Soft Hedgehogs on Coarse Carpets: A Molecular Simulation Study of

Soft Hedgehogs on Coarse Carpets: A Molecular Simulation Study of Capped Nanocrystals Interacting with Self-Assembled Monolayers of Alkylthiols on a G...
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J. Phys. Chem. C 2010, 114, 2531–2537

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Soft Hedgehogs on Coarse Carpets: A Molecular Simulation Study of Capped Nanocrystals Interacting with Self-Assembled Monolayers of Alkylthiols on a Gold (111) Surface Philipp Schapotschnikow and Thijs J. H. Vlugt* Process Energy Laboratory, Delft UniVersity of Technology, Leeghwaterstraat 44, 2628CA Delft, The Netherlands ReceiVed: NoVember 5, 2009; ReVised Manuscript ReceiVed: January 11, 2010

We report atomistic simulations of alkylthiol capped gold nanocrystals (NCs) adsorbed on a gold (111) surface that is covered with a self-assembled monolayer (SAM) of alkylthiols. For these systems, we have computed the potential of mean force as a function of NC-surface distance for alkylthiols of varying tail length (butanethiol-dodecanethiol), temperature (250-400 K), and NC size (1.8-2.7 nm). The length of the thiol in the SAM determines alone the equilibrium distance between the NC and the surface. The equilibrium distance does not depend on the length of the capping molecule on a NC. We explain this finding by the efficient packing of capping molecules on top of the SAM. Conversely, the depth of the potential well only depends on the length of the capping molecule on the NC, and this dependence is linear. The well depth does not depend on the length of the thiol in the SAM. The structure of the SAM and of the capping layer on the NC is described using appropriate order parameters. We also study the behavior of systems where either the capping layer of the NC or the SAM on a flat surface is absent. Unlike capped NCs, bare NCs diffuse a few angstroms into the SAM. When a SAM is not present, the capping layer does not protect a NC from sintering with the surface. 1. Introduction Nanoscale objects are promising building blocks for future materials and devices. Interactions between these microscopic objects determine the thermodynamic stability and mechanical robustness of assemblies. Therefore, knowledge of effective interactions between various nanosized building blocks is crucial for a rational design of composite structures with desired properties. Gold- and semiconductor nanocrystals (NCs) with specific size- and shape-dependent optical and electrical behavior constitute an important class of such nanosized building blocks; they are gaining increasing importance in the fields of optics, electronics, catalysis, magnetic storage, and biophysics.1 NCs are often protected by an organic capping layer of surfactants that prevents aggregation;2 e.g., gold NCs are often capped with alkylthiol molecules.3-6 On planar gold surfaces, alkylthiols are known to form well organized structures called self-assembled monolayers (SAMs). These structures are promising for various applications such as controlled wetting, catalysis, sensoring, and biotechnology.7,8 The gold-thiol interaction is very strong, preventing the surfactants from evaporating or dissolving. Thiol headgroups form a 2D hexagonal overlattice on the flat Au(111) surface with S-S spacing of ≈5 Å.7 The aliphatic tails align parallel to each other making a well-defined angle of 20-30° with the surface normal.7 Due to the high local density and homogeneity, SAMs provide the unique possibility for noncovalent reversible adsorption of nanoparticles to an electrode with a controllable distance to surface. Therefore, SAMs are used as ultrathin insulators between a gold electrode and adsorbed NCs in electron microscopy studies.9,10 Classical atomistic simulations provide a particularly convenient tool for studying nanoscale systems as all thermodynami* To whom correspondence should be addressed. E-mail: t.j.h.vlugt@ tudelft.nl. Tel.: +31-15-2787551. Fax: +31-15-2782460.

cally relevant microscopic details can be accounted for explicitly. Early simulation studies of SAMs on Au(111) have shown that their structure is a result of interactions between aliphatic tails and does not depend on the details of the Au-S bond.11 Finite-size and boundary effects in SAMs were quantified in refs 12 and 13 by studying microscopic islands consisting of a few hundred alkylthiol molecules. Monolayers composed of a binary mixture of alkylthiols with different tail lengths were investigated in refs 14-17. A tendency for segregation was found in ref 14. Preferential adsorption toward the longer thiol was found in refs 15-17, and this preferential adsorption is much stronger in vacuum than in an explicit solvent. Phase transitions during the formation of SAMs at different conditions were characterized in ref 17. The structure and thermodynamics of alkylthiol capping layers on gold NCs were investigated in refs 16-18. Compared to SAMs on extended Au(111) surfaces, significant structural differences were found due to the large surface curvature of the tiny NCs. For instance, adsorption selectivity from a binary alkylthiol mixture in hexane is almost absent on a NC, while it is pronounced on the flat surface.16,17 We are aware of only two earlier atomistic simulation studies of capped nanocrystals at interfaces. Luedtke and Landman studied the diffusion of a capped gold NC on a graphite surface;19 a relatively high mobility was found due to the rolling motion. Tay and Bresme characterized the structure of the capping layer and the solvent for a capped gold NC at the air-water interface.20 The aim of the present work is to compute the free energy, or equivalently, the potential of mean force (PMF), as a function of separation between a capped NC and a flat Au surface covered with a SAM. We systematically investigate the dependence of the PMF on the following parameters: length of the alkyl tail both in the SAM and in the capping layer of a NC, temperature, and NC size. In this manner, a broad range of the parameter space is covered. The knowledge of free energy

10.1021/jp910554e  2010 American Chemical Society Published on Web 01/27/2010

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is of vital importance for the understanding of equilibrium and/ or kinetic system properties of the system. Particularly important are the location and the magnitude of the minimum of the PMF. The former determines the equilibrium geometry at the given thermodynamic conditions, while the latter quantifies the stability and robustness of the system. Based on our simulation results, we formulate the equilibrium distance and the well depth as a function of SAM height and capping length. To show the importance of the SAM and of the capping layer on the NC, systems with a bare NC or with a bare Au(111) surface are studied. We consider here only systems in vacuum. This corresponds to the experimental situation when the solvent is already evaporated. In a previous study, we systematically investigated the PMF between capped gold NCs.21,22 Against expectation, the equilibrium distance for a pair of capped NCs hardly changes with increasing alkyl tail length, and it is always at 1.25 times the core diameter. This result can be explained by a ligand packing model from ref 22. In large NC aggregates, however, many-body interaction may frustrate this universal scaling. We were able to make predictions for which ligand capped NCs self-assemble into highly stable three-dimensional structures, and for which they form high-quality monolayers. The remainder of the paper is structured as follows. Section 2 contains a description of the model and methods used in this work. In Section 3, we present the computed PMFs as a function of various parameters: tail length of the alkylthiol in the SAM and in the capping layer of a NC, temperature, and NC size. Moreover, we investigate the role of the SAM on the flat surface and of the capping layer on a NC. We also give a quantitative description of the structure of the systems. We will show that the length of the thiol in the SAM alone determines the equilibrium distance of the NC to the Au(111) surface, while the strength of the interaction depends only on the length of the capping molecules on the NC. An explanation for these trends is given in Section 4. Section 5 summarizes the main results. 2. Model and Methods 2.1. Model. In all simulations, we apply the united atom model for SH, CH2, and CH3 groups. Alkylthiols are labeled as “SCn”, where n is the number of alkyl chain segments in the linear tail. Beads of different surfactant molecules interact with each other and with gold atoms via truncated and shifted Lennard-Jones (LJ) pair interactions with a cutoff radius of 12 Å. It is important to note that the Au-S interaction is much stronger than other nonbonded interactions. We account for intramolecular bond stretching, bond bending, and torsional interactions.23,24 Additionally, we apply intramolecular LJ interaction between segments that are separated by more than three bonds. The interaction potentials and parameters are summarized in the Supporting Information. To keep computational efforts to acceptable levels, the gold structures are considered as rigid. The gold slab consists of NAu,x × NAu,y × NAu,z ) 36 × 36 × 6 atoms in the face-centered cubic crystal structure with a lattice constant of 4.085 Å.25 The NC cores are modeled as rigid, close-packed icosahedra exposing only (111) facets,26-28 with a Au-Au distance of 2.88 Å. The effective van der Waals interactions between the NC and the slab are modeled via the Hamaker potential.29 As we only consider systems in vacuum, we did not include a solvent in our simulations. 2.2. Methods. Our aim is to determine the free energy, or equivalently, the potential of mean force (PMF), as a function of NC-slab separation. We use constraint molecular dynamics

Schapotschnikow and Vlugt (MD) for the computation of the PMF. For systems with capped NCs, this technique turned out to be more appropriate than unconstrained methods.21 Consider a slab in the (x, y) plane and a NC at a fixed distance r between its center of mass (c.o.m.) and the slab surface. In this case, the mean force Fmean is defined as the average force in the direction of the z axis30,31

1 Fmean(r) ) 〈FzSl - FzNC〉NVT;r 2

(1)

NC where FSl z and Fz are the vertical components of the total force acting on the slab and NC core, respectively. Angular brackets denote ensemble averages in the canonical ensemble with the constraint slab-NC separation r. The potential of mean force is defined as

φMF(r) )

∫r∞ Fmean(s) ds

(2)

The mean force is computed according to eq 1 using constraint MD simulations in the canonical ensemble. Equation 2 is then used to calculate the PMF. The MD simulations are performed using the velocity Verlet (VV) algorithm with the time step of 2 fs.32,33 The temperature is kept constant using the Andersen thermostat.34 The mean force is computed from a 10 ns long MD simulation of an equilibrated system. Unless stated differently, the temperature is T ) 300 K. The gold slab and the NC c.o.m. are fixed during the simulations; the rotation of the rigid NC about its c.o.m. is realized using quaternion rigid body dynamics.35 We have verified that the horizontal component FNC xy of the average force is statistically negligible. This means that, for our systems, the PMF is isotropic in the (x,y) direction, so that it is not necessary to allow the NC to move in this plane. The simulations are performed in a rectangular simulation box in which a gold slab is oriented parallel to the (x,y) plane. Periodic boundary conditions are applied. The box size in the z direction is chosen sufficiently large so that the capped NC does not interact with the periodic image of the slab. 2.3. Sample Preparation. The initial configurations for the constraint MD simulations were generated as follows. First, we prepare a SAM on only one side of the Au(111) slab. The latter is placed at the bottom of the simulation box, and we switch off the periodic boundary conditions in the z direction. Using (semi)grand-canonical Monte Carlo simulations described in ref 17, the desired SAMs are then formed on the upper side of the slab. The equilibrated capped NCs are generated using a protocol from our previous work,21,22 which is summarized in the Supporting Information. The maximum coverage for Au147 and Au561 with 1.8 (3.6) nm diameter, respectively, was found to be 58 (136) alkylthiol capping molecules, respectively. This is in good agreement with experiment.36 Second, we place the capped NC well above the SAM and slowly push it toward the SAM using an artificial spring in a MD simulation until the desired NC-slab separation is reached. The position of the slab remains fixed. Finally, we constrain also the NC c.o.m. and equilibrate the configuration for 500 ps. For a system consisting of a Au(111) slab covered with a SCx SAM and of a AuN NC capped by SCy, we write in our compact notation Au(111)-(SCx)/AuN(SCy), always assuming the maximum coverage both on the slab and on the NC. 3. Results In this section, we present the computed effective NC-slab interactions. The initial configurations were prepared using the

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Figure 1. Snapshots of capped NCs adsorbed to SAMs on Au(111). Gold atoms are represented by large orange spheres; thiol headgroups by small yellow spheres. Carbon atoms in the tails of molecules in the SAM and NC capping layer are represented by small blue and green spheres, respectively. The distance between the NC and the slab corresponds to the equilibrium distance. (a) Au(111)-(SC4)/Au147 (SC8) system. (b) Au(111)-(SC8)/Au147(SC6) system.

Figure 2. Mean force and the potential of mean force as a function of the distance between the center of mass of a Au147 NC and the surface of a Au(111) slab at T ) 300 K. Individual values for the mean force Fmean (symbols) are obtained from separate constrained MD simulations. Integration of Fmean according to eq 2 yields the PMF φMF. The error bars in the mean force data are smaller than the symbol size. The NC is capped by SC8, and the slab is covered with a SC4 SAM. The vertical dashed line denotes the SAM height plus NC radius; i.e., to the left of this line the NC core is immersed into the monolayer. This distance is referred to as the contact distance RC in the text.

procedure described in Section 2.3. Typical snapshots of equilibrated systems are shown in Figure 1; the slab-NC separation corresponds in both cases to the equilibrium distance (17 and 22 Å, respectively). The thiol headgroups are adsorbed to gold structures via the strong Au-S interaction, and they form a hexagonal overlattice on the slab surface. The short carbon tails of SC4 do not show orientational order at T ) 300 K; see Figure 1a. The tails of longer thiols (SC8 and SC12) align parallel to each other and form a well-defined angle with the surface normal of 22° and 26°, respectively; see also Figure 1a. These properties of SAMs on extended Au(111) surfaces are well-known from experimental7,8,37 and simulation11,17 work. It is interesting to note that the structure of the SAM mostly remains intact upon NC adsorption. On the NC, the aliphatic tails form a soft layer around the NC core. This capping layer on the NC is “smeared out” on top of the SAM; molecules adsorbed on the NC hardly penetrate into the SAM. We will come back to this issue later when we describe the local structure quantitatively. Figure 2 shows the potential of mean force between a Au(111) slab covered with a SC4 SAM and a SC8 capped Au147 NC (shorthand: Au(111)-(SC4)/Au147(SC8) system). The PMF is

repulsive at short separations, followed by a very deep well of ≈-75kBT.38 The mean force is nearly constant upon approach to the equilibrium distance (19 Å e r e 27 Å), resulting in a linear attractive part of the PMF. We have observed similar behavior for capped nanocrystals,21,22 and we are not aware of a theoretical explanation for this. The PMF smoothly approaches zero at r ) 29 Å. Note that the contribution of the Hamaker interaction to the total effective interaction is negligible;39 the PMF is dominated by interactions of the surfactant molecules with each other and with gold structures. It is well-known for capped Au NCs that the core-core interaction is very weak compared to the total effective interaction between capping layers.19,21 It is useful to define the contact distance RC as the height of the SAM (defined as the average height of the final segment plus its van der Waals radius) and the NC radius. For the present system, RC ) 9 + 8.7 ) 16.7 (Å) is shown by the vertical dashed line in Figure 2. By definition, the surface of a NC core lies on the level of the SAM if the separation between the slab and the NC c.o.m. is equal to RC. Note that the equilibrium distance in Figure 1 is just slightly larger than RC. In a practical situation, a NC-slab separation close to RC may reveal the false impression that the NC capping is not intact. We computed the PMF for 15 different systems: Au(111)-SCn/ Au147(SCm) with n ) 4, 8, 12 and m ) 4, 6, 8, 10, 12; see Figure 3. The shape of the PMFs is in all cases similar to the one in Figure 2. It is useful to plot the PMF both as a function of NC-slab separation (Figure 3a) and as a function of the NC-SAM separation (Figure 3b). The latter distance is defined as NC-slab separation minus the SAM height. Two trends are important to note. First, let us consider the lines with the same style in Figure 3a. Note that we are now comparing systems with the same SAM height but different capping molecules on the NC. In all cases, the equilibrium distance is close to the corresponding contact distance RC and hardly depends on the length of the capping molecule on the NC. The variation in the equilibrium distance for different NC capping (but the same SAM) is ≈2 Å, whereas the head-to-tail length of the capping molecules varies by 10 Å from SC4 to SC12. The fact that the equilibrium distance is very close to RC is reflected in the snapshots in Figure 1, where the long molecules in the NC capping layer are smeared out on top of the SAM. We have already observed such ligand length independent equilibrium distance for the PMF between capped NCs.21,22 Second, consider the triplets of lines with same color in Figure 3. In this case, we are comparing systems with the same NC capping and with

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Figure 3. (a) Potential of mean force as a function of the distance between the center of mass of a Au147 NC and the surface of a Au(111) slab. The vertical lines denote the corresponding RC (SAM height plus NC radius). (b) Same as in (a), but the PMF is plotted as a function of the distance between the center of mass of a Au147 NC and the SAM. The legend in both figures is the same. The molecule type listed in the left column in the legend is the alkylthiol in the SAM; the one listed in the right column is the capping molecule on the NC. Different colors encode different capping molecules on the NC (SC4-SC12); different line styles encode different alkylthiols in the SAM (SC4, SC8, or SC12).

Figure 4. Well depth of the PMF Umin as a function of the alkyl tail length n in the NC capping molecule SCn. The symbols are the data from Figure 3. The triplets of values of Umin for each n correspond to the three different SAM heights (SC4, SC8, and SC12). The solid line is a linear fit Umin/kBT ) -9.34n.

different SAM heights. In Figure 3b, the lines with same color almost fall on top of each other for distances larger than the equilibrium distance. This means that the interaction strength is determined by the size of capping molecules on a NC, but not by the molecules in the SAM. In particular, the potential well depth Umin only depends on the capping molecule on the NC. We plot Umin versus the ligand tail length (n in SCn) in Figure 4. The three values of Umin for each n in this figure correspond to the three different SAM heights studied for each n. Note that these triplets lie in all cases very close together. Obviously, Umin is proportional to the length of the alkyl tail in the NC capping molecule. This scaling is different from the NC-NC case, where a quadratic relation was found.22 We studied the influence of the temperature on the PMF; in Figure 5, the PMF for the Au(111)-SC8/Au147(SC4) system is shown for temperatures in the range 250-400 K. With increasing temperature, the total interaction becomes less attractive, and the equilibrium distance becomes larger. We also computed the PMF for a system with a larger NC: Au(111)-SC8/ Au561(SC6). To determine the effect of NC size, it is compared to the PMF in a system with the same thiol lengths but smaller NC (Au147); see Figure 6. The potential well for the larger NC is 30% deeper, the repulsive part is steeper, and the PMF is more narrow. The equilibrium distance is again within 2 Å from the corresponding contact distance.

Figure 5. Potential of mean force for the Au(111)-SC8/Au147(SC6) system at different temperatures. All data are divided by kB × 300 K for a fair comparison.

Figure 6. Potential of mean force as a function of the distance between the center of mass of a NC and the surface of a Au(111) slab for two different NC sizes but the same capping molecule and SAM height: Au(111)-SC8/Au561(SC6) and Au(111)-SC8/Au147(SC6). Vertical lines denote the contact distances corresponding to the two different NC sizes.

We have also simulated systems without alkylthiols on one of the gold structures: (1) Au(111)/Au147 (SC4) and (2) Au(111)-(SC8)/Au147 and Au(111)-(SC8)/Au561. The first system, where the Au(111) surface is not covered by a SAM, is unstable at short NC-slab separations. We carried out

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Figure 7. Potential of mean force as a function of the distance between the center of mass of bare NCs of two different sizes and the surface of a Au(111) slab. The vertical lines denote the corresponding RC (SAM height plus NC radius).

unconstraint MD simulation with the initial NC-slab distance of 25 Å. The separation decreased rapidly during the simulation, and after 500 ps the NC core was in direct contact with the slab; a large number of capping molecules jumped from the NC to the slab. In practice, if a gold NC core comes into direct contact with another bare gold structure at ambient conditions, then these two objects start sintering to reduce the surface energy. This means that the capping layer on a NC does not protect it against sintering with a bare Au surface. Accordingly, the computed PMF is strongly attractive without a short-range repulsion (not shown). By contrast, systems with a bare NC adsorbed to a SAM are at least metastable. We have computed the PMF for the Au(111)-SC8/ Au147 and Au(111)-SC8/ Au561 systems; see Figure 7. The shape of these PMFs is different from the one in Figure 2: they have multiple wells to the left of the corresponding contact distance. Thus, uncapped NCs are partially immersed into the SAM upon adsorption. Unlike systems with a capped NC, a bare 1.8 nm (or 2.7 nm) NC core diffuses a few angstroms into the SAM. We carried out an unconstraint MD simulation of 5 ns, which confirmed this behavior. The system remained stable during the simulation time, with the NC slightly immersed into the SAM as expected from the PMF. Note that in the case of the SC8 SAM considered here, the NC core is not expected to come close to the slab surface. To characterize the structure of the capping layer of a NC adsorbed to a SAM, we computed the average segment density along the vertical direction. Figure 8 shows such a density profile for the Au(111)-SC8/Au147(SC6) system at the equilibrium distance (a snapshot of this this system is shown in Figure 1b). One can clearly recognize an interface between the SAM and the capping layer of the NC around z ) 15 Å, where both densities sharply decrease. The reference line in Figure 8 shows the density profile for an isolated capped NC, i.e., when the SAM covered gold slab is absent. Comparing the density profile of the capping layer of the NC with this reference line around the interface, we can see that very few capping molecules penetrate into the SAM. The peak at z ) 18 Å, to the right of the SAM capping layer interface, indicates that tails of many capping molecules are adsorbed directly on top of the SAM, which is consistent with what we see in the snapshots of Figure 1. In Figure 1b, the tail of one single molecule of the NC capping layer lies inside the SAM; the rest of the capping layer lies on top. The segment density profile of the SAM does not change

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Figure 8. Vertical segment density profile Fs(z) for the Au(111)-SC8/ Au147(SC6) system at the equilibrium distance. The densities of the molecules in the SAM and in the capping layer are plotted separately. The reference line represents Fs(z) of an isolated NC that does not feel the presence of a SAM. Note that Fs(z) in this plot is subject to averaging in the (x,y) direction. Therefore, the absolute values for the capping layer depend on the size of the simulation box in the (x,y) direction. Due to this observation, absolute values (and thus units) for Fs(z) of the capping layer are meaningless. It is interesting, however, to compare Fs(z) of the capping layer with the reference line; see the main text.

Figure 9. Plot of the average angle R with the surface normal for thiol molecules in the SAM as a function of the horizontal distance R to the NC c.o.m. The system is Au(111)-SC8/Au147(SC6) at the equilibrium distance (see Figure 1b). The inset clarifies the definition of R and R; see the main text.

notably upon NC adsorption; its individual peaks and wells in Figure 8 are structural properties of the SAM itself. The imprint of a capped NC on a SAM can be quantified by the following orientational order parameter. Consider the average angle R of an alkylthiol in the SAM with the surface normal as a function of the horizontal distance R of this molecule to the NC; see the inset of Figure 9. The plot of this order parameter for the system from Figure 1b can be found in Figure 9. When the NC is very far away, the value of R is constant and equals Req ) 21.9° for SC8 at T ) 300 K. The molecules close to the NC are, as expected, pushed down, so that the angle R increases up to 30°. But the molecules which are further away and not in direct contact with the capped NC form a 1-2° smaller angle than Req. This means that they stand more upright, which is due to the attraction by the capped NC. The range of this frustration is given not by the interaction range of the capped NC with individual molecules, but rather by the orientational correlation within the SAM. For large distances R, the value of R eventually converges toward Req.

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4. Discussion In section 3, we observed two trends for the PMF between a capped NC and a SAM covered gold slab. First, the equilibrium distance between the NC c.o.m. and the Au(111) surface is determined by the SAM height only, and it lies always within 2 Å from the contact distance RC. Second, the value Umin of the PMF well depth does not depend on the SAM height, and it scales linearly with the capping length on the NC. These rather counterintuitive results can be explained using the understanding of NC-NC interactions, which we developed in our earlier work.21,22 We consider the PMF as a sum of effective attractive and repulsive interactions. We will explain that the equilibrium NC-slab distance can bee seen as a sum of four terms: (1) NC radius, (2) SAM height, (3) the maximum smearing out of the capping layer, and (4) the indentation of the SAM upon adsorption of the NC. The first two terms sum up to RC. The third term will be estimated using a chain packing model. The fourth term is small and negative. We begin with discussing the attractive part of the PMF, and in particular why the potential well depth mainly depends on the length of molecules in the NC capping layer. Recall that the PMF is determined mainly by interactions between the capping molecules and the molecules in the SAM. The capping layer of a NC “feels” the SAM as a hydrocarbon slab with certain density that does not depend on the length of the thiol in the SAM. This explains why the magnitude of the attractive force is a function of the separation between the SAM and the NC; see Figure 3b. The linear scaling of the PMF well depth (see Figure 4) can be explained by the same arguments: the number of interaction sites inside the capping layer grows linearly with the length of the capping molecule. In summary, when a capped NC is not too close to the SAM, the PMF is a result of the attractive interaction between the NC capping layer and an effective hydrocarbon slab, so that the PMF only depends on the NC-SAM distance. We continue with a qualitative discussion of the repulsive part of the PMF. As we have seen in the previous section, the molecules of the capping layer hardly penetrate into the SAM. Therefore, the repulsive interaction may be split into two terms: the deformation of the SAM (indentation) and the deformation of the capping layer (of the NC). Each of these deformations is associated with an energy penalty, which is mainly caused by a change in local segment density. The local density inside a SAM is almost the maximum possible for a hydrocarbon at ambient pressure. Thus, already a small indentation is energetically expensive. This explains why frustrations of the structure of SAMs upon NC adsorption are very small. In contrast with the SAM, the local density inside the capping layer is not homogeneous. Instead, it decays quadratically with the distance from the NC center. As the attractive part of the PMF is very strong, we only consider the deformations which yield a comparably strong repulsion. This is the case when the segment density locally exceeds a certain threshold (which can be approximated by the SAM density). If this does not occur, then the elastic response of the capping layer is mainly due to the reduced entropy of some capping molecules. For the relatively small molecules considered in this work, such response may be expected to be small. By contrast, once the local density reaches the threshold we obtain a situation similar to the indentation of a SAM. To predict when this crossover occurs, we consider the density in the contact region between the SAM and the capping layer. Our exact geometrical definition of the contact region is sketched in Figure 10; in analogy with our previous work we dub this

Figure 10. Schematic representation of the overlap cone model for a NC adsorbed to a Au(111) slab covered by a SAM. The slab is represented by the solid horizontal line; the dashed horizontal line represents the upper SAM surface. The NC with diameter dc ) 2rc is shown as a circle; its capping layer is represented by a light gray ring. The thickness of this ring corresponds to the capping length L. The NC-SAM distance is denoted by z; it is identical to rNCTSAM in Figure 3b. The NC-slab separation rNCTslab used other figures is shown to clarify the difference. The overlap cone is shown in dark gray color. Only the capping molecules inside the overlap cone and in the SAM are shown. The thiol headgroups are represented by small black circles; alkyl tails are represented by zigzag lines.

region the oVerlap cone.22 Next we make the following assumptions: (1) The density inside the overlap cone is homogeneous; (2) The volume of the overlap cone is only occupied by the capping molecules whose headgroups are adsorbed inside the overlap cone (and vice versa). Using the exact geometric definition in Figure 10, we can formulate the volume Vlig required by the ligands adsorbed inside the overlap cone and the available volume Vcone as a function of NC core diameter dc, ligand length L, and NC-SAM separation z. The crossover from “weak” to “strong” elastic response occurs then when the two volumes are equal:

Vlig(L, z;dc) ) Vcone(L, z;dc)

(3)

This equation defines (implicitly) the critical separation zcrit(L, dc) as a function of capping length L for a fixed NC size. The equilibrium NC-SAM separation is then close to zcrit. The exact solution of eq 3 for the setup of Figure 10 is given in the Supporting Information. Since the NCs are fully capped in the simulations presented in Figure 3, we consider maximum coverage in the OCM. For all thiols (SC4-SC12) and for the Au147 NC considered here (dc ) 18 Å), the critical separation lies in the narrow range 10 Å < zcrit < 11 Å; see Figure S2. This in excellent agreement with Figure 3b, where the equilibrium distance is always between 9 and 11 Å. Therefore, efficient packing of capping molecules on top of the SAM explains why the equilibrium distance mainly depends on the SAM height but hardly on the length of NC capping molecules. It is important to note that we only consider systems in vacuum in the present work. The effective interactions in systems with solvent are very different. We expect that in an apolar solvent the attractive part of the PMF will (at least partly) diminish due to the good solvent effect, as it is the case for NC-NC interactions.21,40-43 To study the interactions in a solvent using molecular simulations, one has to include the solvent molecules explicitly in the simulations.17 This immensely increases the computational costs and thus limits the size of the systems. For example, only very small Au38 NCs could be

Soft Hedgehogs on Coarse Carpets considered for a study of NC-NC interactions at various solvent conditions in ref 43. Finally, we would like to remark that the suggested trends in equilibrium distance between the capped NC and the SAM capped Au(111) slab can be verified experimentally either using ellipsometry or atomic force microscopy or by measuring the (distance-dependent) energy transfer from a NC to the slab. 5. Conclusions Using atomistic simulations, we have studied alkylthiol capped gold nanocrystals adsorbed on a gold (111) surface that is covered with a self-assembled monolayer of alkylthiols. Upon adsorption, the NC capping layer smears out on top of the SAM, while the structure of the SAM remains intact up to minor orientation frustrations. The capping layer of a NC does not protect it from sintering with a bare (111) surface. Bare NCs diffuse a few angstroms into the SAM, but they do not come close to the gold surface and remain stable. We have systematically investigated the potential of mean force between capped NCs and a gold (111) surface covered with a SAM for a range of capping molecules between butaneand dodecanethiol. The interaction consists of a short-range repulsion and a long-range attraction, with a well depth of the order of tens up to 120 kBT at T ) 300 K. The minimum of the PMF lies always within 2 Å of the contact distance (defined as SAM height plus NC radius); the location of the minimum does not depend on the length of the capping molecule of the NC. This is a result of efficient packing of capping molecules on top of the SAM. When measured experimentally, this result may lead to the false conclusion that the capping layer is not present. On the other hand, the magnitude of the well-depth does not depend on the length of the thiol in the SAM, and it scales linearly with the length of the capping molecule on the NC. This can be understood by considering the SAM as a hydrocarbon slab with a high density that does not depend on the SAM height. With increasing temperature, the PMF becomes less attractive, and the equilibrium distance increases. With increasing NC size, the PMF becomes more attractive and stiffer. Acknowledgment. T.J.H.V. acknowledges The Netherlands Organization for Scientific Research (NWO-CW) for financial support through a VIDI grant. Supporting Information Available: Interaction model and parameters; details of the preparation of capped NCs; detailed description and discussion of the overlap cone model for capped NCs adsorbed to a surface. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Kinge, S.; Crego-Calama, M.; Reinhoudt, D. N. Chem. Phys. Chem. 2008, 9, 20–42. (2) Murray, C. B.; Sun, S.; Gaschler, W.; Doyle, H.; Betley, T. A.; Kagan, C. R. IBM J. Res. DeV. 2001, 45, 47–56. (3) Brust, M.; Bethell, D.; Schiffrin, D. J.; Whyman, R. J. Chem. Soc., Chem. Commun. 1994, 801–802.

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