Line tension, energy

R. Tadmor, Line energy, line tension and drop size. Surface Sci., 602, L108-Llll (2008). 

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... 

We now regard the step as a flexible linear object with a line tension 7. Its deformation h x) at a position. x costs an energy... 

The line can only behave in this way by being flexible. It has an energy per unit length (line tension). This energy per unit length has two parts ... 

Figure 4 The free energy of pore formation in a DPPC bilayer. The dashed line is a quadratic function, while the dotted line is a fit to a model of pore expansion with a line tension of 40 pN, and is close to linear (Adapted from ref. 78 courtesy of O. Edholm).

Fluctuations of an isolated step are also suppressed by the microscopic energy cost to form kinks. On coarse-graining, this translates into an effective stiffness or line tension that tends to keep the step straight. Standard microscopic 2D models of step arrays incorporating both of these physical effects include the free-fermion model and the Terrace-Step-Kink (TSK) model. Both models have proved very useful, though their microscopic nature makes detailed calculations difficult. 

The specific free surface energy of a material is the excess energy per unit area due to the existence of the free surface it is also the thermodynamic work to be done per unit area of surface extension. In liquids the specific free surface energy is also called surface tension, since it is equivalent to a line tension acting in all directions parallel to the surface. 

In Eq. (1) R is the radius of the homogeneously formed sphere, 6 is the wetting angle, yi2, y 13, and y23 are the specific free surface energies at the solution substrate, solution droplet and substrate]droplet interface boundaries, respectively, k is the specific free line energy (or line tension) at the droplet periphery and could be either zero or a positive, or a negative quantity [v, vii-x]. 

The phenomenon of formation of a new NBF when a very small bubble is pressed into the solution/gas surface by buoyancy force, can be used for determining of positive line tension values only. The nascency and expansion of the new contact (NBF) between the bubble and the bulk gas phase are hindered by a force barrier due to the positive linear energy. The buoyancy force (necessary to overcome this force barrier) must be larger than a critical value which depends on the value of k. This principle has been realised in the critical bubble method developed by Platikanov et. al. [474]. The results obtained by this method for 0.05% aqueous solutions of NaDoS are presented in Fig. 3.100 [475]. 

Another theory of the linear energy of the contact line wetting film/bulk liquid drop on a solid surface has been developed by Churaev at al. [478]. These authors also considered both cases of negative and positive line tension. In their interpretation the transition region film/bulk can be presented [478] schematically as shown in Fig. 3.103. The dashed line 1 represents the idealised surface. The real surface is shown for two different cases in case 2 the... 

In systems with phase coexistence, the free energy of the system is discontinuous at the interface between the phases. This discontinuity results in an interfacial tension that drives the system toward a minimization of the interface. In three-dimensional systems, the interfacial tension acts on the area (surface tension) in two-dimensional systems, it acts along the line (line tension) that separates the phases. In both cases, if the interfacial tension is large enough, then it may be expected that... 

Similarly, within the line tension approximation, the energy of the state in which a junction of length Ij has formed is given by... 

Energy of Curved Dislocation in Line Tension Approximation We wrote down the energy of a curved dislocation as... 

In considering the energetics of extended defects, we have repeatedly resorted to locality assumptions as well. In particular, in the context of dislocations we have invoked the line tension approximation to assign an energy of configuration to a dislocation of the form... 

Whereas the line tension was invoked as a way to capture the self-energy of dislocations from an elastic perspective, there are also ways of capturing core effects on the basis of locality assumptions. Recall that in our treatment of dislocation cores we introduced the Peierls-Nabarro model (see section 8.6.2) in which the misfit energy associated with slip displacements across the slip plane is associated with an energy penalty of the form... 

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