Optimal number of faces for fast self-folding kirigami

Kirigami creates 3D structures from cut-and-fold 2D templates. At macroscales, folding is largely deterministic due to energy minimization, but at microscales thermal fluctuations dominate and folding is stochastic. Prior work has focused on selecting 2D nets that maximize the probability (yield) of folding into a target 3D structure, as stochasticity can cause misfolding. However, practical applications also require fast folding, not just high yield. This study targets the folding time of a misfolding-free prototype: a regular pyramid with N lateral faces. The research question is whether and how folding time depends on N, and what mechanisms control this dependence. The authors propose that folding can be decomposed into Brownian first-passage events whose scaling with N determines the overall folding time.

Past studies established that different nets can fold into the same 3D object but with yields strongly influenced by topology and conditions due to stochastic folding (e.g., Pandey et al. 2011; Dodd et al. 2018). Work has thus emphasized identifying optimal nets to maximize yield (Araújo et al. 2018). At macroscale, folding pathways are deterministic via energy minimization, whereas microscale folding is stochastic in fluid environments. Experimental strategies have tuned folding kinetics using cell traction (Kuribayashi-Shigetomi et al. 2012) or graphene-based hinges controlled by temperature and pH (Miskin et al. 2018). These efforts motivate exploring geometrical control—specifically, how the number of faces in a regular pyramid influences folding time under thermal fluctuations.

System: Regular pyramid with N lateral triangular faces connected to a rigid base. The 2D net is an N-pointed star. To eliminate misfolding and constrain dynamics, the base is pinned to a flat substrate and faces can fold only on one side. Modeling: Each lateral face is a rigid body comprised of three point particles at the vertices; the base is a rigid body with N particles at its vertices. All particles have mass m. The height L of faces and vertex masses are chosen so that the pyramid closing angle θ and the moment of inertia I about the edge with the base are constant across N. Dynamics: Particle-based Langevin dynamics integrated via a velocity-Verlet scheme in LAMMPS. Translational motion for face i: M dv_i/dt = -∇U_i - v_i + sqrt(2 M k_B T) ξ_i(t), with M = 3m. Rotational motion: I dω_i/dt = -∇U_i - ω_i + sqrt(2 I k_B T) ζ_i(t). Translational and rotational damping times are set equal (τ_t = τ_r). Stochastic terms satisfy fluctuation-dissipation. The relevant rotational diffusion coefficient is D_rot = k_B T / I. Thermal noise drives stochastic angular motion of faces. Interactions: Binding between faces is modeled by a short-range attractive inverted-Gaussian potential between vertex particles: U_blue/blue(r) = ε exp[-(r/σ)^2], with ε = 20 k_B T. σ is set so that the interaction strength is k_B T at r = L/60, corresponding to an angular window of approximately π/180 around the closing angle. Motion is constrained to one side by a repulsive interaction between all particles and a substrate plane located at distance L/10 below the base, modeled by the repulsive part of a Lennard-Jones potential with prefactor 4 k_B T and cutoff to zero at distance L/50. Simulation protocol: Independent stochastic simulations are run from a flat initial configuration (all face angles θ_i(0) = 0) until complete folding. The attractive interaction is active when faces are within the effective angular region around θ* (e.g., θ* ≈ 3π/4 ± Δ with Δ ≈ π/180), leading to irreversible binding upon contact. Folding time T is measured as the time for all faces to bind and form the pyramid. Results are averaged over 2 × 10^3 independent samples per N. Time is reported in units of Brownian time (the average time for a non-interacting face to diffuse over an angular region of size θ*), and scaling analyses consider τ = θ*^2/(2 D_0), with D_0 a reference angular diffusion coefficient. Analysis framework: The dynamics of face angles θ_i(t) is treated as Brownian processes with reflective boundaries at 0 and π. The first binding event is mapped to a first-passage in a 2D Brownian process with a trap at (θ*, θ*), considering all N(N−1)/2 face pairs in parallel (fastest event sets the time). Subsequent events are mapped to 1D first-passage processes for each remaining free face to reach θ*, with the slowest among N−2 events determining the time from first to last binding. Analytical estimates for mean times leverage order statistics of first-passage times, with assumptions of independent processes and, for later events, a uniform distribution of θ_i(T_F) in [0, θ*]. Parameter sweeps include variations of θ* and D to test scaling.

• The total folding time T exhibits a non-monotonic dependence on the number of faces N, with a minimum at N = 5.
• First binding event time T_F decreases with N, consistent with a 2D first-passage process among N(N−1)/2 competing pairs. Numerical data fit T_F ≈ τ_F / ln[N(N−1)/2] + τ_F0 with τ_F = 1.57 ± 0.02 and τ_F0 = −0.097 ± 0.006 (in Brownian time units).
• Subsequent binding dynamics: After the first binding, the remaining N−2 faces bind one-by-one upon reaching θ*. The time from first to last binding, T_L, is the slowest among N−2 independent 1D first-passage processes and grows approximately as T_L(N) ≈ τ_L ln(N−2) + γ, where τ_L = 4 θ*^2/(π^2 D_0) and γ is the Euler–Mascheroni constant. Numerical data are consistent with this logarithmic growth; an effective τ ≈ 0.474 (in Brownian time units) characterizes the fits in Fig. 4b.
• The total folding time T = T_F + T_L is governed by the competition between the decreasing T_F and increasing T_L with N, explaining the observed minimum at N = 5.
• Data collapse: Rescaling times by θ*^2 and varying D shows collapse for large N, supporting the diffusive scaling τ ∝ θ*^2/D_0. Deviations at small N are attributed to limits of the approximations (e.g., uniform initial angle distribution at T_F and independence).
• Simulations use 2 × 10^3 samples per N, with time expressed in Brownian units and faces’ angular motion behaving as Brownian with reflective boundaries at 0 and π.

The study addresses how geometry (number of faces N) controls folding time in thermally driven self-folding kirigami. By mapping the stochastic face motions to first-passage problems—2D for the first binding between any face pair and 1D for subsequent individual bindings—the authors capture the distinct N-dependencies of early and late folding stages. The analytical forms derived from order statistics quantitatively describe simulation results: T_F decreases with the combinatorial growth of potential binding pairs, while T_L increases logarithmically with the number of remaining faces. Their sum yields a non-monotonic T(N) with an optimal N = 5 for fastest folding in the considered conditions. This framework demonstrates that geometric design can regulate folding kinetics without altering hinge mechanics or external stimuli. The approach generalizes to non-regular pyramids and more complex templates by incorporating different closing angles, diffusion coefficients (face sizes/shapes), and hierarchical binding pathways.

The work shows that the folding time of a self-folding regular pyramid under thermal fluctuations is non-monotonic in the number of faces and is minimized at N = 5. Decomposing folding into a first 2D first-passage binding event among face pairs and subsequent 1D first-passage events for remaining faces provides analytical expressions that accurately match simulations. Times scale diffusively with θ*^2/D_0 and the interplay of decreasing T_F and increasing T_L explains the optimal N. These insights suggest geometry as an effective lever to control folding speed. Future directions include extending the framework to non-regular pyramids with heterogeneous angles and face properties, incorporating correlations between faces, and modeling complex templates requiring hierarchical binding pathways and broader physical effects (e.g., hydrodynamics, temperature-dependent hinge mechanics).

• Misfolding is suppressed by constraining folding to one side and pinning the base, so results pertain to a misfolding-free scenario.
• The analytical treatment assumes independence among face motions and uniform distributions of face angles at the time of the first binding; correlations are neglected.
• Faces are modeled as rigid bodies with simplified short-range interactions; hydrodynamic interactions and detailed hinge mechanics are coarse-grained.
• The study focuses on regular pyramids with identical faces, angles, and diffusion coefficients; generalization to heterogeneous geometries is discussed but not executed.
• Validation is based on simulations; no experimental data are presented here.