Metropolis Monte Carlo Simulations Research Articles

Direct computations of viscoelastic moduli of biomolecular condensates.

Biomolecular condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model that leverages information regarding intra- and inter-chain contacts, which we extract from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations of phase separation. The key ingredient of the generalized Rouse model is a graph Laplacian that we compute from equilibrium MMC simulations. We compute two flavors of graph Laplacians, one based on a single-chain graph that accounts only for intra-chain contacts, and the other referred to as a collective graph that accounts for inter-chain interactions. Calculations based on the single-chain graph systematically overestimate the storage and loss moduli, whereas calculations based on the collective graph reproduce the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two graphs proves to be most accurate. In line with the theory of Rouse and contrary to recent assertions, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic vs dominantly viscous behaviors does not imply a single relaxation time. Instead, it is influenced by the totality of the relaxation modes. Hence, our analysis affirms that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain the relaxation time spectra that underlie the dynamics within condensates. This is of practical importance given advancements in passive and active microrheology measurements of condensate viscoelasticity.

Theory and Monte Carlo simulation of the ideal gas with shell particles in the canonical, isothermal-isobaric, grand canonical, and Gibbs ensembles.

Theories of small systems play an important role in the fundamental understanding of finite size effects in statistical mechanics, as well as the validation of molecular simulation results as no computer can simulate fluids in the thermodynamic limit. Previously, a shell particle was included in the isothermal-isobaric ensemble in order to resolve an ambiguity in the resulting partition function. The shell particle removed either redundant volume states or redundant translational degrees of freedom of the system and yielded quantitative differences from traditional simulations in this ensemble. In this work, we investigate the effect of including a shell particle in the canonical, grand canonical, and Gibbs ensembles. For systems comprised of a pure component ideal gas, analytical expressions for various thermodynamic properties are obtained. We also derive the Metropolis Monte Carlo simulation acceptance criteria for these ensembles with shell particles, and the results of the simulations of an ideal gas are in excellent agreement with the theoretical predictions. The system size dependence of various important ensemble averages is also analyzed.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

A simple model for short-range ordering kinetics in multi-principal element alloys

Short-range ordering (SRO) in multi-principal element alloys influences material properties such as strength and corrosion. While some degree of SRO is expected at equilibrium, predicting the kinetics of its formation is challenging. We present a simplified isothermal concentration-wave (CW) model to estimate an effective relaxation time of SRO formation. Estimates from the CW model agree to within a factor of five with relaxation times obtained from kinetic Monte Carlo (kMC) simulations when above the highest ordering instability temperature. The advantage of the CW model is that it only requires mobility and thermodynamic parameters, which are readily obtained from alloy mobility databases and Metropolis Monte Carlo simulations, respectively. The simple parameterization of the CW model and its analytical nature makes it an attractive tool for the design of processing conditions to promote or suppress SRO in multicomponent alloys.

Direct computations of viscoelastic moduli of biomolecular condensates.

In vitro facsimiles of biomolecular condensates are formed by different types of intrinsically disordered proteins including prion-like low complexity domains (PLCDs). PLCD condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model and information regarding intra- and inter-chain contacts that is extracted from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations. The key ingredient of the generalized Rouse model is the Zimm matrix that we compute from equilibrium MMC simulations. We compute two flavors of Zimm matrices, one referred to as the single-chain model that accounts only for intra-chain contacts, and the other referred to as a collective model, that accounts for inter-chain interactions. The single-chain model systematically overestimates the storage and loss moduli, whereas the collective model reproduces the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two models proves to be most accurate. In line with the theory of Rouse, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic versus dominantly viscous behaviors is influenced by the totality of the relaxation modes. Hence, our analysis suggests that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain distributions of relaxation times that underlie the dynamics within condensates.

Highly efficient spin-orbit torque generation in bilayer WTe2/Fe3GaTe2 heterostructure

The spin-orbit torque (SOT) phenomenon is crucial for advancing spintronics. Therefore, we explore the SOT efficiency in the WTe2/Fe3GaTe2 heterostructure. The WTe2/Fe3GaTe2 heterostructure has a ferromagnetic ground state with a perpendicular magnetic anisotropy. Through the Metropolis Monte Carlo simulations, we find that the WTe2/Fe3GaTe2 heterostructure has a Curie temperature of 345 K. We also calculate the temperature-dependent coercive field HC(T) via the extended Landau-Lifshitz-Gilbert (LLG) equation, and the estimated coercive field is 0.32 T at 300 K. From the electrical and spin Hall conductivity of the WTe2 layer, we obtain a giant spin Hall angle of ∼4 is obtained at a low chemical potential. Besides, the WTe2/Fe3GaTe2 heterostructure exhibits a critical current density of ∼7.40 × 104 A/cm2 at 300 K. This value is superior to that found in most previously reported materials. Finally, we achieve the switching power consumption of approximately 1.80 × 1015 W/m3 at 300 K, and this is comparable to or even lower than the values found in other bulk type or thick thickness SOT materials. Due to this highly efficient SOT performance, the ultrathin WTe2/Fe3GaTe2 heterostructure may offer a promising avenue for advanced room temperature spintronic applications with implications for both fundamental research and technological advancements.

Monte Carlo simulation of the ionization and uptake behavior of cationic oligomers into pH-responsive polyelectrolyte microgels of opposite charge - a model for oligopeptide uptake and release.

External stimuli can tune the uptake and release of guest molecules in microgels. Especially their pH responsiveness makes microgels exciting candidates for drug delivery systems. When both microgel and guest molecules are pH-responsive, predicting the electrostatically driven uptake can be complex since the ionization depends on many parameters. In this work, we performed Metropolis Monte Carlo simulations while systematically varying the pK of the monomers, the concentrations of microgel and guest molecules to obtain a better understanding of the uptake of weak cationic oligomers as a model for oligopeptides into a weak anionic polyelectrolyte microgel. Further, we varied the chain length of the oligomers. The polyelectrolyte networks can take up oligomers when both the network and the oligomers are charged. The presence of both species in the system leads to a mutual enhancement of their ionization. The uptake induces a release of counterions and results in complex formation between the oligomers and the network, leading to the collapse of the networks. Longer oligomers enhance the ionization of the network and, therefore, the complexation. A higher microgel concentration increases the uptake only around the isoelectric point but prevents the uptake due to lower entropy gain at counterion release at higher pH. The results give an insight into the uptake of cationic oligomers into oppositely charged polyelectrolyte microgels and provide hints for the design of anionic microgels as carriers for guest molecules e.g. antimicrobial peptides.

Role of vacancies in structural thermalization of binary and high-entropy alloys

Vacancy assisted atomic self-diffusion is a major structural thermalization mechanism in bulk metal alloys. Depending on alloy composition, the local atomic environments might stabilize vacancies to such extent that the vacancies become trapped and the atomic self-diffusion part of the thermalization process stalls. The consequence is that such alloys get kinetically trapped in disordered structures. In this study, we investigate equimolar AgAu, CuPt, AgPdPtIr, and AgAuCuPdPt alloy thermalizing using Metropolis Monte Carlo simulations in two approaches, one where the alloy structure changes through vacancy migration and one where the structure changes by swapping atomic pairs. By comparing the two approaches, we find that the vacancy is less effective at thermalizing alloys with more elements (i.e. AgPdPtIr and AgAuCuPdPt), more heterogeneous configurational internal energy distributions (i.e. CuPt and AgPdPtIr), and strong interactions between certain elements, e.g. Ir-Ir interactions in AgPdPtIr. In the case of AgPdPtIr, the vacancy cannot thermalize Ir-Ir neighbors even when the vacancy is mobile, because the vacancy has difficulty breaking individual Ir-Ir pairs apart.

Phase transitions of ionic fluids in nanoporous electrodes

In this work, we utilise grand canonical Metropolis Monte Carlo simulations, to establish pore-induced freezing of restricted primitive model fluids. A planar pore model is utilised, with walls that are initially neutral, and either non-conducting or perfectly conducting. The phase of the confined electrolyte (solid/fluid) displays an oscillatory dependence on surface separation, in narrow pores. Conditions are chosen so that the bulk is composed of a stable fluid electrolyte. The tendency for the electrolyte to freeze in narrow pores is somewhat stronger in systems with non-conducting walls. We also demonstrate that an applied potential will, above a threshold value, melt a frozen electrolyte. In these cases, the capacitance, as measured by the average surface charge density divided by the applied potential, will be almost vanishing if the applied potential is below this threshold value. We do not see any evidence for a “superionic fluid”, which has been hypothesised to generate a strong capacitance in narrow pores, due to an efficient screening of like-charge repulsions by image charges.Graphic abstract

Machine Learning of Nonequilibrium Phase Transition in an Ising Model on Square Lattice

This paper presents the investigation of convolutional neural network (CNN) prediction successfully recognizing the temperature of the nonequilibrium phase transitions in two-dimensional (2D) Ising spins on a square lattice. The model uses image snapshots of ferromagnetic 2D spin configurations as an input shape to provide the average output predictions. By considering supervised machine learning techniques, we perform Metropolis Monte Carlo (MC) simulations to generate the configurations. In the equilibrium Ising model, the Metropolis algorithm respects detailed balance condition (DBC), while its nonequilibrium version violates DBC. Violating the DBC of the algorithm is characterized by a parameter −8<ε<8. We find the exact result of the transition temperature Tc(ε) in terms of ε. If we set ε=0, the usual single spin-flip algorithm can be restored, and the equilibrium configurations generated with such a set up are used to train our model. For ε≠0, the system attains the nonequilibrium steady states (NESS), and the modified algorithm generates NESS configurations (test dataset). The trained model is successfully tested on the test dataset. Our result shows that CNN can determine Tc(ε≠0) for various ε values, consistent with the exact result.

Fast, Hierarchical, and Adaptive Algorithm for Metropolis Monte Carlo Simulations of Long-Range Interacting Systems

Fast, Hierarchical, and Adaptive Algorithm for Metropolis Monte Carlo Simulations of Long-Range Interacting Systems

Local cation ordering in compositionally complex Ruddlesden–Popper n = 1 oxides

The Ruddlesden–Popper (RP) layered perovskite structure is of great interest due to its inherent tunability, and the emergence and growth of the compositionally complex oxide (CCO) concept endows the RP family with further possibilities. Here, a comprehensive assessment of thermodynamic stabilization, local order/disorder, and lattice distortion was performed in the first two reported examples of lanthanum-deficient Lan+1BnO3n+1 (n = 1, B = Mg, Co, Ni, Cu, Zn) obtained via various processing conditions. Chemical short-range order (CSRO) at the B-site and the controllable excess interstitial oxygen (δ) in RP-CCOs are uncovered by neutron pair distribution function analysis. Reverse Monte Carlo analysis of the data, Metropolis Monte Carlo simulations, and extended x-ray absorption fine structure analysis implies a modest degree of magnetic element segregation on the local scale. Further, ab initio molecular dynamics simulations results obtained from special quasirandom structure disagree with experimentally observed CSRO but confirm Jahn–Teller distortion of CuO6 octahedra. These findings highlight potential opportunities to control local order/disorder and excess interstitial oxygen in layered RP-CCOs and demonstrate a high degree of freedom for tailoring application-specific properties. They also suggest a need for expansion of theoretical and data modeling approaches in order to meet the innate challenges of CCO and related high-entropy phases.

Height-Switching Dynamics of Mixed Polymer Brushes with Polymers of Different Stiffnesses

Mixed brushes consisting of flexible and semiflexible polymers of the same chain length exhibit a height-switching phenomenon because of rigidity-dependent critical adsorption [Yang et al. Macromolecules 2020, 53, 7369]. Semiflexible polymers stand higher at weak surface attraction (high temperature), but they close to the attractive surface at strong attraction (low temperature). In this work, the height-switching dynamics of the mixed polymer brushes is studied by Metropolis Monte Carlo simulation. The height-switching time is calculated by a sudden change in the surface attraction. Two surface attraction change modes, i.e., the weak-to-strong mode where the attraction is changed from weak to strong and the strong-to-weak mode where it is changed from strong to weak, are investigated. Simulation results show that the height-switching time is related to the grafting density, the polymer stiffness, and surface attraction change mode. We find that the height-switching time is significantly decreased for the strong-to-weak mode. And our results also show that the height switching in the mixed polymer brushes is reversible.

BINARY HARD-SPHERE COLLOID-DROPLET MIXTURES WITH THE PYRITE-TYPE STRUCTURE

We investigate the assembly of a binary mixture of patchy colloids and droplets into crystal structures by use of the space-filling principle and Metropolis Monte Carlo simulations. Here, colloids with six patches in an octahedral symmetry attract droplets, whereas the colloid-colloid and droplet-droplet pairwise interactions are purely hard-core repulsions. Within parameter space regions that allow for stable binary crystals, we find the formation of different structure types, which are colloidal analogs of the NaCl phase and pyrite (FeS2) phase, as a function of the droplet-to-colloid sphere diameter ratio. This finding is consistent with theoretical predictions. Notably, while the NaCl structure type is a commonly known structure of hard-sphere colloids and nanoparticles, the formation of the FeS2 structure type from colloidal dispersion has not been explored. Our approach suggests a potential route to obtain colloidal crystals with more complex structures.

Temperature Dependence of Hydrogen Bond Networks of Liquid Water: Thermodynamic Properties and Structural Heterogeneity from Topological Descriptors.

Topological analyses of hydrogen bond networks were performed based on the complex network and island statistics of liquid water at different temperatures. The influence of temperature on the liquid water structures and the topological properties of the hydrogen bond networks was investigated by Metropolis Monte Carlo simulations with the TIP4P/2005 potential model. The bilinear behavior of the second peak in the radial distribution function with the temperature was properly reproduced by these simulations. The average connectivity also displayed a bilinear behavior consistent with being a local descriptor. The semiglobal average path length (or geodesic distance) descriptor showed an unprecedented trimodal distribution, whose areas were dependent on the temperature. Considering equilibrium between these three sets of networks, standard enthalpy and entropy of equilibrium were determined for the first time, providing new insights into the structural heterogeneities of liquid water with interesting perspectives for modeling these quantitative properties of hydrogen bond networks.

Segregation of solute elements and strengthening effects of CoCrNiCux medium-entropy alloys: A combined experimental and simulation study

Segregation of solute elements and strengthening effects of CoCrNiCux medium-entropy alloys (MEAs) was investigated by a combination of experiments and Metropolis Monte Carlo (MC) simulations. Results revealed that the proper addition of Cu effectively improved the tensile strength and yield strength, which was attributed to the dual action of solid solution strengthening and grain refinement caused by the pinning of nano-size Cu-rich precipitates. Micro-voids readily nucleated at the interface between the Cu-lean and Cu-rich phases due to local uneven stress distribution and dislocation pile-ups, thereby promoting the crack initiation and propagation. Based on Metropolis MC simulations, the annealed structures revealed a typical local ordering character, and the short-range order (SRO) parameter revealed the atomic interactions and local chemical ordering of CoCrNiCu1.0 MEA.

Ionisation and swelling behaviour of weak polyampholyte core-shell networks - a Monte Carlo study.

The network charge of polyampholyte microgels can be tuned by varying the pH of the surrounding solution, and a charge reversal from a positively charged microgel at low pH to a negatively charged microgel at high pH can be achieved. In a titration experiment, it is difficult to tell apart the ionisation of the acidic and basic monomers in the network and to determine the distribution of charges in the network, whereas using Metropolis Monte Carlo simulations, both the degree of ionisation and the distribution of ionised monomers can be determined separately for both species. Building on our earlier work on alternating polyampholyte microgels, we now investigated the pH-dependent ionisation and the swelling behaviour of polyampholyte core-shell microgels under good solvent conditions. For this purpose, we performed Metropolis Monte Carlo simulations for a bead-spring model using the constant-pH method. As in our previous study on alternating microgels, the width of the U-shaped curve of the microgels volume as a function of pH depends on the relative dissociation constants of acid and base, and the microgel volume can be approximated by a linear function of the total network charge. Due to the spatial separation of acid and base in core-shell systems, the ionisation is less enhanced compared to a microgel with an alternating distribution of the two species. Nevertheless, we still see an influence of the presence of one species on the ionisation behaviour of the other species under good solvent conditions. Furthermore, the isoelectric point is shifted towards higher pH, which is caused by a higher charge density in the core compared to that in the shell. Added salt changes the Donnan equilibrium, which determines the counterion distribution within and outside of the microgel. At the same time, it contributes to the electrostatic screening of the network charges, leading to a narrowing of the U-shaped volume transition curve.

Critical properties of the Ising ferromagnet VI[formula omitted] monolayer

Motivated by the experimental observation of two-dimensional van der Waals ferromagnets, we investigated the critical properties of Ising ferromagnet VI3 monolayer using the real-space renormalization group method. It is worth noting that a block-spin construction we choose contains an even number of sites. Thereupon, we present a scheme that combines the majority rule and central rule to determine the block-spin values. The critical temperature TC is calculated to be 46.8 K under the first-order cumulant-expansion approximation, which is in excellent agreement with the recent numerical result of TC = 46 K using the Metropolis Monte Carlo simulations based on the classical Heisenberg model, and is slightly lower than the experimental value TC = 50 K of the corresponding bulk VI3. Furthermore, the critical exponents β and ν are estimated, and compared with the mean-field results and the exact solutions, respectively. Our numerical result β≅0.324 suggests that the magnetic phase transition of the VI3 monolayer appears to lie between the two-dimensional Ising universality class (β=0.125) and three-dimensional Heisenberg universality class (β=0.365). The theoretical results in this paper may stimulate further experimental studies of the VI3 monolayer.

On the formation of ordered protein assemblies in cell–cell interfaces

Crystal structures of many cell-cell adhesion receptors reveal the formation of linear "molecular zippers" comprising an ordered one-dimensional array of proteins that form both intercellular (trans) and intracellular (cis) interactions. The clustered protocadherins (cPcdhs) provide an exemplar of this phenomenon and use it as a basis of barcoding of vertebrate neurons. Here, we report both Metropolis and kinetic Monte Carlo simulations of cPcdh zipper formation using simplified models of cPcdhs that nevertheless capture essential features of their three-dimensional structure. The simulations reveal that the formation of long zippers is an implicit feature of cPcdh structure and is driven by their cis and trans interactions that have been quantitatively characterized in previous work. Moreover, in agreement with cryo-electron tomography studies, the zippers are found to organize into two-dimensional arrays even in the absence of attractive interactions between individual zippers. Our results suggest that the formation of ordered two-dimensional arrays of linear zippers of adhesion proteins is a common feature of cell-cell interfaces. From the perspective of simulations, they demonstrate the importance of a realistic depiction of adhesion protein structure and interactions if important biological phenomena are to be properly captured.

Characterization of Zr-Nb-Fe(-Cr) precipitates in Zr-based alloys using density functional theory

The presence of iron (and chromium) impurities in Zr-Nb based alloys leads to the formation of intermetallic Zr-Nb-Fe(-Cr) precipitates. These precipitates are known to affect the thermomechanical properties of Zr-Nb alloys. This study seeks to model these intermetallic precipitates from first principles using density functional theory (DFT) and ab initio Metropolis Monte Carlo (MC). A wide variety of Laves-type atomic structures were considered, composed of Zr, Nb, Fe and Cr in proportions close to experimental values. An exhaustive ensemble of elemental permutations was taken into account. The calculations show that hexagonal ternary alloys of equiatomic composition can be designated as a Zr(Nb, Fe)2 Laves phase. They also show that hexagonal precipitates found in Canadian Deuterium Uranium (CANDU) pressure tube material can be designated as Zr(Zr, Nb, Fe)2 and Zr(Zr, Nb, Fe, Cr)2 Laves phases. Ab initio MC simulations indicate that the lowest-energy Zr(Nb, Fe)2 phases considered in this work are stable under annealing at 1073 K. The DFT-optimized Zr(Zr, Nb, Fe, Cr)2 atomic positions are consistent with the plane d-spacings observed in TEM images.