Anal. Chem. 1988, 6 0 , 1129-1135
1129
Particle Separation and Characterization by Sedimentat ion/Cyclical-Field Field-Flow Fractionation Seungho Lee, Marcus N. Myers, Ronald Beckett,’ and J. Calvin Giddings*
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The flrst experlmental lmplementatlon of cyclical-fleld fieldflow fractionation (CyFFF), a proposed method in which the strength and/or dkectlon of the applied fieid is cycled during a run, is described. The experimental system was designed for use wlth a gravltational drivlng force. Because gravity cannot be cycled, a system was deslgned In whlch the orlentatkn of the charmel was cycled relative to the gravtlatkmai field. The theory of this method was extended beyond that of earller theoretkal work to include sterk effects due to finite partlcie size. The theory was tested under a variety of condltlons; good agreement was found for all but the largest particles examined In the diameter range 5-26 pm. The method can be used to measure the sedlmentation coefficient of particles and in some cases related parameters such as particle density. Flnaiiy, the separatlon of particles in the 10-20 pm diameter range is demonstrated.
Field-flow fractionation (FFF) is a separation methodology consisting of many diverse branches, as was emphasized in a recent paper (I). The most fundamental division between FFF methods arises in the different possible operating modes including normal FFF (NlFFF), steric FFF (StFFF), hyperlayer FFF (HyFFF), cyclical-field FFF (CyFFF), and secondary-equilibriaFFF (ScFFF). Of these modes the first two (especially the first) have been widely developed (2),the third has only recently been reduced to practice (1, 3), and the fourth and fifth have remained objects of theoretical speculation (4,5). For each of these operating modes there is a further subdivision into subtechniques that depends on the nature of the driving force, normally drawn from a stable including sedimentation, thermal diffusion, electrical fields, cross flow, and magnetic fields. The object of this paper is to demonstrate the experimental viability of the fourth-listed operating mode: cyclical-field FFF. A previous theoretical study of CyFFF has shown that this operating mode is itself subject to a large number of variations including the use of different fields, a number of programming options, and different levels of transverse particle displacement relative to channel thickness w. In this first experimental exploration of CyFFF we have chosen to utilize a simple driving force consisting of gravitational sedimentation, operated without programming but with other variations to be detailed later. According to our previous terminology, this category of FFF should be called sedimentation/cyclical-field FFF, abbreviated to Sd/CyFFF (I). (We do not simplify “Sd” to “S”,as is sometimes done, because of potential confusion with the methodologies of steric FFF and secondary-equilibria FFF.) In CyFFF the magnitude and/or direction of the transverse driving force is subject to periodic variations. Hence, sample particles in the channel are driven back and forth between the walls, cycling from one transverse position (or distribution) Permanent address: Water Studies Centre, Chieholm Institute of Technology, Caulfield East, Victoria, Australia 3145. 0003-2700/88/0360-1129$01.50/0
to another. When the cycle time rCis properly adjusted, the time-averaged distribution of a given particle over the streamlines assumes a significant dependence on the rate of transport df the particle along the transverse axis. Therefore, CyFFF directly reflects transport rates (4). As a consequence, cyclical methods can be assumed to provide a means for measuring transport rates (and parameters related to transport rates) in much the same manner as other FFF techniques can be used for measuring a variety of physicochemical constants of particles and polymers (6). Specifically, in the case of Sd/CyFFF we expect an important connection to exist between separative displacement in the channel and the underlying sedimentation coefficients of constituent particles. Typical of CyFFF operation is the case in which all of the particles start a t a given reference wall (called the accumulation wall) and are then subject to field variations such that in the first phase of the cycle the particles are driven out into the channel and in the second phase back to the accumulation wall. Those particles having the greatest transport (sedimentation) rates will penetrate most deeply into the channel during their outward excursions and will be caught up, on average, in the faster streamlines near the channel center. However, if the particles penetrate well beyond the center line, they will begin to experience the slower streamlines near the opposite wall. Separation takes place, as in all FFF techniques, by the unequal time-averaged distribution of different particles across the flow velocity distribution of the channel. While the concept of CyFFF is fairly straightforward, some experimental difficulty is expected to be encountered in developing practical means for cyclical-field operation. For this reason we have chosen one of the simplest driving forces: gravitational sedimentation. Even here a complication arises because the force of gravity cannot be cycled. To bypass this difficulty we have utilized the strategy of switching the channel from one orientation to another in the fixed field available rather than attempting to cycle the field relative to a fixed channel. This strategy is implemented by the development of a channel capable of rotating around its longitudinal axis as illustrated in Figure 1. Cyclical-fieldoperation is therefore realized by the periodic rotation of this channel system so that the force of gravity is alternately directed at opposite walls. We note here that the cycling conditions can be made asymmetrical by dividing the cycle into unequal intervals so that the particles are driven more persistently toward one wall than the opposite wall. Under these circumstances the particles will tend to spend part of their time in transit between walls and another fraction of their time near the favored wall. During the latter time the particles will tend to migrate downstream according to another FFF mechanism, generally corresponding to one of the other modes of operation outlined above. Thus for the large particles (>2 pm) most readily subject to gravitational cycling, Brownian displacements are small and particles near the wall tend to migrate according to a steric FFF mechanism when subject to ongoing channel flow. Thus asymmetric cycling permits the combination of two mechanisms originating from two different modes of operation; in the present case the combination is that of pure cyclical-field FFF and steric FFF. 0 1988 American Chemical Society
iiao
ANALYTICAL
CHEMISTRY. VOL. 60, NO. 1 1 . JUNE I , 1988
STEP
Ms
Flgure 1. system.
Schematic dlagram of SBdlmentatlonlcyciicLfleld FFF
In theory, the combination of these two basic mechanisms of FFF in different proportions allows us to determine two properties of a narrow particle population. This concept is tested for the standard latex particles utilized in this study; an attempt is made to determine both particle size and density from such a combination. THEORY The velocity of U of a particle sedimenting under the influence of gravity is (7)
where F is the gravitational force acting on the particle, f is ita friction coefficient, m ita mass, pa its density, and p is the density of the fluid. T h e acceleration due to gravity is g. For a spherical particle, Stokes law gives f = 3 ~ 7 dand eq 1 becomes
where Ap = pa - p , d is the diameter of the particle, and r) is the viscosity of the fluid. Using gravity as a field for implementing cyclical FFF requires that the FFF channel be periodically rotated for 180° around ita longitudinal axis to achieve field reversal; channel rotation is a necessity here because, as noted above, the direction of gravitational forces cannot be altered. Following @ample injection, the operation begins with the channel oriented in a fued horizontal position for a sufficient period of time for the particles to reach one wall, termed the accumulation wall. Here our convention is to label the field strength as -1g in this step, with the minus sign arising because particles are driven along the negative direction of the x axis which extends from the accumulation wall out into the channel. Following this initial relaxation the channel is repeatedly rotated or 'flipped" 180°. These successive rotations lead to the effective reversal of the field, which cycles from lg to -1g with succeeding rotations. Thus, despite the fixed nature of gravity, cyclical operation is achieved relative to the frame of reference of the channel. For the square-wave cyclical operation descrihed here, the cycle is split into two operating phases (channel orientations). In general, the cycle time T , is divided into time t,, with acceleration +lg, and t2,with -1s. The times t , and t , are not necessarily equal. During their longitudinal migration in the direction of the channel axis, the particles are continuously cycled hack and forth along the transverse axis in response to the variation of the field strength The nature of the cyclical particle motion depends on the relative and absolute values of times t, and
Mode m: - q + -q
t, < t 2 ,:r
~
I
. ,,
.
.,
Flgum 2. Timsdependem behavki of field and partlcle motion in four different modes of SdlCyFFF.
t,. In general, the technique can be operated in a number of distinct modes according to the magnitudes oft, and t,. T h e four modes utilized in this study are illustrated in Figure 2 and are described in detail below. Mode I: t , = t , (=rc/2)< T ~ * .Here rC*is the time required for the particles, when driven by the field, to traverse the entire thicheaa of the channel. In mode I the particles travel a maximum distance from the accumulation wall of only lo = ut, = U T C / 2 (3) during their outward bound (+lg)motion of duration t,. Since t , < rc*,it follows that 4 < w N Urc*;thus the particles never reach the opposite channel wall at x = w. The retention ratio R is defined as the ratio of the average particle migration velocity V to the average linear velocity of the carrier ( u ) . For CyFFF operatine in mode I. R has been given by (4)
where X, = lo/w and w is the channel thickness. In the derivation of eq 4 the particle was assumed to be a point mass; that is, the effects of the volume of the particle were ignored. However, the finite size of the particle leads to the exclusion of its center of mass from a layer adjacent to each channel wall of thickness a, where a is the particle radius (8). Thus the lateral excursion of the center of mass of the particle is limited to the range between x = a and x = w - a, where x is the coordinate axis extending from the accumulation wall ( x = 0) to the opposite (depletion) wall (x = w). Therefore, the interval of the integration in eq 19 of ref 4 must be modified to [a, lo + a] in place of [O, lo] to account for finite volume effects. Following the same mathematical procedure as used in ref 4, the modified equation for the retention ratio can be obtained as (5)
where a = a / w . The final term of this equation (containing
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
0.6
1
0C 0
under steric conditions (9,lO). Although we observe a variation in y with various experimental parameters (field strength, flowrate, particle size, particle density, carrier viscosity, etc.), values of y in the range 0.5-1.0 can be assumed to reasonably represent steric migration under the relatively low flow conditions used here. Mode IV: tl < tz,T,*. Here the particles migrate by the steric mechanism intermittantly only at the accumulation wall. The retention ratio, again expressed as an average, becomes
//A'
1
I
1
I
0.2
0.4
0.6
0.8
I
1.0
A0
Flgure 3. Theoretical plot of retention ratio R vs retention parameter A,, for mode I cycHcai operation with (a = 0.01, 0.03, 0.05) and without
(a = 0) steric correction. a) represents a correction for finite particle size. As the particle size decreases, a decreases and eq 5 approaches eq 4. The steric correction term is always positive in sign and becomes increasinglyimportant as the particle size increases. The magnitude of the difference between the uncorrected and sterically corrected expressions (eq 4 and 5) for the retention ratio is illustrated in Figure 3. In both expressions, the retention ratio increases as Xo increases and reaches its maximum level when X,is about 0.75, that is, when the lateral displacement of the particles covers about 75% of the channel thickness. In addition, because X, is proportional to the particle diameter squared, the retention ratio increases with particle diameter d in this mode, until Xo approaches about 0.75. Mode 11: tl = tz= T,*. With the steric correction the time T,* becomes the time for the particles to traverse from x = a to x = w - a, a distance of lo = w - 2a. Thus
w - 2a =-
Tc*
U
(SinceT,* depends on velocity U, which can vary from particle to particle, the mode can be different for different particles.) The retention ratio, denoted as R*, is obtained from eq 5 with Xo = (w - 2 a ) / w = 1 - 2a R*
+241-
(7) The value of R* will thus be slightly greater than unity due to the exclusion of the particles from the regions adjacent to the channel walls where the fluid flow velocity has its lowest values. Mode 111: tl = t z > T,*. Here the particles arrive at the depletion wall before the +lg phase elapses and thus migrate by the steric mechanism for the remaining time, tl - T,*. When the field is reversed (to -lg), the particles traverse the channel in the opposite direction and after reaching the accumulation wall migrate sterically at that wall for time t zT,* before the cycle repeats. The retention ratio in this case becomes the weighted average
R=
1131
1
R*T,*
CY)
+ R,,(T,/~ - T,*) 7,/2
(8)
where Ra is the retention ratio for purely steric migration. The value of R* is obtained experimentally by running the sample without cycling the field during the run. Theoretically, Rat is given by (8) 6ya
Rat = - = 6 y a W
(9)
where y is a dimensionless correction factor adopted to account for the hydrodynamiccomplication of particle migration
where R , is the pure cyclical contribution to R and is given by eq 5. Clearly, if t2 = tl, eq 10 is reduced to eq 5 representing purely cyclical migration and if t z >> tl (or tl = O), eq 10 becomes R = Rat representing purely steric migration. With the requirement that tl is fixed at a value smaller than T,*, eq 10 shows that the mechanism by which the particles migrate changes gradually from purely cyclical to purely steric as t 2 increases. It should be noted that eq 8 and 10 are strictly correct only if the particle elution time corresponds exactly to an integral number of cycles. If this condition does not apply, then a correction for the incompleted final cycle can be applied. However, in most cases the number of cycles during the run is high and this correction will thus be negligible. Particle Characterization. We have noted that the cyclical-field FFF techniques have the potential to yield certain physicochemical constants of particles (or distributions of physicochemical parameters for polydisperse particle populations) much like other FFF methods are used for characterization studies. The cyclical techniques will most generally yield transport coefficients and parameters deriving from these coefficients. The results of the sedimentation/ cyclical-field FFF subtechnique utilized here, for example, depend on the sedimentation coefficient s with particle diameter, over and above its direct influence on s, acting as a secondary factor in separation. If we assume mode I operation, and that particle diameter effects are negligible ( a t , (mode N). Four of these are shown in Figure 8. Clearly, the separation time increases with t,. The selectivity for this series is evaluated by plotting (see Figure 9) In R vs In d for each different fractogram. We see that the slope decreases with increasing t,, specifically from 1.36 to 0.809 as t , increases from 8 to 2000 s. Thus the selectivity increases as the Contribution of the cyclical mechanism increases relative to the steric mechanism, as predicted by theory. This trend is further illustrated in Figure 10.
1134
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988 I
~
1
(b)
1
t , = 5 s , t,=14s
1
2.0
1.5
-
c
,
I
I
Experimentol
0.5 1
0
20
40
60
80
t, (SI
Flgure 10. Theoretical and experimental plots of seiecthrl Sdvs time
t , at t , = 5 s.
0
IO
20
10
0
30
20
30
RETENTION TIME ( m i n )
Figure 8. Separation of polystyrene latex bead mixture (d = 10, 15, 20 pm) by WCyFFF with t , = 5 s and f2 changlng from 8 s (a) to 2000 s (d). Flowrate = 1 mL/min in w = 0.254-mm channel.
-6.9
-6.6
-6.3
-6.0
In d Flgure 0. Plot of In R vs In d for fractograms with different values of t , but constant t , (5 s). The slope is the selectlvlty displayed by the
fractogram. While the experimental selectivities (taken from Figure 9) fall somewhat below those predicted by theory (from Figure 4), the two curves are consistent in showing a reduction in Sd with t2.
Modes I11 and IV of cyclical-field operation are in reality hybrids of cyclical-field and steric FFF. Although the selectivity is highest with purely cyclical-field operation, the resolution increases with increasing steric participation (Figure 8). Resolution , of course, has two components, only one of which is selectivity. The other is band sharpness as measured by theoretical plate numbers (14). Clearly, for the fractograms of Figure 8, increased levels of steric FFF give improved resolution because the reduced band broadening offsets the loss of selectivity. However, while the selectivity advantage of the cyclical-field approach is probably intrinsic to the nature of the two approaches, band broadening has not been widely studied and is not clearly understood for either method. Until such studies are carried out, it will be difficult to judge the
relative potential resolution of these two techniques.
CONCLUSIONS It has been shown that sedimentation/cyclical-field FFF is applicable not only to the effective separation of particles in the 10-20-pm range (and undoubtedly beyond) but to the determination of sedimentation coefficients and related parameters such as particle density. While the determination of these constants has been illustrated for well-characterized monodisperse distributions, it should be a simple matter to extend this method to polydisperse particle distributions (e.g., environmental or geological samples) for which a distribution in values of the sedimentationcoefficient should be obtainable. Sd/CyFFF appears to be applicable in the same particle size range as Sd/steric FFF. We have demonstrated that selectivity is highest with purely cyclical-field operation; however, the resolution is shown to increase with increasing steric participation. One advantage of Sd/steric FFF is that centrifugal forces can be readily substituted for gravitational forces. We have shown that this leads to very substantial gains in separation speed (seven well-resolved peaks in 3.5 min) accompanied by high-resolution levels (15). It is not clear how this Sd/CyFFF technique could be subjected to the higher driving forces of a centrifuge short of using a nested system in which one rotor spins inside but off-the-center of another. We note, by contrast, that Sd/CyFFF has some advantages relative to ita steric counterpart. First of all, the particles spend more of their time in the interior of the channel where lift forces and other complicated hydrodynamic effects characteristic of the wall region are minimal. Therefore, the behavior of cyclical-field systems is likely to be more in accord with that of simple theory than that of steric systems. In addition, as we have stated earlier, cyclical-field methods yield transport coefficients, specifically the sedimentation coefficient, which cannot be readily recovered from the results of steric FFF operation. Finally, despite the reasonable agreement between theory and experiment for the present cyclical-field system, more theoretical work is needed to characterize the observed anomalies for the larger particle diameters. In addition, as we have noted, a better understanding should be sought for the nature of band broadening processes in cyclical-field FFF. LITERATURE CITED (1) Glddlngs, J. C.; Chen, X.; Wahlund, K.-G.; Myers, M. N. Anal. Chem. 1987,-59, 1957. (2) Glddlngs, J. C. Sep. Scl. Techno/. 1984, 19, 831. (3) Chmelik. J.: Janca. J. J . L h . Chromatoor. 1986. 9. 55. i4j w i n g s , J. C. Anal. Chem-. 1986, 58,3052. (5) Berth&, A.; Armstrong, D. W. Anal. Chern. 1987, 59, 2410. (6) Glddlngs, J. C.; Karalskakls. G.; Caldwell, K. D.; Myers, M. N. J . CoNoid Interface Sci. 1983, 92, 66.
Anal. Chem. W 8 8 , 6 0 , 1135-1141 (7) Svedberg, T. I n The Ultrecenhlfuge; Svedberg, T., Pederson, K. O., Eds.; Clarendon: Oxford, 1940. (8) Glddings, J. C. Sep. Sci. Teechnol. 1078, 13, 241. (9) Peterson. R. E., 11; Myers, M. N.; W i n g s , J. C. Sep. Scl. Technol.
-
i.n--, ai
.-
1135
(13) Brenner, H. Chem. Eng. Scl. 1081, 16, 242. (14) Gunderson, J. J.; Glddings. J. C. Anal. Chlm. Acta 1088, 189, 1 (15) Koch, T.; Glddings, J. C. Anal. Chem. 1988, 58, 994.
-- . .
10, fin7
(10) C a b 4 K. D.; Cheng, Z.-Q.; Hradecky, P.; Giddings, J. C. Cell 810phys. 1984, 6 , 233. (11) Myers, M. N.; Giddings, J. C. Anal. Chem. 1082, 54, 2284. (12) Batchelor, G. K. J . FlUM Mech. 1072, 52, 245.
RECEIVED for review August7, 1987. Accepted February 12, 1988. This work was supported by Grant No. DE-FGOS86ER60431 from the Department of Energy.
Normal and Reverse Pulse Voltammetry at Microdisk Electrodes Lin Sinru,' Janet Osteryoung, John J. O'Dea, and Robert A. Osteryoung* Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214
Normal and reverse pulse voltammetry have been carrled out on stationary microdlsk electrodes where nonplanar diffusion can be dgniflcant. Boundary conditions are shown to be readHy renewed for both normal and reverse p u b modes by simply waiting for flxed periods of time between pulses. The superpodtion theorem Is shown to be applicable even under conditions where substantial nonplanar diffusion Is present. An equation permlttlng the estimate of the error In the normal pulse lbnitlng current for ratios of pulse width to waiting time for renewal of boundary condltlons Is presented.
For an imbedded circular electrode of small area, referred to here as a microdisk electrode, as a result of nonplanar diffusion, the Cottrell equation no longer adequately expresses the i-t behavior except at short times. Considerable work describing the i-t behavior has been published (10-18),and a general solution, for the case of D, = Dd, has been obtained (16-18). Introducing a dimensionless time variable, P
P = 4Dt/?
i = 4nFDC*rf[P(t)] Pulse voltammetries are useful techniques for the study of electrode processes and for electroanalytical purposes ( I ) . In particular, normal and reverse pulse voltammetry (2) have been shown to be useful in studies at solid electrodes, and, in terms of our own interests, in molten salt systems where mercury electrodes cannot be applied (3-7). In both of these techniques, simple theory assumes that the boundary conditions prevailing at the electrode-solution interface are renewed before each cycle. When these techniques are applied at a dropping (or static) mercury electrode, this renewal is easily accomplished by natural or mechanical drop detachment (2,8).At a solid electrode, however, such is not the case. In general, the electrode must be rotated, or the solution stirred, between each pulse (4-7). This not only necessitates the use of large amounts of solutions, it is also rather time-consuming, although the procedure can be computerized (4-7,9). Electrodes of very small dimensions have significant nonplanar diffusion characteristics. In this work we report on the conditions under which the renewal of the boundary layer for normal or reverse pulse voltammetry can be achieved in shorter times as a result of nonplanar diffusion taking place at electrodes of very small dimensions. The time dependence of the current in response to a potential pulse at a large planar electrode for a diffusion-limited process is given by the Cottrell equation
ic = nFAD1/2C*/(7rt)1/2
(1)
where A is the area of the electrode, t is the duration of the potential pulse, n, F, D, and C* have their usual significance, and ic is the Cottrell limiting current. Permanent address: The Shanghai Institute of Metallurgy, Academy of Sciences of China, Shanghai 200050, China. 0003-2700/88/0360-1135$01.50/0
(2)
the time dependence of the current in response to a potential step at a microdisk electrode can be expressed as
(3)
where the function f [ P ( t ) ]is defined as (18)
f ( P ) = 1 + 0.71835P1/2 + 0.05626F3/2 for P
(4)
> 0.88 and f ( P ) = (7r/4P)'/'
+ ? r / 4 + 0.094P1/2
(5) for P < 1.44. The square root of P can be thought of as the ratio of the time-dependent diffusion layer thickness to the radius of the electrode. A most important feature of these electrodes with significant nonplanar diffusion is that steady-state currents are attained when P becomes large. The Cottrell equation for a disk electrode of area A = rr2 can be rearranged t o yield
ic = 4 n F D C * r ( ~ / 4 P ) l=/ ~4nFDC*rfc[P(t)] (6) The function fc(P) is by definition identical with the f i t term on the right side of eq 5. At short times, P very small, the current at the microdisk electrode approaches the Cottrell limit; at long times f ( P )approaches unity, and the current at the microdisk electrode approaches the steady-state value
i, = 4nFDC*r
(7) In the experiments to be described, normal pulse (NP) and two different modes of reverse pulse voltammetry, one of which is essentially double potential step chronoamperometry, were employed. In normal pulse voltammetry, under conditions where only one form of a redox couple is in solution, pulses of width t , and successively increasing amplitude are applied to an electrode from an initial potential, Ei, where no faradaic reaction takes place, Figure 1. During the waiting time, labeled t , the boundary conditions at the surface are renewed either by detaching a drop, in the case of mercury, or rotating the electrode or s t i r r i i the solution in the case of a solid electrode. 0 1988 American Chemical Society
