Guidelines for selection of chromatographic conditions - Analytical

John A. Thoma and Clarence R. Perisho. Anal. Chem. , 1967, 39 (7), pp 745–755. DOI: 10.1021/ac60251a035. Publication Date: June 1967. ACS Legacy ...
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Guidelines for Selection of Chromatographic Conditions John A. Thoma’ and Clarence R. Perisho2 Deparrment of Chernistr y , Indiana University, Bloomington, Ind., and Department of Chemistry, University of Arkansas, Fayetteoille, Ark. An index of resolution, IR, is defined as the distance between spot centers,, AR, less the sum of the semimajor axes of two close-lying spots. Simulation studies of polar chromatographic systems indicated that IR is a maximum in the locale of R, = 0.25. The theoretical computations were confirmed by chromatographic investigation of the effective ness of various solvents in separating glucose and fructose. On the basis of these studiies recommendations are made for the systematic design of chromatographic conditions of high resolving power.

IN THE PAST as empirica I information about a large number of chromatographic systems accumulated (I+, rules-of-thumb were formulated for the design of solvents t o separate unknown mixtures. These rules have proved very useful as practical guides but are not sufficiently systematic to exploit some of the hidden potential of the chromatographic technique. For example, if the locale on the support where optimum resolution of simjde known and unknown mixtures occurs could be predicted, the selection of chromatographic conditions could be great iy expedited. Because relative band migration rates, and hence resolution, can often be controlled in part by changing solvent proportions, a successful solvent search might simply involve altering the solvent proportions until they migrate the solutes into that locale. The locale of maximum resolution can be predicted if the separation between spot centers a s well as the extent of zone spreading can be quantitatively related to R f . The key relationships useful for these predictions were derived from our earlier studies on the thermodynamic behavior of polar chromatographic systems ( 5 ) and the work of Giddings and Keller (6) relating spot size to the diffusion characteristics, We discovered that for polar solutes and solvents, the ratio of the chemical potentials of transfer of structurally related compounds was constant arid independent of solvent proportions within experimental error (5, 7). Giddings and Keller have shown that spots are noi~nallyelliptical, and their size depends upon several dispersive “actors and the sensitivity of the detection method (2). Because the extent of resolution depends not only upon disengagement of spot centers but also upon band spreading, we have defined an index of resolution, IR, as the distance between spot centers less the sum of the semimajor axes of the elliptical spots. Positive values of 1, accompany distinct Present address, Department of Chemistry, University of Arkansas, Fayetteville, Ark. 72701. Present address, Department of Chemistry, Mankato State College, Mankato, Minn, 56001. (1) J. M. Bobbit, “Thin Layer Chromatography,” Reinhold, New York, 1964. (2) D. E. Durso and W. A. Mueller, ANAL.CHEM.,20, 1366 (1958). (3) E. Lederer and M. Lederer, “Comprehensive Biochemistry,” M. Florkin and E. H. Stotz, eds., p. 151, Elsevier, New York, 1962. (4) J. A. Thoma and D. French, ANAL.CHEM.,29, 1645 (1957). (5) J. A. Thoma, ANAL.(:HEM.,37, 500 (1965). (6) J. C. Giddings and R . A. Keller. J . Chromaroa.. 2 . 626 (1959). (7) C. R. Perish;, A. Rover, and J:A. Thoma, ANAL. CHEM.,39, 737 (1967).

spots, while negative values of I R accompany overlapping spots. We prefer I , to another recommended index (8, 9) of separation, AR f/4u (for symbols consult appendix), because the former index takes cognizance of the spot load and sensitivity of the detection method whereas the latter does not. In this manuscript we have demonstrated that ZR achieves a maximum value a t approximately 1/4 the distance from the origin to the solvent front. Theoretically, this conclusion is valid for a very wide variety of conditions and is supported by experiment. Some suggestions are also afforded for the design of solvents which can achieve the desired migration rates. It must be remembered throughout the following discussion that all dimensions are normalized t o unit support length and that their absolute values will be proportional t o support length; thus their relative values are reported as dimensionless units comparable to R,andARf. MODELS

ARf as a Function of Rfi In preceding articles (5, 7) it has been amply demonstrated that Martin’s relationship (IO) (Equation 1) is extremely useful for describing chromatography of polar solutes with polar solvents on paper.

It is also known that for a given set of solvent components, both Rf and ARf can be regulated within limits by changing solvent proportions. Thus the influence of solvent proportions on ARf can be estimated from Equation 1 if AM/As is reasonably constant and if the dependence of A p on NE,Ofor the two solutes is established. This functional relation is usually not known, but is not necessary if Rf9is a known function of Rfl. Because ARf

=

Rig - R f ,

(2)

the disengagement of spot centers can be calculated by solving Equation 1 for R,, and substituting into Equation 2. This gives ARf

1 =

1 -I-

A,

- Rfl

(3)

ex~Lf(Afii)/RTl

where A ~ = z f(Ap1) is the functional relationship between Ap2 and Apl. For many polar systems Apz is directly proportional to A p l (5, 7). Thus the functional relationship becomes b z = .f(Api) = B Api

(4)

(8) J. C. Giddings, “Dynamics of Chromatography, Part I. Prin-

ciples and Theory ” Marcel Dekker, Inc., New York, 1965. (9) M. J. Johnson, “Manometric Techniques,” W. W. Umbreit, R. H. Burris, and J. F. Stauffer, eds., Chap. 13, Burgess, Min-

neapolis, 1964.

(10) A. J. P. Martin, Biochem. Soc. Symp., 3, 4 (1947). VOL 39, NO. 7, JUNE 1967

a

745

Our model assumes that A p g i A p i i s constant and differs !) which assume that the ratio of the partition coefficients, L Y ~ L Yis~ sonsrant. , Our rather extensive studies in carbohydrate chromatography ( 5 , 7 ) leave little doubt that our model more accurately characterizes these systems. On the basis of our experience and other reports (12,13) there is reason to believe that our model for ARf cs. Rf can sometimes be successfully applied to other systems as well. Nevertheless, before blindly applying the suggestions proposed below to new systems they should be tested for conformity to the model (see below) (5). For the special relation given in Equation 4 for polar systems AR, can be written as an explicit function of Rfi. After dropping the subscript, the formula is

from other models (9,

ARf

=

RrB(1 - Rf) (AM/As)l-’ - R,(1 - Rf)B R~B(A~w/A~)’--B (1 -

+

(5)

where B = Apz/Apl. According to Equation 5, AR, will have the value of zero when Rf

=

zero, one, or

sign when Rf

A.w/As

1

+ A.w/As .

Because ARf changes

AM’As

+

the slower band below this point 1 A.w/As’ becomes the faster band above. This inversion of relative migration rates occurs when the R f values of the two bands are equal and AR,shifts from positive to negative at the critical Rf A MI As value of 1

=

+ A,v/As’

Thus AR, has both a maximum and a minimum, while \ARfl has two maxima. Because in practice AM/As > 1,

iog,

i - tzf ,?

tiuiial to 1 - A p ~ / A p and l is a linear function of log, A M / A s . Thus a change in relative migration rates may be induced by a ~ , latter change in AAMIAsas well as by a change in A P ~ I A Pthe being more effective. These relations are illustrated in Figures 1 and 2. Spot Size as a Function of R p To facilitate discussion of band spreading, we define a zone as an area which contains some definite but arbitrary fraction, say 90 %, of the total solute. A spot, the observed area, corresponds to the portion of the zone containing solute above some visually detectable limit. At the boundary of the spot, the fraction of the solute per unit area will be d,/n where d, is the minimum detectable concentration of solute and n is the total amount of solute. It must be emphasized that, except for initial spot size, reference to zone or spot size always refers to terminal size after irrigation has ceased. We are in no instance concerned with the shape of the band as it moves along the support. Thus, changing the Rf may cause the area occupied by the terminal zone or spot to be larger or smaller than before. If material is allowed to diffuse in one dimension from a point origin, the concentration profile of the solute assumes a Gaussian distribution (15) with a variance of 2Dt. If diffusion from a point is two-dimensional, the concentration profile of the solute assumes a bivariate Gaussian distribution with zero correlation (6, 16). When diffusion in the two directions is unequal, the contour defining a zone or spot is elliptical. When solute distribution is initially Gaussian with standard deviations of uzoand uuo,the effective diffusion coefficients in the two directions are

+ 2D,t

1

+

AM’As lies beyond Rf = 0.5 and the maximum of ARf/ 1 AMAS closest to the origin is the larger of the two and more useful for purposes of separation. This phenomenon is illustrated in Figure 1. Because the laboratory worker can attempt to increase A R f by manipulating both Au/As and Ap2/Ap1 = B, it is important to estimate the relative influence of these parameters on disengagement of spot centers. Rearranging Equation 5 , we have

= 0. For a given value of R,, A R f is propor-

I

u Z 2= uZo2

(84

and

+ 2D,t

u,Q = uYo2

Under these conditions the semimajor axis (in the direction of solvent flow) is a =

[

+

2(aZo2 2 W log,

n

+

I/ 2

+

2 ~ d ~ d ( 2D,t) u ~ (u,,~~ ~ ~ 2G1

(9a) and the semiminor axis (at right angles to solvent flow) is (6)

b =

If the factor of Rf-’ - 1 in the denominator is approximated x-1 by 1, we can apply the approximation (14) -= ‘12 logex

x f l

(9b)

which gives ARf

‘/z (1

- Rf) (1

- Apz/Api) X

This approximation is best when Rf is near

and

(11) M. A. Jermyn and F. A. Isherwood, Biochern. J., 44, 402 (1949). (12) E. Soczewinski, J . Chromatog., 8, 119 (1962). (13) E. Soczewinski and C. A. Wachtmeister, Zbid., 7, 311 (1962). (i4) S. M. Selby, “Handbook of Chemistry and Physics,” 46th cd., p. A-192, The Chemical Rubber Co.. Cleveland. i965. 746

ANALYTICAL CHEMISTRY

These equations will be more useful when D , and D , are expressed as functions of R The effective diffusion coefficients, D , and D , in Equations 8a and 8b, are actually comprised of dispersive effects arising from three different sources. They are ordinary molecular diffusion (Dm), eddy “diffusion” (De), and nonequilibrium “diffusion” (Dne) arising from the kinetics of partitioning. Because variances (in our case, 2 Dt) are additive, the effective diffusion coefficient, D , or D,, is simply the sum of the (15) J. Crank, “The Mathematics of Diffusion,” p. 10, Clarendon Press, Oxford, 1956. (16) A. M. Mood and F. A. Graybill, “Introduction to the Theory a f Statistics,” 2nd ed., p. 198, McGraw-Hill. New York, 1963.

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Figure 1. Theoretical variation of A R , us. R , of solute having largest IApl for different conditions In each figure AM/As remains constant and each curve corresponds to a different value of Apz/Ap1 Figurle and corresponding AM/As: 1-A, 1; 1-By2; 1-C,4; 1-D, 8; 1-E, 16; 1-F, 32 Curvc:number and corresponding A p z / A p ~ :1,0.850; 2,0375; 3, 0.900; 4,0.925; 5,0.950; 6,0.975

VOL 39, NO. 7 , JUNE 1967

747

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-LOG. A S I A #

Figure 2. Theoretical variation of AR, us. log, Aar/As in region of maximum separation In each figure the Rfof one component is held constant and each curve corresponds to a different value of A p t / A p ~ Plot and corresponding Rj of slower component: A, 0.10; B, 0.15; C, 0.20; D, 0.25; E, 0.30; F,0.35 C w e number and corresponding ApdAp, as in Figure 1

748

ANALYTICAL CHEMISTRY

I

1

I

,

..

..,,.I

I

1

lime

Time

Figure 3. Variation of the semiaxis of a zone with time

-

Figure 4. Variation of a semiaxis of a spot with time. Curves of this general shape are generated by Expressions 9a and 9b

a = ku = 4 2 D t

apparent coefficients, D,, De, and D,,, multiplied by an appropriate function or R assigned to each spreading factor. Now assume that molecular diffusion occurs both in the mobile and stationary phases but a t different rates and that eddy diffusion is restricted to the mobile phase. Recall that the fraction of the time that a solute resides in the mobile phase is Rland the fraction of’the time that it resides in the stationary phase is 1 - Rf. Thus, R f a n d 1 - R f a r e the appropriate functions for diffusion in the mobile and stationary phases, respectively. According to the theoretical plate model of Martin and Synge, the nonequilibrium spreading is a binomial distribution (of which the Gaussian distribution is a limiting case) (9, 17). This binomial distribution, when normalized and expressed in units of paper length, has a variance (18) of

If N , the number of thcoretical plates, is sufficiently independent of R f,the approprkte function for nonequilibrium spreading is RX1 - Rl). Assuming nonequilibrium “diffusion” occurs only in the direction of solvent flow, the effective diffusion coefficients are approximately

Dz

=

D,

+ DneRXI - R/)

(114

and

D,

D ~ ( M ) Rt/ Drn(8) (1 - Rf)

+ DeR1

Ulb)

If all scalar quantities are normalized by expressing them in units of paper length, and the time of solvent excursion is set at unity, the effective Liffusion coefficients acquire dimensions of (paper length)2/unit time. Equations l l a and l l b , to be sure, are only approximate for a given set of solvmt components because any manipulation of solvent proportions will influence viscosity, surface tension, time of development, etc., which in turn influence the effective diffusion Coefficients. However, provided these values are not widely variant and the diffusion coefficients are average values, Equations 1l a and 11b are of the proper form and should enable Equations 9a and 9b to predict a reasonable approximation of spol size. We can reduce the number of experimental constants

needed to describe zone spreading by noting that D, is a linear function of R,; thus, on combining, the diffusion coefficients become D,

(1941). (18) R. S. Burlington 2nd D. C . May, Jr., “Handbook of Probability and Statistics with Tables,” p. 68, Handbook Publishers. Sandusky, Ohio, 1958.

D’+ DRf

( W

and Dz

=

Du

+ DneRf (1 - Rj)

(12b)

It will be shown later that D, for paper chromatography is primarily dependent on D,(.,,). Thus, in practice, solute spreading can be defined by two constants, D, = D = DrncM), and D,,, which take on values characteristic of each chromatographic system. Because the time of development is set equal to unity, time disappears from our equations as an explicit variable but in reality is present concealed in the constants D and Dne. Zones and spots exhibit quite different behavior. The semiaxes of a zone are proportional to the standard errors of the solute distribution and expand with the square root of time. This dependence is illustrated in Figure 3. Although zone size increases indefinitely with time, the spot size as defined by Equations 9a and 9b increases with time, passes through a maximum, decreases, and eventually vanishes (Figure 4). This phenomenon becomes intelligible when it is recognized that dispersion of the band constantly reduces the concentration per unit area. When concentration drops below the limit of detectability, the spot vanishes. However, at constant development time, typical of actual experimental conditions where R , is the variable, an entirely different situation is encountered. In this case, diffusion is a quadratic function of Rf so that both zone and spot size may exhibit a maximum or minimum when plotted us. distance traveled. Again, as when time is variable, there is no necessary correlation between zone and spot dimensions (see Figure 5). When the original solute is not concentrated at a point (as is assumed in Equations 9a and 9b) the curve is of the same shape, but the origin of the time axis is shifted to the right so that the spot has finite size at zero time. When the area of a spot is maximum, the concentration at the center of the spot is equal to e = 2.7 times the concentration at the edge of the spot. That is, do,,,,

(17) A. J. P. Martin arid R. L. M. Synge, Biochem. J., 35, 1358

=

=

dux = ed,

= 2.7 do

(13)

where dois the minimum concentration which can be detected. Thus, when a spot reaches maximum size, the concentration profile is very flat so that determining exact dimensions is difficult. VOL. 39, NO. 7, JUNE 1967

* 749

EXPERIMENTAL

0.08

z)

Solutions of glucose and fructose (0.1 were spotted on Eaton and Dikeman No. 613 and irrigated with tert-butanol and water of varying proportions at 25.0 & 0.1 O C for 25.0 f 0.7 hours (5). The papers were spotted with a 0.25- or 0.30-p1 pipet. The papers were dried, and the spots were located by the silver nitrate-sodium hydroxide dip technique (19). To estimate do, the limit of detectability, 1-p1 volumes of solutions of different concentrations of sugars were spotted, the papers dried, and the color was developed with the silver nitrate-sodium hydroxide reagents. Chromatography of amino acids was conducted in various solvents under conditions described elsewhere (5). The effect of molecular diffusion in the stationary phase was ascertained by spotting papers with 0.25 pl of 0.1% sugar and then suspending the paper in a water-saturated atmosphere for 24 hours followed by drying and the silver nitrate-sodium hydroxide treatment. To estimate D m ( M ) , the svzar was spotted as above in the center of a 4-cm filter paper strip which was dipped into a water: tert-butanol solution (&,o = 0.5) until the liquid had just migrated past the sugar. The paper was removed from the solvent and blotted to remove excess liquid, then inverted and inserted into the solvent until the rising liquid reached the existing solvent front. Excess liquid was removed by blotting, and the paper was then hung in a solvent saturated atmosphere for 22 hours before spot developing. The disengagement of spot centers under varying chromatographic conditions was simulated on a digital computer. A computer program was used for a systematic study of the influence of the two chromatographic parameters on AR,. At first it was assumed that AM/A.$and B = Ap2/Ap1were constant. The values of AMjAs used were 2, 4, 8, 16, and 32; Apg/Ap1was varied from 0.85 to 0.975 in increments of 0.025. For given values of the two parameters R,, was varied from 0.02 to 0.98 in increments of 0.02. The computation was as follows: first, Apl was computed from Equation 1 by substituting a value of R,,; second, the value off(Ap1) = B Apl was obtained from Equation 4; and third, R,, and f(Ap,) were substituted into Equation 3 to give AR,. This produced data for one curve. One of the parameters was changed and the process was repeated using all combinations of variables. The graphs of Figure 1 were plotted by the computer under the Michigan State Plot subroutine and the output was printed “on-line.” Another set of computations was performed where the d Ap,. restraint on Ap2/ApLwas relaxed so that Apz = c The constants c and d were selected so that Apz/Apl did not deviate greatly from a constant value. Plots of ARf 6s. log, A M / A swere computed and printed employing the same logic as that described above. In this case each line represents fixed values of ApzIApI and R,,, while log, A M / A swas varied (Figure 2).

+

RESULTS

Separation of Spot Centers. In the laboratory ARj for a given pair of solutes can conveniently be altered by changing solvent components, varying solvent proportions, or altering the stationary support. The effects of these changes on separation of spot centers can be simulated in the digital computer by adjusting the values of the chromatographic and AM/As. The results of this simulation indexes, Ap2/ApL1 over a wide range of conditions are presented in Figure 1. Inspection of this figure reveals that the locale where A R f is maximum lies in a narrow Rf range, 0.2-0.3, over a very wide range of conditions. Remember that the independent (19) W. E. Trevylan? D. P. Procter, and J. S. Harrison, Nufure, 166, 444 (1950?.

750

ANALYTICAL CHEMISTRY

0.06 0.04

0.02 0.000.0 0.2 0.4

0.6

0.8 1.0

1 , , , , 1 0.2 0.4 0.6 0.8 I O

o‘ooO.O

0.06

004

0.04

1000

500 0.02

0.0 0.2 o’ooO.O

0.4

0.6

0.8

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08

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0 06

0.06

0 04

0.04

0 02

0.02 ooO.O

0 00

02

0.4 0.6

Rf

Figure 5. Theoretical variation of semimajor axis (in direction of flow) with R , under a variety of conditions. The numbers on the curves correspond to the value of K The following constants were used in the computations: Graph A B C

D E

F

D,

QO

0

0.002 0.028 0.00144014 m.01 0.002

6

x

10-4

0 0 0 2 x 10-4 2X

=

2rd0/n.

K 0 0.001 0.001

62.5-2300 62.5-2000 62.4-1000

0.001 0.001 0.001

1000

250 62.5-2000

variable is the R f value of the slow migrating compound and not tne average R f . The locale of optimum disengagement of spot centers is more sensitive to the area ratio than to the transfer potential ratio. As the A R f us. Rjcurves have a broad top, it can be shown that, under the conditions examined, A R f will never deviate more than 1 2 z from its maximum value if R f of the slower component is set at 0.25. In the great majority of cases, A R f will be within a few per cent of the maximum. As the effect of zone spreading has not been considered yet, it cannot be concluded that the position of optimum resolution coincides with the position of maximum A R f . It will be shown later, however, that A R f serves as a very useful index of merit for resolution. When the constraint that A ~ z / isAconstant ~ ~ is relaxed so that Ap2 and Apl are related linearly, the plots of A R f US. R f were very similar to those in Figure 1 (so long as Ap?/Apl was approximately constant). The locale in which maximum A R f resides was not significantly altered, but the extent of separation was somewhat larger than when B was constant. Experimental data for A R f us. R f for various pairs of saccharides and amino acids are displayed in Figure 6. The filled circles were calculated by extrapolating or interpolating plots of RM us. NHz0. The similarity between the experimental (Figure 6) and theoretical (Figure 1) curves is striking and further substantiates the underlying assumptions (5) made in developing the formulas. In accord with prediction, AR, is largest when the trailing band has an R , = 0.25 (see dashed vertical line in Figure 6).

I

*f

Figure 6. Experimental variation in ARf us. Rf of slower component for pairs of carbohydrates (7)and amino acids (20) for different conditions For the upper plot, the paper-solvent combination for all curves was acetone:H20

on Whatman 4. The carbohydrate pairs used were: A, lactose-galactose; B, ribose-fructose; C,fructose-galactose; D, mannose-galactow; E, fructose-mannose For the lower plots, the amino acid pair, solvent, and paper used for each curve were: A , leucine-glycine pair using H20 :dimethylformamide on Eaton and Dikeman No. 613; B, phenylalanine-tyrosine pair using H20 :tert-BuOH on Eaton and Dikeman No. 613; C,threonine-serine pair using H2O:acetone on Eaton and Dikeman No. 629; D , phenylalanine-methionine pair using H20 :dimethylformamide on Eaton and Dikeman No. 613 The influence of loge A M / A Son AR? at constant R f in the range 0.1 to 0.35 is illustrated in Figure 2. Changing Ap~/Apl generates a series of curves approximating a pattern of intersecting lines described by the following parametric equation: ARy = 0.22 (1

--

*)

(log.

APl

4 + 2 - 4Rf) AS

/

(14)

which may be compared to the approximation, Equation 7. The lines in Figures ;'-e,2 - 0 , and 2-E, fit by Equation 14, illustrate the accuracy of this approximation. Actually the coefficient of Equation 14 is not constant at 0.22 but is slightly dependent upon R f , so that the fit in plot 2-C and 2-E could be improved if the coefficient were suitably altered. Because the paramders A M / A sand Apz/Apl are normally unavailable, Equation 14 cannot be used for evaluating AR?. Nevertheless, the relation has instructive value because it quantitatively defines the sensitivity of ARf to these parameters. Because ARf is linearly related to the logarithm of the area ratio, a huge change in AM/Asis required to produce even a small change in the absolute value of ARf. On the other hand, a minor decrease in Apz/Apl (which is near unity) can markedly increase ALi f whose absolute value is proportional to [I - (Apz/Ap~)]. For example, a small decrease in the transfer potential ratio from 0.95 to 0.90 would increase AR? by 100%. By way of contrast, a 10-fold increase in A M / A S .From 3 to 30 is needed. to achieve the same result.

Solute Spreading and Spot Sue. The scatter of the experimental data precludes an accurate evaluation of the individual parameters associated with zone enlargement. For this reason, the treatment of our data is not aimed a t the precise determination of diffusion coefficients as much as it is intended to substantiate existing theory and t o aid in locating the region where ZR is maximum. Whenever possible we have grouped terms and combined coefficients to give expressions of simplified form. In order to calculate the terminal spot size as a function of R not only the apparent diffusion coefficients, but also uo and the ratio of the detection limit to the spot load are re2*d0 . quired. We have found it convenient t o use K = - in n place of this ratio. Increasing the sensitivity of the detection method-Le., lowering d,-has the same effect on K as increasing n, the initial spot load. For this reason, K is a more theoretically useful constant than the individual quantities, do and n. Experimentally it is difficult to measure the value of K accurately because the limit of detection of sugar with the silver nitrate-sodium hydroxide reagent is hard to gauge by visual inspection. Although greater precision might be obtained by densitometry, it would yield a value of K inappropriate for laboratory operation where judgments are based (20) D. P Burma and B. Banerjee. ?,, Indim Chem. S O ~ .28, , 135 (1951). VOL. 39,

NO. 7,

JUNE 1767

75'1

upon visual inspection of the chromatogram. When 1 11 of glucose is applied t o paper in decreasing concentrations, it is found that the spot has an average diameter of 0.56 cm and disappears in the range of 0.003-0.005 % sugar. This gives 0.03 mg X 10-3 ml do = a(0.28 cm2)

1 1

% 0.04

x

(15)

.-

Because the solvent traversed 22 cm during the run, the normalized dois 0.06 mg per square paper length. When 0.30 pl of a 0.1 solution is applied, the load, n, is 3 X mg, and K becomes = 1300 paper length2. Certainly the spot load may be increased by a factor of 10 to 15, so K may range from less than 130 to a n upper limit in the neighborhood of

1

=

1.2

10-4- mg cm

4000. Location of Maximum ARf and I R . We have shown that A R , is a maximum when R f is near 0.25. We now demonstrate that the maximum of I R = lARfI - a2 - al frequently occurs in the same region on the R f scale. This point can be

established after the general shape of the curves for the semimajor axes cs. R , under various conditions is calculated. The curves in Figure 5 computed from Equations 9a and 12a and b show the effect of uo(the standard error of the initial distribution of the applied solute), K = 27rd0/n,and the dispersion forces on the semimajor axes. The influence of molecular and eddy diffusion in the mobile phase on final spot size are illustrated in Figure 5-A where, under normal operating conditions, K assumes a value in the neighborhood of 1000. Because of the low velocity of the solvent on papergrams, eddy diffusion does not contribute significantly t o zone enlargement (21). It may be noted that even in this limiting case some of the curves ( K = 500 and 1000) are relatively flat in the region where A R , is a maximum. Figures 5-B and 5-C illustrate how nonequilibrium diffusion and u, control the length of the semimajor axis. Nonequilibrium diffusion tends to maximize the semimajor axis halfway along the support. However, if v0 is initially large, dispersion of the solute may so dilute the band that the spot may actually contract o r even disappear altogether (see Figure 5 4 . In practice, such dilute spots near the vanishing point are very difficult to observe and measure. Dispersion of a solute may lead either to spot enlargement or contraction. For small concentrated areas, dispersion will enlarge spots whenever the bulk of the solute is well above the critical concentration, do,required for detection. On the other hand, when the concentration profile is broad and a substantial amount of the solute is in the concentration range below do, additional dispersion may so dilute the solute that the spot actually diminishes in size. It is possible, therefore, for the initially larger and more dilute spots t o produce smaller terminal spots, a phenomenon demonstrated in Figure 5-L). The effects of a combination of various factors on spot length are shown in Figures 5-E and 5-F. Although most of the graphs in Figure 5 illustrate the effects of extreme conditions, Figure 5-F is typical of many plots that were constructed for combinations of constants that gave spot dimensions consistent with our experience. Many times when K = 1000 spot dimensions increase rapidly until R reaches approximately 0.15 and thereafter change very slowly until R f reaches 0.5. Because I R is the difference between A R and the sums of the semimajor axes, we predict from the simulation studies that the position of maximum resolution will occur in the area of maximum A R , or at a slightly lower R , values. (21) C. L. Deiigny and D. Bax, 2. Anal. Ckern., 205, (11, 333 (1964).

752

0

ANALYTICAL CHEMISTRY

. l

c"

8 0.03

.-E

j

-

0 0.

0.09 -

O.O1

0 .

Y I

0

1 0.2

0.4

0.6

0.8

10

Figure 7. Experimental spot size for glucose and fructose with curves computed from Equation 9a.

Dm = 2.4 x (upper curve); K = 1300, uo = 0.0025, and D,, = 3.0 X D , = 1.8 X and D,, = 9.5 X (lower curve). The symbols used were: 0 , fructose; 0, glucose Constants used were: K = 1300, u0 = 0.0025;

All of the computations were made under the assumption that the loads of the two solutes being separated, as well as their initial distribution, were the same. In practice this condition is frequently not met because the relative amounts of the components may vary from test to test. Although a change in the relative concentrations will produce spots of different sizes, the position of maximum resolution should occur a t about the same position as that for equal sample loads. For unequal loads the smaller size of one of the spots will be Compensated by the smaller AR required for separation. Experimental verification that ZR is a maximum in the R, region of 0.25 was made by studying the chromatographic behavior of glucose and fructose. In chromatographic runs, the radius of the initial spot for both glucose and fructose was found to be 0.17 cm. This value was used to estimate uu. By trial and error using K = 1300 it was found from Equation 9 that uo = 0.0025 gave a = b = 0.17122 = 0.0077. After irrigation with different solvent proportions, the spots assumed elliptical shapes. The semiminor axes perpendicular to the direction of flow were experimentally constant at 0.22 i 0.02 cm, uninfluenced by R,. The semimajor axes displayed a slight dependence on R f (see Figure 7). When the loaded papers were suspended in a watersaturated atmosphere for 24 hours, no expansion of the spot was detected. When the papers were wetted with solvent and hung in a solvent-equilibrated atmosphere, the radius enlarged t o 0.23 i 0.02 cm. On the basis of these results and in agreement with earlier work (21), we conclude that dispersion in the y-direction is primarily caused by molecular diffusion in the mobile phase. In other words, De and Dm(s) are essentially zero. The elongation of the spots in the zdirection must have resulted from nonequilibrium effects. Thus Equations l l a and l l b simplify to Dv

=

D,R,

(164

and De = Dv

+ DneRf(1 - R I )

(16b)

By trial and error, values of D , and D,, were found which gave curves that bracketed the data for the semimajor axes us. R The lower curve in Figure 7 was computed from K = and D,, = 9.5 X 1300, u, = 0.0025, Dm = 1.8 X

The constants for the u?per curve are K = 1300, uo = 0.0025, D,, = 2.4 X 10P, and Dne = 3.0 X The y-intercept gives the original spot :;ize which is a function of both K and go. The constants were chosen so that in both cases the semiminor axis was b = 0.23/22 0.0104 when Rf = 0.3, the center of the R, range of the data. This range of constants not only predicts the correct variation of the semimajor axes with R, but also correctly predicts that the semiminor axes will be essentially constant (0.23 i: 0.02 cm) in the range of R, values siudied. We now have a t OUI' disposal all the necessary constants (A.v/As and Apiruetose/A/ipiuoo~~ from reference 7 and D m , Dm, K , and uo) to calculate the effect of R, on both AR, and IR. Figure 8 compares the computed dependence of AR,and IR on R, with the experimental data. The curve for AR, was computed with A.w/As = 2.2 and Apt/Apl = 0.85 found in reference 7. The curve for IR was computed by subtracting the semimajor axes of the spots from AR,. Terminal spot dimensions were calculated from Equations 9a and 9b. The lines are theoretical and the points are experimental. Although the points are subject t o substantial experimental error, it is quite clear that the data fit the theoretical curves. The plot also supports our previous contention that when Ap2/Ap1is constant A R and IR are maxima in the same region of the support.

0.06

E

DISCUSSION

Relative Influence of' Some Parameters on ARp The evidence given in Figures 1 and 2 , as well as Equations 7 and 14, indicates that B = Ap2jApl is much more effective than p = A.w/Asin determining ARf. An example is given in the Results section. This phenomenon can be clarified by considering the basic roles played by A p and p in determining R, and solvent selectivity. The ratio of solute concentration in the mobile phase t o Because that in the stationary phase is a = exp (-Ap/RT). the ratio of the partition coefficients is a measure of solvent discrimination, it follows that B and not p exercises thermodynamic control over re,jolution and supplies a measure of the potential selectivity. The migration rate, R,, however, is a function of the ratio o f t he amounts of solute in the two phases

and depends upon both A p and p . In fact,

Rl ~

1

- R,'

the ratio of

the amounts of solute i r i the mobile and stationary phases, is p exp (-Ap/RT). Thus p plays a secondary role in the chromatographic process by influencing the number of theoretical plates over which the solutes pass and thereby can affect the ease of achieving resolution when B # 1. What general and practical steps might then be taken to optimize resolution ? On first thought, one is tempted t o choose a solvent mixture with a high degree of thwmodynamic selectivity giving a large RrJR,, ratio. In general, this approach is unsatisfactory because the large R, rs.tios accompany large values of A p . (It can be demonstratcd from Equation 1 that RA/RIl is optimized as AMapproaches its limiting value.) For aqueous O t o zero. As A p mixtures this condition occurs a s N E ~ drops increases with decreasing Ntr,0, the solubility in the mobile phase is lowered to suck. a n extent that the bands will neither migrate nor resolve so that this procedure is self-defeating. High solvent selectivity. then, is no guarantee of separation and may need t o be conipromised with faster migration rates t o ensure band disengagement. There seem to be two general ways to enhance migralion rates. The first involves raising NK,Oand the relative clsncentration of solute in the mobile

0.04

4

0.09

L

$

0

- 0.02

\

\

-0.04

-0.06

1 4 0

0.1

\

0.2

0.3

0.4

0.5

0.6

0.1

0.8

Figure 8. Comparison of theoretical and experimental resolution index for glucose and fructose with a H 2 0 : t e r t butanol solvent K = 1300, uo = 0.0025; D , = 1.8 X and Dn, = 9.5 X (upper curve); I< = 1300, uo = 0.0025, D , = 2.4 X lo-&,and Dee = 3.0 X (lower curve). The symbols used were: 0 , ARf; 0, I R

phase at the sacrifice of selectivity; the second is to enhance the relative volume of the mobile phase and hence the relative amount of solute in the mobile phase. If R, is initially low (below O.l), either or both these changes will tend to increase Rf, increase the number of plates over which the solutes are transported, and probably enhance the resolving power of the system. Because AR, and I R may be regulated by both A p and p , some attention should be given both in the design of chromatographic systems. For aqueous systems A p is largely controlled by the solvent proportions, while p is probably sensitive t o support-solvent interaction but basically dependent upon the support. Comments about the solvent are mentioned below, while comments regarding the support are presented here. The largest values of p accompany finely divided and closely packed particles. For example, Kieselguhr is characterized by a n area ratio a n order of magnitude larger than that of paper, cellulose powder, or silica gel (22, 23). The open structure of these supports probably entraps significant quantities of irrjgating liquid that can act as part of the stationary phase, hindering movement of the zone and probably hindering resolution. Since increases in A R , are linearly related t o lOgep (at constant Rf = 0.25), supports giving the largest values of p are recommended for chromatography. Spot Size. Our index of resolution, IR, depends not only upon the separation of centers but upon spot lengths as well. Because zone spreading in paper partition chromatography, and presumably in related techniques, is basically controlled by D m ( M ) , D,,,and time, depressing any of these parameters will constrain the zone and increase ZR. However, because of competing forces it is often difficult to predict what effect a specific change in one property of the chromatographic system will have on the length of the semimajor axes. For example, (22) C . E. Weill, Department of Chemistry, Rutgers University, Newark, N. J., personal communications, 1964 and 1966. (23) C. E. Weill and P. Hanke, ANAL.CHEM., 34,1736 (1962). VOL. 39, NO. 7 , JUNE 1967

753

increasing solvent viscosity will tend to decrease DmCild) (confine band) while concomitantly increasing the time of irrigation (enlarge band). Unless rather detailed studies have been made on the system, the variation in spot size will depend in a n unpredictable manner on such compensating factors. It is known that the support is the most critical variables controlling spot size. The work presented here indicates that the bulk of diffusion in paper chromatography occurs by ordinary molecular diffusion in the mobile phase. The extent of molecular diffusion ought to be directly proportional to the volume of the mobile phase. Because of the large channel network in paper supports ( 2 4 , they probably imbibe substantially larger amounts of liquid than some of the more finely divided T L C supports and give rise to larger spots. Chromatography using T L C usually offers the additional advantage of a shorter development time, thus limiting the duration of molecular diffusion and zone spreading. The small particle size is also instrumental in constraining zone expansion (8). Although the effect of temperature has not received the attention that it deserves, it is known that increasing the operating temperature can have beneficial effects. It allows larger loads (by increasing solubility), decreases solvent flow time, and produces smaller bands ( 4 , 8, 25-27). A decrease in the transit time between phases lessens nonequilibrium spreading lowering plate height and is a partial explanation of this effect. It would appear intuitively obvious, and indeed it has been recommended, that care should be taken during application of a solute mixture t o minimize initial zone dimensions (6). This recommendation arises from the expectation that the terminal spot size will be proportional to the initial dimensions. Except near the origin the terminal semimajor axis is often independent of the starting radius (Figure 5-E). If the applied spot is near maximum size, the semimajor axis may even be inversely related to the starting radius (Figure 5-D). This phenomenon arises because dilution by dispersion is more effective in reducing spot dimensions for distributions with a large variance than for distributions with a small variance. It is also noteworthy that the region where 0 < R f < 0.15 is characterized by a very rapid increase in size, and it is quite common for the spots t o remain of approximately constant size above that Rf range. Under these circumstances A R , may be an acceptable index of chromatographic merit. Minimizing the spot size a t constant load gives rise t o a practical problem. A large value of n in a confined area leads t o a high solute concentration which overloads the support and induces streaking. This streaking is obviously deleterious t o resolution and must be avoided. It is interesting that simulation studies indicate that attention t o the original spot size may not be as important as suspected. The load, n, is far more critical in controlling terminal dimensions (Figures 5-A, 5-B, 5-C, and 5-F) and merits particular attention. Because the amount of solute applied to the paper has such a profound effect on the final spot size, it should always be minimized commensurate with detectability. This simple expedient may greatly enhance separation and may well save time wasted in search of more selective solvents. Not only is it desirable to have small terminal spots, but it (24) G. H. Steward, Advan. Chromatog., 1,93 (1965). (25) M. Alcock and J. S. Cannell, Nature, 177, 327 (1956). (26) D. French, Department of Biochemistry and Biophysics,

is also important that they have well defined boundaries. In general the dimensions of a spot increase with diffusion, pass through a maximum, decrease, and vanish. i t can be shown from Equations 9a and 9b, that the maximum area of a n elliptical spot occurs when the argument for the logarithm is e = 2.7, and is

,

Because the profile of a solute concentration across a spot that has not reached maximum size is narrow and high while that for a spot that has passed maximum size is broad and flat, spots are much more sharply defined before they reach maximum size. Solvents may also influence the sharpness of spot boundaries. Using this criterion of merit, the water :tert-butanol system was rated excellent while dioxane-water was rated poor, although both produce spots of comparable dimensions. When very volatile components are used, substantial streaking ensues unless the support is pre-exposed t o the vapors. The Solvent. In a n attempt t o optimize resolution the laboratory worker has at his disposal an unwieldy number of variables (temperature, solvent, support, equipment, time of pre-exposure t o vapors, etc.). Major consideration is often given to solvent selectivity, the variable upon which we now wish to focus our attention. In the search for a successful chromatographic system, the decision to vary the solvent is generally guided by previous experience with related compounds. A more systematic approach to this problem is suggested below. First, the choice of solvent components is considered and then the choice of solvent proportions. In the ab;ence of very selective solvent interactions with molecules of highly related structures, it can be predicted that Ap2/Ap1,to a first approximation, is independent of solvent proportions or components. This expectation is fully verified by extensive studies presented earlier ( 7 ) . For nearly identical molecules the task of picking a solvent which produces a transfer potential ratio divergent enough from unity to ensure an adequate AR, is difficult. Careful attention to the chemistry of the solutes will often suggest adjuncts which can promote selective complex formation to increase discrimination potential. Some examples come readily to mind. Borate, Mo, and W o form selective complexes with carbohydrates; p H adjustment may selectively ionize acidic or basic groups; and single or multiple chelating agents form specific complexes with metals. Bush (28) has recommended the use of a n anisotropic component in the solvent mixture-i.e., one that could only donate or accept the proton of a hydrogen bond. Introduction of these association equilibria into the system can selectively and effectively alter the time-average structure of the molecules upon which Ap9/Apl and the ease of resolution is largely dependent. An excellent review on this topic has recently been published (29). Adjuncts which can shift this chromatographic parameter further from unity will enhance the discrimination potential of the system and greatly reduce the search time for an acceptable solvent. However, the introduction of these additional equilibria into the system may so complicate the thermodynamics that Ap2/Ap1may now become a function of the solvent proportions. If Ap2/Ap1 remains reasonably constant, optimum resolution would still be expected in the vicinity of R f = 0.25. This expectation,

Iowa State University, Ames, Iowa, personal communication, 1964. (27) J. A. Thoma, H. B. Wright, and D. French, Arch. Biochem. Biophys., 85,452 (1959).

(28) J. E. Bush, Methods Biochem. Anai.. 13,357 (1965). (29) E. Soczewinski, Adoan. Chromatog., 5, (1967) in press.

though, cannot be made: with the same confidence as it can for simpler systems. Nonetheless, it is still recommended that separation be tried with different proportions because I R SO critically depends upon R,. As long as the achievement of equilibrium between tkle solutes and the adjuncts is rapid, multiple bands (30) wcsuld not be anticipated and the extra reagents should not lead to abnormally large spots. Once the solvent components and the support are chosen, little further control can be exercised over Apz/Ap1 or A,w/As. The practical problem, then, is to achieve the best separation with a fixed set of ratio!;. This goal can often be achieved by picking solvent proportions which migrate the slower component in the range near R f = 0.25 where the maximum of IK occurs. At the bench, the solvent proportions producing this desired R f can quickly be selected by a few trial-and-error experiments. For aqueous mixtures chromatographed on paper, solvents containing 0.4 i 0.2 N E ~ generally O optimize IK on a single pass. For supports having a larger p, less water will be required for the best separation. In order to continuously manipulate the R f of the solutes, two conditions are required. First, a solvent pair should be selected so that the solutes are highly soluble in one and sparingly soluble in the other; second, the two liquids must be miscible in all propcirtions. Frequently systems satisfying the first requirement do not satisfy the second, but a third component can usually be found which will induce formation of a single phase. Typical examples are water:ethanol: nitromethane and water :pyridine :1-butanol (4, 31). Appropriate proportions for these ternary systems lie just above the phase boundary in the region of complete miscibility ( 4 ) . In planning a chromatographic system, development time probably ranks second to solvent selectivity in importance. As mentioned earlier, the support will greatly influence running time, but the investigator may also substitute faster flowing solvents provided they give a suitable transfer potential ratio. Muller and Clegg (32) have shown that the rate of ascent of a solvent depends upon its diffusion coefficient computed from its physical properties such as density, surface tension, and viscosity. Because one organic component will comprise around 50 or more by volume of the solvent, its physical properties car: be used as an approximate index of speed. Thus, a simple computation of the diffusion constant of the potential organic component can serve as a guide to faster running liquids. In very difficult cases, it may be helpful to lengthen the support to enhance tke extent of resolution. This can be accomplished in one 0:-two ways. First, the solvent can be allowed to overflow the support or, second, unidimensional multiple chromatography can be performed (5). Both of these techniques will increase the time of development and will affect spot size as well 2.s A R f . In the former technique, once the solvent overruns the support, liquid flow should continue at an approximately constant rate.. Because A R f increases linearly with time and the semimajor axis with the square root of time, ZR will usually, but not necessarily, increase with time. In the latter technique, spot size is near maximum after one solvent pass (see Figure 3 in reference 5) so that ZR should increase until R f = 0.63 (5). Even though a change in solvent components will frequently change Ap2/Ap1,it may affect resolution by changing spot size. It has been our experience that spot dimensions are much (30) R . A. Keller and J . C . Giddings, J . Chromatog., 3, 205 (1960). (31) T. J. Betts, J . Pharnr. Sci., 53,794 (1964). (32) R . H. Muller and El. L. Clegg, ANAL.CHEM., 23,408 (195:}.

more sensitive to spot load, support, and terminal R than to components and irrigation time (which may vary by a factor of 2 or more). In summary, it is recommended that spot load be minimized and that finely-divided close-packing supports and rapidly migrating solvents be used. The components and adjuncts should be judiciously chosen to favor specific interactions with the solutes. The proportions may be chosen by trial-anderror to carry the bands about 1/4-1/3 way along the support for single pass chromatography. APPENDIX

Symbols

Semiaxis of elliptical spot in z-direction

a

=

A,,, AY

= Maximum area a spot can acquire =

A S

= Cross-sectional area of stationary phase

Cross-sectional area of mobile phase

Semiaxis of elliptical spot in y-direction Apz/Api Minimum amount of solute per unit area that can be detected. D = Diffusion coefficient or apparent diffusion coefficient D' = Constant diffusion term = Index of resolution IR = Total amount of solute in spot n N = Number of theoretical plates = Mole fraction of water in the irrigating solvent NH~O R = Gas constant = Time t T = Absolute temperature = Ratio of concentration of solute in phase M to a that in phase S = Free energy of transfer for one mole of solute (phase S to phase M) = A,w/As P = Standard error of Gaussian distribution U = Standard error of distribution of solute in origg o inal spot

b B do

=

= =

Subscripts 1 = 2 = e = m = M = ne = S = y z

= =

Property of substance 1 Property of substance 2 Eddy Molecular Property associated with mobile phase Nonequilibrium Property associated with stationary phase direction at right angle to solvent flow direction of solvent flow ACKNOWLEDGMENT

Anita Rohrer and Dwight Davis furnished valuable technical assistance and P. L. Phipps gave valuable suggestions concerning the approximate formula for A R

RECEIVED for review January 16,1967. Accepted April 3,1967. This research was supported in part by grants from the U. S. Public Health Service (GM 14289), Corn Industries Research Foundation, and the National Science Foundation Research Participation Programs for College Teachers (EE-3060) and Undergraduates (NSF 521550 and NSF 615M32). VOL. 39, NO. 7,JUNE 1967

755