  Soft gels, formed via the self-assembly of particulate organic materials,
exhibit intricate multi-scale structures that provides them with flexibility
and resilience when subjected to external stresses. This work combines
molecular simulations and topological data analysis (TDA) to characterize the
complex multi-scale structure of soft gels. Our TDA analysis focuses on the use
of the Euler characteristic, which is an interpretable and
computationally-scalable topological descriptor that is combined with
filtration operations to obtain information on the geometric (local) and
topological (global) structure of soft gels. We reduce the topological
information obtained with TDA using principal component analysis (PCA) and show
that this provides an informative low-dimensional representation of gel
structure. We use the proposed computational framework to investigate the
influence of gel preparation (e.g., quench rate, volume fraction) on soft gel
structure and to explore dynamic deformations that emerge under oscillatory
shear in various response regimes (linear, nonlinear, and flow). Our analysis
identifies specific scales and extents at which hierarchical structures in soft
gels are affected; moreover, correlations between structural deformations and
mechanical phenomena (such as shear stiffening) are explored. In summary, we
show that TDA facilitates the mathematical representation, quantification, and
analysis of soft gel structures, extending traditional network analysis methods
to capture both local and global organization.
