Electrodeposition modeling in solid-state batteries

Lithium metal has a theoretical specific capacity per unit mass 10 times that of graphite which is the current anodic material choice in present-day lithium-ion batteries. Lithium metal however cannot be paired with liquid electrolytes due to the detrimental effect of dendrite growth leading to short circuits. Novel solid electrolytes are expected to be the key to meeting energy density requirements necessary for electrification of next generation automobiles and aviation among other critical applications such as portable electronics.

All-solid-state batteries are comprised of lithium metal and a solid electrolyte with lithium metal being plated on at their interface during the charge cycle and stripped off during the discharge cycle. This electrodeposition of lithium involves significant size changes of the lithium anode starkly different from the intercalation mechanism in current lithium-ion batteries. Since this electrodeposition process occurs at an interface buried between 2 solid components on either side, experimental observations of this process presents a challenge. Modeling the electrodeposition process is thus essential to understand the functioning of all-solid-state lithium metal batteries (ASSLMBs) and overcome the concomitant challenges.

We devised a modeling strategy, developed a thermodynamically-consistent continuum mechanical framework and numerically implemented a model for electrodeposition of lithium in ASSLMBs. First, given the difficulty of dynamic addition and removal of  elements using the finite element technique, modeling electrodeposition at a sharp Li/SE interface is a challenge. Instead we introduced a fictitious interphase layer of small but finite thickness, which absorbs and de-absorbs lithium, to replace the sharp interface. As lithium is absorbed, this layer grows to mimic the plating process and similarly as lithium is de-absorbed, this layer shrinks to mimic the stripping process.

With such a strategy in place, we incorporated an elastic-plastic-growth type kinematics for the interphase layer. In our model, we accounted for anisotropic growth via a novel constitutive prescription using growth weights that we introduced. We were able to successfully model the plating-and-stripping cycles using the implicit finite element program ABAQUS by our custom-developed user material (UMAT) subroutine.

Using the described computational tools that we developed, we investigated the role of chemical and mechanical defects. In these cells, the high reactivity of lithium leads to formation of chemical deposits of lithium hydroxide and carbonate at the interface. Accounting for these inhomogeneities, we identified the resultant bending stresses introduced in the brittle ceramic solid electrolyte as the cause for potential fracture of solid electrolyte as well as decohesion at the interface eventually leading to cell failure. Through numerical experiments using our developed computational tools we also parametrically analyzed the effect of electrolyte thickness and plating current helping understand the idea of critical current density observed in contemporary experiments.

The ceramic electrolytes inevitably contain surface defects. We have also numerically studied the role of a surface notch in the ceramic electrolyte on the plating-and-stripping process. Through our simulations, we have identified current intensification and the associated inhomogeneity in the deposition process as a driving force for large stress generation ahead of the notch tip possibly resulting in fracture of the brittle ceramic electrolyte.

Our modeling strategy, constitutive framework and various numerical simulations with results have been reported in our paper which is a first continuum-level modeling effort in the published literature in the context of electrodeposition in ASSLMBs.

Polyelectrolyte gels

Polymeric gels are aggregates of a polymer network and a solvent. Polyelectrolyte gels are a special class of polymeric gels which respond with large volume changes to changes in their chemical environment --- pH and salt concentration, by absorption of solvent, typically water. This differential uptake of water is due to the altering of the chemical states of the polymer chains based on the environmental conditions. These gels swell in more dilute salt solutions and shrink in concentrated salt solutions, since osmosis takes solvent from dilute regions to concentrated regions and solvent flow into the gel results in swelling and vice-versa. Response to pH takes two forms --- swelling at high pH for anionic gels and swelling at low pH for cationic gels. The two kinds of gels differ in their chemical composition resulting in opposing behaviors.​

Our framework described above was specialized to account for the physics driving the aforementioned swelling and de-swelling processes in these gels, which is complex due to the coupled phenomena of large deformations of the gel, simultaneous transport of solvent and ions into and within the gel, and the influence of electric fields on the charged ions. By developing our own user element (UEL) subroutine in Abaqus for our specialized theory, we have validated our model against experiments --- through individual studies of equilibrium swelling of an anionic hydrogel and transient swelling of a cationic hydrogel. We have also shown utility of our model through demonstrations of technologically-relevant devices made of polyelectrolyte gels --- microfluidic flow valves and shape transforming actuators. This work has been published in this paper.

Fracture of amorphous polymers

Following up on the gradient-damage theory that we proposed for fracture of concrete, we extended our framework for application to widely used amorphous polymers such as polymethyl methacrylate (commercially plexiglass), polycarbonate, polystyrene etc. which we presented in our paper in the Journal of the Mechanics and Physics of Solids. The response of these glassy polymers in tension is strongly dependent on the entanglement density. Low entanglement density polymers such as plexiglass, polystyrene show very little shear-yield plastic deformation prior to failure due to crazing. The case of high entanglement density polymers such as polycarbonate is more complex. When under tension polycarbonate does not  exhibit crazing; instead the specimen is drawn out after necking before failing due to disentanglement/chain-scission. However in the presence of sharp notches, internal crazing occurs ahead of the notch tip due to the higher hydrostatic stresses that are developed. This has led to observations of a brittle-to-ductile transition in the fracture response of polycarbonate specimen dependent on the loading scenario.

In our model presented, we accounted for the large shear-yield plastic deformation that these polymers exhibit including the yield peak, strain hardening and strain rate sensitivity features. Further, we introduced dilational inelastic deformation through either a void-nucleated craze type mechanism or a large network-stretch induced disentanglement mechanism. These two inelastic deformations processes are pre-cursors to damage initiation in our model eventually leading to brittle and ductile fracture respectively. The evolution of the damage field following initiation is governed by the gradient-damage formulation akin to our previous work on fracture of concrete.

Once again in this case, numerical implementation of our theory has been carried out through a custom-developed user element subroutine in Abaqus. Using our implementation, we first studied fracture of plexiglass. With our material parameters calibrated using the tension experiments of Gearing and Anand (2004) on notched and unnotched cylindrical bars, our successful prediction of the failure load of a notched rectangular plate in tension from Gearing and Anand (2004) provided quantitative validation of our model. Using the several experiments of Gearing and Anand (2004) on multiple plexiglass specimen, we first calibrated our model and subsequently accurately predicted failure loads thus providing quantitative validation. Additionally we accurately captured both the crack nucleation location as well as the curvilinear crack trajectory observed through compression experiments (Riazia et al. 2015) on disks with varying notch geometries. Similar predictions of crack trajectories in notched compact tension specimen also demonstrated the validity of our proposed damage initiation criteria and evolution constitutive law.

Using the same framework as for the plexiglass, we also investigated fracture of polycarbonate. Unnotched cylindrical polycabonate specimen exhibit large ductility in tension with the specimen necking and drawing out extensively before final fracture. Notched cylindrical polycarbonate specimen exhibit brittle fracture at much smaller strains due to the build up of hydrostatic stresses ahead of the notch. We used the unnotched and notched tension experiments from Gearing and Anand (2004) to calibrate the damage initiation criteria for the disentanglement-based fracture and craze-based fracture respectively. The experiments on polycarbonate also included beam bending studies on specimen with varied notch radius with the fracture behavior transitioning from brittle to ductile with an increase in notch radius. Using our calibrated model, we were able to successfully capture both the brittle fracture response of the sharp-notched beam and the ductile fracture of the blunt-notched beam. The progression of our simulations in either case were coherent with the photomicrographs from the experiments. Not limited to these qualitative observations, we also made satisfactory predictions of the load-displacement histories for either case. These favorable comparisons of results from our simulations against the experiments provided substantial validation of our proposed gradient-damage theory for amorphous polymers.