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When Can the Elastic Properties of Simple Liquids Be Probed Using High Frequency Nanoparticle Vibrations? Debadi Chakraborty, Gregory V Hartland, Matthew Pelton, and John Elie Sader J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09951 • Publication Date (Web): 01 Dec 2017 Downloaded from http://pubs.acs.org on December 19, 2017
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The Journal of Physical Chemistry
When can the Elastic Properties of Simple Liquids be Probed Using High Frequency Nanoparticle Vibrations?
Debadi Chakraborty1, Gregory V. Hartland2, Matthew Pelton3 and John E. Sader1* 1
ARC Centre of Excellence in Exciton Science, School of Mathematics and Statistics, The University of
Melbourne, Victoria 3010, Australia 2
Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556-
5670, USA 3
Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA
Corresponding author *E-mail:
[email protected] ACS Paragon Plus Environment
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ABSTRACT Recent measurements on the gigahertz vibration of nanoparticles immersed in simple liquids, such as glycerol, show that the liquid’s viscoelastic properties can significantly affect the nanoparticle’s mechanical response. Here, we theoretically explore the high-frequency (elastic) limit of this phenomenon where the characteristic time scale for molecular relaxation in the liquid far exceeds the nanoparticle’s vibration period. Paradoxically, we find that the effects of liquid elasticity (and viscosity) may not be visible in the nanoparticle’s dynamic response in this high-frequency elastic limit – the response being identical to that of a macroscopic resonator in an inviscid fluid. A comprehensive mechanistic study reveals that the conditions for this unusual behavior are strongly dependent on the nanoparticle’s vibration mode and the liquid’s properties. A judicious choice of vibration mode is essential for interrogating the viscoelastic properties of simple liquids. Our findings explain recent measurements on nanowires and nanorods immersed in highly viscous liquids.
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The Journal of Physical Chemistry
1. INTRODUCTION Newtonian fluid mechanics is utilized widely to describe the behavior of simple liquids such as water and glycerol.1-5 This framework holds strictly provided the time scale for molecular relaxation in the liquid is small relative to that of the flow. Since the characteristic relaxation times for these liquids typically range from ~1 ps to 1 ns,6-7 this requirement is well satisfied for most flows. However, recent measurements on nanoparticles immersed in simple liquids, using ultrafast laser spectroscopy, show that this traditional framework does not always hold for nanometer scale resonators.8 Strong viscoelastic response in simple liquids is observed and well characterized using (single relaxation time) Maxwell models with no fit parameters for the liquid properties. Both shear and compressional liquid motion have been measured, using the extensional modes of bipyramidal nanoparticles8 and the breathing modes of nanowires,9 respectively. In addition, ambiguity in the choice of constitutive model for compressional flows of these liquids was clarified recently10 and molecular dynamics simulations of the loss and storage moduli of glycerol, in both shear and compression, have been reported.11 A striking observation of these recent measurements is that the quality factor of the fundamental breathing mode of nanowires in pure glycerol is independent of the wire radius (and hence resonant frequency).9 The flows generated in these measurements are well into the high-frequency elastic regime, where the characteristic molecular relaxation time for the liquid (~300 ps) greatly exceeds the vibration period of the nanowire (~15 ps). Similarly, the quality factor for the fundamental extensional mode of bipyramidal (nanorod) particles immersed in glycerol/water mixtures is found to plateau and then increase slightly as the mass fraction of glycerol is increased; this coincides with a simultaneous increase in viscosity and molecular relaxation time of the liquid.8 These extensional mode measurements are also in the high frequency elastic limit for glycerol/water mixtures with high glycerol fraction.
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This measured behavior differs from established dogma stipulating that the quality factor of a resonator will decrease as the viscosity is enhanced and/or the resonator is miniaturized. The difference is explained by the stiffening effect of the liquid’s elasticity.8-9 However, quantitative analysis of the above-mentioned invariance of the quality factor with nanoparticle size and liquid viscosity, in the highfrequency elastic regime (defined above), is yet to be reported. The explanation of this effect is a primary aim of this study. We also elucidate the conditions for such behavior to occur. This information is critical to understanding when the dynamic response of nanoparticles can be used to probe the viscoelastic properties of simple liquids and, conversely, how the viscoelastic properties of liquids affect energy relaxation in nanomaterials.
2. THEORY The oscillation amplitude of the nanoparticle is considered to be much smaller than its size, so that its vibration response and that of the surrounding liquid are linear. The constitutive equation for a linear Maxwell compressible viscoelastic liquid is used,10 which recovers the requisite Newtonian and elastic behavior in the low and high-frequency limits, respectively. The stress tensor is = − + + , (1) where is the thermodynamic pressure, and +
∂ ∂ tr = 2 − , + = (tr ) . (2) ∂t 3 ∂t
Here, and are the shear and compressional relaxation times of the liquid, respectively, and are the shear and bulk viscosities, respectively, and are the shear and compressional
deviatoric stress tensors, respectively, = 1/2( ! + ( !)" ) is the rate-of-strain tensor, ! is the liquid velocity, is the identity tensor and # is time.
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The Journal of Physical Chemistry
To proceed, the explicit time dependence $ %&'( is assumed in all transport variables, where ) is
the imaginary unit; henceforth, we scale time by the angular oscillation frequency, Ω. Spatial dimensions
are scaled by the particle’s hydrodynamic size, +, whereas the maximum particle velocity, ,, scales the liquid velocity. The linearized continuity equation and Cauchy’s (linear) equation of motion for the liquid then become ∇ ∙ ! = )
/0 , (3a) 1
4 1 1 −)1! = − + 3 + 9 ( ⋅ !) + ∇0 !, (3b) 1 − )De 371 − )De 8 1 − )De where is now the pressure perturbation from equilibrium, the normalized wavenumber is / = Ω+/
