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The Statistical Mechanics of Dynamic Pathways to Self-assembly
Stephen Whitelam, Robert L. Jack
TL;DR
Self-assembly pathways exhibit generic physical features across systems with different microscopic details, while existing classical theories do not always explain observed behavior. This review synthesizes their physical characteristics and theoretical descriptions, emphasizing that far-from-equilibrium pathways require explicitly dynamic treatment and will become increasingly important.
Problem
Classical theories are often insufficient for quantitatively observed and directly visualized self-assembly behavior, creating a need for theoretical insights into its control.
Method
The review synthesizes shared physical characteristics, dynamical pathways, and theoretical descriptions of self-assembly across molecular, nanoscale, and micron-scale systems.
Results
Self-assembly pathways share features across microscopically different systems, while far-from-equilibrium pathways require explicit consideration of dynamic effects.
Takeaways & Limitations
Theoretical guidance for far-from-equilibrium self-assembly is likely to become increasingly important as component synthesis and in-situ pathway imaging advance.
Takeaways & Limitations
Near-equilibrium descriptions rely on a small number of collective coordinates, whereas far-from-equilibrium assembly may require accounting for clusters with all possible morphologies.
Abstract
from arXiv · showhide
We describe some of the important physical characteristics of the `pathways', i.e. dynamical processes, by which molecular, nanoscale and micron-scale self-assembly occurs. We highlight the fact that there exist features of self-assembly pathways that are common to a wide range of physical systems, even though those systems may be different in respect of their microscopic details. We summarize some existing theoretical descriptions of self-assembly pathways, and highlight areas -- notably, the description of self-assembly pathways that occur `far' from equilibrium -- that are likely to become increasingly important.
1 Introduction
Self-assembly spans molecular to micron scales and can produce diverse ordered structures, but classical theories often fail to explain observed behavior. This review therefore focuses on generic dynamical pathways and the assumptions behind their theoretical descriptions.
- Scope: Self-assembly organizes inactive components without external direction into ordered patterns across scales from atoms and molecules to colloids.The review focuses on Brownian components in undriven systems, whose motion is driven by thermal fluctuations.
- Motivation: Recent advances in molecular shape design, DNA-mediated interactions, and colloidal particle synthesis have enabled increasingly complex self-assembled structures.Examples include DNA structures with basic functions and chemically patterned patchy particles.
- Motivation: Quantitative experiments and direct visualization increasingly reveal behaviors that classical nucleation-and-growth theories cannot explain sufficiently.These observations motivate new theoretical insights for understanding and controlling self-assembly.
- Review aims: The review examines self-assembly pathways as dynamical processes and identifies features conserved across systems with different microscopic details.It also evaluates the assumptions underlying theoretical descriptions and their validity in practical settings.
- Organization: The discussion is organized around shared physical characteristics, computational modeling, dynamic pathways, and theories describing those pathways.The paper concludes with an outlook on the field.
2 The physical character of self-assembly
Self-assembly pathways reflect a tension between thermodynamic stability and kinetic accessibility. Outcomes depend on interaction strength, specificity, microscopic reversibility, and observation time, with far-from-equilibrium behavior often remaining kinetically trapped.
- Thermodynamic and dynamic factors: Self-assembly evolves from disorder toward ordered states, but the first structure may be metastable or not a free-energy minimum on experimental timescales.The thermodynamic driving force toward lower free energy does not guarantee formation or later relaxation to the equilibrium structure.
- Thermodynamic and dynamic factors: Successful assembly requires balancing thermodynamic impetus with dynamics that let randomly moving components arrange into the desired structure.These requirements commonly oppose one another, so favorable thermodynamic conditions may be dynamically unsuitable, and vice versa.
- Thermodynamic and dynamic factors: Interaction strength and specificity jointly determine whether components form the stable structure, kinetically trapped structures, alternative structures, or no assembly.Excessive strength suppresses error correction, insufficient specificity permits mistakes, and excessive specificity prevents typical encounters from binding.
- Thermodynamic and dynamic factors: Thermodynamically stable assembly typically occurs only within a small parameter-space subset, whereas strong bonds can produce far-from-equilibrium kinetically trapped structures.The same one-component systems can show near-equilibrium or far-from-equilibrium behavior under different conditions.
- Metastable and kinetically-trapped states: In the toy model, strong binding rapidly yields only 1/(1 + M) optimally bound particles when misbound states are dynamically more accessible.The equilibrium-yield timescale is exp (ϵmis/kBT), which becomes large as misbinding energy increases.
- Metastable and kinetically-trapped states: At fixed observation time, yield is non-monotonic in interaction strength because stronger thermodynamic driving is counteracted by slower escape from misbound environments.The observed yield therefore depends on how long the system is observed.
3 Numerical methods for the study of self-assembly
Computational studies separate thermodynamic methods from dynamic methods and use simulations to investigate equilibrium properties, assembly processes, and rare events. These models provide useful qualitative insight but remain approximate representations of experiments.
- Numerical methods: Computer simulations enable precise interaction design and detailed tracking of many particles when corresponding experiments are difficult.The paper surveys computational methods for studying self-assembly.
- Thermodynamic methods: Thermodynamic methods such as Monte Carlo and molecular dynamics estimate equilibrium behavior and free energies, while advanced algorithms improve equilibrium sampling.Specialized algorithms can identify candidate ordered phases or interactions that stabilize selected structures.
- Dynamic methods: Dynamic simulations model Brownian motion using explicit or implicit solvent representations, with implicit treatments reducing computational expense.Brownian dynamics incorporates random forces to represent solvent effects.
- Model limitations: Computational models use coarse-grained particles, approximate effective interactions, and simplified solvents, making fully quantitative dynamical agreement with experiments difficult.Despite this limitation, simple models can provide useful qualitative insight.
4 Dynamic pathways to self-assembly
Self-assembly pathways describe how structures evolve over time, and understanding them requires both thermodynamic and dynamic factors. They range from near-equilibrium routes governed mainly by free-energy surfaces to far-from-equilibrium routes shaped explicitly by microscopic dynamics and competing timescales.
- Pathway framework: A pathway describes the structures that self-assemble as time progresses, often using assembled-structure size as a progress coordinate.Free-energy barriers arise because cluster growth gains volume-related free energy but costs surface-related free energy; spontaneous growth begins beyond a critical size.
- Near- and far-from-equilibrium pathways: Near-equilibrium pathways follow favored routes on a thermodynamic free-energy surface, while far-from-equilibrium pathways require explicit consideration of dynamic effects.Far-from-equilibrium motion is biased by microscopic particle dynamics and competition between several slow timescales.
- Near-equilibrium pathways: Classical single-step assembly reaches a thermodynamically stable structure through nucleation and growth of clusters sharing the stable structure’s properties.Classical nucleation theory treats crossing a single free-energy barrier as the rate-limiting step.
- Near-equilibrium pathways: Multi-step near-equilibrium assembly proceeds through clusters representing metastable bulk phases, whereas other intermediates are selected by structured free-energy surfaces without corresponding directly to bulk phases.These routes include two-step liquid-to-crystal transformations, polymorph changes, faceting-driven intermediates, directional interactions, and hierarchical assembly.
- Far-from-equilibrium pathways: Far-from-equilibrium pathways can produce kinetically trapped structures without special thermodynamic status, including malformed structures, gels, fractal aggregates, and nonperiodic networks.Strong interactions and binding errors can prevent annealing on the observation timescale, although some trapped structures have useful properties.
- Far-from-equilibrium pathways: Far-from-equilibrium assembly can also preserve nonequilibrium component arrangements or internal states when rearrangement and conformational relaxation are slower than structural growth.This includes multicomponent alloys and colloidal crystals, as well as DNA-linked particles and systems whose internal dynamics control assembly pathways.
- Open challenges: Several experimentally observed pathways remain unclassified, including those involving clusters in some mineral phases and proteins.Molecular-scale temporal resolution from newer experiments creates an open challenge for theory.
5 Statistical mechanical descriptions of self-assembly pathways
The paper organizes pathway descriptions around near-equilibrium theories based on collective coordinates and free-energy landscapes, then explains why far-from-equilibrium assembly requires explicit dynamical treatment. These approaches cover classical nucleation, multistep pathways, structured intermediates, and kinetic trapping across diverse systems.
- Near-equilibrium assumptions: Near-equilibrium descriptions assume thermodynamic factors govern assembly and that a small set of collective coordinates, such as cluster size and composition, suffices.Configurations sharing the same collective coordinates are treated as having equilibrium-like probability ratios determined by their energy difference.
- Near-equilibrium assumptions: Quasiequilibrium simplifies pathway analysis when bond formation and breaking are rapid relative to cluster growth, but explicit dynamics are required when this assumption fails.The quasiequilibrium assumption can justify lowest-free-energy-barrier arguments for transformation pathways.
- Classical nucleation theory: Classical nucleation theory models assembly through rare clusters resembling the bulk phase, with a free-energy barrier that can control the nucleation rate.In its simplest form, ∆G(n) = γn^2/3 − n∆µ and knuc = k0 exp[−∆G(n⋆)/kBT].
- Classical nucleation theory: CNT is a useful qualitative starting point, but quantitative rate predictions are limited by barrier uncertainty and non-spherical critical clusters.Even the Ising model requires extensions to simple CNT assumptions for quantitative agreement between theory and simulation.
- Multistep pathways: Two-step pathways are described using free-energy surfaces ∆G(n, m), whose saddle points represent sequential events such as liquid-cluster formation followed by crystallization.Classical density functional theory describes such pathways within the quasiequilibrium assumption, while Ostwald’s rule invokes intermediate bulk phases.
- Far-from-equilibrium pathways: Far-from-equilibrium assembly requires accounting for cluster morphology because rapid growth can prevent structures from relaxing to local equilibrium.Enumerating all morphologies is generally intractable, motivating schematic structural axes as proxies for this complexity.
- Far-from-equilibrium pathways: Strong bonds can prevent ordered structures, while successful assembly typically balances rapid growth against sufficient reversibility and quasiequilibrium.Simulations of model viral capsids associated effective assembly with weak quasiequilibrium deviations and kinetic trapping with its breakdown.
- Far-from-equilibrium applications: Far-from-equilibrium assembly can produce useful gels and nonequilibrium loop structures, but their dynamic effects and aging remain incompletely understood.Gel structures depend strongly on dynamics and can undergo large-scale rearrangements during aging.
6 Outlook
The outlook identifies broad principles shared across microscopically different self-assembly systems, while emphasizing that established near-equilibrium theories need to be complemented by fundamentally dynamic descriptions.
- General principles: Self-assembly pathways exhibit generic features across systems despite differences in microscopic details, and existing theories capture several important general principles.The review summarizes these features and theoretical descriptions as a foundation for continued analysis.
- Future directions: Near-equilibrium ideas remain the foundation of current descriptions, but theories of far-from-equilibrium self-assembly require continued development.The paper highlights dynamic theories as an important direction for the field.