Spreading Law Research Articles

Spreading of nematic liquid crystals on hydrophobic substrates

The spontaneous spreading of nematic liquid crystals on hydrophobic silica substrates has been investigated at various scales by combining ellipsometry, profilometry and interferometry. While anchoring defects play a major role on hydrophilic substrates at the macroscopic scale, making the behaviour of films highly complex, they do not show up on these hydrophobic substrates. Then, the main specificity of the nematic films is the elastic energy associated to the long-range orientational order. Experiment shows that the macroscopic spreading laws differ from the ones of simple wetting liquids. Moreover, at the microscopic scale, a sharp transition between a mesoscopic, nematic film, and a molecularly thin precursor, is observed. The length of the precursor scales as the inverse of the macroscopic velocity, as expected for adiabatic wetting films.

Dynamics of unidirectional gravity-inertial spreading of oil on water

Unidirectional gravity-inertial spreading of oil on calm water is investigated. In this stage of spreading, gravity force is balanced by inertia force as the main resisting force. The use of Fannelop and Waldman boundary condition, based on the analogy with the acoustic limit in gas dynamics, is extended and modified to account for the case of continuous discharge of oil on water. The results yield the spreading laws and the velocity and oil-thickness profiles. The oil spill spreading results are compared with the approximate method based on the dynamics of spreading of constant-volume spill, showing that the approximate method is conservative in a certain range.

Spontaneous Spreading of Nematic Liquid Crystals

The spontaneous spreading of macroscopic drops of nematic liquid crystals on hydrophilic substrates has been investigated by interferometric techniques. There is a complex interplay between the elastic energy, due to antagonist anchoring at the interfaces, and the radial flow in the spreading drop. A relevant parameter appears to be the relative humidity of the atmosphere, because it controls the amount of water molecules adsorbed on the substrate and, therefore, the strength of anchoring defects. The spreading laws differ from the ones of simple wetting liquids, and contact line instabilities coupled to short- (anchoring) or large-scale (disclinations) defects of the nematic film are observed.

PIV observation of instantaneous velocity structure of lock release gravity currents in the slumping phase

The experiments of lock release gravity currents were performed in a rectangular channel with salt water and fresh water. The spreading law of the salt water is validated by using a digital video to record the progress of the current. Detailed instantaneous velocity structure of lock release gravity currents in the slumping phase was studied experimentally with particle image velocimetry (PIV). The time variation of the spatial distribution of velocity and vorticity is obtained, which shows some qualitative characters of the current as well as the effects of the bottom boundary layer and upper mixing layer.

Spreading of block copolymer films and domain alignment at moving terrace steps.

We investigate spreading of phase separated copolymer films, where domain walls and thickness steps influence polymer flow. We show that at early stages of spreading its rate is determined by slow activated flow at terrace steps (i.e., thickness steps). At late stages of spreading, on the other hand, the rate is determined by the flow along terraces, with diffusionlike time dependence t(-1/2). This dependence is similar to de Gennes and Cazabat's prediction for generic layered liquids [P. G. de Gennes and A. M. Cazabat, C.R. Acad. Sci. Paris II 310, 1601 (1990)], as opposed to the classical Tanner's law of drop spreading. We also argue that chain hopping at the spreading terrace steps should lead to the formation of aligned, defect-free domain patterns on the growing terraces.

A molecular view of Tanner's law: molecular dynamics simulations of droplet spreading

By truncating the range of van der Waals forces in our molecular dynamics model we reduce the lengthscale and timescale gap between the outer (wedge) and inner (precursor) regions in droplet spreading simulations. This results in a molecular model which combines atomic-scale resolution with the ability to capture large-scale behaviour as manifested by the Tanner spreading law. Our results show that Tanner's law can be recovered, even if van der Waals effects and the resulting precursor film are limited to distances of the order of three atomic diameters from the substrate. In other words, removal of the singularity is not necessary up to a few atomic diameters from the contact line. The very good quantitative agreement with theory and experiments suggests that the original precursor theory of de Gennes can be generalized to precursors of molecular thickness in which flow is not characterized by the continuum (Stokes) model. Gravity current simulations are also in excellent agreement with the theory of Huppert and recent experimental results showing lubrication scalings at small capillary numbers.

Buoyant mixing of miscible fluids of varying viscosities in vertical tubes

Gravity-induced mixing of two fluids in long vertical tubes is studied experimentally as a function of the density contrast characterized by the Atwood number At (10−5 to 0.2), the fluid viscosity ν (1 to 16×10−6 m2 s−1) and the tube diameter d (2 to 44 mm). At low density contrasts, a stable counterflow is observed over a large fraction of the tube and its region of existence increases at high viscosities and small tube diameters. For larger density contrasts, the flow is either convective or turbulent and the mean concentration profile C̄(x,t) follows a diffusive spreading law characterized by a diffusivity D. An unexpected increase of D and of the characteristic velocity Vf of random fluid motions is observed when ν increases. This results from the coarser mixing in more viscous fluids which increases local density contrasts and buoyancy forces. Dimensionless plots of the diffusion coefficient D/ν as a function of the Reynolds number of the flow indicate a transition between two different diffusive regimes. Scaling arguments are put forward to account for the dependence of Vf and of the characteristic diffusion length in the convective–diffusive regime.

Spreading of non-Newtonian liquids over solid substrates

The spreading of drops of a non-Newtonian liquid (Ostwald–de Waele liquid) over horizontal solid substrates is theoretically investigated in the case of complete wetting and small dynamic contact angles. Both gravitational and capillary regimes of spreading are considered. The evolution equation deduced for the shape of the spreading drops has self-similar solutions, which allows obtaining spreading laws for both gravitational and capillary regimes of spreading. In the gravitational regime case of spreading the profile of the spreading drop is provided.

Spreading of Liquid Drops over Saturated Porous Layers

Spreading of small liquid drops over thin porous layers saturated with the same liquid is investigated from both theoretical and experimental points of view. A theory is presented that shows that spreading is governed by the same power law as in the case of spreading over a dry solid substrate. The Brinkman's equations are used to model the liquid flow inside the porous substrate. An equation of the drop spreading is deduced, which shows that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. The presence of these two phenomena removes the well-known singularity at the moving three-phase contact line. Matching of the drop profile in the vicinity of the three-phase contact line with the main spherical part of the drop gives the possibility to calculate the pre-exponential factor in the spreading law via permeability and effective viscosity of the liquid in the porous layer. Unfortunately, the latter dependency turns out to be very weak. Spreading of silicone oils over different microfiltration membranes is carried out. Radii of spreading on time experimental dependencies confirm the theory predictions. Experimentally found coefficients agree with theoretical estimations.

Experimental Study of Axisymmetric Viscous-Gravity Spreading of Oil on Water.

A new experimental technique is proposed for the study of axisymmetric spread of oil on water. The technique is applied to study the viscous-gravity stage in which oil is spilled on water at a constant rate. Results are found to be in good agreement with the order-of-magnitude analysis. An expression for the spreading law is determined, and the accuracy of using the theoretical constant-volume spreading-law to estimate the spreading-law prefactor for the steady-discharge case is analyzed.

Spreading of Gravity Plumes on an Incline

Gravity plumes are buoyancy-driven underflows in a deep ambient fluid. We develop spreading laws for gravity plumes which propagate laterally as well as longitudinally on a uniform slope. Examples of such flows are turbidity currents in the ocean and ignimbrite flows in the atmosphere. Laboratory experiments with a large tank, employing saline density currents, coupled with dimensional analysis, have been used to develop simple expressions for spreading rates of three-dimensional flows on sloping beds. Characteristic length and time are determined by the flow discharge and the buoyancy flux at the inlet. By knowing the initial width of the flow, the spreading laws can be used to estimate the maximum width of the current at different times as well as the longitudinal spreading rate. Predictions for flows compare favorably against observations.

Spreading dynamics of liquids and surfactant solutions on partially wettable hydrophobic substrates

The drop spreading of water and aqueous solutions of ethanol and nonionic surfactant on hydrophobic substrates (alkylsilane treated glass) have been investigated. For the low viscous liquids and solutions, the spreading on the surface of hydrophobic glass rod was also studied and compared to the drop spreading experiment. In both experiments, care was taken to ensure a minimum impact of inertial forces. The results for the aqueous systems show rapid initial spreading processes that abruptly halts after less than 30 ms, as the interfacial tension forces are balanced. In the case of surfactants solutions, this is followed by slower adsorption driven drift towards equilibrium conditions. During the initial spreading phase, the wetting front exhibits ∼ t 1/2 spreading law. Two more viscous liquids, ethylene glycol and glycerol, were also examined and found to show a weaker time-dependence in the whole spreading regime. An ∼ t 1/10 scaling of the drop radius versus time was for these liquids observed in the asymptotic long-time regime. For the surfactant solution, a slow relaxation towards equilibrium was observed following the initial fast spreading phase. The rate-limiting process in this regimes was in the drop spreading experiment found to be surfactant adsorption from the bulk to the expanding liquid–vapour interface, whereas surface diffusion at the liquid–vapour interface appeared rate-determining in the rod experiment. The reason for this is the differences in aspect ratio between relative expansion of the liquid–vapour and solid–liquid interfaces during spreading in the two experiments. In the study of surfactant solution spreading, the importance of surface relaxation prior to contact of the solution and the solid was also pointed out.

Equilibration and Lateral Spreading of a Strip-Shaped Convective Region*

Abstract The authors investigate the spreading stage of the deep ocean convection for a strip-shaped convective region to study the law of spreading and sensitivity of the eddy exchange efficiency parameter to the shape of the region and the background stratification. To simulate this convection process a two-layer quasigeostrophic model is used and the input of buoyancy due to surface cooling is parameterized through creation of pairs of baroclinic point vortices with opposite signs of potential vorticity (hetons) at a constant rate. It is shown that the eddy exchange efficiency parameter for lateral fluxes of potential density out of the convective region is not universal but essentially depends on the shape of the region and the relative layer thicknesses. The horizontal spreading of the potential density anomaly shows a faster than diffusive, quasi-linear dependence on time at moderate values of the surface buoyancy flux. This behavior is due to the specific dynamics of hetons and heton clusters.

The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid

The problem of lateral heat/buoyancy transport in localized turbulent convection dominated by rotation in continuously stratified fluids of finite depth is considered. We investigate the specific mechanism of the vortex-dominated lateral spreading of anomalous buoyancy created in localized convective regions owing to outward propagation of intense heton-like vortices (pairs of vortices of equal potential vorticity (PV) strength with opposite signs located at different depths), each carrying a portion of buoyancy anomaly. Assuming that the quasi-geostrophic form of the PV evolution equation can be used to analyse the spreading phenomenon at fast rotation, we develop an analytical theory for the dynamics of a population of three-dimensional hetons. We analyse in detail the structure and dynamics of a single three-dimensional heton, and the mutual interaction between two hetons and show that the vortices can be in confinement, splitting or reconnection regimes of motion depending on the initial distance between them and the ratio of the mixing-layer depth to the depth of fluid (local to bulk Rossby radii). Numerical experiments are made for ring-like populations of randomly distributed three-dimensional hetons. We found two basic types of evolution of the populations which are homogenizing confinement (all vortices are predominantly inside the localized region having highly correlated wavelike dynamics) and vortex-dominated spreading (vortices propagate out of the region of generation as individual hetons or heton clusters). For the vortex-dominated spreading, the mean radius of heton populations and its variance grow linearly with time. The law of spreading is quantified in terms of both internal (specific for vortex dynamics) and external (specific for convection) parameters. The spreading rate is proportional to the mean speed of propagation of individual hetons or heton clusters and therefore depends essentially on the strength of hetons and the ratio of local to bulk Rossby radii. A theoretical explanation for the spreading law is given in terms of the asymptotic dynamics of a single heton and within the frames of the kinetic equation derived for the distribution function of hetons in collisionless approximation. This spreading law gives an upper ‘advective’ bound for the superdiffusion of heat/buoyancy. A linear law of spreading implies that diffusion parameterizations of lateral buoyancy flux in non-eddy-resolving models are questionable, at least when the spreading is dominated by heton dynamics. We suggest a scaling for the ‘advective’ parameterization of the buoyancy flux, and quantify the exchange coefficient in terms of the mean propagation speed of hetons. Finally, we discuss the perspectives of the heton theories in other problems of geophysical fluid dynamics.

Inertia–gravity spreading of oil on water

Spreading of oil on calm water is investigated for the stage where gravity and inertia forces act as the major promoting and resisting forces. The treatment of Hoult (1972. Annual Review of Fluid Mechanics, 4, 341–368) is generalized to include the case where the volume of an oil strip is not necessarily constant and varies as a power-law function of time. A similarity solution is developed, and the spreading laws and the velocity and oil thickness profiles are determined, providing a theoretical basis which can be used to test different leading-edge boundary conditions proposed in the literature. In addition, the constant-volume-based approximate spreading-law formula is compared with the results obtained theoretically in this study.

Dynamics of Wetting

We consider the dynamics of spreading of a small drop over a smooth solid surface. The analysis is concerned with complete wetting and accounts for capillary, viscous, and van der Waals effects, with two spreading geometries considered: cylindrical and axisymmetric. A complete description of the drop thickness, slope, and curvature profiles is provided. The model is not based on any fitting parameter, and the spreading laws are found to be in good agreement with published data obtained by different experimental techniques. The study evaluates the relative magnitudes of capillary, viscous, and disjoining-pressure effects and discusses the asymptotic form of the film profile and the existence of a singularity at the three-phase contact line. The model allows determination of the complete distribution of viscous dissipation in the whole drop, and the rate of viscous dissipation is related to the losses of the liquid drop free energy.

Spreading Dynamics of Surfactant Solutions

The spreading behavior of microdrops of surfactant solutions at solid surfaces has been studied. The influence on the spreading of different factors, such as the drop lifetime prior to surface contact, surface tension dynamics, surface energy, and surfactant properties, was systematically investigated. The results obtained suggest the existence of two spreading regimes exhibiting different spreading characteristics: In the first, nondiffusive regime, the spreading is very rapid and controlled to different extents by inertia, gravity, and capillarity, depending on the drop size, impact energy, and interfacial tension balance. It is shown in this study that the initial drop surface tension, which is set by the surface tension decay rate and the drop lifetime prior to the surface impact, strongly influences the maximum spreading distance in the nondiffusive spreading regime. The second, diffusion-controlled regime, is characterized by slower concentration-dependent spreading rates. The spreading rate is, here, mainly controlled by the diffusive transport of surfactant to the expanding liquid−vapor interface. In this regime, the drop base radius exhibits an approximate rb2 ∝ t dependence on time. The spreading kinetics at hydrophobic surfaces has been discussed in the framework of a simplified theory. Depending on the assumptions regarding the drop shape, rb5/2 ∝ t to rb2 ∝ t spreading laws are obtained. This agrees reasonably with the experimentally observed relationship.

Design features for electric communication.

How do the communication discharges produced by electric fish evolve to accommodate the unique design features for the modality? Two design features are considered: first, the limited range of signaling imposed on the electric modality by the physics of signal transmission from dipole sources; and second, the absence of signal echoes and reverberations for electric discharges, which are non-propagating electrostatic fields. Electrostatic theory predicts that electric discharges from fish will have a short range because of the inverse cube law of geometric spreading around an electrostatic dipole. From this, one predicts that the costs of signaling will be high when fish attempt to signal over a large distance. Electric fish may economize in signal production whenever possible. For example, some gymnotiform fish appear to be impedance-matched to the resistivity of the water; others modulate the amplitude of their discharge seasonally and diurnally. The fact that electric signals do not propagate, but exist as electrostatic fields, means that, unlike sound signals, electric organ discharges produce no echoes or reverberations. Because temporal information is preserved during signal transmission, receivers may pay close attention to the temporal details of electric signals. As a consequence, electric organs have evolved with mechanisms for controlling the fine structure of electric discharge waveforms.

Capillary Spreading of Liquid Drops on Prewetted Solid Surfaces

A description of the entire configuration of liquid drops spreading over a previously wetted solid surface is given in the case of negligible evaporation and small Bond and Weber numbers. Two solutions are developed: an outer one which is valid in the bulk of the drop and an inner solution which applies in the vicinity of the macroscopic boundary of the drop. The model accounts for deviations from the constant-curvature profile for the outer solution and, in addition, for deviations of the inner solution from the asymptotic approximation of V. V. Kalinin and V. M. Starov (1986, Colloid J. USSR (English tr.) 48, 907). Both solutions are shown to present an inflexion point. Its location is shown to be very sensitive to one parameter which fully determines the inner solution. The value of this parameter, and the spreading laws for the drop radius, the apex height, and the dynamic contact angle are determined by matching the inner and outer solutions. Results show deviations from the power laws used in the literature. These deviations are discussed in relation to the results obtained by V. M. Starov et al. (1994, Adv. Colloid Interface Sci. 50, 187), and R. Chebbi and M. S. Selim (1997, J. Colloid Interface Sci. 195, 66), compared with the experimental data presented by V. M. Starov et al. for spreading over dry smooth solid surfaces. Moreover, the present analysis allows description of the entire drop configuration and slope and curvature variations.

Kinetic laws of spreading during the initial stages in metallic systems with different types of component interactions. III. Effect of mutual saturation on spreading kinetics in copper—Iron-family-metal systems

Kinetic laws of spreading during the initial stages in metallic systems with different types of component interactions. III. Effect of mutual saturation on spreading kinetics in copper—Iron-family-metal systems