BEH. 462/3.962J Molecular Principles of Biomaterials Spring 2003 Asterisks denote parameters in solution Free energy has 3 components: free energy of mixing, elastic free energy, and ionic free energy =△G+△G+△G At equilibrium, the chemical potential of water inside and outside the gel are equal Eqn 3 u*-1°=1-u0 o Solution contains ions so ui is not equal to ut Eqn 5 (△u1)n=(4u1)hmⅸx+(△u1)+(△)on o The equation we'll try to solve is a rearrangement of this qn 6 (△u1)in-(△μu)on=(△u1)mix+(△u)el o Contributions to the free energy o Free energy of mixing Eqn 7 △Gmx=△Hmix- TASmi o We previously derived the contribution from mixing using the Flory- Rehner lattice model △Gmk= kATIn,in(1-d2.)+xn1中2s (△4a) a(△Gn) kln(1-,)+n+x2=Rn(1-,)++x2 o Second expression puts us on a molar basis instead of per molecule o Elastic free energy Eqn 11 (4A) a△G a(△C =RTi 2M Last equality uses: (on handout) o V=V2/p2 (on handout) o lonic free energy o Term driving dilution of ions diffusing into gel to maintain charge neutrality Lecture 9-polyelectrolyte hydrogels 120f17      BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 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 ∆Gtotal = ∆Gmix + ∆Gel + ∆Gion o At equilibrium, the chemical potential of water inside and outside the gel are equal: Eqn 2 µ1* = µ1 Eqn 3 µ1* - µ1 0 = µ1 – µ1 0 o Solution contains ions so µ1* is not equal to µ1 0 Eqn 4 (∆µ1*) TOTAL = (∆µ1)TOTAL Eqn 5 (∆µ1*) ion = (∆µ1)mix + (∆µ1)el + (∆µ1)ion o The equation we’ll try to solve is a rearrangement of this: Eqn 6 (∆µ1*) ion - (∆µ1)ion = (∆µ1)mix + (∆µ1)el o Contributions to the free energy: o Free energy of mixing: Eqn 7 ∆Gmix = ∆Hmix – T∆Smix o We previously derived the contribution from mixing using the Flory-Rehner lattice model: Eqn 8 ∆Gmix = kBT[n1ln (1-φ2,s) + χn1φ2,s] ( )1 mix =    ∂(∆Gmix )   = kBT[ln(1− φ 2,s Eqn 9 ∆µ ) + φ2,s + χφ 2,s 2,s ∂n1 T,P 2 ] = RT[ln(1− φ2,s) + φ 2,s + χφ2 ] o Second expression puts us on a molar basis instead of per molecule o Elastic free energy: Eqn 10 ∆Gel = (3/2)kBTνe(α 2 – 1 – ln α) Eqn 11 ( ) ∆µ =  ∂(∆Gel) =  ∂(∆Gel)  ∂α   vm,1    φ2,s  1/ 3 − 1  φ 2,s    M 1 el   ∂n1   T ,P   ∂α   T,P   ∂n1   T ,P = RTν   1− 2 Mc  Vr   φ 2rs   2   φ 2rs     vm,1 = RT   vsp,2 Mc       1− 2Mc   φ 2,s  1/ 3 − 1  φ 2,s   M  φ 2,r    φ2rs   2   φ2rs    • Last equality uses: o ν = V2/vsp,2Mc (on handout) o Vr = V2/φ2,r (on handout) o Thus ν/Vr = φ2,r/vsp,2Mc o Ionic free energy: o Term driving dilution of ions diffusing into gel to maintain charge neutrality Lecture 9 – polyelectrolyte hydrogels 12 of 17