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The approach of global isomorphism between the fluid and the Ising model is applied to obtain an expression for the surface tension of the Lennard-Jones fluid on the basis of the information about the Ising model. This is done in a broad interval of temperatures along the phase coexistence, and is valid both in 2D and 3D. The relation between the critical amplitudes of the surface tension of the fluid and the Ising model is derived in the vicinity of the critical point. The obtained theoretical estimates agree well with the literature results for the surface tension. The methodology is demonstrated for the 2D LJ fluid on the basis of the exact solution of the 2D Ising model and is tested for the 3D LJ fluid. As a result, an expression for the surface tension without any fitting parameter is derived., Comment: 3 figures, conference PLMMP 2016, Kiev, May 27-30

International audience; We show that the solvent behaviour in both diffusio-osmosis and Marangoni flow can be derived from a simple model of colloid-interface interactions. We demonstrate that the direction of the flow is regulated by a single value of the attractive parameter covering the purely repulsive and attractive – repulsive interaction cases. The proposed universality between diffusio-osmosis and Marangoni flow is extended further to include diffusio-phoresis. In particular, an object immersed to a colloidal solution moves towards the low concentration of the colloidal particles in the case of colloid-interface repulsion and towards the high concentration of the colloidal particles in the case of colloid-interface attraction. The approach combines the methods of fluid dynamics, molecular physics and transport phenomena and provides a tractable explanation of how the colloid-interface interactions affect the momentum balance and the transport phenomena (interfacially driven transport).

We present a systematic procedure to describe fluid-fluid interfaces of multi-component systems out of equilibrium in a clear macroscopic context on a basis of non-equilibrium thermodynamics. The macroscopic description uses, as Gibbs did for equilibrium interfaces, excess densities. Comparing it with the local description which is based on the non-equilibrium square gradient theory, we show that two descriptions are consistent. This implies that out of equilibrium one can still use interfacial excess densities as if they are functions only of the local temperature and chemical potentials. This justifies that the surface is a separate thermodynamic system and is in local equilibrium.

We show that the equations which describe irreversible evolution of a system can be derived from a variational principle. We suggest a Lagrangian, which depends on the properties of the normal and the so-called "mirror-image" system. The Lagrangian is symmetric in time and therefore compatible with microscopic reversibility. The evolution equations in the normal and mirror-imaged systems are decoupled and describe therefore independent irreversible evolution of each of the systems. The second law of thermodynamics follows from a symmetry of the Lagrangian. Entropy increase in the normal system is balanced by the entropy decrease in the mirror-image system, such that there exists an "integral of evolution" which is a constant. The derivation relies on the property of local equilibrium, which states that the local relations between the thermodynamic quantities in non-equilibrium are the same as in equilibrium.

We present theoretical expressions for the density, strain rate, and shear pressure profiles in strongly inhomogeneous fluids undergoing steady shear flow with periodic boundary conditions. The expressions that we obtain take the form of truncated functional expansions. In these functional expansions, the independent variables are the spatially sinusoidal longitudinal and transverse forces that we apply in nonequilibrium molecular-dynamics simulations. The longitudinal force produces strong density inhomogeneity, and the transverse force produces sinusoidal shear. The functional expansions define new material properties, the response functions, which characterize the system's nonlocal response to the longitudinal force and the transverse force. We find that the sinusoidal longitudinal force, which is mainly responsible for the generation of density inhomogeneity, also modulates the strain rate and shear pressure profiles. Likewise, we find that the sinusoidal transverse force, which is mainly responsible for the generation of sinusoidal shear flow, can also modify the density. These cross couplings between density inhomogeneity and shear flow are also characterized by nonlocal response functions. We conduct nonequilibrium molecular-dynamics simulations to calculate all of the response functions needed to describe the response of the system for weak shear flow in the presence of strong density inhomogeneity up to the third order in the functional expansion. The response functions are then substituted directly into the truncated functional expansions and used to predict the density, velocity, and shear pressure profiles. The results are compared to the directly evaluated profiles from molecular-dynamics simulations, and we find that the predicted profiles from the truncated functional expansions are in excellent agreement with the directly computed density, velocity, and shear pressure profiles.

We have recently realized that some of the equations in this paper contain an error. In the derivation in Sec. VIII we assumed that the viscous pressure tensor παβ is symmetric. In general this is the case for nonpolarizable fluids, if we consider the bulk region, cf. [1], p. 304. This is not correct, however, in the interfacial region even for nonpolarizable fluids. One can have a (parallel) velocity gradient normal to the surface caused by a viscous force along the surface, as well as a (normal) velocity gradient along the surface caused by a viscous force normal to the surface. These phenomena are independent, which causes the viscous pressure tensor to not be symmetric. The error does not affect the results, since we did not study the viscous phenomena in the paper. Therefore the following correction should be made. The second equation of Eq. (7.6) should look like the following

One may think of an alternative definition of an excess quantity $$ \widehat{\phi}_{a}(x^{s}, x_{2}, x_{3}) \equiv \int\limits_{ x^{g,\,s}}^{ x^{\ell,\,s}}{dx_{1}\,{{\mathfrak{h}}}_{1}\,\phi^{\rm ex}({{\mathbf{r}}}; x^{s})} $$ (6.45) We note however, that \(\widehat{\phi}_{a}\) has no physical meaning, while \(\widehat{\phi}\) has. The reason for this is that $$ \Upphi \equiv \int\int\limits_{S}{dx_{2}\,dx_{3}\,{{\mathfrak{h}}}_{2}^{s}\,{{\mathfrak{h}}}_{3}^{s}\,\widehat{\phi}(x^{s}, x_{2}, x_{3})} = \int\int\limits_{S}\int\limits_{ x^{g,\,s}}^{ x^{\ell,\,s}}{dx_{1}\,dx_{2}\,dx_{3}\, {{\mathfrak{h}}}_{1}\,{{\mathfrak{h}}}_{2} \,{{\mathfrak{h}}}_{3}\,\phi^{\rm ex}({{\mathbf{r}}}; x^{s})} $$ (6.46) is the total amount of some physical quantity in the volume which is limited by the surfaces \(S\) at \(x^{g,\,s}\) and \(x^{\ell,\,s},\) while $$ \Upphi_{a} \equiv \int\int\limits_{S}{dx_{2}\,dx_{3}\,{{\mathfrak{h}}}_{2}^{s}\,{{\mathfrak{h}}}_{3}^{s}\,\widehat{\phi}_{a}(x^{s}, x_{2}, x_{3})} = \int\int\limits_{S}\int\limits_{ x^{g,\,s}}^{ x^{\ell,\,s}}{dx_{1}\,dx_{2}\,dx_{3}\,{{\mathfrak{h}}}_{1}\,{{\mathfrak{h}}}_{2}^{s}\,{{\mathfrak{h}}}_{3}^{s}\,\phi^{\rm ex}({{\mathbf{r}}}; x^{s})} $$ (6.47) is not. If the interfacial thickness is small compared to the radii of curvature, the difference between \(\widehat{\phi}\) and \(\widehat{\phi}_{a}\) is small and vanishes for planar interface considered under cartesian coordinates. However, it is \(\Upphi\) but not \(\Upphi_{a}\) which is a physical amount, and thus \(\widehat{\phi}\) is the surface density.

In this paper we apply the general analysis described in our first paper to a binary mixture of cyclohexane and n -hexane. We use the square gradient model for the continuous description of a nonequilibrium surface and obtain numerical profiles of various thermodynamic quantities in various stationary state conditions. Details of the numerical procedure are given and discussed. In the second part of this paper we focus on the verification of local equilibrium of the surface as described with excess densities. We give a definition of the temperature and chemical potentials of the surface and verify that these quantities are independent of the choice of the dividing surface. We verify numerically that the surface in a stationary state of the mixture can be described in terms of Gibbs excess densities, which are found to be in good approximation equal to their equilibrium values at the stationary state temperature and chemical potentials of the surface.

In earlier work a systematic extension of the van der Waals square gradient model to nonequilibrium one-component systems was given. In this work the focus was on heat and mass transfer through the liquid-vapor interface as caused by a temperature difference or an over- or underpressure. We will give an extension of this approach to multicomponent nonequilibrium systems in the systematic context of nonequilibrium thermodynamics. An explicit expression for the pressure tensor is derived valid also for curved surfaces. It is shown how the Gibbs relation should be modified in the interfacial region, in both equilibrium and nonequilibrium. The two-dimensional isotropy of a curved interface is discussed. Furthermore, we give numerically obtained profiles of the concentration, the mole fraction, and the temperature, which illustrate the solution for some special cases.

Validity of local equilibrium has been questioned for non-equilibrium systems which are characterized by delayed response. In particular, for systems with non-zero thermodynamic inertia, the assumption of local equilibrium leads to negative values of the entropy production, which is in contradiction with the second law of thermodynamics. In this paper, we address this question by suggesting a variational formulation of irreversible evolution of a system with non-zero thermodynamic inertia. We introduce the Lagrangian, which depends on the properties of the normal and the so-called "mirror-image" systems. We show that the standard evolution equations, in particular, the Maxwell-Cattaneo-Vernotte equation, can be derived from the variational procedure without going beyond the assumption of local equilibrium. We also argue that the second law of thermodynamics in non-equilibrium should be understood as a consequence of the variational procedure and the property of local equilibrium. For systems with instantaneous response this leads to the standard requirement of the local instantaneous entropy production being always positive. However, if a system is characterized by delayed response, the formulation of the second law of thermodynamics should be altered. In particular, the quantity, which is always positive, is not the instantaneous entropy production, but the entropy production averaged over a proper time interval.

We analyze the local equilibrium assumption for interfaces from the perspective of gauge transformations, which are the small displacements of Gibbs' dividing surface. The gauge invariance of thermodynamic properties turns out to be equivalent to conditions for jumps of bulk densities across the interface. This insight strengthens the foundations of the local equilibrium assumption for interfaces and can be used to characterize nonequilibrium interfaces in a compact and consistent way, with a clear focus on gauge-invariant properties. Using the principle of gauge invariance, we show that the validity of Clapeyron equations can be extended to nonequilibrium interfaces, and an additional jump condition for the momentum density is recognized to be of the Clapeyron type.