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Dewetting during Terahertz Vibrations of Nanoparticles Ching-Chung Hsueh, Reuven Gordon, and Joerg Rottler Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03984 • Publication Date (Web): 08 Jan 2018 Downloaded from http://pubs.acs.org on January 8, 2018

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Dewetting during Terahertz Vibrations of Nanoparticles Ching-Chung Hsueh,† Reuven Gordon,∗,‡ and Jörg Rottler∗,¶ † Department of Physics and Astronomy, University of British Columbia, Vancouver BC V6T 1Z1, Canada ‡Department Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada ¶ Department of Physics and Astronomy and Quantum Matter Institute, University of British Columbia, Vancouver BC V6T 1Z1, Canada E-mail: [email protected]; [email protected]

Abstract We use molecular simulations to demonstrate the formation of a vacuum layer around a vibrating nanoparticle in a liquid. This vacuum layer forms readily for high frequencies with respect to the characteristic vibrational (Einstein) frequency of the fluid, even with small amplitude vibrations. The opposite is true for low frequencies, where large amplitudes are required to demonstrate the vacuum layer. With the vacuum layer forming, the quality factor of the oscillations increases substantially. The findings provide an interpretation of our recent experiments that show the onset of high-quality resonances of nanoparticles in water [Xiang et al., Nano Letters 16, 3638 (2016)] in the GHz to THz range.

Keywords: nanoparticles, terahertz, acoustics, molecular dynamics, simulations, cavitation. 1

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Nanoparticles in solution possess natural vibrations in the GHz to THz range. These oscillations can be used for characterization, 1,2 can be precisely tuned by adding additional layers 3 and are strongly affected by the environment. 4,5 In quantum dots, they can alter the optical emission spectra. 6 For proteins, vibrations are critical to conformational changes, 7 ligand binding, 8 and allostery. 9 For virus particles, these oscillatory motions may be used for resonant excitation 10 (with potential applications in detection and treatment). For DNA fragments, vibrational behavior can give information about the amino-acid sequence. 11 For colloidal particles, acoustical excitations may be used for characterization of structure 12–14 and even in-situ growth monitoring. 15,16 Electrostrictive coupling to nanoparticle vibrations in solution may also provide a strong nonlinear optical response; for example, the enhanced nonlinear response has been studied extensively for nanoparticles in solids, 17 yet the damping in liquids is much lower leading to a stronger resonant response. 18 Motivated by the wide-ranging applications of this basic physical system, there has been significant effort to understand these vibrations, particularly the impact of the surrounding fluid. Continuum models are widely applied in this regard, including Newtonian 19–21 and viscoelastic models, 22 and these show generally good agreement with the associated timedomain experiments. At the same time, the acoustic vibrations of the nanoparticles are often of greater frequency than the characteristic frequency of the fluid, and so relaxation processes that are implicitly assumed in continuum models 21 may be incomplete. It is expected that new physics will emerge beyond continuum theory, and other works have already started to probe this more ballistic regime for the case of diffusion. 23 Here we demonstrate the emergence of new phenomena for acoustic vibrations in a simple Lennard-Jones fluid (LJ) with a deterministically vibrating sphere. In particular, we show that there is a threshold in the vibration amplitude, above which the fluid dewets the surface of the sphere and the damping transitions from continuous to intermittent with a dramatic increase in the quality factor of the vibrations. The threshold is reached much earlier for larger frequencies, which has profound implications for the vibrations in this regime that

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are generally considered to be subject to substantial damping. In several earlier works, we observed in four-wave mixing experiments the sudden onset of a nonlinear signal from nanoparticles in solution. This intensity threshold could not be explained by a stimulated Raman or Brillouin threshold. Furthermore, the vibration frequencies observed lay between the vacuum and viscoelastic values, but slightly closer to the vacuum values. 24 For these reasons, we postulated the formation of a cavity around the nanoparticles at a critical forcing amplitude, but no model was available to support this hypothesis. Here, for the first time, we provide such a model which clearly shows the onset of a cavity in this nanoscale/GHz-THz regime. In the LJ fluid composed of particles of mass m, the fundamental (Einstein) vibrational p timescale is given by τ0 = mσ 2 /, where and σ denote the energy and length scales set by the LJ potential. The potential is truncated at 2.5σ and vertically adjusted for continuity. We seek to understand the interaction of the fluid with a nanoscale object vibrating on the same timescale. To this end, we embed a sphere of radius R0 = 5σ composed of 423 LJ particles forming an fcc lattice (lattice constant 1.7σ) into the fluid. The sphere atoms interact with the fluid atoms through a purely repulsive LJ potential truncated at 21/6 σ. The sphere’s diameter is 10 times larger than that of the fluid particles, which roughly mimicks a nanoparticle of 3 nm diameter immersed in water. Since the hydrostatic pressure is positive at the chosen fluid density, a static sphere is always in contact with the fluid molecules. The fluid is thermostatted to a reduced temperature T ∗ = 1 using a Langevin thermostat with a damping time of τ0 . The equations of motion are integrated for the fluid particles only, while the sphere particles are undergoing radial oscillatory motion with amplitude A about their equilibrium positions ri0 ,

ri (t) = ri0 [1 + A sin(ωt)].

(1)

The entire sphere therefore expands and contracts radially with frequency ω and amplitude

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Figure 1: (a) Snapshot of the simulation box of width 25.7 σ containing a model nanosphere immersed in 13,500 LJ particles. (b) Radial fluid density profiles for ω = 0.1 ω0 and ω = 5 ω0 AR0 . For each value of A, data is recorded over 50 vibration periods after 1τ0 of relaxation. Figure 1(a) shows a snapshot of the simulation setup with a sphere undergoing breathing mode oscillations immersed in a fluid of LJ particles. In panel (b) we analyze the temporally averaged radial fluid density profile with respect to the center of the sphere, which reveals a central result of this work: for small frequencies ω ω0 = 2π/τ0 , the density profile is indistinguishable from that of a static sphere, but a visible gap opens up at larger frequencies ω > ω0 . The emergence of the gap indicates a dewetting phenomenon. Our simulations allow us to directly measure the thickness of the vacuum gap between

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Figure 2: Thickness of the vacuum gap around the sphere vs. vibration amplitude A for different sphere vibration frequencies ω. Amax denotes results for a static sphere with radius R = R0 (1 + Amax ). The uncertainty for all data points is 0.02σ. vibrating sphere and fluid. To this end, we define the position r1/2 (A, ω) of the solvation shell as the distance from the center of the sphere at which the fluid density reaches 1/2 of the bulk value, as indicated by the dashed lines in Fig. 1(b). We then report in Fig. 2 as thickness the difference with respect to the static sphere of radius R0 , i.e. ∆r1/2 = r1/2 (A, ω) − r1/2 (0, 0). Looking first at the curve for low frequencies ω/ω0 = 0.2, we see an increase in gap thickness for A > 0.11. For lower amplitudes, the fluid particles can still completely follow the sphere oscillations and hence completely wet the sphere. Upon further increase of the frequency, however, the onset of gap formation shifts rapidly to lower amplitudes, and we find a maximum gap at ω/ω0 = 5. This behavior suggests that there exists a critical velocity vc ∼ R0 Ac ω, at which the fluid loses the ability to follow the sphere. In the frequency range 0.2 < ω/ω0 < 2, we estimate 0.1 σ/τ < vc < 0.2 σ/τ from Fig. 2. These velocities are about one order of magnitude smaller than the speed of the particles vp ' 1.3 σ/τ in the ballistic regime, which we estimate from the rms distance traveled by the particles in the bulk fluid before colliding with other particles. Values of vc lower, but still comparable to vp seem reasonable when taking into account that the speed of the particles should be reduced at the

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surface due to reflections. A subtle effect occurs at still higher frequencies where the gap thickness decreases once again and saturates at a frequency independent curve. This saturation can be understood from the fact that for high frequencies ω ω0 , the fluid sees essentially a static sphere with a radius given by the maximum vibrational amplitude. Indeed the saturation gap is the same that would be produced by a static sphere of radius R0 (1 + Amax ), as indicated by the corresponding solid black line. At finite frequencies the gap thickness exceeds that limit, and we suggest (but cannot prove explicitly) that this is due to inertial effects. The vacuum gap always vanishes for vanishing oscillation amplitudes. During the simulation, we also record the bulk fluid pressure from the usual virial expression, N kB T + pf = V

P

rij · fij , 3V

(2)

where the summation runs over all fluid particles and the volume is given by the box volume minus the volume occupied by the sphere. rij and fij denote the separation and force vectors between particles i and j. We also compute a local pressure at the surface of the sphere,

ploc =

X fi · n ˆi 4πri2

,

(3)

where the summation runs over all sphere particles, fi denotes the total force on sphere particle i by the fluid, and nbi is the radial unit vector. Figure 3 shows the behavior of these two pressures together with the oscillating sphere radius R(t) over several oscillation periods. We observe qualitatively different behavior depending on whether the sphere oscillates much slower or much faster than the fluid. For ω ω0 , the fluid oscillates essentially in phase with the sphere, while the local pressure always leads the sphere oscillations. The sphere is always in direct contact with the fluid particles. For ω > ω0 by contrast, the local pressure drops to almost zero during most of the period, but spikes dramatically when the radius is maximal. This asymmetric shape of

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the pressure profile shows that the sphere only comes in contact with the fluid during the maxima in the cycle, consistent with the dewetting phenomenon observed in Figs. 1 and 2. The formation of the vacuum layer results in a dramatic decrease of the damping of the sphere vibrations. In order to quantify this effect, we estimate the energy dissipated in one cycle as Z

T

dt

Wdiss =

X

fi · vi ,

(4)

0

where the summation is carried out over all particles in the sphere and vi denote the particle velocities. For simple Stokes friction, we expect Wdiss ∼ ωA. For a given driving amplitude A, we find in Fig. 4(a) that Wdiss has a maximum at ω/ω0 = 2, and decreases strongly for larger frequencies. This frequency is within a factor of 2.5 from the frequency at which we observe the largest vacuum gap in Fig. 2, ω/ω0 = 5. The rise and fall of Wdiss coincides with the growth and shrinkage of the cavity width as observed in Fig. 2. In dissipative systems, the importance of viscous forces relative to inertial forces is often quantified with a quality factor, Q. One possible definition is Q = tan−1 (φ) (assuming a harmonic system, which is not strictly the case here) where φ is the phase angle between the local pressure Ploc (t) and the radius R(t). This picture is motivated by the idea that in a experiment where nanoparticle vibrations are driven by light, the sphere responds to the pressure at the surface. This quantity is shown in Fig. 4(b) and shows a stronger increase for frequencies ω > 2ω0 , also reflecting the reduced dissipation as the sphere loses contact with the fluid over larger and larger fractions of the oscillation period. Of course, it should be noted that we are artificially forcing the oscillations in our experiment and so a more complete model is anticipated where the optical and mass transport are simultaneously considered. 25 We also do not consider heat transfer between the nanoparticle and the fluid, but we believe that the heating is of the order of 0.01K-0.1K at the intensities used in experiments 24 and can therefore be considered negligible. The present results provide a platform for interpreting our recent observation of an unusually strong increase of the acoustic nonlinear response of 2 nm gold nanoparticles suspended 7


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in water at frequencies above 1 Thz. 24 We speculated that this effect results from the formation of a small cavity around the nanoparticle, thus dramatically reducing the damping of the surrounding fluid. Our simulations cover a larger strain range than is likely being explored in this experiment, but show qualitatively that such a mechanism can indeed be operative. It is interesting to note that globular proteins in the 10-100 kDa range (∼ 5 nm in diameter) have their low order vibrational modes in the 50 GHz range and thus could benefit from this mechanism, leading to less dissipation in a normally highly viscous environment. The connection between the sound velocity in matter and the typical timescales associated with molecular dynamics in liquids make this an intrinsically nanoscale effect. In particular, nanoparticles typically have acoustic resonances in the > 10 GHz range because the sound velocity in matter is of the order of 3 km/s in matter. The typical collision rate of molecular interactions in liquids is comparable to this resonance frequency; for example, the low frequency response of water is fit with a Debye scattering rate of around 16 GHz (∼ 1/(2πτ )). 26 We have shown that the threshold for excitation is reduced significantly when the acoustic frequency is comparable to the associated frequency of the fluid. Therefore, in order to achieve natural lowest order resonances that are comparable to or exceed the characteristic frequency of the molecular dynamics of the liquid, it is required to have nanoparticles. Molecular dynamics simulations of Lennard-Jones fluids have been used extensively in related phenomena of cavity nucleation. 27 Our work also has implications for continuum calculations of vibrating nanostructures that model fluid and particles at the level of NavierStokes equations. 22,28 These calculations must make specific assumptions about the coupling of fluid and particle at the interface, such as continuity of velocities and normal stresses. In order to capture the physics revealed by our molecular simulations, these calculations should explicitly introduce a modified boundary layer, where the fluid density is dramatically reduced and the pressure is discontinuous. In conclusion, here we demonstrated via molecular dynamics simulations a dewetting

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phenomena that has not been considered or predicted by previous hydrodynamic investigations. This phenomena is significant because it allows for a cavity to form with reduced overall damping and high quality resonances, even though the nanoparticle is present within a viscous medium. The impact of these higher-quality GHz to THz vibrations of nanoparticles in solution is intriguing for applications in nonlinear optics (nano-optomechanics), the functioning of biomolecules (especially proteins), and potentially mass-loading based sensors that would benefit from solution-based high quality factors at high frequencies. 29–31 The authors gratefully acknowledge funding from the NSERC Discovery Grant program. The authors declare no competing financial interests.

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Figure 3: Pressures Ploc on sphere and Pf on the fluid together with radius R of the vibrating sphere for (a) ω = 0.1ω0 and (b) ω = 5ω0 and driving amplitude A=0.1.

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Figure 4: (a) Dissipated work as a function of frequency for several values of the driving amplitude A. (b) Estimate of the quality factor as Q = 1/ tan(φ), where φ is the phase angle between driving signal eq. (1) and local fluid response eq. (3).

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