During the micro-scale self-assembly process of sol-to-gel transition, nanoparticles or polymer chains do not randomly aggregate in a disordered manner, but instead follow strict physical statistical laws to form an unordered porous network with distinct fractal characteristics. This fractal structure implies that aerogels exhibit statistical self-similarity across different spatial scales. Investigating critical scaling laws and the evolution of percolation thresholds within fractal nanoparticle-aggregated aerogel networks represents a key research direction in statistical physics and the study of complex porous media.
In aerogel growth dynamics, two primary physical mechanisms are typically distinguished: diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA). These mechanisms determine the fractal dimension of the solid skeletal framework of aerogels.
For typical aerogel networks, the fractal dimension generally ranges between 1.7 and 2.5—significantly lower than the Euclidean space dimension (d=3)—which is precisely the geometric origin of their ultra-low density and high porosity.
As precursor concentration or crosslinking density varies, the conductive or mechanical skeleton within the aerogel undergoes a percolation phase transition, evolving from "isolated clusters" to a "long-range macroscopically connected network." When the system's volume fraction reaches the critical percolation threshold ($p_c$), macroscopic physical properties such as electrical conductivity and elastic modulus undergo abrupt changes following power-law scaling relationships.
Here, t denotes the conductivity percolation exponent. Through scaling analysis in statistical physics and Monte Carlo computer simulations, physicists can accurately predict how topological defects and dead-end branches affect the percolation exponent and macroscopic mechanical or electrical transport properties in fractal networks, revealing universal principles governing the transition from microscopic geometric disorder to macroscopic physical connectivity in disordered porous networks.