Anal. Chem. 1994,66, 4215-4228
Theory of Field-Programmed Field-Flow Fractionation with Corrections for Steric Effects P. Stephen Williams and J. Calvin Giddings* Field-Flow Fractionation Research Center, Department of Chemistry, University of Utah, Salt Lake City, Utah 84 1 12
This paper deals with the principal perturbation to ideal normal-mode elution of particles in field-flowhctionation (FFF). This perturbation is due to the finite size of particles undergoing migration in the FFF channel. The effects of a first-order correction for particle size are examined. Equations are derived for retention time, fractionating power, and steric inversion diameter for operation at constant field strength, as well as under conditions of both exponential and power programmed field decay. Useful limiting equations for fractionating power are derived and their validity is c o h e d for typical experimental conditions. The derived equations are necessary for the future development of a systematic optimization strategy for the selection of operating conditions for particle size analysis by FFF. Calculations confirm our previous conclusion that the fractionating power for exponential field programming varies strongly with particle size; this variation is only slightly reduced by steric perturbations. The uniform fractionating power of power programming is slightly disturbed by steric effects although fractionating power remains much more uniform than for exponential programming, It is shown that a higher uniformity in fractionating power can be gained by manipulating the parameters of power programming but that no improvement is possible with exponential programming. Phenomena giving rise to higher order perturbations and to secondary relaxation are discussed and the conditions identified under which these effects are minimized. The field-flow fractionation (FFF) methods are a family of subtechniquescapable of separatingparticulate or macromolecular materials according to various properties such as mass, size, thermal diffusion, or charge.'S2 They are elution methods where the components of a sample mixture are swept selectively out of the FFF system by a carrier fluid; they emerge at times governed by one of the physicochemical properties noted above. Usually the system outlet stream is fed to a detector so that quantitative information may be obtained concerning the distribution of material according to the given property. Since the components are physically separated during elution there is the option of taking fractions from the outlet stream for further analysis. The separation is generally carried out within a thin (-102 pm) parallel-walled channel with a cross section of high aspect ratio ~~
(1) Giddings, J. C. Science 1993,260, 1456-1465. (2) Martin, M.; Williams, P. S. In Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds. NATO AS1 Series C: Mathematical and Physical Sciences 383; Kluwer: Dordrecht, The Netherlands, 1992; pp 513-580. 0003-2700/94/0366-4215$04.50/0 0 1994 American Chemical Society
(commonly around 100:l). A field, of gravitational or electrical nature for example, is applied across the thii dimension. Depending on the strength of interaction with this field, each of the sample components approaches some specific equilibrium concentration profile across the channel thickness, with a maximum concentration generally close to one wall-the so-called accumulation wall. The flow of carrier fluid through the channel provides the means of elution, with the different components migrating at velocities determined by the interaction of the fluid velocity profile with their specific concentration profiles. The high aspect ratio of the channel cross section gives rise to a fluid velocity profile that is dominated by the drag effect of the major (parallel) walls. The profile is therefore uniformly parabolic across the thin dimension with small disturbances toward the channel edges where the velocity must also approach zero. The assumption of a simple parabolic fluid velocity profile has allowed the derivation of relatively simple analytical expressions describing sample elution or retention time and system band spreading. Many different field types have been employed in FFF to drive sample materials toward the accumulation wall. Each field type interacts with sample components on the basis of a different component property. For example, a gravitational or centrifugal field interacts with particles at a level scaled to effective particle mass (particle mass minus the mass of fluid displaced);3,4an electrical field interacts according to effective particle or molecular ~ h a r g e ;a~thermal ,~ gradient induces migration via the thermal diffusion and a cross flow of carrier (through semipermeable channel walls) causes a drag force proportional to the Stokes diameter.gJO Each field type is associated with a particular subtechnique of FFF. The examples mentioned correspond to sedimentation, electrical, thermal, and flow FFF, respectively. Compared to other elution methods, such as size exclusion chromatography, the FFF subtechniques tend to give greater relative differences in retention time for small differences in size (3) Giddings, J. C.; Myers, M. N.; Caldwell, K. D.; Fisher, S. R In Methods of Biochemical Analysis; Glick, D., Ed.; John Wiley: New York, 1980; Vol. 26, pp 79-136. (4) Giddings, J. C.; Caldwell, K D. In Physical Methods of Chemistvy; Rossiter, B. W., Hamilton, J. F., Eds.; John Wiley & Sons: New York, 1989 Vol. 3B, Chapter 8, pp 867-938. (5) Davis, J. M.; Fan, F.-R F.; Bard, A. J. Anal. Chem. 1987,59, 1339-1348. (6) Giddings, J. C. 1.Chromatogr. 1989,480, 21-33. (7)Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976,9, 106-112. (8) Schimpf, M. E. 1.Chromatogr. 1990,517, 405-421. (9) Giddings, J. C.; Yang, F. J.; Myers, M. N. Anal. Chem. 1976,48, 11261132. (10) Ratanathanawongs, S. K.; Giddings, J. C. In Particle Size Distribution II: Assessment and Characterization; Provder, T., Ed.; ACS Symposium Series 472; American Chemical Society: Washington, DC, 1991; Chapter 15, pp 229-246.
Analytical Chemistry, Vol. 66, No. 23, December 1, 1994 4215
or in other propertiesll (Le., they exhibit higher selectivity). While this is desirable for accurate sample characterization,12it can lead to diffculties in the analysis of polydisperse materials. Conditions giving adequate retention and resolution of the smaller components result in excessive retention of the larger components. This problem is exacerbated for those subtechniques having the highest selectivity, such as sedimentation FFF (SdFFF). The selectivity of SdFFF is so high that the detectability of larger components is compromised by their elution over extended time intervals and their consequent dilution in the channel outlet stream.I3 These diffculties may be avoided by field programming whereby the field strength is reduced continuously during elution of the ample.'^,'^ There are an infinite number of possible field decay programs but two have found most extensive use: the exponential decay of field strength with time16-18and the more recently introduced power law decay function known as the power program.lg The FFF of submicrometerparticles is often approximated (see below) by equations that assume the particles to be point masses. The influence of a first-order correction for finite particle size on both retention time and resolution is examined here. The conditions assumed are not only those of constant field but also of exponential and power programmed field decay. It will be shown that simple limiting equations may be derived for retention and resolution that account for this first-order effect. In a future publication it will be shown how these limiting equations may be used to develop a systematic approach to selection of suitable instrumental conditions for the analysis of given samples. The conditions for which higher order corrections are expected to become significant are discussed later. The phenomenon of secondary relaxation, associated with programmed field decay, is also considered. NORMAL-MODE FFF In the initial conception of FFF,20 the nonuniformity in concentration across the channel, brought about by the action of the applied field, is countered by Fickian diffusion. If the particles or molecules are vanishingly small compared to the dimensions of the system, and there are no significant particle-particle or particle-wall interactions, then the concentration profile is described by the simple exponential c (x) = co exp (-x/l)
where c(x) is the concentration at distance x from the accumulation wall, co is the concentration at the wall (x = 0), and 1 is a characteristic length parameter approximately equal to the mean thickness of the exponential layer. This exponential concentration profile adjacent to the wall characterizesthe so-called noma1 mode (11) Gunderson, J. J.; Giddings, J. C. Anal. Chim. Acta 1986,189,1-15. (12) Giddings, J. C.; Williams, P. S . Am. Lab. 1993,25, 88-95. (13) Williams, P. S. In Field-Flow Fractionation Handbook John Wiley: New York Chapter 9, submitted for publication. (14) Yang, F. J. F.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1974,46,19241930. (15) Giddings, J. C.; Caldwell, K D. Anal. Chem. 1984,56, 2093-2099. (16) Yau, W. W.; Kirkland, J. J. Sep. Sci. Technol. 1981,16,577-605. (17) Kirkland, J. J.; Rementer, S . W.; Yau, W. W. Anal. Chem. 1981,53, 17301736. (18) Giddings, J. C.; Williams, P. S.; Beckett, R Anal. Chem. 1987,59,28-37. (19) Williams, P. S.; Giddings, J. C. Anal. Chem. 1987,59,2038-2044. (20) Giddings, J. C. Sep. Sci. 1966,1, 123-125.
4216
Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
of FFF. Since the highest concentration lies in the region next to the accumulation wall, where the fluid velocity approaches zero, the band migrates at a velocity slower than the mean fluid velocity with the result that it is retained relative to the carrier fluid. For the normal mode of elution, the ratio of band velocity %'to mean fluid velocity (v), the retention ratio R, has been shownz1to equal
Y'
R = - = 6A(coth(1/2A) - 22)
(4
where iis the retention parameter defined
2 = l/w
(3)
where w is the thickness of the FFF channel. For strongly retained material, i 0 and coth(l/U) 1so that eq 2 reduces to
-
-
R = 6A(l - 22) which deviates from eq 2 by just -0.011% at 1 = 0.10 and -0.36% at i= 0.15. The good agreement of eq 4 with eq 2 for strong to quite moderate retention is due to the relatively fast convergence of coth(l/W) to unity (for example, coth(1/22) = 1.000 091 and 1.002 549 for 1 = 0.10 and 0.15, respectively). Thus eq 4 is acceptably accurate for R I0.5 or iI0.1056. The field-induced migration toward the accumulation wall, countered by diffusion away from the wall, results in a continuous exchange of material between streamlines within the band thickness. Without this exchange the band would be smeared along the length of the channel by the sheared flow. Equation 1 strictly describes the equilibrium concentration profile, realized only when the carrier fluid is at rest. Channel flow causes the equilibrium to be upset as material at different distances from the wall is displaced at different rates.22 This departure from equilibrium gives rise to nonequilibrium band broadening. In welldesigned FFF systems, other contributions to band broadening are comparatively small whereas the nonequilibrium effect is an unavoidable consequence of elution. Band broadening for FFF is often described in terms of the number N of theoretical plates and the theoretical plate height H = L/N, where L is the length of the channel. For normal-mode elution, nonequilibrium effects have been shownz3to give
H = xwz(v)/D
(5)
where D is the diffusion coefficient of the component and x is the following function of 1 22 = (COth(1/24
- 22)
[ (336A4
+ 242'
- 1) -
(120A3 - 62) coth(1/22) - (122' - 1) c0thz(1/22) 62 coth3(1/22)l (6) (21) Hovingh, M. E.; Thompson, G. E.; Giddings, J. C. Anal. Chem. 1970,42, 195-203. (22) Giddings, J. C. J. Chem. Phys. 1968,49,81-85. (23) Giddings, J. C.; Yoon. Y. H.; Caldwell, IC D.; Myers, M. N.; Hovingh, M. E. Sep. Sci. 1975,10,447-460.
Equation 6 is exactly equivalent to an expression for x published earlierz3but is simpler. For small 1 values, where coth(1/2A) 1, eq 6 reduces to
-
x=
and 6 we find that
xR = 12A2[(336A4+ 242’
- 1) -
(120A3 - 6 4 coth(1/2A) - (12A2 - 1) coth2(1/22) -
+
24A3(1 - 1OA 282’) (1 - 2 4
62 coth3(1/22)l (12)
(7)
which deviates from eq 6 only by -0.11% at A = 0.10 and by +6.1% at 1 = 0.15. Equation 7, like eq 4, is therefore acceptably accurate for R I0.5. The materials studied by FFF are commonly polydisperse, and components are not therefore eluted as discrete peaks with breadths describable in terms of the number of theoretical plates. The resolution R, of two discrete components, as defined for chromatographic separations, may be predicted, but this is not a useful quantity for describing the overall degree of separation for the continuum of components in a polydisperse material. Consequently a form of “specific resolution” called the fiuctionating power was introducedls to describe the separation of continuous size distributions. The particle diameter-based fractionating power Fd is defined by
which, for R
I0.5 where
coth(l/2A)
- 1, reduces to
X R= 144A4(1- 1OA + 28A2)
(13)
Equation 13 deviates from eq 12 by -0.12% at A = 0.10 and +5.8% at 1 = 0.15. An alternative exact expression for xR that is useful in certain instances, such as for the direct numerical solution of eq 11, is obtained by substituting for coth(1/21) in eq 12 using eq 2. The result may be written as
xR =
- R)R2- 4 ( 4 p
3
- 5R + 3)A’
+ 480(1 - R)A4 (14)
The retention parameter 1 is given by the general e q u a t i ~ n ~ - ~
A = D/lUIw where R, is the resolution, 6t,/4ut, for particles differing in diameter by the small relative increment dd/d and whose retention times differ by dt,, and ut is the standard deviation in time units for particles of diameter d. In the limit of dd 0, eq 8 reduces to
-
The diameter-based fractionating power is therefore a uniquely defined continuous function of d that is a measure of the relative resolving power of a system for particles of different sizes. Fractionating powers based on relative molecular or particle mass, density, or any other property influencing retention may be similarly defined. For elution at constant channel flow rate and either a constant or programmed field, the retention time tr of a component may be obtained by solving the integral equation14
to =
htrR dt
(15)
where U is the particle velocity induced by interaction with the applied field. Quantity D is given by the Stokes-Einstein equation
D = kT/3xqd
(16)
where k is the Boltzmann constant, Tis the system temperature, is the carrier fluid viscosity, and d is the hydrodynamic or Stokes particle diameter. In the case of flow FFF (FlFFF), I VI is simply the cross flow velocity, which is given by Vc/U ,where E is the volumetric cross flow rate, L is the channel length, and b the channel breadth. From eqs 15 and 16 we obtain for FlFFF the equation
where VO is the channel void volume that is equal to Lbw. In the case of sedimentation FFF (SdFFF), IUl is the sedimentation velocity and we obtain the following equation for 1
(10)
where to, the void time, is the time for elution of a nonretained material. The standard deviation in retention time due to nonequilibrium effects is given by18
1 = 6kT/njA~GIwd~
where d is here the equivalent spherical particle diameter. An expression for A that is general to all field types (Le., to all subtechniques of FFF) is given by1*
A = A/Swd”
where R, is the value of the retention ratio at the time of elution (Le., at time tr). The determination of Fd, dependent on tr (in the form of dtr/dd) and ut, therefore requires knowledge of the variation of R and the product xR with time. Combining eqs 2
(18)
(19)
where A is a system constant, S is a quantitative measure of the field strength (always positive by convention), and the exponent n is determined by the field type (e.g., n = 1 for FlFFF and n = 3 for SdFFF). The dimensions of S are arbitrary; it is simply necessary that the corresponding dimensions of A yield a Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
4217
dimensionless result for 1. Channel thickness w is retained in the denominator of eq 19 (rather than being incorporated within A) in order to retain consistency with the forms of eqs 3 and 15. The retention parameter 1 is seen to be inversely dependent on field strength and is therefore time dependent during field decay programming. Using the approximate eqs 4 and 13, expressions have been derived that allow determination of t, and Fd as functions of d for normal-mode elution under constant field and under various programmed field decay c o n d i t i o n ~ . ~Relevant ~ J ~ ~ ~equations ~ for constant field strength, and for the commonly used exponential and power programmed field decays, are given below. In the interests of generality, the solutions are given in terms of n, D, and 10, which is the value of 1 at the initial field strength SOfor programmed operation; constant field operation is assumed to be carried out at field strength SO. The variations oft, and Fd with d are then readily obtainable for any specific subtechnique. Constant Field Strength. As mentioned above, we assume for convenience that a field strength equal to SOis used so that the retention parameter is equal to 10. The following equations will then be equally applicable to the elution of material during the initial constant field periods of the two field programs considered. At constant field strength, R is simply the ratio of void time to retention time, so that to a good approximation (for R 5 0.5)
decay constant. For material eluting in purely normal-mode conditions during field decay, it may be shown18 that, based on the approximation of eq 4 for R, retention time is given by
where
and, with the approximation of eq 13 for xR, the fractionating power is given by
where Af is the value of A at time t!. Equations 24 and 26 are accurate for elution at R 5 0.5, and since R increases during elution, it is required that R, 5 0.5. For the special case of tl = d, corresponding to the timedelayed exponential O E ) program proposed by Kirkland and Yau,16,17 and the neglect of the second-order term in 10in Bt, eq 24 reduces to
where the superscript t indicates a value predicted for purely normal-mode elution. The diameter-based fractionating power for normal-mode elution with R 5 0.5 is given by18
Because of the analytical solution that exists for ti (gven by eq 20), eq 21 may be expressed as
We see from eqs 19,20, and 22 that for elution in the normal mode with 10 0 (Le., for strongly retained material), retention time increases with dn and fractionating power increases with d(3n-1)/2.It is this rapid increase of t! with d, particularly in the case of SdFFF where n = 3, that renders constant field analysis of polydisperse materials impractical. Exponential Field Decay. In this case, the field is held constant for a time tl at the initial strength SOafter which it is reduced according to
-
S(t) = Soexp(-(t - tl)/z3
nD1/' F: = -
{to - 62',4!'}
24wA:' {t'(l - 82:
+ (56/3)A!2)}"2
(28)
For conditions of relatively slow field decay, consistent with high fl ,all material will elute with 1: 1,in which case it may be shown that eq 28 reduces to
(23)
where SO) is the field strength at time t and d is the exponential (24) Williams, P. S.; Giddings, J. C.; Beckett, R J Liq. Chromatogr. 1987, 10, 1961-1998.
4218
where e is the base for natural logarithms. The log-normal relationship between d and ti, characteristic of the TDE program, is apparent in this equation. Whether or not the true TDE program is used, some significant delay time tl is commonly employed. If this is so, we may assume that 10 Figure 7. Fractionatingpower and retention time as functions of d for power programmed field decay in SdFFF. Three examples correspond to the following: p = 8, tl = 4.16 min, fa = -33.28 min; p = 10, tl = 4.63 min, ta = -46.33 min; and p = 12, ti = 4.97 min, fa = -59.68 min. Other parameters as for Figure 5.
It was mentioned above that, in the case of exponential field decay programming, once conditions are arranged for the fractionation of some range of d, little can be done to alter the relative variation of Fd with d over this range. The power program offers an additional degree of freedom, however; the power p influences the relative variation of F d with d. It has been shOwnlgthat when p < 3n - 1then fl increases with d and when p > 3n - 1then FA decreases with increase in d. For SdFFF, the use of p > 8 might therefore be expected to yield a more uniform Fd over the diameter range of interest. Figure 7 shows the results of calculations of F d and tr for p = 8, 10, and 12. In each case, tl = -tJp and the particular time constants required for use with each p were obtained by solution of eq 83. The parameters for p = 8 are identical to those used for Figure 5. For p = 10, the parameters were tl = 4.63 min and ta = -46.33 min, while for p = 12, they were tl = 4.97 min and ta = -59.68 min. For this example, the predicted F d is seen to be more uniform when p = 10. A further improvement can be achieved by relaxing the tl = -ta/p relationship. The parameters for the plots shown in Figure 8 were obtained by equating 4 for the three programs at d = 0.06 pm (an arbitrary small diameter where Fd rz and hence solving for (tl - tJ via eq 35. Equation 82 was then solved for
4)
4226
Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
d (pm> Figure 9. Fractionatingpower and retentiontime as functions of d for power programmed cross flow decay in FIFFF. Three examples correspond to the following: p = 2.0, ti = 10.37 min, ta = -20.74 min; p = 1.5, ti = 8.56 min, fa = -12.84 min; and p = 1.0, 9 = 5.88 min, fa = -5.88 min. Other parameters as for Figure 6.
(td - ta) with 4 = 1.0 pm and hence for ta in each case. For p = 10, the parameters obtained were tl = 4.03 min, ta = -47.58 min, and forp = 12, the parameters were tl = 3.91 min and ta = -62.17 min. For the FlFFF case, alternative powers of 1.5 and 1.0 were examined. The results for p = 2, tl = 10.37 min, ta = -20.74 min; p = 1.5, tl = 8.56 min, t a = -12.84 min; and p = 1.0, tl = 5.88 min, t a = -5.88 min are shown in Figure 9. For this particular example, the optimum power appears to be in the region of 1.5.
HIGHER ORDER PERTURBATIONS
Higher order perturbations encompass particle-particle and particle-wall interaction^^-^^ as well as various hydrodynamic effects. At low ionic strengths of carrier fluids, both particleparticle and particle-wall interactions tend to be repulsive in nature, causing early elution of material. The repulsion may be confined to shorter range, and the perturbation to retention (40) Hansen, M. E.; Giddings, J. C.; Beckett, R /. Colloid Intetface Sci. 1989, 132, 300-312. (41)Hansen,M. E.; Giddings, J. C. Anal. Chem. 1989, 61, 811-819. (42) Mori, Y.; Kimura, IC; Tanigaki, M. Anal. Chem. 1990, 62, 2668-2672.
reduced, with increase of ionic strength. Too high an ionic strength may induce wall adhesion or particle aggregation, however, which must be avoided. The composition of the wall must be considered along with that of the carrier since the properties of both iduence particle-wall intera~tion.~~ When compositions have been optimized as far as possible, additional advantage may be found in modifying the method used for initial sample relaxation. During elution, the tendency for wall adhesion is offset to some extent by the hydrodynamic lift forces described below. The risk of adhesion is therefore greatest during a procedure known as stopflow relaxation. This procedure is often carried out immediately following sample introduction to the channel inlet. Its purpose is to allow the sample components to approach their equilibrium concentration profiles before elution takes place. The stopping of channel flow for a suitable period avoids the excessive band broadening that would occur if particles were allowed to relax across the fast flowing streamlines near the channel center plane. The risk of adhesion would be especially great for programmed runs where a relatively high initial field strength is employed. Methods of non-stopflow relaxation using reduced thickness (pinched) inlet regions"%55 or hydrodynamic have been developed to eliminate the inconvenience and time taken for stopflow relaxation. These methods also have the advantage of retaining hydrodynamic lift during relaxation, which reduces the tendency for wall adhesion. Particle-particle interaction gives rise to what is essentially a sample overloading effect. The resultant perturbation to retention may often be reduced to insignificant levels simply by decreasing sample concentration and/or size. There are also certain hydrodynamic interactions that occur between particles within concentrated suspensions undergoing sheared fl0w.4~-~~ There is the potential for an overloading phenomenon in these cases too, but their effect has not been observed under typical conditions of FFF elution. Finally we consider the extension of the hydrodynamic effects that give rise to the mechanism of steric migration into the regime of smaller particles migrating in the normal mode. It is known that a particle of finite size entrained in sheared flow close to a wall migrates at a velocity that is slightly retarded relative to the undisturbed streamline at the position of the particle center.46The channel flow is also known to give rise to hydrodynamic lift f o r ~ e s ~ ~that % ~increase ~ . ~ ~with - ~ flow ~ rate and tend to drive particles away from the accumulation wall. Both the retardation effect and, to a lesser extent, the lift forces attenuate quickly as a (43) Eckstein, E. C.;Bailey, D.G.; Shapiro, A H.J. FluidMech. 1977,79, 191208. (44) Leighton, D.; Acrivos, A J. Fluid Mech. 1987,177,109-131. (45) Leighton, D.; Acrivos, A J. Fluid Mech. 1987,181,415-439. (46) Goldman, A J.; Cox,R G.; Brenner, H.Chem. Eng. Sci. 1967,22,653660. (47) Segre, G.; Silberberg, A Nature 1961,189,209-210. (48) Segrk, G.; Silberberg, A J .Fluid Mech. 1962,14,115-135. (49) Segre, G.; Silberberg, A]. Fluid Mech. 1962,14,136-157. (50) Cox, R G.; Brenner, H.Chem. Eng. Sci. 1968,23,147-173. (51) Ho, B. P.; Leal, L. G. J. Fluid Mech. 1974,65, 365-400. (52) Vasseur, P.; Cox, R G. J. Fluid Mech. 1976,78,385-413. (53) Cox,R G.; Hsu, S. K. Int. J. Multiphase Flow 1977,3, 201-222. (54) Giddings, J. C. Sep. Sci. Technol. 1989,24,755-768. (55) Moon, M. H.;Myers, M. N.; Giddings, J. C.J Chromatop. 1990,517,423433. (56) Lee, S.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1989,61,2439-2444. (57) Giddings, J. C . Anal. Chem. 1990,62,2306-2312. (58) Liu, M.-IC; Williams, P. S.; Myers, M. N.; Giddings, J. C.Ana1. Chem. 1991, 63,2115-2122.
particle moves from the close vicinity of the wall however. It is apparent that high channel flow rates should be avoided if lift forces are not to greatly disturb the particle concentration profile within the channel. At the same time, if the retardation effect is not to have a significant influence on mean particle migration velocity, then the equilibrium layer thickness must be large relative to particle radius. For material eluting close to the steric inversion point this will not be possible, and in the case of programmed field decay, a significant range of particle sizes eluting before steric inversion may start their migration effectively in the steric mode under the high initial field strength conditions. SECONDARY RELAXATION EFFECTS Under conditions of programmed field decay, the effects of secondary relaxation on retention time and fractionating power must be considered. The decay of field strength brings about a continuous change in concentration profile for each sample component. Due to the finite time required for relaxation, the profile for each component tends to lag behind the equilibrium profile corresponding to instantaneousfield strength. The process also leads to a distortion of the profile. The change in concentration profile to accommodate a change in field strength during elution is referred to as secondary relaxation to distinguish it from the primary relaxation that occurs at the inlet of the channel. A solution for the gradual change in band velocity in response to a step change in field strength has been obtained by Yau and Kirkland,59the determination of velocity at arbitrary time following the step change requiring the evaluation of nested sums of infinite series. They applied the approach to exponentialfield decay and determinedthe expected perturbation in retention time for certain examples. Such an approach does not lead in an obvious way to a method of predicting those conditions for which secondary relaxation effects will be significant. Based on an approximate perturbation to concentration profile derived by GiddingsZ7for small departures from equilibrium, Hansen et aLZ8obtained a relatively simple solution for perturbed retention time under conditions of exponentialfield decay. Neither approach took steric perturbations into account. The latter approach does however indicate that, for small departures from equilibrium,the relative error in d incurred in data reduction by ignoring secondary relaxation varies with l/(@2.60 This proportionality holds for both power programmed and exponential field decay. It is therefore a simple matter to impose restrictions on conditions consistent with some for some relatively small or negligible error in d. A more thorough treatment of secondary relaxation including its effect on Fd will be the topic of a future publication. CONCLUSIONS From the foregoing discussionwe conclude that, provided (1) the compositions of the carrier fluid and channel wall are optimized, (2) channel flow rates are not too high, and (3) in the case of programmed field decay, fractionating powers are not too low, then the first-order correction for finite particle size is acceptable for particulate material eluting before, and not close to, steric inversion. The derived equations, in particular the limiting forms for fractionating power, may therefore be used as a basis for the development of a systematic optimization strategy (59) Yau, W. W.; Kirkland, J. J. Anal. Chem. 1984,56, 1461-1466. (60) Williams, P. S., unpublished work.
Analytical Chemistry, Vol. 66, No. 23, December 1, 1994
4227
for selection of operating conditions for particle size analysis by FFF. This will be the subject of a future study. ACKNOWLEDGMENT This work was supported by Public Health Service Grant GM10851-36 from the National Institutes of Health.
b
B Bt c (x) co
d di D e Fd
el G
Go
H k 1
L n
P R Rc Rr
RS S
sd so t ta tr tri
SYMBOLS channel breadth quantity defined by eq 53 quantity defined by eq 25 concentration as a function of x concentration at accumulation wall particle diameter inversion particle diameter diffusion coefficient base for natural logarithms diameter-based fractionatingpower defined by eqs 8 and 9 fractionating power unperturbed by steric effects sedimentation field strength (acceleration) initial sedimentation field strength theoretical plate height Boltzmann’s constant mean layer thickness channel length power dependence of I on d see eq 19 power program parameter; see eq 30 retention ratio core channel retention ratio retention ratio at time tr resolution quantitative measure of field strength diameter-based selectivity constant or initial field strength time time parameter of power program; see eq 30 retention time time to point of inversion
4228 Analytical Chemistry, Vol. 66, No. 23, December 1 , 1994
retention time if unperturbed by steric effects void time delay time before onset of field decay; see eqs 23 and 30 system temperature field-induced velocity of particle mean carrier fluid velocity band velocity channel void volume channel flow rate cross flow rate initial cross flow rate channel thickness distance from accumulation wall, measured across channel thickness ratio of dl(2w) ratio of dil(2w) steric correction factor; see eq 71 small dzerence in d small difference in tr corresponding to dd particle-carrier fluid density difference carrier fluid viscosity retention parameter (=l/w)) core channel value of 1 (=1/(l - 2a)) value of I at time tr value of2 at time t,t value of 1 at field strength SO value of 10for particle of diameter di system parameter; see eq 19 standard deviation in retention time exponential field decay constant; see eq 23 function of I defined by eq 6 core channel value of x Received for review June 15, 1994. Accepted September 14, 1994.@ Abstract published in Advance ACS Abstracts, October 15, 1994.
