Long-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures
LLong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures
Long-Lived Non-Equilibrium Interstitial-Solid-Solutions inBinary Mixtures
Ioatzin R´ıos de Anda, Francesco Turci, Richard P. Sear, and C. Patrick Royall
1, 3, 4 H.H. Wills Physics Laboratory, Tyndall Ave., Bristol, BS8 1TL,UK Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH,UK School of Chemistry, Cantock’s Close, University of Bristol, BS8 1TS,UK Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol BS8 1FD,UK (Dated: 12 November 2018)
We perform particle resolved experimental studies on the heterogeneous crystalli-sation process of two component mixtures of hard spheres. The components have asize ratio of 0.39. We compared these with molecular dynamics simulations of ho-mogenous nucleation. We find for both experiments and simulations that the finalassemblies are interstitial solid solutions, where the large particles form crystallineclose-packed lattices, whereas the small particles occupy random interstitial sites.This interstitial solution resembles that found at equilibrium when the size ratiosare 0.3 [Filion et al. , Phys. Rev. Lett. , 168302 (2011)] and 0.4 [Filion, PhDThesis, Utrecht University (2011)]. However, unlike these previous studies, for oursystem simulations showed that the small particles are trapped in the octahedralholes of the ordered structure formed by the large particles, leading to long-livednon-equilibrium structures in the time scales studied and not the equilibrium in-terstitial solutions found earlier. Interestingly, the percentage of small particles inthe crystal formed by the large ones rapidly reaches a maximum of ∼
14% for mostof the packing fractions tested, unlike previous predictions where the occupancy ofthe interstitial sites increases with the system concentration. Finally, no furtherhopping of the small particles was observed.
I. INTRODUCTION
Solid solutions are some of the toughest and most versatile crystalline materials . Steel isa hugely solid solution , it is an interstitial solid solution (ISS) in which the iron atoms forman ordered crystal lattice, with the smaller carbon atoms in the interstices. Substitutionalsolid solutions (SSS) are also important, there the second component is not in the intersticesbetween atoms of the first component, but substitutes for atoms of the first component, onthe lattice. Very recent developments in electron microscopy have for the first time allowedus to analyse the disorder of an SSS of iron in platinum . Also, High Entropy Alloys(HEAs), which are effectively a multicomponent SSS, are currently very actively studied asthey are among the toughest materials known .Hard spheres, as epitomised by colloids, are widely used as models to study the self-assembly and phase behaviour processes of atoms and molecules. Since the structural evolu-tion of colloidal suspensions can be followed in real space by means of confocal imaging, theirstudy has allowed us to monitor the crystallisation process of different systems . Addition-ally, the equilibrium structures of such processes have also been successfully characterised insimulations, which is crucial for fundamental understanding and for many applications, in-cluding material science, metallurgy, and biotechnology . The most popular example ofthese systems is monodisperse hard spheres, where the particles interact only by hard-corerepulsions and whose phase behaviour has been widely studied, through both simulationsand experiments . These studies have identified the face-centered cubic (fcc) -often in arandom mixture with hexagonal close packing (hcp)-, as the solid stable structure .Crystallisation of multicomponent colloidal mixtures has acquired more interest, as a r X i v : . [ c ond - m a t . s o f t ] S e p ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 2these present a richer and more complex phase behaviour than their monocomponentcounterpart . Amongst these systems, we find binary mixtures of hard spheres, whichare composed of two populations of spheres of differing sizes. Similar to one-componenthard spheres, they are also the simplest multicomponent colloidal system and thus, form areference model to study phase transitions of mixtures . Crystals of these systemswere observed by Sanders and Murray back in 1978, whilst analysing the microscopic struc-ture of gem opals, which are composed of silica colloids and had an structure analogue toAlB and NaZn . So far, both experimental and simulation studies have identified thatthe phase diversity found in binary mixtures depends on the size ratio of the components,the number ratio and total concentration of the particles . These studies have pro-duced structures resembling the ones found in nature for different salts like NaCl, NaZn ,AlB , the Laves structures MgCu , MgNi , MgZn , and most recently, stableISS . This large diversity of equilibrium structures highlights their potential for applica-tions in photonics, optics, semiconductors and structure design . In addition, dueto their simplicity, binary hard sphere mixtures are ideal models to study the kinetics ofcrystallisation in salts, metal alloys, metallic glasses and any other crystallising systemwhere there is more than one species, and so there is a compositional variable .However, only a few experimental studies have focused on the crystallisation process ofbinary hard spheres, which can be due to the difficulty of obtaining close-packed orderedstructures. This is related to slow kinetics, caused by a competition of the growth ofcrystal nuclei and compositional fluctuations. In addition, the differences in sedimentationrates between the particles also play an important role in preventing crystallisation .Similar to one-component systems, the particles in the crystalline structure have a highertranslational entropy than in the metastable fluid, which compensates for the decrease ofentropy as the system becomes more ordered . In addition, it has been proposedthat the theoretically predicted structures will only be thermodynamically stable if theirmaximum close packing fraction exceeds the correspondent 0.7405 for fcc or hcp lattices ofone-component systems .Interestingly, these studies have also shown some discrepancies between the experimen-tal observations and the predicted assemblies for particular size ratios and compositions,especially at concentrations near the glass transition . Furthermore, the kinetic con-tributions to crystallisation, along with microscopic mechanisms that yield the ordering ofmixtures are not yet fully understood , accentuating the complexity of these systems overone-component ones, and limiting their promising applications. Based on the interest inunderstanding such crystallisation processes, our goal is to study experimentally the hetero-geneous crystallisation of a mixture of colloidal hard spheres at the particle-resolved leveland compare our results with previous predictions. We also use simulations to study theevolution of the crystals, in particular the dynamics of the small particles, and proposea mechanism for the crystal formation. We observed that only interstitial solid solutions(ISS) are formed regardless of the different packing fractions tested in both experiments andsimulations, as predicted . In such structures, the crystalline phase –a mixture of fcc andhcp lattices– is formed only by the large particles, whereas the small particles are positionedin a fraction of the octahedral holes of the ordered phase. We compared the composition ofour ISS with the equilibrium composition previously predicted for a related system , andwe observed interesting discrepancies at high packing fractions. By studying the dynamicsof the system we found a trapping effect of the small particles which could be preventingfurther crystallisation and equilibrium filling of the octahedral holes.This paper is organised as follows. In section II we describe the experimental and simula-tions details, along with the techniques used for the analysis of the ordered phases. In sectionIII we first present the results for crystallisation for our experiments and their structuralanalysis. We compare such results with previous work. We continue with the structuralanalysis of the ordered phases found on the simulations and the study of the evolutionand quality of the crystals. We then focus on the dynamics of the species. We finalise byproposing a crystallisation mechanism for our system following the results obtained throughsimulations. Finally, in section IV we present our conclusions.ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 3 Experiment ISS Simulation ISS Simulation fluid
L+LSLS+FL+LS+FL+F φ s m a ll φ large F b c d e f g f e d c b g L S a F ISS+F
Experiment ISS Simulation ISS Simulation fluid φ S φ L 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, andFilion, et al. phase diagram of binary hard spheres with a size ratio γ = 0.4 , in the φ L - φ S plane. Diamonds:ISS from heterogenous nucleation experiments. Purple circles: simulations that did not crystallise. Orangetriangles: ISS from simulations. (b-f) are snapshots of crystalline structures found on experiments fordifferent φ tot , the letters b to f in (a) indicates the position on the phase-diagram. The red and greencolours are the large and small particles, respectively, and the scale bar = 10 µ m. (g) is an ISS found forsimulations with φ tot =0.586, where crystalline large and small particles are coloured purple and yellow,respectively, whereas fluid large and small particles appear in light pink and light yellow, respectively. II. METHODSA. Experiments
The binary system used in the experiments consisted on sterically stabilised poly(methylmethacrylate)(PMMA) particles fluorescently labeled with rhodamine and coumarin dyes to enable sep-arate fluorescent imaging. The sizes of the big particles have a diameter of 1700 nm witha polydispersity of 7%, whereas the small particles are 669 nm in diameter and 7.8% poly-dispersity, obtained by scanning electron microscopy. In spite of their large polydispersity,both particles are able to crystallise on their own. These dimensions yield a size ratio( γ = σ S /σ L ) of 0.39. The dried particles were suspended separately in a solvent mixtureof 27% w/w cis-decahydrophnaphtalene (cis-decalin) and cyclohexyl bromide (CHB) thatmatches the density and refractive index of the particles. Additionally, tetrabutylammo-nium bromide (TBAB) salt was added in order to screen the charges, conditions thatallow the particles to behave in a very close manner to hard spheres .The suspensionswere left to equilibrate for several hours, after which they were mixed together at a fixednumber particle ratio, N L /N S =1, and at several total volume fractions, φ tot , defined as φ tot = φ L + φ S , where φ L and φ L refer to the partial packing fractions of each species.These were calculated based on the amount of particles and solvent weighed, following φ L / S = m L / S / ( m L + m S + m solvent ). After shaking, the samples were confined to squaredglass capillaries of 0.50 x 0.50 mm (Vitrocom) and sealed at each end with epoxy. Thesamples were studied by means of confocal laser scanning microscopy, CLSM (Leica SP5fitted with a resonant scanner) using two different channels at 543 nm and 488 nm andNA 63x oil immersion objective. For particle tracking, 3D data sets were recorded bytaking a full scan of the capillary in the z axis, making sure the pixel size was equal inall axes (0.125 µ m/pixel). We used particle resolved studies to determine the crystallinestructure . The crystallisation time was calculated in Brownian time units for the largeparticles, i.e., the time it takes for the large spheres to diffuse a particle radius, given by τ B = ( σ L / / D = 0 .
963 s, where D is the diffusion constant. The Brownian time τ B thussets our unit of time. Observations were carried out for over a month, which correspondsto 2 . × τ B , and were made all along the length and height of the capillaries, whichwere left standing in a vertical position.ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 4 B. Simulations
Hard-sphere simulations were carried out using the open-source event-driven moleculardynamics DynamO package in isothermal-isochoric (NVT) conditions for a binary mix-ture. The total number of particles is N tot = 10 968. Particles of both species are of equalmass, m = 1. The size and number ratio, as well as the final total particle densities werechosen to match the ones from the experiments, thus γ = 0.39 and φ tot from 0.52 to 0.64,respectively. A fluid system at φ tot = 0.282 was used as the initial configuration, which wasthen linearly compressed following Stillinger and Lubachevsky to the appropriate packingfractions until an equilibrated structure was found. Experiments of colloidal systems haveshown that the particles stick irregularly on the walls of the cell they are contained, thusproducing an irregular surface over which the crystallisation takes place . Due to limita-tions in reproducing said phenomenon and thus the final surface, we elected to conduct oursimulations with periodic boundary conditions. Future work could focus on simulating suchenvironment and its influence over the kinetics of the system presented herein.Finally, simulation times were scaled to the experimental data. We inferred our τ B in thesimulations through the relationship τ α = 2.597 τ B , obtained previously by comparing event-driven MD simulations and colloidal experiments of a one-component hard sphere system at φ =0.38, where the structural relaxation time τ α of the simulations was found to equal 0.404simulation time units . Since this mapping pertains to a one-component system, here weconfirm that the relaxation of the large particles is similar in the binary system of interestherein. Therefore, we calculated τ α of the large particles from the trajectories at the samepacking fraction ( φ tot =0.38). This we did by computing the intermediate scattering function(ISF), F (k , t ) = N − (cid:10) (cid:80) Nj =1 exp[ − i k · ( x j ( t ) − x j (0) (cid:11) . To characterise the mobility in thelength scale of a particle diameter, we evaluated F ( k, t ) at the wavenumber k = 2 π/σ L31,32 .We obtained a value of τ α =0.526 simulation time units, which is comparable to the oneobtained by the mentioned work, thus confirming the validity of our approach, and enablingus to use the aforementioned relationship to infer the τ B for our simulations. C. Location of particles in experiment
In order to determine the local ordering of the crystals formed in our experiments, weanalysed separately the structures formed by each component through particle tracking.To locate the particles in the experiments, 3D data sets were directly analysed using thealgorithm described in Leocmach and Tanaka . D. Identification of local crystal structure
Analysis of the crystalline structure for both experiments and simulations, was done fol-lowing Lechner and Dellago by obtaining the bond orientational order parameters (BOO),based on complex spherical harmonics. This analysis gives information of the degree andtype of ordering, thus enabling us to differentiate between distinct crystalline phases. We fo-cused on the locally averaged order parameters ¯ q and ¯ q for square and hexagonal orders,respectively. These parameters take into account the effect of the second nearest neigh-bours, which distinguish more clearly amongst different arrangements . The methodologyfollowed is detailed elsewhere . Briefly for each particle i and regardless of its small orlarge nature, we identify the nearest neighbours N b ( i ) using a cutoff corresponding to thecontact distance between the closest large particles (obtained through the pair distributionfunction, g ( r )). For the experiments said distance corresponded to 1.38 σ L , whereas for thesimulations this number was 1.25 σ L . Finally for the perfect (by construction) NaCl-typelattice the cutoff was 1.1 σ L . Using said list of neighbours, the local order parameters, orong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 5Steinhardt order parameters can be obtained: q lm ( i ) = 1 N b ( i ) N b ( i ) (cid:88) j =1 Y lm ( r ij ) , (1)and summing over all the harmonics: q l ( i ) = (cid:118)(cid:117)(cid:117)(cid:116) π l + 1 l (cid:88) m = − l | q lm ( i ) | (2)where Y lm ( r ij ) correspond to the spherical harmonics, l and m are integer parameters, thelatter running from m = − l to m = + l , and r ij corresponds to the vector from particle i to particle j . However, these parameters only contain information about the first shellsurrounding particle i . In order to obtain information about the second shell, and thus, thelocally averaged order parameters, we need to sum over the list of ˜ N b ( i ) of particle i andthe particle i itself, giving: ¯ q lm ( i ) = 1˜ N b ( i ) ˜ N b ( i ) (cid:88) k =0 q lm ( k ) , (3)and again summing over all the harmonics we get:¯ q l ( i ) = (cid:118)(cid:117)(cid:117)(cid:116) π l + 1 l (cid:88) m = − l | ¯ q lm ( i ) | (4) E. Determination of the Lengthscale of Crystalline Ordering
Following the analysis described in for identifying the size of domains, we determinedin both experiments and simulations the typical crystalline domain size as N Q = N − (cid:10) (cid:126)Q ∗ · (cid:126)Q (cid:11) , where N is the total number of particles. Here (cid:126)Q = (cid:80) Np =1 (cid:126)q ( p ) is the sum overall particles of the normalised complex vector (cid:126)q ( p ), whose components are the sphericalharmonics with l =6. (cid:126)q ( p ) normalisation is done by setting the average of (cid:126)q ( p ) · (cid:126)q ( p ) ∗ = 1.The values taken by N Q will depend on the amount of particles oriented in the same fashion.The value of N Q is an estimate of the average number of particles in a coherently orderedcrystallite. t / τ B φ total Experiments
FIG. 2.
Experimental results of the heterogeneous crystallisation time of the binary system in terms ofBrownian motion, τ B . The line is a guide to the eye. ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 6 III. RESULTS AND DISCUSSION
We present our analysis in three sections: we first identify the ordering and quality of thestructures found in both our experiments and simulations (Sec. III A). We then focus on thedynamics of our two species in both the fluid and the crystalline regions in our simulations(Sec. III B). Finally, we compare the composition of our ISS with an equilibrium systemwith a comparable size ratio and propose a mechanism for the formation of the structuresfound (Sec. III C). HCP FCC Fluid BCC φ tot = 0.58 φ tot = 0.55 φ tot = 0.60 φ tot = 0.64 φ tot = 0.569 φ tot = 0.557 φ tot = 0.586 φ tot = 0.598 q q q q a b Experiments Simulations
FIG. 3.
Local bond order parameter diagrams for the crystalline phase formed found in (a) experimentsand (b) simulations, showing fcc, hcp and fluid coexistence for different total volume fractions tested forboth studies, all at final times. φ tot =0.557 (simulations) is included to show the presence of only a liquidphase. The regions for perfect bcc, fcc, hcp lattices are shown in the top-left plot. Both parameters werecalculated using only the large particles. A. Analysis of the crystalline structures: identification of Interstitial Solid Solutions (ISS)
Filion and collaborators used computer simulations to study a mixture of binary hardspheres with a size ratio only slightly higher (0.4) than ours (0.39). They calculatedthe equilibrium phase diagram, which we reproduce in Fig. 1 (a). There, the symbolsrepresent our experimental state points for the experiments on heterogeneous nucleation(diamonds) and simulations (circles and triangles). Filion predicted that at equilibrium,there is coexistence between interstitial solid solutions (ISS), and a fluid phase (F) . TheISS consists of a regular close-packed lattice, with the small particles filling some but notall the octahedral holes; the fraction of these holes varies continuously with composition.Indeed we do observe crystalline structures built up by the large particles with interstitialsmall particles positioned in a random fashion (Snapshots Fig. 1), regardless of the φ tot andin coexistence with a fluid phase, for both the experiments and the simulations. Surprisingly,we did not find ordered structures for the simulations below φ tot =0.563 or above φ tot =0.610,where the mixtures did not crystallise (Fig. 1 purple circles).In our experiments, nucleation of the crystals is heterogeneous, the crystals grew parallelto the walls of the capillaries and the fluid phase was present far away from the walls. Thepresence of a flat wall clearly facilitates the nucleation process and thus the mixture is ableto crystallise in a φ tot range larger than the simulations, where the nucleation starts in thebulk.The ordered structures found are stable ISS , on the timescales we consider. In these,the crystals grow from the fluid phase and form ordered structures composed of the largerparticles forming a close packed crystal, and the smaller particles in the interstices. Theong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 7composition of these crystals is much poorer in the small particles than it is in the originalfluid, where the number ratio is one. The heterogeneous crystallisation of our mixtures was studied as a function of the totalvolume fractions φ tot = φ L + φ S and we found that increasing this parameter increasesthe crystallisation rate, as shown in Fig. 2. This observation is in agreement with onecomponent systems .
1. Local structure of the large particles
For all of our samples, the equilibrium structure expected is an interstitial solid solutionin coexistence with a fluid . In such assemblies, the large particles are expected to forma mixture of hcp and fcc lattices, known as random hexagonal close packing (rhcp), sincethe free energy difference between these two crystalline phases is very small . The resultsof the analysis of the crystals formed only by the large particles for the experiments andsimulations are summarised in Fig. 3 (a and b, respectively), where we can observe that inboth cases the crystalline phases do consist of a mixture of fcc and hcp lattices in coexistencewith a fluid phase for all the total volume fractions tested. This random stacking has beenidentified before in experiments of heterogeneous nucleation of one-component systems . g (r) r/ σ S - S L - L S - L g (r) r/ σ S - S L - L S - L g (r) r/ σ S - S L - L S - L
6 7
10 11
Simulations a c b
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10
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 q q F r equen cy q φ tot = 0.569
6 7
9 10 φ tot = 0.586 φ tot = 0.598 q
6 Liquid ISS/Liquid NaCl N Ln g (r) r/ σ Global S - S L - L S - L g (r) r/ σ Global S - S L - L S - L g (r) r/ σ Global S - S L - L S - L
18 8 φ tot = 0.557 r/ σ L FIG. 4. (a) Histograms of the bond orientational parameters q (left) and ¯ q (right) values of the smallparticles at φ tot =0.586 before crystallisation (Liquid) and after crystallisation (ISS/Liquid), and an equi-librated NaCl-type crystal at φ tot =0.64, showing the ability of ¯ q to distinguish the local environment ofthe small particles. (b) Heat maps of the ¯ q values of the small particles and their corresponding number ofneighbouring large particles (N Ln ) for different total volume fractions that crystallised in simulations, calcu-lated following reference 33. The plots show two populations: one below and one above ¯ q = 0.15. Note that φ tot =0.557 does not crystallise. (c) Radial distribution functions of the liquid (top) and ISS/Liquid (mid-dle) both at φ tot =0.586, and NaCl-type crystal (bottom) at φ tot =0.64. Each shows 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. ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 8 a b c N XS t/ τ B N XS / N X L log t/ τ B N XS / N X L φ total N X L t/ τ B Simulations N XS t/ τ B FIG. 5.
The numbers of large (a) and small (b) crystalline particles as a function of time in simulations. (c)evolution of the proportion of small crystalline particles relative to the large crystalline ones in simulationsaccording to their φ tot .The simulation time has been rescaled to Brownian time (see Methods for details).
2. Local environment of the small particles: an analysis on the crystal quality and thevacancies in simulations
In the ISS the small particles are situated in the octahedral holes of the fcc and hcplattices formed by large particles. Filion et al. showed, for their smaller size ratio of 0.3,that the small particles are able to hop at lower pressures– where the system is far fromclose-packing– fill all the octahedral holes and yield an ISS with the NaCl structure . Onthe other hand, for the larger size ratio of 0.4, Filion and coworkers also found that the ISSare the stable assembly, however, for this larger size ratio they do not report the formationof a perfect crystalline NaCl structure, nor they show the dynamics of the small particleslike for γ =0.3 . In our experiments a similar size ratio of 0.39, we also observed only ISS,and we could not observe hopping of the small particles in direct imaging. Due to thelimitations on tracking the motion of the small particles, we decided to conduct hard spheremolecular dynamics simulations in order to quantify both the small sphere ordering anddynamics.Using simulation data, we calculated the BOO q and ¯ q values for the small component,see Fig. 4(a) and (b). In Fig. 4(a), we plot q (left) and ¯ q (right) in the liquid state, in astate that is partially crystallised with remaining liquid, and in a perfect NaCl-type crystal.It is clear that q does not differentiate between the different configurations, but ¯ q doesdistinguish between the liquid and the ISS. So we use the Lechner-Dellago averaged ¯ q inorder to identify the fluid or crystalline nature of the particles. Values of ¯ q above 0.15 arevery rare in the liquid, while in a well-ordered NaCl structure ¯ q is almost always above0.15.In Fig. 4(b), we show heat maps of the probability distribution p ( N Ln , ¯ q ) for 4 states,one liquid (top-left) and three ISS/liquid mixtures. The samples with liquid and ISS showtwo populations. The dominant one, with ¯ q < .
15 is the liquid, but there is a smaller pop-ulation with ¯ q > .
15, which is consistent with values of ¯ q in the crystal. The associationbetween this second population and the crystal is further supported by the fact that smallparticles here have around 6 to 8 large-particle neighbours. In the NaCl structure, eachsmall particle has 6 large-particle nearest neighours, and our cutoff distance for neighbours(dotted line in Fig. 4(c)) is above the small-large nearest neighbour distance, so we wouldfind at least 6 neighbours — precisely what we observed.
3. Composition of the crystalline domains
We followed the crystal growth by estimating N XL and the small particles located at theoctahedral holes (N XS ), according to their ¯ q values. These results are shown in Fig. 5 (a)and (b), respectively. Here, we observe that both quantities reach a plateau, indicating nofurther crystal growth and suggesting no hopping of the small particles within the interstitialsites of the fcc and hcp lattices, as discussed previously . Next, we calculated the amountong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 9of N XS relative to N XL forming the ordered structure. We found two notable phenomena.Firstly, the octahedral holes are filled rapidly as the nucleation takes place (Fig. 5 (c))with a maximum filling of the octahedral sites of around 14%, above φ tot =0.586 (see Fig.8 (b)). Secondly, we observed no further changes in the proportion of crystalline particles,suggesting a trapping effect. Since the number of small crystalline particles also reached aconstant value, we infer that they become trapped (and immobilised) in the the crystallinestructure once it is formed. These phenomena will be investigated in detail in the followingsections.
4. Crystalline domains in experiment and simulation
Notably, see Figs. 1 and 6, the range of packing fractions that crystallised on the sim-ulation timescale was smaller than that of the experiments. But in both experiments andsimulations, there are ranges of volume fraction where maximal crystallisation occurs. Inexperiments we observe regions in the system where there is largely complete crystallisationof the large particles: at φ tot = 0 .
55, 0 .
58 and 0 .
60 —see Fig. 6 (a), the fraction of largeparticles identified as crystalline is around 65% or above. N Q N XL Simulations N Q N XL Experiments a b FIG. 6.
The characteristic size of the crystal domain(s), N Q , as a function of the total number of largecrystalline particles, N XL . For the experiments, we show the final values (as symbols, see key), while forcomputer simulations, we show the evolution with time during crystallisation (as curves). The total numberof large particles in the experiments is 2534, 2645, 2584, 2524 and 2244 for φ tot =0.64, 0.60, 0.58, 0.55 and0.52, respectively. On the other hand, for simulations the total number is 5,484 of large and 5,484 ofsmall particles. N Q shows that the quality of the crystals obtained in the experiments is higher than thesimulations. In our simulations, the total amount of crystalline structure is much smaller, reaching thehighest amount around 18% at φ tot =0.580, after which it started to decrease, as shown inFig. 6 (b). The need for a higher φ tot to observe crystallisation on the simulation timescale( φ tot =0.563 in comparison with φ =0.53 for one component systems ) might be due to ahigher free energy barrier to generate nuclei necessary for crystallisation. The drop of thecrystalline fraction of our system at φ tot = 0 .
598 is similar to the one where one-componentsystems exhibit slow dynamics φ (cid:39) . , and is related to the particles moving more slowlyas their concentration increases, requiring a long time to rearrange in ordered structures.ong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 10A similar trapping phenomenon could be happening here, where the particles do not movefast enough to rearrange and form crystals.To determine if these crystalline large particles are in one large crystalline domain, orif the system is polycrystalline, we follow Klotsa and Jack . These authors defineda parameter N Q to describe the size of coherent crystallites, i.e., the typical number ofparticles in a coherently ordered crystal. We applied this analysis to both our experimentsand simulations to determine which conditions yielded the crystals with the highest quality.In the case of the experiments, we analysed the final crystals formed by the large particles,whereas for the simulations, we studied the evolution of N Q through the crystallisation ofthe different samples.The results are presented in Fig. 6, where we compare this parameter with the numberof large crystalline particles for both cases, N XL . It is evident from these results that thesize of the crystalline domains obtained from heterogeneous nucleation in the experimentsis higher than the homogenous nucleation present in the simulations, as N Q presents sig-nificantly larger values in the former for similar N XL . Furthermore, the N Q values for theexperiments are close to the number of crystalline particles. Reasons for this include thepresence of the flat walls of the capillaries, larger system sizes or the need for longer waitingtimes. Indeed flat walls can serve as a template able to enhance the nucleation and layeringof the particles and thus improve the orientation of the crystal . On the other hand,estimations of the crystallisation time for our experiments for homogenous nucleation werefound to be two orders of magnitude larger than the corresponding crystallisation time onour simulations, suggesting that longer waiting times are required for experimental homoge-nous crystallisation. This might be related to the higher polydispersity in the experimentalsystem. Moreover, said time surpasses our limit for experimental timescale observations ofa month (2 . × τ B ).Finally, the system size in the experiments is of order 10 times bigger than the simu-lations. Hence it seems likely that our experiments can crystallise over a wider range andform crystals with a greater quality due to a combination of the reasons mentioned above.In the case of simulations, the numerical values of N Q are significantly smaller thanN XL , suggesting the presence of several clusters in all the samples. Also, a non-monotonicbehaviour of this value is noted, since the highest values of the former do not correspondto the largest ones for the latter. This is evident for φ tot =0.598, whose maximum N XL surpasses the corespondent of φ tot =0.563 and 0.569, with higher amounts of crystallineparticles (Fig. 6 (b) orange, light pink, and pink lines, respectively). Interestingly, thedependence between these two parameters is not linear, with N Q increasing faster than thecrystal growth. A possible reason for this could be fast coalescence between the formingcrystallites. Indeed, φ tot =0.574 (Fig. 6 (b) dark pink line) shows a faster increase of N Q around N XL ∼ B. The dynamics of both large and small particles are arrested in the crystals
In order to show that our small particles were indeed localised, we computed both themean square displacement (MSD, (cid:10) ∆ r ( t ) (cid:11) = (cid:10) ( r L ( t ) − r L (0)) (cid:11) ) and the intermediatescattering function (ISF), as described previously for the simulations with φ tot =0.586 forgeometrically selected large and small particles in both liquid and crystalline regions. Thiswas done at the final points of the simulation, where the ratio N XS /N XL remained constant.These results are shown in Fig. 7. We observe that both the large and small particles inthe liquid region present movement (Fig. 7, dotted pink and purple lines, respectively), i.e. their MSD (insert) and ISF present a behaviour typical for a confined random walkand fluids, respectively. In contrast, both species located in crystalline regions do not showany movement, identified as a flat line close to zero in the MSD and as a non-decaying lineon the ISF (Fig. 7, continuous lines). We can conclude therefore that indeed, once thecrystals are formed, the small particles are immobile within the crystal formed by the largeong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 11 -1 Large crystalline particles Small crystalline particles Large fluid particles Small fluid particles F ( k ,t ) log t/ τ α -1 Large crystalline particles Small crystalline particles Large fluid particles Small fluid particles F ( k ,t ) log t/ τ α Simulations -2 -1 < ∆ r ( t ) > / σ t/ τ B FIG. 7.
Intermediate scattering functions and mean square displacements (insert) from simulations, at φ tot =0.586. The large and small particles are shown in pink and purple, with continuous lines for particlesidentified as crystal, and dotted lines for particles identified as fluid. The simulations start at the endof a crystallisation simulation. Particles at the liquid-crystal boundary region were discarded in order tocharacterise only the crystalline and fluid particles. particles; unable to hop and fill all the available octahedral holes, and thus our ISS remainas long-lived out-of-equilibrium structures. C. Comparison of our results at a size ratio γ =0.39 to those of Filion et al. at size ratios of0.3 and 0.4 We found that the small particles occupy the octahedral sites of the crystalline lattices inan incomplete fashion, forming a dilute ISS which in our computer simulations consistentlyhas a composition close to seven large particles for every one small particle, much less thanthe one-to-one ratio in the original fluid phase.An ISS has also been identified before in binary mixtures with a smaller size ratio of 0.3compared to 0.39 here . Filion and coauthors found that in order for the small particlesto hop from one octahedral hole to the other, they needed to go through the adjacenttetrahedral hole ( ∼ σ L across). Although slow, they found that their smaller particlescould hop, whereas (see Fig. 7) our larger small particles cannot.Filion also studied the size ratio 0.4, closer to the one used here. For that size ratio,the ISS equilibrium region in the phase diagram is larger than for 0.3. In order to confirmthat our ISS are out-of-equilibrium, we compared our simulation values of X S , with theequilibrium values determined by Filion . As shown in Fig. 8(a), we determined the finalpressure in our simulations, and then using tie lines (which are horizontal here as the twophases have the same pressure) read off the compositions of both phases. The comparisonof these results and the ones corresponding to our ISS are shown in Fig. 8(b), where thedashed curve is the equilibrium X S , and the dotted curve and points are our results. In allcases, the amount of small particles within the crystalline structure of large ones is smallerthan the one predicted for an ISS in 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 equi-librium when they nucleate and grow. As the large particles crystallise into an orderedstructure, some of the small particles become trapped in the octahedral holes, forming theISS. But under the conditions we study, the integration of the small particles into the grow-ing crystal appears to be inefficient, resulting in the low number of small particles in thecrystal. This then causes the composition of the remaining fluid to become increasingly richin the small particles, as the crystal grows.As we can see in Fig. 8(a), increasing X S will ultimately move the fluid out of thecoexistence region and into the one-phase fluid region of the phase diagram. So, we believeong-Lived Non-Equilibrium Interstitial-Solid-Solutions in Binary Mixtures 12 treshold N XS / N X L φ total Out-of-equilibrium Predicted equilibrium 5% stricter q6 treshold N XS / N X L φ total p X S 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’s phase diagram of a binary hard-sphere system with γ = 0 .
4, showing the reducedcoexistence pressure p = βP σ , as a function of composition X S = N S / ( N S + N L . We have added tothis filled symbols to represent our state points where we obtained an ISS and the empty symbols for theconditions were, on our time scales, we did not observe crystallisation. The tie line and arrow are drawnto illustrate the interpolation process to obtain the small particle composition in the equilibrium ISS. (b)The composition as a function of our system’s average volume fraction, φ tot . The dashed line correspondsto the equilibrium composition , and the points are our simulation results. The error bars are the standarddeviation of 8 runs, whereas the dotted line is the error obtained when using a 5% more strict ¯ q threshold. this change in composition contributes to the growth of the crystal slowing and stopping. IV. CONCLUSIONS
Interstitial solid solutions (ISS) were identified in both particle resolved experiments andsimulations of a binary mixture with a size ratio of 0.39. Through particle resolved studiescarried out on the experimental results, we were able to identify that the crystalline struc-ture was made up by the large particles forming a mixture of fcc and hcp lattices, withlow crystallinity, and where the small particles are localised randomly within the octahe-dral holes. Simulations showed the same crystalline structure and allowed us to follow thecrystallisation process and quantify the amount of filling by the small particles and, signif-icantly, the dynamics of the small particles in the ISS. With this information, we were ableto propose a crystallisation mechanism where the large particles form ordered structuresindependent containing small particles, which become rapidly trapped within the growingcrystal to a maximum of ∼ V. ACKNOWLEDGEMENTS
The authors are grateful to John Russo for simulation discussions. IRdA would like tothank Conacyt for financial support. CPR acknowledges the Royal Society for funding andKyoto University SPIRITS fund. FT and CPR acknowledge the European Research Council(ERC consolidator grant NANOPRS, project number 617266). This work was carried outusing the computational facilities of the Advanced Computing Research Centre, Universityof Bristol.
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