3398
J. Phys. Chem. 1985, 89, 3398-3405
than being very broad. In order to investigate this point we have obtained the resonance Raman spectrum of N-methylacetamide in both its deuterio and protio forms in deuterioacetonitrile (Figure 4). The main point is that in this solvent the amide I and I' vibrations are clearly seen with moderate intensity and are quite sharp. We conclude from this that the lack of observation of this band in the aqueous environments is due to a very large line width probably because of a vibrationally heterogeneous distribution of hydrogen-bonding environments not present in acetonitrile.
Acknowledgment. This work was supported by NSF Grant PCM83-08529 and NIH Grant GM32323. L.C.M. was supported by an N I H predoctorial training grant, GM07759. We thank Terry Oas for the gift of the [13C,'5N] N-methylacetamide. A preliminary presentation of this work was made at the 28th Annual meeting of the Biophysical Society, San Antonio, TX (Biophys. J. 1984, 45, 322a.). Registry No. NMA, 79-16-3; N M A N-deuterated, 3669-70-3.
Retention Theory for Fleld-Flow Fractionation In Annular Channels Joe M. Davis and J. Calvin Ciddings* Department of Chemistry, University of Utah, Salt Lake City, Utah 84II2 (Received: January 7, 1985)
The characteristics of field-flow fractionation (FFF) are summarized, and the desirability of expanding the methodology from a flat ribbonlike geometry to an annular geometry is explained. Annular systems are treated subject to the general force law F = A/?, where F is the force on an entrained particle directed along radial coordinate r and A is a constant. The retention ratio R , which describes the relative migration rate, is formulated in terms of complicated integrals involving concentration and velocity profiles. Approximations, which are accurate under most practical conditions, are developed for these integrals. With these approximations, a number of general and limiting expressions for R are obtained for both inner and outer wall retention as a function of n and of the inner-to-outer wall radius ratio. Equations for relaxation time in the annular system are also derived.
Introduction Field-flow fractionation (FFF) is a versatile family of separation The methods related in an operational way to strength of FFF lies in its ability to separate and characterize complex systems of macromolecules. Separation in FFF is achieved in a thin unobstructed flow channel whose narrow dimensions (50-500 pm) promote rapid mass transport and consequently rapid fractionation and measurement. In FFF, a field or gradient is applied in a direction perpendicular to the flow axis, inducing the movement of sample particles toward one wall. A steady-state cloud of particles is formed rapidly a t the wall. The thickness of this cloud depends upon the strength of interaction of the field with the particles. The level of interaction differs for different kinds of macromolecules and particles, leading to the formation of clouds of different thicknesses for different species. When laminar flow is initiated in the channel, the various particle clouds are carried downstream by virtue of their entrainment in the flowing fluid. However, the particle clouds forced closest to the wall by the strongest interactions have their downstream motion relatively impeded because the velocity of the transporting fluid approaches zero at the wall in accordance with normal viscous (generally parabolic) flow. Thus, the different particle types end up with a differentially retarded downstream motion, which leads to separation. The process is illustrated in Figure 1. Because of the open and regular geometry of FFF channels and the simple parabolic flow pattern, the displacement velocity of each particle cloud can be predicted rather exactly in terms of the field-particle interaction forces. Therefore, the rate of migration and the level of separation can be calculated for known particles. More often, particle characteristics are unknown, obscured by the presence of a variety of other particles. In this case each particle type is isolated by the normal separative process of FFF and at the same time, through the observation of the downstream velocity of the particle cloud, the force exerted on the particle by a specified field can be calculated. This calculation provides many avenues for characterizing the different particles (1) J. C. Giddinns. Anal. Chem.. 53. 1170A (1981)
(2j J. C. Giddin&,'M. N. Myers,'and K.D. Chdwdl, Sep. Sci. Technol.,
16, 549 (1981).
of the mixture because particlefield interactions generally reflect some desired property (such as charge, mass, or diameter) of the particles. This approach has been shown to be useful for a great variety of biological, environmental, and industrial macromolecules and particle^.^^^ In the great majority of FFF experiments to date, the channel has consisted of a ribbonlike space between flat parallel plates as shown in Figure 1A. This open parallel-plate channel (OPPC) has a number of advantages, including the simplicity of constructing such units. In such a configuration the sample and carrier fluid streams are simple to introduce and withdraw by means of triangular end pieces. The considerable breadth (1-6 cm) of such channels provides a reasonable sample capacity. More importantly, the OPPC geometry (or something very close to it) can be made compatible with various applied forces. It is preferable that the field vectors intercept the channel walls at right angles, so that the field-induced displacement occurs in a direction perpendicular to the wall rather than along the wall. Furthermore, the field strength should be uniform across the wall area. Most fields and gradients can be conveniently arranged to satisfy these requirements in OPPC systems. At the same time, the channel can be oriented in a direction perpendicular to gravity in order to offset potential convective effects, which may originate in the imposed gradients (such as thermal gradients) or in the local variation of density caused by the formation of the particle clouds. Using OPPC systems, we have implemented FFF using thermal gradients (thermal FFF), electrical fields (electrical FFF), sedimentation forces (sedimentation FFF), and liquid cross flow (flow FFF).4 More recently, magnetic forces have been used.s Each of these fields, by causing migration and separation on the basis of different physicochemical parameters, displays its own unique separation spectrum and selectivity. Furthermore, each field leads to the characterization of particles by a unique set of properties. An alternative geometry for FFF consists of an annulus formed by a cylindrical tube or wire of radius rl (the inner cylinder) centered within a cylindrical cavity of radius r2 (the outer cylinder), (3) J. C. Giddinp, G. Karaiskakis, K. D.Caldwell, and. M. N. Myers, J. Colloid Inrerface Sci., 92,66 (1983). (4) J. C. Giddings, Sep. Sci. Technol., 19,831 (1984). (5) T. C. Schunk, J. Gorse, and M. F. Burke, Sep. Sci. Technol., 19,653 (1984).
0022-3654/85/2089-3398$01.50/0
0 1985 American Chemical Society
Field-Flow Fractionation in Annular Channels
The Journal of Physical Chemistry, Vol. 89, No. 15. 1985 3399
TABLE I: Svamry of Fields Potntidly Useful i. Ammulsr FFF field mrtiele rorocertie~ ~ _ or gradient influencing retention n sedimentation mass, density -1 electrical charge, mobility 1 dielectrical palarizability, dipole moment 3 thermal thermal diffusion coefficient -I cross flow diffusion coefficient.Stokes diameter I shear particle size 5 magnetic magnetic suswptibility 3 r
~
~
zones
A
7 m
~~
-* ~~~
-
/
where n depends on the type of field, A depends (among other things) on the field strength and the relevant physicochemical properties of the components, and r is the radial coordinate. Parameter A can be positive or negative depending on whether the particles are forced to the outer or inner wall. Table I is a summary of various fields potentially useful in annular FFF. The table shows the particle properties influencing the level of interaction with the field and thus influencing retention; these properties (singly or in groups) can therefore be deduced from measured retention levels for particles contained in complex mixtures? The level of retention is expressed through the retention ratio R. Quantity R is the velocity B of the particle cloud divided by the mean carrier velocity ( u ) R =d/(u)
Ira
For the OPPC system (Figure IA) R is expressed by
B Ryre 1. Alternative geometries for FFF (A) open parallel-plate
Rp= 6 A [ ~ t h(2A)-'
channel (OPPC): (B)annular channel (ANNC). as shown in Figure IB. In this system, a radial field or gradient generated across the narmw gap of thickness w = r2- rI will induce the individual components of a mixture to migrate toward either the inner or outer cylinder walls. The components will accumulate at one wall as steady-state layers or clouds of different thicknesses as in the OPPC system; cloud thickness will depend on the interaction of the component particles with the field. The dimerential axial flow will transport the different components through the annular space at unequal speeds depending on their cloud thicknesses. thus separating them. One reason for considering annular channels (ANNCs) is to produce and utilize specific fields which cannot be generated or applied easily in the parallel-plate configuration shown in Figure 1A. These 'fields" include the use of new form, such as shear," dielectmphoretic?JOand magnetophoretic' forces, thus promising to extend the FFF methodology to a wider range of particle properties. In addition, same fields commonly used with the OPPC systems could also be utilized in ANNC systems, thus bypassing the perturbations caused by channel edges" and opening up new experimental approaches. Convection would have to be dealt with on a case-by-case basis, either minimized by thinner channels or low-gravity environments or allowed for by calculation. A comparative study of particle migration rates in both configurations is of interest. since one geometry may be favored over the other for a given sample or type of field. In this paper, we shall develop general equations for the degree of retention and the relaxation time to reach the steady state for ANNC systems. We will assume that the fields display radial symmetry and that the force exerted on a particle of the sample is given by the expression F = A/?
(1)
(6) R. H.Sharer. N. Laikcn, and B. H.Zimm. Biophys. Chrm.. 2. 180 (1974). (7)R. H.Shafcr. Biophys. Chem.. 2. 185 (1974). (8)J. C. Giddingr and S . L. Brantley. Scp. Sei. Techno.. 19.631 (1984). (9)H.A. Pahl, 'Dielatrophoresio". Cambridge University Pres. Cambridge. 1978. (IO) 1. J. Lin. B. 2.Kaplan. and Y. Zimmels.Sep. Sei. Techno/.. IS, 683 (1983). (11) J. C. Giddings and M. R. Schure, Chem. Ens. ScL, accepted for
publication.
(2)
- 2x1 = 6AJ(A)
(3)
where (4)
in which k is Boltzmann's constant, T i s absolute temperature, W is the work done by the constant force Fpin transporting the particle acloss the OPPC of thickness w, andJ(A) is the bracketed expression which approaches unity as A approaches zero Under m a t conditions, eq 3 adequately describes the retention ratio for a component in the OPPC system irrespective of the channel wall toward which the component migrates because the flow profile in the OPPC is generally symmetrical and the force Fpis ordinarily constant over the thickness of the channel. In contrast, the flow profile in an annulus is asymmetrical and the force F, as seen by eq 1, varies with the radial coordinate. Thus, radial symmetry leads to a lack of symmetry around the channel midpoint; as a result, the annular retention ratio R for a component depends on whether thecomponent migrates to the inner or the outer cylinder, which in turn depends on the sign of A and thus of F. Expressions relating the concentration profiles in the annular channel to the form and strength of the applied field are essential to calculate retention ratio R, which is the task undertaken here. to a m u n t for selectivity, band broadening, This theory is n-ry resolution, and the relationship between measured retention times and particle properties, as has been shown for OPPC This treatment will generalize an earlier theory developed specifically for annular shear FFF which dealt only with the limiting case of ANNC systems with channels so thin that they could be treated as OPPC systems? We exclude from consideration here the channels normally used in sedimentation FFF.14 These are ribbonlike channels coiled within a centrifuge basket. While these are technically one form of annular configuration, the extremely small channel thickness ( w ) to coil radius ( r , ) ratio and the use of flow around the coil rather than along the axis of the annulus make these systems equivalent to OPPC systems. (12) J. C. Giddings,I. Chrm. Phyi.. 49.81 (196R). (13)J. C.Giddings. Purc Appl. Chcm.. 51. 1459 (1979). (14)M. Martin and J. C. Giddingr. J . Phyr. C h m . . 85.72 (1981)
~
~
3400 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985
Davis and Giddings
I .6 1.4
Slope
1.2 I.o VI( v )
4
0.8
OT 0
0.6
1
0.1
0.2
0.3
0.4
I
I
I
I
0.5
0.6
0.7
0.8
I
0.9 1.0
s
0.4
Figure 3. Plot of magnitude of limiting slope of normalized velocity ratio u / ( u ) vs. distance from the inner and outer walls as a function of p I .
0.2
0
0.1
0.2
0.3 0.4 0.5 0.6
s= x/w
0.7
0.0 0.9 1.0
Figure 2. Plot of normalized velocity ratio v / ( v ) in an annulus vs. reduced coordinate { for several values of pI.
Theory Retention Ratio. The retention ratio R = e/ ( u ) is obtained as follows. We assume that the concentration distribution c(r,8') of the particle cloud at a reference point (to be specified shortly) along the axis can be approximated by the steady-state distribution c*(r). The azimuthal angle 8' has disappeared from c* because of the radial symmetry of the field and of the channel walls, which determine c*. We have shown elsewhere, for ~hromatography'~ as well as for FFF,lZ that the center of gravity of a thin particle band will migrate along the flow axis at a velocity 8 determined by the steady-state particle distribution c*. Therefore, the point of reference for which we write c = c* is the band center. We have (C*")
d=-
(e*)
We see that the evaluation of R requires that we have expressions both for the velocity distribution u(r) and the steady-state concentration distribution c*(r) within the channel. We start with
"(4.
Velocity Profile. The velocity profile for viscous flow in an annulus is"
where r, is the radial distance to the outside wall and 4 and B are constants. If we replace radius r by the reduced radial coordinate
r/r2
(8)
the velocity profile reduces to (9)
The constants 4 and B can be written in the form
e=-
As p1 (or r l )decreases, it is clear that the asymmetry of the velocity profile (relative to the channel midpoint at { = l/,) increases and that the most rapid flow is shifted toward the inner cylinder. When field strengths are high and the particle cloud is compressed into a thin layer next to the wall, the migration rate is controlled by the velocity profile in the immediate vicinity of the wall. Expansion of u / ( u ) around the coordinate position of either wall yields the following linear limiting expressions for the inside and outside wall, respectively
(5)
where the angled brackets represent cross-sectional averages. Since R is defined as v^/(u), we obtain16
P =
Figure 2 is a plot of the normalized velocity ratio u / ( u ) in an annulus vs. the reduced coordinate { for several values of p l . Quantity {is defined as
1 - PI2
r
where {and are the dimensionless distances from the wall in question. The limiting slopes are shown in Figure 3 as a function of P1. Concentration Profiles. We must next obtain the steady-state concentration profile c*(r). This is the concentration distribution for which the radial flux density J, is everywhere zero dc* J, = UC*- D-dr = 0 This expression integrates to
providing the field-induced radial velocity U and the diffusion coefficient D are independent of concentration, a generally valid assumption for the low concentrations encountered in analytical separations. The term c,* is the concentration at the inner wall. We now express velocity U as U = F/f
(17)
where f is the friction coefficient of the particle. The force F exerted on the particle is assumed to vary in accordance with the simple power law shown in eq 1
In P1 where p 1 = r l / r 2 . (15) J. C. Giddings, 'Dynamics of Chromatography", Marcel Dekker, New York, 1965. (16) M. E. Hovingh, G.H. Thompson, and J. C. Giddings, Anal. Chem., 42, 195 (1970). (17) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, 'Transport Phenomena", Wiley, New York, 1960.
where F2 = A / r z nis the force at the outer wall, and is directed outward or inward depending on whether A is positive or negative. If we express D by
D = kT/f
(19)
then the combination of the last three equations gives the ratio
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3401
Field-Flow Fractionation in Annular Channels
When this ratio is substituted into eq 16 and integrated, we get
providing n # 1 . When n = 1 , we find
The dependence of the c* profile on the term Fzrz, which has the dimensions of energy, suggests reformulating c* in terms of the energy W, which is the work expended in transporting a particle over the channel thickness w. Since X = kT/IW is a parameter of fundamental importance in OPPC systems, as shown by eq 4, the reformulation should lead to expressions for ANNC systems such that rather direct comparison with OPPC systems can be made. To obtain W for n # 1 , we carry out the following integration
However, this equation cannot be evaluated analytically for any value n of interest (for example, n = 5, 3, and -1) but can be computed numerically or evaluated by a power series expansion and subsequent term-by-term integration. This latter approach is ill-conditioned if 1.1 >> 1 (i.e., if A > 1 . Briefly stated, the integral I
I = I A x ) g(x) d x
-
(33)
can be approximated by
I
g(x*)I>x)
dx
(34)
when the relative variation in f is puch greater than the relative variation in g over the interval [a,b],Le., when
thus
d In g df >> dx
When this is substituted into eq 21, we get
fl C* = exp[ C1
*
X
- pll-n
1
- pll-"
]
kT
Iwl
-
(35)
In eq 34, x* is the position of the centroid of the function f over this interval (25)
where the upper (in this case +) sign applies when Wand F a r e positive, that is, force F is directed outwardly along r, leading to the accumulation of the particle cloud at the outer wall; the lower (-) sign implies an inward force. For n = 1 the work integral gives W = -F2rz In p1 (26) A=--
dx
kT IFf2 In PI1
In eq 32, if la1 >> 1 then the relative rate of variation of the function exp(ru) is much greater than that of the other functions. Applying eq 34, we get
in which
and the concentration profile of eq 22 becomes
Retention Ratio for n # I. Equation 6 is the starting point for formulating the retention ratio. For the annular space considered here, the cross-sectional averages in eq 6, represented by terms in angled brackets, are to be obtained generally from
If the species migrates toward the inner cylinder, then x* for small X values can be approximated by a
X*in= -- I PIW1
xZh (39)
For a component which migrates toward the outer cylinder, x* x*,,,~ is approximately x*,,~ = a 1 (40)
+
where y is any locally defined quantity of interest. To obtain R for n 1 , we substitute eq 9 and 25 into eq 6 and then integrate according to eq 29. The integrals containing c*, unfortunately, are quite complicated. Some simplification is found by defining the parameter
+
and the variable With these substituted into the necessary integrals, we obtain
(These latter two approximations are accurate within 3.5% as long as la - a/pI"'I = X-I is greater than (roughly) five, or X < -0.2; we know from OPPC systems that we must have X < 0.1, preferably h 0, eq 45 and 46 are simplified to 2X n = 1, A
