Anal. Chem. 2007, 79, 6334-6340
Influence of Electrophoresis Waveforms in Determining Stochastic Nanoparticle Capture Rates and Detection Sensitivity Ryan J. White and Henry S. White*
Department of Chemistry, University of Utah, 315 South 140 East, Salt Lake City, Utah 84112
Electrophoretic capture and release of charged polystyrene particles at the opening of a membrane pore has been investigated to determine the optimal applied current waveform, iapp(t), for ensuring true stochastic counting rates and to improve detection sensitivity (i.e., ∆(counts per second)/∆(particle concentration)). In capture and release detection, charged particles are electrophoretically driven to the opening of a small pore (∼60 nm diameter) in a membrane; capture of a single particle at the pore opening at time τ is signaled by a decrease in the flux of a redox species (Fe(CN)64-) through the pore. The captured particle is then released by applying an electrophoretic current in the opposite direction, and the process is repeated to acquire sufficient statistical data to determine the solution particle concentration (Cp) based on the relationship between Cp and the average particle counting rate (τavg-1). Both τavg-1 and the method sensitivity are shown, for the detection of 90 nm diameter polystyrene particles, to depend strongly on the applied current waveform. The observed dependence is a consequence of the nonequilibrium distribution of particles in the analyte solution that results from electrostatic forces acting on the particle whenever iapp has a nonzero value. Stochastic capture rates corresponding to an initial uniform distribution of particles are more closely achieved using an applied current waveform that includes an equilibration period (iapp ) 0) prior to electrophoretic capture. An increase in particle detection sensitivity, relative to the previously reported value, results from this equilibration step. In a recent report,1 we introduced a method of particle detection based on the “electrophoretic capture and release” (ECR) technique. In ECR, charged particles are electrophoretically captured at the orifice of a small pore in a synthetic membrane in response to an applied current; the capture event is signaled by an increase in the mass-transport resistance of the pore, as determined by a sudden decrease in the flux of a redox marker species or the ionic conductivity. Preliminary results demonstrated the capability of ECR to discriminate between particles of different size and charge in mixed solutions of particles. The ECR method * Corresponding author. E-mail:
[email protected]. (1) Lee, S.; Zhang, Y.; Harrell, C.; Martin, C. R.; White, H. S. Anal. Chem. 2004, 76, 6108-6115.
6334 Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
has conceptual similarities with electric sensing zone methods used in particle analysis2-4 (e.g., Coulter counters) in which a transient increase in ionic resistance is detected as a particle moves through an opening or pore in glass. However, in ECR, the particle is not required to traverse the length of the pore, greatly reducing the possibility of pore blockage, a problem associated with the detection of particles of nanometer dimensions. ECR particle detection is schematically presented in Figure 1. In step i, a constant current is applied across a porous membrane, using a galvanostat and Ag/AgCl electrodes, to electrophoretically drive charged particles, as well as a redox species of the same charge sign as the particles, toward the pore opening. The specific arrangement shown in Figure 1 corresponds to the experiment presented in the present article, in which negatively charged polystyrene particles and Fe(CN)64- are present in the analyte solution. The purpose of the Fe(CN)64- is to act as a redox marker for measuring the mass-transport resistance of the pore. When the pore is unblocked, Fe(CN)64- is transported, unimpeded, by diffusion and migration, through the pore. The rate of Fe(CN)64transport is measured quantitatively with high precision and accuracy by positioning a small Pt microelectrode at the backside opening of the pore and oxidizing Fe(CN)64- (to Fe(CN)63-) as the molecule exits the pore. The faradaic current recorded at the tip is directly proportional to the transport rate and has been detailed elsewhere.5 Occurring simultaneously with Fe(CN)64transport through the pore, the charged particles are electrically driven to the pore opening and are captured there if the particle diameter is larger than that of the pore diameter. The capture event (step ii) results in a significant decrease in the pore flux of Fe(CN)64-, which is measured by a corresponding decrease in the microelectrode current. The time between applying the electrophoretic current (t ) 0) and the capture event (t ) τ) is the quantity of interest in relating capture rates to particle concentration. Averaged values of τ that have statistical significance in determining particle concentration are obtained by repeating the experiment multiple times. The sequence of events may be repeated by first applying an electrophoretic current of opposite polarity to that used to capture the particle, in order to drive the particles off the membrane opening (step iii), and then (2) Ito, T.; Sun, L.; Henriquez, R. R.; Crooks, R. M. Acc. Chem. Res. 2005, 37, 937-945. (3) Bayley, H.; Martin, C. R. Chem. Rev. 2000, 100, 2575-2594. (4) Youngseon, C.; Baker, L. A.; Hillebrenner, H.; Martin, C. R. Phys. Chem. Chem. Phys. 2006, 8, 4976-4988. (5) Skott, E. R.; White, H. S.; Phipps, J. B. Anal. Chem. 1993, 65, 1537-1545. 10.1021/ac070610i CCC: $37.00
© 2007 American Chemical Society Published on Web 07/20/2007
we have explored the dependence of particle capture rates on the shape of the applied current waveform. The primary interest and goal of this work is to understand how the stochastic nature of single-particle detection is biased when employing external electrostatic forces and to determine if such biases undermine or otherwise influence method performance. The objective, of course, is to improve detection sensitivity and detection limits based on a firm understanding of the principles of the method. The issue of electrostatic bias also appears to apply more generally to stochastic single-molecule detection strategies based on biological or synthetic pores.7,8
Figure 1. Schematic illustration of the capture, detection, and release of particles. In step i, a positive iapp is applied across the membrane to electrophoretically drive negatively charged particles to the small opening of the pore. In step ii, pore blockage results in the reduction the flux of the redox molecule (Fe(CN)64-) and a concurrent decrease in the faradaic current at the SECM tip; the decrease in tip current signals the capture of the particle. In step iii, the polarity of iapp is reversed in order to discharge the particle from the pore opening. In step iv, the current is turned off to allow equilibration of the particle distribution in the analyte solution. The electrolyte (0.1 M KCl on both sides of the membrane) is not shown for clarity.
turning off the applied current (step iv) to allow the particle distribution to relax, by thermal motion, to the initial equilibrium distribution. As will be demonstrated below, the equilibration step is crucial to obtaining high detection sensitivity and true stochastic counting rates. The use of a similar electrophoretic waveform has been employed for the stochastic capture and detection of single DNA hairpin molecules in the protein ion channel R-hemolysin.6 Capture rates in ECR are readily manipulated by adjusting the magnitude of the applied electrophoretic current, the magnitude and duration of the current used to unblock the pore, and the length of the equilibration step. However, particle counting by ECR is a stochastic method in which the arrival of a single particle at the pore opening is inherently variable and determined in part by the initial random particle distribution prior to application of an electrophoretic current, as well as by the random-walk nature of motion as the particle drifts, under the influence of the electric field during the capture process. In principle, averaged stochastic particle capture rates, τavg-1, should vary with particle concentration (Cp) in a well-defined manner based upon transport theory and a statistical description of the random particle distribution. In practice, we previously observed that capture rates were very weakly dependent on particle concentration, i.e., τavg-1 ∝ Cp0.1, for reasons that were not understood.1 With these issues in mind, (6) Vercoutere, W.; Winters-Hilt, S.; Olsen, H.; Deamer, D.; Haussler, D.; Akeson, M. Nature 2001, 19, 248-252.
EXPERIMENTAL SECTION Chemicals and Materials. All solutions were prepared using >18 MΩ‚cm H2O from a Barnstead E-Pure water purification system. K3IrCl6 (Aldrich, Milwaukee, WI), K4Fe(CN)6 (99% Aldrich, Milwaukee, WI), KCl (AR Grade, Mallickrodt, Paris, KY), KH2PO4 (AR Grade, Mallickrodt, Paris, KY), K2HPO4‚3H2O (AR Grade, Mallickrodt, Paris, KY), NaCN (AR Grade, Mallickrodt, Paris, KY), NaOH (AR Grade, Mallickrodt, Paris, KY), and Triton X-100 (LR Grade, Sigma Chemical, St. Louis, MO) were used as received. Polystyrene particles of 90 ( 12 nm diameter with ∼42 000 COOH groups/particle were obtained from Bang’s Laboratories, Fishers, IN. The manufacturer-quoted size was determined by transmission electron microscopy or dynamic light scattering. The average number density of functional groups was determined by conductometric titration. The tracked-etched polycarbonate membranes used in this study were ∼6 um thick and contained conical-shaped pores at a density of 8 × 104 pores/cm2. Large and small openings of the pores, respectively, were 2570 ( 150 and 60 ( 4 nm, determined by scanning electron microscopy. Fabrication and characterization of the membranes have been described elsewhere.9 The pore size and density are sufficient enough that, when the SECM tip is positioned closely enough, events occurring at a single pore can readily be measured. Electrophoresis Cell and Scanning Electrochemical Microscope. The SECM instrumentation and electrophoresis cell for electrophoretic particle capture have been previously described.1 The electrophoresis cell consists of two compartments, the lower (or donor) and upper (or acceptor) compartment, which are separated by the polycarbonate membrane sandwiched between two glass plates with ∼2 mm diameter circular openings. Approximately 2500 pores are contained within the exposed area. The donor compartment contains the particle solution to be analyzed. The acceptor compartment contains 5 mM K3IrCl6 used for negative feedback control on tip approach. The tip is positioned within ∼1 tip radius of the surface by measuring the negative feedback approach curve with IrCl63-/2- as the redox mediator.10 The lower compartment contains a 5 mM solution of K4Fe(CN)6 which acts as the redox marker species to signal particle capture. Both compartments contain 0.1 M KCl and are buffered at pH 7.4 using a 0.01 M phosphate buffer system (PBS). Prior to (7) (a) Gu, L.; Braha, O.; Conlan, S.; Cheley, S.; Bayley, H. Nature 1999, 398, 686-690. (b) Bayley, H.; Cremer, P. S. Nature 2001, 413, 226-230. (8) (a) Sun, L.; Crooks, R. M. J. Am. Chem. Soc. 2000, 122, 12340-12345. (b) Ito, T.; Sun, L.; Crooks, R. M. Anal. Chem. 2003, 75, 2399-2406. (9) Li, N.; Yu, S.; Harrell, C.; Martin, C. R. Anal. Chem. 2004, 76, 2025. (10) Bard, A. J.; Mirkin, M. V. Scanning Electrochemical Microscopy; Marcel Dekker: New York, 2001.
Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
6335
particle analysis, the electrophoretic flux of Fe(CN)64- across the membrane, from the donor to the acceptor compartment, is amperometrically detected at the SECM tip and used to position the tip directly over a large pore opening. A low-noise potentiostat (Chem-Clamp, Dagan, Minneapolis, MN) is used for tip potential control versus a Ag/AgCl (3 M NaCl) reference electrode. The applied electrophoretic current (iapp) is controlled using two large-area Ag/AgCl electrodes in the upper and lower compartments using a galvanostat (RDE-4, Pine Instrument, Grove City, PA). Positive electrophoretic current corresponds to a flux of positive ions from the upper to lower compartment, and negative ions from the lower to upper compartment, and is employed in the capture stage of the analysis. Conversely, negative iapp corresponds to the flux of positive ions from the lower to upper compartment, and negative ions from upper to lower compartments, and is employed in particle release. SECM Tip Preparation. Conical-shaped Pt SECM tips were fabricated as previously described.11,12 Briefly, a 25 µm diameter Pt wire is attached to a 0.01 in. diameter W rod (FHC, Bowdoinham, ME) using Ag composition paint (Dupont). The tip is then electrochemically etched in a 6.0 M NaCN/0.1 M NaOH solution with an applied 4 V peak-to-peak voltage at 60 Hz. Optical microscopy is used to determine if the tip is adequately sharpened. The tip is then sealed using a butane flame in a 5 µL glass capillary (Drummond) and heated until the sharpened tip protrudes from the glass. The tip geometry corresponds approximately to a glassshrouded cone. SECM tip radii (between 300 and 1000 nm) were measured from steady-state voltammetry as previously described.13 RESULTS AND DISCUSSION In the experiments described herein, polystyrene particles are electrophoretically driven to the opening of an individual conicalshaped pore in a polycarbonate membrane that contains a large number of pores exposed to the solution (∼2500). Particle detection at a single pore is made possible by positioning a Pt SECM tip at the backside opening of the pore and monitoring the sudden decrease in the flux of Fe(CN)64- in the pore that occurs when the particle blocks the pore opening. Figure 2 schematically depicts the electrical forces acting on a charged particle during ECR detection. The current, iapp, from the galvanostat, applied across the entire membrane, is partitioned equally through the individual ∼2500 pores of the membrane, such that particle capture occurs with equal probability at every pore. The surfaces of the pores are negatively charged as a result of the pore etching procedure,14 and surface carboxylic acid groups gives the PS particles a negative charge at pH 7.4. The application of a current through an individual pore thus creates an electrophoretic force (Fe) on the particle, as well as generating electroosmotic flow within the pore. The flow exiting the pore exerts a viscous drag force on the particles in the vicinity of the pore opening, which we designate as Feo. When a current is applied across the negatively charged membrane as shown in Figure 2, the forces Fe and Feo acting on a negatively charged particle are (11) Penner, R. M.; Heben, M. J.; Lewis, N. S. Anal. Chem. 1989, 61, 16301636. (12) Ervin, E. N.; Baker, L.; White, H. S. Anal. Chem. 2005, 77, 5564-5569. (13) Zoski, C. G.; Mirkin, M. V. Anal. Chem. 2002, 74, 1986-1992. (14) Apel, P. Y.; Korchev, Y. E.; Siwy, Z.; Spohr, R.; Yoshida, M. Nucl. Intrum. Methods Phys. Rev., Sect. B 2001, 184, 337-346.
6336
Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
Figure 2. Schematic illustration depicting the directions of the electrophoretic (Fe) and electroosmotic forces (Feo) acting on a negatively charged particle (z-) when a positive current is applied (iapp) across the membrane. The current through a single pore (ip) results in electroosmotic flow due to the negatively charged surface. The directions of Fe and Feo are reversed when the current polarity is reversed.
always of opposite directions. For instance, in Figure 2, the application of a positive iapp results a pore current icapt that induces the force Fe pulling the particle toward the pore opening (and eventually pore blockage) and the force Feo pushing the particle away. Conversely, a negative positive iapp to release the particle results in a pore current irel and changes in signs of Fe and Feo. The motion of the particles near the pore opening is thus due to concentration gradients, electric fields, and electroosmotic flow, as described by the Nernst-Planck equation with a linear electroosmotic flow term.15
N ) -D
dφ dC zF DC + Cυ dx RT dx
(1)
In eq 1, D, C, and z are the diffusivity, local concentration, and charge of the particles, respectively. The quantities φ and υ are, respectively, the local electric potential and the solution velocity, F is Faraday’s constant, R is the gas constant, and T is temperature. To the best of our knowledge, no analytical solution to eq 1 has been reported for transport of charged particles (or molecules) to a disk-shaped sink (i.e., the pore opening in ECR). However, using dimensionless analysis, it is instructive to obtain order of magnitude estimates of the individual contributions of diffusion, migration, and electroosmotic transport of particles at the pore opening during electrophoresis. Defining a dimensionless length (x′ ) x/rm, where rm is the radius of a pore opening in the membrane), a dimensionless particle concentration (C′ ) C/Cp, where Cp is the bulk particle concentration), and a dimensionless electric potential (φ′ ) φ/φp, where φp is the potential drop between the pore opening and the bulk solution) yields
N ) -D
dC′ Cp zF dφ′ φp DC′C + C′Cpυ dx′ rm RT dx′ rm
(2)
Because dC′/dx′ and dφ′/dx′ are of the order of unity, and since particle capture does deplete particle concentration, i.e., C′ ∼ 1, eq 2 can be expressed in dimensionless form at room temperature: (15) Bath, B. D.; Lee, R. D.; White, H. S. Anal. Chem. 1998, 70, 1047-1058.
N′ ) (-1 - 40zφp + Pe)
(3)
where N′ ) N/[D(dC′/dx′)(Cp/rm)] and Pe is the Peclet number, Pe ) υrp/D. Substitution of numerical estimates of 40zφ° and Pe provides an order of magnitude analysis of the relative contributions of the three transports processes. During ECR with a typical iapp of 100 µA, the total potential drop across the membrane is ∼0.5 V. About 80% of this voltage occurs between the pore opening and bulk solution, as computed previously using finite-element simulations.1 Thus, taking φ° as equal to ∼0.40 V, and employing a charge z ∼ 1260 (based on 3% ionization8b of the ∼42 000 carboxylic surface groups on the PS spheres), the quantity 40zφ° is ∼20 160. An estimation of Pe requires numerical values of the D and υ. The former is computed to be 5 × 10-8 cm2 s-1 from the StokesEinstein relationship,16 D ) kT/6πηa, where a is the particle radius (45 nm) and η is the solution viscosity (0.01 g cm-1 s-1)). The latter is estimated from the Helmholtz-Smoluchowski equation inside a cylindrical pore, ν ) (0ζp/4πη)(φT/lp), where φT is the potential drop (∼0.50 V) across the membrane, lp is the length of the pore (6 µm), is the dielectric constant of the solution (78), 0 is the vacuum permittivity, and ζp is the zeta potential of the polycarbonate membrane (∼ -40 mV).17 Substitution of these values yields Pe ∼ 55. Thus, although highly approximate, this analysis indicates migration is by far the dominant transport process in particle capture. (The estimation of the electroosmotic contribution is likely to be too large, due to the calculation of ν based on a straight cylindrical pore, rather than conical shaped. The experiments below demonstrate that the migrational flux is significantly larger than the electroosmotic flux.) The fact that diffusion is insignificant is consistent with single-particle capture not generating a gradient in particle concentration. Figure 3 shows ECR data for the detection of 90 nm diameter PS particles (1.02 × 1012 particles/mL, pH 7.4). The itip-t traces are plots of the faradaic current at the Pt microelectrode positioned at the backside of a pore, corresponding to oxidation of Fe(CN)64transported through the pore. The applied electrophoretic current for particle capture and release is (50 µA in this example, with the release current held for 10 s. The bottom part of Figure 3 shows an enlarged region of the trace in top of Figure 3. The steplike features in the itip-t response correspond to capture and release of PS particles. At the beginning of each capture and release cycle, the sudden increase in itip when a +iapp is applied corresponds to the electrophoretically driven flux of Fe(CN)64- through the membrane pores. (In Figure 3, the current increases from ∼1.5 to ∼2.5 nA; the nonzero baseline current of ∼1.5 nA corresponds to the diffusion-limited oxidation of Fe(CN)64that accumulates in the receptor compartment during the measurement. The baseline current slowly increases over the course of the experiment and is a minor inconvenience but does not adversely influence data analysis.) At time τ following application of +iapp, a particle is captured at the pore opening, resulting in a sudden drop in itip, a consequence of increased mass transfer resistance at the pore opening when a particle is captured. At time tcapt, the applied (16) Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, NJ, 1983. (17) Kirby, B. J.; Hasselbrink, E. F. Electrophoresis 2004, 25, 203-213.
Figure 3. (Top) Current-time plot of multiple particle capture events with a capture/release current of (50 µA and the release current held for 10 s. (Bottom) Magnification of current-time plot displaying three subsequent capture and release events. trel indicates the release current period (10 s). The data correspond to 90 nm diameter particles at a concentration of 1.02 × 1012 particles/mL in a pH 7.4 solution.
current is reversed (-50 µA) for the period trel to drive the particles away from the pore opening; the pore is immediately unblocked upon application of irel, and itip returns to the original baseline value. The parameter of interest in this measurement is τ, which provides a statistical measure of the concentration of particles in solution. However, as discussed below, the magnitude and duration (trel) of the release current both strongly influence τ and, thus, the measurement sensitivity. From examination of the itip-t traces in Figure 3, it is clear that itip does not decrease to the baseline value expected for a zero Fe(CN)64- flux through the pore upon particle capture but decreases to a value intermediate between the baseline value and the value observed for the open pore at positive electrophoretic current. The intermediate value of itip reflects the fact that complete blockage of the pore would result in a vanishingly small current and electric field at the pore opening, causing the release of the particle. Thus, a stationary state must exist that corresponds to a partially blocked pore that allows sufficient current flow to maintain the electrophoretic force acting on the particle. Because the forces Fe and Feo always opposed one another (vide supra, Figure 2), the observation that particle capture and release occurs, respectively, upon application of positive and negative currents supports the above conclusion based on dimensionless analysis that migrational transport is the dominant transport mechanism and that electroosmotic convective transport of the particles represents a secondary contribution. If particle capture occurred upon application of a negative current, the opposite conclusion would be reached. Figure 4 shows representative itip-t traces for particle capture (particle concentration of 1.02 × 1012 particles/mL) experiments in which trel is varied from 10 to 20 to 30 s, while keeping other experimental variables constant and equal to the values employed in collecting the data in Figure 3 (icapt ) +50 µA, irel ) -50 µA, and tcapt ) 10 s). The corresponding histograms of τ obtained from Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
6337
Figure 4. (Left) Current-time plots of particle capture events with the release current held for periods (trel) of (a) 10, (b) 20, and (c) 30 s. The capture/release currents were (50 µA. The data correspond to 90 nm diameter particles at a concentration of 1.02 × 1012 particles/mL in a pH 7.4 solution. (Right) Corresponding histograms showing the capture time distributions. The average values of τ are (a) 1.1, (b) 2.3, and (c) 6.5 s.
a series of capture and release events are shown to the right of each example itip-t trace. Examination of the histograms clearly demonstrates that τavg increases with increasing trel. This dependence must arise from migration of the charged polystyrene particles away from the pore opening during the release period, depleting the solution of charged particles near the pore opening. Longer release periods apparently create larger depletion regions and consequently larger τavg values. An analogous increase in τavg is observed when irel is increased while holding trel constant (data not shown). The consequence of creating an electrophoretic depletion zone near the pore opening will be discussed later in considering the sensitivity of the ECR method. The key conclusion from these experiments is that single-particle capture rates are a function of the applied waveform. An analysis of the survival time of an open pore was performed on several ECR data sets to better understand the statistical nature of particle capture. Figure 5 shows plots of survival probability of an open pore as a function of time based on traces of multiple capture and release events at a particle concentration of 1.02 × 1012 particles/mL, using applied current waveforms of (Figure 5A) icapt ) +100 µA and irel ) 0 µA, (Figure 5B) icapt ) +50 µΑ and irel ) -50 µA, and (Figure 5C) icapt ) +100 µA/irel ) -100 µA. Survival probability at a specific time in Figure 5 is defined as the probability of the pore remaining open at that time. Capture and release times were held constant at 10 s for all experiments represented in Figure 5. 6338 Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
The plot of survival probability (P(t)) versus t for the case where no release current is applied (irel ) 0), Figure 5A, yields an exponential survival distribution, P(t) ) A e(-λt), where A ) 1 by definition and λ is a first-order rate constant for particle capture. The first-order rate constant should be equal to τavg-1; the solid line in Figure 5A is the fitted exponential curve using λ ) τavg-1. The finding that these data can be described by an exponential distribution indicates that the particles establish a true random distribution in the solution above the pore opening during the release period when irel ) 0.8a As noted above, the effect of applying a finite irel is to electrophoretically drive the particles away from the pore opening during the release period, creating a zone that is depleted of particles. This depletion layer effect is clearly evident in the survival probability analysis for irel ) 50 (Figure 5B) and 100 µA (Figure 5C), where significantly increases in survival times are observed. Gaussian survival distributions are fitted to the survival probability plots of Figure 5, parts B and C, using the mean values and standard deviations of τ from the raw experimental data. These ECR data cannot be described as purely stochastic because of the deterministic dependence of the τavg on irel. The Gaussian distributions describing these data suggest that there is a relatively well-defined depletion zone, the thickness of which is controlled by irel. The width of these Gaussian distributions for finite irel likely reflects the finite random spread in the particle distribution at the depletion zone boundary prior to application of an electrophoretic current, and the random-walk nature of motion as the particle
Figure 6. (Top) iapp vs time illustrating the three-part current waveform used for the corresponding capture below. (Bottom) Current-time plot of particle capture event after a 1 s release current (-100 µA) followed by a 10 s equilibration (release current ) 0). The data correspond to 90 nm diameter particles at a concentration of 1.02 × 1012 particles/mL in a pH 7.4 solution.
Figure 5. Histograms showing survival probability for (A) capture current ) +100 µA followed by equilibration (release current ) 0), (B) capture/release current ) (50 µA, and a (C) capture/release current ) (100 µA. Release time (tr) ) 10 s for all data. Survival probability is defined as the probability of the pore being unblocked after a given time. (A) The line is an exponential regression fit to the data with the form P(t) ) A e(-λt) where A ) 0.94 and λ ) 0.68. The decay rate of 0.68 s-1 is in agreement with the average τ-1 of 0.67 for that data set. Lines in (B) and (C) are normal survival distributions to the respective data sets using experimental values for average capture time (τavg) and the standard deviation. All experiments were performed with a particle concentration of 1.02 × 1012 particles/mL.
drifts, under the influence of the electric field during the capture process. A complete description of τavg and the survival probability plots is complex and beyond the scope of the current research. For instance, τavg is a function also of icapt, decreasing as icapt is increased due to the larger electrophoretic forces acting on the particle during capture. τavg may also be influenced by capture events at neighboring pores, since the opening or closing of neighbor pores influences the current and field at the pore that is being employed for the analysis. A description of the influence of these effects requires the development of relatively complex theory or numerical simulations. However, the results presented above can be used in a qualitative manner to rationally design waveforms for particle analysis that lead to improved method sensitivity. As reasoned above, we note that the particle capture rate at irel ) 0 corresponds to a random distribution of particles near the pore opening. This random distribution presumably reflects the same concentration as in the bulk solution since there are no forces acting on the particles. Thus, τavg, which is a measure of the time to transport a particle to the pore opening, should decrease with increasing Cp. Conversely, when irel is finite, the
particles are depleted from the pore opening, and τavg reflects the time for particle to migrate a distance determined by the depletion layer thickness, which is independent of concentration and controlled by the magnitudes of irel and trel. Thus, τavg is anticipated to be less strongly dependent on Cp for measurements using a finite irel. To test this hypothesis, the hybrid waveform shown in Figure 6 was employed in an ECR analysis. The waveform consists of a 10 s capture period (icapt ) 100 µA), a 1 s release period where a finite irel (-100 µA) is applied, followed by a 10 s equilibration period (irel ) 0 µA) to allow the particle distribution to relax to its random state. The purpose of the 1 s release period where a finite irel is applied is to ensure that the particle is initially removed from the pore opening. An example of the itip-t response trace using this waveform, with a particle concentration of 1.02 × 1012 particles/mL, is shown in the lower portion of Figure 6, using icapt )100 µA and irel ) -100 µA. The only significant difference in the response function for this waveform (relative to prior examples shown in Figures 3 and 4) is the transient tip current demarking the instant when irel is switched from -100 to 0 µA. Figure 7 shows a plot of log τavg-1 versus log Cp (particles/ 3 cm ) for ECR data obtained using the hybrid waveform (the dashed line represents the least-squares fit) and a waveform where the equilibration period is excluded and irel ) -100 µA is applied for 10 s (solid line). Inspection of these plots clearly demonstrates the sensitivity of the ECR method (i.e., ∆(τavg-1)/∆(Cp)) is significantly improved when employing the equilibration period. The slope of the log τavg-1 versus log Cp plot, for the waveform that does not allow for an equilibration period, is ∼0.4 ( 0.1, a lower-order dependence than anticipated based on a true random distribution of particles. (Note: we previously reported a slope of ∼0.1 for the plot of log τavg-1 vs log Cp in the absence of an equilibration period. The larger value in the current investigation is believed to be more accurate.) The reason for a slope less than unity is that during the period of time that irel is applied, the charged particles are depleted from the solution near the pore Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
6339
Figure 7. Log(τavg-1) vs log(particles/mL) for waveforms with and without an iapp ) 0 equilibrium period. The lines indicate least-squares regressions with slopes of 0.37 ( 0.14 and 1.4 ( 0.1.
opening due to migration. Capture rates are thus effectively determined by the thickness of the depleted region, rather than concentration. This conclusion is reinforced by the data in Figure 4 that demonstrate that τavg-1 increases with the length of time that irel is applied. It should be noted that the two lines plotted in Figure 7 intersect at a particle concentration of 1010 particles/mL. Although lower particle concentrations were not examined in this report using the two-part waveform, the previous report on ECR1 suggests that the linear dependence will continue at concentrations e1010 particles/mL. Interestingly, the particle capture rate for the two-part waveform becomes greater than capture rates for the waveform incorporating an equilibrium step. However, the increased capture rate reflects an increased probability of capturing the same particle. At a concentration of 1010 particles/cm3 we estimate an average distance between particles to be on the order of 5 µm (based on 1/[3x(1010 particles/cm3)]). If we assume that the electric field extends ∼10× the radius of the pore,1 we would expect the particle to migrate on the order of 300 nm from the pore opening during irel. When icapt is then applied, this same particle would be in closest proximity to the pore opening, thus resulting in a higher probability of capture relative to other particles in solution.
6340 Analytical Chemistry, Vol. 79, No. 16, August 15, 2007
The slope of the log τavg-1 versus log Cp plot for the waveform with the 10 s equilibration period is ∼1.4 ( 0.1. Thus, τavg-1 ∝ Cp1.4, a higher-order dependence than anticipated based on the initial assumption that the migrational flux is proportional to Cp (eq 1). A possible reason for the 1.4th power dependence is that the electrical field that drives the particle to the pore opening decreases in proportion to the inverse square of the distance from the pore (dp2/d2).1 Thus, the migrational drift velocity of the particles decreases rapidly with distance from the pore opening. Since in a fully equilibrated but less concentrated solution (i.e., low Cp) the particles are, on average, farther away from the pore opening than particles in a more concentrated solution, it is reasonable to anticipate that the average particle velocity during the capture event is dependent on Cp, increasing with increasing Cp. Thus, τavg-1 reflects not only the random distribution of the particles in the solution but also the particle distribution within a highly nonlinear electric field when icapt is applied. Albeit speculative at this time, this dependency, which will require a simultaneous solution of eq 1 written for the charge particles, counterions, and excess electrolyte ions in order to obtain a quantitative description, may be responsible for the observed 1.4th power dependency. ACKNOWLEDGMENT We thank Dr. L. A. Baker and Professor C. R. Martin (University of Florida) for providing the track-etched polycarbonate membranes, Mr. J. R. Wayment and Professor J. M. Harris (University of Utah) for insights concerning the statistical treatment of particle capture rates, and Professor N. Abbott (University Wisconsin) for suggesting using dimensionless analysis to obtain estimates of the flux contributions. This research was supported by the Defense Advanced Research Project Agency (FA9550-06C-000C). Received for review March 27, 2007. Accepted June 5, 2007. AC070610I
