Multi phase field model for solid state transformation with elastic strain

Steinbach, Ingo; Apel, Markus

doi:10.1016/j.physd.2006.04.001

Cited by 224 publications

(161 citation statements)

References 19 publications

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“…with the concentration flux J f lux related to the Li potential μaccording to [29] where M Li is the Li mobility tensor that is in general a function of Li concentrationĉ, as…”

Section: Phase-field Modelmentioning

confidence: 99%

A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes

Chen

Fan

Hong

et al. 2014

J. Electrochem. Soc.

107

103

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A phase-field model, accounting for large elasto-plastic deformation, is developed to study the evolution of phase, morphology and stress in crystalline silicon (Si) electrodes upon lithium (Li) insertion. The Li concentration profiles and deformation geometries are co-evolved by solving a set of coupled phase-field and mechanics equations using the finite element method. The present phase-field model is validated in comparison with a non-linear concentration-dependent diffusion model of lithiation in Si electrodes. It is shown that as the lithiation proceeds, the hoop stress changes from the initial compression to tension in the surface layer of the Si electrode, which may explain the surface cracking observed in experiments. The present phase-field model is generally applicable to high-capacity electrode systems undergoing both phase change and large elasto-plastic deformation. As a promising anode material for lithium (Li)-ion batteries, 1-3 the theoretical Li capacity of Silicon (Si) is 4200 mAh/g (corresponding to the lithiated phase of Li 4.4 Si), which is one order of magnitude larger than the commercialized graphite anode.1,2 Recent experiments revealed that the lithiation of crystalline Si (c-Si) occurs through a two-phase mechanism, i.e., growth of lithiated amorphous Li x Si (aLi x Si, x ∼ 3.75) phase separated from the unlithiated c-Si phase by a sharp phase boundary of about 1 nm thick.4-8 An abrupt change of Li concentration across the amorphous-crystalline interface (ACI) gives rise to drastic volume strain inhomogeneity. The resulting high stresses induce plastic flow, fracture, and pulverization of Si electrodes, thereby leading to the loss of electrical contact and limiting the cycle life of Li-ion batteries. [9][10][11] Electrochemically driven mechanical degradation in high-capacity electrodes has stimulated enormous efforts on the development of chemo-mechanical models to understand how the stress arises and evolves in lithiated Si electrodes.12-15 These chemo-mechanical models often treated the lithiation-induced stress as the diffusion-induced stress by considering Li diffusion in a solid-state electrode that results in the change of composition from its stoichiometric state. Deviation from stoichiometry usually results in a volume change that generates stress if the Li distribution is non-uniform. Early chemo-mechanical models only involved a unidirectional coupling. Namely, the diffusioninduced mechanical stress was considered, whereas the effect of mechanical stress on diffusion was ignored. Both experimental and computational studies, however, have shown that the mechanical stresses play an important role in the lithiation kinetics of Si electrodes. 4,6,16,17 Recently, fully coupled chemo-mechanical models were developed to incorporate the mechanical stress into the chemical potential. [18][19][20][21][22] In these models, the local stress modulates lithiation kinetics (reaction rate and diffusivity), 4,6,23 and in turn, lithiation kinetics regulates the stress generation in lithiated ...

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“…with the concentration flux J f lux related to the Li potential μaccording to [29] where M Li is the Li mobility tensor that is in general a function of Li concentrationĉ, as…”

Section: Phase-field Modelmentioning

confidence: 99%

A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes

Chen

Fan

Hong

et al. 2014

J. Electrochem. Soc.

107

103

View full text Add to dashboard Cite

show abstract

“…The total free energy function is composed of interfacial energy density, chemical energy density and other energy density. The phase field equation [7], the concentration field equation [8] and the stress-strain field equation [8] can be obtained after a series of derivation.…”

Section: Multi-phase Field Modelmentioning

confidence: 99%

Phase Field Study of Microstructure Evolution in Eutectoid Phase Transformation – I Nucleation

Zhang

Yin

Zhou

et al. 2017

Archives of Foundry Engineering

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Eutectoid growth, as the important reaction mechanism of the carbon steel heat treatment, is the basis to control the microstructure and performance. At present, most studies have focused on lamellar growth, and did not consider the nucleation process. Mainly due to the nucleation theory is inconclusive, a lot of research can support their own theory in a certain range. Based on the existing nucleation theory, this paper proposes a cooperative nucleation model to simulate the nucleation process of eutectoid growth. In order to ensure that the nucleation process is more suitable to the theoretical results, different correction methods were used to amend the model respectively. The results of numerical simulation show that when the model is unmodified, the lateral growth of single phase is faster than that of longitudinal growth, so the morphology is oval. Then, the effects of diffusion correction, mobility correction and ledges nucleation mechanism correction on the morphology of nucleation and the nucleation rate were studied respectively. It was found that the introduction of boundary diffusion and the nucleation mechanism of the ledges could lead to a more realistic pearlite.

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“…No correspondence of material parameters is needed between the phase behavior laws. This is the approach proposed in [37] for incorporating elasticity in a multiphase-field model. For that purpose, the authors resort to a well-known homogeneous stress hypothesis taken from homogenization theory in the mechanics of heterogeneous materials [8].…”

Section: Phase-field Models For Elastoviscoplastic Materialsmentioning

confidence: 99%

Micromorphic vs. Phase-Field Approaches for Gradient Viscoplasticity and Phase Transformations

Forest

Ammar

Appolaire

2011

Advances in Extended and Multifield Theories for Continua

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Abstract. Strain gradient models and generalized continua are increasingly used to introduce characteristic lengths in the mechanical behavior of materials with microstructure. On the other hand, phase-field models have proved to be efficient tools to simulate microstructure evolution due to thermodynamical processes in the presence of mechanical deformation. It is shown that both methods have strong connections from the point of view of thermomechanical field theory. A general formulation of thermomechanics with additional degrees of freedom is presented that encompasses both applications as special cases. It is based on the introduction of additional power of internal forces introducing generalized stresses. The current knowledge in the formulation of physically non-linear constitutive equations is used to develop strongly coupled elastoviscoplastic material models involving phase transformation and moving boundaries.

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Multi phase field model for solid state transformation with elastic strain

Cited by 224 publications

References 19 publications

A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes

A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes

Phase Field Study of Microstructure Evolution in Eutectoid Phase Transformation – I Nucleation

Micromorphic vs. Phase-Field Approaches for Gradient Viscoplasticity and Phase Transformations

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