BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Where△G′ is the activation energy,△H* is activation enthalpy,and△ S"is activation entropy o N= translational oscillating frequency of solute molecule jump rate! o T= temperature o k= Boltzman constant The ratio of diffusion constant in the gel to that in solution is D Eqn 6 o Where As gel is the activation entropy for diffusion in the gel and Aso is the activation entropy in for diffusion in the solvent o This assumes the activation enthalpy and oscillation frequencies for diffusion are approximately the same in the gel and pure solvent (reasonable for dilute and chemically inert systems The activation entropies are Ean 7 △Sae=knP*-knPo △So=knP*- k In Po n Pgel_pgel opering gel, volume. o Popening is the probability that the network has a solute-sized gap to jump through mp into o Where P"volume is the probability that a solute-sized volume of free space exists to jump into volume dr Egn 10 o Where r is the radius of the solute(drug) and E is the network mesh size The probability of a volume to jump into is an exponential of the ratio of the solute size to the available free volume per mole Eqn 11 Ean 12 o Where free is the specific free volume and v* is the volume of the solute(drug) o Refs for free volume theory applied here Lecture 10-Bioengineering Applications of Hydrogels 8of12BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o Where ∆G* is the activation energy, ∆H* is activation enthalpy, and ∆S* is activation entropy o N = translational oscillating frequency of solute molecule (jump rate!) o T = temperature o k = Boltzman constant ƒ The ratio of diffusion constant in the gel to that in solution is: * ∆Sgel k ˆ Eqn 6 D = Dgel = e ∆S0 * D0 e k o Where ∆S*gel is the activation entropy for diffusion in the gel and ∆S*0 is the activation entropy in for diffusion in the solvent o This assumes the activation enthalpy and oscillation frequencies for diffusion are approximately the same in the gel and pure solvent (reasonable for dilute and chemically inert systems) ƒ The activation entropies are: Eqn 7 ∆S*gel = k ln P* - k ln P0 Eqn 8 ∆S*0 = k ln P*0 – k ln P0 * * * Eqn 9 D = Pgel = Pgel,openingPgel,volume ˆ * * P0 P0,volume o Where P*volume is the probability that a solute-sized volume of free space exists to jump into o P*opening is the probability that the network has a solute-sized gap to jump through P*gel,volume P*gel,opening drug drug * ξ − r =1 − r Eqn 10 Pgel,opening = ξ ξ o Where r is the radius of the solute (drug) and ξ is the network mesh size ƒ The probability of a volume to jump into is an exponential of the ratio of the solute size to the available free volume per mole: v* − * Eqn 11 Pgel,volume ~ e v free,gel v* − * Eqn 12 P0,volume ~ e v free,1 o Where vfree is the specific free volume and v* is the volume of the solute (drug) o Refs for free volume theory applied here: Lecture 10 – Bioengineering Applications of Hydrogels 8 of 12