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Classification of drying segregation states by a generalized diffusion model
The Journal of Chemical Physics
Abstract
During drying of binary colloidal mixtures, one colloidal particle component can segregate to the top surface. We investigate conditions where the segregation occurs through the analysis of a linearized diffusion model with Fick’s law generalized for binary colloidal mixtures. The present model is the simplest representation that includes cross-diffusion between different particle components to describe the segregation. Using the analytical solutions of this model, we classify states in terms of which the particle component segregates for the following variables: the mixture ratio of particle components, diffusion coefficients, and drying rates. The obtained state diagrams suggest how to control the segregation by designing material and operation conditions.
Molecular dynamics simulations are used to demonstrate that a binary solvent can be used to stratify colloidal mixtures when the suspension is rapidly dried. The solvent consists of two components, one more volatile than the other. When evaporated at high rates, the more volatile component becomes depleted near the evaporation front and develops a negative concentration gradient from the bulk of the mixture to the liquid-vapor interface while the less volatile solvent is enriched in the same region and exhibit a positive concentration gradient. Such gradients can be used to drive a binary mixture of colloidal particles to stratify if one is preferentially attracted to the more volatile solvent and the other to the less volatile solvent. During solvent evaporation, the fraction of colloidal particles preferentially attracted to the less volatile solvent is enhanced at the evaporation front, whereas the colloidal particles having stronger attractions with the more volatile solvent are driven away from the interfacial region. As a result, the colloidal particles show a stratified distribution after drying, even if the two colloids have the same size.
Colloidal suspensions are coated on substrates and upon drying produce various functional materials. The non-equilibrium structure formation of colloidal particles is induced during coating and drying’ affecting operational efficiency and material performance. To understand the relation between operation conditions and structure formation, we have developed a simulator of colloidal suspensions, SNAP. We discuss the non-equilibrium structure formation through the results calculated by SNAP.
Large scale molecular dynamics simulations for bidisperse nanoparticle suspensions with an explicit solvent are used to investigate the effects of evaporation rates and volume fractions on the nanoparticle distribution during drying. Our results show that “small-on-top” stratification can occur when PesΦs ≥ c with c ∼ 1, where Pes is the Péclet number and Φs is the volume fraction of the smaller particles. This threshold of PesΦs for “small-on-top” is larger by a factor of ~ α² than the prediction of the model treating solvent as an implicit viscous background, where α is the size ratio between the large and small particles. Our simulations further show that when the evaporation rate of the solvent is reduced, the “small-on-top” stratification can be enhanced, which is not predicted by existing theories. This unexpected behavior is explained with thermophoresis associated with a positive gradient of solvent density caused by evaporative cooling at the liquid-vapor interface. For ultrafast evaporation the gradient is large and drives the nanoparticles toward the liquid/vapor interface. This phoretic effect is stronger for larger nanoparticles and consequently the “small-on-top” stratification becomes more distinct when the evaporation rate is slower (but not too slow such that a uniform distribution of nanoparticles in the drying film is produced), as thermophoresis that favors larger particles on the top is mitigated. A similar effect can lead to “large-on-top” stratification for PesΦs above the threshold when Pes is large but Φs is small. Our results reveal the importance of including the solvent explicitly when modeling evaporation-induced particle separation and organization and point to the important role of density gradients brought about by ultrafast evaporation.
Nonequilibrium molecular dynamics simulations are used to investigate the influence of hydrodynamic interactions on vertical segregation (stratification) in drying mixtures of long and short polymer chains. In agreement with previous computer simulations and theoretical modeling, the short polymers stratify on top of the long polymers at the top of the drying film when hydrodynamic interactions between polymers are neglected. However, no stratification occurs at the same drying conditions when hydrodynamic interactions are incorporated through an explicit solvent model. Our analysis demonstrates that models lacking hydrodynamic interactions do not faithfully represent stratification in drying mixtures, in agreement with recent analysis of an idealized model for diffusiophoresis, and must be incorporated into such models in future.
When films are deposited from mixtures of colloidal particles of two different sizes, a diverse range of functional structures can result. One structure of particular interest is a stratified film in which the top surface layer has a composition different than in the interior. Here, we explore the conditions under which a stratified layer of small particles develops spontaneously in a colloidal film that is cast from a binary mixture of small and large polymer particles that are suspended in water. A recent model, which considers the cross-interaction between the large and small particles (Zhou et al., Phys. Rev. Lett., 2017, 118, 108002), predicts that stratification will develop from dilute binary mixtures when the particle size ratio (α), initial volume fraction of small particles (ϕS), and Péclet number are high. In experiments and Langevin dynamics simulations, we systematically vary α and ϕS in both dilute and concentrated suspensions. We find that stratified films develop when ϕS is increased, which is in agreement with the model. In dilute suspensions, there is reasonable agreement between the experiments and the Zhou et al.
Model:
In concentrated suspensions, stratification occurs in experiments only for the higher size ratio α = 7. Simulations using a high Péclet number, additionally find stratification with α = 2, when ϕS is high enough. Our results provide a quantitative understanding of the conditions under which stratified colloidal films assemble. Our research has relevance for the design of coatings with targeted optical and mechanical properties at their surface.
When a rigid colloidal particle is placed in a solution which is not uniform in the concentration of some solute that interacts with the particle, the particle will be propelled in the direction of higher or lower concentration of the solute. The resulting locomotion is called diffusiophoresis. Experimental observations and theoretical predictions of the migration velocity of hydrosoIs are reviewed. Present commercial applications include the formation of rubber gloves and the deposition of paint films onto a steel surface. New applications to the analysis of colloidal mixtures and solid-liquid separation are suggested.
Large scale molecular dynamics simulations are used to study drying suspensions of a binary mixture of large and small particles in explicit and implicit solvents. The solvent is first modeled explicitly and then mapped to a uniform viscous medium by matching the diffusion coefficients and the pair correlation functions of the particles. "Small-on-top" stratification of the particles, with an enrichment of the smaller ones at the receding liquid-vapor interface during drying, is observed in both models under the same drying conditions. With the implicit solvent model, we are able to model much thicker films and study the effect of the initial film thickness on the final distribution of particles in the dry film. Our results show that the degree of stratification is controlled by the Péclet number defined using the initial film thickness as the characteristic length scale. When the Péclet numbers of large and small particles are much larger than 1, the degree of "small-on-top" stratification is first enhanced and then weakens as the Péclet numbers are increased.
Polydisperse smooth and spherical biocolloidal particles were suspended in aqueous media and allowed to consolidate via evaporation-induced self-assembly (EISA). The stratification of the particles at the solid-air interface was markedly influenced but not monotonically, by the drying rate. Cross section imaging via electron microscopy indicated a structured coating morphology that was distinctive from that obtained by using mineral particles with mono- or bimodal distribution. Segregation patterns were found to derive from the interplay of particle diffusion, inter-particle forces and settling dynamics. Supporting our experimental findings, computer simulations showed an optimal drying rate for achieving maximum segregation. Overall, stratified coatings comprising nano and microparticles derived from lignin are expected to open opportunities for multifunctional structures that can be designed and predicted on the basis of experimental Peclet numbers and computational order.
When suspensions containing colloidal particles of two different sizes are coated on substrates to form films by evaporating the host fluids, the smaller particles can segregate to the top surface of the films. We investigate the effects of the evaporation rate on the segregation by use of Langevin dynamics simulations. The evaporation rate is scaled by the Brownian diffusion rate of the particles, yielding a dimensionless number which we define as the particle drying Péclet number. We show that there is a Péclet number at which the segregation is the most enhanced. Our result indicates the need for the regulation of the evaporation rate to control the segregation.
I model the drying of a liquid film containing small and big colloid particles. Fortini et al. [A. Fortini et al, Phys. Rev. Lett. 116, 118301 (2016)] studied these films with both computer simulation and experiment. They found that at the end of drying the mixture had stratified with a layer of the smaller particles on top of the big particles. I develop a simple model for this process. The model has two ingredients: arrest of the diffusion of the particles at high density, and diffusiophoretic motion of the big particles due to gradients in the concentration of the small particles. The model predicts that stratification only occurs over a range of initial concentrations of the smaller colloidal species. At concentrations that are either too low or too high, the concentration gradients due to drying are not enough to push the big particles away and so produce a layer at the top of only small particles. In agreement with earlier work, the model also predicts that large Peclet numbers for drying are needed to see stratification.
A colloidal particle placed in an inhomogeneous solution of smaller non-adsorbing 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 colloidal diffusiophoresis by this mechanism can lead to significant stratification under easily accessible experimental conditions. 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 emphasise 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 which gives rise to much larger but unphysical effects.
During the drying of bimodal colloidal suspensions containing particles of various sizes, smaller particles preferentially migrate to the top surface under particular drying conditions, leading to undesirable drying defects in batteries and in other coating applications. Despite extensive previous studies, the migration mechanism is far from being understood because few in situ observations are available to support the hypotheses. To remedy this, we use real-time photoluminescence (PL) microscopy to investigate the migration of small fluorescent latex particles co-dispersing with large nonemissive latex particles. Comparing the measured PL intensity with that predicted by a model allows us to determine the quantity of small particles near the evaporating surface. The results reveal that the fluorescent particles segregate in two steps: The primary segregation occurs early in the evaporation stage, whereas the secondary stepwise migration occurs when the air–liquid interface invades the particle consolidation layer. The latter migration is attributed to the flow-induced motion of small particles that move through interstitial spaces between large particles.
Stratification in binary colloidal mixtures was investigated using implicit-solvent molecular dynamics simulations. For large particle size ratios and film P?clet numbers greater than unity, smaller colloids migrated to the top of the film, while big colloids were pushed to the bottom, creating an "inverted" stratification. This peculiar behavior was observed in recent simulations and experiments conducted by Fortini et al. (Phys. Rev. Lett. 2016, 116, 118301). To rationalize this behavior, particle size ratios and drying rates spanning qualitatively different P?clet number regimes were systematically studied, and the dynamics of the inverted stratification were quantified in detail. The stratified layer of small colloids was found to grow faster and to larger thicknesses for larger size ratios. Interestingly, inverted stratification was observed even at moderate drying rates where the film P?clet numbers were comparable to unity, but the thickness of the stratified layer decreased. A model based on dynamical density functional theory is proposed to explain the observed phenomena.
When a liquid film of 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 counter intuitive because larger colloids which have smaller diffusion constant are expected to remain near the surface during the drying process, leaving the 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 the diffusion model which accounts for the interaction between the colloids. By studying the evolution equation both numerically and analytically, we derive the condition at which the stratified structures are obtained.
In simulations and experiments, we study the drying of films containing mixtures of large and small colloidal particles in water. During drying, the mixture stratifies into a layer of the larger particles at the bottom with a layer of the smaller particles on top. We developed a model to show that a gradient in osmotic pressure, which develops dynamically during drying, is responsible for the segregation mechanism behind stratification.
Effects of initial and boundary conditions on the distribution of the tracer in time and distance are calculated for several one-dimensional systems (infinite, semi-infinite, and finite) of tagged liquid flowing through a solid matrix, and the effects of hydrodynamic dispersion, diffusion, radioactive decay, and simple chemical interactions of the tracer are included. Exact and approximative solutions show that in most practical conditions the results of steady state experiments are influenced only up to 0.5% by changes in boundary conditions. In nonsteady state, results are influenced up to 5% in the region c/co ≃ 0.5, where c/co is the ratio of the measured concentration and the concentration of the solution entering the porous medium. Results are given in nondimensional form with graphs of representative cases.
Settling of small particles in a fluid; mathematical theory.-Small particles immersed in a liquid experience a motion which is the combination of a steady gravitational drift and a Brownian movement. If there are space variations in the density of distribution of particles, the Brownian movement produces a diffusion which tends to equalize the density. In the steady state the density n of particles is an exponential function of x, the distance below the surface of the liquid. This paper investigates the manner in which the steady state is established. A consideration of the combined effect of fall and diffusion leads to a partial differential equation for the number density of particles as a function of depth and time. A set of special solutions is obtained in terms of which a solution satisfying initial and boundary conditions can be expressed. (1) Liquid of finite depth. The solution is obtained for a liquid of finite depth with an arbitrary initial distribution n0=f(x). For the case of uniform initial distribution a reduced form of the solution is obtained which contains a single parameter. This one parameter family of curves is plotted, and from these curves, either directly or by interpolation, may be obtained the density distribution at any time for a solution of any depth, density, and viscosity, and for particles of any size and density. For small values of t, since the solution obtained converges slowly, an image method is used to obtain an integral formula for the density. (2) Liquid of semi-infinite or infinite depth. In the case of a liquid of infinite depth the solution for an arbitrary initial distribution is expressed by the Fourier integral identity. The case of zero initial density for negative x, and constant initial density for positive x is calculated, as is also the case of particles initially uniformly distributed over a layer of depth h. In the case of a liquid extending from x=0 to x=∞, the boundary conditions are satisfied by assuming a suitable fictitious initial distribution over the range from x=-∞ to x=0. The cases of uniform initial distribution, and initial distribution over a layer, are calculated. The latter case, while derived for a liquid of semi-infinite depth, gives approximately the distribution of density during the settling of a layer of particles initially distributed uniformly over a depth h at the upper end of a very long column of liquid.
We used real-time fluorescence microscopy to investigate the migration of latex particles in drying battery slurries. The time evolution of the fluorescence signals revealed that the migration of the latex particles was suppressed above the entanglement concentration of carboxymethyl cellulose (CMC), while it was significantly enhanced when CMC completely covered the surfaces of the graphite particles. In particular, a two-step migration was observed when the graphite particles flocculated by depletion attraction at high CMC/graphite mass ratios. The transient states of the nonadsorbing CMC and graphite particles in a medium were discussed, and the uses of this novel measurement technique to monitor the complex drying processes of films were demonstrated.
Some interesting aspects of multicomponent diffusion in liquids are discussed. These include the existence of a negative main term diffusion coefficient; the utility of taking different components as the solvent; the change-of-solvent transformation for Fick's law coefficients and Onsager coefficients; the validity of the Onsager reciprocal relations on changing solvents; and the calculation of partial molar volumes from diffusion data. Previous work is surveyed on the irreversible thermodynamic basis for macroscopic diffusion; the importance of reference frames and transformations among them; and second law conditions on the volume-fixed diffusion coefficient matrix. Certain diffusion descriptions in other reference frames do not preserve these second law conditions. 78 references, 5 tables.
Four theoretical aspects of ternary diffusion are investigated. Restrictions imposed on the ternary diffusion process by the entropy inequality are formulated, and their effect on the eigenvalues of the diffusion coefficient matrix is determined. Additional restrictions are formulated by accepting the Onsager reciprocal relations as valid postulates, yielding a relationship that can be regarded as a sufficient condition for the existence of real eigenvalues for the diffusion coefficient matrix. The stability of the ternary system with respect to the equilibrium state is investigated both with and without the Onsager reciprocal relations. The existence of negative concentrations is analyzed, and interpretations of the reasons for the appearance of such nonphysical results are proposed.
Key microstructural properties of particulate coatings such as porosity and particle order are established during drying. Therefore, understanding the evolution of particulate distributions during drying is useful for designing coating properties. Here, a 1D model is proposed for the particle distribution through the coating thickness at different drying times and conditions, including Brownian diffusion, sedimentation, and evaporation. Effects of particle concentration on diffusion and sedimentation rates are included. Results are condensed onto a drying regime map which predicts the presence of particle surface accumulation or sediment based on two dimensionless numbers: the Peclet number and the sedimentation number. Cryogenic scanning electron microscopy (cryoSEM) is used to image the transient particulate distributions during the drying of a model system comprised of monodisperse silica particles in water. Particle size and evaporation rates are altered to access various domains of the drying map. There is good agreement between cryoSEM observations and model predictions. © 2010 American Institute of Chemical Engineers AIChE J, 2010
The mechanism by which the particles in a drying film come into close packing during solvent evaporation has an important role to play in the final film morphology. During drying the particles can develop non-uniform concentrations across the vertical height of the film, depending on their diffusion rate. By applying the principles of classical diffusion mechanics to a hard sphere system, a theory for this novel method of stratification during drying of a two component film has been derived. The model is dependent on the particle Peclet numbers and when one is above unity and the other below, maximum stratification is observed.
In films cast from a colloidal dispersion comprising two particle sizes, we experimentally examine the distribution of particles normal to the substrate. The particle concentrations at various positions in the film are determined through atomic force microscopy and NMR profiling. The results are compared to a previously derived diffusional model. Evidence for diffusional driven stratification is found, but the importance of other flows is also highlighted. The conditions that enhance particle stratification are found to be a colloidally stable dispersion, low initial volume fractions, a low concentration of the stratifying particle, and for the Peclet numbers of the two components to straddle unity.
The movement of some chemicals in soils has been treated theoretically by considering for saturated conditions, the chemical diffusion coefficient, the percolation velocity of the water, the sorbtive properties of the soil, the average particle size of the soil, and the fractional number of sorbing sites. A model has been developed for the movement of chemical in saturated soil, based on Fick's law, conservation of energy, and a sorption isotherm. Theoretical curves for two different boundary conditions are given for realistic values of the water velocity in the pores and the measured diffusion coefficient.
A simple model is proposed for the skin formation in the evaporation process of a polymer solution at a free surface. In this model the skin is regarded as a gel phase formed near the free surface, and the dynamics is described by a diffusion equation for the polymer concentration with moving boundaries. The equation is solved both analytically and numerically. It is shown that the skin phase appears when the evaporation rate is high or when the initial polymer concentration is high. An analytical expression is given for the criterion for the skin phase to be formed.
Ceramic nanoparticle/monodisperse latex coatings with a nanoparticle-rich surface and a latex-rich body were created by depositing aqueous dispersions of monodisperse latex, approximately 550 nm in diameter, and nanosized ceramic particles onto substrates and drying. On the top surface of the dried coating, the latex particles are closely packed with nanoparticles uniformly occupying the interstitial spaces, and along the cross section, nanoparticles fill the spaces between the latex particles in the near surface region; a compacted latex structure, nearly devoid of nanoparticles, lies beneath. Cryogenic scanning electron microscopy images of partially dried coatings at successive drying stages reveal two important steps in forming this structure: top-down consolidation of latex particles and accumulation of nanoparticles in interstitial spaces among latex particles near the surface. A systematic study of the effect of processing conditions, including nanoparticle concentration, nanoparticle size, latex glass transition temperature, and drying conditions, on the final microstructure was carried out. The unique microstructure described above forms when the monodisperse latex is large enough to create pore channels for the transport of nanosized particles and the drying conditions favor "top-down" as opposed to "edge-in" drying.
Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing the new concept of a maximally random jammed state, which can be made precise. Comment: 6 pages total, 2 figures