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I was reading about Solid-Liquid-Air Interfaces pg 484 in Competitive Physics Volume 1 by Wang and Ricardo and we were proving Young's equation regarding the contact angle of the water droplet on this solid surface.

We consider our system as the liquid molecules near the contact point (which is shown in the diagram below as a wedge with infinitesimal width of $d\vec{l}$ pointing into the page).

  • $\gamma_{lv}$ is the cohesive surface tension from the other liquid molecules near our system of molecules
  • $A_{sl}$ is the work per unit area due to the solid liquid adhesion

Force Analysis of Water Droplet near contact angle

I have some confusion on how the direction of surface tension $\gamma_{lv}$ matches up with the following definition of surface tension: enter image description here

As you can see from the 1st diagram, the $\hat{n}$ on the hypotenuse points into the water droplet while $\hat{n}$ on the bottom of the triangle points out of the water droplet, which is not a consistent definition of the direction of our $\hat{n}$ throughout the system.

Is there something I am missing or not understanding correctly?

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    $\begingroup$ Hi Michael. I discussed the details of the Young equation in my answer to Young's equation. Does that help? $\endgroup$
    – John Rennie
    Commented Jun 6, 2023 at 6:15
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    $\begingroup$ Thank you, but I am still wondering why the solid-vapor surface tension tends to pull the droplet outwards while the solid-liquid surface tension tends to pull the droplet inwards? In other words, how did we decipher their directions other than mere intuition? Thanks! $\endgroup$
    – Michael Li
    Commented Jun 6, 2023 at 6:29

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