2410
Anal. Chem. 1907, 59,2410-2413
Theoretical Study on the Use of Secondary Equilibria for the Separation of Small Solutes by Field Flow Fractionation Alain Berthod Laboratoire des Sciences Analytiques, Universitt? Claude Bernard-Lyon 69622 Villeurbanne CBdex, France
1, U.A. CNRS 435,
Daniel W.Armstrong* Department of Chemistry, Texas Tech University, Box 4260, Lubbock, Texas 79409
The dynamic range and selectivity of fleld flow fractlonatlon (FFF) can be Improved by uslng secondary chemical equlllbrla (SCE). SCE are established by Including a macromolecular addltlve, which strongly Interacts with the field, In the carrler. A theoretlcal evaluation of SCE-FFF for ana# d u t e s is presented. The effects of fleid strength, additive concentrallon, solute-addltlve blndlng, and the Interactlon of the addltlve with the external field are evaluated.
Field flow fractionation (FFF)was introduced by Giddings in 1966 (1). This technique used an external field to drive solute molecules, carried by a solvent in a long ribbonlike channel, toward the “lower” wall of the channel. Because of the parabolic flow profile, the molecules forced against the wall travel more slowly and are retained relative to the solvent or solutes that do not interact with the field. Giddinas demonstrated that the use of different fields (e.g. sedimentation FFF, electrical FFF, thermal FFF, flow FFF, and steric FFF) changed the selectivity and increased the number and type of solutes amenable to separation by this technique (2). FFF is particularly useful in analyzing solutes of high molecular weight, such as synthetic polymers, proteins, latex, and a varietv of other macromolecules. The overall ranee ” of FFF is about 12 orders of magnitude, from about lo4 to 10l6molecular weight (2). Secondary chemical equilibrium in the mobile phase was introduced by Armstrong in thin-layer chromatography ( 3 ) . The solutes can interact with the stationary phase (primary equilibrium) but also with micelles added to the mobile phase (secondary chemical equilibrium). Armstrong extended this technique to liquid chromatography ( 4 ) . Micellar liquid chromatography is now utilized and intensively studied in a number of laboratories (5-7). Other substances, such as cyclodextrins, have been utilized to effect chromatographic separations using secondary equilibria (8, 9). Micelles, cyclodextrins, and other additives involved in secondary equilibria have high molecular or aggregational weights. The goal of the present study is to try to extend the range of FFF to low molecular weight solutes by using secondary chemical equilibria (SCE). In SCE-FFF, a liquid carrier is continuously introduced in a rectangular (striplike) channel (Figure 1). This carrier contains a concentration C,, of the product (called “support” in this study) which interacts strongly with the external field. Once the “support” molecule concentration inside the channel is constant, Le., the entering “support” concentration, C,,, is equal to the leaving “support” concentration in the liquid carrier, the sample to be fractionated, a mixture of solutes, which interact weakly with the field but which have a certain affinity for the “support” (secondary equilibria), is injected in the carrier. To use the classical theoretical treatment derived by Giddings ( I , 2, I O , 111, the following assumptions are made: (i)
There is a parabolic flow profile inside the channel. (ii) There is an exponential distribution along the x axis (Figure 1) of the compounds dissolved in the liquid carrier (“support” molecules or solutes). (iii) After the channel equilibration described below, the concentration distribution of the “support” molecules along the x axis is independent of the z position in the channel; i.e., we assume that the edge effects on both sides of the channel have no significant effect. We are aware that this last assumption is questionable and we plan to check this point experimentally. This theoretical study deals with (i) the distribution of the “support” molecules in a rectangular channel, (ii) the distribution of an injected solute in the carrier and the retention of the solute, and (iii) the particular case of micelles, which are dynamic macromolecules, as “support”, and then (iv) a general illustration of the equations derived is given and analytical abilities of SCE-FFF are introduced and (v) some qualitative considerations on efficiency are briefly- presented. _ DISTRIBUTION OF ”SUPPORT” MOLECULES I N A RECTANGULAR CHANNEL Channel Equilibration. In FFF, an external field is apPlied Perpendicular to the direction of flow (Figure 1)- In a rectangular channel, the velocity Profile along the X axis of the flow in the 2 dim3h-1 can be expressed by u = AP/2qLcx(W - x ) (1) where AP is the driving pressure, i.e., the pressure drop across the channel, 7 the mobile phase viscosity, L, the total length of the channel, and W the channel width. One can use the retention parameter R defined by Giddings (2, 10, 11) as
R = V/(u)
(2)
where ( 0 ) is the mean velocity of the mobile phase and V is the average velocity of the solute. Using the ratio d / W , in which d is the mean layer thickness of the “support” molecule, as the basic retention parameter, Giddings (IO, 11) has shown that, in a rectangular channel with a parabolic flow profile (eq l),R was expressed as
R = (6d/W)[coth (W/2d)
-
2d/m
(3)
As the “support” layer is compressed enough so that R < 0.5, eq 3 can be accurately approximated by the simple form (2)
R = 6d/W The retention time, channel is
t,,
(4)
of the “support” molecules in the
t , = L,/V = (L$)/(u)
=
(6dLc)/(W(u))
(5)
As the “support” molecules are retained in the channel, the mean “support” concentration, C,, inside the channel is higher than the “support” concentration in the entering liquid carrier, C,, as long as the external field is turned on. The carrier flow rate can be expressed by A ( u ) , and the channel volume is AL,,
0003-2700/87/0359-2410$01.50/0 ‘01987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59,
NO. 19, OCTOBER 1, 1987
2411
As the channel was equilibrated, this concentration distribution of the “support” molecules along the x axis, in a given layer of thickness dz, can be extended to the whole channel. The lower wall of the channel is covered by a “support” molecule layer which serves as a pseudo stationary phase for solutes having some affinity for the “support” molecules.
Flgure 1. The ribbonlike FFF channel: W, channel width; Y, channel breadth; A, channel area; dz, thickness of a theoretical plate.
with A , the area of the section of the FFF channel (Figure 1). C, is given by
Cs = [(A(u)ts)/(ALJlCsm
(6)
DISTRIBUTION AND RETENTION OF A SOLUTE Distribution of the Solute. A solute is injected in the carrier. This solute has an affimity for the ”support” molecules previously studied. We assume that (i) the solute does not interact with the external field, (ii) the solute does not modify the “support” molecule distribution, and (iii) the exchanges are fast enough so that in each successive layer or theoretical plate, the solute concentrations are a t steady state. The affinity of the solute for the “support” molecules can be represented by the partition coefficient P
P = s,/sb
With eq 2, 4, and 5, we obtain
Cs = (l/R)Csm
(W/6d)Csm
(7)
The difference between the two “support” concentrations , C and C,, outside and inside the channel, respectively, can be compared to the amount of surfactant molecules that is adsorbed on the stationary phase in micellar liquid chromatography (12, 13). “Support” Molecule Distribution. The global concentration of the “support” molecules, C,,, inside the channel, is not homogeneous along the x axis. The external field is supposed to induced an average drift velocity U along the x axis on “support” molecules (IO). The flux J, along the field direction is given by
J, = -D dC/dx - UC
(8)
where D is the diffusion coefficient of the “support” molecule. At steady state, the net flux J, equals zero; then
with S, as the solute concentration in the “support” phase and Sb the solute concentration in the bulk phase. In the first channel layer or the first theoretical plate of volume WY dz (Figure l),the “support” concentration is not uniform but varies along the x axis according to eq 13 or 15. However, if equilibrium is reached, the solute concentrations s, and S b are uniform along the x axis within the “support” phase and the bulk phase, respectively. By introduction of the molar volume p of the “support” molecules, the product Cp is the volume fraction occupied by the “support” phase and 1- C p is the volume fraction of the bulk phase (0 < CP < 1). These fractions vary along the x axis since C varies. In a slice of thickness dx and volume Y dx dz, there is dM moles of solute. dM is given by
dM =
[s,cc
Sb(1 - c n ] Y dx dz
(17)
and with eq 16
dC/dx = -UC/D
(9)
dM = [I + c V ( P - 1)]sbY dx dz (18) The integration of dM between 0 and W will produce the
C = Co exp(-xU/D)
(10)
number of moles of the solute in the studied layer of volume A dz. The mean layer thickness L of that component is given by
and where Co is the “support” concentration on the wall a t x = 0. Inside the channel, the carrier contains an overall concentration, C,, of “support” molecules. In a layer of section A (=WY‘) (Figure 1) and thickness dz, there are C,WY dz moles of “support” and C,WY dz = ( l 0w C dx)Y dz
(11)
L=
W
s,”.
dM
(19)
LWm By using eq 18 and 10, one obtains
xw[l + Co exp(-x/d)P(P - l)]SbYx dx dz
Consequently, we can calculate Co from eq 10 and 11
co = c,
(16)
-
L= Jw[1
i w e x p ( - x U / D ) dx
c,W (D/U)(l - exp(-WU/D)) (I2) The ratio D f U is d, the average distance of the “support” molecules from the wall ( I I ) , so W exp (-x / d ) c = c,d(1 - exp(-W/d)) (13) In the practical case where d is smaller than 0.2 W (2), exp(-W/d) is smaller than 0.007 and can be neglected vs. 1, consequently
C = C,(W/d) exp(-x/d)
(14)
C = C,,(W/6d2) exp(-x/d)
(15)
or, using eq 7
+ COexp(-x/d)p(P
(20) - 1)]sbY dx dz
and by expressing Co using eq 12 and integrating we get - (W/d exp(-W/d))/(I exp(-W/d))lV[1 + C , W - 1)1 (21)
L = ( W / 2 + C,V(P - l)d[I
In the practical case ( 2 ) ,where d
