A Novel Reformulation of the 

Pseudo2D Battery Model Coupling 

Large Deformations at Particle and 

Electrode Levels

Weijie Mai, Andrew Colclasure, Kandler Smith

FEF 2019 MS201

April 2, 2019
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Introduction

 Si anode has high energy density but suffer from large deformation

400% volume expansion of Si Electrode/Cell deformation

Huang et al. Acta Materialia (2013) 

Crack

Daxin (http://www.daxinmat.com)

 Active material (AM) expansion causes electrode deformation and porosity reduction

 A model coupling multi-scale deformations required for better cell design

Wang et al. Advanced Energy Materials (2018) 

Lithiation
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Introduction

 Goal: consistently incorporate deformations based on the P2D framework

 Challenge: infinitesimal deformation assumption inapplicable
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Formulation: large deformation in electrode domain

 Finite strain theory 

: deformation gradient tensor

: displacement vector

multiplicative decomposition

• Deformation composed of elastic and inelastic deformations

• The Jacobian of the deformation gradient –

change of volume

Undeformed

Deformed

• Isotropic inelastic deformation due to Li insertion/extraction

: partial molar volume of Li in electrode

: elastic strain tensor

• Displacement can be solved by 

: Secondary PK stress

: Cauchy stress

: stiffness tensor
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Formulation: conservation law in reference frame 

 Eulerian conservation law

• Volume element (fixed in space)

• Need to include a convection term

• Need to explicitly keep track of the 

deformation

 Lagrangian conservation law

• Approximate field distributions in the 

undeformed geometry

• Material volume: 

• Effect of deformation on conservation is 

embodied in deformation gradient 

tensor 

Undeformed

Deformed
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Formulation: porosity variation

Wang et al. Advanced Energy Materials (2018) 

Lithiation

• AM expansion causes porosity reduction and electrode deformation

Increase rate of 

AM volume

Variation rate of solid 

volume fraction

Electrode 

deformation rate

• Ratio of porosity reduction and electrode deformation depends on fixture condition

• Material frame reformulation

: volume fraction of solid phase

: partial molar volume of Li in electrode

: local electrode velocity vector

: volumetric current source
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Formulation: P2D with large deformation

 Particle deformation

• Particle size change from to after lithiation/delithiation

• Within each particle, the deformation is characterized by the particle deformation 

gradient tensor 

AM expansion affects solid diffusion distance

• In the current model, we assumed that deformation within particle is uniform

• Alternatively the particle deformation can be expressed in terms of electrodelevel variables

• Particle stretch can be expressed as
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Formulation: P2D with large deformation

 Solid diffusion in particle

 Charge conservation in electrolyte

 Charge conservation in electrodes

 Mass conservation in electrolyte

The new model:

• Approximates two additional fields (electrode displacement, porosity)

• Only requires minor modifications of the existing P2D governing equations

electrolyte modeled as incompressible fluid
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Additional multiphysics coupling and assumptions

 Stress-dependent OCP  Porosity-dependent mechanical properties

 Specific surface area

Voltage hysteresis of LixSi system due to the effect of stress Couples particle deformation and porosity reduction

Lu et al. Physical Chemistry Chemical Physics (2016) Kovacik et al. Journal of materials science letters (1999)

 Assumptions

• All deformations are elastic and nondestructive

• Uniform and isotropic deformation within each particle

• Negligible in-plane electrode deformation (electrode is well adhered to 

strong metal foil cc)

• Electrolyte move out/into a material volume only in the out-of-plane direction

• Electrode is composed of only active material and electrolyte
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Low rate performance (0.02C)

Case I: P2D Case II: P2D+deformations Case III: Case II+OCP(σσh)

• Thickness changes: anode (35.6%↑), separator (20.1%↓), cathode (9.3%↓)

• Porosity reductions: anode (43.8%↓), separator (41.8%↓), cathode (18%↓)

• Uniform porosity within each component

• Negligible impact on cell voltage and capacity

 Si anode/NMC532 cathode; ANL Gen2 electrolyte; 5 mAh/cm2 (LLcccccccc = 143.3 μμm) 

 0.02C charge to 4.08 V; both ends of the cell are fixed
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Low rate performance (0.02C, Case II)

• Nonuniform particle expansion and specific surface area increase 

• Magnitude of variation is small due to low charge rate 

• Average particle expansion close to the analytical value

• Both σσxxxx and σσh in anode are uniform due to relatively uniform Li 

insertion rate distribution
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High rate performance (1C)

Si NMC

Sep

Case I: P2D Case II: P2D+deformations Case III: Case II+OCP(σσh)

• Nonuniform porosity reduction and 

deformations due to faster lithiation rate 

near anode/separator interface

• Porosity reduction further decrease the 

utilization of active material near the 

anode/current collector interface

• Case III slightly improve the uniformity of 

field distributions in anode
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High rate performance (1C)

• Increased concentration/potential 

gradients in electrolyte 

• In-plane stress dominates

• Non-uniform σσh distribution: more 

uniform intercalation fraction 

distribution in Case III

Electrolyte depletion
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Effect of anode porosity (1C, Case III)

• Fixed loading (5 mAh/cm2); thicker 

electrode for lower initial porosity

• Optimal volumetric energy density (~900 

Wh/L) obtained for εεaa,0 = 0.5

• Increasing the initial porosity helps to 

reduce σσh in anode at the same sate of 

charge
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Effect of cell loading (1C, Case III) Energy density (Wh/L)

• Fixed initial anode porosity (εεaa,0 = 0.5)

• Higher loading leads to thicker cell

• Classic P2D overpredicts cell energy density especially 

for higher loadings 

• The predicted optimal loading is 4 mAh/cm2
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Effect of cell fixture condition (5 mAh/cm2,εεaa,0 = 0.5,1C)

• Higher cell capacity and lower electrode 

stress when the cell is free to expand 

• Stress in cathode is slightly tensile for P=0 

psi due to NMC contraction

• ~9.1% increase of cell thickness, mainly due 

to Si anode expansion 
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Effect of cell fixture condition (5 mAh/cm2,εεaa,0 = 0.5,1C)

• Smaller porosity variation and thus more 

uniform Si utilization when P=0 psi

• Negligible porosity reduction in cathode 

and separator 

• Separator is compressed more when 

both ends are fixed due to its lower 

Young’s module compared to electrodes
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Conclusion and future work

• The P2D model was reformulated to consistently couple particle and electrode deformations

• Deformations and porosity reduction significantly affects the accessible capacity of the cell

• The proposed model shows notable differences on predicting the optimal cell loading and 

electrode porosity compared with the P2D model

• The model is under further development to resolve particle-level stress and allow simulating 

performances of composite anode (Si/C)
www.nrel.gov

NREL/PR-5400-73575

Q&A

This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy,

LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by U.S. 

Department of Energy Office of Energy Efficiency and Renewable Energy Vehicle Technologies Office, Energy Storage 

Program. The views expressed in the article do not necessarily represent the views of the DOE or the U.S.

Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges

that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce 

the published form of this work, or allow others to do so, for U.S. Government purposes.
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Complementary materials
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Complementary materials
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High rate performance (1C)