Review of Mass Transfer

Fick’s First Law (one
dimensional diffusion)


J flux (moles/area/time)
At steady-state or any
instant in time
dC
J  D
dx

Fick’s Second Law

When concentration
changes with time
dC
d 2C
D 2
dt
dx
1
Example – Fick’s First Law
membrane

Determine amount of
drug to pass through
membrane in one hour

D
h
A
C1
C2
C1
C2

h


SS valid if a high C1 is maintained
and
C2 remains << C1 by removal of
drug or large volume



= 1 x 10-10 cm2/s
= 2 x 10-3 cm
= 10 cm2
= 0.5 mol/L
=0
2
Example Fick’s First Law
3
Partitioning
C1


So far we have
assumed the drug has
equal affinity for
solution and
membrane. This is
unlikely.
Preference is indicated
by partitioning
membrane C1<Cm1
the drug
Cm1
“prefers”
the
polymer
Cm2
h

Flux across membrane
J  D


C2
dCm
 C  Cm1 
 D m2

dx
h


Cannot measure Cm
Partition Coefficient
relates Cm to C
Cm1
Cm 2
K m1 
K m2 
C1
C2
4
Partition Coefficient


A measure of relative concentrations in membrane
vs. solution at equilibrium
If both solvents (1 and 2) are the same, then
K m1  K m2  K m

Flux becomes
dCm
 C2  C1 
J  D
  DKm 
dx
 h 

The term [DKm/h] is the Permeability
5
Partition Coefficient

Sketch the profiles for a high Km and a low Km
C1
C1
C2

C2
h
h
high Km
low Km
Sketch the profile for C2=0

Common situation: body acts as a sink to remove the drug
C1
h
C2=0
6
Example – Transdermal Delivery



Digitoxin (used for heart
failure; ointment)
How much digitoxin can
be delivered
transdermally in one
day
Membrane control –
skin acts as barrier
membrane (stratum
corneum, outermost
layer)

Km1 = 0.014


D


h


A


C1

Between ointment and s.c.
= 5.2 x 10-10 cm2/s
Through the s.c.
= 2x10-3 cm
Typical thickness of s.c.
= 10cm2
Covered by ointment
= 0.01 mg/cm3
Saturation C of drug in
ointment
7
Solution
8
Some K values
Steroid
K
Cortisol
5.5e-3
Estradiol
2e-1
Melengstiol acetate
18.87
Norethindrone
1.22
Norgestrel
3.19
19-Norprogesterone
33.3
Megesterol acetate
35.7
Mestranol
100
Progesterone
22.7
Testosterone
4.31
Reported by Sundaram and Kincl [93] in Kydonieus, Treatise in CDD
9
Fick’s Second Law

For one-dimensional
unsteady-state
diffusion
dC
d 2C
D 2
dt
dx

How many IC’s and
BC’s are needed?
10
Finite source
• IC
Sink
(large volume)
X=L
• At t=0, C=C0
Diffusion
• BC1
X=0
X=-L
Polymer
containing drug
• x=L, C=0
• BC2
• X=0, dC/dx = 0
(symmetric)
11
Finite source, reflective boundary
• IC
Sink
(large volume)
X=L
• At t=0, C=C0
Diffusion
• BC1
• x=L, C=0
X=0
Polymer
containing drug
Impermeable
boundary at x=0
• BC2
• X=0, dC/dx = 0
(symmetric)
12
Solution method

For cartesian (planar) systems



Separation of variables or Laplace Transforms
Error function solution
We will investigate this later!
13