Chiral self-assembly and spontaneous symmetry breaking are fundamental phenomena in supramolecular systems, where molecular components organize into non-superimposable mirror-image structures without external chiral influences. Theoretical models provide a framework to understand the underlying mechanisms, including torque, helical interactions, and amplification processes that drive these systems toward homochirality. This article explores the computational and analytical theories that elucidate these complex behaviors.

At the core of chiral self-assembly is the concept of torque, a rotational force that arises due to asymmetric interactions between molecules. In supramolecular systems, torque can emerge from steric repulsion, electrostatic interactions, or directional bonding. Theoretical treatments often model these interactions using coarse-grained potentials or atomistic molecular dynamics simulations. For instance, the Gay-Berne potential, an extension of the Lennard-Jones potential for anisotropic particles, has been employed to study chiral liquid crystals. The model captures how shape anisotropy and orientation-dependent interactions generate torque, leading to helical twisting. Simulations reveal that even achiral particles can exhibit chiral assemblies when torque exceeds a critical threshold, a phenomenon linked to spontaneous symmetry breaking.

Helical interactions play a pivotal role in stabilizing chiral structures. The Maier-Saupe theory, originally developed for nematic liquid crystals, has been adapted to describe helical ordering in chiral systems. The theory incorporates a chiral term in the free energy expansion, accounting for the energy cost of twist deformation. Analytical solutions show that the pitch of the helix is inversely proportional to the strength of chiral interactions. For example, in systems with a chiral interaction energy of 0.1 kT per molecule, the pitch can range from tens to hundreds of nanometers, depending on the packing density. Computational studies using Monte Carlo or molecular dynamics methods further validate these predictions, demonstrating how helical order propagates through cooperative interactions.

Spontaneous symmetry breaking is a key feature of chiral amplification, where a small initial bias leads to macroscopic homochirality. The Frank model provides a foundational framework, proposing that autocatalytic reactions and mutual inhibition between enantiomers can drive the system toward a single chiral state. The model’s kinetic equations predict that above a critical feedback strength, the racemic state becomes unstable, and the system bifurcates into a homochiral state. Numerical solutions indicate that the critical feedback parameter scales with the rate of enantiomeric cross-inhibition. For instance, when the inhibition rate exceeds the production rate by a factor of two, the system transitions to homochirality within finite time scales.

The role of long-range interactions in chiral amplification has been explored using field-theoretic approaches. The Landau-Ginzburg theory, extended to include chirality, describes how spatial fluctuations and diffusion influence symmetry breaking. The free energy functional includes a gradient term for twist deformation, coupling local chiral order to global structure. Phase diagrams derived from this theory reveal that low temperatures and high chiral interaction strengths favor homochiral phases. Simulations of the time-dependent Ginzburg-Landau equations show that domain growth and defect dynamics further modulate the amplification process. For example, in systems with a chiral coupling constant of 1.0 pN·nm, domains of uniform chirality can span micrometers within milliseconds.

Torque balance theories offer another perspective, particularly for filamentous systems like actin or collagen. These theories consider the mechanical equilibrium between elastic deformation and intrinsic twist. The worm-like chain model, augmented with chiral terms, predicts that the persistence length and twist rigidity determine the equilibrium pitch. For a persistence length of 50 nm and twist rigidity of 100 pN·nm², the predicted pitch is approximately 1 µm, consistent with experimental observations of biopolymer assemblies. Finite element modeling of these systems confirms that mechanical feedback loops can amplify small chiral biases into macroscopic twists.

Nonlinear dynamics models have been applied to study chiral amplification in reaction-diffusion systems. The Selkov model, modified to include chiral intermediates, demonstrates how Turing patterns can evolve into chiral spirals. The model’s parameters, such as diffusion coefficients and reaction rates, dictate the wavelength and stability of these patterns. Numerical simulations reveal that when the chiral species’ diffusion is tenfold slower than the achiral species, stable rotating spirals emerge. These patterns are robust to noise, illustrating how stochastic fluctuations can be harnessed to drive symmetry breaking.

Machine learning approaches are increasingly employed to predict chiral self-assembly outcomes. Neural networks trained on molecular descriptors can classify the chiral phase behavior of supramolecular systems with over 90% accuracy. Feature importance analysis highlights that dipole moment asymmetry and rotational barrier heights are key predictors of chiral ordering. These data-driven models complement analytical theories by identifying non-intuitive correlations between molecular structure and assembly outcomes.

In summary, theoretical models of chiral self-assembly and symmetry breaking integrate concepts from statistical mechanics, nonlinear dynamics, and computational chemistry. Torque and helical interactions provide the microscopic basis for chiral order, while amplification mechanisms explain the emergence of macroscopic homochirality. These frameworks not only deepen our understanding of natural systems but also guide the design of functional chiral materials. Future advances will likely involve multiscale modeling, bridging quantum mechanical details with continuum theories to capture the full complexity of supramolecular chirality.