A thermodynamically consistent gradient theory for diffusion–reaction–deformation in solids: Application to conversion-type electrodes

https://doi.org/10.1016/j.jmps.2021.104368Get rights and content

Highlights

  • Thermodynamically consistent diffusion–reaction–deformation phase-field theory.

  • Captures kinetically or thermodynamically driven sharp interface formation.

  • Kinematical decomposition and thermodynamics based on microstructural RVE.

  • Incorporates and investigates wetting (surface energy) boundary conditions.

  • Three-dimensional FEA application to conversion electrodes for energy storage.

Abstract

We develop a thermodynamically consistent phase-field finite strain theory for problems in solids mechanics that couple transport of species into a host material, sharp interface reactions of the species with the host, mechanical deformation and stress. The theory distinguishes between diffusion-limited and reaction-limited kinetics, resolving the manner in which a sharp reaction front can be developed in either case. The phase field formulation has the added benefit of enabling the application of wetting (surface energy) boundary conditions which are critical in reproducing experimentally relevant reaction front morphologies. The theory is fully coupled with diffusion and reaction phenomena impacting mechanical deformation and subsequent stress generation, and conversely these phenomena are coupled to mechanical stress. We derive thermodynamically consistent driving forces for diffusion and reaction through a continuum treatment of these phenomena. Importantly, the resulting formulation makes precise the nature of the material properties driving these thermodynamic forces and in turn makes it amenable to being specialized and calibrated for application.

While the framework is quite general, we apply it to modeling conversion electrodes for energy storage using a three-dimensional finite element implementation. We demonstrate the manner in which the theory can be specialized and calibrated in straightforward fashion. Simulations are performed of chemical reactions of FeS2 crystals with lithium and sodium ions, both of which proceed through the formation and propagation of a sharp interface, and are compared to experimental observations of the same system. Our simulations show good qualitative agreement with experimental observations, and elucidate the critical role mechanics plays in determining the morphology of the sharp reaction interface and subsequent stress generation which can lead to mechanical deterioration of these materials. Beyond this application, the theoretical framework should serve useful in a number of engineering problems of relevance in which diffusion and sharp interface reactions occur.

Introduction

The development of theoretical frameworks in continuum mechanics which couple chemical stimuli and mechanical deformation – in particular those involving coupling of species diffusion with deformation of the host material – have received significant attention in the recent literature. This drive is largely attributed to the need for understanding the interplay of chemistry and mechanics in a number of engineering problems of relevance where mechanics plays a non-negligible role in the performance of the material. These include energy storage devices (Christensen and Newman, 2006, Zhang et al., 2007, Cheng and Verbrugge, 2008, Cui et al., 2012), chemically active polymers (Bosnjak et al., 2020, Okumura et al., 2020, Mao and Anand, 2018), oxides and corrosion phenomena (Loeffel and Anand, 2011, Konica and Sain, 2020, Cui et al., 2020), as it has become clear that mechanics plays an important role on the chemo-mechanical performance of these systems. Specifically, the design and analysis of energy storage devices required new models that addressed the coupling between transport phenomena and mechanics, and from this need came a group of theories specifically devoted to modeling these problems. The review papers by Zhao et al. (2019b) and Bistri et al. (2020) provide a recent summary of modeling efforts in this area. Among these, the thermodynamics based finite strain models of Bower et al. (2011), Zhao et al. (2011a), Anand and co-workers (Anand, 2012, Di Leo et al., 2014), Levitas and Attariani, 2014, Brassart and Suo, 2013, and Ganser et al. (2019) are notable for their rigorous treatment of the coupling between diffusion and mechanics. These frameworks have been successfully applied to capturing experimentally observed electrochemical phenomena in energy storage devices (cf. Bucci et al., 2014, Di Leo et al., 2015). While rigorous, current theoretical frameworks are largely limited to the class of problems in which the concentration of diffusing species is conserved, i.e. there is no chemical reaction within the solid to convert mass from one compound to another.

There are a number of technologically important applications requiring an understanding of the coupling between chemistry and mechanics including the manner in which reactions occur within the solid. These applications range from oxidation and polymerization (cf. Tolpygo et al., 1998, Minervino et al., 2014, El Kadiri et al., 2008, Gigliotti et al., 2011) to reaction based electrodes for energy storage (cf. Lin et al., 2014, Li et al., 2012). Notable works in thermodynamics based models for modeling reaction–diffusion phenomena including mechanical deformation are the works of Loeffel et al. (2013) for modeling thermal barrier coating oxidation, Zhao et al. (2019a) for SiC fiber oxidation, Xu et al. (2019) for thermal coating corrosion, the general formulation of Svendsen et al. (2018) for phase-field based modeling, and Salvadori et al. (2018) for trapping reactions. The aforementioned theoretical frameworks have made significant progress towards modeling diffusion–reaction problems in solids; however, they lack in a few key areas which we seek to address in this work.

The purpose of this work is to report on a novel, thermodynamically consistent, gradient, theoretical framework for modeling concurrent diffusion, reactions, and deformations in solids with a particular emphasis on problems in which sharp reaction interfaces occur. The framework is general and should be applicable to a number of engineering problems. In particular, our framework includes the following unique features distinguishing it from previous work in this area:

  • The theoretical framework allows for both kinetically and thermodynamically driven sharp interface formation. Kinetically driven interfaces occur in systems in which the reaction kinetics are significantly faster than the diffusion kinetics. Thermodynamically driven sharp interfaces can occur in any system – including those in which the reaction kinetics are sluggish – and are driven by the existence of a thermodynamic energy barrier between reacting phases.

  • The theoretical framework develops a thermodynamically consistent, physically motivated, driving force for chemical reactions that distinguishes the role of various chemical and mechanical driving forces. Particularly useful then is the fact that material properties driving the reaction kinetics can be easily identified from the literature or experiments.

  • The gradient based phase-field formulation allows us to capture surface energy phenomena which significantly affect the morphology of the reaction front. In particular, surfaces with lower energy for being fully reacted will become fully “wetted” (i.e. fully reacted) and this is consistently captured using our gradient formulation (cf. the work of Bazant and co-workers Bazant, 2013, Cogswell and Bazant, 2013, Nadkarni et al., 2018).

To demonstrate the relevance and use of this theoretical framework we specialize it to model the particular engineering problem of reaction electrodes (i.e. conversion electrodes) for energy storage. An accompanying three-dimensional finite element implementation is used to compare numerical simulations using our novel theoretical framework with experimental results. Current state of the art lithium-ion batteries make use of active particles, such as graphite, whose primary mechanism of lithium storage is intercalation. Alternatively, next generation electrode materials made of transition metal oxides which store charge carrying ions via chemical conversion mechanism have recently been given attention as their theoretical capacity dwarfs that of intercalation based electrodes. This superior capacity has been linked to the high number of ions per structural unit that can be stored via conversion vs the lower number that can be intercalated into layered structure of intercalation based electrodes (Poizot et al., 2000, Yu et al., 2016, Li et al., 2012). The conversion reaction, however, is accompanied by structural and chemical phase transformation of the host material (cf. Wang et al., 2012), resulting in a rich chemo-mechanics problem. The exact energy density and the crystalline or amorphous structural phase transformation incurred by the electrode during electrochemical cycling depend on the specific transition metal oxide used (cf. Yu et al., 2016 for a detailed comparison between different compounds). However, nearly all compounds and their nano-structured variations incur very large volumetric expansion and structural changes on the order of 100% strain during cycling (cf. Zhang et al., 2008, Larcher et al., 2002, Ren et al., 2014, Hu et al., 2006). These large reaction-induced deformations can lead to mechanical and electrochemical battery degradation, necessitating the development of a theory that accounts for diffusion of species, chemical reaction and subsequent phase transformation, mechanical deformation and stress, and how these fields are coupled.

We specialize our theoretical framework to modeling the reaction of FeS2 crystals with different ions as experimentally investigated by Boebinger et al. (2018). In doing so, we demonstrate the manner in which mechanics affects the chemical reaction kinetics and morphology of the sharp interface. We further elucidate the role of wetting (surface energy) chemical boundary condition in reproducing reaction morphologies which are consistent with experimental observations. Finally, using our theoretical framework and numerical implementation we provide insight as to how mechanical coupling can explain the counterintuitive experimental observation made by Boebinger et al. (2018), where reactions with larger sodium ions resulted in a more mechanically reliable structure when compared to reactions with lithium ions.

We begin with the conceptual depiction of a generic diffusion–reaction–deformation process as shown in Fig. 1. The figure depicts the reaction A+CBwhich describes the physical phenomena of diffusing species C reacting with the host lattice A to form the new compound B. The reaction itself is treated in a phase-field sense and taken to occur over a diffuse boundary (light gray region) and is tracked through the normalized phase-field parameter ξ̄ which is formally introduced in Section 2. A value of ξ̄=0 represents the original material A, and a value of ξ̄=1 represents the fully reacted and newly formed compound B, while naturally intermediate values of 0<ξ̄<1 represent the reaction zone. The inset in Fig. 1 shows a conceptual illustration of the reaction zone where the phase α is associated with the unreacted material A, while the phase β is associated with the reacted compound B. We note that the reacted phase β may also act as a host for the diffusing species as shown schematically.

The paper is organized as follows. In Section 2 we introduce mass conservation and formally define our physically motivated phase-field parameter ξ governing the extent of reaction. In Section 3 we describe the kinematics of the problem. We introduce a novel decomposition of the velocity gradient to account for chemically induced deformations. Governing balance laws are developed in Section 4 through the use of the principle of virtual power, and the first and second laws of thermodynamics. The constitutive theory is presented in Section 5 and summarized in its general form in Section 6. In Sections 7 Specialization of the constitutive equations, 8 Governing partial differential equations for the specialized constitutive equations. Boundary conditions we present a specialization of our theoretical framework to modeling conversion electrodes for energy storage.

Numerical simulations are presented in Section 9. First, in Section 9.1 we present diffusion–deformation (without mechanics) simulations to elucidate the kinetically driven and thermodynamically driven regimes of sharp interface formation. In Section 9.2 we simulate the reaction of FeS2 crystals with either lithium or sodium ions and compare to experimental results form the literature. Finally, in Section 9.3 we present a series of simulations aimed at elucidating the important role of surface wetting (surface energy) boundary conditions in capturing experimentally relevant reaction front morphologies. We close with concluding remarks in Section 10.

Section snippets

Mass conservation

Considering the diffusion–reaction problem shown schematically in Fig. 1, we wish to write mass conservation for the generic reaction A+CB.Let c(X,t) denote the number of moles of diffusing species per unit reference volume. In addition, let ξ denote the number of moles of reacted species per unit reference volume with ξ̇ the reaction rate. We may then define the quantity ξ̄=ξcmaxR[0,1] as the extent of reaction such that at ξ̄=1 the reaction has led to consumption of cmaxR moles of species.

Kinematics

We introduce here the kinematic formulation for the description of a deformation resulting from coupled chemical diffusion, reactions and mechanics. Starting from a traditional finite deformation framework, consider a body B with an arbitrary material point in B denoted by X. The motion of B is then a smooth one-to-one mapping x=χ(X,t) with deformation gradient, velocity, and velocity gradient given by1

Governing balance laws

In this section, we develop the governing equations for our theoretical framework including macroscopic and microscopic force balance and thermodynamic laws.

Constitutive theory

We divide the section into energetic and dissipative constitutive equations, along with a discussion on isotropy.

Summary of the general constitutive theory

In this section we summarize our general diffusion–reaction chemo-mechanical theory. The theory relates the following fields:

x=χ(X,t),motion;
F=χ,J=detF>0,deformation gradient;
F=FmFc,multiplicative decomposition of F;
Fc,Jc=detFc>0,chemical distortion;
Fm,Jm=detFm>0,mechanical distortion;
Fm=FeFp,elastic–plastic multiplicative decomposition of Fm;
Fe,Je=detFe>0,elastic distortion;
Fp,Jp=detFp=1,elastic distortion;
L=ḞF1=Lm+FmLcFm1velocity gradient;
Lc=ḞcFc1=Dc+Wcwith
Wc=0
chemical velocity

Specialization of the constitutive equations

The theory presented thus far is quite general. We next present special constitutive equations which are: (i) useful for modeling FeS2 conversion electrodes as will be done in Section 9, and (ii) provide a clearer understanding of the specific capabilities of the theoretical framework.

Governing partial differential equations for the specialized constitutive equations. Boundary conditions

The final set of governing partial differential equations consist of:

  • 1.

    The local macroscopic force balance, Eq. (6.12), viz. DivTR+bR=0,where TR given by (7.15)2, and bR is the non-inertial body force.

  • 2.

    The local mass balance for the diffusing species (6.13) which together with the flux (7.17) gives ċ=Div(mμ)ξ̇with the mobility m given in (7.18), the chemical potential in (7.16) and the reaction kinetics governed by the PDE (8.3) below.

  • 3.

    The reaction kinetics are governed by (7.24) which using the

Numerical simulations

In this Section we detail a set of numerical simulations aimed at both highlighting the important features of our theoretical framework and addressing an engineering problem of relevance. The theoretical framework is implemented in Abaqus Standard (Simulia, 2010) with a custom user-element (UEL) subroutine. In order to make the theoretical framework more amenable to numerical implementation, we have utilized the so called “micromorphic” formulation (cf. Forest, 2009, Di Leo et al., 2014,

Concluding remarks

We have formulated a thermodynamically consistent field theory that couples diffusion of species, bulk sharp interface chemical reactions, and mechanical deformations. The framework was specialized to the engineering problem of relevance of modeling conversion electrodes for energy storage. In particular we demonstrated the manner in which material parameters may be calibrated in a straightforward fashion. The simulated results captured the experimental observations including the change in

CRediT authorship contribution statement

Arman Afshar: Design and implementation of the research, Analysis of results, Writing of the manuscript. Claudio V. Di Leo: Design and implementation of the research, Analysis of results, Writing of the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the National Science Foundation, USA under Award No. CMMI-1825132.

References (65)

  • GanserMarkus et al.

    A finite strain electro-chemo-mechanical theory for ion transport with application to binary solid electrolytes

    J. Mech. Phys. Solids

    (2019)
  • GhasemiArman et al.

    A method to predict energy barriers in stress modulated solid–solid phase transitions

    J. Mech. Phys. Solids

    (2020)
  • GigliottiMarco et al.

    Local shrinkage and stress induced by thermo-oxidation in composite materials at high temperatures

    J. Mech. Phys. Solids

    (2011)
  • GurtinMorton E.

    A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

    J. Mech. Phys. Solids

    (2002)
  • HofmannTobias et al.

    Electro-chemo-mechanical simulation for lithium ion batteries across the scales

    Int. J. Solids Struct.

    (2020)
  • HuJin et al.

    Improve the electrochemical performances of Cr2O3 anode for lithium ion batteries

    Solid State Ion.

    (2006)
  • KonicaShabnam et al.

    A thermodynamically consistent chemo-mechanically coupled large deformation model for polymer oxidation

    J. Mech. Phys. Solids

    (2020)
  • LevitasValery I. et al.

    Anisotropic compositional expansion in elastoplastic materials and corresponding chemical potential: Large-strain formulation and application to amorphous lithiated silicon

    J. Mech. Phys. Solids

    (2014)
  • LoeffelKaspar et al.

    A chemo-thermo-mechanically coupled theory for elastic–viscoplastic deformation, diffusion, and volumetric swelling due to a chemical reaction

    Int. J. Plast.

    (2011)
  • LoeffelKaspar et al.

    On modeling the oxidation of high-temperature alloys

    Acta Mater.

    (2013)
  • MaoYunwei et al.

    A theory for fracture of polymeric gels

    J. Mech. Phys. Solids

    (2018)
  • MinervinoM. et al.

    A coupled experimental/numerical approach for the modelling of the local mechanical behaviour of epoxy polymer materials

    J. Mech. Phys. Solids

    (2014)
  • OkumuraDai et al.

    A general expression for linearized properties of swollen elastomers undergoing large deformations

    J. Mech. Phys. Solids

    (2020)
  • RenZhimin et al.

    Preparation of carbon-encapsulated ZnO tetrahedron as an anode material for ultralong cycle life performance lithium-ion batteries

    Electrochim. Acta

    (2014)
  • SalvadoriA. et al.

    A coupled model of transport-reaction-mechanics with trapping. Part I–small strain analysis

    J. Mech. Phys. Solids

    (2018)
  • SvendsenBob et al.

    Finite-deformation phase-field chemomechanics for multiphase, multicomponent solids

    J. Mech. Phys. Solids

    (2018)
  • TaoYiming et al.

    Fes2 microsphere as cathode material for rechargeable lithium batteries

    Solid State Ion.

    (2016)
  • TolpygoV.K. et al.

    Determination of the growth stress and strain in α-Al2O3 scales during the oxidation of Fe–22Cr–4.8 Al–0.3 Y alloy

    Acta Mater.

    (1998)
  • XuG.N. et al.

    A chemo-thermo-mechanically constitutive theory for thermal barrier coatings under CMAS infiltration and corrosion

    J. Mech. Phys. Solids

    (2019)
  • ZhangXiaoxuan et al.

    A reaction-controlled diffusion model for the lithiation of silicon in lithium-ion batteries

    Extreme Mech. Lett.

    (2015)
  • ZhaoYunong et al.

    A diffusion, oxidation reaction and large viscoelastic deformation coupled model with applications to SiC fiber oxidation

    Int. J. Plast.

    (2019)
  • ZhaoYing et al.

    A review on modeling of electro-chemo-mechanics in lithium-ion batteries

    J. Power Sources

    (2019)
  • Cited by (14)

    View all citing articles on Scopus
    View full text