Curvature Dependence Of Surface Tension Research Articles

Abnormal condensation of water vapour at ambient temperature.

The homogeneous condensation of water vapor at ambient temperature is studied using molecular dynamics simulation. We reveal that there is a droplet size at the nanoscale where water droplets can be stabilized in the condensation process. Our simulations show that the growth of water droplets is dominated by collision and coagulation between small water droplets after nucleation. This process is found to be accompanied by exceptionally fast evaporation such that droplet growth is balanced by evaporation when water droplets grow to a critical size, approximately 12.5 Å in radius, reaching a stable size distribution. The extremely high evaporation rate is attributed to the curvature dependence of surface tension. Surface tension shows a significant decrease with decreasing droplet size below 20 Å, which causes the total free energy of nanoscaled water droplets to rise after collision and coagulation. Consequently, water droplets have to shrink via fast evaporation. The curvature dependence of surface tension is related to the dielectric ordering of water molecules near the surface of water droplets. Owing to fast evaporation, secondary condensation occurs, and many small water clusters form, ultimately exhibiting a bimodal distribution of water-droplet size.

Mesoscale perspective on the Tolman length.

We demonstrate that the multiphase Shan-Chen lattice Boltzmann method (LBM) yields a curvature dependent surface tension σ as computed from three-dimensional hydrostatic droplets and bubbles simulations. Such curvature dependence is routinely characterized, at first order, by the so-called Tolman length δ. LBM allows one to precisely compute σ at the surface of tension R_{s} and determine the Tolman length from the coefficient of the first order correction. The corresponding values of δ display universality for different equations of state, following a power-law scaling near the critical temperature. The Tolman length has been studied so far mainly via computationally demanding Molecular Dynamics simulations or by means of Density Functional Theory approaches playing a pivotal role in extending Classical Nucleation Theory. The present results open a hydrodynamic-compliant mesoscale arena, in which the fundamental role of the Tolman length, alongside real-world applications to cavitation phenomena, can be effectively tackled. All the results can be independently reproduced through the "idea.deploy" framework.

Stretch-Induced Cavitation: How Critical Cavity Radius and Barrier Energy, Radius, and Energy of a Stable Cavity Depend on the Stretching Factor.

Characteristics of tension-induced cavitation, such as free energy at the barrier for cavitation, the size of the critical (barrier) cavity, the stable cavity size, and the free energy of the stable cavity, depend on the amount of tension (stretch) and the initial size of the sample. In this work, we study how the characteristics of the cavitation mentioned above scale with the amount of applied tension. We consider two models characterizing the properties of cavitating liquid: (a) a simple model with a linear tension-strain relation and neglect of curvature dependence of cavity surface tension and (b) a more realistic model with a nonlinear tension-strain relation and curvature-dependent surface tension. For both models, we find the relevant scaling relations when we stretch the initial volume of the liquid sample in the interval between 1% and 20% of the initial volume. Specific numerical tests are performed for the case of liquid water when the initial volume of the sample is a sphere with a radius of 100 nm.

Direct measurement of curvature-dependent surface tension of an alcohol nanomeniscus.

Surface tension is a key parameter for understanding nucleation in the very initial stage of phase transformation. Although surface tension has been predicted to vary with the curvature of the liquid-vapor interface, particularly at the large curvature of, e.g., the subnanometric critical nucleus, experimental study still remains challenging due to inaccessibility to such a small cluster. Here, by directly measuring the critical size of a single capillary-condensed nanomeniscus using atomic force microscopy, we address the curvature dependence of surface tension of alcohols and observe that the surface tension is doubled for ethanol and n-propanol with a radius-of-curvature of ∼-0.46 nm. We also find that the interface of larger negative (positive) curvature exhibits larger (smaller) surface tension, which evidently governs nucleation at the ∼1 nm scale and below, indicating more facilitated nucleation than normally expected. Such well characterized curvature effects contribute to better understanding and accurate analysis of nucleation occurring in various fields including materials science and atmospheric science.

Coalescence speed of two equal-sized nanobubbles

In this work, we use molecular dynamics (MD) simulations coupled with continuum-based theoretical analysis to study the coalescence dynamics of two equal-sized nanobubbles (NBs). We first derive a governing equation for the evolution of the capillary bridge radius between two coalescing NBs from the axisymmetric Navier–Stokes equation. To verify the prediction from the governing equation, we carry out MD simulations of the coalescence of two NBs in a Lennard-Jones fluid system and directly measure the bridge radius, rb, as a function of time, t. By varying the bubble diameter, we change the NB Ohnesorge number from 0.46 to 0.33. In all cases, we find the theoretical prediction overestimates the expansion speed of the capillary bridge at early time of NB coalescence. However, once we take into account the curvature-dependent surface tension and restrict the minimum principal radius at the capillary bridge to the size of the atom in the model liquid, the theoretical prediction agrees with the MD data very well in both early time and later time of the coalescence process. From the theoretical model, we find neither liquid viscous force nor liquid inertial force dominates at later time of coalescence of the model NBs. In this case, the MD simulation results show rb(t) ∝ t0.76 ± 0.04 with the scaling exponent considerably higher than that in the scaling law rb(t) ∝ t0.5 for the viscous and inertial dominated regimes. The diameter ratio of fully merged NB to that of the original NB is about 2, which is different from 23 for the coalescence of millibubbles and microbubbles.

Surface tension of supercooled water nanodroplets from computer simulations.

We estimate the liquid-vapor surface tension from simulations of TIP4P/2005 water nanodroplets of size N = 100 to 2880 molecules over a temperature T range of 180 K-300 K. We compute the planar surface tension γp, the curvature-dependent surface tension γs, and the Tolman length δ, via two approaches, one based on the pressure tensor (the "mechanical route") and the other on the Laplace pressure (the "thermodynamic route"). We find that these two routes give different results for γp, γs, and δ although in all cases, we find that δ ≥ 0 and is independent of T. Nonetheless, the T dependence of γp is consistent between the two routes and with that of Vega and de Miguel [J. Chem. Phys. 126, 154707 (2007)] down to the crossing of the Widom line at 230 K for ambient pressure. Below 230 K, γp rises more rapidly on cooling than predicted from behavior for T ≥ 300 K. We show that the increase in γp at low T is correlated with the emergence of a well-structured random tetrahedral network in our nanodroplet cores and thus that the surface tension can be used as a probe to detect behavior associated with the proposed liquid-liquid phase transition in supercooled water.

Some Problems in Simulation of the Thermodynamic Properties of Droplets

In this paper, the Gibbs dividing surface method is used to derive a formula to determine curvature-dependent surface tension in a system with two phases. The well-known Tolman formula is a special case of this formula. The problem of a sessile droplet is considered. The Bashforth–Adams equation analogue (in view of curvature-dependent surface tension) is obtained, and the numerical solution of the equation is carried out. It is shown that if the droplet size is not very large relative to the thickness of the surface layer (micro- or nanodroplets), the dependence of the surface tension on the curvature is very important. In addition, the case is considered where the diameters of cylindrical nanodroplets are shorter than the Tolman length.

Bubbles in water under stretch-induced cavitation.

When a finite sample of water experiences tension, it may develop voids (bubbles). We present here a result for the work (Helmholtz free energy change) that needs to be done for the creation of a bubble in fixed volume of water under tension and show that this result depends on the general form of stress-strain relationship. We observe that it is very important to include the curvature-dependent surface tension into consideration in order to explain bubble stability. The analytical result we obtained for the free energy allows us to make prediction for the values of critical and stable radii of the bubbles. We also performed simulations on the TIP4P/2005 water model and observed creation of bubbles in water under stretch. Combining analytical results obtained from our thermodynamic description with the results from computer simulations allowed us to determine the two parameters that describe the curvature-dependent surface tension and also to find the values of critical and stable bubble radii. We also determined the values of critical bubble radii by using mean first-passage time calculations.

Modeling of a sessile droplet with the curvature dependence of surface tension

In this paper, using the Gibbs dividing surface method, the formula that determines the curvature-dependent surface tension in a system with two phases is deduced. The well-known Tolman formula is a special case for this formula. The problem of a sessile droplet is considered. The Bashforth-Adams equation analogue in view of the curvaturedependent surface tension is obtained and the numerical solution of the equation is carried out. We show that if the droplet size is not so large compared to the thickness of the surface layer (micro- or nanodroplets), the dependence of the surface tension on the curvature is very important.

Measurement of the Curvature-Dependent Surface Tension in Nucleating Colloidal Liquids.

The curvature dependence of the surface tension is central to the nucleation of liquids, but remains difficult to access experimentally and predict theoretically. This curvature dependence arises from the curvature-dependent molecular structure, which, for small nuclei, can deviate significantly from that of the planar liquid interface. Simulations and density functional theory have been used to predict this curvature dependence, however with contradicting results. Here, we provide the first direct measurement of the curvature-dependent surface tension in nucleating colloidal liquids. We employ critical Casimir forces to finely adjust colloidal particle interactions and induce liquid nucleation, and image individual nuclei at the particle scale to measure their curvature-dependent surface tension directly from thermally excited surface distortions. Using continuum models, we elucidate the interplay between nucleus structure, particle pair potential, and surface tension. Our results reveal a 20% lower surface tension for nuclei of critical size compared to bulk liquids, leading to 3 orders of magnitude higher nucleation rates, thus highlighting the importance of surface tension curvature corrections for accurate prediction of nucleation rates.

Direct Evidence for Curvature-Dependent Surface Tension in Capillary Condensation: Kelvin Equation at Molecular Scale

New experiments demonstrate the validity of the Kelvin equation in describing nucleation at molecular scales in a vapor-to-liquid transition, potentially paving the way toward a better understanding of this fundamental process.

Confinement Correction to Mercury Intrusion Capillary Pressure of Shale Nanopores.

We optimized potential parameters in a molecular dynamics model to reproduce the experimental contact angle of a macroscopic mercury droplet on graphite. With the tuned potential, we studied the effects of pore size, geometry, and temperature on the wetting of mercury droplets confined in organic-rich shale nanopores. The contact angle of mercury in a circular pore increases exponentially as pore size decreases. In conjunction with the curvature-dependent surface tension of liquid droplets predicted from a theoretical model, we proposed a technique to correct the common interpretation procedure of mercury intrusion capillary pressure (MICP) measurement for nanoporous material such as shale. Considering the variation of contact angle and surface tension with pore size improves the agreement between MICP and adsorption-derived pore size distribution, especially for pores having a radius smaller than 5 nm. The relative error produced in ignoring these effects could be as high as 44%—samples that contain smaller pores deviate more. We also explored the impacts of pore size and temperature on the surface tension and contact angle of water/vapor and oil/gas systems, by which the capillary pressure of water/oil/gas in shale can be obtained from MICP. This information is fundamental to understanding multiphase flow behavior in shale systems.

A Rigid Stamp Indentation into a Semiplane with a Curvature-Dependent Surface Tension on the Boundary

It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvature-dependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.

Water droplet excess free energy determined by cluster mitosis using guided molecular dynamics.

Atmospheric aerosols play a vital role in affecting climate by influencing the properties and lifetimes of clouds and precipitation. Understanding the underlying microscopic mechanisms involved in the nucleation of aerosol droplets from the vapour phase is therefore of great interest. One key thermodynamic quantity in nucleation is the excess free energy of cluster formation relative to that of the saturated vapour. In our current study, the excess free energy is extracted for clusters of pure water modelled with the TIP4P/2005 intermolecular potential using a method based on nonequilibrium molecular dynamics and the Jarzynski relation. The change in free energy associated with the "mitosis" or division of a cluster of N water molecules into two N/2 sub-clusters is evaluated. This methodology is an extension of the disassembly procedure used recently to calculate the excess free energy of argon clusters [H. Y. Tang and I. J. Ford, Phys. Rev. E 91, 023308 (2015)]. Our findings are compared to the corresponding excess free energies obtained from classical nucleation theory (CNT) as well as internally consistent classical theory (ICCT). The values of the excess free energy that we obtain with the mitosis method are consistent with CNT for large cluster sizes but for the smallest clusters, the results tend towards ICCT; for intermediate sized clusters, we obtain values between the ICCT and CNT predictions. Furthermore, the curvature-dependent surface tension which can be obtained by regarding the clusters as spherical droplets of bulk density is found to be a monotonically increasing function of cluster size for the studied range. The data are compared to other values reported in the literature, agreeing qualitatively with some but disagreeing with the values determined by Joswiak et al. [J. Phys. Chem. Lett. 4, 4267 (2013)] using a biased mitosis approach; an assessment of the differences is the main motivation for our current study.

Vapour-to-liquid nucleation in cone pores

In this work, we perform a systematic study on the vapour-to-liquid nucleation in a cone pore by using comparatively the classical nucleation theory (CNT) and the constrained lattice density functional theory (constrained LDFT). Three different nucleation scenarios are observed depending on the contact angle θ and apex angle α: the spontaneous phase transition, nucleation with a critical nucleus in Cassie state and nucleation with a critical nucleus in Wenzel state. We also found that both the diagram for nucleation mechanisms and the reduced nucleation barriers with respect to the homogeneous nucleation given by the CNT are at least qualitatively consistent with those from the constrained LDFT. For an increasingly small critical nucleus, the difference between nucleation behaviours from two methods becomes significant due to the macroscopic approximations embedded in CNT, which fails to describe the curvature dependence of surface tension, the line tension effect and the space confinement effect.

The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack

A problem of an interface crack between two semi-planes made out of different materials under an action of an in-plane loading of general tensile-shear type is treated in a semi-analytical manner with the help of Dirichlet-to-Neumann mappings. The boundaries of the crack and the interface between semi-planes are subjected to a curvature-dependent surface tension. The resulting system of six singular integro-differential equations is reduced to the system of three Fredholm equations. It is shown that the introduction of the curvature-dependent surface tension eliminates both classical integrable power singularity of the order 1/2 and an oscillating singularity present in a classical linear elasticity solutions. The numerical results are obtained by solving the original system of singular integro-differential equations by approximating unknown functions with Taylor polynomials.

Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.

Manipulation of the Saffman-Taylor instability: A curvature-dependent surface tension approach

A variant of the classic Saffman-Taylor instability problem is reported, in which the surface tension at the fluid-fluid interface depends on the interfacial curvature. We show that the interplay between the variable surface tension and three-dimensional effects connected to the contact angle significantly modifies the scenario of instability formation. This allows the manipulation of the Saffman-Taylor instability, leading to the stabilization (destabilization) of conventionally unstable (stable) situations. This is done analytically through a perturbative mode-coupling approach, providing relevant information about both linear and weakly nonlinear regimes of interface evolution.

Temperature and composition dependence of kinetics of phase separation in solid binary mixtures

We present results on the effects of temperature variation and composition in the kinetics of phase separation in solid binary mixtures (A1 + A2) from Monte Carlo simulations of the Ising model in two dimensions. The simulation results are understood via an appropriate application of the finite-size scaling theory. At moderately high temperatures, for symmetric (50 : 50) compositions of A1 and A2 particles the average size of the domains exhibits power-law growth with the exponent having a Lifshitz-Slyozov value of 1/3 from very early time. However, our analysis shows that for low enough temperatures, the growth exponent at an early time is smaller than the Lifshitz-Slyozov value. For composition dependence, we find that at moderate temperature, even for extreme off-critical composition, the curvature dependent correction to the growth law is weak which is counter-intuitive in case of droplet morphology. This is, however, consistent with the recent understanding on the curvature dependence of surface tension. Results from rather general studies on the finite-size effects with the variations of temperature and composition have also been presented.

Positron Annihilation Parameters in Connection to Surface Phenomenon of Liquids

Surface tension is an important property of liquid, the bulk surface tension concept changes considerably when minute volume of a given liquid (drop let radius<10nm) is considered. Although an extensive thermodynamical treatment and theoretical basis have been considered in the past on the curvature dependence of surface tension, yet its experimental evaluation is deemed necessary and it has been shown hither to that positronium annihilation parameters serve faithfully to this purpose. However, the drastic lowering of surface tension with the dispersion of surfactant in the liquid phase display a different phenomenon altogether and has been dealt separately by positron annihilation results.