The line tension

a is the radius of curvature of the three-phase contact line. For a drop with circular contact area it is the contact radius. 

As in example 2.3, we picture the liquid as being composed of cubic molecules. The size of each cube was calculated from the density of cyclohexane to be a = 0.565 nm. In the bulk each molecule is supposed to directly interact with 6 neighbors. The energy per bond is thus AvapU/6Na- At the rim two bonds less can be formed and the energy loss per molecule is 2AvapU / Na- Thus the energy difference per unit length is 

The result leads to the right order of magnitude. For water, the same calculation results in an estimated line tension of 7.4 x 10-11 J/m. 

Typical line tensions are in the order of 10 1° N [226], In some cases significantly higher effective line tensions have been determined [225], Calculations of the line tension, which are based on the analysis of surface forces, are reviewed in Ref. [227], 

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Very small sessile drops have a shape that depends on the line tension along the circular contact line if large enough it induces a dewetting transition detaching the drop from the surface [84]. 

The free energy of a monolayer domain in the coexistence region of a phase transition can be described as a balance between the dipolar electrostatic energy and the line tension between the two phases. Following the development of McConnell [168], a monolayer having n circular noninteracting domains of radius R has a free energy... 

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... 

Suppose that the line tension for a given three-phase line is 1 x 10 dyn. Calculate ff for drops of radius 0.1, 0.01, and 0.001 cm if the value for a large drop is 56. Assume water at 20°C. 

When a dislocation segment of length L is pinned at the ends under the influence of an applied shear stress t, a balance between the line tension and the applied stress produces a radius of curvature R given by [37]... 

Each process parameter directly affects both the machinery dynamics and the vibration profiles. For example, the line tension, strip width, and hardness of the incoming strip radically affect the vibration profile generated by a continuous process line in a steel mill. With few exceptions, process variations such as these must be considered in the vibration analysis. 

Below roughening, the relaxation is driven by the lowering of the line tension of the curved steps. For evaporation kinetics, continuum theory and simulations show a shrinking of the bumps in the late stages of the decay. At small amplitudes, the radially symmetric profile scales with z r,t) Z(V ct + r )), where r is the distance from the center, and c is a constant. The continuum theory fails to describe the layerwise relaxation monitored in the simulations. ... 

Solution. The line-tension forces acting on a curved differential segment of dislocation having a radius of curvature R due to its line tension will be as shown in Fig. 11.14. The net force exerted on the segment toward the concave side is then... 

The work required to create a new three-phase contact line per unit length is called line tension. It is typically of the order of 0.1 nN. For tiny liquid drops the line tension can significantly influence the wetting behavior. 

According to Eq. (25), the equilibrium radius R of the island is the product of two factors one of them is the equilibrium radius in the absence of other islands (C = 0), and the other one increases exponentially with the area fraction of the islands. R is determined by the line tension (2) and the difference Ap between the two-dimensional dipole moment densities of the LC islands and the LE molecules. Both X and Ap depend on the molecular surface area difference (jAd — Ac) between the LC and LE phases. Since Ac is assumed to be constant and is considered to... 

Therefore, when Ay < 0 is fulfilled everywhere in the transition region between the film and the meniscus, then, in accordance with Eq. (3.55), K< 0. However, if in the transition zone there exist thicknesses with Ay> 0, the positive values of the line tension are not excluded (for more details on the line tension and its sign, see [26,27]). 

The introduction of contact angles gives rise to another interfacial characteristic, the line tension. This is the contractile tension acting in the three-phase contact perimeter around drops as in fig. 1.1, and may phenomenologically be considered the one-dimensioncd analogue of the interfacial tension with SI units of N or J m. As it is a typiccd three-phase characteristic, we shall not treat it here, but in sec. 5.6. 

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