Depletion potential in colloidal mixtures of hard spheres and platelets

0.19 MB·English

by  L. Harnau

Depletion potential in colloidal mixtures of hard spheres and platelets
L. Harnau and S. Dietrich
Max-Planck-Institut fu¨r Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany,
and Institut fu¨r Theoretische und Angewandte Physik,
Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
(Dated: February 6, 2008)
4 The depletion potential between two hard spheres in a solvent of thin hard disclike platelets is
0 investigated by using either the Derjaguin approximation or density functional theory. Particular
0 attention is paid to the density dependence of the depletion potential. A second-order virial ap-
2 proximationisapplied,whichyieldsnearlyexactresultsforthebulkpropertiesofthehard-platelet
fluidatdensitiestwotimessmallerthanthedensityoftheisotropicfluidatisotropic-nematicphase
n
coexistence. Astheplateletdensityincreases, theattractiveprimaryminimumofthedepletionpo-
a
tentialdeepensandanadditionalsmallrepulsivebarrieratlargersphereseparationsdevelops. Upon
J
decreasingtheratiooftheradiusofthespheresandtheplatelets,theprimaryminimumdiminishes
1
and theposition of the small repulsivebarrier shifts to smaller values of thesphere separation.
2

PACSnumbers: 61.20.-p,61.20.Gy,82.70.Dd
]
t
f
o
I. INTRODUCTION theory(Sec. II)tostudythedepletionpotentialbetween
s
. two hard spheres induced by thin hard platelets (Sec.
t
a III). Particularly, we compare the results with the ones
m Depletion interactions between big colloidal particles
obtained for non-interacting platelets.
inducedbysmallerparticles,whichcanbe eitherthesol-
-
d vent particles or a colloidal component of its own, are of
n significantcurrentresearchinterestbecauseoftheimpor-
II. DENSITY FUNCTIONAL THEORY
o tance of these effective interactions in various colloidal
c processes. For example, flocculation of colloids can be
[ We consideraninhomogeneousfluid consisting ofthin
drivenbytheadditionofnon-adsorbingpolymersviathe
platelets of radius R in a container of volume V. The
1 depletionmechanism[1]. Whereasexperimentalandthe- p
platelets are taken to be hard discs without additional
v oretical studies have focussed on binary hard sphere flu-
0 ids as well as on colloidal mixtures of hard spheres and attractive or repulsive interactions. The number density
8 of the centers of mass of the platelets at a point r with
hard rods or polymers, less attention has been paid to
3 an orientation ω = (θ,φ) of the normal of the platelets
hardplatelets acting as depletants, despite the greatim-
1 is denoted by ρ(r,ω). The equilibrium density profile
portanceofcolloidalplateletssuchasblood-plateletsand
0
of the inhomogeneous liquid under the influence of an
4 clay minerals in both biomedicine and geophysics. Very
externalpotential V(r,ω) minimizes the grandpotential
0 recently a colloidal mixture of silica spheres and silica
/ coated gibbsite platelets has been stabilized for the first functional
t
a time [2], and the depletion potential due to the presence
m Ω[ρ(r,ω)]= dr3dωρ(r,ω) k T ln[4πΛ3ρ(r,ω)]−1
of thin hard platelets has been derived theoretically for B
d- non-interactingplateletscorrespondingtothelimitofin- Z−µ+V(r,ω)]+(cid:2) Fex[(cid:0)ρ(r,ω)], (cid:1)
finitedilution[2,3]. IthasbeenfoundthattheDerjaguin
n (1)
approximationforthedepletionpotentialyieldsaccurate
o
c results for non-interacting platelets providedthe ratio of
where Λ is the thermal de Broglie wavelength and µ
: the radius of the spheres and the platelets is large [2].
v is the chemical potential. The free energy functional
i In this paper we focus on the depletion interactionin- F [ρ(r,ω)] in excess of the ideal gas contribution has
X ex
duced by thin hard platelets, taking into account the not been taken into account in previous studies on the
ar steric interactions between the platelets in terms of a depletionforceduetoplatelets[2,3]. Weexpressthe ex-
second-order virial approximation. On the basis of our cessfreeenergyfunctionalasanintegraloverallpossible
recent theoretical studies on fluids of thin hard platelets configurations of two platelets
near hard walls [4, 5], we expect that excluded volume
interactions between the platelets influence the deple- F [ρ(r,ω)] = −kBT dr3dω dr3dω ρ(r ,ω )
tion interaction already at rather low platelet densities ex 2 1 1 2 2 1 1
Z
due to their cumbrous shape as comparedwith spherical ×f (r ,ω ,ω )ρ(r ,ω ), (2)
pp 12 1 2 2 2
or rodlike depletants. Taking excluded volume interac-
tionsintoaccountisparticularlyinterestingbecausecor- wherer =r −r andf (r ,ω ,ω )istheMayerfunc-
12 1 2 pp 12 1 2
relation effects may cause repulsive features of depletion tion of the interaction potential between two platelets.
forceswhichareimportantinthecontextofcolloidalsta- The Mayer function equals −1 if the platelets overlap
bility [6]. In the present paper we use density functional and is zero otherwise. Explicit expressions of the Mayer
2
function for thin platelets are documented in Refs. [4] 2Rp
and [7].
Forthehomogeneousandisotropicbulkfluidthegrand
potential functional [Eq. (1)] reduces to
Ω π2
b =ρ k T(ln[λ3ρ ]−1)−µ + R3ρ2k T , (3)
V b B b 2 p b B
where ρ =(cid:2) V−1 dr3 dωρ(r,ω(cid:3)) is the total particle Rs
b
number density. The equation of state derived from the
grand potential [ERq. (3)R] takes the following form: h
π2
p⋆ =ρ⋆ 1+ ρ⋆ (4)
b b 2 b
(cid:18) (cid:19)
with ρ⋆ = ρ R3 and p⋆ = p R3/(k T). The same equa-
b b p b b p B
tion without the second term in parenthesis holds for
FIG.1: Thesystem underconsideration consists of twohard
the ideal gas limit (i.e., non-interacting platelets). With
spheresofradiusRsimmersedinasolventofhardplateletsof
increasing particle number density the ideal gas equa-
radiusRp. Theseparationbetweenthesurfacesofthespheres
tionofstateononesideandthesecond-termvirialseries
is denoted by h. Only the projection of the spheres on the
[Eq. (4)] as well as computer simulation data [7, 8, 9] on plane of thefigure is shown.
the other side deviate. Thus for ρ⋆ = 0.1 the osmotic
b
pressure p⋆ = 0.15, as calculated from Eq. (4), agrees
b
exactly with simulation data, while the ideal gas equa-
platelets (R ≪ R ) can be calculated from the finite
tion of state underestimates the osmotic pressure by a p s
′
size contribution of the grand potential function ω(h)
factor of 1.5. The comparisonof the calculated equation
of the platelet fluid confined between two parallel hard
of state with computer simulationdata exhibits that the
′
walls at distance h using the Derjaguin approximation
two-term series in Eq. (4) is a good approximation for
ρ⋆b <∼ 0.2, whereas the ideal gas model may be used for [13]
very low particle number densities ρ⋆b <∼ 0.04. For a dis- ∞ ′ ′
cussion of higher order virial terms for fluids consisting WDerj(h)=πRs dh ω(h), (5)
of hard platelets we refer to Refs. [4] and [10, 11, 12]. Zh
In the present study we restrict our attention to parti- where h is the separation between the surfaces of the
cle number densities ρ⋆ ≤0.2 for which the second-order spheres. (For the subtle issue of the range of validity of
b
virialapproximationisappropriateandtheplateletfluid the Derjaguin approximation see Refs. [14] and [15].)
is in the isotropic phase. For comparison, the isotropic- We first consider a hard-platelet fluid confined by two
nematic phase transition is first order with coexistence parallelhardwalls at z =0 and z =h, and calculate the
densities ρ R3 = 0.46 and ρ R3 = 0.5 according to a surface and finite size contributions to the grand poten-
bI p bN p
computer simulation [9]. tial defined via
Ω[ρ(z,θ,φ)] = Vω +2Aγ+Aω(h), (6)
b
III. THE PLATELET-INDUCED DEPLETION where A is the area of a single surface, ω is the bulk
b
POTENTIAL BETWEEN TWO SPHERES
grand canonical potential density, and V is defined as
the volume of the containerwith its surface givenby the
The results of the preceding section show that inter- position of the rim of the particles at closest approach
molecular interactions between platelets increase the os- so that V = Ah. γ is the wall–liquid surface tension in
motic pressure of the bulk fluid already at low particle the absence of the second wall and ω(h) is the finite size
densities. Now we study the influence of intermolecular contribution. Figure2displaysthecalculatedsurfaceand
interactionsonthe depletionpotentialbetweentwohard finite size contributions to the grand potential together
spheresofradiusRs immersedinafluidofhardplatelets with the results for non-interacting platelets [2, 3]
of radius R . The depletion potential W(h) is the free
p
energy difference between the configurations of two big ρ R arcsin h
b p 2Rp
spheres at fixed distance h immersed in the solvent and
at macroscopic separation h=∞ (see Fig. 1). [2γ+kωB(Th)]ideal= +πρ2RhRp(cid:16)r=1−2γi(cid:16)d(cid:16)e2aRhl,p(cid:17)(cid:17)2!,0h≤≥h2R≤2.Rp
A. The Derjaguin approximation As is apparentfromF2igb.2pthe sktBeTricinteractionbeptwe(e7n)
The depletion potential W(h) between two hard the platelets increases the surface contributions with in-
spheres at close distance due to the presence of small creasing density. Within our numerical precision, we
3
0.3
0 (a)
h/R =1.5
p
)
p
T)B 0.2 Rb -1
2/(kp rkTB r *b=0.05
h)] R h/R =0.5 (h)/( rr **b==00..12
( 0.1 p w -2 b
w+

g2
[ 0 1 2 3 4
h/R
p
0
0 0.1 0.2
r R 3 (b)
b p
0
)
FIG. 2: The surface and finite size contribution 2γ +ω(h) b
to the grand potential as obtained from Eqs. (1), (2), and rT
(6) [solid lines] of a fluid consisting of thin hard platelets of (kB -1 r *b=0.05
radius Rp confined in a slit of width h and in contact with h)/ r *b=0.1
an isotropic bulk reservoir at density ρb. The dashed lines f( r * =0.2
representthecorrespondingresultsforanidealgasofplatelets b
[see Eq. (7)]. The width of the slit increases from bottom to
-2
top: h/Rp =0.5,1.5.
0 1 2 3 4
h/R
p
found that an accurate evaluationof the wall–liquidsur-
face tension γ could be achieved for a fixed wall sepa- FIG. 3: (a) The finite size contribution ω(h) to the grand
ration h = 4R at all considered densities ρ R3 ≤ 0.2. potential as obtained from Eqs. (1), (2), and (6) of a fluid
p b p
For much higher densities, a larger value of h might be consistingofthinhardplateletsofradiusRp confinedinaslit
required because of the wetting of the wall–isotropic liq- ofwidthhforthreevaluesofthedensityoftheisotropicbulk
reservoir: ρ⋆ = ρ R3 = 0.05 (dotted line); ρ⋆ = 0.1 (dashed
uid interface by a nematic film of diverging thickness b b p b
line); ρ⋆ = 0.2 (solid line). (b) The solvation force per unit
[5]. On the other hand the wall–liquid surface tension b
area f(h)=−dω(h)/dhof thesame fluid[with thesame line
γ /(k T)=ρ R π/4fornon-interactingplateletsfol-
ideal B b p code as in (a)] as a function of h.
lows from Eq. (7) for a wall separation h = 2R . For a
p
detailed discussion of the surface tension and the excess
coverageaswellasthedensityandorientationalorderpa-
rameterprofilesoffluids consistingofthinhardplatelets pared with those between rods. Moreover f(h) is a con-
near a single hard wall we refer to Ref. [4]. The results vex function for slit widths smaller than two times the
for the finite-size contribution ω(h) are shown in Fig. 3 radius of the platelets, while the solvation force in a sol-
(a). As function of h the finite size contribution corre- vent of hard rods is a concave function for slit widths
sponds tothe solvationfreeenergyforthe immersedtwo smallerthan the length ofthe rods [6]. Figure 4 displays
plates acting as the confining walls for the fluid and, by the depletionpotentialtogetherwiththe resultsfor non-
construction, ω(0)=−2γ and ω(∞)=0. Upon increas- interactingplateletswhichcanbecalculatedanalytically
ing the platelet density, the attractive minimum of ω(h) from Eqs. (5) and (7) [2, 3]:
at h = 0 deepens and a maximum at larger values of
h develops. The corresponding solvation force per unit −πρ R2R h arcsin h
areaf(h)=−dω(h)/dhisattractiveforsmallslitwidths b p s Rp 2Rp
hdeneans.ssiitFsyo,srhthocewomncupisnaprFiosifognt.h3wee(sbon)lv.oatUetipotonhnaftoinrtcchereeaamtshianxg=imt2huRemppslaahttaetrlhpeet- WkD(BiedrTejal) = +34r10−≤(cid:16)h(cid:16)2Rh≤p(cid:17)2R2(cid:16)p1+(cid:16)8hR22p(cid:17)(cid:17)− 2πRhp! ,
cthuaspn ifsormaorceorprreosnpoounndciendg froordthfleuidcon[6fi]ndedueptlaotetlheet flreuliad- 0, h≥2Rp. (8)
tivelylargerstericinteractionsbetweenplateletsascom- The depletion potential due to the presence of interact-
4
energy per sphere is
0
r * =0.02
b f(ρ ,ρ )=f(hs)(ρ )
s b s
∞
r * =0.08
b +2πρ drr2W(r−2R )g(hs)(r,ρ ),(9)
s s s
-1
Z
r * =0.2 2Rs
b
)
T whereg(hs)(r,ρ )istheradialdistributionfunctionofthe
B s
)/(k -2 eprugryepsperhsepreheflrueifdoranthdefs(ahms)e(ρhso)misogtehneeHoueslmfluhiodltozffdreeneseinty-
h
(erj ρs. InthelimitRs ≫Rp the radialdistributionfunction
WD g(hs)(r,ρ ) is almost constant over the range of integra-
s
tion where W(r−2R ) 6= 0 and we can approximate it
s
-3 by its constant contact value g(hs)(2R ,ρ ) [16]. Using
s s
Eq.(8)resulting the integralin Eq.(9) canbe evaluated
R =3.5 R
s p analytically for non-interacting platelets:
∞
(ideal)
-40 1 2 3 4 A(Dideerajl)≡2πρs drr2WDerjk(rT−2Rs)
B
Z
h/Rp 2Rs
64R π R2
=−2π2ρ R3ρ R3 π+ p + p .(10)
b p s s 45R 12R2
FIG. 4: Depletion potential WDerj(h) between two hard s s!
spheres of radius Rs = 3.5Rp due to the presence of thin
hard platelets of radius Rp as obtained from the Derjaguin TheintegratedstrengthofthedepletionpotentialA(Dideerajl)
approximation [Eq. (5)] for various bulk densities ρb of the is negative, reflecting the fact the depletion potential is
platelets: ρ⋆b = ρbRp3 = 0.02 (top curves); ρ⋆b = 0.08 (middle alwaysattractive(Fig.4). Anumericalcalculationofthe
curves);ρ⋆ =0.2 (bottom curves). Thesolid lines (with dots
b correspondingquantityADerj forinteractingplateletsex-
marking the maxima) represent thecalculations for interact-
hibits that the steric interacting between the platelets
ing platelets and thedashed lines denote the results for non-
weakens the integrated strength of the depletion poten-
interacting platelets. For reasons of clarity, the lower four
tial by 22 % at a bulk density ρ R3 = 0.2. Hence the
curves are shifted down by -0.3 kBT and -0.9 kBT, respec- b p
tively. Thedashedlinesarezero for h/Rp ≥2. Althoughthe influence of steric interactions between platelets might
interactionshaveonlyminorinfluenceonthedepthofthepri- be quite visible for phase equilibria. For example, the
maryminimum,thedepletionpotentialbecomessignificantly thermodynamic onset of flocculation of colloidal spheres
less attractive for increasing densities ρp. This weakens the induced by the depletion effect will be reduced due to
platelet induced flocculation of a solution of big spheres. platelet interactions.
B. Density functional approach
A general approach for calculating the depletion po-
ing platelets exhibits a small barrier at larger sphere tentialis basedonadensity functionaltheory(DFT) for
separations h in addition to the primary minimum at a mixture of hardspheres and the particles acting as de-
h=0. Withincreasingplateletdensitythedepletionpo- pletants [17, 18]. This approach avoids the Derjaguin
tentialdeepensandthepositionofthemaximumshiftsto approximation. Translating the particle insertion idea
smallervalues ofh. The smallrepulsivebarrierwill have developed in Refs. [17] and [18] to the present system
minoreffectsonkineticstabilization,althoughtherepul- leads to the following expression for the depletion po-
sivefeaturesmightstillbemeasurable. Forexample,our tential between two spheres due to the presence of hard
numericalcalculationsexhibitamaximumbarrierheight platelets:
of 0.25 k T relative to zero at a density of ρ R3 = 0.2
B b p
in a system of size ratio R /R = 3.5 corresponding to W(r) 1
s p − = dr3dω [ρ(r ,ω)−ρ(∞,ω)]f (r−r ,ω)
the aforementioned mixture of silica spheres and gibb- k T 4π 1 1 sp 1
B
Z
site platelets [2]. In order to examine the influence of (11)
steric interactions between platelets on thermodynamic where f (r−r ,ω) is the Mayer function of the inter-
sp 1
properties of sphere-platelet mixtures, we treat the de- action potential between a sphere and a platelet. The
pletion potential as a perturbation to the hard-sphere Mayer function equals −1 if the particles intersect or
potential. The first-order approximation in this thermo- touch each other and is zero otherwise. ρ(r ,ω) is the
1
dynamic perturbation approach for the Helmholtz free density profile of platelets in the external potential of
5
a single fixed hard sphere located at the origin of the 0.12
coordinatesystemandρ(∞,ω)isthecorrespondingden-
sity profile of the bulk fluid. We emphasize that the
density profile entering Eq. (11) depends only on equi-
librium properties of the depletant fluid in the absence
of the second hard sphere to be inserted at position r. 0.08
This observation simplifies the calculation of W(r) con- 3
p
R
siderably, because the symmetry of the density profile is
)
r
determinedsolelybythesymmetryoftheexternalpoten- (
tialofasinglespherefixedattheoriginofthecoordinate r
0.04
system.
Apart from possible surface freezing at high densities,
non-uniformities of the density depend only on the ra-
dial distance r = |r|, so that ρ(r,ω) = ρ(r,ω). Hence,
calculatingdensity profilesbefore insertionofthe second 0
hardsphereismucheasierthanafterinsertion,whenthe 0 2 4 6 8
presence of the second sphere leads to a more complex r/R
p
spatial variation of the densities. A detailed discussion
of Eq. (11) and its application to the analogous case of
FIG. 5: Orientationally averaged density profile ρ(r) as ob-
non-interactinghardrodsactingasdepletantsisgivenin tainedfromEq.(12)forhardplateletsofradiusRp incontact
Ref. [18]. with a single hard sphere of radius Rs located at r=0. The
Foranidealgasofplateletsincontactwithafixedhard radius of the sphere increases from left to right: Rs/Rp =
spherethedensityprofilereducestoρ(r ,ω)−ρ(∞,ω)= 0.5,1,3,5. Thearrowsmarkthelocation ofthesurfaceofthe
1
fsp(r1,ω) so that the integral in Eq. (11) has a purely sphereatr=Rs. Sincetheplateletsarearbitrarilythintheir
geometrical meaning and measures the excluded volume density is nonzero for r > RS. All curves exhibit a cusp at
of a platelet confined between two hard spheres located r = Rs +Rp followed by a decay towards the bulk density
at the origin of the coordinate system and at position r, ρbR3 = 0.085, which is essentially reached at r = Rs+2Rp.
respectively. WhenthecenterofaplateletislocatedlessthanRp fromthe
spheresurface, thereare fewer possible orientations available
In order to take intermolecular interactions between
to theplatelet.
the platelets into account we first calculate numerically
the density profile ρ(r,ω) of platelets in an external po-
tential of one fixed hard sphere of radius R . There-
s
after the integral in Eq. (11) is evaluated by inserting
upon decreasing the size ratio. These results are due to
this density profile. To our knowledge, this technique
the fact that the number of platelets contributing to the
has not been used before for interacting non-spherical
depletionpotentialdecreasesasthe ratioofthe radiusof
colloids acting as depletants.
the spheres and the platelets becomes smaller at a fixed
The orientationalaverageddensity profile
bulk density (see Fig 5).
In agreementwith a recent theoreticalstudy [2] based
ρ(r)= dωρ(r,ω) (12)
on an evaluation of the excluded volume of a single
Z platelet confined between two hard spheres, we find that
oftheplateletfluidincontactwithonefixedhardsphere the Derjaguin approximation for the depletion potential
isshowninFig.5forvariousradiiR ofthesphere. Upon in the presence of non-interacting platelets yields accu-
s
increasing r ≥R from the surface of the sphere the av- rate results for large size ratios R /R > 1. However,
s s p
eraged number density increases and exhibits a cusp at therearesubstantialdeviationsathigherdensitiesascan
r = R +R where platelets with their normal perpen- be seen from Fig. 6 (a). The absolute value |W(h = 0)|
s p
dicular to the radial direction touch the surface of the of the DFT solution at contact is smaller than the one
sphere with the rim. The maximumatthe cusp is about obtained from the Derjaguin approximation, and the re-
25% above the bulk value ρ R3 = 0.085 for a size ratio pulsive barrier is less pronounced. Hence, for a size ra-
b p
R /R =5andis lesspronouncedforsmallersize ratios. tio R /R = 3 and a platelet density ρ R3 = 0.2 the
s p s p b p
The averaged density close to the surface of the sphere Derjaguin approximation overestimates the depth of the
is larger for a small sphere than for a big one. Figure depletion potential at contact by 37 % and the height of
6 displays the calculated depletion potential for two size the repulsive barrier by 23 %. Increasing the size ratio
ratiosR /R asafunctionofthe separationbetweenthe R /R >3 doesnotleadto a significantimprovementof
s p s p
surfacesofthespheresh=r−2R . Withdecreasingsize the Derjaguin approximation as compared to the results
s
ratio the range and the depth of the primary minimum shown in Fig. 6 (a). For R /R < 1 and higher densi-
s p
shrinks and the position of the small repulsive barrier ties ρ the DFT resultsdeviatestronglyfromthe predic-
b
observedat higher densities shifts to smaller values of h. tions for non-interacting platelets [see ρ⋆ = ρ R3 = 0.2
b b p
Moreover, the height of the repulsive barrier decreases in Fig. 6 (b)].
6
IV. SUMMARY
0
r * =0.02
b
r * =0.08 We have applied a density functional theory to fluids
b
consisting of thin hard platelets confined between two
-1 hardspheres(Fig.1). Withintheframeworkofasecond-
r *b=0.2 ordervirialapproximationoftheexcessfreeenergyfunc-
T)B tional, the depletion potential between the two spheres
k
W(h)/( -2 DDeFrTjaguin dicuaellytoatnhdepcoremsepnacreedofwthitehptlhaetelceotrsreisspdoentedrimnginreedsunlutsmfeorr-
non-interacting platelets. The main conclusions which
ideal gas
emerge from our study are as follows.
(1) Figure 2 demonstrate that steric interactions be-
R =3 R
-3 s p (a) tween thin platelets of radius Rp confined between two
parallelhardwallsincreasethesumofthesurfaceandfi-
nite size contributionto the grandpotentialsignificantly
0 1 2 3 4
h/R already at rather low platelet densities ρbRp3 >∼0.025.
p (2) As function of the slit width h the finite size con-
tribution to the grand potential of a slap of platelets ex-
hibits a minimum at h = 0 [Fig. 3 (a)]. A maximum at
larger values of h is found for higher platelet densities.
0
r * =0.02 The corresponding solvation force is attractive for small
b
slit widths and exhibits a cusp at h=2R [Fig. 3 (b)].
p
-0.1 r *b=0.08 (3)Thedepletionpotentialbetweentwospheresascal-
culated from the Derjaguin approximation exhibits an
r * =0.2 attractive primary minimum at contact which deepens
T) -0.2 b upon increasing the platelet density. Moreover, a small
B
W(h)/(k -0.3 iDdFeaTl gas rwtieoiptnhulbsianivrcerrieebarsarirnerlgiaetrdiveanetstiotlayzre(grFeorigis.spt4yh)p.eriWceaelsleyfipnlaedrsastthtihoanatnsthtdeheevdetelhpoelpers--
mal energy k T, and therefore unlikely to significantly
B
-0.4 alter the kinetics of aggregation of the hard spheres at
Rs=0.5 Rp platelet densities smaller than two times the density of
(b)
theisotropicphaseatbulkisotropic-nematiccoexistence.
-0.5
Nonetheless, with increasing platelet density the inte-
0 1 2 3 4
grated strength of the effective interaction between the
h/Rp spheres becomes significantly weaker and thus reduces
the thermodynamic onset of flocculation.
(4) The orientational averaged density profile of a
FIG. 6: Depletion potential W(h) between two hard spheres
platelet fluid in contact with a single fixed hard sphere
of radius Rs = 3Rp in (a) and Rs = 0.5Rp in (b) due
to the presence of thin hard platelets of radius Rp as ob- decreases towards the surface of the sphere because the
tained from Eq. (11). The bulk densities of the platelets are: range of accessible orientations is reduced when the par-
ρ⋆ = ρ R3 = 0.02 (top curves); ρ⋆ = 0.08 (middle curves); ticle approaches the sphere. It exhibits a cusp at the
b b p b
ρ⋆b = 0.2 (bottom curves). Here h = r −2Rs is the sepa- position where the platelets lose contact with the sur-
ration between the surfaces of the spheres. The solid and face of the sphere (Fig. 5). The maximum at the cusp
dashed lines represent the calculations for interacting (DFT) decreases and the averaged density close to the surface
and non-interacting platelets (ideal gas), respectively. In ad-
increases as the ratio of the radius of the sphere and the
ditionthedepletionpotentialasobtainedfromtheDerjaguin
platelets becomes smaller.
approximation [Eq. (5)] for interacting platelets is displayed
(5) With decreasing ratio of the radius of the spheres
bydottedlinesin(a). Forthesmallest bulkdensitythesolid
and the platelets, the primary minimum at contact and
anddottedlinenearlycoincidein(a). Sincein(b)thespheres
areonlyhalfasbigastheplatelets,inthiscasetheDerjaguin the small repulsive barrier of the depletion potential di-
approximation is unsuitable and therefore not shown. Only minish and the position of repulsive barrier shifts to
for sufficiently large densities a maximum denoted by a dot smaller values of the separation of the surfaces of the
occurs. For reasons of clarity, the lower sets of curves are spheres (Fig. 6). From our numerical results based on
shifted down by -0.3 kBT and -0.9 kBT, respectively in (a) a density functional theory for a mixture of spheres and
and by -0.05 kBT and -0.15 kBT, respectively in (b). platelets we found that the Derjaguin approximation is
valid for large size ratio and very small platelet den-
sity,buttherearesubstantialdeviationsfromthedensity
functionalresultsathigherdensitiesevenforlargesizera-
7
tios[Fig.6(a)]. Thesedeviationsincreasewithincreasing densities the ideal gas approximation for the platelets
platelet density. For small size ratios and high platelet becomes unsuitable [Fig. 6 (b), ρ⋆ =0.2].
b
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