Soft Matter

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aDepartment of Physics, Cornell University,bSchool of Chemical and Biomolecular Eng

York, USA. E-mail: fe13@cornell.edu

Cite this: DOI: 10.1039/c3sm51822a

Received 3rd July 2013Accepted 11th October 2013

DOI: 10.1039/c3sm51822a

www.rsc.org/softmatter

This journal is ª The Royal Society of

Phase behavior of binary mixtures of hard convexpolyhedra

Mihir R. Khadilkar,a Umang Agarwalb and Fernando A. Escobedo*b

Shape anisotropy of colloidal nanoparticles has emerged as an important design variable for engineering

assemblies with targeted structures and properties. In particular, a number of polyhedral nanoparticles

have been shown to exhibit a rich phase behavior [Agarwal et al., Nat. Mater., 2011, 10, 230]. Since real

synthesized particles have polydispersity not only in size but also in shape, we explore here the phase

behavior of binary mixtures of hard convex polyhedra having similar sizes but different shapes.

Choosing representative particle shapes from those readily synthesizable, we study in particular four

mixtures: (i) cubes and spheres (with spheres providing a non-polyhedral reference case), (ii) cubes and

truncated octahedra, (iii) cubes and cuboctahedra, and (iv) cuboctahedra and truncated octahedra. The

phase behavior of such mixtures is dependent on the interplay of mixing and packing entropy, which

can give rise to miscible or phase-separated states. The extent of mixing of two such particle types is

expected to depend on the degree of shape similarity, relative sizes, composition, and compatibility of

the crystal structures formed by the pure components. While expectedly the binary systems studied

exhibit phase separation at high pressures due to the incompatible pure-component crystal structures,

our study shows that the essential qualitative trends in miscibility and phase separation can be

correlated with properties of the pure components, such as the relative values of the order–disorder

transition pressure (ODP) of each component. Specifically, if for a mixture A + B we have ODPB < ODPAand DODP ¼ ODPA � ODPB, then for any particular pressure at which phase separation occurs, the

larger the DODP the lower the solubility of A in the B-rich ordered phase and the higher the solubility

of B in the A-rich ordered phase.

1 Introduction

Interest in material design based on nanocrystal assemblies hasbeen rapidly increasing over the past decade. The choice ofbuilding blocks, size and shape, composition and surfacefunctionalization offers multiple avenues to tailor the proper-ties of these assemblies. Going beyond the prototypical spher-ical nanoparticles, researchers have explored the effect of shapeanisotropy on colloidal self-assembly afforded by the synthesisof cubes, rods, plates, disks and particles with core–shellstructures.1–4 With the advent of new particle synthesismethods5–8 polyhedral nanoparticles are becoming ubiquitousin providing varying extents of shape ‘anisotropy’. Suspensionsof a range of polyhedral shapes have been shown to produceinteresting partially ordered ‘mesophases’9–12 at intermediatevolume fractions. The mixing of nanoparticles of differentchemical compositions could provide a simple way to prepareordered structures with more desirable properties; e.g., if adopant component adds some functionality to the host

Ithaca, New York, USA

ineering, Cornell University, Ithaca, New

Chemistry 2013

component in a matrix. If the particles differ not only incomposition but also in shape, one may gain additional controlover the extent of inltration by the dopant component, howtheir particles orient and what spatial environment theyencounter.

A range of different polyhedral nanocrystals have becomereadily synthesizable by controlling the growth step in a modi-ed polyol process for the formation of nanocrystals made ofgold13,14 and other inorganic materials;15,16 Fig. 1 shows some ofthe particle shapes typically produced this way. This approachalso readily leads to mixtures of these polyhedra (shape

Fig. 1 Figure showing some of the polyhedra synthesized at different steps of amodified polyol process. The arrow shows the order of the shapes produced asthe reaction is allowed for longer times.

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Fig. 2 Snapshots showing the 4 mixtures studied: (a) CS mixture (cubes +spheres), (b) CTO mixture (cubes + truncated octahedra), (c) COC mixture (cubes +cuboctahedra), and (d) COTO mixture (cuboctahedra + truncated octahedra). Therelative sizes accurately describe the size ratios in simulations.

Soft Matter Paper

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bidispersity) by stopping the growth process at different stages.The phase behavior of such binary mixtures (of particles havingsimilar sizes but different shapes) is expected to depend onmultiple factors, including the phase behavior of the individualpolyhedra and in particular their order–disorder transitionpressure (ODP) and the crystal lattice they assemble into. Aspecic class of tessellating mixtures17 has been studied before;in that case the pure components exhibit incompatible crystallattices, but at a precise (stoichiometric) composition and sizeratio, they can assemble into binary crystalline compounds.However, most mixtures of polyhedra with incompatible crystallattices (for the monodisperse systems) would be expected tophase separate at higher volume fractions into ordered phasesresembling those of the pure components.

When a mixture of two polyhedral shapes A + B phaseseparates, it is important to explore the extent to which they mixin the different phases; i.e., how much B is incorporated in theA-rich phase and how much A is incorporated in the B-richphase. Such information will be useful, e.g., in predicting thelimiting compositions at which such bidisperse mixtures canexist without phase separation. Two components that individ-ually form the same type of mesophase may give rise to mixturesthat exhibit a similar mesophase. For example, appreciablemixing of components has been observed in the case of poly-disperse mixtures of rigid rods of different lengths which form anematic phase,18 and of cubes of different sizes that form acubatic mesophase.19

With these considerations in mind, we study here the self-assembly of binary mixtures of convex polyhedra. As a referencecase, we study rst the cube + sphere mixture (CS), taking thesphere as the innite-facet limit of a regular polyhedron.Furthermore, we choose three representative binary mixtures,each made from polyhedra that have been synthesized byvarious experimental methods (Fig. 2):13,14,20 cubes + truncatedoctahedra (CTO), cuboctahedra + cubes (COC), and cubocta-hedra + truncated octahedra (COTO). These mixtures representdifferent degrees of similarity in terms of rotational symmetry,asphericity9 and the crystal lattice they individually assembleinto. The presence of similar mesophases9 may also affect thisdegree of mixing. The large difference in ODPs betweencomponents is associated with the larger difference in sizes andhence is expected to result in more ‘incompatible’mixtures thatreadily phase separate. Conversely, components with similarODPs and similar mesophases could be expected to mix betterwith each other. Our aim in this paper is to determineapproximate pressure–composition phase diagrams for each ofthese mixtures to try to elucidate the relationship between thephase behavior of the mixture and that of the pure components.Although there are many experimental methods prevalent in theliterature that allow synthesis of concave shapes, as a rst step,we aim to keep our analysis simple by restricting ourselves tomore common convex shapes.

The CS system represents a limiting case when one of thecomponents (spheres) has no anisotropy. In this case, theindividual shapes and their crystal structures are quite incom-patible and expected to phase separate above the components'ODPs. In our simulations, we have assumed that the side of the

Soft Matter

cube is equal to the diameter of the sphere. While such a choicedoes fulll the criterion of having components similar in size, itwill necessarily affect the precise location of phase boundariesin the phase diagram and also create an asymmetry in therelative diffusivities of the components in a given phase (withthe smaller component typically having larger mobility).

The other mixtures studied, namely, CTO, COC and COTO,represent cases of components with varying degrees of simi-larities in shape and size. The CTO mixture corresponds tocomponents which are rather distant in the polyol process (seeFig. 1); this mixture also embodies a peculiar feature where thetwo individual components are space tessellating (at closepacking) but a mixture of any composition is not. Such a featureand the fact that their ordered phases have very differentsymmetries suggest that the CTO system should be largelyincompatible (not unlike the CS system). Considering thatcubes and truncated cubes have almost identical phasebehavior and very similar shapes, one can consider the mixtureof cubes and cuboctahedra (COC) as representing shapes whichare essentially neighbors in the polyol process (see Fig. 1).Furthermore, the partial similarity of the ordered structures ofthe pure components suggests that the COCmixture could havean intermediate degree of compatibility. The mixture ofcuboctahedra and truncated octahedral (COTO) can beconsidered as potentially the most compatible given that thecomponents are proximal in the polyol process (Fig. 1) and thatboth form a rotator mesophase. In each of the mixtures, the sizeratios of the components (see next section) are consistent withthose obtained in a typical polyol process.13,14

2 Methodology

In our simulations, we assume the particles to be interacting viahard-core potentials, which amounts to ensuring that particlesnever overlap. Extensive expansion and compression MonteCarlo (MC) runs were performed to map out the equation of

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state of each of the mixtures, at constant pressure and particlenumber (NPT ensemble). Although we observe hysteresisbetween expansion and compression runs, we use expansionruns to estimate the transition pressures because they typicallyneed to surmount a smaller free energy barrier at the transitionpoints and are hence expected to more closely follow thermo-dynamic behavior. Unless otherwise indicated, the mixtureswere simulated at the same composition of 50%. We used atotal of 2560 particles for CS, CTO and COTOmixtures and 2048particles for a COC mixture. As indicated earlier, the size of thecomponents was chosen based on typical results of the modi-ed polyol process. For such a process, Seo et al.13 report theaverage edge lengths of different shapes to be 98, 120 and 145nm for truncated octahedra, cuboctahedra and cubes respec-tively. Of course, synthesis conditions can be altered so that agiven particle shape can be obtained with different sizes, butchoosing sizes consistent with those attainable with a singlegrowth experiment underscores a simple possible strategy toobtain the mixtures studied in this work.

The dimensionless osmotic pressure is dened as

P* ¼ Pl3

3; (1)

where l ¼ 1 is the characteristic length in each of the mixturessimulated here. In mixtures involving the cubic particles, thischaracteristic length is the length of the side of the cube, while inthe COTO mixture, it is the side of the imaginary cube fromwhich the cuboctahedron is cut. 3 is an arbitrary energy param-eter (set to 1). In these reduced units, the edge-lengths, particlevolumes, approximate order–disorder transition pressure (ODP)and mesophase–crystal transition pressure (MCP) for all theparticles studied are shown in Table 1. The volume fraction f ofthe system is just the ratio of the volume occupied by the poly-hedral particles to the total volume of the simulation box.

Each pressure step of the expansion/compression runinvolved a total of 3 � 106 MC cycles (as dened below) for bothequilibration and production. Each MC cycle consisted anaverage of N translational, N rotational,N/10 ip,N/20 swap and2 volume move attempts. Flip moves attempt to rotate a chosenparticle to a random orientation in the plane perpendicular toits present orientation. Swap moves involved picking randomlytwo particles, one of each species, and attempting to swap theirpositions as well as orientations (since such moves are morelikely to succeed with swapped orientations). Swap moves areessential to speed-up the equilibration process by allowing

Table 1 Reference data for the pure components studied. ODP describes order–disorder transition pressure which in most cases is an isotropic–mesophasetransition. MCP denotes the mesophase–crystal transition pressure. The refer-ences used for the data are listed alongside the shape

Shape Edge-length Volume ODP MCP

Cube9 1 1 6.3 8Cuboctahedron19 0.707 0.833 7.1 14Sphere21 1 (diameter) 0.524 11.5 —Truncatedoctahedron9

0.354 0.5 14 28

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particles to move arbitrarily far from their original positionscircumventing the slow diffusion associated with dense phases.Swap moves, however, can only be effective if the particle shapeand size of the two components are similar and densities aremoderate to allow some wiggle room around a lattice site toaccommodate different particle shapes. At pressures where thesystem is ordered (as obtained in trial runs), the volume moveswere allowed to be triclinic.9 The size of the move perturbationswas adjusted so as to get acceptance probability values of 0.4,0.4 and 0.2 for the translation, rotation and volume moves,respectively. Although the size of the pressure steps was notxed, typical values of DP* near the phase transition wereapproximately 0.8, 1.3, 0.8 and 1.6 for CS, CTO, COC and COTOmixtures respectively. All trial moves are accepted according tothe Metropolis criterion22 (which for hard-core interactionsrequires the absence of overlaps, checked via the separatingaxes theorem23 for any two polyhedra or Arvo's algorithm for thecube–sphere case24).

Interfacial simulations where two phases and the inter-vening interfaces are present in the same box were used toestimate coexistence conditions. To facilitate the formation ofdistinct bulk regions, the box was at least four times moreelongated along one direction than the others, so that theinterfaces would form perpendicular to that axis. Once inter-faces form, the longitudinal pressure (i.e., the one acting on aplane parallel to the interfaces) provides a suitable estimationof the coexistence pressure since the transverse pressurecontains a contribution from the surface tension.25 Thus, insuch cases we chose the transverse box length (which sets thedimensions of the interface) to be commensurate with thelattice spacing for the individual phases at the given pressure,only sparingly allowing changes in transverse dimensions tohelp relieve any build-up of stresses. Special care must be takenwhen dealing with interfacial simulations involving two solidphases where the box cross-section (perpendicular to the longaxes) must be chosen so that it can properly accommodate theunit cells of the distinct crystal lattices. Unfortunately, this ishard to achieve over a wide range of pressures. Hence, althoughtriclinic volume moves did allow for box deformations that canrotate the lattices to alleviate internal stresses, our results insuch regions are expected to have larger errors than those atlower pressures.

Unless otherwise indicated, interfacial simulations wereperformed for an equimolar global composition as it wasexpected that, if phase separation occurs, the compositions ofthe coexisting phases would be relatively symmetric (on accountof the particle-size similarity) and hence lead to similaramounts of the two phases. To estimate the equilibrium bulkdensities of the two phases in an interfacial simulation, we useddensity proles along the z-axis to mark the bulk regions.

We calculated the Q4 and Q6 bond-order orientationalparameters26 to probe and monitor translational order. Theseparameters are dened as:

Ql ¼ 4p

2l þ 1

"Xþl

�l

��QlmðrÞ��2#

12

(2)

Soft Matter

Soft Matter Paper

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where �Qlm(r) is given by

QlmðrÞ ¼1

Nb

Xbonds

YlmðrÞ (3)

where Ylm(r) are spherical harmonics for the position vector r.Although the values of these order parameters are sensitive tothe crystal structure, Q6 is generically a good descriptor ofcrystallinity, since its value increases monotonically withorder. The value of the Q4 order parameter gives additionalinformation about the type of crystalline structure present inthe system; i.e., larger values are associated with cubicsymmetry.

To determine the orientational order in the system, wecalculated the P4 cubatic order parameter12 which givesinformation about cubic-like orientational order. To mark theboundary between the rotator mesophase and the crystallinephase, we used a combination of the P4 order parameterand the orientational scatterplots. We set the upper limitingP4 value of the rotationally disordered phase to be 0.3 (notethat the maximum value of P4 for perfect cubic order is 0.583).The scatterplots were generated by plotting the orientationalaxes of each of the particles on a unit sphere. In these scat-terplots, a mostly uniform distribution of points signalsthe absence of orientational order, which along with thepresence of translational order (estimated through the Q6

order parameter) identies a rotator mesophase. A patchypattern gives the signature of a particular orientational-ordersymmetry, which is absent in a rotator mesophase. A set ofrepresentative scatterplots for the systems studied here areshown in Fig. 3.

Fig. 3 Representative orientational scatterplots used for determining theorientational order. From top: (a) truncated octahedra (TO) in a CTO mixture, (b)cuboctahedra (CO) in a COC mixture, (c) CO in a COTO mixture, and (d) TO in aCOTO mixture. For pressures above the phase separation, the scatterplots showthe orientational order for the particles in the ordered phase that is rich in thenamed component only.

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3 Results3.1 Cubes + spheres (CS) mixture

As previously indicated, this system represents a base case ofcomponents with widely different shapes and monodispersephase behavior. Cubes exhibit a mesophase27 for a small rangeof volume fractions (0.51–0.54) which separates isotropic andcubic crystalline phases.9 There is disagreement in the litera-ture28,29 about the precise nature of such a mesophase, whichhinges on the criterion adopted for classifying a phase as beingcrystal-like or liquid-crystal-like. For the range of volume frac-tions specied above, while the average particle positions arecrystal-like (despite a high vacancy content), the variance of theposition uctuations and particle mobilities are liquid-like.19

Since the current taxonomy is not completely satisfactory, forconcreteness we will henceforth refer to this mesophase ascubatic. Spheres, on the other hand, show a single phase tran-sition from isotropic to FCC crystal. While spheres have fullrotational symmetry, cubes do not and possess a relatively largeasphericity g (¼ ratio of circumradius to inradius ¼ 1.732). Inour simulations, the side of a cubic particle was set to be equalto the diameter of the spherical particle.

Extensive MC runs for the equimolar mixture exhibit anisotropic, fully mixed phase, up to P* z 8, close to the MCP forcubes. Thus as phase separation ensues above P*z 8, the cubesare already in a crystalline state, while spheres are still disor-dered. At a pressure P*z 12, the spheres order, a value which isonly slightly higher than the ODP for pure spheres (P*z 11.54).Note that generally, for any A + B mixture, the apparent ODP inthe A-rich phase is expected to be near but slightly above theODP of the pure A system because of the disordering effect ofthe B particles present; our observations are consistent with thisexpectation.

We demarcate the putative cubatic region by a combinationof increased P4 order parameter and high translational mobility.We estimate the latter by calculating the mean squaredisplacement (in a pseudo-dynamic setup) during compressionruns at different pressures. A high value of translationalmobility coefficient9 marks a region of the cubatic phase.Mapping out the region of cubatic behavior was not possible forthe 50% global-composition system as the cube-rich phaseformed by phase separation at even the lowest pressure alreadyhad the cubes in a cubic lattice (with f z 0.59). We thereforeperformed additional simulations at 60% to 90% numbercomposition of cubes at pressures just below the phase sepa-ration pressure. This helped us better dene the isotropic–cubicand isotropic–cubatic boundaries in the phase diagram.

The approximate pressure vs. composition phase diagram isshown in Fig. 4. Note that even the ordered phases of each of theindividual species allow a certain extent of mixing with theother species. Furthermore, the cube-rich phase solvatessignicantly more spheres than the sphere-rich FCC phasesolvates cubes. This asymmetric solvation capacity seems to beprimarily the result of both our choice of particle size ratioswhich gives cubes a larger volume (and excluded volume) thanthat of spheres, and the geometry of the respective latticespacings. Indeed, a sphere can readily replace a cube without

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Fig. 4 Summary of the results for the CS mixture. (a) Equation of state with ODPs for the individual particles shown for reference. Here ODPA is the ODP of sphereswhile ODPB is the ODP of cubes. (b) Plot of Q4 and Q6 order parameters. (c) Pressure vs. composition phase diagramwith different regions marked with the order of theconstituent phases. Here xB is the mole fraction of cubes. (d) Snapshots from the interfacial simulations of 3 representative regimes of the phase diagram (spheres arecyan and cubes are dark red). Bottom: isotropic + cubic phases at P* z 8. Middle: same phases at P* z 11. Top: FCC + cubic phases at P* z 15.

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overlap (if placed at the same center of mass) but not the otherway round. Consequently, mixing entropy favors cubic phaseswhere numerous spheres are allowed provided they do not takeaway the packing entropy gains associated with the overall cubicorder. Conversely, cubes are only allowed in the FCC sphere-richphase as long as they can appear as very dilute localized defectsthat do not compromise the overall structural order.

In the bigger picture, this mixture can be seen to represent a‘base case’, of particles with very distinct shapes and incom-patible lattice structures. In this scenario, although there iscertain amount of mixing at intermediate concentrations, theordered phases and the transition pressures remain quitesimilar to those of the pure components. This is expected sincethe two shapes are different enough that the mixing entropy tobe gained from their intermingling is outweighed by thepacking entropy lost due to the incompatibility of lattices.

3.2 Cubes + truncated octahedra (CTO) mixture

This mixture represents a case of two space-lling polyhedra,which have incompatible crystal structures. While cubes form acubic lattice in their ordered state, truncated octahedra (TOs)form a BCC tessellation. Furthermore, pure TOs exhibit arotator mesophase as an intermediate between the liquid andthe ordered crystal phase. Fig. 5 shows the main results for thissystem with Fig. 3 also showing some relevant scatterplots (inthe top row).

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Upon extensive compression and expansion MC runs, theCTO mixture is observed to form a uniform isotropic phase atlow volume fractions up to P*� 7.7 where two phases emerge: acube-rich cubic crystal and a TO-rich isotropic phase.

At P* z 18, the TO-rich phase is seen to transition into arotator mesophase, a value larger than that for pure TOs whichform a rotator mesophase at P* z 14. This difference isexpected since the presence of cubes in the TO-rich phaseshould make ordering more difficult and drive the orderingpressure upwards. The onset of the positional order in therotator mesophase is pinpointed by calculation of the Q6 orderparameter, which shows a sudden jump in Q6 for the TO-richphase.

At P* z 50, the TO-rich phase achieves a crystalline BCCorder (found through Q4 and Q6 values) and orientational order(found through P4 values and orientational scatterplots), apressure that is again larger than that for pure TOs which gaincrystalline BCC at around P* z 28. To better dene the cubaticregion, we again performed independent runs away from the50% number composition (between 80% and 95%) andassigned a cubatic character to systems exhibiting both high P4values and a high particle translational mobility.

As seen in Fig. 5(c), neither the cube-rich nor the TO-richcrystal phase allows a signicant concentration of the otherspecies. The saturation compositions tend to be symmetric; e.g.,at intermediate pressures the cube-rich phase saturates withapproximately 1% TOs and the TO-rich phase saturates with

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Fig. 5 Summary of results for the CTO mixture. (a) Equation of state with ODPs for the individual particles marked for reference. Here ODPA is the ODP of TOs whileODPB is the ODP of cubes. (b) Plot of P4 and Q6 order parameters. (c) Pressure vs. composition phase diagram with different regions marked with the order of theconstituent phases (xB is the mole fraction of cubes). (d) Snapshots from the interfacial simulations of three representative regimes of the phase diagram (TOs are purpleand cubes are dark red). Bottom: cubic + isotropic phases at P* z 9. Middle: cubic + rotator phases at P* z 25. Top: cubic + BCC phases at P* z 55.

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about 1% cubes. In a way, the CTO mixture shows less inter-species miscibility than the CS mixture. This could be because:(1) a slightly larger size disparity reduces the entropic gainobtained frommixing entropy as the pressure increases, and (2)the space-lling TOs are less tolerant to impurities (cubes) thanspheres (which are non-space lling). This question is revisitedin Section 4.

3.3 Cuboctahedra + cubes (COC) mixture

The components of this mixture have some key similarities anddifferences from those of the CTO mixture. Like TOs in the CTOmixture, cuboctahedra (COs) in the COC mixture also exhibit arotator mesophase over a signicant range of volume fractions,before going into a crystalline phase. The crystalline phase forpure COs, however, is a distorted cubic phase unlike the BCCcrystal exhibited by pure TOs. More importantly, the two speciesin the COC mixture are quite close to each other in size, with avolume ratio of 0.83 (in the CTO mixture this ratio is 0.5). Ourmain results for this system are shown in Fig. 6 (see also Fig. 3for some sample scatterplots).

Compression and expansion MC runs show phase separa-tion of the mixture at P* z 9, just above the MCP for cubes.Hence, phase separation upon compression of the isotropicphase gives rise to a CO-rich isotropic phase and a cube-richcubic crystal. At P* z 14, the CO-rich phase transitions into a

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rotator mesophase, as evidenced by a sudden rise in the value ofthe Q6 order parameter. For pure COs, this transition happensaround P*z 7. As the pressure increases to around P*z 20, theCO-rich phase undergoes another phase transition from rotatorto crystalline phase, while the same transition happens at P*z14 for pure COs. This discrepancy is likely due to frustrationsarising from tting two solid phases in the same box.

An important difference observed in the pressure vs.composition phase diagram for the COC mixture (Fig. 6(c)) ascompared to the earlier mixtures is the relatively large satura-tion concentration of COs in the cube-rich phase (reachingabout 30%) even at high pressures. This enhanced mixing maypartially arise due to the COs gaining packing entropy byfollowing orientations compatible with those of neighboringcubes. Furthermore, near-equal volumes of the two phasesimply that the packing entropy cost of allowing a CO-impurity incubes is minimal. As in the previous systems, the low solubilityof cubes in the CO-rich phase is related to the bigger size of thecubes that makes them hard to be accommodated as guests inthe CO-rich phase at high densities. See further analysis inSection 4.

3.4 Cuboctahedra + truncated octahedra (COTO) mixture

The COTO mixture represents a case where the two shapesinvolved are similar in terms of their individual phase behavior.

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Fig. 6 Summary of results for the COC mixture. (a) Equation of state with ODPs for the individual particles shown for reference. Here ODPA is the ODP of COs whileODPB is the ODP of cubes. (b) Plot of P4 and Q6 order parameters. (c) Pressure vs. composition phase diagram with different regions marked with the order of theconstituent phases (xB is the mole fraction of cubes). (d) Simulation snapshots from three representative regimes of the phase diagram (COs are green and cubes aredark red). Bottom: isotropic mixture phase at P* z 6. Middle: isotropic + cubic phases at P* z 10. Top: distorted cubic + cubic phases at P* z 25.

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Furthermore, they are neighbors in shape evolution (during thepolyol process). While truncated octahedra form a BCC tessel-lation, cuboctahedra arrange in a distorted cubic lattice whichis non-space-lling. However, both shapes show a rotatormesophase for a wide range of volume fractions.

While one could have expected these two shapes to be themost miscible of the mixtures considered here, our simulationresults shown in Fig. 7 (and in the relevant scatterplots of Fig. 3)provide evidence to the contrary. This is primarily becausealthough the similarity between shape and individual phasebehavior could promote miscibility, the size ratios adopted here(based on the polyol process) make cuboctahedra signicantlybigger than truncated octahedra. Both the volume ratio (1.6)and the ratio of the radii of their circumspheres (z1.25)translate into a twofold difference in the components' ODPvalues (see Table 1) and a scant miscibility.

MC compression runs, which start from an isotropic liquidstate, fail to phase-separate and instead get kinetically trappedinto a disordered state. It appears that at pressures where themixture would prefer phase-separation, the mixture is alreadytoo dense for a well-mixed system to demix and phase-separate.However, expansion runs started from a two-phase orderedconguration (with individual shapes ordered in their corre-sponding crystal lattices) are observed to be stable in a two-phase state down to very low volume fractions. We thus useexpansion runs in this case to estimate thermodynamic phasebehavior. The fact that the compression and expansion runs

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yield different results at high pressures signals lack of ergodicity(assumed to bemore severe for the compression process). Whileunphysical MC moves like particle-swaps help overcome diffu-sion barriers to compositional equilibration, more elaboratedmoves may be needed to speed up structural equilibration (andthe nucleation of translational order).

Below P* z 30 (which is slightly above the MCP for TOs),the TO-rich phase goes into a rotator phase while the CO-richphase is still ordered in a distorted cubic lattice. Below P* z18, again slightly above the MCP for COs, the CO-rich phasetoo becomes a rotator phase. In a narrow region between P* ¼13.5 and P* ¼ 11.5, we see the TO-rich phase in an isotropicphase while the CO-rich phase remains as a rotator meso-phase. Below P* ¼ 11.5, the two phases mix to form a singleisotropic phase.

As far as inter-species miscibility is concerned, in the smallregion (between P* of 11.5 and 13.5) where some miscibilityoccurs, COs are seen to be more miscible in the TO-rich phasethan the other way around. This is expected since in thatregion the TO-rich phase is largely isotropic which makes theentropic cost of introducing a CO impurity in the TO-richphase minimal. On the other hand, the CO-rich phase ispositionally ordered and loses some packing entropy byhosting TO-impurities. This mixture demonstrates the factthat along with shape and individual phase behavior, therelative size of the particles can have a dominant effect on themixture phase behavior.

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Fig. 7 Summary of results for the COTO mixture. (a) Equation of state showing the ODPs of the individual particles for reference. Here ODPA is the ODP of TOs whileODPB is the ODP of COs. (b) Plot of P4 and Q6 order parameters. (c) Pressure vs. composition phase diagram with different regions marked with the order of theconstituent phases (xB is the mole fraction of COs). (d) Simulation snapshots of three representative regimes of the phase diagram (COs are green and TOs are purple).Bottom: rotator + isotropic mixture phase at P* z 12. Middle: distorted cubic + rotator mixture phase at P* z 20. Top: distorted cubic + BCC phase at P* z 32.

Fig. 8 Qualitative sketches of pressure–composition phase diagrams for a binarymixture of hard spheres of diameter ratio a ¼ 0.85 (a), and for the CS, CTO, COC,and COTO mixtures (b). I ¼ isotropic, SA, SB ¼ rich A and rich-B solids respectively,with A (B) being the smaller (larger) component.

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4 Discussion of general trends andcomparison to other systems

Systems CS, CTO, and COC constitute similar mixtures of alarge component B, cubes, with a smaller component A that hasmoderate asphericity (g < 1.5) and high rotational symmetry.The COTOmixture is slightly different but can also be cast as a A+ B mixture with the TO and CO corresponding to species A andB. Overall thesemixtures seem to exhibit a eutectic type of phasebehavior, akin to that formed by binary A + B mixtures of hardspheres whose size ratio a (¼diameter of A/diameter of B) isdifferent enough (say a z 0.85) so that at high pressures theytend to phase separate into two incompatible FCC lattices.30,31

Fig. 8 shows a qualitative diagram for such a case along withone that encapsulates the behavior observed in the CS, CTO,and COC systems. Note that our simulations were unable toresolve neither the position of the eutectic point (which in allcases seems to be near zero cube composition) nor the verysmall isotropic + solid A phase coexistence region. Despitethe quantitative disparities in the diagram proportions and thedifferences in the lattice symmetry of the ordered phases,the physics underlying the phase diagrams (a) and (b) in Fig. 8is the same: just like in our systems, in the mixture of hardspheres phase separation at high pressure is driven by themaximization of packing entropy that results from forming twodistinct efficiently packing solid phases. At intermediate

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pressures where (only) the pure B system would solidify (as ithas the lower ODP), a B-rich ordered phase must form whichhence coexists with an isotropic phase within a large two-phaseregion. In this context, the fact that in our systems one or twocomponents have at facets does not fundamentally changethe overall picture, but merely leads to ordered structuresthat depart from the ones favored by spheres. Also, the factthat our systems exhibit mesophases (that precede the perfectcrystal at high pressures) does not add anything fundamen-tally new to the character of these diagrams since most of the

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mesophase–crystal phase transitions are continuous andhence such mesophases simply occupy the lower-pressureportion of the A-rich or B-rich solid regions.

Another general trend relevant to the CS, CTO, and COCmixtures (excluding now the COTO mixture) relates to thecorrelation between component relative sizes and inter-phasesolubility (i.e., howmuch of A can be dissolved in a B-rich phaseand vice versa). Here cubes (the common component) can betaken as the reference whose size sets not only a reference unitlength but also a common baseline to directly compare pressurevalues. In such a case, the pure components rank in order ofdecreasing size (�volumes) as cubes, COs, spheres, and TOs,which is also the order of increasing ODPs. This shows that theODPs are more strongly affected by differences in particle size(than in particle shape). To examine how the saturation solu-bilities compare for the same pressure across mixtures, weoverlay the pressure–composition diagram for these 3 mixturesin the same plot (Fig. 9) and consider (for concreteness) twocases: P* ¼ 10 for the isotropic–cubic coexistence region and P*¼ 18 for the rotator–cubic coexistence region (selecting otherpressures would give similar results).

For a xed P* we will denote as the most compressed A-richphase (among the 3 mixtures) the one that has the lowest ODPA(i.e., the degree of compression is relative to ODPA). This isrooted on the fact that at the ODP, the isotropic phase for allsystems considered has a similar packing fraction and so onecan assume that at the ODP all A-rich (or B-rich) systems arecomparably dense. We see in Fig. 9 that at a given pressure: (1)cubes tend to dissolve more into the A-rich phase (whether it isisotropic or rotator) for the mixture that has the highest ODPA,namely, where the A-rich phase is less compressed (at the givenP*), and (2) conversely, the A component dissolves more into thecubic phase for the mixture that has the lowest ODPA, namely,for the A-rich phase that is more compressed (at the given P*).Both of these trends are consistent with the idea that a

Fig. 9 Pressure–composition phase diagrams for the CS, CTO and COC mixturesoverlaid on top of each other. For simplicity, only the main two-phase envelopesare shown, leaving the boundaries connecting to the pure-species ODPs untraced.Two particular P* values, P* ¼ 10 and P* ¼ 18, are marked for reference. xB is themole fraction of cubes.

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component that is in a more compressed phase has a higherchemical potential (and a higher tendency to escape) and wouldthen have a higher relative proclivity to transfer to the otherphase so that chemical potentials can be equalized. For trend(1) cubes in the cubic phase are always equally compressed (fora given P*) but will transfer to and populate more a coexistencephase that is less compressed and hence more hospitable toguests; for trend (2) the A component experiences differentstates of compression in the A-rich phases across mixtures andhence escapes to different extents into the coexistence cubicphase (that is always equally compressed at a given P*). Becauseour systems are purely entropic, a higher (lower) relative degreeof compression or chemical potential translates to essentially alower (higher) entropy. Of course, these trends are onlyapproximate and expected to hold provided that, amongdifferent systems, the ODPs are sufficiently different but one isstill comparing under conditions where similar phases coexist.Note that other pure-system attributes (besides ODPs) could beused to try to correlate the observed miscibility trends; e.g., theparticle volume which, as Table 1 shows, consistently decreasesas ODP increases and hence it is an equally good descriptor.Furthermore, some particular function of ODP or particlevolume may prove to be better at correlating those trends in amore quantitative way. Our limited sample of mixtures onlyallows us to point out qualitative correlations and to conjecturethat relative values of ODPs, landmarks of pure componentphase behavior that also capture particle size disparities, likelyprovide more robust clues of mixture phase behavior than anysingle geometrical feature of a particle shape. Of course, theprecise particle shapes must also play a role in reinforcing oropposing these broad trends. The COTO mixture does not allowa direct comparison with the other mixtures for a xed pressurebut we expect that similar principles should apply on how therelative compression states of the phases affect the saturationsolubility of the guest component.

5 Conclusions and a road map forequimolar phase behavior

Towards the goal of identifying key factors that govern the self-assembly of mixtures of polyhedral nanoparticles, we studiedhere the phase diagrams of four representative binarymixtures ofhard convex polyhedra, using as components particle shapes thatare readily accessible via well-established synthesis methods. Inparticular, we examined how the properties of the individualcomponents affect the interplay between mixing and packingentropies which ultimately determines the types of phasesformed and the extent of inter-particle mixing in such phases.

We nd that, while the pure-component phase behavior isdetermined by the rotational symmetry and asphericity of theparticle shape, the binary-component phase behavior dependson both the pure-component phase behavior and on the relativesize ratio of the components, which in turn determines theirrelative difference in ODPs. For instance, in the COC mixture acombination of a similarity of size ratio and pure-componentcrystal lattices makes COs particularly more miscible in thecube-rich ordered phase.

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Fig. 10 Tentative sketch of the roadmap of phases at the ODPEM (the ODP of anequimolar binary mixture) of A + B hard particles for components with volumeratio 0.5 < r < 0.85. I ¼ isotropic, LC ¼ liquid crystal, R ¼ rotator solid, S ¼ crystalsolid; a subscript denotes the majority component in the phase. The gray regioncorresponds to single compound solids. Numbered circles denote selected statesstudied in the literature (cases 2 through 5 are from this work) as follows: 1 ¼ ref.30, 6 ¼ ref. 17, 7 ¼ ref. 18 and 32, 8 ¼ ref. 33, and 9 ¼ ref. 34.

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Casting our systems as A + B mixtures where B is thecomponent with the larger size and hence smaller ODP, weexpectedly nd that the relative extent of miscibility of the twoshapes (miscibility of A in the B-rich phase vs. that of B in the A-rich phase) depends on the relative ODP values. In particular, ifDODP ¼ ODPA � ODPB with ODPA > ODPB, then at any partic-ular pressure where phase separation occurs, the larger theDODP the lower the solubility of A in the B-rich ordered phaseand the higher the solubility of B in the A-rich ordered phase.

We attempt now to sketch out a rough phase road map thatidenties the phases formed at the ODP of an equimolarmixture of hard particles (henceforth denoted as ODPEM); i.e.,the rst single- or two-phase state involving at least one orderedphase that arises upon compression of an isotropic equimolarmixture of A + B. We only consider mixtures consisting ofshapes that for any particular asphericity g exhibit (as purecomponents) one or more of the following mesophases orordered states: rotator (R), solid crystal (S), and liquid crystal(LC). If a given particle forms multiple ordered states, it isassumed that a LC occurs at a lower packing fraction (andpressure) than an R phase, which in turn would occur at a lowerpacking fraction than an S phase. Very high g values areassumed to be accessible with prolate or oblate shapes only,which should lead to LC behavior and low ODPs. The tentativeroad map shown in Fig. 10 is based on observations from thiswork and those from selected previous studies on binarymixtures of hard particles (marked by numbers in the plot), andis restricted to intermediate particle volume ratios r between0.5 < r < 0.85, which is the range that our mixtures fall into.

The diagram of Fig. 10 is guided by the following observa-tions. If B is the largest component, then it is expected that forlow to moderate gA, ODPA > ODPB and hence at the ODPEM(slightly above ODPB) a phase separated state should ensuecomprising an ordered B-rich phase (R, S, or LC depending onthe gB value) and an isotropic phase; however, for very large gA

the pure A component would be expected to form a LC withODPA < ODPB in which case at the ODPEM an A-rich LC phaseshould coexist with a B-rich isotropic phase. Of course, acrossover behavior could exist between these small-gA andlarge-gA regimes, where two ordered phases coexist at theODPEM. In Fig. 10 we mark only the ordered phases that couldbe formed, if any such phase will form at all. The alternateoutcome would be the formation of some type of a jammed statewithout a well-dened structural order. The shape and extent ofeach region are only qualitative and meant to guide the eye. Asecondary particle shape parameter besides g would be neces-sary to make more discriminative diagrams (rotationalsymmetry would be a good candidate9). Note that we assumedthat as g approaches 1, particles have higher rotationalsymmetry and we ascribed to crystal phases of spheres a rotatorcharacter since any innitesimal departure from g ¼ 1 wouldlead to a rotational degree of freedom.

Compound crystal phases are known to exist for a numberof binary hard-core particles, like the Laves phases forunequal hard spheres35,36 or polyhedra that form tessellatingcompounds.17 However, these may not be the phases that ariseat the ODPEM (i.e., at equimolar composition) or may arise for

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components with r < 0.5 and so they would not be included inFig. 10. Likewise, a single mixed LC phase would only be apossibility for very specic types of component shapes. Thesetwo scenarios (where a single S or LC phase forms at theODPEM) do not seem to correlate strongly with sphericity (otherthan a loose tendency of components to have similar g) and soare only included in Fig. 10 (as a gray region) for completeness.

Altogether, our observations highlight the fact that althoughmany factors such as relative size, asphericity, individual crystallattices, and mesophase formation determine the phasebehavior of a mixture, the trends in mutual miscibility are bestcaptured by the components' asphericities and their relativeODPs. Towards designing novel nanoparticle superstructureswith desired properties, this study hence provides some guidingprinciples about the phase behavior of the binary mixturesderived from the properties of the constituent shapes. While allsystems studied here are relatively asymmetric in terms of size(and ODP) values, we are currently exploring more symmetricbinary systems where our preliminary results have alreadyrevealed signicant differences with some of the trendsobserved here. Also, while the systems studied here involvedconvex particles only, the use of concave particles, especiallywhen paired with complementary-shaped convex partners,would open the door to much more complex phase behaviors.

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Acknowledgements

Work on mixtures of two polyhedral particles was supported bythe U.S. National Science Foundation Grant no. CBET 1033349.Work associated with mixtures containing hard spheres wassupported by the U.S. Department of Energy, Office of BasicEnergy Sciences, Division of Materials Sciences and Engi-neering under award Grant no. ER46517.

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