification is indicated in the figure by the absence of circles of overlap centered on $ and 6. The X s in the figure indicate the exclusion of r s third nearest neighbor from r s circle of overlap. Because PA2) is the s u m of the probabilities of these pattern formations, one is at a loss as how to assign rigorously a formation probability to each pattern. To determine what fraction of the probability, PA2), is associated with the t,-like pattern and what fraction is associated with the ti-like pattern, 500 computer simulations of 2-Dspot distributions containing m = 500 zones were generated for a variety of a values, as detailed below in the Procedures. The numbers of t,-like and ti-like patterns were determined in each simulation for arbitrarily chosen values of @do,and these numbers were averaged. Figure 6a depicts the number of counts so determined as a function of a. The sum of these counts is clearly approximated by mPf(2),with m = 500 (the agreement is best for small a’s).Note that one distinguishes here between the actual number, m, of zones in the simulation and the statistical approximation, m, to that number. In Figure 6b, the ratio of the number of ti-like patterns to the total number of patterns is graphed against a (open circles). This ratio is remarkably independent of a and is approximately equal to 3/s, except for small a.

2146


For small values of a,500 simulations are simply insufficient to obtain meaningful statistics. For example, the expected number ti of triplets for m = 500 and CY = 0.01 is 0.0748, which corresponds to only 37 counts in 500 simulations. When 5000 simulations are averaged, instead of 500, the ratio of the number of ti-likepatterns to the total number of patterns more closely approaches 3/5 for small a values (filled circles). Some undersampling, albeit less than before, is still apparent at a = 0.01 but not at a = 0.02, where it existed for 500 simulations. The fraction of t,-like patterns is the complement of the fraction 3/5 or Hence, the probability Pf(2)lt8 of forming the t,-like pattern is approximately

and the probability Pc(2)ltiof forming the ti-like pattern is approximately

where the subscripts on Pt(2) indicate that these probabilities are associated with specific patterns of triplets. Calculation of Triplet Number t,. From eq 19, the number t , of straight-chain triplets is straightforwardly calculated. The calculation is straightforward because the additional probabilities prerequisite to its calculation have already been derived in the theory of doublet formation. To wit, the probability that J.'s second nearest neighbor is excluded from +'s circle of overlap, giuen that { overlaps with J. (see Figure 4a), is given by eq 13. Similarly, the probability that f's second nearest neighbor is excluded from Ts circle of overlap, given that {overlaps with [ (see Figure 4a), is also given by eq 13, with the subscript, f , replacing the subscript, J.. The probability p3It,that a straight-chain triplet is formed is approximately the product of these two probabilities and Pr(2)lt.

The reason one does not need to divide this product by the number 3 to obtain p&,, as one divided eq 14 by the number 2 to obtain p z , is the lack of symmetry among zones {, $, and i ; only zone { satisfies the probability expressed by eq 19. The number t , of straight-chain triplets is approximately RP3It.

256

t , = -lii 5

a4e-lZu (1 - e-4~)2

where eqs 5b,c and 8b,c have been used to express t, in terms of CY. By dividing both sides of eq 22 by eq 6 and introducing eq 7, one obtains

By differentiating eq 23 with respect to a, equating the result to zero, and solving this equation numerically, one can show that the maximum value of t,/n, is about 7.12 X and occurs a t a 5 0.338. Calculation of Triplet Number ti. To calculate ti, one must multiply the probability Pf(2)ltiexpressed by eq 20 by two additional probabilities. The first of these is the conditional probability that the third nearest neighbor of $ lies outside $'s circle of overlap, given that $ overlaps with its two nearest neighbors, { and 4, which in turn overlap with each other. The second probability is the conditional probability that the third nearest neighbor of 6 lies outside f ' s circle of

.

/ - /

\

7a)

*---. \

/

7b)

+

Flgure 7. Patterns described by the probabillty, 1 - (1 4a)eA. No third nearest neighbor is excluded here: (a) t,-like pattern; (b) t,-like

pattern.

overlap, giuen that f overlaps with its two nearest neighbors, { and J., which in turn overlap with each other. These probabilities are equal, because of the symmetry of the multiplet, ti. However, in calculating this probability, one encounters difficulties similar to those encountered in determining eqs 19 and 20. Here, one will consider only I)'S interactions with its three nearest neighbors and recognize that E's interactions with its three nearest neighbors are identical. One needs to express these interactions in the form of a conditional probability, which will be represented by P(J.3,sd,l({- E) - J.), where the notation is to be interpreted as before. This conditional probability must be calculated in accordance with eq 12. The appropriate intersection, which is consistent with the exclusion of +'s third nearest neighbor from J.'s circle of overlap, the inclusion of {and 5 within +'s circle of overlap, and the overlap of { and 6 within this circle of overlap, is simply P,(2)lti (eq 20). One must now divide this intersection by the probability that { and f lie within J.'s circle of overlap and also overlap with each other. The probability that {and f lie within J.'s circle of overlap is given by eq 8b with k = 2 1 - (P&O) P,(l)) = 1 - (1 4 ~ ~ ) e - ~ "(24)

+

+

However, one is now confronted by a problem similar to that already faced. The zones { and E may overlap, in which case a ti-like pattern is formed, or they may not overlap, in which case a t,-like pattern is formed. As before, eq 24 accounts for both possible outcomes; Le., it equals the sum of the probabilities of forming the ta-like and ti-like patterns shown in Figure 7. One should note that these patterns are different from those shown in Figure 5, because the exclusion of a third nearest neighbor is not specified here. One is interested here only in that fraction of eq 24 which represents the ti-like pattern in Figure 7b. To determine what fraction of eq 24 is associated with the ti-like pattern, 500 computer simulations, identical with those described above, of 2-D spot distributions containing m = 500 zones were generated for a variety of a values. The numbers of to-likeand ti-like patterns were determined in each simulation for a radius of overlap equal to Od,, and these numbers were averaged. Figure 8a depicts the numbers so determined as a function of a. The sum of these patterns is indeed given by eq 24, multiplied by R = 500. The ratio of the number of +like patterns to the total number of patterns is graphed in Figure 8b as a function of a (open circles), As before, undersampling occurs a t small a's when only 500 simulations are averaged but is reduced when 6000 simulations are averaged (filled circles). In contrast to the results shown in Figure 6b, this fraction is not independent of a and approaches the limiting value, a / 6 (or there about) only as a approaches 0. Interestingly, this variation with a is almost linear over the range 0 5 CY I 0.5. To obtain an approximate expression for ti, one will use here this limiting fraction, 3/5, of eq 24, to calculate the conditional probability. Because of this approximation, the expression

ANALYTICAL CHEMISTRY, VOL. 63,NO. 19, OCTOBER 1, 1991

2147

300 u)

n

250

5

200

E

150 100

50

- 0-

n

0.1

0.2

0.4

0.5

0.6

9a)

9w

(a) Illustration of overlap between ellipses that are not nearest neighbors. (b) Ellipse with semimajor and -minor axes equal to a and b . The radial distance, re, defines the "ellipse of overlap" in accordance with eq 30. Figure 9.

u

e

u-

0.5

0

0.1

0.3

0.2

0.4

0.5

mensionless result ti/n, to maximize. For completeness, however, this maximum is calculated. By dividing both sides of eq 27 by eq 6 and introducing eq 7, one obtains

a (a) Plot of t,-like, t,-like, and total pattern numbers vs a. The patterns shown are identical with those in Figure 7. Points represent the average numbers of patterns determined from 500 simulatiis with m = 500. (b) plot of fraction of t+lke patterns vs a. Open ckcles correspond to 500 slmulations; filled circles at low a's,to 5000 simulations.

-

for ti is expected to apply only in the limit, a 0. The term by which one must divide the intersection P+(2)lti is consequently equal to 3(1 - (P+(O)+ P+(1)))/5. The probability of forming interlocking triplets is proportional to the product of Pr(2)lti and the square of this conditional probability

By differentiating eq 28 with respect to a,equating the result to zero, and solving this equation numerically, one can show that the maximum value of ti/n, is about 6.12 X and occurs at a = 0.403. Expected Number p of Spots. By substituting eqs 10, 16, 22, and 27 into eq 2, one can show that the expected number p of spots is approximately equal to (y2e-8a

(1- eda) 256 5

"Ilim ti a-0

-

I

Pd2)lti

+

lx

pr(2)1ti3(1 - (P+(O) P+(1)))/5

The expected number ti of triplets is approximately mp&

a-0 where the various dependences on a have been explicitly expressed by using eqs 5b,c and 8b,c. It is apparent that the expressions for t, (eq 22) and ti (eq 27) are quite different. Both expressions are quite different from their 1-D analogue, which expresses the sum of the 1-D equivalents of t , and ti as t = t, + ti = lite-2a(1- e-a)2 (I). Because eq 27 is expected to apply only as a approaches zero, caution is suggested in its interpretation. In particular, it may not apply at saturations sufficiently large for the di-

(28)

CY-0

lim

The conditional probability enters the calculation twice, because it applies to [ as well. However, the probability p3Jti of forming interlocking triplets is only one-third of eq 25, because of the symmetry among the various zones comprising ti (cf. discussion on doublets)

a7e-12a

512

ti(n, i= llm 5 (1 - (1 + 4 c ~ ) e - ~ " ) ~

Flgure 8.

512 ~

+-5 (1 - e-4a)2 C

Y

~

~

-

+ ~

~

a6e-12a

(1 - (1

-

+ 4~4e-492

Eq 29 strictly applies only as a 0, not only because of the limitations on the derivation of ti, but because higher order multiplets (e.g., quartets and quintets) have been neglected. As with the SMO ( I ) ,eq 29 can be used to estimate r i i from the observed spot number p and an independent measure of n,, which, together with m, defies a. Unlike in one dimension, however, the estimation requires numerical methods instead of analytical ones. Applicability of Equations to Elliptical Zones. As observed in the beginning of this section, zones in 2-D separations are elliptical, instead of circular. A rigorous theory for the overlap of elliptical zones is much more difficult to derive than that developed here, because elliptical zones which are not nearest neighbors can overlap with each other, as shown in Figure 9a. The basic assumption in the theory developed here, however, is that nearest neighbors overlap. Hence, the theory strictly is not applicable to elliptical zones. In practice, however, it will be applicable to ellipses of small aspect ratio, i.e., ellipses which are almost circular. To gauge the aspect ratio, below which one can apply the theory to elliptical zones, one will calculate the probability that an elliptical zone is a singlet spot and then compare this probability to that calculated for a circular zone. It seems reasonable to conclude that the theory also will apply to ellipses, as long as these two probabilities closely agree. Figure 9b depicts an elliptical zone with semimajor and semiminor axes equal to a and b, respectively. This zone will be free from overlap, if all neighboring zones lie outside the ellipse indicated by the dashed curve in Figure 9b, which, by

2148

ANALYTICAL CHEMISTRY, VOL. 63,NO. 19, OCTOBER 1, 1991

analogy, is designated the "ellipse of overlap". This ellipse can be described by the equation (39) r=re=@ (cos2 0

27b

+ y2 sin2 0)ll2

(30)

where r is the radial coordinate, 0 is the angle defined by Figure 9b, and y is the aspect ratio, alb. The parameter, @, is a scalar, which scales the size of the ellipse of overlap much as it scales the size of the circle of overlap. Because the zone density in a 2-D bed is isotropic (except for the edge regions), one can deduce from the expression fl(r) that the probability of a zone center lying within the differential area element r dr d0 is

140 r

1000 c S

d

800

100

600

ao

400

60

(31) The probability pl(e) that all other zone centers lie outside the ellipse of overlap is obtained by integrating eq 31 from the elliptical boundary re to infinity over all angles

where the substitution, d,2 = 4yb2, has been made to allow one to express eq 32 in terms of a and to facilitate the comparison of the probabilities of singlet formation for circles and ellipses on an equal basis. More specifically, this substitution implies that the areas of a circle of diameter do and an ellipse with semimajor and -minor axes equal to a and b are the same. Because these areas are equal, the peak capacity n,, as defined by eq 6, is the same for both zone shapes. The numerical value of eq 32 must be unchanged by the substitution of 1/ y for y, because one must determine the same result, if a b. As the aspect ratio, y, approaches 1, eq 32 approaches exp(-4a), as expected. PROCEDURES Computer programs were written in FORTRAN 77 to generate 2-D simulations of random zone distributions in square and rectangular beds and to count the numbers of singlets s, multiplets d , t,, and ti, and the t,-like and ti-like patterns shown in Figures 5 and 7. The coordinates of each zone center were generated by two sequential subroutine calls on an in-house random number generator adapted from ref 40. The sequence of pseudorandom numbers so generated was tested by computing from them the cumulative distribution function, which essentially equalled a straight line of zero intercept and unit slope. The distances between the first, second, and third nearest neighbors of each zone were computed and compared to a series of @dovalues, which defined a series of a values, in accordance with eq 1 and the definition of a , eq 5b. These programs were executed on the IBM 3081-GX computer at Southern Illinois University. Graphs of these spot distributions were generated by the application KaleidaGraph (Synergy Software, Reading, PA) on a Macintosh SE microcomputer and scaled, if necessary. The maximum values of d/n,, t,/nc, ti/nc,and (t, + ti)/n,were determined numerically by the bisection method (41). Equation 32 was integrated numerically by using Simpson's rule (41). RESULTS AND DISCUSSION Figure 10 is a graph of s, d, t,, and ti vs a. The circles represent the average numbers of spots counted in 500 simulations, each of which contained m = lo00 zones distributed randomly in a square area. The solid curves are graphs of the singlet and multiplet equations, eqs 10, 16, 22, and 27, with ni = 1000. (A large value of m was chosen to ensure that enough events were included for the statistics to be meaningful.) A good agreement between simulation and theory is found, when a is less than 0.1 or so, i.e., when only 10% or

500 rlmulollonr

40

200

20

0

0.1

0.3

0.2

0.4

0.5

0

U

35

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

U

r

i* 3 0

1# 20

25 15

20

10

15 10

5

XeTPr dr d0

120

5

"

n

0

0.1

0.2

a

0.3

04

0.5

0

0.1

0.2

U

Figure 10. Plot of s, d , t,, and t , vs a. Curves are graphs of eqs 10, 16, 22, and 27 for f i = 1000. Circles represent the average numbers of spots determined from 500 simulations with m = 1000.

less of the bed is used. Surprisingly, in spite of the limitations on its derivation, ti agrees with simulation over a larger a range than does t,. The expression for s describes the simulation results well over the full a range, although the agreement is best for small a's. A clear discrepancy exists between simulation and the expressions for multiplets, however, when a is greater than 0.1 or so. More specifically, the expressions for d, t,, and ti are in error by more than lo%,when a is greater than 0.20, 0.12, and 0.24, respectively. The observed numbers of multiplet spots are substantially larger than predicted by theory, although their variations with a roughly parallel theory. In part, this behavior can be attributed to "edge effects" (34), in which spots are partially shielded from overlap, because they lie near the border of the bed. Because no zones exist on the other side of the border, the likelihood that zones in this vicinity are overlapped is reduced and the observed spot numbers rise above theoretical expectation. This perturbation can be gauged by the following calculation. The average separation distance ( r ) between nearest neighbors is calculated from fl(r) as

( r ) = Jmrfl(r) d r = 2aX

Jm

r2e-*xf d r = (2h1iz)-l

(33)

An approximate value of a , above which edge effects can be expected, is determined by equating @doto (r). In this case, the circles of overlap about zones near the border roughly overlay the border, and the zone density about them is no longer isotropic. For larger @d,,'s,the anisotropy is even larger. The magnitude of a so defined is a116 = 0.20, which is in rough accordance with the a value at which the discrepancy begins. Yet, one would be unwise to conclude that edge effects account for the entirety of the error in the multiplets, which is rather pronounced a t high saturations. Some systematic error in the analysis may affect the predictions at high saturations. These errors are presently being studied. Figure 11 is a plot of sln,, dln,, and t/nc vs a (those who object to the definition of n, as 4A/r(@dJ2may identify the denominator of the ordinate with this latter expression). The bold curves in Figure l l a , b are graphs of eqs 11 and 17, whereas the solid curves are their 1-D analogues (I). In Figure l l c , the dashed curves are graphs of eqs 23 and 28, the bold curve is their algebraic sum, and the solid curve is the 1-D analogue to this sum (1). The coordinates of the maximum values of the bold curves are indicated in the figures. Figure l l a shows that only e-'/4 = 0.092 of the available peak capacity in a 2-D separation can be utilized in resolving

ANALYTICAL CHEMISTRY, VOL. 63, NO. 19, OCTOBER 1, 1991 0.2

Table I. Nearest Integral Estimates of Singlets s, Doublets d , and Triplets t , and ti Expected in 1-D and 2-D Separations of m = 100 Components at a = 0.05 and 0.W

So

3j

2149

0.16 0.12

a = 0.05

multiplet type

0.08

a = 0.10

multiplet

multiplet no.

components

S

82 (90)

82 (90)

67 (82)

67 (82)

tot.

90‘(94)

99 (98)

80 (91)

95 (101)*

0.04

no. of

no.

no. of

components

0 0 0.1

1

0.5

r

1.5

2

U

The numbers in parentheses correspond to 1-D separations. The number of 1-D triplets t reported equals t , + ti. *Total exceeds m because of roundoff. (0 3 1 2 , 0.028i)

0.02 O

’

O

4

K

,

, ‘

0 0

0.5

0.06r

1

1.5

2

U

/Q$,/t

tot,I,

2-D

0.01

,

n

0

0.5

1

1.5

2

M Figure 11. Plot of sln,, dln,, and t l n , vs a. Bold curves are graphs of .eqs 11, 17, and the algebraic sum of eqs 23 and 28. Regular curves are predictions of the SMO.

singlet spots, even under the best of conditions. This fraction is exactly half of the maximum number of singlets resolvable per unit peak capacity in 1-D chromatograms, which is e-’/2 = 0.184 (I). For all a’s, the value of s/n, is smaller for 2-D separations than for 1-D separations. This observation leads one to the somewhat startling conclusion that fewer singlets per unit capacity can be resolved in two dimensions than in one dimension. This statement should not be interpreted as a contradiction of the obvious fact that a 2-D separation is superior to a 1-D separation, other factors being equal. One obtains this superiority, simply because one typically has so much more capacity in two dimensions than in one dimension that one can “waste” a good fraction of it by ineffective utilization and still achieve a better separation. The following explanation may help one to understand this relative loss of efficiency in 2-D separations. To make the explanation, the author has assigned numerical values to the 1-D resolution RE*and 2-D 0factors, such that singlet peaks and spots are completely free from overlap. These assignments, however, are not integral to the explanation. In a 1-D separation, in which the baseline width of a single-component peak of standard deviation u is x , = 6u, one must provide for x, units of component-free space on either side of the peak maximum to ensure that the peak is a singlet (RE*= 1.5) (I). In other words, two x , units of empty space are required for each singlet of width x,. In a 2-D separation, in which the area of a single-component spot is 1rd,2/4, one must provide for rd,2 = 4(1rd,2/4)units of component-free space (corresponding to the area of the circle of overlap) to ensure that the spot is a singlet (0= 1). In other words, four nd,2/4 units of empty space are required for each singlet of area 1rd,2/4. Thus, for every two component-free zone widths required to form singlets in 1-D separations, four component-free zone areas are required in 2-D separations. The space demands

for singlet formation simply are more stringent in two dimensions than in one, leading to the results described above. In Figure llb,c, the functions din, and (especially) t/ncare larger than their 1-D analogues at small a’s. This behavior is observed because a smaller fraction of peak capacity is available to resolve 2-D zones into singlets than to resolve 1-D zones into singlets (see Figure lla), and this relative loss of efficiency appears as an increase in the number of low-order multiplets per unit peak capacity at small a’s. The trends in these figures are best emphasized by specific examples. Table I reports to the nearest integer the numbers of singlet, doublet, and triplet spots expected by the partial resolution of m = 100 components in beds of capacity 2000 and 1000 ( a = 0.05 and 0.10). Also reported to the nearest integer are the numbers of components represented by these three spot types. For purposes of comparison, the numbers of singlet, doublet, and triplet peaks expected in 1-D chromatograms of the same saturation are reported in parentheses, as are the numbers of components represented by these three peak types. (For 1-D separations, only the algebriac sum t = t, + ti is reported.) A t these saturations, over 95% of the components appear as one of these spot (peak) types. In both cases, fewer singlets are resolved in two dimensions than in one (e.g., 82 [2-D] vs 90 [1-D] for a = 0.05) and more doublets and triplets are found in two dimensions than in one (e.g., 11 doublets [2-D] vs 8 [1-D] for a = 0.10). In general, the number of singlets developed in a 2-D separation should equal or exceed the number of singlets developed in the f i s t dimension, as long as the second dimension is orthogonal to the first and the band broadening in the second dimension does not mix zones resolved in the first dimension. This statement is essentially equivalent to the assertion of Giddings, who observed that orthogonal displacements in a 2-D separation will lead to separation if either displacement produces separation (22). From these considerations, one might argue that the use of 2-D Poisson statistics is inappropriate to the modeling of 2-D separations. For example, if a zone is resolved as a singlet during its first displacement, then the probability that it is a singlet after its second displacement is unity. In other words, no statistical consideration exists for the second dimension; the zone is undeniably a singlet. This argument breaks down, however, because one does not know a priori that any peak resolved in the first displacement is a singlet. In fact, one can only speak of the probability, s l p , that any observed peak is a singlet, which is exp(-a) (I). If the complementary probability, 1- exp(-a), is realized (i.e., if the peak indeed is a multiplet), then the separation of its components along the second dimension is subject to statistical constraints, although the severity of these constraints certainly differs from that imposed by the first dimension. Since one does not know whether any particular peak developed in the first dimension is a singlet, one must treat both dimensions statistically.

2150


A

a 0.8

v

r

0.6

0.4 0.2

, '/= 1.5,

Figure 12. Approximation of the circle of overlap by a square, whose length equals the diameter of the circle of overlap.

'42

0C

0

0.1

0.2

a

0.3

0.4

0.5

Figure 14. Plot of probability p,(e) of forming an elliptical singlet zone vs a. Bold curve is a graph of eq 9; other curves were determined by integration of eq 32 for various aspect ratios y.

0

83) 'A0 0

gg ,.' 0 Flgure 15. Computer simulation of 180 randomly spaced elliptical zones, for which y = 2. The varlous multiplet spots are indicated. The numbers of spots predicted (found) are: s = 122 (122), d = 19 (21), t , = 2 (3), and t, = 1 (1). One quartet q is also observed. a = 0.0977 (P = 1).

Figure 13. Computer simulation of 180 randomly spaced circular zones. The various mulplet spots are indicated. The numbers of spots predicted (found) are: s = 122 (1 19), d = 19 (21), t , = 2 (4), and t , = 1 (1). One quartet q is also observed. a = 0.0977 (@ = 1).

T o emphasize these arguments, the expression for s can be derived by an alternative (but only approximate) method. Figure 12 approximate a zone's circle of overlap by a square centered about this circle. The centers of the zone, circle of overlap, and square lie a t the origin of the n and y axes indicated in the figure. If the indicated zone is to be a singlet, its nearest neighbor in the positive x direction must lie beyond @doand its nearest neighbor in the negative x direction must lie beyond $do. The probability that either of these independent events is realized along this 1-D axis is e- (I). Similar arguments can be made for neighboring zones along the y axis. The probability that all four events happen simultaneously is = e-4a, in agreement with eq 9. T o test the predictions of theory against the numbers of spots found in a single simulation, 180 circular zones were randomly distributed throughout the square 2-D space shown in Figure 13. The diameter doof these circles was measured to three significant digits with calipers. The radius of the circle of overlap was chosen to equal do (i.e., p = 1). By this convention, zones are considered to be overlapped if they overlay or touch one another. The a value so determined was 0.0977. For this a and f i = 180, theory predicts that 122 singlets, 19 doublets, 2 straight-chain triplets, and 1 interlocking triplet should be observed. These predictions are in very good agreement with the observation of 119 singlets, 21 doublets, 4 straight-chain triplets, 1 interlocking triplet, and 1 quartet (p = 146 spots). The various multiplet spots in the figure are appropriately labeled. Others may question the assignment of a few of these spots (e.g., t,4), because some subjective judgment is required here.

This simulation clearly illustrates some of the painful realities of 2-D separations. Although more than 10 times the needed peak capacity is available, approximately one-third of the zones are lost in overlap. (Admittedly, not all of the overlaps are severe ones.) This loss is comparable to that found in the 1-D gas chromatography of complex mixtures, in which only 2 times or so more peak capacity (as defined for maxima) is available than is needed (36). One really does lose resolving power, per unit peak capacity, when one moves from one dimension to two dimensions. Figure 14 is a graph of the probability pl(e) that an elliptical zone is a singlet vs a for various aspect ratios y. The bold curve corresponds to y = 1, for which p,(e) = p1 = exp(-4a), in accordance with eq 9. The probability of singlet formation clearly increases with increasing y. This behavior can be rationalized, if one thinks of an ellipse of finite area and very large aspect ratio as approximately equivalent to a straight line of infinitesimal thickness. Such an ellipse would not overlap with any of the infinitesimally thin ellipses paralleling it, because any small differences among the positions of the zone centers would be enough to prevent overlap. In this limiting case, pl(e) would approach unity. One finds a similar behavior with large but finite aspect ratios. Equation 32 indeed is unchanged by the substitution of l l y for y. Over the range 0 < a < 0.1, in which theory is largely free of errors, the difference between eqs 9 and 32 is less than 2%, when y < 2.0. Hence, one would expect the equations for doublets and triplets also to apply to elliptical zones with aspect ratios less than 2 over this a range. To confirm that these equations indeed apply to such zones, the simulation presented as Figure 13 was replotted as Figure 15, in which the circular zones have been replaced by elliptical ones with aspect ratios equal to 2. Otherwise, the figures are identical (e.g., a = 0.0977). In this bed, 122 singlets, 21 doublets, 3 straight-chain triplets, 1 interlocking triplet, and 1 quartet are found (the various multiplets are labeled in the

ANALYTICAL CHEMISTRY, VOL.

figure). As expected, these numbers do not substantially differ from those predicted above for circular zones. The only changes are that spot t,4 in Figure 13 has evolved into the doublet d l and a singlet in Figure 15, and doublet d9 in Figure 13 has evolved into two singlets in Figure 15. CONCLUSIONS The principal conclusion one must draw from this study is that 2-D separations, although very powerful, are not as good as one would like to think, a t least when m is large and statistical considerations apply. Per unit peak capacity, they are actually worse than what one can accomplish in a single dimension. The examples shown in Figures 13 and 15, in which roughly a third of all zones are lost by overlap even though 10 times the needed separation space is available, are most probably the norm and not the exception. This study consequently should dampen our spirits somewhat and force us to examine reality with a more sober perspective. I t also should challenge us to find more systematic means for introducing order into such separations to avoid the unpleasant consequences of statistics. As with the SMO, one must recognize (and I clearly do) that these predictions of theory are a “worst case” senario and that not all 2-D multicomponent separations will be subject to the limitations suggested here. Indeed, when m is relatively small, the positions of zone centers can be fairly ordered by choosing careful separation conditions (25). This order tends to disappear, however, as more components are added to the bed (25,26,28). Each separation must be evaluated on an individual basis. It will be argued by some that the deliberate experimental development of a uniform zone density in two dimensions is not easily achievable. In such cases, the theory developed here will not apply to a full bed but only to (at best) several small sections in the bed, in which zone densities are roughly constant. One suspects that this objection is rather immaterial for large m, because in practice the theory can be applied independently to each section, as long as each section contains several zones. This approach has been used successfully in one dimension (IO)and one foresees no difficulty in its use in two dimensions. From another perspective, whether or not the theory is rigorously applicable to any real-world bed is not as important as the overall trends suggested by the theory. It should be clear that chaotic and disordered distributions of zones in 2-D beds will lead to substantial overlap, regardless of the actual uniformity of zone density in the bed. I anticipate that in the future someone may theoretically justify the assignment of the relative probabilities, 3/a and 2/5, to the t,-like and t,-like patterns shown in Figures 5 and 7. Here, these assignments were based on empirical evidence. The simplicity with which the fractions of ti-like patterns vary with a in Figures 6b and 8b suggests that this behavior can be interpreted analytically. For the moment, however, one is simply pleased to have the expressions for t , and ti. Because of the length of the Theory section, only a few representative applications could be incorporated into this work. Additional work presently is underway to test the application of this theory to a variety of m values, including small ones (e.g., m = 26)) to estimate R by fitting values of p and n, to theory, to investigate the overlap of elliptical zones of various aspect ratioe, and to consider other practically useful scaling parameters 8. Prior to closing, I should observe that, after the submission of this work, one of my graduate students (Frances J. Oros) found a short book entitled The Theory of Random Clumping (Methuen & Co., Ltd., London, 1968) and authored by S. A. Roach. This work, which was unknown to me, was published 23 years ago in the health hygienics community and contains

63,NO. 19, OCTOBER 1, 1991

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several thearetical studies of phenomena closely resembling the overlap of zones in 2-D beds. Most of these theories are inferior to that developed here. A theory developed by Roach himself, however, shows more promise in correctly describing overlap at high saturations (e.g. a > 0.1) than does the theory derived here. Future studies will make ample use of Roach’s theoretical efforts. GLOSSARY A bed area scaled area of circular zone A0 d number of doublets d0 diameter of circular zone distribution function for distance to kth nearest fk(r) neighbor m number of detectable zones ni statistical approximation to m nc peak capacity P spot number probability of forming an n-tet spot P” probability of forming a singlet spot, which is elpl(e) liptical probability that n zones lie within the circle of P(n) overlap r radial coordinate radial coordinate defining ellipse of overlap re distance to kth nearest neighbor rk (r) average distance between zones resolution factor in 1-D separations R,* S number of singlets t total number of triplets ti number of interlocking triplets 4 number of straight-chain triplets XO width of peak in 1-D separation (Y r W d o ) 2 14 scaling parameter for circle or ellipse of overlap P aspect ratio Y 8 angle defined by Figure 9b x zone density ACKNOWLEDGMENT I thank Frances Oros of Southern Illinois University for help with the computer simulations. LITERATURE CITED (1) Davls, J. M.; Qlddlngs, J. C. Anal. Chem. 1083, 55, 418. (2) Rosenthal, D. Anal. Chem. 1082, 54, 63. (3) Nagels, L. J.; Creten, W. L.; Vanpeperstraete, P. M. Anal. Chem. 1883, 55, 216. (4) Martln, M.; Herman, D. P.; Gulochon, G. Anal. Chem. 1006, 58, 2200. (5) Fellnger, A.; Pastl, L.; Dondl, F. Anal. Chem. 1000, 6 2 , 1846. (6) Glddlngs, J. C.; Davls, J. M.; Schure, M. R. I n Uhahlgh Resolution Chromatography; AhuJa, S.,Ed.; ACS Symposlum Serles 250; Amerlcan Chemlcal Society: Washlngton, DC, 1984; p 9, (7) Davls, J. M.; GMdlngs, J. C. J . Chromatogr. 1084, 289, 277. (8) Herman, D. P.: Qonnard, M. F.; Qulochon, Q. Anal. Chem. 1084, 58, 995. (9) Davls, J. M.; Qlddlngs, J. C. Anal. Chem. 1085, 5 7 , 2168. (10) Davls, J. M.; Glddlngs, J. C. Anal. Chem. 1085, 5 7 , 2176. (11) Dondl, F.; Kahle, Y. D.; Lodl, Q.; Remelll, M.; Reschlgllan, P.; Blghl, C. Anal. Chlm. Acta 1088, 191, 261. (12) Coppl, S.; Bettl, A.; Dondl, F. Anal. Chlm. Acta 1088, 212, 165. (13) Davls, J. M. J. Chromatogr. 1008, 440, 41. (14) Dellnger, S. L.; Davls, J, M. Anal. Chem. 1000, 6 2 , 436. (15) Martln, M.; Qulochon, Q. Anal. Chem. 1085, 5 7 , 289. (16) Creten, W, L.; Nagele, L. J. Anal, Chem. 1087, 5Q, 822. (17) Schure, M. R., J. Chromatogr. 1001, 550, 51. (le) El Fallah, M. 2.; Martln, M. 17th Internatlonal Sympoilum on Chromatography, Vlenna, Sept 25-30, 1888. (19) Kleln, P. D.: Tyler, 5. A. Anal. Chem. 1086, 3 7 , 1280. (20) Connors, K. A. Anal. Chem. 1074, 46, 53. (21) Schure, M. R. Personal communloatlon, 1991. (22) Qlddlngs, J. C. HRC 6 CC, J . H@h Re8olutkm Chromtogr., C h m m togr. Commun. 1087, 10, 318. (23) Qlddlngs, J. C. UnlfM Separatbn Scknce; WLy-Intersolenor: New York, 1991. (24) Martln, M. Personal communlcatlon, 1991. (25) Zakarla, M.; Qonnord, M.-F.; Qulochon, G. J . Chromatogr. 1083, 271, 127. (28) O’Farrell, P. H. J. Blol. Chem. 1075, 250, 4007. (27) Scheele, Q. A. J. Bbl. Chem. 1075, 250, 5375. (28) Anderson, N. 0.; Anderson, N. L. Anal. Blochem. 1078, 85, 331. (29) Qulochon, 0.; Beaver, L. A.; Gonnord, M.-F.; Slouffl, A. M.; Zakarla, M. J . Chromatogr. 1083, 255, 415.

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(30) Gubchon, 0.; Gonnord, M. F.; Zakarla, M.; Beaver, L. A,: Siouffi, A. M. Chmmatographia 1983, 17, 121. (31) Bushey, M. M.; Jorgenson, J. W. Anal. Chem. 1990, 62, 978. (32) Llu, Z.; Phillips, J. J. Chromatogr. Scl. 1991, 29, 227. (33) Vanmarcke, E. Random Fields: Analysls and Synthesis; MIT Press: Cambridge, MA, 1983. (34) Tuckwell, H. C. Nementary Applications of Probablllfy Theory: Chapman and Hall: London, 1968. (35) Giddlngs, J. C.; Keller, R . A. J. Chromatogr. 1959, 2 , 626. (36) Oros, F. J.; Davis, J. M. J. Chromatogr. 1991, 550, 135. (37) Green, J. R.: Margerlson, D. Statistical Treatment of Experimental Data ; Elsevier: Amsterdam, 1978.

(38) Ross, S. M. Introduction to FTobabiiIfy Models, 2nd 4.:Academic Press: New York. 1980. (39) Thomas. G. B., Jr. Calculus end Analytic Geomeby, alternate ed.;Addison-Wesley: Reading, MA, 1972. (40) Nyhoff, L.; Leestma, S. FORTRAN 77 for Engineers and Scientists; Macmillan: New York, 1965. (41) Dahiqulst, G.; BJGrck, A. Numerical Methods; PrentlcaHall, Inc.: Englewood Cliffs, NJ. 1974.

RECEIVED for review March 8,1991. Accepted June 28,1991.

Simultaneous Supercritical Fluid Derivatization and Extraction J. Ward Hills and Herbert H. Hill, Jr.*

Department of Chemistry, Washington State University, Pullman, Washington 99164-4630 Tsuneaki Maeda

DKK Company, Ltd., Kichijojikitamachi, Musashino, Tokyo 180,Japan

Slmultaneous supercrltlcal fluld derlvatlratlon and extractlon (SFDE) was Investlgated as a rapld alternatlve to llquld solvent extractlon followed by derlvatlzatlon for analysis by gas chromatography (GC). AddRlon of a commercially avallable sllylatlon reagent, trl-sll concentrate, dlrectly to the sample matrlx, enables extractlon of analytes from samples that were prevlously exhaustlvely extracted wlth conventlonal supercrltlcal fluld extractlon (SFE). The SFDE extracts were collected, ready for GC analysls without addltlonal derlvatizatlon, not only resultlng In tlme savlngs but also In Improved extractlon yleld of both derlvatlzed and underlvatlzed species. I n addltlon to maklng polar compounds more soluble In the supercrltlcal extractlng fluld through derlvatlratlon, the derlvatlzatlon reagent Is thought to compete wlth the analyte for actlve Snes of the matrix, thus displaclng compounds from the matrlx. Appllcatlon of SFDE for the extractlon and analysls of roasted coffee beans, roasted Japanese tea, and marine sediment Is demonstrated.

INTRODUCTION The unique chemical and physical properties of supercritical fluids have attracted considerable attention during the past decade for use in analytical chemistry. Supercritical fluid chromatographic methods have extended the high-resolution separation of semivolatile and thermally labile compounds beyond the limit of gas chromatography, entering the domain of liquid chromatography (I).In addition, analytical-scale supercritical fluid extraction (SFE) has emerged as a rapid and efficient method for extracting organic compounds for separation and detection by gas or supercritical fluid chromatography (1-4).The primary advantage of SFE over liquid-phase extractions is the lower viscosity and variable density of the supercritical fluid (SF). Mass transfer occurs more quickly and efficiently in supercritical fluids than in liquids. Driven by the desire to limit toxic solvent waste, reduce analysis times, and increase extraction efficiencies, the practice of SFE is growing a t such a rapid rate that within several years the use of traditional liquid-phase extractions may be the exception rather than the norm.

* To whom correspondence should be addressed. 0003-2700/91/0363-2152$02.50/0

Although carbon dioxide SFE is rapidly becoming recognized as an efficient and time-saving alternative to Soxhlet extraction for trace organic analysis, carbon dioxide is limited as an extracting solvent by its nonpolar character. C o t SFE extraction efficiency of polar and ionic organic species can be poor. More polar supercritical fluids require critical conditions that are too extreme for practical utility. For example, water with a critical pressure of 226.3 atm and a critical temperature of 374.4 O C has proved technically too difficult for practical analytical-scale application. Generally, SFE conditions have been empirically determined with emphasis placed on the widely variable solvent strength of supercritical fluids (3-6). In contrast, a recent review by Hawthorne (1)and results reported by Alexandrou and Pawliszyn (7) call attention to the matrix from which the extraction is being made. Alexandrou and Pawliszyn’s extractions of tetrachlorodibenzo-p-dioxins(TCDDs) were made from fly ash with SF C02 and with SF C02 after treatment of the fly ash with acid. The acid treatment displaced adsorbed TCDD from polar active sites on the fly ash, resulting in improved extraction efficiencies (9% without acid treatment, 100% with acid treatment). Some compounds in the study could not be extracted at pressures of 400 atm before treatment with acid. They concluded that the extraction was limited, not by the solubility of the solute in supercritical COZ but rather by the desorption process of the compound from the active site. Improved solvent characteristics and extraction efficiencies can be obtained if polar solvents such as methanol are added to the supercritical carbon dioxide (8). Several investigators have used polar modifiers (most commonly methanol) mixed in the supercritical fluid, increasing the solvent polarity and extraction efficiency. Modifiers are thought to aid extraction in two ways: increasing the solvent strength of the fluid as well as competing with adsorbed compounds for active sites of the matrix. Extraction is a partitioning of the compound between the matrix and the solvent. Clearly the equilibrium is driven to favor solvation of the compound either if the compound is made more soluble in a relatively nonpolar supercritical fluid or if the compound is driven off the matrix. Like adding modifiers, addition of derivatizing reagents may drive the equilibrium in both of these ways. Covering polar moieties by derivatization will make the compound more soluble in SF 0 1991 American Chemical Society

Statistical theory of spot overlap in two-dimensional separations

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