Determination of the origin and magnitude of logarithmic finite-size effects on interfacial tension: Role of interfacial fluctuations and domain breathing
The ensemble-switch method for computing wall excess free energies of condensed matter is extended to estimate the interface free energies between coexisting phases very accurately. By this method, system geometries with linear dimensions $L$ parallel and $L_z$ perpendicular to the interface with various boundary conditions in the canonical or grandcanonical ensemble can be studied. Using two- and three-dimensional Ising models, the nature of the occurring logarithmic finite size corrections is studied. It is found crucial to include interfacial fluctuations due to "domain breathing".
A mean-field density-functional model for three-phase equilibria in fluids (or other soft condensed matter) with two spatially varying densities is analyzed analytically and numerically. The interfacial tension between any two out of three thermodynamically coexisting phases is found to be captured by a surprisingly simple analytic expression that has a geometric interpretation in the space of the two densities. The analytic expression is based on arguments involving symmetries and invariances. It is supported by numerical computations of high precision and it agrees with earlier conjectures o…
Predictions of the extended de Gennes-Fisher local-functional theory for the universal scaling functions of interfacial tensions near critical endpoints are compared with experimental data. Various observations of the binary mixture isobutyric acid $+$ water are correlated to facilitate an analysis of the experiments of Nagarajan, Webb and Widom who observed the vapor-liquid interfacial tension as a function of {\it both} temperature and density. Antonow's rule is confirmed and, with the aid of previously studied {\it universal amplitude ratios}, the crucial analytic ``background'' contributio…
By means of extensive equilibrium molecular dynamics simulations we have investigated, the behavior of the interfacial tension $γ$ of two immiscible symmetrical Lennard-Jones fluids. This quantity is studied as function of reduced temperature $T^{*}={{k_{_B} T}\over ε}$ in the range $0.6 \leq T^{*} \leq 3.0$. We find that, unlike the monotonic decay obtained for the liquid-vapor interfacial tension, for the liquid-liquid interface, $γ(T)$ has a maximum at a specific temperature. We also investigate the effect that surfactant-like particles has on the thermodynamic as well as the structural pro…
Parametric scaling representations are obtained and studied for the asymptotic behavior of interfacial tensions in the \textit{full} neighborhood of a fluid (or Ising-type) critical endpoint, i.e., as a function \textit{both} of temperature \textit{and} of density/order parameter \textit{or} chemical potential/ordering field. Accurate \textit{nonclassical critical exponents} and reliable estimates for the \textit{universal amplitude ratios} are included naturally on the basis of the ``extended de Gennes-Fisher'' local-functional theory. Serious defects in previous scaling treatments are rectif…
The concept of a non-equilibrium interfacial tension, defined via the work required to deform the system such that the interfacial area is changed while the volume is conserved, is investigated theoretically in the context of the relaxation of an initial perturbation of a colloidal fluid towards the equilibrium state. The corresponding general formalism is derived for systems with planar symmetry and applied to fluid models of colloidal suspensions and polymer solutions. It is shown that the non-equilibrium interfacial tension is not necessarily positive, that negative non-equilibrium interfac…
Excess contributions to the free energy due to interfaces occur for many problems encountered in the statistical physics of condensed matter when coexistence between different phases is possible (e.g. wetting phenomena, nucleation, crystal growth, etc.). This article reviews two methods to estimate both interfacial free energies and line tensions by Monte Carlo simulations of simple models, (e.g. the Ising model, a symmetrical binary Lennard-Jones fluid exhibiting a miscibility gap, and a simple Lennard-Jones fluid). One method is based on thermodynamic integration. This method is useful to st…
We study the phase behavior and the interfacial tension of the screened Coulomb (Yukawa) restricted primitive model (YRPM) of oppositely charged hard spheres with diameter s using Monte Carlo simulations. We determine the gas-liquid and gas-solid phase transition using free energy calculations and grand-canonical Monte Carlo simulations for varying inverse Debye screening length k. We find that the gas-liquid phase separation is stable for k s <= 4, and that the critical temperature decreases upon increasing the screening of the interaction (decreasing the range of the interaction). In additio…
The aim of this work is to study the problem of the existence of a fundamental relation between the interfacial tension of a system of two partially miscible liquids and the surface tensions of the pure substances. It is shown that these properties cannot be correlated from the physical point of view. However, an accurate relation between them may be developed using a mathematical artifact. In the light of this work, the basis of the empirical formula of Girifalco and Good is examined. The weakness of this formula as well as the approximation leading to it are exposed and discussed and, a new …
In passive fluid-fluid phase separation, a single interfacial tension sets both the capillary fluctuations of the interface and the rate of Ostwald ripening. We show that these phenomena are governed by two different tensions in active systems, and compute the capillary tension $σ_{cw}$ which sets the relaxation rate of interfacial fluctuations in accordance with capillary wave theory. We discover that strong enough activity can cause negative $σ_{cw}$. In this regime, depending on the global composition, the system self-organizes, either into a microphase-separated state in which coalescence …
The surface tension of rough interfaces between coexisting phases in 2D and 3D Ising models are discussed in view of the known results and some original calculations presented in this paper. The results are summarised in a formula, which allows to interpolate the corrections to finite-size scaling between two and three dimensions. The physical meaning of an analytic continuation to noninteger values of the spatial dimensionality d is discussed. Lattices and interfaces with properly defined fractal dimensions should fulfil certain requirements to possibly have properties of an analytic continua…
We present a microscopic theory for the nonequilibrium interfacial tension $γ_{\rm gl}$ of the free interface between gas and liquid phases of active Brownian particles. The underlying square gradient treatment and the splitting of the force balance in flow and structural contributions is general and applies to inhomogeneous nonequilibrium steady states. We find $γ_{\rm gl}\geq 0$, which opposes claims by Bialké et al. [Phys. Rev. Lett. $\bf{115}$, 098301 (2015)] and delivers the theoretical justification for the widely observed interfacial stability in active Brownian dynamics many-body simul…
A theory is presented to account for the increase in surface tension of water in the presence of electrolyte. Unlike the original ``grand-canonical'' calculation of Onsager and Samaras, which relied on the Gibbs adsorption isotherm and lead to a result which could only be expressed as an infinite series, our approach is ``canonical'' and produces an analytic formula for the excess surface tension. For small concentrations of electrolyte, our result reduces to the Onsager-Samaras limiting law.
We study the electrical double layer at the interface between a protein crystal and a salt solution or a dilute solution of protein, and estimate the double layer's contribution to the interfacial tension of this interface. This contribution is negative and decreases in magnitude with increasing salt concentration. We also consider briefly the interaction between a pair of protein surfaces.
Accurate and useful analytic approximations are developed for order parameter profiles and interfacial tensions of phase-separated binary mixtures of Bose-Einstein condensates. The pure condensates 1 and 2, each of which contains a particular species of atoms, feature healing lengths $ξ_1$ and $ξ_2$. The inter-atomic interactions are repulsive. In particular, the effective inter-species repulsive interaction strength is $K$. A triple-parabola approximation (TPA) is proposed, to represent closely the energy density featured in Gross-Pitaevskii (GP) theory. This TPA allows us to define a model, …
One of the long standing challenges in molecular simulation is the description of interfaces. On the molecular length scale, finite size effects significantly influence the properties of the interface such as its interfacial tension, which can be reliably investigated by molecular dynamics simulation of planar vapor-liquid interfaces. For the Lennard-Jones fluid, finite size effects are examined here by varying the thickness of the liquid slab. It is found that the surface tension and density in the center of the liquid region decreases significantly for thin slabs. The influence of the slab t…
We develop a methodology for the calculation of surface free energies based on the probability distribution of a wandering interface. Using a simple extension of the NpT sampling, we allow the interface area to randomly probe the available space and evaluate the surface free energy from histogram analysis and the corresponding average. The method is suitable for studying systems with either continuous or discontinuous potentials, as it does not require explicit evaluation of the virial. The proposed algorithm is compared with known results for the surface tension of Lennard--Jones and Square W…
We study numerically a model for active suspensions of self-propelled repulsive particles, for which a stable phase separation into a dilute and a dense phase is observed. We exploit that for non-square boxes a stable "slab" configuration is reached, in which interfaces align with the shorter box edge. Evaluating a recent proposal for an intensive active swimming pressure, we demonstrate that the excess stress within the interface separating both phases is negative. The occurrence of a negative tension together with stable phase separation is a genuine non-equilibrium effect that is rationaliz…
Interfaces involving coexisting phases in condensed matter are essential in various examples of soft matter phenomena such as wetting, nucleation, morphology, phase separation kinetics, membranes, phase coexistence in nanomaterials, etc. Most analytical theories available use concepts derived from mean field theory which does not describe adequately these systems. Satisfactory numerical simulations for interfaces at atomistic to mesoscopic scales remains a challenge. In the present work, the interfacial tension between mixtures of organic solvents and water is obtained from mesoscopic computer…
The computation of interfacial free energies between coexisting phases (e.g.~saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface, and due to interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering $d$-dimensional systems with interfacial area $L^{d-1}$ and linear …
As a generic example for crystals where the crystal-fluid interface tension depends on the orientation of the interface relative to the crystal lattice axes, the nearest neighbor Ising model on the simple cubic lattice is studied over a wide temperature range, both above and below the roughening transition temperature. Using a thin film geometry $L_x \times L_y \times L_z$ with periodic boundary conditions along the z-axis and two free $L_x \times L_y$ surfaces at which opposing surface fields $\pm H_{1}$ act, under conditions of partial wetting, a single planar interface inclined under a cont…
We study the density-density correlation function $G({\bf r},{\bf r}')$ in the interfacial region of a fluid (or Ising-like magnet) with short-ranged interactions using square gradient density functional theory. Adopting a simple double parabola approximation for the bulk free-energy density, we first show that the parallel Fourier transform $G(z,z';q)$ and local structure factor $S(z;q)$ separate into bulk and excess contributions. We attempt to account for both contributions by deriving an interfacial Hamiltonian, characterised by a wavevector dependent surface tension $σ(q)$, and then recon…
The shape of a liquid drop placed in a cone is analyzed macroscopically. Depending on the values of the cone opening angle, the Young angle and the line tension four different interfacial configurations may be realized. The phase diagram in these variables is constructed and discussed; it contains both the first- and the second-order transition lines. In particular, the tricritical point is found and the value of the critical exponent characterizing the behaviour of the system along the line of the first-order transitions in the neighbourhood of this point is determined.
Interfacial profiles and interfacial tensions of phase-separated binary mixtures of Bose-Einstein condensates are studied theoretically. The two condensates are characterized by their respective healing lengths $ξ_1$ and $ξ_2$ and by the inter-species repulsive interaction $K$. An exact solution to the Gross-Pitaevskii (GP) equations is obtained for the special case $ξ_2/ξ_1 = 1/2$ and $K = 3/2$. Furthermore, applying a double-parabola approximation (DPA) to the energy density featured in GP theory allows us to define a DPA model, which is much simpler to handle than GP theory but nevertheless…
We present a generalized theory for studying phase separation in blends of polymers containing dipoles on their backbone. The theory is used to construct co-existence curves and to study the effects of dipolar interactions on interfacial tension for a planar interface between the coexisting phases. We show that a mismatch in monomeric dipole moments, or equivalently a mismatch in the dielectric constant of the pure components, leads to destabilization of the homogeneous phase. Corrections to the Flory-Huggins phase diagram are predicted using the theory. Furthermore, we show that the interfaci…
In this paper we review simulation and experimental studies of thermal capillary wave fluctuations as an ideal means for probing the underlying disjoining pressure and surface tensions, and more generally, fine details of the Interfacial Hamiltonian Model. We discuss recent simulation results that reveal a film-height-dependent surface tension not accounted for in the classical Interfacial Hamiltonian Model. We show how this observation may be explained bottom-up from sound principles of statistical thermodynamics and discuss some of its implications.
Due to the simultaneous presence of bulk-like and interfacial fluctuations the understanding of the structure of liquid-vapour interfaces poses a long-lasting and ongoing challenge for experiments, theory, and simulations. We provide a new analysis of this topic by combining high-quality simulation data for Lennard-Jones fluids with an unambiguous definition of the wavenumber-dependent surface tension $γ(q)$ based on the two-point correlation function of the fluid. Upon raising the temperature, $γ(q)$ develops a maximum at short wavelengths. We compare these results with predictions from densi…
We study the equilibrium of a liquid film on an attractive spherical substrate for an intermolecular interaction model exhibiting both fluid-fluid and fluid-wall long-range forces. We first reexamine the wetting properties of the model in the zero-curvature limit, i.e., for a planar wall, using an effective interfacial Hamiltonian approach in the framework of the well known sharp-kink approximation (SKA). We obtain very good agreement with a mean-field density functional theory (DFT), fully justifying the use of SKA in this limit. We then turn our attention to substrates of finite curvature an…
In this paper we use molecular dynamics (MD) to answer a classical question: how does the surface tension on a liquid/gas interface appear? After defining surface tension from the first principles and performing several consistency checks, we perform a dynamic experiment with a single simple liquid nanodroplet. At time zero, we remove all molecules of the interfacial layer, creating a fresh bare interface with the bulk arrangement of molecules. After that the system evolves towards equilibrium, and the expected surface tension is re-established. We found that the system relaxation consists of …
Several model fluids in contact with planar, spherical, and cylindrical walls are investigated for small number densities within density functional theory. The dependence of the solid-fluid interfacial tension on the curvature of spherical and cylindrical walls is examined and compared with the corresponding expression derived within the framework of morphometric thermodynamics. Particular attention is paid to the implications of the choice of the interface location, which underlies the definition of the interfacial tension. We find that morphometric thermodynamics is never exact for the consi…
Attempts to extend the capillary-wave theory of fluid interfacial fluctuations to microscopic wavelengths, by introducing an effective wave-vector ($q$) dependent surface tension $σ_\text{eff}(q)$, have encountered difficulties. There is no consensus as to even the shape of $σ_\text{eff}(q)$. By analysing a simple density functional model of the liquid-gas interface, we identify different schemes for separating microscopic observables into background and interfacial contributions. In order for the backgrounds of the density-density correlation function and local structure factor to have a cons…
We study the effect of surface fields on the interfacial properties of a binary polymer melt confined between two parallel walls. Each wall attracts a different component of the blend by a non-retarded van der Waals potential. An interface which runs parallel to the surfaces is stabilized in the center of the film. Using extensive Monte Carlo simulations we study the interfacial properties as a function of the film thickness, the strength of the surface forces and the lateral size over which the profiles across the film are averaged. We find evidence for capillary wave broadening of the appare…
Within the effective interfacial Hamiltonian approach we evaluate the excess line free energy associated with cylinder-shaped droplets sessile on a stripe-like chemical inhomogeneity of a planar substrate. In the case of short-range intermolecular forces the droplet morphology and the corresponding expression for the line tension - which includes the inhomogeneity finite width effects - are derived and discussed as functions of temperature and increasing width. The width-dependent contributions to the line tension change their structure at the stripe wetting temperature T_W1: for T<T_W1 they d…
A phenomenological mean-field theory is used to investigate the properties of solvent-diluted di-block copolymers (BCP), in which the two BCP components (A and B) form a variety of phases that are diluted by a solvent (S). Using this approach, we model mixtures of di-block copolymers and a solvent and obtained the corresponding critical behavior. In the low solvent limit, we find how the critical point depends on the solvent density. Due to the non-linear nature of the coupling between the A/B and BCP/solvent concentrations, the A/B modulation induces modulations in the polymer-solvent relativ…
