Cross Interaction Drives Stratification in Drying Film of Binary Colloidal Mixtures

When a liquid film of a colloidal solution consisting of particles of different sizes is dried on a substrate, the colloids often stratify, where smaller colloids are laid upon larger colloids. This phenomenon is counterintuitive because larger colloids which have a smaller diffusion constant, are expected to remain near the surface during the drying process, leaving a layer of larger colloids on top of smaller colloids. Here we show that the phenomenon is caused by the interaction between the colloids, and can be explained by a diffusion model accounting for the interaction between the colloids. By studying the evolution equations both numerically and analytically, we derive the condition at which the stratified structures are obtained.

Figure 1

Drying of a binary colloidal solution in a film makes a stratified film with small colloids on top of large colloids.

Figure 2

Time variation of concentration profile of colloidal particles having size ratio $\alpha =4$: (a) ${\mathrm{Pe}}_1=0.1$, ${\mathrm{Pe}}_2=0.4$; (b) ${\mathrm{Pe}}_1=1$, ${\mathrm{Pe}}_2=4$; (c) ${\mathrm{Pe}}_1=5$, ${\mathrm{Pe}}_2=20$. The initial concentrations are ${\varphi }_{01}={\varphi }_{02}=0.02$. The curves from bottom to top correspond to time $\tau =0.005$, 0.1, 0.2, 0.3, 0.4, 0.5.

Figure 3

(a) State diagrams in the ${\varphi }_{01}\text{−}\alpha$ plane. Parameters are ${\mathrm{Pe}}_1=2$, ${\varphi }_{02}=0.01$ (left), and ${\varphi }_{02}=0.05$ (right). (b) State diagrams in the ${\varphi }_{01}\text{−}{\mathrm{Pe}}_1$ plane. Parameters are $\alpha =3$, ${\varphi }_{02}=0.01$ (left), and ${\varphi }_{02}=0.05$ (right). The solid curve corresponds to ${\alpha }^2(1+{\mathrm{Pe}}_1){\varphi }_{10}=1$.

Figure 4

Master plot of the state diagram in the ${\varphi }_{01}–{\alpha }^2(1+{\mathrm{Pe}}_1)$ plane. The solid curve corresponds to ${\alpha }^2(1+{\mathrm{Pe}}_1){\varphi }_{01}=1$. The states labeled by (filled square) and (filled circle) are shifted slightly in the ${\varphi }_{01}$ axis for a better view.

Figure 5

State diagram when close packing is also considered. The size ratio is $\alpha =7$. The symbols are results taken from Ref. [13]. The labeling is the same as in Figs. 3 and 4 and discussed in [15].