Theoretical models of self-assembly under shear flow or extensional forces provide critical insights into the dynamic behavior of soft matter systems, including polymers, colloids, and surfactants. These models elucidate how external flow fields influence the alignment, breakup, and reformation of self-assembled structures, offering predictive capabilities for designing functional materials. Computational approaches, particularly molecular dynamics (MD), Brownian dynamics (BD), and mesoscale simulations, have been instrumental in advancing this understanding.

In polymer systems, shear flow induces chain alignment and deformation, which can be modeled using coarse-grained MD or bead-spring representations. Under shear, polymer chains transition from coiled to stretched conformations, with the degree of alignment dependent on the Weissenberg number (Wi), the ratio of shear rate to polymer relaxation time. At low Wi, chains exhibit minor deformation, while at high Wi, they undergo significant stretching, leading to anisotropic orientation. The Rolie-Poly model, a simplified constitutive equation, captures these dynamics by incorporating reptation, contour length fluctuations, and convective constraint release. Simulations show that shear flow can also induce phase separation in polymer blends, where the interplay between shear-induced alignment and thermodynamic forces dictates the morphology.

Colloidal systems under shear exhibit rich phase behavior, including the formation of strings, layers, or gels. Theoretical frameworks such as Stokesian dynamics or lattice Boltzmann methods model hydrodynamic interactions and Brownian motion. At low shear rates, colloids form ordered crystalline structures, while increasing shear disrupts long-range order, leading to shear-induced melting or the formation of anisotropic clusters. The Mason number (Mn), which compares viscous to thermal forces, governs these transitions. For example, simulations of hard-sphere colloids reveal a critical Mn of approximately 0.1, above which ordered phases break down. Shear also affects depletion interactions in colloid-polymer mixtures, where flow can either enhance or suppress phase separation depending on the relative strength of shear and entropic forces.

Surfactant self-assembly under flow is often studied using dissipative particle dynamics (DPD) or coarse-grained MD. Micelles, vesicles, and bilayers respond to shear by aligning, breaking, or merging. The capillary number (Ca), representing the ratio of viscous to interfacial forces, determines the stability of these structures. For wormlike micelles, shear can induce a transition from entangled networks to aligned cylindrical structures, with scission and recombination events governed by the interplay between flow and micellar kinetics. Simulations predict a critical shear rate beyond which micelles break into smaller fragments, followed by reformation into larger structures when shear is reduced. In bilayer systems, shear can induce vesicle deformation or rupture, with the bending modulus and membrane tension playing key roles in stability.

Extensional flows, such as those in elongational or converging geometries, produce distinct effects compared to shear. Polymers in extensional flow experience strong stretching, leading to coil-to-stretch transitions at a critical Deborah number (De). The FENE-P model, which incorporates finite extensibility, accurately predicts these transitions. Colloids in extensional flow exhibit clustering along the extension axis, with simulations showing that the cluster size distribution depends on the Peclet number (Pe). Surfactant systems under extension form elongated micelles or bilayers, with breakup occurring at higher strain rates due to increased hydrodynamic stress.

Theoretical models also address the interplay between flow and thermodynamic driving forces. For example, dynamic density functional theory (DDFT) combines free energy landscapes with flow fields to predict self-assembly pathways. In polymer-colloid mixtures, DDFT simulations reveal shear-induced microphase separation, where flow competes with thermodynamic demixing. Similarly, in surfactant systems, flow can shift the equilibrium phase diagram, stabilizing metastable states such as shear-induced lamellar phases.

Breakup and reformation dynamics are central to understanding self-assembly under flow. For polymers, the Doi-Edwards theory describes chain scission and recombination in entangled solutions, with simulations showing that breakup occurs at specific chain orientations relative to the flow direction. In colloids, hydrodynamic forces dominate breakup, with clusters fragmenting when viscous stress exceeds interparticle adhesion. Surfactant systems exhibit reversible breakup, where micelles or vesicles reassemble upon flow cessation due to thermodynamic driving forces.

Alignment under flow is often quantified using order parameters, such as the nematic order parameter for liquid crystalline systems or the orientation tensor for polymers. Simulations show that alignment is non-monotonic with shear rate, peaking at intermediate rates before decreasing due to flow-induced disorder. In extensional flow, alignment is more pronounced and persists at higher rates, reflecting the uniaxial nature of the flow field.

Theoretical advances in machine learning have enabled the prediction of self-assembly outcomes under flow. Neural networks trained on simulation data can map flow conditions to structural properties, bypassing expensive computations. For example, graph neural networks predict colloidal cluster distributions under shear by learning from BD simulations. These approaches accelerate the exploration of parameter space, identifying optimal conditions for desired self-assembled structures.

Challenges remain in modeling polydisperse systems, where size or composition variations introduce additional complexity. Multiscale methods, coupling atomistic details with continuum descriptions, are being developed to address these challenges. For instance, hybrid MD-continuum models capture the dynamics of polymer nanocomposites under flow, accounting for both molecular-scale interactions and macroscopic rheology.

In summary, theoretical models of self-assembly under flow provide a powerful toolkit for understanding and predicting the behavior of soft matter systems. By integrating molecular-level interactions with macroscopic flow fields, these models reveal the underlying physics of alignment, breakup, and reformation, guiding the design of materials with tailored properties. Computational advancements continue to expand the scope and accuracy of these predictions, enabling deeper insights into nonequilibrium self-assembly phenomena.