Cluster Dynamics Model Research Articles

A reduced cluster dynamics modeling of radiation damage in tungsten

A reduced cluster dynamics modeling of radiation damage in tungsten

Cluster dynamics modeling of hydrogen retention and desorption in tungsten with saturation and multi-trapping effect of sinks

Hydrogen (H) retention and desorption in tungsten (W)-based plasma-facing materials are still not well understood, largely due to the limitations of ex-situ observations in experimental detection methods like thermal desorption spectroscopy (TDS). In order to reveal the fundamental mechanisms behind H retention and desorption, we developed a cluster dynamics model, IRadMat-TDS, for theoretical modeling of depth distribution and TDS of deuterium (D) in polycrystalline W. The model newly includes the saturated absorption and emission of D in inherent sinks like grain boundaries (GBs), as well as the multi-trapping effect of D in various types of GBs with different trapping energies. The simulated TDS spectra are in agreement with experimental ones. For polycrystalline W under D ion irradiation within keV-energy range, two typical thermal desorption peaks in TDS at around 490 and 550 K are explicitly attributed to D emission from GBs and vacancies, respectively. And GBs play a major role in D retention. Moreover, the broad peaks in TDS come from the convolution of multi-trapping of D in sinks with different types of trapping sites rather than a single-site approximation.

Cluster dynamics study on nano damage of RPV steels under proton irradiation at 290°C

Irradiation-induced defects such as dislocation loops, cavities or solute clusters and chemical composition segregation of reactor pressure vessel (RPV) steel are the root causes of irradiation embrittlement. Combining two nucleation mechanisms, namely, the uniform nucleation and non-uniform nucleation of solute clusters (such as Cu-rich phase), a cluster kinetic simulation was established based on the reaction rate theory, and the co-evolution of matrix damage and Cu-rich phase in low-copper RPV steel was simulated under irradiation. And the average size and number density of defective clusters and solute clusters were established with irradiation dose. Compared with the average size and number density of dislocation loops observed by transmission electron microscopy (TEM) of proton irradiated RPV steel at 290°C, the verification results show that the cluster dynamics model considering both the nucleation mechanism of interstitial dislocation loops and vacancy clusters can well simulate the irradiation damage behavior of materials.

Spatially-resolved cluster dynamics modeling of irradiation growth

We develop here a spatially resolved, three-dimensional continuum model coupling cluster dynamics (SR-CD) and crystal plasticity to investigate irradiation growth in zirconium. The model uses scale separation to divide the population of the irradiation cluster into mobile and immobile families. Small interstitial and vacancy clusters are modeled using anisotropic reaction–diffusion equations. Among the immobile clusters, an atomistically-informed vacancy cluster to vacancy loop transition is taken into account. The coupling between the evolution equation of CD and the plastic deformation of the material is two-fold, with stress-informed bias factors and local inelastic strains computed from the evolution of the evolving cluster population. The numerical implementation of the model utilizes the finite element method to analyze both single-crystal and polycrystalline samples. The growth strains that are computed align well with the experimental data provided by Carpenter for single-crystal Zr. Furthermore, the transformation of a vacancy cluster into a complete vacancy loop, occurring at a size of 14 nm, is in agreement with experimental observations and atomistic simulations. The density, size, and growth rate of the dislocation loops, denoted as 〈c〉 and 〈a〉, also exhibit good agreement with transmission electron microscopy (TEM) analysis of irradiated Zr and its alloys. Our findings demonstrate that there is a spatial correlation between the growth of these dislocation loops and growth strains, significantly influenced by the crystal size. To explain the expansion of the 〈a〉 axis and the contraction of the 〈c〉 axis in irradiated Zr, it is necessary to consider the diffusion anisotropy difference (DAD) of mobile interstitial species. We show that the PWR Kearns parameters, specifically fr = 0.63, ft = 0.32, fa = 0.05, confer enhanced irradiation resistance to Zr along the principal directions when compared to single crystals. Additionally, reducing the grain size to nanograins further enhances the resistance to irradiation-induced growth, particularly along the direction with the highest volume fraction of basal poles [0001].

Cluster dynamics of proton irradiated RPV steel at room temperature

The life of reactor pressure vessels in nuclear power plants will be extended to 60 years, while the irradiation data of materials from commercial and experimental reactors are rare and expensive. There is an urgent need to solve how to accurately predict the changes in microstructure and mechanical properties of high-dose irradiated materials of RPV. The cluster dynamics method can be used to establish the relationship between nanostructure and macro properties and to study its evolution over a long period of years. Based on the average rate field theory, a cluster dynamics model was developed for the evolution of irradiated microstructure of RPV steel, which considers both homogeneous and heterogeneous nucleation of solute clusters. The model was successfully used to simulate the influence of the proton irradiation-induced cluster size and number density of RPV model steel with a low copper content at room temperature.

Study of xenon evolution in UO[formula omitted] using multi-grain cluster dynamics modeling

The behavior of fission gas is a key issue for the performance of UO2 fuel, which is the most widely used fuel type in GEN-III reactors. Fission gas will lead to lower thermal conductivity and affect the microstructure of fuel, gradually cause swelling and other phenomena. It seriously affects the performance and safety of nuclear reactor. The numerical simulation method based on physical model is an important means to study fission gas. In this paper, in order to study the key parameters affecting the behavior and evolution of fission gas xenon in UO2 fuel, a multi-grain parallel cluster dynamics model is designed and implemented. The model can describe the multi-grain structure characteristics of UO2, and establish the correlation for the physical process of intra- and inter-granular bubbles. Furthermore, the model is extended to a larger temporal and spacial scale through parallelization, so as to simulate the fission gas behavior closer to the actual situation, and expand the applicable conditions and range of the model. Based on the modeling work, the long-term evolution processes of xenon under different initial grain radius and temperature conditions are analyzed, the results show that the initial grain radius has a much bigger effect on the inter-granular gas concentration than the intra-granular gas, while temperature has a large effect on both intra- and inter-granular gas concentrations. The model and analysis results provide theoretical support for understanding fission gas behavior and improving UO2 fuel performance.

J-GAIN v1.1: a flexible tool to incorporate aerosol formation rates obtained by molecular models into large-scale models

Abstract. New-particle formation from condensable gases is a common atmospheric process that has significant but uncertain effects on aerosol particle number concentrations and aerosol–cloud–climate interactions. Assessing the formation rates of nanometer-sized particles from different vapors is an active field of research within atmospheric sciences, with new data being constantly produced by molecular modeling and experimental studies. Such data can be used in large-scale climate and air quality models through parameterizations or lookup tables. Molecular cluster dynamics modeling, ideally benchmarked against measurements when available for the given precursor vapors, provides a straightforward means to calculate high-resolution formation rate data over wide ranges of atmospheric conditions. Ideally, the incorporation of such data should be easy, efficient and flexible in the sense that same tools can be conveniently applied for different data sets in which the formation rate depends on different parameters. In this work, we present a tool to generate and interpolate lookup tables of formation rates for user-defined input parameters. The table generator primarily applies cluster dynamics modeling to calculate formation rates from an input quantum chemistry data set defined by the user, but the interpolator may also be used for tables generated by other models or data sources. The interpolation routine uses a multivariate interpolation algorithm, which is applicable to different numbers of independent parameters, and gives fast and accurate results with typical interpolation errors of up to a few percent. These routines facilitate the implementation and testing of different aerosol formation rate predictions in large-scale models, allowing the straightforward inclusion of new or updated data without the need to apply separate parameterizations or routines for different data sets that involve different chemical compounds or other parameters.

Single-size and cluster dynamics modeling of intra-granular fission gas bubbles in UO[formula omitted

We perform simulations of intra-granular fission gas bubble evolution in UO2 using both a relatively simple, computationally inexpensive single-size model and a detailed cluster dynamics model. Simulations encompass 36 experimental cases from 4 different databases, covering various temperature and burnup levels. We systematically compare results from the two models to each other and to post-irradiation experimental data of bubble average size and number density. Overall, the model-to-model comparisons reveal an excellent agreement across the set of simulations. This outcome indicates that, in spite of the underlying assumptions, the single-size model provides a good approximation of the complex physical behavior that is more rigorously described by the cluster dynamics model. Qualitatively, both models reproduce the trends of the experimental data with temperature and burnup correctly. Quantitatively, calculated results are either in good agreement with the data or within errors that appear consistent with the inherent uncertainties. Moreover, for the single-size model, we demonstrate and assess a multiscale approach whereby values for the fission gas atom diffusion coefficient from separate atomistic calculations are used. Systematic comparisons to experimental data point out a credible accuracy of the multiscale model.

Theoretical Study of the Gas-Phase Hydrolysis of Formaldehyde to Produce Methanediol and Its Implication to New Particle Formation.

Aldehydes were speculated to be important precursor species in new particle formation (NPF). The direct involvement of formaldehyde (CH2O) in sulfuric acid and water nucleation is negligible; however, whether its atmospheric hydrolysate, methanediol (CH2(OH)2), which contains two hydroxyl groups, participates in NPF is not known. This work investigates both CH2O hydrolysis and NPF from sulfuric acid and CH2(OH)2 with quantum chemistry calculations and atmospheric cluster dynamics modeling. Kinetic calculation shows that reaction rates of the gas-phase hydrolysis of CH2O catalyzed by sulfuric acid are 11-15 orders of magnitude faster than those of the naked path at 253-298 K. Based on structures and the calculated formation Gibbs free energies, the interaction between sulfuric acid/its dimer/its trimer and CH2(OH)2 is thermodynamically favorable, and CH2(OH)2 forms hydrogen bonds with sulfuric acid/its dimer/its trimer via two hydroxyl groups to stabilize clusters. Our further cluster kinetic calculations suggested that the particle formation rates of the system are higher than those of the binary system of sulfuric acid and water at ambient low sulfuric acid concentrations and low relative humidity. In addition, the formation rate is found to present a negative temperature dependence because evaporation rate constants contribute significantly to it. However, cluster growth is essentially limited by the weak formation of the largest clusters, which implies that other stabilizing vapors are required for stable cluster formation and growth.

Accelerating cluster dynamics simulation of fission gas behavior in nuclear fuel on deep computing unit–based heterogeneous architecture supercomputer

High fidelity simulation of fission gas behavior is able to help us understand and predict the performance of nuclear fuel under different irradiation conditions. Cluster dynamics (CD) is a mesoscale simulation method which is rapidly developed in nuclear fuel research area in recent years, and it can effectively describe the microdynamic behavior of fission gas in nuclear fuel; however, due to the huge cost of computation needed for CD model solution, the application scenario of CD has been limited. Thus, how to design the acceleration algorithm for the given computing resources to improve the computing efficiency and simulation scale has become a key problem of CD simulation. In this work, we present an accelerating cluster dynamics model based on the spatially dependent cluster dynamics model, combined with multi optimization methods on a DCU (deep computing unit)-based heterogeneous architecture supercomputer. The correctness of the model is verified by comparing with experimental data and Xolotl—a software of SciDAC program from the U.S. Department of Energy’s Office of Science. Furthermore, our model implementation has a better computing performance than Xolotl’s GPU version. Our code has gained great strong/weak scaling performance with more than 72.75%/84.07% parallel efficiency on 1024 compute nodes. This work developed a new efficient model for CD simulation of fission gas in nuclear fuel.

Modeling of irradiation growth of Zr single crystals with diffusion anisotropy

To predict the irradiation deformation of hexagonal materials, a cluster dynamics model is proposed in this study describing the evolutions of point defects and defect clusters with the diffusion anisotropy of self-interstitial atoms involved. Nucleation and growth models for 〈c〉 vacancy, 〈a〉 interstitial, and 〈a〉 vacancy dislocation loops are developed. Irradiation growth strains are obtained considering the defect formation, growth of dislocation loops, and climb of dislocations. Simulated growth strains along a- and c-axes agree well with the experimental data of annealed and pre-deformed Zr single crystals irradiated at different temperatures. It is found that the dominant contributions to the growth strains by different deformation mechanisms are changed by varied dislocation densities. The irradiation growth could be greatly affected by diffusion anisotropy due to the variation in the absorption efficiency of point defects by sinks. This study reveals the deformation mechanisms in the irradiation growth of Zr single crystals and could provide an effective calculation model for the irradiation deformation of hexagonal materials.

A combined theoretical-experimental investigation of thermal transport in low-dose irradiated thorium dioxide

During reactor operation, nuclear fuels are subject to extreme temperature and irradiation conditions which can significantly degrade the fuel's thermal transport properties. The reduction in thermal conductivity of the fuel as a result of irradiation-induced lattice defects is arguably the most important fuel performance metric in regard to reactor efficiency and safety. Because thorium dioxide (ThO2) is suitable as a model system for more complex materials such as UO2 and its mixed oxides, we present a theoretical investigation of thermal conductivity reduction seen in defect-bearing thorium dioxide and compare directly to experimental measurements. Phonon-mediated thermal transport of the fuel is modeled by a solution to the Boltzmann transport equation (BTE) for phonons. A cluster dynamics (CD) model for lattice defect evolution during irradiation predicts defect densities which are used as input to the BTE for modeling phonon-defect scatterings. Phonon scatterings by lattice defects include those from point defects and vacancy clusters and interstitial clusters of various sizes. The CD model is benchmarked against structural defect characterization of irradiated thorium dioxide using electron microscopy. Thermal conductivity predicted by the BTE model is compared to measured values for irradiated thorium dioxide specimens below room temperature to isolate effects of phonon-defect scattering from intrinsic 3-phonon processes, which dominate at higher temperatures. The computed conductivity values are in partial agreement at temperatures close to room temperature while slight deviations are observed at the lowest measured temperatures, suggesting that implemented phonon-defect scattering cross-section expressions may not be adequate for low temperatures. The presented work provides a necessary investigation of the influence of irradiation induced defects on fuel performance and represents a first step toward a full characterization of phonon mediated thermal transport in irradiated materials with complex defect microstructure.

Critical Role of Iodous Acid in Neutral Iodine Oxoacid Nucleation.

Nucleation of neutral iodine particles has recently been found to involve both iodic acid (HIO3) and iodous acid (HIO2). However, the precise role of HIO2 in iodine oxoacid nucleation remains unclear. Herein, we probe such a role by investigating the cluster formation mechanisms and kinetics of (HIO3)m(HIO2)n (m = 0–4, n = 0–4) clusters with quantum chemical calculations and atmospheric cluster dynamics modeling. When compared with HIO3, we find that HIO2 binds more strongly with HIO3 and also more strongly with HIO2. After accounting for ambient vapor concentrations, the fastest nucleation rate is predicted for mixed HIO3–HIO2 clusters rather than for pure HIO3 or HIO2 ones. Our calculations reveal that the strong binding results from HIO2 exhibiting a base behavior (accepting a proton from HIO3) and forming stronger halogen bonds. Moreover, the binding energies of (HIO3)m(HIO2)n clusters show a far more tolerant choice of growth paths when compared with the strict stoichiometry required for sulfuric acid–base nucleation. Our predicted cluster formation rates and dimer concentrations are acceptably consistent with those measured by the Cosmic Leaving Outdoor Droplets (CLOUD) experiment. This study suggests that HIO2 could facilitate the nucleation of other acids beyond HIO3 in regions where base vapors such as ammonia or amines are scarce.

Atomistically-informed modeling of point defect clustering and evolution in irradiated ThO2

A cluster dynamics (CD) model has been developed to investigate the nucleation and growth of point defect clusters, i.e., interstitial prismatic loops and nanoscale and sub-nanoscale voids, in ThO2 during irradiation by energetic particles. The model considers cluster off-stoichiometry due to the asymmetry of point defect generation on the O and Th sublattices under irradiation, as well as the point defect diffusivities and the defect binding energies to clusters. The energies were established using detailed molecular dynamics simulations considering the statistical variability of cluster configuration. A high-order adaptive time-integration has been used to solve the model. The predicted loop density and their average size is in good agreement with reported experimental observations for proton irradiated ThO2 at 600°C. The model did not predict void evolution due to the sluggish kinetics of cation vacancies, explaining the absence of voids in proton irradiated ThO2 (and other oxides) at relatively low temperatures.

Cluster dynamics modeling of Mn-Ni-Si precipitates coupled with radiation-induced segregation in low-Cu reactor pressure vessel steels

Formation of precipitates enhanced or induced by irradiation causes hardening and embrittlement of nuclear structural materials. Post-irradiation microstructure characterization of reactor pressure vessel (RPV) steels has shown that precipitation can be strongly associated with radiation-induced segregation (RIS) of solutes on dislocations. However, the effect of RIS on the kinetics of Mn-Ni-Si precipitation has not been quantified and coupled in previous model predictions. In this study, a new hybrid and spatially-dependent precipitation model is developed that couples cluster dynamics with RIS with transport coefficients of diffusion mediated by vacancies and interstitials. The new approach provides a unique way to account for concurrent evolution of heterogeneous cluster densities as well as solute and point defect concentration profiles. The model is applied to study the segregation of Mn, Ni, and Si on dislocations and heterogeneous nucleation of Mn-Ni-Si rich precipitates (MNSPs) in a low-Cu RPV steel. The RIS result shows that the segregation tendency is dominated by the vacancy mechanism for Mn, Ni, and Si. The coupled cluster dynamics modeling result shows that the onset of MNSP nucleation on dislocations depends strongly on RIS and occurs at the fluence of 2×1023n..m−2. The number density and mean radius can reach ∼1024m−3 and 1−2 nm at high fluence, respectively. These observations suggest that the role of RIS and heterogeneous nucleation can be significant for RPV steels under high neutron fluence. Results of simulations adopting various dose rates and dislocation densities show an increased fraction of MNSPs on dislocations at the condition of low dose rate, high dislocation density, and high fluence.

Pyruvic acid, an efficient catalyst in SO<sub>3</sub> hydrolysis and effective clustering agent in sulfuric-acid-based new particle formation

Abstract. The role of pyruvic acid (PA), one of the most abundant α-keto carboxylic acids in the atmosphere, was investigated both in the SO3 hydrolysis reaction to form sulfuric acid (SA) and in SA-based aerosol particle formation using quantum chemical calculations and a cluster dynamics model. We found that the PA-catalyzed SO3 hydrolysis is a thermodynamically driven transformation process, proceeding with a negative Gibbs free-energy barrier, ca. −1 kcal mol−1 at 298 K, ∼ 6.50 kcal mol−1 lower than that in the water-catalyzed SO3 hydrolysis. Results indicated that the PA-catalyzed reaction can potentially compete with the water-catalyzed SO3 reaction in SA production, especially in dry and polluted areas, where it is found to be ∼ 2 orders of magnitude more efficient that the water-catalyzed reaction. Given the effective stabilization of the PA-catalyzed SO3 hydrolysis product as SA⚫PA cluster, we proceeded to examine the PA clustering efficiency in a sulfuric-acid–pyruvic-acid–ammonia (SA-PA-NH3) system. Our thermodynamic data used in the Atmospheric Cluster Dynamics Code indicated that under relevant tropospheric temperatures and concentrations of SA (106 molec.cm-3), PA (1010 molec.cm-3) and NH3 (1011 and 5 × 1011 molec.cm-3), PA-enhanced particle formation involves clusters containing at most one PA molecule. Namely, under these monomer concentrations and 238 K, the (SA)2⚫PA⚫(NH3)2 cluster was found to contribute by ∼ 100 % to the net flux to aerosol particle formation. At higher temperatures (258 and 278 K), however, the net flux to the particle formation is dominated by pure SA-NH3 clusters, while PA would rather evaporate from the clusters at high temperatures and not contribute to the particle formation. The enhancing effect of PA was examined by evaluating the ratio of the ternary SA-PA-NH3 cluster formation rate to binary SA-NH3 cluster formation rate. Our results show that while the enhancement factor of PA to the particle formation rate is almost insensitive to investigated temperatures and concentrations, it can be as high as 4.7 × 102 at 238 K and [NH3] = 1.3 × 1011 molec.cm-3. This indicates that PA may actively participate in aerosol formation, only in cold regions of the troposphere and highly NH3-polluted environments. The inclusion of this mechanism in aerosol models may reduce uncertainties that prevail in modeling the aerosol impact on climate.

Atomistic and cluster dynamics modeling of fission gas (Xe) diffusivity in TRISO fuel kernels

TRISO fuel particles are candidates for use in next generation reactors including gas reactors, fluoride salt-cooled high temperature reactors, and micro-reactors. The UCO fuel kernel consists of a uranium dioxide (UO2) and uranium carbide mixture. The addition of UC2 helps suppress the formation of carbon monoxide gas, which led to failures during initial TRISO development. The addition of uranium carbide alters the chemistry of the UO2 kernel, which is known to influence performance parameters such as fission gas diffusivity, although the impact has not been quantified and no models exist that take the change in chemistry into account. Therefore, better understanding and more accurate models of the impact of chemistry on fuel performance are of high priority. In this paper, a first-principles density functional theory (DFT) and empirical potential based multi-scale study has been carried out to model the diffusivity of fission gas xenon (Xe) in UCO TRISO fuel kernels. The focus is on the UO2 component in the UCO fuel kernels, as that represents the largest volume fraction of the fuel kernels. The study relies on DFT and empirical potential calculations to determine Xe and point defect properties, which are then used in thermodynamic and kinetic models to predict diffusion for intrinsic conditions. In addition, the information is utilized in cluster dynamics simulations using the Centipede code to estimate the impact of irradiation on defect transport. The presence of UC2 or UC2−x in the UCO fuel kernels is shown to have a substantial impact on the UO2 non-stoichiometry by inducing oxygen vacancies and driving UO2 sub-stoichiometric, which causes much slower Xe diffusion in UCO compared to light water reactor UO2 fuel. The application of this model in fuel performance simulations using the Bison code is also demonstrated.

Evolution mechanisms of irradiation-induced helium bubbles, C15 clusters and dislocation loops in ferrite/martensite steels: A cluster dynamics modeling study

• Evolution of helium bubbles , C15 clusters and dislocation loops is studied by a cluster dynamics model. • The model considers the formation of C15 clusters and followed transformation into loops. • Irradiation temperature and dose effects on the microstructural evolution are analyzed. • Helium bubble evolution upon irradiation is governed in sequence by three phases. • Loop growth driven by long-term defect migration follows a power law. Using a cluster dynamics model, we studied the evolution of helium bubbles, C15 clusters and dislocation loops produced by Fe 3+ /He + dual ion beam irradiation in reduced activation ferrite/martensite Eurofer97 steels. The model reasonably reproduced size distributions of experimentally detected helium bubbles and dislocation loops. We show that bubble evolution upon irradiation was governed by three phases in sequence: bubble nucleation dominated phase controlled by irradiation cascade, coupled phase of bubble nucleation and growth, and bubble growth dominated phase. As a first attempt, the model considers the formation of widely reported C15 clusters and develops a scheme for C15 clusters transformation into loops. We show that the loop growth driven by long-term defect migration follows a power law f(n) ~ a/n S . The increase of irradiation temperature from 603 K to 673 K promoted the growth of C15 clusters resulting in pronounced loop evolution. While at 773 K, the high flux of vacancies and low fluxes of single, di- and tri-interstitials suppressed loop nucleation and growth, leading to no loop formation confirmed by experiment. The effectiveness of the model developed in this study demonstrates its potential to be used in similar irradiation cases.

Multi-scale fracture probability analysis of tungsten monoblocks under fusion conditions

Plasma facing components inside future nuclear fusion reactors are subjected to a high heat load and intense irradiation conditions. Using advanced computational material models, several problems can be solved that reflect tungsten monoblocks under fusion relevant loading scenarios. This allows for the identification of the conditions under which material failure is probable. The material model and parameters are identified such that the mechanical behaviour is in accordance with the homogenized behaviour of a previously developed crystal plasticity model on the microscopic scale. The heterogeneous stress field that follows is analysed in order to assess the probability of material failure, which is typically reflected by unstable crack propagation. Since fracture is an inherently multi-scale problem, critical regions are analysed in detail by means of a representative volume element. The resulting analysis reveals that in case the stress relaxation in the monoblock under the applied static heat load is complete, the probability of unstable crack propagation can reach values close to 35%. Finally, the impact of prolonged neutron irradiation is simulated by means of a cluster dynamics model. Although irradiation drastically increases the brittleness of tungsten, its impact on the overall monoblock performance remains limited. • The performance of tungsten monoblocks under fusion conditions is investigated. • Failure is predicted by application of the weakest-link theory on two length scales. • Stress relaxation increases the risk of failure upon shut down of the heat source. • The impact of neutron irradiation on the failure probability is shown to be limited.

Cluster dynamics modeling of niobium and titanium carbide precipitates in α-Fe and γ-Fe

Kinetic behaviors of niobium and titanium carbide precipitates in iron are simulated with cluster dynamics. The simulations, carried out in austenite and ferrite for niobium carbides, and in austenite for titanium carbide, are analyzed for dependences on temperature, solute concentration, and initial cluster distribution. The results are presented for different temperatures and solute concentrations, compared to experimental data available. They show little impact of initial cluster distribution beyond a certain relaxation time and that highly dilute alloys with monomers only present a significantly different behavior from denser alloys or ones with different initial cluster distributions.