Sequential Self-Folding Structures by 3D Printed Digital Shape Memory Polymers

Folding is ubiquitous in nature with examples ranging from the formation of cellular components to winged insects. It finds technological applications including packaging of solar cells and space structures, deployable biomedical devices, and self-assembling robots and airbags. Here we demonstrate sequential self-folding structures realized by thermal activation of spatially-variable patterns that are 3D printed with digital shape memory polymers, which are digital materials with different shape memory behaviors. The time-dependent behavior of each polymer allows the temporal sequencing of activation when the structure is subjected to a uniform temperature. This is demonstrated via a series of 3D printed structures that respond rapidly to a thermal stimulus, and self-fold to specified shapes in controlled shape changing sequences. Measurements of the spatial and temporal nature of self-folding structures are in good agreement with the companion finite element simulations. A simplified reduced-order model is also developed to rapidly and accurately describe the self-folding physics. An important aspect of self-folding is the management of self-collisions, where different portions of the folding structure contact and then block further folding. A metric is developed to predict collisions and is used together with the reduced-order model to design self-folding structures that lock themselves into stable desired configurations.


S1. Description of Multi-branch Model
The multi-branch model (also referred as generalized standard linear solid model) has been shown to be able to capture the shape memory effects of the polymers 1 . Figure S1 shows a schematic representation of the model, where the number of branches depends on the width of glass transition temperature range and the structure of the polymers.
where e m (t) is the total strain, E eq and E i are the elastic moduli in the equilibrium where α T (T ) is the time-temperature superposition (TTSP) shifting factor and 0 i τ is the relaxation time at the reference temperature when ( ) 1 T T α = . At temperatures around or above a reference temperature T s , the WLF equation is applied, where C 1 and C 2 are material constants and T M is the WLF reference temperature.
When the temperature is below T s , α T (T ) follows the Arrhenius-type behavior: A finite deformation constitutive model can be developed based on the above linear viscoelastic model 5 . Eq. S1-S4 can be implemented into a Matlab code.

S2. Material parameters
All the material parameters are obtained for the seven digital materials tested in this work.. The method to calibrate corresponding material parameters for all equilibrium and non-equilibrium branches can be found in our previous work 1 Poly.3 Poly.4

S3. Scaling rule between bending deformation and linear strain deformation
To demonstrate the scaling relation between bending deformation and linear strain deformation, we conduct FEA simulation of bending model and compare with linear strain model. In addtion, bending deformaiton can be considered as combination of linear deformation of each minor layer along thickness (as shown in Figure S3b). When considering there is no deformation of the central layer in bending process, the average linear strain change can be simplifilly defined as Here where is the thickness of the sample. The above angluar change rate relationship is applied in the following bending model. All the material parameters for the bending model are the same with that of the linear strain model. Linear strain model: In the programming step, the SMP is stretched to a target strain e max (20%) with a constant loading rate ! e (0.01s -1 ) at the programming temperature T d followed by a specified holding time at T d before being cooled to the shape-fixing temperature T L (10 o C ) at the rate of (2.5 o C min -1 ). Once T L is reached, the specimen is held for 1 hour then the tensile force is removed. Finally, the temperature is increased to the recovery temperature at the same rate of cooling and subsequently stabilized for another 50 mins. In the programming step, stress can obtained directly from Eq. S1 when the displacement controlled is used (as in our case); in the recovering step, the total strain ( ) e t is obtained by solving Eq. S1 with a zero stress on the left-hand-side. Then the recovery ratio is obtained as where r e is the strain measured at the start of the revovery process. To have a same loading rate with that of linear strain model, the angle change rate of ! θ = 0.04 π h is applied to arrive target deformation. The following holding condition and recover conditions are set the same with that in the linear strain model. The general shape memory cycle is shown in Figure S4a. The shape fixity is defiend as max f r R θ θ = , and the shape recovery ratio is defined as Here r θ is the angle at the start of the free recovery process, and max θ is the maxiumum angle change. From the Fig. S4c, the bending deformation has a same shape fixity with that of linear strain deformation when under same mechanical and thermal conditions. This is because the shape memory behavior is mainly affected by loading strain rate rather than the final deformation magnitude if the deformation occur in the rubbery region. Fig. S4d shows that the recovery behaviors are the same for linear strain and bending as long as they have the same fixity. It can be concluded bending deformation has a same shape memory behavior with that of linear strain case, when the same loading strain rate as well as other mechanical and thermal conditions are applied.