Table Of ContentFrom Lumps to Lattices:
Crystallized Clots Made Simple
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n P. Ziherl∗,†,‡ and Randall D. Kamien∗,¶
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0 FacultyofMathematicsandPhysics,Universityof Ljubljana,Jadranska19,SI-1000 Ljubljana,
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Slovenia,JožefStefanInstitute,Jamova39, SI-1000Ljubljana,Slovenia,andDepartmentof
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Physics andAstronomy,Universityof Pennsylvania,209S. 33rdSt.,PhiladelphiaPA
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t. 19104-6396,USA
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d E-mail: [email protected];[email protected]
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Towhomcorrespondenceshouldbeaddressed
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†FacultyofMathematicsandPhysics,UniversityofLjubljana,Jadranska19,SI-1000Ljubljana,Slovenia
‡JožefStefanInstitute,Jamova39,SI-1000Ljubljana,Slovenia
¶DepartmentofPhysicsandAstronomy,UniversityofPennsylvania,209S.33rdSt.,PhiladelphiaPA19104-6396,
USA
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Abstract
Usingaminimalmodelbasedonthecontinuum theory ofa2Dhard-core/square-shoulder
ensemble,wereinterpretthemainfeaturesofclustermesophases formedbycolloidswithsoft
shoulder-like repulsive interactions. We rederive the lattice spacing, the binding energy and
the phase diagram. Wealso extend the clustering criterion [Likos, C.N.,etal.Phys. Rev. E,
2001,63,031206; Glaser, M.A.,etal.EPL2007,78, 46004] toinclude theeffectofthehard
cores,whichprecludes theformationofclustersatsmalldensities.
Introduction
Generically, neutral colloidal particles attract each other via van der Waals and Casimir forces,
and measures must be taken to prevent aggregation of clusters in both experimental systems and
technologicalapplications. One mightbe led to concludethat purely attractivehard-core particles
could only form close-packed lattices or glassy messes. But the phenomenon of aggregation in
colloids is not restricted to particles that attract each other. Over the past two decades clustering
in purely repulsive pair potentials has been explored in some detail to find that it is distinguished
byemergingordernotseeninattractiveparticles. Inparticular,ithasbeenestablishedthatatlarge
enough density, clumping repulsive colloids can form superstructures with large voids on the or-
der of many particle diameters.1 Indeed, Malescio and Pellicane2 demonstrated that even simple
hard-core, square-shoulder potentials led to clustering purely on energetic grounds. The cluster
morphologies and the clustering criterion itself have been studied theoretically using a range of
approachesincludingliquid-statetheory,3 latticetheory,4,5 densityfunctionaltheory,6 andcontin-
uummodels,5,7 andthepredictionsofthedifferentapproachesareremarkablyconsistent. Equally
unequivocal are the results of numerical studies, mostly using Monte Carlo methods5,8 and direct
search ofminimal-energyconfigurationsusinggeneticalgorithms.9–11
It is worthwhile at this juncture to step back and develop a minimal set of simple rules to ex-
posethemechanismoflatticeformationinthesesystems. It isourhopethattheseruleswillmake
clear the essential ingredients needed for pattern formation in this large class of purely repellent
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systems. Moreover,we developtheseideas in thecontextof real-space potentialsand interactions
yet recapitulate the clustering criterion of the more technical (though precise) Fourier-based anal-
yses.1,3,5 Designing potentials in real space offers a more intuitiveroute to rational self-assembly
and connectsdirectly with,for instance,laser-trap and depletionbased potentials.
In this note, we do this by using the continuum T = 0 model of the stripe phase formed by
particlesinteractingviathehard-core/square-shoulderpairpotential
¥ , r<s
U(r)= e , s <r<l (1)
0, r>l
where s and l > s are the diameters of the core and the shoulder, respectively, and e is the
shoulder height.12,13 Characterized by a one-dimensional density modulation, the stripe phase is
mathematically the most transparent of all cluster morphologies and our real-space analysis may
beeasierto visualizethan reciprocal-spacearguments.3–5 Moreover,by focusingonenergy rather
than on free energy we emphasize that the entropy does not promote clustering. These results
are then extended to explore the generic phase diagram of the cluster-forming system at finite
temperatures.
Clusters at T = 0
Inthefollowing,wefirstevaluatetheenergy ofatwo-dimensionalhard-core/square-shouldersys-
tem with given hard-core diameter s and shoulder diameter l at a fixed average number density,
and we minimizeit with respect to intra-cluster density, cluster size, and lattice spacing. We con-
sider the simplest cluster morphology, the one-dimensional square-wave of uniformly populated
parallel stripes of width d and particle-free gaps of width ℓ d; the lattice spacing is ℓ. In the
−
continuummodel suitablein thelargeshoulder-to-corelimitwhere l is sufficientlylargerthan s ,
3
theenergy perparticlecan berelated totheaverageoverlapareaper particledefined as
1 d
w = dy dr Q (l r r ), (2)
1 2 2 1
2d 0 −| − |
Z Z
where Q (r) is the Heaviside step function describing the shape of the shoulder potential and r
1
and r are the locations of particles 1 and 2, respectively, measured from the origin at an edge of
2
the stripein question. The first integrationovery , thedistance ofparticle 1 from a stripeedge, is
1
overthestripecontainingparticle1andtheintegrationoverr goesoverallstripes. w capturesthe
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interaction oftheshouldersof particles, whereas thehard-core repulsionis treated in a mean-field
approximationby demandingthat thenumberdensitywithinstripesr benolarger than
stripes
2
r = (3)
cp √3s 2
corresponding to the close-packed hexagonal arrangement of the particles’ hard-disk cores. This
approximationisapplicableinthebroadshoulderregimel s andatdensitieslargeenoughthat
≫
thestripesare sufficientlywiderthan thecore diametersothat speakingof“intra-clusterpacking”
ofparticleshas meaning.
In termsofw , theaverage energy perparticlereads
E =er w . (4)
stripes
Butthenumberdensityofparticleswithinstripesdependsontheirwidthrelativetolatticespacing
d/ℓ, which represents the fraction of the total area that is occupied by the stripes. In terms of the
averagedensityr ,
r ℓ
r = . (5)
stripes
d
Since the energy is to be minimized at fixed r rather than at fixed r it is convenient to intro-
stripes
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ducethescaled averageoverlaparea
w
W = (6)
d/ℓ
whichincludesall dependenceof
E =er W (7)
ond andℓ. Forthehard-core/square-shoulderpairinteraction,W canbecomputedanalyticallybut
theresultistoo cumbersometo beofinteresthere.
We willnowestablishsomerules ofthumb.
I. Clusters are close-packed In [figure][1][]1 we plot the reduced scaled average overlap area
W¯ = W /l 2 as a function of reduced lattice spacing ℓ¯= ℓ/l for several stripe widths d/ℓ. The
cluster-free, unmodulated phase corresponds to ℓ¯= d¯=0 and its nature depends on density: As
argued below,itsphasesequenceincludestheexpandedfluid,theexpandedhexagonalcrystal,the
condensed fluid, and the condensed hexagonal crystal. But since we treat the hard-core part of
the pairinteraction in a mean-field fashion, worrying onlyabout theaverage numberof neighbors
within the reach of a particle’s shoulder, the exact nature of the unmodulated phase is not crucial
tounderstandwhyclusteringtakes place.
The uniform, unmodulated phase can be interpreted as a stripe morphology with a very fine
densitymodulationsuchthatthelatticespacingandthestripewidtharemuchsmallerthanthetwo
characteristiclengthscalesofthepairpotential,l ands . Inthislimit,W¯ =p /2irrespectiveofd/ℓ
whichtellsusthatford,ℓ l theaverageenergyperparticleisE =perl 2/2. Asℓ¯isincreased,
≪
W¯ oscillates around p /2, reaches a global minimum at ℓ¯ 1.2, and then grows monotonically to
≈
saturate at a valueof p ℓ/2d. The large ℓ¯behavior is a signatureof macroscopic phase separation:
At fixed d/ℓ, states with reduced lattice spacing ℓ¯beyond 1 correspond to thick stripe widths d¯
∼
withaneversmallernumberofparticlesresidingattheboundaryofthestripes. Theaverageenergy
perparticleisgraduallydominatedbythatoftheparticleswellwithinthebulkofthestripes,which
reads E = per l 2/2 = perl 2ℓ/2d because the density within the stripes is larger than the
stripes
averagedensitybya factorofℓ/d. ThisresultgivesW¯(ℓ ¥ )=p ℓ/2d.
→
5
Figure1: Reducedscaledaverageoverlapareaasafunctionofthereducedlatticeconstantℓ¯=ℓ/l
ford/ℓ=0.1(0.1)0.9;forthesakeofclarity,onlycurves correspondingtod/ℓ=0.1,0.5,and0.9
are labeled. The global minimum of the reduced scaled overlap area W¯ at ℓ¯ 1.2 is deepest for
≈
vanishinglysmalld/ℓ,which suggeststhattheequilibriumstripesareas compact as possible,i.e.,
close-packed. The square density waves representing stripes with d/ℓ = 0.1,0.5, and 0.9 (top)
schematically depict the stripe morphologies at ℓ¯= 0.5,1,1.5, and 2; the second column where
ℓ¯=1 correspondsto alatticespacingexactlyequal to theshoulderwidth.
The most important feature of this diagram is that the depth of the minimum decreases with
increasing relative stripe width d/ℓ. This means that at any given average density r =r d/ℓ,
stripes
the system will select the state with the smallest possibled/ℓ at the expense of the density within
stripes. For example, the curves corresponding to d/ℓ=0.1 and 0.2 represent two possible states
of the system of a fixed average density r . According to [figure][1][]1, the absolute minimum of
the former is lower than that of the latter, which means that of the two stripe phases in question,
the d/ℓ=0.1 state minimizes the total energy E =erl 2W¯. But since the relative stripe width of
the d/ℓ=0.1 state is half of that of the d/ℓ=0.2, the corresponding density within stripes must
be twice as large as in the latter state. In other words, the ground state geometry of the stripes
minimizes d/ℓ, restricted only by the close-packing limit forbidding stripes with r = r ℓ/d
stripes
beyond the close-packed density r . We conclude that at T =0 the optimal stripes are made of
cp
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close-packed particlesso thatr =r and thereduced averagenumberdensity
stripes cp
r
n= (8)
r
cp
coincideswithd/ℓ.
II.Latticespacingweaklydependsondensity Anadditionalfeatureofthegraphsin[figure][1][]1
isthattheequilibriumreducedlatticespacing,ℓ¯ ,dependsonlyweaklyond/ℓ. Itisapparentthat
eq
ℓ¯ is largest at half-filling where it reaches 1.217, which nicely agrees with the value of 1.223
eq
predicted by the lattice theory.5 It can be shown that the dependence of ℓ¯ on n is well described
eq
by a parabola symmetric about n=0.5. Even in the infinitely-dilute and close-packed limits, ℓ¯
eq
tends to 1 — but this result is of no physical consequence. As shown below, the binding energy
vanishes in these two limits so that the clustering mechanism is not acting and the corresponding
equilibriumreduced latticespacingsare irrelevant.
ThisobservationismadeevenmorecompellingbyemployingobservationI:Sincetheground-
statestripesareclose-packed,wemayaswellswitchbacktotheaverageoverlapareaperparticle,
w and write the energy as E =er w ; w¯ =w /l 2 can thus be regarded as the reduced energy. In
cp
[figure][2][]2weplotw¯ asafunctionofreducedlatticespacingℓ¯. Itisapparentinthispresentation
that the location of the global minimum of w¯ is little changed over the range 0.1 n 0.9 and
≤ ≤
againweconcludethatℓ¯ is roughlyindependentofn.
eq
From [figure][2][]2 we also extract thereduced bindingenergy D w¯ defined as thedepth of the
global minimum of w¯(n) at ℓ¯ relative to the unmodulated phase at ℓ=0. The reduced binding
eq
energy shown in the inset to [figure][2][]2 is a skewed U-shaped function of reduced average
density which vanishes for n = 0 and n = 1. In an infinitely dilute system, the energy of the
unmodulated phase itself tends to 0 and no spatial modulation of the density profile can reduce it
furtherso thatthebindingenergy is 0too. Ontheotherhand,asystemofaveragedensitycloseto
r cannot undergo but avery restricted spatial modulation(because thedensity withinthestripes
cp
shouldnotexceed r )and thustheenergy gaineduponclusteringapproaches 0 whenr r .
cp cp
→
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Figure 2: Reduced energy of the close-packed stripe morphology, w¯, as a function of reduced
lattice spacing ℓ¯for n = 0.1(0.1)0.9. In the unmodulated phase at ℓ¯= 0, w¯ = np /2. The inset
showsthereducedbindingenergyD w¯ =w¯(ℓ¯ ) w¯(ℓ¯=0)[asillustratedinthew¯(n=0.5)curve]
eq
−
whichis askewedU-shaped functionofreduced averagedensityandvanishesat n=0 and n=1.
Although w¯ and W¯ are closely related, they convey a somewhat different message. From
[figure][1][]1 welearnedthattheparticleswithinthestripesareclose-packed,whichenabledusto
directly relate the average density r to relativestripe width d/ℓ. On the other hand, [figure][2][]2
exposes the binding energy of the equilibrium stripe morphology more clearly. But as far as the
magnitudeoftheequilibriumlatticespacing isconcerned, bothquantitiesare equallytelling.
Phase diagram
Our two observations can be used to qualitatively outline the phase diagram of the cluster phases
in the temperature-density plane. To this end, we need to study the difference of the free energies
ofthestripephaseand theunmodulatedphase,whichconsistsofanenergy termD E =er l 2D w¯
cp
andofanentropicterm. Theentropyofthetwophasesdependsontheirstructureelaboratedbelow
ina semi-quantitativefashion.
At absolute zero, one of the hallmark features of the stripe morphology is its compact intra-
stripe structure where the impenetrable hard cores of the particles are packed together as tightly
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as possible at any reduced average density n. This state is materialized by the hexagonal lattice.
Because of the robust, density-independent nature of this behavior, we posit that at finite temper-
atures the intra-stripe density r should not depend strongly on the reduced average density
stripes
either(thoughitmustbesmallerthantheclose-packingdensityr )andthattheintra-stripeorder
cp
remainshexagonal.
The structure of the unmodulated phases of hard-core/square-shoulder particles is more com-
plicatedanddespitedecadesofefforts(see,e.g.,Refs.15–17),theirthermodynamicsremainsonlya
partlysolvedproblem. Forthepurposeofpresentdiscussion,itsufficestonotethatatlowtemper-
atures where they competewith cluster morphologies,the phase sequence of unmodulatedphases
consists of 4 variants, the 2 low-density and high-density phases being characterized with little
andsizableoverlapofparticles’shoulders,respectively. Atverysmalldensities,theparticlesform
an expandedfluidofdisksofshoulderdiameterl (schematicallyshownin[figure][3][]3). Asthis
fluidiscompressed,itundergoesatransitiontotheexpandedhexagonalcrystalofdisksofdiameter
l . Thelocation of the transitioncan be estimated by rescaling thephase diagram of the hard-disk
system: r 0.792(s /l )2r .20 Upon further compression, the expanded hexagonal phase
ef ec cp
− ≈
remelts to avoid close-packing; since the overlap of shoulders is increasingly less unfavorable at
elevated temperatures, the phase transition density should decrease with temperature. From this
transition on, the particles behave essentially as hard spheres of diameter s and the transition to
thecondensedhexagonalphasetakesplaceapproximatelyat r 0.792r .20
cf cc cp
− ≈
Understanding the main features of the sequence of unmodulated phases helps us to construct
semi-qualitativelytheentropicpart oftheirfree energy. Wefirstnotethatthepressureofthehard-
disk hexagonal crystal can be roughly regarded as a continuation of the hard-disk fluid branch20
and we approximate the excess entropic free energy per particle in both the fluid and the crystal
unmodulatedphaseby
a n
Fcond(n)=k T ln(1 a n) (9)
ex B 1 a n− −
(cid:20) − (cid:21)
wherea =p /2√3 0.907. Thispredictionisbased onthe2D Carnahan-Starling typetheory for
≈
thefluidphase.21,22 Admittedlyaroughapproximation—itdoesnotdistinguishbetweenthefluid
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and the crystalline phase, and the hexatic phase is disregarded altogether — this excess entropic
free energy provides a simple and adequate model across a broad range of reduced densities con-
sistentwiththescopeofthisanalysis. Itsmain deficiency isthepoordescriptionofthecrystalline
phase (whose excess entropic free energy should diverge at reduced average density of n=1 and
not at n = 1/a = 1.103). Yet we note that in the phase diagram of hard disks, the condensed
crystallinephaseisstableat n>0.79220 sothatthediscrepancyis limitedonlytolarge densities.
Using the model F [Eq. (??)], we can outline the excess entropic free energy for the un-
ex
modulated phases. In [figure][3][]3, we plot the excess free energy per particle of the expanded
and the condensed phases, Fexp(n) = Fcond (l /s )2n and Fcond(n). The former diverges at
ex ex ex
n = (s /l )2/a which corresponds to close-p(cid:0)acked dis(cid:1)ks of diameter l . This divergence is, of
course, unphysical because the expanded hexagonal lattice remelts upon compression so that in
thisregime,thetrueexcess entropicfree energy interpolatesbetween Fexp andFcond.
ex ex
Figure 3: Excess entropic free energies per particle of the expanded and the condensed unmodu-
lated phases derived using a Carnahan-Starling type theory (solid lines; the expanded phase cor-
responds to l /s = 2). The divergence of Fexp at n = 0.276 is unphysical; instead of approach-
ex
ing the shoulder-to-shoulderclose-packed structure, the system melts to form the condensed fluid
phase. The dashed line represents a qualitatively correct interpolation between the expanded and
thecondensedbranch. —Theschematicsillustratethestructureofthe4unmodulatedphases: The
expanded fluid, the expanded crystal, the condensed fluid, and the condensed crystal. Full circles
indicatethehard cores ofparticleswhereas theshaded coronas represent theshoulders.
TheexactshapeoftheinterpolatingF isnotknown. Butourmean-fieldmodelisdesignedto
ex
work best for broad shoulders and in this case, the existence of the expanded phases is restricted
toreducedaveragedensitiesbelow(s /l )2/a ,i.e.,toverysmalln. Beinginterestedintheoverall
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