Anal. Chem. i9a4, 56, 1461-1466
1461
Nonequilibrium Effects in Sedimentation Field Flow Fractionation W. W. Yau* and J. J. Kirkland
E. I. d u Pont de Nemours & Company, Central Research & Development Department, Experimental Station, Wilmington, Delaware 19898
Perturbatlon of the sedlmentatlon field flow fractionatlon (SFFF) retention theory of Glddlngs can occur at the hlgher flow rates requlred for fast separatlons (e.g., 15 mln), partlcularly when tlme-delayed exponentlal force-fleld decay (TDE-SFFF) Is used wlth relatively high lnltlal force fields. While sample relaxatlon-tlme requirements normally are short In TDE-SFFF because of the hlgher lnltlal force fields, slow partlcle mass transfer during rapld force-field decay can resuit In a dlstortlon of the normal exponentlal partlcle concentratlon proflles. Thls In turn can cause longer retention tlmes than predlcted by the equilibrium retention theory. To account for thls effect, a quantltative theory has been developed to describe retention effects caused by both Incomplete partlcle relaxation and nonequillbrlum mass transfer. By use of thls klnetlc theory H Is posslble to carry out very fast quantltatlve analyses with a rapld programmed decrease of the operatlng force fleld. The quantitatlve nonequlllbrlum theory also should be applicable to all of the other FFF methods.
Sedimentation field flow fractionation (SFFF) is the most highly developed of the various FFF subtechniques, which have unique capabilities for the high-resolution separation of particulates and large macromolecules. SFFF separates species on the basis of differences in mass, and has the highest intrinsic resolving power of any of the existing FFF methods ( I ) . It has been used for fractionating a wide variety of organic and inorganic particulates (2-6) and organic macromolecules (7, 8). In addition, SFFF has been developed to carry out quantitative particle-size-distribution analyses and molecular weight measurements that are difficult or impossible to perform by other techniques (4, 5, 9). In a recent work we have described the use of time-delayed exponential force field decay SFFF (TDE-SFFF) to resolve particles over a wide size or mass range with a single experiment in a few minutes (5,10,11). After the particles relaxed to their equilibrium distance from the analytical wall with no flow, the SFFF separation is initiated with flow by maintaining the initial force field constant for a time period T. Following this, the force field is decayed exponentially also with a time constant 7. In this TDE-SFFF approach all monodisperse particles are eluted as peaks of about the same widths and in an even-resolution format. Peak detection is good for all particle sizes, and uniformly high resolution is maintained throughout the relatively fast separation. We have discovered, however, that fast SFFF separations, particularly in the TDE-SFFF mode, sometimes exhibit retention behavior that is no longer in exact agreement with an equilibrium theory developed according to the concept of particle concentration profile (1,12). We found in TDE-SFFF that large particles sometimes elute later than expected and some small particles elute earlier than predicted by the equilibrium theory. We noted that the deviation of experimental data from the equilibrium theory became greater with the shorter exponential force-fielddelayldecay time constant 0003-2700/84/0356-1481$01.50/0
7 values. A similar trend was found for fast- vs. slow-flow experiments. At higher flow rates in SFFF, experimental data essentially follow that predicted by the equilibrium theory. However, at slow flow rates a significant deviation of experimental points is found, relative to that predicted at equilibrium. Similar effects were also found with changes in channel thickness W , where a greater deviation from the equilibrium theory was found in TDE-SFFF experiments with increasing channel thicknesses. These and other data strongly suggested a kinetic or nonequilibrium factor influencing retention in fast SFFF separations. When fast SFFF separations are to be used for exact particle-size-distributionanalyses, it is important that any such kinetic effects on retention be properly taken into account to permit accurate particle-size measurements. This report describes the causes for kinetic effects in SFFF and presents a theory to account for these effects in SFFF experiments. In another publication, we will provide the detailed derivation of this nonequilibrium theory, together with appropriate data confirming its application in a wide variety of SFFF operating conditions, including those involving steric effects (13).
EXPERIMENTAL SECTION SFFF data reported in this study were obtained on a research instrument constructed in a Sorvall Model RC-5 Superspeed centrifuge (Du Pont Biomedical Products, Wilmington, DE) or a modified Model L5-50B ultracentrifuge (Beckman Instruments, Fullerton, CA). “Floating”plastic channels usually were utilized in both instruments (11). The rest of the apparatus used for the separations has been described previously ( 4 , 5 ) . A MINC-023 computer (DigitalEquipment Corp., Maynard, MA) was used to control rotor speed, collect elution data, and calculate sample particle size data. General procedures for equipment operation including sample injection, relaxation and other features have been previously reported (4). Aqueous solutions of 0.1% F1-70 cationic surfactant or 0.1% Aerosol-OT anionic surfactant (Fisher Scientific Co., Pittsburgh, PA) were used as aqueous mobile phases. Polystyrene latex standards were obtained from Dow Diagonstics (Dow Chemical Co., Midland, MI). CONCEPTS AND THEORY Movement of particles in an SFFF channel is very complex and is influenced by three major factors, as illustrated in Figure 1: the sedimentation velocity U, operating in the Y direction of the channel, i.e., in the direction of channel thickness; the mobile phase velocity profile Voin the X direction down the channel; and the natural diffusion of the sample particles. The magnitude of Vovaries as a function of the Y scale in the direction of channel thickness. Particle elution is subject to the combined effect of convective diffusion, the combined effects of the mobile phase flow and the natural particle diffusion. To account for mass transfer in SFFF, the differential expression in eq 1previously has been developed (14) to describe the change in solute concentration with respect to time ( t )and X , Y location as governed by the balance of three mass transfer terms involving: sedimentation velocity, U,, which affects the concentration gradient in the Y direction, d C / d v mobile phase velocity Vowhich affects 0 1984 American Chemical Society
1462
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984 (CHANNEL-SIDE VIEW)
I A = p / w = 0.11
(END VIEW)
1
Y.
NO RELAXATION
_.__-__--__
A
Flgure 1. Forces influencing movement of particles in SFFF channel. See text for discussion. TIME (MIN.)
FORCE FIELD
1
IA c
'
= 0.11
I
'.,::5:
:::::;.: ::., :, .:..... ....., ...,,
I
,
I
*.
"
0.21 I
I
I
0
5
10
15
RETENTION TIME, t ~ MINUTES ,
Flgure 3. Effect of relaxatlon on solute migration velocity in constant-field SFFF.
Flgure 2. Relaxation of solute concentration profile under sedimentation force field. mass transfer and the concentration gradient in the X direction, aC/aX; and normal diffusion in both the X and Y directions where D is the particle diffusion coefficient.
The solution to eq 1 (13)gives the concentration profile C at any point down the channel. The average particle migration velocity along the channel can be predicted by averaging the product of the linear mobile-phase velocity weighted by this particle concentration profile, as shown in eq 2.
vp
Vp(t) = JC(t,X,Y).Vo(Y) dX d Y
(2)
The resulting particle concentration and particle velocities are time-dependent in fast SFFF experiments, as discussed in the following sections. Relaxation of Solute Concentration Profile under a Force Field. The schematic diagrams in Figure 2 illustrate how the particle concentration profile relaxes in an SFFF channel under the influence of a sedimentation force field, as calculated by the new kinetic theory developed from eq 1. Initially at time zero, particles are homogeneously distributed across the channel. As time progresses under the force field, particles are compacted toward the bottom wall, as represented by the dotted line denoting the particle concentration profile predicted by the new kinetic theory. With the typical operating conditions used in Figure 2, at 5 min relaxation or longer, particle concentration approaches the steady state or equilibrium condition and will no longer vary with increasing relaxation time. At this point the mean of the exponential particle concentration profile expressed as the characteristic particle layer thickness 1 determines the equilibrium velocity of the particle as it moves down the channel. Retention can be described by the dimensionlessretention parameter h which equals 1/ W, where W is the the thickness of the channel (typically, 250 Fm). In Figure 2, the h value was arbitrarily selected as 0.1. Different particles relax differently under
varying conditions such as force field strength, channel thickness, etc. Therefore, the time scale of 5 min represented in Figure 2 is used for illustration purposes only and merely represents a typical case. While the required relaxation times increase with increasing channel thickness, decreasing particle-mobile p h e density differences,and increasing diffusion coefficient, exact details of a particular relaxation effect now may be quantitatively predicted by the kinetic theory. By integration of particle concentration profiles at different times such as those illustrated in Figure 2 over the channel thickness, corresponding particle velocities can be predicted at various times by eq 2. The plots in Figure 3 qualitatively illustrate particle velocities determined in this manner. In this figure the average particle velocity Vp is plotted in the Y axis (in units of cm/s) as a function of elution time, in this case from 0 to 15 min, as plotted on the X axis. Figure 3A illustrates an SFFF experiment in which no relaxation is used before attempting to elute the particles. Particles without relaxation initially assume a velocity that is two-thirds that of the maximum mobile phase velocity which occurs along the center line of the channel. As time progresses particles are pushed closer and closer to the velocity predicted by the equilibrium theory (dotted line, Figure 3A) and ultimately reach this velocity somewhere along the channel. The effect is that unrelaxed particles travel a certain length of the channel in higher-than-predicted velocity streams before they are relaxed to their equilibrium 1 values, with the result that these particles are eluted more quickly than predicted by the equilibrium theory. In the case illustrated in Figure 3A, the particles must move down the channel for about 5 min before they reach steady-state concentration profile and the equilibrium particle velocity predicted by the equilibrium theory (dashed line). The area between the solid and dotted lines in Figure 3A reflects the deviation of retention behavior from the equilibration theory-particles elute faster than predicted. It should be apparent that the elution error resulting from incomplete relaxation would be less in a slow SFFF separation as compared to a fast analysis. Conversely, as illustrated in Figure 3B, by relaxing the particles in a force field prior to starting the flow of mobile phase, retention errors are significantly reduced. In the case illustrated, a 5-min relaxation places the particles at the approximate 1 value predicted by the equilibrium theory. A fast (15 min) separation then can be carried out essentially without
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
-
I
5 MIN. RELAXATION
the reduced parameters accounting for force field and time, respectively, in the elution stage, 1 the characteristic mean or center of gravity distance for particle concentration profile away from the wall (cm), W the channel thickness (cm), k the Boltzmann constant (1.38 X (gcm2)'/(s2.deg)),T the absolute temperature (Kelvin), d, the particle diameter (cm); A p the density difference between sample component and mobile phase (g/cm3),w the centrifuge speed (rad/s), r the the radial distance from the centrifuge rotating axis to the SFFF channel (cm), D the particle diffusion coefficient (cm2/s),and t the time (min). Equation 4 describeskhe velocity of particles as predicted by the kinetic theory to account for nonequilibrium processes in the relaxation and elution steps
I
W = 0.025 cm
1483
A
-
V, = 6hV[coth ( 1 / 2 h ) - 2 h ] LequilibriumI
5 10 RETENTION TIME, f ~ MINUTES ,
0
15
Flgure 4. Effect of field strength and channel thlckness on constant-field SFFF retention: (A) channel thickness W = 0.025 cm; (6) channel thickness W = 0.050 cm.
elution error. It is apparent, therefore, that approximately complete particle relaxation is an important step in fast SFFF separations, if nonequilibrium effects are not taken into account in the retention particle-size calibration. Useful insights on the total effect of the relaxation process can be obtained with the kinetic theory. For example, in Figure 4A we see that as higher force fields are used, the nonequilibrium effect of incomplete relaxation on fasterthan-predicted retention is reduced, until no effect is obtained when complete relaxation is is affected (A = 0.05, in this case). On the other hand, as illustrated in Figure 4B, an increase in channel thickness has a very pronounced effect of increasing the relaxation error, causing particles to elute much earlier than is predicted by the equilibrium theory. Clearly, when thick channels are utilized, it is especially important that proper relaxation is carried out to minimize the effect of nonequilibrium on retention. General Solution of SFFF Particle Migration Velocity. While the insights just discussed are important for carrying out accurate constant-field SFFF experiments, there is a particular need for a general kinetic theory so that nonequilibrium effects can be accurately predicted for TDE-SFFF where the force field is continuously reduced during the separation. As a result, a general solution to particle migration velocity has been developed to account for retention errors in both relaxation and nonequilibrium effects, particularly when programmed force-field decrease is used. As shown in the general-form eq 3, the average particle migration velocity Vp is a function of both relaxation and elution force field conditions where A = - -1
w
-
6kT ad,3WApw2r
and 4Dt with X, and 4, being reduced parameters accounting for force field and time, respectively, in the relaxation stage, X and 4
.
-t
kinetic theory
where the parameter A,,, is a function of retention parameters A,, A, a,,,, and a,, and a involves the reduced retention parameter X, which is proportional to the force field am =
( r n ~ / 2+) ~(1/4X,)2
+
a, = ( n ~ / 2 ) (1/4X)' ~
Combination of the equilibrium and nonequilibrium processes in eq 4 then allows the general description of SFFF particle migration velocity under any separation conditions for a step change in the force field. By use of the approach in eq 4, the average velocity of particles can be predicted at any time as they pass down the channel. Equation 4 describes the general case of SFFF in which particles are relaxed at a different force field which may be high, equal to, or less than the force field used for elution. Note in eq 4 that as C$ becomes large-as with a thin channel, or for small particles with large diffusion coefficients, or for long separation times-the nonequilibrium portion of the equation approaches zero and is no longer a factor. With very slow SFFF separations, C$ becomes very large and the second term of the kinetic theory in eq 4 sharply decreases due to the large negative exponent. In this case the average particle migration velocity Vp approaches equilibrium (the first term of eq 4), as expected. Conversely, for fast SFFF separations the nonequilibrium portion of eq 4 becomes more prominent, and increasingly larger errors in retention occur, compared to retention predicted by the equilibrium theory. The fact that the kinetic theory expresses the average particle migration velocity Vp in terms of several uniquely interrelated physical parameters included in the reduced retention parameters X and C#I (eq 3 and 4) provides significant practical insight for manipulating SFFF separations. With the relationships in eq 4, we can now determine the relative importance in interrelationships among various operating parameters. For example, in considering parameter 4, we can see that the effect of decreasing channel thickness W by 2-fold is equivalent to increasing analysis time by 4-fold, for the same level of kinetic effect on retention. Note also that the terms a,,, and a, in eq 4 are functions of the reduced retention parameters X and A,, representing the effects of force field in the elution and relaxation states, respectively. With the general kinetic theory in eq 4 we can also predict how an equilibrated or relaxed particle Concentration profile recovers as a function of time when the force field is decreased. This provides a basis useful for predicting retention in the
1464
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984 TIME (MIN.)
I
5 MIN. RELAXATION
W
I
:0025
cm
P
I
t u
s
I
I
I
I
W
I
W = 0.050cm
W
d 0.6 k L
'..,
0.P
....................................................
OA
5
...... ....... .....
;,:.I..
t I I ( o.2
Flgure 5. Recovery of relaxed particles with sudden force-fled de-
crease.
0
5
10
15
RETENTION TIME, IR, MINUTES
more complicated case of a continuous force-field decrease, for example, in a TDE-SFFF experiment. Figure 5 shows the recovery of a fully equilibrated (relaxed) particle-concentration profile when the force field is suddenly removed (step change) as calculated by the kinetic theory. The calculated particle-concentration recovery profile shapes are considerably different than those associated with the relaxation stages depicted in Figure 2. For example, in Figure 5 we see that when the force field is suddenly removed there is a very quick movement toward their proper concentration level for particles closest to the wall. This effect is typical of a diffusion-controlled recovery process, as compared to the relaxation stage depicted in Figure 2 where gradient changes in concentration near the wall are evidence of a sedimentation-velocity controlled process. Note that the concentration profiles in Figure 5 during recovery are no longer in the usual approximate exponential form predicted for the equilibrium state. With eq 4 the effect of reducing the SFFF force field on particle velocity also can be predicted. In Figure 6A we see that in a 0.025-cm channel, particles that have been slightly under-relaxed (A, = 0.2) will elute somewhat faster than predicted by the equilibrium theory (dotted line). On the other hand, particles that have been increasingly over-relaxed by higher initial force fields (A, = 0.1 and 0.05) spend increasingly longer times at small 1 values before they reach the equilibrium 1 value under the suddenly reduced field, resulting in longer elution times relative to that predicted by the equilibrium theory. Note from the calculated plots in Figure 6B that even larger kinetic effects are predicted for thicker channels under the condition of a constant relaxation time. It is also likely with thicker channels that particles will elute earlier than expected, relative to the equilibrium theory, even when particles are relaxed in a higher force field than that used for elution stage. The two upper plots in Figure 6B represent earlier-than-expected elution, while the bottom curve represents laterthan-expected elution. The plots in Figure 6 predicted by the kinetic theory of eq 4 show that the rate of recovery of relaxed particles by diffusion actually can be the rate-determining step in a fast TDE-SFFF separation. Thus, if the force field is decreased too rapidly, particles may not be able to follow their recovery to the expected larger 1 value, and they are eluted later than expected by slower flow strbams. It is apparent that we now can explain why-in fast TDE-SFFF experiments-small particles can elute too early and large particles can elute later than predicted by the equilibrium theory. Early elution can
Flgure 6. Particle relaxation and subsequent elutlon with sudden forcbflelddecrease: (A) channel thickness W = 0.025 cm; (6)channel thickness W = 0.050 cm.
result from incomplete relaxation, and late elution from the inability of particles to diffuse to their proper 1 values under a rapid force-field decrease. Such nonequilibrium effects are especially magnified in thick channels and in fast programmed separations where the force field is removed quickly. It is interesting to note that when higher flow rates are used in TDE-SFFF separations, small A values result; consequently, kinetic effects of the type just described generally are smaller at higher flow rates for a given value of time constant 7. This effect is opposite from that intuitively anticipated from kinetic effects in other separation processes such as chromatography. The reason is that at higher flow rates the particles are forced to migrate downstream faster and elute at a small final 1 value at the channel exit, as a result of the time dependence of the force-field decay process. Conversely, at low flow rates in TDE-SFFF, particles must diffuse from the initial 1 value to a much large I value at the exit. In such cases, particles may not have sufficient time to diffuse to the expected 1 value at the exit, and the result is a longer-than-expected retention. General Solution for Particle Retention. By integration of the average particle migration velocity described in eq 4 over time to the exit of the particle a t t R at the end of the channel, a general solution for particle retention in SFFF is obtained, as illustrated in eq 5
L=
1"" V,, 0
dt
where L is the channel length (cm). The analytical solution to eq 5 can be numerically integrated (13) to obtain SFFF elution data discussed in the following section. RESULTS Theoretical retention values accounting for nonequilibrium by the relationship described in eq 5 now produce retention values very close to those observed experimentally, as illustrated by the data in Figures 7 and 8. In Figure 7 calibration plots for equilibrium and kinetic values are calculated for the particular TDE-SFFF experiments for two different time delay/decay 7 values. The effect on nonequilibrium is greater with the small 7 values, resulting in larger-than-expected retention time and particle size values, as illustrated by the experimental points. Some crossover values at the smaller particle sizes are the result of under-relaxation, causing early elution also as suggested by the experimental data for small
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
DECAYIDECAY TIME CONSTANT, T
2 MIN
4 MIN
Y
E i
h
mmmcooo ??r'"c?LD t-at-mmhl
000000
-EQUILIBRIUM THEORY
I 0.01
O
EXPERIMENTAL DATA
-- KINETIC THEORY 10
20
30
40
50
60
RETENTION TIME, tR. MINUTES
Figure 7. Effect of time delay/decay 7 values on TDE-SFFF retention: channel, 40.4 X 1.91 X 0.025 cm (ref 11): mobile phase, 0.1% Aerosol-OT; flow rate, 1.5 mL/min: initial rotor speed, 10 000 rpm; sample,0.1%of0.085pm,0.1%of0.091pm,0.04%of0.176pm, 0.04% of 0.220 pm, 0.04% of 0.312 pm, and 0.025% of 0.481 pm; sample volume, 50 pL each; injection from loop, 1.0 min at 0.5 mL/min; relaxation, 5.0 min; time delay/decay constant 7 as shown, detector: UV, 300 nm.
particles. Figure 7 shows the calculated retention values from the nonequilibrium theory of eq 5. These predicted nonequilibrium values now closely correspond to experimental data. Similar nonequilibrium effects are predicted for flow rate changes, as illustrated in Figure 8. Under the operating conditions shown, higher flow rates result in little kinetic effect since the particles are kept near the wall during the TDESFFF separation and have to diffuse only a very short distance to elute to a relatively small Zexit value. However, a t low flow rates the large difference between initial and final Z values requires more diffusion time than is available before the particles elute. Therefore, a t the flow rate of 1.5 mL/min in this experiment, particles elute later than expected from the equilibrium theory and appear larger than actual. Early elution of small particle sizes primarily is the result of under-relaxation. Again, retention values calculated from the nonequilibrium theory of eq 5 closely correspond to experimental retention values. Our studies have suggested that the apparent slight retention deviation of some of the polystyrene latex particles from a strict log linear d, vs. retention time relationship at equilibrium is likely related to inaccurate values assigned by the manufacturer to these particles. Table I lists the manufacturer's particle diameter values for a series of polystyrene latex standards that were determined by transmission electron microscopy (TEM) (column A). Also shown in Table I are values determined by TEM in this laboratory (column B), together with values measured by constant-field SFFF by Giddings (column C), constant-field SFFF in this laboratory (column D),and three different TDE-SFFF measurements in this laboratory using different channels or operating parameters (column E, F, G). Particle diameters calculated from the values in columns B-G are shown in column H with standard deviations. Clearly, the actual diameters for smaller particle-size standards (0.085,0.091, and 0.176 pm, nominal) are somewhat smaller than those reported by the manufacturer.
A
D
aco r l m
NLD 0
rlwo
CDt-W
091 000
0
1485
1466
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
t
FLOWRATE. ML/MIN
-EOUI
i 1
0.01 0
-I
5
I
LIBRI UM THEORY EXPERIMENTAL DATA KINETIC THEORY
I
I
I
10 15 20 25 RETENTION TIME, tA. MINUTES
30
Flgure 8. Effect of flow rate on TDE-SFFF retention. Conditions same as for Figure 7,except 7 = 2.0 and flow rate as indicated.
As an additional check on the accuracy of these values, measurements on the two smallest standards were carried out by quasi-elastic light scattering. While particle diameter values by this technique are known to be of a somewhat higher average in this particle size range, the results in column I in Table I still show smaller values than those reported by the manufacturer. It is interesting to note in Table I that the manufacturer’s values for the “0.220 and 0.312” pm standards closely correspond with SFFF results. On the other hand, the manufacturer’s value for the “0.481 pm” sample is somewhat smaller than that found by the variety of SFFF measurements. Another feature of the data in Table I is that the TEM values measured in this laboratory closely correspond to those of SFFF measurements. Particle size values predicted by the nonequilibrium theory of eq 5, shown in column I of Table I, now agree well with the more accurate particle-size values in column H. This excellent cross-check gives strong evidence of the effectiveness of the nonequilibrium theory to provide accurate mass or particle size measurements over a wide range of SFFF operating variables.
CONCLUSIONS As a result of this study, several important features regarding retention in SFFF have been recognized. First, the kinetic or nonequilibrium effects in field-decay SFFF have
been identified and quantitatively described. The effects of under-relaxation in both constant-field and TDE-SFFF have been characterized. Second, a quantitative understanding of the effect of operating variables is now available to enable optimization of SFFF separations. The effects of relaxation and a programmed decrease of force field on SFFF retention are now clearly understood. Third, absolute molecular weight or particle size measurements now can be performed accurately over a wide range of operating conditions. By application of the quantitative kinetic theory it is possible to carry out very fast quantitative analyses involving a rapid programmed decrease of the operating force field (e.g., TDESFFF). Very small delay/decay time constant 7 values may be used and appropriate corrections made for nonequilibrium effects to produce accurate results. Also, it appears feasible to reduce or eliminate the relaxation step and save total analysis time, since nonequilibrium effects from relaxation can be calculated by the quantitative kinetic theory and accounted for in the final results. Finally, the general nonequilibrium or kinetic theory herein described should be applicable to all of the other field flow fractionation methods. Therefore, it might now be anticipated that practical fast separations by the other FFF methods can now be comtemplated without concern about kinetic effects on expected retention.
ACKNOWLEDGMENT We sincerely thank C. H. Dilks, Jr., for assistance with the experiments, M. J. Van Kavelaar for transmission electron microscopy measurements, and R. B. Flippen for quasi-elastic light scattering data. LITERATURE CITED Giddings, J. C.; Yang, F. J. F.; Myers, M. N. Anal. Chem. 1974, 4 6 , 1917. Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Fisher, S. R. Methods Biochem. Anal. 1980, 2 6 , 79. Caldwell, K. D.; Karalskakis, G.; Giddlngs, J. C. Colloids Surf. 1981, 3 , 223. Klrkland, J. J.; Yau, W. W.; Doerner, W. A.; Grant, J. W. Anal. Chem. 1980. 52, 1977. Klrkland, J. J.; Rementer, S. W.; Yau, W. W. Anal. Chem. 1981, 53, 1730. Kirkland, J. J.; Yau, W. W.; Szoka, F. C. Sclence 1982, 725, 296. Caldwell, K. D.; Nguyen, T. T.; Glddings, J. C.; Mazzone, H. M. J . Virol. Methods 1980, 7 , 241. Klrkland, J. J.; Yau, W. W. Science 1982, 278, 121. Yang, F. J. F.; Caldwell, K. D.; Giddlngs, J. C. J. Colloid Interface Sci. 1983, 92, 81. Kirkland, J. J.; Yau, W. W. US. Patent 4285610. Klrkland, J. J.; Dilks, C. H., Jr.; Yau, W. W. J. Chromatogr. 1983,
---. Glddlngs, J. C.; Karaishakls, G.; Caldwell, K. D.;Myers, M. N. J. Callold 255. -255. --
Interface Sci. 1983, 92, 66. Yau. W. W.; Kirkland, J. J., manuscript In preparation. Krlshnamurthy, S.; Subramanlan, R. S. Sep. Sci. 1977, 12, 347.
RECEIVED for review November 8, 1983. Accepted March 19, 1984. Given in part at the Sixth International Symposium on Column Liquid Chromatography, Baden-Baden, May 3-6, 1983.
