Programmed Crystal Assembly: Design & Kinetics

• Programmed Crystal Assembly (PCA) is a strategy that encodes target crystal structures through controllable variables such as interaction specificity, patch geometry, and kinetic pathways.
• It leverages design rules—including patchy interactions and SAT-based inverse design—to finely tune nucleation, growth, and defect suppression in diverse systems.
• Experimental implementations range from DNA-coated colloids and acoustic fields to wafer-scale stacking of 2D materials, showcasing PCA’s versatility in controlling free-energy landscapes.

Programmed Crystal Assembly (PCA) denotes, across several research literatures, the deliberate encoding of crystal structure and crystallization pathway into controllable microscopic variables such as interaction specificity, patch geometry, sequence complementarity, field landscapes, confinement, or layer-by-layer stacking order. In this usage, the target architecture is specified a priori, and nucleation, growth, annealing, and defect suppression are treated as design variables rather than as consequences of an unconstrained search. The term has been used for protein crystallization recast as patchy-colloid self-assembly, DNA-programmed colloidal crystals, topological-soliton lattices in chiral nematics, acoustically and electro-osmotically organized colloids, SAT-designed and torsionally specific patchy particles, TPMS-like assemblies from addressable triangular units, controlled salt spherulites, and wafer-scale van der Waals stacking of monolayer crystals (Fusco et al., 2015, Jacobs et al., 2024, Yang et al., 30 Jul 2025).

1. Scope and conceptual structure

Across these works, PCA has a common architecture: one first identifies the structural degrees of freedom that distinguish the desired crystal from competing assemblies, and then one couples those degrees of freedom to experimentally adjustable “knobs.” In protein and colloidal settings, those knobs are effective pair interactions, valence, angular specificity, range, stoichiometry, and supersaturation. In field-programmed systems, they are pressure-node landscapes, electro-osmotic flow fields, patterned anchoring, electric fields, or curvature-selected defect sites. In atomically thin materials, they are layer-resolved composition, interlayer sequence, and in-plane orientation or twist angle (Fusco et al., 2015, Jacobs et al., 2024, Duzgun et al., 2020, Yang et al., 30 Jul 2025).

This broad usage implies that PCA is not restricted to equilibrium structure selection. Several of the cited works define PCA explicitly through pathway control: suppressing excessive nucleation, maintaining reversible local rearrangements, separating seeding from growth, or exploiting two-step pathways only within a narrow operating window. A plausible implication is that PCA is best understood as a control problem over both free-energy landscapes and kinetic accessibility, rather than as a purely thermodynamic inverse-design exercise (Hensley et al., 2021, Klotsa et al., 2012).

The same conceptual template appears in systems that are physically very different. Proteins are modeled as patchy colloids with short-range anisotropic attractions; DNA-coated colloids use sticky-end hybridization and linker concentration to encode interactions; liquid-crystal skyrmions are positioned by attractive and repulsive walls in the director field; acoustic assembly uses standing-wave minima or maxima; electro-osmotic pumps use inward transport fields; and van der Waals assembly of graphene or monolayer hBN uses standardized monolayer building blocks whose stacking sequence and twist are set layer by layer (Fusco et al., 2015, Jacobs et al., 2024, Duzgun et al., 2020, Owens et al., 2015, Niu et al., 2016, Yang et al., 30 Jul 2025).

2. Encoding target structure in interactions, geometry, and symmetry

A large fraction of PCA theory is built on patchiness. In the protein-crystallization literature, a canonical coarse-grained description is the Kern–Frenkel-type pair potential

$$U(r,\Omega_i,\Omega_j)=U_{\text{sw}}(r)\,f(\Omega_i,\Omega_j,\hat r)+U_{\text{HS}}(r),$$

where a hard core is decorated by discrete attractive patches, and the selector $f$ enforces angular alignment and matched patch pairs. In that framework, patch number programs valence, patch geometry programs lattice directionality and space-group selection, angular half-width $\theta_c$ sets the specificity–kinetics tradeoff, and attraction strength $\varepsilon$ with range $\delta$ sets phase behavior and reversibility (Fusco et al., 2015).

DNA-based PCA realizes an experimentally programmable analogue of this patchy logic. Sticky-end sequences set hybridization thermodynamics through

$$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$

while spacer length, brush architecture, linker concentration, particle size, and grafting density determine the effective pair potential

$U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$

Sequence specificity can produce SC, BCC, FCC-like and binary lattices for isotropic spheres, while anisotropic building blocks and DNA origami enable diamond, clathrates, gyroid-related phases, quasicrystals, and lattices lacking atomic analogs (Jacobs et al., 2024).

Two inverse-design lines make this structural encoding explicit. One recasts the interaction-design problem as a Boolean satisfiability problem: given a target unit cell, particle geometry, and allowed orientations, a SAT solver returns patch colors, a color-compatibility matrix, and species assignments such that every target contact is allowed and all non-target contacts are forbidden. That construction was demonstrated for cubic diamond, pyrochlore, and clathrate lattices (Romano et al., 2020). A related scheme introduces a torsional component to the patchy-particle potential so that bond formation requires not only correct patch alignment but also the correct relative in-plane orientation. In the reported examples, this torsional specificity was decisive for selecting BC8 over alternative tetrahedral networks, while a more complex clathrate still assembled even after torsional restrictions were removed because of its greater remaining specificity (Tracey et al., 2019).

A different kind of symmetry encoding appears in size-controlled triply-periodic polyhedra. There, a CK-like triangulation index

$T=h^2+hk+k^2$

classifies Primitive, Diamond, and Gyroid polyhedra, and the assembly energy penalizes deviations from target edge lengths and dihedral angles:

$$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$

The same symmetry principles that reduce species count also admit generalized disclinations, so economy and fidelity become coupled design variables (Duque et al., 2023).

At the atomic-thickness limit, PCA no longer refers to self-assembly from mobile particles but to programmable layer-by-layer construction. Wafer-scale PCA of graphene and monolayer hBN uses epitaxially grown monolayers on Ge(110) as standardized building blocks, then stacks them under all-dry, vdW-assisted conditions to control thickness, layer sequence, composition, twist angle, and crystal symmetry. The reported system achieves >99% monolayer uniformity, near-unity stacking yield, and twist-angle accuracy within $\pm 1^\circ$ in multilayer stacks (Yang et al., 30 Jul 2025).

3. Nucleation, growth, and real-time control

A central PCA theme is that the same design variables that select a lattice also reshape nucleation barriers and growth laws. Classical nucleation theory remains a common starting point. In the protein literature,

$f$0

with $f$1. Shorter interaction ranges and sharper angular specificity typically increase $f$2 but also increase crystal stability relative to disordered states, so PCA becomes a balance between selectivity and interface formation (Fusco et al., 2015).

For open protein crystals, two widely used heuristics are explicitly qualified. First, the reduced second virial coefficient

$f$3

can locate a rough crystallization window, and many proteins crystallize for $f$4 in a narrow negative range, commonly cited as roughly $f$5 to $f$6; however, patchiness can shift the optimal range and orientation-averaging can mask decisive directional features. Second, proximity to a metastable liquid–liquid or liquid–vapor critical point does not generically help open, anisotropic crystals. For the square-lattice protein model patterned after SbpA, good assembly required a thermodynamic driving force $f$7–$f$8 per particle and interactions as nonspecific as possible without promoting liquid–vapor phase separation; assembly was not predicted by the second virial coefficient alone and was not enhanced by the critical point (Haxton et al., 2011).

DNA-coated colloids provide an experimentally resolved case in which CNT becomes quantitatively predictive only after adding an interfacial kinetic bottleneck. The measured nucleation rate takes the form

$f$9

but for micrometer-scale particles the prefactor is set by rolling-limited attachment:

$\theta_c$0

This rolling-limited kinetics explains why nucleation rates vary by orders of magnitude within a temperature window of about $\theta_c$1, why diffusion-limited prefactors overestimate measured rates by at least $\theta_c$2, and why early growth is reaction-limited before crossing into late diffusion-limited growth with crystal mole fraction rising as $\theta_c$3 (Hensley et al., 2021).

Process programming follows directly from this kinetic view. A robust two-step DNA protocol separates nucleation from growth: monodisperse droplets plus staircase cooling produce monodisperse single-domain seeds, then those seeds are transferred into a bath of “weak” particles and grown by seeded diffusion-limited growth with $\theta_c$4. This circumvents the extremely narrow nucleation window of optical-scale DNA colloids, where nucleation can vary by $\theta_c$5 over only $\theta_c$6 and viable crystallization windows can be as narrow as $\theta_c$7 (Hensley et al., 2023, Jacobs et al., 2024).

Real-time control has also been formalized directly. A feedback protocol for short-ranged attractive colloids measures a correlation function $\theta_c$8 and linear response $\theta_c$9 during assembly, compares them to the equilibrium fluctuation–dissipation relation, and updates bond strength according to

$\varepsilon$0

with target reversibility $\varepsilon$1. The practical rule is to maintain small but finite deviations from equilibrium: strong enough to drive growth, but weak enough to preserve bond breaking and defect correction (Klotsa et al., 2012).

A different nonclassical kinetic regime appears in evaporating sulfate mixtures. There, divalent ions induce sodium sulfate spherulites through a two-step pathway involving dense liquid or pre-nucleation clusters, followed by nanocrystal aggregation and radial self-assembly. Growth follows the parabolic law

$\varepsilon$2

with $\varepsilon$3, consistent with diffusion-limited growth in highly viscous solutions; for Na$\varepsilon$4SO$\varepsilon$5–MgSO$\varepsilon$6 at $\varepsilon$7, the extrapolated viscosity at the onset of spherulitic nucleation is $\varepsilon$8 (Heeremans et al., 1 Jun 2025).

4. Experimental platforms and realized architectures

The most extensively quantified PCA demonstrations are in DNA-coated colloids. A microfluidic droplet platform established a complete crystallization pathway from thermally activated nucleation to reaction-limited growth and then diffusion-limited growth. Under slow linear cooling, 600-nm particles at $\varepsilon$9 (v/v) with a ramp of $\delta$0 over $\delta$1 hours produced more than $\delta$2 single crystals, and 400-nm particles at $\delta$3 (v/v) with a ramp of $\delta$4 over $\delta$5 hours yielded $\delta$6 single crystals comprising approximately $\delta$7 particles each; the latter displayed pronounced structural coloration (Hensley et al., 2021).

The same kinetic logic was pushed to macroscopic optical-scale crystals by seeded growth. In that implementation, very small droplets of about $\delta$8 nL and a slow ramp of $\delta$9 per $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$0 hours produced near-$$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$1 single-crystal seeds, which were then grown in a weak-particle bath into macroscopic single-domain photonic crystals containing millions of particles. The method was demonstrated for 600 nm, 430 nm, and 250 nm particle systems spanning CuAu-like, intermediate BCT, and CsCl-like habits, with strong structural coloration emerging around $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$2 linear size and final crystal volumes exhibiting coefficient of variation below $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$3 (Hensley et al., 2023).

Acoustic PCA realizes a different class of ordered structures by programming an energy landscape rather than a bond network. In a square chamber driven by two orthogonal bulk standing waves, microparticles migrate to pressure nodes or antinodes according to the sign of the acoustic contrast factor. The Gor’kov potential is

$$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$4

This platform forms thousands of size-limited assemblies in minutes, with assembly time around $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$5 minutes at $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$6 kPa and less than $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$7 minute above about $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$8 kPa. Increasing pressure amplitude and local occupancy drives a monolayer-to-multilayer transition, with stacks observed above roughly $$\Delta G_{\mathrm{hyb}}(T,c_{\mathrm{salt}})=\Delta H-T\Delta S,$$9 kPa and about $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$0 particles/mm$U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$1 (Owens et al., 2015). A related 3D ultrasonic geometry uses three orthogonal standing waves to build orthorhombic lattices with programmable lattice constants

$U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$2

and reports millisecond-scale reconfiguration estimates for 90 $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$3m polystyrene spheres at MHz frequencies (Caleap et al., 2015).

Electro-osmotic PCA programs crystals by transport fields. Ion-exchange micro-pumps on charged glass generate inward electro-osmotic flow and outward electrophoretic drift; colloids stall at a finite assembly distance and crystallize there. Under optimized low-flow conditions, monodisperse colloids form defect-free single-domain crystals outside a colloid-free void, with facetted inner crystal boundaries centered on the ion-exchange particle, and binary mixtures exhibit radial size sorting with larger particles assembling closer to the pump (Niu et al., 2016). An inverted configuration, with the ion-exchange resin fixed on the top slide, extends the flow range to millimeter scales and yields seedless millimeter-sized monolayer single crystals; two pumps separated by about $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$4 impose a predefined orientation along the connecting line, and patterned pump arrays can inscribe hexagons, stars, and more complex motifs (Niu et al., 2018).

In chiral nematics, PCA acts on topological textures rather than on material particles. Skyrmions in a homeotropic background can be confined, repelled, and trapped by alignment-induced attractive or repulsive walls. The range of the wall–skyrmion interaction is set primarily by the cholesteric pitch $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$5: the reported finishing distance is $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$6 for 2D skyrmions and $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$7 for 3D skyrmions. Arrays of weak-anchoring spots act as lattice sites, while electric fields, patterned electrodes, and optically modified chirality provide reconfigurable “walls” and moving traps for chains, lattices, and periodically patterned skyrmion arrays (Duzgun et al., 2020). More broadly, a curvature–topology–chirality framework shows how genus, Euler characteristic, anchoring, and chirality can program defect sites, elastic multipoles, and localized solitons that assemble into lattice-like textures or colloidal crystals in liquid crystals (Smalyukh et al., 8 Jun 2026).

At the 2D-material end of the spectrum, wafer-scale PCA of graphene and monolayer hBN uses all-dry vdW-assisted assembly to produce pristine interfaces, layer-resolved composition, and programmed twist or rotational polarity. The reported stack quality includes interlayer spacing $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$8 Å for graphene/hBN, $U(r)=U_{\mathrm{attr}}(r)+U_{\mathrm{rep}}(r).$9–$T=h^2+hk+k^2$0 Å for twisted graphene/graphene, coherent multilayer XRD thickness of about $T=h^2+hk+k^2$1 nm for $T=h^2+hk+k^2$2, and a sharp C1s peak with FWHM $T=h^2+hk+k^2$3 eV. Device arrays show zero-bias resistance–area products modulated over orders of magnitude by single-layer hBN thickness increments, and large-area CTG and ATG stacks exhibit programmable minibands and mini-gaps in ARPES (Yang et al., 30 Jul 2025).

5. Design rules, figures of merit, and recurrent misconceptions

Several quantitative rules recur across otherwise unrelated PCA platforms. For protein crystals, the cited soft-matter literature recommends modeling the protein as a particle with valence at least $T=h^2+hk+k^2$4, using short interaction ranges $T=h^2+hk+k^2$5–$T=h^2+hk+k^2$6, angular half-widths $T=h^2+hk+k^2$7–$T=h^2+hk+k^2$8, and moderate patch strengths of order a few $T=h^2+hk+k^2$9 to enable reversible bonding and defect annealing. The goal is to enter the crystallization gap between the solubility line and the metastable critical point without crossing into spinodal decomposition or gelation (Fusco et al., 2015).

For open protein crystals, a recurrent misconception is that the second virial coefficient alone predicts optimal assembly. It does not. Different balances of specific and nonspecific attraction can produce the same $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$0 yet give high-yield crystals, poor crystals, or no crystals. The square-lattice model based on SbpA instead identifies the key control parameter as the per-particle thermodynamic driving force, with best assembly at $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$1–$$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$2 and supersaturation $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$3–$$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$4; proximity to a metastable critical point, often helpful for close-packed isotropic spheres, was found to hinder the assembly of low-coordination open lattices (Haxton et al., 2011).

For micrometer-scale DNA colloids, another recurrent misconception is that equilibrium structural selection is sufficient. In practice, kinetics dominate. Successful protocols require precise thermal control, high grafting density, and explicit control over when nucleation is allowed to occur. The review literature recommends grafting densities above approximately $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$5 for micrometer colloids, notes that crystallization is suppressed below approximately $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$6, and emphasizes that slower linear cooling, strong-seed/weak-growth protocols, and two-step droplet-seeding plus bulk diffusion-limited growth are practical ways to navigate temperature windows as narrow as $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$7–$$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$8 (Jacobs et al., 2024).

For TPMS-like programmable assemblies, the principal design rule is a tradeoff between economy and fidelity. The same symmetry constraints that reduce the number of distinct species to $$E=\sum_{\langle ij\rangle}\left\{-\varepsilon_{ij}+\frac{k}{2}|\ell_{ij}-\ell_{ij}^{(0)}|^2+\kappa\left[1-\cos(\theta_{ij}-\theta_{ij}^{(0)})\right]\right\}.$$9 also permit generalized disclinations at high-symmetry sites. In the reported grand-canonical simulations, high-fidelity and efficient growth required an intermediate degree of geometric flexibility, approximately $\pm 1^\circ$0 and $\pm 1^\circ$1 for $\pm 1^\circ$2; too flexible produced off-target disclination-rich assemblies, and too rigid stalled growth (Duque et al., 2023).

For dynamic control protocols, the feedback literature makes the same point in a different language: optimal assembly sits near, but not at, equilibrium. The target value $\pm 1^\circ$3 was chosen specifically to maintain reversible microscopic bonding while preserving net growth. This suggests a broader PCA principle: design should target a finite reversibility window rather than maximal bond strength or maximal driving force (Klotsa et al., 2012).

6. Limitations and emerging directions

Despite its breadth, PCA remains limited by model-to-material translation. In protein crystallization, patch locations, angular widths, and energetic heterogeneity are difficult to infer from sequence or structure alone; non-crystallographic contacts, transient complexes, and internal flexibility complicate coarse-graining, and data scarcity limits predictive parameterization (Fusco et al., 2015). In DNA systems, more explicit sequence-resolved models of $\pm 1^\circ$4, $\pm 1^\circ$5, and $\pm 1^\circ$6 at colloidal interfaces are still needed, particularly outside the narrow windows where current reduced models have been validated (Jacobs et al., 2024).

Scalability remains heterogeneous across platforms. Droplet-based DNA seeding is predictive and robust, but translating single-crystal droplet assembly into bulk synthesis required an additional seeded-growth step, and controlling deep-quench gelation remains difficult (Hensley et al., 2023). Wafer-scale 2D PCA has already reached near-unity stacking yield for graphene and monolayer hBN, but broader industrialization will require further automation in alignment, transfer, and heterogeneous material integration (Yang et al., 30 Jul 2025).

Several directions suggest that PCA is moving beyond static equilibrium design. The DNA-colloid review explicitly points to kinetic programming for retrieving multiple distinct target structures from one suspension and to dissipative schemes that stabilize structures by energy consumption rather than equilibrium free-energy minimization (Jacobs et al., 2024). In liquid crystals, curvature, topology, and chirality are increasingly treated as programmable variables that define defect budgets, elastic traps, and soliton lattices, implying a form of PCA in which “binding sites” are topological rather than chemical (Smalyukh et al., 8 Jun 2026).

A plausible synthesis is that PCA is converging on a multiscale control framework. At one end are chemically specific bonds, torsional offsets, and sequence thermodynamics; at another are transport fields, patterned boundaries, and vdW stacking operations; and between them lie kinetic controllers, seeding strategies, and topology-guided defect engineering. The unifying problem is the same in each case: how to encode a target ordered structure while keeping the assembly pathway inside a reversible, defect-correcting, experimentally accessible window.