Key Concepts for this section
1: Lorentz force law, Field, Maxwell’s equation
2: Ion Transport, Nernst-Planck equation
3: (Quasi)electrostatics, potential function,
4: Laplace’s equation, Uniqueness
5: Debye layer, electroneutrality
Goals of Part II:
(1) Understand when and why electromagnetic (E and B)
interaction is relevant (or not relevant) in biological
systems.
(2) Be able to analyze quasistatic electric fields in 2D
and 3D.
r
E = −∇Φ
∇2Φ = −
ρe
ε
r
∇ ⋅ (ε E ) = ∇ ⋅ (−ε∇Φ ) = ρe
( Poisson ' s Equation)
However, biomolecules in the system do not generate E-field, since they
are shielded by counterions (electroneutrality)…….
It all comes down to solving…..
∂c
∇ c=
(Fick’s second law)
∂t
2
∇ 2 Φ = 0 ( Laplace ' s Equation)
∇ 2c = 0
r
q = −k ∇T (Fourier’s law for heat conduction)
r
∇ ⋅ q = 0 (conservation law for heat)
(steady-state diffusion)
∇ 2T = 0
(steady heat flow)
Electrostatics
Steady state diffusion ∇ 2 c = 0
Φ2=0
Φ1=0
Φ=?
Φ5=0
c2=0
c1=0
c=0
Φ3=0
c5=0
c4=0
Φ4=0
Thermal conduction
T2=0
T1=0
∇ 2T = 0
T=0
T5=0
T4=0
T3=0
c3=0
Uniqueness of Solution
Let’s assume two different solutions, Φa and Φb
ρe
; Φ a = Φ i on Si
ε
ρ
∇ 2 ⋅ Φ b = − e ; Φ b = Φ i on Si
ε
∇2 ⋅ Φa = −
Then define Φ d = Φ a − Φ b
∇ 2 ⋅ Φ d = 0; Φ d = 0 on Si
S2
S1
Answer:
Φd = 0
(satisfy Laplace Eq.)
for everywhere
∴ Φ a − Φb = 0
∇2 ⋅ Φ d = 0
S5
Φ d = 0 on all
S4
Si
S3
Gel Electrophoresis
Plastic (σ =0)
Pt electrode
Gel (ε, σ)
Φ=V0
biomolecules
r
r
∂B
∇× E = −
= 0 (electrostatics)
∂t
r
E = −∇Φ
ur
∂ρ
∇⋅ J = −
= 0 (steady state, no charge accumulation)
∂t
ur
ur
r
2
∇
Φ=0
∇⋅ J = ∇⋅ σ E = 0
∇⋅E = 0
( )
y
W
∂Φ
∂y
=0
y =W
Φ=0 when x=0
Φ=V0 when x=L
∂Φ
∂y
ur
∇⋅ J = 0
ur
J = J x xˆ
J=0 (insulator)
=0
L
x
y =0
(no charge accumulation)
J y = σ Ey = 0 =
∂Φ
∂y
y = 0 or W
Boundary Conditions (For EQS approximation)
ur ρ
∇⋅E =
ε
r
∇× E = 0
ur
∂ρ
∇⋅ J = −
∂t
uur
uur
nˆ ⋅ (ε1 E1 − ε 2 E2 ) = σ s
uur
uur
r
nˆ × E1 = nˆ × E2 ( E1
tangential
r
= E2
tangential
)
uur
uur
∂σ
nˆ ⋅ (σ 1 E1 − σ 2 E2 ) = − s
∂t
Figure 5.3.1 (a) Differential contour intersecting surface supporting surface charge density. (b) Differential
volume enclosing surface charge on surface having normal n.
Courtesy of Herman Haus and James Melcher. Used with permission.
Source: http://web.mit.edu/6.013_book/www/
Electrostatics
Steady Diffusion
Φ=V0
Φ=0
Gel or tissue
(σ,ε)
Φ=0
2
∇ Φ=0
r
J e = −σ∇Φ
C=C0
Φ=0
C=0
Gel or tissue
(D)
C=0
∇ 2C = 0
r
J i = − Di ∇ci
C=0
Figure 5.5.1 Two of the infinite number of potential functions having the form of
(1) that will fit the boundary conditions Φ = 0 at y = 0 and at x = 0 and x = a.
Courtesy of Herman Haus and James Melcher. Used with permission.
Source: http://web.mit.edu/6.013_book/www/
Solution
Known Solutions for Laplace equations
Cylindrical Coordinates
∂ 2 Φ 1 ∂Φ 1 ∂ 2 Φ ∂ 2 Φ
∇ Φ( ρ , ϕ , z ) = 0 ⇒
+
+ 2
+ 2 =0
2
2
ρ ∂ρ ρ ∂ϕ
∂ρ
∂z
Φ ( ρ , ϕ , z ) = R( ρ )Ψ (ϕ ) Ζ( z )
R( ρ ) ⇒ Bessel Functions ( J n , N n , I n , K n )
2
Ψ (ϕ ) ⇒ Trigonometric (sin, cos,sinh, cosh)
Ζ( z ) ⇒ Trigonometric (sin, cos,sinh, cosh)
Spherical Coordinates
∇ Φ(r ,θ , ϕ ) = 0 ⇒
2
1 ∂ ⎛ 2 ∂Φ ⎞
1
∂ ⎛
∂Φ ⎞
1
∂ 2Φ
=0
⎜r
⎟+ 2
⎜ sin θ
⎟+ 2 2
2
2
r ∂r ⎝ ∂r ⎠ r sin θ ∂θ ⎝
∂θ ⎠ r sin θ ∂ϕ
Φ (r ,θ , ϕ ) = R(r )Θ(θ )Ψ (ϕ )
R(r ) ⇒ Spherical Bessel Functions
Θ(θ ) ⇒ Legendre Functions ( Pn (cos θ ))
Ψ (ϕ ) ⇒ Trigonometric (sin ϕ , cos ϕ )
Solving Laplace’s Equation (Numerically)
2
d Φ
= 0 → Φ ( x) = ax + b
1D case:
2
dx
Φ (n − 1, m)
2
2
∂
Φ
∂
Φ
2D case:
+
=0
2
2
∂x
∂y
∂Φ
1
(n + , m) = Φ (n + 1, m) − Φ (n, m)
∂x
2
Φ (n, m + 1)
Φ ( n, m )
Φ (n + 1, m)
Φ (n, m − 1)
y (m)
x (n)
∂Φ
1
(n − , m) = Φ (n, m) − Φ (n − 1, m)
∂x
2
1
1
∂ 2Φ
∂Φ
∂Φ
(
n
,
m
)
(
n
,
m
)
(
n
, m) = Φ (n + 1, m) + Φ (n − 1, m) − 2Φ (n, m)
=
+
−
−
2
2
2
∂x
∂x
∂x
Φ (n, m + 1)
Laplace’s equation
In discretized form
Φ (n − 1, m)
Φ ( n, m )
Φ (n + 1, m)
Φ (n, m − 1)
y (m)
∂ 2Φ
∂ 2Φ
( n, m ) + 2 ( n , m ) =
2
∂x
∂y
Φ (n + 1, m) + Φ (n − 1, m) + Φ (n, m + 1) + Φ (n, m − 1) − 4Φ (n, m) = 0
x (n)
Φ (n + 1, m) + Φ (n − 1, m) + Φ (n, m + 1) + Φ (n, m − 1)
Φ ( n, m ) =
4
Value in the middle = average of surrounding values
Finite Element Method