Vertical drying of a suspension of sticks: Monte Carlo simulation for continuous two-dimensional problem

The vertical drying of a two-dimensional colloidal film containing zero-thickness sticks (lines) was studied by means of kinetic Monte Carlo (MC) simulations. The continuous two-dimensional problem for both the positions and orientations was considered. The initial state before drying was produced using a model of random sequential adsorption with isotropic orientations of the sticks. During the evaporation, an upper interface falls with a linear velocity in the vertical direction, and the sticks undergo translational and rotational Brownian motions. The MC simulations were run at different initial number concentrations (the numbers of sticks per unit area), $p_i$, and solvent evaporation rates, $u$. For completely dried films, the spatial distributions of the sticks, the order parameters, and the electrical conductivities of the films in both the horizontal, $x$, and vertical, $y$, directions were examined. Significant evaporation-driven self-assembly and stratification of the sticks in the vertical direction was observed. The extent of stratification increased with increasing values of $u$. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of $p_i$ and $u$.

Figure 1

To the calculation of the distributions of local number density, $\rho \left(y\right)$, and local order parameter, $S\left(y\right)$, along the vertical axis $y$. Here $y_i$ is the vertical coordinate of the center of the stick, $\mathrm{\Delta }y=H/32$ is the width of the strip, $H$ is the running height of the film, ${\theta }_i$ is the angle between axis of the stick and the horizontal axis $x$.

Figure 2

Evolution of drying patterns at different values of the relative height of the film, $h$, for a constant drying rate, $u={10}^{-3}[{\rho }_c=\infty$ in Eq. (5)] (a), and for a variable drying rate $[{\rho }_c=7$ in Eq. (5)] (b). The initial concentration of sticks was ${\rho }_i=1.0$.

Figure 3

Drying rate, $u$ (a), relative height of the film, $h$, and average order parameter, $S$ (b), for constant and variable drying rates vs MC steps. The initial concentration of sticks was ${\rho }_i=1.0$.

Figure 4

Profiles of number density, $\rho (y/H_i)$ at different values of relative height of the film, $h$, for initial number density ${\rho }_i=0.5$ and drying rate $u={10}^{-5}$ (a), ${\rho }_i=0.5,u=5\times {10}^{-3}$ (b), ${\rho }_i=2.0,u={10}^{-5}$ (c), and ${\rho }_i=2.0,u=5\times {10}^{-3}$ (d).

Figure 5

Profiles of number density, $\rho (y/H_i)$ (a), and local order parameter, $S(y/H_i)$ (b), at different values of initial number density, ${\rho }_i$, at fixed value of the relative height of the film, $h=0.2$, for the relatively high drying rate, $u={10}^{-3}$.

Figure 6

Comparison of the drying patterns at different values of the initial number density, ${\rho }_i$, at a fixed value of the relative height of the film, $h=0.2$, for the relatively high drying rate, $u={10}^{-3}$. See Ref. [50].

Figure 7

Mean order parameter, $S$, versus the relative height of the film, $h$, at different drying rates, $u$, and initial number densities, ${\rho }_i=0.5$ (a), 1.0 (b), 2.0 (c), and 5.0 (d). The corresponding densities, $\rho$, are also presented on the upper horizontal axis.

Figure 8

Electrical conductivity in the horizontal, ${\sigma }_x$, and vertical, ${\sigma }_y$, directions versus the relative height of the film, $h$, at an initial density of ${\rho }_i=0.1$ and different drying rates, $u={10}^{-4}$ (a) and $u={10}^{-3}$ (b). The data are presented for different sizes of the supporting square lattice, $m=256,512,$ and 1024. The dashed lines correspond to the evolution of order parameter, $S$, during the drying.

Figure 9

Example of a supporting lattice.

Figure 10

Example of transformation of cells into sets of conductors.

Figure 11

Mean order parameter, $S$, versus the relative height of the film, $h$, at fixed drying rate $u={10}^{-3}$ and initial number density ${\rho }_i=1.0$. The corresponding densities, $\rho$, are also presented on the upper horizontal axis.