arXiv subjects

M. Elimelech

Publications and source records attributed to M. Elimelech.

5 recordsLinked to original sources

arXiv2025

A concise tutorial review of reverse osmosis and electrodialysis

Reverse osmosis (RO) and electrodialysis (ED) are the two most important membrane technologies for water desalination and treatment. Their modes of operation and transport mechanisms are very different, but on a closer look also have many similarities. In this concise version of our tutorial review, we describe state-of-the-art theory for both processes, focusing on simple examples that are helpful for the non-specialist and useful for classroom teaching. Both processes are described by solution-friction (SF) theory which combines ion and water transport across membranes with chemical and mechanical equilibrium at membrane/solution interfaces. We present a derivation of SF theory based on force balances on water and ions and show how the various terms, convection, diffusion, and electromigration, are derived, and how solute partitioning is implemented. Finally, we demonstrate how SF theory accurately describes the osmosis experiment where water and ions are transported in opposite directions across a membrane.

physics.chem-ph↗

arXiv2024

Extended Donnan model for ion partitioning in charged nanopores

Membranes consist of pores and the walls of these pores are often charged. In contact with an aqueous solution, the pores fill with water and ions migrate from solution into the pores until chemical equilibrium is reached. The distribution of ions between outside and pore solution is governed by a balance of chemical potential, and the resulting model is called a Donnan theory, or Donnan equation. Including a partitioning coefficient that does not depend on salt concentration results in an extended Donnan equation `of the first kind'. Recently, an electrostatic model was proposed for ions in a pore based on the arrangement of ions around strands of polymer charge, including also ion activity coefficients in solution. That framework leads to an extended Donnan equation `of the second kind', which has extra factors depending on ion concentrations in the pores and salt concentration in solution. In the present work, we set up another Donnan model of the second kind by evaluating the Coulombic interactions of ions in a cylindrical pore, including the interaction of ions with the charged pore walls and between the ions. We assume that counterions are near the pore wall while coions distribute over the center region. Starting from a complete analysis, we arrive at an elegant expression for the chemical potential of ions in such a pore. This expression depends on coion concentration, pore size, and other geometrical factors, but there is no additional dependence on counterion concentration and charge density. This model predicts the Coulombic contribution to the chemical potential in the pore to be small, much smaller than predicted by the electrostatic model from literature. Instead, we predict that up to around 1 M salt concentration, activity effects of ions in solution are more important.

physics.chem-ph↗

arXiv2024

Analysis of concentration polarization in reverse osmosis and nanofiltration: zero-, one-, and two-dimensional models

Reverse osmosis and nanofiltration are membrane-based methods that remove solutes from solvent, for instance they remove salts from water (desalination). In these methods, an applied pressure is the driving force for solvent to pass the membrane, while most of the solutes are blocked. Very important in the theory of mass transport is the concentration polarization layer (CP layer), which develops on the upstream side of the membrane. Because of the CP layer, the solvent flux through the membrane is reduced while leakage of solutes through the membrane increases, and both these effects must be minimized. So it is very important to understand and describe the nature of the CP layer accurately, especially to find a good estimate of the CP layer mass transfer coefficient, $k$. This is also important for the accurate characterization of membranes in a test cell geometry. We theoretically analyze the structure of the CP layer using three levels of mathematical models. First, we present a modification of an equation for $k$ by Sherwood et al. (1965) and show that it works very well in a zero dimensional model. Second, we evaluate a one-dimensional model that is more accurate, which can incorporate any equation for the flow of solvent and solutes through the membrane, and which also makes use of the new modified Sherwood equation. Finally, we fully resolve the complete channel in a two-dimensional geometry, to validate the lower-order models and to illustrate the structure of the CP layer. The overall conclusion is that for typical test cell conditions, the modified Sherwood equation can be used to characterize the CP layer, also when solvent flux through the membrane changes between inlet and outlet of the test cell. Furthermore, the one-dimensional model accurately describes solute removal (for instance water desalination) not just in a short test cell but also in a longer module.

physics.chem-ph↗

arXiv2023

General validity of the exponential law for the effect of concentration polarization in reverse osmosis in a stirred-cell geometry, including an activity correction for 1:1 salt solutions

Reverse osmosis (RO) is a method to desalinate water with membranes and an applied pressure. Very important in the theory of mass transport in RO is the concentration polarization (CP) layer, which develops on the upstream side of the membrane because of a combination of salt convection and diffusion. Because of the CP-effect, the salt concentration at the membrane surface is higher than in the channel, and this increases the osmotic pressure there, and thus transmembrane water flux is reduced (the osmotic pressure acts against water flux), while salt leakage through the membrane increases. So it is very important to understand and describe the CP-layer accurately. We analyze a one-dimensional geometry, which is of relevance for a typical lab-scale RO setup using small membrane coupons where the solution on the feed side of the membrane is stirred. For this geometry, the standard film layer approach is often used that assumes a stagnant film layer of a defined thickness, which however does not exist in reality. We set up a model without that assumption but including refreshment of solution because of the flow of water along the membrane due to stirring. We show that the `exponential law' for the CP-layer that is predicted by the the film model, also applies for this more accurate model. We further improve the model by including the activity coefficient of salt ions, as described by the Bjerrum theory that is based on ion-ion Coulombic interactions. We evaluate the original linearized Bjerrum theory as well as an extended Bjerrum equation that is valid up to 1.5 M salt concentration. We show how including this activity correction leads to a reduction of the diffusional driving force at high concentration, and thus the salt concentration at the membrane further increases. However, the effect can be easily included by reducing the CP-layer mass transfer coefficient by a fixed percentage.

physics.chem-ph↗

arXiv2023

Theory of expansion and compression of polymeric materials

We extend classical Flory-Rehner theory for the expansion and compression of porous materials such as cross-linked polymer networks. The theory includes volume exclusion, affinity with the solvent, and finite stretching of the polymer chains. We also modify this equilibrium theory -- that applies to equal expansion of a material in all directions -- to the situation that a material can only expand in a single direction, as is the case when a thin layer is tightly bound to a support structure. We extend this equilibrium model to the case that a pressure is applied across such a thin layer of the polymer material, for instance a membrane, and liquid flows across this layer. The theory describes how in the direction of liquid flow the membrane is increasingly compacted (becomes less porous), and the more so at higher applied pressures. We provide results of example calculations for a thick membrane with significant changes in compaction across its thickness, and a thin membrane for which compaction due to flow is minor. In the last section we model the dynamics of the change of size of a porous material in time after a step change in the solvent-polymer attraction parameter.

physics.chem-ph↗