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 tlhorng123@gmail.com

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

•  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 …

美國加州乾旱 (California  mega-­‐drought)…

NF membrane

Flow is driven by pressure

If driven by electric potential, will be EOF.

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).  

Mathematical model: PNP-steric and Navier-Stokes

equations in axisymmetric coordinate

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

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

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.

Domain decomposition

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 →∞,

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( )+ ,

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 ,

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 ,

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

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 = =⎜ ⎟∂ ∂⎝ ⎠

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.

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.

Streamlines and velocity profiles:

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.

!! 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.

)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)

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

)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))

)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:

)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)

)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)

!!

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)

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(−∞)

.

Comparison with DSPM-DE model

Generally, salt rejection increases with flow velocity (pressure).

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.

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

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).

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).

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.

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

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).

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).

Steric effect and dielectric exclusion

Fail to agree with DSPM-DE at εp=40, since dielectric exclusion (related to solvation energy described by Born model) dominates in DSPM-DE.

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]

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)

Thank  you  for  your  a,en.ons.  Ques.ons?