Micellization as a connectivity transition: a topological Ising model with a hydrophobic constraint

Micellization is commonly described as a collective response driven by the hydrophobic effect. Here we propose and study a topological Ising model that abstracts this effect as a solvent-exclusion constraint defined purely by local connectivity. On a square lattice with binary occupancy (amphiphiles/water), we characterize neighborhood by a topological kernel of radius R and metric (Chebyshev or Manhattan). A water site becomes "restricted" when the local overlap with amphiphiles, computed via convolution with the kernel, exceeds a fixed threshold. The system energy is F = N_restr; we set α = 1 by design, working in dimensionless units that prevent interpreting α as carrying any metric information. Dynamics are explored with Metropolis updates at temperature T. Control parameters are amphiphile density ρ, temperature T, the metric, and R. As an order parameter we use S_max/N_a, the fraction of amphiphiles in the largest connected cluster. In the surveyed ranges we observe, for more connective kernels (e.g., Chebyshev with R ≥ 3), the emergence of a giant component in finite regions of (ρ, T), while less connective configurations (e.g., Manhattan with R = 1) do not aggregate in the same window. These results support the view that micellization, in this framework, is a connectivity transition governed by the topology of local interactions rather than by explicit metric scales. We discuss implications and routes for quantitative comparisons with experiments and more detailed simulations.