Thermodynamics of hydrogel swelling Applications of hydrogels in bioengineering Last Day: Structure of hydrogels Today: bioengineering applications of hydrogels Thermodynamics of hydrogel swelling Reading: N.A. Peppas et al., ‘Physicochemical foundations and structural design of hydrogels in medicine and biology,’ Annu. Rev. Biomed. Eng., 2, 9-29 (2000). Supplementary Reading: P.J. Flory, ‘Principles of Polymer Chemistry,’ Cornell University Press, Ithaca, pp. 464469, pp. 576-581 (Statistical thermodynamics of networks and network swelling) P.J. Flory, ‘Principles of Polymer Chemistry,’ Cornell University Press, Ithaca, pp. 495507 (Entropy of polymer-solvent mixing) Announcements: Lecture 7 Spring 2006 1 hydrogels Lecture 7 Spring 2006 2 Thermodynamics of hydrogel swelling polymerize Move to a new, larger aqueous bath Vr Lecture 7 Spring 2006 swelling V3s Thermodynamics of hydrogel swelling Competing driving forces determine total swelling: Vr swelling Vs Lecture 7 Spring 2006 4 Description of cross-linked network Cross-linking (relaxed) Vr Vs Expansion factor: α αxαyαz = α3 = Vs/Vr = (V2 + n1vm,1)/Vr φ2,s = V2/(V2 + n1vm,1) φ2,r = V2/Vr swelling volume fraction of polymer in swollen gel volume fraction of polymer in relaxed gel Lecture 7 Spring 2006 5 Free energy of mixing in the network: Starting point: thermodynamic description of simple polymer-solvent mixing: Seek to derive an expression for the free energy of mixing: ∆Gmix Lecture 7 Spring 2006 6 Free energy of mixing in the network: Lattice model description of polymers: (Flory/Huggins) ENTHALPY OF MIXING: Energy of contacts: ∆ω12 Lecture 7 Spring 2006 7 Free energy of mixing in the network: Lattice model description of polymers: (Flory/Huggins) ENTHALPY OF MIXING: Lecture 7 Spring 2006 8 Free energy of mixing in the network: Lattice model description of polymers: (Flory/Huggins) ENTROPY OF MIXING: Lecture 7 Spring 2006 9 Free energy of mixing in the network: Lattice model description of polymers: (Flory/Huggins) ENTROPY OF MIXING: Image removed due to copyright reasons. Please see: Figure 110 in Flory, P. J. Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press, 1953. Lecture 7 Spring 2006 10 Free energy of mixing in the network: Lattice model description of polymers: (Flory/Huggins) Lecture 7 Spring 2006 11 Description of cross-linked network A Assume cross-links are randomly placed; on average, all are equidistant: ν = number of subchains in cross-linked network νe = number of ‘effective’ subchains: tethered at both ends M = MW of original chains Mc = MW of subchains = MW between cross-links Example: assume polymer chains have a molecular weight M = 4A and each ‘subchain’ has molecular weight A: Two useful relationships: ν= V2/vsp,2Mc νe = ν(1 - 2(Mc/M)) Lecture 7 Spring 2006 12 Elastic contribution to hydrogel free energy: ∆Gel •Account for entropic retraction force that restrains swelling: ∆Gel > 0 Lecture 7 Spring 2006 13 Elastic contribution to hydrogel free energy: ∆Gel Lecture 7 Spring 2006 14 Elastic contribution to hydrogel free energy: ∆Gel Lecture 7 Spring 2006 15 Complete expression for the free energy of the gel: Lecture 7 Spring 2006 16 Complete expression for the free energy of the gel: Lecture 7 Spring 2006 17 Complete expression for the free energy of the gel: Lecture 7 Spring 2006 18 Complete expression for the free energy of the gel: Lecture 7 Spring 2006 19 Complete expression for the free energy of the gel: Lecture 7 Spring 2006 20 Predictions of Flory/Peppas theory 120 0.2 0.2 0.1 0.05 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.2 0.1 0.05 100 S = Vs/Vdry volume fraction polymer in swollen state Varying φ2,r: 0.04 80 60 40 20 0.02 0 0 0 10000 20000 30000 40000 50000 0 10000 20000 30000 40000 Mc (g/mole) Mc (g/mole) Lecture 7 Spring 2006 21 50000 Predictions of Flory/Peppas theory Varying χ: hydrogel swelling vs. solvent quality 0.2 0.5 0.4 0.2 0.18 0.16 120 0.5 0.4 0.2 100 0.14 80 0.12 0.1 S vo lu m e frac tio n p o lym er in s w o llen s tate hydrogel swelling vs. solvent quality 60 0.08 40 0.06 0.04 20 0.02 0 0 0 0 10000 20000 30000 40000 50000 10000 20000 30000 40000 Mc (g/mole) Mc (g/mole) Lecture 7 Spring 2006 22 50000 Model parameters bath 0 chemical potential of water in external bath ( = µ1 ) µ1 chemical potential of water in the hydrogel µ1 0 chemical potential of pure water in standard state µ1 pair contact interaction energy for polymer with water ∆w12 z model lattice coordination number x number of segments per polymer molecule M Molecular weight of polymer chains before cross-linking Molecular weight of cross-linked subchains Mc number of water molecules in swollen gel n1 χ polymer-solvent interaction parameter Boltzman constant kB T absolute temperature (Kelvin) molar volume of solvent (water) vm,1 molar volume of polymer vm,2 specific volume of solvent (water) vsp,1 specific volume of polymer vsp,2 total volume of polymer V2 total volume of swollen hydrogel Vs total volume of relaxed hydrogel Vr ν number of subchains in network number of ‘effective’ subchains in network νe volume fraction of water in swollen gel φ1 volume fraction of polymer in swollen gel φ2,s volume fraction of polymer in relaxed gel φ2,r Lecture 7 Spring 2006 23 Key properties of hydrogels for bioengineering applications: Lecture 7 Spring 2006 24 Further Reading 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Flory, P. J. & Rehner Jr., J. Statistical mechanics of cross-linked polymer networks. II. Swelling. J. Chem. Phys. 11, 521-526 (1943). Flory, P. J. & Rehner Jr., J. Statistical mechanics of cross-linked polymer networks. I. Rubberlike elasticity. J. Chem. Phys. 11, 512-520 (1943). 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). Flory, P. J. Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953). An, Y. & Hubbell, J. A. Intraarterial protein delivery via intimally-adherent bilayer hydrogels. J Control Release 64, 205-15 (2000). Brannonpeppas, L. & Peppas, N. A. Equilibrium Swelling Behavior of Ph-Sensitive Hydrogels. Chemical Engineering Science 46, 715-722 (1991). Chiellini, F., Petrucci, F., Ranucci, E. & Solaro, R. in Biomedical Polymers and Polymer Therapeutics (eds. Chiellini, E., Sunamoto, J., Migliaresi, C., Ottenbrite, R. M. & Cohn, D.) 63-74 (Kluwer, New York, 1999). Hubbell, J. A. Hydrogel systems for barriers and local drug delivery in the control of wound healing. Journal of Controlled Release 39, 305-313 (1996). Jen, A. C., Wake, M. C. & Mikos, A. G. Review: Hydrogels for cell immobilization. Biotechnology and Bioengineering 50, 357-364 (1996). Nguyen, K. T. & West, J. L. Photopolymerizable hydrogels for tissue engineering applications. Biomaterials 23, 4307-14 (2002). Peppas, N. A., Huang, Y., Torres-Lugo, M., Ward, J. H. & Zhang, J. Physicochemical foundations and structural design of hydrogels in medicine and biology. Annu Rev Biomed Eng 2, 9-29 (2000). Lecture 7 Spring 2006 25