DIFFUSION COEFFICIENT
description
Transcript of DIFFUSION COEFFICIENT
DIFFUSION COEFFICIENTAREA VELOCITY (m2/s)
1) MUTUAL (“i” in “j”): Dij
i
j
i
i
i
i
i
j
j
j
jj
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
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
ii i*
i*
i*
i*
i*
i*
i*
i
ii
ηi
*i
RTD
i
i*ii lnd
lnd
C
aDD
R = universal gas constantT = temperaturei = resistance coefficientai = “i” activityci = “i” concentration
i
GEL: D0, DS, D
Drug
Solvent
POLYMERIC CHAINS
D0, DS, D EVALUATION
MOLECULAR THEORIES STATISTICAL MECHANICALTHEORIES
Atomistic simulationsMathematical models of the GEL network
Hydrodynamic
Kinetics
Obstruction
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
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
D(drug diffusion coefficient in the swollen gel)
Obstruction theories
1 CARMAN
Polymeric chains
drug
LMIN L1
L2
L3
2
0 τ
1
D
D1
*nτ
MIN
n
1ii
L
L
Polymer chains as rigid rods
2 Mackie Meares
Drug molecules of the same size of polymer segments
Polymer
Drug
Lattice Model
2
0 1
1
D
D
= polymer volume fraction(fraction of occupied sites inthe lattice)
3 Ogston
Diffusing molecules much bigger than polymer segments
Polymeric chains:- Negligible thickness - Infinite length
Drug
2 rs
21
f
fs
e0
r
rr
D
D = polymer volume fractionrs = solute radiusrf = polymer fibre radius
4 Deen
Applying the dispersional theory of Taylor
Drug
2 rs
2 rfPolymer
21α
0
e D
D = polymer volume fractionrs = solute radiusrf = polymer fibre radius
= 5.1768-4.0075+5.43882-0.60813
= rs/rf
5 Amsden
Openings 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
4
π
0
err
rr
D
D
radiusaverageopeningskr s5.0
Hydrodynamic theories
1 Stokes-Einstein
All these theories focus the attention on the calculation of f, the friction drag coefficient
f
KT
R
KTD
H0 πη6
Drug
Polymer Solvent
2 Cukier
Strongly crosslinked gels (rigid polymeric chains)
21
fcf
Ac
2ln
3
0
e
srrLM
NL
D
D
Weakly crosslinked gels (flexible polymeric chains)
75.0ce
0
srk
D
D
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
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
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
1 Eyring
According to this theory step 1 (interactions break up) is the most important
kD 20 λ
KTVKTm
KTk
ε
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
V
V
D
Dε'-ε
3
1
'f
f
2
0
eλ
λ'
Gel
superscript refers to solvent-polymer
properties
2 Free Volume
According to this theory step 2 (voids formation) is the rate determining step
f
*
γ
h eV
V
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 eλV
V
vD
vT = solute thermal velocity = jump length
Solution
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
q
P
D
D
1
0
e
FujitaIt holds for small value of the polymer volume fraction
p and q are two independent parameters
Lustig and Peppas
1s
0
e2
1Yr
D
D
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
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
D
D Cukier
1s
0
e2
1Yr
D
DLustig Peppas
PAAM (polyacrylamide),PVA (polyvinylalcohol),PEO (polyethyleneoxide),PHEMA (polyhydroxyethylmethacrylate)
Y = k2*rs2 rs <<
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
4
π
0
err
rr
D
Dradiusaverageopeningskr s5.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.05 0.1 0.15(-)
D/D
0
Cukier
Peppas
Amsden
BSA CASE
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
Temperature 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
4 The following relation hold
PT,s
sssss ρ
μρ
RT
DD
FH
*pp
*s ξωω
γ
0sss eV
VVs
DD
RT
E
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
γ
ξωω
0s2
sFH
*pp
*s
e2χ-1-1V
VVs
DD
ps
ss ρ1ρ
ρω
sp ω1ω
g222212
g121111FH
γγγTTK
KTTK
KV
(K11/, K12/, (K21-Tg1) and (K22-Tg2)), for several polymer – solvent systems, can be
found in literature