Symmetry-breaking in patch formation on triangular gold nanoparticles by asymmetric polymer grafting

Synthesizing patchy particles with predictive control over patch size, shape, placement and number has been highly sought-after for nanoparticle assembly research, but is fraught with challenges. Here we show that polymers can be designed to selectively adsorb onto nanoparticle surfaces already partially coated by other chains to drive the formation of patchy nanoparticles with broken symmetry. In our model system of triangular gold nanoparticles and polystyrene-b-polyacrylic acid patch, single- and double-patch nanoparticles are produced at high yield. These asymmetric single-patch nanoparticles are shown to assemble into self-limited patch‒patch connected bowties exhibiting intriguing plasmonic properties. To unveil the mechanism of symmetry-breaking patch formation, we develop a theory that accurately predicts our experimental observations at all scales—from patch patterning on nanoparticles, to the size/shape of the patches, to the particle assemblies driven by patch‒patch interactions. Both the experimental strategy and theoretical prediction extend to nanoparticles of other shapes such as octahedra and bipyramids. Our work provides an approach to leverage polymer interactions with nanoscale curved surfaces for asymmetric grafting in nanomaterials engineering.


Supplementary Notes 1-9 Supplementary Note 1: dependent excluded volume and chain scaling behaviors of polystyrene-bpolyacrylic acid (PS-b-PAA) adsorbed onto nanoprisms
Previous theories 1,2 developed for chain partitioning on the surface of anisotropic nanoparticles (NPs) explained the driving force localizing chains to the vertices of anisotropic NPs, using athermal systems without chain-chain interactions. Briefly, locations of higher curvature on the surface of anisotropic NPs afford more volume per solid angle with increasing radial distance away from the NP surface as in Fig. 2a. Thus, polymers anchored to NPs surface of equivalently high curvature have similarly less entropic cost of grafting, leading to symmetric grafting. However, such energetic cost only, cannot explain why symmetrybreaking occurs regardless of similar local curvature on the nanoprism vertices.
To incorporate intermolecular enthalpy into PS-b-PAA in solvent system, we made two assumptions. PS-PS interaction dominates over that of PAA-PAA and PAA-blocks minimizes their contact with PSblocks and faces the solvent in equilibrium. These assumptions are based on the fact that the volume fraction of PS-blocks is 0.86 and their interaction with solvent molecules is unfavorable compared to that of PAA, consistent with established observations that highly asymmetry (low PS-block fraction) block polymers form micelles in solution 3 . Making the above assumptions allow us to split the polymeric grafts into two layersinternal PS-block and external PAA-block. The outer PAA-block governs patch-solvent stability and the internal PS-block drives chain partitioning.
We additionally note that DMF-a good solvent for PS-constitutes the 82% of the solvent composition. This suggests that perturbations about athermal grafting theories can provide a good first order approximation 1,2,4 . Taking a page from developed theories for planar brushes, we choose to account for deviations away from good solvent behaviors via modifying the Flory interaction parameter χ of PSsolvent 5,6 . More specifically, since the presence of water decreases the solubility of the PS-block, we work in the limit of χ > 0. From a scaling perspective, the effect of nonzero χ is incorporated into the excluded volume such that ~ o (1 − 2χ), where o is the unperturbed good solvent excluded volume. While this relation is well-known in the polymer physics community, we provided its derivation here to better help with clarity. Our starting point is the traditional polymer extension to regular solution theory, where the mean free energy of mixing m ̅̅̅̅ can be written as β m ̅̅̅̅ = ϕ P ln ϕ P + 1 − ϕ P S ln(1 − ϕ P ) + χϕ P (1 − ϕ P ) (1) where ϕ P is the polymer volume fraction, is the polymer chain length, S is the solvent chain length (typically S = 1, but we keep it as a variable for generalizability), and β = 1/ B . Taylor expansion of ln(1 − ϕ P ) in the ϕ P ≪ 1 limit gives β m ̅̅̅̅ = ϕ P ln ϕ P + ϕ P (χ − 1 S ) + 1 2 ϕ P 2 ( 1 S − 2χ) + 1 6 We then define the osmotic pressure Π such that Π = − m ̅̅̅̅ / , with = 3 P /ϕ where is the Kuhn length of the polymer of interest and P is the number of chains within the system. The relation for is simply the definition of the volume fraction. Applying the derivative and accounting for change of variables gives (dropping terms of order ϕ P 4 and higher): We then note that ϕ P −3 is simply the polymer number density η P , substituting into Eq. 3  (4) Eq. 4 is the virial expansion in density: βΠ = 1 η p + 2 −1 2 η p 2 + ⋯ = ∑ −1 η P ∞ =1 . As such, the excluded volume is the second virial coefficient 2 = 3 (1/ S − 2χ) . 3 is set to be a reference, unperturbed excluded volume o ~ 3 . Setting = 1 for typical solvents gives ~ o (1 − 2χ).
Similar to our works for grafted chain behaviors 7 , we define a swelling parameter κ for chains in the Θ ("theta") and good solvent limit.
Eq. 6, however, operates in the limit of free chains in solution. Extending to grafting on the surface of an anisotropic particle (such as triangular prisms), we employ four geometrical scaling balances. This set of derivations is analogous to that for linear and branching chains with χ = 0 previously developed by our group 1,7,8 .
Geometric Balance 1: Local monomer density = global monomer density Geometric Balance 2: Correlation size swells relative to Θ-solvent behavior Geometric Balance 3: Unswollen correlation size = Θ-solvent chain size Geometric Balance 4: Chain swelling (Eq. 6) where ξ is the correlation blob size, above which the intramolecular interaction vanishes due to the presence of other polymer chains, is the monomer count at distance from the core's surface, ψ is the monomer volume fraction, Ω is a core shape parameter, and is the number of grafts. is related to the grafting density σ via σ ~ / o 2 with o representing the nanoprism in-sphere radius. Physically geometric balances 1 -4 (Eq. 7 -10) define volume filling constraints for all grafted chains. More specifically, they make the critical assumption that chains will swell/shrink to fill all possible space available. Relative degrees of swelling are quantified relative to the gaussian, Θ-solvent limit since this is the chain's ideal state. Enforcing equality between local and global density ensures that the chemical potential is the same at all radially equivalent positions relative to a particle's surface. In other words, we assume that chains are free to reorganize and seek to predict their equilibrium behavior. Solving the coupled set of equations gives ( ) ~ ( ) Eq. 11 -14 operate in the regime where solvent quality plays a role in controlling the amount of swelling and number of monomers present in each correlation blob. Geometry sets the correlation blob size at a given distance from the core (Eq. 11). Solvent qualityo [1 − 2χ]-dictates how many monomers fill the given correlation blob ξ (Eq. 12), which then controls both the relative degree of swelling (Eq. 13) and monomer density (Eq. 14).
Moving closer to the core, there exists a crossover distance where the monomer density becomes so high that no solvent can penetrate the grafted layer. This means that monomers sitting within this region effectively sees only monomers from either the same or neighboring chains. In essence, monomers within this regime will behave as if they are sitting in a melt and has no driving force to swell. Setting the swelling parameter κ( ) ~ 1, defines this crossover distance θ to be The corresponding relations for ( ), ξ( ), and ψ( ) are then Similar to Eq. 11, the correlation size (Eq. 16) is defined purely based on geometry and is therefore identical. However, since no solvent can penetrate the region where < θ , the effect of solvent interactions (χ-dependence) necessary disappear in Eq. 17 -18.
To be rigorous, there exists an additional regime where there is high crowding between monomers that chains are forced to fully stretch. Here, the monomer density becomes uniform. Setting ψ( ) ~ 1 gives Trivially, ( ) ~ 1, κ( ) ~ 1, and ξ( ) ~ within this regime ( < c ). To obtain the chain size, we integrate ψ( ) across all regimes using the following volumetric balance ( ⃑ ~ 2 Ω 3 ) Expanding over the relevant regimes gives Integrating and keeping the relevant terms with χ dependency gives the end-to-end distance for a polymer chain anchored to different locations on the nanoprism surface In the limit of long chains ( ≫ 1) Eq. 22 simplifies to In our limit of interest, where χ > 0, Eq. 18 is such that the chain shrinks since [1 − 2χ] 1/5 < 1. Physically, this captures the phenomenon that the chains slightly prefer themselves over the solvent. The grafted layer partially shrinks to enhance monomer-monomer interactions. In effect, Eq. 18 performs a perturbation about Eq. 24 where the size of the grafted layer either expands or contracts to account for changes between solvent-chain and chain-chain interactions.
Here, one could argue heuristically that Eq. 23 can be derived by replacing with o [1 − 2χ] in Eq. 24. While this does capture the physical picture of perturbations in solvent quality, deriving Eq. 11 -23 enables us to derive a cross-over χ value where chain-chain interactions become strong enough to induce the symmetry breaking in patch formation on NPs. This is of particular interest in guiding experimental selection of solvent/grafting conditions.
Our logical scaling argument is as follows. With small to zero changes in solvent quality, the driving force favoring chain-chain interactions is small and the system should still exhibit good solvent chain scaling as derived in Eq. 22 (or Eq. 23). Conversely, in the high reduction in solvent quality limit, solvent molecules are squeezed out of the grafted chain layer in favor of more chain-chain interaction. If χ becomes 0.5, is zero and create a Θ-like environment for polymer chains in solution. This results in a majority of the grafted chain layer transitioning to exhibit a "melt-like" state as only chain-chain interactions are present. The transition from solvent wetting to melt-like directly mirrors the emergence of symmetry breaking patches we seek to take advantage of in experiments. We derive the "melt-like" chain size using Eq. 25 Taking the limit of ≫ 1, we solve Eq. 24 for the end-to-end chain distance at Θ-like environment Θ to give Θ ~ o σ 1/4 1/2 1/2 Ω −3/4 ( / o ) 1/2 . Equating Θ with Eq. 22 and solving for χ gives the crossover χ *

Supplementary Note 2: End-attached polymer chain model validation
The polymers attached on the prism surface can be modeled using the end-attached configuration. The major experimental evidence of the polymer chains adopting an end-attached configuration as the patches on the NP surface is the patch thickness ( m ), which is up to 18.9 ± 4.0 nm (Fig. 1e) across the conditions we have sampled in our work. This thickness matches with the expected extended chain length based on the polymers' molecular weight (~22.9 nm, in good solvent). Furthermore, we note that the solvent quality and the PAA block fraction ϕ A used in our polymer is within the micellar formation regime (i.e., low ϕ A regime on the block co-polymer phase diagram) 9,10 . On surfaces, this will result in the equilibrium conformation of a double layer-PS inner layer and PAA outer layer. The chains are mobile on the particle surface (see Supplementary Note 4) and thus we assume that they can reorganize into the "end attachment" double layer motif shown in the schematic of Fig. 2a. Since our theory is equilibrium by design, this assumption of the final "end attachment" conformation becomes appropriate for our geometrical scaling balances.
It is also true that when the polymer concentration on the prism surface (i.e., grafting density) is low, several PS units could lie flat on the NP surface, as shown in the NPs collected at 10% reaction procession ( Supplementary Fig. 5a). In those cases, the patch thickness is smaller ( m = 5.8 ± 2.3 nm) due to the more flattened polymer configuration. However, as the system is in equilibrium, PS-b-PAA polymers form a double layer with their PS end attached on the prism surface. This configuration in polymer scaling theory still holds the polymer chains to be effectively attached by one end (PS), although the "end" becomes larger as the number of attached PS units is increased.

Supplementary Note 3: Monte Carlo grafting simulation and collapsing into phase diagram
We develop a Monte Carlo grafting algorithm to chain-chain interactions driving the symmetry breaking during the grafting process. Using the chain size , the free energy of each chain is Eq. 27 is then employed to compute a Boltzmann weight probability of chain grafting at a given nanoparticle (NP) surface position (defined in Ω) such that graft ~ −β , which is then used as the acceptance probability in the standard Metropolis criterion. To graft chains to an anisotropic particle of interest, we start by constructing a mesh of points occupying the particle surface. For each surface point, Ω is defined such that Ω = ‖ ⃑‖ {min‖⃑⃑⃑‖} −1 . Here, ⃑ is the vector pointing from the point of interest to the center-of-mass of the surface mesh and min‖⃑⃑⃑‖ operates over all points on the mesh. To capture chainchain interaction, a occupancy matrix is employed to keep track of which surface points is occupied and updated with every new graft addition. All surface points whose distance is within ξ( ) of occupied points as defined in feels the effect of chain-chain interaction (i.e., χ ≠ 0). Specifically, for graft points within ξ( ) of an occupied point Eq. 23 is used for in Eq. 27. For all points outside this cutoff distance, Eq. 24 is plugged in for in Eq. 27. This process repeats until the number of grafted chains matches that defined by a targeted grafting density σ.
The crossover phase boundary in Fig. 2j is derived by defining the condition when the enthalpic gain for adding another chain to the growing patch is on the order of thermal noise (i.e. unity). At this transition, chains are stretched outward due to an already high degree of confinement. As such, ~ σ 1/3 1/3 . The scaling balance takes the form χ 2 −3 ~ 1 . From earlier, χ ~ σ 1/4 −1/2 −1 . Plugging in χ and rearranging terms give ~ σ −3/4 1/2 . We employed this relationship to draw the theoretical phase boundary as well as collapsing 3D data onto a 2D phase diagram in Fig. 2j.

Supplementary Note 4: Interaction range of polymer chains on the prism tip
We note that the distance between two neighboring tips (62.5 nm) is greater than the coiled chain size of 9.3 nm (assuming good solvent). However, there are two things worth noting here. Firstly, the size of a chain decorating the tip of the prism is generally more extended than that of a free coil. A higher extension implies that the interaction range between a free chain and chain decorating the tip effective increases. Secondly, chains can reorganize. Therefore, stochastically adsorbed chains on a second tip could detach into the solution or move around the particle surface to find the most energetically favorable sites. In other words, as long as the maximal extension length of one chain falls within the maximal range of the other, the chains can reorganize to an equilibrium single patch formation, as confirmed via experimental observations of high-yield single-patch prisms (Fig. 1) and simulations of the polymer adsorption process (Fig. 2h-i).
To help with clarity, we plot the monomer density distribution, ( ) (predicted from theory), as a function of distance from the tip ( Supplementary Fig. 9). One chain is placed at a reference tip ( = 0) and the other at a distance equal to the edge length of the prism ( ). Single dashed lines mark the equilibrium, hydrodynamic size of the tip-decorating chain ( ℎ ) and double dashed lines indicate the fully extended size of the chain ( ). Colors correspond to the same chain (i.e., blue: chain at reference tip, orange: chain at adjacent tip) and the x-axis is scaled by ℎ . The gray shading region indicates the region of space where the chains monomer densities overlap with each other. This means that any monomer within this region will experience strong monomer-monomer interactions that ultimately drive chain reorganization and subsequent merging to a single tip.

Supplementary Note 5: Quantitative Analysis of the TEM images
The projected area of a polymer patch (Fig. 3c) was obtained from TEM images using built-in "Binary" function of ImageJ with low intensity thresholds to segment polymer patches, assisted by manual correction of the patch contour. The local thickness of a polymer patch (Fig. 3a) is defined as the diameter of the largest circle that incorporates the outmost pixel of the patch boundaries and remain within the polymer patch. We obtained local thickness using the built-in function "local thickness" on binary images in opensource software ImageJ, as detailed in our previous work 11,12 . The local curvature at each coordinate of the contours of the prism (Supplementary Fig. 4) and patch (Figs. 3e, 4c, Supplementary Fig. 14g, and 16f) is defined as the inverse of the radius of the best-fitted circle to the local arc at the coordinate. The contour coordinates of prism and polymer patches are obtained from the segmented images using the MATLAB built-in function "bwboundaries.m". The coordinates are then plugged into our customized MATLAB code to calculate curvatures with optimized parameters (e.g, 12 pixels, 5.0 nm; fitted by 25 pixels, 10.3 nm), as detailed in our previous work 12 . For the local curvature of the merged patches (Fig. 4c, Supplementary Fig. 16f), we use the multi-point function in ImageJ to manually annotate the contour of the merged patches between a pair of single-patch nanoprisms. The coordinates of the annotated contours are saved in .csv files and then output into the customized MATLAB code to compute the local curvature values as described above.
The patch width ( ) (Fig. 3f) was measured based on segmented images using our customized codes written in MATLAB. Prior to the measurement, Gaussian filter (20 pixels) is applied on the images to decrease the outlier pixels and noise in the raw image. In the code, the binarized images of segmented patches are first rotated so that the line connecting the center of the prism and one of the prism tips aligns with -axis. The tip of the prism is assigned as = 0. Note that ( ) at < 0 region was measured by subtracting the prism width ( ) from the width of patch contour at . The 10 representative ( ) profiles are averaged along the -axis and standard deviation was obtained for each .
The tip-to-tip distance and interparticle angle γ between nanoprism pair were measured as following. The "oval" function (in circle mode) in ImageJ is used to fit six rounded tips from two nanoprisms in one nanoprism pair connected by the merged patches. The center coordinates and radii of the fitted circles are the inputs into MATLAB to compute the interparticle angle γ between the nanoprism pair using our customized codes. We first connect the center of one prism, which is defined as the mean of three centers of the circles fitted from each tip, with the center of the circle fitted from the tip with the merged polymer patch in the same nanoprism, to draw a nanoprism orientation vector. The angle between two lines from two nanoprisms in a pair (annotated in Fig. 4a scheme and Fig. 4d -axis ticks) is then calculated and plotted as the interparticle angle γ as shown in Fig. 4d. The tip-to-tip distance is measured as the centerto-center distance between two circles fitted from two tips connected by the merged polymer patches in a nanoprism pair subtracted by the sum of two radii.

Supplementary Note 6: Scaling behavior for patch size and shape
The patch area can be described as ~ δ , where δ is the fractional coverage of polymer chains over the adsorption sites and is the corresponding interchain separation. Fractional coverage δ exhibits a grafting temperature dependence δ ~ −9/2 . This scaling exponent stems from a combination of the temperature dependence of adsorption on a flat surface (−3/2) and the effect of surface curvature (−3). From our scaling theory, ~ α 1/5 −1 and ~ α −1/2 . −1 reflects the Arrhenius behavior of χ. Scaling exponents for α of 1/5 and −1/2 reflect good solvent and standard interchain polymer brush spacing scaling, respectively. Combining all terms results in a scaling of ~ α −3/10 −11/2 .
In addition to the crossover interaction parameter χ * , we additionally employ Eq. 11 -23 to predict the shape of the patches that form from symmetry breaking grafting. Due to differences in chain-chain and chain-solvent interactions, we can leverage traditional wetting relationships to predict the surface curvatures of the resulting patchy grafts as a function of distance from the prism core's surface . Our governing balance can be derived as follows. Starting from the Young-Laplace wetting relationships we have SG = SL + LG cos θ c where θ c is the contact angle and SL , LG , and SG are the surface tension costs of core-chain, chainsolvent, and core-solvent contacts, respectively. SG is taken to be negligible since particles are freely where as we assume that the contact area are of the same order of magnitude (within a constant scaling constant). Therefore, the contact angle can be approximated as the ratio of the energy gain of chainchain/core contact versus the energy cost of chain-solvent contacts for a given layer at distance from the prism core surface . ε SL ( ) and ε LG ( ) can be readily computed using the surface area and volume of a conical frustum at every radial layer at from the surface.
where ℎ is the height of the frustrum and 1 and 2 are the radii of the larger and smaller circular faces, respectively. Here, our physical picture is of increasingly bigger correlation blobs ξ with larger radial distance from the core's surface. Thus, both radii and ℎ are given by the size of the blob, giving The relevant energies are then ε SL ( ) ~ ( )ψ( ) and ε LG ( ) ~ ( )ψ 2/3 ( ). Using Eq. 32 -33, we can then solve for the contact angle θ c as a function of distance from the nanoparticle's surface and construct the patch shape as shown in Fig. 3f.

Supplementary Note 7: Grafting density estimation from TEM images
We calculate the grafting density of polymers on each patch 13 based on patch width ( ) and tip of the prism width ( ) in Supplementary Note 5, to investigate the patch shape-dependent grafting density. First, the volume of grafted polymers per patch patch at each reaction condition was obtained based on TEM images of the patches following Eq. 34, assuming that The molar volume of polymer chain pspaa is estimated to be 4.15×10 25 nm 3 /mol from Eq. 35 by plugging in the molar weight of PS block n(ps) and PAA block n(paa) as 1.6×10 4 g/mol and 3.7×10 4 g/mol, respectively, and the density of PS ρ ps and PAA as ρ paa as 1.05×10 -21 g/nm 3 Next, the surface area of nanoprisms covered by a patch patch are calculated for each reaction condition using Eq. 37 -38 and average patch chord length in Fig. 3d, average prism thickness prism of 27.2 nm, and the average prism tip radius tip of 5.5 nm as detailed in Supplementary Fig. 1. In detail, patch can be approximated as a combination of three geometric shapes: two equal triangles 1, four trapezoids 2 , and a one third of the surface area of a prolate spheroid prolate 3 as illustrated in Supplementary Fig. 15 where the surface area of the prolate spheroid prolate is 783.3 nm 2 from the Eq. 39, the semi-minor axis is the prism tip radius tip , a semi-major axis is the half of the prism thickness prism , and the eccentricity is defined as Eq. 40. prolate = 2π tip 2 ( 1 + prism 2 tip · arcsin ) (39) Last, the grafting density σ′ is obtained as dividing the number of polymer chains per patch chain by the surface area covered by the patch patch as Eq. 41.
For example, at reaction condition of = 90 °C and α = 25 nM, the volume of the patch patch is 11862.9 ± 5725.0 nm 3 by using the Eq. 34 and the average chord length is 16.8 nm. Therefore, the number of polymer chains chain is 172.2 ± 83.1 by the Eq. 36. The surface area of nanoprism covered by the patch patch is 606.0 nm 2 from the Eq. 38. Thus the grafting density σ′ of the single-patch prisms obtained is 0.28 ± 0.14 nm −2 . Similarly, patch and σ′ are obtained for the patches from different reaction conditions and summarized in Supplementary Table 5.

Supplementary Note 8: Patch-patch merging and binary assembly scaling
Here, we posit that patch-patch merging is facilitated by free ends of chains contacting with those from another grafted prism. As such, merging can only occur at a sufficient local chain concentration * . This is readily given using the overlap concentration crossover * ~ For concentrations greater than * , grafted chains enter the semi-dilute regime. Here, grafted chains are split into two regionsouter and inner. The inner region near prism core is dominated by chain-chain interactions between grafted on the same core. Conversely, the outer region is influenced by chains from other grafted particles. The crossover between the two regimes can be defined by where the monomer concentration ψ( ) is equal to the bulk concentration in semi-dilute regime 3 , giving ψ( ) ~ 3 . Solving for the crossover radial distance L gives Solving for c gives The correlation size of outer regime chain x is a function of the local concentration and can be readily computed using classical semi-dilute scaling relation: where the number of monomers in x is x ~ − c . Therefore, the chain size in the limit at semi-dilute region sd where patches are interacting and overlapping with each other is sd ~L + x , giving Eq. 46.
Taking the ratio of Eq. 47 to Eq. 23 allows us to predict interpenetration ratio λ between two interacting polymer patches shown in Fig. 4f as following: λ ∼ 1 − sd / .
Note that in terms of the PAA conformation between the tips, we model the merging PAA blocks as a spherical packing of correlation blobs to fill up the space available between the tips. This picture is a manifestation of the idea that introducing charge moieties results in a reorganization of the charged PAA monomers into neutral correlation blobs that then interact via steric repulsion with each other. As discussed above (Supplementary Note 8), this assumption enables the usage of a classical semi-dilute scaling

between sterically interacting correlation blobs to model
both the correlation size as well as the effective width of the merging regime between the two interacting tips.

Supplementary Note 9: Localized surface plasmon resonance (LSPR) properties of the patch-topatch assembled prisms
We calculated LSPR properties of four types of patch merging driven assemblies of symmetry breaking patchy nanoprisms, namely the nano-bowties, quasi-linear and branched trimers, and tetramer as shown in Fig. 4a, Supplementary Figs. 17 and 18. The extinction spectra (Supplementary Figs. 17q , 18y) and electric field distribution maps for the corresponding structures show that nanoprisms have configurationdependent extinction energies and "hot spots", where the electric fields are a few orders of magnitude more intense than the surroundings. The nano-bowties ( Supplementary Fig. 17a,e,i,m) present the face, side and tip modes 14,15 upon excitation with visible light. The dimer in Supplementary Fig. 17a, for example, exhibit the face, edge, and tip modes at 2.81 2.72, and 2.33 eV respectively ( Supplementary Fig. 17b-d). As the particles are brought closer together with larger γ (Supplementary Fig. 17i,m), the respective modes redshift and the electric field intensity is shown to spread in between the space of the assembled particles, visualizing a stronger interaction between particles. The hot spots visualized in the electric field maps possess a relative field intensity enhancement close to ~ 8×10 7 , compared to the near vanishing intensity ~200 nm away from the nanoprisms. The corresponding optical extinction spectra of structures in air, to mimic the dried-state assembly are compared to the spectrum of a single-patch nanoprism before merging ('single' in Fig.  Supplementary Fig. 17q).
Likewise, the quasi-linear trimer (Supplementary Fig. 18a) has the LSPR peaks at energies corresponding to 2.81, 2.69, 2.31 and 2.27 eV ( Supplementary Fig. 18b-e), with "hot spots" showing up over the face, edge, and tip of each nanoprism. The dipolar tip mode further splits into the antisymmetric antibonding and symmetric bonding modes present at higher and lower energies, respectively. As the particles are brought closer ( Supplementary Fig. 18f), the edge and tip modes show an appreciable interaction with the electric field intensity being confined in the space between the nanoprisms ( Supplementary Fig. 18i,j), with a redshift in the antibonding mode from 2.31 eV to 2.29 eV compared to the structure in Supplementary Fig. 17a. We further calculate the electromagnetic enhancements of branched trimer and the tetramer shown in Fig. 4a (Supplementary Fig. 18o,s, respectively). The branched trimer ( Supplementary Fig. 18o) has one particle pointing up, resulted in the shift of face modes from 2.81 to 2.85 eV, while edge mode remains unchanged at 2.69 eV, and the tip mode blue-shifted from the nondegenerate energies at 2.31 and 2.27 eV to 2.29 and 2.25 eV, respectively. The tetramer ( Supplementary  Fig. 18s) exhibits hot spots around the truncated particle present near the top of the assembly, in addition to the usual face, edge and tip modes. The extinction peak at 2.47, 2.29, and 2.27 eV (Supplementary Fig.  18v,w,x, respectively) shows the electric field localized at the non-truncated face of the particle, the tips of non-truncated particles, with some delocalization occurring in between these particles. The LSPR energy shifts of various structures are represented in the extinction spectra obtained from the structures in air ( Supplementary Fig. 18y), with the spectrum corresponding to the trimer shown in Fig. 4g. Tables 1-6   Supplementary Table 1. Reaction conditions for synthesizing patchy nanoprisms with varied patch count per prism and shapes as shown in Fig.2d,f, and 3a. The reaction procedures and definition of stock solution II are detailed in Methods. The α is the ratio of 2-NAT molar concentration to the optical density of the prism suspension at its maximum extinction wavelength.   Table  5). Supplementary Fig. 17. Spatial electric field distribution maps of bowties. (a,e,i,m) The electron microscope images of the gold nanoprisms assembled into dimer configuration of different angles. The spatial electric field distribution maps for each configuration correspond to near field enhancements at extinction maxima are represented using false color maps on the same row (b-d, e-h, i-l, m-p). The excitation energies are noted in the maps. (q) The respective optical extinction spectra of structures in air compared with the spectrum corresponds to the single-patch prism before merging (noted as "single prism"). Scale bars: 50 nm (a,e,i,m). Supplementary Fig. 18. Spatial electric field distribution maps of trimers and tetramers. (a,f,k,o,s) TEM images of the gold nanoprisms assembled into trimers with (a,f) quasi-linear, (k) angled, and (o) branched configuration, and (s) tetramer. The spatial electric field distribution maps for each configuration correspond to near field enhancements at extinction maxima are represented using false color maps on the same row (b-e, g-j, i-n, l-n, p-r, t-x). The excitation energies are noted in the maps. (y) The respective optical extinction spectra of the structures in air compared with the spectrum corresponds to the trimer in