Can Confinement-Induced Variations in the Viscous Dissipation be Measured?
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Liquids confined to molecular scales become anisotropic and often show pronounced self-organization such as layering. Although this effect is well accepted, it is still debated if confinement induces measurable changes of viscous friction. We use molecular dynamics to address this issue by simulating a Lennard-Jones liquid confined between a solid cylinder and an atomically smooth surface. The simulations reproduce the well-established variations of normal force, density, and diffusivity with the distance between wall and cylinder. We find high diffusivity and low density when the numbers of layers is in between integers. This observation seems to contradict most experimental results on the effective damping between atomic force microscope tips and substrates when interpreting them within continuum hydrodynamics used to connect liquid viscosity and diffusivity. This contradiction is resolved by directly extracting the damping that the tip experiences, which we achieve by using the fluctuation-dissipation theorem; as in experiment, we find local minima in the damping near integer numbers of molecular layers and maxima in between. These variations correlate with distinct structural changes in the microscopic order of the fluid. We reconfirm that constitutive equations valid at macroscopic scales cannot be used to interpret confined liquids and finally conclude that viscous friction displays measurable, non-monotonic behavior with the degree of confinement.
KeywordsNanotribology Lubrication Molecular dynamics
The dissipation in liquid layers confined to a few nanometers plays a crucial role for friction experienced by lubricated contacts [1, 2]. While there is overwhelming evidence that simple liquids can be described by standard continuum models using the bulk viscosity and density down to length scales of 10 nm, deviations occur when the liquid thickness is reduced to a few molecular diameters . Seminal work using the Surface Forces Apparatus (SFA) established that the presence of solid walls induces a layered structure in the liquid leading to the so-called oscillatory solvation forces . X-ray reflectometry studies  as well as theory and numerical simulations [6, 7] demonstrated the existence of an oscillatory density profile, which decays approximately exponentially with a decay length of a few molecular diameters as a function of the distance from the wall. While this behavior of the conservative forces is well understood and has also been reproduced for nanoscale contacts in early Atomic Force Microscopy (AFM) experiments , the dissipative behavior of nano-confined liquid films is heavily debated. In the SFA community, slightly different measurement techniques for nominally identical systems produced highly incompatible experimental results with interpretations ranging from confinement-induced solidification  or a glassy response  to bulk-like dynamics for liquid films of even just a few molecular layers thickness [11, 12, 13]. SFA measurements are limited to film thicknesses of integer numbers of molecular layers due to a mechanical (“jump-to-contact”) instability. In contrast, dynamic AFM allows for a more complete characterization of the conservative forces and the dissipation including in particular film thicknesses corresponding to non-integer numbers of molecular layers. Notwithstanding early controversies [14, 15, 16], there is a growing consensus that the dissipation experienced in lubricated single-asperity nano-contacts varies in a non-monotonic manner as a function of the thickness of the confined liquid film: Qualitatively similar behavior was observed for a variety of liquids including the simple Lennard-Jones liquid Octamethylcyclotetrasiloxane (OMCTS) [17, 18, 19], dodecanol , and water  (also depending on the ion concentration ). Although, chemical details are known to qualitatively alter the friction forces [23, 24], this similarity in observations using a large variety of liquids with rather disparate molecular properties and solid–liquid interactions  suggests the presence of a generic underlying principle.
In this letter, we describe equilibrium molecular dynamics (MD) simulations of a confined Lennard-Jones liquid that mimics the experimental conditions of AFM experiments with OMCTS and graphite surfaces. Early non-equilibrium simulations [26, 27], applied to study the dissipative properties, were often performed at high shear rates while only for sufficiently low velocities the experimentally examined Stokes friction can be extracted . Instead, we analyze the temporal fluctuations of the force and make use of the fluctuation-dissipation theorem to extract the dissipation from our equilibrium simulations. Therefore, and in contrast to previous equilibrium MD simulations of confined liquids [29, 30], our study is not limited to the equilibrium properties such as density profiles, oscillatory solvation forces, and diffusivities. In this manner, we reproduce for the first time the distance dependence of the damping measured in AFM experiments.
2 Background and Technical Aspects
The tip is placed in the middle of the simulation box creating two gaps of equal width at the top and at the bottom of the graphite surface, respectively. While the tip is cylindrically symmetric, the simulation box is cubic (lateral dimensions ~45 × 45 nm) with periodic boundary conditions in x and y. The simulations are carried out as follows: the system containing approximately 100.000 liquid molecules is first equilibrated for 60 ns at a constant temperature and pressure of T = 0.88 ε/k B and P = 3.3 ε/σ 3, respectively. Since confinement can move the liquid–solid transition line in the phase diagram , the vicinity to the melting point may affect our results, in the simulations and experiments alike. Temperature and pressure are kept constant using Toxvaerds implementation of the Nosé-Hoover thermostat and barostat . Integration of the equations of motion is carried out using the leap-frog Verlet algorithm . Following equilibration, barostat and thermostat are turned off and we let the system evolve at constant particle number N, volume V, and energy E for another 16 ns. The forces acting on the tip are calculated from these runs at constant N, V, E. Simulations are carried out for various spacings between the solid surfaces leading to gap widths D ranging from 0.5 to 3.9 nm.
To quantify the variations of the atomic mobility we trace the mean square displacements of the atoms in the region underneath the spherical tip. This analysis reveals two aspects: first, the mean square displacements (MSDs) increase linearly in time within the time window of the simulations, both for the x–y and for the z-direction. The linear behavior of the MSDs allows for the definition of diffusion coefficients D xy and D z, being the slope of the means square displacement versus time. Second, the MSDs are highly anisotropic (see Fig. 4a). While D xy and D z are small compared to the bulk diffusion coefficient, D xy is still substantially larger than D z. Qualitatively, a reduction of the diffusivity is of course expected. Already macroscopic fluid dynamics predicts a reduction of the diffusivity due to the interaction of the diffusing atom with its image on the opposite side of the wall [37, 38, 39]. Yet, the variations observed here are non-monotonic and reflect the layered structure of the liquid. The increase in density under the tip—and hence the decrease in space available for the site-exchange processes required for molecular diffusion—lead to a dramatic reduction of the diffusivity as the layers are compressed. Once the gap width is reduced to such an extent that the number of layers is reduced, the average density decreases again and molecules can diffuse faster again (Fig. 4e).
The results presented here provide the first numerical confirmation of the rather generic experimental observation that the dissipation in confined liquid films shows distinct maxima and minima as a function of the gap width. Moreover, the maxima in the damping are found at transition distances 3 ↔ 2, 2 ↔ 1 and 1 ↔ 0, very similar as observed experimentally. Throughout all our simulations, we found no indication that the system can get arrested in non-equilibrium states. The system always converged towards the structures shown above, irrespective of the initial conditions. From a physical perspective, this may not seem surprising given the typical time scales of molecular motion and the small size of the system. Yet, various experimental studies in the literature reported a strong dependence of the mechanical response of the liquid under investigation on the (very slow) approach rate of the cantilever (~Å/s) [17, 21]. A qualitatively similar dependence was reported in SFA experiments . We suspect that this dependence is related to the rather large lateral dimensions of the confinement region in these experiments: The tips in the two AFM experiments were rather blunt with tip radii of order >500 nm . Under such conditions, it is easier to conceive that the molecules get “jammed” into some non-equilibrium configuration than for the system studied here, which displays nano-scale dimensions in all directions.
It is also interesting to compare the present results to previous interpretations of so-called confinement-induced solidification. For the present system, the material in the gap clearly crystallizes provided that the gap width corresponds to one of the plateau regions. At the same time, the dissipation due to the confined film remains at a very low level—and is in fact compatible with the same (noise) level that is also found at the largest gap widths: The dissipation of a thin solid film is just as low as for a thicker liquid film of small viscosity. This indicates that the dissipation experienced by an AFM cantilever oscillating in the vertical direction is not sensitive to a possible solidification of the layer. Similar conclusions apply for the tangential xy motion. Distinguishing between the two states seems to require a different experimental technique.
These results indicate the behavior of the liquid is more complex than just a viscous liquid or an elastic solid. However, further research and especially longer simulations are required.
In summary, we showed that MD simulations of simple confined Lennard-Jones fluids are able to reproduce the characteristic peaks of excess dissipation that were found in a series of recent AFM experiments on confined fluids. The simulations suggest a generic relation between the dissipation measured in AFM experiments and the molecular scale packing of the molecules in the confined liquid layers. Highly ordered layers display little dissipation—a less ordered structure entails excess dissipation.
The authors thank M. H. Müser for the discussions and his help on the manuscript. This work has been supported by the Foundation for Fundamental Research of Matter (FOM), which is financially supported by the Netherlands Organization for Scientific Research (NWO).
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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