arXiv subjects

P. M. Biesheuvel

Publications and source records attributed to P. M. Biesheuvel.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of the Rankine attraction term in an equation of state based on the London dispersion force

The attraction term in an equation of state for gases, $-a c^2$, proposed by Rankine in 1854, is generally related to the London dispersion force via the equation for the second virial coefficient, $B_2$, given by $B_2 = 2πN_0 \int_0^\infty \left(1- \exp \left(-ω/ kT\right)\right) r^2 \text{d}r$, where $ω$ is the attraction energy between two molecules in the gas. This equation works very well to describe $B_2$ and thus $a=B_2 RT$ as function of temperature, but the derivation is complicated. Here we present two other methods to derive $B_2$ and thus $a$ from the London equation, which have a more intuitive background. (The more simple of the two we expect must be available in literature.) We analyze these three models for the gas argon at temperatures between 150 and 900 K. All three methods fit the data quite closely while an analytical approximation fits data even better. The temperature dependence of $a$ is well described by a proportionality of $a$ with a term $1+T^*/T$ where $T^*$ is a constant that depends on the type of gas.

physics.chem-ph↗

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↗

arXiv2025

Tutorial on the chemical potential of ions in water and porous materials: transport, isotherms, and electrical double layer theory

In this tutorial we discuss the chemical potential of ions in water (i.e., in a salt solution, in an electrolyte phase) and inside (charged) nanoporous materials such as porous membranes. In water treatment, such membranes are often used to selectively remove ions from water by applying pressure (which pushes water through the membrane while most ions are rejected) or by current (which transports ions through the membrane). Chemical equilibrium across a boundary (such as the solution-membrane boundary) is described by an isotherm for neutral molecules, and for ions by an electrical double layer (EDL) model. An EDL model describes concentrations of ions inside a porous material as function of the charge and structure of the material. There are many contributions to the chemical potential of an ion, and we address several of these in this tutorial, including ion volume and the effect of ion-ion Coulombic interactions. We also describe transport and chemical reactions in solution, and how they are affected by Coulombic interactions.

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

Derivation of an extended Bjerrum equation for the activity coefficient of ions based on an analysis of Coulombic forces

The activity coefficient of ions in solution was proposed by Bjerrum (1916) to depend on the cube root of concentration, because of a good fit with data for low and moderate salt concentration. However, Debye and Hückel (DH) later developed a theory that prevailed, and this theory has a square root dependence. The derivation of the DH equation is not easy or intuitive, and has various uncertain elements, and the fit to data is not very good for salts with non-unity valencies. We develop a model for the activity coefficient based on the Coulombic forces between an ion and the nearest ions of opposite charge, including the distribution of separations between these ions. For a symmetric salt, we only have to analyse the interactions between one anion and one cation, and we can derive an expression for the dilute limit that depends on the Bjerrum length and the cube root of salt concentration, the same as an expression put forward by Bjerrum. Our theory also analytically describes higher salt concentrations and a non-zero ion size, not only for 1:1 but also for 2:2 and 3:3 salts. For a 1:1 salt, the cube root scaling law describes data of the activity coefficient very well, both at low and intermediate salt concentrations, while the extended Bjerrum equation describes data even better and up to higher concentrations, and also includes the effect of ion size. For asymmetric salts (2:1 and 3:1 salts), a numerical procedure can be used which considers all possible orbits of two or three monovalent ions around a central multivalent ion of opposite charge, which also fits data accurately.

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↗

arXiv2021

The difference between Faradaic and non-Faradaic electrode processes

Both Faradaic and non-Faradaic processes can take place at an electrode. The difference between the two processes is clearly discussed in several classical sources, starting with Grahame (1952). However, later reference to charge transfer across the metal-solution interface as a defining feature of a Faradaic process, has led to ambiguities. Following Grahame, in a Faradaic process, charged particles transfer across the electrode, from one bulk phase to another. Thus, in a Faradaic process, after applying a constant current, the electrode charge, voltage and composition go to constant values. Instead, in a non-Faradaic (capacitive) process, charge is progressively stored. We characterize the intercalation material nickel hexacyanoferrate by two electrochemical methods and compare with theory. Data for the capacitance of this material is well described by the extended Frumkin isotherm. This data, and the correspondence with theory, demonstrates that this is a capacitive material and ion and charge storage in this material a non-Faradaic electrode process. Cyclic Voltammetry (CV) diagrams for this material have broad peaks for certain potential windows, and rectangular shapes for other conditions, both experimentally and in theoretical calculations based on a RC network model that includes how capacitance is a function of charge. Measured and predicted CV diagrams are in perfect agreement with one another. This shows that (broad) peaks in CV diagrams do not establish whether an electrode material is Faradaic or not.

physics.chem-ph↗

arXiv2020

Theory of expansion of porous carbon electrodes in aqueous solutions according to the Donnan model

The Donnan model describes the electric double layer structure inside carbon micropores and is an essential element of larger-scale models for capacitive porous carbon electrodes for energy storage and water desalination. The Donnan model was not yet developed to describe the expansion of carbon electrodes upon charging. Here we provide a simple theory which predicts that the one-dimensional expansion of a carbon particle scales with electrode charge density according to a 4/3$^\text{rd}$ power.

physics.chem-ph↗

arXiv2020

Nickel Hexacyanoferrate Electrodes for Continuous Cation Intercalation Desalination of Brackish Water

Using porous electrodes containing nickel hexacyanoferrate (NiHCF) nanoparticles, we construct and test a device for capacitive deionization in a two flow-channel device where the intercalation electrodes are in direct contact with an anion-exchange membrane. Upon negatively charging NiHCF, cations intercalate into it and the water in its vicinity is desalinated; at the same time water in the opposing electrode becomes more saline upon positively charging the NiHCF in that electrode. In a cyclic process of charge and discharge, fresh water is continuously produced, alternating between the two channels in sync with the direction of applied current. We present proof-of-principle experiments of this technology for single salt solutions, where we analyze various levels of current and cycle durations. We analyze salt removal rate and energy consumption. In desalination experiments with salt mixtures we find a threefold enhancement for K$^+$ over Na$^+$-adsorption, which shows the potential of NiHCF intercalation electrodes for selective ion separation from mixed ionic solutions.

physics.chem-ph↗

arXiv2020

Nernst-Planck transport theory for (reverse) electrodialysis: III. Optimal membrane thickness for enhanced process performance

The effect of the thickness of ion exchange membranes has been investigated for electrodialysis (ED) and reverse electrodialysis (RED), both experimentally and through theoretical modeling. By developing a two-dimensional model based on Nernst-Planck theory, we theoretically find that reducing the membrane thickness benefits process performance only until a certain value, below which performance drops. For ED, an optimum thickness can be identified in the range of 10-20 microns, while for RED the maximum power density is found for membranes that are three times as thick. Model calculations compare well with experimental data collected with a series of homogeneous membranes with the same chemical composition and a thickness in the range of 10-75 microns. Our results show that the classical picture that membranes should be as thin as possible (as long as they remain pinhole-free and structurally stable) is insufficient, and must be replaced by a more accurate theoretical framework.

physics.chem-ph↗

arXiv2020

Two-section reactor model for autothermal reforming of methane to synthesis gas

A one-dimensional and stationary reactor model is presented to describe the catalytic conversion of a gaseous hydrocarbon fuel with air and steam to synthesis gas by autothermal reforming (ATR) and catalytic partial oxidation (CPO). The model defines two subsequent sections in the reactor, namely an upstream oxidation section, and a downstream reforming section. In the oxidation section all of the oxygen is converted, with partial conversion of the fuel. An empirical fuel utilization ratio is used to quantify which part of the fuel is converted in the oxidation section as function of the relative flows of air and steam. In the oxidation section, the gas temperature rapidly increases toward the toptemperature at the intersection with the reforming section. In this section the temperature decreases while the fuel is further converted with water and CO2 as oxidant. For methane as fuel, simulation results are presented and compared with experiments. For multicomponent fuels such as natural gas and naphtha, it is described how the two-section model can be applied.

physics.chem-ph↗

arXiv2019

Theory of Ion and Electron Transport Coupled with Biochemical Conversions in an Electroactive Biofilm

Bioelectrochemical systems are electrochemical cells that rely on conductive biofilms covering an electrode. We consider the example of a microbial fuel cell, and we derive a dynamic model of ion transport, biochemical reactions and electron transport inside such a biofilm. After validating the model against data, we evaluate model output to obtain an understanding of the transport of ions and electrons through a current-producing biofilm. For a system fed with a typical wastewater stream containing organic molecules and producing 5 A/m$^{2}$, our model predicts that transport of the organic molecules is not a limiting factor. However, the pH deep within the biofilm drops significantly, which can inhibit current production of such biofilms. Our results suggest that the electronic conductivity of the biofilm does not limit charge transport significantly, even for a biofilm as thick as 100 $μ\mathrm{m}$. Our study provides an example of how physics-based modelling helps to understand complex coupled processes in bioelectrochemical systems.

physics.chem-ph↗

arXiv2019

Theory of ion and water transport in electron-conducting membrane pores with pH-dependent chemical charge

In this work, we develop an extended uniform potential (UP) model for a membrane nanopore by including two different charging mechanisms of the pore walls, namely by electronic charge and by chemical charge. These two charging mechanisms will generally occur in polymeric membranes with conducting agents, or membranes made of conducting materials like carbon nanotubes with surface ionizable groups. The electronic charge redistributes along the pore in response to the gradient of electric potential in the pore, while the chemical charge depends on the local pH via a Langmuir-type isotherm. The extended UP model shows good agreement with experimental data for membrane potential measured at zero current condition. When both types of charge are present, the ratio of the electronic charge to the chemical charge can be characterized by the dimensionless number of surface groups and the dimensionless capacitance of the dielectric Stern layer. The performance of the membrane pore in converting osmotic energy from a salt concentration difference into electrical power can be improved by tuning the electronic charge.

physics.chem-ph↗

arXiv2019

Electrostatic cooling at electrolyte-electrolyte junctions

Electrostatic cooling is known to occur in conductors and in porous electrodes in contact with aqueous electrolytes. Here we present for the first time evidence of electrostatic cooling at the junction of two electrolyte phases. These are, first, water containing salt, and, second, an ion-exchange membrane, which is a water-filled porous layer containing a large concentration of fixed charges. When ionic current is directed through such a membrane in contact with aqueous phases on both sides, a temperature difference develops across the membrane which rapidly switches sign when the current direction is reversed. The temperature difference develops because one water-membrane junction cools down, while the other heats up. Cooling takes place when the inner product of ionic current $\textbf{I}$ and field strength $\textbf{E}$ is a negative quantity, which is possible in the electrical double layers that form on the surface of the membrane. Theory reproduces the magnitude of the effect but overestimates the rate by which the temperature difference across the membrane adjusts itself to a reversal in current.

physics.chem-ph↗

arXiv2019

Theory of water desalination with intercalation materials

We present porous electrode theory for capacitive deionization (CDI) with electrodes containing nanoparticles that consist of a redox-active intercalation material. A geometry of a desalination cell is considered which consists of two porous electrodes, two flow channels and an anion-exchange membrane, and we use Nernst-Planck theory to describe ion transport in the aqueous phase in all these layers. A single-salt solution is considered, with unequal diffusion coefficients for anions and cations. Similar to previous models for CDI and electrodialysis, we solve the dynamic two-dimensional equations by assuming that flow of water, and thus the advection of ions, is zero in the electrode, and in the flow channel only occurs in the direction along the electrode and membrane. In all layers, diffusion and migration are only considered in the direction perpendicular to the flow of water. Electronic as well as ionic transport limitations within the nanoparticles are neglected, and instead the Frumkin isotherm (or regular solution model) is used to describe local chemical equilibrium of cations between the nanoparticles and the adjacent electrolyte, as a function of the electrode potential. Our model describes the dynamics of key parameters of the CDI process with intercalation electrodes, such as effluent salt concentration, the distribution of intercalated ions, cell voltage, and energy consumption.

physics.chem-ph↗

arXiv2019

Analysis of electrolyte transport through charged nanopores

We revisit the classical problem of flow of electrolyte solutions through charged capillary nanopores or nanotubes as described by the capillary pore model (also called "space charge" theory). This theory assumes very long and thin pores and uses a one-dimensional flux-force formalism which relates fluxes (electrical current, salt flux, fluid velocity) and driving forces (difference in electric potential, salt concentration, pressure). We analyze the general case with overlapping electric double layers in the pore and a nonzero axial salt concentration gradient. The 3$\times$3 matrix relating these quantities exhibits Onsager symmetry and we report a significant new simplification for the diagonal element relating axial salt flux to the gradient in chemical potential. We prove that Onsager symmetry is preserved under changes of variables, which we illustrate by transformation to a different flux-force matrix given by Gross and Osterle (1968). The capillary pore model is well-suited to describe the nonlinear response of charged membranes or nanofluidic devices for electrokinetic energy conversion and water desalination, as long as the transverse ion profiles remain in local quasi-equilibrium. As an example, we evaluate electrical power production from a salt concentration difference by reverse electrodialysis, using an efficiency vs. power diagram. We show that since the capillary pore model allows for axial gradients in salt concentration, partial loops in current, salt flux or fluid flow can develop in the pore. Predictions for macroscopic transport properties using a reduced model where the potential and concentration are assumed to be invariant with radial coordinate ("uniform potential" or "fine capillary pore" model), are close to results of the full model.

physics.chem-ph↗