# Self-Assembly of Gradient Concentric Rings via Solvent Evaporation from a Capillary Bridge

Jun Xu, Jianfeng Xia, Suck Won Hong, Zhiqun Lin, Feng Qiu and Yuliang Yang, Self-Assembly of Gradient Concentric Rings via Solvent Evaporation from a Capillary Bridge, PRL 96 066104 (2006) [1]

### Brief Summary

The dewetting of a liquid forming a capillary bridge between a plane and a sphere is studied. The evaporating liquid carries MEH-PVV (a non-volatile polymer with optical applications) which in a sense acts to record the drying dynamics of the carrier liquid (toluene). In fact, the polymers also perturb the dewetting process via a ``slip-stick" mechanism similar to the effect observed when a drop dries off of an unclean surface. The evaporation process produces a pattern concentric rings of polymer on the Si substrate, with a non-trivial radial scaling of ring spacing: an artifact of the drying geometry.

### Capillarity Phenomena

When the toluene (containing MEH-PVV) evaporates from the capillary bridge (see figure 1), the footprint of the liquid on the Si wafer is determined by a competition between pinning and capillary forces.

An agglomeration of polymer just inside of the triple line results in a low-energy interface between a ring of polymer and solvent. This produces a pinning forces which prevents the contact line from receding when the volume of the capillary bridge is reduced by evaporation. This pinning persists until the initial contact angle of the bridge <math>\theta_i</math> reaches a critical value <math>\theta_c</math> at which point the capillary force at the triple line overcomes the pinning force. As a result, the liquid recedes to a new radius where the pinning force is high enough to halt the decrease in triple line radius. This leaves behind a ring of polymer which was previously pinning the toluene to the Si substrate.The authors develop a rather involved mathematical description of this effect by considering the change in contact angle after the evaporation of a volume <math>\delta V</math> of liquid. The non-trivial geometry of the capillary bridge renders this somewhat complex to attack specifically, however the salient features can be explained rather simply. Firstly, the pinning force must clearly be proportional to the circumference of the contact circle <math>2\pi X</math>. However, the radial component of the capillary force <math>F_{cap}</math> does not simply scale with the circumference of the footprint. <math>F_{cap}</math> consists of a force per unit length directed towards the centre of the sphere, which itself if radius dependent because <math>d\theta/dX \neq 0</math>. The author's state that <math>F_{cap} \propto X \atan{4aR/X^2} </math>