Anal. Chem. 2006, 78, 4998-5005
Electric Circuit Model for Electrical Field Flow Fractionation Joseph J. Biernacki,*,† P. Manikya Mellacheruvu,‡ and Satish M. Mahajan‡
Department of Chemical Engineering and Department of Electrical and Computer Engineering, Tennessee Technological University, Cookeville, Tennessee 38505
In electrical field flow fractionation (EFFF or ElFFF), an electric potential is applied across a narrow gap filled with a weak electrolyte fluid. Charge buildup at the two poles (electrodes) and the formation of an electric double layer shields the channel, making the effective field in the bulk fluid very weak. Recent computational research suggests that pulsed field protocols, however, should improve retention and may enhance separation in EFFF through systematic disruptions of the double layer resulting in a stronger effective field in the bulk fluid. Improved retention has already been demonstrated experimentally. Accurate modeling and subsequent device optimization and design, however, depends, in part, on formulating a suitable model for the capacitative response of the channel and double layer at the electrode surfaces. Early models do not correctly describe experimentally observed current-time response and are not physically meaningful even when accurate mathematical fits of the data are realized. A new model and conceptual framework based on electrical resistance and capacitance variations of the double layer is suggested here. Physical interpretations of the electrical response have been developed and compared to published experimental data sets. Electrical field flow fractionation (EFFF) (The acronym ElFFF is used by some authors for electrical field flow fractionation.) is a promising technology for the separation of nanoparticles including macromolecules, cells and organelles, enzymes and proteins, colloidal matter, and other electrically charged particles. In electric field flow fractionation, an electric potential is used to separate particles according to differences in their electrophoretic mobility and molecular diffusivity. The potential is applied normal to the flow direction forcing particles to migrate toward the channel walls (the electrodes), the rate of migration being a function of the particles’ electrophoretic mobility, molecular diffusivity, and effective field strength in the channel. Like particles accumulate preferentially into gradients. Since flow is streamlined, particles located at different distances from the channel wall are thus swept through the channel at different velocities, resulting in differential retention and hence, separation. Polarization near the electrode surfaces, e.g., formation of an electric double layer, * To whom correspondence should be addressed. E-mail: jbiernacki@ tntech.edu. Phone: 931-372-3667. Fax: 931-372-6352. † Department of Chemical Engineering. ‡ Department of Electrical and Computer Engineering.
4998 Analytical Chemistry, Vol. 78, No. 14, July 15, 2006
however, has been reported to dramatically reduce the effective field within the channel1,2 and thus limits separation efficiency. Double layer formation is the consequence of the interaction between the applied electric potential and charged particles present in the channel fluid. This double layer can shield as much as 99% of the applied potential resulting in a very weak effective field inside the channel.1,2 Chemicals such as quinine-hydroquinone and pulsed field protocols have been used to disrupt the electric double layer and produce a stronger average effective field.3-5 Pulsed field protocols create a dynamic response wherein the electric double layer polarity is switched, thus, disturbing the steep field gradient near the wall (electrode) that would otherwise result under constant applied potential. To optimize device performance, it is important to understand the governing phenomena and kinetics of the formation of the electric double layer. Efforts have been made in the past to model the EFFF channel using an electric circuit approach.2,5 Palkar and Schure2 obtained the values of solution resistance and double layer capacitance experimentally. They proposed a series resistance/capacitance (RC) circuit having a double layer capacitance (Cdl′) in series with the solution resistance (Rs) and reported a time constant (τ ) RsCdl) of 40 s for an applied potential of between 1 and 2 V. Their experiments were conducted for a macroscale channel with a channel gap (electrode separation) of 100-200 µm. More recently (2001), Gale et al. proposed a circuit model (Figure 1a and b) and offered reasonable physical interpretations.6 In this model, Gale et al. described EFFF as a series of three resistor-capacitor branches that represent the two double layers (one at each electrode) and the solution between them. For Cs , Cdl′ and double layer resistance Rdl′ . Rs, their model simplifies to Palkar and Schure’s2 series RC model. Gale et al. used their model to assess the scalability of EFFF and, in 2002, performed experiments using a graphite electrode (channel gap of 176 µm) in the macrodomain and gold and platinum electrodes (channel gaps of 28 and 28.5 µm, respectively) in microdomain.7 For an applied (1) Caldwell, K. D.; Kesner, L. F.; Myers, M. N.; Giddings, J. C. Science 1972, 176, 296-298. (2) Palkar, S. A.; Schure, M. R. Anal. Chem. 1997, 69 (16) 3223-3229. (3) Schimpf, M. E.; Russel, D. D.; Lewis, J. K, J. Liq. Chromatogr. 1994, 17 (14, 15), 3221-3238. (4) Giddings, J. C. Anal. Chem. 1986, 58, 2052-2056. (5) Lao, A. I. K.; Trau, D.; Hsing, I. M. Anal. Chem. 2002, 74 (20). (6) Gale, B. K.; Caldwell, K. D.; Frazier, A. B. Anal. Chem. 2001, 73 (10), 23452352. (7) Gale, B. K.; Caldwell, K. D.; Frazier, A. B. Anal. Chem. 2002, 74 (5), 10241030. 10.1021/ac0600961 CCC: $33.50
© 2006 American Chemical Society Published on Web 06/21/2006
Figure 2. Double-exponential current response.
Figure 1. (a) Electric circuit model,6 (b) simplified circuit model,2,5 (c) Laplace transform of circuit shown in (a), and (d) electric circuit model with time varying parameters.
voltage of 1.5 V, they reported an RC time constants of between 1 and 4 s for the macrochannel and of 1 s for the microchannel. Lao et al.5 conducted similar experiments on a microchannel (channel gap, 40 µm) and proposed an RC circuit similar to that of Palkar et al.2 Lao’s current versus time data were characterized by a two-stage process with an initial rapid current decay (stage I) followed by a much slower decrease in current with time (stage II). They reported a time constant of 0.02 s for the initial current decay and suggested a “delayed” buildup of the double layer, due to the shearing action of the carrier fluid flow, as being responsible for the second relatively slower process. In the context of Gale’s circuit model (Figure 1a), Lao et al. explained the observed electrical performance of an EFFF channel by assuming the double layer resistance (Rdl′) to be very large and solution capacitance (Cs) to be very small; e.g., Rdl′ . Rs and Cs , Cdl′ (the double layer capacitance). They then suggested that the initial current decay can be explained using the time constant RsCdl (see corresponding equivalent circuit in Figure 1b). Using deionized water and an applied voltage of 1.73 V dc, Lao et al. measured the experimental current for their EFFF channel. Applying their simple circuit model, they demonstrated good agreement between the theoretical channel resistance of 493 Ω and their experimentally determined value of 423 Ω. From their current versus time data, they estimated the initial current decay time constant (0.02 s) from which they also estimated the double layer capacitance to be 46.9 µF (Cdl′ ) 93.8 µF).
Based on experimental data of Lao et al., Biernacki and Vyas8 suggested an empirical double-exponential model and estimated the time constants for the initial current decay as well as the presumed delayed buildup of the double layer. Their initial time constant (τ1) was found to be 0.017 s, consistent with Lao’s value of 0.02 s. Biernacki and Vyas’ time constant for the delayed buildup of the double layer was found to be 1.2 s (τ2). Their empirical current values closely match with Lao’s experimental data as can be observed from Figure 2. It should also be noted from Figure 2 that the transition time at which τ1 changes to τ2 is roughly 0.12 s. In pulsed EFFF, the capacitative response of the channel will dictate, to some extent, the frequency of pulsation necessary to have an influence on retention and hence separation. To optimize channel performance, models that accurately describe this response are sought. Such models will incorporate RC time constants or similar parameters that control the current decay and concomitant change in effective field strength. Since crossflow movement, and hence separation, of particles in the channel is a function of the effective field strength, it is imperative that this field be accurately modeled. While interpretation of the experimental data of Lao et al. seems plausible and Biernacki and Vyas’s mathematical model is a reasonably good empirical fit to the data, their models are either incomplete or without a physical basis. An effort has been made here to understand the physics behind the double layer formation by assuming an entirely electrical parameter-based model. An equivalent electric circuit model with time-dependent parameters has been proposed here in an effort to offer physical interpretations of experimentally observed EFFF channel behavior. This approach is a significant departure from previous electrical models, which used constant circuit parameters. Changes were made to Gale’s model6 and validity of the new electrical model has been assessed using the Debye-Huckel theory. It is notable to point out that in review of the literature in search of data sets, two principle citations were found, Lao et al.5 and Palkar and Schure.2 As described here, Lao et al. exhibited a twostage current decay, suggesting rapid initial double layer charging followed by a slower dynamic. Analysis of Palkar and Schure’s data set, however, indicates a slow response characterizable by a (8) Biernacki, J. J.; Vyas, N. Electrophoresis 2005, 26, 18-27.
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single RC time constant. Palkar and Schure gathered current versus time data at 5-s intervals apparently between 5 and 80 s and extrapolated their data through t ) 0, suggesting an initial current of ∼180 µA (based on a best approximation read from their Figure 4). However, using a fluid resistivity of 1 MΩ‚cm and their channel geometry (length 64 cm, width 2 cm, gap 178 µm), an initial current of more than 7000 µA would be expected. On the basis of this calculation, it is expected that Palkar and Schure missed the two-stage dynamics of the double layer and captured instead only the diffusive response of the channel and long-time double layer assembly due to electromigration of ions. This data set was not analyzed further since the short-term stage I behavior was not included. THEORETICAL BASIS Electric Circuit Theory. Figure 1c represents a Laplace circuit equivalent to that shown in Figure 1a. The Laplace transform technique was used to find the current I(s) in the circuit shown in Figure 1c:
I(s) )
V(s) ) Z(s)
1.73/s (1) Rs Rdl′ Rdl′ + + sRdl′Cdl′ + 1 sRsCs + 1 sRdl′Cdl′ + 1
Using Lao et al.’s channel dimensions (channel length l ) 90 mm, channel width b ) 10 mm, and channel gap w ) 40 µm) the value of Cs was estimated to be 15.4 nF. Since this value of Cs , Cdl′ ) 93.8 µF, Cs was neglected in eq 1 yielding
I(s) )
1.73Rdl′Cdl′ sRsRdl′Cdl′ + (2Rdl′ + Rs)
+
1.73 (2) s(sRsRdl′Cdl′ + 2Rdl′ + Rs)
Rearranging eq 2, the following expression for I(s) was obtained:
I(s) )
1.73 Rs
1 + (2Rdl′ + Rs) s+ RsRdl′Cdl′
[
1.73(RsRdl′Cdl′)
1 RsRdl′Cdl′(2Rdl′ + Rs) s
]
1 (3) (2Rdl′ + Rs) s+ RsRdl′Cdl′
Taking the inverse Laplace Transform of eq 3 and simplifying gives the resulting expression for the current as a function of time:
(
I(t) ) exp -
I(t) )
)[
]
t(2Rdl′ + Rs) 3.46Rdl′ u(t) + RsRdl′Cdl′ Rs(Rs + 2Rdl′)
(
)
1.73 u(t) (4) Rs + 2Rdl′
t(2Rdl′ + Rs) 1.73 exp u(t) + Rs RsRdl′Cdl′
(
(
τ)
RsRdl′Cdl′ 2Rdl′ + Rs
(6)
Using values of Rs ) 423 Ω and Cdl′ ) 93.8 µF, Lao et al. found that τ ) 0.02 s. However, these values can only be used to model the initial current decay (up to 0.12 s). Furthermore, eq 5 cannot explain the double-exponential behavior observed by Lao et al. when the circuit parameters are constants. Either the circuit is incorrect or the circuit parameters Rdl′ and Cdl′ are time dependent. Since Gale’s circuit appears to have some physical interpretations, it was used as the basis for model development. However, Rdl′ and Cdl′ were treated as time-dependent variables rather than constants. The following three time-dependent approaches were considered: (1) Rdl′ varying from stage I to stage II and Cdl′ constant, where stage I is from 0 to 0.12 s and stage II is from 0.12 to 2 s; (2) Rdl′ constant and Cdl′ varying from stage I to stage II; (3) Rdl′ and Cdl′ varying simultaneously and linearly with respect to time. In approach I, a step variation of Rdl′ from stage I to stage II was considered. If Rdl′ were to vary continuously with time, the Laplace transform could not be used. Therefore, a step variation in Rdl′ was used in order to obtain appropriate upper and lower bounds consistent with eq 6. The same explanation holds for approach II. Once the Rdl′ and Cdl′ were bounded (limits defined by approaches I and II), continuous variation of Rdl′ and Cdl′ was considered (approach III). Debye-Huckel Theory and the Thickness of the Double Layer. Validation of any circuit parameters that characterize the double layer should be reconciled in view of the well-established Debye-Huckel double layer theory. The Poisson-Boltzmann (PB) equation is the starting point of the Debye-Huckel theory. However, since the PB equation does not have an explicit solution, it was solved for limiting cases. When the exponent in the PB equation is expanded, it gives rise to the linearized PB equation. This linearization results in a parameter k, the inverse Debye length, where k-1 is a physical length characteristic of the doublelayer to which independent measures of thickness can be judged large or small,9
k)
(
1000e2Na KBT
∑z i
2 i
)
Mi
1 2
(7)
where e is the electron charge, 1.6 × 10-19 C, Na is Avogadro’s number, ) 6.023 × 1023/mol, is the permittivity of the double layer (F/m), T ) 298 K, KB is the Boltzmann constant, 1.38 × 10-23 J/K, zi is charge of the species i (unitless), and Mi is moles per cubic meter. Upon expansion, the concentration weighted squared charge summation for aqueous systems becomes
∑z
2 i
Mi ) z2H+MH+ + z2OH- MOH- + z2pMp + ...
(8)
i
) )
t(2Rdl′ + Rs) 1.73 u(t) - exp u(t) (5) Rs + 2Rdl′ RsRdl′Cdl′ 5000
where u(t) is the unit step function u(t) ) 0, t < 0, u(t) ) 1, t g 0. From the exponential part of eq 5, the time constant was found to be
Analytical Chemistry, Vol. 78, No. 14, July 15, 2006
where MH+ ) [H+] ) 10-7 mol/L ) 10-4 mol/m3 ) [OH-] ) MOH- (assuming pH neutrality), Mp is the concentration of analyte
Figure 4. Comparison of ln(i) versus t from electrical model (approach I) with experimental data sets.
Figure 3. Asymptotic values of time constants.
particles and “...” represents the zi2Mi for all other analyte or ionic species in the system, i.e., added electrolyte or buffer. RESULTS AND DISCUSSION Approach I: Rdl′ Varying from Stage I to Stage II and Cdl′ Constant. Initially, τ ) τ1 ) 0.02 s. If Rs ) 423 Ω and Cdl′ ) 93.8 µF5 then, from eq 6, it can be shown that, τ is bounded between 0.0 and 0.0198 s (approximate value of 0.02 s), with 0.0198 s being the asymptotic upper limit. A value of τ2 ) 1.2 s is physically impossible if Rdl′ is the only parameter varying, and in fact, eq 6 can only take on the values of τ >0.0198 s when Rdl′ is less than zero for Rs ) 423 Ω and Cdl′ ) 93.8 µF as a plot of τ versus Rdl′ illustrates; see Figure 3. Substituting Rs ) 423 Ω, and Cdl′ ) 93.8 µF5 into eq 6 gives a value of τ approaching 0.0198 for large positive values (i.e., 10 000 Ω) or large negative values of Rdl1′. To achieve a value of τ ) τ2 ) 1.2 s, Rdl2′ must be equal to -215 Ω. Although there is no physical interpretation for negative values of double layer resistance, nonetheless, it is possible to generate a current versus time response mathematically. Plots of ln(i) versus t are shown in Figure 4. When the transition time is taken to be 0.12 s, consistent with Lao et al.’s experimental data, and when Rdl′ is changed from 10 000 to -215 Ω, reasonable doubleexponential behavior is observed; however, the percentage error is nearly 25% for stage II. However, if the transition time is changed from 0.12 to 0.08 s, the resultant current curve, consistent with eq 5, was found to matching closely with experimental data (see Figure 4). While approach I can provide a reasonable fit to the observed double-exponential current decay, it is still empirical and has no physical interpretation since negative values of Rdl′ are an integral part of it. (9) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1997.
Figure 5. Comparison of ln(i) versus t from electrical model (approach II) with experimental data sets.
Approach II: Cdl′ Varying from Stage I to Stage II and Rdl′ Constant. Since the stage I time constant must be 0.02 s and since Lao et al. found that Cdl1′ was on the order of 94 µF, it can be shown that any large value of Rdl′ can be used since τ becomes invariant with respect to Rdl′ as Rdl′ becomes large; see eq 6. Substituting τ ) τ1 ) 0.02 s into eq 6 gives
0.04Rdl′ - 423Rdl′Cdl1′ + 8.46 ) 0
(9)
Likewise, substituting τ ) τ2 ) 1.2 s in eq 6 gives
2.4Rdl′ - 423Rdl′Cdl2′ + 507.6 ) 0
(10)
Equations 9 and 10 were solved for various assumed values of Rdl′. A value of 1 MΩ was chosen for Rdl′ since it makes Cdl1′ equal to 94 µF, which is close to the Cdl′ of 93.8 µF reported by Lao;5 note that Cdl′ ) 2Cdl ) 2(46.9 µF) ) 93.8 µF. This makes the corresponding Cdl2′ during stage II equal to 5600 µF, per eq 8. The circuit in Figure 1c was simulated using these values of Cdl′ for the two stages. Figure 5 is a comparison of ln(i) versus time for approach II and Lao et al.’s experimental data. Again, a reasonable double-exponential behavior is achieved; however, the fit error is excessive reaching as much as 35% for portions of stage Analytical Chemistry, Vol. 78, No. 14, July 15, 2006
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II. As with approach I, it was observed that when the transition time is changed from 0.12 to 0.08 s the model current fits more accurately with the experimental values (Figure 5). Though, the model fits well to the data when the transition time is taken to be 0.08 s, as with approach I, the transition is abrupt and unphysical. Unlike approach I, however, variations in Cdl′ are physically meaningful and may lead to insights about the mechanisms involved in polarization at the double layer. To further improve the model, continuous variations in Rdl′ and Cdl′ were considered per approach III. Approach III: Rdl′ and Cdl′ Varying Simultaneously and Linearly with Respect to Time. Approaches I and II assumed that either Rdl′ or Cdl′ will change abruptly. This, of course, is unphysical, though in both cases can produce i versus t results that empirically follows the observed short-time and long-time experimental trends. In approach III, the circuit resistances and capacitances were varied continuously. A conceptual circuit is illustrated in Figure 1c. Equation 5 no longer applies since eq 5 was developed by assuming constant coefficients (constant circuit parameters Rdl′ and Cdl′ at least over specified time domains). Use of Laplace transforms for the product of two time varying functions, i.e., Rdl(t) and Cdl(t), becomes complicated and requires use of the convolution property. Therefore, an alternate approach using Kirchoff’s voltage law (KVL) was pursued in the time domain followed by a numerical integration. The circuit shown in Figure 1b was considered for this purpose. The parallel combination of Rdl′ and Cdl′ branches in Figure 1a were lumped together into a single branch where Rdl ) 2Rdl′ and Cdl ) Cdl′/2 (Figure 1b). This can be justified since Rs , Rdl′. The bounds for Rdl′ and Cdl′ were already established in approaches I and II. Applying KVL for the outer loop gives
-V + Rdl(t)(i(t) - i1(t)) + Rsi(t) ) 0 i1(t) )
(11)
i(t)(Rs + Rdl(t)) - V
(12)
Rdl(t)
and, for the inner loop
i1(t)
∫C t
0
dl(t)
dt + Rdl(t)(i1(t) - i(t)) ) 0
(13)
Substituting eq 10 into eq 11 gives
[
t
0
dl
]
i(t) V dt + Rdl(t) dl(t)
∫ C 1(t) i(t) + R R s
[
]
i(t) V Rdl(t) i(t) + Rs - i(t) ) 0 (14) Rdl(t) Rdl(t) i(t) dt + Rs dl(t)
∫C t
0
∫R t
0
i(t) dt - V (t)C dl dl(t)
∫R t
0
1 dt + (t)C dl dl(t)
Rsi(t) - V ) 0 (15) Finally, differentiating eq 13 on both sides produces the following differential equation: 5002
Analytical Chemistry, Vol. 78, No. 14, July 15, 2006
i(t) di(t) i(t) V )0 + Rs + Rs dt Cdl(t) Rdl(t)Cdl(t) Rdl(t)Cdl(t)
(16)
Equation 14 was solved using the Runge-Kutta method, and the resulting values of current were plotted. It should be noted that in eq 14 Rs ) 423 Ω and V ) 1.73 V; Cdl′ was chosen to vary linearly according to the expression
Cdl′ ) mct + bc
(17)
where mc and bc are the capacitance slope and capacitance intercept, respectively, and Rdl′ was also varied linearly according to the expression
Rdl′ ) mrt + br
(18)
where mr and br are the resistance slope and resistance intercept, respectively. Lao’s experimental data were then fit by varying mr, br, mc, and bc systematically. Table 1 summarizes the outcomes of various fit trials, and Figure 6 illustrates the respective fit of eq 16 to the experimental data. The limiting case of Rdl′ constant was first explored, i.e., with mr ) 0.0. In addition to this constraint, the values of br, mc, and mr were limited to positive only. A least-squares error optimization was conducted. Rdl′ was found to be equal to 41.5 kΩ, and Cdl′ was found to range from 64.6 µF at t ) 0.0-4460 µF at t ) 2 s. In theory, the Cdl′ value should be “zero” at t ) 0.0, which means that there is no double layer at t ) 0.0 s. Numerically, however, a finite Cdl′ value must be taken at t ) 0.0 to avoid division by zero. Figure 6 compares the experimental data set and the fit and gives the fit residual (the difference between the observed experimental values and the respective model predictions). Limiting Rdl′ to a constant value and permitting Cdl′ to vary linearly produced a fit that has the general double-exponential character of the experimental data set but, however, does not fit the data well over the entire time range. An effort was then made to find a set of parameters that further reduces the sum of squared errors. In this case, all four parameters, mr, br, mc, and bc, were permitted to vary in the optimization to obtain the best fit. Not surprisingly, optimization resulted in multiple local minima being found. Table 1 compares various optima thought to be close to the global minimum, each located by changing the initial conditions. Notably, small changes in initial conditions lead to different minima; however, all initial conditions must be in the neighborhood of the minima to obtain convergence. From these results it appears that the double layer resistance increases from ∼10 to ∼60 kΩ. The behavior of the capacitance, however, is less certain. The best fit (trial 4) suggests that the capacitance increases from ∼19 to ∼72 µF. Other local optima, with nearly equivalent least-squared error, suggest that a constant (trial 2) or decreasing (trial 3) capacitance also provides a reasonable model. Statistically, it is difficult to favor a particular fit since the uncertainty in the data set is sufficient to argue that one cannot tell the difference between these three minima. In these simulations, a constant bulk solution resistance of 423 Ω was assumed, consistent with the value reported by Loa et al.
Table 1. Tabulated Values of Slopes and Intercepts for the Double Layer Resistance and Capacitance and Estimated Double Layer Thickness Based on Capacitance Values trial no.
mr′ 104
br′ 104
mc′ 10-5
bc′ 10-5
∑2
approach III-1
0.0
4.15
2.2
6.46
1.96
approach III-2 approach III-3
2.5 2.48
1.11 1.10
0.0 -4.76
0.75 .74
approach III-4
2.64
.97
2.64
10.6 10.8 1.87
.59
Rdl′ (kΩ)
Cdl′ (µF)
double layer thickness (nm)
41.5
64.6-4460
11.2-61 11.1-61
106 108-13
0.75-0.011 ( ) 6) 10-0.14 ( ) 80) 0.45 ( ) 6)-6.01 ( ) 80) 0.44-3.68 ( ) 6) 5.90-49.0 ( ) 80) 2.52-0.66 ( ) 6) 33.5-8.85 ( ) 80)
9.88-61.8
19-72
Figure 6. Comparison of ln(i) versus t from electrical model (approach III, trial 1) and (approach III, trial II) with experimental data sets.
for their data set. This value is based on the mean fluid concentration and so must represent the lowest value of resistance observed since, as ions migrate, the resistivity in the bulk fluid will become a function of position, yet the mean resistance will increase. Though some elements of the fluid will become less resistive as the local ionic concentration increases, the remaining bulk of the fluid, however, will have a resistance much greater than that computed from the bulk mean ionic concentration (>423 Ω); thus, the overall resistance will be greater than 423 Ω. While the overall bulk fluid resistance must increase, it was shown that even a change as large as 1 order of magnitude has little effect on the fit error or fit parameters. For this reason, the solution resistance was held constant in all simulations. Finally, it may be possible to suggest a best fit based upon double layer characteristics implied by circuit parameters. For the fit values of Cdl′ and Rdl′ to be physically meaningful, the thus
derived values of resistance and capacitance should predict a physically meaningful double layer geometry, e.g., thickness, and resistivity. Thickness of Double Layer. It was observed that Rdl′ varies linearly between about 10 and 60 kΩ and that various Cdl′ behaviors, including increasing, constant, and decreasing, can produce a current response that matches closely with the experiment, e.g., provides an acceptable least-squares best fit to the experimental data set. In this section, an attempt has been made to justify and discriminate between these values of double layer resistance and capacitance by estimating the resistivity and thickness of the double layer corresponding to these values. Double layer thickness is a function of the concentration of charged particles in solution. For the experimental data used here, that of Lao et al., the bulk mean carrier concentration was 10 µM NaCl and current measurements were made in the absence of Analytical Chemistry, Vol. 78, No. 14, July 15, 2006
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analyte particles. In this case, the double layer thickness is dominated by the electrolyte. The local concentration in the volume near the electrode, however, changes with time. Initially, the concentration is equal to the bulk mean value and, hence, the charge sum is
∑z
2 i
Mi ) 12 × 10-2 + 12 × 10-2 ) 0.02
(19)
i
and the resulting Debye length is between 0.84 and 3.06 nm depending upon the value of the relative permittivity of the double layer ( has been shown to vary between 6 and 80 × 8.85 × 10-12 F/m10). As ions migrate due to the applied field, the concentration of ions near the electrodes will increase. For Na+ and Cl- ions and the geometry and effective potential realized in Lao et al.’s experiment, the surface concentration can be as much as 10× the bulk mean value. (The dimensionless steady-state concentration at the electrode C ˜ (0) can be obtained from C ˜ (0) ) e1/λ/λ(e1/λ - 1), where λ ) D/µΦeff, D is molecular diffusivity, µ is electrophoretic mobility, and Φeff is the effective field.8) In this case, the charge sum would increase to 0.2 and the Debye length would be between 0.27 and 0.97 nm. Based on Debye-Huckel theory alone, it is expected that the double layer thickness becomes thinner as it assembles due to two effects: decreasing permittivity and increasing local ionic concentration. Interpretations based on double layer resistance, Rdl′, were made using the definition of resistance:
Rdl′ ) Fxdl/A
(20)
where F is the resistivity of the double layer and A is the area of the electrodes, 9 × 10-4 m2.5 Since the resistivity of the double layer is unknown, it was computed from nominal values of the Debye length and from the capacitance analysis. In this case, a nominal value of xdl of 0.5 nm was chose as a basis. This results in a double layer resistivity ranging between 2000 and 11 000 GΩ‚ cm. While these values are extremely high, it is apparent that they must be high in order to account for the observed shielding effect that the double layer has. By comparison, reported values of resistivity for deionized water vary between 0.1 and 10 MΩ‚ cm,12 and the data of Lao et al. produce a value of 9.5 × 105 Ω‚cm for 10 µM NaCl solution. Apparently, the rate of transport of electrons and or electron carriers is very slow through the double layer and becomes increasingly slow as the double layer assembles under the influence of increasing ionic strength at the electrode surface. Next, the double layer thickness was estimated from the determined double layer capacitance (fit parameters), Cdl ) Cdl′/2 using the capacitance definition given by
Cdl′ ) A/xdl
(21)
where A is the capacitor area ) w (width) × l (length) ) (10 × (10) Kovacs, G. T. A. Micromachined Transducers; McGaw-Hill: New York, 1998. (11) Vyas, N. Characterization and Modeling of a Micro Electrical Field Flow Fractionation Device, M.S. Thesis, Department of Chemical Engineering, Tennessee Technological University, 2004. (12) http://www.elgalabwater.com/?id)503, date accessed: March 08, 2005.
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10-3) × (90 × 10-3).5 Table 1 summarizes and compares double layer thickness values for each of the simulation cases (fit trails). A range reflecting the uncertainty in the double layer permittivity was computed for each value of Cdl′. Double layer thicknesses in the ranging from 0.44 to 49.0 nm were computed from capacitance datasoverlapping with the Debye length in the range 0.27-3.06 nm. This implies that the values of double layer capacitance reported are physically realizable; however, the multiplicity of possible capacitance behaviors must be reconciled against other observations and interpretations. The objective of this work was to offer a more rigorous circuitbased interpretation of the observed capacitative response of EFFF electrodes. Though the outcome suggests that a model based on variable double layer capacitance and resistance provides a remarkably good fit to experimental data, a plausible physical interpretation is now required to substantiate this finding. A number of events occur upon application of a potential across the EFFF electrodes. Initially, the double layer will assemble relative to the bulk mean concentration of the fluid. Since the concentration of ions in the bulk fluid above the double layer will initially be low and though the permittivity is unknown, it is likely to vary from high to low as the double layer assembles and ion migration increases the concentration of the bulk fluid near the electrode surface. Under these conditions, the double layer will be thick, relative to some later time. As the ions migrate under the influence of the electric field, the concentration near the electrode surface will increase and hence the double layer thickness will decrease accordingly. These changes will likely be accompanied by concomitant changes in double layer properties including permittivity and resistivity. These arguments and estimates of Debye length would suggest that the double layer might vary in thickness from about 3 to 0.3 nm (3.06 to 0.27 nm per eq 20) over the course of the observed buildup, ∼2 s in the case of the present experimental data set. This appears to be consistent with reported permittivity values for pure water relative to the double layer, wherein the permittivity may change by more than 1 order of magnitude due to saturated dipole orientations.10 Trial 2 (constant capacitance, variably increasing resistance) and trial 3 (decreasing capacitance with variably increasing resistance) both predict that the double layer thickness would actually be growing and so should be rejected. Trial 4 (increasing capacitance with variably increasing resistance) predicts double layer behavior that is at least somewhat consistent in magnitude and becoming thinner with time consistent with the Debye-Huckel estimates. In this case, the double layer thickness varies between about 2.5 and 0.7 nm when the permittivity is 6 F/m, though higher values are suggested when the permittivity is greater. Furthermore, predictions of the electrodiffusive time constant based on earlier work suggest a time constant on the order of 1/10 s, similar to that observed for the delayed double layer assembly indicated from Lao et al.’s8 experiment. This suggests that the long-time delayed double layer buildup is likely due to ionic transport within the channel rather than the fluid shearing as suggested previously.5 Yet, some explanation is forthcoming regarding the linearity of the capacitance and resistance changesone might expect some form of exponential change or at least something nonlinear. The model used here is still somewhat empirical. It does not directly
link properties to the local nanoscale environment, nor is it linked to the long-time diffusive transport of ions. What it does, however, is suggests that various competing phenomena including changes in the geometry and local properties of the fluid result in a net observable (apparent), empirically linear change in resistance and capacitance. The present results reflect the changes in nanoenvironment and diffusive transport, but at this time cannot explain the observed linear behavior based upon more fundamental phenomena. Finally, some interpretations regarding the double layer in systems containing analyte particles can also be made. In pH neutral systems with no added carrier electrolyte, the presence of the analyte particles will dominate the double layer thickness. Consider, a nominal bulk mean particle concentration of 3.75× 10-7 mol/mol of water11 ) 0.021 mol/m3. In this case, the effective charge sum is
∑z
2 i
Mi ) 10-4 + 10-4 + 676 × 0.0208 ≈ 14
(22)
i
and so the Deby-Huckel theory would predict a double layer thickness with characteristic length between about 0.03 and 0.12 nm. These values, however, are likely erroneous since typical analyte particles are large colloidal or macromolecular, have mean diameters on the order of 101 nm, and should not be treated as point charges. Though, it appears that the charge sum is dominated by the presence of analyte particles and so they likely play some role in dictating the double layer thickness during separations. It is also notable to point out that separations occur for impulse injections of analyte and so the effective double layer travels with the analyte impulse. This further complicates the problem for “real” applications of EFFF. Since it appears that particles dominate the double layer charge sum, why then does ionic strength of the carrier have a significant effect on retention as shown by Caldwell and Gao?13 Palkar and Schure2 offered a plausible explanation. The current through the EFFF must be (13) Caldwell, K. D.; Gao, Y.-S. Anal. Chem. 1993, 65, 1764-1772.
equal to the quotient of the applied potential ∆V and the sum of the double layer resistance Re and the bulk solution resistance Rs (∆V/(Re + Rs)). The effective potential in the channel is given by I × Rs, the voltage drop across the bulk fluid exclusive of the double layer. Adding electrolyte ions to the bulk decreases Rs, but since Re . Rs, I is effectively unchanged and thus I × Rs decreases. The decrease in I × Rs is reflected in lower retention resulting from the presence of electrolyte ions in solution. CONCLUDING REMARKS Experimental data5 and subsequent mathematical analysis8 show that current across the electrodes of an EFFF channel decays according to an empirical double-exponential model. Though others have suggested an electric circuit description for EFFF, their description with constant electrical parameters of capacitance and resistance cannot explain the observed two-state experimental behavior. As an alternative, Gale’s electric circuit model6 for EFFF was investigated as the possible basis for describing observed current versus time behavior; however, time variant electrical parameters were used rather than constants. Assuming linear time-dependent models for both resistance and capacitance, it was found through optimization that a time variant increasing resistance with increasing capacitance can provide an excellent fit to the experimental data. Furthermore, the double layer thickness computed from capacitance data matches well with that derived from a classical Debye-Huckel analysis. Double layer resistance values suggest that the double layer resistivity is on the order of 1012-1013 Ω‚cm. Such is yet to be substantiated by other means and must be further explored. ACKNOWLEDGMENT The authors acknowledge financial support from the Ivanhoe Foundation, California, and from the Center for Management, Utilization and Protection of Water Resources at Tennessee Technological University. Received for review January 13, 2006. Accepted May 18, 2006. AC0600961
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