BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 o The mesh size is related to the network swelling Q and the end-to-end distance between cross-links (≤r2>)12=N212a between cross-links qn o. assuming a polymer chain that has 2 carbon-carbon bonds per repeat unit o derived from random walk chain statistics Where /is the bond length in the polymer backbone Mc is the molecular weight between cross-links Mo is the molecular weight per repeat unit Where Cn is the characteristic ratio for the polymer chain 70) Ean 4 (o)=Cm0 Q is the degree of swelling = swollen polymer/var ry polyme N is the degree of polymerization between cross-links The mesh size is related to the diffusion constant of a solute in the network Eyring theory of diffusion Egn 5 Where△G′ is the activation energy,△H* is activation enthalpy,and△ S* is activation entropy o N=translational oscillating frequency of solute molecule (ump rate!) o T= temperature o k= Boltzman constant The ratio of diffusion constant in the gel to that in solution is Eqn 6 o Where As gel is the activation entropy for diffusion in the gel and As o 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 ent Ean 7 △sga=klnP*- kIn Po Lecture 10-Bioengineering Applications of Hydrogels 20f4BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o The mesh size is related to the network swelling Q and the end-to-end distance between cross-links: (<r0 2>)1/2=Nc 1/2a statistical segment length Number of segments between cross-links Eqn 3 r0 2 1/ 2  2M c  1/ 2 C1/ 2 ( ) = l    M 0    n o …assuming a polymer chain that has 2 carbon-carbon bonds per repeat unit o derived from random walk chain statistics ƒ Where l is the bond length in the polymer backbone ƒ Mc is the molecular weight between cross-links ƒ M0 is the molecular weight per repeat unit ƒ Where Cn is the characteristic ratio for the polymer chain ( )2 1/ 2 Eqn 4 ξ = r0 1/ 3 = Q1/ 3 ( ) r0 2 1/ 2 = Cn 1/ 2 Q1/ 3 N1/ 2 l φ2,s ƒ Q is the degree of swelling = Vswollen polymer/Vdry polymer ƒ N is the degree of polymerization between cross-links ƒ The mesh size is related to the diffusion constant of a solute in the network ƒ Eyring theory of diffusion: − ∆G* − ∆H * ∆S* Eqn 5 D = Tνe kT = Tνe kT e k 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 Lecture 10 – Bioengineering Applications of Hydrogels 2 of 4