Design principles for self assembly with short ranged interactions
directed self-assembly, functionalised particles, colloidal clusters, random energy model
Self-assembly is an attractive method for fabricating micro-structures. It is free of the complexity and cost of traditional top-down approaches such as microfabrication; however, at its current stage of development, it suffers from low yield. In order to direct the formation of pre-designed structures with self-assembly, the constituent particles are frequently functionalised (by means of DNA coating, for example), so that their interactions can be controlled. Previous work on the design criteria for these interactions was based on local particle properties, such as the short-range interaction strength. While these criteria are important, the authors claim that global thermodynamic quantities need also be considered in order to design systems with high yield.
The authors describe their model in terms of a system with constituent particles labeled with two types of labels: geometric labels, which denote the resulting pattern, and energetic labels, which denote the interaction energy between a particle and its neighbors. These concepts are illustrated schematically in figure 1, where the numbers denote the geometrical label and the colors denote the energetic label. Note that in both cases a checkerboard pattern is created, however the energy landscape across the two patterns is different.
As an example of a desired structure, the authors have chosen the cluster of eight colloidal spheres shown below. If all the spheres are identical, the yield of each cluster formation is prescribed by entropy. However, if different types of spheres are used, the fractional yield of each eight-sphere cluster can be altered; the question the authors aim to answer is how many types of spheres (i.e. geometrical labels) and with what interaction strengths (i.e. energetic labels) are necessary in order to enhance the yield of the highlighted cluster above all others.
In more strictly mathematical terms, this problem amounts to finding the optimal interaction matrix. According to the authors, three parameters need be specified for this problem to be solved: the number of energetic labels, A (which specifies the matrix dimensions), the number of desired interactions, nf (i.e. how many pairs of sphere types should be encouraged to connect), and the standard deviation of the interaction energies between all pairs of sphere types, σ (i.e. the width of the spectrum of all available interaction energies). This problem can be simplified by two convenient facts: a. it is always possible to choose A and nf that are large enough so that there are no unfavorable interactions in the target configuration (which, in this optimization problem, is the ground state of the system). This follows from the Graph Coloring Theorem. b. It suffices to consider only two interactions strengths, one for all favorable interactions and one for all unfavorable ones. This follows from the fact that the more the interaction energies, the wider their standard deviation, and the more the system will behave like a randomly mixed system; hence it is preferable to equate all favorable interactions and all unfavorable ones.
These considerations allow the determination of the values of the interaction matrix and its form in terms of the three variables listed earlier; the matrix dimension A, the standard deviation of the interaction energies σ, and the number of favorable interactions nf. nf can be determined by the desired configuration (since it has been argued that all interactions in the ground state should be favorable). Given nf, a lower bound on A can be imposed by demanding that the free energy of the ground state be less than the free energy of the same configuration in a model with random energy assignments. Upper and lower bounds on σ can be respectively imposed by demanding that all interaction energies are below the bath temperature T and that the free energy of the target configuration is low enough to surpass the entropic cost of its preferential assembly. Note that all these limits depend directly on the desired structure and can be appropriately taylored depending on the target of self-assembly.
The design criteria described are applied to the physical example of the eight-sphere cluster shown above in Figure 1. The yields from two systems with alphabet sizes A=2 and A=8 are compared, and values for the favorable and unfavorable interaction energies for the better alphabet size A=8 are suggested. The assumption on the equality of all favorable and unfavorable interaction strengths is validated by a perturbation calculation of the yield in a system with randomly assigned variations in the values of same-type interactions, which shows that the yield is maximal when the variations go to zero. The suggested values for the interaction energies are experimentally achievable with temperature-tuning of DNA interactions.