TIRM is a technique for monitoring the instantaneous separation distance *h* between a microscopic sphere and a flat plate (often a glass microscope slide). This page describes how we obtain the potential energy (PE) profile f(*h*) of interactions between the sphere and the plate.

After taking one measurement, we wait for the distance to change by Brownian motion before taking a second measurement. We then repeat this process a statistically large number of times: typically we take at least 50,000 measurements of the separation at 10 ms intervals. The figure below shows some typical raw data observed with a 10 **m**m PS latex sphere undergoing Brownian motion in 0.5 mM NaCl solution.

We then form a histogram of these 50,000 measurements.

If the particle has had time to sample all elevations a statistically large number of times, the shape of this histogram converges to the shape of the probability density function *p*(*h*) appearing in *Boltzmann’s equation*:

where *p*(*h*)*dh* is the probability of finding the sphere between *h* and *h*+*dh*, f(*h*) is the PE of the sphere at elevation *h*, *kT* is the thermal energy and *A* is a normalization constant whose value is chosen such that = 1.

In essence, TIRM is then capable of directly measuring this probability density function. Knowing *p*(*h*), we can turn Boltzmann’s equation "inside-out" to deduce the PE profile f(*h*): to eliminate *A*, we divide this equation by itself evaluated at some reference position denoted *hm* before solving for the PE. This leaves

Usually *hm* is chosen as the elevation corresponding to the minimum in f(*h*). Since Boltzmann’s equation defines the mean potential in statistical mechanics, we claim that TIRM directly measures the PE profile.