# Explaining the low-frequency shear elasticity of confined liquids

^{a}Department of Physics “A. Pontremoli,” University of Milan, 20133 Milan, Italy;^{b}Department of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge CB3 0AS, United Kingdom;^{c}Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom;^{d}School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom

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Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved July 15, 2020 (received for review May 27, 2020)

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## Abstract

Experimental observations of unexpected shear rigidity in confined liquids, on very low frequency scales on the order of 0.01 to 0.1 Hz, call into question our basic understanding of the elasticity of liquids and have posed a challenge to theoretical models of the liquid state ever since. Here we combine the nonaffine theory of lattice dynamics valid for disordered condensed matter systems with the Frenkel theory of the liquid state. The emerging framework shows that applying confinement to a liquid can effectively suppress the low-frequency modes that are responsible for nonaffine soft mechanical response, thus leading to an effective increase of the liquid shear rigidity. The theory successfully predicts the scaling law

The elasticity of liquids is well understood in the high frequency limit of the mechanical response, where pioneering work by Frenkel (1) has shown that the response of a liquid is basically indistinguishable from that of an amorphous solid, provided the frequency of mechanical oscillation is sufficiently high. The idea here is that at short times (high frequency) the diffusive component of the liquid motion is absent and liquids behave as solids. This has become an accepted view (2). However, later experiments have challenged this view (3⇓⇓⇓–7) and found a remarkable solid-like property of liquids to support shear stress at very low frequency, albeit in confinement. This phenomenon is not currently understood. This is a limitation for the full development of small-scale nano-, micro-, and submillimeter flow technologies.

High-frequency mechanical response of liquids is typically measured with ultrasonic techniques in the megahertz range corresponding to shear elastic moduli of the order of gigapascals (8). The behavior is well described by Frenkel’s theory, which links it to transverse acoustic phonons and their vanishing at a characteristic internal time scale, the Frenkel time, which is related to the viscoelastic Maxwell time. Conversely, low-frequency shear elasticity has been identified fairly recently (in view of the long history of liquid research), starting with the pioneering work of Derjaguin et al. (3, 4) and of Noirez and coworkers (6, 7). The low-frequency elasticity of liquids is weaker, on the order of

Here we provide a description of liquid elasticity inspired by Frenkel’s ideas on the phonon theory of liquids, combined with recent developments in the microscopic theory of elasticity of amorphous materials. The resulting framework allows us to decompose the various contributions to liquid elasticity based on wavevector k, and thus to identify how the shear modulus of a liquid changes upon varying the confinement length L.

Following previous literature (9), we introduce the Hessian matrix of the system

As shown previously, the equation of motion of atom i, in mass-rescaled coordinates, can be written (9, 10)

The above equation of motion can be derived from a model particle-bath Hamiltonian as shown in previous work (9). Furthermore,

In liquids, a microscopic expression for **2** runs over all

As usual when dealing with eigenmodes, the sum over n (labeling the eigenmode number) can be replaced with a sum over wavevector k, with

We thus rewrite Eq. **2** in terms of a sum over k as follows:

In isotropic media, eigenmodes can be divided into longitudinal (L) and transverse (T) modes. Therefore we can split the sum in Eq. **3** into a sum over L modes and a sum over T modes,

One should note that while k is in general not a good quantum number in amorphous materials (as the connection between energy and wavevector is no longer single-valued as it is in crystals where Bloch’s theorem holds), it still can be used to provide successful descriptions of the properties of amorphous materials and liquids (13).

We now discuss the dispersion relations for longitudinal and transverse excitations in liquids. For the longitudinal modes, one can resort to the Hubbard–Beeby theory of collective modes in liquids (14), which has been shown to provide a good description of experimental data, and use equation 43 in ref. 14. As will be shown below, the final result for the low-frequency

Differently from the gapless longitudinal dispersion relations and generally from phonon dispersion relations in solids, liquids have the gap in k-space in the transverse phonon sector. This follows from the dispersion relation (15),

Eq. **6** follows from the Maxwell–Frenkel approach to liquids where the starting point of liquid description includes both elastic and viscous response (15) and implies that transverse modes in liquids propagate above the threshold value **6** becomes gapless and solid-like.

In a large system, **7**. We take the real part of **7**) in controlling the infrared cutoff of the transverse integral. In the experiments where the size effect of confinement is seen, **7**, leading to

We now compare Eq. **10** to available experimental data of low-frequency **2** has been successfully tested also for polymer melts in ref. 9). In Fig. 1 we compare the trend for the storage modulus **10**, with well-controlled experimental data of confined LC-polymer (**10** recovers the well-known result for liquids, that is,

In conclusion, we have developed an analytical theory of the shear modulus of liquids based on nonaffine atomic deformations. This approach allows us to decompose the nonaffine elasticity of the liquid into different phonon-like contributions in terms of their momentum k. Since the overall nonaffine/relaxational contribution to the low-frequency shear modulus is negative, and is expressed as an integral over k, the effect of confinement leads to an infrared (long-wavelength) cutoff of the k-integral. which is inversely proportional to confinement size L. This explains why reducing the confinement size L effectively increases the shear rigidity by suppressing long-wavelength nonaffine relaxations that soften the response. The predicted

## Data Availability.

All study data are included in the article.

## Acknowledgments

A.Z. acknowledges financial support from US Army Research Office, contract W911NF-19-2-0055. Prof. L. Noirez is gratefully acknowledged for input and discussions.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: alessio.zaccone{at}unimi.it.

Author contributions: A.Z. and K.T. designed research; A.Z. performed research; A.Z. and K.T. analyzed data; and A.Z. and K.T. wrote the paper.

The authors declare no competing interest.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

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