Shape-Specific Patterning of Polymer-Functionalized Nanoparticles

Transcription

1 Supporting Information Shape-Specific Patterning of Polymer-Functionalized Nanoparticles Elizabeth Galati, 1 Moritz Tebbe, 1 Ana Querejeta-Fernández, 1 Huolin L. Xin, 2 Oleg Gang, 2,3 Ekaterina B. Zhulina, 4,5 and Eugenia Kumacheva *,1,6,7 1 Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada 2 Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA. 3 Department of Chemical Engineering and Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA 4 Institute of Macromolecular Compounds of the Russian Academy of Sciences, Saint Petersburg, , Russia 5 Saint Petersburg National University of Informational Technologies, Mechanics and Optics, Saint Petersburg, , Russia 6 Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario M5S 3G9, Canada 7 Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario M5S 3E5, Canada 1

2 1. Evolution of the number of patches with time Figure S1a shows the distribution of the number of patches on the NCs in the course of the etching process, that is, in the course of their transformation into NSs. The analysis was performed based on the analysis of TEM images (top view). We followed the change in the number of patches localized on the top vertices and edges of the NCs. In this approach, the largest fraction of the NCs had four vertex patches, which corresponded to 20 patches (8 vertex and 12 edge patches) of the NCs. With etching time, the fraction of nanoparticles (NPs) that exhibited four patches decreased, while the fraction of NPs with three patches increased. Figures S1b-f show representative images of the NPs for different stages of the NC-to-NS shape transformation. The images show the gradual change in the NP shape, as well as the change in the patch position and number. Figure S1. Patch evolution with NP shape transformation. (a) Distribution of the number of patches at different etching times. Inset shows etching times. (b-f) Corresponding TEM images 2

3 (top view) of the NPs subjected to etching for 0 (b), 10 (c), 30 (d), 60 (e), and 120 (f) min. Scale bars are 50 nm. 2. Extinction spectra of NCs and NSs For the purpose of comparison, the surface plasmon resonance peak is shown for the NCs with an edge length of 59 ± 2 nm and for as-synthesized NSs with a diameter of 60 ± 2 nm. The NCs exhibit a peak position at 548 nm, while for NSs the peak is centered at 541 nm. These resonance peak positions are comparable with those observed for the NCs before and after etching, as shown in Figure 3 in the main text. Figure S2. UV-Vis spectroscopy of 60 nm NPs (red line NCs, blue line NSs) functionalized with PS-50K in THF. 3. Discussion of polymer grafting density on NCs and NSs The difficulty in quantifying the average polymer grafting density on the NCs is caused by the copious amounts of non-recoverable material needed for analytical characterization, which becomes a challenge due to the relatively low polymer-to-nc concentration ratio and low (< 1 nm) NC concentration used in our experiments. Secondly, the polymer is not evenly grafted throughout the NC surface: the grafting density of high surface curvature regions is higher than on the planar facets. Using previously published results, 1 we estimated the grafting densities of the polymer on the vertices, edges and facets of the NCs. Figure S3 shows a geometric breakdown of the NC surface, based on the local site-specific curvature. According to Figure 4 in the main text, the 3

4 edge rounding ratio of the original NCs prior to the etching process is 25%, that is, a rounding radius of 15 nm. In our previous work, 1 we showed that for the similar grafting conditions of PS-50K, the grafting density on 20 nm is 0.03 chains/nm 2. We speculate that the NC vertices (red regions in Figure S3) have a similar grafting density as 20 nm NSs, that is, 0.03 chains/nm 2. Similarly, we assumed that for identical PS-50K grafting conditions, NC facets (blue regions in Figure S3) have a similar polymer grafting density as 80 nm-diameter gold NSs, 1 that is, 0.01 chains/nm 2. The second assumption is based on the results of our work 1 and the report of Mirkin et al., who showed that the grafting density of oligonucleotides resemble that of the planar substrate, when the NS diameter exceeds 60 nm. 2 Since the NC edges (green regions in Figure S3) have a similar curvature as the red regions in one direction and a zero-curvature in the other direction, similar to the blue regions, we assume the polymer grafting density on the edges is an average of those on the vertices and facets, that is, 0.02 chains/nm 2. With these assumptions, we found that the ratios of the grafting densities on the facet (, edge ( and vertex (, are and = 3. These values are comparable with the threshold grafting density ratios that lead to preferential formation of patches on NC vertices and edges (see section 5). Figure S3. Schematic of the various regions of a NC. The blue, green and red colors show the facets, edges and vertices, respectively. Following the NC-to-NS transformation during the etching process, the polymer grafting density changes, as the high curvature regions on the NCs are removed. We assumed that the grafting density of the polymer on the facets of the NCs and the NSs is similar, as adsorption of additional polymer on the facets is kinetically limited. However as stressed in the main text, the number of patches for the NSs does not depend on the grafting density, 1 thus validating 4

5 conclusions regarding the role of in situ change on the number and distribution of the polymer patches. 4. Discussion of the NC etching process 4.1 N,N-Dimethylformamide as a reducing agent N,N-Dimethylformamide (DMF) is a reaction solvent as well as a reducing agent. 3 Although DMF is more frequently used as a reducing agent at higher temperatures, it can reduce metal ions at room temperature in the presence of water. 3 Note that in the present work, NC etching was performed at room temperature in DMF and in the DMF/water mixture at C w = 1 vol%. In both cases, a minute amount of water was added with the aqueous solution of HAuCl 4. In particular, DMF can reduce AuCl - 4, if Au 0 atoms are present, 4 as in the case of our experiments, that is, the reaction solvent is reducing the etching agent, thus decreasing the etching agent s potency in etching the NCs. 4.2 Verification of oxidative states of the Au ions in the redox reaction Figure S4. In-situ extinction spectra of the Au 3+ -CTAB complex in DMF. (a) reduction of Au 3+ to Au 1+ ions; (b) further reduction of Au 1+ ions to Au 0 with subsequent addition of NaBH 4. It has been reported that in DMF the disappearance of the Au 3+ charge-transfer-to-solvent (CTTS) absorption band at 325 nm takes 20 h, 4 while for AuCl 4 -, it takes up to 5 days. To evaluate the rate of the reduction of Au 3+ in DMF and to confirm the oxidative states of Au in the redox reaction in our work, the change in Au 3+ -CTAB complex CTTS absorption band was monitored by ultraviolet-visible spectroscopy (Figure S4a). More specifically, HAuCl 4 ( M) was added to a M solution of CTAB in DMF. Figure S4a shows the temporal decrease in the absorption peak of AuCl 4 - at 324 nm and the formation of Au 3+ -CTAB complex 5

6 at 398 nm, followed by a decrease of the intensity of the latter as a result of the reduction of Au 3+ to Au 1+. After 4 h, the Au 3+ -CTAB complex peak disappeared and the solution was colorless, as depicted by the spectrum. To confirm that this colorless solution contained Au 1+ (rather than only Au 0 ), a stronger reducing agent, NaBH 4 ( M), was added to the system. The resulting spectra are shown in Figure S4b. Immediately after the addition of NaBH 4, a new peak appeared at 548 nm, thus indicating further reduction of the Au 1+ ions to Au 0 and confirming the oxidative states due to the DMF reduction. The peak broadening observed within minutes (Figure S4) indicates NP aggregation. 4.3 Optimization of the etching conditions The etching procedure using Au 3+ in the presence of CTAB was adapted from the previously reported methods. 5,6 In particular, it has been shown that in water, selective etching of metal NPs can be carried out to different degrees by varying the Au 3+ /Au 0 molar ratio, where Au 3+ is added to the solution of NPs (carrying Au 0 atoms). Stoichiometrically, two moles of Au 0 are consumed for each mole of Au 3+, in accordance with the following redox reaction 5 : AuCl Au Cl 3 AuCl 2 A NS is approximately half in volume of a cubic NP when the NS diameter is equal to the NC edge length. Thus, to transform a NC to a NS, the molar ratio of 0.5:1 (Au 3+ /Au 0 ) is sufficient to dissolve half of the Au 0 atoms comprising the NC. In this process, selective etching occurs in the areas of high NC curvature, that is, at the edges and vertices. 5 While this molar ratio requirement applies to the etching process in water, it changes in the presence of DMF, which acts as reducing agent, as previously discussed in Section 4.1 and 4.2. As a result, a competitive reaction occurs between the reduction of Au 3+ by DMF vs. the intended reaction of oxidation of Au 0 atoms of the NCs to Au 1+. To account for this competition, the expected Au 3+ /Au 0 molar ratio was increased two-fold and seven-fold for the etching of NCs in DMF and in the DMF/water mixture at C w = 1 vol%, respectively (that is, it was increased from 0.5:1 to 1:1 and to 3.5:1 for DMF and DMF/water mixture conditions, respectively). The significantly increased ratio for the C w = 1 vol% is caused by the increase in kinetics of the Au 3+ reduction in the water/dmf mixture. 6

7 4.4 Redeposition of Au onto etched NPs As discussed in the main text, the in situ UV-visible spectra acquired following the NCsto-NSs transformation (taking 80 min in the DMF/water solution at C w = 1 vol%), showed a blue shift in the surface plasmon resonance peak. Following the completion of the etching process, with storage time, the UV-Visible spectra revealed a subsequent red shift of the surface plasmon resonance peak, as well as an increase in the peak intensity, as shown in Figure S5. This effect was attributed to the redeposition of Au ions onto the NPs, as shown in Figure S5 inset. To avoid this effect, it was imperative to remove Au ions via a centrifugation cycle, when the etching reaction is complete. Figure S5. Variation in extinction spectra of patchy NCs during the course of etching process in the DMF/water mixture at C w = 1 vol%. Inset shows the TEM image of a representative NP taken 500 min after the addition of the etching agent (representative of Au redeposition). Scale bar is 50 nm. 5. Theoretical analysis of the polymer state on nanocube surface 5.1 Assumptions of the theoretical model Consider a NC with a side length L, which is capped with flexible polymer molecules composed of N >> 1 Kuhn segments with a length b. The polymer molecules are end-tethered with a grafting density 1, 2, and 3 to the NC facets, edges and vertices, respectively. The solvent is poor for polymer, and its volume fraction in the globular state is 1 > (T- )/T > 0, where T and are the temperature and the -temperature, respectively. The surface tension at the polymer-solvent interface is = C 2 /b 2, where C is a numerical coefficient on the order of unity and k is Boltzmann s constant. The surface tensions at the NC-solvent and NC-polymer 7

8 interfaces are assumed to be equal, thus giving the contact angle of 90 o for the polymer patch on the NC surface, independently of the position of its center on a facet, an edge, or a vertex. A pinned micelle (surface patch) is composed of P chains collected from the micelle footprint area A (Figure S6). It has a condensed core that is composed of densely packed thermal blobs with the size = b/, and P legs (strings of the thermal blobs). The number of thermal blobs per polymer chain is N 2 > 1. Later in the text, we use subscripts 1, 2 and 3 for the micelles located at the faces, edges and vertices of the NCs, respectively. (1) (2) (3) 1 A A 2 2 L L L A 3 3 Figure S6. Schematics of the pinned micelle on the NC facet (1), edge (2) and vertex (3). 5.2 Equilibrium conditions of micelle formation on NCs Facet micelle A facet micelle with a footprint radius ρ 1 and area A 1 = πρ 2 1 contains P = A 1 σ 1 = πρ 1 ²σ 1 molecules. On a planar NC facet with L >> A 1/2, the free energy F₁ per polymer chain is 1 F 1 (P) = F 1,leg (P) + F 1,core (P) = Bρ 1 (P)τ b + Cτ2 R1(P) 2π[ b ]2 P (1) where R 1 is the radius of the core of the facet micelle related to the number of chains P as R 1 /b = (3/2 ) 1/3 (PN/ ) 1/3, ρ 1 = (P/ 1 ) 1/2, and B is a numerical coefficient on the order of unity. Minimization of F 1 with respect to P yields the equilibrium aggregation number of polymer molecules in a micelle P 1 P 1 (σ 1 ) = π ( 2C 5 3B ) N 4 5 τ 2 5 (σ 1 b 2 ) 3 5 (2) The equilibrium free energy per polymer molecule in a micelle is F 1 (P 1 ) = ( 5B ) 5 2 (2C 3B ) N 2 5 τ 6 5 (σ 1 b 2 ) 1 5 (3) 8

9 Edge micelle Similarly, to the footprint of a pinned micelle on a NC facet, the footprint area A 2 of the micelle on the NC edge is a circle with a radius ρ 2 = (A 2 /π) 1/2 = (P/πσ 2 ) 1/2. The equation for the free energy of polymer leg stretching is F 2,leg = Bρ 2 (P) /b. The core of the edge micelle constitutes 3/4 of a sphere with radius R 2 and is related to the number of polymer molecules P as R₂ = b(pn/πτ) 1/3. The polymer-solvent contact area of the core is then 3πR₂ 2, and the free energy F 2 per molecule is F 2 (P) = F 2,leg (P) + F 2,core (P) = Bρ 2 (P)τ b + Cτ2 R2(P) 3π[ b ]2 P (4) Minimization of F 2 with respect to P yields the equilibrium aggregation number P 2 P 2 (σ 2 ) = ( 2C 5 3B ) N 4 5 τ 2 5 (σ 2 b 2 ) 3 5 = ( )2 ( σ 2 ) 3 5 P σ 1 (σ 1 ) = ( σ 2 ) 3 5 P 1 σ 1 (σ 1 ) 1 The equilibrium free energy per molecule in the edge micelle is (5) F 2 (P 2 ) = (5B) (2C 3B ) N 2 5 τ 6 5 (σ 2 b 2 ) 1 5 = (3 2 )1 ( σ 1 5 σ2 )1 F 1 (P 1 ) (6) Vertex micelle In a vertex micelle the core constitutes 7/8 of a sphere with a radius R₃, which is related to the number P of molecules in the micelle as R₃ = b(6/7π) 1/3 (PN/τ) 1/3. The polymer-solvent contact area of the core becomes 7πR3 2 /2, while the area A 3 of the micelle footprint constitutes 3/4 of the circle with a radius ρ 3 = (4A 3 /3π) 1/2 = (4P/3πσ 3 ) 1/2. The free energy F 3 per molecule is F 3 (P) = F 3,leg (P) + F 3,core (P) = Bρ 3 (P)τ b + Cτ2 ( 7π 2 )[R 3 (P) b ]2 P (7) Minimization of F 3 with respect to P gives the equilibrium aggregation number P 3 P 3 (σ 3 ) = π ( 2C 5 3B ) N 4 5 τ 2 5 (σ 3 b 2 ) 3 5 = ( 3 ) 5 4 (7 3 )2 ( σ 3 ) 3 5 P σ 1 (σ 1 ) = ( σ 3 ) 3 5 P σ 1 (σ 1 ) (8) 1 and the equilibrium free energy per molecule in the vertex micelle is 9

10 F 3 (P 3 ) = (5B) (2C 3B ) N 2 5 τ 6 5 (σ 3 b 2 ) 1 5 = (7 3 )1 ( σ 1 5 σ3 )1 F 1 (P 1 ) (9) Although the numerical values of coefficients B and C are unspecified, eqs (3), (6) and (9) enable comparison of free energies of three types of micelles and the evaluation of threshold ratios of polymer grafting densities (σ₃/σ₁)* and (σ2/σ₁)*, above which the vertex and the edge micelles become more favorable than the facet ones. Based on eqs (6) and (9), the vertex micelles are more favorable than the facet micelles, that is, F 3 (P 3 ) < F 1 (P 1 ) when ( σ 3 σ 1 ) > ( σ 3 σ 1 ) = 7 3 (10) and the edge micelles are more favorable than the facet micelles, that is, F 2 (P 2 ) < F 1 (P 1 ) when ( σ 2 σ 1 ) > ( σ 2 σ 1 ) = 3 2 (11) Comparison of the free energies of the vertex and edge micelles in eqs (6) and (9) suggests that the vertex micelles are more favorable than the edge micelles when ( σ 3 ) > ( σ 3 ) = 14 σ 2 σ 2 9 (12) As we show below, the range of predicted threshold values (σ₃/σ₁)* and (σ2/σ₁)* can be reached under experimental conditions. As follows from eqs (3), (6), (9), when 1 = 2 = 3 and micelle formation is not restricted by the NC size, the facet micelles have the lowest free energy per molecule, F 1 (P 1 ) < F 2 (P 2 ) < F 3 (P 3 ), and are thus thermodynamically most favorable. However, a higher grafting density is expected for NC edges and vertices, due to a larger volume available to polymer chains grafted to the curved NC regions. 7 A scaling-type estimate for the distribution of polymer grafting density on the NCs can be obtained by setting the chemical potential of the molecules on the NC surface constant. Below we discuss average grafting densities σ₃ and σ 2 of the polymer ligands within the footprint areas A 3 (σ₃) and A2(σ 2 ), of the vertex and edge micelles, respectively. Under -solvent conditions, the chemical potential of a chain in a planar brush is 1 / N( 1 b 2 ); in a cylindrical brush (with the radius of the grafting surface ρ 2 A 1/2 2 [P 2 (σ 2 )/σ 2 ] 1/2 is 2 / N 1/3 ρ 2 2 b) 2/3, and in a spherical brush with the total number 10

11 of chains P 3 ( 3 ), 3 / P 3 ( 3 ) 1/2. 7 Here the symbol indicates the equality with the accuracy of a numerical coefficient on the order of unity. The equality 1 = 2 = 3 allows one to present the ratios σ₂/σ₁ and σ₃/σ₁ for a NC with the size L significantly larger than the thickness of a planar brush in a -solvent 1) bn( 1b 2 ) 1/2 as ( σ 2 ) (σ σ 1 b 2 ) 7 8 N 3 4 τ 1 4 (13) 1 ( σ 3 ) (σ σ 1 b 2 ) 7 3 N 2 τ 2 3 (14) 1 By substituting experimentally realistic values of σ₁ = 0.01 chains/nm², b = 1.8 nm, N = 70, τ = 0.35, 1 and implementing unity as prefactor in eqs (13) and (14), one estimates σ₃/σ₁ 3.3 and σ2/σ₁ 1.56 and thus σ₃/σ For this choice of the system parameters, σ₃/σ₁ > (σ₃/σ₁)* = 7/3 2.3, σ₃/σ2> (σ₃/σ2)* =14/9 1.6, and F 1 (P 1 ) > F 2 (P 2 ) > F 3 (P 3 ), and the vertex micelles are thermodynamically most favorable. Here the superscript "*" indicates the threshold polymer grafting density ratios, at which a transition is expected from the facet to vertex micelles and from the edge to vertex micelles. The estimated values σ₃/σ₁ 3.3 and σ2/σ₁ 1.56 are expected to be almost constant, when the micelle aggregation number P is not significantly different than its equilibrium value. We note that the ratios (σ₃/σ₁)* and (σ3/σ2)* are estimated with the accuracy of the numerical coefficients on the order of unity, and the calculations merely point to the possibility of preferential micelle formation on the NC vertices and edges, rather than facets. Although at σ₃/σ₁ > (σ₃/σ₁)* and σ3/σ2 > (σ3/σ2)* the vertex and edge micelles are more favorable than the facet micelles, on large NCs with L >> A 1/2, micelles will still form in the centre of the facets, because most of the polymer molecules cannot reach the vertices and edges. However, for L/2, the facet micelles disappear in favor of the vertex and edge micelles. The latter situation is illustrated in Figure S7, where the normalized free energy per molecule, F i (P)/F 1 (P 1 ), is plotted as a function of the relative number of molecules P/P 1, for the facet (i = 1), edge (i = 2), and vertex (i = 3) micelles, based on eqs (3), (6), and (9), respectively. The reduced variables F i (P)/F 1 (P 1 ) and P/P 1 allow us to avoid the use of unknown numerical coefficients B and C in these equations, and directly compare the free energies of molecules in different types of pinned micelles. Figure S7 shows that at higher polymer grafting densities at the NC vertices and edges, the corresponding micelles have a lower normalized free energy per 11

12 molecule and are thus preferred. For scaling estimates σ₃/σ₁ 3.3 and σ2/σ₁ 1.56, the minima on the F 3 (P)/F 1 (P 1 ) and F 2 (P)/F 1 (P 1 ) solid curves corresponding to the equilibrium aggregation numbers P 3 /P and P 2 /P , respectively, are indicated with the solid green and blue triangles, respectively. The minimum in the normalized free energy per molecule for the facet micelle located at P/P 1 = 1 is marked by the red solid triangle. Dashed blue and green lines show the variations in the normalized free energy per molecule for the edge and vertex micelles, respectively, at σ₃ = σ2 = σ₁ with minimum values marked by empty blue and green triangles, respectively. Figure S7. (a) Illustration of pinned micelles formed on (top-to-bottom) the NC face, edge, and vertex and on the NS. (b) Normalized free energies F i (P)/F 1 (P 1 ) per molecule in the facet (i = 1, red color), edge (i = 2, blue color), and vertex (i = 3, green color) micelles, plotted as a function of the relative aggregation number P/P 1. The colors of the lines correspond to the colors of the frames in (a). Solid green and blue curves correspond to the ratios of grafting densities σ 3 /σ and σ 2 /σ Dashed green and blue lines correspond to the edge and vertex micelles at σ 3 = σ 2 = σ 1. The black dotted line shows the variation in the normalized free energy per molecule for a micelle on a NS with the grafting density σ s σ 1 and equilibrium aggregation number P s P 1. Triangles indicate the equilibrium values of aggregation number for each curve. Pinned micelles on the surface of NSs with a radius L/2 are characterized by grafting density s, footprint radius s and the radius of the micelle core R s. Since etching removes polymer-capped NC vertices and edges, we assumed that the remaining pinned micelles are composed of the ligands capping NC facets, and that they have the grafting density σs σ₁. The approximation s 1 (which holds at H << L/2), and the relationship R s << L/2 allow the use of 1 and P 1 as the estimates for the grafting density of ligands and the aggregation number in the 12

13 NS micelle. The NS curvature leads to the increase in size of pinned micelle, in comparison with that on a planar NC facet. However, since R s << L/2, NS micelles can be considered as quasiplanar with the aggregation number P s P 1. Thus, the equilibrium normalized free energies per molecule in the NS and facet pinned micelles are also close, that is, F s (P s ) F 1 (P 1 ), and F s (P) follows F 1 (P), at least, in the vicinity of the aggregation number P P 1 (Figure S7b). 5.3 Comparison of the theoretical model with experimental results The estimation of the thickness of the planar brush in a -solvent, bn( 1b 2 ) 1/2, with b = 1.8 nm, N = 70, σ₁ = 0.01 chains/nm², and the numerical prefactor equal to unity gives 21 nm, which is smaller than 30 nm (the length L/2 of a NC and the NS radius), and can be even smaller due to the numerical prefactor below unity. 8 In the following analysis, we use the scaling estimates σ₃/σ₁ 3.3 and σ2/σ₁ 1.56 for which F3( 3) < F2( 2) < F1( 1), and the vertex micelles are most favorable. A higher grafting density of polymer molecules on the NC edges and vertices is expected, due to the reduction in steric repulsion between the polymer molecules on higher curvature surfaces. 2,9,10 The experimental results agree with the relationship between the normalized free energies shown in Figure S7, given that a high grafting density of the polymer is expected for high curvature regions. In the experiments, micelles (patches) formed preferentially on the NC vertices and edges. Because the free energies per chain in the vertex and edge micelles exhibit shallow minima around optimized aggregation numbers, P 3 and P 2, these micelles can readily change their aggregation numbers and optimize the total free energy of the NC with the surface area 6L 2. Etching the NC with the edge length L = 60 nm transformed it into a NS with a diameter of 60 nm and the surface area L 2, which is by a factor of 6/ 2 smaller than the NC surface. The distribution of the number of patches on the NSs obtained by NC etching and on the assynthesized NSs were similar, with the average number of patches per NS of m 3 (Figure 5 in the main text). Each pinned micelle on the NS occupied the surface area 2 s = L 2 /3 with a radius of a footprint s ( s ) L/3 1/2 and a radius of the core R s < s. The number of micelles, m, was determined by the balance between the gain in the polymer solvent interfacial energy and the penalty in the energy of stretching of end-tethered polymer molecules. 13

14 In summary, we conclude that the asymptotic scaling theory developed for the polymer ligands with N >> 1, and >> R, predicts the appearance of the vertex and edge micelles, when the polymer grafting densities on these sites exceed the threshold ratios of (σ₃/σ₁)* = 7/3 and (σ2/σ₁)* = 3/2. At these threshold ratios, the vertex micelles are larger than the edge micelles by 20%, that is (qualitatively) consistent with experimental observations (Figure 1c', main text). In this theoretical framework, NC etching leads to the decrease in the total number of polymer ligands and the transformation of the NC into a NS with the grafting density σ₁ σ s. As a result, the average number of patches on the NSs is smaller than the average number of micelles on the NCs. 6. References (1) Choueiri, R. M.; Galati, E.; Thérien-Aubin, H.; Klinkova, A.; Larin, E. M.; Querejeta- Fernández, A.; Han, L.; Xin, H. L.; Gang, O.; Zhulina, E. B.; Rubinstein, M.; Kumacheva, E. Surface Patterning of Nanoparticles with Polymer Patches. Nature 2016, 538, (2) Hill, H. D.; Millstone, J. E.; Banholzer, M. J.; Mirkin, C. A. The Role Radius of Curvature Plays in Thiolated Oligonucleotide Loading on Gold Nanoparticles. ACS Nano 2009, 3, (3) Pastoriza-Santos, I.; Liz-Marzán, L. M. N, N -Dimethylformamide as a Reaction Medium for Metal Nanoparticle Synthesis. Adv. Funct. Mater. 2009, 19, (4) Gaikwad, A. V; Verschuren, P.; Kinge, S.; Rothenberg, G.; Eiserz, E. Matter of Age: Growing Anisotropic Gold Nanocrystals in Organic Mediaw. Phys. Chem. Chem. Phys. 2008, 10, (5) Rodríguez-Fernández, J.; Pérez-Juste, J.; Mulvaney, P.; Liz-Marzán, L. M. Spatially- Directed Oxidation of Gold Nanoparticles by Au(III)-CTAB Complexes. J. Phys. Chem. B 2005, 109, (6) Khanadeev, V. A.; Khlebtsov, N. G.; Burov, A. M.; Khlebtsov, B. N. Tuning of Plasmon Resonance of Gold Nanorods by Controlled Etching. Colloid J. 2015, 77, (7) Birshtein, T. M.; Zhulina, E. B. Scaling Theory of Supermolecular Structures in Block Copolymer-Solvent Systems: 1. Model of Micellar Structures. Polymer. 1989, 30, (8) Zhulina, E. B.; Adam, M.; Larue, I.; Sheiko, S. S.; Rubinstein, M. Diblock Copolymer 14

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