Ionic in

Matt Krems and Massimiliano Di Ventra

Department of Physics, University of California, San Diego, La Jolla, CA 92093 (Dated: December 6, 2018) Understanding the dynamics of ions in nanopores is essential for applications ranging from single- molecule detection to DNA sequencing. We show both analytically and by means of molecular dynamics simulations that under specific conditions ion-ion interactions in nanopores lead to the phenomenon of ionic Coulomb blockade, namely the build-up of ions inside a with specific capacitance impeding the flow of additional ions due to Coulomb repulsion. This is the counterpart of electronic Coulomb blockade observed in mesoscopic systems. We discuss the analogies and differences with the electronic case as well as experimental situations in which this phenomenon could be detected.

Recently, there has been a surge of interest in takes the same ion to propagate unhindered by the pore. nanopores due to their potential in several technologi- In other words, the “contact” ionic resistance of the pore cal applications, the most notable being DNA sequencing with at least one reservoir has to be much larger than and detection [1–7]. In addition, nanopores offer unprece- the resistance offered to ion flow without the pore. From dented opportunities to study several fundamental physi- previous work [8, 9] we know that such a situation may cal processes related to ionic transport in confined geome- be realized when the pore opening is in the “quantum” tries. For example, due to the tightly bound hydration regime, namely when the pore radius is small enough that layers surrounding an ion, it may be possible to observe an ion has to shed part of its hydration layers to cross steps in the ionic conductance due to the shedding of the pore aperture. water molecules in the individual layers as the ion-water The analogy with the electron case therefore seems “quasi-particle” passes through the pore opening [8, 9]. complete safe for a notable exception. In the ionic case, This can be viewed as the classical analog of the electron conductance quantization in nanoscopic/mesoscopic sys- tems (see, e.g., Ref. [10, 11]). In this letter, we discuss another fundamental physical process which may occur in nanopores due to screened ion-ion interactions. This many-body effect originates 40 when ions flow under the effect of an electric (or pressure) field and build up in nanopores of specific capacitances, 20 )

so that the flow of additional ions is hindered. To be ◦ A

( 0

specific, Fig. 1 shows a typical pore geometry we have in z

mind with an opening wider than the other so that ions of 20 one polarity can easily accumulate inside the pore, while

ions of opposite polarity will accumulate outside the nar- 40 row opening of the pore. The effect we consider would then occur when the kinetic energy of an ion in solution, 0.002 -0.004 3 given approximately by the sum of the thermal energy Charge Density (e/A◦ ) and M(µE)2/2 (where µ is the ion mobility, M its mass and E the electric field) is smaller than the capacitive 2 Figure 1. (Color online) Right panel: a snapshot of a molec-

arXiv:1103.2749v4 [cond-mat.soft] 6 Jun 2011 energy Q /2C, with Q the average number of ions in the ular dynamics simulation of 1M KCl solution subject to an pore, and C the capacitance of the pore. electric field and translocating through a cone-shaped Si3N4 This effect is similar to the Coulomb blockade phe- pore (indicated by the yellow and blue atoms) at a pore slope nomenon experienced by electrons transporting through angle of 45◦ with a bottom opening of 7 Å radius and thickness − + a quantum dot weakly coupled to two electrodes - in the of 25 Å. Cl ions and K ions are colored aqua and brown, sense that the tunneling resistance between the electrodes respectively. The symbols γt and γb indicate the rates of ion transfer at the bottom and top openings, respectively. The and the dot is much larger than the quantum of resis- field used to generate this figure is 5.0 kcal/(molÅe) which tance [12]. Like the quantum case, we also expect ionic allows an easier visualization of the ions build-up. Left panel: Coulomb blockade to occur when the nanopore is “weakly the net charge density corresponding to the right panel config- coupled” to at least one ion reservoir. In this case, this uration. It is clear that in this case there is an accumulation of means that the average time it takes an ion to “cross” the Cl− ions inside the pore while the K+ ions are located mostly weak contact link is much longer than the average time it outside the pore. 2 the Pauli exclusion principle need not be satisfied: we can rection of ion motion. If we reversed the bias direction we fit as many ions inside the pore as its capacitance allows would obtain similar results but with the opposite charge without limitations from Fermi statistics. We thus expect sign accumulation inside and outside the pore. ionic Coulomb blockade to occur as a function of the ionic We assume that the rate through the bottom opening concentration - which could be thought of as controlled is related to the top one by by a “gate voltage” - but not as a function of bias, namely we expect the current to show a strongly non-linear be- γb = αγt 0 < α < 1 , (1) havior as a function of molarity - within the limits of ion precipitation - at fixed voltage, and essentially linear be- with havior with bias - for values below electrolysis - at fixed γ = A µn E, (2) molarity [13]. In addition, nanopores as those shown in t t o Fig. 1 can be easily fabricated [1]. We thus expect our where A is the top area of the pore, µ is the ionic mo- predictions to be readily accessible experimentally. t bility, n0 is the ionic density, and E = V/d is the electric Note also that the phenomenon we consider here is field, where V is the voltage and d is the length of the fundamentally different from other nonlinear ionic trans- pore [19]. The parameter α takes into account the dif- port effects that have been discussed previously in the ference in the two rates due to the partial shedding of literature [8, 14–17]. For instance, ions in solution can the hydration layers around the ions when they cross the form an “ionic atmosphere”, namely a region around a bottom neck, and the fact that the radius of the pore given ion in which ions of opposite charge are attracted opening is smaller at the neck than at the top. electrostatically [14]. However, the ionic atmosphere is a In order to be able to solve the rate equations analyti- dynamic phenomenon with typical lifetimes on the order cally we assume only two possible “ionic states”. The first of 10−8 s and it can be easily destroyed at field strengths state with transition probability P0 corresponds to only of 104 V/cm [14]. Therefore, for the fields and molarities the background charge in the pore, namely to bn0Ωpc we consider in this work the ionic atmosphere is not rel- ions, where n0 is the background density, Ωp is the vol- evant [14]. Another deviation from Ohmic behavior has ume of the pore, and the symbol b· · · c represents the been observed experimentally in Ref. [15]. In this work, floor function which will make the product n0Ωp an in- the addition of a small amount of divalent cations to an teger. The second state with probability P1 is that of ionic solution leads to current oscillations and negative- the background charge plus one single extra ion, namely incremental resistance. The phenomenon has been ex- bn0Ωpc + 1 ions in the pore. While this may seem like plained as due to the transient formation and breakup of an oversimplified situation, it captures the main trends nano-precipitates [15]. The nano-precipitates temporar- observed with the molecular dynamics simulations as we ily block the ionic current through the pore thus causing show below. The Markovian equation for the transition current oscillations. Again, this effect does not pertain probability P0 is then to the present work. Let us then start by presenting first a simple analytical dP 0 = Γ P − Γ P , (3) model that captures the main physics of ionic Coulomb dt 1→0 1 0→1 0 blockade. We will later corroborate these results with all-atom classical molecular dynamics simulations. Like where Γ0→1 is the transition rate of going from state 0 to the electronic Coulomb blockade case (see, e.g., [18]), we state 1, and Γ1→0 is the reverse process. The equation of can model the dynamics of ionic conduction using a rate motion for P1 is easily obtained by interchanging 0 with equation approach by calculating the transition proba- 1 in Eq. 3. bilities for the pore to accommodate a certain amount of The transition rate Γ0→1 contains the capacitive bar- charges in addition to the background charge. Referring rier experienced by an ion entering the pore already oc- again to Fig. 1 we consider the rates of conduction in and cupied by bn0Ωpc ions. Using the steady-state Nernst- out of the wide opening of the pore (“top opening”) and Planck equation it can then be written as [9] the neck of the pore (“bottom opening”), as indicated by Γ0→1 = γt exp (−C /kT ), (4) γt and γb, respectively. For clarity, we refer to the ions with the charge sign that favors accumulation inside the where  is a single-particle capacitive energy barrier, k − C pore. In the case of Fig. 1 these are Cl ions, as also the Boltzmann constant, and T the temperature. We can demonstrated by the density profile on the left panel of write the single-particle capacitive energy as [18] the same figure. The opposite-charge ions accumulate on the outskirts of the pore entrance (see Fig. 1) and the 2bn Ω c + 1  = e2 0 p (5) current blockade occurs when ions from the reservoir ad- C 2C jacent to the neck of the pore attempt to enter it. The considerations we make below would then be similar, but where e is the elementary charge and C is the capacitance clearly with different values of the parameters and the di- of the pore. Since the capacitance of the pore is linearly 3

related to the ratio of the surface areas of the top and 30.0 r =5 A◦ bottom openings, it must vary according to the param- 10 20 ◦ 8 15 r =6 A eter α we have introduced in Eq. (1). We then assume 25.0 6 ◦ 10 r =7 A 4 ◦ C = αC0 with C0 some reasonable experimental value 2 5 r =10 A 20.0 Current (nA) 0 Current (nA) 0 for the capacitance. The rate Γ1→0 is instead simply 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 Molarity (M) Voltage (V) 15.0 Γ1→0 = γb, (6) because there is no capacitive barrier for this process. We Current (nA) 10.0 also note that, in principle, there is a finite rate for an ion to move in the direction opposite to that determined 5.0 by the electric field. This rate would contribute to both 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Γ0→1 and Γ1→0. However, this process is exponentially Molarity (M) suppressed for the biases we consider here, and we can thus safely ignore it [20]. At steady state, dP/dt = 0, and from Eq. (3) - and the Figure 3. (Color online) Current as a function of KCl molarity equivalent equation for P - we obtain the steady-state for various neck radii and at a fixed bias of 1.0 V. For relevant 1 experimental molarities, we observe an almost linear increase probabilities of the current for a radius r = 10 and a non-linear behavior approaching a saturation at a given molarity for smaller radii. Γ1→0 γb P0 = = (7) The current saturation and decrease is shown explicitly in the Γ0→1 + Γ1→0 γt exp (−C /kT ) + γb left inset for a 7 Å neck radius pore. We also show the increase in current with voltage in the other inset.

Γ0→1 γt exp (−C /kT ) P1 = = (8) Γ0→1 + Γ1→0 γt exp (−C /kT ) + γb with γt and C given by Eqs. (2) and (5), respectively. Finally, the current at steady state is the same every- This equation shows indeed what we were expecting: the where and we evaluate it across the neck of the pore current is linearly dependent on bias (via the parame- ter γt) for a fixed ion concentration, and it saturates - b b  Ib = e Γ1→0P1 − Γ0→1P0 , (9) and eventually decreases - with increasing concentration according to the difference between the top and bottom where the superscript b indicates that we retain only the openings (parameter α). This is shown in Fig. 2 where terms corresponding to the bottom part of the transition 3 we have used the parameters At = 7.8 nm , d = 25 Å, rates. In the present case, this gives µ = 7 × 10−8 m2V−1s−1, T = 295 K, and C/α = 1.0 fF.   The latter is a reasonable value for a nanopore of our exp (−C /kT ) Ib = eαγt , (10) size [21]. With these parameters we indeed find that the exp (− /kT ) + α C typical drift contribution to the kinetic energy of an ion in solution is some fraction of 1 meV, which (as the ther-

50.0 mal energy) is smaller than the capacitive barrier (on the 10.0 order of several tens of meVs). 8.0 α = 0.2 We now show that this model captures the main 40.0 6.0 4.0 α = 0.8 physics of this phenomenon by performing all-atom 2.0 Current (nA) 0.0 0.0 0.5 1.0 1.5 2.0 molecular dynamics simulations using NAMD2 [22]. The 30.0 Voltage (V) α = 0.6 pores are made of 25 Å thick silicon nitride material in the β-phase and they have a conical shape with a slope 20.0 angle of 45◦ (see Fig. 1). We vary the bottom opening Current (nA) α = 0.4 from a radius r = 5 Å to a radius r = 10 Å. We then 10.0 introduce a given concentration of KCl while keeping the α = 0.2 temperature fixed at 295 K. By introducing an external 0.0 0.0 2.0 4.0 6.0 8.0 10.0 constant electric field we can then probe the ionic con- Molarity (M) ductance. Other details of the simulations are reported in Ref. [21]. Figure 2. (Color online) Current from Eq. (10) as a function The results of these simulations are plotted in Fig. 3. of the ion concentration at a fixed voltage of V = 1.0 V for As predicted by Eq. (10) our calculations show an almost various pore neck openings as represented by the parameter linear behavior of the current as a function of bias, and α (larger α implies larger neck opening). The inset shows the for a fixed bias a saturation of the current as a func- current, at a concentration of 1.0 M, as a function of bias for tion of ion concentration which is more pronounced for α = 0.2. See text for all other parameters. pores with smaller neck radius. In the inset of Fig. 3 we 4 also show explicitly current saturation and decrease for a 7 Å neck radius pore, which has to be compared with the results reported in Fig. 2. Note, however, that current 1 M. Zwolak and M. Di Ventra, Rev. Mod. Phys. 80, 141 saturation and decrease occur at molarities well above (2008). the ionic precipitation limit of KCl of about 3.5 M. We 2 D. 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