Liquid Flow through Defective Layered Membranes: A Phenomenological Description

We present a realistic phenomenological description of liquid transport through defective, layered membranes. We derive general expressions based on conventional models of laminar flow and extend the formalism to accommodate slip flow. We consider different types of defects including in-layer vacancies that provide an activation-free tortuous path through the membrane. Of the many factors that affect flow, the most important is the radius of in-layer vacancy defects, which enters in the fourth power in expressions for the flux density. We apply our formalism to water transport through defective multilayer graphene oxide membranes and find that the flow remains in the laminar regime. Our results show that observed high water permeability in this system can be explained quantitatively by a sufficient density of in-layer pores that shorten the effective diffusion path.

Figure 1

Mechanism of liquid flow through a defective layered membrane. (a) Schematic route of a molecule, proposed by Boehm [1] for the representative case of ${\mathrm{H}}_2\mathrm{O}$ passage through GO. The entrance to interstices occurs at the edge of a finite flake and proceeds through slit pores in-between layers of thickness $t$, separated by the average distance $d$. Here $L_p$ is the average in-layer pore separation, with $L_p\to \mathrm{\infty }$ in a defect-free membrane. In-layer pores may be connected (b) in parallel and form channels that are separated by $L_p$ and normal to the membrane, or (c) in series, forming tortuous channels in a staggered arrangement with lateral offset $L_{\mathrm{sp}}$.

Figure 2

Liquid flow through defective membranes. (a) Perspective view of a membrane containing $N_l$ layers, separated by the distance $d$. Each layer contains round pores of radius $R$ and average separation $L_p$. Straight flow through a “pinhole,” shown on the left, results from a straight arrangement of pores. Tortuous flow shown on the right results from a staggered arrangement of pores. (b) Liquid flow through defective layered membranes as a function of pore radius $R$. Specifics of each membrane are discussed in the text. Relative flux densities $q/q_0$ are presented in units of the flux density $q_0$ across a reference membrane containing pores with $R=R_0=0.5$ nm. All results are for liquid water with viscosity $\eta ={10}^{-3}\mathrm{Pa}\mathrm{s}$.

Figure 3

Fraction of pores with specific radii represented by a Gaussian distribution defined in Eq. (23). Distributions are presented for the average radius $⟨R⟩=7.5$ nm and different values of $\sigma$.

Figure 4

Fraction of pores with specific radii represented by the Cauchy distribution defined in Eq. (25). Distributions are presented for $⟨R⟩=7.5$ nm and different values of $\nu$.