Line energy and the relation between advancing, receding, and young contact angles

The line energy associated with the triple phase contact line is a function of local surface defects (chemical and topographical); however, it can still be calculated from the advancing and receding contact angles to which those defects give rise. In this study an expression for the line energy associated with the triple phase contact line is developed. The expression relates the line energy to the drop volume, the interfacial energies, and the actual contact angle (be it advancing, receding, or in between). From the expression we can back calculate the equilibrium Young contact angle, theta0, as a function of the maximal advancing, thetaA, and minimal receding, thetaR, contact angles. To keep a certain maximal hysteresis between advancing and receding angles, different line energies are required depending on the three interfacial energies and the drop's volume V. We learn from the obtained expressions that the hysteresis is determined by some dimensionless parameter, K, which is some normalized line energy. The value of K required to keep a constant hysteresis (thetaA-thetaR) rises to infinity as we get closer to theta0 = 90 degrees.     

Anisotropic wetting characteristics on submicrometer-scale periodic grooved surface

Submicrometer-scale periodic structures consisting of parallel grooves were prepared on azobenzene-containing multiarm star polymer films by laser interference. The wetting characteristics on the patterned surfaces were studied by contact angle measurements. Macroscopic distortion of water drops was found on such small-scale surface structures, and the contact angles measured from the direction parallel to the grooves were larger than those measured from the perpendicular direction. A thermodynamic model was developed to calculate the change in the surface free energy as a function of the instantaneous contact angle when the three-phase contact line (TPCL) moves along the two orthogonal directions. It was found that the fluctuations, i.e., energy barriers, on the energy versus contact angle curves are crucial to the analysis of wetting anisotropy and contact angle hysteresis. The calculated advancing and receding contact angles from the energy versus contact angle curves were in good agreement with those measured experimentally. Furthermore, with the groove depth increasing, both the degree of wetting anisotropy and the contact angle hysteresis perpendicular to the grooves increased as a result of the increase in the energy barrier. The theoretical critical value of the groove depth, above which the anisotropic wetting appears, was determined to be 16 nm for the grooved surface with a wavelength of 396 nm. On the other hand, the effect of the groove wavelength on the contact angle hysteresis perpendicular to the grooves was also interpreted on the basis of the thermodynamic model. That is, with the wavelength decreasing, the contact angle hysteresis increased due to the increase in the number of energy barriers. These results may provide theoretical evidence for the design and application of anisotropic wetting surface.     

A quantitative experimental study of wetting hysteresis on discrete and continuous chemical heterogeneities

Chemically heterogeneous surfaces are well known to induce contact angle hysteresis due to the local energy barriers that oppose contact line movement. In many cases, the surface heterogeneity is discontinuous, i.e. discrete regions of different wettability exist, which leads to pinning of the contact line at boundaries between regions. Pinning on individual rows of microscopic defects arranged in a square lattice can be sensed using a Wilhelmy balance to reveal discrete stick-slip motion. For defects more wettable than the matrix with a lattice spacing of 28 μm, the advancing contact line slips over ～10 rows in a single slip step, while the receding contact line stick-slips between individual rows of defects. Single, millimetre-scale defects were used to assess the energy involved when a contact line advances or recedes over a hydrophilic (more wettable) defect. Quantitative information about defect-induced hysteresis in relation to defect dimensions is obtained. The crucial importance of wetting boundaries is highlighted with an experimental example of a surface that is heterogeneous yet, due to the continuously changing pattern, does not exhibit contact angle hysteresis.     

Impact of pinning of the triple contact line on electrowetting performance

Pinning of the triple contact line adversely affects electrowetting on dielectric. Electrowetting response of substrates with contact angle hysteresis ranging from 1° to 30° has been characterized, and the results are interpreted within the framework of electromechanics corrected for pinning. The relationship between contact angle hysteresis, threshold potential for liquid actuation, and electrowetting hysteresis is quantified. Our results demonstrate that a modified electrowetting equation, based on balance of forces (including the pinning forces) acting on the triple contact line and on the drop, describes the electrowetting response of substrates with significant contact angle hysteresis. Finally, the surface properties of PDMS Sylgard 184 were found to be influenced by the electric field.     

Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces

We investigate contact angle hysteresis on chemically patterned and superhydrophobic surfaces, as the drop volume is quasistatically increased and decreased. We consider both two (cylindrical drops) and three (spherical drops) dimensions using analytical and numerical approaches to minimize the free energy of the drop. In two dimensions, we find, in agreement with other authors, a slip, jump, stick motion of the contact line. In three dimensions, this behavior persists, but the position and magnitude of the contact line jumps are sensitive to the details of the surface patterning. In two dimensions, we identify analytically the advancing and receding contact angles on the different surfaces, and we use numerical insights to argue that these provide bounds for the three-dimensional cases. We present explicit simulations to show that a simple average over the disorder is not sufficient to predict the details of the contact angle hysteresis and to support an explanation for the low contact angle hysteresis of suspended drops on superhydrophobic surfaces.     

An empirically validated analytical model of droplet dynamics in electrowetting on dielectric devices

Explicit analytical models that describe the capillary force on confined droplets actuated in electrowetting on dielectric devices and the reduction in that force by contact angle hysteresis as a function of the three-dimensional shape of the droplet interface are presented. These models are used to develop an analytical model for the transient position and velocity of the droplet. An order of magnitude analysis showed that droplet motion could be modeled using the driving capillary force opposed by contact angle hysteresis, wall shear, and contact line friction. Droplet dynamics were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The first four parameters describe the device geometry and fluid properties; the remaining parameters were determined experimentally. Images of the droplet during motion were used to determine the evolution of the shape, position, and velocity of the droplet with time. Comparisons between the measured and predicted results show that the proposed model provides good accuracy over a range of practical voltages and droplet aspect ratios.     

Surface free energy of a solid from contact angle hysteresis

Nature of contact angle hysteresis is discussed basing on the literature data (Colloids Surf. A 189 (2001) 265) of dynamic advancing and receding contact angles of n-alkanes and n-alcohols on a very smooth surface of 1,1,2,-trichloro-1,2,2,-trifluoroethane (FC-732) film deposited on a silicon plate. The authors considered the liquid absorption and/or retention (swelling) processes responsible for the observed hysteresis. In this paper hysteresis is considered to be due to the liquid film left behind the drop during retreating of its contact line. Using the contact angle hysteresis an approach is suggested for evaluation of the solid surface free energy. Molecular spacing and the film structure are discussed to explain the difference in n-alkanes and n-alcohols behaviour as well as to explain the difference between dispersion free energy gamma(s)(d) and total surface free energy gamma(s)(tot) of FC-732, as determined from the advancing contact angles and the hysteresis, respectively.     

Fabrication of surfaces with extremely high contact angle hysteresis from polyelectrolyte multilayer

High contact angle hysteresis on polyelectrolyte multilayers (PEMs) ion-paired with hydrophobic perfluorooctanoate anions is reported. Both the bilayer number of PEMs and the ionic strength of deposition solutions have significant influence on contact angle hysteresis: higher ionic strength and greater bilayer number cause increased contact angle hysteresis values. The hysteresis values of ~100° were observed on smooth PEMs and pinning of the receding contact line on hydrophilic defects is implicated as the cause of hysteresis. Surface roughness can be used to further tune the contact angle hysteresis on the PEMs. A surface with extremely high contact angle hysteresis of 156° was fabricated when a PEM was deposited on a rough substrate coated with submicrometer scale silica spheres. It was demonstrated that this extremely high value of contact angle hysteresis resulted from the penetration of water into the rough asperities on the substrate. The same substrate hydrophobized by chemical vapor deposition of 1H,1H,2H,2H-perfluorooctyltriethoxysilane exhibits high advancing contact angle and low hysteresis.     

Modeling contact angle hysteresis of a liquid droplet sitting on a cosine wave-like pattern surface

A liquid droplet sitting on a hydrophobic surface with a cosine wave-like square-array pattern in the Wenzel state is simulated by using the Surface Evolver to determine the contact angle. For a fixed drop volume, multiple metastable states are obtained at two different surface roughnesses. Unusual and non-circular shape of the three-phase contact line of a liquid droplet sitting on the model surface is observed due to corrugation and distortion of the contact line by structure of the roughness. The contact angle varies along the contact line for each metastable state. The maximum and minimum contact angles among the multiple metastable states at a fixed viewing angle correspond to the advancing and the receding contact angles, respectively. It is interesting to observe that the advancing/receding contact angles (and contact angle hysteresis) are a function of viewing angle. In addition, the receding (or advancing) contact angles at different viewing angles are determined at different metastable states. The contact angle of minimum energy among the multiple metastable states is defined as the most stable (equilibrium) contact angle. The Wenzel model is not able to describe the contact angle along the three-phase contact line. The contact angle hysteresis at different drop volumes is determined. The number of the metastable states increases with increasing drop volume. Drop volume effect on the contact angles is also discussed.     

Constitutive modeling of contact angle hysteresis

We introduce a phase field model of wetting of surfaces by sessile drops. The theory uses a two-dimensional non-conserved phase field variable to parametrize the Gibbs free energy of the three-dimensional system. Contact line tension and contact angle hysteresis arise from the gradient term in the free energy and the kinetic coefficient respectively. A significant advantage of this approach is in the constitutive specification of hysteresis. The advancing and receding angles of a surface, the liquid-vapor interfacial energy and three-phase line tension are the only required constitutive inputs to the model. We first simulate hysteresis on a smooth chemically homogeneous surface using this theory. Next we show that it is possible to study heterogeneous surfaces whose component surfaces are themselves hysteretic. We use this theory to examine the wetting of a surface containing a circular heterogeneous island. The contact angle for this case is found to be determined solely by the material properties at the contact line in accord with recent experimental data.     

Comparison of contact angle hysteresis of different probe liquids on the same solid surface

Advancing and receding contact angles of water, formamide and diiodomethane were measured on 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) layers deposited on three different solid supports—glass, mica and poly(methyl methacrylate). Up to five statistical monolayers were deposited on the surfaces by spreading DPPC solution. It was found that even on five statistical DPPC monolayers, the hysteresis of a given liquid depends on the kind of solid support. Also on the same solid support the contact angle hysteresis is different for each probe liquid used. The AFM images show that the heights of roughness of the DPPC films cannot be the primary cause of the observed hysteresis because the heights are too small to cause the observed hystereses. It is believed that the hysteresis is due to the liquid film present right behind the three-phase solid surface/liquid drop/gas (vapour) contact line and the presence of Derjaguin pressure. The value of contact angle hysteresis depends on both the solid surface and liquid properties as well as on intermolecular interactions between them.     

Contact angle hysteresis on polymer substrates established with various experimental techniques, its interpretation, and quantitative characterization

The effect of contact angle hysteresis (CAH) was studied on various polymer substrates with traditional and new experimental techniques. The new experimental technique presented in the article is based on the slow deformation of the droplet, thus CAH is studied under the constant volume of the drop in contrast to existing techniques when the volume of the drop is changed under the measurement. The energy of hysteresis was calculated in the framework of the improved Extrand approach. The advancing contact angle established with a new technique is in a good agreement with that measured with the needle-syringe method. The receding angles measured with three experimental techniques demonstrated a very significant discrepancy. The force pinning the triple line responsible for hysteresis was calculated.     

Pressure dependence of the contact angle

When a liquid and its vapor contact a smooth, homogeneous surface, Gibbsian thermodynamics indicates that the contact angle depends on the pressure at the three-phase line of an isothermal system. When a recently proposed adsorption isotherm for a solid-vapor interface is combined with the equilibrium conditions and the system is assumed to be in a cylinder where the liquid-vapor interface can be approximated as spherical, the contact-angle-pressure relation can be made explicit. It indicates that a range of contact angles can be observed on a smooth homogeneous surface by changing the pressure at the three-phase line, but it also indicates that the adsorption at the solid-liquid interface is negative, and leads to the prediction that the contact angle increases with pressure. The predicted dependence of the contact angle on pressure is investigated experimentally in a system that has an independent mechanism for determining when thermodynamic equilibrium is reached. The predictions are in agreement with the measurements. The results provide a possible explanation for contact angle hysteresis.     

How Wenzel and cassie were wrong

We argue using experimental data that contact lines and not contact areas are important in determining wettability. Three types of two-component surfaces were prepared that contain "spots" in a surrounding field: a hydrophilic spot in a hydrophobic field, a rough spot in a smooth field, and a smooth spot in a rough field. Water contact angles were measured within the spots and with the spot confined to within the contact line of the sessile drop. Spot diameter and contact line diameter were varied. All of the data indicate that contact angle behavior (advancing, receding, and hysteresis) is determined by interactions of the liquid and the solid at the three-phase contact line alone and that the interfacial area within the contact perimeter is irrelevant. The point is made that Wenzel's and Cassie's equations are valid only to the extent that the structure of the contact area reflects the ground state energies of contact lines and the transition states between them.     

Anisotropic drop morphologies on corrugated surfaces

The spreading of liquid drops on surfaces corrugated with micrometer-scale parallel grooves is studied both experimentally and numerically. Because of the surface patterning, the typical final drop shape is no longer spherical. The elongation direction can be either parallel or perpendicular to the direction of the grooves, depending on the initial drop conditions. We interpret this result as a consequence of both the anisotropy of the contact line movement over the surface and the difference in the motion of the advancing and receding contact lines. Parallel to the grooves, we find little hysteresis due to the surface patterning and that the average contact angle approximately conforms to Wenzel's law as long as the drop radius is much larger than the typical length scale of the grooves. Perpendicular to the grooves, the contact line can be pinned at the edges of the ridges, leading to large contact angle hysteresis.     

Drops down the hill: theoretical study of limiting contact angles and the hysteresis range on a tilted plate

The limiting inclination angle (slip angle), for which a two-dimensional water drop may be at equilibrium on a chemically heterogeneous surface, is exactly calculated for a variety of cases. The main conclusion is that, in the cases studied, the contact angles at the upper and lower contact line do not always simultaneously equal the receding and advancing contact angles, respectively. On a hydrophobic surface, the lowest contact angle (at the upper contact line) tends to be approximately equal to the receding contact angle, while the highest contact angle (at the lower contact line) may be much lower than the advancing contact angle. For hydrophilic surfaces, the opposite is true. These conclusions imply that the hysteresis range cannot in general be measured by analyzing the shape of a drop on an inclined plane. Also, the limiting inclination angle cannot in general be calculated from the classical equation based only on the advancing and receding contact angles.     

Dynamic contact angles and hysteresis under electrowetting-on-dielectric

By designing and implementing a new experimental method, we have measured the dynamic advancing and receding contact angles and the resulting hysteresis of droplets under electrowetting-on-dielectric (EWOD). Measurements were obtained over wide ranges of applied EWOD voltages, or electrowetting numbers (0 ≤ Ew ≤ 0.9), and droplet sliding speeds, or capillary numbers (1.4 × 10(-5) ≤ Ca ≤ 6.9 × 10(-3)). If Ew or Ca is low, dynamic contact angle hysteresis is not affected much by the EWOD voltage or the sliding speed; that is, the hysteresis increases by less than 50% with a 2 order-of-magnitude increase in sliding speed when Ca < 10(-3). If both Ew and Ca are high, however, the hysteresis increases with either the EWOD voltage or the sliding speed. Stick-slip oscillations were observed at Ew > 0.4. Data are interpreted with simplified hydrodynamic (Cox-Voinov) and molecular-kinetic theory (MKT) models; the Cox-Voinov model captures the trend of the data, but it yields unreasonable fitting parameters. MKT fitting parameters associated with the advancing contact line are reasonable, but a lack of symmetry indicates that a more intricate model is required.     

Sliding variability of droplets on a hydrophobic incline due to surface entrained air bubbles

The ability of a liquid droplet to move on an incline has important ramifications in discrete volume fluidic devices. By taking advantage of the spontaneous and copious formation of visible air bubbles within water droplets on a polytetrafluoroethylene (PTFE) surface, we uncovered a direct correlation between their presence and the ability of droplets to slide down an incline. We forward two possible mechanisms to account for this behavior. The first is attributed to the air bubbles creating regions where additional solid-liquid-vapor phase interfaces are present; wherein due to the buoyancy force acting upwards, the orientation of the contact angles of each bubble (which should also be in hysteresis but in the opposite direction of the hysteresis at the droplet rim contact lines) dictate that the net force of the bubbles in the droplet act down an incline. We show here that this mechanism cannot fully account for the bubble enhanced sliding behavior. The second mechanism is based on the occurrence of the droplet front advancing first, causing the droplet to elongate and thus allowing the receding contact line to partially sweep inwards over the bubbles. This causes a series of point-wise disruptions on the contact line that permits the droplet to slide down more readily. The relatively short time of ～180s during which these micron sized bubbles decrease in size indicates a possibility of this mechanism contributing to a transient means to reduce the retention force of droplets that reside on hydrophobic surfaces.     

Anomalous contact angle hysteresis of a captive bubble: advancing contact line pinning

Contact angle hysteresis of a sessile drop on a substrate consists of continuous invasion of liquid phase with the advancing angle (θ(a)) and contact line pinning of liquid phase retreat until the receding angle (θ(r)) is reached. Receding pinning is generally attributed to localized defects that are more wettable than the rest of the surface. However, the defect model cannot explain advancing pinning of liquid phase invasion driven by a deflating bubble and continuous retreat of liquid phase driven by the inflating bubble. A simple thermodynamic model based on adhesion hysteresis is proposed to explain anomalous contact angle hysteresis of a captive bubble quantitatively. The adhesion model involves two solid–liquid interfacial tensions (γ(sl) > γ(sl)′). Young’s equation with γ(sl) gives the advancing angle θ(a) while that with γ(sl)′ due to surface rearrangement yields the receding angle θ(r). Our analytical analysis indicates that contact line pinning represents frustration in surface free energy, and the equilibrium shape corresponds to a nondifferential minimum instead of a local minimum. On the basis of our thermodynamic model, Surface Evolver simulations are performed to reproduce both advancing and receding behavior associated with a captive bubble on the acrylic glass.     

Theoretical contact angles on a nano-heterogeneous surface composed of parallel apolar and polar strips

Neumann-Good's parallel strip model (J. Colloid Interface Sci. 1972, 38, 341) was used to analyze the contact angle hysteresis for a liquid on a heterogeneous surface composed of alternatively aligned horizontal apolar (theta = 70 degrees ) and polar (theta = 0 degree ) strips. The critical size of the strip width, below which the contact angle hysteresis disappears, was determined on the basis of the analysis of the activation energy for wetting to be from 6 to 12 nm. This calculated value of the critical strip size is 1 order of magnitude smaller than that of 0.1 microm, which has been commonly considered as the limit of heterogeneity size causing the appearance of the contact angle hysteresis.