U ( x ) profiles inferred from measured g ( r ) profiles for particles with positively charged (pink curve) and negatively charged coatings (blue and green curves) with layer numbers in parentheses (Supplementary Fig. 23 and Supplementary Tables 9 11 ). At long range ( x5^{-1} ), negatively charged particles (charge density < 0) display an attraction which implies that the interparticle force is attractive, that is, -dU(x)/dx < 0 , whereas the converse is true for positively charged particles ( > 0) where we infer -dU(x)/dx > 0 for all separations, which agrees qualitatively with DLVO theory. All experiments were performed at c 0 = 15 M. Error bars denote estimated uncertainties of 100 nm in particle diameter and 1.5 k B T in w (Supplementary Fig. 23 ).
