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### Transcript of Lecture 1. The Poisson{Boltzmann bli/presentations/Taiwan2015_ Lecture 1. The Poisson{Boltzmann

• Lecture 1. The Poisson–Boltzmann Equation

I Background I The PB Equation. Some Examples I Existence, Uniqueness, and Uniform Bound I Free-Energy Functional. Variations I Free-Energy Functional. Minimizers and Bounds I PB Does Not Predict Like-Charge Attraction I References

• Background

Coulomb’s Law

I potential: U21 = 1

4πε0

q1q2 r

I force:

F21 = −∇U21(r) = − 1

4πε0

q1q2 r2

r21

2

r q

1

q

Poisson’s equation: ∇ · εε0∇ψ = −ρ I ψ: electrostatic potential

I ρ: charge density

I ε0: vacuum permittivity

I ε: dielectric coefficient or relative permittivity (εmin ≤ ε ≤ εmax)

• The Poisson–Boltzmann Equation

∇ · εε0∇ψ + M∑ j=1

qjc ∞ j e −βqjψ = −ρf

Poisson’s equation:

Charge density:

Boltzmann distributions:

Charge neutrality:

∇ · ε(x)ε0∇ψ(x) = −ρ(x) ρ(x) = ρf (x) +

∑M j=1 qjcj(x)

cj(x) = c ∞ j e −βqjψ(x)∑M

j=1 qjc ∞ j = 0

I ρf : Ω→ R: given, fixed charge density I cj : Ω→ R: concentration of jth ionic species I c∞j : bulk concentration of jth ionic species

I qj = zje : charge of an ion of jth species (zj : valence, e: elementary charge)

I β: inverse thermal energy (β−1 = kBT )

• PBE ∇ · εε0∇ψ + M∑ j=1

qjc ∞ j e −βqjψ = −ρf

I The Debye–Hückel approximation (linearized PBE)

∇ · εε0∇ψ − εε0κ2ψ = −ρf

Here κ > 0 is the ionic strength or the inverse Debye screening length (κ = λ−1D ), defined by

κ2 = β

εε0

M∑ j=1

q2j c ∞ j

I The sinh PBE for 1:1 salt (q1 = −q2 = q, c∞2 = c∞1 = c∞)

∇ · εε0∇ψ − 2qc∞ sinh(βqψ) = −ρf

• Some Examples

Example 1. A negatively charged plate at z = 0 with constant surface charge density σ < 0 and with 1:1 salt solution in z > 0. εψ

′′ = 8πec∞ sinh eβeψ for z > 0

ψ′(0) = −σ ε > 0

The solution is

ψ(z) = − 2 βe

ln

( 1 + γe−z/λD

1− γe−z/λD

)

γ2 + 2lGC λD

γ − 1 = 0 (γ > 0)

λD =

( 8πβc∞e2

ε

)−1/2 (Debye screening length)

lGC = e

2π|σ|lB (Gouy–Chapman length)

lB = βe2

ε (Bjerrum length)

• Example 2. A spherical solute with a point charge at center immersed in a solution with multiple species of ions.

Debye–Hückel approximation:{ ∇ · εε0∇ψ − χ{r>R}εwε0κ2ψ = −Qδ ψ(∞) = 0

O

ε

R

m εw

Q

ψ(r) =

 Q

4πεmε0

( 1

r − 1

R

) +

Q

4πεwε0R(1 + κR) for r < R,

Q

4πεwε0(1 + κR)

e−κ(r−R)

r for r > R

The Yukawa potential

Y (r) = e−κr

4πr

Solution of { −∆u + κ2u = δ u(∞) = 0

• Existence, Uniqueness, and Uniform Bound

Consider the boundary-value problem of PBE:

PBE ∇ · εε0∇ψ − B ′(ψ) = −ρf in Ω BC ψ = g on ∂Ω

B(ψ) = β−1 ∑M

j=1 c ∞ j

( e−βqjψ − 1

) o

ψ

B

Define

I [φ] =

∫ Ω

[εε0 2 |∇φ|2 − ρf φ+ B(φ)

] dV

H1g (Ω) = {φ ∈ H1(Ω) : φ = g on ∂Ω}

Theorem (Li, Cheng, & Zhang. SIAP 2011).

I The functional I : H1g (Ω)→ R has a unique minimizer ψ. I The minimizer is bounded in L∞(Ω) uniformly in ε ∈ [εmin, εmax].

I The minimizer is the unique solution to the boundary-value problem of PBE.

• ∇ · εε0∇ψ − B ′(ψ) = −ρf

I [φ] =

∫ Ω

[εε0 2 |∇φ|2 − ρf φ+ B(φ)

] dV

Proof. Step 1. Existence and uniqueness of minimizer.

First, the lower bound by Poincaré inequality

I [φ] ≥ C1‖φ‖2H1(Ω) − C2 ∀φ ∈ H 1 g (Ω).

Let α = infφ∈H1g (Ω) I [φ]. Then α is finite. There exist ψk ∈ H 1 g (Ω)

(k = 1, 2, . . . ) such that I [ψk ]→ α. By the lower bound, {ψk} is bounded in H1(Ω). Hence it has a subsequence (not relabeled) such that ψk → ψ weakly in H1(Ω) and a.e. in Ω for some ψ ∈ H1g (Ω). The weak convergence and Fatou’s lemma lead to

α = lim k→∞

I [ψk ] ≥ I [ψ] ≥ α.

Uniqueness of minimizer ψ follows from the strict convexity of I [·] :

I [λu + (1− λ)v ] ≤ λI [u] + (1− λ)I [v ] (0 < λ < 1).

• Step 2. The L∞-bound for ψ uniform for ε ∈ [εmin, εmax].

Let φg ∈ H1g (Ω) be such that ∇ · εε0∇φg = −ρf . Then φg is bounded in L∞(Ω) uniformly in ε. Let ψ0 ∈ H10 (Ω) be the unique minimizer in H10 (Ω) of

J[φ] =

∫ Ω

[εε0 2 |∇φ|2 + B(φg + φ)

] dV .

Then ψ = ψ0 + φg . Prove ‖ψ0‖L∞(Ω) ≤ C uniform in ε.

Since B ′(±∞) = ±∞, there exists λ > 0 with B ′(φ0 + λ) ≥ 1 and B ′(φ0 − λ) ≤ −1 a.e. in Ω. Note λ is uniform in ε. Define ψλ by

ψλ(x) =

 − λ if ψ0(x) < −λ, ψ0(x) if |ψ0(x)| ≤ λ, λ if ψ0(x) > λ.

Then ψλ ∈ H10 (Ω). We have J[ψ0] ≤ J[ψλ] and |∇ψλ| ≤ |∇ψ0|. Hence ∫

Ω B (φg + ψ0) dV ≤

∫ Ω B (φg + ψλ) dV .

• Consequently, we have by the convexity of B : R→ R that

0 ≥ ∫ {ψ0>λ}

[B(φg + ψ0)− B(φg + λ)] dV

+

∫ {ψ0λ}

B ′ (φg + λ) (ψ0 − λ)dV

+

∫ {ψ0λ}

(ψ0 − λ)dV − ∫ {ψ0λ}

(|ψ0| − λ)dV

≥ 0.

Hence |{|ψ0| > λ}| = 0 and |ψ0| ≤ λ a.e. Ω.

• Step 3. The minimizer is the unique solution to the boundary-value problem of PBE.

Routine calculations:

δI [ψ][η] := d

dt

∣∣∣∣ t=0

I [ψ + tη] = 0 ∀η ∈ C 1c (Ω).

Since ψ ∈ L∞(Ω), we have∫ Ω

[ εε0∇ψ · ∇η − ρf η + B ′(ψ)η

] dV = 0 ∀η ∈ H10 (Ω).

So ψ is a weak solution to the boundary-value problem of PBE. Uniqueness again follows from the convexity. Q.E.D.

• Free-Energy Functional. Variations

Electrostatic free-energy functional of ionic concentrations c = (c1, . . . , cM)

G [c] =

∫ Ω

12ρψ + 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)

I Λ : the thermal de Broglie wavelength

I µj : chemical potential for the jth ionic species

Observations

I ψ = L(ρf + ∑M

j=1 qjcj) is affine in c. So, ρψ is linear and quadratic in c .

I G [c] is strictly convex in c :

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

• G [c] =

∫ Ω

12ρψ + β−1 M∑ j=1

cj [ ln(Λ3cj)− 1

] −

M∑ j=1

µjcj

 dV First variations:

d

dt

∣∣∣∣ t=0

G [c + tdjej ] =

∫ Ω

(δG [c])jdj dV ,

where (δG [c])j = qjψ + β

−1 ln(Λ3cj)− µj . Equilibrium conditions

(δG [c])j = 0 ∀j ⇐⇒ cj(x) = c ∞ j e −βqjψ(x) ∀j .

(c∞j = Λ −3eβµj ) These are the Boltzmann distributions.

Minimum value of G is the electrostatic free-energy, the PB free energy; given by (note the sign!):

Gmin =

∫ Ω

−εε0 2 |∇ψ|2 + ρf ψ − β−1

M∑ j=1

c∞j

( e−βqjψ − 1

) dV .

• Second variations:

δ2G [c][u, v ] = d

dt

∣∣∣∣ t=0

δG [c + tv ][u]

=

∫ Ω

 M∑ j ,k=1

qjqkujLvk + M∑ j=1

ujvj βcj

 dV . In particular, if u = v then

δ2G [c][u, u] =

∫ Ω

 M∑ j

qjuj

 L  M∑

j

qjuj

+ M∑ j=1

u2j βcj

 dV > 0. So, G is convex.

• Free-Energy Functional. Minimizers and Bounds

Define

X =

c = (c1, . . . , cM) ∈ L1(Ω,RM) : M∑ j=1

qjcj ∈ H−1(Ω)

 . Theorem (B.L. SIMA 2009).

I The functional G has a unique minimizer c ∈ X . I There exist constants θ1 > 0 and θ2 > 0 such that

θ1 ≤ cj(x) ≤ θ2 a.e. x ∈ Ω ∀j = 1, . . . ,M. I All cj are given by the Boltzmann distributions.

I The corresponding potential is the unique solution to the PBE.

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

• Proof. Existence and uniqueness of minimizer by the direct method in the calculus of variations.

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

I By de la Vallée Poussins criterion, a minimizing sequence c(k)

has a subsequence that converges weakly to c in L1.

I Convexity and continuity imply the weak l