Size Selectivity in Field-Flow Fractionation: Lift Mode of Retention with

A simple theoretical model for the size selectivity, Sd, in the lift mode of retention in field-flow fractionation (FFF) is developed on the basis of ...
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Size Selectivity in Field-Flow Fractionation: Lift Mode of Retention with Near-Wall Lift Force Michel Martin† and Ronald Beckett*,‡ †

Ecole Supérieure de Physique et de Chimie Industrielles, Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH UMR 7636 CNRS - ESPCI-ParisTech - Université Pierre et Marie Curie 6 - Université Paris Diderot), 10 rue Vauquelin, 75231 Paris Cedex 05, France ‡ Water Studies Centre, School of Chemistry, Monash University, Clayton, Victoria 3800, Australia ABSTRACT: A simple theoretical model for the size selectivity, Sd, in the lift mode of retention in field-flow fractionation (FFF) is developed on the basis of the nearwall lift force expression. Sd is made up of two contributions: the flow contribution, Sd,f, arising from the variation of the flow velocity at center of particle due to a change in particle position with particle size, and a slip contribution, Sd,s, arising from the concomitant change in the extent of retardation due to the presence of a nearby channel wall. The slip contribution is minor, but not negligible, and amounts to 10−20% of the overall size selectivity. It contributes to reduce Sd in sedimentation FFF but to enhance it in flow FFF. Sd would steadily increase with particle size if the flow profile was linear (Couette flow). Because of the curvature of the flow profile encountered in the classical Poiseuille flow, Sd exhibits a maximum at some specific particle size. The model predicts a significant difference in Sd between sedimentation FFF and flow FFF, arising from the different functional dependences of the field force with particle size between these two methods. The predictions are in good agreement with the various Sd values reported in the literature in both sedimentation and flow FFF. On the basis of the model, guidelines are given for adjusting the operating parameters (carrier flow rate and field strength) to optimize the size selectivity. Finally, it is found that Sd generally decreases with decreasing channel thickness.

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INTRODUCTION This paper explores the factors influencing the selectivity of separations of micrometer size particles using the lift mode of field-flow fractionation (FFF). FFF was introduced about 45 years ago as an analytical scale separation method suitable for macromolecules and fine particles.1 It has the advantage of high selectivity and accuracy but is only capable of separating small amounts of sample ( 300 μm, from bottom to top: d = 5, 10, 20 μm.

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A similar analysis for the corresponding optimum size selectivity, Sd,opt, gives Sd,opt = 1.82 − 10.8α + 47α 2

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In both cases the correlation coefficient is better than 0.99. However, it should be noted that the ultimate goal in the optimization of the separation conditions usually consists of reaching the largest possible resolution (or fractionating power) in an acceptable overall run time. Selectivity is not the only component of the resolution. The dependence of band broadening on operating parameters may lead to a shift of the values of the optimal conditions. This dependence has been recently investigated for lift mode operation in gravitational FFF.47 Effect of Channel Thickness on Size Selectivity. Decreasing the FFF channel thickness is often advocated as an approach for improving separation performances. This approach was shown to provide improved efficiency or reduced analysis time in the Brownian mode of thermal FFF.63 The influence of the channel thickness, w, on size selectivity in the lift mode is not as easy to predict as that of the other operating parameters entering the expressions of φeq and κeq in eqs 24

for SdFFF and FlFFF, respectively, the field strength being kept constant in all cases. The thick and light curves correspond to constant ⟨vf⟩ and constant Q cases, respectively. The calculations are performed for three particle diameters, 5, 10, and 20 μm such that φeq = 0.01 in SdFFF and κeq = 25 in FlFFF for a 250 μm thick channel (this corresponds to intermediate values for curves shown in Figures 2−6). Whether velocity or flow rate is kept constant, Sd is seen to steadily increase with increasing channel thickness in SdFFF. As w increases, for a given particle size, α decreases and φeq decreases (as 1/w2 at constant ⟨vf⟩, and as 1/w3 at constant Q, according to eq 24). As seen from Figure 3a, the former effect tends to decrease Sd (as long as α remains lower than its optimum value), whereas the latter tends to increase it. Figure 9a indicates that the latter effect dominates, especially when the 6549

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flow rate is kept constant. Hence, reducing the channel thickness would be detrimental to size selectivity in SdFFF, as was experimentally observed.45 The situation is more contrasted in FlFFF. At constant flow average velocity, according to eq 26, κeq is unchanged, and an increase in w corresponds to a decrease in α. Figure 9b shows that Sd goes then to a maximum as w increases. The maxima of Sd for the three thick curves of Figure 9b are the same and are equal to the maximum Sd value of the intermediate thick curve of Figure 4. Because that maximum corresponds to a given α, for a fixed κeq, the larger is the particle size, the larger is the optimum channel thickness in Figure 9b. However, after reaching this maximum, Sd decreases rather slowly. Therefore, from the sole point of view of size selectivity improvement, the interest of reducing the channel thickness at constant average flow velocity is moderate. In implementing this approach, care should be paid to limit this reduction to the optimal value of w. That value is not easy to predict in practice as it depends of the operating parameters which control κeq, as well as of the midparticle size of the sample at hand. Decreasing w at constant flow rate induces not only an increase in α but also an increase in κeq, which is proportional to 1/w, which is associated to an increase in the maximum Sd, as seen in Figure 4. Figure 9b shows that the overall effect is such that, in the practical range of interest, Sd essentially decreases with decreasing w. Again, there is no interest to decrease the channel thickness in this case. It has been recently observed that the size selectivity in SdFFF increases with increasing carrier viscosity.47 This is expected because, according to eq 24, an increase in viscosity leads to an increase in φeq, hence, as seen in Figure 3a, to an increase in Sd, whatever the particle size.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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