2328
Anal. Chem. 1988, 60,2328-2333
(9) HjertGn, S. Chromatogr. Rev. 1967, 9 , 122-219. (10) Herren, B. J.; Shafer, S.0.;Alstine, J. V.; Harris, J. M.; Snyder, R. S. J. ColloM Inferface Sci. 1987, 115, 46-55. (11) Jorgenson, J. W.; Lukacs, K. D. Science (Washington, D.C.) 1983, 222, 266-272. (12) HjertGn, S. J. Chromafogr. 1985, 347, 191-198. (13) Lauer, H. H.; McManigHI, D. Anal. Chem. 1986, 58, 166-170. (14) Glajch, J. L.; Kirkiand, J. J. U S . Patent 4705725, 1987. (15) Regnier, F. E. J. Chromafogr. 1987, 418, 115-143. (16) Iler, R. K. The Chemistry of Silica; Wlley: New York, 1979; p 42. (17) Lukacs, K. D.;Jorgenson, J. W. HRC CC,J . High Res. Chromafogr. Chromafogr. Commun. 1985, 8 , 407-411.
Parks, G. A. Chem. Rev. 1985, 65,177-198. Mltsyuk, B. M. Russ. J. Inorg. Chem. 1972, 17, 471-473. Metcalfe, L. D. Nature (London) 1960, 188, 142-143. Emerick, R. J. J. Nutr. 1987, 117, 1924-1928. Fausnaugh, J. L.; Regnler, F. E. J. Chromarogr.1986, 418, 131-146. , Kohler, J. Chromafographis 1966, 2 1 , 573-582. i24) Jorgenson, J. W. TrAC, Trends Anal. Chem. (Pers. Ed.) 1984, 3 , 51-54. (25) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298-1302.
RECEIVED for review April 29, 1988. Accepted August 2,1988.
Experimental Observation of Steric Transition Phenomena in Sedimentation Field-Flow Fractionation Seungho Lee and J. Calvin Giddings* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The steric transMan region of field-flow fractionation (FFF) Is described as that part of a fractogram, found at very high retention volumes, in which m a l FFF undergoes a tradlon to steric FFF by virtue of increasing particle diameter. The steric transition region Is treated theoretically by assuming first that the steric factor y is constant and second that It Is related by a -le power law to particle diameter. We then report the first experiments in which fractograms dtsplay the characteristic “signature” predicted by theory for the steric transltion: a narrow terminal peak fobwed by a rapid dropoff to base line. I t is shown that the steric transinon polnt, which coincides with the dropoff, Is displaced to higher retention volumes with Increasing field strength, approximately as expected. The expected steric transhion phenomena are fwther confirmed by collecting narrow fractions of a polydisperse po@(vlnylchloride) sample near the steric tradion point and sublectlng them to electron microscopy. The particle size distrlbution of the fractions is found to be sharply bimodal, in accordance with steric transition theory. However, satlsfactory agreement between the measured partlcle diameters and the theoretical expressions Is found only by application of the more complicated equations in which y is assumed to be size dependent.
The methodology of field-flow fractionation (FFF) is based on the action of an external field or gradient whose direction is perpendicular to the axis of flow in a thin channel (1-3). The field forces particulate and macromolecular species to accumulate in narrow zones such that each is intercepted by different flow laminae and thus displaced at different velocities down the flow channel. The technique divides into a number of categories depending on the nature and distribution of the narrow zone and upon the field applied ( 4 ) . Normal FFF is defined as that group of techniques in which species are forced to one wall (the accumulation wall) by the field. Their mean distance from the wall is determined by the force exerted on the particles by the field and by diffusion (Brownian motion), which counteracts the buildup of particles at the wall. For components having the same density, those with the highest molecular mass or size have the greatest force exerted on them and they equilibrate closest to the wall. Here
the downstream fluid motion is highly retarded by the frictional drag of the wall and component particles are displaced only slowly along the flow axis. Species of smaller size occupy laminae positioned further from the wall where the flow displacement is more rapid. A trend is thus established in which particle velocity decreases with increasing particle size. This is illustrated by the leftihand branch of Figure 1in which velocity is expressed in terms of retention ratio R, the velocity of the component particles relative to the mean velocity ( u ) of the carrier fluid. In steric FFF the species are also pushed toward the wall. However, in steric FFF the nearness of approach to the wall is determined more by the size of the particles than by the competition between the applied force and Brownian motion. In the simplest model we imagine that the field-induced transverse motion of a particle is halted once it touches the wall, leaving the particle to protrude out into the flow stream by a distance equal to its diameter. Since large particles, by virtue of their size, extend more deeply into the flow channel than small, they are swept more rapidly downstream than the small particles. Thus particle velocity tends to increase with size, as shown by the right-hand branch of Figure 1. The opposing trends shown by normal FFF and steric FFF in Figure 1 are joined smoothly in a transition region. Thus, as particle size increases from ita most miniscule level, particle velocity decreases (retention time t, or volume V , increases) until one reaches a size such that the mean distance of the particle from the wall is little more than the particle diameter. A t this stage the particle size begins to exert a significant influence on displacement velocity by virtue of its physical extension in space. This occurs a t the beginning of the transition region. As the particle size increases further, the size-based effects increase and the particle velocity goes through a minimum (called the inversion point) and begins increasing. For still larger particles the steric effect is fully dominant, with the resultant velocity closely linked to particle diameter. The transition noted above has been characterized theoretically for both sedimentation FFF and flow FFF (5). The transition is generally expected to occur for particles of diameter from 0.1 to 1.0 hm, depending on the field strength. The retention ratio R at the inversion point is very small, of the order of which means that the inversion particles (those eluting at the inversion point) are retained roughly 100
0003-2700/88/0380-2328$01.50/0 0 1988 American Chemical Society
ANALYTICAL CHEMISTRY. VOL. 60. NO. 21. NOVEMBER 1. 1988
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ticles to adhere to the wall, we have found it possible to explore
the transition region in some detail. Not only have fractograms been generated that demonstrate the expected tran-
sition properties, but the foldhack phenomenon has also been observed and characterized by collecting fractions and identifying the two-component particle populations hy electron microscopy. These studies have been accomplished with sedimentation F F F other subtechniques would be expected to yield similar results.
THEORY F%RTICLE SIZE Flgun 1. Schematic Illustration of opposing be& in component ml@a wbHy (measured by r e m ralb R ) with particle sire f a m l FFF and steric FFF. The two bWhg cases are Mned s " l y in a transklon region.
times longer than the void peak. Once the inversion particles have been eluted from the channel, no other particles are expected to appear because both larger and smaller particles have already been washed through by the steric and normal FFF mechanisms. Thus an abrupt termination of particle elution is expected a t the inversion point. Insofar as analytical objectives are concerned, the primary importance of the transition phenomenon is that the R versus particle size curve folds hack along the R axis. This foldback means that the particles associated with an experimentalvalue of R (i.e.. eluted at a specified retention volume V.) may in theory consist of one or both of two populations of particles, one described by the normal branch of the curve in Figure 1and the other related to the steric branch. Since the partide diameters of the two hypothetical populations will normally differ by an order of magnitude or more, external evidence concerning the origin or nature of the sample may well rule against the presence of one of the two populations. In cases in which particles span a very large diameter range, however, different sized particles may actually elute in the m e volume element of carrier. This coelution phenomenon introduces an ambiguity into the interpretation of the fractognun. Thus, the double-valued problem originating in the steric foldhack must be thoroughly understood and characterized in order to maximize the useful information to he gained from FFF runs. Another practical consequence of the transition phenomenon is that the selectivity, the ability to distinguish between particles of different sizes, vanishes a t the inversion point. This is a consequence of the flat minimum of the curve shown in Figure 1;at the minimum there is no change in R or V, with a small change in particle diameter and thus there is no resolution a t this point. I t is consequently necessary to adjust experimental parameters (such as field strength) so that important particle populations do not fall near the minimum. A secondary consequence of the existence of the minimum and the associated loss of selectivity is that all of the particles occupying a finite diameter range near the minimum essentially coelute. The simultaneousappearanceof differenbsized particles in the m e small elution volume element is expected to give rise to a false peak on the fractogram immediately preceding the anticipated dropoff of the particulate concentration to zero. The assorted phenomena predicted above to he associated with the steric transition region have never been documented experimentally. The region of the fraetogram in which the transition is expected to occur, a t an elution volume approximately lo2times the void volume, is further out along the elution axis than that normally encountered in FFF work and has been experimentally inaccessible in previous efforts to extend the operating range to this extreme. However, with due precautions taken to reduce the tendency of larger par-
Under the high retention conditions that prevail in the vicinity of the steric transition, the retention ratio R can be approximated ( 5 ) as
R = Vo/Vr = 6h + 6 y a (1) where the ratio of channel void volume VD to retention volume V, is equivalent to the particle/carrier velocity ratio stated earlier as defining R. The steric factor y is a dimensionless parameter (see later) of order unity and the terms a and A are given by a = a/w = d/2w
(2)
h = l/w
(3)
and where w is the channel thickness, LI is the particle radius, and 1 is the mean displacement from the wall caused by Brownian motion. For sedimentation FFF
l = - 6kT rd3ApG
(4)
where k is the Boltzmann constant, T i s the temperature, Ap is the density difference between particle and carrier, and G is the field strength m e a s d as amleration (I). For spherical particles, d is the particle diameter; otherwise, d is the effective spherical diameter. When eq 2,3,and 4 are substituted hack into eq 1,we get
(5) In this equation the first term accounts for normal FFF retention and is the dominant term for small particle diameter d. The inverse dependence on d is illustrated schematically hy the left-hand branch of Figure 1. The second term in eq 5, proportional to d , is dominant for large particles. Thii term gives rise to the right-hand or steric branch of Figure 1. The steric inversion point, represented by the minimum of the curve in Figure 1,is defined by the condition dR/dd = 0. By use of eq 5 and asauming y to be constant. this condition gives for the particle diameter at the inversion point
and for the inversion retention ratio
The dimensionlesssteric factory. which represents the complication of lift forces and related hydrodynamic effects (6). can be expressed in terms of the retention ratio Ri a t the inversion point hy the rearrangement of eq 7. We obtain = (w;)( -
.A~G)"" 36kT
(8)
2330
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988
There is now strong experimental evidence that y is a function of particle diameter d (6, 7). In this case the simplified expressions above, which describe the inversion point, must be modified. While the exact dependence of y upon d is not known, we can obtain a first-order correction to the above equations by using the approximation (6) where C is a constant and the exponent /3 is a fraction whose value was found in earlier work (6) dealing with much larger particles to be approximately0.4. If eq 9 is used in conjunction with the steric inversion condition, dR/dd = 0, we obtain the modified inversion particle diameter
v, /V"
The corresponding inversion retention ratio is
Ri =
(4 - (3)Cdil-D W
=
[(
-
36kT
rApGw ,3)3/(4-@)
)'-@(
y]i'(4-8)
[(I -
+ 3(1 - ,3)(@-1)/(4-8)] (11)
These equations are obviously more awkward than the expressions given by eq 6 and 7. (Note that eq 10 and 11reduce to eq 6 and 7, respectively, if /3 = 0 and y = C.) Aside from the location of the steric transition described by eq 7 or 11, we wish to examine theoretically the shape of the elution profile in the transition region. In particular, we wish to examine the "false" peak expected to elute near the inversion point. We described in the introduction how such a peak is expected to arise in association with the loss of selectivity in the transition region. To describe this phenomenon in mathematical terms, we must relate the particle size distribution curve n(d) and the FFF elution concentration profile c( V,) by using the expression (8) n(d) dd = c(V,) dV, (12) where n(d) dd equals the number of particles between diameter d and d + dd, c( V,) is the number-based concentration (number of particles per unit carrier volume) eluting at volume V,, and dV, is the increment in elution volume corresponding to the small particle diameter increment dd. This equation applies individually to both the normal FFF and steric FFF branches of the particle population. The concentration profile of particles occupying each branch found in the eluting stream, based on eq 12, is given by
where the right-hand side consists of two contributions, one each from the normal and steric branches. In that the diameter-based selectivity is given (5) by
The concentration profile of the eluting stream assumes the form
I
0
,
,
30 45 VJV"
,
#
60
75
Figure 2. Comparison of effluent concentratian profiles and fractogram for G = 28 gravities and y = 1.35 for PVC latex beads (Ap = 0.40 g/mL): (A) calculated intrinsic steric spiking function based on uniform particle size distribution; (B) calculated concentration profile for particle size distribution described by eq 17; (C) calculated fractogram of PVC sample.
by the particle size distribution curve n(d) and by the dependence of S d on d and will be marked particularly by the intrinsic spiking phenomenon that accompanies the reduced selectivity in the transition region. The intrinsic spiking function is expressed by dividing the right-hand side of eq 15 by n(d), which gives
This spiking function is illustrated in Figure 2A for the specific case (corresponding to subsequent experiments) in which G = 28 gravities, w = 0.254 mm, T = 293 K, Ap = 0.40 g/mL, and y = 1.35. The position of the "spike" will shift, as shown by eq 7 (or eq l l ) , as the product of the two parameters G and Ap varies. Thus Figure 2A provides only an example of the spiking profile. The experimental fractogram would approximate Figure 2A provided n(d)was uniform (constant with respect to d), the detector response was proportional to n(d), and peak broadening was minimal. We note that the response of most detectors depends on particle size as well as concentration, so that close agreement between calculated c( V,) curves and observed fractograms is not expected without correction (9, 10). Since n(d) will not be uniform for real particulate systems, it is useful to see how c(V,), and thus roughly the shape of the fractogram, responds to nonuniform particle distributions. We illustrate this by showing in Figure 2B the c( V,) curve calculated from eq 15 for the case in which the particle distribution is described by n ( d ) = const
At the exact point of inversion both dRldd and dV,/dd go to zero, which means that the selectivity S d also goes to zero. Thus by the above formulation, the concentration is expected to spike to infinity at the inversion point. Obviously, band broadening will moderate this effect to produce a finite peak. From the above discussion it is clear that the experimental elution profile, which follows c(V,), will be influenced both
,
15
X
exp[-12.5(ln d
+ 0.76)2]
(17)
where the unit of the particle diameter d is micrometers. The above distribution approximates that of a PVC sample that was studied.
EXPERIMENTAL SECTION The basic sedimentation FFF system used in this work has been described elsewhere ( I ) . The channel dimensions were thickness
ANALYTICAL CHEMISTRY, VOL. 80. NO. 21. NOVEMBER 1. 1988 2331
w = 0.254 mm, tip-to-tip length L = 95.0 em, and breadth b = 2.00 cm;the void volume V' measured by means of the elution volume of an unretained peak (sodium benzoate) was 3.95 mL. The radius of rotation wan 15.5 cm. After the channel wall had been coated with polyimide tape (CHRIndustries, Inc.) to reduce particle adhesion, Vo was reduced to 2.83 mL because of distortions in the tape. The monodisperse colloids used in thii study were polystyrene latex beads (Seragen Diagnostics, Indianapolis, IN) with mean diameters of 0.868,0.945, and 2.020 pm and standard deviations of 0.0104, 0.0064, and 0.0135 pm, respectively. The reported density for these particles is 1.05 g/mL. The polydisperse sample consisted of a poly(viny1 chloride) (PVC)latex obtained from David Milenius at B. F. Goodrich. The diameter range of these particles was reported to be 0.2-2.0 pm and the density was 1.4 g/mL. Further details on the size distribution were obtained by transmission electron microscopy (JEOL Model JEM 1OOCXII). The carrier liquid for all experiments was douhly distilled water containing 0.1% FL70 detergent (Fisher ScientiIicCo.)and 0.02% sodium azide used as a hacteriocide. AU experiments were mid out at ambient laboratory temperatures, 293 f 1 K. A 1C-15-pL sample was injected into the channel directly through a septum by use of microspinge while carrier was introduced at a very low flow rate by a Gilson Minipuls 2 pump. The syringe was then withdrawn slowly and the flow was completely stopped. The centrifuge was turned on at the desired setting for a period (typically2C-30 min) of relaxation, following which flow was resumed. The emerging sample was monitored by a UV detector (254 nm) (Altex Model 153) and a strip chart recorder.
RESULTS AND DISCUSSION The observed size distribution of the polydisperse PVC sample was fit to a log normal distribution function, which yielded eq 17. We would therefore expect a fractogram of this sample, when run a t 28 gravities, to display a shape similar to that of Figure 2B providing the detector response was independent of particle diameter. An example of the observed fractogram for the PVC sample under the calculated conditions is shown in Figure 2C. While parts B and C of Figure 2 do not agree in detail, as expected, the similarities are evident. The narrow peak followed by a rapid return to base line, common to both curves. may be considered the characteristic "signature" of the steric transition. It is unlikely that this unusual profile would arise for broad particle distributions except as a consequence of steric inversion. Thus Figure 2C appears to represent a successful experimental observation of the steric transition phenomenon. Although parts B and C of Figure 2 me of the same general form, we observe several differences. The sharpness of the distinctive peak (spike) and of the subsequent falloff is clearly moderated by band broadening. The difference in the shape of the broad peak maximum can he attributed in part to the failure of the detector to give a number-based response. However, profile variations may also arise because of shifts in the steric factory with particle diameter; Figure 2B is based on a constant y. We note that the close agreement in the positions of the steric inversion points in the two figures can be ascribed to the fact that the value of y was extracted from the steric inversion point of the observed fractogram (Figure 2C) by means of eq 8 and used for the calculation of the theoretical fractogram (Figure 2B). However, this empirical value, y = 1.35, is in the expected range near unity, leading us to believe that another y value determined by another method would not produce a significantly different result. Equation 7 predicts that the inversion R value decreases slowly with increasing field strength G. Thus an increase in G is expected to cause a shift of the inversion retention volume (VJi to higher values. This prediction is tested hy obtaining fractograms for a series of runs made a t different G values. Four of these are illustrated in Figure 3 (solid lines). This figure shows a distinctive shift in the steric inversion point
Y
Fractograms of polydisperse PVC sample showing shin in steric inversion point to !a* V , values wim increasing Reld st'ength. Flow rate was 2.05 mllmin. m e points in A and B represent experimentally measured particle diameters while the broken line rep resents a plot of dvs VJV" according to eq 5 w M omstant y values. The numbered shaded regions represent collected fractions. FIgum 3.
Table I. Variation of the Steric Inversion Retention Ratio, 7 Factor, and Particle Diameter with Field Strength' field strength (rpm), G (gravities)
R?
r'
di? pm
400, 21.8 500, 43.4 600, 62.4 100, 85 800,111
0.0163 0.0146 0.0118 0.0101 0.00178
1.35 1.29 1.19 1.07 0.824
0.767
0.678 0.632 0.600 0.599
'Flow rate wan 2 mL/min. *Observed. 'Calculated from eq 8. dCalculatedfrom eq 6.
to higher V,values as the revolutions per minute (rpm) increase from 500 to 800. (If we consider Figure 2C as part of the series, the increase is from 400 to 800 rpm, a kfold increase in G.) The shift in V, with G is actually somewhat more pronounced than that predicted by the one-fourth power dependence of eq 7; the observed dependence appears to be govemed by an exponent of approximately 0.55. We note that eq 11 predicts an opposite trend the exponent would be expected to drop to for 6 = 0.4. The discrepancy (0.55 va 0.25) can be explained by the observation that 7 decreases with increasing G (7). We observe some degradation of the fractograms in Figure 3 as we proeeed to higher rpm values. This is not unexpected as we generally have difficultyin obtaining reproducible results a t these very high retention levels. The difficulties could be related to an increase in particlewall interactions for increasing particle diameter and field strength or to overloading effects. We note that as we proceed through the series of fractograms from Figure 2C to Figure 3D, the particle diameter di a t the inversion point is expected to shift along with the position of the inversion point. The shift in di is predicted by eq 6 or the first part of eq 7. Table I shows a compilation of the observed steric inversion retention ratios Ri, the value y calculated by substituting the observed Ri into eq 8, and the value of the inversion particle diameter di calculated by substituting this value of y into eq 6. While the overall trends in the observed fractograms are consistent with theoretical expectations, the fractograms provide no detail about the underlying microscopic distribution of particles. In order to confirm the postulated microscopic basis of the steric transition, we have collected narrow
2332
ANALYTICAL CKMISTRY. VOL. 60. NO. 21, NOVEMBER 1, 1988
LL
r
1.0-
W
5n aaasY a4-
8 azI
L
q
20
0
€0
40
BO
100
120
V,/V’
A
B
Flwo 4. Electron mlaoqaphs conanadpated bmodal she dlshibutlon of particles collected from emuent fractions near me steric inversion point: (A) fraction 4 of fractogram A In Figure 3: (8) frectlon 5 of the Same fractogram.
Flgure 5. Fractogram of Figure 3A shown along with exparlmental values of particle dlemetw measured for differentfractlons and wee particb diameter c w e s plotted acmding to eq 5 using eq 9 with the 0 values indicated.
I
fractions from the runs shown in Figure 3A.B and have examined the particle size of the fractions by electron microscopy. The positions of the cuts are shown along with the fraction identification number in Figure 3A,B. The observations hy the electron microseope show that the fractions taken in advance of the steric inversion point have a bimodal particle size distribution much as predicted. Two of these distributions, taken from fractions 4 and 5 of Figure 3A, are illustrated in Figure 4. The larger particles correapond to the steric FFF population while the smaller particles have undergone normal FFF migration. We observe that the particle diameter of the two populations in Figure 4B are much closer together than t h c e shown in Figure 4A. This is hecause cut 5 of Figure 3A, represented in Figure 4B, was collected a t a position close to (specifically at 93% 00the steric inversion elution volume; the bimodality is expected to vanish at the inversion point. Figure 4A shows particles collected further hack from the inversion point, namely from cut 4 of Figure 3A, where the elution volume is 85% of its inversion value. Consequently, Figure 4A demonstrates a much greater difference in particle diameter between the steric and normal FFF populations. While the electron micrographs of Figure 4 confirm the trends expected in particle diameter in the steric transition region, we can take this study a step further hy measuring the two mean particle diameters from each of a series of electron micrographs and comparing the values against the theoretical R versus d expression of eq 5. The major difficulty of the comparison is again that of finding an appropriate y value; we have chosen as before to use the y value obtained from eq 8 at the observed steric inversion point. With this y value (equal to 1.29 for Figure 3A and 1.19 for Figure 3B), the d values obtained from eq 5 can be plotted as a function of R or V,. These plots (broken lines) are superimposed on the fractograms shown in Figure 3A,B. The diameter scale is shown a t the left of these figures. The diameters obtained from electron microscopy are shown as points on these plots. Again, while the trends are generally correct, we observe that the diameters taken from electron microscopy are consistently higher (by an average of approximately 14%) than the diameters predicted by theory. The presumed decrease in y with increasing particle diameter may explain the discrepancy between predicted and observed particle diameters. The lower branch of the particle diameter line, corresponding to particles of diameter less than di, is expected to have y values larger than those a t the inversion point, which means (eq 5) that the smaller particles will migrate faster and elute at an earlier V,than predicted.
o’16i 0.12
01 0
i
4
I 20
40
60
80
io0
G (gravities) Figure 6. plot of experimemal R value versus fled shecgth fw p&ystyrene beads of two diameters. The two curves cross at G = 37 gra*s due to steric bansMan effects. The Ww rate was 1.4 mUmln (33 cmlmin).
A shift to lower elution volumes for the small particles is indeed confirmed hy Figure 3A,B. The opposite trend observed for the large particles (upper branch) can be explained hy the smaller y values and the corresponding increase in the retention volume. T o investigate this poasihility more closely, we have allowed y to vary with particle diameter according to eq 9. When this y is substituted hack into eq 5 and plotted in terms of V,/ VO (=l/R) versus d for several values of exponent 8, we get the plots shown in Figure 5. We find reasonable agreement between the experimental results and the plot hy using 8 = 0.4, which is consistent with the earlier study (6). Since some of the discrepancy between theory and experiment noted above can likely be attributed to variations in y , i t is useful to investigate a system for which y can he followed in detail. For this purpose we chose two monodisperse polystyrene populations with particle diameters of 2.022 and 0.868pm, respectively. The experimental retention ratio R(exp) was measured for these two samples as a function of field strength. The results are shown in Figure 6. We observe that the two curves representing the two populations cross one another a t G = 37 gravities and R = 0.039. This c m v e r point corresponds to a fixed value of R or V , in which the two fractions would coelute if part of a broad particle distribution. In theory, any two components with different particle diameters can he made to coelute with the proper choice of G. Conversely, any coeluting species can he separated if desired by changing G. By way of a physical model, the crossover in Figure 6 can he explained by noting that the larger particles are pushed
ANALYTICAL CHEMISTRY, VOL. 60, NO. 21, NOVEMBER 1, 1988 4.0 I
I .o
I
I
I
1
I
\
c
I I
2.022pm
I I I
I
0
I
I
I
I
2333
Fm (7). If the size dependence is described by eq 9, the best value of the exponent /3 is found to be 0.46, in reasonable agreement with the value of 0.4 deduced earlier. For reference purposes we note that y is also sensitive to the carrier flow velocity ( u ) , as has been shown for larger particles (6). A plot showing the dependence of y on ( u ) for the PVC sample is shown in Figure 8. The plot was obtained by observing the shift in the steric inversion point with changes in ( u ) . The dependence of y on ( u ) is such that the steric transition region of the fradogram and the associated inversion point shift to lower elution volumes at higher flow rates (see eq 5).
CONCLUSIONS Flgun 7. Variation of y wlth field strength for two polystyrene beads. The values of y were obtained from Figure 6 used in conjunction wlth eq 5. Flow rate was 1.4 mL/min.
2.0 1
I
I
0.2
0.4
I
I
0.6
0.8
I
I
This study has shown that it is possible to observe steric transition phenomena in FFF by their distinctive influence both on the experimental fractogram and on the particle size distribution of collected fractions. It is possible to correlate both kinds of observations with theory. Some small residual discrepancies between the theoretical and experimentalresults may in many cases be attributed to the variation of the dimensionless steric factor y, defined by eq 1. Although y is described moderately well by eq 9, y is unfortunately a complicated hydrodynamic parameter whose full range of variations has not yet been fully described. As y becomes better characterized, it should be possible to extract useful analytical information from fractograms in the steric transition region related to the density and density distribution of simple and complex colloidal materials.
LITERATURE CITED
0
0
1.0
1.2
(v) (cm/s) F w e 8. Variation of y for polydisperse PVC sample with flow velocity ( v ) and fleld strength. The IndIvMual y values were determined by measuring the sterlc Inversion polnt for PVC at each flow velocity ( v ) and applying eq 8 to the results.
closer to the wall (and thus have lower R values) than the smaller particles under normal FFF conditions at low G values. As G increases, however, the large particles are eventually blocked from a closer approach to the wall by their own finite diameter. Meanwhile, the smaller particles continue to approach the wall more closely as G increases, leading eventually to the crossover. The experimental R values shown in Figure 6 can be combined with eq 5 to obtain a series of y values. The latter are plotted in Figure 7. This figure shows that y decreases with both increasing field strength and increasing particle diameter, as observed previously for particles in the diameter range 2-45
(1) Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Fisher, S. R. I n Methods of Biochemical Analysis; Glick, D., Ed.; Wlley: New York, 1980; Vol. 28, p 79. (2) Giddlngs, J. C. Anal. Chem. 1081, 53. 1170A. (3) Wings, J. C. Sep. Scl. Technd. 1084, 831. (4) Glddlngs, J. C. In Chemical Separetlons; Navratll. J. D.. King, C. J., Eds.; Litarvan: Denver, CO. 1988; Vol. 1, p 3. (5) Myers, M. N.; Glddings, J. C. Anal. Chem. 1082, 54, 2284. (8) Peterson, R. E., 11; Myers, M. N.; Glddings. J. C. Sep.Sci. Techno/. 1984, 19, 307. (7) Koch, T.; Glddlngs, J. C. Anal. Chem. 1988, 58, 994. (8) Wings, J. C.; Myers, M. N.; Yang, F. J. F.; Smith, L. K. In CoMdand Interface Science; Kerker, M.,Ed.; Academic: New York, 1976 Vol. I V , p 381. (9) Kirkland, J. J.; Rementer, S. W.; Yau, W. W. Anal. Chem. 1081, 53,
1730. (10) Yang, F.-S.; Caldwell, K. D.; Giddings, J. C. J . COlloM Interface Scl. 1083, 92, 81.
RECEIVED for review December 23,1986. Resubmitted July 22,1988. Accepted August 3,1988. This work was supported by Grant No. DE-FG02-86ER60431from the Department of Energy.
