Uploaded to arXiv.org Controlling ion transport through nanopores: modeling transistor behavior Eszter Madai´ 1, Bartłomiej Matejczyk2, Andras´ Dallos1,Monika´ Valisko´ 1, Dezso˝ Boda1,3* Abstract We present a modeling study of a nanopore-based transistor computed by a mean-field continuum theory (Poisson-Nernst-Planck, PNP) and a hybrid method including particle simulation (Local Equilibrium Monte Carlo, LEMC) that is able to take ionic correlations into account including finite size of ions. The model is composed of three regions along the pore axis with the left and right regions determining the ionic species that is the main charge carrier, and the central region tuning the concentration of that species and, thus, the current flowing through the nanopore. We consider a model of small dimensions with the pore radius comparable to the Debye- screening length (Rpore/λD ≈ 1), which, together with large surface charges provides a mechanism for creating depletion zones and, thus, controlling ionic current through the device. We report scaling behavior of the device as a function the Rpore/λD parameter. Qualitative agreement between PNP and LEMC results indicates that mean-field electrostatic effects determine device behavior to the first order. Keywords nanopore — transistor — Poisson-Nernst-Planck — Monte Carlo 1Department of Physical Chemistry, University of Pannonia, P. O. Box 158, H-8201 Veszprem,´ Hungary 2Department of Mathematics, University of Warwick, CV4 7AL Coventry, United Kingdom 3Institute of Advanced Studies Koszeg¨ (iASK), Chernel u. 14, H-9730 Koszeg,¨ Hungary *Corresponding author: [email protected] 1. Introduction ity of PNP in such nano-scale confinements. We show that transistor behavior scales with the pore radius relative to the In this work, we present a modeling study for a nanopore- screening length, thus making it possible to design nanoflu- based transistor throughwhich ionic currentcan be controlled idic devices of varying length scales by adjusting the elec- via manipulatingthe chargepattern on the porewall. Nanopores trolyte concentration applied in the device. are nano-scale holes in synthetic membranes made of, for Nanopores with controlled ion flow are related to semi- example, silicon, graphene, or plastic [1, 2]. Their natural conductor transistors that modulate the current between the relatives are called ion channels that, as parts of a complex emitter and collector via tuning the availability of charge car- physiological machinery, facilitate controlled ion transport riers in a certain region of the device. This control can be throughthe cell membrane [3]. Similarly, synthetic nanopores realized by injecting charge carriers or by manipulating the are essential building blocks of nanofluidic circuitries that electric field, and, thus, the probability of charge carriers in- aim to control the behavior of fluids in the nanometer scale habiting the device (e.g., their concentrations). Control of [1]. Furthermore, nanoporesin proton-conductingmembranes electric field is commonly done by setting the electrical po- are the key elements for microfuel cells, which can convert tential at a third electrode, the gate. chemical energy directly into electricity [1], most effectively In nanopores, there are several ways to regulate the elec- from alternative fuels that has very high energy content by tric field inside the pore. Using embedded electrodes, elec- arXiv:1806.10438v1 [cond-mat.mes-hall] 27 Jun 2018 weight because of the high hydrogen content in their molecu- trically tunable nanopore-based transistors can be fabricated lar structures. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] similarly to field- The radius of our nanopore model is comparable to the effect semiconductor transistors. Control of cation/anion con- characteristic screening length of the electrolyte leading to centration in nanopores, however, can also be achieved by behavior different from micropores such as formation of ex- manipulating the surface charge pattern on the wall of the tended depletion zones and correlations between neighbor- nanopore [17, 18, 19, 20, 21, 22, 23] just as the density of ing regions. Electrostatic and excluded volume correlations electrons and holes can be controlled with doping in semi- between ions are expected to be important requiring special conductor devices. Moreover, if the membrane is made of computational techniques beyond the mean-field level of the a semiconductor material, the two techniques can be com- Poisson-Nernst-Planck (PNP) theory. We use a Monte Carlo bined [24, 9, 25, 26, 27], namely, ion accumulation can be (MC) particle simulation method in comparison with PNP controlled both with doping the semiconductor material con- to compute these correlations and to assess the applicabil- fining the pore and with the electrical potential imposed on Controlling ion transport through nanopores: modeling transistor behavior — 2/14 it. 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 concentra- tion 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.