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
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)
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 )
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
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)
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
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.
∇ · 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).
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 .
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.
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.
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).
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 .
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.
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.
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.
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
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.
References
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