Semi-grand Canonical Ensemble Research Articles

Quantifying free energies through local composition fluctuations in binary solid solutions: A proof-of-concept study using atomistic embedded-atom simulations

While thermodynamic fluctuation theory has been extensively utilized in liquids for decades to obtain direct thermodynamic information, such as Gibbs free energies, from local composition fluctuations, this study presents an extension of the theory for its application in solids. It is demonstrated that composition fluctuations in solids are influenced by an additional elastic work term, which is specific to solids and encompasses liquids as a special case. To verify the extended fluctuation model, atomistic simulations were conducted on an exemplary Cu–Ni embedded-atom system. Monte Carlo simulations were performed in the semi-grandcanonical ensemble at a fixed temperature, covering the entire composition range. Composition fluctuations within a subvolume were monitored over time and statistically analyzed in terms of the variance. The atomistic results exhibit perfect agreement with the predictions derived from the extended model. This methodology has primarily been developed for the analysis of experimental atom probe tomography data, which provides three-dimensional chemical information with sub-nanometer precision. This technique enables the direct measurement of local composition fluctuations, facilitating the extraction of thermodynamic information from any allowed temperature–composition pair in the phase diagram.

Machine-learning-accelerated simulations to enable automatic surface reconstruction.

Understanding material surfaces and interfaces is vital in applications such as catalysis or electronics. By combining energies from electronic structure with statistical mechanics, ab initio simulations can, in principle, predict the structure of material surfaces as a function of thermodynamic variables. However, accurate energy simulations are prohibitive when coupled to the vast phase space that must be statistically sampled. Here we present a bi-faceted computational loop to predict surface phase diagrams of multicomponent materials that accelerates both the energy scoring and statistical sampling methods. Fast, scalable and data-efficient machine learning interatomic potentials are trained on high-throughput density-functional-theory calculations through closed-loop active learning. Markov chain Monte Carlo sampling in the semigrand canonical ensemble is enabled by using virtual surface sites. The predicted surfaces for GaN(0001), Si(111) and SrTiO3(001) are in agreement with past work and indicate that the proposed strategy can model complex material surfaces and discover previously unreported surface terminations.

A new method for reactive constant pH simulations.

We present a simulation method that allows us to calculate the titration curves for systems undergoing protonation/deprotonation reactions-such as charged colloidal suspensions with acidic/basic surface groups, polyelectrolytes, polyampholytes, and proteins. The new approach allows us to simultaneously obtain titration curves both for systems in contact with salt and acid reservoir (semi-grand canonical ensemble) and for isolated suspensions (canonical ensemble). To treat the electrostatic interactions, we present a new method based on Ewald summation-which accounts for the existence of both Bethe and Donnan potentials within the simulation cell. We show that the Donnan potential dramatically affects the pH of a suspension. Counterintuitively, we find that in suspensions with a large volume fraction of nanoparticles and low ionic strength, the number of deprotonated groups can be 100% larger in an isolated system, compared to a system connected to a reservoir by a semi-permeable membrane-both systems being at exactly the same pH.

Charge fluctuations in charge-regulated systems: dependence on statistical ensemble.

We investigate charge regulation of nanoparticles in concentrated suspensions, focusing on the effect of different statistical ensembles. We find that the choice of ensemble does not affect the mean charge of nanoparticles, but significantly alters the magnitude of its fluctuation. Specifically, we compared the behaviors of colloidal charge fluctuations in the semi-grand canonical and canonical ensembles and identified significant differences between the two. The choice of ensemble-whether the system is isolated or is in contact with a reservoir of acid and salt-will, therefore, affect the Kirkwood-Shumaker fluctuation-induced force inside concentrated suspensions. Our results emphasize the importance of selecting an appropriate ensemble that accurately reflects the experimental conditions when studying fluctuation-induced forces between polyelectrolytes, proteins, and colloidal particles in concentrated suspensions.

Atomistic investigation on radiation-induced segregation in W-Re alloy under cumulative flux irradiation: A hybrid simulation with MD and VC-SGC-MC

Due to its great performance, tungsten (W) and W-based alloys have been considered as the candidates for plasma-faced materials in future fusion reactors. However, the radiation-induced segregation under the reactor environment is one of the key issues to limit the long-term service properties of W-based alloys, and the understanding of nucleation and growth mechanisms of Re-rich precipitates is still incomplete. In the present study, both molecular dynamics (MD) and variance-constrained semi-grand canonical ensemble Monte Carlo (VC-SGC-MC) methods are employed to investigate the nucleation and growth mechanisms of Re clusters precipitated in W-Re alloys under cumulative flux irradiation. It is found that all segregated Re-rich clusters in the model alloys without irradiation are a plate-like structure (χ-phase) which orientates along the <110>, while they change to a needle-like structure (σ-phase) orienting along the <111> direction after irradiation. The combination of defects and Re atoms in W-Re alloys are considered with the most probable nucleation precursors. The Re-rich clusters will pin the interstitials and form stable nucleation sites. During the process of cumulative flux irradiation, newborn defects will be trapped by the nucleation site promoting the growth of Re clusters.

Semigrand-canonical Monte-Carlo simulation methods for charge-decorated cluster expansions

Monte-Carlo sampling of lattice model Hamiltonians is a well-established technique in statistical mechanics for studying the configurational entropy of crystalline materials. When the species to be distributed on the lattice carry charge, the charge balance constraint on the overall system prohibits single-site Metropolis exchanges in MC. In this article, we propose two methods to perform MC sampling in the semigrand-canonical ensemble in the presence of a charge-balance constraint. The table-exchange method (TE) constructs small charge-conserving excitations, and the square-charge bias method (SCB) allows the system to temporarily drift away from charge neutrality. We illustrate the effect of internal hyper-parameters on the efficiency of these algorithms and suggest practical strategies on how to employ these algorithms in real applications.

Fluctuation-based fracture mechanics of heterogeneous materials.

We present results of a hybrid analytical-simulation investigation of the fracture resistance of heterogeneous materials. We show that bond-energy fluctuations sampled by Monte Carlo simulations in the semigrand canonical ensemble provide a means to rationalize the complexity of heterogeneous fracture processes, encompassing probability and percolation theories of fracture. For a number of random and textured model materials, we derive upper and lower bounds of fracture resistance and link bond fracture fluctuations to statistical descriptors of heterogeneity, such as two-point correlation functions, to identify the origin of toughening mechanisms. This includes a shift from short- to long-range interactions of bond fracture processes in random systems to the transition from critical to subcritical bond fracture percolation in textured materials and the activation of toughness reserves at compliant interfaces. Induced by elastic mismatch, they connect to a number of disparate experimental observations, including toughening of brittle solids by deformable polymers or organics in, e.g., gas shale, nacre; stress-induced transformational toughening in ceramics; and toughening of sparse elastic networks in hydrogels, to name a few.

Sampling lattices in semi-grand canonical ensemble with autoregressive machine learning

Calculating thermodynamic potentials and observables efficiently and accurately is key for the application of statistical mechanics simulations to materials science. However, naive Monte Carlo approaches, on which such calculations are often dependent, struggle to scale to complex materials in many state-of-the-art disciplines such as the design of high entropy alloys or multi-component catalysts. To address this issue, we adapt sampling tools built upon machine learning-based generative modeling to the materials space by transforming them into the semi-grand canonical ensemble. Furthermore, we show that the resulting models are transferable across wide ranges of thermodynamic conditions and can be implemented with any internal energy model U, allowing integration into many existing materials workflows. We demonstrate the applicability of this approach to the simulation of benchmark systems (AgPd, CuAu) that exhibit diverse thermodynamic behavior in their phase diagrams. Finally, we discuss remaining challenges in model development and promising research directions for future improvements.

Reactive Monte Carlo simulations for charge regulation of colloidal particles.

We use a reactive Monte Carlo simulation method and the primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different numbers of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium chloride, the titration curves are compared with the experimental measurements obtained using potentiometric titration. A good agreement is found between simulations and experiments. In the case of lithium chloride, the specific ionic adsorption is taken into account through the partial dehydration of lithium ion.

Database-driven semigrand canonical Monte Carlo method: Application to segregation isotherm on defects in alloys.

The application of existing semigrand canonical ensemble Monte Carlo algorithms to alloys requires the chemical potential difference values between pairs of atomic species in the alloys as inputs. However, finding the appropriate values for a target system at a desired temperature and bulk composition is a time-consuming task consisting of multiple test runs to determine the chemical potential differences. This problem becomes more serious when dealing with systems containing three or more atomic species, such as medium- and high-entropy alloys, due to the increase of the number of chemical potential differences that need to be calculated. Here we propose a method for sampling from the semigrand canonical ensemble that relies on energy databases acting as an external atomic reservoir at the desired temperature and composition. Given these energy databases, the desired bulk composition and corresponding chemical potential differences can be satisfied in a "single" Monte Carlo simulation. Moreover, the energy databases shed light on the underlying energetics of alloys, reflecting their local chemical ordering. We demonstrate the validity of this method using analyses of segregation isotherms at grain boundaries and dislocations in two alloy systems: Fe-1-at.-%-Si and NiCoCr medium-entropy alloy. We also discuss the possibly relevant information contained in such energy databases.

Influence of magnetic excitation and vibrational entropy on the phase diagram of Fe1−xCrx alloys

We study the thermodynamic properties of iron-chromium alloys in the framework of the Magnetic Cluster Expansion, a model of effective interactions explicitly taking into account the atomic magnetic moments. Monte Carlo simulations are used to study the magnetic properties and measure the mixing enthalpies of disordered solid solutions. The phase diagram of the Fe-Cr system is established by simulations performed in the canonical and semi-grand canonical ensembles, which require the relaxation of the chemical and magnetic configurations. The effect of magnetism on thermodynamic properties is highlighted via simulations with a fixed spin temperature, in order to impose specific magnetic states. We also propose a simple method to introduce the vibrational entropy in the framework of the Magnetic Cluster Expansion. These simulations allow us to estimate the respective weights of the chemical, magnetic and vibrational contributions and we conclude that they have all a significant effect on the thermodynamics of iron-chromium alloys.

Computer simulations of lipid regulation by molecular semigrand canonical ensembles

The plasma membrane is the interface between cells and exterior media. Although its existence has been known for a long time, organization of its constituent lipids remain a challenge. Recently, we have proposed that lipid populations may be controlled by chemical potentials of different lipid species, resulting in semigrand canonical thermodynamic ensembles. However, the currently available molecular dynamics software packages do not facilitate the control of chemical potentials at the molecular level. Here, we propose a variation of existing algorithms that efficiently characterizes and controls the chemical nature of each lipid. Additionally, we allow coupling with collective variables and show that it can be used to dynamically create asymmetric membranes. This algorithm is openly available as a plugin for the HOOMD-Blue molecular dynamics engine.

Direct evaluation of the phase diagrams of dense multicomponent plasmas by integration of the Clapeyron equations.

Accurate phase diagrams of multicomponent plasmas are required for the modeling of dense stellar plasmas, such as those found in the cores of white dwarf stars and the crusts of neutron stars. Those phase diagrams have been computed using a variety of standard techniques, which suffer from physical and computational limitations. Here we present an efficient and accurate method that overcomes the drawbacks of previously used approaches. In particular, finite-size effects are avoided as each phase is calculated separately; the plasma electrons and volume changes are explicitly taken into account; and arbitrary analytic fits to simulation data as well as particle insertions are avoided. Furthermore, no simulations at "uninteresting" state conditions, i.e., away from the phase coexistence curves, are required, which improves the efficiency of the technique. The method consists of an adaptation of the so-called Gibbs-Duhem integration approach to electron-ion plasmas, where the coexistence curve is determined by direct numerical integration of its underlying Clapeyron equation. The thermodynamics properties of the coexisting phases are evaluated separately using Monte Carlo simulations in the isobaric semigrand canonical ensemble (NPTΔμ). We describe this Monte Carlo-based Clapeyron integration method, including its basic physical and numerical principles, our extension to electron-ion plasmas, and our numerical implementation. We illustrate its applicability and benefits with the calculation of the melting curve of dense carbon-oxygen plasmas under conditions relevant for the cores of white dwarf stars and provide analytic fits to implement this new melting curve in white dwarf models. While this work focuses on the liquid-solid phase boundary of dense two-component plasmas, a wider range of physical systems and phase boundaries are within the scope of the Clapeyron integration method, which had until now only been applied to simple model systems of neutral particles.

A semi-grand canonical kinetic Monte Carlo study of single-walled carbon nanotube growth

Single-walled carbon nanotubes exist in a variety of different geometries, so-called chiralities, which define their electronic properties. Chiral selectivity has been reported in catalytic chemical vapor deposition synthesis experiments, but the underlying mechanisms remain poorly understood. In this contribution, we establish a simple model for the prediction of the growth rates of carbon nanotubes of different chiralities as a function of energies characterizing the carbon nanotube–catalyst interface and of parameters of the synthesis. The model is sampled efficiently using kinetic Monte Carlo simulations in the semi-grand canonical ensemble, uncovering the interplay of the external experimental conditions and the configuration and energetics of the interface with the catalyst. In particular, the distribution of chiral angle dependent growth rates follows non-monotonic trends as a function of interface energies. We analyze this behavior and use it to identify conditions that lead to high selectivity for a variety of chiral angles.

Combining DFT and CALPHAD for the development of on-lattice interaction models: The case of Fe-Ni system

We present a model of pair interactions on rigid lattice to study the thermodynamic properties of iron-nickel alloys. The pair interactions are fitted at 0 K on ab initio calculations of formation enthalpies of ordered and disordered (special quasirandom) structures. They are also systematically fitted on the Gibbs free energy of the $\ensuremath{\gamma}$ Fe-Ni solid solution as described in a CALPHAD (CALculation of PHAse Diagrams) study by Cacciamani et al. This allows the effects of finite temperature, especially those of magnetic transitions, to be accurately described. We show that the ab initio and CALPHAD data for the $\ensuremath{\gamma}$ solid solution and for the $\mathrm{Fe}{\mathrm{Ni}}_{3}\text{\ensuremath{-}}\mathrm{L}{1}_{2}$ ordered phase can be well reproduced, in a large domain of composition and temperature, using first and second neighbor pair interactions which depend on temperature and local alloy composition. The procedure makes it possible to distinguish and separately compare magnetic, chemical, and configuration enthalpies and entropies. We discuss the remaining differences between the pair interaction model and CALPHAD, which are mainly due to the treatment of the short-range order and configurational entropy of the solid solution. The FCC phase diagram of the Fe-Ni system is determined by Monte Carlo simulations in the semigrand canonical ensemble and is compared with experimental studies and other models. We especially discuss the stability of the $\mathrm{Fe}\mathrm{Ni}\text{\ensuremath{-}}\mathrm{L}{1}_{0}$ phase at low temperature.

Analytic description of grain boundary segregation, tension, and formation energy in the copper–nickel system

In this theoretical study, a recently proposed segregation model is applied to segregation data of an exemplary Σ5 grain boundary (GB), which is investigated using a copper–nickel embedded-atom method potential. Segregation in the semi-grandcanonical ensemble is systematically studied by varying the chemical potential in order to explore the full composition range for temperatures from 500 K to 1000 K. As a major thermodynamic feature, the mentioned segregation model avoids the usage of interface compositions, for which an arbitrary volume must be defined, but rather models the thermodynamically unambiguous solute excess. It was shown that the solute excess and the interface formation energy could be described very accurately over a wide range of temperatures and over the entire composition range based on a composition-dependent, but temperature-independent effective energy of segregation. However, the model was initially derived for systems without lattice mismatch so that interface tensions were neglected. Since the copper–nickel system exhibits a moderate lattice mismatch of roughly 2.7%, the copper–nickel system is chosen in this study to further extend the segregation model by a linear-elastic theory to also account for the interface tensions. Using this extended model, it will be shown that the solute excess, GB tensions, and GB formation energies can be derived from the effective energy of segregation for all temperatures and over the whole composition range. As in principle only one single segregation isotherm is sufficient to determine the effective energy of segregation and derived thermodynamic interface quantities, the application of this model is especially interesting for experimental evaluations.

Regulating Lipid Composition Rationalizes Acyl Tail Saturation Homeostasis in Ectotherms

Cell membranes mainly consist of lipid bilayers with an actively regulated composition. The underlying processes are still poorly understood, in particular, how the hundreds of components are controlled. Cholesterol has been found to correlate with phospholipid saturation for reasons that remain unclear. To better understand the link between cell membrane regulation and chemical composition, we establish a computational framework based on chemical reaction networks, resulting in multiple semigrand canonical ensembles. By running computer simulations, we show that regulating the chemical potential of lipid species is sufficient to reproduce the experimentally observed increase in acyl tail saturation with added cholesterol. Our model proposes a different picture of lipid regulation in which components can be regulated passively instead of actively. In this picture, phospholipid acyl tail composition naturally adapts to added molecules such as cholesterol or proteins. A comparison between regulated membranes with commonly studied ternary model membranes shows stark differences: for instance, correlation lengths and viscosities observed are independent of lipid chemical affinity.

Tight-binding Ising modeling of the interplay between bulk ordering and surface segregation in Pt-Ag nanoalloys

We use a Tight-Binding Ising Model (TBIM) with effective pair interactions depending on concentration able to reproduce the bulk phase diagram of Pt-Ag system and in particular its strong asymmetry as a function of composition. This system presents both an L11 ordered structure around equiconcentration and a wide miscibility gap on the Pt-rich side. Using Monte Carlo simulations in canonical and semi-grand canonical ensembles, we succeed in reproducing qualitatively the bulk phase diagram and we study the equilibrium configurations of the (111) and (100) surfaces, and truncated octahedron clusters. The infinite surfaces reproduce sensibly the bulk phase diagram except that they present a strong silver surface segregation limited to the top surface layer. The clusters behave differently due to the reduction of the miscibility gap when decreasing the cluster size and present two typical ordered phases related to the L11 phase: one characterized by pure concentric atomic layers starting from silver surface layer which extends up to the center of the cluster, and the other one at higher silver concentration where the L11 phase is perfectly oriented along one (111) direction of the cluster. The later one has already been reported by recent experiments [1].

Genetic algorithm-guided deep learning of grain boundary diagrams: Addressing the challenge of five degrees of freedom

Grain boundaries (GBs) often control the processing and properties of polycrystalline materials. Here, potentially transformative research is represented by constructing GB property diagrams as functions of temperature and bulk composition, also called “complexion diagrams,” as a general materials science tool on par with phase diagrams. However, a GB has five macroscopic (crystallographic) degrees of freedom (DOFs). It is essentially a “mission impossible” to construct property diagrams for GBs as a function of five DOFs by either experiments or modeling. Herein, we combine isobaric semi-grand canonical ensemble hybrid Monte Carlo and molecular dynamics (hybrid MC/MD) simulations with a genetic algorithm (GA) and deep neural network (DNN) models to tackle this grand challenge. The DNN prediction is ∼108 faster than atomistic simulations, thereby enabling the construction of the property diagrams for millions of distinctly different GBs of five DOFs. Notably, excellent prediction accuracies have been achieved for not only symmetric-tilt and twist GBs, but also asymmetric-tilt and mixed tilt-twist GBs; the latter are more complex and much less understood, but they are ubiquitous and often limit the performance properties of real polycrystals as the weak links. The data-driven prediction of GB properties as function of temperature, bulk composition, and five crystallographic DOFs (i.e., in a 7D space) opens a new paradigm.

Development of a segregation model beyond McLean based on atomistic simulations

In the present atomistic study, layer-specific segregation to the surface is investigated for an exemplary (100) surface in iron–chromium alloys using an embedded-atom potential. Through a continuous variation of the chemical potential difference in the semi-grandcanonical ensemble, the full composition range is explored at temperatures between 600 K and 1400 K. The obtained layer-specific segregation curves demonstrate the well-known limitations of the widely used McLean model for interface segregation, i.e. a monolayer model imposing ideal behavior. However, keeping the original idea of McLean, the segregation model is extended in two ways in order to provide a complete analytical description of segregation: Firstly, the entire surface is hypothetically replaced by an equivalent single layer with identical segregation properties, which accounts for all subsurface layer effects, but also enables the application of an effective monolayer model with only one single energy of segregation. Secondly, directly following from the general treatment assuming non-ideal solution behavior, the energy of segregation becomes composition-dependent. The validity and accuracy of the proposed analytical model is confirmed by fitting the composition-dependent energy of segregation to the thermodynamically unambiguous solute excess. The change in surface formation energy according to the interfacial adsorption equation can be described excellently over the entire composition range for all investigated temperatures. The model can be applied to experimental data and directly transferred to grain boundaries.