Diffusion through complex confining environments: fluctuating triply periodic minimal surfaces

The transport of individual entities through interconnected structures is a process of practical relevance both in biology and technology. Examples are given by diffusive dynamics of molecules in porous structures. In soft environments, this transport can be strongly influenced by fluctuations of the porous structure itself. Here, we focus on triply periodic membrane structures found both in cell organelles and in synthetic amphiphilic systems. We theoretically study the effect of a complex three-dimensional fluctuating environment on the diffusive motion of a test object, using a phase field approach. The rigid spherical test object is energetically forced to not penetrate the membrane. Generally, the pores of the membrane structure can be smaller than the diffusing object. Yet, fluctuations of the membrane can intermittently widen its pores, still allowing for the motion of the larger particles through them. Thus, the object stays trapped for a while inside one cavity formed by the membrane, before an appropriate fluctuation event widens a membrane pore in the right moment so that the object can jump into the next cavity. The process is reflected by a pronounced plateau in the time evolution of the mean squared displacement. We think that the described scenario should be directly observable, for instance, in protein diffusion through biological environments.

💡 Research Summary

The paper investigates how a rigid spherical tracer diffuses through a three‑dimensional, thermally fluctuating, triply periodic minimal surface (TPMS) that separates two interpenetrating labyrinthine domains. Using a phase‑field representation of the membrane, the authors construct a free‑energy functional that combines Helfrich curvature elasticity, a global area constraint, a volume‑balance term, and a short‑range repulsive interaction between the tracer and the membrane. The membrane is modeled as the zero‑level set of a scalar field ϕ(r); its equilibrium shape minimizes the functional, yielding a surface of constant mean curvature (zero for a minimal surface). Thermal fluctuations are introduced via stochastic Langevin dynamics for both the phase field and the tracer position, with noise amplitudes set by the temperature kBT.

First, the authors validate the model by deriving the height‑fluctuation spectrum of a flat membrane patch from the curvature term and confirming the k‑4 dependence (⟨h_k h_{−k}⟩ ∝ k_B T / κ|k|⁴) against spectral simulations. This establishes that the phase‑field approach reproduces the known Helfrich statistics.

Next, they generate a P‑type TPMS (the Schwarz P surface) using the Cahn‑Hilliard‑type initialization of Ref.