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Anal. Chem. lQ86, 58, 161-164
General Retention Theory for Sedimentation Field-Flow Fractionation Joe M. Davis Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The prlnclples of sedlmentatlon fleld-flow fractlonatlon (SdFFF) are reviewed and an lncentlve for developlng m a l l SdFFF systems Is reported. A general retention theory for SdFFF Is then developed that accounts for the radlal symmetry of the sedlmentatkn field and of the flow channel. The SdFFF retentlon ratio, R , Is expressed as a compllcated Integral, the value of which must generally be determined numerlcally and depends on the SdFFF parameter X and the Inner-to-outer radius ratlo. Analytical expressions for R are calculated for the strong-field llmH of SdFFF. For practical systems, the R predlcted by thls theory does not dlffer slgnlflcantly from the retention ratlo tradltknally used In SdFFF, which Is based on a parallel-plate flow channel.
Sedimentation field-flow fractionation (SdFFF) is a subtechnique of the general separation method, field-flow fractionation (FFF), in which the components of a mixture are localized by centrifugal forces near the walls of a spinning flow channel, which is usually an annular ring (AR) formed from a coiled rectangular conduit, and are separated along the channel length by flowing viscous fluid (I),as shown in Figure 1. The mathematical simplicity of the approximate equations describing the spatial distribution of components and the velocity profile (or field) in the AR has facilitated the derivation of approximate equations describing (among other things) the average rate of migration of components (or zones) through the AR and the dispersion by flow of zones along the channel length (I).Such equations are useful for estimating select physicochemical properties of a mixture’s components from their observed experimental behavior in FFF systems, thus utilizing FFF as an analytical tool. The average migration velocity, u, of a component, for any subtechnique of the FFF family, is usually described by retention ratio, R (2)
where c * is the equilibrium concentration of the component, and u and ( v ) are the velocity profile and average linear velocity of the fluid, respectively. (The brackets in eq 1 indicate that c* and u are averaged over the cross section of the flow channel.) A heuristic theory predicts that the retention ratio R, for SdFFF is (1,2)
R , = 6X(c0th (2X)-’
- 2X)
(2)
where
and where k is Boltzmann’s constant, T i s absolute temperature, W is the work required to transport a particle of mass m and density p s across a channel of width w , A p is the difference in densities between particle and fluid, d is Stokes
particle diameter, and G = w2r is the centrifugal acceleration acting on a particle a t radial coordinate r (see Figure 1)and arising from spinning the AR with angular velocity w. (In eq 3, G is approximated by the constant w2rl = w2r2, where rl and r2 are the inner and outer radii of the AR, respectively.) By relating eq 2 and 3 to the experimental migration velocities of species fractionated by SdFFF, the mass, density, and Stokes diameter of the species’ constituent particles can be estimated (1,3-15). In the derivation of eq 2, it was assumed that the conduit forming the AR can be mathematically treated as two parallel plates. More specifically, it was supposed that a given species is localized near a planar wall by a constant force and the velocity profile in the AR is identical to that in a parallel-plate channel. These assumptions are quite good when the ratio of channel width, w ,to the mean radius of the AR is small, as is the case for present-day instrumentation, for which typically w = r2 - rl = m and rl = r2 = 0.2 m. However, a departure from eq 2 is expected as this ratio is increased, either by expanding the channel width (which, as discussed below, has undesirable consequences) or by reducing the mean radius of the AR. Commercial interest in SdFFF systems formed from ARs having smaller mean radii than traditionally used has recently arisen (16),most likely from considerations of marketability. Since such systems may be available in the near future, it is important to determine the magnitude of error in eq 2 arising from the above assumptions. Without an estimate of this error, the accuracy of the physicochemical properties estimated via eq 2 and 3 cannot be properly gauged. Accordingly, a general equation for the SdFFF retention ratio that accounts for the radial symmetry of the channel is derived below.
THEORY To determine an expression for the general SdFFF retention ratio, R, the equilibrium concentration, c*, of a zone (or component) and the velocity profile, u, in the AR must be known, as dictated by eq 1. An equation for c* accounting for the variation of the centrifugal acceleration, G = w2r, over the channel width was recently developed in a study of annular FFF channels (17,18). An approximate expression for v is now derived below. An exact description of v is complicated by secondary flow, which is a component of flow perpendicular to the axis of any curved channel. Secondary flow arises because the rapidly flowing fluid in the channel center is subject to a greater centrifugal acceleration than the slowly flowing fluid near the channel walls (1,9, 19-22). Early attempts to fractionate mixtures using SdFFF channels formed from circular tubes were unsuccessful principally because of the circulation by secondary flow of species over the channel cross section (I). The present-day SdFFF channel is a coiled rectangular conduit characterized by a large breadth-to-width (or aspect) ratio (typically, =1@),which largely inhibits secondary flow because of substantial frictional drag (1, 9). Since the resolving power and practical utility of SdFFF greatly decreases in the presence of significant secondary flow, it is reasonable to consider first the case in which this com-
0003-2700/86/0358-0161$01.50/0 0 1985 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 1, JANUARY 1986
i
Flow in
O
L
L
0
L
L
02
L
-
-
+
d
0.4
l
0.6
C.8
.o
Figure 2. Plot of v o / ( v o ) vs. { for select values of p,.
Figure 2 is a plot of uo/ (u 8 ) vs. the reduced channel width, 3; for select values of pl. The quantity {is defined as (17,18)
r-rl P-Pl (10) r2 - r1 1 - P1 The most rapid flow shifts toward the inner wall of the AR as p1 approaches zero because the smaller surface area of this wall exerts less frictional drag than the greater surface area of the outer wall. (A similar plot of reduced velocity profile vs. reduced channel width for various p1 values was developed for annular FFF channels, in which flow is parallel to the axial coordinate of the annulus (17, It?).) An analysis of the equation of continuity leads to the following expression for the SdFFF equilibrium concentration, {=-=--
Flgure 1. Schematic of AR for SdFFF. Front and top views are drawn to different scales.
ponent of flow is negligible. The general equation for u = u(u,,uo,u,) is obtained by solving the Navier-Stokes equations for the radial ( U J , angular (ut), and axial (u,) velocity components, whose directions are indicated in Figure 1. When secondary flow is negligible, u, and u, are (very nearly) zero and u = uo is greatly simplified. Considering here only the case in which the flow rate through the AR is constant, the Navier-Stokes equations reduce to (23)
where pf is fluid density, 7 is viscosity,p is pressure, and ap/M is the constant pressure gradient arising from pumping fluid through the AR. Equation 5 was solved, subject to the boundary conditions (23-25)
u&r = rl) = 0
c* (17, 18) c* = cl* ex.(
to obtain uo, which is
where p = r/r2, p1 = rl/rz, 6 is a constant equal to
and ( u o ) is the average fluid velocity in the 0 direction, defined by the integral
1 P2-P1
(11)
l-p12
where cl* is the concentration at the inner wall of the AR and X = kT/IW. This X differs from the third and fourth identities of eq 3, in which G is assumed constant, and is calculated by evaluating W via a force-distance integral (17,18). Because G is not symmetrical with respect to the channel’s midsection, the c* profiles for the outer wall (indicated by the upper sign in eq 11)and the inner wall (the lower sign) are different (17). Combining eq 1,7, and 11,the general retention ratio, R , for SdFFF in the absence of secondary flow can be written as
R=
11
exp(Fw2)(p12(ln pl)(p - P - ~ ) / u - p12) &&:P
(6)
uo(r = r2) = 0
*-( 2))
+p
1n p) dp
exp(Fw2) dp (12)
where (17) a = (X(p12 - 1))-’ (13) The integral comprising the numerator of eq 12 cannot be evaluated analytically and, in general, must be computed numerically. Alternatively, using a procedure previously described (17, 18, 26), the retention ratios Ri, and Rout of species that migrate to the inner and outer walls, respectively, of the AR can be shown to equal approximately, for X
