Experimental Assessment and Model Validation on How Shape

Article Cite This: Langmuir XXXX, XXX, XXX−XXX

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Experimental Assessment and Model Validation on How Shape Determines Sedimentation and Diffusion of Colloidal Particles Rouven Stuckert,† Claudia Simone Plüisch,† and Alexander Wittemann* Colloid Chemistry, University of Konstanz, Universitaetsstrasse 10, 78464 Konstanz, Germany

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ABSTRACT: Understanding the hydrodynamics of colloids with complex shapes is of equal importance to widespread practical applications and fundamental scientific problems, such as gelation, crystallization, and phase behavior. Building on previous work, we present a comprehensive study of sedimentation, diffusion, intrinsic viscosities, and other shapedependent quantities of clusters built from spherical nanoparticles. Cluster preparation is accomplished by assembling surface-modified polystyrene particles on evaporating emulsion droplets. This results in supracolloids that exhibit welldefined configurations, which are governed by the number of constituent particles. Sorting into uniform cluster fractions is achieved through centrifugation of the cluster mixture in a density gradient. Sedimentation coefficients are elucidated by differential centrifugal sedimentation. Rotational and translational diffusion of the clusters are investigated by polarized and depolarized dynamic light scattering. The experimental results are compared to data obtained via a bead-shell model suitable for predicting hydrodynamic quantities of particles with arbitrary shapes. The experimental data are in excellent agreement with the predictions from hydrodynamic modeling. The variety of investigated shapes shows the robustness of our approach and provides a complete picture of the hydrodynamic behavior of complex particles.

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INTRODUCTION In nature, colloidal particles are found in organisms, soils, waters, and air as biological matter, fine mineral particles, dust, or volcanic ash.1 On the application side, colloidal particles are exploited in wide-ranging industrial sectors that benefit society.2 Prominent examples are nanomaterials as part of coatings, paints, adhesives, lubricants, catalysts, pharmaceuticals, foods, and cosmetics.3−5 The remarkable development of nanoscale materials was boosted by intense efforts on the research side. In particular, studies of structure formation and phase behavior of colloidal particles have led to a fundamental understanding of a broad range of physical phenomena such as nucleation, glass transition, mixing, melting, freezing, and gelation.6−9 Due to their sizes at the nanoscale, colloidal particles exhibit large specific surface areas, leading to a considerably large number of surface atoms. This can make the ultrafine particles chemically more reactive than microparticles.10 Moreover, they diffuse much faster and are less prone to sedimentation. For these reasons, colloidal particles can assume vital functions in many environments, but they may also cause adverse effects such as allergies and diseases.11 Colloidal particles diffuse by Brownian motion and sediment by gravity, given that their density is higher than the density of the medium.12 The interplay of Brownian diffusion and gravitational forces determines solution behavior of colloidal particles as well as their distribution in different environments. © XXXX American Chemical Society

For instance, diffusion and sedimentation are the major deposition mechanisms for nanoparticles in the lower parts of the respiratory system.13 In the absence of strong gravitational fields, Brownian motion prevails over gravitational settling as long as the particle size is kept at the nanoscale. However, one should bear in mind that attractive interactions among nanoparticles may produce agglomerates beyond the Brownian regime.12 In addition, gravitational settling is the precondition for sorting colloidal mixtures in centrifugal separations.14 Sedimentation in centrifugal fields is also a proven method for the assembly of colloids into various ordered and disordered phases.15 In the case of nonspherical particles, the dependence of the rates of diffusion and sedimentation on particle size and shape is nontrivial. For example, hydrodynamic friction acting on a particle dimer differs considerably from friction acting on a sphere or an ellipsoid of equal volume.16 In view of the solution behavior of complex colloids, it is thus important to explore in depth the interrelation among particle shape at high resolution and shape-related characteristics such as sedimentation and diffusion. Experimental studies in this direction may substantially benefit from progress in hydrodynamic modeling of rigid particles of arbitrary shape.16−18 Garcı ́a de la Torre and Received: September 3, 2018 Revised: October 10, 2018 Published: October 10, 2018 A

DOI: 10.1021/acs.langmuir.8b02999 Langmuir XXXX, XXX, XXX−XXX

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and balanced understanding of hydrodynamic behavior by also exploring in depth the sedimentation of colloidal molecules in gravitational fields and their intrinsic viscosities. Experiments probing diffusion and sedimentation are combined with hydrodynamic modeling in consideration of the exact particle shape. In this way, various particle shapes can be related to the corresponding hydrodynamic quantities.

co-workers have established model building and hydrodynamic calculation routines based on theories developed by Kirkwood, Bloomfield, and their co-workers.19,20 For a long time, scientific inquiry has focused on colloids with simple geometries such as spheres, platelets, or ellipsoids. Yet it is precisely the shape of the particles used as elementary units that opens up new horizons for hierarchical organized materials.21−24 Colloidal particles and small assemblies built thereof bridge the gap between the atomic scale and the macroscopic domain and thus allow for an unprecedented degree of manipulation and control.25,26 The phase behavior of colloids with nonspherical symmetries is essentially rich, but at present, it is still scarcely explored.27−29 For example, diverse hierarchical organized phases were predicted for superstructures made of tetrahedral- and octahedral-shaped clusters of spherical particles.22 In recent times, it has become possible to synthesize such colloids.30−32 Efforts should be now geared toward (i) making them available at larger scales (ii) and exploring viable routes for their organization into supracolloidal materials.33 The return would be exceptionally high because of exciting new physical properties and practical applications, which are to be expected from joining shapetailored particles into organized materials.23,24 Their well-defined shapes make clusters of uniform spherical particles equally suited as building units for organized superstructures and as model colloids. They can be used to explore translational and rotational diffusion of particles with complex shapes.34−40 Most of work is based on microscale clusters whose diffusion can be recorded using particle tracking microscopes.41 In parallel to the experimental works, hydrodynamic calculations and molecular dynamics simulations were pursued.18,42 The colloidal clusters studied were found to optimize specific packing criteria, resulting in well-defined configurations, which mimic those of molecular structures.43,44 The idea of considering their constituent particles as a sort of “big atoms”9 has lent particle clusters their alternative name “colloidal molecules”.45,46 The pioneering work by Pine and co-workers showed that the fabrication of colloidal molecules can be assisted by toluene-in-water emulsions with the particles confined by surface tension to the droplet surfaces.43 When the toluene is removed by evaporation, the particles pack into dense clusters. This emulsion-assisted strategy has proven extremely reliable and versatile for joining microparticles.47 Size and dispersity of the emulsion droplets can be decreased by ultrasound. This in turn allows the preparation of clusters from nanoscale particles.48 In recent studies, we have reported on colloidal clusters made of polymer latex particles,49 plasmonic nanoparticles,50 and binary nanoparticle mixtures.51 The deliberate use of nanoscale particles as elementary units keeps the global cluster dimensions in the regime of Brownian motion.38 Knowing their shapes facilitates model building and prediction of hydrodynamic properties. This is achieved by building clusters from uniform nanospheres. As a result, cluster configurations can be reduced to packings of spheres.52 In most studies, diffusion of complex particles and their sedimentation in gravitational fields have been considered separately.34,53,54 In this manner, we specifically reported on the translational and rotational diffusion of Brownian particle clusters some time ago.38 Meanwhile, significant improvements in sorting colloidal clusters and hydrodynamic modeling could be achieved. This makes it possible to include additional cluster species. In this work, we will strive for a comprehensive

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EXPERIMENTAL SECTION

Materials. Styrene (99%, Merck) was passed through a tertbutylcatechol inhibitor remover (Sigma-Aldrich) column before use. N-Isopropylacrylamide (NIPAM, 97.0%), divinylbenzene (DVB, 80.0%), potassium persulfate (99%), sodium dodecyl sulfate (SDS, 99%), and Pluronic F-68 were obtained from Sigma-Aldrich and used as received. D(+)-Sucrose (99.5%) and sodium chloride (99.5%) were purchased from Carl Roth. Deionized water (resistivity >18 MΩ) obtained from a reverse osmosis water purification system (Millipore Direct 8) was used throughout the entire studies. Synthetic Procedures. Polystyrene-co-poly(N-isopropylacrylamide) (PS-co-PNIPAM) Particles. Narrowly dispersed surfacemodified polystyrene latex particles were prepared by emulsion polymerization in a batch process. A three-necked flask was charged with styrene (208.3 g, 2.0 mol), NIPAM (11.32 g, 0.1 mol), SDS (0.88 g, 3.0 mmol), and water (840 g). The mixture was heated up to 80 °C under nitrogen atmosphere and stirred continuously at 300 rpm. Polymerization was initiated by adding potassium persulfate (0.819 g, 3.0 mmol) dissolved in 20 g of water. Right after the initiation, DVB (13.0 g, 0.8 mmol) was added over a period of 60 min. The cross-linker was added under starved conditions to facilitate homogeneous cross-linking. The polymerization was allowed to continue for further 6 h. The suspension of polystyrene-co-poly(Nisopropylacrylamide) (PS-co-PNIPAM) particles was purified by exhaustive ultrafiltration against deionized water. Preparation and Fractionation of Colloidal Clusters. The assembly of PS-co-PNIPAM particles into clusters was accomplished along the lines given in refs 48, 49. Briefly, an aqueous suspension of the cross-linked PS-co-PNIPAM particles was freeze-dried and resuspended in toluene at concentrations of 4 wt %. These suspensions (3 mL) were added to 27 mL of a 0.5 wt % aqueous solution of Pluronic F-68. Emulsification was carried out in a rosette cell (RZ 2, Bandelin Electronic) using an ultrasonic homogenizer (Sonoplus HD 3200, 200 W, 20 kHz, probe KE 76, Bandelin Electronic). Three sonication steps, each for 5 min with 2 min rest in between, were performed at an amplitude of 20%. Assembly into clusters was achieved by gentle evaporation of the droplet phase using a rotary evaporator (RV 10 digital, IKA-Werke). The mixture of clusters of varying number of constituents was separated into uniform cluster fractions. Sorting according to the sedimentation coefficients was achieved by rate-zonal density gradient centrifugation.55 Linear sucrose density gradients (37 mL), ranging from 5 to 10 wt %, were prepared in centrifuge tubes (Ultra-Clear Centrifuge Tubes, Beckman Instruments) at a flow rate of 18 μL s−1 using a piston gradient pump (DURATEC Analysentechnik). The cluster suspension (2 mL) was layered on top of the density gradient. Centrifugation was performed using an Optima XPN-90 Ultracentrifuge equipped with an SW 32 Ti swinging-bucket rotor (Beckman Coulter). Typical centrifugation conditions were 28 min at 25 000 rpm and 20 °C. Cluster fractions were collected using a self-built fraction recovery system enabling recovery from the top. The tip of a drain tube was placed right into the zone of banded particles to be recovered. A slight negative pressure was used for the extraction. Because clusters with N = 5 and 6 are banded in deeper layers of the gradient, a self-built centrifuge tube slicer was used to remove the zones lying above prior to the extraction. All cluster fractions were purified by exhaustive dialysis (5 days), first against deionized water and finally against 10−4 M NaCl solution. Methods. Differential Centrifugal Sedimentation (DCS). Measurements of sedimentation coefficients and particle size distributions B


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superior to other bead modeling strategies.16,17 Finely shaped details can be considered by using a large number of small beads. In fact, we found an excellent agreement of experimental diffusion coefficients and predictions based on the shell model in a preliminary study.38 It is for this reason that the shell model is also used in this work to not only predict rotational and translational diffusion coefficients but also evaluate further hydrodynamic properties such as sedimentation coefficients and intrinsic viscosities. Computing. Calculations of hydrodynamic properties were carried out with the public domain program HYDRO++.17 The variable parameters that entered the calculation are temperature (25 °C) and solvent viscosity (0.891 g m−1 s−1). The particle mass (1.65 × 10−15 g) was calculated from the number-average TEM radius (72 nm) and particle density (1.057 g cm−3). The computing routine utilizes “stick boundary conditions”, i.e., the solvent molecules near the surface translate and rotate with the particle clusters. Stick boundary conditions are commonly assumed for small macromolecules such as proteins.58,59 Recently, we have found that stick boundary conditions may also apply for submicronsized particle assemblies.38 The colloids described herein are bearing a thin hydrogel layer at the surface that further promotes stick boundary conditions. Aside from this, slip boundary conditions cannot apply, merely due to the fact that DDLS experiments reveal a finite value of the rotational diffusion coefficient for the spherical PS-co-PNIPAM particles. A sphere, which is subject to slip boundary conditions, does not have to displace any fluid during rotation. Hence, the rotational diffusion coefficient should diverge because one should not expect any viscosity dependence on particle reorientation.60 Another indicator supporting stick boundary conditions is the intrinsic viscosity of the PS-co-PNIPAM particles. If stick boundary conditions are assumed, one gets an intrinsic viscosity of 2.463, which is close to the value of 2.5 that was proposed by Einstein for hard spheres at infinite dilution (Table 2).61 In line with standard practice, sedimentation coefficients, diffusion coefficients, and intrinsic viscosities were calculated for various bead sizes σ. The calculated hydrodynamic properties converge into the exact quantities by increasing the number of beads NB, while decreasing their radius σ. The limit of a continuous shell is approached by extrapolation to zero bead radius (σ → 0) and infinite bead number (NB → ∞). In doing so, any discontinuities arising from the positional arrangement of finite beads such as inherent roughness and small gaps are eliminated.

were carried out in an ultra-high-resolution disk centrifuge (DC 24000 UHR, CPS Instruments), which is equipped with a high-speed motor operating at up to 24 000 rpm and an optical detector calibrated at 405 nm. The particles are separated within a rotating hollow disk, which is optically clear. The disk is loaded with a density gradient (Figure 4). The gradient wards off bulk sedimentation (streaming) and thus makes the particles settle individually at rates according to their sedimentation coefficients s (Table 3). The experiment is initialized by placing a minute amount of the suspension to be analyzed on top of the gradient. When the particles approach a fixed position near the edge of the rotating disk, they scatter a portion of a light beam passing through the disk (Figure 4). At any time, particles of a particular size or, more precisely, sedimentation coefficient are detected. For this reason, the method is classified as a differential sedimentation technique. The measured absorption is defined as the natural logarithm of the ratio of the initial light intensity at the detector to the light intensity reaching the detector during the centrifugal run. It is continuously recorded and used to derive the concentration of particles arriving at the detector position with regard to the settling time of the species. The latter is related to the apparent particle size and thus provides access to the particle size distribution. Detailed information on the method and its application to particle sizing is given in ref 56. (Depolarized) Dynamic light scattering ((D)DLS). Translational and rotational motion of colloidal clusters were investigated with polarized and depolarized dynamic light scattering (DLS and DDLS). All measurements were carried out at 25 °C on an ALV/CGS-3 Compact Goniometer system (ALV-Laser Vertriebsgesellschaft). The setup was equipped with a 632.8 nm wavelength He−Ne laser. An ALV-7004 Multiple Tau Digital Real Time Correlator was utilized to produce the autocorrelation function of the scattered intensity. The particles to be probed were dispersed in 10−4 M NaCl at volume fractions ranging from 10−4 to 10−5 and passed through a 0.8 μm syringe filter. The quartz glass cuvette was placed in an index matching vat filled with toluene. Each measurement comprised five runs of 90 s each. The scattered light passed through a GlanThompson prism (B. Halle Nachfl., extinction ratio 10−7), filtering out either the vertically or horizontally polarized part of the scattered light. Further Methods. Transmission electron microscopy (TEM) analyses were performed on a Zeiss Libra 120 microscope at an acceleration voltage of 120 kV. Field emission scanning electron microscopy (FESEM) images were recorded on a Zeiss CrossBeam 1540 XB microscope operating at 3 kV. Electrophoretic mobilities were measured on a Malvern Zetasizer Nano ZSP. Conversion into ζ potential was done using the approximation of Henry’s function implemented by Ohshima.57 The nitrogen content of the PS-coPNIPAM particles was quantified in a CHN(S) elemental analyzer (vario MICRO cube, Elementar). The density of the PS-co-PNIPAM particles was determined using a digital density meter (DMA 5000 M, Anton Paar). Hydrodynamic Modeling. Predictions of sedimentation coefficients, diffusion coefficients, and intrinsic viscosities were made using the hydrodynamic bead-shell strategy established by Garcı ́a de la Torre and co-workers.16,17,58 In this approach, rigid particles of arbitrary shape are represented as assemblies of spherical frictional elements, which are termed beads. The global size and shape of such an array of frictional elements should be made coherent with the colloidal particle to be probed. Stokes’ law friction coefficients are assigned to each bead. The hydrodynamic interactions among the frictional elements are described by using modified Oseen tensors along the lines of ref 17. Shell Model. Excellent overviews of the various types of bead models and their elaboration along with their strengths and limitations are given in refs 16, 17. Shell models have proven as a general and accurate method of predicting hydrodynamic properties of rigid particles with complex shapes. Friction occurs at the particle surface. This applies in particular to rigid particles, as these are inaccessible to solvent. This makes the shell model, which treats the particle surface as a dense array of identical frictional elements,

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RESULTS AND DISCUSSION A fundamental study on the hydrodynamic behavior of complex colloids greatly benefits from model colloids with shapes that are precisely defined, particularly if a tight connection between experiment and theoretical modeling is to be achieved. The first priority should thus be given to the colloidal clusters that pave the way for extensive hydrodynamic studies. Model Colloids with Tailored Configurations. Cluster Constituents. The clusters are made up of spherical polymer particles. The latter are prepared by emulsion polymerization of styrene as the major monomer, DVB as a cross-linking agent, and NIPAM as an auxiliary monomer. The use of a cross-linker is vital to the fabrication of colloidal clusters, which proceeds in the presence of an organic solvent. The watersoluble comonomer NIPAM participates in the free-radical polymerization of styrene and thus becomes covalently incorporated in the polymer chains. Elemental analysis gives a nitrogen content of 0.6%, a carbon content of 90.7%, and a hydrogen content of 7.9%. This is in line with complete incorporation of NIPAM. Due to their polar nature, the Nisopropylamide groups tend to be at the particle surface. The thin hydrophilic surface layer contributes to the colloidal stability of the polymer particles. It reinforces surface hydration C


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Langmuir and therefore influences the hydrodynamic behavior. In addition, sulfate groups originating from the persulfate initiator provide electrostatic stabilization, which is mirrored in the ζ potential of −70 ± 5 mV. Due to the excellent colloidal stability, the polymer particles as well as clusters based on them may be subjected to exhaustive purification by ultrafiltration without detrimental effects. The hydrodynamic diameter of the PS-co-PNIPAM particles (145 nm, measured by DLS) is close to the number-average particle size (144 nm, determined by TEM), owing to the low dispersity of the particles. They can be regarded as nearly monodisperse because the polydispersity index given by the weight-average diameter divided by the number-average diameter is 1.001. TEM analysis confirmed the very narrow size distribution of the spherical particles, which is indispensable for their assembly into defined cluster configurations (Figure S1). In earlier studies, we have noted that virtually monodisperse spherical particles with average diameters of about 150 nm are ideally suited to (i) prepare clusters with varying number of constituents and global dimensions in the Brownian regime38 and (ii) allow for proper fractionation into uniform clusters.48 Cluster Fabrication. Assembly of the polymer particles into stable clusters was accomplished along the lines given in earlier publications.48,49 In brief, cross-linked polystyrene latex particles are trapped at the surface of toluene droplets. The confinement is driven by minimization of interfacial energy, but does not constitute a Pickering effect in the stricter sense because the polymer particles are considerably swollen with toluene.52 Evaporation of the droplet phase creates strong capillary forces that make the particles on a droplet pack into a dense cluster.43 Owing to the random distribution of particles on the droplets, various clusters that differ in the number of constituent spheres N are obtained. This paves the way for exploring species with distinct aggregation numbers and configurations, but requires efficient sorting into uniform species. Before looking on fractionation, attention should be first directed to morphologies and configurations of the clusters. Cluster Morphology and Stability. It should be noted that the clusters differ from assemblies of touching spheres. This is particularly visible in TEM images (Figure 1), which indicate partial fusion of the particles when being packed into clusters. At this stage, the particles are deformable to a certain extent because they are partially swollen with organic solvent.52 Deformability along with capillary forces, which act during particle assembly, produces clusters that are denser than packings of hard spheres. Surprisingly, evaluation of TEM images revealed that partial fusion of the cluster constituents during the assembly is associated with a loss in volume. It amounts to 7.7% of the volume of a single constituent per interparticle contact. The volume shrinkage may be explained by a partial loss of the diffuse surface layer when the particles pack into clusters. The difference in volume was taken into account when calculating volumes and intrinsic viscosities. On account of the broadened contact areas among the particles, a strong cohesion is achieved resulting in robust assemblies. This strong cohesion is mediated by entangled polymer chains at the junctions between the cluster constituents. In fact, we found that the clusters withstand even repeated treatments with high-frequency ultrasound (Figure S3).

Figure 1. Clusters assembled from 144 nm-sized surface-modified polystyrene latex particles having well-defined geometries and underlying Brownian diffusion. (A) Field emission electron microscopy (FESEM) images, (B) transmission electron microscopy (TEM) images, and (C) bead-shell models used for calculating hydrodynamic properties. The scale bars are 200 nm.

Cluster Configurations. Centrosymmetric particles are used as elementary units of the clusters to avoid directional interactions during particle assembly. The configurations of the resulting assemblies are thus regulated by packing criteria for spheres, such as the formation of energetically favorable states or optimization of packing efficiencies.52,62 The fact that the assembly proceeds from a spherical template as well as pair interactions among the particles are additional factors, albeit packing criteria are predominant for clusters with N ≤ 6 (Figure 1). Needless to say that clusters made up of two particles (N = 2) exhibit a single configuration, manifested in a dumbbell-shaped geometry. Clusters with N = 3 have also just one configuration with a triangular planar symmetry. A tetrahedral coordination is the prevalent configuration of a cluster with N = 4. However, we could also detect few clusters D


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species have the highest aspect ratios and related to this, the highest anisotropies among the clusters. With respect to sphericity, they differ little from the first group by virtue of their compact shapes. Fractionation. Sorting of the clusters into species with the same number of constituent particles N was accomplished by rate-zonal density gradient centrifugation (Figure S4).14,64 The mixture of clusters is placed on top of a density gradient. Through centrifugation, different cluster populations migrate as discrete zones because of the differences in their sedimentation coefficients (Table 3). Building on earlier work based on this method,48,49 we have continuously enhanced fractionation protocols striving to achieve an optimum of sorting accuracy. In parallel, different ways to isolate zones of banded clusters have been tested to minimize turbulent flows during fraction recovery. A brief summary of the procedure that has turned out to be most efficient is found in the Experimental Section. The intensity of scattered light increases disproportionately with particle size.65 Hence, even traces of a second population can falsify diffusion coefficients derived from scattering experiments if the size of the impurities exceeds the dimensions of the species to be probed. Consequently, varietal purity of the isolated cluster fractions had to be validated (Figure S4). Measuring the particle size distribution of each cluster fraction by differential centrifugal sedimentation has proven as an efficient and reliable way to do this. Fractions with uniform populations could be obtained for clusters made from two to four elementary units (Figure 2). With rising aggregation

that did not assume a tetrahedral symmetry. These assemblies can be best described as folded-square configurations (Figure S4). In this context, it should be noted that an open cluster with four interparticle contacts was predicted in computer simulations as a byproduct, when distinct pair interactions are operative among the particles during their assembly into clusters.52 However, to date, we have not found such assemblies in previous works,33 which suggests that their presence might be related to the use of NIPAM as a comonomer, the only significant difference to earlier studies.33,49 All in all, the number of folded-square-shaped clusters is however virtually negligible, in line with predictions by computer simulations.52 For clusters with N = 5, we observe triangular dipyramidal and square pyramidal configurations, which are present in comparable shares. Two isomers are predominant for clusters with N = 6: the octahedral configuration and a polyhedral packing with C2v symmetry, which can be described as pentagonal dipyramid with one particle missing in the fivemembered ring (Figure 1). The latter configuration presents a minimal-energy packing of hard spheres,44 albeit it was not found in computer simulations that reproduce the templatebased assembling strategy.52 In addition to the two major isomers, a rather small number of trigonal prismatic clusters is occasionally found in FESEM images (Figure S4). However, their share of the total quantity of six-particle clusters is negligible. Shape-Related Descriptors. Building colloidal molecules from defined elementary units simplifies the assignment of shape parameters to distinct cluster configurations. The aspect ratio AR is defined as the ratio of the longest dimension of a colloidal cluster to its shortest dimension. As AR equals 1 for spherical particles, deviations from 1 correlate with deviations of a distinct cluster from a sphere. According to Wadell, sphericity Sw = 3 Vc/Vs can be specified as the third root of the ratio between the cluster volume Vc and the volume Vs of a sphere circumscribed around the cluster.63 The sphericity of a perfect sphere equals 1, whereas it is less than 1 for anisotropic particles. Relative anisotropies A can be calculated from the eigenvalues of the translational diffusion tensor along the lines given in ref 17. A equals 0 for spherical particles and converges to 0.75 for infinite rods. The shape parameters for the various clusters shown in Figure 1 are gathered in Table 1. The three parameters allow for a rough classification into two categories: tetrahedral clusters and octahedral clusters have aspect ratios and relative anisotropies, which are equal or close to 1. However, they are distinguished from spheres by lower values of Sw. The counterpart to these is set by the particle doublets, particle triplets, and triangular dipyramidal clusters. These

Figure 2. Fractionation of clusters varying in the number of constituent particles N was accomplished by rate-zonal density gradient centrifugation. Particle size distributions of cluster fractions reveal the excellent sorting accuracy that could be achieved. The analyses were performed by differential centrifugal sedimentation (DCS). The inset FESEM image highlights a fraction of clusters built from five constituents (N = 5). FESEM images of the other fractions are provided in Figure S4.

Table 1. Aspect Ratios AR, Wadell Sphericities Sw, and Relative Anisotropies A of Partially Fused Particle Clusters N

configuration

AR

Sw

A

1 2 3 4 5 5 6 6

single particle doublet triplet tetrahedron triangular dipyramid square pyramid octahedron pentagonal dipyramid minus one

1.00 1.76 1.66 1.00 1.65 1.35 1.00 1.24

1.00 0.71 0.75 0.76 0.73 0.78 0.82 0.75

0.000 0.098 0.071 0.008 0.068 0.013 0.002 0.056

number N, sorting becomes significantly more difficult because differences in the sedimentation coefficients are getting small. With the refined procedure of particle recovery, however, it was possible to get almost pure fractions of clusters with N = 5 and 6. The presence of traces of smaller colloids (predominantly single particles) is secondary in this case because the largest species will dominate in scattering experiments. Separation has E


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Langmuir reached its limit with the six-particle clusters. The differences in sedimentation coefficients of clusters with N > 6 are too small to allow for proper isolation. Clusters as Brownian Particles. According to the classification made by the IUPAC, a colloid “should have at least in one direction a dimension roughly between 1 nm and 1 μm”.66 If transferred to a dense cluster, this criterion applies to all three directions. As long as the global dimensions of the clusters are kept below the size limit of 1 μm, Brownian diffusion will prevail over gravitational sedimentation. To pinpoint this aspect, the gravitational length lg is an appropriate measure67 lg =

kT DT = B vS g ΔρVc

(1)

where kBT is the thermal energy, g is the acceleration due to gravity, Δρ is the difference between cluster density and solvent density, and Vc is the cluster volume. The gravitational length is defined as the ratio between the single-particle diffusion coefficient DT and the Stokes’ sedimentation velocity vs, and can be regarded as the minimum distance an object has to settle before the sedimentation drift exceeds the root-meansquare displacement of Brownian motion.67 By way of illustration, the gravitational length of a polystyrene microsphere with diameter 1.94 μm is already as short as its own diameter, whereas the gravitational lengths of the clusters shown in Figure 1 exceed their hydrodynamic diameter by factors ranging from 3271 to 30 234 (Table S1). Modeling of Spherical Particles and Clusters Built Thereof. The particular suitability of clusters built from uniform spheres is also reflected in the simplicity of model building. It has turned out that it is not necessary to construct a surface model of each cluster species and place the beads on top of it. A single model representing the cluster constituents is sufficient. The clusters can then be treated as combinations of several spherical shells (Figure 1C). For this purpose, the center-point coordinates of the individual shells need to be aligned with the configuration of the cluster being explored. Each cluster constituent is represented as a smooth spherical shell of densely packed spherical frictional elements. Equally sized beads are placed as stacking rings on a sphere with a diameter of 72.5 nm. This size is based on the hydrodynamic radius of the particles and consequently considers the hydration layer at the particle surface. The beads within the stacking rings of varying radii are positioned by specification of their coordinates so that each bead is nearly tangent to its neighbors. This way, a maximum amount of frictional beads can be placed onto the surface. In line with earlier works and the explanations given above,18,38 partial fusion of its constituents has to be considered when designing the model for a particle cluster. This partial fusion is reflected by (i) a broad contact area among the particles and (ii) a center-to-center distance below the value expected for touching spheres. This can be easily taken into account by an overlap between the spherical shells representing the constituents of the cluster (Figure 3). Evaluation of TEM images of particle doublets showed that the center-to-center distance is on average reduced by 34 ± 3 nm compared to touching spheres. However, there is no volume overlap in the experimental clusters; instead, a portion of the material is pushed to the side. This is reflected by pronounced contact areas between adjacent constituents.

Figure 3. Prediction of hydrodynamic properties: calculations of the translational diffusion coefficients DT based on bead-shell models representing a particle doublet are shown as an example. Variable model parameters are the bead radius σ and the overlap among the spherical shells onto which the beads are located. The values for DT converge into the limit of a smooth particle surface when the bead size is decreased, accompanied by an increase in the number of beads. The model calculations were performed in steps of 5 nm for overlaps ranging from 15 to 55 nm (from top to bottom).

Moreover, the degree of overlap may also differ among the various cluster configurations. It is for this reason that hydrodynamic properties were calculated for various overlaps, ranging from 15 to 55 nm reductions of the specified center-tocenter distance for touching spheres, which is equal to the particle diameter of 145 nm (Figure 3). Parallel to this, the calculations were carried out for various bead radii. For example, bead radii were decreased from 15 to 10 nm, which, in turn, increases the bead number representing a particle doublet from 184 to 624 (Figure 3). Finally, extrapolation of the hydrodynamic data to σ → 0 and NB → ∞ converges into the continuous shell limit, which allows for a prediction of the exact values.16,17 A comparison of the predicted hydrodynamic quantities to the experimental data gathered in Tables 3 and 4 revealed that a center-to-center distance of 105 nm is particularly suitable and universally applicable to the various cluster configurations shown in Figure 1. In other words, the hydrodynamic properties of all cluster species studied in this work can be accurately predicted using an array of overlapping smooth spherical shells, in which adjacent shells overlap by 14% of their diameter. For the sake of comparison, we also considered the scenario of nonoverlapping spherical shells representing a dimer of touching spheres (Table S2). The deviations of the predicted quantities to the experimental values for s, DT, and DR were −5.3, −4,4, and −28.1%, respectively, whereas they amount to only −0.4, +0.8, and −3.4% when considering overlapping shells. This underlines the importance of looking at the clusters as partially fused particle assemblies. Intrinsic Viscosities of Colloidal Clusters. Einstein showed that there is a linear relationship between the relative viscosity ηrel and the volume fraction ϕ of a dilute dispersion,61 which reads as F


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Langmuir ηrel =

η ≈ 1 + [η ]ϕ η0

directly measure sedimentation coefficients of shape-anisotropic colloids such as gold nanorods.53 The combined method of particle separation and detection can be also applied to determine sedimentation coefficients within a multimodal mixture of particles in a single centrifugal run. It is therefore not necessary to have the colloidal clusters sorted into uniform species at this stage. The sedimentation coefficient is given by the effective mass meff = mp − mf and the friction coefficient f of a given particle. The effective mass denotes the difference between the particle mass mp and the mass of the fluid mf that is displaced by the particle. In centrifugal fields, the sedimentation coefficient can also be expressed as the settling velocity vN = (dR/dt) divided by the centrifugal acceleration ωR2, where ω denotes the angular velocity and R is the distance to the rotor axis.

(2)

where η is the viscosity of the dispersion and η0 is the viscosity of the solvent. The intrinsic viscosity [η] is a dimensionless quantity that measures the contribution of the suspended particles to the viscosity of the dispersion. [η] depends sensitively on particle shape. It equals 2.5 for hard spheres at infinite dilution.61 Garcı ́a de la Torre and co-workers established improved calculation routines based on bead-shell modeling that enable the prediction of intrinsic viscosities for particles of arbitrary shape. In particular, calculations were made for a broad range of model structures built of touching spheres.17 Some of these assemblies display configurations, which are similar to the particle clusters shown in Figure 1. However, there is a substantial difference. As stated above, the experimental clusters differ from assemblies of touching spheres by partial fusion of their elementary units. To investigate the influence on intrinsic viscosity, calculations based on models of overlapping and touching spherical shells of frictional elements were performed. Table 2 presents intrinsic viscosities predicted for various cluster configurations together with the values reported by

sN = 1 =

configuration

[η]fc

[η]ts

[η]lit

1 2 3 4 5 5 6 6


2.463 2.840 2.810 2.651 2.863 2.755 2.673 2.784

2.463 3.389 3.466 3.484 3.618 3.538 3.402 3.509

2.473 3.413 3.462 3.487

f

=

vN ω 2R

=

(dR /dt ) ω 2R

(3)

For spherical particles, the friction coefficient from Stokes’ law can be used, which is expressed as f St = 6πηrh. The clusters, however, have geometries to which Stokes’ law is not applicable. This issue can be solved if the single particles (N = 1) present in the cluster mixture are taken as an internal reference. Due to their spherical shape, their sedimentation coefficient can be calculated by

Table 2. Comparison of Intrinsic Viscosities [η] Calculated for Partially Fused Clusters (fc) and Clusters of Touching Spheres (ts) with Values Taken from Ref 17 (lit) N

mp − mf

sN = 1 =

4 3 (ρp − ρf ) πr 3 6πηrh

(4)

where η = 0.891 g m−1 s−1 is the fluid viscosity, ρf = 0.997 g cm−3 is the fluid density, and ρp = 1.057 g cm−1 is the particle density at 25 °C. Please note that we prefer to distinguish between the volume-related particle radius r (experimentally derived by TEM) and the hydrodynamic particle radius rh (experimentally obtained from DLS), albeit the difference is relatively small in the present case. As stated above, the PS particles are bearing a hydrophilic surface layer that is partially swollen with water resulting in a hydrodynamic radius of 72.5 nm, which slightly exceeds the TEM radius of 72.0 nm. Taking this into account, the sedimentation coefficient of single PS particles reads as sN=1 = 762 Sv. The sedimentation coefficient of a cluster assembled from N constituent particles sN can be directly obtained by DCS because sN is inversely proportional to the settling time tN, which is the primary quantity measured53

3.402

Garcı ́a de la Torre and co-workers. As expected, the intrinsic viscosities predicted for clusters of touching constituents are virtually identical to the values found in the literature. This reflects the high reliability of hydrodynamic modeling based on bead-shell models. Partial fusion among the constituent particles has a major impact on intrinsic viscosity. It raises the level of compactness as expressed through a higher sphericity and a lower aspect ratio (Table 1). For this reason, intrinsic viscosities of fused particle clusters are closer to the boundary value of 2.5 set by rigid spheres (Table 2). A particularly high level of sphericity is realized in clusters with tetrahedral and octahedral symmetries. Indeed, fused particle clusters with these symmetries have intrinsic viscosities, which hardly exceed 2.5. However, if the degree of fusion among the particle is reduced, intrinsic viscosities of tetrahedral and octahedral clusters will clearly distinguish from 2.5. Apart from this, a general trend toward higher intrinsic viscosities is found with lower sphericities and higher aspect ratios. This is particularly reflected in intrinsic viscosities of linear particle assemblies (Table S3). Sedimentation in a Centrifugal Field. The sedimentation behavior of a colloidal particle is captured in a physical quantity known as the sedimentation coefficient. In a recent publication, it has been shown that DCS is ideally suited to

sN =

ln(RD/R 0) ω 2tN

=

k tN

(5)

This relationship results from eq 3 after separation of the variables and integration. Within the DCS run, each cluster is migrating at a specific settling rate from the starting position R0 to the detector position RD within the hollow disk rotating at an angular velocity ω (Figure 4). The parameters R0, RD, and ω are identical to all particulate species and can thus be merged into a single constant k. This constant can be accessed via the settling time of nonaggregated spherical PS particles left in the cluster mixture. Their sedimentation coefficient sN=1 is given above. Consequently, an immediate and reliable access to the sedimentation coefficients of the various cluster species is achieved. G


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values derived by calculation of a shape-related friction coefficient using the bead-shell model. Diffusion Studies. With regard to a comprehensive understanding of the hydrodynamic behavior of complex particles, we are now addressing the diffusion of colloidal clusters, albeit certain aspects in this direction had been already explored in an earlier study on similar colloids.38 We believe that the progress achieved with regard to sorting accuracy and hydrodynamic modeling justifies an enhanced look into rotational and translational diffusion of colloidal molecules, particularly as the diffusion of nanoscale clusters is poorly studied to date. Experiments were carried out using a combination of farfield dynamic light scattering (DLS) and depolarized dynamic light scattering (DDLS). DLS is the most common method to study translational diffusion of colloidal particles.68 Mean translational diffusion coefficients are obtained if the particles exhibit anisotropic shapes.60,69,70 DLS is, however, not capable of sensing rotational motion of polymer colloids because decorrelation of the scattered light is almost exclusively due to translational diffusion. DDLS corresponds to a polarizer/analyzer version of DLS that can be used to explore translational and rotational motion of optically anisotropic colloids.71 With regard to rotational diffusion, the rotation around the axis perpendicular to the main symmetry axis is probed.38 A schematic representation of the experimental setup is found in Figure 5A. Particles that are anisotropic in shape or composition can depolarize the incident light, which is usually vertically polarized (v). Using a polarizer, the vertically (IvV) and horizontally (IvH) polarized components of the scattered light can be measured separately. As a result, two intensity autocorrelation functions g(vV) and 2 g(vH) were obtained for each cluster species. Since the dilute 2 suspensions behave ergodic, the measured autocorrelation functions g2 are related to the autocorrelation functions of the scattered electric field g1 via the Siegert relation, which reads as

Figure 4. Sedimentation analysis: a mixture of clusters is placed on a density gradient, which is located within a hollow disk rotating at 24 000 rpm (see inset illustration). During their migration from the starting position R0 to the detector position RD, the different cluster populations are getting sorted according to their sedimentation coefficients. The latter are reciprocal to the sedimentation time at which a given species reaches R0. Owing to the high resolving power of the centrifugal method, the cluster populations are recorded separately from one another, as detected by light absorption at 405 nm (main graph).

Figure 4 shows that DCS permits high resolution, even for cluster species whose sedimentation coefficients differ just by a few percent. This is a particular strength of using a centrifugal fractionation method in nanoparticle analytics. In Table 3, experimental sedimentation coefficients obtained by DCS are compared to predicted values based on the shell model. The exact match of the values for single particles refers to their use as a calibration standard. The differences between experimental and predicted values for the various cluster species are less than 4%, which is within the limits of experimental errors. Yet it is striking that the predicted values are systematically somewhat higher. This may be probably due to fact that we kept the overlap among the spherical shells of frictional elements constant for all species (Figure 1C). In the experimental clusters, the number of contacts among the particles grows with the aggregation number. The capillary forces that occur during particle assembly are spread across the particles, which may result in a lower degree of deformation at the junctions with growing aggregation number. However, it is difficult to ascertain this idea safely from electron micrographs (Figure 1B). Apart from these minor deviations, the experimental sedimentation coefficients are in good agreement with the

g2 = 1 + β ·|g1|2

where β is a constant of order 1, which is referring to the alignment of the laser in the experimental setup. In the vV operational mode with the incident and scattered light having parallel polarization directors, the autocorrelation functions g1 and g2 follow a single exponential decay (blue curve in Figure 5B). g1(vV) = A exp( −ΓvV ·t )

configuration

sexp (Sv)

spred (Sv)

1 2 3 4 5 5 6 6


770 1170 1522 1850 2145 2145 2440 2440

770 1166 1536 1911 2155 2164 2601 2571

(7)

where ΓvV = DT·q is the decay rate of the polarized (vV) autocorrelation function, which provides access to the translational diffusion coefficient DT. The absolute value of the scattering vector q is given as q = (4πns/λ) sin(Θ/2), where ns is the refractive index of the medium, λ is the wavelength, and Θ is the scattering angle. Equation 7 applies also to a DLS experiment carried out in the absence of a polarizer. In principle, the autocorrelation curve in vV operation could contain an additional contribution attributed to a fast process that is seen in the vH operational mode. The latter plays a significant role in noble-metal nanoparticles that show plasmon-enhanced scattering.72 In the present case of probing dielectric particles, this contribution to the decorrelation of the scattered light is however negligible. The autocorrelation functions measured in the vH configuration can be described as the sum of two exponentials 2

Table 3. Experimental and Predicted Sedimentation Coefficients s of Colloidal Clusters N

(6)

H


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depolarized signal that is sufficient to allow for a reliable analysis could be detected for single particles and tetrahedral clusters, but not for octahedral clusters. The latter species contain minor quantities of foreign clusters with higher relative anisotropies A. The single particles have shapes that slightly deviate from a perfect sphere (see Figure S2). Additional optical anisotropy could arise from an inhomogeneous comonomer distribution within the particles and at the surface. Optical anisotropy of tetrahedral clusters should primarily arise from the polydispersity of their constituents, although they are narrowly dispersed. In contrast, particle doublets, triplets, triangular dipyramids, and square pyramids show a much higher tendency to depolarize light. This is due to the anisotropic shape of these clusters, which is the prevailing factor for depolarization in this case. The considerations set out above are only valid if rotational diffusion is decoupled from translation diffusion. This can be assessed from the absolute value of the scattering vector q and the mean cluster size d. Given that q·d < 5, coupling of rotational diffusion with translational diffusion is essentially absent.69 Since this is the case for the experiments shown here, higher-order translational−rotational terms could be neglected. It should be noted, however, that data fitting was rigorously limited to the areas in which this condition holds true. The use of stretched exponentials in eqs 7 and 8, as employed in other work on nanoparticle dynamics,72 was not necessary because of the uniform size of the clusters. The diffusion coefficients for translation DT and rotation DR are obtained by plotting ΓvV and ΓvH against q2 and performing linear regressions using the above-mentioned equations (inset in Figure 5B and Figure S5). Linear regression was carried out over scattering angles from 25 to 135° in vV configuration and from 25 to 67.5° in vH operational mode. An exemption had to be made for single particles (N = 1). Reliable depolarized autocorrelation functions were only obtained for scattering angles of 30° and above.38 DLS and DDLS studies were done in the dilute regime. This ensures that the particles underlie Brownian short-time selfdiffusion. Moreover, multiple scattering may be excluded.69 All experimental work was carried out in 10−4 M NaCl solutions, unless noted otherwise. As a complementary measure, consideration was given to a salt dependence of the hydrodynamic behavior. For this purpose, the diffusion of particle doublets was studied at varying concentrations of added salt (10−2−10−4 M NaCl). No salt effect on rotational and translational diffusion was observed (Figure S6). This indicates that electrostatic repulsions among the clusters do not prevail in the dilute regime, notwithstanding the longrange nature of electrostatic interactions. Table 4 shows the experimental diffusion coefficients in comparison to the values predicted for the different cluster species. In all cases, an excellent agreement between experiment and hydrodynamic modeling is found, which underlines that the clusters offer a symbiotic relationship between complexity and shape accuracy. Deviations from the predicted values are less than 2% in the case of DT and less than 5% for DR. The larger deviations for DR are explained by the strong dependence of rotational motion on particle size. For spherical particles, DT scales with rh−1, whereas DR is proportional to rh−3. Despite their complex shapes, a similar trend is envisaged for dense particle clusters.

Figure 5. (A) Schematic representation of polarized (vV) and depolarized (vH) dynamic light scattering configurations using a polarizer/analyzer scheme. The vV experiment gives a signal that is governed by the translational diffusion coefficient DT, whereas the signal of the vH experiment provides access to DT as well as to the rotational diffusion coefficient DR. See text for further explanations. (B) Intensity autocorrelation functions g2 of doublet clusters, as measured in polarized (vV) configuration (blue squares) and depolarized (vH) configuration (red circles). The gray solid lines represent fits of eqs 6−8 to the experimental data measured in vV and vH configuration. Inset: linear relationships are obtained when the decay rates of the polarized (ΓvV: blue squares) and depolarized (ΓvH: red circles) autocorrelation functions are plotted against the square of the magnitude of the scattering vectors q. The slopes of both relaxation modes are equivalent to DT. The blue solid line representing the data points of the slow (vV) relaxation mode is passing through the origin, in line with ΓvV = DT· q2. The intercept of the red solid line representing the fast (vH) relaxation mode corresponds to a sixfold of DR, pursuant to ΓvH = DT·q2 + 6DR.

g1(vH) = A exp( −ΓvV ·t ) + B exp( −ΓvH·t )

(8)

where the second exponential is the actual contribution from depolarized scattering. Its decay rate is defined as ΓvH = DT·q2 + 6DR. Plotting ΓvH against q2 and performing a linear regression give access to DT and DR (Figure S5). The first term in eq 8 arises from the limited extinction ratio (10−7) of the polarizer. It yields a residual portion of polarized scattering, which is not negligible because of the rather low intensity of the depolarized scattering. The intensity of the depolarized component of the scattered light scales with the mean-squared optical anisotropy of the particles. In the present case, optical anisotropy is linked to shape anisotropy. For this reason, the capability of a distinct cluster to depolarize light is connected to its relative anisotropy A (Table 1). A and, as a result, depolarized scattering amount to zero for optically isotropic particles. Apart from the spherical particles, this should also apply to clusters with tetrahedral and octahedral symmetries. Nevertheless, a poor I


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Langmuir Table 4. Experimental and Predicted Diffusion Coefficients for Translation (DT) and Rotation (DR) of Colloidal Clusters DT,exp N

configuration

1 2 3 4 5

single particle doublet triplet tetrahedron triangular dipyramid tetragonal pyramid octahedron pentagonal dipyramid minus one

5 6 6

(10

−12

2

DT,pred −1

m s ) (10

−12

2

−1

m s )

DR,exp

DR,pred

−1

(s )

(s−1)

3.35 2.54 2.22 2.04 1.91

3.38 2.56 2.24 2.05 1.93

472 178 157 131 70

482 172 151 132 71

1.91 1.86 1.86

1.90 1.90 1.88

70

75 70 67

Figure 6. Assignment of diffusion coefficients (red: DR; blue: DT) to the corresponding sedimentation coefficients s for the colloidal clusters shown as inset illustrations. Filled symbols show experimental results from DLS and DDLS. Data predicted from hydrodynamic modeling are given by open symbols. Predicted values for the five- and six-particle clusters with two isomers are represented by their arithmetic averages because differences are small. Lines are guide to the eyes without physical significance.

Cross-Validation of Hydrodynamic Quantities. In the dilute regime, the translational diffusion coefficient is given by the Einstein relation, which states that DT is inversely proportional to the friction coefficient f.73 The sedimentation coefficient s is inversely proportional to f as well. Hence, both hydrodynamic quantities are linked to one another via the same parameter. It should be noted that a different friction coefficient f R applies to rotational diffusion, which scales with the third power of particle size, whereas f increases linearly with particle size. Division of s by DT gives the Svedberg equation, which reads as73 mp(1 − ρf /ρp ) s = DT kT

and 6 are rather small and do not reflect the size increase. This is a clear hint that shape has a major impact on the rotational diffusion of the colloidal clusters. To gain further insights into this, the hydrodynamic quantities are compared to the corresponding values of spheres of equal volume (Figure 7). Based on this normalization, deviations from unity indicate the degree to which the hydrodynamic behavior of a given cluster specified by s, DT, and DR differs from the behavior of a spherical particle. In other words, the data gathered in Figure 7 permits a quick evaluation to what extent shape determines the hydrodynamic behavior of a given cluster species. As expected, s and DT values come closest to the behavior of spheres because of the high sphericities of the clusters (Figure 7A,B). However, all values are clearly below the ones expected for spheres. This is explained by the larger surface areas of the clusters, which is associated with a higher degree of friction. Due to its high sphericity (Table 1), the mass distribution of an octahedral cluster comes closest to the one of a sphere. This is evidenced by a convergence of s and DT toward the spherical limit. For the sake of comparison, we also predicted s and DT values for linear clusters built of a variable number of constituents N. In this regard, we are continuing the lines of an earlier experimental study.39 Model building was accomplished by considering linear chains of overlapping spheres with frictional elements at the surface (Figure S8). The overlap between adjacent shells was again set at 14% of the diameter of an individual spherical shell. It turns out that the hydrodynamic behavior of linear assemblies is rather different to dense clusters. Aspect ratios AR, relative anisotropies A, and total surface areas gradually increase with N, whereas Wadell sphericities Sw significantly decrease (Table S3). As a result, sedimentation and translational motion is significantly slowed down compared to spheres of equal volume (Figure 7A,B). As indicated above, rotational motion is much more sensitive to shape than translational motion. Clear evidence of this is found in Figure 7C. In the light of the above, linear clusters show continually growing deviations from spherical

(9)

where kT is the thermal energy. The right-hand side of the equation depends on parameters that are known. This enables a consistency check against the data gathered in Tables 3 and 4. The cross-validation including data from different experiments and theoretical modeling is shown in Figure S7. The experimental values as well as the predicted values of (s/DT) from theoretical modeling are in excellent agreement to the values calculated according to eq 9. This clearly demonstrates the robustness of the hydrodynamic data in Tables 3 and 4, irrespective of the path over which it was collected. It shows once again that clusters made of spherical elementary units are ideal models for particles with complex shapes. Impact of Shape on Hydrodynamic Properties. Based on the results in the previous sections, it is now possible to define a specific set of hydrodynamic values (s, DT, and DR) for each cluster species. This is expressed in a graphical form in Figure 6. The two diffusion coefficients are plotted against the respective sedimentation coefficient for each cluster species. Again, an excellent agreement between experimental and predicted data gets apparent. The evolution of DT with increasing N behaves largely inversely proportional to the mean cluster size (Table 4). A parallel can thus be drawn to the Stokes−Einstein equation, albeit it is only valid for spheres. In this context, it should be noted that the aspect ratios of the clusters gathered in Figure 1 are below 2 (Table 1). Consequently, shape has a secondary impact on the translational motion of complex colloids, provided that they have low aspect ratios, and therefore high sphericity values. The situation is different from rotational motion. DR decays less regular with rising N (Table 4). For example, deviations among rotational diffusion coefficients of clusters with N = 5 J


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Figure 7. Cross-comparison of hydrodynamic quantities of densely packed (red spheres) and linear (blue spheres) particle assemblies with spheres of equal volumes (black stars). Filled symbols represent experimental data from DCS, DLS, and DDLS. Quantities predicted by hydrodynamic modeling are given by open symbols. All data are normalized to values to be expected for spheres of equal volume. Deviations from 1 thus show the impact of shape on the hydrodynamic behavior of a cluster with N constituents. (A) Relative sedimentation coefficients, (B) relative translational diffusion coefficients, (C) relative rotational diffusion coefficients, and (D) intrinsic viscosities normalized to the value of 2.463 for spherical particles at infinite dilution. The lines are guide to the eye without physical significance.

behavior with increasing N. The rodlike shape drastically reduces rotational motion because a substantial part of the mass is located away from the center of gravity. With regard to dense particle clusters, the picture is more complex, as the influence of a distinct geometry becomes apparent. Apart from the particle doublet, marked deviations from spheres of equal volume are observed for clusters with N = 3 and 5 (Figure 7C). This is due to the fact that particle triplets and triangular dipyramids have high aspect ratios among the experimental clusters. Consequently, deviations should be small for particle tetrahedrons and octahedrons because they have aspect ratios of unity and high sphericities (Table 1). Indeed, minor deviations from rotational motion of spheres are observed for clusters with N = 4 and 6 (Figure 7C). This is also encouraged by the low aspect ratio of the six-particle clusters with C2v configuration (pentagonal dipyramid minus one). Similar conclusions can be made for intrinsic viscosities. The same dependence on cluster configurations is found, albeit deviations among individual cluster species are smaller (Figure 7D). Consequently, a coherent overall picture of the hydrodynamic behavior, sorted into four individual quantities, is observed.

analytical methods. Experiments and hydrodynamic modeling thus complement each other in the best manner. This is particularly advantageous in cases where a certain strategy can be only realized with difficulties. The experimental methods and the modeling strategy are ready to be transferred directly to other particles and assemblies built thereof. For example, this may include finite structures built from cube-shaped particles. Finding ways to extend the studies on colloidal clusters from the dilute to the semiconcentrated regime remains a challenge for the future. Making colloidal molecules available at larger scales is an important prerequisite on the experimental side. In addition, suitable hydrodynamic models are required for the semiconcentrated or concentrated regime. This could, in turn, contribute to a better understanding of the formation of crystalline, glassy, or gel-like superstructures of complex particles, which is intimately tied to diffusion, sedimentation, and viscosity.

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ASSOCIATED CONTENT

S Supporting Information *

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b02999.

CONCLUSIONS We explored the hydrodynamics of colloidal clusters as model systems for complex nanoparticles. Colloidal clusters combine a high level of structural diversity with well-defined geometries and symmetries. As a result of their defined shapes, model building of colloidal clusters can be streamlined through a modular layout on the basis of the constituent particles. Hydrodynamic quantities predicted from these modular models reproduce experimental data determined from different

Hydrodynamic quantities for clusters of touching spheres and linear particle assemblies; size distribution and sphericity analysis of PS-co-PNIPAM particles; stability analysis of colloidal clusters; FESEM images of cluster fractions; (D)DLS measurements; crossvalidation of hydrodynamic quantities; and model building for linear assemblies (PDF) K


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alexander Wittemann: 0000-0002-8822-779X Author Contributions †

R.S. and C.S.P. contributed equally to this work.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors benefited from the people and equipment of the Particle Analysis Center (PAC) and the Nanostructure Laboratory (Nanolab) at the campus. In particular, they acknowledge Brigitte Bö ssenecker for assistance during nanoparticle fractionation. Marina Krumova is thanked for her support with TEM. Philipp S. Menold and Maxim Schlegel are thanked for synthesis of PS-co-PNIPAM particles. The financial contribution of the Struktur- und Innovationsfonds für die Forschung in Baden-Württemberg (SI-BW) facilitated the acquisitions of the DLS/DDLS instrument and the preparative ultracentrifuge. Financial support from the Deutsche Forschungsgemeinschaft (DFG) within SFB 1214/ A10 is gratefully acknowledged.

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ABBREVIATIONS DCS, differential centrifugal sedimentation; DDLS, depolarized dynamic light scattering; DLS, dynamic light scattering; DVB, divinyl benzene; FESEM, field emission scanning electron microscopy; NIPAM, N-isopropylacrylamide; PNIPAM, poly(N-isopropylacrylamide); PS-co-PNIPAM, polystyrene-co-poly(N-isopropylacrylamide); SDS, sodium dodecyl sulfate; TEM, transmission electron microscopy

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REFERENCES

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