Transport dynamics of charged colloidal particles during directional drying of suspensions in a confined microchannel

Directional drying of colloidal suspensions, experimentally observed to exhibit mechanical instabilities, is a nonequilibrium procedure that is susceptible to geometric confinement and the properties of colloidal particles. Here, we develop an advection-diffusion model to characterize the transport dynamics for unidirectional drying of a suspension consisting of charged particles in a confined Hele-Shaw cell. We consider the electrostatic interactions by means of the Poisson-Boltzmann cell approach with the viscous flow confined to the cell. By solving the nonequilibrium transport equations, we clarify how the multiple parameters, such as drying rate, confinement ratio, and the monovalent slat concentration, affect the transport dynamics of charged colloidal particles. We find that the drying front recedes into the cell with linear behavior, while the liquid-solid transition front recedes with power law behaviors. The faster evaporation rate creates a rapid formation of the drying front and produces a thinner transition layer. We show that confinement is equivalent to raising the effective concentration in the cell, and, accordingly, the drying front appears earlier and grows more rapidly. Under geometric confinement, a longer fully dried film is created while the total drying time is shortened. Moreover, we have theoretically illustrated that low salt loadings cause a large collective diffusivity of charged colloidal particles, which results in a colloidal network by aggregation. Thus, the drying behavior alters dramatically as salt loadings decrease, since the resulting compacted clusters of charged particles eventually convert the suspension into a gel-like material instead of a simple fluid. Our model is consistent with the current experiments and provides a simple insight for applications in directional solidification and microfluidics.

Figure 1

(a) The schematic illustration of the directional drying of suspensions with the dispersed charged colloidal particles in a confined Hele-Shaw cell. (b) The sketch of transport dynamics of the charged particles through a cross section of the cell, which is directly used for our 1D configuration model. In the liquid-solid transition front, particles collide with each other, hindering transport. The drying front (solidification) and the liquid-solid transition front recede together towards the interior cell (indicated by black arrows), while the particles are carried towards the drying end (indicated by red arrows). (c) The Poisson-Boltzmann cell model, an electrically neutral cell is conceived to surround an individual charged particle influenced by the interplay between the ions at the surface and in the ambient solution.

Figure 2

The effective diffusion coefficient quantified by Eq. (20) for different colloidal particles: (a) $a=100$ nm and $\gamma =0.033\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}$; (b) $a=50$ nm and $\gamma =0.132\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}$. The black line signifies the diffusivity of uncharged particles ($f_q=0$). The confinement parameter is $\varepsilon =100$ for both of the cases.

Figure 3

The transport dynamics of the charged particles for different Peclet numbers with $\varepsilon =100$ ignoring the confinement. The solid lines in the insets presents particle velocity $v_p$, and the dashed lines are for suspension velocity $v_s$. The parameters used for calculations are $a=100$ nm, $\gamma =0.033\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}, n_0=1$ mM, and ${\varphi }_0=0.1$.

Figure 4

(a) The time evolution of drying front and (b) liquid-solid transition layer in the log-log plot for different Peclet numbers. Other parameters used here are the same as those in Fig. 3.

Figure 5

The transport dynamics of the charged particles for various confinement effects at the moderate Peclet number $\mathrm{Pe}=50$. The other parameters are used by $a=100$ nm, $\gamma =0.033\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}, n_0=1$ mM, and ${\varphi }_0=0.1$. The orange lines present the concentrations when a dried solid film is achieved.

Figure 6

The velocity of the particles (dashed lines) and the suspension (solid lines) at time $\tau =0.2$ for various confinement parameters $\varepsilon$. The insets present that the particle velocity ${\tilde{v}}_p$ varies as a function of concentration $\varphi$ with the given suspension velocity ${\tilde{v}}_s$ for different $\varepsilon$. The given ${\tilde{v}}_s$ in the insets are calculated by the initial concentration ${\varphi }_0=0.05$ (pink) and ${\varphi }_0=0.1$ (black), respectively.

Figure 7

(a) The evolution of the drying front and (b) the evolution of the liquid-solid transition layer for different confinement parameters. The inset in (b) shows the log-log plot of the liquid-solid transition layer over time. The growth of the liquid-solid transition font $L_{tf}$ before the formation of the drying front can hold the same linear behavior. Other parameters are the same in Fig. 5.

Figure 8

Time evolution of concentration of the charged particles ($a=50$ nm, $\gamma =0.132\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}$) for various monovalent salt loadings. Here, $\varepsilon =200$ is used to keep the same cell height for the large particles ($a=100$ nm). Other parameters are $\mathrm{Pe}=50$ and ${\varphi }_0=0.1$.

Figure 9

The evolution of the particles velocity and the suspension velocity when drying the suspension with low salt concentration $n_0=0.1$ mM. The yellow dashed line connects the liquid-solid transition front $L_{tf}$ over time. Other parameters are the same in Fig. 8.

Figure 10

(a) The evolution of drying front and (b) the evolution of liquid-solid transition layer for different salt loadings. The inset is the log-log plot of the main profiles in (b). The time guided by the arrow in (b) is for the occurrence of the drying front. Other parameters are the same as those in Fig. 8.

Figure 11

Reproducing the experiments by the theoretical model. (a) The experiments for colloidal silica spheres with $a=14$ nm, $\gamma =0.5\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}$, and the salt concentration $n_0=0.5$ mM. The diffusion constant of an individual particle is given by $D_0=15.3\mu {\mathrm{m}}^2/\mathrm{s}$ and the average evaporation is $v_f=0.41\mu \mathrm{m}/\mathrm{s}$. The parameters used for calculation by our model are estimated roughly as $\mathrm{Pe}=320, \tau =0.15$, and ${\varphi }_0=0.193$. (b) The experiments for colloidal silica spheres with $a=8$ nm, $\gamma =0.5\mathrm{e}\mathrm{n}{\mathrm{m}}^{-2}$, and salt concentration $n_0=5$ mM. The diffusion constant of an individual particle is given by $D_0=26.8\mu {\mathrm{m}}^2/\mathrm{s}$ and the average evaporation is $v_f=0.28\mu \mathrm{m}/\mathrm{s}$. The parameters used for calculation by model are estimated roughly as $\mathrm{Pe}=105, \tau =0.19$, and ${\varphi }_0=0.195$.