diffusion coefficient

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

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

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 τ1

DD

1*n

τMIN

n

1ii

L

L

Polymer chains as rigid rods

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)

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

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

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

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

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

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

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

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

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

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

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 <<

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

00.10.20.30.40.50.60.70.80.9

1

0 0.05 0.1 0.15(-)

D/D

0

CukierPeppasAmsden

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

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

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

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