Dynamic Permeability in Metastable Droplet Interfacial Bilayers

TL;DR

This work develops a mechanistic theory linking transient pore growth in droplet interfacial bilayers (DIBs) to dynamic, size-selective permeability, enabling inference of membrane structure from transport measurements. By formulating a pore-size distribution n(R,t/τ_P) and a dimensionless permeability κ̄^α, the authors connect transport to specific growth mechanisms, notably Ostwald ripening, coalescence, and surfactant desorption, with a two-dimensional Lifshitz–Slyozov–Wagner (LSW) framework for Ostwald ripening yielding R_c ∼ t^{1/3}. They derive explicit expressions for κ̄^α(t/τ_P) and show that the crossing time t_{50} for dye particles scales as t_{50} ∝ x_0^3 for large x_0 and depends inversely on Pe for small Pe, providing experimentally testable signatures to identify the dominant pore-growth pathway. The theory predicts strong size selectivity in early times and a universal scaling function for the pore distribution, and it outlines concrete experimental tests using DIBs to extract pore-area fractions, mobility, and line tension, with implications for understanding metastable membranes. Overall, the framework links transport measurements to evolving membrane microstructure, offering guidance for designing responsive, size-selective DIBs and for probing membrane mechanics in synthetic systems.

Abstract

Membrane pores are implicated in several critical functions, including cell fusion and the transport of signaling molecules for intercellular communication. However, these structural features are often difficult to probe directly. Droplet interfacial bilayers offer a synthetic platform to study such membrane properties. We develop a theory that links size-selective transport across a metastable membrane with its transient structural properties. The central quantity of our theory is a dynamic permeability that depends on the mechanism of pore growth, which controls the transient distribution of pore sizes in the membrane. We present a mechanical perspective to derive pore growth dynamics and the resulting size distribution for growth \textit{via} Ostwald ripening and discuss how these dynamics compare to other growth mechanisms such as coalescence and growth through surfactant desorption. We find scaling relations between the transported particle size, the pore growth rate, and the time for a given fraction of particles to cross the membrane, from which one may deduce the dominant mechanism of pore growth, as well as material properties and structural features of the membrane. Finally, we suggest experiments using droplet interfacial bilayers to validate our theoretical predictions.

Figures (10)

• Figure 1: Dye transport across a nanoparticle--surfactant droplet interface bilayer (NPS--DIB) Wu2025Submitted. Optical images show two aqueous droplets—one containing Rhodamine 6G dye (pink) and one initially dye-free—at the moment of contact ($t_{\rm{contact}} = 0~\rm{hr}$) and after 48 hours. Dye transport across the interfacial barrier is slow; little is visible in the initially empty droplet by eye after two days. Confocal fluorescence microscopy confirms that dye transfer does occur, as shown by the red emission signal detected in the receiving droplet (upper right inset). Imaging the droplet junction after six hours of contact (upper left inset, scale bar $100~\upmu \rm m$) shows a bridge connecting the droplets, consistent with the formation of a dense NPS DIB that impedes but does not completely block transport.
• Figure 2: Schematic of a DIB formed between two droplets each coated with a surfactant monolayer (green) and with one droplet containing dye particles of different sizes. Droplets are brought together and a DIB forms upon contact. Over time, the DIB grows and critical nuclei form. The smallest dye particles cross the bilayer through these small pores. As the pores grow, the membrane becomes permeable to larger particles. Pore growth causes the bilayer to degrade until, eventually, the two droplets coalesce. The zoomed-in panel depicts pore growth through a mechanism that conserves the total pore area faction (top and side view are of the same time points). The top panel highlights that the pore distribution shifts to larger pores. The bottom panel emphasizes that the membrane becomes permeable to larger particles over time.
• Figure 3: Plots of the pore distribution for the Ostwald ripening mechanism, normalized so that the area under the curves is unity, $n(R,t)/C$ with ${C = \int n(R,t) d R/R_0}$, for various times non-dimensionalized by the pore-growth time scale, $t/\tau_p$. At early times the bilayer features several small pores and as $t/\tau_P$ increases the distribution widens such that the bilayer contains fewer but larger pores.
• Figure 4: The non-dimensional membrane permeability, $\overline{\kappa}^{\alpha}$, plotted as a function of time in units of the pore-growth time scale, $t/\tau_p$, for dye sizes measured against the initial pore radius $x_0=r^{\alpha}/R_0$ increasing across four orders of magnitude. In the Ostwald ripening mechanism, the area fraction of pores remains fixed. Therefore, since $\overline{\kappa}^{\alpha}$ is non-dimensionalized by its value if all initial pores are permeable, all curves plateau at unity.
• Figure 5: The non-dimensional dye density in the acceptor droplet, $\bar{\rho}^{\alpha}_a$, plotted as a function of dimensionless time, $t/\tau_D$ where $\tau_D$ is the dye diffusion time scale. In the left panel, $\tau_D$ is varied relative to the pore growth time-scale via$\mathrm{Pe} = \tau_D/\tau_P$, with the scaled dye size relative to the initial pore radius $x_0 = r^{\alpha}/R_0$ fixed at $x_0 = 1.5$. Here, we see that increasing $\mathrm{Pe}$ for fixed particle size shifts the density profile to earlier times. In the right panel, the scaled dye sizes, $x_0$, are varied while $\mathrm{Pe} = 1.0$ and we see that increasing dye size shifts the density profile to later times.