Optimal number of faces for fast self-folding kirigami

There is an increasing body of research studying how to obtain 3D structures at the microscale from the spontaneous folding of planar templates, using thermal fluctuations as the driving force. Here, combining numerical simulations and analytical calculations, we show that the total folding time of a regular pyramid is a non-monotonic function of the number of faces (N), with a minimum for five faces. The motion of each face is consistent with a Brownian process and folding occurs through a sequence of irreversible binding events between faces. The first one is well-described by a first-passage process in 2D, with a characteristic time that decays with N. By contrast, the subsequent binding events are first-passage processes in 1D and the time of the last one grows logarithmically with N. It is the interplay between these two different sets of events that explains the non-monotonic behavior. Implications in the self-folding of more complex structures are discussed. Inspired by the Japanese art of Kirigami, microscopic self-folding structures are gaining interest due to the possible implementation of controlled drug encapsulation and release driven by thermal fluctuations. Here, the authors show that folding time scales can be accurately predicted by mapping the dynamics into a set of competing Brownian processes.

Kirigami is the art of cutting two-dimensional templates and fold them into three-dimensional structures. Nowadays, there is a growing interest on extending this ancient idea to design materials that fold spontaneously into targeted 3D structures. The driving mechanism depends on the lengthscale. At the macroscale, folding is driven by energy minimization (e.g. stress relaxation), and thus the folding pathway is deterministic [1][2][3][4][5][6][7][8][9][10][11][12]. By contrast, at the microscale, since folding occurs usually in suspension, the fluctuations in the fluid-structure interaction dominate and folding is stochastic [13,14]. This challenges the use of Kirigami at the microscale as, for example, in encapsulation, drug delivery, and soft robotics [15][16][17][18].
To design self-folding Kirigami, one first needs to identify what are the two-dimensional templates (nets) that fold into the desired structure. For shell-like structures of rigid panels connected by edges, these nets are obtained by edge unfolding, i.e., by cutting edges and opening the structure [19]. In principle, different nets can fold into the same three-dimensional structure. However, recent experiments and numerical simulations show that the stochastic nature of folding might lead to misfolding. By performing independent samples, they found that the probability for a given net to fold into the desired structure (yield) strongly depends on the topology of the net and experimental conditions [13,14,20,21]. Thus, the focus has been on identifying what are the optimal nets that maximize the yield [13,21]. But, what about the folding time? For practical applications, it is not only critical to reduce misfolding but also to guarantee that folding occurs in due time. Here, we address this question. To focus on the folding time, we consider as a prototype the spontaneous folding of a pyramid, where misfolding is not possible.
Let us consider a pyramid with N lateral faces (see Fig. 1). The 2D net is a N -pointed star, obtained by cutting the edges of the lateral faces and unfolding them. To simulate the folding dynamics, as explained in detail in . The 2D template of microscopic panels (center) is obtained by cutting the edges between the lateral faces and unfolding the faces. To simulate the folding dynamics, we developed a coarse-grained numerical model where each face is described as a rigid body of three particles (right) at the vertices. The base is described by N particles at the vertices. The interaction between particles is considered pairwise and attractive. To suppress misfolding, the base is pinned to a flat substrate and the lateral faces can only fold in one side.
the Supplemental Material [22] and summarized in Fig. 1, we performed particle-based simulations. We are interested in the limit where the interaction between faces is short-ranged (contact like) and the edge closing irreversible. Thus, each face is described as a rigid body of three particles at the vertices. The attractive interaction along the edges is modeled by a strong inverted-Gaussian potential between particles. so the faces can only fold in one side (see Supplemental Material [22] for further details). We performed independent simulations for different numbers of lateral faces N , starting from a flat (2D) configuration and running until the final pyramid is obtained. As shown in Fig. 2, we find that, the total folding time T is a non-monotonic function of the number of faces N , with an optimal time for five faces. To characterize the dynamics, we define θ i as the angle between the face i and the substrate (see scheme in the top of Fig. 3). Since the motion is constrained by the substrate, θ i ∈ [0, π].
As an example, we consider now the folding of a pyramid of three lateral faces (N = 3). The time dependence of the three angles is shown in Fig. 3(a). Due to thermal fluctuation, each face jiggles around until the first two faces (A and B in the figure) meet at time t 1 st and bind irreversibly, closing the edge between them. The third face (C) also binds to the first two at a later time t 2 nd . Thus, folding occurs through a sequence of irreversible edge closings. Below, we discuss the first and subsequent edge closings independently.
As shown in the Supplemental Material [22], the statistics of the three time series θ i (t) in Fig. 3 is consistent with a 1D Brownian process with reflective boundaries at θ i = 0 and θ i = π. The short-ranged (attractive) interaction between faces is only effective in a small region of the angular space, θ * = 3π/4 ± ∆, with ∆ ≈ π/180 as estimated from the properties of the potential (see Supplemental Material [22]). For the first edge closing to occur, the angle of two faces need to be at θ * at the same time and, once there, they get trapped. Thus, if we map the motion of each pair of faces j and k into a 2D Brownian process, with coordinates (θ j , θ k ) and a trap at (θ * , θ * ), the edge closing between j and k occurs when the corresponding 2D Brownian process hits the trap (see Fig. 3(b)). In the general case of N lateral faces, since there are N (N − 1)/2 possible pairs of faces, the time of the first edge closing is the fastest of N (N − 1)/2 firstpassage processes.
To estimate the average time T F of the first edge closing for a pyramid of N lateral faces, we define g(t) as the first-passage time distribution of a 2D Brownian process. There are N (N − 1)/2 pairs of faces and so the same number of competing Brownian processes. The first edge closing is the fastest of all possible ones and thus T F = min{t 1 , t 2 , t 3 ...t N (N −1)/2 }, where t i are random values following the distribution g(t). If we neglect any correlations between the motion of the different faces, from the theory of order statistics [23], we estimate that, where the term with the square brackets corresponds to the probability that, provided that a first-passage process occurs at time t, all the remaining N (N − 1)/2 − 1 occur at a later time. g(t) depends on the geometry and initial conditions [24][25][26][27][28]. For a set of N (N − 1)/2 Brownian processes [25,29] starting at the origin (θ i (0) = 0), The dynamics of the subsequent edge closings is fundamentally different. While for the first edge closing, two faces need to meet at a particular angular θ * , the remaining faces will close edges one-by-one as soon as they reach θ * . The folding is complete when all faces reach this value. Thus, each of the subsequent (N − 2) edge closings is a 1D first-passage process (see Fig. 3(c)). We define T as the total folding time and T L = T − T F as the time from the first to the last edge closing. Each free face i binds when θ i (T F + t) = θ * (with t ≥ 0) for the first time. To estimate T L , we assume that θ i (T F ) < θ * for all i and that θ i (T F + t) is well described by a 1D Brownian process, with one reflective boundary at θ i = 0 and a trap at θ * . T L is then the slowest of the (N − 2) 0 π/2 π 0 0.5 1D first-passage processes. Thus, (3) where f (t) is the 1D first-passage time distribution and the term with square brackets is the probability that, provided that a first-passage process occurs at time t, all the remaining ones were faster. Assuming that θ i (T F ) is uniformly distributed in [0, θ * ], the first-passage time distribution is f (t) ≈ e −t/τ L , with τ L = 4θ * 2 /D 0 π 2 [30], where D 0 is the diffusion coefficient of the Brownian process. This gives, and thus, T L (N ) ≈ τ L [ln(N − 2) + γ], where γ is the Euler-Mascheroni constant. Figure 4(b) depicts T L obtained numerically for different N . The numerical data is consistent with Eq. (4) (solid line). The dependence on the number of lateral faces N of T F and T L is significantly different. While T F decreases with N , T L grows. The total folding time T is the sum of the two. Thus, for low values of N , the total folding time is dominated by the time of the first edge closing, whereas for large N is the last closing that sets the overall timescale. It is the interplay between these two timescales that explains the minimum observed in Fig. 2.
So far, we considered always the same closing angle θ * and diffusion coefficient D 0 . Since the motion of the faces is diffusive, all timescales should scale with τ = θ * 2 /2D 0 , which is the average time for a non-interacting 1D Brownian process to diffuse in an angular region of size θ * . Conclusions. Under thermal fluctuation, a N -pointed star template of rigid panels and flexible hinges folds into a 3D pyramid of N lateral faces. Folding occurs through a sequence of edge closings, but the nature of the first and subsequent edge closings is significantly different. For the first edge closing, two jiggling faces need to meet at a particular angle, whereas for the subsequent edge closings, only one face needs to reach that angle. We hypothesized that the first edge closing can be mapped into a firstpassage event of a 2D Brownian process [26,27,31,32], obtaining an expression for the corresponding time. This expression predicts that the time for the first edge clos- ing decreases with N , what describes quantitatively the numerical data. By contrast, to estimate the time for the subsequent edge closing, we mapped them into a set of first-passage events in 1D and derived the time for the slowest of them all. We predict that this time should rather grow logarithmically with (N − 2), which is also observed numerically. Since the total folding time is the sum of the two times, a non-monotonic dependence on N is found.
Spontaneous folding at the microscale is an intricate process that might depend on the physical properties of the structure, fluid-structure interactions, and thermostat temperature [13,14,21]. Nevertheless, our approach shows that, by mapping folding into a set of competing Brownian processes and binding events, one can predict accurately the relevant time scales. For simplicity, we considered a pyramid, a structure with equivalent folding panels. In general, the template for a given polyhedral structures has different types of panels. They differ not only in shape and size, but also in their position relative to the panel of reference (e.g. base). To extend our framework to those structures, it is critical to consider that folding evolve through a hierarchy of edge closing events that depend on the kinetic pathway of folding.