Open Access Article

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Matt P.
Milner
^{a},
Lihua
Jin
^{b} and
Shelby B.
Hutchens
*^{a}
^{a}Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign, Urbana, IL, USA. E-mail: hutchs@illinois.edu; Tel: +1-217-300-0412
^{b}Department of Mechanical and Aerospace Engineering, University of California Los Angeles, Los Angeles, CA, USA

Received
26th June 2017
, Accepted 10th August 2017

First published on 22nd August 2017

We report on crease morphology and evolution at the surface of contracting cavities embedded within elastomeric solids of varying composition (Sylgard 184: pre-polymer to crosslinker mixing ratios of 10:1, 12:1, 17.5:1, and 25:1). Cavity contraction is achieved through evaporation of an embedded 10 μL liquid droplet. In validation of recent theoretical predictions, strain-stiffening modeled via the Gent constitutive relation [Jin and Suo, JMPS, 2015, 74, 68–79] is found to govern both crease onset and crease density. Specifically, crease onset matches prediction using only experimentally-measured parameters. Neo-Hookean solids are found to prefer initiating creasing with many short creases that join to form a collapsed state with only a few creases, whereas creasing in Gent solids initiates with a few creases that propagate across the cavity surface. These experimental observations are explained by energy minimization using finite element simulation of a cylindrical crease geometry.

This study was motivated by both theoretical predictions for osmosis-driven collapse^{12} as well as an aim toward understanding biological structural changes due to fluid loss from an enclosed cavity. For example, in the animal kingdom, the female reproductivity cycle involves regression of the post-egg-release corpus luteum (a fluid filled transient endocrine gland essential to fertility) to corpus albicans before eventual re-absorption.^{13} In the plant kingdom, structures in organ-based plants couple water flux to geometric changes in order to achieve motion without muscle.^{14,15} The latter coupling is found in hygromorphs;^{16,17} surface tension propulsion;^{18} and drying-associated, hydrostatic-pressure-driven energy storage that precedes failure^{19} or cavitation^{15,20} events. As a result of the large deformations in these soft systems, creases are observed at the inner liquid–solid interface. We begin to explore the formation of these creases in a spherically symmetric geometry by embedding a droplet within a large elastomeric solid (Fig. 1).

Briefly, creasing in soft solids is due to the application of a globally compressive strain that, after reaching a critical value, results in the onset of localized, self-contacting deformations on the solid's surface.^{1} An overview of the extensive creasing literature is available in a number of important publications.^{1,2,21} Here we will focus on three key findings most relevant to describing our observations of creasing in spherical cavities within elastomeric solids. First, recent theory finds that hyperelastic strain-stiffening delays the onset of creasing as a function of applied strain.^{22} For reference, incompressible, neo-Hookean elastomers, which exhibit no strain-stiffening, crease at a uniaxial plane-strain of 0.35.^{1,2} This behavior has been validated by experimental observation in several soft materials.^{2,10} Crease onset occurs at the same strain value, regardless of the material's shear modulus, μ. To our knowledge, most creasing experiments are performed with constant composition materials, i.e., constant modulus, with one notable exception finding a dependence of crease-onset-strain on cross-linker in polyacrylamide gels.^{10} These authors attributed this observation to possible variation in compressibility with changes in the gel composition; verification was left for future study. More recently however, Jin and Suo^{22} calculated that for a strain-stiffening Gent material,^{23} a larger strain is required to induce creasing for materials with earlier onset of strain-stiffening. This behavior is predicted to result in crease suppression for materials with early onset of strain-stiffening (J_{lim} = 3.1, where J_{lim} determines the onset of strain-stiffening and is the first strain invariant minus 3 evaluated at the limiting stretch state). The second key finding is that after creasing has occurred, crease density decreases with the increasing application of strain. Quantitative predictions of crease density, dependent on geometry,^{2,12,24} are based on energetic arguments comparing strain energy between finite element models of differing crease number or number density. Third, creasing behavior can be altered by surface energy, γ, which may act as an energy barrier similar to that found in nucleation theory. Chen et al.^{3} showed that creases, which nucleate from pre-existing defects then grow via channeling,^{9} can be induced to grow/channel further when surface energy is decreased. This phenomenon was shown to relate to a film-thickness-normalized elastocapillary number, γ/μt, where t is the thickness of the creased film. The results presented here validate and build upon each of these three findings for a spherical geometry.

Our observations of evaporating liquid droplets embedded within elastomeric solids of varying composition experimentally verify Jin and Suo's prediction of strain-stiffening governed crease onset. We also find, in agreement with the second characteristic described above, that increasing deformation leads to a decrease in crease density for neo-Hookean materials. However, this crease density dependence breaks down for strain-stiffening materials. We qualitatively describe this phenomena using finite element simulation of a cylindrical cavity (Fig. 1a). Further, we find that the morphology of surface creases is largely unaffected by decreased surface energy at large deformations, but near crease onset, surface energy plays a small role. For the macroscale cavities (∼1 mm) studied here, this suggests that at low strains an elastocapillary number, γ/μL, that uses crease size as L rather than system geometry (cavity size) is more appropriate in the deformation regime near crease onset.

This article is organized as follows. We outline the materials used and describe methods for sample fabrication and finite element simulation. Contraction of the liquid-filled cavity in the breathing mode,^{12} during which the cavity remains spherical, is modeled using mass transport. Dependence of cavity deformation at the onset of creasing is demonstrated to have quantitative agreement with Jin and Suo^{22}'s predictions via independent characterization of the elastomer's constitutive behavior. We then show results for crease morphology evolution as a function of decreasing cavity volume, drawing parallels with a simulated cylindrical geometry. All creasing results are interpreted assuming a biaxial compressive state at the liquid/cavity interface as in Cai et al.^{12} We show evidence for a small surface energy dependence of crease morphology near onset. Finally, we touch on cavitation of the embedded liquid droplet.

Upon curing, samples are placed either in front of the camera for observation or in a sealed container containing an open water source. During observation, a hotplate maintains a temperature of 60 °C via constant heat output (Fig. 2a). Images are taken at 30 minute intervals over the course of evaporation (∼3 days) until cavitation occurs. Radii are determined using ImageJ. Images of post-creased droplets are characterized manually, using Matlab to record data (ESI†).

Fig. 2 Droplet evaporation. (a) A photograph of the experimental setup for scale (left) and a schematic (right) showing the 2.5 cm cubic elastomer with embedded droplet. The heating element increases evaporative flux. (b) Modeling of breathing mode deformation dynamics using quasi-steady state diffusive transport (red line, eqn (4)). The scale bar in the inset images is 1 mm. |

(1) |

(2) |

(3) |

(4) |

The droplet dynamics modeled by eqn (4) accurately describe the breathing mode deformation observed experimentally. Radii extracted from time-lapse images of the droplet are fit using a solubility value of S = 7.1 × 10^{−4} (as described above) keeping A and D_{WP} as fitting parameters. Fig. 2b shows a representative fit converted to normalized droplet volume versus time, V/V_{0} = (a/A)^{3}. Fitted values for A differ less than 0.15 ± 0.3% from initial radius values taken from final, unloaded cavities, illustrating consistency between the fit and experiments. Diffusivity did not show a trend as a function of mixing ratio; the mean diffusivity across all samples was 4.9× 10^{−9} ± 1.3 × 10^{−9} m^{2} s^{−1}, within the range of previously measured values.^{26,28} Given the close description of the change in droplet size viaeqn (4) we use an extrapolation of this fit in order to calculate the magnitude of applied deformation, i.e., decrease in cavity volume, to creased cavities that can no longer be assumed to be spherical.

For an incompressible neo-Hookean material, creasing has been predicted^{1} and observed^{10,29} to occur under a uniaxial plane strain of ε = 0.354, independent of modulus. Following the parallel made by Cai et al.^{12} between uniaxial plane strain and deformation at the surface of a spherical cavity,^{12} we relate the uniaxial strain, ε to the cavity stretch ratio, a/A, as

(5) |

Fig. 4 Verification of delayed crease onset due to early onset of strain-stiffening as quantified by J_{lim}. The y-axis is the normalized cavity radius, a/A, at crease onset, which is equivalent to the critical biaxial stretch ratio at the cavity surface, λ_{c}. Red, blue, and black symbols correspond to mixing ratios of 10:1, 12:1, and combined 17.5:1 and 25:1 data, respectively. Error bars are the standard deviation from four or more samples. The black curve corresponds to Gent model creasing onset predicted by Jin and Suo,^{22} modified for a spherical geometry using eqn (5). The dashed curve represents crease onset for neo-Hookean behavior. For measured values of J_{lim} > ∼40 strain stiffening occurs at such a high stretch that the behavior is indistinguishable from neo-Hookean behavior, as depicted in the gradient region. Inset: Results re-plotted as critical biaxial strain, ε_{c}, versus J_{lim}. |

Mixing ratio is often oversimplified to a change in linear shear modulus, μ, for Sylgard 184 under the assumption of neo-Hookean behavior at all compositions.^{30,31} However, changes in modulus for a neo-Hookean solid would have no effect on crease onset or morphology,^{1} failing to describe the composition dependence we observe in Fig. 3. We show that this composition dependence is primarily due to strain-stiffening, in validation of Jin and Suo's predictions.^{22} These results illustrate the importance of accounting for material constitutive behavior in creasing systems.

3.2.1 Strain-stiffening material behavior.
Validating Jin and Suo's^{22} predictions requires evaluation of each composition as a strain-stiffening, Gent hyperelastic solid.^{23} The strain energy function for the Gent model is given by

where μ is the small strain shear modulus, J = I − 3 (I is the first strain invariant of the Cauchy–Green strain tensor), and J_{lim} determines the stretch value at which the strain energy asymptotes to infinity. We experimentally determine J_{lim} from uniaxial compression tests as shown in Fig. 5a. True stress (assuming incompressibility) under uniaxial load, σ, and stretch, λ_{u} = l/L (Fig. 5a), data are fit to the Gent hyperelastic model under uniaxial loading:

Four representative compression curves and their corresponding fits are shown in Fig. 5a. For the lowest mixing ratios (10:1 and 12:1), Fig. 5 illustrates the improved fit of the Gent model (solid lines) in comparison to a neo-Hookean model fit using the low stretch response (thin dashed lines). For the largest mixing ratios used (17.5:1 and 25:1), behavior is indistinguishable from neo-Hookean within the limitations of this testing geometry. Resulting fit parameters μ and J_{lim}, plotted in Fig. 5b, quantify the increase in J_{lim} with increasing mixing ratio. Characterization under two environmental conditions, hydrated at 60 °C and room temperature (Table S1, ESI†), revealed a weak dependence of J_{lim} and a moderate dependence of μ on environment. We conclude from these findings that gradients in either temperature or water concentration within the polymer have negligible effect on J_{lim} and therefore no effect on the interpretation to follow.

(6) |

(7) |

Fig. 5 Strain-stiffening material behavior of hydrated Sylgard 184 at 60 °C. (a) Selected uniaxial compression curves for 10:1 (red diamonds), 12:1 (blue circles), 17.5:1 (green squares), and 25:1 (orange ×'s) mixing ratios. Gent model (eqn (7)) fits are given with solid lines. Black, dashed curves denote neo-Hookean fits to the initial loading data (0.9 ≤ λ_{u} ≤ 1). (b) Gent model fit parameters, J_{lim} and μ determined from (a). J_{lim} and experimentally determined crease onset values comprise the experimental data in Fig. 4. |

3.2.2 Effect of strain-stiffening on crease onset.
Jin and Suo^{22} predict that decreased J_{lim} delays the onset of creasing for a plane strain geometry. Numerical interpolation of these predictions yields a relation between J_{lim} and the critical strain, ε_{c}. (ESI,† eqn (S1) is used for interpolation.) The latter is converted to a critical radius a/A|_{c} using eqn (5). The resulting prediction of a/A|_{c} as a function of J_{lim} (solid black line, Fig. 4) describes the experimentally-measured dependence of crease onset on experimentally-determined J_{lim} well, with no fitting parameters. Notably, the measured J_{lim} of the 10:1 ratio falls near the asymptotic region (J_{lim} = 3.1) and the scatter in the crease onset measurements reflect this behavior.

A trend toward decreasing number of creases (decreased density) accompanying an increase in applied strain has been both predicted^{12} and observed^{2,12} for cylindrical and planar geometries in neo-Hookean materials. A similar trend is followed in the spherical geometry characterized here for the low modulus, neo-Hookean materials as shown in Fig. 6c. In contrast, strain-stiffening elastomers initiate few creases at low applied deformation and those creases channel across the surface with increasing deformation. Visually, the strain-stiffening creases appear deeper than those in the neo-Hookean materials. This supposition is supported by the observation that the total length of all creases combined is lower in strain-stiffening solids versus neo-Hookean solids under the same deformation, 1 − V/V_{0} (Fig. 6a). The strain must be accommodated through increased crease depth. To compare ‘near-onset’ effects we use an overstrain term, V_{c}/V_{0} − V/V_{0}, similar to that in Chen et al.^{3} (Fig. 6e and f). Doing so, we observe that creases in 25:1 samples initiate approximately ten times more frequently with around one tenth the length of creases in 10:1 samples (Fig. 6e and f). The mechanism leading to these crease morphology changes with increasing strain-stiffening will be explained through a combination of existing models of crease evolution.

3.3.1 Crease spacing in Gent and neo-Hookean materials.
Finite element analysis has been used previously to predict crease onset and model crease morphology in planar^{1,2} and radially symmetric geometries.^{7,12} Here, we show qualitative similarity between the energetically preferred number of creases in a simulated cylindrical cavity and our experimental observations as a function of material behavior. These calculations provide evidence that the observed morphological variation is primarily due to the material's strain-stiffening response.

where A in this case refers to the initial cylindrical cavity radius and S is the cylindrical cavity's area. f(S) is obtained from subtracting the analytically obtained strain energy of homogeneous deformation (Appendix B) from the strain energy calculated in ABAQUS. Fig. 7a compares f(S) between a neo-Hookean and Gent material behavior (J_{lim} = 5) plotted as a function of the cylindrical overstrain, i.e., the departure from the area at which crease onset occurs, S_{c} − S. Neo-Hookean results agree qualitatively with earlier simulations by Cai et al.^{12} (who used a different ratio of outer to inner boundary radii). Gent curves are cut-off at high overstrains due to an exponential increase in the homogeneous strain energy as the limiting stretch is approached; this leads to imaginary values of ΔΠ at overstrains available to the creased state, but unavailable to the homogeneous state. The inset in Fig. 7a illustrates an energetic preference for the maximum number of creases simulated in neo-Hookean materials (purple inverted-triangle, solid line). In contrast, for a Gent material having J_{lim} = 5, the preferred number of creases is two just after onset. Note that in order to construct this comparison, the strain energy at crease onset is set to zero via

Crease onset in the cylindrical geometry is determined using previously calculated critical strains for plane strain^{1,22} (eqn (3)), following Cai et al.^{12} (Appendix A). Without performing this zeroing operation, small numerical errors due to the presence of an initial defect lead to crease-number-related error near crease onset. This numerical error originates prior to creasing due to the presence of the defect and becomes similar in magnitude to free energy differences just after crease onset. This numerical variation would be eliminated for an infinitesimally small initial defect. (ESI,† Fig. S6 demonstrates the trend toward zero ΔΠ at crease onset for decreasing defect size.) Fig. 7b summarizes the preferred number of creases among those values simulated (2, 3, 5, 7, 10) for a neo-Hookean solid (black squares), Gent material with J_{lim} = 10 (blue circles), and Gent materials with J_{lim} = 5. In agreement with experimental observation of spherical cavities, materials initiate creasing with fewer number of creases the earlier onset of strain-stiffening occurs, i.e., lower J_{lim}. This result is visualized in Fig. 8. Given the lengthy deformation zone in which the maximum number of creases simulated is energetically preferred for the neo-Hookean solid, we believe an energetic minimum likely occurs at an even larger number of creases.

We performed finite element simulations of cylindrical cavities in ABAQUS using neo-Hookean and Gent^{22} constitutive models in order to calculate strain energy. This cylindrical cavity approximation provides a qualitative description of the observed phenomena while avoiding complexity associated with fully three-dimensional crease simulation^{8,32} or the selection of experimentally inapplicable boundary conditions associated with a planar simulation (ratio of film width to height). The energetically preferred number of creases is determined by finding the minimum strain energy among simulations containing periodically-spaced numbers of creases ranging from 2 to 10. Creases are initiated at small geometric defects. Following Cai et al.,^{2,12} the strain energy per unit thickness of the creased state relative to a homogeneously deformed uncreased state undergoing the same outer surface displacement (eqn (7)) takes the form

ΔΠ = μA^{2}f(S), | (8) |

(9) |

Fig. 7 Simulated results for crease number trends for a cylindrical tube. (a) f(S − S_{c}) from eqn (8) and (9) as a function of normalized cylindrical overstrain, S_{c}/S_{0} − S/S_{0} where S_{0} = πA^{2}, for neo-Hookean (solid lines) and Gent (J_{lim} = 5, dotted lines) material behaviors. The inset illustrates that near crease onset, S_{c} − S = 0, a free energetic preference for the largest number of creases simulated (10) exists for a neo-Hookean response. This is in opposition to a preference for the fewest number of creases simulated (2) for the J_{lim} = 5 response. At high overstrain, both behaviors prefer the fewest number of creases. (b) The progression of free energetically-preferred crease number among those simulated for neo-Hookean (black squares), J_{lim} = 10 (blue circles), and J_{lim} = 5 (red triangles). These trends for the cylindrical geometry match those observed in spherical cavities (Fig. 3 and 6). |

3.3.2 Limited role of surface energy.
Crease growth via channeling has been previously shown to arrest or propagate based on elastocapillary control.^{3} Low elastocapillary numbers, γ/μL, and therefore creases that more readily channel are achieved with either high moduli or low surface energies. Estimation of the elastocapillary number for this system, where the relevant length scale L is taken to be the initial cavity radius, A = 1.5 mm, yields a range from 4.1 × 10^{−5} (10:1) to 3.7 × 10^{−4} (25:1). Within this regime, surface energy should play a negligible role and surface energy should provide no barrier to channeling. However, we observe little channeling in the softest compositions. Even an order of magnitude change in surface energy should have no effect. To test the assumption that initial cavity radius, A, was the appropriate length scale, we changed the surface energy while keeping composition constant, similar to Chen et al.^{3}

Liquid drops embedded in 25:1 Sylgard were altered through the addition of the surfactant 3-[hydro(polyethyleneoxy) propyl] heptamethyltrisiloxane (Gelest) above its critical micelle concentration.§^{33} Interestingly, as reported by Chen et al.,^{3} creases formed under the effect of surfactant channeled further than those in non-surfactant cavities at low overstrain (Fig. 6e, f and Fig. 9). The morphology of these 25:1/surfactant systems more closely resembles that in the 10:1 and 12:1 mixing ratios immediately after crease onset. To understand this effect, we estimate the elastocapillary length scales for 25:1/water, 25:1/surfactant, and 10:1/water as 550 nm, 6.5 nm, and 61 nm, respectively (μ_{10:1} = 0.65 ± 0.04 MPa, μ_{25:1} = 0.073 ± 0.004 MPa, γ_{H2O} = 40 mN m^{−1}, γ_{surf} = 0.5 mN m^{−3}). It appears that this length scale in the surfactant cavity approaches that of the 10:1/water system, mirroring the approach of surfactant cavity morphology to that of the 10:1 cavity.¶ This result indicates that crease size, as opposed to system geometry, may be the governing length scale for calculation of the elastocapillary number in the initial stages of creasing. Using previously determined experimental values for the crease depth (d ∼ 20 μm from Cai et al.^{2}) as estimate for the length scale, L, the elastocapillary numbers for 25:1/water, 25:1/surfactant, and 10:1/water systems are 0.07, 9 × 10^{−4}, and 5 × 10^{−3}, respectively. The elastocapillary number for the 25:1/water system is within the range reported by Chen et al.^{3} as having elastocapillary governed channeling. Therefore, while surface energy does not predominately account for the morphology differences observed here, it is interesting to note its role in a system for which it would typically be considered negligible.

Another phenomena observed during cavity contraction is the formation of ‘microbubbles’ at the PDMS-water interface (Fig. S4, ESI†). These structures grow in size as deformation increases, eventually appearing to coalesce, and are more prominent in the stiffer elastomer formulations. Such microbubbles would lead to cavitation if they were water vapor or would require any gas within them to be under tension if they were air. We therefore suggest that they might be formed from squeezing small volumes of un-crosslinked silicone^{35} out of the bulk of the PDMS. Similarly, silicone nanobubbles have been observed to coalesce in PDMS-water mixtures.^{36} Given our primarily continuum mechanics, negligible surface energy description of the observed creasing phenomena, these fluid bubbles are expected to have minimal effect on the results reported herein.

• The deformation at which crease onset occurs depends on strain-stiffening.

• Neo-Hookean solids have an energetic preference for many creases near crease onset.

• The number of creases near onset decreases with decreasing J_{lim}. (In the Gent model, J_{lim} characterizes strain at onset of stiffening.)

• All material behaviors tend toward similar crease numbers at large overstrain.

These results experimentally validate the important role that strain-stiffening plays in determining crease morphology in this contracting cavity system. It follows that strain-stiffening may be a critical consideration in predicting biologically relevant crease morphologies in other geometries.^{7,8,13}

(A1) |

(A2) |

Similarly, in a cylindrical geometry under plane strain conditions, the principal stretches are λ_{1} = r/R, the circumferential stretch, λ_{2} = 1, the longitudinal stretch, and λ_{3} = R/r, the radial stretch. Following the same procedure as above, a/A at crease onset in the cylindrical geometry is

(A3) |

(B1) |

r^{2} − a^{2} = R^{2} − A^{2} | (B2) |

λ_{θ} = r/R = λ. | (B3) |

λ_{r} = R/r = 1/λ. | (B4) |

J = λ^{2} + λ^{−2} − 2 | (B5) |

r = (a^{2} − A^{2})^{1/2}(1 − λ^{−2})^{−1/2}. | (B6) |

(B7) |

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm01258f |

‡ Water within a few samples cavitated prior to crease onset, see ESI.† |

§ Concerns of surfactant migration into the uncured necessitated an increase in curing temperature and reduction in curing time, 150 °C and 10 min. As a control, pure water cavities in rapidly cured 25:1 polymer behaved identically to those created using the slower curing conditions reported above. |

¶ Large deformations in surfactant cavities were not observed, likely due to early cavitation facilitated by the presence of surfactant. |

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