Hysteresis Due to Strong, Sparse Defects

Under these conditions, it is straightforward to evaluate the threshold force per unit length required for the line tO advance, which is given (at the macroscopic level) by the expression 

Assume now that we displace the line by a distance Ax. If n is the number of defects per unit area, the total number of defects that the line snaps by during its sweep is nAx (per unit length of the line). Each of these defects dissipates a snapping energy W, which is lost as viscous dissipation in the fluid. It follows that 

FIGURE 3.6. A vertical interface yz plane) is deformed by pulling at the level of point P over a width 2ro fixed by the size of the defect. 

The minimum size tq will be comparable to the size of the defect. The waist radius tq can be related to the total force / exerted by the defect. At the waist radius, the surface tension 7 pulls on a circle of perimeter 27rro. It follows that the total force is 27T ro. Note, however, that we use only half the surface described by equation (3.18), namely, the half corresponding to z 0. As a result, the force of the defect is only half that derived above, or 

The equation of the triple line of contact u y) is obtained by means of equation (3.18) after replacing xby u and r by y (since we then have z = 0)  

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