Anal. Chem. 1987, 59, 2038-2044
2038
Power Programmed Field-Flow Fractionation: A New Program Form for Improved Uniformity of Fractionating Power P. Stephen Williams and J. Calvin Giddings* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
We propose here a new power-law field-decay program for field-flow fractionation (FFF) in which the field strength S is hekl constant at an lnltlal level Sofor a thre-lag period tl and then decayed with time t according to the expression S ( t ) = So[(t, t , ) / ( t f,)P where t , and p are arbltrary parameters. Equations are derived which express both the fractionating power and the retention time for particles subJected to such a program. Various limiting expressions are also obtained. I t is shown that for ail well-retained particles the fractionating power F d is uniform over the particle diameter range provkllng p is properly chosen. For sedhrentatbn FFF, p = 8. Plots of fractlonatlng power (and retention time) vs. particle diameter are developed to verlfy this conclusion and to examine the effects of various system and programming parameters on performance.
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Field-flow fractionation (FFF) is a family of separation techniques widely applicable to macromolecules and colloidal particles ( I , 2). The optimization of FFF is complicated by its large intrinsic diversity, requiring choices to be made relative to field type, field strength, channel dimensions, carrier type, flowrate, and programming. Programming itself represents a broad subclass of operating conditions with a near-infinite number of programming options (3). Both field and carrier (solvent) programming were implemented in the first comprehensive study of programming in FFF ( 4 ) . Flow programming was added later (5). Field programming has been most extensively developed. In this technique the field strength is reduced with time, generally according to some specific mathematical function. Parabolic field decay (linear rpm decay) and parabolic decay combined with an abrupt reduction of field strength were used in the earliest programmed sedimentation FFF experiments (4, 6). Linear and parabolic field decay combined with an initial time lag period were utilized for thermal FFF (7). Subsequently, Kirkland, Yau, and co-workers introduced and widely applied exponential field decay with and without time lag (8-11). The general theory of retention in field programmed FFF was developed in conjunction with the f i s t experimental work (4).Since then, program choices have been made on the basis of the unique distributions of retention times for each program rather than on the basis of differences in resolving power. The Kirkland-Yau time-delayed exponential program was chosen to yield a linear relationship between time and the logarithm of particle diameter (8). Only recently have the necessary theoretical tools been developed to examine the details of resolving power in programmed FFF (12). In this study the resolving power was characterized by the fractionating power Fd, which was expressed as a mathematically continuous function of particle diameter d. Fd is the resolution between two close lying particles divided by their relative diameter difference. Plots of Fd vs. d for exponential programming displayed a maximum or "peak" in Fd at a low value of d followed by a gradual decline
in Fd. In more recent work, even more severe nonuniformities (sharper peaks) in the Fd curves have been shown to characterize linear and two forms of parabolic decay (13). In some instances these Fd peaks can be assumed to be useful in focusing the maximum fractionating power on some small diameter range of particular interest. However, for those cases in which particle resolution and characterization are equally important a t all diameters, it can be argued that a program providing a uniform level of fractionating power over a substantial diameter range would be advantageous. It is the object of this study to present a new mathematical form for field decay which, with the proper adjustment of parameters, yields uniform Fd vs. d plots. The use of such a program form will, in view of its simple power-law nature, be termed power programming. Below we examine the properties of power programmed FFF in some detail. THEORETICAL BACKGROUND The power program for the case of programmed-field operation takes the general form
with the requirement that t 2 t l > t , and p > 0. Here S ( t ) is the field strength at time t , Sois the initial field strength, t , corresponds to an optional constant-field (time-lag) period preceding field decay, and t, and p are variable program parameters. An example of field decay according to this program is given by the full line in Figure 1. It is seen that S ( t )approaches infinity as time t approaches t,; therefore t , corresponds to the position on the time axis of the asymptote of S ( t ) as it approaches infinite values. The field strength decays inversely with time, measured from t,, raised to the power p . The program parameter t , (or its ratio to t o ,the nonretained peak elution time) may be positive, negative, or zero, but it is necessary that tl > t , in order that the program be meaningful. Figure 1 also shows an exponential decay program for comparison with the power program. The particular power program illustrated is for p = 8 and t,/tl = -8, which, as discussed later, is consistent with the recommendations for sedimentation FFF where close to constant fractionating power is desired over a range of particle sizes. The exponential decay program illustrated has decay constant 7' set equal to t1/4. Although the programs appear very similar, the predicted variations of fractionating power with particle diameter differ greatly. As already mentioned, this particular example of a power program results in close to constant Fd over a wide particle size range, the lower limit being determined for a given system by the initial field strength and the upper limit by the onset of steric effects. The latter point will be discussed more fully in a later publication. It has been shown (12),however, that with an exponential decay from a sufficiently high initial field strength the predicted Fd decreases with the reciprocal of the square root of particle diameter. A 10-fold increase in particle diameter would therefore result in a 68% decrease in Fd. Such a range of particle size, or a wider range, is
0003-2700/87/0359-2038$01.50/0 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 17, SEPTEMBER 1, 1987
-t$1
H = xw2(u)/D
(7) where x is a function of the retention parameter A, w is the thickness of the FFF channel, and D is the particle diffusion coefficient. It follows that
c r' (I
i:
2039
a
For all calculations we use the following approximation for retention ratio R R = 6h(l - 2X) (9) which is good to within 0.37% up to X = 0.15. For x we use the approximation (14)
5M B 4
LI)
n
24h3(1 - 1OX X = I
00
05
10
15
20
25
30
35
40
45
50
55
t / t, Figure 1. Comparison of fielddecay curves for exponential programming (broken line, T' = t 1/4) and power programming (solid Ilne, p = 8, t,lt = -8). Inset: schematic plot showing asymptotic behavior of power program function to t = t,.
commonly encountered. The relatively large variation of Fd with d inherent to exponential field decay may be considered undesirable, in which case a suitable power program would be superior. Before we describe the derivation of expressions for retention time and fractionating power specific to the power program, we shall set out the general definitions and assumptions on which our approach is based. Our measure of resolving power is termed the fractionating power F d , defined by (12)
+ 28X2)
( 1 - 2X)
(10)
which is valid to within 6.2% up to X = 0.15. The retention parameter X is related to field strength S by (12)
X = A/Swd"
(11)
where the exponent n is determined by the field type (e.g., n = 3 for sedimentation FFF and n = 1 for flow FFF), and the constant A is given by
A = kTdn/(F/S) = kTd"/4
(12)
where k is the Boltzmann constant, T i s the system temperature, F is the force exerted on a particle of diameter d by the field and 4 = F / S is the field-particle interaction parameter. In the case of sedimentation FFF, cp is equivalent to the effective particle mass, that is r$
= (7r/6)d3 Ap
(13)
where Ap is the difference in density between the particle and the carrier fluid. For sedimentation FFF it follows that where R, is the resolution, 6tr/40,, for particles whose diameters differ by the small relative increment 6d/d, and whose retention times consequently differ by 6tr. Quantity ut is the standard deviation in retention time for particles of diameter d. In the limit 6d 0 we have
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A =6kT/~Ap
Finally, it is assumed that particle diffusion coefficients are given by the Stokes-Einstein equation
D = kT/37r?d where
where Sd is the diameter-based selectivity. Retention time in programmed FFF is given as the upper limit of the general integral equation (4)
L = r t r R ( u ) dt JO
(4)
where L is the channel length. In general, the retention ratio R (for field programming) or the mean fluid velocity ( u ) (flow programming) may vary with time. We are here concerned only with field programmed FFF, in which R varies with field strength S but ( u ) remains constant. We have
where t o is the elution time of a nonretained peak. The standard deviation in retention time for programmed field operation is given by (12)
where H i s the theoretical plate height and R, is the retention ratio at the point of elution. In FFF the plate height H may be expressed as (2, 14)
(14)
(15)
is the carrier viscosity.
POWER PROGRAMMING THEORY Isocratic Elution with t ,I t 1. The introduced sample is assumed to undergo primary relaxation at zero flow velocity and at the initial field strength. The initiation of flow starts the process of elution and corresponds to time zero of the field program. We first consider those components that are completely eluted during the period elapsing up to time tl, during which the field strength is held constant at So. For this case retention time is given by LO
t , d - --
Ro
CO
c
6X0(l - 2X0)
(16)
where Ro and X,are the retention ratio and retention parameter, respectively, for the component at the initial field strength So; Ro is replaced by X, by using eq 9. We can differentiate with respect to d (remembering that Xo is inversely proportional to d" according to eq 11)to give dt, - nto (1 - 4%) _ (17) dd d 6X0(l - 2 ~ ~ ) ~ When eq 9 and 10 are substituted for R and x in eq 8, we obtain 2wtr112 X o ( l - 10x0 + 28AO2)'I2 a, = (18) DIP (1- 2x0)
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 17, SEPTEMBER 1, 1987
We now obtain an expression for Fd by the substitution of eq 16, 17, and 18 into eq 3
Fd =
n (3 Dto)llz 24w
1 -4ho h,3/2(1 - 2X0)1/2(1 -
2
loxo + 28X02)1/2
This in turn corresponds to the condition
x I(2x0)-1/p
(28)
Substituting the condition 28 into eq 23 gives us the following requirement for the existence of a meaningful solution:
(19)
Programmed Elution with t , > t,. When t, exceeds tl, we see from eq 5 that to =
JfiR0 d t
+ I t r Rd t 1
The second integral is obtained by writing R in terms of X by using eq 9. A consideration of eq 1 and 11 shows that the retention parameter X is the following function of time t when t > t,
x
=
xo(
-)
Thus eq 20 becomes
+ 6AoJ
t-t,
tr(
(30)
t-ta
tl
to = Rotl
Fractionating Power (Fd).We can differentiate eq 23 with respect to d. After rearranging the result and making the substitution (see eq 21)
)
dt, _ dd
- dt tl - ta
we obtain the following expression for dt,/dd:
- ta
-
Ltr(
I-’.) tl- ta dt
12XO2
(22)
The integrals are readily evaluated and the resultant expression rearranged to the form
where R , is the value of the retention ratio at the point of elution. The standard deviation at in retention time is given by eq 8. Substituting for R and x via eq 9 and 10, respectively, and for h via eq 21, and subsequently solving the integral and rearranging the result give the following expression for ot:
Equation 23 may be treated as an equation in the unknown x = (t,- ta)/(tl - t,), and solved by using the Newton-Raphson iterative technique. We write
~-1
(P + 1)
t l ( i - 2x0j (tl - tJ
+ 6&(t1t o- tal
(24)
The first derivative of F ( x ) with respect to x is then given by
F’(x) = xP(2XOxP - 1)
The substitution for dt,/dd and for at in eq 3 by using eq 31 and 32, respectively, results in an expression for Fdof the form
(25)
The iteration, which converges on the solution of F ( x ) = 0, is given by (26)
Test for a Meaningful Solution. The derivation of eq 23 involved the use of an approximate expression for R (eq 9). This expression yields meaningless negative values for R when X exceeds 0.5. It follows that there cannot be a meaningful solution to eq 23 corresponding to A, > 0.5, where X, is the value of X a t the point of elution. In practice, as explained later, solutions corresponding to A, > 0.15 are discarded for reasons of accuracy. Nevertheless it is essential to test for the existence of a meaningful solution before the iteration is initiated. We see from eq 21 that the condition A, I 0.5 corresponds to
-.
(33) Computation. A FORTRAN 77 computer program was written to carry out the calculation of retention time and fractionating power and to plot the results as functions of particle diameter d. The program was implemented on a DEC 20 computer and the plots were produced by an Apple Laserwriter. The approach was the same as that taken for the different program types discussed previously (12, 13). The range of
ANALYTICAL CHEMISTRY, VOL. 59, NO. 17, SEPTEMBER 1, 1987
particle diameters of interest is divided into a large number of small increments, the calculation commencing with the smallest discrete particle size. Retention is first determined for isocratic elution a t the initial field strength according to eq 16. If t,, so calculated, does not exceed the initial hold time t,, then Fdis determined by using eq 19. Once a particle size is encountered for which retention under initial conditions exceeds tl, then it and all remaining particles are eluted from the channel under conditions of programmed field decay. Numerical solution for x (= (t,- t,)/(tl - ta) = (X,/XO)'/~) and hence for t, is therefore necessary. Being an iterative technique, the Newton-Raphson method requires an initial estimate to the solution for x . For the first discrete particle size to elute under programmed decay, t, will just exceed tl and a suitable estimate for 3c will be unity. For successive particle sizes the initial estimate is equated to the solution for the preceding particle size. Before the program iteratively solves for x , the test (eq 29) for existence of a meaningful solution is applied. If the result turns out to be negative, then a simple branch from the calculation loop is taken. Iteration is continued until the relative difference between results of consecutive iterations is less than The approximate expression assumed for x (eq 10) deteriorates rapidly in accuracy for X > 0.15, whereas that assumed for R (eq 9) is accurate to 2.2% up to A = 0.2 and falls only to 6.9% at X = 0.25. As for the types of program previously examined (12,13),calculation and plotting of retention times and F d are carried out only for those particles for which Xo 4 X, I0.15. Only toward the extreme ends of discontinued plots does A, approach 0.15, and we can therefore have confidence in the validity of all tabulated and plotted results. (Only simple exponentially programmed decay may be considered a possible exception to this rule. For this program A, quickly approaches a constant level independent of d (12).) RESULTS AND DISCUSSION A comparison of the power program and an exponential program is shown in Figure 1. While the two curves have been assigned parameters to bring them as close together as possible, there is a small residual difference that has a substantial impact on F d values. Most significantly, s for the power program remains higher a t long times and thus avoids the declining F d vs. d curves characteristic of exponential programming (12). Simplified Equations for Fdand t ,. The analysis of the role of various parameters in influencing the dependence of F d and t , on particle diameter d is greatly simplified by assuming Xo < A, (tl - t,)/9, the F d curve rises above the horizontal in the lower region of d, and when t , < (tl - ta)/9, the curve lowers in this region. The setting of tl equal to (tl - ta)/9 is particularly useful in extending the range of uniformity of F d with respect to d. As far as overall analysis time is concerned, the employment of a nonzero tl simply increases the time for well-retained samples by the increment tl. Influence of Other Parameters. We now examine the influence of the remaining system parameters on the Fdcurves. First we see from eq 23, or from the simplified forms given by eq 46 and 47, that, for a particle of given size (d),the ratio of t,/tois dependent on the ratios of the program time constants tl and t , to to (i.e., tl/to and t a / t o together ) with the initial field strength So, the value of p, and the system parameters w, Ap, t, and T. It follows from eq 33 that Fd is directly proportional to (t0)ll2providing the ratios of time constants tl and t , to t o ,along with all other system parameters, remain constant. This is a perfectly general result, valid for all field decay programs as well as for isocratic FFF (12). We are particularly interested in the effect of various parameters on Fdin the case where either the condition Xo 0 P R retention ratio R a t initial field strength So RO
LITERATURE CITED (1) (2) (3) (4)
Giddings, J. C. Anal. Chem. 1981, 53, 1170A. Giddings, J. C. Sep. Sci. Technol. 1984, 79, 831. Giddings, J. C.; Caldwell, K. D. Anal. Chem. 1984, 56, 2093. Yang, F. J. F.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1974, 4 6 , 1924. (5) Giddings, J. C.; Caldwell, K. D.; Moellmer, J. F.; Dickinson, T. H.; Myers, M. N.; Martin, M. Anal. Chem. 1979, 57,30. (6) Yang, F. J.; Myers, M. N.; Giddings, J. C. J . Colloid Interface Sci. 1977, 60, 574. (7) Giddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chern. 1978, 48, 1587. ( 8 ) Kirkland, J. J.; Yau, W. W.; Downer, W. A. Anal. Chem. 1080, 52, 1944. (9) Kirkland, J. J.; Rementer, S.W.; Yau, W. W. Anal. Chem. 1981, 53, 1730. (IO) Kirkland, J. J.; Yau, W. W. Science 1082, 218, 121. (11) Kirkland, J. J.; Yau, W. W. Macromolecules 1985, 78.2305. (12) Giddings, J. C.; Williams, P. S.; Beckett, R. Anal. Chem. 1987, 59, 28. (13) Williams, P. S.; Glddings, J. C.; Beckett, R . submitted for publication in J . Liq. Chromatogr. (14) Giddings, J. C.; Yoon, Y. H.; Caldwell, K. D.; Myers, M. N.; Hovingh, M. E. Sep. Sci. Technol. 1975, 1 0 , 447. (15) Yau, W. W.; Kirkland. J. J. Sep. Sci. Technol. 1981, 76, 577.
RECEIVED for review December 23, 1986. Accepted April 29, 1987. This work was supported by Grant No. GM10851-29 from the National Institutes of Health.
