J. Chem. Phys. 155, 194106 (2021); https://doi.org/10.1063/5.0066194 155, 194106

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An inverse averaging finite elementmethod for solving three-dimensionalPoisson–Nernst–Planck equations innanopore system simulationsCite as: J. Chem. Phys. 155, 194106 (2021); https://doi.org/10.1063/5.0066194Submitted: 09 August 2021 • Accepted: 25 October 2021 • Accepted Manuscript Online: 25 October2021 • Published Online: 15 November 2021

Qianru Zhang, Qin Wang, Linbo Zhang, et al.

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An inverse averaging finite element methodfor solving three-dimensional Poisson–Nernst–Planck equations in nanopore system simulations

Cite as: J. Chem. Phys. 155, 194106 (2021); doi: 10.1063/5.0066194Submitted: 9 August 2021 • Accepted: 25 October 2021 •Published Online: 15 November 2021

Qianru Zhang,1,2 Qin Wang,1,2 Linbo Zhang,1,2 and Benzhuo Lu1,2,a)

AFFILIATIONS1 CEMS, LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

a)Author to whom correspondence should be addressed: bzlu@lsec.cc.ac.cn

ABSTRACTThe Poisson–Nernst–Planck (PNP) model plays an important role in simulating nanopore systems. In nanopore simulations, the large-sizenanopore system and convection-domination Nernst–Planck (NP) equations will bring convergence difficulties and numerical instabilityproblems. Therefore, we propose an improved finite element method (FEM) with an inverse averaging technique to solve the three-dimensional PNP model, named inverse averaging FEM (IAFEM). At first, the Slotboom variables are introduced aiming at transformingnon-symmetric NP equations into self-adjoint second-order elliptic equations with exponentially behaved coefficients. Then, these exponen-tial coefficients are approximated with their harmonic averages, which are calculated with an inverse averaging technique on every edge ofeach tetrahedral element in the grid. Our scheme shows good convergence when simulating single or porous nanopore systems. In addition, itis still stable when the NP equations are convection domination. Our method can also guarantee the conservation of computed currents well,which is the advantage that many stabilization schemes do not possess. Our numerical experiments on benchmark problems verify the accu-racy and robustness of our scheme. The numerical results also show that the method performs better than the standard FEM when dealingwith convection-domination problems. A successful simulation combined with realistic chemical experiments is also presented to illustratethat the IAFEM is still effective for three-dimensional interconnected nanopore systems.

Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0066194

I. INTRODUCTION

The nature of ion transport in nanopore systems has receivedattention with continuous breakthroughs in experimental and the-oretical research of nano-materials.1–4 Numerical simulations canhelp researchers further understand ion transports and ionic distri-butions inside the nanopore under different conditions.5–12 More-over, numerical simulation results can provide strong theoreticalsupport and interpretation for experimental data.

The Poisson–Nernst–Planck (PNP) model, which couples theelectrostatic potential equation with convection–diffusion equa-tions, is one of the most successful continuous physical models tocharacterize the electro-diffusion process of ions in an electrolytesolution. It has been widely used in biological ion channels.13–16

At present, it is also useful in simulating nanopore systems.17–20

A variety of finite element methods (FEMs) have been proposed tosolve PNP equations in biological systems16,21–24 and semiconductor

devices.25,26 In general situations, the biological systems are small,and the Nernst–Planck (NP) equations are not convection domina-tion or with less severe convection-domination effect.27 However,different from simulations of biological systems, overcoming theconvergence difficulty and numerical instability caused by large-sizenanopore systems and convection dominance is a challenging taskin nanopore simulations. People have developed many stabilizationtechniques aiming at improving the convergence and stability of thestandard finite element method (FEM).22,23,28–30 Some stabilizationtechniques, such as the streamline upwind/Petrov–Galerkin (SUPG)method,28 not only eliminate negative oscillations (negative valuesof ion concentrations) but also lead to a piling up of remaining pos-itive oscillations.31,32 In addition, the stabilization approaches thatenhance the stability of the numerical algorithm by adding diffusionterms or interior penalties (IP) cannot guarantee the conservationof computed total currents.23 To get an efficient scheme for solvingPNP equations, we refer to our previous work,33 which introduces

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a series of FEMs with different averaging techniques for solvingthe three-dimensional drift-diffusion (DD) model in semiconduc-tor device simulations. The inverse averaging techniques have beensuccessfully employed in finite element/difference methods34–37 tosolve the DD model. In Ref. 33, we also introduce several inverseaveraging techniques aiming at improving the numerical stabil-ity of our schemes. In addition, comparing with previous works,the derivation of our methods33 and corresponding final stiffnessmatrices are much simpler, which greatly reduces the complex-ity of parallel implementations. In this paper, we extend the lastscheme with the best performance in Ref. 33 to nanopore simula-tions, which is the inverse averaging FEM (IAFEM) introduced. Thismethod possesses good convergence performances when simulatingnanopore systems. It can solve the non-physical spurious oscillationproblems caused by the convection domination and guarantee theconservation of computed total currents.

In nanopore system simulations, distributions of ion concen-trations cannot be directly approximated with piecewise polyno-mials when a strong electrostatic field exists. However, flux densi-ties, in general, vary moderately. Thus, we treat the flux density asa whole by employing the Slotboom variables.38 The introductionof Slotboom variables eliminates the cross term in the flux den-sity, and it transforms the non-symmetric NP equations into self-adjoint second-order elliptic equations with exponentially behavedcoefficients. Thus, the main difficulty lies in dealing with exponen-tial coefficients. Due to the moderate variation of flux densities, weapproximate them with constant vectors on each tetrahedral ele-ment. In addition, the corresponding exponential coefficients arealso approximated with constants on every tetrahedron of the gridin the finite element discretizations of NP equations. By follow-ing our previous work,33 the exponential coefficients are substitutedwith their harmonic averages, which are calculated on each edge ofeach tetrahedral element with an inverse averaging technique. Then,we propose the IAFEM suitable for nanopore system simulations,which possesses the upwinding property. In addition, it is stablewhen dealing with convection-domination problems. We also provethat our method can maintain the conservation of computed cur-rents well, which is the advantage that many stabilization schemesdo not possess.23 Moreover, the derivation of this method is sim-ple, and the stiffness matrix can be readily built by looping overthe tetrahedral elements of the grid, which greatly simplifies parallelimplementations of the method.

In the numerical experiments, we first use two benchmarkproblems to verify the accuracy and robustness of our IAFEM. Inaddition, numerical results show that our scheme performs betterthan the standard FEM. Then, a series of numerical experimentson synthetic nanopore systems illustrates that the IAFEM can accu-rately depict the electrostatic potential and ion concentration pro-files in the nanopore and further produce reasonable current charac-teristics. At last, a successful example of combining our method withrealistic chemical experiments8 is presented, which implies that thescheme is still effective for three-dimensional (3D) interconnectednanopore systems.

The rest of this paper is organized as follows. In Sec. II,we introduce the mathematical model, PNP model, employed innanopore simulations and its corresponding boundary conditions.In Sec. III, we present our IAFEM, in which the finite element dis-cretizations of the Poisson equation and NP equations are separately

given in detail. The evaluation of approximate currents using ourmethod is also discussed in this section. Numerical experimentsabout different kinds of nanopore systems are conducted in Sec. IVto verify the accuracy and effectiveness of the IAFEM. This paperends with Sec. V.

II. THE MATHEMATICAL MODELIn this work, the Poisson–Nernst–Planck (PNP) model is used

to study ion transports in nanopore systems. We consider thesteady-state PNP equations that consist of the Poisson equation (1)and the Nernst–Planck (NP) equations (2) in the nanopore systemΩ (see Fig. 1). The Poisson equation determines the electrostaticfield induced by charge distributions containing mobile charges inΩ, space charges inside the channel region, and surface charges onthe surface of the channel region, which is described as

−∇ ⋅ (εε0∇ϕ) =K

∑k=1

qkck + λσb in Ω. (1)

Here, ε0 is the permittivity of vacuum, ε is the dielectric constantof the electrolyte solution, ϕ is the electrostatic potential, ck is theconcentration of the kth ion species carrying charges qk = zkec, andK is the number of diffusive ion species. The characteristic func-tion λ is equal to 1 in the channel region and 0 in the reservoirregion, which illustrates that body charges whose density is denotedwith σb only exist inside the channel. The NP equations are currentdensity equations employed to model the flows of ions driven bytheir concentration gradients and the electrostatic field, which areas follows:

⎧⎪⎪⎨⎪⎪⎩

−∇ ⋅ Jk = 0 in Ω,

Jk = −Dk(∇ck + βqk∇ϕck), k = 1, 2, . . . , K,(2)

where Dk is the spatial-dependent diffusion coefficient of thekth ion species. The constant β = 1/(kBT) is the inverse Boltz-mann energy with the Boltzmann constant kB and the absolutetemperature T.

The boundary of the nanopore system is subdivided in two dis-joint parts ΓD and ΓN , respectively. The first one is the Dirichletboundary, on which boundary conditions are given by

⎧⎪⎪⎨⎪⎪⎩

ϕ∣ ΓD= Vapp,

ck∣ ΓD= cb

k, k = 1, 2, . . . , K.(3)

FIG. 1. Illustration of the nanopore system Ω. Take the single nanopore system asan example.

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The applied voltage Vapp and ion bulk concentrations cbk may take

different values on different parts of the Dirichlet boundary ΓD.The second part is the Neumann boundary, which can be subdi-vided into disjoint portions pertaining to the channel region and thereservoir region, respectively, so that we have ΓN = ΓNc ∪ ΓNr . Thecorresponding boundary conditions are

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εε0∂ϕ∂n∣ΓNr

= 0,

εε0∂ϕ∂n∣ΓNc

= σs,

Jk ⋅ n∣ ΓN= 0, k = 1, 2, . . . , K.

(4)

Here, n is the unit normal vector on ΓN and σs is the surface chargedensity on the surface of the channel.

For the computational convenience, an appropriate scaling forthe PNP model is necessary. Let ϕ← ecβϕ, which is dimensionless.Then, the scaled PNP equations are as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

−∇ ⋅ (ε∇ϕ) =K

∑k=1

e2c βε0

zkck + λecβε0

σb in Ω,

−∇ ⋅ Jk = 0 in Ω,

Jk = −Dk(∇ck + zk∇ϕck), k = 1, 2, . . . , K.

(5)

The corresponding Dirichlet and Neumann boundary conditions (3)and (4), respectively, become

⎧⎪⎪⎨⎪⎪⎩

ϕ∣ ΓD= ecβVapp,

ck∣ ΓD= cb

k, k = 1, 2, . . . , K(6)

and⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ε∂ϕ∂n∣ΓNr

= 0,

ε∂ϕ∂n∣ΓNc

=ecβε0

σs,

Jk ⋅ n∣ ΓN= 0, k = 1, 2, . . . , K.

(7)

III. FINITE ELEMENT METHODThe PNP model is a strongly coupled system. We employ the

Gummel iteration method39 to decouple it, which is a nonlinearblock Gauss–Seidel iteration. In this section, we use the FEM todiscretize the above scaled PNP equations (5). In the Poisson equa-tion, the electrostatic potential ϕ is directly discretized with linearLagrangian basis functions. For NP equations, we refer to the lastscheme in our previous work33 and propose a finite element dis-cretization suitable for nanopore system simulations. We enhanceour scheme’s numerical stability by introducing the Slotboom vari-ables and an inverse averaging technique.

A. Finite element discretizationof the Poisson equation

Let H1(Ω) be the Sobolev space of weakly differentiable

functions, and H1D(Ω) = {v ∈ H1

(Ω) ∣ v = 0 on ΓD}. The variational

form of the first Poisson equation in (5) is to find ϕ ∈ H1(Ω)

satisfying the first Dirichlet boundary condition in (6) such that

∫Ω

ε∇ϕ ⋅ ∇vdΩ =∫ΓNc

ecβε0

σsvds +K

∑i=1

e2c βε0∫

ΩzicivdΩ

+ λecβε0∫

ΩσbvdΩ, ∀v ∈ H1

D(Ω). (8)

𝒯 h denotes a tetrahedral mesh over the whole region Ω, Xh = {qi}Nvi=1

denotes the set of all vertices of 𝒯 h, and T∈ 𝒯 h denotes a tetrahedralelement in 𝒯 h. The test function v is chosen in the piecewise linearfinite element space Vh ⊂ H1

D(Ω), and it is denoted by vh. The elec-trostatic potential ϕ is discretized with ϕh = ∑i ϕh(qi)φi, where φidenotes the linear Lagrangian basis function at qi. Then we get thediscrete system of the Poisson equation in the variational form (8) asfollows:

∑T∈𝒯 h

∫T

ε∇ϕh ⋅ ∇vhdT

= ∑T∈𝒯 h

⎛

⎝∑qj∈T

ϕh(qj)∫T

ε∇φj ⋅ ∇vhdT⎞

⎠

= ∑T∈𝒯 h

(∫∂T⊂ΓNc

ecβε0

σsvhds +e2

c βε0

× ∫T

K

∑k=1

zkckvhdT + λecβε0∫

TσbvhdT).

On each tetrahedral element T, through letting vh = φi, we obtain thefollowing linear system related to ϕh:

∑qj∈T

ϕh(qj)∫T

ε∇φj ⋅ ∇φidT =∫∂T⊂ΓNc

ecβε0

σsφids

+e2

c βε0∫

T

K

∑k=1

zkckφidT

+ λecβε0∫

TσbφidT, qi ∈ T.

B. Finite element discretizations of NP equationsIn NP equations, ion concentrations may vary rapidly if there

exists a strong electrostatic field. Obviously, piecewise polynomialsare not appropriate for approximating the ion concentrations. How-ever, the flux densities vary moderately in the region Ω. Therefore,we treat the flux density Jk (concerning the kth ion species) as awhole by introducing the scaled Slotboom variable,38

Φk = ck exp(zkϕ).

Then, the flux density Jk becomes

Jk = −Dk(exp(−zkϕ)∇Φk).

In the later processing, we approximate the flux density Jk with aconstant vector on each tetrahedral element due to its moderatevariation in the region Ω.

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The employment of the scaled Slotboom variable eliminates thecross term in the kth NP equation of (5) and transforms it intoa self-adjoint second-order elliptical differential equation with theexponential coefficient exp(−zkϕ),

0 = −∇ ⋅ Jk = ∇ ⋅Dk(exp(−zkϕ)∇Φk) in Ω. (9)

Equation (9) is coupled with the following Dirichlet–Neumannboundary conditions:

Φk∣ ΓD= cb

k exp(zkVapp),

Jk ⋅ n∣ ΓN= 0.

(10)

We first present the variational form of Eq. (9), which is to findΦk ∈ H1

(Ω) satisfying the first Dirichlet boundary condition in (10)such that

∫Ω

Dk exp(−zkϕ)∇Φk ⋅ ∇wdΩ = 0, ∀w ∈ H1D(Ω). (11)

The test function w is chosen in the piecewise linear finite elementspace Wh ⊂ H1

D(Ω), and it is denoted by wh. The Slotboom vari-able Φk is also discretized with Φkh = ∑i Φkh(qi)φi, where φi denotesthe linear Lagrangian basis function at qi. As stated before, the fluxdensity Jk is approximated with a constant vector on each tetra-hedral element. Hence, we also approximate the exponential coef-ficient exp(−zkϕ) with the piecewise constant EkT on each tetra-hedral element T. The diffusion coefficient Dk is a non-constantcontinuous function in space. We use its value at the center ofgravity of the tetrahedral element DT

k to represent its value onthe whole tetrahedron T. Then, the discrete form of (11) is asfollows:

0 = ∑T∈𝒯 h

∫T

Dk exp(−zkϕ)∇Φkh ⋅ ∇whdT

∑T∈𝒯 h

DTk EkT∫

T∇Φkh ⋅ ∇whdT

= ∑T∈𝒯 h

DTk EkT∑

qj∈TΦkh(qj)∫

T∇φj ⋅ ∇whdT. (12)

Next, we introduce in detail the computation of the element-wise stiffness matrix of the kth ion species’ NP equation, i.e., Ak

= (akTij )T∈𝒯 h

. On each tetrahedral element T, let wh take the asso-ciated linear Lagrangian finite element basis function φi. Then, wehave

DTk EkT∫

T∇Φkh ⋅ ∇φidT = DT

k EkT∑qj∈T

Φkh(qj)∫T∇φj ⋅ ∇φidT

≜ DTk EkT∑

qj∈TΦkh(qj)eT

ij . (13)

Note that eTij = ∫T∇φj∇φidT reflects some geometric information

of the tetrahedron T, and it holds for linear Lagrangian finite

element basis functions that eTii = −∑j≠ie

Tij . Therefore, Eq. (13) can

be rewritten as

DTk EkT∫

T∇Φkh ⋅ ∇φidT = −DT

k ∑qj∈T,qj≠qi

EkT(Φkh(qi) −Φkh(qj))eTij .

(14)

From Eq. (14), we see that the integral on the tetrahedral element canbe written in a form related to the difference between the Slotboomvariable at two adjacent points. To get a better approximation of theexponential coefficient exp(−zkϕ), we let EkT take different valuesEkℰ ij

on different edges of the tetrahedral element T. In addition,Eq. (14) becomes

DTk EkT∫

T∇Φkh ⋅ ∇φidT = −DT

k ∑qj∈T,qj≠qi

Ekℰ ij(Φkh(qi) −Φkh(qj))eT

ij .

Here, Ekℰ ijis calculated through an inverse averaging technique on

the edge ℰ ij = qiqj of the tetrahedral element T, which is describedas follows:

Ekℰ ij=⎛

⎝

∫qj

qiexp(zkϕ)ds∣ℰ ij∣

⎞

⎠

−1

. (15)

Notice that Ekℰ ijis the harmonic average of the exponential coef-

ficient exp(−zkϕ) on the edge ℰ ij. The harmonic average is com-monly used in finite element/difference methods34–37,40,41 for semi-conductor device simulations. In addition, it has been proved to beable to provide a better result than the general mean value in theone-dimensional case, especially when dealing with the coefficientthat exhibits sharp variations or is even discontinuous on mesh ele-ments. As stated before, the electrostatic potential ϕ is discretizedwith linear Lagrangian basis functions on each tetrahedral elementT in the solution of the Poisson equation, which implies that ϕ islinear on each edge of the tetrahedron, namely,

ϕ(x) = (ϕj − ϕi

∣ℰ ij∣)(x − xqi) + ϕi, x ∈ [xqi , xqj]. (16)

By substituting (16) into (15), we get

Ekℰ ij=⎛

⎝

∫qj

qiexp(zkϕ)ds∣ℰ ij∣

⎞

⎠

−1

=⎛⎜⎝

∫xqj

xqiexp(zk(

ϕj−ϕi∣ℰ ij ∣ )(x − xqi) + ϕi)

∣ℰ ij∣

⎞⎟⎠

−1

=zk(ϕj − ϕi)

ezkϕj − ezkϕi= e−zkϕi B(zkϕj − zkϕi),

where the Bernoulli function is defined as

B(t) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

tet − 1

, t ≠ 0,

1, t = 0.

Our numerical experience shows that the Slotboom Φk is not suit-able for practical computations. Therefore, we employ the follow-ing discrete form for the variational form of the NP equation withrespect to the original variable ck. The corresponding discrete form

J. Chem. Phys. 155, 194106 (2021); doi: 10.1063/5.0066194 155, 194106-4

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of the concentration ck is ckh. In addition, the value of ckh at thepoint qi is denoted as ckh(qi) = e−zkϕi Φkh(qi),

0 = ∑T∈𝒯 h

DTk EkT∫

T∇Φkh ⋅ ∇φidT

= − ∑T∈𝒯 h

DTk ∑

qj∈T,qj≠qi

Ekℰ ij(Φkh(qi) −Φkh(qj))eT

ij

= − ∑T∈𝒯 h

DTk ∑

qj∈T,qj≠qi

e−zkϕi B(zkϕj − zkϕi)(ckh(qi)ezkϕi − ckh(qj)ezkϕj)eTij

= − ∑T∈𝒯 h

DTk ∑

qj∈T,qj≠qi

(B(zkϕj − zkϕi)ckh(qi) − B(zkϕi − zkϕj)ckh(qj))eTij

= ∑T∈𝒯 h

⎧⎪⎪⎨⎪⎪⎩

⎛

⎝− ∑

qj∈T,qj≠qi

DTk B(zkϕj − zkϕi)eT

ij⎞

⎠ckh(qi)

+ ∑qj∈T,qj≠qi

(DTk B(zkϕi − zkϕj)eT

ij)ckh(qj)

⎫⎪⎪⎬⎪⎪⎭

.

The nonzero entries of the element-wise stiffness matrix Ak= (akT

ij )T∈𝒯 hare given by

akTij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

DTk B(zkϕi − zkϕj)eT

ij , j ≠ i,

−∑m≠i

DkB(zkϕm − zkϕi)eTim, j = i.

We call the above algorithm the inverse averaging finite elementmethod (IAFEM). At last, we briefly analyze the upwinding propertyof the IAFEM. The flux density Jk on the edge ℰ ij is

Jk∣ℰ ij= −DT

k B(zkϕj − zkϕi)eTij ckh(qi) +DT

k B(zkϕi − zkϕj)eTij ckh(qj).

In addition, because

B(t) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1, t → 0,0, t → +∞,− t, t → −∞,

we have

Jk∣ℰ ij=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

DTk eT

ij (ckh(qj) − ckh(qi)), when ϕi = ϕj,

DTk eT

ij ckh(qj)(zkϕj − zkϕi), when ϕi ≪ ϕj,

DTk eT

ij ckh(qi)(zkϕj − zkϕi), when ϕi ≫ ϕj.

(17)

From (17), we see that the flux density Jk∣ℰ ijcalculated with

the IAFEM is similar to that derived from the central differencescheme when the electrostatic field is zero (ϕi = ϕj). When ϕi ≪ ϕjor ϕi ≫ ϕj, the flux density obtained with the IAFEM is similar tothe first-order upwinding form of the finite difference method.42

C. Evaluation of approximate currentsIn this subsection, we present a scheme to evaluate approxi-

mate currents using the IAFEM. We also prove that our method canmaintain the conservation of currents well. This property will alsobe verified with later numerical experiments.

In a nanopore system, the Dirichlet boundary ΓD is composedof a finite number of separated parts ΓDm , and the end-points ofeach part ΓDm are just mesh nodes of the tetrahedral mesh 𝒯 h.Let Sh ≜ span{φi} ⊂ H1

(Ω). If wh ∈ Sh satisfies wh∣ ΓD= 0, then

wh ∈Wh. For any part of the Dirichlet boundary ΓDm , let φDm be apiecewise constant function satisfying

φDm =

⎧⎪⎪⎨⎪⎪⎩

1, x ∈ ΓDm ,

0, x ∈ ΓD/ΓDm .

Through multiplying (9) with φDm and integrating by parts, we get

0 = −∫Ω(∇ ⋅ Jk)φDm dΩ = −∫

ΓDm

Jk ⋅ nds + ∫Ω

Jk ⋅ ∇φDm dΩ.

If φIDm is the linear interpolation of φDm , then φI

Dm ∈ Sh and φIDm

= ∑NDmi=1 φi. The current IDm

kh flowing through ΓDm is approximated as

IDmkh = qk∫

ΓDm

Jkh ⋅ nds = qk∫Ω

Jkh ⋅ ∇φIDm dΩ

= −∫Ω

qkDk exp(−zkϕ)∇Φkh ⋅ ∇φIDm dΩ

= −

NDm

∑i=1∑

T∈𝒯 h

qkDTk EkT∫

T∇Φkh ⋅ ∇φidT

=

NDm

∑i=1∑

T∈𝒯 h

qkDTk ∑

qj∈T,qj≠qi

(B(zkϕj − zkϕi)ckh(qi)

− B(zkϕi − zkϕj)ckh(qj))eTij .

Finally, the total current IDmtotal flowing out of ΓDm is the sum of all ion

species’ currents, namely,

IDmtotal =

K

∑k=1

IDmkh .

Remark 1. The computed total current flowing through ΓD isconservative,

∑ΓDm⊂ΓD

IDmtotal = 0.

Proof 1. By referring to (12), we get the sum of the kth ion species’currents on all parts of ΓD,

∑ΓDm⊂ΓD

IDmkh = − ∑

ΓDm⊂ΓD

∑T∈𝒯 h

∫T

qkDk exp(−zkϕ)∇Φkh∇φIDm dT

= − ∑T∈𝒯 h

∫T

qkDk exp(−zkϕ)∇Φkh ⋅ ∇φIdT

= − ∑T∈𝒯 h

∫T

qkDk exp(−zkϕ)∇Φkh∇(φI− 1)dT = 0.

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Here, φI= ∑ΓDm⊂ΓD

φIDm ∈ Sh and φI

∣ΓD = 1; thus, φI− 1 ∈Wh. There-

fore,

∑ΓDm⊂ΓD

IDmtotal =

K

∑k=1∑

ΓDm⊂ΓD

IDmkh = 0.

IV. NUMERICAL EXPERIMENTSIn this section, we use different kinds of numerical examples to

test the accuracy and robustness of our IAFEM. At first, two bench-mark problems are presented to verify the method’s accuracy. Then,we compute ion transports in a synthetic nanopore system and showthat the method can accurately describe electrostatic potential andion concentration profiles in the nanopore. Finally, the method isapplied to studying improved conductance and high energy conver-sion for a 3D interconnected nanopore system, which is a simplifiedstructure of the sophisticated bioinspired hydrogel membrane. Thiswork is also a successful example of combining our method withrealistic chemical experiments.8 The IAFEM is implemented basedon the open-source finite element toolbox Parallel Hierarchical Grid(PHG).43 The computations were done on the high performancecomputers of State Key Laboratory of Scientific and EngineeringComputing, Chinese Academy of Sciences.

A. Benchmark problemsTwo numerical benchmark problems are introduced in this

subsection to verify the accuracy of our IAFEM. An aqueoussolution of KCl is always considered in the following benchmark

TABLE I. Comparisons between the standard FEM and our IAFEM for different cubesizes in a fixed mesh with 24 576 tetrahedral elements.

U 10−7 m 10−8 mCube size [−100, 100 nm]3 [−10, 10 nm]3

τ πL 5.63 5.63 × 10−2

IAFEM ✓ ✓

Standard FEM × ✓

problems, the dielectric constant ε is 80, and the diffusion coeffi-cients are DK+ = 1.96 × 10−9m2

/s and DCl− = 2.03 × 10−9m2/s.

Benchmark problem 1: In this benchmark problem, we con-sider a dimensionless PNP system (18) with source terms, which hasanalytical solutions (19) and (20) in a cube region Ω = [− L

2 , L2 ]

3. Thedimensionless PNP system,

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−∇ ⋅ (ε∇ϕ) =e2

c βε0(cK+ − cCl−) in Ω,

∇ ⋅Di(∇ci + zici∇ϕ) + fi = 0 in Ω, i = K+, Cl−,(18)

is augmented with Dirichlet boundary conditions on ∂Ω, namely,

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ϕ(x, y, z)∣∂Ω = ecβVappz + L/2

L,

cK+ ∣∂Ω = cb,

cCl− ∣∂Ω = cb.

FIG. 2. Our IAFEM’s numerical errors in the L2-norm of (a) the electrostatic potential ϕ, (b) K+ ion concentration cK+ , and (c) Cl− ion concentration cCl− when U = 10−7 mand Vapp = 0 V.

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Analytic solutions for K+ and Cl− concentrations are separatelydefined as

cK+ = cb+

cb

2cos(

πxL) cos(

πyL) cos(

πzL),

cCl− = cb−

cb

2cos(

πxL) cos(

πyL) cos(

πzL).

(19)

The analytic solution for the dimensionless electrostatic potential isdefined as

ϕ = τ cos(πxL) cos(

πyL) cos(

πzL) + ecβVapp

z + L/2L

. (20)

Here, τ = L2

3π2ε cb e2c βε0

, which is used to make the Poisson equationin (18) satisfied. The substitution of ion concentrations’ analyticsolutions (19) into NP equations in (18) produces source termsfi, i = K+, Cl−. Referring to (20), we get the analytic electrostaticfield,

Ð→E = −∇ϕ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

τπL

sin(πxL) cos(

πyL) cos(

πzL)

τπL

cos(πxL) sin(

πyL) cos(

πzL)

τπL

cos(πxL) cos(

πyL) sin(

πzL) −

ecβVapp

L

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

which controls the convection-domination effect of NP equations.We notice that the electrostatic field is influenced by coefficients

τ and Vapp. In this example, let cb= 0.5 mM and L = 2U, where

U is used to denote the unit length of the cube. Then, we haveτ = L2

3π2ε cb e2c βε0= 3.583 × 1014 U2

m2 .We use this cube benchmark problem to test the performance

of the IAFEM and the standard FEM.At first, we let Vapp = 0 and only consider the effect of τ.

From Table I, it is found that the standard FEM does not work inthe convection-domination case for a mesh with 24 576 tetrahe-dral elements. However, our method can handle the not-too-severeconvection-domination situation (U = 10−7 m), and the corre-sponding L2-norm numerical errors decrease with mesh refinements(see Fig. 2). If the convection is not dominating, both the standardFEM and our method perform well, and their L2-norm numericalerrors are pretty close (see Fig. 3).

In general nanopore system simulations, the diameter andlength of the nanopore are on the order of nanometers and microm-eters, respectively. Then, we mainly consider the effect of the appliedvoltage Vapp in a [−100, 100 nm]3 cube. If Vapp = 3.0 V, then ecβVapp

L= 58.41. In addition, the value range of the z-direction componentof the electrostatic field is [−64.04,−52.78], which illustrates that theapplied voltage (Vapp = 3.0 V) enhances the convection-dominationeffect in the z-direction. The IAFEM can still work in this case, andthe L2-norm of the numerical errors can reach relatively small valuesif the mesh size is small enough (see Fig. 4).

The above numerical results show that the standard FEM isonly suitable for the case that the convection is not dominating.

FIG. 3. Numerical errors in L2-norm of (a) the electrostatic potential ϕ, (b) K+ ion concentration cK+ , and (c) Cl− ion concentration cCl− when U = 10−8 m and Vapp = 0 V.

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FIG. 4. Our IAFEM’s numerical errors in the L2-norm of (a) the electrostatic potential ϕ, (b) K+ ion concentration cK+ , and (c) Cl− ion concentration cCl− when U = 10−7 mand Vapp = 3.0 V.

However, our IAFEM can solve the cube benchmark problem whenthe convection-domination is not too severe. Therefore, we preferthe IAFEM in nanopore system simulations. The following bench-mark problem also verifies the accuracy of our method from theperspective of the nanopore system.

Benchmark problem 2: This is a nanopore benchmark problemwhose simplified two-dimensional case has been introduced in Ref.18. The computational region is a cylindrical nanopore of height H= 0.5 μm and diameter d = 0.1 μm (see Fig. 5).

First, we fix the surface charge density σs to be −10−3 C/m2

on the top of the cylindrical nanopore ΓNc. On the Dirichlet bound-ary ΓD, a zero voltage is applied, and three different bulk concen-trations, cb

KCl = cbK+ = cb

Cl− = 10−4, 10−3, 10−2M, are considered. Inthis nanopore system, the exact surface potential ϕ is given by theGrahame equation44

ϕ =2kBT

ecsinh−1⎛

⎝

βσs

(8εε0cbKCl)

12

⎞

⎠. (21)

Table II compares the exact and computed values of the surfacepotential ϕ for different bulk concentrations. In addition, it showsthat the surface potential calculated from PNP equations using ourIAFEM is in good agreement with that obtained from the Grahameequation (21).

Then, we let the top of the cylindrical nanopore satisfy theDirichlet boundary conditions, i.e., the Neumann boundary ΓNc issubstituted by the Dirichlet boundary ΓD. On the bottom of the

nanopore, the voltage is always set to zero. In addition, a nonzerovoltage Vapp is applied on the top of the nanopore. The ion bulkconcentrations are set as 10−4M. In this nanopore system with-out surface charges, distributions of the electrostatic field and ionconcentrations in the x − y direction are negligible, and this sys-tem can be approximated as a one-dimensional system in the

FIG. 5. Computational region for Benchmark problem 2.

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TABLE II. Comparisons of exact and computed values of the surface potential withdifferent bulk concentrations.

cbKCl(M) ϕ(exact)(mV) ϕ(computed)(mV)

10–4−39.53 −39.52

10–3−13.60 −13.58

10–2−4.34 −4.29

z-direction. In the flux density Jk, the gradient of the ion concen-tration in the z-direction is assumed to be zero, and the electro-static field in the z-direction is assumed to be a constant value ofVapp/H. Hence, the total current flowing out of the top/bottom ofthe nanopore can be directly given by

Ittotal = ∑

k=K+ ,Cl−It

k = ∑k=K+ ,Cl−

(−Dkz2ke2

c βVapp

Hck

πd2

4),

Ibtotal = −It

total = ∑k=K+ ,Cl−

Dkz2ke2

c βVapp

Hck

πd2

4.

Table III compares the exact and computed values of the totalcurrents with different applied voltages and shows that differencesbetween them are negligible. It also implies that our method canguarantee the conservation of the computed total currents flowingthrough all parts of the Dirichlet boundary.

B. Ion transports in a synthetic nanoporeIn this subsection, we want to test the effectiveness of the

IAFEM by studying the effects of applied voltage, surface chargedensity, and ion bulk concentration on ion transports in a syntheticnanopore (see Fig. 6).

The diameter of the channel region dc is set to be 30 nm, whichcorresponds to the Debye length of 10−4M KCl electrolyte solu-

tion. The Debye length is calculated as λD =

√εε0kBT2e2

c cbKCl= 30.81 nm,

and cbKCl is the bulk concentration of the electrolyte solution. The

height of the channel region Hc is 1 μm. The diameter dr and the

TABLE III. Comparisons of computed and exact total currents for different appliedvoltages.

Vapp(V) Ittotal(A) Ib

total(A)

Absolute valuesof exact

currents (A)

2.0 −4.676 326 × 10−11 4.676 326 × 10−11 4.678 205 × 10−11

4.0 −9.352 653 × 10−11 9.352 653 × 10−11 9.356 410 × 10−11

6.0 −1.402 898 × 10−10 1.402 898 × 10−10 1.403 461 5 × 10−10

8.0 −1.870 531 × 10−10 1.870 531 × 10−10 1.871 281 9 × 10−10

10.0 −2.338 163 × 10−10 2.338 163 × 10−10 2.339 103 × 10−10

12.0 −2.805 796 × 10−10 2.805 796 × 10−10 2.806 923 × 10−10

14.0 −3.273 428 × 10−10 3.273 428 × 10−10 3.274 744 × 10−10

16.0 −3.741 061 × 10−10 3.741 061 × 10−10 3.742 563 × 10−10

18.0 −4.208 694 × 10−10 4.208 694 × 10−10 4.210 384 × 10−10

20.0 −4.676 326 × 10−10 4.676 326 × 10−10 4.678 205 × 10−10

FIG. 6. Computational region of a synthetic nanopore.

height Hr of the reservoir region are all 300 nm. In this subsection,an aqueous solution of KCl is used to study ion transports. Becausediffusion coefficients of K+ and Cl− are very close, we can focus onthe effects of applied voltage, surface charge density, and ion bulkconcentration.

Effect of applied voltage: We let the surface charge densityσs = −10−3 C/m2 and ion bulk concentrations be 10−4M. As for theapplied voltage, in the left reservoir region, Vapp∣ΓD = 0V , and Vappvaries from 0 to 5 V at an interval of 1 V in the right reservoir region.Figure 7 shows the electrostatic potential and ion concentration pro-files along the symmetry axis of the nanopore. By integrating the

FIG. 7. Effect of applied voltage on (a) electrostatic potential profiles and (b)ion concentration profiles along the symmetry axis of the nanopore (along thex-direction) for five different applied voltages. The bulk concentration of the KClaqueous solution is 10−4M, and the surface charge density is −10−3 C/m2.

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FIG. 8. Current–voltage (I–V) curves of the charged (blue lines) and uncharged(red lines) nanopore. The bulk concentration of the KCl aqueous solution is 10−4M,and the surface charge density is −10−3 C/m2.

Poisson equation (1) on any cross section of the channel region, weget the difference between ion concentrations of K+ and Cl− insidethe channel,

Δc = cK+ − cCl− = −4σs

ecdc.

In this case, the theoretical value of Δc is 1.38 mM. In Fig. 7(b), wesee that the ion concentration differences between K+ and Cl− insidethe channel are all about 1.1 mM with different applied voltages,which are close to the theoretical value. This also verifies that theion concentration difference is fully independent from the appliedvoltage.

The bulk concentration is obviously less than the ion concen-tration difference. Therefore, there exists a large potential barrier forCl− ions to enter the channel region, and the K+ ions accumulatein the channel region to neutralize negative surface charges on thesurface of the channel. As the applied voltage increases, more K+

ions accumulate in the channel, but the electrostatic field strengthdoes not increase proportionally [see Fig. 7(a)]. This will cause thecurrent–voltage (I–V) characteristics of a nanopore with surfacecharges to be nonlinear (see Fig. 8).

Effect of surface charge density: To study the effect of sur-face charge density, we let σs vary from 0 to −2 × 10−3 C/m2.

FIG. 9. Effect of surface charge density on ionic currents. The bulk concentrationof the KCl aqueous solution is 10−4M, and the applied voltage in the right reservoirregion is 5 V.

FIG. 10. Effect of ion bulk concentration on ionic currents. The ionic current of thecharged nanopore I is normalized by that of the uncharged nanopore I0. The sur-face charge density is −10−3 C/m2, and the applied voltage in the right reservoirregion is 5 V.

FIG. 11. Two simulated models. (a) Model 1 consists of two parallel cylindricalchannels. (b) Model 2 is the simplified model of the HEMAP hydrogel membrane.

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Other conditions are the same as that in the above case exceptfor the applied voltage in the right reservoir region, which is fixedat 5 V. Figure 9 shows the effect of surface charge density onionic currents. As the magnitude of the negative surface chargedensity increases, the ionic current of K+ keeps increasing, but theCl− current has a downward trend. This is because more negativesurface charges will attract more K+ ions to accumulate inside thechannel region and prevent Cl− ions from entering into the channel.However, the high concentration gradient will produce an opposingdiffusion current when K+ ions accumulate in the channel, so thegrowth rate of K+ current slows down as the magnitude of negativesurface charge density increases.

Effect of ion bulk concentration: In this case, the bulk concen-tration of the KCl aqueous solution ranges from 10−1 to 103 mM.The surface charge density σs is always fixed at −10−3 C/m2 forthe charged nanopore. In addition, the applied voltage Vapp is stillset as 5 V in the right reservoir region and 0 V in the left reser-voir region. In Fig. 10, the ionic current of the charged nanoporeI is normalized by that of the uncharged nanopore I0. In addition,Fig. 10 shows that the normalized ionic currents I/I0 of K+ and Cl−

both approach unity as the bulk concentration of the KCl aqueoussolution increases. This illustrates that the increase in the bulk con-centration leads to more accumulation of K+ ions inside the channelregion, which can effectively neutralize the negative surface chargeson the surface of the channel region. Then, the range affected bysurface charges is much smaller than the channel diameter, and thecharged nanopore behaves much like an uncharged one.

C. Simulation of a three-dimensional interconnectednanopore system

In this subsection, we introduce a successful application exam-ple using our IAFEM to realistic chemical experiments. Ournumerical simulation results provide a strong support for experi-ment observations that the 3D interconnected nanopore system canachieve improved conductance.8

The 3D structure provides sufficient channels for ion trans-ports and shortens the paths for ions transporting. Therefore, it canenhance the ion shuttle and improve conductance. Chen et al.8 real-ized a bioinspired hydrogel membrane, 2-hydroxyethyl methacry-late phosphate (HEMAP) hydrogel membrane, with 3D intercon-nected and space-charged nanopores. For more details about theexperiments, please refer to Ref. 8. We verify the above conclusionabout the 3D interconnected and space-charged nanopores from theperspective of numerical simulations based on the PNP model.

The model of the HEMAP hydrogel membrane is simplified toFig. 11(b) (model 2) and compared with the model consisting oftwo parallel cylindrical nanopores, as shown in Fig. 11(a) (model1). The channel length Hc and diameter dr of both models are setto 25 and 5 nm, respectively. In this nanopore system, we onlyconsider the effect of the space charge density. Therefore, thereis no surface charge on the surface of the channel region, that is,ε∂ϕ∂n ∣ ΓN

= 0. From our numerical experience, the standard FEM andthe SUPG method28 cannot work for the interconnected nanoporesystem (model 2) with nonzero space charge density, so we adoptour IAFEM in the following 3D interconnected nanopore systemsimulation experiments.

According to the reported work,45 the space charge densityσb of the HEMAP hydrogel membrane usually ranges from 10 to20 C/cm3. We set it to 10, 15, and 20 C/cm3, respectively. Then,we design some numerical experiments to verify that the 3D inter-connected structure of the nanomembrane can improve its con-ductance. In these numerical experiments, the 0.1M KCl electrolytesolution is selected and different applied voltages are considered.The length of the reservoir region Hr is set to 3 nm. Figure 12shows the I–V curves of the above two models with different spacecharge densities. All currents increase linearly with the appliedvoltage increases, and they also increase when the space charge den-sity increases. Moreover, the currents of model 2 are always higherthan those of model 1, which indicates that the ion transport in theinterconnected nanopores of model 2 is more efficient than that in

FIG. 12. Simulated I–V curves of the two models with different space charge densities. Left: Model 1. Right: Model 2.

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FIG. 13. Conductance of two models with different space charge densities.

the parallel nanopores of model 1. Figure 13 directly shows that theconductance of model 2 is always higher than that of model 1 withdifferent space charge densities. This once again verifies that the3D interconnected structure of the nanomembrane can improve itsconductance.

V. CONCLUSIONSSolving the PNP model in nanopore system simulations is a

challenging task because of the convergence difficulty and numer-ical instability caused by the large-size nanopore system and theconvection-domination problem. In this work, we extend the lastscheme with the best performance in our previous work33 tonanopore simulations and propose the IAFEM to solve the 3DPNP equations. This scheme first transforms NP equations intoself-adjoint equations with exponential coefficients by employingthe Slotboom variables. In addition, because of the moderate vari-ation of flux densities, we use constant vectors to approximateflux densities on each tetrahedral element of the grid. Correspond-ingly, the exponential coefficients are also substituted with theirharmonic averages, which are calculated with the inverse aver-aging technique on every edge of the tetrahedral element. Ourscheme possesses good convergence performances when simulat-ing single or porous nanopore systems. Moreover, it can handle theconvection-domination NP equations and preserve the conservationof computed currents with our current evaluation approach. Thisovercomes the disadvantages of some stabilization methods thatcannot maintain the current conservation. Our numerical experi-ments with benchmark problems verify the accuracy and robustnessof the IAFEM. In addition, the numerical results further show thatour method performs better than the standard FEM when handlingconvection-domination NP equations. A series of successful numer-ical calculations on synthetic and 3D interconnected nanopore sys-tems in chemical experiments also implies that the IAFEM is stilleffective for real nanopore systems.

ACKNOWLEDGMENTSThis work was supported by the National Key R & D Program

of China (Grant Nos. 2019YFA0709600 and 2019YFA0709601) andthe National Natural Science Foundation of China (Grant Nos.11771435 and 22073110).

AUTHOR DECLARATIONSConflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are avail-able within the article. The source code is available fromhttps://github.com/bzlu-Group/Nanopore.

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J. Chem. Phys. 155, 194106 (2021); doi: 10.1063/5.0066194 155, 194106-13

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