Non-idealities in charged membranes: how interactions between ions and solvent shape membranes’ mechanics and electrochemistry

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Membranes are selective barriers that separate two solutions with different characteristics, letting some solutes pass through and blocking others. From cell membranes that allow life by creating a unique environment for biochemical reactions to the membranes that we use every day to filter impurities from our drinking water, there are countless examples of these critical components in physical and chemical systems. Membranes are everywhere.

The types of solutes that a membrane can block depend on its microstructure. Of particular importance are charged membranes, which hosts charges of one sign in their backbone. These fixed charges interact with ions in solution, repelling charges of the same sign and attracting those with opposite sign. Cell membranes are an emblematic example of charged membranes, as they bear a fixed negative charge on their surface proteins that maintains the homeostatic balance within the cell. In the energy industry, artificial charged membranes called ionic polymers are widely utilized in devices for energy storage and production, such as electrolyzers and fuel cells. These applications are becoming more and more critical with the transition to renewable energy sources.

Despite their importance toward understanding the rules of life and energy storage, our comprehension of the physics of charged membranes is still limited. In particular, we know very little about the interactions between charge movement in the membranes (electrochemistry) and their deformations (mechanics). Even the most sophisticated models cannot capture the complexity of the membranes’ physics; only very recent efforts have started to explicitly include the motion of solvent within the membrane.

The inclusion of the motion of solvent poses a variety of questions regarding its role on the mechanics and electrochemistry of charged membranes. Electrochemical solutions of ions are profoundly different from solutions of uncharged species. In fact, the strong interactions between ions and solvent molecules modify how the solution behaves. Specifically, the way in which solutes and solvent mix together is different from ideal gas mixtures, such that an "excess entropy" is generated. The effects of this non-ideal mixing are further exacerbated in charged membranes, due to their microstructure.

In our manuscript, we established a modeling framework to study the effect of the interactions between ions, solvent, and microstructure of charged membranes on their mechanics and electrochemistry. We considered four sources of non-ideal mixing due to these interactions (see Fig. 1): i) solvent molecules dispose themselves around ions due to ion-dipole or ion-induced dipole interactions (solvation); ii) ions of one sign tend to be surrounded by ions of the opposite sign, and vice versa (electrostatic interactions); iii) ions of one sign may collide with ions with an opposite charge (physical interactions); and iv) pores of the membranes have a finite volume, limiting the maximum concentration that can be reached by ions and solvent (pore steric effects). While these effects on electrochemistry have been studied in rigid membranes, their role on the interplay between mechanics and electrochemistry has never been investigated.

Figure 1. Illustration of the four types of sources of non-idealities. Blue particles are the fixed charges, while green particles are the mobile ones. The solvent (in this case water) is illustrated as a red-gray molecule (oxygen and hydrogen, respectively). Orange lines indicate the interactions considered by each source of non-ideality.

Our model lays its foundations in continuum mechanics and thermodynamics for the description of mechanical deformations and charge dynamics. We adopt a thermodynamically consistent approach, which automatically derives the constitutive equations from the Helmholtz free-energy density. The novelty of our formulation stands in the inclusion of an excess free-energy, which accounts for the interactions between ions, solvent, and membrane microstructure.

As a practical application of our theory, we studied the actuation of ionic polymer-based actuators. Actuation is driven by the stress generated in the material by osmotic pressure and Maxwell stress, following the application of an external electric field. Typically, to apply the electric field, metal layers are deposited on each side of the membrane through an electroless plating process, forming a so-called ionic-polymer metal composite (IPMC). These actuators hold promise in soft robotics and biomedical engineering applications, due to their low driving voltage and the ability to operate underwater.

We simulated the response of ionic polymer-based actuators through numerical simulations. We investigated how the electrochemistry and mechanics of the actuators are affected by non-idealities. We found remarkable quantitative and qualitative effects of non-idealities on both electrochemical and mechanical variables. We discovered that all of the sources of non-idealities affect the free-energy, with the contributions related to solvation and electrostatic interactions being on the same order of magnitude of that associated with ideal mixing. We further studied how non-idealities are mediated by the type of mobile ions in the membrane. We compared the actuation of membranes with small (lithium) and large (cesium) mobile ions, which affect differently the four sources of non-idealities.

Overall, our work paves the way to physically accurate models of the mechanics and electrochemistry of charged membranes. These models are critical for the analysis and design of charged membranes, in a variety of applications in science and technology. In particular, they can improve our understanding of mechanosensory receptors in cell membranes, which are at the basis of the sense of touch. Accurate models are also of interest to the electrochemical energy storage and production community, especially within the development of flexible batteries that may be used in wearable devices. Finally, physically based models can be utilized for the inverse design of soft actuators based on ionic membranes, which find application in soft robotics and biomedical engineering.

You can read our work here: https://doi.org/10.1038/s41524-022-00827-2

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