Neumann triangle

Neumann triangle. Graphical representation of equilibrium of three surface tensions at point of contact of two immiscible liquids with air. 

The balance of forces between surface tensions at the contact line results either in the Neumann triangle for a liquid/liquid/liquid or liquid/liquid/gas system or in the Young-Dupre equation on a liquid/liquid/solid or a liquid/gas/solid system (Fig. 1). While the Neumann triangle represents a true balance of forces, the Young-Dupre equation is little more than a definition of the (o As ctbs) term, a difference between the respective solid/fluid surface free energies and not truly solid/fluid interfacial tensions. 

Fig. 1 Balance of forces between surface tensions at the contact line Neumann triangle and Young-Dupre equation. (View this art in color at www.dekker.com.)...

This equation is equivalent to the static force balance where each of the y terms is considered as the force applied to the vertex. This condition of equilibrium is often called a Neumann triangle. Equation (233) is the basis of Young s equation, which is used in the contact angle determination of liquid drops on flat substrates (see Sections 5.5 and 9.1). 

Surface tension. Assuming that a membrane stretches over each interface, the magnitudes of the interfacial tension between each pair of phases are the fluid-fluid interfacial tension Ogt, the wetting fluid-solid interfacial tension oft, and the nonwetting fluid-solid interfacial tension ogs. When in static equilibrium, the vectorial force balance at the line of contact (the law of Neumann triangle, Ref. 87) gives... 

Fig. 10.3 Surface tension in a three-phase system. Illustration of the Neumann triangle (a) and Young law (b) and the three cases of wetting phenomena non-wetting (c), partial wetting (d) and complete wetting (e)...

This is a so-called Neumann triangle valid for any three phases. 

PA-6/PP-g-MA Melt mixing/SEM/TEMfinterfacial tension studied using Neumann Triangle method Zhaohui et al. 2001... 

In the next section we deduce the macroscopic condition for the existence of such a line of three-phase contact. The condition is that each of the three interfadal tensions be less than the sum of the other two that is, that the three tensions be related to each other as are the sides of a triangle (the Neumann triangle). We shall see how the interfacial tensions are related to the contact angles, which are the dihedral angles occupied by the three phases at their line of mutual contact. The discussion in 8.2 is wholly macroscopic and therefore not subject to the uncertainties that arose in 2.5 and 4.8 when mechanical arguments were used at the microscopic level. 

When the Neumann triangle collapses to a line, that is, when the largest of the three two-phase tensions is equal to the sum of the two smaller, there is no longer a line of three-phase contact. Instead, the equilibrium configuration of the three phases is that in which one of them— the one whose interfaces with the other two are those of lowest... 

When one of the inequalities (8.7) becomes an equality, that is, when the largest of the three tensions equals the sum of the two smaller, the Neumann triangle degenerates to a line the vertex that was previously opposite the longest side comes to lie on that side, as the altitude of the triangle, measured from that vertex to that side, vamshes. Suppose is the largest tension. Then in the limit we are now contemplating... 

We turn now to a consideration of the tension and microscopic structure of the line in which three phases meet— when there is such a line that is, when the three interfacial tensions satisfy the Neumann-triangle conditions as expressed by the inequalities (8.7) or by Fig. 8.2. 

If we assume a spherical cap geometry for the heavy DCM drop, the forces acting on the triple contact line can be represented by the Neumann triangle as shown in Fig. 5. The corresponding young equation reads 7w/a = 7o/a+7w/o cos 0. 

Chen P, Gaydos J, Neumann AW. The contaet line quadrilateral relation a generalization of the Neumann triangle relation to inelude line tension. Langmuir 19% 12 5956-5962. 

---