Table Of ContentHydrodynamic radius approximation for spherical particles suspended in a viscous
fluid: influence of particle internal structure and boundary
Bogdan Cichocki
Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, Hoz˙a 69, 00-681 Warsaw, Poland
Maria L. Ekiel-Jez˙ewska∗ and Eligiusz Wajnryb
Institute of Fundamental Technological Research,
Polish Academy of Sciences, Pawin´skiego 5B, 02-106 Warsaw, Poland
(Dated: January 27, 2014)
4
1 Systems of spherical particles moving in Stokes flow are studied for a different particle internal
0 structure and boundaries, including the Navier-slip model. It is shown that their hydrodynamic
2 interactions are well described by treating them as solid spheres of smaller hydrodynamic radii,
n which can be determined from measured single-particle diffusion or intrinsic viscosity coefficients.
a Effective dynamics of suspensions made of such particles is quite accurately described by mobility
J coefficients of the solid particles with the hydrodynamic radii, averaged with the unchanged direct
4 interactions between theparticles.
2
] I. INTRODUCTION show how to apply it to account for the hydrodynamics
t ofparticleswithadifferentinternalstructureandbound-
f
o aries.
Hydrodynamic effects of surface layers or surface
s The plan of the paper is the following. With the ul-
. roughness on micro or nanoparticles are important for
t timate goal to understand hydrodynamics of spherical
a dynamics and rheology of systems, which contain many
m particles, we start in Sec. II from analyzing a simplified
such particles. Examples are micro and nanogels, which
problem: a fluid separatedfroma permeable medium by
- has been recently investigated in many contexts, espe-
d a planar interface or bounded by a flat rough solid sur-
cially withthe perspective ofusing themas “containers”
n face. Itisshownthatthefluidflowcanbe approximated
which can be used in the processes of drug or protein
o by the flow of the fluid bounded by an effective planar
c delivery [1, 2].
smooth solid surface. In Sec. III, the system of isolated
[ Anderson et al. analyzed theoretically hydrodynamic
spherical particles, permeable or with rough surfaces, is
1 thickness of thin polymer layers at solid-liquid sur- considered.
v faces [3], anddetermined the effect ofpolymer layersad- The hydrodynamic radius model is constructed and
9 sorbedto colloidalparticlesontheirbasichydrodynamic
justified, based on the multipole method of solving the
0 properties [4].
Stokes equations. In Sec. IV, hydrodynamic interactions
4
Inthis paper,the resultsfromRefs.3 and4aregener- betweenmany particles areanalyzed. The method ofre-
6
. alized for a wide class of the boundary conditions at the flections(scatterings)isdescribed,andtheRotne-Prager
1
particlesurfaces,andforarbitraryambientflows,includ- approximation is derived for the mobility of many par-
0
ing flows generated by other suspension particles. The ticles described by the hydrodynamic radius model. In
4
1 goalistoshowthathydrodynamicpropertiesofspherical Sec.V, the dynamics ofsuspensions is analyzed,andthe
: particles with a different internal structure and bound- role of direct interactions in averaging the effective mo-
v
ariescanbewell-approximatedonlyintermsoftheirhy- bility coefficients is discussed. Conclusionsare presented
i
X drodynamicradiianddirectinteractions,suchase.g. the in Sec. VI.
r no-overlapcondition or electrostatic repulsion.
a
Theconceptofthehydrodynamicradiusiswidelyused
intheliteratureasthebasicfeaturededucedfromsingle- II. BASIC CONCEPT: SHEAR FLOW CLOSE
TO A FLAT INTERFACE OR SOLID WALL
particle experiments, e.g. the DLS measurements of the
translational self-diffusion coefficients of dilute suspen-
sions, combined with the Stokes-Einstein relation [6], or Before focusing on hydrodynamics of spherical parti-
viscometric measurements of the intrinsic viscosity, sup- cles made of a given material, we analyze an auxiliary
plemented by the Einstein’s theory [5]. This concept is problem of a flat boundary between such a material and
oftenalsoappliedtonon-sphericalparticles[7]. However, the fluid. We assume that the half-space z > 0 is filled
inthisworkweanalyzetheconceptofthehydrodynamic with a viscous fluid, which at z = 0 is separated from a
radius only in the context of spherical particles, and we certainmedium,andmovesasashearflowwhenz →∞,
as shown in Fig. 1. The plane z = 0 represents the
boundary of a permeable medium, which fills the half-
space or forms a layer, or a flat surface which touches
∗Electronicaddress: [email protected] the highest peaks of the random or regular (periodic)
2
z
with the following boundary conditions,
vov xx ((zz))~∼=αα zz + β z >> Lh=β/α v(z)∼v0(z)=αzex for z →∞, (3)
and v(z) → 0 at the rough solid surface, or continuity
ofthe fluid velocityandtangentialstressatthe interface
z = 0 of a permeable medium, with v(z) → 0 at z →
−∞or at the other boundary of the permeable layer. In
particular, such boundary conditions exclude fluid-fluid
boundaries.
For z →+∞, the solution has the form,
v(z)≈αze [1+L /z+o(L /z)], (4)
z=0 x h h
L
h where Lh depends on the medium at z <0.
For example, consider a uniformly permeable medium
with the hydrodynamic screening length κ−1, described
by the Brinkman-Debye-Bueche equations [8, 9],
η(∇2−κ2)v(r)−∇p(r) = 0, (5)
FIG. 1: A simplified system: fluid bounded by an interface. ∇·v(r) = 0. (6)
The slip length Lh determines the position of the effective
hydrodynamicboundary. Ifthemediumfillsthehalf-spacez <0,thenthesolution
of the above problem gives
solid asperities, as illustrated in Fig. 2. L =κ−1. (7)
h
z=0
If the medium forms a layer, with −δ <z <0, then
L =κ−1tanh(κδ), (8)
h
seeEq.(20a)inRef. 3. The explicitformofL has been
h
alsoderivedforalayermadeofnon-uniformlypermeable
permeable half−space medium,withtheexponentialprofileofthesegmentden-
sity of the adsorbed polymer, see Eq. (20b) in Ref. 3.
For the Navier boundary conditions at the solid sur-
z=0
face z = 0 [12] (also called the stick-slip or mixed stick-
slip [13]), given in terms of the stress tensor σ =
ij
η(∂ v +∂ v )−δ p by the following expressions,
i j j i ij
λ
v = σ , for i=x,y, (9)
i iz
η
permeable layer
v = 0, (10)
z
z=0 the slip length is equal to λ,
L =λ. (11)
h
For a corrugated wall, L has been evaluated in
h
Ref. 10.
rough solid
FIG. 2: Examples of fluid boundaries, which are hydrody-
III. SPHERICAL PARTICLE IN AN INCIDENT
namicallywell-approximatedbyaneffectiveflatsolidsurface.
FLOW
The fluid velocity v(r) and pressure p(r) in the half-
A. Mutipole method
space z ≥0 satisfy the Stokes equations,
η∇2v(r)−∇p(r) = 0, (1) Now we consider a new system: a spherical particle of
∇·v(r) = 0, (2) radius a, immersed in an arbitrary incident flow v (r),
0
3
p (r)whichsatisfiestheStokesequations(1)-(2). Owing incident flow with a given l,m, scattered by a spherical
0
to the presence of this particle, the fluid velocity v (r)is particle, leads to the disturbance which is characterized
0
modified, by the same values of l,m.
ThecoefficientsX(l) areexpressedintermsoftheso-
v=v +v , (12) σ,σ′
0 1 called scattering coefficients A and B , which depend
lσ l2
on the boundary conditions at the particle surface and
with the disturbance v (called “the reflected flow”)[11]
1 its internal structure, as shown in Table I.
vanishing far from the particle center, which is located
at r =0, The coefficients Xσ(l,)σ′ (and therefore also the scatter-
ing coefficients A , B ) are evaluated from Eq. (18),
lσ l2
v (r)→0 for r →∞. (13) supplemented by the appropriate boundary conditions.
1
The scattering coefficients A , B are dimensional and
lσ l2
The disturbance depends on the particle internal struc- scale with the particle radius a as
ture,describedintermsoftheboundaryconditionsatthe
particlesurfaceandtheflowequationsinsidetheparticle. A (a)∼a2l−1, A (a),A (a)∼a2l+1, B (a)∼a2l+3.
Calculation of the flow v “reflected” by the particle l0 l1 l2 l2
1 (19)
immersed in the incident flow v is the basic procedure
0
in the method of multiple reflections used to determine
For example, for a hard solid sphere with the stick
fluid flows in many-particle systems. [11]
An arbitrary incident flow v (r) can be expressed as boundary conditions (i.e., v=0 at its surface), they are
0
given by the following expressions,
a linear combination of the elementary solutions, intro-
duced by Lamb [14], and expressed by him in terms of
2l+1
the solid harmonics, which form the basic set of the so- Ahs(a)=a2l−1 , Ahs(a)=a2l+1, (20)
lutions to the Laplace equation and are labeled by the l0 2 l1
indices Ahs(a)=a2l+12l+3, Bhs(a)=a2l+32l+1. (21)
l2 2 l2 2
l = 1,2,3,..., (14)
m = −l,...,l. (15) The scattering coefficients have been evaluated for a
wide class of spherical particles; among others, for the
EachLamb’selementarysolutionbelongsto oneofthree particles with the stick-slipboundary conditions attheir
families [14]. surfaces [16], the uniformly permeable particles [19, 20],
Linear combinations of the Lamb’s elementary solu- and the core-shell particles with uniformly permeable
tions, the multipole functions v+ (r), have been con- shells and solid cores [21].∗
lmσ
structed by Felderhof and Schmitz [15, 16] with the use The coefficients Alσ with l = 1,2 have a special phys-
of the irreducible representation of the group of rota- ical meaning, because they specify how does a particle
tions [17]. Three families of solutions have been labeled moveinanarbitraryincidentflow. InRef. 23,theFaxen
by an additional index, laws have been expressed in terms of the scattering co-
efficients (see also Ref. 21 for the expressions using the
σ =0,1,2. (16) presentnotation). Aparticlecenteredatr whichunder-
k
goesthe externalforceFandtorqueT, andthe ambient
The multipole functions vl+mσ(r) form a complete set flow v0, moves with the translational and rotational ve-
of regular elementary solutions to the Stokes equations, locities, UandΩ,andexertsonthefluidthe stressletS,
therefore,
v (r)= c+ v+ (r). (17)
0 lmσ lmσ
lXmσ TABLE I: The coefficients [−Xσ(l,)σ′/(2l+1)].
σ′ 0 1 2
Due to linearity of the problem, it is sufficient to focus
σ
on a single term of the above series. Each elementary
flow v+ (r) is “scattered” [18] by the sphere centered
at r =lm0,σand results in the following flow disturbance, 0 2l(l2+l−11)Al0 0 (2l−1)(2l+1)Al2
2 1 0 l(l+1)Al1 0
v0(r)=vl+mσ(r) ⇒ v1(r)=σ′=0Xσ(l,)σ′ vl−mσ′(r), (18) 2 (2l−1)(2l+1)Al2 0 (l+1)(2l+2l1)2(2l+3)Bl2
X
wherev− (r)formacompletesetofsuchelementaryso-
lmσ
lutionstotheStokesequations,whicharesingularatthe
sphere center [15, 16, 18, 22]. The advantage of choos-
ing v± (r) rather than the Lamb’s solutions is that the ∗ ThemisprintsfromRef. 21havebeencorrectedinRef.34.
lmσ
4
which satisfy the following relations, Analogical relations hold for B . Therefore, within the
l2
linear approximation, all the scattering coefficients are
A
F=4πηA U− 1+ 12 ∇2 v (r ) , (22) determined by a single parameter: the hydrodynamic
10 0 k
10A
(cid:20) (cid:18) 10 (cid:19) (cid:21) radius, no matter what are the details of their internal
1 structure and boundary conditions.
T=8πηA Ω− ∇×v (r ) , (23)
11 2 0 k Experimentally, the hydrodynamic radius can be de-
(cid:20) (cid:21)
termined by diffusion, sedimentation or viscosity mea-
4π A
Sαβ =− ηA20 1+ 22 ∇2 [∂αv0β(rk)+∂βv0α(rk)]. surementsforasingle-particle,withthe useofEqs.(25)-
3 14A
(cid:18) 20 (cid:19) (27). For example,
(24)
k T
The factors A10, A11 and A20 arerelatedto the coeffi- Dt = B . (31)
6πηa
cients of the particle translational, rotational and dipole eff
mobility, respectively. In addition, with the use of the
Within the linear approximation, the hydrodynamic
fluctuation-dissipation theorem, they also determine the
radii, given by Eqs. (25)-(27), measured in the trans-
translational and rotational self-diffusion, Dt and Dr,
lational diffusion, rotational diffusion and intrinsic vis-
and the intrinsic viscosity [η],
cosity experiments are the same; the difference is of the
k T higher order, O(L2/a2), with L =a−a .† Therefore,
Dt = B , (25) h h eff
4πηA inthelinearapproximation,allthescatteringcoefficients
10
are determined by a single hydrodynamic radius a .
k T eff
Dr = B , (26) In particular, within the hydrodynamic radius model,
8πηA
11 the Faxen laws (22)-(24) take the form,
A
20
[η]φ = φ, (27)
a3 a2
F=6πηa U− 1+ eff∇2 v (r ) , (32)
with the volume fraction φ = 4πna3/3 and the particle- eff 6 0 k
(cid:20) (cid:18) (cid:19) (cid:21)
number concentration n. 1
T=8πηa3 Ω− ∇×v (r ) , (33)
eff 2 0 k
(cid:20) (cid:21)
B. Hydrodynamic radius S =−10πηa3 1+a2eff∇2 [∂ v (r )+∂ v (r )]
αβ 3 eff 10 α 0β k β 0α k
(cid:18) (cid:19)
In this section, we consider a single spherical particle (34)
of radius a, made of the material and with the interface
describedinSec.IIbythesliplengthL ,seee.g. Eqs(7), Thehydrodynamicradiusmodelisapplicabletoawide
h
(8) or (11). The slip length L evaluated for the flat class of particles with a different internal structure and
h
geometry [3] determines also the slip on the surface of boundary conditions. Droplets and bubbles (and fluid-
the spherical particle [4], if fluid interface) are excluded, as discussed in Sec. II.
L /a<<1. (28)
h
Theflatgeometrycanbeunderstoodasthelimitingcase IV. MANY-PARTICLE HYDRODYNAMIC
INTERACTIONS
of a particle with a → ∞. The normalized scattering
coefficients, A (a)/Ahs(a) and B (a)/Bhs(a), are ex-
lσ lσ l2 l2
panded in 1/a→0 up to the linear terms, keeping fixed A. Method of reflections (scattering expansion)
all the other parameters of the boundary, the interior
medium and the way it extends inside from the bound- The fluid flow in many-particle systems is the sum of
ary. For the media and interfaces described by Eqs (7), the ambientflow andits disturbance causedby the pres-
(8) or (11), what correspondsto the particle types listed enceoftheparticles. Thedisturbancecanbeconstructed
after Eq.(21), the same structure of the limiting expres- by the same procedure as in Eqs. (17)-(18), repeated in
sions is obtained. For example, in case of σ = 0, the theprocessofmultiplereflections(orscatterings)[11],for
linear terms are given as eachflowv− (r)reflectedbyaparticlek withitscenter
lmσ
atr andtreatedasthe incidentflow v (r) incomingon
L L2 k 0
A (a) = Ahs(a) 1−(2l−1) h +O h . (29) a particle n.
l0 l0 a a2
(cid:20) (cid:18) (cid:19)(cid:21) The many-particle mobility can be also written as a
The slip length L is now used to define the hy- multiple scattering series. For example, consider the
h
drodynamic radius of an effective solid smooth sphere,
a =a−L . With this definition, we obtain for all val-
eff h
ues of l and σ the same universal formula,
† For the stick-slip boundary conditions, the difference is even
Alσ(a) = Ahlσs(aeff) 1+O La22h . (30) Rsmefa.ll1e3r., i.e. O(ξ2), where ξ = (a−aeff)/a = λ/(a+3λ), see
(cid:20) (cid:18) (cid:19)(cid:21)
5
translational-translationalmobility matrix µ , given by [27], it is described by the following, positive-definite,
ij
the relation, translational-translationalmobility matrix,
Ui = µij ·Fj, (35) µRiiP = 6π1ηaI, (39)
j
X
which determines velocity U of a particle i in a system
i
of particles j which undergo external forces Fj. µRP = 1 (I+rˆijrˆij)+2a2 (I−3rˆijrˆij) for i6=j,
The translational-translational mobility matrix µ ij 8πη r 3 r3
ij " ij ij #
canbewrittenasthefollowingsuperpositionofoperators
(40)
[24],
These expressions correspond to the first and the sec-
µ =µ (i)δ +(1−δ )µ (i)Z (i)G(ij)Z (j)µ (j)
ij 0 ij ij 0 0 0 0 ondterms inthe scatteringexpansion(36), i.e. the lead-
−µ (i)Z (i) G(ik)Zˆ (k)G(kj)Z (j)µ (j)+... ing terms in the inverse inter-particle distance, up to
0 0 0 0 0
O(1/r3).
k6=Xi,k6=j ij
The expressions (39)-(40) are therefore easily general-
(36)
ized for particles with a different internal structure and
This superposition is the multiplication and summation boundary conditions, and in general unequal radii. By
of the corresponding matrix elements, labeled by the in- inserting the appropriate scattering coefficients into Ta-
dices l, m ,σ. For example, the matrix elements of the bleIandEqs.(37),andtakingthefirstandsecondterms
single-particle friction operator Z (i) (which depends on in Eq. (36), we obtain the generalized Rotne-Prager ex-
0
theparticleiinternalstructureandboundaryconditions pressions,
at its surface) have the form [22],
1
µRP = I, (41)
Z0(lmσ,l′m′σ′) = −ηδll′δmm′Xσ(l,)σ′, (37) ii 4πηA10(ai)
and G(ik) represents multipole elements of the Oseen (I+rˆ rˆ ) 1 A (a ) A (a ) (I−3rˆ rˆ )
tensor T0(rik) [11], where rik is the relative position of µRijP= 8πηijr ij +5 A12(ai)+A12(aj) 8πηirj3 ij ,
the centers of particles i and k, see e.g. Ref. [24, 25] for ij (cid:18) 10 i 10 j (cid:19) ij
the explicit expressions. for i6=j. (42)
The operator µ (i) shown in Eq. (36), involves only
0 The scattering coefficients, which appear in the above
onemultipoleelementofthesingle-particletranslational-
equations,havebeenexplicitlyevaluatedforawiderange
translational mobility,
of the particle models [22, 28].
µ (1m0,1m0)= Z (1m0,1m0)−1. (38) The question is how to relate the generalized Rotne-
0 0
Prager mobility to experiments. The self-mobility, given
The interpretation of the second term in Eq. (36) is by Eq. (41), is determined e.g. from Eq. (25), with the
the following. G(ij)Z (j)µ (j) represents the flow gen- measured value of the translational self-diffusion coeffi-
0 0
eratedbythepresenceofaparticlej 6=iandincomingon cient. Such an experiment allows to define the hydro-
theparticlei. Whenµ (i)Z (i)isappliedtothisflow,it dynamic radius according to the Stokes law, given by
0 0
results in a change of velocity of the particle i after this Eq. (31), with separate values of Dit and aeff,i for each
single scattering. species i. Therefore,
The interpretation of the third term in Eq. (36) is the
a =2A (a )/3. (43)
following. Each multiple scattering from a particle k 6=i eff,i 10 i
adds an additional contribution to velocity of particle i,
The key point is that the Rotne-Prager mobility µRP,
now with the requirement that no additional force nor ij
given by Eq. (42), is well-approximated with the use of
torque on the particle k is induced as the result of the
scattering event (this requirement leads to Zˆ (k) rather thehydrodynamicradiionly. Indeed,takingintoaccount
than Z (k), with the definition and explicit e0xpressions the expansion of the scattering coefficients, given by
0 Eq.(30),andusingEqs.(20)-(21),werewriteEq.(42)as
given e.g. in Ref. [26]).
1 (I+rˆ rˆ ) a2 +a2 (I−3rˆ rˆ )
µRP= ij ij + eff,i eff,j ij ij +...
B. Rotne-Prager approximation ij 8πη" rij 3 ri3j #
(44)
First,considerasystemofhardsolidsphericalparticles
of radii a with the stick boundary conditions, with the In this way we obtain the Rotne-Prager approximation
distance between the centers of spheres with labels i6=j for the translational-translational mobility of the solid
denoted as r . Within the Rotne-Prager approximation particles with the hydrodynamic radii a .
ij eff,i
6
The first term in Eq. (44) scales as ∼ 1/r and it bythepresenceofacloseparticlesurfaceneedstobeeval-
ij
is exact and independent of the particle internal struc- uatedbymultiplereflections,describedbythescattering
ture andboundaryconditionsonits surface. The second coefficientswithhighervaluesofl. Thelargerl,thelarger
term scales as ∼1/r3 and it is approximate; within the isthe O[(L /a)2]correction,andthesmalleraccuracyof
ij h
O([(a−a )/a]2) accuracy,it depends onthe the particle the hydrodynamic radius approximation to the scatter-
eff
internalstructureandboundaryconditionsonlythrough ing coefficients, given by Eq. (30). These coefficients are
the hydrodynamic radius. As it follows from Eq. (30), needed to evaluate and average the mobility matrix for
the dots in Eq. (44) mean O([(a−a )/a]2). systems of close particles. Therefore, the larger volume
eff
fraction, the smaller accuracy of the hydrodynamic ra-
dius approximationof the transport coefficients.
V. EFFECTIVE HYDRODYNAMIC
Examplesofthenumericalestimatesoftheaccuracyof
PROPERTIES OF SUSPENSIONS
thehydrodynamicradiusapproximationtothetransport
coefficients can be found in Refs. [29–31, 34]. The direct
A. Theoretical description
interactions between particles considered there are given
by the no-overlap condition, with the no-overlap radius
We now move on to the description of suspension equal to the geometrical radius a.
transport-properties. They aregivenin termsof suchef-
Refs. [29–31, 34] contain reference data for the trans-
fective coefficients as e.g. the sedimentation, short-time
port coefficients of suspensions made of solid particles
diffusion and high-frequency viscosity coefficients. They
withhydrodynamicradiia ,with largerno-overlapradii
ff
are evaluated by averaging the corresponding mobility
a. These results can be used to model suspensions made
coefficients for systems of many particles. For example,
ofparticlesofdifferentinternalstructureandboundaries.
the translational self-diffusion coefficient is given by the
The example of the no-overlap direct interactions be-
following expression,
tween permeable or core-shell particles in suspensions,
<Trµ > discussedinRefs[29–31,34],indicatesthatinbothcases,
D = k T ii , (45)
s B 3 the typical inaccuracy of the hydrodynamic radius ap-
where T is the temperature, k denotes the Boltzmann proximationisverysmall: comparabletoorlessthan5%,
B
constant, and Tr stands for the trace of a Cartesian ma- even for very condensed systems with φ=0.45 [29, 34].
trix. D is obtained by averaging coefficients of the di- In particular, it is worthwhile to discuss here the sys-
s
agonaltranslational-translationalmany-particlemobility temsofparticlesofradiiccoveredbythinpolymershells
matrix µ , with respect to the directions in space, par- of thickness δ (the outer radii are a = c + δ), with
ii
ticle labels i, and positions of all the particles, with the ǫ = δ/c < 1. Ref. [31] is focused on the dynamics
equilibrium distribution. of dilute suspensions of non-overlapping core-shell par-
From the analysis of the scattering coefficients, per- ticles with thin, uniformly permeable shells, and the ra-
formedintheprevioussections,itfollowsthatthemany- tio x = cκ of the inner radius c to the hydrodynamic
body-mobilitycoefficientsforparticlesofadifferentinter- screening length κ−1 of the permeable medium. In this
nalstructureandboundariescanbeapproximatedbythe reference,thevirialcoefficients,whichcorrespondtopair
many-body-mobilitycoefficients for solidparticles atthe hydrodynamicinteractions,havebeenevaluatedanddis-
same configurationof their centers, but with the smaller cussed.
hydrodynamic radii, determined from single-particle ex- Surprisingly, even for quite thick shells, with the rela-
periments. tive thickness ǫ=δ/c≤0.3 and x≥10, the uncertainty
The averaging of the mobility coefficients depends on ofthemodelissmallerthan0.2,1,2,3%forthesedimen-
theprobabilitydistributionoftheparticleconfigurations. tation,translationalself-diffusion,high-frequencyviscos-
Thisdistributionfollowsfromdirectinteractionsbetween ity and rotational self-diffusion, respectively. This un-
particles. Examples of direct interactions are Yukawa or certainty is significantly smaller than ǫ2. The explana-
Debye-Hu¨ckel external forces, or the no-overlap require- tion of this high precision is simple: the relative shell
ment. thickness ǫ is not the relevant parameter. The relevant
Therefore, the appropriate approximation of suspen- parameter is the relative hydrodynamic shell thickness,
sion effective transport properties is to evaluate the mo- Lh/a = (a−aeff)/a, with Lh given by Eq. (8). Lh/a is
bility coefficients for the solidparticles with the effective typically much smaller than ǫ. For example, for x = 10
radii, and average them with the correlation functions and ǫ = 0.3, the relative hydrodynamic shell thickness,
based on the unchanged direct interactions between the L /a≈0.08−0.09,is3-4timessmallerthanthegeomet-
h
particles. rical one.
This example illustrates the essential difference be-
tweenthe strategypresentedinthispaperincomparison
B. Discussion to Ref. [4]. In the last reference, ǫ (called there λ) was
used as the expansion parameter, and the O(ǫ2) correc-
Thequestionariseshowaccurateistheaboveapproxi- tions had to be evaluated, because they were not small.
mation. For very dense systems, the fluid flow disturbed Here, we use a much smaller expansionparameter L /a,
h
7
andastheresult,thetermsO((L /a)2)canbeneglected, encebetweentheparticlegeometricalandhydrodynamic
h
and no details of the particle internal structure and sur- radii.
face are needed, except the value of the hydrodynamic Dynamics of suspensions of permeable, core-shell or
radius. rough spherical particles, or model particles with the
Navier stick-slip boundary conditions, can be described
with a reasonable accuracy by evaluating many-particle
VI. CONCLUSIONS hydrodynamic mobility coefficients within the hydrody-
namic radius approximation, and averaging them with
Fromtheanalysisdescribedinthispaperitfollowsthat theconstraintthatdirectinteractionsbetweenthesepar-
spherical particles of radius a, characterized by a wide ticles are kept unchanged.
rangeofdifferentinternalstructuresandboundaries,dis- It has been checked that for a wide range of particle
turbtheincidentflowinapproximatelythesameway: as internal structures and boundaries, and the no-overlap
effectivesolidparticlesofasmallerradiusa ,calledhere direct interactions, deviations from the above approxi-
eff
the hydrodynamic radius. The relative differences are of mation are very small [29, 31, 32, 34] and as such, hard
the order of [(a−a )/a]2. to be detected in experiments, even for relatively dense
eff
Thereasoningpresentedheredoesnotrefertodroplets suspensions.
nor bubbles. It applies, for example, to suchsystems are
permeable, core-shell or rough spherical particles, or the
particles with the Navier stick-slip boundary conditions. Acknowledgments
The results obtained for the last model of the particle
internal structure are especially meaningful: the Navier M.L.E.-J.andE.W.weresupportedinpartbythePol-
slip length λ cannot be arbitrarily chosen or fitted to ish NCN Grant No. 2011/01/B/ST3/05691. M.L.E.-J.
modelrealistic particlesof a complexstructure; its value acknowledges the scientific benefits from the COST Ac-
follows from the scattering theory as a−a , the differ- tion MP1106.
eff
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