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

i i* i i* i i i i i* i i i i* i* i* i i i i i i* i* i i 2) INTRINSIC: Di  It depends only on “i” mobility 3) SELF: Di*  It depends only on “i” mobility R = universal gas constant T = temperature sih = resistance coefficient ai = “i” activity ci = “i” concentration i i* i i* i i i i i* i i i i* i* i* i i i i i i* i* i i

GEL: D0, DS, D POLYMERIC CHAINS Drug Solvent

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

Hydrodynamic Theory: Stokes Einstein 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 K = Boltzman constant T = temperature RH = drug molecule hydrodynamic radius h = solvent viscosity

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 Polymeric chains drug LMIN L1 L2 L3

2 Mackie Meares Drug molecules of the same size of polymer segments Lattice Model Polymer Drug 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

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

5 Amsden Openings size distribution: Ogston 2 r Drug 2 rs 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)

Hydrodynamic theories 1 Stokes-Einstein Drug Polymer Solvent All these theories focus the attention on the calculation of f, the friction drag coefficient

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

Kinetics theories Existence of a free volume inside the liquid (or gel phase) Solvent molecule Liquid environment 1) Holes volume is constant at constant temperature 2) Holes continuously appear and disappear randomly in the liquid Free volume Vmolecules < Vliquid Liquid environment

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

superscript refers to solvent-polymer properties 1 Eyring 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 molecule energy with respect to 0°K Gel superscript refers to solvent-polymer properties

2 Free Volume vT = solute thermal velocity l = jump length According to this theory step 2 (voids formation) is the rate determining step Solution 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

Vfs = solvent free volume ws = solvent mass fraction 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

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) Y = k2*rs2 It is a parameter not far from 1 It holds for small polymer volume fraction

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

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

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 coefficients 1 2 Ideal solution: no mixing effects upon solvent – polymer meeting 3 The solvent chemical potential ms is given by Flory theory

The following relation hold 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

(K11/g, K12/g, (K21-Tg1) and (K22-Tg2)), for several polymer – solvent systems, can be found in literature