Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary MixturesIoatzin Ŕıos de Anda,1 Francesco Turci,1 Richard P. Sear,2 and C. Patrick Royall1, 3, 41)H.H. Wills Physics Laboratory, Tyndall Ave., Bristol, BS8 1TL, UK2)Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH,UK3)School of Chemistry, Cantock’s Close, University of Bristol, BS8 1TS, UK4)Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol BS8 1FD,UK

(Dated: 13 September 2017)

We perform particle resolved experimental studies on the heterogeneous crystallisation process of two compo-nent mixtures of hard spheres. The components have a size ratio of 0.39. We compared these with moleculardynamics simulations of homogenous nucleation. We find for both experiments and simulations that thefinal assemblies are interstitial solid solutions, where the large particles form crystalline close-packed lattices,whereas the small particles occupy random interstitial sites. This interstitial solution resembles that foundat equilibrium when the size ratios are 0.3 [Filion et al., Phys. Rev. Lett. 107, 168302 (2011)] and 0.4[Filion, PhD Thesis, Utrecht University (2011)]. However, unlike these previous studies, for our system sim-ulations showed that the small particles are trapped in the octahedral holes of the ordered structure formedby the large particles, leading to long-lived non-equilibrium structures in the time scales studied and not theequilibrium interstitial solutions found earlier. Interestingly, the percentage of small particles in the crystalformed by the large ones rapidly reaches a maximum of ∼14% for most of the packing fractions tested, unlikeprevious predictions where the occupancy of the interstitial sites increases with the system concentration.Finally, no further hopping of the small particles was observed.

I. INTRODUCTION

Solid solutions are some of the toughest and mostversatile crystalline materials1. Steel is a hugely solidsolution2, it is an interstitial solid solution (ISS) in whichthe iron atoms form an ordered crystal lattice, with thesmaller carbon atoms in the interstices. Substitutionalsolid solutions (SSS) are also important, there the sec-ond component is not in the interstices between atoms ofthe first component, but substitutes for atoms of the firstcomponent, on the lattice. Very recent developments inelectron microscopy have for the first time allowed us toanalyse the disorder of an SSS of iron in platinum3. Also,High Entropy Alloys (HEAs), which are effectively a mul-ticomponent SSS, are currently very actively studied asthey are among the toughest materials known4–8.

Hard spheres, as epitomised by colloids, are widelyused as models to study the self-assembly and phasebehaviour processes of atoms and molecules. Sincethe structural evolution of colloidal suspensions can befollowed in real space by means of confocal imaging,their study has allowed us to monitor the crystallisa-tion process of different systems9. Additionally, theequilibrium structures of such processes have also beensuccessfully characterised in simulations, which is cru-cial for fundamental understanding and for many ap-plications, including material science, metallurgy, andbiotechnology10,11. The most popular example of thesesystems is monodisperse hard spheres, where the par-ticles interact only by hard-core repulsions and whosephase behaviour has been widely studied, through bothsimulations and experiments6. These studies have iden-tified the face-centered cubic (fcc) -often in a randommixture with hexagonal close packing (hcp)-, as the solid

stable structure5,12.

Crystallisation of multicomponent colloidal mixtureshas acquired more interest, as these present a richerand more complex phase behaviour than their monocom-ponent counterpart5,12–17. Amongst these systems, wefind binary mixtures of hard spheres, which are com-posed of two populations of spheres of differing sizes.Similar to one-component hard spheres, they are alsothe simplest multicomponent colloidal system and thus,form a reference model to study phase transitions ofmixtures5,12,14,15. Crystals of these systems were ob-served by Sanders and Murray back in 1978, whilstanalysing the microscopic structure of gem opals, whichare composed of silica colloids and had an structure ana-logue to AlB2 and NaZn13

18. So far, both experimen-tal and simulation studies have identified that the phasediversity found in binary mixtures depends on the sizeratio of the components, the number ratio and total con-centration of the particles5,6,19,20. These studies haveproduced structures resembling the ones found in na-ture for different salts like NaCl, NaZn13, AlB2

5,14–16,19,the Laves structures MgCu2, MgNi2, MgZn2

5,6,18, andmost recently, stable ISS5,6. This large diversity of equi-librium structures highlights their potential for applica-tions in photonics, optics, semiconductors and structuredesign5,14,15,19. In addition, due to their simplicity, bi-nary hard sphere mixtures are ideal models to study thekinetics of crystallisation in salts, metal alloys, metallicglasses and any other crystallising system where there ismore than one species, and so there is a compositionalvariable4–8.

However, only a few experimental studies have focusedon the crystallisation process of binary hard spheres,which can be due to the difficulty of obtaining close-

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 2

packed ordered structures. This is related to slowkinetics, caused by a competition of the growth ofcrystal nuclei and compositional fluctuations. In ad-dition, the differences in sedimentation rates betweenthe particles also play an important role in preventingcrystallisation15,21–24. Similar to one-component sys-tems, the particles in the crystalline structure have ahigher translational entropy than in the metastable fluid,which compensates for the decrease of entropy as the sys-tem becomes more ordered12,14–16,24. In addition, it hasbeen proposed that the theoretically predicted structureswill only be thermodynamically stable if their maximumclose packing fraction exceeds the correspondent 0.7405for fcc or hcp lattices of one-component systems12,14,15,24.

Interestingly, these studies have also shown some dis-crepancies between the experimental observations andthe predicted assemblies for particular size ratios andcompositions, especially at concentrations near the glasstransition5,12,24. Furthermore, the kinetic contributionsto crystallisation, along with microscopic mechanismsthat yield the ordering of mixtures are not yet fullyunderstood24, accentuating the complexity of these sys-tems over one-component ones, and limiting their promis-ing applications. Based on the interest in understandingsuch crystallisation processes, our goal is to study ex-perimentally the heterogeneous crystallisation of a mix-ture of colloidal hard spheres at the particle-resolved leveland compare our results with previous predictions. Wealso use simulations to study the evolution of the crys-tals, in particular the dynamics of the small particles,and propose a mechanism for the crystal formation. Weobserved that only interstitial solid solutions (ISS) areformed regardless of the different packing fractions testedin both experiments and simulations, as predicted6. Insuch structures, the crystalline phase –a mixture of fccand hcp lattices– is formed only by the large particles,whereas the small particles are positioned in a fraction ofthe octahedral holes of the ordered phase. We comparedthe composition of our ISS with the equilibrium compo-sition previously predicted for a related system6, and weobserved interesting discrepancies at high packing frac-tions. By studying the dynamics of the system we founda trapping effect of the small particles which could bepreventing further crystallisation and equilibrium fillingof the octahedral holes.

This paper is organised as follows. In section II wedescribe the experimental and simulations details, alongwith the techniques used for the analysis of the orderedphases. In section III we first present the results forcrystallisation for our experiments and their structuralanalysis. We compare such results with previous work.We continue with the structural analysis of the orderedphases found on the simulations and the study of theevolution and quality of the crystals. We then focus onthe dynamics of the species. We finalise by proposinga crystallisation mechanism for our system following theresults obtained through simulations. Finally, in sectionIV we present our conclusions.

II. METHODS

A. Experiments

The binary system used in the experiments con-sisted on sterically stabilised poly(methylmethacrylate)(PMMA) particles fluorescently labeled with rhodamineand coumarin dyes to enable separate fluorescent imag-ing. The sizes of the big particles have a diameter of1700 nm with a polydispersity of 7%, whereas the smallparticles are 669 nm in diameter and 7.8% polydisper-sity, obtained by scanning electron microscopy. In spiteof their large polydispersity, both particles are able tocrystallise on their own. These dimensions yield a sizeratio (γ = σS/σL) of 0.39. The dried particles were sus-pended separately in a solvent mixture of 27% w/w cis-decahydrophnaphtalene (cis-decalin) and cyclohexyl bro-mide (CHB) that matches the density and refractive in-dex of the particles. Additionally, tetrabutylammoniumbromide (TBAB) salt was added in order to screen thecharges, conditions that allow the particles to behave ina very close manner to hard spheres19,25,26.The suspen-sions were left to equilibrate for several hours, after whichthey were mixed together at a fixed number particle ratio,NL/NS=1, and at several total volume fractions, φtot,defined as φtot = φL + φS, where φL and φL refer tothe partial packing fractions of each species. These werecalculated based on the amount of particles and solventweighed, following φL/S = mL/S/(mL + mS + msolvent).After shaking, the samples were confined to squared glasscapillaries of 0.50 x 0.50 mm (Vitrocom) and sealed ateach end with epoxy. The samples were studied by meansof confocal laser scanning microscopy, CLSM (Leica SP5fitted with a resonant scanner) using two different chan-nels at 543 nm and 488 nm and NA 63x oil immersion ob-jective. For particle tracking, 3D data sets were recordedby taking a full scan of the capillary in the z axis, makingsure the pixel size was equal in all axes (0.125 µm/pixel).We used particle resolved studies to determine the crys-talline structure9. The crystallisation time was calcu-lated in Brownian time units for the large particles, i.e.,the time it takes for the large spheres to diffuse a particleradius, given by τB = (σL/2)

2/6D = 0.963 s, where D isthe diffusion constant. The Brownian time τB thus setsour unit of time. Observations were carried out for overa month, which corresponds to 2.6 × 106 τB , and weremade all along the length and height of the capillaries,which were left standing in a vertical position.

B. Simulations

Hard-sphere simulations were carried out using theopen-source event-driven molecular dynamics DynamOpackage27,28 in isothermal-isochoric (NVT) conditions fora binary mixture. The total number of particles is Ntot=10 968. Particles of both species are of equal mass,m = 1. The size and number ratio, as well as the fi-

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 3

0.45 0.50 0.55 0.60 0.650.02

0.03

0.04

0.05

0.06 Experiment ISS Simulation ISS Simulation fluid

L+LS

LS+F

L+LS+F

L+F

φ small

φlarge

F

b c d

e f g

f e d c b g

L

S

a

0.45 0.50 0.55 0.60 0.650.02

0.03

0.04

F

ISS+F

Experiment ISS Simulation ISS Simulation fluid

φ S

φL

a

b c

d e f

g

FIG. 1. (a) The state points we study (symbols) at a size ratio γ = 0.39, and number ratio of one, and Filion, et al. phase diagram ofbinary hard spheres with a size ratio γ = 0.46, in the φL-φS plane. Diamonds: ISS from heterogenous nucleation experiments. Purplecircles: simulations that did not crystallise. Orange triangles: ISS from simulations. (b-f) are snapshots of crystalline structures foundon experiments for different φtot, the letters b to f in (a) indicates the position on the phase-diagram. The red and green colours are thelarge and small particles, respectively, and the scale bar = 10 µm. (g) is an ISS found for simulations with φtot=0.586, where crystallinelarge and small particles are coloured purple and yellow, respectively, whereas fluid large and small particles appear in light pink and lightyellow, respectively.

nal total particle densities were chosen to match the onesfrom the experiments, thus γ= 0.39 and φtot from 0.52to 0.64, respectively. A fluid system at φtot = 0.282 wasused as the initial configuration, which was then linearlycompressed following Stillinger and Lubachevsky29 to theappropriate packing fractions until an equilibrated struc-ture was found. Experiments of colloidal systems haveshown that the particles stick irregularly on the walls ofthe cell they are contained, thus producing an irregularsurface over which the crystallisation takes place30. Dueto limitations in reproducing said phenomenon and thusthe final surface, we elected to conduct our simulationswith periodic boundary conditions. Future work couldfocus on simulating such environment and its influenceover the kinetics of the system presented herein.

Finally, simulation times were scaled to the experi-mental data. We inferred our τB in the simulationsthrough the relationship τα= 2.597 τB , obtained pre-viously by comparing event-driven MD simulations andcolloidal experiments of a one-component hard spheresystem at φ=0.38, where the structural relaxation timeτα of the simulations was found to equal 0.404 sim-ulation time units31. Since this mapping pertains toa one-component system, here we confirm that the re-laxation of the large particles is similar in the binarysystem of interest herein. Therefore, we calculated ταof the large particles from the trajectories at the samepacking fraction (φtot=0.38). This we did by comput-ing the intermediate scattering function (ISF), F (k, t) =

N−1〈∑N

j=1 exp[−ik · (xj(t)−xj(0)〉. To characterise the

mobility in the length scale of a particle diameter, we

evaluated F (k, t) at the wavenumber k = 2π/σL31,32.

We obtained a value of τα=0.526 simulation time units,which is comparable to the one obtained by the men-tioned work, thus confirming the validity of our approach,and enabling us to use the aforementioned relationshipto infer the τB for our simulations.

C. Location of particles in experiment

In order to determine the local ordering of the crys-tals formed in our experiments, we analysed separatelythe structures formed by each component through parti-cle tracking. To locate the particles in the experiments,3D data sets were directly analysed using the algorithmdescribed in Leocmach and Tanaka33.

D. Identification of local crystal structure

Analysis of the crystalline structure for both experi-ments and simulations, was done following Lechner andDellago34 by obtaining the bond orientational order pa-rameters (BOO), based on complex spherical harmon-ics. This analysis gives information of the degree andtype of ordering, thus enabling us to differentiate be-tween distinct crystalline phases. We focused on thelocally averaged order parameters q̄4 and q̄6 for squareand hexagonal orders, respectively. These parameterstake into account the effect of the second nearest neigh-bours, which distinguish more clearly amongst different

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 4

arrangements34. The methodology followed is detailedelsewhere34–36. Briefly for each particle i and regard-less of its small or large nature, we identify the nearestneighbours Nb(i) using a cutoff corresponding to the con-tact distance between the closest large particles (obtainedthrough the pair distribution function, g(r)). For the ex-periments said distance corresponded to 1.38σL, whereasfor the simulations this number was 1.25σL. Finally forthe perfect (by construction) NaCl-type lattice the cut-off was 1.1σL. Using said list of neighbours, the localorder parameters, or Steinhardt order parameters can beobtained:

qlm(i) =1

Nb(i)

Nb(i)∑j=1

Ylm(rij) , (1)

and summing over all the harmonics:

ql(i) =

√√√√ 4π2l + 1

l∑m=−l

|qlm(i)|2 (2)

where Ylm(rij) correspond to the spherical harmonics, land m are integer parameters, the latter running fromm = −l to m = +l, and rij corresponds to the vectorfrom particle i to particle j. However, these parametersonly contain information about the first shell surround-ing particle i. In order to obtain information about thesecond shell, and thus, the locally averaged order param-eters, we need to sum over the list of Ñb(i) of particle iand the particle i itself, giving:

q̄lm(i) =1

Ñb(i)

Ñb(i)∑k=0

qlm(k), (3)

and again summing over all the harmonics we get:

q̄l(i) =

√√√√ 4π2l + 1

l∑m=−l

|q̄lm(i)|2 (4)

E. Determination of the Lengthscale of CrystallineOrdering

Following the analysis described in37,38 for identifyingthe size of domains, we determined in both experimentsand simulations the typical crystalline domain size as

NQ = N−1〈 ~Q∗ · ~Q〉, where N is the total number of parti-

cles. Here ~Q =∑Np=1 ~q6(p) is the sum over all particles of

the normalised complex vector ~q6(p), whose componentsare the spherical harmonics with l=6. ~q6(p) normalisa-tion is done by setting the average of ~q6(p) · ~q6(p)∗ = 1.The values taken by NQ will depend on the amount ofparticles oriented in the same fashion. The value of NQis an estimate of the average number of particles in acoherently ordered crystallite.

0.52 0.54 0.56 0.58 0.60 0.62 0.64101

102

103

104

105

106

t/τ B

φtotal

Experiments

FIG. 2. Experimental results of the heterogeneous crystallisationtime of the binary system in terms of Brownian motion, τB . Theline is a guide to the eye.

III. RESULTS AND DISCUSSION

We present our analysis in three sections: we first iden-tify the ordering and quality of the structures found inboth our experiments and simulations (Sec. III A). Wethen focus on the dynamics of our two species in boththe fluid and the crystalline regions in our simulations(Sec. III B). Finally, we compare the composition of ourISS with an equilibrium system with a comparable sizeratio and propose a mechanism for the formation of thestructures found (Sec. III C).

A. Analysis of the crystalline structures: identification ofInterstitial Solid Solutions (ISS)

Filion and collaborators used computer simulations tostudy a mixture of binary hard spheres with a size ratio6

only slightly higher (0.4) than ours (0.39). They calcu-lated the equilibrium phase diagram, which we reproducein Fig. 1 (a). There, the symbols represent our experi-mental state points for the experiments on heterogeneousnucleation (diamonds) and simulations (circles and trian-gles). Filion predicted that at equilibrium, there is co-existence between interstitial solid solutions (ISS), anda fluid phase (F)6. The ISS consists of a regular close-packed lattice, with the small particles filling some butnot all the octahedral holes; the fraction of these holesvaries continuously with composition.

Indeed we do observe crystalline structures built up bythe large particles with interstitial small particles posi-tioned in a random fashion (Snapshots Fig. 1), regard-less of the φtot and in coexistence with a fluid phase, forboth the experiments and the simulations. Surprisingly,we did not find ordered structures for the simulations be-low φtot=0.563 or above φtot=0.610, where the mixturesdid not crystallise (Fig. 1 purple circles).

In our experiments, nucleation of the crystals is het-erogeneous, the crystals grew parallel to the walls of the

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 5

0.6

0.5

0.4

0.3

0.2

0.1

102 101 100

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

HCP FCC

Fluid

BCC

φtot = 0.58 φtot = 0.55 φtot = 0.60 φtot = 0.64

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3

φtot = 0.569 φtot = 0.557 φtot = 0.586 φtot = 0.598

102 101 100

q6

q6

q4

q4

a

b

Experiments

Simulations

FIG. 3. Local bond order parameter diagrams for the crystalline phase formed found in (a) experiments and (b) simulations, showingfcc, hcp and fluid coexistence for different total volume fractions tested for both studies, all at final times. φtot=0.557 (simulations) isincluded to show the presence of only a liquid phase. The regions for perfect bcc, fcc, hcp lattices are shown in the top-left plot. Bothparameters were calculated using only the large particles.

capillaries and the fluid phase was present far away fromthe walls. The presence of a flat wall clearly facilitatesthe nucleation process and thus the mixture is able tocrystallise in a φtot range larger than the simulations,where the nucleation starts in the bulk.

The ordered structures found are stable ISS39, on thetimescales we consider. In these, the crystals grow fromthe fluid phase and form ordered structures composed ofthe larger particles forming a close packed crystal, andthe smaller particles in the interstices. The compositionof these crystals is much poorer in the small particlesthan it is in the original fluid, where the number ratio isone.5,40

The heterogeneous crystallisation of our mixtures wasstudied as a function of the total volume fractions φtot= φL + φS and we found that increasing this parameterincreases the crystallisation rate, as shown in Fig. 2.This observation is in agreement with one componentsystems10.

1. Local structure of the large particles

For all of our samples, the equilibrium structure ex-pected is an interstitial solid solution in coexistence witha fluid6. In such assemblies, the large particles are ex-pected to form a mixture of hcp and fcc lattices, knownas random hexagonal close packing (rhcp), since the freeenergy difference between these two crystalline phases isvery small5,6. The results of the analysis of the crystalsformed only by the large particles for the experimentsand simulations are summarised in Fig. 3 (a and b, re-spectively), where we can observe that in both cases the

crystalline phases do consist of a mixture of fcc and hcplattices in coexistence with a fluid phase for all the totalvolume fractions tested. This random stacking has beenidentified before in experiments of heterogeneous nucle-ation of one-component systems10.

2. Local environment of the small particles: an analysison the crystal quality and the vacancies in simulations

In the ISS the small particles are situated in the octa-hedral holes of the fcc and hcp lattices formed by largeparticles. Filion et al. showed, for their smaller size ratioof 0.3, that the small particles are able to hop at lowerpressures– where the system is far from close-packing– fillall the octahedral holes and yield an ISS with the NaClstructure5. On the other hand, for the larger size ratio of0.4, Filion and coworkers also found that the ISS are thestable assembly, however, for this larger size ratio theydo not report the formation of a perfect crystalline NaClstructure, nor they show the dynamics of the small par-ticles like for γ=0.36. In our experiments a similar sizeratio of 0.39, we also observed only ISS, and we could notobserve hopping of the small particles in direct imaging.Due to the limitations on tracking the motion of the smallparticles, we decided to conduct hard sphere moleculardynamics simulations in order to quantify both the smallsphere ordering and dynamics.

Using simulation data, we calculated the BOO q6 andq̄6 values for the small component, see Fig. 4(a) and (b).In Fig. 4(a), we plot q6 (left) and q̄6 (right) in the liquidstate, in a state that is partially crystallised with remain-ing liquid, and in a perfect NaCl-type crystal. It is clear

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

2

4

6

8

10

12

14

16

18

g(r)

r/σ

S - S L - L S - L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

6

7

8

g(r)

r/σ

S - S L - L S - L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

6

7

8

9

10

g(r)

r/σ

S - S L - L S - L

4 5 6 7 8 9 10 11

0.30 0.25 0.20 0.15 0.10 0.05

Simulations a c

b

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

103 102 101

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Liquid ISS/Liquid

NaCl

Liquid ISS/Liquid

NaCl

q6 q6

Freq

uenc

y

0.30 0.25 0.20 0.15 0.10 0.05

q6

φtot = 0.569

4 5 6 7 8 9 10 11

φtot = 0.586 φtot = 0.598

q6

Liquid

ISS/Liquid

NaCl

NLn

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

6

7

8

9

10

g(r)

r/σ

Global S - S L - L S - L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

6

7

8

9

10

g(r)

r/σ

Global S - S L - L S - L

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

1

2

3

4

5

6

7

8

9

10

g(r)r/σ

Global S - S L - L S - L

18

8

φtot = 0.557

r/σL

FIG. 4. (a) Histograms of the bond orientational parameters q6 (left) and q̄6 (right) values of the small particles at φtot=0.586 beforecrystallisation (Liquid) and after crystallisation (ISS/Liquid), and an equilibrated NaCl-type crystal at φtot=0.64, showing the ability ofq̄6 to distinguish the local environment of the small particles. (b) Heat maps of the q̄6 values of the small particles and their correspondingnumber of neighbouring large particles (NLn) for different total volume fractions that crystallised in simulations, calculated followingreference 33. The plots show two populations: one below and one above q̄6 = 0.15. Note that φtot=0.557 does not crystallise. (c) Radialdistribution functions of the liquid (top) and ISS/Liquid (middle) both at φtot=0.586, and NaCl-type crystal (bottom) at φtot=0.64. Eachshows the small-small (S-S), large-large (L-L) and small-large (S-L) interactions. The dashed lines indicate the minimum correspondingto the small and large contact cutoff used for the BOO analysis.

that q6 does not differentiate between the different con-figurations, but q̄6 does distinguish between the liquidand the ISS. So we use the Lechner-Dellago averaged q̄6in order to identify the fluid or crystalline nature of theparticles. Values of q̄6 above 0.15 are very rare in the liq-uid, while in a well-ordered NaCl structure q̄6 is almostalways above 0.15.

In Fig. 4(b), we show heat maps of the probabilitydistribution p(NLn, q̄6) for 4 states, one liquid (top-left)and three ISS/liquid mixtures. The samples with liquidand ISS show two populations. The dominant one, withq̄6 < 0.15 is the liquid, but there is a smaller popula-tion with q̄6 > 0.15, which is consistent with values ofq̄6 in the crystal. The association between this secondpopulation and the crystal is further supported by thefact that small particles here have around 6 to 8 large-particle neighbours. In the NaCl structure, each smallparticle has 6 large-particle nearest neighours, and ourcutoff distance for neighbours (dotted line in Fig. 4(c))is above the small-large nearest neighbour distance, so

we would find at least 6 neighbours — precisely what weobserved.

3. Composition of the crystalline domains

We followed the crystal growth by estimating NXL andthe small particles located at the octahedral holes (NXS),according to their q̄6 values. These results are shown inFig. 5 (a) and (b), respectively. Here, we observe thatboth quantities reach a plateau, indicating no furthercrystal growth and suggesting no hopping of the smallparticles within the interstitial sites of the fcc and hcplattices, as discussed previously5. Next, we calculatedthe amount of NXS relative to NXL forming the orderedstructure. We found two notable phenomena. Firstly,the octahedral holes are filled rapidly as the nucleationtakes place (Fig. 5 (c)) with a maximum filling of theoctahedral sites of around 14%, above φtot=0.586 (seeFig. 8 (b)). Secondly, we observed no further changes in

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 7

a b c

0 20000 40000 60000 80000 100000 1200000

50

100

150

200

250

300

0.563 0.569 0.574 0.580 0.586 0.592 0.598

NXS

t/τB1000 10000 100000

0.00

0.05

0.10

0.15

0.20

0.563 0.569 0.574 0.580 0.586 0.592 0.598

NXS/N

XL

log t/τB

0.56 0.57 0.58 0.59 0.600.10

0.12

0.14

0.16

NXS/N

XL

φtotal0 20000 40000 60000 80000 100000 1200000

400

800

1200

1600

2000

0.563 0.569 0.574 0.580 0.586 0.592 0.598

NXL

t/τB

Simulations

0 20000 40000 60000 80000 100000 1200000

50

100

150

200

250

300

0.563 0.569 0.574 0.580 0.586 0.592 0.598

NXS

t/τB

FIG. 5. The numbers of large (a) and small (b) crystalline particles as a function of time in simulations. (c) evolution of the proportion ofsmall crystalline particles relative to the large crystalline ones in simulations according to their φtot.The simulation time has been rescaledto Brownian time (see Methods for details).

the proportion of crystalline particles, suggesting a trap-ping effect. Since the number of small crystalline par-ticles also reached a constant value, we infer that theybecome trapped (and immobilised) in the the crystallinestructure once it is formed. These phenomena will beinvestigated in detail in the following sections.

4. Crystalline domains in experiment and simulation

Notably, see Figs. 1 and 6, the range of packing frac-tions that crystallised on the simulation timescale wassmaller than that of the experiments. But in both ex-periments and simulations, there are ranges of volumefraction where maximal crystallisation occurs. In exper-iments we observe regions in the system where there islargely complete crystallisation of the large particles: atφtot = 0.55, 0.58 and 0.60 —see Fig. 6 (a), the fractionof large particles identified as crystalline is around 65%or above.

In our simulations, the total amount of crystallinestructure is much smaller, reaching the highest amountaround 18% at φtot=0.580, after which it started to de-crease, as shown in Fig. 6 (b). The need for a higherφtot to observe crystallisation on the simulation timescale(φtot=0.563 in comparison with φ=0.53 for one compo-nent systems41) might be due to a higher free energybarrier to generate nuclei necessary for crystallisation.The drop of the crystalline fraction of our system atφtot = 0.598 is similar to the one where one-componentsystems exhibit slow dynamics φ ' 0.588, and is relatedto the particles moving more slowly as their concentra-tion increases, requiring a long time to rearrange in or-dered structures. A similar trapping phenomenon couldbe happening here, where the particles do not move fastenough to rearrange and form crystals.

To determine if these crystalline large particles are inone large crystalline domain, or if the system is polycrys-talline, we follow Klotsa and Jack37,38. These authorsdefined a parameter NQ to describe the size of coher-ent crystallites, i.e., the typical number of particles in

a coherently ordered crystal. We applied this analysisto both our experiments and simulations to determinewhich conditions yielded the crystals with the highest

0 500 1000 1500 2000 25000

200

400

600

800

1000

1200

1400

1600

1800

0.563 0.569 0.574 0.580 0.586 0.592 0.598N

Q

NXL

Simulations

0 500 1000 1500 2000 25000

200

400

600

800

1000

1200

1400

1600

1800

0.52 0.55 0.58 0.60 0.64

NQ

NXL

Experiments

a

b

FIG. 6. The characteristic size of the crystal domain(s), NQ, asa function of the total number of large crystalline particles, NXL.For the experiments, we show the final values (as symbols, seekey), while for computer simulations, we show the evolution withtime during crystallisation (as curves). The total number of largeparticles in the experiments is 2534, 2645, 2584, 2524 and 2244for φtot=0.64, 0.60, 0.58, 0.55 and 0.52, respectively. On the otherhand, for simulations the total number is 5,484 of large and 5,484 ofsmall particles. NQ shows that the quality of the crystals obtainedin the experiments is higher than the simulations.

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 8

quality. In the case of the experiments, we analysed thefinal crystals formed by the large particles, whereas forthe simulations, we studied the evolution of NQ throughthe crystallisation of the different samples.

The results are presented in Fig. 6, where we comparethis parameter with the number of large crystalline parti-cles for both cases, NXL. It is evident from these resultsthat the size of the crystalline domains obtained fromheterogeneous nucleation in the experiments is higherthan the homogenous nucleation present in the simula-tions, as NQ presents significantly larger values in theformer for similar NXL. Furthermore, the NQ values forthe experiments are close to the number of crystallineparticles. Reasons for this include the presence of theflat walls of the capillaries, larger system sizes or theneed for longer waiting times. Indeed flat walls can serveas a template able to enhance the nucleation and layeringof the particles and thus improve the orientation of thecrystal10,42. On the other hand, estimations of the crys-tallisation time for our experiments for homogenous nu-cleation were found to be two orders of magnitude largerthan the corresponding crystallisation time on our simu-lations, suggesting that longer waiting times are requiredfor experimental homogenous crystallisation. This mightbe related to the higher polydispersity in the experimen-tal system. Moreover, said time surpasses our limit forexperimental timescale observations of a month (2.6×106τB).

Finally, the system size in the experiments is of order105 times bigger than the simulations. Hence it seemslikely that our experiments can crystallise over a widerrange and form crystals with a greater quality due to acombination of the reasons mentioned above.

In the case of simulations, the numerical values ofNQ are significantly smaller than NXL, suggesting thepresence of several clusters in all the samples. Also, anon-monotonic behaviour of this value is noted, sincethe highest values of the former do not correspond tothe largest ones for the latter. This is evident forφtot=0.598, whose maximum NXL surpasses the core-spondent of φtot=0.563 and 0.569, with higher amountsof crystalline particles (Fig. 6 (b) orange, light pink, andpink lines, respectively). Interestingly, the dependencebetween these two parameters is not linear, with NQ in-creasing faster than the crystal growth. A possible reasonfor this could be fast coalescence between the formingcrystallites. Indeed, φtot=0.574 (Fig. 6 (b) dark pinkline) shows a faster increase of NQ around NXL∼ 1000,which could be a consequence of two clusters merging toform a larger one.

B. The dynamics of both large and small particles arearrested in the crystals

In order to show that our small particles were in-deed localised, we computed both the mean square dis-placement (MSD,

〈∆r2L(t)

〉=〈(rL(t) − rL(0))2

〉) and

10-1 100 1010.0

0.2

0.4

0.6

0.8

1.0

Large crystalline particles Small crystalline particles Large fluid particles Small fluid particles

F(k,t)

log t/τα

10-1 100 1010.0

0.2

0.4

0.6

0.8

1.0

Large crystalline particles Small crystalline particles Large fluid particles Small fluid particles

F(k,t)

log t/τα

Simulations

0 100 200 300 400 500 60010-2

10-1

100

101

/σ2 L

t/τB

FIG. 7. Intermediate scattering functions and mean square dis-placements (insert) from simulations, at φtot=0.586. The largeand small particles are shown in pink and purple, with continuouslines for particles identified as crystal, and dotted lines for particlesidentified as fluid. The simulations start at the end of a crystalli-sation simulation. Particles at the liquid-crystal boundary regionwere discarded in order to characterise only the crystalline and fluidparticles.

the intermediate scattering function (ISF), as describedpreviously32 for the simulations with φtot=0.586 for geo-metrically selected large and small particles in both liquidand crystalline regions. This was done at the final pointsof the simulation, where the ratio NXS/NXL remainedconstant. These results are shown in Fig. 7. We ob-serve that both the large and small particles in the liquidregion present movement (Fig. 7, dotted pink and pur-ple lines, respectively), i.e. their MSD (insert) and ISFpresent a behaviour typical for a confined random walkand fluids, respectively. In contrast, both species locatedin crystalline regions do not show any movement, iden-tified as a flat line close to zero in the MSD and as anon-decaying line on the ISF (Fig. 7, continuous lines).We can conclude therefore that indeed, once the crystalsare formed, the small particles are immobile within thecrystal formed by the large particles; unable to hop andfill all the available octahedral holes, and thus our ISSremain as long-lived out-of-equilibrium structures.

C. Comparison of our results at a size ratio γ=0.39 tothose of Filion et al. at size ratios of 0.3 and 0.4

We found that the small particles occupy the octahe-dral sites of the crystalline lattices in an incomplete fash-ion, forming a dilute ISS which in our computer simula-tions consistently has a composition close to seven largeparticles for every one small particle, much less than theone-to-one ratio in the original fluid phase.

An ISS has also been identified before in binary mix-tures with a smaller size ratio of 0.3 compared to 0.39here5. Filion and coauthors found that in order for the

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 9

0.56 0.57 0.58 0.59 0.600.10

0.12

0.14

0.16

0.18

0.20

0.22

Out-of-equilibrium Predicted equilibrium 5% stricter q6 treshold

NXS/N

XL

φtotal0.56 0.57 0.58 0.59 0.60

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Out-of-equilibrium Predicted equilibrium 5% stricter q6 treshold

NXS/N

XL

φtotal

0.0 0.2 0.4 0.6 0.8 1.0

30

40

50

60

p

XS

0.563 0.569 0.574 0.580 0.586 0.592 0.598 0.610 0.616 0.622

ISS

ISS+Fluid

Fluid

a b

FIG. 8. (a) Filion’s6 phase diagram of a binary hard-sphere systemwith γ = 0.4, showing the reduced coexistence pressure p = βPσ3,as a function of composition XS = NS/(NS+NL. We have added tothis filled symbols to represent our state points where we obtainedan ISS and the empty symbols for the conditions were, on our timescales, we did not observe crystallisation. The tie line and arroware drawn to illustrate the interpolation process to obtain the smallparticle composition in the equilibrium ISS. (b) The compositionas a function of our system’s average volume fraction, φtot. Thedashed line corresponds to the equilibrium composition6, and thepoints are our simulation results. The error bars are the standarddeviation of 8 runs, whereas the dotted line is the error obtainedwhen using a 5% more strict q̄6 threshold.

small particles to hop from one octahedral hole to theother, they needed to go through the adjacent tetrahe-dral hole (∼0.225σL across). Although slow, they foundthat their smaller particles could hop, whereas (see Fig.7) our larger small particles cannot.

Filion also studied the size ratio 0.4, closer to the oneused here. For that size ratio, the ISS equilibrium re-gion in the phase diagram is larger than for 0.3. In orderto confirm that our ISS are out-of-equilibrium, we com-pared our simulation values of XS, with the equilibriumvalues determined by Filion6. As shown in Fig. 8(a),we determined the final pressure in our simulations, andthen using tie lines (which are horizontal here as the twophases have the same pressure) read off the compositionsof both phases. The comparison of these results and theones corresponding to our ISS are shown in Fig. 8(b),

where the dashed curve is the equilibrium XS, and thedotted curve and points are our results. In all cases, theamount of small particles within the crystalline structureof large ones is smaller than the one predicted for an ISSin equilibrium. This difference is large for systems athigher volume fractions.

Thus we conclude that our ISS are out of equilibrium,and presumably are out of equilibrium when they nu-cleate and grow. As the large particles crystallise intoan ordered structure, some of the small particles becometrapped in the octahedral holes, forming the ISS. But un-der the conditions we study, the integration of the smallparticles into the growing crystal appears to be ineffi-cient, resulting in the low number of small particles inthe crystal. This then causes the composition of the re-maining fluid to become increasingly rich in the smallparticles, as the crystal grows.

As we can see in Fig. 8(a), increasing XS will ulti-mately move the fluid out of the coexistence region andinto the one-phase fluid region of the phase diagram. So,we believe this change in composition contributes to thegrowth of the crystal slowing and stopping.

IV. CONCLUSIONS

Interstitial solid solutions (ISS) were identified in bothparticle resolved experiments and simulations of a binarymixture with a size ratio of 0.39. Through particle re-solved studies carried out on the experimental results,we were able to identify that the crystalline structurewas made up by the large particles forming a mixtureof fcc and hcp lattices, with low crystallinity, and wherethe small particles are localised randomly within the oc-tahedral holes. Simulations showed the same crystallinestructure and allowed us to follow the crystallisation pro-cess and quantify the amount of filling by the small par-ticles and, significantly, the dynamics of the small par-ticles in the ISS. With this information, we were ableto propose a crystallisation mechanism where the largeparticles form ordered structures independent containingsmall particles, which become rapidly trapped within thegrowing crystal to a maximum of ∼ 14%, a smaller fill-ing from the predicted in equilibrium conditions, partic-ularly for packing fractions above 0.586. This exclusionof the small particles depletes the fluid around the grow-ing crystal and we believe that this compositional changeprevents further crystalline growth. Finally, we were ableto show that the small particles are not able to penetratethe formed crystalline structure nor to move within theavailable octahedral holes, thus producing long-lived out-of-equilibrium ISS.

V. ACKNOWLEDGEMENTS

The authors are grateful to John Russo for simulationdiscussions. IRdA would like to thank Conacyt for finan-

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 10

cial support. CPR acknowledges the Royal Society forfunding and Kyoto University SPIRITS fund. FT andCPR acknowledge the European Research Council (ERCconsolidator grant NANOPRS, project number 617266).This work was carried out using the computational facil-ities of the Advanced Computing Research Centre, Uni-versity of Bristol.

VI. REFERENCES

1R. W. Cahn, The Coming of Materials Science, vol. 5 of Perga-mon Materials Series. Pergamon, Oxford, 2009.

2Verhoeven John D., Steel metallurgy for the non-metallurgist.Materials Park, Ohio : ASM International, 2007.

3Yang Yongsoo, Chen Chien-Chun, Scott M. C., Ophus Colin, XuRui, Pryor Alan, Wu Li, Sun Fan, Theis Wolfgang, Zhou Jihan,Eisenbach Markus, Kent Paul R. C., Sabirianov Renat F., ZengHao, Ercius Peter, and Miao Jianwei, “Deciphering chemical or-der/disorder and material properties at the single-atom level,”Nature, vol. 542, no. 7639, pp. 75–79, 2017.

4B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P.George, and R. O. Ritchie, “A fracture-resistant high-entropyalloy for cryogenic applications,” Science, vol. 345, no. 6201,pp. 1153–1158, 2014.

5L. Filion, M. Hermes, R. Ni, E. C. M. Vermolen, A. Kuijk,C. G. Christova, J. C. P. Stiefelhagen, T. Vissers, A. vanBlaaderen, and M. Dijkstra, “Self-assembly of a colloidal intersti-tial solid with tunable sublattice doping,” Physical Review Let-ters, vol. 107, pp. 168302–, 10 2011.

6L. Filion, Self-assembly in colloidal hard-sphere systems. PhDthesis, Utrecht University, 2011.

7K. Zhang, W. W. Smith, M. Wang, Y. Liu, J. Schroers, M. D.Shattuck, and C. S. O’Hern, “Connection between the packingefficiency of binary hard spheres and the glass-forming ability ofbulk metallic glasses,” Physical Review E, vol. 90, pp. 032311–,09 2014.

8C. P. Royall and S. R. Williams, “The role of local structure indynamical arrest,” Physics Reports, vol. 560, pp. 1–75, 2 2015.

9A. Ivlev, H. Löwen, G. Morfill, and C. P. Royall, ComplexPlasmas and Colloidal Dispersions: Particle-resolved Studies ofClassical Liquids and Solids. World Scientific Co., SingaporeScientific, 2012.

10K. Sandomirski, E. Allahyarov, H. Lowen, and S. U. Egelhaaf,“Heterogeneous crystallization of hard-sphere colloids near awall,” Soft Matter, vol. 7, no. 18, pp. 8050–8055, 2011.

11A. E. Saunders and B. A. Korgel, “Observation of an ab phasein bidisperse nanocrystal superlattices,” ChemPhysChem, vol. 6,no. 1, pp. 61–65, 2005.

12A. B. Schofield, P. N. Pusey, and P. Radcliffe, “Stability of thebinary colloidal crystals $a{b} {2}$ and $a{b} {13}$,” PhysicalReview E, vol. 72, pp. 031407–, 09 2005.

13A. T. Pham, R. Seto, J. Schonke, D. Y. Joh, A. Chilkoti, E. Fried,and B. B. Yellen, “Crystallization kinetics of binary colloidalmonolayers,” arXiv:1605.05389, 2016.

14E. Trizac, M. D. Eldridge, and P. MaMadden, “Stability of theab crystal for asymmetric binary hard sphere mixtures,” Mol.Phys., vol. 90, p. 675, 1997.

15N. Hunt, R. Jardine, and P. Bartlett, “Superlattice formationin mixtures of hard-sphere colloids,” Physical Review E, vol. 62,no. 1, 2000.

16P. Bartlett, R. H. Ottewill, and P. N. Pusey, “Superlattice for-mation in binary mixtures of hard-sphere colloids,” Phys. Rev.Lett., vol. 68, p. 3801, 1992.

17M. Leocmach, C. P. Royall, and H. Tanaka, “Novel zone forma-tion due to interplay between sedimentation and phase ordering,”EPL (Europhysics Letters), vol. 89, no. 3, p. 38006, 2010.

18J. V. Sanders and M. J. Murray, “Ordered arrangements ofspheres of two different sizes in opal,” Nature, vol. 275, pp. 201–203, 09 1978.

19E. C. M. Vermolen, A. Kuijk, L. C. Filion, M. Hermes, J. H. J.Thijssen, M. Dijkstra, and A. van Blaaderen, “Fabrication oflarge binary colloidal crystals with a nacl structure,” Proceedingsof the National Academy of Sciences, vol. 106, no. 38, pp. 16063–16067, 2009.

20C. Christova, Binary colloidal crystals. PhD thesis, Utrecht Uni-versity, 2005.

21C. P. Royall, E. C. M. Vermolen, A. van Blaaderen, andH. Tanaka, “Controlling competition between crystallization andglass formation in binary colloids with an external field,” Journalof Physics: Condensed Matter, vol. 20, no. 40, p. 404225, 2008.

22E. Vermolen, Manipulation of Colloidal Crystallization. PhDthesis, Utrecht University, 2008.

23K. P. Velikov, C. G. Christova, R. P. A. Dullens, and A. vanBlaaderen, “Layer-by-layer growth of binary colloidal crystals,”Science, vol. 26, no. 5565, 2002.

24M. D. Eldridge, P. A. Madden, P. N. Pusey, and P. Bartlett,“Binary hard-sphere mixtures: a comparison between computersimulation and experiment,” Molecular Physics, vol. 84, pp. 395–420, 02 1995.

25C. P. Royall, W. C. K. Poon, and E. R. Weeks, “In search ofcolloidal hard spheres,” Soft Matter, vol. 9, no. 1, pp. 17–27,2013.

26C. P. Royall, A. A. Louis, and H. Tanaka, “Measuring colloidalinteractions with confocal microscopy,” J. Chem. Phys., vol. 127,p. 044507, 2007.

27M. N. Bannerman, R. Sargant, and L. Lue, “Dynamo: A freeo(n) general event-driven simulator,” J. Comp. Chem., vol. 32,pp. 3329–3338, 2011.

28M. N. Bannerman and L. Lue, “Transport properties of highlyasymmetric hard-sphere mixtures,” The Journal of ChemicalPhysics, vol. 130, no. 16, 2009.

29F. H. Stillinger and B. D. Lubachevsky, “Crystalline—amorphousinterface packings for disks and spheres,” Journal of StatisticalPhysics, vol. 73, no. 3

30C. P. Royall, M. E. Leunissen, and A. van Blaaderen, “A newcolloidal model system to study long-range interactions quanti-tatively in real space,” Journal of Physics: Condensed Matter,vol. 15, no. 48, p. S3581, 2003.

31C. P. Royall, S. R. Williams, and H. Tanaka, “The nature of theglass and gel transitions in sticky spheres,” ArXiv e-prints, Sept.2014.

32A. J. Dunleavy, K. Wiesner, R. Yamamoto, and C. P. Royall,“Mutual information reveals multiple structural relaxation mech-anisms in a model glass former,” Nature Communications, vol. 6,pp. 6089 EP –, 01 2015.

33M. Leocmach and H. Tanaka, “A novel particle tracking methodwith individual particle size measurement and its application toordering in glassy hard sphere colloids,” Soft Matter, vol. 9, no. 5,pp. 1447–1457, 2013.

34W. Lechner and C. Dellago, “Accurate determination of crystalstructures based on averaged local bond order parameters,” J.Chem. Phys., vol. 129, p. 114707, 2009.

35S. Jungblut and C. Dellago, “Crystallization of a binary lennard-jones mixture,” The Journal of Chemical Physics, vol. 134,no. 10, p. 104501, 2011.

36I. Rios de Anda, A. Statt, F. Turci, and C. P. Royall, “Low-density crystals in charged colloids: comparison with yukawa the-ory,” Contributions to Plasma Physics, vol. 55, no. 2-3, pp. 172–179, 2015.

37C. J. Fullerton and R. L. Jack, “Optimising self-assembly throughtime-dependent interactions,” ArXiv e-prints, Sept. 2016.

38D. Klotsa and R. L. Jack, “Controlling crystal self-assembly usinga real-time feedback scheme,” The Journal of Chemical Physics,vol. 138, no. 9, p. 094502, 2013.

39N. DeCristofaro and R. Kaplow, “Interstitial atom configurationsin stable and metastable fe-n and fe-c solid solutions,” Metallur-

Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 11

gical Transactions A, vol. 8, no. 140F. C. Campbell, ed., Elements of Metallurgy and EngineeringAlloys, ch. Solid Solutions. Materials Park, OH: ASM Interna-tional, 2008.

41T. Schillng, H. J. Schoepe, M. Oettel, G. Opletal, and I. Snook,“Precursor-mediated crystallization process in suspensions ofhard spheres,” Phys. Rev. Lett., vol. 105, p. 025701, 2010.

42R. P. A. Dullens and W. K. Kegel, “Reentrant surface melt-ing of colloidal hard spheres,” Physical Review Letters, vol. 92,pp. 195702–, 05 2004.