Modeling mechano-chromatic lamellar gels

Transcription

1 Modeling mechano-chromatic lamellar gels ei Hong ( 洪伟 ), * and Xiao ang ( 王宵 ) epartment of Aerospace Engineering, Iowa State University, Ames, IA 500, USA epartment of Materials Science and Engineering, Iowa State University, Ames, IA 500, USA *whong@iastate.edu Abstract Consisting of alternating swelling and nonswelling polymeric layers (SLs and NLs), lamellar gels are photonic crystals with tunable optical properties. The lamellar structure induces a constraint between the SLs and the NLs, resulting in an anisotropic swelling behavior coupled with deformation. The coupling gives rise to the mechano-chromatic effect, and quantitative understanding of it is the key to many applications. This letter formulates a nonlinear continuum model for lamellar gels by considering the constrained swelling of SLs and the anisotropic deformation in both types of layers. A finite-element method is further developed to simulate the response to non-uniform deformation. Photonic crystals are materials baring stop-bands in visible or infra-red range of light. The periodic microstructure of a photonic crystal intensifies the reflection at the particular wavelength satisfying the Bragg condition and results in a brilliant appearance. The same mechanism is commonly seen in colorful biological features such as feathers and scales -4. Numerous applications could be foreseen: color display, cosmetics, rapid mechanical and chemical characterization, etc 5-9. Polymer-based soft photonic crystals or photonic gels 6,7,9-, with tunable stop-bands through deformation, have shed light on another application full-field measurement and visualization of large deformation. The recently reported lamellar gels, which consist of aligned nonswelling polymer layers (NLs) and swelling gel layers (SLs), have been demonstrated to have mechano-chromatic responses 9-. Unlike regular photonic crystals with three-dimensional lattice structures, a lamellar gel contains unidirectional layers and exhibits a one-dimensional optical lattice. Showing a linear correlation between the characteristic wavelength and the though-thickness strain, a lamellar gel can readily be used to measure the uniaxial or equi-biaxial strain on a uniformly deformed body 0. However, the measurement of a nonuniform strain field cannot be done without a material model. The nonlinearity associated with large deformation prohibits conventional linear composite theory to accurately predict the coupling behavior of the material. This letter aims at formulating a simple yet effective model for the anisotropic property of lamellar gels, by accounting for both the swelling of the SLs and constraint from the NLs. The discussion

2 in this letter will be limited to cases of the most technical interest: the reversible deformation and swelling of lamellar gels. If needed, the theoretical framework may be extended to describe damage evolution or time-dependent behavior of the materials 9,. Swelling Layer h λ h λ Nonswelling Layer Reference State (As-synthesized) λ eformed / Swollen State λ Fig. (Color online) Sketch of the SL-NL structure of a lamellar gel. The as-synthesized state is taken to be the reference, in which the HPN already contains solvent. The overall stretches are λ, λ, and λ, while the through-thickness stretch of the SLs is λ. As sketched in Fig., the lamellar structure of the composite gel consists of alternating SLs and NLs. The deformation and swelling of the soft SLs made from hydrophilic polymer networks (HPN) is constrained by the stiff NLs, made from glassy polymer or surfactant bilayers. The particular ingredients and microstructure varies between different systems 0,. This letter focuses on an idealized microstructure, in which both SLs and NLs are flat, infinitely large, and perfectly aligned. For simplicity, the principal directions of deformation are assumed to be either parallel or perpendicular to the lamellae. In a representative volume much larger than each individual layer yet much smaller than sample size, the material appears to be homogeneous but anisotropic. The Helmholtz free energy per unit homogenized volume is, in general, a function of the principal stretches λ i and the number of solvent molecules per unit reference volume C. The free energy is further divided into three packages to account for the special microstructure: ( C) ( λ ) + ( λ ) ( C) λ,, () i n i b i + m

3 in which n is the contribution from stretching the HPN, b that from deforming the NLs, and m that from mixing the HPN and the solvent. The interaction energy between the NLs and the solvent hardly changes during swelling or deformation, and is thus omitted from the free-energy function. hile the choice of reference state is arbitrary, the as-synthesized state is selected to be the reference for ease of description. The NLs are assumed to be freely extended in this state. The HPN, on the other hand, is stretched isotropically and homogeneously due to the presence of a small amount of solvent during synthesis 0,. Let λ0 be the initial stretch of the HPN with respect to the dry state. For simplicity, the effect of ions is excluded and the HPN is modeled as a neutral polymer. Let the energy of stretching HPN be neo-hookean: 4 ( λ ) NkT ( λ λ + λ λ + λ λ ln λ λ λ ) n i 0 0 0, () and the energy of mixing be that of a polymer with simple solvent: 4 kt φ ( C) [ ( φ ) ln( φ) + χφ] m. () v λ 0 Here, N is the number of HPN chains per reference volume, kt the temperature in the unit of energy, v the volume per solvent molecule, and χ the Flory-Huggins parameter. Let λ, λ, and λ be the principal stretches in the SLs, and specifically λ be that in the through-thickness direction of a lamella. Since NLs are usually much stiffer and much thinner than SLs, the through-thickness strain of the NLs is neglected, and λ is related to the overall stretch φ, where λ as λ ( λ φ ) ( ) [ ] volume fraction in the reference state. In Eq. (), + λ ( φ ) HPN in SLs. 0 φ is the NL φ vc is the volume fraction of the Each NL acts as an elastic membrane embedded in the gel. It is assumed that the deformation is affine and the in-plane stretches of the NLs match the corresponding macroscopic stretches, λ and λ. ith each NL modeled as an incompressible hyperelastic membrane with modulus µ, the elastic energy of deforming the bilayers is ( ) φ µ ( λ + λ + λ λ ). (4) b λ i For simplicity, some constant terms are omitted from the energy functions in Eqs. () (4).

4 Furthermore, to introduce the coupling between the mechanical deformation and the polymer volume fraction, we assume molecular incompressibility by insisting that ( φ ) λ0 + λλ φ + vc λ, (5) i.e. the total volume of the gel-solvent system is conserved during swelling. Utilizing Eq. (5), the HPN φ φ λ λ λ φ λ. The volume fraction can also be expressed in terms of the stretches, ( ) ( ) 0 elastic energies in Eq. () and (4) and the energy of mixing in Eq. () are coupled through constraint (5). The calculations here are limited to equilibrium states, in which chemical potential is uniform, µ µ 0, and solvent migration has equilibrated. The mathematical representation of such states is facilitated by introducing an alternative free-energy function through a Legendre transform: 5 ( λ, µ ) ( λ, C) C ˆ. (6) i i µ ith the aid of ˆ, the equilibrium stress-stretch relation is just like that of a hyperelastic material: s i ˆ. (7) λ i The solvent chemical potential µ acts as a parameter rather than a field variable. Substitution of Eqs. () (6) into Eq. (7) yields the explicit constitutive relation of the gel. The in-plane stresses are kt where v ln( φ) s s ( λ0λ λ ) + φµ ( λ λ λ ) pλ NkT λ ( λ0λ λ ) + φµ ( λ λ λ ) pλ NkT λ [ + φ χφ ], (8), (9) p µ + is the osmotic pressure. The out-of-plane stress is: v s λ φ NkT λ0 pλλ. (0) ( φ ) λ φ The model for lamellar hydrogels is verified through comparison with existing experiments. Let us first look at the free swelling of a lamellar hydrogel. Equating the in-plane and out-of-plane stresses (8)- (0) to 0, we arrive at the nonlinear equations for the stretches of free-swelling, λ i. Take the chemical 4

5 potential of pure solvent as the reference µ 0, and utilizing the symmetry, λ λ, we can simplify the equations as 5 ( λ λ λ ) + φ µ ( λ λ ) pλ λ 0 NkT, () 0 λ φ NkT λ0 0 pλ, () ( φ ) λ φ kt where p [ ln( φ ) + φ + χφ ] and φ ( φ ) ( ) λ λ φ λ0 v are the osmotic pressure and HPN volume fractions, respectively, in a free-swelling state. Solving the nonlinear algebraic equations () and () simultaneously, we can obtain the equilibrium stretches of a free-swelling lamellar gel. The material parameters are estimated as follows for a specific lamellar gel, in which the SLs consist of polyacrylamide (PAAm) hydrogel and the NLs are polymeric dodecylglyceryl itaconate (GI) bilayers, to enable comparison with the available experimental data 0. Taking the room temperature and using the size of a water molecule, we estimate kt v 00MPa T 00K. Taking the molecular weight of GI to be 7 and assuming a density similar to that of water, we relate the NL volume fraction φ to the molar concentration c of GI monomer approximately as 0.7mol φ L c. In general, the stiffness of the HPN NkT, the swelling parameter χ, and the NL shear modulus µ can be determined from independent experiments. Lacking further experimental data, we use these them as fitting parameters. In the following calculations, we use NkT 400Pa, χ 0. 5, and µ MPa, all within the range of commonly accepted values for the materials 0,6. The resulting free-swelling stretches parallel ( λ ) and perpendicular ( λ ) to the lamellae are plotted in Fig. a, as functions of the GI concentration. The results exhibit good agreement with the experimental measurements 0. At low NL concentration, the lamella structure may be less ordered and the deformation may not be affine due to the large distance between neighboring bilayers, the relatively large discrepancy is thus expected. 5

6 a swelling ratio λ.5.5 λ λ c (mol/l) b Modulus E (kpa) E E GI concentration c (mol/l) Fig. (Color online) (a) Equilibrium stretches of a free-swelling PAAM-PGI lamellar gel, paralell ( λ ) and perpendicular ( λ ) to the lammellae, as functions of the GI monomer concentration. (b) Tensile moduli of the PAAM-PGI gel paralell ( E ) and perpendicular ( E ) to the lammellae as functions of the GI concentration. Squares and circles are experimental results from Ref. 0. The behavior of the gel under uniaxial tension/compression parallel to the bilayers ( s s 0 ) or 6 perpendicular to the bilayers ( s s 0) can be obtained by solving the nonlinear Eqs. (8) (0). The tensile modulus, namely the initial slope of the stress-strain curve in uniaxial tension/compression, can also be evaluated through linearizing the stress-stretch relation at the small strain limit. Using the same set of material parameters as in the swelling calculation, we compute the tensile modulus in the direction of the bilayers, E, and that in the perpendicular direction, E. The results are plotted in Fig. b, and compared to the available experimental data 0. The prediction of the model shows good agreement at a GI concentration higher than 0.06 mol/l. Finally, to demonstrate the mechano-chromatic capability, we simulate the pseudo color map of a cracked specimen of lamellar hydrogel under tension. The material parameters are the same as those used

7 in previous calculations, and the strain-wavelength relation is directly taken from experimental measurements 0. The numerical results are obtained by implementing the constitutive relations, Eqs. (8)- (0), in the finite element software, COMSOL Multiphysics 4.. Although no experimental result is available for comparison at the time of writing, we expect our model prediction to deviate from the actual observation because of the inevitable material damage at the crack tip. However, the difference between the model prediction and the experimental observation could actually provide valuable insights towards the fracture process zone of the material. Since the chromatic response is independent of the nonlinearity in deformation, the highly stretchable lamellar gels are most suitable for measuring the large deformation of soft materials, on which conventional strain gauges are deemed to fail. The capability of full-field measurement would also enable the visualization of fracture or localized damage of soft materials and structures. Fig. (Color online) Simulated result of a cracked specimen of lammelar hydrogel under tension. The shading indicates the through-thickness stretch, λ. It should also be pointed out that the mechano-chromatic mechanism would not be effective when the lammellar gel is subject to an in-plane compression. The large difference in stiffness between the SL and the NL makes the composite gel very similar to the structure of a hard film on a soft substrate. hen compressed beyond a critical strain, the deformation will no longer be affine and wrinkles are expected to form in the hard film or NLs. Taking the modulus ratio between the NL and SL to be ~00, and the 7

8 thickness ratio to be ~0.05, we can estimate the critical compressive strain for wrinkle instability to be ~0.5%, and the wrinkle wavelength to be ~0 times of the NL thickness 7. The wavelength of the wrinkles in the NL is still comparable to that of visible light, but is insensitive to the applied strain. Such a prediction is awating to be confirmed experimentally. In summary, in this letter a continuum model has been developed for mechano-chromatic lamellar gels consisting of alternating SLs and NLs. ith the coupling accounted for, the model successfully recovers the experimental measurements on free-swelling and anisotropic deformation. Upon calibration, the model can assist experiments in full-field strain measurement of the large deformation on soft materials. The damage and fracture processes in soft materials may also be visualized with the material and the supporting theory. The model may well be used on other systems with similar structures. It is also suggested that the mechano-chromatic mechanism of the lamellar gel would fail if an in-plane compression is applied, under which wrinkles will form in the stiff NLs. H acknowledges the support from the NSF through Grant No. CMMI-09004, and Prof. J. P. Gong and r. M. A. Haque for providing the experimental details. J.. Joannopoulos, P. R. Villeneuve and S. Fan, Nature 86 (), 4-49 (997). S. Kinoshita, S. Yoshioka and J. Miyazaki, Rep. Prog. Phys. 7, (008). S. M. oucet and M. G. Meadows, J. R. Soc. Interface, 6 S5-S (009). 4 P. Vukusic and J. R. Sambles, Nature 44 (4), (00). 5 J. Ge, J. Goebl, L. He, Z. Lu and Y. Yin, Adv. Mater. (4), (009). 6 J. J. alish, Y. Kang, R. A. Mickiewicz and E. L. Thomas, Adv. Mater. (4), (009). 7 J. H. Holtz and S. A. Asher, Nature 89 (), 89-8 (997). 8 Y. Y. Li, F. Cunin, J. R. Link, T. Gao, R. E. Betts, S. H. Reiver, V. Chin, S. N. Bhatia and M. J. Sailor, Science 99 (565), (00). 9 M. A. Haque, T. Kurokawa, G. Kamita, Y. Yue and J. P. Gong, Chem. Mater. (), (0). 8

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