Large Deformation of Hydrogels Coupled with Solvent Diffusion Rui Huang

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1 Large Deformation of Hydrogels Coupled with Solvent Diffusion Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin July 5

2 Hydrogel = polymer + water (solvent) Crosslinked polymer Solvent molecules Ambrosio Large and reversible deformation facilitated by solvent migration Response to mechanical/chemical/electrical stimuli Applications: biomedical, soft machines/robots

3 Transient behaviors of hydrogels Tanaka et al, 987 Swelling Indentation Fracture Bonn et al, 998 Hu et al,

4 Linear Poroelasticity 3 material parameters: G, ν, k/η i j j i ij x u x u i j ij b x x k x k p k t k k x p k J kk ij ij kk ij ij p G Biot, 94; Scherer, 989; Suo et al.,. Linear constitutive relation Linear kinematics Linear kinetics Inertia ignored

5 A Nonlinear Continuum Theory for Gels Free energy density function: Nominal stress: Chemical potential: s U ( F, C) ik U F U C ik Nonlinear constitutive relations Nonlinear kinematics Geometrically nonlinear kinetics Inertia ignored Mechanical equilibrium: s X ik K B i Diffusion kinetics: J K DC k T B X x K k X x k L X L M KL X L Conservation of solvent molecules: C t J X K K X K M KL X L Hong, Zhao, Zhou, and Suo, 8.

6 Flory Rehner (943) free energy density function: Neo Hookean rubber elasticity: Flory Huggins polymer solution theory: A Material Model U U e m U ( F, C) U ( F) U ( C) e F Nk T F F 3 lndet( F) ( C) kbt B C ik ln ik C C m C C Volume of gel:: det(f) C U ( F, C) U ( F) U ( C) C det( F) e m 4 material parameters: Nk B T, NΩ, χ, D

7 Linearization near a swollen state Isotropic swollen state: ) ( N N T Nk G B ln N k B T ij ij kk ij ij G k k B k x M x T k D c j 3 3 T k D M B 3 c c c kk k k k k x x M x j t c p c 3 3 T k D k B The linearized equations are identical to the theory of linear poroelasticity. The 3 parameters in the linearized equations depend on the swelling ratio. Bouklas and Huang, Soft Matter.

8 Yoon et al., Soft Matter 6, 64 6 (). Poisson s ratio Poisson's ratio, = N = N = -4, -3, -, - N 3 ( ) N 5 Instantaneously incompressible (ν =.5) The equilibrium Poisson s ratio depends on the initial state (λ or μ ) as well as the intrinsic material properties (χ and NΩ). Poisson's ratio good solvent poor solvent Bouklas and Huang, Soft Matter.

9 A nonlinear, transient finite element method Weak form for the coupled fields: t ij i, J K, K i i i i s u C tj dv bu rt C dv Tu ds itds V V S S Discretization u n u N u n N μ δu u n N δu n N δμ For numerical stability, the D Taylor Hood elements (8u4p) are implemented. A slightly different material model: U ( F, C) U ( F) U ( C) U ( F, C) Legendre transform e m c Uˆ (F, ) U(F, C) - C U c ( F, C) K det( F) C Bouklas et al., JMPS 5

10 Numerical stability for mixed finite element method Nearly incompressible behavior at the instantaneous response limit for K >> Nk B T Discontinuity of the boundary condition at with respect to the initial conditions Choice of appropriate interpolations: LBB condition (Ladyzhenskaya Babuska Brezzi) UNSTABLE Equal order 8u8p STABLE Taylor Hood 8u4p Incompressible linear elasticity Stokes flow Linear poroelasticity

11 Constrained swelling of hydrogel layers IC s u X, X X X, BC s S : S : u, J S : u, J 3 UNSTABLE Equal order 8u8μ element Oscillations in the early stage μ/k B T time

12 Evolution of chemical potential oscillations UNSTABLE 8u8μ /(k B T) t/ = -4 /(k B T) t/ = X /H X /H -. t/ = - -. t/ = - /(k B T) /(k B T) X /H X /H H D.4, N. 3.4, K B Nk T

13 Effect of compressibility UNSTABLE 8u8μ /(k B T) K = 4 Nk B T /(k B T) K = 3 Nk B T X /H X /H.4, N..4 /(k B T) K = Nk B T X /H /(k B T) K =Nk B T X /H t / 4 The oscillations decrease with increasing compressibility (K decreasing)

14 STABLE Taylor Hood 8u4μ element Use of stable interpolation t/ = -4 K = 3 Nk -. t/ = -3 B T -. t/ = - -. t/ = - -. K = 4 Nk B T K = 3 Nk B T K = Nk B T K = Nk B T t/ = -4 /(K B T) -.3 /(K B T) , N. K Nk T 3.4, B X /H X /H Oscillations are significantly reduced and do not depend on compressibility. The remaining oscillations are due to discontinuity of the boundary condition at. Bouklas et al., JMPS 5

15 Swelling induced surface instability h/h n e = 5 n e = 3 n e = n e = 4 =.4.5 = t/ Swell induced compressive stress could lead to surface instability For a poor solvent (χ =.6), the surface remains flat and stable. For a good solvent (χ =.4), the homogeneous hydrogel layer is instantaneously unstable upon swelling; numerically, divergence occurs at a finite time that depends on the mesh size. Bouklas et al., JMPS 5

16 Swell induced surface wrinkling.6 s.5.4 A/h t/ Regularization of the surface instability by using a stiff skin layer. Wrinkle amplitude grows over time, with a selected wrinkle wavelength. Bouklas et al., JMPS 5

17 Two types of surface instability A B Type A: soft-on-hard bilayer, critical condition at the short wave limit, forming surface creases; Type B: hard-on-soft bilayer, critical condition at a finite wavelength, forming surface wrinkles first. Wu, Bouklas and Huang, IJSS 5, (3).

18 A soft on hard gel bilayer (Type A)

19 A hard on soft gel bilayer (Type B)

20 Flat Punch Indentation F (δ) S σ/nk B T µ/k B T S 5 a a S x x S 3 S 4 h b σ/nk B T µ/k B T F t ( F( t) F( ))/( F() F( )) D (F(t)-F()) / (F()-F()) /h = -3 /h = - /h =. /h =. fitting t/ σ/nk B T σ/nk B T µ/k B T µ/k B T

21 Fracture Properties of Hydrogels Reported values of fracture toughness range from to J/m Fracture mechanism may vary over different types of hydrogels Potential rate dependence may relate to viscoelasticity or solvent diffusion Experimental methods to measure fracture properties of hydrogels are not well developed. Kwon et al.,. Sun et al.,.

22 A center crack model Consider a center crack in a hydrogel plate under remote strain A nonlinear, transient finite element method is employed Sharp and notched crack geometries are used for small to large deformation Bouklas et al., JAM 5

23 Immersed and not immersed cases t t Hydrogel is immersed in solvent Assume solvent permeable boundaries and crack faces.5 N N 3 5 ( ) t t Hydrogel is not immersed in solvent Assume solvent impermeable boundaries and crack faces Overall volume is conserved, but solvent redistribution within the gel may lead to local volume change.

24 LEFM Predictions Crack tip stress field: ij K I I r, fij T i j r K I a T G G Nk B T Stress would relax over time Crack opening: uˆ x a x Crack opening would remain constant, independent of modulus or Poisson s ratio

25 Crack tip stress fields 3 t.5 t.45 The square root singularity is maintained at both the instantaneous and equilibrium states, but the stress intensity factor is lower in the equilibrium state due to poroelastic relaxation.

26 Evolution of crack opening Immersed 3 x -3 immersed not immersed LEFM Not immersed Opening at the center: /a.5 = -3 h/a t/ Bouklas et al., JAM 5

27 Solvent diffusion around crack tip Instantaneous limit: Homogeneous solvent concentration, but inhomogeneous chemical potential. Equilibrium state: Homogeneous chemical potential, but inhomogeneous solvent concentration. Bouklas et al., JAM 5

28 Evolution of Solvent Concentration Immersed Not immersed (C-C ) - -4 t/ = - t/ = t/ = t/ = 4 = -3 (C-C ) - -4 t/ = - t/ = t/ = t/= 4 = r/a r/a A singular concentration field is approached in the equilibrium state for both the immersed and not immersed cases. Bouklas et al., JAM 5

29 Energy release rate Rate of change of potential energy: d du dxi dx dv b dv T i ds rdv ids V i i dt dt V dt S dt V S Rate of dissipation due to diffusion: d J V K dv dt X K Total energy: d dt Rate of total energy change due to crack extension Conserved throughout the transient evolution (without crack growth). d du di dx di da V da X da S da da K i K dv Ti NK ds K

30 A modified J integral Define a nominal energy release rate: (per unit length of crack in the reference state) J * d da x C * i J UN s S ij NJ ds dv X V X ˆ x * i J UN s S ij NJ ds CdV X V X J* is path independent. The second form is more convenient for numerical calculations. A domain integral method is used to calculate J*. Bouklas et al., JAM 5

31 Evolution of J integral J * /(ank B T) 8 x not immersed immersed LEFM ( =.5) LEFM ( =.45) h/a =,, J * = t/ LEFM predictions at the instantaneous and equilibrium limits: * t 4 Nk T a J t Nk T a B B

32 Dependence on material parameters.5.4 =. = N = - = -3 J * /(4G a).3.. N = -3 N = t/ * N = - J * /(4G a).3.. =. =.4 = t/ * The J* is well predicted by LEFM at the instantaneous limit (t ) The equilibrium J* depends sensitively on χ (good/poor solvent) The time scale is captured effectively by the poroelastic diffusivity Delayed fracture possible. Bouklas et al., JAM 5

33 Under moderately large remote strain. /(Nk B T) ~ r - ~ r -/ t/ = -4 t/ = 5 = r/a Similar behavior for crack opening (and J* too, not shown) Stress distributions show transition of singularity: with a stronger /r singularity due to hyperelasticity and a weaker singularity closer to the crack tip due to solvent diffusion; the typical square root singularity is approached at the equilibrium limit. Bouklas et al., JAM 5

34 Under large remote strain.5 Immersed Not immersed The /r singularity becomes more prominent in both cases The singularity weakens over time for the immersed case Bouklas et al., JAM 5

35 J integral.7.6 J * /(4G a) immersed not-immersed = t/ The J* is considerably lower than the LEFM prediction, while the evolution trends are similar to the small strain behaviors. Bouklas et al., JAM 5

36 Fracture Criterion J * t * J c? The right hand side of the fracture criterion has to be determined by well designed experiments. Could the right hand side depend on time due to the evolving, inhomogeneous field of solvent concentration around the crack tip?

37 Summary A nonlinear, transient finite element method was implemented for large deformation of hydrogels coupled with solvent diffusion. The transient processes of constrained swelling (with surface instability) and flat punch indentation were studied in some details. Effects of solvent diffusion on the crack tip fields and driving force for fracture of hydrogels were studied. Apathindependent, modified J integral was derived as the transient energy release rate of crack growth in hydrogels, although a complete fracture criterion is yet to be established.

38 Min Kyoo Kang Nikolas Bouklas Zhigen Wu Chad Landis Acknowledgments Funding by NSF