Theory of Gel Filtration (Permeation) Chromatography J. CALVIN GlDDlNGS and KANA1 L. MALLIK Departmenf of Chemistry, University o f Utah, Salt lake City, Utah
The theory of zone broadening in gel filtration chromatography is formulated. Starting with a general plate height equation, particular contributions are evaluated in the light of the unique characteristics of this technique. Experimental values, taken from several literature sources, are shown to often exceed the hoped for limits characteristic of high efficiency columns. Reasons for this, including the large ratio of column to particle diameter, are discussed. It is concluded that the excess plate height is not due to stationary phase nonequilibrium because this term is almost negligibly small in gel filtration. This, along with coupling, permits a relatively high speed of operation.
I
x RECENT YEARS a general technique variously known as gel filtration (18) or gel permeation (12) chromatography has proved itself highly effective in the separation of high molecular weight components, particularly solutes of biological origin and polymeric materials. This method is unusual in that the primary mechanism of retention is an incomplete penetration into a swollen gel; separation occurs as each solute is excluded from different fractions of the gel volume by steric effects (19). Large molecules are generally excluded to the greatest degree, and therefore appear first in elution order. The theory of retention in gel filtration chromatography has been considered especially by Porath (17) and by Laurent and Killander ( 1 1 ) , the latter employing a model first developed by Ogston (15). These theories have been quite successful in correlating retention volume with molecular dimensions. The theoretical problem that is complementary to that of retention-dealing with zone dimensions--has not yet been considered in detail. As in most chromatographic systems, zone spreading causes the overlap of adjacent zones and therefore fixes a lower limit to differences in retention which must be achieved for adequate separation (2). It is therefore important t o understand the factors which lead to zone spreading and to discuss how this spreading can be reduced in practical systems. Such is the object of the present paper.
THEORY
Although the retention mechanism of the gel filtration column is rather unique, the dynamic processes leading to zone dispersion are basically the same as for other chromatographic columns. It is therefore desirable to start from the general dynamic theories of chromatography ( d ) , and then to show the special modifications characteristic of gel filtration materials. Zone spreading is universally characterized by the plate height, H ( 2 ) . This is a rather arbitrary parameter which adequately characterizes zone width despite the misleading implication that there are “plates” or units resembling “plates” in chromatography. Experimentally one can determine the value of the plate height in elution chromatography by the equation
H
=
L
()’
where L is the column length, Ti, the component’s elution volume, and w the width of the zone measured in volume units (w is defined as four standard deviations, 4a, and can be obtained as the distance between the base line intercepts of lines drawn tangent to the zone’s points of inflection). The plate height of a chromatographic column can be written in the following quite general form (2)
H
B
= V
+ CU + i
1
1/Ai
+ lC,iV
(2)
where v is the flow velocity of the mobile phase-Le., the velocity of a totally excluded component. An exception to this form occurs when column diameter is excessively large, as with preparative scale columns ( 7 ) . The first term on the right accounts for longitudinal molecular diffusion; the second for nonequilibrium (or “mass transfer”) effects in the stationary phase; and the third for flow pattern and nonequilibrium effects in the mobile phase. The nature of these terms in a gel filtration column will be discussed shortly. First it is important to estimate diffusivity within the gel since this is a crucial component of several terms in Equation 2. Gel Diffusion Coefficient. The diffusion coefficient for solute within
the stationary phase (the gel) can be denoted by D,. Diffusion within gels occurs for the most part as diffusion in the pure mobile phase substance since the latter swells the polymer and occupies the space between the fixed chains. Hence D , would be equal to the mobile phase value, D,, except for obstruction by the gel network and for adsorption effects when they exist. These obstructions lead to a slightly reduced diffusivity giving an obstructive factor for the gel, ya = D./Dm, somewhat less than unity. The theoretical basis for such obstruction for small molecules has been considered ( 1 ) . Horowitz and Fenichel (8) have measured diffusion coefficientsin swollen dextran gels. Upon comparing diffusivities between numerous waterswollen gels and the corresponding dilute aqueous solutions (gel-free), they concluded that D,/D, was usually in the range 0.6-0.7 and relatively constant with respect to size, chemical grouping, and temperature. While these authors did not examine particularly large solute molecules, nor extreme cases where adsorption might interfere, it seems safe to conclude from their results that y a may ordinarily be assumed equal to about 2/3. Longitudinal Molecular Diffusion. Ordinary diffusion along the tube axis, which appears as the B term of the last equation, occurs in both mobile and stationary phases. The sum of the two contributions is given roughly by the following equation ( 2 ) . [A more rigorous means of combining the two, developed initially for electrophoresis is also applicable here ( I ) . ]
B
=
2yDm
= mn[y
+ 27.Dm
+
n/d(l
(1 - R ) / R - R)/RI ( 3 )
where y is the obstruction factor for diffusion through the interstitial space (mobile phase) of the granular bed and R is the ratio of zone velocity to mobile phase velocity. The value of y will in fact depend somewhat on y., but to a first approximation it is constant. The gas chromatographic experiments (which are equally applicable here) by Knox and McLaren (10) show that this parameter can also be approximated by 2 / 3 . In the ideal case, then, where adsorption is absent and no unusual obstruction effects present themselves, VOL. 38, NO. 8, JULY 1966
e
997
a good approximation for B exists in the fo m
2 yields a final approximation for plate height in gel filtration columns. (7)
Nonequilibrium in the Stationary Phase. For spherical gel particles of diameter d,, in which a given solute has the diffusion coefficient y8D,, the nonequilibrium term, C , is given by the following equation.
If the particles are irregular, the constant 1/30 must be replaced by some other parameter, the value of which can be determined by a Poisson-type equation integrated over the particle volume (2, 5 ) . If the particles are unequal in size and shape, the C term must be calculated by the general combination law, which accomplishes the necessary “averaging” ( 5 ) . For the most part Equation 5 should suffice if particle dimensions are not too variable and if d p is taken as the mean particle diameter. Flow Pattern and Nonequilibrium in the Mobile Phase. The summation term in Equation 2 involves two kinds of quantities. The Ai’s describe an “eddy diffusion” effect which occurs in the absence of any diffusion lateral to flow, and the Crnl.’s describe the role of nonequilibrium in the absence of the “eddy” effect. These two terms are given individually by
where the xi’s and the w,’s are geometrical factors of order unity. Giddings has suggested the existence of five different mobile phase terms (d), four of which mould exist in gel methods. (Diffusion within the gel particle, rather than in the interstitial space, is not considered, as the entire particle is regarded as part of the stationary phase.) This means that there are four A, and four terms, all different. These eight parameters, when substituted into the summation term of Equation 2, yield a rather complicated expression for plate height. [Knox (9) has suggested an alternate espression, but it is in an integral form and is even more complex.] A numerical example mill be given later. The eight terms taken as a wholei.e., the discussed summation term, depends on column-to-particle diameter ratio and on the structure of the packed bed. Consequently it varies with such miscellaneous factors as column diameter, particle diameter, particle shape and softness, and the method of packing the column. These factors are at present rather difficult to account for in a quantitative theoretical way. They can perhaps best be dealt with in a semiempirical manner-Le., by studying the extremes of experimental variables in order to isolate the individual parameters and their limits. Unfortunately very little has yet been achieved in this direction. The substitution of component terms-Equations 4, 5 , 6 and 7-back into the general expression of Equation
4 D, H=--+3 Rv
1
20
dp2v R(l-R)-+ Dm
1
1/2x,d,
+ Dm/widP2 (8)
The final term involves the greatest uncertainty although its general form is known. The above equation can be used to select operating parameters for optimal performance. The optimization of equations of this form, both for minimum plate height and maximum speed, has been discussed quite fully elsewhere (8) and need not be elaborated on here. COMPARISON OF EXPERIMENTAL A N D THEORETICAL PLATE HEIGHT
There has been no careful and quantitative experimental study of the variables influencing plate height in gel filtration chromatography. There are very few cases in which the plate height has even been reported. Our principal approach in view of this was to calculate plate heights from published elution profiles. These isolated values, as they first appear, reveal no general relationships as they were obtained from miscellaneous experiments carried out with different solutes, a variable particle size, etc. They can be examined in relationship to one another however, if reduced coordinates are used (3). Thus we focus on two dimensionless parameters, the reduced plate height, h h
=
H/d,
(9)
and its dependence on the reduced velocity, Y
Literature Data on Gel Filtration and Parameters Derived from Published Reports Reported parameters Pln-, Calculated Darameters Ill = Col. dimension ~ 0 1 . Particle diam. (em.) rate Reduced Reduced Solute (cm.) diam./d, Dry Swollen (ml./h.) velocity ( B ) Plateht,.H (cm.) plateht. h Col. material and ref. 290 0.005. 0,014 12 13.9 Peak I 0.239 17.3 Sephadex G-200 (19) Human plasma 4 X 128 Peak I1 0.228 16.5 3 X 130 199 0.0068 0.015 ... ... Peak I 1.01 66.9 G-100 (19) P-Hy. st. de.* Peak I1 0.731 48.3 507 0.005. 0,014 68 25.7 Peak I 0.259 18.8 G-200 ( 1 6 ) Human plasma 7 X 50 Peak I1 0.710 51.5 Peak 111 0.246 17.8 X 110 350 0.0052 0.014 40 30.7 Peak I 0.253 17.6 G-200 (16) Serum albumin 5 Peak I1 0.167 11.7 1 . 5 X 38 99.3 0.0068“ 0.015 ... ... 0.088 5.82 G-100 (16) Protein and enzyme 0.0112 ... 25 ... 0.116 Human serum 2.5 X 17.5 ... Dextran gel (18) protein 0.064c 1.89 27.7 120 0.0223 0.034 3 . 5 X 3 9 . 4 103 Sephadex G-25 (19) Glycine 0 . 0 1 1 ~ 1.40 3.66 0.008 18 228 0.0052 1.8 X 49 G-25 (19) Glycine 6.92 172 U.0Sc 160 0.0052 0.012 328 3.8 X 40 G-100 (19) Albumin 15.5 129 0.48 0.031 50 0.0112 129 4 x 42 G-200 (19) Albumin 0.08 4.53 37.1 25 0.018 0.0064 226 4 X 42 G-200 (19) Albumin Obtained from out,side information. b 6-hydroxysteroid dehydrogenase. c Reported by the authors. Table 1.
I
5
998
0
ANALYTICAL CHEMISTRY
_
L I”W
50
t
I
0
O
0
0
0
b
I
20
O
60
60
100
Reduced velocity,
Figure 1.
1
1
I
43
123
I
140
1
lh0
150
y
Reduced plate height vs. reduced velocity
The experimental points a r e from Table 1. The theoretical curve is from present theory using reasonable parameters which should give typical experimental valuer
v = d,v/D, In terms of these new parameters the plate height expressed by Equation 8 becomes 4 1 1 h = s u + zR ( l - R ) v +
The reduced plate height not only transforms various experimental measurements to a common basis but it also provides a criterion of excellence for the chromatographic system. At moderate values of Y (say 0.2-10) the h value for an inherently efficient column should be less than ten and in some cases as low as one or two. Experimental Values from the Literature. The pertinent data obtained from various literature sources (derived from the work of Porath and coworkers) are reported in the left hand part of Table I. The right hand side shows the calculated parameters related t o plate height. The estimation of reduced velocity requires knonledge of the diffusion coefficient D,. Gosting (6) has reported D , = 1.06 X cm.2/'second for glycine and 6.6 X 10-7 cm.2/second for albumin, both in water at 25' C. The buffered solutions commonly used in this work are probably sufficiently close in properties to water to make these approximations valid. In addition we have assumed that D , is essentially the same for components of human plasma and serum protein as reported above for albumin. In several cases reduced velocities are not given because adequate approximations could not be made. Since compiling these data a paper by Smith and Kollmansberger (80) has appeared with numerous plate height values reported on a gel permeation (hydrophobic) system. These results are not summarized here because
the column consisted of coupled segments filled with different materials (a heterogeneous column) and also because pertinent information is missing on particle size, flow velocity, etc. Nonetheless these results are significant in a general way and will be mentioned again later. In Figure 1 are shown some of the reduced plate height values from Table I. These values scatter very badly and show little correlation with reduced velocity, Y . Some of the h values are unreasonably high, in most cases greater than 10. It is very much worthwhile inquiring why some of these columns are intrinsically poor, compared with known high efficiency columns (particularly GC columns) since means of improvement may stem from such inquiry. Unfortunately the present paucity of data leaves little concrete evidence to base conclusions on. Note that in comparing some gel filtration results unfavorably with other methods we do not mean to detract from the technique but to improve it-its immense value is already well established in the literature of biochemistry. It is always possible in chromatographic systems that the stationary phase term, Cv, leads to excessive plate height, particularly a t high velocities. This term appears as the center one on the right hand side of Equation 11. It reaches a maximum for components with R value 0.5, a t which point its contribution to h is u/80. Thus even a t Y = 160, near the most extreme experimental point in Figure 1, this term would contribute only a small amounta value of two-to the reduced plate height. Its contribution to H would be less than 0.5 for most points. This is in agreement with the general conclusion that stationary phase diffusion is rarely a rate-controlling step in liquid chromatography as long as the stationary phase diffusion coefficient is
m'
(column diem.) (part. diam.)
Figure 2. Reduced plate heights vs. the ratio of column diameter to particle diameter, m The solid lines arbitrarily delineate the general range of the experimental points, obtained from Tobie I
comparable to that of the mobile phase (2). Our present calculation is based on the conclusions by Horowitz and Fenichel (8) that D,/D, is usually -2/3, a conclusion that could be in error occasionally but not sufficiently often to explain the relatively large experimental plate height values of Figure 1. Another possible source of scatter and of the frequently high values of plate height could reside in effects outside the column-Le., in large dead volumes and times connected with the processes of injection and detection. This is difficult to determine without a detailed investigation of apparatus. It is very possible that occasional loss of resolution results from the microscopic packing characteristics of the particles. This could also explain the erratic nature of some results. It has been shown quite conclusively that gas chromatographic columns become more efficient as a reduction is made in the column-to-particle diameter ratio (81). Theory indicates that this trend may be true for all forms of chromatography (8). We note from the column dimensions reported in Table I that fairly wide columns have been used-from 1.5 to 4 em. These diameters may be needed for sample capacity, but they may a t the same time be excessive for achieving a high degree of resolution. To test this hypothesis, a plot of reduced plate height us. m (column diameter/particle diameter) was made. As shown in Figure 2, there is a tendency for h to increase with m despite the great amount of scatter. The points seem to fall in the broad upswinging region between the solid lines. Part of the scatter may result from the fact that h also depends on velocity and other factors. The m VOL. 38, NO. 0,
JULY 1966
999
values (Table I) are exceedingly large, ranging from 100 to 500. I n columns of excessive diameter (sometimes referred to as preparative columns) some unusual phenomena occur which require unusual modification to the usual plate height expression of Equation 2 (7). For one thing plate height acquires a dependence on column length. I n order to test whether the present results fall in this category, we have calculated the “transition length,” Ld, for several columns, where (4)
and d, is the column diameter (the criterion presented by Ld is adequate despite the high v values). With one exception these lengths are the order of lo4 cm., much greater than the column length. Thus the columns do indeed fall into the special category, characterized by a lack of equilibrium across the lateral dimensions of the column and by unusual plate height effects. I n order to make some comparisons with the column efficiency which should be obtainable in a small bore tube, we have calculated theoretical plate height values assuming m = 20 (a column of roughly 1 mm. diameter). Equation 11 has been used, neglecting the first two terms on the right because of their small contributions. Estimates of the X, and wI parameters are those obtained from random walk theory ( 2 ) . The results of this calculation are shown by the solid cuve in Figure 1. This curve in no sense represents a conceptual minimum; it should roughly reflect typical experimental values. The experimental work reported here usually, but not always, exhibits a higher plate height than the theoretical curve. Evidence from Gel Permeation Columns. The previously cited work of Smith and Kollmansberger (ZO), while not directly comparable to the above data fcr the reasons stated, is nevertheless indicative on several points. First of all it offers evidence in support of the plate height equations used here; specifically, a plot of plate height us. velocity is concave down as predicted by the coupling theory of Equation 2 (the authors also note a correspondence with film diffusion theory). Second, rough values of reduced plate height can be calculated by estimating particle
1000
ANALYTICAL CHEMISTRY
size. Assuming d, = 60p (a value, often used in gel permeation work), and noting that their plate height is typically about 500p,it is concluded that the reduced plate height is of the order of 10. This is roughly consistent with several, total plate numbers mentioned by Moore and Hendrickson (13). These values are comparable with those obtained from good-but not the very best-gel filtration columns. Conclusions. There is no inherent reason why a majority of gel filtration columns cannot be improved in efficiency if enough is understood about the fundamental mechanisms in operation. One element of improvement may involve a reduction in column diameter. This requires a compromise with sample capacity that must ultimately be judged by the operator. Packed columns down to a fraction of a millimeter have been used successfully in gas chromatography (14). With proper detection methods a like diameter could perhaps be applied to gel filtration. Figure 1 shows that, in contrast to most forms of gas chromatography, plate height is not greatly increased at high relative velocities-Le., velocities greatly exceeding that corresponding to minimum plate height a t about v = 1. The coupling phenomenon ( 2 ) , described in the last term of Equation 2, is partly responsible for a slow increase in plate height with velocity in both gas and liquid forms. I n contrast to gas chromatography, gel permeation (perhaps in common with most liquid chromatography) has a relatively small stationary phase term which makes no significant contribution until Y >> 100. Thus high relative velocities can be, and have commonly been used leading to an increased speed separation. The absolute flow velocity is, of course, less than that in gas chromatography, but not as much smaller as indicated by the ratio of those velocities which yield a minimum in the plate height curve. LITERATURE CITED
( 1 ) Boyack, J. R., Giddings, J. C., ANAL.CHEM.36, 1229 (1964). ( 2 ) Giddings, ,J; C., “Dynamics of Chro-
matography,
Marcel Dekker, Inc.,
New York, 1965. (3) Giddings, J. C., J. Chromatog. 13,
301 (1964). (4) Giddings, J. C., J. Gas Chromatog. 1. 12 (1963). (5) ’Giddings,’J. C., J . Phys. Chem. 68, 184 (1964). (6) Gosting, L. T., “Advances in Protein
Chemistry,” M. L. Anson, K. Bailey, J. T. Edsall, eds., Vol. 11, p. 429-554, Academic Press, New York, 1956. (7) Hawkes, S. J., Giddings, J. C., ANAL. CHEM.36, 2229 (1964). ( 8 ) Horowitz, S. B., Fenichel, I. R., J . Phys. Chem. 68,3378 (1964). (9) Knox, J. H., ANAL.CHEM.38, 253 (1966). (10) Knox, J. H., McLaren, L., Ibid., 36, 1477 (1964). (11) Laurent, T. C., Killar‘der, J., J. Chromatog. 14, 317 (1964). (12) Moore, J. C., J . Polymer Sci. A2, 835 (1964). (13) Moore, J. C., Hendrickson, J. G., J . Polymer Sci. C3, 223 (1965). (14) Myers, M. N., Giddings, J. C., ANAL.CHEM.38, 294 (1966). (15) Ogston, A. G., Trans. Faraday SOC. 54, 1754 (1958). (16) Porath, J., “Advances in Protein
Chemistry,” C. B. Anfinsen, M. L. Anson, K. Bailey, and J. T. Edsall, eds., Vol. 17, p. 209, Academic Press, New York, 1962. (17) Porath, J., Pure A p p l . Chem. 6, 233
f\____,. 1 Qfi.?) (18) Porath, J., Flodin, P., Nature 183, 1657 (1969). (19) Porath, ‘ J., Flodin, P., “Protides of the Biological Fluids,” H. Peeters, ed., 10th ed., p. 290, American Elsevier, New York, 1963. (20) Smith, W. B., Kollmansberger, A., J . Phys. Chem. 69, 4157 (1965). (21) Sternberg, J. C., Poulson, R. E., ANAL.CHEY.36, 1492 (1964).
RECEIVED for review January 31, 1966. Accepted May 2, 1966. This investigation was supported by Public Health Service Research Grant GM 10851-09 from the National Institutes of Health.
Correction Cont in uo us Ohmic Pola r iza tion Compensator for a Vo It a mmetric A p p a rat us Utilizing Operational Amplifiers I n this article by Dirk Pouli, James
R. Huff, and James C. Pearson [ANAL. CHEW 38, 383 (1966)] on page 383 a n error appears in the legend for Figure 2 . “RL 100 kiloohms” should read “RL Load resistor.”
