Photo-origami. Author: Matjaº Li en Adviser: Assoc. Prof. Dr. Irena Dreven²ek Olenik. Ljubljana, December Department of Physics

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1 Department of Physics Seminar I b - 1 st year, nd cycle program Photo-origami Author: Matjaº Li en Adviser: Assoc. Prof. Dr. Irena Dreven²ek Olenik Ljubljana, December 013 Abstract Photo-Origami is a process of folding two dimensional polymer sheets into three dimensional objects using light to create hinges by stimulating folding at the desired locations. The approaches described rely on creating a stress gradient throughout a uniform material, which then bends to one side to reduce the tension. We rst demonstrate how to calculate the stress needed to deform a cross-linked polymer and then proceed to two examples of how light can be used to actuate bending in polymers. The rst uses light to heat up the polymer over its glass-transition temperature in specic locations, while the second uses light to rearrange chemical bonds inside a specially prepared polymer.

2 Contents 1 Introduction Elasticity of polymers 3 Folding actuated by heating 5 4 Folding actuated by splitting of chemical bonds 8 5 Conclusion 10 1 Introduction Origami is an old Japanese art of folding a sheet of paper into objects. The principle of folding two dimensional sheets of material into three dimensional object has evolved through mathematical analysis and is now often used to solve problems that involve folding objects into more convenient shape, so that they can be stored and unfolded for use later. Examples range from solar arrays in space structures to shopping bags and biomedical applications. When folding objects on smaller scales, one usually wishes to create simple geometrical shapes like boxes or pyramids, but the word origami is still used to describe the process. In order to fold a two-dimensional sheet of material one must chose a material and a method of folding. To determine where the material is going to be folded, one can either construct a uniform material and use a non-uniform stimulus, such as irradiating through a mask, or the other way around, where the material comes prepared with inbuilt hinges and folding is actuated by exposing the entire sheet to the same stimulus. Many dierent approaches have been demonstrated, where materials were dierent combinations of metals, polymers, liquid crystals and so on, while actuation of the hinges was achieved by changes in temperature or ph, capillary forces, magnetic forces, ultrasonic pulses and so forth [1,, 3, 4]. In this seminar, we take a look at methods for folding polymers that use light to control the process. The methods we discuss here rely on creating a stress gradient throughout a uniform material, which then bends to one side to reduce the tension. We describe two dierent approaches, one based on heating the material with light and the other based on using light to split chemical bonds inside the material. Elasticity of polymers Both examples that will be discussed later involve stretching the polymer to induce stress, which is later reduced only at specic locations. Polymers are large molecules constructed by combining a large number of smaller molecules called monomers, typically to form chains. Depending on which monomers are present in the polymer and how they are combined, polymers can have a wide variety of properties. In addition, polymeric materials can typically be observed in two states or phases - glass state and rubber state. At lower temperatures, polymers are found in the glass state and are solid and brittle, while at higher temperatures, polymers are in the rubber state and are viscous uids. If polymer chains are linked to one another the rubber phase becomes visco-elastic instead. We call such polymers cross-linked. Examples of polymers are various plastic materials: poly(methyl methacrylate) is commonly used in its glass phase as Plexiglass, while cross-linked polymers like natural rubber are used

3 for their elasticity in their rubber phase. Here we demonstrate how to calculate the relationship between the stress and strain of a cross-linked polymer [5]. Let us imagine a prism-shaped chunk of natural rubber with sides L x = L y and L z and call the relative deformation along the z axis λ. λ = L z + L z L z (1) Because rubber is nearly incompressible (V = const.) and we usually observe it at a constant temperature, we can calculate the force f via the Helmohltz free energy F : ( ) F f = = 1 ( ) F. () L z L z λ V,T Since free energy is a sum of the internal energy of the rubber and its entropy, we can divide the force into two parts, f ɛ and f S, pertaining to energy and entropy respectively. At this point we make an assumption that does not have a clear theoretical basis but concurs with experimental evidence, namely that f ɛ = 0. We can now write the entire force as equal to entropy's contribution to equation (). Let us rst derive the force needed to stretch a freely jointed chain with a mean square distance between its two ends being R0. Because the joints are freely rotating, there is no energy contribution to the force and equation () is written as [6] f = T S z. (3) Here we chose to strain the polymer along the z axis. If we assume that the distance of the end of the polymer from its origin is distributed according to a Gaussian distribution then the partition function of a freely jointed chain with distance z between its two ends can be written as ) Z (z) exp ( 3z R0. (4) Inserting this result into Boltzman's equation for entropy V,T we write the entropy as and the force as S = k B ln Z (z) (5) ( ) S (z) = S (0) + k B 3z R0, (6) f = 3k BT R0 z. (7) The required force is linearly dependant on the distance between the start and the end of the polymer chain, which means the polymer acts as a spring. In order to apply this result to a cross-linked polymer like rubber, we will assume the rubber is constructed of many adjoining polymers with equal values of R0, while still retaining that the distance between one end and the other is distributed according to a Gaussian distribution. In addition, we are going to assume that the junctions between the polymers are xed and can only move due to 3

4 deformations of the entire material. The entropy is now written as a sum of the entropies of each of the polymer chains. S = i s i (8) If V is the volume of the rubber and c p is the density of the polymer chains, we can rewrite equation (8) in an integral form: S = V c p ˆ s (x, y, z) p (x, y, z) dxdydz. (9) Here p (x, y, z) is the Gaussian distribution of the location of a chain's nal link. ( ) 3/ 3 p (x, y, z) = πr0 exp ( 3 ( x + y + z ) ) R0 (10) Inserting equations (6) and (10) into equation (9) we can write down the entropy of the rubber as a function of deformations along x, y and z axis, labelling the relative deformations λ x, λ y and λ z respectively. ( S (λ x, λ y, λ z ) = V c p s (0, 0, 0) k B ( λ x + λ y + λ ) ) z (11) The entropy of a non-deformed polymer can be written as ( S 0 = V c p s (0, 0, 0) k ) B 3 and the dierence in entropy between a deformed and a non-deformed polymer as (1) S (λ x, λ y, λ z ) = V c p k B ( λ x + λ y + λ z 3 ). (13) In the case of λ z = λ, λ x = λ y and λ x λ y λ z = 1, which describes a deformation along z axis while the volume of the rubber is conserved, equation (13) can be simplied. k B S (λ) = V c p (λ + λ ) 3. (14) From here, we can use equation (3) to calculate the force needed to deform the rubber. f = T S L z λ = T S L z λ = V c pk B T (λ 1λ ) L (15) z We can also express this as the ˆσ zz element of the nominal stress tensor. ˆσ zz = f = c p k B T (λ 1λ ) L x L y A comparison of values of ˆσ zz predicted by equation (16) and experimental data for rubber is shown in gure 1. We observe that our model is accurate for deformations up to λ = 1.5, but fails when trying to predict larger deformations. Despite this, the model is still an adequate description of the examples discussed below, since the strain there never exceeds 0% of the total length. 4 (16)

5 Figure 1: Values of ˆσ zz as predicted by equation (16) (dashed line) and measurements for natural rubber (points) as a function of the relative deformation λ. The chart on the right shows the chart on the left in greater detail for λ <. We see that for small deformations, our result corresponds well to the experimental data [5]. This derivation holds for polymers above glass-transition temperature, i.e. polymers in their rubber phase. In the glass phase, polymer chains can no longer move freely, which is contrary to the assumption of a freely-jointed chain. As a result, the stiness of a polymer is much higher in its glass phase than in its rubber phase. An illustration of this change in stiness is shown in gure. E[Pa] E 1 E T g T m T Figure : Illustration of the drop in stiness of a polymer when it is heated across its glasstransition temperature T g. Heating the polymer further, causes it to melt and its stiness drops to zero [7]. 3 Folding actuated by heating Let us rst take a look at a method that was used by Ying Liu et al. to fold polymer sheets into dierent 3D shapes [8]. They used a polymer from a children's toy called Shrinky-Dinks, which contracts when heated, the idea being that large sheets could be coloured and then shrunk down uniformly to form small, but very detailed objects. The sheets themselves are prepared by rst heating a sheet of polystyrene over its glass-transition temperature, so that it 5

6 can be stretched without tearing, and then cooling it down again, hardening it in its stretched state. Reheating the polymer causes it to return to its original dimensions. Hinges are printed on the sheet in black ink with a conventional ink-jet printer and the entire sheet is irradiated with infrared light. The polymer itself is almost transparent for infrared light so that the absorption occurs only where it had been coated with black ink. Thus, we only heat up the hinges and locally create a temperature gradient between the irradiated side of the material and the opposite, unirradiated side. Where the polymer is heated above T g, the stress in the polymer relaxes causing the polymer to shrink. The sheet bends towards the light source if the irradiated side undergoes a phase transition and the unirradiated side remains in the glass phase, because one side of the hinge is now wider than the other. After cooling below T g, the entire polymer reverts to its glass phase, preserving the fold that had been created. It is also possible to bend the polymer away from the light source by printing the ink on the other side of the polymer sheet. The process of preparing and bending the polymer is illustrated in gure 3. The angle at which the polymer bends is controlled by the duration of exposure to infrared light and by the width of the pattern that is printed on the sheet: longer exposures and wider hinges result in larger bending angles. Results of simulations exploring dierent hinge widths and irradiation times are presented in gure 4. Figure 3: Preparation of a Shrinky-Dink polymer and its subsequent bending by local light absorption. The original polymer is heated above its glass-transition temperature, brining it into a rubbery state, and stretched along one of its sides. While strained, it is cooled below its glass-transition temperature, making it harden in its stretched state. The polymer is now printed with the desired pattern that denes the hinges and then irradiated with IR light, which is only absorbed in printed areas. Hinges printed on the top of the sheet cause the polymer to bend towards the light, hinges printed on the bottom, cause the polymer to bend away from the light. The thickness of the ink is not drawn to scale [8]. This approach utilises commercially available tools and materials, i.e. IR light bulb, ink-jet 6

7 printer, and Shrinky-Dink polymer, and can be used to fold 3D objects from a D sheet of polymer. Additionally, heating the entire polymer over the glass-transition temperature causes the 3D object to unfold and shrink in the plane of the original stretched sheet; however, the polymer cannot be folded again. Some examples of objects created this way are shown in gure 5. Figure 4: Simulations of temperature prole generated by irradiating the polymer under different circumstances. The thermal conductivity of polysytrene is around 0. W/mK. Results of dierent durations of irradiation are displayed on the left side; the polymer bends when the top of the sheet is heated above the glass-transition temperature, which is around 100 C for the Shrinky-Dink polymer, while the bottom remains in the glass phase. Charts on the right side show results of simulations of dierent hinge widths along with dierent overall temperatures of the polymer. The top two charts (a, b) are result from a simulation at room temperature. In the case of the narrower hinge, the top of the polymer is above T g over the entire area of the hinge, while the bottom remains below T g. In the case of a wider hinge, the bottom of the sheet reaches the glass-transition temperature before the entire width of the hinge is above it at the top. The bottom two charts (c,d) are results from a simulation done at a higher overall temperature. In this case, even the top of the wider hinge can be brought above T g before the bottom of the hinge softens. Narrower hinges take more time to start folding because a greater fraction of the heat is transferred from the hinge to the bulk of the sheet [8]. 7

8 Figure 5: Example object created by irradiating a patterned Shrinky-Dink polymer with IR light. The hinges are about 1 cm in length, with the longer hinges in gure d) being cm long. a) Single line resulting in a hinge; b) two lines printed on opposite sides of the sheet, resulting in a zig-zag shape; c) lines on alternating sides of the sheet cause the sheet to bend into an M shape; d) rectangular box; e) tetrahedral box; f) tetrahedral box with two adjacent double hinges [8]. 4 Folding actuated by splitting of chemical bonds The second approach, which was used by Ryu et al., is based on UV light splitting chemical bonds inside the polymer [9]. They used a polymer previously described by Scott et al., which showed stress relaxation upon exposure to UV light without loss of the initial material properties [10]. The initial mixture from which they created the polymer contained three monomers (Pentaerythritol tetra(3-mercaptopropionate) (PETMP), -methylenepropane-1,3-di(thioethylvinylether) (MDTVE), and ethylene glycol di(3-mercaptopropionate) (EGDMP)), two photo-initiators commercially known as Irgacure 819 and Irgacure 184, and a UV light absorbing chemical named Tinuvin 38. Irradiating the mixture with nm light causes Irgacure 819 to break down into radicals, which facilitate the polymerisation of the monomers, while trapping Irgacure 184 and Tinuvin 38 inside the material. The sheet is then strained along a single axis and irradiated with light at 365 nm through a mask that determines where the folding will occur. Irradiation causes the second photo-initiator, Irgacure 184, to break down into radicals, which allow the chemical bonds inside the stretched polymer to shift, locally lowering the stress. We can understand this as locally reducing the deformation of the material, which, according to equation (16), lowers the stress. This does not happen uniformly throughout the entire depth of the material however, due to light being absorbed by Tinuvin 38. More bonds are rearranged at the irradiated side of the sheet than on the unirradiated side, making the polymer bend away from the light source after removing the stress. Polymer construction and rearrangement of chemical bonds is illustrated in gure 6, an example of creating a self-folding box with this process, along with an image of an experimentally folded box and its simulated equivalents, is shown in gure 7. By manipulating the magnitude of the stress applied on the polymer sheet and the width of the irradiated area, the bending angle can be controlled. Greater stress and wider areas result in larger bending angles as shown in gure 8. 8

9 Figure 6: Polymer creation and rearrangement of chemical bonds after straining and irradiating with UV light. The monomers needed for the fabrication of the polymer are shown on the left, followed by the photo-initiators and the photo-absorber needed in the formation and manipulation of the polymer. Irradiation with nm light breaks down the Irgacure 819 molecules into radicals, which bind the monomers into a cross-linked polymer. After stretching, the polymer is irradiated with 365 nm light causing the Irgacure 184 molecules to break down into radicals, which in turn facilitate the rearrangement of the chemical bonds in the polymer, causing the stress to be lowered in the irradiated area [9]. Figure 7: Construction of a self-folding cube. a) Individuals steps required to create a box. The polymer is strained along the x-axis and irradiated through a mask to create three of the ve hinges required. The load is then removed and the polymer strained in along the y axis and irradiated to create the remaining two hinges. The perimeter of the box is then cut out and the box folds into its nal shape. b) Image of the box created in the experiment. c) Simulated closed box [9]. 9

10 Figure 8: Measured and simulated results of the dependence of curvature on the strain of the polymer (a) and hinge angle on mask width (b). Measured results are displayed as black dots, simulated results are displayed as a red line. Insets show images of experiment and simulation. We see that the hinge angles decrease faster for small mask widths than for large ones. This is due to an edge eect: when stress is relaxed inside the irradiated area it is also decreased in the nonirradiated area to maintain equilibrium. This pulls the irradiated material from the edge of the mask out of the irradiated region, eectively increasing the width of the hinge. [9]. 5 Conclusion We have presented two possible ways of folding two dimensional sheets of a cross-linked polymer into three dimensional objects, using light to stimulate the folding. In both cases the initial polymer was uniform so that it could be folded into any shape, regardless of the production process. The processes involved in creating the folded object were also relatively simple: in the rst case printing and uniformly irradiating, in the second case stretching and irradiating through a mask. Though these experiments were done at the millimetre scale, there is no direct limitation for a similar process on the micrometre scale. The use of polymers makes this sort of approach useful in medicine, because the human body is better equipped for processing organic compounds than, for example, metals. That means that drugs can be encapsulated into vessels folded from polymers and then released by unfolding the polymer at a specic point in the body. One could also imagine boxes with holes, which allow for gradual release of the encapsulated substance. However, manipulating these particular polymers inside the human body seems impractical, since UV light is harmful to the tissue as is the temperature needed for glass-transition in a Shrinky-Dink polymer. Other drawbacks should be mentioned as well. These approaches do not allow for multiple folding and unfolding around the same hinges: once the hinge is straightened, it cannot be folded again. Nonetheless, these experiments serve as a proof of concept. 10

11 References [1] L. Ionov, Soft Matter 7, (011). [] C. Py, P. Reverdy, L. Doppler, J. Bico, B. Roman, and C. N. Baroud, Phys. Rev. Lett. 98, (007). [3] J.-H. Cho, M. D. Keung, N. Verellen, L. Lagae, V. V. Moshchalkov, P. Van Dorpe, and D. H. Gracias, Small 7, (011). [4] T. G. Leong, A. M. Zarafshar, and D. H. Gracias, Small 6, n/an/a (010). [5] R. Podgornik, rudi/sola/rubbel.pdf (01). [6] P. Ziherl, ziherl/smt.pdf (013). [7] (4/1/013). [8] Y. Liu, J. K. Boyles, J. Genzer, and M. D. Dickey, Soft Matter 8, (01). [9] J. Ryu, M. D'Amato, X. Cui, K. N. Long, H. Jerry Qi, and M. L. Dunn, Applied Physics Letters 100, (01). [10] T. F. Scott, A. D. Schneider, W. D. Cook, and C. N. Bowman, Science 308, (005). 11