Spherical probes for simultaneous measurement of rotational and translational diffusion in 3 dimensions

Real time visualization and tracking of colloidal particles with 3D resolution is essential for probing the local structure and dynamics in complex fluids. Although tracking translational motion of spherical colloids is well-known, accessing rotational dynamics of such particles remains a great challenge. Here, we report a novel approach of using fluorescently labeled raspberry-like colloids with an optical anisotropy to concurrently track translational and rotational dynamics in 3 dimensions. The raspberry-like particles are coated by a silica layer of adjustable thickness, which allows tuning the surface roughness. The synthesis and applicability of the proposed method is demonstrated by two types of probes: rough and smoothened. The accuracy of measuring Mean Squared (Angular) Displacements are also demonstrated by using these 2 probes dispersed in 2 different solvents. The presented 3D trackable colloids offer a high potential for wide range of applications and studies, such as probing crystallization dynamics, phase transitions and the effect of surface roughness on diffusion.


Introduction
Studying colloidal dynamics via time-resolved locations of individual particles can provide microscopic insights to a variety of physical phenomena in different phases of matter [1][2][3][4][5] . Especially in systems with an intrinsic inhomogeneity, correlating the dynamics with the location inside the material can provide unique information that cannot be obtained with techniques that take an ensemble average over all particles, or any other bulk methods. Examples of research areas where local translational dynamics have been analyzed for this purpose include dense suspensions 6 , glassy materials [7][8][9] , polymer networks 10,11 , spatially confined materials [12][13][14][15] , food products 16 ,cell biology 17,18 and virus infection mechanisms 19 .
In contrast to the many studies on translational dynamics, research related to rotational dynamics is rather scarce. This has been ascribed to a lack of experimental approaches for capturing rotational motion 20 . Rotational dynamics can shed a unique light onto various dynamic phenomena that cannot be accessed only with translational degrees of freedom, such as motion in glassy and supercooled states (where decoupling between translational and rotational diffusion emerges) [21][22][23] ; particle adsorption and self-assembly at fluid interfaces 24,25 ; interfacial dynamics at solid-liquid interfaces 26 and biological interactions; such as viruses binding to membranes 27 Ensemble averaged rotational diffusion of colloids has been studied by techniques such as fluorescence recovery after photobleaching (FRAP) 28,29 , depolarized dynamic light scattering 30,31 and nuclear magnetic resonance (NMR) spectroscopy 32 . These bulk methods fall short in identifying local (dynamic) heterogeneities. Measuring rotational diffusion via individual probe particles can provide valuable local information in terms of dynamic length scales and structural signatures of complex fluids, such as local defects and crosslink densities in polymer networks 33 , local rheology of soft materials 34 , or intrinsic features of active fluids in biochemical processes 35 . In recent years, different strategies have been used towards tracking the rotational motion of diffusive probes of both spherical and anisotropic particle shapes such as rods, ellipsoids, and particle clusters [36][37][38][39][40] . Here, geometrical anisotropy is widely utilized because it naturally provides an identifiable optical axis to track angular displacements 20 . For spherical colloids, the lack of such a natural frame of reference requires a design where the optical isotropy is broken. A common type of such probes is the modulated optical nanoprobes (MOON) 41,42 . This type of Janus particles usually consists of a fluorescent sphere that is half coated with a metal layer. Although these rotational probes are attracting interest in various fields 43 , they have some drawbacks too. Due to the metal coating on one side, the surface chemistry is no longer uniform, and refractive index mismatches with the surrounding medium may compromises the image quality, especially for biological systems and high volume concentrations. Recently a new type of spherical rotational probes was introduced 23,44 . These probes are bicolor or multicolor colloidal spheres with an eccentric core(s) shell structure, requiring multiple excitation wavelengths to be used. Although they provide homogenous surface chemistry, the centers of the core and the shell may not coincide precisely. In the case where multiple cores are utilized, they have to be overgrown to a rather large size, to attain a (near) spherical overall shape.
In this work, we introduce a novel and simple method for i) synthesizing fluorescently labelled raspberry-like spherical probes and ii) using them for simultaneously accessing rotational and translational dynamics in 3D. Our probes are made by densely covering a large silica core with many small SiO2 particles, a fraction of which is fluorescently labeled. By coating these raspberries with a layer of silica of controllable thickness, we obtain particles of variable roughness while maintaining uniform surface chemistry. Optical anisotropy, that is introduced via the fluorescent tracers, allows for simultaneous tracking of the translational and rotational motion of each probe in 3D, using just one fluorescent dye. We demonstrate a proof of concept by tracking 2 types of probes (smoothened and rough) in the dilute regime.

Results
From Raspberries to Rotational Probes. Raspberry particles are synthesized by coating the surface of positively functionalized SiO2 particles (core, r=1.04 ± 0.033 m) with a dense layer of negatively charged small SiO2 particles (berry, r=0.164 ± 0.021 m) via electrostatic heteroaggregation. To preserve mechanical integrity and to modify the surface roughness, a silica layer is overgrown on to these colloids via seeded growth. Figures 1b and 1c   Extracting Rotation and Translation. For optimal visualization, both type of probes are dispersed in refractive index matching solvent mixtures (n=1.45). Solvents are water-glycerol (S1, 1:4 by weight, =59 mPa.s) and water-glycerol-Dimethyl sulfoxide (DMSO) (S2, 2:4:3 by wt., =20 mPa.s). Particle volume fractions are chosen around 0.3%, to approach the dilute limit while keeping enough particles in the image volume for statistical analysis later on. Confocal Scanning Laser Microscopy (CSLM) in fluorescence mode is used to visualize only the labeled berries (see Supporting Information for details of experimental conditions). Their Cartesian coordinates are extracted using well-known particle locating algorithms 45,46 . The located berries are then grouped in clusters to identify to which raspberry they belong; this is achieved using a maximum distance criterion (see Supporting Information for details). Only raspberries that contain 4 or more non-coplanar tracers are kept. This minimum number is required for simultaneously finding the center location (x,y,z) and optical radius (Rfit) of the raspberry particle, which is achieved via least-squares fitting to a sphere. Each obtained center location then provides an origin in a 3D Cartesian coordinate system that allows defining the spatial orientation of the raspberry probes based on the angular displacement of the tracer berries. In this scheme, the translation of each raspberry is extracted from the time-dependent center location, leaving the rotational motion to be measured from the changes in orientation. The latter is achieved using a modified algorithm, based on ref 38 , by calculating angular displacements from a rotational transformation matrix in terms of 3 Euler angles. The key steps involved in dissecting translational and rotational motion are illustrated in Figure 2.
Construction of trajectories from the time-dependent coordinates is achieved via publicly available tracking routines 46,47 . The accuracy of the codes was tested with simulated data (mimicking typical experimental conditions) and gave good agreement ( Figure S3, Supporting Information). The coupled displacement of a group of berries on a raspberry is illustrated in Figure 3a  The mean squared angular displacement (MSAD) is calculated in an analogous manner, but without the need for a drift correction. The rotational diffusion coefficient Dr is obtained from equation (2) below (the detailed explanation of calculation from rotations around 3 principle axis is given in the experimental section and Supporting Information): Validation of Brownian behavior. We now examine the accuracy of measuring the diffusion coefficients for our raspberry probes, in different stages of the data analysis. First, we consider the fitted sphere's radius (Rfit) as extracted from the center locations of its berries. Given the dispersities of the core and berry systems, Rfit should be close the sum of the typical radii: Rfit≈<Rcore>+ <Rberry>, where the brackets indicate an average. Using transmission electron microscopy (TEM), we find Rcore = 1041 ± 33 nm and Rberry = 160 ± 14 nm. Figure 4a shows Rfit to be peaked at ≈1280 nm, giving a fairly close correspondence. The calculated standard deviation of 35 nm is the resultant of the two poly-dispersities and the typical uncertainty in Rfit, which is estimated to be 44 nm.
For the translational trajectories, the removal of drift is an essential correction, unless the lag time τ << Dt/v 2 , with v the drift velocity 48 . In our case, drift analysis also contributes to validation, because the vertical motion is dominated by sedimentation. The latter is illustrated in Figure 4b

Rotational and translational diffusion coefficients.
To assess the accuracy of the measured diffusion coefficients, we compare MSDs and MSADs for the two probe systems (RP, SP) in solvents with different viscosities: S1 (59 ± 3 mPa.s) and S2 (20 ± 1 mPa.s). Both probes are synthesized using the same core and berry particles, but for the SP system, a thicker silica layer has been overgrown to achieve a smoothened surface (Figure 1e). This thicker layer also gives a significant increase in the final probe size (Figure 5a). The MSAD measurements are shown in Figure 5b. Regardless of the solvent, both the RP and SP probes demonstrate a purely diffusive behavior, i.e. a linear increase in MSAD with lag time. Extracted numerical data are summarized in Table 1
where kb is the Boltzmann constant, T is temperature and R is radius. For rough spheres, Dr and Dt should be inversely proportional to the solvent viscosity η. Calculating Di(S1)/Di(S2) with i ∈ , for RP and SP separately, we obtain η(S1)/η(S2)=3.2 ± 0.3, in good agreement with the 3.0 ± 0.2 obtained from the measured viscosities.
Comparing the cases where the same probe system is dispersed in 2 different solvents, a different accuracy assessment can be made by calculating Dt/Dr. Now the viscosity effect is 'divided out' because of the inverse proportionality of both Dt and Dr to η. The results in Table 1  The Rh values in Table 1 are comparable to those from the TEM measurements. Some slight differences between the values obtained from S1 and S2 are found; these might be because each of the 4 raspberry/solvent combinations was explored with a fresh solvent mixture (possibility of slight differences in viscosities) and a limited number of particles (possibly introducing a sampling effect).
A striking observation is that for both RP and SP, the Rh values obtained from Dr correspond better with TEM measurements. Considering the R 3 proportionality of Dr for smooth spheres (while it is ~R -1 for Dt) it is likely that Dr provides a more precise measure for Rh. Considering how close <Rh [Dt]> and <Rh [Dr]> are to those of the TEM measurements, we conclude that the surface roughness (being larger for RP) does not have a discernable effect on either of the two diffusion coefficients in the dilute limit.
Application Scope . Our novel probes can be useful in various cases where simultaneous 3D tracking of individual particle locations and orientations is needed. Due to tunability of the outer layer thickness, surface roughness can be altered. This offers a broad application potential for our particles in many colloidal dynamics studies. At low particle concentrations the effects of roughness on the two diffusion coefficients were too subtle to be measured. However at high concentrations, rough probes can be employed for studying the relation between roughness and diffusion. In the dense regime, particle-particle interactions play an important role in colloidal dynamics 50 , and strong correlations between roughness and jamming have already been found 51-53 . In the smooth limit, our probes can be used for shedding more light on the effect of colloidal interactions on rotational diffusion 54,55 or on transient phenomena like glassy dynamics and crystallization. Also in complex fluids whose structure is not dictated by particles, our (rough or smooth) probes could provide information about local (mechanical) properties. Here the surrounding 'bulk' material could e.g. be polymer solutions/gels or liquid interfaces.
Lastly, the synthesis and utilization of raspberry probes are not limited to the demonstrated methodology. Due to the simple synthesis and the use of only one fluorescent label for simultaneous tracking of two different motions, the 'berry platform' can pave the way for designing similar probes with additional shape isotropy. Applying the same concept while using different materials is yet another direction. Probes could be functionalized as active colloids, soft compressible particles, or serve as probes in bio-mimicking studies to resolve dynamics at biological processes 19,56 .

Conclusions.
We developed a new type of colloidal probes with a near-spherical (i.e. raspberry) shape and homogenous surface chemistry. The use of fluorescent tracers in the shell allows to resolve both the center location and the orientation of each probe. As a proof of concept we demonstrated the simultaneous tracking of translational and rotational motion using time-resolved 3D confocal microscopy. The probes exhibit purely diffusive behavior in the dilute regime, with diffusion coefficients that are similar to theoretical values for smooth spheres of the same size. We envision that our probes can be employed in various cases where simultaneous 3D tracking of individual particle locations and orientations is needed.

Synthesis of Raspberry Probes with Optical Anisotropy.
Dense coverage of the cores by the berries is achieved via electrostatic hetero aggregation. While the presence of (+) amine groups on the cores, and (-) silanol ones on the berries already favors such aggregation, an optimization of the pH is needed to obtain interparticle bonds that are strong enough to prevent detachment by stirring forces. Meanwhile, also the stability (against homo-aggregation) has to be preserved for both cores and berries. Using HNO3 to adjust the pH, Zeta potential measurements (with the Zetasizer) are conducted at varying pH for the aqueous dispersions of the particles. The results (shown in SI Figure   S1) indicate an optimal pH of 4.5, where, coremV and berrymV, fluo.berrymV.
Besides the zeta potentials, also the mixing ratio of the cores and berries has to be considered. The number of berry particles needed to ensure dense coverage on a core particle is estimated by calculating how many berry particles can be fitted into the shell space between a hypothetical sphere with a radius of [Rcore+2Rberry] and a core particle. For our system, this calculation gives 386 berries per core. In practice more berries are needed to ensure colloidal stability throughout the selfassembly process. Especially at the initial stages where the cores are only partially covered by berries, 'bridging aggregation' (berries binding to two cores) must be avoided. To prevent this a 10 times Rotational Tracking. For the analysis of rotational displacements we followed an adapted methodology as described in ref. 38 As reported in detail before 38 Where [x, y, z] coordinates denotes a location of an individual berry tracer at i th time step and R is the rotational transformation matrix. The calculation of R is as given in 38 . After obtaining this transformation matrix, we calculate the rotations around each principle axes. A general rotational matrix has a form of: Where andare the rotations around the principle X, Y, Z axes respectively. This rotation matrix is used to calculate the Euler angles directly 58 (see Supporting Information for details of extraction of angles). In a spherical system, due to the bounded nature of rotation angles, calculation of angular displacements relative to the axis of rotation will yield in displacements greater than a factor of 3/2 of the actual displacements 38,59 .For that reason MSAD values are multiplied for a factor of 2/3 before calculating Dr by using eq n (2).

Supporting Information
S.mov1.: Brownian motion of Rough Probes dispersed in water and gravity settled onto a glass substrate.

S.mov2.:
Reconstruction of the 3D motion of a typical RP dispersed in S1.
-Materials and Experimental conditions -Zeta potential vs pH for the core and fluorescent berry particles.
-TEM size distributions for the core and berry particle systems.