Critical Onset of Layering in Sedimenting Suspensions of Nanoparticles

A. V. Butenko,1 P. M. Nanikashvili,2 D. Zitoun,2 and E. Sloutskin1*1Physics Department and Institute for Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel

2Chemistry Department and Institute for Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel(Received 3 September 2013; published 7 May 2014)

We quantitatively study the critical onset of layering in suspensions of nanoparticles in a solvent, wherean initially homogeneous suspension, subject to an effective gravity a in a centrifuge, spontaneously formswell-defined layers of constant particle density, so that the density changes in a staircaselike manner alongthe axis of gravity. This phenomenon is well known; yet, it has never been quantitatively studied underreproducible conditions: therefore, its physical mechanism remained controversial and the role of thermaldiffusion in this phenomenon was never explored. We demonstrate that the number of layers forming in thesample exhibits a critical scaling as a function of a; a critical dependence on sample height and transversetemperature gradient is established as well. We reproduce our experiments by theoretical calculations,which attribute the layering to a diffusion-limited convective instability, fully elucidating the physicalmechanism of layering.

DOI: 10.1103/PhysRevLett.112.188301 PACS numbers: 47.57.ef, 47.55.pb, 82.70.Kj

Sedimentation of micro- and nano-particles in a solventunder gravity is common in bio- and nano-technology [1],occurring in a wide range of geophysical systems [2]and limiting the shelf life of food products and pharmaceut-icals [3]. Sedimentation is also widely used as an analyticaltool for industrial, medical [4], and research applications[5–7]. Under most common experimental conditions, thedensity of particles in a sedimenting fluid suspension is acontinuous function of time and spatial coordinates [8].However, occasionally, the density of particles developsmultiple (roughly) equispaced plateaus, thus adopting astaircaselike appearance along the axis of sedimentation.This phenomenon, called “layering” or “stratification,”has been known for more than a century [9–11]. Yet,most previous experimental realizations of this effectemployed micron-sized particles [10,12,13], for which thelayer structure is highly sensitive to tiny temperature gra-dients [10,12,14], prohibiting extraction of quantitativeexperimental information. Other experiments employedparticles with high or unknown polydispersity [6,9], whichlimited the availability of interpretable experimental data. Asa result, the physical mechanism of layering in sedimentingsuspensions remained ambiguous [15], with several com-peting theoretical scenarios attributing the layering to eitherBurger’s shock formation [16], spinodal decomposition [12],vertical streaming flows [2], spontaneous formation ofmagicnumber clusters [6,14,17], long-range hydrodynamic inter-actions [18,19], or convective instability [10,20]. Quantitativeexperimental information, which would allow the truemechanism of layering to be unequivocally identified, wasmissing.We follow the full dynamics of layer formation in

sedimenting suspensions of several different types ofnanoparticles in various organic solvents subjected to an

effective gravity in a centrifuge, employing light trans-mission (LT) through the samples. We demonstrate that byusing nanoparticles, the layering phenomena are muchmore robust than in the previous studies [10,20]; thissystem allows quantitative and reproducible measurementsto be collected with our experimental setup. Furthermore,this setup allows the effective gravity a, measured in theunits of g ¼ 9.8 m=s2, to be varied; we use it to study thecritical onset of the layering effect, where pattern formationby layering overcomes the significant thermal diffusion ofthe nanoparticles. We demonstrate that in this regime, thenumber of layers N in a sample exhibits a unique power-law scaling as a function of a and the height H of initialsuspensions; the dependence on H of the critical effectivegravity ac, below which the layers do not form, is exploredas well. We reproduce most of our observations by numericalcalculations, employing a hydrodynamical model that attrib-utes the layering to a convective instability [10,20]. Wesuggest that the spontaneous layering in suspensions ofnanoparticles may serve as a basis for future analyticaltechniques for nanoscale colloids, and may have importantapplications in self-assembly of metamaterials.We prepare Cu@Ag and pure Ag nanoparticles, stabi-

lized by either an oleylamine or a dodecanethiol surfacemonolayer [21], and suspend them in pure hexane orheptane at a low volume fraction c0 ¼ 10−4 − 10−3; theseparticles form promising inks for inkjet printing [21]. Theaverage diameter σ and size distributions PðσÞ of theparticles are measured by transmission electron microscopy(TEM). Most samples exhibit a simple Gaussian PðσÞ,peaking between 10 and 20 nm, with a width of ∼4 nm (seeSupplemental Material [22]). We load the initially homo-geneous fluid suspension into an analytical tabletop cen-trifuge (Lumifuge), where a is in the range 200 < a < 2500.

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Our centrifuge measures LT profiles IðxÞ through thesuspension in real time during the centrifugation, as shownin Fig. 1. The cooling element of our setup, positioned atthe bottom, sets the direction of the temperature gradientz [23], so that the temperature drop over the sample is~Δ ≈ 0.4 K [22]. We fill our suspensions into polyamidecuvettes to a height 5 < H < 35 mm; the cuvettes are thenloaded horizontally into the centrifuge.The suspension is initially homogeneous and opaque for

LT. Sedimentation of particles by effective gravity forms aparticle-free supernatant region in the sample (on the lefthand side of Fig. 1); LT through this region is >95%. Forsilica colloids [24,25] in ethanol, the supernatant is sep-arated from the sedimenting suspension by a relativelysharp interface, or sedimentation front, which propagatesalong the effective gravity at a constant speed v0, for whichthe centrifugal force is balanced by the Stokes drag [7], seeFig. 2(a). With nanoparticles, the sedimentation front getsincreasingly smeared at short times due to their significantpolydispersity [green dotted curve in Fig. 2(b)]. Strikingly,a staircaselike variation of transmission is then developed[Fig. 2(b)], indicating formation of distinct layers ofconstant density [10,12,13]. This effect cannot be attributedto particle size or shape segregation [9], as PðσÞ of ourparticles is single peaked (Fig. S9) and the particles appearrounded by TEM [22]. Similarly, particle clustering[6,14,17] is excluded, as it does not produce uniform stepsin LT; also, sedimentation velocities of compact n-particleclusters scale as n2=3 [26], while the experimental velocitiesof plateau boundaries are all very close together [10,12].Varying the interparticle potentials by changing the par-ticles’ surface layer from oleylamine to dodecanethiol, orreplacing hexane with heptane, does not significantly alterthe appearance of layers; increasing particle density makesthe layering vanish. Both of these observations rule out theaggregation scenario.More recent studies [10,20] suggest that the layering is

driven by a convective instability. In particular, a tinythermal gradient normal to the sedimentation axis wasconjectured [10] so that the two sides of the sample are at a

temperature difference ~Δ; this increases the average gravi-metric density of the suspension on one side of the cuvetteby δρ, resulting in an increased sedimentation rate on thatside [inset to Fig. 3(a)]. For a homogeneous sample, anindividual convection roll should form, spanning the wholesample. However, when the initial particle density cðxÞ issloped, the average gravimetric density of the suspension(with the particles included) ρðxÞ is sloped as well. As aresult, the densities on both the cold and the hot sides of thecuvette will match again [20] if the sides are mutuallyshifted by a distance Λ ¼ δρðdρ=dxÞ−1 along g. This setsthe width of the layers to be roughly equal to Λ; a moreaccurate estimate requires the full details of roll formationdynamics to be taken into account [20]. Once the con-vection rolls form, the particle density within each roll ishomogenized and levels of equal density flatten out, givingrise to a staircaselike appearance of transmission profiles,such as in Fig. 2(b). According to this model, the layeringphenomenon is an example of a spontaneous symmetrybreaking, akin to the Belousov-Zhabotinsky patterns inchemistry or the Liesegang layers in geology [11,27]. Sinceeven very small ~Δ may be sufficient for the layering tooccur, a direct measurement of ~Δ inside the centrifuge ischallenging; yet, as in previous works [10], we couldalmost completely eliminate the layering by thermal shield-ing of the samples, which were wrapped for that purposewith a copper foil (see Fig. S1 [22]). More interestingly, thelayering phenomenon is notoriously sensitive to the shapeofPðσÞ. We could selectively eliminate the layering in a partof the suspension by truncating the low-σ wing of particlesize distribution [22]. Remarkably, the asymmetry of PðσÞis difficult to measure in nanoparticles by either classicallight scattering [28,29] or modern analytical centrifugation

FIG. 1 (color online). Experimental setup. An optically trans-parent cuvette (2 × 8 × 50 mm), loaded with the suspension(orange) is centrifuged in the xy plane; the centrifugal accel-eration is a≡ ax. The sample is illuminated with a planar sheet oflight at 870 nm (vertical red lines). LT profiles are measured witha position sensitive detector (PSD). Thick down-oriented arrowsindicate the heat flow direction.

FIG. 2 (color online). LT profiles along the sedimenting sus-pensions, obtained for a suspension of (a) silica colloids(σ ≈ 0.5 μm) and (b) Cu@Ag nanoparticles. Profiles correspond-ing to different time points after the beginning of the centrifugationare overlayed such that the time separation between subsequentcurves is 30 sec in (a) and 112 sec in (b);H ¼ 29 mm. While theprofiles in (a) are monotonic, they adopt a steplike shape in (b)at long centrifugation times, indicating the onset of layering.A solid sediment is formed for x=H > 0.95, blocking the LTin this region. The same data are shown animated in theSupplemental Material [22].

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methods [5], while TEM can only be carried out with driedparticles. Therefore, the unique sensitivity of layering tothis asymmetry suggests that new analytical techniquesexploiting the layering patterns at preset temperaturegradients may be developed, which would allow fullcharacterization of PðσÞ in fluids.As a more direct and quantitative test of the convective

instability model, we exploit the high reproducibility of thelayering transition and the variability of sedimentationvelocity in a centrifuge to study the critical conditionsfor the onset of layering; such a test was impossible inearlier experiments, where even the radiated body heat ofthe experimenter was sufficient to destroy the layer pattern[10,12], making it very challenging to collect quantitativeexperimental results. As the simplest quantitative measureof layering, we count the number of layers N appearing in

the sample of initial height H. Strikingly, the number oflayers exhibits a critical scaling N ∝ ja − acj0.39�0.05, asshown in Fig. 3(a); the overlapping open red and solidmagenta circles [Fig. 3(a)], obtained for different samples,demonstrate good reproducibility. N is directly propor-tional to H, except for the fine staircaselike structure due tothe integer nature of N [see Fig. 3(b)]; this indicates that thecritical thickness of an individual layer, before the nextlayer starts forming, is independent of the sample size. ForH < Hc, the distance passed by the sedimentation front issmall, so its broadening is negligible and the resultingparticle concentration gradient dc=dx is too steep for thelayering to occur [22]. More surprising is the acðHÞ scaling[Fig. 3(c)]; as the maximal broadening of the sedimentationfront for a givenH does not depend on a, we carry out a fullnumerical solution of a system of partial differential equa-tions (PDEs) describing the convective instability [10,20], inan attempt to account for the observed scaling.For nanoparticles in a solvent, the Reynolds numbers are

small, and the equations of motion for an incompressiblefluid in Stokes approximation are ∇p ¼ ρðν▵uþ gaÞ and∇ · u ¼ 0, where u is the convection velocity and p is thepressure [20,30]; here the suspension is treated as an effectivemedium of kinematic viscosity ν. For the particles, the massconservation is [20] _cþ u ·∇cþ v0 ·∇c ¼ D▵c, where thehindrance of dynamics by particle crowding [31] wasneglected in our range of c and D is the diffusion coefficientof the nanoparticles. Retaining the c and T dependence in theforcing term [20], we obtain (to the leading order) from theequationofmotion∇p ¼ ρν▵uþ Δρcga − ð2dÞ−1zβρ ~Δga,where β ≈ 10−3 °C−1 is the coefficient of thermal expansionand 2d ¼ 2 mm is the thickness of our cuvettes (Fig. 1). Toreduce the problem to one spatial dimension, an approximatesolution along z conforming with the geometry of rollformation is guessed [20]; this is known as the Galerkinmethod. The dimensionless version of the resulting PDE[22] includes only four parameters: α≡ jv0j=U, γ ≡ σ=2d,δ≡D=jv0jd, and c0, where U ≡ β ~Δgad2=ν sets the scaleof convection velocities, jv0j ¼ Δρσ2ga=18νρ0, and Δρ isthe excess gravimetric density of the particles, compared tothat of the solvent ρ0; the dimensionless time is tr ≡ tjv0j=d.We solve the equations for ~Δ ¼ 0.1 K, employing the finitedifference method, with hard boundary conditions set at thehigh-x end of the cuvette. For the layering to occur, we use alinearly sloping cðxÞ ¼ c0½1þ pðx −H=2Þ� for the initialconditions, where we choose p ¼ 0.16 cm−1; this is com-parable to the experimental cðxÞ, which is significantlybroadened due to particle polydispersity prior to the onset oflayering [Fig. 2(b)]. The PDE exhibit formation of densitylayers; the layers move along the effective gravity and dis-appear as they reach the bottom of the sample, where a densesediment forms [22]. The total number of layers formingin a sample N scales as N ∝ ja − acj0.30�0.01 [Fig. 4(a)], ingood agreement with the experiment, which validates thetheoretical model. To collapse together data obtained for

FIG. 3 (color online). (a) The experimental number of layers Nobserved in samples of different initial height H (see labels)varies as a function of the effective gravity a; no layering isobserved for a < ac. Inset: Transverse temperature gradientinduces a sedimentation velocity difference between sample edges.Velocities, in the frame comoving with the center of mass of thesuspension, are represented by white arrows; color map representsthe temperature. (b) For a constant a ¼ 1.6 × 103, N increaseswith H; no layering occurs for H < Hc ¼ 8 mm. Inset: Thesedimentation front velocities of Ag nanoparticles at short cen-trifugation times, as a function of the particle mass fractionCm0 ≈ 15C0, normalized by the sedimentation velocity of a free

particle v0. (c) The critical threshold for layering scales asac ∝ H−4=3; the fitted exponent accuracy is �0.06. Inset: Exper-imental (open squares) and theoretical (solid rhombii) N coincide,plotted as a function of dimensionless offset τ of the temperaturegradient.

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different values of H, we scale each NðaÞ curve by anarbitrary Nmax, as shown in Fig. 4(a). These Nmax valuesare linear in H, see Fig. 4(b), much like in the experiment[Fig. 3(b)]. According to our theoretical model, the criticalsample size for layering is H ¼ 0; this is reasonable, as afinite distance Hc is required in the experiment for thesedimentation front propagation to develop the linear cðxÞprofile, used as the initial condition for our PDE. Thus,overall, the agreement of this model with the experiment isvery good; the only exception is the theoretical acðHÞscaling [Fig. 4(c)], which is quantitatively different fromthe experimental one; this suggests that a more elaboratetheoretical model, possibly taking into account the spreadof experimental sedimentation velocities due to polydisper-sity, both before and after the onset of layering, may beneeded to reproduce the experimental observations in fulldetail.The good agreement of these PDE with the experiment

allows the role of thermal diffusion in this system to beinvestigated. The calculated average layer width λ⋆ ≡ λðt⋆Þat the onset of layering t ¼ t⋆ exhibits a diffusive scalingλ⋆ ∼

ffiffiffiffiffiffiffiffi

Dt⋆p

[22], suggesting that t⋆ may be determined by acompetition between convection and diffusion. For a giventime t, structural details finer than

ffiffiffiffiffiffi

Dtp

are smeared bydiffusion, while the typical length scale for structure for-mation by the convection rolls is proportional toUt. At t ¼ t⋆both effects are balanced, so that t⋆ ¼ DU−2. In our PDE,Dand a appear only through the ratio D=a, which leads us toassume that the scaling of the dimensionless t⋆ is

t⋆r ∼ ðD=aÞμ. In such case, t⋆ ∼ a−1ðD=aÞμ, which is indeedobtained in Figs. S8 (b),(c) for μ ≈ −0.3 [22]. Combinedwith the above, we obtain λ⋆ ∼ ðD=aÞðμþ1Þ=2 ∼ ðD=aÞ0.35,so that N ∼ 1=λ⋆ ∼ ða=DÞ0.35; this is in perfect agreementwith the experiment [Fig. 3(a)] and emphasizes the role ofD,which is much higher in our nanoparticles compared toemulsions employed in earlier work [10]. Finally, wesystematically vary N by tuning of ~Δ; for this purpose,we introduce a static electrically heated copper plate abovethe sample, controlling the temperature offset ~Δexpbetween this plate and the cooling element (Fig. 1). Theresults are shown together with the theoretical N in theinset to Fig. 3(c), where τ≡ ð ~Δ − ~ΔcÞ=ð ~Δc þ BÞ; ~Δcis the value of ~Δ at the onset of layering, and B ¼ 0for the theoretical data. To account for the differencebetween ~Δexp and ~Δ, we fit B ¼ 120 K as a free para-meter for the experimental data, replacing ~Δ → ~Δexp and~Δc → ~Δc;exp ≈ 0.9 in the expression for τ. The perfectagreement thus obtained for NðτÞ, which necessitated a fullsolution of the PDEs and could not be guessed from theabove-mentioned simplistic scaling of Λ, is a strongsupport for our theoretical model. Though in othersedimentation instabilities, driven by hydrodynamic inter-actions, finite wave number structures form for D > 0[18,19], the observed ~Δ dependence is unique; futureexperiments should allow λ to be measured at the steadystate conditions (t ≫ t⋆) and compared with the predic-tions of our model and other theoretical scalings [18,19].The formation of convection rolls, demonstrated to be

the physical mechanism of layering, modifies the sedimen-tation hydrodynamics. In particular, the sedimentationvelocity, which is supposed to follow the classicalvðcÞ ¼ v0ð1 − 6.55c0Þ Batchelor’s law [26] at c0 ≪ 1, ismodified in our case even for t ≪ t⋆; see inset to Fig. 3(b),where the slope jdvðcÞ=dcj exceeds the Batchelor’s lawprediction by a factor of ∼20. Other hydrodynamicinstability has recently been demonstrated to increasethe absolute value of vðcÞ for macroscopic spheres underconfinement [32]. These observations call for moreadvanced theoretical models to describe the early stagesof sedimentation.In conclusion, we have employed suspensions of nano-

particles to obtain a reproducible and controllable layering,and have followed the critical scaling laws and comparedthem with theory, demonstrating a semiquantitative agree-ment; this proves that the formation of layers is driven by aconvective instability, which competes with thermal dif-fusion. Other mechanisms, suggested in earlier works, areincompatible with our observations. Finally, the achievedunderstanding of the basic physics of layering, and also theobserved reproducibility of this effect with nanoparticles,open a broad perspective for future research exploringsimilar phenomena in the presence of particle crowding, innon-Newtonian solvents, in complex temperature fields,and in cuvettes of nontrivial geometry; a wide range of

FIG. 4 (color online). (a) The theoretical number of layers N insamples of different height H (see legend) exhibits a criticalbehavior as a function of the effective gravity a. Inset: Same dataon a log-log scale. N is scaled by Nmax to make all data collapsetogether; Nmax values are shown in (b). (c) The scaling of thecritical threshold for layering ac ∝ H−0.24�0.06 is less steep than inexperiment [see Fig. 3(c)]. Our PDE are stiff and numericaldivergences occur when the layers become exceedingly sharp; toavoid this numerical instability, we set d ¼ 0.2 mm [22].

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objectives can thus be pursued, from the basic science offluid and solid [7] sediments to the development ofanalytical methods for nanoparticle characterization andnanopatterning technologies.

We are grateful to T. Sobisch, M. Shmilovitz, N. Shnerb,D. A. Kessler, and S. Shatz for insightful discussions.The authors thank A. Muzikansky for synthesis of earlyCu@Ag samples and D. Fridman for technical assistance.This research is supported by the Israel Science Foundation(Grants No. 85/10 and No. 1668/10). P. M. N. acknowl-edges Ministry of Absorption for funding.

*eli.sloutskin@biu.ac.il[1] E. C. Cho, Q. Zhang, and Y. Xia, Nat. Nanotechnol. 6, 385

(2011).[2] W. H. Bradley, Science 150, 1423 (1965).[3] S. Calligaris and L. Manzocco, in Shelf Life Assessment of

Food, edited by M. C. Nicoli (CRC Press, Boca Raton,USA, 2012).

[4] H. L. Haber, J. A. Leavy, P. D. Kessler, M. L. Kukin, S. S.Gottlieb, and M. Packer, N. Engl. J. Med. 324, 353 (1991).

[5] R. P. Carney, J. Y. Kim, H. Qian, R. Jin, H. Mehenni,F. Stellacci, and O.M. Bakr, Nat. Commun. 2, 335 (2011).

[6] J. Boom, W. S. Bont, H. P. Hofs, and M. De Vries,Molecular biology reports 3, 81 (1976).

[7] S. R. Liber, S. Borohovich, A. V. Butenko, A. B. Schofield,and E. Sloutskin, Proc. Natl. Acad. Sci. U.S.A. 110, 5769(2013).

[8] K. Benes, P. Tong, and B. J. Ackerson, Phys. Rev. E 76,056302 (2007).

[9] W. H. Brewer, Clays Clay Miner. 13, 395 (1884).[10] D. M. Mueth, J. C. Crocker, S. E. Esipov, and D. G. Grier,

Phys. Rev. Lett. 77, 578 (1996).[11] T. Shinbrot and F. J. Muzzio, Nature (London) 410, 251

(2001).[12] D. Siano, J. Colloid Interface Sci. 68, 111 (1979).

[13] N. B. Uriev and I. I. Bardyshev, Colloids Surf. A 225, 25(2003).

[14] W. S. Bont, H. P. Hofs, and M. De Vries, Colloid Polym.Sci. 257, 656 (1979).

[15] R. Piazza, S. Buzzaccaro, and E. Secchi, J. Phys. Condens.Matter 24, 284109 (2012).

[16] W. van Saarloos and D. A. Huse, Europhys. Lett. 11, 107,(1990).

[17] W. S. Bont, Mol. Biol. Rep. 36, 959 (2009).[18] D. Saintillan, E. S. G. Shaqfeh, and E. Darve, Phys. Fluids

18, 121503 (2006).[19] É. Guazzelli and J. Hinch, Annu. Rev. Fluid Mech. 43, 97

(2011).[20] A. E. Hosoi and T. F. Dupont, J. Fluid Mech. 328, 297

(1996).[21] A. Muzikansky, P. Nanikashvili, J. Grinblat and D. Zitoun,

J. Phys. Chem. C 117, 3093 (2013).[22] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.112.188301 fordependence of layering on ~Δ, d, and particle polydispersity;details on PDE solution, scaling laws, and dynamics arediscussed and animated.

[23] T. Sobisch, Lumifuge (private communication).[24] W. Stöber, A. Fink, and E. Bohn, J. Colloid Interface Sci.

26, 62 (1968).[25] N. M. Abrams and R. E. Schaak, J. Chem. Educ. 82, 450

(2005).[26] R. J. Hunter, Foundations of Colloid Science, (Oxford

University Press, New York, 2009), 2nd ed.[27] S. Thomas, I. Lagzi, F. Molnár, and Z. Rácz, Phys. Rev.

Lett. 110, 078303 (2013).[28] B. J. Frisken, Appl. Opt. 40, 4087 (2001).[29] P. N. Pusey and W. van Megen, J. Chem. Phys. 80,3513

(1984).[30] L. D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon,

New York, 1987), 2nd ed.[31] G. K. Batchelor, J. Fluid Mech. 52, 245 (1972).[32] S. Heitkam, Y. Yoshitake, F. Toquet, D. Langevin, and

A. Salonen, Phys. Rev. Lett. 110, 178302 (2013).

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Supplemental Material: Critical onset of layering in sedimenting suspensions of

nanoparticles

A. V. Butenko,1 P. Nanikashvili,2 D. Zitoun,2 and E. Sloutskin1

1)Physics Department and Institute for Nanotechnology and Advanced Materials,

Bar-Ilan University, Ramat-Gan 52900, Israel

2)Chemistry Department and Institute for Nanotechnology and Advanced Materials,

Bar-Ilan University, Ramat-Gan 52900, Israel

(Dated: 1 April 2014)

1

I. LAYERING VARIATION AS A FUNCTION OF ∆T AND d

The layering is almost completely diminished by shielding of the sample (see Fig. S1,

where Ag nanoparticles, such as in Fig. S2, were used). For that purpose, we wrapped the

sample with a copper foil, leaving a thin window for the light transmission measurements,

as discussed in the main part of the article. The temperature drop over the sample ∆,

in absence of shielding, was estimated by attaching a thermocouple to the sample, within

seconds after the full stop of sample rotation. We obtain ∆ ≈ 0.4 K; the values do not vary

significantly with either the centrifugation velocity or the time (5−10 sec) between the stop

of rotation and the ∆ measurement.

The width of the experimental cuvette d may influence the experiment through long range

hydrodynamic interactions. However, when d is reduced, the temperature drop across the

sample ∆ is reduced accordingly. Assuming that the temperature gradient stays unchanged,

we obtain ∆ ∝ d. To test the influence of d on our experiments, we repeated the same

experiment in cuvettes of d = 1, 0.85, and 0.5 mm; other dimensions of the cuvettes were

identically the same in all cases. The cuvettes were obtained by cutting of the commercial

d = 1mm cuvettes; the parts were then glued together, using an epoxy glue, into which

a mixture of rosin in xylene was added. This combination was found to be sufficiently

strong to avoid leaking of the suspensions at elevated effective gravity in the centrifuge.

The resulting profiles are shown in Fig. S3, where the number of layers slightly increases for

lower d. We reproduce this dependence by computer simulations, employing the theoretical

model described below and in the main text. In these simulations (see Fig. S4), we vary d

for a constant ∆ (blue circles), vary both d and ∆ so that their ratio is kept constant (red

squares), and vary ∆ for a constant d (olive triangles). Note, that while N(d) is sloping,

for either a constant or a varying ∆, N(∆) is almost constant for this range of parameters

[cf. Fig. 3(c, inset) of the main text], suggesting that the variation of N with d in our

experiments is of a hydrodynamic origin, rather than being simply a result of the ∆(d)

dependence. Importantly, while d changes the number of layers, it also has a dramatic

impact on the steepness of the stairs in the concentration profile. As a result, numerical

instabilities are encountered for high d values; therefore, we do all our calculations for lower

d-values than those used in the actual experiments.

2

(b)I / I 0

x / H

1

2

0.6

0.8

1.0

6

54

32

I / I 0

(a)

1

FIG. S1. Transmission profiles through sedimenting suspensions of Ag nanoparticles for (a) un-

shielded and (b) thermally-shielded samples. Note the significantly diminished layering in the

sample which was thermally shielded; only 1-2 layers form in this sample, while there are 6 layers

in the unshielded one. The time separation between subsequent profiles is 360 sec and 600 sec in

(a) and (b) respectively; a = 2.5× 103, and H = 22 mm. The layer numbers are on the right.

II. SENSITIVITY TO THE PARTICLE SIZE DISTRIBUTION

The layering phenomenon is notoriously sensitive to the shape of P (σ). We could se-

lectively eliminate the layering in a part of the suspension by truncating the low-σ wing

of particle size distribution (solid green circles in Fig. S5(a)); with P (σ) being steep on its

low-σ side, the low-x side of the sedimentation front does not broaden significantly with

time, so that the particle density gradients in this part of the sample are too steep to al-

low formation of multiple rolls. As a result, the layering occurs only at the high-x side of

the sedimentation front [see Fig. S5(c)], while it takes place at both sides of the front for

a symmetric P (σ) [Fig. S5(b)]. Remarkably, the asymmetry of P (σ) is hard to measure

in nanoparticles by classical experimental techniques, suggesting that the layering patterns

at pre-set temperature gradients, may be used in the future as a basis for new analytical

3

FIG. S2. A typical TEM image of the Ag nanoparticles1.

techniques for full characterization of P (σ).

III. NUMERICAL MODEL

A. Partial Differential Equations (PDE)

The dimensionless form of the PDE used in the main part of this work is:

2∂2u∂x2 = 1

3∂4u∂x4 +

212u− 81

4α2b+ 7

8

∂c∂t

= δ ∂2c∂x2 +

∂c∂x

(kcc0 − 1)− 1235

γ2

c0

∂(ub)∂x

+ 6835

kc0γ4α2b ∂b

∂x

∂b∂t

= δ ∂2b∂x2 + kc0

∂(bc)∂x

− 4217δb− ∂b

∂x− 3

17c0

γ2α2u∂c∂x

(1)

Here c(x, t), t and x are the dimensionless equivalents of particle volume fraction c, time

t, and spatial coordinate x, obtained by the following rescaling: c/c0 → c, tv0/d → t, and

x/d → x. u(x, t) describes the x-dependence of the convection velocity component in x

direction, u · x = u(x, t) ∂∂z

[

14(z2 − d2)

2]

; in the dimensionless version of the PDE, u(x, t) is

rescaled: ud3/U → u. b(x, t) is an auxiliary dimensionless function2, used to describe the

variation of density along z. α, γ, δ and c0 are dimensionless parameters described in the

main part of the text. k ≈ 6.55 is a constant, used in the Batchelor’s formula to describe

the hindrance of sedimentation velocity at a finite particle density; particle densities in our

current work are very low, such that setting k = 0 does not change any of the results.

4

0.4 0.5 0.6 0.7 0.80.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

d = 0.5 mm

(c)

I/I0

x/L0

12

4

7

d = 0.85 mm

(b)

I/I0

10

4

7

d = 1 mm

8I/I

0

4

7

(a)

FIG. S3. Transmission profiles through the sedimenting suspensions of nanoparticles (average

diameter: 〈σ〉 ≈ 18 nm), obtained with cuvettes of different widths d. Labels inside the plots show

the corresponding d values; labels on the right show the sequential number of the plateau, from the

top of the plot. Here H = 24.5 mm and a ≈ 2.5 × 103. Note the increase in the number of layers

for thinner cuvettes. The time separation between subsequent curves is 100 sec, in all sections.

To solve the equations, we employ the pdepe Matlab function, with the ode parameters

set to: MaxOrder=2, RelTol=1e-2, AbsTol=1e-5; changing these parameters does not alter

any of the solutions, yet may significantly increase the calculation time and even make

the calculation stop due to convergence issues. The length of the simulated cuvette x was

discretized into 400 or 800 equidistant points. The time scale, chosen so as to cover the

5

0.12 0.14 0.16 0.18 0.20 0.22

25

30

35

40

45

50

0.12 0.14 0.16 0.18 0.20 0.22

= 0.1K; d - variable /d = 1 K/mm - variable; d = 0.15mm

~

~

/ K

d / mm

~

~

FIG. S4. The theoretical number of plateaus in a sample N is shown for a range of different

cuvette widths d (bottom horizontal scale) for a constant ∆ = 0.1K (blue circles) and for a

constant temperature gradient ∆/d = 1 K/mm (red squares). For comparison, we also show the

N for a constant d = 0.15 mm (green triangles), for a range of different ∆ (top horizontal scale).

In all plots, a = 2000, H = 2.5 mm; other parameters are the same as elsewhere in this work.

sedimentation process, was typically discretized into 100-400 steps; longer calculations were

done where needed, to provide sufficient time-resolution for an accurate determination of N

(see main text). The viscosity of the simulated solvent was set to ν = 0.3 cSt, the relative

density mismatch of the nanoparticles with the solvent was set to ∆ρ/ρ = 13, the average

particle volume fraction was c0 = 10−4 and the diameter of the particles was set to σ = 10

nm; all these values are very close to the experimental ones. The overall dynamics of roll

formation in the sample is best visualized by a color map of u(x, t), such as in Fig. S6.

The central region of this plot is composed of green stripes, corresponding to the layers

of constant particle density, separated by red and yellow boundaries, where the flow is at

a normal to the gravity. The large red region on the left corresponds to the supernatant,

where no splitting into finite-length rolls occurs. The large red region on the right is the

high-density suspension at the bottom of the experimental cell. The complexity of roll

formation dynamics in Fig. S6, where pairs and triplets of rolls occasionally merge, calls for

future experimental studies to be carried out with much longer samples, where the approach

6

0.4

0.6

0.8

1.0

(a)

I / I 0

(b)

0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

I / I

0

(c)

x / H

5 10 15 200.0

0.1

0.2

P(

)

/ nm

1

2

FIG. S5. (a) Diameter distributions in two different samples of Cu@Ag nanoparticles; note the

symmetric and highly-skewed P (σ) for sample 1 (diamonds) and 2 (circles), respectively. Light

transmission I/I0 profiles through sedimenting suspensions are shown for both samples in (b) and

(c); the time separation between subsequent profiles is 50 sec in (b) and 240 sec in (c), a = 2.1×103,

and H = 23 mm. (b) For sample 1, which exhibits a symmetric P (σ), the plateaus are observed

even at high I/I0, corresponding to particles smaller than the average, moving slower than the

sedimentation front. (c) For sample 2, where the low-σ wing of P (σ) is truncated, the layering

occurs only for the low I/I0 (corresponding to the high-σ-end of the particle size distribution),

demonstrating the unique sensitivity of the layering phenomenon to the full shape of P (σ).

to steady state would be more clearly visible; for that purpose, an analytical centrifuge of a

much larger radius must be constructed, dedicated to these measurements.

B. Dependence on particle concentration gradient

The average width λ⋆ of the simulated layers at t⋆ decreases as p−1 [see Fig. S7(a)], where

p is the initial particle concentration gradient (normalized by the average concentration c0).

7

/H

t/ hr

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

FIG. S6. The maximal value of the x-component of computer simulated convection velocity u · x,

in the cold (−d < z < 0) region of the sample, shown as a color map; here z = −d/2, a = 210,

H = 2.5 cm, and ∆ = 0.17 K. Numerical values in the color scale (on the right) were multiplied by

a factor of (8/3)U−1, for non-dimensionalization; note, with this normalization, the data simply

represent the dimensionless u(x, t) (see text). The green regions of the plot correspond to layers

of constant c(x), where convection velocity along the gravity axis is maximal (in absolute value).

The velocity is normal to the gravity at the boundaries between the convection rolls; thus, the

boundaries appear red in the color map. Note the initial formation of transient rolls at low t; some

of these rolls merge together at later times.

This decay law is consistent with our estimate (see main text) of the roll size being roughly

equal to Λ ∝ (dρ/dx)−1, confirming the convective instability as a mechanism for layering.

Instabilities due to hydrodynamic interactions3,4 exhibit a different scaling, allowing these

mechanisms to be ruled out in the present case.

Interestingly, a more complex behavior occurs for the initial gradient of particle concen-

tration being non-uniform [Fig. S7(b)]. In particular, a wide layer of a constant particle

concentration forms at the top of the sedimenting suspension, at all centrifugation times.

8

0.08 0.12 0.160.05

0.10

0.15

0.00 0.25 0.50 0.75 1.000.8

1.0

1.2 (b)

p-1

p-1/4* /H

p cm

p-2/5

(a)

0 hr0.2hr 2.8hr

c/c 0

x/H

6.9hr

FIG. S7. (a) The average width of the layers λ⋆ at t⋆ is shown for a range of different initial

particle concentration gradients p; here H = 2.5 cm and a = 210. A fitted p−1 decay (solid line)

perfectly matches the simulated data (symbols), as predicted in the main text. Instabilities due

to hydrodynamic interactions3,4 exhibit other scalings, as shown in dashed and dash-dotted lines

(see labels). (b) For a non-uniform initial gradient of particle concentration (black dashes), getting

smaller with x, a wide plateau (marked by an arrow) forms at the top of the sedimenting suspension

at all t (see labels).

This behavior takes place in spite of the initial particle concentration c(x) gradient being

the highest at the top of the suspension, such that the local Λ is there the lowest. This

further supports the validity of our current theoretical model, as a similar effect is observed

in experiments, see Fig. 2(b) of the main paper.

C. Scaling laws

The scaling of the simulated layer width λ⋆ at the layering onset time t = t⋆ is shown in

Fig. S8(a). Here, t⋆ is defined as the time at which the layers in the middle of the sample

are fully visible (t⋆ ≈ 2.5 hrs in Fig. S6). The scaling of t⋆ for a range of a and D values

9

200 400 600 800 100010-2

10-1

100

0.5 1.0 1.5 2.0

3

6

9

0 2 4 6 8

6

9

12

(c)

(b)

t*

(hrs

)

a

(a)

t* (h

rs)

D(m2/s) 1010

* 1

00

Dt* / m2 106

FIG. S8. (a) The simulated layer width at t⋆ (symbols), for a range ofD values, is fitted by (Dt⋆)1/2,

suggesting that the onset of convection instability is limited by diffusion. (b) The calculated

time t⋆ associated with the onset of layering (symbols) decreases with the effective gravitational

acceleration in the centrifuge a. The fitted power law t⋆ ∼ a−0.7±0.1 is shown in a solid line.

(c) Diffusion D smears the layering, so that the calculated layering onset times increase with D,

diverging for the present set of parameters at Dc ≈ 2.2× 10−10 m2/s. The solid curve corresponds

to t⋆ ∼ |D −Dc|−0.3.

is shown in Fig. S8(b-c). The theoretical scaling laws, discussed in the main part of the

article, are shown in solid lines, confirming that the onset of layering is determined by the

competition between formation of convection rolls and structure smearing by diffusion.

10

4 8 12 16 20 24 280.00

0.05

0.10

0.15

0.20

0.25

0.30

P(

)

/ nm

FIG. S9. Size distribution of Cu@Ag nanoparticles used for the supplementary animation, as also

for Fig. 2(b) in the main text.

IV. LAYERING ANIMATION

In the supporting animation file layering movie.avi we demonstrate the formation of

layers in experimental Cu@Ag suspensions (see size distribution in Fig. S9). Each of the

coloured columns in this animation visualizes the light transmission through the correspond-

ing sample (see labels at the bottom); the corresponding transmission profiles are shown to

the right. Note the striking difference between the monotonic profiles obtained for the sus-

pensions of silica colloids (on the left) and the layered profiles in the suspensions of Cu@Ag

nanoparticles (center). The simulated transmissions are shown at the right. Note that the

simulation does not take into account the spread of sedimentation velocities due to the

polydispersity of the particles. Therefore, the finite slope of c(x), achieved after the initial

stage of centrifugation, prior to the onset of layering, is introduced into the corresponding

PDE through the initial conditions (see main text). Consequently, we start the animation

of simulated data with a delay, corresponding to the development of the sloping c(x).

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11

3D. Saintillan, E. S. G. Shaqfeh, and E. Darve, Phys. Fluids 18, 121503 (2006).

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12