Microscale theory of surface tension

The fourth law of thermodynamics: steepest entropy ascent

When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose 'greatness' derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper, we show that in the past four decades, an enormous body of scientific research devoted to modelling the essential features of non-equilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the 'fourth law of thermodynamics'. We state it as follows: every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-non-equilibrium realm within the framework of the rate-controlled constrained-equilibrium approximation (also known as the quasi-equilibrium approximation). This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Cusps and cuspidal edges at fluid interfaces: Existence and application

One of the intriguing questions in fluid dynamics is on the interrelation between dynamic singularities in the solutions of fluid dynamic equations - unboundedness of the velocity field in an appropriate norm - and the geometric ones - divergence of curvature at fluid interfaces. The present work focuses on two generic interfacial singularities - genuine cusps and cuspidal edges - found here in both two and three dimensions thus establishing a relation between real fluid interfaces and geometric singularity theory. The key finding is the necessary condition for the existence of geometric singularities, which is a variation of surface tension. It is also established here that the dynamic and geometric singularities entail each other only in the case of three-dimensional cusps. Explicit asymptotic solutions for the flow field and interface shape near steady-state singularities at fluid interfaces are developed as well. The practical motivation for the present study comes from the fundamental role interfacial singularities play in sustaining self-driven conversion of chemical into mechanical energy.

Extending the nonequilibrium square-gradient model with temperature-dependent influence parameters

Nonequilibrium interface phenomena play a key role in crystallization, hydrate formation, pipeline depressurization, and a multitude of other examples. Square gradient theory extended to the nonequilibrium domain is a powerful tool for understanding these processes. The theory gives an accurate prediction of surface tension at equilibrium, only with temperature-dependent influence parameters. We extend in this work the nonequilibrium square gradient model to have temperature-dependent influence parameters. The extension leads to thermodynamic quantities which depend on temperature gradients. Remarkably the Gibbs relation proposed in earlier work is still valid. Also for the extended framework, the "Gibbs surface" described by excess variables is found to be in local equilibrium. The temperature-dependent influence parameters give significantly different interface resistivities (∼9%-50%), due to changed density gradients and additional terms in the enthalpy. The presented framework facilitates a more accurate description of transport across interfaces with square gradient theory.

Effects of quenching rate and viscosity on spinodal decomposition

Spinodal decomposition of deeply quenched mixtures is studied experimentally, with particular emphasis on the domain growth rate during the late stage of coarsening. We provide some experimental evidence that at high Péclet number, the process is isotropic and the domain growth is linear in time, even at finite quenching rates. In fact, the quenching rate appears to influence the magnitude of the growth rate, but not its scaling law. In the second part of the work we analyze the effect of viscosity on the growth rate. As predicted by the diffuse interface model, we do not find any effect of viscosity on the growth rate of the nucleating drops, although, as expected, the viscosity of the continuous phase does influence the settling speed and thus the total separation time.

Condensation of a vapor bubble in submicrometer container

Condensation of a spherically symmetric submicrometer size vapor bubble is studied using diffuse interface hydrodynamic model supplemented by the van der Waals equation of state with parameters characteristic for argon. The bubble, surrounded by liquid, is held in a container of constant volume with temperature of the wall kept fixed. The condensation is triggered by a sudden rise of the wall temperature. We find that in the same container and subjected to a similar increase of the wall temperature the condensation process is totally different from the opposite process of droplet evaporation. In particular, the rapid change of the wall temperature excites the wave, which hits the interface and compresses the bubble, leading to a considerable increase of the temperature inside. The condensation of the submicrometer size bubble takes tens of nanoseconds, whereas evaporation of the same size droplet lasts roughly 50 times longer. In contrast to evaporation the condensation process is hardly quasistationary.

Smoothed particle hydrodynamics model for phase separating fluid mixtures. I. General equations

We present a thermodynamically consistent discrete fluid particle model for the simulation of a recently proposed set of hydrodynamic equations for a phase separating van der Waals fluid mixture [P. Español and C.A.P. Thieulot, J. Chem. Phys. 118, 9109 (2003)]. The discrete model is formulated by following a discretization procedure given by the smoothed particle hydrodynamics (SPH) method within the thermodynamically consistent general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework. Each fluid particle carries information on the mass, momentum, energy, and the mass fraction of the different components. The discrete model allows one to simulate nonisothermal dynamic evolution of phase separating fluids with surface tension effects while respecting the first and second laws of thermodynamics exactly.

Evaporation of a Sub-Micrometer Droplet

Evaporation of a spherically symmetric sub-micrometer size liquid droplet is studied using a diffuse interface hydrodynamic model supplemented by the van der Waals equation of state with parameters characteristic for argon. The droplet, surrounded by saturated vapor, is held in a container with the temperature of the walls kept fixed. The evaporation is triggered by a sudden rise of the temperature of the walls. Time and space evolution of the basic thermodynamic quantities is presented. The time and space scales studied range from picoseconds to microseconds and from nanometers to micrometers, respectively. We find that the temperature and chemical potential are both continuous at the interface on the scale larger than the interfacial width. We find that at long times the radius R of the droplet changes with time t as R(2)(t) = R(2)(0) - 2tkappa(v)(T(w) - T(l))/ln(l), where kappa(v) is the heat conductivity of the vapor, n(l) and T(l) are the density and the temperature of liquid inside the droplet, respectively, l is the latent heat of transition per molecule, and T(w) is the temperature of the ambient vapor.

Evaporation of a thin liquid film

Evaporation of a thin (submicrometer size) liquid film confined between two solid substrates is studied using diffuse interface hydrodynamic model supplemented by the van der Waals equation of state. The time and space evolution of the basic thermodynamic quantities such as temperature, density, entropy, chemical potential, and entropy production is presented. The values of numerical parameters chosen correspond to those of argon. The time and space scales studied range from picoseconds to microseconds and from nanometers to micrometers correspondingly.

Kinetic derivation of the hydrodynamic equations for capillary fluids

Based on the generalized kinetic equation for the one-particle distribution function with a small source, the transition from the kinetic to the hydrodynamic description of many-particle systems is performed. The basic feature of this interesting technique to obtain the hydrodynamic limit is that the latter has been partially incorporated into the kinetic equation itself. The hydrodynamic equations for capillary fluids are derived from the characteristic function for the local moments of the distribution function. Fick's law appears as a consequence of the transformation law for the hydrodynamic quantities under time inversion.

Density change effects on crystal growth from the melt

When a crystal grows from its undercooled melt the local density changes, driving a convective flow in the liquid phase. Then, the purely diffusional description of the process ceases to be satisfactory. Moreover, the dynamic pressure associated with the flow field may affect the melting temperature (and the effective undercooling) of the system. Both these effects have been addressed in recent experimental work. In the present study we derive a thermodynamically consistent phase-field model that accounts for the density change effects in the solidification of a pure substance. Starting from a thermodynamic potential that includes squared gradient terms for both the order parameter and the density, the field equations are derived assuming positive local entropy production. The model is numerically solved in one dimension to show deviations from the classic phase-field description of the same phenomenon.