Statistical mechanics of DNA-mediated colloidal aggregation.

We present a statistical mechanical model of aggregation in colloidal systems with DNA-mediated interactions. We obtain a general result for the two-particle binding energy in terms of the hybridization free energy DeltaG of DNA and two model-dependent properties: the average number of available DNA bridges <N> and the effective DNA concentration c(eff). We calculate these parameters for a particular DNA bridging scheme. The fraction of all the n-mers, including the infinite aggregate, are shown to be universal functions of a single parameter directly related to the two-particle binding energy. We explicitly take into account the partial ergodicity of the problem resulting from the slow DNA binding-unbinding dynamics, and introduce the concept of angular localization of DNA linkers. In this way, we obtain a direct link between DNA thermodynamics and the global aggregation and melting properties in DNA-colloidal systems. The results of the theory are shown to be in quantitative agreement with two recent experiments with particles of micron and nanometer size.