Controlled Release
Estimation of the Diffusion Coefficient
1
Stokes-Einstein Relation
n
For free diffusion
kT
D=
6πrη
q
Assumes a spherical
molecule
n
i.e., not valid for a longchain protein
n
k = Boltzman Constant
q
n
n
n
1.38 x 10-23 J/K
? = solvent viscosity
(kg/ms)
T is temperature (K)
r is solute molecule
radius
q
related to molecular
weight
2
Stokes-Einstein Relation
n
Radius, r, is related to MW
4

( MW ) = NρV = Nρ  πr 3 
3

n
 3( MW )  3
r=

 4πNρ 
Substitute into S-E eqn
D=
q
n
n
Solve for r
1
n
n
N is Avagadro’s Number
V is the molal volume of
the solute.
r is the hydrodynamic
radius, which considers
solvent bound to solute
kT
 3( MW ) 
6πη 

 4πNρ 
1
3
Note D is not a strong fn of MW
3
Diffusion Coefficient
n
Many drugs have a low
molecular wt
q
q
n
100 < MW < 500 g/mol
Not true for proteins and
larger molecules
Examples
Drug
MW (g/mol)
D(cm2/s)
Aqueous
Note
Caffeine
194.2
4.9 x 10-6
Medium size
Insulin
41,000
8.3 x 10-7
Huge size
4
Sutherland-Einstein Correlation
n
Sutherland-Einstein Equation
q
q
More reliable prediction for smaller solutes.
Modification of Stokes-Einstein, which considers sliding
friction between solute and solvent.
RT  βr + 3η 
D=


6πNrη  β r + 2η 
q
For large molecules, ß approaches infinity to reflect “no
slip” conditions, and the equation reduces to the StokesEinstein relation
5
Wilke-Chang Correlation
n
n
n
Semi-empirical modification of the Stokes-Einstein
relation. 10-15% error.
Widely used for small molecules in low MW
solvents, at low concentrations.
Involves
q
q
q
q
q
MW of solvent
Absolute temperature
Solvent viscosity
Solute molal volume at normal BP
Association factor of solvent (2.6 for water, 1.9 for
methanol, 1.5 for EtOH)
6
Diffusion Coefficients in Polymers
n
For large pores
q
q
n
For small pores
q
q
n
Diffusion through liquid filled pores
Steric hindrance, friction
For non-porous polymer networks
q
q
n
diffusion through liquid filled pores
Porosity, tortuosity, partition coefficient
Complicated; various mechanisms proposed
Depends on crystallinity, swelling, crosslinking, rubbery vs.
glassy state
Predictions based on semi-empirical or empirical
approaches
7
Measurement of D: rotating disk
From Kydonieus, A., Treatise on Controlled Drug Delivery
8
Measurement of diffusion coefficient
n
Rotating disk for measurement of D in liquids
q
Considers the rate of dissolution of a drug from a disk spinning
in a liquid
Q = 0.62 AD 2 / 3v0 − 1.6ω 0.5C s
q
q
q
q
q
n
Q is rate of dissolution
A is surface area of drug on disk
vo is the kinematic viscosity of the solvent,
Cs is the drug solubility in the solvent
? is the rotational speed of the disk
Perform experiments at different ? . What plot will give a
slope of D?
q
Q vs. ?
0.5
9
Measurement of D: permeation cell
From Kydonieus, A., Treatise on Controlled Drug Delivery
10
Measurement of diffusion coefficient
n
Two Cell Permeation
System
q
q
For D in liquids or
polymers
Two stirred cells divided
by a membrane. Initially
one contains the drug in
suspension.
DKCs
 dM 

 =
l
 dt  ∞
n
n
n
LHS is the steady-state
permeation rate
K is the partition
coefficient
(membrane/solution)
L is the membrane
thickness
11
Measurement of diffusion coefficient
n
n
More methods for polymers
Direct release method – curve fitting from a
release experiment.
q
n
Usually preferable to determine through
independent experiments.
Sorption and desorption from stirred finite
volume
q
q
Simple experiment
Mathematics of diffusion through a solid will be
investigated later
12
D values for small solutes in liquids
From Kydonieus, A., Treatise on Controlled Drug Delivery
13
D values for small solutes in polymers
Solute
Polymer
Hydrocortisone
Silicone rubber
EVA
PVA terpolymer
Poly(EVA)
PMMA
PVA
Salicylic acid
Diffusion
Coefficient
(cm2/s)
4.5 x 10-7
1.18 x 10-11
4.31 x 10-12
2.8 x 10-9
9.55 x 10-15
4.37 x 10-11
From Kydonieus, A., Treatise on Controlled Drug Delivery
14