1. Yuhang Hu Advisor: Zhigang Suo May 21, 2009 Based on Zhigang’s notes, ucsb talk and an on going paper by Zhigang, Wei and Xuanhe

2. 2 Introduction (microstructure and applications) A field theory of gels coupling large deformation and electrochemistry of ions and solvent Homogeneous field in the interior of a gel and solution Inhomogeneous field near interface between gel and external solution

3. 3 Network Network Solvent Solvent Fixed ions Fixed ions Mobile ions Mobile ions + http://en.wikipedia.org/wiki/Polyelectrolyte strong polyelectrolyte + or weak polyelectrolyte hydrogel solution polyelectrolyte + + - - + - + - + - + -

4. Negatively charged proteins 4 Repulsion retain water during compression. And thus maintain small friction.

5. 5

6. 6

7. 7 battery,  pump,  electrons,  q Ions,  M standard electrolyte electrolyte electrode neutrality equilibrium work done by the pump,  M work done by the battery,  q Helmholtz free energy, F Electrochemical potential: Mechanical work done by bringing one ion from a standard state to a specified concentration and electric potential Gibbs (1878)

8. 8 Helmholtz free energy of the gel work done by the weight work done by the pumps work done by the battery Applicable to a single macromolecule, a cell or a large system l

9. 9  deformation gradient Reference state (Dry state) Current state X x(X, t) Marker  nominal concentrations  nominal electric field x(X+dX, t) ground Free-energy density

10. 10 in volume on interface Define the stress s iK, such that holds for any test function  i ( X ) Apply divergence theorem, one obtains that

11. 11 in volume on interface Define the electric displacement, such that holds for any test function  ( X ). Apply divergence theorem, one obtains that + + - + + - -

12. 12 in volume on interface Number of ions is conserved: The above two equations is equivalent to holds for any test function

13. in volume on interface by battery by pumps fixedmobile 13

14. 14  Work done by the weights  Work done by the batteries  Work done by the pumps a field of weights, pumps and batteries

15. 15 work done by weights Free energy change of the composite system: work done by pumpswork done by batteries Free energy density change of the gel element: Thermodynamics:

16. Local Equilibrium: Kinetic law: 16

17. 17

18. 18  Free energy of stretching  Free energy of mixing Free-energy function  Free energy of polarization (Ideal dielectric material) microscopic effect  Swelling increases entropy by mixing solvent and polymers, but decreases entropy by straightening the polymers.  Redistributing mobile ions increases entropy by mixing, but increase polarization energy  Free energy of dissolving ions Flory-Rhner

19. 19 + = V dry + V sol = V gel Assumptions:  Individual solvent molecule and polymer are incompressible.  Gel has no voids.  An ion occupies a same volume in the solvent or in the gel v a – volume per particle of species a + + - - + + - -

20. 20 ions solvent

21. 21 In liquid far from interface between gel and solvent In liquid near interface In gel far from interface In gel near interface

22. 22 In equilibrium Infinity in liquid Π=E=D=0 & State equations

23. 23 Debye length: When |Φ| << kT/e Stress near the interface Negative surface tension! x0 LDLD solution + + + + + + _ _ _ _ _ _ _ _ + + gel infinity General solution: fast decay electric field in liquid near interface

24. 24 Infinity in liquid infinity in gel Electric field vanishes and electric charge neutral gel swells uniformly incompressibility

25. 25 nonionic gel Swelling ratio Concentration of ions in external solution Concentration of fixed ions

26. 26 x0 LDLD solution + + + + + + _ _ _ _ _ _ _ _ + + gel infinity Inhomogeneous field incompressibility Deep in gel, electric filed vanishes and no net charge External solution +

27. 27 x0 LDLD solution + + + + + + _ _ _ _ _ _ _ _ + + gel infinity

28. 28 Hydrogel : poroelasticity Li battery : field theory coupling large deformation and electrochemistry of ions and solvent Electrolyte Li-ion A discharging Li-ion cell. Load

29. 29