The Shape of Chocolate: A Topological Perspective on Food Microstructure

TL;DR

This paper introduces a topological data analysis framework to characterize cocoa butter microstructure during chocolate tempering, revealing distinctive topological signatures for optimal crystallization phases.

Contribution

It applies persistent homology and entropy measures to molecular simulations, providing a novel, non-invasive method to identify phase transitions in chocolate tempering.

Findings

Form V shows a local minimum in persistent entropy and beta_1, indicating optimal tempering.

Topological metrics successfully distinguish between different cocoa butter phases.

Persistent entropy correlates with phase transitions, validating TDA as a quality indicator.

Abstract

We present a computational framework for characterizing the molecular self-organization of cocoa butter (Theobroma cacao) during dark chocolate tempering through the lens of Topological Data Analysis (TDA). A physics-inspired particle simulation models N=100 triglyceride molecules across the full temperature range 15--60 degrees C, spanning all six crystalline polymorphs of cocoa butter (Forms I--VI) as well as the melt and superheating regimes. At each temperature tick, we construct a Vietoris-Rips filtration and compute the persistent homology groups H0 (connected components), H1 (independent cycles), and H2 (3D voids). The resulting persistence diagrams are analyzed via persistent entropy E = -sum_i p_i log2(p_i), where p_i = l_i / sum_j l_j and l_i = death_i - birth_i denotes feature lifetime; essential classes are assigned death = m+1 (m = eps_max) following the standard persistent entropy convention (Rucco 2026, arXiv:2602.09058). Our results demonstrate that Form V (the optimal tempering polymorph, 29.5--34 degrees C) is characterized by a distinctive topological signature: a local minimum in the H0 persistent entropy (E0 = 5.74 +/- 0.04 bits), a pronounced depression in the first Betti number beta_1 (1562 +/- 35), and a global minimum in the H2 entropy (E2 = 12.29 +/- 0.25 bits) reflecting coherent inter-bilayer lamellar cavities. Via Theorem 1 and Corollary 1 of Rucco (2026), persistent entropy is proven to separate the ordered and disordered phases by an asymptotically non-vanishing gap whenever a phase transition induces the creation or destruction of topological mass at macroscopic scales -- a condition we verify empirically across all eight cocoa butter regimes. These findings suggest that TDA-based metrics could serve as non-invasive quality indicators for industrial chocolate tempering processes.