Classical density functional theory for nanoparticle-laden droplets

TL;DR

Droplets of pure liquids evaporate in air due to Laplace pressure, but dissolved nanoparticles can stabilize them by shifting internal pressures. The authors develop a continuum density functional theory framework for a binary square-well mixture to model nanoparticle laden droplets, treating the solvent grand-canonically and nanoparticles canonically, and solving for equilibrium density profiles with a staged minimization strategy. They find oscillatory, concentric-shell density structures inside droplets, with interior shelling emerging prominently at large size ratios up to $q=10$, and show that a simple mechanical and chemical equilibrium model reproduces the stabilized droplet sizes under different humidities. These results quantify a stabilization mechanism relevant to aerosol lifetimes and potential virus transmission, while the approach can be generalized to other fluids lacking associative interactions.

Abstract

Droplets of a pure fluid, such as water, in an open container surrounded by gas, are thermodynamically unstable and evaporate quickly. In a recent paper [Archer et al. J. Chem. Phys. {\bf 159}, 194403 (2023)] we employed lattice density functional theory (DFT) to demonstrate that nanoparticles or solutes dissolved in a liquid droplet can make it thermodynamically stable against evaporation. In this study, we extend our model by using continuum DFT, which allows for a more accurate description of the fluid and nanoparticle density distributions within the droplet and enables us to consider size ratios between nanoparticles and solvent particles up to 10:1. While the results of the continuum DFT agrees well with those of our earlier lattice DFT findings, our approach here allows us to refine our understanding of the stability and structure of nanoparticle laden droplets. This is particularly relevant in light of the recent global COVID-19 pandemic, which has underscored the critical role of aerosol particles in virus transmission. Understanding the stability and lifetime of these viron-laden aerosols is crucial for assessing their impact on airborne disease spread.

Figures (7)

• Figure 2: Schematic picture of a nanoparticle-laden droplet in a spherical system of radius $R_m$ containing a droplet (dark blue) of radius $R$ in equilibrium surrounded by vapor (light blue). A fraction $\xi$ of the nanoparticles (green) are inside the droplet.
• Figure 3: Density profiles $\rho_l(r)$ (left) and $\rho_n(r)$ (right) of a 2:1 droplet with $\beta\epsilon_{nn}=0,\,\beta\epsilon_{ln}=3.78,\,\lambda_{nn}=1.01$ and $N_n=92281$. For $H_r=50\%$ (red) the droplet has a radius $\tilde{R}=80$ which increases as the humidity is raised to $H_r=60\%$ (green), $H_r=70\%$ (orange) and $H_r=80\%$ (blue). The insets show a heatmap plot of the density in a 2:1 droplet for $H_r=50\%$.
• Figure 4: Density profiles $\rho_l(r)$ (left) and $\rho_n(r)$ (right) of a 10:1 droplet with $\beta\epsilon_{nn}=0,\,\beta\epsilon_{ln}=3.69,\,\lambda_{nn}=1.01$ and $N_n=1983$ with an offset represented by blue lines. For $H_r=50\%$ (red) the droplet has a radius $\tilde{R}=80$, which increases as the humidity is raised to $H_r=60\%$ (green), $H_r=70\%$ (orange) and $H_r=80\%$ (blue). The insets show a heatmap plot of the densities of the 10:1 droplet for $H_r=50\%$.
• Figure 5: The droplet radius $R$ is plotted as a function of nanoparticle number $N_n$ with a fixed interaction strength $\beta\epsilon_{ln}=3.69$ of Fig. \ref{['fig::10_to_1']} for several humidity values considered here. The inset displays the dependence of the number of liquid particles $N_l$ on the nanoparticle number $N_n$.
• Figure 6: Density profiles $\rho_l(r)$ (left) and $\rho_n(r)$ (right) of a 10:1 droplet with $\beta\epsilon_{nn}=6.96,\,\epsilon_{ln}=\sqrt{\epsilon_{ll}\epsilon_{nn}},\,\lambda_{nn}=1.01$ and $N_n=2063$ with an offset represented by blue lines. For $H_r=50\%$ (red) the droplet has a radius $\tilde{R}=80$ which increases as the humidity is raised to $H_r=60\%$ (green), $H_r=70\%$ (orange) and $H_r=80\%$ (blue). The insets show a heatmap of the 10:1 droplet for $H_r=50\%$.