Screened and Unscreened Phases in Sedimenting Suspensions

4 Pages·0.13 MB·English

by  Alex Levine

Screened and Unscreened Phases in Sedimenting Suspensions
Alex Levine1, Sriram Ramaswamy2, Erwin Frey3, and Robijn Bruinsma4
1Exxon Research and Engineering, Rte 22E Clinton Township, Annandale NJ 08801, USA,
2Department of Physics, Indian Institute of Science, Bangalore 560 012, India,
3Institut fu¨r Theoretische Physik, Physik-Department der Technischen Universit¨at Mu¨nchen, D-85747 Garching, Germany,
4Department of Physics, University of California, Los Angeles CA 90095, USA.
(February 7, 2008)
8
9 A coarse-grained stochastic hydrodynamical description of velocity and concentration fluctuations
9 insteadilysedimentingsuspensionsisconstructed,andanalyzedusingself-consistentandrenormal-
1 ization group methods. We find that there exists a dynamical, non-equilibrium phase transition

n from a“unscreened”phasein which werecovertheCalflisch-Luke(R.E.Calflisch andJ.H.C. Luke,
a Phys. Fluids 28, 759 (1985)) divergence of the velocity variance to a “screened” phase where the
J velocityfluctuationshaveafinitecorrelationlengthgrowingasφ−1/3 whereφistheparticlevolume

6 fraction,inagreementwithSegr`eetal. (Phys.Rev.Lett. 79,2574(1997))andthevelocityvariance
1 is independent of system size. Detailed predictions are made for the correlation function in both
phases and at thetransition.

]
t
f PACS numbers: 05.40+j 05.45.+b 82.70.Dd
o
.s Sedimentation [1] is a rich and complex phenomenon of k for kz = 0 and k⊥ →0. Detailed, experimentally
t in suspension science and a frontier problem in nonequi- testable expressions for the structure factor and velocity
a
m librium statistical mechanics. The average sedimenta- correlationsin the screenedphase are presented after we
- tion speed vsed of solute particles drifting down in a sol- outline our calculations. The two phases are separated
d vent is determined by balancing the driving force (grav- inour “phase-diagram”(Fig.1)by a striking continuous
n ity) against the dissipative force (viscous drag). Giant nonequilibrium phase transition whereξ divergesatleast
o non-thermal fluctuations in the velocity and concentra- as rapidly as (K−K )−1/3 as a control parameter K is
c c
[ tionfieldsinasteadilysettlingsuspension,observedeven decreased towards a critical value Kc.
for non-Brownian systems, have been a puzzle for some Thehydrodynamicequationsweusedtoarriveatthese
1
years. CaflischandLuke(CL)[2]showed,forsteadysed- results are
v
imentationina containerofsmallestlinear dimensionL,
4 ∂c
6 thattheassumptionofpurelyrandom localconcentration ∂t +v·∇c=[D0⊥∇2⊥+D0z∇2z]c+∇·f(r,t) (1)
1 fluctuations led to velocity fluctuations with a variance
1 hv2i ∼ L. Most experiments, however, find no depen- and
0 dence of hv2i on L [3–5], althoughLadd’s simulations [6]
8 η∇2v (r,t)=m gP c(r,t), (2)
and the data of Tory et al. [7] appear to be consistent i R iz
9
/ with CL. where c(r,t) and v(r,t) are the fluctuations about the
t
a InthisLetterweproposearesolutionofthispuzzleby
mean concentration c0 and the mean sedimentation ve-
m meansofasetofcoarse-grained,fluctuatingnonlinearhy-
locity −vsedzˆ respectively. We justify these equations
- drodynamic equations for the long-wavelengthdynamics briefly below; for a more detailed discussion we refer
d
ofconcentrationandvelocityfluctuationsinasuspension
n the reader to Ref. [9]. Eq. 1 is the anisotropic ran-
settlingsteadilyinthe−z direction,atvanishinglysmall
o domlyforcedadvection-diffusionequationwithbareuni-
c Reynolds number. Our theory is similar in spirit to the axialdiffusivities (D0z,D0⊥)andarandomstirringforce
: Koch-Shaqfeh (KS) [8] “Debye-like” screening approach
v f(r,t) [10]. The Stokes equation, Eq. 2, which expresses
i but differs in severalimportant details and predictions. the balance between the driving by gravity and the dis-
X
The central conclusion of our study is that there are
sipation by the viscosity η, describes how the concen-
r two qualitatively distinct nonequilibrium phases for a
a tration fluctuations produce velocity fluctuations. Here
sedimenting suspension. In the “unscreened” phase hv2i
m g is the buoyancy-reduced weight of a particle, while
R
diverges as L, as in CL and, in addition, concentration
the pressure field has been eliminated by imposing in-
fluctuationswithwavevectork=(k⊥,kz)relaxatarate compressibility via the transverse projection operator
∝ k1/2. The “screened” phase is characterizedby a cor- P =δ −∇ ∇ (∇2)−1.
ij ij i j
relation length ξ similar to that predicted by KS such
Hydrodynamic equations such as Eqs. 1 and 2 arise
that hv2i ∼ L for L ≪ ξ and hv2i ∼ ξ for L ≫ ξ. Deep
from a coarse-graining of the microscopic equations of
in the screened phase we predict ξ ∼ φ−1/3 where φ is
motion. The latter, for the main case of interest here,
the particle volume fraction. This is in agreement with
viz., non-Brownian suspensions at zero Reynolds num-
the experiments of Segr`e et al. [5], but not with KS [8].
ber,are the deterministic equations ofStokesiandynam-
The relaxationrate in the screened phase is independent
ics for N hydrodynamically coupled particles, and are
1
known to be chaotic [11]. The noise, or random stirring ThequantitiesDz,⊥(q)andNz,⊥(q)representrenormal-
current f(r,t) and the diffusivities in Eq. 1 represent a ized diffusivities and noise amplitudes [13]. But, most
phenomenological description of the deterministic chaos importantly, the advective nonlinearity to lowest-order
atlengthscalesbelowthecoarse-graininglengthℓ(which perturbation theory leads to an additional term in the
must be large compared to the particle radius a). We renormalization of the relaxation rate which is of the
use these hydrodynamic equations to predict the veloc- form Γ(q) = γ(q)q2/q2. Starting from the stochastic
⊥
ity and concentration fluctuations at length scales large hydrodynamic equations, Eqs.1-3, it turns out that the
compared to ℓ driven by the random stirring at short amplitude of this singular contribution becomes a con-
distances. stant, limq→0γ(q) ∝ I(βN,βD), which depends on the
We assume, as is reasonable, that f(r,t) is Gaussian anisotropy ratios of the noise and diffusivity coefficients
white noise with uniaxial symmetry:
N⊥ D⊥
β = , and β = . (9)
hfi(r,t)fj(r′,t′)i=2c0N0ijδ(r−r′)δ(t−t′) (3) N Nz D Dz
with an anisotropic noise amplitude N0ij = N0⊥δi⊥j + In particular I(βN,βD) is proportionalto βN −βD. and
N0zδizj, where δizj and δi⊥j are the projectors along cdoiffnusesqivuietyntrlyatmioasy. cFhoarnIg(eβsig,nβup)o<nv0artyhiinsgwtohuelndoliesaedantdo
and normal to the z axis, respectively. Because of N D
exponentially growing concentration fluctuations in the
the nonequilibrium origin of the noise and diffusion
limit of long wavelength. Here we do not pursue this
constants, we may not [12] assume that N0⊥/N0z = intriguing possibility further but instead restrict our at-
D0⊥/D0z as would be true for the Langevin equation tentiontoI(β ,β )≥0,forwhichthe modelcaneither
of a dilute suspension at thermal equilibrium. Note that N D
be treatedwithin dynamic renormalizationgrouptheory
no correlations have been fed in via the noise: any that
or using self-consistency methods.
emerge in the long-wavelength properties are a result of
We start our discussion at the borderline of stability,
the interplay of advection and diffusion.
β = β . For these parameter values it can be shown
Let us now consider the nature of the spatio-temporal N D
that the fluctuating hydrodynamic equations describe a
correlations implied by Eqs. 1 and 2. We will focus on
dynamics which obeys detailed balance [15]: the advec-
the structure factor for concentration fluctuations
tive nonlinearity does not affect the equal-time correla-
S(q)≡c−1 ddrhc(0)c(r)ie−iq.r (4) tions,andS(q)inparticularisjusttheconstantN⊥/D⊥.
0 Z There are singularities in N⊥,z and D⊥,z which we dis-
cuss later.
from which the velocity structure factor can be derived
For β ≥ β , detailed balance is violated and a sin-
N D
through Eq. 2. If we ignore the advective nonlinearity gular diffusion term Γ(q) is generated within perturba-
v·∇c, then S(q) can be computed by straightforward
tion theory. In order to analyze the dynamics in this
Fourier transformation of Eq. 1, resulting in
regimeweuseone-loopself-consistenttheory(modecou-
S(q)=S0(q)≡ DN00⊥⊥qq⊥⊥22 ++DN00zzqqzz22. (5) pling theory) and armrivegat2theqexPpre(sks)ikonP (q)
Using Eq. 5 in Eq. 2 we can compute hv2i as a function Γ(q)=c0(cid:18) ηR (cid:19) Zk i iz k2qj2 jz
of the system size L with the result: [S(q−k)−S(k)]
× (10)
R(k)+R(q−k)
S(q)
2
hv i∼ ∼L. (6)
Zq>1/L q4 with R(q) given by (7), and similar self-consistent in-
tegral equations for D⊥(q), Dz(q), N⊥(q), and Nz(q).
In other words, neglecting large-scale advection by the We find that there are two types of iteratively stable so-
velocity fluctuations leads to the CL [2] result. lutions to these coupled self-consistent equations: those
To include the effect of the advective nonlinearity we with γ(q →0)>0, which we obtain below the solid line
have performed a self-consistent mode coupling calcula- in the phase diagram spanned by the two anisotropy ra-
tion [14] on Eqs. 1-3. Our results can be expressed in tios (“screened” phase in Fig. 1), and those with γ(q →
terms of a renormalized relaxation rate 0) = 0, which arises for values of the anisotropy param-
eters that lie above the solid line and below the dashed
R(q)=D⊥(q)q⊥2 +Dz(q)qz2+Γ(q) (7) line of the same figure, i.e., in the “unscreened” phase.
Notethatwithintheself-consistenttheorythelineinthe
and a renormalized structure factor of the form
phase diagram where γ(q = 0) changes sign (solid line)
S(q)= N⊥(q)q⊥2 +Nz(q)qz2. (8) htuarsbsahtiifotnedthweiothryredsipsceucststeodtahbeorveesu(dltaosfhethdelionnee)-.loopper-
R(q)
2
100
Screened Phase: Inthe screenedphase,Γ(q)is ofthe 0.5
2 2
formγq /q inthesmallqlimit,withγ afiniteconstant.
⊥
80
Thisimplies thatthe structurefactoratsmallwavenum-
Unscreened
ber becomes
60 0
S(q)≃ N⊥q⊥2 +Nzqz2 (11) D D 0 0.5
D⊥q⊥2 +Dzqz2+γq⊥2/q2 z40
withN⊥,zandD⊥,z constants. FromEq.11wecandefine
a correlation length ξ ≡ (D⊥/γ)1/2 such for q⊥ ≫ 1/ξ 20
Screened
the structure factor is not significantly affected by ad-
0
vection. On the other hand, for q⊥ ≪ 1/ξ the in-plane 0 20 40 60 80 100
structure factor reads S(q⊥,qz =0)≃(N⊥/γ)q⊥2, while N
S(q⊥ =0,qz)≃(Nz/Dz). Physically,thismeansthatat N
z
long wavelength advection strongly suppresses in-plane
FIG.1. Dynamical phase diagram for sedimentation. Be-
concentration fluctuations.
low the solid line in the parameter space spanned by the
Using Eq.11 in conjunction with Eq.2, one finds that
anisotropy factors for the noise and diffusivities velocity and
for length scales L less than ξ, hv2i∝L, consistent with
concentrationfluctuationshaveafinitescreeninglengthinthe
CL, while for L large compared to ξ, hv2i ∝ ξ. Veloc-
limit vanishing wavevector. This region is called “screened”
ity fluctuations on length scales small compared to ξ are above. Intheupperregion called “unscreened”thescreening
thushighlycorrelatedwhiletheybecomeuncorrelatedat length becomes infinite. The dashed line represents the set
larger length scales. ofvaluesfortheanisotropy factors wherethehydrodynamics
Deep inside the screened phase, i.e., for large γ, the obeys detailed balance. The inset shows the behavior of the
renormalization of the diffusion and noise parameters is phase boundary in the limit of large noise and diffusivity in
negligible and we can explicitly compute γ, and thus ξ, thevertical direction as compared to thehorizontal plane.
by inserting Eq.8 in Eq.10 usingthe barevalues for the
N’s and D’s. We find
exactly, z = d/2 − 1. This implies that the diffusiv-
ξ =8(mRg)−2/3c0−1/3 1− 2 −1/3, (12) liotinegs awnadvenleonigsetha.mEpvlietnudtehsouscgahlethaesreq−aǫr/e2 n=owq−si3n/g2ufloarr
ηD (cid:18) β (cid:19)
N correctionstoDz,⊥(q)andNz,⊥(q), theanomalousΓ(q)
where for simplicity we have set D0⊥ = D0z = D. Ac- termiszero. Forparametervaluesintheregimebetween
cording to Eq. 12, the correlation length increases as we the dashedline (detailed balanceline) andthe solidline,
decrease the β parameter (which could be done by in- which marks the location of the nonequilibrium phase
N
creasing the thermal noise amplitude) and diverges at transition, renormalization group methods may be used
β = 2. Strictly speaking, as β → 2, the diffusiv- to determine the renormalization of the noise and diffu-
N N
ity corrections are no longer negligible, and the actual sivity amplitudes. In view of the results from the above
divergence of ξ is probably stronger than (12), and oc- self-consistency calculation (γ = 0 in the unscreened
curs at a larger value of β . An explicit analytical (but phase) and the exact results at the detailed balance line
N
lengthy) result for the correlation length ξ can also be itis quite likely thatthe resulting renormalizationgroup
obtained throughout the screened phase as a function of flowwilltend towardsafixedpointwhichobeysdetailed
both anisotropy parameters [9] and the phase boundary balance. Weleavethe detailsofsuchaninvestigationfor
can also be computed. The phase boundary resulting a future publication [9].
from this result is shown in Fig. 1 as the solid line sep- The analysisofourhydrodynamicequations thus con-
arating the screened from the unscreened phases. The firms that screening can suppress the CL divergence of
2
dashedlineinthefigurecorrespondstothesetofparame- hv i with L, as argued by KS, while it allows for a sec-
tervalueswherethehydrodynamicequationscorrespond ond, unscreened phase. This result may help explain
to a Langevin dynamics in thermal equilibrium. theconflictingresultsonhv2iobtainedbydifferentwork-
Unscreened Phase: As already noted above, the hy- ers [3–7]. The self-consistent structure factor, Eq. 5 we
drodynamic equations obey detailed balance [15] along obtained differs significantly from the one proposed by
the line β = β in the phase diagram. As a conse- KS. Experimental test will thus be of considerable im-
N D
quence the ratio of noise to diffusivity can be identified portance. Measurements of S(q), for example by PIV
as a direction-independent “noise-temperature”. Fur- [16](ParticleImagingVelocimetry),wouldconstitutethe
thermore, the structure factor S(q) becomes a constant most direct test of the theory since our prediction that
D⊥/N⊥ and we recover the CL result. In conjunction S(q⊥,qz =0)∝q⊥2 does not hold in the KS description.
withanexponentidentityresultingfromGalileaninvari- DetailedmeasurementsofS(q)forsedimentingsolutions
ance this is enough to determine the dynamic exponent are not yet available. However, Segr`e et al. [5] do report
3
that the size-dependence of the amplitude hv2i of the lowship. E.F. acknowledgessupport by a Heisenberg fel-
velocity fluctuations depends on a characteristic length lowship (FR 850/3-1) from the Deutsche Forschungsge-
scale ξ such that hv2i ∝ ξ for length scales L ≫ ξ meinschaft.
S S S
while for L ≪ ξ , hv2i grows with L. They report that
S
ξ ∼aφ−1/3 with φ the particle volume fraction.
S
Ourcorrelationlengthξ,inEq.12,hasthesamephys-
icalinterpretationasξ . Deepinthescreenedphase,i.e.,
S
for I(β ,β )≫0, ξ can be written as:
N D
ξ(φ)∼(m g/ηD)−2/3aφ−1/3I(β ,β )−1/3 (13) [1] R. Blanc and E. Guyon,La Recherche, 22, 866 (1991).
R N D [2] R.E. Caflisch and J.H.C. Luke, Phys. Fluids 28, 759
(1985).
On scaling grounds, we expect that D ∝ δv ξ with
RMS [3] H. Nicolai and E. Guazzelli, Phys. Fluids 7, 3 (1995).
δv the root mean square of the velocity field fluctua-
RMS [4] J.Z. Xue,et al.,Phys.Rev. Lett.69, 1715 (1992).
tions. Experimentally,δv ξisfoundtobeindependent
RMS [5] P.N. Segr`e, E. Herbolzheimer, and P.M. Chaikin, Phys.
ofvolumefractionφ. Inthatcase,Eq.13reproducesthe Rev. Lett.79, 2574 (1997).
experimentally observed volume-fraction dependence, in [6] A.J.C. Ladd,Phys. Rev.Lett. 76, 1392 (1996).
contrast to KS [8]. It should be noted that this volume [7] E.M.Tory,M.T.Kamel,andC.F.ChanManFong,Pow-
fractiondependenceofthecorrelationlengthimpliesthat der Technology 73, 219 (1992).
there is a fixed number of colloids within a correlation [8] D.L. Koch and E.S.G. Shaqfeh,J. Fluid Mech. 224, 275
volume independent of volume fraction. (1991).
The observation of a transition from the screened to [9] A.Levine,S.Ramaswamy,E.Frey,andR.Bruinsma, in
preparation.
the unscreened phase would obviously be the most con-
[10] InEqs.(1-2)otherpossiblenonlinearities,e.g.thosearis-
clusive evidence supporting our theory, in particular if
ing from theconcentration dependenceof mobilities and
the transition were accompanied by a divergence of the
viscosities, as well as multiplicative noise terms of the
velocity fluctuation correlation length. Even in the ab- form ∇·(ch), where his a spatio-temporally white vec-
senceofsuchdirectevidence,theobservationofscreened
tornoise,canreadilybeshown,bypower-counting,tobe
behavior combined with our theory requires that the subdominantatsmallwavenumberrelativetotheadvec-
anisotropies in the noise and diffusivity lie in the lower tionterm∇·vc,ascanadvectionbythermal fluctuations
region of our dynamical phase diagram, Fig. 1. A com- in thefluid velocity field (S. Ramaswamy, unpublished).
pletetestofourtheorythusrequiresmeasurementofthe [11] I.M. J´anosi et al., Phys. Rev. E 56, 2858 (1997); J.F.
N and D parameters. These could be obtained from the Brady and G. Bossis, Ann. Rev. Fluid Mech. 20, 111
measurement of the steady-state static structure factor (1988).
[12] See e.g. B. Schmittmann and R.K.P. Zia, in Phase tran-
S(q), e.g. by particle imaging or light scattering experi-
sitions and critical phenomena, Vol. 17, ed. byC. Domb
ments both along the z direction and in the x−y plane,
and J. Lebowitz (Academic Press, London,1994).
coupled with tracer diffusion measurements.
[13] Including frequency-dependence will not alter the criti-
Finally, it would be interesting to vary the effective
cal exponents and the structure of the scaling variable;
noise and diffusion constants in a controlled manner in
it may, however, affect the functional from of the full
an experiment. While there is, as yet, no method to cal- relaxation rates.
culatetheseconstantsdirectlyfromamicroscopictheory [14] The“Galilean”invarianceofEqs.1and2underv(r,t)→
it is reasonable to expect that by decreasing the Peclet v(r,t)+U,r → r−Ut guarantees that the nonlinear
number (i.e., increasing the role of isotropic thermal dif- coupling will not renormalize. This means we need to
fusion) one could drive the sedimenting system into the worry only about the corrections to the noise strength
unscreenedphase. Thus by repeating the experiments of and relaxation rate; A. Levine, S. Ramaswamy, E. Frey,
andR.Bruinsma(unpublished);seealsoD.Forster,D.R.
Segr`eet al. [5]withcolloidsthataremorenearlydensity
Nelson, and M.J. Stephen,Phys.Rev.A16, 732(1977).
matchedtothe solventone couldtestour predictionofa
[15] For nonequilibrium random processes the stationary
transition to an unscreened phase.
probability distribution function is not known a priori,
We would like thank M. Rutgers, P. Chaikin, and P.
exceptforcertaincaseswheresocalled“potentialcondi-
Segr`eforcommunicatingunpublishedresultsandforuse- tions” (see e.g. R. Graham, in Springer Tracts in Mod-
ful discussions. We wouldalsoliketo thank J.Brady,D. ern Physics Vol. 66, Springer Verlag, Berlin, 1973) are
Durian, E. Herbolzheimer, S. Milner, R. Pandit,J. Rud- fulfilled. Then the random process has detailed balance
nickandU.C.T¨auberforusefuldiscussions. S.R.thanks propertyandthestationarydistribution functioncanbe
F.PincusandC.SafinyaandtheMaterialsResearchLab- calculatedexplicitly.Inthepresentcasedetailedbalance
oratory, UCSB (NSF DMR93-01199 and 91-23045), as holdsforβN =βDwiththeequilibriumdistributionfunc-
well as the ITP BiomembranesWorkshop(NSF PHY94- tion given by Pst[c]∝exp − ddx(D⊥/N⊥)c2(x) .
07194)forpartialsupportintheearlystagesofthiswork [16] R.J. Adrian,Annu.Rev.F(cid:8)luidRMech. 23, 261 (199(cid:9)1).
. A.L.acknowledgessupportbyanAT&TGraduateFel-
4