Modeling transport and filtration of nanoparticle suspensions in porous media

Recently membrane filters have gained in significance due to the need to provide protection against airborne pollution. A question of importance, and some controversy, is the efficiency of filters for small nanoparticles with diameters below 100 nm as these are considered particularly dangerous due to possible penetration into the lungs. The efficiency is measured by the number of particles blocked by the pore structure after passing though the filter. To study the penetration into pores by nanoparticles suspended in a fluid, a stochastic transport theory based on an atomistic model is used to calculate particle density and flow within the pores, resulting pressure gradient, and filter efficiency. The importance of pore size relative to particle diameter and of the parameters of the pore wall interactions are investigated. The theory is applied to aerosols in fibrous filters and found to reproduce common trends in measurements. As particles enter the initially empty pores on relaxation to the steady state the small penetration measured at the onset of filtration increases faster in time the smaller the nanoparticle diameter. Control of pollution by filtration is achieved by strong repulsion of pore walls for particle diameters greater than twice the effective pore width. For smaller nanoparticles the steady-state efficiency decreases as the pore wall interactions weaken. Effective efficiency is increased when the suspended nanoparticles inside the pores combine into clusters of sizes greater than the filter channel width.

Figure 1

Schematic representation of the geometry of the slit pore. The number density of suspended particles of diameter $\sigma$ is held fixed at the pore entrance at $x=0$ and at the exit at $x=L$. The arrows mark the flow direction in the steady state. The walls of the pore are located at $z=0$ and $z=H$.

Figure 2

The efficiency E as a function of $2\sigma /H$ (particle diameter $\sigma$ to channel width $H$) using the solution in wide pores for $\sigma /H\to 0(\cdots )$ and in narrow pores for $\sigma /H\gg 1(-)$. The Lennard-Jones wall interaction parameter is $\epsilon /kT=0.8$. The tungsten oxide nanoparticle filter efficiency measured in the nanofibrous polyvinylalcohol filter F5 from Fig. 4 in Ref. [31] is shown for comparison $\left(\blacksquare \right)$ for an effective channel width $H=4$ nm.

Figure 3

The interaction $W\left(z\right)/kT(\cdots )$ of the channel wall with suspended particles of diameter $\sigma$ and the local suspended particle density profile $n\left(z\right)/n_0(-)$ along the width ($H=4\sigma$) of the channel at the entrance of the channel for wall interaction $w\left(z\right)=\epsilon (\sigma /z)$ with $\epsilon /kT=2$. The density of particles before entering the channel is $n_0$.

Figure 4

The efficiency $E$ for a channel of length $L$ as a function of time $t/\tau$ for three values of particle size $\sigma (-),\sigma /2(--),\sigma /5(\cdots )$ corresponding to three values of characteristic time $\tau ,\tau /2,\tau /5$ of decreasing particle size. The characteristic time $\tau =L^2/\left({\pi }^{2}D\right)$ is determined by the size dependent diffusion coefficient D, 2D, 5D of the suspended particles in the channel. Steady-state efficiency is taken from Fig. 5 for $w\left(z\right)=\epsilon (\sigma /z)$ at $(2\sigma /H)=0.5$

Figure 5

The efficiency E as a function of $(2\sigma /H)$ for repulsive wall potential $\epsilon (\sigma /z)$ (–) and $\epsilon {(\sigma /z)}^3$ ($\cdots$). The channel width is H, the particle diameter is $\sigma$ and $\epsilon /kT=1$. The efficiency measured for NaCl aerosol particles in polycarbonate filters in Fig. 4 of Ref. [31] are shown for channel widths $H=1\mu (•)$ and $H=3\mu (\square )$.

Figure 6

The efficiency $E$ as a function of $(2\sigma /H)$ due to formation of clusters $(--)$, due to repulsive wall interaction ($\cdots$), and total apparent efficiency (–). Channel width is $H$, particle diameter is $\sigma$.