Chemical Subsystem
Governing Equations:
∂C
= −∇ ⋅ N i + Rv
∂t
(2) Nernst-Planck Flux Equation (Fick’s diffusion + convection + migration)
z
N i = − D∇Ci + vCi + i ui Ci E
zi
Boundary conditions come from
1. flux matching
0 = n1 ⋅ ( N 1 − N 2 ) + RS
2. concentration matching
c( x+ ) = K ⋅ c( x− )
3. symmetry
Ni = 0
(1) Conservation Law (Rv=Rate of Formation)
Solution Methods
Scale first to get more easily solved equations, and more general solutions. You should
already have the solutions to most scaled equations you might encounter in your notes,
homework solutions, and textbook examples…
Linear ODE’s
1.
Direct Integration
2.
Tables
3.
Green’s Functions
Linear PDE’s
1.
Linear Operator Theory
Sturm-Liouville Operators
2.
3.
L=
1 ∂ 
∂ 

 p ( x )  + q( x )

w( x )  ∂x 
∂x 

Separation of Variables
Assume
φ ( x, y ) = X ( x )Y ( y )
Finite Fourier Transform (Chapter 4 Deen, tables)
Dimensionless Groups (Generate by scaling governing equations)
kL2 / D hom og.rxn
(1) Dahmkohler (reaction/diffusion)
Da = 
heterog .rxn
kL / D
(2) Peclet
ρUL
Re =
(3) Reynolds (inertial/viscous)
µ
2nd Order Chemistry
A+ B ⇔ C
RvA = RvB = −k on AB + k off C
RvC = k on AB − k off C
Kd =
Rapid Equilibrium
k off
k on
=
AB
C
∂A ∂B ∂C
=
=
=0
∂t
∂t
∂t
CTA = A + C
C=
CTA B
Kd + B
Mechanical Subsystem
Governing Equations
(1) Conservation of Mass
Incompressible Fluid
(2) Conservation of Momentum
(3) Navier-Stokes
∂ρ
+ ∇⋅ ρv = 0
∂t
∇⋅v = 0
Dv
ρ
=F
Dt
Dv
ρ
= −∇P + ρ e E + µ∇ 2 v + Fother
Dt
Poiseulle Pipe Flow
2
R 2 ∂p   r  
v z (r) = −
1 −   
4 µ ∂z   R  
R 2 ∂p
U =−
8µ ∂z
Stress
Stress and stress tensor (units F/A)
τ ij = n ij ⋅ T
F = ∇ ⋅T
Viscous stress
Generalized Hooke’s Law
1  ∂vi ∂v j 
+
2  ∂x j ∂xi 
2 

Tijv = 2 µe&ij +  λ − µ δ ij e&kk
3 

e&ij =
Electrical Subsystem
Electroquasistatic approximation: Assumes that magnetic field (B) is negligibly small, so
the change with time is also negligible. This is applicable when L (the characteristic
length) is much smaller than the wavelength.
Governing Equations:
(1) Gauss’s Law
(2) Faraday’s Law
(3) Conservation of Charge
(4) Current Constitutive Law
∇ ⋅ε E = ρ
∇ × E = 0 ⇒ E = −∇Φ
∂ρ
= −∇ ⋅ J
∂t
J = ∑ z i F N i = σ E + ()∇ci + ()v fluid
i
Boundary Conditions:
(1) n ⋅ (ε 1 E 1 − ε 2 E 2 ) = σ s
∂ρ
= 0( steady − state)
∂t
(3) n × (E 1 − E 2 ) = 0 ⇒ Φ 1 = Φ 2
(2) n ⋅ (σ 1 E 1 − σ 2 E 2 ) = −
Modification to Species Flux:
z

N i = − Di ∇ci +  i u i ci  E where u is the mobility of species I
 zi

Charge relaxation:
τ=
ε
σ
In most bio-systems, relaxation time is on the order of nanoseconds.
Diffusion and migration occur through very different mechanisms, leading to very
different relaxation profiles.
Conductivity of a solution can be described by: σ = ∑ u i ci
i
Electrical Double Layer:
ci ( x) = c0 e − zi FΦ / RT
Boltzmann distribution
Æ equilibrium distribution of ions, obtained from setting flux = 0 and using the Einstein
RT Di
=
Relation
zi F ui
Debye length:
1
κ
=
εRT
2c 0 z 2 F 2
as solved in HW 3, Problem 4.
The debye length has several interpretations: a.) length over which the potential of a
charged surface decays, b.) length scale over which diffusion and charge relaxation time
scales are on the same order, c.) length scale over which diffusion and migration
compete.
In bio-systems, this is generally on the order of nanometers.
For Lcharacteristic >>
1
κ
, electroneutrality applies:
ρ e = ρ m + ∑ z i Fci = 0
i
This simplifies Gauss’s Law. This, combined with Faraday’s Law gives Laplace’s
Equation: ∇ 2 Φ = 0
Donnan Potential:
Often used as a boundary condition. This equation describes the equilibrium
concentration gradient and potential due to a charged material immersed in an ionic
solution.
∆Φ D = −
1 RT  ci 
ln
z i F  ci 
Examples:
• Non-equilibrium/non-steady transport of ions across neutral material
• Donnan equilibrium problems of 2 classes: a.) given fixed charge of material,
concentration in bath, find internal concentrations and potential, b.) given bath
concentration and constitutive equation describing fixed charge of material, find
self-consistent material charge, internal ion concentration, and potential.
• IGF Diffusion/Reaction through tissue
• Tendon swelling experiment
• Electrodiffusion through the glycocalyx
• Minority carrier phenomenon – majority carriers shield the fixed charges.
Minority carriers travel through the material as if charges aren’t present.(See
example 2.6.4 in AJG manuscript)
Electrokinetic Phenomena – Electrical and Mechanical coupling
Navier-Stokes with electrical forces:
ρ
Dv
= −∇P + ρ e E + µ∇ 2 v + Fother
Dt
Assumptions/Boundary Conditions:
(1) Frequently used, E r >> E 0 z . This allows decouple of fields in two directions
when solving equations
(2) Potential at “wall” (really the slip boundary) is ζ (zeta potential).
(3) Models for the Debye layer (see AJG p.357):
a. Diffuse Double layer Φ ( x ) = Φ wall e −κx
Φ( x ) = ζe −κ ( x−δ )
ζ
∂Φ
=
linear potential profile
∂x x=δ 1κ
impulse model for charge
ρ e = −σ d δ f x − 1
b. Guoy-Chapmann
c. Helmholtz
(
σd =
(4) No-slip boundary at x = δ
(5) Symmetry
ε
1
κ
ζ
κ
Electrokinetic Phenomena:
Solved for the pipe problem:
2
(
−ε
R − δ ) − r 2  ∆P 
(ζ − Φ(r ))E0 z −
vz (r ) =


µ
4µ
 L 
R −δ
R −δ
∫ 2πrρ (r )v (r )dr + ∫ 2πrσ (r )E
i=
e
0z
z
0
dr
0
R −δ
Q=
∫ 2πrv (r )dr
z
0
Using Helmholtz model, and assuming R >>
 πR 4
−
Q p   8µL
 i  =  πR 2εζ
 p 
 µL

1
κ

πR 2εζ
  ∆P 
µL
⋅
 πR 2σ 2πRε 2ζ 2  ∆V 

− 
+
µ ( 1κ )L 
 L
Special Cases:
• Darcy’s Law
U z ∆V =0 = −k11∇P
•
Streaming Current
•
Electroosmotic Flow U z ∆P=0 = L12 ∆V
J z ∆V =0 = L21∆P
we get:
)
•
Streaming Potential
∆V
•
Filtration
Uz
J =0
J =0
=−
L21
∆P
L22

L L 
=  L11 − 12 21 ∆P
L22 

Electrophoresis (see ALG section 6.4):
εζ
Electrophoretic mobility =
•
U
µ
=
εζσ m
E0 z 1 + µσR +
First correction term:
εζσ m
τ
≈ relax considers relative importance of counterµσR τ convection
ion flow
•
σm u
σR
Second correction term:
σm u
where u is the mobility of counter-ion describes
σR
effects of surface current
•
εζ
For R >> 1 , σ m =
κ
1
κ
Problems/Examples/Applications:
• Comparison of macroscale and microscale model
• Continuous flow PCR on a chip
• Capillary electrophoresis