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Lecture 7 Spring 2006 1
Thermodynamics of hydrogel swelling
Applications of hydrogels in bioengineering
Announcements:
Last Day: Structure of hydrogels
Today: bioengineering applications of hydrogelsThermodynamics of hydrogel swelling
Reading: N.A. Peppas et al., Physicochemical foundations and structural design of hydrogels inmedicine and biology,Annu. Rev. Biomed. Eng., 2, 9-29 (2000).
Supplementary Reading: P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, pp. 464-469, pp. 576-581 (Statistical thermodynamics of networks and network swelling)
P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, pp. 495-507 (Entropy of polymer-solvent mixing)
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Lecture 7 Spring 2006 2
hydrogels
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Lecture 7 Spring 2006 3
Thermodynamics of hydrogel swelling
Vr
Vsswelling
Move to a
new, larger
aqueous
bath
polymerize
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Lecture 7 Spring 2006 4
Thermodynamics of hydrogel swelling
Vr
Competing driving forces determine total swelling:
Vs
swelling
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Lecture 7 Spring 2006 5
Description of cross-linked network
Vs
VrCross-linking
(relaxed)
swelling
Expansion factor:
x
y
z= 3 = V
s/Vr = (V
2+ n
1v
m,1)/V
r
2,s = V2/(V2 + n1vm,1) volume fraction of polymer in swollen gel
2,r= V2/Vr volume fraction of polymer in relaxed gel
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Lecture 7 Spring 2006 6
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
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Lecture 7 Spring 2006 7
Free energy of mixing in the network:
Lattice model description of polymers: (Flory/Huggins)
ENTHALPY OF MIXING:
12
Energy of contacts:
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Lecture 7 Spring 2006 8
Free energy of mixing in the network:
Lattice model description of polymers: (Flory/Huggins)
ENTHALPY OF MIXING:
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Lecture 7 Spring 2006 9
Free energy of mixing in the network:
Lattice model description of polymers: (Flory/Huggins)
ENTROPY OF MIXING:
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Lecture 7 Spring 2006 10
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 ofPolymer Chemistry. Ithaca, NY:
Cornell University Press, 1953.
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Lecture 7 Spring 2006 11
Free energy of mixing in the network:
Lattice model description of polymers: (Flory/Huggins)
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Lecture 7 Spring 2006 12
Description of cross-linked network
Assume cross-links are randomly placed; on average, allare equidistant:
= number of subchains in cross-linked network
e
= number of effective subchains: tethered at both
ends
M = MW of original chains
Mc = MW of subchains = MW between cross-links
A
Example: assume polymer chains have a molecular
weight M = 4A and each subchain has molecular weight
A:Two useful relationships:
= V2/vsp,2Mc
e = (1 - 2(Mc/M))
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Lecture 7 Spring 2006 13
Elastic contribution to hydrogel free energy:
GelAccount for entropic retraction force that restrains swelling:
Gel > 0
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Lecture 7 Spring 2006 14
Elastic contribution to hydrogel free energy:
Gel
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Elastic contribution to hydrogel free energy:
Gel
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Complete expression for the free energy of the
gel:
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Complete expression for the free energy of the
gel:
C l i f h f f h
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Lecture 7 Spring 2006 18
Complete expression for the free energy of the
gel:
C l t i f th f f th
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Lecture 7 Spring 2006 19
Complete expression for the free energy of the
gel:
C l t i f th f f th
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Lecture 7 Spring 2006 20
Complete expression for the free energy of the
gel:
P di ti f Fl /P th
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Predictions of Flory/Peppas theory
Varying2,r:
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000
Mc (g/mole)v
olumefraction
polymerinsw
ollen
state
0.2
0.1
0.05
0
20
40
60
80
100
120
0 10000 20000 30000 40000 50000
Mc (g/mole)
S=
Vs/Vdry
0.20.1
0.05
Predictions of Flor /Peppas theor
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Predictions of Flory/Peppas theory
Varying :
hydrogel swelling vs. solvent quality
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10000 20000 30000 40000 50000
Mc (g/mole)
volum
efraction
polym
erin
swollen
state
0.5
0.4
0.2
hydrogel swelling vs. solvent quality
0
20
40
60
80
100
120
0 10000 20000 30000 40000 50000
Mc (g/mole)
S
0.5
0.4
0.2
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Model parameters1
bath chemical potential of water in external bath ( = 1
0)
1 chemical potential of water in the hydrogel1
0 chemical potential of pure water in standard state
w12 pair contact interaction energy for polymer with waterz model lattice coordination numberx number of segments per polymer moleculeM Molecular weight of polymer chains before cross-linking
Mc Molecular weight of cross-linked subchainsn1 number of water molecules in swollen gel
polymer-solvent interaction parameterkB Boltzman constantT absolute temperature (Kelvin)vm,1 molar volume of solvent (water)
vm,2 molar volume of polymervsp,1 specific volume of solvent (water)vsp,2 specific volume of polymerV2 total volume of polymerVs total volume of swollen hydrogelVr total volume of relaxed hydrogel
number of subchains in networke number of effective subchains in network
1 volume fraction of water in swollen gel
2,s volume fraction of polymer in swollen gel
2,r volume fraction of polymer in relaxed gel
Key properties of hydrogels for bioengineering
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Key properties of hydrogels for bioengineering
applications:
Further Reading
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Further Reading
1. Flory, P. J. & Rehner Jr., J. Statistical mechanics of cross-linked polymer networks. II. Swelling.J. Chem. Phys.11, 521-526 (1943).
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. Peppas, N. A. & Merrill, E. W. Polyvinyl-Alcohol) Hydrogels - Reinforcement of Radiation-Crosslinked Networksby Crystallization. Journal of Polymer Science Part a-Polymer Chemistry14, 441-457 (1976).
4. Flory, P. J. Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953).5. An, Y. & Hubbell, J. A. Intraarterial protein delivery via intimally-adherent bilayer hydrogels.J Control Release64,
205-15 (2000).
6. Brannonpeppas, L. & Peppas, N. A. Equilibrium Swelling Behavior of Ph-Sensitive Hydrogels.ChemicalEngineering Science46, 715-722 (1991).
7. 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).
8. Hubbell, J. A. Hydrogel systems for barriers and local drug delivery in the control of wound healing.Journal ofControlled Release39, 305-313 (1996).
9. Jen, A. C., Wake, M. C. & Mikos, A. G. Review: Hydrogels for cell immobilization.Biotechnology and
Bioengineering50, 357-364 (1996).10. Nguyen, K. T. & West, J. L. Photopolymerizable hydrogels for tissue engineering applications.Biomaterials23,4307-14 (2002).
11. Peppas, N. A., Huang, Y., Torres-Lugo, M., Ward, J. H. & Zhang, J. Physicochemical foundations and structuraldesign of hydrogels in medicine and biology.Annu Rev Biomed Eng2, 9-29 (2000).