Size dependent thermodynamics and kinetics in electric field mediated colloidal crystal assembly

Abstract

We compare measurements and models of the system size dependent assembly of quasi-two dimensional (2D) colloidal crystals within interfacial quadrupole electrodes. Perturbation theory (PT) and Monte Carlo (MC) simulations are used to make thermodynamic predictions of the voltages required to obtain all particles within a crystalline phase for different system sizes. These results show good agreement with video microscopy (VM) measurements of colloidal crystals containing 75-300 particles. Colloidal assembly kinetic trajectories for the same system sizes are quantified from VM experiments using order parameters (reaction coordinates) to capture real-space local and global hexagonal ordering. These trajectories are fit with a Smoluchowski model to extract configuration dependent free energy and diffusivity landscapes (FEL, DL) that capture the microscopic details of assembly dynamics. Landscape features, such as the number and heights of barriers and local gradients, capture the formation, diffusion, and migration of grain boundaries in polycrystalline states, which are increasingly prevalent for n > 200 but vanish for n < 150. These results demonstrate the importance of system size effects in the formation and annealing of polycrystalline states in colloidal crystallization processes.

Colloidal crystal grain boundary formation and motion

“…, applied voltage) is effectively a surrogate for pressure and hence acts as a global thermodynamic variable that determines the relative free energy of all particle configurations for fixed voltage, number, and temperature. In the following, we report non-dimensional voltages, where V * = 1 is the voltage required for N particles to produce a hexagonal close packed crystal with hexagonal morphology (as demonstrated in previous work with agreement between microscopy experiments, MC simulations, and perturbation theory 19 , see Supplemental Information for additional details).…”

“…Our previous characterization of electric field mediated colloidal interactions and assembly has yielded kT -scale potentials 20 21 , feedback control over system size 18 , and conditions to crystallize N particles 19 . Based on these findings, we set N = 210 in Fig.…”

“…1b–f ) including: the radius of gyration, R g 18 , to capture condensation from fluid to crystal states, average local hexagonal order, 〈 C 6 〉 25 , to capture the onset of crystallization, and global hexagonal order, ψ 6 26 , to capture the degree of polycrystallinity. R g and 〈 C 6 〉 are normalized by their N particle single crystal values (see Supplementary Methods ), so that 〈 C 6 〉 goes from 0–1 for fluids to complete locally ordered states, and R g decreases from arbitrarily high numbers to 1 for complete condensation 19 . ψ 6 is 0 for fluids and 1 for single-domain crystals like 〈 C 6 〉, but in contrast, depends strongly on relative domain size and misorientation ( e.g.…”

“…(1) from trajectories like those in Figs. 1 and 2 , we analyzed Brownian Dynamic (BD) simulations that were matched to experiments (by capturing all equilibrium and dynamic properties of the quadrupole experiment on the particle scale 19 20 21 , see Supplementary Methods ). This approach was used because statistics on the particle scale are easily obtained to match experiments and simulations, but BD simulations are better suited to generating large numbers of grain boundary trajectories ( e.g.…”

Controlling colloidal crystals via morphing energy landscapes and reinforcement learning

“…In general, interactions and transport of colloids in electric fields at different frequencies ( 23 ) and in media with varying properties (e.g., polar, nonpolar, and salt) ( 24 , 25 ) are sufficiently well understood to manipulate colloidal assembly for diverse materials. More recently, direct measurements have quantitatively connected dipole-field and dipole-dipole interaction potentials to local phase behavior ( 26 ) and morphology including different field shapes ( 27 ). The ability to tune electric field shape, amplitude, and frequency in space and time has a number of promising features for achieving high fidelity control over colloidal assembly processes.…”