Supporting Information
Synthesis of {111}-faceted Au Nanocrystals Mediated by Polyvinylpyrrolidone: Insights from Density-Functional Theory and Molecular Dynamics Shih-Hsien Liu,† Wissam A. Saidi,‡ Ya Zhou,† and Kristen A. Fichthorn∗,†,¶ Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States, and Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States E-mail: fi
[email protected] Phone: +1-814-863-4807. Fax: +1-814-865-7846
∗
To whom correspondence should be addressed The Pennsylvania State University ‡ University of Pittsburgh ¶ Also in the Department of Physics †
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Details of the DFT Calculations The Au surfaces were modeled using six-layer (4 × 4 × 14) supercells for Au(111) and Au(100) and a (5 × 4 × 14) supercell for (5 × 1) Au(100)-hex. The gas-phase 2P molecule was studied using a cubic supercell with side length of 20 Å. Wave functions were expanded using a kinetic energy cut-off of 400 eV. The Brillouin zone was sampled with a (4 × 4 × 1) Monkhorst-Pack k-point grid and with a Methfessel-Paxton smearing of 0.1 eV. We also conducted a projected density-of-states (PDOS) analysis, where we used an (8 × 8 × 1) k-point grid. We adsorbed 2P on only one side of the slab, we kept the bottom 3 layers of the Au slabs fixed, and we relaxed all the other atomic coordinates using a force-convergence criterion of 0.01 eV/Å. As described below, the computed bulk Au lattice constant and the surface energies of Au(111), Au(100), and (5 × 1) Au(100)-hex are in good agreement with previous theoretical results. 1 In a previous study, 2 we showed that the computed structure of gas-phase 2P is consistent with experiment. 3
Au Lattice Constant and Surface Energy with DFT The bulk Au lattice constant a0 was calculated using a face-centered cubic (fcc) primitive cell built with different lattice parameters ranging between 4.00 and 4.28 Å. We used a (12 × 12 × 12) k-point grid for this primitive cell. We plotted the total energy vs. a0 and then fit the curve with a third-degree polynomial. We found a0 = 4.17 Å at the location where the derivative of the polynomial was zero. The corresponding bulk Au cohesive energy is Ebulk = -3.273 eV. These values are in exact agreement with previous PAW-PBE theoretical results. 1 For calculating the surface energies of the three Au surfaces (γAu ), we first built a supercell with the smallest repeat unit for each surface in the x− and y−directions parallel to the surface. In the z−direction perpendicular to the surface, we had 23 total layers: 8 vacuum and 15 Au. We relaxed the 3 surface layers at each of the Au-vacuum interfaces in this 2
periodic cell and the 9 central layers were fixed at the bulk termination. We used (16 × 16 × 1), (16 × 16 × 1), and (4 × 20 × 1) k-point grids for Au(111), Au(100), and (5 × 1) Au(100)-hex, respectively. After geometry optimization, we calculated γAu using
γAu =
EAu − N Ebulk 2Asurf
,
(S1)
where EAu is the energy of the optimized supercell, N is the number of Au atoms in the supercell, Ebulk is the bulk Au cohesive energy, and 2Asurf is the sum of the areas at the top and the bottom of the Au slab. The surface energies of Au(111), Au(100), and (5 × 1) Au(100)hex were found to be 0.044, 0.054, and 0.050 eV/Å2 , respectively, in excellent agreement with previous theoretical values, 1 although PBE may underestimate surface energies. 4
2P Conformations on Au Surfaces with DFT
Figure S1: Top-down view of the 5 unique conformations of 2P on Au(111). Oxygen is red, nitrogen is blue, carbon is black, and hydrogen is white.
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Figure S2: Top-down view of the 4 unique conformations of 2P on Au(100). Oxygen is red, nitrogen is blue, carbon is black, and hydrogen is white.
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Figure S3: Top-down view of the 20 unique conformations of 2P on (5 × 1) Au(100)-hex. Oxygen is red, nitrogen is blue, carbon is black, and hydrogen is white.
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PDOS Analysis of 2P/Au Binding We can gain further insight into the binding energies on the various surfaces by examining the angular-momentum resolved PDOS projected onto the Au atoms and the 2P molecule. In Figs. S4, S5, and S6, we choose three different conformations each for 2P on Au(111), Au(100), and (5 × 1) Au(100)-hex, to show the PDOS sensitivity with respect to the binding conformation. The binding energies of these conformations are shown in Fig. 3 in the main text and the conformations are depicted in Figs. S1, S2, and S3. For Au(111) we show the PDOS for conformations 1, 3, and 5 (Fig. S4), for Au(100) we show the PDOS for conformations 1, 2, and 4 (Fig. S5), and for (5 × 1) Au(100)-hex, we show the PDOS for conformations 1, 8, and 20 (Fig. S6). Since the PDOS of the Au d-band does not vary appreciably between different conformations, this is only shown for conformation 1 on each surface. For contrast, the insets in these figures show PDOS in the non-interacting limit, where 2P is far from the surface. Comparing the PDOS for the three different conformations on Au(111), it can be seen that these are fairly similar, which corroborates their similar binding energies. On the other hand, the PDOS for the three conformations on Au(100) and (5 × 1) Au(100)-hex show more variation. Here, we see a shift in the 2P PDOS toward higher energies and less overlap between the 2P sp-band and the Au d-band edge as the binding energy decreases. Examining the PDOS for the different conformers and their binding energies, we see that the overlap between the 2P sp-band and the metal d-band correlates positively with the short-range interaction, which agrees with our previous findings for Ag. 5 For instance, for Au(111), and in contrast to the Ag(111) surfaces, there is more overlap between the 2P sp-band and the metal d-band in all of the conformations, which explains why Eshort-range contributes more to Ebind for 2P on Au(111) than that on Ag(111). 5
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Figure S4: PDOS for conformations 1, 3, and 5 of 2P on Au(111). The inset is the PDOS in the non-interacting limit where the 2P molecule is about 10 Å from the Au surface.
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Figure S5: PDOS for conformations 1, 2, and 4 of 2P on Au(100). The inset is the PDOS in the non-interacting limit where the 2P molecule is about 10 Å from the Au surface.
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Figure S6: PDOS for conformations 1, 8, and 20 of 2P on (5 × 1) Au(100)-hex. The inset is the PDOS in the non-interacting limit where the 2P molecule is about 10 Å from the Au surface.
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Figure S7: Correlation between short-range binding energies and distances from the oxygen atom of 2P to the Au surface for all conformations of 2P on Au(111), Au(100), and (5 × 1) Au(100)-hex. The numbers next to the markers are the conformation numbers of 2P that are shown in Figs. S4-S6.
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2P Conformations on Au Surfaces Using DFT with VASPsol
Figure S8: Top-down view of the optimized conformations of 2P on (a) Au(111), (b) Au(100), and (c) (5 × 1) Au(100)-hex in implicit solvent. Oxygen is red, nitrogen is blue, carbon is black, and hydrogen is white.
Details of Developing Force Field for PVP-Au Interactions In our force field, atoms (M) in PVP molecules interact with the Au surfaces via a pair potential. Additionally, the O atoms in PVP can modify the electron density of atoms in the Au surfaces, which we describe using an embedded-atom method (EAM) potential, 6 modified as described below. Thus, the potential for the PVP-Au interaction is given by
E=
Au ∑ i
FAu
Au ∑
ρAu-Au (rij ) +
j̸=i
O ∑ j̸=i
ρO→Au (rij )
Au-M ∑ ∑ 1 Au-Au ϕAu-Au (rij ) + ϕAu-M (rij ) + 2 i̸=j i̸=j
(S2) ,
where FAu is the Au embedding energy, which is a function of electron density ρ, and ρA-B (rij ) and ϕA-B (rij ) are the electron density and pairwise potential between atomic species A and B separated by a distance of rij , respectively. The superscript on the sum indicates the sum 11
runs over the specified species or species pairs. We note that FAu , ρAu-Au and ϕAu-Au are given by the EAM potential. 7,8 To account for the many-body aspects of the Au-M interactions, the additional electron density supplied one-way from O to Au (ρO→Au ) is included and defined as ρO→Au (rij ) = fO ρAu-Au (rij ) ,
(S3)
where a scaling factor fO is used to adjust the amount of electron density supplied from O to Au, which is critical in surface-sensitive adsorption of organic molecules on Au and Ag nanoparticles. 6,9 The EAM potential and the one-way electron-density function are implemented in LAMMPS via the Finnis-Sinclair option. 10 The pair potentials describing Au-M interactions (ϕAu-M ) are defined as [
ϕAu-M (rij ) = D0,Au-M e−2αAu-M (rij −r0,Au-M ) − 2e−αAu-M (rij −r0,Au-M ) −s6 fdamp (rij , R0,Au , R0,M )C6,Au-M rij−6
]
(S4) ,
where the first term is a Morse potential to account for short-range, direct-bonding interactions with parameters D0,Au-M , αAu-M , and r0,Au-M , and the second term accounts for the long-range vdW interactions in the form prescribed by Grimme. 11 Here, s6 is a global scaling factor, fdamp is a damping function associated with vdW radii R0,Au and R0,M , and C6,Au-M is the dispersion coefficient for Au-M interactions. The values of C6 and R0 for atomic species in the organic molecules are those given by Grimme, 11 and those by Ruiz 12 were employed for Au atoms to account for screening effects. We use a cutoff radius of 5.5 Å and 12 Å for Morse potentials and vdW interactions, respectively. To parameterize the force field based on Eqs. (S2), (S3), and (S4), we adjusted D0,Au-M , αAu-M , r0,Au-M , fO , and s6 using the simulated annealing protocol that we used in previous work. 9 In this protocol, we use force- and energy-matching to to reproduce the binding energies for various conformations of 2P on Au(111) and (5 × 1) Au(100)-hex observed in our DFT calculations. To better reproduce the striped binding-energy pattern of 2P on (5 ×
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1) Au(100)-hex (cf., Fig. 6 in the main text), we found it necessary to classify the Au surface atoms into 4 different groups. Each unique group (AL, AM, AN, and AH) corresponds to each unique stripe (first, second, third, and fourth stripe from the left to right in Fig. 6 in the main text). The atoms in each of the 4 groups are allowed to have a separate set of potential parameters for the Au-M interactions. With s6 = 1.5, and Morse potential parameters and fO given in Table S1, we show the results of our force-field-based calculations in Tables 2, S2, and Fig. 7 in the main text. Table S1: Morse potential parameters (D0 , α, and r0 ) and the scaling factors for the one-way O-Au electron-density function [cf., Eq. (S3)] used for modeling short-range Au-M interactions. All atoms in the Au(111) slab and atoms in the bottom 5 layers of (5 × 1) Au(100)-hex are denoted as Au. D0 (eV) α (Å−1 )
Interaction Au, AH, AM, AN, AL – aliphatic H Au, AH, AM, AN, AL – aliphatic C in backbone Au, AH, AM, AN, AL – aliphatic C in ring Au, AH, AM, AN, AL – amide N Au, AH, AM, AL – C bonded to O AN – C bonded to O Au – carbonyl O AH – carbonyl O AM – carbonyl O AN – carbonyl O AL – carbonyl O
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0.00960 0.00488 0.00460 0.00984 0.01444 0.02888 0.00650 0.00650 0.00650 0.00650 0.00650
1.30 1.01 1.03 1.76 1.14 1.14 3.34 3.34 3.34 3.34 3.34
r0 (Å)
fO
1.94 4.92 4.92 3.26 4.85 4.85 2.65 2.65 2.65 2.65 2.65
– – – – – – 0.9 1.0 0.9 0.3 0.2
Table S2: Binding energies for the 20 conformations of 2P on (5 × 1) Au(100)hex predicted by DFT and the force field, arranged in descending order, and corresponding percent errors of the force field with respect to DFT. Conformation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
DFT Ebind (eV) Force Field Ebind (eV) Percent Error 1.129 1.078 1.075 1.054 1.048 1.041 1.040 1.024 1.016 1.011 1.002 1.001 0.994 0.980 0.977 0.975 0.970 0.948 0.945 0.936
1.132 1.089 1.018 1.018 1.005 1.005 1.005 1.003 1.000 0.993 0.990 0.986 0.985 0.980 0.976 0.969 0.958 0.953 0.951 0.920
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0.3 1.0 5.3 3.4 4.1 3.5 3.4 2.1 1.6 1.8 1.2 1.5 0.9 0.0 0.1 0.6 1.2 0.5 0.6 1.7
Details of the MD Simulations The Au surfaces were modeled using rigid (40 × 46 × 60 Å) and (40 × 40 × 60 Å) supercells for Au(111) and (5 × 1) Au(100)-hex, respectively. It is noted that these Au surfaces are flat, with periodic boundary conditions to account for the relatively large nanocrystals grown experimentally. 13,14 The rigid surface consists of 6 layers, where the bottom 3 layers are fixed at the bulk termination, and the top 3 layers are fixed after being optimized using the FIRE algorithm 15 with a force-convergence criterion of 0.01 eV/Å. As described below, the bulk Au lattice constant and its corresponding cohesive energy are in good agreement with previous theoretical results. 7 We built two initial orientations of 24 atactic PVP icosamer chains on each Au surface: parallel and perpendicular to the (5 × 1) Au(100)-hex unit cell and two perpendicular orientations on Au(111). Initially, the PVP backbones were all in the trans conformation. The MD simulations were conducted in the canonical ensemble at 550 K 16 with a time step of 1 fs. We used the Nosé-Hoover thermostat to maintain constant temperature and the instantaneous potential energies and configurations were saved every 0.1 ns.
Au Lattice Constant and Surface Energy with EAM We use the bulk Au lattice constant a0,EAM = 4.08 Å, determined by experiment. 17 We computed the bulk cohesive energy using the EAM potential using a (15 × 15 × 15) facecentered cubic (fcc) cell. The bulk Au cohesive energy is Ebulk,EAM = -3.924 eV, in exact agreement with previous EAM theoretical results. 7 For calculating the surface energies of the two Au surfaces (γAu,EAM ), we built (18 × 20 × 160 Å) and (20 × 20 × 160 Å) supercells for Au(111) and (5 × 1) Au(100)-hex, respectively. The surface consists of 15 layers, where 3 surface layers at each of the Auvacuum interfaces were relaxed and 9 central layers were fixed at the bulk termination.
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After geometry optimization, we calculated γAu,EAM using
γAu,EAM =
EAu,EAM − N Ebulk,EAM 2Asurf
,
(S5)
where EAu,EAM is the energy of the optimized supercell, N is the number of Au atoms in the supercell, Ebulk,EAM is the bulk Au cohesive energy, and 2Asurf is the sum of the areas at the top and the bottom of the Au slab. The surface energies of Au(111) and (5 × 1) Au(100)-hex were found to be 0.075 and 0.101 eV/Å2 , respectively. The value of γAu(111),EAM is in excellent agreement with previous theoretical value 7 and also close to experiment. 18
Equilibrated Configurations of PVP Layers on Au Surfaces with MD
Figure S9: Top-down and side views of the equilibrated configurations of a PVP monolayer on Au(111). The initial orientation of the PVP chains is (a) horizontal, and (b) vertical with respect to the page. Oxygen is red, nitrogen is blue, and carbon is black.
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Figure S10: Top-down and side views of the equilibrated configurations of a PVP monolayer on (5 × 1) Au(100)-hex. The initial orientation of the PVP chains is (a) parallel, and (b) perpendicular to the long axis of the (5 × 1) Au(100)-hex unit cell. Oxygen is red, nitrogen is blue, and carbon is black.
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