imide could also act as an oxidizing agent, and perhaps as a chlorinating agent. A wide range of aromatic amines reacts with quinone chloroimides; the reaction is affected little by methyl or chlorine substituents. A carbonyl group, in 4,4 ’-bis(dimethylamino)benzophenone, and a sulfonamido group, in sulfanilamide, did not prevent color formation. It was previously shown (7) that dibromoquinone chloroimide gave color reactions with a variety of primary arylamines having two bromines or two or three chlorines on the ring; nitroanilines gave weak colors which were intensified by ammonia treatment. It was stated that attempts to use dibromoquinone chloroimide for the differentiation of chloronitro-, dichloronitro-, and dinitro-anilines, as well as halo- and nitro-benzene derivatives were without success. In the present work nitrotoluene gave negative results. Thus, it appears that as long as an amino group is present on the ring, many compounds with halogens alone, or with one nitrogroup or similar electron attracting group, may be detected. On the other hand, N-acyl groups, in acetanilide and diethyldiphenylurea, and a m-chlorobenzoyl group on a carbazole ring, prevent color formation. With regard to steric effects, methyl groups at both orthopositions, which can reduce interaction of the electrons on nitrogen with the ring of dimethylanilines by decreasing coplanarity (30), do not greatly affect reaction. Although the results were mostly poor with aliphatic tertiary amines, reaction of primary amines was not affected by branching of the alkyl group. Likewise, the hindered 2,2,6,6-tetramethylpiperidine reacted well. The poor results with N-oxides do not correlate in a simple way with basicity. Trimethylamine oxide and dimethylaniline oxide are bases (reacting at oxygen) of strength comparable (30) G. S . Hammond in “Steric Effects in Organic Chemistry,” M. S. Newman, Ed., John Wiley and Sons, New York, N. Y., 1956, p 437.
with anilines (31). Because conversion of an aniline to an N-oxide makes the unshared electrons of nitrogen no longer available to the aromatic ring, it would be expected that both indophenol formation and pi-complexing would be diminished; such an effect may be responsible for the low reactivity of acetanilide. Pyridine oxide, however, which is more reactive than pyridine toward electrophilic reagents (32), produced a weak color reaction, as did pyridine. Pyrene not only produced an intense color with dichloroquinone chloroimide, but had such a strong fluorescence that it could be seen against the fluorescing agents in the Silica Gel HF264+366 coating. It was used, therefore, in the form of a solution containing 2 grams per 100 ml of benzene, as a reference compound in the development of weakly adsorbed aromatic amines with solvents of low polarity. The purest grade (No. 3627) from Eastman Kodak and a practical grade from Chem-Service gave the same results. A recent communication (33) stated that dibromoquinone and dichloroquinone chloroimides decomposed exothermally on heating, and that the former exploded on a reagent shelf. It is recommended, therefore, that the compounds, which have been in use as analytical reagents for many years, be handled and stored with caution. In a personal communication, Mr. Taranto has suggested storage of small quantities in corked (not screw-capped) bottles in a steel safety jacket in a refrigerator. Dibromoquinone chloroimide has also been stated to be less stable on storage if moist (1).
RECEIVED for review June 14, 1968. Accepted September 9, 1968. Some of the results have been presented at the Central Regional Meeting, American Chemical Society, Akron, Ohio, May 10, 1968. (31) D. D. Perrin, “Dissociation Constants of Organic Bases in Aqueous Solution,” Butterworths, London, 1965. (32) C. C. J. Culvenor, Reu. Pure and Appl. Chem., 3, 108 (1953). (33) B. J. Taranto, Chem. Eng. News, 45 (52), 54 (1967).
Resolution and Optimization in Gel Filtration and Permeation Chromatography J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Here we present a theoretical study of resolution in exclusion chromatography and the factors which influence it. Optimum parameters are suggested on the basis of general chromatographic theory and a recent entropy-based formulation for partition coefficients. I t is concluded that highest resolution and speed will be associated with long, narrow columns with fine particles and high pressure drops. Temperature and solvent should be chosen to minimize viscosity. Pores should be relatively small, leading to early elution, and their total volume should be large. Grounds are presented for optimizing pore shape as well as size. Finally, these results are used to predict the size and molecular weight increments needed for satisfactory resolution in columns of different efficiencies.
GEL FILTRATION (1) and permeation (2) chromatography have established themselves as exceedingly useful techniques in separating macromolecules. Their relative usefulness derives more from the inherent difficulty of large-molecule chromatography (3) than it does from their basic separating
efficacy, which is rather bad compared to other chromatographic methods. Here we examine the latter aspect, with the view that resolution must be improved and optimum conditions established if these methods are ever to make the highly refined separations which are so needed in the next stage of macromolecular studies. Since gel filtration and permeation chromatography depend on the same underlying phenomena, we group them both together for theoretical purposes, and lend them the common title, ‘‘excIusion” chromatography. In an earlier paper (41, the total number of peaks resolvable by exclusion chromatography was deduced. By contrast to total peak capacity, we focus on the resolution of close(1) J *Porath and p. F1odin, Nature$ lS3,1657 (Ig59). (2) J. C . Moore, J. Polymer Sei., Part A , 2, 835 (1964); Chem. Eng News, 40, 43 (Dec. 19, 1962). (3) J, c,Giddings, J , G~~ Chromafogr.,5,413 (1967). 39, 1027 (1967). (4) J. C . Giddings, ANAL.CHEM., VOL. 40, NO. 14, DECEMSER 1948
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lying pairs of peaks to arrive at criteria for their adequate and optimum separation. The present status of optimization studies has been well reviewed by Johnson et al. (5). Partition coefficients in the exclusion methods are determined by the configurational entropy change which RCcompanies the transfer of solute into the porous particle (6). This entropy may be written AS0
= (R
n
In - = QO
(R
In Q ,
dz s -
where
O1
=
CYAK 4 1+01K VJV,
A L -
(1)
where (R is the gas constant and n, = n/Qo is the fraction of, for example, rotational and translational states remaining free in the pore network-i.e., that fraction not lost because of interference at the pore walls. The partition coefficient, K, equals Qf. The latter was calculated as a function of pore and molecule size for several geometrical models of pores and molecules. As expected, K and Q E are ~ determined by the relative size of molecule and pore. The dominance of entropy effects in exclusion chromatography leads to certain unique characteristics not previously recognized. In particular, the role of solvent and temperature is unlike that applying to the usual enthalpy-dominated retention in chromatography, and leads to different (and simpler) criteria for optimization. The choice of pore size also presents rather special problems. These matters are explored below. The resolution of close-lying peaks in chromatography is given rather generally by the approximation (7)
R
where L can be interpreted as a mean diameter or length for the molecule (as will be discussed), or as any arbitrary length parameter for geometrically similar molecules. Quantity c, inversely proportional to pore size, is constant for a given porous material. Substitution of Equation 5 into 4 leads to
(2)
in which N is the number of theoretical plates, AK is the difference in partition coefficients, KII-KI, for the two species, and Y J V , is the ratio of mobile phase (interparticle) volume to stationary phase (intraparticle or pore) volume in an exclusion column. Quantities N and K represent average values for the two species. If a certain degree of resolution, R,,is required (usually R, l), the increment in partition coefficients must be
-
(3)
-
as can be deduced from Equation 2. Because (AKjK) (KIIiKI) - 1, and because 52, = K, the fundamental requirement for resolution is that the ratio must be a certain minimum in accord with (4)
The significance of this expression in separating molecules and molecular depends on the basic relationship between size and shape. This relationship, while differing somewhat between various pore models, can be approximated by the random-plane model (below) of a pore network which yields (6) QI - K = e-OL (5)
L z
(6)
InK
where AL = LII-LI. (All incremental quantities will be expressed as that for trailing peak I1 minus that for leading peak I, even though some such quantities like AL are negative.) Equations 4, 6, and related expressions can be used to deduce minimum resolution requirements, as will be shown. A numerical example will help illustrate the magnitude of terms in fundamental Equation 4. Not uncommonly we might find values in the vicinity of N 1000, 01 1, K and Rs In this case,
-
-
-
-
(7) This ratio is sufficiently large that some error in our results can be anticipated because we assume the peaks to be closelying. One can avoid the latter assumption in a straightforward mathematical manner, but the gain in accuracy is not ordinarily worth the additional complexity of the results. The simpler and more direct approach used here is accurate enough to establish valid guidelines for most laboratory work, and is particularly applicable to well designed columns with large plate numbers. NATURE OF OPTIMUM PARAMETERS
The resolution expression, Equation 2, can be broken into two parts: the part that depends on column efficiency, N”’/ 4, and the part that depends on equilibrium and retention, aAK/(l aK). Most column and operating parameters which influence column efficiency but not retention should be optimized by much the same criteria applicable to other forms of chromatography (7). Any parameter which strongly influences relative retention must be fixed by unique criteria because the retention mechanism is unique. In the former category are such common parameters as column length, column diameter, flow velocity, pressure and pressure drop, particle size, and dead volume. AS emphasized previously (7, a), optimum conditions are most likely to entail long, narrow columns packed with fine particles and an absolute minimum of dead pockets and free connections. Flow velocity and pressure drop should be fairly high. Presently the latter is often restricted by the softness and, thus, the variable compactness of ordinary gel beads. This poses a serious dilemma because the success of the long column-high velocity-fine particle combination hinges crucially upon high pressure drops. If ‘[flexible” gel particles cannot be adapted to these conditions, it is probable that superior results will eventually be achieved with rigid porous particles, such as the porous glasses, which can be used in high-pressure columns.
+
ROLE OF SOLVENT AND TEMPERATURE (5) J. F. Johnson, R. S. Porter, and M. J. R. Cantow, Rev. Macromol. Chem., 1, 393 (1966). (6) J. C. Giddings, E. Kucera, C. P. Russell, and M. N. Myers, J. Pliys. Chem., in press. (7) J. C. Giddings, “Dynamics of Chromatography. Part 1.
Principles and Theory,” Dekker, New York, 1965. 2 144
e
ANALYTICAL CHEMISTRY
Solvent (mobile phase) properties, within limits, mainly influence efficiency too. (In this sense they are analogous to gas properties in GC, but are not comparable to any
-
(8) J. C. Giddings and K. L. Mallik, ANAL.CHEM., 38,997 (1966).
conventional form of liquid chromatography.) The limits to this categorization are important and must be made clear: Changes in solvent properties must not cause precipitation, adsorption, or changes in molecular or pore dimensions. Within these limits, one may beneficially choose a solvent system with low viscosity and high diffusivity. The role of temperature in the exclusion methods of chromatography is much like that of the solvent. Within the same limits outlined for solvent properties, temperature changes d o not influence retention (9). This result is indeed unique, and is not characteristic of any other form of chromatography, gas or liquid. This is a consequence of the dominant role of entropy, as opposed to enthalpy, in partitioning equilibrium, as discussed. This can be best illustrated by substituting Q f for K i n Equation 2 :
Because the fraction of free or allowed states, Of,is a function only of molecular and pore dimensions, this is temperature independent as long as dimensions remain constant and sorption is absent. The lack of temperature dependence means that a reduction in temperature has no beneficial effect on selectivity, contrary to most other forms of chromatography. The only role of temperature derives from its influence on the number of theoretical plates, N . If it is assumed once again that all dimensions remain constant, temperature affects only one basic parameter-diffusivity-out of all those which help determine plate height. Because mobile and stationary phases are composed of the same liquid, changes only in the diffusion coefficient, Dm, for solute in the mobile phase determine the temperature dependence of IV. PLATE HEIGHT 4ND OPTIMUM VELOCITY, TEMPERATURE, AND SOLVENT
A plot of reduced plate height, h = H/d,, against reduced velocity, v = dpu/Dm,yields curves roughly in superposition for all chromatographic systems (7, 10). (By contrast, a plot of ordinary plate height, H , us. linear velocity, u yields curves displaced from one another by factors of up to lo6.) This correspondence principle, while mainly used to relate diverse chromatographic systems, can also predict the gradual change in optima caused by changes in, for example, temperature, and solvent. The common curve alluded to above can be represented by (7)
where 7, wi, and X8 are structure-retention parameters (related mainly to packing geometry) which can vary moderately from column to column. Quantity p , is a term for longitudinal diffusion in the stationary phase; it depends upon R and the ratio of diffusion coefficients in the two phases, Ds/Dm,along with geometrical factors. Quantity Q’ accounts for stationary phase mass transfer and, in the partition forms of chromatography, is also a function of D,/D, and structural parameters. The “eddy” diffusion coefficient, At, depends primarily on packing structure. (9) M. J. R. Cantow, R. S. Porter, and J. F. Johnson, J. Polym. Sci., Part A - I , 5, 987 (1967).
(10) J. C. Giddings, J. Chwmatogr., 113, 301 (1964).
The “universal” curve concept is an approximation in most cases, if packing variations, structural differences in support particles, and the ratio D,/ D, are ignored. (Neither does it allow for high-velocity turbulent flow.) If chromatographic experiments are conducted with the same column, and changes are allowed only in the temperature or solvent, all factors except D,/Dm and retention will remain fixed and the h us. v curves will approach one another even more closely. Exclusion Chromatography is again unique because changes in temperature (or solvent) alter neither the D,/Dm ratio nor retention. Therefore, h us. v curves obtained at different temperatures (or with different solvents) should coincide rather exactly. This coincidence means that the same minimum value of h would be found at all temperatures. Because particle diameter dp remains essentially constant, Hmi,would, therefore, be independent of temperature, and a velocity-optimized column would have the same number of plates at any temperature (or with any solvent). While it is generally important to maximize resolution, as above, a more practical optimization in liqiud chromatography (which is inherently slow compared to GC) is that which minimizes the time required to achieve a given resolution. For this reason, laboratory velocities generally exceed those corresponding to Hmin,even though resolution is lost. We may now assume that suitable conditions (which may be based on resolution, speed, or other criteria) for the operation of a column at a given temperature have been determined, and inquire about the possible role of temperature in speeding up the operation. Clearly a temperature increase If flow velocity is increased by a correwill increase D,. sponding amount, v and resolution will remain unchanged (because v is proportional to u/D,), but there will be faster elution. In general, then, elution time is inversely proportional to D,. If we consider liquid diffusion an activation process, we find the following relationship between elution time and temperature: t = C’ exp (W@T)
(10)
where Wl, the activation energy, is 2.5 kcallmole for many common liquid solvents (11). This corresponds to a 10 to 2.5z reduction in elution time for each 10 ‘C temperature rise. The pressure drop in our variable-temperature column will be determined both by viscosity and flow velocity. In most liquid systems viscosity is inversely proportional to D,. Consequently, viscosity would decrease as flow is made to increase (to keep pace with Dm);one effect would be exactly balanced by the other, making it possible to use a fixed pressure drop as temperature and speed increase. The above principle is valid for changes in solvent as well as temperature if, as stressed before, adsorption and dimensional changes do not occur. Generally, we may state that speed increases in inverse proportion to viscosity, whether viscosity is manipulated thermally or chemically. Viscosity is the critical parameter and, within the stated limits, should be minimized. An interesting prediction results from the above arguments. Chromatograms showing concentration us. elution volume should be identical for a given pressure drop applied (11) S . Glasstone, K. J. Laidler, and H. Eyring, “The Theory of Rate Processes,” McGraw-Hill, New York, 1941, Chapter IX. VOL. 40, NO. 14, DECEMBER 1968
0
2145
to a given column, even with arbitrary variations in solvent and temperature. The chromatograms showing concentration us. time however, would stretch out in proportion to viscosity. All the above arguments apply solely to entropy-controlled exclusion mechanisms. In some systems we may expect a mixture of entropy and enthalpy (adsorption, two-phase partitioning, etc.) effects. Enthalpy effects have been demonstrated in certain gel systems by the excessive retention of some components (12). In any such mixed case we must write the elution volume as a sum of terms corresponding to the different retention mechanisms (13): V = Vnl
+ KV, + ZK’V,’
(11)
The last (summation) term accounts for various enthalpic effects. Both kinds of effects are reasonably well understood, thus making it possible to synthesize an approximate description of mixed sorption-exclusion chromatography. The principal uncertainty revolves around the question of whether some molecules are kept away from “sorption” sites by the exclusion phenomenon; in this case, the extent of the sorptive phase or surface, Vs’, is a function of molecular dimension-roughly, a function of K. THEORY OF OPTIMUM PORE SIZE
Retention in the idealized exclusion column is a function only of the dimensions of the pores and of the molecules; temperature, solvent, sorption, and related effects are not directly involved. If the separation problem revolves around a certain group or pair of molecules of given size, the pore dimensions will be the only retention-controlling variables. The specific effect of changes in pore size and shape hinges on the precise relationship between K and the dimensions of both molecules and pores. Generally we can assume that K is some function of a characterizing group of size and shape parameters: the Ll’s for molecules and the 8 s for pores-i.e.,
K
=
K(L1, Lz, . . . Lm; di, dz
. , . dn)
(12)
The increment AK in the partition coefficient of close-lying molecules can be approximated by AK
=
- ALj
dK a2K ALj - a + ffK)c bdi bdi dLj j
1
=1
dK - ALj dLj
=
0 (14)
whose solutions are the n optimum pore-size parameters, di. The above treatment is not presently applicable in all its generality because the precise form of K has not been obtained for arbitrary systems. However, the exact theory which has been worked out for a number of models provides broad direction (6). Model calculations indicate that a single length parameter will reasonably characterize a molecule’s partitioning behavior irrespective of molecular shape. In earlier work, (12) D. Eaker and J. Porath, Separ. Sei., 2, 507 (1967). (13) R. A. Ke!ler and G. H. Stewart, ANAL.CHEM., 34, 1834 (1962). 2146 *
ANALYTICAL CHEMISTRY
(6). The problem of characterizing “pore size” is perhaps even more difficult. For one thing molecules, even complex ones, are limited in extent, whereas a pore is part of a continuous, interconnected network which must be considered as a whole. There is no way known to characterize rigorously a pore network’s partitioning behavior short of describing its detailed geometry. Experimental pore size distribution, while not uniquely specifying geometry, may approximately characterize exclusion partitioning (16) if some assumption is made about pore shape. Until better theoretical and experimental tools are available for studying the effects of pore geometry, pore size is perhaps best characterized by the simplest and most direct of experimental parameters, surface area per unit pore volume, s. Beyond the advantage of simplicity, s is a choice made more rational by the realization that exclusion stems from the restraints imposed by pore surfaces on molecular motion (6). Our previous work with various models confirms the broad correlating ability of quantity s. If, then, we reduce the multiplicity of parameters in Equation 12 to just two, E and s, we can write K = K (2, s), and the optimum pore size, which is proportional to s-l, is obtained from bZK bK bK (1 Ka) - ff & = 0
+
as opposed to the more involved group of expressions represented by Equation 14.
bK
j = 1 bL,
This can be substituted directly into the resolution expression, Equation 2. The latter can be optimized with respect to the pore dimensions by making bR,/bdi = 0 for all i. This gives n equations of the form (1
the radius of gyration (14) was suggested as a characteristic length; more recently, the hydrodynamic radius has been proposed (15). Our calculations for rigid macromolecules show that the former is the best of the two, particularly in providing a common denominator for comparing rodshaped and spherical molecules. However, the best single parameter is apparently the mean external diameter, 2. Quantity I’, is obtained by noting the maximum length of the molecule projected along a given axis, then averaging this length over all possible molecular rotations. This is the only proposed length parameter which applies rigorously to all arbitrary molecular shapes within any chosen model of the pore network. In our case, the model network was created by intersecting random planes (the “random-plane” or “box-pile” model), but even in other networks E has been shown to characterize most reasonably diverse molecules
OPTIMUM PORE SIZE AND RETENTION BASED ON GEOMETRIC SIILULARITY
For the moment, our hypothetical problem will be the separation of two geometrically similar molecules--e.g., spheres. We are to choose a pore network from a group having the same shape characteristics, but differing in pore size. Any geometric theory of partitioning (including the earlier mentioned entropic theory) in such geometrically similar systems will conclude that K is a function of only one dimensionless parameter, x, the ratio of molecular size to pore size. It does not matter which measure of size is taken so long as the measure is consistent in the series. If we extend the theory, if only in approximation, to molecules and pore networks not exactly geometrically (14) S. Porath, Pure Appl. Chem., 6, 233 (1963). (15) H. Benoit, Z . Grubisic, P. Rempp, D. Decker, and J. G. Zilliox, J. Chin?.Phys., 63, 1507 (1966). (16) A. J. deVries, M. LePage, R. Beau, and C. L. Guillemin, ANAL.CHEW,39, 935 (1967).
t
0.9
on our conclusions). We will compare hypothetical columns, all with the same efficiency, N , each attempting to separate a pair of molecules with constant AL/E, Therefore, we will compare the magnitude of the function
tI
-R, -
const.
x dKldx _.--1 S K
where Rs const.
const. = NO. CYLINDRICAL OR SQUARESHAPED PORES ERICAL CAVITIES
Figure 1. Relative resolution for two unequal spherical molecules in different kinds of pore networks and at various pore sizes (proportional to s-l)
similar, we should choose the most “universal” measures of size possible. In view of the earlier discussion, we will henceforth use and s as parameters characterizing molecule and pore size, respectively, and therefore give x the meaning
x
=
‘ioL
(1 6)
Our theoretical development will not depend on these particular choices, but our calculated examples will. [Surface area, s, appears in the numerator because its reciprocal, s-l, is directly proportional to pore size. The coefficient, lJz, is an arbitrary constant. We note that quantity for spherical molecules is equal to the sphere diameter; for thin-rod molecules it is one-half the rod length, etc. (6).] Once again we start with the basic resolution expression, Equation 2. We can write
The latter equality makes use of Equation 16. With this expression, resolution becomes
As an initial observation about the above expression, we note that resolution increases with a: (= Vs/Vm)-i.e., the optimum a is the largest possible value. Because the porosity of randomly packed spheres is about 0.40, a could be no larger than 0.6/0.4 = 1.5 in most present systems even if the pore walls were infinitely thin and the solid portion of the porous particle therefore vanishingly small. More typically, about one third of the particle may be assumed solid, leading to a 1. There may be some virtue in partially compressing or fusing together such porous particles (carefully avoiding the destruction of pores) to increase a , This would aid total peak capacity as well as the resolution of individual pairs. For the following examples we will assume a: to be unity (slight variations from unity will have only a minor effect
- V%AL/4L
at various values of x and for different models which have been theoretically evaluated. (Recall that, by convention, AL is negative. dK/dx is also negative, so that both R , and const. are positive quantities.) Figure 1 shows RJconst. us. x for spherical molecules in various pore geometries. From these plots one can identify the optimum x for each pore type--i.e., that value giving maximum resolution. From xoptone can calculate the optimum pore size, SI, using Equation 16. Besides optimizing pore size, these plots show how various pore types compare in their ability to separate spherical molecules. Uniformly spaced planes clearly offer the best resolution (by a factor of 2.3), but the resolution curve is more sharply peaked. This means that one must adjust the spacing or pore size more carefully than in the other cases so that it is neither too large nor too small. Because R,/const. falls to zero for x > 1, the largest of the two molecules, I, should lie at x = 1 (space between planes equals molecular diameter). The smaller molecule, 11, would then lie on the downward sloping left branch of the curve (Figure 1) at x = (diameter>,,/(diameter)I. This is preferable to straddling the peak-ie., making the peak correspond to the average of LI and LII-because absolutely no resolution is gained beyond x > 1. (The placement of two molecules jndividually on the curve recognizes that the separated peaks are not infinitesimally close-lying. Resolution is calculated as the mean of the two positions.) While carefully spaced parallel planes offer the highest resolution, as we have seen, the narrowness of their resolution peak makes this model pore system inferior for separating a broad spectrum of molecular sizes--i.e., a wide range of E’s. As expected, the best capability for broad spectrum separation comes from the random-plane model. Clearly, any pore network having a strong element of randomness will have, among its diverse pores, a size highly selective for almost any pair of molecules. Thus, for the randomplane model the resolution, while not extremely high at any one point, stays within 5 0 2 of its maximum value over a IO-fold range of x values. This means that molecular size, L, can range over a 10-fold scale without a severe loss of resolution for any of the pairs. By contrast, the x-ratio at half-height for uniform planes is only ( X I / X I I ) ~ / ~ = 1.5, reflecting its narrow-spectrum characteristics. (For comparing with the 10-fold random-pore range, the x ratio 4.0 is more meaningful because over this range one has R,/ const. > 0.14, the half-height resolution for random pores.) The x ratio at half height for cylindrical or square-cross section pore is 4; for spherical cavities it is 5 . Resolution for all models peaks at nearly the same value of x (and, thus, surface area s). The variation is only 28% between extremes. Better still, all resolution is well within 90% of optimum at x = 1. This suggests that surface area alone can be adjusted to optimize the separation of known molecules irrespective of unknown pore-shape factors. Furthermore, we would hope that s could be generally VOL. 40, NO. 14, DECEMBER 1968
e
2147
o’6
1
O’gl 1
SPHERICAL CAVITIES CYLINDRICAL PORES SQUARE CROSS SECTION
0.8
Figure 2. Relative resolution for two thin-rod molecules in different pore networks and pore sizes
chosen in advance, without trial and error methods, on the above theoretical grounds. Figure 2 shows RJconsl. for two thin-rod molecules, with the same group of pore models employed as above. The general results are similar, although the change from spherical to thin-rod molecules changes the relative advantages of the various pore types in making the separation. For example, the uniformly spaced planes now offer the poorest “peak” resolution of the uniform (nonrandom) pores, but the broadest spectrum. The optimum now lies at x = 0.5, for which the plane spacing equals the (average) rod length. The random-plane model still offers the best broad-spectrum characteristics. In fact, its curve is identical in Figures 1 and 2, reflecting the universal nature of E as a measure of molecular “size” for this particular model (recall that E equals rod length for thin rods). The similarity of curves for the other models shows that E is a reasonable sine measure for most porous networks. In most cases “peak” resolution lies near x = 1. Because K is a unique function of x for each model, the data of Figures 1 and 2 can be alternatively presented as plots of relative resolution us. K (Figures 3 and 4). This is more meaningful to the experimentalist because K[= (I/.)( Ye/ V , - l)],limited to the range from zero to one, relates directly to elution volume, V,. Maximum resolution is generally achieved toward the left side of these figures, in the K range 0 to 0.5. This means that the best resolution is with pores which are small enough to strongly exclude solute and force its early elution. Resolution falls off strongly for all models for K > 0.5, whether the molecules are spherical or rod shaped. The mutual agreement of these results would
Table I. Optimum x(=
s
0.6 I
RS
const.
I/ 01
‘0
0.2
03
04
06
05
K
(
~
0.8
0.9
$
~
$
~
~
-
(a: = 1)
0.28
ae-2 - x
1.oo
0.00
x-1=0
Long cylindrical pores
1.17
0.17
Long square pores
1.17
0.17
Equations can be used to obtain xopt and KO,$for any a.
~
~
s
suggest that, in practice, pore size be chosen so that difficult pairs will elute early ( K 0.25), and that an attempt be made to avoid difficult separation work at K > 0.5 or 0.75. While Figures 3 and 4 are for extreme molecular typesspheres and thin rods, respectively-the plots are roughly comparable. It is likely that similar results and conclusions would apply to other types. Clearly Figures 1 through 4 could be used to tailor pore shape as well as size to fit specific separation problems. The
1.28
Spherical cavities
07 Q
Figure 4. Relative resolution us. K for two thin rods
(a = 1)
ANALYTICAL CHEMISTRY
01
Partition Coefficient,
Kopt
e
SPHERICAL CAViTlES SYLINDRICAL PORES
-
Xopt
2148
.
L/2) and K Values for Spherical Molecules in Different Types of Pores at a:
Pore type Random planes (“box pile”) Parallel planes, uniform spacing
a
Figure 3. Relative resolution us. K for two spheres
Eq for x o p t a
- 401- ( a :
VJV, = 1 Gen, eq for Ka
+- 1 = 0
+ 1 ) ( x - 1) = 0 x2 - ( a + 1) ( x - 1) = 0 a: x2
=
,-x
1-x
(1 (1
;y - ;y -
)
capability of uniform, parallel planes for spherical molecules and their unsuitability for rod-shaped molecules is but one example of many which could be suggested. Full implementation of this proposal depends on the development of better methodology for the control of pore shape. The exact location of maximum resolution can be found, much as before, by using the condition b R , / b x = 0 in combination with resolution Equation 19. Thus, the optimum pore size is that which yields an x satisfying the equation (1
+ Ka) (
x g + 2)- (2) ax
=
0
EXAMPLES OF RESOLUTION AS RELATED TO MOLECULAR SIZE DIFFERENCES We will use the random-plane model to estimate the increment in molecular size needed to achieve resolution at a given level. Using K = exp ( - x ) , we obtain from Equation 18 the expression
Here we have completed a full circle back to Equation 6, which is equivalent to the above for close-lying peaks-e.g., large N , low R,. The divergence in the two expressions for small N stems from the different use of the close-lying assumption in their respective formulations-Le., Equation 4 and 17. Setting a = 1 and using the previously derived optimum, K = 0.28, we get
If we require unit resolution (we may often find it necessary to settle for less, perhaps using sharper methods of data interpretation, as the following calculations show), this becomes =
-14.314s
Xopt
Pore type Random planes (“box pile’’) Parallel planes, uniform spacing Long cylindrical pores Long square pores Spherical cavities
(a! =
Xopt
1)
1.2s
(a = 1)
0.50 0.71
0.28 0.50 0.33
0.90 0.97
0.27 0.17
(21)
This equation has been applied to the previously mentioned models. Calculated values of x and K optima are shown in Tables I and 11, assuming a = 1. These tables confirm our earlier conclusion that superior resolution is associated with low K values and, thus, early elution. Also shown in Table I are the equations for K for the various models as derived in the previous work (6). One can get Koptfor any a value by first solving for x in the equations for xOptshown in the table. This solution, equal to xopt,is then substituted into the K expression to yield KO,%.We have not presented the comparable thin-rod expressions because of their greater complexity (6). The optimum K values obtained above emphasize high resolution as opposed to high speed. From theoretical and experimental work on other chromatographic systems, it can be concluded that speed-optimization will shift the Ks to lower values (earlier elution). However, as our optimum K s are already low, speed-optimization would not be expected to alter the above conclusions to any significant degree.
AEiE
Table 11. Optimum x(= s L/2) and K Values for Thin-Rod Molecules in Different Types of Pores at a = V J V m= 1
(24)
Table I11 shows the relative size difference, -AL/E, which must exist between molecules for their separation at different column efficiencies. Also shown are the minimum molecular weight ratios of two barely separated species. The first of these, with subscript s, refers to geometrically similar space-filling molecules-e.g., spheres-of constant density. The second, subscript p , refers to random-chain polymers.
Table 111. Relative Sizes and Molecular Weight Ratios of Molecules Barely Separable (R,= 1) on Columns of Different Numbers of Theoretical Plates, N AT
100 300 1000 3000
10000 100000
-A t / L -1.43 -0. 83
[(MW)II(iMW)II18 [(MW)II(MW)II, -200 -40 -14 4
0.26 0.14
3.9 2.2 1.5
0.05
1.15
0.45
2.5
1.7 1.3 1.10
For the latter, molecular weight is assumed lo increase with the square, as opposed to the cube, of mean molecular length E . The approximate signs (-) are used for the first two rows because the poor resolution severely stretches our assumption that the two peaks are close-lying. The AE/E figures are valid for molecules of any fixed shape, or even for two molecules of different shape, because of the generality of the random-plane model. However, any pore network would be expected to yield comparable results. Note that ALiL, a dimensionless ratio, can be replaced by A(diameter)/diameter, A(length)/length, or any other related measure for geometrically similar molecules. The message of Table 111 is clear. One must have a great number of theoretical plates to separate molecules which are at all comparable in size when using the exclusion methods of chromatography. At a typical 1000 plates, for instance, one could not separate spherical molecules whose diameters did not differ by almost 50% relative to the mean diameter and whose molecular weights were not almost fourfold different. Only at 10,000 plates do we get down to the separation of reasonably comparable molecules: They can differ by as little as 14% in diameter and 50% in molecular weight. We would need 104 to 106 and more plates to achieve the kind of fine separations normally expected with other chromatographic systems. Much of the virtue of exclusion chromatography lies in its broad spectrum separations. Nonetheless, the method would be immensely more useful if macromolecules of comparable size could be resolved. It would certainly be worth seeking the high plate numbers suggested here, perhaps using long, high pressure columns as suggested earlier, to improve an already invaluable technique. ACKNOWLEDGMENT The author acknowledges the assistance of C. P. Russell in checking derivations and for numerical calculations. RECEIVED for review April 9, 1968. Accepted September 6, 1968. Investigation supported by Public Health Service Research Grant G M 10851-11 from the National Institutes of Health. VOL. 40, NO. 14, DECEMBER 1968
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