Can we define a unique microscopic pressure in inhomogeneous fluids?

Estimating water transport in carbon nanotubes: a critical review and inclusion of scale effects

The quasi-frictionless water flow across graphitic surfaces offers vast opportunities for a wide range of applications from biomedical science to energy. However, the conflicting experimental results impede a clear understanding of the transport mechanism and desired flow control. Existing literature proposes numerous modifications and updated boundary conditions to extend classical hydrodynamic theories for nanoflows, yet a consensus or definitive conclusion remains elusive. This study presents a critical review of the proposed modifications of the pressure driven flow or the Hagen-Poiseuille (HP) equations to estimate the flow enhancement through carbon nanotubes (CNTs). For such a case, we performed (semi-)classical molecular dynamics simulations of water flow in various sizes of CNTs, applied the different forms of boundary definitions from the literature, and derived HP equation models by implementing these modifications. By aggregating seven distinct experimental datasets, we tested various flow enhancement models against our measurements. Our findings indicate that including the interfacial layering-based dynamic slip-definition in the proposed HP equations yields accurate estimations. While considering interfacial viscosity predicts the individual CNT experiments well, using the experimental viscosity yields better estimations of measurements for the water flow enhancement through membranes of CNTs. This critical review testing existing literature demonstrates how to refine continuum fluid mechanics to predict water flow enhancement at the nanoscale providing holistic multiscale modeling.

Measuring line tension: Thermodynamic integration during detachment of a molecular dynamics droplet

The contact line (CL) is where solid, liquid, and vapor phases meet, and Young's equation describes the macroscopic force balance of the interfacial tensions between these three phases. These interfacial tensions are related to the nanoscale stress inhomogeneity appearing around the interface, and for curved CLs, e.g., a three-dimensional droplet, another force known as the line tension must be included in Young's equation. The line tension has units of force, acting parallel to the CL, and is required to incorporate the extra stress inhomogeneity around the CL into the force balance. Considering this feature, Bey et al. [J. Chem. Phys. 152, 094707 (2020)] reported a mechanical approach to extract the value of line tension τℓ from molecular dynamics (MD) simulations. In this study, we show a novel thermodynamics interpretation of the line tension as the free energy per CL length, and based on this interpretation, through MD simulations of a quasi-static detachment process of a quasi-two-dimensional droplet from a solid surface, we obtained the value τℓ as a function of the contact angle. The simulation scheme is considered to be an extension of a thermodynamic integration method, previously used to calculate the solid-liquid and solid-vapor interfacial tensions through a detachment process, extended here to the three-phase system. The obtained value agreed well with the result by Bey et al. and showed the validity of thermodynamic integration at the three-phase interface.

Anisotropic pressure effects on nanoconfined water within narrow graphene slit pores

There is an increasing interest toward disclosing and explaining confinement effects on liquids, such as water or aqueous solutions, in slit pore setups. Particularly puzzling are the changes of physical and chemical properties in the nanoconfinement regime where no bulk-like water phase exists between the two interfacial water layers such that the density profile across the slit pore becomes highly stratified, ultimately leading to bilayer and monolayer water. These changes must be quantified with respect to some meaningful reference state of water, the most natural one being bulk water at the same pressure and temperature conditions. However, bulk water is a homogeneous liquid with isotropic properties, whereas water confined in slit pores is inhomogeneous, implying anisotropic properties as described by the perpendicular and parallel components of the respective tensors. In the case of pressure, the inhomogeneous nature of the setup results in a well-defined difference between the perpendicular and parallel pressure tensor components that is uniquely determined by the interfacial tension being a thermodynamic property. For bilayer water constrained in graphene slit pores that are only about 1 nm wide, we demonstrate that there exists a thermodynamic point where the pressure tensor of the inhomogeneous fluid, nanoconfined water, is effectively isotopic and the pressure is thus scalar as in the homogeneous fluid, bulk water. This specific point of vanishing effective interfacial tension is proposed to serve as a well-defined reference state to compare the properties of nanoconfined liquids to those of the corresponding bulk liquid at the same (isotropic) pressure and temperature conditions. In future work, this idea could be applied to assess confinement effects on chemical reactivity in aqueous solutions as well as to other nanoconfined liquids in other pores such as layered minerals.

Impact of the Force Field on the Calculation of Density and Surface Tension of Epoxy-Resins

The molecular simulation of interfacial systems is a matter of debate because of the choice of many input parameters that can affect significantly the performance of the force field of reproducing the surface tension and the coexisting densities. After developing a robust methodology for the calculation of the surface tension on a Lennard-Jones fluid, we apply it with different force fields to calculate the density and surface tension of pure constituents of epoxy resins. By using the model that best reproduces the experimental density and surface tension, we investigate the impact of composition in mass fraction on uncured epoxy resins and the effects of degree of cross-linking on cured resins.

A perspective on the microscopic pressure (stress) tensor: History, current understanding, and future challenges

The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this Perspective, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and nonequilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties with many-body and long-range potentials; the insufficiency of software and computational tools; and the lack of experimental routes to probe the pressure tensor at the nanoscale. Possible future directions are suggested.

Interfacial Forces in Free-Standing Layers of Melted Polyethylene, from Critical to Nanoscopic Thicknesses

Molecular dynamics simulations of ultrathin free-standing layers made of melted (373.15–673.15 K) polyethylene chains, which exhibit a lower melting temperature (compared to the bulk value), were carried out to investigate the dominant pressure forces that shape the conformation of chains at the interfacial and bulk liquid regions. We investigated layer thicknesses, tL, from the critical limit of mechanical stability up to lengths of tens of nm and found a normal distribution of bonds dominated by slightly stretched chains across the entire layer, even at large temperatures. In the bulk region, the contribution of bond vibrations to pressure was one order of magnitude larger than the contributions from interchain interactions, which changed from cohesive to noncohesive at larger temperatures just at a transition temperature that was found to be close to the experimentally derived onset temperature for thermal stability. The interchain interactions produced noncohesive interfacial regions at all temperatures in both directions (normal and lateral to the surface layer). Predictions for the value of the surface tension, γ, were consistent with experimental results and were independent of tL. However, the real interfacial thickness—measured from the outermost part of the interface up to the point where γ reached its maximum value—was found to be dependent on tL, located at a distance of 62 Å from the Gibbs dividing surface in the largest layer studied (1568 chains or 313,600 bins); this was ~4 times the length of the interfacial thickness measured in the density profiles.

Equation of state for confined fluids

Fluids confined in small volumes behave differently than fluids in bulk systems. For bulk systems, a compact summary of the system’s thermodynamic properties is provided by equations of state. However, there is currently a lack of successful methods to predict the thermodynamic properties of confined fluids by use of equations of state, since their thermodynamic state depends on additional parameters introduced by the enclosing surface. In this work, we present a consistent thermodynamic framework that represents an equation of state for pure, confined fluids. The total system is decomposed into a bulk phase in equilibrium with a surface phase. The equation of state is based on an existing, accurate description of the bulk fluid and uses Gibbs’ framework for surface excess properties to consistently incorporate contributions from the surface. We apply the equation of state to a Lennard-Jones spline fluid confined by a spherical surface with a Weeks–Chandler–Andersen wall-potential. The pressure and internal energy predicted from the equation of state are in good agreement with the properties obtained directly from molecular dynamics simulations. We find that when the location of the dividing surface is chosen appropriately, the properties of highly curved surfaces can be predicted from those of a planar surface. The choice of the dividing surface affects the magnitude of the surface excess properties and its curvature dependence, but the properties of the total system remain unchanged. The framework can predict the properties of confined systems with a wide range of geometries, sizes, interparticle interactions, and wall–particle interactions, and it is independent of ensemble. A targeted area of use is the prediction of thermodynamic properties in porous media, for which a possible application of the framework is elaborated.

Molecular interactions at the metal-liquid interfaces

We reported molecular simulations of the interactions among water, an epoxy prepolymer diglycidic ether of bisphenol A (DGEBA), and a hardener isophorone diamine (IPDA) on an aluminum surface. This work proposes a comprehensive thermodynamic characterization of the adhesion process from the calculation of different interfacial tensions. The cross-interactions between the atoms of the metal surface and different molecules are adjusted so as to reproduce the experimental work of adhesion. Water nanodroplets on the metal surface are then simulated to predict their contact angle. Liquid-vapor surface tensions of the epoxy prepolymer (DGEBA) and hardener (IPDA) and the solid-vapor surface tension of the aluminum surface are also calculated to provide the solid-liquid interfacial tension that remains very difficult to obtain from the mechanical definition.

Soret separation and thermo-osmosis in porous media

When a temperature difference is applied over a porous medium soaked with a fluid mixture, two effects may be observed, a component separation (the Ludwig-Soret effect, thermodiffusion) and a pressure difference due to thermo-osmosis. In this work, we have studied both effects using non-equilibrium thermodynamics and molecular dynamics. We have derived expressions for the two characteristic parameters, the Soret coefficient and the thermo-osmotic coefficient in terms of phenomenological transport coefficients, and we show how they are related. Numerical values for these coefficients were obtained for a two-component fluid in a solid matrix where both fluid and solid are Lennard-Jones/spline particles. We found that both effects depend strongly on the porosity of the medium and weakly on the interactions between the fluid components and the matrix. The Soret coefficient depends strongly on whether the fluid is sampled from inside the porous medium or from bulk phases outside, which must be considered in experimental measurements using packed columns. If we use a methane/decane mixture in bulk as an example, our results for the Soret coefficient give that a temperature difference of 10 K will separate the mixture to about 49.5/50.5 and give no pressure difference. In a reservoir with 30% porosity, the separation will be 49.8/50.2, whereas the pressure difference will be about 15 bar. Thermo-osmotic pressures with this order or magnitude have been observed in frost-heave experiments.

How to produce confidence intervals instead of confidence tricks: Representative sampling for molecular simulations of fluid self-diffusion under nanoscale confinement

Ergodicity (or at least the tantalizing promise of it) is a core animating principle of molecular-dynamics (MD) simulations: Put simply, sample for long enough (in time), and you will make representative visits to states of a system all throughout phase space, consistent with the desired statistical ensemble. However, one is not guaranteed a priori that the chosen window of sampling in a production run is sufficiently long to avoid problematically non-ergodic observations; one is also not guaranteed that successive measurements of an observable are statistically independent of each other. In this paper, we investigate several particularly striking and troublesome examples of statistical correlations in MD simulations of nanoconfined fluids, which have profound implications on the quantification of uncertainty for transport phenomena in these systems. In particular, we show that these correlations can lead to confidence intervals on the fluid self-diffusion coefficient that are dramatically overconfident and estimates of this transport quantity that are simply inaccurate. We propose a simple approach—based on the thermally accelerated decorrelation of fluid positions and momenta—that ameliorates these issues and improves our confidence in MD measurements of nanoconfined fluid transport properties. We demonstrate that the formation of faithful confidence intervals for measurements of self-diffusion under nanoscale confinement typically requires at least 20 statistically independent samples, and potentially more depending on the sampling technique used.

Local stress tensor calculation by the method-of-plane in microscopic systems with macroscopic flow: A formulation based on the velocity distribution function

In this work, we developed a calculation method of local stress tensor applicable to non-equilibrium molecular dynamics (NEMD) systems, which evaluates the macroscopic momentum advection and the kinetic term of the stress in the framework of the Method-of-Plane (MoP), in a consistent way to guarantee the mass and momentum conservation. From the relation between the macroscopic velocity distribution function and the microscopic molecular passage across a fixed control plane, we derived a method to calculate the basic properties of the macroscopic momentum conservation law including the density, the velocity, the momentum flux, and the two terms of the stress tensor, i.e., the interaction and the kinetic terms, defined on a surface with a finite area. Any component of the streaming velocity can be obtained on a control surface, which enables the separation of the kinetic momentum flux into the advection and stress terms in the framework of MoP, and this enables strict satisfaction of the mass and momentum conservation for an arbitrary closed control volume (CV) set in NEMD systems. We validated the present method through the extraction of the density, velocity, and stress distributions in a quasi-one-dimensional steady-state Couette flow system and in a quasi-2D steady-state NEMD system with a moving contact line. We showed that with the present MoP, in contrast to the volume average method, the conservation law was satisfied even for a CV set around the moving contact line, which was located in a strongly inhomogeneous region.