Order-to-Disorder Transition in Ring-Shaped Colloidal Stains

A colloidal dispersion droplet evaporating from a surface, such as a drying coffee drop, leaves a distinct ring-shaped stain. Although this mechanism is frequently used for particle self-assembly, the conditions for crystallization have remained unclear. Our experiments with monodisperse colloidal particles reveal a structural transition in the stain, from ordered crystals to disordered packings. We show that this sharp transition originates from a temporal singularity of the flow velocity inside the evaporating droplet at the end of its life. When the deposition speed is low, particles have time to arrange by Brownian motion, while at the end, high-speed particles are jammed into a disordered phase.

Figure 1

Order-to-disorder transition in the particle stain left by an evaporating drop. (a) A $3\mathrm{\text{−}}\mu \mathrm{L}$ sessile water droplet evaporates from a glass substrate. (b) Ring-shaped stain of red particles (the coffee stain) left on the substrate after evaporation. (c) A close-up by an optical microscope of the bottom layer of the stain, taken from the white square in (b), shows that the outermost lines of the stain (left) have an ordered, crystalline structure. Towards the center of the drop (right), a transition to a disordered particle arrangement is observed. (d) A top view of the ring stain, taken from the red square in (c) with a scanning electron microscope (SEM), shows that the first lines of particles (left) are arranged in a hexagonal array, while the next lines (brighter in the image) are arranged as square, followed by again hexagonal array.

Figure 2

Analysis of the particle ordering in the stain. (a) Overview of the different patterns observed in the ring stain. The position of the contact line is on the left side of the figure (the first layers of particles are not shown). (b1)–(b3) A close-up reveals the sequence of patterns in the stain: square packing close to the contact line (b1), followed by hexagonal packing (b2), and finally disordered packing (b3). The Voronoi cells belonging to the particles are shown by blue lines. (c) The area of the Voronoi cells plotted versus the distance $x$ from the contact line. The order-to-disorder transition at $x=x_c$ is determined by the increase in the mean Voronoi area and the dispersion, due to the lack of order. This transition is indicated by the gray (red) dotted line.

Figure 3

Origin of the order-to-disorder transition. Plot of the radial particle velocity versus time, measured at a distance of $174\mu \mathrm{m}$ from the contact line. The fluid velocity predicted by the model (1) is plotted as a solid line. Both model and experiment show the dramatic velocity increase at the end of the droplet’s life (the rush hour). The inset shows the direction of the radial velocity in the droplet. The time $t_c$ at which the order-to-disorder transition occurs is determined from the videos taken for the $\mu \mathrm{PIV}$ measurements; the corresponding critical velocity $u_c$ is read from this graph (indicated by the black line in the figure).

Figure 4

The experimentally determined critical velocity $u_c$ versus its theoretical prediction (3).