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How structure-directing agents control nanocrystal shape: PVP-mediated growth of Ag nanocubes Xin Qi, Tonnam Balankura, Ya Zhou, and Kristen A. Fichthorn Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b04204 • Publication Date (Web): 28 Oct 2015 Downloaded from http://pubs.acs.org on November 1, 2015

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Nano Letters

How structure-directing agents control nanocrystal shape: PVP-mediated growth of Ag nanocubes Xin Qi,† Tonnam Balankura,† Ya Zhou,† and Kristen A. Fichthorn∗,†,‡ Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States E-mail: [email protected]

Abstract The importance of structure-directing agents (SDAs) in the shape-selective synthesis of colloidal nanostructures has been well documented. However, the mechanisms by which SDAs actuate shape control are poorly understood. In the PVP-mediated growth of {100}-faceted Ag nanocrystals, this capability has been attributed to preferential binding of PVP to Ag(100). We use molecular dynamics simulations to probe the mechanisms by which Ag atoms add to Ag(100) and Ag(111) in ethylene glycol solution with PVP. We find that PVP induces kinetic Ag nanocrystal shapes by regulating the relative Ag fluxes to these facets. Stronger PVP binding to Ag(100) leads to a larger Ag flux to Ag(111) and cubic nanostructures through two mechanisms: enhanced Ag trapping by more extended PVP films on Ag(111) and a reduced free-energy barrier for Ag to cross lower-density films on Ag(111). These flux-regulating capabilities depend on PVP concentration and chain length, consistent with experiment. ∗

To whom correspondence should be addressed The Pennsylvania State University ‡ Department of Physics †

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Keywords: nanocrystal, Ag, polyvinylpyrrolidone, molecular dynamics, density-functional theory


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Noble-metal nanocrystals hold great promise for a wide range of applications, including active and selective catalysts and photocatalysts, 1–6 high-precision spectroscopy and imaging, 7–11 as well as efficient solar cells. 12–14 The beneficial properties of these nanocrystals are sensitive to their size and shape and there is a great impetus to achieve precise control in this arena. While there have been many reports of successful solution-phase syntheses of nanocrystals with various shapes, 15–24 the fundamental understanding of these syntheses is still in its elementary stages. Central to these syntheses are structure-directing agents (SDAs), also known as capping agents, or solution-phase additive molecules that help to guide the formation of certain crystal shapes. These molecules could have multiple functions in shape-selective syntheses. For example, SDAs prevent aggregation by shielding nanocrystal surfaces. By having an energetic preference to bind to a particular facet, an SDA can alter the relative surface energies of the facets in a nanocrystal, leading to thermodynamic Wulff shapes that are unique to the SDA/solvent/metal combination. Away from equilibrium, SDAs may promote kinetic nanocrystal shapes by affecting atom deposition rates to various facets, and also inter- and intra-facet atom diffusion. 25–27 It is a current challenge to unravel the extent to which these mechanisms are operative during nanocrystal growth. Atomic-scale simulations can be beneficial in efforts to understand and optimize the workings of SDAs. First-principles calculations based on dispersion-corrected density-functional theory (DFT) predict the binding energies of SDA molecules to various metal facets. 28–34 These studies are consistent with experimental trends 35–37 and indicate that the crystal facets to which SDA molecules bind the most strongly are those that are expressed in experimental studies of the corresponding systems. The successes of these studies indicate that the DFT binding energy might be an appropriate descriptor in the search for an optimal SDA for a given synthesis. While they reveal trends in binding, DFT calculations are typically done in vacuum and at zero temperature so they are not suitable to directly unravel the mechanisms by which SDAs promote selective shapes in solution. Atomic-scale molecular dynamics (MD)


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simulations can be useful in this regard, provided that these are based on force fields that have high fidelity to first principles and experiment. In this paper, we report on the results of MD simulations based on force fields with high fidelity to experiment that we use to resolve the mechanisms by which SDAs achieve facet-selective growth in solution-phase syntheses. Our work is partially inspired by an experimental study of Xia et al., 19 who probed the growth of cubic Ag nanocrystal seeds in ethylene glycol (EG) solvent using polyvinylpyrrolidone (PVP). They found that {100}-faceted seed cubes continue to grow as cubes, if the concentration of PVP is sufficiently high. Below a critical PVP concentration, the cubic seeds evolve into cuboctahedra and {111}-faceted octahedra. Both thermodynamics and kinetics likely play a role in the formation of these nanostructures. Thermodynamic shapes are achieved via kinetics – specifically, the kinetics of inter- and intra-facet atom diffusion on the crystal, as well as by the kinetics of atom deposition on the various crystal facets. Because these nanocrystals are relatively large (in the 40-200 nm size range 19 ), we consider kinetic origins of their shapes. When the size of a nanocrystal is sufficiently large, the diffusion of deposited atoms between facets becomes slow compared to their deposition rate. Regarding intra-facet diffusion, we note that the facets are relatively smooth in the experiments. 19 A similar scenario was observed experimentally by Liao et al. in their study of the solution-phase growth of Pt nanocrystals with liquid-phase transmission electron microscopy, where they observed layerby-layer-like growth of the Pt facets. 27 With negligible inter-facet diffusion and relatively smooth facets, we can look to the kinetics of atom deposition on smooth facets and the kinetic Wulff construction 38,39 to understand the origins of these nanocrystal shapes. To reveal Ag atom deposition mechanisms, we first carry out atomic-scale MD simulations to investigate how solution phase Ag atoms add onto PVP-covered Ag surfaces in the EGPVP solution environment. Since the reported nanostructures are cubes, cuboctahedra and octahedra, we study atom deposition onto Ag(100) and Ag(111). The {100} and {111} facets are represented by periodic Ag slabs, to represent the large crystal facets in the experimental


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study. 19 To obtain high fidelity to experiment, it is essential that MD simulations are based on a force field that correctly captures the interatomic interactions in the system. Our MD simulations are based on a combination of three force fields that describe the preferential binding of PVP to Ag(100) with high fidelity to first-principles DFT and experiment. We use the embedded-atom-method (EAM) potential for Ag-Ag interactions 40,41 and the CHARMM force field for organic-organic (EG and PVP) interactions. 42–46 The EAM potential has high credibility for metallic systems, 40 and the parameters developed by Williams et al. 41 can reproduce many physical properties of Ag, including the lattice constant, bulk cohesive energy, surface energies, and various defect energies. Similarly, the CHARMM force field has been tested against various physical properties of the organic molecules. 42–45 The interactions between Ag and organic species are described by the force field developed by Zhou et al. 46 In addition to pair-wise additive potentials to describe short-range Pauli repulsion, direct chemical bonding and dispersion interactions, our potential contains a many-body term to enhance the pair-wise description of the bond between O atoms in PVP and Ag surface atoms. Regarding the many-body term, experimental spectroscopic studies of Ag nanoparticles in EG solution with PVP 47–51 indicate that the O atom in PVP forms a chemical bond with Ag-nanoparticle atoms. DFT studies 28,29 show that the O-Ag bond is stronger on Ag(100) than on Ag(111). Although pairwise-additive potentials have been successfully applied in studies of peptide adsorption on metal surfaces in solution, 52,53 they cannot model this type of surface-selective interaction. As described elsewhere, 46,54 the many-body term in our potential describes the contribution of the O atoms in PVP to the electron density of the Ag surfaces via the EAM formalism, it endows the O atom with a preference to bind to Ag atoms with lower coordination in the surface, and it predicts stronger binding of PVP to Ag(100) than to Ag(111). The initial configuration for atom deposition is shown in Figure 1(a). After equilibration, PVP chains form an adsorbed layer that strongly binds to the Ag surfaces. To initiate deposition, we insert a free Ag atom into the EG solution phase above the adsorbed PVP


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(a)

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(b)

Figure 1: Snapshots from MD simulations. For clarity, EG solvent molecules are shown as being smaller than PVP in (a) and are removed in (b). (a) An example initial configuration for Ag deposition on Ag(100). A free Ag atom is inserted at the bulk-PVP interface at z (dashed blue line) with an initial velocity toward the surface. A successful deposition is counted when the atom reaches the surface at zf – although sometimes the atom turns around and progresses to the interface between the near-surface region and the bulk solution at z0 (dashed red line). (b) Snapshots of the Ag atom deposition mechanism: when an Ag atom (yellow) arrives at the bulk-PVP interface (A), it can be attracted into the PVP layer by interacting with O atoms (red) on PVP (B); the Ag atom is retained in the PVP layer but direct deposition is hindered by the PVP surface layer (C). The deposition finishes when a “hole” of suitable size opens so the Ag atom can reach the surface (D).


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layer with a thermal velocity at 433 K in the direction of the slab and record the trajectory until the atom reaches the Ag surface. From these simulations, we identify the characteristic Ag deposition pathways, as can be seen in a sequence of snapshots from an MD trajectory in Figure1(b). When a Ag atom arrives at the PVP-bulk solution interface, the strong interaction between Ag and O atoms in PVP 28 attracts and retains (i.e., traps) the Ag atom in the PVP layer (B). The Ag atom is held a few ˚ A above the surface by O atoms on PVP (C). At this point, the Ag atom is held essentially irreversibly in the PVP layer and we do not observe Ag escape from the PVP layer to the bulk solution. Ag atoms join the substrate when a hole opens up in the PVP layer (D). We observe two routes through which the Ag atom finds a deposition site. In the first mechanism, PVP molecules rearrange themselves on the surface to open a hole underneath an Ag atom. We also observe that Ag atoms are mobile in the PVP layer and they diffuse by being passed between the O atoms in PVP. In this way, Ag atoms also reach open sites via diffusion in the PVP layer. To characterize the observed Ag adsorption mechanism, the potential of mean force (PMF) is helpful because it reveals the free-energy landscape an Ag atom experiences as it diffuses from the bulk solution to the Ag surfaces. To obtain the PMF for Ag adsorption, we use umbrella sampling 55 with a harmonic bias potential, along with umbrella integration. 56 The reaction coordinate of the PMF is the orthogonal axis to the Ag surface. Details of these calculations are provided in the Supporting Information (SI). Figure 2(a) shows the PMF profile of an Ag atom as it approaches Ag(100) and Ag(111) surfaces covered with a monolayer (ML) of PVP decamers (PVP10mer). This profile qualitatively represents the PMF for all the PVP-covered Ag surfaces in our studies. Overall, the PMF profile indicates that the role of PVP is to attract Ag atoms to the surfaces. Near the surfaces, the Ag atom is bound in a deep minimum, indicating strong and irreversible adsorption of Ag atoms on the Ag surfaces. Further away from the surfaces, there is a secondary free-energy minimum in the PMF. To progress from this minimum to the Ag surface, Ag atoms must surmount a free-energy


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(b)

Figure 2: (a) PMF for a Ag atom to approach Ag(100) and Ag(111) slabs containing a ML of PVP10mer along the orthogonal axes to the facets and (b) the density profile of oxygen atoms in the corresponding PVP10mer film. The inset is a blow-up of the density profiles far from the surface. A detailed discussion of the density profile is given in the SI. barrier of 0.134 eV and 0.072 eV for Ag(100) and Ag(111), respectively. The smaller barrier for Ag(111) leads to a larger flux there, which then grows the Ag(100) facets. We also note that the barrier for an Ag atom to go to either surface is considerably smaller than the freeenergy difference between the secondary minimum and the bulk solution, which explains the observation in the deposition simulations that atoms trapped in this minimum always progress to the surface instead of desorbing to the bulk solution. Interestingly, the location where the PMF attains its bulk value of zero is further from the surface for Ag(111) than for Ag(100), reflecting the more extended PVP layer on Ag(111). The density profile of oxygen atoms on PVP [Figure 2(b)] indicates the origin of the PMF shape. As we discuss below and in the SI, each peak in Figure 2(b) is associated with a mechanism that affects atom deposition. Given a PMF profile, we can obtain the flux of Ag atoms to the Ag surfaces by solving the Smoluchowski equation to obtain the mean-first passage time (MFPT) tm . 57 The flux of Ag atoms to a surface is inversely proportional to the MFPT. For an atom beginning at a height of z from the surface [see Figure 1(a)], that is absorbed onto the surface at a height


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of zf , tm is given by ∫

zf

∫

z′

tm (z, zf ) = z

exp[−W (z ′′ )/kT ]dz ′′

z0

exp[W (z ′ )/kT ] ′ dz D(z ′ )

,

(1)

where W (z) is the PMF and D(z) is the z-dependent diffusion coefficient of the Ag atom. The reflecting boundary at z0 represents the upper boundary of the bulk-solution region and z separates the bulk-solution and near-surface regions. Equation 1 can be solved for tm if we know D(z). Alternatively, we can find the MFPT from MD simulations and we choose this approach here. We can decompose tm into two times: a diffusion time tD and a reaction time tR . tR is the average time for the Ag atom to diffuse through the PVP layer to the Ag surface at zf before it reaches z0 and tD is the average time to diffuse from z to z0 . As we elaborate in the SI, tm can be written in terms of tD and tR as

tm =

(1 − p) 2tD + tR p

,

(2)

where p is the reaction probability, or the probability that an atom beginning at z will proceed to zf before z0 . The distinction between tD and tR (i.e., the location of z) is somewhat arbitrary, although z should lie above the PVP surface layer. When this is the case, tD is a bulk diffusion time and is given by tD =

(z0 − z)2 2D

,

(3)

where D is the diffusivity of a Ag atom in the bulk EG solution. To obtain tD , we note that the value of z0 relative to z can be chosen to represent the bulk concentration of Ag for given lateral cell dimensions, so that our simulation cell contains one Ag atom on the average, while matching the experimental bulk Ag concentration in solution. For this, we need an estimate of the concentration of Ag atoms in bulk solution. The source


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of Ag atoms in the experimental system is AgNO3 19 and the Ag+ atoms from the salt must be reduced to obtain neutral Ag. We assume that Ag reduction occurs in the bulk solution and that a concentration of neutral Ag resides there. At any given time, this concentration is likely a fraction of the salt concentration in the bulk solution. We obtain an upper bound on the neutral Ag atom concentration in solution, by assuming it is equal to the bulk salt ˚3 per Ag atom. Based on concentration in experiment, 19 which gives a volume of 30,174 A the lateral simulation cell dimensions we use for most of the simulation cells, this yields a height (in ˚ A) of z0 = z + 19. We also need to have the bulk diffusivity D to calculate the diffusion time using equation (3). We calculate D = 3.3 ± 0.2 × 10−5 cm2 /s for an Ag atom in EG solvent from MD simulation. Details on the calculation of D are given in the SI. From our estimates of D and z0 − z, we arrive at a diffusion time of tD = 0.55 ns. From eq 2, the effective diffusion time t′D is given by t′D =

(1 − p) 2tD p

.

(4)

To obtain tR from MD simulations, we insert an Ag atom at z with a thermal velocity at 433 K towards the slab and follow its trajectory until the atom either diffuses to the Ag surface at zf and binds there or it diffuses to the bulk-solution interface at z0 [cf., Figure 1(a)]. If the atom ends at the surface, we record the time for the trajectory and increment a counter NR for reactive events. For each of the surfaces studied, we ran a total of N trajectories such that NR = 100. We obtain the reaction probability as p = NR /N and the reaction time tR as the average time to achieve a reactive trajectory. To study how PVP concentration and chain length affect Ag nanocrystal shape, we study atactic PVP oligomers of three different lengths, namely pentamers (PVP5mer), decamers (PVP10mer), and icosamers (PVP20mer). PVP is a semi-flexible polymer chain in EG solvent and we calculate 58 an effective Kuhn length of ∼5 repeat units (see the SI). If the chain length is much greater than the Kuhn length, PVP oligomers can effectively represent the statistical behavior of longer chains and this is the case for the PVP20mer. For each of 10 ACS Paragon Plus Environment

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these lengths, we probe PVP films with full ML coverage, as well as a fraction of the full ML. We also consider the case of zero PVP coverage (i.e., only EG solvent). In this way, we probe the effect of PVP chain length and PVP surface coverage, as was done experimentally. 19 Due to the relatively strong binding affinity of PVP to both facets, we expect 59 (and observe) that long-chain PVP primarily adopts a “train” conformation, in which it tends to lie flat against the surface. PVP does not form extended multi-layer films in these systems and when we have PVP in an amount that is in excess of that required to saturate the surface in a full ML, the excess PVP is free in solution. Solution-phase Ag atoms can associate with solution-phase PVP and this significantly hinders their bulk diffusion beyond the MD time scale. Thus, we do not consider systems with excess PVP here. We presume that excess PVP resides in the bulk solution in the analogous experimental system, 19 that solution-phase Ag atoms associate with it, and that free Ag atoms in solution are in equilibrium with the atoms held by PVP. We ran MD simulations on all of the above systems. For each system, we obtain the reaction time tR , the reaction probability p, and the effective bulk diffusion time t′D using eq 4. We obtain tm using eq 2. The results are summarized in Figure 3, which shows tm , the contribution of tR and t′D to tm [Figure 3(a)], and the values of p for the different systems [Figure 3(b)]. Both diffusion and reaction contribute significantly to tm in Figure 3(a). Interestingly, tm is the largest for the surfaces with no PVP at all (i.e., for the surfaces with just EG solvent), indicating the smallest Ag flux. From Figure 3(b), we see that p is the smallest for these surfaces, which is the origin of the large t′D . As we will discuss elsewhere, EG solvent forms a tight network near the Ag surfaces, which leads to a free-energy barrier that inhibits Ag atoms from adding to them. The monomers of PVP have significantly stronger binding to the Ag surfaces than the EG solvent 46 and they easily displace EG solvent molecules from the surface. When PVP adsorbs to the Ag surfaces, it breaks up the EG solvent network and allows for higher Ag fluxes (smaller tm ). Because of the relatively strong binding of PVP,


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(a)

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(b)

Figure 3: (a) The MFPT tm and the contributions of tR and t′D , as well as (b) the reaction probability p for each system studied. EG solvent plays a minor role in dictating Ag fluxes to the Ag surfaces in PVP solution. For PVP-covered surfaces, we note that tm is greater for Ag(100) than for Ag(111) and the difference is especially pronounced for the longer chains. The higher fluxes (smaller tm ) to Ag(111) in Figure 3(a) can be ascribed to the less attractive PVP-Ag(111) interaction in two different ways that affect both tR and p. First, in the O-atom density profile in Figure 2(b) and in those in the SI, the two largest peaks near the surface arise from 2-pyrrolidone side rings that reside close to the surface and, due to variations of the side-ring distribution in atactic PVP, away from the surface. These peaks (and similar peaks in the density profiles in the SI) are associated with the hole-opening mechanism by which Ag accesses the surfaces. We see a larger O-atom density near the surface on Ag(100), where PVP binds more strongly. The higher density on Ag(100) leads to a higher hole-opening barrier, as can be seen in the PMFs in Figure 2(a). This effect is more pronounced at higher coverages, where vacancies are less likely to occur within the PVP layers, as well as for longer chains, which have longer segment trains and, hence, stronger binding at the surface. Overall, the lower segment density on the {111} facets leads to smaller tR for these surfaces. A second way that the stronger attraction of PVP to Ag(100) influences the Ag surface flux is through its effect on Ag-atom trapping, the first step in the process of Ag-atom


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addition to the Ag surfaces [cf., Figure 1(b)]. Differences in the abilities of different PVP chains to trap solution-phase Ag atoms can be seen as differences in the reaction probability p between various systems. The weaker PVP-Ag(111) attraction leads to more loosely packed PVP films on Ag(111), such that these films can extend further into solution than those on Ag(100) [cf., Figure 2(b) and those in the SI]. These “fluffier” PVP films on Ag(111) tend to more effectively trap solution-phase Ag atoms (higher p) in a way that depends on both chain length and coverage. In Figure 3(b), we see that p111 and p100 are similar for PVP5mer films regardless of the coverage, as well as for PVP10mer and PVP20mer films at low coverage, but p111 > p100 for surfaces fully covered with PVP10mers and PVP20mers. The rod-like PVP5mer, whose length is equal to the Kuhn length, does not have conformational freedom at any surface coverage, so PVP5mer films are flat and there is essentially no difference in p on the two surfaces. When we progress to PVP10mer and PVP20mer films, both of which are long enough to be semi-flexible, we begin to see larger differences between the two surfaces. To understand differences between Ag atom trapping on Ag(111) and Ag(100) for PVP10mer and PVP20mer films, we turn to the O-atom density profiles on these surfaces. In particular, we scrutinize the third peak, between 9 and 20 ˚ A in the density profiles in Figure 2(b) (and similar peaks in the density profiles in the SI). This peak arises from conformational irregularities, as well as small tail segments in the PVP film, and represents PVP segments that trap incoming Ag atoms. The density on Ag(111) in this third peak is higher than that on Ag(100) so that Ag atoms more readily encounter PVP segments as they approach Ag(111). We further discuss this issue in the SI. We can quantify the effect of PVP film “fluffiness” on Ag trapping by using eq 1 to describe the MFPT for an Ag atom to reach an irreversibly trapped state in the PVP film at a height of zf′ above the Ag surface, where zf < zf′ < z < z0 in Figure 1(a). This irreversibly trapped state is the “point of no return” as the Ag approaches the surface – an Ag atom that has reached zf′ will not return to the bulk solution. Assuming a constant, bulk diffusivity


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D, eq 1 yields a mean trapping time of

τ =−

1 (z − zf′ )(z − 2z0 + zf′ ) , 2D

(5)

where z and z0 are defined above. For two surfaces with the same z and z0 , the trapping time τ is shorter for larger zf′ (more extended films). Although the values of zf′ are not precisely known, we can estimate them using eq 4. When the difference between z and zf′ is small, as it is in the simulations, τ = t′D and we can calculate the value of zf′ using this equality. For A for Ag(100) and zf′ = 16.0 ˚ A for Ag(111), which a ML of PVP10mer, this yields zf′ = 12.2 ˚ is consistent with the density profiles and PMFs shown in Figure 2. To extrapolate our results to experiments (PVP of Mw ≥ 10 000), 19 we note that in polymer adsorption, the conformations consist of train, loop and tail segments. Coarsegrained Monte Carlo simulation studies of polymer adsorption suggest that, for semi-flexible polymers adsorbed on strongly attractive surfaces in good solvent (such as the Ag-PVP-EG system), the fraction of loops is not significant and the fraction of segments in trains is approximately three times the fraction in tails. 60 The PVP20mer, which is the closest to a long chain, exhibits trains and tails in our simulations. As we elaborate in the SI, tails can trap incoming Ag atoms and constitute a slow growth channel. However, considering that the time scale of this channel is much longer than other deposition events, that the fraction of tails is insignificant compared to trains, and that the trapped Ag are in equilibrium with bulk Ag, we do not consider this to be a significant source of the Ag surface flux. Thus, in experimental systems, fluxes towards various facets should be predominantly determined by the train conformation regions predicted here. We can use the MFPT from our simulations to predict nanocrystal shapes through the framework of the kinetic Wulff construction, which predicts nanocrystal shapes given linear facet growth rates G. 38,39 With irreversible Ag adsorption [cf., Figure 2(a)], smooth, layer-by-layer-like growth and insignificant inter-facet transport (all of which are likely for


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the analogous experimental system, 19 as we discussed above), the linear facet growth rate G is proportional to the deposition flux. G is inversely proportional to tm and we have G111 /G100 = tm,100 /tm,111 . If we begin with an arbitrary initial seed with a well-defined crystal shape, the structure goes through a dynamic shape evolution and reaches a final shape at steady state. The geometry of the steady-state structure is defined by

¯i x¯i = G

i = 1, . . . , N − 1 ,

(6)

¯ i is the relative where x¯i is the dimensionless distance of facet i from the shape center, G growth rate of facet i, and N is the number of facets in the shape. Equation 6 is normalized to a reference facet. As we elaborate in the SI, we can begin with a cubic, {100}-faceted nanocrystal, as was done in the analogous experiments, 19 and this shape will evolve to reach a steady-state shape that depends on the relative fluxes (inverse MFPTs) to the {100} and {111} facets.

Figure 4: The kinetic Wulff construction for Ag nanocrystal shapes ranging from octahedra to cubes. Predicted shapes are shown for example growth-rate ratios. The {100} facets are green and the {111} facets are orange. Figure 4 indicates expected kinetic Wulff shapes as a function of the relative growth 15 ACS Paragon Plus Environment

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rate G111 /G100 in the growth of an fcc crystal, where cubes, cuboctahedra, or octahedra can be formed. As we see in Figure 4, octahedra are expected when

G111 G100

≤ 0.58. As G111

increases relative to G100 , we observe a shape progression to cuboctahedra, truncated cubes, √ G111 and eventually to cubes for G ≥ 3. Our calculated shape progression is the same as that 100 given by Wang. 61

Figure 5: The predicted shapes of nanocrystals with different PVP coverages and chain lengths.

Using the kinetic Wulff plot (Figure 4), we now predict Ag nanocrystal shapes corresponding to the relative fluxes from the MFPTs in Figure 3(a). The results are shown in Figure 5, where we see that both PVP10mer and PVP20mer films at full coverage produce {100}-faceted cubes with tiny {111}-faceted corners. As the PVP concentration decreases, these cubic shapes progress to cuboctahedra and truncated octahedra. As we discussed above, the PVP5mer does not exhibit the same trend as the PVP10mers and PVP20mers due to its rod-like conformation. The apparent convergence of the results for PVP10mer and PVP20mer films supports the assumption that PVP oligomers of multiple Kuhn lengths can be used in simulations to represent long-chain behavior in experiment. Although our results suggest truncated octahedra form in EG solvent only, we expect unstructured aggregates in experiment, because small-molecule EG solvent is not a capping agent and does not adsorb 16 ACS Paragon Plus Environment

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strongly enough to these surfaces to prevent nanocrystal aggregation. Our simulation results agree well with the shape progression demonstrated by Xia et al. 19 We also note that with emerging experimental capabilities in liquid-phase transmission electron microscopy, it has become possible to measure nanocrystal facet growth rates in situ 27 and confirm growth mechanisms experimentally. In summary, the capabilities of SDAs to affect the shape-selective formation of colloidal nanocrystals have been demonstrated in many experimental studies 15,16 and, in the case of PVP-mediated growth of {100}-faceted Ag nano-objects, this capability has been attributed to the stronger binding of PVP to Ag(100) facets. 19,23,24,35 Here, we show that the stronger binding of PVP to Ag(100) leads to a larger flux of Ag atoms to Ag(111) facets through two mechanisms: enhanced trapping of incoming Ag atoms by the more extended PVP films on Ag(111) and a reduced free-energy barrier for Ag atoms to access the surface on the lower-density films on Ag(111). The larger flux of Ag to Ag(111) leads to {100}-faceted cubes. We note that the quantitative trends established here in our MD simulations mirror qualitative trends that can be observed in the PVP segment-density profiles near the Ag(100) and Ag(111) surfaces. Similar trends were observed in MD simulations of PVP segment densities on Au surfaces, which indicate that Au nanoparticles are likely {111}-faceted in the presence of PVP 32 – in agreement with experiment. 37 Understanding the mechanisms of shape selectivity will allow for better design and control of nanocrystal syntheses, which will benefit many existing and emerging applications in catalysis, sensing, imaging, and energy.

Supporting Information Available Details of MD simulations, PMF calculations, MFPT derivations, calculation of the diffusivity of Ag atoms in bulk EG solution, calculation of the Kuhn length of PVP in EG, discussion of PVP adsorption conformations on Ag surfaces, and PVP density profiles on Ag surfaces are provided in the Supporting Information. This material is available free of charge via the


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Internet at http://pubs.acs.org.

Notes The authors declare no competing financial interest.

Acknowledgement This work is funded by the Department of Energy, Office of Basic Energy Sciences, Materials Science Division, grant number DE-FG02-07ER46414. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) supported by NSF/OCI-1053575.

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