Table Of ContentCubatic phase for tetrapods
Ronald Blaak∗
Institut fu¨r Theoretische Physik II, Heinrich-Heine-Universit¨at, D-40225 Du¨sseldorf, Germany
Bela M. Mulder† and Daan Frenkel‡
FOM Institute for Atomic and Molecular Physics,
Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
4
0 We investigate the phase behavior of tetrapods, hard non-convex bodies formed by 4 rods con-
0 nectedundertetrahedralangles. Wepredictthat,dependingontherelativelengthsoftherodsthese
2 particlescanformauniaxialnematicphase,andmoresurprisinglytheycanexhibitacubaticphase,
a special case of the biaxial nematic phase. These predictions may be experimentally testable, as
n
experimental realizations of tetrapods haverecently become available.
a
J
1 I. INTRODUCTION cubatic phase in the case that the three rods have ap-
proximately equal lengths, but they can also show rod-
]
t like and plate-like behavior by forming uniaxial nematic
f Thesimplestliquidcrystallinephaseisthenematic. It
o phases if one, respectively two, rods are dominant in de-
is a spatially homogeneous phase in which the orienta-
s termining the shape of the particle. Moreover,at higher
. tions of the non-spherical component particles, e.g. rod-
t densities,theseuniaxialnematicphasesbecomeunstable
a likeordisk-likecolloids,aredistributedinananisotropic
anddifferenttypesofbiaxialnematic phases areformed.
m fashion. More precisely, they are oriented around a pre-
Unfortunately these lower-symmetry liquid-crystalline
- ferred axis yielding a phase with macroscopic uniaxial
d phases have, thus far, not been observed in experiment.
optical anisotropy. However, this represents only the
n The main problem seems to be that, unlike the rod-like
simplest form of rotational symmetry breaking. When
o colloidalparticlesthatformnematics,cross-likeparticles
inadditionthe cylindricalsymmetryaroundthenematic
c
that are both rigid and sufficiently monodisperse could
[ director is broken, the phase that results is the so-called
not be made in a sufficient quantities to allow a system-
biaxial nematic phase. As the name biaxial suggests,
1 atic study of their phase behavior. In particular the cu-
there are now two preferred axes, which are mutually
v batic phase has not yet been observed in experiments,
perpendicular. Biaxialphasescanbeexpectedifthecon-
3
although simulations have suggested that a phase with
0 stituentparticlesthemselvesdonothave(effective)cylin-
this symmetry may exist in a system of disk-like parti-
0 drical symmetry, but are only invariant under a limited
cles [9].
1 number of discrete rotations [1, 2]. Biaxial phase may
0 alsoappearinmixturesofrod-likeanddisk-likeparticles. Recently, however, Alivisatos et al. have reported
4 Each of the two components individually will form uni- the synthesis of colloidal CdTe tetrapods [10]. These
0 axialnematicphasesatsufficientlyhighdensities. When particles could be made with a high yield and with
/
t mixedtheywilldothesame,buttheirmutualinteraction well-controlled nanoscale dimensions. The experimental
a
m is such that the preferred orientation axis for the rods is tetrapodsconsistsofasmallcrystallinebodyfromwhich
perpendicular to that of the disks [3, 4, 5, 6]. four arms grow under tetrahedral angles. Since these
-
arms are also crystalline, the tetrapods are fairly rigid
d Thisstilldoesnotexhaustallpossibilitiesforspatially
n and, with a suitably chosen solvent (and proper steric
homogenous liquid-crystalline phases. Frenkel [7] pro-
o stabilization),shouldbehaveasrigidhard-coreparticles.
posed that particles consisting of three identical rods,
c In this paper we consider the liquid crystalline behav-
: connected at right angles at their center, should form a
v ior of tetrapods. For simplicity we work in the Onsager
stable high-density phase with cubic orientational sym-
i limit of large aspect ratios and only take into account a
X metry. This liquid crystalline phase is referred to as a
hard-core interaction. We assume that the particles are
cubatic phase. It is a special case of the general biaxial
r
a phase, since there are now three mutually perpendicular monodisperse, but we treat all possible combinations of
relative lengths for the arms of the tetrapod. We focus
axes of symmetry that are equivalent.
here on a bifurcation analysis, which gives us an upper
Thismodelhassubsequentlybeengeneralizedtocross-
limit to the stability of the isotropic phase and yields an
likeparticles(”Onsagercrosses”),inwhichthethreerods
indication of the nature of the more stable liquid crys-
canhaveunequallengths[8]. Theseparticlesshowasur-
talline phases. Using this analysis, we argue that the
prisingly rich phase behavior. Not only do they form a
tetrapods of ref. [10] should, under certain conditions,
form cubatic phases.
In section II we justify the main assumptions of the
model and derive an expression for the Helmholtz free
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] energy. We make use of rotation matrix elements, of
‡Electronicaddress: [email protected] which the main properties and conventions are briefly
2
described in the appendix. In section III we perform the is an order D/L smaller than the second virial contribu-
stability analysisofthe isotropicphase andinterpretthe tionandcanthereforebeneglectedinthelimitofinfinite
results,andconcludewithadiscussionofthemainresults aspect ratios.
in section IV. In summary then, the assumptions that virial expan-
sion of the free energy can be truncated at the second-
viriallevelandthattheinteractionbetweenparticlescan
II. THE MODEL be considered as a sum of pair interactions between the
”arms”ofthe particles,become exactinthe limit ofinfi-
niteaspectratios. Forlargebutfiniteaspectratios,these
To analyze the phase behavior of hard tetrapods, we
assumptions should constitute excellent approximations
need expression for the free energy of this system. In
Thetruncationofthevirialseriesleadstothefollowing
general, this is an intractable problem. However, for
free-energy functional for homogeneous systems
tetrapodswithsufficientlyslenderarmswecanmakethe
same assumptions that were introduced by Onsager in
βf[ψ]=βfˆ+ dΩψ(Ω)lnψ(Ω)
the context of the isotropic-nematic transition of thin
hard rods [11]. Onsager showed that for a fluid of par- Z (1)
1
ticles with large (strictly speaking, infinite) length-to- + ρ dΩ dΩ ψ(Ω )ψ(Ω )E(Ω ,Ω )
1 2 1 2 1 2
2
width ratio, the excess free energy can be truncated at Z Z
thesecondvirialcoefficientlevel. Inthecaseofhard-core Here f is the Helmholtz free energy per particle, which
interactionsthisisequivalenttoassumingthatifoneran- is a functional of ψ the orientational distribution func-
domly places particles with a given density in space, the tion (ODF). This ODF is a measure for the fraction of
probability that three particles mutually overlap is neg- particles with an orientation Ω, which is shorthand for
ligibly small. the three Euler angles (α,βγ) required to specify an ar-
Although we have assumed large aspect ratios for the bitraryorientationina fixedreferenceframe,andisnor-
armsthatconstituteatetrapod,itisnotimmediatelyob- malized to unity. β = (k T)−1 is the inverse tempera-
B
viousthatthesecondvirialapproximationisvalid. How- ture,ρthe number densityandfˆthe idealgastermthat
ever,sincetetrapodsareessentiallyobjectswithanopen doesnotexplicitlydependontheODF.Thesecondterm
structureandconsistoffourconnectedrod-likeparticles, corresponds to the orientationalentropy, while the third
one would expect that if two particles overlap with each term is associated to the translational entropy through
other this is mainly due to a single arm of one particle the kernelE(Ω ,Ω )describingthe overspaceintegrated
1 2
that overlaps with a single arm of the other particle. interactionoftwotetrapodswithorientationsΩ andΩ .
1 2
The validity of this plausibility argument is confirmed The interaction of hard-core objects is taken into ac-
explicitely for Onsager crosses with three equally long count via the excluded volume E(Ω ,Ω ). This volume,
1 2
arms in the isotropic phase [12]. In a detailed analysis is defined as the volume around particle 1 (with orien-
it is shown that for a length-to-width ratio of about one tation orientations Ω ) that is inaccessible to particle 2
1
thousand the probability that, under the constraintthat (with orientation orientations Ω ). For two slender par-
2
twoparticlesoverlap,morethanasinglepairofthearms ticles, with lengths L and L and diameters D and
1 2 1
are overlapping, is less then a percent and decreases for D respectivelythe excludedvolumeis,to leadingorder,
2
increasing aspect ratios. In other words, in the limit of given by
large aspect ratios one can describe the particle-particle
L L (D +D )|sinγ|, (2)
interaction in terms of independent pairs of arm-armin- 1 2 1 2
teractions only. where γ is the angle between the long axes of the par-
What Onsager showed for elongated particles in the ticles. Corrections to this expression are of order D/L.
isotropic phase is that the asymptotic limit of the third Sincewearemainlyinterestedinthelimitoflargeaspect
virial coefficient B3 can be expressed in terms of the ratios, we restrict ourselves to the leading order only.
second virial coefficient and the aspect ratio by B3 = In the case of tetrapods the excluded volume is, of
B22O(D/Llog(L/D)) [11]. This has been confirmed by course, more complicated. First of all we now have four
the calculation of virial coefficients for long spherocylin- rod-like arms. We will assume that the arms can be ap-
ders [13]. The fourth and fifth virial showed a similar proximated by cylinders with identical diameter D, but
dependence Bn = B2(n−1)O(D/L). Since the interaction possibly different lengths Li, where i = 1,2,3,4. With
between Onsager crosses in leading order is determined the assumption that the arms of tetrapods, whilst con-
by single rod-rod interactions the same behavior should nected, interact independently, the excluded volume of
be observedfor Onsagercrosses,as indeed is found, pro- twotetrapodsbecomesasumoverthe pairwiseexcluded
vided aspect ratios are of the order thousand [12]. In volumes of the arms
addition, the main contribution to the higher virial co-
E(Ω ,Ω )= 2L(1)L(2)D|sinγ12|, (3)
efficients stems from the so-called ring diagrams, which 1 2 i j ij
i,j
wouldleadtosamescalingbehaviorforhigherordervirial X
coefficients. Hence corrections to the free energy due to where the superscript refers to the particle and the sub-
the simultaneous interaction of three or more particles script to the arm of the tetrapod, hence γ12 is the angle
ij
3
between the ith and jth arm of particle 1 and 2 respec- has no effect on the final result.
tively. For the isotropic phase this leads to a simple ex- It is obvious that the excluded volume (3) of two
pressionforthesecondvirialcoefficientB2. Itisequalto tetrapods cannot depend on both orientations indepen-
half the excluded volume averaged over all orientations, dently,butonlyontherelativeorientationΩ−1Ω . Hence
1 2
as it is simply equal to the sum of second virials for all we can rewrite the excluded volume as
pairs of rods
E(Ω−1Ω )≡E(Ω ,Ω )
2 1 2 1 2
B2 = πD Li . (4) = 2L(i1)L(j2)D|Ω1eˆ(i1)×Ω2eˆ(j2)|
4 Xi ! Xi,j (5)
= 2L(1)L(2)D|zˆ(1)×g−1Ω−1Ω g zˆ(2)|.
We note that we can expand any ODF depending on i j i 1 2 j
Ω in terms of a linear combination of Wigner rotation Xi,j
matrixelements Dl (Ω). Inwhatfollows,we adoptthe
m,n
With the aid of this form we can now expand
conventionusedbyBrinkandSatchlerinthedescription
the excluded volume in terms of rotation matrix ele-
of the Wigner matrices [14].
mentsDl (Ω),byintroducingtheexpansioncoefficients
In order to make use of the free energy functional (1), m,n
E
we need to rewrite the interaction (3) in terms of these l,m,n
functions. To this end, we introduce an arbitrary refer-
ence orientationof a tetrapod. We denote the directions E(Ω)≡ E Dl (Ω), (6)
l,m,n m,n
of the arms with length Li of the reference tetrapod by l,m,n
X
theunit-vectorseˆ. Forsimplicityweassumethateˆ =zˆ
i 1
is the positive z-direction and eˆ2 is lies in the xz-plane where l = 0,1,··· ,∞, and −l ≤ m,n ≤ l. Strictly
withapositivex-component. Sincebydefinitionthemu- speaking, the rotation matrix elements also defined for
tual directions are under tetrahedral angles this fixes all half-integer ”spin” values. However, for reasons of sym-
directions. metry these can be omitted [14].
This allows us to interpret the orientation Ω of a par- The expansion coefficients can be evaluated by using
ticle, as the one we wouldobtainif we takethe reference the orthogonality relation (A2) for the rotation matrix
particle and rotate it over the Euler angles denoted by elements
Ω. Additionally we can also interpret Ω as the actual
rotation matrix, so the directions of the arms of a par- E = 2l+1 dΩDl ∗ (Ω)E(Ω). (7)
ticle become Ωeˆ. Finally, we introduce the rotations g , l,m,n 8π2 m,n
i i
Z
such that eˆ ≡ g zˆ. Note that these rotations g are not
i i i
uniquely defined, since only two of the Euler angles are Substitutingtheexpression(5)andchangingtheintegra-
requiredinordertosatisfythe restriction. However,this tion variables we get
2l+1
E = 2D L L dΩDl ∗ (g Ωg−1)|zˆ(1)×Ωzˆ(2)|
l,m,n 8π2 i j m,n i j
i,j Z
X
(8)
2l+1
= 2D L Dl ∗ (g ) L Dl (g ) dΩDl∗ (Ω)|sinβ|
8π2 i m,p i ! j n,q j p,q
p,q i j Z
X X X
where we made use of the symmetry relation (A1) and ducing the following shorthand notations
closure relation (A3) and replaced the cross product by
its representation in Euler angles |sinβ|. E ≡ L Dl (g ) (9)
l,m i m,0 i
i
X
2l+1 π
µ ≡ dβP (cosβ)sin2β (10)
l l
2
Z0
In order for the integral to be non-zero, it is required the expansion coefficients of the excluded volume can be
thatbothindicesoftherotationmatrixelementarezero. written in a compact form as
This is a special case for which the function reduces to
a Legendre polynomial Dl (Ω) = P(cos(β)). By intro- E =(2D)µ E∗ E . (11)
0,0 l l,m,n l l,m l,n
4
The integral that remains can be readily evaluated (See the stability equation that is obtained as the functional
[15]Eq. 7.132.1)andis onlynon-zeroforevenvalueofl derivative of the free energy with respect to the ODF
2 δ
π(4l+1) 2l
βf[ψ]−λ dΩψ(Ω) =0, (17)
µ =−
2l (l+1)(2l−1)24l+2(cid:18)l(cid:19) (12) δψ(Ω)(cid:26) Z (cid:27)
µ2l+1 =0. whereλisaLagrangemultipliertotakecareoftheproper
normalization of the ODF. Evaluating this expression
Finally we introduce another shorthand notation by
and using the definition (13) this gives us
usingthekernel(3)asafunctionalactingonanarbitrary
function ψ(Ω) ln(ψ(Ω))+ρE[ψ](Ω)=λ. (18)
E[ψ](Ω)≡ dΩ′E(Ω′−1Ω)ψ(Ω′). (13) For both the ODF ψ and number density ρ we take
the formal expansion in a small parameter ǫ
Z
In particular we allow it to operate on a rotation matrix ψ =ψ +ǫψ +ǫ2ψ +··· (19)
0 1 2
element. Usingtheexpansion(6)andtheproperties(A1) ρ =ρ +ǫρ +ǫ2ρ +··· . (20)
0 1 2
and (A3) this can be manipulated to yield
Hereweuseψ = 1 astheODFfortheisotropicphase.
E[Dml ,n]= El,n,pDml ,p(Ω). (14) By inserting t0hese8πe2xpansions in the stability equation
p (18) and grouping terms for each power in ǫ, we obtain
X
the bifurcation equations. By solving these we can find
Note thatthis generatesalinearcombinationofrotation
thelowestdensityρ atwhichasymmetrybreakingmode
0
matrixelementswiththesamevalueforlandm. Inother
exists that lead to a lower free energy than that of the
wordseachsetDl withn=−l,··· ,lformsasubsetof
m,n isotropicphase. Inthecaseofanon-zerovalueforρ this
1
functions that is invariant under the functional operator
is sign ofa first orderphase transitionat a lowerdensity
of the excluded volume. We can go one step further by
and hence ρ is the upper limit for the meta-stability of
0
evaluating eigenfunctions of the excluded volume. Using
the isotropic phase.
the special form of the coefficients (11), one can easily
The zeroth-order bifurcation equation merely states
check that for each combination of l and m at most a
that the isotropic solution ψ = ψ is a solution of the
singleeigenvectorχl existwithanon-zeroeigenvalueλ 0
m l stationarityequation. Thefirstorderbifurcationhasthe
form of an eigenvalue problem
χlm(Ω)= El,pDml ,p(Ω) (15) ψ1
+ρ E[ψ ]=0, (21)
Xp ψ0 0 1
wherewe havealreadyeliminatedthe constantcontribu-
λ =(2D)µ E∗ E . (16) tions. Since we are interested in a non-trivial solution
l l l,p l,p
Xp that leads to the lowest possible positive value of ρ0, we
only need to consider linear combinations of eigenfunc-
Note thatthe eigenvalueisindependent ofmandinspe- tions(15). Inparticularweneedtofindtheonethathas
cial cases also might become zero as for instance for odd thelargestabsolutevalueamongallnegativeeigenvalues.
values of l. Onecanshowthat,forthecaseoftetrapods,thereare
only two eigenvalues that can fulfill that requirement,
namely the ones corresponding to l =2 and l=4
III. BIFURCATION ANALYSIS
π3
λ =− DL2(4R2−1) (22)
The thermodynamically stable phase of our model, is 2 6
describedbytheODFthatminimizesthefreeenergy(1).
Usually this free energy is not known exactly and one
uses a truncated expansion as an approximate function. π3
λ =− DL2(80R2+1). (23)
There is however one exception: the isotropic phase. In 4 1296
thelimitofinfinitedilution,particlesdonotinteractand
For practical purposes we used here L = L , and
thereforeeachorientationhasthesameprobability,hence i i
R2 = ( L2)/L2. Since λ = λ for R2 = 31 and
the ODF is merely a constant. i i 2 4 P 112
Withtheaidofabifurcationanalysiswecandetermine by constPruction 41 ≤ R2 ≤1, we need to distinguish two
an upper limit to the stability of the isotropic phase. types of tetrapods. The ones with R2 > 31 for which
112
To this end, we make an expansion of the ODF around we need to consider modes related to l=2 and the ones
the stable isotropic solution. Rather than inserting this with R2 < 31 for which the modes with l = 4 are the
112
into the free energy (1), we use this to find solutions of important ones. For the special case R2 = 31 we would
112
5
300 the orientation Ω, gives us a set of coupled non-linear
l=2 equations in the coefficients cm and constant ρ1
250 l=4
l=6 1
l=8 ρ λ c dΩχl∗χl = dΩχl∗ψ2. (26)
200 1 l nZ n n 2ψ02 Z n 1
Wecanalsoextractthevalueofρ ,ifweusethecomplete
2 1
ρB 150 function ψ1∗ in stead of χln∗
100 1 dΩψ3
ρ = 1. (27)
1 2λψ2 dΩψ2
l 0 R 1
50
ItisimportanttorealizethattRhereisarestrictiononρ .
1
The reason is that a non-zero value of ρ is associated
0 1
0 0.2 0.4 0.6 0.8 1 with a first order phase transition and that in order to
L /L
1 followthesolutiontowardslowerdensitiesweneedanon-
positive value, hence ρ ≤0.
1
FIG. 1: The density at which the isotropic phase becomes Let us now consider the case R2 > 31 with l = 2
unstable with respect to different modes characterized by l 112
in the preceding equations. The set of equations (26)
for a subset of particles with equal volume and arm lengths
L2 = L3 = L4 = (L−L1)/3. The solid line (l = 2) denotes can be solved analytically and yields only a single non-
trivialsolution,which is degeneratesince allrotationsof
the isotropic-nematic instability and the dotted line (l = 4)
indicates the isotropic-cubatic instability. a solution are also solutions of the set of equations. For
a conveniently chosen reference frame the solution is
ψ (Ω)=c χ2(Ω) (28)
actually need to consider combinations of both, which 1 0 0
makes the analysis somewhat more involved, but since
It bifurcates at a reduced density
this is notgoing to leadto new insights we willnot treat
it separately. 12
ρ B = . (29)
AsanillustrationweshowinFig.1the locationofthe 0 2 4R2−1
four lowest instabilities of the isotropic phase for a spe-
cific class of particles, i.e. those for which L = L = Note that in the limit of a single arm (R2 = 1) this
2 3
L = (L−L )/3. This set includes the fully symmet- reduces to the correct result for uniaxial rod-like parti-
4 1
ric tetrapod (L /L = 1/4) and a limiting tetrapod with cles. Thisparticularsolutionisinvariantunderrotations
1
only a single arm (L /L = 1). Note that the volume of aboutthe z-axis,andhence one canexpect that the sys-
1
each particle is the same v =(π/4)D2L, but the density tem shows a phase transition from an isotropic to an
at which the isotropic phase becomes unstable for the uniaxial nematic phase. Inserting the solution in the ex-
symmetrictetrapodis approximatelythirtytimes higher pression (27) for ρ1 we obtain
than that of a tetrapod with a single arm. The volume
fraction φ=ρv, however, is proportional to D/L, which −π3Dc0
ρ = [L +L −L −L ]×
means that in the limit of large aspect ratios the transi- 1 14ψ2λ2 1 2 3 4 (30)
0 2
tion takes place at small volume fractions. [L −L +L −L ][L −L −L +L ].
1 2 3 4 1 2 3 4
Althoughwenowknowtheupperlimittothestability
of the isotropic phase, we do not yet know which are From this result it follows that if the sum of the lengths
the symmetry breaking modes. To find these, we need of two arms equals the sum of the length of the two re-
to perform a second order bifurcation analysis, i.e. solve maining arms we find ρ = 0. These special particle
1
the equation configurationscouldthereforepossiblyleadtoacontinu-
ousphasetransitionandbethesourceofabiaxialphase.
ψ 1 ψ 2 ψ2 Whether this is scenario really applies, cannot be deter-
2 − 2 +ρ E[ψ ]+ρ E[ψ ]=− dΩ 1 (24)
ψ 2 ψ 0 2 1 1 2ψ3 mined from this analysis. One could resolve this issue
0 (cid:18) 0(cid:19) Z 0 either by solving higher order bifurcation equations or
employing the general solution of the first order bifurca- by afull numericalminimization ofthe free energy. This
tion equation given by (21) falls outside the scope of the present paper. For other
particle configurations the requirement of non-positive
ψ1(Ω)= cmχlm(Ω), (25) values for ρ1 will fix the sign of the coefficient c0, which
canbepositiveornegative,andhencefullydeterminethe
m
X solution(29),becausethemagnitudewillonlydependon
where the c are some complex constants and l is ei- the choice of normalization.
m
ther 2 or 4. Substitution in the second-order bifurcation Similar to the case for Onsager crosses [8], we can in-
equation (24), multiplying with χl∗ and integrating over terpretthisphenomenonintermsofrod-likeanddisk-like
n
6
FIG. 2: Phase diagram of the nature of symmetry breaking
modes leading to the instability of the isotropic phase. Each
point inside the tetrahedron corresponds to a given shape of FIG. 3: Cross-sections of the phase diagram for constant
thetetrapod. Avertexcorrespondstoasinglearm,theoppo- value of li, from left to right, top to bottom for decreasing
site planetoa tetrapod with threearms. For particles inside values. The dark circular areas correspond to the cubatic
the sphere a cubatic phase is expected, outside the sphere a solution,thelightandwhiteareastothetwotypesofnematic
nematicphasewhichiseitherrod-like,ifitcontainsavertex, solutions, being rod- and disk-likerespectively.
or platelet-like if it does not. Both species are separated by
planes denoting the particles that might have a continuous
phase transition and show biaxial behavior. isthe particularsolutioninvariantunderrotationsabout
the z-axis, and c < 0 in order for ρ to be negative.
0 1
Contrary to the previous case c does not change sign.
0
behavior. For each of the four arms of the tetrapod, we A particular solution of the second family of solutions
candetermine the nematic orderparameter,whichis de- is given by
finedastheaveragevalueof 3cos2(θ)−1 withθtheangle
2 2
betweenthenematicaxisandthedirectionofthearm. It 5
ψ (Ω)=c χ4(Ω)+ χ4(Ω)+χ4 (Ω) . (33)
canbeshownthat,uptoapositivenormalizationfactor, 1 0 0 14 4 −4
(cid:26) (cid:27)
thisisproportionaltoc0(4Li−L)/3forallarms. Making (cid:0) (cid:1)
This solution is only invariant under discrete rotations
use of permutations of arms it follows that for positive
over π/2 about the x-, y-, and z-axes. It therefore cor-
values of c the longest arm has the largest nematic or-
0
responds to the cubic symmetry group and hence to a
der,whilefornegativec itwouldbetheshortestarm. A
0
cubatic phase. Also in this case, the constraint on ρ
special limit of the former, is the case where L is much 1
1
results in a negative value for c .
larger than the other three. It is obvious that for such 0
This analysis allows us to sketch a tentative phase di-
particlesthislongestarmwilldominatethebehaviorand
agram (see Figure 2). In this a schematic figure, we in-
the tetrapod behaves as a single rod-like particle. The
dicate the nature of the symmetry-breaking modes that
other extreme occurs when L is much smaller than the
1
lead to the instability of the isotropic phase. We can
other three. In that case the competition among those
characterize the particle shape by the normalized val-
three arms does not allow any of them to dominate and
ues l = L /L with the constraint l +l +l +l = 1.
there is a preference for them to be perpendicular to the i i 1 2 3 4
Each possible particle shape, i.e. combination of rela-
nematicaxisand,asinthecaseofdisks,itisthesmallest
tive lengths of the arms, corresponds to a point inside a
dimensionthatdeterminestheorientationoftheparticle.
tetrahedron,whichistheprojectionofthe4-dimensional
The set of equations (26) can also be solved for the
case of R2 < 31 , but then we need to put l = 4 in the ”shape” space. A vertex of the tetrahedron represents
112 the limit of a tetrapod with only a single arm, the tri-
equations (25)-(27). The bifurcation density can easily
angularplane oppositeto the vertexcontains allparticle
be determined from the proper eigenvalue (23)
shapesforwhichthatsamearmhaszerolengthwhilethe
2592 remaining three arms have non-zero lengths. In Figure
ρ B = . (31)
0 2 80R2+1 3, we have made cross-sections corresponding to planes
with constant value for one of the l .
But instead of having a single family of solutions that i
Nine distinct regions can be identified: a spherical re-
solve the equations (26), we now have two. The first
gion corresponding to R2 ≤ 31 , where there are two
family correspondsagainto auniaxialnematic phase for 112
modes that lead to the instability of the isotropic phase,
which
oneofwhichyieldsthecubaticsolution. InFigure3these
ψ (Ω)=c χ4(Ω) (32) regions are indicated as the dark circular areas. Outside
1 0 0
7
theseregions,wefindtheparticlesforwhichthereisonly confirm the existence of these phases and to ascertain
amodewithuniaxialsymmetrythatmakesthe isotropic whethera transitionfromisotropicphaseto anyofthese
phase unstable. There are four equivalent areas close four phases is possible.
to a vertex, shown as white in Figure 3, that represent The formation of a cubatic phase even for fully sym-
tetrapodswherethesolution(28)hasapositivesign,and metric tetrapods may seem surprising. Naively, one
fourequivalentareas,indicatedbylightgrayinFigure3, might expect to observe a ”tetrahedratic” liquid crys-
where the solution has a negative sign. They are sepa- talline phase. The reason why the latter phase does
rated by the planes(lines) leading to “biaxial” particles. not appear here is related to the fact that in the On-
sager approximation presented here the arms of a parti-
cle effectively interact independently with those of other
IV. DISCUSSION particles. Essentially we are therefore insensitive to the
details on how the arms are connected. Although the
relative orientation of the arms within each particle is
The present work suggests that the availability of
fully accounted for, we cannot specify that the arms are
nanocrystallinetetrapods with a wellcontrolledsize and
connected end-on. The same results would therefore be
shape [10],maymakeitpossibletoobservecubaticliquid
obtained for particles in which the four rods were con-
crystallinephases in experiments. Of course,the present
nected at their mid-points. Such particles would in fact
”Onsager-style”analysisonly becomes exactin the limit
havecubic symmetryandarehence areunable toforma
of very slender, rigid arms. Within this approximation,
phase with tetrahedral symmetry.
wehavedeterminedtheupperlimittothestabilityofthe
Theresultspresentedhereareonlyvalidinthelimitof
isotropicphasebymeansofabifurcationanalysis. Inad-
infinite aspect ratios. For finite aspect ratios we expect
dition,wehavedeterminedthenatureofthefluctuations
twotypesofcorrections: (i)contributionsduetosimulta-
that cause the instability.
neousoverlapofthreeormorearmsand(ii)dependencies
Roughly speaking we can distinguish two types of
onthe detailedconstructionofthe particles. Inpractice,
tetrapods, the ones for which the arms have approxi-
one would presumably require aspect ratios of the order
matelythe samelengthsandtheoneswhereoneormore
one thousandor more in orderto makethese corrections
armsaresignificantlylongerthantheothers. Forthefirst
negligible (see Ref. [8]). Nevertheless one could expect
groupthe isotropicphase becomes unstable with respect
thatourqualitativefindingsremainvalidevenforsmaller
to a distortion with either nematic or cubatic symme-
aspectratios,asisthe caseforsinglerods. Thefactthat
try,while forthe secondgrouponlyanematic symmetry
particles might not be perfectly monodisperse is proba-
comes into play.
bly not a problem, since there is a rather broad range of
In general the transitions will be first order since we
particle shapes that gives rise to the cubatic instability
obtainedanon-zerovalueofthefirstordershiftindensity
of the isotropic phase. However, finite aspect ratios im-
ρ alongthebifurcatingsolutions,indicatingthepresence
1 ply that also non-homogeneousphases should be consid-
of a v.d. Waals loop. An exception might be formed by
ered, in which the isotropic-to-cubatic transition might
the particles located in the planes in Figure 2. Although
be preempted by crystallization. Finally, there may be
the results of the bifurcation analysis cannot guarantee
kinetic limitations to the formation of cubatic phases of
that the symmetry of the fluctuations that lead to the
tetrapods, as the tetrapods are likely to become entan-
instability will also be the symmetry of the more sta-
gled at high densities, which could lead to kinetically
ble phase, the experience in a similar study of Onsager
arrested glass-like phases.
crosseshasshownthatthebifurcationanalysishasahigh
predictivevalue[8]. Inaddition,italsostronglyindicates
that a system of tetrapods with approximately identical APPENDIX A: ROTATION MATRIX ELEMENTS
arms will have an isotropic to cubatic phase transition,
whichinprinciplecouldbeverifiedbyafullminimization
Herewelistthemainpropertiesoftherotationmatrix
of the Helmholtz free energy functional.
elements. For a more extended discussion we refer the
Based on the results of the Onsager crosses, we pre-
reader to [14].
dict that at higher densities a system of tetrapods will
A rotation Ω = (α,β,γ) is obtained by successive ro-
undergo additional transitions to phases with yet lower
tations of angles α, β, and γ about the z-,y-, and z-axis
symmetry, ultimately arriving at the phase where only
respectively. The invariant measure of the rotation is
arms with identical lengths are aligned. In particular
given by dΩ = sin(β)dαdβdγ, with α,γ ∈ [0,2π] and
three intermediate phases might appear that are invari-
β ∈[0,π].
ant under 2-fold, 4-fold, and 6-fold rotations from the
Symmetry relation
cubic group. In some cases these phases could actually
preempt the isotropic-to-nematic or isotropic-to-cubatic Dl ∗ (Ω)=Dl (Ω−1) (A1)
n,m m,n
phase transition. This is most likely to happen for par-
Orthogonality relation
ticle shapes close to the planes and/or surface that sep-
anruamteertihcaeldfriffeee-reennetrgreygmioinnsiminiztahteionphwaoseulddiabgerraemqu.irAedfutlol dΩDml′′∗,n′(Ω)Dml ,n(Ω)= 28lπ+21δl,l′δm,m′δn,n′ (A2)
Z
8
Closure relation
l
Dl (Ω Ω )= Dl (Ω )Dl (Ω ) (A3)
m,n 2 1 m,p 2 p,n 1
p=−l
X
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