The goal of our research is to fabricate an autonomic material system that provides compartmentalization and multi-bilayer networks for enabling collective biomolecular functionality, as is found in living cells and tissues. The material system is based on biomolecular unit cells, which consist of synthetic lipid bilayers formed at the interfaces of lipid-coated aqueous droplets submerged in oil and contained in a solid material. This paper focuses on microfluidic encapsulation of unit cells within a solid material and tuning the amount of contact between droplets, two approaches aimed at increasing the functional density of the droplet-based material system. Hydrodynamic traps within microfluidic platforms have shown to be a promising method to capture single droplets within microfluidic devices. Herein, we develop a resistive flow model to design hydrodynamic traps for collecting pairs of droplets in a direct trapping mode to form unit cells. We also compare to the model the results of droplet trapping in a prototype microfluidic device fabricated prior to model development. In addition to flow techniques for assembling unit cells in solid materials, we examine the use of mineral oil as the hydrophobic oil phase that surrounds the droplets to increase the area of the lipid membrane formed between neighboring droplets. Compared to hexadecane, mineral oil produces larger contact areas between droplets and more-tightly packed multi-bilayer networks. The total free energies of formation for droplet arrays in mineral oil and hexadecane indicate that connected droplets in mineral oil exhibit a greater decrease in free energy upon formation (i.e. they exist at a lower energy state compared to those in hexadecane) and that hexagonal packing provides the maximum amount of decrease in free energy per droplet for droplets in large arrays. Electrical measurements of unit cells formed in mineral oil initially show gigaohm resistances typical of unit cells, however these unit cells exhibit increasing values of conductance as the bilayer areas grow.