Anal. Chem. 1995,67, 1179-1185
Concentration Effects in Field=FlowFractionation with Secondary Chemical Equilibria Mauricio Hoyos and Michel Martin* Laboratoire de Physique et Mecanique des Milieux Hbtbroghnes (URA CNRS 857), Ecole Superieure de Physique et Chimie Industrielles, 10, rue Vauquelin. 75231 Paris Cedex 05, France
In field-flow fractionation with secondary chemical equilibria (SCE-FFF), small solutes can be separated according to their distribution coefficient between a carrier additive species affected by the applied field and the bulk carrier. Due to its interaction with the applied field, the average additive concentration within the FFF channel is generally significantly larger than in the carrier reservoir and becomes so high that additive-additive interactions cannot be neglected. A recent retention model of FFF at finite concentrations is applied to SCE-FFF. In contradistinction with previous theories in which additiveadditive interactions were not accounted for, it is found that the retention time of a given solute goes to a maximum when the additive concentration in the carrier increases. Published SCE-FFF experimental data are examined in light of this model. The potential of this FFF operating mode and guidelines for optimizing the separation are discussed. The separation in field-flow fractionation (FFF) results from the differential sampling of the streamlines of a carrier flow in a thin ribbon-like channel by the various sample components (which, in the following, will be referred to as the “solutes” since they will most often be soluble species, but the topic of this article may also apply, at least in principle, to small particulate species in suspension). The nonuniformity of the velocity of the flow streamlines is a consequence of the viscous forces arising in a pressure-driven flow system. The differential distributions of the various solutes in the flow streamlines result from the application of an external force field perpendicularly to the direction of the flow. In most FFF operation modes, the nonuniform distribution of a given solute across the channel thickness arises from the direct action of the forces involved (field force, thermodynamic diffusive force and, possibly, hydrodynamic lift force) on the solute molecules. An alternative possibility to create a nonuniform solute distribution across the channel thickness is to associate the solutes with a mediating agent which itself is nonuniformly distributed across the channel thickness as a result of the above-described forces acting on its molecules or particles. In this case, the retention of a solute depends on its degree of association with the mediating agent. This mediating agent is permanently present in the separation channel, and because it is flowing and is therefore nonstationary in the channel, it is continuously introduced with the camer liquid in the channel. For this reason, the mediating agent is referred to in the following as the “camer additive” or, simply, “additive”. Since the separation between solutes arises from variations in their degree of association with the mediating 0003-2700/95/0367-1179$9.00/0 0 1995 American Chemical Society
agent, it is much like a chromatographic process. This FFF operation mode was developed by Berthod and Armstrong’ and implemented by Berthod et aL2 Because this mode involves a partitioning of the solutes between the bulk camer and the additive species, in addition to the primary normal, or Brownian, FFF operating mode acting on the additive molecules or particles, it is termed FFF with secondary chemical equilibria (SCE-FFF) .3 A retention model of this FFF mode was developed by Berthod and Armstrong.’ This model appears to be a limiting law that is correct only when the solutes have a very high affinity for the additive species and when this additive species is highly compressed near the accumulation wall. The retention model was later extended by one of us to any value of the solute distribution coefficient between the additive species and the continuous bulk carrier liquid and to any level of compression of the additive species toward the accumulation wall.4 However, in both models, it is assumed that the additive species is exponentially distributed across the channel thickness as the result of the competition of the field and diffusive forces. This implies that the diffusion coefficient and the field-induced velocity of the additive are constant and do not depend on the distance to the accumulation wall. This hypothesis is correct when the additive species is diluted since there is then no interaction between the additive particles (or molecules, but for clarity, one will assume in the following that the additive is a particulate species). In addition, this dilution assumption must then be satisfied at any position across the channel, especially in the immediate vicinity of the accumulation wall, where the additive concentration is the largest. In practice, because the compression of the additive layer near the accumulation wall may reach relatively high levels, the additive concentration, or volume fraction, at the accumulation wall may be considerably larger than the mean concentration, or volume fraction, in the channel. Furthermore, because the additive is retained by the applied field, this mean fraction may itself be quite larger than the volume fraction additive in the incoming carrier. Thus, the range of additive volume fraction in the incoming carrier for which the previous model can satisfyingly be applied is likely to be narrow and limited to a rather low value. We recently developed a model describing the sedimentation equilibrium distribution for suspensions of hard spheres at moderately large concentrations up to about 40% in volume (1) Berthod, A; Armstrong, D. W. Anal. Chem. 1987,59, 2410-2413. (2) Berthod, A; Armstrong, D. W.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1988,60, 2138 2141. (3) Martin, M.; Williams, P. S. In Theoretical Advancement in Chvomatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; pp 513-580. (4) Martin, M. Chromatographia 1992,34, 325-330.
Analytical Chemistry, Vol. 67, No. 7, April 1, 1995 1179
fraction,j and we used the resulting concentration profile in a retention model for sedimentation FFF.6 In the following, we apply this model to describe the behavior of the additive particles in the FFF channel, at any concentration, and to compute the retention of the solute in SCE-FFF. THEORY The SCE-FFF mode of operation has been conceived as a means to extend the separation capability of FFF to small solute It is also perceived as a physicochemical tool for obtaining partition coefficients of solutes between a dispersed phase and a bulk phase and/or for obtaining information on the physicochemical properties of the a d d i t i ~ e . ~Small , ~ amounts of solute are normally required to these objectives so that one can reasonably assume that the distribution of the additive particles in any cross section of the channel is not influenced by the amount of solute or by the extent of its partitioning between the additive and the bulk carrier liquid. Furthermore, since the solute is generally a species with a low molecular weight, one can assume that it does not directly interact with the applied field. Solute Retention. Let c , , ~and c,,b be the concentrations of a given solute in the additive and in the bulk carrier liquid, respectively, at equilibrium in a given channel cross section. The overall concentration c,(x) of the solute at a distance x from the accumulation wall is then equal to cS(X)
= cS,a(~)#a(~)
+ cS,b(x) [ 1 - 4a(x)l
(1)
where 4a(x) is the volume fraction of the additive particles at position x . With the above hypotheses, since the solute is not affected by the field, c,,b is constant in the whole channel thickness. Similarly, c,,,, which represents the amount of solute per unit volume of additive particles, does not depend on x. Introducing the solute distribution coefficient, P, as one gets cs(x>
= 11 $- 4a(X)(p-
l>Ics,b
(3)
The mean solute concentration, (c,), in the channel cross section is given as
1
(c,) = , ~ w c ~ ( x & ) = [ 1 + (4a)(p - l)Ic,,b
(4)
where w is the channel thickness (distance between the two walls) and (+a) the mean additive volume fraction in the channel (since the additive is introduced continuously in the channel, its mean volume fraction in steady-state conditions is, unlike the mean solute concentration, independent of the position of selected cross section along the channel). Elimination of c,,b between eqs 3 and 4 provides the relative solute concentration profile:
(5) Let Ra and R, be the retention factors of the additive species and of the solute, respectively. They are defined as
and ( 5 ) Martin, M.; Hoyos, M.; Lhuillier, D. Colloid Polym. Sci. 1994,272,15821589. (6) Hoyos, M.; Martin, M. Anal. Chem. 1994,66,1718-1730.
1180 Analytical Chemistry, Vol. 67, No. 7, April 1, 1995
(7) In these equations, ca and (ca) are the additive concentration, at position x , and mean additive concentration, respectively. They are both proportional to the corresponding volume fractions +a and u / ( u ) is the relative velocity profile, which may not have the classical parabolic shape if the additive concentration is large enough to selectively modify the viscosity of the carrier according to x. Combining eqs 5-7, one gets the desired retention factor of the solute:
It is interesting to note that this equation is identical to that obtained previously by assuming that the velocity profile is parabolic and the additive concentration profile exponential.* It is therefore quite general and applies whatever the shapes of the carrier velocity profile and additive concentration profile. As a consequence, the conclusions previously drawn on limiting or specific values of Rs,for P going to 0 or infinity, P equal to 1, +a,inc going to 0, and Ra going to 0 or 1,also apply to the present model.
Additive Volume Fraction Distribution in the Channel. The average additive volume fraction in the channel, (Oa), is not equal, but larger than the additive volume fraction in the incoming carrier, +,,in,, since the additive is retained in the channel by the applied field. It is easily shown1,*that one has
so that the solute retention factor is given, in terms of
@,,inc,
by
The three independent parameters that control the solute retention in a SCE-FFF run are the solute partition coefficient, P, the additive volume fraction in the incoming carrier, &inc, and a parameter expressing the extent of the additive retention. This parameter is classically described by the basic FFF parameter, %a, defined as
A, = kT/Faw
(11)
where k is the Boltzmann constant, T the absolute temperature, and Fathe force exerted by the field on a single additive particle, which depends on the field intensity and on the additive suscep tibility to the field. Although 1, does not explicitly appear in solute retention eq 8 or 10, it affects R, through the additive retention factor R,. When the additive volume fraction is small at any position in the channel, additive particle-particle interactions are negligible and the additive is exponentially distributed through the channel thickness, as assumed in the previous model.* When the additive concentration is large enough for the surface-to-surface interparticle mean distance to become of the order of a few additive particle diameters, steric and possibly other interactions between particles start to significantly influence the transport properties of the additive. Its diffusivity and its fieldinduced migration become concentration-dependent and its concentration profile is no longer exponential. In addition, the viscosity of the additive suspension then also depends on concentration and the carrier velocity profile is no longer parabolic. In this situation, the additive retention factor depends not only on Aa but also on (4,).
Hard-SphereModel for Particle Interactions. We recently developed a model for the sedimentation equilibrium of hardsphere particles at finite concentration^.^ It is based on the Camahan-Starling semiempirical expression of the osmotic pressure of a hard-sphere f l ~ i dwhich , ~ is considered to satisfactorily describe the physical properties of monodisperse sols of spherical particles.* The resulting additive volume fraction profile is given by
where ro is the viscosity of the suspending carrier liquid. Since the additive volume fraction depends on x , through eq 12, the carrier viscosity also depends on x. Then, the relative velocity profile is no longer parabolic but is given by3
with C equal to where is the additive volume fraction at the accumulation wall. This volume fraction is generally not known. It can be expressed in terms of the average additive volume fraction, (@a), through
where
@ a , ~is
the additive volume fraction at the depletion wall
(x/w = l ) , which according to eq 12 is given by
The computation of the relative velocity profile was performed as a function of (@a) and Av6 It was found that as (@a) increases and as ,la decreases the distortion of the flow profile from the highdilution parabolic shape increases and that the carrier velocity in the vicinity of the accumulation wall decreases. Therefore, the effect of the steric particle interactions on the velocity profile tends to increase the particle retention. The overall additive retention factor depends on both the additive concentration profile and the carrier velocity profile, as seen from eq 6. It can be calculated using eqs 6, 12, 15, and 16 with the help of eqs 13, 14 and 17. The numerical computation has been performed for various (da)and lavalues6 It was found that, for a given average volume fraction, the retention factor increases with increasing A,, as normally observed in the Brownian FFF mode, but that for a given ;la the retention factor increases with increasing volume fractiom6 This indicates that the effect of the excluded-volume interactions on the mean distance from the accumulation wall is stronger than its effect on the viscosity. RESULTS AND DISCUSSION
Elimination of @a,o and @ a , ~between eqs 12-14 provides, at least in principle, the expression of @a vs x / w in terms of (da)and 1,. The numerical computation of the concentration profile was performed for various (@a) and ;la valuesG5It was found that the distortion of the additive concentration profile from the infinite dilution exponential shape increases with increasing average additive volume fractions for a given A, value and with decreasing 1, for a given average volume fraction. The effect of the excludedvolume interactions between particles is to increase the mean distance of the particle distribution from the accumulation wall. This effect alone tends to repel the particles toward the fast moving streamlines in the center core of the channel and thus contributes to decreasing the retention factor. However, the viscosity, 7,of the carrier depends on the additive volume fraction. The semiempirical expression of this dependence was derived and found to be in good agreement with experimental viscosity data on hard-sphere suspension^.^ It is given by (7) Carnahan, N. F.; Starling,IC E. J. Chem. Phys. 1969, 51, 635-636. (8) Lyklema, J. Fundamentals of Intetface and Colloid Science. Vol. I: Fundamentals; Academic Press: London, 1991; Chapter 6. (9) Graham,A L.; Steele, R. D.;Bird, R B. Ind. Eng. Chem. Fundam. 1984, 23, 420-425.
Additive Volume Fraction and Retention. The solute retention depends of course on the amount of additive particles contained in the channel or, more exactly, on the average additive volume fraction in the channel, which itself depends on the additive volume fraction in the incoming carrier as well as on the additive field-interaction parameter, 1,. For a given field strength, as long as the additive is highly diluted in the incoming carrier, its average volume fraction in the channel increases linearly with its volume fraction in the incoming carrier. However, when the particle-particle steric interactions start to become significant, the retention of the additive decreases and the additive average volume fraction in the channel does not increase as fast as the volume fraction in the incoming carrier, as shown in Figure 1for various A, values. The curves are limited to situations for which the additive volume fraction at the accumulation wall does not exceed about 40% as the present hard-sphere model does not correctly describe the steric interactions between particles at larger concentration^.^ The slopes of the curves at the origin (@ahc = 0) correspond to the hypothetical situations where the additive distribution in the channel is exponential whatever its average volume fraction. The differences between the exponential model and the present one are quite large. For instance, in the conditions selected by Berthod and Armstrong in their initial paper (La = 0.015 and &inc = 0.01), the exponential model would give Analytical Chemistry, Vol. 67, No. 7, April 1, 1995
1181
0.10
1
0 0.00
0.02
0.06
0,OR
6 a,inc
0 a,inc
Figure 1. Average volume fraction of the additive in the channel, (@a), as a function of the additive volume fraction in the incoming carrier, @a,,nc. From upper to lower curves, respectively: l a = 0.01, 0.015, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1.
0.04
Figure 2. Mean relative retention time of an additive particle in a SCE-FFF run, l/&, vs the additive volume fraction in the incoming = 0.01, carrier, @,,inc. From upper to lower curves, respectively: ia 0.015, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1. i
(4,) = 0.115 (this is the value on which their retention calculations are based) while one gets (4,) = 0.048 with the hard-sphere model.
l/R,
f
This corresponds to differences in additive retention factor that are, according to eq 9, in the same ratio (R, (exponential) = 0.087, R, (hard sphere) = 0.207). This illustrates how strong is the effect of hard sphere interactions on the distortion of the concentration profile and on the deviation from the ideal retention expression. When the solute is equally distributed between the dispersed phase and the continuous phase (P = l), its concentration distribution within the channel cross section is uniform and its elution time is equal to the channel void time, to = L/(u), where L is the channel length and ( u ) the mean carrier velocity. (It is 0 0.00 0.02 0.04 0.06 0.08 interesting to note that it is so because the solute samples equally all the flow streamlines whatever the flow profile and its eventual Qa,inc deviation from the parabolic shape due to concentration effects Figure 3. Solute relative retention time, tr/fo= l/& vs additive in the carrier medium). When the solute partition coefficient volume fraction in the incoming carrier, @a,,nc for a given solute with increases, its retention time, tr, increases up to a maximum value, P = 1000. From upper to lower curves, respectively: I., = 0.01,0.015, to/%, reached when the solute is totally distributed on or within 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1. the additive particles (P -). The relative time window, (tr(e-) - t,(P=l))/t,, within which solutes with distribution coefficients given field strength, does not increase steadily with the additive varying between 1 and infinity are eluting is then equal to (1/RJ volume fraction in the incoming carrier as was found in previous - 1 and is controlled by the incoming volume fraction, @,,inc, and but instead goes to a maximum for a rather low @,.inc the field-interaction parameter, 1,. This parameter is easily value. This value can be shown from eq 10 to be the solution of determined once the additive relevant physicochemical parameters (particle size and density in sedimentation FFF) and the field strength are known. Figure 2 shows the variations of l/Ra with the additive incoming volume fraction, for various 1, values. Obviously, for a given @,,inc value, l/Ra, which is the mean time which, using eq 9, gives spent by an additive particle in the channel, increases with 1 - R, decreasing %a, i.e., with increasing field strength. However, while (19) @ a , i n c o ~ t - (p_lj( 1 (d In Ra/d In (@a))- R,) in the previously described exponential model Ra was independent This equation cannot be used to calculate +a,inc,,?, the @a,inc of the additive volume fraction, Figure 2 indicates that, because providing the maximum solute retention time, as R, and (@a) of the steric interactions between additive particles, l/Ra decreases depend in a complex way of +,,inc. Nevertheless, the Ra function relatively strongly with increasing 4,. This effect has a direct in parentheses is likely to vary only slightly with +,,inc and ia so consequence on the solute retention. that one expects that the optimum +%inc value does not signscantly Influence of Additive Volume Fraction on Solute Retenchange when the field strength for a given solute (given P value) tion. The variations of the solute relative retention time, tJt, = is modified. This is confirmed by Figure 3 in which, for P = 1000, l/Rs,lying in the 1to l/Ra range, with +,,inc are shown in Figures 3 and 4, at constant P = 1000 for various 1,values and at constant the incoming additive volume fraction providing the maximum solute retention is seen to be approximately equal to 0.3%whatever 1, = 0.015 for various P values, respectively. The major feature of these figures is that the retention time of a given solute, at a La Then, eq 19 indicates that, for a given la, this optimum volume
'I
-
1182 Analytical Chemistry, Vol. 67, No. 7, April 7, 7995
“
0 ) 0
I
o.noo
0.0 i n
0.005
I
I
0.025
0.05
0.075
0.1
0.015
‘il,inc
Figure 4. Solute relative retention time, fr/b = l/&, vs additive volume fraction in the incoming carrier, &,nc, for a given field strength with La = 0.015. From lower to upper curves, respectively: P = 30, 100, 300, 1000, 3000, and 10000.
fraction is approximately proportional to 1/(P - 1) or to 1/P as P is generally large. This is in agreement with Figure 4, which shows that #,,incopt decreases approximately as 1/P as P increases. Ultimately, when P is going to infinity, this optimum volume fraction is approaching zero, Le., that l/Rs is then continuously decreasing with increasing &inc, as seen on Figure 2, since then the solute moves at the velocity of the additive. The maximum relative retention time, (l/R&=, is then given as
(;
- =
‘4 7.5
2.5
1 - R, (d In R,/d In (4,)) - Ra)/
As the fraction in the right-hand-side term of this equation can be considered as roughly constant, one expects (1/Rs),,=- 1 to be approximately proportional to l/Ra. This factor is then seen from Figure 2 to increase with decreasing A, at constant P, since then 4,bc is nearly constant, and to increase with decreasing da;inc at constant La, since then varies roughly as UP. These expectations are in agreement with the optimum values of 1/R, observed in Figures 3 and 4. The apparently surprising finding that l/Rs goes to a maximum for a given &inc can be explained on physical grounds as follows. When @,,inc is equal to 0, l/Rs is of course equal to 1 since there is no additive in the channel. When &inc increases but is still small, hard sphere interactions between additive particles in the channel are still negligible, R, remains nearly constant and l/Rs increases with increasing 4,,incsince there are more and more additives in the low-velocity streamlines near the accumulation wall. When hard sphere interactions begin to be significant, the additive volume fraction near the accumulation wall does not increase as fast as the volume fi-action in the incoming carrier, but the additive volume fraction in the high-velocity streamlines near the channel center still increase proportionally to qja,inc. At some point in the process of increasing +,,inc, the volume fraction near the accumulation wall becomes nearly constant at a value approaching the packing volume fraction (around 0.6), so the capacity of the additive particles to trap the solute molecules in this low-velocity region of the channel thickness ceases to increase
\
-
01 0
0.05
0.1
Figure 6. Solute relative retention time, fr/b = l/& vs additive basic FFF parameter, ,la, for a given additive volume fraction in the incoming carrier, Qa,inc = 0.001. From lower to upper curves, respectively: P = 30, 100, 300, 1000, 3000, and 10000.
while the additive volume fraction in the region close to the channel center and behind continues to build up. Then, the solute retention time starts to decrease with increasing &,inc. Ultimately, for relatively large the additive volume fraction in the channel becomes nearly uniformly distributed and the additive loses its capacity to retain solute molecules. The solute relative retention time is then approaching unity. One should note that it may appear surprising that the retention time of a solute that is entirely distributed on or within the additive particles (i.e., a solute with an infinite P value) is steadily decreasing with increasing #,,inc. Obviously, when there is no additive, the retention factor of this solute is equal to 1, not to l/Ra as apparently implied by the above discussion. This paradox can be understood by noting that an infinite P value cannot be approached when &inc is vanishingly small and that the l/Rs curve for an infinite P value cannot be extrapolated to 4a,inc
= 0.
Influence of Field Strength on Solute Retention. If one increases the field strength, which amounts to a decrease in A,, and maintains constant the additive incoming volume fraction, the additive becomes more strongly compressed near the accumulation wall and thus more retained. Then, the retention time of a given solute increases. This can be seen in both Figures 5 and 6 Analytical Chemisfty, Vol. 67, No. 7, April 1, 1995
1183
‘
but for different incoming additive volume fractions, are not constant but decrease with increasing @,,in,. Since both the abscissa of the inflection point and the upper limit of the ordinate decrease with increasing @,,in,, the curves 1/R, vs log(P - 1) for various @,,inc appear to cross each other. The relative retention times of various solutes vary in the range of 1 (for small P ) to l/Ra - 1 (for large P ) . If one normalizes the 1/R, - 1 values, one gets from eq 10
A
2t
2
i
3
4
5
6
log(P-1)
Figure 7. Solute relative retention time, &/to= l/&, vs log(P - l), where P i s the solute distribution coefficient, for a given field strength with La = 0.015.From upper to lower curves, respectively, at large P values ( P lo5): @a,lnc = 0.0000873,@a,lnc = 0.000568,@a,” = 0.001 34,@a,lnc = 0.00322,and @a,lnc = 0.0141.These @a,lnc values correspond to (@a) = 0.001,(@a) = 0.005,(@a) = 0.01,(@a) = 0.02, and (4a) = 0.05,respectively.
’
I 1
2
3
4
5
6
log(P-I)
Figure 8. Solute relative retention time, fr/fo = l/&, vs log(P - l), where P is the solute distribution coefficient, for a given additive volume fraction in the incoming carrier, @a,inc = 0.001,From upper to lower curves, respectively: 1, = 0.01,0.015,0.02,and 0.03.
in which the solute relative retention time is plotted vs i, for various incoming volume fractions of the additive at constant P = 1000 and for various distribution coefficients at constant @,,inc = 0.001, respectively. The upper curve of Figure 5 is found to correspond to a value of @,.in, close to the optimum of the curves in Figure 3, for the same P value. In a relatively large i, range, the solute appears to have the same retention time whether the incoming volume fraction of the additive is 0.1% or 1%.This is related to the presence of a maximum in Figure 3, which implies that the same retention time is obtained for two different @,,in, values, one smaller and the other larger than @a,inCopt. The fact that these two values remain approximately identical in a relatively large Iba range is likely to be related to the fact that @,,incop, is approximately independent of I.,. Figure 6 shows a monotonous change of the &/tovs 1, curves when P is modified. Influence of Solute Distribution Coefficient on Retention. The relative solute retention time is plotted as a function of log@ - 1) in Figure 7 for various values of the incoming additive volume fraction. As in the case of the previous exponential model, the curves appear to have a sigmoidal shape. It is easy to show, from eq 10, that the curves 1/R, vs log(P - 1) are symmetrical around the intlection point for which (P- 1) = 1/@,,inc and 1/R, = (l/Ra + 1)/2. These characteristics are true whether there are interactions between additive particles or not. However, in the present case, the upper limiting values of l/R,, at a given field strength 1184
Analytical Chemisfry, Vol. 67, No. 7, April 7, 7995
Equation 21 shows that, whatever the field strength and hence La, this normalized retention time function vs log(P - 1) curve is unique when @, is fixed. Therefore, the range of P values for which a significant variation of the retention time occurs is strictly controlled by the volume fraction of the additive in the incoming carrier. But the amplitude of these variations depends on R,, Le., on both @a,lnc and &. This is seen in Figure 8, in which 1/R, is plotted vs log(P - 1) for various La values. For a fixed value of @, the range of relative retention time values is seen to increase with decreasing i, Le., , increasing field strength. As discussed above, this range is significantly reduced when hard sphere interactions are present. For instance, for the value selected by Berthod and Armstrong (A, = 0.015), this range is reduced from 10.45 without interactions to 7.97 with steric interactions for = 0.1%. For a larger volume fraction (@,, = l%),this reduction is larger (from 10.45 to 3.83). Examination of Published Experimental Results. The SCE-FFF experiments performed by Berthod et aL2 can be examined at the light of the present model. Using a microemulsion as the additive system, they obtained a slight separation between a polar solute and an apolar solute for which the distribution coefficients were assumed to be zero and infinity, respectively. One notes that a solute which does not distribute itself within or on the additive particles cannot be used as a marker for the channel void volume because it is partly excluded from the low-velocity streamlines occupied by the particles in the vicinity of the accumulation wall4 Using eq 10, one can determine the retention factors R,, (= RJ and Rs,pof the apolar and polar solutes from the knowledge of the experimental values of @,, and of the retention volumes, TIR,~ and T/KP, of these two solutes:
and %,p
= Ra(VR.a/VR,d
(23)
From the data reported in Tables 1 and 2 of ref 2, one gets R, = 0.88 and 0.85 for the microemulsion systems 1and 2, respectively. In both cases, Rs,pis close to 1.05, which means that the polar solute moves faster than the void volume marker. The mean additive volume fraction in the channel, (4,) given by eq 9, was then very large (0.29 and 0.23 for systems 1 and 2, respectively) and exceeded by far the largest volume fraction for which we performed retention computations in our hard sphere interaction model.6 Although a hard sphere model cannot generally be directly applied to real suspensions in which electrostatic interactions add themselves to steric interactions to influence the suspension structure, it has been found that it can correctly
describe the behavior of emulsions.1° Therefore, extrapolating our R vs (4) curves to the above values, one estimates that the corresponding 1,values for the systems 1 and 2 are about 0.04 and 0.05, respectively. Even if these values are only rough approximations, they are 6-8 times smaller than the values that would have been obtained by applying the classical retention equation to the above R, data. This implies that the emulsion droplets are 6-8 times bigger and their size 80-100% larger than would have been estimated from the classical model. This huge difference is due to the very large additive volume fraction used by Berthod et al. in their experiments. From Figure 2, it can be estimated that, under similar field strength conditions, the retention time of the apolar solute would have been 3 times larger (instead of only 200/0 larger) than that of the polar one if the additive volume fraction in the incoming carrier was only about 0.5%, assuming that the droplet size does not change when the emulsion is diluted. CONCLUSION The initial retention model of the SCE-FFF mode of separation derived by Berthod and Armstrong implicitly assumed large P and low I., values.' It was later extended to any value of the distribution coefficient and of the basic FFF parameter of the additive, A,.* In these two studies, it was assumed that the concentration of the additive was exponentially decreasing with increasing distances from the accumulation wall and that the velocity profile of the carrier was parabolic. This implicitly amounted to a limitation of the study to low additive volume fractions. The retention model of FFF at finite concentrations recently developed on the basis of hard sphere interactions between particles6 allows one to extend the previous models to rather large additive volume fractions. The striking effect arising from these interactions is that there is an optimum value of the additive volume fraction in the incoming carrier which leads to a maximum retention time of a given solute for a given field strength. One should not forget that the present model, like the previous ones, relies on some hypotheses that have already been discussed.* It is assumed that the solute is not influenced by the applied field and that its partition within or on the additive particles does not intluence the interaction of these particles with the field. These two hypotheses are likely to be correct for solutes with relatively low molecular weight. In addition, the present model, like the others, assumes that the solute partition and migration along the channel occurs under quasi-equilibrium conditions. To fuliill this condition may require that the solute injection point is located downstream of the carrier introduction point if the additive has a finite relaxation time. (10) Tadros, T. F. Plenary Lecture presented at the First World Congress on Emulsion, Paris, October 19-22, 1993.
In spite of the fact that this condition may be difficult to implement with existing instruments, the results of the present model as well as the examination of the experimental fractograms obtained by Berthod et al.2 allows one to derive guidelines for optimizing SCE-FFF separations. Three independent parameters appear to control solute retention: the additive volume fraction in the incoming carrier, #a,hc;the field-interaction parameter of the additive, I.,, which depends on both solute characteristics (size and density in sedimentation FFF) and field strength; and solute distribution coefficient, P. For a given additive system, for instance an emulsion, and a given sample, the additive volume fraction in the incoming carrier must be first adjusted to be equal to 1/(Pl), where P is an appropriate average value of the distribution coefficients of the sample components. This will ensure that the P-based selectivity is near the optimum. Then, the retention time window and the maximum relative retention time, l/R,, can be adjusted by the proper selection of la, i.e., of the field strength. Obviously, the highest possible field strength will give the longest retention time, but one can get an indication of the practically achievable retention time by looking at Figure 2. Such an approach for selecting the experimental conditions obviously requires an approximate knowledge of the average solute distribution coefficient and of the size and density of the additive particles (or emulsion droplets) if a centrifugal field is used. If this is not the case, then optimization can be done by examinating the outcome of an initial experimental run in light of the model and figures presented above. It has already been stressed that the potential of the SCE-FFF mode of retention does not lie in the possibility of separating small molecular solutes, which are more efficiently separated by chromatographic methods, but instead in the determination of the distribution coefficients between a dispersed additive phase and the bulk solution as well as, using solutes with very large Pvalues, in the determination of the physicochemical characteristics of the additive particles or droplets via ;la. In order to get relatively large tr/to ratios for accurate determinations, one will preferably choose low values of 4,,incwhich will allow the analysis of solutes with rather large P coefficients. However, the proper interpretation of the retention data to get P or 1,estimates imperatively requires that the influence of concentration effects on retention are correctly taken into account. It is believed that the present model, based on hard sphere interactions, can be used for such a purpose either directly in hard-spheretype suspensions, like emulsions, or by means of an effective volume approach when electrostatic interactions between particles are significant. Received for review October 3, 1994. Accepted January 24, 1995.@ AC9409703 @
Abstract published in Adounce ACS Abstracts, February 15, 1995.
Analytical Chemistry, Vol. 67, No. 7, April 1, 1995
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