Abstract

Deformation and stress in battery electrode materials are strongly coupled with diffusion processes, and this coupling plays a crucial role in the chemical and structural stability of these materials. In this work, we performed a comparative study of the mechanical characteristics of two model materials (lithiated and sodiated germanium (Ge)) by nanoindentation. A particular focus of the study was on the indentation size effects and harnessing them to understand the chemo-mechanical interplay in these materials. While the quasi-static measurement results showed no significant size dependence, size effects inherent in the nanoindentation creep response were observed and utilized to investigate the deformation mechanism of each material. Supplemented by computational chemo-mechanical modeling, we found that lithiated Ge creeps through a stress-gradient-induced diffusion (SGID) mechanism but a model combining the SGID and conventional shear transformation deformation (STD) mechanisms was needed to capture the creep behavior of sodiated Ge. Broadly, this work reveals the importance of stress-diffusion coupling in governing the deformation of active electrode materials and provides a quantitative framework for characterizing and understanding such coupling.

1 Introduction

For the goal of improving capacity of alkali-ion batteries (AIBs) such as lithium-ion batteries (LIBs) and sodium-ion batteries (NIBs), there are promising high-capacity anode materials, consisting of a range of group 14 (Si, Ge, and Sn) [1,2] and group 15 elements (P, As, Sb, and Bi) [37]. A major downside to the high energy density of these electrode materials is that during reaction with alkali species they suffer from extreme volumetric expansions [8,9]. These large expansions, coupled with alkali ion diffusion, can create irreversible morphological and chemical changes that severely compromise battery performance. Therefore, understanding the mechanical behavior of these active materials plays a vital role in designing and making rechargeable batteries with high cyclability.

In an effort to address the aforementioned mechanical issue, experimental investigations have been performed regarding the specifics of electrochemical reaction kinetics [1015], reaction-induced deformation [8,1618] and stress [1927], stress-diffusion coupling [28], and chemical-concentration-dependent mechanical properties [2931]. The experimental methods employed run the range from microscale optical measurement [3234] and X-ray tomography [8,17,35,36] to nanoscale in situ scanning electron microscopy [37] and transmission electron microscopy [38]. Depth sensing nanoindentation, known for its robustness and versatility in testing small-volume samples, has also been increasingly applied to the mechanical characterization of electrochemically active battery materials. There are multiple examples for its use in measuring Young’s moduli [3942], hardness [41,42], time-dependent mechanical properties [43,44], and fracture toughness [30,45] of high-capacity anode materials. Despite these efforts, the fundamental deformation mechanisms of these materials, and how they are coupled to electrochemical processes are still not fully understood.

In our earlier work, we showed that it is possible to characterize the stress-diffusion coupling in lithiated germanium (Ge) by nanoindentation [44]. In the current work, we extended our study to sodiated Ge and performed a comparative study of the mechanical characteristics of lithiated and sodiated Ge as two model high-capacity anode materials. Unlike the more widely studied silicon anode for LIBs, Ge can readily be lithiated or sodiated, making it suitable for both LIB and NIB applications. A comparative study of lithiated and sodiated Ge would allow us to examine the influence of reaction with different alkali species on chemo-mechanical properties. Another important aspect considered in this work is the presence of any size effects in nanoindentation experiments, which was not considered in our earlier work. Nanoindentation size effects refer to variations in mechanical characteristics with varying indentation depth and have been extensively studied for single-crystal and polycrystalline metals [4648], ceramics [49], and polymers [50]. To the best of our knowledge, such size effect phenomenon has not been previously reported for battery electrode materials. Here, we conducted indentation experiments at a range of peak forces to quantify and take advantage of size effects for identifying quasi-static and creep deformation mechanisms in the model materials. The creep experiments were also supplemented with continuum chemo-mechanical modeling to characterize and understand the strong coupling between mechanical stress and alkali ion diffusion. In addition, variable-temperature nanoindentation testing was performed to investigate the effects of temperature on chemo-mechanical behavior.

2 Experimental Methods

2.1 Sample Fabrication.

A 99.9% pure titanium (Ti) sheet of 0.5 mm thickness was cut into 10 × 10 mm squares and was single-side polished to a surface finish of 0.5 nm RMS. The polished Ti specimens were used as a supporting substrate on which a 20 nm thick adhesion promoting layer of Ti was coated by radio frequency (RF) sputtering in an argon (Ar) environment, followed by RF sputter coating of a Ge electrode layer of 1.06 μm thickness. The sputtering was performed at a power of 100 W under 1.33 Pa pressure of Ar. Earlier work has shown that Ge sputtered under such condition is amorphous [51]. The Ti and Ge film thicknesses after sputtering were characterized using a Tencor P15 profilometer (KLA Corp., Milpitas, CA, USA).

For lithiation of the amorphous Ge (a-Ge) thin-film electrodes, a custom-made electrochemical cell was assembled in an argon-filled glovebox with both oxygen and moisture levels kept below 0.1 ppm. The cell used Li foil as the counter electrode and 1 mol/l of lithium hexafluorophosphate in 1:1 (wt%) ethylene carbonate (EC):dimethyl carbonate (DMC) as the electrolyte. Sodiation of Ge was performed with the same electrochemical cell using Na foil as the counter electrode and 0.667 mol/l of sodium perchlorate (NaClO4) in 1:2 (wt%) EC:DMC as the electrolyte. The assembled cell was galvanostatically cycled with a battery tester (UBA 5, Vencon Technologies, ON, Canada) to reach a targeted Li or Na to Ge molar ratio of 0.6. The cycling current density used (12.5 μA/cm2) was equivalent to a low charge/discharge rate of 1/63 C for lithiation and 1/17 C for sodiation, which were necessary for a uniform distribution of Li or Na within the reacted Ge electrodes.

2.2 Nanoindentation.

After electrochemical reaction, the lithiated or sodiated Ge electrode was submerged in anhydrous DMC for five hours and was then wipe cleaned with cotton balls inside an argon-filled glovebox. In this way, the solid electrolyte interphase (SEI) and any remaining electrolyte salt on the electrode surface could be removed. Both lithiated Ge and sodiated Ge are highly reactive on exposure to air, so nanoindentation testing was conducted with a modified commercial nanoindenter (Hysitron Ubi-1). The nanoindenter was equipped with a variable-temperature sample stage and was situated inside an argon-filled glovebox for environmentally controlled testing of air sensitive materials. More details on the nanoindentation system can be found in Ref. [52].

Two lithiated Ge (Li0.6Ge) samples were prepared, one for room temperature testing and the other for elevated temperature testing at 80 °C using a resistive heating stage. One sodiated Ge (Na0.6Ge) sample was used for room temperature testing. Elevated temperature tests of Na0.6Ge at 80 °C were attempted but proved problematic. At this temperature, side reactions with trace amounts of moisture and oxygen (<0.1 ppm) in the glovebox were found to occur rapidly (<15 min), which changed the material properties of the surface and resulted in sullied indentation results.

Two separate load functions were used for nanoindentation testing. First, a triangular load function (Fig. 1(a)) was used for characterizing the quasi-static mechanical properties, including the hardness and reduced elastic modulus. Second, a trapezoidal Nanoscale Dynamic Mechanical Analysis (NanoDMA) load function (Fig. 1(b)) was used for creep testing. The NanoDMA function consists of a linear loading portion, a load holding portion in which the indentation load oscillates sinusoidally around a mean load with a fixed frequency f and small amplitude ΔP, and a linear unloading portion. The hold portion allows for continuous measurement of the contact stiffness at the mean load, from which the creep displacement can be deduced with the minimum influence of thermal drift.

Fig. 1
Nanoindentation load functions employed in this study: (a) quasi-static triangular load function to characterize elastic modulus and hardness and (b) NanoDMA load function where the function oscillates around a mean load at a fixed frequency of 210 Hz to characterize creep behavior
Fig. 1
Nanoindentation load functions employed in this study: (a) quasi-static triangular load function to characterize elastic modulus and hardness and (b) NanoDMA load function where the function oscillates around a mean load at a fixed frequency of 210 Hz to characterize creep behavior
Close modal

3 Results and Discussion

Figure 2 shows the voltage profiles for electrochemical lithiation and sodiation of a-Ge. The portion of each curve, before the spike, was a discharge/charge cycle to allow the initial formation of SEI on the electrode surface. This step was necessary for accurately controlling the amount of lithium or sodium inserted into the Ge electrode because the SEI formation could consume some active material. After the SEI formed, the electrode was charged to 1.5 V to remove possible mobile ions from the electrode material, and then discharged for 10 h to the desired molar ratio of 0.6. During the final discharge step, the lithiation voltage curve continued to decrease over time while the sodiation voltage curve eventually slightly increasing trend near the end of the process. This anomalous voltage increase was consistently observed in multiple sodiated Ge samples in our experiment, and is believed to be due to a combined effect of stress-potential coupling and stress relaxation in the electrode film. The mechanical stress in a large-volume-change electrode is coupled to its electric potential, as shown by earlier work on lithiated silicon [19,20]. This stress-potential coupling has a positive coupling coefficient, meaning that an increase in stress causes an increase in electric potential. Sodiation of a Ge thin-film electrode results in large compressive biaxial stress, which lowers the electrode potential compared to its stress-free state. During prolonged sodiation process, the magnitude of the compressive film stress may decrease due to viscoplastic stress relaxation as well as chemical-concentration-dependent yield strength. The decrease in magnitude of the compressive stress translates to an increase in stress value, and therefore a possible overall increase in electrode potential.

Fig. 2
(a) Electrochemical charge and discharge curves for lithiation and sodiation and (b) film thicknesses of pure a-Ge, Li0.6Ge, and Na0.6Ge. The initial portion of each curve in (a) corresponds to an SEI pre-formation step set to minimize the error of controlling how much lithium or sodium was inserted.
Fig. 2
(a) Electrochemical charge and discharge curves for lithiation and sodiation and (b) film thicknesses of pure a-Ge, Li0.6Ge, and Na0.6Ge. The initial portion of each curve in (a) corresponds to an SEI pre-formation step set to minimize the error of controlling how much lithium or sodium was inserted.
Close modal

Lithiation or sodiation of the a-Ge films resulted in increases in film thickness, which was determined by finding edges of the reacted Ge films and measuring their surface morphologies with the in situ tip scanning function of the Hysitron nanoindenter. The difference between the original a-Ge film and the lithiated and sodiated films is depicted in Fig. 2(b). For the same molar concentration, Li0.6Ge saw a lower increase in film thickness, with a 42.7% increase, than Na0.6Ge, which had a 64.3% increase. The thickness difference arose because Na has a larger atomic volume (23.7 cm3/mol) than Li (13.1 cm3/mol). Assuming the film thickness linearly varies with the molar ratio, Li0.6Ge and Na0.6Ge have theoretical expansions of 48% and 75.6%, respectively. The experimentally measured thicknesses compare reasonably well with these theoretical volumetric expansions.

To investigate the elastic and plastic properties of the pristine and reacted a-Ge films, quasi-static nanoindentation testing was performed using the triangular load function shown in Fig. 1(a). Figure 3 shows representative indentation load versus displacement curves for each material system tested at room temperature (RT, 25 °C). Under the same peak load of 2000 μN, Li0.6Ge has a slightly larger maximum indentation displacement than pristine Ge. In contrast, Na0.6Ge is significantly softer, experiencing around three times more indentation depth than pristine Ge. Raw indentation curves were obtained for three different cases (Li0.6Ge at RT and 80 °C, and Na0.6Ge at RT) and varying peak loads (Pmax = 1000, 1500, 2000, and 2500 μN). These curves were processed with the Oliver–Pharr method [53], resulting in the plots of reduced modulus and hardness versus indentation depth shown in Fig. 4. The flatness of all the curves indicates that there is no significant size dependence in each case. Lithiation or sodiation of Ge usually results in amorphous reaction phases, and crystallization has only been observed for LixGe at fully lithiated state (x = 3.75) [54]. Li0.6Ge and Na0.6Ge tested here are hence expected to be amorphous and deform plastically by the conventional shear transformation deformation (STD) mechanism, in which a group of atoms move together under an applied shear stress. The STD mechanism has an intrinsic length scale defined by the size of shear transformation zones, which is on the order of a few nanometers [55]. This size is very small compared to the range of indentation depths (>75 nm) obtained in our nanoindentation tests. Therefore, the mechanical properties measured from nanoindentation are essentially size independent, as shown in Fig. 4.

Fig. 3
Representative indentation load versus displacement curves of pure Ge, Li0.6Ge, and Na0.6Ge at room temperature. Na0.6Ge displaces much further than Ge and Li0.6Ge.
Fig. 3
Representative indentation load versus displacement curves of pure Ge, Li0.6Ge, and Na0.6Ge at room temperature. Na0.6Ge displaces much further than Ge and Li0.6Ge.
Close modal
Fig. 4
Quasi-static nanoindentation results for sodiated and lithiated Ge electrodes tested at peak indentation loads of 1000, 1500, 2000, and 2500 μN: (a) reduced modulus and (b) hardness measurement results
Fig. 4
Quasi-static nanoindentation results for sodiated and lithiated Ge electrodes tested at peak indentation loads of 1000, 1500, 2000, and 2500 μN: (a) reduced modulus and (b) hardness measurement results
Close modal

The reduced modulus values of sodiated Ge shown in Fig. 4(a) are roughly half those of lithiated Ge. Figure 4(b) shows a similar trend for hardness, with sodiated Ge’s hardness being only one-fourth of lithiated Ge. This comparison is consistent with what is anticipated from the raw indentation curves in Fig. 3. Concerning temperature dependence, lithiated Ge shows relatively little change in mechanical behavior with elevated temperature. Figure 4(a) shows that the room temperature and 80 °C values of reduced modulus of Li0.6Ge are nearly overlapping, while in Fig. 4(b) there is a slight decrease in hardness with increase in temperature. The near temperature independence of the reduced modulus values is attributed to the small temperature variation from 25 °C to 80 °C. Both temperatures are well below the melting point, and the temperature variation is too small to cause an appreciable change in elastic modulus that can be detected by nanoindentation measurement.

In predicting the properties of a reacted electrode material, a simple linear rule of mixtures is often assumed. By this rule, the material’s properties are weighted average of the properties of each constituent. Figure 5 compares the room temperature experimental values of reduced modulus and hardness for pure Ge, pure Li, pure Na, Li0.6Ge, and Na0.6Ge, and those predicted by the rule of mixtures. The data points used for Li0.6Ge and Na0.6Ge were obtained by averaging the reduced modulus and hardness values across all indentation depths in Fig. 4. Figures 5(a) and 5(b) show a comparison using the atomic fraction of Li or Na, fA = x/(1 + x). It can be seen that the hardness of lithiated Ge is close to the linear line, but none of the others. Conversely, Figs. 5(c) and 5(d) were generated based on the volume fraction, fV = (xVL/N)/(VGe + xVL/N), which is a more commonly used measure of mixing. Here, x is the molar ratio of Li/Na to Ge, and VL/N and VGe are the partial atomic volumes of Li/Na and Ge, respectively. While the reduced modulus of lithiated Ge agrees with the linear rule of mixtures, sodiated Ge does not follow it closely. Concerning the hardness data in Fig. 5(d), both lithiated and sodiated Ge are positioned well below the linear line. Earlier work on the effects of lithiation, sodiation, and potassiation of a conversion-type electrode (FeS2) demonstrated that alkali ion insertion resulted in non-linear concentration dependence of the mechanical properties, and that sodiated and potassiated structures were less likely to fracture [56]. A similar non-linear behavior is observed in this work for alloying-type Ge electrodes.

Fig. 5
Rule of mixtures for comparing lithiated and sodiated Ge with their constituent components: (a) and (c) reduced modulus and (b) and (d) hardness measurement results. (a) and (b) are presented by atomic fraction, while (c) and (d) are by volume fraction.
Fig. 5
Rule of mixtures for comparing lithiated and sodiated Ge with their constituent components: (a) and (c) reduced modulus and (b) and (d) hardness measurement results. (a) and (b) are presented by atomic fraction, while (c) and (d) are by volume fraction.
Close modal
In addition to the quasi-static mechanical properties of lithiated and sodiated Ge, dynamic nanoindentation was conducted to study their time-dependent behavior, using the NanoDMA load function from Fig. 1(b) with a fixed oscillation frequency of 210 Hz and varying peak loads. Figure 6(a) shows representative room temperature curves of indentation load versus displacement of Li0.6Ge and Na0.6Ge at a peak load of 2000 μN. Both materials exhibit a significant increase in indentation displacement over time during the load-hold period, which is indicative of large creep deformation. The creep portion of each measured indentation curve, defined as the additional displacement that occurs during the load-hold period, was found to fit well to the following analytical formula as follows:
Δh(t)=Δho[1e(t/tc)]
(1)
Fig. 6
(a) Nanoindentation load versus displacement curves of Li0.6Ge and Na0.6Ge at room temperature and (b) analytical fitting of an experimental creep curve versus time of Li0.6Ge using an exponential equation (Eq. (1)). The quantities of characteristic time tc and overall creep displacement Δho are represented.
Fig. 6
(a) Nanoindentation load versus displacement curves of Li0.6Ge and Na0.6Ge at room temperature and (b) analytical fitting of an experimental creep curve versus time of Li0.6Ge using an exponential equation (Eq. (1)). The quantities of characteristic time tc and overall creep displacement Δho are represented.
Close modal

Here, the fitting variables are the amount of creep experienced, Δho, and the characteristic time, tc, at which the creep displacement reaches 63.2% of its total amount. Figure 6(b) shows a representative creep curve applying this fitting, and how the parameters correspond to the shape of the curve. For the purpose of analysis, Δho was normalized by the indentation depth at which the creep portion began, ho, resulting in a dimensionless term, Δho¯=Δho/ho, that characterizes the total amount of creep deformation.

Figure 7 summarizes all the creep parameters determined by NanoDMA testing of lithiated and sodiated Ge. The data points are plotted against indentation depth in order to examine possible size effects. Figure 7(a) shows that the change in temperature does not significantly alter the Δho¯ values of lithiated Ge. However, the role of temperature on the tc values of lithiated Ge is clearly shown in Fig. 7(b), with the 80 °C values distinctly lower those at the room temperature. Concerning size effects, the Δho¯ values of lithiated and sodiated Ge show no significant dependence on indentation depth. On the other hand, the characteristic times, tc, of both materials are strongly dependent on indentation depth.

Fig. 7
Characteristic creep parameters of Li0.6Ge and Na0.6Ge determined from raw nanoindentation curves
Fig. 7
Characteristic creep parameters of Li0.6Ge and Na0.6Ge determined from raw nanoindentation curves
Close modal

An amorphous solid may creep through atomic diffusion via free volume at low stress or STD at high stress [57]. However, none of these conventional mechanisms has an intrinsic length scale that can be used to explain the measured strong size dependence of tc. To understand the experimentally observed creep behavior, we employed a chemo-mechanical finite element model to simulate the response of lithiated and sodiated Ge to nanoindentation. This model accounts for finite strain, elastoplastic flow, as well as full two-way coupling between ion diffusion and mechanical stress (see the  Appendix for model details). The previously characterized elastic and plastic properties were incorporated into the model, and the diffusivity of the mobile species, D, and stress-diffusion coupling coefficient, α, were treated as free constitutive parameters. Here, α is a dimensionless parameter that is related to the composition dependence of the activity coefficient and modulates the ion flux due to hydrostatic stress gradient (see Eqs. (A5) and (A7)). For an ideal solid solution, the activity coefficient is equal to one and therefore α = 1. In presence of solution non-ideality, the α value would deviate from unity.

Figure 8(a) shows the simulated nanoindentation creep curves of lithiated Ge at two different peak loads (Pmax = 1000 and 2500 μN), assuming rate-independent J2 plasticity and D = 10−10 cm2/s, and α = 0.1. The characteristic times of the two cases differ significantly at 3.15 s and 6.86 s, respectively. As shown in Fig. 8(c), when the sharp indenter tip is pressed into the electrode material, it creates a high gradient in hydrostatic stress that drives an outward Li flux. This causes volumetric shrinkage beneath the tip and therefore induces creep strain. It is also observed that, due to the geometric self-similarity of the indenter tip, the stress fields for Pmax = 1000 and 2500 μN at the beginning of load holding are similar to each other. However, the gradient in hydrostatic stress for Pmax = 1000 μN has a larger magnitude because of the smaller indentation depth, leading to higher initial Li flux and consequently larger creep strain rate around the indent. Figure 8(d) shows that both the Li flux and creep strain rate are reduced as time progresses, due to the increased diffusion impedance caused by the growing Li concentration gradient beneath the indenter tip. We can also see that at t = 8 s, the creep strain rate for Pmax = 1000 μN becomes much lower than that for Pmax = 2500 μN. From these modeling results, the origin of the size dependence/independence of tc and Δho¯ is clear. The indentation depth sets an important characteristic length for both stress gradients and mobile ion diffusion. A smaller indentation depth intensifies the gradient in hydrostatic stress to drive a higher initial diffusion flux, and it also sets a shorter diffusion path, leading to a shorter characteristic time for diffusion-induced creep. On the other hand, the value of Δho¯ is dictated by the balance between the diffusion driving force arising from the gradient in hydrostatic stress and the resisting force due to the chemical concentration gradient. This balance does not exhibit a size dependence, since both gradient terms are inversely related to the indentation depth.

Fig. 8
(a) Simulated creep curves of Li0.6Ge under two different indentation peak loads (Pmax = 1000 and 2500 μN) using the SGID model with a diffusivity value of D = 10−10 cm2/s and a stress-diffusion coupling coefficient of α = 0.1, (b) schematic of the 3D computational model showing a cross-section used in (c) and (d) for visualization of computational results, (c) vector plots of ion flux and contour maps of hydrostatic pressure (p, the negative of hydrostatic stress) and equivalent strain rate (ε˙eq) at t = 0.01 s and (d) similar plots to those in (c) at t = 8 s. The equivalent strain rate is defined as ε˙eq=2ε˙ijε˙ij/3. The horizontal arrows in (c) and (d) represent reference fluxes of 5000 and 150 nm−2/s, respectively.
Fig. 8
(a) Simulated creep curves of Li0.6Ge under two different indentation peak loads (Pmax = 1000 and 2500 μN) using the SGID model with a diffusivity value of D = 10−10 cm2/s and a stress-diffusion coupling coefficient of α = 0.1, (b) schematic of the 3D computational model showing a cross-section used in (c) and (d) for visualization of computational results, (c) vector plots of ion flux and contour maps of hydrostatic pressure (p, the negative of hydrostatic stress) and equivalent strain rate (ε˙eq) at t = 0.01 s and (d) similar plots to those in (c) at t = 8 s. The equivalent strain rate is defined as ε˙eq=2ε˙ijε˙ij/3. The horizontal arrows in (c) and (d) represent reference fluxes of 5000 and 150 nm−2/s, respectively.
Close modal

Figure 9(a) shows the experimental values of Δho¯ for lithiated Ge compared to the simulated values of a fixed diffusivity (D) value of 10−10 cm2/s and a sweep of stress-diffusion coupling (α) values. The simulated Δho¯ values do not show significant dependence on ho, in consistent with the experimental observation. Meanwhile, with increasing α, there is a monotonic trend of increasing Δho¯. Figure 9(b) depicts the experimental values of tc for lithiated Ge, which are in comparison to the simulated values of a fixed α value of 0.1 and a sweep of D values. The simulated curves see an inverse trend, where with increasing diffusivity the tc values decrease. This trend is expected since increasing the diffusivity allows the stress-gradient-induced diffusion (SGID) and the associated creep deformation to occur in less time. The experimental and simulated characteristic times for creep both show a strong size effect, with a parabolically increasing trend with indentation depth. If a self-similar indenter tip is used, as it is in this case, the indentation depth is the only length scale. Through a simple dimensional analysis, we can show that tc scales with the square of the indentation depth and inversely with the diffusivity. This scaling behavior is seen to be well represented by the experimental and computational curves in Fig. 9(b).

Fig. 9
Comparison between the experimental creep parameters and the simulated ones using the SGID model: (a) and (b) depict Δho¯ and tc for lithiated germanium at room temperature and 80 °C, and (c) and (d) depict the same for sodiated germanium at room temperature
Fig. 9
Comparison between the experimental creep parameters and the simulated ones using the SGID model: (a) and (b) depict Δho¯ and tc for lithiated germanium at room temperature and 80 °C, and (c) and (d) depict the same for sodiated germanium at room temperature
Close modal
By performing a two-dimensional (2D) least-squares fitting of the simulated creep parameters (tc and Δho¯) to the experimental ones, the stress-diffusion coupling coefficient (α) and diffusivity (D) values could be obtained. An α value of 0.084 was determined for room temperature Li0.6Ge. The increase in temperature to 80 °C leads to a slightly higher α value of 0.096. The least-squares fitting also resulted in diffusivity measurement of 1.01 × 10−10 cm2/s for room temperature and 2.22 × 10−10 cm2/s for 80 °C. This difference in diffusivity with temperature can be used to determine the activation energy of Li in a-Ge according to the Arrhenius’s equation:
D(T)=Dpe(Ea/kT)
(2)
where D is diffusivity (cm2/s), Dp is the temperature-independent pre-exponent (cm2/s), Ea is the activation energy (eV), k is the Boltzmann constant (eV/K), and T is the temperature (K). Based on this equation, the activation energy measured for Li in a-Ge is 0.1299 eV.

Figures 9(c) and 9(d) depict the experimental values of Δho¯ and tc for sodiated Ge at the room temperature, together the computational values obtained for a sweep of diffusivity values and a fixed α value of 0.1. Δho¯ is seen to be not significantly affected by the changes in diffusivity. As contrasted with Fig. 9(b), where the experiments and simulations showed the matching parabolically increasing trend, the size effect of the experimental tc values for sodiated Ge is much weaker than that predicted by the SGID model. Thus, the SGID creep mechanism cannot fully explain the experimentally observed behavior of Na0.6Ge.

Given the amorphous nature of the sodiated Ge electrode, we hypothesize that the conventional STD creep mechanism may additionally be at play, giving rise to the relative size independence of the creep characteristic time. To examine this hypothesis, we implemented the STD mechanism into the earlier SGID model by incorporating a viscoplastic strain rate function as follows:
ε¯˙pl=ε˙0(σ¯σY)m
(3)
where ε˙0 is a reference strain rate, σ¯ is the von Mises stress, σY is the yield stress, and m is the stress exponent. The aforementioned power-law equation replicates the limiting case of strain rate-independent plasticity (i.e., SGID creep only) when m approaches infinity. The implementation of the STD mechanism added another fitting step, as the stress exponent (m) could be tuned to obtain the right mix between SGID and STD where the size dependence of the computational curves lines up with that of the experimental curves. After identifying a suitable m value, values of α and D were obtained in an identical procedure as before, employing a 2D parametric study. Figure 10 demonstrates how changing values of m significantly affects the Δho¯ and tc results, with a trend of decreasing Δho¯ and increasing size dependence of tc with increasing m. A stress exponent of 25 is seen to result in computational Δho¯ and tc curves which are clearly in good agreement with the experimental curves. Room temperature testing of sodiated Ge yielded a stress-diffusion coupling coefficient of 0.0217 and a diffusivity of 6.08 × 10−10 cm2/s, with power-law viscoplastic creep parameters of ε˙0=0.5s1 and m = 25. The size effect of Na0.6Ge is captured more precisely with these parameters in place, demonstrating the cooperative roles of both SGID and STD mechanisms in the creep behavior of the sodiated Ge system. We noted that lithiated Ge in general should also exhibit viscoplastic behavior through the STD mechanism. However, the experiments for lithiated Ge and the SGID simulations assuming rate-independent plasticity showed the same parabolic dependence of tc on ho (Fig. 9(b)). This agreement shows that SGID is the dominant mechanism responsible for nanoindentation creep in lithiated Ge and the contribution from viscoplastic deformation is negligible, thus supporting the assumption of rate-independent plasticity.
Fig. 10
Comparison between the experimental creep parameters of sodiated germanium and the simulated ones incorporating both the SGID and STD creep mechanisms: (a) normalized amount of creep Δho¯ and (b) characteristic time tc
Fig. 10
Comparison between the experimental creep parameters of sodiated germanium and the simulated ones incorporating both the SGID and STD creep mechanisms: (a) normalized amount of creep Δho¯ and (b) characteristic time tc
Close modal

The room temperature diffusivity values for Li and Na in a-Ge obtained in this work are one to two orders of magnitude higher than those typically reported for other AIB anode materials (e.g., ∼10−12 cm2/s for Li diffusion in Si [58,59] and ∼10−11 cm2/s for Na diffusion in hard carbon [60]), indicating a great potential of a-Ge as a high-rate anode material. Counterintuitively, despite the larger size of Na atoms compared to Li atoms, the room temperature diffusivity of Na in a-Ge (6.08 × 10−10 cm2/s) was found to be much higher than that of Li in a-Ge (1.01 × 10−10 cm2/s). This large difference in diffusivity is likely attributed to weaker dopant-host bonds and/or more favorable dopant-dopant interactions in sodiated Ge than in lithiated Ge, which result in a lower energy barrier for Na mobility. It is also noteworthy that the obtained stress-diffusion coupling coefficient values are much less than the unity value of an ideal mixture. Nevertheless, they are significant enough to contribute to the size-dependent creep behavior found in our nanoindentation experiments. Battery electrode materials are usually used in nanoparticle or microparticle form to produce composite electrodes. The strong nano/microscale heterogeneity inherent to these composites can promote SGID creep deformation as observed in nanoindentation to affect their overall chemo-mechanical response. Our work here reveals the importance of two-way stress-diffusion coupling and provides a quantitative method for characterizing such coupling.

4 Conclusions

We performed a combined experimental and chemo-mechanical continuum modeling study to understand the nanoindentation response of lithiated and sodiated Ge. The quasi-static measurement results of elastic modulus and hardness showed no significant size effect. The clear difference between the mechanical properties of lithiated and sodiated Ge demonstrated the importance of alloying materials in affecting mechanical properties. The size effects inherent in the experimental creep response were employed to investigate the creep mechanism of each system. The size dependence of characteristic creep time of lithiated Ge matched well with that predicted by the SGID mechanism, but the SGID model failed to capture the creep behavior of sodiated Ge. Resolving this discrepancy required a creep model that combines SGID with the conventional size-independent STD mechanism. After the creep mechanism of each material system was determined, parametric computational studies were carried out to characterize the fundamental constitutive parameters, including the stress-diffusion coupling coefficient and diffusivity. The activation energy of lithiated Ge was also determined by time-dependent nanoindentation at room and elevated temperatures. These findings offer fundamental insights into the strong stress-diffusion coupling in high-capacity electrode materials and can enable quantitative and predictive modeling of their chemo-mechanical performance.

Acknowledgment

The authors acknowledge the support of National Science Foundation Grant No. NSF-CMMI-1554393. This work was performed in part at the Georgia Tech Institute for Electronics and Nanotechnology, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant No. ECCS-2025462).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Appendix: A Stress-Diffusion Coupling Computational Model

The diffusive flux of a mobile species in an electrode material, J, obeys the law of mass conservation through:
ct=J
(A1)
in which c is the concentration of the mobile species. Taking into account the stress effect, the flux is given by Fick’s law as:
J=D0ckT(μchemΩσh)
(A2)
Here, μChem is the stress-free chemical potential, Ω is the partial molar volume of the mobile species, σh is the hydrostatic stress, D0 is the tracer diffusivity, k is the Boltzmann constant, and T is the temperature. The chemical potential is related to the concentration and a reference potential μ0 according to:
μchem=μ0+kTln(γc)
(A3)
where γ is the activity coefficient. Combining Eqs. (A2) and (A3) leads to:
J=D0[1+ln(γ)ln(c)]c+D0cΩkTσh
(A4)
The above equation can be rewritten in a more convenient form as follows
J=D(cαcΩkTσh)
(A5)
in which D is the intrinsic diffusivity and α is the stress-diffusion coupling coefficient. These constitutive parameters are defined by
D=D0[1+ln(γ)ln(c)]
(A6)
α=[1+ln(γ)ln(c)]1
(A7)

The electrode material is assumed to be isotropic and deform in an elastic-perfectly plastic manner. The plastic deformation of the material is modeled by the classical rate-independent or viscoplastic J2 flow theory. This chemo-mechanical computational model was implemented as a user element subroutine in commercial software abaqus to simulate nanoindentation testing. The Berkovich indenter tip was treated as a rigid body meshed using rigid shell elements (R3D4), and the deformable electrode material was meshed using eight-node linear brick elements. Because of the large face angle of the Berkovich tip, the tip-electrode contact was assumed to be frictionless.

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