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 1-3 , refined for treatment of ionic hydrogels by Brannon-Peppas and Peppas 4,5 ? Other theoretical treatments 6 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) c s concentration of electrolyte c 2 concentration of ionizable repeat units in gel (moles/volume) * μ 1 chemical potential of water in solution μ 1 chemical potential of water in the hydrogel μ 1 chemical potential of pure water in standard state M Molecular weight of polymer chains before cross-linking M c Molecular weight of cross-linked subchains n 1 number of water molecules in swollen gel χ polymer-solvent interaction parameter o Asterisks denote parameters in solution k B Boltzman constant T absolute temperature (Kelvin) v m , 1 molar volume of solvent (water, volume/mole) v m,2 molar volume of polymer (volume/mole) v sp , 1 specific volume of solvent (water, volume/mass) v sp,2 specific volume of polymer (volume/mass) V 2 total volume of polymer V s total volume of swollen hydrogel V r total volume of relaxed hydrogel ν number of subchains in network ν e number of ‘effective’ subchains in network ν + stoichiometric coefficient for eletrolyte cation ν ? stoichiometric coefficient for eletrolyte anion φ 1,s volume fraction of water in swollen gel φ 2,s volume fraction of polymer in swollen gel φ 2,r volume fraction of polymer in relaxed gel x 1 mole fraction of water in swollen gel x 1 * 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 ?G total =?G mix +?G el +?G ion 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: Eqn 2 μ 1 * = μ 1 Eqn 3 μ 1 * - μ 1 0 = μ 1 – μ 1 0 o Solution contains ions so μ 1 * is not equal to μ 1 0 Eqn 4 (?μ 1 *) TOTAL = (?μ 1 ) TOTAL Eqn 5 (?μ 1 *) ion = (?μ 1 ) mix + (?μ 1 ) el + (?μ 1 ) ion o The equation we’ll try to solve is a rearrangement of this: Eqn 6 (?μ 1 *) ion - (?μ 1 ) ion = (?μ 1 ) mix + (?μ 1 ) el o Contributions to the free energy: o Free energy of mixing: Eqn 7 ?G mix = ?H mix – T?S mix o We previously derived the contribution from mixing using the Flory-Rehner lattice model: Eqn 8 ?G mix = k B T[n 1 ln (1-φ 2,s ) + χn 1 φ 2,s ] () 1 mix = ? ? ? ?(?G mix ) ? ? ? = k B T[ln(1?φ 2,s ) +φ 2,s +χφ 2,s 2,s Eqn 9 ?μ ?n 1 T ,P 2 ] = RT[ln(1?φ 2,s ) +φ 2,s +χφ 2 ] o Second expression puts us on a molar basis instead of per molecule o Elastic free energy: Eqn 10 ?G el = (3/2)k B Tν e (α 2 – 1 – ln α) Eqn 11 () ?μ = ? ?(?G el ) ? = ? ?(?G el ) ? ? ?α ? ? ? v m,1 ? ? ? φ 2,s ? 1/3 ? 1 ? φ 2,s ? ? ? M 1 el ? ? ?n 1 ? ? T ,P ? ? ?α ? ? T ,P ? ? ?n 1 ? ? T ,P = RTν ? ? ? 1? 2 M c ? V r ?? ? φ 2rs ? ? 2 ? ? φ 2rs ? ? ? ? v m,1 = RT ? ? v sp,2 M c ? ? ? ? ? ? 1? 2M c ? ? ? φ 2,s ? 1/3 ? 1 ? φ 2,s ? ? ? M ? ? φ 2,r ? ?? ? φ 2rs ? ? 2 ? ? φ 2rs ? ? ? ? Last equality uses: o ν = V 2 /v sp , 2 M c (on handout) o V r = V 2 /φ 2 , r (on handout) o Thus ν/V r = φ 2 , r /v sp,2 M c 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 ?μ 1 11 ion =μ * ?μ 0 = RT ln a 1 * ? RT ln x 1 * = 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 all _ ions all _ ions all _ ions all _ ions Eqn 13 () ?μ 1 * ion ??RT ∑ x * j =? RT ∑ n * j =? v m,1 RT ∑ n * j ??v m,1 RT ∑ c * j j n j v m,1 n j j o The first approximation holds if Σx j * 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 v m , 1 n = V, the total volume of the system o Chemical potential change in gel: all ?ions Eqn 14 (?μ 1 ) ion = μ 1 ?μ 1 0 = RT ln a 1 ? ?v m,1 RT ∑ c j j all?ions () * () * Eqn 15 ?μ 1 ion = v m,1 RT ∑( c j ? c j ) 1 ion ??μ 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 C ν + z?z + A ν ? →ν + C z + +ν ? A z? ? e.g. CaCl 2 : ν + = 1, ν - = 2, z + = 2, z - = 1 Eqn 17 ν + +ν ? =ν ? …for a 1:1 electrolyte ? Eqn 18 ν + =ν ? = ν …for a 1:1 electrolyte 2 * * * ? * Eqn 19 c + + c ? = (ν + +ν ? )c s =νc s …total concentration of ions o We will derive equations for an anionic network o Assuming activities ~ concentrations o Inside gel: Eqn 20 c + = ν + c s Eqn 21 c - = ν - c s + ic 2 /z - o c 2 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 K a = [ RCOO ? ] H + [ RCOOH ] Eqn 23 [ RCOO ? ] K a H + i = [ RCOO ? ] [RCOOH] [] K a K a 10 ? pK a = = = = = H + [ RCOOH ] + [ RCOO ? ] [ RCOO ? ] 1+ K a [] + K a 10 ? pH + K a 10 ? pH + 10 ? pK a H +1+ [ RCOOH ] [] Lecture 9 – polyelectrolyte hydrogels 3 of 6 BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 o Outside gel: Eqn 24 c + * = ν + c s * Eqn 25 c - * = ν - c s * o Our relationship for the ionic chemical potentials is now: all?ions () * () * Eqn 26 ?μ 1 ion = v m,1 RT ∑( c j ? c * j ) =v m,1 RT ( 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 = v m,1 RT ? ? ν + c s +ν ? c s + ic 2 ? ? c s * ? ? = v m,1 RT ? ? ν c s + ic 2 ? ? c s ? ? 1 ion ??μ ν ? ν z ? z ? ? ? ν(c s = v m,1 RT ? ? ic 2 ? ? * ? c s ) ? ? z ? o How can we relate c s and c s *? 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? = a + Eqn 32 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 + c s ? ν+ ? ν ? c s ? ? ? ν + c * s ? ? = ? ? ? ν ? c s + ic 2 ? ? ? z ? ? ν? ? ? ν+ ? * ? Eqn 35 ? ? ? ? c c s * s ? ? ? ? = ? ? ? c s + c s ic 2 ? ? ? ν ? z ? ? ? Lecture 9 – polyelectrolyte hydrogels 4 of 6 ? ? ? ? ? ? ? ? ? ? ? ? BEH.462/3.962J Molecular Principles of Biomaterials Spring 2003 ν? ? ?ν+ * ? * ? 1 ? ? ?? ic * 2 ? ? ? 2 Eqn 36 c * s c ? * s c s = 1? ? ? ? ? c s + c s ic 2 ? ? ? = 1 ? c s + c s ic 2 = ν ic 2 * ? ? ? 2z + z ? ν 2 ? ? c s ? z ? c s ? ? ν ? z ? ? ν ? z ? o Derivation of this equation in appendix o Now Eqn 27 becomes: ? i 2 c 2 2 ? () * Eqn 37 ?μ () 1 ion = v m,1 RT ? ? 2z + z ? ν * ? ? 1 ion ??μ ? c s o But definition of ionic strength I is: all _ ions ? c s * Eqn 38 I = 1 ∑ z i 2 c i = z + z ? ν …for a 1:1 electrolyte 2 i 2 null Where z i is the charge on ion i o Therefore: ? i 2 φ 2 ? () * () 2,s Eqn 39 ?μ 1 ion = v m,1 RT ? ? ? i 2 c 2 2 ? ? ? = v m,1 RT ? ? 4Iv sp,2 M 0 2 ? ? 1 ion ??μ 4I 2 φ 2,s o (Using relation c 2 = v sp,2 M 0 =moles ionizable groups/volume) o Eqn 39 can be re-cast in terms of the solution pH: 2 ? () 1 * ion ?? () μ = v m,1 RT ? K a ? 2 ? φ 2,s ? 2 ? K a ? 2 ? φ 2,s 2 Eqn 40 ?μ 1 ion 4I ? ? 10 ? pH + K a ? ? ? ? z ? v sp,2 M 0 ? ? = v m,1 RT ? ? 10 ? pH + K a ? ? ? ? 4Iv sp,2 M 0 2 ? ? o Returning to the equilibrium criterion: Eqn 41 2 ? ? 10 ? pK a ? 2 ? φ 2,s 2 ? v m,1 ? ? ? ? 1? 2 M c ? ? ? ? ? ? ? φ 2,s ? ? 1/3 ? 1 ? ? φ 2,s ? ? ? ? 2 v m,1 ? ? 10 ? pH + 10 ? pK a ? ? ? ? 4Iv sp,2 M 0 2 ? ? = ln(1 ?φ 2,s ) +φ 2,s +χφ 2,s +φ 2,r ? ? v sp,2 M c ? ? M ? ? φ 2,r ? 2 ? φ 2,r ? ? o Brannon-Peppas paper analyzes Polyacrylates/polymethacrylates: o In water pH 7.0 with I = 0.35 o χ = 0.8 o pK a = 6.0 o v sp,2 = 0.8 cm 3 /g o M = 75,000 g/mole o M c = 12,000 g/mole o M 0 = 90 g/mole o φ 2,r = 0.5 Lecture 9 – polyelectrolyte hydrogels 5 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 B- Polymer Physics 38, 3309-3317 (2000). Lecture 9 – polyelectrolyte hydrogels 6 of 6