Origami Inspired Mechanics: Measuring Modulus and Force Recovery

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Origami Inspired Mechanics: Measuring Modulus and Force Recovery with Bent Polymer Films Theresa Elder,† Damith Rozairo,† and Andrew B. Croll*,†,‡ Materials and Nanotechnology and ‡Department of Physics, North Dakota State University, Fargo, North Dakota 58102, United States

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ABSTRACT: Origami, the art of paper folding, has recently seen an upsurge of interest due to its use in guiding the design of lightweight deployable structures. Despite the heavy use of thin films in origami designs, comprehensive mechanical understanding lags behind. This is partly because origami structures are often made from new materials for which bulk material properties are not available. In this work, we show how bending can be used to gather broad mechanical information from thin films, and we show how that information can be applied to more complex structures. Explicitly, we use the technique to measure the Young’s modulus and monitor the force recovery of polydimethylsiloxane, polystyrene, and polycarbonate films. Our force recovery data are consistent with the sparse published data available but reveal a previously unreported film thickness dependence. We hypothesize that the thickness dependence is related to the strain gradient present in bending.

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films are made of flat segments and bent segments as described above, but also d-cones and stretching ridges. D-cones, or developable cones, occur when a film that is bent along one axis is bent along a second orthogonal axis.14 The film cannot accommodate the two bends without stretching and when the film is relatively rigid the stretching is focused to a point (leaving the majority of the film unstretched or developable). If two d-cones are created in a sheet, they are linked by a stretching ridge.15 In this case, the two cones cannot be created without causing additional stretching in the sheet. The longtime dynamics of d-cones, ridges, and crumples are also not well understood, but measurements often show logarithmic or stretched exponential changes as a function of time.12,16 In this work we take a step toward understanding the underlying mechanics of origami by mechanically probing a single bend in a sheet as well as a doubly bent sheet (which contains a single d-cone). With the singly bent films we focus on low curvatures, avoiding highly collapsed folds.17−19 Our primary interest is in showing that a simply bent sheet can be used to measure the static and dynamic mechanical response of a thin film and that more complex origami structures may not deviate much from this basic material response. Specifically, force−displacement and force−recovery curves are collected as a singly or doubly bent sample is crushed between two parallel plates. Samples are imaged with a confocal microscope during the entire process to clarify the geometric details during the experiment. Three different polymeric materials are used polydimethylsiloxane (PDMS), polystyrene (PS), and poly-

INTRODUCTION Interest in slender materials has greatly increased in recent years due to advances in origami inspired structural design,1−4 the creation of new materials for coatings and barrier layers,5,6 and the development of thin electronic platforms.7−10 Slender structures are easily bent, but not easily stretched, which leads to unique advantages (structures designed around stretching are strong and lightweight) or disadvantages (an inextensible sheet cannot easily wrap a sphere) depending on application. Despite the conceptual simplicity of origami and wrapping, understanding the basic geometry and mechanical response of slender structures in different configurations remains challenging. Part of the difficulty in predicting the mechanical response of origami structures is that designs require focused bending along certain predefined paths to function. This scheme creates rigidity when a load is directed along the unbent segments of film but allows configurability as bent locations can easily be deformed. Such “weakened” segments may be created through variations in cross-link density, variations in thickness, or through plasticity. The crux of the problem is that energy is not distributed uniformly; it varies with location in the sheet as do the regions of high curvature. Additionally, origami designs constructed with real materials (such as paper) often require bending to be driven to extreme scales where radii of curvature approach the film thickness, leading to plasticity and material failure. Predicting long time behavior requires a careful accounting of the different energetic modes and their relationships to one another. The complexity of thin film mechanics is probably best seen in a crumpled ball; this origami structure is quite stiff but is made of at least four different substructures.11−13 Crumpled © XXXX American Chemical Society

Received: September 15, 2018 Revised: December 21, 2018

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DOI: 10.1021/acs.macromol.8b02002 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematic experimental setup. This setup is the “large-scale” experiment. The bottom plate is stationary atop a scale while the top plate is attached to a motor and is mobile. A telescoping lens and mirrors allow for the film and scale numbers to be in focus, allowing accurate timing between plate position and recorded force. The small scale experiment is conceptually similar, replacing the camera with a confocal microscope.

dependence to the long-time dynamics for both PC and PDMS. We organize this article around a basic scaling description of the mechanical response of a singly or doubly folded thin film. After describing the experiment, we outline the model and then show how it can be used to determine a film’s bending modulus, which, with a measured thickness, (h), gives a quantitative measure of Young’s modulus. We then go on to discuss several nonlinear features of the modulus which emerge from the measurement. Notably, we evaluate the long time response of the modulus which is easily measured in this geometry and of great importance in predicting the long time stability of origami-based structures. Ultimately, we show that our measurements are equivalent to published results collected with more traditional testing of bulk materials, proving that thin film bending is a comprehensive and useful mechanical testing method. Our method can be adapted to measure bending in other origami systems so long as uniform layers can be made of the “weakened” material (for example, a uniform sheet with a low cross-link density).

carbonate (PC)to highlight the versatility and accuracy of the technique. Over the range of thicknesses (1 μm−1 mm) tested, no deviation from our simple analytic model was noted. The results suggest that bending mechanics could be used on much smaller length scales complementing existing techniques.17,20−25 The PDMS used, Sylgard 184, is a heavily used commercial silicon elastomer, filling roles in microfluidic,26 adhesion,27 and biological research.28 As such, the material is fairly well studied mechanically,28−34 although much less information is available regarding its long time behavior (creep or force recovery).28,31,32,34 PS is an amorphous glassy polymer with a Young’s modulus of 3.0−3.5 GPa and a glass transition temperature of 100 °C.35−41 The loose packing of phenyl groups along the C backbone cause failure at relatively low strains in the form of crazing and cracks.38,39 PC is also a glassy polymer (glass transition of 145 °C) with failure typically taking the form of plastic deformation rather than fracture.41−58 PC at room temperature has a Young’s modulus of 2.0−2.6 GPa.41−58 Our results largely agree with this existing body of mechanical research, however, we note several deviations related to long-time analysis. First, our results indicate that long-time processes in PDMS are better described by a single logarithmic function of time rather than the commonly used Maxwell models. Our more accurate measurement and modeling of the long time behavior will impact the many uses of Sylgard (for example, adhesion measurements are directly related to material modulus).27 Second, we note that over the time range explored here PC is better fit by a logarithmic function rather than the more firmly established stretched exponential function. Finally, the more accurate logarithmic fits reveal a previously unreported film thickness

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EXPERIMENTAL SECTION

Fabrication of PDMS Films. Sylgard 184 silicone elastomer base and curing agent (cross-linker) were mixed in a 10:1 weight ratio with a glass pipet for 5−10 min prior to film preparation. Thicker films were created by pouring a desired weight of mixed solution into polystyrene (PS) sample containers. Thinner films were produced by flow coating solution onto a mica surface or spin coating on a mica surface. Sample containers were placed in a vacuum at 20−25 in.Hg for 5 min and then brought back to room pressure. The cycle was repeated four times to speed degassing. After cycling, the film was placed in a vacuum oven to anneal for 90 min at 15 in.Hg and ∼85 °C. Films were removed promptly from the oven and allowed to cool a minimum of 30 min prior to use. Nile red dye was often added to films to facilitate fluorescent imaging. B

DOI: 10.1021/acs.macromol.8b02002 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Fabrication of Polystyrene and Polycarbonate Films. Solutions of different weight percent polystyrene (PS) in toluene (or tetrahydrofuran) were created and thoroughly mixed. Polycarbonate (PC) and chloroform solutions were also created to various weight percent. Solutions were left to sit for 48−96 h before use, allowing polymer solutions to fully mix. Freshly cleaved mica was then placed on a glass microscope slide and held in place by the capillary force of a small amount of water. Drop-cast films were created on the mica substrate by adding solution dropwise until the substrate was uniformly covered. Films were then placed in a chamber saturated with solvent vapor. Films slowly formed over 24−48 h. Similar mica/glass substrates were also used in homebuilt flow-coating apparatus. In this case, solution was placed behind a blade which was then moved at constant velocity and fixed height from one side of the mica to the other. Varying speed, blade height, and solvent concentration allowed for a range of thicknesses to be created. Additionally, mica/microscope slide substrates were placed on the vacuum stage of a spin coater. Solution was added dropwise into the center of the substrate, which was then spun at various speeds, creating a range of film thicknesses. Subsequent to film preparation, thermoplastic samples were annealed on a hot plate at 150 °C for 90 min (PS) or at 180 °C for 60 min (PC). Films were allowed to cool for 1−2 h at room temperature, and then each film was cut into a rectangular shape at the scale needed. Dimensions of length, width, and thickness were recorded with a calipers, confocal microscopy, or atomic force microscopy depending on the scale. The experiments we report range over thickness from ∼1 μm to 1 mm, and the error as determined by standard deviation was typically 40%. The data essentially point out the need for a more detailed nonlinear model to be properly interpreted. More pragmatically, they show the strain limits for which eqs 2 and 4 can be safely applied. Force Recovery. The large error of the average logarithmic slope, ⟨β⟩, also suggests that other features are influencing the measurement. The bending measurement is very sensitive to film thickness which may also play a role in microscopic dynamics. Plotting β as a function of film thickness in Figure 11 reveals a clear relationship between β and h. Empirically, we find that the data are well described by the function β = A / h where A is a constant of ∼1 × 10−4 m1/2.

Figure 11. β as a function of thickness for PDMS (circles) and PC (triangles).

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Despite being a heavily used elastomer, very little information is available in the literature for PDMS (in particular Sylgard 184) dynamics. Schneider and co-workers performed tensile tests on bulk (thick) Sylgard 184 samples and analyzed the creep behavior of the material.31 They found creep over 30 min to be well fit by a Burger model and report a time constant of 236 s at room temperature under a stress of 3.125 N/mm2. Zhang and collaborators conducted indentation force recovery experiments at room temperature over 100 s as well as dynamic mechanical analysis of bulk material and force recovery of elastomer post indentation (over 6.5 s) in a second work.28,32 They fit the data with a generalized Maxwell model which assumed more than a single time constant. Neither group showed data on a linear/log axis so it is difficult to compare with our experiment or judge the model fit on different time scales. For the sake of comparison, a logarithmic trend can be fit to the Maxwell model fit reported by the Zhang group (admittedly with poor overlap), resulting in a fit of β = 4.4 × 10−3 ± 8 × 10−3. This puts their data within error of the average β measured here, highlighting the reliability of our bending measurements. Remarkably, the polycarbonate data fall along a similar trend when β is plotted against thickness as in Figure 11. The similarity, despite the great physical differences in the two

CONCLUSIONS In this work we have shown how simple bending can be used to measure detailed mechanical properties of thin polymer films, and we show how these intrinsic material properties are applicable even in more complex origami scenarios. We have used the technique to determine Young’s modulus for a polydimethylsiloxane elastomer (Sylgard 184), polystyrene, and polycarbonate. Our measurements compare well with accepted values. More importantly, we have also conducted force recovery experiments on all three materials. Here we find the brittle polystyrene difficult to reliably measure due to limitations in our current setup but collect reliable data for polycarbonate and polydimethylsiloxane. Both polycarbonate and polydimethylsiloxane are well fit in the 1−10000 s time range by a single logarithmic function, which we find to depend on film thickness. There are no signs that the method cannot be scaled down below the current 1 μm limit set by the force resolution of our current force transducer. Our measurements highlight how easy it is to collect material properties from thin films before collecting system properties of more complex structures such as origami designs or crumpled matter. H

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AUTHOR INFORMATION

Corresponding Author

*(A.B.C.) E-mail: [email protected]. ORCID

Andrew B. Croll: 0000-0002-6890-3084 Author Contributions

A.B.C. conceived the experiments; T.E., D.R., and A.B.C. conducted the experiments. The manuscript was written through contributions of T.E. and A.B.C. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support from the AFOSR under the Young Investigator Program (FA9550-15-1-0168). T.E. thanks the NDSU Materials and Nanotechnology program for support.

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ABBREVIATIONS PC, polycarbonate; PDMS, polydimethylsiloxane; PS, polystyrene; LSCM, laser scanning confocal microscopy.

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