Effects of Light Dispersed Particles on the Stability of Dense

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B: Fluid Interfaces, Colloids, Polymers, Soft Matter, Surfactants, and Glassy Materials

Effects of Light Dispersed Particles on the Stability of Dense Suspended Particles against Sedimentation Yung-Jih Yang, Elias I. Franses, and David S. Corti J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b10172 • Publication Date (Web): 03 Jan 2019 Downloaded from http://pubs.acs.org on January 4, 2019

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The Journal of Physical Chemistry

Effects of Light Dispersed Particles on the Stability of Dense Suspended Particles Against Sedimentation Yung-Jih Yang+, Elias I. Franses, and David S. Corti* Davidson School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-2100 USA

Abstract A novel method in which vesicular dispersions of the double-chain cationic surfactant DDAB (didodecyldimethylammonium bromide) stabilize suspensions of high density titania particles was recently presented (Yang, Y.-J; Corti, D.S.; Franses, E. I. Langmuir 2015, 31, 8802-8808). At high enough DDAB concentration, the vesicles form a closepacked structure, providing strong resistance to the sedimentation of the titania particles, while the dispersions remain highly shear-thinning with moderate limiting viscosities. Here, to elucidate the key factors of the mechanism by which vesicles or other non-settling particles stabilize high density particles against sedimentation, we use Brownian dynamics simulations (BDS) to examine the sedimentation behavior of mixtures of “dense particles” that settle rapidly on their own and “light particles” that represent non-settling “rigid vesicles”. BDS confirm that for large enough values of the volume fraction

of the light

particles, the dense particles should remain suspended. The rheological behavior of the mixtures is also computed with BDS. The observed shear-thinning behavior of the light particle dispersion suggests that the suspensions of the dense particles are still flowable at high shear stresses. Furthermore, the local viscosity of light particles around the dense 1 ACS Paragon Plus Environment

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particles significantly increases with increasing

, particularly when the same

gravitational force applied in the BDS is exerted on a dense particle. The arrangement of light particles around the moving dense particles is an important factor in determining the stability of the dense particles against sedimentation. The BDS results indicate that dispersions of non-settling particles provide a general method for the stabilization against sedimentation of high density particles.

+Current address: Intel Corporation, 2501 Northwest 229th Avenue, Hillsboro OR 97124

*Author to whom correspondence should be addressed: [email protected]

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1.

Introduction Colloidal dispersions, such as suspensions (solid particles in a liquid) and emulsions

(liquid droplets in an immiscible liquid), are key components in various products, such as inks, paints, consumer products, foods, and pharmaceuticals.1 To maintain homogeneity of suspension properties, either during storage or in flow, particles or droplets should be prevented from agglomerating or coalescing, and from sedimenting or creaming. If the particles agglomerate, or become “unstable” against agglomeration (U-SAA), then clusters will form. These clusters tend to sediment or cream in gravity faster than the primary particles. Then, the suspensions also become “unstable” against sedimentation (U-SAS). Many studies have already been performed on the prevention of particle agglomeration.2-10 Approaches for keeping a suspension stable include: (a) controlling the effective Hamaker constant of the particles or the suspension medium, thereby minimizing the attractive interparticle forces; (b) reducing the particle sizes; (c) increasing the particles’ surface charges or maintaining a low ionic strength of the suspension medium, thereby increasing the repulsive electrostatic forces among the particles; or (d) adding certain polymers and surfactants at low to moderate concentrations to modify further the interparticle attractive and repulsive forces. In certain cases of inks or paints, however, preventing the sedimentation of the suspended pigment particles before the suspension is used to coat a substrate, and the liquid medium has evaporated, is more important than preventing particle agglomeration. In other examples of aqueous suspensions containing high-density colloidal particles, such as titania, silica, or iron oxide, the suspended particles can quickly settle without agglomeration and even with diameters smaller than 1000 nm.4,11



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In contrast to the problem of agglomeration, the development of strategies for preventing particle sedimentation has received little attention in the literature. To prevent dispersed particles from sedimenting, one may reduce their diameter, , or the density difference, Δ , between the particles and the suspension medium, by using hollow particles, or by attaching, physically or chemically, a polymer or a surfactant to the particle surfaces. While each of these approaches can lead to lower sedimentation rates, they may nevertheless fail to be of practical use. For example, utilizing very small particles may yield inks or paints that are more stable against sedimentation (SAS), but the resulting optical quality of the deposited ink or paint layers may be poor. Similarly, smaller particles with encapsulated drugs may have too large dissolution rates. Reducing the density difference may yet again reduce the optical qualities of the ink and paint layers, or affect negatively other properties of the suspensions. Another way to slow down the particle sedimentation is by increasing the viscosity of the suspension medium, or by generating a gel-like material with a yield stress. But then higher bulk viscosities may significantly reduce the bulk flowability or injectability of the suspensions in inkjet printing, painting, or other applications. We recently presented a novel method of using close-packed fluid vesicular dispersions for making stable suspensions of dense titania particles in water with Δ g/cm3 and

3.2

280±100 nm.12,13 Many double-chain surfactants form lamellar liquid

crystals, multilamellar vesicles or “liposomes”, and unilamellar vesicles, or simply “vesicles”.14-16 A vesicle is a hollow surfactant-based particle in which a thin layer (usually a bilayer) of surfactant contains and completely surrounds an aqueous core.


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In

our

previous

work,

we

generated

aqueous

vesicular

dispersions

of

didodecyldimethylammonium bromide (DDAB), a cationic double-chain surfactant. At concentrations from 6.5×10-4 to over 2.0 wt%, DDAB in water can form liposomes and vesicles with mild stirring, and only vesicles with stirring and sonication.17 Since the bilayer chain melting phase transition temperature of DDAB is about 16

,18 at the higher

the bilayer interior is expected to be fluid-like and the vesicles are

temperature of 25

deformable. Moreover, these vesicles contain so much water that their overall density difference is practically zero, much less than 0.01 g/cm3. Their gravitational Peclet number, , is

1. This dimensionless number is defined as the ratio of the sedimentation flux

to the macroscopic diffusion flux, and is given by19 ≡

∆

⁄

⁄6

where is the particle concentration, equal to

/3

),

force on a particle, Boltzmann constant,

Δ

(1) is the Stokes-Einstein diffusion coefficient (and is /18 is the sedimentation velocity for Stokes’ drag

is the gravitational acceleration, is the absolute temperature, and

is the sample height,

is the

is the bulk viscosity of the

is so small, the vesicles remain stable for

assumed Newtonian fluid medium. Since

months (SAS), with little size growth. At surfactant concentrations of 2 wt%, the DDAB vesicular dispersions prepared by stirring were found to form close-packed structures. Consequently, when dense titania particles, with high values of

, were added to these

close-packed DDAB dispersions, they remained suspended for more than 18 months. In contrast, in the absence of DDAB, or at DDAB concentrations less than 1 wt%, the titania particles sediment quite rapidly, without agglomeration, by as much as 0.5 cm in 30 h.5 The close-packed DDAB vesicular dispersions at high enough DDAB concentrations 5 ACS Paragon Plus Environment


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provide therefore a significant resistance against sedimentation of the titania particles. Moreover, the close-packed vesicles do not significantly impede the flow of the bulk suspension, allowing for their potential uses in ink-jet printers.13 In this article, we aim to elucidate computationally several fundamental factors of this novel stabilization mechanism and provide general guidelines on the use of vesicles or other non-sedimenting, or “light”, colloidal particles for stabilizing or slowing down drastically the sedimentation of high-density, or “dense”, particles. For such a problem, we use Brownian dynamics simulations (BDS), a computational approach that has been widely used to study other colloidal suspension properties, such as the dynamics of coagulation, flocculation, sedimentation, deposition processes, the microstructure of agglomerates, and suspension rheology.20-30 The goals of the BDS are to (1) test the hypothesis that light, nonsettling dispersed particles can slow down or prevent the sedimentation of dense, settling particles; (2) examine the effect on goal (1) of the volume fraction,

, of the light particles;

(3) study the effect of

on the bulk viscosity of the suspension of light particles; (4)

examine the effect of

on the resistance to settling, or “local viscosity”, felt by a dense

particle; and (5) determine the physicochemical mechanisms by which the light particles hinder the motion of the dense particles under the influence of gravity and Brownian motion. In Section 2, we review and summarize some of our previously published experimental data that are most relevant to the BDS and which also help to outline further the scope of this article.12 In Sections 3 and 4, we present the computational methods used and the results for mixtures of non-deformable light (non-settling) and dense (settling) particles. Using BDS, we determine the sedimentation behavior of the dense particles for


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various volume fractions of the light particles. We also examine as a function of volume fraction the bulk shear viscosities of suspensions of light particles, as well as their viscosities at the microscopic scale, or their “microrheology”. The local viscosities “felt” by the dense particles, and which arise from the surrounding vesicles,23,31-32 provide the basis for the mechanism by which the motion of the dense particles is significantly hindered by the light particles. Even though the actual DDAB vesicles are deformable, as evidenced from their rheological behavior, for simplicity, and to a first approximation, only rigid light particles are used here to model the behavior of the vesicles. Such rigid light particles effectively still describe well several aspects of high volume fraction vesicular dispersions, thereby providing important insights of the effects of vesicular dispersions on the sedimentation behavior of dense particles.

2. Experimental Section 2.1. Materials and Methods 2.1.1. Materials. Titania particles were obtained from Huntsman, type TR52, and used as received. Their density is 4.1 g/cm3, and their diameter is 280±100 nm. The particles were characterized as detailed previously.4 Ultrapure water was obtained from a Milli-Q water system (from Millipore). Didodecyldimethylammonium bromide (DDAB), in powder form, was purchased from Sigma-Aldrich, and was used as received. 2.1.2. Sample Preparation. DDAB aqueous dispersions with concentrations

of

0.5 wt% or 2.0 wt% were first shaken by hand, and then stirred magnetically for 30 min. The titania suspensions (1.0 wt%) were prepared by first mixing titania particles and



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DDAB powders. The mixed powder was dispersed in water with the above procedure. All samples were prepared at room temperature, which was 25±2

.

2.1.3. Cryo-Transmission Electron Microscopy. Cryo-TEM images were taken at the Purdue Cryo-EM Facility/Department of Biological Sciences at Purdue University. A 3.5 μL drop of the sample was put on a carbon-coated “holey film” (film with holes) supported by a TEM copper grid in a controlled environment vitrification chamber. The relative humidity in the chamber was kept close to 100% to prevent water evaporation form the sample. The drop was first gently blotted with filter paper to create an ultrathin liquid film (< 1μm) over the grid. Then, the samples were vitrified by being plunged into liquid ethane cooled to -114

. The photomicrographs were taken under magnification of 22500X with

a Titan FEG microscope and an acceleration voltage of 300 kV. 2.1.4. Rheological Measurements. The rheological data were obtained with a concentric-cylinder geometry and a TA DHR-2 rheometer. The steady-state shear-ratedependent viscosities, , of 0.5 wt% and 2.0 wt% DDAB dispersions at 25

with shear

rates, , ranging from 10-4 to 103 s-1 were measured with the flow sweep procedure. Each measurement was done twice, and the averages of the shear stress, τ, or the viscosity as a function of shear rate were obtained. 2.2. Experimental Results and Discussion The sedimentation stability of 1 wt% titania particles either in DDAB vesicular dispersions with 0.5 wt% (10.8 mM) or in 2.0 wt% (43.4 mM) surfactant, prepared with stirring only, was examined. A marked difference was observed in these two cases; see Figure 1(a). At

0.5 wt%, the particles settled substantially after two weeks, and

settled completely after about two months. At

2.0 wt%, the particles remained


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suspended for at least 18 months.12 A schematic diagram of the stabilization mechanism is shown in Figure 1(b). At low surfactant concentrations, the effective volume fractions of the vesicles were low. Hence, the dense particles were still able to move around quite easily, and therefore eventually settle under normal gravity. On the other hand, when the surfactant concentration was high enough to yield an effective volume fraction of the vesicles near the close-packed limit of rigid particles, or even higher than that for the deformable vesicles, the particles became trapped by the vesicles, and their motion was severely limited. Then, the particles were unable to settle or agglomerate at all, or any settling would not be noticeable until extremely long times had passed. The cryo-TEM technique was used to examine directly the microstructures of the DDAB vesicular dispersions in water at 0.5 wt% and 2.0 wt% to test the stabilization mechanism hypothesis; see Figure 1(c). At 0.5 wt%, while some much larger liposomes (multilamellar vesicles) may have also formed, most of the surfactant appeared to be in the form of vesicles (unilamellar vesicles). The vesicles were found to be quite spherical and polydisperse with an average diameter of 300 nm, and were not in contact with each other. At 2.0 wt%, however, in addition to the vesicles, a few vesicles with multiple layers were observed. (The much larger liposomes may have again also formed, with sizes of at least several microns, and would not appear in the resulting cryo-TEM images.) These observations are consistent with those by Regev and Khan,14 who prepared their DDAB dispersions by only shaking them for several days. The effective volume fraction of the vesicles was estimated from their size and weight fraction to be higher than 0.7, exceeding the close-packed limit for rigid spheres. Therefore, they did not retain their spherical shape, but were deformed into foam-like polyhedral shapes. Several other lines of evidence,



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including the autocorrelation function from the dynamic light scattering technique and the electrophoretic mobility from the zeta potential measurements, also supported the notion that the vesicles were quite mobile at 0.5 wt%, but had a very low mobility at 2.0 wt%, at which the vesicles were trapped or immobilized by each other and appeared to be in a tightly packed configuration.12 The cryo-TEM technique also captured certain arrangements of the titania particles within the close-packed vesicular dispersion microstructure. At

2.0 wt%, a titania particle, the black particle in Figure 2, was

found to be completely surrounded by vesicles, which impeded both the diffusive and the gravity-induced motion of the titania particle. By encountering a high resistance of the close-packed vesicles microstructures, the titania particles could no longer settle or agglomerate. The effective shear-viscosities

of 2.0 wt% DDAB vesicular dispersions were also

measured over a range of shear rates, , from 10‐ to 10 s , or a range of shear stresses, , from 10

to 10 Pa, where

≡ / .12 The results are shown in Figure 3. The vesicular

dispersion was found to be a shear-thinning fluid, with a very high viscosity of 105 mPa⋅s at shear stresses below 0.02 Pa, but with low viscosities of around 10 mPa⋅s at shear stresses exceeding 0.5 Pa.12 Such strong shear-thinning behavior was also reported previously for DDAB in water.33,34 Under normal gravity, the local shear stress exerted by a sedimenting titania particle should, to a first approximation, be equal to its effective weight (accounting for buoyancy effects) divided by its projected area, or m/s2, this local shear stress is about 0.012 Pa for

∆

/

. With g = 9.8

= 300 nm. Under this low shear stress,

the local viscosity “felt” by a settling particle should be about 105 mPa⋅s or higher (Figure 3), which apparently provides an explanation for why the titania particles remain suspended 10 ACS Paragon Plus Environment

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for at least 18 months. In contrast, the wall shear stresses in gravity-driven bulk flows in capillary tubes with diameters of 0.3 to 0.7 mm are about 1 to 10 Pa. For these high shear stresses, the viscosity is much lower, ca. 10 mPa s. Thus, the close-packed DDAB vesicular dispersions provide an optimal resistance against the sedimentation of the titania particles with diameters of about 280±100 nm, while also not significantly impeding the flow of the bulk dispersion through a large enough capillary tube with a diameter of over 0.3 mm. We do note that the rapid rise of the viscosity seen in Figure 3 at a shear stress ~0.01 Pa may indicate that the DDAB vesicular dispersion forms a yield stress fluid, with a yield stress ~0.01 Pa. Additional experiments were not, however, performed to fully confirm the onset of a yield stress at very low shear stresses. The sudden increase in the viscosity at low shear stresses nonetheless provides a clear indication that the vesicles themselves are providing a strong resistance to the motion of the titania particles (either due to an “infinite”-resistance from a yield stress fluid or a very high resistance due to a high “local viscosity”). The minimum particle size that a yield stress fluid can support can be calculated from the exact results of Ref. [35], which indicates that titania particles of radii ~3 m or less will remain suspended indefinitely in a fluid with a yield stress of 0.01 Pa. The DDAB vesicles have radii around 0.1-0.5 m, and our previous experiments considered titania particles with radii around 0.3 m. Since these two sets of radii are approximately the same, it is the close-packing of the vesicles themselves that serve to prohibit the motion of the titania particles (which can be equivalently viewed on this length scale as providing a very high local resistance or a local yield stress).



3.

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Brownian Dynamics Simulations

3.1. Introduction and Methodology Colloidal dispersions have been studied widely with various computer simulation methods, including the Monte Carlo method (MC), the molecular dynamics method (MD), and the Brownian dynamics method (BD). With these methods, one avoids certain complications encountered in experimental investigations of colloidal dispersions, while one can study the physics of several aspects of their structure and dynamics. The MC method is relatively easy to implement, but provides predictions only of equilibrium properties. By contrast, both the MD and BD methods allow simulations of nonequilibrium states. If the MD method were used for describing the motion of large colloidal particles and solvent molecules, and since the characteristic times of the former are much larger than those of the latter, then a very large number of time integration steps would be needed to probe the colloidal dynamics. The BD method is more useful in this case, since it uses a random (Brownian) force to describe one of the effects of the solvent molecules. Hence, BD simulations (BDS) can be used with moderate computational efforts to study the structure and dynamics of colloidal dispersions. In a BDS, the motion of a colloidal particle is described by the Langevin equation24,36 ∙ where

(2)

is the mass of the particle,

external forces acting on the particle,

is its velocity vector,

is the vector sum of all

is the friction tensor, which is related to the

hydrodynamic drag on the particle, and in turn to the diffusivity tensor

/ , and

is a stochastic force term that is designed to mimic the force due to the Brownian fluctuations exerted on a particle as a result of the random fluctuations of the solvent 12 ACS Paragon Plus Environment

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molecules. In these simulations, the rapid motion of the solvent molecules is not explicitly determined, as stated above, but the influence on the motion of the colloidal particles is accounted for through the combination of the random and frictional force terms in Eq. (2). The trajectory of the motion of the particles, or its position vector

, can be

determined by integrating the Langevin equation twice, using a method developed by Ermak and McCammon.37 For an isotropic fluid, the friction tensor

for a spherical

particle with no hydrodynamic interactions between the particles is equal to 3 the diffusivity tensor

is therefore equal to

/3

, where

, and

is the identity tensor.

Then, the trajectory of the particle can be expressed as follows37 Δ

Δ

(3)

where Δ is the integration time step and

is the particle’s scalar diffusion coefficient. The

second term on the right hand side is the “drift” term, and is due to the sum of the forces acting on the particle. The last term,

, is the contribution of the Brownian

fluctuations, a random Gaussian-distributed vector that must satisfy various constraints resulting from the fluctuation-dissipation theorem.38 This theorem provides the relation between the response of the system to an external perturbation and the internal fluctuations of the system at thermal equilibrium. Consequently, the average value of each Cartesian component  ( , , or ) is zero, 〈

〉

0, and the standard deviation, 〈

〉, is 2 Δ .

Equation (3) is valid when the chosen value of Δ is at least an order of magnitude larger than the particle’s momentum relaxation time, or “Brownian time,” ≡ where

(4)

is the density of the particle and

is the medium viscosity, where the medium is

assumed to behave as a Newtonian fluid. The time step Δ should be small enough so that 13 ACS Paragon Plus Environment


the force on a particle is (nearly) constant during each time step. An in-house code was developed for the BDS using Eq. (3) for studying the sedimentation of the dense colloidal particles and the rheological properties of the suspensions of light particles. It is noted that Eq. (3) only accounts for the viscous drag on a particle due to the solvent, and does not consider the hydrodynamic interactions that may arise between the particles. While these additional hydrodynamic effects could be included in a modified form of Eq. (3), via a more complicated form of the diffusivity tensor37, we nonetheless chose to use Eq. (3) to minimize the computational time needed to analyze the behavior of the suspensions and dispersions of interest. First, our sedimentation studies are based on BDS that have reached a steady-state limit, and neglecting these hydrodynamic interactions may affect little these steady-state results (which are equivalently the equilibrium behavior of the suspensions in the chosen gravitational field). The timescales at which these steady-state limits are reached, or the specific sedimentation rates of the suspensions, should not be relevant for our labeling of a given suspension as “stable” or “unstable” (which as discussed in the next section is based on the steady-state concentration of dense particles located near the very top of the simulation cell). Second, the specific results of our rheological studies would change if hydrodynamic effects were included. But we nevertheless expect Eq. (3) to yield qualitative agreement, if not semi-quantitative agreement, with those BDS that include hydrodynamic interactions, if particle packing effects are assumed to provide the dominant contribution to the rheological behavior (although more specifically the microrheological behavior) of close-packed vesicular dispersions. The analyses provided in the next section of the arrangement of light particles that develop locally around the dense particles at high volume fractions of light particles indicate that the net interparticle forces on the dense


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particles are quite large. Hence, these packing effects should provide the dominant contribution to the resistance of the motion of, or the local viscosities felt by, the dense particles.

3.2. Brownian Dynamics Simulation Details 3.2.1. Simulations of Dense Particle Sedimentation Mixtures of two types of colloidal spherical particles are considered: (a) “dense”, or high-

particles with

12000,

250 nm, and Δ

0.01,

mimicking the titania particles; (b) “light”, or zero200 to 500 nm, and Δ

0.53, vesicles;

and

particles with

3.2 g/cm3, 0, or 0.15 to

0 g/cm3, representing non-settling “rigid” DDAB

are the volume fractions of the dense and the light particles,

respectively. Since the total height L of the simulation cell within the BDS is typically only around two orders of magnitude larger than a particle diameter, a large value of g, or

,

is needed to observe in a BDS any significant settling of the dense particles in the absence of the light particles. Since Brownian time,

0 for the light particles, they do not settle at all. The

, of the dense particles is 1.63×10-8 s, and

of the light particles

ranges from 2.33×10-9 to 1.46×10-8 s, with the ratio of the two times ranging from 1.12 to 7.00. The particles are assumed to be spherical and monodisperse, an approximation of the actual systems, since the titania particles are non-spherical and both the titania and DDAB vesicle sizes are polydisperse. The vesicles are spherical at low DDAB concentrations. To ensure adequate particle displacements in each time step for both types of particles, Δ is chosen to range from 20

to 70

. A rectangular cell with a 3:1:1 ratio

of height-to-length-to-width is used under periodic boundary conditions in the x and y 15 ACS Paragon Plus Environment


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directions. For the z direction, impenetrable walls are set up at the bottom and at the top of the cell. Two sets of simulations have been performed with the following system sizes: (a) for

, the number of the dense particles

particles

is 1000, and the number of the light

ranges from 5000 to 18000; (b) for

,

200 and

12000. The simulation cell volume, , is determined from the values of

4000 to and

, in

/6 .

which

Three types of forces acting on the particles are considered: the net gravitational force , the overall DLVO interparticle force

, and the forces

due to the confinement

of the particles by the upper and lower wall boundaries. The net gravitational force on a particle is given by (5) in which the acceleration constant value of

to be used in the simulations is determined for a given

. Because L is small, a large value of 12000,

; i.e., for

1000, and

is required to reach a large value of 0.01,

10

, where

is the

normal gravity acceleration of 9.8 m/s . The DLVO potential Φ

, which is the sum of the van der Waals attractive and

the electrostatic repulsive potential energies,2,29,39 is used to determine the DLVO Φ

interparticle forces

Waals attractive potential energy Φ diameters

and

Φ

,

, where

is the gradient operator. The van der

between two spherical particles i and j with

is29 ,

,

ln


(6)

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where

is the Hamaker constant for particles i and j and

is the center-to-center

distance between the particles. For a symmetric z:z electrolyte solution with a molar concentration

, for which the ionic strength is equal to

, the electrostatic repulsive

potential energy Φ , under the condition of constant surface potential according to the Hogg-Healy-Fuerstenau equation,29 is Φ

,

,

,

,

, ln

where

is the vacuum permittivity,

ln 1

(7)

is the relative permittivity of the solvent,

and

are the surface electrical potentials of particles i and j, the surface-to-surface separation distance is which

, and the inverse Debye length is

is Avogadro’s number and

, in

is the elementary charge. To prevent particle

agglomeration in the simulations, a weak vdW attractive interaction and a strong electrostatic repulsive interaction are used. For simplicity, the three Hamaker constants, ,

, and

, are set to 0.5

. The surface potentials of both types of particles are

set to -25 mV. The ionic strength is set at 0.15 or 1 mM, for which the Debye length 24.6 or 9.6 nm, which is less than 15% or 5% of the

and the

is

values. With these

parameters, the DLVO potential energy yields a very high barrier of 40

to 100

,

and the particles do not agglomerate in the simulations, consistently with the actual system of interest, since for the cases studied previously both the titania particles and the DDAB vesicles are stable against agglomeration or coalescence.12



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Finally, the following repulsive force portion of the Lennard-Jones-type potential was chosen to describe the confinement forces,

, between the particles and the walls, acting

in the z-direction only 24 where

(8)

is the separation distance between the particle surface and the wall. The positive

and negative signs are for the bottom and the top walls, respectively, and

is the

characteristic energy of interaction between the particle and the wall, and is taken to be equal to

for convenience.

For the low volume fractions of

0.25, the initial configurations used to begin

a BDS were generated by randomly placing all the particles within the simulation cell, ensuring that no two particles overlapped and do not reside in the close-range potential well of the DLVO potential. For

0.25, randomly placing all of the particles without

any overlaps is found to be difficult to achieve. Instead, at these higher volume fractions, the light particles are first placed at the positions of a face-centered-cubic (fcc) lattice structure at the smaller

of 0.20. Then the dense particles are randomly placed at various

locations in the fcc lattice to yield the appropriate volume fractions of both particles. The fcc lattice configuration is then “melted”, or fluidized, in a separate simulation at which there are no gravitational forces. The pair correlation function,

, which is the ratio of

the local particle number density of the bulk particle number density, is then checked to ensure that the mixture had indeed become liquid-like.40 The cell volume is then slowly decreased in a subsequent simulation in order to reach the desired higher volume fraction, . Upon reaching the final volume fraction, an arbitrary configuration of all the particles is then used as the initial configuration for the main simulation. Five (or more) different 18 ACS Paragon Plus Environment

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initial configurations are used for five (or more) separate simulations, and the averages are reported. A case of only dense particles, with

0, is simulated first to obtain the

benchmark sedimentation time. Then, the effects of

and

on the sedimentation rate of

the dense particles are investigated. The particle concentrations of the light and dense particles are computed as a function of height, , and are discussed further in Section 4. 3.2.2. Computation of the Shear Viscosities or Macroviscosities of Dispersions of Light Particles The bulk shear viscosities of the dispersions containing only the light particles with 500 nm and

0.01-0.50 are computed from the BDS detailed in Section 3.2.1.

For these simulations, a cubic cell is used with

2500 under periodic boundary

conditions in the x, y, and directions, and with no gravitational forces. Under the zero-shear limit, the Green-Kubo formula41 is used to determine the zeroshear viscosity 0 〉d

〈 where

is the

interparticle forces,

(9)

component of the instantaneous stress tensor resulting from the is the “infinite-shear-rate” viscosity, which is given by

1

and describes the contribution to the shear viscosity from the hydrodynamic flow of isolated single particles, and

is the solvent viscosity. Because of the Brownian motion

of the particles, the shear stress,

, fluctuates around its average value of zero in these

simulations, with periodic boundary conditions in all directions. The shear stress autocorrelation function, 〈

0 〉, in Eq. (9) is not equal to zero at short times, and

is a measure of how fast these fluctuations become uncorrelated. The autocorrelation



functions of

and

Page 20 of 52

are also calculated similarly, and are expected to be the same as

, since there are no preferred directions in these equilibrium simulations. Thus, the average value of these three components is used in Eq. (9) to help reduce its statistical noise or the standard deviation. For computing the bulk shear viscosity beyond the zero-shear limit, a steady linear velocity profile, with a velocity gradient, or shear rate

d

/d , in the x-direction is

applied along with the Lees-Edwards (LE) boundary conditions,42-45 (with periodic boundary conditions applied as before in the y and z-directions only). The LE boundary conditions allow for the introduction of a simple shear flow in the simulations. Then, the shear viscosity

of the dispersion is calculated as the sum of the contribution

from the

flow of the isolated single particles, as done above, and the contribution

from the

interparticle force,28, 46 (10) Again, the hydrodynamic interactions between the particles are not included in the BDS. In these simulations, the shear stress

resulting from the interparticle potential energy is

determined from28 ∑

∑

(11)

where Φ is the potential energy between particles i and j, distance, and

and

are the

and

is their center-to-center

components of the center-to-center distance vector.

The instantaneous shear stress is calculated every 100 time steps. The block average value 〈

〉 is determined every 105 steps. The system is deemed to have reached a steady state

when five or more successive block averages differed by 5% or less. The final steady-state


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shear stress, 〈

〉 , is calculated as the average of these successive block averages. The

shear viscosity contribution 〈

was then determined using28

〉 /

(12)

To further describe the relative importance of the shear and Brownian forces, the shearrate-based Peclet number,

, is used, and is defined as the ratio of the shear rate to the

diffusion rate of a single particle, or27,28 ≡

6

/

(13)

3.2.3. Computation of the Microviscosities of the Dispersions of Light Particles The rheology at the microscopic scale, or “microrheology,” describes the local or small scale viscous or elastic properties of the dispersion.23,29,31,47-50 A standard experiment to probe the microrheology involves tracking the motion of one or more colloidal particles, and therefore their resistance to motion, due to either thermal fluctuations (passive microrheology) or an external force (active microrheology).31,47-50 Here, we perform BDS for active microrheology with particles with DLVO-type interactions, as previously done by Carpen and Brady23 and by Gnann et al.30 for suspensions of hard spheres. A “probe” particle with diameter

250 nm (the same as for the dense particle) is set in motion

through a collection, or a “bath,” of “light particles” with 0.01-0.45. A constant force,

2500,

500 nm, and

, acting in the x-direction only is applied to the probe

particle, with its resulting velocity in the same direction varying after each BDS time step. Since there are large fluctuations in the probe particle’s velocity after each individual BDS time step of Δ , the average velocity is computed over a larger number of BDS time steps. For example, the -displacement of the probe particle, Δ , is determined every

BDS

time steps, and is used to calculate the probe particle’s average velocity in the -direction, 21 ACS Paragon Plus Environment


in which 〈

〉

depending on

Δ / and

Δ . The value of

is chosen to range from 1×103 to 1×108,

. The average of these successive block average values of 〈

then used to calculate the effective local viscosity /3

〈

Page 22 of 52

〉 are

according to23

〉

(14)

The simulation setup procedures are the same as in Section 3.2.2. For these simulations, the relevant Peclet number is defined as follows23 ≡

(15)

The microrheology can also be analyzed by moving the probe particle with a constant velocity, and determining the resulting average interparticle force on the probe particle in the same direction as its motion.23 This case was considered before,51 but is not presented here, because it is less relevant to the sedimentation experiments. The steady state pair correlation functions, PCFs, of the light particles in front of, , and behind the probe particle,

, are also calculated for two circular cones with

an angle of 60 degrees emanating from the center of the dense particle. Since the probe particle is biased to move in a particular direction (the

direction), the arrangement, or

local density, of bath particles in front of and behind the probe particle (in its direction of motion) will not necessarily be the same, and should be the most asymmetric in these two volumes. Hence, the net forces exerted on the probe particle from those bath particles within these regions should yield the most important contribution of the interparticle forces to the resistance of the probe particle’s motion. These PCFs therefore provide valuable information and insights on the mechanism by which the light particles respond and rearrange around the moving probe particle, and on why and how the local viscosity


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depends upon the volume fraction of the light particles and the magnitude of the external force.

4.

Simulation Results and Discussion

4.1. Results on Particle Sedimentation 4.1.1. Suspensions of Dense Particles For suspensions with

250 nm,

0.01,

≅ 12000, the

= 1 mM, and

particle positions were tracked throughout the simulations. Representative snapshots of the particles at various times, projected onto the

plane, are shown in Figure 4. The particles

sediment completely from the top 5% of the simulation cell, from dimensionless time

≡

/

1.0 to

0.95, at a

7.4×104. This timescale is used as the benchmark for

the analyses of the simulations described in Section 4.1.2. The particles sediment completely from to

1.0 to

0.05 (95%) at

0.50 (50% or the half-height) at ≡ /

5.6×105, and

1.5×106. The concentration profiles vs. and are in agreement

with the predictions from the Mason-Weaver equation, which is the unsteady-state sedimentation-diffusion equation for a dilute suspension of monodisperse spherical particles.5,39 The sedimentation velocity of the particles is about 0.113±0.002 cm/s, which is consistent with the value of 0.115 cm/s predicted from the Stokes law. Hence, as expected for

≅ 12000, particle diffusion effects have little influence on the settling

velocity. At steady state, the particles in the bottom of the simulation cell have an average volume fraction of about 0.43, which is slightly less than the value of 0.49 at which monodisperse hard spheres (without electric double layer effects) undergo a fluid-solid 23 ACS Paragon Plus Environment


Page 24 of 52

transition.52-54 If the electrical double layer around a particle is considered, then the 2

effective particle diameter is larger, and is estimated to be around

. Then,

the effective volume fraction is 0.54, which is closer to the random packing limit for softrepulsive colloidal dispersions.55 4.1.2. Mixtures of Dense Particles (1) and Light Particles (2) 250 nm,

Mixtures of dense particles (1) with 200 to 500 nm,

and

0.01, and

0, were examined at = 1 mM.

0.15 to 0.53, and

Each simulation started with uniform initial concentrations, ,

particles. The subsequent concentration profiles, /

calculate the dimensionless concentrations 1000

≅ 12000,

and

, for both types of

,

and

, were monitored to /

and

(Figure 5(a)). At

and 3000 , the concentration profiles were nearly identical; the suspension 1000 .

was deemed to have reached its steady state limit no longer than

According to the resulting sedimentation behavior, the suspensions were classified into three groups, depending on the value of (1) “unstable” suspensions at which

5% at the top 5% of the simulation cell:

5% was less than 0.2, implying significant

sedimentation (U-SAS); (2) “stable” suspensions at which

5%

“somewhat unstable” suspensions in a transition region at which example, shown in Figure 5(b), for a mixture with

0.7 (SAS); and (3) 5%

300 nm,

0.2-0.7. In one 5% ≅ 0 at

0.21; hence, the suspension is unstable. Moreover, all the dense particles (black filled circles) sedimented to the bottom of the cell, while the light particles (blue open circles) at the cell bottom were pushed away by the dense particles. A snapshot of a stable suspension with

0.43 is shown in Figure 5(c), in which the concentration of the dense particles

is nearly uniform throughout the simulation cell. 24 ACS Paragon Plus Environment

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The values of

5% are affected mainly by the volume fraction of the light particles

and secondarily by the size of the light particles, as shown in Figure 6. Projections of plane for

Figure 6 onto the

1 mM and

1 mM and 0.15 mM are provided in Figure 7. For

200 nm, the suspensions are unstable (U-SAS) at low

0.26 (and presumably also lower than 0.15). At ϕ and it is quite stable (SAS) at

, 0.15 to

0.30, the suspension is more stable,

0.35 to 0.50. The same trends with

are observed for

other sizes of the light particles. At sufficiently high volume fractions

, the dense

particles are stabilized by the light particles. For larger light particles, a higher volume fraction

is needed to provide good stability against sedimentation. For the lower

stability bound of the suspension, a linear empirical relationship between found:

6.81

10

1.66

is

10 . This equation may be used for some limited

interpolation or extrapolation purposes, to predict the values of stabilize particles of size

and

and

needed to

.

To examine the possible effects of the DLVO forces on the sedimentation stability of the dense particles, separate simulations were run at the lower ionic strength of 0.15 mM, for which the Debye length increased to 24.6 nm; see Figure 7(b). Although the trends of and of

were found to be similar, the

slightly downwards to

5.83

line for the lower stability bound shifts

10

1.02

10

, because of the higher

effective size of the light particles. 4.2. Results on Bulk Shear Viscosities or Macroviscosities The computed zero-shear bulk viscosities, particles with

500 nm and

, of dispersions containing only light

0.01 to 0.48 (with

in Table 1. The viscosities increase sharply for

,

0.01 to 0.57) are shown

≳ 0.40 and suggest the onset of a



Page 26 of 52

colloidal glass transition, which for hard-sphere suspensions starts at a volume fraction of about 0.58.56 The steady-state shear-viscosities were also computed, for

0.01 to 70, and 0.10, the dispersion

compared to the zero-shear viscosities in Figure 8. For

essentially behaves as a Newtonian fluid, with a relative shear viscosity / which is close to the zero-shear-limit value of 1.34. For slightly for ≅

for

of about 1.28,

0.20 (or 0.25), /

decreases

0.70 to 70 from 1.72 to 1.59 (or 2.06 to 1.75). At these concentrations, 0.7. For

0.35 to 0.5, the results indicate a stronger shear-thinning

behavior. At very low shear rates, . Then, at higher values of

≪ 1, the steady-state shear viscosities are close to

, they decrease, and for

≫ 10, the viscosities reach a

limiting value, smaller than 10, that depends on the volume fraction. This is because the Brownian motion is negligible relative to the particle motion caused by the shear force. At high volume fractions

, the bulk viscosities are quite low, a result that is

consistent with the experimental data of the DDAB vesicular dispersions. Since the contributions of the dense particles to the bulk shear viscosities can be neglected, the low shear viscosities at high values of

imply a significant flowability for the suspensions,

even at high volume fractions of the light particles. 4.3. Results on Microviscosities: Constant Force Simulations In order to examine the effect of a “bath” of dispersed light particles on the net motion of a dense particle experiencing a certain force, or to estimate the microviscosity “felt” by a Brownian particle settling under gravity, a single-probe particle ( with a constant force for microviscosity

/

500 nm,

0.01 to 0.45, and

250 nm) is pulled 10 . The reduced

, which is the effective microviscosity normalized by the solvent 26 ACS Paragon Plus Environment

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bulk viscosity, is plotted in Figure 9 against PCFs of the light particles in front of, provided in Figures 10, 11, and 12 for For low values of ranging from 10

for various , and behind,

-values. Examples of the , the probe particle are

0.30, 0.35, and 0.40, respectively.

0.01 and 0.10), the reduced microviscosities are ~1.0 for

to 10 . Clearly, there are too few light particles close to the probe

particle to have a significant impact on its motion. (Note: If hydrodynamic interparticle interactions were included in the BDS, they could still have some effect on the microviscosities.57) For the higher volume fractions

0.20, 0.25 and 0.30, but low

0.2, the

relative microviscosity is also about 1.0. The corresponding PCFs for

0.30 and

0 in Figure 10(a) show that the light particles in front of and behind the probe particle are arranged similarly, as expected, since no external force is exerted on the probe particle. (Similar results, not shown here, are also obtained for low values of 0.45.) For these volume fractions and low values of

and for

, the light particles have time

to reorganize themselves symmetrically about the probe particle. Hence, the interparticle forces on the probe particle in front of and behind it are about the same and the net interparticle force on the probe particle is, on average, quite low. Since the external force on the probe particle is much smaller than the Brownian force, the motion of the probe particle is affected primarily by the Brownian fluctuations, and hence, For

/

≅ 1.

∼ 0.5 to 5, the average velocity of the probe particle varies linearly with the

applied force, and the effective microviscosity is larger than 1 and constant (and so exhibits a plateau). For

≳ 5, the microviscosity decreases with increasing

, or shows a

“force-thinning” behavior, analogous to a “shear-thinning” behavior. This trend suggests 27 ACS Paragon Plus Environment


Page 28 of 52

that as the probe particle moves, it causes a rearrangement of the microstructure of the surrounding light particles. At For

0.30 and at

1000, a second plateau appears. = 25 and 100 (Figures 10(b) and 10(c)), the PCFs indicate

that a “wall” (represented by a sharp peak) of bath particles develops in front of the probe particle. The local density of the bath particles in front of the probe particle is much greater than the bulk density. Behind the probe particle, the PCF for a local density nearly equal to the bulk density. At

= 25 is structureless with

= 100, the “wall” is denser (with a

higher peak in the PCF for the particles in front of the probe), and a “depletion zone” develops behind the probe particle, in which the local density of light particles is lower than the bulk density. These two sets of PCFs are still representative of fluid-like microstructures. The bath particles can move around the probe particle, but not rapidly enough to prevent their net accumulation in front of the probe particle. Furthermore, the large differences in the local densities in front of and behind the probe particle indicate that the net interparticle force on the probe particle is, on average, large. Thus, the bath particles provide a strong resistance to the probe particle’s motion, which explains the high microviscosity results in Figure 9. While the PCFs provide a more detailed explanation of the microviscosity predictions, the observed trend for

0.30 in Figure 9 (a plateau followed by “force-thinning”)

ultimately depends, however, upon the relative importance of the net interparticle force, the Brownian force, and the applied external force. Even though the PCFs at

= 100

indicate that the net interparticle force that resists the probe particle motion is larger than at

= 25, the larger applied force is nonetheless sufficient to overcome this greater

resistance, giving rise to a smaller microviscosity.


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For

0.35, 0.40 and 0.45, the effective microviscosities show a qualitatively

different behavior (Figure 9). At

∼ 1, the microviscosities are large, around 1000, with

large standard deviations, suggesting that the probe particle motion is severely hindered by the presence of the light particles. For

0.45, in particular, the microviscosities are

quite large, of O(104), and have large standard deviations. The reduced microviscosities at 0.35, 0.40 and 0.45 first drop to ca. 30, 100, and 3000, respectively, at to ca. 10 or less at

1000. For

25, and

1000, the probe particle would eventually

agglomerate with a bath particle within the BDS. This apparently unphysical prediction may have resulted from the large external force on the probe particle, and the discreteness of the BDS integration algorithm. At these conditions, the probe particle was found to “jump” over the high repulsive energy barrier between it and another light particle. The PCFs for

0.35 and at

= 10, 100 and 1000, shown in Figure 11, are

qualitatively similar to those obtained at

0.30. A “wall” of light particles develops in

front of the probe particle, and its density increases as

increases, while the local density

of the light particles behind the probe particle decreases as the “wall” is greater for

0.35 than at

increases. The density of

0.30 for the same

-values. Hence, as

expected, the net interparticle force, and so the resistance to the motion of the probe particle due to the light particles, increases with an increase in the volume fraction. This is consistent with the observed increase in the effective microviscosity with an increase in . At

0.40 and

= 25, the PCFs are qualitatively different from those at

0.35. Instead of a “wall” and a depletion zone, a “solid-like” microstructure develops in front of and behind the probe particle. Apparently, the light particles close to the probe



Page 30 of 52

particle do not have sufficient time to reorganize into their prior “fluid-like” structure. A “cage”-like solid structure of light particles develops around the probe particle, causing a very high resistance to the probe particle motion. At

= 1000, however, fluid-like PCFs

develop again, with a dense “wall” in front of and a depletion zone behind the probe particle. Evidently at this high value of

the force applied to the probe particle is so strong that

the “cage” of light particles is disrupted. Because the external force is so large, the probe particle motion persists and is able to “deconstruct” its local environment, pushing the densely-packed bath particles out of its way, resulting in a smaller microviscosity Such behavior has been inferred by confocal microscopy data obtained by Habdas et al.58 and confirmed by BDS by Carpen and Brady23 and by Gnann et al.30 Finally, we have also calculated the net probe-bath interparticle force,

, due to all

bath particles within both conical regions, and compared it to the external dragging force, ; see Table 2. At

= 0.30, the ratio of

/

increases from 0.114 at

at

= 25; it then decreases to 0.0846 and 0.0157 at

At

= 0.35, the

at

= 0.40, the

= 10, 25, 100,

/ -values are 0.107, 0.349, 0.289, and 0.105

= 10, 25, 100, and 1000, respectively. The maximum value of the ratio of

occurs at ca. for

= 100 and 1000, respectively.

/ -values are 0.100, 0.163, 0.114, and 0.0637 at

and 1000, respectively. At

= 10 to 0.130

= 25 as

/

= 25 at all volume fractions, and there is a sudden increase in this ratio increases from 0.35 to 0.40. While these results suggest that the bath

particles provide the highest resistance against the motion of the probe particle around = 25, the highest microviscosities for a given volume fraction in Figure 9 are not located at

= 25. Nevertheless, large microviscosites still arise around

increase in the microviscosity is observed at

= 25 as


= 25, and a large

increases from 0.30 to 0.40.

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These microrheological results are consistent with the BDS results (in which 12000) of the sedimentation behavior of the dense particles in Section 4.1.2, where the dense particles remain suspended in the suspensions containing the light particles with 500 nm and

0.5. With the gravitational force acting on the dense particles

considered as the applied force on any one of the dense (or probe) particles, around 25. For and at

25, at

is therefore

0.30, the reduced microviscosities are smaller than 10,

0.35 or 0.40, the reduced microviscosities are about 33 or 100. Hence, the rate

of particle sedimentation is smaller by at most 100-fold, compared to that in the absence of the light particles, which is consistent with the dense particles sedimenting eventually (U-SAS) in the BDS. At

0.45,

/

is around 3000 with a large standard deviation;

the dense particles can remain suspended for times larger than their original sedimentation times by around 3000 times. Hence, no significant sedimentation (SAS) should be observed during a BDS. This conclusion may be modified if much longer simulation times, 1000 , are used to ensure that the system has reached its steady-state limit. The large standard deviations suggest, however, that a suspension at

0.45 could be somewhat

unstable, or in the transition region. Although no microviscosities could be calculated reliably by BDS for the stable suspensions (SAS) at

0.50, for computational reasons,

we speculate that they would be even higher than those at

0.45, and that the dense

particles would remain suspended for even longer times, as observed in Section 4.1.2. Hence, the combination of several computational results provides useful insights on the mechanism by which nearly closepacked dispersions of light particles reduce the sedimentation rate of dense particles.



5.

Page 32 of 52

Conclusions We recently presented a novel method of using close-packed vesicular dispersions for

stabilizing commercial titania particles with

3.2 g/cm3 and

300 100 nm.12 A

double-chain cationic surfactant, didodecyldimethylammonium bromide (DDAB), was chosen for forming the vesicular dispersions. At high enough concentrations of DDAB, for which the volume fraction of vesicles exceeds the close-packed limit for rigid spheres, the vesicles, being deformable, trap themselves into a foam-like microstructure. Such closepacked vesicular dispersions provide a high resistance against both the diffusive and the gravity-induced motion of the titania particles. Nonetheless, the vesicular dispersions are highly shear-thinning. At high shear rates or shear stresses, the motion of the overall bulk suspension is not significantly impeded. The bulk viscosity remains low, allowing easy flow in ink-jet printers and other applications. We have used BDS to elucidate theoretically the key factors of the mechanism by which vesicles or non-settling particles can stabilize dense particles. To mimic certain conditions of the experimental results, we considered suspensions consisting of dense particles ( nm,

250 nm, 0.15-0.55, and

0.01, and

12000) and light particles (

200-500

0). At these conditions, the dense particles are predicted

to sediment rapidly in the absence of the light particles, as occurs for the titania particles. The light particles remain suspended indefinitely under gravity, and represent the vesicles, which are assumed to be rigid for simplicity. The particles are assumed to interact with DLVO potentials, the parameters of which are chosen to represent strong interparticle repulsive interactions between the non-agglomerating particles. The simulation results show that the stability of the dense particles against sedimentation improves markedly with


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increasing volume fraction of the light particles,

, as observed experimentally. When

exceeds a certain threshold, the dense particles are predicted to remain suspended for very long times. Furthermore, this threshold was found to be higher for larger light particles, and increased as the ionic strength increased. For deformable light particles, such as DDAB vesicles, the volume fraction needed for stabilizing the dense particles could be higher than predicted by the BDS of rigid light particles. The macroviscosities (or shear viscosities) of dispersions of light particles were also computed as a function of rates

for

500 nm and a wide range of dimensionless shear

. They provide a measure of whether the light particle dispersions can maintain

their flowability, while still stabilizing the dense particles against sedimentation. At low values of

(≪ 1), the dispersions are predicted to behave as Newtonian fluids, with bulk

viscosities approaching their zero-shear-limit values. At moderate shear rates, 0.5 10, the dispersions show a shear-thinning behavior. At even higher shear rates, the dispersions reach another viscosity plateau, at which the viscosities depend only on

,

and their values are lower than 10 times the solvent’s bulk viscosity. These low values suggest that the suspensions retain their ability to flow under moderate shear rates (or stresses), even though they provide a large resistance to the settling of the dense particles. The microviscosities

of the suspensions were also computed for

500 nm and

ranging from 0.01 to 0.45 by applying a fixed external force on a probe particle ( 250 nm) immersed in a collection of the light particles, and determining the resulting average velocity of the probe particle. These simulations also showed that the probe particle significantly alters the local microstructure of the surrounding light particles, which is an important factor in determining the stability of the dense particles against 33 ACS Paragon Plus Environment


Page 34 of 52

sedimentation. For the chosen volume fractions of light particles, the microviscosities were found to first increase with an increase in the force-based Peclet number, a plateau at larger values of

, and subsequently decreased upon a further increase in

, showing a force-thinning behavior. For each limiting value at

, then reached

, large values of

≪ 1), are predicted to occur for

(compared to its

25, which is the value of

that corresponds to the gravitational force experienced by the dense particles in the BDS sedimentation study. For example, at

0.40,

25 and

/

100,

suggesting that the dense particles should still settle, but more slowly than in the absence of the light particles. This is what is observed in the BD sedimentation simulations, as these suspensions were characterized as U-SAS. The local microstructure of the light particles around the probe particle is also found to drastically change at

25 as

is increased

from 0.35 to 0.4, changing from the usually observed “fluid-like” structures of a “wall” in front and a “depletion zone” behind to a more “cage-like” solid structure that fully surrounds the probe particle. On the other hand, at

0.45,

/

3000 with a large

standard deviation. This explains why the dense particles suspensions are somewhat unstable against sedimentation at

0.45. Based on the observed trends of

microviscosities vs. volume fraction, we infer that at

0.50,

/

should be much

greater than 3000. The dense particles should therefore remain suspended for much longer times, or even indefinitely, a prediction that is again consistent with the BDS sedimentation results since these suspensions are characterized as SAS. The BDS results presented here are consistent with the DDAB experimental data, in that the concentrated dispersions of the light, or non-settling, particles can provide a moderate to strong resistance against the sedimentation of the dense particles. The key


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factor in this stabilization mechanism is the volume fraction of the light particles, which controls the local viscosity around the dense particles. The size of the light particles is also important, though it has a smaller effect on the stabilization mechanism. Finally, the simulation results also suggest that the bulk suspensions retain their ability to flow under moderate shear rates or stresses.

Acknowledgements YJY is grateful to Purdue University for a Bilsland Fellowship. EIF and DSC are grateful to the National Science Foundation for partial support of this work (Award # 1706305). We are also thankful to Prof. You-Yeon Won of Purdue University for the use of the TA DHRS rheometer and to Mrs. Valorie D. Bowman of Purdue University for expert sample preparation for obtaining the cryo-TEM images.



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(a)

(b)

(c)

Figure 1. (a) Photographs of aqueous DDAB dispersions with 1.0 wt% titania particles 0.5 wt% after 2 months (top), and at 2.0 wt% after 18 months at (bottom). (b) Schematic diagrams a vesicular dispersion at a low volume fraction (top), and at a high volume fraction (bottom), where there is a stabilization against sedimentation. (c) 0.5 wt% (top), and at Cryo-TEM micrographs of DDAB vesicles in water at 2.0 wt% (bottom).



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Figure 2. Cryo-TEM micrograph of a titania particle in a 2.0 wt% DDAB-water vesicular dispersion. Adapted from Fig. 2 in Ref. [12].


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Figure 3. Shear viscosity of 2.0 wt% DDAB vesicular dispersion vs. shear stress and vs. shear rate (▲). Adapted from Fig. 6 in Ref. [12].

(■)



Figure 4. Snapshots of the particle positions at 1.5×106 as projected onto the (dimensionless)

Page 42 of 52

0, 7.4×104, 5.6×105, and ≡ / plane. The dashed lines are at = 0.95.


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(a)

(b)

(c)

Figure 5. (a) Dimensionless concentrations of the dense particles at the top 5% of the cell, (5%) for mixtures with 300 nm and various values of . Snapshots of the 300 and (b) 0.21, a particle position at 1000 for the mixtures with typical snapshot of an unstable dispersion; or (c) 0.43, a typical snapshot of a stable suspension.



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Figure 6. The dense particle concentration at top 5% of the simulation box, c1(5%), at τ/τ = 1000 with various values of d ranging from 200 to 500 nm, and various values of ϕ ranging from 0.15 to 0.55. The ionic strength used in the systems was 1 mM.


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(a)

(b)

Figure 7. Stability diagram of the dense particles in the binary mixtures with various diameters d2 and volume fractions ϕ2 with an ionic strength of (a) 1 mM and (b) 0.15 mM. A square (■) denotes a stable suspension (SAS); a circles (●) denotes a suspension that is in the transition region (T) and somewhat unstable; a triangles (▲) denotes an unstable suspension (U-SAS). The dashed straight lines, ( ), are the lower bounds of the stable suspensions. The ( ) equations are shown in the text.



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Figure 8. Zero-shear viscosities normalized by the solvent viscosity (dashed lines on the left) and the normalized steady-state shear viscosities (solid symbols) for dispersions 500 nm and various volume fractions , as containing light particles with diameter indicated.


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Figure 9. The reduced effective microviscosities vs. the force-based Peclet number, $%& ≡ '( )* +, -

, for a constant force system and various volume fractions. The vertical dashed line is

at $%&

25, which corresponds to the gravitational force imposed in the BDS.



(a)

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(b) $%& = 0

$%& = 25

(c) $%& = 100

Figure 10. The pair correlation functions, .' (/) and .0 (/), of the light particles in front 0.30 and a) of (blue dashed line) and behind (red dashed line) the probe particle for $%& = 0, b) $%& = 25, and c) $%& = 100.


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(a)

(b) $%& = 10

$%& = 100

(c) $%& = 1000

Figure 11. Same as Figure 10, for 1000.

0.35 and a) $%& = 10, b) $%& = 100, and d) $%& =



(a)

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(b) $%& = 1000

$%& = 25

Figure 12. Same as Figure 10, for

0.40 and a) $%& = 25 and b) $%& = 1000.


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Table 1. Bulk zero-shear viscosities 12 /13 as a function of the volume fraction 500 nm. effective volume fraction 455 for

12 /13 1.03 1.13 1.34 1.78 2.20 4.68 7.94 21.89 119.49 661.69 3083.00

455

0.01 0.05 0.10 0.20 0.25 0.30 0.35 0.40 0.43 0.45 0.48

or the

0.01 0.06 0.11 0.22 0.28 0.34 0.42 0.48 0.51 0.54 0.57

Table 2. The ratio of the net probe-bath interparticle forces, 6789 , to the imposed external force, 6: , at various values of $%& and the volume fraction . $%&

10 25 100 1000

0.30 0.114 0.130 0.0846 0.0157

6789 /6: 0.35 0.100 0.163 0.114 0.0637

0.40 0.107 0.349 0.289 0.105



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TOC Graphic


Effects of Light Dispersed Particles on the Stability of Dense

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