Mesoscopic Research Articles

{101¯4} twin boundary in calcite: Structure and physical properties

Calcite, a CaCO3 polymorph, is one of the most significant materials in nature. In addition to its fundamental role in the global carbon cycle, it fills an unparalleled range of applications, both as a natural biomineral and more directly engineered materials. One of the most notable characteristics of calcite is its tendency to form twins, the boundaries of which are expected to determine key attributes of the host mineral at mesoscopic scales. Using a classical molecular dynamics simulation performed with a thermodynamically consistent rigid-ion force field, we show that the walls between the two twin domains aligned along the hexagonal (101¯4) plane of calcite have some unexpected properties: (1) rather than monoatomic planes, these twin walls are composed of two layers with fundamentally different distortions, (2) atomic shifts within the twin walls create strong polarity while the bulk remains centrosymmetric, and (3) the temperature evolution of the structural order parameter is mostly, but not completely, related to the tilt and rotation of CO32− molecules. Molecular dynamics simulations using thermodynamically accurate interatomic potentials demonstrate that the temperature evolution of twin walls is different from that of the bulk material. We show that kinks in twin walls have a low energy and generate high structural disorder near the kink position. Our methods are applicable to other molecular systems and emphasize the dominance of twin walls in materials. Published by the American Physical Society 2024

A Simulation Study on the Effect of Supersonic Ultrasonic Acoustic Streaming on Solidification Dendrite Growth Behavior During Laser Cladding Based on Boundary Coupling

Laser cladding has unique technical advantages, such as precise heat input control, excellent coating properties, and local selective cladding for complex shape parts, which is a vital branch of surface engineering. During the laser cladding process, the parts are subjected to extreme thermal gradients, leading to the formation of micro-defects such as cracks, pores, and segregation. These defects compromise the serviceability of the components. Ultrasonic vibration can produce thermal, mechanical, cavitation, and acoustic flow effects in the melt pool, which can comprehensively affect the formation and evolution for the microstructure of the melt pool and reduce the microscopic defects of the cladding layer. In this paper, the coupling model of temperature and flow field for the laser cladding of 45 steel 316L was established. The transient evolution laws of temperature and flow field under ultrasonic vibration were revealed from a macroscopic point of view. Based on the phase field method, a numerical model of dendrite growth during laser cladding solidification under ultrasonic vibration was established. The mechanism of the effect of ultrasonic vibration on the solidification dendrite growth during laser cladding was revealed on a mesoscopic scale. Based on the microstructure evolution model of the paste region in the scanning direction of the cladding pool, the effects of a static flow field and acoustic flow on dendrite growth were investigated. The results show that the melt flow changes the heat and mass transfer behaviors at the solidification interface, concurrently changing the dendrites’ growth morphology. The acoustic streaming effect increases the flow velocity of the melt pool, which increases the tilt angle of the dendrites to the flow-on side and promotes the growth of secondary dendrite arms on the flow-on side. It improves the solute distribution in the melt pool and reduces elemental segregation.

Multidimensionally ordered mesoporous intermetallics: Frontier nanoarchitectonics for advanced catalysis.

Ordered intermetallics contribute to a unique class of catalyst materials due to their rich atomic features. Further engineering of ordered intermetallics at a mesoscopic scale is of great importance to expose more active sites and introduce new functions. Recently, multidimensionally ordered mesoporous intermetallic (MOMI) nanoarchitectonics, which subtly integrate atomically ordered intermetallics and mesoscopically ordered mesoporous structures, have held add-in synergies that not only enhance catalytic activity and stability but also optimize catalytic selectivity. In this tutorial review, we have summarized the latest progress in the rational design, targeted synthesis, and catalytic applications of MOMIs, with a special focus on the findings of our group. Three strategies, including concurrent template route, self-template route, and dealloying route, are discussed in detail. Furthermore, physicochemical properties and catalytic performances for several important reactions are also described to highlight the remarkable activity, high stability, and controllable selectivity of MOMI nanoarchitectonics. Finally, we conclude with a summary and explore future perspectives in the field to contribute to wider applications.

Toward High-Performance Li-Rich Mn-Based Layered Cathodes: A Review on Surface Modifications.

Lithium-rich manganese-based layered oxides (LRMOs) have received attention from both the academic and the industrial communities in recent years due to their high specific capacity (theoretical capacity ≥250 mAhg-1), low cost, and excellent processability. However, the large-scale applications of these materials still face unstable surface/interface structures, unsatisfactory cycling/rate performance, severe voltage decay, etc. Recently, solid evidence has shown that lattice oxygen in LRMOs easily moves and escapes from the particle surface, which inspires significant efforts on stabilizing the surface/interfacial structures of LRMOs. In this review, the main issues associated with the surface of LRMOs together with the recent advances in surface modifications are outlined. The critical role of outside-in surface decoration at both atomic and mesoscopic scales with an emphasis on surface coating, surface doping, surface structural reconstructions, and multiple-strategy co-modifications is discussed. Finally, the future development and commercialization of LRMOs are prospected. Looking forward, the optimal surface modifications of LRMOs may lead to a low-cost and sustainable next-generation high-performance battery technology.

Recent developments on multiscale simulations for rheology and complex flow of polymers

AbstractThis review summarized the multiscale simulation (MSS) methods for polymeric liquids. Since polymeric liquids have multiscale characteristics of monomeric, mesoscopic, and macroscopic flow scales, MSSs that relate different hierarchical levels are adequate to reproduce flow properties accurately. Our review includes pioneering studies to the most advanced MSS studies on rheology predictions and flow simulations of polymeric liquids. We discuss two major types of MSS methods: the bottom-up and model-embedded MSS methods. The former method mainly connects all-atom molecular dynamics models and mesoscopic models to predict rheological properties. In contrast, the latter method, where a microscopic or mesoscopic model is embedded in a macroscopic computational domain, is designed to predict macroscopic flow properties. Finally, we also discuss MSS methods using machine learning techniques. Graphical abstract

Recent studies on multiscale modeling of carbon fiber-reinforced composites

Multiscale modeling has garnered attention as an analytical method for investigating the mechanical behavior of composite materials. This paper introduces a comprehensive multiscale modeling approach developed by the authors’ group, which enables comprehensive analysis of the material properties from the atomic-scale reaction characteristics to laminate-scale fracture analysis; an overview of relevant literature is included as well. First, the methods of analysis at the atomic/molecular, microscopic, and mesoscopic scales are separately explained. Finally, a multiscale modeling approach that integrates these models and bridges the gaps between the scales is presented. The multiscale model is used to predict the fracture behavior of carbon fiber-reinforced composites based on the mechanical properties of epoxy resin.

Closed-loop fMRI at the mesoscopic scale of columns and layers: Can we do it and why would we want to?

Technological advances in fMRI including ultra-high magnetic fields (≥ 7 T) and acquisition methods that increase spatial specificity have paved the way for studies of the human cortex at the scale of layers and columns. This mesoscopic scale promises an improved mechanistic understanding of human cortical function so far only accessible to invasive animal neurophysiology. In recent years, an increasing number of studies have applied such methods to better understand the cortical function in perception and cognition. This future perspective article asks whether closed-loop fMRI studies could equally benefit from these methods to achieve layer and columnar specificity. We outline potential applications and discuss the conceptual and concrete challenges, including data acquisition and volitional control of mesoscopic brain activity. We anticipate an important role of fMRI with mesoscopic resolution for closed-loop fMRI and neurofeedback, yielding new insights into brain function and potentially clinical applications.This article is part of the theme issue 'Neurofeedback: new territories and neurocognitive mechanisms of endogenous neuromodulation'.

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

A clearer understanding of the dynamic structuring of different natural rubber genotypes on a macroscopic and mesoscopic scale by asymmetrical-flow field-flow fractionation (A4F) analysis

Unlike synthetic elastomers, the structure of natural rubber (NR) evolves (dynamic structuring) and so do its properties during the storage before reaching an industrial mixer. In the rubber industry this is known as storage hardening. NR samples from three genotypes (GT1, RRIM600 and PB235) were subjected to different levels of structuring by varying the structuring time (t) on phosphorus pentoxide (0 < t < 28 h). Storage hardening (ΔP) of the samples was then determined by measuring the increase in Wallace plasticity (P) (macro-scale) and by analyzing their mesostructure (meso-scale) using asymmetrical flow field flow fractionation (A4F). Monitoring ΔP as a function of structuring time revealed a diversity of behaviors specific to the genotype from which the rubber originated. For example, NR samples from genotypes GT1 and PB235 exhibited different kinetics for t < 12 h, an increase in ΔP with structuring time, but reached the same final plateau (t > 12 h). An A4F analysis of the samples was used to quantify the fraction of microaggregates smaller than 1 μm (microgel<1μ). The microgel<1μ rate decreased with structuring time to varying extents depending on the genotype. A very significant negative relationship was found between ΔP and the microgel<1μ rate, indicating that the NR samples that hardened the most contained the lowest microgel<1μ rate, but the highest macrogel rate.

A clustering coefficient structural entropy of complex networks

The structural entropy is a quantification of the topological structural complexity of the static complex networks, which is defined based on the structural characteristics and the Shannon entropy. For instance, the ’degree structural entropy’ is based on the network’s ’first-order’ topological properties: the degree of each node. The ’betweenness structural entropy’ is based on the betweenness centrality of nodes, which is a global topological structural property of static complex networks. The two different structural entropy give two completely different views of the network’s topological structural complexity. However, a ’mesoscopic’ structural entropy is still missing in the network theory. In this work, a clustering coefficient structural entropy of complex networks is proposed to quantify the structural complexity of static networks on the mesoscopic scale. The effectivity of the proposed ’mesoscopic’ structural entropy is verified in a series of networks that grow from two different seed networks under the Barabási–Albert and Erdős–Rényi rules. We also find that the quantification of structural entropy effectively reflects the impact of structural heterogeneity on the growth rule in the early stages of seed network growth. Finally, we observe that the structural ratio of the clustering coefficient structural entropy and degree structural entropy remains stable and unchanged when network growth reaches maturity. We also note that the convergence rate of the network’s structural entropy ratio varies under different guiding rules. These findings suggest that the differences in structural entropy can serve as a novel tool for measuring the stability of complex networks and provide fresh insights into achieving a ’balanced’ state in the dynamic evolution of complex networks.

P09.02.A A WINDOW TO THE BRAIN: MONITORING THE HUMAN BRAIN USING FUNCTIONAL ULTRASOUND (FUS)-IMAGING IN FREELY MOVING SUBJECTS WITH AN ACOUSTICALLY FAVORABLE SKULL IMPLANT

Abstract BACKGROUND To this day, the mainstay of human neuroimaging remains (functional) Magnetic Resonance Imaging (fMRI), where subjects are asked to lie completely still and silent in a noisy scanner. Many key questions in clinical care and neuroscience remain poorly studied because of fMRI’s inability to facilitate natural behavior such as locomotion or even verbal speech. What is more, due to the cost and size of an (f)MRI-machine, monitoring of morphological (i.e. tumor regrowth) or functional changes in the human brain, pre- and post-tumor resection, is often spread out over intervals of many months, leaving patients in suspense about possible tumor regrowth.With the advent of ultrafast ultrasound imaging, a new high-resolution, depth-resolved and mobile neuro-imaging technique called functional Ultrasound (fUS) has emerged. In the last decade, fUS has been applied successfully in awake neurosurgical contexts to map out hemodynamics-based functional brain activity as well as tumor tissue at a mesoscopic scale. However, given the significant aberration of ultrasound signal through human skull bone, fUS is not available outside of OR as of yet. As a first step outside of the OR, we aimed to apply fUS in freely moving subjects with a skull bone defect covered by PEEK, an acoustically favorable material. MATERIAL AND METHODS Two subjects with PEEK-implants were repeatedly imaged over a period of &amp;gt;1 year. One of the subjects received a PEEK-implant after infection of the autologous bone, post-resection of an astrocytoma. Each subject received a custom-fitted 3D-printed helmet based on their MRI to fixate the probe and to attach the geometry needed for optical tracking. Using an optical tracking camera, the position of the Power Doppler Image (PDI) relative to CT/(f)MRI-volumes of the subject could be determined in real-time. Images were acquired using an experimental research system (Vantage-256, Verasonics) interfaced with a 9L-D linear array (GE, 5.3 MHz). All scans were acquired with a PRF of 800 Hz. The PDIs and angle compounded beamformed frames were stored for offline processing, consisting of 3D-vascular reconstructions and functional mapping. Functional tasks focused on the sensorimotor area of the lip. For measurements outside of the lab, the acquisition system was transformed into a mobile cart, which could be pushed by the subject itself. RESULTS Trans-PEEK imaging resulted in PDIs with highly detailed vasculature of healthy brain tissue and the resection cavity, up to several centimeters in-depth, reproduced reliably over a period of &amp;gt;12 months. In sedentary, standing and walking conditions, we were able to produce consistent functional maps of the human brain. CONCLUSION Our current work demonstrates how fUS has the potential to become a powerful window to the brain, both in terms of vascular monitoring of tumor regrowth, as well as functional brain mapping, outside of the operating room.

Mesoscopic coupled plasticity-damage model of rough surface considering size-dependent plasticity

Metallic materials exhibit significant size effect at mesoscopic scale have been revealed by many experiments. Thus, it is essential to consider size effect when conducting mesoscopic scale studies in tribology. However, there is no constitutive model considering both ductile fracture and size effects of rough surface in the literature. At the same time, the simulation of friction and wear at the mesoscopic scale suffer from computational inefficiency using the conventional theory of mechanism-based strain gradient (CMSG) plasticity theory. To address the aforementioned challenges, we propose a mesoscopic coupled plasticity-damage model by combining the coupled plasticity-damage model and the simplified CMSG plasticity theory. This model can take into account not only the effect of stress state, temperature, strain rate on plasticity and fracture, but also the effect of the effective plastic strain gradient. Meanwhile, the model is able to solve the effective plastic strain gradient by a simplified method, thereby enhancing the computational efficiency. Then, model parameters for bearing bushing material of an engine are calibrated. Finally, scratch simulation and scratch tests of bearing bushing material at the mesoscopic scale are conducted to validate the effectiveness of the mesoscopic constitutive model. The residual scratch depth and width, coefficient of friction at different normal loads are investigated by experiment and simulation, respectively. Results proved the effectiveness of mesoscopic coupled plasticity-damage model, which is applicable for investigating issues related to wear, friction, and contact.

Thermal reaction based mesoscale ablation model for phase degradation and pyrolysis of needle-punched composite

Needle-punched composites are highly valued for their exceptional resistance to interlaminar properties, ablation, and design flexibility, making them increasingly popular in aerospace thermal protection systems. This work investigates the mesoscale structural characteristics and thermophysical properties of needle-punched composites in ablation process. Oxyacetylene ablation experiments were carried out at different temperatures, and a mesoscopic needle-punched structure model was established based on the results of CT characterization. Further, Abaqus custom subroutine was used to reveal the ablation evolution mechanism of carbon fiber reinforced phenolic resin-based needle-punched composites. The results show that, at mesoscopic scale, the acicular fiber bundle perpendicular to the ablative surface accelerates the heat conduction to the interior of the material and promotes the thermal damage and performance degradation of the composite.

Chemically Transferable Electronic Coarse Graining for Polythiophenes.

Recent advances in machine-learning-based electronic coarse graining (ECG) methods have demonstrated the potential to enable electronic predictions in soft materials at mesoscopic length scales. However, previous ECG models have yet to confront the issue of chemical transferability. In this study, we develop chemically transferable ECG models for polythiophenes using graph neural networks. Our models are trained on a data set that samples over the conformational space of random polythiophene sequences generated with 15 different monomer chemistries and three different degrees of polymerization. We systematically explore the impact of coarse-grained representation on ECG accuracy, highlighting the significance of preserving the C-β coordinates in thiophene. We also find that integrating unique polymer sequences into training enhances the model performance more efficiently than augmenting conformational sampling for sequences already in the training data set. Moreover, our ECG models, developed initially for one property and one level of quantum chemical theory, can be efficiently transferred to related properties and higher levels of theory with minimal additional data. The chemically transferable ECG model introduced in this work will serve as a foundation model for new classes of chemically transferable ECG predictions across chemical space.

An extended mesoscaled M-K model for predicting discrete ultimate strains and evaluating the effect of normal stress on forming limits

Due to the size effect, the formability of metal sheets diminishes and the ultimate strain points exhibit a more discrete distribution. This poses a huge challenge to the process design and manufacture of microparts. This paper presents a modified mesoscopic scale forming limit model based on the M-K model. The model extends the forming limit curve (FLC) into forming limit bands to predict the discrete nature of the ultimate strain. In addition, a method for applying normal stresses to increase the forming limit of thin metal sheets is proposed. The effects of normal stresses on surface roughening and flow stresses in thin metal sheets were evaluated and quantified by uniaxial tensile experiments under normal stresses. The results show that the normal stress suppresses the increase of surface roughness and improves the work hardening rate. The effect of normal stresses on forming limits was obtained using a constructed model. The results show that the predictions of the proposed model are able to include almost all the ultimate strain points. The forming limit bands are more inclusive and safer than the single forming limit curve. In addition, the forming limits of thin plates under normal stresses are significantly increased, which is attributed to the suppression of surface roughening and the increase of strain hardening index. The forming limit bands proposed in this paper provide a more reliable form of forming limit assessment for thin metal plates. The proposed method of applying normal stresses provides a robust and inexpensive process route to address the challenge of poor formability of thin metal sheets.

Effect of the advancing contact angle on coal particle–bubble detachment at the mesoscopic scale

Improving particle–bubble adhesion stability and reducing particle–bubble detachment are crucial in froth flotation processes. In this study, a nonequilibrium molecular dynamics approach is applied to investigate the dynamic detachment process of low-rank coal particles adhering with/without a collector from bubbles under different pull forces at the mesoscopic scale. Results indicate that a collector oil film on the surface of a low-rank coal particle can considerably increase the advancing contact angle and counteract the contraction of the three-phase contact line, enabling the particle to withstand a greater detachment force. Conversely, the residual water clusters on the particle surface tend to cause contraction of the contact line. Force analysis of the liquid surrounding the particle and the contact line indicates that the maximum static capillary force between the particle and the bubble is equivalent to the capillary force when the bending angle of the liquid surface reaches the advancing contact angle. When the bending angle of the liquid surface exceeds the forward contact angle, the force on the liquid around the contact line becomes unbalanced, causing the contact line to move to decrease the bending angle of the liquid surface and restore balance to the surrounding liquid. Hence, on mesoscopic scales of ten to tens of nanometers, the advancing contact angle still determines particle–bubble detachment. The study of particle–bubble detachment mechanisms at the mesoscopic scale is expected to establish a relation between microscopic mechanisms at the molecular scale and macroscopic experimental studies at the micrometer to millimeter scale

Multiscale study of interfacial properties of carbon fiber reinforced polyphthalazine ether sulfone ketone resin matrix composites

In view of the limitations of traditional research tools on interfacial failure mechanisms in fiber/PPESK composites, this work proposes a multiscale research tool to carry out an in-depth study of the interfacial behavior between fibers and matrix. Based on microdroplet debonding tests, at the mesoscopic scale, the influence of residual thermal stress on the interface damage mode is explored through finite element (FEM) simulations. The evolution mechanism of composite material interfaces in spatial and temporal dimensions is examined based on changes in interfacial stress distribution, energy dissipation, and damage morphology during the debonding process, which can be summarized as follows: accompanied by elastic-plastic deformation and friction effects, the progressive process from localized to complete failure presents a dominant Type II damage mode at the interface. To further explore the interface failure mechanism at the molecular level, an interface model of CF/PPESK composite materials was established using molecular dynamics (MD) method. By monitoring the atom movement trend, the "fiber-matrix displacement synergistic effect" in the interfacial shear damage process was revealed, thereby establishing a multiscale mapping relationship of composite material interface. Based on this, the combination of FEM and MD was utilized to investigate the interface damage process of composite materials under different service conditions and to reasonably predict the initiation and expansion of microcracks. This study provides a pioneering perspective on interface damage research in composite materials with a "top-down" multiscale approach.

A practical multiscale analysis for nonlinear RC structures using a smart substructure replacement strategy

This paper presents a novel practical multiscale analysis (MSA) method for simulating local damages in large reinforced concrete (RC) structures during seismic events. Traditional techniques like the finite element (FE) method often struggle to balance computational costs with detailed simulation. The proposed method facilitates sequential local refinement from macroscopic to mesoscopic scales of RC structures, focusing strategically on critical areas prone to damage and cracking. Key features include: (1) 'Correction forces' enable integration and collaboration among models of different refinement levels, keeping models unchanged (i.e., without additions, deletions, or further refinement of elements), streamlining analysis and independent implementation across methods, scales, and platforms. (2) A smart replacement strategy ensures smooth transitions and prevents abrupt force changes between models at different refinement levels. (3) A ‘training process’ before performing the replacement and a semi-explicit method guarantee the accuracy and efficiency of the MSA method. (4) Parallel and distributed computing is seamlessly applied, significantly accelerating the analysis at each level. Implemented in the open-source software OpenSees, this method is illustrated through three examples that efficiently capture both the macroscopic mechanical responses and detailed local damage behaviors. This approach provides a valuable tool for the refined analysis of large-scale RC structures.

Percolation versus depinning transition: The inherent role of damage hardening during quasibrittle failure.

The intermittent damage evolution preceding the failure of heterogeneous brittle solids is well described by scaling laws. In deciphering its origins, failure is routinely interpreted as a critical transition. However at odds with expectations of universality, a large scatter in the value of the scaling exponents is reported during acoustic emission experiments. Here we numerically examine the precursory damage activity to reconcile the experimental observations with critical phenomena framework. Along with the strength of disorder, we consider an additional parameter that describes the progressive damageability of material elements at mesoscopic scale. This hardening behavior encapsulates the microfracturing processes taking place at lower length scales. We find that damage hardening can not only delay the final failure but also affect the preceding damage accumulation. When hardening is low, the precursory activity is strongly influenced by the strength of the disorder and is reminiscent of damage percolation. On the contrary, for large hardening, long-range elastic interactions prevail over disorder, ensuring a rather homogeneous evolution of the damage field in the material. The power-law statistics of the damage bursts is robust to the strength of the disorder and is reminiscent of the collective avalanche dynamics of elastic interfaces near the depinning transition. The existence of these two distinct universality classes also manifests as different values of the scaling exponent characterizing the divergence of the precursor size on approaching failure. Our finding sheds new light on the connection between the level of quasibrittleness of materials and the statistical features of the failure precursors. Finally, it also provides a more complete description of the acoustic precursors and thus paves the way for quantitative techniques of damage monitoring of structures-in-service.

Collective oscillations in a three-dimensional spin model with non-reciprocal interactions

Abstract We study the onset of collective oscillations at low temperature in a three-dimensional spin model with non-reciprocal short-range interactions. Performing numerical simulations of the model, the presence of a continuous phase transition to global oscillations is confirmed by a finite-size scaling analysis, yielding values of the exponents β and ν compatible with both the three-dimensional XY and Ising equilibrium universality classes. By systematically varying the interaction range, we show that collective oscillations in this spin model actually result from two successive phase transitions: a mean-field phase transition over finite-size neighborhoods, which leads to the emergence of local noisy oscillators, and a synchronization transition of local noisy oscillators, which generates coherent macroscopic oscillations. Using a Fokker–Planck equation under a local mean-field approximation, we derive from the spin dynamics coupled Langevin equations for the complex amplitudes describing noisy oscillations on a mesoscopic scale. The phase diagram of these coupled equations is qualitatively obtained from a fully-connected (mean-field) approximation. This analytical approach allows us to clearly disentangle the onset of local and global oscillations, and to identify the two main control parameters, expressed as combinations of the microscopic parameters of the spin dynamics, that control the phase diagram of the model.