Mean-Field Description of Ionic Size Effects

Bo LiDepartment of Mathematics and

NSF Center for Theoretical Biological PhysicsUniversity of California, San Diego

Work Supported by NSF, NIH, CSC, CTBP

ICMSEC, Chinese Academy of Sciences, BeijingJune 17, 2011

Biomolecular Interactions

Charge effect. Left: no charges. Right: with charges.

Some existing works on special cases of size effects.

V. Kralj-Iglic and A. Iglic. A simple statistical mechanicalapproach to the free energy of the electric double layerincluding the excluded volume effect. J. Phys. II (France),6:477-491, 1996.

I. Borukhov, D. Andelman, and H. Orland. Steric effects inelectrolytes: A modified Poisson-Boltzmann equation. Phys.Rev. Lett., 79:435-438, 1997.

V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag, and S. Doniach.Evaluation of ion binding to DNA duplexes using asize-modified PoissonBoltzmann theory. Biophys. J,93:3202-3209, 2007.

Some existing works on special cases of size effects (cont’d)

G. Tresset. Generalized PoissonFermi formalism forinvestigating size correlation effects with multiple ions. Phys.Rev. E, 78:061506, 2008.

X. Shi and P. Koehl, The geometry behind numerical solversof the Poisson–Boltzmann equation, Commun. Comput.Phys., 3,1032–1050, 2008.

A. R. J. Silalahi, A. H. Boschitsch, R. C. Harris, and M. O.Fenley, Comparing the predictions of the nonlinearPoisson–Boltzmann equation and the ion size-modifiedPoisson–Boltzmann equation for a low-dielectric chargedspherical cavity in an aqueous salt solution, J. Chem. TheoryComput. 6, 3631-3639, 2010.

Sizes of ions are different! Sodium: 3.34 A, Chloride: 2.32 A.

Main references

Bo Li, Minimization of electrostatic free energy and thePoisson–Boltzmann equation for molecular solvation withimplicit solvent, SIAM J. Math. Anal., 40, 2536–2566, 2009.

Bo Li, Continuum electrostatics for ionic solutions withnonuniform ionic sizes, Nonlinearity, 22, 811–833, 2009.

Shenggao Zhou, Zhongming Wang, and Bo Li, Mean-fielddescription of ionic size effects with non-uniform ionic sizes: Anumerical approach, Phys. Rev. E, 2011 (in press).

Bo Li, Xiaoliang Cheng, and Zhengfang Zhang, Dielectricboundary force in molecular solvation with thePoisson–Boltzmann free energy: A shape derivative approach,2011 (submitted to SIAM J. Applied Math.).

Outline

1. The Classical Poisson–Boltzmann Theory

2. Mean-Field Models with Ionic Size Effects

3. Non-Uniform Sizes: Constrained Optimization

4. Numerical Results

5. Conclusions

1. The Classical Poisson–Boltzmann Theory

Consider an ionic solution occupying a region Ω.

ρf : Ω → R: given, fixed charge density

cj : Ω → R: concentration of jth ionic species

c∞j : bulk concentration of jth ionic species

qj = zje : charges of an ion of jth species

zj : valence of ions of jth species

e : elementary charge

β: inverse thermal energy

Poisson’s equation:

Charge density:

Boltzmann distributions:

∇ · ε(x)ε0∇ψ(x) = −ρ(x)

ρ(x) = ρf (x) +∑M

j=1 qjcj(x)

cj(x) = c∞j e−βqjψ(x)

The Poisson–Boltzmann Equation (PBE)

∇ · εε0∇ψ +M

∑

j=1

qjc∞

j e−βqjψ = −ρf

PBE ∇ · εε0∇ψ +

M∑

j=1

qjc∞

j e−βqjψ = −ρf

The Debye–Huckel approximation (linearized PBE)

∇ · εε0∇ψ − εε0κ2ψ = −ρf

Here κ > 0 is the ionic strength or the inverse Debye–Huckelscreening length, defined by

κ2 =β

∑Mj=1 q2

j c∞

j

εε0

The sinh PBE (q2 = −q1, c∞2 = c∞1 )

∇ · εε0∇ψ − 2qc∞1 sinh(βqψ) = −ρf

Electrostatic free-energy functional

G [c] =

∫

Ω

1

2ρψ + β−1

M∑

j=1

cj

[

ln(Λ3cj) − 1]

−M

∑

j=1

µjcj

dV

ρ(x) = ρf (x) +∑M

j=1 qjcj(x)

∇ · εε0∇ψ = −ρ (+ B.C., e.g., ψ = 0 on ∂Ω)

Λ : the thermal de Broglie wavelength

µj : chemical potential for the jth ionic species

Equilibrium conditions

(δG [c])j = qjψ + β−1 ln(Λ3cj) − µj = 0 ⇐⇒ cj(x) = c∞j e−βqjψ(x)

Minimum electrostatic free-energy (note the sign!)

Gmin =

∫

Ω

−εε0

2|∇ψ|2 + ρf ψ − β−1

M∑

j=1

c∞j

(

e−βqjψ − 1)

dV

Theorem (B.L. 2009).

The functional G has a unique minimizer c = (c1, . . . , cM).

There exist constants θ1 > 0 and θ2 > 0 such thatθ1 ≤ cj(x) ≤ θ2 ∀x ∈ Ω ∀j = 1, . . . ,M.

All cj are given by the Boltzmann distributions.

The corresponding potential is the unique solution to the PBE.

Remark. Bounds are not physical! A drawback of the PB theory.

Proof. By the direct method in the calculus of variations, using:

Convexity.G [λu + (1 − λ)v ] ≤ λG [u] + (1 − λ)G [v ] (0 < λ < 1);

Lower bound. Let α ∈ R. Then the function s 7→ s(ln s + α)is bounded below on (0,∞) and superlinear at ∞;

A lemma (cf. next slide). Q.E.D.

G [c] =

∫

Ω

1

2ρψ + β−1

M∑

j=1

cj

[

ln(Λ3cj) − 1]

−

M∑

j=1

µjcj

dV

Lemma (B.L. 2009). Given c = (c1, . . . , cM). There existsc = (c1, . . . , cM) that satisfies the following:

c is close to c ; G [c] ≤ G [c]; there exist constants θ1 > 0 and θ2 > 0 such that

θ1 ≤ cj(x) ≤ θ2 ∀x ∈ Ω ∀j = 1, . . . ,M.

Proof. By construction using the fact that the entropic change isvery large for cj ≈ 0 and cj ≫ 1. Q.E.D.

O s

slns

PBE: ∇ · εε0∇ψ − B ′(ψ) = −ρf

Define: I [φ] =

∫

Ω

[εε0

2|∇φ|2 − ρf φ + B(φ)

]

dV

Notation

B(ψ) = β−1M

∑

j=1

c∞j

(

e−βqjψ − 1)

H1g (Ω) = φ ∈ H1(Ω) : φ = g on ∂Ω o ψ

B

Theorem.

The functional I : H1g (Ω) → R has a unique minimizer ψ.

The minimizer is the unique solution to the PBE.

PBE: ∇ · εε0∇ψ − B ′(ψ) = −ρf

I [φ] =

∫

Ω

[εε0

2|∇φ|2 − ρf φ + B(φ)

]

dV

Proof. Step 1. Existence and uniqueness by the direct method.Step 2. Key: The L∞-bound. Let λ > 0 and define

ψλ(x) =

− λ if ψ0(x) < −λ,

ψ0(x) if |ψ0(x)| ≤ λ,

λ if ψ0(x) > λ.

I [ψ] ≤ I [ψλ], |∇ψλ| ≤ |∇ψ|, the properties of B, and theuniqueness of maximizer =⇒ ψ = ψλ for large λ.

Step 3. Routine calculations. Q.E.D.

2. Mean-Field Models with Ionic Size Effects

Electrostatic free-energy functional

G [c] =

∫

Ω

1

2ρψ + β−1

M∑

j=0

cj

[

ln(a3j cj) − 1

]

−M

∑

j=1

µjcj

dV

ρ(x) = ρf (x) +∑M

j=1 qjcj(x)

∇ · εε0∇ψ = −ρ (+ B.C., e.g., ψ = 0 on ∂Ω)

c0(x) = a−30

[

1 −∑M

i=1 a3i ci (x)

]

aj (1 ≤ j ≤ M): linear size of ions of jth species

a0: linear size of a solvent molecule

c0: local concentration of solvent

Remarks.

Derivation from a lattice-gas model only for the case of auniform size: a0 = a1 = · · · = aM .

G [c] is convex in c = (c1, . . . , cM).

Theorem (B.L. 2009). The functional G has a unique minimizer(c1, . . . , cM), characterized by the following two conditions:

Bounds. There exist θ1, θ2 ∈ (0, 1) such that

θ1 ≤ a3j cj(x) ≤ θ2 ∀x ∈ Ω ∀j = 0, 1, . . . ,M;

Equilibrium conditions (i.e.,(δG [c])j = 0 for j = 1, . . . ,M)

(

aj

a0

)3

log(

a30c0

)

− log(

a3j cj

)

= β (qjψ − µj) ∀j = 1, . . . ,M.

Proof. Similar to the case without the size effect. Q.E.D.

Remark. The bounds are non-physical microscopically!

Lemma (B.L. 2009). Given c = (c1, . . . , cM). There existsc = (c1, . . . , cM) that satisfies the following:

c is close to c ;

G [c] ≤ G [c];

there exist θ1 and θ2 with 0 < θ1 < θ2 < 1 such that

θ1 ≤ a3j cj(x) ≤ θ2 ∀x ∈ Ω ∀j = 0, 1, . . . ,M.

Proof. By construction in two steps. First, take care of c0. Then,take care of cj (j = 1, . . . ,M). Q.E.D.

(

aj

a0

)3

log(

a30c0

)

− log(

a3j cj

)

= β (qjψ − µj) ∀j = 1, . . . ,M.

The case of a uniform size: a0 = a1 = · · · = aM = a.The generalized Boltzmann distributions

cj =c∞j e−βqjψ

1 + a3∑M

i=1 c∞i e−βqiψ, j = 1, . . . ,M.

The generalized PBE

∇ · εε0∇ψ +

∑Mj=1 qjc

∞

j e−βqjψ

1 + a3∑M

j=1 c∞j e−βqjψ= −ρf

A variational principle: ψ minimizes the convex functional

I [φ] =

∫

Ω

εε0

2|∇φ|2 − ρf φ + β−1a−3 log

1 +M

∑

j=1

a3c∞j e−βqjφ

dV

(

aj

a0

)3

log(

a30c0

)

− log(

a3j cj

)

= β (qjψ − µj) ∀j = 1, . . . ,M.

The general case: Implicit Boltzmann distributions

Set DM = u = (u1, . . . , uM) ∈ RM : uj > 0, j = 0, 1, . . . ,M

u0 = a−30

(

1 −∑M

j=1 a3j uj

)

fj(u) = a3j a

−30 log

(

a30u0

)

− log(

a3j uj

)

, j = 1, . . . ,M.

Lemma (B.L. 2009). The map f : DM → RM is C∞ and bijective.

Proof. It is clear that f is C∞.

f is injective. det∇f 6= 0, use det(I + v ⊗ w) = 1 + v · w .

f is surjective. Note: fj(u) = rj ⇐⇒ ∂jz = ∂z/∂uj = 0, where

z(u) =M

∑

j=0

uj

[

log(

a3j uj

)

− 1]

+M

∑

j=1

rjuj .

Construction: minDMz < min∂DM

z . So all ∂jz = 0. Q.E.D.

Set g = (g1, . . . , gM) = f −1 : RM → DM

Bj(φ) = gj (β(q1φ − µ1), . . . , β(qMφ − µM))

B0(φ) = a−30

[

1 −∑M

j=1 a3j Bj(φ)

]

Define B(φ) = −M

∑

j=1

qj

∫ φ

0Bj(ξ) dξ ∀φ ∈ R

Assume∑M

j=1 qjBj(0) = 0 (electrostatic neutrality)

Lemma (B.L. 2009). The function B is strictly convex. Moreover,

B ′(φ) = −M

∑

j=1

qjBj(φ)

> 0 if φ > 0,

= 0 if φ = 0,

< 0 if φ < 0,

and B(φ) > B(0) = 0 for all φ 6= 0. o ψ

B

Proof. Direct calculations using the Cauchy–Schwarz inequality toshow B ′′ > 0. Also, use the neutrality. Q.E.D.

G [c] =

∫

Ω

1

2ρψ + β−1

M∑

j=0

cj

[

ln(a3j cj) − 1

]

−M

∑

j=1

µjcj

dV

Theorem (B.L. 2009).

The equilibrium concentrations (c1, . . . , cM) andcorresponding potential ψ are related by the implicitBoltzmann distributions

cj(x) = Bj(ψ(x)) x ∈ Ω, j = 1, . . . ,M.

The potential ψ is the unique solution of the boundary-valueproblem of the implicit Poisson–Boltzmann equation

∇ · εε0∇ψ − B ′(ψ) = −ρf .

This is the Euler–Lagrange equation of the convex functional

J[φ] =

∫

Ω

[εε0

2|∇φ|2 − ρf φ + B (φ)

]

dV . Q.E.D.

3. Non-Uniform Sizes: Constrained Optimization

Electrostatic free-energy functional in (ψ, c)-formulation

Minimize F [ψ, c] =

∫

Ω

[εε0

2|∇ψ|2 + β−1Q(c)

]

dV

Q(c) =

∑M

j=1cj

[

log(

Λ3cj

)

− 1]

without size effect∑M

j=0cj

[

log(

a3j cj

)

− 1]

with size effect

with the constraints

Conservation of mass:

∫

Ωcj dV = Nj , j = 1, . . . ,M

Charge neutrality:∑M

j=1Njqj +

∫

Ωf dV +

∫

Γσ dS = 0

Poisson’s equation: ∇ · εε0∇ψ = −(

f +∑M

j=1qjcj

)

in Ω

Boundary condition: εε0∂nψ = σ on ∂Ω

Notations Nj : total number of ions of the jth species

f : volume charge density

σ : surface charge density

Electrostatic free-energy functional in (E, c)-formulation

Minimize F [ψ, c] =

∫

Ω

[εε0

2|E|2 + kBTQ(c)

]

dV

Q(c) =

∑M

j=1cj

[

log(

Λ3cj

)

− 1]

without size effect∑M

j=0cj

[

log(

a3j cj

)

− 1]

with size effect

with the constraints

Conservation of mass:

∫

Ωcj dV = Nj , j = 1, . . . ,M

Charge neutrality:∑M

j=1Njqj +

∫

Ωf dV +

∫

Γσ dS = 0

Gauss’s Law: ∇ · εε0E = f +∑M

j=1qjcj in Ω

Boundary condition: − εε0E · n = σ on Γ

Compatibility: ∇× E = 0 in Ω

Nondimensionalization

The Bjerrum length: lB =βe2

4πεε0

c ′j = 4πlBcj N ′

j = 4πlBNj

Λ′ = (4πlB)−1/3Λ a′j = (4πlB)−1/3aj

f ′ =4πlB f

eσ′ =

4πlBσ

e

ψ′ = βeψ E′ = βeE

Nondimensionalized (ψ, c)-formulation

Minimize F ′[ψ′, c ′] =

∫

Ω

[

1

2|ψ′|2 + Q ′(c ′)

]

dV

Q ′(c ′) =

∑M

j=1c′

j

(

log c ′j − 1)

without size effect∑M

j=0c′

j

(

log c ′j − 1)

with size effect

with the constraints

∫

Ωc ′j dV = N ′

j , j = 1, . . . ,M

∑Mj=1N

′

j zj +

∫

Ωf ′ dV +

∫

Γσ′ dS = 0

∆ψ′ = −(

f ′ +∑M

j=1zjc′

j

)

in Ω

∂nψ′ = σ′ on Γ

Nondimensionalized (E, c)-formulation

Minimize F ′[E′, c ′] =

∫

Ω

[

1

2|E′|2 + Q ′(c ′)

]

dV

Q ′(c ′) =

∑M

j=1c′

j

(

log c ′j − 1)

without size effect∑M

j=0c′

j

(

log c ′j − 1)

with size effect

with the constraints

∫

Ωc ′j dV = N ′

j , j = 1, . . . ,M

∑Mj=1N

′

j zj +

∫

Ωf ′ dV +

∫

Γσ′ dS = 0

∇ · E′ = f ′ +∑M

i=1zjc′

j in Ω

− E′ · n = σ′ on Γ

∇× E′ = 0 in Ω

A Lagrange multiplier method for the case without size effect

min(E,c)

max(ψ,λ)

L(E, c , ψ, λ)

L(E, c , ψ, λ) = F (E, c) −

∫

ΩψK (E, c)dV +

∑Mj=1λjHj(cj)

K (E, c) = ∇ · E −∑M

j=1zjcj − f

Hj(cj) =

∫

Ωzjcj dV − zjNj , j = 1, . . . ,M

Conditions for a saddle point

∂EL = E + ∇ψ = 0

∂ψL = −K (E, c) = 0

∂ciL = log cj + zjψ + λjzj = 0, j = 1, . . . ,M

∂λjL = Hj(cj) = 0, j = 1, . . . ,M

Finally, solve numerically

cj =Nje

−zjψ

∫

Ω ezjψdVin Ω, j = 1, . . . ,M

− ∆ψ =M

∑

j=1

zjNje−zjψ

∫

Ω e−zjψdV+ f

+ Boundary Conditions

Algorithm

0. Initialize ψ0, k = 0, ω ∈ (0, 1), and tol > 0.

1. Find ψ∗ that solves the boundary-value problem of

−∆ψ∗ =M

∑

j=1

zjNje−zjψk

∫

Ω e−zjψk dV+ f .

2. If |ψ∗ − ψk | < tol, then stop. Otherwise, set

ψk+1 = ωψk + (1 − ω)ψ∗ and k ← k + 1,

and go to Step 1.

An augmented Lagrange multiplier method for the case withthe size effect

min(E,c)

max(ψ,λ)

L(E, c , ψ, λ, r)

L(E, c , ψ, λ, r) = F (E, c) −

∫

ΩψK (E, c)dV

+M

∑

j=1

λjHj(cj) +M

∑

j=1

rj

2[Hj(cj)]

2

∂EL = E + ∇ψ = 0 ⇐⇒ E = −∇ψ

∂ψL = −K (E, c) = 0 ⇐⇒ −∆ψ =∑M

j=1zjcj + f

∂cjL = −

a3j

a30

log(

1 −∑M

i=1a3i ci

)

+ log cj + (λj + ψ)zj + rjzjHj(cj)

= 0, j = 1, . . . ,M

∂λjL = Hj(cj) = 0, j = 1, . . . ,M

Algorithm

0. Initialize c(0), ψ(0), λ(0), and r (0). Fix β > 1. Set k = 0.

1. Solve the boundary-value problem of Poisson’s equation with

c(k)i to get ψ(k+1).

2. Use Newton’s method to solve for c(k+1) with ψ(k+1), λ(k),and r (k).

3. Update the Lagrange multipliers and the penalty parameters

λ(k+1)j = λ

(k)j + r

(k)j Hj(c

(k+1)j ), j = 1, . . . ,M,

r(k+1)j = βr

(k)j , j = 1, . . . ,M.

4. Test convergence. If not, set k ← k + 1 and go to Step 1.

Newton’s method for solving

−a3j

a30

log(

1 −∑M

i=1a3i ci

)

+ log cj + (λj + ψ)zj

+ rjz2j

(∫

Ωcj dV − Nj

)

= 0 in Ω, j = 1, . . . ,M.

Uniform discretization with N grids and cell volume ∆v

θmj ≡ −

a3j

a30

log(

1 −∑M

i=1a3i c

mi

)

+ log cmj + (λj + ψm)zj

+ rjz2j

(

∆v∑N

i=1cij − Nj

)

, m = 1, . . . ,N, j = 1, . . . ,M.

Iteration for k = 1, 2, . . .for j = 1, . . . ,M

Newton’s method for θmj = 0 (m = 1, . . . ,N)

end forend for

Fix j . Denote Θ = (θ1j , θ

2j , · · · , θN

j )T and c = (c1j , c2

j , · · · , cNj )T .

The system of equations: θmj = 0 (m = 1, . . . ,N) ⇐⇒ Θ(c) = 0.

∂Θ

∂c= diag

(

1

ξ1, . . . ,

1

ξN

)

+ rjz2j ∆ve ⊗ e

ξm =

(

1

cmj

+a6j

a30 − a3

0

∑Mi=1 a3

i cmi

)

−1

det∂Θ

∂c=

1 + rjz2j ∆v

∑Nm=1 ξm

∏Nm=1 ξm

> 0

(

∂Θ

∂c

)

−1

= diag(

ξ1, . . . , ξN)

−rjz

2j ∆v

1 + rjz2j ∆v

∑Nm=1 ξm

ξ ⊗ ξ

Newton’s iteration

cmj ← cm

j − γξm

(

θmj −

rjz2j ∆v

∑Np=1 θp

j ξp

1 + rjz2j ∆v

∑Np=1 ξp

)

, m = 1, . . . ,N.

4. Numerical Results

Computational setting

The Bjerrum length lB = 7 A

Ω = (0, L) × (0, L) × (0, L)

Ball Bc of radius R centered in Ω

Fixed surface charges σ = Ze on ∂Bc

Example 1.

M = 2, z1 = −1, z2 = +1,

a1 = 3.34 A, a2 = 2.32 A, a0 = 2.75 A,

N1 = 120, N2 = 60, Ze = 60 e,

R = 8 A, L = 80 A, 256 × 256 × 256 grid points.

Ionic concentrations in the mid-plane z = 40 A.

0 500 1000 1500 2000119.996

119.998

120

120.002

120.004

120.006

Iteration steps

To

tal p

art

icle

nu

mb

er

of co

un

terio

n

(a)

0 500 1000 1500 200059.996

59.998

60

60.002

60.004

60.006

Iteration stepsT

ota

l p

art

icle

nu

mb

er

of co

ion

(b)

Convergence of total numbers of counterions (left) and coions(right) in iteration.

0 500 1000 1500 2000

−1

0

1

x 10−3

Iteration steps

To

tal ch

arg

e o

f th

e s

yste

m

(c)

Convergence of total charges in iteration.

103

104

105

106

107

108

100

101

102

103

104

105

Number of grid nodes: N

CP

U t

ime

(s)

AugLagMulti

SMPBmove

O(Nlog(N))

Log-log plot of the CPU time vs. the number of grid points. TheO(N log N) complexity results from FFT.

Example 2. M = 2, z1 = −1, z2 = +1, N1 = 2Z , N2 = Z ,

R = 14 A, L = 160 A, 256 × 256 × 256 grid points.

5 10 15 20 25 30

0.5

1

1.666

2.5

3.2543.5

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

Classical PB theory

a0=10A, a1=10A, a2=10A

a0=10A, a1=10A, a2=2A

a0=8A, a1=10A, a2=2A

a0=8A, a1=8A, a2=2A

Concentration of counterions with linear size a1, and Z = 60.Note that 1/a3

1 = 1.666 M when a1 = 10 A and 1/a31 = 3.254 M

when a1 = 8 A. Observation: Maximal packing!

5 10 15 20 25 30

0.5

1

1.5

1.666

2

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

σ=0.0325 e/A2

σ=0.0244 e/A2

σ=0.0162 e/A2

σ=0.0081 e/A2

Variation of surface charges with a0 = 8 A, a1 = 10 A, a2 = 2 A.Observation: (1) Threshold. (2) Higher charge, wider saturation.

Example 3. M = 3, z1 = +3, z2 = +2, z3 = +1, Z = −200,

z1N1 = z2N2 = z3N3 = −Z/3,

R = 10 A, L = 80 A, 128 × 128 × 128 grid points.

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(a)

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(b)

(a) The classical PB solution: no size effect. (b) The size effectwith a0 = a1 = a2 = a3 = 5 A.Observation: non-monotonocity and stratification!

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(c)

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(b)

(c) a0 = 4 A and a+1 = a+2 = a+3 = 5 A.(d) a0 = 2 A and a+1 = a+2 = a+3 = 5 A.Observation: Smaller solvent molecules, larger discrepancy.

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(a)

5 10 15 20 250

5

10

15

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(b)

Denote αi = zi/a3i (i = 1, . . . ,M), the valence-to-volume ratios.

(a) a0 = 2 A, a+3 = 7 A, a+2 = 6 A, a+1 = 5 A.α+2 : α+3 : α+1 = 1.163 : 1.088 : 1.(b) a0 = 2 A, a+3 = 7 A, a+2 = 5 A, a+1 = 6 A.α+2 : α+3 : α+1 = 3.478 : 1.891 : 1.Observation: The valence-to-volume ratios are key parameters!

5 10 15 20 250

5

10

15

20

25

30

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(c)

5 10 15 20 250

5

10

15

20

25

30

Distance to a charged surface

Co

nce

ntr

atio

n o

f co

un

terio

n (

M)

+3+2+1

(d)

(c) a0 = 2 A, a+3 = 7 A, a+2 = 6 A, a+1 = 4 A.α+1 : α+2 : α+3 = 1.793 : 1.069 : 1.(d) a0 = 2 A, a+3 = 8 A, a+2 = 6 A, a+1 = 4 A.α+1 : α+2 : α+3 = 2.644 : 1.576 : 1.Observation: The valence-to-volume ratios are key parameters!

5. Conclusions

Summary

Minimization of electrostatic free-energy functional: uniqueset of equilibrium concentrations and electrostatic potential.

Uniform size: generalized PBE. Non-uniform sizes: implicit PBE.

Constrained optimization methods for the case of non-uniformionic sizes.

Predictions and discoveries: Counterion saturation near the charged surface; Counterion stratification near the charged surface; The ionic valence-to-volume ratios

αj =zj

a3j

(j = 1, . . . ,M)

not just the valences zj(j = 1, . . . ,M), are the key parametersin the stratification.

Discussions

Not included and studied: optimal packing; the Stern layer;and charge inversion.

Analytical studies of the differences between a uniform sizeand non-uniform sizes.

Any consequences of the discovery of the importance of thevalence-to-volume ratio?

Mean-filed models still can not predict the ion-ioncorrelations. New, consistent, and efficient models?

Applications to variational implicit solvation.

Thank you!