1314
Anal. Chern. 1981, 53, 1314-1317
Verification of Retention and Zone Spreading Equations in Sedimentation Field-Flow Fractionation George Karaiskakis,' Marcus N. Myers, Karin
D. Caldwell,
and J. Calvin Glddlngs"
Department of Chernistty, University of Utah, Salt Lake City, Utah 84 7 72
I t Is argued that accurate characterlzatlon of complex partlcle and macromolecule populatlons by field-flow fractionation (FFF) hinges, in large part, on the agreement between FFF theory and experlment with respect to retentlon and zone Spreading. After briefly reviewing past comparlsons of theory and experlment, we summarlze the relevant theoretlcal equatlons. Two sedlmentatlon FFF systems are descrlbed and results are presented for polystyrene latex beads that are presumably well characterized. Very close ( 1% ) agreement In retention measurements and theory for some samples contrasts strongly with cases havlng serlous dlscrepancies. I t is suggested that the dlscrepancles are due to the inaccurate speclflcatlons glven for partlcle dlameters. The zone spreadlng (plate helght) measurements verify this, demonstratlng rather uniform consistency wlth the retention measurements but not wlth some reported dlameters. I t ls concluded that theory and experiment agree very well (a few percent) for retention parameters and withln 5-10% for plate height parameters.
-
THEORY The theory of FFF has been developed and refined in several papers (6, 9-11). Below we summarize the basic relationships which govern retention and zone spreading in FFF systems, with particular focus on sedimentation FFF. Retention. The basic retention equation for FFF shows retention ratio R to be uniquely related to the reduced layer thickness of the zone, X, in the following manner
R = 6X[c0th 1/2h - 2h]
Retention ratio R is determined experimentally as the ratio of the column void volume Vo to the retention volume V, of the component zone
R = Vo/Vr
A-0
'Present address: Department of Chemistry,Univeristy of Patras, Patras, Greece.
(2)
Under conditions of high retention the solute layer is strongly compacted, leading to small values for parameter A. Under these circumstances the value of the bracketed term in eq 1 approaches unity, and we have lim R = 6X
One of the analytical strengths of field-flow fractionation (FFF) is that separation occurs in a channel of such simple geometry, within which there is flow of such uniform profile, that exact equations can be developed relating such component's retention and zone spreading to underlying physicochemical parameters. Presumably, then, the measurement of a solute's behavior in an FFF channel can be used to deduce these physicochemicalparameters, thus helping characterize or identify the solute material. It is therefore important to determine the closeness of the agreement between theory and experiment in FFF to assess the potential accuracy of this procedure. Consequently, a number of studies of retention and zone spreading have been made by using various subtechniques of FFF (1-8). Generally, the retention equations have been well verified, often accurate to within a few percent. Zone spreading measurements are subject to larger relative uncertainties than retention measurements. In the most meticulous of previous studies, using linear polystyrene polymers in 11different thermal FFF channels, the departure from theory averaged approximately 30% (8). SedimentationFFF, using polystyrene latex beads as sample probes, would appear to be an ideal combination to test the agreement between theory and experiment. Such beads are used as standards in electron microscope measurements and the necessary size, density, and polydispersity parameters are available from the manufacturer. Earlier experiments with such beads verified the applicability of the FFF retention equations but encountered large discrepancies in zone spreading (2). In this study we will reconfirm the retention studies and then attempt to improve zone spreading measurements to a level of accuracy where meaningful comparisons with theory can be made.
(1)
(3)
In the special case of sedimentation FFF the reduced layer thickness takes the form (4)
where G is the acceleration (gravitational) field imposed on solute particles of mass m and density pa, p is the carrier density, w is the thickness of the flow channel, T i s the temperature, and k the Boltzmann constant. In the case of spherical solute particles, X can be expressed in terms of particle diameter d
X=
6kT
nd3ApGw
where Ap is the density difference between solute and solvent, Ps - P.
Plate Height. Zone spreading is measured by plate height H exactly as in chromatography. The factors influencing plate height in field-flow fractionation can be summarized in a relationship similar to the equation governing chromatography in capillary tubes (12)
where D is the diffusion coefficient of the solute in the solvent or carrier fluid, ( u ) the mean linear fluid velocity, and x the A-dependent nonequilibrium coefficient which has the following limit at high retention (6): lim A-0
x = 24X3
(7)
The substitution of eq 5 into 7 and the latter into eq 6, combined with the Stokes-Einstein equation for D, leads to the following coefficient of ( u ) for the second term:
0003-2700/81/0353-1314$01,25/00 1981 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
1315
// The first term on the right-hand side in eq 6 describes zone spreading due to longitudinal diffusion, the second accounts for nonequilibrium or mass transfer effects, and the third is a composite term which includes any disturbances due to imperfect channel design, relaxation processes (initial establishment of the zonal equilibrium distribution), width of the injection slug, end effects, polydispersity, etc. In practical work, the first term of eq 6 is negligible because of the sluggish diffusion of large solute species. If stop-flow procedures are used, the relaxation contribution to plate height is negligible. With proper equipment design and careful procedures, all contributions to CHi except polydispersity can be practically eliminated, leaving CHi as a measure of the polydispersity of the sample. If the sample is indeed polydisperse, its different components experience different, degrees of retention in the channel due to differences in maw, and as a result the composite zone is broadened. The polyldispersity contribution to peak variance gP2can be calculated as (2)
where X is the distance migrated by a particle of diameter d in a given time, and od2is the variance in the particle diameter within the sample. This equation can be converted to a plate height expresision based on FFF parameters in a series of simple steps (2). For a well-retained peak (A 0) in a uniform channel of length L the expression becomes +
H = gL(ad/d)' (10) If we assume that polydispersity and nonequilibrium are the only important plate height terms, H assumes the following simple form, linear in velocity ( u ) : H = C ( u ) Hp (11)
+
By using eq 8 and 10 in conjunction with eq 11,it is possible to make quantitative comparisons between experimental and theoretical nonequilibrium and polydispersity contributions to the zone spreading in sedimentation FFF. Secondary Flow. The foregoing theoretical equations hinge on the presence of ~t uniform, axially directed parabolic flow in the channel in which edge effects are neglected. However, it is known that fluid flowing in a spinning duct is subject to a nonaxially directed secondary flow (13),which, if it is strong enough, will circulate the solute over the channel cross section. Significant circulation would be expected to hinder retention and increase plate height, causing important departures from the theoretical equations noted above. However, as noted in an early paper (2),secondary flow is inhibited by a rectangular channel of large breadth to thickness ratio (- lo2),such as we routinely use. .Nonetheless, some secondary flow will exist. Its effects are expected to be quite different if the flow and spin directions are the same (parallel) rather than acting in opposition (antiparallel). If the former, there will be a gradual motion of solute (in the case where solute is compressed toward the outer wall of the channel) toward the channel edges If the latter, the solute will move toward the midway plane dividing the channel breadth a. Figure 1 shows a schematic representation of the former. The true situation can be more complex, with the lateral flow profie dividing into additional cells under some circumstances. There is some possibility that a mild secondary flow under antiparallel flow-spin conditions would remove solute from the edges of the channel where, normally, a perturbation from parabolic flow occurs. However, this benefit may be offset by the "updraft" at the center tending to carry solute to higher
p
-I
Solute layer
Figure 1. Secondary flow in a sedimentation FFF channel in which the direction of flow and spin are the same. The arrows are reversed when the directions of flow and spin are opposite.
than calculated elevations and by an undue concentration of solute near the midway plane. These factors require theoretical attention. Our emphasis here is on qualitative aspecb of secondary flow as a prelude to some empirical tests to follow.
EXPERIMENTAL SECTION The general characteristics of our experimental sedimentation FFF device have been described (14). Two sedimentation FFF systems which differed mainly with respect to channel thickness were used in this study. The dimensions of channel I were w = 0.0254 cm, length L = 83.3 cm, and breadth a = 2.00 cm; channel I1 had w = 0.0127 cm, L = 79.4 cm, and a = 2.00 cm. The rotor radius was 7.7 cm for each of the two instruments. Control of rotation for the channel I system was provided by equipment designed and built in this laboratory, while the channel I1 system was controlled by a D-C motor speed control, type DPM-6130 E, from Bodine Electric Co., Denver, CO. The rates of rotation were determined in both cases by an in-house built digital counter. The carrier fluid was degassed at 80 "C and pumped through channel I1 by a Model Minipuls 2 metering pump from Gilson Medical Electronics, Middleton, WI, and through channel I by a metering pump of type CMP-1 from Laboratory Data Control Inc., Riviera Beach, FL. Following channel I1 was a UV monitor Model 1205 also from Laboratory Data Control, Inc., Riviera Beach, FL, while the effluent from channel I was monitored by an Altex Analytical UV detector, Model 153, from Beckman Instruments, Inc., Berkeley, CA. Both detectors used a wavelength of 254 nm, and their respective signals were fed to Omniscribe recorders from Houston Instrument, Inc., Austin, TX. The samples employed in this study were spherical polystyrene latex beads from Dow Diagnostics Indianapolis, Inc., nominal particle diameters of 0.220,0.357,0.481, 0.620, and 0.945 pm, respectively. The standard deviations in these diameters were given as 0.0065, 0.0056, 0.0018, 0.0076, and 0.0064 pm, respectively. All beads were assumed to have a density of 1.05 g/cm3. The samples, as supplied by the manufacturer, were all suspensions of 10% (by weight) solids in an aqueous detergent. The carrier solutions were doubly distilled water containing FL-70 detergent from Fisher Scientific Co., Pittsburgh, PA, 0.5% FL-70 in the case of channel 11, and 0.1% in the case of channel I. To both carriers was added a small amount (0.02%) of sodium azide as a bacterocide. The samples were introduced by means of microsyringes (10 and 25 LLL) from Hamilton Co., Reno, NV. In all exper-
1316
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
Table I. FFF Retention Data for Polystyrene Latex Spheresa d (nominal value, in fim)
0.220 0.357 0.357 0.481 0.620 0.945
rpm
v, mm/s
1440
0.3-3.3 1.8-7.4
w ,vm 254 127 254 2 54 127 127
0.032 0.014 880 0.1-3.4 0.021 500 0.3-3.3 0.020 600 0.8-7.6 0.017 400 1.4-7.7 0.01 3 a The calculated values of R, Rcdcd are based on nominal (manufacturer’s)d values using eq 5 runs were made at 25 “C in a carrier with density p = 1.00 g/mL. 1500
riments the injected sample was 1 pL. The injection was immediately followed by a 12-min stop-flow period to allow for relaxation. Retention ratio R was measured as the volume of a nonretained peak (acetone) divided by the elution volume recorded for the sample peak. Both volumes were corrected for the dead volume encountered between channel exit and detector cell. These dead volumes were 0.15 mL for channel I1 and 0.20 mL for channel I. Plate height calculations were based on the measured peak width at half-height on the recorder tracing.
i: 0.004 0.085 i. 0,001 0.124 ? 0.002 0.122 i: 0.003 0.101 i: 0.002 0.080 ? 0.001
0.191
I
+ 2.7
0.186 0.084 0.1 20 0.150
-18.7
0.100
t1.0
1-1.2 t 3.3
0.064 t 25.0 combined with eq 1. The
/
I
RESULTS AND DISCUSSION Retention. The results from retention measurements are reported in Table I. The two different channels employed are indicated by their different thickness, w, 254 and 127 pm, respectively. The experimental R values were obtained as an average of five to ten individual values calculated at different flow velocities extending over the range shown under the ( v ) column. The overall agreement between experimentaland calculated R values is reasonably good, and in some specific cases, excellent. However, in other cases the departures are substantial. Because the various FFF retention experiments appeared to be quite consistent with one another, we have investigated the possibility that the reported particle diameters might be in error, thus leading to the observed discrepancy. We have noted substantial discrepancies in previous studies with latex beads from another manufacturer (3). The diameter and polydispersity of the latex spheres were determined by the manufacturer from electron microscopic observations. However, as indicated by independent electron microscopy studies on other polystyrene latex preparations from DOW,there are, in some instances, substantial discrepancies between results obtained by different observers (15, 16). The impact of the electron beam apparently causes a noticeable shrinkage of latices, which in turn makes recorded particle diameters highly dependent on the microscopic procedure used to obtain such data. The size distribution for any given sample is recorded from micrographs of 102-103beads per picture. As these beads are all exposed to the electron beam in a uniform manner, it is reasonable to assume a uniform shrinkage of all particles. The relative standard deviation in size should thus be a reasonably reliable number, particularly for samples where the number of measured beads approaches lo3. Because of the excellent agreement between measured and calculated retention in three of the six systems studied, and the self-consistency of the results reflected in the low standard deviations in R values and the very close agreement of two separate channels with the calculated values for 0.357-pm beads, it appears that retention in sedimentation FFF can be predicted quite accurately under the experimental conditions used here. The annoying discrepanciesfound for three other systems can, we have found, possibly be traced to errors in reported bead diameters.
(v) (rnrnlsec)
Flgure 2. Plate height vs. flow velocity plots for nominal 0.481-pm polystyrene latex beads.
Fortunately, the study of peak spreading, which we initially undertook as an independent investigation, can be used to produce additional evidence bearing on this matter. This is a consequence of the fact that both retention (eq 1 and 3) and the nonequilibriumcontribution to plate height (eq 7) depend solely on A, thus linking the two together with a common, size-dependent (eq 5) parameter. Zone Spreading. Equation 11 suggests that the plate height-flow velocity relationship is linear, with slope C and intercept Hp’ This conclusion is substantially verified by our data, an example of which we have plotted in Figure 2. In that Hpand C are calculable for a well-defined particle sample using eq 8 and 10, the experimental plot can be compared with a theoretical plot, the latter shown as “calculated, A” in Figure 2. The figure shows that the discrepancy between the two is substantial, the slope of the experimental line being very much lower. Since all C H i terms add to the plate height, neglect of important velocity-dependent terms would result in too low a value for the calculated slope. Our observed plate heights are in this case substantially lower than the calculated values, indicating that some other problem must exist. Next we examine the possibility that the discrepancy is caused by an incorrect particle diameter. The slope C would reflect this error dramatically because of the 8 power dependence of C on d (eq 8). In order to check this possibility, we have used A values from the retention data of Table I to calculate C (C = 24A3w2/D with D calculated from D = k T / 3 ~ q ( 6 k T / ~ A w A p G )rather ~ / ~ ) than eq 5 which requires some preassumed d value. When C is calculated on this basis, the slope is in better agreement with the data, as shown by the line “calculated, B” in Figure 2. This lends very strong support to the contention that the reported particle diameter is in error and is responsible for the discrepancy. The con-
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
1317
Table 11. Plate Height Parameters for Polystyrene Latex Spheres (Temperature 25 ‘C,carrier viscosity 0.01 P)
d (nominal value, in pm)
exptl
w ,pm 2 54 127 1254 1254 2 54 !154 1127 1127
0.481 0.620 0.945
‘Parallel spin flow. nominal d values.
6o
slope C, s
-
0.220 0.357 0.357
28.27 f 0.93 f 11.93 (11.72 t (11.71 ?: 14.26 ?: 2.80 t 1.781t
*
1.78 0.04 0.15 0.15)~’ 0.94 0.12 0.05
Antiparallel spin flow,
..-.
E
E ..I
I
2.0
A d
30.70 0.96 12.53
33.53 0.88 11.38
17.48 2.84 1.99
30.80 2.68 1.05
3.0
2.8 f 3.5 1.5 f 0.2 2.5 f 0.3 (2.5 f 0 . 3 ) ~ (3.5 f 0.3)a 3.0 + 1.9 1.8 ?r. 0.5 1.4 f 0.2
Values calculated from retention values.
1
1.0
BC
0.14)ab
t
0 0
intercept, mm expected from manufacturer’s exptl data
4.0
(v) (rnm/sec) Figure 3. Experimental plate height vs. flow velocity plot for 0.357-pm
nominal beads under condltioiis of parallel (open circles)and antiparallel (solid circles) spin flow. sistency of the two top liines in Figure 2 also verifies that the observed plate height is in good agreement with theoretical predictions. Table I1 shows the relevant results for all of the latex beads measured. Columns A and B under the “slope C” heading correspond to calculated lines A and B in Figure 2. A comparison of the “experimenltal” column and B shows agreement with 10% in all cases and down to about 5% in several cases. In all but two of the six cases the “B” calculations lie within 2 standard deviations of the experimental slope. Thus the discrepancy between the two is neither large nor particularly systematic. The results suggest that observed plate height values are indeed calculahle, if one has the correct parameter for the particles, within 51-10%. This is considerably better than previous results. The experimental intercept values to Table I1 show considerably more scatter than the slope values. This is a consequence of the smallness of the intercepts, which reflects the extremely low polydispersity of the latex beads. The reliability of the intercepts appears to be -1 mm, which corresponds to a polydispersity, ad/d of only about 1%. The tendency for the intercepts to be somewhat larger than predicted may indicate a small plate height contribution due to dead volume effects, etc., or it may reflect an error in the reported polydispersity of the latex spheres. Unfortunately, there is no second approach for determining the reliability of the manufacturer’s polydispersity ‘values,as there is for mean particle diameter values. With respect to evaluating the influence of secondary flow, we obtained plate height measurements for the 0.357-pm
6.5 1.8 1.8 0.1 1.1 0.3
Values calculated from
nominal beads under both parallel and antiparallel spin-flow conditions. The results are shown in Figure 3 and summarized in the terms in parentheses in Table 11. No systematic difference between parallel and antiparallel conditions can be discerned from the figure. The slopes reported in Table I1 are also in near perfect accord, although some noticeable difference is found in the intercepts. We conclude that secondary flow does not have a very significant effect under present experimental conditions, but we leave open the question of its importance under other circumstances. We note that we have found conditions under which there are signficant departures of theory and experiment. The problem is particularly severe for the larger beads at high flow rates. We are presently studying these departures in order to better define the range over which good theoretical-experimental agreement can be found. In conclusion, we find that we can obtain very good selfconsistency between theoretical and observed retention and plate height values in sedimentation FFF in our normal range of operations despite the lack of well-characterized particles. In fact, our results suggest that FFF data may be more reliable and exacting than data from other sources (such as the electron microscope) in reflecting the properties of particle populations, and thus in characterizing particles. We intend to pursue this avenue of investigation in future work.
LITERATURE CITED Smith, L. K.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1977, 49, 1750. Giddlngs, J. C.; Yang, F. J. F.; Myers, M. N. Anal. Chem. 1974, 46, 1817 . _ .. . Giddings, J. C.; Myers, M. N.; Moeiimer, J. F. J. Chromatogr. 1978, 149, 501. Giddings, J. C.; Lln, G. C.; Myers, M. N. J. Colloid Interface Sci. 1978, 65.67. Myers, M. N.; Caldwell, K. D.; Giddings, J. C. S e p . Sci. 1974, 9, 47. Hovingh, M. E.; Thompson, G. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195. Giddings, J. C.; Martin, M.; Myers, M. N. J. Polym. Sci. 1981, 19, 815. Martin, M.; Myers, M. N.; Giddings, J. C. J. Llq. Chromatogr. 1979, 2, 147. Giddings, J. C. J. Chem. Phys. 1988, 49, 1. Giddings, J. C. J. Chem. Educ. 1973, 50, 667. Giddings, J. C. Sep. Sci. 1973, 8, 567. Giddings, J. C. “Dynamics of Chromatography, Part I”; Marcel Dekker: New York, 1965. Benton, G. S.;Boyer, D. J. Fluid Mech. 1966,26, 69. Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Fisher, S.R. I n “Methods of Biochemical Analysls”; Glick, D., Ed.; Wiiey: New York, 1980; p 79. Heard, M. J.; Weiis, A. C.; Wiffen, R. D. Atmos. Environ. 1970, 4, 141. Porstendorfer, J.; Heyder, J. Aerosol Sci. 1972, 3 , 141.
RECEIVED for review January 19, 1981. Accepted April 30, 1981. This project was supported by National Science Foundation Grant No. CHE 79-19879.
