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

Abstract

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 <i>R</i> and metric (Chebyshev or Manhattan). A water site becomes "restricted" when the local overlap with amphiphiles, computed <i>via</i> convolution with the kernel, exceeds a fixed threshold. The system energy is <i>F</i> = <i>N</i>_restr; we set <i>α</i> = 1 by design, working in dimensionless units that prevent interpreting <i>α</i> as carrying any metric information. Dynamics are explored with Metropolis updates at temperature <i>T</i>. Control parameters are amphiphile density <i>ρ</i>, temperature <i>T</i>, the metric, and <i>R</i>. As an order parameter we use <i>S</i>_max/<i>N</i>_a, the fraction of amphiphiles in the largest connected cluster. In the surveyed ranges we observe, for more connective kernels (<i>e.g.</i>, Chebyshev with <i>R</i> ≥ 3), the emergence of a giant component in finite regions of (<i>ρ</i>, <i>T</i>), while less connective configurations (<i>e.g.</i>, Manhattan with <i>R</i> = 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.