The exponential power law: partial wetting kinetics and dynamic contact angles

An equation for the kinetics of partial drop spreading is proposed. This equation was empirically derived from experimental data for the spreading kinetics of partially wetting liquids in terms of the wet area versus time. The equation has the form of an exponential power law (EPL), and transforms into the well-known power law for complete wetting, when the equilibrium contact angle approaches zero. The EPL fits very well available experimental data. To lend additional support to the validity of this generalized equation, it will be demonstrated that when it is transformed to present the dynamic contact angle (DCA), it fits very well DCA experimental data for other wetting processes, such as capillary flow and tape coating.     

Collapse kinetics of vibrated granular chains

The kinetics of the collapse of the coil state into condensed states is studied with vibrated granular chain composed of N metal beads partially immersed in water. The radius of gyration of the chain, R(g) is measured. For short chains (N < 140), disk-like condensed state is formed and R(g) decreases with time such that the function ΔR(g)(2) (≡ R(g)(2) - R(g)(2)(∞)) = A e(-t/τ), where the relaxation time τ follows a power-law dependence on the chain length N with an exponent γ = 1.9 ± 0.2. For the chains with length N ≥ 300, rod-like clusters are observed during the initial stage of collapse and R(g)(2) = R(g)(2)(0) - Bt(β), with β = 0.6 ± 0.1. In the coarsening stage, the exponential dependence of ΔR(g)(2) on time still holds, however, the relaxation time τ fluctuates and has no simple dependence on N. Furthermore, the time dependence of the averaged radius of gyration of the individual clusters, R(g,cl) can be described by the theory of Lifshitz and Slyozov. A peak in the structure function of long chains is observed in the initial stage of the collapse transition. The collapse transition in the bead chains is a first order phase transition. However, features of the spinodal decomposition are also observed.     

Spreading of liquid drops over dry surfaces

Spreading of thin, axisymmetric, non-volatile, Newtonian liquid drops over a dry, smooth, flat solid surface is considered both theoretically and experimentally in the case of complete wetting. The drop profile is solved analytically by matching the “outer” solution for large film thicknesses, where only the capillary effects are important, with the “inner” solution for small film thicknesses, where the viscous and disjoining pressure effects are comparable to capillary effects. It is shown that the apparent radius of the wetted spot, the apex height of the drop, and the apparent advancing dynamic contact angle follow different power laws in time and the advancing dynamic contact angle follows a power law in capillary number. Both the prefactor and the exponent of each power law are derived theoretically. Good agreement between the theory predictions and experimental measurements is shown for both the prefactor and exponent of each power law. It is necessary to emphasize that the theory suggested does not include any fitting parameters.     

Fast dynamic wetting of polymer surfaces by miscible and immiscible liquids

The initial stages of spontaneous spreading of a solvent drop (toluene) on the surface of a soluble polymer (polystyrene) have been studied with a high-speed camera. For drops of 1–4 μL volume, the increase in contact radius r can be described by a power law r μ t^a r \propto {t^{\alpha }} , with the spreading exponent α = 0.50 and for the first ≈8 ms. Thereafter, the three-phase contact line was pinned leading to a macroscopic static contact angle of Θ₀ = 12–15°. The insoluble liquids ethanol (α = 0.47, Θ₀ = 0) and water (α = 0.35, Θ₀ = 90°) showed a slower spreading. We attribute the fast spreading of toluene to the strong interaction with the polymer, like in reactive wetting. The finite macroscopic contact angle indicates the formation of a ridge by softening of polystyrene due to permeated toluene and the subsequent plastic deformation by the surface tension of the liquid. This interpretation is supported by experiments on polymers grafted from a silicon wafer. Toluene completely wets polymer brush surfaces. Transport of toluene through the vapor phase plays a significant role.     

Experimental investigation of the spontaneous wetting of polymers and polymer blends

The dynamics of polymeric liquids and mixtures spreading on a solid surface have been investigated on completely wetting and partially wetting surfaces. Drops were formed by pushing the test liquid through a hole in the underside of the substrate, and the drop profiles were monitored as the liquid wet the surface. Silicon surfaces coated with diphenyldichlorosilane (DPDCS) and octadecyltrichlorosilane (OTS) were used as wetting and partial wetting surfaces, respectively, for the fluids we investigated. The response under complete and partial wetting conditions for a series of polypropylene glycols (PPG) with different molecular weights and the same surface tension could be collapsed onto a single curve when scaling time based on the fluid viscosity, the liquid-vapor surface tension, and the radius of a spherical drop with equivalent volume. A poly(ethylene glycol) (PEG300) and a series of poly(ethylene oxide-rand-propylene oxide) copolymers did not show the same viscosity scaling when spread on the partially wetting surface. A combined model incorporating hydrodynamic and molecular-kinetic wetting models adequately described the complete wetting results. The assumptions in the hydrodynamic model, however, were not valid under the partial wetting conditions in our work, and the molecular-kinetic model was chosen to describe our results. The friction coefficient used in the molecular-kinetic model exhibited a nonlinear dependence with viscosity for the copolymers, indicating a more complex relationship between the friction coefficient and the fluid viscosity.     

Kinetics of wetting of liquid on a solid surface

To consider a sessile drop on an ideal solid surface in equilibrium with a vapor phase, the classic Young equation was given. The derivation of the Young equation was based on both the mechanics and the energy knowledge. According to the constant volume of the liquid in the wetting process of the liquid on a smooth and homogeneous solid surface and the low energy law, Young equation was ob-tained through the mathematic method in this paper. The previous work indicated that the contact angle θ was a function...     

Wetting Film Dynamics

The spreading of a tiny macroscopic drop of a nonvolatile, completely wetting liquid over a flat solid is considered, assuming no gravitation. A liquid, in creeping, is subjected to capillary forces and van der Waals forces. This nonstationary and nonlinear problem in the dynamics of the wetting film from a droplet is studied using numerical modeling. The precursor wetting film motion is described by an evolution equation with conditions at the moving boundaries. The wetting line is regarded as an unknown boundary to be determined in the course of solution. A simplified equation for the wetting line dynamics is analyzed. The difference between the wetting line radius and a fixed (nonzero) radius is described by a diffusion time law. Results of numerical experiments show the simplified law of wetting to be valid over a wide range of spreading times (or a wide range of radii of the wetting line). Copyright 2000 Academic Press.     

Geometry-driven wetting transition

Wetting states are quantitatively described by the number of inflection points on the liquid-vapor interface and by the macroscopic contact angle. The number of inflection points required for complete, partial, and pseudopartial wetting is determined for geometries with positive, zero, and negative capillary pressures. The wetting state of a material system is not always independent of the magnitude of the capillary pressure; for example, the wetting state of a fluid inside a capillary tube may depend on the capillary radius. In particular, a fluid that pseudopartially wets the inside of a tube exhibits a transition to partial wetting (or complete wetting) as the capillary radius is decreased.     

Competitive spreading versus imbibition of polymer liquid drops in nanoporous membranes: scaling behavior with viscosity

The way a liquid drop that is in contact with a nanoporous substrate evolves essentially depends on the competition between imbibition and spreading. Although the scaling behavior of this competitive process with liquid viscosity is important for various applications requiring the filling of nanoporous substrates (template-assisted fabrication, storage, and controlled release of liquids), they appear to be poorly investigated and insufficiently understood. We developed a model study to investigate the wetting and spontaneous imbibition of silicon oil drops of viscosities ranging from 1 to 100 Pa s on nanoporous alumina membranes (pore size of 200 nm). Our results show that the drop radius essentially follows the power law t1/10 time dependence as expected by Tanner's law. However, the scaling of the spreading velocity with the viscosity (approximately eta-n) was found to display an exponent that is comparable on both the reference (impermeable) and nanoporous substrates (n = 0.55) but notably higher than theoretically expected (0.1). More surprisingly, we show that despite the confinement, the rate of imbibition into the nanopores displays a weaker dependence on the viscosity, as compared to the spreading velocity on both the reference and nanoporous substrates. On the basis of Darcy's law for capillary-driven imbibition, this result was discussed in the context of the scaling behavior of the contact angle with the viscosity.     

A review of the role of wetting and spreading phenomena on the flotation practice

Wetting and spreading phenomena are the most important parameters for understanding of froth flotation practice. The wetting and spreading of fluids on the solid surface should be considered in the high efficiency flotation process. These phenomena involve surface tension forces, contact line dynamics, surface roughness and heterogeneity, contact angles, bubble–particle interactions and other factors. This review highlights the various concepts of contact angles and well-known equations in this respect and compares these equations. Based on this review, flotation selectivity and efficiency are highly dependent on solid–liquid contact angles and collision, collection, attachment, and stability efficiency could be predicted by wetting and spreading roles. In order to control flotation performance, efforts should be made to determine wetting characteristic of the flotation process. It is imperative that an improved understanding of wetting and spreading phenomena in the phase's interfaces will provide an improved and efficient flotation practice. It is proposed that future research should focus on the scientific and engineering aspect of wetting and spreading phenomena on flotation and on the development of a method to enhance flotation performance by controlling these phenomena.     

Critical behavior of trifunctional randomly branched polycyanurates

The behavior near the gelation threshold of trifunctional randomly branched polycyanurates is studied by static and dynamic light scattering. By static measurements the critical exponents γ, σ and η were obtained, which describe the divergence of the weight average (M_w) and the cutoff (M*) molecular weights and the radius of gyration (R_g) respectively. All these independently measured exponents together with τ, characterizing the power law behavior of the molecular weight distribution and measured by size exclusion chromatography coupled with light scattering, confirm the predictions of the three-dimensional percolation theory. With the help of size exclusion chromatography coupled with a light scattering and a viscosity detector, a fractal dimension D = 2.24 is obtained. On the other side, from the corresponding exponent for the whole unfractionated samples a fractal dimension D = 2.21 results, using a theory of Daoud. This suggests that the fractal dimension of the polycyanurates in dilute solution lies between the theoretical predictions D = 2.5 for the unswollen and D = 2.0 for the completely swollen state. Furthermore, it is shown by dynamic light scattering that the power law behavior over some decades in time of the time autocorrelation function and the divergence of the mean relaxation time are characteristics of the gelpoint. The development with increasing reaction time of the time correlation function of the gelling system from the pregel through the gelpoint into the gel state is analyzed quantitatively by a hybrid of a stretched exponential and a power law function.     

Simultaneous spreading and evaporation: Recent developments

The recent progress in theoretical and experimental studies of simultaneous spreading and evaporation of liquid droplets on solid substrates is discussed for pure liquids including nanodroplets, nanosuspensions of inorganic particles (nanofluids) and surfactant solutions. Evaporation of both complete wetting and partial wetting liquids into a nonsaturated vapour atmosphere are considered. However, the main attention is paid to the case of partial wetting when the hysteresis of static contact angle takes place. In the case of complete wetting the spreading/evaporation process proceeds in two stages. A theory was suggested for this case and a good agreement with available experimental data was achieved. In the case of partial wetting the spreading/evaporation of a sessile droplet of pure liquid goes through four subsequent stages: (i) the initial stage, spreading, is relatively short (1–2 min) and therefore evaporation can be neglected during this stage; during the initial stage the contact angle reaches the value of advancing contact angle and the radius of the droplet base reaches its maximum value, (ii) the first stage of evaporation is characterised by the constant value of the radius of the droplet base; the value of the contact angle during the first stage decreases from static advancing to static receding contact angle; (iii) during the second stage of evaporation the contact angle remains constant and equal to its receding value, while the radius of the droplet base decreases; and (iv) at the third stage of evaporation both the contact angle and the radius of the droplet base decrease until the drop completely disappears. It has been shown theoretically and confirmed experimentally that during the first and second stages of evaporation the volume of droplet to power 2/3 decreases linearly with time. The universal dependence of the contact angle during the first stage and of the radius of the droplet base during the second stage on the reduced time has been derived theoretically and confirmed experimentally. The theory developed for pure liquids is applicable also to nanofluids, where a good agreement with the available experimental data has been found. However, in the case of evaporation of surfactant solutions the process deviates from the theoretical predictions for pure liquids at concentration below critical wetting concentration and is in agreement with the theoretical predictions at concentrations above it.     

Review of non-reactive and reactive wetting of liquids on surfaces

Wettability is a tendency for a liquid to spread on a solid substrate and is generally measured in terms of the angle (contact angle) between the tangent drawn at the triple point between the three phases (solid, liquid and vapour) and the substrate surface. A liquid spreading on a substrate with no reaction/absorption of the liquid by substrate material is known as non-reactive or inert wetting whereas the wetting process influenced by reaction between the spreading liquid and substrate material is known as reactive wetting. Young's equation gives the equilibrium contact angle in terms of interfacial tensions existing at the three-phase interface. The derivation of Young's equation is made under the assumptions of spreading of non-reactive liquid on an ideal (physically and chemically inert, smooth, homogeneous and rigid) solid, a condition that is rarely met in practical situations. Nevertheless Young's equation is the most fundamental starting point for understanding of the complex field of wetting. Reliable and reproducible measurements of contact angle from the experiments are important in order to analyze the wetting behaviour. Various methods have been developed over the years to evaluate wettability of a solid by a liquid. Among these, sessile drop and wetting balance techniques are versatile, popular and provide reliable data. Wetting is affected by large number of factors including liquid properties, substrate properties and system conditions. The effect of these factors on wettability is discussed. Thermodynamic treatment of wetting in inert systems is simple and based on free energy minimization where as that in reactive systems is quite complex. Surface energetics has to be considered while determining the driving force for spreading. Similar is the case of spreading kinetics. Inert systems follow definite flow pattern and in most cases a single function is sufficient to describe the whole kinetics. Theoretical models successfully describe the spreading in inert systems. However, it is difficult to determine the exact mechanism that controls the kinetics since reactive wetting is affected by a number of factors like interfacial reactions, diffusion of constituents, dissolution of the substrate, etc. The quantification of the effect of these interrelated factors on wettability would be useful to build a predictive model of wetting kinetics for reactive systems.     

The phase dynamics and wetting layer formation mechanisms in two-step surface-directed spinodal decomposition

The two-step quench process of surface-directed spinodal decomposition is numerically investigated by coupling the Flory-Huggins-de Gennes equation with the Cahn-Hilliard-Cook equation. The phase dynamics and formation mechanisms of the wetting layer in two-step surface-directed spinodal decomposition have been concerned in detail. The results demonstrate that a parallel strip structure forms near the wetting layer and propagates into the bulk, when the first quench depth is very shallow and the bulk does not undergo phase separation, and the second quench depths are various points with deeper quench depths. In this case, the wetting layer turns to be unchangeable at the intermediate and later stages of the second quench process, compared to the growth with a time exponent 1/2 during the first quench process. When the first quench depth is deeper and phase separation occurs in the bulk during the first quench process, it is found that a deeper second quench depth can stimulate a more obvious secondary domain structure, and the formation mechanism of the wetting layer changes from logarithmic growth law to Lifshitz-Slyozov growth law.     

Calculating the hopping times of confined fluids: two hard disks in a box

The dynamical transition between the anomalous single file diffusion of highly confined fluids and bulk normal diffusion can be described by a phenomenological model involving a particle hopping time tau(hop). We suggest a theoretical formalism that will be useful for the calculation of tau(hop) for a variety of systems and test it using a simple model consisting of two hard disks confined to a rectangular box with hard walls. In the case where the particles are moving diffusively, we find the hopping time diverges as a power law in the threshold region with an exponent of -(3/2). Under conditions where the particles move inertially, transition state theory predicts a power law behavior with an exponent of -2. Molecular dynamics simulations confirm the transition state theory result for inertial dynamics, while Brownian dynamics simulations suggest the scaling exponent is highly sensitive to the details of the algorithm.     

Confinement free energy and chain conformations of homopolymers confined between two repulsive walls

Lattice Monte Carlo simulations of polymer solutions confined between two parallel plates were performed. The confinement free energy Deltamicro(conf) per chain and the radius of gyrations of the chains parallel and perpendicular to the plates were obtained. When the concentration of the confined solution is above the overlap concentration, Deltamicro(conf) is found to scale with Na/D in a power law, betaDeltamicro(conf) approximately (Na/D)(m), with an exponent m=1.10+/-0.02 for athermal walls where N is the number of monomers in a chain, D is the slit width, and a is the lattice spacing. The presence of a weak attractive polymer/wall interaction epsilon(w) does not change the scaling variable, but the exponent m increases slightly. Extrapolating the results to melt would suggest that the predictions made by de Gennes [C. R. Acad. Sci. Paris II 305, 1181 (1987)] about the confinement free energy cost per chain in polymer melt is correct as far as the scaling variable is concerned, but is incorrect about the exponent m observed. The implication of this result on the predicted force between plates immersed in polymer melt is discussed. The parallel dimensions of the confined chain is expanded when the slit width D is narrow, however, the expansion is reduced at high concentration. It is conceivable that in melt the chain is not expanded when confined in a repulsive slit.     

Flux-assisted wetting and spreading of Al on TiC

The effect of a K-Al-F-based flux on the spreading of Al on TiC, at temperatures up to 900 degrees C, in Ar and in air has been studied. Whilst obtuse contact angles were observed without flux, the flux facilitated rapid spreading to a perfect wetting condition, in both Ar and in air. The atmosphere was found to have a weak effect on the spreading kinetics as the liquid flux provides a locally protective atmosphere by spreading over the TiC surface and also on the solid surface of Al. The flux dissolves the aluminium oxide, covering Al, so that when Al melts, and the oxide layer has been removed or weakened, intimate contact occurs between liquid Al and the TiC substrate facilitating spontaneous spreading and instantaneous wetting of liquid Al on TiC. Since flux-assisted spreading is very rapid and occurs without the formation of a reaction layer at the Al/TiC interface, this process is very different to the reactive wetting behaviour previously reported in the Al-TiC system.     

Ion-specific and reversible wetting of imidazolium-based minigels

Cross-linked imidazolium-based [poly(ViEtIm +Br -)] microparticles were synthesized, and their wetting properties were studied by optical microscopy, after addition of aqueous solutions of sodium halides. Particle wetting showed ion specificity due to counterion binding, described by Desnoyer's model. The interaction between anions and the microparticles allowed exchanging halogenides between them in a reversible way. A salt-independent characteristic wetting time was found as well as a decreasing power law with salt concentration, for the network diffusion coefficient. It modified the polymer network elasticity as ion concentration increased, making the network softer.     

Manipulation of droplets by dynamically controlled wetting gradients

The reversible transportation of droplets was realized by spatiotemporal control of the wetting gradient. The surface wetting was reversibly regulated by using electrochemical reactions of the ferrocenyl (Fc) alkanethiol monolayer, and the wetting gradient was generated by the application of the in-plane bias voltage to the substrate. The back-and-forth motion of the wetting boundary, where the surface changed from wetting to repulsive, sequentially caused a droplet unidirectional spreading and shrinking on the surface. These unidirectional deformations resulted in the net transport of the droplet in an inchwormlike manner. The droplet moved backward when the direction of the in-plane bias voltage was reversed.     

Dynamics of wetting

We review the mechanisms controlling the dynamics of wetting in partial and complete wetting regimes. It is shown that the behaviour in several timescales may characterize the dynamics since different channels of energy dissipation have to be considered within spreading.