Reconfiguration of Multistable 3D Ferromagnetic Mesostructures Guided by Energy Landscape Surveys

Reconfigurable three-dimensional (3D) structures that can reversibly change their geometries and thereby their functionalities are promising for a wide range of applications. Despite intensive studies, the lack of fundamental understanding of the highly nonlinear multistable states existing in these structures has significantly hindered the development of reconfigurable systems that can realize rapid, well-controlled shape change. Herein we present a systematic, integrated experimental and computational study to control and tailor the multistable states of 3D structures and their reconfiguration paths. Our energy landscape analysis using a discrete shell model and minimum energy pathway methods leads to design maps for a controlled number of stable states by varying geometry and material parameters, and energy-efficient reconfiguration paths among the multistable states. Concurrently, our experiments show that 3D structures assembled from ferromagnetic composite thin films of diverse geometries can be rapidly reconfigured among their multistable states, with the number of stable states and reconfigurable paths in excellent agreement with computational predictions. In addition, we demonstrate a wide breadth of applications including reconfigurable 3D light emitting systems, remotely- controlled release of particles/drugs from a reconfigurable structure, and 3D structure arrays that can form desired patterns following the written path of a magnetic “pen”. Our results represent a critical step towards the rational design and development of well-controlled, rapidly and remotely reconfigurable structures for many applications.


Abstract
Reconfigurable three-dimensional (3D) structures that can reversibly change their geometries and thereby their functionalities are promising for a wide range of applications. Despite intensive studies, the lack of fundamental understanding of the highly nonlinear multistable states existing in these structures has significantly hindered the development of reconfigurable systems that can realize rapid, well-controlled shape change. Herein we present a systematic, integrated experimental and computational study to control and tailor the multistable states of 3D structures and their reconfiguration paths. Our energy landscape analysis using a discrete shell model and minimum energy pathway methods leads to design maps for a controlled number of stable states by varying geometry and material parameters, and energy-efficient reconfiguration paths among the multistable states. Concurrently, our experiments show that 3D structures assembled from ferromagnetic composite thin films of diverse geometries can be rapidly reconfigured among their multistable states, with the number of stable states and reconfigurable paths in excellent agreement with computational predictions. In addition, we demonstrate a wide breadth of applications including reconfigurable 3D light emitting systems, remotely-controlled release of particles/drugs from a reconfigurable structure, and 3D structure arrays that can form desired patterns following the written path of a magnetic "pen". Our results represent a critical step towards the rational design and development of well-controlled, rapidly and remotely reconfigurable structures for many applications.
Despite intensive studies, realizing robust and efficient architectural reconfiguration without the need for persistent external stimuli, especially for asymmetric and complicated modes, remains a challenge. Reconfigurability based on the multistability of structures can potentially realize shape changes without the requirement of persistent external stimuli [53][54][55][56] .
However, there is still a lack of fundamental understanding of the principles that control the multistable states, where each stable state corresponds to a local minimum point in a highdimensional and complex energy landscape. One challenge is associated with the existence of a large number of local minimum configurations, which could affect the stability of the targeted stable state under perturbations (e.g., environmental noises) or trap the structure in an intermediate state during the process of reconfiguration. For example, it is recently recognized by researchers that hidden local minimum configurations can destroy the designed pathway of deploying origami structures 28,30,57 . Thus, it is crucial to be able to manipulate the energy well depth of targeted configurations and eliminate unfavorable local minima in the design of reconfigurable structures. Another challenge stems from the complicity of the transition paths from one local minimum state to another, especially when the two states are separated by other stable and transition states [58][59][60][61][62] . A recent work 44 shows that the transition from a uniformly deformed cylindrical shell to a 9-dimple buckled pattern needs to pass 7 local minimum configurations, which cannot be achieved by directly applying local probes to the shell.
Complicated transition paths among local minima are also found in mechanical metamaterials, which further provides new design parameters to program the deformed configurations of the structures 61 . In addition, it is also very challenging to realize on-demand, locally controlled, and rapid reconfiguration of the deformed, highly nonlinear structures 63,64 . These challenges call for a new means of systematically surveying the energy landscapes of the multistable structures to probe and tailor the energy barrier height among different local minima.
In this work, we present a set of strategies and design concepts to address these challenges. 3D structures are assembled from ferromagnetic composite thin films using schemes that reply on strain release from biaxially prestrained elastomer platforms 38,40 . We show that the assembled 3D structures can be remotely and rapidly deformed into multiple distinct states under external magnetic forces and maintain the deformed shapes even after the magnetic field is removed. To identify the spectrum of existing multistable states, we conduct fundamental studies of the multistablity of the 3D structures through exploring their energy landscapes using a discrete shell model 65,66 . We develop phase diagrams showing how the available stable states sensitively depend on the material and geometrical parameters of the elastic structures, which are validated by our experiments. Reconfiguration paths between the stable states are then identified using a combination of the string 67 and binary image transition state search methods and realized by magnetically reconfiguring 3D structures assembled from ferromagnetic composite thin films.
A wide range of geometries, including ribbons and structures that resemble tables, baskets, flowers, boxes, single and double beams, and their multistable states are demonstrated in both simulations and experiments. In addition, 3D reconfigurable light emitting systems illustrate the capacity to integrate other materials and functional components into these 3D reconfigurable structures. Magnetically actuated particle delivery in different modes of a table structure highlights applications in reconfigurable 3D systems of potential relevance in biomedical devices. Furthermore, a structure array of reconfigurable 3D ferromagnetic structures, which can display well-controlled patterns formed by reconfiguring the structures following the written path of a magnetic "pen", serves as an additional example in reconfigurable systems. Figure 1a schematically illustrates the assembly process of a 3D structure from ferromagnetic composite thin films via compressive buckling, followed by architectural reconfiguration via magnetic actuation. The ferromagnetic composite is prepared by homogeneously embedding hard NdFeB (neodymium-iron-boron) microparticles with an average diameter of 5 μm into a soft elastomer, polydimethylsiloxane (PDMS) (Figure S1, Supporting Information). NdFeB is chosen for our study because its high residual magnetic flux density over a wide range of applied magnetic fields allows complex modes of reconfiguration 68 .

Stability of multistable states of 3D buckled structures
The assembly of 3D ferromagnetic structure starts with patterning the ferromagnetic composite thin films (125 μm -150 μm thick) into 2D layouts using a CO2 laser (VLS 3.50, University Laser System, Norman, OK), followed by their axial magnetization using impulse magnetic fields (about 2.7 T) generated by an impulse magnetizer. The 2D patterns are then laminated onto a prestretched elastomer (Dragon skin, Smooth-on, Chicago, IL), with a very thin layer of superglue applied at selective locations (bonding sites) of the 2D patterns to generate strong bonding between the pattern and the elastomer substrate. The interfacial interactions at all other locations are dominated by comparatively weak van der Waals forces. Releasing the prestrain in the substrate leads to compressive forces on the 2D ferromagnetic pattern at the bonding sites and transforms the pattern into a 3D structure through controlled compressive buckling.
Throughout the paper, we adopt a local coordinate system for the 3D structure such that the 2D layout is always on the x-y plane with its normal vector as the z-direction. The assembled 3D structure can have multiple stable shapes, and the reconfiguration among them is achieved remotely and rapidly by leveraging external magnetic forces applied to the ferromagnetic films.
Due to structural stability, the deformed shape can be well maintained after the magnetic force is removed. ferromagnetic table structure (film thickness: 150 µm) and its multiple stable states. This table structure is oriented such that gravity is in the negative z-direction, perpendicular to the substrate, which is the case for most of the structures in this study unless specified otherwise.
To efficiently explore the possible stable states existing in the table structure, we use the discrete shell model to locate the stable configurations, where the energy of the buckled structures is locally minimized. Three distinct states (state 1-3) with a total of nine configurations are discovered. By magnetically deforming the assembled table structure using a portable disk magnet (Neodymium magnet, K&J Magnetics), in a way that one leg becomes flat and in contact with the substrate while the other three legs are in a buckled status, the table structure (state 1) can be reconfigured into distinct stable configurations with four different orientations (state 2: shape I-IV). Unlike previously reported reconfigurable 3D structures that rely on persistent external stimuli to maintain their deformed shape 39,48,49,[68][69][70] , the deformed configurations of the table structure shown here can well maintain their shapes after the applied magnetic field is removed, due to structural stability (will be discussed in detail later).
Magnetically deforming the structure with an additional leg of the table structure becoming flat and in contact with the substrate results in another stable state with four different orientations (state 3: shape I-IV). The discrete shell model (see description in the Methods section) captures all the stable states and their different orientations in experiments, with the color in the results denoting the displacement along the z direction in the structure.
Using the same experimental and computational strategy, we further study the effect of geometry and material properties on the entire spectrum of stable states, resulting in the phase diagram shown in Figure 1c. It illustrates the number of stable states according to two dimensionless parameters that correspond to the ratios between three relevant energy scalesgravitational ( ), stretching ( ), and bending ( ). These energy scales can be expressed in terms of the material and geometric parameters of the 3D structure: where is the material density, is the Young's modulus, is the Poisson ratio, is the film thickness, and represents the in-plane size of the structure (the width of legs is used for this study). The cubic term of L in comes from the fact that the volume of the structure scales with L 2 and the displacement in the direction of gravity is proportional to L. Three regimes can be identified in

Transition paths among distinct stable states of 3D buckled structures
Identifying the optimal transition pathways between stable states of a 3D structure is critical for well-controlled, energy-efficient reconfiguration. We further examine the energy landscape of the table structure to identify the so-called minimum energy pathways, corresponding to steepest-descent pathways between metastable states in which the maximum example, the energy barrier to progress from state 3 to state 1 is much smaller than that from the inverse direction, so state 1 is more stable. Comparing paths 1 and 2 also suggests that a smaller input energy is needed when reconfiguring the table structure along path 2 (passing two saddle points) than that along path 1. However, the finite energy barrier ∆ 3 also indicates that the transition can be potentially trapped in state 2 if insufficient external energy is provided for reconfiguration. In addition, reconfiguration along path 2 requires more variance in the direction and the strength of external magnetic forces, compared to the direct pathway following path 1. These complexities clearly show the importance of harnessing the energy landscape analysis for guiding the choice of transition paths based on specific applications and the nature of available external magnetic fields (or other external stimuli). In addition, it is also worth noting that the same pathways are observed in experiments when an external magnetic field is introduced to reconfigure the structure from state 3 to state 1, which indicates the high fidelity and reliability of the approach used in this study. Moreover, the reconfigurations in our experiment can be completed within a few seconds (Supplementary Movies 1 and 2). Such fast and remotely controlled transitions are desired for numerous applications, such as in soft robotics [13][14][15]71 and multifunctional metasurfaces 72,73 .
As we decrease L, all the energy barriers become smaller monotonically, as shown in  Figure   2c. However, we use the superscript * in path 2* in Figure 2d to reflect that state 2 is no longer stable, unlike for path 2 in Figure 2c. Interestingly, we also find that there is a crossover in the

3D multistable structures of diverse configurations
The assembly strategy and the energy landscape analysis for multistability described above are versatile and can be extended to other geometries and types of structures, as illustrated in Figure 3 and Figure S2. Figure 3a shows an assembled ferromagnetic basket structure (shape I; film thickness: 125 µm) and its multiple stable states (shapes II-IV) that can be achieved via magnetic control. In addition, by introducing creases to selective locations of a ribbon structure (crease thickness: 70 µm; non-crease thickness: 125 µm), we can create an origami structure, which can be further tuned to display multiple stable configurations by using magnetic forces.
The elastomer substrate used for 3D assembly in Figure 1, Figure 2 and Figure 3a provides essential support for the assembled 3D structures, but it also poses some limitations for the reconfiguration in the out of plane direction. To allow more freedom for spatial reconfiguration, the substrate underneath the 3D table structure is removed, while the substrate adjacent to the bonding sites is maintained to support the 3D structure (see Figure S3). Such 3D structure with hollow substrates allows bending up/down deformations across the plane of the substrate. Figure 3b and Figure S2 demonstrate a rich library of reconfigurable 3D structures on hollow substrates, including basket, single-and double-beam structures, as well as their distinct, multi-stable shapes.
It is very interesting to note that the multi-stable configurations on hollow substates are sensitive to the orientation of the structures. Take the table structure for example, it is shown that the structure only displays two stable states, buckled up and down states, when placed horizontally (i.e., the direction of gravity is perpendicular to the in-plane direction of the substrate, Figure S2d). However, when the table structure is placed vertically with the direction of gravity in the x-direction (Figure 3b), in addition to the buckled up (Shape I) and down (Shape IV) states, it can deform into an asymmetric, twisted shape (Shape II), similar to state 3 of the table structure with an intact substrate. This deformed shape can be further magnetically deformed into shape III, with the four bonding sites attached to the substrate and the rest of the structure self-supported in the air. We also carry out energy landscape analysis for this structure, the details of which are presented in Figure S4. Similar to the case with an intact substrate, we can distinguish several regimes with different numbers of stable states, depending on the three relevant energy scales: gravitational, stretching, and bending energies. Furthermore, the dimension of the structure on intact and hollow substrates can be further reduced using CO2 laser (Figure S5), and the structures are shown to maintain their shapes after the applied magnetic field is removed. Structures at an even smaller scale can be achieved using more advanced microfabrication techniques. However, miniaturization of these structures is not the focus of this study and will be pursued elsewhere.
In addition to tuning the strength and the direction of external magnetic forces for reconfiguration, an alternative and potentially more versatile means to realize magnetically controlled reconfiguration is through tuning the distribution of magnetically active materials within hybrid 3D structures. Figure 3c shows reconfigurable hybrid 3D table and flower structures by locally integrating ferromagnetic composite film patterns (125 µm thick) onto a PDMS layer (180 µm thick) (Method I, Figure S6). Both hybrid 3D structures are magnetically tuned to display five stable configurations using a portable magnet and maintain them after the magnetic force is removed, the capabilities of which are similar to those made of pure ferromagnetic materials. Hybrid 3D structures at a smaller scale can be achieved by directly patterning a bilayer or multilayer 2D structure using the raster mode of a laser or reactive ion etching (Method II in Figure S6) to remove ferromagnetic films at undesired regions. In addition, hybrid 3D structures can potentially allow more local magnetic control by varying the direction and the strength of the residual magnetic field in each ferromagnetic pattern through magnetization in different angles and mixing different concentrations of magnetic particles in the composite film, respectively.

Applications of 3D multistable structures
Integrating functional materials and components onto 2D ferromagnetic composite patterns provides access to 3D tunable, functional devices. As an example, we demonstrate reconfigurable light emitting systems in Figure 4a. connected to the two poles of the LED serve as interconnects for subsequent LED activation.
Formation of 3D architectures from these 2D functional precursors follows the 3D buckling schemes introduced earlier. Magnetically tuning the assembled 3D structure to deform from shape I to II (or III) enables a firm contact between the two poles of the LED (located at one beam) with two corresponding copper films (on the substrate) that are connected to an external power supply for power input, and therefore activates the LED. Furthermore, the LED can maintain its "on" status after the magnetic force is removed due to the stability of the deformed configurations (shapes II and III). When the two beams are simultaneously deformed to form stable configuration IV using a magnet, both LEDs are turned on. This simple example suggests a broad range of possibilities in other types of electronic and optoelectronic devices.
The reconfigurable structures presented in this work also have a lot of potential applications in soft robotics, metasurfaces, and biomedical devices. Based on the magnetically actuated 3D reconfiguration strategy, we also assemble a functional structure in a table shape for the delivery of particles/drugs in a well-controlled manner. As shown in Figure 4b and To demonstrate the scalability of the fabrication process as well as more capabilities of reconfigurable structures, we assembled a 4×4 structure array of reconfigurable table structures with a unit cell size of 3×3 mm 2 . Each structure unit has the identical original shape, as shown in the magnified image (image A) in Figure 5. Using a portable magnet to tune the 3D structure array following a defined path will consecutively and rapidly reconfigure the 3D structure along the path into a deformed shape (image B) that can be maintained after the magnet passes.
Following different paths leads to varied magnetically drawn patterns, like the "V" and line patterns shown in Figure 5a and Figure S8, respectively. Also, each structure unit is allocated sufficiently far to avoid interference during magnetic control, i.e., the structure unit is actuated consecutively along the path. Figure 5b shows a 4×5 structure array of table structures on hollow substrates. The structure can be reconfigured from its "popped down" (image C) configuration to its "popped up" (image D) one following the path of the magnet to display desired patterns like letter "M" (Supplementary Movie 7).

Discussion
To conclude, we have tightly integrated experiments and modelling to control and tailor multi-stable states existing in highly nonlinear 3D ferromagnetic structures as well as energy- Fabrication of a robot for particle release. A thin layer of a pure PDMS film (thickness: 60 µm) was laser cut and attached to the top of the table structure to form a semi-spherical shape for holding particles, as can be seen in Figure S7.
Fabrication of reconfigurable structure array. 3D table structures were assembled into an array with a distance of 2 cm between two table structures to ensure that magnetic manipulation of one structure unit does not affect its adjacent structures. The magnet was manually moved from one targeted structure to another following the specified trajectory including a line and an alphabet pattern to form the desired patterns.

Modeling:
Discrete shell model 65,66 . The structures are modelled as a thin elastic sheet using a 2D Delaunay triangulation. The structure energy has four contributions, corresponding to stretching, bending, gravity, and a repulsive substrate interaction. The stretching energy is given by the summation of the stretching energies of the individual bonds in the triangulation, each with a stiffness related to the Young's modulus, , and material thickness, , where and 0 are the length and relaxed length for bond . The bending energy is obtained by considering pairs of adjacent faces to be connected by elastic hinges, where the optimal dihedral angle is π. The stiffness of the hinge is again related to the material parameters, but also upon the size and shape of the two faces so that it accurately approximates the flexural rigidity of continuum elastic models. The result is where is the Poisson ratio, ℎ is the length of hinge ℎ, ℎ, is the area of each triangular face, and ℎ is the dihedral angle of the hinge. The gravitational energy is simply obtained by a summation of the gravitational energies of each of the triangular segments. The heights of which are measured relative to the center of the flat substrate in a direction dependent upon the orientation of the structure. Finally, in the cases with a substrate the repulsive interaction is modelled using a Lennard-Jones 9-3 potential that is cutoff and shifted to remove the attractive region. Consequently, the model does not take into consideration the effect of adhesion which may become significant at small scales.
In our simulations, we set the Young's modulus of the ferromagnetic film to be =  where the first two terms are the energies of the two states, and the third and fourth are constraints that are parametrized by and . The energy constraint penalizes any difference between the two states to ensure that neither state passes over the division between the basins of attraction for the two minima. The second constraint prescribes the distance between the states to be 0 , which is steadily decreased to zero so that the two states converge towards the transition state. Once a transition state for a given structure is found, the method of continuation can be exploited to locate the corresponding transition state for structures with different material properties. The pathways themselves are then found using the string method, initialized using the known minima and transition states. While the string method itself can in principle be harnessed to find the transition states, we find the BITSS method is faster and more accurate.