Capillary Wave Dynamics Research Articles

Dynamics of thermally driven capillary waves for two-dimensional droplets

Capillary waves have been observed in systems ranging from the surfaces of ordinary fluids to interfaces in biological membranes and have been one of the most studied areas in the physics of fluids. Recent advances in fluorescence microscopy and imaging enabled quantitative measurements of thermally driven capillary waves in lipid monolayers and bilayers, which resulted in accurate measurements of the line tension in monolayer domains. Even though there has been a considerable amount of work on the statics and dynamics of capillary waves in three dimensions, to the best of our knowledge, there is no detailed theoretical analysis for two-dimensional droplet morphologies. In this paper, we derive the dynamic correlation function for two-dimensional fluid droplets using linear response theory and verify our results using a novel particle-based simulation technique for binary mixtures.

Thermal capillary waves relaxing on atomically thin liquid films

Atomistic simulations are used to investigate the relaxation dynamics of thermal capillary waves on thin flat liquid films. Short Lennard-Jones polymers (n=2, 4, and 8) model the liquid in films of thickness 6σ to 96σ, where σ is the Lennard-Jones atomic length-scale parameter. Assuming no-slip boundary conditions on the solid wall and constant surface tension and viscosity, the standard continuum model predicts that capillary waves decay with rates ω that scale with wavenumber q as ω∼q4 for long wavelengths and ω∼q for short wavelengths. The atomistic simulations do indeed show these scalings for ranges of q, and, of course, this model must fail for large q as wavelengths approach atomic scales. However, before a complete breakdown of the continuum description, an unexpected intermediate regime is found. Here the decay rates follow an apparent ω∼q2 power law. The behavior in this range collapses for all the cases simulated when q is scaled with the radius of gyration of the polymers, indicating that a molecular-scale effect underlies the relaxation mechanics of these short waves.

Hydrodynamics of Nanoscopic Capillary Waves

The dynamics of nanoscopic capillary waves on simple liquid surfaces is analyzed using molecular dynamics simulations. Each Fourier mode of the surface is obtained from the molecular positions, and its time behavior compared with the hydrodynamic prediction. We trace the transition from propagating to overdamped modes, at short wavelengths. The damping rate is in very good agreement with the hydrodynamic theory up to surprisingly small wavelengths, of about four molecular diameters, but only if the wave number dependent surface tension is considered. At shorter scales, surface tension hydrodynamics break down and we find a transition to a molecular diffusion regime.

Finite system size effects in the interfacial dynamics of binary liquid films

We study the relaxation dynamics of capillary waves in the interface between two confined liquid layers by means of molecular dynamics simulations. We measure the autocorrelations of the interfacial Fourier modes and find that the finite thickness of the liquid layers leads to a marked increase of the relaxation times as compared to the case of fluid layers of infinite depth. The simulation results are in good agreement with a theoretical first-order perturbation derivation, which starts from the overdamped Stokes' equation. The theory also takes into account an interfacial friction, but the difference with no-slip interfacial conditions is small. When the walls are sheared, it is found that the relaxation times of modes perpendicular to the flow are unaffected. Modes along the flow direction are relatively unaffected as long as the equilibrium relaxation time is sufficiently short compared to the rate of deformation. We discuss the consequences for experiments on thin layers and on ultralow surface tension fluids, as well as computer simulations.

Dynamics and critical damping of capillary waves in an ionic liquid

The dynamics of thermal capillary waves (CWs) on an ionic liquid's surface are studied at the transition from propagating to overdamped CWs by x-ray photon correlation spectroscopy. The analysis considers both homodyne and heterodyne contributions, and yields excellent full line-shape experiment-theory agreement for the structure factor. The CWs' Brillouin scattering becomes extinct at a critical temperature Tc JK approximately 10 K above Tc LL , the propagating modes' hydrodynamic limit, in agreement with linear response theory. Surprisingly, the same power law applies at both Tc. The results rule out the presence of a suggested surface dipole layer.

Erratum: Capillary wave dynamics on supported viscoelastic films: Single and double layers [Phys. Rev. E.75, 021604 (2007)

We study the capillary wave dynamics of a single viscoelastic supported film and of a double layer of immiscible viscoelastic supported films. Using both simple scaling arguments and a continuum hydrodynamic theory, we investigate the effects of viscoelasticity and interfacial slip on the relaxation dynamics of these capillary waves. Our results account for the recent observation of a wavelength-independent decay rate for capillary waves in a supported polystyrene/brominated polystyrene double layer [X. Hu {\em et al.}, Phys. Rev. E {\bf 74}, 010602 (R) (2006)].

Capillary wave dynamics on supported viscoelastic films: Single and double layers

We study the capillary wave dynamics of a single viscoelastic supported film and of a double layer of immiscible viscoelastic supported films. Using both simple scaling arguments and a continuum hydrodynamic theory, we investigate the effects of viscoelasticity and interfacial slip on the relaxation dynamics of these capillary waves. Our results account for the recent observation of a wavelength-independent decay rate for capillary waves in a supported polystyrene/brominated polystyrene double layer [X. Hu, Phys. Rev. E 74, 010602(R) (2006)].

A Mean-Field Pressure Formulation for Liquid-Vapor Flows

A nonlocal pressure equation is derived from mean-field free energy theory for calculating liquid-vapor systems. The proposed equation is validated analytically by showing that it reduces to van der Waals’ square-gradient approximation under the assumption of slow density variations. The proposed nonlocal pressure is implemented in the mean-field free energy lattice Boltzmann method (LBM). The LBM is applied to simulate equilibrium liquid-vapor interface properties and interface dynamics of capillary waves and oscillating droplets in vapor. Computed results are validated with Maxwell constructions of liquid-vapor coexistence densities, theoretical relationship of variation of surface tension with temperature, theoretical planar interface density profiles, Laplace’s law of capillarity, dispersion relationship between frequency and wave number of capillary waves, and the relationship between radius and the oscillating frequency of droplets in vapor. It is shown that the nonlocal pressure formulation gives excellent agreement with theory.

Structure and dynamics of a free liquid crystal surface at the N-to-SmA phase transition

Capillary wave dynamics at the free liquid crystal surface depends upon the viscosity which is anisotropic in the nematic phase and related to the structural order. We have compiled the dispersion relations governing the capillary wave dynamics on a liquid crystal surface. The surface-induced structural ordering of the liquid crystal compound 8OCB was determined by grazing incidence X-ray diffraction from the free surface in the vicinity of the nematic (N)-to-smectic A (SmA) phase transition. This allows us to identify the relevant dispersion relation, which is of decisive importance for the interpretation of dynamic X-ray photon correlation spectroscopy (XPCS) data taken at the N-to-SmA phase transition.

Investigation of surface dynamics on micro- and nanometer scales

A kinematical scattering theory for the interpretation of X-ray photon correlation spectroscopy (XPCS) measurements from fluctuating surfaces is discussed in detail. First a general formulation without approximations is given. Then as an example XPCS measurements of capillary wave dynamics on glycerol surfaces will be presented and compared to the calculations applying the usual approximations. It turns out that these approximations work well if the perpendicular wave-vector transfer is small. The measurements confirm on micrometer length scales the prediction from classical hydrodynamics that all waves are overdamped due to the high viscosity of glycerol. In addition, the measurements show that thermally activated capillary wave motion is slowed down exponentially when the sample is cooled below 273 K. Further, the extension of XPCS measurements down to nanometer scales is discussed.

Capillary waves in slow motion

Capillary wave dynamics on glycerol surfaces has been investigated by means of x-ray photon correlation spectroscopy performed at grazing angles. The measurements show that thermally activated capillary wave motion is slowed down exponentially when the sample is cooled below 273 K. This finding directly reflects the freezing of the surface waves. The wave-number dependence of the measured time constants is in quantitative agreement with theoretical predictions for overdamped capillary waves.

Interaction of capillary waves with longer waves. Part 2. Applications to waves in two surface dimensions and to waves in shallow water

In Part 1 of this work we presented a new mathematical formulation for numerical investigation of capillary wave dynamics. This permitted calculations to be readily performed on a desktop computer. Applications given were primarily to waves in one surface dimension. In Part 2 we describe further applications to waves in two surface dimensions and also to waves in shallow water (not, however, shallow for the capillary waves). Wavenumber–frequency spectra for wind waves are calculated. As was observed in tank experiments by Hara et al. (1997), our calculations show both a frequency spread and a frequency up-shifting which suggests that capillary waves are ‘dragged along’ by longer waves. Fine-scale roughness near wind wave crests shows a transitory nature, changing with the wave pattern. We discuss implications of this for microwave remote sensing. The propagation of short gravity waves in shallow water is studied. As these waves develop bore-like forward faces, generation of parasitic capillary waves is observed. Generation rates for these are substantially greater than observed for waves in deep water.

Interaction of capillary waves with longer waves. Part 1. General theory and specific applications to waves in one dimension

A Hamiltonian formulation is used to investigate irrotational capillary wave dynamics. Dissipation is accounted for by putting the wave system in contact with a ‘heat bath’. The generation of short waves by longer waves is studied. It is found that millimetre-wavelength waves tend to be created on the forward face of a steep longer wave, while centimetre waves tend to form near the crest. Generation of capillary waves by wind waves is investigated. The results are compared with predictions of the Hasselmann transport equation. It is found that off-resonance interactions lead to significant corrections to the transport theory. The relative importance of three-wave and four-wave interactions is studied, as well as the role of triad resonances. For the capillary phenomena studied here, the four-wave terms in most cases lead to quantitative, but not qualitative, corrections to the three-wave only calculations. However, restricting interactions to the neighbourhood of triad resonances can give quite erroneous results. Use of a canonical transformation to pseudo-wave variables can greatly reduce numerical computation times.

Dynamics of capillary waves on a bubble performing nonlinear pulsations in a lowviscosity liquid

UDC 541.24:532.5 We examine the dynamics of capillary waves of small amplitude on a bubble, performing initially spherosymmetric pulsations in a liquid of low viscosity. We study in the shortwave approximation the characteristics of the growth of surface disturbances with arbitrary pressure differentials and polytropic exponents. We determine the asymptotic time dependences of the mean-square amplitude of the disturbances and obtain approximate formulas for the amplitude growth characteristics for a wide range of parameters. The existence of a single universal dependence of the wave index is discovered. A strong influence of the polytropic exponent near the isotherm on the wave growth characteristics is found. We establish an analogy between the growth of inertial-capillary waves on the surface of a nonlinearly pulsating bubble and on a plane surface with constant acceleration. It is shown that in the case of large-amplitude pulsations the low viscosity approximation is capable of describing the nonsmall effects of change of the wave growth characteristics. We determine the influence of viscosity on the dynamics of the disturbances, and note the viscosity-associated stratification of the universal relation for the wave growth characteristic. 1. Asymptotic Description of Nonstationary Short Waves on a Pulsating Bubble. We shall examine the small perturbations of the spherically symmetric pulsations of a gas bubble in a liquid that is at rest at infinity. We denote the initial gas pressure by P0 and the pressure at infinity by Po.. The dynamics of the disturbances on the pulsating gas bubble depends on the polytropic exponent k and the pressure ratio parameter e and the capillarity parameter a: t = po/p** , cr = a'/p| a o = a/e (or' is the surface tension coefficient, R o' is the initial radius, the primes indicate dimensional quantities). We shall introduce the viscosity influence parameter later, after accounting for the inertial-capillary effects in the ideal fluid formulation. We examine the limit a < < 1, when the capillary forces do not influence the change of the bubble radius, defined in dimensionless form by the equation 2 RR + (3/2)R = eR -a- 1, R(0) = I,R(0) = 0, t R = R'/Ro, t = t'(p| (I.1) The amplitudes of the small perturbations of the surface, represented by a series in the spherical harmonics

Influence of acoustic fields on drop dynamics in an acoustic resonator

Capillary waves on a liquid sphere may be excited by acoustic fields, especially due to the higher harmonics of the resonant acoustic fields. The growth of the capillary waves has been analytically discussed. For a manipulation of weightless molten material without contacts, rotation of a liquid drop by acoustic waves is important. Approximate solutions to govern the dynamics of capillary waves on a rotating liquid drop has been analytically derived. Instabilities of multi-lobed waves due to a rotational flow in the perturbed flow field and due to the acoustic fields have been discussed.