Mechanics of Soft Active Materials (SAMs)

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1 Mechanics of Soft Active Materials (SAMs) Zhigang Suo Harvard University Soft: large deformation in response to small forces (e.g., rubbers, gels) Active: large deformation in response to diverse stimuli (e.g., electric field, temperature, ph, salt) 1

2 Dielectric elastomer: electrostatics meets mechanics Reference State Current State L A Compliant Electrode Dielectric Elastomer l a Q Q Φ Pelrine, Kornbluh, Pei, Joseph Highspeed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000). 2

3 Insertion reaction: electrochemistry meets mechanics Positive electrode Load electron Negative electrode Liion Electrolyte Lithiumion battery Conducting polymers 3

4 gel = network solvent solvent network reversible gel Solidlike: long polymers crosslink by strong bonds. Retain shape Liquid like: polymers and solvent aggregate by weak bonds. Enable transport 4

5 Valve in microfluidics David Beebe Responsive to Physiological variables: ph Salt Temperature light 5 Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)

6 Gels regulate flow in plants Missy Holbrook Zwieniecki, Melcher, Holbrook, Hydrogel control of xylem hydraulic resistance in plants Science 291, 1095 (2001) 6

7 Swelling packers in oil wells 7 A project with Schlumberger

8 Gels as transducers A swelling gel is blocked by a hard material. Blocking force. Inhomogeneous deformation. Need to formulate boundaryvalue problems 8

9 2 ways of doing work Equilibrium condition δ F = Pδl µδm Reference state M=0 L A Current state M µ pump Solvent, µ δf AL = P δl δm µ A L AL l gel δm W (,C ) δ W = sδ µδc Helmholtz free energy per volume = s = l L P A δl P weight C = M AL 9

Inhomogeneous field Deformation gradient Concentration Freeenergy function F ik xi = X ( X, t) K C ( X,t) W ( F,C ) Gibbs (1878) Condition of equilibrium δwdv BiδxidV TiδxidA = µ δcdv µ

10 Inhomogeneous field Deformation gradient Concentration Freeenergy function F ik xi = X ( X, t) K C ( X,t) W ( F,C ) Gibbs (1878) Condition of equilibrium δwdv BiδxidV TiδxidA = µ δcdv µ = constant solvent Gel 1. How to solve boundaryvalue problems? 2. How to prescribe W ( F,C )? 3. What boundaryvalue problems to solve? Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008) Weight 10

11 Finite element method Equilibrium condition Legendre transform δwdv BiδxidV Tiδx ida = µ δcdv ( µ ) = W C W ˆ F, µ WdV BiδxidV δ ˆ = T δx da i i A gel in equilibrium is analogous to an elastic solid ABAQUS UMAT 11 Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)

12 Freeenergy function Swelling decreases entropy by straightening polymers. Swelling increases entropy by mixing solvent and polymers. Paul Flory Freeenergy function W ( F, C) = W ( F) W ( C) s m Free energy of stretching W ( F) 1 NkT[ F F 3 2 log( det F) ] s = 2 ik ik Free energy of mixing Adding volume Flory, Rehner, J. Chem. Phys., 11, 521 (1943) kt 1 χ ( C ) vc log 1 W m = v vc 1 vc 1 vc = detf 12

13 13 Stressstrain relation ( ) Π = σ NkT ( ) Π = σ NkT ( ) Π = σ NkT ( ) = Π log χ µ v kt v

14 Anisotropic swelling L Free swelling 0 L analytical FEM µ 0 /kt Unidirectional swelling s 0 H 1 H µ/kt A gel imbibes a different amount of solvent under constraint. Treloar, Trans. Faraday Soc. 46, 783 (1950). 3 ( free ) unidirectional 14

15 Inhomogeneous swelling Empty space Core R A r A 0 Dry polymer B Gel b (a) (b) Reference State Equilibrium State Concentration is inhomogeneous even in equilibrium. Stress is high near the interface (debond, cavitation). Sternstein, J. Macromolol. Soc. Phys. B6, 243 (1972) 15 Zhao, Hong, Suo, APL 92, (2008 )

16 Swellinginduced buckling R O / H = 15 R i / H = 5 / H = 7 R i R / H = 10 i / H = 12 R i 16

Gel and nanorods Joanna Aizenberg θ 0.3 0.

17 Gel and nanorods Joanna Aizenberg θ = 1.5 χ = 0.1 Nv = 10 3 RH = 80% vg/kt RH = 95% Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007). Theory: Hong, Zhao, Suo, JAP104, (2008). RH = 100% θ (rad) 17

18 Critical humidity can be tuned = 1.7 θ (rad) χ = 0.1 Nv = Relative humidity (%) 18

19 Swellinginduced bifurcation Shu Yang 19 Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008).

20 Swellinginduced bifurcation Experiment: Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008). Simulation: Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009) 20

21 Crease Liang Fen ( 凉粉 ), a starch gel A food popular in northern China 21

22 Dough Zhigang, I was making bread this weekend, and realized that when the rising dough was constrained by the bowl it formed the creases that you were talking about in New Orleans. An from Michael Thouless 22

23 Brain 23

24 Face 24

46 Maurice Biot Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999)

25 Crease: theory and experiment Biot, Appl. Sci. Res. A 12, 168 (1963). Theory: linear perturbation analysis ε Biot 0.46 Maurice Biot Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999) Ghatak, Das, PRL 99, (2007) Experiments: bending rods of rubber and gel ε Gent 0.35 Alan Gent 25

26 What s wrong with Biot s theory? (a) L L A' O A reference state Homogeneous deformation ε (b) A' O A homogeneous state Mahadevan ε ε (c) A/A' O creased state ε Biot s theory: infinitesimal strain from the state of homogeneous deformation Crease: Large strain from the state of homogeneous deformation Hohlfeld, Mahadevan (2008): crease is an instability different from that analyzed by Biot. 26

27 An energetic model of crease (a) A' L O L A U = L 2 Gf ( ε ) reference state 0.8 ε (b) A' O A homogeneous state ε 0.6 NeoHookean material Plane strain conditions ε (c) A/A' O creased state ε U/GL ε c ε ε c 27 Hong, Zhao, Suo,

28 Crease under general loading crease Incompressibility 1 23 = Critical condition for crease Biot 3 / 1 = 3 / 1 = Hong, Zhao, Suo,

29 Crease of a gel during swelling Toyoichi Tanaka Tanaka et al, Nature 325, 796 (1987)

30 Crease of a swelling gel Ryan Hayward 1 gel swell η substrate Theories Experimental data η c = 2.4 η biot = 3.4 Southern, Thomas, J. Polym. Sci., Part A, 3, 641 (1965) Tanaka, PRL 68, 2794 (1992) exp = Trujillo, Kim, Hayward, Soft Matter 4, 564 (2008) η exp = 2. 0 η η exp = Hong, Zhao, Suo,

31 31 Polyelectrolyte gels A network of polymers Solvent (e.g., water) Gel External solution Fixed charges Counterions Coions Crosslinks

32 3 ways of doing work to a gel x(x, t) pump µ a a µ δm a gel δq δq battery Φ P weight δx δwdv = BiδxidV TiδxidA ΦδqdV ΦδωdA a µ a δc a dv free energy of gel W = volume in reference state ( X, t) C a = #of ions of species a volume in reference state 32

33 Swelling regulated by concentration of ions Nv = χ = 0.1 Swelling ratio vc s vc 0 = 0.01 Concentration of fixed ions s vc * vc 0 Concentration of ions in external solution nonionic gel 33

34 Constrained swelling Swelling ratio vc s vc 0 = 0.01 Nv = χ = = 1.5 Concentration of fixed ions gel External solution vc * s vc 0 Concentration of ions in external solution Hong, Zhao, Suo, 34

35 phsensitive hydrogel RCOOH RCOO fixed H mobile [ ][ ] RCOO H = K a [ RCOOH] 35

36 ph 10 [ ] = log H Polymers are charged Polymers are neutral 36

37 Valve in microfluidics 37

38 Crease 38

39 Timedependent process Shape change: shortrange motion of solvent molecules, fast Volume change: longrange motion of solvent molecules, slow 39 Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

40 Concurrent deformation and migration Deformation of network F ik = x i ( X, t) X K Conservation of solvent J Nonequilibrium thermodynamics ( X, t) J ( X, t) r( X t) C K =, t X t K N K K ( X, t) i = t δwdv BiδxidV Tiδx ida µδrdv µδida pump Maurice Biot JAP 12, 155 (1941) Gel Weight Local equilibrium Rate process s ik = W F W = C ( F, C) ik ( F, C) (, ) s X t ik X K B s N = T µ ik K i i = 0 J K = M KL µ X ( X, t) L 40 Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

41 ideal kinetic model Solvent molecules migrate in a gel by selfdiffusion J K = M KL µ X L Diffusion in true quantities j i = cd kt µ x i Conversion between true and nominal quantities j i = FiK det F J K µ = µ F X x K i ik M KL D = H ik H il F vkt ( det 1) 41 Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

42 Finite element method for concurrent deformation and migration Dt L 2 = 0 Dt L 2 = 10 Dt 2 L = 42 Zhang, Zhao, Suo, Jiang, JAP 105, (2009)

43 Alginate Hydrogels Ionic crosslinks Ca( ) Na Covalent crosslinks AAD Irreversible 43

44 Stressrelaxation test PBS Gel disk Impermeable Rigid plates strain time stress time 44 Zhao, Huebsch, Mooney, Suo, submitted.

$Elasticity, plasticity, fracture$ state 45% compressive strain 50%

45 Elasticity, plasticity, fracture Ionic crosslinks 10mm Swollen state 45% compressive strain 50% compressive strain Covalent crosslinks Swollen state 15% compressive strain 20% compressive strain 45

46 Stress Relaxation 4 stress (kpa) Strain: 15% Radius: 6mm Covalent ~10 2 Ionic time (s) 46

47 Size effect R 47

48 Gel with covalent crosslinks relaxes stress by migration of water σ ( t,r) = f R t D ~ R 2 t relax ~ 10 8 m 2 /s 48

49 Outlook Soft Active Materials (SAMs) have many uses (microfluidics, artificial muscles, drug delivery, tissue engineering, water treatment, packers in oil wells). Mechanics of SAMs is interesting and challenging (large deformation, mass transport, multiple thermodynamic forces, many modes of instability). The field is wide open (fabrication, computation, devices, phenomena). 3 lectures on SAMs: Dielectric elastomers Gels Polyelectrolytes 49

Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load Wei Hong a,#, Zishun Liu b, Zhigang Suo a,* a School of Engineering and Applied Sciences, Harvard University, Cambridge,