Detection of polystyrene sphere translocations using resizable elastomeric nanopores
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Industrial Research Limited Report No. 2385, December 2009
Detection of polystyrene sphere translocations using resizableelastomeric nanopores
Geoff R. Willmott : and Lara H. Bauerfeind Industrial Research Limited, 69 Gracefield Rd, PO Box 31-310, Lower Hutt 5040, New Zealand : Corresponding authorEmail: [email protected]: (64) (0)4 931 3220Fax: (64) (0)4 931 3117
Resizable elastomeric nanopores have been used to measure pulses of ionic current caused by car-boxylated polystyrene spheres of diameter 200 nm and 800 nm. The nanopores represent a noveltechnology which enables nanoscale resizing of a pore by macroscopic actuation of an elastomericmembrane. Three different pores were employed with variable applied strain, transmembrane poten-tial, particle concentration and sphere radius. Theory describing current pulse magnitude has beenextended to conical pore geometry. A consistent method for interpretation of data close to the noisethreshold has been introduced, and experimental data has been used to compare several methods forefficient, non-destructive calculation of pore dimensions. The most effective models emphasize theabsolute pulse size, which is predominantly determined by the opening radius at the narrowest partof the roughly conical pores, rather than the profile along the entire pore length. Experiments werecarried out in a regime for which both electro-osmotic and electrophoretic transport are significant. Introduction
Resizable elastomeric nanopores represent one of the most interesting new technologies in the burgeoningfield of nanopore science. Individual nanopores in thin membranes are attracting interest due to theirapplication to sensing of single molecules or small particles, their simplicity, and the growing capabilityin related fabrication and characterisation techniques. Resizable nanopores are each fabricated in anelastomeric membrane, so that the nanoscale dimensions of the pore can be altered by stretching andrelaxing the membrane on macroscopic scales. An initial study demonstrated that individual double-stranded DNA molecules could be gated using this mechanism [1]. More recent work has includedsystematic ionic current measurements and analysis of pore actuation [2, 3], efforts to characterize thepores using AFM and SEM [2, 4], and an initial compilation of theory, experiments and ideas relatingto resizable nanopores [5]. Resizable nanopore technology is being uniquely developed by Izon Science(Christchurch, New Zealand), who supplied specimens, analysis software and the qNano ™ actuationplatform for the present work.The fundamental, nanoscale functionality of resizable nanopores lends itself to a wide range of po-tential applications. Entirely new processes could arise from this concept, such as mechanical trappingof small particles or molecules, controlled mechanical tuning at the nanoscale, or localisation and con-finement of reaction chemistry. However, the most immediate interest has been generated by the newfunctionality added to applications which have been studied using conventional (static) pores, such astranslocation of particles [6–10]. Work towards fast genomic sequencing, perhaps the most exciting po-tential application of nanopores [11–16], has arisen from translocation experiments using nucleic acids[17]. With a resizable pore, translocation rates can be controlled, particles can be gated, and the physicsof translocation can be studied in novel ways. For translocation measurements, nanopores are filledand surrounded by an aqueous electrolyte, allowing measurement of electric current when a potential isapplied across the membrane.Resizable nanopores have further practical advantages over static pores, which have employed arange of membrane materials and nanopore fabrication methods [6–8, 18, 19]. The most widely-usedconventional pores are ‘solid-state’ nanopores fabricated in thin, rigid, silicon-based [6] or polymer [7, 8]2embranes and ‘biological’ pores [20], especially the α -haemolysin pore [21] supported by a phospholipidmembrane [6, 14, 15, 22]. In comparison with such pores, resizable nanopores are less likely to irreversiblyclog, as the elastomer can be stretched to allow the pore to clear. The elastomer is light and the effectivepore size for a single specimen can be varied over approximately one order of magnitude [2], enablingit to perform the tasks of several static pores. The fabrication method is quick, simple and relativelyinexpensive [1, 5]. Elastomeric pores are chemically, mechanically and thermally stable in comparisonwith biological pores, which must be carefully supported under laboratory conditions. The major presentdrawback of solid state pores, including resizable nanopores, is poor reproducibility of geometry below (cid:18)
50 nm diameter [23]. It is hoped that studies such as the present one will pave the way for themanufacture, characterization and widespread use of reliable, molecular-scale resizable pores.In this article, we present experiments in which polystyrene nanospheres have been detected usingresizable nanopores. The work demonstrates that this technology can be used for translocation experi-ments, in a manner similar to static nanopores. The experiments build on initial work [5] by obtainingdata using various nanopore specimens, applied strains, applied potentials, particle sizes and particleconcentrations. The investigation is the first to consider the relationship between current signals, noise,particle size, particle type and pore geometry with relation to resizable pores. We address the fact thatparticle-induced current signals may not always indicate passage of the particle from one side of themembrane to the other. An important feature of this work is a direct comparison of existing theoreti-cal models for relating the resistance pulse size to pore dimensions [9, 23–28], including a conical-poreapproach developed in the theoretical background. An efficient, non-destructive model for determiningpore size would be valuable for many envisaged applications of this technology. Particle transport is alsospecifically addressed, both in experiments and the identification of relevant aspects of the Space-Chargemodel (e.g. [24, 29]) in the theoretical background.Resizable nanopore specimens are currently available for sensing particles in the hundreds of nanome-tres range; the size of the polystyrene particles used here (200 and 800 nm diameter) indicates the sizeof the narrowest part of the pore. This size range is relevant to many viruses of interest in human andveterinary medicine, agriculture and environmental studies. Many of the recent developments in virus3etection technology have required fluorescent molecular tagging of the virus. The most relevant previ-ous work includes recent nanopore studies using particles of similar size and material [25, 27, 30–32] aswell as earlier work relating to the development of Coulter counters [23, 26, 31, 33, 34], which includeddetection of particles as small as 60 nm [34]. The membranes used in experiments are relatively thick(200 µ m when unstretched) in comparison with other studies. Pores are conical [2, 4, 5], and somecurrent rectification, as studied in some depth elsewhere [35–40], has been observed previously [2]. Thepores used do not close entirely when the membrane is relaxed. In this Section, theory relating to two major topics is presented. Firstly, we recount descriptions oftransport and apply them in the context of the present work. The approaches used have been widelyapplied to both ions [30, 31, 41–43] and particles [8, 10, 23, 25, 30, 31] in experimental nanopore workelsewhere. Secondly, we present theory developed by deBlois and Bean [26] and widely used in recentstudies [9, 23–25, 27, 28] to describe the size of a resistive pulse generated by a spherical particle passingthrough a nanopore. This theory is reviewed, so that assumptions can be tested by the experimentalwork, and we explicitly extend the existing theory to conical geometry.
Several papers [24, 29, 30, 38, 43] summarise theoretical approaches that have been used to studytransport of ions and particles in pores, ranging from biological ion channels to micron-sized syntheticpores. We employ a simple approach [44] devised for assessing the relative contribution of varioustransport mechanisms, and their functional dependencies, for experimental work. This approach is asimplification of the comprehensive, but computationally expensive Space-Charge model [24, 29].Transport of aqueous ions or larger charged particles is described by the Nernst-Planck equation. Theparticle flux J is given in terms of diffusive J diff , electrophoretic J ep and convective J conv contributions,by 4 (cid:16) J diff (cid:0) J ep (cid:0) J conv (cid:16) (cid:1) D ∇ C (cid:0) ξek B T DC E (cid:0) C v . (1)Here, D is the diffusion coefficient, C is concentration, k B is Boltzmann’s constant, T is temperature, E isthe electric field and v is the convective flow velocity. The electronic charge magnitude e is multiplied by ξ to find the total effective charge on the ion or particle; ξ (cid:16) (cid:1) R (cid:16) k B N A and the Faraday constant F (cid:16) N A e , where N A is Avogadro’snumber. At 293 K, a dimensionless ratio for determining the significance of the diffusion term is [44], (cid:12)(cid:12)(cid:12)(cid:12) J diff J ep (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) . ξV . (2)Generally, it is expected that | ξ | Á
100 for carboxylated polystyrene nanospheres of O (100) nm diameter,in which case the ratio in Eq. 2 is
5% for any potential above (cid:18) .
01 V. Therefore, diffusion shouldnot significantly contribute to particle transport in the present work. Electrolyte concentration is thesame on both sides of the pore, so ionic diffusion is also not important.In the absence of other external forces, such as a pressure gradient, electro-osmosis is the solemechanism for convective transport. Referring to the geometry defined in Fig. 1, and using the simplifiedmodel of a long, thin cylindrical pore ( a (cid:16) b (cid:16) a , l " a ) to model the electro-osmotic flux ( J z (cid:16) | J | ),we find [44, 45] J z,eo (cid:16) (cid:1) ǫψ πη CE z (cid:2) (cid:1) I p κa q κaI p κa q (cid:10)(cid:16) (cid:1) ǫψ πη E z A. (3)Here, η is the dynamic viscosity of the fluid, ǫ is the dielectric constant, ψ is the potential of the porewall and κ (cid:1) is the characteristic thickness of the electrical double layer at the pore wall, and I n is the n th-order modified Bessel function of the first kind. The subscript eo refers to electro-osmotic flow. Thevalue of A increases asymptotically towards 1, and is greater than 0.9 for a Á
20 nm when using 0.1 MKCl ( κ (cid:1) (cid:16) .
62 x 10 (cid:1) m (cid:1) [46]). The ratio of electro-osmotic to electrophoretic particle fluxes is5IG. 1. A schematic cross-section through a truncated circular conical pore, showing geometrical quan-tities used in the theoretical analyses. Cone entry radii are a and b and the pore length (equal to themembrane thickness) is l . A translocating sphere of radius a is shown in the conical passage and acylindrical polar co-ordinate system is defined. J z,eo J z,ep (cid:16) (cid:1) a ǫψ ξe A, (4)where the particle is assumed to be a sphere of hydrodynamic radius a . For water at 293 K, this ratiois (cid:1) . A , when calculated using values of a
100 nm, ψ (cid:16) (cid:1)
75 mV and ξ (cid:16) (cid:1) ψ (cid:16) (cid:1)
75 mV is equivalent to surface charge density of 0 . e nm (cid:1) in 0.1 M KCl, consistentwith literature values for track-etched polymers [35, 37, 38]. Note that J z,ep (cid:0) J z,eo (cid:16) CE z (cid:2) ξe πηa ǫψ πη A (cid:10) , (5)so both contributions are linearly dependent on particle concentration and potential difference.A corresponding comparison of electro-osmotic and electrophoretic contributions to ionic currentcarried by ions [44, 45] reveals that electrophoresis dominates in the experimental regime considered6ere. If the second (electrophoretic) term dominates in Eq. 1, pore size can be estimated from ioniccurrent measurements by treating the pore as an isotropic, homogeneous conical conductor with bulkconductance ρ equal to that of the electrolyte (see Eqs. 6 and 11 below). This approach has been widelyused [9, 25, 26, 28, 30, 32], including with conical pores [2, 7, 8, 47].Current rectification has been observed using resizable nanopores, and should be studied further,although the polarity of applied potential was not varied during the present experiments. Studies haveshown [35–40] that rectification is determined by asymmetric geometry, non-uniform surface chargedistributions and ionic concentration gradients. In previous work, the current anisotropy observed usingelastomeric nanopores [2] was at most 25% for a transmembrane potential of (cid:8)
200 mV.
When a particle travels through a nanopore at constant applied potential, a momentary change inpore resistance is observed - a resistance ‘pulse’ caused by a translocation event. The characteristicsof this pulse yield information on the size, shape and nature of the particle. Qualitatively, it is easyto understand that an insulating polystyrene sphere will ‘block’ the pore, increasing its resistance, asit passes through the membrane; that a smaller sphere will produce a smaller increase in resistance;and that the resistance increase is greatest when the particle passes through the narrowest part of thepore. Nearly forty years ago, deBlois and Bean [26] used classical electrostatics to derive quantitativeexpressions for pulse magnitudes that have been widely applied in recent studies. In this section,approaches detailed by these researchers and others are recounted, and in some cases extended toincorporate a linear conical geometry which is relevant in the present work. It is of interest to comparethe various models using experimental data in order to probe the functional relation between particlesize, pore size and pore geometry.DeBlois and Bean’s approaches are approximations, each of which assumes that the particle is spher-ical and that the surfaces of the membrane, pore and translocating sphere are uncharged. The fullsolution of Laplace’s equation, which is required in order to solve the classical electrostatic problem, isnot even available for the simple geometry of an insulating sphere within a conducting tube (Fig. 1).7oreover, a real system introduces additional complications such as non-cylindrical pore geometry,solution and surface chemistry effects and aspherical particles.
Recent studies of spherical proteins [9, 28] and other particles [9, 23–25, 27] have based their interpreta-tion of translocation pulses on the simplest of the approaches presented in [26], which assumes that thesphere radius a is small in comparison with the cylindrical pore radius a . For the geometry defined inFig. 1, we initially consider a pore which is cylindrical ( a (cid:16) b (cid:16) a ) and very long ( l " a ). In this case,the electrical resistance of a pore filled with electrolyte of resistivity ρ is R cyl (cid:16) ρlπa . (6)Maxwell’s expression for the resistivity of a solution containing insulating spheres ρ eff with volumefraction f is ρ eff (cid:16) ρ (cid:2) (cid:0) f (cid:0) ... (cid:10) , (7)where f is just the ratio of the single particle volume to the pore volume, f cyl (cid:16) a a l . (8)The absolute change in resistance during a translocation event, or the amplitude of the resistance pulse,is ∆ R cyl (cid:16) R cyl (cid:1) R cyl (cid:16) ρa πa , (9)where R cyl is the equivalent resistance when the pore contains one spherical particle. Dividing Eq. 9by Eq. 6, the fractional change in resistance is∆ R cyl R cyl (cid:16) a a l . (10)8or spheres of constant size, the absolute resistance change varies as a (cid:1) and the fractional resistancechange varies as a (cid:1) . Therefore, current pulses are most clearly observed when the pore size matchesparticle size as closely as possible.The same approach can be extended to different pore and particle geometries, while still assumingthat a " a . If l " a does not hold, end effects are significant, in which case the solution given inEqs. 6 to 10 is simply modified by replacing the pore length l with the approximate factor l (cid:0) . a [26].For the present work, we explicitly consider a linear conical pore with end radii a and b , in which casethe pore resistance is R con (cid:16) ρlπab , (11)the volume fraction is f con (cid:16) a p a (cid:0) ab (cid:0) b q l , (12)and ∆ R con (cid:16) ρa πab p a (cid:0) ab (cid:0) b q . (13)Equations 11 to 13 are derived using volume integrals over a conical pore, and apply when a , b and a are much smaller than l . For a truncated cone in which end effects are significant, note that the poreresistance is simply the series sum of the contributions from the ends and central part of the pore. Byanalogy with the approximation for the cylindrical pore, we can replace l with l (cid:0) . p a (cid:0) b q in Eqs. 11and 12. We obtain a conical version of Eq. 10 by dividing Eq. 13 by Eq. 11,∆ R con R con (cid:16) a l p a (cid:0) ab (cid:0) b q . (14) Other methods may be more applicable in the large-sphere cylindrical-pore limit. An approach presentedin [26] and attributed to Gregg and Steidley [48] has been used in some experiments with latex spheres927]. In this method, pore resistance is calculated by integrating the annular section surrounding aspherical particle over the length of the pore. This approach does not require that the particle issignificantly smaller than the pore size, but necessarily underestimates the pore or blockage resistance,because a uniform current distribution is assumed, whereas any nonuniformity creates a larger resistance.The absolute change in pore resistance is∆ R cyl (cid:16) ρπa (cid:4)(cid:6)(cid:6)(cid:6)(cid:5) sin (cid:1) (cid:1) a a (cid:9)(cid:2) (cid:1) (cid:1) a a (cid:9) (cid:10) (cid:1) a a (cid:12)ÆÆÆ(cid:13) . (15)In the limit a " a , this result differs from the previous small sphere result (Eq. 9) by a factor oftwo-thirds. However, Eq. 15 is asymptotically valid when the sphere size approaches the pore size [26].DeBlois and Bean more rigorously approximated the problem using a different method, in whichthe calculated electric field bulges around the insulating sphere contained within the pore. Deflectionof field lines is minimal, especially for a small sphere, allowing calculation of an upper limit for ∆ R cyl .This method applies for any sphere size when l " a , and gives a more rigorous result than Eq. 15 inthe regime a a , asymptotically approaching the result from Eq. 9 in the small sphere limit. Theresult may be most appropriate for the intermediate region in which the sphere is smaller than, but ofcomparable dimensions to, the pore size. It can be approximated using the series expansion [26]∆ R cyl (cid:16) ρa πa (cid:2) (cid:0) . a a (cid:0) . a a (cid:10) . (16)Zimmermann and Jelsch [33] considered resistance pulses generated by cells, and therefore modelledaspherical particles of finite conductivity. Using their approach, the fractional volume change for a singleparticle in a cylinder is ∆ R cyl R cyl (cid:16) f s f c a a l , (17)where f s and f c are shape and charge factors respectively. This result reduces to Eq. 10 for a perfectlyinsulating spherical particle, when f s (cid:16) and f c (cid:16)
1. The value f s (cid:16) f c (cid:16) (cid:1) p f s (cid:1) q(cid:1) , so there is a continuum of possible fractional resistance amplitudes whichincludes zero. This analysis gives some simple insight into the dependence of translocation current peakson charge distributions, and the associated solution chemistry. Polystyrene is an effective insulator, butsurface charges, polarization charges and associated screening charges are present in any nanofluidicsystem. For example, screening can reduce the value of ξ by more than two orders of magnitude forcarboxylated polystyrene spheres [49]. Experiments were carried out using a 0.1 M KCl solution, prepared using deionised water (18.2 MΩ)and buffered at pH 8.0 using 0.01 M tris base (p K a µ m-thick central septum using a specially etched, sharpened tungsten tip. The geometry of each pore isroughly a truncated circular cone, tapering from a large hole on the ‘cis’ surface to a smaller aperturenear the ‘trans’ surface. For pores manufactured under the same conditions as the specimens used here,SEM images have revealed cis surface openings of characteristic radius 15 µ m [2, 4, 5]. AFM imaging[2, 5] suggests that the narrowest part of the pore, where the electronic sensing is most sensitive, iswithin (cid:18) µ m of the trans surface opening, rather than on the surface itself.Pores are actuated by extending the distal ends of the cruciform legs, causing the central membraneto stretch radially, with azimuthal symmetry [2]. Each specimen undergoes stress-softening prior to use,so actuation is largely reversible and reproducible outside the near-pore region, which is catastrophicallyoverstretched during pore formation [5]. Following actuation, the cis side pore radius, and the thickness11 a) (b) (c) FIG. 2. Apparatus used in the experiments. The circular central septum of a TPU cruciform (Fig. 2(a),with a millimetre scale), where nanopores are located, is 200 µ m thick. The remainder of the cruciformis predominantly (cid:18) (cid:1) was identified by the qNano system. All such events had magnitude between 0.05 and0.06 nA. When nanospheres are introduced to one half of the fluid cell, increased noise is generatedby particle activity, such as partial blockages of the pore and interactions of spheres with each otherand the pore walls. Fig. 3 demonstrates that events near the noise threshold may be more frequentthan a histogram peak caused by translocations at greater event magnitude, or less frequent, as inFigs. 3(c) and 3(d) respectively. This translocation peak can be dominated by noise (as in Fig. 3(e)),13 .0 0.5 1.0 1.5 2.090.090.190.290.390.490.5 . n A C u rr e n t / n A Time / s
10 ms (a)
10 ms C u rr e n t / n A Time / s . n A (b) C oun t s Event Size / nA (c) C oun t s Event Size / nA (d) C oun t s Event Size / nA (e)
FIG. 3. Examples of raw data and event histograms. Figures 3(a) and 3(b) show excerpts from typicalraw current records using cruciform B (see Table I). Typical events, indicated by a dashed ring onthe main trace, are expanded in the inset traces. In Fig. 3(a), from series 1, events were caused by800 nm spheres and the applied potential was 0.42 V; Fig. 3(b) is from series 2, using 200 nm spheresand the potential was 0.44 V. Figures 3(c), 3(d) and 3(e) are event size distribution histograms from,respectively, cruciform C series 1 at 0.40 V (1 minute duration), cruciform B series 1 at 0.50 V andcruciform B series 1 at 0.30 V (both 5 minutes duration). Event sizes were characterised relative to thesmoothed background current in the vicinity of each event [5] and collated into 10 pA bins.14specially for smaller particles, due to the dependence of event size on particle volume in the simplemodel (Section 2.2). In such cases, the tally of events is taken to represent particle ‘activity’, ratherthan just translocations; this approach is consistent with similar work elsewhere [50].The two key parameters extracted from the raw data for the Results are (i) the size of translocationblockage signals and (ii) the frequency of events. The histograms demonstrate that it is not useful tocalculate the translocation event magnitude using a mean of all data. It is difficult to separate translo-cation events from noise events when the tails of the distributions overlap. Therefore, the characteristicsize of a translocation current blockage is defined here as the modal peak once the peak at the noiselevel is discarded. When there is no discernable translocation peak (e.g. Fig. 3(e)), the data is usuallydiscarded, but the peak at the noise threshold can be interpreted as an upper bound on the actualtranslocation blockage magnitude. This approach is comparable to the identification of ‘clusters’ usedelsewhere [9].In order to ensure consistency in frequency data, all identified events above the 0.05 nA noise limitare included. Therefore, frequency data represents all activity, not just translocations. Use of thisapproach is supported by the smooth linear relationship between event frequency and concentrationobserved in the Results. The smoothed baseline current adjacent to each event was also recorded inorder to calculate the fractional event size, and for estimation of pore size using the bulk conductivitymethod.Event duration can be determined by considering the current signal around the event in detail, as inthe inset to Figs. 3(a) and 3(b). This parameter is not analysed in the present work due to inconsistencyof the data. As discussed above, it is difficult to discriminate translocations from other events. Thebeginning and end of an event can also be ill-defined, especially when it is considered that the spheres arepassing through a long cone. Further work is required to address these issues, or to set up experimentsso that pulse durations are more regular.Experiments are grouped in specific ‘series’ for each cruciform, with each series corresponding to astable cruciform setting at which translocations were observed, as shown in Table I. Within each series,one variable, usually voltage, was altered while a number of separate data files (‘runs’) were recorded15able I. Summary of cruciform specimens, experimental ‘series’ and the associated experimental condi-tions. As described in the main text, concentration is relative to as-received solutions and the cruciformstretch is measured relative to a resting length of 42 mm, which is also used to calculate strain.Cruciform Series Stretch Strain Concentration Sphere Radius Voltage Rangemm x 10 (cid:1) nm mVA 1 10.3 0.25 2 400 0.16-0.302 10.4 0.25 2 400 0.16-0.303 16.0 0.38 2 400 0.16-0.304 19.1 0.45 2 400 0.16-0.30B 1 19.2 0.46 1 400 0.34-0.502 19.2 0.46 1 100 0.38-0.463 10.0 0.24 2 400 0.12-0.504 10.0 0.24 2 100 0.16-0.495 9.51 0.23 Varied 400 0.30 onlyC 1 13.5 0.32 2 400 0.12-0.462 13.5 0.32 2 400 0.14-0.44for analysis. For cruciforms A and B (series 1-4), each data point at a particular voltage represents arun of typical duration between 5 and 10 minutes. For cruciform B series 5 and cruciform C, each datapoint represents the average value from several runs (typically four or five), each of typical duration 30 sto 1 minute. Voltage was generally varied in random chronological order to avoid systematic error.16 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10510152025303540455055 Experiment 1 Experiment 2 E ve n t F r e qu e n cy / s - Concentration in Electrolyte / x 10 -3 (a) Series 1 Series 2 E ve n t F r e qu e n cy / s - Transmembrane Potential / V (b)
FIG. 4. Figure 4(a) is a plot of event frequency as a function of concentration (relative to as-receivedcolloidal solutions) at 0.30 V applied transmembrane potential (cruciform B, series 5). The two experi-ments represent data captured on consecutive days. R (cid:16) .
973 and 0.997 for linear fits to experiments1 and 2 respectively. Figure 4(b) shows event frequency plotted over a working range of voltages fortwo series using cruciform C. R values for a linear fit were less than 0.5 for these series, which wererecorded at constant particle concentration and pore resistance. Plotted points and error bars representthe mean and standard error in the mean for multiple recordings.17 Results
The data in Fig. 4(a) are consistent with the predicted electro-osmotic and electrophoretic particle flux(Eq. 5), as well as previous experiments [5]. It is notable that the trend does not pass through theorigin, suggesting that the simple transport model of Section 2.1 does not extend to low concentrations.Data plotted in Fig. 4(b), which are typical of other experiments, suggest a weak positive trend betweenevent frequency and applied voltage, especially if the outlying data with relatively large error bars isdiscounted. The transport model suggested in Section 2.1 predicts that the event frequency shouldincrease linearly with applied voltage. There is little evidence for such a trend; any linear positive trendwould appear to intercept above the origin. In particular, the significant scatter in the data appearsinconsistent with the result from Fig. 4(a).There are several possible reasons for the inconsistency between Figs. 4(a) and 4(b). In a broadsense, it is most likely that the data reflects differences between the two specimens used. One other likelyexplanation is that pore morphology changes unpredictably with applied voltage, producing the markeddifference in the scatter of the data in the two figures. The polymeric internal pore surface and adjoiningelectrolyte include regions of aggregated charge that could experience a force in an applied electric field.Parts of the pore surface near the constriction can be ragged and mechanically unconstrained due to thefabrication method.Other factors could explain the nonlinear relation or generally poor data consistency observed inFig. 4(b), but not necessarily both. The simplifying assumptions required in order to arrive at Eq. 5should not explain the scatter in the data. Relevant issues would include any electric field leakage nearthe tip of the cone, the suitability of the Space-Charge approach for a conical geometry, and possiblevariations in particle concentration when the pore diameter approaches the particle size. Three furtherexperimental factors could be significant. Firstly, the number of recorded events represents all activity,rather than just translocations (discussed above). Secondly, apparently spontaneous changes betweendistinct, stable ‘modes’ are observed when using this device, evidenced by step changes in baselinecurrent and possibly caused by adhesion of particles to the polymer surface. However, switches between18
Cruciform A Series 1 Cruciform A Series 2 Cruciform A Series 3 Cruciform A Series 4 Cruciform B Series 3 Cruciform B Series 4 Cruciform C Series 1 Cruciform C Series 2 R / M R / M (a) Cruciform C Series 1 Cruciform C Series 2 Linear Fit: Cruciform C Series 1 Linear Fit: Cruciform C Series 2 B ase li n e C u rr e n t / n A Transmembrane Potential / V (b)
FIG. 5. Figure 5(a) plots the modal resistance change of events against baseline nanopore resistance.Each data point represents the average of data collected at a particular applied potential. 200 nmparticles are used in cruciform B series 4, for which the data represent an upper bound on translocationresistance pulse magnitude (see Section 3); 800 nm particles are used for the other data. In Fig. 5(b),baseline current is plotted against applied potential for experiments using cruciform C. R (cid:16) .
97 forlinear fits to the data in both cases.such modes are typically identified if they are significant. Thirdly, it is possible that a small pressuredifference is introduced across the membrane as it is sealed to the lower half of the fluid cell.
Figure 5(a) shows the modal size of the resistance pulses associated with translocation events. Overall,there is a rough positive trend between the size of a resistance pulse and the baseline resistance. Theexact relation is inconsistent between different series due to geometrical variations relating differentpores, stretch states and particle sizes. These observations do not depend strongly on the applied19ransmembrane potential, consistent with the analyses in Section 2.2. It is apparent in Fig. 5(a) that therange of R values is limited for any given series. Further, Fig. 5(b) demonstrates that baseline resistanceis typically characterised by an Ohmic I - V plot over the range of interest. Although rectification was notspecifically studied here, the Ohmic characteristic suggests that any current asymmetry is not significantin the interpretation of results.Experiments using cruciform B demonstrate smaller resistance pulses for smaller particles in twoseries at the same cruciform strain. This trend is intuitive and qualitatively in line with the theoreticalapproaches. Quantitatively, there is a factor of (cid:18)
10 difference between the modal resistance peak for800 nm particles and the upper bound on that peak for 200 nm spheres, using a cruciform held at thesame strain (Fig. 5(a)). This is lower than the difference predicted (e.g. Eq. 9) if the blockade magnitudescales with the sphere volume. Taking into account that the peak for 200 nm spheres is an upper bound,the data therefore suggest that improved resolution is required to accurately study particles of suchdisparate sizes using a static pore. A key method of improving the signal-to-noise ratio is to matchthe particle size more closely to the pore size, a process which is directly enabled by resizable nanoporetechnology and which will be the subject of further work.
The estimated pore radius derived from experimental data, denoted a est , is heavily dependent on themodel used for the calculation. At present, this value should be treated as an indication of effective poresize for comparative purposes, rather than necessarily an accurate determination of pore radius. Typicaldata showing the calculated estimated radius a est as a function of applied potential are shown in Fig. 6.These data are calculated using the modal fractional event size (Eq. 10) at each potential. In eachcase, a est is calculated with the nanopore in a stressed state, with changes of the membrane thickness l estimated using the simple approach described in [3]. This plot further explores the assertion of voltageindependence and an effectively Ohmic relationship observed in relation to Fig. 5, and consistent withthe analyses in Section 2.2.For most of the data series in Table I, results are similar to those plotted in Fig. 6 for cruciform B20 .0 0.1 0.2 0.3 0.4 0.5 Cruciform B Series 3 Cruciform C Series 1 Cruciform C Series 2 a es t / n m Transmembrane Potential / V
FIG. 6. Estimated pore radius a est , calculated from the fractional event size using Eq. 10, is plottedas a function of applied potential. Stretch-induced membrane thinning has been accounted for in thecalculation, and each data point represents the average of all events recorded at a particular voltage. Forthe cruciform C series, error bars represent the standard error in the mean of several, relatively shortruns at the same potential. 21able II. Comparison of the theoretical approaches to calculating a est , as applied to two data seriesobtained under similar experimental conditions (see Table I). Thinning of the stretched membrane istreated as described in [3]. All of the models assume that a est ! l , and uncertainties are dominated bychoice of the model used for calculations, as discussed in the text. Estimates refer to the smaller poreradius in the conical case, and to the narrowest part of the pore when using the event resistance change.Model for a est Assumes a a est ? Cruciform C Series 1 Cruciform C Series 2 µ m µ mFrom Baseline Resistance:Cylinder (Eq. 6) Yes 3 . . . . . . . . . . . . a est from current measurements are comparedin Table II. Three models, using absolute resistance data with Eqs. 9, 15 and 16, give very similarresults. It follows that the assumptions differentiating these approaches are relatively insignificant inthe pore-sizing process. These assumptions are (i) a a , (ii) the electric field is uniformly distributedfor Eq. 15, and (iii) the electric field bulges as required for Eq. 16. In contrast, the method of dealingwith pore geometry (including end effects) significantly varies between these three models and each ofthe other models.DeBlois and Bean’s simplest analysis can be applied to both absolute (Eq. 9) and fractional (Eq. 10)resistance measurements. The absolute case, which assumes cylindrical pore geometry and does notdepend on l or b , gives a significantly smaller pore size than the cylindrical result from baseline resistance,because the absolute pulse is dominated by the signal when the sphere is within the narrowest part ofthe conical pore - at the small pore end. Values calculated using Eqs. 15 and 16 use a similar geometricapproach. Results calculated using the fractional resistive pulse data are two to three times smaller thanthe absolute resistance data. This is expected, because a est R / ∆ R q for the fractional data and a est R q for the absolute data; the measured value of R , averaged over the length of the pore,is relatively low, whereas the value of ∆ R is mostly dependent on the narrowest part of the pore.As expected, baseline resistance data gives the largest value for the cylindrical case, because thatmodel assumes consistent resistive losses along the pore length. For the conical model, resistive lossesare concentrated near the smaller end of the truncated cone. The conical calculation uses estimatedvalues of the larger pore radius b , determined using SEM images of typical pores at rest and adjusted23o reflect membrane stretching using the approach described in [3].Typically, the standard error in measurement of resistance or modal event size is a est
10 nm) because of theirdependence on b . The probable explanation for this observation is that the profile of the pore, ratherthan being a linear truncated cone, is convex (trumpet-shaped) [5]. It follows that the volumetric ratio inEq. 8 is an underestimate. If this explanation is correct, the calculation of a est using the conical, baselineresistance method should also be an underestimate; this is entirely reasonable given that the smallerpore radius should accommodate the translocating particles. It should further be noted that the modelfor membrane thinning and pore resizing [3] uses linear elastic material parameters. The approacheswhich use the experimental value ∆ R have the disadvantage of introducing greater fractional randomerror associated with this differential measurement. However, the greatest measurement uncertaintyin the present experiments is associated with the larger pore radius b , which is typically (cid:18) (cid:8) µ m.Improvement to techniques for imaging these pores is ongoing [4]. Other issues, which are less consistentwith observed differences between model calculations, could arise from the method of electronic samplingand from significant charge effects.There would be great practical benefits associated with determining pore size and shape using simplecurrent measurements. A reliable method would enable expedient use of the technology, without the needfor expensive, time-consuming and potentially destructive methods of specimen characterisation. Of theapproaches considered, the most reliable are likely to be those models using absolute pulse data withcylindrical geometry. These models estimate the pore size where the particle is most constricted withinthe pore, so the value of a est should refer to the characteristic pore size at or near this constriction.These approaches do not require detailed knowledge of the pore profile, or the values of b and l , somodelling of stretch effects is not required. Improved control and understanding of pore geometry willbe key to developing pore-sizing techniques, while charge effects are likely to become more significantfor smaller pores. 24urther development of the analytic current blockade analyses could extend the models for which a a does not hold to a conical pore, in which case the effective solution resistance is not constantdown the length of the pore. A range of more realistic, non-conical profiles such as those used byRam´ırez et al. [38] could be considered. End effects might be important when a particle is near themost active, narrow end of the nanopore. The usual assumption for neglecting the pore ends is a ! l , sosignificant effects are likely when the particle is within a few microns of the narrow pore opening. Thetotal population of spheres within the cone of the pore could also be considered. The presence of multiplespheres could significantly impact the effective solution conductivity and current noise characteristics.Finally, surface charge effects should be included for application to future experiments using smallerresizable pores. This paper has presented initial experimental work using novel resizable nanopore technology. Vari-ation of a number of experimental variables has been demonstrated using theory and experiments.Experimental trends agree with those established using other apparatus. For example, the absolutesize of experimental translocation resistance pulses increases with the baseline resistance, meaning thattranslocation signals are clearest for a particular particle set when the resizable pore radius is as smallas possible. The increase in pulse size between signals for 200 nm and 800 nm particles is consistentwith volumetric scaling of pulse size. There is little dependence of pulse size on applied voltage, a resultconsistent with a predominantly Ohmic I - V relationship, which was independently verified. The exper-imental data also demonstrated some inconsistency, manifested as scatter, reflecting that measurementswere carried out using a dynamically variable system.A method for accurately estimating nanopore size from current measurements alone would be valu-able for applications of this technology. For that purpose, several theoretical models were directlycompared using experimental data, including a novel approach in which truncated cone geometry wasapplied to resistance pulse magnitudes. The most accurate models rely on absolute resistance pulseheight, emphasizing data recorded when the sphere is at the narrowest part of the pore. Such models do25ot heavily depend on the pore profile through the entire membrane thickness. Overall, the spread ofvalues calculated using different models demonstrated that internal pore geometry is the most important,least well-understood experimental parameter.Several issues relating to experimental configuration and data collection have been directly addressedin this work. It was established that both electrophoretic and electro-osmotic flow were probably sig-nificant for particle transport in the experiments described. Key functional relationships presented inthe theory section allowed effective handling of this experimental regime. In particular, it was predictedthat particle flux should be proportional to concentration and electric field. Experiments supportedthe former relationship, while the latter was less certain. Discrimination of translocations from noiseand other particle activity was addressed by specifically identifying modal translocation peaks in theevent size distribution, consistently applying a noise threshold across all experiments, and including allactivity in the event frequency statistics.Future experiments will probe the theoretical models and experimental variables more closely, andmove towards reliable miniaturization of the technology. Improvement of experimental characterizationprocedures will provide the most significant advances towards efficient, accurate pore-sizing. Importantareas for development include assessment of the internal profile of pores and studies of the charge-based solution chemistry, including determination the surface potential on the interior surface of apore. As Henriquez et al. [23] have suggested, only particle size and concentration can be determinedfrom nanopore experiments when geometry and chemistry are poorly defined. Ongoing theoreticalwork relating to blockade sizes should concentrate on incorporating charge effects and pore actuation.Further developments relating to geometrical aspects are likely to provide gains in accuracy which areinsignificant when compared with the experimental spread of values arising from simple models. Acknowledgements
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