Fig. 3: Calculations and simulations of capillary and magnetic forces.

a Capillary forces induced by single (\({\hat{F}}_{{capi}}\)) and merged (\({F}_{{capi}}\)) menisci at the three-phase-contacting lines dependent on the angles (\(\hat{\theta }\) and \(\theta\)) between the according menisci and the horizontal direction as the interactive distance (\({x}_{d}\)) changes; the resultant lateral force for assembly is \({F}_{{MSA}}\). The height functions of \(Z(x)\) to display the meniscus contours (\({Z}_{1}\): single meniscus; \({Z}_{{total}}\): merged meniscus) depending on the density (\(\rho\)), surface tension (\(\gamma\)) of the liquid, the original meniscus height (\({Z}_{0}\) = \(\hat{{Z}_{0}}=3\,{mm}\)) at \(x=0\); \(g\) is the gravitational acceleration. The calculated/simulated contours are shown in the meniscus photos (i, ii). Scale bars: 5 mm. Correlations between the capillary/magnetic forces dependent on the interactive distances between b A-B and c A-A, which exhibit additive and competitive effects, respectively. d The calculated free energy changes (\(\triangle U\)) with the interactive distance (\({x}_{d}\)) by integration of \({F}_{{MSA}}\). The insets show the simulated energy maps during the stepwise approaching between A-B as the interactive distance was varied (\({x}_{d}={{{{\mathrm{15,10,5,0}}}}}\,{mm}\)). Simulated results of the distribution of local magnetic field between e A-B and f A-A. Source data are provided as a Source Data file.