Anal. Chem. 1001, 63, 2115-2122
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Hydrodynamic Relaxation in Flow Field-Flow Fractionation Using Both Split and Frit Inlets Min-Kuang Liu, P. Stephen Williams, Marcus N. Myers, and J. Calvin Giddings* Field-Flow Fractionation Research Center, Department of Chemistry, Uniuersity of Utah, Salt Lake City, Utah 84112
Two means are descrlbed for achlevlng hydrodynamlc relaxation and thus avoldlng the stopflow Injection procedure in f W l o w fracthatlon (FFF): spllt flow lnlectlon and frH Inlet InJectbn. The advantages, dlsadvantages, and the theoretical bask of these procedures are dlscussed. Incremental band broadening due to the final relaxatlon step Is examined theoretlcalty and shown to be negllglble when the flow rate of the sample Inlet substream Is small compared to the total channel flow rate. The optknlzatlon of the sample Inlet flow rate Is discussed. Experlmental results for both Injection procedures are reported for flow/sterlc (or hyperlayer) FFF applled to latex standards, conflrmlng the expected trends. However, c h r examlnatlon shows that the observed Incremental band broadenlng associated wHh hydrodynamlc relaxation Is somewhat larger than the value predicted.
INTRODUCTION In field-flow fractionation (FFF), particles and macromolecules are separated by virtue of the fact that different species are driven into different transverse distributions in a thin channel by an external driving force and the different distributions are carried at unequal rates by the nonuniform (usually parabolic) flow in the channel (1-5). Generally, the distributions that are displaced in this manner by flow are equilibrium distributions established as a balance involving diffusion, steric effects, and field-driven transport. However, when the sample is first carried into the channel, the sample particles are distributed over the entire flow cross section, obviously far out of equilibrium. Upon experiencing the field-based forces, particles relax toward equilibrium. Unfortunately, the particle band can broaden substantidy during the relaxation process unless the flow is halted (stopflow procedure) during relaxation (6). Consequently, injection followed by stopflow has become standard practice in FFF. However, the stopage of flow has its own disadvantages including an increased run time, baseline instabilities, and the risk of particle adhesion to the channel wall. Hydrodynamic relaxation is an alternative to field-driven relaxation. In hydrodynamic relaxation, sample particles are carried to the vicinity of their equilibrium positions by flow rather than by field-driven transport. Hydrodynamic relaxation, where feasible, is rapid and does not require a stopflow procedure. Two special channel inlet configurations have been proposed recently for achieving a major part of sample relaxation hydrodynamically. One is a split-flow inlet in which a thin flow splitter divides the inlet region into two slitlike flow spaces (7,8). The sample suspension is introduced into one of the flow spaces through a sample inlet while a substream of carrier liquid enters a second inlet (see Figure la). When the flow rate of the carrier substream is significantly larger than that of the sample substream, the sample substream is compressed into a thin lamina near the accumulation wall of the channel
* Corresponding author. 0003-2700/91/0363-2115$02.50/0
upon the merging of the substreams. Since the accumulation wall is, by definition, the wall near which equilibrium distributions accumulate, the relaxation process is largely completed by the flow process alone. The remaining relaxation to final equilibrium occurs rapidly and generally unobtrusively by field-driven transport without an interruption of flow (referred to as stopless flow injection). The second or frit inlet configuration utilizes a small piece of permeable wall material (a frit element) imbedded in the depletion wall (opposite the accumulation wall) of the channel near the inlet (9). The system is designed to allow an independently controlled flow stream to permeate into the channel through this frit element. The sample substream, introduced into the channel inlet tip upstream of the frit element, is again compressed toward the accumulation wall upon merging with the second (frit inlet) substream. This method of achieving hydrodynamic relaxation is illustrated in Figure lb. The advantages and disadvantages of this inlet compared to the split inlet have been summarized in a preceding article (9). In both parts a and b of Figure 1an inlet splitting plane is shown. This is the streamplane dividing the fluid elements entering from the two substreams. Since the sample enters by means of the sample inlet substream, the sample particles are almost entirely confined to the region below the inlet splitting plane. As this plane is driven toward the accumulation wall by the relatively high flow of carrier liquid entering through the carrier inlet (in case a) or the frit inlet (case b), the sample is driven to the vicinity of the accumulation wall and hydrodynamic relaxation is realized. Despite the promise of hydrodynamic relaxation, only one experimental study has been done to test its effectiveness (8). It was shown in this study that a split inlet system could be used in both the normal and steric modes of sedimentation FFF to resolve different-sized latex particles. However, the effectiveness of hydrodynamic relaxation has not been tested for any other subtechnique of FFF, nor has it been previously examined by using a frit inlet system (except in both instances for a single example shown in ref 9). The main purpose of this study is to broaden this limited experience base by means of an experimental investigation of hydrodynamic relaxation in flow FFF. Both split inlet and frit inlet configurations are used in this study. Important elements of background theory are also developed.
THEORY When FFF is carried out with hydrodynamic relaxation, the separation process in the main body of the channel is expected to proceed much as it does in ordinary stopflow injection FFF. Retention and band broadening should be governed by the same principles and mathematical expressions in the two cases. The corresponding theory has been covered at some length (2-6). Where hydrodynamic relaxation FFF and stopflow injection FFF are expected to differ is in the microscopic flow processes occurring at the inlet end of the channel. An important factor is that injected particles are kept in motion during hydrodynamic relaxation. The hydrodynamic displacement and the lift forces generated by the ongoing fluid flow are expected 0 1991 American Chemical Society
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a
ANALYTICAL CHEMISTRY, VOL. 63,NO. 19, OCTOBER 1, 1991 carrier inlet
idealized starting band
U
---inlet splitting plane
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I I I\
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relaxation traiectories
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c---initial
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hydrodynamically relaxed sample pulse
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particles hydrodynamic undergoing relaxation final relaxation
b
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Flgwe 2. Residual fieM driven relaxation of a hydrodynamically relaxed band confined beneath the splitting plane at elevation x,.
initial sample pulse
hydrodynamically relaxed sample pulse
accumulation wall
Flgure 1. Schematic illustration of hydrodynamic relaxation achieved in (a) a split inlet system and (b) a permeable-wall (frit) inlet system.
to strongly inhibit particle adsorption a t the channel wall. The
widespread occurrence of such adsorption is supported by the fact that most FFF channels have to be cleaned occasionally, with most of the residue found at the accumulation wall near the inlet where stopflow relaxation takes place. The continual (or stopless) fluid motion used in hydrodynamic relaxation not only will discourage the adhesion of any particle touching the wall but also will keep particles having a size beyond a certain threshold diameter (almost certainly those above 1pm in diameter and perhaps significantly smaller particles as well) from ever making contact with the wall. Such contact-free FFF systems should be valuable for highly interactive species, especially those of biological origin. A potential disadvantage of hydrodynamic relaxation is that it is associated with incremental band-broadening effects. First, the ubiquitous broadening of component bands caused by finite sample size and dispersion processes in the inlet tubing and the inlet tip of the channel is amplified by the reduced flow rate of the sample inlet substream (9). Second, even after hydrodynamic relaxation, a small residue of fielddriven relaxation must normally occur before equilibrium is reached. Some band broadening will be generated in this final relaxation step. The overall band broadening of a monodisperse particle pulse in an FFF channel using hydrodynamic relaxation and the role of the above increments can be summarized by the plate height equation where H,is the contribution due to nonequilibrium and other effects encountered in normal migration through the channel, Hendis the contribution of end effects, and H , is the contribution of the residual relaxation effects. The term Hendis always present in FFF but is amplified, as noted above, by hydrodynamic relaxation. I t can be expressed by (9)
where L is the channel length, t, the component retention time, av2is the volume-based variance of the incoming component peak, and V, is the flow rate of the sample inlet substream. Amplification results from the inverse dependence on V:, which, for a given overall flow rate, decreases with increasing hydrodynamic relaxation. It is expected that both of the incremental band-broadening terms (the last two terms in eq 1)can be reduced and perhaps made inconsequential through proper inlet design and flow
control. Some essential features of inlet design related to Hed have already been discussed (9). In this paper we will examine the theoretical basis of the second effect, the band broadening induced by the final relaxation step, giving rise to H,. The origin of the band broadening associated with final relaxation is illustrated in Figure 2. This figure shows relaxation trajectories (the path followed by nondiffusing particles) for sample material starting down the FFF channel as a narrow band. In the absence of hydrodynamic relaxation, the band will extend across the thickness of the channel. Without stopflow, as the different segments of the band descend toward the accumulation wall, they will be simultaneously carried downstream. Those segments (or particles) starting a t the highest elevations will obviously be carried furthest downstream, while those starting near the accumulation wall will be displaced very little. The uppermost particles in the starting band will be carried downstream a distance ho given by (6, IO) ho = w ( u > / U (3) where w is channel thickness, ( u ) is the mean flow velocity down the channel, and U is the transverse velocity induced by the applied field. For flow FFF, U is simply the crossflow velocity. If now all particles are subject to hydrodynamic relaxation, in which case they enter beneath a splitting plane at elevation x,, then in effect the starting band becomes the truncated shaded band shown in Figure 2. The uppermost particles in this case, because of their low starting elevation ( x J , are carried a much shorter distance downstream than found in the absence of hydrodynamic relaxation. Thus, when relaxation is completed, the hydrodynamically relaxed sample occupies a band length in the channel of h, compared to the much greater length for the nonhydrodynamically relaxed band. Clearly, the band-broadening contribution of this final relaxation will be much smaller in the first case than in the second. Below we develop the theory necessary to evaluate this bandbroadening effect. For this treatment we make the following assumptions. First, we assume that negligible field driven relaxation takes place within the region of hydrodynamic relaxation so that the initial configuration of the sample band is, as shown in Figure 2, a homogeneous layer of thickness x, a t the head of the channel, adjacent to the accumulation wall. Any fielddriven relaxation within this region is beneficial and our treatment therefore results in an upper bound to expected band broadening. Second, we assume that the velocity profile across the thickness of the channel immediately following hydrodynamic relaxation, with either inlet type, is parabolic. This assumption is quite reasonable, since parabolic flow will be established within a distance equal to a few channel thicknesses downstream from any flow disturbance such as that caused by merging substreams. We note also that the at-
ANALYTICAL CHEMISTRY, VOL. 63, NO. 19, OCTOBER 1, 1991
tainment of the final equilibrium position x , (designated as in ref 9) by the splitting plane is contingent on the existence of a parabolic velocity profile. Third, we assume that after field-driven relaxation is accomplished, the mean particle elevation above the accumulation wall, and thus the velocity of displacement by flow, is negligible compared to the elevation x , of the splitting plane and its corresponding velocity. When this assumption does not hold, the band broadening will again be less than that calculated here. In order to determine the contribution to plate height H, due to field-driven relaxation, it is necessary first to establish the band concentration profile along the flow (or z ) axis resulting from the relaxation of an infinitely narrow (in the z direction) initial sample band of height x , lying next to the accumulation wall. Hovingh et al. (6) showed that a particle or molecule starting at position z = 0 and at elevation x above the accumulation wall would follow a trajectory such that it would reach the accumulation wall at a distance z along the length of the channel, where z is given by z = h0(3(X/w)~- 2(x/43) (4) xd
where ho is defined by eq 3. The uppermost particles or molecules in a hydrodynamically relaxed band will therefore migrate during final field-driven relaxation a distance h, along z given by h, = h0(382- 2 ~ 9 ~ ) (5) where 0 is defined as x , / w . By combining this equation with eq 5 of ref 9, we find that the ratio h,/hois simply V,/ V , the ratio of the sample inlet flow rate and the total channel flow rate. Hovingh et al. (6) also showed that following relaxation, the normalized concentration profile along z is given by the following function of x / w :
For an initial band height of x , we know that the width (extension along the flow axis) of the band following relaxation will be h, as given by eq 5. The area of the concentration profile as given by eq 6 up to position h, is given by
where u,2 is the variance in the distance travelled in the z direction by the band of particles or molecules during relaxation. Once again, changing the variable of integration to x / w results in
which, on evaluating, reduces to
The contribution to plate height due to residual relaxation following the hydrodynamic relaxation process is then given by
Where there is no hydrodynamic relaxation and 0 is equal to unity, H, is seen to be equal to (17/140)hO2/L,which is consistent with the findings of Hovingh et al. (6). For values of 0 less than unity, H, decreases rapidly. When 8
