Deformation Localization in Molecular Layers Constrained between

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Deformation localisation in molecular layers constrained between self-assembled Au nanoparticles Guillaume Copie, Moussa Biaye, Heinrich Diesinger, Thierry Mélin, Christophe Krzeminski, and Fabrizio Cleri Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00237 • Publication Date (Web): 21 Feb 2017 Downloaded from http://pubs.acs.org on February 23, 2017

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Deformation localisation in molecular layers constrained between self-assembled Au nanoparticles G. Copie, M. Biaye, H. Diesinger, T. Melin, C. Krzeminski, F. Cleri∗ Institut d’Electronique, Microlectronique et Nanotechnologie (IEMN), CNRS UMR 8520, Universit´ e de Lille 1 Sciences et Technologies, Avenue Poincar´ e, 59652 Villeneuve d’Ascq, France

Abstract The localised deformation of molecular monolayers constrained between the spherical surfaces of Au nanoparticles is studied by means of molecular dynamics simulations. Alkyl or polyethylene glycol long-chain molecules were homogeneously distributed over the curved Au surface, pushed against each other by repeated cycles of force relaxation and constant-volume equilibration, at temperatures increasing from 50 to 300K, before being slowly quenched down to near-zero temperature. Plots of minimum configurational energy can be obtained as a function of the nanoparticle distance, according to different directions of approach; therefore, such simulations describe a range of deformations, from perfectly uniaxial compression, to a combination of compression and shear. Despite the relative rigidity of molecular backbones, the deformation is always found to be localised at the interface between the opposing molecular monolayers. We find that shorter ligands can be more densely packed on the surface, but do no interdigitate upon compression; they respond to the applied force by bending and twisting, thus changing their conformation while remaining disjointed. On the other hand, longer ligands attain lower surface densities and can interprenetrate when the nanoparticles are compressed against each other; such molecules remain rather straight and benefit from the increased overlap, to maximise the adhesion by dispersion forces. The apparent Young and shear moduli of a dense nanostructure, composed of a triangular arrangement of identical MUDA-decorated Au nanoparticles, are found to be smaller than estimates indirectly deduced by atomic-force experiments, but quite close to previous computer simulations of molecular monolayers on flat surfaces, and of bulk nanoparticle assemblies.

1

Introduction

Self-assembled monolayers of long-chain molecules provide an approach to tune the interaction between surfaces with well-defined size, shape, and composition, notably the spherical surfaces of colloidal noble-metal nanoparticles. Alkane-thiol ligands (of the type SH–(CH2 )n –T , with T a terminal group) have found many applications in chemistry, optics, electronics, biology and medicine (for reviews see e.g. [1, 2]), also because their properties can be modified by selectively changing specific functional headgroups. For example, by changing the terminal groups of alkanethiols from CH3 to OH it is possible to modify the surface chemistry from hydrophobic to hydrophilic. ∗ To

whom correspondence should be addressed. E-mail: [email protected]

Preprint submitted to Langmuir

January 26, 2017

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Among their many interesting properties, colloidal nanoparticles (NPs) can be self-assembled into dense microstructures to realise strain gauges [3–6], with potential applications in touchsensitive panels on flexible substrates. The ligand monolayer plays two major roles in such a system: firstly, it assists the self-assembly technique to precipitate the NPs from the liquid suspension into a dense nanostructure; and secondly, it determines the strain-dependent tunnelling current, as measured by an atomic-force microscope (AFM) tip, put in contact with the NP nanostructure. A clear understanding of the behaviour under mechanical stress of such complex systems, self-assembled colloidal NPs with their ligand monolayers, is mandatory in order to correlate the properties of the self-assembled structure with the macroscopic physical response, such as piezoelectric currents. In a recent work [7], conducting-AFM current-force spectroscopy was performed on assemblies of colloidal Au NP, coated by thiol- and phosphine-based ligands [8]. Such systems are thin films made of a few layers of NPs, arranged in a dense close-packed configuration with typically triangular stacking. By applying the tip force on a localised area of the two-dimensional film, the AFM measurements allowed to address the behaviour of a small group of NP-NP junctions at each time. Force-dependent transition voltage spectroscopy was performed in parallel to the application of the localised force, to study the effect of mechanical compression on the tunnel barrier height, or band offset. In this way different types of NPs and ligands could be compared, with respect to their suitability for realising resistive strain gauges for touch-sensible applications. Following up to those experimental results, in this paper we will focus our attention on the computational study of the structural modifications occurring under compression, in the molecular monolayers that coat the Au nanoparticle surfaces, since this information was impossible to access directly in experiments. It is suggested that the mechanical response of the long-chain ligands under different compression and shear regimes, could greatly influence the correlation between the measured electrical response and the mechanical force actually applied. To this end, we simulated by Molecular Dynamics (MD) fully atomistic modelling the mechanical response of Au NPs coated with different types of molecules: a 1.7 nm long, 11-carbon linear organic molecule terminated by a thiol and one end and by a -COOH group at the other end (11-mercaptoundecanoic acid, SH-(CH2 )10 -COOH, or MUDA); and a 3.5 nm long, 7-monomer polyethylene glycol molecule (SH-PEG7-COOH, or PEG-7), with the same end terminations (Figure 1). Both types of molecules were used as ligands on Au NPs of fixed size (14 nm). We performed mechanical deformations of the ensemble by means of molecular dynamics simulations, by creating variable strains along different directions of the interface between pairs of NPs. Among other interesting results, the effective Young’s moduli of the molecular layers could be estimated, and where possible compared to experimental observation.

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Simulation Methods

To study the mechanical properties of the self-assembled nanostructures of ligand-coated Au NPs, we restricted the consideration to the regions of contact between pairs on NPs, by presuming that all the deformation would be concentrated in those regions and would not affect significantly the overall shape of the Au core. In this way, we could decompose the deformation into elementary modes, namely pure compression and pure shear. To construct the NP-NP interface, we initially cut a sphere of 14 nm in diameter from a block of bulk Au. Then a spherical cap, or ”dome”, of four Au planes was taken from the sphere, corresponding to a surface of approximately 27.3 nm2 . We choose to take the dome axis aligned with the [111] direction of the Au lattice, which is known to give the surface of lowest energy [9]. All over the curved surface of the Au dome, enough ligand molecules were randomly placed

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Figure 1: Ball-and-stick representation of the isolated molecules used in this study. (a): the 11mercaptoundecanoic acid (MUDA). (b): the 7-ethylene glycol acid (PEG-7). Yellow, cyan, red and white spheres respectively indicate S, C, O and H atoms.

up to achieving a coverage density close to the experimentally measured values [10]. Each ligand molecule was anchored to a unique Au atom of the surface via a thiol bond, in the configuration which shows the highest stability according to our previous works [11]. In this way, a total of 148 ligand molecules were densely arranged on the Au dome. A symmetrical copy of this system was then constructed, reversed along the perpendicular [111] direction, and placed facing to the first one. In this way, we obtained a system with two naturally curved molecular layers face to face, at a given distance between the respective outermost atoms of the ligand layers (see Fig. 2, for the case of the MUDA ligands). These will be the initial configurations of the NP-NP interface for all the foregoing atomistic simulations. Molecular dynamics (MD) simulation on these systems were performed with the DLPOLY4 code, using the MM3 force field for the description of the molecules. This force field, based on allelectron quantum chemical calculations of the molecular forces [12], includes bonding (stretching, bending, torsion), and non-bonding (van der Waals, hydrogen-bond, Coulomb) force terms. The Van der Waals (VdW) terms for the interaction between Au and other atoms are described by a Born-Huggins-Meyer potential: D C − 8 (1) r6 r with the parameters given in Table 1 [13]. Hydrogen bonds between C or O (acceptor) and O (donor) atoms linked via a H are included via an approximate function: ( 6 ) RHX r UHB (r) = HB A exp [−12(RY H /r)] − B cos β (2) 0 RHX RY H U (r) = A exp [B(σ − r)] −

where RY H and RHX are the acceptor-H and H-donor distances, and β is the HXY angle. According to generally accepted criteria, a valid H-bond is accounted if the donor-acceptor distance is ≤3 ˚ A, and the angle β ≤30 deg. The constants A=1.84×105 and B=2.25 give values in kcal/mol. Note that the H-bond energy in DLPOLY is counted together with the VdW energy (as any other non-bonded term, besides the Coulomb term), therefore it will be quite difficult to separately account for the two contributions. However, the number of H-bonds can be estimated, according to the geometrical parameters defined above. The Au atoms of the NP surface were kept frozen during the MD simulation, by considering that the very small vibrations of Au atoms compared to the molecular motion at T ≤ 300K

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Figure 2: Ball-and-stick representation of the initial configuration of the MUDA-coated Au nanoparticle system. Yellow spheres correspond to Au atoms. Cyan, red, yellow and white sticks in the molecules correspond to C, O, S, and H atoms.

Atom

A (kcal/mol)

B (˚ A−1 )

σ (˚ A)

C (kcal.˚ A6 /mol)

D (kcal.˚ A8 /mol)

Au C S H O

3.035 0.064 9.033 0.041 0.048

4.009 3.370 4.474 3.894 3.535

2.993 3.561 2.682 3.082 3.395

4363.440 261.001 6723.798 70.276 146.997

0.000 0.000 0.000 0.000 0.000

Table 1: Parameters for the Born-Huggins-Meyer potential for Au-atoms vdW interaction in Eq. 1.

should have a very minor effect on the mechanical response of the system (the Debye-Waller factor of Au is of the order of 0.01 ˚ A at room temperature [14]). The actual validity of such approximation can only be verified a posteriori. On the other hand, such a simplification greatly reduces the computational complexity, notably by avoiding the need to introduce an explicit AuAu interaction, and by drastically reducing the number of atoms in the system, since considering only the NP dome allows to exclude about 4/5 of the Au atoms in the whole bulk NP. The formal charges of atoms in both the MUDA and PEG-7 molecules, to be used in the Coulomb force calculations, were determined by two separate Gaussian03 calculations [15] for the free molecules, by using the Mulliken definition of the effective atomic charge. Electrostatic interactions between point charges in the MD simulations were calculated using the Ewald sum, with a real-space cutoff of 9 ˚ A. The long-range cutoff for van der Waals interactions was set as well at 9 ˚ A. All the MD simulations were performed by using an integration time step of 1 fs, to accurately describe the vibration of the H atoms, and with the constant-{NVT} ensemble, by using a NoséHoover thermostat [16] with a relaxation time of 5 ps. Periodic boundary conditions were applied in the three directions, with a small layer of vacuum to isolate the two symmetric halves of the system along the z direction, perpendicular to the Au dome surfaces. The simulated mechanical experiments were performed by progressively adjusting the relative 4


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Figure 3: Zero-temperature snapshots of the system of two Au NP coated with ligands. (a): MUDA molecules at the equilibrium distance d0 = 2.7 nm between the outermost Au NP surfaces. (b): PEG-7 molecules at the equilibrium distance d0 = 3 nm. Yellow spheres represent Au atoms; ligand molecules are shown with red and blue sticks on either side of the NPs, to highlight the degree of interdigitation. (Note that the scale is not the same in the two panels, since PEG-7 is about twice as long as MUDA.)

position of the rigid domes in small distance steps (typically 0.5-1 nm), along different directions, so as to simulate perfect uniaxial compression or a mix of compression and shear stress. After each step, all-atom force relaxation by conjugate gradient was performed, followed by several steps of finite-temperature MD equilibration, allowing the molecular system to find the optimal configuration at each new NP-NP distance. This procedure is described in more details in the next Sections, for the different modes of deformation.

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Results and Discussion

3.1. Equilibrium distance To find the equilibrium distance of perpendicular interacting NP-NP pairs, we proceed by approaching the center of mass of the two NP in successive steps of ∼0.01-0.2 nm, by starting with a distance d = 3.8 nm between the top of the two Au surfaces. For each step, a short MD relaxation of 10 ps at 10K was done, to avoid close contact between atoms during the approach. This procedure was repeated until a distance of 1.9 nm between the NP surfaces was achieved. Then for 13 further decreasing steps of the distance d, spaced by 0.1 nm each, a sequence of force-relaxation followed by MD equilibration was performed, at temperatures increasing from 50K to 300 K, each temperature step lasting for 120 ps. Such variable-temperature simulations were performed in order to let the NP-NP system explore different configurations of the ligand molecular networks at the interface, and to possibly find the most convenient configuration in terms of interaction energy, for each NP-NP separation distance d. After each series of relaxation-equilibration simulations, the system temperature was slowly quenched down to near-zero temperature (T ≈ 1 K, and the potential energy was recovered, allowing to obtain a value of system configurational energy at each NP-NP distance. Figure 3 shows two sample snapshots of the two molecule-coated NP systems, from long MD runs equilibrated at T =77 K and subsequently quenched down to zero temperature. Firstly, we present the case of the MUDA molecule, for which a more complete study was carried out. Given the size constraints, we could obtain a best density of 5.4 molecules/nm2 on the Au NP surfaces, to be compared to the experimental value of about 5.7 [10]. From the plot of total energies as a function of the NP-NP distance d Figure 4(a), it can be estimated an equilibrium distance of d0 = 2.7 nm, which is in fairly good agreement with the experimental value deduced from transmission-electron microscope images, of ∼2.5 nm. By inspecting the 5


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Figure 4: (a) Potential energy as a function of the NP-NP distance d from the MD simulations, after quenching down to T ≈ 1 K (blue squares). The red continuous line corresponds to the fit by a quadratic equation (see text). A well defined minimum can be observed at d ∼ 2.7 nm. (b) Plot of the average number of hydrogen bonds across the NP-NP contact surface, at the relative distance d=2.5, 2.7, and 2.9 nm, during quite long (25 ns), constant-{N V T } MD simulations at T =77 K. It can be seen that the contribution of H-bonds is maximized at the equilibrium distance.

molecular configurations of the NP-NP system at the equilibrium distance, it can be seen that in the case of the MUDA ligands, the molecules from both NPs are not at all interdigitated, but rather bent, and repelling each other. This morphology could be explained by the concurrence of two phenomena: (1) the density of molecules on the surface is so high that there is little free space around each molecule to allow a proper interdigitation; (2) the configuration of MUDA molecule itself, which does not remain straight but, upon interacting with the nearby molecules, tends to assume a kind of helical structure. Such a behaviour is very much opposite compared to the one we observed in our previous study, with longer molecules [17]. As cited above, the MM3 force field includes an approximate functional to describe the hydrogen bonding between various pairs of donor-acceptor species. In the case of the MUDA, the H-bonds are always formed between the OH donor of a molecule on a NP, and the C=O acceptor of another molecule on the opposite NP. Figure 4(b) shows the time evolution of the number of H-bonds, during MD simulations at constant-{N V T } at the temperature of 77 K. It can be seen that the average number is maximum for the optimal bonding distance of d=2.7 nm, while it is typically lower for both shorter and longer NP-NP separation. An interesting result of the present study, is that the deformation is completely constrained within the two molecular layers, and does not impact the structure of the Au surfaces. This localisation of the deformation was verified by looking at the force exerted on the S atoms, which represent the terminal link between the end of the molecule (for both the MUDA and PEG-7) and the fixed Au surface. In Figure 5 we display the plot of the vertical component of the force acting on the S atoms (fz ), and the modulus of the total force |f | = |fx2 + fy2 + fz2 |1/2 , averaged over about 30 molecules in the centre of the NP-NP contact area. The black set of plots corresponds to a NP-NP distance d=3 nm, when there is not contact between the surfaces and only some residual vdW and electrostatic attraction works between the molecules; the red set of plots corresponds to d=2.4 nm, when the two NP are already under compression. The fact that the force plots are practically identical both at large NP separation and in compression, confirms that there is no recall force on the S atoms, possibly originating from the rebounding

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Figure 5: Plot of the fz component and of the |f 2 |1/2 modulus of the total force acting on the S atoms linking the MUDA molecules to the Au surface. The four plots represent the values averaged over about 30 molecules at the center of the contact area. Black plots correspond to values measured at a NP-NP distance d=3.0 nm, red plots to d=2.4 nm.

against the rigid Au surface. Therefore, the terminal parts of the molecules in direct contact with the NP surface are practically not perturbed by the overall state of deformation induced by the compression. As a consequence, we deduce that all the deformation must be localised at the opposite end of the molecules, about the region of NP-NP contact. The energy of NP-NP interaction can be decomposed into its different contributions. In Figure 6 the different energy contributions for the case of the MUDA molecule are spelled out, as a function of the NP-NP distance d, by taking the zero of the energy at the equilibrium distance d0 . By looking at the differences in the compression regime (d < d0 ), it is observed that the intermolecular repulsive part is dominated by the electrostatic contribution within each molecular layer (a), accounting for approximately 50 to 75% of the total; the VdW contribution (b) is the second most important, contributing about 30-15%. Such repulsive forces are also connected to structural changes in the configuration of the molecules: in fact, ∼15% of the repulsive energy comes from the modifications in the dihedral angles, while just a few per-cent are due to the angle-bending and bond-stretching terms. At the same time, attractive forces between the two opposing molecular layers (panel (c) in the Figure) appear to be roughly equally shared among the electrostatic and van der Waals contributions. (Note that the electrostatic contribution goes to zero at about d > 3 nm, while the longer cut-off makes the VdW term to be still different from zero at the same NP-NP distance.) The same kind of analysis can be performed for the PEG-7 ligand monolayer. For this much longer molecule, the highest density that we could obtain on the Au curved surface was smaller than for the MUDA, being at best ∼ 4.3 molecules/nm2 . This effect of reduction of surface density with increasing chain length is known, and has been documented for a range of similar ligands of varying length [10], with an experimental value of just 4.3 molecules/nm2 for the PEG-7. By comparing to the MUDA higher-density monolayer, in this case it is seen that the opposing monolayers are much more interdigitated, with a consequent increase of the contact 7


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Figure 6: Different contributions to the total potential energy after the quench to T ≈ 1 K, for the data shown in Fig. 4(a). (a) Electrostatic. and (b) vdW contributions within each molecular layer (blue and red symbols refer to molecule colours in Fig. 3). (c) vdW (red symbols) and electrostatic (blue symbols) contributions to the NP-NP interaction energy (or ”adhesion energy”, equal to the difference between the total and the values in (a,b)). (d) Dihedral-, (e) angle-bending, and (f) bond-stretching energy contributions to the total molecular energy.

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surface between the NPs. The equilibrium distance is estimated to be 3.0 nm, indeed even shorter than the length of the isolated molecule (3.3 nm). This means that the molecules are not only very much interdigitated, but also somewhat bent and twisted, giving rise to some degree of entanglement like in a polymer mixture. By looking at the different energy contributions, we observe in this case that the repulsive part is essentially due to structural changes in the configuration of the molecules, with 85% of the repulsive energy coming from the distortions of the dihedral angles, and the remaining 15% mostly involving angle-bending. 3.2. Stiffness of the molecular layer One of the main objectives of the experiments described in Ref.[7], was to deduce a correlation between the applied force and the resulting AFM tunnel current. The underlying assumption was that by increasing the force, the amount of interfacial contact between the molecular layers covering the NP surfaces would increase, thereby inducing a variation in the overall electrical response (conductance) of the self-assembled NP system. However, it was impossible from the experiments to correlate the observed changes in electrical behaviour with the possible restructuring of the molecular layers under stress. Therefore, the deformation of the molecules at the interfaces could be accounted for only with effective models, in which the two extreme approaches of the ”Hooke-like” and ”Hertz-like” description could be adapted to the case at hand. Rigid displacements were assumed, without considering the possibility of shear and dissipation during more complex deformation modes. On the other hand, the present MD simulations give access to a direct measurement and structural characterisation of the molecular deformations during the simulated experiment, and allow to shed light on the possible mechanisms underlying the variation of electrical coupling between the coating monolayers. The range of data points in the compression regime (d < d0 ), from the previous study of the equilibrium NP-NP distance for both types of ligands, can be further analysed to extract an estimate of the effective Young’s modulus in compression. It is worth recalling that such compression occurs in practice only within the molecular layers, since the NP surfaces are sufficiently distant to be considered as rigid. 3.2.1. Hooke model In the Hooke model, the deformation ∆d of a uniform body of size d and cross section A, is taken to be simply proportional to the intensity of the force applied, F = k · ∆d. The eventual, very small variation of the contact surface, is neglected. Therefore a value of Young’s modulus E can be estimated as: ∆d F =E· (3) A d This would be the case for a stiff material, or for a very low intensity of the applied force. For the case of our NP-NP contact experiments, the equivalent stiffness constant k is related to the effective Young modulus E of the ensemble NP+molecules, the contact surface area A, and the distance between NPs at the equilibrium value d0 . The compression force in our MD simulations can be calculated by fitting the curve of the potential energy E vs. distance d with a polynomial equation, and by taking the derivative of this fitting equation: ∂E F = (4) ∂d d=d0 For the MUDA molecule the best fit of the energy curve gives the quadratic polynomial E(d) = 1280.6d2 − 6494d + 741.2. For the PEG-7 the same equation gives E(d) = 568.5d2 − 3453.2d − 2760.1. 9


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To estimate the contact surface A between the two opposing MUDA molecular layers, we take the extent of the region including all the atoms from various molecules, fully contained in a layer of thickness 4 ˚ A, centered at the middle point between the two Au NPs. This gives a contact surface area A ' 15 nm2 . The 4 ˚ A value was chosen empirically, as the one giving a reasonable representation of the NP-NP contact; small variations about this value would give an error of the order of ±10% on the estimate of E. In the limit of a small variation about the equilibrium distance d0 , the compression force is estimated to be 2 nN, and with A = 15 nm2 , Eq. 3 gives E ' 4 GPa. For the PEG-7 case, a similar analysis gives a value of E ' 0.7 GPa. While the above value of 4 GPa appears to be about 7 times smaller than the one experimentally deduced in our previous work [7], it is however of the same order of magnitude as previous estimations for similar molecular monolayers [18, 19]. Notably, such values were obtained on flat ligand monolayers between Au tips, to simulate AFM indentation of the monolayer. In the present case, instead, two opposing monolayers are sandwiched between rounded NP surfaces, therefore the E value refers to the mutual interaction of monolayers, rather than to a singlemonolayer compression. The present situation is closer to the large-scale MD simulations of Landman & Luedtke [20], in which values of E ' 1 GPa was estimated for 3-dimensional dense NP arrays, and to the nanoindentation measurements on 3-dimensional assemblies of Au NPs coated with long alkane-dithiols, which gave again values E ' 3 GPa [21]. Experimental values along these lines were also reported in studies of contact-AFM on 2-dimensional ligand-coated Au NP membranes, for which average values of 6 GPa were measured [22]. In the present simulation model, the role of surface curvature has the effect of changing the geometrical factor for the effective contact area, and cannot significantly modify the order of magnitude of our results, while it might possibly change the agreement with the experimental values. Notably, in the above cited MD simulations of AFM-monolayer contact, possible sources of discrepancy with experimental results were also identified, most likely coming from the difficulty of estimating an appropriate value of the contact surface area, independently on the force-current measurement. Also, the effect of curvature is such that the molecules at the outer perimeter of the contact surface should experience a different value of compressive force, compared to those near the center of the contact surface. It should be noted that E values in the few GPa range represent upper bounds for the Young’s modulus of all polymeric materials, including polyethylene and poly-methyl-metacrylate. In bulk polymers the apparent Young’s modulus is a direct result of the entanglement of long-chain molecules. The MUDA ligands on the NP surfaces are too short to entangle, and the relatively large value of E is a consequence of the confinement of the ligands to the NP surfaces. On the other hand, when ligands are longer and experience substantial interdigitation, as in the case of PEG-7 (probably an intermediate configuration, between full separation and complete entanglement), the value of E correspondingly decreases, both because of increase in contact area and reduction in the compressive force. 3.2.2. Hertz model In the Hertz model, by contrast, the contact surface is a function on the intensity of the force applied. It is the case for soft materials, or for a larger value of the compression force. In this model, spherical object of radius R is assumed to push on a flat surface with a force F . The sphere is deformed over a flat region of radius r, which gives a contact area A = 4πr2 . In this case, a linear relationship is found between the force applied and the cube of the radius of the contact area: F =

4Er3 3R

10


(5)

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Figure 7: Plot of the contact area radius r3 (nm3 ) as a function of the compression force (nN). Blue squares represent the calculated values. The green line shows the linear fit according to Eq. (5). The orange line shows the best fit without imposing the condition of d = 0 at zero force.

For the case of two identical spheres in contact, the NP radius R is replaced by the value R/2. By plotting the curve of the radius of the contact region to the power three, r3 , vs. the applied force F , we should obtain a linear plot in which the leading coefficient is directly related to the effective Young’s modulus. Note that in principle the value of E in the above model equations could include a correction for the lateral deformation in the plane perpendicular to the applied force, as E ∗ = E/(1 − ν 2 ), with ν the Poisson’s ratio of the system. While some particular values can be found in the literature, estimating a Poisson’s ratio in the present case is extremely complicated, therefore the values of E obtained from the two models (Hooke vs. Hertz) will be taken as upper bounds to the real value. The Figure 7 shows the plot r3 = f (F ), again for the MUDA molecular monolayer. We observe a straight line, in agreement with the Hertz model. From the fitted slope, an effective Young’s modulus E ' 3 GPa can be deduced, very close to the value of 4 GPa previously estimated with the Hooke model. 3.2.3. Adhesive contact models In Figure 7 it is evident that, for the best fit curve (orange line) at zero compression force, the value of surface contact area is not equal to zero. This can be qualitatively understood by considering the nature of our NP-NP system, in which a residual molecular attraction (”stickyness”) is present, mostly because of the long-ranged dispersion and electrostatic forces, notably around the equilibrium distance d0 . The overall cohesive interaction makes the contact surface non-zero at the equilibrium NP separation. A more quantitative assessment of this effect is provided by the Johnson, Kendall and Roberts model (or JKR), of contact between adhesive surfaces [23]. The expression for the radius of the sticky contact region in the JKR theory is:

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i p 3R h F + 6γπR + 12γπRF + (6γπR)2 (6) 4E The γ represents the work of adhesion between the two surfaces. In our MD simulations, this quantity could be identified with the depth of the NP-NP interaction potential, Fig. 4, measured from the equilibrium distance d0 to infinite separation. For γ = 0, the Hertz limit Eq.(5) is recovered. On the other hand, for a compressive force F → 0, the above expression turns into the zero-force limit: r3 =

9γπR2 (7) E that gives a finite contact radius, if γ > 0. Therefore, this last equation provides a way of extracting the effective work of adhesion from the non-zero intercept of the plot in Fig. 7. By using r = 2.35 nm, E = 3 GPa, R = 9.5 nm (including the molecular layer thickness), a value γ=15.3 mJ/m2 is obtained. When integrated over the equilibrium contact area A=15 nm2 , we obtain a total energy of ∼1.4-1.5 eV, corresponding to ∼30-35 kcal/mol. Such a value does not compare well to the depth of the minimum in Fig. 4, which is about 200 kcal/mol, a factor of 6 larger (or more, considering that, while disjointed at d=3.1 nm, the two NP are not yet at infinite separation). If instead of the JKR model we consider the Derjaguin-Muller-Toporov (DMT) model, which describes additional adhesion in the area around the contact [24], its zeroforce limit is the same as Eq.(7) but with a prefactor of 3/2 instead of 9. With this correction, the new estimate of γ ' 92 mJ/m2 is deduced, and a total adhesion energy of 200 kcal/mol, nicely coherent with the estimate of the NP-NP potential depth. For the PEG-7 case, the contact surface spreads across a much larger area, almost entirely covering the xy plane of the simulation cell. Also, the considerable degree of interdigitation makes the overall NP-NP to get softened by about a factor of 2, thereby leading to a much lower estimate of E in the range of 80-150 MPa, even smaller than the previously estimated value by the Hooke’s model. However, given the fact that the contact surface practically covers the entire cell, the use of the Hertz model seems inappropriate in this case, therefore the value of 0.7 GPa can be retained as a kind of upper bound to the estimate of the Young’s modulus. r3 =

3.3. Structural perturbations of the molecular layer A number of structural factors could indeed influence the experimental NP arrangements, in a way that is quite difficult to control. Therefore, another series of MD simulations was carried out with the aim of looking at possible sources of variability, under three different scenarios: (1) extra water molecules randomly dispersed within the molecular monolayers, to simulate the effect of solvent that may eventually survive after the drying and vacuum processing; (2) a combination of compression and shear strain, to represent the alternative loading configurations that can occur, depending of the relative position of the NPs within the compressed layers; and (3) the role of a variable density of molecules on the NP surfaces, following the preliminary results of the comparison between MUDA and PEG-7, whose surface densities differ by about 25%. 3.3.1. Influence of residual solvent molecules In order to evaluate the influence of adding residual water molecules from the solvent, we placed 20 H2 O molecules at the interface between nanoparticles. At the atomistic level, water molecules were described by the TIP3P model, with 5 parameters from the MM3 library and 3 point charges. The same NP-NP step-approach procedure was repeated, as previously explained. The number of water molecules was estimated as being representative of the residual solvent concentration, as a function of the free space remaining at the interstices around the 12


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Figure 8: Left: Schematic representation of a compact assembly of the Au nanoparticles. A compression force (black arrow) apply on the top NP, involves a mixed stress (green arrow) shared between the two NPs below. Right: Representation of the initial step for the simulation with mixed compressive-shear stress. The top NP was displaced in steps according to the direction indicated by the green arrow.

nanoparticles (≈ 100 nm3 ), and of the level of vacuum which is experimentally obtained prior to the measurements (10−7 mbar). The MD simulation results demonstrate that the energy vs. distance curve remains practically unchanged after the addition of water molecules. This can be explained by the configuration reached by the water molecules, that, at equilibrium, are situated at the contact surface between the two NPs. However, upon increasing the compression, i.e. following the decrease of the NP-NP distance, water molecules move to the free space remaining in the empty interstices between the NP, thereby not contributing in any way to the apparent stiffness of the compressed molecular layers. 3.3.2. Combined compression-shear loading Up to this point we always assumed a uniaxial compression strain, which is the case for NPs self-assembled one on top of each other. However, in a dense multi-layered structure, such as those realised in our published experiments ([7], see also [21]), or in 3-dimensional closepacked bulk assemblies [20], the NPs can exert mutual stresses along other directions than purely perpendicular. In particular, if we consider a close-packed triangular or hexagonal arrangement (see Fig. 8) the resulting force should be directed at an angle of 30◦ with respect to the direction √ of the AFM applied force, therefore with compressive and shearing components in the ratio 3. For a body-centered cubic assembly, the angle is rather 45◦ , and the ratio of compression to shear is 1. In order to evaluate the effect of a mixed compression/shear stress between pairs of NPs, we could change the approaching path of the applied force (Fig 8). We chose the bcc configuration, in order to estimate the worst coupling condition, the results for the fcc or hcp packing being proportionally less affected by the shear component. In the foregoing simulations, one NP was displaced in equal steps with respect to the other along the x− and z−directions, so as to make steps of length comparable to the case of pure compression. The same procedure above of forcerelaxation, followed by finite-temperature equilibration up to 300 K, and final quenching to T ≈ 1 K was carried out, for each NP-NP distance, and only for the case of MUDA ligands. In this case, we observed a more noisy energy-distance plot (Fig. 9a), making a simple polynomial fit more difficult. However, the more straightforward estimation of the Young modulus according to 2-dimensional version of the Hooke’s model can be applied in this mixed-stress case. In the most general case, dissipative effects should be included via a two-dimensional Maxwell’s model [25], by including

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Figure 9: (a): Potential energy vs. relative distance for an uniaxial (orange), and mixed (blue) compressive+shear stress. The ∆E values with respect to the respective minima at d0 are plotted, to make easier the comparison between the two calculations. The curvature about the minimum is smaller for the mixed stress, which means a reduced effective stiffness of the molecular layer. (b): Plot of the principal stress components along x and z, as a function of the strain = ∆d/d0 .

compressive and shear viscosities, aside of the elastic and shear moduli. The situation can be simplified by dropping the dissipative terms, which is a usual approximation well supported by experimental findings in systems like the ones describe here [26]. By defining 1 and 2 the principal directions (z and x respectively), and by assuming 2-dimensional symmetry in the y direction (perpendicular to the page, in Fig. 8), the stress-strain equations read: τ1 = (E + G)1 + (E − G)2 (8) τ2 = (E − G)1 + (E + G)2 +τ2 2 and G = 21 τ11 −τ in which E = 12 τ11 + −2 are the elastic (Young’s) and shear moduli, respectively, 2 and τ and are the stress and strain tensors. It is worth noting that dissipative terms could be added to the above linear model, by including the compressive and shear viscosity as coefficients of the strain rate. In our case the MD simulations are quasi-static (i.e., an infinite relaxation time is assumed between each successive displacement), such that dissipative effects are not included. Such effect, however, could affect the experimental samples, in which the rates of force application are within the range of the relaxation time-scales. Figure 9b reports the plot of the two components τ1 , τ2 as a function of the µ1 = µ2 strain components. Despite a very large error bar, coming from the statistical fluctuation of the force calculations, the best fit to the Eqs.(5) in the limit of infinitesimal compression, allows to estimate values of E=0.3-0.4 GPa, and G ∼ 0.1 GPa. Such values, and notably the estimate of E, are smaller than the ones calculated in the case of a pure strain, meaning that the shearing component has an important effect in the redistribution of the effective force. If, in interpreting the experiments, it is assumed that all the applied force is used for only compression, while instead a fraction goes into shearing, the measured force should be scaled roughly by the ratio τ2 /(τ1 + τ2 ), to extract the apparent value of E.

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3.3.3. Influence of the surface molecular density We finally looked at the influence of the molecular density on the NP surface. Already the comparison between the MUDA and PEG-7 ligands, the former being shorter and higher density, the latter being about twice longer and with a 25% lower density, demonstrated a completely different behavior under pure compression. To complete our analysis, we built two MUDA-covered surfaces, at the reduced coverage of 4 molecules/nm2 (i.e. 75% of the full, nearexperimental density), or 2.7 molecules/nm2 (i.e. half the full density), and carried out again the pure compression MD-simulated experiment. Figures 10(a) to (d) show a sequence of snapshots of the half-density system, after a long equilibration at T =77 K and subsequent quenching to zero temperature. From a purely qualitative standpoint, it can be seen that at the shorter distances the MUDA molecules from opposite sides stand straight, and form an adhesion patch; at the longer distances, molecules from opposite NPs detach, and tend to form a point-like contact between smoothly curved surfaces. The plots of the potential energy vs. distance are shown in Fig. 10(e) and (f), respectively for the 75% and 50% surface density. Each panel displays also the breakdown of potential energy components, into Coulomb, Van der Waals (including H-bonds), and bonding terms (sum of pair, angular and torsion energy). (Note that in (e), the Coulomb and total are given on the left y-axis, while the VdW and bonding are on the right y-axis). The 75% system still displays a minimum (total potential energy, square symbols and red curve fit), albeit very shallow, at the equilibrium distance d0 =2.3 nm, that is about 0.4 nm closer than for the full-density case. The molecules are slightly interdigitated, at variance with the MD simulations for the full density case. We can interpret this result, as the reduced surface density in this case leaving more free space around each ligand, thus allowing the interdigitation. The reduced density of this system provides also a reason for the shallow minimum lying at a smaller d0 : less molecules means less interaction, at a given NP-NP distance. Therefore, the new equilibrium distance (the balance between repulsive and attractive components of the effective force) is obtained by slightly increasing the compression (i.e., reducing the NP-NP distance). At the even reduced density of 2.7 mol/nm2 , that is 50% with respect to the near-experimental value, the system appears unable to provide a true adhesion minimum. The total potential energy steadily decreases to a nearly constant value, suggesting a soft-sphere behavior, i.e. short range repulsion without actually sticking. From the energy components break-up, it is seen that the VdW term provides the soft-sphere short range repulsion, while the Coulomb and bonding components remain nearly constant over the whole distance range. Concerning H-bonds, as already said, it is difficult in the present set-up to clearly separate the H-bond energy from the total of non-bonded energy; however, the trend of the total number of H-bonds across the contact region appears to follow closely the shape of the VdW energy component, in Fig. 10(e)-(f): it remains practically constant with d for the 75% density, and it decreases monotonically to a plateau value at large d for the 50% density. By applying the Hooke’s model to the energy-distance plot (e) for the 75% density, it can be estimated a value of E ' 0.4 GPa, about one order of magnitude lower than what found for the high-density configuration, and closer to the value estimated for the better interdigitated configurations of the PEG-7. Once again, the increased free space around molecules can be invoked as the main explanation for this huge decrease of the stiffness. Indeed, with more free space, molecules can move more easily in the direction perpendicular to the compressive force, without being restrained by neighbour molecules, and without bending or twisting (which makes for strongly positive contributions to the conformational energy, i.e. repulsive force terms). Instead, a better intermolecular overlap is attained, thanks to the gain in the attractive van der Waals interactions. Therefore, the easier mutual accommodation of the molecules against the compressive force implies a decrease of the apparent stiffness. At the same time, the much shallower minimum indicates a much reduced ”stickyness” of the NP assembly.

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Figure 10: Effect of molecular density variation on the NP surface. (a)-(d): Zero-temperature snapshots of the system at half density (two Au NP coated with 2.7 mol.nm−2 MUDA ligands), at distances increasing from 2.4 to 2.7 nm. (e): Total potential energy, and split up contributions (Coulomb, VdW, bonding), for the 75% density (4 mol.nm−2 ). (f): Total potential energy, and split up contributions (Coulomb, VdW, bonding), for the half density (2.7 mol.nm−2 ). The red curves in the plots (e)-(f) represent a numerical fit by a third-order polynomial.

4

Summary and Conclusions

In summary, in this work we reported the results of molecular dynamics simulations of the compression of monolayers of organic long-chain ligands covalently attached on the spherical surface of Au nanoparticles. The ligands were a 1.7-nm long 11-mercaptoundecanoid acid (MUDA), and a 3.5-nm long 7-ethylen-glycol (PEG-7), linked to the Au surface by thiol (SH) groups. At the experimental surface density of ligands (5.4 molecules/nm2 for the MUDA and 4.3 for the PEG-7) we found and equilibrium NP-NP distance in close agreement with the available experimental data. Because of the potential interest of such self-assembled nanoscale systems in the area of touch-sensitive materials, we performed pure compression, as well as mixed compressionshear MD simulated experiments, to estimate the apparent Young’s modulus of the assembly. The deformation is found to be concentrated in the molecular layers, while the NP surface remained stress-free, thus justifying the approximation of considering the Au-Au interactions as frozen during the simulations. 16


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The estimated values of E for the molecular layers fall in the range between ∼ 0.5 and 4 GPa, depending on the chain length, surface density, and compression mode, the lower end belonging to low-density coverage and/or longer ligands, the upper end to the high-density shorter ligands. Such values are in good agreement with previous atomistic simulations of both selfassembled NP systems, covered by various ligands, as well as simulations of direct compression of dense ligand monolayers, attached on a flat surface. However, a disagreement with our recently published experiments of contact-AFM was also apparent. In those experiments [7], we reported values of E about one order of magnitude larger, as deduced from the tunnel current under an increasing contact force applied by the AFM tip on the outer surface of a self-assembled NP thin film. Possible sources for the discrepancy were identified in: (i) the different relaxation time-scale between simulations and experiments, which could imply a role of dissipative effects in the experiments; (ii) the difficulty of measuring a NP-NP contact area in the experiment, independently on the tunnel current measurement from which the apparent E is deduced; (iii) the non-ideal conditions of the experiment, in which multiple layers on top of each other are compressed, with a resulting complex redistribution of the stress, compared to the idealised pair-contact model of the molecular simulations. All such factors combined could make the experimentally applied force to appear larger than needed to produce the observed displacement, therefore leading to a larger estimate of E. On the other hand, we found that the effect of including residual solvent (water) molecules in the model has but a negligible effect on the results. At the equilibrium NP-NP distance, we find that the higher-density molecules are not interdigitated upon compression, while this becomes possible when the surface density of ligands is reduced. This can be obtained either by using longer ligands, such as the PEG-7, or by deliberately reducing the density of shorter ligands, for example to 75% of the maximum density. In both cases, the less dense coverage leads to an increased overlap of the molecules from opposite NP surfaces, thereby increasing the role of attractive forces and making for a larger NP-NP effective contact surface. Both effects go in the direction of reducing the apparent stiffness of the ensemble. At the opposite, fully-dense monolayers cannot accommodate any interdigitation: upon compression the molecules from opposite NP surfaces repel each other, with the mechanical result of changing their conformation by bending and twisting. Such molecular-scale structural modifications imply a strong positive contribution to the conformational energy, thereby contributing a repulsive component to the apparent force, and an increased stiffness (or larger E) of the ensemble. The NP-NP adhesion in this case is mediated only by the mix of dispersion and Coulomb attractive forces, acting over the relatively small contact surfaces. When the density is much reduced with respect to the ideal (i.e., experimental) value, up to 50% lower, makes the adhesion minimum disappear, and the molecule-coated NPs behave as a system of soft, nonsticky particles, with only a short range repulsion. The final message could be that in order to fabricate nano-assembled materials with a large range of elastic moduli, nanoparticles densely covered with proportionally shorter ligands, should be a preferable choice.

Acknowledgements GC acknowledges support from IEMN and University of Lille postdoctoral grants. MB, HT and TM acknowledge support from the French funding agency (ANR, Nanoflexitouch project, grant number 11BS1000401) and the ReNaTech network. GC, CK and FC acknowledge support from the French HPC resources of CINES and IDRIS, made available by GENCI (”Grand Equipement National de Calcul Intensif”) under project x2016-077225.

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Table of Contents GRAPHICS

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Deformation Localization in Molecular Layers Constrained between

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