the dynamics of ions in electrolyte solutions 
 is receiving renewed 
 interest due to the appearance of novel sensing applications , such 
 as single - molecule approaches to sequencing based on micro / nanofluidic 
 devices . as part of this general platform , several studies , for instance , 
 measured the tunneling current across individual nucleotides using 
 a mechanically controllable break junctions ( mcbj ) . other recent works developed ionic current rectifiers 
 that operate by exploiting surface charge effects , including devices 
 that can be reconfigured by the application of external fields . moreover , another pioneering work investigated 
 the ionic response to ac fields . from 
 another perspective , there are many possibilities for constructing 
 narrow channels to obtain various nonlinear ionic transport characteristics . 
 despite a long history , however , the relation between the time and spatial 
 scales of the electrical response of ionic solutions at the nanoscale 
 remains unsolved . this is a key issue in controlling ionic transport 
 by externally applied dc or ac fields . its study is also of relevance 
 for any nanofluidic technology that employs electrodes in solution , 
 e.g. , for enhancing the energy storage of electrochemical capacitors and for solid - state nanopore technologies in which electrical probes interrogate 
 translocating molecules . in such complex systems , current noise is also inevitable 
 due to the presence of ions and this noise obscures the electrical 
 signals of interest  the ones from the individual molecules . 
 a full understanding of the mechanisms responsible for the behavior 
 of these  background currents is therefore needed , 
 as it will help the development of single - molecule approaches to sequencing 
 based on electronic transport . in this study , we develop a combined experimental and theoretical 
 understanding of the transient electrical response of ions in the 
 vicinity of biased electrodes that have several associated length 
 scales , such as the mcbj electrodes shown in figure 1a . this system displays a slowly decaying transient current 
 and an associated noise that increases with the ionic concentration . we demonstrate a useful computational method to treat the various 
 length and time scales involved in this complex phenomenon . the resulting 
 theoretical understanding reveals the rich behavior in the electrochemical 
 response of ions between nanoelectrodes . parts a and b of figure 1 show a schematic 
 illustration and scanning electron microscopy ( sem ) image , respectively , 
 of a mcbj made with au electrodes . the lithographically defined nanojunction 
 is fabricated on a polyimide - coated phosphor bronze substrate . it 
 consists of 30 nm thick au / cr wires of approximately 10 mm end - to - end 
 length and width tapered from 10 to 1 m along the length . we 
 create a pair of nanoelectrodes using a self - breaking technique , which forms a 1 nm electrode gap with atomically 
 sharp au tips sensitive enough to detect single molecules by a tunneling 
 current . in our experimental setup , therefore , large surfaces 
 ( 10 m ) of the lead line are 
 exposed to the electrolyte solution , which cause the background current 
 with noise , as shown in figure 1c . nacl aqueous 
 solution is prepared by dissolving nacl salt in milli - q water . to 
 investigate the dependence of transient ionic currents on applied 
 voltages and molarities , nacl solutions ( 1 , 10 , 50 , 100 , and 200 mm ) 
 are prepared and time transition profiles are measured under a constant 
 dc voltage ranging from 0.1 to 0.9 v. here , the electrical response 
 of nacl solution is measured for various salt concentrations under 
 constant dc voltage with a sampling frequency of 10 khz by using a 
 custom - built logarithmic current amplifier and a pxi-4071 digital 
 multimeter ( national instruments ) . in 
 this study , nacl solution is employed to clearly separate the transport 
 of na and cl , although kcl may be better 
 as an electrolyte to yield high conductivity of ionic currents . from 
 figure 1c , d , the transient response displays 
 a rapid increase and subsequent slow decay of the ionic current . the 
 time constant is evaluated by least - squares fit and summarized in 
 table 1 . particularly , the time constant increases 
 with increasing molarity and applied potential and is on the order 
 of 1 s for the parameter regimes studied . assuming an equivalent circuit , 
 this system would then consist of a 1 g resistance evaluated 
 from the current  voltage characteristics and a 1 nf capacitor . however , such a huge resistance and small capacitance in the narrow 
 space can not be explained by the conventional macroscopic models . ( a ) schematic illustration and ( b ) sem 
 image of mcbj electrodes , 
 ( c ) transient current response of nacl solution under an applied potential 
 of 0.4 v for several salt concentrations , and ( d ) current response 
 for a concentration of 200 mm nacl at various voltages . data from 
 applied voltages above 0.5 v are omitted here for clarity , although 
 all data ranging from 0.1 to 0.9 v are summarized in table 1 . after an initial rapid increase , the ionic current 
 then slowly decays ( with a time scale of about 1 s , table 1 ) . a large number of previous works sought to understand electrochemical 
 reactions at electrode surfaces , ionic motions in solutions , and dynamical response . in recent years , the nonequilibrium behavior of ions confined to 
 micro / nanoscale spaces is becoming increasingly important . here , we 
 focus on phenomena outside the strongly adsorbed ( stern ) layer . ionic motion in aqueous solution is expressible 
 in terms of newton s equation of motion with fluctuations and 
 dissipation , which gives mv = v + f + r , where m is the mass , v is the velocity ,  is the 
 friction coefficient , f is the external force ( due 
 to an applied potential ) , and r is the random force 
 from the solvent molecules . in this scenario , ions will rapidly respond 
 to the application of an applied potential , strongly screening it 
 by building up near the electrode surfaces . we elucidate the scale 
 of this phenomena ( and hence illuminate 
 the underlying physics ) by considering one - dimensional ionic current 
 densities ji(x , t ) due to the migration and diffusion 
 of each of the ionic species i in aqueous solution . 
 taking into account the stochastic process in electrolyte solutions planck equation for 
 each species:1where i(x , t ) is the charge density expressed 
 by the valence zi , the 
 number density ni(x , t ) , and the elementary charge e : i = zieni , and di is the 
 diffusion coefficient . the coordinate x is defined 
 in an interval of x  ( 0 , l ) where the electrode surface is at x = 0 and the 
 thickness of the diffusion layer is l. ionic migration 
 is driven by the electrostatic force due to the applied potential 
 and thus fi can be represented 
 by the gradient of potential  such that fi = zie. the relation between 
  and i is expressed by the 
 poisson equation:2where  is the dielectric 
 constant . the solution with respect to charge distributions is given by3where (0 ) and (l ) are the potentials 
 at x = 0 and the 
 summation is taken for all species 
 that contribute to . the ion density and the electrostatic 
 potential should be determined self - consistently to solve the nonlinear 
 partial differential equation . in previous works , sokalski et al . and lingenfelter et al . developed numerical 
 methods for the coupled nernst planck and poisson equations 
 and obtained useful results for the ion selective membrane potentials . 
 in particular , they discuss a long - period response at the interface 
 of the electrolyte solution and an ion - exchange membrane . here , focusing 
 on mcbj electrodes , we introduce another method that correctly predicts 
 the rapid response and noise found in experiments ( figure 1c , d ) . incorporating source terms into the dynamics , 
 eq 1 becomes4where fir are constants depending 
 on the current density and noise , 
 and ir are the phase shifts at t = 0 . in the numerical solution ,  depends only on 
 the displacement of ions and thus it is treated as independent of 
 time in the short interval . the smallest time step , which is large 
 enough to represent the stochastic process , should be determined properly 
 to maintain constraints at boundaries and electroneutrality . the source 
 terms can express noise generated at x = 0 , which 
 induces external flux in the domain . as a first step , we consider 
 gaussian white noise , although it is known that noise detected in 
 micro / nanofluidic devices usually also show flicker noise . the parameters ir and ir are generated by gaussian and uniform probability distributions , respectively . rescaling x , t , and  to 
 make them dimensionless by5and , considering 
 a unit surface , replacing 
 i by6eq 4 becomes7where fir * = lfir / ziedi and ir * = lir / di . for the other species , the equation 
 can be derived in the same manner . here , ni * and i * are expanded by the fourier series:89where k = 0 , 1 , 2 , 
 ... , (x * ) is also expanded similarly:10each basis function is orthogonal 
 on x *  [ 1 , + 1 ] . in this model , ions 
 are adsorbed or reflected at x * = 0 and the concentrations 
 correspond to those of bulk at x *  1 , conserving 
 the electroneutrality in the domain . on the basis of the description 
 above , eq 7 becomes11where12i*are real functions 
 and the variable transform of eq 6 results in equations amenable to analytic treatment . for the homogeneous 
 case , we can solve eq 11 via its eigenvalues 
 ij and eigenvectors . using these 
 solutions if ij are nonzero , then13where ukj are 
 the conjugate transpose of ujk , cil0 are constants determined at the 
 initial condition and14at equilibria , ij should be zero . 
 as a consequence , we obtain15and , from eq 1,16the nonlinear eq 7 can be numerically solved according to the procedure 
 shown in figure 2 . the one - dimensional coordinate 
 space is divided into 2 points where n  7 is employed 
 to maintain the numerical accuracy . computations are carried out for l = 2 , 5 , 10 , 15 , and 20 nm with a 128 grid and for l = 30 and 40 nm with a 256 grid to evaluate eigenvalues 
 of the homogeneous equation of eq 7 . furthermore , 
 for the inhomogeneous case with the source and noise , the computations 
 are carried out for l = 40 nm with a 256 grid . in 
 each case , the time step is fixed at dt = 1  
 10 s to maintain the computational accuracy of 
 i and . from the viewpoint 
 of stochastic processes , the condition of dt < 
 10 s tends to cause the divergence of computations . 
 on the other hand , large dt can not represent electronic 
 properties precisely . in the numerical analysis , ir are determined randomly by the box  muller algorithm and ir by a uniform random number in ( /2 , + /2 ) . the variance of gaussian white noise is set to ( 1  10 ) . in this study , we apply this model to evaluate 
 the ionic current near an au electrode 
 in nacl solution . at the negatively charged cathode surface , na is highly concentrated regardless of the anion species . in 
 the numerical analysis the applied potential is restricted to a practical 
 range of potentials in aqueous solution : (0 ) = 0.01 
 to 1 v and (l ) = 0 v. the valence 
 is + 1 and 1 for na and cl , 
 respectively . the diffusion coefficients are known : dna = 1.35  10 m / s and dcl = 
 2.03  10 m / s . the dielectric constant and temperature are  = 78.40 , where 0 is 
 the dielectric constant of vacuum and t = 298.15 
 k , respectively . as a first approximation , the time constant  
 = l / d due to 
 the slowest process within the transient response is determined by 
 the homogeneous equation of eq 7 for the two 
 component system . figure 3 shows  
 for na and cl as a function of the 
 ratio of v = (l ) (0 ) to l for the molarity of 1 , 10 , 50 , and 
 100 mm , where v / l merely expresses 
 a fraction of the 
 applied potential and the length , but not the actual electric field 
 in the system . it is found that  tends to be proportional to 
 ( v / l ) , where  
 = 2.15 for na and 2.05 for cl resulting from the fits of the computational data at 1 mm . we can 
 then predict that   ( v / l ) in the dilute limit , because the second term 
 on the left - hand side of eq 7 dominates as ni *  0 . on the other hand , the gradient 
 decreases as the molarity increases due to the strong screening of 
 the electrode surface . these results suggest that ions rapidly respond 
 to the strong electric field near the electrode surface and that  
 becomes large in the weak electric fields far away from the electrode . the magnitude of  is expected to be on the order of 1 s for 
 the electric field of 10 v / m . thus , l is estimated to be 100 m ( 1 mm ) , when the applied potential 
 is on the order of 0.1 v ( 1 v ) . recently , a similar trend of time 
 and spatial scales of the ionic response was reported by using a nanofluidic 
 field effect transistor . in experimental 
 systems , as shown in figure 1a , several length 
 scales are usually present , but the slowest process is likely the 
 result of the longest length scale . to evaluate  from the experimental 
 results ( figure 1c , d ) , l is 
 fixed at 1.57 mm assuming that the longest length along the electric 
 field line in the experimental system is a quarter of 1 mm radius 
 circle , although the actual diffusion layer thickness is possibly 
 less than this value . the series of  from the experiment with 
 200 mm solution is expected to show a negative gradient if l becomes larger with increasing applied potential . consequently , 
 the theoretical evaluations are in reasonable agreement with the experimental 
 results . time constant  of the transient response of ( a ) na and ( b ) cl in aqueous solution as a function 
 of ( v / l ) where v = (l ) and v / l show   ( v / l ) for na and 
   ( v / l ) for cl fitting the computational data at 1 mm . experimental results ( figure 1c , d ) are also 
 shown in insets . to discuss the spatial 
 scale of the ionic response , the time evolution 
 of the ionic current and density at the surface are calculated for 
 various l with respect to applied potentials . a constant 
 source term , on the right - hand side of eq 7 , 
 is applied only for na to define a steady current condition . einstein relation , where n0 is the number density of bulk solution and  is 
 a constant . here , we demonstrate a case of  = 1 . ji(xl ) ) , which is simply 
 the difference of ion flux at both ends without any electrochemical 
 reaction at the electrode surface . figure 4 shows the time evolution of net current density and normalized number 
 density of na at the cathode surface , resulting from the 
 condition of l = 5 and 20 nm under the applied potential 
 of v = 0.05 v. in both lengths of l , the current density increases and converges to a constant value 
 that becomes higher with the molarity , as shown in figure 4a , c . the response of the current density is apparently 
 affected by the density of na . as shown in figure 4b , it is interesting that the fractional density 
 of na relatively increases up to 100 mm and then decreases 
 with increasing the molarity when l = 5 nm . on the 
 other hand , the normalized density seems to uniformly decrease with 
 increasing molarity when l = 20 nm ( figure 4d ) . this means that there is an absolute limit of 
 the surface density with respect to v and l. the response time becomes longer with increasing l. it is then suggested that it takes long times when there 
 is a large space for ions to stabilize the electric field in solution . figure 5 also shows similar computational results 
 for the applied potential of v = 0.3 v. as shown 
 in figure 5a , the current density quickly increases 
 and is saturated at a steady state , when the bulk solution is defined 
 at l = 5 nm . the response time becomes shorter with 
 increasing the molarity . as shown in figure 5b , the response of surface density is correlated with the current 
 density and the difference from the bulk tends to be apparent at low 
 concentrations . on the other hand , clear peaks can be observed in 
 the ionic current response as l increases . j(l ) ) and normalized number density of 
 na at x = 0 under applied electric potential 
 of v = (l ) (0 ) 
 = 0.05 v. ( a ) and ( b ) are respectively the current density and the 
 normalized density ( magnfied view in inset ) for l = 5 nm , and ( c ) and ( d ) are similarly for l = 20 
 nm . molarity dependence is presented ranging from 10 to 200 mm . as shown in figure 5c , a peak point can 
 be recognized at 40 mm and turns to be a sharp one as the molarity 
 increases . it is found that the 
 small l causes a sufficient provision of na near the cathode that results in quickly charging and stabilizing 
 the surface . this property may be preferable to make a good capacitor , 
 but such a trend was not observed in experiments . on the other hand , 
 the large l results in the appearance of maximum 
 peak in the response , in which na highly concentrates 
 at the electrode surface immediately after applying an electric potential 
 and successively reduces to relax the excessive concentration . additionally , 
 it is clear that the magnitude of applied potentials also causes the 
 response characteristics , in comparison with figure 4 . time evolution of net current density ( j(0 )  j(l ) ) and normalized number density of 
 na at x = 0 under applied electric potential 
 of v = (l ) (0 ) 
 = 0.3 v : ( a ) , ( b ) for l = 5 nm and ( c ) , ( d ) for l = 20 nm , in the same manner as figure 4 . in the next step , to replicate 
 noise associated with electrochemical 
 reactions , we then apply other source terms 
 on the right - hand side of eq 7 . perturbations 
 at the cathode surface ( x = 0 ) due to charge transfer 
 via electrochemical reactions are mimicked by the frequency - dependent 
 sources . the amplitude of noise is determined to be proportional to 
 the square root of the bulk density based on the surface charge density such that fnar = n0 with a constant  common 
 to each molarity . is determined from fnar=0 of 10 mm 
 such that  = fnar=0/n0 , because in this study , the molarity of 10 
 mm is the minimum and available as a reference . details of electrochemical 
 reactions have not been explicitly represented in the framework of 
 nernst planck equation . here , we suggest a mathematical model 
 to mimic such perturbations , although the quantitative evaluation 
 of noise remains to be solved . we perform computations for a case 
 of  = 1 ,  = 100 , l = 40 nm , and v = 0.3 v. figure 6 shows current 
 density and normalized number density of each species obtained from 
 10 , 50 , 100 , and 200 mm nacl solutions as a function of time . as shown 
 in figure 6a , the net current density rapidly 
 increases at the moment when the electric potential is applied . the 
 maximum peak appears to be higher as the molarity increases , associated 
 with the increase of na at the cathode surface ( figure 6b ) . such tendencies are similar to those in figure 5c , d . furthermore , the time constant  of na tends to become large as the molarity increases as shown 
 in the inset of figure 6a . this trend has never 
 been observed with short l. the time scale  
  0.1 s , obtained from l = 40 nm and v = 0.3 v ( v / l this result explains well the experimental observations 
 ( figure 1c , d and table 1 ) . cl also shows large  but does not contribute 
 to the current density at the cathode side due to the extremely low 
 density near the surface . as shown in figure 6b , the response in the current density is obviously caused by the 
 prominent increase and subsequent decrease of na at the 
 surface . furthermore , the noise propagating through the solution is 
 suppressed due to the highly screened surface . this is a reason why 
 the effect of noise seems to be weakened as the molarity increases , 
 even though the noise is proportional to the square root of bulk concentration . time evolution 
 of ( a ) current density of 10 ( purple ) , 50 ( blue ) , 
 100 ( orange ) , and 200 ( red ) mm nacl solution where  of na is also shown in the inset and ( b ) normalized number density 
 of na and cl ( inset ) at x = 0 . time transition of number densities , electrostatic potential , and 
 electrostatic field resulting from ( a)(c ) 10 mm , ( d)(f ) 
 50 mm , ( g)(i ) 100 mm , and ( j)(l ) 200 mm nacl solutions , 
 applying v = 0.3 v on the length of l = 40 nm . figure 7 shows the density profiles of na , cl , the electrostatic potential , and 
 the electric field obtained from the simulations . l of figures 7 present results from 10 , 50 , 100 , and 200 mm nacl solutions , 
 respectively . resulting from the density of na , the electrode 
 surface is strongly screened and the diffusion layer is gradually 
 formed as time passes . on the other hand , the distribution of cl rapidly decreases near the electrode surface and exhibits 
 uniform increase as a function of x in each molarity . the electrostatic potential shows a drastic increase near the electrode 
 surface and the screening effect is apparent as the molarity increases . as previously discussed for figures 4 and 5 , high salt concentrations may cause 
 a rapid response to the applied field especially near the electrode 
 surface and thus weak fields remain behind the strongly screened surface , 
 which cause the whole solution to take a long period to relax . normalized 
 density distribution of ( a ) na and ( b ) cl for 10 ( purple ) , 50 ( blue ) , 100 ( orange ) , and 200 
 ( red ) mm and ( c ) the electrostatic potential ( solid line ) and the 
 electric field strength ( dashed line ) that is also shown in logarithmic 
 scale in the inset . figure 8a shows density distributions 
 of 
 na at t = 1.0 s . for each concentration , 
 a minimum peak of na this peak 
 implies that part of the ions tends to adsorb on the electrode surface 
 and the others separate , forming density gradients . in the case of cl as shown in figure 8b , 
 the concentration is depleted near the electrode surface and approaches 
 its bulk density as x increases . due to these distributions , the electrostatic potentials show extremely steep gradients near the 
 electrode surface , as shown in figure 8c . in 
 the 100 and 200 mm solutions , the electric field strength is 2 orders 
 of magnitude different between the two ends . despite the strong screening , however , weak fields also seem to exist widely in the solution , which 
 drives the transport of electrolytes . in 
 this study we developed a theoretical model and carried out 
 numerical calculations to explain the dynamical behavior of electrolytes 
 near biased electrodes in nanofluidic devices . we have found that 
 the ionic response strongly depends on the applied potential and the 
 length of the nanogap , from which an effective power law arises in 
 the numerical calculations . moreover , multiple time and length scales 
 are involved in the ionic response to a biased nanogap with large 
 electrode surfaces as is present in mcbj systems . the power law behavior 
 shown in figure 3 will allow for the prediction 
 of the response times in devices and future experiments . furthermore , 
 this work is an important building block to investigate multiple electrode 
 systems  a problem that requires a solution in higher dimensions 
 where efficient computational techniques are required . these findings 
 will also be helpful in understanding the background currents in novel 
 single - molecule sequencing approaches , as well as the behavior of 
 electrochemical capacitors for high - energy density storage
novel experimental techniques allow 
 for the manipulation and interrogation 
 of biomolecules between metallic probes immersed in micro / nanofluidic 
 channels . 
 the behavior of ions in response to applied fields is a 
 major issue in the use of these techniques in sensing applications . 
 here 
 , we experimentally and theoretically elucidate the behavior of 
 background currents in these systems . 
 these large currents have a 
 slowly decaying transient response , as well as noise that increases 
 with ionic concentration . using mechanically controllable break junctions 
 ( mcbj ) 
 , we study the ionic response in nanogaps with widths ranging 
 from a few nanometers to millimeters . 
 moreover , we obtain an expression 
 for the ionic current by solving time - dependent nernst 
 planck 
 and poisson equations . 
 this expression shows that after turning on 
 an applied voltage , ions rapidly respond to the strong fields near 
 the electrode surface , screening the field in the process . 
 ions subsequently 
 translocate in the weak electric field and slowly relax within the 
 diffusion layer . 
 our theoretical results help to explain the short- 
 and long - time behavior of the ionic response found in experiments , 
 as well as the various length scales involved .