Controlling Ion Transport Through Nanopores: Modeling Transistor Behavior

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We couple this method to the Nernst-Planck (NP) equation The chargeof the nanoporecan be manipulated with chem- (NP+LEMC method) to compute flux just as PNP does. PNP ical methods by anchoring functional groups to the pore wall is a continuum theory that, sometimes combined with the [28, 29, 30, 31, 32, 33, 34, 16]. Surface charge can also be Navier-Stokes equation to describe water flux, is commonly modulated with pH if protonation/deprotonation of the func- used to study nanopores [61, 62, 63, 64, 10, 65, 12, 66, 67, tional groups is pH sensitive [35, 36, 37]. Sensors can be de- 68, 69, 22, 23, 70, 71, 72, 73, 9, 25, 27, 74, 75, 76, 77]. signed if molecules attached to the pore wall can bind other PNP studies that consider transistor models similar to ours molecules selectively so that binding these molecules mod- will be discussed in the Discussions in relation to our results ulates the current of the background elecrolyte through the [9, 25, 27, 76, 62, 67, 68, 23]. pore in a characteristic and identifiable way [38, 39, 40, 41, This work belongs to a series of publications [57, 58, 59] 42, 43, 44, 45, 46, 47]. In the special case of charged biopoly- in which we use a multiscale modelingapproachto study nan- mers (such as DNA), electric current and/or field changes odevices using models of different resolutions computed by during the translocation of the polymer through the nanopore the appropriate computational method. In a previous publica- that can make sequencing possible [48, 49]. tion [58], we compared results for an implicit water (studied The difference between nanopores and their microfluidic by NP+LEMC) and an explicit water (studied by molecular counterparts [50] is that the radial dimension of the nanopore dynamics) for a bipolar nanopore model (“−+” charge dis- (pore radius, Rpore) is comparable to the Debye screening tribution on the pore wall along the axis) and justified the length, λD of the electrolyte that is defined as applicability of the implicit water model. We showed that the reduced model properly captures device behavior (recti- −1/2 fication), because it includes those degrees of freedom (ions’ q2c λ = i i , (1) interaction with pore, pore charges, and applied field) that D ∑ ε εkT i 0 ! are necessary to reproduce the axial behavior of concentration profiles and ignores those (explicit water molecules) that where qi is the charge and ci is the bulk concentration of ionic determine the radial behavior. As it turned out, radial be- species i, respectively, ε0 is the permittivity of vacuum, ε is havior had secondary importance in reproducing the device the dielectric constant of the electrolyte, k is Boltzmann’s behavior. constant, and T is the absolute temperature. The Debye- In another study [59] we showed that even mean-field length is loosely identified with the thickness of the electrical electrostatics captures those effects that provide the quali- double layer that is formed by the ions near the charged wall tative axial behavior of the bipolar nanopore by comparing of the nanopore. If the pore is narrow enough or the double PNP to NP+LEMC. That work justified using PNP in com- layer is wide enough (at lower concentration), the diffuse part putational studies of nanopores by calibrating PNP to a par- of the double layer extends into the pore preventing the for- ticle simulation method (LEMC), at least, for the case of 1:1 mation of a bulk electrolyte in the central region of the pore electrolytes. In this work, we continue this study by cre- [51, 52, 53]. ating a transistor model that can be viewed as two bipolar The double layers are generally depleted of the coions diodes combined head-to-head (“− + −”). For that reason, whose charge has the same sign as the pore’s surface charge. it is often called a bipolar transistor. In this symmetric three- If we apply large enough surface charge, depletion zones (re- regionmodel, the two “−”regionsare used to definethe main gions of very low local concentration) for the coions can be charge carrier ionic species (cations), while the central region established in certain sections of the nanopore. Since consec- is used to control the concentration of cations by tuning the utive sections of the nanopore along its axis behave as resis- surface charge of that region. tors connected in series, any of these elements with a deep depletion zone (with large resistance) makes the resistance of the whole pore large. Transistor behavior can be produced 2. Model and methods if we can adjust the depth of the depletion zone in such a Model of nanopore section. The device studied here is composed of two baths separated This paper explorestransistor behavior for a model nanopore by a membrane. The two sides of the membraneis connected for different charge patterns, pore geometry, and bulk con- by a single cylindrical pore that penetrates the membrane. centration. We focus on small Rpore/λD ratios by using rel- The system has a rotational symmetry around the axis of the atively small pore radii below 2.5 nm. In such a narrow pore, therefore, the solution is presented in terms of cylin- pore, we expect that ion size effects are important, there- drical coordinates z and r (the simulation cell in the LEMC fore, we model ions as charged hard spheres immersed in simulation is three-dimensional, however). The solution do- a continuum background dielectric. To compute ionic cor- main is a cylinder of 30 nm width and 9 nm radius for a relations beyond mean-field, we use the Local Equilibrium pore with Hpore = 10 nm length and Rpore = 1 nm radius. For Monte Carlo (LEMC) simulation method that is an adapta- longer and wider pores, these dimensions are proportionately tion of the Grand Canonical Monte Carlo (GCMC) technique larger. Fixed values of the concentrations and potential are to a non-equilibrium situation [54, 55, 56, 57, 58, 59, 60]. prescribed on the half-cylinders on the left and right hand Controlling ion transport through nanopores: modeling transistor behavior — 3/14

larly, water is modeled as an implicit continuum background that has two kinds of effect on ions. Water molecules screen the charges of ions. This “ener- getic” effect is taken into account by a dielectric constant, ε, in the denominator of the Coulomb potential acting between the charged hard spheres with which we model the ions:

∞ for r < Ri + R j ui j(r)= qiq j (2) for r ≥ Ri + R j, ( 4πε0εr

where Ri is the radius of ionic species i and r is the distance Figure 1. Schematics of the cylindrical nanopore that has between the ions. Water molecules hinder the diffusion of ions with friction. three regions of lengths Hn, Hx, and Hn. These regions carry This “dynamic” effect is taken into account by a diffusion co- σn, σx, and σn surface charges, respectively. The radius of efficient, Di(r), in the Nernst-Planck (NP) transport equation the nanopore is Rpore. The simulation cell is larger than this domain of this figure, but also rotationally symmetric; the for the ionic flux: three-dimensional model is obtained by rotating the figure −kTj (r)= D (r)c (r)∇µ (r), (3) about the z-axis. The electrolyte inside the pore and on the i i i i two sides of the membrane is represented as charged hard where ji(r) is the particle flux density of ionic species i, ci(r) sphere ions immersed in a dielectric continuum of dielectric is the concentration, and µi(r) is the electrochemical poten- constant ε = 78.5. The dielectric constant is the same tial profile. everywhere including the interior of the membrane. The The diffusion coefficient profile, Di(r), is a parameter to PNP model closely mimics this model as described in the be specified by the user. In the baths, we can use experi- main text. mental values. We can adjust its value inside the pore to experiments (as in the case of the Ryanodine Receptor calcium channel [55, 78]) or to results of molecular dynamics side. simulations (as in the case of bipolar nanopores [58]). It can The membrane and the pore is confined by hard walls. also be just an arbitrary parameter as in this study, because its The thickness of the membrane is the length of the pore, value inside the pore tunes the current practically linearly and Hpore. A symmetric charge pattern is created on the wall of its precise value is inconsequential from the point of view of the nanopore as shown in Fig. 1. There are regions of widths understanding the behavior of the system. Hn on the two sides of the pore carrying σn surface charges, To solve the NP equation, we need a closure between the while there is a central region of width Hx and charge σx. concentration profile, ci(r), and the electrochemical poten- Here, the σ regions set the main charge carrier. In this n tial profile, µi(r). Such a closure is provided by statistical study, we typically use negatively charged regions (hence the mechanics. We apply two kinds of methods in this work, a notation n), so the main charge carriers are the cations be- particle simulation method (LEMC) and a continuum theory cause the σ surface charges produce depletion zones of an- n method (PNP). Once the relation between ci(r) and µi(r) is ions in these regions. available, a self-consistent solution is obtained iteratively in The task of the central region with the adjustable surface which the conservation of mass, namely, the continuity equa- charge σ is to regulate the flow of cations (this is the inde- x tion, ∇ · ji(r)= 0,is satisfied. pendent variable of the device, hence the notation x). If σx is positive, it produces a depletion zone for cations, so the Local Equilibrium Monte Carlo pore contains depletion zones for both ionic species. The to- LEMC is an adaptation of the GCMC technique to a non- tal current, therefore, is small. This corresponds to the OFF equilibrium situation [54, 56, 55, 78]. The independent state state of the device. We distinguish special cases for combina- function of the LEMC simulation is the chemical potential tions of σn and σx when these surface charges are -1, 0, or 1 profile, µi(r), while the output variable is the concentration 2 − e/nm . We denote these charges by symbols “ ”, “0”, and profile, ci(r). Chemical potential is constant in space in equi- 2 2 “+”, respectively. So if σn = −1 e/nm and σx = 1 e/nm , librium for which GCMC simulations were originally designed. − − the nanopore is characterized by the string “ + ” (as in Out of equilibrium, however, µi(r) is a space-dependentquan- Fig. 1). tity. The transition from global equilibrium to non-equilibrium Reduced model of electrolyte and ion transport is possible by assuming local equilibrium (LE). We divide We describe the interactions and transport of ions with a re- the solution domain into small volume elements, Bα , and as- duced model, in which the degrees of freedom of solvent sume that the chemical potential is constant in this volume, α molecules is coarse-grained into response functions. Particu- µi . This value tunes the probability that ions of species Controlling ion transport through nanopores: modeling transistor behavior — 4/14 i occupy this volume. LEMC applies ion insertion/deletion in unit mol/dm3 in order to make it easier to relate to usual steps that are very similar to those used in global-equilibrium concentration units. GCMC with the differences that (1) the electrochemical po- The solution domain is different in the case of PNP and α tential value, µi , assigned to the volume element in which NP+LEMC. In NP+LEMC, the interior of the membrane is the insertion/deletion happens, Bα , is used in the acceptance part of the simulation cell, where electric field lines can pro- probability, (2) the volume of the volume element, V α , is trude (it is an empty continuum with dielectric constant ε). used in the acceptance probability, and (3) the energy change In PNP, this region is excluded from the solution domain. associated with the insertion/deletion, ∆U, contains all the Neumann boundary conditions are prescribed on the surface interactions from the whole simulation cell, not only from of the membrane as detailed in our previous study. On the volume element Bα . A self-consistent solution is found by nanopore’s wall, we prescribe Neumann boundary condition α an iterative process, in which µi is changed until conserva- that produces the desired surface charge: ∂Φ(r)/∂nW = σpore(z), tion of mass (∇ · ji = 0) is satisfied. Details can be found in where σpore(z) is the prescribed surface charge at coordinate our earlier publications [54, 56, 55, 78]. z (σn or σx) and nW is the normal vector on the surface of the The advantage of this technique (coined as NP+LEMC) pore wall. The boundary condition ji · n = 0 for the impene- is that it correctly computes volume exclusion and electro- trable membrane surface is also set. static correlations between ions, so it is beyond the mean- On the membrane’s surfaces that are perpendicular to the field level of the PNP theory. Its advantage compared to the z-axis at z = ±Hpore/2 we impose the boundary conditions app Brownian Dynamics method [79, 80, 81] is that sampling of ji(r) · nM = 0 and ∂Φ(r)/∂nM = ∂Φ (r)/∂nM where nM ions passing the pore is not necessary: current is computed is the outer normal and Φapp is the applied field used in the with the NP equation. Sampling of passing ions can be poor LEMC model. This solution mimics the LEMC case where especially when these events are rare due to the small current there is an electric field across the membrane. associated with the depletion zones of ions. The transistors In the case of the two half-cylinders confining the solu- studied here belong to this category because their behavior is tion domain, the same boundary conditions are prescribed as governed by these depletion zones. The NP+LEMC method in the case of the NP+LEMC model. Bulk concentrations L R has been successfully applied for membranes [54, 82], ion ci and ci are set on the left and right hand side cylinders. channels [56, 55, 57, 78] and nanopores [58, 59, 60]. Dirichlet boundary conditions for the electrical potential are In the three-dimensional LEMC model, the pore charges set: ΦL and ΦR. In practice, ΦL = 0 (left side is grounded) are placed on the pore wall as point charges on a grid. The and ΦR = U (U is the voltage). size of a grid surface element is about 0.2 × 0.2 nm2. The To solve the two-dimensional PNP system we use the magnitude of point charges was calculated so that the surface Scharfetter–Gummelscheme which is based on a transformed charge density agrees with the preset values σn or σx. This formulation of the system in entropy variables [83]. We usea solution was chosen to mimic the continuous charge distribu- two-dimensional finite element method for the actual imple- tion used in the PNP calculations. mentation and a triangular mesh containing 100 − 200 thou- sand elements. The mesh is non-uniform in order to ob- Poisson-Nernst-Planck theory tain high accuracy, especially close to the pore entrances and In this work, we apply a two-dimensionalversion of the steady charged pore walls. state PNP system as described in Matejczyk et al. [59]. PNP is a mean field method that does not consider the particles as individual entities, but investigates the concentration profiles 3. Results as the probability of finding a center of a particle in a cer- This paper studies the quantitativeeffect of changingthe charge tain location. The concentration depends on the interaction pattern (the values of σn, σx, Hn, and Hx) on the nanopore’s energy of the ion with the average (mean) electrical potential, wall. We introduce special cases that we denote by strings Φ(r), produced by all the charges in the system, including all “− + −”, “−0−”, “−−−” and so on as introduced earlier. the ions. The electrochemical potential in PNP reads as Some of these patterns are defined as ON states of the transistor (“−0−” and “−−−”), while “− + −” is defined as the µPNP(r)= µ0 + kT lnc (r)+ q Φ(r). (4) i i i i OFF state. This way, we can define a switch whose device The excess term describing to two- and many-body correla- function is the ratio of currents in the ON and OFF states, tions between ions and hard-sphere exclusion, therefore, is ION/IOFF. The larger this number is, the better the device absent. These correlations are sampled naturally in LEMC. works as a switch. The concentration profiles are related to the mean electri- In this work, we use a 1:1 electrolyte with the same ionic cal potential through Poisson’s equation, namely diamaters for the cation and the anion (0.3 nm). This choice makes a more straightforward comparison with PNP that can- 2 1 ∇ Φ(r)= − qici(r). (5) not distinguish between ions of different sizes. The dielectric ε ε ∑ 0 i constant is ε = 78.5, the temperature is T = 298.15 K. The The above equations are valid if we measure concentration in bulk diffusion constant of both ion species is 1.334 · 10−9 m−3 (number density). The results, however, will be shown m2/s, while the value inside the pore is ten times smaller Controlling ion transport through nanopores: modeling transistor behavior — 5/14

[59, 60], a choice that does not qualitatively affect our con- clusions. In the case of 0.1 M concentration, this corresponds to about 800 ions in the LEMC simulations. An NP+LEMC calculation contained 80 iterations with LEMC simulations sampling 30 million configurations in an iteration. Such a simulation lasted about 3 days. This resulted in small statistical uncertainties for the currents; the error bars are smaller than the symbols with which the current data are plotted in the figures. The PNP calculations, on the other hand, took only a few minutes. In this work, we will show cross-averaged axial concentration profiles computed as

1 Reff c (z)= c (z,r)2πrdr, (6) i A i eff Z0 2 where Aeff = Reffπ is the effective cross section of the pore that is accessible to the centers of the ions. For the hard sphere ions in LEMC Reff = Rpore − Ri, while for the point ions in PNP Reff = Rpore. This deﬁnition makes proﬁles more comparable between LEMC and PNP.

Effect of charge pattern: changing surface charges

As a first step, we vary the charge densities σx and σn and examine the resulting effect on the ionic current through the nanopore for a fixed geometry (Hn = 3.4 nm and Hx = 3.2 nm). This current is driven by voltage 200 mV; the concentration of the electrolyte is c = 0.1 M on both sides of the membrane. These parameters are valid for all figures unless otherwise stated. 2 Figure 2A shows results for a fixed σn = −1 e/nm and varying σx. The negative value of σn makes the nanopore cation-selective due to the large surface charge and small pore radius. Therefore, the main charge carrier is the cation. The current of the anion remains below 0.5 pA. The anions have depletion zones in the two n regions as seen in Fig. 2B. Whether the anions have depletion zones in the central x region dependson the value of σx, but this is irrelevant, because they already have depletion zones in the n regions. In this model, the value of σx tunes the depletion zones of the cations, and, thus, the cation current. In the case of 2 σx = −1 e/nm (“−−−”), cations do not have a depletion zone in the middle, so they carry electrical current. This is an ON state of the device (top panel of Fig. 2B). Increasing σx Figure 2. (A) Current as a function of σx while σn = −1 towards positive values, the depletion zone of cations grad- e/nm2 is kept fixed. Selected charge patterns are indicated ually appears (see Fig. 2B) and the cation current gradually with “−−−” (ON), “−0−”, and “− + −” (OFF). decreases (see Fig. 2A). Increasing σx makes the x region more positive, so the I(−σx) function is monotonically decreasing. (B) Effect of charge pattern: changing region widths Concentration profiles for these selected charge patterns. Next, we fix the charge densities and change the geometry, Widths of the regions are Hx = 3.2 and Hn = 3.4 nm, namely, the widths Hx and Hn for a fixed pore radius. Par- electrolyte concentration is c = 0.1 M, voltage is 200 mV. ticularly, we examined the effect of changing the relative Symbols and lines denote NP+LEMC and PNP results, widths of the x and n regions while keeping the total width respectively, here and in all the remaining figures unless Hpore = 2Hn +Hx = 10 nm fixed. In the ON state (“−−−”), otherwise stated. Controlling ion transport through nanopores: modeling transistor behavior — 6/14

Figure 3. Currents in the OFF state (“− + −”) through nanopores with varying region lengths. The total length, Hpore = 2Hn + Hx = 10 nm, is kept ﬁxed. The results are shown as functions of Hx. Top panel shows the total current, while the bottom panel shows the cation and anion currents. The inset of the top panel shows the ION/IOFF ratio, where the charge pattern of the ON state is “−−−” (its current is independent of Hx).

Figure 4. (A) Total currents as functions of pore length, there is no difference between these regions, so we need to Hpore, for various charge patterns with Hn/Hx = 1.0625 kept examine the OFF state (“− + −”) only. We plot the currents fixed. The currents are normalized with the values at in the OFF state as functions of Hx in Fig. 3. Hpore = 10 nm. The inset shows the ION/IOFF ratio for the The top panel showing the total current exhibits a mini- two cases where the ON states are defined either with mum that is better observed in the inset that shows ION/IOFF. “−0−”or“−−−”. (B) Concentration profiles of the Because ION does not depend on Hx, the ratio is proportional anions (the charge carriers) for Hpore = 10 nm (black) and to the reciprocal of IOFF. The minimum in IOFF, therefore, Hpore = 25 nm (red) for charge patterns “− + −” (solid) and corresponds to a maximum in the ratio characterizing the “−0−” (dashed) as obtained from NP+LEMC simulations. quality of the device as a switch. The explanation of this extremum can be depicted from the bottom panel of Fig. 3. For small Hx values, the pore cases (ON and OFF). is largely negatively charged, so the main charge carrier is Figure 4A shows that the currents decrease faster in the the cation. For large Hx values, the situation is reversed: the OFF state than in the ON states. This results in an increasing main charge carrier is the anion. The minimum of the current ION/IOFF ratio as shown in the inset of Fig. 4A. The expla- occurs at a Hx value, where both regions have sufficient size nation is the deepening depletion zones with increasing Hpore to produce sufficiently deep depletion zones for both ionic (Fig. 4B). species: for cations in the n regions, while for anions in the x The inset of Fig. 4A also shows that the ON/OFF ra- region. This value is somewhere around Hx = 5 nm. tio exhibits a saturation behavior so we can extrapolate to large Hpore values that are more common in experiments, but Effect of pore length harder to attain with particle simulations such as LEMC. Sum- Figure 4 shows the result for the case, where the Hn/Hx ratio marized, increasing pore length promotes the formation of is kept fixed (at the value of 1.0625) and the total pore length, depletion zones due to weakening electrostatic correlations Hpore = 2Hn + Hx is changed. Figure 4A shows the relative between neighboring zones. currents for the “−−−”, “−0−” (ON), and “−+−” (OFF) states. We plot relative currents (normalized by the values at Effect of pore radius and concentration Hpore = 10 nm) because we are rather interested in how fast We discuss the effect of nanopore radius and concentration the currents decrease as functions of Hpore in the different together, because concentration determines λD (see Eq. 1), Controlling ion transport through nanopores: modeling transistor behavior — 7/14

Figure 6. (A) Concentration dependence of the current in the ON (“−−−”) and OFF (“− + −”) states. The inset Figure 5. (A) Total currents as functions of pore radius, shows the ION/IOFF ratio. (B) Ratio of cation concentration Rpore, for various charge patterns for Hn/Hx = 1.0625 and profiles in the OFF and ON states for different bulk Hpore = 10 nm. The currents are normalized with the values concentrations. at Rpore = 1 nm. The inset shows the ION/IOFF ratio for the two cases where the ON states are defined either with − − −−− “ 0 ”or“ ”. (B) Axial concentration profiles of the Figure 5B shows the cross-section-averaged axial concen- cations (the charge carriers) for various Rpore values for tration profiles of the cations, c (z), in the OFF state for dif- − − + charge pattern “ + ” (OFF state). The insets show the ferent pore radii. As R decreases, the depletion zones in ≈− pore radial concentration profiles for z 3.5 nm (at a peak) and the middle get deeper, so the current decreases as Fig. 5A ≈ z 1 nm (at the depletion zone). shows. The two insets show the radial concentration profiles, c+(r), at two characteristic axial positions: z ≈−3.5 nm is a peak, while z ≈ 1 nm is a depletion zone. The profiles at so Rpore and c influence the Rpore/λD ratio that distinguishes z ≈−3.5 nm show that the cations are attracted to the pore nanopores from micropores as discussed in the Introduction. wall and their concentrations decline approaching the pore In this work, we study the effect of changing Rpore/λD in centerline (r ∼ 0). The absence of a bulk electrolyte along three ways. First, we keep λD constant by fixing the concen- the centerline is more apparent from the c+(r) profiles for tration at c = 0.1 M and vary Rpore, then we do the reverse. z ≈ 1 nm showing that concentrations never reach the bulk Finally, we change both Rpore and c while keeping Rpore/λD value (0.1 M). fixed. Next, we study the effect of changing Rpore/λD by keep- Figure 5A shows the normalized currents as functions of ing Rpore fixed at 1 nm and changing λD through varying con- Rpore for the OFF (“− + −”) and the two ON (“−0−” and centration from c = 0.05 M to c = 1 M (it corresponds to “−−−”) cases. Here, we normalize with the currents at changing the Debye length from λD = 1.36nm to λD = 0.304 Rpore = 1 nm. The relative current in the OFF state decreases nm). Figure 6A shows the currents for the “−−−” (ON) faster with decreasing Rpore than in the OFF state, which, and “− + −” (OFF) states. Both currents decrease with de- in turn, results in increasing ION/IOFF ratios with decreasing creasing concentration, but the OFF-state current decreases Rpore as shown by the inset. faster than the ON-state current. This results in a increasing Controlling ion transport through nanopores: modeling transistor behavior — 8/14

Finally, we performed simulations for two ﬁxed values of Rpore/λD (1.56 and 2.6) by using various combinations of Rpore and c (see caption of Fig. 7). These ways of studying Rpore/λD dependence are summarized in Fig. 7A by plotting the ION/IOFF ratio against the Rpore/λD ratio. The factthat the data are located along a single curve shows a scaling behavior: we can either use a wide pore with small concentration (if fabrication of a narrow pore is the limiting factor), or a narrow pore with large concentration (if using small concentrations is the limiting factor due, for example, to detecting small currents). Figure 7B shows the cOFF(z)/cON(z) cation proﬁles for those combinations of Rpore and λD (changed via changing c) that provide the 1.56 and 2.6 values for the ratio. The coin- cidence of the curves shows that scaling is valid not only for current ratios, but also for concentration ratios. Such scaling behavior is always advantageous in designing devices for a given response function.

4. Discussion Controlling with pH Manipulating charge pattern on the nanoporesurface is a non- trivial chemical treatment for which, generally, the nanopore needs to be removed from the measuring cell. There is, however, a way of altering charge pattern during the measurement by changing the pH of the bath electrolytes in the measuring cell. If there are differentchemicalgroupson the pore surface Figure 7. (A) The ION/IOFF ratio as a function of the in the x and n regions that respond differently to pH (proto- Rpore/λD variable for the cases, when we change Rpore at nation vs. deprotonation), their charge can be changed with fixed λD (c = 0.1 M, red), and when we change λD by varying pH. changing concentration for a fixed Rpore = 1 nm (black). The numbers near symbols indicate pore radii (red) or For example, if the surfaces of the n and x regions are concentration (black). (B) Ratio of cation concentration functionalized by carboxyl and amino groups, respectively, they become negative and positive, respectively, at neutral profiles in the OFF and ON states for combinations of Rpore pH (“−+−”, OFF state). Changing the pH to acidic, the car- and λD for fixed Rpore/λD = 1.56 (solid lines and open symbols) and 2.6 (dashed lines and closed symbols) ratios. boxly groups in the n regions get protonated and become neu- From bottom to top, the curves correspond to the following tral. The amino groups of the x region, in the meantime, remain positive, so this results in a “0 0” (ON) state. Chang- (Rpore/nm; c/M) pairs: (1.924; 0.0563) (blue), (1.5;0.1) + ing the pH to basic, the amino groups in the x region get de- (red), (1;0.225) (black) for Rpore/λD = 1.56 and (3.5;0.0511) (blue), (2.5;0.1) (red), (1;0.626) (black) for protonated and become neutral. The carboxyl groups of the n regions, in the meantime, remain negative, so this results in Rpore/λD = 2.6. The ION/IOFF values for these points are indicated by blue triangles in Fig. 7A. a“−0−” (also ON) state. The results are shown in Fig. 8. Currents are shown as functions of a quantity depicted as “total pore charge”. This

ION/IOFF ratio with decreasing c (see inset). The explanation is practically the sum of the magnitudes(with sign) of surface again follows from the behavior of depletion zones. charges in the three regions. This figure is closely related to Figure 6B shows the cation concentration profiles in the Fig. 3, where this “total pore charge” was controlled with OFF state divided by the profiles in the ON state. The be- Hx. There, the minimum of the curve was at Hx ≈ 5 nm, that havior of these curves for different bulk concentration re- corresponds to zero “total pore charge”. In that case, there veals that the cations have deeper depletion zones compared are both positive and negative regions in a balanced ratio so to the ON state for smaller bulk concentrations. Because that depletion zones of both cations and anions form in an optimal way so that current is minimized. Here, the OFF the cON(z)/cOFF(z) ratio is a first-order determinant of the state (“+ − +”) appear at neutral pH, while the pore can be ION/IOFF ratio, this ratio increases with decreasing c due to deepening depletion zones in the OFF state relative to the ON switched ON with changing pH in any direction [10]. state at same c. Controlling ion transport through nanopores: modeling transistor behavior — 9/14

Figure 8. Demonstration of the effect of pH by plotting the current against the “total pore charge” characterizing the asymmetry of the pore’s charge distribution. Assuming that the n and x regions have about equal lengths, this 3 dimensionless number is obtained by ∑k=1 σk/σ0, where σk 2 is the surface charge of region k and σ0 = 1 e/nm . OFF states of the transistor are present in cases when this number is close to zero, namely, when depletion zones for both ionic species are present (“− + −”). For the example given in the main text (carboxyl and amino groups), this charge pattern is present at neutral pH. ON states are present when depletion zones for one of the ionic species are absent. The charge patterns “0 + 0”or“−0−” can be produced by tuning the pH towards acidic or basic, respectively. Figure 9. (A) The value of the mean electrical potential on the surface of the pore wall (r = Rpore) for three selected charge patterns as obtained from NP+LEMC calculations. Controlling with charge vs. potential (B) The value of this potential in the center of the pore Controlling surface charge is quite different from controlling (z = 0, r = Rpore shown with larger symbols in panel A) as a the electrical potential from a practical point of view, but function of σx. The figure demonstrates the monotonic from a modeling pointof view, they are similar becausecharge relation between surface charge density, σx, and surface is always related to electrical potential through Poisson’s equa- potential, Φ(z = 0,r = Rpore). tion (Eq. 5). To show this, we plot the electrical potential profile on the surface of the nanopore, r = Rpore, for different values of σx in Fig. 9A. The potential profile changes in duced charges or the electrical potential produced by them is zone x, because it is not an imposed quantity. The magni- a time consuming process compared to the homogeneous di- tude of the potential characterized by its value in the center, electric model and precalculated applied potential used here, z = 0, depends unambiguously on σx. As Fig. 9B shows, because the ion-ion interactions are not additive any more there is a monotonic relation between charge, σx, and poten- [84, 85, 86]. We refer studying this important case to future tial, Φ(0,Rpore). Therefore, to a first approximation, control- studies. ling the surface charge can mimic controlling the electrical potential, so the results of this study can be informative re- It is common to include electrodes (through imposing garding the case of field effect nanofluidic transistors too. Dirichlet boundary conditions) and dielectric boundaries in mean field calculations (such as PNP). These calculations, Using an electrode to control the electrical potential near however, include the effect of polarization charges only on the nanopore leads to the presence of dielectric interfaces the average electrical potential. Electrostatic correlations re- between materials of different polarization properties (elec- sulting from the effect of induced charges on individual ionic trolyte vs. metal, for example). Polarization charges are in- configurations is ignored. If the electrodes are far from the duced at these dielectric boundaries that are different in every nanopore, this approximation can be sufficient, however. configuration of the ions, therefore, their presence influences the outcome of the calculations through influencing the prob- abilities of the individual configurations. Calculation of in- Controlling ion transport through nanopores: modeling transistor behavior — 10/14

Comparison of PNP and NP+LEMC our model shows reasonable switching behavior and consid- One of the motivationsof this work was to produceresults for erable current response to changing σx for such short pores is the model nanopore transistor using both a mean-field con- that we use relatively large surface charge. Its value, ±1e/nm2, tinuum theory (PNP) and a hybrid method including particle however, is typical in experiments for PET nanopores. The simulations (NP+LEMC) that can compute ion size effects other reason is that we use small Rpore/λD values. If this ra- and electrostatic correlations beyond the mean-field treatment. tio is small (narrow pore or low concentration), the double In the light of the results we can conclude that PNP is able layer formed at the pore wall does not reach a bulk behavior to capture the qualitative behavior of the device as shown by in the pore center and depletion zones of coions can form in Figs. 2–8. the respective regions. This indicates that the behavior of ionic profiles (as the A similar parameter domain was considered by Gracheva first-order determinant of current) mainly depends on the in- et al. [9, 25, 27] who considered a nanopore through a semi- teraction of ions with pore charges and applied field, while conductor membrane and by Park et al. [76] who considered interaction of ions beyond interaction with the mean electric a model called double-well nanofluidic channel. The qualita- field is secondary. Interaction with pore charges tunes the tive behavior reportedby these authors is similar to the model depth of depletion zones and directly modulates the electric considered by us. Extended depletion zones created by gate current. The applied potential makes the profiles asymmetric voltages of appropriate sign were found. along the axial dimension and produces the driving force of Other studies considered wider and longer pores at lower the steady-state current. concentrations keeping the Rpore/λD ratio close to 1 [62, 67, The approximate nature of the PNP theory appears in 68, 23]. Surface charge was quite low in these studies that did quantitative disagreement between PNP and NP+LEMC re- not make it possible to exclude the coions from the pore. De- sults. This can be seen both in the current data (Figs. 2A, 3, pletion zones, therefore, were not formed in the entire posi- 4A, 5A, 6A, 7A, and 8) and in the concentrationprofiles (Figs. tively or negativelychargedregions of the nanopore. Whether 2B, 5B, 6B, and 7B). Sources of this quantitative disagree- the scaling behavior found in our model holds in this parame- ment are the following. (1) Ions modeled as point charges ter regime is unclear. This question will be addressed in later in PNP can approach the charged wall infinitely close, so studies. their interaction with the surface charge is overestimated near Instead, a different mechanism worked that produced the the wall. PNP, therefore, overestimates concentration profiles depletion zones at the junctions of the differently charged re- at the peaks (Figs. 2B and 5B). (2) Lack of hard sphere ex- gions. This behavior can be observed in Fig. 4B (solid red clusion in PNP also tends to cause overestimation compared curve). The concentration of cation is made asymmetric by to NP+LEMC. (3) Lack of electrostatic correlations in PNP, the applied field. The profile decreases in the central x re- on the other hand, tends to decrease concentration profiles gion from z ∼−4nm to z ∼ 4 nm reaching quite a low value in the depletion zones compared to NP+LEMC, where ions at z ∼ 4 nm at the junction of the x and the right n regions. that have peaks (counterions) tend to draw the ions of oppo- This junctional depletion zone can be made deeper by mak- site charges (coions) into the depletion zones through pair- ing the pore longer or the voltage larger. Fig. 4(c) of Daiguji correlations. et al. [62], for example, clearly shows this behavior for 0.015 2 Qualitative agreement, however, indicates that PNP is a e/nm surface charge, 5 mM concentration, 2 µm x region, proper tool to study the behavior of this system and those and 5 V voltage. even larger in dimensions as demonstrated by several computational studies [62, 10, 87, 67, 21, 68, 74, 75, 88, 23]. Basi- 5. Summary cally, PNP works well for these systems (also in the case of To summarize, we identified two different mechanisms for bipolar diodes), becase the behavior of these devices is pri- creating depletion zones dependingon the values of nanopore marily driven by the depletion zones caused by mean-field parameters, especially, the magnitudes of the surface charges effects (interaction with pore charges and applied field). (σ0). If σ0 is large (our case), depletion zones are formed in Qualitative disagreement is expected in cases where ionic the entire n and x regions (not only at the junctions) if the correlations casue asymmetic behavior such as electrolytes Rpore/λD ratio is low enough (overlapping double layers and containing multivalent ions (e.g., 2:1 and 3:1 electrolytes). excluded coions). In this case, interaction with the surface Furthermore, PNP cannot compute cases where the size of charges (σx and σn) is the primary effect that creates the de- ions and specific interactions with binding sites are impor- pletion zones in the radial dimension (see the insets of Fig. tant such as in the case of sensors [60]. In general, particle 5B). This mechanism works even if the pore is short and volt- simulations are better suited for modeling sensors based on age is low. specific interactions and geometries. If surface charge is low enough and/or Rpore/λD is large enough, bulk electrolytes are formed at the centerline of the Comparison of PNP studies from literature pore so coions are not excluded. Depletion zones for both The behavior of the device studied here depends on all the pa- ions at one of the junctions are created by the applied field rameters used in the model. One basic reason of the fact that along the axial dimension. The two mechanisms can com- Controlling ion transport through nanopores: modeling transistor behavior — 11/14 bine [12]. The second mechanism, for example, can enhance [11] E. B. Kalman, O. Sudre, and Z. S. Siwy. Control of the effect of the first mechanism as pore length increases as ionic transport through an ionic transistor based on gated seen in Fig. 4. single conical nanopores. Biophys. J., 96(3):648a, 2009. We identified a scaling behavior of device function (ION/IOFF)[12] L.-J. Cheng and L. J. Guo. Ionic current rectifica- in relation to the Rpore/λD ratio. This scaling works for both tion, breakdown, and switching in heterogeneous oxide PNP and NP+LEMC. The two methods provide qualitatively nanofluidic devices. ACS Nano, 3(3):575–584, 2009. similar results indicating that device behavior is governedby [13] mean-field effects (interaction with mean potential produced K. Tybrandt, R. Forchheimer, and M. Berggren. 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