Anal. Chem. 1997, 69, 3230-3238
Mechanistic Study of Electrical Field Flow Fractionation. 2. Effect of Sample Conductivity on Retention Saurabh A. Palkar and Mark R. Schure*
Theoretical Separation Science Laboratory, Rohm and Haas Company, 727 Norristown Road, Spring House, Pennsylvania 19477
Experimental work presented in this paper demonstrates that retention in electrical field-flow fractionation shows a very strong dependence on the amount of injected sample, even for very low concentrations. This concentration dependence on retention is shown to result from conductivity differences between the colloidal particles and the carrier fluid. As a result of this concentration dependence, the electric field is a function of position inside the channel. This fact is important when electrical FFF is used for determining electrophoretic mobilities and particle sizes because the analyst must be careful in interpreting the physical origin of retention changes. Furthermore, it is shown through experiment that residual salt in the injected sample buffer can affect retention. A transport model is developed that characterizes the relaxed zone profile; computation of these developed equations shows the extent of deviation from the standard transverse exponential concentration profile. The separation and characterization of colloidal material has been carried out by field-flow fractionation (FFF) methods for a number of years.1 As the methodology and applications of FFF expand in maturity and as new FFF methods come into existence, careful evaluation of the mechanism of separation and investigation of possible secondary effects have typically taken place. It appears that each subtechnique of FFF has unique operational aspects, as is the case with other polymer separation techniques, such as gel permeation chromatography. However, as these FFF methodologies mature and the operational aspects are elucidated, the suitability of these methods for solving problems in colloid and polymer science seems to increase. Therefore, it is worthwhile to characterize these techniques and understand their unique properties so that these can be controlled and modified to yield the highest separation quality, speed, and accuracy of physical parameter estimation when possible. The “new” form of electrical FFF2,3 where the electrode is the channel wall is no exception to this pattern. In part 1 of this series of papers,4 we discuss how the electric field inside the channel is dependent on the electrode characteristics, carrier ionic strength, and flow rate. The channel is represented as an electrical circuit, and different methods are used to characterize its different elements. The studies reported in part 1 focus on matters that (1) Caldwell, K. D. Anal. Chem. 1988, 60, 959A-971A. (2) Caldwell, K. D.; Gao, Y. S. Anal. Chem. 1993, 65, 1764-1772. (3) Caldwell, K. D.; Schimpf, M. Am. Lab. 1995, 27, 64-68. (4) Palkar, S. A.; Schure, M. R. Anal. Chem. 1997, 69, 3223-3229 (preceding paper in this issue).
3230 Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
are internal to the channel and do not include how the field couples with the colloidal particles. Lightfoot and co-workers5 studied hollow fiber electropolarization chromatography of proteins and noticed a decrease in retention with an increase in protein loading. This effect was attributed to the concentration dependence of viscosity and diffusivity. No detailed model was developed that could explain the experimental observations. In this paper, we report on mechanistic studies that describe the migration of injected colloid particles in the electric field and how the concentration of colloid particles can influence the electric field through a mechanism based on particle conductivity. We demonstrate these effects through particle retention studies and through a simple theoretical model of particle relaxation that includes the particle conductivity and its effect on the effective electric field. We also study the effects of zone relaxation with and without stopped-flow and demonstrate significant effects from the presence of residual salt in the injected zones as viewed through elution fractograms. EXPERIMENTAL METHODS Colloid. The polystyrene latex particles used in this study are obtained from Interfacial Dynamics Corp. (Portland, OR) and are reported to be surfactant free and monodisperse. Particles of various diameters with sulfate, carboxylate, and amidine surface functionalities are used for conductivity measurements. For the fractograms shown in this paper, only the particles with sulfate surface functionality are used. These particles have a mean diameter reported by the manufacturer of 140 nm ( 2.3%, and the reported surface charge density (determined by titration) is 0.79 µC cm-2. The effect of sample concentration is studied by injecting a constant volume of particles and fluid (1 µL) and varying the percentage of solids within this volume. Note that all percentages are reported as weight percentages. Deionized (DI) water is used as described in the first paper of this series4 with no other pH buffers present in solution. Conductivity. The conductivity of latex suspensions is measured using a Model 32 conductance meter (Yellow Springs Instrument Co., Yellow Springs, OH). The frequency is 1000 Hz, and the electrode material is platinized platinum-iridium. Dialysis Procedures. The particle standards are dialyzed using Slide-A-Lyzer dialysis cassettes (Pierce Chemical Co., Rockford, IL). These have a 10K molecular weight cutoff (5) Chiang, A. S.; Kmiotek, E. H.; Langan, S. M.; Noble, P. T.; Reis, J. F. G.; Lightfoot, E. N. Sep. Sci. Technol. 1979, 14, 453-474. S0003-2700(97)00014-0 CCC: $14.00
© 1997 American Chemical Society
membrane. The samples are allowed to equilibrate with DI water for ≈24 h. A description of the fractionator, detector, pump, and electronic components used here is reported in the first paper of this series.4 THEORETICAL TRANSPORT MODEL FOR STOPPED-FLOW RELAXATION The transport paths of colloidal particles inside the electrical FFF channel depend on the strengths of the electric field (transverse transport) and the pressure-driven flow (axial transport). Since we are interested in determining the effect of sample concentration on retention, we follow a continuum approach here. The concentration of particles inside the channel is governed by the convection-diffusion equation6 Figure 1. Section of the channel showing variation of the field with position.
∂C(x, z, t) ∂C(x, z, t) ∂ + uz + (uxC(x, z, t)) ) ∂t ∂z ∂x
(
D
∂2C(x, z, t) ∂x2
)
∂2C(x, z, t) +
∂z2
(1)
where C(x, z, t) is the particle concentration, uz is the velocity due to the pressure-driven flow, ux is the velocity due to the field, and D is the particle diffusion coefficient. The x and z coordinates are the transverse and axial coordinates, respectively, and are shown in Figure 1 of the preceding paper.4 It is assumed that the solute concentration is so low that the viscosity and axial velocity at any position in the channel is equal to that of the carrier fluid. It also is assumed that the parallel plate system has a large aspect ratio (breadth to width ratio) so that the breadth, b, and y coordinate can be neglected in discussing retention theory. The velocity profile due to the flow is given as
uz ) 6〈uz〉
[
]
x2 x - 2 w w
(2)
where w is the channel thickness and 〈uz〉 is the average flow velocity. The transverse velocity of particles due to the field, ux, depends on the strength of the field at any point and can be written as
ux ) µE
(3)
where E is the local electric field and µ is the electrophoretic mobility of the particles.7 The electrophoretic mobility depends on the particle size, shape, and surface charge density and the ionic strength of the solution. For the model developed in this paper, we assume that µ is independent of the particle concentration. When a vanishingly small concentration of particles is present between the channel walls, the electric field is constant at all locations, and the velocity due to the field is independent of the position in x. In this case, the retention theory of FFF, based on a linear model is valid. By linear we denote that an increase of injected sample gives a linearly proportional increase in the concentration of the material at all points in the zone. In addition, a linear zone implies that the mean layer thickness of the zone is (6) Krishnamurthy, S.; Subramanian, R. S. Sep. Sci. 1977, 12, 347-356. (7) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1991.
concentration invariant. The assumption of linear zones in FFF has been discussed by Hoyos and Martin8 in the context of zone compression due to transverse particle movement. With a large concentration of particles, the electric field within the particle population changes with the local concentration of these particles, leading to a nonlinear field effect which is the subject of this paper. This model of zone movement has been discussed in the context of isotachophoresis theory,9 where the conductivity of zone and buffer are critical in establishing the zone shape. The big difference here is that the theory to isotachophoresis is essentially a one-dimensional problem whereas the theory to electrical FFF requires two dimensions and if performed rigorously is extremely complex. For simplicity in explanation, we consider a small axial cut of the channel shown in Figure 1a. In this cut we place three compact zones of colloidal particles which are spatially isolated in the transverse (x) coordinate and of finite length in the axial (z) coordinate, as shown in Figure 1b. This figure is conceptual and used for illustration; i.e., we do not actually expect the electrical FFF experiment to form compact focused zones under the conditions described in this paper. The electrical potential difference applied to the channel, ∆V, is equal to the total sum of transverse voltage drops across the particles and the fluid. The local electric field, E(x, z, t), i.e., the gradient of the electric potential,10 is given as
E(x, z, t) ) I(z, t)
∆R(x, z, t) ∆x
(4)
where R(x, z, t) is the resistance of material in the channel as a function of position and I(z, t) is the local current. The total current in the external circuit, IT(t), can be obtained by integrating the local current density, J(z, t), defined as J(z, t) ) I(z, t)/A, where A is the cross-sectional area (A ) b∆z). Assuming that the current density is uniform in breadth
IT(t) ) b
∫
L
0
J(z, t) dz
(5)
where L is the channel length. At any location, z the local current (8) Hoyos, M.; Martin, M. Anal. Chem. 1994, 66, 1718-1730. (9) Everaerts, F. M.; Beckers, J. L.; Verheggen Theo P. E. M. Isotachophoresis: Theory,Instrumentation, and Applications; Elsevier: Amsterdam, 1976. (10) Purcell, E. M. Electricity and Magnetism; McGraw-Hill: New York, 1985.
Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
3231
depends on the average concentration, C h(z, t), which is defined as
C h (z, t) )
1 w
∫
w
0
C(x, z, t) dx
(6)
contributes little to zone broadening here and neglecting this term helps to simplify the problem. In addition to eq 11, the following no flux boundary conditions at the walls must be included
-D where w is the channel width. Equation 4 demonstrates that as the resistance changes, due to the presence of particles, the local electric field will also change. When deionized water is used as the carrier fluid, the conductivity of the particles is higher than the conductivity of the carrier. This is typical for electrical FFF and results in the local electric field, E(x, z, t), being lower in regions of particle zones, as given in Eq 4, and as depicted in Figure 1c. The local resistance difference ∆R(x, z, t) can be expressed as
∂C + uxC ) 0 ∂x
(12)
The long time solution of eq 11 with boundary conditions given by eq 12 gives the required steady-state concentration profile. This can be written implicitly as
C(x) e(a1/a0)C(x) ) e-(U*/D)x C0 e(a1/a0)C0
(13)
U* ) µJ/a0
(14)
where
1 ∆x ∆R(x, z, t) ) σ*(x, z, t) A
(7)
where σ*(x, z, t) is the local specific conductivity with units of reciprocal resistance per unit distance. Since we are working in a dilute concentration region, the specific conductivity at any point inside the channel can be approximated as a linear function such that
and C0 is the concentration at the accumulation wall where x ) 0. When the velocity is independent of position, eq 13 reduces to the standard linear theory of FFF:
(8)
Although stopped-flow relaxation has not apparently been used in previous electrical FFF studies, it is useful to know how fast the injected zone relaxes. In addition, the calculation of the steadystate concentration profiles under the proposed nonlinear conditions can help interpret the behavior of the experiment. Numerical Procedures. The time-dependent equation, eq 11, is evaluated by using an implicit finite difference scheme.11,12 The space derivatives are approximated by a fourth-order biased upwind difference formula.12 In electrical FFF, and most other forms of FFF, we typically encounter transverse Peclet numbers, Pex ) U*w/D, greater than 200. For such high fields the zone is concentrated very close to the wall. To assure numerical stability with such high concentration gradients, it is necessary to use a nonuniform grid in x. In our approach, the grid is split into two regions with different mesh sizes depending on the value of Pex. Typically, 50 mesh points in each region work well, when the domain is divided into two regions x/w ) 0-0.1 and x/w ) 0.11, for Pex > 50. Equation 13 is written in analytical form for the steady-state concentration profile, which is implicit in concentration and needs to be solved numerically. This is done by expressing the problem as a root problem:
σ*(x, z, t) ) a0 + a1C(x, z, t)
where a0 is the specific conductivity of the carrier phase and a1 depends on the surface charge and radius of the particles. We will show experimental data below that suggests eq 8 is valid as an approximation. By combining eqs 4, 7, and 8, the local electric field can now be written as
E(x, z, t) )
J(z, t) a0 + a1C(x, z, t)
(9)
By combining eqs 1 and 9 and excluding the explicit spatial and temporal dependence notation
(
) (
)
∂ ∂C C ∂2C ∂2C ∂C + uz + µJ )D 2 + 2 ∂t ∂z ∂x a0 + a1C ∂x ∂z
(10)
Many FFF techniques make use of stopped-flow relaxation to obtain steady-state concentration profiles prior to the application of flow. During stopped-flow relaxation, the electric field is applied but flow is not (uz ) 0). The above equation then becomes
(
)
2
∂ ∂C C ∂C + µJ )D 2 ∂t ∂x a0 + a1 C ∂x
Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
C(x) e(a1/a0)C(x) - e-(U*/D)x ) 0 C0 e(a1/a0)C0
(15)
(16)
(11)
The diffusion term for the z direction is not considered further in eq 11 since its contribution to zone broadening is known to be minimal with the diffusion coefficients found in the colloidal domain. In addition, the length scale for the z direction is very large compared to that for the x direction. Hence, axial diffusion 3232
C(x) ) C0 e-(U*/D)x
A value of C0 is first chosen, and then C(x) is estimated at each x by utilizing Newton’s method of root finding13 on eq 16. The (11) ) Lapidus, L.; Binder, G. F Numerical Solution of Partial Differential Equations in Science and Engineering John Wiley and Sons: New York, 1982. (12) Schiesser, W. E. The Numerical Method of Lines; Academic Press: San Diego, 1991. (13) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes; Cambridge University Press: New York, 1992.
Figure 3. Conductivity as a function of particle concentration for various IDC particle standards as supplied.
Figure 2. Fractogram showing detector response as function of time for sulfate polystyrene latex: applied potential difference 1.2 V, current 0.2 mA, flow rate 1 mL/min; (a) high sample concentrations (1-0.1 wt %), (b) low sample concentrations (0.1-0.01 wt %).
injected sample concentration, Cin, can then be found by integrating the transverse concentration profile such that
Cin )
1 w
∫
w
0
C(x) dx
(17)
The C0 term is readjusted manually and the procedure repeated until Cin is equal to or near the requested value. All computer programs are written in FORTRAN and executed on RS/6000 Model 591 workstations (IBM, Armonk, NY). RESULTS AND DISCUSSION Concentration Dependence. Electrical FFF has the potential of separating larger sample amounts as compared to capillary electrophoresis (CE) because the channel volume is larger for electrical FFF than CE. Preparative-mode separations based on charge could be very useful for studying a number of colloidal phenomena. For this purpose, and to study analytical utilization, it is important to examine the effect of sample concentration on the retention and zone broadening behavior. Figure 2 shows typical fractograms for high sample concentrations. From these fractograms, it is apparent that the retention time and peak width are strong functions of the concentration of the injected sample. Retention decreases and peak width increases with increasing injection concentration. This effect is
observed even at a very low particle concentration of 0.01% (0.1 µg µL-1). The concentration effect appears to be more dominant here as compared to sedimentation FFF,14 thermal FFF,15 and flow FFF.15 However, running charged samples without salt to screen the particle double layer charge16 suggests that this form of fractionation would be prone to particle-particle repulsion upon zone compression and some form of concentration effect should be expected. This has recently been shown in hollow fiber flow FFF separations of charged polymers to be a dominating effect.17 Nonetheless, data presented below suggest that other effects may predominate over the particle-particle repulsion in explaining the results shown in Figure 2. To understand these concentration effects, we need to elucidate the dependence of the electric field on the system parameters. The body force term in eq 10, µJ(∂/∂x)(C/(a0 + a1C)), is a function of the electrophoretic mobility of the particles, µ, the current density, J, and the local conductivity at any position in the channel. All of these depend on the amount of salt present in the sample, and the local conductivity also depends on the charge and local concentration of the particles. Hence, if the injected sample contains a high concentration of salt and/or particles, the local electric field will be very low. This can lead to a decrease in the retention of the sample. The conductivities of the sample and other particles with different surface functionalities are shown in Figure 3. This conductivity data illustrates that the conductivity as a function of particle concentration in the concentration range pertinent to these studies is mostly linear. The results of Figure 3 also contrast the conductivities of the sulfate particles used here with other surface functionalities. In the vicinity of 0.5 wt %, the 140 nm sulfate latex particles have a conductivity of ≈30 µS cm-1. This is much higher than the value expected due to particles alone.18 After synthesis, Interfacial Dynamics Corp. dialyses these particles, and these are then stored for long periods of time prior (14) Hansen, M. E.; Giddings, J. C.; Beckett, R. J. Colloid. Interface Sci. 1989, 132, 300-312. (15) Caldwell, K. D.; Brimhall, S. L.; Gao, Y., Giddings, J. C. J. Appl. Polym. Sci. 1988, 36, 703-719. (16) Hunter, R. J. Foundations of Colloid Science; Oxford: New York, 1986; Vol. I. (17) Wijnhoven, J. E. G. J.; Koorn, J.-P.; Poppe, H.; Kok, W. Th. J. Chromatogr. A 1996, 732, 307-315. (18) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge: New York, 1989.
Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
3233
Figure 5. Fractograms in Figures 2b and 4b plotted together. The fractograms of the regular sample (Figure 2b) are displaced along the time axis by 2.5 min.
Figure 4. Fractogram showing detector response as a function of time for dialyzed sulfate polystyrene latex: applied potential difference 1.2 V, current 0.2 mA, flow rate 1 mL/min; (a) high sample concentrations (0.5-0.05 wt %), (b) low sample concentrations (0.1-0.01 wt %).
to sales. As a result, there is a long equilibration period between the particles and solution. This apparently leads to a higher concentration of charged species in solution and to higher solution conductivity. After 24 h of dialysis, the conductivity is ≈4 µS cm-1, a value we expect due to particles in the absence of conducting buffer.18 We do not report the conductivity of dialyzed samples as a function of particle concentration here since the conductivities are very small and the conductivity meter used is not very accurate in the more dilute concentration region. The same experiment as shown in Figure 2 is carried out with dialyzed samples, and the corresponding fractograms are presented in Figure 4. The concentration effect is still present, even at very low particle concentrations. Thus, the observed variation in retention with changes in sample concentration is due to the presence of salt as well as the differences in particle concentration. Salt present in the sample reduces the effective field only in the initial parts of the channel. This is because it is not retained and hence gets separated from the particles. This is illustrated in Figure 5, where fractograms of nondialyzed (Figure 2b) and dialyzed (Figure 4b) samples are plotted together. In Figure 5, fractograms of the nondialyzed samples are artificially displaced along the time axis to show that the shapes of the fractograms are almost identical for these two cases. In a sense, the presence of salt reduces the effective length of the channel, as shown in Figure 5. This is undesirable for 3234
Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
separation efficiency, although the zone broadening appears to be nearly invariant in this case. More importantly, Figure 5 demonstrates that there will be differences in retention which the analyst may inadvertently ascribe to differences in particle size and/or charge when in effect the only difference is in the sample injection environment. For this reason, we routinely dialyze samples for electrical FFF analysis. The transport model developed in the previous section can be used to explain this behavior. We first start with eq 10 and consider two limiting cases. In the first case, which we will denote as case I, the conductivity of the carrier phase is very high, for example 10-2 M KCl, or the concentration of sample is very low. For this case a0 . a1C so that σ* ≈ a0 as denoted by eq 8. The transverse particle velocity due to the field is independent of the position for this case. Hence we get the single particle limit. The linear retention theory is valid in this case, and there will not be any effect of sample concentration on the retention time. Case II includes the scenario where a large amount of sample is injected and the carrier conductivity is low so that a0 , a1C and from eq 8 we note that σ* ≈ a1C. As a result, the term due to the electric field in eq 10 drops out and there will not be any retention of particles. The physics of the electrical FFF process lies between these two extreme cases where an increase in the sample particle concentration leads to a zone with higher mean layer thickness, λ, because the local field is decreased. This zone elutes earlier from the channel as compared to a zone with smaller λ. This is why we observe concentration-dependent retention in the electrical FFF experiment. These two cases can also be deduced through the application of the retention theory of FFF,19,20 which uses the transverse concentration profiles, C(x), of the relaxed zones to compute the retention ratio, Rr, through application of
Rr )
〈C(x)v(x)〉 〈C(x)〉〈v(x)〉
(18)
where the brackets again indicate transverse averages. Note that (19) Giddings, J. C. J. Chem. Phys. 1968, 49, 81-85. (20) Hovingh, M. E.; Thompson, E. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195-203.
Figure 6. Steady-state concentration profiles for a1/a0 ) 0, Pex ) 200, and various injected sample concentrations.
Figure 7. Steady-state concentration profiles for a1/a0 ) 5, Pex ) 200, and various injected sample concentrations.
we use the symbol Rr to distinguish the retention ratio from the symbol for resistance, R. As explained above, these relaxed zones are given by eq 13 for the nonlinear case and by eq 15 for the linear case. Figures 6 and 7 show such relaxed zones for the linear and nonlinear cases, respectively, for various values of the injection concentration. When the field is independent of the concentration, any increase in sample amount leads to an increase in concentration at any location by a constant multiple. Hence the retention ratio remains the same for any sample concentration. This is not true when the the field is dependent on the particle concentration. In this case, as the amount of sample is increased the zone becomes more and more uniform, and thus the nondimensional mean layer thickness, λ, is increased. The mean layer thickness is now defined here as the first moment of the transverse concentration distribution:
∫ xC(x) dx ∫ C(x) dx w
le 1 λe ) ) w w
0
w
(19)
0
where the subscript “e” on λ indicates the exact mean layer thickness rather than the usual asymptotically approximate mean layer thickness. The quantity le is the analogous exact dimensional mean layer thickness. Increasing the mean layer thickness will of course result in reduced retention.
Figure 8. Exact mean layer thickness, λe, as a function of sample concentration, Cin, for linear and nonlinear retention models, a1/a0 ) 5, Pex ) 200.
In Figure 8, we plot λe as a function of sample concentration, Cin, for a Pex of 200. The linear retention theory gives a constant mean layer thickness irrespective of the sample concentration, while the nonlinear theory presented in this paper demonstrates how the mean layer thickness will increase with increasing concentration. As seen from Figure 8, the effect of concentration on λe is quite striking and shows the sensitivity of λe to Cin. Since it is well-known that λ and the retention ratio, Rr, are approximately linear for well-retained zones, Figure 8 implies that large changes in Rr can take place even at small injection concentration differences. Stopped-Flow Relaxation. For colloidal particle separations using techniques such as flow and sedimentation FFF, it is common to use a stopped-flow relaxation period of the order of 10 min where the flow is absent but the field is present. This allows particles to obtain steady-state concentration densities in the transverse direction and avoid large amounts of zone broadening because of exposure of zones to fast flow velocities in the middle of the channel. In electrical FFF, the current in the channel and subsequently the internal field strength depends on the carrier flow rate.4 This effect is significant at higher voltages where the current is mass transport limited.4 Hence, if a stoppedflow relaxation is attempted, the field drops considerably compared to its value when flow is applied. This implies that stopped-flow relaxation may be ineffective if used. Toward determining the importance of these considerations, a number of experiments utilizing stopped-flow relaxation are reported here. Figures 9 and 10 show fractograms of the polystyrene sulfate particles where an 11 min stopped-flow relaxation is utilized without dialysis (Figure 9) and with dialysis (Figure 10). Here we show data only after the relaxation period is over and the flow is turned on again. The first peak seen in these fractograms is due to some reacting species that gets transferred from the graphite electrodes to the bulk solution. During normal operation of the electrical FFF channel, this species elutes at a steady concentration, but during the stopped-flow period, it builds up inside the channel, and that is why it appears as a peak in the fractogram. Concentration-dependent retention is observed regardless of the sample treatment for the 1% particle concentration case. However, for lower concentrations, the stopped-flow technique produces a minimal change in the elution time of the peak maxima. Careful examination of the elution fractograms shown Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
3235
Figure 9. Fractogram showing detector response as a function of time after stopped-flow relaxation of 11 min: applied potential difference 1.2 V, current 0.2 mA, flow rate 1 mL/min; (a) high sample concentrations (1-0.1 wt %), (b) low sample concentrations (0.10.01 wt %).
Figure 10. Fractogram showing detector response as a function of time for dialyzed latex after stopped-flow relaxation of 11 min: applied potential difference 1.2 V, current 0.2 mA, flow rate 1 mL/ min; (a) high sample concentrations (0.5-0.05 wt %), (b) low sample concentrations (0.1-0.01 wt %).
in Figures 9 and 10, however, reveals that there is an elevated baseline between the apparent void peaks and the particle elution peaks when dialysis is not used (Figure 9) and that this effect is smaller when dialysis is used prior to sample injection (Figure 10). Hence, in certain cases, stopped-flow may be desirable when the elution time is to be correlated with particle size and/or charge. Also note that zones with dialysis show a larger retention that zones without dialysis, as expected. The oscillations seen in Figure 10b near the peak maxima may be due to flocculation phenomena. This effect is suppressed when the sample is not dialyzed because salt or other low molecular weight conducting impurities will lower the effective field. The lower field will disfavor flocculation at the accumulation wall (x ) 0) because the lower field would result in lower local concentrations of particles at the wall. However, the poorer zone shape shown in Figure 10b may be due to small amounts of slow desorbing particles on the accumulation wall whereby a larger percentage of the lower concentration zones is affected. The larger concentration zones may simply show less of this effect because the amount adsorbed is small relative to the total amount of particles injected. Figure 11 shows the dynamics of zone relaxation as a function of time for both linear and nonlinear zone behavior. As discussed above, these zone profiles are obtained by numerically evaluating eq 11. Figure 11a shows the concentration profile away from the
accumulation wall, where 0.1 < x/w < 1, and Figure 11b shows the concentration profile in the vicinity of the accumulation wall. As seen from Figure 11, the zone relaxes very quickly in ≈1 min. The difference between the linear and nonlinear models initially appears to be small with the linear model relaxing faster than the nonlinear model as expected. At 35 s, under the conditions specified in this calculation, Figure 11b shows that the concentration at the wall is just starting to relax toward the exponential concentration distribution that is expected for the linear model. Comparison of the 35 s result with the result from 100 s suggests that the zone is still not sufficiently relaxed to obtain good fractionation. Although not shown in Figure 11, comparison of the 100 s concentration profile for both the linear and nonlinear cases with those obtained from the steady-state theories shows no difference between these curves. Hence, under the conditions used here, the zone profiles at 100 s are fully relaxed. Thus, the relaxation step appears to be fast if the analytical run is on the order of tens of minutes and the relaxation step is complete before the zone has had a chance to travel a significant amount down the channel. These calculations suggest that it is not critical to stop the flow at the beginning to obtain reasonably relaxed zones. This supports the comparison between Figure 2 and Figure 9 and between Figure 4 and Figure 10 which demonstrates that stopped-flow relaxation does not substantially
3236 Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
that can relax particles on time scales of the order of 2 min. This is due to the strength of the electrical force on colloidal particles. In sedimentation FFF, relaxation times on the order of 10 min are needed at very high field strengths to retain particles on the order of 100 nm. In the electrical FFF experiment, much higher forces are easily achieved with much less instrumental difficulty. Finally, we note that the transport model developed in this paper through eq 10 can be used to calculate the resulting peak shape of the zones as they elute from the channel. If the field is independent of the concentration, then one can find the transverse concentration profile for the relaxed zone by using eq 15, as has been done for previous cases of FFF.1 Application of eq 18 then yields the retention ratio Rr. Using a Gaussian peak shape model, the zone-broadening equations21 can then be applied to give a temporal concentration profile which suffices for most conventional forms of FFF. Unfortunately, this scheme cannot be used for electrical FFF with concentration effects because we need to solve the complete 2-D equation to determine how the zones will migrate. This is an extremely difficult computational task since there are very high concentration gradients in both the x and z directions. Standard and exotic finite difference numerical schemes fail because of these high gradients. Future simulation work along these lines may help to elucidate the zone-broadening characteristics of the electrical FFF technique and help bring more complete understanding to the electrical FFF process. Figure 11. Stopped-flow relaxation of particles. Scaled concentration vs scaled position for various time: Pex ) 183, injected sample concentration, Cin ) 0.02; (a) bulk solution, (b) near the accumulating wall. For the nonlinear case, a1/a0 ) 5.
increase retention under the conditions used here. As shown in Figure 10, with dialyzed samples it may even introduce new distortions, although the origin of this may be the sample particles themselves. The results presented in this series of papers suggest that the electrical FFF experiment should be run in a carefully controlled manner where small sample concentration and sample dialysis are mandatory if retention times are to be correlated to physical properties. Although these suggestions seem constraining, the use of sensitive detectors like the evaporative light scattering detector or the multiangle laser light scattering detector should partly compensate for these restrictions since they are far more sensitive than the simple detector that we have utilized here. One method that may be utilized in the future is the use of redox couples in the carrier solution, as demonstrated previously.2 If the use of a redox couple makes it possible to use a higher ionic strength carrier, then some of the concentration sensitivity can be reduced and the experiment will be less critical toward injection solvent conditions. Future work is now in progress to elucidate the mechanisms of the redox couple and how the electric field is mediated in that mode of operation. Perhaps one of the more interesting aspects of this technique is how little voltage is truly needed to mediate a high field strength. This is in contrast to CE, where high voltages, typically 20 kV, are routinely used across capillaries of lengths between 10 and 50 cm. In the electrical FFF experiment, a few volts across a space of hundreds of micrometers are utilized and produce fields (21) Giddings, J. C.; Yoon, Y. H.; Caldwell, K. D; Myers, M. N.; Hovingh, M. E. Sep. Sci. 1975, 10, 447-460.
ACKNOWLEDGMENT We thank Nancy Cohlberg, of Cohlberg Analytical, Blue Bell, PA, for editorial assistance. We also thank Dave Strumfels for developing the data acquisition software for our system. SYMBOLS A
cross-sectional area
b
channel breadth
C
concentration of particles
C0
concentration of particles at the accumulation wall where x ) 0
Cin
concentration of injected particles
D
diffusion coefficient of particles
E
electric field
I
current
J
current density
L
channel length
Pex
transverse Peclet number
R
resistance
Rc
resistance due purely to the carrier fluid
Re
solution-electrode interfacial resistance
Rp
resistance due purely to the solute particles
Rr
retention ratio equal to V0/Vr
Rs
total solution resistance
t
time
t0
void time
tr
retention time
U*
quantity µJ/a0
ux
solute transverse velocity Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
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V(x)
electrical potential
µ
electrophoretic mobility
V0
void volume
σ*
specific conductivity
Vr
retention volume
le
exact dimensional mean layer thickness
Vs
potential drop in the solution
〈v〉
fluid average velocity
w
channel width
x
transverse coordinate of the channel
Received for review January 2, 1997. Accepted May 21, 1997.X
y
breadth coordinate of the channel
AC970014W
z
axial coordinate of the channel
λe
exact nondimensional mean layer thickness
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Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
X
Abstract published in Advance ACS Abstracts, July 1, 1997.
