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ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
(9) K . A. Connors, "Reaction Mechanisms in Organic Analytical Chemistry ' , Wiley-Interscience, New York, 1973, Chapter 13. (10) K. A. Connors and K . S. Albert, J . Pharm. S o . , 62, 845 (1973) (11) E. L. Rowe and S. M . Machkovech. J . Pharm Sci , 66. 273 (1977). (12) K. A. Connors and N. K. Pandit, Anal Chem., 5 0 , 1542 (1978) (13) N. K. Pandit, M. S. Thesis, University of Wisconsin. Madison, Wis., 1978. (14) C. S. Hudson and J. K . Dale, J . A m . Chem. Soc., 37, 1264 (1915). (15) C. S. Hudson and J. M. Johnson, J . Am. Chem. Soc., 37. 2748 (1915).
(16) J.
K. N. Jones and M. B. Perry. Can.
J . Chem., 40, 1339 (1962).
RECEIYEII for review August 31, 1978. Accepted October 12, 1978. Supported in part by National Science Foundation Grant C H E 78-06603 and in part by Exchange Visitor Program C-5-261, US.Food and Drug Administration.
Flow Programmed Field-Flow Fractionation J. Calvin Giddings," Karin D. Caldwell, John F. Moellmer, Thomas H. Dickinson, Marcus N. Myers, and Michel Martin Department of Chemistry, University of Utah, Salt Lake City. Utah 84 112
The concept of flow programming in field-flow fractionation (FFF) is introduced and justified. It is shown that the method can preserve or improve the resolution of early peaks and at the same time significantly hasten the elution of late peaks, thus increasing analysis speed. Flow programming is compared briefly to the related options, field-strength and solvent programming. The background theory is then presented, and is used to show that very abrupt velocity programs can be employed without undue degradation of resolution for late peaks. Finally, the utility of flow programming is illustrated using three sizes of polystyrene latex beads in a sedimentation FFF system.
T h e programming of column flow (or inlet pressure) is an accepted but not very widely used technique to improve the spacing of peaks and increase the speed of analysis in gas chromatographic systems ( 1). One drawback of the approach is t h a t the increased flow rate realized in the latter parts of the run may increase the plate height excessively and thereby hinder the resolution of late peaks. One usually ends up with a system where the resolving power available for the late peaks has significantly deteriorated relative to the early peaks. Field-flow fractionation ( F F F ) is distinctly different from most chromatographic systems in the sense that the plate height, H . is expected on theoretical grounds to decrease rapidly with increasing retention (2). Therefore, the resolving power available to work with late peaks is normally a good deal better than t h a t for early peaks. If the excess resolving power in the latter part of a run is unnecessarily large for practical goals, it can be "traded" for increased separation speed. This tradeoff can be realized through a gradually increasing flow velocity, which hastens the elution of late peaks a t t h e expense of resolution. As a corollary to the prediction that late peaks will emerge with greater relative sharpness than early peaks, it was shown in a previous paper that the optimum flow velocity is greater for late peaks than it is for those which first emerge (2). On this basis, the utility of a flow programming system for F F F was first predicted. T h e expected contrast between early and latp peaks has never been experimentally demonstrated in a very conclusive way, partly because the intrinsic sharpness produced by the FFF column is often obscured by the polydispersity of the sample material. T h e object of this paper is to demonstrate the expected trend in peak sharpness, and to show how a flow programming system can be used to increase the separation 0003-2700/79/0351-0030$01 O O / O
speed of a solute system subject to this trend. For these purposes, we use the subtechnique of sedimentation F F F for our experimental uork, although we expect similar results for the other subtechniques of F F F . Sl'e note t h a t the flow programming proposed here is intended to complement field-strength programming, which we ha\ e discussed in several papers (3-5). It is also related closely t o solvent programming (31, because both entail changes in the state of the incoming solvent. Of the three approaches, field-strength programming is perhaps the most versatile and direct. However. as we shall see, flow programming is potentially very effective, and may ultimately have an important analytical role, either used alone or used in combination with field-strength or solvent programming In fact, flow programming shows rather general theoretical advantages o\ er field-strength programming. as will be demonstrated shortly.
THEORY In an ideal F F F system, the plate height is given by ( 3
where D is the diffusion coefficient. ( r ) the mean flow velocity, u the channel thickness. R the retention ratio (equivalent to Vq/ V,, the ratio of the column void volume to the peak retention volume) and x a coefficient which in most practical cases can be approximated by the limiting form ( 6 )
x
= R3/9
(2)
In Equation 1. the first term accounts for longitudinal diffusion and the second for nonequilibrium. If we make the reasonable assumption that longitudinal diffusion is negligible. and use Equation 2 for x in the second term, we have
H
= (R'/S)u.'(c)/D
(3)
or, with R =
H
=
v , ) ~) /gn (
(4)
This equation illustrates a striking dependence of H on elution volume V,,showing that H decreases rapidly as V, increases. A closer scrutiny of the situation shows that D generally has a weak inverse dependence on V,. which somewhat subtracts from the cube dependence of the first term. T h e amount subtracted from the exponent three on account of D will generally fall in the range 0.33-1, so that i f we write Equations 3 and 4 in the forms
H
= A ( c ) / V , ' = A'(c)R7
6 1978 American Chemical Society
(5)
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1 , JANUARY 1979
where A and A 'are constants, the exponent *, will lie between 2 and 2.67. This, of course, reflects a dramatic overall decrease of H with increasing retention. Increasing Velocity vs. Decreasing Field. \Vith this theoretical background, it is useful t o resolve t h e general question of whether a decreasing field strength or an increasing flow velocity leads to the greater speed of elution of highllr retained components, given t h e same minimum permissible degradation of resolution. For t h e moment we assume that n o programming is employed. T h e fixed plate height corresponding to minimum acceptable resolution can be maintained a t any number of ( t i ) values provided the follouing product in Equation 5 is not allowed to exceed a certain constant value
( r ) R ' = constant
(6)
In other words, both ( L ' ) and R must vary simultaneously in a manner dictated by Equation 6 to maintain a constant plate height. However. ( L ' ) a n d R both influence retention time. t,, in accordance with t h e simple equation t, = L / R ( r ?
(8)
which shows t h a t for *, > 1. t , is best minimized by pushing (i.) t o the highest feasible value. According to Equation 6, high ( i.) values are associated with l o a R values. that is. high retention. T h e theoretical superiority of high retention conditions have long been recognized for FFF W , and so this simply extends a n old conclusion into a new dimension. T h e situation is somewhat more complex for programming systems, although we expect. based on the conclusion above, t h a t a n increasing flow program will prove superior t o a decreasing field program. T h e simplest hypothetical requirement for flow programming would be of such a nature t h a t t h e plate height would be forced t o remain constant throughout t h e elution spectrum. thus providing a constant value for relative peak sharpness. Inasmuch as H normally falls off with elution volume according to I/\',-,, the maintenance of a constant H would require a compensating increase in velocity. T h e presumed form of this increase can be seen by letting H in Equation 5 equal some constant. Ho, and solving for ( I')
sL 0
.
HdZ/ f 0i d Z
(10)
where Z is the position of the zone center measured from the beginning of the column. This distance-based average properly accounts for t h e constancy of peak variance, 2,at t h e exit of the channel since the local plate height, H , is du2/dZ. A small increment d Z in zone migration is related to retention ratio R and mean carrier velocity ( c ) by
dZ
=
R(c)dt
(11)
LVhen integrated over column length L,this equation gives
L
= S '0' R ( c j d t
uhich is the general equation for field or flow programming. In the case of field programming, R changes continuously with time as a result of a varying field strength. For flow programming, R remains constant and ( L ) varies. In this case we have
(7)
where L is column length. Thus, t h e optimization question posed above reduces to determining the way in which ( t ' ) and R must be varied to reduce t, to a minimum subject to the restraint of Equation 6. If we solve for R from Equation 6 and substitute this into Equation 7 . we get t, = c o n s t a n t ' / ( ~ ' ) " - "
H =
31
L = R I t0 r ( i > ) d t
(13)
If we replace R by the ratio of the column void volume t o the retention volume, P/ V,, and s o h e for V,, we get
Equation 11 provides a means of converting t h e distance-based integrals of Equation 10 into t h e time-based integrals suitable to describe flow programming. Substituting d Z from Equation 11 into Equation LO (remembering that R is constant insofar as integration is concerned) and employing retention time t , for t h e upper limit in place of L yields
Quantities H and ( t ), of course, must ultimately be expressed in terms of time, the latter through the basic programming function. ( L ' ) 3 ( u ) ( t ) . Plate height H is first written in terms of ( t ) using Equation 5 . Equation 15 then assumes the form
If we replace V , by the integral expression given in Equation 14. we get
4St7(1 0 )?,St
A,
(17) ( P / L ) - ( Jtr(L1)di)*'+'
Equation 9 shows t h a t (i') must increase rapidly with V,in order t o maintain constant column efficiency. The of Equation 9 is calculated as if H , were constant throughout the column. b u t in fact, in the case of programming. plate height varies with velocity. Hence. Equation 9 shows only the increasing velocity requirements for keeping plate height constant in a series of individual runs with different components having increasing V,/;'s. Obviously. with proper averaging, one would find a similar tendency for programmed systems. T h a t trend clearly requires higher velocities for components with high retention volumes. This is dealt with quantitatively in t h e material below. Plate Height and Peak Migration in Flow Programming. In t h a t velocity and consequently plate height vary continuously in the programmed passage of a zone through a n FFF channel. our model requires that the appropriate acerage value of plate height be constant from one zone t o another. T h e average we use is ( t i )
which is t h e general expression for the dependence of t h e average plate height on retention time t,. If we were t o require t h a t H stay constant for all eluted peaks, then H in Equation li would have to be independent of 1,. To investigate this possibility, we select the arbitrary programming function (i') =
(r)@
(18)
and seek a value of n that would yield a constant H. T h e substitution of Equation 18 into Equation 17, followed by integration, yields
n
~
Quantity H will be independent o f t , only if the exponent, y n y,is zero. This requires t h a t n be ~
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 1 , JANUARY 1979
n = y / ( l - Y)
(20)
which has no finite, positive solution for y 2 1, and thus none for the common forms of FFF. In a similar vein, we find t h a t the assumption of an exponential program
yields
A=
A(kL/VO)Y ( e Z k t-r 1) 2 ( ~ ) ~ ~(ektr - 1-
I)?+I
(22)
I
which after a brief transient period of duration - 1 j h approaches the form
(23)
2
3
c
j
Tl',)E
which again fails t o become independent o f t , for any finite. positive value of h when y > 1. T h e failure t o t u r n up a program in which ( L ' ) increases rapidly enough with time to yield a constant H may seem surprising on physical grounds. However, one must keep in mind that although a rapidly increasing velocity does subject later peaks to considerably higher average velocities and therefore increased plate heights, the escalating velocity also sweeps peaks quickly out of the column before the increases they are subjected to are substantial. Mathematically, this restraint appears in the V,? term of Equation 16, which subsequently appears as the ( S o f T ( c ) d t )term ' by virtue of Equation 14. With a rapidly increasing program, the t , from this integral will increase only slowly with V,, which means t h a t highly retained peaks do not stay in the column for enough additional time to be subjected to the enormous velocity increases t h a t one might intuitively expect. In a practical sense, one may find that the theoretical decrease in plate height with increasing V , is not realized because of peak broadening outside of the column. This disturbance cannot be treated in the general case because i t depends upon each specific apparatus. Clearly, for well designed equipment, the velocity can increase rather rapidly without unfavorable effects. T h e normal situation, therefore, is for increasing relative peak sharpness with increasing elution volume whether one has programming or not. Therefore, the major challenge of flow programming is simply that of finding a velocity function that will provide adequate resolution in the least possible time within the most demanding region of the size spectrum.
EXPERIMENTAL The basic system was similar to that described previously ( 3 , 7). Channel thickness LL' was 0.127 mm, width was 10 mm, and length was 794 mm. The void volume was 2.02 mL. A new seal was designed to isolate the inflow from the outflow. In the programmed run, the flow from the Laboratory Data,Control CMP IV pump was increased according to the equation V = 4 + 0.0189 t 2 ,where V is the volumetric flow rate in mL/h, and t is time in minutes. Manual adjustments to follow this program were made a t 1-min intervals during a period of 1 h. The aqueous solution contained 0.1% FL 70 detergent from Fisher Scientific and 0.02'70 sodium azide. The polystyrene latex beads were supplied by Dow Diagnostic. The speed of the centrifuge was maintained using a frequency meter from Berkeley Division, Beckman Instruments Model FR-67/U. This instrument measured the frequency of the interruption of a light beam by a slotted disk attached to and rotating with the centrifugal system. The flow of carrier was adjusted to its starting value (4 mL/h in the programmed run), and then the sample was injected using a microliter syringe. The flow was stopped approximately 15 s after injection t o allow the sample to flow into the head of the
j
.
5
5
\ ;
iirg'~
Figure 1. Separation of a mixture of Dow polystyrene latex beads of mean diameters 0.220, 0.357, and 0.620 pm by nonprogrammed
sedimentation field-flowfractionation at 1000 rpm (89.5gravities). Solute relaxation time = 10 min. Flow rate = 4.0 mL/h channel, then spinning was initiated to allow the material to relax into its characteristic equilibrium position. After 10 min relaxation time, the carrier flow was resumed. In the case of the flow programmed run. the programming started immediately with the resumption of carrier flow.
RESULTS AND DISCUSSION We have chosen to illustrate the operation of flow programming by using polystyrene latex beads of three respective diameters, 0.220 pm, 0.357 pm, and 0.620 pm. We note t h a t the diameter of the middle-sized beads, 0.357 pm, is near the geometric mean of the diameters of the beads a t the two extremes, 0.369 pm. Thus the ratios of diameters are roughly equal when comparing the large beads to the middle-sized beads (1.74) and the middle-sized beads to the small beads (1.61). Consequently, the separation of the middle-sized beads from either the small or large beads would be of roughly equal difficulty providing column efficiency (number of theoretical plates) remained constant in a run. However, since column efficiency increases with retention, as we have noted earlier, we expect a better separation between the large and middle-sized beads than between the middle-sized and small beads. Figure 1 bears out this conclusion in a rather dramatic way. It clearly illustrates the trend to increasing relative peak sharpness with increasing retention. Figure 1 is a fractogram produced in a nonprogrammed run employing a flow rate of 4 m L / h . T h e low value of the flow rate ensures a rather complete resolution of the two smaller groups of beads, b u t the total elution time exceeds 10 h. Any substantial increase in flow velocity beyond that used in Figure 1 would seriously interefere with the resolution of the first two peaks, as previous work has shown ( 8 ) . This is illustrated in Figure 2 where the elution time has been reduced to roughly 1.5 h through the use of a flow rate of 32 m L / h , but at the expense of a serious overlap of the first two peaks. Figure 3 illustrates the value of a flow program for this system of particles. In this case the flow rate was started a t 4 m L / h and was increased above t h a t level in proportion to the square of time. This increase continued until the terminal value of 5 2 m L / h was reached, as shown by the plot of the program in Figure 3. T h e run was then completed a t this terminal value. Elution of the final peak could then have been hastened even more had we continued to increase flow velocity beyond its terminal value. T h e programmed run of Figure 3 has clearly served the function of maintaining an adequate resolution between the first two peaks and, a t the same time, reducing the total
ANALYTICAL CHEMISTRY, VOL 5 1 , NO 1, JANUARY 1979 C220iiv
33
3357ii~n
220 p m
0357prn
0620pm
Sta.1 -
Figure 4. Separation of the mixture of Figures 1-3 by means of a stepped program at 1 5 0 0 rpm. The time scale has been expanded relative to that shown in Figures 1-3 to illustrate peak asymmetry Figure 2. Separation of the mixture of Figure 1 under the same experimental conditions except that the flow rate has been increased from 4 to 32 mL/h 3357 i i m
0
io
20
TIME (hrs)
Figure 3. Separation of the mixture of Figures 1 and 2 in a flow programmed run at 1000 rpm. The form of the program IS superimposed on the fractogram
duration of the run to less than 1.5 h. Even better results could be anticipated with a flow program rising more abruptly near the end where there is still a n excess of resolution to be traded for speed. In general, nearly any program with a fairly rapid increase in flow rate is expected to have a beneficial effect in most F F F runs. T h e optimal program will depend on the nature of the solute mixture, the varying demands for resolution, and equipment limitations. Clearly, it is advantageous to push both maximum flow rate and retention to rather extreme values when it is practical to d o so. As a n example of program versatility, Figure 4 illustrates the use of a stepped program, a different form from any of the programs so far discussed. While our work has shown that this program, too, is advantageous compared to nonprogrammed runs, Figure 4 cannot be directly compared with
Figures 1-3 because the former was operated at 1500 rpm, a higher rotational speed than the 1000 rpm used in the earlier figures. Clearly, however, we have achieved the same kind of results with a rather complete resolution of the first two peaks within an overall time span of
