Three-phase line/perimeter, contact

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. 

Upon the formation of two-dimensional phases a new interfacial quantity, the line tension enters the analysis. Phenomenologically the line tension is the onedimensional analogue of the Interfacial tension. It has the dimensions of a force and acts in the perimeter of three phase contacts. When it is positive it tends to... 

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. 

Any mismatch resulting from overlap between these three profiles in the three-phase contact zone leads to a positive or negative excess in 2° per unit perimeter length and hence to a (positive or negative) contribution to the line tension. 

For small holes (R < Aq), viscous dissipation within the film due to radial and ortho-radial deformations dominate. This leads to an exponential growth of the hole, a consequence of a continuously increasing length of the three-phase contact line, i.e., the perimeter of the hole. During this regime, no rim is formed. For R > Aq interfacial friction dominates. At this later stage, the relative increase in hole perimeter is small and so the increase in driving force is smaller than the increase in frictional force. Thus, a rim of width Aq starts to buUd up. 

It is also worth emphasizing that the interatomic bonds are not fully compensated at the three-phase contact line. This results in a free energy excess and a linear tension, ae, acting along the perimeter of a three-phase boundary. This linear tension can be either positive or negative and does not exceed 10 dyn/cm. While the linear tension can in most cases be neglected, it plays an essential role in the case of very small droplets, particularly in nucleation. 

Linder eqnihbrium conditions, the excess free energy should reach its minimum value. The mathematical expressions for this requirement are the following conditions (1) the first variation of the free energy, 8 , should be zero, (2) the second variation, 8 , should be positive, and (3) the transversahty condition at the drop perimeter at the three-phase contact line — that is, at r = P — should be satisfied. In Section 2.2, these conditions are discussed in more detail, and it... 

The presence of the transition zone between a drop or a bubble and thin liquid interlayers can be described in terms of line tension, x, a concept first introduced by Gibbs (see for example [22]). In the case of surface tension, the transition zone between the liquid and vapor is replaced by a plane of tension with excess surface energy, y. By analogy, the transition zone between a drop or a bubble and the thin liquid interlayer may be replaced by a three-phase contact line with an excess linear energy, x. In contrast to surface tension defined always as positive, the value of the line tension may be positive and negative. When positive, it contracts the wetting perimeter, whereas the perimeter expands if the line tension is negative [33-36]. 

FIGURE 3.2 Spreading of a spherical droplet. At h > fr, the spherical droplet profile is not distorted by the hydrodynamic flow t < is the radins of action of the disjoining pressure r t) is the macroscopic wetting perimeter (the apparent three-phase contact line) R(t) is the tme microscopic wetting perimeter 0(f) is the dynamic contact angle and H(t) is the drop apex. 

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