Rony A.
Waheibi
and
Lilian C.
Hsiao
*
Department of Chemical and Biomolecular Engineering, North Carolina State University, 911 Partners Way, Raleigh, NC 27606, USA. E-mail: lilian_hsiao@ncsu.edu
First published on 5th November 2024
We report the ensemble-averaged and pairing-specific network microstructure formed by short-range depletion attractions in hard sphere-like colloidal systems. Gelation is induced by adding polystyrene molecules at a fixed concentration to colloids with different colloid bidispersity ratios (α = 1, 0.72, and 0.60) across a range of volume fractions (0.10 ≤ ϕ ≤ 0.40). 3D confocal microscopy imaging combined with a scale-invariant feature transform algorithm show that monodisperse colloids pack more efficiently, whereas increasing the size disparity leads to looser, more disordered, and sub-isostatic packings. Categorizing the structures formed by small and large particles reveal that certain cluster configurations may be favored due to the complex interplay between the differences in particle surface areas and attractive potentials. These pairwise bonds assemble to affect the density of tetrahedral and poly-tetrahedral clusters in bidisperse systems. With the exception of non-percolating samples at ϕ = 0.10, increasing the gel volume fraction leads to an increase in the number of nearest neighbors. However, the internal density within each cluster decreases, possibly due to kinetic arrest from the deeper potential wells of tetrahedral clusters at low volume fractions in which vertices are primarily made out of larger particles.
The prevalence of size polydispersity in most technological applications has led to an increasing number of studies that focus on the microstructure, dynamics, and rheology of colloidal materials prepared from bidisperse particles.17–22 It is now understood that bidisperse and polydisperse packings are almost always denser than that of monodisperse particles due to geometrical factors that change the jamming fraction.23–28 In bidisperse systems, changing the size ratio of the particles and the overall volume fraction produce pronounced effects on the colloidal phase behavior. Even in non-Brownian hard sphere suspensions, the viscosity decrease seen when small particles are added to large ones is due to an increase in the jamming fraction with an increasing size ratio.25 Incorporating attractive interactions between particles generate additional complexity in the free energy landscape because of the preferential contact between particles with the largest surface areas and the deepest potential wells.29–31 The resultant structural changes are further convoluted by the differing electrostatics for particles of different sizes. Research investigating varying ratios of these attractive and repulsive forces has revealed changes in the phase diagram behavior, percolation, and clustering.32–38 For instance, Pandey and Conrad found that increasing the ratio of small to large particles in a colloidal depletion gel reduces the particle displacement significantly, an effect that is exacerbated by confinement.18 Increasing the volume fraction of large particles softens gels but only when the density of large particles exceeds a critical value.19 This softening effect is attributed to large particles that change the effective range of the attraction, which consequently shifts the gel line.
Early theoretical work into the phase diagram of colloid–polymer mixtures began with insights into the demixing of colloid-rich and colloid-poor regions.39–41 Identification of spinodal decomposition as a mechanism towards transient gelation (originally predicted from polymers42) and thus non-equilibrium dense clusters was identified by Verhaegh et al. through small angle X-ray scattering.43 Subsequent attempts to characterize the resulting cluster size distributions used a second virial coefficient equation of state to model the cluster size distributions, where the gelation phase boundary obtained from experiments coincides with predictions from spinodal decomposition.44,45 These studies imply that under sufficient polymer concentration gels form by first demixing into colloid-rich and colloid-poor regions, followed by subsequent kinetic arrest of dense clusters. Evidence of such clusters are provided in several other investigations where confocal microscopy, computer simulations, and scattering techniques are used to extract a microstructural cluster correlation length.9,46–51 Contrary to earlier observations where the internal volume fraction of gel clusters is constant at ϕg ≈ 0.57 regardless of the overall volume fraction or attraction strength,44 Whitaker and coworkers find that stronger gels contain a larger number of clusters with lower ϕg that is representative of a looser local structure.9 Others have shown the importance of self-limiting structural rearrangements of particles to the kinetic arrest of gels above the spinodal line.52–55 The precise mechanism for how colloidal gels differ from glasses in their dynamic arrest, examined through the lens of localized tetrahedral-shaped clusters,54,56–59 remains an active area of research.
Bidisperse colloids can have a strong impact on the phase behavior of gels because they change the underlying cluster size distributions through favoring specific types of local order.22,60 Consider the elastic packing of a set of large and small particles in a dense cluster, where the size difference between particles impacts local packing. From the perspective of a small particle, the solid angle occupied by a large particle means that, on average, small particles will have less space for contacting neighbors, 〈zi〉, than vice versa, leading to an expectation of 〈zL〉 > 〈zavg〉 > 〈zs〉.61 The geometric variations in the local environment, in addition to the available surface area and excluded volume changes that gives rise to different attractive and repulsive forces in depletion interactions, cause bi- and polydisperse gels to assemble with a higher variability in the local microstructure.35,62–64 As the size disparity increases, small particles can also fit into the interstitial spaces of contacting large particles and lead to non-monotonic changes in the local energy landscape and cluster sizes.35,60,65,66
A full characterization of the cluster structure requires moving away from ensemble-averaged parameters such as the radial distribution function and 〈z〉 to metrics that more closely represent the cluster-level bond orientation and structural ordering. In this work, we suspend monodisperse and bidisperse colloids in a density and refractive index-matched solvent containing trace amounts of salt and a fixed concentration of non-adsorbing polymer. Care is taken to account for the swelling of particles in the solvent. While hydrodynamic interactions such as near- and far-field lubrication forces play a part in the phase behavior and in the gravitational sedimentation of depletion gels,67–71 here, we only consider the nature of mechanically rigid clusters within the colloidal network. The 3D microstructure of the gels is fully captured using confocal laser scanning microscopy and an image processing algorithm with proven accuracy in identifying the centroids of particles in contact with one another. Ensemble-averaged and pairing-specific structural quantities are used to determine the relative degree of disorder as a function of size ratio and overall volume fraction. We find that gels with a moderate size disparity appears to generate the most disordered structure, as seen from parameters that include the number density of tetrahedral and poly-tetrahedral clusters within mechanically rigid subunits.
Monodisperse colloids were combined to generate bidisperse systems of different size ratios (α = aS/aL, aS = radius of smaller particles, aL = radius of larger particles) as listed in Table 1. Colloidal depletion gels with α = 0.60, 0.72, 1.00 were prepared by suspending PHSA-PMMA colloids (refractive index nPMMA = 1.49, density ρPMMA = 1.18 g cm−3) at volume fractions of ϕ = 0.10, 0.20, 0.30, 0.40 in a density- and refractive index-matched solvent containing 66:34 vol% of cyclohexylbromide (nCHB = 1.50, ρCHB = 1.32 g cm−3) and cis-decalin (ndecalin = 1.48, ρdecalin = 0.90 g cm−3). Tetrabutyl ammonium chloride salt was added at a concentration of 1 μM to provide charge screening (zeta potential ζ < 0.01 V73). Linear polystyrene (PS) (number average molecular weight Mn = 900000 g mol−1, weight average molecular weight Mw = 1047500 g mol−1, radius of gyration Rg = 41 nm, c* = Mw/4π3/2Rg(0)3NAΨ* = 4.72 g L−1 where Rg(0) is the zero-concentration radius of gyration measured from static light scattering, NA is Avogadro's number, and Ψ* ≈ 0.24 is a correction factor for linear polymer74–76) was added as a non-adsorbing depletant to induce gelation. The concentration of PS, c, varies with colloidal volume fraction from c/c* = 1.02 to 1.05. Because the polymer concentration is close to the overlap concentration, the polymer itself cannot be treated as non-interacting due to the osmotic compressibility exerted between the chains. As a result, we recalculate the non-ideal radius of gyration for semi-dilute polymers, Rg(c), from the zero concentration radius of gyration, Rg(0), using the framework proposed by Zhou et al.74 The resulting Rg value is approximately 21 nm. In Table 1, we report our attractive range, ξ = Rg/a, as varying from 0.02 to 0.04 and calculate the net potential U/kT at contact by summing the Asakura-Oosawa and Yukawa potentials. Symbols with “S” and “L” subscripts indicate properties of small and large particles. The average particle size in bidisperse gels was obtained using aavg = aSxS + aLxL, where xS and xL are the number fractions of small and large particles. All gels were allowed to equilibrate for 60 minutes prior to imaging and structural analysis.
Parameter | Monodisperse | α = 0.72 ± 0.02 | α = 0.60 ± 0.02 |
---|---|---|---|
2as (μm) | N/A | 1.12 ± 3% | 1.10 ± 3% |
2aL (μm) | N/A | 1.56 ± 4% | 1.82 ± 4% |
2aavg (μm) | 1.30 ± 4% | 1.34 ± 4% | 1.55 ± 4% |
ξ S | N/A | 0.038 | 0.038 |
ξ L | N/A | 0.027 | 0.023 |
ξ | 0.032 | 0.031 | 0.027 |
U S/kT | N/A | 10.0 | 10.2 |
U L/kT | N/A | 14.2 | 16.5 |
U avg/kT | 11.8 | 12.2 | 14.1 |
x S/xL | N/A | 1 | 0.7 |
Fig. 1(a) shows a representative 2D image of a monodisperse colloidal gel prepared at ϕ = 0.20. In colloidal gels where particles are in near contact, accurate particle centroid detection is challenging due to the large point spread function (PSF) in the z-axis of confocal microscopy images in addition to overlapping and convolved PSFs of neighboring particles in contact. We used a two-step image processing protocol involving deconvolution and a scale invariant feature transform (SIFT) technique,65 rather than the conventional algorithm that uses brightness-weighted intensities.77 Deconvolution suppresses the optical blurring induced by the overlapping PSFs and enhances the particle detection accuracy by reducing misplacement at the risk of introducing artifacts. These artifacts are removed as their DoG response is often much less intense, which indicates a spurious detection, or an otherwise inappropriate size. SIFT determines the particle position and size using a difference of Gaussians (DoG) procedure, where Gaussian blurs of increasing kernel sizes are applied to the deconvolved 3D image volume. Image volumes with different Gaussian kernels are subtracted from one another (Fig. 1(b)) to enhance the pixel intensity gradients at particle edges. Fig. 1(c) and (d) demonstrate that SIFT generates particle centroids that are much more accurate than that obtained from the brightness weighted intensity method.77
Fig. 1 (a) A representative 2D raw image of a monodisperse colloidal gel (ϕ = 0.20). Representative overlay images show detected particles (blue circles) obtained using (b) a conventional brightness-weighted intensity algorithm77 and (c) the SIFT method. Scale bar represents 5 μm. (d) SIFT is used to process 3D image volumes through a difference of Gaussians method where blurs of increasing kernel sizes are applied to the raw image. Subtraction of processed images is used to identify particle size and position. |
Structural parameters were extracted from particle centroids found at locations beyond two particle diameters of the image boundaries. Using these centroids, we obtained a variety of ensemble-averaged structural parameters and categorized them based on their pairings with neighboring particles, namely: small–small (SS), small–large (SL), large–small (LS), and large–large (LL) pairings. Each particle was assigned a unique identifier to facilitate the tracking and analysis of the contact network microstructure.
Fig. 2 Phase diagram for monodisperse gels. The solid line represents the gas–liquid coexistence derived from generalized free volume theory,41,78 blue representing the fluid phase and green representing the gas-crystal coexistence. Dotted line represents the gel line above which spinodal decomposition occurs.78 Blue squares represent bulk conditions explored in this study. |
Fig. 3 shows representative images of colloidal gels across the range of ϕ and α explored in this study. Regardless of α, visual inspection shows that all samples prepared at ϕ = 0.10 (Fig. 3(a), (e) and (i)) do not percolate and remain as heterogeneous fluid clusters. Here, we define percolation as the complete tracing of a strand from one end of the 3D image volume to the opposite end. This discrepancy may arise from two mechanisms. First, it is possible that while parts of the sample has gelled, the highly heterogeneous nature of a dilute gel prepared at ϕ = 0.10 led to a lack of visualization of any space-spanning clusters.71,80 Second, the long range electrostatics (Debye length, κ−1 = 68 nm) may have slowed the formation of percolated clusters such that they were not found within the experimental timeframe. Other studies on colloidal depletion gels with much lower colloid volume fractions have shown that clusters can percolate with sufficient waiting time.33,37,81
Above ϕ = 0.10, contact networks are formed by an increasing density of colloids in near contact, concomitant with a reduction in the overall spatial heterogeneity. For monodisperse colloids (α = 1) (Fig. 3(a)–(d)), there is a subtle local order in which colloids are arranged in a Bernal-like spiral and where the spiral ordering decreases with increasing ϕ. This feature likely emerges due to the increased ergodicity of colloids at low ϕ, or in other words, the lower packing density enables monodisperse colloids to pack more efficiently by allowing them to explore their local energy landscape within experimental time scales.82
The observation of local order is significantly reduced in bidisperse colloidal gels (α = 0.72 and 0.60), because the non-equilibrium and attractive nature of colloidal depletion gels kinetically traps individual colloids at higher ϕ and therefore limits them to only explore the local free-energy minima. Moreover, outside of known crystalline phases of specific size ratios and stoichiometries, bidisperse gels have no known crystalline phases to most efficiently pack into.24,60 Decreasing the bidispersity ratio (α = 0.60) results in gel networks (Fig. 3(i)–(l)) with finer strands and branches. The strand-like morphology is likely due to the greater packing density of larger particles around smaller ones, which reduces the excluded volume and leads to a higher local attraction strength. Finer strands are typically seen in depletion gels of higher attraction strengths,9 in gels of anisotropic colloids where certain packing configurations are thermodynamically more favorable,83–85 and in the morphogenesis of bacterial colonies when depletion attractions are present.86 The finer strands found in bidisperse systems may correlate with a decrease in larger locally favored structures, which are associated with both the mechanical stability and resistance to thermal fluctuations of colloidal gels.81
As α decreases, the sharpness of both the first and second peaks decrease in magnitude and increase in broadness, with an increased splitting of the first peak into multiple peaks clustered around r/2aavg = 1. These results show that the presence of different particle sizes disrupts the emergence of short-range order. Once the size disparity is sufficiently large (α = 0.60) (Fig. 4(c)), the split first peaks emerge at distances equivalent to the separation between SS, SL, and LL particle contacts. The magnitude of the SL peak is the largest, indicating that pairings between particles of different sizes are the most common.
Fig. 5(b) shows the breakdown of 〈z〉 into 〈zs〉 and 〈zL〉 for bidisperse gels as a function of ϕ. For both bidisperse systems, small particle contacts are less frequent and show no significant change with respect to ϕ (〈zs〉 = 5.0 for α = 0.72, 〈zs〉 = 4.2 for α = 0.60). The lack of dependence implies that 〈zs〉 has reached a maximum value based on its available surface area and is not a function of ϕ. The lower 〈zs〉 value for α = 0.60 relative to 0.72 is a geometric result from large particles occupying more surface area than small particles. This trend is known in dense suspensions, where small particles have fewer neighbors than large particles in the same sample.61,87 Unlike small particles, large particles show a ϕ-dependence where 〈zL〉 increases with ϕ. At α = 0.72, 〈zL〉 rises from a value of 4.2 at ϕ = 0.10 to 〈zL〉 = 7.0 at ϕ = 0.40. This increase is less pronounced when α = 0.60, where 〈zL〉 increases from 5.7 to 6.2 due to stoichiometry. Gels at ϕ = 0.60 have an average of 10 large particles to every 7 small particles. Since the surfaces of small particles are fully occupied by contacts, large particles only find new contacts with other large particles that still have unoccupied spaces on their surfaces, leading to new LL contacts.
Bidispersity plays a significant role in the particle pairings due to a greater amount of surface area available for large particles to connect with more neighbors. Fig. 5(c) and (d) further separate the contact numbers into the average number of contacts by pairing, 〈zij〉, for a colloid i of radius ai with another colloid j of radius aj, calculated as zij = zizjxi/z, where xi is the number density of colloid i.23Fig. 5(e) visually depicts the relationship between average contact number by size (Fig. 5(b)) and the different contact pairing types (Fig. 5(c) and (d)). In both bidisperse systems, the number of SS and SL contacts (green) remain constant across all ϕ, while the number of LS and LL contacts (magenta) increase with ϕ. These results suggest that contacts between large particles are limited by stoichiometry and surface area in finer strands found at low ϕ. A moderate bidispersity of α = 0.72 appears to spatially constrain all large particle contacts, while a smaller bidispersity ratio of α = 0.60 allows small particles to fit between interstitial spaces regardless of ϕ.
For α = 0.72 (Fig. 6(b)), the P(θ) peaks at θ = 58° and 110° are smaller in magnitude and shifted as compared to that of α = 1. As ϕ increases, the first maximum decreases slightly in magnitude and there is a general smoothing of the curve beyond the first maxima, indicating that the distribution of angles between bonded particles and hence the degree of disorder increases with ϕ. At the largest particle size disparity of α = 0.60 (Fig. 6(c)), the first maximum splits into two peaks at θ = 52° and 74°, corresponding to the angle between two small particles with a large reference particle and two large particles with a small reference particle (inset in Fig. 6(c)). As ϕ increases, the magnitude of the peak at θ = 52° decreases, suggesting that smaller particles become less likely to pair with one another. The curves are generally smooth at θ > 74° with the pronounced presence of small secondary maxima and minima at α = 0.60. This shows that gels formed with a higher bidispersity ratio have a greater degree of local order. A small dip in P(θ = 180°) at α = 0.60 and ϕ = 0.10 indicates that particles tend not to align in a straight chain in this specific set of gels.
Because the primary g(r) peaks for gels of α = 0.60 (Fig. 4(a)) are at r/2aavg values that are characteristic of SS, SL and LS, and LL pairs, we separate P(θ) into pairing-specific distributions as shown in Fig. 7, which plots the frequency of three-body configurations normalized by the total number of particles. For any three-body configuration where particles A and C are in contact with the reference particle B, θABC indicates the angle formed by particles A and C at vertex B. An example is shown in Fig. 7(a). There are six unique permutations of pairings including SSS, LLL, SLL, SLS, SSL, and LSL. Once separated, each triplet pairing has the geometrically predicted peaks expected due to excluded volume effects.
The reduction in the local order observed in Fig. 4(b) is a consequence of competition of different triplet interaction types. For moderately bidisperse gels (α = 0.72), triplet pairings of SSS and LLL are the least frequent pairing, followed by LSL and SLS configurations, and finally by the most common LSS and SLL configurations. These probabilities are not fully explained by random independent events that would otherwise give rise to equal probability distributions. For pairings that are inversions of each other, such as SSL and SLL, the reason for the increased prevalence of SLL could be due to the increased surface area of large particles. Because we are working with non-equilibrium structures above the gel line in Fig. 2, equilibrium arguments such as those proposed by Meng and coworkers88 do not fully explain why triplet pairings favor rotationally asymmetric structures. The variety of angles at which peaks emerge suggest an increased disorder, but further confirmation would be required through higher-order cluster analysis. This is especially true for the secondary peaks, which becomes relevant in the analysis of structures such as pentagonal bipyramids which are associated with fivefold symmetry that are known to be associated with the emergence of gel solidity.59
The highest frequency peaks in Fig. 6 align with the expected peaks of tetrahedral structures, suggesting that 1T is the lowest free energy configuration in colloidal depletion gels with short-range attractions. There is an increase in the number of all tetrahedral structures from ϕ = 0.1 to ϕ = 0.2 which corresponds to the formation of percolating networks, where the subpopulation of single tetrahedra is always the largest (ρ1T > ρ2T > ρ3T > ρ5T). All tetrahedral densities decrease when ϕ > 0.2, that is, the number density of tetrahedral structures decreases with increasing ϕ and therefore contain looser packing. This counterintuitive observation implies that the overall particle volume fraction is directly correlated with the volume fraction of a cluster (〈z〉 increases as ϕ increases), but that the density of a cluster decreases (ρ1T decreases as 〈z〉 and ϕ both increase) beyond the gelation threshold of ϕ = 0.20.
Fig. 6 and 8 show that there are fewer regular tetrahedral structures in bidisperse colloids due to the existence of higher order, spatially disordered configurations where the bond angle θ deviates from 60°.60 The value of ρ1T is slightly higher for gels at α = 0.60 as compared to α = 0.72. This may be due to the stoichiometric ratio of 7 small particles to 10 large particles in α = 0.60 and a ratio of 1:1 in gels at α = 0.72. Due to the the differences in both stoichiometry and occupied volume of large particles, α = 0.60 gels have a higher partial volume fraction of large particles. As a result, gels at α = 0.60 can be considered to have a higher degree of local order than ones prepared at α = 0.72.
Fig. 9 Schematic of the structural packing observed in gels with size ratios (a) α = 1, (b) α = 0.72, and (c) α = 0.60. |
Bidispersity introduces re-entrant behavior in that signatures of local order are greatest for the monodisperse system (α = 1) and for colloids with the largest size ratio (α = 0.60). All structural parameters used in this study indicate that the gels at α = 0.72 have the most disordered microstructure. Stoichiometric reasons notwithstanding, the reason why this moderate size ratio produces a locally and globally amorphous packing could be due to competing mechanisms where thermodynamics favor certain cluster structures, but mechanical caging effects hinder the particles from forming densely packed tetrahedra. The number of tetrahedral and poly-tetrahedral structures is further controlled by the geometrical constraints on the translational degrees of freedom, which are different depending on the size ratio.
Let us first consider what the cluster volume fraction should be from a purely geometry-based jamming perspective. For a monodisperse Brownian hard sphere system with no net external stress, as in the case of our samples under discussion, the quasi-static jamming point for a random close packed system is ϕRCP = 0.64–0.66 depending on the size polydispersity of the particles.89 In 3D systems, the isostatic point for hard spheres is 〈z〉 = 6. Adding bidispersity increases both ϕRCP and 〈z〉 in ways that depend on the size and the volumetric mixing ratio (β = 0.28 for α = 0.72 and β = 0.13 for α = 0.60) of the small and large particles. Singh and coworkers performed computer simulations on non-Brownian particles with various mixing ratios and report the variation of ϕRCP with α and β.25 Their data show that in hard sphere systems with no attractive interactions where 0.60 ≤ α ≤ 1 and 0.15 ≤ β ≤ 0.40 (note that Singh et al. use the symbols Δ for size ratio and α for the stoichiometric ratio, with slightly different definitions from ours), ϕRCP should be between 0.66–0.67. The value of ϕRCP can be thought of as the internal cluster volume of a gel, but only if the mechanism of gel formation were to be purely mechanical in nature, driven only by entropic caging and nothing else, as equilibrium spinodal decomposition might suggest.45
Yet the true cluster volume fraction ϕg in depletion gels is likely much lower than the expected value of ϕRCP = 0.64–0.67, meaning that jamming or glass transition is not necessarily the mechanism for gelation. Our data are also supported by other evidence in the literature, for example in the cluster volume fractions and size distributions reported by Lu et al. (ϕg ≈ 0.57), Whitaker et al. (0.35 ≤ ϕg ≤ 0.5), and by Tsurusawa and Tanaka (ϕg ≈ 0.30, albeit defined as the fraction of clustered pentagonal bipyramids).9,45,59 While we do not report ϕg in this study, the parameters ρ1T, ρ2T, ρ3T, and ρ5T serve as indirect measures of the densest clusters that are formed in a gel, since tetrahedra and poly-tetrahedra are defined only when four particles make contact with one another. Overall isostatic rigidity is unnecessary (〈z〉 can be ≤6) because tetrahedral structures can form with hypostatic packings as long as the local cluster coordination number is 〈zc〉 ≥ 3. Our results are therefore in support of the notion that the mechanism of glass formation is completely different from that of depletion gelation, in which the kinetic arrest in a gel is not necessarily driven by the formation of dense clusters except at the lowest attraction strengths and highest volume fractions. This mechanism only happens very close to the phase boundaries in weak and highly concentrated gels.59 A similar phenomenon is seen in the depletion-driven gelation of anisotropic colloidal discoids.84,90
In the case of bidisperse systems, these arguments about the nature of gelation remain valid, with the exception that pairings between small and large particles influence both the jamming threshold and the pairwise attractive and repulsive potential. We observe that large particle vertices are more frequent than equivalent pairings with small particle vertices where P(θSLL) > P(θSSL), as seen in Fig. 7(c). This observation could be a consequence of the increased surface area availability of the large particles, or due to an increased attractive strength of the bond, or from a combination of the two. In addition, the different electrostatic contributions and stronger repulsive forces for large particles compared to small particles may be contributing to the anisotropic structures and favorability of asymmetric clusters seen in Fig. 7.19,34,35 These factors strongly influence not only the initial formation of the gel but also the rate of bond rupture and reformation in depletion gels.55,91 SS bonds (U = −10.2 kT) have a higher likelihood of rupturing as opposed to SL/LS (U = −14.1 kT) or LL (U = −16.5 kT) bonds with deeper potential wells. Once a bond ruptures, freely diffusing or hopping particles are more likely to find a large particle due to both the unoccupied surface area and larger excluded volume compared to small particles.23,61,87,92 These rearrangements during the aging process increase the frequency of large bond vertices which may influence the ordering process of neighbors and secondary structures in bidisperse systems. Although tetrahedral and polytetrahedral clusters play a key role during glass formation,57,58 this process proceeds homogeneously and at higher volume fractions than explored in this study, whereas the gels here are formed primarily through kinetic arrest. Our results show that cluster formation occur even in non-percolated samples and suggest that locally favored structures do not form by vitrification, similar to the results shown by Tsurusawa et al.59 Further validation through algorithms such as the topological cluster classification system may be able to further reveal hidden higher order local structures,53,56 especially when investigating the differences between glass and gel identities.
It would be interesting to investigate the dynamics of the particles within gel clusters as measuring the particle mobility within clusters may provide additional clarity on how bidisperse gels differ from bidisperse glasses. Although not being considered in this study, the effects of lubrication forces, osmotic pressure of concentrated colloids, and many-body interactions will become important when studying particle dynamics in Brownian colloidal systems especially if gravitational settling is present.67,93,94 Future work could also investigate why LL contacts are the only ones that increase in frequency with increasing ϕ, since cluster structure and bond lifetime are both important in connecting to the gel viscoelasticity.10,95 The findings here were somewhat analogous to those reported by Singh et al.,25 where the colloid stoichiometry and size ratio are both important parameters in differentiating the microstructure of bidisperse systems from their monodisperse counterparts. A larger set of bidispersity ratios and stoichiometry, perhaps combined with different types of colloids and depletants, would more definitively determine if these results are generalizable to a broad set of colloidal materials.
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