Colloidal brazil nut effect in sediments of binary charged suspensions

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by  Ansgar Esztermann

Colloidal brazil nut effect in sediments of binary charged suspensions
Ansgar Esztermann, Hartmut Lo¨wen
Institut fu¨r Theoretische Physik II, Heinrich-Heine-Universit¨at Du¨sseldorf,
Universit¨atsstraße 1, D-40225 Du¨sseldorf, Germany
(Dated: January 2nd, 2004)
Equilibrium sedimentation density profiles of charged binary colloidal suspensions are calculated
bycomputersimulationsanddensityfunctionaltheory. Fordeionizedsamples,wepredictacolloidal
4 “brazil nut” effect: heavy colloidal particles sediment on top of the lighter ones provided that
0 their mass per charge is smaller than that of the lighter ones. This effect is verifiable in settling
0
experiments.
2

n PACSnumbers: 82.70.Dd,61.20.Ja,05.20.Jj
a
J
Binarysystemsofgranularmatterseparateuponshak- are treated explicitly at constant temperature T. With
2
ingingravity,sothatthelargerparticleslieontopofthe Z1e, Z2e, −qe and σ1, σ2, σc denoting the charges and

] smalleroneseveniftheyareheavieranddenserthanthe the diameters of the two colloidal species and the mi-
t
f latter. This is due to a sifting mechanism in which tiny croions, the interaction between the charged particles is
o
grainsfilterthroughtheintersticesbetweenthelargepar- given as a combination of Coulomb forces and excluded
s
. ticles which is well-known as “brazil nut” effect: in a jar volume of the hardparticle cores. Here, e is the electron
t
a ofmixednutsorinapackageofcereal,thelargestspecies charge. For simplicity we assume that the co- and coun-
m risestothe top[1,2]. This clearlydistinguishes granular terionsofthesaltsolutionhavethesamevalencyandthe
matter from ordinary fluids where the rising species is same hardcore diameter σ . We considera finite system
- c
d controlled by Archimedes’ law. Understanding the full of N , N chargedmacroions and correspondingnumber
1 2
n details of the brazil nut effect is still a problem; recently of counterions (fixed by global charge neutrality) plus
o
even a reverse brazil nut effect of large light grains sink- salt ions of bulk concentrationc . The simulation box is
c s
ing in a granular bed has been predicted [3, 4, 5] and rectangularwith lengths L =L and L =32L in the
[ x y z x
verified in experiments [6]. three different spatial directions with periodic boundary

1
Inthis letter,wereportonequilibriumdensityprofiles conditions in x and y direction and finite length in z-
v
0 of binary charged colloidal fluids (“macroions”) under direction. Hardwallsareplacedatz =0,Lz andgravity
1 gravity. Using extensive Monte-Carlo computer simula- with acceleration g points along the −z direction. Only
0 tionsofthe“primitive”model[7]ofstronglyasymmetric the colloidalparticles with their buoyantmasses m1 and
1 electrolytes and a density functional theory, we predict m2 aresubjecttogravity,whereasthemicroionsarenot.
0 that the heavier particles sediment on top of the lighter A Monte-Carlo simulation is performed in the canon-
4
ones provided the charge per mass of the heavier parti- ical ensemble with the long-ranged Coulomb interaction
0
clesishigher. Inanalogytogranularmatter,wecallthis treatedviaLeknersums[17]. Typically103 MonteCarlo
/
t counter-intuitive phenomenon a colloidal brazil nut ef- moves per particle were performed for equilibration and
a
m fect. It is generated by the entropy of the microscopic ittookanadditional104 MonteCarlomovesperparticle
counterions in the solution, which are coupled to the to gather statistics. Finite system size effects were care-
-
d macroions by strong Coulomb binding. Clearly, though fullycheckedbychangingalllaterallineardimensionsby
n this effect is qualitatively similar to the granular brazil afactorof4. This meansthatwehavechangedthetotal
o nut effect insofar as heavy particles are on top of lighter numberofcolloidalparticlesN +N inthe rangeof12–
1 2
c ones, its physical origin is different: first, the particle 200. Wehavecalculatedtheinhomogeneousz-dependent
:
v charge (and not the size) is crucial. Second, the col- averageddensity profiles ρ (z), ρ (z), ρ (z) and ρ (z) of
1 2 3 4
i loidalbrazilnuteffectis apure equilibriumphenomenon thetwomacroionsandthecounter-andcoions. Dataare
X
while the granular brazil nut effect happens intrinsically shown for the largest system size where N =N =100.
1 2
r
a innon-equilibrium. Thecolloidalbrazilnuteffectcanbe Besides the ratios Z /q, Z /q, m /m , σ /σ , σ /σ ,
1 2 1 2 1 2 c 2
verified,e.g.,indepolarized-lightscatteringorreal-space the system is characterized by two partial area den-
experiments on sediments of strongly deionized binary sities n = N /L L (i = 1,2), the Bjerrum length
i i x y
chargedsuspensions [8, 9]. Similar techniques have been λ = q2e2/ǫkT (kT denoting the thermal energy), the
B
usedtomeasureone-componentcolloidaldensityprofiles gravitational length ℓ = kT/m g of the second parti-
2 2
[9, 10] where deviations from the ideal barometric law cle species, and the salt concentration. In order to re-
[11] are still an ongoing debate [12, 13, 14, 15, 16]. duce the parameter space, we have assumed throughout
We simulate the asymmetric “primitive” model of bi- the simulations monovalent microions (q = 1), and the
narychargedsuspensionsinwhichthesolventonlyenters same hard core diameter of the macroions σ = σ = σ
1 2
via a continuous dielectric background with permittiv- which serves as a natural length scale. We further fixed
ity ǫ but all charged particles (two species of negatively Z = 15, λ = σ = σ/128 and n = n = 0.1/σ2. This
2 B c 1 2
chargedmacroionsand microscopic counter- and coions) corresponds to typical parameters for low-charge aque-
2
tive hard interactioncores,the high-chargeparticles will
0.004
have a larger diameter and hence Archimedes’ principle
3) -3 will lift the high-chargeparticles in the sea of small ones
0.003 σz) -4 provided their mass density is smaller [18]. For our pa-
ρ(i -5 rameters,however,the latter effect is smallandconfined
(10 -6 to regions close to the wall. More importantly, as re-
3 g
σz) 0.002 lo -7 vealed by the semi-logarithmic plot in the inset of Fig-
ρ(i 0 200 400 600 800 1000 ure 1, there is an exponential decay of the two colloidal
z/σ profiles for intermediate heights associated with two dif-
Z =45
0.001 1 ferent decay lengths for the two colloidal species. The
Z1=30 decay length for the heavier but high-charge particles is
larger than that for the low-charge particles. This gives
Z =25
1
0 the most significant contribution in the integral (1) of
0 200 400 600 800 1000 the mean height. Finally, for large z, there is cross-over
z/σ towardsanotherexponentialdecayinvolvingthegravita-
tional length ℓ =kT/m g for the high-charge particles.
1 1
FIG. 1: Density profiles of macro- and counterions for Z1 = Itisimportanttonotethatthedensityprofilesfulfilllocal
45,30,25 (solid lines; top to bottom). The second colloidal
charge neutralitythroughoutthe whole sample exceptat
component is shown as a dashed line. Counterion densities
thecontainerbottom(z =0)andatverylargeheightsz.
(thinlines) havebeendividedbyZ1+Z2. Forclarity,curves
But even there the excess charge separated is small [19].
pertaining to different simulation runs have been shifted by
5×10−4 with respect to each other. The parameters are: Ourtheoreticalexplanationforthe colloidalbrazilnut
m1/m2 = 1.5, ℓ2/σ = 10, cs = 0. Inset: Semi-logarithmic effect is based on a simple density functional approach.
plots of the same colloidal density profiles with the slopes The free energy F per unit area, which is a functional
predicted from theory. of the inhomogeneous density fields ρ (z),ρ (z),ρ (z)
1 2 3
of the macro- and counterions, splits into the gravi-
tational energy, the entropy of the three species and
ous suspensions. We varied the colloidal charge Z , the
1 all Coulomb contributions. The latter are approxi-
colloidal mass ratio m /m (with m > m ), the gravi-
1 2 1 2 mated within a mean-field-type Poisson-Boltzmann the-
tational length ℓ , and the salt concentration.
2 ory [12, 13, 14, 15, 16]. Hence:
In the salt-free case, density profiles for the two
macroions and the counterions are shown in Figure 1
2 ∞
for m1/m2 = 1.5 and three different macroion charges F[ρ1(z),ρ2(z),ρ3(z)]= dzmνgzρν(z)
Z1 = 45,30,25. For large heights z, the heavy particles νX=1Z0
(solidcurve)areontopofthelighterones(dashedcurve).
3 ∞
This colloidal brazil nut effect is getting stronger for in- + dzkTρ (z)(ln(Λ3ρ (z))−1) (2)
creasing charge asymmetry between the colloidal parti- νX=1Z0 ν ν ν
cles: for the highest charge Z1 = 45, the two macroion 1 ∞ ∞ ρ (z)ρ (z′)
speciesarealmostcompletelyseparated. Toquantifythe + d2r′ dz dz′ t t
2Z Z Z ǫ r′2+(z−z′)2
brazil nut effect, we define – in analogy to the ordinary 0 0
brazilnutproblem[5]–amean(orsedimentation)height p
Here, Λ ,(ν = 1,2,3), are Lagrange multipliers which
of the two profiles via ν
ensure that the overall colloidal densities per unit area
∞ ∞
zρ (z)dz equaltheprescribednumberdensities,i.e. dzρ (z)=
hi = R0∞ρi(z)dz i=1,2 (1) ni (i = 1,2). Λ3 is fixed by global charRg0e neutirality,
0 i and ρ (z) = Z eρ (z)+Z eρ (z)−qeρ (z) is the total
R t 1 1 2 2 3
Bydefinition,thecolloidalbrazilnuteffectoccursifh > local charge density of the system. The functional F is
1
h , but there is no brazilnut effect (or a “reverse”brazil minimal for the physically realized equilibrium density
2
nut effect) in the opposite case, h ≤h . profiles.
1 2
Let us now describe the properties of the density pro- We discuss two different cases of weak and strong
filesforincreasingz qualitativelyandputforwardasim- Coulombcouplingsubsequently. ForweakCoulombcou-
ple theory to describe the basic features. Close to the pling, which is realized for very large heights z where
hardcontainer bottom atz =0, small correlationeffects the densities are extremely small, one may neglect the
are visible as a density shoulder of the low-charge par- third term on the right hand side of Eq. (2). Then, the
ticles while high charge particles are depleted from the minimization of F yields colloidal density profiles which
wall. The reason is a combination of a pure interface ef- followthe traditionalbarometriclawρ (z)∝exp(−z/ℓ )
i i
fect and Archimedes’ principle. First, we have checked (i=1,2). Strong Coulomb coupling, on the other hand,
by simulation that depletion of high-charge particle per- will impose local charge neutrality ρ (z) = 0 which im-
t
sists for zero gravity. Second, by crudely mapping the plies that the counteriondensity field is enslavedto that
interacting binary colloidal mixture onto one with effec- ofthemacroions. MinimizingF inthislimitwithrespect
3
400
brazil nut effect
no brazil nut effect
3
300
1) 2.5
+
2
1)/(Z 2 σh/i 200
+
1
Z
(
100
1.5
1 0
1 1.2 1.4 1.6 1.8 2 2.2 2.4 1.5 2 2.5 3
m /m Z /Z
1 2 1 2
FIG. 2: Crossover towards the brazil nut effect. Shown are FIG.3: Mean heightshi of heavy(+)andlight (⊙)particles
differentchargeasymmetries(Z1+1)/(Z2+1)andmassratios as a function of Z1 for the parameters of Fig. 1. The lines
m1/m2 forwhichthebrazilnuteffectwas(circles)orwasnot show thetheoretical predictions.
observed (crosses). The straight line shows the theoretical
prediction. The parameter combinations were obtained by
changing ℓ2 and Z1 is the range of 4...20σ and 16...45, for a large range of intermediate heights; the theoreti-
respectively. cal predictions for the slopes as given by Eqs. (3) and
(4) are shown as thick lines. The crossover to the bare
gravitationallengthforlargez is confirmedfor the high-
to the two colloidal densities only, again yields an expo-
charge particles and was found to be pretty sharp. Con-
nential decay ρ (z) ∝ exp(−γ z) (i = 1,2) but with in-
i i sistently with the theoretical assumption, local charge
versedecaylengthsγ whicharesmallerthan1/ℓ . These
i i neutralityinthe intermediateregimeis fulfilled. Second,
decay lengths γ turn out to be as follows: Let α be the
i we have tested the locationof the transitiontowardsthe
index i for which the mass per charge ratio mi is
Zi/q+1 colloidal brazil nut effect by systematically varying the
minimal and let β be the index complementaryto α, i.e. mass and charge ratio. The results are summarized in
β =1 if α=2 and β =2 if α=1. Then Figure 2. The theory, which predicts the transition at
mα g (Z1+1)/(Z2+1) = m1/m2, is shown as a straight line
γα = (3) there. All parameter combinations simulated are indeed
Z /q+1kT
α separated by this theoretical prediction, confirming our
and simple theory. This is remarkable as any wall or bulk
correlationeffects are neglected in the theory.
γ = m −m Zβ/q g >γ (4) In order to elucidate this further, we have compared
β β α α
(cid:18) Z /q+1(cid:19)kT thetheoreticalpredictions1/γ ,1/γ fortheheightswith
α 1 2
the simulation data in a situation where the transition
with γβ ≥ kT(Zmββ/gq+1) ≥ γα. This second case of strong line was crossed. In Figure 3 the heights are shown as
Coulombcouplingwillberealizedforheightszwherecor- a function of a varied charge Z . The simulation data
1
relations between the ions are small (which justifies the reveal that the variation of both heights h with Z is
i 1
mean-field approximation) but where local charge neu- largeclosetothe transitionandbecomesmaximalatthe
trality is still valid. The physical reason for the much transition. In the theory, this feature is reproduced and
slower decay of the colloidal density profiles in the sec- accompaniedbyagenericcuspatthetransition. Though
ond regime results from the counterion entropy which the cusp is smeared out and in general the heights are
tends to delocalize the counterions. However, since the largerinthesimulationdataduetothedensityreduction
macroionsarecoupledtothecounterionsduetothecon- close to the wall, there is still semi-quantitative agree-
straintof localchargeneutrality, they are lifted upwards ment. The rapid variation of the heights at the transi-
togetherwiththeircounterions. Assumingthatthemain tion implies that the location of the transition towards
contributionintheintegrandoftheright-hand-sideof(1) thebrazilnuteffectisveryrobustexplainingthevalidity
comesfromthesecondregime,themeanheightsaregiven of the theory in Figure 2.
by h =1/γ and h =1/γ . A brazil nut effect occurs Finally we address the case of added monovalent salt.
α α β β
whenα=1andβ =2. Hencethe transitiontowardsthe Density profiles for the two macroion species and the
brazil nut effect happens at m1 = m2 . counter- and coions are presented in Figure 4. Ad-
Z1/q+1 Z2/q+1
Letusnowtestthepredictionofthetheoryagainstour dition of salt reduces the brazil nut effect; in the in-
simulationdata. First,theslopesinthe insetofFigure1 set of Figure 4 the heights are plotted versus added
confirm the inverse decay lengths γ and γ perfectly salt concentration. The threshold salt concentrationc(0)
α β s
4
salt, it will only show up for deionized solutions. Highly
0.002
charged suspensions in non-polar solvents with a small
160 dielectricconstantǫandlowimpurityionconcentrations
140 [20] are promising candidates to exhibit a strong brazil
120 nut effect.
σ
3σ h/i 100 In conclusion, we predict an analog of the granular
ρ(z)i 0.001 80 brazilnut effect in equilibriumsediments ofchargedsus-
60 pensions, which is generated by an entropic charge lift-
0 0.01
ing due to the Coulomb coupling to the counterions. An
csσ3 experimentalverificationofthe levitationshouldbe pos-
sible employing depolarized light scattering [9] or con-
focal microscopy [20]. The simulated charge asymme-
0
0 200 400 600 800 1000 tries correspond to micelles and proteins rather than
z/σ highly charged colloids but the simple theory which was
confirmed by the simulations is applicable for arbitrary
FIG. 4: Density profiles for added salt. The counter- and charges.
coiondensitieshavebeendividedbyZ1+Z2,thecoiondensity
is smaller than that of the counterions. The parameters are The lifting effect is immediately generalizable to sed-
Z1 = 45, m1/m2 = 2, ℓ2 = 20σ, and csσ3 = 2.5×10−3. iments of solutions which are polydisperse in mass and
Inset: Sedimentation heights hi of heavy (+) and light (⊙) charge [21]. Therefore the colloidal brazil nut effect has
particles asafunction ofreducedsalt concentration forZ1 = important biochemical implications for the separationof
32,m1/m2 =2,andℓ2 =5σ. Thetheoreticalestimateofthe polydispersebiologicalmatter, suchas proteinsolutions.
threshold salt concentration is marked byan arrow. Analyticalsedimentationistypicallyusedinanultracen-
trifuge to separate different species [22]. The colloidal
brazil nut effect implies that the separation is sensitive
needed to reverse the brazil nut effect is estimated as
to the mass per charge but not to the mass itself.
c(0) ≈ Z ρ (z )+Z ρ (z ) which is the counterion con-
s 1 1 0 2 2 0
centrationatthepositionz wherethetwocolloidalden- WearegratefultoT.Palberg,F.Scheffold,A.Philipse,
0
sity profiles cross in the salt-free case. This salt concen- C.N.Likos,R.vanRoij,andR.Piazzaforhelpfuldiscus-
tration is indicated as an arrow in the inset of Figure 4, sions. FinancialsupportwithintheDeutscheForschungs-
confirmingthevalidityoftheestimate. Thoughthebrazil gemeinschaft(wettingpriorityprogramandSFBTR6)is
nuteffectisproventoremainstablewithrespecttoadded gratefully acknowledged.
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