A scaling behaviour of the transport of multivalentions through rectifying nanopores

Dávid Fertig1, Mónika Valiskó1, Bartlomiej Matejczyk2, Dirk Gillespie3 and Dezső Boda1

1 University of Pannonia, Veszprém, Hungary 2University of Warwick, England 3 Rush University Medical Center, Chicago, USA

IntroductionRectification is an important property of nanopores that have asymmetric behaviour. We studied ionic transport through bipolar nanopores where the pore’s surface isasymmetrically charged. Our research group previously showed that nano-transistors show similar switching behavior for a given value of RP/λD, where RP is the radius ofthe nanopore and λD is the Debye-length of the electrolyte. The device function (switching) scales with the parameter RP/λD, namely, all the points are located along asingle curve for a 1:1 electrolyte. The question rose: is it possible to generalize scaling for electrolytes containing ions with higher valencies and if yes, how is it possible?

Model+method

-3 0 3z / nm

-2

0

2

r /

nm

σp

σn

cL c

RΦ

L

LP

L

Rpore

ΦR

LN

Nernst-Planck equation computes the ionic flux:

ji(r) = − 1kT

Di(r)ci(r)∇µi(r),

Two different modeling levels were used:• Poisson-Nernst-Planck (PNP) is a continuum theory: uses

the mean-field Poisson-Boltzmann theory to relate ci(r) toµi(r)

• Local Equilibrium Monte Carlo (LEMC) is a particle sim-ulation method: computes ion-correlations correctly

Radial behaviour

0.01

1

0.01

1

0.01

1

-1 0 1

r / nm

0.01

1

-1 0 1

r / nm

-3 -2 -1 0 1 2 3

r / nm

Cation

Anion

ξ = 2.00 Rp = 1 nm R

p = 2 nm

ξ=1.50

ξ=2.00

ξ=2.50

ON

OFF

ξ=1.05

Overlap of double layers inside the pore behaves similarly at a given ξivalue, but at different RP and concentration values.

Scaling parameter

0 1 2 3 4R

P/λ

D

10

100

1000

| I O

N/I

OF

F |

Solid: PNPSymbols: LEMC

1:1

2:1

0 1 2 3 4

ξi

1:1 2:1

ξi = RP/λi/√z+|z−|

PNP λi=λD, where λ2D=

εε0kBT

e20NA∑i

ciz2i

is the Debye-length

LEMC λi=1/2Γ, where 4Γ2= e20

εε0kBT

∑i

ρi(

zi1+Γσi

)2 is the screening

length given by the Mean Spherical Approximation (MSA)

The effect of general and individual properties on scaling

0 1 2 3 4

ξi

10

100

1000

| I O

N/I

OF

F |

Solid: PNP

Symbols: LEMC

1:1

2:1

3:1

2:2

• ON state: +200 mV, OFF state: -200mV• Rectification scales with the parameter ξi• Deviations present (confinement, ion correlations)• Axial profiles: Device behaviour• ON state: Charge neutralization• OFF state: Formation of depletion zones• LEMC: Charge inversion

-3 -2 -1 0 1 2 3

z / nm-3 -2 -1 0 1 2 3

z / nm

2:1 3:1

-3 -2 -1 0 1 2 3

z / nm

0.01

0.1

1

10

c(z

)/c

bu

lk

0

5

10

15

20

25

c(z

)/c

bu

lk

1:1 2:2

-3 -2 -1 0 1 2 3

z / nm

ON

OFF

Cation

Anion

Symbols: LEMC Solid: PNP

Rp = 2 nm0.12238 M

0.09064 M 0.07343 M 0.15639 M

0.09283 M 0.06189 M 0.04641 M 0.09283 M

ξ = 2.00

r = r+SOFF+ + r−S

OFF−

ri=∣∣∣∣ ION

i

IOFFi

∣∣∣∣ ,

SOFFi = IOFF

i

IOFF+ + IOFF

−

• Asymmetric electrolytes:anion-selectivity

• Individual rectifications scalewith ξi if they are weightedwith OFF-state selectivities

10

100

SO

FF

-*r-

0 1 2 3 4

ξi

0 1 2 3 4

ξi

10

100

r -

0.5

0.6

0.7

0.8

0.9

1

SO

FF

,-

Symbol:LEMC

Solid: PNP

1:1

2:1

3:1

2:2

Conclusion• Introducing ξi as a scaling parameter it is possible to relate the device function for systems with

different geometries and electrolytes.• Higher valencies can be taken into account by scaling with

√z+|z−|.

• Ion correlations can be accounted for by using the MSA scrreening length.

AcknowledgementThis work was supportedby the Hungarian NationalResearch, Developement andInnovation Office (NKFIHK124353).

References• Mádai et al. Controlling ion transport through nanopores:

modeling transistor behavior Phys. Chem. Chem. Phys.20(37):24156-24167, 2018.

• Boda et al. Steady state electrodiffusion from the Nernst-Planck equation coupled to Local Equilibrium Monte Carlosimulations. J. Chem. Theor. Comp., 8, 824, 2012.

• D. Fertig et al. Scaling behavior of rectification of bipolarnanopores as functions of pore radius, concentration, andion valences, targeted paper J. Chem. Phys. C.