Evaluating Dispersion in Gel Permeation Chromatography Axial Dispersion of Polymer Molecules in Packed Beds of Nonporous Glass Beads R. N. Kelley and F. W. Billmeyer, Jr. Departments of Chemistry and Materials, Rensselaer Polytechnic Institute, Troy, N . Y. 12181 The causes of zone broadening in GPC columns can be attributed to free-stream dispersion (outside the gel) and broadening due to the permeation process itself. Free-stream dispersion in the absence of permeation was studied utilizing nonporous lass beads and solutes covering a wide range o diffusivity. The major variables examined were flow rate, solute diffusivity, and particle size and size distribution of the packing beads. The experimental data are in good agreement with the theoretical model previously proposed, which incorporates molecular diffusion, eddy diffusion, and velocity-profile effects as the major causes of axial dispersion. Particle segregation effects were shown to be relatively unimportant.
9
THEEFFICIENCY of a chromatographic process depends upon the nature and extent of the dispersive mechanisms operative in the packed bed. As a result of such processes, not all the molecules of a single solute species are eluted at the same time. This leads to a spread of elution or retention times about a mean, and determines the column efficiency. An understanding of the factors affecting the distribution of retention times is a prerequisite for the successful design of chromatographic columns and packed beds. It has been established (1-5) that the gel permeation chromatography (GPC) separation is dependent upon the volumes of the solute molecules, but relatively little is known about the detailed mechanisms controlling the permeation process itself. Various models for predicting elution times in GPC have been proposed (5-12), and several approaches (4, 12-21) (1) J. C. Moore, J. Polym. Sci., Part A , 2, 835 (1964). (2) J. C. Moore and M. C. Arrington, “The Separation Mech-
anism of Gel Permeation Chromatography: Experiments with Porous Glass Column Packing Materials,” 3rd International Seminar on Gel Permeation chromatography, Geneva, Switzerland, May 1966. (3) H. Benoit, Z. Grubisic, P. Rempp, D. Decker, and J. G . Zilliox, J. Chim. Phys., 63, 1507 (1966). (4) W. B. Smith and A. Kollmansberger, J. Phys. Chem., 69, 4157 (1965). (5) E. F. Casassa, J. Polym. Sci., Part B, 5,773 (1967). (6) J. Porath, Pure Appl. Chem., 6, 233 (1963). (7) T. C. Laurent and J. Killander, J. Chromatogr., 14, 317 (1964). (8) P. G. Squire, Arch. Biochem. Biophys., 107,471 (1964). (9) W. W. Yau and C. P. Malone, J. Polym. Sci., Part B, 5 , 663 (1967). (10) W. W. Yau, H. L. Suchan, and C. P. Malone, J. Polym. Sci., Part A-2, 6, 1349 (1968). (11) J. B. Carmichael, J. Polym. Sci., Part A-2,6,572 (1968). (12) J. B. Carmichael, Macromolecules, 1, 526 (1968). (13) J. C. Giddings and K. L. Mallik, ANAL.CHEM., 38,997 (1966). (14) F. W. Billmeyer, Jr., G. W. Johnson, and R. N. Kelley, J. . Chromatogr., 34; 316 (1968). (15) F. W. Billmever. Jr.. and R. N. Kelley, ibid., P 322. i16) J. Coupek and W. Heitz, Makromol: Che& la, 286 (1968); W. Heitz and J. Coupek, “Column Efficiency in GPC,” 5th
International Seminar on Gel Permeation Chromatography, London, England, May 19-22,1968. (17) W. Heitz and W. Kern, Angewandte Makromol. Chem., 1, 150 (1967). 874
ANALYTICAL CHEMISTRY
have been taken to evaluate peak broadening, which is probably the major factor limiting the accuracy with which molecular-size distributions can be determined by GPC at the present time. Our own approach (14) proposes that broadening during flow through a GPC column arises from free-stream dispersion effects plus mass transfer of solute molecules into and out of the pores (the permeation process). We feel that the detailed mechanisms controlling the permeation process can best be elucidated when the free-stream dispersion characteristics of macromolecular solutes are better understood. Many of the concepts developed to explain chromatographic dispersion phenomena may be applied, in some part, to the GPC process, and many excellent reviews of dispersion in gaseous (22-24) and liquid (25-28) systems have been published. Conventional studies of dispersion in liquid systems have usually employed conductivity devices for determining solute concentrations within or at the exit of a packed bed. Correspondingly, most data in the literature have been obtained for aqueous systems with ionic solutes. Comparatively few data are available for organic systems and almost none have been reported for high-polymer solute molecules. The recent development of a continuous recording differential refractometer, by Waters Associates, Inc., Framingham, Mass., has facilitated dispersion measurements with organic solvents and macromolecular solutes, and has made this investigation possible. In this paper, the results of axial dispersion measurements using nonporous glass beads are presented and discussed in light of the approach previously described (14,15). The main purposes of these axial measurements are to gain an understanding of the dispersion characteristics of macromolecules as compared to those of small molecules; to determine the major variables (mechanisms) affecting free-stream dispersion in GPC; and to establish whether our theoretical approach (18) J. G. Hendrickson, “Basic Gel Permeation Chromatography
Studies VI. Peak Spreading Causes and Evaluation,” 4th International Seminar on Gel Permeation Chromatography, Miami Beach, Fla., May 22-24, 1967. (19) M. LePage, R. Beau, and A. J. deVries, Polymer Preprints, ACS Div. Polymer Chem., 8, (2), 1211 (1967). (20) L. H. Tung, J. Appl. Polym. Sci., 10,375,1271 (1966). (21) J. J. Hermans, J. Polym. Sci., Part A-2,6, 1217 (1968). (22) K. B. Bischoff, Znd. Eng. Chem., 58, 18 (1966). (23) 0. Levenspiel and K. B. Bischoff, Ado. Chem. Eng., 4, 95 (1963). (24) G. W. Johnson, Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, N. Y., Feb. 1967. (25) S. F. Miller and C. J. King, University of California Radiation Laboratory Report 11951, May 1965. (26) A. Hennico, G . Jacques, and T. Vermeulen, University of California Radiation Laboratory Report 10696, March 18, 1963. (27) T. K. Perkins and 0. C. Johnston, Soc. Petrol. Eng. J., 3, 70 (1963). ( 2 8 ) M. C. Hawley, Ph.D. Thesis, Michigan State University, 1964.
-
I
I
I I
1.0
I
I 10
8
I I
I
I
I 1
100
1000
Figure 1. Variation of dispersion parameter ( h P ) with aspect ratio ( 6 ) as a function of p for uniform packing From Johnson (24); p = eddy diffusivity/(molecular diffusivity eddy dmusivity)
+
adequately describes dispersion in such systems. With these goals achieved, permeation effects may be isolated from freestream dispersion effects and studied independently.
cross section. A nonuniform velocity profile causes a spread in retention times whose magnitude is determined primarily by radial (transverse) diffusion. Johnson (24) applied this approach to gaseous systems and stated that a major portion of the observed broadening in liquid systems results from velocity-profile effects. Sie and Rijnders (32) found this approach valuable for describing band broadening in packed chromatographic columns for gaseous systems, and point out that it may also be useful in liquid chromatography. We have previously indicated (14, 15) the possible value of utilizing a velocity-profile approach to explain broadening in GPC. Most chromatographic column-packing materials have a relatively wide particle-size range. During column packing, the particles can segregate, producing some cross sections having a small-particle-diameter packing and others having large-diameter packing. Such particle-size segregation, coupled with variations in packing density, presents variable resistance to the flowing fluid and leads to a nonuniform velocity profile. Because the column void fraction is greater near the wall, the flowing fluid encounters less resistance there, and the average velocity near the wall is correspondingly greater than in the center of the bed. Several investigators have reported this behavior (24, 34-37). The overall longitudinal-dispersion number D is therefore assumed to be the sum of contributions from molecular diffusion, eddy diffusion, and velocity-profile effects operating in the column: D =
THEORY The theoretical-plate concept as expressed by the height equivalent to a theoretical plate (HETP) has proved to be extremely useful for evaluating the separation efficiency of chromatographic columns. Several theories have been proposed (13, 21, 29-32) to predict HETP in gaseous or liquid chromatographic systems. In general, it is postulated that HETP
=
2D/U
+ (a mass transfer contribution)
(1)
where D is an overall dispersion number and U is the average linear flow velocity. The assumptions and equations involved in deriving Equation 1 have been discussed in a previous paper (14).
Van Deemter (29) assumed that the dispersion number was composed of a longitudinal molecular-diffusion term and an eddy-diffusion term and that the mass-transfer contribution was a linear function of velocity. For some chromatographic systems, this approach did not correlate well with experimental results. It was to explain these deviations that Giddings developed his coupling theory (30, 33). In this theory, the eddy diffusivity is coupled in a nonadditive manner with the mobile-phase resistance to nonequilibrium mass transfer. Another approach (32) which helps to explain the origin of coupling has been to extend van Deemter’s theory by incorporating an additional term to account for velocity-profile effects caused by nonuniform velocity over the column (29) J. J. van Deemter, F. Zuiderweg, and A. Klinkenberg, Chem. Eng. Sei., 5,271 (1956). (30) J. C. Giddings, ANAL. CHEM., 34, 1186 (1962). (31) H. F. Walton, in “Chromatography,” E. Heftmann, Ed., Reinhold Publishing Corp., New York, N. Y., 1961, p 299. (32) S. T. Sie and G. W. A. Rijnders, Anal. Chim. Acfa, 38, 3 (1967). (33) J. C. Giddings, ANAL.CHEM.,35, 1338 (1963).
402 molecular diffusion
+
XUdp
eddy diffusion
+
hRZU2/Dr
(2)
velocity-profile effects
is a tortuosity factor, Dz is diffusivity in the where 4 = mobile-phase, X = is a eddy-diffusion proportionality constant, d p is the effective particle diameter, h is a velocityprofile constant, R is the column radius, and D , is an average radial diffusivity. This form of the expression for the overall dispersion number was developed by Taylor (38) and Aris (39).
In cases where a mass transfer process such as adsorption or permeation is absent or negligible, as in the use of nonporous glass beads with suitable solute molecules, Equation 1 becomes : HETP
=
24Dz/U
+ 2Xdp + 2hR2U/D,
(3)
The utility of this equation for describing mobile-phase dispersion in actual liquid chromatographic systems depends greatly on the evaluation of the average radial diffusivity 6, and the appropriate velocity-profile constant h for the experimental column used. Previous work (24, 27, 32, 40) has shown that the radial diffusivity is determined by radial gradients existing within the column which come about from both molecular-diffusion and eddy-diffusion processes. (34) R. Rhoades, Ph. D. Thesis, Rensselaer Polytechnic Institute, Troy, N.Y., 1963. (35) C. E. Schwartz and J. M. Smith, Znd. Eng. Chem., 45, 1209 (1953). (36) M. Morales, C. W. Spinn, and J. M. Smith, ibid., 43, 225 (1951). (37) E. J. Cairns and J. M. Prausnitz, ibid., 51, 1441(1959). (38) G. I. Taylor, Proc. Roy. SOC.,A219, 186 (1953); A223, 446 (1954); A225,473 (1954). (39) R. Ark, Proc. Roy. Soc., h 3 5 , 67 (1956). (40) V. P. Dorweiler and R. W. Fahien, Am. Znsf. Chem. Eng. J., 5, 139 (1959). VOL. 41, NO. 7, JUNE 1969
875
Table I. Packed Columns Used in This Study Av particle diameter for Reynoldsnumber calculations, Void Mesh range fraction Plates/ftQ P 40-45 80-100 100-120 120-140 200-230 80-10/10G120 1/1 by
150 475 650
0.39 0.36 0.38 0.39 0.33 0.38
900 480
385 163 137 115 68 147
860
130
1400
\
0'5[ 0.4
O.'t
volume
0
80-100/100-120/120-140 1/1/1 by volume
0.39
I
, , , I
,
0.0001 O.Ool
Measured at 1 cc/rnin with cyclohexane solute and toluene solvent.
1
1
1
1
,
1
1
1
1
I
REYNOLD^ NUMBER
I
,
,
.O
Figure 2. Theoretical variation of HETP with Reynolds number as a function of molecular diffusivity (cm2/sec) Calculated using Equation 11, toluene data, 40-micron uniform particle diameter, 0.305 in i.d. column, and h P taken from Figure 1
These in turn depend upon the variation of velocity across the column: D ~ P =) +Dl
+W~(P)
(4)
where p is a dimensionless radial position variable: p = r / R , where r is the actual radial position variable. From Equation 4,the average radial diffusivity is
B, = +Dj
+ Ad,U
(5)
Sie and Rijnders (32) have pointed out that incorporation of this expression for D, into the velocity-profile term of Equation 3 leads to a coupling which is very similar to that of Giddings (30). Such coupling assumes that molecular diffusion and eddy diffusion are independent. By definition (39),
Use of this graph permits evaluation of the velocity-profile constant. Incorporating these concepts, the overall HETP equation describing mobile-phase dispersion may be written : HETP
=
2(+Dt/U
+ Ad,) + 2hR2/(+D1/U+ Ad,)
(11)
Such phenomena as eddy diffusion, molecular diffusion, velocity-profile effects in the mobile phase, dead-volume effects, sorptive effects, viscosity effects, and dispersion due to diffusion into and out of the pores (the permeation process) are expected to be encountered to varying degrees in the GPC process. In this study, the permeation effect has been eliminated; viscosity effects have been minimized by using the lowest practical solute concentration; sorptive effects have been minimized by using relatively-nonpolar solutes ; and dead-volume effects have been neglected. EXPERIMENTAL
where pt and p" are dummy integration variables, f i ( p ) is a velocity variation function, and h(p) is a radial-dispersion variation function. Both f i ( p ) and f i ( p ) may be obtained directly from the velocity profile according to the relationships
fib) =
U(P)/(U
- 1)
(7)
and z(P) = Dr(P)/Dr = 1
+ Pfi(P)
(8)
where
p
+ 1)
=
a/(a
=
AUd,/+Dz
(9)
and Q!
(10)
Johnson (24) evaluated the velocity-profile constant h for packed beds with uniform packing as a function of aspect ratio (6 = column diameter/particle diameter) and p utilizing the basic definition given in Equation 6 and a modified theoretical velocity profile which extended the work of Rhoades (34) to include the radial variation of void fraction in a packed bed. The resulting data are plotted in Figure 1. 876
*
ANALYTICAL CHEMISTRY
These studies were carried out utilizing a Waters Associates (Framingham, Mass.) Model 100 Gel Permeation Chromatograph with toluene as the solvent at room temperature. A microrefractometer detection cell having a volume of 10 pl was used with a l/&n. null glass. The flow system of the chromatograph was slightly modified as previously described (15) to permit accurate dispersion measurements with single 4-ft length columns. Recorder-chart speeds up to 60 in./hr were used to allow precise measurement of peak widths at half-height and of retention times. Standard GPC columns, 4 ft in length and having an inside diameter of 0.305 in., with Waters Associates end fittings, were packed with nonporous glass beads. The beads ("Superbrite," 3M Company) were carefully sieved into desired mesh cut ranges with U. S. Standard Sieve Screens and a Rotap Sieve Shaker. The sieving procedure was repeated at least six times to ensure that the amount of fines in the desired mesh range was minimal. Prior to column packing, the beads were washed in sequence with toluene, tetrahydrofuran, water, acetone, and anhydrous ether to remove surface impurities. They were dried and then immersed in a beaker containing pure toluene. The empty column with the bottom end fitting in place was completely filled with toluene to remove all air. The packing was slowly added at the top until the column was completely packed. As packing was added to the column, a small stream of solvent was continually withdrawn from the bottom and the column was inter-
Table 11. Diffusivities of Materials Used in This Study Diffusivity Concentration, in toluene,a Schmidt Substance Supplier Wt. cm2/sec numberb Hexane Fisher Scientific 0.25 1.955 X 326 Cyclohexane Matheson Coleman & Bell 0.25 2.177 x 10-5 293 Humphrey & Wilkinson Inc. 0.50 7.031 X 907 n-C36H74 2000 PSC ArRo Laboratories 0.25 4.65 X 1,370 0.25 3.33 x 10-6 1.920 3600 PS ArRo Laboratories 0.25 1.8247 x 3,500 10,300 PS Waters Associates 0.25 1.2438 x 5,130 19,800 PS Waters Associates 0.125 5.134 x lo-’ 12,400 97,200 PS Waters Associates 0.125 3.710 x 10-7 17,200 160,Ooo PS Waters Associates 4 Diffusivitiesof hexane, cyclohexane, and n-Ca6H7a were estimated from references 42 and 43; those of the other materials were taken or extrapolated from the data of reference 18. b Schmidt number NSC = v / D l . c PS = polystyrene.
z
mittently vibrated. The packing material in a toluene slurry was slowly inserted into the column with a 10-ml hypodermic syringe (with no needle), in about 5-ml quantities, making sure that no air passed into the column. After the column was completely filled, the upper end fitting was put on and the packed column was purged overnight at a high flow rate to settle the packing. The end fitting was then removed to determine if the packing had settled, and more packing material was added if necessary. In most cases, none was needed. The characteristics of the packed columns employed in the study are given in Table I. Because one of the primary objectives of this investigation was to determine the effects of diffusivity upon the resulting mobile-phase dispersion, a number of different solute molecules ranging from hexane to a low-polydispersity polystyrene (PS) of 160,000 molecular weight were employed. These solutes with pertinent supplier, concentration, and diffusivity data are listed in Table 11.
2.0
lft 1
Id, 0.8
0.4
I
\
1.2
HETP -
-
I
L \
2
0 ,
4 0 ~
/
,
I
1
1
1
1
, ! , I
1
, , , I
,
-
, , , ,
THEORETICAL RESULTS The variation (according to Equation 11) of HETP with molecular diffusivity and Reynolds number (NRE = dpU/v where v is the kinematic viscosity) is shown in Figure 2 for a system typical of those encountered in GPC. The calculations were carried out for toluene solvent, a column i.d. of 0.305 in., and a uniform packing material of 40-microns diameter. The dispersion parameter hs2 was taken as 0.48 from Figure 1 . Typical Reynolds numbers encountered in GPC range from about 0.01 to 1.0. In this region, the lower the molecular diffusivity, the higher the HETP. At high Reynolds numbers, the dispersion appears to become independent of the molecular diffusivity. On the other hand, at very low flow rates, molecular diffusion becomes controlling and the HETP increases very rapidly. In the intermediate regions, the predicted HETP curves bend concave downward with decreasing flow rate, reaching a minimum below which molecular diffusion dominates. These curves are very similar to those predicted by Giddings’ coupling theory (13, 41). The minimum is determined by the magnitude of the molecular dsusivity, and as this is lowered by a factor of 10, the minimum occurs at a Reynolds number a decade lower. Therefore, if one were to replot Figure 2 as HETP us. (Reynolds number times Schmidt number) = UdJDl, a single curve would result. This product is analogous to the “reduced velocity” which is often employed in chromatographic studies. The predicted effect of particle size on HETP as a function
of Reynolds number is shown in Figure 3. These calculations were also carried out for a toluene system, with a solute species having a molecular diffusivity of 1 O-5cm2/sec, a column i.d. of 0.305 in. and uniform particle-size packings. In each case, the dispersion parameter ha2 was estimated using Figure 1. At a given Reynolds number, the higher the particle diameter of the packing (the lower the aspect ratio) the greater the predicted dispersion. The curves show the same overall shape as in Figure 2. However, it is interesting to note that in the Reynolds number region between 0.01 and 1.0, the slope of the curve becomes much greater with increasing particle diameter. EXPERIMENTAL RESULTS AND DISCUSSION Diffusivity and Flow Rate Effects. A series of solute molecules representing a Schmidt-number range from 293 to 17,200 was used to determine the effect of molecular diffusivity on dispersion in the mobile phase as a function of Reynolds number for a column packed with nonporous glass beads. The HETP data presented in Figure 4,obtained with a column packed with 120-140 mesh nonporous glass (42) C. R. Wilke and P. Chang, Amer. Inst. Chem. Eng. J., 1,
(41) J. C. Giddings, “Dynamics of Chromatography,” Part I, Marcel Dekker, New York, 1965.
264 (1955). (43) R. C. Reid and T. K. Sherwood, “The Properties of Gases and Liquids,” McGraw-Hill, New York, 1958, p 51. VOL. 41, NO. 7 , JUNE 1969
877
20
I
1
,
I
,
I
I
,
I
8
IC n
1
A 16-
12.0 HETP
(It
lo3) 8 .O
08
4 .O
-
0 4 -0
0
! I 1
I
I
I
I
1
0 01
I
I
100
Figure 4. HETP us. Reynolds number for 120-140 mesh nonporous glass bead column 0 Hexane v 10,300PS 0 Cyclohexane 0 19,800PS 0 n-CssH74 A 97,200PS a Z,OOOPS A 160,~PS V 3,600PS Toluene solvent at room temperature; solute concentrations and diffusivity data are given in Table II beads, show that the molecular diffusivity of the solute species and the Reynolds number are important parameters determining dispersion. The dispersion of high-Schmidtnumber (low-diffusivity) solutes such as 160,000 PS in flow through the packed bed is independent of flow rate over the entire range of Reynolds numbers studied. On the other hand, HETP drops with decreasing Reynolds number for low-Schmidt-number solutes such as hexane, cyclohexane, and n-C3BH74.At a low Reynolds number, as the molecular diffusivity is decreased, the HETP increases from 0.4 X ft for 160,000 PS. The HETP of ft for hexane to 1.6 X low-Schmidt-number systems more closely approaches that of solutes with high Schmidt numbers as the Reynolds number is increased. These experimental data closely follow the predicted curves shown in Figure 2, except that the overall dispersion is greater. The broadening should be greater for the experimental case since the particle diameter is larger and there is a distribution of particle sizes which leads to a larger velocityprofile constant. It is significant, however, that the theoretically-predicted trends are closely followed experimentally. A series of runs was carried out at varying flow rates with a column packed with 40-45 mesh (-385 microns) nonporous glass beads. Cyclohexane, nC3~H74, 3600 PS, 19,800 PS, and 97,200 PS were the solutes, giving a relatively
I
I I I I
loo
"dp'pc
'
' " 1
KKK,
'
'
1
1
1
loo00
'
'
ANALYTICAL CHEMISTRY
I
I
,
I
I
I
,
10.0
1.0 REYNOLDS NUMBER
Figure 5. HETP us. Reynolds number for 40-45 mesh nonporous glass bead column Legend and conditions same as in Figure 4 wide spread (293-12,400) in Schmidt number. Because of the very low aspect ratio of 20, the dispersion was greatly increased as shown in Figure 5. The experimental curves show similar trends with decreasing diffusivity as obtained in Figure 4 and further substantiate the utility of the theoretical model. The data of Figure 5 were replotted in Figure 6 as a function of Udp/DI. For comparison purposes, the data obtained for the 120-140 mesh column are represented as a single line near the bottom of Figure 6. Almost all experimental points for the 40-45 mesh column system fall on a single line, within experimental error, as theoretically predicted. Figure 7 presents dispersion data obtained with a column packed with 200-230 mesh (-68 microns) nonporous glass beads. The HETP curves again follow the predicted trends as a function of solute diffusivity and Reynolds number, except that the expected increase in efficiencyover the 120-140 mesh column was not obtained. Microscopic examination revealed the beads to have many small surface depressions in contrast to the smooth surface of the 120-140 mesh packing material. Such depressions could account for the lower efficiency observed. Particle S u e Effects. Figure 3 shows that the mathematical model predicts a much higher dispersion level with decreasing aspect ratio. Therefore, higher efficiencies should
I.6
t
I
"
1OOOOO
Figure 6. HETP us. U D p / D 1for 40-45 mesh nonporous glass bead column Legend and conditions same as in Figure 4. Data obtained with the 120-140 mesh column are represented as a line near the bottom of the figure 878
I
0.1
,
10
REYNOLDS NUMBER
0
0
0
1
0.01
0. I
1.0
REYNOLDS NUMBER
Figure 7. HETP us. Reynolds number for 200-230 mesh nonporous glass bead column Legend and conditionssame as in Figure 4
, ,
4.0
#
I
f
'
[
I
I
t
I
(
3,2
80- IOOMESH
c
mo 2.4
-td
#
"2
I
100-l2OMESH
J
100-120MESH
2.4
120-140MESH
0.8
I
I l l
I
I
I l l
I I
I
, I
I
I
I l l
I
I
1.0
0.1
I
I
10.0
REYNOLOS NUMBER
Figure 9. Effects of particle size and segregation on dispersion of 3,600 polystyrene Legend and conditions same as in Figure 8 10
be attained as the particle size of the packing is lowered with a given diameter column. Achieving such benefits with small-diameter particles is, however, often difficult because of problems involved in obtaining a uniform particle diameter. In addition, Figure 3 shows that the slope of the HETP curve over the Reynolds-number region of interest is greatly increased with increasing particle size. Data in Figure 6 show that as the aspect ratio of the system for a fixed column diameter is lowered, the HETP is increased and the slope of the HETP curve as a function of flow rate is increased. To further illustrate particle size effects, a series of runs was made using three successive mesh cuts; 80-100, 100-120, and 120-140 mesh. The results in Figure 8 and 9 again show that the higher the particle size, the greater the dispersion in the same diameter column. The effect of aspect ratio could also be explored using columns of different diameters and a single packing material. This would eliminate problems associated with preparation of uniform packing materials at different size levels, but would introduce other problems such as end effects (15). Segregation Effects. Segregation has been shown to be a major factor contributing to dispersion in gaseous systems by greatly increasing the magnitude of the velocity-profile constant (24). Three consecutive mesh ranges were mixed to give column packings with a wide distribution of particle sizes. Figures 8 and 9 show that greatly increasing the range of the particle size distribution does not significantly increase the resulting broadening. Figure 10 compares the theoretically-predicted HETP curves for uniform particles and segregated particles with experimental data obtained with the 120-140 mesh column. The dispersion parameter h6* was evaluated for both cases from work on gaseous systems by Johnson (24). From these data, it is also evident that segregation effects are relatively unimportant in these liquid systems; this supports the conclusions of Horne, Knox, and McLaren (44). In summary, our experimental data generally follow the theoretically-predicted trends, indicating that this model should have great utility in predicting mobile-phase dispersion in liquid as well as gaseous systems. If one assumes that (44) D. S. Horne, J. H. Knox, and L. McLaren, Separation Sci., 1 (9, 531 (1966).
9-
B
8 -
0.01
0.1
1.0
10.0
REYNOLDS NUMBER
Figure 10. Experimental data for 120-140 mesh compared to theoretically predicted dispersion of segregated and uniform particles Conditions same as in Figure 4, dispersion parameter h6* taken from Ref. 24 0 Cyclohexane A 97,200 Polystyrene A Theoretical curves for cyclohexane B Theoretical curves for 97,200 polystyrene mobile-phase dispersion and broadening due to permeation into and out of the gel are statistically independent processes, it should ultimately be possible to determine the functional form of the relationship governing broadening arising from the permeation process itself. Such information would be extremely useful in developing a comprehensive model describing gel permeation chromatography. ACKNOWLEDGMENT We are grateful to John C. Moore and Glenn W. Johnson for encouragement and many helpful discussions. RECEIVED for review December 10, 1968. Accepted March 17, 1969. Work supported jointly by the Texas Division of the Dow Chemical Company and Waters Associates, Inc. In addition, one of us (R.N.K.) wishes to thank Hercules, Inc., for fellowship support. The research was carried out at Rensselaer's Materials Research Center Laboratory, a facility supported by the National Aeronautics and Space Administration, VOL. 41, NO. 7, JUNE 1969
879
