Nature utilizes amphiphilic assemblies to compartmentalize different chemical environments, modulate interfacial properties and reaction rates, and create complex materials with mesoscopic order. Many models explain the phase behavior of amphiphiles, but our understanding of the non-equilibrium behaviors of amphiphilic assemblies is limited. There exists a vast and rich space of structural and functional variations unexplored by equilibrium methods and models. A recent set of amphiphilic self-assembly experiments have suggested that non-equilibrium pathways of amphiphilic assemblies are complex and poorly understood\cite{Griffith2014}. In the experiments, assembly of 2-oxooctanoic acid was photoinitiated and eventually yielded monodisperse aggregates of long-term stability. In this dissertation, I explore various states of amphiphilic assemblies and the non-equilibrium processes that interconnect these states in the context of the photoinitiated assembly model system. In Chapter 2, I used molecular dynamics simulations to study structural variations and mechanisms of formation of the early nuclei from aqueous solution. Importance sampling and classical nucleation theory were employed to estimate the rate of nucleation. I found that the kinetics of coaggregation, where a mixture of more than one amphiphilic species, could differ significantly from that of single-species aggregation. Specifically, the participation of a second species could open up new pathways of growth and introduce microscopic phase separation of the two species within the aggregates and as a result modulate the growth rates. In Chapter 3, I computed the solutions to master-equation chemical kinetic models of amphiphilic aggregation pathways. I found that a large critical nucleus size is an important factor that contributes to the production of a narrow aggregate size distribution, a highly desirable characteristic in the preparation of self-assembled nanoparticles. I incorporate the elementary steps of nucleation, growth, and a source of precursor molecules to observe the effects of competing elementary rates on the aggregate size distribution. In Chapter 4, I furthered the development of the charge-frustrated Ising model to represent amphiphilic species with non-zero spontaneous curvatures. I also built in the correct interfacial roughness by using two lattice spacings in the same lattice model.

Fabrication of materials through bottom-up, noncovalent self-assembly has the potential to revolutionize many areas of science and engineering that depend upon the precise arrangement and interaction of nanometer-scale particles. Theory and computation provide a useful route to studying self-assembly of these materials due to their ability to study the kinetic and thermodynamic features of simplified systems at a level of detail not possible in experiments. In this thesis we study the self-assembly of two bio-inspired systems using a combination of theory and computation. The first is comprised of archaeal chaperonin proteins that form in-vitro two remarkably different structures: filamentous chains and layers of sheets. Using a quasi-dynamical Monte Carlo algorithm, we show that the binary decision for sheet or string formation can be explained by allowing for conformational changes between a sheet-favoring state and a string-favoring state. Using advanced sampling techniques, we find that the energy gap for this conformational change controls structure formation. The second system is a self assembling cyclic peptide inside a block copolymer matrix. We develop a computationally efficient pseudospectral technique to simulate a Langevin dynamics derived from the block copolymer field-theoretic Hamiltonian and demonstrate two different processes by which nanoparticles may be incorporated into this framework.

In support of the peptide-polymer work, we develop a new algorithm for Metropolis Monte Carlo simulations on high-performance graphics processing units (GPUs) that relies on the local equilibration of non-interacting regions of a lattice system. We show how the technique can better exploit the GPU memory hierarchy resulting in over 100-fold speedups. This technique is well-suited for lattice systems with couplings beyond nearest neighbors and systems with complicated local or dynamical constraints.

Finally, we investigate the implications for excluding volume in a $\phi^4-\phi^2$ field theory, representative of a block copolymer theory. We provide a simplified derivation for the response of a Gaussian liquid to a volume-excluding solute. We apply this technique to a discrete $\phi^4-\phi^2$ theory and show that the effects of volume exclusion act independently of the $\phi^4$ term. Finally, we show using Monte Carlo simulations that the stabilization provided by the $\phi^4$ term may be replaced by a hard constraint on density fluctuations that imparts stability, while making the model's connections to well-studied Gaussian models of liquids more transparent.

Rich understanding of a complex system can often emerge from simple but carefully constructed models. With an appropriate model, we can ask questions about how tuning the parameters of the model or modifying the constraints of the system changes the system behavior. My research involves applying such an approach to actin, an essential biopolymer in the cell. In this work, we explore how forces affect actin at the filament and network length scales.

In the first part, we investigate how different forces modulate the interaction between actin filaments and actin-binding proteins. One such protein complex, Arp2/3, can cause filaments to form branches. Experiments indicate that branches preferentially form on the convex side of bent filaments. Using a coarse-grained model discretized at the monomer pair level, we show that binding is dependent upon a high local curvature fluctuation of the filament. The results indicate that actin can sense and respond to mechanical environmental cues to regulate the binding of Arp2/3. We further believe such a picture can serve as a useful framework for studying the effects of force on the binding and function of other proteins. In a follow-up project, we derive analytical expressions for the nanoscale curvature distribution of a worm-like chain and membrane as a function of applied tension. These expressions can be used to understand the force dependence of protein binding on actin filaments and membranes within a biological context.

In the second part, we focus on actin network elasticity. Specifically, we explore how actin networks respond to large external forces. However, the theoretical toolkit for such a task is incomplete. First we develop a constant-stress framework to apply large forces on soft but strongly nonlinear materials. Additionally, we create a toy model of a soft elastic solid with a nonlinear elastic response on which we test our constant-stress method. Finally, we utilize the constant-stress method and a coarse-grained model for short, semiflexible chains to explore actin network elasticity under compression. We consistently observe stress softening under compression, which we analyze from a single filament perspective and using normal mode analysis.

An overview of five studies is presented in two parts. The first part presents two studies

of supercooled fluids. The second part presents three studies of water and aqueous solutions.

Each study seeks a minimal model of a condensed matter system. In the first study,

kinetically constrained models (KCM’s) are compared to alternative theories of the glass

transition in high dimensions. Dimensionality is used as a parameter to tune the connectivity

of a lattice, where a higher dimensional model has more interactions between neighboring

sites. This study finds that KCM’s outperform alternative theories in high dimensions. The

second study explores the possibility that bacteria have evolved to exploit the glass transition

to enter a dormant state when environmental conditions are unfavorable. Although the

available evidence shows that the bacterial cytoplasm does not meet the strict definition of a

fragile glass former, much of its behavior is similar to and can be described using close analogies

with the glass transition. In the second part, the third study describes the molecular

mechanisms that gives rise to large electric field fluctuations, which in turn cause autoionization

and ion dissociation. The fourth study analyzes several candidate order parameters

as the basis for a Gaussian field theory of ion solvation. Finally, the fifth study discusses the

most popular current explanation for observed charge asymmetry at liquid-vapor interfaces.

This explanation, based on linear response of the surface polarization to the presence of an

ion, is incorrect. Instead, the surface polarization responds non-linearly to the presence of an

ion. Incorporating these non-linear fluctuations is essential to predict solvation free energies.

The surface of liquid water is characterized by large fluctuations that are too complex to describe exactly but, like with other aspects of molecular liquids, can be described in a useful way when approached statistically. The water liquid/vapor interface is composed of a sharp drop in density and reverberations from that drop that extend about 1 nm from the surface. Transient but characteristic structure forms as water molecules realign to maximize hydrogen bonding without ready neighbors in vapor. Much of the structure of water’s surface can be accounted for as a broadening of the bilayer structure of the basal plane of the ice crystal, particularly the layering in density and the orientational trends.The mean field surface model predicts the average orientational behavior of structured molecular liquids, which is demonstrated in two dimensions with the water-like Mercedes-Benz (MB) model and in three dimensions with comparisons to the SPC/E and TIP5P models of water. Liquids with strong directional bonding tend to orient predictably when potential bonding neighbors are removed. The mean field surface model assumes that bonds are most likely to form in regions of high density, then uses a self-referencing iterative approach to determine whether neighbors are also likely to be oriented in a way that is favorable to bonding. Physically motivated changes to the reference density, bonding rules, and molecular structure clarify which aspects of molecular water are actually essential to the orientational structure at the interface. It is shown that moderate changes to bonding rules, like flexibility, strength, and asymmetry, have little impact on the main surface patterns. The layering in the density profile of water is by contrast necessary to accurately predict water’s surface structure. All of these observations can be rationalized through liquid water’s relationship to the structure of the basal plane of ice. The dielectric response of water is also affected by the introduction of an interface. The Gaussian field model known as dielectric continuum theory[1–3] (DCT) makes predictions for the effect of an interface on the dielectric behavior of water. Using the method of images[4] and correcting for the dielectric saturation of a rigid water model, it is shown that polarization fluctuations near the air-water interface largely follow these predictions at scales as small as 5 ̊A despite the macroscopic character of DCT.

Using tools of statistical mechanics, it is routine to average over the distribution of microscopic configurations to obtain equilibrium free energies. These free energies teach us about the most likely molecular arrangements and the probability of observing deviations from the norm. Frequently, it is necessary to interrogate the probability not just of static arrangements, but of dynamical events, in which case analogous statistical mechanical tools may be applied to study the distribution of molecular trajectories. Numerical study of these trajectory spaces requires algorithms which efficiently sample the possible trajectories. We study in detail one such Monte Carlo algorithm, transition path sampling, and use a non- equilibrium statistical mechanical perspective to illuminate why the algorithm cannot easily be adapted to study some problems involving long-timescale dynamics. Algorithmically generating highly-correlated trajectories, a necessity for transition path sampling, grows exponentially more challenging for longer trajectories unless the dynamics is strongly-guided by the “noise history,” the sequence of random numbers representing the noise terms in the stochastic dynamics. Langevin dynamics of Weeks-Chandler-Andersen (WCA) particles in two dimensions lacks this strong noise guidance, so it is challenging to use transition path sampling to study rare dynamical events in long trajectories of WCA particles. The spin flip dynamics of a two-dimensional Ising model, on the other hand, can be guided by the noise history to achieve efficient path sampling. For systems that can be efficiently sampled with path sampling, we show that it is possible to simultaneously sample both the paths and the (potentially vast) space of non-equilibrium protocols to efficiently learn how rate constants vary with protocols and to identify low-dissipation protocols.

When high-dimensional molecular dynamics can be coarse-grained and represented by a simplified dynamics on a low-dimensional state space, the trajectory space may also be analytically studied using methods of large deviation theory. We review these methods and introduce a simple class of dynamical models whose dynamical fluctuations we compute exactly. The simplest such model is an asymmetric random walker on a one-dimensional ring with a single heterogeneous link connecting two sites of the ring. We derive conditions for the existence of a dynamic phase transition, which separates two dynamical phases—one localized and the other delocalized. The presence of distinct classes trajectories results in profoundly non-Gaussian fluctuations in dynamical quantities. We discuss the implications of such large dynamical fluctuations in the context of simple stochastic models for biological growth.

Systems driven out of equilibrium display a rich variety of patterns and surprising response behaviors. There exist different types of non-equilibrium processes, for instance a system that has been prepared in a non-Boltzmann initial state and is relaxing back to equilibrium, or a system that adopts a non-equilibrium steady state distribution when it is driven by an external field. In these different cases, the main characteristic that distinguishes these systems as non-equilibrium is that they are constantly dissipating heat, or likewise producing entropy. This entropy production is often the starting point for developing a systematic theory to describe such non-equilibrium processes.

Entropy production can be related to the irreversible processes occurring within a system. Particularly strong statements can be made about non-equilibrium systems when a local equilibrium assumption can be made, that is, when smaller subsets of a large system can be considered to be in equilibrium. This turns out to be justified for a wide variety of systems under different conditions. When this holds, the entropy production can be written as a generalized thermodynamic force (often the gradient of some intensive variable of the system) multiplied by a flux. When the thermodynamic force is small, the fluxes can be written as linear combinations of the thermodynamic forces, connected by response coefficients–this is known as linear irreversible thermodynamics. The full extension of equilibrium thermodynamic concepts to dissipative processes beyond this linear regime, including the development of microscopic principles justifying irreversible thermodynamic theories (as equilibrium statistical mechanics justifies equilibrium thermodynamics), is still a work in progress.

In this thesis, we work towards advancing the thermodynamic theory of non-equilibrium phenomena by studying models of driven-diffusive systems, growth processes, and active matter. We use developments from stochastic thermodynamics, large deviation theory, and irreversible thermodynamics to characterize the non-equilibrium phases and properties exhibited by these systems. We question to what extent equilibrium approximations are valid for predicting pattern formation in these systems and whether there exist general unifying features describing these non-equilibriums processes. In the process we develop trajectory sampling methods to investigate the statistics of dynamical order parameters distinguishing these phases. We show how the first and second laws of thermodynamics, including consistent expressions for entropy production, can be extended to active systems, where microscopic reversibility is broken at the level of individual particles. Additionally we derive fluctuation relations, exact analytical results for the fluctuations of entropy production in the form of equalities, for the entropy production in active systems. We also extend the Irving-Kirkwood procedure to active systems, deriving the balance laws of mass, momentum, and energy. Consequently we obtain expressions for the stress and couple stress tensors in the system as functions of the microscopic variables. This provides a foundation to extend the framework of irreversible thermodynamics to active systems.