1/F Pink Chaos in Nanopores

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1/f pink chaos in nanopores

a b Cite this: RSC Adv.,2017,7, 46092 Vishal V. R. Nandigana * and N. R. Aluru

Nanopores have been used for myriad applications ranging from water desalination, gas separation, fluidic circuits, DNA sequencing, and preconcentration of ions. In all of these applications noise is an important factor during signal measurement. Noisy signals disrupt the exact measuring signal in almost all of these applications. In this paper, we rationalize whether current oscillations should be classified only as noise or the physical disturbance in ionic charges has some other meaning. We infer that the physical disturbance in ionic charges and the current oscillations are not noise but can be chaos. Chaos is present in the system due to depletion of the ions, created by nonequilibrium anharmonic distribution in the electrostatic potential. In other words, multiple electric potential wells are observed in the nanoporous system. The multiple electric potential wells leads to bi-directional hopping of ions as the ions transport through the pore. The bi-directional hopping results in current oscillations. This paper suggests that chaos exists from a deterministic perspective and that there is no stochastic element leading to current oscillations. We prove this case by considering a simple oscillator model involving the electrostatic and dissipative forces in order to model ionic current. We observed current oscillations even in the absence of a stochastic noise force. Hence, we state that current oscillations in nanopores Received 6th June 2017 can be due to chaos as well and not necessarily due to noise. Furthermore, the color associated with the Accepted 4th September 2017 chaotic spectrum is not brown but pink, with 1/f type dynamics similar to the 1/f type pink noise DOI: 10.1039/c7ra06323g presented by theorists and experimentalists. However, the 1/f type pink chaos exists due to deterministic rsc.li/rsc-advances current oscillations and not due to a stochastic fluctuating noise force.

I. Introduction of the ions. Not only were the current oscillations perceived as noise but the resulting current spectra had a 1/f type pink noise Ionic logic-based circuits have witnessed a tremendous surge in in the low frequency regime. What causes the ionic membranes recent years, ranging from single molecule sensing,2,3 DNA in electrolyte cells to exhibit pink noise is a subject for inspec- sequencing,1,4–6 nanopower generators7 and electrolyte cells.8 tion. Several of these notions employ uctuations in the The unsupported electrolyte is actively impelled across the ion inherent ionic mobility of the electrolyte.9 Another reason membrane under an electric potential. One of the measuring pertains to the stochastic variation in the protonation and signals is the current as ions are transported through the pore. deprotonation reactions near the charge site of the membrane Recently, it was found that the current signal has built-in surface. Finally, experimentalists have also argued that the oscillations as the ions traverse through the pore. These oscil- membrane structural constituents also prompt the low lations have been conrmed as noise by earlier theorists.9–13 The frequency pink noise.10,12 Now, to what extent colored pink noise is perceived as random statistical uctuation in the elec- noise is the true state representation of the current dynamics trical current signal. Variations in noise were explained earlier under a dynamical driving/dissipative potential is a challenging by the mechanistic motion of the charges due to surface defects, question. Is it possible that physical disturbance in the ionic or else they were perceived as thermal uctuations within the charges and current oscillations in the ion-selective membranes purview of the classical uctuation–dissipation theorem.14 can be triggered by some phenomenon other than statistical Several instances of noise in membrane phenomenology occur uctuation, namely noise? In this paper, we state that the due to the cooperative motion of ions due to random memory physical disturbance in ionic charges is due to chaos, and force. In other words, a noise force is used to model the mobility current oscillations can be perceived as chaos as well and we need not restrict ourselves in claiming current oscillations as noise. We note that chaos occurs when the potential of mean aDepartment of Mechanical Engineering, Indian Institute of Technology, Madras, force on a charged ion is anharmonic in the nonequilibrium Chennai 600036, India. E-mail: [email protected]; Web: https://mech.iitm.ac.in/ state space. In other words, a multi-well potential acts on the meiitm/personnal/dr-vishal-v-r-nandigana/; Tel: +91-44-2257-4668 charged ion when the ion is driven under an electric eld. The bDepartment of Mechanical Science and Engineering, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana – Champaign, Urbana, Illinois anharmonic or multi-well potentials arise due to the depletion 61801, USA of ions owing to the charge conservation. The depleted ions

46092 | RSC Adv.,2017,7, 46092–46100 This journal is © The Royal Society of Chemistry 2017 Paper RSC Advances create an electric eld barrier that prevents other ions from transfer of the bound charge was directionally dissimilar to the entering into the nanopore. This causes the ions to hop in a bi- free charge because of the dielectric polarization. In a linear, directional manner before the ions enter into the nanopore. The homogeneous and isotropic dielectric with the electric eld bi-directional hopping of ions results in current oscillations instantaneously varying with time, the polarization density is ~ ~ leading to chaos. We argue that the oscillations observed in the constituted by P ¼ e0cE,wherec is the degree of polarization current signal have a deterministic origin and do not arise due in conjunction with the electric eld. The electric eld caused to stochastic uctuations postulated by earlier theorists. In this the bound and free charges to polarize in space constituting study we also observe that the spectrum of the current oscilla- dr ’ V$e ~ ¼ f “ ” Gauss stheorem, rE .Theequationwasarrivedatby tions is pink in color as it exhibits 1/f type dynamics similar to e0 pink noise. substituting the polarization density ~P in the previous V$~E

In this work, we understand that the origin of chaos occurs equation. er is the dielectric permittivity of the medium when the nanomembrane is selective to ions. More precisely, (water) and correlated with the degree of polarization, c,by the membrane is ideally selectivetooneparticulartypeofion er ¼ 1 þ c. The notion of Maxwell’selectricdisplacementeld which is opposite to the membrane charge. The dispropor- and electric ux conservation was conferred upon the ion tionate opposing charge has the space to conne a linear and gating membrane resulting in the governing equation anharmonic potential under an external electric eld given by dr ∭ V$e ~ ¼ ∭ f   dU rE dU , where the electric eld is related to the the mean eld approximation. That means a multi-well e0 potential exists in an ideal ion-selective nanoporous Coulomb potential (f), ~E ¼Vf,anddU is the gating membrane when an electric eld is applied on the ion. In the membrane. We radially averagedthegoverningequationto case of a non-ideal membrane, we cease to observe chaos. v vf^ Dð Þ s ð Þ Dð Þe ¼ z dr^ þ 2 s z  Now, the transport of charge over an electrostatic force in an give z r f a er invoking vz vz e0 RðzÞ ideal-membrane results in electrostatic potential instability s ð Þ ’ e Vf$~ ¼ s z ~ due to the existence of multiple potential wells. We propose Gauss slaw, r n e ,wheren is the unit normal to the ‘ 0 this mechanism as the Potential-charge momentum surface and z is the axial direction of the membrane system. ’ theorem . We further demonstrate a simple anharmonic The above area-averaged model was congured across the ‘ ’ oscillator model, calling it the VN Oscillator model which inhomogeneous system, namely along the nanopores and stands for Variational Non-equilibrium potential oscillator, to micropores. D(z) is the cross-sectional area, accounting for the explain the chaos mechanism. The current oscillations in the micro and nanogating membrane radius given by R(z), and ss(z) ideally selective nanoporous membrane can be applied in the is the variational surface charge density of the nanogating  eld of nanopore sensing for DNA sequencing and chemical membrane along with the microporous membrane. Now, we can reactions. Furthermore, the current oscillations can be inspect the transfer of charges with the mass transfer of ions. The    understood to manipulate uid ow in uidic logic circuits in dynamic mass transfer of ions is correlated by the diffusive ionic areas of combinatorial chemistry, biology and electronic gradients along with the electric mobility polarization density applications. formulated by cU~E,wherecU is the degree of polarization of the ion required to translocate the ion in the presence of an electric II. Theory eld. We call this model the ‘charge oscillator model’. The rate of A. Charge oscillator model change of the ionic charge that was radially averaged was given as v^ v v^ v ci ci ^ Before looking at the VN oscillator model, we rst disclose DðzÞ ¼ DðzÞD ðU DðzÞ^c z EÞ,whereU is the vt vz i vz vz i i i i a unique charge-transfer and potential-dissipation model to mobility of the individual ions which is related to the diffusion uncover the notion of chaos that is exhibited by ion-selective ffi ’ U ¼ Di nanogating channels. To demonstrate this phenomenon, we coe cient of each ion (Di) given by Einstein srelation, i . K T V$~ ¼ dr ~ B discerned the idea of the Maxwell model, D f, where D is K T is the thermal energy with K being the Boltzmann constant  ff B B the electric displacement vector eld accounting for the e ects and T being the absolute thermodynamic temperature. The of free and bound charge within the open-ended state space and concentration of ions within the control volume was radially dr  f is the free charge in space (ionic charge) con ned within ^ averaged and dened as ci. zi is the valence of each ion ð ð p a given volume. The free charge is a derivative of the total 1 r 2 dr ^ ¼ ð ð ; qÞÞ q charge, , which is the sum total of the free and bounded and f Dð Þ f r rd dr denotes the area-averaged z 0 0 charges (dr ); dr ¼ dr + dr . The electric displacement eld was b b f mapping function. The current density in the system was given ~ ¼ e ~ þ ~ demarcated using the constitutive relationship D 0E P, Xn v^ ¼ G G ¼ ci þ U ^ where P is the polarization density whose divergence envisions as Itot F zi i,where i Di v iziciE is the total area- ~ i¼1 z the density of bound charges, V$P ¼ dr , and e0 is the dielectric b  permittivity of free space. The stretching and compressive averaged ux of each ion. The total current across the nanogating nature of the electric eld (~E) triggered the total charge in the membrane was manifested from the current density by multi- dr V$~P plication with the membrane cross-sectional area.  V$~ ¼ f  con ned ion gating regime, E e e . The negative The coupled electric displacement eld and charge transfer 0 0  divergence in polarization density implied that the charge governing equations were numerically solved using the nite

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15 _ € volume method in OpenFOAM (Open Field Operation and calculating the velocity ~xðtÞ and acceleration ~xðtÞ we had to Manipulation). The details regarding the solver implementation – substitute the right length dl and surface charge s to account for are discussed elsewhere.16 19 s the micro and nanopore membranes.

B. VN oscillator model III. Simulation details We now discuss another simple oscillator model proposed by ¼ The simulated domain consisted of a nanopore of length Ln us to understand the notion of chaos. We call this oscillator 0.5 mm and diameter dn ¼ 10 nm. The diameter of the micro- model the ‘variational non-equilibrium potential oscillator’ pore was dmicro ¼ 500 nm, while the length of the micropore was or VN oscillator. In this model, we assumed that the nano- Lmicro ¼ 4 mm. Phosphate buffer of a constant concentration gating membrane was ideally selective to one type of ion, i.e., (0.39 mM NaH2PO4 and 0.61 mM Na2HPO4) was used in all of the membrane allowed only ions of opposite polarity to that of the simulations. The temperature was T ¼ 300 K. The diffusiv- the membrane charge. The model states that the mass + 2 9 2 1 ities of Na ,H2PO4 and HPO4 are 1.33 10 m s , 0.879 ð~_ Þ transfer of each dissimilar ion ni with respect to the charge 10 9 m2 s 1, and 0.439 10 9 m2 s 1, respectively. We of the ion (qi) constituted the equation of motion of the assumed the dielectric constant of the aqueous solution to be XN e ¼ 21 ~ ~_ r 80. We also assumed zero surface charge density on the charged particle, manifesting current: I ¼ niqi.Themass walls of the micropore, s ¼ 0, as they were away from the i m nanopore membrane and could not have an inuence on the _ dV ¼ transfer rate was mapped to the charge transfer rate qn transport. The charge on the walls of the nanopore was s ¼ 2 s ¼ 2 ðq dV ðt þ DtÞq dV ðtÞÞ assumed to vary from n 0mCm to n 30 mC m . The n n which was again mapped to the ¼ ¼ Dt voltage V0 was varied from V 0VtoV 100 V. ¼ 12 ~ In the VN oscillator model, the mass mdV 7 10 kg to Eulerian space inferred in the net ion motion I ¼ nd dV V match the charge oscillator model and the distance x ¼ ð dV ð þ D Þ dV ð ÞÞ d~ 0 qn t t qn t dV 1 xn ~ ~ ¼ ¼ ¼ ndV dVqn ðt þ DtÞ limd~/~ , Lmicro,|xb2 xb1| Ln +2Lmicro. At time t 0, the initial charge D ~ l Lnano d t dl t dV was assumed to be equal to the surface charge, qn |t¼0 ¼ sndS, where ndV is the number of ionic charges in the control d ¼ p d with S dnLn. The time step used in the simulation was 1 ns. volume dV, varying across the nanopore and micropore. q V is n The total time of the simulation was 3 ms. the charge of the ionic particle, Dt isthetimestepanddl is the length of the gating membrane accounting for either the micro or nanomembrane. We propose a theorem called the IV. Results ‘potential charge-momentum theorem (PCM)’ to rationalize Schematics of the dynamic charge oscillator model and VN the displacement of a charged particle,~x .Thetheoremsolves n oscillator model are shown in Fig. 1. On solving the charge Newton’s law of motion for a coarse grained charged particle oscillator model, we observed that the current oscillated with v2~ v~ xn _ xn time. The oscillations were bi-directional, indicating that the given by mdV þ l ¼ F| ,wheremdV is the mass of the v 2 v tn t t ions hop back and forth as they enter into the nanopore. The l_ ¼ U  ionic particle, 1/ is the mass transfer rate of the con ned results are shown in Fig. 2(a). Interestingly, the VN oscillator KBT momentized particle given as , Db is the bulk diffusion model also showed similar current oscillations where the ion Db hops bidirectionally similar to the charge oscillator model (see ffi coe cientoftheionicparticle,(KBT)isthethermalenergyof Fig. 2(a)). The magnitude of the oscillations was similar to that the ionic particle, and F| is the mean force harnessed at time tn of the charge oscillator model, thus indicating that our notion tn by the ionic particle. The mean interaction potential force of representing the charge dynamics as a simple potential- produced by electrostatic interaction is dissipative oscillator model is sufficient to capture the physics. dV dV qsqn qn V0 F| ¼ þ ,whereqs ¼ ssdS is the tn pe e j~ ~ j2 j~ ~ j 4 0 r xn x0 xb2 xb1 A. Origin of chaos charge constituted by the surface of the nano and micro We are now interested in answering the question ‘why do these membrane and dS is the surface area of the membrane. s is the s oscillations arise’? The oscillations are due to the depletion of surface charge density of the micro or nanogating membrane. An ions near the micro-nano interface region, owing to the charge electrostatic potential, V , was applied across the micro- 0 conservation. To elaborate, let us consider an ideal cation nanoporous architecture within a distance of |~x ~x |. ~x is b2 b1 0 selective nanogating membrane as considered in this study. In the position of the charged nano-membrane. The PCM model an ideal cation selective nanopore, only cations are allowed to was marched in time for Dt using a well known velocity-verlet enter into the nanopore. When an electric eld was applied _ 1€ algorithm: ~x nðt þ DtÞ¼~x nðtÞþ~xðtÞDt þ ~xðtÞðDtÞ2.20 The ionic 2 from an anode to cathode, the anions were repelled from the particle acceleration at (t + Dt)wascongured using the inter- cathode and from the cation-selective membrane (since the action potential force at ~xn(t + Dt). Finally, the velocity is membrane allows only cations). Thus, the anions were accu- mulated or enriched near the cathode-nanogating membrane _ _ 1 € € ~xðt þ DtÞ¼~xðtÞþ ~xðtÞþ~xðt þ DtÞ ðDtÞ. Note, while interface. To maintain neutrality, the cations were also enriched 2

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Fig. 1 A schematic physical view of the charge oscillator model (on the left) and the VN oscillator model (on the right). In the charge oscillator model, the following geometry for the micro-nanopore is considered. The nanogating membrane is a cylindrical nanopore of length Ln ¼ 500 nm, and diameter dn ¼ 10 nm. The nanopore is leveraged with two microporous reservoirs of length Lmicro ¼ 4 mm and diameter dmicro ¼ 500 nm. A 1 mM phosphate buﬀer electrolyte cell is considered in the reservoir. The thermodynamic temperature is T ¼ 300 K. In the VN 12 oscillator model, the reduced mass mdV is considered as 7 10 kg to match the charge oscillator model.

at this interface. At the anode-membrane interface the anions limit ourselves to analyzing current oscillations from a noise were repelled from the cation selective membrane due to elec- perspective. trostatic repulsion and attracted towards the anode due to When the physics of chaos occurs in nanogating membranes electrostatic attraction, thus the anions were depleted at the is the next interesting question one has to argue. The key to this anode-nanogating membrane interface. To maintain neutrality, lies in our proposed non-dimensional Nandigana number, the cations were also depleted at this interface. We observed ssdS ℕ v ¼ . The Nandigana number is arrived at by that this depletion region was very strong for ideal ion-selective 4pe0erxDV0 nanopores, resulting in non-equilibrium anharmonic potential non-dimensionalizing the VN oscillator model. We considered and an electric barrier. The potential and the electric barrier the following terms to non-dimensionalize the VN oscillator creates an instability in the ionic charges near the interface. model: distance x is normalized by xD ¼ 1 nm; time t by tD ¼ 1 2 1 This instability results in bi-directional hopping of ions leading ns; diﬀusion coeﬃcient D by DD ¼ 1nm ns ; charge q by qD ¼ to chaos. From our understanding, this observation of bi- ssdS; and potential V by V0. The derived Nandigana number directional hopping of ions is made for the rst time in the correlates the inuence of the surface charge density of the literature. A schematic of ion enrichment and ion depletion is membrane with the applied electric potential. The scale of the shown in Fig. 2(b). We note that such oscillations were observed Nandigana number should be of the order of 0.17 to visualize when the nanopore was ideally ion selective to one particular potential instability instigating chaos. At this Nandigana type of ion. Furthermore, we observed a stable current when the number, the surface charge density of the nanopore was 10 membrane was non-ideal, i.e., when the membrane allowed mC m 2, when the nanopore was ideally ion-selective. The both cations and anions. We played with the ideality of the relationship of the current with the Nandigana number is nanopore using the surface charge density. For nanoporous shown in Fig. 2(c) using the charge oscillator model. The gure 2 surface charge densities between sn ¼ 0mCm and sn ¼ –3 shows that the ion dynamics were stable until a Nandigana 2 mC m , we did not observe current oscillations nor bi- number of 0.05294 and we see unstable ion dynamics for Nv > directional hopping of the ions. Only for the nanoporous 0.05294. Under such a Nandigana number, the nanoporous s $ 2 2 surface charge density of n 10 mC m did we observe an surface charge density was greater than :3mCm . To truly ideal ion-selective nature of the nanopore and current oscilla- note that the current oscillations were indeed chaotic in nature, tions and chaos. Such large surface charge density nanopores we calculated the positive Lyapunov exponent using the current can be assumed to be present for aluminum oxide nanopores.22 oscillations. The notion of determining the Lyapunov exponent We would like to stress the fact that neither the charge oscillator infers upon the dimensionality of the system. In this study, we model nor the simple VN oscillator model takes into account focused our attention towards the calculation of the maximum any statistical uctuating force to observe these oscillations. Lyapunov exponent, l*, from the current–time map. We formed ’  ~ ~ Even the earlier theorists con rmation of necessitating the two state vectors, In1 and In2, from the current signal at an uctuating ionic mobility or nano membrane surface charge ~ ~ ¼ d initial time with a distance of ||In1 In2|| 0 1. The ion  uctuating force to observe current oscillations is not consid- trajectories from the initial distance travelled in time Dt were ered in either of these models. The oscillator models manifest d ¼ ~ used to map the new state vector distance, Dt ||In1+Dt current oscillations and chaos in view of dynamic potential ~ l* In2+Dt||. The maximum Lyapunov exponent ( ) in this vector instability. Thus, in our study, we state that current oscillations * d z d l Dt d D [ l* can also be viewed from a chaos perspective and we need not space was formed as Dt 0e ; Dt 1 and t 1. When was positive, we observed an exponential divergence of the current time series data, resulting in dynamic instability or

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Fig. 2 (a) The dynamic similarity of the charge oscillator and VN oscillator at Nn ¼ 0.5294. (b) Schematic of the ion enrichment and ion depletion zone in an ideal nanoporous membrane, and (c) variation of the computational current with the Nandigana number. The inset reveals the ideal hIþi cation-selective nature L ¼ of the nanomembrane with the Nandigana number. L ¼ 1 refers to the ideal cation-selective membrane. (d) hItotali Simulation results of the variation of the maximum Lyapunov exponent (l*) with Nn.

chaos. The maximum Lyapunov exponent was calculated from 0.05294, the majority of the current was contributed by the the current dynamics using TISEAN soware.23 The transition cation, as the transference number was almost equal to 1 (see from a stable current signal to a chaotic current signal in terms Fig. 2(c)), thus indicating that the pore was ideally selective to of the maximum Lyapunov exponent is shown in Fig. 2(d). We one type of ion. At this transference number and Nandigana observed chaos when the pore was ion selective. The trans- number, we observed the transition from stable current ference number is dened as the current due to positive ions dynamics to unstable or chaotic current dynamics (see Fig. 2(c)). compared to the total current. We considered the positive ions Using the VN oscillator model, we could track the position of since the pore was cation selective. We observed that when Nv > the charged ionic particle. Fig. 3(a) shows the ionic particle near

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Fig. 3 Characterization of the position of a charged ionic particle. The ﬁgure illustrates (a) ionic oscillations due to electrostatic repulsion at Nn ¼ 0.5294 and the linear transport of the ion at Nn ¼ 0, and (b) the charge of the ionic particle at Nn ¼ 0.5294. The charge oscillates as the ion nears the nano membrane. The power spectral density demonstrating pink chaos with 1/f type dynamics for diﬀerent Nandigana numbers (c and d).

the entrance of the micro-nanopore, i.e., when the particle micropore to the nanopore and to the other end of the reservoir. reached x ¼ 4 mm, the particle was repelled from the cation The dynamics are shown in Fig. 3(a). Hence, we could control selective nanopores due to electrostatic interaction of the the stability of the particle motion using our VN oscillator particle with the charged membrane and applied electric model. Using our VN oscillator model we also showed how the potential. There is electrostatic repulsion resulting in the ionic charge acted on the particle. Fig. 3(b) shows that as the particle particle tracing back to its initial position, i.e., the particle neared the nanogating membrane, i.e. at x ¼ 4 mm, the particle moved back to x ¼ 0. The charged particle tracing back to its was repelled from the nanomembrane. Furthermore, the charge initial position led to depletion of the ions near the micro- of the particle decreased due to electrostatic repulsion. Thus we nanogating membrane junction, i.e.,atx ¼ 4 mm as shown in see a decrease of the charge at x ¼ 4 mm (see Fig. 3(a) and (b) Fig. 3(a). This physics was observed when the Nandigana together). When the particle reached the entrance, i.e.,atx ¼ 0, number ¼ 0.5294. The action was repeated every time the the charge increased on the particle due to electrostatic inter- charged particle moved near to the micro-nanogating action of the particle with the voltage, V0. Hence, as the particle membrane junction. Hence, the particle oscillated with time oscillated between x ¼ 4 mm and x ¼ 0, the charge decreased giving rise to instability and chaos. To get a better idea of the and then increased periodically. The results are shown in physics, we looked at the dynamics at a low Nandigana Fig. 3(b). The charge oscillations resulted in current oscilla- number ¼ 0. At this Nandigana number, the charged particle tions, leading to instability and chaos. The dynamics shown in did not interact with the membrane surface, since at this Fig. 3(b) were at Nv ¼ 0.5294. Thus, these two gures illustrate number the surface charge density of the nanomembrane was the depletion of ions and electrostatic repulsion experienced by zero, allowing both cations and anions to enter into the pore. the charged particle near the nano membrane. Thus, the electrostatic repulsion was absent at this Nandigana number, resulting in transport of the charged particle through B. Power spectral density the pore. We see a linear transport of the charged particle with Finally, we need to resolve the conundrum of whether chaos has time revealing that the particle was transported through the color. To answer this query, we probed the current dynamics by

This journal is © The Royal Society of Chemistry 2017 RSC Adv.,2017,7, 46092–46100 | 46097 RSC Advances Paper reviewing the power spectra of the current dynamics. The we observed the particle change its direction, revealing the bi- temporal power spectral density (SI) was analyzed in a frequency directional hopping nature of the ionic particle. The phase ¼ D range of 0 to fs 1/ t and was obtained using the Welch map between the charged ionic particle and the potential (see method.24 The current signal was divided into the longest Fig. 4(c)) shows a chaotic distribution of ions in space at a high sections to obtain close to, but not exceeding, 8 segments with Nv. The current-charge phase map in Fig. 4(d) shows that the 50% overlap. Each section was windowed with a Hamming current oscillated between 20 pA to 100 pA when the charge window. The spectral estimate was evaluated by averaging the was between 471.5 aC and 472 aC. The oscillations were chaotic modied periodograms for a frequency range [0–fs/2]. The and bi-directional due to the depletion of ions near the micro- power spectral density in Fig. 3(c) and (d) shows a power-law nanogating membrane junction. The periodical bi-directional a dependence (1/f ) with the frequency. The exponent a was hopping of ions and the electric potential are shown in close to 1 in this frequency range, similar to that of the pink Fig. 4(e) and (f), respectively. These oscillations were observed at ¼ noise phenomenon. However, we rationalized that the a relatively higher value of Nv 1.058. The charge-potential dynamics are pink chaos and not pink noise. This is because the phase map in Fig. 4(g) and the current-charge phase map in pink chaos originates due to instability in the electrostatic Fig. 4(h) show two circles indicating that the potential and potential which is a deterministic phenomenon in contradic- current in these phase maps change their signs periodically tion to needing a random memory force which was postulated resulting in bi-directional hopping of ions leading to instability by earlier theorists and experimentalists9–12 when observing and chaos. pink noise. Thus, the dynamics of the current are pink chaos We now think back to the VN oscillator model to according to our understanding. derive analytical expressions for the charge transfer. We considered the VN oscillator model without taking into account the ion acceleration. The model reduced down C. Current-charge phase maps v~ _ xn to l ¼ F| . Solving this equation analytically and Similar to the VN oscillator model, the charge oscillator model vt tn v~x also depicted interesting physics. The anharmonic distribution substituting the velocity, n, in the ionic current expression, in the electrostatic potential leading to multiple potential wells vt dV ¼ is shown in Fig. 4(a). A section of the potential was sliced along we obtained the charge on the ionic particle using qn D j j2j j the time plane, and we see the potential-time dynamics in 1 xn x0 xb2 xb1 2 2 , Fig. 4(b). We observed oscillating potential dynamics due to tð4pe0erx0jxn x0j þssdSjxb2 xb1jÞþðD1D2jxn x0j jxb2 xb1jÞ _ depletion of ions near the micro-nanogating junction. The where D1 ¼ð4pe0erldlÞ, dl varied along the ion particle oscillations changed their sign, indicating that the particle displacement motion accounting for micropores and nano- ¼ D changed its direction as it oscillated in the pore. At Nv pores and 2 was obtained by invoking the initial particle dV ¼ s d ¼ 0.52942, the oscillations were not exactly bi-directional at every charge condition, qn Z n S at time t 0 and the position time interval, i.e., the oscillations were not periodical but still

Fig. 4 (a) The space-time potential obtained from the charge oscillator model, (b) the potential-time dynamics, (c) the charge and potential phase map, and (d) the charge–current phase map at Nn ¼ 0.5294. (e) to (h) The above plots at Nn ¼ 1.0588.

of the charged particle at the origin, xn ¼ 0 in the above 2 S. Howorka and Z. Siwy, Nanopore analytics: sensing of analytical equation. Z was the valence of the ion transporting single molecules, Chem. Soc. Rev., 2009, 38, 2360. through the pore. Note, if the pore is selective for monovalent 3 O. A. Saleh and L. L. Sohn, An articial nanopore for ions, Z is 1, and similarly for divalent ions, Z ¼ 2. This results in molecular sensing, Nano Lett., 2003, 3, 37. 1 4 J. J. Kasianowicz, E. Brandin, D. Branton and D. W. Deamer, D2 ¼ . This model also resulted in the bi-directional ZsndS Characterization of individual polynucleotide molecules hopping of charge and current when we assume the ions are using a membrane channel, Proc. Natl. Acad. Sci. U. S. A., displaced in a bi-directional manner. This illustrated that the 1996, 93, 13770. acceleration of the ionic particle was not necessary to observe 5 H. Chang, F. Kosari, G. Andreadakis, M. A. Alam, current oscillations and chaos. Only the electrostatic interaction G. Vasmatzis and R. Bashir, DNA-mediated uctuations in of the ions with the membrane surface and the dissipative ionic ionic current through silicon oxide nanopore channels, particle force were necessary to observe the current oscillations. Nano Lett., 2004, 4, 1551. 6 J. B. Heng, C. Ho, T. Kim, R. Timp, A. Aksimentiev, V. Conclusion Y. V. Grinkova, S. Sligar, K. Schulten and G. Timp, Sizing DNA using a nanometer-diameter pore, Biophys. J., 2004, In conclusion, we show that for an ideally ion-selective nano- 87, 2905. pore, current oscillations can also exist due to chaos rather than 7 J. Feng, M. Graf, K. Liu, D. Ovchinnikov, D. Dumcenco, noise. We observed chaos along with bi-directional hopping of M. Heiranian, V. V. R. Nandigana, N. R. Aluru, A. Kis and ions due to depletion of ions at the micro-nanopore junction. A. 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