Monte Carlo simulations of phase separation in chemically reactive binary mixtures

Interplay of phase segregation and chemical reaction: Crossover and effect on growth laws

By combining the nonconserved spin-flip dynamics driving ferromagnetic ordering with the conserved Kawasaki-exchange dynamics driving phase segregation, we perform Monte Carlo simulations of the nearest-neighbor Ising model. This kind of mixed dynamics is found in a system consisting of a binary mixture of isomers, simultaneously undergoing a segregation and an interconversion reaction among themselves. Here, we study such a system following a quench from the high-temperature homogeneous phase to a temperature below the demixing transition. We monitor the growth of domains of both the winner; the isomer, which survives as the majority; and the loser, the isomer that perishes. Our results show a strong interplay of the two dynamics at early times, leading to a growth of the average domain size of both the winner and loser as ∼t^{1/7}, slower than a purely phase-segregating system. At later times, eventually the dynamics becomes reaction dominated and the winner exhibits a ∼t^{1/2} growth, expected for a system with purely nonconserved dynamics. On the other hand, the loser at first show a faster growth, albeit, slower than the winner, and then starts to decay before it almost vanishes. Further, we estimate the time τ_{s} marking the crossover from the early-time slow growth to the late-time reaction-dominated faster growth. As a function of the reaction probability p_{r}, we observe a power-law scaling τ_{s}∼p_{r}^{-x}, where x≈1.05, irrespective of the temperature. For a fixed value of p_{r} too, τ_{s} appears to be independent of the temperature.

Segregation disrupts the Arrhenius behavior of an isomerization reaction

Coexistence of segregation and interconversion or isomerization reaction among molecular species leads to fascinating structure formation in the biological and chemical worlds. Using Monte Carlo simulations of the prototype Ising model, we explore the chemical kinetics of such a system consisting of a binary mixture of isomers. Our results reveal that even though the two concerned processes are individually Arrhenius in nature, the Arrhenius behavior of the isomerization reaction gets significantly disrupted due to an interplay of the nonconserved dynamics of the reaction and the conserved diffusive dynamics of segregation. The approach used here can be potentially adapted to understand reaction kinetics of more complex reactions.

Collective catalysis under spatial constraints: Phase separation and size-scaling effects on mass action kinetics

Chemical reactions are usually studied under the assumption that both substrates and catalysts are well-mixed (WM) throughout the system. Although this is often applicable to test-tube experimental conditions, it is not realistic in cellular environments, where biomolecules can undergo liquid-liquid phase separation (LLPS) and form condensates, leading to important functional outcomes, including the modulation of catalytic action. Similar processes may also play a role in protocellular systems, like primitive coacervates, or in membrane-assisted prebiotic pathways. Here we explore whether the demixing of catalysts could lead to the formation of microenvironments that influence the kinetics of a linear (multistep) reaction pathway, as compared to a WM system. We implemented a general lattice model to simulate LLPS of a collection of different catalysts and extended it to include diffusion and a sequence of reactions of small substrates. We carried out a quantitative analysis of how the phase separation of the catalysts affects reaction times depending on the affinity between substrates and catalysts, the length of the reaction pathway, the system size, and the degree of homogeneity of the condensate. A key aspect underlying the differences reported between the two scenarios is that the scale invariance observed in the WM system is broken by condensation processes. The main theoretical implications of our results for mean-field chemistry are drawn, extending the mass action kinetics scheme to include substrate initial "hitting times" to reach the catalysts condensate. We finally test this approach by considering open nonlinear conditions, where we successfully predict, through microscopic simulations, that phase separation inhibits chemical oscillatory behavior, providing a possible explanation for the marginal role that this complex dynamic behavior plays in real metabolisms.

Enzyme-Enriched Condensates Show Self-Propulsion, Positioning, and Coexistence

Enzyme-enriched condensates can organize the spatial distribution of their substrates by catalyzing nonequilibrium reactions. Conversely, an inhomogeneous substrate distribution induces enzyme fluxes through substrate-enzyme interactions. We find that condensates move toward the center of a confining domain when this feedback is weak. Above a feedback threshold, they exhibit self-propulsion, leading to oscillatory dynamics. Moreover, catalysis-driven enzyme fluxes can lead to interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division.

A systematic study on immiscible binary systems undergoing thermal/photo reversible chemical reactions

In this work, we systematically study an immiscible binary system undergoing thermal/photo reversible reactions in theory. For the thermal reaction case, no dissipative structures can be formed and only uniform equilibrium states are observed but the dynamical evolution to these trivial states witnesses a new type of sophisticated phase amplification phenomenon-temporary phase separation (TPS). Linear analysis and light-scattering calculations confirm that TPS is predominated either by spinodal decomposition or nucleation and growth mechanism, or by both successively. For the photo reaction case, steady dissipative patterns exist and are maintained by the external energy input of lights. Linear analysis together with simulations reveals that the characteristic wavelength (ξ) of these structures shortens as the input energy density increases and they obey the relation of ln ξ∝ 1/T_b with T_b the effective temperature of lights. The TPS phenomenon and length-scale dependency of dissipative patterns observed in this simple binary system might have rich implications for the non-equilibrium thermodynamics of biological systems.

Topology-driven spatial organization of ring polymers under confinement

Entropic repulsion between DNA ring polymers under confinement is a key mechanism governing the spatial segregation of bacterial DNA before cell division. Here we establish that "internal" loops within a modified-ring polymer architecture enhance entropic repulsion between two overlapping polymers confined in a cylinder. Interestingly, they also induce entropy-driven spatial organization of polymer segments as seen in vivo. Here we design polymers of different architectures in our simulations by introducing a minimal number of cross-links between specific monomers along the ring-polymer contour. The cross-links are likely induced by various bridging proteins inside living cells. We investigate the segregation of two polymers with modified topologies confined in a cylinder, which initially had spatially overlapping configurations. This helps us to identify the architectures that lead to higher success rates of segregation. We also establish the mechanism that leads to localization of specific polymer segments. We use the blob model to provide a theoretical understanding of why certain architectures lead to enhanced entropic repulsive forces between the polymers. Lastly, we establish a correspondence between the organizational patterns of the chromosome of the C.crescentus bacterium and our results for a specifically designed polymer architecture. However, the principles outlined here pertaining to the organization of polymeric segments are applicable to both synthetic and biological polymers.

Polymer architecture orchestrates the segregation and spatial organization of replicating E. coli chromosomes in slow growth

The mechanism of chromosome segregation and organization in the bacterial cell cycle of E. coli is one of the least understood aspects in its life cycle. The E. coli chromosome is often modelled as a bead spring ring polymer. We introduce cross-links in the DNA-ring polymer, resulting in the formation of loops within each replicating bacterial chromosome. We use simulations to show that the chosen polymer-topology ensures its self-organization along the cell long-axis, such that various chromosomal loci get spatially localized as seen in vivo. The localization of loci arises due to entropic repulsion between polymer loops within each daughter DNA confined in a cylinder. The cellular addresses of the loci in our model are in fair agreement with those seen in experiments as given in J. A. Cass et al., Biophys. J., 2016, 110, 2597-2609. We also show that the adoption of such modified polymer architectures by the daughter DNAs leads to an enhanced propensity of their spatial segregation. Secondly, we match other experimentally reported results, including observation of the cohesion time and the ter-transition. Additionally, the contact map generated from our simulations reproduces the macro-domain like organization as seen in the experimentally obtained Hi-C map. Lastly, we have also proposed a plausible reconciliation of the 'Train Track' and the 'Replication Factory' models which provide conflicting descriptions of the spatial organization of the replication forks. Thus, we reconcile observations from complementary experimental techniques probing bacterial chromosome organization.

Phase transitions affected by natural and forceful molecular interconversion

If a binary liquid mixture, composed of two alternative species with equal amounts, is quenched from a high temperature to a low temperature, below the critical point of demixing, then the mixture will phase separate through a process known as spinodal decomposition. However, if the two alternative species are allowed to interconvert, either naturally (e.g., the equilibrium interconversion of enantiomers) or forcefully (e.g., via an external source of energy or matter), then the process of phase separation may drastically change. In this case, depending on the nature of interconversion, two phenomena could be observed: either phase amplification, the growth of one phase at the expense of another stable phase, or microphase separation, the formation of nongrowing (steady-state) microphase domains. In this work, we phenomenologically generalize the Cahn–Hilliard theory of spinodal decomposition to include the molecular interconversion of species and describe the physical properties of systems undergoing either phase amplification or microphase separation. We apply the developed phenomenology to accurately describe the simulation results of three atomistic models that demonstrate phase amplification and/or microphase separation. We also discuss the application of our approach to phase transitions in polyamorphic liquids. Finally, we describe the effects of fluctuations of the order parameter in the critical region on phase amplification and microphase separation.

Interconversion-controlled liquid-liquid phase separation in a molecular chiral model

Liquid-liquid phase separation of fluids exhibiting interconversion between alternative states has been proposed as an underlying mechanism for fluid polyamorphism and may be of relevance to the protein function and intracellular organization. However, molecular-level insight into the interplay between competing forces that can drive or restrict phase separation in interconverting fluids remains elusive. Here, we utilize an off-lattice model of enantiomers with tunable chiral interconversion and interaction properties to elucidate the physics underlying the stabilization and tunability of phase separation in fluids with interconverting states. We show that introducing an imbalance in the intermolecular forces between two enantiomers results in nonequilibrium, arrested phase separation into microdomains. We also find that in the equilibrium case, when all interaction forces are conservative, the growth of the phase domain is restricted only by the system size. In this case, we observe phase amplification, in which one of the two alternative phases grows at the expense of the other. These findings provide novel insights on how the interplay between dynamics and thermodynamics defines the equilibrium and steady-state morphologies of phase transitions in fluids with interconverting molecular or supramolecular states.

Phase separation in fluids with many interacting components

Fluids in natural systems, like the cytoplasm of a cell, often contain thousands of molecular species that are organized into multiple coexisting phases that enable diverse and specific functions. How interactions between numerous molecular species encode for various emergent phases is not well understood. Here, we leverage approaches from random-matrix theory and statistical physics to describe the emergent phase behavior of fluid mixtures with many species whose interactions are drawn randomly from an underlying distribution. Through numerical simulation and stability analyses, we show that these mixtures exhibit staged phase-separation kinetics and are characterized by multiple coexisting phases at steady state with distinct compositions. Random-matrix theory predicts the number of coexisting phases, validated by simulations with diverse component numbers and interaction parameters. Surprisingly, this model predicts an upper bound on the number of phases, derived from dynamical considerations, that is much lower than the limit from the Gibbs phase rule, which is obtained from equilibrium thermodynamic constraints. We design ensembles that encode either linear or nonmonotonic scaling relationships between the number of components and coexisting phases, which we validate through simulation and theory. Finally, inspired by parallels in biological systems, we show that including nonequilibrium turnover of components through chemical reactions can tunably modulate the number of coexisting phases at steady state without changing overall fluid composition. Together, our study provides a model framework that describes the emergent dynamical and steady-state phase behavior of liquid-like mixtures with many interacting constituents.

Controlling composition of coexisting phases via molecular transitions

Phase separation and transitions among different molecular states are ubiquitous in living cells. Such transitions can be governed by local equilibrium thermodynamics or by active processes controlled by biological fuel. It remains largely unexplored how the behavior of phase-separating systems with molecular transitions differs between thermodynamic equilibrium and cases in which the detailed balance of the molecular transition rates is broken because of the presence of fuel. Here, we present a model of a phase-separating ternary mixture in which two components can convert into each other. At thermodynamic equilibrium, we find that molecular transitions can give rise to a lower dissolution temperature and thus reentrant phase behavior. Moreover, we find a discontinuous thermodynamic phase transition in the composition of the droplet phase if both converting molecules attract themselves with similar interaction strength. Breaking the detailed balance of the molecular transition leads to quasi-discontinuous changes in droplet composition by varying the fuel amount for a larger range of intermolecular interactions. Our findings showcase that phase separation with molecular transitions provides a versatile mechanism to control properties of intracellular and synthetic condensates via discontinuous switches in droplet composition.

Lattice Boltzmann simulation for phase separation with chemical reaction controlled by thermal diffusion

This study investigates phase separation behavior and pattern formation in a binary fluid with chemical reaction controlled by thermal diffusion. By incorporating the Arrhenius equation into the lattice Boltzmann method (LBM), the coupling effects of the pre-exponential factor K, viscosity [Formula: see text], and thermal diffusion D on phase separation were successfully evaluated. The effect of the competition between thermal diffusion and concentration on the phase separation morphology and dynamics of binary mixtures under a chemically reacting controlled by slow cooling is assessed based on the extended LBM. The calculations indicated that increases in viscosity and thermal diffusion can obtain interconnected structures (ISs) and lamellar structures (LSs) for cases with small K. However, concentric phase-separated structures (CSs) were observed in cases with large K. The increase in the degree and efficiency of phase separation were significantly greater in cases with decreased viscosity and increased thermal diffusion.

Surface wetting by kinetic control of liquid-liquid phase separation

Motivated by the observations of intracellular phase separations and the wetting of cell membranes by protein droplets, we study the nonequilibrium surface wetting by Monte Carlo simulations of a lattice gas model involving particle creation. We find that, even when complete wetting should occur in equilibrium, the fast creation of particles can hinder the surface wetting for a long time due to the bulk droplet formation. Performing molecular dynamics simulations, we show that this situation also holds in colloidal particle systems when the disorder density is sufficiently high. The results suggest an intracellular control mechanism of surface wetting by changing the speed of component synthesis.

Hierarchical microphase separation in non-conserved active mixtures

Non-equilibrium phase separating systems with reactions, such as biomolecular condensates and bacteria colonies, can break time-reversal symmetry (TRS) in two distinct ways. Firstly, the conservative and non-conservative sectors of the dynamics can be governed by incompatible free energies; when both sectors are present, this is the leading-order TRS violation, captured in its simplest form by 'Model AB'. Second, the diffusive dynamics can break TRS in its own right. This happens only at higher order in the gradient expansion (but is the leading behaviour without reactions present) and is captured by 'Active Model B+' (AMB+). Each of the two mechanisms can lead to microphase separation, by quite different routes. Here we introduce Model AB+, for which both mechanisms are simultaneously present, and show that for slow reaction rates the system can undergo a new type of hierarchical microphase separation, which we call 'bubbly microphase separation'. In this state, small droplets of one fluid are continuously created and absorbed into large droplets, whose length-scales are controlled by the competing reactive and diffusive dynamics.

Effect of configuration-dependent multi-body forces on interconversion kinetics of a chiral tetramer model

We describe a reformulation of the four-site molecular model for chiral phenomena introduced by Latinwo et al. ["Molecular model for chirality phenomena," J. Chem. Phys. 145, 154503 (2016)]. The reformulation includes an additional eight-body force that arises from an explicit configuration-dependent term in the potential energy function, resulting in a coarse-grained energy-conserving force field for molecular dynamics simulations of chirality phenomena. In this model, the coarse-grained interaction energy between two tetramers depends on their respective chiralities and is controlled by a parameter λ, where λ < 0 favors local configurations involving tetramers of opposite chirality and λ > 0 gives energetic preference to configurations involving tetramers of the same chirality. We compute the autocorrelation function for a quantitative chirality metric and demonstrate that the multi-body force modifies the interconversion kinetics such that λ ≠ 0 increases the effective barrier for enantiomer inversion. Our simulations reveal that for λ > 0 and temperatures below a sharply defined threshold value, this effect is dramatic, giving rise to spontaneous chiral symmetry breaking and locking molecules into their chiral identity.

Phase amplification in spinodal decomposition of immiscible fluids with interconversion of species

A fluid composed of two molecular species may undergo phase segregation via spinodal decomposition. However, if the two molecular species can interconvert, e.g., change their chirality, then a phenomenon of phase amplification, which has not been studied so far to our best knowledge, emerges. As a result, eventually, one phase will completely eliminate the other one. We model this phenomenon on an Ising system which relaxes to equilibrium through a hybrid of Kawasaki-diffusion and Glauber-interconversion dynamics. By introducing a probability of Glauber-interconversion dynamics, we show that the particle conservation law is broken, thus resulting in phase amplification. We characterize the speed of phase amplification through scaling laws based on the probability of Glauber dynamics, system size, and distance to the critical temperature of demixing.

Spinodal decomposition of chemically fueled polymer solutions

Out-of-equilibrium phase transitions driven by dissipation of chemical energy are a common mechanism for morphological organization and temporal programming in biology. Inspired by this, dissipative self-assembly utilizes chemical reaction networks (CRNs) that consume high-energy molecules (chemical fuels) to generate transient structures and functionality. While a wide range of chemical fuels and building blocks are now available for chemically fueled systems, so far little attention has been paid to the phase-separation process itself. Herein, we investigate the chemically fueled spinodal decomposition of poly(norbornene dicarboxylic acid) (PNDAc) solution, which is driven by a cyclic chemical reaction network. Our analysis encompasses both the molecular level in terms of the CRN, but also the phase separation process. We investigate the morphology of formed domains, as well as the kinetics and mechanism of domain growth, and develop a kinetic/thermodynamic hybrid model to not only rationalize the dependence of the system on fuel concentration and pH, but also open pathways towards predictive design of future fueled polymer systems.

Surface-directed spinodal decomposition on morphologically patterned substrates

This paper is the second in a two-part exposition on surface-directed spinodal decomposition (SDSD), i.e., the interplay of kinetics of wetting and phase separation at a surface which is wetted by one of the components of a binary mixture. In our first paper [P. Das, P. K. Jaiswal, and S. Puri, Phys. Rev. E 102, 012803 (2020)2470-004510.1103/PhysRevE.102.012803], we studied SDSD on chemically heterogeneous and physically flat substrates. In this paper, we study SDSD on a chemically homogeneous but morphologically patterned substrate. Such substrates arise in a vast variety of technological applications. Our goal is to provide a theoretical understanding of SDSD in this context. We present detailed numerical results for domain growth both inside and above the grooves in the substrate. The morphological evolution can be understood in terms of the interference of SDSD waves originating from the different surfaces comprising the substrate.

Nonequilibrium Theory of Epigenomic Microphase Separation in the Cell Nucleus

Understanding the spatial organization of the genome in the cell nucleus is one of the current grand challenges in biophysics. Certain biochemical-or epigenetic-marks that are deposited along the genome are thought to play an important, yet poorly understood, role in determining genome organization and cell identity. The physical principles underlying the interplay between epigenetic dynamics and genome folding remain elusive. Here we propose and study a theory that assumes a coupling between epigenetic mark and genome densities, and which can be applied at the scale of the whole nucleus. We show that equilibrium models are not compatible with experiments and a qualitative agreement is recovered by accounting for nonequilibrium processes that can stabilize microphase separated epigenomic domains. We finally discuss the potential biophysical origin of these terms.

Pattern formation of skin cancers: Effects of cancer proliferation and hydrodynamic interactions

We study pattern formation of skin cancers by means of numerical simulation of a binary system consisting of cancer and healthy cells. We extend the conventional model H for macrophase separations by considering a logistic growth of cancer cells and also a mechanical friction between dermis and epidermis. Importantly, our model exhibits a microphase separation due to the proliferation of cancer cells. By numerically solving the time evolution equations of the cancer composition and its velocity, we show that the phase separation kinetics strongly depends on the cell proliferation rate as well as on the strength of hydrodynamic interactions. A steady-state diagram of cancer patterns is established in terms of these two dynamical parameters and some of the patterns correspond to clinically observed cancer patterns. Furthermore, we examine in detail the time evolution of the average composition of cancer cells and the characteristic length of the microstructures. Our results demonstrate that different sequence of cancer patterns can be obtained by changing the proliferation rate and/or hydrodynamic interactions.

Physics of active emulsions

Phase separating systems that are maintained away from thermodynamic equilibrium via molecular processes represent a class of active systems, which we call active emulsions. These systems are driven by external energy input, for example provided by an external fuel reservoir. The external energy input gives rise to novel phenomena that are not present in passive systems. For instance, concentration gradients can spatially organise emulsions and cause novel droplet size distributions. Another example are active droplets that are subject to chemical reactions such that their nucleation and size can be controlled, and they can divide spontaneously. In this review, we discuss the physics of phase separation and emulsions and show how the concepts that govern such phenomena can be extended to capture the physics of active emulsions. This physics is relevant to the spatial organisation of the biochemistry in living cells, for the development of novel applications in chemical engineering and models for the origin of life.

Field-Embedded Particles Driven by Active Flips

Systems of independent active particles embedded into a fluctuating environment are relevant to many areas of soft-matter science. We use a minimal model of noninteracting spin-carrying Brownian particles in a Gaussian field and show that activity-driven spin dynamics leads to patterned order. We find that the competition between mediated interactions and active noise alone can yield such diverse behaviors as phase transitions and microphase separation, from lamellar up to hexagonal ordering of clusters of opposite magnetization. These rest on complex multibody interactions. We find regimes of stationary patterns, but also dynamical regimes of relentless birth and growth of lumps of magnetization opposite of the surrounding one. Our approach combines Monte Carlo simulations with analytical methods based on dynamical density functional approaches.

Physical principles of intracellular organization via active and passive phase transitions

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.

Structuring polymer gels via catalytic reactions

We use computer simulations to investigate how a catalytic reaction in a polymer sol can induce the formation of a polymer gel. To this aim we consider a solution of homopolymers in which freely-diffusing catalysts convert the originally repulsive A monomers into attractive B ones. We find that at low temperatures this reaction transforms the polymer solution into a physical gel that has a remarkably regular mesostructure in the form of a cluster phase, absent in the usual homopolymer gels obtained by a quench in temperature. We investigate how this microstructuring depends on catalyst concentration, temperature, and polymer density and show that the dynamics for its formation can be understood in a semi-quantitative manner using the interaction potentials between the particles as input. The structuring of the copolymers and the AB sequences resulting from the reactions can be discussed in the context of the phase behaviour of correlated random copolymers. The location of the spinodal line as found in our simulations is consistent with analytical predictions. Finally, we show that the observed structuring depends not only on the chemical distribution of the A and B monomers but also on the mode of formation of this distribution.

Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies

We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to reinitiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes almost periodic such that after each redispersion event the next colony forms in a random location. These results, which hint at some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.

Efficient Application of Continuous Fractional Component Monte Carlo in the Reaction Ensemble

A new formulation of the Reaction Ensemble Monte Carlo technique (RxMC) combined with the Continuous Fractional Component Monte Carlo method is presented. This method is denoted by serial Rx/CFC. The key ingredient is that fractional molecules of either reactants or reaction products are present and that chemical reactions always involve fractional molecules. Serial Rx/CFC has the following advantages compared to other approaches: (1) One directly obtains chemical potentials of all reactants and reaction products. Obtained chemical potentials can be used directly as an independent check to ensure that chemical equilibrium is achieved. (2) Independent biasing is applied to the fractional molecules of reactants and reaction products. Therefore, the efficiency of the algorithm is significantly increased, compared to the other approaches. (3) Changes in the maximum scaling parameter of intermolecular interactions can be chosen differently for reactants and reaction products. (4) The number of fractional molecules is reduced. As a proof of principle, our method is tested for Lennard-Jones systems at various pressures and for various chemical reactions. Excellent agreement was found both for average densities and equilibrium mixture compositions computed using serial Rx/CFC, RxMC/CFCMC previously introduced by Rosch and Maginn (Journal of Chemical Theory and Computation, 2011, 7, 269-279), and the conventional RxMC approach. The serial Rx/CFC approach is also tested for the reaction of ammonia synthesis at various temperatures and pressures. Excellent agreement was found between results obtained from serial Rx/CFC, experimental results from literature, and thermodynamic modeling using the Peng-Robinson equation of state. The efficiency of reaction trial moves is improved by a factor of 2 to 3 (depending on the system) compared to the RxMC/CFCMC formulation by Rosch and Maginn.

Phase transition of a symmetric diblock copolymer induced by nanorods with different surface chemistry

We investigate the phase transition of a symmetric diblock copolymer induced by nanorods with different surface chemistry. The results demonstrate that the system occurs the phase transition from a disordered structure to ordered parallel lamellae and then to the tilted layered structure as the number of rods increases. The dynamic evolution of the domain size and the order parameter of the microstructure are also examined. Furthermore, the influence of rod property, rod-phase interaction, rod-rod interaction, rod length, and polymerization degree on the behavior of the polymer system is also investigated systematically. Moreover, longer amphiphilic nanorods tend to make the polymer system form the hexagonal structure. It transforms into a perpendicular lamellar structure as the polymerization degree increases. Our simulations provide an efficient method for determining how to obtain the ordered structure on the nanometer scales and design the functional materials with optical, electronic, and magnetic properties.

Patterning in systems driven by nonlocal external forces

This work focuses on systems displaying domain patterns resulting from competing external and internal dynamics. To this end, we introduce a Lyapunov functional capable of describing the steady states of systems subject to external forces, by adding nonlocal terms to the Landau Ginzburg free energy of the system. Thereby, we extend the existing methodology treating long-range order interactions, to the case of external nonlocal forces. By studying the quadratic term of this Lyapunov functional, we compute the phase diagram in the temperature versus external field and we determine all possible modulated phases (domain patterns) as a function of the external forces and the temperature. Finally, we investigate patterning in chemical reactive mixtures and binary mixtures under irradiation, and we show that the last case opens the path toward micro-structural engineering of materials.

Spinodal decomposition of chemically reactive binary mixtures

We simulate the influence of a reversible isomerization reaction on the phase segregation process occurring after spinodal decomposition of a deeply quenched regular binary mixture, restricting attention to systems wherein material transport occurs solely by diffusion. Our theoretical approach follows a diffuse-interface model of partially miscible binary mixtures wherein the coupling between reaction and diffusion is addressed within the frame of nonequilibrium thermodynamics, leading to a linear dependence of the reaction rate on the chemical affinity. Ultimately, the rate for an elementary reaction depends on the local part of the chemical potential difference since reaction is an inherently local phenomenon. Based on two-dimensional simulation results, we express the competition between segregation and reaction as a function of the Damköhler number. For a phase-separating mixture with components having different physical properties, a skewed phase diagram leads, at large times, to a system converging to a single-phase equilibrium state, corresponding to the absolute minimum of the Gibbs free energy. This conclusion continues to hold for the critical phase separation of an ideally perfectly symmetric binary mixture, where the choice of final equilibrium state at large times depends on the initial mean concentration being slightly larger or less than the critical concentration.

Molecular dynamics study of phase separation in fluids with chemical reactions

We present results from the first d=3 molecular dynamics (MD) study of phase-separating fluid mixtures (AB) with simple chemical reactions (A⇌B). We focus on the case where the rates of forward and backward reactions are equal. The chemical reactions compete with segregation, and the coarsening system settles into a steady-state mesoscale morphology. However, hydrodynamic effects destroy the lamellar morphology which characterizes the diffusive case. This has important consequences for the phase-separating structure, which we study in detail. In particular, the equilibrium length scale (ℓ(eq)) in the steady state suggests a power-law dependence on the reaction rate ε:ℓ(eq)∼ε(-θ) with θ≃1.0.

Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard equation

We investigate microphase separation patterns on curved surfaces in three-dimensional space by numerically solving a nonlocal Cahn-Hilliard equation for diblock copolymers. In our model, a curved surface is implicitly represented as the zero level set of a signed distance function. We employ a discrete narrow band grid that neighbors the curved surface. Using the closest point method, we apply a pseudo-Neumann boundary at the boundary of the computational domain. The boundary treatment allows us to replace the Laplace-Beltrami operator by the standard Laplacian operator. In particular, we can apply standard finite difference schemes in order to approximate the nonlocal Cahn-Hilliard equation in the discrete narrow band domain. We employ a type of unconditionally stable scheme, which was introduced by Eyre, and use the Jacobi iterative to solve the resulting implicit discrete system of equations. In addition, we use the minimum number of grid points for the discrete narrow band domain. Therefore, the algorithm is simple and fast. Numerous computational experiments are provided to study microphase separation patterns for diblock copolymers on curved surfaces in three-dimensional space.

Suppression of Ostwald ripening in active emulsions

Emulsions consisting of droplets immersed in a fluid are typically unstable since they coarsen over time. One important coarsening process is Ostwald ripening, which is driven by the surface tension of the droplets. Stability of emulsions is relevant not only in complex fluids but also in biological cells, which contain liquidlike compartments, e.g., germ granules, Cajal bodies, and centrosomes. Such cellular systems are driven away from equilibrium, e.g., by chemical reactions, and thus can be called active emulsions. In this paper, we study such active emulsions by developing a coarse-grained description of the droplet dynamics, which we analyze for two different chemical reaction schemes. We first consider the simple case of first-order reactions, which leads to stable, monodisperse emulsions in which Ostwald ripening is suppressed within a range of chemical reaction rates. We then consider autocatalytic droplets, which catalyze the production of their own droplet material. Spontaneous nucleation of autocatalytic droplets is strongly suppressed and their emulsions are typically unstable. We show that autocatalytic droplets can be nucleated reliably and their emulsions stabilized by the help of chemically active cores, which catalyze the production of droplet material. In summary, different reaction schemes and catalytic cores can be used to stabilize emulsions and to control their properties.

Disrupted coarsening in complex Cahn-Hilliard dynamics

Predicting the pattern formation in a system maintained far from equilibrium is a complex task. For a given dynamics governed by the evolution of a conservative order parameter, recent investigations have demonstrated that the knowledge of the long time expression of the order parameter is sufficient to predict the existence of disrupted coarsening, i.e., the pinning of the inhomogeneities wavelength to a well defined value. However, there exists some dynamics for which the asymptotic form of the order parameter remains unknown. The Cahn-Hilliard-like equation used to describe the stability of solids under irradiation belongs to this class of equations. In this paper, we present an alternative to predict the patterning induced by this equation. Based on a simple ansatz, we calculated the form factor and proved that a disrupted coarsening takes place in such dynamics. This disrupted coarsening results from the bifurcation of the implicit equation linking the characteristic length of the dynamics (k_{m}^{∞})^{-1} to a control parameter describing the irradiation. This analysis is supported by direct simulations. From this paper, it clearly appears that the bifurcation of k_{m}^{∞} is a criterion for disrupted coarsening.

Entropy-mediated patterning of surfactant-coated nanoparticles and surfaces

We perform atomistic and mesoscale simulations to explain the origin of experimentally observed stripelike patterns formed by immiscible ligands coadsorbed on the surfaces of gold and silver nanoparticles. We show that when the conformational entropy gained via this morphology is sufficient, microphase-separated stripelike patterns form. When the entropic gain is not sufficient, we instead predict bulk phase-separated Janus particles. We also show corroborating experimental results that confirm our simulational predictions that stripes form on flat surfaces as well as on curved nanoparticle surfaces.

Modeling self-assembly and phase behavior in complex mixtures

Using a variety of computational techniques, I investigate how the self-assembly of complex mixtures can be guided by surfaces or external stimuli to form spatially regular or temporally periodic patterns. Focusing on mixtures in confined geometries, I examine how thermodynamic and hydrodynamic effects can be exploited to create regular arrays of nanowires or monodisperse, particle-filled droplets. I also show that an applied light source and chemical reaction can be harnessed to create hierarchically ordered patterns in ternary, phase-separating mixtures. Finally, I consider the combined effects of confining walls and a chemical reaction to demonstrate that a swollen polymer gel can be driven to form dynamically periodic structures. In addition to illustrating the effectiveness of external factors in directing the self-organization of multicomponent mixtures, the selected examples illustrate how coarse-grained models can be used to capture both the equilibrium phase behavior and the dynamics of these complex systems.

Dynamics of ternary mixtures with photosensitive chemical reactions: creating three-dimensionally ordered blends

Using computer simulations, we establish an approach for creating defect-free, periodically ordered polymeric materials. The system involves ABC ternary mixtures where the A and B components undergo a reversible photochemical reaction. In addition, all three components are mutually immiscible and undergo phase separation. Through the simulations, we model the effects of illuminating a three-dimensional (3D) sample with spatially and temporally dependent light irradiation. Experimentally, this situation can be achieved by utilizing both a uniform background light and a spatially localized, higher intensity light, and then rastering a higher-intensity light over the 3D sample. We first focus on the case where the higher-intensity light is held stationary and focused in a distinct region within the system. The C component is seen to displace the A and B within this region and replicate the pattern formed by the higher-intensity light. In effect, one can write a pattern of C onto the AB binary system by focusing the higher-intensity light in the desired arrangement. We isolate the conditions that are necessary for producing clearly written patterns of C (i.e., for obtaining sharp interfaces between the C and A/B domains). We next consider the effect of rastering a higher-intensity light over this sample and find that this light "combs out" defects in the AB blend as it moves through the system. The resulting material displays a defect-free structure that encompasses both a periodic ordering of the A and B domains and a well-defined motif of C. In this manner, one can create hierarchically patterned materials that exhibit periodicity over two distinct length scales. The approach is fully reversible, noninvasive, and points to a novel means of patterning with homopolymers, which normally do not self-assemble into periodic structures.

Exploiting photoinduced reactions in polymer blends to create hierarchically ordered, defect-free materials

Computer simulations reveal how photoinduced chemical reactions can be exploited to create long-range order in binary and ternary polymeric materials. The process is initiated by shining a spatially uniform light over a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and phase separation. We then introduce a well-collimated, higher intensity light source. Rastering this secondary light over the sample locally increases the reaction rate and causes formation of defect-free, spatially periodic structures. These binary structures resemble either the lamellar or hexagonal phases of microphase-separated diblock copolymers. We measure the regularity of the ordered structures as a function of the relative reaction rates for different values of the rastering speed and determine the optimal conditions for creating defect-free structures in the binary systems. We then add a nonreactive homopolymer C, which is immiscible with both A and B. We show that this component migrates to regions that are illuminated by the secondary, higher intensity light, allowing us to effectively write a pattern of C onto the AB film. Rastering over the ternary blend with this collimated light now leads to hierarchically ordered patterns of A, B, and C. The findings point to a facile, nonintrusive process for manufacturing high quality polymeric devices in a low-cost, efficient manner.

Orientational order transition of the striped microphase structure of a copolymer-homopolymer mixture under oscillatory particles

Based on the three-order-parameter model, we investigate the orientational order transition of striped patterns in microphase structures of diblock copolymer-homopolymer mixtures in the presence of periodic oscillatory particles. Under suitable conditions, although the macrophase separation of a system is almost isotropic, the microphase separation of the system will be significantly perturbed by the oscillatory field, and composition fluctuations are suppressed anisotropically. The isotropy of the microphase will be broken up. By changing the oscillatory amplitude and frequency, we observe the orientational order transition of a striped microphase structure from the isotropic state to a state parallel to the oscillatory direction, and from the parallel state to a state perpendicular to the oscillatory direction. We examine, in detail, the microstructure and orientational order parameter as well as the domain size in the process of orientational order transition under the oscillatory field. We study also how the microphase structure changes with the composition ratio of homopolymers and copolymers in mixtures. The results suggest that our model system may provide a simple way to realize orientational order transition of soft materials.

Effect of hydrodynamic interactions on the evolution of chemically reactive ternary mixtures

We investigate the structural evolution of an A/B/C ternary mixture in which the A and B components can undergo a reversible chemical reaction to form C. We developed a lattice Boltzmann model for this ternary mixture that allows us to capture both the reaction kinetics and the hydrodynamic interactions within the system. We use this model to study a specific reactive mixture in which C acts as a surfactant, i.e., the formation of C at the A/B interface decreases the interfacial tension between the A and B domains. We found that the dynamics of the system is different for fluids in the diffusive and viscous regimes. In the diffusive regime, the formation of a layer of C at the interface leads to a freezing of the structural evolution in the fluid; the values of the reaction rate constants determine the characteristic domain size in the system. In the viscous regime, where hydrodynamic interactions are important, interfacial reactions cause a slowing down of the domain growth, but do not arrest the evolution of the mixture. The results provide guidelines for controlling the morphology of this complex ternary fluid.