Stratification Dynamics in Drying Colloidal Mixtures - Langmuir (ACS

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Stratification dynamics in drying colloidal mixtures Michael P. Howard, Arash Nikoubashman, and Athanassios Z. Panagiotopoulos Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00543 • Publication Date (Web): 28 Mar 2017 Downloaded from http://pubs.acs.org on March 31, 2017

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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 of America, and Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz, Germany E-mail: [email protected]

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 ∗

To whom correspondence should be addressed Princeton University ‡ Johannes Gutenberg University Mainz †

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the thickness of the stratified layer decreased. A model based on dynamical density functional theory is proposed to explain the observed phenomena.

Introduction Drying colloidal dispersions 1–3 are encountered in many technological processes, including applying latex paints, 4,5 ink jet 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, differential 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 to accumulate in a layer on top of the film. True-

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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 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 mix-

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ture 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.

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 . Netural colloids were modeled using the purely repulsive Weeks-Chandler-Andersen (WCA) potential, 32  σ 6 1 12 σ  ij ij  4ε r < 21/6 σij − + r r 4 Uij (r) = ,   1/6 0 r ≥ 2 σij

(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

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modeled with a repulsive potential,  " W 3 √ # W 9  2 σ 10 σi  i  − + z < (2/5)1/6 σiW  εW 15 z z 3 UiW (z) = ,    0 z ≥ (2/5)1/6 σiW

(2)

where z is the height of the colloid’s center above the substrate, εW sets the strength of the repulsion and σiW = (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. This simulation method neglects solvent flow and solvent-mediated hydrodynamic forces between colloids. The Langevin dynamics equation of motion for the k-th colloid is

mi¨ rk = Fk − γi r˙ k + η k (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 η k is a random force with independent Gaussian-distributed components. Each component α of the random force has zero-mean and variance hηk,α (t)ηk′ ,α′ (t′ )i = 2γi kB T δk,k′ δα,α′ δ(t − t′ ).

(4)

A planar solvent interface was modeled using a purely repulsive harmonic potential, 26,36,37

UiE (z) =

    0    

z 0 indicates that big colloids have travelled farther than small colloids, and suggests the formation of a stratified layer. We observed that, for the model considered here, ∆hzS i − ∆hzB i correlated well with the apparent height of the stratified layer in the density profiles, and that the height of the stratified layer grew larger and faster for larger dB . For a given diameter, the height of the stratified layer increased with time, and at a given time, the size of the stratified layer was larger for larger colloid diameters. This finding is in agreement with recent experiments and simulations performed by Mart´ın-Fabiani et al., 53 who also observed an increase in the height of the stratified layer when they increased the size asymmetry from roughly dB /dS = 4 to dB /dS = 7. We also note that when we performed additional simulations for dB = 2 dS (Figure S3 of the Supporting Information), the height of the stratified layer further decreased to the point that it became challenging to define the layer, which is consistent with our other findings and the results of Mart´ın-Fabiani et al. 53 40

∆〈zS〉 − ∆〈zB〉 [dS]

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dB = 4 dS dB = 6 dS dB = 8 dS

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5000

7500

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t [τ]

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

Effect of drying rate on stratification For a given particle size ratio, reducing the drying rate is expected to decrease the extent of stratification because fewer colloids accumulate at the interface, and the colloids remain distributed closely to their equilibrium profiles. Our simulations and the work of Fortini et al. showed that inverted stratification is exhibited in the regime where PeS ≫ 1 and 14


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PeB ≫ 1. In our simulations for dB = 8 dS and v = 0.02 dS /τ , the film Péclet numbers were PeS = 15 and PeB = 120, If the drying rate is reduced by a factor of 10 (v = 0.002 dS /τ ), the film Péclet numbers are PeS = 1.5 and PeB = 12, and primarily big colloids are expected to accumulate at the interface. Reducing the drying rate further to v = 0.0002 dS /τ reduces the Péclet numbers to PeS = 0.15 and PeB = 1.2. For this slowest drying speed, essentially no accumulation of small colloids is expected at the interface, and only a small number of big colloids should accumulate. It is conceivable in these small Péclet number regimes that inverted stratification might not occur anymore because primarily big colloids are expected to accumulate at the interface. In order to test this hypothesis, we performed additional simulations at the slower drying rates, and computed the number density profiles at the final film height H = 100 dS , as shown in Figure 7. As the drying rate was reduced, the small colloids approached a more uniform distribution. For v = 0.0002 dS /τ , there was nearly no gradient in nS , which is 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 15


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0.6 0.4 0.2 0.0 -100

-80

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z − H [dS]

-20

0

Figure 7: Number density profiles at final film height H = 100 dS for dB = 8 dS after drying at different interface speeds. simulation. For v = 0.02 dS /τ , drying was stopped at t = 5 000 τ (intermediate film height H = 200 dS ) and at t = 10 000 τ (final film height H = 100 dS ). 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 Zimmerman 21 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 experiments 28 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 ≈ d3i (∂P/∂z). The drag force on the sphere was FD = −Ki γi ui , 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

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ui ≈ d3i (∂P/∂z)/Ki γi , and so the difference in velocity ∆u = uB − uS is ∆u/uS ≈

dB dS

2

KS KB

− 1,

(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 ∆u/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 = 0.02 dS /τ . Representative velocity profiles are shown in Figure 8 for dB = 8 dS for the initial (t = 0 τ ) and final (t = 2500 τ ) 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 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 ∆u after a given drying time for each particle diameter. For each set of velocity profiles, ∆u was averaged in a window z ∗ ± 5 dS . We confirmed that this choice of the averaging window had a negligible impact on the calculated ∆u by decreasing the size of the window by a factor of 2. For a given big colloid diameter, the measured migration velocity

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uS, t = 2500 τ uB, t = 2500 τ

0.00

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∆u

-0.02

-140

-120

-100

-80

-60

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z − H [dS]

Figure 8: Small and big colloid velocity profiles for dB = 8 dS . The interface speed v = 0.02 dS /τ is shown as a dotted line. The velocity difference ∆u is computed at z ∗ (Figure 5), and is indicated by an arrow. appeared to approach a steady-state 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 ∆u/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; 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

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1

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1.5

t = 500 τ t = 1000 τ t = 1500 τ

t = 2000 τ t = 2500 τ

1.0 4

6

8

dB / dS

Figure 9: Measured migration velocity at z ∗ scaled according to eq. (9) for varied colloid size ratios and drying times with v = 0.02 dS /τ . Scaling exponent predicted by eq. (9) in the high-density limit is also indicated for comparison. In the typical dynamical density functional theory formulation, the number density flux of species i, ji , is given by 58 ji = −

ni (r, t) ∇µi (r, t), Γi

(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 = ni ui . In one dimension, this gives a difference in migration velocity:

∆u/uS =

∂µB /∂z ∂µS /∂z

ΓS ΓB

− 1.

(12)

The effective friction coefficient can be approximated by Γi ≈ Ki γi . The approximate scaling of Ki proposed by Fortini et al. 26 then predicts Γi ∼ di in the low-density regime and Γi ∼ d2i in the high-density regime. When dB > dS , there is then a competition between the relative

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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 ∆u > 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 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 approximation 47 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

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8

Figure 10: Measured migration velocity at z ∗ scaled according to eq. (12) for varied colloid size ratios and drying times with v = 0.02 dS /τ . Scaling exponent predicted by eq. (12) in the high-density limit is also indicated for comparison. 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 Boubl´ıkMansoori-Carnahan-Starling-Leland equation of state, 61,62 a generalization of the accurate Carnahan-Starling hard-sphere-fluid equation of state 63 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.

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

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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 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 the 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 this 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 even 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 hydrodynamicor 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 fullly hydrodynamics resolved can be used to assess the quantitative accuracy of the model and importance of these contributions. This is left as a subject of future work.

Supporting Information Number density profiles during drying for dB = 2 dS , 4 dS , and 6 dS at v = 0.02 dS /τ .

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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 Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI1238993) 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 number NI 1487/2-1.

References (1) Russel, W. B.; Seville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (2) Russel, W. B. Mechanics of Drying Colloidal Dispersions: Fluid/Solid Transitions, Skinning, Crystallization, Cracking, and Peeling. AIChE J. 2011, 57, 1378–1385. (3) Routh, A. F. Drying of thin colloidal films. Rep. Prog. Phys. 2013, 76, 046603. (4) Keddie, J. L. Film formation of latex. Mater. Sci. Eng., R 1997, 21, 101–170. (5) Keddie, J. L.; Routh, A. F. Fundamentals of Latex Film Formation: Processes and Properties; Springer: Dordrecht, 2010.

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