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DIFFUSION COEFFICIENT
AREA VELOCITY (m2/s)
SOLUTION
1) MUTUAL (“i” in “j”): Dij
j
i
i
DEPENDS ON
1) “i” intrinsic mobility
2) The presence of “j”
j
i
j
i
i
j
j
j
Unless “I” and “j” have the same mass and
size, a hydrostatic pressure gradient arises.
This is balanced by a mixture bulk flow.
i
Dij is the result of molecules random motion
and bulk flow
2) INTRINSIC: Di  It depends only on “i” mobility
3) SELF: Di*  It depends only on “i” mobility
i
i*
i
i*
i
i
i
i
i*
i
i*
i
i
i*
i*
i
i
i
i
i
i*
i*
i
i
RT
D 
 iη
* d ln ai 
Di  Di
dln Ci 
*
i
R = universal gas constant
T = temperature
ih = resistance coefficient
ai = “i” activity
ci = “i” concentration
GEL: D0, DS, D
Drug
Solvent
POLYMERIC CHAINS
D0, DS, D EVALUATION
MOLECULAR THEORIES
Mathematical models of
the GEL network
Obstruction
Hydrodynamic
Kinetics
STATISTICAL MECHANICAL
THEORIES
Atomistic simulations
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
KT
D0 
6 πηRH
K = Boltzman constant
T = temperature
RH = drug molecule hydrodynamic radius
h = solvent viscosity
D0*106
T
rs
Solute
(cm2/s)
(°C)
(Å)
urea
18.1
37
1.9
glucose
6.4
23
3.6
theophylline
8.2
37
3.9
sucrose
7.0
37
4.8
caffeine
6.3
37
5.3
phenylpropanolamine
5.5
37
6.0
vitamin B12
3.8
37
8.6
PEG 326
4.9
25
7.5
PEG 1118
2.8
25
13.1
PEG 2834
1.8
25
20.4
PEG 3978
1.5
25
24.5
ribonuclease
0.13
20
16.3
myoglobin
0.11
20
18.9
lysozyme
0.11
20
19.1
pepsin
0.09
20
23.8
ovalbumin
0.07
20
29.3
bovine serum albumin
0.06
20
36.3
immunoglobulin G
0.04
20
56.3
fibrinogen
0.02
20
107
Diffusion coefficient
D0 in water and radius
rs of some solutes
D(drug diffusion coefficient in the swollen gel)
Obstruction theories
1 CARMAN
Polymer chains as rigid rods
LMIN
L1
L2
drug
L3
Polymeric chains
n
τ
L
i 1
i
n * LMIN
1
D 1
 
D0  τ 
2
2 Mackie Meares
Drug molecules of the same size of polymer segments
Lattice Model
Polymer
Drug
D 1 

 
D0  1   
2
f = polymer volume fraction
(fraction of occupied sites in
the lattice)
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
D
e
D0
 rs  rf 1 2 

 


r
f


4 Deen
Applying the dispersional theory of Taylor
2 rf
Polymer
Drug
2 rs
f = polymer volume fraction
rs = solute radius
rf = polymer fibre radius
D

 α 1 2 
e
D0
= 5.1768-4.0075l+5.4388l2-0.6081l3
l = rs/rf
5 Amsden
Openings size distribution: Ogston
Polymer
2r
Drug
2 rs
f = polymer volume fraction
rs = solute radius
rf = polymer fibre radius
ks = constant (it depends on the
polymer solvent couple)
D
e
D0

 π  rs  rf
 
 4  r  rf





2





r  0.5ks   openings average radius
Hydrodynamic theories
1 Stokes-Einstein
Polymer
Solvent
KT
KT
D0 

6πηRH
f
Drug
All these theories focus the attention on the calculation
of f, the friction drag coefficient
2 Cukier
Strongly crosslinked gels (rigid polymeric chains)
D
e
D0
  3Lc N A
 12
 


  M ln  L 2 r   rs  
c
f 
  f

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)
D
D0


k r 
e
0.75
c s
kc = depends on the polymer
solvent couple
Kinetics theories
Existence of a free volume inside the liquid (or gel phase)
Solvent molecule
Free volume
Vmolecules < Vliquid
Liquid environment
Liquid environment
1) Holes volume is constant at constant
temperature
2) Holes continuously appear and
disappear randomly in the liquid
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
1 Eyring
According to this theory step 1 (interactions break up) is the most important
Solution
D0  λ k
2
KT
k
Vf
2 π mr KT
 ε 


1/3  KT 
e
l = 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
1
3
D  λ'   Vf 
    '  e
D0  λ   Vf 
2
 ε -ε' 


KT


superscript refers to
solvent-polymer
properties
2 Free Volume
According to this theory step 2 (voids formation) is the rate determining step
Solution
ph  e
 V* 
 γ 
 V 
f 

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
D0  vT λ e
 V* 
 γ 
 V 
f 

vT = solute thermal velocity
l = jump length
Gel
Assuming negligible mixing effects, the free volume Vf of a mixture
composed by solvent, polymer and drug is be given by:
Vf  Vfd ωd  Vfs ωs  Vfp ω p
Vfd = drug free volume
wd = drug mass fraction
Vfs = solvent free volume
ws = solvent mass fraction
Vfp = polymer free volume
wp = polymer mass fraction
Fujita
D
e
D0


 1

q
 P








p and q are two f independent parameters
Lustig and Peppas
D  2rs 
 1  e
D0   
It holds for small value of the polymer
volume fraction f
   
 Y 
 

1



 
They combine the FVT with the idea that
diffusion can not occur if solute diameter
is smaller than crosslink average length ()
It holds for small polymer volume fraction
Y = k2*rs2 It is a parameter not far from 1
Polymer
kc (Å-1)
Solute
D
 e kcrs  
0.75
D0
urea
rs (Å)
k2 (Å-2)
rs (Å)
Hydrodynamic theory
Free Volume theory
(eq.(4.121))
Cukier
(eq.(4.130))
Lustig
Peppas
1.12
1.9
0.774
D  2rs 
e
 1 
D0 
 
1.9
sucrose
1.06
4.75
0.281
4.75
ribonuclease
0.55
16.3
0.060
16.6
bovin serum albumin
0.45
36.3
0.023
36.3
lysozyme
0.57
19.1
0.038
19.4
bovin serum albumin
0.58
36.3
0.021
36.3
immunoglobulin G
0.66
56.3
0.016
56.5
vitamine B12
0.62
8.7
0.061
8.7
lysozyme
0.40
19.1
0.044
19.4
PEO
caffeine
0.88
5.25
0.179
5.25
PHEMA
phenylpropanolamine
1.10
6.0
0.081
6.0
Y = k2*rs2
   
 Y 
 

  1  
rs << 
PAAM
Dextran
PVA
Cukier and Peppas equations bets fitting (fitting
parameters kc and k2, respectively).
(polymer concentration f is the independent variable).
PAAM (polyacrylamide),
PVA (polyvinylalcohol),
PEO (polyethyleneoxide),
PHEMA (polyhydroxyethylmethacrylate)
D
e
D0

 π  rs  rf
 
 4  r  rf

Polymer




2





r  0.5ks   openings average radius
Solute
ks (Å)
rs (Å)
Obstruction theory
(eq.(4.118))
Amsden
alginate
bovin serum albumin
5.73
36.3
myoglobin
11.63
18.9
bovin serum albumin
12.45
36.3
agarose
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
BSA CASE
D/D0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Cukier
Peppas
Amsden
0
0.05
0.1
f(-)
0.15
CONSIDERATIONS
1) Free Volume and Hydrodynamic theories
should be used for weakly crosslinked networks
2) Obstruction theories
should better work with highly crosslinked networks
DS(solvent diffusion coefficient in the swelling gel)
The only available theory is the free volume theory of Duda and Vrentas
HYPOTHESES
1
Temperature independent thermal expansion coefficients
2
Ideal solution: no mixing effects upon solvent – polymer meeting
3
The solvent chemical potential ms is given by Flory theory

μ s  μ  RT ln 1       χ
0
s
2

4
The following relation hold
Dss ρ s
Ds 
RT
Dss  D0s e
 ωs Vs*  ωp Vp*ξ 
 γ



VFH


  μs 


 ρ s  T, P
D0s  D0sse
 E 


 RT 
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
g = 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
Ds  1 -  1 - 2χD0s e
2
ρ s
ωs 
ρ s 1     ρ p
 ωs Vs* ωp Vp*ξ 




VFH γ


ω p  1  ωs
VFH K11
K12

w1 K 21  T  Tg1  
w2 K 22  T  Tg2 
γ
γ
γ
(K11/g, K12/g, (K21-Tg1) and (K22-Tg2)), for several polymer – solvent systems, can be
found in literature