Optimal number of faces for fast self-folding kirigami

The field of kirigami, the art of cutting and folding 2D templates into 3D structures, is experiencing a surge in interest, particularly in the creation of materials that self-fold spontaneously. At the macroscale, deterministic folding is driven by energy minimization, such as stress relaxation. However, at the microscale, where folding often occurs in suspension, the stochastic nature of fluid-structure interactions dominates the process. This stochasticity presents a significant challenge in using kirigami at the microscale for applications such as encapsulation, drug delivery, and soft robotics. Designing effective self-folding kirigami at this scale requires identifying 2D templates (nets) that reliably and quickly fold into the desired 3D structures. While previous research has focused on optimizing nets to maximize the yield (probability of successful folding), this study addresses the equally crucial aspect of minimizing the folding time. The authors choose the self-folding of a regular pyramid as a prototype system, simplifying the analysis by eliminating the possibility of misfolding. This allows a focused investigation of the relationship between the number of faces and the folding time. By characterizing the motion of the faces as Brownian processes and the binding events as first-passage processes, the study aims to establish a quantitative understanding of the folding dynamics and identify the optimal number of faces that results in the fastest folding time.

Existing literature extensively explores self-folding techniques at both macro and micro scales. Macroscale folding often relies on deterministic mechanisms such as stress relaxation in thin films [1-12], while microscale folding is characterized by stochasticity due to thermal fluctuations and fluid-structure interactions [13, 14]. The design of self-folding kirigami has focused on identifying optimal nets that maximize the yield of successful folding [13, 21], considering the inherent challenges of stochasticity and potential for misfolding [13, 14, 20, 21]. Studies have demonstrated the use of self-folding kirigami in various applications, including encapsulation and drug delivery [15-17], and soft robotics [18]. The understanding of unfolding processes and the generation of 2D templates (nets) from 3D structures, using edge unfolding techniques [19], are also established concepts. However, the relationship between the geometric parameters of the template and the folding time has not been systematically investigated, which this research aims to address.

The study uses a combination of numerical simulations and analytical calculations to investigate the folding dynamics of regular pyramids. The simulations employ a particle-based approach where each lateral face of the pyramid is represented by a rigid body of three particles at its vertices. The base of the pyramid is also modeled as a rigid body of N particles, pinned to a substrate to constrain the folding to one side and eliminate misfolding. The interactions between particles are modeled using a strong inverted-Gaussian potential, representing short-ranged attractive forces that mimic the irreversible binding between faces observed at the microscale. The stochastic motion of the faces under thermal fluctuations is simulated by solving the Langevin equations for both translational and rotational degrees of freedom, parameterized by a rotational diffusion coefficient D₀. The Langevin equations of motion were integrated numerically using a velocity Verlet scheme within the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [34]. The simulations track the angles (θᵢ) between each face and the substrate, which are constrained to the range [0, π]. The total folding time (T) is defined as the time required for all faces to reach their final positions in the folded pyramid. The key parameters in the simulations include the rotational diffusion coefficient (D₀) and the closing angle (θ) of the pyramid. To maintain consistency across different numbers of faces (N), the height (L) of the lateral faces and the mass of the vertices were adjusted to keep θ and the moment of inertia (I) constant. An exception is Figure 4, where both D₀ and θ are varied. A repulsive Lennard-Jones potential was included between particles and the substrate to enforce the one-sided folding constraint. The analytical approach involves modeling the first binding event as a 2D first-passage process and subsequent binding events as 1D first-passage processes. Statistical methods, including the theory of order statistics [24], are applied to estimate the average time for these events.

The numerical simulations reveal a non-monotonic relationship between the total folding time (T) and the number of lateral faces (N), with a minimum folding time observed for N=5. The folding process is characterized by a sequence of irreversible binding events. The time for the first binding event (TF) is shown to decrease with increasing N, consistent with a 2D first-passage process where the probability of two faces reaching a critical angle simultaneously increases with the number of available pairs. This time dependence is quantitatively described by Equation (2) and Figure 4a, demonstrating that TF decreases with increasing N(N-1)/2, the number of possible face pairs. In contrast, the time for the subsequent binding events (TL), representing the time from the first to the last binding event, is shown to increase logarithmically with (N-2), consistent with a series of 1D first-passage processes where the slowest event determines the total time. This logarithmic growth is described by Equation (4) and Figure 4b. For large N, the total folding time T is dominated by TL. The data collapse shown in Figure 4c supports the assumption of a scaling relationship with τ = (θ*)²/2D₀ for large N. The non-monotonic behavior of the total folding time is attributed to the opposing trends of TF and TL: for small N, TF dominates, while for large N, TL dominates. The interplay between these timescales results in the minimum observed at N=5.

The findings demonstrate that the folding time of self-folding kirigami structures is not simply a monotonic function of the number of faces. The non-monotonic behavior, with an optimal number of faces for fast folding, arises from the interplay between the 2D first-passage process governing the initial binding event and the series of 1D first-passage processes governing subsequent bindings. This insight provides a valuable design principle for optimizing the folding time of self-folding kirigami structures. The model accurately predicts the observed timescales, highlighting the effectiveness of mapping the folding dynamics to Brownian first-passage processes. The study's focus on regular pyramids simplifies the analysis and allows for a clearer understanding of the fundamental relationship between geometry and folding time. The results suggest that adjusting the geometry of the template, specifically the number of faces, provides a simple and effective method for controlling the folding time, offering an advantage over altering the material properties or driving forces.

This research establishes a quantitative link between the geometry of a self-folding kirigami structure and its folding time. The key finding is the non-monotonic relationship between the number of faces and the total folding time, with an optimal number of five faces for fastest folding in the case of regular pyramids. This is explained by the competing timescales of initial and subsequent binding events, modeled as 2D and 1D first-passage processes, respectively. The methodology employed, combining numerical simulations and analytical calculations, offers a powerful framework for designing self-folding structures with optimized folding characteristics. Future studies could explore the impact of irregular geometries, different hinge mechanics, and the incorporation of hierarchical binding events on the folding dynamics.

The study focuses on regular pyramids, simplifying the system and limiting the generalizability of the findings to more complex structures. While the model effectively describes the folding dynamics of regular pyramids, the applicability to irregular pyramids or other shapes would require modifications to account for varying face sizes, shapes, and closing angles, potentially affecting the diffusion coefficients and timescales of the first-passage processes. The assumption of irreversible binding might not hold in all cases, especially when considering the influence of environmental conditions or material properties. Further investigation is needed to explore the effects of reversible binding on the folding time.