Fig. 2: Direct measurements of the D‒I interaction force.

a Determination of the equilibrium position of \({{{{{{\rm{P}}}}}}}_{2}\) in water. b Measurement of \({F}_{{{{{{\rm{D}}}}}}-{{{{{\rm{I}}}}}}}\). Red dots indicate the trap positions (not drawn to scale). c Schematic side view for the force measurement and the confirmation procedure of dimer formation (not drawn to scale). d Measured D‒I interaction force profile (\({F}_{{{{{{\rm{D}}}}}}-{{{{{\rm{I}}}}}}}\)) and comparison with the DLVO forces (\({F}_{{{{{{\rm{el}}}}}}}\) and \({F}_{{{{{{\rm{vdW}}}}}}}\)). The water phase contains 10 mM NaCl. The error bar in each force profile indicates thermal fluctuations of a trapped particle. The x axis of the graph is on a logarithmic scale. The red solid line represents a fitted curve that uses the mean values of the two fitting parameters, \(\left\langle {F}_{0}\right\rangle=-0.44\) and \(\left\langle b\right\rangle=7.18\). The black dotted lines indicate the guideline for \(F \sim {r}^{-7}\). The force profiles display before the paired particles come into contact. The left two inset plots display histograms of the values of the two fitting parameters for 72 pairs. The inset on the right represents a log–log plot of \(\left|F\right|\) versus \(r/d\) for short-range separation. e Magnified force profiles in the short-range separation, and the x axis is on a logarithmic scale.