Imaging the polymerization of multivalent nanoparticles in solution

Numerous mechanisms have been studied for chemical reactions to provide quantitative predictions on how atoms spatially arrange into molecules. In nanoscale colloidal systems, however, less is known about the physical rules governing their spatial organization, i.e., self-assembly, into functional materials. Here, we monitor real-time self-assembly dynamics at the single nanoparticle level, which reveal marked similarities to foundational principles of polymerization. Specifically, using the prototypical system of gold triangular nanoprisms, we show that colloidal self-assembly is analogous to polymerization in three aspects: ensemble growth statistics following models for step-growth polymerization, with nanoparticles as linkable “monomers”; bond angles determined by directional internanoparticle interactions; and product topology determined by the valency of monomeric units. Liquid-phase transmission electron microscopy imaging and theoretical modeling elucidate the nanometer-scale mechanisms for these polymer-like phenomena in nanoparticle systems. The results establish a quantitative conceptual framework for self-assembly dynamics that can aid in designing future nanoparticle-based materials.


Supplementary Tables
Supplementary Table 1. Number-average degree of polymerization ( ). The values measured from experimental data shown in Fig. 1e is compared with those derived from the extent of polymerization reaction in Supplementary Fig. 5b. The R-squared values in the last column are from the fitted curves in Supplementary Fig. 5b. Fig. 5b The and are fitted using the exponential decay function ( ) = e / at the range of of 0 -1 nm. 5,6 Here is the measured hydration interaction per area, is the hydration repulsion energy per area when in contact ( = 0), is the decay length and is the distance between two surfaces. Under electron beam, radiolysis of pure water gives rise to radiolysis products including charged species (e.g., e  , H + , OH  , HO2  , O  , O2  , and O3  ), which increase ionic strength and decrease pH.
The degree of radiolysis is dependent on the dose rate. A rough estimation of the dependence of ionic strength and pH on dose rates was shown in Supplementary Fig. 2  μM for cyclic chains were rationalized based on our theoretical calculations for total interaction energy between a pair of prisms, which would allow the two prisms to overcome a repulsion barrier and assemble into the prism chains (see more details in Supplementary Fig. 8 and Table 2).

Supplementary Note 2. Evaluation of pH effects on the surface charge density of prisms
The portion of negatively charged/neutral thiol ligands is determined by the pH of the solution, which determines the surface charge density of the prism. The ratio of the two states of the thiol ligands at different pH conditions was calculated by the Henderson-Hasselbalch equation, pH = pK a + log [HA] , where [A  ] is the molar concentration of negatively charged ligands (-COO  ), [HA] is that of neutral ligands (-COOH), and we use 3.5 as pKa (pKa = 3.5-3.7, provided by the manufacturer). The thiol ligand (> 1%) begins to be protonated below pH = 5.5. At pH = 4, the surface charge density decreases to 76% of the initial value ( Supplementary Fig. 2b).

Supplementary Note 3. Time-lapse liquid-phase TEM image processing protocol
The time-lapse liquid-phase TEM images shown in Figs. 1c, 2b-d and 5b were processed via frame average, background subtraction and contrast adjustment. An example set of TEM images after the above image processing procedures are presented in Supplementary Fig. 3. Each frame was first extracted from TEM movies (in the format of .dm3) using an open source software ImageJ, and six sequential frames of interest were averaged by use of our customized MATLAB script.
Background subtraction was performed with the averaged TEM image via ImageJ, followed by 22 manual adjustment of contrast. The pixel size for the background subtraction was chosen as 40, which is slightly larger than the dimension of a single prism.

Supplementary Note 4. Calculation of the step-growth polymerization parameters
The assembly rate constant ( ) for the tip-to-tip prism assembly into linear chains ( Fig. 1e and Supplementary Fig. 3) was derived using the rate equation for step-growth polymerization after the linear fitting of the data points shown in Fig. 1e Supplementary Fig. 3. Thus, the value was calculated as 1.1 × 10 3 M -1 ·s -1 .
The extent of reaction was derived via Flory-Schulz distribution, 8,9 which is defined as ⁄ = where is the number of -mers containing prisms, is the total number ofmers, and is the extent of reaction at time . The fraction ( ⁄ ) at each time calculated from the data for the experiment presented in Supplementary Fig. 3 was plotted as a function of ( Supplementary Fig. 5a). The data points were fitted with the function of Flory-Schulz distribution, from which we derived the extent of reaction, , at each time ( Supplementary Fig. 5b). The number-average degree of polymerization ( ) measured from Supplementary Fig. 3 (see Fig. 1e) was compared with those calculated from the extent of reaction ( ), verifying the success of fitting via Flory-Schulz distribution (Supplementary Table 1). The PDI values were calculated and shown as a good fit with PDI = 2 − (Supplementary Fig. 5c).

Supplementary Note 5. Particle tracking and automatic analysis
Using a custom-developed MATLAB script, we captured not only the position of the prism, but also their anisotropic shape details from a liquid-phase TEM movie ( Fig. 4c and Supplementary   Fig. 6a). The automatic analysis of a TEM movie about 50 s long resolved temporal traces of two 23 interacting prisms (Supplementary Fig. 6b). A TEM movie with high resolution is desirable for automatic tracking of nanoparticle motions together with their shape details.

Supplementary Note 6. Prism tip contour curvature analysis
For the local curvatures of prism tips analyzed in Fig. 4c and Supplementary Fig. 10, they were measured from the TEM images of individual prisms using ImageJ and our customized MATLAB codes. The contour of the triangular surface of a single prism is determined by the image intensity gradient using customized MATLAB codes, and the corresponding local curvature was calculated from the inversed radius (1/ ) of locally best-fitted circles (Fig. 4c). 10,11 The prism geometry parameters measured from TEM images is what we used in the calculation model (Supplementary Table 3, the prism side length was rounded to 90 nm for calculation convenience).

Supplementary Note 7. Estimation of hydration interactions
Hydration interactions occur when two surfaces immersed in water come into a sufficiently close distance, where structured water layers adsorbed or adjacent to the surfaces start to generate surface-surface interactions. 6 The range of hydration interactions is thus about a few water molecule size, about 0.1-1.4 nm. 5 Previous studies have utilized direct force measurements (surface force apparatus, atomic force microscopy, etc.) to measure the magnitude and mathematical form of hydration interactions for various solid surfaces. [2][3][4][12][13][14] Empirically, the hydration repulsion for two hydrophilic surfaces per unit area is written as ( ) = e / . 5,6 Here is the hydration repulsion energy per area at a distance , is the hydration repulsion energy per area when the surfaces are in contact, is the decay length, and is the distance  Table 4), 2-4 and estimated the corresponding hydration interactions following the empirical equation. We chose a gap distance ranging from 0.5 nm to 2 nm for the estimation of hydration interaction in our system, based on the gap distance values we observed (0.5-2.4 nm, the inset distribution in Supplementary Fig. 11). We found that the hydration interaction between two tip-to-tip assembled prisms are constantly below 1 , much smaller than the electrostatic repulsion ( Supplementary Fig. 11). Thus, in our interaction calculation that explains the bimodal bond angle distribution, we considered only van der Waals attraction and electrostatic repulsion, which indeed correctly predicted our experimental observation.
Our estimation that the hydration interaction is negligible for the gold prisms coated with charged organic ligands is consistent with previous studies. Previous force measurements 2,15,16 have shown that if the solid surfaces are coated with charged ligands like in our work (other than H + or OH  ions, generic for oxides), hydration interactions are negligible due to three reasons: the ligands strongly interact with surface water layers and disturb the liquid structuring; the ligands prevent the surfaces from getting close enough to fall into the short range of hydration interaction; the charged ligands bring strong electrostatic repulsion into the system which obscures hydration interaction. Likewise, previous work on metallic nanoparticle growth and coalescence 17,18 have shown that when nanoparticles are coated with charged ligands (cetyltrimethylammonium bromide), 17 hydration interactions are negligible in determining assembly configurations, but when the nanoparticles are naked (no ligand coating), hydration interactions are significant. 17,18 Note that in our literature survey, we did not find hydration interaction measurements for gold surfaces coated with the same charged ligands as the thiol ligands in our sample. We used the available literature values for either naked gold 2 or nonionic ligand coated gold surfaces, 3,4 which actually render our estimated hydration interactions ( Supplementary Fig. 11) higher than the actual values for our gold surfaces coated with charged ligands, which further suggests that hydration interaction does not influence the assembly configurations in our system.

Supplementary Note 8. Analysis of branched prisms in assembled chains
The branched prisms were counted from the linear chain assemblies ( Fig. 1 and Supplementary   Fig. 3) and cyclic chain assemblies (Supplementary Movie 4). We defined the branched prism as a prism that is connected with more than two neighboring prisms in chains. At different assembly times, we counted the total number of prisms and branched prisms in chains, and obtained the fraction of branched prisms by dividing the number of branched prisms by the total number of prisms in chains ( Supplementary Fig. 15c).