Download - DIFFUSION COEFFICIENT
![Page 1: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/1.jpg)
DIFFUSION COEFFICIENTAREA VELOCITY (m2/s)
1) MUTUAL (“i” in “j”): Dij
i
j
i
i
i
i
i
j
j
j
j j
DEPENDS ON1) “i” intrinsic mobility2) The presence of “j”
Unless “I” and “j” have the same mass andsize, a hydrostatic pressure gradient arises.This is balanced by a mixture bulk flow.
Dij is the result of molecules random motionand bulk flow
SOLUTION
![Page 2: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/2.jpg)
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
ii
i
ii i*
i*
i*
i*
i*
i*
i*
i
ii
ηi*i
RTD
i
i*ii lnd
lndCaDD
R = universal gas constantT = temperaturei = resistance coefficientai = “i” activityci = “i” concentration
i
![Page 3: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/3.jpg)
GEL: D0, DS, D
Drug
Solvent
POLYMERIC CHAINS
![Page 4: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/4.jpg)
D0, DS, D EVALUATION
MOLECULAR THEORIES STATISTICAL MECHANICALTHEORIES
Atomistic simulationsMathematical models of the GEL network
Hydrodynamic
Kinetics
Obstruction
![Page 5: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/5.jpg)
D0(mutual drug diffusion coefficient in the pure solvent)
Hydrodynamic Theory: Stokes Einstein
1 It holds for large spherical molecules ….
… in a diluted solution2
H0 πη6 R
KTD
K = Boltzman constantT = temperatureRH = drug molecule hydrodynamic radius = solvent viscosity
![Page 6: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/6.jpg)
Solute D0*106
(cm2/s)
T
(°C)
rs
(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
![Page 7: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/7.jpg)
D(drug diffusion coefficient in the swollen gel)
Obstruction theories
1 CARMAN
Polymeric chains
drug
LMIN L1
L2
L3
2
0 τ1
DD
1*n
τMIN
n
1ii
L
L
Polymer chains as rigid rods
![Page 8: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/8.jpg)
2 Mackie MearesDrug molecules of the same size of polymer segments
Polymer
Drug
Lattice Model
2
0 11
DD
= polymer volume fraction(fraction of occupied sites inthe lattice)
![Page 9: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/9.jpg)
3 OgstonDiffusing molecules much bigger than polymer segments
Polymeric chains:- Negligible thickness - Infinite length
Drug
2 rs
21
f
fs
e0
r
rr
DD
= polymer volume fractionrs = solute radiusrf = polymer fibre radius
![Page 10: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/10.jpg)
4 DeenApplying the dispersional theory of Taylor
Drug
2 rs
2 rfPolymer
21α
0
e DD
= polymer volume fractionrs = solute radiusrf = polymer fibre radius
= 5.1768-4.0075+5.43882-0.60813
= rs/rf
![Page 11: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/11.jpg)
5 AmsdenOpenings size distribution: Ogston
Drug
2 rs
Polymer
2 r
= polymer volume fractionrs = solute radiusrf = polymer fibre radiusks = constant (it depends on the polymer solvent couple)
2
f
fs
4π
0
err
rr
DD
radiusaverageopeningskr s5.0
![Page 12: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/12.jpg)
Hydrodynamic theories
1 Stokes-Einstein
All these theories focus the attention on the calculation of f, the friction drag coefficient
fKT
RKTD
H0 πη6
Drug
Polymer Solvent
![Page 13: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/13.jpg)
2 Cukier
Strongly crosslinked gels (rigid polymeric chains)
21
fcf
Ac
2ln3
0
e
srrLMNL
DD
Weakly crosslinked gels (flexible polymeric chains)
75.0ce
0
srk
DD
Lc = polymer chains lengthMf = polymer chains molecular weightNA = Avogadro numberrf = polymer chains radiusrs = drug molecule radius= polymer volume fraction
kc = depends on the polymer solvent couple
![Page 14: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/14.jpg)
Kinetics theories
Existence of a free volume inside the liquid (or gel phase)
Solvent molecule
Liquid environment
Vmolecules < Vliquid
Liquid environment1) Holes volume is constant at constant temperature2) Holes continuously appear and disappear randomly in the liquid
Free volume
![Page 15: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/15.jpg)
Solute
1) Energy needed to break the interactions with surrounding molecules
DIFFUSION MECHANISM
2) Probability of finding a sufficiently big hole at the right distance
![Page 16: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/16.jpg)
1 Eyring
According to this theory step 1 (interactions break up) is the most important
kD 20 λ
KTVKTm
KTkε
1/3f
r
eπ2
Solution
= mean diffusive jump length k = the jump frequency
K = Boltzman constantT = temperaturemr = solvent-solute reduced massVf = mean free volume available per solute molecule = solute molecule energy with respect to 0°K
KT
VV
DD ε'-ε3
1
'f
f2
0
eλλ'
Gel
superscript refers to solvent-polymer
properties
![Page 17: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/17.jpg)
2 Free VolumeAccording to this theory step 2 (voids formation) is the rate determining step
f
*γ
h e VV
pProbability 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 < < 1 => it accounts for the overlapping of the free volume available to more
than one molecule
f
*γ
T0 eλ VV
vD
vT = solute thermal velocity = jump length
Solution
![Page 18: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/18.jpg)
Gel
Assuming negligible mixing effects, the free volume Vf of a mixture composed by solvent, polymer and drug is be given by:
pfpsfsdfdf ωωω VVVV
Vfd = drug free volumed = drug mass fractionVfs = solvent free volumes = solvent mass fractionVfp = polymer free volumep = polymer mass fraction
![Page 19: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/19.jpg)
q
P
DD
1
0
e
Fujita It holds for small value of the polymer volume fraction
p and q are two independent parameters
Lustig and Peppas
1s
0
e21Yr
DD
They combine the FVT with the idea that diffusion can not occur if solute diameter is smaller than crosslink average length ()
Y = k2*rs2 It is a parameter not far from 1
It holds for small polymer volume fraction
![Page 20: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/20.jpg)
Cukier and Peppas equations bets fitting (fitting parameters kc and k2, respectively).(polymer concentration is the independent variable).
Polymer Solute kc (Å-1) rs (Å) k2 (Å-2) rs (Å)
Hydrodynamic theory
(eq.(4.121))
Free Volume theory
(eq.(4.130))
urea 1.12 1.9 0.774 1.9
sucrose 1.06 4.75 0.281 4.75
ribonuclease 0.55 16.3 0.060 16.6 PAAM
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 Dextran
immunoglobulin G 0.66 56.3 0.016 56.5
vitamine B12 0.62 8.7 0.061 8.7 PVA
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
75.0ce
0
srk
DD
Cukier
1s
0
e21Yr
DD
Lustig Peppas
PAAM (polyacrylamide),PVA (polyvinylalcohol),PEO (polyethyleneoxide),PHEMA (polyhydroxyethylmethacrylate)
Y = k2*rs2 rs <<
![Page 21: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/21.jpg)
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
Polymer Solute ks (Å) rs (Å)
Obstruction theory
(eq.(4.118))
alginate bovin serum albumin 5.73 36.3
myoglobin 11.63 18.9 agarose
bovin serum albumin 12.45 36.3
Amsden
2
f
fs
4π
0
err
rr
DD
radiusaverageopeningskr s5.0
![Page 22: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/22.jpg)
00.10.20.30.40.50.60.70.80.9
1
0 0.05 0.1 0.15(-)
D/D
0
CukierPeppasAmsden
BSA CASE
![Page 23: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/23.jpg)
CONSIDERATIONS
1) Free Volume and Hydrodynamic theories should be used for weakly crosslinked networks
2) Obstruction theories should better work with highly crosslinked networks
![Page 24: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/24.jpg)
DS(solvent diffusion coefficient in the swelling gel)The only available theory is the free volume theory of Duda and Vrentas
HYPOTHESESTemperature independent thermal expansion coefficients1
2 Ideal solution: no mixing effects upon solvent – polymer meeting
3 The solvent chemical potential s is given by Flory theory
20ss χ1lnμμ RT
![Page 25: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/25.jpg)
4 The following relation hold
PT,s
sssss ρ
μρ
RT
DD
FH
*pp
*s ξωω
γ
0sss eV
VVs
DD
RTE
DD e0ss0s
s, s, s, Vs* = solvent density, chemical potential, mass fraction and specific critical free
volume
p, 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 ≤ ≤ 1) (dimensionless)
VFH = specific polymer-solvent mixture average free volume
= ratio between the solvent and polymer jump unit critical molar volume
![Page 26: DIFFUSION COEFFICIENT](https://reader036.vdocument.in/reader036/viewer/2022062316/56816851550346895dde5bc0/html5/thumbnails/26.jpg)
γ
ξωω
0s2
sFH
*pp
*s
e2χ-1-1V
VVs
DD
ps
ss ρ1ρ
ρω
sp ω1ω
g222212
g121111FH
γγγTTKKTTKKV
(K11/, K12/, (K21-Tg1) and (K22-Tg2)), for several polymer – solvent systems, can be
found in literature