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DIFFUSION COEFFICIENT

DIFFUSION COEFFICIENT. AREA VELOCITY (m 2 /s). SOLUTION. 1) MUTUAL (“i” in “j”): D ij. 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.

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DIFFUSION COEFFICIENT

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  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. R = universal gas constant T = temperature sih = resistance coefficient ai = “i” activity ci = “i” concentration 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

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

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

  5. D0(mutualdrug 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) LMIN L1 1 CARMAN L2 drug L3 Polymeric chains Obstruction theories Polymer chains as rigid rods

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

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

  10. 4 Deen Polymer 2 rf f = polymer volume fraction rs = solute radius rf = polymer fibre radius • = 5.1768-4.0075l+5.4388l2-0.6081l3 l = rs/rf Applying the dispersional theory of Taylor Drug 2 rs

  11. 5 Amsden 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) Openings size distribution: Ogston

  12. 1 Stokes-Einstein Polymer Solvent Drug Hydrodynamic theories 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. Free volume Solvent molecule Liquid environment 1) Holes volume is constant at constant temperature 2) Holes continuously appear and disappear randomly in the liquid Kinetics theories Existence of a free volume inside the liquid (or gel phase) 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. 1 Eyring Gel superscript refers to solvent-polymer properties 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 moleculeenergy with respect to 0°K

  17. 2 Free Volume 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 According to this theory step 2 (voids formation) is the rate determining step Solution

  18. 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) It holds for small polymer volume fraction Y = k2*rs2Itis a parameter not far from 1

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

  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. 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 DS(solvent diffusion coefficient in the swelling gel) Theonly available theory is the free volume theory of Duda and Vrentas HYPOTHESES

  25. 4 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

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