Anisotropic functionalized platelets: percolation, porosity and network properties

Anisotropic functionalized platelets are able to model the assembly behaviour of molecular systems in two dimensions thanks to the unique combination of steric and bonding constraints. The assembly scenarios can vary from open to close-packed crystals, finite clusters and chains, according to the features of the imposed constraints. In this work, we focus on the assembly of equilibrium networks. These networks can be seen as disordered, porous monolayers and can be of interest for instance in nano-filtration and optical applications. We investigate the formation and properties of two dimensional networks from shape anisotropic colloids functionalized with four patches. We characterize the connectivity properties, the typical local bonding motives, as well as the geometric features of the emerging networks for a large variety of different systems. Our results show that networks of shape anisotropic colloids assemble into highly versatile network topologies, that may be utilized for applications at the nanoscale.

For calculating certain network properties only the bonding information of the particle network matters, while physical properties of the bonded particles -the particle positions and orientations and bonding angles -can be ignored.Thus, the aggregate can be treated as a non-physical network, where each particle becomes a network node and each bond between two particles becomes an edge connecting two nodes.With this network representation we calculate the bond degree distribution, the percolation line and the percentage of particles within network loops of a certain size.We obtained these network properties with our own software -partially published here [1] -that relies on Pythons Networkx module [2].

Network degree distribution
We used Networkx' degree() method to return the list of node degrees and subsequently calculated the degree distribution, the average degree and its standard deviation.The average bond degrees of all particle types are featured in Fig. 7 and the bond degree distribution in Fig. 8 within the main text.

Percolation line
A percolating cluster is defined as cluster of bonded particles that extends beyond the periodic simulation box to infinity in at least one direction.To test for percolation in practice, we calculate the largest particle cluster over all nine periodic images of the simulation box and define the cluster as percolating if more than 0.33 of all particles are in this largest cluster.We chose this threshold of 0.33, because in that case the percolating cluster spans across the periodic images in at least one direction.To calculate the largest cluster of all periodic images, we first enhance the network to include the particles and bonds -i.e., the nodes and edges -of all nine periodic images.Note that, to build this large network efficiently, the only additional information needed is the connectivity between two periodic images, as for non boundary particles we can just copy the edge connectivity from the original simulation box.We calculate the largest cluster of this network of periodic images by calling Networkx' method connected components(), which returns all connected components, i.e., all clusters, as lists of nodes; the largest of these lists, is then the size of the largest cluster.If the relative size of this largest cluster of all periodic images, R N * 9 -where R is the size of the largest cluster, and N is number of particles within the original simulation box -is larger than than 0.33 the cluster is defined as percolating.We define the percolation locus for each density as the temperature where configurations in at least half of the parallel runs contain a percolating cluster; The percolation line is the line that connects the percolation points across all densities.See Fig. 6 in the main text for a depiction of the percolation lines for all studied systems.The first step in pore analyses must be the identification of the pore space -i.e., the area of the simulation box occupied by pores, where a pore is defined as an isolated subspace surrounded by particles.For this purpose, we developed a volumetric-element (voxel) based method, where the simulation box is split into small squares (voxels) of a side length l v , where l v is significantly smaller than the side length of a rhombus particle l r .We chose l v ≈ 0.25, as this size enables us detect the smallest pores, while still allowing us to evaluate the pores for all state points efficiently.Here it is important to note that l v varies slightly for different packing fractions, with an estimated variation of ≈ 0.005 as the voxels always need to tile space.The voxels are arranged in a square lattice, with eight neighbours per voxel, where four neighbours share an edge and four share a corner.Periodic boundary conditions apply and next neighbours of voxels at the simulation box edge are given through minimum image convention.For each voxel vx i , we estimate the area occupied by rhombus particles.Here, cell-lists ensure efficiency in calculating potentially overlapping rhombi rh j , where rhombus-center-to-voxel-center distances are small enough for the shapes to possibly overlap.For each of these overlap candidates, we estimate the area of overlap a ij by randomly drawing m = 100 points from the voxel vx i and checking if they overlap with a given rhombi rh j .This overlap-detection procedure is repeated for all neighbouring rhombi of voxel vx i , and the in total occupied area for vx i is obtained by summing up a i = N bi j=0 a ij where N b i is the number of overlap candidates for voxel vx i .The given voxel vx i is set to a filled state if the occupied area ai exceeds a threshold t occ = 0.25 and remains in an empty state otherwise.
Note, that we evaluated the pore space also for m = 500 and m = 1000 sample points instead of m = 100, and also for different overlap thresholds t occ = [0, 0.1, 0.25, 0.3] and found no significant difference.
The pore space can be finally calculated by forming a network of empty voxels, where every empty voxel is a node and every connection with another empty neighbouring voxcel is an edge; from this empty voxel network we obtain the pores as lists of voxels by invoking Networkx's connected components() method.The snapshots in Fig. 9 in the main text highlight the pores in different colors, while the insets show voxel space with all voxels framed in blue.

Pore area
From the voxel method, we obtain the pores as lists of voxels and the pore area (pore size) distribution can be obtained by calculating the length of these list and multiplying it with the voxel area a v = l 2 v ≈ 0.625.For Fig. 10 and 11 in the main text, we further divide each pore area A p by the rhombus area A r = 0.866025, as this gives us the pore area in units of the particle area.

Pore circumference
The circumference Cf m of a pore p m built from K voxels vx k can be obtained by summing up all exposed edges per voxel, where N edge,k is the number of edge-to-edge voxel neighbours of voxel vx k within that pore, l v is the side length of a voxel and l r is the particle side length.The pore circumference of all particle types at patch position ∆ = 0.2 and temperature T = 0.01 can be appreciated in Fig. 10, Panels (a)-(d).

Pore asymmetry measures
To estimate the asymmetry of the pores we use two measures, with one measuring the convexity and the other one the elongation of the pores.
In contrast to the pore area and the circumference, that were evaluated algebraically, for the pore asymmetry measures the actual pore shape matters.Therefore, as pre-evaluation step, pores that extend beyond the periodic boundaries, must be stitched together.We perform the stitching of a boundary pore p m by randomly selecting a voxel n within the pore, and then chose the periodic images of all other voxels to be closest to that voxel.
As this method breaks when the extent of a pore grows beyond half of the simulation box, we restrict ourselves to pores smaller than 2000 voxels.
To evaluate the convexity of a pore p m , we compare the actual pore area A p,m , to the area of its convex hull A convex,m .We calculate the convex hull A convex,m of a pore p m by forming a point cloud of center and corner positions of all voxels contained in that pore and then use this point could as input calculation of the convex hull with the qhull package [4,5].We obtain the convexity measure, A p,m /A convex,m as a number between 0 and 1, where values close to 0 indicates highly non-convex pores, while 1 represents convex pores.Generally, pores with a high convexity tend towards elliptic or circular shapes, while pores with a high convexity exhibit many dents and fingers.
To evaluate the pore elongation we perform a principal component analysis (PCA) with the aid of Pythons Machine Learning package scikit-learn [6].The input data for each pore p m is the point cloud of all voxel positions and corners contained in p m .We use the percent of explained variance of the first principal component, p P CA1 , as measure for the pore elongation: if p P CA1 is close to 1, most of point-cloud's variance can be explained by the first -the dominant -component, i.e the clusters are very elongated, while if p P CA1 is closer to 0.5, the point-cloud's shape is more spherical.
In The order parameter 1 − 2P np , with P np as the fraction of non-parallel bonds, indicates the average bonding affinity of the system, where a value of -1 represents system with only non-parallel bonds and 1 designates a completely parallel assembly.While the main text discusses the bonding affinity of particle topologies with ∆ = 0.2 (see main text, Fig. 7, Panels (a)-(d), second row), here in the Supporting Information we show the results for ∆ = 0.3 in Fig. 1 and for ∆ = 0.4 in Fig. 2. For dma-as1, dmo-as1, and dmo-s2, we observe similar trends as in ∆ = 0.3 and ∆ = 0.4 as we do for ∆ = 0.2: For dma-as1 (Panel (a) of Figs. 1 and 2), we observe a dominance in non-parallel bonding already for high temperatures, that becomes even more prevalent in the broad intermediate temperature range around the percolation point.Well below the percolation point, non-parallel bonding diminishes and we observe more mixed bonding.In dmo-as1 (Panel (b) of Figs. 1 and 2), we generally observe more mixed bonding througout all temperatures, with only a slight predominance of parallel bonds at higher temperatures.For dmo-s2 (Panel (d) of Figs. 1 and 2), we observe a slight tendency towards parallel bonding for high temperatures.For intermediate temperatures, this parallel tendency becomes stronger for ∆ = 0.3, while for ∆ = 0.4 it remains minor, and for low temperatures the bonding affinity becomes more mixed again for both ∆ = 0.3 and ∆ = 0.4.
For dmo-s1, on the other hand we observe a qualitative change in the bonding affinity for ∆ = 0.3 (Panel (c), of Fig. 1), and ∆ = 0.4 (Panel (c), of Fig. 2), compared to ∆ = 0.2.In the main text (see Fig. 7, Panel (c), scond row), we had observed for ∆ = 0.2 more mixed bonding for high temperatures that transforms to more non-parallel bonding for intermediate temperatures, and then back again to more mixed bonding at low temperatures, where the affinity towards non-parallel bonding is stronger for higher densities, where it peaks at -0.5 for intermediate temperatures.For ∆ = 0.3 the tendency towards non-parallel bonding becomes significantly weaker, especially for the high densities, with a -now diminished peak -peak at −0.2 for high densities.For lower densities the propensity to bind non-parallel is even even further diminished and at the highest density only a slight tendency towards non-parallel bonding is visible.For ∆ = 0.4, the tendency towards non-parallel bonding is only visible for higher temperatures, at intermediate temperatures, this tendency reverses, and parallel bonding becomes more prevalent instead.While this additional positive peak can we observed for all densities, it is strongest for low packing fractions, where it peaks at 0.3.At low temperatures, we again spot more mixed bonding across all densities.
B. Particle Loops for ∆ = 0.3 and ∆ = 04 In the main text we analyse particle loops, i.e loops within the network graph, for ∆ = 0.2 and find that in the vicinity of the percolation points, a majority of particles are bound within loops of size three to six (see Fig. 8 .Generally we find that for the dma-as1, the dmo-as1 and dmo-s2 system, the percentage and type of the dominant loops is qualitatively similar to ∆ = 0.2.More specifally, in the dmo-as1 system (Figs. 3 and 4, Panel (a)) non-parallel (np) particle loops of size three -boxes -make up the majority of present loops for all ∆-values and packing fractions.Their occurrence at high temperatures is density dependent, and for high densities they already occur to large amount at the highest temperatures.For intermediate temperatures near the percolation points boxes occur at 60-80% for all densities and even far below the percolation line, about 20% of particles remain bonded in boxes across all packing fractions.For both dmo-as1 (Figs. 3 and 4, Panel (b)) and dmo-s2 (Figs. 3 and 4, Panel (d)), we observe either parallel (p) or mixed (p-and np-bonds) six particle loops as the dominant loop motives; at high temperatures we detect that about 20% of particles are bonded these motives for high densities, with the percentage decreasing progressively as the density becomes lower; for intermediate temperatures, be observe up to 80% of p-bonded and mixed loops for all densities and for low temperatures this amount drops to 20% for all packings.In contrast, in dmo-s1, ∆ = 0.3 (Fig. 3, Panel (c)) and ∆ = 0.4-systems (Fig. 4, Panel (c)) differ qualitatively from the ∆ = 0.2 case.While for ∆ = 0.2, the dominant loop motives are 5-np and 6-np loops -so called 5-stars and 6-stars -for ∆ = 0.3 mixed (with p-and np-bonds) six particle loops become more prevalent than stars for all temperatures and densities, and for ∆ = 0.4 it is p-bonded as well as mixed loops that overshadow the occurrence of stars.At high temperatures, mixed and p-bonded/mixed loops occur at a significant percentage for higher densities for ∆ = 0.3 and 0.4 respectively, but remain insignificant for lower densities; at intermediate temperatures 60-80% percent of particles reside within mixed and p-boned/mixed loops across all densities, while for low temperatures their amount drops to 10-20%.To illustrate these qualitative changes further, we show the percentages of box loops, star loops, parallel loops and mixed loops for dmo-s1 ∆ = 0.2 (Fig. 5), ∆ = 0.3 (Fig. 6) and ∆ = 0.4 (Fig. 7).
C. Bond degree distribution for ∆ = 0.3 and ∆ = 0.4 In the main text, we report the bond degree distribution of all patch topologies for the patch position ∆ = 0.2 and the packing fraction φ = 0.2 as well as three temperatures T = 0.14, 0.10 and 0.01 Here, in the Supporting Information we additionally discuss the bond degree distribution for ∆ = 0.3 in Fig. 8 and of ∆ = 0.4 in Fig. 9.We find that the bond degree distribution for ∆ = 0.3 qualitatively exhibits same behaviour as ∆ = 0.2: for high temperatures at T = 0.14 (Fig. 8, Panel (a)), we observe all bonding scenarios from zero to fully bonded (four), with the majority of particles bonded to only one neighbour particle.While dmo-as1, dmo-s1 and dmo-s2 show a similar bonding distribution, in dma-as1 we observe a higher percentage of two-bonds, which indicates the presence of the three particle box clusters already at high temperatures.At intermediate temperatures, at T = 0.10 (Fig. 8, Panel (b)), we observe that the majority of particles has two or three neighbours, which -as with ∆ = 0.2 -hints at the presence of connected clusters of loops described in Fig. 3.For the lowest studied temperatures, T = 0.01 (Fig. 8, Panel (c)), we find that two-particle bonds become the prevalent bonding type, which indicates the increased chaining of the network.In ∆ = 0.4, for T = 0.14 and T = 0.01 (Fig. 8, Panel (a) and (d)), the bond degree distribution remains qualitatively similar to ∆ = 0.2 and ∆ = 0.3.On the other hand for intermediate temperatures T = 0.10 (Fig. 8, Panel (b), while the majority of particles of all patch topologies still has two-or three neighbours, we spot a significantly higher amount of four-bonded particles for dmo-s1 and dma-as1.At visual inspection of simulation snapshots we find that some of these four-bonded particles reside in fully-bonded finite clusters.We conclude that for the more symmetric ∆ = 0.4 patch positions, the bonding strain due to the patch asymmetry is already reduced to such that fully bonded finite clusters become more favorable and do co-occur with two-and three-bonded finite clusters.

Pore circumference
Besides the pore area, which is discussed in the main text, we also evaluated the pore circumference in units of the rhombi side length l r according the formula in Eq. 1 at the lowest temperature T = 0.01 for the patch position ∆ = 0.2 and for different packing fractions φ = [0.125,0.15, 0.2, 0.3, 0.5].Analogously to the pore area, the scatter plot of pore circumferences of all patch topologies at the lowest packing fraction of φ = 0.125 (Fig. 10, Panels (a)-(d) top row, blue scatter) shows a split between pore circumferences smaller than 200 -representing the small and mid-size pores and pore circumferences of > 4000, representing the void space.As the density increases, the difference in circumference between void space and pore space becomes smaller, as the large void space is cut into smaller pieces by bonded particles sharing a smaller simulation box, and at φ = 0.3 (Fig. 10, Panels (a)-(d) first row, red scatter), no significant difference is visible anymore between the circumference of void and pores.The kernel density estimation (KDE) -a parameter free estimation of the probability density -of the pore circumference (see Fig. 10, Panels (a)-(d), second row) shows that for low packing fractions of φ = 0.125 and φ = 0.15 the pore circumferences of dmo-s1 with an KDE-estimated length of around ≈ 50 and -to a lesser degree -dmo-s2 with a length of ≈ 40 are longer than the circumferences of dma-as1 and dmo-as1 with lengths of ≈ 30.

Dependency of asymmetry measures on pore size
In the main text we evaluated the convexity and the elongation of pores with pore areas smaller than 1000 (see Fig. 11, Panels (a)-(d), third and fourth column in the main text) voxels and found that, for both convexity and elongation, every particle type has its own typical peak patterns, discern-able by the relative height and sharpness of the peaks.Here, we investigate in detail which pore sizes are responsible for these unique peak patterns by evaluating the convexity and the pore elongation for pores of different size ranges.The results are shown in Fig. 11, Panels (a)-(d) for the convexity measure and in Fig. 12, Panels (a)-(d) for the elongation measure.It is important to note that, for this analysis the pore area is given as the number of voxels within a pore, as this enables us to better trace the asymmetry measures back to the pore shapes.Generally we observe that it is mainly the small pores, that are smaller than 20 voxels, that are responsible for the unique peak patterns in both measures.More specifically in the convexity measure (see Fig. 11, Panels (a)-(d), the peaks very close to one, indicating convex pores, are due to pores of sizes < 10, the peaks in the vicinity of 0.68 are due to pores between sizes 5 and 20, while the peaks near 0.85 appear for pores of size 10 − 20.While these three peaks do occur in all patch topologies, the relative height of these three peaks differs and we argue that is this difference that largely defines the unique peak patterns for each system.For pores beyond the size of 20, the peaks at 0.68 and 0.85 broaden and become one wider peak, while the peak close to one -signifying rather convex pores -disappears.Finally, we observe that at the highest density ( , an analogous picture forms: for pores of sizes < 5, the elongation measures peaks at 0.5 -indicating rather symmetrical pore shapes and for pores between 5 and 10, this 0.5-peak vanishes, and instead two peaks -a larger one at 0.65 and a smaller one at 0.75 appear.While all patch topologies and densities show these exact distinct peaks for pores of size 0 − 5 and size 5 − 10 -albeit with different heights -for pore sizes between between 10 and 20 more distinguished peaks appear between 0.65 and 0.9, but in these cases, the exact number of the peaks as well as their position and width differs for different patch-topologies and densities.Here it is interesting to note, that despite these peak pattern differences for different densities in each system for pore sizes 10 − 20, when evaluated over all pores sizes, a unique peak pattern is retained for each systems across all densities.Moving to pores > 20, we find a severe broadening of the distributions for all systems, to extent where no distinct peak can be identified.We interpret the results such that, while pores with sizes < 20 do have distinct shapes with unique peak patterns per particle type and density, larger pores do occur in all elongations. Fig. 11 in the main text, the pore convexity (Panels (a)-(d), third column) and the pore elongation (Panels (a)-(d), fourth column) are plotted for all particle types at patch position ∆ = 0.2 and for different packing fractions φ = [0.125,0.15, 0.2, 0.3, 0.4, 0.5] at temperature T = 0.01.Additionally we show the convexity and elongation binned per size ranges here in the Supporting Information in Fig. 11 and Fig. 12. II.ADDITIONAL DATA A. Average bonding affinity for ∆ = 0.3 and ∆ = 0.4 , Panels (a)-(d) in the main text).Here, in the Supporting Information we report the occurrence of the most dominant loops as function of temperature for ∆ = 0.3 (Fig 3, Panels (a)-(d)) and ∆ = 0.4 (Fig. 4, Panels (a)-(d)) Fig 11, Panels (a)-(d), last column), the pore convexity is significantly lower for larger pores 200 − 1000, representing a trend that can be spotted for all patch topologies.Taking a closer look at the elongation at different size ranges (see Fig 12, Panels (a)-(d))

5 FIG. 4 : 5 FIG. 5 :
FIG. 4: Fraction of dominant loops type for each particle type as function of temperature T, at patch position ∆ = 0.4 and packing fraction φ = 0.2.Panel (a): fraction of dma-as1 particles bonded in open box loops.Panel (b): fraction of dmo-as1 particles bonded in parallel and mixed (parallel and non-parallel) 6-particle loops.Panel (c): fraction of dmo-s1 particles bonded in non-parallel 5-and 6-particle loops.Panel (d): fraction of dmo-s2 particles bonded in parallel and mixed (parallel and non-parallel) 6-particle loops.Red stars highlight the state point at which percolation occurs.