BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Brannon-Peppas theory of swelling in ionic hydrogels Original theory for elastic networks developed by Flory and Mehrer, refined for treatment of ionic hydrogels by Brannon-Peppas and Peppas Other theoretical treatments Derivation of ionic hydrogel swelling Model structure of the system Model of system norganic anion, e.g. CI e Inorganic cation e.g. Na* 8() () System is composed of permanently cross-linked polymer chains, water, and salt We will derive the thermodynamic behavior of the ionic hydrogel using the model we previously developed for neutral hydrogels swelling in good solvent · Model parameters activity of cations in gel Boltzman constant activity of cations in solution bsolute temperature( Kelvin activity of anions in gel Vm,1 molar volume of solvent(water, volume/mole) activity of anions in solution molar volume of polymer (volume/mole) C+ concentration of cations in gel(moles/volume) Vsp,1 specific volume of solvent(water, volume/mass) C+. concentration of cations in solution(moles/volume vsp, 2 specific volume of polymer(volume/mass) concentration of anions in solution(moles/volume) total volume of polymer concentration of anions in solution( moles/volume) vs total volume of swollen hydrogel concentration of electrolyte total volume of relaxed hydrogel concentration of ionizable repeat units in gel number of subchains in network (moles/volume) number of 'effective subchains in network H1 chemical potential of water in solution stoichiometric coefficient for eletrolyte cation chemical potential of water in the hydrogel H1 chemical potential of pure water in standard state stoichiometric coefficient for eletrolyte anion volume fraction of water in swollen gel M Molecular weight of polymer chains before cross-linking olume fraction of polymer in swollen gel M Molecular weight of cross-linked subchain number of water molecules in swollen gel 22x volume fraction of polymer in relaxed gel polymer-solvent interaction parameter mole fraction of water in swollen gel mole fraction of water in soluti o Asterisks denote parameters in solution o Free energy has 3 components: free energy of mixing, elastic free energy, and ionic free energy Eqn 1 △G=△G+AG+AC Lecture 9-polyelectrolyte hydrogels 1 of 6BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 Brannon-Peppas theory of swelling in ionic hydrogels • Original theory for elastic networks developed by Flory and Mehrer1-3 , refined for treatment of ionic hydrogels by Brannon-Peppas and Peppas4,5 • Other theoretical treatments6 Derivation of ionic hydrogel swelling • Model structure of the system: Model of system: Inorganic anion, e.g. Cl￾Inorganic cation, e.g. Na+ (-) (-) (-) (-) (-) (-) (-) (-) (-) water • System is composed of permanently cross-linked polymer chains, water, and salt • We will derive the thermodynamic behavior of the ionic hydrogel using the model we previously developed for neutral hydrogels swelling in good solvent • Model parameters: a+ activity of cations in gel a+* activity of cations in solution a- activity of anions in gel a-* activity of anions in solution c+ concentration of cations in gel (moles/volume) c+* concentration of cations in solution (moles/volume) c- concentration of anions in solution (moles/volume) c-* concentration of anions in solution (moles/volume) cs concentration of electrolyte c2 concentration of ionizable repeat units in gel (moles/volume) * µ1 chemical potential of water in solution µ1 chemical potential of water in the hydrogel µ1 chemical potential of pure water in standard state M Molecular weight of polymer chains before cross-linking Mc Molecular weight of cross-linked subchains n1 number of water molecules in swollen gel χ polymer-solvent interaction parameter o Asterisks denote parameters in solution kB Boltzman constant T absolute temperature (Kelvin) vm,1 molar volume of solvent (water, volume/mole) vm,2 molar volume of polymer (volume/mole) vsp,1 specific volume of solvent (water, volume/mass) vsp,2 specific volume of polymer (volume/mass) V2 total volume of polymer Vs total volume of swollen hydrogel Vr total volume of relaxed hydrogel ν number of subchains in network νe number of ‘effective’ subchains in network ν+ stoichiometric coefficient for eletrolyte cation ν− stoichiometric coefficient for eletrolyte anion φ1,s volume fraction of water in swollen gel φ2,s volume fraction of polymer in swollen gel φ2,r volume fraction of polymer in relaxed gel x1 mole fraction of water in swollen gel x1* mole fraction of water in solution o Free energy has 3 components: free energy of mixing, elastic free energy, and ionic free energy Eqn 1 ∆Gtotal = ∆Gmix + ∆Gel + ∆Gion Lecture 9 – polyelectrolyte hydrogels 1 of 6 0