A Variable Pressure Method for Characterizing Nanoparticle Surface

Article pubs.acs.org/ac

A Variable Pressure Method for Characterizing Nanoparticle Surface Charge Using Pore Sensors Robert Vogel,*,† Will Anderson,‡ James Eldridge,§,⊥ Ben Glossop,# and Geoff Willmott§ †

School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD 4072, Australia The Center for Biomarker Research and Development, Australian Institute for Bioengineering and Nanotechnology, The University of Queensland, St. Lucia, QLD 4072, Australia § The MacDiarmid Institute of Advanced Materials and Nanotechnology, Industrial Research Limited, Lower Hutt 5040, New Zealand ⊥ School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6140, New Zealand # Izon Science Ltd., Bishopdale, Christchurch, 8543, New Zealand ‡

S Supporting Information *

ABSTRACT: A novel method using resistive pulse sensors for electrokinetic surface charge measurements of nanoparticles is presented. This method involves recording the particle blockade rate while the pressure applied across a pore sensor is varied. This applied pressure acts in a direction which opposes transport due to the combination of electro-osmosis, electrophoresis, and inherent pressure. The blockade rate reaches a minimum when the velocity of nanoparticles in the vicinity of the pore approaches zero, and the forces on typical nanoparticles are in equilibrium. The pressure applied at this minimum rate can be used to calculate the zeta potential of the nanoparticles. The efficacy of this variable pressure method was demonstrated for a range of carboxylated 200 nm polystyrene nanoparticles with different surface charge densities. Results were of the same order as phase analysis light scattering (PALS) measurements. Unlike PALS results, the sequence of increasing zeta potential for different particle types agreed with conductometric titration.

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calculated from the transit times of the nanoparticles through the pore. Two types of 60 nm carboxylated polystyrene particles with different degrees of surface group densities were investigated. CNCCs were shown to provide single particle analysis of electrophoretic mobility, giving rise to a narrower zeta potential distribution compared to phase analysis light scattering (PALS), the current industry standard for colloidal surface charge measurements. All previous nanopore studies have calculated particle electrophoretic mobility from particle transit times through the pore. Here we report on a novel resistive pulse (blockade) frequency-based approach to measure the electrokinetic surface charge of nanoparticles using variable pressure. Although pressure-driven flow has been used previously in several studies to control particle movement through pores and determine particle concentration,2a,c,d,j,3 it has not been used to measure particle charge. In this paper, electrokinetic surface charges were measured for a range of carboxylated polystyrene nanoparticles by gradually varying pressure until the forces on the nanoparticles due to pressure and electrokinetics were in equilibrium. The

n the last 10 years, nanopore/micropore-based resistive pulse sensors have been shown to be useful tools for sensitive, high throughput particle-by-particle characterization of biomolecules and nanoparticle suspensions.1 These sensors measure the change in electrical resistivity as individual particles pass through a single pore filled with aqueous electrolyte, when an electric potential is applied across the pore. The use of these sensing devices for size and concentration analysis is well established.1a,2 However, quantitative surface charge analysis of nanoparticles has been reported less frequently. Quantitative surface charge analysis, i.e., determination of electrophoretic mobility and zeta potential, is an integral part of nanoparticle analysis and sheds light on the forces that stabilize colloidal suspensions. DeBlois et al.2c were the first to report electrokinetic measurements using resistive pulse sensing. They used cylindrical nanopores made in polycarbonate to simultaneously measure the size and electrophoretic mobility of latex spheres (91 nm in diameter) and a range of viruses. Particle velocities were extracted from the transit times through the pore and the respective electrophoretic mobilities calculated. Ito et al.1a,2f,i reported on the use of carbon nanotube-based coulter counters (CNCCs) for the simultaneous determination of size and surface charge of carboxylated polystyrene nanoparticles. Similar to DeBlois et al.,2c electrophoretic mobilities were © 2012 American Chemical Society

Received: November 21, 2011 Accepted: February 27, 2012 Published: February 27, 2012 3125

dx.doi.org/10.1021/ac2030915 | Anal. Chem. 2012, 84, 3125−3131

Analytical Chemistry

Article

point of minimum blockade frequency represents force equilibrium and gives an indication of the electrokinetic surface charge of these particles. Although this simple methodology has only been demonstrated with an elastomeric tunable pore sensor, it can also be applied to any fixed pore sensor with pressure control.

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MATERIALS AND METHODS Polystyrene Particles. Carboxylated polystyrene particle standards with nominal diameters of approximately 200−220 nm were purchased from Bangs Laboratories and Polysciences. Uncarboxylated polystyrene particles (NIST traceable size standards) with nominal diameters of 200 and 4000 nm were purchased from Polysciences. The 200−220 nm carboxylated particles are denoted as B121, B86, B47, and PSCOOH, respectively. The 200 and 4000 nm uncarboxylated NIST traceable particles are denoted as PS and PS4000, respectively. The specific surface charges of B121, B86, and B47 as determined by the manufacturer using titration were 120.8, 86, and 47 μequiv/g (with equiv being the equivalent moles of titrant per gram of sample required to reach the equivalence point) and their nominal diameters 200, 220, and 220 nm, respectively. Electrolytes and Buffers. Electrolyte consisting of 0.1 M KCl, 15 mM Tris buffer, 0.01% v/v Triton X-100, 3 mM EDTA, and HCl to adjust to pH 8, was used in all nanopore experiments. Nanoparticles were immersed in electrolyte at concentrations of approximately 109−1010 particles/mL. Phase Analysis Light Scattering (PALS). Zeta potentials of all 200−220 nm polystyrene particles were measured on a Malvern Zetasizer Nano ZS. PALS analysis was used to determine the average zeta potential of the different nanoparticle types dispersed in Tris-buffered 0.1 M KCl electrolyte. Data were obtained from 15 measurement cycles and repeated twice. The ionic strength of the dispersing media (electrolyte) was high, so the instrument selected a fast measurement process (fast field reversal mode: FFR), in order to prolong the life of the measurement cell. Control measurements for PSCOOH were carried out on another Malvern Zetasizer Nano ZS instrument by a different user and also repeated twice. qNano and Variable Pressure System. Tunable nanopores, which are fabricated in thermoplastic polyurethane membranes, were used in combination with IZON proprietary software to record current pulse signals, as detailed by Willmott et al.2a and Vogel et al.7 Briefly, a flexible polyurethane membrane containing a single conical nanopore was placed between top and bottom fluid cells, each containing a measurement electrode, as shown in Figure 1b. By applying a constant potential across the pore (typically less than 1 V), the translocation of particles through the pore can be detected as a brief change in the background current (details concerning event detection are discussed in the Supporting Information). Changing the distance between the geared jaws connected to the four arms of the membrane, the geometry of the pore can be tuned in real time to better suit different particle sizes. The dimensions of the pore openings and membrane thickness were determined using SEM (see Supporting Information Figure S3) and a digital micrometer, respectively. Experiments using 4 μm particles (PS 4000) were conducted at a voltage of 0.02 V with particles only in the top fluid cell, while measurements for 200−220 nm polystyrene particles were carried out at a voltage of 0.3 V with equal concentration of particles in both fluid cells. Two different membranes were used for measuring the 4 μm

Figure 1. Schematic of the variable pressure set up used with the qNano pore sensor (a). Pressure in the top fluid cell (b) is precisely controlled via a flexible tubing connection (c) by varying the height difference between the water level in a partially submerged buret (d) and the water level of a large water reservoir (e). The buret was equilibrated with atmospheric pressure by opening a valve (f). Progression of an experiment where pressure is varied from positive to negative (g).

and the 200−220 nm particles. The stretch of the membrane and hence the pore dimensions were kept constant during all measurements of the 220−220 nm polystyrene particles. The pressure applied across the membrane was precisely regulated by a simple manometer arrangement (Figure 1). In a typical experiment, the valve (Figure 1f) was closed, sealing the buret and tubing, following equilibration of the fluid cell with atmospheric pressure. In this state, the pressure across the pore was that of the gravitational pressure head (Figure 1b). The pressure in the top fluid cell was then reduced by gradually releasing water at a constant flow rate from the beaker, so that the height of fluid in the buret was greater than that in the beaker, with the difference in heights creating an applied pressure. Applied pressures were known precisely, because the weight of fluid displaced for a given change in height of the water in the main beaker relative to that in the buret was calibrated. Due to the lowering of the water level in the beaker, the rate at which the fluid is displaced decreases slowly with time. This small deviation from the linear relationship between pressure and time was considered when calibrating the system (see Supporting Information). Electro-osmosis Measurement. Electro-osmotic flow measurements were made using the current monitoring method.4 Rectangular microchannels (dimensions 0.022 × 0.1 × 30 mm) joining 2 mm diameter wells were fabricated from the same thermoplastic polyurethane used to make tunable elastomeric nanopores (Ellastollan 1160D, BASF) by hot embossing. Channels were initially filled with electrolyte to replicate the pore-sensing experiments (Tris-buffered 100 mM KCl, adjusted to pH 8 by adding concd HCl). With 500 V applied via platinum electrodes and a steady current established, the electrolyte in one well was replaced with electrolyte at a slightly lower concentration (98 mM). Electroosmotic flow rates were measured by monitoring the linear current change as the high concentration electrolyte was displaced. Channels were wetted with methanol and washed with DI water before and after each measurement. 3126



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RESULTS AND DISCUSSION Theoretical Basis. We use measurements of blockade rate to derive the zeta potential of nanoparticles. Particle transport can be summarized using the Nernst−Planck approach,5 (ζ particle − ζ pore) Q J =ε E+ P C η A

membrane surfaces is taken to be half the pressure drop across a thin circular orifice.12 The same assumption contributes negligible error to the pressure drop between a half space and the end of a cylindrical tube of finite13 or infinite12b length. P = Pin + Pends ⎞ ⎛ d ⎛1 8η 1 ⎞⎟⎟ ⎜⎜ d z 1.5 = Q P ⎜⎜ + η + ⎟ 3 z 0 πλ(z)4 lh3 ⎠⎟⎠ ⎝ sh ⎝

(1)

∫

where J is the total particle flux, C is particle concentration, ε and η are the permittivity and kinematic viscosity of the electrolyte, respectively, ζ indicates zeta potential, E is the electric field, QP is the pressure-driven volumetric flow rate, and A is the cross-sectional area of the pore constriction. The equation includes contributions due to electrophoresis6 (term including ζparticle), electro-osmosis6 (term including ζpore), and hydrodynamic transport7 (pressure). Diffusion has been omitted, because its contribution is negligible compared to the other terms.2a Electro-osmotic and electrophoretic contributions are estimated using the Smoluchowski equation.8 This is a valid approximation for a monovalent electrolyte concentration of 0.1 M and particle sizes of approximately 200 nm, where the product of particle radius and inverse Debyelength is much larger than 1.8 Calculation of zeta potential is achieved by setting J = 0 in eq 1, so that the applied pressure exactly opposes the combined effect of electro-osmosis, electrophoresis, and inherent pressure. Use of eq 1 requires measurement of ζpore and determination of the inherent pressure head. Of the other terms, ε and η are known under standard laboratory conditions,9 while QP, A, and E were calculated using a conical profile based on SEM imaging of the pore (not shown) under the same stretch conditions used in the experiments. Effective radii s and l of the small and large pore openings were used to calculate the electric field, while hydrodynamic radii sh and lh were used to derive pressure variation. Effective radii were calculated, such that the area of circles of radii s and l is the same as the actual area of the openings, while hydrodynamic radii were determined by dividing area by half the pore perimeter.10 The pore wall zeta potential was determined by measuring electro-osmosis in a well-defined channel (see Materials and Methods). The electric field within the conical pore was calculated by deriving the pore resistance, including access resistance between the pore openings and the electrodes,11 Ez(z) = −I0

V ⎛ ρ ⎞ dR ⎟⎟ = 0 ⎜⎜ R 0 ⎝ πλ(z)2 ⎠ dz

⎞ ⎛ ⎞ ⎛ ⎟ ⎜ 8ηd⎜ 1 − 1 ⎟ 3 3 ⎛ ⎞ lh ⎠ ⎟ ⎜ ⎝ sh 1 1 = QP⎜ + 1.5η⎜⎜ 3 + 3 ⎟⎟⎟ z 3π(l − s ) l h ⎠⎟ ⎝ sh h h ⎜ ⎟ ⎜ ⎠ ⎝

Calculations of both E and QP assume that transport is dominated by the vector component parallel to the pore axis z, which is appropriate for a pore with width that varies slowly with length, as is the case for the tunable elastomeric nanopores used in this study. Pore Wall Zeta Potential and Electro-osmosis. We work in high-salt regime, where the Debye length does not exceed 3 nm, so electro-osmotic transport is described by plug flow.14 The velocity of the electro-osmotic flow can be calculated by applying the Helmholtz−Smoluchowski equation,2c,8,15 veo =

(2)

l−s z d

and R0 =

−εζ poreE η

(4)

The description of electro-osmotic flow rate through the conical nanopore assumes that the electro-osmotic velocity veo is proportional to the local electric field E, an appropriate assumption for present channels and 0.1 M KCl electrolyte.16 Zeta potentials of pore walls were calculated by evaluating electro-osmotic flow within microchannels and using eq 4. A value of ζpore = −8 ± 3 mV was obtained for the specific electrolyte (Tris-buffered 0.1 M KCl) and elastomeric material used in nanopore experiments. Uncertainty in the quoted value (37.5%) is based on numerous measurements using different channels of the same material. The pressure which produces an equivalent volumetric flow rate to the electro-osmotic flow (denoted as electro-osmotic pressure) was calculated to be 43 ± 27 Pa (using eqs 2−4 and considering that the fluid flow due to electro-osmosis is πsh2veo). Inherent Pressure. The inherent pressure across the pore is the pressure due to gravitational head and meniscus effects when the system is equilibrated with atmospheric pressure, so that no pressure is externally applied via the variable pressure system (Figure 1g(ii)). We make a clear distinction between inherent pressure and externally applied pressure, which is due to height difference of water levels in the buret and the beaker (Figure 1g). The sum of both is denoted as ‘net pressure’ which gives rise to the total volumetric flow rate QP. A precise knowledge of inherent pressure is important for measurement of the zeta potential, particularly in vertical fluid cell arrangements where the inherent pressure can be significant. The inherent pressure in the top fluid cell was determined using the variable pressure method described above. In this case, a large pore (small pore opening diameter of approximately 10 μm) was used with 4 μm uncarboxylated polystyrene particles (diluted to 0.1 wt % in Tris-buffered 0.1

where

λ (z ) = l −

(3)

ρ(d + 0.8(s + l)) πsl

Ez, I0, V0, ρ, and R0 are the electric field component along the pore axis, electric current, voltage, resistivity of the electrolyte, and resistance of the electrolyte-filled pore, respectively. s, l, and d are the pore dimensions with d being the pore length. The pressure-driven volumetric flow rate parallel to the central axis (QPz) is calculated by considering the flow resistance through the pore. The contribution beyond the 3127



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Figure 2. A representative trace from an approximately 11 min long variable pressure experiment used to determine the inherent pressure head in the top fluid cell. From left to right the insets show single particle pulse traces initially traveling from the small to large pore openings when the pressure head is large, slowing as the net pressure approaches zero, and then the direction of the particle translocation reversing as the net pressure becomes negative in the top fluid cell.

M KCl), to ensure that particle transport was dominated by pressure, i.e., electrokinetic forces were negligible. The inherent pressure was measured by increasing the negative pressure until particle translocations stopped. A trace from one of the replicates of this experiment is presented in Figure 2. The insets depict representative single particle blockade events at different times during the experiment, all displayed over a 0.3 s time interval. As the pressure in the top fluid cell was reduced, thereby reducing average particle velocity, the duration times of blockade events increased. When the net pressure in the top fluid cell became negative, fluid and particles began to flow from the bottom to the top fluid cell. This change in particle flow direction is indicated by a change in the shape of the blockade event, which is a result of the resistance gradient in the pore due to its conical shape.2h,11 When approaching the point of zero net pressure, the blockade frequency is near zero and the sparse blockade events are up to 100 times longer in duration than blockade events with no externally applied pressure. The minimum in blockade frequency goes hand-inhand with an increase in background current. A significant increase was only observed for large pores and can vary significantly between similarly sized pores, with the increase ranging between 3 and 40 nA. It is thought to be due to the changing pore shape when reversing the pressure. Further studies are in progress to elucidate the causes of this effect. The pressure in the top fluid cell for a 40 μL sample was measured at 46 ± 3 Pa. Previous studies of nanoparticle surface charge using nanopores1a,2c,f,i have not accounted for inherent pressure. Although this pressure is relatively small, it can overcome electrophoretic forces on nanoparticles which carry a low surface charge,2a and can play a significant role in both horizontal and vertical sample cell arrangements. Fluid flow rates through nanopores are negligible, so pressure differences across a pore do not equalize over practical time scales. Therefore, the pressure difference between both sample cells needs to be either measured or equalized to enable an accurate measurement of particle zeta potential. Having determined the inherent pressure in the fluid cell, it was possible to accurately measure the electrokinetic forces on nanoparticles using the variable pressure method with smaller pores. Variable Pressure Measurements and Determination of Inflection Point. Figure 3 shows the cumulative counts versus applied pressure for a range of 200 nm polystyrene particles (PS, B47, B86, B121, and PSCOOH). When applied

Figure 3. S-curves for various polystyrene particles, acquired using the variable pressure method fitted with cubic (PS, B47, B86, B121) and parabolic (PSCOOH) curves.

pressure is gradually varied from atmospheric to negative pressure (see Figure 1g), the cumulative counts describe an Scurve for each particle type. Because blockade frequency (counts/time) scales linearly with pressure,2c the two halves of the S-curves are expected to be parabolic, with one branch having negative and the other positive curvature. The parabolic dependency of accumulated counts on pressure is demonstrated in eq 5 with P being the applied pressure, a, b, and c being constants which depend on particle type and concentration, and BF being blockade frequency (i.e., counts/ time). counts = a + bP + cP 2

BF =

d(counts) dP = (b + 2cP) dt dt

(5)

The pressure at the inflection point of each S-curve allows calculation of the particle electrokinetic surface charge. The gradient is expected to be zero at this inflection point. However, peculiarities such as nonthrough and multiple blockade events (Figure 4 a,b) result in a nonzero blockade frequency. Hence, the inflection points of S-curves were initially determined using third-order polynomial fits to the exper3128



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Figure 4. Traces from high surface charge B121 (a) and low surface charge PS (b) variable pressure experiments. Indicative traces for other particle types are shown in Figure S1 of the Supporting Information. The current baseline is shifted to 0 nA to remove baseline drift and for simplified event analysis. Insets show blockade events at different times in the pressure titration experiments. Parts a(i′) and b(i′) show regular blockade events for both the high and low charged particle types under pressure. Parts a(ii′) and b(ii′) show representative blockade events when the forces on the particles due to pressure and electrokinetics are in equilibrium.

because the zeta potential of nanoparticles scales linearly with surface charge density.8 Surface charge densities were calculated by converting the specific charges 121, 86, and 47 μequiv/g of B121, B86, and B47, respectively (as given by Bangs

imental data. Both branches of the S-curve were then fitted with parabolas, omitting the data within P = 50 Pa (∼0.5 cm H2O) of the initial inflection point (see fit to PSCOOH in Figure 3). This approach was sufficient to exclude all atypical events. From the parabolic fits, the inflection point pressures for the different particle types were equal to −b/2c (eq 5). Averaging triplicate measurements, the calculated externally applied pressures at the inflection point were 76 ± 8 Pa, 132 ± 9 Pa, 155 ± 10 Pa, 213 ± 24 Pa, and 132 ± 12 Pa for PS, B47, B86, B121, and PSCOOH, respectively. Inflection point pressures are independent of particle concentration, because b and c scale linearly with concentration. Interestingly, along with longer duration times, blockade events close to the inflection point typically had very low magnitudes (Figure 4b(ii′)). Low magnitude signals are thought to be due to particles entering the pore but not traversing it. Similar events were recently described in detail by Bacri et al.17 and could be due to Brownian motion playing a significant role in the movement of low charge particles. For B121 particles, both low magnitude events and multiple events are observed. Multiple events are thought to be due to particles being trapped in the pore constriction for up to 40 ms while moving backward and forward. The example in Figure 4a(ii′) shows a particle apparently entering the pore and becoming trapped before translocating under electrokinetic forces. Figure 5 shows that for a variety of polystyrene particles the relationship between the inflection point pressure and surface charge density is linear. This linear dependence is expected,

Figure 5. Dependence of the inflection point pressure (cm H2O) on surface charge density. B47, B86, and B121 were fitted by a straight line. Nonzero surface charge of uncarboxylated PS particles is indicated by the arrow. The dashed horizontal line at −3 Pa is the sum of electro-osmotic transport and inherent pressure. 3129



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Laboratories using conductometric titration) into C/m 2 through multiplication by the Faraday constant. Weight was converted into surface area by considering nominal particle radii and assuming a density of polystyrene of 1.06 g/cm3. Calculated surface charge densities correspond to surface areas per carboxylate group of 0.91 nm2, 0.50 nm2, and 0.39 nm2 for B47, B86, and B121, respectively. Uncarboxylated PS particles have an unknown, nonzero surface charge due to sulfate surface functionality. The sum of electro-osmotic and inherent pressure (dashed line in Figure 5) is subtracted from the inflection point pressure to obtain the total pressure (‘total inflection pressure’) required to oppose electrophoresis and extract the zeta potential of the polystyrene particles. It is expected that the total inflection pressure required to oppose electrophoresis scales with the inverse of blockade duration. Therefore, the product of duration and total inflection pressure for various particle types should be constant. In Table 1, pressures are compared

Table 2. Comparison of Zeta Potentials Measured Using the Variable Pressure Method, PALS, and Surface Charge Densities Determined by Acid/Base Titrationa

B121 B86 B47 PSCOOH PS a

fwhm [ms] 0.213 0.264 0.282 0.298 0.622

(0.041) (0.070) (0.048) (0.052) (0.128)

inflection pressures [Pa] 210 (24) 152 (10) 129 (9) 129 (9) 73 (8)

ζparticle (PALS) [mV]b

surface charge density (C/m2)

B121 B86 B47 PSCOOH PS

−39 (13) −28 (9) −24 (8) −24 (8) −14 (5)

−26.7 −26.8 −23.0 −27.8 −17.4

−0.409 −0.320 −0.175 −c −c

effective radius, so the percentage error is three times the measurement uncertainty in the pore opening radii. The zeta potentials measured using PALS are very similar to those obtained from the variable pressure method (Table 2). The variable pressure method suggests that surface charge increases in the sequence PS, PSCOOH, B47, B86, and then B121. In contrast, PALS suggests a different order of increasing charge, being PS, B47, B121, B86, and then PSCOOH. Titration results provided by Bangs for B47, B86, and B121 (Table 2) support the results obtained using the variable pressure method.

fwhm × pressure [ms·Pa] 45 40 36 38 45

ζparticle (variable pressure) [mV]

a Figures in parentheses represent the measurement uncertainty. bZeta potentials were averaged over two measurements, and repeats agreed to within 3%. Control measurements for PSCOOH on another Zetasizer instrument were −29.7 mV, within 7% of the recorded values. cNot provided by the manufacturer.

Table 1. For Different Particle Types, the Mean FWHM Duration for Events with Dominant Electrophoresis Is Listed, along with Inflection Pressures and the Product of These Two Measuresa sample

sample

(14) (13) (9) (10) (14)

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CONCLUSIONS A novel and highly accurate method for measuring nanoparticle zeta potentials using pore sensors has been demonstrated. This resistive pulse method is based on measuring the blockade rate while varying the applied pressure. With precise control of applied pressure, this method is an extremely sensitive way of analyzing particle surface charge. Zeta potentials of a range of unmodified and carboxylated 200 nm polystyrene particles were measured and showed a linear relationship with the nominal surface charge densities, as expected according to zeta potential theory.

Numbers in parentheses represent the standard deviation.

with the blockade durations extracted near the point where applied pressure is equal to the sum of inherent and electroosmotic pressures, so that electrophoresis is the sole driving force of particles through the pore. For traces displayed in Figure 4, this equilibrium point was reached after approximately 5 s. Blockades in the first 10−12 s of each run were evaluated and full width half-maximum (fwhm) durations extracted. The durations of at least 45 blockades were averaged for each sample. The final column of Table 1 shows a reasonably constant value, as expected. Zeta Potential of Nanoparticles. The tunable elastomeric nanopores used in our experiments have been previously characterized and are conical in shape.2e,18 From analysis of the SEM images, s, l, sh, and lh were 470 ± 70 nm, 15.3 ± 2.3 μm, 460 ± 70 nm, and 13.6 ± 2.0 μm, respectively. The membrane thickness d at the strain used was 260 ± 15 μm. Having determined the zeta potential of the pore, the inherent pressure, and the applied pressure required to minimize the event rate, we used the Nernst−Planck equation (eq 1) and a simple geometric model, customized for these pores (eqs 2 and 3),11 to derive the particle zeta potential of the particles (Table 2). The quoted error of ζ is dominated by uncertainty in the geometric dimensions of the pore, because pressure-driven flow is highly dependent on pore geometry. We accounted for 15% random uncertainty in both cone opening radii. This very conservative estimate covers factors such as nonconical shape, calculation of 'equivalent’ radius (i.e., not circular), matching of stretch states, and viscoelasticity. As a first approximation, this calculation depends on the cube of the

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ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +61 7 3365 3463. Fax: +61 7 3365 1242. E-mail: vogel@ physics.uq.edu.au. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Rod Stanley (Industrial Research Ltd) for microchannel fabrication, Linda Groenewegen (IZON Science Ltd) for SEM imaging, Nadiah Ali (IZON Science Ltd.) for doing some supporting experiments, Darby Kozak (The University of Queensland) for his suggestions, and Bryn Mc Donagh (ATA Scientific) for his help and advice. Support was provided by New Zealand’s Ministry for Science and Innovation (NERF contract C08X0806). 3130



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W.; Broom, M. F.; Vogel, R.; Trau, M. Small 2010, 6 (23), 2653− 2658.

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