# Joint effect of advection, diffusion, and capillary attraction on the spatial structure of particle depositions from evaporating droplets

## TL;DR

This paper presents a simplified simulation model to analyze how advection, diffusion, and capillary forces influence the formation of particle chains and deposits in evaporating droplets, aligning with experimental observations.

## Contribution

It introduces a computational model that captures the combined effects of advection, diffusion, and capillary attraction on particle deposition patterns during droplet evaporation.

## Key findings

- Chains form due to capillary attraction near the fixing radius.
- Close packing occurs if evaporation time exceeds diffusion ordering time.
- Chains develop at the final evaporation stage when the fixing radius approaches the droplet center.

## Abstract

A simplified model is developed, which allows us to perform computer simulations of the particles transport in an evaporating droplet with a contact line pinned to a hydrophilic substrate. The model accounts for advection in the droplet, diffusion and particle attraction by capillary forces. On the basis of the simulations, we analyze the physical mechanisms of forming of individual chains of particles inside the annular sediment. The parameters chosen correspond to the experiments of Park and Moon [Langmuir 22, 3506 (2006)], where an annular deposition and snakelike chains of colloid particles have been identified. The annular sediment is formed by advection and diffusion transport. We find that the close packing of the particles in the sediment is possible if the evaporation time exceeds the characteristic time of diffusion-based ordering. We show that the chains are formed by the end of the evaporation process due to capillary attraction of particles in the region bounded by a fixing radius, where the local droplet height is comparable to the particle size. At the beginning of the evaporation, the annular deposition is shown to expand faster than the fixing radius moves. However, by the end of the process, the fixing radius rapidly outreaches the expanding inner front of the ring. The snakelike chains are formed at this final stage when the fixing radius moves toward the symmetry axis.