pubs.acs.org/Langmuir © 2009 American Chemical Society
Prolonging Density Gradient Stability Huda A. Jerri, William P. Sheehan, Charles E. Snyder, and Darrell Velegol* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received September 24, 2009. Revised Manuscript Received November 10, 2009 For bottom-up particle fabrication, separation of complex particle assemblies from their precursor colloidal building blocks is critical to producing useable quantities of materials. The separations are often done using a density gradient sedimentation due to its simplicity and scalability. When loading density gradients at volume fractions greater than 0.001, however, an inherent convective instability arises. By translating the Rayleigh-Benard instability from the heattransfer literature into an analogous mass-transfer problem, the variables affecting the critical stability limit were effectively catalogued and examined. Experiments using submicrometer particles loaded onto sucrose and Ficoll density gradients matched theoretical trends and led to a series of useful heuristics for prolonging density gradient stability. Higher particle loading heights, lower volume fractions, and smaller gradient material diffusion coefficients were found to improve stability. Centrifugation was useful at short times, as particles were removed from top of the gradient where the stable density profile degrades to unstable, and the resulting density inversion arises as the sucrose diffuses upward.
Introduction Upon completion of a chemical reaction, one usually has the desired product, some unreacted monomer, and also undesirable side products. Likewise, upon completion of a bottom-up colloidal assembly process, several species often exist in suspension. One species consists of the properly formed, desired colloidal “molecules”1 such as doublets,2-6 colloidal “water”,7,8 colloidal stars,7 or precise grapelike clusters.9 A second species consists of improperly formed, side-product assemblies. Yet a third species that remains consists of the unassembled singlet particles, after the assembly step. Sorting this mixture of proper assemblies, side products, and unassembled singlets is critical to producing useable quantities of materials, but currently there are no reliable, scalable methods for sorting many particle mixtures. As we and other researchers work to build colloidal devices, isolation of the desired assemblies is a major bottleneck to production, in part since the other particles often have identical components, such as when doublets are assembled from singlets. Numerous techniques are currently used to sort mixtures of two or more types of particles. These include field flow fractionation,10 hydrodynamic chromatography,11-13 centrifugal elutriation,14 electro- and magnetophoretic separations,15 and even *To whom correspondence should be addressed: Ph (814) 865-8739; Fax (814) 865-7846; e-mail
[email protected]. (1) van Blaaderen, A. Science 2003, 301, 470-471. (2) Yin, Y.; Lu, Y.; Xia, Y. J. Am. Chem. Soc. 2001, 123, 771-772. (3) Gu, H.; Zheng, R.; Zhang, X.; Xu, B. J. Am. Chem. Soc. 2004, 126, 5664-5665. (4) Johnson, P. M.; van Kats, C. M.; van Blaaderen, A. Langmuir 2005, 21, 11510-11517. (5) Yake, A. M.; Panella, R. A.; Snyder, C. E.; Velegol, D. Langmuir 2006, 22, 9135-9141. (6) McDermott, J. J.; Velegol, D. Langmuir 2008, 24, 4335-4339. (7) Yin, Y.; Lu, Y.; Gates, B.; Xia, Y. J. Am. Chem. Soc. 2001, 123, 8718-8729. (8) Snyder, C. E.; Ong, M.; Velegol, D. Soft Matter 2009, 5, 1263-1268. (9) Manoharan, V. N.; Elsesser, M. T.; Pine, D. J. Science 2003, 301, 483-487. (10) Barman, B. N.; Giddings, J. C. Langmuir 1992, 8, 51-58 51 [Giddings was the inventor of the field flow fractionation technique]. (11) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: New York, 1989; pp 220-222. (12) DosRamos, J. G.; Silebi, C. A. J. Colloid Interface Sci. 1989, 133, 302-320. (13) DosRamos, J. G.; Silebi, C. A. J. Colloid Interface Sci. 1990, 135, 165-177. (14) Pretlow, T. Cell Biochem. Biophys. 1979, 1, 195-210. (15) Giddings, C. Unified Separation Science; Wiley: New York, 1991.
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osmotic techniques.16,17 While some of these techniques work well for lab scale analysis, they are less effective for preparative scale, often having throughput less than 1 mL/h. A seemingly simple and scalable way to separate suspended particles by size or shape is centrifugation or sedimentation. Indeed, the technique has advantages in robustness and cost over other techniques for most separations. Since the suspension behaves as a single complex fluid with a density different from the suspending fluid by itself, one cannot simply place a drop of suspension at the top of a tube and allow the faster-settling particles to sort from the slower-settling particles. Instead, the droplet of suspension simply sinks through the underlying fluid (Figure 1). As a result, a density gradient sedimentation process is often used, in which the underlying fluid;often a solution of sucrose, Ficoll, cesium chloride, or some other species;has a density that is greater than the suspension. Density gradients are typically made by layering solutions with decreasing densities upward from the bottom of a centrifuge tube, either in steps or in a continuous fashion (Figure 2). A simple and inexpensive gradientmaking device enables the user to set a higher solute concentration at the bottom of the tube, graded to a lower concentration at the top. Density gradient sedimentation18 exploits subtle differences in particle sizes or densities to sort particle mixtures. The technique has been used extensively by biologists to separate cell organelles or proteins,19,20 sometimes by isopycnic sorting.21 To separate particles based on differing densities, isopycnic density gradients allow particles to fall until they reach a section of the gradient where the fluid density is matched to the fluid, such that the particles rest in a fluid of the same density. This technique can be used to sort singlets from trimer assemblies.8 One challenge with (16) Mason, T. G. Phys. Rev. E 2002, 66, 060402-1-060402-4. (17) Imhof, A.; Pine, D. J. Adv. Mater. 1998, 10, 697-700. (18) Price, C. A. Centrifugation in Density Gradients; Academic Press: New York, 1982. (19) Meselson, M.; Shahl, F. W.; Vingrograd, J. Proc. Natl. Acad. Sci. U.S.A. 1957, 43, 581-588. (20) Rickwood, D.; Graham, J. M. Biological Centrifugation; Springer-Verlag: Berlin, 2001. (21) Rickwood, D., Ed. Centrifugation: A Practical Approach, 2nd ed.; IRL Press: Washington, DC, 1984.
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Figure 1. Adding a 100 μL droplet of 0.01 volume fraction suspension containing 200 nm polystyrene particles in water to 5 mL of water. The suspension (particles plus water) has a greater density than the pure water and simply drops through the water convectively.
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In conducting our own density gradient sorting of particles, we have often found density gradients to appear finicky. The classic book by Price18 mentions that the maximum stable volume fraction for the loaded material is roughly 10-3 or even 10-4, but the factors that affect the sedimentation stability have not been clearly addressed in the literature. Here we aim to identify the variables that affect density gradient stability as well as their significance. Our analysis is based on the theory for a RayleighBenard instability, and it consists of both experiments and modeling using the transport equations. Our examination of density gradient instability has given us a set of heuristics, some not initially intuitive to us, which prolong the stability of gradients so that we may obtain a useful separation. The instability at the top of the density gradient is due to the higher density of the fluid there, and not the settling rate; hence, the experimental analysis was simplified using submicrometer particles to remove the settling parameter.
Materials and Methods Materials. Ultrapure bioreagent sucrose (C12H22O11, MW
Figure 2. Schematic of sucrose density gradient loaded with particles. The sucrose solution is graded in concentration from bottom to top in a centrifuge tube, and the fluid has a density profile with a gradient γ. The densest solution is placed in the bottom of the tube, and a 6 cm deep density gradient is built off that platform. Sucrose density gradients of γ = 100, 305, and 980 kg/m4 are used in this research. A particle suspension with a height h0 and volume fraction j is loaded on top of the gradient.
isopycnic sorting arises, however, when the particles are much more dense than available suspending media. For example, silica has a specific gravity of 2.2 and gold has a specific gravity of 19.3, much greater than is readily achieved for a laboratory density gradient. A second challenge is that isopycnic sorting fails to separate singlets from assemblies composed of the same particle. In these cases, density gradient separation is used as a rate separation where particles are separated by size and mass. A beautiful example of multicomponent rate sorting9 has appeared in the literature for sorting clusters of various sizes. Density gradient rate separation is a valuable technique for sorting singlet particles from doublets and other assemblies. The suspension medium (sucrose, Ficoll, cesium chloride) initially supports the loaded particle solution at the solution/gradient interface and the separation process begins. When loading density gradients with volume fractions greater than 10-4 or 10-3, however, a notorious convective instability often occurs during sedimentation, such that the particles simply mix throughout the sedimentation vessel.18,22-24 At first the instabilities start as tendril-like structures (Figure 3b) but then proceed to produce mixing throughout the entire separation vessel (Figure 3c). (22) Sheeler, P. Centrifugation in Biology and Medical Science; Wiley: New York, 1981. (23) McDermott, J. J.; Velegol, D. Langmuir 2008, 24, 4335-4339. (24) Velegol, D.; et al. Ind. Eng. Chem. Res. 2009, 48, 2414-2421.
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342.3) was bought from J.T. Baker. Ficoll400 was obtained from Sigma-Aldrich Chemicals, USA, along with potassium chloride (KCl, Mw 74.5) and used without further purification. 15 mL glass centrifuge tubes were purchased from Kimble Glass, Inc., and sprayed with commercially available Rain-X glass treatment to reduce the meniscus effect prior to use. The gradients were formed in the centrifuge tubes using a gradient maker purchased from CBS Scientific (model #GM-100). An assortment of particles was used in this research. We used sulfated polystyrene particles ranging in size from 110 nm to 7.7 μm. We quickly realized that the use of the smaller particles avoided the complication of having the particles settle, and so we mostly used the 110 nm to establish stability criteria. Larger particles were used when settling was desired. Particles were loaded using micropipets with J-shaped needle attachments. Millipore Corp. Milli-Q deionized (DI) water (resistivity greater than 18.2 MΩ 3 cm) was used in all experiments. Methods. 15 mL glass tubes were coated with Rain-X glass treatment liquid using soaked Kim-wipes, allowed to dry, coated again, and rinsed thoroughly with deionized water. Control experiments showed that the Rain-X did not affect the stability of the density gradient; but the Rain-X does simplify the experimental analysis since it prevents a meniscus of aqueous solution at the glass surface, giving a flat profile as shown in Figure 3a. The centrifuge tubes were then air-dried. Density gradients were prepared by measuring equal volumes (5 mL each) of a dense sucrose solution and less dense sucrose solution into a density gradient maker. The density gradient maker comprises two connected fluid chambers such that the lighter phase flows into the chamber containing the heavier phase which is first to drain from the gradient maker into the centrifuge tube. 2 mL of the denser solution were carefully pipetted into the centrifuge tubes prior to layering of the density gradient to plug the conical part of the centrifuge tube. Density gradients of γ = 100, 305, and 980 kg/m4 were created using 5 wt % sucrose as the less dense top fluid and 6.5, 9.5, and 19 wt % sucrose, respectively, as the denser fluids, spanning a distance of 6 cm. All sucrose solutions were made in 10 mM KCl. The salt was added to reduce any electrophoretic effects in the sample,25 although little sedimentation of particles actually occurred for most cases. Various particle suspensions were carefully loaded on top of the uppermost layer of the density gradient using a J-shaped needle which was luer-locked onto a 1 mL syringe or attached to a micropipet for more accurate metering. Particle suspension volumes of 100, 300, 1000, and 3000 μL were added, with the larger (25) Rasa, M.; Philipse, A. P. Nature 2004, 429, 857-860.
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Figure 3. Convective instability at the top of a density gradient. The suspension has a volume fraction of 0.01, and it is supported above a sucrose density gradient with γ=305 kg/m4. (a) At a time t = 0, the interface is flat. (b) At t = 15 min, tendril-like structures have formed. (c) At t = 1 h, the test tube holds a well-mixed suspension. The images were taken with a Fujifilm Finepix S1000 digital camera. volumes added in smaller increments of 200 μL to avoid disturbing the upper gradient surface. A loading volume of 100 μL corresponds to a loaded height of 0.65 mm for our tubes. Particle volume fractions (volume of particles/volume of total suspension) between 10-2 and 10-6 were investigated for various particle sizes and materials, and these solutions were made in DI by considering the density and % solids of the purchased particle solutions. A series of centrifuge tubes were placed in plastic or Styrofoam holders and imaged using a black background to emphasize the opaque particle bands. A Damon/IEC Division International Centrifuge model K swing arm centrifuge was used with 15 mL plastic centrifuge tubes that were also treated with Rain-X prior to use. Images were captured immediately (roughly within 15 s) after loading particles, and sequential snapshots were taken to determine the onset of instability. A laser pointer was used to scan the tubes to determine the times for preliminary destabilization or formation of small flocculates and position of faint bands which are characteristic of the 10-5 volume fractions.
Modeling The phenomena of spontaneous suspension instability, observed in part by sedimentation at rates that greatly exceed predicted values, can be explained by translating the well-known RayleighBenard (RB) instability26,27 from heat transfer theory into an analogous mass-transfer problem. By examining the convective instability in sedimentation from the RB perspective, we were able to map a number of experimental variables to examine. Usually the RB instability is analyzed as a convective instability arising when a fluid layer is confined between a lower heated plate and an upper cooled plate, such that a temperature-induced density inversion arises with a cooler layer of fluid resting above a warmer layer in a continuous fashion. Although the density gradient is inverted, Henri Benard’s experimental work28 in 1900, and 16 years later Lord Rayleigh’s modeling work, showed that no fluid instability occurs until the Rayleigh number (R) exceeds a “critical Rayleigh number” (Rc). The Rayleigh number for the heat transfer problem is given by 1 DF DT 4 d g FDT Dz < Rc ð1Þ R ¼ Rν 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 (26) Lord Rayleigh Philos. Mag. 1916, 32, 529-546. (27) Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Dover: New York, 1961. (28) Benard, H. Rev. Gen. Sci. Pures Appl. 1900, 11, 1261-1271 and 1309-1328.
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lower warm surface, d is the thickness of the fluid, R is the thermal diffusivity, and ν is the kinematic viscosity of the fluid. The critical Rayleigh number (Rc) is 1707.8 for liquid between two rigid, no slip surfaces and drops to 657.5 for two stress-free, slip surfaces. The case of a fluid between two stress-free surfaces was long considered to be strictly an academic problem, as it is puzzling how to confine a liquid between two miscible liquid surfaces. And yet, just such a case arises in the sedimentation problem, as a mass-transfer analogue for the density gradient. In eq 1, it is apparent that the Rc value, and hence convection swirls, can be minimized when thermal diffusion allows the heat to quickly dissipate to the surrounding fluid (large R value) and also when greater fluid viscosity impedes the movement of an unstable rising region of fluid. The parallel case we investigated involved a sucrose density gradient where the density of the gradient varied along the distance down the centrifuge tube instead of between two plates and depended on the concentration of the solute instead of on temperature. The gradient density might also be varied by controlling temperature, since the density of water and other fluids depends on temperature sufficiently; however, while the use of temperature gradients to give density gradients would yield cleaner particle assemblies and circumvent the use of sucrose, Ficoll, etc., this method is currently harder to control experimentally. The analogous barriers to instabilities for the loaded particles are the diffusivity of the sucrose (Ds) and the viscosity of the local sucrose solution (ηs), which supports the particles, and the depth of the density inversion (h). Thus, the analogous masstransfer Rayleigh equation for particles loaded onto a density gradient can be rewritten as DF -g h4 Dz < R R ¼ c Ds ηs
ð2Þ
The pieces that go into the Rayleigh number provide a catalog of variables to examine for density gradient instability. We decided to simplify the experiments, without losing generality, by making gravitational settling negligible in our system by using submicrometer particles; that is, the particles simply rested atop the sucrose gradient, while the sucrose diffused into the particle suspension, and the particles diffused into the sucrose solution. In order to circumvent particle settling, we used polystyrene particles (Fps = 1050 kg/m3) with radius a = 55 nm. We calculated the settling velocity U0 using Stokes’ law: U0 ¼
2a2 ðFp - Ff Þg 9η
ð3Þ
According to eq 3, for 110 nm particles in water with a viscosity of η = 0.001 cP and density FH2O = 1000 kg/m3, the particles settle ∼1.2 μm/h in DI and 0.6 μm/h in a 5 wt % solution of sucrose with a density Fsucrose = 1019 kg/m3. Roughly 800 h (32 days) would have to pass before a readily perceptible 0.5 mm drop in particle height could be observed if the submicrometer particles were to remain stable atop the density gradient and settle only due to gravity. The Stokes-Einstein diffusion coefficient for these 110 nm particles in water is 3.9 10-12 m2/s, so 18 h would elapse before the particles would diffuse that same 0.5 mm distance. That leaves the diffusive movement of sugar as dominant in changing the density profile with time while the submicrometer particles remain relatively stationary. Having accounted for Stokes’ law, settling, differential settling, and Brownian motion, we expected the base of the loaded particle band to be flat and smooth if stable. By virtually eliminating sedimentation of the particles, we identified a DOI: 10.1021/la903616p
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common culprit of the density inversion, namely, the sucrose diffusion of the density gradient into the suspension and therefore the inversion of the gradient. The modified RB model accounts for the upward diffusion of sucrose from the gradient into the suspension as well as the diffusion of the particles downward from the suspension into the sugar gradient. To account for these concentrations, we solve Dci D 2 ci ¼ Di 2 Dt Dx
ð4Þ
for species i = 1 (sucrose) and 2 (particles), with appropriate boundary and initial conditions. The initial conditions are that in the loaded suspension region, c10 = 0 and c20 = cp0, the initial particle number concentration. In the density gradient, we specify the gradient of the sucrose, and c20 = 0. By knowing the local concentrations of particles and sugar, we calculate the local density as FðzÞ ¼ Ff ð1 - φÞ þ Fp φ ¼ ðFH2 O þ γzÞð1 - φÞ þ Fp φ
ð5Þ
and therefore find dF/dz as a function of position. The particles initially rest above the gradient, and then a density inversion occurs with time as sugar begins diffusing into the particle solution and the top of the gradient erodes. As the gradient changes with time, the model reflects the burgeoning instability, and the R value increases from extremely negative values (very stable) toward Rc = 658. The loaded height of the particles is h0. The depth (h) of the inversion is important to the Rayleigh instability and encompasses the particle density, particle volume fraction, and gradient density terms. In the case of particles which settle due to gravity or diffuse downward, the maximum inversion depth is reached as the density of the gradient increases and supports the particle solution. The inversion depth is approximated by hmax ≈
ðFp - Ff0 Þφ γ
ð6Þ
This relationship agrees with our experimental findings that lower volume fractions and larger density gradients result in tighter bands and thus smaller inversion depths as shown in Figures 4 and 5. The mass-transfer Rayleigh number can be calculated using the model once the density gradient profile is determined and the height of the region where a density inversion occurs is known. Rather than resolving all the original equations and finding the eigenvalues numerically, we make an approximation that we find dR from eq 2 and then integrate dR over the density inversion to estimate R. The bottom of the inversion we call z=0, and the top of the inversion we call z = h. Z h 4g DF 3 z dz ð7Þ R ¼D s η Dz 0
Results and Discussion Convective stability was examined under conditions of various particle diameters (2a), loading volumes (h0), volume fractions (φ), and density gradients (γ=100, 305, and 980 kg/m4). Initially, a survey of different particle sizes ranging from 2a=110 nm to 7.7 μm was performed at a relatively high loading volume fraction of φ=0.01. At this volume fraction, with a standard loading volume of 100 μL, all samples showed the fingering effects and band spreading illustrated in Figure 3b. The 110 nm particles had the 4728 DOI: 10.1021/la903616p
Figure 4. Higher volume fractions crash fastest as seen in the time sequence for volume fractions ranging from φ = 0.01 to 0.000 01 imaged after an initial time of (a) 30 min and (b) 4.5 h after loading 100 μL of 110 nm polystyrene latex particles onto γ = 305 kg/m4 density gradients. Higher particle volume fractions have considerably wider bands, with fingering effects at the base of the bands compared with lower volume fractions which remain thin and flat.
tightest bands. The experiment was repeated at a lower volume fraction of 0.0001 for particles with a similar size range. After ∼3 h, the larger microparticles had dropped uniformly down in the centrifuge tubes whereas the submicrometer and smaller microparticles widened slightly but remained high in the gradient. After 22 h, the 2.4, 4.9, and 7.7 μm microparticles had dropped even lower due to expected settling. No flocculates were found below the particle bands for any of the samples. The larger microparticles had the thinnest and most stable bands as they dropped lower while the submicrometer particle bands continued to widen. Part of the widening for large particles was due to differential settling, resulting from the slight polydispersity as indicated by the coefficients of variance for the purchased particle solutions. Typical polydispersities are 5% of the particle diameter. From the two different studies with particle diameters and lab experience showing that only dilute particle solutions can be separated, we targeted volume fraction as a critical parameter to study. Particle volume fractions ranging from 0.01 to 0.000 01 were studied using the 110 nm diameter particles, with replicates to determine the consistency of the density gradients as shown in Figure 4. The three replicates for φ = 0.003 16 gave nearly identical results (see Figure 5). Figure 4 demonstrates that volume fraction has a significant effect on time for instability or particle band widening; the φ = 0.01 sample widened after 30 min and went unstable with obvious fingering at the bottom of the band after 60 min. In contrast, the φ = 0.0001 to φ = 0.000 01 samples had thin bands after 20 h while the remaining tubes had Langmuir 2010, 26(7), 4725–4731
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Figure 5. Effect of density gradient on prolonging convective instability. Increasing the gradient better supports submicrometer particles, as bands remain thinner with flatter bases. 100 μL of 110 nm polystyrene latex particles were loaded onto three density gradients and examined after t=180 min. The fingering effects and band widening corresponding to (a) γ=100 kg/m4 gradients are clearly more noticeable than those for higher density gradients of (b) γ = 305 kg/m4 and (c) γ = 905 kg/m4.
Figure 6. Density profile through the tube. The diffusive transport of sucrose and 110 nm polystyrene particles, loaded at a depth of 0.65 mm, were modeled with Mathematica as a function of time. The figure shows four times. At 12 min the first density inversion occurs; at 56 min the Rayleigh number first exceeds 658.
convective instabilities and “crashed” with tiny flocculates throughout the tubes. Although the bands crash in order of decreasing volume fraction, it should be noted that the φ = 0.000 01 bands are extremely faint and difficult to see. The effects of volume fraction are important, as going from γ= 100 kg/m4 to γ=980 kg/m4 density gradients increases the time of instability, but cannot prevent the bands from widening, even at lower volume fractions as shown in Figure 5. The 0.01 volume fraction went unstable after ∼15 min in the 100 and 305 kg/m4 but lasted 2 h in the higher 980 kg/m4 gradient before showing signs of fingering. The times for the initial fingering and widening effects characteristic of instability were compared to theoretical times using our model, written in Mathematica. As shown above in Figure 5, a band of 100 μL of 100 nm PSL particles loaded onto γ=305 kg/m4 sucrose gradients at j=0.01 will display distinct fingering effects after 15 min. Modeling the problem requires some assumptions about the loading, making an exact comparison difficult. The main approximation is that some mixing occurs between the top of the sucrose solution and the particle suspension as the particle suspension is loaded onto the underlying sucrose gradient. We will estimate that this cuts the upper sucrose concentration to 2.5%, from 5.0%. It is this upper sucrose concentration that determines stability during short times. At short times the gradient in fact plays little role, other than the self-created gradient that develops between the top of the sucrose and the loaded suspension. The Rayleigh model predicts that an inversion will occur after 12 min, while the critical Rayleigh number is Langmuir 2010, 26(7), 4725–4731
reached after 56 min. The density profiles of the samples are generated using the Rayleigh model. The majority of experiments were performed on 305 kg/m4 density gradients using various particle sizes, densities, and volume fractions. Different loading volumes of 100, 300, 1000, and 3000 μL were tested on γ = 305 kg/m4 density gradients at volume fractions of φ = 0.001 (Figure 7). Although all bands show evidence of widening with time, the bottoms of bands for higher loading volumes remain flat longer. Volume fraction and loading height have much to do with avoiding convective instabilities, indicating that the interaction between the upper density gradient and the loaded particle suspension has a greater influence than the type or size of particles. The experiments investigating loading volume contribute to dF/dz as expected where increased loading height leads to increased stability as the density gradient is extended through the particle solution instead of quickly degrading. The modified Rayleigh model supports the relationship between increased loading height (a loading volume of 100 μL gives a loaded height of 0.065 cm) and theoretical time to reach Rc with higher loading volumes corresponding to longer times needed to broach instability. The model supports the experimental findings in terms of trends, with higher loading volumes and lower volume fractions yielding more stable density gradients. The times required to develop inversions and reach the critical R value are always longer than the experimental times. We expect this, since the loading process is disrupting the top of the gradients, especially since the particles are manually loaded, and the particle solution is not ejected from the J-tip in a perfectly reproducible manner. Mixing of the particle solution with the top of the sucrose gradient during the loading process should induce an inversion faster experimentally, as we observe in the test tubes. Experiments were also conducted to examine the material used to form the density gradients, as suggested by the modified Rayleigh equation. We found that experiments comparing sucrose density gradients to Ficoll density gradients appeared to support the relationship between viscosity and stability. As shown below in Figure 8, using a Ficoll gradient leads to improved stability as particles remain in tighter bands for up to a week, contrasted with several hours for the 200 nm polystyrene particles loaded onto a 305 kg/m4 sucrose density gradient. The viscosity of Ficoll is roughly 10 times greater than sucrose at a concentration of 20% w/v (ηrel = 1.97 for sucrose and 20 for Ficoll). An additional benefit of using Ficoll as the gradient material is the small diffusion coefficient. The Stokes radius is ∼5 nm for Ficoll versus 0.5 nm for sucrose. It follows then that the diffusion DOI: 10.1021/la903616p
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Figure 7. Higher loading volumes result in flatter bands and less flocculation. Loading volumes ranging from 100 to 3000 μL were imaged after (a) 10 min, (b) 4.5 h, and (c) 22 h. 110 nm polystyrene particles were loaded onto γ = 305 kg/m4 density gradients at volume fractions of φ = 0.001, and higher loading volumes demonstrate the least amount of band expansion and fingering.
Figure 8. Ficoll density gradients support thinner, tighter bands than sucrose gradients after 24 h. 100 μL of 200 nm polystyrene particles was loaded onto γ = 305 kg/m4 sucrose density gradients and γ = 417 kg/m4 Ficoll density gradients at volume fractions of j=0.001 and imaged (a) initially and (b) after 24 h. One week later, the Ficoll density gradient remained stable, while the sucrose gradient crashed after several hours.
coefficient of Ficoll is approximately an order of magnitude smaller than that of sucrose.29 While is might appear that having a smaller diffusion coefficient would increase the Rayleigh number, it is important to recognize that the smaller diffusion coefficient of Ficoll plays a prominent role in the dF/dz term, which determines the overall sign of the critical Rayleigh value. Thus, a smaller diffusion coefficient has been found to enhance gradient stability, by avoiding the density inversion. The magnitude of the Rayleigh number is larger with Ficoll, but the sign remains negative for a much longer time. A final set of experiments was based on an additional observation we made, which at first seemed paradoxical. In several experiments using much larger particles (7.7 μm polystyrene), we observed that the settling even under 1g seemed to cause the bands to remain stable for φ = 0.0001. We hypothesized that the larger particles were escaping the upper portion of the disintegrating density gradient due to settling. We therefore tried various smaller particles down to 200 nm but at higher g values so that the particles could still escape the upper region before the density gradient disappeared. As long as we kept the volume fraction at φ=0.0001, we found that we were able to maintain stable bands. One of the experiments we ran used particles of intermediate size (810 nm). Test tubes filled with 305 kg/m4 density gradients and loaded with 100 μL of 810 nm polystyrene particles at volume fractions of φ = 0.0001 were imaged at 1g and again after centrifuging the samples for 2 h at 90g using a swinging bucket (29) Lavrenko, P. N.; Mikryukova, O. I.; Didenko, S. A. Polym. Sci. U.S.S.R. 1986, 28, 567-584.
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Figure 9. Centrifugation at 90g for an intermediate time of 2 h results in distinct, thin bands with flat edges and no flocculation. 100 μL of 810 nm polystyrene particles were loaded onto γ = 305 kg/m4 density gradients at volume fractions of j=0.0001 and imaged (a) initially and (b) after 2 h of centrifugation in a swing arm centrifuge at 90g.
centrifuge. As seen in Figure 9, centrifugation at higher g results in tight bands with clean leading edges. Thus, while higher g values might seem to increase R, they again keep the density gradient with a favorable negative sign. The modified Rayleigh equation predicts that increased g forces would push the system toward instability. From the experiments performed, the centrifugation enhances “separation”, but the true benefit comes after pushing the particles out of the upper portion of the gradient.
Conclusions Heuristics were found for prolonging the stability of density gradients. The factors were found by analogy to the Rayleigh-Benard instability of heat transfer. Theses experiments were simplified using submicrometer particles, which initially rest above the sucrose gradient in a thin band. We found that higher particle loading heights, lower particle volume fractions, and smaller diffusion coefficients for the density-making solute all improve stability. Centrifugation helped stability at short times by enabling particles to escape the top of the gradient where the density profile degrades and the inversion arises as the sucrose diffuses upward. Although theoretical times to reach instability were longer than observed experimental times, the trends were consistent with the modeling. The research described here was motivated by an urgent need for sorting kilogram quantities of colloidal particle mixtures. One conclusion from this research is that density gradient sedimentation is a challenging method to use for sorting significant quantities of particles, since stability degrades rapidly even at volume fractions of 0.001. We note that the same challenges will Langmuir 2010, 26(7), 4725–4731
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be encountered with any sorting method that applies a force monopole on the particles, including magnetophoresis, dielectrophoresis, and possibly electrophoresis. In addition, the solute in the density gradient can contaminate the particles, a significant problem in colloidal assembly operations, although this problem might be circumvented by using temperature gradients to produce the
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density gradients. Our lab continues to search for scalable, clean sorting methods for colloidal assemblies. Acknowledgment. The authors thank the National Science Foundation for funding this work through CBET Grant #0651611.
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