Letter pubs.acs.org/NanoLett
Fluctuation-Driven Anisotropic Assembly in Nanoscale Systems Behnaz Bozorgui, Dong Meng, and Sanat K. Kumar* Department of Chemical Engineering, Columbia University, New York, New York, United States
Charusita Chakravarty Department of Chemistry, Indian Institute of Technology, New Delhi, India
Angelo Cacciuto Department of Chemistry, Columbia University, New York, New York, United States S Supporting Information *
ABSTRACT: We demonstrate that the self-assembly of spherical nanoparticles (NPs), grafted isotropically with polymeric ligands, into anisotropic structures is a manifestation of the fluctuations inherent in small number statistics. Computer simulations show that the organization of ligand atoms around an individual NP is not spatially isotropic for small numbers of grafts and ligand monomers. This inherent, spatially asymmetric ligand distribution causes the effective, two-body inter-NP potential to have a strong orientational dependence, which reproduces the anisotropic assembly observed ubiquitously for these systems. In contrast, ignoring this angular dependence does not permit us to capture NP self-assembly. This idea of fluctuation-driven behavior should be broadly relevant, and, for example, it should be important for the assembly of ligand-decorated quantum dots into arrays. KEYWORDS: Self-assembly, ligand grafted nanoparticles, fluctuation effects, anisotropic assemblies, computer simulations
R
different origin. Indeed, we show that the anisotropy does not arise as a result of competing interactions or from complex many-body effects, but that it is inherently encoded at the single NP level, as a consequence of large fluctuations in the small number statistics associated with the low-density polymer grafting of the NPs. To understand the origin of this anisotropy, it is instructive to consider an old problem in statistics: the random division of a circular interval.24 The probability that at least k of the n arcs, constructed by sequentially linking (say clockwise) n points randomly set along the perimeter of a circle of radius R, are longer than a specific value ξ is pk(ξ) = [1 − kξ/(2πR)]n. This problem can be considered as a two-dimensional analog of our system, where the circle is the NP and the n “points” are the grafting loci of the polymeric ligands each having a radius of gyration, Rg. Now pk(ξ) = [1 − k[ξ + 2Rg]/(2πR)]n can be interpreted as the probability of having at least k regions of size ξ on the particle perimeter not covered by the polymer. If we choose ξ = 4(RRg)1/2, that is, the smallest polymer-free arc required for two particles to be in contact without interacting with the ligands (Figure 1), we have a simple way of predicting
ecent experiments and simulations on spherical, uncharged NPs sparsely, but uniformly, grafted with polymer chains show that they assemble into anisotropic structures such as sheets and strings.1−6 This assembly is thought to be driven by the dislike between the grafted chains and the particle cores. Thus, the grafted NP acts akin to an amphiphile and assembles into a range of morphologies that are reflective of its solvophilic−solvophobic balance. The idea that a spherically symmetric entity forms nonisotropic structures is somewhat surprising since conventional wisdom suggests that NP shapes and interactions determine their self-assembled superstructures.7−13 For instance, it has been shown that spheres organize into anisotropic assemblies when they interact through multipole interactions, for example, embedded magnetic dipoles.14 Similarly, molecular recognition events can also drive the assembly of objects into anisotropic structures.15,16 Nevertheless, over the past decade several studies17−23 have indicated that anisotropic aggregates can form when, for instance, the interactions between NPs involve two different length scales. While the anisotropy in these systems develops because of the frustration induced by competing energetic interactions coupled to many-body effects, we demonstrate conclusively that the physical mechanism behind the anisotropic morphologies formed by NPs sparsely, but randomly, grafted with polymeric ligands has a fundamentally © XXXX American Chemical Society
Received: April 17, 2013 Revised: May 16, 2013 A
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
Letter
the likelihood of having independent, bare “patches” on the NPs. Clearly, in the large grafting density limit (n ≫ 1) or in the long ligand limit, pk(ξ) becomes very small and the interaction potential between NPs is properly described by a spatially isotropic function. However, at low grafting densities and for short ligands, pk(ξ) is non-negligible, and one or more bare patches, the source of spatial anisotropy, are likely present on every NP. While this simple model accounts for fluctuations in the local grafting density, these are not the only source of anisotropy. In fact, even if we were to place the grafting points in a symmetric manner, for instance, at the vertices of an inscribed Platonic solid, significant fluctuations in the polymer conformations occur, and for small n, this will also lead to nonnegligible anisotropy (see below, especially the discussion of Figure 4e). Here, we show that we can systematically incorporate this fluctuation-driven idea into an effective angle-dependent interaction between NPs and demonstrate that the resulting anisotropic potential provides an elegant representation of experiments and simulations over a wide range of physical
Figure 1. Two circular disks of radius R decorated by grafted chains of length N (of radius of gyration Rg) at a “kissing distance” where the cores are just in contact. ξ is the size of the smallest arc (patch) required to hold this configuration while preventing any contact between the ligands. Simple geometric arguments lead to ξ = 2αR ∼ 4(RRg)1/2, for R ≫ Rg.
Figure 2. The assembly behavior of nanoparticles, each grafted with six chains of length N. The first column represents snapshots from simulations where the chains are modeled in full detail, while the last column shows typical configuration of chains on a single NP. Note that the particles become noticeably Janus-like for N > 3. The second column shows results obtained from an isotropic, effective potential of mean force. The fourth column was derived by assuming an angularly dependent potential, while the third column compares the distribution of nearest-neighbor NPs as derived from the exact simulations (black lines) with the angularly dependent potential (red lines). B
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
Letter
parameters.1−6 Our ideas also shed light on the behavior of arrays of quantum dots each of which is densely grafted with ligands; here, the ligand organization around a NP represents a competition between inherent (fluctuation-driven) and emergent (due to the interplay between interligand attractions, solvent,25 and the distances between the NPs) anisotropies. Our crucial point is that the anisotropic assembly of NPs can be driven by fluctuation effects and need not be only due to nonspherical particle shape, intrinsically directional interaction potentials, emergent many body effects, or due to the interplay between two lengths in the interaction potential. We used computer simulations to gain insights into the range of nanostructures that can be formed by “surfactant” NPs where the core and the corona polymer chains energetically dislike each other. Monte Carlo simulations were conducted on 100 NPs whose centers are placed on the z = 0 plane with periodic boundary conditions in all three directions. We model these systems in implicit solvent and include the effect of solvent in the effective potentials between the different moieties in the system. The cell is a cube of side 158, where distance is measured in units of the monomer diameter on the tether ligands. The NP centers are confined to this plane, but the chains are free to move in all three dimensions subject to bonding constraints. Each NP is modeled as a sphere of diameter 7.5σ, randomly grafted with six polymer chains. Polymer chains are modeled as hard spheres connected by strings; a chain is comprised of N monomers each of diameter σ connected by strings that can freely fluctuate in the range 1.02− 1.1 σ. (By assuming that a Kuhn length of a typical polymer is ∼1 nm, it follows that these NPs are ∼7.5 nm in diameter.) Each grafted polymer is connected to the NP surface via one of its terminal monomers. These grafted terminal monomers are positioned randomly on the NP surface with the constraint that they overlap no other monomer. They remain fixed on the surface during the simulation while the other groups move freely, subject to bond distance constraints. Note that a random distribution of grafting sites does not necessarily imply symmetric placement of the sites on the sphere surface; in fact the probability that these six graft points are placed symmetrically on the NP surface is very low. The bare inter-NP potential is modeled through an attractive square-well interaction of depth 5kT between any two NPs whose separations are in the range 7.5−8σ; the inter-NP interaction potential is ∞ for any shorter distance. The normalized NP density is set to ϱσ2NP = 0.63, where ϱ is the areal twodimensional number density of NPs. We perform Metropolis Monte Carlo simulations using a suite of moves: particle rotations, translations, chain regrowth, and cluster moves to ensure equilibration. We find that the NPs form circular, ordered clusters when there are no grafted chains; this reflects the minimization of contact area between the particle-rich and the implicit solventrich phases formed by phase separation (Figure 2, first column). The grafting of six randomly spaced chains with N = 3 on the particle surface causes them to assemble into “clumps” that eventually evolve into one-dimensional stringlike aggregates for long grafted chains (N = 8). Apparently, the clusters begin to break apart with larger N resulting in increasingly more dispersed NP arrangements. To rationalize the formation of noncircular, self-assembled structures for 2 ≤ N ≤ 8 we construct the potential of meanforce (PMF) between two polymer-grafted NPs under the assumption that the interactions are isotropic, that is, they are
independent of the relative orientations of the NPs: U(r) = −kT ln g(r), where U(r) is the effective potential of mean force, k is Boltzmann’s constant, T is temperature, and g(r) is the inter-NP radial distribution function. Simulations of two NPs with their grafted chains are used to construct g(r) between NP centers from which the potential follows. The effective interaction between NPs derived in this manner has a short-ranged attraction and a longer-ranged repulsion.21 The brush−brush interactions are responsible for the longranged repulsion, and the core−core potential accounts for the short-range attraction. Figure 3a, which only shows the
Figure 3. (a) Plot of the isotropic potential of mean force derived from two NP simulations. Only the repulsive part of the potential is shown. (b) The angular part of the potential between two NPs as a function of the angular parameter, cos θ, as described in the text. The results are for a variety of chain lengths N, in the situation where each NP has six grafted chains.
isotropic repulsion (i.e., the derived PMF minus the known core−core square-well attractive potential), demonstrates that this part of the effective potential becomes larger, both in magnitude and range, with increasing N, in line with the expectation that the longer grafts offer increased steric stabilization.26 As such, this model belongs to the class of isotropic potentials with two competing length scales, which are known to lead to NP assembly into a variety of anisotropic structures, including the linear aggregates described above.17−23 To understand if this model can explain the results obtained with an explicit representation of the chains, we have conducted a series of simulations at a NP number density ϱσ2NP = 0.63 using the derived effective U(r) between two NPs. Figure 2 (second column) shows the results obtained for a range of chain lengths going from N = 2 to N = 8. We remind ourselves that the simulations with the explicit consideration of the grafted chains show phase separation for N = 2 but selfassembly for 2< N ≤ 8 (Figure 2 first column). On the other hand, the effective potential model, which incorporates the C
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
Letter
Figure 4. (a) Schematic representation of the Δ vector and the ligand asymmetry in grafted NPs. (b) The most favorable configuration of two grafted NPs demonstrated with Δ vectors pointing away from each other. (c) The least favorable configuration of two grafted NPs. (d) Plots of the distribution of “dipole” vectors Δ for NPs of diameter 5σ each grafted with 30 chains of length N = 10 in a full three-dimensional calculation;27 the distances are normalized by the particle diameter, σNP, D corresponds to the inter-NP separation, and M represents the chain length of the matrix polymer into which the two NP are dispersed. Note that the distributions are independent of all of these variables, pointing to the fact that the anisotropy is a single NP property. Inset: Corresponding results from our quasi-two-dimensional system comprised of NPs of diameter 7.5σ grafted with six chains each of length N = 10. The NPs are placed in an implicit solvent. (e) The root-mean squared values of ⟨Δ2⟩ as function of the inverse chain length from our quasi-two-dimensional system. The circles correspond to a random distribution of graft sites, while the diamonds are for a case where the six grafting sites are perfectly ordered on the surface of the corresponding Platonic solid, in this case an inscribed cube.
average asymmetrically grafted (see also Figure 2 last column). We have empirically found that these distributions can be described by a Gaussian form, P(Δ) ∝ Δ2 exp [−(Δ2/⟨Δ2⟩)] where ⟨Δ2⟩ is the mean-squared value of Δ. Note that the rootmean-squared value of Δ increases with increasing chain length (Figure 4e). We have also recomputed this distribution in the case of the three-dimensional analog [Figure 4d main figure] in the presence of a polymer melt with chains of length ranging from M = 10 to M = 70 and in the presence of a second grafted NP placed in close proximity to the one under consideration.27 The overlap of all these distributions strongly argue that the asymmetry of ligand monomers is an inherent property of a single NP and that the presence of a second NP or a surrounding polymer matrix, which mimics the effect of manybody terms in the interactions, do not affect this property. The NPs thus appear to have an in-built asymmetry in terms of their spatial ligand distributions and hence presumably interactions. This behavior is very reminiscent of Janus particles, whose selfassembly behavior has been the focus of much recent work.28 Given this asymmetry, we now recalculate the radial distribution function, which now is not exclusively dependent on the interparticle distance r, but also on a second angular
effects of the chains through an isotropic repulsive PMF, always predicts phase separation between a dilute gas phase and a crystal for all N ≤ 7 (Figure 2 second column). It is apparent that this isotropic model does not capture the self-assembly of grafted NPs in this regime. One reason to explain this failure, as has been conjectured by several workers, is that the observed self-assembly is an emergent property due to multibody physics not captured in a two-body PMF.1,4,25 Another line of thought, as discussed above, is that the anisotropy in the chain grafting process can lead to directional inter-NP interactions through the presence of exposed attractive patches of their surface. To delineate the second argument, we quantify the asymmetry in polymer grafting by calculating the center of mass of polymer segments relative to the center of mass of the NP core, Δ (Figure 4a). Figure 4d (inset) shows the probability of finding different Δ values for a fixed value of N = 10. (The main plot on Figure 4d shows results for a closely related three-dimensional analog of the systems described in the work). This model was previously used to compute the potential of mean force between two NPs.27 Note that these distributions have a peak at a nonzero value of Δ, implying that the NPs at low grafting density are on D
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
Letter
conformations assumed by the grafted chains. In agreement with this notion, our previous work1 has shown that NPs with uniformly grafted ligands also give rise to anisotropic assemblies. We return to the role of many-body effects. It was previously thought that the assembly of isotropically grafted NPs into anisotropic structures represents emergent, multibody physics.1−6 That is, although a single NP was thought to be isotropically grafted, the grafted chains rearrange when two particles come in contact due to favorable intercore interactions. Such rearrangements were thought to lead to an emergent “dipolar” character to the inter-NP potential, and thus the formation of stringlike structures. Our results clearly show that such emergent effects play a secondary role to the inherent anisotropy effects in the limit of low grafting densities, which are probably the most relevant to the experiments on the polymer grafted NP.1 On the other hand, emergent effects are likely to play a significant role in situations with much higher ligand graft density, for example, quantum dots densely passivated by alkane ligands.25,29,30 In these cases, simulations have shown that the arrangement of ligands can be affected significantly by interactions (or temperature), solvent, and presumably the distance between the NPs. We therefore believe that many-body physics should dominate, not only the ligand distributions, but presumably also the assembly of these more densely decorated quantum dots.30 We end by reiterating the unusual phenomena that can occur at the nanoscale. Sparsely grafted NPs are dominated by fluctuation effects caused by the relatively small number of chains that are grafted onto a typical particle surface. Thus, even though the grafts are placed randomly on the surface, the particles, on average, naturally come out with an anisotropic distribution of ligand monomers. This anisotropy, which is an inherent single NP property, then directly manifests itself in the inter-NP potential and leads to self-assembly phenomena. While several isotropically interacting models can be constructed to yield similar self-assembly phenomena, the range of parameter values where such phenomena occur do not map into the experimental/simulated systems. An outstanding challenge here is to validate the simulation derived fact that these NPs are anisotropically grafted, a fact that appears intuitively clear, but one that is hard to characterize from an experimental viewpoint. To understand, and more importantly control, this anisotropy so as to obtain NP assemblies with desired morphologies remains an open challenge in this field.
variable, which is the cosine of the angle between the vector joining the centers of the NPs and the asymmetry vector, Δ. From here we calculate the PMF as a function of r and cos θ and express it as a sum of two contributions: U(r, cos θ) = U(r) + Ur(cos θ). The first term corresponds to the isotropic potential discussed above, while the second term accounts for the angular dependence of the PMF at each given value of r. Figure 3b shows how the PMF depends on cos θ at one chosen distance (r = 7.5 σ) for different values of N. First of all, note that the y-axis on these plots is comparable to the y-axis in Figure 3a, implying that the anisotropy in the potential is comparable in magnitude to the isotropic brush induced repulsion between NPs. The potential has a negative value when cos θ = −1 corresponding to the fact that the brush on particle 1 is not located in the region where the NPs face each other (Figure 4b) but in the diametrically opposite pole. In contrast, the potential has the most unfavorable value when cos θ = 1 (Figure 4c). The magnitude of this potential increases with N. Using this new set of interactions we have conducted a series of simulations at the same conditions of those performed with the purely isotropic potential. Figure 2 (fourth column) shows the results obtained for a variety of chain lengths going from N = 2 to N = 8, while the third column of the same figure compares the distribution of nearest neighbor NPs from the full simulations to this anisotropic coarse grained model. (Nearest neighbors are those NPs within the attractive range of the bare square-well intercore attraction, namely between 7.5σ and 8σ. A low-density gas phase system would have a peak at 0, i.e., essentially no neighbors within the range of attraction.) Remarkably, this angularly dependent potential yields nearly quantitative agreement with self-assembly behavior seen in the full simulations for N > 4. Clearly, the anisotropy in ligand distributions about the NPs is a necessary factor to reproduce the essential physics for these values of N. For longer chains, we expect increasingly uniform NP dispersion indicating that the isotropic strength of the repulsion dominates over this anisotropic ligand-driven behavior. It is also apparent that the anisotropic model fails for the N = 3 and 4 where it predicts phase separation rather than the structures observed in the full simulations. We believe that this is because the effective inter-NP interaction no longer has a simple dipolar dependence but rather acquires a multipole character with decreasing N of the grafts. We do not discuss this issue further since we believe that most experiments do not fall in this regime, but it is clear that the particles become more “patchy” with decreasing N. As such, this discrepancy is merely the result of an inaccurate mapping of our grafted NPs into a dipolar model and not into a more pertinent multipatch model. However, this fact does not undermine our statement on the physical mechanism behind the anisotropic aggregation of these NPs. It is also important to stress that to this point the ligand chains were randomly grafted on the particle surface with the only condition that no two monomers overlap. We have also examined situations where the graft points were uniformly placed on the NP surface, that is, on the vertices of an inscribed Platonic solid. Figure 4e shows that these ordered distribution of grafting sites demonstrate no asymmetry for N = 1 but that Δ2 grows with chain length, although it is always smaller than the case with a random distribution of graft sites. Apparently, the asymmetry in ligand distributions has its origins in both the distribution of the graft sites and also the random
■
ASSOCIATED CONTENT
S Supporting Information *
Additonal information and figure. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The authors thank the National Science Foundation (CBET1028299) for partial support of this work. A.C. acknowledges support from the National Science Foundation under Career Grant DMR-0846426. We thank Chris Iacovella, Peter Cummings, and Jack Douglas for many useful discussions. E
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters
■
Letter
REFERENCES
(26) Striolo, A.; Egorov, S. A. Steric stabilization of spherical colloidal particles: Implicit and explicit solvent. J. Chem. Phys. 2007, 126 (1), 014901. (27) Meng, D.; et al. Effective interactions between grafted nanoparticles in a polymer matrix. Soft Matter 2012, 8 (18), 5002− 5010. (28) Sciortino, F.; Giacometti, A.; Pastore, G. Phase Diagram of Janus Particles. Phys. Rev. Lett. 2009, 103, 237801. (29) Ghorai, P. K.; Glotzer, S. C. Molecular Dynamics Simulation Study of Self-Assembled Monolayers of Alkanethiol Surfactants on Spherical Gold Nanoparticles. J. Phys. Chem. C 2007, 111 (43), 15857−15862. (30) Geyer, T.; Born, P.; Kraus, T. Switching Between Crystallization and Amorphous Agglomeration of Alkyl Thiol-Coated Gold Nanoparticles. Phys. Rev. Lett. 2012, 109, 128302.
(1) Akcora, P.; et al. Anisotropic particle self-assembly in polymer nanocomposites. Nat. Mater. 2009, 8, 354−359. (2) Bedrov, D.; Smith, G. D.; Li, L. W. Molecular Dynamics Simulation Study of the Role of Evenly Spaced poly(ethylene oxide) Tethers on the Aggregation of C-60 Fullerenes in Water. Langmuir 2005, 21 (12), 5251−5255. (3) Hooper, J. B.; Bedrov, D.; Smith, G. D. Supramolecular SelfOrganization in PEO-Modified C-60 Fullerene/Water Solutions: Influence of Polymer Molecular Weight and Nanoparticle Concentration. Langmuir 2008, 24 (9), 4550−4557. (4) Gao, B.; Arya, G.; Tao, A. R. Self-orienting nanocubes for the assembly of plasmonic nanojunctions. Nat. Nanotechnol. 2012, 7 (7), 433−437. (5) Quan, Z. W.; et al. Tilted Face-Centered-Cubic Supercrystals of PbS Nanocubes. Nano Lett. 2012, 12 (8), 4409−4413. (6) Padmanabhan, V. Effect of grafting on nanoparticle segregation in polymer/nanoparticle blends near a substrate. J. Chem. Phys. 2012, 137, 094907. (7) Glotzer, S. C.; Solomon, M. J. Anisotropy of building blocks and their assembly into complex structures. Nat. Mater. 2007, 6 (8), 557− 562. (8) Iacovella, C. R. Icosahedral packing of polymer-tethered nanospheres and stabilization of the gyroid phase. Phys. Rev. E 2007, 75, 040801. (9) Starr, F. W.; Douglas, J. F.; Glotzer, S. C. Origin of particle clustering in a simulated polymer nanocomposite and its impact on rheology. J. Chem. Phys. 2003, 119 (3), 1777−1788. (10) Tang, Z. Y.; et al. Self-assembly of CdTe nanocrystals into freefloating sheets. Science 2006, 314 (5797), 274−278. (11) Zhang, X.; Zhang, Z. L.; Glotzer, S. C. Simulation Study of Dipole-Induced Self-Assembly of Nanocubes. J. Phys. Chem. C 2007, 111 (11), 4132−4137. (12) Zhang, Z. L.; et al. Tethered Nano Building Blocks: Toward a Conceptual Framework for Nanoparticle Self-Assembly. Nano Lett. 2003, 3 (10), 1341−1346. (13) Stradner, A.; et al. Equilibrium cluster formation in concentrated protein solutions and colloids. Nature 2004, 432 (7016), 492−495. (14) Van Workum, K.; Douglas, J. F. Symmetry, equivalence, and molecular self-assembly. Phys. Rev. E 2006, 73, 031502. (15) Caswell, K. K.; et al. Preferential End-to-End Assembly of Gold Nanorods by Biotin-Streptavidin Connectors. J. Am. Chem. Soc. 2003, 125 (46), 13914−13915. (16) Wang, Y.; et al. End-to-end assembly of gold nanorods by means of oligonucleotide-mercury(II) molecular recognition. Chem. Commun. 2010, 46 (8), 1332−1334. (17) Norizoe, Y.; Kawakatsu, T. Monte Carlo simulation of stringlike colloidal assembly. Europhys. Lett. 2005, 72 (4), 583−589. (18) Norizoe, Y.; Kawakatsu, T. Particle Monte Carlo simulation of string-like colloidal assembly in two and three dimensions. J. Chem. Phys. 2012, 137, 024904. (19) Groenewold, J.; Kegel, W. K. Anomalously Large Equilibrium Clusters of Colloids. J. Phys. Chem. B 2001, 105 (47), 11702−11709. (20) Grousson, M.; Tarjus, G.; Viot, P. Phase diagram of an Ising model with long-range frustrating interactions: A theoretical analysis. Phys. Rev. E 2000, 62 (6), 7781−7792. (21) Mossa, S.; et al. Ground-State Clusters for Short-Range Attractive and Long-Range Repulsive Potentials. Langmuir 2004, 20 (24), 10756−10763. (22) Gelbart, W. M.; et al. Array formation in nano-colloids: Theory and experiment in 2D. Faraday Discuss. 1999, 112, 299−307. (23) Malescio, G.; Pellicane, G. Stripe phases from isotropic repulsive interactions. Nat. Mater. 2003, 2 (2), 97−100. (24) David, A. D.; Nagaraja, H. N. Order Statistics; Wiley: New York, 2004. (25) Lane, J. M. D.; Grest, G.S. Spontaneous Asymmetry of Coated Spherical Nanoparticles in Solution and at Liquid-Vapor Interfaces. Phys. Rev. Lett. 2010, 104, 235501. F
dx.doi.org/10.1021/nl401378r | Nano Lett. XXXX, XXX, XXX−XXX
