Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The...

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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

Transcript of Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The...

Page 1: Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The Poisson{Boltzmann Equation I Background I The PB Equation. Some Examples I Existence,

Lecture 1. The Poisson–Boltzmann Equation

I BackgroundI The PB Equation. Some ExamplesI Existence, Uniqueness, and Uniform BoundI Free-Energy Functional. VariationsI Free-Energy Functional. Minimizers and BoundsI PB Does Not Predict Like-Charge AttractionI References

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Background

Coulomb’s Law

I potential: U21 =1

4πε0

q1q2

rI force:

F21 = −∇U21(r) = − 1

4πε0

q1q2

r2r21

2

rq

1

q

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

I ρ: charge density

I ε0: vacuum permittivity

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

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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)∑Mj=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 )

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PBE ∇ · εε0∇ψ +M∑j=1

qjc∞j e−βqjψ = −ρf

I The Debye–Huckel approximation (linearized PBE)

∇ · εε0∇ψ − εε0κ2ψ = −ρf

Here κ > 0 is the ionic strength or the inverse Debyescreening length (κ = λ−1

D ), 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

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Some Examples

Example 1. A negatively charged plate at z = 0 with constantsurface charge density σ < 0 and with 1:1 salt solution in z > 0. εψ′′ = 8πec∞ sinh eβeψ for z > 0

ψ′(0) = −σε> 0

The solution is

ψ(z) = − 2

βeln

(1 + γe−z/λD

1− γe−z/λD

)

γ2 +2lGC

λDγ − 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)

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

Debye–Huckel approximation:∇ · εε0∇ψ − χr>Rεwε0κ

2ψ = −Qδψ(∞) = 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)

rfor r > R

The Yukawa potential

Y (r) =e−κr

4πr

Solution of −∆u + κ2u = δ

u(∞) = 0

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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 inε ∈ [εmin, εmax].

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

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∇ · 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 Poincare inequality

I [φ] ≥ C1‖φ‖2H1(Ω) − C2 ∀φ ∈ H1

g (Ω).

Let α = infφ∈H1g (Ω) I [φ]. Then α is finite. There exist ψk ∈ H1

g (Ω)

(k = 1, 2, . . . ) such that I [ψk ]→ α. By the lower bound, ψk isbounded in H1(Ω). Hence it has a subsequence (not relabeled)such that ψk → ψ weakly in H1(Ω) and a.e. in Ω for someψ ∈ H1

g (Ω). The weak convergence and Fatou’s lemma lead to

α = limk→∞

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).

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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 andB ′(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 ∫ΩB (φg + ψ0) dV ≤

∫ΩB (φg + ψλ) dV .

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

0 ≥∫ψ0>λ

[B(φg + ψ0)− B(φg + λ)] dV

+

∫ψ0<−λ

[B(φg + ψ0)− B(φg − λ)] dV

≥∫ψ0>λ

B ′ (φg + λ) (ψ0 − λ)dV

+

∫ψ0<−λ

B ′ (φg − λ) (ψ0 + λ) dV

≥∫ψ0>λ

(ψ0 − λ)dV −∫ψ0<−λ

(ψ0 + λ)dV

=

∫|ψ0|>λ

(|ψ0| − λ)dV

≥ 0.

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

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Step 3. The minimizer is the unique solution to theboundary-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 ∈ H1

0 (Ω).

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

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Free-Energy Functional. Variations

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

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)

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 andquadratic in c .

I G [c] is strictly convex in c :

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

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G [c] =

∫Ω

1

2ρψ + 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 freeenergy; given by (note the sign!):

Gmin =

∫Ω

−εε0

2|∇ψ|2 + ρf ψ − β−1

M∑j=1

c∞j

(e−βqjψ − 1

) dV .

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Second variations:

δ2G [c][u, v ] =d

dt

∣∣∣∣t=0

δG [c + tv ][u]

=

∫Ω

M∑j ,k=1

qjqkujLvk +M∑j=1

ujvjβcj

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.

Page 15: Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The Poisson{Boltzmann Equation I Background I The PB Equation. Some Examples I Existence,

Free-Energy Functional. Minimizers andBounds

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 thatθ1 ≤ 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.

Page 16: Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The Poisson{Boltzmann Equation I Background I The PB Equation. Some Examples I Existence,

Proof. Existence and uniqueness of minimizer by the directmethod 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 Vallee Poussins criterion, a minimizing sequence c(k)

has a subsequence that converges weakly to c in L1.

I Convexity and continuity imply the weak lower semicontinuity.

I Uniqueness follows from the convexity.

Bounds follow from a lemma (cf. next slide).

Routine calculations to obtain the Boltzmann distributions andfurther to show that the potential is the unique solution to theboundary-value problem of PBE. Q.E.D.

Page 17: Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The Poisson{Boltzmann Equation I Background I The PB Equation. Some Examples I Existence,

G [c] =

∫Ω

1

2ρψ + β−1

M∑j=1

cj[ln(Λ3cj)− 1

]−

M∑j=1

µjcj

dV

Lemma Given c = (c1, . . . , cM). There exists c = (c1, . . . , cM)that satisfies the following:

I c is close to c ;I G [c] ≤ G [c];I 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

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The PB Theory Does Not Predict theLike-Charge Attraction

∆ψ = V ′(ψ) inside walls/outside balls

ψ = const. on the walls

ψ = const. on bdry of balls

V ′′ > 0 and V ′(0) = 0

The electrostatic surface force is given by

F =1

2

∫∂(balls)

(∂nψ)2n dS

F · (the unit horizontal vector toward the center) < 0.

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References

I M. Gouy, Sur la constitution de la charge electrique a lasurface d’un electrolyte, J. Phys. Theor. Appl., 9, 457–468,1910.

I D. L. Chapman, A contribution to the theory ofelectrocapillarity, Philos. Mag., 25, 475–481, 1913.

I P. Debye and E. Huckel, Zur Theorie der Elektrolyte, Physik.Zeitschr., 24, 185–206, 1923.

I F. Fixman, The Poisson–Boltzmann equation and itsapplication to polyelecrolytes, J. Chem. Phys., 70, 4995–5005,1979.

Page 20: Lecture 1. The Poisson{Boltzmann Equationbli/presentations/Taiwan2015_Lecture1.pdfLecture 1. The Poisson{Boltzmann Equation I Background I The PB Equation. Some Examples I Existence,

I K. A. Sharp and B. Honig, Calculating total electrostaticenergies with the nonlinear Poisson–Boltzmann equation”, J.Phys. Chem., 94, 7684–7692, 1990.

I K. A. Sharp and B. Honig, Electrostatic interactions inmacromolecules: Theory and applications, Annu. Rev.Biophys. Chem., 19, 301–332, 1990.

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I D. Andelman, Electrostatic properties of membranes: ThePoisson–Boltzmann theory, in Handbook of Biological Physics,ed. R. Lipowsky and E. Sackmann, 1, 603–642, Elsevier, 1995.

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I D. Andelman, Introduction to electrostatics in soft andbiological matter, in Proceedings of the Nato ASI & SUSSPon “Soft condensed matter physics in molecular and cellbiology” (2005), ed. W. Poon and D. Andelman, 97–122,Taylor & Francis, New York, 2006.

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

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