Spatiotemporally Programmable Surfaces via Viscoelastic Shell Snapping

Many species can dynamically alter their skin textures to enhance their motility and survivability. Despite the enormous efforts on designing bio‐inspired materials with tunable surface textures, developing spatiotemporally programmable and reconfigurable textural morphing without complex control remains challenging. Herein, a design strategy is proposed to achieve surfaces with such properties. The surfaces comprise an array of unit cells with broadly tailored temporal responses. By arranging the unit cells differently, the surfaces can exhibit various spatiotemporal responses, which can be easily reconfigured by disassembling and rearranging the unit cells. Specifically, viscoelastic shells as the unit cells is adopted, which can be pneumatically actuated to a concave state, and recover the initial convex state sometime after the load is removed. It is shown computationally and experimentally that the recovery time can be widely tuned by the geometry and material viscoelasticity of the shells. By assembling such shells with different recovery times, surfaces with pre‐programmed spatiotemporal textural morphing under simple pneumatic actuation is built, and temporal evolution of patterns, such as digit numbers and emoji, and spatiotemporal control of friction are demonstrated. This work opens up new avenues in designing spatiotemporal morphing surfaces that could be employed for programming mechanical, optical, and electrical properties. A preprint version of the article can be found at: https://www.authorea.com/doi/full/10.22541/au.164020946.62560710.

DOI: 10.1002/aisy.202100270 Many species can dynamically alter their skin textures to enhance their motility and survivability. Despite the enormous efforts on designing bio-inspired materials with tunable surface textures, developing spatiotemporally programmable and reconfigurable textural morphing without complex control remains challenging. Herein, a design strategy is proposed to achieve surfaces with such properties. The surfaces comprise an array of unit cells with broadly tailored temporal responses. By arranging the unit cells differently, the surfaces can exhibit various spatiotemporal responses, which can be easily reconfigured by disassembling and rearranging the unit cells. Specifically, viscoelastic shells as the unit cells is adopted, which can be pneumatically actuated to a concave state, and recover the initial convex state sometime after the load is removed. It is shown computationally and experimentally that the recovery time can be widely tuned by the geometry and material viscoelasticity of the shells. By assembling such shells with different recovery times, surfaces with pre-programmed spatiotemporal textural morphing under simple pneumatic actuation is built, and temporal evolution of patterns, such as digit numbers and emoji, and spatiotemporal control of friction are demonstrated. This work opens up new avenues in designing spatiotemporal morphing surfaces that could be employed for programming mechanical, optical, and electrical properties. A preprint version of the article can be found at: https://www.authorea.com/doi/full/10.22541/au.164020946.62560710.
buckle into the concave state when subjected to a pressure load, and recovers the convex state after a certain amount of time when the load is removed. [29][30][31][32][33] The recovery time can be widely tuned by the geometry and viscoelasticity of the shell. [31] As a result, a surface comprising the shell units with different recovery times can exhibit preprogrammed temporal texture evolution, which can be easily reconfigured by rearranging the shell units (Figure 1c-d). The proposed surfaces are used to display the temporal evolution of patterns, such as digit numbers and emoji, and programmed spatiotemporal friction control.

Design of Viscoelastic Shells with Tunable Recovery Time
Consider viscoelastic shells of revolution with the following profile of the internal surfaces ( Figure 2a) where H is the height, d is the thickness, and R is the radius of the shell. Such geometry ensures dh=dr ¼ 0 and d 2 h=dr 2 ¼ 0 at both the center (r ¼ 0) and the edge (r ¼ R) of the shell to facilitate the implementation of the boundary conditions in both simulations and experiments. Subjected to clamped boundary conditions and a pressure load Δp, the shell can buckle into a concave shape (dashed lines), yielding a displacement w at its center. We first conducted finite element analyses (FEA) to study the viscoelastic responses of such shells upon pressure loads using the commercial package Abaqus/Standard (Method). The instantaneous constitutive behavior of the shells is modeled as an incompressible neo-Hookean material with the instantaneous shear modulus μ 0 (Supplementary Text 1). The relaxation of the shear modulus over time t, μðtÞ is described by the Prony series where n is the number of the series terms, g i is the dimensionless relaxation modulus, and τ i is the relaxation time constant. Only the first term of the Prony series is considered in the FEA with the relaxation parameter g and relaxation time constant τ.
We first investigate the instantaneous responses of the shells. The normalized pressure Δp/μ 0 increases, decreases, and increases again with the normalized displacement w/H for the shells with H/R ¼ 0.4 and different d/R ranging from 0.105 to 0.16 ( Figure 2b). We find that the Δp/μ 0 -w/H curves for thick shells with d=R ≥ 0.11 stay above the horizontal line of zero pressure, indicating that these shells are monostable. As d/R decreases below 0.11 (d/R ¼ 0.105), the Δp/μ 0 -w/H curves intersect with the horizontal line of zero pressure, indicating that the shells are bistable. The monostable shells can instantaneously recover from their concave shapes once the pressure load is removed, whereas the bistable shells can stay concave without a pressure load.
Monostable viscoelastic shells are capable of temporarily staying concave for a certain amount of time before recovering to their convex state, even though the pressure load is removed. This phenomenon is called pseudo-bistability, [30][31][32][33] and will be utilized to build surfaces with programmable spatiotemporal textural morphing. To quantify the phenomenon in FEA, we instantaneously imposed a pressure load Δp/μ 0 ¼ 0.03 on a viscoelastic shell with H/R ¼ 0.4, d/R ¼ 0.11, and g ¼ 0.2, and released the pressure after holding it for 2τ ( Figure 2c). The corresponding displacement w-time t relation was computed. As a result, the shell immediately snaps into a concave shape once the pressure is applied, and creeps with a small displacement increase w/H ¼ 0.0326 for t creep ¼ 2τ during the loading. As the pressure is removed, the shell temporarily stays concave for t rec /τ ¼ 7.1 with w/H slightly decreasing prior to snapping back to the convex state ( Figure 2c). Reproduced with permission. [3] Copyright 2013, Wiley Periodicals, Inc. b) As the unit cell of the surfaces, a viscoelastic shell can have two states: convex and concave, analogous to the extended and retracted states of the papillae of cephalopods. The convex shell can be pneumatically actuated to the concave state, and recover the initial convex state after a certain amount of time when the load is removed. c-d) By arranging the shells of different recovery times, surfaces can be formed to exhibit pre-programmed spatiotemporal textural morphing. The surfaces can be easily disassembled and rearranged to new ones with different spatiotemporal responses. A "heart" texture shown in (c) can be reconfigured into a "tilted square" texture in (d) by simply reorganizing the shell units.
Next, we investigate the effect of loading history, geometry, and viscoelasticity on the recovery time t rec /τ. As a result, for the shell with H/R ¼ 0.4 subjected to a step pressure load Δp/μ 0 ¼ 0.03, t rec /τ increases with t creep /τ, and saturates when t creep /τ ¼ 2 for a range of d/R and g (Figure 2d). For a given t creep /τ, a larger g leads to a longer t rec /τ. We also find that t rec /τ is considerably reduced when the shell becomes thicker ( Figure 2d). These trends can be seen more clearly in the contour plot of t rec /τ with respect to d/R and g when t creep /τ ¼ 2 ( Figure 2e); t rec /τ increases as g increases or d/R decreases. The growth rate of t rec /τ with respect to g is dramatically increased as d/R approaches 0.11, which is the boundary demarcating the monostable and bistable shells. The shells corresponding to the region underneath the white line in this contour have a zero recovery time, indicating that they snap back immediately once the pressure load is removed. We note that Δp/μ 0 can increase t rec /τ, but this increase is negligible compared to the effects of t creep /τ, g, and d/R ( Figure S2, Supporting Information).

Spatiotemporally Programmable Surfaces and Experimental Characterization
Having identified how the recovery time is tuned by the loading history, geometry, and viscoelasticity, we now demonstrate the spatiotemporally programmable surfaces composed of viscoelastic shells with different recovery times.
We selected two materials that have similar moduli but very different viscoelastic properties for the fabrication: silicone rubber (Dragon Skin 30, D30) and urethane rubber (VytaFlex 20, V20). We conducted stress relaxation tests for these two rubbers and fitted the two-term Prony series to the data, the results of which are given in Table 1 (Method and Supplementary Text 1). We find that the silicone rubber D30 is highly hyperelastic whereas the urethane rubber V20 is highly viscoelastic, since their moduli decay by g 1 þ g 2 ¼ 5.08% and 36.71%, respectively, as the time approaches infinity. The two terms in the Prony series for both materials have quite different time constants. Since the creeping time, we apply in the experiments is comparable to τ 1 , the pseudo-bistable behavior of the shells is also dominated by the relaxation over a time scale comparable to τ 1 . Therefore, in the following, we only consider the first term in the Prony series.  Viscoelastic shells are fabricated by molding, and are bonded onto a hollow substrate made of polylactic acid (PLA), forming a shell unit, as shown in Figure 3a. To diminish the influence of geometric imperfections on the pseudo-bistability, [30] the molds for shells were 3D printed with high resolution (0.06 mm), and a step structure was used to ensure the concentricity between the shell and the substrate (Method, Figure S3-S5, Supporting Information). We prepared the following four types of shell units: bistable shells and shells with zero, medium, and long recovery time. All the shells have the same radius R ¼ 10 mm and height H ¼ 4 mm. The thicknesses and materials for the shells of these four types are summarized in Table 2. The predicted recovery time for the shells with zero, medium, and long recovery time by FEA is marked by a circle, square, and pentagram in Figure 2e, respectively.
Spatiotemporally programmable surfaces are created by assembling the shell units with different recovery times (Figure 3b and S6, Supporting Information). The shell units are interconnected via tubing so that the pressure loads exerted on them are always the same (Figure 3b and S6, Supporting Information). A pneumatic actuation system is used to extract, hold, and release air (Figure 3b). This system contains three vacuum pumps connected in parallel, two 3-way solenoid valves, and a pressure sensor (Method and Figure S7, Supporting Information). All the shell units are connected to this system in a way that the tubing lengths from all the shell units to the pumps are equal ( Figure S6b, Supporting Information). Initially, the two solenoid valves are deactivated, and the pump can rapidly extract air from all the shell units, yielding a pressure load. Once the pressure reaches a targeted value, valve 2 is activated so that the pressure can be held. After a certain amount of creeping time t creep , valve 1 is activated to release the pressure.  As a result, the shells first buckle into the concave states at a critical pressure, and then start creeping during the pressure holding period (Figure 3c). After the pressure load is removed, different shells recover to their convex states at different predetermined recovery time t rec , yielding spatiotemporal texture morphing (Figure 3c). With appropriate choices of shell units, we can create a surface with desired spatiotemporal textural morphing. Next, we experimentally characterize the recovery time t rec of shells on the spatiotemporally programmable surfaces. As an example, a surface is created by arranging the four types of shell units in a 4-by-3 grid pattern with each row corresponding to one type of shell unit (Figure 3d and S6, Supporting Information), and is connected to the pneumatic actuation system. In each measurement, we fixed the average pressure load to be around 7 kPa during the holding process (Δp/μ 0 ¼ 0.0249 for D30, Δp/μ 0 ¼ 0.0407 for V20) ( Figure S8, Supporting Information), while changing the creeping time t creep . We measured the average t rec in five trials for t creep ¼ 1, 5, 10 s using a camera (Method). Using the middle shells in each row as examples, we find that t rec for Row 2 keeps nearly zero despite the increase in t creep , whereas t rec for Rows 3 and 4 increase with an increasing t creep (blue solid lines in Figure 3e). Figure 3f summarizes the average t rec for Rows 2, 3, and 4 when the surface is actuated for t creep ¼ 10 s (Movies S1, Supporting Information), and compares the results with the corresponding numerical predictions. The experimental results show that the shells in Rows 2, 3, and 4 have almost zero, medium, and long t rec , respectively, which agrees with the predictions from FEA. From Movie S1, Supporting Information, we can clearly see that the shells in row 1 stay concave, whereas the shells in Rows 2, 3, and 4 snapback sequentially after the pressure is released. To check whether t rec of a shell in a surface is the same as that when it is actuated individually, we actuated the middle shells in Rows 2-4 individually, and measured t rec for different t creep (red dashed lines in Figure 3e). We find that the t rec -t creep curves only slightly shift down for the shells actuated individually. This is caused by the shorter time for the air to flow into a single shell unit than that into the surface after the pressure is released, given that the surface has a chamber volume 12 times larger than a shell unit.

Surfaces Exhibiting Programmable Temporal Evolution of Patterns
So far, we have identified via simulations and experiments four types of shell units: bistable shells and the shells with zero, medium, and long recovery time t rec . Next, we use these shell units as building blocks to create surfaces, which can display programmed patterns that evolve over time. To make the shells in the convex and concave states look more different, a piece of black thin paper with a hole is placed on the top of the shells ( Figure S5, Supporting Information). This cover is so flexible that it does not affect the snap motion of the shell. We first assembled 15 shell units into a 5-by-3 surface (Figure 4a), in which 2 shell units are bistable, 11 shell units have zero t rec , 1 shell unit has medium t rec , 1 shell unit has long t rec . We instantaneously applied an average pressure around 7 kPa, held this pressure for t creep ¼ 10 s, and quickly released the pressure (similar to Figure 3d). At t ¼ 0.33 s after the pressure is released, the shells with a zero t rec snap from the concave state to the convex state, showing a digit number "5". At t ¼ 2.57 s, the shell with a medium t rec recovers, showing a digit number "6". At t ¼ 9.33 s, the shell with a long t rec recovers, showing a digit number "8" (Figure 4a and Movies S2, Supporting Information). The surface can be easily reconfigured into a 5-by-5 surface (Figure 4b), in which 11 shell units are bistable, 9 shell units have zero t rec , 2 shell units have medium t rec , 3 shell units have long t rec . We used the same pressure load to actuate this new surface, and observed sequentially a smiley emoji at 0.30 s, a winking emoji at 2.97 s, and an astonished emoji at 11.03 s (Figure 4b and Movies S3, Supporting Information).

Surfaces with Programmable Spatiotemporal Friction
Besides displaying temporally programmable patterns, the surface can also exhibit spatiotemporal control of friction. When a single shell unit is in contact with a flat rigid plate, the friction at the interface depends on the convexity state of the shell. If the shell is concave, the plate is in contact with the PLA substrate, yielding a low friction. Otherwise, if the shell is convex, the friction is relatively high since the plate is in contact with the rubber (Figure 5b). Therefore, a surface comprising multiple shell units is capable of spatiotemporally tuning its friction by switching the states of the shell units.
We assembled a 4-by-3 surface composed of the four types of shell units. An acrylic board together with a weight (total weight of 0.424 kg) was placed on this surface, and pulled forward at a constant velocity of 2 mm s À1 using an Instron testing machine (Model 5944) (Figure 5a, S9, Supporting Information). The effective frictional coefficient μ is given by the pulling force F divided by the normal force (4.16 N) exerted on the surface. While the board is pulled forward, the surface is subjected to a pressure load history in which an average pressure around 7 kPa is applied instantaneously, then held for t creep ¼ 10 s, and released (similar to that in Figure 3d). Accordingly, μ is high before the pressure load is applied when all the shell units are in the convex state. Then μ drops sharply and maintains low for t creep ¼ 10 s, since all the shells become concave due to the pressure load ( Figure 5c). Depending on the types of shell units on the surface, μ evolves quite differently over time after the pressure is released. When all the shell units have zero recovery time t rec , they snap immediately after the pressure is released. Correspondingly, μ quickly recovers the value before the pressure load is applied (black line in Figure 5c). When all the shell units are bistable, they stay concave even though the pressure is released. Thus, μ remains low (blue line in Figure 5c). When the surface is a mixture of the four types of shell units, where each row corresponds to one type ( Figure S9a, Supporting Information), μ exhibits a multistep function of time after the pressure is released (red line in Figure 5c), in which the first, second, and third steps correspond to the recovery of the shell units with zero, medium, and long t rec , respectively. Since the bistable shell units stay concave, the number of shell units in contact with the acrylic board is less than that before the pressure load is applied, and thus μ does not fully recover. We also find that the first step has a larger increase in μ than the other two steps, because the 3 www.advancedsciencenews.com www.advintellsyst.com  www.advancedsciencenews.com www.advintellsyst.com shell units snapping back first bear the whole weight, leading to a larger contact area and thus a higher friction at the interface.

Conclusion
In summary, we proposed a new design strategy for reconfigurable surfaces that can exhibit spatiotemporally programmable textural morphing with simple control. The surfaces were created by assembling an array of unit cells with tunable temporal responses, and can be reconfigured on demand by disassembling and reorganizing the unit cells. Here, we adopted viscoelastic shells as the model unit cells, which can be pneumatically actuated to a concave state, and snapped back to the convex state after a certain amount of time when the load is removed. Combining numerical simulations and experiments, we found that the recovery time of those shells can be broadly tuned by the geometry and viscoelastic property of the materials. Using the shell units with different recovery time, we created surfaces that can display preprogrammed temporal evolution of patterns, such as digit numbers and emoji. We also demonstrated that the surfaces can exhibit spatiotemporal evolution of friction. The proposed spatiotemporally programmable surfaces opened the door to a wide range of potential applications. For example, the surfaces could be employed as next-generation intelligent reflective roofs. [34] The roof made of the surfaces can change its solar reflectance over time by tuning its surface texture such that it reflects less light in the morning and evening, and more at noon. Besides, the surfaces could be programmed to have spatiotemporal wetting and adhesion properties. [35] The surfaces could also be used to achieve spatiotemporally programmable electrical or thermal conductivity at interfaces by manipulating contact.

Experimental Section
Finite Element Simulations: To investigate the behaviors of our viscoelastic shells in response to the pressure loads, we performed finite element analysis using the commercial software Abaqus/Standard. We first conducted the simulations for instantaneous loading. The incompressible Neo-Hookean material was used to define the hyperelastic behavior of the shells. Riks method was implemented to capture the complete equilibrium pressure-displacement responses. We then conducted simulations for time-dependent loading using the dynamic implicit method. The single-term Prony series was used to describe the viscous behavior of the shells. Numerical damping with moderate dissipation was applied to reduce the noise to the solution caused by the rapid snap motion. In all the simulations, we constructed axisymmetric shell models and meshed them using hybrid quadratic rectangular elements (Abaqus type CAX8H). We imposed a fixed boundary condition on the edge of the shells.
Stress Relaxation Tests: Stress relaxation tests were performed to determine the viscoelastic properties of the silicone rubber (Dragon Skin 30) and urethane rubber (VytaFlex 20). A thin-film specimen of length 80 mm, width 20 mm, and thickness 2 mm was fabricated by molding for each material, with 1% by weight of Ignite orange pigment. An Instron testing machine (Model 5944) equipped with a 50 N loading cells was used for the tests. In the relaxation tests, a 20% tensile strain was applied within 0.5 s and maintained for 300 s. The relations between the resultant forces and time were recorded. A two-term Prony series was fitted to the experimental data using the least-square approach to determine the viscoelastic properties of the materials. We find that the two-term Prony series is sufficient for an accurate fit (the root-mean-square error is less than 0.6%). More details on curve fitting are provided in Supplementary Text 1.

Fabrication of Viscoelastic Shell Units:
The shells were made of silicone rubber (Dragon Skin 30) and urethane rubber (VytaFlex 20), with 1% by weight of ignite orange pigment. The materials were cast into PLA two-part molds, which were printed by an Ultimaker S5 printer with a resolution (layer height) of 0.06 mm. The inner surfaces of the molds for urethane shells were coated with Universal Mold Release to facilitate demolding. All the shells have a radius R ¼ 10 mm and height H ¼ 4 mm. Their thickness d varied from 1.05 to 1.25 mm. The shells at their edges had a flange of width 3 mm. The change in volume of a shell from its concave state to convex state is 528.66 mm 3 when d ¼ 1.05 and 492.82 mm 3 when d ¼ 1.25 mm. After the shells are made, they were glued onto a hollow substrate by applying super glues onto the flanges of the shells. The substrate was made of PLA and 3D printed in the same way as the molds. The chamber in the substrate is a cylindrical void of radius 10 mm and height of 34 mm, yielding a volume of 10 681 mm 3 . The change in volume caused by the snap motion is less than 5% of the volume of the chamber, and thus, the snapping of one shell does not influence the recovery of others. More details on the fabrication of the shell units are provided in Supporting Information.
Pneumatic Actuation System: A pneumatic actuation system was built for extracting air from the shell units, and holding and releasing the pressure. This system consists of three vacuum pumps connected in parallel (2.1 L min À1 ), two 3-way miniature solenoid valves, an Arduino microcontroller, a power supply, and a pressure sensor (Panasonic, ADP5101) ( Figure S7, Supporting Information). A camera (Sony Alpha a6000, 60 fps) was used to record the snap motion of the shells. When the pressure is released, a 5 mm light-emitting diode red lamp is turned on to start timing.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.