Young equation validity

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]. 

Real solid surfaces never satisfy completely the conditions for the Young equation to be valid, namely chemical homogeneity and perfect smoothness. Several phenomena result from this deviation, most importantly ... 

From the standpoint of electrowetting contact angle saturation, the Lippmann-Young equation (Eq. (1) above) is valid up to V = Vsat at contact angle saturation. Assuming a composite insulator with a fluoropolymer layer over an underlying oxide layer, then at contact angle saturation Eq. (1) can be rewritten as... 

As stated, the Young equation is only valid for solid, ideal, and smooth surfaces thus the wetting behavior of a rough surface is described by the Wenzel equation (Equation 10.18) ... 

The Young equation cannot be used directly to explain the effect of surface roughness on the wettability of a material because it is valid only for ideal smooth solid surfaces. There are two wetting models that are proposed when a water droplet sits on rough surfaces, these are the Wenzel model and the Cassie-Baxter model. 

The discussion of the van Oss-Good and Neumann theories is compheated by many factors, e.g. the validity range of the Young equation, the role of spreading pressure, the accuracy of experimental contact angle data and others. 

The Young equation is a well-known special case of Eq. (5.75) valid for the contact of an amorphous, deformable phase (1) with a rigid phase (s) and a gas phase (g)... 

Since —1 < cos < 1, Young s equation is valid only when the right hand side of Eo 3 or Eq. 3a lies between these limits, i.e. the observed contact angle is finite. In le event that the measured contact angle is 0°, i.e. full spreading occurs, one may conclude only that... 

A requirement underlying the validity of Zisman plots is that there be no specific interactions, such as acid-base interactions, between the solid surface and the probe liquids. Such interactions, however, can, in principle, be taken into account by Young s equation, provided the contact angle remains finite. Their... 

Good-GLrifalco-Fowkes (GGF) equation Using ysi = ysv + yiv - 20(ysvyiv) in Young s equation leads to 1+COS0 2 Uvj Yggf obtained from a plot COS0 versus 4> is solid-liquid interaction parameter 0 = 1 if the interactions are purely dispersive. Based on Berthelot relation for attractive constants valid only when the solid-liquid interactions are dominantly dispersive. [77-82]... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

This gives the value of 0 = 50° when using the measured values of ys.octane - Ywater-sV Yoctane-water The experimental 0 is the same. This analysis showed that the assumptions made in derivation of Young s equation are valid. 

It must be concluded, therefore, that the kinetic scheme proposed in Equations (2) and (3) cannot be valid for these systems. Although the seeming correspondence between the half-order kinetics and the dimeric association in the case of styrene might validate the above kinetic scheme, even this hypothesis has been recently contradicted (18). Thus Fetters and Young... 

Since the polymer-filler interaction has direct consequence on the modulus, the derived function is subjected to validation by introducing the function in established models for determination of composite modulus. The IAF is introduced in the Guth-Gold, modified Guth-Gold, Halpin-Tsai and some variants of modified Halpin-Tsai equations to account for the contribution of the platelet-like filler to Young s modulus in PNCs. These equations have been plotted after the introduction of IAF into them. 

Both roughness and heterogeneity may be present in real surfaces. In such a case, the correction factors defined by Equations (45) and (46) are both present. Although such modifications adapt Young s equation to nonideal surfaces, they introduce additional terms that are difficult to evaluate independently. Therefore the validity of Equation (44) continues to be questioned. 

These considerations are valid for any small part of the liquid surface. Since the part is arbitrary the Young-Laplace equation must be valid everywhere. 

One should keep in mind that Young s equation is only valid in thermodynamic equilibrium, hence in the presence of a saturated vapor of the liquid. In most practical applications this is not the case. 

Young s equation is also valid if we replace the gas by a second, immiscible liquid. The derivation would be the same, we only have to replace 7l and 7sl by the appropriate interfacial tensions. For example, we could determine the contact angle of a water drop on a solid surface under oil. Instead of having a gas saturated with the vapor, we require to have a second liquid saturated with dissolved molecules of the first liquid. 

In the classical treatment of surface tensions, it is intuitively assumed that the surface tension of a solid, 7s, can be assigned as if it is a material constant. In a practical sense, Eq. (25.3) is valid if the surface tension of the solid does not change after the contact with the liquid (sessile droplet) is made. While Young s equation describes the force balance at the three-phase line, it does not give information relevant to the true interfacial tension at the interface that is beneath the droplet, which is the major concern of surface dynamics. In general cases, 7s and 7sl are... 

There has been no shortage of attempts to estimate surface (Helmholtz-) energies from contact angles, by invoking some model. A controversial issue is Neumann s equation of state method ) which is based on the assumed validity of a second relationship between interfacial tensions, so that and y can be individually estimated. Another route starts by assuming [5.7.5] to be valid. For an apolar liquid on a solid S (y = y ), combination with Young s law gives... 

Problem 7-14. The Young-Goldstein-Block Problem Revisited. Let us reconsider the Young-Goldstein Block problem, but in this case we directly seek the solution for the case in which the temperature gradient has the value that causes the velocity of the bubble, U, to be exactly equal to zero. Starting with the governing Stokes equations and boundary conditions, nondimensionalize and re-solve for this particular case. It may be useful to remember that this solution is valid when the hydrodynamic force on the bubble is exactly equal to its buoyant force. 

We next provide an approximate method to estimate the effects of particle flexibility (the fact that the elastic moduli of the dispersed particles are not infinite). Assuming that the effects of flexibility can be described completely by making 4>m0 a function of flexibility, calculated so that as y—the flexibility effects vanish in the same manner as < > —> d>ocp, we use Equation 13.43 where P is a flexibility parameter ranging from 0 in the limit of infinite rigidity to 1 in the limit of complete flexibility, with Equation 13.44 where E is Young s modulus (in MPa, for consistency with how the equation was calibrated to obtain the factor of 22.5) which is valid for both cylindrical fibers of aspect ratio Af>l and cylindrical platelets of aspect ratio (1/Af)[Pg.574]

When the radius of the capillary tube is appreciable, the meniscus is no longer spherical and also 9> 0°. Then, Equation (329) requires correction in terms of curvatures and it should give better results than those from the rough corrections given in Equations (330)-(332) for almost spherical menisci. Exact treatment of the capillary rise due to the curved meniscus is possible if we can formulate the deviation of the meniscus from the spherical cap. For this purpose, the hydrostatic pressure equation, AP = Apgz (Equation (328)), must be valid at each point on the meniscus, where z is the elevation of that point above the flat liquid surface (see Figure 6.1 in Chapter 6). Now, if we combine the Young-Laplace equation (Equation (325)) with Equation (328), we have... 

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