Conformational Mobility in Monolayer-Protected Nanoparticles: From

Article pubs.acs.org/JPCC

Conformational Mobility in Monolayer-Protected Nanoparticles: From Torsional Free Energy Profiles to NMR Relaxation Andrea Piserchia,† Mirco Zerbetto, Marie-Virginie Salvia, Giovanni Salassa, Luca Gabrielli, Fabrizio Mancin, Federico Rastrelli,* and Diego Frezzato* Dipartimento di Scienze Chimiche, Università degli Studi di Padova, via Marzolo 1, I-35131 Padova, Italy S Supporting Information *

ABSTRACT: In this work, we develop a self-consistent route to inspect the conformational mobility of single chains in gold nanoparticles passivated with a monolayer of decanethiols. The approach is based on the match between a theoretical modeling under a coarse-grained level (i.e., system definition, buildup of the free energy profiles along relevant coordinates, modeling/parametrization of the friction, production of stochastic trajectories) and experimental spectroscopic investigations. The agreement between calculated and experimental values of 13C NMR longitudinal relaxation times supports the theoretical assumptions and the model parametrization. On physical grounds, it emerges that the mobility of the single chains resembles that of an ideal chain made of connected n-butane-like bonds.

1. INTRODUCTION Matter reduced to the nanoscale acquires properties that are not commonly observed in bulk materials and that can be exploited for several relevant applications. In the case of nanoparticles (NPs), such relevant features are complemented by the multifunctional and multivalent nature of the surface coating monolayer. Indeed, in order to be chemically and physically stable, nanoparticles need in many cases to be passivated by a monolayer typically composed of organic molecules. Such molecules are much more than mere surfactants or stabilizers since they can confer additional properties to the NP by modifying its interaction with the environment. Besides modulating the NPs solubility in polar/apolar solvents, monolayer modifications can provide the NPs with recognition, self-assembly, translocation, and transformation abilities, enabling their use in fields ranging from nanomedicine, sensing, catalysis, and materials development.1−3 A deep understanding of the monolayer dynamics is necessary for the successful rational design of NPs for such applications. Indeed, the examples listed above fall into two classes of phenomena. In some cases, the mobile part of the system (i.e., the organic molecules in the monolayer) undergoes thermal (unspecific) fluctuations at the equilibrium. In other cases, the system is involved in specific out-ofequilibrium processes, where an external stimulus (a signal, a mechanical stress, a controlled energy exchange, a coupling with a chemical reaction, etc.) produces a response of the system itself or vice versa. In any case, in order to rationalize the dynamic behavior of such a “soft part” of a nanosized system, a focus on its fluctuations at the equilibrium is required © XXXX American Chemical Society

in terms of modes and related time scales. In fact, the nonequilibrium thermodynamics paradigm,4 here brought to the nanoscopic scale, states that a weakly perturbed system likely “adopts” some of the equilibrium fluctuation modes when performing its nonequilibrium process. This view is the same as that which implicitly lays beneath the modern understanding of molecular machines (see, for example, Kolomeishy and Fisher, ref 5 and references therein). Thus, at first instance, a thorough investigation of conformational mobility at thermal equilibrium is mandatory. The term “mobility” itself involves two closely related concepts. The first one is the configurational distribution at the thermal equilibrium, namely the mobility in the “static” sense of accessible configurations, and their Boltzmann statistical weights. The second one is the stochastic dynamics that take place within such a distribution (mobility in a “dynamic” sense). To explore the configurational distribution at the thermal equilibrium, it would suffice to employ equilibrium sampling techniques such as the basic Importance Sampling Monte Carlo (IS-MC) or more efficient biased sampling routes.6,7 To investigate also the fluctuations, Molecular Dynamics (MD) of the whole system is the best known tool, provided that a suitable parametrization of the force field is supplied and that the required time window can be covered in an acceptable computational time. In the context of monolayer-protected gold NPs, since the early works of Luedtke and Landman8,9 MD simulations have gained an increasing relevance10,11 Received: May 21, 2015 Revised: July 27, 2015

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DOI: 10.1021/acs.jpcc.5b04884 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

complex functionalized NPs. Our aim is to characterize the conformational distribution and dynamics, at the equilibrium, of a single thiolate chain in the coating monolayer of 2 nm gold core diameter NPs selected as representative of the nanoparticles produced by the most popular Brust and Shiffrin protocol and still extensively studied (see SI-Sec3). As we shall briefly discuss in the final section, the methodology described here can be adapted and applied to other systems. The following section is devoted to describing the methodological framework that comprises the synthesis of monolayer-protected AuNPs, NMR relaxometry, and modeling/computation. In section 3, we illustrate the single methodological blocks and show that the computed and measured spectroscopic observables (longitudinal 13C NMR relaxation times) agree well; this will globally validate both the model and the adopted parametrization. Most technicalities are provided in a structured Supporting Information file whose specific sections are cited throughout the text.

culminating in the simulation of very complex phenomena such as aggregation.12 To mention a few recent studies on functionalized gold NPs, trajectories of hundreds of nanoseconds can now be calculated at the atomistic level with explicit inclusion of the solvent, the charges on functional groups, and counterions.13,14 Very recently, force fields have been parametrized even for the bilayer environment of lipid membranes in order to simulate spontaneous permeation into the cell.15 In the MD practice, the deterministic evolution of the largest manageable portion of system is simulated. The downside of this approach comes from the fact that the simulated time window may not be extended enough to capture very slow processes or to allow a reliable averaging of the calculated properties. An alternative approach to overtake the time scale problem is to consider from the beginning the stochastic nature of the dynamics for a relevant part of the system in contact with the thermal bath. One adopts a reduced description of the system in terms of few relevant variables, while the dismissed degrees of freedom bear on the unstructured environment/thermal bath, which exerts a damping effect on the motion and confers the stochastic character to dynamics (fluctuations) via random interactions with the systems. Fluctuations can range from single-moiety motions to collective processes (e.g., flickering of the soft coating, collective bending modes, etc.) depending on the system extension and time scale of observation. The advantage of adopting such a low-dimensional description is that the problem of covering long time intervals is overtaken, yet at the price of a sensible modeling/parametrization of the reduced system. Namely, the theoretical treatment of the stochastic dynamics requires a priori full characterization of the equilibrium distribution, plus an additional information about the friction that damps the motion while keeping it alive at the same time.16 The few relevant configurational variables are chosen to collectively originate a Markovian (memory-less) stochastic process.16 The solution of the corresponding theoretical model yields a series of dynamic observables which can be also measured by suitable experiments. This closes a methodological pathway wherein the measures can validate (or reject) the physical picture underneath the modeling and the phenomenological parameters involved. On the basis of a validated model for equilibrium fluctuations, the buildup of the higher level description of nonequilibrium responses can be faced. Stochastic simulations represent a well-established tool in the context of monolayer passivated nanoparticles. To mention a few recent examples, Brownian dynamics (BD) have been applied to estimate the transport coefficients of polymerstabilized NPs,17 and the dissipative particle dynamics (DPD) method, which bridges MD and BD,18 enabled to reproduce complex processes like the self-assembly of NPs,19 targeting to cell surfaces with receptors,20 translocation of NPs through membranes,21 and release from microgel capsules.22 In most cases, one tries to handle the whole system by modeling the drift forces on the basis of the force field that would be adopted in the corresponding MD simulation or by trusting some basic descriptions like the bead−spring model for the flexible parts. In our opinion, there is still the need of a methodological route to handle a few properly selected degrees of freedom and explore the related mobility in the twofold sense of energetics and dynamics discussed above. In this work, we apply the stochastic approach to an archetype system which captures the essential traits of more

2. METHODOLOGY The theoretical/computational treatment requires a structural model able to catch the essential traits of the target system. In essence, such a reference model has to display the average representative features found in the real sample of synthesized NPs, which is inevitably inhomogeneous in terms of number of gold atoms per NP, shape and dimensions, number of tethered chains, and surface roughness. A reference thiolate chain, hereafter referred to as the “probe chain”, is then selected. To limit the computational effort, only a minimal cluster of surrounding chains can be taken into account, as indeed we shall do in our treatment. The guess (which, however, can be validated by increasing the cluster size) is that such an environment exerts a genuine confinement on the probe chain; that is, no artifacts due to the finite-size are produced. Starting from such an underlying structure, the modeling begins with the selection of the lowest number of configurational variables. The choice must obey the following requisites: (1) the dynamics of such set of variables constitutes a memoryless Markov stochastic process,16 and (2) the set of variables is adequate to describe the dynamics in a given time window.23 Here, we shall focus on dynamics in the so-called diffusive regime of motion (or “overdamped regime”) in which inertial effects are negligible.16 This would correspond to observe the system by means of a “slow” enough experimental technique, that is, at sparse enough time points (roughly speaking, at intervals in the nanosecond time scale) such that memory of the momenta at the preceding instant is lost. Accordingly, only structural variables need to be considered. As we shall detail in the next section, a coarse grained representation (connected beads) is here adopted for the ensemble of chains. The conformation of each chain will be described by a set of torsion angles, θ, which specify the conformation of each bond. Bond lengths and bond angles are here fixed. The ingredients of the model for the stochastic conformational dynamics of the probe chain are (i) the energy landscape over the set of torsion angles and (ii) the viscous friction which generally depends on the actual chain conformation. It is crucial to recognize that “energy” here stands for a Helmholtz free energy, Achain(θ), since all momenta and inessential fast-relaxing variables have been averaged out. Formally, Achain(θ) should be constructed from first principles by projecting out the inessential variables according to a mean-field-construction procedure6,24 up to achieve the θ-dependent canonical partition B


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The Journal of Physical Chemistry C function Zchain(θ). Thus, Achain(θ) = C(T) − kBT ln Zchain(θ), with C(T) being a constant at fixed temperature. Such a formal route is, however, impracticable, and direct numerical strategies are demanded. In addition, the evaluation of Zchain(θ) by means of standard tools like Molecular Dynamics (MD) simulations and basic Monte Carlo (MC) samplings is expected to be inappropriate. In fact, very long MD trajectories or MC chains are required a priori to sample correctly the configurational distribution of the whole system at any fixed conformation θ of the probe chain, since the configurational energy landscape likely features several local minima, possibly separated by high energy barriers. Our approach consists of constructing the torsion free-energy for each bond of the probe chain and then inferring a likely form of Achain(θ). The single-bond profiles will be also compared to that for the central bond of liquid n-butane; we choose such a profile as representative of an ideal “free-chain” energetics. To obtain the torsion free energy on each bond, we employ a nonequilibrium tool based on Jarzynski’s equality (JE)25 and which implements an expedient recently proposed by some of us26,27 to improve the accuracy of the outcomes. Such a strategy has been recently successfully applied in a preparatory study on clusters of −S−C10H21 chains tethered to a planar array of gold atoms.28 Concerning the θ-dependent friction due to the viscous drag of the surrounding chains, we shall adopt a basic model to take into account that the friction magnitude depends on the size of the chain portion which is displaced in a small conformational change.29,30 As later discussed, the effective viscosity used in calculations is of central importance. Once Achain(θ) and θ-dependent friction are defined, one has to choose between two alternative but equivalent descriptions of the stochastic dynamics: a probabilistic description (i.e., solution of the multidimensional Smoluchowski equation which yields the time evolution of the nonequilibrium probability density p(θ, t)) or the simulation of an ensemble of stochastic trajectories (or a single long one) via Langevin equation for Brownian-like motion in the conformational space.16 The former approach is impracticable when the number of stochastic variables is indicatively more than 3 or 4. On the contrary, the latter approach is easier to implement in computer codes and remains manageable even in the present case (see the next section). From a simulated long trajectory, one can compute those time-correlation functions and related spectral densities which are required to evaluate the dynamic observables achieved from spectroscopic experiments suited to probe the conformational motion. In our case, the T1 13C NMR relaxation times for each carbon atom of the chains have been measured. These quantities are representative of the chain mobility since 13C relaxation occurs mainly via modulation of the C−H dipolar interactions.31 If the modeling is designed on sound assumptions and the required input parameters are well estimated/assigned, the matching is good; otherwise, one goes back to revise the assumptions and/or to tune the parameters in a sort of fitting route based on a search, step-by-step, for best matching. The schematic of the methodological approach above outlined is depicted in Figure 1. At last, one ends up with an experimentally supported picture about the underlying model for energetics and friction, that is, in essence, a picture of the single-chain mobility at equilibrium.

Figure 1. Outline of the experimental/computational methodological approach.

3. EXPERIMENTAL DETAILS AND DISCUSSION 3.1. Synthesis and NMR. Synthesis of C10 AuNPs. Natural abundance decanethiol is commercially available. The synthesis of the 13C labeled decanthiol 1 was performed in four steps starting from decanoic acid-1-13C. This was first reduced with borane in THF, affording the 13C-labeled decanol 2 (98% yield) and then reacted with mesyl chloride in dry dichloromethane and triethylamine as the base, giving the corresponding mesylate 3 (96% yield). Subsequent substitution with potassium thioacetate was performed in acetone, giving compound 4 (90% yield), which was finally deprotected under Zemplèn conditions, affording the final decanthiol-1-13C 1. Au NPs protected with a monolayer of decanethiol (5) or of 1 (6) were prepared according to a previously reported twostep procedure (Scheme 1).32 A water solution of gold(III) chloride trihydrate was extracted with a solution of tetraoctylammonium bromide in toluene. Then the appropriate amount of dioctylamine was added in order to obtain 2 nm nanoparticles. Subsequent reduction with NaBH4 led to nanoparticle formation, and finally, dioctylamine was displaced by addition of decanethiol, affording the desired monolayerprotected AuNP. TEM and TGA analysis indicates that the nanoparticles have an average diameter of 2.0 ± 0.3 nm and the average molecular formula Au247(SR)116, based on the different nanoparticles’ batches described in SI-Sec3. As reported, such figures correspond to a mixture of nanoparticles of different formulas, where Au102(SR)44, Au130(SR)50, Au144(SR)60 Au187(SR)68, and Au330(SR)79 are relevant members. Samples Preparation and NMR Setup. Solutions of both 5 and 6 NPs were prepared in CD2Cl2 at a concentration of 700 μM (50 mM in thiols). A 600 μL portion of each solution was C


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The Journal of Physical Chemistry C Scheme 1. Synthesis of Decanthiol-1-13C from Decanoic Acid-1-13C and Atom Numbering of Unlabeled and 13C1 Labeled AuNPs

Figure 2. 13C NMR spectra of both 5 (A) and 6 (B) C10 nanoparticles. *Signal of C1 in the disulfide originating from detachment of 6. °Solvent signal.

The T1 values obtained by data fitting for each experiment were averaged, and the resulting T1 are summarized in Table 1 for the different carbon atoms. The errors were estimated as the sum in quadrature of statistical uncertainties (maximum semidispersion) and fitting errors.

then introduced in screw cap 5 mm NMR tubes for analysis. Sample degassing was found not to affect the estimated T1 values within the experimental errors. All of the spectra were acquired on a Bruker 500 AVANCE III spectrometer equipped with a z-gradient BBI probe and running Topspin 3.1 software. The spectrometer frequencies for 1H and 13C were 500.13 and 125.77 MHz, respectively. T1 values for 13C nuclei were estimated by means of the inversion recovery experiment. Transient spectra were obtained using power-gated decoupling. The 13C NMR spectra of NPs 5 and 6 are reported in Figure 2. Inspection of the two spectra reveals that, as known for alkanethiol-protected small nanoparticles, the signals of carbons 3−10 are relatively sharp and similar to those observed for decanethiol in solution. On the other hand, signal arising from carbons 1 and 2 are extremely broad and barely detected in the spectrum of 5. In the spectrum of 6, a very broad signal from C1 is observed at 38 ppm. The 5-NPs were used to estimate T1 for carbons C3−C10. The T1 values for C1 and C2 were tentatively determined also on this sample; however, as expected from the low signal-tonoise ratio, the estimates are not accurate. In order to obtain a better estimate for T1, the inversion recovery experiment was repeated on the 6-NP labeled with 13C1, the conditions being detailed in SI-Sec4. The T1 values were obtained by fitting the signal intensities versus inversion time data obtained from the inversion recovery experiments with the Bruker Dynamic Center 2.2.1 software package, using the partial inversion recovery model eq 133 St = S0(1 − α e−t / T1)

Table 1. T1 Estimates for the Different Carbon Nuclei Obtained with C10 Nanoparticles

a

carbon atom

chemical shift (ppm)

C10 C9 C8 C7 C3−6 C1−2 C1a

13.90 22.76 32.06 29.60 29.94 36.11 38.00

T1 (s) 3.70 2.97 1.70 0.87 0.65 0.25 0.20

± ± ± ± ± ± ±

0.55 1.04 0.44 0.10 0.05 0.12 0.06

Obtained from 6-NPs.

A relaxation-ordered 2D spectrum displaying T1 as a function of the chemical shift is shown in Figure 3. Inspection of the Figure, as well as of the data reported in Table 1, reveals that T1 values apparently undergo a small increase along the chain from C1 (0.2 s) to C7 (0.87 s), after this point the T1 values increase by about 1 s for each carbon. 3.2. Modeling and Computations. Model System. As a reference structure we have adopted the model recently proposed by Häkkinen for Au144(SR)60.34−36 This is the largest gold nanoparticle whose structure has been modeled in detail and corresponds to one of the components of the samples synthesized. The metallic surface is known to exhibit “staple motifs” where an S atom is linked both to a gold atom of the core and to a gold adatom, which elevates above the surface.34,35 Such a bridge hampers the rotation of the −R group about the Au(core)−S bond. With reference to the underlying rhombicosidodecahedron model for the Au atoms of the core, we focused on the surface portion shown in Figure

(1)

where St is the signal intensity at the time point t, S0 is the signal intensity at the steady state, T1 is the longitudinal relaxation time, and α is the amplitude of recovered magnetization. Three different inversion recovery experiments were performed for 5-NPs. D


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ment on the central probe chain; an argument in support of this methodological choice will be provided later. Coarse-Grained Description, Configurational Variables, and Internal Energy. Similar to what has been done in ref 28, we employ a coarse-grained description of the system: sulfur atoms, methylenes, and methyls groups are treated as spherical beads, while the NP surface is approximated by an interpolating smooth spherical surface. Bond lengths and bond angles are fixed, i.e., the vibrational degrees of freedom are ignored. The geometrical parameters employed, and the related bibliographic sources are provided in the SI-Sec5. Possible penetration of the solvent (CD2Cl2) into the alkyl layer is assumed not to significantly affect the conformational distribution of the chains. Indeed, the high-packing density of chains (as it appears from the atomistic representation with van der Waals radii, see panel d in Figure 4) supports the idea that solvent penetration should play a minor role. We anticipate that such an assumption is supported, a posteriori, by the good match between calculated and experimental 13C NMR relaxation times, as it will be shown later (if the solvent would strongly perturb the layer energetics, also dynamic properties would be markedly affected). In the final section, we will revisit this issue with some hints for future inclusion of the solvent. A snapshot of the system is shown in Figure 4, panels c and d. To specify the actual configuration of the whole system, a set of molecular frames rigidly attached to each bead are introduced as described in detail in the SI-Sec6. Ultimately, the global configuration is specified by the collection Φ of the sets (one set per each chain) of torsion angles θ = (θ1, θ2, ..., θ9); the geometrical meaning of the generic angle θi is shown in Figure 5 in terms of Newman projection (panel a). With reference to a nanoparticle frame whose origin is placed at the center-of-mass of the NP core (and with known but

Figure 3. 2D relaxation-ordered spectrum of the C10 AuNP sample, displaying T1 (s) as a function of the chemical shift (13C, ppm).

4, as delimited by the green border. Panel b displays the actual model NP proposed in ref 35, as obtained by DFT calculations

Figure 4. (a) Sketch of the rhombicosidodecahedron describing the approximate geometry of the Au144 core according to refs 34 and 35. The thick green line delimits the surface portion of the NP under consideration; cyan spots are the chain-tethering points; and the asterisk marks the probe chain. (b) View of the surface portion of the relaxed structure35 used in this work. Key: orange, surface Au atoms; red, Au adatoms; yellow, sulfur atoms; gray, carbon atoms. (c) Snapshot of the coarse-grained representation for the 15-chains cluster under consideration: the probe chain is displayed in blue. For clarity, all of the beads are represented as small spheres. (d) Atomistic view with van der Waals radii.

Figure 5. (a) (Left) Bead representation of a portion of chain to highlight internal coordinates. The bond length, dCC, and the bond angle, ϕCCC, are fixed, while the torsion angles, θi, are the only internal degrees of freedom. (Right) Newman projection showing our choice for the angle set to 0°, i.e., the trans conformation. (b) Sketch of the bead representation of the 1-decanethiol chain forming a “staple motif”, showing the labeling of beads and torsion angles. The blue arrow depicts the torsion about a controlled angle θc (the case θc = θ5 is just an example).

for the case R = CH3. The probe chain is tethered to the vertex marked by an asterisk and is surrounded by 14 other chains tethered to the other centers of the surface portion (cyan spots in Figure 4, panel a). Such a minimal environment has been adopted to lower the computational effort as much as possible, on assuming that it suffices to describe the genuine confineE


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Figure 6. Calculated torsion free energies at 300 K for rotation about the bonds S−C1 (a); C1−C2 (b); C2−C3 (c); C3−C4 and C4−C5 (d); C5− C6 and C6−C7 (e); C7−C8 and C8−C9 (f). profiles are shifted with respect to the value at angle 0°, and error bars are statistical uncertainties on the outcomes. ● = torsion free energies at 300 K for the central bond of liquid n-butane at 300 K.

positions of the 15 Au0 atoms, resulting in a 7.04 Å radius. In addition to the nonbonded LJ interactions, intrinsic intrachain torsion potentials have been included in the force field.28 In particular, a sum of Ryckaert−Bellemans terms38 has been employed for dihedral bonded terms. Details and parameters of the force field are provided in the SI-Sec8. Ultimately, one ends up with the energy function Vθc (Φ) where the controlled angle θc (c = 1, ..., 9) is the selected dihedral angle belonging to the probe chain along which the torsion free energy profile is going to be constructed as described hereafter. Torsion Free Energy Profiles on Single-Chain Bonds. The torsion free energy profile about each bond of the probe chain has been constructed by employing the nonequilibrium statistical tool described in refs 26 and 39 and implemented in the C++ library JEFREE, recently developed by some of us27

arbitrary orientation), given the position vectors of the gold atoms tethering the chains (Au0) and the orientations of the molecular frames attached to them (see SI-Sec6), the Cartesian coordinates of each bead are unequivocally determined by consecutive rotations of the molecular frames plus rigid translations. Once the Cartesian coordinates of the beads are specified, the total internal energy is computed as sum of bead−bead interactions (intra- and interchain) modeled as Lennard-Jones (LJ) 12−6 potential terms and bead-surface interactions of the LJ 12−3 type;37 mathematical expressions and parameters are given in SI-Sec7. To speed up the calculations, a spatial cutoff of 12 Å has been applied after a preliminary check (in a trial calculation) to ensure that no artifacts are introduced. The NP core has been modeled as a sphere whose surface fits at best the F


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progression steps was taken the same for the morphing/ steering stages of the transformation, namely 5 × 106 + 5 × 106 steps for θ2, θ3, and 106 + 106 steps for θ1 and for θ4 to θ9. The lower progression rate (i.e., slower and “less dissipative” transformation) for the bonds closer to the gold surface was a necessity to reduce the statistical uncertainty on the free energy profiles. The results are shown in Figure 6, where the torsion potential for the liquid n-butane, Vbutane(θ), is also reported with black circles. For a direct comparison, all profiles are shifted with respect to the value at 0° for the angle under consideration. Regardless of the likelihood indicators (see SISec9), the accuracy can be directly assessed by looking at the mismatching between the values at 0° and 360° (or between the two values at 180° for the angles θ2 and θ3). What it appears is that “chemical accuracy” (variations within 1 kBT unit) is not attained, but the quality of the profiles allows one to infer about the torsion mobility on each bond of the probe chain. Inspection of Figure 6a reveals that the rotation about the S− C1 bond is confined into a potential well centered at 0°. By looking at the whole free energy excursion, and considering the large mismatch of the values at 0° and 360°, the profile for θ1 is clearly not meaningful by itself (we report it only for completeness), but it gives the message that an extremely high energy barrier prevents rotation about such a bond. This is much likely due to the presence of the NP surface which impedes the rotation of the whole chain. A similar feature has been found also for alkyl thiols tethered to planar gold surfaces with hexagonal array √3 × √3 − R30°;28 in that case, it had been proved that the high barrier was entirely due to the presence of the impenetrable surface rather than the presence of the surrounding chains. For angles θ2 (rotation about C1− C2) and θ3 (rotation about C2−C3), the profiles closely resemble that of the n-butane, although with slightly higher barriers for the direct conversion between the gauche conformations (precise statements cannot be made with the present level of accuracy). Such an outcome is rather different from that obtained in our previous study on the coated planar surface.28 In that case, the torsion potentials about the first two carbon−carbon bonds appeared to be quite different from that of n-butane, also in terms of qualitative features. Namely, we found that for the C1−C2 bond, the two gauche conformations appear to be much more stable (of about 20 kBT units at room temperature) with respect to the trans one, while for the C2− C3 bond the gauche conformations are less stable than the trans but they are separated by a large energy barrier of about 30 kBT. In passing from the planar situation to the nanoparticle arrangement, these peculiar features are lost. For angles θ4 to θ9 (see the right panels of Figure 6), the match of the torsion free energy profiles with that of n-butane is even more pronounced. As anticipated in subsection Model System, it has to be assured that considering only a reduced number of alkyl thiols surrounding the probe chain, instead of the full coating, does not introduce artifacts (e.g., due to a “fall out” of the peripheral chains of the domain). A possibility would be to consider a more extended cluster around the probe chain, as done in ref 28 to check/search for convergence on the single-bond free energy profiles. However, the computational handling of the 15-chains cluster was very demanding,41 hence such a cluster sets the actual treatable size for this kind of flexible NP coatings. Thus, we followed a different strategy. Only for the most critical angles, θ2 and θ3, we opted to perform calculations

and used to perform the calculations. Details about this specific implementation are given in the SI-Sec9. The tool has been recently applied to the preparatory case-study of alkyl thiols tethered to a planar gold surface;28 several issues presented there are of use in the present context. The method is based on Jarzynski’s equality (JE)25 applied to simulated two-stage steered transformations performed on the thermostated system, namely: (1) guided morphing of the internal energetics of the whole system (intrachain, interchain, chain−surface interactions) starting from a “flat” energy profile and (2) subsequent steered rotation of 360° about the selected bond of the probe chain (variation of angle θc). Constant progression-rate protocols are applied for both morphing and steering stages. In each progression step, the relaxation phase on the uncontrolled configurational variables Φ is realized via collective Importance-Sampling Monte Carlo moves6 (see SISec9 for details). From the statistical ensemble of work values, produced by repetitions of the steered transformation, the JE extracts the free energy profile versus the controlled coordinate(s) (see SI-Sec9 for an overview). For each bond, the outcome of a JEFREE calculation is the profile ΔA(θc) = A(θc) − Aid(θc,init) vs θc, that is the difference between the Helmholtz free energy of the system for the selected torsion angle at the value θc, and the free energy of the virtual “ideal” system in which all interactions among the constituting moieties are absent (but connections are established) and the bond angle is set to a value θc,init arbitrarily chosen. Since Aid(θc,init) can be taken as an immaterial reference value, A(θc) versus θc gives the torsion free energy profile we are interested in. The input information is the internal energy function Vθc(Φ) presented in the previous section (see eq 7.2 in SI-Sec7). As discussed in ref 26, the energy morphing eliminates the a priori error due to poor sampling of starting system configurations. Since the transformations begin from a “flat” energy landscape on Φ, the starting configurations are produced by generating chains conformations where the value of each angle (apart from the controlled angle θc) is randomly drawn from a uniform distribution between 0° and 360°. LJ contributions are submorphed by adopting the modified form proposed in ref 40 such that the repulsive interactions develop smoothly (see SI-Sec9); this improves accuracy/reproducibility of the outcomes. Beads−surface LJ 12−3 interactions are further modified by introducing a penalty term to expel those possible chains initially “wrapped” so that one or more beads fall inside the NP core (see SI-Sec10 for details and demonstration of the effectiveness of such a strategy). All these modified (soft-core) nonbonded potentials become the real LJ interactions as the morphing stage is concluded. The temperature of the simulated system is 300 K. For each angle θc, a total number of 5000 repeated transformations has been performed; trajectories are then divided into 50 blocks of 100 runs each (taken in the order as they are generated) for the estimation of the statistical uncertainty on the outcome.26 For angle θ1 and for θ4 to θ9, the initial value was set to θc,init = 0° corresponding to the trans conformation of the specific bond, while for angles θ2 and θ3 it has been preferred to start from the eclipsed conformation at θc,init = 180° (with subsequent rotational steering up to 540°); such a choice proved to yield better outcomes since the steering stage appears to be less “dissipative” (in the terminology of thermodynamic transformations25) yielding smaller systematic errors. The number of G


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T2 relaxation times. The key-relations are (see, for example, section II.B of ref 45 and references therein)

by inserting a confining barrier constituted by a conical surface with vertex on the NP center, such that the first carbons of all the 15 chains are enclosed in its interior (see the SI-Sec11 for details). The lateral cone and the delimited portion of NP surface are treated as a whole by applying what follows: beads interact with such a surface by taking the point of closest approach and applying the same LJ 12−3 parameters for both the cone and the NP surfaces. The resulting torsion free energy profiles are provided in the SI-Sec11, where it can be seen that the “pinning” conical barrier does not introduce significant changes. Being such a “pinning” barrier immaterial we can infer that the surrounding 14 chains already exert, from the point of view of the central chain, the genuine confinement in the NP coating. Free Energy of the Chain. The global outcome about singlechain conformational mobility may be summarized as follows: apart from the S−C1 bond, whose rotation appears “blocked”, at 300 K all the carbon−carbon bonds display a torsional energetics similar to that of liquid n-butane. From this evidence it is straightforward to adopt for the chain free-energy Achain(θ) the following model based on independent contributions Vbutane(θi)

b2 1 = NH CH [J(ωH − ωC) + 3J(ωC) + 6J(ωH + ωC)] T1 10 b2 1 = NH CH [4J(0) + J(ωH − ωC) + 3J(ωC) + 6J(ωH) T2 20 + 6J(ωH + ωC)] (4)

where the dynamics is accounted for by the spectral densities J(ω) of time-correlation function of the second-rank Legendre polynomial whose argument is the cosine of the angle between the magnetic field and the actual orientation of one of the 13 C−H bonds. In eq 4, bCH = μ0γHγCℏ/(4πr3CH) = 1.466 × 105 −1 s with μ0 the magnetic permeability in vacuum, γH and γC the gyromagnetic ratios of the nuclei, and rCH the length of the C−H bond; ωH and ωC are the Larmor frequencies of 1H and 13 C, respectively; finally, multiplications by the factor NH = 2 is required to account for the two equivalent 1H nuclei of the methylene group. Calculations have been performed with ωH = 500.13 MHz, ωC = 125.77 MHz (see section 3.1), and rCH = 1.09 Å. Computation of J(ω) is simplified by assuming a decoupling between the rotational motion of the NP and the conformational dynamics; that is, the NP tumbling does not produce a conformational change of the chains and vice versa. Accordingly, the contribution from internal dynamics is computed from a Brownian θ(t) trajectory, while the tumbling contribution is accounted for analytically once a value for the rotational diffusion coefficient Drot of the coated NP is given (see SI-Sec14 for details). Estimation of the coefficient Drot is made via the Debye− Stokes−Einstein relation

9

Achain (θ) ≡ C +

∑ Vbutane(θi) i=2

(3)

where C is a constant term at a given temperature. The S−C1 torsion angle is taken as completely frozen with the angle θ1 fixed to 0°; hence, its energetic contribution to Achain(θ) is absorbed into C. Brownian Dynamics Simulations of Single-Chain Conformational Dynamics. Conformational dynamics of the probe chain are here modeled as Brownian motion in the θ-space by adopting a Langevin equation in the so-called overdamped (or “high friction”) regime.16 Details are given in SI-Sec12. The drift force is related to the gradient of the chain free energy, Achain(θ). The viscous drag is accounted for by considering a conformation-dependent friction matrix which is computed, in the present coarse-grained framework, by employing the model of one-side tethered chain without hydrodynamic interactions.29,30 We have checked that inclusion of hydrodynamic interactions42,43 brings negligible quantitative changes, at the price of a much higher computational cost. Expressions of the friction matrix elements and values of the parameters are provided in SI-Sec13. The only mediumdependent parameter to be assigned is the effective viscosity, ηlayer. As a trial effective viscosity we used that of liquid 1decanethiol (which likely mimics the properties of the alkyl coating) at 300 K, which is 1.347 × 10−3 Pa·s.44 Simulations of Brownian trajectories have been performed with a Fortran77 code developed by us (see SI-Sec12 for some technicalities). Calculation of 13C NMR Relaxation Times and Comparison with Experiments. For a single 13C-probed nucleus, the main interactions whose time-modulation drives the longitudinal (T1) and the transverse (T2) relaxation times of the macroscopic magnetization are the dipolar interactions with the bonded 1H nuclei; modulation of the chemical shift anisotropy is negligible and has been ignored. The stochastic motions responsible for the modulation of the dipolar interactions are the rotation of the NP as a whole and the internal conformational dynamics of the alkyl chain. Analysis in the fast motional (or Redfield) limit is sufficient to estimate T1 and

Drot =

kBT 3 8πηsolv R eff

(5)

The whole NP was approximated as a sphere of effective radius Reff in the solvent of viscosity ηsolv. The effective radius was taken as Reff ≡ ⟨RC10⟩, where RC10 is the distance of C10 from the NP center, and the average is meant to be taken over the ensemble of chain conformations. We computed Reff by performing the average ⟨RC10⟩ over the conformational distribution corresponding to the model eq 3 for the chain energetics and considering the Au144 core structure. The result was Reff = 15 Å. Taking the viscosity of liquid CH2Cl2 at 300 K as 4.17 × 10−4 Pa·s,44 the result was Drot = 1.18 × 108 rad/s. Note that under the assumptions of an average spherical geometry and stick boundary conditions one finds that Dtransl = (4/3)DrotR2eff = 3.52 × 10−10 m2/s,46 a value close to the experimental one Dtransl,exp = 5.4 × 10−10 m2/s as estimated by diffusometry (see SI-Sec4). Outcomes for T1 from calculations and experiments are shown in Figure 7 (in SI-Sec15 we report the calculated T2 together with preliminary experimental outcomes). The experimental relaxation times for carbons C1 and C2, and for carbons C3 to C6, are merged into single values (see the horizontal lines) since the specific resonances cannot be disentangled. Only for carbon C1, however, has the T1 been determined by selectively enriching with isotope 13C; the value is also displayed in the figure (see the blue triangle). The T1 for H


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4. CONCLUSIONS In this work, we have presented the methodological steps to undertake the study of conformational mobility (in the twofold sense of equilibrium distribution and dynamics) of flexible coatings of gold nanoparticles. We have shown that, upon choice of suitable structural variables and likely coarse-grained representation, the free energy of a subpart of the whole system can be constructed versus some relevant coordinates. In this “pilot application”, the torsion free energy of a selected chain of the coating has been computed, bond by bond, by employing Jarzynski’s equality in a novel nonequilibrium sampling tool. By using such a static information in the construction of a stochastic model with likely parametrization of the dissipative terms (friction), we have seen that experimental 13C NMR relaxation times could be well reproduced. Moreover, the effective radius (of the reference Au144 coated NP) estimated by taking into account the resulting flexibility of the thiols, allowed the calculation of a translational diffusion coefficient in good agreement with that obtained by NMR diffusometry. In the end, the nanoparticles coating monolayer behaves as a highly “mobile” pseudophase and the thiols chains resemble a liquid phase of molecules of the same chemical nature (alkanes in this case). This may appear quite counterintuitive, but it is supported by the good agreement between calculated and experimental T1 along with the diffusion coefficient. We recall that the solvent has been here excluded in the torsion free energy computations; its inclusion in the simulation box is scheduled for future work. We shall stress that the computational/theoretical integrated approach presented in section 2, proved able to catch all features (but only the relevant ones) of the system under study, so that physical observables of the material (here, the T1 and T2 NMR relaxation times and the diffusion coefficient) could be calculated without free parameters in an ab initio fashion. Thus, the proposed computational protocol, while being quite general, showed to be predictive when tailored to the study of the dynamics of flexible coatings of gold nanoparticles when employing ad hoc theoretical tools suited for the problem. Specially, the coupling of the energy morphing strategy to the standard steering methods turned out to be efficient to construct torsion free energy profiles which may display barriers even of several kBT units. Our belief is that the capability of the proposed protocol of being predictive may become of particular help in the design of functionalized nanomaterials, especially in the fine-tuning/enhancement of those properties (e.g., the sensitivity of a nanosensor) that are somehow affected by the complex internal dynamics of the system. Such an objective could be much more efficiently pursued if, relying on the method here presented, a library of “free energy force fields” for Brownian dynamics simulations (as analogous of those existing for standard molecular dynamics calculations) are produced and made available, thus avoiding the necessity to repeat the parametrization for classes of similar systems. We conclude by stressing that the case of NPs coated with alkyl thiols is an archetype of more complex systems which can be treated with the same methodological approach outlined here. Indeed, future work will encompass more complex ligands that include functional groups, wherein the interchain noncovalent interactions are expected to strongly affect the dynamics and the relaxivity of the nuclear spins.

Figure 7. Experimental (empty squares, triangle) and calculated (filled circles) 13C NMR longitudinal relaxation times T1 for different labeled carbon atoms. The experimental T1 values for the methyl group (not reported in the figure) is 3.70(±0.55) s. Dashed lines are guides for the eye.

C10 has been not computed since its evaluation requires a separate modeling. Globally the agreement is satisfactory, especially considering that no physical parameters have been adjusted in the calculations to fit the experimental data: conformational energetics are inferred from calculated torsion free energies on the model structure, and friction is provided by the viscous drag acting on the beads (with an a priori likely parametrization of the model). Moreover, it must be stressed that the NMR technique probes the overall contribution to relaxation from carbon spins with slightly different magnetic environments both due to inhomogeneity of the synthesized NPs and to the possible subtle differentiation of the single thiol chains within the same NP. This implies that the longitudinal relaxation monitored in the NMR practice bears an intrinsic inhomogeneous contribution (i.e., a broadening in terms of spectral lineshapes) that is not accounted for in the present calculations. A comment is due about the increase of T1 from the NP surface to the periphery. This is related to the fact that methylene groups close to the surface are essentially immobile in the nanoparticle-fixed frame, and modulation of C−H dipolar interactions is due only to the tumbling motion. Moving toward the periphery, the methylene groups acquire conformational mobility which contributes (together with tumbling) to the spectral densities in eqs 4. The isotropic distribution of C− H bonds (with respect to the Laboratory Frame) is gradually more efficiently explored by the global dynamics, also taking into account that the intralayer viscous friction decreases due to the fact that a shorter outer portion of chain has to be moved. Emerging Picture of Conformational Mobility. On physical grounds, the following physical picture has emerged about the mobility of the single chain: apart from the first S−C bond, whose rotation is essentially “blocked”, the equilibrium distribution for the subsequent bonds resembles that of the central bond of n-butane in the liquid phase at the same temperature; moreover, the emerged conformational dynamics is simply a Brownian motion of an ideal chain made of connected butane-like bonds, tethered to one end, and moving in a homogeneous medium with the viscosity of layer. I


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(9) Luedtke, W. D.; Landman, U. Structure and thermodynamics of self-assembled monolayers on gold nanocrystallites. J. Phys. Chem. B 1998, 102, 6566−6572. (10) 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, 15857− 15862. (11) Tian, P. Molecular Dynamics simulation of nanoparticles. Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. 2008, 104, 142−164. (12) de Moura, A. F.; Bernardino, K.; Dalmaschio, C. J.; Leite, E. R.; Kotov, N. A. Thermodynamic insights into the self-assembly of capped nanoparticles using molecular dynamic simulations. Phys. Chem. Chem. Phys. 2015, 17, 3820−3831. (13) Heikkilä, E.; Gurtovenko, A. A.; Martinez-Seara, H.; Häkkinen, H.; Vattulainen, I.; Akola, J. Atomistic simulations of functional Au144(SR)60 gold nanoparticles in aqueous environment. J. Phys. Chem. C 2012, 116, 9805−9815. (14) Van Lehn, R. C.; Alexander-Katz, A. Structure and dynamics of mixed-monolayer-protected nanoparticles in aqueous salt solution from atomistic molecular dynamics simulations. J. Phys. Chem. C 2013, 117, 20104−20115. (15) Heikkilä, E.; Martinez-Seara, H.; Gurtovenko, A. A.; Javanainen, M.; Häkkinen, H.; Vattulainen, I.; Akola, J. Cationic Au nanoparticle binding with plasma membrane-like lipid bilayers: potential mechanism for spontaneous permeation to cells revealed by atomistic simulations. J. Phys. Chem. C 2014, 118, 11131−11141. (16) Zwanzig, R. Nonequilibrium Statistical Mechanics; Oxford University Press: New York, 2001. (17) Almusallam, A. S.; Sholl, D. S. Brownian Dynamics study of polymer-stabilized nanoparticles. Nanotechnology 2005, 16, S409− S415. (18) Groot, R. D.; Warren, P. B. Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 1997, 107, 4423−4425. (19) Hu, S.; Sheng, Y.; Tsao, H. Self-Assembly of organophilic nanoparticles in a polymer matrix: depletion interactions. J. Phys. Chem. C 2012, 116, 1789−1797. (20) Djohari, H.; Dormidontova, E. E. Kinetics of nanoparticles targeting by Dissipative Particle Dynamics simulations. Biomacromolecules 2009, 10, 3089−3097. (21) Ding, H.; Tian, W.; Ma, Y. Designing nanoparticle translocation through membranes by computer simulations. ACS Nano 2012, 6, 1230−1238. (22) Masoud, H.; Alexeev, A. Controlled release of nanoparticles and macromolecules from responsive microgel capsules. ACS Nano 2012, 6, 212−219. (23) Concerning requisite (2), pure geometrical variables are insufficient to describe very rapid librational motions with correlation times on the order of picoseconds; that would be probed by fast techniques like Incoherent Quasi-Elastic Neutron Scattering. See, for example: Leadbetter, A. J.; Richardson, R. M. The Molecular Physics of Liquid Crystals; Gray, G. W., Luckhurst, G. R., Eds.; Academic Press: London, 1979. In such a time window, the dynamics would partially retain inertial components and a set of conjugate momenta have to be included amongst the relevant variables. (24) Huang, K. Statistical Mechanics; John Wiley & Sons: Singapore, 1987. (25) Jarzynski, C. Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 5018− 5035. (26) Zerbetto, M.; Piserchia, A.; Frezzato, D. Looking for some free energy? Call JEFREE (...). J. Comput. Chem. 2014, 35, 1865−1881. (27) Library JEFREE is freely distributed under the GPL v2.0 license. See ref 26 for the web repository of software and related documentation. (28) Piserchia, A.; Zerbetto, M.; Frezzato, D. Probing the conformational energetics of alkyl thiols on gold surfaces by means

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b04884. Experimental procedures; synthesis of 13C-labeled thiols; synthesis and characterization of monolayer protected gold nanoparticles; additional NMR experiments; geometrical parameters; molecular frames; Lennard-Jones interactions and employed parameters; Ryckaert−Bellemans torsion potential and employed parameters; nonequilibrium strategy based on Jarzynski’s equality with initial internal energy morphing; energy penalty to expel “wrapped chains” from the metallic core during the morphing stage; outcomes with conical confinement; employed Langevin equation and simulation details; conformation-dependent friction matrix and employed parameters; 13C NMR relaxation: frame-transformations and time-correlation functions; experimental and calculated 13C NMR T2 relaxation times (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. (F.R.) *E-mail: [email protected]. (D.F.) Present Address †

Scuola Normale Superiore, piazza dei Cavalieri 7, I-56126 Pisa, Italy. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. H. Häkkinen (University of Jyväskylä, Finland) for the reference structure of the coated gold nanoparticle. All simulations have been run on the C3P (“Centro di Chimica Computazionale di Padova”) HPC facility of the Dipartimento di Scienze Chimiche of the Università degli Studi di Padova. This work was funded by Starting Grant 2010 MOSAIC (259014) granted to F.M. by ERC and by FIRB 2012 granted to M.Z. by MIUR.

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REFERENCES

(1) Giljohann, D. A.; Seferos, D. S.; Daniel, W. L.; Massich, M. D.; Patel, P. C.; Mirkin, C. A. Gold nanoparticles for biology and medicine. Angew. Chem., Int. Ed. 2010, 49, 3280−3294. (2) Bonomi, R.; Scrimin, P.; Mancin, F. Phosphate diesters cleavage mediated by Ce(IV) complexes self-assembled on gold particles. Org. Biomol. Chem. 2010, 8, 2622−2626. (3) Daniel, M.-C.; Astruc, D. Gold nanoparticles: assembly, supramolecular chemistry, quantum-size-related properties, and applications toward biology, catalysis, and nanotechnology. Chem. Rev. 2004, 104, 293−346. (4) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover Publications, Inc.: New York, 1984. (5) Kolomeisky, A. B.; Fisher, M. E. Molecular motors: a theorist’s perspective. Annu. Rev. Phys. Chem. 2007, 58, 675−695. (6) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987. (7) Frenkel, D.; Smit, B. Understanding Molecular Simulations − From Algorithms to Applications, 2nd ed.; Academic Press: New York, 2002. (8) Luedtke, W. D.; Landman, U. Structure, dynamics, and thermodynamics of passivated gold nanocristallites and their assemblies. J. Phys. Chem. 1996, 100, 13323−13329. J


Article

The Journal of Physical Chemistry C of a morphing/steering non-equilibrium tool. Phys. Chem. Chem. Phys. 2015, 17, 8038−8052. (29) Ferrarini, A.; Moro, G.; Nordio, P. L. Conformational kinetics of aliphatic tails. Mol. Phys. 1988, 63, 225−247. (30) Moro, G. J.; Ferrarini, A.; Polimeno, A.; Nordio, P. L. Models of conformational dynamics,107−139. Reactive and Flexible Molecules in Liquids; Kluwer Academinc Plublishers: Dordrecht, 1989. (31) Kowalewski, J.; Mäler, L. Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications; CRC Press: Boca Raton, 2006. (32) Manea, F.; Bindoli, C.; Polizzi, S.; Lay, L.; Scrimin, P. Expeditious synthesis of water-soluble, monolayer-protected gold nanoparticles of controlled size and monolayer composition. Langmuir 2008, 24, 4120−4124. (33) Canet, D.; Levy, G. C.; Peat, I. R. Time saving in 13C spin-lattice relaxation measurements by inversion-recovery. J. Magn. Reson. 1975, 18, 199−204. (34) Häkkinen, H. The gold-sulfur interface at the nanoscale. Nat. Chem. 2012, 4, 443−455. (35) Lopez-Acevedo, O.; Akola, J.; Whetten, R. L.; Grönbeck, H.; Häkkinen, H. Structure and bonding in the ubiquitous icosahedral metallic gold cluster Au144(SR)60. J. Phys. Chem. C 2009, 113, 5035− 5038. (36) Häkkinen, H.; Walter, M.; Grönbeck, H. Divide and protect: capping gold nanoclusters with molecular gold-thiolate rings. J. Phys. Chem. B 2006, 110, 9927−9931. (37) Hautman, J.; Klein, M. L. Simulation of a monolayer of alkyl thiol chains. J. Chem. Phys. 1989, 91, 4994−5001. (38) Ryckaert, J. P.; Bellemans, A. Molecular dynamics of liquid alkanes. Faraday Discuss. Chem. Soc. 1978, 66, 95−106. (39) Zerbetto, M.; Frezzato, D. Towards bulk thermodynamics via non-equilibrium methods: gaseous methane as a case study. Phys. Chem. Chem. Phys. 2015, 17, 1966−1979. (40) Steinbrecher, T.; Mobley, D. L.; Case, D. A. Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations. J. Chem. Phys. 2007, 127, 214108. (41) Calculations have been performed on the Cluster “Avogadro” of the C3P facility. Main characteristics of the cluster are: 71 nodes (284 CPU), 4 processors per node with 2.6 GHz and 8.0 Gb RAM each. Generation of the torsion free energy profiles for angles θ2 and θ3 required 20 days (parallelizing over 32 processors), while those for angles θ1 and θ4 to θ9 required 7 days (same operative conditions). (42) Nigro, B.; Di Stefano, D.; Rassu, A.; Moro, G. J. Cranklike conformational transitions in polyethylene. J. Chem. Phys. 2004, 121, 4364−4376. (43) Rotne, J.; Prager, S. Variational treatment of hydrofobic interaction in polymers. J. Chem. Phys. 1969, 50, 4831−4837. (44) Yaws, C. L. Handbook of Viscosity, Vol. 1; Gulf Publishers: Houston, 1995. (45) Kotsyubynskyy, D.; Zerbetto, M.; Soltesova, M.; Engström, O.; Pendrill, R.; Kowalewski, J.; Widmalm, G.; Polimeno, A. Stochastic modeling of flexible biomolecules applied to NMR relaxation. 2. Interpretation of complex dynamics in linear oligosaccharides. J. Phys. Chem. B 2012, 116, 14541−14555. (46) We have also calculated both the average and the mode of the distribution of the Dtransl values as obtained by the Gaussian distribution of the core diameters reported in Figure S1 of the Supporting Information, assuming the same average contribution to Reff from decanethiol chains. The two outcomes were similar and close to 3.0 × 10−10 m2/s.

K


Conformational Mobility in Monolayer-Protected Nanoparticles: From

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