Diffusiophoresis in nonadsorbing polymer solutions: The Asakura-Oosawa model and stratification in drying films

A colloidal particle placed in an inhomogeneous solution of smaller nonadsorbing polymers will move towards regions of lower polymer concentration in order to reduce the free energy of the interface between the surface of the particle and the solution. This phenomenon is known as diffusiophoresis. Treating the polymer as penetrable hard spheres, as in the Asakura-Oosawa model, a simple analytic expression for the diffusiophoretic drift velocity can be obtained. In the context of drying films we show that diffusiophoresis by this mechanism can lead to stratification under easily accessible experimental conditions. By stratification we mean spontaneous formation of a layer of polymer on top of a layer of the colloid. Transposed to the case of binary colloidal mixtures, this offers an explanation for the stratification observed recently in these systems [A. Fortini et al., Phys. Rev. Lett. 116, 118301 (2016)]. Our results emphasize the importance of treating solvent dynamics explicitly in these problems and caution against the neglect of hydrodynamic interactions or the use of implicit solvent models in which the absence of solvent backflow results in an unbalanced osmotic force that gives rise to large but unphysical effects.

Figure 1

Schematic of a drying liquid (cyan) film containing colloidal particles (dark blue) and smaller polymer molecules (orange). The evaporation speed is $v_{\mathrm{ev}}$.

Figure 2

Schematic of a large particle (blue) immersed in a solution of AO-model polymer (orange) of radius $R$, in a solvent of smaller molecules (green). The particle excludes the polymer from the layer of width $R$ (indicated by a dotted line). Along the horizontal direction, there are gradients of both the polymer contribution to the pressure ($\mathrm{\Pi }$) and the solvent contribution ($p_{\mathrm{s}}$).

Figure 3

Schematic of a flat wall (blue) in contact with AO-model polymer (orange) in a solvent of smaller molecules (green). The wall excludes the polymer from the layer of width $R$; the top of this layer is indicated by a dotted line. There are gradients of both the polymer and solvent concentrations, parallel to the wall along the $x$ axis. The flat wall is assumed to be stationary and then the fluid flows to the left. Here $\gamma$ is the wall-solution surface free energy.

Figure 4

Plots of excess AO-model polymer $\mathrm{\Delta }(z,\tau )$ accumulating in front of a wall moving at speed $v_{\mathrm{ev}}$ (see Appendix pp2). The red curves are the exact solution of Fedorchenko and Chernov [58], while the blue curves are the approximate solution in Eq. (14). The solid curves are at the early time $\tau =t{v}_{\mathrm{ev}}^2/D_p=0.25$ when the gradient is being established, while the dashed curves are at the later time $\tau =5$ when the exponential $z$ dependence is well established. At still later times, the height of the exponential continues to increase linearly, while the width remains constant.

Figure 5

Plot of the $({\mathcal{P}}_{\mathrm{film}},{\varphi }_0)$ plane, separated into the region where the large particles accumulate at the descending interface at the top of the drying film (yellow) and the region where the large particles are excluded from the region at the top of drying film just below the interface (blue). Large values of ${\mathcal{P}}_{\mathrm{film}}$ and ${\varphi }_0$ give rise to strong diffusiophoresis and drive particles away from the top of the film.

Figure 6

Trajectories $z\left(t^{*}\right)$ of a large particle as a function of reduced time $t^{*}$ (red curves). Two of the trajectories have arrows to indicate the direction of the movement. The position of the top interface $z_{\mathrm{int}}$ is shown in blue. The green dashed curve is the separatrix that divides trajectories that touch the top interface from those that do not. Colloidal particles are excluded from the triangular region between the green dashed curve and the interface. This exclusion zone is shaded in pink. Calculations are for ${\mathcal{P}}_{\mathrm{film}}=20$ and ${\varphi }_0=0.3$.