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Self-assembly is organization of interacting building blocks into a larger structure without individually placing every component. Static self-assembly approaches an equilibrium free-energy minimum under fixed conditions; a resulting structure may subsequently become kinetically trapped. Dynamic self-assembly maintains a nonequilibrium organized state through continuing energy or matter throughput and dissipation, and can change or disappear when that drive is removed.

For nanofabrication, parallel assembly of many small components can generate fine periodicity, hierarchical structures and adaptive or self-repairing arrangements with limited direct patterning. Building-block chemistry encodes interactions. However, access to a desired morphology depends on kinetics and competing states; assembly does not automatically eliminate defects or provide arbitrary device geometry.

Brownian motion is the fluctuating motion of a suspended particle produced by thermal molecular collisions. Using the standard Brownian displacement conventions, in a uniform fluid without external drift, mean displacement is zero but mean-square displacement grows with time. For overdamped diffusion, $⟨\Delta x^2⟩=2Dt$ in one coordinate and $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$ in three dimensions, with $D=k_{B}T/\zeta$ and Stokes friction $\zeta =6\pi \eta a$ for a no-slip sphere.

Brownian transport brings nanoparticles into encounters and allows them to explore configurations during self-assembly. Reversible interactions permit rearrangement toward low-free-energy structures; interactions that are too strong can instead trap disordered aggregates. Thermal fluctuations also oppose localization in shallow traps. The symbol $\zeta$ avoids confusing friction with an unrelated chemical potential.

For this self-assembly into a colloidal crystal, four useful elements are building blocks, interactions, environment and reversible dynamics. The building blocks here are nearly monodisperse polymer spheres; their shape and size distribution constrain the accessible packing. Their interactions include excluded volume, Van der Waals forces and electrostatic or steric stabilization, which set the free energy landscape. The aqueous environment supplies temperature, viscosity, ionic strength, confinement and concentration controls. Finally, Brownian motion and rotation must let particles encounter one another and rearrange or detach from incorrect contacts. Attractions that bind every collision irreversibly favour an amorphous aggregate rather than an annealed colloidal crystal. These four categories include both thermodynamic selection and the transport/kinetic access needed to realize it.

Brownian motion is the irregular thermal displacement produced by fluctuating molecular forces from the surrounding liquid. In the overdamped regime its long-time displacement is diffusive, with mean squared displacement proportional to time. It brings nanoscale building blocks into contact and lets them sample orientations and escape weak incorrect contacts. Interparticle forces bias this exploration toward lower-free-energy structures; thermal motion alone does not select a particular crystal. Strong irreversible attraction, high viscosity or glassy arrest can stop the defect correction needed for successful self-assembly.

Full title: Theory of self-assembly of hydrocarbon amphiphiles into micelles and bilayers. J. Chem. Soc., Faraday Trans. 2, 72, 1525–1568, DOI 10.1039/F29767201525. Used only for the general physical surfactant-packing description; no biological membrane mechanics researched.