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A charged nanopore model for nanofiltration (NF)
帶電奈米孔洞模式研究奈米過濾
By
Dept. of Applied Math. Feng Chia University
Taichung, Taiwan [email protected]
http://newton.math.fcu.edu.tw/~tlhorng
Allen T.-L. Horng (洪子倫)
IMA Hot Topics Workshop Mathematics of Biological Charge Transport: Molecules and Beyond, July 20-24, 2015
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• From the point of view of continuum model, ion channel is actually a charged nanopore immersed in electrolyte with complicated geometry (structure) and solvated environment.
• Model is also complicated (steric effect, solvation situation, …). Charge distribution is complicated. 3D Computation is difficult.
• Have we fully understood the physical mechanism of a simple charged nanopore? If not (at least for myself), let us study a simple cylindrical nanopore with uniform surface charge density first. Avoid complex geometry and charge distribution, and focus on effectiveness of model.
• Easier to conduct experiments to check model. • Charged nanopore nowadays has important applications: desalination,
supercapacitor (quick current for TESLA motor), DNA translocation, electro-kinetic battery …
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美國加州乾旱 (California mega-‐drought)…
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NF membrane
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Flow is driven by pressure
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If driven by electric potential, will be EOF.
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However, the sizes of ions are smaller than a nanopore. How does a nanopore sieve ions? The answer is by electrostatics (overlapping electric double layer).
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Mathematical model: PNP-steric and Navier-Stokes
equations in axisymmetric coordinate
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Poisson-Nernst Planck equations with steric terms (PNP-steric)
01
( ) ,N
i ii
e z ecε φ ρ=
−∇⋅ ∇ = +∑
∂ci
∂t+∇⋅
!Ji = 0,
!Ji =!uci − Di∇ci −
Dici
kBTzie∇φ −
Dici
kBTgij∇cj
j=1
N
∑ , i = 1,", N ,
based on variation of free energy:
Eδ =ρ2!u2+ kBT ci logci +
12 ρ0e+ zieci
i=1
N
∑"
#$$
%
&''φ
i=1
N
∑"
#$$
%
&''∫ d!x +
gij2 ci
!x( )c j
!x( )∫ d!x
i , j=1
N
∑ ,
T.-‐L. Horng, T.-‐C. Lin, C. Liu and B. Eisenberg*, 2012, "PNP equaFons with steric effects: a model of ion flow through channels", Journal of Physical Chemistry B, 116: 11422-‐11441
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Non-dimensionalization:
and obtain:
!c =ci
c0
, !ρ0 =ρ0
c0
, !φ = φkBT / e
, !s = sL
, !t = tL2 / DK
, !Di =Di
DK
, !gij =gij
kBTc0
,
"!Ji ="Ji
c0DK / L, !I = I
c0DK L, "!u =
"uDK / L
.
20
1,
N
i iiz cφ ρ
=
−Γ∇ = +∑
∂ci
∂t+∇⋅
!Ji = 0,
!Ji =!uci − Di ∇ci − cizi∇φ − ci gij∇cj
j=1
N
∑⎛
⎝⎜⎞
⎠⎟, i = 1,", N ,
where , and the Debye length . 2
2LλΓ =
λ =
εkBTc0e
2
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Considering axisymmetric nanopore with binary electrolyte:
2
021
1 ,N
i ii
r z cr r r z
φ φ ρ=
⎛ ⎞∂ ∂ ∂−Γ + = +⎜ ⎟∂ ∂ ∂⎝ ⎠∑
( ), ,1 0,i r i zirJ Jc
t r r z∂ ∂∂ + + =
∂ ∂ ∂
Ji,r = urci − Di
∂ci
∂r+ cizi
∂φ∂r
+ ci gij
∂cj
∂rj=1
N
∑⎛
⎝⎜⎞
⎠⎟, i = p,n,
Ji,z = uzci − Di
∂ci
∂z+ cizi
∂φ∂z
+ ci gij
∂cj
∂zj=1
N
∑⎛
⎝⎜⎞
⎠⎟, i = p,n.
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Domain decomposition
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Boundary conditions at reservoirs:
Boundary conditions at wall:
Γ ∂φ∂n
=σ , !J p ⋅!n =!Jn ⋅!n = 0,
0.p nr r rφ∂ ∂ ∂= = =∂ ∂ ∂
Boundary conditions at r=0 and rmax:
φ = φL , as z →−∞; ∂φ∂z
= 0, as z →∞,
p = pL , as z →−∞; ∂p∂z
= 0, as z →∞,
n = nL , as z →−∞; ∂n∂z
= 0, as z →∞.
zp p + znn = 0, z →∞,
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Interface (between pore and reservoirs) conditions:
where the subscripts + and – stand for two sides adjacent to the interface.
φ− = φ+ , Γ ∂φ∂z
⎛⎝⎜
⎞⎠⎟ −
= Γ ∂φ∂z
⎛⎝⎜
⎞⎠⎟ +
,
p− = p+ , !J p ⋅!n( )
−=!J p ⋅!n( )
+,
n− = n+ , !Jn ⋅!n( )− =
!Jn ⋅!n( )+ ,
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Navier-Stokes equations:
0,z r ru u uz r r
∂ ∂+ + =∂ ∂
2 2
2 2
1 1 ,z z z z z zz r z
u u u u u upu u Ft z r z z r r r
νρ
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂+ + = − + + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠2 2
2 2 2
1 1 ,r r r r r r rz r r
u u u u u u upu u Ft z r r z r r r r
νρ
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂+ + = − + + + − +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
, ,e ez rF F
z rρ ρφ φρ ρ
∂ ∂= − = −∂ ∂
e i iiz c eρ =∑
!u = ur
!er + ur!er ,
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Non-dimensionalization:
Uref =
DK
L, !p = p
ρUref2 , !ui =
ui
Uref
, !r = rL
, !z = zL
, !t = tL2 / DK
.
0,z r ru u uz r r
∂ ∂+ + =∂ ∂
2 2
2 2
1 ,z z z z z zz r x i i
i
u u u u u upu u Sc G z ct z r z z r r r z
φ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂ ∂⎛ ⎞+ + = − + + + −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠∑
2 2
2 2 2
1 ,r r r r r r rz r x i i
i
u u u u u u upu u Sc G z ct z r r z r r r r r
φ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂ ∂⎛ ⎞+ + = − + + + − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠∑
Sc = ν
DK
= 446,
Gx =
c0kBTρUref
2 ,
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Vorticity transport equation:
!ω = ∇× !u = !eθ
∂ur
∂z−∂uz
∂r⎛⎝⎜
⎞⎠⎟= !eθω ,
2 2
2 2 2
1
.
irz r x i
i
ix i
i
cuu u Sc G zt z r r z r r r r z r
cG zr z
ωω ω ω ω ω ω ω φ
φ
⎛ ⎞ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ + − = + + − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠∂ ∂⎛ ⎞+ ⎜ ⎟∂ ∂⎝ ⎠
∑
∑
1 1, ,z ru ur r r z∂Ψ ∂Ψ= = −∂ ∂
Stokes stream function:
∂E2Ψ∂t
+ 1r∂Ψ∂z
∂E2Ψ∂r
− 1r∂Ψ∂r
∂E2Ψ∂z
+ 2r 2
∂Ψ∂z
E2Ψ = ScE4Ψ
+rGx zi
∂ci
∂zi∑⎛⎝⎜
⎞⎠⎟∂φ∂r
− rGx zi
∂ci
∂ri∑⎛⎝⎜
⎞⎠⎟∂φ∂z
,
2 22
2 2
1 ,Er r r z∂ ∂ ∂= − +∂ ∂ ∂
21 .Er
ω = − Ψ
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Boundary conditions:
0, =0, as ,z ru U u z= →∞
=0, at wall,z ru u=
max0, at 0, .zr
u u r rr
∂ = = =∂
20
1 , =0, as ,2 zU r zΨ = Ψ →∞
20 max
1 , =0, at wall,2U r
n∂ΨΨ =∂
0, =0, at 0,rr
∂ΨΨ = =∂
20 max max
1 1, =0, at .2U r r r
r r r∂ ∂Ψ⎛ ⎞Ψ = =⎜ ⎟∂ ∂⎝ ⎠
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Decoupling of PNP-Steric and Navier-Stokes equations:
ˆ ,p p p= +
ˆ0 ,x i i
i
p G z cz z
φ∂ ∂⎛ ⎞= − − ⎜ ⎟∂ ∂⎝ ⎠∑
0 = − ∂ p̂
∂r−Gx zici
i∑⎛⎝⎜
⎞⎠⎟∂φ∂r
,
2 2
2 2
1 ,z z z z z zz r
u u u u u upu u Sct z r z z r r r
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂+ + = − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
2 2
2 2 2
1 .r r r r r r rz r
u u u u u u upu u Sct z r r z r r r r
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂+ + = − + + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
∂E2Ψ∂t
+ 1r∂Ψ∂z
∂E2Ψ∂r
− 1r∂Ψ∂r
∂E2Ψ∂z
+ 2r 2
∂Ψ∂z
E2Ψ = ScE4Ψ.
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Numerical method
• MulF-‐block Chebyshev pseudopectral method together with the method of lines (MOL) to solve governing equaFons with the associated boundary/interface condiFons .
• Governing equaFons are first semi-‐discreFzed in space together with boundary condiFons.
• The resulFng equaFons are a set of coupled ordinary differenFal algebraic equaFons (ODAEs).
• The algebraic equaFons come from the Poisson equaFon and those boundary/interface condiFons which are all Fme-‐independent.
• This ODAE system is index 1, which can be solved by many well-‐developed ODAE solvers. ode15s in MATLAB is a variable-‐order-‐variable-‐step index-‐1 ODAE solver, that can adjust the Fme-‐step to meet the specified error tolerance, and integrate with Fme efficiently. The numerical stability in Fme is automaFcally assured at the same Fme.
• The spaFal discreFzaFon is performed by the highly-‐accurate Chebyshev pseudospectral method with Chebyshev Gauss-‐Loba_o grid and its associated collocaFon derivaFve matrix.
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Streamlines and velocity profiles:
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How good is our model? It will be compared with the most popular 1D NF model: Donnan steric pore model with dielectric exclusion (DSPM-DE), which was developed by chemists.
Reference: A. A. Hussain, M. E. E. Abashar, and I. S. Al-Mutaz, Influence of ion size on the prediction of nanofiltration membrane systems, Desalination, 214 (2007) 150-166.
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!! Interface!partition!coefficient:!ki = [steric]×[electrostatic!(Donnan)]×[solvation!(Born)]×…
Only computing extended Nernst-Planck equation inside pore with interface conditions related to concentrations and electric potential at reservoirs.
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)1(⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
dxd
RTDc
ucKj iipiiici
µ
)2()441.0988.0054.00.1)(2( 32iiiiicK λλλφ +−+−=
)3()1( 2ii λφ −=
)4(p
ii rr=λ
)5(ηηo
iidip DKD ∞=
)6(224.0154.1304.20.1 32iiiidK λλλ ++−=
(extended Nernst-Planck equation)
(steric partition coefficient)
(related to viscosity by Stokes-Einstein equation)
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ji = Ki,cci (x)u−Dipci (x)∂x lnγ i −Dip∂xci (x)
− 1RT
ViDipci (x)∂xP −FRT
ziDipci (x)∂xψ(10)
(9)iii ca γ=
ηηo
=1.0+18 drp
!
"##
$
%&&− 9
drp
!
"##
$
%&& (7)
(8)constantln +++= ψµ FzPVaRT iiii
∂xP =ΔPeΔx
=8ηurP2 (11)
(d: thickness of the oriented swater layer, 0.28 nm)
((10) is from differentiating (8) and substituting into (1))
(Hagen-Poiseuille equation)
0 (Debye-Huckel)
ai: activity, γi: activity coefficient
2
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)12(πΔ−Δ=Δ PPe
)13(82 dx
dcDzRTF
dxdcDucVD
RTrKj iipi
iipiiip
pici
ψη −−⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
)14()( uCj ii+= δ
)15()(82 dx
dczRTF
DuCcVD
RTrK
dxdc
iiip
iiiipp
ici ψδη −
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−= +
( )∑=
−=Δn
ipiwi ccRT
1,,π(Van’t Hoff )
((13) is from substituting (11) into (10), convection+diffusion+electro-migration)
((15) is from substituting (14) into (13))
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)16(
)(8
1
2
12
1
dxdcz
RTF
DuzCcVD
RTrK
dxdcz
n
iii
n
i ip
iiiip
pic
n
i
ii
ψ
δη
⎟⎠⎞⎜
⎝⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∑
∑∑
=
=
+
=
)17(0)(,0)0(11
== +
=
−
=∑∑ δin
iii
n
ii CzCz
!!zi
i=1
n
∑ ci(x)= −χd , 0< x <δ , (18)
Electro-neutrality at external solutions:
Electro-neutrality inside the pore:
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)19(
)(8
1
2
12
∑
∑
=
=
+
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
= n
iii
n
i ip
iiiip
pic
czRTF
DuzCcVD
RTrK
dxd
δηψ
!!
ki = [steric]×[electrostatic!(Donnan)]!!!!!!!×[solvation!(Born)]×…
(20)
((19) is from differentiating (18) and substituting into (16))
(partition coefficient)
(iterating on Donnan potential to satisfy electro-neutrality inside the pore) !!
ki 0 = !Ci(0+ )Ci(0− ) =φi exp −
FziRT
Δψ D(0)⎛
⎝⎜⎞
⎠⎟
!!!!!!!exp −ΔWi(0)kT
⎛
⎝⎜⎞
⎠⎟, !!! ziCi(0+ )
i=1
n
∑ = ziCi(0− )i=1
n
∑ φi
!!!!!!!exp −FziRT
Δψ D(0)⎛
⎝⎜⎞
⎠⎟exp −
ΔWi(0)kT
⎛
⎝⎜⎞
⎠⎟= −χd !
(21)
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)23(permeate)(
)feed()()()(
)0()0()0(+−
−+
−=Δ
−=Δ
δψδψδψψψψ
D
D
)24(118
22
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=Δ
bpio
ii r
ezWεεπε
(a jump in electrical potential can be understood from EDL or Poisson equation)
(dehydration when ions entering pore, change of dielectric constant, solvation energy based on Born model)
!!
ki δ = !Ci(δ − )Ci(δ + ) =φi exp −
FziRT
Δψ D(δ )⎛
⎝⎜⎞
⎠⎟
exp −ΔWi(δ )kT
⎛
⎝⎜⎞
⎠⎟, !!! ziCi(δ − )
i=1
n
∑ = ziCi(δ + )i=1
n
∑ φi
exp −FziRT
Δψ D(δ )⎛
⎝⎜⎞
⎠⎟exp −
ΔWi(δ )kT
⎛
⎝⎜⎞
⎠⎟= −χd !
(22)
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!!
εp =
2πrεb dr +0
rp−d
∫ 2πrε *drrp−d
rp
∫πrp
2
!!!!! = εb −2(εb −ε * )drp
⎛
⎝⎜
⎞
⎠⎟ +(εb −ε * )
drp
⎛
⎝⎜
⎞
⎠⎟
2(25)
!!Ri =1−
Ci(δ + )Ci(0− ) (26)
(the wall of pore covered by one layer of oriented water molecules of thickness d and dielectric constant ε*)
(rejection coefficient)
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Results: a case of NF
Parameters: [KCl]=0.011982M, r0=2nm, Uref=0.97850m/s, 4 dielectric situations inside pore are considered: (1) εp=80, λb=4nm, Γp=4, (2) εp=40, λb=2.8284nm, Γp=2, (3) εp=20, λb=2nm, Γp=1, (4) εp=10, λb=1.4142nm, Γp=0.5, Surface charge density σ=-2 (ζ=-17.945mV), only distributed inside pore. Diffusion coefficient in pore reduced to 0.25 bulk value (from DSPM-DE). Input: a bunch of U0’s with various gpn=gnp (gnn=gpp =0). Output: salt rejection rate R=
ci(−∞)− ci(∞)ci(−∞)
.
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Comparison with DSPM-DE model
Generally, salt rejection increases with flow velocity (pressure).
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Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
Γp=4, U0=0.003, gpn=0.
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Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
Γp=4, U0=0.003, gpn=0.5
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Γp=2, U0=0.003, gpn=0.
Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
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Γp=2, U0=0.003, gpn=0.5
Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
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Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
Γp=1, U0=0.003, gpn=0.
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Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
Γp=1, U0=0.003, gpn=0.5
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Γp=0.5, U0=0.003, gpn=0.
Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
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Γp=0.5, U0=0.003, gpn=0.5
Steady-state whole domain distributions of (a) p, (b) n, (c) ϕ, (d) ρe, (e) distributions of p, n versus r at z=9 (center location of pore) and (f) distribution of p and n along axis (r=0).
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Steric effect and dielectric exclusion
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Fail to agree with DSPM-DE at εp=40, since dielectric exclusion (related to solvation energy described by Born model) dominates in DSPM-DE.
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Conclusions and future works
• Convection breaks symmetry and causes salt rejection when passing a charged nanopore.
• In NF, NS-PNP-steric model agrees well with DSPM-DE for εp=80, but poorly at εp=40.
• High salt rejection in DSPM-DE is chiefly due to strong dielectric exclusion (solvation energy barrier modeled by Born model), which can not be fit by NS-PNP-steric model (without extra solvation energy added) no matter how gpn is adjusted.
• With solvation energy, eg. Born model, added into energy of present model, jump condition on ionic concentrations at interfaces happens. It has been derived from continuity of flux. Computations based on it will be conducted in the future to compare with DSPM-DE again.
• Large gpn with bi-Laplacian diffusion (single-file diffusion) will be applied when the pore size is further reduced (more significant finite-size effect). [Q. Chen, J. D. Moore, Y.-C. Liu, T. J. Roussel, Q. Wang, T. Wu, and K. E. Gubbins, 2010, Transition from single-file to Fickian diffusion for binary mixture in single-walled carbon nanotubes, J. Chem. Phys., 113, 094501]
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Over-screening will not happen in 2:1 electrolyte without large gpn here.
Importance of large gpn with bi-Laplacian diffusion, eg. charged wall problem (EDL): result compared with Boda et al. (2002) (a MC simulation)
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Thank you for your a,en.ons. Ques.ons?