The contact angle

The second approach to obtaining the surface energies of solids involves the study of wetting and non-wetting liquids on a smooth, clean solid substrate. Let us examine the situation for a non-wetting 

From simple geometry, d/ = d/cos0 and hence at equilibrium, where dG/dl = 0, it follows that 

In water the wetted solid is termed hydrophilic , whereas the non-wetted solid is hydrophobic . Naturally hydrophobic minerals, such as some types of coal, talc and molybdenite are easily separated from the unwanted hydrophilic quartz sand (referred to as gangue ). However, surfactants and oils are usually added as collectors . These compounds adsorb onto the hydrophilic mineral surface and make it hydrophobic. 

Frotliers are also added to stabilize the foam at the top of the chamber, so that the enriched mineral can be continuously scooped off. The selective flotation of a required mineral depends critically on surface properties and these can be carefully controlled using a wide range of additives. Throughout the world a large quantity (about 10 tons annually) of minerals are separated by this method. 

Modern photographic-quality inkjet papers have a surface coating comprising either a thin polymer film or a fine porous layer. In either case the material is formed using a high-speed coating process. This process requires careful control to obtain the necessary uniformity together with 

The Contact Angle When the substrate is a solid, the spreading coefficient is usually evaluated by indirect means, since surface and interfacial tensions of solids cannot easily be measured directly. The method of doing this involves measuring the contact angle the substrate makes with the liquid in question. 

Equation 6.3 is generally called Young s equation and the quantity yM cos 0 the adhesion tension (Bartell, 1934). Note that ySA, the interfacial tension in equilibrium with the gas and liquid phases in the system, is not ys, the free energy per unit area of the solid in a vacuum, but ys — n, where n is the reduction in interfacial free energy per unit area of S resulting from adsorption of vapor of L that is, n = ys-isA- 

When 0 is hnite, (cos 0 — 1) is always negative, and Sl/S, too, is always negative. If the contact angle is 0°, then Sl/s may be zero or positive. In either case here, complete spreading wetting occurs. 

When the solid substrate is a nonpolar, low-energy surface, the contact angle can be used to determine the surface (excess) concentration of the surfactant at the solid-liquid interface rS . 

If we assume that, for a low-energy surface, yS4 does not change with change in the liquid phase surfactant concentration, i.e., d ySA)/d n C = 0, then 

Young s equation is the basis for a quantitative description of wetting phenomena. If a drop of a liquid is placed on a solid surface there are two possibilities the liquid spreads on the surface completely (contact angle 0 = 0°) or a finite contact angle is established.1 In the second case a three-phase contact line — also called wetting line — is formed. At this line three phases are in contact the solid, the liquid, and the vapor (Fig. 7.1). Young s equation relates the contact angle to the interfacial tensions 75, 7l, and 7sl [222,223]  

If the interfacial tension of the bare solid surface is higher than that of the solid-liquid interface (7s Isl), the right hand side of Young s equation is positive. Then cos has to be positive and the contact angle is smaller than 90° the liquid partially wets the solid. If the solid-liquid 

A small change in the contact radius a, leads to a change in the liquid surface area of 

Unfortunately, the change in surface area depends on two variables a and h (we do not have to consider a change in 0, it is of second order). However, these two variables are not independent because the volume of the drop is constant. The volume of a spherical cap is 

Now we can write the total change in the Gibbs free energy as dG = (7sl 7s) dASL + 7L dAL 

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Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

Perhaps the best discussions of the experimental aspects of the capillary rise method are still those given by Richards and Carver [20] and Harkins and Brown [21]. For the most accurate work, it is necessary that the liquid wet the wall of the capillary so that there be no uncertainty as to the contact angle. Because of its transparency and because it is wet by most liquids, a glass capillary is most commonly used. The glass must be very clean, and even so it is wise to use a receding meniscus. The capillary must be accurately vertical, of accurately known and uniform radius, and should not deviate from circularity in cross section by more than a few percent. 

This very simple result is independent of the value of the contact angle because the configuration involved is only that between the equatorial plane and the apex. 

A liquid of density 2.0 g/cm forms a meniscus of shape corresponding to /3 = 80 in a metal capillary tube with which the contact angle is 30°. The capillary rise is 0.063 cm. Calculate the surface tension of the liquid and the radius of the capillary, using Table II-l. 

Equation 11-30 may be integrated to obtain the profile of a meniscus against a vertical plate the integrated form is given in Ref. 53. Calculate the meniscus profile for water at 20°C for (a) the case where water wets the plate and (b) the case where the contact angle is 40°. For (b) obtain from your plot the value of h, and compare with that calculated from Eq. 11-28. [Hint Obtain from 11-15.]... 

There are some subtleties with respect to the physicochemical meaning of the contact angle equation, and these are taken up in Section X-7. The preceding, however, serves to introduce the conventional definitions to permit discussion of the experimental observations. 

Ruch and Bartell [84], studying the aqueous decylamine-platinum system, combined direct estimates of the adsorption at the platinum-solution interface with contact angle data and the Young equation to determine a solid-vapor interfacial energy change of up to 40 ergs/cm due to decylamine adsorption. Healy (85) discusses an adsorption model for the contact angle in surfactant solutions and these aspects are discussed further in Ref. 86. 

The axisymmetric drop shape analysis (see Section II-7B) developed by Neumann and co-workers has been applied to the evaluation of sessile drops or bubbles to determine contact angles between 50° and 180° [98]. In two such studies, Li, Neumann, and co-workers [99, 100] deduced the line tension from the drop size dependence of the contact angle and a modified Young equation... 

The capillary rise on a Wilhelmy plate (Section II-6C) is a nice means to obtain contact angles by measurement of the height, h, of the meniscus on a partially immersed plate (see Fig. 11-14) [111, 112]. Neumann has automated this technique to replace manual measurement of h with digital image analysis to obtain an accuracy of 0.06° (and a repeatability to 95%, in practice, of 0.01°) [108]. The contact angle is obtained directly from the height through... 

Yaminsky and Yaminskaya [114] have used a Wilhelmy plate to directly measure the interfacial tension (and hence infer the contact angle) for a surfactant solution on... 

Fig. X-9. Zisman plots of the contact angles of various homologous series on Teflon O, RX , alkylbenzenes (f), n-alkanes , dialkyl ethers , siloxanes A, miscellaneous polar liquids. (Data from Ref. 78.)...

Contact angle will vary with liquid composition, often in a regular way as illustrated in Fig. X-13 (see also Ref. 136). Li, Ng, and Neumann have studied the contact angles of binary liquid mixtures on teflon and found that the equation of state that describes... 

The extensive use of the Young equation (Eq. X-18) reflects its general acceptance. Curiously, however, the equation has never been verified experimentally since surface tensions of solids are rather difficult to measure. While Fowkes and Sawyer [140] claimed verification for liquids on a fluorocarbon polymer, it is not clear that their assumptions are valid. Nucleation studies indicate that the interfacial tension between a solid and its liquid is appreciable (see Section K-3) and may not be ignored. Indirect experimental tests involve comparing the variation of the contact angle with solute concentration with separate adsorption studies [173]. 

The microscopic complexity of the contact angle is illustrated in Fig. X-14, which shows the edge of a solidified drop of glass—note the foot that spreads out from the drop. Ruckenstein [176] discusses some aspects of this, and de Gennes [87] has explained the independence of the spreading rate on the nature of the substrate as due to a precursor film present also surrounding a nonspread-... 

Bikerman [182] criticized the derivation of Eq. X-18 out of concern for die ignored vertical component of On soft surfaces a circular ridge is raised at the periphery of a drop (see Ref. 67) on harder solids there is no visible effect, but the stress is there. It has been suggested that the contact angle is determined by the balance of surface stresses rather than one of surface free energies, the two not necessarily being the same for a... 

There is no reason why the distortion parameter should not contain an entropy as well as an energy component, and one may therefore write 0 = 0q-sT. The entropy of adsorption, relative to bulk liquid, becomes A5fi = sexp(-ca). A critical temperature is now implied, Tc = 0o/s, at which the contact angle goes to zero [151]. For example, Tc was calculated to be 174°C by fitting adsorption and contact angle data for the -octane-PTFE system. 

Why do you think the Cassie equation Eq. X-27 might work better than Eq. X-28 for predicting the contact angle as a function of surface polarity ... 

Using the data of Table X-2, estimate the contact angle for benzene on aluminum oxide and the corresponding adhesion tension. 

Fowkes and Harkins reported that the contact angle of water on paraffin is 111° at 25°C. For a O.lAf solution of butylamine of surface tension 56.3 mJ/m, the contact angle was 92°. Calculate the film pressure of the butylamine absorbed at the paraffin-water interface. State any assumptions that are made. 

Discuss the paradox in the wettability of a fractal surface (Eq. X-33). A true fractal surface is infinite in extent and a liquid of a finite contact angle will trap air at some length scale. How will this influence the contact angle measured for a fractal surface ... 

Derive from Eq. XU-24 an expression for the maximum work of adhesion involving only and 7c. Calculate this maximum work for 7c = 22 dyn/cm and 0 = 0.030, as well as 7/. for this case, and the contact angle. 

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