Debye-Hückel theory for interfacial geometries

Roland R. Netz

Introduction – DH theory

PB theory describe the potential distribution of the electrolyte solution near the interface

DH potential - Solution of PB equation in linearized case

http://depts.washington.edu/solgel/images/courses/MSE_502/Ch_2/Figure_2.14.JPG

2

01

iz enkT

i ii

F c z eψ

ψε

→ −

=

∇ = − ∑ Where, F is Faraday constant

For uniform, diluted monovalent ion case: Spherical symmetry

2 2 21 2e eFc ceψ ψ→ −∂ 2 02 2

1 2( ) ( )kT kTFc cer e ekTr r

ψ ψ ψε ε

∂∇ = = − − +∂

Introduction – DH theory

Solve this inhomogeneous PDE,

22re ceκ−0

1 3 04Α0

2,e cer kT

ψ ψ κε

= = 1 3.04D cλ κ − Α= =

Because DH theory has linear relation, superposition principle is valid.

But with interface, DH theory should be modified due to polarization charges at the interface.

Modification Mobile salt ion distributions at the upper, lower, interface are given as,

> >j j different type of n ions with charge q (j=1,2,..., )M M> >

= =j j different type of n ions with charge q (j=1,2,..., )M M= =

00

zz>=

< <j j different type of n ions with charge q (j=1,2,..., )M M< <0z <

And satisfy the Electroneutrality condition in each regionAnd locate test charges at carry i iR Q

Partition function of system produced by electrostatic potential is,

condition in each region

Modification In equation, coulomb operator v(r,r’) is,

‘Classical electrodynamics, John David Jackson Chapter4’

1ε 2ε

1 ( 0)E zε ρ→ →∇ = >i 1 1

li liz zE E

E Eε ε⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬

q

1

2

( )

0( 0)

0( )

E z

E h

ε→ →

→ →

∇ = <

∇×

i0 0

lim limx xz z

y y

E EE E

→ + → −

⎪ ⎪ ⎪ ⎪→ =⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

<B d diti >z 0( )E everywhere∇× =

By using image charge method (in cylindrical coordinates)

<Boundary condition>

'1 ( ) ( 0)q q fΦ + >

By using image charge method,(in cylindrical coordinates)

1( )E zρ φ→ → ∧ ∧ ∧∂ ∂ ∂= ∇Φ ∝ + + Φ

1 1 2

( ) ( 0)4

q q for zR Rπε

Φ = + >

1 ''( ) ( 0)q for zΦ= <

( )E zz

ρ φρ ρ φ

= −∇Φ ∝ + + Φ∂ ∂ ∂

Takes the limit and compare

2 2

( ) ( 0)4

for zRπε

Φ= <component by component in B.C

Modification ' '' ' ''

1 2

1 1, ( )q q q q q qε ε

− = + =

2

: Bjerrum length4B

elkTπε

=

And charge density operator is given by, ( r )ρ∧ →

Modification After Hubbard-Stratonovich transformation, the partition function is up to second order fluctuating field is given as,φ

0where Z is the partition function of inverse coulomb operator,

0Z det( ), is ideal entropy mixing v S∝

, , : screening length about upper, lower, in planeκ κ κ> < =

2 2 22 2 2'4 ( ) , 4 ( ) , 4 ( )B j j B j j B j jj j j

l q c l q c l q cεκ π κ π κ πε

> > < < = <> < == = =∑ ∑ ∑

After shifting the fluctuating field , linear terms of in (6) is removed and the effective free energy term of n test particle can be separated

φ φ

be separated.

Modification

effective free energy of test particle

After Fourier transformation of in plane direction and adjust the canonical relation between and it’s inverse we can set the differential equation and solve it for each regions.

DHV ( , ') r r DHV ( , ') r r

After transformation the equation is 2nd order partial Differential equations contain source termDifferential equations contain source term

Modification Result

'F 0≥ ≥For z 0 z≥ ≥

'And for z 0,z 0≥ ≥'

where εηε

=

These results are can be applied in various situationEx) metallic half plane, air water interface,

lipid bilayer immersed in salt solution….

Application (limit case)⊙ First check the conversion to the classical DH theory (bulk solution)

'( , 0, and no dielectiric jump, = )κ κ κ ε ε> < == =

' 2 2

DH 2 2

2V ( , ', ) z z pBlz z p ep

κπκ

− − +=+

'( )DH DH2V ( , ') V ( , ', )

(2 )i r r pd pr r e z z p

π

→→ →→

−= ∫ i

It converse to classical result

⊙ In metallic half space ,η →∞

2' 2 2 2( ')DH 2 2

2V ( , ', ) z z p z z pBlz z p e ep

κ κπκ

> >− − + − + +⎧ ⎫= −⎨ ⎬

⎩ ⎭+ From here we can define the self energy of test chargeself energy of test charge

Application (Self energy)

So, interface strongly attracts any kind of charges

2 22

(2 )2 2 2

22(2 )

zz pB

Bld p ee l

zp

κκπ

π κ

>>

→−

− +⎡ ⎤−⎢ ⎥= = −⎢ ⎥+⎣ ⎦

∫

y g

⊙ In low dielectric half space ,η →∞

2' 2 2 2( ')DH 2 2

2V ( , ', ) z z p z z pBlz z p e ep

κ κπκ

> >− − + − + +⎧ ⎫= +⎨ ⎬

⎩ ⎭+

2 2(2 )

2 2 2 2

2( , , ) [ ]2

z pself BDH

lV z z p ep p

κπ κκ κ κ

− ++ −=+ + +

Interface strongly repel any kind of charges

Application (salty interface)a) Thin lipid bilayer immersed in solution , 1 and η κ κ κ> <= = =

2 2(2 )2( ) [ ]z pself BlV z z p e κπ κ − ++ −=2 2 2 2

( , , ) [ ]2

DHV z z p ep pκ κ κ

=+ + +

2 2(2 )22( ) ( ) [ ]z pself self Bld p pdpV z V z z p e κππ κ→

− ++ −∫ ∫ ( )

2 2 2 2 2 2( ) ( , , ) [ ]

(2 ) (2 ) 2pf f

DH DHV z V z z p ep pπ π κ κ κ

= =+ + +∫ ∫

※ By definition of incomplete Gamma function,

1 2 2" (0, ) ", set ( 2 , ( 2 ))t

xx t e dt t z p x zκ κ κ κ

∞ − −= =Γ = − = + = +∫

1for z ( 2 )κ κ −= +

1for z ( 2 )κ κ −= +

Always attractive

Application (salty interface)b) Monolayer at air / water interface , 0η =

2 22 2

(2 )

2 2 2 2

2( , , ) [1 ]z pself BDH

plV z z p ep p

κκ κπκ κ κ

− +=

=

→

+ −+= ++ + +

2 22 2

(2 )2 2 2 2 2 2

2 2

22( ) ( , , ) [1 ](2 ) (2 )

z pself self BDH DH

pld p pdpV z V z z p ep p

l l

κκ κπππ π κ κ κ

→

− +=

=

+ −+= = +

+ + +∫ ∫

2 22 2

(2 )2 2 2 2 2

22 [1 ]2(2 )

z pB Bpl lpdp ep p

κκ κπππ κ κ κ

− +=

=

+ −+= + =

+ + +∫ (2 )(2 ) 2 (0,2 ( )zzBe l e z

zκκ κ κ κ=−

= =− Γ +

It has some different behavior with ratio between ,κ κ=

Application (salty interface)

) Bil la) Bilayer case : alwaysattractive

b) Monolayer air/waterb) Monolayer,air/water interface : have some minima if κ κ= >

Application (polymer)Polyelectrolyte polymer can be regarded as line charge

llRecall

Make continuous distribution and do transformation, ,then self energy for polymer is given as,

For large and small separation from wall, self energy is asymptotically,

1for z κ −>

Attraction and repulsion rely on the charge >

1for z κ −

y gdensity ratio

Attraction and repulsion for z κ> rely on the dielectric constant ratio

Conclusion

⊙ Debye-Hückel theory can be modified if we consider screening effect at the boundary

⊙ Separate partition function with test charge and system by field⊙ Separate partition function with test charge and system by fieldtheoretical approach, define the self energy of test charge

⊙ For high dielectric constant ratio, in the case of metallic half space, there is attractive interaction with interface and test chargethere is attractive interaction with interface and test charge

⊙ For low dielectric constant ratio, in the case of air / water interface, attraction and repulsion depend on salt concentration ratio betweeninterface and bulkinterface and bulk

Limitation

⊙ Debye-Hückel theory fit well in low concentration of salt.It means that higher correction PB equation should be considered.

⊙ The system was restricted that summation of charge in each region is⊙ The system was restricted that summation of charge in each region iszero and the salt ions are confined.

⊙ Mutual interaction between test charges should be considered carefully if the interface strongly attract themif the interface strongly attract them