Neumann s triangle

Coghill and Anderson found that the angles, and probably the surface tensions, of various liquids resting on water changed with time, so that the experimental verification of Neumann s triangle was not very accurate. 

When a lens of one liquid floats on the surface of another (Fig. 9.VIII G) there are three contact surfaces, gas /liq. 1, gas /liq. 2, and liq, 1 /liq. 2, The three interfacial tensions, Oi, 02, and oi2 will be in eqiMbrium if lines representing them in magnitude and direction form a closed triangle, usually called Neumann s triangle. The relation ... 

Neumann s triangle, 164 normal liquids, 159 to a plane curve, 425 to a surface, 433... 

Figure 5.6. Floating drop (a) with Identification of the three angles and interfacial tensions (b). Construction (c) is known as Neumann s triangle in the stricter sense.

For basic studies, model experiments have to be carried out and analyzed. We shall mostly consider liquid (L) drops on horizontal solids (S) in a vapour phase (G), because in such a system hysteresis is most manifest. A simple set-up is sketched in fig. 5.29. The droplet on a tilting plate (fig. 5.32) is an alternative. In fluid-fluid systems (LjLjG), say an oil drop floating on water, or a water droplet on mercury, hysteresis may be absent, but then the interpretation is more complex because the vertical component of the surface tension ly sin a) has also to be considered because the liquid substrate cannot support a vertical stress. In other words, analysis requires Neumann s triangle (5.1.3] rather than Young s law [5.1.2]. 

Line tensions (r) are forces acting in the three-phase contact perimeter. Their SI-unit is N. When r > 0 the line tension tries to shorten the perimeter, for r < 0 the trend is the other way around. Line tensions are typiccdly excess qucuitities in that their action comes on top of that of the three interfacial tensions constituting the contact angle, and which are related through Young s law or Neumann s triangle. 

Eq 8.5 provides agood explanation for the morphology evolution in three-component systems. A more precise discussion can be based on the concept of Neumann s triangle, shown in Figure... 

By the Neumann s triangle arguments, not only the encapsulation, but also another morphology in three-component systems, that of stuck hybrid particles (Figure 8.26c), can be explained as well. 

Figure 8.26. Schematic illustration of the interfacial tensions balance at the interface between three liquids. The Neumann s triangle, and the morphologies to appear in the melt-blends of three polymers are also shown. Four morphologies are predicted for dispersions of liquids 1 and 2 in liquid 3 (a) encapsulated hybrid particles (1 in 2), (a ) encapsulated hybrid particles (2 in 1), (b) stuck hybrid particles, (c) isolated particles [Nakamura and Inoue, 1990].

The angles in the Neumann s triangle can be directly measured by a droplet-sandwich experiment. For this purpose, small pieces of C polymer are placed between two layers of polymers A and B, then annealed at a melt temperature. After quenching, the cross-section of the sandwich is observed under a microscope. From the value of cos , in Neumann s triangle, the results in Figure 8.25 can be justified. Also note that values of two Vy coefficients can be obtained, if one of them is known [Nakamura and Inoue, 1990]. 

PVDF poly(vinylidene fluoride) 0 angle in Neumann s triangle... 

Neumann s Triangle Nakamura and Inoue [8] applied the concept of Neumann s triangle to discuss the multi-phase blend morphology developed by meltprocessing of a ternary blend consisting of polymers (A, B and C) which are immiscible with each other in the forms of homopolymers. For a C polymer... 

Fig. 1.3. Equilibrium contact angles at a triple line at rest, (a) Solid substrate Young s construction. (b) Liquid substrate Neumann s triangle. If the line moves, the dynamical contact angle is 9d-If 6d < E, the line recedes in the B direction and if > fe, the line advances in the A direction.

If all three phases are fluid (La, Lb, and air), the condition for mechanical equilibrium, known as the Neumann s triangle construction, can be written in vector form... 

Neumann s Triangle. At the junction where three phases meet, three vectors representing the forces of interfacial tension among pairs of phases can be drawn. At equilibrium, the sum of these vectors of Neumann s triangle will equal zero. 

FIGURE 3.1 Contact angles in pseudoemnlsion film illustrating Neumann s triangle and definition of angle 0 in this context. 

Neumann s triangle and application of the cosine rule means that we can write [72], for example, that... 

FIGURE 4.13 Neumann s triangle and oil lens at air-water surface. Definition of symbols given in text. 

Existence of a line tension contributes an additional force term to Neumann s triangle of forces in the plane of the air-water surface. That term is the one-dimensional analogue of the Laplace equation, xIR, where R is the radius of the line. If then we resolve the horizontal forces in a lens as depicted in Figure 4.21, we can write... 

FIGURE 4.21 Relation of line tension, t, to Neumann s triangle in oil lens where R is radius of the line at three-phase contact line. 

Once the pseudoemulsion films break, then the possible configurations of the bridging drop in a Plateau border are determined by Neumann s triangle and three surface tensions—0 0, Oqw, and a w There has been no analysis of the stability of the resulting configurations. It is, however, possible to speculate. We show a... 

Angles determining shape of lens of liquid i at the surface of fluids j and k by the surfaces ij and ki according to Neumann s triangle 0 Angle formed at a lens of liquid i at the surfaces of fluids J and k by the interfaces... 

As a matter of historical interest. Buff and Saltsburg, in their paper on the theory of the three-phase contact line and Neumann s triangle (ref. 3), also reported, briefly and qualitatively, their observation that in the three-liquid system water-aniline-heptane, the middle, aniline-iich phase is transformed from a spread layer to a lens upon the addition of a detergent to the system. That was an early observation of a Cahn transition. 

Contact Angle, Dupre s Equation, and Neumann s Triangle... 

This triangular equilibrium condition for the three vectors is known as Neumann s triangle. If a lens of liquid is sandwiched between two different fluids, the equilibrium of the fluids will appear as shown in Fig. 1.22 and the... 

---