Binary Collision Approximation Research Articles

Probing radiation resistance in simulated metallic core–shell nanoparticles

We present molecular dynamics (MD) simulations of radiation damage in Fe nanoparticles (NP) and bimetallic FeCu core–shell nanoparticles (CSNP). The CSNP includes a perfect body-centered cubic (bcc) Fe core coated with a face-centered cubic (fcc) Cu shell. Irradiation with Fe Primary Knock-on Atoms (PKA) with energies between 1 and 7 keV leads to point defects, without clustering beyond divacancies and very few slightly larger vacancy clusters, and without interstitial clusters, unlike what happens in bulk at the same PKA energies. The Fe-Cu interface and shell can act as a defect sink, absorbing radiation-induced damage and, therefore, the final number of defects in the Fe core is significantly lower than in the Fe NP. In addition, the Cu shell substantially diminishes the number of sputtered Fe atoms, acting as a barrier for recoil ejection. Structurally, the Cu shell responds to the stress generated by the collision cascade by creating and destroying stacking faults across the shell width, which could also accommodate further irradiation defects. We compare our MD results to Monte Carlo Binary Collision Approximation (BCA) simulations using the SRIM code, for the irradiation of an amorphous 3-layer thin film with a thickness equal to the CSNP diameter. BCA does not include defect recombination, so the number of Frenkel pairs is significantly higher than in MD, as expected. Sputtering yield (Y) is underestimated by BCA, which is also expected since the simulation is for a thin film at normal incidence. We also compare MD defect production to bulk predictions of the analytic Athermal Recombination Corrected Displacements Per Atom (arc-dpa) model. The number of vacancies in the Fe core is only slightly lower than arc-dpa predictions, but the number of interstitials is reduced by about one order of magnitude compared to vacancies, at 5 keV. According to the radiation resistance found for FeCu CSNP in our simulations, this class of nanomaterial could be suitable for developing new radiation-resistant coatings, nanostructured components, and shields for use in extreme environments, for instance, in nuclear energy and astrophysical applications.

Al Implantation in Sic; Where Will the Ions Come to Rest?

In all implantations into crystalline targets, quite a few ions find a path along a crystal channel or plane, so called channeling, and these ions travel deep into the crystal. This paper treats aluminum (Al) implantation in 4H-SiC and show how the crystal lattice will guide incoming ions deep into the target and modify the final dopant distribution. 4H-SiC samples have been implanted with 100 keV Al-ions, in a “random” direction using the wafer miscut angle of 4°, as well as with the impact beam aligned anti-parallel to the [0001] direction. Aluminium concentration versus depth profiles has been recorded by secondary ion mass spectrometry (SIMS). To track the most probable ion paths during stopping process, SIIMPL, a Monte Carlo simulation code based on the binary collision approximation (MC-BCA) has been used. In addition, the remaining ion energy has been extracted from SIIMPL at various depth along the ion path. Our results show that, independent of the used impact angle, some ions will be steered by crystal planes predominantly into the direction and also along the six directions. The energy loss is smaller along these low index axes. Therefore, at a depth of 1.2 μm, some Al ions along a path may still have kinetic energy, more than 40% of the original 100 keV, and continues to move deep into the SiC sample. The mean projected range of 100 keV ions in 4H-SiC is about 120 nm.

Tritium-titanium target degradation due to deuterium irradiation for DT neutron production

In the present article, we have investigated tritium removal from tritium-titanium targets during fusion neutron production and the impact of tritium degradation on neutron production. The removal of tritium from the target is predicted for deuterium ion irradiation with the SDTrimSp code. We adopt the binary collision approximation method to simulate the recoils and projectile trajectories and concentration of constituents in the target. We have modelled four phenomena in our simulations; ion exchange, sputtering, outgassing of tritium, and thermal diffusion of hydrogen isotopes in the target caused by deuterium irradiation. Insignificant contributors such as burn-up of tritium in neutron production and loss of tritium due to radioactive decay are not included in our model. This tritium removal results in the nonuniform distribution of tritium in the target. A Python-based script is developed to investigate the effects of tritium removal on neutron production with these pristine and irradiated targets. This script uses the layered composition of the constituents’ atoms, DT reaction cross-section, and stopping power of deuterium ions in the target. The script is validated with the NeuSdesc code for the pristine target. Using the layered composition of tritium atoms in the target obtained from the SDTrimSp simulations, the script predicts the degradation in neutron production for different irradiation scenarios.

New Compound and Hybrid Binding Energy Sputter Model for Modeling Purposes in Agreement with Experimental Data

Rocky planets and moons experiencing solar wind sputtering are continuously supplying their enveloping exosphere with ejected neutral atoms. To understand the quantity and properties of the ejecta, well-established binary collision approximation Monte Carlo codes like TRIM with default settings are used predominantly. Improved models such as SDTrimSP have come forward, and together with new experimental data, the underlying assumptions have been challenged. We introduce a hybrid model, combining the previous surface binding approach with a new bulk binding model akin to Hofsäss & Stegmaier. In addition, we expand the model implementation by distinguishing between free and bound components sourced from mineral compounds such as oxides or sulfides. The use of oxides and sulfides also enables the correct setting of the mass densities of minerals, which was previously limited to the manual setting of individual atomic densities of elements. All of the energies and densities used are thereby based on tabulated data, so that only minimal user input and no fitting of parameters are required. We found unprecedented agreement between the newly implemented hybrid model and previously published sputter yields for incidence angles up to 45° from surface normal. Good agreement is found for the angular distribution of mass sputtered from enstatite MgSiO3 compared to the latest experimental data. Energy distributions recreate trends of experimental data of oxidized metals. Similar trends are to be expected from future mineral experimental data. The model thus serves its purpose of widespread applicability and ease of use for modelers of rocky body exospheres.

Numerical framework for multi-scale modeling planar DC magnetron sputtering

In this study, we propose a numerical framework for multi-scale modeling planar Direct Current (DC) magnetron sputtering for a single alloy target. In the model, we account for electromagnetics, fluid dynamics, discharge chemistry and thermal transport across multiple length scales. Specifically, we employ Lattice Boltzmann Method (LBM) for transport phenomena, Finite Elements Method (FEM) for magnetron discharge transport, and binary collision approximation Monte-Carlo method for sputtering and atomic deposition. Our modeling framework could predict the composition, uniformity, and deposition flux of deposited thin film alloys. Furthermore, the simulation also provides the information of plasma in the chamber such as electron, ion densities, argon ion energy and argon ion flux bombarding on the target. Through parametric analyses, we found that substrate film composition is scaled with both chamber pressure and inverse chamber temperature, although the effect of chamber temperature is considerably marginal under low pressure conditions. We also found that magnetic fields do not influence the film composition, but it significantly affects the substrate deposition flux and film uniformity. Our work provides useful insights to operational heuristics involved in magnetron sputtering and other physical chemical processes.

Establishing a Best Practice for SDTrimSP Simulations of Solar Wind Ion Sputtering

Solar wind (SW) ion irradiation on airless bodies can play an important role in altering their surface properties and surrounding exosphere. Much of the ion sputtering data needed for exosphere studies come from binary collision approximation sputtering models such as TRansport of Ions in Matter and its more recent extension, SDTrimSP. These models predict the yield and energy distribution of sputtered atoms, along with the depth of deposition and damage of the substrate, all as a function of the incoming ion type, impact energy, and impact angle. Within SDTrimSP there are several user-specific inputs that have been applied differently in previous SW ion sputtering simulations. These parameters can influence the simulated behavior of both the target and sputtered atoms. Here, we have conducted a sensitivity study into the SDTrimSP parameters in order to determine a best practice for simulating SW ion impacts onto planetary surfaces. We demonstrate that ion sputtering behavior is highly sensitive to several important input parameters including the ion impact angle and energy distribution and the ejected atom surface binding energy. Furthermore, different parameters can still result in similarities in the total sputtering yield, potentially masking large differences in other sputtering-induced behaviors such as the elemental yield, surface concentration, and damage production. Therefore, it is important to consider more than just the overall sputtering behavior when quantifying the importance of different parameters. This study serves to establish a more consistent methodology for simulations of SW-induced ion sputtering on bodies such as Mercury and the Moon, allowing for more accurate comparisons between studies.

Ion Irradiation Effects on Two-Dimensional MXene Ti2C for Applications in Extreme Conditions: Combined Ab Initio and Monte Carlo Simulations

As an emerging category of two-dimensional (2D) materials, transition metal carbides and/or nitrides (MXenes) have attracted considerable interest for various disciplines. They exhibit unique properties typical of both metals and ceramics, rendering significant potential for harsh conditions such as in outer space and nuclear systems. In this work, using Ti2C as a prototype probe, we investigate the ion irradiation effects by combining ab initio and Monte Carlo simulations. We first consider the structures and energetics of various atomic vacancies in Ti2C. Based on binary collision approximation, a Monte Carlo approach is developed to explore the defects induced by ion bombardment. Systematic Monte Carlo (MC) simulations of different incident ions (i.e., H, He, Li, Be, B, C, N, O, F, and Ne) and selected isotopes reveal that defects are formed via primary collision and in-plane recoils, and the number of displaced atoms decreases with incident energy and increases with the atomic number of incident ions. Interestingly, Ti atoms are much easier to be knocked out than C atoms, beneficial for selective etching for nanoelectronics. We show that Ti2C remains metallic with vacancies of large size, indicating the high irradiation tolerance of electrical conductivity. This work not only provides fundamental insights into the irradiation response of MXenes but also paves the way for future exploration and implementation of 2D nanostructures in extreme conditions, e.g., as coatings or structural materials for nuclear reactors.

Modeling experimental low energy ion scattering multi-angle maps with molecular dynamics FAN

Multi-angle maps obtained with low energy ion scattering (LEIS) are not only a convenient way to visualize the atomic-level surface structure of single crystals, but also provide a comprehensive data set that is useful for quantitative fitting with models. However, modeling these maps is often difficult due to the complexity of the scattering process, as well as the number of simulations needed to accurately reproduce a single experimental map. For example, traditional molecular dynamics (MD) and binary collision approximation (BCA) simulations typically require on the order of one billion simulated ion trajectories to obtain sufficient statistics. This problem is exacerbated when modeling impact-collision ion scattering spectroscopy (ICISS) maps, as backscattering cross sections are much smaller than forward-scattering cross sections. A workaround to this problem was given by Neihus and Spitzl in the form of FAN (Niehus and Spitzl, 1991), a simulation that approximates the angular dependence of the backscattering signal by only simulating ion trajectories that reach and emanate from lattice atoms, using a BCA code. This greatly reduces the number of trajectories that must be calculated, though it can oversimplify some of the more complex collision sequences. In this work, we develop a fully three-dimensional FAN code that uses MD to simulate trajectories rather than the BCA. Our approach exploits the improved speed of FAN, while retaining the more accurate ion trajectory calculations of MD. We apply the MD-FAN simulation to model LEIS measurements on two surfaces. First, we benchmark our MD-FAN simulation in a comparison to a BCA simulation for 3 keV He+ incident on W(111) that has been previously reported in Wong et al. (2020). The MD-FAN map had stronger agreement with the experimental ICISS measurements than did the BCA map, despite containing nearly 100-fold fewer ion trajectories. Second, to obtain experimental data sets for a more complex surface, we perform experimental ICISS measurements using a 3 keV He+ beam on a magnetite single crystal, Fe3O4(001) to generate a multi-angle map. The simulated MD-FAN map for the Fe3O4(001) surface once again agreed well with the experimental map, allowing for quantitative comparisons. The MD-FAN simulation also provided insight into the scattering processes, revealing that some of the observed pattern arises from 3 keV He+ that have backscattered into the detector from Fe atoms more than 2 nm below the surface.

Investigation of stepped InP(001) surface by the method of low energy ion scattering

5 keV - energy Ne+ ions are scattered from stepped InP(001)<110> surfaces in grazing incidence. The study of a stepped surface using the ion scattering method is of great interest, since ions are scattered at large angles. For the stepped surface calculated the simple trajectories channeled ions using binary collision approximation method. Calculated energy and angle of ions, which exits from this surface channel. It indicates that at the energy distribution observe intensively peaks, which corresponded to the ions leaved surface channels formed, stepped atoms.

HPIC2: A hardware-accelerated, hybrid particle-in-cell code for dynamic plasma-material interactions

The exascale era of high performance computing promises to bring the field of computational plasma physics ever closer to the goal of accurate multiscale modeling. Such computers will rely on hardware acceleration to offload work to dedicated components, notably general-purpose graphics processing units (GPUs). However, devices from different manufacturers require software to be written with different parallel programming models, greatly increasing the code maintenance burden of applications designed to perform on more than one such device. hPIC2 is a hybrid plasma simulation code developed with the Kokkos performance portability framework to target the architectures that will drive exascale computing for the foreseeable future. As a hybrid simulation code, hPIC2 investigates the simultaneous use of various plasma models on the same domain, at the same time. hPIC2 also optionally couples to RustBCA, which accurately models ion-material interactions using the binary collision approximation (BCA) method [1]. hPIC2 therefore achieves scalable performance on a variety of computing architectures when simulating complex and diverse plasmas, particularly near plasma-material interfaces.

SAFARI – A low and hyperthermal energy ion scattering simulation

SAFARI [1][2] is a classical molecular dynamics simulation designed to assist with analysis of low and hyperthermal energy ion spectra (1 eV - 1 keV) for ion-surface scattering experiments. SAFARI is optimized for this specific energy range to reduce the required runtime, and to produce more precise spectra. This paper discusses the algorithms used in SAFARI, as well as various optimizations implemented for the specific ranges involved. Program summaryProgram Title: SAFARICPC Library link to program files:https://doi.org/10.17632/t5kxgnmjwv.1Developer's repository link:https://github.com/SINS-Lab/SEA-SAFARICode Ocean capsule:https://codeocean.com/capsule/3787536Licensing provisions: MITProgramming language: C++Nature of problem: SAFARI models low and hyperthermal energy (1 eV - 1 keV) ion scattering spectra. This energy range requires molecular dynamics techniques to adequately reproduce experimental spectra, as the energies are too low for appropriate use of the sequential binary collision approximation. Several approximations and optimisations are used to reduce the runtime of the simulation, to allow generating larger statistical datasets.Solution method: SAFARI utilises a predictor-corrector integrator, and classical Hamiltonian mechanics to model the trajectory of the ions. Ion-lattice forces are computed by a combination of image parameters, optional frictional losses, and binary pair potentials. The pair potentials are only computed for nearby lattice sites to reduce the computational complexity of the simulation.

Simulation of the evolution of rough surfaces by sputtering using the binary collision approximation

The evolution of triangular surface structures with different sizes, the roughening of initially smooth surfaces, and the smoothing of initially rough surfaces were studied by computer simulations for sputtering by 5 keV argon on copper at normal incidence as a function of incident fluence. The program SDTrimSP-2D was used in binary collision approximation, diffusion was neglected. As already predicted by Sigmund [J. Mater. Sci. 1973, 8, 1545], a triangular ridge is relatively stable under bombardment at low fluences. However, at fluences above a critical fluence, it eventually flattens out and finally disappears. Initially smooth surfaces get rougher during bombardment, the roughness increases as the square root of incident fluence. This effect is more pronounced for larger metaparticle sizes. Initially rough surfaces get smoother by sputtering, the final smoothness depends on the initial degree of roughness. The smoothing process follows a characteristic pattern with smaller-scale roughness disappearing first and larger-scale roughness remaining stable until higher fluences. This effect gets visible in the Fourier-spectrum of the rough surface as damping of higher frequencies.

Comparative study regarding the sputtering yield of nanocolumnar tungsten surfaces under Ar+ irradiation

Nanostructured tungsten has been proposed as a promising option for plasma facing materials in future fusion reactors, because compared to conventional tungsten it shows advantages such as a better radiation resistance and, in particular, a retardation of tungsten-fuzz growth. Besides these aspects, the sputtering yield of nanostructured tungsten under ion bombardment is of interest, since it would affect the atomic density of tungsten emitted into the fusion plasma, which leads to radiative heat losses. In this work, we present a multiscale approach for investigating the sputtering yield of nanocolumnar tungsten surfaces under 1 keV and 2 keV ${\mathrm{Ar}}^{+}$ irradiation. Our results cover experiments and also computational simulations, which operate either on the basis of the binary collision approximation and ray tracing or use a full molecular dynamics implementation. In our studied case, both computational approaches can predict the sputtering yield of nanocolumnar tungsten surfaces very well. In comparison to flat W, we observe a much reduced dependence on the ion incidence angle, similar as reported for conventional rough surfaces in literature. However, an additional global reduction of the sputtering yield was identified, which can be attributed to geometrical redeposition effects between the separated nanocolumns. These results support the applicability of nanocolumnar tungsten as a first wall coating.

Formation of core-shell nanostructure through wrapping of cuprous oxide nanowires by hydrogen titanate nanotubes

Recent developments of heterojunction-based devices through bottom-up approach such as sensors, light-emitters, energy generation and storage have emerged with great interest due to their wide range of operation and application related flexibilities. This work demonstrates how as-grown hydrogen titanate nanotubes (HTNT) bend and wrap on pristine curpous oxide nanowires (CONW) when mixed together. The unique architecture of wrapping followed by junction formation enhances the active surface area and reduces the contact resistance between the adjacent CONW and HTNT. Such a film upon further ion beam irradiation produces a large-scale network of hetero- and homo-junctions. This newly formed thin film surface upon irradiation shows strong water repelling properties and higher electrical conductivity. The wrapping mechanism, bond formation and the change of conductivity are explained using first principles calculations. The ion beam modifications and large-scale joining are predicted by state-of-the-art TRI3DYN simulation, which is based on binary collision approximation and simulated in a Monte Carlo approach. The observed wrapping and heterojunction are expected to provide excellent mechanical strength and flexibility, which are suitable for fabrication of flexible electronic devices.

Binary collision approximation simulations of ion solid interaction without the concept of surface binding energies

The sputtering yield of atoms removed from a solid upon ion irradiation can be calculated using Monte Carlo simulations based on the binary collision approximation. These simulations rely on the assumption of a planar surface potential barrier similar to the work-function for electron emission. Although this model yields good agreement with experimental data, the surface binding energy often has to be arbitrarily adjusted. We propose the bulk binding energy model to calculate sputter yields using sublimation enthalpies and a spherically symmetric potential barrier at the position where the sputtered atoms are generated in the bulk. A planar potential barrier at the surface no longer exists. We also consider the energy density deposited in nuclear collisions, which may lead to a lower sublimation enthalpy compared to standard room temperature conditions. The new bulk binding energy model is able to quantitatively explain many experimental sputtering data.

Deuterium ion irradiation impact on the current-carrying capacity of DI-BSCCO superconducting tape

In the present work, we have irradiated the DI-BSCCO superconducting tapes with the 100 keV deuterium ions to investigate the effect of ion irradiation on their critical current (Ic). The damage simulations are carried out using the binary collision approximation method to get the spatial distribution and depth profile of the damage events in the high temperature superconducting (HTS) tape. The point defects are formed near the surface of the HTS tape. These point defects change the vortex profile in the superconducting tape. Due to the long-range interaction of vortices with each other, the Ic of the tape degrades at the 77 K and self magnetic field. The radiation dose of 2.90 MGy degrades the 44% critical current of the tape. The results of the displacement per atom (dpa) and dose deposited by the deuterium ions are used to fit an empirical relation for predicting the degradation of the Ic of the tape. We include the dpa, dose and columnar defect terms produced by the incident particles in the empirical relation. The fitted empirical relation predicts that light ion irradiation degrades the Ic in the DI-BSCCO tape at the self field. This empirical relation can also be used in neutron irradiation to predict the lifetime of the DI-BSCCO tape. The change in the Ic of the DI-BSCCO tape due to deuterium irradiation is compared with the other second-generation HTS tape irradiated with energetic radiation.

Threshold Ion Energies for Creating Defects in 2D Materials from First-Principles Calculations: Chemical Interactions Are Important.

The characteristics of two-dimensional (2D) materials can be tuned by low-energy ion irradiation provided that the ion energy is correctly chosen. The optimum ion energy is related to Ethion, the minimum kinetic energy the ion should have to displace an atom from the material. Ethion can be assessed using the binary collision approximation (BCA) when the displacement threshold of the atom is known. However, for some ions the experimental data contradict the BCA results. Using density functional theory molecular dynamics (DFT-MD), we study the collisions of low-energy ions with graphene and hexagonal boron nitride and demonstrate that the BCA can strongly overestimate Ethion because energy transfer takes a finite time, and therefore, chemical interactions of the ion with the target are important. Finally, for all projectiles from H up to Ar, we calculate the values of Ethion required to displace an atom from graphene and h-BN, the archetypal 2D materials.

Solar Wind Ion Sputtering of Sodium from Silicates Using Molecular Dynamics Calculations of Surface Binding Energies

Abstract For nearly 40 yr, studies of exosphere formation on airless bodies have been hindered by uncertainties in our understanding of the underlying ion collisional sputtering by the solar wind (SW). These ion impacts on airless bodies play an important role in altering their surface properties and surrounding environment. Much of the collisional sputtering data needed for exosphere studies come from binary collision approximation (BCA) sputtering models. These depend on the surface binding energy (SBE) for the atoms sputtered from the impacted material. However, the SBE is not reliably known for many materials important for planetary science, such as plagioclase feldspars and sodium pyroxenes. BCA models typically approximate the SBE using the cohesive energy for a monoelemental solid. We use molecular dynamics (MD) to provide the first accurate SBE data we are aware of for Na sputtered from the above silicate minerals, which are expected to be important for exospheric formation at Mercury and the Moon. The MD SBE values are ∼8 times larger than the Na monoelemental cohesive energy. This has a significant effect on the predicted SW ion sputtering yield and energy distribution of Na and the formation of the corresponding Na exosphere. We also find that the SBE is correlated with the coordination number of the Na atoms within the substrate and with the cohesive energy of the Na-bearing silicate. Our MD SBE results will enable more accurate BCA predictions for the SW ion sputtering contribution to the Na exosphere of Mercury and the Moon.

Simulation of neutron irradiation damage in lead lanthanum zirconate titanate by Monte Carlo method

Lead lanthanum zirconate titanate (PLZT) has a broad application prospect for energy storage devices with high energy density, since it possesses excellent dielectric and energy storage properties. To investigate the irradiation damage to the PLZT induced by neutrons with different energy, the primary energetic recoil spectra of each kind of element are first extracted from the transportation simulations of neutrons with energy ranging from 1 to 14 MeV, respectively. Then, the displacement damages (including vacancies and interstitial atoms) induced by each type of recoil with different energy are simulated based on the binary collision approximation method. Finally the number of defects in PLZT produced by neutrons with an energy range from 1 to 14 MeV is calculated based on the recoil energy spectra and the defect number produced by the recoils. The results show that the number of defects produced in the PLZT material with a thickness of 3 cm is approximately independent of the neutron energy for the fast neutrons with energy in a range from 1 to 14 MeV, even though the primary recoil energy spectra from neutrons with different energy are completely different. The average number of defects produced in 3-cm-thick PLZT is about 460 ± 120 vacancies/neutrons. For neutrons with energy ranging from 1 to 14 MeV, the produced defect concentration in PLZT decreases slightly with the depth increasing within a thickness of 3 cm. The difference in defect concentration in this 3 cm is in a range of 50%. This decrease is caused mainly by the fact that some of neutrons are back-scattered during transport. The average defect concentration produced by neutron irradiation in the PLZT with a thickness of 3 cm is slightly(～20%) higher than that in the PLZT with a thickness of 1 mm. The reason for the higher defect concentration in a thicker (3 cm) PLZT can be attributed to the following facts: (i) the (n, 2n) reactions between neutron and material can make the number of neutrons increase during transport; (ii) the scattering can make the path of neutron longer; (iii) the inelastic scattering can lead to a smallnumber of moderated neutrons, which have a slightly larger interaction cross section with materials. This indicates the damage produced in thick PLZT is quite complicated and closely related to the process of neutron transport. This work presents a method of calculating the displacement damage of neutrons in materials, and the simulation results can provide guidance for studying the neutron irradiation effects of PLZT-based electronic devices.

Collisions between solitary waves propagating in the two-dimensional hexagonal packing granular system

In this paper, we investigate head-on and overtaking collisions of two solitary waves propagating in two-dimensional (2D) hexagonal packing confined in a rectangular region, using the fifth-order Runge–Kutta numerical scheme. In the head-on collision, numerical results indicate that the velocity amplitude of the solitary waves and the second solitary waves is proportional to the impacting velocity of the solitary waves. The relationship between phase shift and impacting velocity satisfies the equation [Formula: see text]. For head-on collisions of the waves with an equal amplitude of velocity, this power exponent is [Formula: see text], which can be well predicted by the binary collision approximation theory. For the case of head-on collisions between the waves with unequal amplitudes of solitary waves, this power exponent becomes [Formula: see text]. During an overtaking collision, the fast solitary wave has a positive phase shift, and the slow solitary wave has a negative phase shift. The phase shift of the fast solitary wave also has a power exponential relationship with the impacting velocity. The two waves either form a single peak in transition, or remain separated throughout the encounter with a double-peak formation depending on the ratio of the impacting velocity of solitary waves. The critical condition between the two regimes is consistent with those obtained by the Korteweg–de Vries model.