BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 o Outside gel Ean 24 Eqn 25 o Our relationship for the ionic chemical potentials is now (AHon -(Au)on =Vm, RT 2(,-C)=vm, RT(++c.-c*-c") o Using Eqn 20, Eqn 21, Eqn 24, and Eqn 25, Eqn 26 becomes Ean 27 (4)-(△4) lonImrT vc,+v-cs+ ic:=vm, RTvC,+2 V,RTl ( c o How can we relate cs and cs*? o We can make simplifications for a 1: 1 cation anion electrolyte o The chemical potentials of the mobile ions must also be equilibrated inside/outside the gel o Add Eqn 29 to Eqn 28 Ean 31 RTIna++ rTina-= RIna++tina v o Therefore we can write Ean 32 Assuming dilute solutions where the activities are approximately equal to the concentrations Ean 3 Ean 34 vc+ Eqn 35 Lecture 9-polyelectrolyte hydrogels 4 of 6BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o Outside gel: Eqn 24 c+* = ν+cs* Eqn 25 c-* = ν-cs* o Our relationship for the ionic chemical potentials is now: all−ions ( )* ( ) * Eqn 26 ∆µ 1 ion = vm,1RT ∑(c j − c * j)=vm,1RT(c+ + c− − c+ − c− * 1 ion − ∆µ ) j o Using Eqn 20, Eqn 21, Eqn 24, and Eqn 25, Eqn 26 becomes:     * Eqn 27 ( ) ∆µ * ( )1 ion = vm,1RT ν +cs + ν −cs + ic2 − ˆcs *   = vm,1RT ν cs + ic2 − ˆcs  1  ion − ∆µ ν ˆ ν z− z−   ν(c = s vm,1RT  ic2 − ˆ * − cs)   z− o How can we relate cs and cs*? o We can make simplifications for a 1:1 cation:anion electrolyte: o The chemical potentials of the mobile ions must also be equilibrated inside/outside the gel: Eqn 28 µ+ = µ+* Eqn 29 µ- = µ-* o Add Eqn 29 to Eqn 28: Eqn 30 µ+ + µ-= µ+* + µ-* ν + Eqn 31 RT ln a+ + RT ln a− ν − = RT ln a+ *ν + + RT ln a− *ν − o Therefore we can write: ν + a− ν − Eqn 32 a+ = a+ *ν + a− *ν − • Assuming dilute solutions where the activities are approximately equal to the concentrations: ν + ν −  c+   c− *  Eqn 33   c+ = *     c−     ν − *  Eqn 34 ν +cs  ν +  ν −cs    ν +c * s   =   ν −cs + ic2    z−  ν −   ν +  *  Eqn 35     c c s * s     =    cs + cs ic2    ν − z−   Lecture 9 – polyelectrolyte hydrogels 4 of 6