Sedimentation equilibrium and the generalized Archimedes' principleAlberto Parola, Stefano Buzzaccaro, Eleonora Secchi, and Roberto Piazza Citation: J. Chem. Phys. 138, 114907 (2013); doi: 10.1063/1.4795427 View online: http://dx.doi.org/10.1063/1.4795427 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i11 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 138, 114907 (2013)

Sedimentation equilibrium and the generalized Archimedes’ principleAlberto Parola,1,a) Stefano Buzzaccaro,2 Eleonora Secchi,2 and Roberto Piazza2

1Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy2Politecnico di Milano, Dipartimento CMIC, via Ponzio 34/3, 20133 Milano, Italy

(Received 20 December 2012; accepted 28 February 2013; published online 21 March 2013)

The buoyancy concept is critically re-examined for applications to dispersions of nano-particles,such as colloids, proteins, or macromolecules. It is shown that when the size of the buoyant parti-cle is not too different (say, at most a factor of ten) from the size of the dispersed particles, newintriguing phenomena emerge, leading to the violation of the Archimedes’ principle. The result-ing buoyancy force depends not only on the volume of the particle and on the mass density of thedispersion, but also on the relative size of the particles, on their geometry, and on the interactionsbetween the buoyant particle and the fluid. Explicit expressions for such a generalized Archimedes’principle are obtained and the results are tested against targeted experiments in colloidal dispersions.© 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795427]

I. INTRODUCTION

The understanding of the physical mechanism leading tothe buoyancy force goes back to Archimedes’ works datingmore than 23 centuries.35 Macroscopic objects in molecularfluids obey the so-called Archimedes’ principle both in com-pressible gases and in incompressible liquids, giving rise toa buoyancy force which equals the weight of the fluid “dis-placed” by the object. The sedimentation process governs avariety of phenomena: from the deposition of materials ingeology1 to plankton settling in the oceans,2 to planetesimalaccretion.3 The same concepts also find numerous applica-tions in other frameworks, like the use of centrifuges in thechemical, mining, and nuclear industry4 or protein and nu-cleic acids fractionation in biochemistry.5 However, when thehost molecular fluid is substituted by a colloidal dispersion,for instance, in current separation techniques based on den-sity gradient ultracentrifugation,6 and the size of the floatingbody is not too different from the size of the particles in thesuspension, the whole subject becomes somehow fuzzy andseveral heuristic arguments have been suggested in the liter-ature to correct or integrate Archimedes’ principle, althougha thorough thermodynamic approach is available since a longtime.7 The two most popular viewpoints can be summarizedby saying that buoyancy depends only on the solvent prop-erties or that solvent and colloidal solute together define themass density of the host fluid.8, 9 In fact, to our knowledge,no targeted experiments in controlled systems have so farbeen performed to clarify the issue, although some discrep-ancy between experimental results and expectations based onArchimedes’ principle have been already highlighted.10

We have recently tackled this problem both from the the-oretical and the experimental side showing that, starting froma microscopic perspective, the buoyancy force experienced bya body in solution can be properly defined and calculated insimple models.11 In this paper, we provide a detailed discus-sion of our approach also providing experimental tests of our

a)Electronic mail: alberto.parola@uninsubria.it.

predictions. We aim at determining the buoyancy force actingon a nano-particle (a colloid, a protein, or a macromolecule)floating in a host fluid in the presence of a slowly varying ex-ternal potential. Starting from a microscopic description ofthe system, we derive explicit expressions for the equilib-rium density profile, which can be tested against experiments.We will thoroughly discuss the case of the gravitational field,leading to sedimentation phenomena, but the formalism is de-veloped for a more general situation, for instance, the cen-trifugal potential. The important requirement, which allowsto obtain closed expressions, is the occurrence of density gra-dients on a characteristic range larger than any microscopiclength-scale (particle size, correlation length, etc.). In the spe-cific case of gravity, the scale of density modulations is givenby the gravitational length lg = kT/(mg) which, for the nano-particles considered here, lies in the millimeter range.

We will first consider the case of a one component corre-lated fluid (Sec. II), extending then the results to the moreintriguing case of a binary mixture, where one of the twocomponents (say component “1”) is modeled as an ideal gas,thereby describing the low concentration limit (Sec. III). Sev-eral limiting cases will be discussed and general expressionsfor the density profile and buoyancy force will be obtained.Applications to selected models of fluids, ranging from hardsphere mixtures to charged colloids will be presented inSec. IV. Sedimentation experiments are shown in Sec. V,while in Sec. VI we offer some final perspective on thesubject.

II. ONE COMPONENT FLUIDS

The theory of sedimentation for one component corre-lated fluids is well known.12–14 Here, for completeness, weprovide a derivation particularly suitable for generalizationsto the case of mixtures. We start from the formal expressionof the grand potential (divided by −kT) �[ζ (r)] of a classi-cal inhomogeneous fluid as a functional of the local fugac-ity ζ (r) = �−3 ζ e−β�(r), where � is the thermal wavelength,

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114907-2 Parola et al. J. Chem. Phys. 138, 114907 (2013)

ζ = eβμ is the fugacity, β = (kT)−1 and �(r) is the externalpotential. For a homogeneous fluid, the Gibbs-Duhem equa-tion relates � to the pressure P: �(ζ ) = β P (ζ, T ) V . In thepresence of a slowly varying external field, we can assumethat the isothermal fluid is locally uniform. In such a localdensity approximation14 the grand potential becomes

�[ζ (r)] = β

∫dr P (ζ (r), T ). (1)

From this expression, the local density immediately followsvia functional differentiation:15

n(r) = δ�[ζ (r)]

δ log ζ (r)= β

∂P (ζ, T )

∂ log ζ

∣∣∣ζ=ζ (r)

. (2)

Equation (2) provides the explicit form of the density profilein terms of the equation of state P(ζ , T). An equivalent expres-sion is obtained by multiplying both sides of (2) by ∇log ζ (r)= −β∇�(r),

n(r) ∇�(r) = −∇P (n(r), T ). (3)

This is indeed the familiar hydrostatic equilibrium condition,expressing the balance between pressure gradients and exter-nal forces. In the case of gravity �(r) = m g z and we obtain

∇P (n, T )

n= −m g z, (4)

where z is the unit vector along the (vertical) z axis. Introduc-ing a dimensionless length x = z/lg, with lg = kT/(mg), Eq. (4)becomes

dβP

dx= −n, (5)

which can be easily integrated if the equation of state P(n, T)is known. In the particularly important case of hard spheres,we can use the Carnahan-Starling parametrization of theequation of state,15

βP = n1 + η + η2 − η3

(1 − η)3, (6)

where η = π6 nσ 3 is the packing fraction (or volume fraction).

Direct integration of Eq. (5) gives the sedimentation equilib-rium profile for hard spheres η(x) in implicit form

x = x0 − log η − 1

(1 − η)2− 2

(1 − η)3, (7)

where x0 = log η0 + (1 − η0)−2 + 2 (1 − η0)−3 if η0 is thevolume fraction at the bottom of the cell (x = 0).

III. BINARY MIXTURES

We now examine the case of a binary mixture, wherecomponent 1 is treated as a low-density ideal gas interactingthrough a potential w(r) with component 2, which is insteada generic correlated fluid. External (gravitational) potentials�i(r) act onto both species. The hard core radii of the twospecies of colloids are denoted by R1 and R2, respectively.

A. Heuristic argument

Before providing a rigorous derivation of the equilibriumdensity profile of a binary mixture, it is instructive to antici-pate a heuristic argument leading to the correct result, withoutresorting to subtle technicalities. The argument11 is a gener-alization of the elementary derivation of the buoyancy forceexperienced by an object immersed in a molecular fluid.

Let us first consider a one component fluid of type-2 (host) particles in sedimentation equilibrium. FollowingArchimedes, in this “reference case” the weight of an ar-bitrary portion of fluid must exactly balance the buoyancyforce acting on the same volume. Now we place an addi-tional type-1 (guest) sphere at height z in this inhomoge-neous fluid: The presence of such an intruder acts as a per-turbation on the density profile of the host fluid, leading toa rearrangement of the particles in the neighborhood of theguest sphere. The excess number of type-2 particles aroundthe added sphere, with respect to the reference case, is givenby δN2 = n2(z)

∫dx h12(x), where n2(z) is the local density

of the host fluid and we took advantage from the known rela-tion between the density profile of type-2 particles as seen bya type-1 particle and the pair correlation function in the mix-ture: n2(x) = n2 (1 + h12(x)).15 As a result, the buoyancy forceon a type-1 particle equals the weight of the excess particlesδN2 “displaced” by the type-1 particle,

Fz = −m2 g δN2 = −m2 g n2(z)∫

dx h12(x). (8)

B. General analysis

Starting from the general definition of the grand partitionfunction of a binary mixture, the exact form of the grand freeenergy, in the low-density limit of type-1 particles, is

�[ζ1(r1), ζ2(r2)] = �1[ζ1(r1)] + �2[ζ2(r2)]

+∫

dr1 ζ1(r1)[e�(r1) − 1], (9)

where ζi(ri) = �−3i eβμi−β�i (ri ), �i is the thermal wavelength,

and �i is the grand potential (divided by −kT) of a gas ofparticles of type i. �1 reduces to the ideal gas contribution:�1 = ∫

dr1 ζ 1(r1). Here, �(r1) is the difference between thegrand potential of a fluid of type-2 particles with and withouta single type-1 particle located in r1,

�(r1) = �2[ζ2(r2)e−βw(r1−r2)] − �2[ζ2(r2)]. (10)

In Eq. (9), we just neglected terms quadratic in the fugacityof type-1 particles, which is correct in the low concentrationlimit. The equilibrium density profile of type-1 particles isthen given by

n1(r1) = ζ1(r1)δ�[ζ1(r1), ζ2(r2)]

δζ1(r1)

= ζ1(r1) e�(r1) ∝ e−βm1g z1+�(r1), (11)

where we used the ideal gas nature of type-1 particles. Thisequation formally solves the problem, but we need to evaluatethe function �(r1).

114907-3 Parola et al. J. Chem. Phys. 138, 114907 (2013)

Note that in Eq. (10), the fluid of type-2 particles is sub-ject to two distinct “external” potentials: the gravitational one(�2(r2)) and that induced by the presence of a single type-1 particle in r1 (w(r2 − r1)). These two contributions haveconsiderably different ranges. While the gravitational term�2(r2) leads to a slowly varying density profile, the secondone acts on the scale of the colloid diameter and affects the lo-cal density in a small neighborhood of r1. Following the anal-ysis carried out in Sec. II, it is possible to separate the problemin two steps adopting a local density approximation. First, weevaluate the effect of the local potential w(x) induced by theguest (type-1) particle on a fluid of uniform density n2 (or fu-gacity ζ 2),

�(n2) = �2[ζ2 e−βw(x)] − �2[ζ2]. (12)

Then, we determine the sedimentation profile n2(r2) due tothe presence of the gravitational potential alone, according tothe procedure discussed in Sec. II, i.e., via Eq. (4) substitut-ing n2(z2) into Eq. (12) to obtain �(n2(z2)). Equation (11)together with (4) and (12) formally provide the sedimenta-tion profile of a diluted gas of particles in a correlated solventwhose equation of state is known.

The formal expression of the density profile (11) also al-lows to evaluate the buoyancy force experienced by a guestparticle,

F = kT ∇[�(r1)]. (13)

Notice that, at a microscopic level, this force is due to theinteractions between the type-1 particle and the surroundingtype-2 fluid, which can be written as

F = −∫

dr2 〈N∑

i=1

δ(r2 − xi)〉 ∇r1w(r2 − r1)

= −∫

dr2 n2(r2) ∇r1w(r2 − r1)

=∫

dr2 n2(r2) ∇r2w(r2 − r1). (14)

It is straightforward to check that the expression (13) where�(r1) is given by its definition (10) coincides with this micro-scopic derivation of the buoyancy force (14).

In the special but rather common case of a pure hardsphere interaction w(x), the latter equation can be also writ-ten as the flux of the local pressure tensor of the host fluidP

αβ

2 (x) through the particle surface (see the Appendix). It isnow interesting to investigate few limiting cases where theseexpressions can be further simplified.

C. Continuum limit: Archimedes’ principle

Let us consider the limit of small size ratio q = R2/R1

between type-2 and type-1 particles, and an interspecies po-tential w(r) of pure hard-sphere form. In this case, Eq. (12)provides the excess grand potential of the host fluid due to thepresence of the cavity induced by a single type-1 particle,

�(n2) = −β v1 P2(n2), (15)

where v1 is the cavity volume, i.e., the volume of the guestparticle, and P2(n2) is the pressure in the host fluid at localdensity n2 (and temperature T). The connection with the cel-ebrated Archimedes’ principle is easily seen by substitutinginto Eq. (15) the non-uniform density profile of type-2 par-ticles, implicitly given by (4), and evaluating the buoyancyforce according to Eq. (13),

Fz = kT∂�

∂z= −v1

∂P2

∂z= v1 n2(z) m2 g. (16)

In this limit, the buoyancy force is always directed upwardsand equals the weight of the host molecules contained into thevolume occupied by a single guest particle. In an incompress-ible fluid of mass density ρ2, this force amounts to substitutethe actual mass of the guest particles m1 by their buoyant massm∗

1 = m1 − ρ2v1.

D. Equal size limit

In the opposite limit, where host and guest particles haveidentical interactions (but possibly different masses) the dif-ference in the grand potentials due to a particle insertion canbe calculated exactly (see Ref. 15),

�(r) = ln

[n2(r)

ζ2(r)

], (17)

which, when used into Eq. (11), gives

n1(r) = n2(r)ζ1(r)

ζ2(r)∝ n2(r) eβ(�2(r)−�1(r)), (18)

where, in a sedimentation experiment, �i(z) = mig z. If theguest particles behave as an ideal gas, their density profileis exponential and n1(z) is unaffected by the presence of thehost fluid. In the special case m1 = m2, the system is effec-tively one component and the density profile of type-1 andtype-2 particles coincide. However, the last term provides acorrection in the case of different masses.

E. Buoyancy force and the correlation hole

The general expression of the buoyancy force (13) previ-ously derived can be also written in equivalent forms, shed-ding light on its physical meaning and making contact withthe heuristic derivation of Sec. III A. At first we note that thespatial dependence of �(r) is entirely due to the presence ofa non-uniform density profile n2(r),

F(r) = kT ∇�(n2(r)) = kTd�

dn2

∣∣∣n2=n2(r)

∇n2(r). (19)

From Eq. (2), the gradient of the density profile can be ex-plicitly evaluated in terms of the long wavelength limit of thestructure factor of the host fluid S22(0),

kT ∇n2(r) = −β∂2P2(ζ, T )

∂ log ζ 2

∣∣∣ζ=ζ2(r)

∇�2(r)

= −n2 S22(0)∣∣∣n2=n2(r)

∇�2(r). (20)

114907-4 Parola et al. J. Chem. Phys. 138, 114907 (2013)

While the density derivative of � is

d�

dn2=

[∂�2[ζ2 e−βw(x)]

∂ log ζ2− ∂�2[ζ2]

∂ log ζ2

]∂ log ζ2

∂n2

=∫

dx n2(x) − ∫dx n2

n2 S22(0)

=∫

dr h12(r)

S22(0), (21)

where we used the thermodynamic identities

δ�2[ζ2(x)]

δ log ζ2(x)= n2(x)

∂n2

∂ log ζ2= β

∂2P2(ζ2, T )

∂ log ζ 22

= n2 S22(0)

and we expressed the density profile of type-2 particles asseen by a type-1 particle in terms of the pair correlation func-tion in the mixture: n2(x) = n2 (1 + h12(x)).15 Equation (21)together with Eq. (3) provides a simple expression of thebuoyancy force F acting on the guest particle in r,

Fz = −m2 g n2(z)∫

dx h12(x), (22)

where h12(x) is evaluated at the local density n2(z). This re-sult coincides with that obtained by the heuristic argumentput forward in Eq. (8) of Sec. III A. Note that the integral hasa negative contribution coming from excluded volume effects,but also a positive part close to contact, which might even giverise to a “reversed buoyancy” (pointing downwards) if the at-tractive correlations prevail.

To point out the consequences of this expression for thebuoyancy force, we first examine the case of hard core interac-tions, keeping only the excluded volume effects. The simplestapproximation for h12(r) which takes into account the hard-core repulsion between type-1 and type-2 particles is givenby

h12(x) ={−1 for x < R1 + R2

0 elsewhere. (23)

By substituting this form in the expression (22) for the buoy-ancy force, we obtain

Fz = m2 g n2(z)4π

3(R1 + R2)3 (24)

showing that the force acting on the colloid closely re-minds the known expression of Archimedes’ principle Fz

0= m2 g n2(z) 4π

3 R31 with an important caveat: the “volume

occupied by the guest particle” must be replaced by the “ex-cluded volume for the host particles.” This leads to an en-hanced buoyancy

Fz = Fz0 (1 + q)3, (25)

where q = R2/R1 is the size ratio. While this correction isnegligible for macroscopic objects in molecular fluids whereq � 1, it gives rise to sizable effects even for q � 0.1.

Our general expression depends on the mutual correla-tions between the two species but can be equivalently writtenin terms of purely thermodynamic quantities. Regarding thesystem as a binary mixture where component 1 is extremely

diluted, the Ornstein-Zernike relation allows to express the in-tegral of h12(r) in terms of the integral of the direct correlationfunction c12(r) and the long wavelength limit of the structurefactor of a type-2 (host) one component fluid S22(0),∫

dr h12(r) = S22(0)∫

dr c12(r). (26)

Both terms at right-hand side can be written as thermody-namic derivatives of the Helmholtz free energy of the mixture(divided by −kT) A via the compressibility sum rules15

n2S22(0) = −[∂2(A/V )

∂n22

]−1

∫dr c12(r) = ∂2(A/V )

∂n2∂n1. (27)

In the limit n1 → 0, we can express the free energy derivativesappearing in Eq. (27) in terms of the pressure

βP = A

V− n2

∂(A/V )

∂n2− n1

∂(A/V )

∂n1(28)

leading to

Fz(z) = ∂2(A/V )

∂n1∂n2

[∂2(A/V )

∂n22

]−1

m2 g

=[

∂P

∂n1− kT

] [∂P

∂n2

]−1

m2 g. (29)

In experiments, both type-1 and type-2 particles are usuallysuspended in a molecular liquid of mass density ρ (e.g., wa-ter). In this case, we should rigorously deal with a ternarymixture. However, according to McMillan-Mayer theory ofsolutions, the degrees of freedom of the molecular solventcan be integrated out, giving rise to an additional volume termin the free energy and to effective interparticle interactions.16

Equation (29) remains valid in this case if the total pressureP is substituted by the osmotic pressure of the solution � andthe particle mass m2 by the buoyant mass m∗

2 = m2 − ρv2.The buoyancy force obtained in this way just includes the con-tribution due to the presence of type-2 particles, which mustbe added to the usual term induced by the solvent of massdensity ρ: F0 = ρ v1g.

Remarkably, in both the equivalent expressions for thebuoyancy force (22) and (29) any reference to the sedimenta-tion equilibrium profile of the host particles, originally presentin the formal derivation, disappeared. We may thus wonderwhether our generalization of the Archimedes’ principle hasa more general validity and can be applied also in cases wherethe sedimentation equilibrium had not been fully reached yet.We argue that this is indeed the case, whenever two condi-tions are satisfied: (i) local equilibrium between the guest par-ticle and the host fluid is attained and (ii) viscous stresses inthe host solvent are efficient enough to balance the externalforces, thereby maintaining all the particles floating in the sol-vent in a stationary state. These requirements can be satisfiedin real systems if the corresponding time-scales are faster thanthe sedimentation rates.

114907-5 Parola et al. J. Chem. Phys. 138, 114907 (2013)

F. The density profile

From the general expression of the buoyancy force (22),it is straightforward to obtain �(z) via integration, accord-ing to Eq. (13). By substituting the resulting expression intoEq. (11), we obtain an alternative form for the density profile

n1(z) ∝ e−βm1gz−β m2 g∫ z

0 dz′ n2(z′)∫

dx h12(x). (30)

By employing the excluded volume approximation (23), weget a useful explicit expression

n1(z) ∝ e−βg∫ z

0 dz′ [m1−m2n2(z′) 4π3 (R1+R2)3]. (31)

The term in brackets at the exponential is the difference be-tween the mass of a guest particle and the mass of the samevolume would have if it were occupied by host particles. Be-ing n2(z) a decreasing function of z, we conclude that at smallz (at the bottom of the cell) this quantity is negative, while be-comes positive at the top of the cell. The exponent in Eq. (31)has then a maximum at z* satisfying

m1 = m2 n2(z∗)4π

3(R1 + R2)3, (32)

which defines the equilibrium point. By introducing the massratio s = m2/m1 and recalling the definition of the size ratio q= R2/R1, we can express this result in dimensionless form

η2(z∗) = q3

s (1 + q)3. (33)

A better approximation to Eq. (31) is obtained by expandingthe integrand around the point z* and keeping only the firstorder term. The final result is simply

n1(z) = n1(z∗) e− 1

2 βgm1|n′

2(z∗ )|n2(z∗ ) (z−z∗)2

, (34)

where primes represent spatial derivatives. The colloid den-sity profile is then described by a gaussian centered at z*whose width is determined by the local shape of the sedimen-tation profile of the small particles.

IV. PHYSICAL MODELS

In this section, we examine few models of fluids wherethe expressions previously obtained can be quantitativelyevaluated.

A. Hard sphere mixtures

Here, we will investigate buoyancy phenomena in a hardsphere binary mixture, which is often considered as a modelfor colloidal particles.17 As previously noticed, nanoparticlesare usually suspended in a solvent, which can be modeled asan incompressible continuum, whose effects can be taken intoaccount by replacing the actual colloidal masses with theirbuoyant masses, as shown in Sec. III C.

1. Sedimentation profile

The equation of state of a mixture of hard sphereswith not exceedingly different radii (say 0.1 � q � 10)is accurately represented by the analytical form provided

by Mansoori-Carnahan-Starling18 (MCS). This allows, viaEq. (9), to obtain a closed expression for �(n2) and then toexplicitly evaluate the density profile and the buoyancy force.

The first step is to recognize that the function �(n2) isin fact directly related to the free energy of the hard spheremixture in the limit of low density of type-1 particles. This ismost easily seen by using the asymptotically exact expression(9) for the grand potential in the limit of uniform system

�(ζ1, ζ2) = �2(ζ2) + ζ1 V e�(ζ2) (35)

and performing a Legendre transform

A(N1, N2) = �(ζ1, ζ2) − N1 log ζ1 − N2 log ζ2

= Aid1 (N1) + A2(N2) + N1�(n2) (36)

to lowest order in N1. This result shows that �(n2) can beobtained from the free energy of a homogeneous mixture inthe low density limit of type-1 particles. The excess free en-ergy Aex(N1, N2) for a hard sphere mixture is accurately rep-resented by

Aex(N1, N2)

(N1 + N2)

=[3

2(1 − y1 + y2 + y3) − 3y2 + 2y3

(1 − η)

− 3 − 3y1 − 3y2 − y3

2(1 − η)2− (y3 − 1) log(1 − η)

], (37)

where η = η1 + η2 and yi are defined in the originalreference.18 Taking the limit η1 → 0 in this expression wecan easily identify �,

�(η2) = q−3[3(1 − q2)

1 − η2+ 2 − 3q

(1 − η2)2− 2

(1 − η2)3

+ (1 − q)2(2 + q) log(1 − η2)]. (38)

In the limit q → 1, this expression coincides with Eq. (17)when z is expressed in terms of the density profile (7). In theother known limit q → 0 we expect Archimedes’ law to hold,considering the host molecules as a solvent. Thus, we foreseeto recover the result (15) with βP(n2) given by Eq. (6),

�0(η2) = −q−3 η2 + η22 + η3

2 − η42

(1 − η2)3. (39)

Instead, we find

�0(η2) = q−3[ 3η2

1 − η2+ 2

(1 − η2)2

− 2

(1 − η2)3+ 2 log(1 − η2)

](40)

besides an irrelevant additive constant. The discrepancy be-tween these formal expressions must be ascribed to the ap-proximate nature of the MCS equation of state. However, thedifference between the two results is always smaller than 1%in the whole density range, as shown in Fig. 1.

Inserting the general result (38) into Eq. (11) and express-ing the particle coordinate z in terms of the density profile ofthe host fluid (7), we obtain our final formula for the density

114907-6 Parola et al. J. Chem. Phys. 138, 114907 (2013)

FIG. 1. (Upper panel) logarithmic plot of �0(η) as obtained in Eqs. (39)and (40). The two curves are superimposed. The difference between the twoexpressions is plotted in the lower panel.

profile of the guest particles

log n1(η2(x)) = s−1[

log η2 + 1

(1 − η2)2+ 2

(1 − η2)3

]+�(η2) + const., (41)

which depends on the two-dimensionless parameters s= m2/m1 and q = R2/R1. This expression has always a max-imum whenever s > q3, i.e., whenever the mass density ofthe host particles ρ2 is larger than the mass density of theguest particle ρ1. Defining their ratio as ρ12 = ρ1/ρ2 = q3/s, inFig. 2 we plot the host particles volume fraction correspond-ing to the position where the guest particle distribution is cen-tered. The curves are all monotonically decreasing and forsmall q they tend to the limiting value (33) as expected. Thewidth of the density distribution is also shown by dashed lines.

FIG. 2. Equilibrium volume fraction of host particles denoting the pointwhere the guest particle distribution is peaked, as a function of the size ratioq for different mass density ratios ρ21. The dashed lines bound the standarddeviation of the guest particles density distribution.

2. Buoyancy force

From Eq. (19), we can immediately read off the contribu-tion to the buoyancy force Fz acting on the guest particle, dueto the presence of the host fluid

Fz = kTd�(z)

dz= m2g

d�(x)

dx= m2g

d�(η2)

dη2

dη2

dx, (42)

where in the second equality we introduced the dimensionlessunits and in the third we realized that � depends on x onlyvia the local density of host particles, whose spatial profileis given in Eq. (7). According to Archimedes’ law, we wouldexpect that the buoyancy force is given by

Fz = 4

3π R3

1ρ∗g, (43)

where ρ* is the effective mass density of the surroundingmedium (minus the solvent mass density ρ). If the medium isidentified as the whole suspension of host particles, we wouldobtain ρ* = m2n2. Therefore, it is convenient to evaluate theratio between the effective mass density and the actual massdensity of the solution

ρ∗

m2n2= −q3 d�

dη2

(1 − η2)4

(1 − η2)4 + η2(8 − 2η2), (44)

where use has been made of the definitions of ρ* and Fz, asgiven by Eqs. (43) and (42). Employing the limiting expres-sion (39) for �(η2), we can check that this ratio tends to unityfor q → 0. Analogously, in the low density limit for the hostparticles (η2 → 0), the ratio tends to

ρ∗

m2n2→ (1 + q)3, (45)

where use has been made of the general result of Eq. (38).This expression agrees with the simplified form (25), whereonly excluded volume effects are taken into account, as canbe verified by substituting this result into Eq. (43).

The “exact” ratio for few values of q as given by the MCSequation of state is shown in Fig. 3 as a function of the volumefraction of the host particles.

The buoyancy force acting on a guest particle can be de-picted by plotting the ratio ρ*/ρ2 = Fzq3/(m2g) as a functionof η2 as shown in Fig. 4. The dependence of buoyancy on thelocal density of the “solvent” becomes non-monotonic if thehost particles are larger than the guest particle, i.e., if q > 1as shown in Fig. 5. An optimal volume fraction appears wherebuoyancy is strongly enhanced. It is instructive to examine theformal limit q → ∞ of Eq. (44),

Fz

m2g−→

q→∞η2 (1 − η2)3

(1 − η2)4 + η2 (8 − 2η2)(46)

showing that the buoyancy force on a (small) particle does notdepend on the size of the particle but only on the mass of the(large) surrounding bodies and displays a maximum at η2 ∼0.1546, where

Fz ∼ 0.05495 m2g. (47)

114907-7 Parola et al. J. Chem. Phys. 138, 114907 (2013)

FIG. 3. Density of the effective medium defining the particle buoyancy asa function of the local density of the host particle for few size ratios. Thedashed line marks the prediction of Archimedes’ principle, recovered in theq → 0 limit.

FIG. 4. Effective mass density of the surrounding medium, relative to themass density of host particles as a function of η2 for few q ≤ 1.

FIG. 5. Same as Fig. 4 for q > 1.

FIG. 6. Buoyancy force acting on a small particle floating on larger spheres.

The full dependence of the buoyancy force on the local vol-ume fraction of the fluid, in the small particle limit, is shownin Fig. 6. To better understand the physical meaning of the re-sults shown in Fig. 5, let us consider a solution of hard spherenanoparticles in sedimentation equilibrium characterized bya packing fraction profile η2(z). We now insert an additionalsmall (dense) hard sphere colloid with size ratio q and massdensity ρ1. The intersections of the curve shown in Fig. 5 withthe line ρ* = ρ1 defines the equilibrium densities where thegravitational force is balanced by the buoyancy force. If themass density of the guest particle is too large no intersec-tion is present: the gravitational force prevails and the par-ticle falls down. For a sufficiently small mass of the addedparticle, two equilibrium positions are possible: the large η2

intersection (η*) corresponds to an unstable solution, becausebuoyancy decreases if the particle is pushed to larger pack-ing fractions, i.e., to the bottom of the cell. Conversely, thelow η2 solution (η◦) is stable, and we expect that the colloidwill perform Brownian motion centered at the correspondingheight z◦. We conclude that placing a uniform distribution oftype-1 colloids in a solution of type-2 particles in sedimen-tation equilibrium, the fraction of colloids initially located inan environment with η2 > η* falls to the bottom of the cell,while the rest of the colloids eventually settle to an equilib-rium distribution centered at the height z◦ corresponding toη2(z◦) = η◦.

The equilibrium sedimentation profile of the guest col-loids when their radius is much smaller than that of the hostparticles can be readily evaluated within our approach. Bysubstituting �(η2) from Eq. (38) in the general expression(11) and taking q → ∞ we obtain

n1(z) ∝ [1 − η2(z)] e−βm1gz. (48)

This form leads to a simple physical interpretation: the smallcolloids settle according to the barometric law in the availablevolume not occupied by the (large) host particles.

B. Platelets and needles

The analysis of excluded volume effects, performed inSec. III F, shows that, for spherical particles, the size of the

114907-8 Parola et al. J. Chem. Phys. 138, 114907 (2013)

guest colloid strongly affects its buoyancy, contrary to the ex-pectations based on Archimedes’ principle. On this basis, wemay wonder whether buoyancy also depends on the geometryof the particles. As an illustration, it is instructive to evaluate,in the simplest approximation, the buoyancy of a disk and ofa rod in a fluid of spheres. Physical systems mimicking thesereduced dimension geometries have already been synthesizedand studied in other frameworks.19, 20 Sedimentation exper-iments in gibbsite platelets/spherical colloids mixtures havebeen recently performed and interpreted on the basis of den-sity functional theory.21 According to the excluded volumeapproximation, we just evaluate the volume V ex inaccessibleto the host spherical particles due to the presence of a disk ofradius R1 and vanishing thickness (and then vanishing mass).A simple geometrical calculation, together with Eq. (22) givesthe result

V exdisk = 4π

3

[R3

2 + 3π

4R1R

22 + 3

2R2

1R2

](49)

leading to a buoyancy force

Fzdisk = m2gη2

[1 + 3π

4q−1 + 3

2q−2

], (50)

where q = R2/R1. Analogously, the presence of a one-dimensional rod of length R1 in a fluid of spheres reduces theavailable volume by V ex

rod = 4π3

[R3

2 + 34R1R

22

]leading to

Fzrod = m2gη2

[1 + 3

4q−1

]. (51)

Note, however, that the reduced thickness of disk or rod-shaped colloids implies that their mass may be extremelysmall, emphasizing the excluded volume effects in the cal-culation of the buoyancy force. While, on the basis ofArchimedes’ principle, particles of reduced dimensionalityshould feel essentially no buoyancy, according to the Gen-eralized Archimedes’ principle (GAP), in these systems, theforce on disks and rods in a fluid of hard sphere should be sig-nificantly larger. As an example, let us consider LaponiteTM,a synthetic clay disk-like particles with R1 = 15 nm: accord-ing to our analysis, the buoyancy force felt by such a parti-cle in a water solution of spherical globular polymers, likedextran (molecular weight Mw = 40 kDa and radius of gy-ration R2 = 6 nm), at volume fraction η2 = 0.1 is abut 10%larger than in pure water. The effects are more striking for rod-like shape molecules. For instance, if tobacco mosaic virus(TMV), which is shaped as a rigid rod of length R1 = 300 nm,is dispersed in a water suspension of poly-methylmethacrylate(PMMA) particles with radius R2 = 50 nm and densityρ2 = 1.19 g cm−3, the buoyancy force equals the weight ofthe particle at a tiny volume fraction of the host fluid, η2

= 0.06.

C. Particles in electrolytes

An electrolyte can be modeled as a mixture of positiveand negative ions uniformly distributed in a solvent. If anexternal colloid is inserted in such a solution, the buoyancyforce is simply given by Archimedes’ principle, due to thelarge size difference between the colloid and the molecules

of the electrolyte. The sedimentation equilibrium of a suspen-sion of charged colloids in a solution of point-like anions andcations has been theoretically investigated.22 Here, we justdiscuss the single particle limit taking into account the massof all the charged particles. If the colloid has a charge Q = Ze,a screening cloud quickly develops on the scale of the Debyelength of the electrolyte. This region is characterized by anexcess of ions of opposite charge and a depletion of ions ofequal charge. In the simplest Debye-Hückel approximation,the total number of ± ions displaced in a neighborhood of thecolloid is equal to |Z|/2 for both ionic charges in a monova-lent electrolyte. However, the masses m± of the two ions aregenerally different, leading to a different buoyancy force forpositive or negatively charged colloids. The excess buoyancy,besides that obtained by use of Archimedes’ principle, is justgiven by the elementary formula, which follows from chargeconservation

Fz = Z

2(m+ − m−)g. (52)

Note that the sign of the excess buoyancy depends of thecharge Z of the particle, leading to an enhanced buoyancywhen the heavier ions are depleted in the screening cloud.This effect, clearly negligible for macroscopic objects, islikely to be small too for moderately charged macromoleculeslike proteins. Consider for instance lysozyme, a globular pro-tein with molecular weight Mw = 14.7 kDa, a molecular vol-ume of about 3 × 10−20 cm3, and a density in solution ofabout 1 − 37 g cm3. In sodium acetate buffer at pH � 4.7),lysozyme bears a net charge Z � 10. Since in this case (m+− m−) = 6 × 10−23 g, Eq. (52) yields an additional contribu-tion to the buoyancy which is less than 1% of its bare value.However, the situation noticeably changes if we consider ahighly charged nanoparticles such as a sodium dodecyl sul-fate (SDS) micelle. SDS micelles have an aggregation number(thus a bare charge) which is about 100, and a size comparableto lysozyme. Hence, in the presence of a salt such as sodiumacetate, the effect would be an order of magnitude larger. Ifwe also take into account that SDS has a density which ismuch closer to the density of water than lysozyme, the neteffect would be even more pronounced.

D. Yukawa mixtures

The explicit expression of the buoyancy force (22) canbe evaluated in simple models of fluid mixtures by taking theη1 → 0 limit of the Helmholtz free energy, as already dis-cussed for the case of hard spheres in Sec. IV A. As an ex-ample, we consider a hard core Yukawa mixture where theparticles of the fluid are characterized by purely repulsiveinteractions

v22(r) ={

+∞ for r < σ22

ε2e−z2(r−σ22)

r/σ22elsewhere

. (53)

Here σ 22 = 2 R2 is the hard sphere diameter of the fluid parti-cle and z2 is the range of the Yukawa potential. The amplitudeof the potential is parameterized by ε2 > 0. The added colloidinteracts with the host particles by a similar Yukawa potential,

114907-9 Parola et al. J. Chem. Phys. 138, 114907 (2013)

with different parameters

w(r) ={

+∞ for r < σ12

ε e−z(r−σ12)

r/σ12elsewhere

, (54)

where now σ 12 = R1 + R2 is the minimum distance betweenguest and host particles, z is the range and ε is the amplitudeof the interaction, which may be either positive (repulsive) ornegative (attractive).

This simple model is usually adopted to represent adispersion of charged colloids in an electrolyte: within theDebye-Hückel approximation, screened Coulomb interac-tions can in fact be described by Yukawa potentials. In thiscase, the range of both interactions are given by the inverseDebye length κD,

z2 = z = κD =√

8πlBNAI, (55)

where lB is the Bjerrum length, NA is the Avogadro number,and I is the ionic strength, i.e., the molar charge concentrationin the solution. The amplitudes ε2 and ε depend on the col-loidal charges but tend to a well-defined value in the limit ofhigh structural charges23

ε2 = 8 kTR2

lB,

ε = 16 kTR1 R2

lB (R1 + R2). (56)

By use of the Hypernetted Chain integral equation15 we haveevaluated the equation of state and the mutual correlationfunction h12(r) for this mixture. Inserting the results intoEqs. (5) and (22), the density profile of the fluid and thebuoyancy of the added type-1 particle have been calculated.The sedimentation profile turns out to be very similar to thatof hard spheres, provided the volume fraction η2 = π

6 σ 32 is

defined in terms of an effective hard sphere radius σ ∗2 , ac-

cording to the Barker-Henderson prescription.15 The buoy-ancy force, expressed in terms of the effective mass densityof the medium ρ* as defined in Eq. (43) is shown in Fig. 7.Buoyancy is strongly reduced by increasing the range of therepulsive interaction. This effect is enhanced if the size of theguest particle is increased.

We also investigated the effects of attractive interactionson the buoyancy properties of a particle in a hard sphere fluid.Within the hard core Yukawa model, we chose ε2 = 0 andz = 2/σ 22 examining the dependence of buoyancy upon thevalue of ε/kT < 0. The results are shown in Fig. 8: attractiveinteractions oppose buoyancy, possibly leading to reversedbuoyancy, i.e., a “buoyancy” force pointing downwards, asshown for strong attraction |ε| � kT at low densities of thehost fluid. This phenomenon is due to the growth of the peakin the mutual correlation function h12(r) which overcomes thenegative contribution coming from excluded volume effectsin the evaluation of the total buoyancy, according to the GAPequation (22).

V. EXPERIMENTAL

In this section, we collect together and discuss in moredetail the set of results, obtained for colloidal systems behav-

FIG. 7. Effective mass density of the surrounding medium, relative to themass density of host particles as a function of η2 for particle radii appropri-ate for describing a PMMA particle in a fluid of MFA colloids R1 = 220nm and R2 = 90 nm.11 Different curves correspond to different inverse De-bye lengths κD (in units of R2). The upper curve, labeled HS, shows thehard sphere result. The effective hard sphere diameter used for the normaliza-tion of the volume fraction is provided by the Barker-Henderson prescription:σ ∗

22/σ22 = 24.28, 6.83, 4.14, 2.67 for κD = z = 0.2, 1, 2, 4, respectively.

ing to a good approximation as binary hard-sphere mixtures,we already presented in our first communication,11 togetherwith additional measurements on similar systems.24 In addi-tion, we present new results that further support the generalapproach to buoyancy we have developed in non-equilibriumconditions too. The general strategy we have adopted to testEq. (22) and its consequences discussed in Sec. IV A can besummarized as follows. First of all, we focused on equilib-rium sedimentation measurement. In principle, valuable infor-mation can be extracted by measuring the steady-state settlingvelocity of the guest particle, which is experimentally mucheasier. Yet, although small particles settling in a creeping flow

FIG. 8. Effective mass density of the surrounding medium, relative to themass density of host particles as a function of η2 for particle radii R1 = 220nm and R2 = 90 nm and attractive interactions. Different curves correspondto different interaction strengths. The upper curve is the reference hard sphereresults, the lower curves refer to ε/kT = −0.4, − 0.6, − 0.8, − 1, respectively.Below the dashed line interactions lead to reversed buoyancy.

114907-10 Parola et al. J. Chem. Phys. 138, 114907 (2013)

regime are still expected to be subjected to the buoyant forcegiven by Eq. (22), evaluating their friction coefficient is amuch harder task (note, in particular, that the host particlesare settling too, in general with a different speed). Equilib-rium measurements, although much more time-consuming,conversely allow for a quantitative test of the theoretical pre-dictions without any assumptions concerning hydrodynamiceffects. A typical experiment consists then in adding to a sus-pension of host (type-2) particles, made of a material withdensity ρ2, a small amount of guest (type-1) particles withmaterial density ρ1. Notice that, because Eq. (22) holds onlyprovided that the self-interactions between the guest particlescan be neglected, the latter should be very diluted: At finiteconcentration, the interactions between guest particles mayinduce phase transitions, as shown in Ref. 25. The mixtureis then let settle until equilibrium is reached. At equilibrium,the host particles distribute according to a concentration pro-file η2(z) that decreases monotonically from the bottom of thecell (z = 0) upwards, and which for ideal hard spheres wouldbe given by Eq. (7). The guest particles are then expected toconcentrate in a thin layer around the vertical position wherethe buoyant force they feel equates their weight. Accordingto the standard Archimedes’ principle, this layer would thenfloat around the so-called “isopycnic level,” namely, the levelz* where the local density ηiso

2 of the suspension of particles2 is equal to ρ1,

ρ + ρ2[1 − ηiso2 ] = ρ1 =⇒ ηiso

2 = ρ1 − ρ

ρ2 − ρ, (57)

where ρ is the density of the bare solvent. The prediction ob-tained by the GAP can be noticeably different. For instance,even in the simple excluded-volume approximation, Eq. (33)can be simply recast to yield

η2(z∗) = ηiso2

(1 + q)3, (58)

which, even for a size ration as small as q = 0.1, is about 30%smaller than ηiso

2 .This kind of experiment requires the equilibrium sedi-

mentation profiles of both kinds of particles to be indepen-dently measured: this means that, to set apart one kind of par-ticles from the other, the two concentration profiles shouldbe obtained from different physical properties. To avoid per-turbing the delicate sedimentation equilibrium, measurementsmust be strictly non-invasive, hence optical methods are themost natural choice. To be direct probes of the local particleconcentration, the selected optical properties should possiblybe weakly dependent on interparticle interactions, which forthe host suspension are in general not negligible. For instance,extracting the concentration profile from standard measure-ments of the scattered intensity, which depends both on parti-cle concentration and on the local structure factor of the sus-pension, would be far from being trivial. At the same time,the optical property used to extract the concentration profileof the guest particles should be very sensitive, because of thenegligible concentration of the latter.

A. A big particle in a “sea” of smaller ones

We first discuss the results obtained for a very dilute sys-tem of hard spherical colloids dispersed in a suspension ofsmaller spheres. This system presents a major advantage: asformerly noticed, provided that the size ratio q is sufficientlysmall, the simple excluded volume approximation is reason-ably accurate even when the suspension of host particles ismoderately concentrated. Hence, a straightforward quantita-tive comparison with the theoretical predictions can be made,essentially on the basis of geometrical parameters. Then, wewill show how a full theoretical calculation improves theseapproximate results.

1. Colloidal system and experimental methods

As guest particles we have used monodisperse spheri-cal (PMMA, density ρ1 = 1.19 g cm−3) particles (microParti-cles GmbH, Berlin), with four different sizes (R1 = 220, 300,400, and 500 nm). Because of their high refractive index (n1

= 1.49), PMMA particles in this size range scatter light verystrongly and can be easily detected, as we shall see, even bysimple transmittance measurements.

Whereas PMMA particles are just standard latex colloids,the host particles, made of MFA, a copolymer of tetrafluo-roethylene (TFE) and perfluoro-methylvinylether (PF-MVE),were selected because of their rather peculiar optical prop-erties. The addition of PF-MVE to TFE, bestowing a largerflexibility to the polymer chains, yields spherical, monodis-perse particles, which can be envisioned as an assembly ofcrystallites (closely resembling those formed by the tetraflu-oroethylene homopolymer, PTFE), embedded into an amor-phous matrix.26 This partial crystallinity yields a residual op-tical anisotropy of the particles (i.e., MFA particles display aresidual birefringence), and consequently a noticeable depo-larized contribution IV H

27 to the scattered light, which nei-ther depends on the refractive index of the solvent, nor oninterparticle interaction.26 By contrast, the polarized compo-nent can be written as IV V = Icoh + (4/3)IV H , where the co-herent contribution Icoh is proportional, as in the usual caseof isotropic particles, to the suspension structure factor. Thelatter can be fully suppressed by adding to water a moder-ate amount of a simple solute such as urea or glycerol, so tomatch the average particle refractive index n = 1.352. Thisnoticeably reduces the total scattering cross section, allowingto study very concentrated dispersions with negligible multi-ple scattering effects. Because IV H is fully incoherent (actu-ally, it fully parallels the signal obtained in incoherent neutronscattering), it is strictly proportional to the particle numberdensity in the scattering volume, and is therefore an accu-rate probe of the concentration profile. A further advantageof MFA for the present study is its high material density ρ2

= 2.14 g/cm3. Since the pre-factor in Eq. (22) is propor-tional to the buoyant mass of the host particles, this fea-ture allows indeed to emphasize deviations from the standardArchimedes’ law.

The specific MFA particle we used have an average ra-dius of 90 nm and a polydispersity of less than 4%. Theoriginal sample batch, kindly gifted us by Solvay Specialty

114907-11 Parola et al. J. Chem. Phys. 138, 114907 (2013)

Polymers (Bollate, Italy), was extensively dialyzed, and itsparticle volume fraction calibrated by density measurements.Colloidal mixtures were prepared by adding a minute quantity(η1 = 10−6 − 10−5) of PMMA particles to MFA suspen-sions at volume fraction η0

2 = 0.03 in aqueous suspensionsof urea at a weight fraction of 15% (mass density ρ = 1.04,viscosity 1.07 cP), which ensures very good index match-ing of the MFA colloid. In this solvent, the MFA particleshave a sedimentation length lg2 = kBT/mg � 130 μm and aStokes settling velocity vS � 1.5 mm/day. Hence, to reachsedimentation equilibrium starting from a sample height ofh = 2 cm, MFA particles require typically a few Stokes timestS = h/vS � 14 days. The sedimentation lengths for PMMAparticles increase from lg1 � 5 μm for R1 = 500 nm up to lg1

� 60 μm for R1 = 220 nm. The corresponding Stokes settlingvelocities decrease from vS � 6 mm/day for R1 = 500 nm tovS � 1.2 mm/day for R1 = 220 nm. Hence, whereas in the lat-ter case equilibration takes place on a very similar time scalefor PMMA and MFA, the largest PMMA particles settle morethan four times faster than MFA. Nonetheless, as we shallsee, at equilibrium they float in the upper part of the MFAsediment. We finally point out that both PMMA and MFAparticles bear a weak negative surface charge. Hence, besidespure hard sphere interactions, moderate repulsive electrostaticcontributions to the self and mutual interactions are actuallypresent. However, since urea display a little dissociation inthe acidic conditions provided by the particle counterions,the Debye-Hückel screening length does not exceed a fewnanometers. Hence, the PMMA + MFA can be safely re-garded as a mixture of effective hard spheres, provided thatwe allow for a slightly larger effective size (see below).

The equilibrium sedimentation profiles of MFA particleswere obtained by depolarized light scattering measurements.The experimental apparatus basically consists of a custom-made light scattering setup,28 operating at a fixed scatteringangle ϑ = 90◦. Selection of the incident and detected polar-ization of the scattered intensity is made by means of twoGlan-Thomson polarizers (Bernhard Halle, Berlin) with anextinction ratio better than 1 × 10−6. The cell is mounted on aDC-motorized micrometric translator (Physik Instrumente M-150.11) allowing cell positioning with a resolution of 0.1 μmand an absolute accuracy and bi-directional repeatability ofabout 3 m. A 17 mW incident He–Ne laser beam is mildly fo-cused in the cell to a spot size w � 45 μm, corresponding toa Rayleigh range of about 10 mm (fixing the maximum usefuloptical path in the cell). A telescope system, embodied in thephotodetection systems, allows imaging the sample and lim-iting the scattering volume using horizontal and vertical slits.The whole setup is enclosed into a removable hood temper-ature controlled to better than 0.5 ◦C. Sedimentation profilesare obtained by scanning the beam through the profile andmeasuring the depolarized scattering intensity as a functionof the vertical distance from the cell bottom, acquiring thedata with a typical scan speed of 5–20 μm/s. Cell translationand data acquisition are controlled by a 16 bit A/D interfaceboard using custom-made software.

Due to their extremely low concentration, the PMMAparticles are barely detectable in the initially homogeneousmixture, their weak scattering being masked by the incoher-

ent scattering of MFA. However, as we mentioned, while sed-imentation proceeds they progressively concentrate in a thinlayer, where they became clearly visible as a whitish band.Nonetheless, even within the layer, the PMMA peak concen-tration does not exceed a volume fraction η1 � 10−4. In theseconditions, interaction between the PMMA particles are stillnegligible, so that their concentration profile could be in prin-ciple obtained by standard (polarized) light scattering. How-ever, even at such low concentration, they scatter so muchthat, in the layer region, light extinction can reach up to 40%–50% of the incident beam intensity. For these turbidity valuesintensity, multiple scattering effects are not negligible, so thatthe scattered intensity at a given angle is not anymore pro-portional to particle concentration, even for a dilute system.Multiple scattering effects have, however, no influence on theamount of transmitted intensity, which is still related to the to-tal single scattering cross section (once a photon is scatteredout of the beam, it does not matter if it undergoes further scat-tering events). The distribution of the guest particles can thenbe simply obtained by monitoring the intensity of the trans-mitted beam, collected by a small area photodiode placed farfrom the cell,29 while scanning through the profile. The par-ticle distribution in the accumulating layers can then be nor-malized by calibration to the total amount of guest particlesadded to the initially homogeneous mixture. Since the layer israther thin, care was taken of de-convoluting the signal fromthe gaussian profile, with a beam waist w0 = 45 μm at the cellcenter, of the incident laser beam. Even for the thinnest layer,corresponding to the PMMA particles with R1 = 500 nm, thiscorrection was anyway found to be very small.

2. Particle distributions and effective buoyancy

The equilibrium sedimentation profile, obtained by de-polarized light scattering, of a suspension containing onlyMFA particles at the same initial volume fraction η0

2= 0.03 used for the mixtures with added PMMA is shown inFig. 9. Describing a suspension of charged particles as an ef-fective hard sphere system is known to be reasonably accu-rate, provided that the screening length λDH is much smallerthan the particle size. The profile is indeed very well fitted byEq. (7), provided that a slightly larger effective particle radiusR∗

2 = 1.09R2 � 98 nm is assumed.30 The radius of the guestparticles (PMMA) has also been consistently increased by thesame amount (8 nm) to take care of the screening cloud. Theconcentration profiles for the 4 samples prepared with the ad-dition of PMMA particles of the four investigated sizes showalmost indistinguishable equilibrium profiles, if the exceptionis made of the thin regions, with a thickness varying in therange 100 − 200 μm (see below) where the PMMA particlesconcentrate. The inset in Fig. 9 shows an expanded view of theregion where guest particle accumulate, re-plotted as a func-tion of the local suspension mass density ρs(z) = ρ + (ρ2

− ρ)η2(z).The full dot shows, in both panels, the position of the

isopycnic point, namely, the level where the local suspensiondensity is about 1.19 g cm−3 (corresponding to ηiso

2 = 0.136)where, according to the simple Archimedes’ law, the guest

114907-12 Parola et al. J. Chem. Phys. 138, 114907 (2013)

FIG. 9. Equilibrium sedimentation profile, as a function of the distance zfrom the cell bottom, of a suspension of MFA particles, originated from aninitially uniform suspension at volume fraction η0

2 = 0.03, filling a rectangu-lar cuvette up to an initial height h = 18 mm. The full line is the theoreticalprofile corresponding to the Carnahan-Starling e.o.s. for HS, calculated for aneffective particle size R∗

2 = 98 nm. The inset shows an expanded view of thecorresponding local density profile in the region within the dashed box. Notethan, in the experimentally interesting region, the density profile is approxi-mately linear in z. On both panels, the average position of the PMMA layers,together with the isopycnic point expected from Archimedes’ principle, areshown by full symbols specified in the legend.

particles should accumulate. Yet, as evident from the fig-ure, all layers lie around positions z* well above this level,the more the larger the size ratio q. The analysis of thetransmittance profile from the sample allow for a detailedreconstruction of the full distribution of the guest particles.Since in the region where the guest particle accumulate, thevolume fraction η2(z) of the host particles, and therefore thelocal suspension density, varies almost linearly with z (seeFig. 9), this distribution is expected to have approximately agaussian shape (in other words, Eq. (34) is expected to holdin a rather wide range of z). Because the experimental “effec-tive” isopycnic points correspond to rather dilute suspensionsof the host particles (η2(z*) � 0.09), and since for all the in-vestigated samples the size ratio is moderately small, we mayexpect the simple excluded volume model to hold. In terms ofexperimental quantities, the effective isopycnic concentrationof the host particles is then expected to be given by Eq. (33),

η2(z∗) = ηiso2

(1 + q)3= ρ1 − ρ

ρ2 − ρ

1

(1 + q)3. (59)

The standard deviation of the gaussian distribution can be in-ferred from Eq. (34),

σ =√

η2(z∗)lg1

∣∣∣∣dη2

dz

∣∣∣∣−1

z=z∗. (60)

Note that, when the host particles are very dilute, so that theyshow a barometric sedimentation profile η2(z) = η2(z*)exp[−(z − z*)/lg2], the gaussian width simply becomes the har-

TABLE I. Experimental (e) values for the effective isopycnic volume frac-tions η2(z*), local density of the host particle suspension ρs(z*) (in g cm−3),and standard deviation σ of the gaussian fits to the PMMA profiles, com-pared to the theoretical values (t) obtained according to the exact expressionfor hard spheres, Eq. (41). The volume fraction corresponding to the maxi-mum density of guest particles in the excluded volume approximation (ex) isalso given. All lengths are given in microns.

R1 q �g1 ηt2(z∗) ηex

2 (z∗) ηe2(z∗) ρe

s (z∗) σ t σ e

A 0.22 0.43 63 0.049 0.042 0.052 1.098 123 113B 0.30 0.32 25 0.065 0.055 0.072 1.120 84 80C 0.40 0.24 10 0.079 0.067 0.083 1.131 58 58D 0.50 0.19 5 0.088 0.077 0.086 1.135 43 54

monic mean of the sedimentation lengths of the two species

σ = (lg1lg2)1/2 = lg1

√ρ1 − ρ

q3(ρ2 − ρ). (61)

The experimental results, presented in Table I, show thatboth the position, in terms of volume fraction or density, andthe width of the guest particle distributions are in rather goodagreement with the predictions of the full hard sphere model,according to Eq. (41). Indeed, the excluded volume approxi-mation provides a simple, although less accurate, estimate ofthe position of the maximum of the guest particle distribution.

As the results show, at fixed R2, the width of the gaussianincreases with q. Note also that σ crucially depends on theslope dη2/dz of the concentration profile of the host suspen-sion at the effective isopycnic level. To appreciate the size-selectivity of the host medium we have chosen, we plot inFig. 10 the normalized probability densities for the guest par-ticle distribution as a function of the local suspension densityρs(z).31 The plot shows that the distribution corresponding toguest particles differing in size by of a factor of two wouldbe barely set apart. In terms of size-fractionation efficiency,the model system we choose performs rather poorly, basically

FIG. 10. Normalized probability densities for the position of the guest par-ticles for the investigated samples, compared to the theoretical fits. (Inset)Normalized probability densities for the position of the guest particles forsamples A, C, and for a 1 : 1 mixture of them. The broken line is obtained as[pA(η2) + pC(η2)]/2.

114907-13 Parola et al. J. Chem. Phys. 138, 114907 (2013)

because the MFA particles setting the overall scale of the den-sity gradient, have a small value of lg2. A practical separationtechnique would then require to generate the gradient usingmuch smaller particles (large lg2) with larger values of ρ2

− ρ. To avoid an extremely long equilibration time, experi-ments should however be performed in a centrifuge. Nonethe-less, the layering we observe does not amount to a genericsegregation of the guest species, but is really size-dependent.To confirm this, we have prepared a ternary mixture by addingto the MFA suspension both 220 nm and 400 nm PMMA par-ticles, in the same amount contained in samples A and C,respectively. The inset of Fig. 10 shows the normalized dis-tribution pA + C(η2) of the PMMA particles obtained for theA+C sample basically coincides with the average of pA(η2)and pC(η2).

3. Dynamic buoyancy

As we have discussed, we expect that the expression forthe generalized buoyancy is valid not only in equilibrium con-ditions, but also during settling, provided that the times ittakes for the local distribution of the host suspension aroundthe guest particle to form, which is of the order of the dif-fusion time of the host particles over the range of the paircorrelation function h12, is sufficiently short compared to thetime it takes for the latter to settle over the same lengthscale.This requirement is surely satisfied for the mixtures we haveconsidered. Hence, we may expect the guest particle layer tobuild up during settling around the position corresponding, inthe kinetic settling profile of MFA, to the equilibrium valueof the effective isopycnic point, retaining the same position interms of the local suspension density, while the MFA profileevolves in time. This is fully confirmed by the kinetic settlingdata for sample C shown in Fig. 11, where the layer position isseen to follow rigidly the settling MFA profile, keeping fixedaround the position where η2 � η2(z*).

FIG. 11. Time-evolution of the concentration profile of the MFA particlesfor sample C, expressed in units of the Stokes settling time tS = h/vS . Theinset, where the boxed area is enlarged, shows that the average position, interms of local MFA volume fraction, of the PMMA layer (full squares) isbasically time-independent and coincides with the equilibrium value η2(z*)(broken line).

B. A small particle in a sea of bigger ones

As discussed in Sec. IV A 2, however, more striking ef-fective buoyancy effects are predicted when the guest particlesettles in a suspension of larger particles (q > 1). For suffi-ciently high concentration of the latter, the effective buoyantforce becomes indeed non-monotonic, reaching a peak valuethat can be a sizable fraction of the weight of a host parti-cle (see Fig. 6). Even if the latter has a material density ρ1,larger than the density ρ2 of type-2 particles, the total forceon particle 1 can be directed upwards, so that a noticeablefraction of the guest particles is pushed towards the top of asuspension whose average density can be consistently smallerthan ρ1. The physical interpretation of this curious effect, dis-cussed in detail in Sec. IV A 2, is rather simple, being di-rectly related to the wide range of the perturbation in thehost fluid (namely, of h12) induced by the guest particle. Inour original communication,11 we presented a striking evi-dence of this effect, by showing that gold particles (densityρ1 = 19 g cm−3) with an average radius R1 � 16 nm can floatatop of a dispersion of MFA particles which has a suspensiondensity ρs � 1.2 g cm−3. Here, we present another evidencethat, albeit not conclusive, qualitatively supports the observa-tions made for gold particles in MFA suspensions. As a hostfluid, we have used an aqueous suspension of PMMA parti-cles with radius R2 = 400 nm) (those used as guest particlesin sample C) at average volume fraction η0

2 � 0.1, to whichwe added a small amount of spherical silica particles of ra-dius R1 � 170 nm, with a structure characterized by a fluores-cent rhodamine-labeled (red) core surrounded by a pure silicashell, synthesized in our lab.32 These particles have a den-sity ρ1 � 1.8 g cm−3, and therefore a single-particle settlingspeed vS � 0.05 μm s−1 which is about half the value for thePMMA particles. Figure 12 m, which summarizes the timeevolution of the mixture over about two weeks, shows that thered silica particle, mostly segregate to the top of the sediment.This qualitatively resembles, with much more striking visualevidence, what we observed for gold. Yet, there is a specificreason why this evidence may not be considered as conclu-sive. Indeed, the average PMMA concentration in the bottomlayer, as judged from the position of the meniscus, is η2 � 0.5− 0.55. For this concentration values, most of the PMMA

FIG. 12. Time-evolution of a mixture of rhodamine-labeled silica particleswith radius R1 = 170 nm, dispersed in a suspension of PMMA particles withradius R2 = 0.5 μm.

114907-14 Parola et al. J. Chem. Phys. 138, 114907 (2013)

suspension must be in a colloidal crystal33 phase, whereas,due to the very short PMMA sedimentation length (lg1

� 5 μm) the fluid phase of the host suspension has to be verythin. Stating that the silica particle “float” atop of a lightersuspension (the average density of the PMMA sediment is ρs

� 1.1 g cm3 would then be misleading, because the lower ly-ing layer probably behaves as a solid with finite elastic modu-lus. Nonetheless, while the settling proceeds and PMMA con-centrates in the lower part of the cell, the red particles lying inthat region seem to rise, apparently leaving the PMMA sed-iment devoid of them. Furthermore, a careful observation ofthe picture taken after 15 days, which is basically the equi-librium profile, shows that not all the silica particles lie overthe PMMA sediment. A faint red hue can indeed be detectedclose to the cell bottom, consistently with what discussed inSec. IV A 2.

VI. DISCUSSION

The theoretical and experimental investigation of thesedimentation phenomenon in thermodynamical equilibriumprovided clear-cut evidence of the inadequacy of the standardArchimedes’ principle in the nano-world. The required gen-eralization leads to a closed expression for the effective buoy-ancy (8) with a simple physical interpretation. Experimentsin controlled systems quantitatively agree with the theoreticalpredictions thereby confirming the validity of our approach,and suggest an even wider applicability of the GAP to non-equilibrium situations. A careful and critical investigation ofsuch a scenario is under way.

Even limiting the validity of GAP to equilibrium condi-tions, it is possible to envisage a wide class of applications:the development of new protocols for particle fractionaliza-tion by use of density gradient untracentrifugation may be anextremely promising route in biochemistry and pharmacol-ogy; the huge effects predicted for disk-like or rod-like par-ticles may find applications in the realm of carbon nanotubesor graphene sheets; anomalous buoyancy in electrolytes maybe relevant also in other frameworks, such as geophysics ormarine ecology.

ACKNOWLEDGMENTS

We thank Solvay Specialty Polymers (Bollate, Italy) forthe kind donation of the MFA sample batch.

APPENDIX: BUOYANCY FORCE ANDPRESSURE TENSOR

Here, we express the buoyancy force (14) in a binarymixture in terms of the pressure tensor36 of the host fluidof type-2 particles. The hydrostatic equilibrium condition ofthe host fluid can be derived from first principles in statisticalphysics34

∂β Pαβ

2 (x) = −n2(x) ∂α vext (x), (A1)

where summation over repeated indexes is understood. Pαβ

2is the pressure tensor of the fluid, which acquires off diago-nal components due to the presence of a non-uniform density

profile, and vext (x) is the full external potential acting on thefluid. In our problem it is due to the combined effect of theexternal potential �(x) and of the interactions with the type-1particle placed in x,

vext (x) = w(x − r) + �2(x). (A2)

By substituting Eq. (A2) into Eq. (14), we obtain

Fα = −∫

dx[∂β P

αβ

2 (x) + n2(x) ∂α �2(x)], (A3)

where the integration domain is the whole space, but the re-gion inside the type-1 particle does not contribute becauseboth density and pressure vanish.

It is convenient to consider also the pressure tensor ofthe host fluid under the external potential �2(bx) but in theabsence of the type-1 particle

∂β Pαβ

2 (r) = −n2(r) ∂α �2(r). (A4)

In general, both the pressure tensor and the density profile willbe different. By use of this identity, we can write Eq. (A3) inthe form

Fα = −∫

dx ∂β

[P

αβ

2 (x) − Pαβ

2 (x))]

−∫

dx [n2(x) − n2(x)] ∂α �2(x). (A5)

The first term can be converted to a surface integral by use ofGreen’s lemma: the surface at infinity gives no contributionbecause asymptotically the effect of the single type-1 particleis negligible and we obtain

Fα = −∫

�

dσβ[P

αβ

2 (x) − Pαβ

2 (x)]

+∫

dx [n2(x) − n(x)] ∂α�2(x), (A6)

where � is the inner integration boundary (if any), whose sur-face element dσα is oriented inwards. This expression is for-mally exact. If the fluid is incompressible and the fluid-sphereinteraction w(x) is purely hard core, we can choose the in-ner surface just inside the sphere. In this case P

αβ

2 (x) = 0 onthe surface, while n2(x) = n2(x) = n0 in the bulk. As a conse-quence the buoyancy force is given by the integral of the pres-sure of the external fluid, which is unchanged by the presenceof the particle

Fα =∫

�

dσβPαβ

2 (x). (A7)

Otherwise we can avoid the presence of a inner surface alto-gether and integrate Eq. (A3) on the whole space. The surfacecontribution in Eq. (A6) then disappears, while the secondterm gives

Fα = n0

∫V

dx ∂α�2(x), (A8)

where the integration is limited to the volume of the type-1particle, where n(x) = 0. This is the well known Archimedes’principle.

More generally, for a compressible fluid and a genericcontinuous potential w(r), Eq. (A6) must be integrated in the

114907-15 Parola et al. J. Chem. Phys. 138, 114907 (2013)

whole space. The first term does not contribute and we obtain

Fα =∫

dx [n2(x) − n2(x)] ∂α�2(x), (A9)

which coincides with the generalized Archimedes principle(22) whenever the range of correlations is shorter than thegravitational length.

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