The Importance of Oscillatory Structural Forces in the Sedimentation of

Ind. Eng. Chem. Res. 2009, 48, 6641–6651

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The Importance of Oscillatory Structural Forces in the Sedimentation of a Binary Hard-Sphere Colloidal Suspension Jan Sudaporn Vesaratchanon,† Alex Nikolov, and Darsh Wasan* Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

Douglas Henderson Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602

Concentrated colloidal dispersions are complex systems comprised of colloidal particles of various sizes. This paper summarizes recent experimental and theoretical findings on understanding the interactions between microspheres in the presence of submicrometer particles; the submicrometer particles can be solvent molecules, surfactant micelles, or other nanometer-sized particles. In a hard-sphere suspension containing microspheres in the presence of submicrometer particles, oscillatory structural (entropic) forces between large particles, caused by small particles, arise as a result of the collective particle-particle interactions. The magnitude of the oscillatory structural forces is theoretically estimated based on the statistical mechanics approach, Monte Carlo simulations, and stochastic particle dynamics methods. Also, sedimentation experiments using spherical low-charged, binary hard-sphere particle suspensions with a large size ratio are presented to elucidate the importance of the oscillatory structural forces (i.e., attractive depletion and repulsive structural forces) in particle dispersion stability. Micronmeter-sized, low-charge latex particle concentration profiles in the presence of nanoparticles (nonionic micelles) were monitored using a nondestructive Kossel diffraction technique. To rationalize the experimental observations, the particle settling dynamics were simulated by considering the hydrodynamic interactions, structural forces, and Brownian motion in a gravity field. The theoretical predictions were determined to be in satisfactory agreement with the experimental observations. The effect of the particle size ratio (i.e., micrometer to submicrometer) on interparticle interaction energy is highlighted. Introduction Colloidal dispersions have technological importance because of their widespread applications in consumer products, foodstuffs, agrochemical and pharmaceutical formulations, and coatings, as well as in the manufacturing of nanostructured materials. Colloidal dispersions are complex systems that are comprised of colloidal particles of various sizes, shapes, and concentrations in a supporting fluid medium. The microstructure formation and stability of colloidal dispersions, as well as the related phenomena of flocculation, sedimentation, and separation, are governed by the collective particle-particle interactions. This is particularly true of concentrated systems where particles are in close proximity. Here, we first present a brief literature survey on the role of interparticle interactions in bidispersed suspensions. Sanyal et al.1 reported the flocculation of binary almost-hardsphere latex dispersions using nanoparticles and attributed it to the Asakura-Oosawa depletion attraction.2 Kaplan et al.3 found an ordered phase near the wall of the bottom of the container in a binary mixture of polystyrene spheres with large-to-small size ratios varying from 3:1 to 5:1. The formation of the ordered phase of the larger particles near the wall was explained as an entropically driven phenomenon promoted by the small particles. Roth et al.4 used the density function approach to calculate the * To whom correspondence should be addressed. E-mail: wasan@ iit.edu. † Currently with BASF, Germany.

depletion interaction potential between two large hard spheres in a sea of small hard spheres and for various concentrations of small spheres with a size ratio of 10:1. Tolpekin et al.5 observed the aggregation under the shear of bidisperse sphere mixtures. Tohver et al.6 and Martinez et al.7 observed that colloidal microspheres with negligible surface charges in the presence of highly charged nanoparticles can transform a colloidal gel into a stable fluid and subsequently into a colloidal gel with an increasing nanoparticle concentration (due to the so-called “halo effect”). Chan and Lewis8 investigated the size ratio effect on the interparticle interactions and phase behavior of microspherenanoparticle mixtures. They observed that a binary mixture, in which the nanoparticle size is reduced at a fixed microsphere diameter, exhibits a narrow window of stability that ultimately disappears with the increasing ionic strength. Baird and Walz9 studied the effect of silica nanospheres and an electrolyte on the structure of an aqueous suspension of disk-shaped kaolinite particles. The authors suggested that the observed structural transition in kaolinite suspensions is due to the sphere-plate interactions. Xu et al.10 observed that, in a bidisperse suspension (size ratio of 10:1), the sedimentation velocity first increases and then decreases as the small particle concentration increases. The authors rationalized their experimental observations in terms of the depletion and structural forces between the large particles in the presence of small particles. Figure 1 depicts the effective interaction potential between two macroparticles in the presence of small particles; the small

10.1021/ie8019856 CCC: $40.75  2009 American Chemical Society Published on Web 06/12/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Figure 1. Structural force between two large particles immersed in colloidal suspension. Reprinted with permission from ref 11. Copyright 2007 American Institute of Chemical Engineers.

particles can be solvent molecules, surfactant micelles, or other nanometer-sized particles.11 When the separation distance between two large particles is smaller than the diameter of the small particles, so that no small particles can fit in the gap between the large particles, the small particle concentration is locally depleted; the attractive force between the larger particles appears as a result of the osmotic pressure gradient between the gap and the bulk.2 When the separation distance between two large particles is on the order of several small-particle diameters, the small particles had a tendency to form a layering structure between or around the large particle and a higher repulsive energy barrier is expected. Generally, because of the attractive depletion and the repulsive structural potentials, the effective interaction (i.e., pair potential of mean force between the large particles) is oscillatory. Crocker et al.12 conducted measurements using a line optical tweezer of entropic attraction and the repulsion effects between two micrometer-sized colloidal hard spheres in a fluid of muchsmaller spheres (with a size ratio of 13:1). The large particle-particle interaction potential was attractive (i.e., the attractive depletion force) at a low concentration of small spheres, while an oscillatory potential was observed at higher concentrations. Piech et al.13 used atomic force microscopy to measure the depletion and structural forces in a polydispersed charged-particle fluid and highlighted the oscillatory-decay nature of particle-particle interactions. During the past decade, we have been investigating the importance of particle collective interactions in particle selfassembly in a confined geometry (i.e., a film) and the stability of colloidal suspensions, using both theory and experiments. We used film interferometry to investigate particle self-assembly in a liquid film.14,15 The suspension stability was monitored by observing particle self-structuring using the backscattered technique.16,17 Our theoretical studies involved the use of the statistical mechanics approach, Monte Carlo simulations, and stochastic particle dynamics methods.16-20 The goals of the present study are (i) to elucidate the importance of oscillatory structural forces between large particles that are caused by smaller particles and (ii) to examine how strong the effects are by measuring sedimentation profiles and rates.

This paper begins with a brief review of the theory of complex colloidal fluids, followed by our recent experimental and theoretical work on the importance of oscillatory structural forces arising because of the collective particle interactions in a hard-sphere suspension that contains microspheres in the presence of nanoparticles. Sedimentation experiments using the nondestructive Kossel diffraction technique are presented, and complementary stochastic particle dynamics simulations are performed to verify the experimental findings. The effect of the size ratio of large particles to small particles on the effective interaction energy is highlighted. Theory of Complex Colloidal Fluids The interaction between spherical colloidal particles of different sizes consists of repulsion at short separations between them. For simplicity, we can approximate the particle interaction as hard-core repulsion, UHS(r) )

{

∞ 0

(r (r

< d) > d)

(1)

where r is the separation distance between the particles and d is the particle diameter. At short separations, there is usually a dispersion interaction that is attractive,UD(r) ) Cr-6, where C is a constant. The particles may be charged and interact in a long-range Coulomb interaction: Ucc(r) )

qiqj εr

(2)

where qi and qj are charges on the particles i and j, respectively, and ε is the dielectric constant. The colloidal mixture would also contain solvent molecules. For simplicity, we include the interactions with and among the solvent molecules by means of the dielectric constant. The effective interaction between the pair of particles i and j is obtained from the potential of the mean force, Wij(r) ) -kT ln gij(r)

(3)

where gij(r) is the radial distribution function (RDF), which gives the local density of the particles of species i at a distance r


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from a particle of species j so that it has a value of 1 at large r. The parameters k and T are the Boltzmann constant and temperature, respectively. However, we use a linearized version of eq 3 and replace ln(1+ x) with x so that eq 3 becomes Wij(r) ) -kThij(r)

(4)

where hij(r) ) gij(r) -1. If the well-known linearized Debye-Hu¨ckel (LDH) theory is employed, gij(r) ) g0(r) -

[

]

exp[-κ(r - dij)] r ε(1 + κdij) βqiqj

2

(5)

where g0(r) is the portion of the RDF that comes from the hardcore and dispersion forces. Here, dij ) (di + dj)/2, κ is the Debye inverse screening length, κ)

4πβ ε

∑q

Fi

2 i

(6)

i

and Fi is the density (number of particles divided by the volume) of the particles of species i. The parameter β is defined as β ) 1/(kT). Equation 5 is not very satisfactory because it treats the charge diameters asymmetrically. It is better to symmetrize eq 5 and write gj(r) ) g0(r) -

{

}

βqiqj exp[-κ(r - dij)] ε[1 + (κdii /2)][1 + (κdjj /2)] r

(7)

Having separate terms in dii and djj allows us to consider both solvent molecules and large colloidal particles. To illustrate the application of our approach, consider the interactions between a pair of large spheres of radius a dispersed in a hard-sphere fluid mixture comprised of solvent molecules (s) and colloidal particles characterized by large-sized ratio (with a colloid to solvent size ratio of 10:1). The contribution of the hard-core repulsions can be obtained using the result of Henderson,21 which was derived from the expressions of Lebowitz22 obtained from the Percus-Yevick theory for a mixture of hard spheres in the limit where two of the spheres are exceedingly large but very dilute. The Laplace transform is analytical; the result in real space can be obtained by numerical inversion or by means of an explicit truncated expansion. In either case, the result for the hard-core portion of the interaction energy is

[

WHS(r - 2a) hHS(r - 2a) ) kT a a

]

(8)

where a is the radius of the large spheres. Figure 2 shows the interaction energy (i.e., potential of mean force) between a pair of large spheres immersed in onecomponent (monodisperse) and two-component (bidisperse) hard-sphere fluids.23 We see that the interaction between two large spheres in a one-component fluid is governed by the depletion attraction at short separations and exhibits a repulsive barrier at a separation of approximately one colloidal particle diameter, and the potential of interaction versus the separation distance is an oscillatory decay curve for separations larger than the colloidal particle diameter. Note that this potential of mean force is oscillatory, which is a feature that is not inherent in the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory, a

Figure 2. Effective interaction energy (potential of mean force, W, between two extremely large spheres of diameter d dissolved in one-component and two-component hard-sphere fluids. One-component fluid (short-dashed line) consists of the colloidal particles of diameter d2 and has a volume fraction of φ2 ) 0.20. The two-component fluid (thick solid line) is composed of the same colloid particles as in a one-component fluid plus the small particles (solvent) of diameter d1 ) 0.1d2 and volume fraction φ1 ) 0.15. Reprinted with permission from ref 23. Copyright 2003 American Institute of Physics.

classical model of dispersion stability for charged particles, which is limited to dilute systems and systems that consist of ions and solvent molecules without size in the supporting fluid.24,25 The depletion attraction is the result of the absence of colloidal particles in the space between the surfaces of the large spheres (see Figure 1). The structural repulsion (at separations larger than one colloidal particle diameter) results from the layering of the colloidal particles in the space between the two large spheres. We have extended this result to a bidisperse supporting fluid, a multicomponent supporting fluid and a polydisperse supporting fluid.9,10,23,26,27 The potential of mean force between two large spheres in a two-component (bidisperse) supporting fluid is also plotted in Figure 2. The potential of mean force between two large spheres in a bidisperse fluid is quite different from that for the monodisperse fluid, with additional oscillations due to the selflayering of smaller particles around the larger particles. The periodicity of these new oscillations is governed by the diameter of the smaller particles; this underlines the importance of the colloidal or solvent component. The smaller particles in colloidal suspensions have a tendency to layer around the large particle, and the layering phenomenon is dependent on the volume and the particle size ratio. The smaller particles also contribute to the increase in the amplitude of the depletion and stabilization barriers. Adding a particular concentration and certain size of smaller particles can control these barriers (and thereby the dispersion stability). In this paper, we will discuss how the concentration and size ratio of small to large particles govern the dispersion stability by monitoring the sedimentation rates of large particles in the presence of small particles. In addition to the interactions between particles in a bidisperse fluid, the interactions between plane parallel surfaces (i.e., film) confining a bidisperse fluid are also important for the microstructure formation and stability of fluid dispersions such as foams and emulsions.28,29 Figure 3a presents a comparison of the film structural disjoining pressure (i.e., an excess pressure in the film relative to the bulk) and film structural energy (see Figure 3b) for films formed from a monodisperse suspension of only large particles (short dashed lines) and from a bidisperse suspension of the large particles (volume fraction ) 0.20) and small particles (volume fraction ) 0.15), at a particle size ratio

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Figure 3. (a) Structural disjoining pressure and (b) structural interaction energy between film surfaces of films formed from monodisperse (dashed line) and bidisperse (solid line) suspensions. The particle volume fractions in both cases are 0.35. Reprinted with permission from ref 29. Copyright 2001 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine.

of 10:1. Note the significant differences in the disjoining pressure and structural film energy for monodisperse and bidispersed fluids. For both types of film, the structural disjoining pressure and structural film energy show oscillatory decay curves with the increasing film thickness; the period of oscillation is on the order of one diameter of a larger particle. The value of the amplitude and the position of the maxima and minima for monodisperse and bidisperse systems are reversed (see Figure 3). For the film with a bidisperse suspension and thickness corresponding to approximately one large particle diameter, the structural film energy has a minimum, whereas the structural film energy for a monodiperse system has a maximum. Therefore, the bidisperse film is expected to be thermodynamically stable, while the monodisperse should be unstable. To verify the model predictions of the effect of particle bidipersity on the colloidal suspension stability, we conducted a simple sedimentation experiment using an aqueous suspension of colloidal particles in the presence of a nanofluid (micellar solution). The reference system was an aqueous suspension of only colloidal particles. The unique feature of this experiment is that, as time proceeds, the colloidal particles have a tendency to settle in the lower portion of the container, and the concentration ratio of large particles to small particles increases. We monitored the concentration profiles of large particles with time, as well as the sediment microstructure. We also studied the effects of the particle size ratio (large to small) and small particle concentration. The results for the dependencies of particle interactions on both the concentration and particle size ratio were compared with the theoretical model predictions. Sedimentation Experiment We conducted gravity sedimentation experiments using a concentrated binary mixture of low-charge latex particles and nonionic micelles (as model hard-sphere nanoparticles) with two large-to-small particle size ratios (50:1 and 80:1). The hydroxylate latex dispersions were produced by Polysciences, Inc., as monodisperse suspensions. The particles were 500 and 800 nm in diameter, with a standard deviation of r*)

(D (r

(11)

where wdep(r) and wstr(r) represent the depletion-type attraction and repulsive structural forces, respectively. D is the diameter of the large sphere, and r* is the distance between two large particles where the depletion and structural forces are matched. This position lies within the separation distance of one smallparticle diameter and is dependent on the small-particle concentration.4 The depletion interaction energy (wdep(r)) can be expressed as

( kT1 )w

dep

(x) )

( D 2d+ d )(a + bx + cx

2

+ dx3)

(12)

where a ) -2.909φSmall b ) 6.916φSmall - 4.616φSmall2 + 78.856φSmall3 c ) -4.512φSmall + 15.860φSmall2 - 93.224φSmall3 d ) -φSmall exp(-1.734 + 8.957φSmall + 1.595φSmall2) This equation is an analytical equation that is based on the density functional approach for binary hard-sphere mixtures. In the low concentration limit, the equation was determined to recover the classical Asakura-Oosawa result.2 The coefficients shown in eq 12 are described as a function of the nanoparticle volume concentration.4,35 At high concentrations of nanoparticles, the structural forces (which are oscillatory in nature with the period of the nanoparticle diameter) are discussed elsewhere4,35,36 and can be described with the expression

( kT1 )w

(x) ) u0 cos(ω0x + φ0) exp(-k0x)

str

(13)

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∏

The frequency (ω0) and decay length coefficient (k0) are given in terms of the nanoparticle volume fractions: ω0 ) 4.4516 + 7.10586φSmall - 8.30671φSmall2 + 8.29751φSmall3 κ0 ) 4.78366 - 19.64378φSmall + 37.37944φSmall2 30.59647φSmall3

(14)

r - 2a d

The parameter r is the separation distance where the depletion and structural forces match. The position of r is within the surface-to-surface distance of one nanoparticle diameter and changes with the small-particle concentrations. When the large particles approach to an interparticle distance (e.g., 10 nm or less), van der Waals interactions dominate and lead to strong attraction forces. Thus, the total energy of interaction between large particles (WT) is the combination of entropic and van der Waals interactions: (17)

when Wvdw )

-aA 12r

(18)

where A is the Hamaker constant, which is estimated for hydroxylate latex particles in an aqueous medium to be 4.4 × 10-19 J,36,37 and r is the surface-to-surface separation distance. Interparticle Interactions between Only Large Hard-Sphere Particles We used 15 vol % large latex particles in our sedimentation experiments; therefore, the interactions between only large particles also must be evaluated. The interparticle interactions between large particles can be quantified by the second virial coefficient. The osmotic pressure (Π) of a large particle can be calculated as a function of the particle concentration using the CarnahanStarling equation of state:36,38 Π ) cRT

[

1 + φ + φ 2 - φ3 (1 - φ)3

]

(19)

where c is the particle concentration (in mol/L), φ the volume fraction of particles, R the gas constant, and T the temperature. The reduced osmotic pressure (denoted as Π/(RTc)) versus concentration (c) has a linear dependence.39 The slope of the curve is related to the value of the second virial coefficient (B) and the intercept is an inverse function of the molecular weight (M):

1 M

(20)

The second virial coefficient (B′) (in units of cm3 mol g2) is related to B from the aforementioned equation:

( )

(15)

(16)

WT(r) ) WEN(r) + Wvdw(r)

) Bc +

B′ ) B

Trokhymchuk et al. determined the amplitude (ω0) and phase coefficients (φ0) at a volume fraction of small particles in the dispersed phase equal to 0.30 (the value used in our experiments).35 The parameter x is the surface-to-surface separation distance between two large particles scaled by the nanoparticle diameter and is related to the center-to-center distance by the relation x)

RTc

M2 NA

(21)

when NA is Avogadro’s number (NA ) 6.02 × 1023). McQuarrie40 proposed the calculation of the potential of interaction using the second virial coefficient (B′) as B′ ) 2π

∫

∞

0

r dr [1 - exp(- W(r) kT )] 2

(22)

The effective potential of interaction for hard spheres (W(r)/ (kT)) is an oscillatory decay curve with a period of oscillation of one particle diameter and can be calculated by solving the integral described in eq 22. First, the value of W(r) was obtained by Monte Carlo simulation at a certain particle size (2a) and volume fraction (φ). It then was substituted into eq 22 and the value of B′ was calculated. After obtaining W(r), it was used to calculate the sedimentation velocity in eq 9. Results and Discussion Equations 11-18 are used first to calculate the total effective potential of interaction between a pair of large hard spheres in the presence of nanoparticles, including the van der Waals interaction. The volume fractions of nanoparticles are 0.15 and 0.30, respectivelysthe same as those used in our experiments. The potential of interaction is the oscillatory decay. The effective interaction energy between a pair of large hard-sphere particles using the particle dynamic simulations was also compared with Trokhymchuk et al.’s method of calculation41 by solving the Ornstein-Zernike integral equation of statistical mechanics. Good agreement between the results obtained by the two different methods was observed. Equations 19-22 were then used to evaluate the multiparticle interactions (self-depletion) between the larger particles alone for two different particle sizes (500 and 800 nm). When the large particle size is reduced from 800 nm to 500 nm, the particle self-depletion attraction (∼3kT) may also need to be considered. However, the self-depletion attraction of ∼3kT is much less than the depletion attraction between large particles in the presence of small particles, which is estimated to be 40kT. Hence, in this study, we neglect the self-depletion effect between only large particles. The sedimentation experiments were performed using the backscattered technique. The accuracy of the measurements in the sedimentation experiments was confirmed by the comparison of the experimental structure factor, S(Q), as a function of the scattering vector Q, scaled by the large particle diameter D, with the theoretical predictions from the Monte Carlo simulations using the hard-sphere potential for latex suspensions of 0.5 and 15 vol %. The Monte Carlo simulations were performed in a fixed cubic box with dimensions of 15 × 15 × 15 (scaled by particle diameter). The number of particles in the simulated box varied over a range of 50-1000 particles, depending on the particle concentration. An acceptable agreement was observed between the theoretical simulations and the experimentally measured values of the structure factor.


Figure 4. Comparison of the final settling stage of sedimentation with and without micelles.

Figure 5. Sedimentation rate as a function of time. Particle size, 800 nm; initial volume fraction of latex particles, 0.15 (bidisperse); effective micellar volume fraction, 0.30. Table 1. Calculated Dimensionless Force Parameters (FD/(kT)): Large Particle Size, 800 nm; Particle Size Ratio, 80:1; Proportion of Small Particles, 30 vol % force parameter

value

gravity hydrodynamic Brownian motion depletion well structural barrier

0.026 0.02 0.00004 0.02 0.007

The two photos shown in Figure 4 depict the final stage of the sedimentation process of colloidal suspension with and without nonionic micelles. It was observed that, in the presence of the nanofluid (micelles), the large-particle separation is more complete, because the particles are more likely to stay in contact with other particles, resulting in the colloid crystal formation indicated by an iridescent color (Figure 4). This becomes apparent when we plot the sedimentation rate, determined as a function of time, as shown in Figure 5. We observed three different regimes. At the beginning of the sedimentation process, the separation rate remains almost constant, as indicated by a constant slope (up to about 40 days). Because the initial particle suspensions contain 15 vol % large particles, the interparticle distance between the particles is relatively large and there is no significant interaction between the large particles (regime 1). At this interparticle distance, the effects of the depletion and structural forces do not play a significant role. The reduction in the settling rate is observed when the interparticle distance between the large particles approaches several diameters of the nanoparticles; this occurs

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Figure 6. Comparison of the sedimentation velocity normalized by the hydrodynamic backflow as a function of the particle volume fraction. Latex particle size, 800 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.30.

when the large particles are prevented from approaching each other because of the layered structure that has formed around them by the small particles. The settling process is reduced. The repulsive structural barrier (the effective pair potential of interactions between large particles) is not sufficient (FD/(kT) ) 0.007) to oppose the gravitational force (FD/(kT) ) 0.026; see Table 1) and stop the sedimentation. However, it significantly reduces the sedimentation rate (regime 2). As settling proceeds and the large particles approach an interparticle distance of less than one nanoparticle diameter, the depletion attraction promoted by the small particles accelerates the sedimentation rate (regime 3). To elucidate the role of structural and depletion forces promoted by the presence of small particles (micelles), we compare the sedimentation of binary systems with monodisperse (without the presence of small particles) systems (see Figure 5). At the beginning (i.e., when the surface-to-surface distance between the large particles is large), the sedimentation rates for both the monodisperse and bidisperse suspensions (after considering the effect of micellar solution viscosity) show the same settling velocity. In the monodisperse suspension, we do not observe the regions where depletion and structural forces play a significant role (as seen in the bidisperse system). Figure 6 illustrates the comparison of the sedimentation velocity normalized by the hydrodynamic backflow, as calculated from eq 9 using K ) -6.55 against the particle volume fraction. The accuracy of the sedimentation rate measurement ((15%) is shown by an error bar. There is reasonable agreement between the theoretical model and experimental results. This becomes clear when we calculate the dimensionless force parameters used in the simulation model (see Table 1). The measured viscosity of the 30 vol % micellar solution is 4.7 cP. The repulsive structural barrier is ∼3 times less than the gravitational force; the gravitational force is higher and moves a particle down past the energy barrier into the attractive depletion region. The sedimentation experiments were also conducted to study the effects of the gravitational force (by varying the colloidal particle size) and structural depletion forces (by varying the micellar concentration) on the sedimentation rates. Effect of Gravitational Force. The sedimentation experiment was conducted for 500-nm latex (instead of 800 nm) particles in the dispersion of 30 vol % (instead of 20 vol %) micelles. Figure 7 shows how the colloidal particle size and the micellar volume fraction affect the normalized sedimentation rate (i.e.,

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Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 Table 3. Calculated Dimensionless Force Parameters (FD/(kT)): Large Particle Size, 800 nm; Particle Size Ratio, 80:1; Proportion of Small Particles: 15 vol % force parameter

value


0.026 0.02 0.00004 0.0048 0.0009

Table 4. Parameters Used in the Modela 30 vol % Micelles parameter Figure 7. Comparison of sedimentation rate normalized by the backflow effect. Latex particle size, 500 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.30. Table 2. Calculated Dimensionless Force Parameters (FD/(kT)): Large Particle Size, 500 nm; Particle Size Ratio, 50:1; Proportion of Small Particles, 30 vol % force parameter

value


0.004 0.0028 0.00006 0.02 0.007

dynamic stability) of the suspension. By reducing the particle size, we decrease the gravitational force; by increasing the micellar volume fraction, we enhance the structural repulsive force. As shown in Table 2, the gravitational force of the 500nm particles (FD/(kT) ) 0.004) is ∼2 times less than the structural force (0.007). Indeed, after the colloidal particle concentration reaches a volume fraction of ∼50% (or the distance between large particles measured from surface to surface, corresponding to one small particle diameter), the structural stabilization barrier dominates the gravitational force and the sedimentation stops. The corresponding dimensionless force parameter used in the particle dynamic simulation is also shown in Table 2. The viscosity of the 30 vol % micellar solution is 4.7 cP. We found good agreement between the model prediction of the sedimentation velocity normalized by the backflow and the experimental observations.

Figure 8. Comparison of sedimentation rate normalized by the backflow effect. Latex particle size, 800 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.15.

particle size, d viscosity, η (at 25 °C) density, F particle interaction, U0 diffusion coefficient, D0 a

15 vol % Micelles

size ratio ) 80/1

size ratio ) 50/1

size ratio ) 80/1

800 nm 4.7 cP 1.05 g/cm3 4.18 cm/s 1.2 × 10-9 cm2/s

500 nm 4.7 cP 1.05 g/cm3 1.63 cm/s 1.9 × 10-9 cm2/s

800 nm 1.5 cP 1.05 g/cm3 12.8 cm/s 3.6 × 10-9 cm2/s

Obtained using eq 9.

Figure 9. Comparison of concentration profile after 40 days. Latex particle size, 800 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.30.

Effect of Nanoparticle (Micellar) Concentration. We also conducted the sedimentation experiments for the 800-nm latex particles in the presence of small particles (15 vol %). The experimental results are shown in Figure 8. No reduction in the sedimentation rate was observed when the concentration of the small particles was reduced; the repulsive structural stabilization significantly decreased from 20kT to 3kT. However, upon decreasing the small particle concentration, both the structural and depletion forces are expected to decrease. The change in the depletion energy has less of an effect, compared to that of the structural barrier, so we observed an increase in the settling rate that was due to the depletion attraction force. The dimensionless force parameter used in the model prediction is summarized in Table 3. The viscosity of the 15 vol % micellar solution is 1.5 cP. The sedimentation rate normalized by the backflow effect is in satisfactory agreement with the model, with consideration to the effective interaction in the presence of small particles. Settling Concentration Profiles. We also model the settling concentration profiles using particle dynamic simulations with the parameters listed in Table 4. Figure 9 illustrates the settling concentration profile (obtained by the diffraction experiments) of the sedimentation process of 800-nm latex particles in the


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Figure 10. Comparison of concentration profile after 50 days. Particle size ratio, 80/1; initial large particle, 15 vol %; small particle concentration, 30 vol %.

suspension of 30 vol % micelles after 40 days. The accuracy of the diffraction measurement of the large particle concentration was within (5%. Based on the trend of the data for the latex particle concentration presented in Figure 9, four sedimentation concentration regions are established. The first region is at the top of the containersthe “blanket zone”, where a clear liquid interface can be seen. Below the first region is the second region, the “free-settling zone”, where the particle concentration is constant during the sedimentation process. The large particle concentration in this region is the same as the initial particle concentration. In the third region, the particle concentration increases sharply, resulting in a large gradient. Finally, the particles accumulate at the bottom of the container, until the particle concentration approaches the maximum packing structure in the fourth region (see Figure 10). This figure shows the particle sedimentation profile at the final settling stage (after 60 days) where the coexistence of the liquid- and colloidalcrystal structure was observed. The sedimentation process occurs slowly, so the particle crystallization accumulates at the bottom of the container with a final volume fraction of 0.64. We observed a coexistent phase of a liquidlike and solidlike structure. The model results agree with the experimental observations. We also model the settling concentration profile using 500nm latex particles in a suspension of 30 vol % micelles (nanoparticles). The data are compared with the modeling results presented in Figure 11. The model results are in good agreement with the experimental observations. The concentration profile of the sedimentation process using 800-nm latex particles in the suspension of 15 vol % micelles (nanoparticles) is shown in Figure 12. Using the parameters presented in Table 3, it was determined that the model results are again in good agreement with the experimental observations. Effect of Particle Size Ratio. To investigate the role of collective particle interactions in a bidisperse fluid that contains particles with a large particle size ratio (such as 80:1) when the large particles are surrounded by other neighboring particles, as shown in the sketch in Figure 13, two scenarios are analyzed: two large particles (pair) surrounded by nanoparticles and many large particles (collective effect) surrounded by nanoparticles. The concentration of small particles is fixed at 20 vol % with a diameter of 10 nm. Both cases are expected to have the same structural barrier promoted by the nanoparticles. However, it is determined that, in the presence of 15 vol % larger particles and 20 vol % small particles, because of the additional confinement of small particles between the large particles, the small

Figure 11. Comparison of concentration profile after 40 and 60 days. Latex particle size, 500 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.30.

Figure 12. Comparison of concentration profile after 40 and 60 days. Latex particle size, 800 nm; initial latex volume fraction, 0.15; effective micellar volume fraction, 0.15.

Figure 13. Structural stabilization barrier in the unit of kT energy, as a function of particle size ratio. Small particle concentration ) 20 vol %. The large particle concentration is 15 vol % when considering the collective interaction curve.

particles have a tendency to layer around the large particles, instead of remaining free in the bulk phase. This collective effect leads to higher structural stabilization (i.e., increase in structural barrier), as shown in Figure 13.

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Figure 14. Plot of interaction energy as a function of the particle size ratio (large to small).

the concentration of smaller particles. At larger size ratios (R′ > 20), larger particles do not affect ln g22. In other words, the packing structure of larger particles is controlled by the smaller particles, except in the limit of small concentrations of larges particles. Increasing the size ratio leads to an increase in the structural barrier, as ln g22 (d22) is directly proportional to the energy of interaction between large particles. These results for the effect of the size ratio on the structural energy barrier are consistent with our theoretical results, based on the stochastic particle dynamics method (see Figure 13). Increasing the particle size ratio leads to an increase in the structural barrier, because the structural stabilization barrier promoted by small particles is directly related to the layering phenomenon of the small particles attached to the surface of the large particles. Hence, the total surface area of the large particles (AT) is a key parameter. AT ) 4N

πd222 ) Nπd222 4

(25)

and φ)

( )

3 N πd22 V 6

(26)

then AT ) Figure 15. Number of small particles required to stabilize the large particles, as a function of particle size ratio (large to small). Large particles, 15 vol %; small particles, 20 vol %.

We verified the particle size ratio effect using theoretical calculations based on the Percus-Yevick (PY) theory for interactions between large particles in a hard-sphere mixture of small particles as a function of their size ratio, R′ ) d22/d11, where d22 is the diameter of large particles and d11 is the diameter of small particles. Following Lebowitz,22 the contact values of the radial distribution function of large particles g22 (d22) can be obtained from the Laplace transform. The resulting contact value between a pair of large particles is obtained from the Laplace transform: g22(d22) )

(

)

δ2 1 3 + d R′ 1 - δ3 2 1 - δ3 11

(23)

where δn )

π F(φ1dn11 + φ2dn22) 6

(24)

Here, φ is the volume fraction, N/(N1 + N2) ) Ni/N, and F ) N/V (where V is the volume of the system). Figure 14 is a plot of ln g22 (d22) as a function of the size ratio R′ for δ3 ) 0.3 and φ2 ) 0, 0.005, 0.01, and 1. The upper dash curve represents the packing density of a pair of large particles (when φ2 f 0) that are completely surrounded by small particles. The value of ln g22 (d22) increases the size ratio increases. However, the magnitude of ln g22 (d22) is strongly dependent on the concentration of large particles in the system. As the concentration of large particle decreases, ln g22 increases and the curves have a maximum at a particle size ratio of ∼5. At smaller size ratios, ln g22 (d22) is dependent appreciably on

6φV d22

or AT ∝

1 d22

(27)

where N/V is the number density, φ the volume fraction, and d22 the diameter of the large particles. By increasing the particle size ratio (d22/d11), the total surface area clearly decreases as the number of large particles (N) decreases; the number of small particles required to stabilize the large particles is small (see Figure 15). The calculation represents the system with a large particle concentration at 15 vol % with 20 vol % small particles present. Concluding Remarks We have presented the results of our theoretical and experimental investigations on the non-DLVO (entropic) oscillatory structural forces of interaction between microscopic particles and nanoparticles in a hard-sphere fluid. In addition, we have explored the importance of both the attractive depletion and structural forces in particle dispersion stability. The dispersion stability experiments involved the sedimentation of a suspension of latex microspheres in the presence of nonionic micelles (i.e., hard nanoparticles) with large size ratios of large particles to small particles of 50:1 and 80:1. The theoretical approach is based on statistical mechanics, Monte Carlo simulations, and stochastic particle dynamic methods. The effects of the particle size ratio and nanoparticle concentration on the interparticle interaction potential due to the collective effect (and, thereby, on the dispersion stability) are highlighted. The results of the present study clearly indicate that the amplitude and positions of the maxima and minima in energy oscillations due to the collective interactions between the colloidal particles (i.e., microspheres) in the presence of nanosized particles are greatly dependent on the concentration and particle size ratios (large to small). At a fixed concentration and size ratio of colloidal particles, higher concentrations of nanoparticles yield higher structural energy barriers, resulting


in more-stable colloidal dispersions. Also, at a fixed concentration of colloidal particles and nanoparticles, greater size ratios result in more-stable colloidal dispersions. The stability of the colloidal dispersion can be controlled by manipulating both the volume concentration of the particles and their size ratios. We verified the role of surface forces (the attractive depletion and structural energy barrier) in the stability of a bidisperse fluid system by conducting a sedimentation experiment using a suspension of low-charge latex particles dispersed in a nanofluid (nonionic micelle solution). During sedimentation, we monitored the colloidal particle concentration profile, microstructure formation, and rate of sedimentation. We compared the experimental results with the theoretical model predictions and found good agreement. Literature Cited (1) Sanyal, S.; Easwar, N.; Ramaswamy, S.; Sood, A. K. Phase Separation in Binary Nearly-Hard-Sphere Colloids: Evidence for the Depletion Force. Europhys. Lett. 1992, 18, 107–110. (2) Asakura, S.; Oosawa, F. Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci. 1958, 38, 183. (3) Kaplan, P. D.; Rouke, J. L.; Yodh, A. G. Entropically Driven Surface Phase Separation in Binary Colloidal Mixtures. Phys. ReV. Lett. 1993, 72, 582–585. (4) Roth, R.; Evan, R.; Dietrich, S. Depletion Potential in Hard-Sphere Mixtures: Theory and Applications. Phys. ReV. E 2000, 62, 5360–5377. (5) Tolpekin, V. A.; Duits, M. H. G.; van den Ende, D.; Mellema, J. Stability Ratio in Binary Hard Sphere Suspensions, Measured via TimeResolved Microscopy. Langmuir 2003, 19, 4127–4137. (6) Tohver, V.; Smay, J. E.; Braem, A.; Braun, P. V.; Lewis, J. A. Nanoparticle halos: A new colloid stabilization mechanism. Proc. Natl. Acad. Sci., U.S.A. 2001, 98, 8950–8954. (7) Martinez, C. J.; Liu, J.; Rhodes, S. K.; Luijten, E.; Weeks, E. R.; Lewis, J. A. Interparticle Interactions and Direct Imaging of Colloidal Phases Assembled from Microsphere-Nanoparticle Mixtures. Langmuir 2005, 21, 9978–9989. (8) Chan, A. T.; Lewis, J. A. Size ratio effects on interpaticle interactive and phase behavior of microsphere-nanoparticle mixtures. Langmuir 2008, 24, 11399–11405. (9) Baird, J. C.; Walz, J. Y. The effects of added nanoparticles on aqueous kaolinite suspensions: I. Structural effects. J. Colloid Interface Sci. 2006, 297, 161–169. (10) Xu, W.; Nikolov, A. D.; Wasan, D. T. Role of Depletion and Surface-Induced Structural Forces in Bidisperse Suspensions. AIChE J. 1997, 43, 3215–3222. (11) Wasan, D.; Nikolov, A.; Henderson, D. New Vistas in Dispersion Science and Engineering. AIChE J. 2007, 49 (3), 550–556. (12) Crocker, J. C.; Maheoo, J. A.; Dinsmore, A. D.; Yodh, A. G. Entropic Attraction and Repulsion in Binary Colloids Probed with a Line Optical Tweezer. Phys. ReV. Lett. 1999, 82, 4352–4355. (13) Piech, M.; Weronski, P.; Wu, X.; Walz, J. Y. Prediction and Measurement of the Interparticle Depletion Interaction Next to a Flat Wall. J. Colloid Interface Sci. 2002, 247, 327–341. (14) Nikolov, A. D.; Wasan, D. T. Dispersion Stability Due to Structural Contributions to the Particle Interaction as Probed by Thin Liquid Film Dynamics. Langmuir 1992, 8, 2985. (15) Henderson, D. J.; Nikolov, A. D.; Trokhymchuk, A.; Wasan, D. T. Confinement-Induced Structural Forces in Colloidal Systems. In Encyclopedia of Surface and Colloid Science, Second Edition; Somasundaran, P., Ed.; Taylor and Francis: New York, 2006; pp 1485-1494. (16) Xu, W.; Nikolov, A. D.; Wasan, D. T.; Gonsalves, A.; Browankar, R. Particle Structure and Stability of Colloidal Dispersions as Probed by the Kossel Diffraction Technique. J. Colloid Interface Sci. 1997, 191, 471– 481. (17) Vesaratchanon, S.; Nikolov, A. D.; Wasan, D. T. Sedimentation in Nano-Colloidal Dispersions: Effects of Collective Interactions and Particle Charge. AdV. Colloid Interface Sci. 2007, 134-135, 268–278.

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ReceiVed for reView December 23, 2008 ReVised manuscript receiVed May 14, 2009 Accepted May 14, 2009 IE8019856

The Importance of Oscillatory Structural Forces in the Sedimentation of

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