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# Dense colloidal fluids form denser amorphous sediments

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 25, 2013 (received for review August 29, 2012)

## Abstract

We relate, by simple analytical centrifugation experiments, the density of colloidal fluids with the nature of their randomly packed solid sediments. We demonstrate that the most dilute fluids of colloidal hard spheres form loosely packed sediments, where the volume fraction of the particles approaches in frictional systems the random loose packing limit, *φ*_{RLP} = 0.55. The dense fluids of the same spheres form denser sediments, approaching the so-called random close packing limit, *φ*_{RCP} = 0.64. Our experiments, where particle sedimentation in a centrifuge is sufficiently rapid to avoid crystallization, demonstrate that the density of the sediments varies monotonically with the volume fraction of the initial suspension. We reproduce our experimental data by simple computer simulations, where structural reorganizations are prohibited, such that the rate of sedimentation is irrelevant. This suggests that in colloidal systems, where viscous forces dominate, the structure of randomly close-packed and randomly loose-packed sediments is determined by the well-known structure of the initial fluids of simple hard spheres, provided that the crystallization is fully suppressed.

Physical mechanisms that determine structure and density of noncrystalline solids remain controversial after several decades of intense experimental (1⇓⇓⇓⇓–6) and theoretical (7⇓⇓–10) research. In granular systems of hard spheres, for a wide range of experimental and theoretical protocols, particle motion is arrested or “jammed” when the volume fraction of the spheres reaches *φ*_{RCP} ≈ 0.64, known as the random close packing (RCP) density. However, the notion of random close packing is ill-defined; denser packings, up to the limit of ∼0.7405, are readily achieved by increasing the crystallinity of the structure, whereas the randomness of the RCP state must still be quantified (11). Moreover, some experiments (12⇓–14) and simulations (15, 16) indicate that the system can become solid-like at a much lower volume fraction, down to the so-called random loose packing (RLP) limit of *φ*_{RLP} ≈ 0.55. In particular, experimental packings of macroscopic spheres, gently sedimented in a buoyancy-matched fluid, closely approach the RLP limit (13, 17, 18). The RLP limit, related to the friction between the constituent particles (14, 16), is even more controversial than the RCP (11, 17). Thus, the notion of both the RLP and the RCP states remains ill-defined (11, 19).

We form sediments out of a fluid suspension of hard micrometer-sized spheres in a solvent, known as colloids, and demonstrate that the structure of the initial fluid uniquely determines the density of the sediments. This is the case for a wide range of sedimentation rates, which are sufficiently high to prohibit crystallization. In particular, the most disordered suspensions, where the initial volume fraction *φ*_{0} tends to zero, form loosely packed sediments; the volume fraction of these sediments *φ*_{j} approaches *φ*_{RLP}. Denser suspensions form denser sediments, where *φ*_{j} approaches *φ*_{RCP}. The structure of fluids of simple hard spheres in thermodynamic equilibrium, such as our initial colloidal suspensions, is well established (20). Therefore, the observed relationship between the structure of these simple fluids and the density of their nonergodic sediments suggests that the thermodynamics of the initial fluids may possibly be used to fully understand the physics of the RLP and the RCP states in amorphous sediments. This deeper understanding of sediments, in addition to its fundamental importance, may contribute to abundant industrial processes, such as the slip-casting (21) of ceramics, in which water is rapidly drawn from a suspension of clay particles to form solid ceramic objects.

## Materials and Methods

To form the sediments, we suspend poly(methyl methacrylate) (PMMA) colloidal spheres in mixed decahydronaphthalene (≥98%; Sigma-Aldrich). The sediment is formed by centrifugation in a thermally regulated centrifuge (Advanced LF-110 LUMiFuge), with the amplitude of the centrifugal acceleration set to a value between 130*g* and 2,080*g*, where *g* = 9.8 m/s^{2}. We use direct confocal microscopy, in three dimensions, to measure the radial distribution function of the colloids (20) *g*(*r*) and the crystalline local bond-order parameters (11); these metrics confirm absence of any significant crystalline domains within the sediments. The particles are sterically stabilized by poly-12-hydroxystearic acid (22), such that the interactions in a fluid suspension are best described by a hard potential (23, 24). The dynamic viscosity of our solvent at *T* = 22°C, obtained using a Cannon-Manning semimicro viscometer, is *η*_{s} = 2.4 ± 0.05 mPa⋅s. The average diameter of our particles is *σ* = 2.4 ± 0.05 μm and their polydispersity is <5%, as detected by static and dynamic light scattering, confocal microscopy, and scanning electron microscopy of dry particles under vacuum; this very low polydispersity of our particles allows any variation in the density of the sediments due to possible segregation of particle sizes to be ruled out completely. The diameter of our particles is sufficiently small so that the particles undergo Brownian motion. The time for a free particle in a solvent to diffuse its own diameter is *t*_{D} = (2*k*_{B}*T*)^{−1}*πσ*^{3}*η*_{s} = 12.8 s at *a* = 130*g*; this time scale corresponds to a free particle displacement of ∼1 mm along the effective gravity, so thermal structural reorganization of solid sediments during the centrifugation is unlikely.

All our samples are prepared by dilution (or removal of supernatant) from the same initial batch of suspension; the volume fraction of colloids in this batch *φ*_{00} was initially estimated as 0.3−0.4. The sample is homogenized and randomized by vortex mixing after either dilution or removal of the supernatant. With the gravimetric density of our solvent *ρ*_{s} = 0.868 g/cm^{3} measured by pycnometry, the volume fractions of all our samples, as a function of *φ*_{00}, are known to a high precision. We use analytical centrifugation to obtain the value of *φ*_{00}, as detailed below. About 0.4 mL of the suspension is loaded into an optically transparent polyamide cell, which has a rectangular cross-section of 2 × 8 mm, so that the initial height of the sample is *L*_{0} ≈ 25 mm. The cell is then vigorously shaken on a vortexer and placed into the centrifuge (Fig. 1 *Inset*). Our LUMiFuge analytical centrifuge measures light transmission profiles through the sample (*Supporting Information*) at a wavelength of 870 nm, in situ, during the centrifugation.

## Results

To accurately calibrate the volume fraction of particles in the suspensions, we measure the velocity of colloidal sedimentation (25). As the particles sediment, a colloid-free region (supernatant) is formed in the topmost part of the sample, as shown in Fig. 1 *Inset*. We track the position *x* of the boundary between the supernatant at the top of the sample and the fluid colloidal suspension below. First, *x*(*t*) is linear in time; then, it saturates when all of the particles are fully arrested within the sediment, as shown in Fig. 1. Interestingly, the linear regime survives even for the densest suspensions, during most of the centrifugation process, indicating that the densities and the structures of our fluid suspensions do not significantly change during the centrifugation. Although significant spatial fluctuations in microscopic sedimentation velocities of the particles were previously observed in similar systems (26), these fluctuations do not necessarily change the structure of the suspensions. Indeed, the fluctuations were interpreted in the past in terms of an effective temperature (26); in our system of hard spheres the energy scale is missing, such that the average structure of the fluid does not depend on the temperature. The slope of the linear region of *x*(*t*) in Fig. 1 yields the velocity of the sedimentation front *v* = *dx*/*dt*. For a very dilute suspension, *v* is determined by the simple Stokes’ law. At high concentrations of colloids, front velocities are slowed by interparticle interactions that increase the fluid drag acting on each colloidal sphere.

To focus on the hydrodynamics of our fluids, we divide our front velocities, obtained in the linear range of *x*(*t*), by the amplitude of the centripetal acceleration *a* = (2*π**f*)^{2}*R*, where *R* and *f* are the radius and the frequency of rotation in the centrifuge, respectively. For Stokesian sedimentation of a sphere in a fluid, its velocity is *v*_{0} = Δ*ρσ*^{2}*a*/(18*η _{s}*), where Δ

*ρ*=

*ρ*

_{p}−

*ρ*

_{s}is the mismatch between the density of the sphere

*ρ*

_{p}and that of the surrounding fluid

*ρ*

_{s}, which has a dynamic viscosity

*η*

_{s}; thus, for a given sphere in a given fluid,

*v*

_{0}/

*a*is constant. Similarly, all our front velocities

*v*(

*φ*

_{0}) collapse together, when normalized by the corresponding values of

*a*, as shown in Fig. 2. The collapse of the data is quite remarkable, given that the amplitudes of our vary by a factor of 16 and the average separation between the particles in the fluid at

*φ*

_{0}≈ 0.35 is only 1.5

*σ*. With the

*v*/

*a*collapsed together, we can describe our front velocity by the Stokes formula for the velocity of an individual sphere, as above, where

*η*

_{s}and

*ρ*

_{s}are replaced by the viscosity

*η*and the density

*ρ*of the suspension. Importantly, these

*η*and

*ρ*now depend on the volume fraction of colloids

*φ*

_{0}. To match this effective medium approximation to our experimental data, we use the experimental static

*η*(

*φ*

_{0}) and directly measure the density of our suspensions ρ by pycnometry, so that .* This allows our experimental

*v*/

*a*, for a wide range of centrifugation rates and volume fractions, to be matched, with the only free-fitting parameter being

*φ*

_{00}. Note the very nice fit to the experimental data, as shown by a dashed curve in Fig. 2, with a fitted value of

*φ*

_{00}= 0.35. The fitted

*φ*

_{00}allows the

*ρ*

_{p}value to be obtained as 1.045 g/cm

^{3}, which is smaller than the bulk density of solid PMMA (27) (1.17–1.20 g/cm

^{3}), yet significantly denser than

*ρ*

_{s}= 0.868 g/cm

^{3}of the pure solvent, indicating that some absorbtion of decahydronaphthalene into the particles may have possibly occurred. Most importantly, the obtained value of

*φ*

_{00}allows the absolute

*φ*

_{0}values to be known with a high accuracy, such that the volume fraction

*φ*

_{j}of colloids within the sediments is obtained as

*φ*

_{j}=

*φ*

_{0}

*L*

_{0}/

*x*(

*t*→ ∞).

A very common, yet questionable (25), assumption in colloidal physics is that the particle volume fraction in sediments prepared by centrifugation is *φ*_{j} = *φ*_{RCP} ≈ 0.64. In particular, *φ*_{j} is typically assumed to be independent of the colloidal volume fraction in the initial suspension (25). This assumption was questioned in recent experimental (28) and theoretical (29) studies, demonstrating a decrease in *φ*_{j} with *φ*_{0} in packings of macroscopic objects. However, in these studies the inertial forces were significant. Our sediments are prepared at low Reynolds numbers, typical for colloidal systems, where the inertial effects are negligible.^{†} Strikingly, the measured colloidal volume fractions of our sediments increase with the density of the initial fluid suspensions, as shown in Fig. 3*A* (solid symbols). The measured *φ*_{j} are independent of the initial height *L _{0}* of the suspension (Fig. 3

*B*). This indicates that the potential energy of our colloids with respect to gravity is irrelevant. In addition, this demonstrates that

*φ*

_{j}are not sensitive to the macroscopic shape of the top of the sediment, as also to the density of particles in the topmost region of the sediment, which is formed in the nonlinear regime of

*x*(

*t*) (Fig. 1). The measured

*φ*

_{j}are also independent of the centripetal acceleration, in our range 130

*g*<

*a*< 2,080

*g*, which corresponds to Péclet numbers ranging from 10 to 10

^{3}. The independence of

*φ*

_{j}on

*a*and L

_{0}stays in contrast with the fluidized bed experiments (12, 30), where a packing of granular spheres is fluidized by multiple water flow pulses, followed by formation of a new solid packing. In the fluidized bed experiments, the densities of solid packings decrease monotonically with the flow rate, so that the most expanded fluidized beds form highly expanded solid packings. Unfortunately, the volume fraction of the fluidized beds was not measured and the homogeneity of the fluidized state was not tested; this and the dependence of these results, obtained at Reynolds numbers of order unity, on sample height and on the rate of sedimentation (12, 30) complicates the comparison with our work, motivating additional studies in both fields.

We confirm the increase of *φ*_{j} with *φ*_{0}, observed in Fig. 3*A*, independently, by measuring the mass of the supernatant in some of these samples. The observed scaling of *φ*_{j} with *φ*_{0} in our system allows the density of our packings to be tuned in a controllable way. This is impossible with the classic granular packings (1, 18), where various, rather uncontrolled tapping protocols were used to increase the packing density (17). Importantly, our experimental *φ*_{j} tend to 0.55 for the most dilute suspensions, where *φ*_{0} → 0. This value is very close to the well-known, yet highly controversial, RLP limit of the granular packings (13, 14, 16⇓–18). Moreover, a sediment can never be less dense than the original suspension, *φ*_{j} > *φ*_{0}; thus, all our data must fall above the *φ*_{j} = *φ*_{0} dashed-and-dotted line in Fig. 3*A*. This obvious geometrical argument implies that there must be an upper limit on the density of amorphous colloidal sediments, set by the intersection between the experimental *φ*_{j}(*φ*_{0}) scaling (fitted by a solid line) and the *φ*_{j} = *φ*_{0} dashed-and-dotted line. This limiting volume fraction is obtained as *φ*_{j} = 0.64, which is very close to the well-known, yet strongly debated, RCP limit (11). The observed *φ*_{j}(*φ*_{0}) in our system, where Péclet numbers are sufficiently high so that structural thermal relaxation within the solid sediments is prohibited, suggests that the states of our solid sediments may possibly be related to the thermodynamics of the initial fluids.

## Discussion

To test the relation between the density of the sediments and the thermodynamic states of the initial fluids of hard spheres, we carry out simple computer simulations, where we neglect the hydrodynamic interactions and any possible structural reorganizations within the sediments and within the fluid suspensions. We choose these, somewhat oversimplified, conditions to focus on the most basic physical mechanisms underlying the experimental *φ*_{j}(*φ*_{0}) scaling. We simulate a fluid of simple hard spheres, thermodynamically equilibrated at an initial volume fraction *φ*_{0}. As mentioned earlier, the velocity fluctuations in experimental sedimenting suspensions, while raising the effective temperature of the suspension (26), do not necessarily alter the sample-averaged local structure in our fluids of hard spheres with respect to the thermally equilibrated structure, which is dictated solely by the entropy and therefore does not depend on the temperature. Therefore, modeling the structure of our sedimenting fluid suspensions by that of a thermodynamically equilibrated fluid is a reasonable approximation. The simulated cell, where the number of particles was chosen to be between 4 × 10^{3} and 4 × 10^{5}, is subject to periodic boundary conditions in *x* and *y* directions. To simulate the structure of the sediment, we make the particles fall, one by one, along the −*z* direction, to the bottom of the cell (*z* = 0). The particles that are the closest to the bottom are the first to fall. To stop falling, the particle must either contact the bottom of the cell or contact *Z*_{s} particles that already belong to the sediment. If *Z*_{s} > 1, once a falling particle meets its first contact with a sphere belonging to the sediment, it slides along the circumference of that sphere; then it either meets another contact or falls again (*Supporting Information*). Once it has stopped falling, the particle is considered to belong to the sediment; its position is then fixed during the rest of the simulation. Importantly, the sedimentation process is nonrandom, such that for a given structure of the initial fluid, the structure of the final sediment is fully determined. Surprisingly for such a simplistic simulation, the densities of our simulated sediments increase (roughly) linearly with φ_{0} (Fig.4*A*), as in the experiments. The slope of the simulated *φ*_{j}(φ_{0}) varies with the number of supports *Z*_{s}, which are necessary to stabilize a particle under gravity. By symmetry, if a particle is supported on average by *Z*_{s} underlying particles, it must support (on average) *Z*_{s} overlying particles. The data in Fig. 4*A* are labeled by the total number of contacts per particle *Z* = 2*Z*_{s} in each of the simulations. When the number of contacts is low *Z* = 2, the sediments are very dilute. This is the case with the sediments of cohesive particles (18) and packings prepared by random ballistic deposition (31) of sticky spheres. In these systems (31), *φ*_{j} ≃ 0.15 for *φ*_{0} = 0, in perfect agreement with our simulations. For frictionless spheres at the isostatic conditions, the number of contacts is *Z* = 6 and the slope of *φ*_{j}(*φ*_{0}) is very low (15). We show the slopes of the simulated *φ*_{j}(*φ*_{0}) in Fig. 4*B* (open symbols), where the values decay exponentially with *Z*. The slope of the experimental data is consistent with the number of contacts being (slightly smaller than) ∼4, as in an isostatic frictional system, where the spheres are unable to slip past each other. Whereas our colloids behave as perfect hard spheres in their fluid state, the friction in colloidal systems is difficult to obtain by experimental techniques (5). However, the arrested rotational diffusion in our sediments (*Supporting Information*) suggests that the frictional forces between the particles may be significant, in agreement with the obtained value of *Z*. Although direct measurement of contacts between the colloids is a very challenging task, the simulated coordination numbers are in good agreement with the experimental ones, for all *φ*_{j} (*Supporting Information*).

Our simulations, where structural relaxation is prohibited and the sedimentation process is deterministic, link between the structures of solid sediments and the fluid states of initial suspensions: Each fluid state, specified by the position of all particles in the simulation, forms a sediment with a certain predetermined structure. When the initial fluid is very dilute (*φ*_{0} → 0) and the correlations between particles are negligible, all mechanically stable structures of sediments are equally probable. The number of different mechanically stable structures varies with the sediment density *φ*_{j}, peaking (11) for the frictional particles (32) at *φ*_{j} ≈ 0.55 (Fig. 4*A* *Inset*). However, by simple geometry, when φ_{0} is finite, the only states of the sediment that can form are the ones that are denser than the initial fluid, *φ*_{j} > *φ*_{0} (Fig. 4*A* *Inset*, hatched area). This shifts the average density of the sediment to a higher value of *φ*_{j}, as experimentally observed.

To test the physical mechanism for *φ*_{j}(*φ*_{0}) dependence, we measure the degree of structural order within the colloidal sediments formed from fluids of different *φ*_{0}, using direct confocal microscopy. The degree of structural order *χ* is quantified by the deviation of the experimental radial distribution functions from their value for an ideal gas *g*(*r*) = 1, where correlations are missing. The value of *χ* increases for sediments prepared from denser fluids, as demonstrated in Fig. 4*C*, indicating that sediments prepared from denser fluids are more ordered (*Supporting Information*). Whereas *χ* measures the positional order of the particles, which has a complex dependence on *φ*_{j}, another measure of structural order is provided by the orientations of geometrical bonds between the particles. The excess of this bond-orientational order with respect to an ideal gas is given by , where averaging is carried out over the whole sample and is the correlation between bond orientations of the *i*-th and the *j*-th particles, normalized by to have the correlation of the *i*-th particle with itself be identically equal to unity. The definition of *q*_{ij} makes use of the sixfold-symmetric local bond-orientational parameter of the *i*-th particle , where are the spherical harmonics (of degree 6 and order *m*) for the orientation of the bond between the *i*-th particle and the *k*-th one; the *k*-index summation is carried out over all nearest neighbors of the *i*-th particle. With this definition of *q*_{ij}, which was widely used in many previous studies (33, 34), the 〈*q*_{ij}〉 does not completely vanish even in a fluid or in an ideal gas; therefore, we measure the excess orientational correlation *ψ*_{6}, where the ideal gas value is subtracted. The resulting *ψ*_{6} values increase for those sediments that were formed from high-density fluids, as shown in circles in Fig. 4*C*; thus, both the bond-orientational order and the positional order indicate that sediments formed by denser fluids are more ordered, suggesting that microscopic low-density states are excluded in these sediments.

In our range of Péclet numbers, crystallization is prohibited and the sediment structure is amorphous. The number of amorphous sediment states drops at very high *φ*_{j}, as for the uniformly compressed systems at the jamming transition (7, 35); this possibly sets the highest limit for the density of fluids and amorphous sediments in colloidal suspensions, denoted above by *φ*_{RCP}. Clearly, a more elaborate theoretical model is necessary to fully account for the details of our experimental and simulated observations.

In addition to structural measurements, we demonstrate that sediments prepared from dense colloidal fluids exhibit higher mechanical stability. We prepare three sediments from three initial suspensions, which are identical except for having different *φ*_{0}. We choose the amounts of the initial suspensions such that the heights of these three sediments are identical; supernatant is added, where needed, after the formation of the sediment, to equalize the total height of the three samples. We compare the mechanical stability of these three sediments by placing them horizontally on a vortex mixer. After a couple of minutes, the sediment that was prepared from a low-*φ*_{0} fluid is almost completely molten, whereas the sediment that was prepared from a high-*φ*_{0} fluid is almost intact (see the relative heights of the sediments in Fig. 5). Similar observations were previously reported for the macroscopic fluidized bed packings (36), which suggests that these results may have an important impact on abundant processes based on sedimentation, in nature (37) and industry.

Preliminary results (22) suggest that the observed *φ*_{j}(*φ*_{0}) scaling is not unique for systems of sterically stabilized colloids. The same type of behavior takes place for other types of colloids, for example for 400-nm, charge-stabilized silica particles, provided that the sedimentation is sufficiently fast to avoid any possible crystallization. Future studies in which sedimentation is slowed down by many orders of magnitude by fine-tuning the density of the solvent to approximate that of the colloids should allow the influence of crystallization on density and mechanical properties of the sediments to be detected.

In conclusion, our observations suggest that the structure of noncrystalline sediments may possibly be fully understood, based on the well-known microscopic structure of thermodynamically equilibrated fluids. This should possibly allow the jammed states formed in nonequilibrium particle systems, such as sand, to be characterized by comparison with our sediments, leading to a deeper understanding of noncrystalline solids.

## Acknowledgments

We thank Y. Roichman, Y. Shokef, M. Wyart, D. A. Weitz, Y. Rabin, M. Schwartz, and S. Torquato for fruitful discussions. We thank M. Shmilovitz (Tetra Sense Scientific Tools Ltd.) for assistance with the LUMiFuge experiments and P. Nanikashvili and D. Zitoun for experimental assistance. Part of the equipment used in this work was funded by the Kahn Foundation. This research is supported by Israel Science Foundation Grants 85/10 and 1668/10.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: eli.sloutskin{at}biu.ac.il.

Author contributions: A.V.B. designed research; S.R.L., S.B., and A.V.B. performed research; A.B.S. contributed new reagents/analytic tools; S.R.L., S.B., A.V.B., and E.S. analyzed data; and A.V.B. and E.S. wrote the paper.

The authors declare no conflict of interest.

↵*We interpolate the experimental

*η*(*φ*_{0})/*η*_{s}[Segrè PN, Meeker SP, Pusey PN, Poon WCK (1995) Viscosity and structural relaxation in suspensions of hard-sphere colloids.*Phys Rev Lett*75(5):958–961] by [*A*/(*A*−*φ*_{0})]^{α}, where*A*= 0.53 and*α*= 1.53, which perfectly matches the experimental data up to*φ*_{0}= 0.45.↵

^{†}The Reynolds and Stokes numbers (14) in our experiments are very low, below 3 × 10^{−4}and 4 × 10^{−5}, respectively. Thus, compaction mechanisms discussed by Farrell et al. (14) are irrelevant in our case; in our range of parameters, particles entering the packing do not have the ability to rearrange the structure of the packing. The Péclet number (5) ranges in our studies from 10 to 10^{3}.This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1214945110/-/DCSupplemental.

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