Growth Mechanism of Five-Fold Twinned Ag Nanowires from

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Growth Mechanism of Five-Fold Twinned Ag Nanowires from Multi-Scale Theory and Simulations Xin Qi, Zihao Chen, Tianyu Yan, and Kristen A. Fichthorn ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.9b00820 • Publication Date (Web): 14 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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Growth Mechanism of Five-Fold Twinned Ag Nanowires from Multi-Scale Theory and Simulations Xin Qi,† Zihao Chen,† Tianyu Yan,† and Kristen A. Fichthorn∗,†,‡ †Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States ‡Department of Physics E-mail: [email protected]

Abstract Five-fold twinned metal nanowires can be synthesized with high aspect ratios via solution-phase methods. The origins of their anisotropic growth, however, are poorly understood. We combine atomic-scale, meso-scale, and continuum theoretical methods to predict growth morphologies of Ag nanowires from seeds and to demonstrate that high aspect-ratio nanowires can originate from anisotropic surface diffusion induced by the strained nanowire structure. Nanowire seeds are similar to Marks decahedra, with {111} “notches” that accelerate diffusion along the nanowire axis to facilitate one-dimensional growth. The strain distribution on the {111} facets induces heterogeneous atom aggregation and leads to atom trapping at the nanowire ends. We predict that decahedral Ag seeds can grow to become nanowires with aspect ratios in the experimental range. Our studies show that there is a complex interplay between atom deposition, diffusion, seed architecture, and nanowire aspect ratio that could be manipulated experimentally to achieve controlled nanowire syntheses.

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Keywords: nanocrystal, nanowire, molecular dynamics, kinetics, growth, Markov chain

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Five-fold twinned nanowires (NW) can be synthesized from various fcc metals via solutionphase methods. These anisotropic nanostructures can be grown with aspect ratios (AR) beyond 1000 and they possess unique electrical, optical, and mechanical properties that show promise for many applications (e.g., see refs. 1 and 2). Despite considerable interest in their synthesis and properties, the kinetic mechanisms that promote the high-AR growth of metal NW are not well understood. The commonly assumed structure for five-fold twinned NW can be inferred from Fig. 1. Here, we see that these wires consist of ten {111} “end” facets and five {100} “side” facets. In experimental studies, 3 it was suggested that the selective flux of Ag atoms/ions to the {111} ends relative to the {100} sides results in high-AR growth. This flux selectivity has been attributed to the facet-selective binding of capping molecules (e.g., PVP, in the case of Ag NW 3 ) to the {100} facets. While recent experiments 4 and theoretical studies based on quantum density-functional theory (DFT) 5,6 confirm facet-selective binding of PVP to Ag(100), they also indicate that it is unlikely to be sufficient to produce NW. 4,7

Figure 1: NW at various relative linear facet growth rates. The inset shows the AR. {100} facets are light and {111} facets are dark. Steady-state, kinetic nanocrystal shapes can be described by the kinetic Wulff construction. 8,9 In Fig. 1, we show a kinetic Wulff construction that describes NW shapes for various linear facet growth rates Gi of the {111} ends and the {100} sides. Here, we see that to 3


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achieve AR around 100 (seen in some experiments 1,3,10–13 ), the flux to the {111} facet (G{111} ) needs to be nearly 100 times larger than the flux to the {100} facet (G{100} ). Our DFTbased calculations 14–16 do not bear this out for Ag NW capped solely with PVP. In these studies, 15,16 we found G111 /G100 ≈ 1.7 − 2.0, which is sufficient to produce {100}-faceted nanocubes, but would lead to NW with AR around 2. In recent studies of the growth of Cu NW in the presence of HDA capping molecules and chloride, 2 we found that chlorine adsorption can lead to facet-selective depletion of capping molecules, such that the ends of Cu NW are free of HDA, while the sides are covered. This can promote a selective flux of Cu ions to NW ends to facilitate high-AR growth. Meena and Sulpizi reached a similar conclusion regarding CTAB/CTAC micelles and their influence on the growth of Au nanorods. 17,18 It is possible that such halide-enhanced NW growth could occur for Ag NW, as Li et al. found that these can be grown with AR up to 2000 in the presence of bromide. 19 Da Silva et al. also found NW with AR greater than 1000 in their studies of Ag NW growth in the presence of bromide. 1 However, several groups 1,3,10–13 have grown Ag NW (albeit with lower AR around 100) in the absence of halide. Thus, a mechanism other than facet-selective ion deposition can be operative in producing NW. Here, we examine the possibility that this mechanism involves surface diffusion. In this work, we consider the diffusion of Ag atoms on the surfaces of NW seeds and the possibility that their accumulation on the {111} facets can lead to growth of the seeds into NW. Five-fold twinned NW grow from decahedral seeds that are almost always depicted as shown in the lower left inset of Fig. 1. Seeds consist of five subunits with {111} twin planes between each unit. The five-fold geometry is not space filling and there is a gap after fitting the five sections together. Closure of this gap results in lattice strain. 20,21 As the seeds grow into wires, they develop five {100} facets along the ⟨110⟩ axis, which are typically assumed to produce an Ino decahedron (Dh), 22 as shown in Fig. 2(a). Marks showed that the addition of {111} “notches” to the corners of the twin boundaries, similar to the structure in Fig. 2(c), reduces the strain and lowers the potential energy relative to the Ino Dh. 23 Our calculations

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also indicate that a Marks-like Dh is energetically favored for five-fold twinned Ag seeds. Moreover, this seed shape is conducive to the growth of NW.

Results and Discussion Structure of Five-Fold Twinned Seeds

Figure 2: Front and side views of (a) an Ino Dh seed, (b) annealed seed, and (c) idealized annealed seed with {111} notches and {110} corners. Each color in the zoomed-in sections represents an atomistic layer in the stepped corner.

We used molecular dynamics (MD) simulations and energy minimization to probe possible structures for five-fold twinned seeds in the same size range as those in experiment. 1,3,10–13 Our calculations begin with an Ag Ino Dh, shown in Fig. 2(a), in which the {100} side has 60 atoms perpendicular to and 20 atoms along the ⟨110⟩ axis. This Ino Dh consists of 2.6×105 5


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atoms and has an apparent diameter of 28 nm. We performed MD annealing simulations to achieve possible transformations of the initial Ino Dh seed. Figure 2(b) shows the resulting structure, which clearly possesses {111} notches, characteristic of a Marks Dh. In addition, the corners are smoothed by a series of steps that descend from the notches to the {111} end facets [see the inset of Fig. 2(b)]. The stepped corners seen in Fig. 2(b) exhibit an irregular structure. To obtain a characteristic structure for the corners, we constructed various facets (e.g., {110}, {311}, {511}, etc.) on the corners of multiple Marks Dh. We found that the {110} facet yields the lowestenergy structure. To see if there could also be an alternative facet between the {100} and {111} facets, we tested various structures and observed no energy-lowering effect. Thus, we conclude that an alternative facet between the {100} side and the {111} end is not preferred. Figure 2(c) shows an idealized seed structure. Alpay and Marks argued that sharp corners present a singularity that should not occur in nanoshapes 24 and our observations are consistent with this. The Marks-like Dh structure is also consistent with the rounded pentagonal cross sections and the rounded axial tip profiles observed experimentally for Ag Dh seeds 1 and NW. 20,24,25 Five-fold twinned NW can be synthesized from other fcc metals and we are aware of (high-resolution) transmission electron microscope and scanning electron microscope images of Dh Pd 26 and Au 25,27 seeds with rounded, pentagonal cross sections similar to those we find here. Rounded pentagonal cross sections in Cu, 28 Pd, 26 and Au 25 NW have been observed that are also consistent with our results. We also performed an energetic analysis of various Dh, with a focus on examining how notches contribute to the stability of the structures. For energetic analysis, we adopted the quantity ∆ introduced by Baletto et al. 29 to evaluate the energetics of the Ino Dh, the Marks Dh, and the rounded Marks Dh found in this study. The quantity ∆ has the form

∆=

Etot − N Ecoh N 2/3

6

,


(1)

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Figure 3: The values of ∆ for an Ino Dh, a Marks Dh with a notch size of 2, and a rounded structure based on the Marks Dh with {110} corners that are 3 unit cells wide. The insets depict a view of the five {111} facets (purple) and {110} facets (red). where Etot is the total energy of the system, N refers to the total number of atoms, and Ecoh is the cohesive energy of a bulk atom. We compare the values for the three structures in Fig. 3. Here, the notch size refers to the number of layers of atoms removed from the twin edge of an Ino Dh. We see that ∆ can be lowered significantly by having a notch and further lowered by having the {110} corner, which confirms the effect of these modifications in stabilizing the structure. We used the ∆ value to further evaluate the optimal notch and {110} facet size of the NW. These calculations are discussed in the Supporting Information (SI). We briefly mention here that we adopt a notch size of 2 and a {110} facet width of 3 unit cells in our simulations, which is the size we obtained upon annealing in Fig. 2. Though these are minimal notch and {110} facet sizes, we will demonstrate below that they can still influence NW growth kinetics significantly.

Island Nucleation The first step in NW growth is island nucleation and it is possible that island nucleation could determine the NW growth rate. To understand island nucleation and the growth environment 7


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Figure 4: The atom distribution on the {111} end with (a) high atom density and (b) low atom density and the {100} side with (c) high atom density and (d) low atom density. The insets depict the geometry of the end [(a) and (b)] and side [(c) and (d)] facets. {111} facets are purple, {100} facets are blue, and {110} facets are red.

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in this system, we ran MD simulations of Dh seeds with various atom coverages at T = 160◦ C – the experimental temperature in ref. 1. The atoms are initially distributed randomly on the surface and they can diffuse to form aggregates over time. Our MD simulations reveal that atoms diffuse rapidly on Ag(111), so that all of them form aggregates within ns – a time scale much shorter than the deposition time. However, atoms diffuse relatively slowly and remain essentially isolated on Ag(100) over the same time window. As we will show below, these observations are consistent with the small/large diffusion barriers that we find for Ag(111)/Ag(100), respectively. To track Ag-atom aggregates, we obtained atom density maps. Figure 4 shows atom density maps for two different atom coverages: a high-density group, which represents 25 − 50% atom coverage, and a low-density group for the 1 − 5% atom coverage range. Regarding the aggregates on Ag(111), in Fig. 4(a) and (b), we see a relatively low atom density in the center of Ag(111) and a high density near the {111}-{110} interface, parallel to the center of the twin edge, and around the NW tip where the five segments meet. Atoms accumulate near the tip due to a higher energy barrier to cross the twin edge than to diffuse on {111} (cf., Table 1), which limits translation. The tendency to have high local-atom densities on {111} is also correlated with the strain distribution on the surfaces of the seed. We calculated the strain on the surfaces of the Ino Dh shown in Fig. 2(a), the Dh in Fig. 2(c), and a Marks Dh of the same size. Figures 5 and 6 show the strain distributions on these three structures. We see on both the {100} and {111} facets that tension and compression alternate on bonds that are perpendicular and parallel to the ⟨110⟩ axis, respectively. This tension-compression pattern has also been observed experimentally 20 in X-ray and electron diffraction studies of five-fold twinned Ag nanowires. It is evident that the notch releases stress near the twin edge on the {100} side and the {110} facet lowers the strain at the corner between the {100} side and the {111} end. It is also notable that areas near the twin edge (on both {100} and {111}) are mostly compressed. The area of the purely compressed region can be reduced through introduction of a notch and the {110}

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Figure 5: The strain on a single {100} side surface of an Ino Dh [(a) and (b)], a Marks Dh [(c) and (d)] and the structure found in this study [(e) and (f)]. Figures (a), (c) and (e) show the strain distribution with scales given in the legends. Figures (b), (d) and (f) illustrate the bonds with tension (labelled as red dots) vs. compression (labelled as blue dots). The inset depicts the geometry of the side facet for (e) and (f). {111} facets are purple, {100} facets are blue, and {110} facets are red.

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Figure 6: The strain on a single {111} end surface of an Ino Dh [(a) and (b)], a Marks Dh [(c) and (d)] and the structure found in this study [(e) and (f)]. Figures (a), (c) and (e) show the strain distribution with scales given in the legends. Figures (b), (d) and (f) illustrate the bonds with tension (labelled as red dots) vs. compression (labelled as blue dots). The inset depicts the geometry of the end facet for (e) and (f). {111} facets are purple, {100} facets are blue, and {110} facets are red.

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facet, and this effect is more significant for the {111} end. The strain distribution shows an interesting pattern on the {111} facet. Here, we see two stripes that show pure tension running parallel to and near the middle of the twin edges [Fig. 6(a), (c) and (e)]. There is also a purely compressed region near the {111}{110} interface, as shown in Fig. 6(b), (d) and (f). This region becomes smaller when the {110} facets are introduced, but it still leaves one row of fcc sites purely compressed parallel to the {110} facet. This strain distribution on the {111} end, particularly the interplay of compression and tension near the {110} facet, greatly affects atom aggregation on this surface, and is important for achieving NW growth. We will elaborate further on this below. We find that the atom-density distribution shown in Fig. 4(a) and (b) mirrors the trend suggested by the strain distribution. The stripes of tension that run parallel to the twin edges in Fig. 6(c) attract and hold aggregates in Fig. 4(a) and (b). The areas with less anisotropic distortion near the {110}-{111} border also attract aggregates. Interestingly this area is not immediately adjacent to the {110} facet and is only about one atom wide; therefore, when an atom transits from {110} to {111}, it is likely to meet and join an aggregate nearby. The binding of atoms one row away from the {110} facet ensures that this facet will persist when a new {111} layer is complete. Thus, island nucleation occurs rapidly – within ns – on Ag(111), but not on Ag(100). Aggregates on Ag(111) form in preferred locations that can be correlated to strain on the surface, as well as to the geometry of the {111} facet. As we will discuss below, the aggregates on Ag(111) play a key role in promoting NW growth and our observations are consistent with the experimentally observed growth of Ag NW.

NW Growth Five-fold twinned NW growth is a complex system due to the solution-phase environment of the synthesis, as well as the strained nature of the NW. Despite these difficulties, there are simplifying features in this system. First, it is observed that Ag NW retain the diameters 12


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of the initial seeds. 1 This indicates that the strain state of the NW facets is approximately constant, since the strain depends on the NW diameter. Strain can influence the diffusion processes that occur on the NW surfaces, 30 so here we can assume that the diffusion processes do not change with NW size. A second simplifying feature is that NW growth is slow. Based on the data in ref. 1, the average time per deposition is ∼ 10−4 s in the growth of Ag NW. As we will elaborate below, this time is comparable to or longer than atom diffusion times from {100} to {111} facets and implies a low Ag atom coverage on the {100} facets. A low Ag(100) coverage is expected if NW grow to a high AR while retaining the diameter of the seeds. Finally, we consider the diffusion of Ag atoms on Ag surfaces in vacuum. Interestingly, the vapor-phase growth of five-fold twinned Ag NW has been observed experimentally. 31 In solution, Ag NW can be grown in various solvents, with or without capping molecules, and in the presence of various reductant molecules. 1,3,10–13 These results suggest that once a seed is established, its intrinsic structure possesses features that promote one-dimensional (1D) growth – though solvent, capping molecules, and other additives could work in synergy in this regard. We can describe NW growth in terms of atom deposition and diffusion between the {100} and {111} facets using dN{111} = Rdep,{111} + R{100}→{111} − R{111}→{100} dt

,

(2)

dN{100} = Rdep,{100} + R{111}→{100} − R{100}→{111} dt

.

(3)

and

Here, Rdep,i is the deposition rate on facet i and Ri→j is the rate to transit from facet i to facet j via surface diffusion. The time derivatives of the numbers of atoms accumulating on the facets provide the linear facet growth rates ( i.e., dNi /dt = Gi ) in Fig. 1. When deposition occurs much faster than diffusion, the linear facet growth rates G{111} and G{100} are essentially equal to the facet deposition rates. The kinetic Wulff construction 13


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in Fig. 1 can describe NW morphology in this scenario and this has been considered in previous studies. 7,15,16 If the inter-facet diffusion rates are much greater than the deposition rates, we expect each deposited atom to achieve steady-state occupancy of the {100} and {111} facets. The kinetic Wulff construction in Fig. 1 can also describe NW morphology in this scenario. As we will discuss below, it is unlikely that the growth of Ag NW fits into either of these limiting cases. To apply Eqs. 2 and 3 to describe NW growth, we calculate mean first-passage times (MFPT) for inter-facet diffusion. The MFPT to go from facet i to j, ⟨ti→j ⟩ is related to Ri→j in Eqs. 2 and 3 by Ri→j ∝ ⟨ti→j ⟩−1 – the inverse of the MFPT can be regarded as a first-order rate constant for the dilute limit considered here and it becomes the rate in the single-atom limit. We obtain MFPT using the theory of absorbing Markov chains. 32,33 As we discuss in the SI (and as is discussed elsewhere 34–38 ), these calculations involve enumerating all n possible states for an atom on facet j – the transient states, as well as m neighboring states on facet i – the absorbing states. The transient and absorbing states in this system are illustrated in Fig. S2 in the SI. Here, we note that atom binding sites in the notch are counted as part of the {100} facet, implying that the notch grows with the {100} facet. Similarly, we include the {110} facet with both the {100} and {111} facets. Thus, we assume this facet is a “bridge” that maintains a constant size with a zero growth rate. A square transition matrix A of rank n + m is constructed that contains the rates of all the j → j and j → i transitions (cf., Eq. S2 in the SI) and we obtain the Markov matrix M from A, 32,33,37 as we elaborate in the SI. We obtain the MFPT, as well as the probability that an atom beginning on facet j will end up at a particular absorbing state on facet i from submatrices of M. 32,33,37 The MFPT and exit probabilities can also be obtained via kinetic Monte Carlo (kMC) simulations. 39,40 Absorbing Markov-chain calculations yield exactly the same MFPT as kMC simulations with the same rate processes. 34–38 When all states of the system can be enumerated with relative ease, as is the case here, Markov-chain calculations are more efficient and easier to implement than kMC.

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Surface Diffusion To obtain MFPT for inter-facet diffusion, we need to construct A and M and we need to find transition rates between various states on the {100}, {111}, and {110} facets, as well as the notch. We obtained these using harmonic transition-state theory, in which the transition rate between two neighboring sites is given by k = ν0 exp(−EB /kT ). We assumed that ν0 = 1013 s-1 for all moves. A table of key diffusion barriers is given in Table 1. To obtain the values in Table 1, we considered that diffusion can occur via hopping between nearest-neighbor sites, exchange between an on-surface and in-surface atom, as well as by concerted many-atom moves. In the case that diffusion could occur via more than one mechanism (e.g., hopping or exchange), we reported only the lowest-energy barrier in Table 1. In Table 1, we see that on the {100} facet, atoms diffuse by hopping between nearestneighbor, four-fold hollow binding sites. The diffusion barrier is slightly lower for diffusion along the NW long axis than for diffusion perpendicular to the long axis. This is a consequence of strain, which is compressive and tensile parallel and perpendicular to the NW long axis, respectively (cf., Fig. 5). From the {100} facet, atoms can exchange to the notch, the {111} facet, and to the {110} facet via two different channels parallel and perpendicular to the ⟨110⟩ direction [cf., Fig. S2(a)]. The diffusion barriers for hopping between three-fold hollow sites on Ag(111) are low (0.08 eV) compared to the barriers to diffuse between four-fold hollow sites on Ag(100) (∼0.5 eV), which is consistent with the relatively rapid aggregation of atoms on Ag(111) compared to Ag(100), discussed above. Atoms can jump from one {111} end facet to another by crossing a twin edge with a relatively high barrier (0.38 eV). To account for atom aggregation on the {111} facets, we characterized the barrier for an atom to leave a dimer, which is 0.42 eV. Based on the atom-density maps for nucleation on the {111} facet in Fig. 4(a) and (b), we consider aggregates in three locations on the {111} facets: near the tip, along the twin edge and near the {110}-{111} interface. These locations are indicated in Fig. S2(b). The barriers for atoms to leave these sites are those to dissociate from a dimer. We note that this 15


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Table 1: Key Diffusion Barriers

Position

Area

Description

100 plane

in facet

perpendicular to

0.53

0.53

along

0.51

0.51

facet to notch

exchange to notch

0.62

0.76

facet to {110}

perpendicular to

0.36

0.69

facet to {110}

along

0.39

0.72

across twin

hop to another segment

0.38

0.38

in facet

hop from fcc to hcp

0.08

0.08

dimer formation

aggregate

0.08

0.42

Interfacet {100} and {111}

interfacet edge middle

exchange from {100} to {111}

0.69

0.7

In the notch

along

0.16

0.16

dimer along

0.21

0.21

111 plane

On {110}

Forward barrier (eV)

Backward barrier (eV)

notch to {110}

as a single atom

0.33

0.57

notch to {110}

was a dimer in notch, only one atom dissociates

0.3

0.31

across channel

towards {111}

0.31

0.31

in channel

0.33

0.33

to {111}

0.33

0.59

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is a conservative estimate of the effects of atom aggregation on the MFPT, as most of the aggregates we observed on the ns time scale contain more than two atoms. Despite this, we show that even such minimal energetics can induce long-time atom trapping on the {111} facet and lead to 1D NW growth.

Figure 7: The energy profile for an atom to transit from the {100} facet (pink) to the {111} facet (violet) through the notch (white) and {110} facet (multi-colored). Key diffusion barriers in the most prominent pathway from {100} to {111} are shown in Fig. 7. Here, we see that the barrier for an atom to go from the {100} facet to the bottom of the “V” that makes up the notch (state 1 → 2) is lower than the reverse barrier. Moreover, the barrier for an atom to diffuse along the bottom of the notch (0.16 eV) is significantly lower than the barriers on the {100} facet (0.51-0.53 eV - cf., Table 1). This indicates that an atom reaching the notch will tend to stay there and exhibit rapid, 1D diffusion along the NW axis. Thus, the notch is a “super highway” for rapid transport of atoms from {100} to {111}. Since atom accumulation in the notch would promote aggregation, we investigated dimer-diffusion barriers in the notch. We found that dimers in the notch (state 17


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3) are particularly stable and they diffuse relatively rapidly, with a barrier of 0.21 eV (state 3 → 4). Interestingly, at the end of the notch, state 3 in Fig. 7 is relatively more stable than state 4. This energy preference aides in propagation of the {110} facets as the NW grow longer. When an atom in the notch reaches the edge of the {110} facet (state 4), it proceeds along the {110} facet (states 5, 6) with relatively low barriers until it reaches the edge of the {111} facet, at state 6. The final transition from state 6 to the {111} end is energetically preferred by the atom joining a dimer (state 7) to form a trimer. Finally, the trimer rotates to bring the newly joined atom from state 7 → 8 in Fig. 7. In addition to the pathway shown in Fig. 7, another low-energy pathway occurs when atoms access the {110} facet directly from the {100} facet, as the barriers for this transition are around 0.36 eV (cf., Table 1). However, there is limited access to this pathway compared to access from the notch, especially as the NW grows to achieve a high AR. We will return to this point below.

MFPT We now combine all of the elements discussed above: seed structure, nucleation, and diffusion to find MFPT for inter-facet transport. Using the framework of harmonic transition-state theory, we incorporated the energy barriers in Table 1 into atom diffusion rates with the form k = ν0 exp(−EB /kT ), with ν0 = 1013 s-1 and at a temperature of T = 160◦ C – the temperature used in ref. 1. As described in the SI, we constructed the transition matrix A and the Markov matrix M from the transition rates to obtain MFPT values. In our MFPT calculations, we assumed that each site on a facet has an equal probability of being occupied initially. Figure 8 shows ⟨t{100}→{111} ⟩ as a function of the NW length (or AR, since the NW diameter is constant here). We see that there is a rapid rise in the MFPT with increasing length for seed-length NW, which falls off when the NW length reaches ∼500 sites. We can

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Aspect Ratio 40 60

80

0.00014 < t{100}→{111} > (s)

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0.00010 < t{111}→{100} > = 0.4555s 0.00006

0.00002 0

2000

4000 6000 Length (sites)

8000

Figure 8: The MFPT for an atom to transit from {100} to {111} as a function of the NW length (lower axis) and AR (upper axis). understand the trends in Fig. 8 in terms of the three different exit channels from {100} to {111}: (1) direct transitions {100} → {111}; (2) {100} → {110} → {111} and; (3) {100} → notch → {110} → {111}. Figure 9 illustrates these three channels and shows the probability for an atom to exit to the {111} facet via the {110}-{111} interface, which encompasses exit channels (2) and (3), as a function of NW length. For an initial seed, a randomly deposited atom on {100} will encounter the {100}-{111} interface many more times than the notch-{110} region as it diffuses over the {100} facet. Thus, even though the energy barrier to access the {111} facet directly from {100} is high (cf., Table 1), a significant fraction of atoms accesses {111} via the {100}-{111} interface. At such low AR, an atom is also the most likely to encounter the {100}-{110} interface, where we expect the {100} → {110} transition to be rapid, due to the low barrier for this transition (cf., Table 1). Since Fig. 9 does not distinguish {100} → {110} → {111} transitions from {100}→ notch → {110} → {111} transitions, we ran a set of calculations in which we suppressed {100} → 19


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Figure 9: The probability p for an atom to access the {111} facet via the {110} facet as a function of NW length. The inset depicts three possible pathways for an atom to access {111} (purple) from {100} (blue) and p is the sum of probabilities for two of these pathways, in which the atom accesses {111} via {110} (red). {110} → {111} transitions. As we see in Fig. S3 in the SI, ⟨t{100}→{111} ⟩ is somewhat longer for short NW lengths when {100} → {110} → {111} transitions are suppressed. However, {100} → {110} → {111} transitions have a negligible influence on the MFPT for long NW. As the NW length increases, the {100}-notch interface increases and exits to {111} via the notch become increasingly likely. By the time the wire is in the nanorod length range (length of ∼ 1000 sites, AR around 17) 99% of exits to {111} occur via the {110}-{111} interface. As we see in Figs. 9 and S3, almost all of these exits occur through the notch. It may be possible to tune the {100}-notch transition rate by varying the notch size and wire diameter (via strain). This rate would decisively affect the NW AR. For a seed with all of the features on the {111} facets discussed above (i.e., trapping at the {110}-{111} interface, the NW tip, and parallel to the twin edge), we find ⟨t{111}→{100} ⟩ = 0.4555 s. This is significantly longer than the values of ⟨t{100}→{111} ⟩ in Fig. 8 and it is also much longer than the deposition time ( 10−4 s 1 ), indicating accumulation will occur on

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the NW ends. With regard to Eqs. 2 and 3, we have R{100}→{111} & Rdep ≫ R{111}→{100} . Thus, Eqs. 2 and 3 become dN{111} = Rdep,{111} + R{100}→{111} dt

,

dN{100} ≈ 0 ⇒ Rdep,{100} = R{100}→{111} dt

(4)

.

(5)

Since Rdep = Rdep,{111} + Rdep,{100} , we have dN{111} = Rdep dt

.

(6)

Thus, using a multi-scale approach we predict that (nearly) all of the atoms deposited onto the surfaces of a growing NW are channeled to the NW ends via surface diffusion.

Figure 10: The AR predicted using Eq. 8 as a function of the deposition rate. The upper and lower curves represent upper and lower bounds on the AR. Figure 8 shows that ⟨t{100}→{111} ⟩ is less than the experimentally estimated deposition time for NW lengths of up to ∼4000 sites, or AR of around 40. This can be regarded as

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a lower bound on the AR – the single-atom limit for which R{100}→{111} = k{100}→{111} = ⟨t{100}→{111} ⟩−1 . For dilute atom concentrations on Ag(100), we have

R{100}→{111} = k{100}→{111} N{100}

.

(7)

By selecting an arbitrarily low atom number(concentration) on {100} as dilute and taking Rdep,{100} ≈ Rdep for a long NW, we can obtain an upper bound on the AR using Eq. 5

Rdep = k{100}→{111} N{100}

.

(8)

Thus, NW will grow to a length L at which k{100}→{111} (L) satisfies Eq. 8. Using one atom as a lower bound and five atoms as an upper bound, we used Eq. 8 to obtain a lower and upper limit on the AR as a function of the deposition rate. This is shown in Fig. 10, where we see that for an experimental 1 deposition rate of 104 atoms/s, we predict AR between 40 and 180. These AR cover the range of those observed in experimental studies of NW growth in the absence of halides. 1,3,10–13 For NW lengths (AR) far outside the upper bound, we have R{100}→{111} < Rdep , atom accumulation begins to occur on the {100} facets, and the NW will grow thicker. Figure 10 indicates that long NW can be grown with low deposition rates. Here, it is important to point out that factors affecting the deposition and/or diffusion rates on Ag(100) can impact the AR. These factors include highly selective deposition on the ends (e.g., due to halide adsorption 2 ) or the strained NW diameter and its effects on diffusion as NW thicken. In our analysis, we implicitly assume that the deposition rate and MFPT are the sole factors determining the AR. However, other factors can determine the AR in an experimental system, including the depletion of metal salt and a diminishing coverage capping molecules on the surfaces as the NW grow.

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Conclusions In conclusion, we demonstrated, using a multi-scale approach, that the 1D growth of NW can originate from surface diffusion. A major factor that promotes 1D growth is the strained NW structure. NW seeds are similar to Marks Dh, with {111}-faceted “notches” at the twin edges along the NW axis and {110} facets in the {100}-{111}-notch region. Such a structure relieves stress and is consistent with the rounded pentagonal NW cross-sections observed in experiments. The strain distribution over the {111} facets induces heterogeneous atom aggregation and leads to atom trapping there. Trapping at aggregates contributes to relatively long times for atoms to transit from the {111} to the {100} facet and leads to atom accumulation at the NW ends. The “super highway” notches running along the NW axis act as conduits that funnel atoms to the {111} ends relatively rapidly. For slow and experimentally realizable deposition rates, NW can grow to significant lengths without atom accumulation on their sides and this promotes the growth of NW with high AR, in the experimental range. 1,3,10–13 It is important to emphasize that our calculations address NW that grow from seeds with diameters of 28 nm. While this is an experimentally relevant diameter, we recognize that the diffusion barriers involved in NW growth depend on the NW diameter because strain affects the values of these barriers and strain depends on the NW diameter. In addition, it seems likely that structural attributes such as the size of the notch and the {110} facet can impact NW growth and morphology. Finally, there is a distinct possibility that NW seeds themselves are non-equilibrium structures. Drawing on knowledge from studies of the minimum-energy structures of small Ag nanocrystals in vacuum, 29 it seems that Dh nanocrystals are the thermodynamically preferred shapes when the crystals are in the 100– 1000 atom size range. Above these sizes, single crystals with truncated octahedral shapes are preferred thermodynamically. However, the seeds of five-fold twinned Ag NW with the smallest diameters synthesized to date (∼20 nm 1,19 ) possess ∼105 atoms. While the solutionphase environment can lower interfacial free energies, 7,41 to possibly alter the size range (from 23


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vacuum values) over which Dh nanocrystals are the thermodynamically favored structure, it is possible that the experimental Dh seeds are kinetic structures. The possibility of kinetic seeds provides impetus to study synthesis and processing routes by which their morphology could be tuned. Future studies aimed at probing the interplay between deposition, diffusion, seed architecture, and NW AR would be beneficial in achieving controlled NW syntheses.

Methods The MD simulations are done using the LAMMPS code. 42 We use an embedded-atommethod (EAM) potential for Ag-Ag interactions. 43,44 We use MD to simulate annealing of the initial seed structure in Fig. 2(a) by slowly raising the temperature to 1100 K, then cooling to 300 K. The system is first equilibrated at 300 K in the NVT ensemble using the Nosé-Hoover thermostat 45,46 for 5 ns, and then heated to 1100 K with a heating rate of 10 K/ns. After the set temperature is reached, the system is kept at 1100 K for an additional 10 ns to relax the structure, and then slowly cooled to 300 K at a rate of 2.5 K/ns. Finally, the structure is stabilized at 300 K for 10 ns. To understand island nucleation on the nanowire surfaces, we ran MD simulations of Dh seeds with various atom coverages. In these simulations, we randomly place atoms on the surface of a 60 × 20 Dh and simulate their motion for 10 ns in the NVT ensemble at T = 433 K to reach a steady-state distribution. For each surface, we obtain the final distribution for two atom-density groups. We consider 25 − 50% atom coverage as the high-density group and 1 − 5% atom coverage as the low-density group (cf., Fig. 4). In each density group and facet type, we run 15 individual simulations. Each distribution map in Fig. 4 shows an averaged density over the 15 simulations. We use the climbing-image nudged elastic-band (CI-NEB) method 47 with the quick-min minimization algorithm described in ref. 48 to calculate values of diffusion-energy barriers EB . This is implemented in the LAMMPS code. 42 We consider adatom motion mediated by

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hopping of single atoms between nearest-neighbor surface sites, concerted moves involving several on-surface atoms, and exchange of adatoms with surface atoms. Diffusion-energy barriers were obtained from a 60×20 Dh with a notch size of 2 and a {110} facet size of three unit cells. All of the matrix operations and calculations of MFPT (described in the SI) were performed using Mathematica.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website.

Acknowledgement This work is funded by the Department of Energy, Office of Basic Energy Sciences, Materials Science Division, grant number DE-FG02-07ER46414. Z.C. acknowledges training provided by the Computational Materials Education and Training (CoMET) NSF Research Traineeship (DGE-1449785). The authors acknowledge helpful comments from Ben Wiley.

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Table of Contents Graphic

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Growth Mechanism of Five-Fold Twinned Ag Nanowires from

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