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 6
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 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. ClInorganic 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
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
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 (△4)2 RT V,RT Eqn 13 E-RT>x:= ∑ R o The first approximation holds if Ex is small o Fourth equality holds because we assume in the liquid lattice model that the molar volume of all species is the same, thus Vm n=V, the total volume of the system o Chemical potential change in gel all-ions Eqn 14 (△A)mn=H-A0= RT In a≡-VmRT∑c Eqn 15 (A)m-(A4)m=mRr∑(-c) o The electrolyte dissolved in water provides mobile cations and anions in the solution and in the gel o E.g. NaCl: Na vcr v+()->v Na(ag) vcr(ag) o v=v=1 stoichiometric coefficients CA→vC+vA eg.CaCl2:v+=1,V=2,z=2,z=1 v*+v=v . for a 1: 1 electrolyte for a 1: 1 electrolyte Eqn 19 =(v+v)c total concentration of ions o We will derive equations for an anionic network o Assuming activities- concentrations o Inside gel Egn 20 Ean 21 C.=v. Cs+ ic lz. o C2 is the moles of ionizable repeat groups on gel chains per volume o First term comes from electrolyte anions in gel, second term from ionized groups on the polymer chains o The degree of ionization i can be related to the pH of the environment and the pKa of the network groups K= IRCOOH'] [RCOOHT rcoo RCOO RCOOH K K 10 [ RCOOH+[FoO],风R o0 1+K ph 10PH+10-pa Lecture 9-polyelectrolyte hydrogels 3 of 6
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 all _ ions all _ ions all _ ions all _ ions Eqn 13 ( ) ∆µ1 * ion ≅ −RT ∑x * j = − RT ∑n * j = − vm,1RT ∑n * j ≅ −vm,1RT ∑c * j j n j vm,1n j j o The first approximation holds if Σxj * is small o Fourth equality holds because we assume in the liquid lattice model that the molar volume of all species is the same, thus vm,1n = V, the total volume of the system o Chemical potential change in gel: all−ions Eqn 14 (∆µ1 )ion = µ1 − µ1 0 = RT ln a1 ≅ −vm,1RT ∑c j j all−ions ( )* ( ) * Eqn 15 ∆µ 1 ion = vm,1RT ∑(c j − c 1 ion − ∆µ j) j o The electrolyte dissolved in water provides mobile cations and anions in the solution and in the gel: o E.g. NaCl: Na+ ν+Cl- ν+ (s) → ν+ Na+ (aq) + ν- Cl- (aq) o ν+ = ν- = 1 stoichiometric coefficients Eqn 16 Cν + z+ z− Aν − →ν +Cz+ + ν −Az− • e.g. CaCl2: ν+ = 1, ν- = 2, z+ = 2, z- = 1 Eqn 17 ν+ + ν− = νˆ …for a 1:1 electrolyte ˆ Eqn 18 ν+ = ν− = ν …for a 1:1 electrolyte 2 * * * ˆ * Eqn 19 c+ + c− = (ν+ + ν− )cs = νcs …total concentration of ions o We will derive equations for an anionic network o Assuming activities ~ concentrations o Inside gel: Eqn 20 c+ = ν+cs Eqn 21 c- = ν-cs + ic2/zo c2 is the moles of ionizable repeat groups on gel chains per volume o First term comes from electrolyte anions in gel, second term from ionized groups on the polymer chains o The degree of ionization i can be related to the pH of the environment and the pKa of the network groups: [ ] Eqn 22 Ka = [RCOO− ] H+ [RCOOH] Eqn 23 [RCOO− ] Ka H+ i = [RCOO− ] [RCOOH] [ ] Ka Ka 10− pKa = = = = = H+ [RCOOH]+ [RCOO− ] [RCOO− ] 1+ Ka [ ]+ Ka 10− pH + Ka 10− pH +10− pKa H 1+ + [RCOOH] [ ] Lecture 9 – polyelectrolyte hydrogels 3 of 6
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 6
BEH.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
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 Eqn 36 C Derivation of this equation in ap o Now Eqn 27 becomes (44)n-(△x)n o But definition of ionic strength I is Egn 38 . for a 1: 1 electrolyte Where z, is the charge on ion i (△A)。-(△A4)n=vmR 4/=mP7, 4lvs2Mo moles ionizable groups/volume) o Eqn 39 can be re-cast in terms of the solution pH Eqn 40 (4A4)n-(A4) V. 4(10 41v52M6 o Returning to the equilibrium criterion Eqn 41 10-pka 2M =-,)+,十 M 2(φ2 o Brannon-Peppas paper analyzes Polyacrylates/polymethacrylates o In water pH 7.0 with 1=0.35 oX=0.8 pKa =6.0 o Vsp. 2=0.8 cm"/g M=75,000 g/mole o Mc=12,000 g/mole o Mo= 90 g/mole dr=0.5 Lecture 9-polyelectrolyte hydrogels 5 of 6
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 ν − ν + * * 1 ic * 2 2 Eqn 36 c * s c − * s cs =1− cs + cs ic2 =1 − cs + cs ic2 = ν ic2 * − 2z+z−ν2 cs ˆz−cs ˆ ν −z− ν −z− o Derivation of this equation in appendix o Now Eqn 27 becomes: i 2 c2 2 ( )* Eqn 37 ∆µ ( )1 ion = vm,1RT 2z+z−ν * 1 ion − ∆µ ˆcs o But definition of ionic strength I is: all _ ions ˆcs * Eqn 38 I = 1 ∑zi 2 ci = z+z−ν …for a 1:1 electrolyte 2 i 2 Where zi is the charge on ion i o Therefore: i 2 φ2 ( )* ( ) 2,s Eqn 39 ∆µ 1 ion = vm,1RT i 2 c2 2 = vm,1RT 4Ivsp,2 M0 2 1 ion − ∆µ 4I 2 φ2,s o (Using relation c2 = vsp,2 M0 =moles ionizable groups/volume) o Eqn 39 can be re-cast in terms of the solution pH: 2 ( )1 * ion − ∆( ) µ = vm,1RT Ka 2 φ2,s 2 Ka 2 φ 2,s 2 Eqn 40 ∆µ 1 ion 4I 10− pH + Ka z−vsp,2 M0 = vm,1RT 10− pH + Ka 4Ivsp,2 M0 2 o Returning to the equilibrium criterion: Eqn 41 2 10− pKa 2 φ 2,s 2 vm,1 1− 2 Mc φ 2,s 1/ 3 − 1 φ2,s 2 vm,1 10− pH +10− pKa 4Ivsp,2 M0 2 = ln(1 − φ2,s) + φ2,s + χφ2,s + φ2,r vsp,2 Mc M φ 2,r 2 φ2,r o Brannon-Peppas paper analyzes Polyacrylates/polymethacrylates: o In water pH 7.0 with I = 0.35 o χ = 0.8 o pKa = 6.0 o vsp,2 = 0.8 cm3 /g o M = 75,000 g/mole o Mc = 12,000 g/mole o M0 = 90 g/mole o φ2,r = 0.5 Lecture 9 – polyelectrolyte hydrogels 5 of 6
BEH.462/3. 962J Molecular Principles of Biomaterials Spring 2003 o 250 g1600 9d2日0:0d 20o1/2/3 1200 150 900 md 1234567日101112 pH of Swelling Medium 5678101112 pH of Swelling Medium Fig. 3. Theoretical swelling predictions at comparable ionic strength conditions for an anionic network with Fig 4. Theoretical swelling predictions at comparable ionic (1)pKn=202)pK。=40(3)pKn=60.(4pkn=80.andstrengthcondtionstrananionienetworkwith(r=005 pKa=10.0 (2)I=0.1,(3)I=0.25,(4)I=0.5,(5)I=0.75,(6)I=10, and(7)I=20. References 1. James, H M.& Guth, E Simple presentation of network theory of rubber, with a discussion of other theories. J. Poym.Sc.4,153-182(1949) Flory, P J. Rehner Jr, J. Statistical mechanics of cross-linked polymer networks. L. Rubberlike elasticity. J. chem.Phys.11,512-520(1943) 3. Flory, P J& Rehner Jr, J. Statistical mechanics of cross-linked polymer networks. IL Swelling. J. Chem. Phys 11,521-526(1943 Brannonpeppas, L& Peppas, N. A Equilibrium Swelling Behavior of Ph-Sensitive Hydrogels. Chemical Engineering: pvatpololyvinyl-Alcohol)Hydrogels-Reinforcement of Radiation-Crosslinked Networks 715-722(1991) N.A. by Crystallization. Journal of Polymer Science Part a-Polymer Chemistry 14, 441-457(1976) Ozyurek, C, Caykara, T, Kantoglu, O& Guven, O. Characterization of network structure of poly(N-vinyl 2- pyrrolidone/acrylic acid) polyelectrolyte hydrogels by swelling measurements. Journal of Polymer Science Part B Polymer PhysicS 38, 3309-3317(2000) Lecture 9-polyelectrolyte hydrogels 6 of 6
BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 References 1. James, H. M. & Guth, E. Simple presentation of network theory of rubber, with a discussion of other theories. J. Polym. Sci. 4, 153-182 (1949). 2. Flory, P. J. & Rehner Jr., J. Statistical mechanics of cross-linked polymer networks. I. Rubberlike elasticity. J. Chem. Phys. 11, 512-520 (1943). 3. Flory, P. J. & Rehner Jr., J. Statistical mechanics of cross-linked polymer networks. II. Swelling. J. Chem. Phys. 11, 521-526 (1943). 4. Brannonpeppas, L. & Peppas, N. A. Equilibrium Swelling Behavior of Ph-Sensitive Hydrogels. Chemical Engineering Science 46, 715-722 (1991). 5. Peppas, N. A. & Merrill, E. W. Polyvinyl-Alcohol) Hydrogels - Reinforcement of Radiation-Crosslinked Networks by Crystallization. Journal of Polymer Science Part a-Polymer Chemistry 14, 441-457 (1976). 6. Ozyurek, C., Caykara, T., Kantoglu, O. & Guven, O. Characterization of network structure of poly(N-vinyl 2- pyrrolidone/acrylic acid) polyelectrolyte hydrogels by swelling measurements. Journal of Polymer Science Part BPolymer Physics 38, 3309-3317 (2000). Lecture 9 – polyelectrolyte hydrogels 6 of 6