Patchy Particles Design: From Floppy Modes to Sloppy Dimensions

TL;DR

Patchy-particle design space is explored with end-to-end differentiable MD to link design parameters such as patch size $a/r$, patch opening angle $\theta$, and binding energies $E_{AA}, E_{AB}, E_{BB}$ to finite-cluster outcomes. The authors show that the relationship between these parameters is strongly dictated by the target structure's floppiness, with Hessian analysis revealing stiff versus sloppy directions in the design landscape. By contrasting self-assembly and stabilization tasks, the work highlights the role of kinetics and parameter coupling in gradient-based inverse design. The approach provides practical design guidelines for constructing patchy particles to yield specific finite clusters and offers an information-geometry-inspired diagnostic framework for high-dimensional design spaces.

Abstract

Patchy particles have proven to be a prominent model for studying the self-assembly behavior of various systems, ranging from finite clusters to bulk crystal assemblies, and from synthetic colloidal particles to viruses. The patchy particle model is flexible, but it also comes with its own pitfalls -- the potential design space is infinite. Many efforts have been put into building inverse-design frameworks that efficiently design patchy particles for targeted assembly behaviors. In contrast, little work has been done on investigating the interplay between different types of parameters that can be optimized for patchy particles, such as patch location, patch size, and patch binding energies. Here, by utilizing molecular dynamics with automatic differentiation, we elucidate the relationships between different potential optimization parameters and provide general guidelines on how to approach patchy particle design for various types of finite clusters. Specifically, we find that the design parameter landscape is highly dependent on the floppiness of the target structure, and we can identify stiff and sloppy parameters by computing the Hessians of all optimization parameters.

Figures (5)

• Figure 1: System design (a) We use a particular patchy particle model with one big sphere (radius $r$) as the body and two patches (radius $a$) A (green) and B (orange) as the attractive binding sites. The locations of the two patches are determined by the opening angle $\theta$, and the binding energies of the two patches are determined by the binding energy matrix $E_b$. We define a loss function $\mathcal{L}$ to find the best parameter combinations to assemble either a finite triangle or square robustly. (b) Parameter evolution for a typical self-assembly optimization with three different learning rates. At the beginning of every new learning rate, we select the best parameters from the previous learning rate as the starting point. The insert simulation snapshots show the final states of a sample batch. As the optimization converges, more squares start to form.
• Figure 2: JAX-MD optimization framework: the flowchart shows how the parameters for optimization are threaded through the simulation, allowing us to retrieve the gradients after a complete forward MD simulation.
• Figure 3: Self-Assembly Optimization Results: (a) Optimal patch opening angle with respect to different patch sizes for triangle and square assembly. The square markers indicate results for square assembly, while the triangle markers indicate results for triangle assembly. The color of the markers corresponds to the yield $\eta$ of a given optimization. We classify an optimization to be successful for $\eta>50\%$ and we plot three $\eta$ cutoffs: [50%, 75%, 90%]. (b) Four sample snapshots of the forward simulation for the last step of the optimization. We note successful triangle and square formations for both small and large patch sizes.
• Figure 4: Stabilization Optimization Results: (a) Optimal patch opening angle with respect to different patch sizes for triangle stabilization. Each data point represents one successful stabilization optimization. The darker regions indicate multiple optimizations converged to the same patch opening angle. The data points are categorized into three colors: (i) pink, (ii) green, and (iii) purple, indicating three different stable triangle morphologies. (b) Optimal patch opening angle with respect to different patch sizes for square stabilization. Each data point represents a successful stabilization optimization. The darker regions indicate multiple optimizations converged to the same patch opening angle. The data points are categorized into two types: (iv-v) pink and (vi-vii) purple, indicating the two different square conformations. (c) Interior angle distribution for squares and triangles. While the interior angle distribution is agnostic of patch sizes, the angle distribution for squares is dramatically bigger than that of triangles.
• Figure 5: Relationship Between Optimized Patchy Particle Parameters: (a) Scatter plot of optimized patch opening angle as a function of on-target binding and off-target binding energy ratio ($E_{AB}^2/(E_{AA}E_{BB})$). Each data point represents one successful self-assembly optimization (square marker for square optimization and triangle marker for triangle optimization). The binding energy ratios are plotted with an upper limit of 50. (b-c) Hessians for the four optimized parameters (patch opening angle, $E_{AA}$, $E_{AB}$, and $E_{BB}$) and the corresponding eigenvalues for two different successful square and triangle optimizations. The optimized parameters are listed at the top of the Hessian matrices.