Determining the peak interaction force between an oscillating nanoscale tip and a sample surface has been a fundamental yet elusive goal in amplitude-modulated atomic force microscopy. Closed form analytical expressions are derived using nonlinear asymptotic theory for the peak attractive and repulsive forces that approximate with a high degree of accuracy the numerically simulated peak forces under ambient or vacuum conditions. Scaling laws involving van der Waals, chemical forces, nanoscale elasticity, and oscillator parameters are identified to demonstrate approximate similitude for the peak interaction forces under practical operating conditions.

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