BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 o At equilibrium, the chemical potential of water inside and outside the gel are equal Ean 2 o Solution contains ions so ui is not equal to u Ean 4 (△u1+) TOTaL=(△)roAL (△1*)ion=(△u)mⅸ+(4u1)e+(4u1)n o The equation we'll try to solve is a rearrangement of this Egn 6 (△u41+)ion(4gu1)on=(41)mx+(△u1)e Contributions to the free energy: FI gy of mi Ean 7 △Gmn=△Hmx-T△Smx o We previously derived the contribution from mixing using the Flory- Rehner lattice model △Gmⅸ= kBT[n,In(1-czs)+zn1中zs] a△Gnn) Eqn 9 m=k27T1-,)+,+=Rm(-)++z o Second expression puts us on a molar basis instead of per molecule o Elastic free energy: AGel=(3/2)kBTve(a--1-In a) Eqn 11 aAGe) x△G))aa =RTN1 2M.vm. o M 2M R Last equality uses 2/vsp, 2Mc (on handout) V.=V2/p2 (on handout) Thus v/,=p2. /vsp. 2M o lonic free enere o Term driving dilution of ions diffusing into gel to maintain charge neutrality o Chemical potential change in solution: Eqn12(4)1=-A= retina≡ RTInx=RThm(-∑x) o approximation in third equality is used for dilute solutions Lecture 9-polyelectrolyte hydrogels 20f6      BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 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 o Chemical potential change in solution: all _ solutes ( )* Eqn 12 ∆µ1 ion = µ1 1 * − µ0 = RT ln a1 * ≅ RT ln x1 * = RT ln(1− ∑x * j) j o approximation in third equality is used for dilute solutions Lecture 9 – polyelectrolyte hydrogels 2 of 6