On certain applications of gradient nanochemomechanics: deformation and fracture of LIB and SGS

Abstract The term gradient nano-chemo-mechanics was introduced to encompass models incorporating higher order couplings between deformation and chemistry at the nanoscale. Along these lines, the article first reviews the basics of a robust theoretical framework developed for such processes focusing on elasticity and diffusion. The classical laws for Hookean deformation and Fickean transport are modified to include extra Laplacian terms and corresponding internal lengths modeling nonlocal interactions. Then, special cases are considered to describe deformation and fracture aspects of new energy materials; namely Li-ion battery (LIB) nanostructured anodes and disclinated metallic microcrystals (DMC). Both of these material systems are characterized by a high degree of spatial gradient structures (SGS) with extended surface for energy storage and catalysis applications.


Introduction
The terms "nanomechanics" and "chemomechanics" were introduced by the last author in 1995 [1] and 1980 [2] to point out the need for extending existing continuum mechanics models to the nanoscale, as well as the need for developing coupled deformation-di usion models when mechanical stress and chemistry are present on an equal footing. Recent advances on nanoscience and nanotechnology have tremendously accelerated the development of such models, mainly due to the exponential growth of sophisticated multiscale computer codes and corresponding laboratory probes enabling to compare theory, simulation, and experiment at the same scale of observation.
A most notable contribution was the extension of classical Hooke's law of elasticity to include an extra gradient term (the Laplacian of Hookean stress) multiplied by a corresponding internal length parameter to account for weak deformation nonlocality. The resulting model [3], commonly known as gradient elasticity (or GradEla model) was used to eliminate stress singularities in dislocation lines and crack tips, as well as to interpret size e ects in elastic structural components, not captured by classical elasticity theory. A recent review with a long list of related references can be found in [4]. Along similar lines, the classical Fick's law of di usion was extended to include an extra gradient term (the Laplacian of Fickean ux) multiplied by a corresponding internal length parameter to account for weak di usion nonlocality. The resulting model [5], commonly known as higher order di usion, was used to interpret experimental data for grain boundary di usion and nanopolycrystals, not captured by classical di usion theory. A recent review with a long list of related references can be found in [6].
The aforementioned models of gradient elasticity and higher order di usion have not been su ciently considered to address coupled deformation-di usion problems at the nanoscale where the interaction between elastic internal lengths and di usion internal lengths need to be accounted for. This issue becomes especially important in considering chemomechanical processes at small volumes, such as those occurring in microstructured components used in advanced applications for energy storage and catalysis devices. Such a coupled nanochemomechanics framework will be outlined here for chemoelastic deformations. It will be used then to discuss chemomechanical damage in rechargeable lithium-ion battery (LIB) nanostructured anodes, as well as deformation and fracture of pentagonal disclinated microcrystals (DMC). LIBs is an important class of advanced energy storage devices (ESD). DMCs is an important class of objects with extended sur-face and spatially gradient structure (SGS). In this connection, it is noted that some background on related aspects can be found in published joint work by the rst and the last authors [7][8][9][10].
The plan of the paper is as follows. In Section 2 a brief review of open problems in LIB anodes is presented and the same is done for SGS materials. In Section 3 the basics of gradient elasticity and higher order di usion are presented, along with the governing equations accounting for higher order elastodi usion couplings. In Section 4 representative results on both LIB and SGS chemomechanical aspects are presented. Finally, in Section 5 some closing remarks for this initial e ort on modeling deformation and fracture in LIB and SGS components are given, along with some open remarks for future directions.

Open problems in LIB and SGS
(i) LIB Anodes: In Figure 1, a representative con guration of a LIB device is shown, along with electrochemical reactions taken place in the anodes and cathodes. Commercially available LIBs use graphite (C) as anode. Current battery developers focus on replacing C with Sn and Si which lead to higher capacities. In Figure 2, the cyclic response of nanostructured anodes and electron microscopy images are provided. It is noted, however, that during lithiation/delithiation the active nanoparticles (Si, Sn or Al) embedded in an inert matrix (C, ceramic or polymer) undergo large volumetric expansions/contractions (theoretically up to 400% for freely expanding and fully lithiated Si active particles) which result to fracture and capacity fade. These volume changes may be suppressed by reducing the size of active nanoparticles to prevent chemomechanical damage and capacity fade upon electrochemical cycling. Obviously, this is a size e ect problem which can be conveniently addressed within the coupled nanochemomechanical framework outlined in Section 3. Some representative results of such considerations are provided in Section 4. (ii) SGS Objects: In Figure 3, various SGS objects as produced by electrodeposition under mechanical steering are presented. These structures are characterized by pentagonal symmetry (not predicted by classical crystallography) due to internal disclination e ects that emerge during processing and result to an extended surface with unusually high chemical activity. In Figure 4, a pentagonal Cu microparticle is shown, along with an internal void revealed by etching. The pentagonal symmetry can be described in terms of a disclination defect in an initially solid microparticle, the stress eld of which can act as a vacancy accumu- lator along the disclination line, resulting to the formation of an internal void. The competition of the stress eld generated by the disclination and that due to the external surface can lead to a fracture criterion as shown in Figure 5 by using a previous purely classical elasticity framework. Again, this is a size e ect problem which can be conveniently addressed within the coupled nanochemomechanical framework outlined in Section 3. Some representative examples of such considerations are provided in Section 4.   The critical parameter ξ (the ratio of the internal over external radius) of a hollow ISP particle as determined by classical elasticity theory. For details, the reader can consult [10,20,24] and references quoted therein.

Gradient elastodi usion theory
For elastic deformations, the term ε ∇ [λεmm δ ij + Gε ij ] -where ε denotes an elastic internal length (IL), ε ij is the elastic strain, and (λ, G) are the Lamé constants -is incorporated into classical Hooke's law. Previous results (see [4] and references quoted therein) show that the re-sulting internal length gradient (ILG) model can eliminate stress/strain singularities from dislocation/disclination lines and crack tips and interpret elastic size e ects. For elastic deformations at the atomic scale (near dislocation lines in crystals), ε relates to the subatomic con guration and electronic state (through DFT calculations), while at the microscale ε relates to particle size/spacing (through MD simulations). For di usion problems the ILs enter through the additional term d ∇ j, which generalizes the classical Fick's law ( d is a di usional internal length and j denotes the di usion ux), in a manner similar to the Cahn-Hilliard theory [11] for spinodal decomposition.
The above ideas can be extended in a straightforward manner for coupled elastodi usion processes accounting for higher order internal length couplings. The standard equations that are usually employed to model coupled elasto-di usion processes (without accounting for internal length couplings) are of the form σ ij = λεmm δ ij + Gε ij − αρδ ij , j = −D∇ρ + Mρ∇σ ii for the chemostress σ ij and the mechanodi usive ux j, where the coe cients (α, M) denote chemomechanical coupling constants and D is the di usivity. The elds (ρ, ε ij ) denote concentration of the di using chemical agent and mechanical strain, respectively. Since these constitutive equations do not contain higher order ILs, related chemomechanical size e ects and pattern formation may not be captured.
Within our Laplacian-based ILG formulation, it turns out that the above constitutive equations are generalized by replacing σ ij with σ ij − σ ∇ σ ij ; ε ij with ε ij − ε ∇ ε ij ; and ρ with ρ − ρ ∇ ρ, with ( σ / ε , ρ) denoting stress/strain and di usional ILs. Under suitable assumptions, it is possible to uncouple the deformation and chemical elds by rst computing a "ground" hydrostatic stress component σ h from a conventional or a gradient elasticity theory, and then derive the concentration ρ from a stress-assisted di usion equation of the form N is a new phenomenological constant accounting for the e ect of hydrostatic stress on di usivity. This model with ρ = has been used extensively to model hydrogen embrittlement and stress corrosion cracking in metals [12], and it is compatible with a mechanodi usive ux of the form j = − D + Nσ h ∇ρ + (M + N)ρ∇σ h provided that ∇ σ h = . This constraint is removed here and the model is adopted to consider chemomechanical damage and failure in LIB anodes. As a "ground" hydrostatic stress, the constitutive equation σ h = σ ij ε ij is used, where σ ij = σ c ij − ε ∇ σ c ij ; σ c ij = λεmm δ ij + Gε ij − α ij ρ; α ij = λε mm δ ij + Gε ij and the tensor ε ij relates to the elastic mis t strains ρε ij , which are assumed transversely isotropic for the Li/Si sys-tem. In this connection, it is noted that for isotropic materials, α ij reduces to α ij = αδ ij shown above in the standard form for the chemostress.

Benchmark problems for LIB & SGS
In this section, we present some results on model con gurations pertaining to LIB nanostructured anodes and SGS pentagonal objects. We focus, in particular, on size e ects in Si-nanowires (NWs) since these objects are currently explored as potential candidates for LIB anodes, as well as on size e ects in hollow icosahedral small particles (ISPs) since these objects are currently explored as potential candidates for nanocatalysis. (i) Size e ect on failure of Si NW: Failure during lithiation of a single Si nanowire has been addressed by developing a concentration-dependent modi cation of the so-called "universal fracture criterion" [13] to account for the strength decrease as the lithiation proceeds due to the weakening of Si-Si covalent bonding and the formation of metallic or mixed ionic-covalent bonds [14,15]. An equivalent stress measure σeq has been de ned for this criterion, with the onset of failure corresponding to values σeq /τ ≥ . The quantity τ is given by τ = ασ F T , where the intrinsic parameter α depends on the compression-tension strength ratio σ F C /σ F T . This ratio varies as a linear function of Li concentration between the values α ∼ = . for ρ = and α = for ρ = with the latter corresponding to the boundary between metallic and ionic or covalent bonding (see Figure 4 in [13]). For the tensile strength, the theoretical limit approximation σ F T ∼ = E/ is adopted and the dependence of σ F T on lithium concentration is incorporated by employing the linear rule of mixtures GPa, E ∼ = GPa are the Young's modulus values found in [15] for Si and Li . Si, respectively. Moreover, f (ρ) = υρ/(υρ + −ρ) is the volume fraction with υ ∼ = . denoting the molar volume ratio between the fully lithiated and the unlithiated phase.
It has been found that smaller specimens exhibit lower equivalent stresses during the entire lithiation process, i.e. they are less susceptible to failure ( Figure 6). More important, there is a critical value d/ ε cr ∼ = . at which the ratio max(σeq)/τo becomes smaller than 1 and thus, the theoretical limit fracture level is suppressed. Employing the value ε ∼ = . nm, which is estimated by tting the predictions of the model to experimental results on the thickness of the lithiated/unlithiated Si interface, a critical specimen diameter dcr ∼ = nm is obtained, which is quite small (~19 times the Van der Waals diameter of silicon atoms). Following the same procedure used for single silicon specimens, the theoretical failure stress of the active material in a nanocomposite anode made of silicon nanowires (active sites) of diameter d Si embedded in a glass matrix can be estimated. In the absence of any data concerning the elastic internal length m of the matrix, the inactive material is assumed to obey classical linear elasticity, i.e. m = .
The maximum equivalent stress for the entire lithiation process does not only depend on the ratio d Si / ϵ, but also on the ratio dm /d Si between the diameters of the unit cell and the active site which, in turn, is directly related to the density of silicon nanowires in the nanocomposite material. More important, as shown in Figure 7, the critical value (d Si / ϵ)cr for fracture suppression is quite larger than that of a free expanding nanowire and it is increased as dm /d Si is decreased, i.e. the density of the active sites becomes larger. Employing the aforementioned value of ϵ ∼ = . nm, it gives a range between (d Si )cr ∼ = . nm for dm /d Si = and (d Si )cr ∼ = . nm as dm /d Si → . It is noted that while theoretical fracture of silicon is suppressed for any pair of values below the curves of Figure 7, the failure of both the active site and the glass matrix is suppressed only in the grey region thereof. This gives a minimum value of~44% for the Si volume fraction in the closest (hexagonal) packing arrangement of the unit cells, combined with a maximum of d Si ∼ nm.

(ii) Size e ect on failure of a hollow Cu ISP:
In this section, a stress gradient model of the form σ − ∇ σ = σ c is employed to discuss size e ects occurring during the fracture of hollow ISPs produced by electrodeposition under mechanical steering. Under radial symmetry, this leads to the following di erential equations for the components σrr  . (1) This system is solved for a hollow disclinated ISP, as well as for a spherical shell subjected to uniform internal pressure P. For the former, the classical elastic stress eld reads [16] σ c rr = Gκ +ν −ν ln r where κ is the power of the Marks-Yo e disclination, G is the elastic shear modulus, ν is the Poisson's ratio, R is the outer radius of the particle and R is the radius of the void.
For an internally pressurized spherical shell the respective classical eld reads For both problems, the homogeneous natural boundary condition ∂nσ := σ ij,m nm = is taken at the inner and outer surfaces. This choice is the most widely used when dealing with gradient elasticity [17,18]. The same boundary condition has been assumed in [19] by employing suitable thermodynamics arguments for the Eringen's nonlocal elasticity or stress gradient model. In this connection, it is noted that this di ers from the conditions employed in [10] to determine the stress pro les for both solid and hollow ISP. The derived solutions of Eq. (1) are not provided here due to their complicated form but in both cases, they can be recast into the forms σrr = σ c rr + σ g rr , σ θθ = σ c θθ + σ g θθ . As shown in [10], these, in turn, lead to the following expression for the total energy of an ISP with a void in its center where γ is the surface energy of the material and E g d is the part of the strain energy introduced by the presence of the gradient term in the constitutive equation. For the aforementioned homogeneous boundary condition, E g d may be calculated from the relation R R σ g rr ε c rr + σ g θθ ε c θθ r dr, where ε c rr , ε c θθ is the classical strain eld. Following a similar analysis with that given in [20] in the framework of classical elasticity, the pressure on the inner surface of the ISP particle may be calculated by using the relation P ISP = E ISP /V, where V = π(R − R ) is its volume. The respective pressure P MAX at the point of fracture is determined by the solution of Eq. (1) for the internally pressurized shell, yielding where σ is the ultimate stress of the material making up the spherical shell and ξ := R /R ,¯ := /R , whileÎ n/ (x) := π/ xI n/ (x) with I n/ (x) denoting the respective modied Bessel functions. Moreover, C (¯ , ξ ) and C (¯ , ξ ) are integration constants determined by the aforementioned boundary conditions but they are not given here due to their lengthy form. It is also noted that unlike [20], the thick-shell solution has been adopted here rather than its thin-shell simpli cation.
For electrolytic copper the material parameters found in [20] have been used apart from κ which has been misprinted therein although the approximation κ = . of the correct value κ = . [21] has been used in their calculations. In Figure 8, the functions P ISP (ξ ) and P MAX (ξ ) for di erent internal over geometrical length ratios¯ := /R are depicted. As shown, larger gradient e ects (i.e. greater¯ values) lead to larger ξc values at the point of fracture P ISP = P MAX and thus, larger voids are attainable.

Conclusions
A preliminary e ort has been described to show that coupled chemomechanical models are necessary to model elastodi usion processes in small volume material components used in current energy storage and catalysis technologies. The focus was on nanostructured LIB anodes and SGS objects with pentagonal symmetry. The results indicate that such models can capture size e ects that are observed in these components and can potentially lead to protocols and design criteria to prevent failure and optimize their performance in related applications.