First-order virial expansion of short-time diffusion and sedimentation coefficients of permeable particles suspensions

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by  Bogdan Cichocki

First-order virialexpansionofshort-timediffusion andsedimentation coefficients ofpermeable
particles suspensions
Bogdan Cichocki,1 Maria L. Ekiel-Jez˙ewska,2,∗ GerhardNa¨gele,3 and Eligiusz Wajnryb2
1InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw, Hoz˙a69,00-681Warsaw, Poland
2InstituteofFundamentalTechnologicalResearch,PolishAcademyofSciences,Pawin´skiego5B,02-106Warsaw,Poland
3Instituteof ComplexSystems, ICS-3, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
(Dated:January25,2011)
Forsuspensionsofpermeableparticles,theshort-timetranslationalandrotationalself-diffusioncoefficients,
and collective diffusion and sedimentation coefficients are evaluated theoretically. An individual particle is
modeledasauniformlypermeablesphereofagivenpermeability,withtheinternalsolventflowdescribedby
theDebye-Bueche-Brinkman equation. The particlesareassumed tointeract non-hydrodynamically by their
excluded volumes. The virial expansion of the transport properties in powers of the volume fraction is per-
formeduptothetwo-particlelevel.Thefirst-ordervirialcoefficientscorrespondingtotwo-bodyhydrodynamic
interactionsareevaluatedwithveryhighaccuracybytheseriesexpansionininversepowersoftheinter-particle
1 distance.Resultsareobtainedanddiscussedforawiderangeoftheratio,x,oftheparticleradiustothehydro-
1 dynamicscreeninglengthinsideapermeablesphere. Itisshownthatforx >∼ 10,thevirialcoefficientsofthe
0
transportpropertiesarewell-approximatedbythehydrodynamicradius(annulus)modeldevelopedbyusearlier
2
fortheeffectiveviscosityofporous-particlesuspensions.

n
a PACSnumbers:82.70.Dd,66.10.cg,67.10.Jn
J

3
2 I. INTRODUCTION radiustothehydrodynamicpenetrationdepthinsideaperme-
able sphere. Large (low) values of x correspond to weakly

] (strongly)permeableparticles. Whilethemodelofuniformly
t One of the theoretical methods to analyze transport prop-
f permeablehardspheresignoresaspecificintra-particlestruc-
o erties in suspensions of interacting colloidal particles is the
ture,itisgenericinthesensethatthehydrodynamicstructure
s virial expansion in terms of the particle volume fraction φ.
. of more complex porous particles can be approximately ac-
t Forsuspensionsofnon-permeablehardsphereswithstickhy-
a counted for in terms of a mean permeability. In our related
drodynamicboundaryconditions,virialexpansionresultsfor
m previous publications, using a hydrodynamicmultipole sim-
short-time properties are known to high numerical precision
- up to the three-particlelevel, i.e., to quadraticorderin φ for ulation method of a very high accuracy[21], encodedin the
d
n diffusion and sedimentation coefficients [1, 2], and to third HYDROMULTIPOLEprogrampackage[1],wehavecalculated
theshort-timetranslationaldiffusionproperties[17, 18], and
o order in φ for the effective viscosity [3]. The concentration
thehigh-frequencyviscosity[19,20]ofthepermeablespheres
c range of applicability of these hard-sphere virial expansion
[ results in comparison to simulation data has been discussed model, as functionsof φ and x. These results coverthe full

1 in[4]. Ourknowledgeonvirialexpansioncoefficientsofcol- range of permeabilities, with volume fractions extending up
totheliquid-solidtransition.
v loidaltransportpropertiesislessdevelopedwhensuspensions
6 ofsolvent-permeableparticlesareconsidered.Thetheoretical While the simulation results are importantfor the general
9 descriptionof their dynamicsis more complicatedsince one understanding of permeability effects in concentrated sys-
3 needstoaccountforthesolventflowalsoinsidetheparticles. tems,forpracticaluseinexperimentaldataevaluationandas
4 Permeableparticlesystemsarefrequentlyencounteredinsoft inputin long-timetheories, virialexpansionresultsbasedon
.
1 matter science. Prominent examples of practical relevance arigoroustheoreticalcalculationarestillstronglyondemand.
0 which are the subject of ongoing research, are dendrimers Infact,theknowledgeoftheleading-ordervirialcoefficients
1 [5, 6], microgel particles [7–9], a large variety of core-shell canbe a goodstartingpointin derivingapproximateexpres-
1
particles with a dry core and an outer porous layer [10–14], sionsfortransportproperties,whichmaybeapplicableatcon-
:
v fractalaggregates[15],andstar-likepolymersoflowerfunc- centrationsmuchhigher than those where the original(trun-
Xi tionality[16]. cated)virialexpansionresultisuseful. Anexampleincaseis
providedin our recent derivation[20] of a generalizedSaitoˆ
Inaseriesofrecentarticles[17–20],wehaveexploredthe
r
a genericeffectofsolventpermeabilityontheshort-timetrans- expression for the effective high-frequency viscosity η∞ of
permeable spheres, based on the second-orderconcentration
portusingthemodelofuniformlypermeablecolloidalspheres
expansionresult,i.e. aHugginscoefficientcalculation,ofthis
with excluded volume interactions. This simple model is
property.
specifiedbytwoparametersonly,namelytheparticlevolume
fraction φ = (4π/3)na3, where n is the number concentra- In[19],wehaveperformedvirialexpansioncalculationsfor
tionandaistheparticleradius,andtheratioxoftheparticle η∞.Wehaveinvestigatedthereinasimplifyinghydrodynamic
radius model (HRM), where a uniformly permeable sphere
of radius a is described by a spherical annulus particle with
an inner hydrodynamic radius a (x) < a, and unchanged
eff
∗Electronicaddress:[email protected] excluded-volumeradiusa. Inthisannulusmodel(HRM),the
2
Huggins coefficient describing two-body viscosity contribu- In this paper, the two-particle translational-translationalmo-
tions has been evaluated and shown to be in a remarkably bility matrices µtt(r ,r ) are evaluated using the multipole
ij 1 2
good agreement with the precise numerical data for porous method of solving the Stokes an DBB equations [21]. The
particles, characterizedbya widerangeofpermeabilitiesre- clusterexpansionoftheabovemobilitymatricesreads,
alizedinexperimentalsystems.
Inthe presentarticle, the aforementionedtheoreticalwork µtt(r ,r )=µtδ 1+µtt(2)(r), (6)
ij 1 2 0 ij ij
onthevirialexpansionofthehigh-frequencyviscosityisgen-
eralized to short-time diffusion properties. In Sec. II, the wherer=r2−r1 and
virial expansion is performed for the translational and rota-
1
stieodniamlesnetlaf-tidoinffucosieofnficcioeenftfiKci,eanntsdDthteaansdsoDcira,terdescpoelcleticvteivlye,dthife- µt0 = 4πη0A10 (7)
fusioncoefficientD =KDt/S(0).Here,S(0)isthesmall-
C 0 isthesingleporous-particletranslationalmobility. HereA
wavenumberlimitofthestaticstructurefactor,andDt isthe 10
0 isasingleporous-particlescatteringcoefficient[26],givenex-
single-particle translationaldiffusioncoefficient. In Sec. III,
plicitlyinAppendixA. Foranon-permeablehardspherewith
weprovidehighlyaccuratenumericalvaluesforthefirst-order
stickboundaryconditions,Ahs =3a/2.
(i.e.,two-particle)virialcoefficients,λ (x),λ (x),λ (x)and 10
t r C The single particle scattering coefficients A , with l =
λ (x), ofD ,D ,D andK,respectively,inthefullrange lσ
K t r C 1,2,3,4,... and σ = 1,2,3 [26], are essential to perform
ofpermeabilities. InSec.IV,wealsorecalculatethesevirial
the multipole expansion. They determine the corresponding
coefficientsapproximatelyonthebasisofthesimplifyingan-
multipoles of the hydrodynamic force density on a particle
nulusmodel(HRM).InSec.V,weconcludethatintherange
immersedinanambientflow;examplesareEqs.(7)and(24).
x>∼10typicalofmanypermeableparticlesystems,thevirial ThesamecoefficientsA specifyalsothecorrespondingmul-
coefficientsarewellapproximatedbytheannulusmodel. lσ
tipolesofthefluidvelocity,reflected(scattered)byaparticle
immersedinagivenambientflow.ThisiswhyA arecalled
lσ
“scattering coefficients”. In the multipole approach, differ-
II. THEORY
encesin the internalstructureof particles(e.g. solid, liquid,
gas,porous,core-shell,stick-slip),arefullyaccountedbydif-
We consider a suspension made of a fluid with shear vis-
ferentscatteringcoefficients. Theotherpartsofthemultipole
cosityη andidenticalporousparticlesofradiusa. Thefluid
0 algorithmneednottobechanged. Thescatteringcoefficients
flowischaracterizedbyReynoldsnumberRe<< 1. Outside
are the matrix elements of two single-particle friction oper-
the particles, the fluid velocity v and pressure p satisfy the ators, Z and Zˆ , which determine the hydrodynamic force
Stokesequations[22,23], 0 0
densityexertedbyagivenambientflowona motionlessand
η ∇2v(r)−∇p(r) = 0 afreelymovingparticle,respectively.
0
∇·v(r) = 0, (1) Inthemultipoleexpansionmethod,thetwo-particlemobil-
ityµ(2)(1,2)(e.g.translationalone,asinEq.(6),orrotational
and inside the particles, the Debye-Bu¨che-Brinkman(DBB) one,asinEq.(23))isexpressedintermsofthesingle-particle
equations[24,25], friction operators, Z (i) and Zˆ (i), with i = 1,2, and the
0 0
η ∇2v(r)−η κ2[v(r)−u (r)]−∇p(r) = 0 Green operator G(1,2). The later relates the flow outgoing
0 0 i
fromparticle2andincomingonparticle1.Wecanwriteµ(2)
∇·v(r) = 0, (2)
asaninfinitescatteringseries,
whereκ−1isthehydrodynamicpenetrationdepth.Theskele-
ton of the particle i, centered at ri, moves rigidly with the µ(2) = µ0Z0(1+GZˆ0)−1GZ0µ0
localvelocityui(r)=Ui+ωi×(r−ri),determinedbythe = µ Z GZ µ −µ Z GZˆ GZ µ +.... (8)
translational and rotational velocities U and ω of the par- 0 0 0 0 0 0 0 0 0
i i
ticle i, respectively. The fluid velocity and stress tensor are Since the multipole matrix elements of the Green tensor G
continuousacross a particle surface. The effectof the parti- scaleasinversepowersoftheinterparticledistancer,Eq.(8)
cleporosityisthereforedescribedbytheratioxoftheparti- correspondsto a power series in 1/r. Truncatingthe expan-
cleradiusatothehydrodynamicscreeninglengthκ−1 ofthe sionat order1/r1000, we obtaina veryhighprecisionof the
porousmaterialinsidetheparticle,i.e. mobility calculation, actually much higher than needed for
anypracticalapplications.
x = κa. (3)
In the presentwork, we investigatethe short-timedynam-
OwingtolinearityoftheStokesandDBBequationsandthe ics,attimescalest<<a2/Dt,where
0
boundaryconditions,theparticlevelocityU dependslinearly
i
ontheexternalforcesFj exertedonaparticlej. Inparticular, D0t =kBTµt0 (9)
fortwointeractingsphericalparticles, 1,2, intheabsenceof
externaltorquesandflows, is the single-particle translational diffusion coefficient, with
theBoltzmannconstantk andtemperatureT. Ontheshort-
U1 = µt1t1(r1,r2)·F1+µt1t2(r1,r2)·F2 (4) timescale,thesystemisdBescribedbytheequilibriumparticle
U = µtt(r ,r )·F +µtt(r ,r )·F . (5) distribution [27]. In the further analysis, we will need only
2 21 1 2 1 22 1 2 2
3
the small-concentration limit g (r) of the pair distribution Forthenon-overlappingspheres[27],
0
function, where r is the interparticle distance. For particle-
particledirectinteractionsdescribedbyapairpotentialV(r), S(0)=1−8φ+O(φ2). (21)
thispairdistributionisg (r) = exp(−V(r)/k T). Fornon-
0 B Inthiscase,
overlappingspheresofradiusa,
0 for r ≤2a, λC =λK +8. (22)
g (r) = (10)
0 1 for r >2a.
(cid:26) Therelation(22)followsfromEqs.(14),(19)and(20).
We proceed by analyzing the short-time rotational self-
Thefirst-ordertermsinthevirialexpansionoftheshort-time
diffusion coefficient. In the absence of external forces and
transport coefficients are obtained by averaging the corre-
flows,thetwo-particlerotational-rotationalmobilitymatrices
sponding two-particle mobility elements. As the result, the
virialcoefficientsareobtainedasintegrals,whichinvolvethe µrijr(r1,r2)satisfytherelationanalogicaltoEqs.(4)-(5),with
thetranslationalvelocitiesreplacedbytherotationalones,and
mobilityelementsandg (r).
0
the forces replaced by the torques. The two-particle cluster
The first-order virial expansion of the short-time transla-
expansionnowreads,
tionalself-diffusioncoefficienthastheform,
D =Dt(1+λ φ+O(φ2)). (11) µrr(r ,r )=µrδ 1+µrr(2)(r), (23)
t 0 t ij 1 2 0 ij ij
Thecoefficientλtisgivenbytherelation[28], with
+∞ 1
λt =8 g0(R)Jt(R)R2dR, (12) µr0 = 8πη A . (24)
Z1 0 11
withR=r/2aand The scattering coefficient A11 for a porous particle [26] is
giveninAppendixA. Foranon-permeablehardspherewith
Jt(R)= µ1tTrµt1t1(2)(r). (13) theTshteicvkibrioaulnedxaprayncsioonndoitfiotnhse,rAoh1tas1ti=onaa3l.self-diffusioncoeffi-
0
cientis
Here, Tr denotes the trace operation. For the sedimentation
coefficient,oneobtains, D =Dr(1+λ φ+...), (25)
r 0 r
K =1+λ φ+... (14)
K where
where
Dr =k Tµr (26)
0 B 0
2 8 +∞
λK = 5a3A12+ aA10 [g0(R)−1]RdR (15) and
+∞ Z0 +∞
+8 g0(R)JK(R)R2dR, (16) λr =8 g0(R)Jr(R)R2dR, (27)
Z1 Z1
and with
1 1
J (R)= Tr µtt(2)(r)+µtt(2)(r)−T (r) . (17) J (R)= Trµrr(2)(r). (28)
K µt 11 12 0 r µr 11
0 0
h i
Intheaboveexpression,
1+rr III. RESULTS
T (r)= , (18)
0
8πη r
0
Toevaluatethefirstordervirialcoefficientsλ=λ ,λ ,λ ,
istheOseentensorandr=r/r. bb wecalculatetwo-particlemobilitymatrixelements,tperKformr-
ThescatteringcoefficientA12 fora porousparticle[26] is ingaseriesexpansioninpowersof1/ruptotheorder1000,
given explicitly in Appendix A. For a non-permeable hard asdescribedinSec.II. Integrationwithrespecttotheparticle
b
spherewiththestickboundaryconditions,Ahs =5a3/2. positionshasbeenperformedanalyticallytermbyterm,using
12
Thecollectivediffusioncoefficientisgivenby theexpressionsgivenintheprevioussection. Thevirialcoef-
ficientsλhavebeenevaluatedforawiderangeofx. Selected
D =DtK/S(0), (19)
C 0 resultsarelistedinTableI. Alldisplayeddigitsaresignificant.
(Moredataareavailableonrequest.)
where S(0)is the zero-wavenumberlimit of the static struc-
In Fig. 1, the first order viral expansion of the short-time
turefactor,S(0)=limq→0S(q). Thefirst-ordervirialexpan-
translationalself-diffusioncoefficientofaporous-particlesus-
sionofEq. (19)hastheform,
pensioniscomparedwiththeaccuratesimulationresultsper-
D = Dt(1+λ φ+...). (20) formedinRef.[17]forvolumefractionsφ ≤ 0.45. Thefirst
C 0 C
4
spheres. For uniformly porous particles, the boundary
TABLE I: First virial coefficients λt, λK and λr for the short- collocation method was applied by Chen and Cai [30]
time translational self-diffusion, sedimentation and rotational self-
to evaluate λ = −3.46, −5.50, −6.23, −6.44 for
diffusion,respectively. K
x2 = 10α, with α = 1,2,3,4, respectively. Com-
x λ λ λ
t K r paring their results with our very accurate values, λ =
K
3 -0.2497 -3.4451 -0.03257 −3.5723, −5.5480, −6.2504, −6.4563, we conclude that
4 -0.4159 -4.1066 -0.06336 theuncertaintyoftheirresultsisdecreasingfrom3%atα=1
to0.3%atα = 4. Theaccuracyoftheboundarycollocation
5 -0.5692 -4.5539 -0.09682
methodisworseatsmallervaluesofx,i.e. forlargerperme-
6 -0.7021 -4.8722 -0.12956
abilities.
7 -0.8149 -5.1084 -0.16012
In Figs. 2-4, the first-order virial coefficients λ are plot-
8 -0.9102 -5.2898 -0.18802
ted versus the porosity parameter 1/x, for 1/x ≤ 0.1, i.e.
9 -0.9909 -5.4328 -0.21327 for x ≥ 10. For 1/x = 0, i.e. for x = ∞, the limit
10 -1.0598 -5.5480 -0.23606 of a non-permeable hard sphere with radius a is recovered,
11 -1.1190 -5.6426 -0.25662 λhs = λ(∞). AsshowninFig.2,foralowpermeability,the
13 -1.2151 -5.7884 -0.29208
16 -1.3202 -5.9380 -0.33426 −4.5
18 -1.3730 -6.0095 -0.35692
20 -1.4161 -6.0662 -0.37628
−5
30 -1.5499 -6.2335 -0.44202
40 -1.6190 -6.3149 -0.48007
50 -1.6610 -6.3628 -0.50497 −5.5
65 -1.7001 -6.4064 -0.52959 K
λ
100 -1.7460 -6.4563 -0.56075
−6
∞ -1.8315 -6.5464 -0.63054
−6.5
1
−7
0 0.02 0.04 0.06 0.08 0.1
0.8 1/x
FIG. 2: Two-particle sedimentation virial coefficient λK(x). Our
tD00.6 preciseresultsforporousparticles(solidline)arewell-approximated
D/t bytheannulusmodel(dashedline).
x=5 coefficient λ ≈ λhs + 10/x is approximately a linear
K K
0.4 x=10 function of 1/x. The coefficients λ and λ as functions of
t r
x=30 1/xareshowninFigs.3and4.
x=∞
0.2
0.05 0.15 0.25 0.35 0.45
φ
FIG.1: Translational self-diffusioncoefficient Dt forasuspension
ofporousparticles. Symbolsconnectedbysplines(solidlines): ac-
curatesimulationresultsfromRef.[17]. Dashedstraightlines:first-
ordervirialexpansioncalculatedinthiswork.
ordervirialexpansion,seeEq.(11),canbeusedasanaccurate
approximationofD inaverywiderangeofvolumefractions,
t
evenforrelativelylargevaluesofx(i.e. lowpermeabilities).
In contrast, values of the sedimentation coefficient K
differ significantly from the first-order virial estimation al-
ready at rather small volume fractions, see Ref. [17].
Our values of λ (∞) reproduce with a higher accuracy
K
the classic Batchelor’s result [29] for non-permeable hard
5
0 ThemethodusedtodetermineλAisdescribedinAppendixB.
ThecalculatedvaluesarelistedinTableII.
−0.5 TABLE II: First-order virial coefficients λAt , λAK and λAr for the
short-time translational self-diffusion, sedimentation and rotational
self-diffusion,respectively,forasuspensionoftheannulusparticles.
λt −1 ǫ λAt λAK λAr
0.00 -1.8315 -6.5464 -0.63055
0.01 -1.7523 -6.4601 -0.56666
−1.5 0.02 -1.6793 -6.3769 -0.51671
0.03 -1.6109 -6.2962 -0.47417
0.04 -1.5466 -6.2179 -0.43699
−2 0.05 -1.4860 -6.1419 -0.40402
0 0.02 0.04 0.06 0.08 0.1
1/x 0.06 -1.4286 -6.0680 -0.37451
0.07 -1.3743 -5.9962 -0.34791
FIG. 3: Two-particle translational self-diffusion virial coefficient 0.08 -1.3228 -5.9263 -0.32381
λt(x). Ourpreciseresultsforporousparticles(solidline)arewell- 0.09 -1.2739 -5.8582 -0.30189
approximatedbytheannulusmodel(dashedline).
0.10 -1.2274 -5.7918 -0.28187
0.11 -1.1832 -5.7272 -0.26354
0 0.13 -1.1008 -5.6027 -0.23122
0.18 -0.9253 -5.3166 -0.16974
−0.1
0.24 -0.7595 -5.0135 -0.12034
0.31 -0.6111 -4.7051 -0.08296
−0.2
0.45 -0.4093 -4.1986 -0.04242
−0.3 0.66 -0.2401 -3.6278 -0.01775
λr
−0.4
Now we are going to comparethe first-order virial coeffi-
cients,calculatedintheprevioussectionforporousparticles,
−0.5
withthecorrespondingresults,obtainedinthissectionforthe
annulus model, called also the hydrodynamic radius model
−0.6
(HRM).AsimilarcomparisonhasbeendoneinRef.[19]for
the effective viscosity. The key concept in this procedure is
0 0.02 0.04 0.06 0.08 0.1 thehydrodynamicradiusofaporousparticle. Forthetransla-
1/x tionaldiffusion(self-diffusionandsedimentation),thehydro-
dynamicradiusat isobtainedfromthesingle-particletrans-
eff
FIG. 4: Two-particle rotational self-diffusion virial coefficient lationaldiffusioncoefficient,withtheuseoftherelation,
λr(x). Ourpreciseresultsforporousparticles(solidline)arewell-
approximatedbytheannulusmodel(dashedline). k T
Dt = B . (30)
0 6πη at
0 eff
IV. ANNULUSMODEL The dependence of at on the porosity parameter x follows
eff
fromEqs. (7) and(9), whichdeterminethetranslationaldif-
Intheannulusmodel[31],aparticlesuspendedinaviscous fusioncoefficientofasingleporousparticle[24,25,32],and
fluid is characterizedby two radii, a< and a>. Its hydrody- Eq. (A1), whichspecifiesthescatteringcoefficientA10. Ex-
namicinteractionsaregovernedbythesmallerradiusa . In plicitly,
<
addition, there exist also direct pair interactions. Two parti-
clescannotcometooclosetoeachother,withtheno-overlap at (x) = a 2x2(x−tanh(x)) . (31)
radius equal to a . For such a model, the first-order virial eff 2x3+3(x−tanh(x))
>
expansion of transport coefficients has been performed with
respecttothevolumefractionφ = (4π/3)na3. Thecorre- For the rotational self-diffusion, the hydrodynamicradius
> >
spondingfirst-ordervirialcoefficientsλA =λA,λA,λAhave ar is obtained from the rotational diffusion coefficient of a
t K r eff
beenevaluatedasfunctionsofǫ,where singleporousparticle[32,33],withtheuseoftherelation,
ǫ = a>−a<. (29) Dr = kBT . (32)
a< 0 8πη0(areff)3
6
The dependence of ar on the porosity parameter x follows low. Systematically, the annulus approximation slightly un-
eff
from Eqs. (24) and (26), with the scattering coefficient A derestimatesthevirialcoefficientsofporousparticlessuspen-
11
givenbyEq.(A2). Astheresult, sions. For rotationaldiffusion, a reasonable 5% accuracy of
this modelis reached at x >∼ 20. For translationaldiffusion
3 3coth(x) 1/3 (collectiveandself),thecomparableorevenbetter3-5%pre-
areff(x) = a 1+ x2 − x . (33) cision is obtained already for x >∼ 10. For the sedimenta-
(cid:20) (cid:21)
tion coefficient, the accuracy is even higher (a 3% precision
Forlargex, already at x = 5), owing to much larger absolute values of
λ =8+λ .
ar (x) = at (x)+O 1/x2 . (34) K C
eff eff Theannulusmodelis expectedto workwellalso atlarger
volume fractions, if the porosity parameter x is sufficiently
Aporousparticleofradiusaandth(cid:0)eporo(cid:1)sityparameterx
large. The accuracy of this approximation at larger volume
ismodeledasanannulusparticle,seeFig.5,bymatchingits
fractionswillbeinvestigatedinaseparatepublication.More-
geometricalradiusatotheannulusno-overlapradiusa =a.
> over, the annulus model is also useful to estimate diffusion
The smaller annulus radius a = a is determined by the
< eff andshearviscositycoefficientsofsuspensionsmadeofparti-
effectivehydrodynamicradiusa ,giveninEqs. (31)-(33).
eff clescharacterizedbyadifferentinternalstructure,suchasthe
core-shell.Thisproblemwillbethesubjectofafuturestudy.
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)a(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(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aeff a The scatterAinpgpencdoiexffiAc:iSencatsttefroinrgacoeufnfiicfioenrmtsly permeable
POROUS ANNULUS
sphere with x = κa, where a is the sphere radius and 1/κ
isthehydrodynamicpenetrationdepth,havetheform[32],
FIG. 5: The annulus (or hydrodynamic radius) model of a porous
particle. (2l+1)g(x) l(2l−1)(2l+1)g(x) −1
A = l 1+ l a2l−1,
In this way, the annulusparameter ǫ, defined in Eq. (29), l0 2gl−2(x) (cid:20) (l+1)x2gl−2(x) (cid:21)
(A1)
becomesthefollowingfunctionofx,
g (x)
a−a (x) Al1 = l+1 a2l+1, (A2)
ǫ(x) = eff , (35) gl−1(x)
a (x)
eff 2l+3 2(2l+1)(2l+3) 2l+3
A = + a2A − a2l+1,
witha (x)determinedbyEqs.(31)and(33). l2 2l−1 (l+1)x2 l0 2l−1
eff (cid:20) (cid:21)
InFigs.2-4,theannuluscoefficientsλA(ǫ),withǫ(x)spec- (A3)
ified by Eq. (35) (dashed lines) are comparedto the porous- 2(2l−1)(2l+1)
particlevirialcoefficientsλ(x)(solidlines).Forthesedimen- Bl2 = 1+ (l+1)x2 a2Al2−a2l+3, (A4)
tationcoefficient,theannulusmodelisaccurate,witha half- (cid:20) (cid:21)
percentrelativeaccuracyalreadyatx = 20andareasonable
wherel =1,2,...andg (x)= π/2xI (x)isthemodi-
l l+1/2
3%precisionatx=5.Forthetranslationalself-diffusion,the
fiedsphericalBesselfunctionofthefirstkind.
annulusmodelislessaccurate,butstillitgivesonlya2%er- p
rorforx=20,anda5%errorforx=10,anda7%errorfor
x=5. Theleastaccurateistheannuluspredictionforthero-
AppendixB:Theannulus(hydrodynamicradius)model
tationalself-diffusion,witha5%errorforx = 20anda11%
errorforx=10.Summarizing,theannulusmodel(HRM)ap-
Forasuspensionofparticlesdescribedbytheannulus(hy-
proximateswellthefirstvirialcoefficientsofporousparticles
drodynamicradius)model,thevirialcoefficientsλAarefunc-
suspensions, in the range of intermediate and small particle
tionsofthemodelparameterǫ,definedbyEq.(29)andlisted
permeability(i.e. formoderateandlargevaluesofx).
in Table II. In this Appendix, we explain how these values
havebeenevaluated.
Thefirst-ordervirialexpansionofD /Dt,D /Dt,K and
V. CONCLUSIONS t 0 C 0
D /Dr,canbeperformedwiththeuseofφ =(4π/3)na3,
r 0 < <
orφ definedbytheanalogicalexpression,
>
In this paper, the short-time diffusion properties of dilute
suspensionsofuniformlyporoussphericalparticleshavebeen 1+λA(ǫ)φ +O φ2 = 1+λ¯A(ǫ)φ +O φ2 . (B1)
> > < <
investigated. The first-order virial coefficients λ of the dif-
fusionandsedimentationcoefficientshavebeenevaluatedas Therefore, (cid:0) (cid:1) (cid:0) (cid:1)
functionsofpermeability. Valuesofλarewell-approximated
by the annulus(hydrodynamicradius)model, if the parame- λ¯A(ǫ)
λA(ǫ) = . (B2)
ter x is sufficientlylarge, i.e. the permeabilityis sufficiently (1+ǫ)3
7
To evaluate λ¯A(ǫ), we now introduce the dimensionless in- λ¯A(ǫ)=8 +∞J (R)R2dR
terparticle distance as R = r/2a , and we replace the pair r r
< Z1+ǫ
distribution function from Eq. (10) by the following expres-
1+ǫ
sion, = λhs−8 J (R)R2dR, (B6)
r r
Z1
0 for R≤1+ǫ,
g (R) = , (B3)
0
(1 for R>1+ǫ.
where
whichcorrespondstotheno-overlapconditionata largerra-
diusa . Then,weapplytheEqs.(12),(16)and(27),takenin
>
thenon-permeablehard-spherelimit,x = ∞. We obtainthe 1
J¯ (R)= Tr µtt(2)(R)+µtt(2)(R) . (B7)
followingexpressions, K µt 11 12
0
h i
+∞
λ¯A(ǫ)=8 J (R)R2dR
t t
Z1+ǫ InthisAppendix,allthemobilitycoefficients,theassociated
= λhs−8 1+ǫJ (R)R2dR, (B4) functionsJ, and the superscript hs, refer to the hard spheres
t t withthestickboundaryconditions.
Z1
2 4
λ¯A(ǫ)= Ahs − Ahs(1+ǫ)2 Theaboveformulashavebeenappliedtocomputethefunc-
K 5a3 12 a 10
< < tions λA(ǫ), λA(ǫ) and λA(ǫ), listed in Table II. The func-
+∞ tions λ¯At (ǫ), λ¯AK(ǫ) and λ¯Ar(ǫ) have been also evaluated and
+ 8 J (R)R2dR t K r
K listedinRef.[34],wheretheyhavebeenappliedtomodeldif-
Z1+ǫ
1+ǫ fusionandrheologyofparticleswith a hardsolidcoreanda
= λhs −8 J¯ (R)R2dR, (B5) thinporousshell.
K K
Z1
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