text: With improvements in measuring techniques such as atomic force microscopy, confocal microscopy, and scanning electron microscope, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental observations of wetting. These observations confirmed that the modified Young's equation does not hold at the micro-nano scales. Jasper proposed that including a  term in the variation of the free energy may be the key to solving the contact angle problem at such small scales. Given that the variation in free energy is zero at equilibrium:
  The variation in the pressure at the free liquid-vapor boundary is due to Laplace pressure, which is proportional to the mean curvature. Solving the above equation for both convex and concave surfaces yields:
  where 
  This equation relates the contact angle, a geometric property of a sessile droplet to the bulk thermodynamics, the energy at the three phase contact boundary, and the mean curvature of the droplet.  For the special case of a sessile droplet on a flat surface ():
  In the above equation, the first two terms are the modified Young's equation, while the third term is due to the Laplace pressure. This nonlinear equation correctly predicts the sign and magnitude of , the flattening of the contact angle at very small scales, and contact angle hysteresis.