Stratification in drying films: a diffusion–diffusiophoresis model

Abstract

This research is motivated by the desire to control the solids distribution during the drying of a film containing particles of two different sizes. A variety of particle arrangements in dried films has been seen experimentally, including a thin layer of small particles at the top surface. However, it is not fully understood why this would occur. This work formulates and solves a colloidal hydrodynamics model for (i) diffusion alone and (ii) diffusion plus excluded volume diffusiophoresis, to determine their relative importance in affecting the particle arrangement. The methodology followed is to derive partial differential equations (PDEs) describing the motion of two components in a drying film. The diffusive fluxes are predicted by generalising the Stokes–Einstein diffusion coefficient, with the dispersion compressibility used to produce equations valid up until close packing. A further set of novel equations incorporating diffusiophoresis is derived. The diffusiophoretic mechanism investigated in this work is the small particles being excluded from a volume around the large particles. The resulting PDEs are scaled and solved numerically using a finite volume method. The model includes the chemical potentials of the particles, allowing for incorporation of any interaction term. The relative magnitudes of the fluxes of the differently sized particles are compared using scaling arguments and via numerical results. The diffusion results, without any inter-particle interactions, predict stratification of large particles to the top surface. Addition of excluded volume diffusiophoresis introduces a downwards flux on the large particles, that can result in small-on-top stratification, thus providing a potential explanation of the experimental observations.

Suppression of self-stratification in colloidal mixtures with high Péclet numbers

“…Models developed by Zhou et al , 31 Sear and Warren, 30 Sear, 32 and Rees-Zimmerman and Routh 33 make predictions for the parameter values for which stratification is observed. Rees-Zimmerman and Routh and Zhou et al both used models in which the large and small colloids each obey diffusion-like equations.…”

“…Sear and Warren 30 approximated the ratio as 1. Thus, they and others 31,33 neglected the effects that occur when the width of the accumulation zone for small particles becomes comparable to the size of the larger colloids. This neglect is typically reasonable for Pe values of order ten or less.…”

Microfluidic free interface diffusion: Measurement of diffusion coefficients and evidence of interfacial-driven transport phenomena

“…In the case of neutral solutes, small molecules but also polymers [54][55][56][57][58][59] , studies are scarcer and comparisons with theoretical models still raise issues 56,59 . This last case is also key to the understanding of other phenomena such as the stratification observed during the drying of colloidal films [60][61][62][63][64] .…”

Dynamic density functional theory for drying colloidal suspensions: Comparison of hard-sphere free-energy functionals

“…4,5 Exploring the many chemical and physical parameters that affect film formation is a time-consuming experimental task, 3,8 so accurate computational models are highly desired. Most models in prior work on drying colloidal suspensions can be divided into two categories: particle-based simulations 9,14-25 and continuum models 7,9,18,19,[26][27][28][29][30] (Fig. 1).…”

“…However, a good approximation for the particle flux resulting from density gradients is required for the continuum model to be accurate. Flux models for drying suspensions have often been chosen empirically, e.g., from phenomenological equations, 26,30,34 leading to spirited debate over what is required to describe experimental observations, particularly for multicomponent suspensions. 21,28,30,34,35 Dynamic density functional theory (DDFT) is a convenient framework to systematically construct continuum models for colloidal suspensions from knowledge of the particle-level interactions and dynamics using statistical mechanical principles.…”