Lecture 1. The Poisson{Boltzmann bli/presentations/Taiwan2015_ Lecture 1. The Poisson{Boltzmann

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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