Anisotropic growth of silver nanoparticles is kinetically controlled by

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Anisotropic growth of silver nanoparticles is kinetically controlled by polyvinylpyrrolidone binding Zhifeng Chen, Ji Woong Chang, Choumini Balasanthiran, Scott T. Milner, and Robert M. Rioux J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b11295 • Publication Date (Web): 18 Jan 2019 Downloaded from http://pubs.acs.org on January 18, 2019

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Journal of the American Chemical Society

Anisotropic growth of silver nanoparticles is kinetically controlled by polyvinylpyrrolidone binding Zhifeng Chen,a Ji Woong Chang,a,b,* Choumini Balasanthiran,a Scott T. Milner,a,* Robert M. Rioux a,c,* aDepartment

of Chemical Engineering, Pennsylvania State University, University Park, PA 16802

(USA) bDepartment

of Chemical Engineering, Kumoh National Institute of Technology, Gumisi, Gyeongsangbuk-do, 39177, South Korea cDepartment

of Chemistry, Pennsylvania State University, University Park, PA 16802 (USA)

*[email protected], [email protected], [email protected]

KEYWORDS: polyvinylpyrrolidone, silver nanoparticles, equilibrium constant, thermodynamic control, kinetic control, binding free energy

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ABSTRACT

Polyvinylpyrrolidone (PVP) is used in the synthesis of Ag nanoparticles (NPs) with controlled shape, most commonly producing cubes. The mechanism for shape control is unclear, but believed by many to be caused by preferential binding of PVP to Ag(100) facets compared to Ag(111) facets, and assumed by most to be the result of thermodynamic control, whereby facets with lower interfacial free energy predominate. To investigate this mechanism, we measured adsorption isotherms of PVP on different-shaped Ag NPs, to determine the thermodynamics of PVP adsorption to Ag(100) and Ag(111) facets. The equilibrium adsorption constant is independent of PVP molecular weight, and depends only weakly on NP shape (and thus Ag facet).

The

equilibrium adsorption constant for PVP on Ag(111) (2.8 M-1) is about half that on Ag(100) (5 M1).

From a Wulff construction this difference is not nearly enough to produce cubes via

thermodynamic control.

This result indicates the importance of kinetic control of the Ag

nanoparticle shape by PVP, as has recently been proposed.


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INTRODUCTION Solution-based synthesis is a versatile method to produce metal nanoparticles (NPs) with size,1–5 shape1,6–8 and composition1,3,9–11 control. For synthetic approaches that do not utilize direct deposition on substrates, capping agents such as halogens,12,13 citrate14–16 and polymers17–20 are used to stabilize the NPs against aggregation. These capping agents are thought to serve a dual role during synthesis, acting not only as a stabilizing agent but also as structure-directing agents (SDA), since their addition often influences the shape of the NPs.12,13,16,21–24 Among polymers, poly(vinylpyrrolidone) (PVP) is widely used for the synthesis of coinage metal NPs (Ag, Au, Cu), to produce different shapes including cubes,12,17,22,2526 octahedra,23,25,27,28 prisms28–30 and wires.21, 31,32

PVP is also prevalently used for the synthesis of many other metal NPs, such as Pd,33 Pt,34

and Ru.35

Xia’s group has published extensively on Ag NP synthesis in the presence of PVP.7,14,15,22,36–55 In 2002, his group employed a polyol method for the shape-controlled synthesis of Ag nanocubes38 and Ag nanowires,37 in which the solvent ethylene glycol reduced AgNO3 in the presence of PVP under controlled reaction conditions. Experimental variables such as temperature,37,38,56 AgNO3 concentration,7,36,38 AgNO3/PVP ratio7,37,38,42 and PVP molecular weight57–59 all affect the morphology of the Ag products, suggesting a complex mechanism underlies shape controlled synthesis of Ag nanoparticles. Additives such as Cl-/O2,41 HCl,22,54 Na2S/NaHS,53,48,51 Fe3+/Fe2+ species44 and Br- 46,55,60 combined with PVP during synthesis promote formation of monodisperse Ag nanocubes. Not all coordination polymers work equally well; replacement of PVP with other commonly used polymers such as poly(ethylene oxide) and poly(vinyl alcohol) failed to produce Ag shapes mainly composed of (100) facets.36


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Various hypotheses as to how PVP directs Ag nanoparticle growth have been proposed, based on parametric studies of process variables (such as PVP concentration and molecular weight) on NP shape. A common element in these hypotheses is the preferential binding of PVP to Ag(100) facets compared to Ag(111) facets, resulting in Ag nanostructures mainly composed of Ag(100) facets; the origin of this control, whether thermodynamic or kinetic is unknown.7,14,36,40,61–63 For Ag nanocrystals, the vacuum surface free energy of Ag(111) is less than Ag(100).23,64 Thus for thermodynamic shape control to work, the preferential binding of PVP to Ag(100) must be strong enough to reduce the free energy of the (100) facet well below that of (111), in order for a thermodynamic Wulff construction to result in cube-shaped NPs. In contrast, for kinetic control to be effective, preferential binding of PVP to Ag(100) must reduce the growth rate of the (100) facet compared with the (111) facet. Xin and Fichthorn65 recently demonstrated with molecular dynamics simulations that PVP forms a thicker layer on Ag(100) facets, impeding the diffusion of Ag ions to the surface. Although many authors have invoked preferential binding of PVP to explain observed anisotropic growth, few studies have quantified the role of PVP in shape-controlled synthesis of Ag nanoparticles. The first attempt to qualitatively assess whether PVP has a facet-dependent binding energy for Ag was carried out by Xia’s group through a comparison of the reactivity of a dithiol for the sides versus the ends of the Ag nanowires.40 Significant difference in reactivity of a dithiol group with the side (100) and end (111) surfaces of Ag nanowires indirectly suggests PVP binds more strongly to (100) compared to (111). Either a more facile ligand exchange or an apparent lower coverage of PVP on the (111) facets led to a greater density of dithiol ligands on the ends of


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the nanowire, as evidenced by greater accumulation of Au nanoparticles (because their association with the thiols) at the ends of the nanowire. In a more recent study, Xia’s group developed a procedure to indirectly determine the surface density of PVP on Ag(100) facets, using seed mediated growth of Ag nanocrystals at various PVP concentrations.23 At low PVP concentration, there is not enough PVP to saturate the surface of the Ag nanocubes, enabling the production of particles with exposed Ag(111) facets. At higher PVP concentration, with PVP adsorbed at full coverage on Ag(100) facets, larger Ag nanocubes formed. The authors argue the stability of cubes during secondary growth suggests shape control is controlled by thermodynamics, since Ag(100) facets remained stable at high PVP coverage while Ag(111) remained stable at low PVP coverage. The difference in shape evolution during seed-mediated growth of the Ag nanocubes was attributed to differences in the surface density of PVP. Several groups have asserted PVP binds to the Ag surface via its highly polar carbonyl group66–68, citing evidence from Raman spectroscopy and X-ray photoelectron spectroscopy (XPS). Raman spectroscopy has also been used to investigate preferential PVP adsorption on (100) versus (111) facets. Xia employed surface enhanced Raman spectroscopy (SERS) to compare the characteristic C=O peak intensity on Ag nanocubes and Ag octahedra.23 The higher C=O peak intensity on Ag nanocubes was interpreted as a higher surface density of PVP on Ag(100) than Ag(111) facets. Theoretical investigations utilizing PVP monomer and oligomers have shed light on the role of PVP in the anisotropic growth of Ag nanoparticles. Using density functional theory (DFT) calculations, the adsorption energy of 2-pyrrolidone (2P) on Ag(100) and (111) surfaces in vacuum was calculated. The binding energy on Ag(100) was 0.78 eV, 0.08 eV higher than on Ag(111),62,63


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consistent with the prevailing idea that PVP binds more strongly to Ag(100) than Ag(111). However, these binding energies are calculated in vacuum, and are thus much larger than binding free energies in solution, because solvation effects are not included. In subsequent work, Qi and co-workers calculated surface free energies of Ag(100) and (111) surfaces at various surface density of PVP,64 using a multi-scheme thermodynamic integration method.69 With the surface energy of Ag(100) (γ100=710 mJ/m2) larger than that of Ag(111) (γ111=642 mJ/m2) even at full coverage of PVP , a thermodynamic Wulff construction predicts truncated octahedron as the prevailing shape. The preferential binding of PVP to (100) was not significant enough to reverse the order in facet-dependent Ag surface energy.

This work

challenges the idea that PVP exerts thermodynamic control, since the predicted truncated octahedron is inconsistent with the cubes obtained experimentally. More recently, molecular dynamics (MD) simulations were used to investigate the growth rate on Ag(100) and Ag(111) surface with various surface density of PVP oligomers. The results suggest the role of PVP in Ag nanocube growth is kinetic in nature.65 The increased coverage of PVP on Ag(100) surfaces reduced the Ag flux relative to the Ag(111) facet, leading to faster growth of (111) and hence a (111) to (100) facet transformation. With a growth rate of Ag(111)/Ag(100) larger than 1.73, Ag cubes were predicted through a kinetic Wulff construction,70,71 which predicts particle shape through differences in growth rate, rather than differences in surface free energy as for thermodynamic Wulff construction. To summarize prior work in the field, there is presently no agreement as to whether the role of PVP in shape control is thermodynamic or kinetic in nature. Likewise, there are no quantitative experimental measures of the surface density of PVP on various shaped silver NPs.

The


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measurement of the equilibrium constant for PVP adsorption on different facets of Ag under realistic conditions is essential to determine the mechanism by which PVP influences nanoparticle growth. Here we develop a quantitative method to measure the PVP adsorption isotherm on Ag surfaces, by conducting equilibrium adsorption measurements on NPs with either Ag(100) and Ag(111) facets. We investigate the adsorption of PVP adsorption with isothermal titration calorimetry to experimental estimates of adsorption enthalpies with included solvation effects. We vary the molecular weight of PVP, to explore its influence on adsorption thermodynamics. Our quantitative approach is essential to evaluate hypotheses regarding preferential binding of PVP to different Ag facets as the underlying mechanism of shape-controlled synthesis of Ag NPs.


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RESULTS AND DISCUSSION Equilibrium Adsorption of PVP on Ag Nanocubes To measure adsorption isotherms of PVP on Ag surfaces, we varied the PVP concentration during Ag NPs synthesis, and separately measured the amount of adsorbed PVP on the particles and the equilibrium concentration of PVP in the supernatant solution based on HPLC calibration curves (Figure S1). The available surface area for adsorption was determined by measuring the total mass of particles produced, and using SEM to measure the particle edge length. For this series of measurements we used PVP with a molecular weight of 55 kg/mol, and a synthesis protocol that produced Ag nanocubes with (100) facets.22 Different concentrations of PVP monomer (6.125 mM, 12.25 mM, 24.5 mM, 36.75 mM, 73.5 mM and 147 mM) were used during the synthesis, while all other synthesis parameters were held constant. Scanning electron microscopy (SEM) was used to image the resulting Ag nanocubes (see Figure 1A-F). The edge length of Ag nanocubes obtained at different PVP concentration is shown in Figure S2A. With no PVP used in the synthesis, large Ag aggregates form, demonstrating the importance of PVP as a stabilizing agent. To determine the amount of PVP adsorbed on Ag surface and the concentration of PVP in solution in equilibrium with adsorbed PVP, we separated the silver NPs from the supernatant by centrifugation, and measured both the amount of PVP associated with Ag and the concentration of PVP in the supernatant (See experimental section for details). The presence of PVP on Ag NPs was further confirmed using X-ray photoemission spectroscopy (XPS) through the detection of nitrogen (Figure S3).


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The total surface area of the particles produced was calculated from the total mass of particles measured from inductively coupled plasma optical emission spectroscopy (ICP-OES) and their size as measured by SEM. We report the adsorbed surface concentration of PVP normalized by the concentration of surface Ag atoms, i.e., PVP monomers per Ag atom on the surface. This can be easily converted to mass per area, as 1 PVP monomer per surface Ag atom equals 2.25 mg/m2. Over the range of PVP synthesis concentration explored (6-150 mM), the adsorbed PVP surface concentration varies linearly with the PVP monomer concentration in solution (Figure S4). This linear variation indicates a relatively weak, non-saturating interaction between PVP and the surface of the Ag nanocubes. Such a linear region is consistent with the Langmuir adsorption isotherm in the Henry’s region, where the product KPVPCPVP is much less than unity. Attempts to synthesize Ag nanocubes at higher PVP monomer concentration failed; at higher PVP concentration (220.5 mM), Ag nanowires and large irregular particles formed (Figure S5). PVP is also reported to be a reducing agent,72,47 which indicates higher concentration of PVP accelerates the reduction of AgNO3 during the nucleation step, resulting in the incomplete etching of multiple twinned seeds.22 To extend our measurement of the equilibrium isotherm to higher concentrations and observe the isotherm approach to saturation (full coverage), we employed a different strategy. To further increase the surface density of adsorbed PVP, we mixed a well-characterized suspension of postsynthesis Ag nanocubes with a concentrated PVP solution. The adsorbed amount of PVP and the solution concentration in equilibrium with the nanoparticles were measured with the same protocol as before. To verify adsorption was reversible, so that our measurements truly represent equilibrium


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adsorption, the solution concentration of PVP was also varied post-synthesis, by mixing a suspension of post-synthesized Ag nanocubes with either pure ethylene glycol, or a more concentrated PVP solution. We verified consistent results were obtained for different paths to the same equilibrium condition. Ag nanocubes were synthesized with PVP 55 K (at monomer concentration of 125 mM postsynthesis) with all other synthetic parameters held constant. To dilute the solution concentration of PVP, the synthesized products were diluted twice and four times, respectively, with ethylene glycol to final monomer concentrations of 31.25 mM and 62.5 mM and stirred for over 48 h to ensure equilibrium. To increase the solution PVP concentration, synthesized products were mixed with higher PVP concertation solutions (375 mM, 775 mM and 1475 mM) and similarly equilibrated, to reach final monomer concentrations of 250 mM, 450 mM and 800 mM respectively. Figure 1 (G-I) are SEM images of the Ag nanocubes synthesized at 125 mM, the diluted sample at 31.25 mM and the concentrated sample at 800 mM, confirming the dilution and concentration procedure does not change the shape of the Ag nanocubes. Figure 2 demonstrates the saturation limit of the isotherm was reached as the concentration of PVP increased up to 800 mM. Over the measured range, the data is phenomenologically consistent with the Langmuir isotherm, which we use below to extract values of the effective equilibrium constant and saturation coverage. The samples diluted to 31.25 mM and 62.5 mM fit in the linear region of the Langmuir isotherm as shown in Figure S6, indicating our measurement of PVP


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concentration in solution and on the Ag NP surface represent equilibrium quantities. The amount of PVP adsorbed on Ag NP surface conforms to a Langmuir adsorption isotherm (eqn. 1) 𝜏 = τ𝑚𝑎𝑥 ×

KPVP × [PVP]𝑠𝑜𝑙

(1)

1 + KPVP × [PVP]𝑠𝑜𝑙

where [PVP]𝑠𝑜𝑙 is the equilibrium concentration of PVP monomer in solution, KPVP is the equilibrium constant for PVP monomer adsorption to the Ag NP surface, τ is the measured coverage of PVP monomer per surface Ag atom and τ𝑚𝑎𝑥 represents the saturation coverage of PVP monomer per surface Ag atom. Qualitatively, the saturation coverage must be governed by steric crowding of the polymers as they compete for limited surface area; how this takes place, and the characteristics of the resulting adsorbed layer, is discussed further below. Fitting Eqn (1) to the data of Figure 2 yields an equilibrium adsorption constant of about 5 M-1, and a saturation coverage τ𝑚𝑎𝑥 of about 10 monomers per surface Ag atom, which is equivalent to a maximum surface concentration of 140 monomers/nm2. For a complementary approach to investigating PVP adsorption, we conducted competitive adsorption

experiments

using

isothermal

titration

calorimetry

(ITC),

in

which

3-

mercaptopropionic acid (3-MPA) (high affinity ligand)73 progressively displaces PVP (low affinity ligand) from the surface of Ag NPs (see details in Supporting Information and Figure S7). The apparent binding constant of 3-MPA on Ag nanocubes yields an estimate of the equilibrium constant of PVP in the range of 5 to 50 M-1 (Figure S8), where the low boundary is close to our equilibrium adsorption isotherm result. The uncertainty spans an order of magnitude because 3MPA binds so strongly to Ag that its adsorption is only weakly perturbed by PVP. A more weakly binding ligand might provide narrower bounds on the PVP equilibrium adsorption constant.


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Figure 1. SEM images of silver NPs at different concentrations of PVP (MW = 55 kg/mol) synthesized by the HCl method, at post-synthesis PVP concentrations of (A) 6.125 mM (A), (B) 12.25 mM, (C) 24.5 mM, (D) 36.75 mM, (E) 73.5 mM, (F) 147 mM, and (G) 125 mM. In (H), the synthesized product was diluted to 31.25 mM PVP, and in (I) concentrated to 800 mM PVP.


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Figure 2. Adsorption isotherm at room temperature for PVP 55K on Ag nanocubes, formed by direct synthesis with varying concentrations of PVP (blue triangles) and post-synthesis combining with different concentration of PVP (red squares). The insert shows the linear form the adsorption isotherm. The linear fit yields τ𝑚𝑎𝑥 ≈ 10 PVP monomers per surface Ag atom and KPVP ≈ 5 M-1.


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Binding Energy and Proposed Adsorption Configuration of PVP 55K on Ag Nanocubes The adsorption of PVP is well described by an adsorption isotherm of the Langmuir form, characterized by two parameters: the saturation coverage, max and the equilibrium constant KPVP. The simple Langmuir adsorption model posits a species adsorbing from dilute solution onto a surface with a limited “carrying capacity”, and no interactions between adsorbate molecules other than steric hindrance. Correspondingly, the Langmuir model is most applicable to adsorption of small molecules or colloidal particles; it is not immediately apparent this model appropriately describes adsorption of long polymer chains. Polymers are not compact particles and may adopt a variety of conformations in an adsorbed layer. To ask if the Langmuir adsorption model is appropriate to describe the adsorption of PVP, we first consider the possible structure of the adsorbed polymer layer. From the measured adsorption isotherm, the saturation coverage is about 10 PVP monomers per Ag atom on the (100) surface. The (100) surface has two atoms per unit cell with a linear dimension of 4.1 Å, so the area per Ag atom, AAg is 0.084 nm2. A useful benchmark to compare to the experimental saturation coverage is a monolayer of monomers. Sweeping aside molecular details, we can approximate a monolayer as a 2D square array of cubes, each of volume Ω = L3 equal to that of one monomer at melt density. With a melt density of 1.2 g/cm3 and a monomer molecular weight of 114 g/mol, the volume Ω equals 0.16 nm3. Therefore the cube has linear dimension L = 0.54 nm, and area A = 0.29 nm2. Thus the number of monomers per Ag atom at monolayer coverage is AAg/A = 0.3. The saturation coverage of 10 monomers per Ag atom is 30 times larger than monolayer coverage. Hence for almost the entire adsorption isotherm, the adsorbed polymer chains are not typically lying close to the surface like snakes on the ground, but must extend away from the surface in some sort of carpet of loops and tails. This is qualitatively


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consistent with Kyrychenko’s recent MD simulations of PVP chains (816 and 1440 monomers) adsorbed onto Ag NPs.74 As the number of monomers per chain increased from 816 to 1440, the percentage of monomers bound to the Ag surface decreased from 62.3% to 45.8%, and the thickness of PVP layer increased from 0.65 nm to 0.95 nm, corresponding to multilayer adsorption. Similarly, Zhang et al. found out that 45% of poly(allylamine hydrochloride) adsorption on 5 nm nanodiamond retains high mobility in the form of loops and free tails instead of trains confirmation through complementary NMR based method.75 Polymers weakly adsorbed to an interface (i.e., the binding free energy per monomer to the surface is assumed to be a fraction of the thermal energy kT) have been studied theoretically by de Gennes.76–78 Under these conditions, and assuming a good solvent (i.e., chains in dilute solution have a positive virial coefficient), de Gennes developed a scaling picture of the adsorbed layer as a self-similar structure of semi-dilute “blobs". In this structure, the blob size (correlation length) is on the order of the distance to the substrate. Correspondingly, the monomer concentration falls off away from the substrate, as a power law for z > D: 𝑧 ―4/3 𝑐(𝑧) = 𝑐𝐷( ) 𝐷

(2)

Here D is the thickness of the dense layer near the surface, and cD is the monomer concentration within this layer; c(z) evidently equals cD at z = D, and becomes more dilute farther from the substrate. The thickness D and concentration cD are set by a different scaling argument that balances chain entropy and adsorption energy in a concentrated confined layer. The key result is that for sufficiently strong binding per monomer (but still small compared to kT), this proximal dense layer has a thickness D of the order of the interaction range, and a concentration cD comparable to a melt. Again sweeping aside molecular details, we expect the range of the


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attractive interaction between the monomers and the substrate to be of the order of the monomer size itself. The proximal dense layer is basically a monolayer, estimated above to be about 0.54 nm thick. The power-law density profile of eqn. (2) is cut off at the characteristic size of polymer coils, set by the mean-square end-to-end distance R2. Essentially, chains are unwilling to stretch beyond their natural size to make contact with the surface; the surface affinity for monomers is satisfied as long as it is covered in monomers, regardless of whether these monomers come from many chains or few. The physical reason for an extended layer is that this structure maximizes the entropy of the adsorbed chains, freeing them from the requirement of staying close to the substrate, allowing them to adopt nearly random-walk conformations. The power-law density profile extends far from the surface, but that does not mean it contains a ∞

vast amount of material. If we integrate c(z) from z = D to infinity, we obtain ∫𝐷 𝑐(𝑧)𝑑𝑧 = 3𝐷𝑐𝐷. The entire fluffy layer beyond z = D only contains three times the mass of the dense proximal layer. Thus the entire adsorbed layer contains only about four monolayers. Evidently, this model cannot describe the data of Figure 2, for which the saturation coverage is more like 30 monolayers. Although these values depend on precise choices of coefficients of order unity in our various estimates, it seems unlikely this conclusion would be reversed by slight changes in how we estimate the properties of the proximal dense layer. We posit another possible adsorbed layer structure for PVP below. If the solvent quality is marginal (i.e., if chains in solution do not repel each other), de Gennes’ model is no longer valid. Qualitatively, we may expect such a layer to be denser.

From literature reports79–81, we

hypothesize at high temperature and high PVP concentration, ethylene glycol becomes a marginal


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solvent for PVP. This assumption is supported by the observation that it is rather difficult to dissolve PVP in EG. Then an adsorbed layer might be bound by substrate contacts, but rather dense throughout, compared to the fluffy power-law concentration profile. To further explore this idea, we suppose the adsorbed layer is at melt density throughout, and estimate the thickness of the fully saturated layer. Since there are 30 monolayers total with a monolayer thickness of 0.54 nm, the fully saturated layer has a thickness of 16 nm. We compare this value to the mean-square end-to-end distance R2 for PVP chains of molecular weight Mw = 55 kg/mol. We estimate R2 using the characteristic ratio of R2/M, which is measured for polystyrene (PS) as 0.434 Å2/(g/mol).82 PS is reasonably close structurally to PVP, with a pendant phenyl ring on alternating carbons, and a similar monomer mass of 104 g/mol. We estimate R for PVP of 55 kg/mol as 15.4 nm. Thus, the minimum thickness of the fully saturated layer (i.e., assuming melt density throughout) is just slightly larger than R, which does not rule out this structure for the adsorbed layer. Absent any alternative that explains such a high saturation coverage and reversible adsorption, we assume the adsorbed layer is dense, and attribute this to marginal solvent quality for PVP in ethylene glycol. The adsorption constant KPVP in the Langmuir model can be related to the binding free energy of the adsorbing species:

KPVP =

eβ∆F 𝐶0𝑃𝑉𝑃

(3)

where β∆F is the binding free energy of PVP (an entire chain) and 𝐶0𝑃𝑉𝑃 is the PVP monomer concentration in the fully saturated layer. From this relation, we can infer the binding free energy per chain, and ultimately the binding free energy per adsorbed monomer. To proceed, we compute


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the monomer concentration in the saturated layer as that of a PVP melt, about 10.5 M. The experimental value for KPVP is ~5 M-1; hence, we have β∆F = log(K𝐶0𝑃𝑉𝑃) = 3.89

(4)

The binding free energy per chain implied by the data is slightly less than 4 kT, which is consistent with our observation that PVP adsorption is reversible. To relate this per-chain binding free energy to the binding per monomer, we need to know how many monomers per chain are actually bound to the surface. We assume only monomers in the first monolayer are bound, and that each chain has the same average share of these monomers. Since the fully saturated layer is 30 monolayers, only 1/30 of the monomers bind, which is about 16 monomers per chain. Hence the binding free energy per monomer is about -0.23 kT (-0.57 kJ/mol). As shown in Figure S9, with 𝐾𝑃𝑉𝑃 of 5 M-1, we can further extract the enthalpy of adsorption as -4.6 kJ/mol based on bound monomers (see details in Supporting Information). With the calculated monomer binding free energy of -0.57 kJ/mol, the monomer entropy of adsorption is -14 J/(K·mol) or about 1.7R, hence 1-2 degrees of freedom lost per adsorbed monomer. The origin of the binding free energy of PVP monomers to the metal substrate is electrostatic. PVP monomers have a large dipole of about 4.1 D, in the plane of the ring, pointing along the carbonyl oxygen. When a dipole approaches a metal substrate, a surface charge distribution arises, sufficient to cancel electric fields within the metal. The net result is that the dipole is attracted to its reflected “image” in the metal substrate.


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In the present case, the PVP monomer dipoles are not in vacuum, but in a medium of their own melt, which has a dielectric constant of about 45 at 25°C.83 The adsorbed layer may contain some EG, but this has a similar dielectric constant, about 40 at 25°C; also, the relevant dielectric constant may be smaller than this because the chains may be hindered to some extent near the interface in response to applied fields. The corresponding image attraction is between a pair of dipoles in a dielectric medium, roughly 40 times weaker than in vacuum. To estimate the binding energy per monomer, we again make simplifying assumptions while retaining the essence of the molecular situation. We represent the dipole and its image as point dipoles, but restricted by steric hindrance in how close they can approach the metal surface. The PVP monomer side group can approach the surface in different orientations. The ring can lie on the surface, so the dipoles are oriented transverse to their separation vector; alternately, the ring can stand up on the surface, so the dipoles are aligned with their separation vector. We make a rough guess of the distance of closest approach between the dipole and its image, as 4 Å in the “flat” orientation and 6 Å in the “standing” orientation. Evaluating the dipole-dipole interaction:

𝑈=

1

(

4𝜋𝜖𝑟3

𝑑1 ∙ 𝑑2 ―

3(𝑑1 ∙ 𝑟)(𝑑2 ∙ 𝑟) 𝑟2

)

(5)

for each case, we find a binding energy for the flat orientation equal to -0.16 kT and -0.10 kT for the standing orientation. The standing orientation is more favorable because the dipoles are aligned, but hampered because the distance of closest approach is larger. Given the simplicity of the estimate and the sensitivity to the distance of closest approach, it is remarkable that the


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estimated binding energies are rather consistent with the value of -0.23 kT inferred from our simple analysis of the Langmuir isotherm.


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Surface Density of PVP on Ag Nanocubes at Various Molecular Weights To study the influence of PVP molecular weight on Ag NP synthesis, a series molecular weights (10 K, 29 K, 40 K, 55 K, 360 K and 1300 K) were used at a monomer concentration of 36.75mM, while keeping all other experimental parameters constant. After synthesis, particles were imaged by SEM (Figure 3). As previously observed,36,59 large aggregates and irregular particles formed when a Mw of 10K was used (Figure 3A). At all other Mw, the final products were cubes with high yield and a narrow size distribution (Figure S2B). The concentration of adsorbed PVP was measured utilizing the analytical approach documented above. For synthesis using PVP with an Mw of 10K, no measurable PVP was detected on Ag NPs, indicative of an affinity of PVP 10K to Ag surface lower than PVP 55K. For all other PVP Mw, the adsorbed amount was measurable by our method. As shown in Figure S10, the number of polymers per nanocube decreased as the molecular weight increased. However, in terms of monomers per area, the surface concentrations for different Mw are similar, ~20 monomers/nm2 (Figure 4). This is inconsistent with the simple notion the adsorbed layer is always a melt layer of thickness comparable to Rg. However, polydispersity complicates the matter, because shorter chains may preferentially adsorb onto the Ag surface.76 The commercial PVP we used has polydispersity Mw/Mn greater than 2.5.84,85

Our observation that the surface coverage is

independent of Mw beyond a certain critical value, is consistent with Nunnery’s work on adsorption of PMMA on Al2O3.86 We may also compare our measured adsorbed concentrations to work of Xia, who estimated a surface density of 100 monomers/nm2 for PVP 55 K on Ag nanocubes formed from seed-mediated growth.23 This value is considerably larger than the 20 monomers/nm2 we measured, but does fall within our saturation surface density of 140 monomers/nm2. We found no detectable adsorbed


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PVP for Ag NPs synthesized with PVP 10 K, but Xia’s group estimated the surface density of 2030 monomers per nm2 using PVP 10 K in seed-mediated growth. However, the decrease of surface density from PVP 55 K to 10 K found by Xia is at least qualitatively consistent with our findings. The apparent high number density of the PVP monomers (~20 monomers/nm2) is most likely a result of the three-dimensional nature of the adsorbed polymer. The high number density calculated in this work is consistent with the results of Zhang et al75 who calculated that the nitrogen atom density (31 atoms/nm2) from adsorbed poly(allylamine) chains is greater than the surface density of carbon atoms at the close-packed (111) surface of diamond (~18 atoms/nm2). It is noted that observed number densities of molecular adsorbates are reported to be significantly lower.

Smith and co-workers87 have

summarized ligand densities for Au nanoparticles, and they range from 0.4 ligands/nm2 for large bulky calixarene derivatives, while thiolate monolayers densities were on average 2-6 ligands/nm2.

For the other values of PVP Mw (29 K, 40 K, 55 K, 360 K and 1300 K), the similarity in adsorbed monomer concentration at fixed synthesis concentration indicates a similar binding constant (5 M1)

for these molecular weights. The full equilibrium adsorption isotherm was also measured for

PVP 1300 K, and the same equilibrium constant (5 M-1) was obtained (Figure S11), which suggests


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the molecular weight of PVP does not influence its monomer binding constant to Ag NP surface for Mw larger than 29 K.


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Figure 3. SEM images of silver NPs synthesized with different PVP molecular weights (g/mol). (A) Mw = 10000; (B) Mw = 29000; (C) Mw = 40000; (D) Mw = 55000; (E) Mw = 360000; (F) Mw = 1300000. The HCl method was used for all syntheses at a PVP monomer concentration of 36.75 mM.


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Figure 4. Surface density of PVP on Ag NPs surface at various PVP molecular weight and a constant PVP monomer concentration of 36.75 mM.


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Equilibrium Constant on Different Ag Facets To determine whether the free energy of binding of PVP is different on different Ag facets, we measure the equilibrium adsorption constant for NPs with different shapes. Seed-mediated synthesis was carried out using nanocubes as seeds, and the intermediate cuboctahedron product and final octahedron product were isolated during the syntheses (Figure 5 and Figure S12). We measured the equilibrium adsorption constant of PVP 55K on Ag cuboctahedra and octahedra, using the same analytical methods described previously. For octahedral NPs, the linear portion of the adsorption isotherm has a slope of 18 M-1 (Figure 6), which is about half the initial slope for adsorption onto cubes (32 M-1). For cuboctahedra, the initial slope of 27 M-1 lies between these two values as expected (Figure 6). For a perfect cuboctahedron, (100) facets comprise 66% of the total surface area, with rest being (111) facets (see Supporting Information and Figure S13-S16 for details). In fact, the initial slope for cuboctahedra (27 M-1) equals the area-weighted average of the initial slopes for (100) facets (cubes) and (111) facets (octahedra).

This indicates adsorption onto each facet occurs

independently. From the analysis of the full adsorption isotherm, the equilibrium constant for adsorption onto (111) facets is found to be 2.8 M-1, slightly more than half the value of 5 M-1 for (100) facets. PVP does bind more strongly to (100) facets than to (111) facets. However, this small difference in equilibrium constant between facets is not nearly enough to yield cubic particles from a purely thermodynamic analysis. Similarly, Vivek et al. demonstrate that quaternary ammonium bromide surfactant’s facet dependent adsorption does not lead to the thermodynamic shape control of Au nanoparticles.88 Recent DFT calculations of Fichthorn et al. show the vacuum interfacial energy of (100) facets is about 900 mN/m, about 70 mN/m larger than the value for (111) facets (830 mN/m).69 We can estimate the reduction in interfacial tension ∆𝛾 for each facet from PVP adsorption, as the binding free energy per monomer (-0.23 kT for (100)


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facet) divided by the area per adsorbed monomer (0.29 nm2). For the (100) facet, this gives ∆𝛾 = 3.3 mN/m. From the slightly smaller equilibrium constant for (111) of 2.8 M-1 compared to 5 M-1 for (100), we can use Eqn. (4) to infer a binding free energy per monomer on (111) slightly smaller by a factor of log(2.8)/log(5) = 0.64. Thus, the shift in interfacial tension from adsorption on (111) would be slightly smaller, ∆𝛾 = 2.1 mN/m. Evidently these tiny shifts are much too small to make the (100) interfacial tension smaller than the (111) tension. For the (100) shift itself to be of order 100 mN/m so that it substantially altered the vacuum interfacial energy, the monomer binding free energy would need to be 30 times larger, or about 17 kT per monomer – in which case PVP chains would be bound so strongly that we could not observe reversible equilibrium as we do. We must seek elsewhere for the mechanism by which PVP controls particle shape. Kinetic control of Ag particle shape by PVP has been proposed as an alternate mechanism. Recent MD simulations by Fichthorn et al. to model the growth of different Ag facets65 found PVP forms a thicker layer on Ag(100) facets, which results in 0.58 times slower growth rate on Ag(100) facets compared to Ag(111) facets.65 Indeed, the effect of PVP layer thickness on the growth rate they observed was strong enough to markedly change the shape of nanoparticles. The so-called “kinetic Wulff construction” posits that particle shape is governed by the growth speed of each facet, such that the distance from the particle center is proportional to the growth speed.70,71 Relatively modest differences in growth speeds between facets yield large changes in particle shape. Faster-growing facets tend to “grow themselves into extinction”; if the growth speed along (111) is 3 = 1.732 times larger than along (100), the (111) facet disappears entirely. Fichthorn et al. found for their largest adsorbed coverage, the ratio of (111) to (100) growth rates was nearly large enough (~1.5) to give cubes.65


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Although the simulation results of Fichthorn et al. are very encouraging, in a real experiment other species such as Cl-, O2and Cu2+ are present in solution, which may impact the overall growth mechanism. A recent study by Sangaru et al. combined high-resolution transmission electron microscopy (HRTEM) and energy dispersive X-ray spectroscopy (EDS). They found evidence for a AgCl layer on the surface of Ag nanocubes synthesized in the presence of added HCl, a typical additive of the polyol recipe.89 Similarly, for Ag octahedron synthesis CuCl2 was added27, in which case both AgCl and Cu might be found on the surface. These species may also play a role, whether thermodynamic or kinetic, in controlling the shapes of Ag NPs under the synthesis protocol. However, as we have shown, PVP adsorption has a minuscule effect on the interfacial tension of Ag facets, and hence no chance to affect the equilibrium crystal shape determined by a thermodynamic Wulff construction. In contrast, a simple estimate shows the measured difference in adsorption isotherms between (100) and (111) facets can affect the growth rates enough to change particle shape. Recall the initial slope of the isotherm for octahedral particles (18 M-1) was about half that found for cube-shaped particles (32 M-1).

Hence under identical synthesis

conditions in the linear region of the isotherm, the PVP coverage on (100) facets will be about 32/18 = 1.78 times as large as on (111) facets. If the PVP layers are thick enough that facet growth is limited by diffusion through the layer, the incoming flux of Ag ions from solution will be inversely proportional to the adsorbed amount. Hence the (111) growth rate is expected to be 1.78 times larger than the (100) growth rate – just enough to yield cubic particles, as observed.


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Figure 5. SEM images of silver NPs synthesized with different PVP concentration using CuCl2 method.27 (A) Ag cuboctahedron with PVP concentration of 56 mM; (B) Ag cuboctahedron with PVP concentration of 86 mM; (C) Ag octahedron with PVP concentration of 78 mM; (D) Ag octahedron with PVP concentration of 138 mM.


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Figure 6. Adsorption isotherms of PVP 55K on different shapes of Ag NPs: cube, cuboctahedron and octahedron. The slope (maxKPVP) in the Henry’s law regime is equal to 32 M-1 for cube, 27 M-1 for cuboctahedron and 18 M-1 for octahedron.


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Conclusion We have quantified the thermodynamics of PVP adsorption to (100) and (111) terminated Ag nanoparticles, by measuring adsorption isotherms of particles of different shapes.

Our

measurements indicate that the thermodynamics of adsorption are largely independent of molecular weight, and depend only weakly on the shape of the Ag nanoparticle. We find an equilibrium adsorption constant of 5 M-1 on Ag nanocubes and saturation coverage of 10 monomers per Ag atom. Assuming only the first monolayer of PVP binds to the surface of Ag(100), about 16 monomers per chain (out of 500 monomers for PVP 55K) bind to Ag surface with a binding free energy per monomer of -0.23 kT. For PVP molecular weights larger than 10kg/mol, similar surface densities were determined, indicating the equilibrium adsorption constant depends at most weakly on molecular weight. Comparing the equilibrium constant for PVP adsorption on Ag(100) and Ag(111) facets, we find that PVP exhibits only a weak preference for (100) facets, with a binding affinity only a factor of two larger. Applying the thermodynamic Wulff construction, this small difference in binding energy between facets is not large enough to result in particles with exclusively (100) facets, as is observed experimentally. In contrast, we estimate the two-fold higher coverage of PVP on Ag(100) facets compared to Ag(111) facets will affect Ag deposition rates strongly enough to induce a cubic shape under the kinetic Wulff construction, as recently suggested by simulation results of Qi et al.68 Our quantitative assessment of how PVP directs the growth of Ag nanoparticles under typical conditions, enables a mechanistic understanding of colloidal nanoparticle synthesis. Our general approach can be applied to other structure-directing agents, to determine the mechanism underlying shape controlled synthesis of metal nanoparticles.


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ASSOCIATED CONTENT Supporting Information. Materials and experimental methods, HPLC calibration curves for various Mw of PVP, control experiment to validate the quantification method, edge length of Ag nanocubes synthesized at various PVP monomer concentration and molecular weight, XPS for PVP detection on Ag NPs, SEM image of the Ag products synthesized with 220.5 mM PVP 55 K, comparison of adsorption isotherm for PVP 55 K on as-synthesized Ag nanocubes and Ag nanocubes after dilution, ITC to demonstrate the weak binding of PVP to Ag nanocubes, details of equilibrium constant calculation for PVP 55 K, calculations of PVP surface density on Ag NPs and error analysis, influence of Mw on the coverage of PVP polymer per Ag nanocubes, adsorption isotherm for PVP 1300 K on Ag nanocubes, SEM images for shape evolution from Ag nanocubes to octahedron and models for calculation of total Ag atoms and surface Ag atoms for Ag nanocubes, cuboctahedron and octahedron. AUTHOR INFORMATION Corresponding Author * Email: [email protected], [email protected], [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENT This work was funded by the Department of Energy, Office of Basic Energy Sciences, Materials Science Division, grant number DE-FG02-07ER46414.


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Anisotropic growth of silver nanoparticles is kinetically controlled by

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