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Ind. Eng. Chem. Res. 2009, 48, 2414–2421
Rayleigh-Be´nard Instability in Sedimentation Darrell Velegol,* Shailesh Shori, and Charles E. Snyder Department of Chemical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802
Instabilities often arise in the sedimentation of colloidal particles, even when using the density gradient technique. These instabilities occur at volume fractions as low as 10-3 or 10-4, causing mixing of particles throughout the suspension, rather than the smooth sedimentation of particles. Here, we show that the mixing is due to a classical Rayleigh-Be´nard instability. The sedimentation process is modeled in an approximate manner, and experiments are done to test whether the approximate model gives maximum stable volume fractions in the correct range. Sedimentation is a well-studied problem, and yet, to our knowledge, this is the first time that the well-known instability has been described in terms of a Rayleigh-Be´nard instability. Introduction In the preparation of colloidal assemblies,1-6 a key bottleneck to scalable production is the separation of desired assemblies from unreacted precursor particles and undesired side products. The desired assemblies, unreacted precursors, and side products often have different sizes or shapes. A seemingly simple and scalable way to separate suspended particles by size or shape is by gravity, that is, by sedimentation or centrifugation (hereafter called simply “sedimentation” unless otherwise noted). However, a notorious instability problem occurs during sedimentation, such that the particles simply mix throughout the separation vessel.7-9 As a result, scientists often use a suspending medium having a density gradient to promote a well-behaved separation. However, the density gradient sedimentation is still limited to a low maximum allowable particle volume fraction (φmax < 10-3 or 10-4).10 The purpose of this Article is to show that the origin of this difficulty is a classic Rayleigh-Be´nard instability. In our own laboratory, we have a need to separate properly formed colloidal assemblies from unreacted colloidal precursors and side products.4-6 Many other research groups around the world have the same need.1-3 As one specific example, we want to separate mixtures of roughly 80% single spheres and 20% doublets of those spheres into a stream of greater than 95% doublets, at a throughput of 10-100 kg/day. Using particle volume fractions Ff0 is the density of the particles; and γ is the suspending fluid density gradient (dFf/dz, assumed linear here for simplicity). As long as ∂F/∂z > 0 everywhere in the system, the system is stable. The density changes as a function of particle volume fraction, or gradients due to sugar, Ficol, temperature, or other means. The challenge of course is that for a time t > 0, the dense particles settle into the underlying gradient. At some point it
10.1021/ie800720k CCC: $40.75 2009 American Chemical Society Published on Web 11/06/2008
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Figure 2. Schematic of the Rayleigh-Be´nard problem. The fluid heated from below has a lower density, and thus the situation shown in the schematic will invert by gravity, once the temperature gradient is high enough. Gravity is pointing in the negative z direction. Figure 1. Schematic of the sedimentation problem. The top is loaded with particles, such that the density of the loaded complex fluid is less than the upper density of the fluid below. This makes loading the suspension on top easier. In the absence of a density gradient (γ), the sedimentation process becomes unstable, with the particle suspension mixing into the underlying fluid in a rapid manner. However, the instability can occur even with a density gradient (as shown), if the volume fraction of particles is too high. The particle suspension is miscible with the density gradient fluid beneath, giving a Rayleigh-Be´nard instability.
will occur, especially for significant volume fractions of particles, that ∂F/∂z < 0. Two important questions arise at this point: (1) For a given initial volume fraction (φ0), what will be the maximum density inversion? (2) Is it possible that even if ∂F/∂z < 0, the suspension could be stable? Answering these questions by modeling is one primary objective of this Article. The other objective is to provide some experimental testing of the model. We note that for particles of different density, one can often use an isopycnic separation, with the suspending fluid having a spatially varying density, such that a given particle type stops settling when the particle density equals the local fluid density. However, isopycnic separations are not possible if the particles have a greater density than do the feasible fluids. For example, silica has a specific gravity (SG) of 2.0, silicon has SG ) 2.33, and gold has SG ) 19. For assemblies containing these materials, isopycnic separations are seldom possible. When particles have the same density, polystyrene suspensions for example, even if one has 80% single polystyrene spheres and 20% polystyrene doublets, a rate separation is used for gravity separations. Rate separations are subject to instabilities. We also note that other techniques are currently used to separate two or more types of particles by size. These include field flow fractionation,16 hydrodynamic chromatography,17,18 and even osmotic techniques.19,20 While these techniques obtain excellent separation for laboratory scale analysis, they are less effective for commercial scale preparation. For example, in hydrodynamic chromatography, the throughput is usually less than one milliliter per hour. Field flow fractionation (FFF) has a related basis as hydrodynamic chromatography, except that a field (e.g., gravity) is used to pull one type of particle selectively toward the wall, causing it to move more slowly. Commercial field flow fractionation units exist (e.g., the Wyatt Eclipse FFF sells for about $75,000 in 2007), but the throughput is still analytical scale. Modeling the Rayleigh-Be´nard Instability In the year 1900, Henri Be´nard found that a fluid heated from below, such that the density of the lower fluid was lower than the fluid above it, becomes unstable (Figure 2). Yet he also
found that this occurs only above a certain temperature gradient.21 Over a decade later, Lord Rayleigh developed the definitive model describing the instability.22,23 By combining the energy balance, the heat equation, the fluid dynamics equations, and a perturbation scheme, Rayleigh showed from a normal modes analysis that the system would be stable between two surfaces as long as the Rayleigh number (R) is less than a critical Rayleigh number (Rc):23
(
gR)
1 ∂F ∂T 4 d F ∂T ∂z < Rc Rν
)
(4)
where g is the gravitational constant; F ) F(T) is the fluid density as a function of temperature (T); z is the vertical distance from the lower warm surface; d is the thickness of the fluid region between the surfaces; R is the thermal diffusivity (R ) k/FCp depends on the fluid conductivity k; Cp is the fluid heat capacity); and ν is the momentum diffusivity (ν ) η/F depends on the fluid viscosity η). If both surfaces (i.e., the hot temperature surface and the cold temperature surface) are rigid, Rc ) 1707.8. If instead there is one rigid surface and one stress-free surface, Rc ) 1100.7, while if there are two free surfaces (e.g., a fluid-fluid interface), Rc ) 657.5. The key question from Figure 2 is not so much, “What causes the instability?” It makes sense that when a more dense fluid is on top, the system will be unstable. Rather, the question is: What is the physics that allows a small density inversion to remain stable? The essential physics of stability is this: Say a perturbation develops, such that a warm region of fluid begins to rise. Two rates are set into competition: (a) the time required to rise (depends on the kinematic viscosity ν), and (b) the time required to re-equilibrate the temperature so that the region ceases to rise (depends on thermal diffusivity R). If heat conducts very quickly (high R) and the fluid droplet rises very slowly (high ν), then the warm region re-equilibrates quickly relative to the risetime, and the system restabilizes. The point of this Article is that a temperature gradient is not the only thing that can produce the adverse density gradient. Gradients of particle volume fraction (φ) can also produce density gradients, as can gradients of sugar, salt, or polymer concentration in the suspending fluid. Also, whereas heat (i.e., the diffusion of thermal energy, and thus T) relieves the adverse density profile in Rayleigh’s problem, the diffusion of mass (e.g., of sugar, with a diffusion coefficient D) can relieve an adverse density profile in the sedimentation problem. We hypothesize that the mass transfer version of the Rayleigh-Be´nard instability model explains the sedimentation instability quantitatively. The Rayleigh number can be rewritten as
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∂F 4 d ∂z < Rc Dη
-g R)
(5)
where D is the diffusion coefficient of the material increasing the fluid density (in this Article, we will call it “sugar”,24 although it can be Ficoll, CsCl, or even temperature-induced density gradients), and η is the fluid viscosity. By the Stokes-Einstein equation, the combination Dη ≈ D0η0 over a small temperature or sugar concentration range, where the “0” indicates the values at some temperature and other conditions; that is, the product is roughly unchanging. When the Rayleigh number in a system exceeds a critical value for the case of a fluid between two stress-free surfaces, well-studied convection rolls will occur. As described above, if the adverse density profile is linear, the critical value Rc ) 657.5. We see no reason why the critical value should be different for the heat transfer and mass transfer case, because the variable that matters is the density. We are not aware of the Rayleigh-Be´nard explanation being used before for sedimentation separation, either qualitatively or quantitatively. The tasks before us then are to estimate d and dF/dz. To solve the problem properly, we would need to conduct Brownian dynamics simulations on an arbitrary distribution of particle sizes, account for the diffusion of sugar, and then solve numerically for the instability eigenvalue. We proceed in a simpler manner, which we expect will reveal the underlying physics. Before the start of the sedimentation process, one simply has the density gradient. A typical gradient of ours might have the upper fluid density at 1018 kg/m3 (due to 5% sucrose added), and 5 cm below, the lower fluid density at 1039 kg/m3 (10% sucrose). Once a height (h0) of particle suspension is added on top of the density gradient, a mixing process occurs. Three processes contribute to this mixing: (a) Sugar diffuses into the colloidal suspension above; (b) Brownian motion causes a net diffusion of the particles into the fluid below; and (c) the particles settle into the fluid below. The diffusion of sugar into the particles quickly gives rise to a density inversion (Figure 3). That is, although the sugar gradient has ∂F/∂z > 0 (i.e., stable), one sees that after the particles are loaded onto the gradient with some depth and with a uniform volume fraction, there arises a small region in which an adverse density gradient occurs. However, the depth of this density inversion is limited, because below the particles, the fluid density continues to increase due to the sugar gradient.
The condition that gives the maximum depth (dmax) of the inversion is approximately F(z, φ) ≈ F(z + dmax, φ ) 0)
(6)
That is, the fluid directly below the particles has a lower density because it has only fluid. Yet the fluid density increases because the sugar gradient exists. This equation gives the depth over which these two effects balance. The result, when eq 3 is used, is (Fp - Ff0)φ (7) γ For particles to immerse a distance (dmax) into the fluid, some time (t0) is required. We can estimate this time by summing the contributions of Brownian motion (with a particle diffusion coefficient Dp) and settling. dmax ≈
dmax ≈ (2Dpt0)1⁄2 + U0t0
(8)
We can examine whether Brownian motion or settling is more important for particle transport, by examining the Peclet number: U0dmax 4πa3(Fp - Ff)2gφ Pe ) ) Dp 3kTγ
(9)
For the experiments examined later in this Article, Pe > 10, and thus settling is the dominant mechanism by which particles immerse into the sugar gradient. We can therefore estimate t0 as t0 )
dmax 9ηφ ) U0 2γa2g
(10)
During this time, the leading front of the particles has time to spread out, due to both Brownian motion and differential settling. Differential settling occurs because even though the particles have an average radius a, some particles are slightly larger, and others are slightly smaller. We can estimate the differential settling by approximating that the particles have a Gaussian distribution of radii. We let σ be the coefficient of variation of the particle size (assuming a Gaussian distribution), such that 68% of the particles are in a size range between a(1 ( σ). The difference in settling speed (δU) between those particles with radius a(1 ( σ) and those with radius (a) is 4 2 2 2 [a (1 + σ)2 - a2](Fp - Ff)g a σ(Fp - Ff)g 9 9 ≈ (11) δU ) η η where the final estimate assumes a small value for σ (e.g., σ < 0.2). For particles with other shapes, one can substitute the appropriate expression for settling speed into eq 11. Just as we defined a Peclet number for particle transport, we can define a Peclet number for particle spreading as Pespread ) δUdmax/Dp ) 2σPe. For the experiments shown later in this Article, differential settling dominates, but both effects are important for spreading. The distance of spreading (s) can be estimated by simply adding the differential settling and the Brownian motion of the particles, giving
Figure 3. The density profile in a sedimentation cell, at two different times. Initially (solid line), the density of the particle suspension (density at position a) and the fluid gradient (starting at position b) are stable. With time, as the sugar mixes into the suspension and the particles mix into the sugar, a small density inversion (∂F/∂z < 0) occurs, over a distance (d). The particles spread primarily over a distance (s). One finds that d > s, because the diffusion of the sugar also contributes to the changing density gradient.
4 2 a σ(Fp - Ff0 - γz)g 9 t s ≈ (2Dpt) + η 1⁄2
(12)
Substituting t0 for t and using the Stokes-Einstein equation for Dp gives
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(
)
3kTφ 1⁄2 2σ(Fp - Ff)φ (13) + γ 2πa3γg Note if the particles are large and all of identical radius, s ≈ 0. That is, the spreading distance (s) is not the same as the immersion distance (d). Next, we estimate dF/dz. Recall that in the first instant after the particles are loaded onto the density gradient, dF/dz ) -∞. Immediately, the particles start to spread as described previously, reducing the magnitude of this gradient. Differentiation gives s)
∂F ∂φ ) (Fp - Ff0 - γz) + γ(1 - φ) ∂z ∂z We can then estimate
(14)
∂φ φ ≈(15) ∂z s This estimate comes from assuming a linear gradient, so that ∂φ/∂s ≈ ∆φ/∆s ) (φ - 0)/(0 - s), because the top of the region (0) has a volume fraction (φ) and the bottom has φ ) 0. For small values of z and φ, which are typical in a density gradient sedimentation, these equations give ∂F ) ∂z
(
-φ(Fp - Ff0) +γ 3kTφ 1⁄2 2σ(Fp - Ff)φ + γ 2πa3γg
(16)
)
Therefore, our final expression for R is g R)
[
(Fp - Ff0)φ γ D0η0
]
4
[(
]
φ(Fp - Ff0) -γ 3kTφ 1⁄2 2σ(Fp - Ff)φ + γ 2πa3γg
)
(17)
If the value of R exceeds 657.5, then a sedimentation instability will result. It is helpful to remember the assumptions and approximations used to obtain eq 17. We estimated the value of d on the basis of the maximum distance over which a density inversion would develop. By finding the time required for particles to settle this distance, because the particles are ever settling through the tube, we estimated a characteristic time for spreading, accounting for Brownian motion and differential settling. The value of t0 led us to a distance (s) for spreading. We assumed small z and φ values, usually good approximations for our sedimentation experiments. Another important assumption is that we assumed a linear density gradient, as Rayleigh used in solving the original eigenvalues problem, so that we can use Rc ) 657.5. It is unlikely that we will have a linear density gradient for our mass transfer problem. To do the problem in full, not only would we have to detail the density profile using a numerical simulation (e.g., Brownian dynamics for the particles, diffusion for the sugar), but we would also have to numerically search for the instability eigenvalues, rather than obtaining it analytically as Rayleigh did. Experimental Materials and Methods Materials. Ultrapure bioreagent sucrose for the density gradients was obtained from J.T. Baker, lot number E10628. Potassium chloride GR ACS crystals were obtained from EMD, lot number 44322531. Three types of particles were used: 4.0 µm surfactant-free sulfate polystyrene latex microspheres (Invitrogen-Molecular Probes, 4.1% solids, lot number 53607A),
Table 1. Summary of Gravity Experiments Using Singlet Particlesa particle d (µm) σ (%) γ (kg/m4) g/g0 silica PSL PSL PSL PSL PSL
3.16 3.95 5.18 3.95 5.18 3.95
1.9 4.4 5.3 4.4 5.3 4.4
2924 288 288 288 288 288
1 1 1 15 15 25
RB φmax
expt φmax
0.81 × 10-4 × 4.3% 27.9 × 10-4 × 11% 28.5 × 10-4 × 11% 13.5 × 10-4 × 33% 12.1 × 10-4 × 33% 11.8 × 10-4 × 33%
1.5 × 10-4 1.7 × 10-4 3.6 × 10-4 8.9 × 10-4 18 × 10-4 8.9 × 10-4
a Various values of particle diameter, density gradient, and g values were used. The value of g0 ) 9.8 m/s2. The PSL particles are sulfated polystyrene latex particles. The uncertainties on the experimental values reflect one-half the difference between the last stable and first unstable volume fraction.
4.9 µm surfactant-free sulfate polystyrene latex microspheres (Invitrogen-Molecular Probes, 4.1% solids, lot number 51933A), and 3.0 µm silica microspheres (Bangs Laboratories, 9.8% solids, lot number 5411). The 15 mL snap-cap centrifuge tubes were obtained from Kimble Glass, Inc. Rain-X glass cleaner was obtained from a local automobile parts store; the Rain-X was used to reduce meniscus effects. J-shaped and fine-tipped syringes were used for loading and extracting particles, respectively. The gradient maker was obtained from CBS Scientific (model #GM-100). Our Sorvall Biofuge Primo centrifuge (Kendro Laboratory Products) is equipped with a swing-bucket rotor that can carry two capped tubes simultaneously. The imaging equipment included a scanning electron microscope (Hitachi SEM model S-3500N) and an optical microscope (Nikon Eclipse TE2000-U). A simple eight-tube holder was placed in a dark box, lighted by a fluorescent lamp, and viewed with a video camera. The images were captured with MovieStar software and analyzed using ImageJ software. Methods. Sucrose solutions were chosen to give the desired density gradient (γ). We found γ as the difference between densities in the bottom and top layers divided by the length of the density gradient. The density solutions are made using a standard sucrose concentrations table25 by weighing out the sucrose, diluting with prepared 10 mM KCl, and mixing thoroughly. Next, 15 mL glass tubes were cleaned and coated with Rain-X glass cleaner, which prevented significant meniscus effects. We found no chemical effects resulting from the use of Rain-X. The selected colloidal microspheres, consisting mostly of singlet particles with some flocculates that were simply sonicated or vortexed apart, were imaged and sized using SEM for an overall average diameter and resulting coefficient of variation of 100 particles to compare to the provided manufacturer’s specifications. Before the density gradient was made, 3 mL of a dense sucrose solution (specific gravity SG ) 1.080) was added to the bottom of each centrifuge tube to prevent the particles from settling to the very bottom of the tube. The density gradient, consisting of a determined volume (5 mL) of both the light (SG ) 1.018) and the heavy (SG ) 1.036) phases, was then placed on top of this denser solution in each tube using a density gradient maker. This gradient varied approximately linearly from the heavy sucrose solution at the bottom to the light solution at the top of the test tube. For our ImageJ analysis, the tubes filled with the sucrose solutions were imaged within a dark box for subtracting background effects from images taken with the separating colloidal particles. Using a J-shaped syringe, 1 mL of the colloidal microspheres at specified volume fractions was carefully loaded just below the surface of the density gradient into each glass tube. The tubes were then imaged. The suspensions either settled under the influence of gravity by sedimentation or were centrifuged to expedite the sedimentation (and any separation) process. As
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Figure 4. Stable and unstable sedimentation. The particles were 3.95 µm PSL singlet at 1g and volume fractions (from left to right) of 1.15 × 10-4, 1.53 × 10-4, 1.91 × 10-4, 2.30 × 10-4, 3.06 × 10-4, 4.59 × 10-4, 6.12 × 10-4, and 7.65 × 10-4. The density gradient γ ) 288 kg/m4. (a) Initial loading. (b) After 48 h of sedimentation. The two tubes on the left are stable, while the others are not.
the particles settled down the density gradient at 1g, snapshots were taken for the stability analysis and for the formation of any separation bands that may appear. Tubes that were centrifuged could not be examined during the centrifugation process, and so images were taken after the centrifugation process was completed. If definitive bands appeared within the gradient, especially for our experiments where we intentionally started with some aggregates (e.g., doublets, triplets, etc.), colloidal particle types were sometimes extracted and imaged. The resulting separations were viewed using optical microscopy for determining particle type (e.g., singlets, doublets, triplets) and could be sized with SEM for evaluating singlet separations by measuring particle diameters.26 Results and Discussion Sedimentation. Table 1 summarizes the results from the sedimentation and centrifugation experiments. Singlet microspheres consisting of 3.95 and 5.18 µm in diameter sulfated polystyrene latex (PSL) and 3.16 µm silica particles were used. The sizes and coefficients of variation (σ) listed in Table 1 were determined using SEM. The density gradient values (γ) were 288 or 2924 kg/m4 for these experiments. A significantly higher gradient value was chosen for the silica singlets to account for the higher density of the particles, and thus to increase the volume fraction and visibility of the particles. Using these parameters, a maximum theoretical critical volume fraction RB (φmax ) was calculated, and then compared to the experimental expt maximum (φmax ). Figure 4 shows the 3.95 µm PSL singlet spheres settling by sedimentation. The initial loaded tubes are shown in Figure 4a,
Figure 5. Suspensions at 15g. 3.95 µm PSL singlet particles. (a) Initially, (b) after 4.5 h of continuous centrifugation, and (c) then after 21 h of sedimentation at 1g following centrifugation. The tube on the left has a volume fraction of 5.93 × 10-4, and the tube on the right has 1.19 × 10-3. Both have a density gradient γ ) 288 kg/m4.
while Figure 4b shows the particles after settling. As expected, with increased volume fraction, the spread of particles increases, indicating an instability. Faint, yet noticeable, bands of singlets can even be seen at the lower volume fractions, indicating a separation. Interestingly, when these suspensions were allowed to settle for a significantly longer time, distinct bands formed in all of the tubes, including the higher volume fractions, signifying that the observed instabilities eventually were resolved. Bands were produced equivalent to that of the lower concentrations of singlet particles at earlier times in the sedimentation process. The dynamics behind the mechanism for restabilization are not clear at this point, but similar findings have been observed previously in the literature.15 Comparing stable volume fractions from the model and from experiments was not a trivial procedure. Experimental challenges existed, such as visualizing the instabilities for low volume
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Figure 7. Doublets at two different g values. 3.95 µm PSL particles consisting of fused doublets (a) after 4.75 h of continuous centrifugation at 15g, and (b) then after an additional 22 h of sedimentation at 1g following centrifugation with 5.05 × 10-4 and 1.44 × 10-3 volume fractions (from left to right in both) and γ ) 288 kg/m4.
Figure 6. Suspensions at higher g values. 3.95 µm PSL singlet particles (a) after 3.5 h of continuous centrifugation at 25g, (b) after 2.5 h of continuous centrifugation at 50g, and (c) after 1.5 h of continuous centrifugation at 100g. The tube on the left has a volume fraction of 5.93 × 10-4, and the tube on the right has 1.19 × 10-3. Both have a density gradient γ ) 288 kg/m4.
fractions (φ < 10-3), especially over short (micrometer scale) distances. Another important experimental difficulty, perhaps the most important, was loading the particles, because some mixing of the loaded suspension and the existing density gradient occurs during the loading process. From a modeling perspective, our assumptions have been noted previously. Another challenge was in experimentally characterizing the distribution of particle sizes as an input to our model. In our model for differential settling, we use a coefficient of variation (σ), which assumes a Gaussian particle size distribution. In fact, the particle size distribution was sometimes bimodal, as if we had two populations of particles, each with a smaller value of σ. Nevertheless, for various values of particle sizes, g values, and gradient values, the observed differences between the predictions from the stability model and the experimental results were within at most
a factor of 17. We would not call this “good agreement”, but nor is it entirely disappointing, and, indeed, our model has provided useful guidance in our laboratory for stable particle loadings in density gradient sedimentation. Centrifugation. Density gradient centrifugation experiments above 1g were also performed (Table 1). Figure 5 illustrates results from centrifuging 3.95 µm ((0.17 µm, from Table 1) sulfate PSL singlet particles continuously at 15g after 4.5 h. Various volume fractions of particles were used. Figure 5b shows a relatively stable suspension on the left and an unstable suspension on the right. After these tubes were centrifuged, they were allowed to undergo sedimentation under 1g. We often found a greater tendency for instabilities to develop when the suspensions were returned to low g values (Figure 5c, after 21 h). The current model would predict the opposite trend, that lower g values would be more stable, and further study will be required here. To assess the effect of higher gravitational fields, the identical particle volume fractions used at 15g were also centrifuged at 25g, 50g, and 100g (Figure 6). At higher g values, the singlet microspheres mixed almost completely within the tube, even at volume fractions 90% singlets), (b) band second from top (typically >80% doublets), and (c) third from top band.
unjoined singlets and/or higher aggregates. That is, the separations are very good, but not perfect. Conclusions The Rayleigh-Be´nard instability theory has been translated from the temperature diffusion case to mass diffusion and has been applied to the sedimentation process of microparticles (eq 17). A number of approximations were used for this first analysis. Rayleigh’s original problem also differs in that it involved an unchanging temperature gradient, whereas here in the sedimentation problem, the density gradient is ever forming at lower and lower positions within the sedimentation tube. Yet at 1g the model gives predicted stability results within a factor of 17 (Table 1). While not giving precise agreement between the predicted maximum stable volume fraction and experiments, the Rayleigh-Be´nard model has been very helpful in guiding
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experiments in our laboratory for the separations of desired colloidal assemblies. The model gives the maximum allowed volume fraction, in terms of the particle’s buoyant density and the density gradient. Importantly, at least for larger particles, the model predicts that differential settling, and therefore the coefficient of variation in particle size, controls φRB max. This means that if one chooses to separate a mixture twice, such that σ is smaller for the second run than the first, the separation will require a lower volume fraction of particles (φ). Equation 17 also reveals that gold (SG ) 19) has a lower allowable volume fraction than does polystyrene latex (SG ) 1.055), for the same coefficient of variation in particle size. At high g values, the sedimentation process becomes unstable, as predicted by the model. We have found that starting and stopping the centrifugation process promotes mixing as well. Curiously, even for unstable suspensions, with longer settling times one often finds a restabilization forming bands; furthermore, the bands contain distinct types of particles, meaning that a useful separation has occurred. Although not done in this work, the Rayleigh-Be´nard model holds promise for explaining the convection rolls seen in sedimentation experiments found in the literature.14,15 However, to calculate results to explain the experiments, which had various geometries and other conditions, it is likely that the full model would be needed. Such a model would account for particle movement with Brownian dynamics, sugar diffusion, and the instability eigenvalues with numerical root searching. We have also begun to examine experiments in which the value of d from the Rayleigh-Be´nard model is kept to a minimum. Although this is difficult in an open solution, it is simple and scalable in a packed column. Initial experiments have demonstrated success. Furthermore, for those systems that do not permit a “contaminant” like sugar to be added to the particles, temperature gradients can be used to effect small density gradients within the fluid. It is important to recognize that the instabilities described in this paper will apply in separating suspensions using forces on particles other than gravity, such as magnetic forces. Acknowledgment D.V. thanks Professor Anderson for his magnificent example of scholarship and leadership, as well as for his friendship. D.V. also thanks Mrs. Linda Angus, whose insights into the instability phenomenon preceded his own by several years. Finally, we thank the National Science Foundation for funding this research through NIRT grant CCR-0303976 and CBET grant 0651611. Literature Cited (1) Manoharan, V. N.; Elsesser, M. T.; Pine, D. J. Dense Packing and Symmetry in Small Clusters of Microspheres. Science 2003, 301, 483– 487. (2) Yin, Y.; Lu, Y.; Gates, B.; Xia, Y. Template-Assisted Self-Assembly: A Practical Route to Complex Aggregates of Monodispersed Colloids with Well-Defined Sizes, Shapes, and Structures. J. Am. Chem. Soc. 2001, 123, 8718–8729. Yin, Y.; Lu, Y.; Xia, Y. A Self-Assembly Approach to the Formation of Asymmetric Dimers from Monodispersed Spherical Colloids. J. Am. Chem. Soc. 2001, 123, 771–772. (3) Johnson, P. M.; van Kats, C. M.; van Blaaderen, A. Synthesis of Colloidal Silica Dumbells. Langmuir 2005, 21, 11510–11517.
(4) Snyder, C. E.; Yake, A. M.; Feick, J. D.; Velegol, D. Nanoscale Functionalization and Site-Specific Assembly of Colloids by Particle Lithography. Langmuir 2005, 21, 4813–4815. (5) Yake, A. M.; Panella, R. A.; Snyder, C. E.; Velegol, D. Fabrication of Doublets by a Salting Out - Quenching - Fusing Technique. Langmuir 2006, 22, 9135–9141. (6) Yake, A. M.; Snyder, C. E.; Velegol, D. Site-Specific Functionalization on Individual Colloids: Size Control, Stability and Multi-Layers. Langmuir 2007, 23, 9069–9075. (7) Price, C. A. Centrifugation in Density Gradients; Academic Press: New York, 1982. This book describes many of the instabilities that occur, and how to counteract them using density gradients. (8) Sheeler, P. Centrifugation in Biology and Medical Science; Wiley: New York, 1981. (9) Rickwood, D. Centrifugation: A Practical Approach, 2nd ed.; IRL Press: Washington, DC, 1984; Chapter 3 in particular describes different types of centrifuges. (10) Hinton, R.; Dobrota, M. Density Gradient Centrifugation; Elsevier: North-Holland, 1976. (11) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Kluwer: Boston, 1991. (12) Batchelor, G. K. Sedimentation in a Dilute Suspension of Spheres. J. Fluid Mech. 1972, 52, 245. (13) Batchelor, G. K. Sedimentation in a Dilute Polydisperse System of Interacting Spheres. J. Fluid Mech. 1982, 119, 379. (14) Segre`, P. N.; Herbolzheimer, E.; Chaikin, P. M. Long-Range Correlations in Sedimentation. Phys. ReV. Lett. 1997, 79, 2574. (15) Tee, S.-Y.; Mucha, P. J.; Cipelletti, L.; Manley, S.; Brenner, M. P.; Segre, P. N.; Weitz, D. A. Nonuniversal Velocity Fluctuations of Sedimenting Particles. Phys. ReV. Lett. 2002, 89, 054501-1–054501-4. (16) Barman, B. N.; Giddings, J. C. Kinetics and Properties of Colloidal Latex Aggregates Measured by Sedimentation Field-Flow Fractionation. Langmuir 1992, 8, 51–58. Giddings was the inventor of the field flow fractionation technique. (17) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: New York, 1989; pp 220-222 for an introduction to hydrodynamic chromatography. (18) DosRamos, J. G.; Silebi, C. A. An Analysis of the Separation of Submicron Particles by Capillary Hydrodynamic Fractionation (CHDF). J. Colloid Interface Sci. 1989, 133, 302–320. (19) Mason, T. G. Osmotically Driven Shape-Dependent Colloidal Separations. Phys. ReV. E 2002, 66, 060402-1–060402-4. (20) Imhof, A.; Pine, D. J. Uniform Macroporous Ceramics and Plastics by Emulsion Templating. AdV. Mater. 1998, 10, 697–700. (21) Be´nard, H. Les Tourbillons cellulaires dans une nappe liquide. ReV. Gen. Sci. Pures Appl. 1900, 11, 1309–1271. (22) Rayleigh, L. On Convective Currents in a Horizontal Layer of Fluid when the Higher Temperature Is on the Under Side. Philos. Magn. 1916, 32, 529–546. (23) Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Dover: New York, 1961. Chapter 2 of this book lays out the full derivation of the Rayleigh-Be´nard instability, using the fluid mechanics and heat equations, and the perturbation analysis into the normal modes. (24) For sucrose, the molecular weight ) 342 Da and D ) 4.59 × 10-10 m2/s at 20 °C. See: Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; p 1100; Table 22.2. The combination of Dη should be roughly constant with temperature. By the Stokes-Einstein equation, the diffusion coefficient of a 1 µm diameter sphere in water at 20 °C is 4.29 × 10-13 m2/s, almost 1000× smaller. (25) Weast, R. C.; Astle, M. J.; Beyer, W. H. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1986. (26) Schope, H. J.; Marnette, O.; van Megen, W.; Bryant, G. Preparation and Characterization of Particles with Small Differences in Polydispersity. Langmuir 2007, 23, 11534–11539.
ReceiVed for reView May 3, 2008 ReVised manuscript receiVed September 20, 2008 Accepted September 24, 2008 IE800720K
