Dmitry I. Kopelevich

NONLINEAR EVOLUTION OF WAVES ON A VERTICALLY FALLING FILM

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ 1 travels slower than infinitesimally small waves of the same wavelength while the other family γ 2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ 1 with wavenumber α s (or α 2 ) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber α m and approaches the wave on γ 1 with the highest flow rate at low Reynolds numbers. On the fast γ 2 family, however, multiple bands of near-solitary waves bounded below by α f are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α 2 of the slow γ 1 family. They then approach the long stable waves on the γ 2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

Laminarizing effects of dispersion in an active-dissipative nonlinear medium

Abstract The Kuramoto-Sivashinsky equation has become a popular prototype for systems which exhibit spatial-temporal chaos. We show here that a linear dispersion term δ h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures, as is consistent with prior numerical and experimental observations. The study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation. It is shown that by δ > 1.1, the infinite families of stationary periodic waves of the Kuramoto-Sivashinsky equation, each ending in a solitary wave, have been annihilated successively such that only a lone family of periodic waves, consisting of one-hump KdV pulses for δ > 3.7, remains as the only periodic stationary wave attractors of the system. These periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial-temporal chaos in an extended domain with significant dispersion.

A multiscale model for kinetics of formation and disintegration of spherical micelles

Dynamics of self-assembly and structural transitions in surfactant systems often involve a large span of length and time scales. A comprehensive understanding of these processes requires development of models connecting phenomena taking place on different scales. In this paper, we develop a multiscale model for formation and disintegration of spherical nonionic micelles. The study is performed under the assumption that the dominant mechanism of micelle formation (disintegration) is a stepwise addition (removal) of single monomers to (from) a surfactant aggregate. Different scales of these processes are investigated using a combination of coarse-grained molecular dynamics simulations, analytical and numerical solution of stochastic differential equations, and a numerical solution of kinetic equations. The removal of a surfactant from an aggregate is modeled by a Langevin equation for a single reaction coordinate, the distance between the centers of mass of the surfactant and the aggregate, with parameters obtained from a series of constrained molecular dynamics simulations. We demonstrate that the reverse process of addition of a surfactant molecule to an aggregate involves at least two additional degrees of freedom, orientation of the surfactant molecule and micellar microstructure. These additional degrees of freedom play an active role in the monomer addition process and neglecting their contribution leads to qualitative discrepancies in predicted surfactant addition rates. We propose a stochastic model for the monomer addition which takes the two additional degrees of freedom into account and extracts the model parameters from molecular dynamics simulations. The surfactant addition rates are determined from Brownian dynamics simulations of this model. The obtained addition and removal rates are then incorporated into the kinetic model of micellar formation and disintegration.

Diffusion of inert gases in silica sodalite: Importance of lattice flexibility

It is known that some zeolites are able to adsorb sorbates with diameters larger than their pore size. This ability is usually explained by the flexibility of zeolite lattices. In this paper we quantify this explanation by incorporating lattice flexibility into transition state theory to compute transport rates of inert gas molecules (Ne, Ar and Kr) through narrow windows of silica sodalite cages. We find that the diffusion rates obtained with the flexible zeolite lattice are in much better agreement with experiment than those with a rigid lattice, as stretching of the zeolite window considerably reduces the activation barrier.

Does lattice vibration drive diffusion in zeolites

A method of estimation of the effect of lattice vibration as a driving force for sorbate diffusion in zeolites is proposed. A realistic lattice model is employed to cut off unrealistic long vibrational modes and eliminate feedback due to lattice periodicity. A generalized Langevin equation for sorbate motion is then derived with the magnitude of the lattice vibration captured by two parameters, μ and ν, which can be readily computed for any system. The effect of lattice vibration is then estimated for a variety of sorbate–zeolite pairs. Lattice vibration is found to be a negligible driving force for some systems (e.g., methane and xenon in silicalite) and an important driving force for other systems. In the latter case, the lattice vibration can provide either linear stochastic Langevin-type force (e.g., for benzene in silicalite) or nonlinear deterministic force (e.g., for argon in sodalite).

One-dimensional potential of mean force underestimates activation barrier for transport across flexible lipid membranes.

Transport of a fullerene-like nanoparticle across a lipid bilayer is investigated by coarse-grained molecular dynamics (MD) simulations. Potentials of mean force (PMF) acting on the nanoparticle in a flexible bilayer suspended in water and a bilayer restrained to a flat surface are computed by constrained MD simulations. The rate of the nanoparticle transport into the bilayer interior is predicted using one-dimensional Langevin models based on these PMFs. The predictions are compared with the transport rates obtained from a series of direct (unconstrained) MD simulations of the solute transport into the flexible bilayer. It is observed that the PMF acting on the solute in the flexible membrane underestimates the transport rate by more than an order of magnitude while the PMF acting on the solute in the restrained membrane yields an accurate estimate of the activation energy for transport into the flexible membrane. This paradox is explained by a coexistence of metastable membrane configurations for a range of the solute positions inside and near the flexible membrane. This leads to a significant reduction of the contribution of the transition state to the mean force acting on the solute. Restraining the membrane shape ensures that there is only one stable membrane configuration corresponding to each solute position and thus the transition state is adequately represented in the PMF. This mechanism is quite general and thus this phenomenon is expected to occur in a wide range of interfacial systems. A simple model for the free energy landscape of the coupled solute-membrane system is proposed and validated. This model explicitly accounts for effects of the membrane deformations on the solute transport and yields an accurate prediction of the activation energy for the solute transport.

Secondary and tertiary excitation of three-dimensional patterns on a falling film

The primary instability of a falling film selectively amplifies two-dimensional noise down-stream over three-dimensional modes with transverse variation. If the initial three-dimensional noise is weak or if it has short wavelengths such that they are effectively damped by the capillary mechanism of the primary instability, our earlier study (Chang et al . 1993 a ) showed that the primary instability leads to a weakly nonlinear, nearly sinusoidal γ 1 stationary wave which then undergoes a secondary transition to a strongly nonlinear γ 2 wave with a solitary wave structure. We show here that the primary transition remains in the presence of significant three-dimensional noise but the secondary transition can be replaced by a selective excitation of oblique triad waves which can even include stable primary disturbances. The resulting secondary checkerboard pattern is associated with a subharmonic mode in the streamwise direction. If the initial transverse noise level is low, a secondary transition to a two-dimensional γ 2 solitary wave is followed by a tertiary ‘phase instability’ dominated by transverse wave crest modulations.

Molecular modeling of surfactant covered oil-water interfaces: Dynamics, microstructure, and barrier for mass transport.

Mass transport across surfactant-covered oil-water interfaces of microemulsions plays an important role in numerous applications. In the current work, we use coarse-grained molecular dynamics simulations to investigate model systems containing flat hexadecane-water interfaces covered by monolayers of nonionic surfactants of various lengths. Several properties of the surfactant monolayers relevant to the mass transport are considered, including the monolayer microstructure, dynamics, and a free energy barrier to the solute transport. It is observed that the dominant contribution of a surfactant monolayer to the free energy barrier is a steric repulsion caused by a local density increase inside the monolayer. The local densities, and hence the free energy barriers, are larger for monolayers composed of longer surfactants. Since it is likely that the solute transport mechanism involves a sequence of jumps between short-lived pores within a monolayer, we perform a detailed analysis of structure, size, and lifetime of these pores. We demonstrate that the pore statistics is consistent with predictions of percolation theory and apply this theory to identify the characteristic length scale of the monolayer microstructure. The obtained pore structures are sensitive to minute changes of surfactant configurations occurring on the picosecond time scale. To reduce this sensitivity, the pores are averaged over short time intervals. The optimal duration of these time intervals is estimated from analysis of dynamics of pores with diameters comparable to or exceeding the characteristic percolation length scale. The developed approach allows one to filter out transient events of the pore dynamics and to focus on events leading to substantial changes of the monolayer microstructure.

Molecular transport through surfactant-covered oil-water interfaces: role of physical properties of solutes and surfactants in creating energy barriers for transport.

Mechanisms of molecular transport across oil-water interfaces covered by nonionic surfactants are investigated using coarse-grained molecular dynamics simulations. Resistance of the surfactant monolayer to the solute transport is shown to be controlled by dense regions in the monolayer. The dense regions are formed on both sides of the dividing surface and the barrier to the solute transport is created by those of them experiencing unfavorable interactions with the solute. Resistance to the transport of a hydrophobic (hydrophilic) solute increases with the excess density of the head (tail) group region of the monolayer, which in turn increases with the length of the surfactant head (tail) group. Barriers for solute transport through surfactant monolayers are also influenced by the solute size. However, the extent of this influence is determined by the monolayer thickness and the solute structure and composition. For example, it is shown that resistance offered by thin monolayers to transport of linear oligomers is relatively insensitive to the solute length. The barrier sensitivity to the length of these solutes increases with the monolayer thickness. In addition to the static barriers, the solute transport is shown to be affected by dynamic barriers due to a nonadiabatic coupling of the monolayer surface with the solute position and configuration. This coupling leads to deviations of the system dynamics from the minimum energy path. The deviations are most significant in the neighborhood of the static energy barrier, which effectively leads to an increase of the barrier for the solute transport.

Molecular modeling of key elastic properties for inhomogeneous lipid bilayers

Fusion and fission of biological membranes play a crucial role in intracellular transport. Until recently, it was believed that membrane shape transformations involved in these processes are driven by proteins. However, recent evidence shows that lipids, by themselves, can drive membrane deformations. It has been hypothesized that the localized formation of certain lipids changes elastic properties of a membrane in such a way that the membrane deforms spontaneously. This study represents a step towards a systematic investigation of the role of various lipids in local changes of membrane elastic properties. We use coarse-grained molecular dynamics (CGMD) simulations to determine possible effects of addition of phosphatidylinositol-4-phosphate (PI4P) lipids on elastic properties of dipalmitoyl phosphatidyl choline (DPPC) lipid bilayers. We investigate the splay (bending) and the molecular tilt moduli of mixed DPPC/PI4P bilayers, as well as the line tension between domains of pure DPPC and mixed DPPC/PI4P bilayers. Although our results indicate negligible effects of PI4P on elastic properties of DPPC bilayers, the developed methodology can be applied to a wide range of lipid systems.