Adiabatic Simulations Research Articles

Nonadiabatic Excited-State Molecular Dynamics with an Explicit Solvent: NEXMD-SANDER Implementation.

In this article, the nonadiabatic excited-state Molecular dynamics (NEXMD) package is linked with the SANDER package, provided by AMBERTOOLS. The combination of these software packages enables the simulation of photoinduced dynamics of large multichromophoric conjugated molecules involving several coupled electronic excited states embedded in an explicit solvent by using the quantum/mechanics/molecular mechanics (QM/MM) methodology. The fewest switches surface hopping algorithm, as implemented in NEXMD, is used to account for quantum transitions among the adiabatic excited-state simulations of the photoexcitation and subsequent nonadiabatic electronic transitions, and vibrational energy relaxation of a substituted polyphenylenevinylene oligomer (PPV3-NO2) in vacuum and methanol as an explicit solvent has been used as a test case. The impact of including specific solvent molecules in the QM region is also analyzed. Our NEXMD-SANDER QM/MM implementation provides a useful computational tool to simulate qualitatively solvent-dependent effects, like electron transfer, stabilization of charge-separated excited states, and the role of solvent reorganization in the molecular optical properties, observed in solution-based spectroscopic experiments.

Molecular-Level Investigation of Binary Fluid Droplets Impacting Surfaces

Droplets impacting with surfaces are commonly encountered processes in the field of protective coatings. The behavior of a colliding binary liquid droplet is sensitive to the impact velocity, surface wetting properties, and the droplet composition. Modeling molecular dynamics and classical density functional theory studies of impacting droplets as well as interfacial-surface free energies was reported on. The presence of two components in the liquid drop makes the surface collision a complicated problem. During the collision the kinetic energy of the drop is converted into heat. Thus, the temperature varies during the collision and throughout the droplet. Two extreme situations were captured by performing both adiabatic and isothermal simulations. Molecular dynamics and classical density functional theory were used to explore the effects of the mixing parameter on the phase diagram of the binary AB mixed droplets. The location of liquid–vapor and liquid–liquid phase separation was determined. In addition, the value of the interfacial tensions of all interfaces was computed. These can be used to predict when an A-rich and B-rich droplet will stay attached and when it will detach.

Nonadiabatic Hydrogen Tunneling Dynamics for Multiple Proton Transfer Processes with Generalized Nuclear-Electronic Orbital Multistate Density Functional Theory.

Proton transfer and hydrogen tunneling play key roles in many processes of chemical and biological importance. The generalized nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed in order to capture hydrogen tunneling effects in systems involving the transfer and tunneling of one or more protons. The generalized NEO-MSDFT method treats the transferring protons quantum mechanically on the same level as the electrons and obtains the delocalized vibronic states associated with hydrogen tunneling by mixing localized NEO-DFT states in a nonorthogonal configuration interaction scheme. Herein, we present the derivation and implementation of analytical gradients for the generalized NEO-MSDFT vibronic state energies and the nonadiabatic coupling vectors between these vibronic states. We use this methodology to perform adiabatic and nonadiabatic dynamics simulations of the double proton transfer reactions in the formic acid dimer and the heterodimer of formamidine and formic acid. The generalized NEO-MSDFT method is shown to capture the strongly coupled synchronous or asynchronous tunneling of the two protons in these processes. Inclusion of vibronically nonadiabatic effects is found to significantly impact the double proton transfer dynamics. This work lays the foundation for a variety of nonadiabatic dynamics simulations of multiple proton transfer systems, such as proton relays and hydrogen-bonding networks.

Accelerated convergence via adiabatic sampling for adsorption and desorption processes.

Under isothermal conditions, phase transitions occur through a nucleation event when conditions are sufficiently close to coexistence. The formation of a nucleus of the new phase requires the system to overcome a free energy barrier of formation, whose height rapidly rises as supersaturation decreases. This phenomenon occurs both in the bulk and under confinement and leads to a very slow kinetics for the transition, ultimately resulting in hysteresis, where the system can remain in a metastable state for a long time. This has broad implications, for instance, when using simulations to predict phase diagrams or screen porous materials for gas storage applications. Here, we leverage simulations in an adiabatic statistical ensemble, known as adiabatic grand-isochoric ensemble (μ, V, L) ensemble, to reach equilibrium states with a greater efficiency than its isothermal counterpart, i.e., simulations in the grand-canonical ensemble. For the bulk, we show that at low supersaturation, isothermal simulations converge slowly, while adiabatic simulations exhibit a fast convergence over a wide range of supersaturation. We then focus on adsorption and desorption processes in nanoporous materials, assess the reliability of (μ, V, L) simulations on the adsorption of argon in IRMOF-1, and demonstrate the efficiency of adiabatic simulations to predict efficiently the equilibrium loading during the adsorption and desorption of argon in MCM-41, a system that exhibits significant hysteresis. We provide quantitative measures of the increased rate of convergence when using adiabatic simulations. Adiabatic simulations explore a wide temperature range, leading to a more efficient exploration of the configuration space.

CatchyCFDEM: Open-source microkinetic modeling of heterogeneous catalysis in Eulerian-Lagrangian CFD applied to oxidative coupling of methane

This work introduces catchyCFDEM, an OpenFOAM-based open-source CFD-DEM framework designed for simulating reactive gas–solid fluidized beds with both gas-phase chemistry and heterogeneous catalytic chemistry. A convective gas–solid heat transfer model is implemented and validated using experimental data gathered in a non-reactive pseudo-2D fluidized bed. Catalytic chemistry including different levels of complexity are included: pseudo-homogeneous, heterogeneous without mass transfer, and heterogeneous with diffusive mass transfer. Verification studies are performed by simulating a packed bed reactor using catchyCFDEM and comparing mass fraction profiles to a 1D plug flow simulation in Cantera. To enhance the computational efficiency of the resource-intensive reactive CFD-DEM simulations, speed-up algorithms such as Particle Agglomeration (PA) and in-situ adaptive tabulation (ISAT) are integrated in catchyCFDEM. As a case study, a gas–solid vortex reactor (GSVR) is simulated using the validated framework. The applicability of the GSVR for the oxidative coupling of methane (OCM) is evaluated based on an isothermal analysis evaluating the influence of several operational and geometrical parameters on the process. Finally, a proof-of-concept adiabatic simulation of the GSVR is presented. The catchyCFDEM framework is made publicly available on GitHub.

Aerosol mixing state, new particle formation, and cloud droplet number concentration in an urban environment

Anthropogenically derived aerosols have been hypothesized to influence convective precipitation by increasing the available pool of cloud condensation nuclei. Here, we present a synthesis of aerosol size distribution and subsaturated hygroscopicity measurements between 15 and 250 nm diameter particles during the TRacking Aerosol Convection interactions ExpeRiment (TRACER). We found that the aerosol is externally mixed and can be described by a quasi-two-component description comprising a more and less hygroscopic mode. The mean hygroscopicity parameters for these modes across all sizes were 0.03 ± 0.04 and 0.22 ± 0.08 with no significant dependence on particle size. The number fraction of the more hygroscopic mode is 40 % for particles between 15 and 40 nm and gradually increases to ~70 % for particles >100 nm. Winds from the southerly direction feature particles with larger hygroscopicity parameters and have a larger fraction of particles in the more hygroscopic mode. The hygroscopicity parameter exhibits diurnal cycles that are consistent with condensation of a species with a hygroscopicity parameter ~0.1 which corresponds to values expected for secondary organic aerosol. We also identified nine small particle events that were attributed to particle formation by nucleation. The data are consistent with new particle formation having occurred aloft, followed by downward mixing with daytime turbulence. The species that are responsible for modal growth had hygroscopicity parameters varying between 0.05 and 0.34. These values systematically depended on the wind sector, suggesting that the chemical composition of the precursors differed. Hourly cloud condensation nuclei (CCN) and cloud droplet number concentration (CDNC) values derived from the aerosol size distribution, subsaturated hygroscopicity measurements, and adiabatic parcel model simulations showed a dynamic range of a factor of 2–3 in CDNC depending on the wind sector, with lower values associated with southerly onshore flow.

Study on the impact of heat loss on premixed hydrogen/air deflagration in closed pipelines

This study developed a one-step chemical reaction model for hydrogen combustion and investigated the effect of heat loss on premixed hydrogen/air deflagration in closed pipelines using numerical simulations with the CFD code GASFLOW-MPI. The simulation results were compared and verified with experimental data. The numerical simulations accurately predicted the flame front structure, pressure, and flame tip position during the deflagration of premixed hydrogen/air. Furthermore, a comparison of adiabatic simulation and heat transfer simulation results revealed that large-scale vortices within the turbulent flow were closely linked to heat loss during the tulip flame stage. Simultaneously, heat loss reduced pressure and flame tip position by diminishing the flame wrinkling factor caused by thermodiffusive instability. Finally, convective heat transfer identified as the most critical factor contributing to heat loss during the deflagration of premixed hydrogen/air in closed pipelines. Specifically, during the tulip flame stage, heat loss was mainly due to convective heat transfer (73.8%∼76.3%), with radiation playing a minor role (23.7%∼26.2%). Therefore, accurate prediction of characteristic parameters during the deflagration of premixed hydrogen/air requires numerical simulations that account for both convective and radiative heat transfer.

Isotope effect suggests site‐specific nonadiabaticity on Ge(111)c(2×8)

AbstractEnergy transferred in atom‐surface collisions typically depends strongly on projectile mass, an effect that can be experimentally detected by isotopic substitution. In this work, we present measurements of inelastic H and D atom scattering from a semiconducting Ge(111)c(2×8) surface exhibiting two scattering channels. The first channel shows the expected isotope effect and is quantitatively reproduced by electronically adiabatic molecular dynamics simulations. The second channel involves electronic excitations of the solid and, surprisingly, exhibits almost no isotope effect. We attribute these observations to scattering dynamics, wherein the likelihood of electronic excitation varies with the impact site engaged in the interaction.Key Points Previous work revealed that H atoms with sufficient translational energy can excite electrons over the band gap of a semiconductor in a surface collision. We studied the isotope effect of the energy transfer by H/D substitution and performed band structure calculations to elucidate the underlying excitation mechanism. Our results suggest a site‐specific mechanism that requires the atom to hit a specific surface site to excite an electron‐hole pair.

Better together: the complex interplay between radiative cooling and magnetic draping

ABSTRACT Rapidly outflowing cold H i gas is ubiquitously observed to be cospatial with a hot phase in galactic winds, yet the ablation time of cold gas by the hot phase should be much shorter than the acceleration time. Previous work showed efficient radiative cooling enables clouds to survive in hot galactic winds under certain conditions, as can magnetic fields even in purely adiabatic simulations for sufficiently small density contrasts between the wind and cloud. In this work, we study the interplay between radiative cooling and magnetic draping via three dimensional radiative magnetohydrodynamic simulations with perpendicular ambient fields and tangled internal cloud fields. We find magnetic fields decrease the critical cloud radius for survival by two orders of magnitude (i.e. to sub-pc scales) in the strongly magnetized (βwind = 1) case. Our results show magnetic fields (i) accelerate cloud entrainment through magnetic draping, (ii) can cause faster cloud destruction in cases of inefficient radiative cooling, (iii) do not significantly suppress mass growth for efficiently cooling clouds, and, crucially, in combination with radiative cooling (iv) reduce the average overdensity by providing non-thermal pressure support of the cold gas. This substantially reduces the acceleration time compared to the destruction time (more than due to draping alone), enhancing cloud survival. Our results may help to explain the cold tiny rapidly outflowing cold gas observed in galactic winds and the subsequent high covering fraction of cold material in galactic haloes.

Buoyancy response of a disc to an embedded planet: a cross-code comparison at high resolution

ABSTRACT In radiatively inefficient, laminar protoplanetary discs, embedded planets can excite a buoyancy response as gas gets deflected vertically near the planet. This results in vertical oscillations that drive a vortensity growth in the planet’s corotating region, speeding up inward migration in the type-I regime. We present a comparison between pluto/idefix and fargo3D using 3D, inviscid, adiabatic numerical simulations of planet–disc interaction that feature the buoyancy response of the disc, and show that pluto/idefix struggle to resolve higher-order modes of the buoyancy-related oscillations, weakening vortensity growth, and the associated torque. We interpret this as a drawback of total-energy-conserving finite-volume schemes. Our results indicate that a very high resolution or high-order scheme is required in shock-capturing codes in order to adequately capture this effect.

Computational study of the adamantane cation: simulations of spectroscopy, fragmentation dynamics, and internal conversion

Diamondoids, of which adamantane (C_{10}H_{16}) is the simplest representative, constitute an intriguing class of carbon based nanomaterials with interesting chemical, mechanical and opto-electronic properties. While neutral diamondoids have been extensively studied for decades, their cationic counterparts were a subject of recent experimental investigations motivated by their potential role in astrochemistry. Here, we perform a computational study of the adamantane cation (C_{10}H_{16}^+) complementing the recent experimental findings. Specifically, we extend earlier theoretical work on vibrationally resolved electronic spectroscopy by accounting for the higher lying electronically excited states of the cation in the absorption and photoelectron spectra. We also perform adiabatic and nonadiabatic (surface hopping) molecular dynamics simulations to study (fast) fragmentation processes and electronic relaxation of C_{10}H_{16}^+. Our simulations reveal that after excitation with near-infrared–ultraviolet photons, the adamantane cation undergoes an ultrafast internal conversion to the ground (doublet) state (on a time scale of 10–100 fs depending on initial excitation energy) which can be followed by a fast fragmentation, predominantly H loss. The yield of the ultrafast hydrogen dissociation as a function of excitation energy is reported.

Water harvesting system in greenhouses with liquid desiccant technology

The study aimed to investigate the feasibility of utilizing atmospheric water generation systems (AWG) in the agricultural sector by implementing a liquid desiccant-based moisture harvesting system. The system was modeled using an adiabatic simulation approach to evaluate the performance of the system in terms of moisture removal rate during the dehumidification process. Additionally, an economic analysis was conducted to compare the costs and payback period of three different water extraction scenarios for the agricultural sector in Jordan. The scenarios considered the Levelized Cost of Water (LCOW) and payback period over a 20-year life cycle. The results of the study showed that the average annual water extracted during the tomato cultivation period was 2,198 m3 and 1,451 m3 for the strawberry cultivation period. The payback period for non-subsidized water selling prices was between 6 and 12 years and between 3 and 4 years if water was purchased from artesian wells. The LCOW for the different scenarios ranged from 1.92 USD/m3 to 2.91 USD/m3, depending on the type of crop and the desiccant solution used. Overall, the study demonstrated that the implementation of an AWG system in the agricultural sector can be a viable option with a reasonable payback period and cost-competitive LCOW.

Quasi-spiral solution to the mixed intracluster medium and the universal entropy profile of galaxy clusters

ABSTRACT Well-resolved galaxy clusters often show a large-scale quasi-spiral structure in deprojected density ρ and temperature T fields, delineated by a tangential discontinuity known as a cold front, superimposed on a universal radial entropy profile with a linear K(r) ∝ Tρ−2/3 ∝ r adiabat. We show that a spiral structure provides a natural quasi-stationary solution for the mixed intracluster medium (ICM), introducing a modest pressure spiral that confines the locally buoyant or heavy plasma phases. The solution persists in the presence of uniform or differential rotation, and can accommodate both an inflow and an outflow. Hydrodynamic adiabatic simulations with perturbations that deposit angular momentum and mix the plasma thus asymptote to a self-similar spiral structure. We find similar spirals in Eulerian and Lagrangian simulations of 2D and 3D, merger and offset, clusters. The discontinuity surface is given in spherical coordinates {r, θ, ϕ} by ϕ(r, θ) ∝ Φ(r) , where Φ is the gravitational potential, combining a trailing spiral in the equatorial (θ = π/2) plane and semicircles perpendicular to the plane, in resemblance of a snail shell. A local convective instability can develop between spiral windings, driving a modified global instability in sublinear K(r) regions; evolved spirals thus imprint the observed K ∝ r onto the ICM even after they dissipate. The spiral structure brings hot and cold phases to close proximity, suggesting that the observed fast outflows could sustain the structure even in the presence of radiative cooling.

Excited-state molecular dynamics simulation based on variational quantum algorithms

We propose an excited-state molecular dynamics simulation method based on variational quantum algorithms at a computational cost comparable to that of ground-state simulations. We utilize the feature that excited-states can be obtained as metastable states in the restricted variational quantum eigensolver calculation with a hardware-efficient ansatz. To demonstrate the effectiveness of the method, molecular dynamics simulations are performed for the S1 excited-states of H2 and CH2NH molecules. The results are consistent with those of the exact adiabatic simulations in the S1 states, except for the CH2NH system, after crossing the conical intersection, where the proposed method causes a nonadiabatic transition.

Probing many-body effects in harmonic traps with twisted light

We explore the potential of twisted light, a structured beam carrying orbital angular momentum, as a tool to unveil many-body effects in parabolically confined systems. According to the generalized Kohn theorem, the dipole response of such a multiparticle system to a spatially homogeneous probe is indistinguishable from the response of a system of noninteracting particles. Twisted light however can excite internal degrees of freedom, resulting in the appearance of new peaks in the multipole spectrum which are not present when the probe is a plane wave. We also demonstrate the ability of the proposed twisted light probe to capture the transition of interacting fermions into a strongly correlated regime in a one-dimensional harmonic trap. We report that, by suitable choice of the probe's parameters, the transition into a strongly correlated phase manifests itself as an approaching and ultimate superposition of peaks in the second-order quadrupole response. These features are observed in exact calculations for two electrons and well reproduced in adiabatic time-dependent density-functional theory simulations.

The effect of hydrogen addition on methane/air explosion characteristics in a 20-L spherical device

The addition of hydrogen to methane changes its deflagration characteristics and increases the combustion rate. However, studies on the effect of hydrogen on methane deflagration remain insufficient. Therefore, based on the CFD code GASFLOW-MPI, a four-step combustion-mechanism model was established for methane/hydrogen mixtures. The deflagration characteristics of a premixed combustible gas in a 20-L spherical device was numerically simulated using a methane/hydrogen/air equivalence ratio of 1 and hydrogen addition in the range of 0–50%; subsequently. The results were compared with experimental data. The four-step methane/hydrogen combustion-mechanism could effectively reproduce the methane/hydrogen deflagration process on considering the heat losses. With an increase in hydrogen addition, the laminar burning velocity increases, and the deflagration duration reduces. It decreases the explosion heat loss and increased the maximum deflagration pressure. Under adiabatic simulation, the maximum deflagration pressure decreased with an increase in hydrogen addition, in contrast with the experimental results. This indicates that the heat-loss effect of the methane/hydrogen/air-mixture deflagration process should not be ignored. Moreover, the heat loss during the methane/hydrogen/air-mixture deflagration was mainly caused by thermal radiation. Thus, the influence of the thermal-radiation and convective heat-transfer mechanisms should be considered in the numerical simulations of methane/hydrogen/air-mixture deflagration.

A data-driven optimization method for coarse-graining gene regulatory networks

One major challenge in systems biology is to understand how various genes in a gene regulatory network (GRN) collectively perform their functions and control network dynamics. This task becomes extremely hard to tackle in the case of large networks with hundreds of genes and edges, many of which have redundant regulatory roles and functions. The existing methods for model reduction usually require the detailed mathematical description of dynamical systems and their corresponding kinetic parameters, which are often not available. Here, we present a data-driven method for coarse-graining large GRNs, named SacoGraci, using ensemble-based mathematical modeling, dimensionality reduction, and gene circuit optimization by Markov Chain Monte Carlo methods. SacoGraci requires network topology as the only input and is robust against errors in GRNs. We benchmark and demonstrate its usage with synthetic, literature-based, and bioinformatics-derived GRNs. We hope SacoGraci will enhance our ability to model the gene regulation of complex biological systems.

Decoding the mechanical conductance switching behaviors of dipyridyl molecular junctions.

Dipyridyl molecular junctions often show intriguing conductance switching behaviors with mechanical modulations, but the mechanisms are still not completely revealed. By applying the ab initio-based adiabatic simulation method, the configuration evolution and electron transport properties of dipyridyl molecular junctions in stretching and compressing processes are systematically investigated. The numerical results reveal that the dipyridyl molecular junctions tend to form specific contact configurations during formation processes. In small electrode gaps, the pyridyls almost vertically adsorb on the second Au layers of the tip electrodes by pushing the top Au atoms aside. These specific contact configurations result in stronger molecule-electrode couplings and larger electronic incident cross-sectional areas, which consequently lead to large breaking forces and high conductance. On further elongating the molecular junctions, the pyridyls shift to the top Au atoms of the tip electrodes. The additional scattering of the top Au atoms dramatically decreases the conductance and switches the molecular junctions to the lower conductive states. Perfect cyclical conductance switches are obtained as observed in the experiments by repeatedly stretching and compressing the molecular junctions. The O atom in the side-group tends to hinder the pyridyl from adsorbing on the second Au layer and further inhibits the conductance switch of the dipyridyl molecular junction.

Theoretical study on mechanical evolution process of interface between gold electrode and pyridyl anchor group

Pyridyl-ended molecular junctions show high and low breaking forces successively in formation process and at the same time exhibit intriguing conductance switching behaviors. To understand the forming process of pyridyl-ended molecular junctions, the interaction between 4,4′-bipyridine molecule and gold electrode is studied by the <i>ab initio</i>-based adiabatic simulation method. The processes that the molecule moves away from electrode tip with different contact configurations are simulated, and the molecule-electrode interface evolutions, energy of the molecule-electrode system and the force between molecules and electrode are calculated in the simulations. The numerical results show that during the molecule moving away from the pyramid-shaped electrode, the pyridyl is easy to vertically adsorb on the second gold layer of the electrode tip. In this contact configurations, the tip Au atom deviates from the original position due to the lateral pushing force of the pyridyl. It needs about 1.3–1.5 nN stretching force for the pyridyl breaking from the second gold layer and switching to the tip Au atom, which is evidently larger than the force of 0.8–1.0 nN for the molecule breaking from the tip Au atom. This result is well consistent with the experimental observations, which thus reveals the relationship between the interface structures and the stretching force in the formation process of bipyridyl molecular junction in the experiment. The interaction between 4,4′-bipyridine molecule and plane-shaped gold electrode is very weak. It needs no more than 0.5 nN for the molecule breaking from the plane-shaped gold electrode. However, when the molecule adsorbs on the single Au atom which is adsorbed on the surface of plane-shaped electrode, the molecule can sustain 1.7 nN stretching force. Our study shows that the <i>ab initio</i>-based adiabatic stretching simulation method can not only reveal the geometric evolution process of molecule-electrode systems, but also identify the specific contact configurations between molecule and electrode.

Phase-field modeling of coupled spall and adiabatic shear banding and simulation of complex cracks in ductile metals

In this work, a unified phase field theory for two coupled fracture types is proposed. Compared with the existing unified phase field theory for single fracture type, it considers the coupling behavior between different fracture types and is consistent with the single fracture type theory when one type of fracture is suppressed. Based on a new form of energy decomposition, that is, the strain energy is decomposed into deviatoric, tensile volumetric and compressive volumetric parts, and specifying the driving energy for each fracture type, a double phase-field model for coupled spall and adiabatic shear banding, which are two kinds of typical dynamic ductile fracture, is realized with the proposed unified phase field theory for two coupled fracture types. This coupled model can be used to predict the potential spall and adiabatic shear banding failures in ductile metals under dynamic loading, revealing the corresponding fracture types under different loadings without any additional criteria. The entire damage and fracture evolution (that is, from damage evolution, through crack expansion, and to fragmentation) are captured with this unified phase-field modeling framework. In addition, as an advantage of the double phase-field model, the complex multicrack distributions of spall and adiabatic shear bands in ductile metals can be presented with the proposed model. Based on these features, the expanding characteristics of complex cracks in metallic expanding shells are numerically studied. It is found that, if multilayer spall occurs, the specimen tends to fracture into three layers, that is, a complete spall layer, fragmentations in multilayer areas, and the main body, which is consistent with the experimental observations under certain conditions and is not the regular multiple layers predicted by the multilayer spall theory with ideal material assumption. For multiple adiabatic shear bands (ASBs) in the shell, the spiral patterns of ASBs’ distributions, the initial positions, and the propagating path are clarified for the collapsed and expanding shells. The collapsed shell tends to form a single-direction spiral pattern whereas the expanding shell tends to form a double-direction spiral pattern. In addition to the inner surface, the intersection points of two ASBs can initiate cracks, and, as the intersection points have the chance of being distributed on the outer surface, the expanding shell has a greater potential to initiate a crack on the outer surface than the collapsed shell. The propagating path will be deflected after interactions of two ASBs, and the quantitative relationship between the deflection angle and the stress state before the interactions is given, which explains why the deflection angle is significantly larger in the collapsed shell compared with that in the expanding shell. Based on the above results and analysis, four typical stages of fractures in the expanding shell under internal explosive loading are identified.