In situ microscopy of the self-assembly of branched nanocrystals in solution

Solution-phase self-assembly of nanocrystals into mesoscale structures is a promising strategy for constructing functional materials from nanoscale components. Liquid environments are key to self-assembly since they allow suspended nanocrystals to diffuse and interact freely, but they also complicate experiments. Real-time observations with single-particle resolution could have transformative impact on our understanding of nanocrystal self-assembly. Here we use real-time in situ imaging by liquid-cell electron microscopy to elucidate the nucleation and growth mechanism and properties of linear chains of octapod-shaped nanocrystals in their native solution environment. Statistical mechanics modelling based on these observations and using the measured chain-length distribution clarifies the relative importance of dipolar and entropic forces in the assembly process and gives direct access to the interparticle interaction. Our results suggest that monomer-resolved in situ imaging combined with modelling can provide unprecedented quantitative insight into the microscopic processes and interactions that govern nanocrystal self-assembly in solution.

The images show movement of octapods in this area. In particular, a large particle (bright) can be clearly distinguished in image (b). This particle is not present in the FOV initially (at time t, panel (a)) and has moved out of the FOV 9 s later (panel (c)). d. High-resolution TEM image of a particle similar to that observed in (b), shown among ordinary octapods. The size of these particles, which constitute a small percentage of nuclei that did not grow into octapods, is ~120 nm. The uninhibited movement of particles of such size confirms that the liquid thickness is >120 nm, consistent with the thickness calculated from lowloss EELS (Supplementary Fig. 3).  (1) and (2) lie within the region with low octapod concentration; positions (3) to (5) are within the high-concentration region. Note that lower contrast corresponds to more electron scattering in the inverted dark-field STEM image. The analysis confirms the visual impression of darker contrast within the highconcentration area near the corner of the viewing window. c. Inverted dark-field STEM image, extracted from supplementary movie 1, for time t = 200 s. Scale bar: 500 nm. d. Image contrast measured at 8 positions marked in c. Positions (1) and (2) lie within the original region with low octapod concentration; positions (6) to (8) are within the remaining high-concentration region at the lower edge of the viewing window; positions (3) to (5) are in areas between longer octapod chains. Note that the STEM contrast at these positions is nearly identical to that in the low-concentration region. Figure 10. Bending flexibility of interlocked 1D octapod chains. a., b. Out-of-plane bending of octapod chains located primarily near the SiN x membrane of the liquid cell. Scale bars: 200 nm. a. Bending of one end of the chain into the bulk toluene solution, a process that facilitates the capturing of short chain segments that assemble in the 3D liquid environment. b. Downward bending near the center of an octapod chain, demonstrating the relatively weak interaction of the chain with the membrane. c. Bending of chains in the plane of the membrane, which is observed to change as a function of time, demonstrating the freedom of the 1D chains to change conformation in the plane. Measured bending radii are: 482 nm (blue), 608 nm (red), and 718 nm (orange). Hard octahedral core of the octapod, consisting of 18 Molecular Dynamics (MD) beads (gold) that interact via a Weeks-Chandler-Anderson (WCA) interaction potential. b. The hard part of the octapod: core (gold) and pods (green, red). Each pod consists of 34 MD beads, which also interact via a WCA interaction potential. The beads are arranged in a triangular configuration that tapers towards the tip, to reproduce the triangular shape observed in the experiment. The use of color is to differentiate between pods that (may) have different physical properties, e.g., positive and negative charge. c. Interaction backbone of the model. The backbone consists of 57 MD beads, which may be endowed with different generic Lennard-Jones interaction potentials or charges; the central bead is indicated in transparent magenta and the pod beads using transparent cyan. d. The full model for octapods, which interact via the calculated van der Waals interactions. The radius of the soft beads is chosen according to the range of the beads' interaction potential.

Supplementary
c. d.
Supplementary Figure 18. The distance-dependence of the total interaction potential between two model octapods for various configurations representative of those found in experiment. The total interaction potential from our simulation model A (red) and from the theoretical calculation of van der Waals (vdW) interaction U vdW (r) (blue) are shown, also see Ref. 1 . Here, the interaction between the octapods is given as a function of the distance between the surfaces r of two octapods that are initially in contact (r = 0) (thick black line) and move away from each other along the line connecting their centers.
The contact point of the simulation model A is defined as the value of the octapod separation, for which the interaction potential is +1 k B T, with k B Boltzmann's constant and T the temperature; set to r = 0 here. H is the length of the ligands that form the capping layer (2.5 nm), which covers the octapods and induces steric repulsion, see Ref. 1 . The thick gray lines indicate the distance for which the octapods' surfaces are separated by one and two capping lengths, respectively. We considered three configurations for which we checked the correspondence between the vdW calculation and the simulation result: interlocking (solid curves), pod-pod contact (dashed), and tip-tip contact (dotted).
Note that agreement between the simulation model A and the theory is reasonable when r > 2H, i.e., outside the range of steric interactions.
Supplementary Figure 19. The interaction potential required to achieve interlocking-chain self-assembly in our molecular dynamics simulations. The total interaction potential U tot (r) between two octapods from our simulation model B (red, solid) is compared to the theoretically calculated van der Waals (vdW) interaction (blue, dashed). The notation is the same as in supplementary Fig. 18.

Supplementary Note 1. Additional Evidence for a Wet Environment during 1D Chain Self-Assembly
The continued diffusion of monomers in all parts of the FOV throughout the observation period provides evidence that the liquid cell remains filled with toluene solvent at all times (Supplementary Fig. 8; Supplementary Movie 7). The STEM contrast between the high-and low-concentration regions in Fig. 2 of the main text could be interpreted as a liquid-gas interface, however, such an interface was not observed anywhere in the viewing window of the liquid cell, neither before nor after the appearance of the octapod chains. Indeed, a quantitative analysis of low-loss electron energy loss (EELS) spectra obtained at different points of the liquid cell show that all parts of the viewing window are filled with liquid, whose thickness varies between 90 nm and 190 nm (Supplementary Figs. [3][4][5][6] and in which diffusion and rotation of suspended octapods is observed even in the thinnest areas ( Supplementary Fig. 3). In the areas in which the self-assembly of linear octapod chains is observed the liquid thickness was greater than 140 nm, both before and after the assembly process. The fact that the liquid thickness remained unchanged over the period of observation of assembly in several areas of the cell (> 1 hour) strongly supports the notion that our experiments do not involve any significant drying of the solvent ( Supplementary Fig. 5).
Hence, the transition seen in Fig. 2 reflects merely a boundary between areas of high and low octapod density. Analysis of the contrast in STEM further supports this conclusion: initially, there is a clear difference in contrast (as perceived visually) between the highand low-concentration regions ( Supplementary Fig. 9 a, b). As octapod chains form in the high-concentration region, the STEM contrast in their vicinity changes sharply and becomes nearly identical to that in the low-concentration region (Supplementary Fig. 9 c, d), because the density of high-Z objects suspended in the solution drops as octapods self-assemble into chains. The residual area in which individual suspended octapods can still be found thus shrinks progressively toward the lower boundary of the FOV. In absence of any significant drying of the solvent, the overall directionality of the selfassembly may be due to the drop in the local concentration of free octapods accompanying chain formation. As newly formed chains deplete their surrounding solution of octapods, this apparently provides an environment that facilitates selfassembly in their vicinity. This, together with our observation that no chains form in lowconcentration areas, suggests the existence of an optimal concentration at which selfassembly proceeds most efficiently.

Supplementary Note 2. Brownian Motion of Individual Octapod Nanocrystals in Toluene
By effectively creating a '2.5D' (i.e., half-space) observation environment, the SiN x membranes of the liquid cell enable high-resolution imaging of self-assembly without significantly altering the underlying solution-phase processes. We used real-time observations of the Brownian motion of individual octapods near the membrane to further characterize this environment. Following individual octapods over long distances is difficult due to frequent jumps outside the field of view, 2,3 or jumps resulting in attachment to chain segments. To suppress these events, we reduced the liquid thickness in the cell compared to the experiments discussed previously. We expect the measured diffusion coefficient to represent a lower bound to that in the thicker cells used for studying self-assembly. Individual octapods move both by translation and rotation. From the mean-square displacement of the translational motion, which is clearly Brownian (Supplementary fig. 16), we estimate a diffusion coefficient for octapods in toluene, D t ≈ 1⋅10 -15 m 2 s -1 . The expected translational diffusion coefficient for an octapod with an estimated hydrodynamic radius of 40 nm in a bulk fluid is D t ≈ 7⋅10 -12 m 2 s -1 , i.e., 3 orders of magnitude larger than the experimentally determined one. This difference can be explained by a reduced diffusivity due to the interaction of the octapods with the SiN x membrane. However, in our case these membrane interactions are relatively weak, compared for example to the significantly dampened dynamics and much lower (10 10 times) diffusion coefficients found previously for the movement of particles in ultrathin liquid layers. 4

Supplementary Note 3. Statistical Mechanical Model
The relative abundance of -octapod chains in equilibrium can be determined from the grand canonical distribution function: Here the integration goes over translational and rotational degrees of freedom of octapods (which are internal degrees of freedom on the chain, with representing the result of integration over all Euler angles of an unconstrained free octapod. H is the interaction Hamiltonian of the system, and is the chemical potential of single (unbound) octapods that depends logarithmically on their density: .
It is a formidable task to calculate the above quantity precisely, even if the interaction potential is known. Instead, we will use a number of simplifications, which, however, will preserve the crucial physical aspects of the problem. First, let us consider a case where the interactions are so strong that the system has a maximal possible number of contacts between octapods, . This implies that out of internal degrees of freedom, are strongly localized due to an interaction potential acting between the pods. First, we can conduct integration over those degrees of freedom, which can be simply represented by a set of relative separations of the pairs of pods in contact, . The result of this integration can be expressed as Here -is the depth of the interaction potential and is a characteristic localization length that arises from the integration around that minimum. Since the integration variables, , are different from the original degrees of freedom, one needs to introduce a proper Jacobean. In order to estimate it, we note that replacing one translational degree of freedom with one bond length results in a factor of the order of 1. It can be estimated in the following manner. If one shifts a specific octapod by a vector and rotates it by a small angle, , the bond length between that octapod and its neighbor changes by the amount where is the position of the i-j contact with respect to the center of the i-th octopod, and is a unit vector along the bond between the two pods in contact. Based on this expression, if we replace one translational or one rotational degrees of freedom with the bond length , the overall expression for the abundance of the -chains gains a factor of the order of 1 or 1/r, respectively. This leads to the following result:

(4)
Here the factor represents an integration of each translational degree of freedom not suppressed by the binding. Another important aspect of this calculation is that due to the cubic symmetry of an octapod, it has equivalent orientations (just like a cube). Since the integration over bond length only accounts for one specific binding topology, the overall distribution has to be multiplied by . This correction largely compensates the role of the prefactor in the original expression. The remaining factor of the order of 1 per degree of freedom, can be absorbed e.g. within the model parameter .
Our final step is to account for the fact that the chain formation does not require individual bonds to be very strong. Instead, each of them can break and re-form without the octapod chain losing its integrity. In fact, this increases the chain stability due to an additional entropy. In the approximation that each of the breaking events is independent of the others, this gives rise to a minimal modification of our previous result: we need to replace each factor with . Here again represents an integration over a translational degree of freedom that emerges after the bond is broken. This leads to the result presented in the main text of the paper:

Supplementary Note 4. Interlocking Chains using Molecular Dynamics Simulations
In this section, we discuss the results of the MD simulation models A and B (see Supplementary Methods 1 below) and relate these to the experimental results, as well as the theoretical calculations performed in the main manuscript.
Model A, Van der Waals Interactions. We consider the interlocking configuration, to relate model A to the theoretical calculations performed in our manuscript. The dimer (interlocking) configuration has 8 pod-pod contacts, because we used a perfectly symmetric octapod with perfect alignment between two octapods to determine the vdW interaction energy. Therefore, the value of the energy per contact at 2.5 nm separation (r = H) is about 0.5 k B T. If we assume that this configuration corresponds to a four contact situation, we obtain about 1 k B T of energy per contact in the simulation model. This is too low to observe self-assembly according to the theoretical model. Indeed, using the procedures described in Supplementary Methods 1, we did not observe self-assembly in any of the simulations that we performed. We performed 10 simulations for each of the number of octapods in the box, thus allowing us to obtain reasonable statistics for a range of particle densities. It should be noted that the particle density φ = N/L 3 for each of the N that we simulated, is in the range of the preparation concentrations used in the experiment, namely: 10 -8 mol l -1 . A single simulation of 20 octapods at a density comparable to the one in the corner of the cell, where the self-assembly was experimentally observed, did not result in interlocking-chain formation either. The 30% difference between the energy of interlocking configuration calculated in Ref. 5 at r = H and the one used in our model A, see supplementary Fig. 18, is insufficient to effect the conclusion that vdW-type interactions are sufficient to induce self-assemlby of the octapods.

Model B, Short-Ranged Beyond-vdW Interactions.
For this interaction, we observed self-assembly into interlocking chains. Our samples contained mostly interlocking chains and only occasionally branched structures, which did not appear long-lived. For greater numbers of particles in the box, mixtures of short and long chains with minimal branching were observed. The maximal chain length for these parameters was found to be 9, longer chains showed a tendency to break. It is therefore likely that these interactions are of sufficient range and strength to account for the self-assembly observed in the experiment. However, we did not observe preferential self-assembly of trimers and limited self-assembly of dimers using this model. In fact, dimers are the first step in the formation of longer oligomers of octapods in these simulations. This indicates that the essential physics of the system is not captured by our simulation model, presumably due to a mismatch in directionality of the interaction potential compared to the experimental system, as well as difficulties in achieving the time scales of the experiment. In summary a substantially longer-ranged interaction potential is required to achieve selfassembly in our simulations than the vdW interaction predicted by our Hamaker-de-Boer calculations. 1 In fact, the decay length of the potential had to be increased by a factor of three for the interlocking configuration to occur, see supplementary Fig. 19. The interaction potential also had to be substantially deeper than we estimated from our previous predictions. This is in agreement with the theoretical model put forward in the main text, which predicts a contact-point interaction strength of ~2.7 kT for the experimentally observed self-assembly. In the experiment a preferential self-assembly of trimers into the interlocking configuration was observed, while stable interlocking dimers were not found. This observation was explained by our theoretical model, and it, together with the above simulation results, allowed us to exclude vdW interactions as the main driving force behind the observed self-assembly. However, excluding a vdW-only based mechanism leaves significant freedom on how to achieve the desired self-assembly within the simulation model. Indeed, a combination of vdW, steric, Coulomb, and other interaction potentials may be at play in our system.

Supplementary Methods. Molecular Dynamics Simulations
In this section, we present the simulation methods used to study the self-assembly observed in the in-situ transmission electron microscopy (TEM) study of octapod-shaped branched nanocrystals (bNCs) described in the main manuscript. We implemented two simple coarse-grained models for the octapod bNCs, labelled A and B, as detailed below. These models are based on Molecular Dynamics (MD) simulations with Langevin dynamics, which were performed using the MD software package Extensible Simulation Package for Research on Soft Matter (ESPResSo). 5 Generic Coarse-Grained Octapod Model. Anisotropic hard-core interactions and shortranged attractive van der Waals (vdW) interactions are considered crucial for the formation of the interlocked structures in bulk suspension, based on our previous study of the octapod self-assembly. 1 Therefore, we coarse-grained the shape and theoretically predicted interactions of the octapods to achieve a model suited for MD simulations. In Refs. 1,6,7 we approximated the shape of the octapod by a triangular mesh and a spherocylindrical model, respectively, in order to endow the model with a hard-core interaction. In this study it proved more convenient to construct our model using a combination of isotropic interaction potentials arranged in an anisotropic way to achieve the same effect, see Supplementary Fig. 17. These interaction potentials are as follows In the representation shown in Supplementary Fig. 17 the various parts of our model have been color coded to indicate differences in the employed interaction potentials. Our model consists of 347 so-called MD beads. The hard core, see supplementary Fig. 17 (a,  b) is composed of 290 MD beads which interact with beads of the same type via a Weeks-Chandler-Anderson (WCA) potential, which in our case is defined by Eq. (6), where r is the inter-bead distance in units of s, k B is Boltzmann's constant, and T the temperature. Note that we cap our potential at 20 k B T in order to allow for the WCA beads to overlap to better match the triangular cross section of the pods, without this leading to extremely high energies. The 57 beads belonging to the ranged-interaction backbone, see Supplementary Fig. 17 c, interact with each other according to the generic Lennard-Jones (LJ) interaction potential specified in Eq. (7), where i, j are the indices of the bead types; e ij gives the strength of the interaction potential in units of k B T; R ij is a measure for the size of the particle in units of s; m ij and n ij are exponents that give the decay of the repulsive and attractive part of the potential, provided m ij > n ij > 3, respectively -the latter constraint implying a short-ranged and truncateable potential; s ij is a shift in the value of minimum of the potential and is dimensionless; and finally c ij is the range at which the potential is cut, which also has units of s. To ensure that Newton's third law is obeyed all coefficients are symmetric under exchange of i and j. The values of the above parameters were tuned to ensure that the theoretically predicted vdW attractions are accurately reproduced, as we will detail shortly. All beads are connected to a central bead via a rigid virtual bond, 5 such that the entire octapod retains its shape during the simulation. The forces and torques are translated back to this central bead. This creates a rigid structure that is free to translate and rotate. MD beads from one octapod do not bond with the central bead of another. Our change in modeling from triangular-tessellation (Refs. 1,6,7 ) to cluster-based should not lead to an oversimplification of the hard-core interaction between octapods, since we have already demonstrated that the use of spherocylindrical elements leads to qualitatively similar behavior for this interaction compared to that of a triangular-meshbased model. 6,7 Parameter Mapping. In order to compare simulation models to the vdW calculations of Ref. 1,6,7 , we mapped the MD units onto the physical units using experimentally reasonable parameters. The effective hard-core arm length of our model is L = 5.5 σ and the length of the arms in the experimental system is ~60 nm, therefore the MD base unit of length σ ≈ 11 nm. The liquid in which the octapods are suspended is assumed to be at room temperature, i.e., 300 K, from which we obtain the value for the MD base unit of energy of ε ≡ k B T ≈ 4.1⋅10 -21 J. The final ingredient required to fix all the MD unit conversions is the mass of a cube with volume σ 3 , or, equivalently, the density of the material. We chose the density to be that of CdSe: ρ = 5.8⋅10 3 kg m -3 , which sets the MD base unit of mass to m = 7.7⋅10 -21 kg. These three units, length, energy, and mass, allow us to calculate the MD base unit of time: τ = mσ 2 ε ≈1.5 ⋅10 −8 s. Using this mapping we were able to quantify the difference between our simulation model and the vdW interaction calculated in our previous work. 1 Matching Molecular Dynamics Parameters to Experiment. We tuned the parameters for the simulations to model the experimental system. We started by introducing a mass and inertia tensor, in order to accurately capture the short-time dynamics of the particles. The mass was computed by integrating the density over the volume covered by the WCA beads (accounting for the overlaps); resulting in a total mass of M = 38 m. We used standard integral expressions to determine the moment of inertia over the same volume. This resulted in a diagonal inertia tensor with equal entries on the diagonal of I = 2.0⋅10 2 m σ 2 . We also calculated the hydrodynamic radius of the octapod in order to estimate the diffusion constant of these particles. The expression for the hydrodynamic radius of an assembly of spheres 8 was used to approximate the value of the octapod's hydrodynamic radius and we found R H ≈ 3.7 σ. The translational diffusion constant that follows from R H is D T = k B T/(6πη R H ) = 1.2⋅10 -3 σ 2 τ -1 , where we used the value of the viscosity of toluene η = 5.5⋅10 -4 kg m -1 s -1 or equivalently η = 12 mσ -1 τ -1 in MD units. N.B. This is the bulk value of the translational diffusion coefficient; the diffusion coefficient under confinement may be substantially reduced, as in the experiment the translational diffusion coefficient was determined to be 1000 times lower. From the above, we can determine the friction that is required in the Langevin thermostat to ensure that the long-time diffusion matches the estimated experimental value. A backof-the-envelope calculation indicated that only thermostating the central bead, as is typical for virtualized (rigid) models, would lead to a prohibitively large friction coefficient ζ ≈ 8⋅10 2 mτ -1 . Such a high value negatively impacts the stability of the MD simulation, thus necessitating an extremely small time step. Moreover, it effectively stretches the range of the ballistic regime of the particle diffusion, giving rise to erroneous short-time behavior. We employed the features VIRTUAL_SITES_THERMOSTAT and THERMOSTAT_IGNORE_NON_VIRTUAL of ESPResSo 5 to thermostat the 346 virtual n > 0 beads instead of the single central bead to reduce the friction coefficient. The friction coefficient was tuned to achieve the desired diffusion coefficient by performing short simulations, wherein we calculated the mean square displacement (MSD) and velocity auto-correlation function (VACF) of a single octapod in a cubic simulation box of length (100 σ 3 ) with periodic boundary conditions. A Langevin thermostat 'temperature' of 1.0ε and a friction coefficient of ζ = 1.75 mτ -1 resulted in a long-time diffusion of D T = 1.2⋅10 -3 σ 2 τ -1 . We were restricted to using an MD time step of 0.01τ to ensure stability of the MD integration. This implies that 6.7⋅10 6 integration steps go into simulating 1 second of 'real' time. N.B. Henceforth, we will express the number of integration steps used in the simulation in terms of the effective real time, e.g., 6705 steps correspond to 1 ms. This also allows for direct comparison to the acquisition time given in the main manuscript when we discuss our results.
Simulation Approach. To verify if our vdW-based theoretical model could produce chains, we considered self-assembly in three-dimensional (3D) systems, rather than in the more complex quasi-two-dimensional (quasi-2D) environment of the octapods confined in the in-situ TEM fluid flow cell. In particular, we focused on a small number of octapods N = 2 to 6 in a cubic simulation box with periodic boundary conditions and edge length of L = 40σ. These octapods were initialized at random positions throughout the box and were allowed to reposition and orient randomly to prevent overlaps via an initial warm-up integration (100 ms). During this warm up, the generic LJ potentials were not activated. We then switched on the generic LJ potentials and equilibrated the system for an additional 100 ms. The typical production run length was 1 s, which is substantially shorter than the experimental time scale, but was close to the limit of what is acceptable in terms of CPU hours.
Model A. We matched the coefficients (ε ij , R ij , m ij , n ij , s ij , and c ij ) to most accurately reproduce the vdW interactions obtained in Ref. 1. Supplementary Fig. 18 shows the result as a function of the distance for three configurations, where we show both our simulation model and the result of our vdW calculations. In Ref. 1 we argued that the repulsive interaction between octapods at close separation is governed by a steric contribution. However, we were uncertain about the local surface coverage of polymers that effect this repulsion. We therefore estimated the depth of the minimum from the value of the vdW interaction at surface separations comparable to the length of the fully stretched polymer, H = 2.5 nanometers (see Ref. 1); where we assumed that for larger distances the steric repulsion would be dominated by vdW attraction. We arrived at a potential well depth of around 3 to 4 k B T in toluene for the interlocking configuration.