Stratification Dynamics in Drying Colloidal Mixtures - ACS Publications

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Stratification Dynamics in Drying Colloidal Mixtures Michael P. Howard,† Arash Nikoubashman,‡ and Athanassios Z. Panagiotopoulos*,† †

Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz, Germany

‡

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S Supporting Information *

ABSTRACT: Stratification in binary colloidal mixtures was investigated using implicit-solvent molecular dynamics simulations. For large particle size ratios and film Péclet 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 Péclet 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 Péclet 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.

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INTRODUCTION Drying colloidal dispersions1−3 are encountered in many technological processes, including applying latex paints,4,5 inkjet printing,6,7 manufacturing polymer LED displays,8,9 and spraying pesticides.10 In these technologies, colloidal particles are typically initially dispersed in a liquid solvent such as water. The dispersion may have a complex formulation, containing particles of different sizes and materials,11 surfactants,12 or polymers.13 The solvent is evaporated from the dispersion, the colloid concentration increases until a close-packed structure is formed, and a solid coating or film is deposited.4,5 The drying process can significantly influence the structure of the deposited solid. Key parameters of this process include the evaporation rate and drying angle with respect to the substrate. For example, in the well-known “coffee-ring” effect, capillary flows within a pinned drying droplet force colloids outward to form a concentrated ring-like deposit.14−19 Complex dynamics are observed during drying parallel to a substrate, including the formation of solidification, cracking, and dewetting fronts, which significantly influence the final morphology of the film.3,20 Fast drying perpendicular to a substrate leads to an accumulation of colloids at the drying interface,21−25 often referred to as skin-layer formation. In colloidal mixtures with multiple particle types, different rates of accumulation at the interface can induce a separation into distinct stratified layers under certain conditions,3 as schematically illustrated in Figure 1 for a binary mixture of particles with different sizes. Such stratification may be advantageous, e.g., to create a multilayer coating during a single drying step, or undesirable, e.g., if a uniform particle coating is required. Nikiforow et al.27 showed that mixtures of charged and neutral particles can undergo a stratification process that causes the neutral particles © 2017 American Chemical Society

Figure 1. Schematic illustration of stratification of a binary colloid mixture as observed by Fortini et al.26 The initially well-dispersed mixture (left) stratifies on drying into a layer enriched in small colloids on top of a layer of big colloids (right).

to accumulate in a layer on top of the film. Trueman et al. observed segregation due to stratification in binary mixtures of latex particles.28 Typically, the large particles accumulated at the interface, but it was found that the stratification behavior depended on the evaporation rate, initial volume fractions, and particle size ratio. It is an important challenge to model drying colloidal mixtures in order to effectively engineer their properties. There have been several prior theoretical and simulation studies of drying of colloidal suspensions. Routh and Zimmerman modeled skin-layer formation for a suspension of monodisperse colloidal particles.21 Central to their model is the film Péclet number, Pe = H0v/D, which quantifies the relative contributions of advection and diffusion to particle motion. Received: February 16, 2017 Revised: March 27, 2017 Published: March 28, 2017 3685

DOI: 10.1021/acs.langmuir.7b00543 Langmuir 2017, 33, 3685−3693

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Langmuir Here, H0 is the initial film height, v is the typical speed of the interface, and D is the particle diffusion coefficient. When Pe ≪ 1, diffusion dominates, and particles remain uniformly distributed in the film. When Pe ≫ 1, however, particles accumulate at the drying front faster than they diffuse away, and a skin layer forms. Routh and Zimmerman’s theory predicts that the magnitude of the particle concentration gradient in the film should scale with Pe1/2. Ekanayake et al. tested this prediction experimentally using magnetic resonance profiling25 and found modest agreement (the gradient scaled with Pe0.8). Trueman et al. extended the theory of Routh and Zimmerman to binary mixtures of colloidal particles of different sizes.29 Each particle species i was assigned a Péclet number, Pei, that depended on the species diffusion coefficient Di. Larger particles have larger Pei since Di is expected to decrease with increasing particle diameter according to the Stokes−Einstein relation. A stratified layer of bigger colloids was predicted to form on top of a layer of smaller colloids, with optimal stratification occurring when the big particles had PeB > 1 and the small particles had PeS < 1. Atmuri et al. further extended this work to mixtures of charged particles.30 Reasonable qualitative agreement was obtained between model predictions and experiments in this regime.28,30 Recently, Fortini et al. studied the stratification of a binary mixture of neutrally buoyant, purely repulsive spheres with PeS > 1 and PeB > 1 in both experiments and computer simulations.26 They observed a peculiar inverted stratification with a layer of the smaller colloids segregated to the top of the dried film and the larger colloids pushed to the bottom. The experiments of Trueman et al. suggested similar stratification behavior only for large particle size ratios (roughly 4:1 by diameter).28 However, such a stratification of small colloids on top of big colloids was not predicted by their theoretical model,29 and Trueman et al. hypothesized that this effect may have stemmed from depletion forces and sedimentation in the experiments that were not accounted for in the theory.28 Cheng and Grest observed a similar inverted stratification in molecular simulations of the dispersion of colloids into a polymer film, with a polymer skin layer formed on top of the colloids at fast drying rates.31 In order to explain the observed inversion, Fortini et al. proposed an alternative model based on a force balance between the osmotic pressure gradient and the drag force on a colloid.26 Their model predicts that the migration velocity away from the drying interface is faster for larger colloids and that the difference between the migration velocities of big and small colloids in a binary mixture is a power law of the particle diameter ratio. Qualitatively, this model correctly predicts that the small colloids should be found on top of the larger colloids. However, no quantitative validations of the model’s predictions, such as the power-law scaling of the migration velocities with the particle diameter, were reported, and the mechanism for the observed inverted stratification remains an open question. In this work, the dynamics of stratification were investigated for a binary colloid mixture of varying diameter ratios in order to provide insight into the inverted stratification observed by Fortini et al.26 Initially, size ratios, drying rate, and initial colloid concentration comparable to the work of Fortini et al.26 were studied using implicit-solvent molecular dynamics simulations, and inverted stratification with a layer of small colloids segregated on top of a layer of big colloids was observed. The particle size ratio and drying rate were then systematically varied over a significantly larger range of values than studied by

Fortini et al.26 Detailed dynamics of the inverted stratification were quantified for the first time from the colloid density and velocity profiles measured in the simulations. The stratified layer of small colloids was found to grow faster and to larger sizes for larger diameter ratios. Inverted stratification was found to occur even at moderate Péclet numbers (slower drying rates), PeS ≪ 1 and PeB ∼ 1, although the extent of stratification was much less compared to larger Péclet numbers (faster drying rates), PeS ≫ 1 and PeB ≫ 1. A model based on dynamical density functional theory is proposed to capture the observed phenomena.

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SIMULATION MODEL AND METHODS A binary mixture of small (S) and big (B) spherical colloids with diameters dS and dB and masses mS and mB was studied. The masses were set proportionally to the colloid volumes, mB/ mS = (dB/dS)3. Neutral colloids were modeled using the purely repulsive Weeks−Chandler−Andersen (WCA) potential32 ⎧ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 1⎤ ij ij ⎪ ⎜ ⎟ ⎜ ⎟ + ⎢ ⎥ r < 21/6σij 4 − ε ⎪ ⎝ ⎠ ⎝ ⎠ 4 r r ⎣ ⎦ ⎨ Uij(r ) = ⎪ ⎪0 r ≥ 21/6σij ⎩

(1)

where r is the distance between the centers of colloids of types i and j, ε sets the strength of the repulsion, and σij = (di + dj)/2. The mixture was supported on a smooth substrate modeled with a repulsive potential ⎧ ⎡ ⎛ W ⎞ 9 ⎛ W ⎞3 ⎪ ε ⎢ 2 ⎜ σi ⎟ − ⎜ σi ⎟ + ⎪ W W ⎝ z ⎠ Ui (z) = ⎨ ⎢⎣ 15 ⎝ z ⎠ ⎪ ⎪0 ⎩

⎤ 10 ⎥ z < (2/5)1/6 σi W 3 ⎥⎦ z ≥ (2/5)1/6 σi W

(2)

where z is the height of the colloid’s center above the substrate, εW sets the strength of the repulsion, and σW i = (dS + di)/2. The solvent was modeled implicitly using Langevin dynamics simulations,33−35 which incorporate the effects of Brownian motion and drag from the solvent on the colloids. The employed simulation method neglects solvent flow and solventmediated hydrodynamic forces between colloids. The Langevin dynamics equation of motion for the kth colloid is mi rk̈ = Fk − γi rk̇ + ηk(t )

(3)

where rk is the position of the colloid, Fk is the conservative force acting on the colloid, γi is the friction coefficient with the implicit solvent, and ηk is a random force with independent Gaussian-distributed components. Each component α of the random force has zero mean and variance ⟨ηk , α (t )ηk ′ , α ′(t ′)⟩ = 2γikBTδk , k ′δα , α ′δ(t − t ′)

(4)

A planar solvent interface was modeled using a purely repulsive harmonic potential26,36,37 ⎧0 z 0 indicates that big colloids have traveled farther than small colloids and suggests the formation of a stratified layer. We observed that for the model considered here Δ⟨zS⟩ − Δ⟨zB⟩ correlated well with the apparent height of the stratified

Figure 7. Number density profiles at final film height H = 100dS for dB = 8dS after drying at different interface speeds.

Figure 6. Relative center-of-mass displacements during drying at v = 0.02dS/τ. 3689


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Langmuir consistent with the small colloid film Péclet number being reduced from PeS ≫ 1 to PeS ≪ 1. Interestingly, inverted stratification was still observed for the big colloids at all drying rates. The extent of this stratification was noticeably lessened as the drying rate decreased and PeB approached unity. For the fastest drying rate, there were essentially no big colloids in the stratified layer. At the slower drying rates, more big colloids were found closer to the interface. However, in all cases, there was still an apparent gradient in nB, with the big colloid density decreased near the interface, suggesting that the mechanism driving inverted stratification at high Péclet numbers is also relevant even when PeS ≲ 1. Comparison to Theoretical Models. As pointed out by Fortini et al., the observed stratification appears to be a purely nonequilibrium separation.26 We can estimate from the nominal volume fractions of the final state that an equivalent bulk mixture would be a single, dispersed phase using the phase diagram for a binary hard-sphere mixture.54,55 We confirmed this estimate for our films by direct simulation. For v = 0.02dS/ τ, drying was stopped at t = 5000τ (intermediate film height H = 200dS) and at t = 10000τ (final film height H = 100dS). The small and big colloids mixed in both cases and adopted density profiles qualitatively similar to those shown in Figure 2. Structuring was increased near the interfaces due to the increased confinement. However, no phase separation was observed on the length scales obtained in Figure 4. Although the drying model of Routh and Zimmerman21 shows reasonable agreement with experiments of a monodisperse suspension,25 the extension to binary mixtures by Trueman et al.29 was unable to qualitatively capture the inverted stratification observed in their experiments28 or our simulations. Fortini et al. proposed a model that qualitatively predicts inverted stratification.26 Briefly reprised, the drying process is modeled using a balance of forces due to the osmotic pressure gradient and the drag force on a particle. At a given height in the film position with pressure gradient ∂P/∂z, the net force exerted on the particle is approximated as FP ≈ di3(∂P/ ∂z). The drag force on the sphere was FD = −Kiγiui, where ui is the colloid velocity along the z-axis and Ki is the sedimentation coefficient that depends on the particle volume fractions and is equal to 1 in the dilute limit.1 Balancing forces gives ui ≈ di3(∂P/∂z)/Kiγi, and so the difference in velocity Δu = uB − uS is ⎛ d ⎞2 ⎛ K ⎞ Δu/uS ≈ ⎜ B ⎟ ⎜ S ⎟ − 1 ⎝ dS ⎠ ⎝ KB ⎠

Figure 8. Small and big colloid velocity profiles for dB = 8dS. The interface speed v = 0.02dS/τ is shown as a dotted line. The velocity difference Δu is computed at z* (Figure 5) and is indicated by an arrow.

qualitatively expected because there was no flux of colloids through the substrate or drying interface. It is apparent that during drying the velocity of the big colloids was faster than the velocity of the small colloids. This behavior is qualitatively expected because, on average, the big colloids must migrate faster than the small colloids in order for inverted stratification to occur. Note that there is also a region where the big colloids move downward faster than the interface speed. This region was larger and had a faster average downward speed for larger colloid size asymmetries, enhancing the stratification. The scaling prediction of eq 9 was tested by computing the difference in migration velocity Δu after a given drying time for each particle diameter. For each set of velocity profiles, Δu was averaged in a window z* ± 5dS to improve sampling. We confirmed that this choice of the averaging window had a negligible impact on the calculated Δu by decreasing the size of the window by a factor of 2. For a given big colloid diameter, the measured migration velocity appeared to approach a steadystate value as the drying time increased. For completeness, data are reported here for all time points measured. The migration velocities were scaled according to eq 9 and are shown plotted on logarithmic axes in Figure 9. For a given drying time, the scaled migration velocity increased as the particle size ratio increased, which is qualitatively consistent with the model of Fortini et al.26 However, the measured Δu/uS + 1 appears to scale with an exponent smaller than predicted by the theory. In fact, the scaling appears to be sublinear and to remain roughly constant for all drying times considered here;

(9)

making use of the fact that γB/γS = dB/dS. The authors argue that at low densities KS/KB ≈ 1, while at high densities KS/KB ≈ dS/dB.26 Their model then predicts that Δu/uS + 1 should scale as a power law of dB/dS with an exponent between 1 and 2. In order to test this prediction, the average colloid velocity profiles, ui, were computed during the early stages of drying for v = 0.02dS/τ. Representative velocity profiles are shown in Figure 8 for dB = 8dS for the initial (t = 0τ) and final (t = 2500τ) times for which the velocity profile was measured. Initially, the average velocities were essentially zero everywhere in the film, as expected at equilibrium. During drying, both small and big colloids at the interface had an average velocity consistent with the interface speed v, shown as a dotted line in Figure 8. Below the interface, the colloids also migrated downward as expected, and the migration speed decayed to zero farther into the film. This behavior at the boundaries was

Figure 9. Measured migration velocity at z* scaled according to eq 9 for varied colloid size ratios and drying times with v = 0.02dS/τ. Scaling exponent predicted by eq 9 in the high-density limit is also indicated for comparison. 3690


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Langmuir however, there is insufficient data to numerically quantify this exponent. Hence, although the model qualitatively predicts the correct inversion, it does not appear to quantitatively predict the rate of stratification. One possible source of error in the models by Trueman et al.29 and Fortini et al.26 is that their approximations for the diffusive driving force (fundamentally, the chemical potential gradient) might not be sufficiently accurate. Since DFT accurately predicts the equilibrium density profiles (Figure 2), the equilibrium free energy functional may also provide a reasonable approximation for the chemical potential gradients during out-of-equilibrium drying. We propose a model for stratification in the framework of dynamical density functional theory,56,57 which can be considered a correction to the form of the chemical potential employed by Trueman et al.29 In the typical dynamical density functional theory formulation, the number density flux of species i, ji, is given by58 n (r , t ) ji = − i ∇μi (r, t ) Γi

Figure 10. Measured migration velocity at z* scaled according to eq 12 for varied colloid size ratios and drying times with v = 0.02dS/τ. Scaling exponent predicted by eq 12 in the high-density limit is also indicated for comparison.

et al.26 for the high-density regime, ΓS/ΓB ∼ (dB/dS)−2. This finding suggests that the proposed dynamical density functional theory may be able to quantitatively predict the evolution of the colloid density profiles, including the inverted stratification. Unfortunately, solution of eqs 10 and 11 for direct comparison with the density profiles obtained in simulation is numerically challenging and is left as the subject of future work. After the submission of this article, Zhou et al.59 proposed a similar stratification model based on chemical potential gradients. They employed a simpler form of the free energy functional using a local-density approximation47 and the second-order virial expansion for a hard-sphere mixture. This approximation is expected to be reasonable for a dilute mixture with weak variations in density.60 Zhou et al. showed that their model is able to qualitatively capture the inverted stratification behavior, and their predictions for regimes where stratification can occur are in reasonable agreement with previous experiments and simulations.59 However, the drying density profiles predicted by their model appear to be missing some qualitative features observed in our simulations, such as the trapping of colloids initially layered near the interface. The reason for these differences is again likely related to the approximate chemical potential gradients derived from the free energy functional. The authors note that their model can be generalized to a wider range of densities with a more accurate equation of state. The model that we have proposed does precisely that. The White Bear version of fundamental measure theory is based on the Boublı ́k−Mansoori−Carnahan−Starling−Leland equation of state,61,62 a generalization of the accurate Carnahan−Starling hard-sphere-fluid equation of state63 to hard-sphere mixtures. Moreover, our proposed model employs a nonlocal free energy functional that should be more accurate near interfaces where density variations occur. The results of Zhou et al.59 fully support our proposed model.

(10)

where μi(r, t) = δF/δni(r,t) is the chemical potential obtained by functional differentiation of F and Γi is the effective friction coefficient. The number density profiles should then evolve according to the conservation law:

∂ni = −∇·ji ∂t

(11)

For noninteracting particles, eqs 10 and 11 reduce to the familiar diffusion equation. The diffusive flux in eq 10 can be equivalently represented as a net velocity through ji = niui. In one dimension, this gives a difference in migration velocity: ⎛ ∂μ /∂z ⎞⎛ Γ ⎞ ⎟⎟⎜ S ⎟ − 1 Δu/uS = ⎜⎜ B ∂ ∂ μ / z ⎝ S ⎠⎝ ΓB ⎠

(12)

The effective friction coefficient can be approximated by Γi ≈ Kiγi. The approximate scaling of Ki proposed by Fortini et al.26 then predicts Γi ∼ di in the low-density regime and Γi ∼ di2 in the high-density regime. When dB > dS, there is then a competition between the relative magnitudes of the chemical potential gradients and the friction coefficients to determine which species migrates faster. Typically, the chemical potential gradient for the big colloids is larger than for the small colloids because more volume is excluded to the big colloids, and so the driving force for migration is larger. But, the big colloid friction coefficient is also larger, reducing the net migration velocity induced. The extent of inverted stratification should increase with dB when ∂μB/∂z increases relative to ∂μS/∂z as a function of dB faster than the size ratio increases. We computed the chemical potential gradients at the peak position z* using the density profiles obtained from the simulations and the equilibrium Helmholtz free energy functional and found that eq 12 correctly predicted Δu > 0, which is necessary for inverted stratification. The measured migration velocities were then scaled by the computed chemical potential gradients in order to test eq 12 against the simulations. These data are shown in Figure 10 on logarithmic axes. It appears that the scaled migration velocities follow a power law of the size ratio with an exponent of approximately −2, which is in agreement with the scaling proposed by Fortini

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CONCLUSIONS We performed implicit-solvent molecular dynamics of drying binary colloidal mixtures. In agreement with a recent study by Fortini et al.,26 we showed that an inverted stratification occurs at large film Péclet numbers, with a layer of small colloids segregated to the top of the film, and big colloids pushed toward the bottom. The size of this stratified layer depended on the particle size ratio, with larger stratified layers forming for larger size ratios. We also showed that a similar inverted stratification occurs for slower drying rates, but to a lesser 3691


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Waters sustained-petascale computing project, which is supported by the National Science Foundation (Awards OCI0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana− Champaign and its National Center for Supercomputing Applications. A.N. received support from the German Research Foundation (DFG) under Project NI 1487/2-1.

extent. The inverted stratification effect appeared to persist even as the big colloid Péclet number approached unity. We tested the drying model proposed by Fortini et al.26 and found that their model did not predict the correct scaling of the particle migration velocities as a function of size ratio. We proposed an alternative model based on dynamical density functional theory (DDFT), which showed reasonable scaling agreement with the measured migration velocities. Qualitatively, inverted stratification occurs in our DDFT model because the chemical potential gradients driving diffusion of the big colloids are larger than for the small particles, and the magnitude of this driving force increases with particle diameter faster than the mobility decreases with diameter, resulting in a net faster migration velocity for larger particles. This model may provide useful quantitative predictions for engineering the extent of segregation in drying colloidal mixtures. In the current work, we have neglected the influence of solvent-mediated hydrodynamics. Although experiments by Fortini et al. suggested that inverted stratification still occurs in the presence of solvent,26 an investigation of the extent to which solvent flow and hydrodynamic forces influence stratification is warranted. We have also neglected any hydrodynamic- or local-density-dependent contributions to the friction coefficients in the DDFT. Direct solution of the density profiles from the proposed DDFT and comparison to simulations with fully resolved hydrodynamics can be used to assess the quantitative accuracy of the model and importance of these contributions. This task is left as a subject of future work.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00543. Number density profiles during drying for dB = 2dS, 4dS, and 6dS at v = 0.02dS/τ (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.Z.P.). ORCID

Michael P. Howard: 0000-0002-9561-4165 Arash Nikoubashman: 0000-0003-0563-825X Athanassios Z. Panagiotopoulos: 0000-0002-8152-6615 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge use of computational resources supported by the Princeton Institute for Computational Science and Engineering (PICSciE) and the Office of Information and Technology’s High Performance Computing Center and Visualization Laboratory at Princeton University. Financial support for this work was provided by the Princeton Center for Complex Materials (PCCM), a U.S. National Science Foundation Materials Research Science and Engineering Center (grant DMR-1420541). Additionally, M.P.H. received Government support under Contract FA9550-11-C-0028 and awarded by the Department of Defense, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 FR 168a, and the Blue 3692


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Stratification Dynamics in Drying Colloidal Mixtures - ACS Publications

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