1. DIFFUSION COEFFICIENT
AREA VELOCITY (m2/s)
SOLUTION
1) MUTUAL (“i” in “j”): Dij
DEPENDS ON
“i” intrinsic mobility
The presence of “j” j i i j j i
Unless “I” and “j” have the same mass and
size, a hydrostatic pressure gradient arises.
This is balanced by a mixture bulk flow. j i i j j i
Dij is the result of molecules random motion
and bulk flow

2. i i* i i* i i i i i* i i i i* i* i* i i i i i i* i* i i
2) INTRINSIC: Di  It depends only on “i” mobility
3) SELF: Di*  It depends only on “i” mobility
R = universal gas constant
T = temperature
sih = resistance coefficient
ai = “i” activity
ci = “i” concentration i i* i i* i i i i i* i i i i* i* i* i i i i i i* i* i i

3. GEL: D0, DS, D
POLYMERIC CHAINS
Drug
Solvent

4. D0, DS, D EVALUATION MOLECULAR THEORIES STATISTICAL MECHANICAL
Mathematical models of the GEL network
Atomistic simulations
Obstruction
Hydrodynamic
Kinetics

5. Hydrodynamic Theory: Stokes Einstein
D0(mutual drug diffusion coefficient in the pure solvent)
Hydrodynamic Theory: Stokes Einstein 1
It holds for large spherical molecules …. 2
… in a diluted solution
K = Boltzman constant
T = temperature
RH = drug molecule hydrodynamic radius
h = solvent viscosity

6. Diffusion coefficient D0 in water and radius rs of some solutes

7. D(drug diffusion coefficient in the swollen gel)
Obstruction theories 1
CARMAN
Polymer chains as rigid rods
Polymeric chains
drug
LMIN L1 L2 L3

8. 2 Mackie Meares Drug molecules of the same size of polymer segments
Lattice Model
Polymer
Drug
f = polymer volume fraction
(fraction of occupied sites in
the lattice)

9. 3 Ogston Diffusing molecules much bigger than polymer segments
Polymeric chains:
- Negligible thickness
- Infinite length
Drug
2 rs
f = polymer volume fraction
rs = solute radius
rf = polymer fibre radius

10. 4 Deen Applying the dispersional theory of Taylor Polymer 2 rf Drug
f = polymer volume fraction
rs = solute radius
rf = polymer fibre radius
= l l l3
l = rs/rf

11. 5 Amsden Openings size distribution: Ogston 2 r Drug 2 rs
Polymer
2 r
Drug
2 rs
f = polymer volume fraction
rs = solute radius
rf = polymer fibre radius
ks = constant (it depends on the polymer solvent couple)

12. Hydrodynamic theories1
Stokes-Einstein
Drug
Polymer
Solvent
All these theories focus the attention on the calculation of f, the friction drag coefficient

13. 2 Cukier Strongly crosslinked gels (rigid polymeric chains)
Lc = polymer chains length
Mf = polymer chains molecular weight
NA = Avogadro number
rf = polymer chains radius
rs = drug molecule radius
f = polymer volume fraction
Weakly crosslinked gels (flexible polymeric chains)
kc = depends on the polymer solvent couple

14. Kinetics theories
Existence of a free volume inside the liquid (or gel phase)
Solvent molecule
Liquid environment
1) Holes volume is constant at constant temperature
2) Holes continuously appear and disappear randomly in the liquid
Free volume
Vmolecules < Vliquid
Liquid environment

15. DIFFUSION MECHANISM
Solute
2) Probability of finding a sufficiently big hole at the right distance
1) Energy needed to break the interactions with surrounding molecules

16. superscript refers to solvent-polymer properties1
Eyring
According to this theory step 1 (interactions break up) is the most important
Solution
= mean diffusive jump length
k = the jump frequency
K = Boltzman constant
T = temperature
mr = solvent-solute reduced mass
Vf = mean free volume available per solute molecule
e = solute molecule energy with respect to 0°K
Gel
superscript refers to solvent-polymer properties

17. 2 Free Volume vT = solute thermal velocity l = jump length
According to this theory step 2 (voids formation) is the rate determining step
Solution
Probability that a sufficiently large void forms in the proximity of the diffusing solute
V* = critical free volume (minimum Vf able to host the diffusing solute molecule)
0.5 < g < 1 => it accounts for the overlapping of the free volume available to more than one molecule
vT = solute thermal velocity
l = jump length

18. Vfs = solvent free volume ws = solvent mass fraction
Gel
Assuming negligible mixing effects, the free volume Vf of a mixture composed by solvent, polymer and drug is be given by:
Vfd = drug free volume
wd = drug mass fraction
Vfs = solvent free volume
ws = solvent mass fraction
Vfp = polymer free volume
wp = polymer mass fraction

19. Fujita
It holds for small value of the polymer volume fraction f
p and q are two f independent parameters
Lustig and Peppas
They combine the FVT with the idea that diffusion can not occur if solute diameter is smaller than crosslink average length (z)
Y = k2*rs2 It is a parameter not far from 1
It holds for small polymer volume fraction

20. Cukier Lustig Peppas Y = k2*rs2 rs << z PAAM (polyacrylamide),
Cukier and Peppas equations bets fitting (fitting parameters kc and k2, respectively).
(polymer concentration f is the independent variable).
Cukier
Lustig Peppas
PAAM (polyacrylamide),
PVA (polyvinylalcohol),
PEO (polyethyleneoxide),
PHEMA (polyhydroxyethylmethacrylate)
Y = k2*rs2
rs << z

21. Amsden
Amsden best fitting (fitting parameter ks) on experimental data referred to different polymers and solutes (polymer concentration  is the independent variable). Fitting is performed assuming rf = 8 Å

22. BSA CASE

23. CONSIDERATIONS 1) Free Volume and Hydrodynamic theories
should be used for weakly crosslinked networks
2) Obstruction theories
should better work with highly crosslinked networks

24. DS(solvent diffusion coefficient in the swelling gel)
The only available theory is the free volume theory of Duda and Vrentas
HYPOTHESES
Temperature independent thermal expansion coefficients 1 2
Ideal solution: no mixing effects upon solvent – polymer meeting 3
The solvent chemical potential ms is given by Flory theory

25. The following relation hold4
The following relation hold
rs, ms, ws, Vs* = solvent density, chemical potential, mass fraction and specific critical free volume
wp, Vp* = polymer mass fraction and specific critical free volume
D0ss = pre-exponential factor
= accounts for the overlapping of free volume available to more than one molecule (0.5 ≤ g ≤ 1) (dimensionless)
VFH = specific polymer-solvent mixture average free volume
 = ratio between the solvent and polymer jump unit critical molar volume

26. (K11/g, K12/g, (K21-Tg1) and (K22-Tg2)), for several polymer – solvent systems, can be found in literature