Pattern formation in binary colloidal assemblies: hidden symmetries in

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Pattern formation in binary colloidal assemblies: hidden symmetries in a kaleidoscope of structures Valeria Lotito, and Tomaso Zambelli Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01411 • Publication Date (Web): 10 Jun 2018 Downloaded from http://pubs.acs.org on June 10, 2018

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Pattern formation in binary colloidal assemblies: hidden symmetries in a kaleidoscope of structures Valeria Lotito*, Tomaso Zambelli Laboratory of Biosensors and Bioelectronics, Institute for Biomedical Engineering, ETH Zurich, Gloriastrasse 35, 8092 Zurich, Switzerland * [email protected]; [email protected]

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Abstract In this study we present a detailed investigation of the morphology of binary colloidal structures formed by self-assembly at air/water interface of particles of two different sizes, with a size ratio such that the larger particles do not retain a hexagonal arrangement in the binary assembly. While the structure and symmetry of binary mixtures in which such hexagonal order is preserved has been thoroughly scrutinized, binary colloids in the regime of non-preservation of the hexagonal order have not been examined with the same level of detail due also to the difficulty in finding analysis tools suitable to recognize hidden symmetries in seemingly amorphous and disordered arrangements. For this purpose, we resorted to a combination of different analysis tools based on computational geometry and computational topology in order to get a comprehensive picture of the morphology of the assemblies. By carrying out an extensive investigation of binary assemblies in this regime with variable concentration of smaller particles with respect to larger particles, we identify the main patterns that coexist in the apparently disordered assemblies and detect transitions in the symmetries upon increase in the number of small particles. As the concentration of small particles increases, large particle arrangements become more dilute and a transition from hexagonal to rhombic and square symmetries occurs, accompanied also by an increase in clusters of small particles; the relative weight of each specific symmetry can be controlled by varying the composition of the assemblies. The demonstration of the possibility to control the morphology of apparently disordered binary colloidal assemblies by varying experimental conditions and the definition of a route for the investigation of disordered assemblies are precious for future studies of complex colloidal patterns to understand selfassembly mechanisms and to tailor physical properties of colloidal assemblies. Keywords Air/water interface self-assembly, amorphous colloidal assemblies, colloidal pattern analysis, disordered binary assemblies, persistent homology, order parameter, Voronoi tessellation, shape factors


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Introduction Two-dimensional (2D) binary colloidal assemblies, i.e. 2D mixtures of colloidal particles of two different sizes, are attracting increasing attention among scientists. The possibility to tune the geometry of the final patterns by varying the ratio between the number of smaller (S) and larger (L) quasi-spherical colloidal particles ( ⁄ ) as well as their size ratio (the ratio between the particle diameters ⁄ ) offers the potential for an abundant gamut of structures in comparison to single-sized colloids that typically assemble in an energetically favourable hexagonal closely packed (hcp) structure (or in a hexagonal non-closely packed arrangement under certain experimental conditions, e.g. as usual at oil/water interfaces), although with different characteristics in terms of order related to the experimental implementation of colloidal selfassembly [1]. The rich variety of patterns that can arise in quasi-bidisperse suspensions has been proposed for diverse applications from light propagation engineering in photonic glasses [2] to phononic materials with tailored sound wave propagation [3,4] or cell culture substrates with enhanced or inhibited cell spreading [5]. In addition, binary colloids can be used as templates in colloidal lithography for the fabrication of nanostructures, for instance plasmonic arrays [6]. Given the outstanding potential of colloidal self-assembly and the wide range of techniques available also for self-assembly of binary colloids [1], binary colloidal assemblies are expected to play a major role in the coming years. Understanding pattern formation in a binary assembly is essential due to the intimate relationship between the structural morphology and the properties of interest for the different specific applications (e.g. optical or elastic properties). Restricting the analysis to experiments based on co-self-assembly of binary colloids, i.e. in which L and S colloids are simultaneously assembled (in contrast to two-step assembly procedures whereby assembly of L particles is followed by assembly of S particles) [1], a basic distinction can be traced between arrangements in which a hexagonal pattern is retained by L particles and those in which it is lost (Fig. 1).


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Figure 1 – Scanning electron micrographs of different regimes for binary assemblies: (a) preservation (scale bar = 2 μm) and (b) loss (scale bar = 1 μm) of hexagonal arrangement for L particles.

In the first case (Fig. 1(a)), a periodic and regular structure is observed, where S particles, in a well-defined pattern, fill the interstices between L particles or are located along the bridges between L particles, i.e. along the lines connecting the centers of two neighbouring L particles. This regime has been thoroughly investigated to individuate the variegate range of possible configurations that differ in the number and arrangement of S particles present in the interstices and in the bridges and that can be properly engineered upon suitable selection of size ratio ⁄ and number ratio ⁄ . In addition, a careful quantitative investigation of the order of the binary assemblies has been carried out, by using different functions and metrics aimed at assessing how close the actual experimental pattern is to the theoretical pattern predicted for given values of ⁄ and ⁄ , such as the pair correlation function and the bond orientational order parameter [7,8]. The patterns obtained in the second regime (Fig. 1(b)) have generally not been examined with the same level of detail as the first one. Although transitions to different types of ordered periodic structures (e.g. square or rhombic patterns) have been deemed feasible and theoretically predicted on the grounds of geometric considerations concerning sphere packing and numerical analyses performed, for instance, with genetic algorithms and Monte Carlo simulations [9-11], in experiments such patterns have been observed more rarely only in specific experimental conditions (for example involving the use of magnetic colloids or 4 ACS Paragon Plus Environment

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AC electric fields) and are typically limited to regions smaller than those encountered in case of preservation of hexagonal order for L particles [12,13]. More frequently, L and S particles typically mix together in more complex configurations without a clear periodic repetition of specific patterns and result in disordered and amorphous arrangements in contrast to the more crystalline arrangements of the first regime, with coexistence of more phases or random tiling in which unit cells of different structures combine in a random rather than in a periodic fashion [2,9,14-16]. Phase separation in which L and S particles separately arrange in a hexagonal pattern can even occur [9,17]. A precise experimental evaluation of the evolution of dominating shapes and patterns is missing, despite the importance it has not only to cast light on particle assembly process, but also to understand how the level of symmetry and disorder can be tuned and exploited for some specific applications. For example, optical properties of binary colloidal assemblies change dramatically upon transition from a crystalline regime to a disordered regime, switching from an optical response characterized by sharp diffraction resonances to one dominated by scattering of single particles and Mie resonances with intermediate behaviours that can be achieved by varying the composition of binary assemblies [2]. Great effort has been devoted to the development of tools for the analysis of the morphology of colloidal assemblies which represents a research theme on its own, as it is of great importance to compare different self-assembly techniques and experimental conditions or to cast light onto the mechanisms underlying colloidal self-assembly [18-23]. However, traditional analysis techniques are still inadequate to recognize hidden symmetries in seemingly amorphous and disordered arrangements. The lack of an accurate quantitative morphological description of binary colloidal assemblies in the regime of loss of hexagonal order for L particles can be mainly ascribed to such insufficiency. More common analysis tools based, for instance, on the pair correlation function and other related metrics are suitable to characterize ordered systems consisting of a periodic repetition of a given unit cell and to identify the transition from a crystalline regime to non-crystalline ones [24-27], but they fail to provide a quantitative morphological characterization in a more disordered regime based on the coexistence of different symmetries. For example, the inadequacy of the pair correlation function to highlight structural transitions in disordered binary colloidal assemblies has been pointed out in [28,29]. Other analysis approaches, such as those based on the computation of the fractal dimension, have been adopted for the identification of structural changes in more disordered systems especially in case of more sparse structures consisting of fractal-like aggregates [30,31], but they still do not provide clues to recognition and quantification of dominant symmetries concealed in an amorphous assembly and require care in the selection of the computation method and parameters [27]. More sophisticated approaches based on integral geometry have been adopted for the identification of transitions to increasing levels of disorder in binary colloidal assemblies of superparamagnetic particles, but they have not been used for a quantification of the different types of symmetries, an issue that has been only partially addressed by the use of the bond orientational order parameters to identify few regular patterns (e.g. an S particle in the interstice between L particles arranged in a square pattern) [14,32]. In brief, a more comprehensive quantitative characterization of the diverse patterns and symmetries hidden even in an apparently disordered system is not feasible with traditional analysis tools, although it would beneficial not only for the analysis of binary colloids in the regime of loss of hexagonal order, but also for other types of more amorphous colloidal assemblies. In fact, complex patterns characterized by the coexistence and transition between different symmetries have been encountered experimentally, envisaged theoretically and studied for the intrinsic interest in colloidal interactions and for potential applications also in different systems. For instance, recently, such complex arrangements have been predicted for 2D systems in molecular dynamics simulations of particles interacting via a core-softened potential [33]. 5 ACS Paragon Plus Environment

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The transition between different morphologies within the regime of preservation of hexagonal order as well as the transition between the preservation of hexagonal order for L particles and its loss occur for certain threshold values of ⁄ or ⁄ , which are related to both experimental conditions and geometric constraints. Among the different experimental techniques available for colloidal self-assembly, air/water interface selfassembly is gaining widespread diffusion since the seminal demonstration of interfacial trapping of colloidal particles [34] and the observation of diverse patterns attainable under different experimental conditions [35], due to reduced sensitivity of the characteristics of the self-assembled colloidal monolayer to the substrate on which it is transferred in comparison to techniques in which the monolayer is directly assembled on the desired substrate [1]: binary colloidal assemblies in the regime of preservation of hcp order have been obtained in this way even with simple and straightforward implementations such as the one based on surface confinement and water discharge [7]. Approximate values of ⁄ or ⁄ for the transitions between different hcp-preserving morphologies and between preservation and loss of hcp order for L particles have been found under specific assumptions [17,36]. The goal of this article is twofold: •

the investigation of pattern evolution in binary colloidal monolayers obtained by self-assembly at air/water interface in the regime of loss of hexagonal order for large particles; we will demonstrate that, by tuning the composition of the binary assembly (particle number ratio ⁄ ), it is possible to control the morphology of a binary colloidal assembly even in the regime of loss of hexagonal order so as to get dominant patterns under specific conditions;

•

the definition of a combined approach based on computational geometry and computational topology which allows an accurate and comprehensive characterization and quantitative description of the morphology of complex colloidal assemblies and is able to detect hidden symmetries and dominant patterns in ostensibly disordered assemblies in contrast to traditional analysis tools which are typically suitable for the analysis of more ordered quasi-periodic structures but fail to identify symmetries in more disordered patterns.

In particular, we present a detailed investigation of the structure of binary colloidal assemblies of polystyrene (PS) particles obtained at air/water interface for a particle size ratio ⁄ slightly above the threshold for the transition from preservation of hcp order for L particles to its loss and for variable number ratio of S and L particles between 0 and 2. The analysis aims mainly at assessing how an increasing population of S particles affects the arrangement of L particles and at identifying recurrent patterns attainable for increasing number ratios. A careful quantitative evaluation of such transitions is carried out based on image processing of scanning electron micrographs of binary colloidal assemblies and image analysis techniques with tools borrowed from computational geometry and computational topology. Our study, which includes the computation of the pair correlation function, Voronoi tessellation, shape factors and persistence diagrams, reveals a decrease in the six-fold symmetry in favour of a combination of different structures formed by particles with four, five and six nearest neighbours together with the appearance of an increasing variety of arrangements from square to rhombic and more complex shapes. Our first experimental quantitative assessment of patterns of nonmagnetic colloids can shed light on structure formation in the non-hcp regime at air/water interface. In addition, we establish an effective and robust approach for the analysis and comparison of binary colloidal assemblies and, more in general, for the detection of hidden symmetries in complex and even seemingly disordered colloidal patterns obtained 6 ACS Paragon Plus Environment

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in different experimental conditions. In the future this approach for pattern identification can be used in fundamental studies aimed at understanding interactions between colloidal particles under given experimental conditions and in practical applications to help scientists to correlate the physical properties of complex colloidal assemblies and, in particular, binary colloidal assemblies with their structure. Experimental Section Materials PS particles of diameters = 220 nm and = 500 nm were purchased from Thermo Scientific (5000 Series Polymer particles packaged as aqueous suspensions at 10 wt % solids); the coefficient of variation (CV) was specified to be less than 3%. Fabrication of colloidal assemblies Single-sized and binary colloids have been assembled at air/water interface using the technique of surface confinement and water discharge and the apparatus described in [7,37]. In brief, the experimental set-up consists of a glass container provided with a hole and a tap and a polytetrafluoroethylene (PTFE) apparatus, made up of a circular base located on the bottom of the glass container into which three cylindrical rods and a sample holder are inserted; the glass container is filled with water and a small amount of surfactant (0.1 mM of sodium dodecyl sulfate); subsequently, the target substrate is placed on the sample holder beneath the water surface and a nitrile butadiene rubber (NBR) ring is put on the water surface within the area confined by the three rods. The colloidal suspension mixed 1:1 with ethanol and containing either only L particles or both L and S particles is dispensed on a tilted glass slide that lets them flow gently to the air/water interface within the surface area confined by the NBR ring kept in a fixed position by the three cylindrical rods; after self-assembly, the colloidal monolayer is transferred onto the target substrate placed beneath, by opening the tap of the glass container and letting water flow out. For all the experiments carried out at different ⁄ values, the particles covered an area of the water surface smaller than the area encircled by the NBR ring, so as to ensure uniform experimental conditions for all particle assemblies independently of the specific symmetry and packing fraction obtained for the different particle number ratios. The rubber ring placed on the water is not used for compression of the particle monolayer, but for the localization of the monolayer over the target substrate placed beneath the water surface and for the isolation of the monolayer from the walls of the glass container, beneficial during transfer of the monolayer onto the target substrate by water discharge, as it prevents free floatation of the monolayer and its possible sticking to the walls. Hence, the monolayer can be smoothly transferred onto the target substrate without any disturbance or damage. In this way, we could investigate the assemblies spontaneously formed at air/water interface for variable composition of the binary mixture. Binary assemblies with particle ratios ⁄ from 0 to 2 with a step of 0.5 have been produced. This parameter can be estimated using the following expression:

=

(1)

where and represent the volumes of S and L particle suspensions. A correction factor needs to be applied to take into account particle loss due to partial sticking to the glass slide and partial sinking of particles. The correction factor can be estimated by computing separately for S and L particles the ratio between the actual area / covered by a single-sized monololayer of S (L) particles and the ideal area 7 ACS Paragon Plus Environment

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covered by S (L) particles / when a given particle suspension volume is dispensed; the latter can be computed as:

/ = /

/ 6 ∙ 10 ∙ ∙ = / ! ∙ " ∙ /

! /

(2)

where the number of particles contained in a volume / of particle suspension and the particle diameter are expressed in mL and μm, respectively, is the volume fraction of solids of the suspension, ! is the suspension density (g/mL), ! is the density of the particle material (g/mL), / is the area covered by a single S (L) particle, = "

√ is $

the packing density for a hcp pattern. The ⁄ values computed with this

approach have been confirmed by those extracted from scanning electron micrographs. Characterization Scanning electron microscopy (SEM) images of the colloidal assemblies transferred onto the substrates have been obtained using Zeiss Ultra 55 and Zeiss Ultra Plus scanning electron microscopes, after electron beam evaporation of a thin conductive layer. Image analysis The structure of the resulting colloidal assemblies has been investigated using different numerical tools. SEM micrographs have been first processed to identify separately the coordinates of the centroids of L and S particles relying on algorithms for circle detection based on the circular Hough transform. Such coordinates have been used for the construction of the Voronoi diagrams (i.e the tessellation based on the use of particle centroids as seeds to partition the space into polygonal cells, each containing points closer to one seed than to the others), the determination of the number of nearest neighbours %%& (also called coordination number) for each of the L particles, the evaluation of the pair correlation function '()*, the computation of the persistence diagrams. The entropy of conformation +, has been calculated as:

+, = − . /011 ln /011

(3)

011

where /011 represents the probability of finding particles with %% nearest neighbours for a given configuration. (&*

We have proceeded to the computation of the bond orientational order parameter 30 defined as: (&* 30

0117

(4)

1 = . 4 05678 %%& 9:

where ;&9 is the angle of the line between the center of particle j and each of its %%& nearest neighbours with respect to a reference axis and is an integer chosen according to the specific symmetry under investigation (for example = 6 to assess the presence of six-fold hexagonal symmetry and = 4 for (&*

(&*

four-fold square symmetry); for a particle exhibiting a perfect N-fold symmetry =30 = = 1 (e.g. =3$ = = 1


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(&*

for a particle with nearest neighbours located at the vertices of a regular hexagon and =3> = = 1 for a particle with nearest neighbours located at the vertices of a square). The Minkowski structure metric for a particle ? has been computed as: (&* 30@

1 = A&

0117(CDEF*

.

9:

B&9 405678

(5)

where A& is the perimeter of the Voronoi cell associated to particle ? composed of %%&(5G* edges or sides, B&9 is the length of the side of the Voronoi cell corresponding to the bond between particles ? and H and ;&9 is the angle of the line connecting the centers of particles ? and H with respect to a reference axis. Starting from the Voronoi tessellation, for each of the L particles, the so-called shape factor I(&* has been computed as:

I(&* =

A&

(6)

4"&

where & represents the surface area of the Voronoi cell associated with the particle j. Such metric provides information about the shape of the cell; in particular, one has I(&* = 1 for a circle and I(&* > 1 for all the other shapes. Matlab and JavaPlex have been used for the analyses [38]. The different parameters have been computed on sets of SEM images collected for each ⁄ value. For an accurate characterization, on the grounds of previous studies on amorphous arrangements of colloidal particles [32], approximately 2 ∙ 10K L particles have been examined for each particle number ratio; this amounts to a total of 2 ∙ 10$ L and S particles for all the data sets. Results and Discussion Coordination number and entropy of conformation For our experiments, we wished to operate in a regime of lack of hcp order for L particles. The conditions for the appearance of specific patterns, in particular hcp patterns and square closely packed patterns, for large hard spheres in binary mixtures have been studied in [9], under the assumption that the small spheres have to fit exactly into the interstices between the closely packed large spheres, with a consequent reduction of the problem to the study of small and large discs; for instance, under this hypothesis, for an hcp pattern with one S particle exactly inscribed in each of the interstices between three L particles with centroids located at the vertices of an equilateral triangle (triangular lattice) one should have ⁄ = 0.155 and ⁄ = 2; on the other hand, a square packing for L particles with one S particle precisely fitting in each of the interstices has been predicted for ⁄ = 0.414 and ⁄ = 1. For ⁄ and ⁄ that do not precisely satisfy such conditions, different structures have been hypothesised, for example by distorting the lattice or replacing one S particle with several smaller S particles or combining different unit cells [9]. In experiments, the scenarios get even more complicated because S and L particles do not necessarily have the same equatorial plane, as S particles can actually “sit” on top of L particles, which means that the projection of S particles on the equatorial plane can intersect L discs and the problem cannot be reduced to 9 ACS Paragon Plus Environment

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a 2D disc filling analysis. Indeed, such a behaviour has been experimentally detected in numerous works, with the observation of several morphologies where hcp order is preserved for L particles and one or more S particles fill the interstices or the bridges between L particles even if not satisfying the filling conditions in terms of 2D projections [7,17,36]. Although the exact value of the threshold for the transition is sensitive to diverse experimental conditions (e.g. surface charge, pH, surfactant that can affect the contact angle of L and S particles as well as the experimental technique used for self-assembly [1]), approximate values have been computed: up to ⁄ = 0.414, hcp preservation should theoretically be feasible [36,39]. Despite the fact that even larger limits for ⁄ have been predicted on the grounds of purely geometric considerations on sphere packing [17], if we limit the analysis to co-self-assembly experiments at air/water interface, hcp preservation has been experimentally observed up to ⁄ = 0.32 [36]. In this study, we have compared the morphology of binary assemblies obtained with particles L and S of nominal diameters and equal to 500 nm and 220 nm, respectively. These diameters have been chosen in such a way to result in a particle ratio ⁄ = 0.44 which is expected to be close to a transition between preservation and loss of hcp order for L particles. Fig. 2 (left column) shows examples of SEM micrographs obtained for increasing number ratio. As clearly visible, when no S particles are present, L particles assemble in an hcp pattern, whereby each L particle is surrounded by six L particles with centers approximately located at the vertices of a regular hexagon; as ⁄ increases, the hcp pattern progressively deteriorates, in that the hexagonal patterns become more irregular and the number of particles surrounded by six particles tends to decrease, with the appearance of new arrangements involving also a number of nearest neighbours different from six. Before proceeding to the quantification of the number of nearest neighbours, it is worth to observe that its definition is inherently controversial. One option consists in considering as nearest neighbours of a given particle those particles placed within a certain threshold distance from that particle [40]. In case of nearly regular structures, such as with a mainly closely packed hexagonal or square symmetry, such value lies in the interval between the theoretical minimum distance between a particle and its nearest neighbours and the theoretical distance between a particle and its second nearest neighbours (i.e. between and √3

for an hcp arrangement and between and √2 for a square closely packed arrangement); however, for more disordered structures with local variations in the symmetry and spacing (i.e. structures characterized by the coexistence of different patterns with a different number of nearest neighbours or with variable inter-particle distance), the choice of the cut-off value can become more critical. To circumvent the issue of the choice of the cut-off radius, the number of nearest neighbours can be computed starting from the Voronoi diagrams as the number of sides of the Voronoi cell associated to a given particle [40,41]; yet, this method is sensitive to small perturbations and inaccuracies in the computed positions of the particles, thereby frequently resulting in an overestimation of the number of nearest neighbours and failing to capture dominant features of a given arrangement [41]. An approach has been proposed for the evaluation of nearest neighbours in dendritic nanostructures, which discards from the computation of the nearest neighbours those corresponding to a side of the Voronoi cell below a certain fraction of the total perimeter of the cell A& [41]. In our case, we have adopted the modified Voronoi approach, including in the computation of nearest neighbours only those particles that are associated to a side above or equal to a variable threshold A& ⁄2%%&(5G* where %%&(5G* is the number of sides of the original Voronoi cell. The morphology of the assemblies as well as the number of nearest neighbours is easily visualized using the Voronoi diagram as depicted in Fig. 2 (right column), in which cells have been coloured according to the number of nearest neighbours computed with the modified Voronoi approach. As apparent, without S 10 ACS Paragon Plus Environment

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particles, the dominant cell type is a hexagon, resulting in a honeycomb pattern. For increasing concentration of S particles, the morphology departs from the regular honeycomb pattern and different cell shapes are present; besides distorted hexagons, other polygons are present, in particular pentagons and quadrilaterals. Fig. 3(a) reports the percentage of L particles with 4, 5 and 6 L nearest neighbours for variable ⁄ : the analysis confirms that a coordination number of 6 is dominant for ⁄ = 0, with only fractional presence of particles with a different coordination number, mainly due to grain boundaries (e.g. resulting in alternating particles with 5 and 7 nearest neighbours), vacancies (missing particles), impurities (e.g. bigger particles due to the non-perfect monodispersity), whereas, for higher ⁄ , almost half of the particles tend to be surrounded by 5 particles, with a simultaneous decrease in the number of particles with a coordination number of 6 and an increase of those with a coordination number of 4. The so-called entropy of conformation +, , used in previous studies to investigate breath figures [42-44], provides insight into how a given morphology departs from a pattern made up of L particles surrounded by the same number of nearest neighbours; in fact, if all the particles are surrounded by the same number of nearest neighbours (/011 = 1 for %% = and 0 otherwise), then +, = 0; otherwise +, > 0. In our assemblies +, is low in the absence of S particles; a steep increase in +, is observed when the number of S particles is half of the number of L particles followed by a steady and slower increase for higher concentrations of S particles (Fig. 3(b)).


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Figure 2 – Binary assemblies for variable small particle concentration. Left column: SEM micrographs for variable ⁄ (scale bar = 2 μm); right column: corresponding Voronoi diagrams of L particles with cells coloured according to the number of nearest neighbours %%& computed with the modified Voronoi approach (yellow: %%& = 3; red: %%& = 4; cyan: %%& = 5; blue: %%& = 6; magenta: %%& = 7).

Figure 3 – Coordination number and entropy of conformation of L particles in binary assemblies: (a) percentage of L particles with a number of L nearest neighbours %%& equal to 4, 5 and 6 for variable ⁄ ; (b) entropy of conformation of L particles for variable ⁄ .

Pattern identification in binary assemblies: qualitative analysis As ⁄ increases, the hexagonal pattern is gradually replaced by other morphologies. We have first carried out a qualitative identification of the morphologies recurring for the different compositions of binary assemblies. Fig. 4(a-c) illustrate examples of cells commonly encountered in our assemblies, i.e. hexagonal, square and rhombic patterns. It is noteworthy to observe that, although all such patterns exhibit the potential for a periodic packing, except for the case of absence of S particles, where a hexagonal periodic packing is dominant, no other morphology prevails to the same extent in other configurations; on the contrary, the coexistence of different types of cells in more random arrangements is typically observed as highlighted in Fig. 4(d-k).

Figure 4 – Patterns of colloidal assemblies: (a) hexagonal; (b) square; (c) rhombic; (d-k) coexistence of hexagonal, square, rhombic patterns; (l) hexagonal LS1; (m) transition between square and hexagonal; (n) transition between square and rhombic; (o) transition between squares with different orientation.

The hexagonal arrangement is dominant in the absence of S particles, while it gradually reduces for increasing S particle concentration. Square cells start to appear for ⁄ = 0.5, are encountered more frequently for ⁄ = 1 and persist also for ⁄ = 1.5, while they tend to occur just in single unit cells composed of four L particles and 13 ACS Paragon Plus Environment

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more seldom for ⁄ = 2; such a configuration is characterized by a closely packed arrangement for L particles, with one S particle located in each of the interstices (even if S particles can sometimes be missing); it resembles closely the square S1 morphology predicted in [9] for ⁄ = 0.414 and ⁄ = 1 with a disc approximation; however, no distortion into a non-closely-packed square arrangement is found due to the fact that, as previously observed, the S particles can sit on the L particles. We should observe that this square pattern occurs in arrangements exhibiting the same orientation that typically involve a limited number of L particles, even if, for ⁄ = 1, several square arrangements occur in spite of their different orientation. Also a rhombic packing is a potential candidate for a periodic regular packing: in this case, two S particles are located in the gap between four L particles arranged in a rhombus; the unit cell of this configuration involves the presence of one L particle and two S particles, even if it does not generate a truly periodic pattern but several arrangements with different orientations (typically along one spatial direction and more rarely in small clusters appearing as sheared hexagons); sometimes such rhombi also appear distorted. This geometry resembles the triangular T1 morphology predicted for ⁄ > 0.155 [9]. Even if other unit cells with the potential of a periodic space filling have been found (for instance, the one depicted in Fig. 4(l) known as LS1 according to the notation commonly used in the regime of hexagonal symmetry-preserving binary assemblies consisting in an L particle surrounded by six L particles with centroids placed at the vertices of a regular hexagon and S particles present in alternating interstices between L particles), typically they occur much more occasionally in comparison to the previously discussed ones. More interesting for our analysis due to their repeated occurrence are those structures that result from a transition between the different most frequent cells (in particular hexagonal and square or rhombic) (Fig. 4(m-n)), or also between two cells of the same type (for example both square) but characterized by a different orientation or a shift (Fig. 4(o)). As we will see, frequently Voronoi cells associated to such transitions are pentagonal; as binary arrangements are characterized by the coexistence of hexagonal, square and rhombic symmetry, in combination with geometries of transition between them, this provides a qualitative justification of the predominance of the coordination number 5 in the presence of S particles. Furthermore, for increasing S particle concentration, we observe the presence of larger voids between L particles, that tend to be filled by clusters of S particles. In the following sections, a more quantitative analysis is presented to verify our observations based on a qualitative inspection of our assemblies. Pattern identification in binary assemblies: pair correlation function, inter-particle distance, packing fraction The distances between particles in arrays exhibiting perfect hexagonal close packing or square close packing are exactly defined. Hence, as a next step, in order to get a comprehensive picture of the dominance of specific symmetries, we have computed the pair correlation function which quantifies the probability of finding a particle at a distance ) from a given particle relative to the case of randomly distributed particles. For a random arrangement, '()* approaches 1 for distances above the particle diameter; for particles arranged in a regular pattern, '()* tends to exhibit peaks for some specific values corresponding to the most probable inter-particle distances. Such a tool has been used for the analysis of the order of colloidal assemblies with a nearly hexagonal pattern under different experimental conditions [1], in particular for self-assembly at air/water interface of single-sized colloids [37,45,46] and of binary colloids in the regime of preservation of hexagonal order for L particles [7]. In this case, we apply it to the more complex regime of loss of hexagonal order for L particles characterized by the coexistence of different symmetries. Fig. 5(a-b) report examples of the pair correlation function '()* computed from the centroids 14 ACS Paragon Plus Environment

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of L particles for the different ⁄ values and compared against the theoretical positions of an ideal hcp pattern and an ideal closely packed square pattern (pointed out as black lines). As visible, without S particles, the peaks in the pair correlation function are sharp and close to the expected positions for an hcp pattern; for increasing ⁄ , the individual peaks tend to become broader and merge; more interestingly, new peaks appear corresponding to the emergence of new patterns. In particular, we can observe the presence of a peak at lower distances visible already at ⁄ = 0.5 and getting progressively more pronounced; such peak corresponds to the emergence of square and rhombic arrangements. In fact, the second peak predicted for a rhombic arrangement is sensitive to the angle O of a rhombic arrangement and to the inter-particle distance , i.e. the minimum distance between the centers of two L particles (see the inset in Fig. 5(c) for a clarification of the notation): assuming an angle O in the range P60°, 90°P, it will be at √2 − 2 cos O (degenerating to √2 for O = 90° , i.e. a square pattern) (Fig.

5(c)); for 60°, the arrangement ends up in a hexagonal pattern (where the second peak is located at √3) (Fig. 5(d)); is assumed to be variable in order to take into account both the non-perfect monodispersity of the particles and the possibility of a non-close packing. The coexistence of several different morphologies for increasing concentration of S particles and the small extent of repetition of the same unit cell with the same orientation as well as the variability in O for rhombic patterns prevent us from using the position of higher peaks as a clue to the identification of the most recurrent rhombic morphology; the transition to non-closely-packed arrangements represents an additional factor that hinders an exact identification. As evident in Fig. 5(a-b), higher peaks become decreasingly discernible for raising ⁄ ratio. Nonetheless, the comparison of the second peak position in the pair correlation function with the computations of Fig. 5(c-d) confirms a trend compatible with a scenario of presence of a gradual transition from hexagonal symmetry to coexistence of hexagonal patterns with square and rhombic ones. As mentioned and as apparent from SEM images, for increasing concentration of S particles, L particles tend to become more spaced. The presence of voids and larger clusters of S particles turn into an increase in the mean value of the distance between nearest neighbours for L particles which varies approximately from 510 nm to 590 nm, by switching from a regime of absence of S particles to a number of S particles twice as much as the one of L particles; a decrease in the packing fraction (computed only for L particles) is correspondingly observed (Fig. 5(e-f)). Although the computed values of the packing fraction are compatible with the theoretical values computed for a hexagonal pattern of particles of 500 nm diameter and average distance equal to those computed for the different ⁄ values (Fig. 5(g)), the previous analysis of the coordination number and the increase in the variability of the distance between nearest neighbours for increasing particle ratio (from about 40 nm to 100 nm) suggest that indeed the computed packing fraction for higher ⁄ values is the result of the combination of different patterns (hexagonal, rhombic, etc.) with variable inter-particle spacing (i.e. more or less close one to the other) rather than the offspring of the dominance of a hexagonal pattern with an inter-particle spacing much larger than the nominal particle diameter; on the contrary, in case of absence of S particles, the hexagonal pattern is actually dominant and the packing fraction value is very close to the ideal one, taking into account the effect of defects and polydispersity occurring in the experimental self-assembly process. These considerations are confirmed by the analysis of the shape factors reported in the next paragraph.


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Figure 5 – Pair correlation function, average inter-particle distance and packing fraction for L particles in binary assemblies: (a) pair correlation function of L particles for variable ⁄ compared against ideal peak positions for a periodic closely packed hexagonal arrangement (black lines); (b) pair correlation function of L particles for variable ⁄ compared against ideal peak positions for a periodic closely packed square arrangement (black lines); (c) theoretical position of the second peak in the pair correlation function for a rhombic pattern for variable O and for variable inter-particle distance normalized to the nominal value of ; (d) theoretical position of the second peak in the pair correlation function for a hexagonal pattern for variable inter-particle distance normalized to the nominal value of ; (e) packing fraction for L particles in binary assemblies for variable ⁄ (the dotted lines indicate the

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values of the packing fraction for closely-packed hexagonal and square arrangements); (f) average L inter-particle distance in binary assemblies for variable ⁄ ; (g) theoretical packing fraction for variable O and for variable interparticle distance .

Pattern identification in binary assemblies: shape factors and Minkowski structure metric As observed in the previous paragraphs, binary assemblies are characterized by a variability in the interparticle distance, which prevents a precise identification of dominant patterns based on the pair correlation function or on the packing fraction. In order to get a quantitative insight into the patterns originated in binary assemblies, we have investigated the shape factor distributions of the Voronoi cells for the different particle ratios ⁄ . The shape factors I(&* are dimensionless quantities describing the shape of the cell, independently of its size (which is sensitive to the inter-particle distance), and are defined in such a way that I(&* = 1 for a circle and I(&* > 1 for any other polygonal shape. For regular polygons (e.g. equilateral triangles, squares, regular pentagons, hexagons and heptagons), the shape factor can be expressed as a function of the number of sides W as

% ; X YZ[(X⁄%*

for example one gets I(&* = 1.2732 for a square Voronoi

cell, I(&* = 1.1563 for a regular pentagon, I(&* = 1.1027 for a regular hexagon and I(&* = 1.0730 for a regular heptagon. Rhombic patterns are characterized by a distorted hexagonal cell, that degenerates in a regular hexagon for O = 60° and in a square cell for O = 90°, with shape factors included in the range between the two extremes. As each pattern encountered in a colloidal assembly results in a Voronoi cell of a given shape and, hence, a certain I(&* , the analysis of the shape factor distributions can shed light on the dominant patterns and structural transitions occurring for specific experimental conditions. As such, it has been used to investigate crystallization dynamics and structural transitions of granular assemblies of singlesized spherical particles and disks for variable particle densities both theoretically and experimentally [47,48] and to study the behaviour of single-sized microgel particles under variable temperature [49]. Here we apply this analysis technique to investigate the arrangements of L particles in binary colloidal assemblies with variable composition. Figure 6(a) shows the shape factor distribution for increasing ⁄ . As visible, without S particles, a sharp maximum is observed close to the value expected for a regular hexagon; for ⁄ = 0.5, the primary maximum loses its sharpness and exhibits a shoulder located towards higher values; a secondary maximum at about 1.175 appears; for ⁄ > 1, a bimodal distribution is observed, with a dip roughly at 1.159 very close to the shape factor value of a regular pentagon with a gradual shift upwards for the distribution for higher concentrations of S particles accompanied by a broadening and smoothening of the overall shape. Interesting considerations can be drawn by comparing our distributions obtained for binary assemblies with the trends observed in a single-sized granular system with different levels of dilution, where a bimodal distribution was observed for packing fractions higher than 0.65, approaching a monomodal distribution for increasing packing fraction, i.e. as the arrangement tends to a regular hexagonal pattern [47,48]; the good qualitative agreement can be explained by reckoning that the eventual effect of S particles is that of making the overall assembly of large particles more dilute, as previously observed in the computation of the packing fraction.


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Figure 6 – Shape factor analysis for L particles in binary assemblies: (a) probability distribution of shape factors; the probability distribution for ⁄ = 0 has been divided by a factor 5 for the sake of a better comparison with distributions attained for other ⁄ values; coloured crosses on the top of the graph correspond to the values of the shape factor for regular polygons; (b) Voronoi diagrams of SEM images in Fig. 2 with cells coloured according to


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the shape factor class; (c) percentage of shape factors belonging to classes A, B, C for variable ⁄ ; (d) shape factors computed for hexagonal, rhombic and square patterns; (e) examples of Voronoi cells associated to transition patterns.

As a further step, for the sake of a more precise characterization and comparison with single-sized assemblies with different levels of dilution, we have grouped particles into three classes, i.e. class A cells (exhibiting I(&* ≤ 1.159 below the dip of the bimodal distribution), class B cells (with 1.159 < I(&* ≤ 1.25) and class C cells (having I(&* > 1.25); the upper bound 1.25 was selected as the value for which the population of class B cells was equal to the population of class C cells according to Monte Carlo simulations [47]. Figure 6(b) illustrates examples of Voronoi diagrams computed for the SEM images of Figure 2 and with cells coloured according to the belonging to a specific class. As visible, in the absence of S particles, class A cells with nearly regular hexagonal shape are dominating. Already at ⁄ = 0.5, class A cells tend to deviate from the regular hexagonal shape and class B cells start to appear accompanied by a marginal presence of class C cells; for increasing S particle concentration, class B cells increase to the detriment of class A cells and class C population becomes more significant. Figure 6(c) plots the fraction of cells belonging to the three classes for L particles in binary assemblies with variable composition; recalling, from the previous paragraph (Figure 5(e)), that the packing fraction of L particles plummets from roughly 0.85 to 0.65 upon a variation of ⁄ from 0 to 2, the trend in the population of Voronoi cells belonging to the different classes for L particles in our binary assemblies is akin to the one of single-sized granular systems with variable packing fraction, where class A population decreased from about 100% to nearly 50% in that packing fraction range, with a corresponding increase in B population from 0 to nearly 50% and a gradual increase in C population [48]. An analysis of the dominating patterns linked to a given probability distribution of the shape factor has not been carried out in previous studies. Even if a systematic association is not viable, in an attempt to provide a correlation with the morphology of our binary assemblies, we have proceeded to identify patterns in our visual inspection of binary assemblies that can potentially fit the observed trends in the probability distribution of the shape factors. As noticed, the increase in S particles results in a transition from regular hexagonal (corresponding to regular hexagonal Voronoi cells) to rhombic (distorted hexagonal Voronoi cells) and square patterns (square Voronoi cells). Figure 6(d) sketches the Voronoi cell of a generic rhombic pattern that degenerates in a hexagonal cell for O = 60° and in a square cell for O = 90° and reports the shape factor computed for variable O; in a first approximation, we could classify as nearly hexagonal patterns those with 60° ≤ O ≤ 67.5°, as nearly rhombic those exhibiting 67.5° < O < 82.5° and as nearly square those with 82.5° ≤ O ≤ 90°. The computed values span through the three different shape factor classes (the plot has been splitted in sections of different colours according to the class to which they belong); their comparison with our probability distribution could confirm a transition towards rhombic and square patterns with increasing ⁄ . In addition, we should remember that square and rhombic patterns in our assemblies are typically repeated to a small extent in arrangements involving few particles, which means that they do not often give rise to the Voronoi cell typical of a periodic rhombic or square packing; for example at least 3x3 particles in a square arrangement would be necessary to result just in one square Voronoi cell associated with the central particle, while the other eight particles located at the periphery would give rise to cells of different shapes depending on the neighbouring patterns. Hence, this would lead to an inherent underestimation of rhombic and square patterns. As we noticed, some transition patterns are recurrent in our assemblies, for example square-to-hexagonal transition, as in the first sketch of Figure 6(e), or transition between square symmetries with different orientation, as depicted in the second sketch, or between square and rhombic, as in the third sketch, or between rhombic cells with different orientation, 19 ACS Paragon Plus Environment

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as in the fourth sketch. Such transitions typically result in distorted hexagonal cells or pentagonal cells with shape factors falling in the different classes. For instance, the first three cells belong to class B and the forth one to class A and they exhibit values close to the peaks of the probability distributions. Hence, a good qualitative correlation can be found between the visual morphological inspection and the overall trends of the shape factors distribution. In order to get better quantitative insight into the symmetry of L particles in our binary assemblies using an approach not sensitive to inter-particle distance, we have computed the distribution of the absolute value (&*

(&*

of the Minkowski structure metric =3>@ = and =3$@ = for each value of ⁄ (Figure 7) and the distribution (&*

(&*

of the absolute value of the bond orientational order parameters =3> = and =3$ = (Figure S1). We point

out that both the sets of parameters cast light on the orientational order of a colloidal assembly; however, (&*

according to its definition, the bond orientational order parameter 30 is affected by the aforementioned (&*

ambiguity in the definition of the nearest neighbours %%& , while the Minkowski structure metric 30@ circumvents this issue by considering all the %%&(5G* particles that surround particle ? and result in a side

of the Voronoi cell associated to particle ?, but gives them a different weight according to the length B&9 of the side of the Voronoi cell (originated by the bond between particle ? and the surrounding particle H) relative to the total perimeter A& of the Voronoi cell associated to particle ? [40,50]. For the computation of (&*

the distribution of =30 =, we used two different definitions of the nearest neighbours, one based on the

original number of sides of the Voronoi cell (i.e. %%& = %%&(5G* ) and the other according to the modified approach, that discards sides below a certain fraction of the Voronoi cell diameter A& .

(&*

(&*

Figure 7 – Probability distribution of Minkowski structure metric _3>@ _ and _3$@ _ for L particles; the probability distribution for ⁄ = 0 has been divided by a factor 5 for the sake of a better comparison with distributions attained for the other ⁄ values. (&*

(&*

For ⁄ = 0, the distributions of the Minkowski structure metric =3>@ = and =3$@ = are essentially

monomodal with a sharp peak close to 1 for = 6 and close to 0 for = 4. For ⁄ equal to 0.5 and 1, (&*

both the distributions tend to become broader: the peak of =3>@ = gets shifted to higher values and exhibits 20 ACS Paragon Plus Environment

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(&*

a shoulder, while the distribution of =3$@ = presents two peaks and a shoulder; for even higher ⁄ (&*

values, the distribution of =3>@ = tends to become monomodal and the peaks and shoulder in the (&*

distribution of =3$@ = get fainter. This trend marks the conversion from an arrangement with a dominant

hexagonal symmetry to assemblies characterized by the coexistence of different symmetries (hexagonal, square, rhombic) and transition geometries, which tend to become more disordered as the concentration of small particles increases. Features of transition patterns can also be recognized: for example, the transition geometry reported in the first sketch of Figure 6(e), frequently encountered especially for ⁄ (&*

equal to 0.5 and 1, is compatible with the shoulder observed in the probability distribution of =3>@ = and (&*

(&*

with the peak slightly close to 0.5 in =3$@ = for these ⁄ values. A comparison with =30 = computed with the Voronoi approach and with the modified Voronoi approach reveals that, in the former, peaks and (&*

shoulders are generally less visible (for example the distribution of =3> = is always monomodal), whereas, (&*

in the latter, they appear more distinct. The good qualitative agreement between =30 = computed with the (&*

modified Voronoi approach and =30@ = suggests that the modified approach gives also a more reasonable

estimate of %%& and catches more closely the overall symmetry of the assemblies. For instance, (&*

fingerprints of transition patterns can be more easily recognized in the probability distribution of =30 = computed with the modified Voronoi approach: referring, e.g., to the transition geometry in the first sketch of Figure 6(e), we observe the correspondence of this pattern with the second peak in the distribution of (&*

(&*

=3> = and with the peak close to 0.6 in the distribution of =3$ = for ⁄ equal to 0.5 and 1; such (&*

features appear hidden in the probability distribution of =30 = computed with the Voronoi approach. Pattern identification in binary assemblies: persistence diagrams

The analysis based on Voronoi tessellation, shape factors, Minkowski structure metric and bond orientational order parameters poses some challenges for the investigation of disordered colloidal assemblies as is the case for our binary assemblies. As previously observed, while truly periodic patterns, consisting, e.g. of hexagonal, rhombic or square arrangements repeated periodically over large areas, give (&*

(&*

rise to cells with clearly identifiable characteristics and values of I(&* , =30@ = and =30 =, patterns with a lower degree of order exhibit coexistence of different unit cells with a low extent of repetition and (&*

transition patterns from one geometry to the other which can be very diverse in terms of I(&* , =30@ = and (&*

=30 =. Although, as previously highlighted, some transition geometries are more recurrent and give rise to

specific features, a more systematic identification and quantification is not practical. This prevents also the recognition of the dominance of specific patterns: for instance, as mentioned, arrangements of 3x3 particles in a square pattern would result in only one square Voronoi cell and other eight cells with diverse shape depending on the neighbourhood and transition patterns; an arrangement of 2xM particles in a square pattern would result in no square Voronoi cell; therefore, the quantification of occurrence of square patterns would not be viable. To this end, the detection and implementation of analysis tools able to recognize features with a lower extent of periodicity and repetition is essential. Persistent homology (PH) has recently been used to investigate topological features of granular media and colloidal assemblies, for example to investigate the influence of the effective temperature on the 2D 21 ACS Paragon Plus Environment

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assembly of paramagnetic colloids or to analyse order-to-disorder transitions in 3D granular crystallization or to study the disorder of colloidal assemblies formed by spin coating on patterned substrates [50-53]. A detailed description of the theory behind PH is beyond the scope of this article and can be found in [54,55]. PH is constructed on building blocks, called H-simplices, with H being the dimension of the simplex, which consists of points (H = 0), edges (H = 1), triangles (H = 2), etc. Here we provide an intuitive, even if not rigorously mathematic, picture of its use for the analysis of our colloidal assemblies. Let us consider a set of points on a plane (e.g. a point cloud made up of the L particle centroids). One can start building a network of connections (edges) between these points; the variable parameter for the construction of such a network is the length of these edges B; as long as the edge length is below the minimum distance between the points (e.g., for the case of L particles, the L particle diameter ), the points cannot be connected; once this value is reached, some of the points will start being connected, giving rise to triangles or other polygonal shapes, for example quadrilaterals; as the edge length B is increased further, non-triangular holes will start to disappear as they will be covered by triangles. Persistence diagrams in dimension H=1 show how such holes form and disappear as B varies, by plotting in the abscissa the value of the variable edge length B for which they are generated (birth) and in the ordinate the value of B for which they disappear (death). For example, a square hole formed by four closely packed particles of diameter “is born” when B = (as the four vertices are all connected) and “dies” when B = √2 (as the two diagonals will connect the two couples of non-connected points); more in general for

a rhombus of angle O, the birth will occur for B = and the death for B = √2 − 2 cos O. Therefore, the representation of a rhombic pattern of closely packed L particles in a persistence diagram will consist in a point of coordinates à& , & b = ` , √2 − 2 cos Ob. In this way, the persistence diagram can be used to discriminate dominant patterns in a given set of points. Figure S2 shows the persistence diagrams for hexagonal, rhombic (for O = 75°) and square patterns for increasing level of noise; the colour scale for each ⁄ value indicates the frequency of occurrence of each couple of coordinates of birth and death

normalized to the peak: as visible, the initial narrow distribution of points around ` , √2 − 2 cos Ob spreads for an increasing level of noise, while still preserving the dominant character. Birth and death coordinates are normalized to . We have computed the persistence diagrams for L particles in our binary assemblies and variable particle number ratio ⁄ (Figure 8(a)). As visible, without S particles, the point distribution is very close to a low-noise hexagonal pattern, with a small dispersion close to ( , * that is due to defects occurring in a real assembly experiment due to particle polydispersity, presence of voids or defects (e.g. heptagonal/pentagonal defects). For ⁄ = 0.5, beside the highly populated spot close to ( , *, a

second area of population of the diagram in the proximity of ` , √2b is present, which denotes the appearance of a nearly square pattern close to the hexagonal one; such a population corresponding to a nearly square distribution further increases for ⁄ = 1, with a corresponding reduction in the nearly hexagonal population. Upon a further rise in the concentration of S particles, an overall broadening in the distribution of points is observed, accompanied by a shift towards higher birth and death values, in agreement with the observed increase in the mean value and standard deviation of the inter-particle distance, due to the more dilute nature of L particles for rising ⁄ . While for ⁄ = 1.5 a main

component close to the line ` , √2 − 2 cos Ob is still observed, indicating the prevalence of the coexistence of hexagonal, square and rhombic patterns with a nearly closely packed arrangement, for


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⁄ = 2 the longer and broader tail towards higher values of à& , & b indicates the less regular and less closely-packed arrangement for L colloids. We have used PH to quantify the fraction of holes with a nearly-hexagonal, nearly-rhombic and nearlysquare closely-packed arrangement by counting the frequency of holes falling in regions around ` , √2 − 2 cos Ob with O = 60° (nearly hexagonal), O = 75° (nearly rhombic) and O = 90° (nearly square) (regions enclosed by white edges in Figure S2). Figure 8(b) reports the plots for variable ⁄ : the fraction of hexagonal arrangements plunges quickly for increasing S particle content, while the fraction of square arrangements quickly increases until reaching a maximum above 40% for ⁄ = 1 before declining again for higher ⁄ values. As previously observed, square packing has been theoretically predicted for ⁄ = 0.414 and ⁄ = 1 [9] and appears dominant in our binary assemblies for ⁄ = 0.44 and ⁄ = 1. Rhombic patterns are present at a fraction between 10% and 20% for all number ratios; their frequency is comparable with the one of square patterns for ⁄ = 2. For ⁄ values up to 1, hexagonal, rhombic and square patterns account for more than 50% of the holes; this does not occur for higher ⁄ values. As observed, for increasing S particle concentration, patterns become less regular and more complex due also to the increasing frequence of S particle clusters. In comparison to methods for the identification of square patterns based on the computation of the bond orientational order parameter and on the bond length deviation, i.e. the variability in the distance between a particle and its nearest neighbours [14], the proposed approach is more robust with respect to the definition of the nearest neighbours and feasible for the identification of particle arrangements of different symmetries; moreover, it allows the identification of square patterns of L particles, even when the S particle is missing.


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Figure 8 – Persistent homology analysis: (a) persistence diagrams for L particles for variable ⁄ ; (b) percentage of nearly hexagonal, nearly square and nearly rhombic arrangements of L particles for variable ⁄ ; (c) persistence diagrams for S particles for variable ⁄ .

These trends are confirmed by the persistence diagrams of S particles computed for variable ⁄ > 0 (Figure 8(c)). For the sake of comparison, we have reported the persistence diagrams calculated for regular periodic arrangements of S particles in the interstices between L particles for different geometries (Figure S3). As visible, in our binary assemblies, points in the persistence diagrams of S particles are more broadly scattered in comparison to those obtained for L particles, with a large diversity in the birth and death coordinates, which suggests their less regular arrangement in comparison with L particles and the lack of prevalence of specific patterns. In fact, as we observed, even in case of regular arrangements of square or rhombic patterns for L particles, periodic patterns with a uniform orientation have a limited extension as typically tiny arrangements with different orientation coexist; this factor, in combination with the fact that S particles can even be missing within the regular arrangement of L particles (e.g. square arrangement of four L particles without S particle in the interstice) contributes to the more random character of the arrangement of S particles. However, some interesting considerations can still be drawn. For ⁄ = 0.5, despite the disordered appearance of S particle distribution (due to the fact that for such a low S particle concentration, the hcp pattern is still dominant and S particles start to appear either in random positions within the L particle interstices or in isolated square arrangements of L particles), it is noteworthy that the scattered spots are mostly located at birth coordinates a& > , which means that S particles are relatively distant one from the other and do not tend to form clusters. For ⁄ = 1, a main spot of higher concentration of population appears, corresponding to a nearly square pattern (Figure S3(d)); another area of concentration appears close to the diagonal at higher à& , & b coordinates; the area of the graph at birth coordinates a& < is still poorly populated. Upon a further increase in S particle concentration, the population at a& < starts to become more significant; the presence of clusters of S particles is apparent for ⁄ = 2, with the emergence of a clear spot close to a& = ; a partial shift of the square patterns towards more rhombic arrangements (resembling the type of Figure S3(c)) is observed. From this analysis, one can draw the conclusion that the pattern closer to the ideal square pattern of L particles with S particles placed in the interstices is achieved for ⁄ = 1 and that a significant presence of S particle clusters is observed at ⁄ = 2. In brief, the PH based analysis confirms the prevalence of more regular square arrangements for ⁄ = 1 not only for L particles, but also for S particles which tend to be more regularly present in the interstices between square L particle arrangements. It should be observed that PH analysis complements the aforementioned analysis techniques based, e.g. on Voronoi tessellation, shape factors and Minkowski structure metric: PH analysis can cast light on single repetitions of a specific pattern (which can be useful in more disordered arrangements), while the other techniques can give a picture of the order on a larger scale. Summary and Conclusions Colloidal assemblies formed at air/water interface and consisting of binary mixtures of L and S polystyrene particles have been studied for variable ratio of the number of small and large particles ⁄ in the regime of loss of hexagonal order for L particles.


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Such a regime poses more challenges from the point of view of the analysis in comparison to the regime of preservation of hexagonal order because traditional analysis tools are not sufficient alone for its investigation. Using an innovative approach based on the combination of different tools borrowed from computational geometry and computational topology, we have detected hidden symmetries and dominant patterns that coexist in the ostensibly more disordered arrangement of L particles. In particular, the construction of the Voronoi tessellation from L particle centroids and the evaluation of different metrics such as coordination number, entropy of conformation, shape factors, Minkowski structure metric and bond orientational order parameter have confirmed the transition from a hexagonal pattern in the absence of S particles to a greater variability in patterns for increasing S particle concentration. Hexagonal, rhombic and square arrangements coexist with a diverse variety of transition patterns between different types of symmetries, e.g. hexagonal-square transitions, rhombic-square or between rhombic or square patterns with different orientations. Due to the limits of computational geometry approaches based on Voronoi tessellation in detecting hexagonal, rhombic or square geometries in disordered patterns, persistent homology analysis has been carried out. We have performed the first experimental identification and quantification of patterns of different symmetries in nonmagnetic binary colloidal assemblies using this tool. Hence, our analysis based on the synergistic use of computational geometry and computational topology has demonstrated how dominant patterns and symmetries can be found even in more disordered binary assemblies and tailored by tuning the composition of the binary assembly. Colloidal particles are important building blocks or templates for the design of novel materials [1,56,57]. An accurate characterization of the morphology of colloidal assemblies can be beneficial in tailoring the properties of such novel materials. In fact, physical properties of colloidal assemblies are governed by two sets of parameters, i.e. those inherently related to the colloids (such as material, functionalization, shape) and those related to the morphology of the assembly (such as surface coverage, inter-particle distance, level of order, dominant symmetry) [58,59]. A correlation between physical properties and assembly morphology has been demonstrated for singlesized and binary periodic structures [3,60,61]; differences in the optical properties of plasmonic structures with square and hexagonal symmetry have also been observed and such plasmonic patterns are frequently obtained using colloidal particles as a template [1,62-64]. The introduction of disorder has a dramatic impact on the properties of a colloidal assembly: structural disorder can affect, for example, vibrational and optical properties of a colloidal material and, hence, for instance, its interactions with sound and light [2,4,28,29,58,65-67]. Restricting the attention to optical properties, different types of light-matter interactions have been pointed out for ordered systems, completely disordered systems and intermediate regimes [2]. Such effects can be related not only to the inter-particle distance, but also to the specific arrangements, which exhibit distinctive features and affect local and collective coupling effects [58,68]. Therefore, the demonstration of the possibility to tailor the geometry of binary colloidal assemblies to result in specific dominant patterns is of utmost relevance. Shedding light on the structural evolution and prevailing patterns upon variation in the concentration of S particles in comparison to L particles, even if still in a regime of coexistence of different symmetries, can be important in understanding the reason of the transition in physical properties of assemblies that could be classified as disordered at first sight without any further analysis of their morphology. In this way, a controlled form of disorder can be 26 ACS Paragon Plus Environment

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introduced to steer physical properties of such assemblies into specific directions for particular applications due to the intimate relationship between morphological structure and physical properties. As we pointed out, an essential role in understanding morphological properties of binary assemblies, especially the amorphous ones considered in our study, is played by the development of proper image processing and analysis tools. While this represents a general issue for the investigation of the structure of all the types of colloidal assemblies, even those exhibiting a well-defined prevailing symmetry, the problem is still more puzzling for more disordered assemblies. As we have seen, even the definition of nearest neighbours can be problematic in this case, while, for more ordered colloidal assemblies, different definitions substantially entail similar results. Only the simultaneous use of different analysis techniques can provide a more reliable and comprehensive description of the different features. The combined procedure developed in our study can be applied to the analysis of disordered colloidal assemblies not only via scanning electron microscopy, but also with other characterization tools such as atomic force microscopy, optical microscopy and scanning near field optical microscopy in both topographical or optical imaging mode thanks to advances in this field [12,18,45,69-75]. Using our robust quantitative approach, in the current study, we have focused our attention on binary colloidal monolayers spontaneously assembled at air/water interface without compression as this selfassembly approach has attracted the attention of scientists over the past years because of its relative simplicity and control on the final monolayers in comparison with other techniques [1]. As a rich variety of self-assembly approaches has flourished throughout the years [1], a possible future application could consist in the analysis of binary colloidal monolayers in the regime of lack of hexagonal order assembled with other self-assembly techniques exploiting different self-assembly mechanisms and harnessing different external forces, such as electric fields, as occurs in dielectrophoretic self-assembly, or compression, as is the case for air/water interface self-assembly in a Langmuir trough. For instance, in [76,77], a distortion of the hexagonal pattern into a rhomboedral structure has been observed for monolayers of single-sized particles at oil/water interface due to the anisotropic compression imposed by the Langmuir trough. More complex patterns could be expected in case of use of anisotropic compression of binary colloidal monolayers by varying the surface pressure. The resulting patterns obtained by this and other different techniques could, hence, be scrutinized using our comprehensive analysis approach and contrasted with those examined in the current study. Our innovative pattern analysis approach represents a powerful analysis tool for the investigation and a comprehensive and quantitative characterization of different coexisting patterns in complex colloidal assemblies obtained under different experimental conditions and can be used to accurately determine the morphology of variegated and involved colloidal systems in order to better understand colloidal assemblies. In this sense, with our work we have not only contributed to detect and describe the hidden symmetries in disordered binary assemblies, but also indicated a route for the investigation of colloidal assemblies of different types and different levels of disorder.


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Supporting information Bond orientational order parameters, persistent homology analysis of simulated binary arrangements. Acknowledgements This work was supported by the Swiss innovation promotion agency KTI-CTI (Contract 14336.1 PFNM).We are grateful to Stephen Wheeler and Martin Lanz for their technical help.


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Pattern formation in binary colloidal assemblies: hidden symmetries in

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