Theoretical Perspectives on the Influence of Solution-Phase Additives

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Theoretical Perspectives on the Influence of SolutionPhase Additives in Shape-Controlled Nanocrystal Synthesis Xin Qi, Tonnam Balankura, and Kristen A. Fichthorn J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00562 • Publication Date (Web): 17 Apr 2018 Downloaded from http://pubs.acs.org on April 24, 2018

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The Journal of Physical Chemistry

Theoretical Perspectives on the Influence of Solution-Phase Additives in Shape-Controlled Nanocrystal Synthesis Xin Qi,† Tonnam Balankura,† 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]

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Abstract Shape-selective solution-phase nanocrystal growth is facilitated by capping agents, or structure-directing agents (SDAs), which guide shape evolution. It is often stated that these chemical additives impart shape selectivity by promoting nanocrystals with a majority of facets to which they bind most strongly. However, little is known on the mechanisms through which they impart shape selectivity. In this Feature Article, we highlight our recent studies aimed at understanding the thermodynamic and kinetic influences of SDAs using theory and computational tools. We review our studies of the PVP-mediated synthesis of {100}-faceted Ag nanocubes in ethylene glycol solution, which has been studied experimentally. Our studies of the interfacial free energy of Ag-PVP solution interfaces show that while solution-phase PVP does bind more strongly to Ag(100) than to Ag(111), this selectivity is not sufficient to thermodynamically change the Wulff shape of a PVP-covered Ag nanocrystal in solution from that of the bare metal in vacuum. These studies indicate that a strong facet binding selectivity is needed for an SDA to thermodynamically alter the solution-phase crystal shape from that of the bare metal. Interestingly, the binding selectivity of PVP for Ag(100) is sufficient to regulate the atom deposition fluxes to Ag(100) and Ag(111), so that cubic Ag(100) nanocrystals form kinetically. Altogether, our studies indicate that kinetic control of metal nanocrystal shapes is likely more prevalent than thermodynamic control. We outline some current challenges in understanding shape-selective solution-phase nanocrystal syntheses.

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Introduction Nanocrystals with well-defined shapes have the capability to advance many modern technologies, including selective catalysis, 1–6 electronic devices, 7 energy storage 8 and medicine. 9,10 Noble metals have attracted attention because they uniquely enhance light scattering and absorption within the visible light range via localized surface-plasmon resonance (LSPR). 11–13 This fascinating property allows their application in areas such as effective photovoltaics 14–18 and high-precision imaging and sensing. 11,19,20 Because these applications are highly sensitive to the sizes, shapes and symmetry of the nanocrystals, numerous shape-controlled synthesis techniques have been developed in the last decade, among which solution-phase syntheses appear to be especially successful for a variety of metals, such as Ag, 21–23 Au, 24,25 Cu, 26–28 Pt, 29–31 and Pd. 32,33 Despite many reports of successful syntheses and some quantitative efforts to understand them, it is still challenging to predict the outcome of a chemical synthesis of metal nanocrystals. Major challenges include understanding the complex chemistry involved when a metal salt is reduced, how metal atoms and ions nucleate seeds, and how seeds grow into welldefined shapes, possibly with the help of solution-phase additives called capping agents, or structure-directing agents (SDAs). Theory and simulation can be beneficial in efforts to understand these syntheses. However, a complete theoretical description of an entire synthesis is hampered by numerous complexities. First, the chemical complexities in these systems practically mandate that any theoretical description should be based on quantum mechanics. Though quantum mechanical studies, particularly those based on density-functional theory (DFT), have been useful in understanding how solution-phase species interact with metal crystals as they grow, 34–42 such calculations are typically done in a zero-temperature and vacuum environment, which is far removed from the solution-phase environment of experimental syntheses. DFT-based ab initio molecular dynamics (AIMD) studies can and have been used to study small, capped nanoparticles at finite temperatures. 43 However, AIMD simulations are still limited to ∼1000 3



atoms and ps time scales. These difficulties make it unlikely that quantum calculations alone will lead to significant advances in understanding syntheses. Classical molecular-dynamics (MD) simulations can extend the range of first-principles DFT calculations, to further advance our understanding of solution-phase syntheses. The accuracy of these simulations is governed by the underlying force field, which is typically fit to results from quantum calculations and/or experiment. Typical length and time scales that can be reached in MD simulations are currently in the nanoscale and it is not uncommon to use such simulations to probe microscale lengths and times. With MD simulations, we can rigorously compute free energies, which is difficult in DFT without resorting to approximations for the liquid phase. Though MD simulations may still not reach experimental scales, many different theoretical techniques have been developed to allow for enhanced MD sampling of thermodynamic and kinetic quantities, such as interfacial free energies and reaction rates. These quantities can be incorporated into continuum theories and coarse-grained simulations to extend the range of lengths and times that can be probed. Thus, MD simulations can be useful in evaluating various aspects of solution-phase metal crystal growth. In this Feature Article, we review our MD studies 44–48 aimed at understanding the origins of {100}-faceted Ag nanostructures that are observed experimentally 21,22,49–51 in the solutionphase polyol process. In particular, we investigated the role of polyvinylpyrrolidone (PVP), a commonly used SDA in solution-phase syntheses, in directing the growth of monocrystalline Ag seeds into {100}-faceted Ag cubes. Our studies are based on the metal-organic many-body (MOMB) force field we developed to bridge DFT and MD. 52–54 This force field encompasses both metal-organic direct-bonding interactions through a Morse potential and a many-body term based on the embedded-atom method potential, as well as pair-wise van der Waals interactions. By including a many-body term for direct chemical bonding of certain organic atoms to metal surfaces, it captures the facet-selective binding of PVP on Ag(100), as well as facet-selective binding of alkylamines on Cu(100) 55 that are seen in DFT calculations. We examine the commonly held idea that SDAs promote shapes with a predominance of

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a particular crystal facet by binding the most strongly to that facet. We note that our first-principles studies with DFT indeed confirm this idea, as these studies indicate that PVP does bind more strongly to Ag(100) than to Ag(111). 36,37 However, DFT studies do not indicate the thermodynamic and kinetic ramifications that this selective binding may have for crystal shapes. We note that other MD studies have proven useful in resolving the anisotropic growth of Au nanorods, 56–59 peptide-mediated growth of Pt nanocubes, 60 etc. 61 A hallmark of our work is that we rigorously evaluate free energies associated with the kinetics and thermodynamics of nanocrystal growth at liquid-solid interfaces. Below, we review our studies.

Thermodynamic Influence of Capping Molecules Under thermodynamic control, the solid-liquid interfacial free energy γsl is a key parameter in determining the equilibrium nanocrystal shape. When crystal growth reaches equilibrium, the total interfacial free energy is minimized and the Wulff construction yields the equilibrium shape. 62,63 For bare, face-centered cubic (fcc) metals, we have γ{111} < γ{100} < γ{110} and the Wulff shapes primarily express {111} facets. 64 Various adsorbed species, including solvent and capping molecules, can alter γsl of crystal facets in the solution phase and potentially influence the equilibrium crystal shape. Consequently, a prevailing hypothesis is that preferential binding of a capping molecule to a particular crystal facet can lead to a nanocrystal shape that predominantly expresses that facet. In the Wulff construction, facets in the equilibrium crystal shape follow the relation hi =λ , γsl,i

(1)

where hi is the orthogonal distance from the crystal center to facet i, γsl,i is the solidliquid interfacial free energy of facet i, and λ is a constant for the structure. Thus, given γsl,i for each relevant facet i, we can predict the equilibrium shape of a crystal in liquid 5



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solution. Experimentally, γsl can be estimated from the contact angle of a liquid drop on a solid surface 65,66 or back-calculated from nucleation or crystal growth rates. 67–70 Interfacial energies obtained via such means tend not to be facet-specific, 67–70 so they would be of little use in the Wulff construction. Theoretical and computational tools can be an excellent way to evaluate γ and its facet dependence. 71–75 First, we should clarify that although the interfacial tension σ is occasionally used to represent the interfacial free energy γ, this mathematical equivalence only holds true for fluid systems and not for fluid-solid interfaces. The interfacial free energy is related to the interfacial tension via σ = γ + A(dγ/dA) ,

(2)

where A is the interfacial area. 76,77 This equation indicates that for stress-sensitive surfaces, such as the surfaces of solids, γ varies with A such that dγ/dA is non-zero. It is evident that γsl cannot be approximated using σsl for such surfaces and that it is the interfacial free energy that is relevant in determining crystal shape. We recently introduced an MD-based multi-scheme thermodynamic integration (TI) method 44 to estimate γsl . Our method is inspired by prior “cleaving-wall” methods 72–74 and it is targeted to resolve previous difficulties in applying cleaving-wall methods to all-atom simulations in multi-component systems. 44 This is a six-step method in which we reversibly transform a solid-liquid interfacial system to individual solid and liquid bulk phases, as is outlined in Fig. 1. In each step i, the free-energy change ∆Fi is calculated from TI, in which ∆Fi is given by Z ∆Fi =

∂U (λ) ∂λ

dλ .

(3)

Here λ is a scaling parameter that varies between 0 and 1. λ is used to manipulate the interaction(s) that accomplish the needed transition. U (λ) is the potential energy of the system at an intermediate state, whose energy is influenced by λ, and the angled brackets denote an ensemble average. We briefly summarize our method below and note that a

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detailed description can be found in ref. 44. The multi-scheme TI method begins with a solid-liquid interface, in which the solid is represented using a slab and the rest of the simulation cell is filled with liquid/solution molecules at the bulk liquid density. In Step 1, we gradually switch on two superimposed, short-range repulsive walls at each of the two solid-liquid interfaces that occur in a simulation cell with periodic boundary conditions (PBC). One of the superimposed walls interacts only with the solid and the other interacts only with the liquid. The positions of these walls are fixed throughout all steps of the simulation. The solid-liquid interaction is gradually turned off in Step 2. Since the solid and liquid are confined by repulsive walls, they behave as non-interacting solid and liquid slabs shown in Fig. 1(2b) at the end of Step 2. In Steps 3 and 4, we recover a bulk liquid by changing the PBC of the liquid cell in the direction normal to the confining walls, then we remove the walls. In Steps 5 and 6 we recover a bulk solid. We note that we employ different schemes to recover the bulk liquid and solid and that these differences are discussed in ref. 44. The interfacial free energy per unit area is then given by γsl = −

∆F1 + ∆F2 + ∆F3 + ∆F4 + ∆F5 + ∆F6 2Lx Ly

,

(4)

where, Lx and Ly are the x- and y-dimensions of the simulation box, the product of which gives the interfacial area, and the factor of 2 in the denominator indicates there are two interfaces in a simulation cell with PBC. We applied the multi-scheme TI method to understand the role of PVP in the solutionphase growth of {100}-faceted Ag nanocrystals 45 by calculating γsl for Ag(100) and Ag(111) – the most relevant facets in this synthesis. We calculated the surface free energy γ of the bare metal facets (i.e., Ag in contact with vacuum), as well as γsl of the surfaces in pure solvent (ethylene glycol – EG), and in PVP-EG solution with various PVP concentrations, where we use PVP icosamer (PVP20) to represent long-chain PVP in experiment. The results for vacuum-Ag, EG-Ag and PVP-EG-Ag interfaces are summarized in Fig. 2. Our results for Ag surfaces in contact with vacuum confirm the expectation that fcc metals 7



(1)

(2)

(2b)

(3)

(4)

(5)

(6) 6)

Figure 1: Schematic illustration of the multi-scheme TI method. Details on Steps 1 through 6 are given in the text and in ref. 44. Reprinted with permission from ref 44, with the permission of AIP Publishing. Copyright 2016 American Institute of Physics.

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Figure 2: The interfacial/surface free energy of the studied interfaces. The predicted thermodynamic equilibrium shape for each system is nearly identical and is shown as the structure in the figure. Reproduced from Ref. 45 with permission from the Royal Society of Chemistry. Copyright 2017 Royal Society of Chemistry.

have a lower γ for {111} facets than {100} facets. Ag facets in contact with EG solvent have significantly lower values of γsl . Although the adsorption PVP can further lower γsl from the pure-solvent values, the ratio of interfacial free energies between Ag(100) and Ag(111) remains relatively unaffected. Using the Wulff construction (Eq. 1), we predict that nanocrystals in EG solvent or PVP-EG solution will have essentially the same shape as crystals in vacuum – a truncated octahedron or cuboctahedron. We find that the stronger binding of PVP to Ag(100) facets only lowers γsl,Ag(100) by 5.3 mJ/m2 more than γsl,Ag(111) , whereas γAg(100) is 72.4±1.1 mJ/m2 higher than γAg(111) . Therefore, it is unlikely that PVP induces thermodynamic Ag nanocubes. Although it is evident that PVP is incapable of altering the equilibrium shape of an Ag crystal in solution, this result raises the question of how facet-selective the binding of a capping molecule needs to be to effectively induce an equilibrium crystal-shape change. To probe this question, we performed an analysis using generic capping molecules, represented as Lennard-Jones (LJ) particles with tunable affinities for {100} and {111} facets. 45 By varying the LJ interaction-strength parameters for these model capping agents and calculating γsl for

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Figure 3: A crystal-shape map that relates the thermodynamic equilibrium shape of a nanocrystal to the SDA binding energies on Ag(100) and Ag(111). Shapes on the transition curve indicates the predicted shape in each color region. Reproduced from Ref. 45 with permission from the Royal Society of Chemistry. Copyright 2017 Royal Society of Chemistry.

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Ag(100) and Ag(111) facets, we can predict equilibrium crystal shapes for a range of bindingaffinity combinations. From the results of these calculations, we constructed a crystal-shape map (Fig. 3) to link equilibrium, solution-phase crystal shapes to zero-temperature SDA binding energies. We see in Fig. 3 that a relatively large difference in the binding energies of SDAs to {100} and {111} surfaces is needed to promote {111}-faceted octahedra (upper left corner of Fig. 3) or {100}-faceted cubes (lower right corner of Fig. 3). Moderate or weak differences in facet binding energies lead to truncated octahedra, cuboctahedra, or truncated cubes, which fall on the diagonal that runs between the cubic and octahedral regions of Fig. 3. We note that Fig. 3 was constructed for SDA binding to Ag, so the surface energies of facets in a solution environment reflect the bare-surface energies of Ag (cf., Fig. 1). Since Ag(111) has a lower energy than Ag(100), the map in Fig. 3 is somewhat skewed toward {111}-faceted crystals, as discussed in ref. 45. The crystal-shape map in Fig. 3 can be helpful in predicting equilibrium shapes of solution-phase crystals when the solvent-surface and solvent-SDA interactions are similar to those in our studies and when SDA binding energies are known from DFT calculations. To test the veracity of our predictions, we examine two chemical species whose DFT binding energies and the resulting crystal morphologies in solution-phase (EG) syntheses are known. One is 1-methyl-2-pyrrolidone (1M2P), 52 an analog of PVP monomer. The DFT binding energies for this monomer, along with its predicted surface energies / morphology, are indicated in Fig. 3. These predictions fall in line with experimental observations that 1-ethyl-2-pyrrolidone is not an effective structure-directing agent in the shape-selective synthesis of Ag nanostructures 78 and is also in agreement with our full calculation with PVP20, indicated in Fig. 2. The second species tested is citric acid (CA), 34 the acid form of sodium citrate which helps to make Ag octahedra in experiments with EG solvent. 79 As we can see in Fig. 3, the strong binding of CA to Ag(111) relative to Ag(100) places it in the range where an octahedron with slight corner truncation is expected. This consistency with experiment suggests that CA induces the formation of octahedra via thermodynamic control.

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Kinetic Shape Control via SDA Regulation of Deposition Flux The thermodynamic equilibrium of a nanocrystal is achieved via intra- and inter-facet atom migration. If atom diffusion is slow on the time scale of deposition, we expect to observe steady-state, kinetic nanocrystal shapes. These kinetic shapes can be predicted by the kinetic Wulff construction, in which the relative linear growth rates Gi of various facets i determine the shape. 80 The linear facet growth rate of facet i can occur via several different processes: the nucleation of facet i on a facet of a different type, diffusion onto facet i from another facet, or deposition onto facet i from solution. In our studies to date, 46–48 we considered facet growth due to deposition. Such a scenario would occur if inter-facet diffusion is slow, as would be the case for the relatively large nanocrystals observed in relevant growth experiments, 22 and if the nucleation/growth of new facets on top of existing ones is slow or does not occur. If Gi is determined by the solution-phase atom deposition flux to crystal facet i, the ¯ i normalized to a reference facet is equivalent to the normalized deposition flux relative G F¯i . Then, the normalized distance x¯i along a vector, which originates at the center of the crystal and is normal to facet i, is given by

x¯i = G¯i = F¯i

,

i = 1 ,...,N − 1 ,

(5)

where N is the number of facets in the crystal. For an fcc crystal with two possible facets, √ G111 ≤ 1/ 3, and a cube is predicted {100} and {111}, an octahedron is predicted when G 100 √ 111 when G ≥ 3. 46,47 Below, we discuss our investigations with MD simulations on how G100 the adsorption of capping molecules can influence kinetic nanocrystal shapes by regulating relative deposition fluxes. It is helpful to qualitatively characterize the deposition of an Ag atom onto PVP-covered Ag(100) and Ag(111) facets in EG solution using direct MD simulations. 46 We initiated

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(a)

(b)

Figure 4: Snapshots from MD simulations of Ag deposition on Ag(100) depicting the key steps in the deposition mechanism. For clarity, EG solvent molecules (pink) are downsized in (a) and removed in (b). (a) An example initial configuration. (b) Key steps in deposition. A solution-phase Ag atom (yellow) arrives at the PVP-EG interface (A). The Ag atom further reaches the PVP layer through interaction with extended segments, especially through AgO (red) interaction (B). The immediate deposition is blocked by PVP and the Ag atom can diffuse inside the PVP layer (C). The deposition is accomplished when Ag can channel through a “hole” with suitable size (D). Reprinted with permission from ref 46. Copyright 2015 American Chemical Society.

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deposition in these simulations by inserting a free Ag atom with a randomly chosen velocity from the Maxwell-Boltzmann distribution near the PVP-EG interface at z [cf., Fig. 4(a)] and following its trajectory. We find that deposition occurs in four steps. First the Ag atom diffuses from the EG solution to the PVP film. When the atom reaches the film, it is attracted to the PVP segments and it diffuses into the PVP layer. The atom is held in the PVP layer for some time, during which it can diffuse parallel to the surface, until a hole opens in the PVP layer that allows the atom to access the Ag surface below. Details of each of these stages are illustrated in Fig. 4(b).

(a)

(b)

Figure 5: (a) PMF for a Ag atom to approach PVP10-covered Ag(100) and Ag(111) surfaces and (b) the density profile of O atoms in the corresponding PVP10 film. The inset shows a zoom-in feature at the “tail” part of the PVP layer. Reprinted with permission from ref 46. Copyright 2015 American Chemical Society. The potential of mean force (PMF), or the free-energy profile that characterizes the process by a solution-phase atom accesses the metal substrate, provides quantitative insight into the observed mechanisms. We obtained the PMF using umbrella sampling 81 and umbrella integration 82 by sampling the deposition path normal to the surface. The PMFs are shown in Fig. 5(a), where we notice three main features. First, as the Ag atom approaches the PVP layer, the free energy begins to decrease at a certain height above the surface, corresponding to Ag-atom trapping in the PVP layer. We note that the decrease in the free energy commences further from the surface for Ag(111) than for Ag(100). The free energy continues to decrease until it reaches a local minimum, which corresponds to the Ag atom 14


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being trapped in the PVP layer. The atom must surmount a free-energy barrier to access the Ag surface. We note that the barrier to access the surface is significantly lower than the free-energy difference for moving back to the solution phase. Thus, we expect Ag atom adsorption to be virtually irreversible. We also note that the energy barrier to reach the surface is lower for Ag(111) than Ag(100). Differences between the free-energy profiles for Ag(100) and Ag(111) can be attributed to the stronger binding of PVP to Ag(100) and than to Ag(111). These differences are also reflected in the PVP segment density profiles near the two surfaces. We show density profiles of O atoms in PVP (which represents the segment density) on the two surfaces in Fig. 5(b). At the interface between the PVP layer and the bulk solution, PVP has a greater segment density far from the surface on Ag(111) than on Ag(100) because PVP is less strongly bound on Ag(111) and it forms a looser layer there. These extended segments can more easily attract solution-phase atoms, so that the free energy begins its decrease further from the surface on Ag(111) than on Ag(100). Similarly, due to the higher binding affinity of PVP for Ag(100), PVP segments are denser and closer to the surface on Ag(100), which leads to a higher free-energy barrier for Ag to access Ag(100). These differences in the PMFs and PVP segment (O-atom) density profiles imply that we will have different deposition rates on the two facets, which could lead to steady-state kinetic shapes that are different from thermodynamic Wulff shapes. We obtain the ratio of deposition rates, FAg(111) /FAg(100) , that will enable us to calculate kinetic Wulff shapes via Eq. 5. Here, we utilize the reciprocal relation between the deposition rate and the mean first-passage time (MFPT) tm for an atom to reach the surface from the bulk solution phase, such that FAg(111) /FAg(100) = tm,Ag(100) /tm,Ag(111) . For an Ag atom starting at a height of z from the surface [Fig. 4(a)] and ending at the absorbing boundary at zf at the Ag surface, tm is given by solution of the one-dimensional Smoluchowski equation 83 Z

zf

Z

z0

exp[−W (z 00 )/kT ]dz 00

tm (z, zf ) = z

z0

15

exp[W (z 0 )/kT ] 0 dz D(z 0 )


,

(6)


where W (z) is the PMF, D(z) is the z-dependent diffusion coefficient of the Ag atom, and z0 represents the reflecting boundary that accounts for the bulk-solution region. We obtained zdependent diffusion coefficients from the umbrella-sampling trajectories using the simplified Woolf-Roux equation. 84,85

Figure 6: Mean first passage times obtained from Eq. 6 for various adsorbing films on Ag(100) and Ag(111) surfaces. 47 For the PVP systems, the MFPTs are partitioned into diffusion and reaction times. Reprinted with permission from ref 47, with the permission of AIP Publishing. Copyright 2016 American Institute of Physics.

The MFPTs obtained from Eq. 6 for deposition on Ag(100) and Ag(111) are shown in Fig. 6. To provide insight into the deposition mechanism, we divided the MFPTs for the PVP-covered surfaces into diffusion and reaction times, as seen in Fig. 6. The diffusion time corresponds to the average time required for the atom to diffuse from bulk solution into the PVP film, and the reaction time characterizes the average time required for the atom to reach the Ag surface once it has entered the PVP film. We note that both diffusion and reaction contribute significantly to the MFPT; therefore, Ag atom deposition is neither diffusion-limited nor reaction-limited. In Fig. 6, we see that the MFPTs for the systems with solvent only (i.e., no PVP) are significantly larger than those for systems with PVP. This is because PVP attracts incoming 16


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Ag atoms and, in the absence of PVP, EG solvent molecules form a tight network on the surface that results in a free-energy barrier for solution-phase Ag atoms to reach the surface. 47 The MFPTs are longer for Ag(100) than for Ag(111) for the PVP10 and PVP20 systems, which mostly results from the higher energy barrier in the PMF for Ag(100) surfaces [cf., Fig. 5(a)]. We note that the results in Fig. 6 are within a factor of two of values obtained from direct MD simulations, 46 which can be regarded as being exact for the system simulated. Thus, Eq. 6 provides a satisfactory description of the MFPT for these systems.

Figure 7: Kinetic Wulff shapes of Ag nanocrystals in different solution environments predicted from MFPT calculations. The {100} facets are green and the {111} facets are orange. Reprinted with permission from ref 47, with the permission of AIP Publishing. Copyright 2016 American Institute of Physics.

The kinetic Wulff shapes that correspond to computed growth-rate ratios for each PVP system are shown in Fig. 7. For Ag surfaces with a full monolayer coverage of PVP10 and PVP20, {100}-faceted cubes with no {111} corner facets are predicted, while we predict truncated octahedra for PVP5 films. Our results suggest the formation of truncated cubes in EG solvent only; however, the kinetic Wulff construction does not take into account nanocrystal aggregation. We would expect irregular nanostructures to be formed experimentally in the 17



absence of PVP because EG solvent molecules do not adsorb strongly enough to Ag surfaces to protect the nanocrystals from aggregating. These trends are also observed from the direct MD approach. 46 Further, in our direct MD study, we observed that cubes were not formed with less-than-monolayer PVP coverage. 46 Experimentally, Ag nanocubes are obtained with sufficient PVP concentration, 49 so our predictions are in agreement with experiment. We note that Meena and co-workers have also proposed for the growth of Au nanorods that the more dense packing of CTAB molecules on the (110) and (100) surfaces with respect to the (111) surface could result in greater growth of the nanorod in the (111) direction. 56

Conclusions and Outlook In this Feature Article, we summarized our recent theoretical studies aimed at understanding the role of solution-phase additives in shape-controlled metal nanocrystal synthesis, with a focus on the role of PVP in colloidal Ag nanocube formation in EG solution. We examined both the thermodynamic and kinetic influences of PVP on nanocrystal shape by using appropriate computational tools. To examine the thermodynamic influence of SDAs in nanocrystal synthesis, we developed a method to calculate the interfacial free energy for multi-component molecular solid-liquid systems. 44 We applied this method to resolve the effect of PVP in altering the interfacial free energies of solution-phase, PVP-covered Ag surfaces. 45 Using our calculated interfacial free energies to predict Wulff shapes of solution-phase Ag nanocrystals, we found that PVP does not significantly influence the equilibrium shapes of these nanocrystals, so they essentially assume the Wulff shape of bare Ag in vacuum. 45 We then considered the question of how facet-selective SDA binding needs to be to influence equilibrium nanocrystal shapes, by employing a general model. We found that only a strong binding preference to a particular facet can induce an equilibrium shape that is significantly different from that of the bare metal. This is because the formation of the bare metal surface has the largest contribution to the metal-solvent-SDA interfacial free energy, in which the

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contribution of the SDA-metal interaction is small in comparison. 45 We also investigated the kinetic influence of PVP on Ag nanocrystal shapes, by assuming that the linear facet growth rates of Ag nanocrystals are determined by the Ag atom deposition fluxes to the various facets. 46,47 We obtained MFPTs for Ag atoms to reach Ag(100) and Ag(111) surfaces with several PVP coverages. By using MFPT ratios on the two surfaces to predict kinetic Wulff shapes, we found that {100}-faceted PVP-covered Ag nanocubes are the preferred kinetic shapes when the PVP coverage is sufficiently high and the PVP chains are sufficiently long. This result is in agreement with experiment. 49 Thus, we find that SDA binding must exhibit a considerable selectivity to a certain facet to thermodynamically alter the shape of a nanocrystal from that of the bare metal. However, the binding selectivity needed to kinetically influence nanocrystal shape is much weaker. There has been much discussion regarding possible thermodynamic nanocrystal shapes, as has been evidenced in two recent reviews. 86,87 Our results indicate that kinetic control of metal nanocrystal shapes is likely more prevalent than thermodynamic control and provide a rational basis for understanding the requirements for thermodynamic control. We note that there is much more to be done in the quest to understand shape-selective nanocrystal growth. Our work has shown that SDA-induced differences in facet deposition rates can alter kinetic nanocrystal shapes and this idea has been embraced by others. 56 However, we note that the differences in atom deposition rates that we observe here cannot explain the shapes of fivefold-twinned Ag nanowires that grow from decahedral seeds in similar synthesis conditions. 88 Preliminary calculations indicate that we would need flux ratios (cf., Fig. 7) of more than 1000 to achieve experimentally observed nanowire aspect ratios. There are other possible kinetic steps that can determine nanocrystal shapes in certain synthesis protocols, such as the surface diffusion of adsorbed atoms on buried interfaces and nanocrystal aggregation. More research is needed to understand the role of these phenomena in promoting nanocrystal shapes. It is still not fully understood which attributes of SDA or solvent molecules lead to certain nanocrystal shapes for a given initial seed. Recent work

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from Wall and co-workers 59 has shown that surfactants may not be needed at all in the growth of Au nanocrystals. They developed a theoretical framework for predicting linear facet growth rates based on atom diffusion and surface nucleation kinetics, which may be a promising approach for future studies. Since the complexity of the solution-phase chemistry that accompanies nanocrystal nucleation and growth rivals that of gas-phase combustion reaction networks, it is still difficult to ascertain key species or intermediates in certain synthesis protocols. It is a current theoretical challenge to describe the solution-phase chemistry associated with these syntheses in an accurate and efficient way, so understanding how seed crystals nucleate and grow into selective shapes – an enormously important problem, as the shape of the final crystal is often determined by the shape of the seed – is still beyond our current theoretical grasp. Continued progress along these lines, with the help of detailed experimental investigations, will be fruitful in efforts to grow nanocrystal shapes by design.

Acknowledgement This work is funded by the Department of Energy, Office of Basic Energy Sciences, Materials Science Division, grant number DE-FG02-07ER46414. TB acknowledges training provided by the Computational Materials Education and Training (CoMET) NSF Research Traineeship (grant number DGE-1449785). This work used the Extreme Science and Engineering Discovery Environment (XSEDE) supported by NSF/OCI-1053575.

Notes The authors declare no competing financial interest. The Table of Content figure is reproduced from ref 46 and ref 45 with permissions from American Chemical Society and Royal Society of Chemistry. Copyright 2015 American Chemical Society. Copyright 2017 Royal Society of Chemistry.

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Biographies

Xin Qi graduated from the University of Iowa with a B.S.E in Chemical and Biochemical Engineering with distinction and is currently pursuing a Ph.D. in Chemical Engineering at the Pennsylvania State University. Her Ph.D. work under Dr. Kristen Fichthorn focuses on investigating the growth mechanisms of monocrystalline and multiply-twinned nanocrystals using computational methods. She is interested in using various theoretical approaches to unravel complex growth phenomena from multiple perspectives, as well as method development.

Tonnam Balankura received a B.Eng. from Sirindhorn International Institute of 33



Technology, Thailand, in 2013 and a Ph.D. with Prof. Kristen A. Fichthorn at the Pennsylvania State University in 2018, both in Chemical Engineering. His area of research during his Ph.D. studies includes shape-control phenomena in nanocrystal synthesis, interfacial phenomena, rare-event sampling, and molecular simulation.

Kristen Fichthorn is the Merrell Fenske Professor of Chemical Engineering and a Professor of Physics at the Pennsylvania State University. She received a B.S. in Chemical Engineering from the University of Pennsylvania in 1985 and a Ph.D. (with Robert M. Ziff and Erdogan Gulari) in Chemical Engineering from the University of Michigan in 1989. She spent one year as an IBM Postdoctoral Fellow (with W. Henry Weinberg) in the Department of Chemical Engineering at the University of California at Santa Barbara before joining the Department of Chemical Engineering at Penn State as an Assistant Professor in 1990. She was promoted to Associate Professor and Professor in 1996 and 1999, respectively. Professor Fichthorn’s research is primarily in multiscale simulation of fluid-solid interfaces, in which she applies theoretical techniques ranging from quantum density functional theory to molecular dynamics, Monte Carlo methods, and continuum theories to a diverse array of applications in nanoscale materials, thin-film and crystal growth, colloidal assembly, and wetting.

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Theoretical Perspectives on the Influence of Solution-Phase Additives

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