Poisson-Boltzmann Research Articles

Comparison of a new type of Dark Matter with the Milky Way and M31 grand rotation curves

In the electron Born self-energy (eBse) model, free electrons are of finite-size and possess both a rest mass, me, as well as, a Born mass, meB = 74,000 me. The Born mass, which originates from the energy contained within the electric field that surrounds a finite-sized electron, serves as a Dark Matter (DM) particle in this theory (designated eBDM, electron Born Dark Matter). The equation of state for meB is w = -1, which implies that two Born masses experience a repulsive gravitational interaction. This repulsive gravitational interaction stabilizes the formation of a DM halo of meB particles, of typical halo size ~ 100 kpc, around a central mass M (e.g. a galaxy), where this gravitational stability arises from the competing attractive M - meB and repulsive meB - meB interactions. A solution of the linearized Poisson-Boltzmann equation, for this system, allows one to derive an expression for the rotational velocity VeBDM(R), as a function of radius R from the galactic center. A composite model composed of rotational velocity contributions from the galactic bulge, galactic disk, as well as, VeBDM(R) is found to provide a good description of the Grand Rotation Curves for the Milky Way and M31 galaxies.

The Poisson–Boltzmann equation in micro- and nanofluidics: A formulary

The Poisson–Boltzmann (PB) equation provides a mean-field theory of electrolyte solutions at interfaces and in confinement, describing how ions reorganize close to charged surfaces to form the so-called electrical double layer (EDL), with numerous applications ranging from colloid science to biology. This formulary focuses on situations of interest for micro- and nanofluidics, and gathers important formulas for the PB description of a Z:Z electrolyte solution inside slit and cylindrical channels. Different approximated solutions (thin EDLs, no co-ion, Debye–Hückel, and homogeneous/parabolic potential limits) and their range of validity are discussed, together with the full solution for the slit channel. Common boundary conditions are presented, the thermodynamics of the EDL is introduced, and an overview of the application of the PB framework to the description of electrokinetic effects is given. Finally, the limits of the PB framework are briefly discussed, and Python scripts to solve the PB equation numerically are provided.

Buffer Screening of Protein Formulations Using a Coarse-Grained Protocol Based on Medicinal Chemistry Interactions.

In drug and vaccine development, the designed protein formulation should be highly stable against the temperature, pH, buffer, excipients, and other environmental settings. Similarly, in a sensing unit, one needs to know how strongly two biomolecules bind to guide the design of the biorecognition unit accordingly. Typically, the community performs a series of experiments to thoroughly examine the parameter space, the so-called design-of-experiment (DoE) method, to identify the optimal formulation conditions. Unfortunately, extensive physical testing entails high costs, repeatability issues, and a lack of in-depth knowledge of the underlying mechanisms that affect the final outcome. To address these challenges, we developed a physics-based simulation protocol for buffer screening of protein formulations. We are introducing a coarse-grained molecular simulation protocol that consists of six different interactions. The so-called medicinal chemistry interactions (electrostatics, hydrophobicity, hydrogen bonding propensity, disulfide bonding, and water-water) are based on the physical nature of the protein's amino acid and the partitioning/polarity of any other chemical constituent. The protocol is applied in immunoglobulin-based monoclonal antibodies. We have analyzed the protein behavior as a function of acidity (pH) to discover the isoelectric point by solving the Poisson-Boltzmann equation in a mesoscale grid. To identify the conditions under which the protein oligomerizes in a given buffer, pH, temperature, and ionic strength, we are performing dissipative particle dynamics (DPD) simulations. The protocol allows researchers to reach the high time/space scales required to study protein formulations in their full complexity. Combined with the disruptive protein folding artificial intelligence (AI) algorithms that have been recently developed, the protocol creates a powerful digital framework for cultivating advanced pharmaceutical and biological applications.

Evaluation of inhibition effect and interaction mechanism of antiviral drugs on main protease of novel coronavirus: Molecular docking and molecular dynamics studies

The outbreak of pneumonia caused by the novel coronavirus (SARS-CoV-2) has presented a challenge to public health. The identification and development of effective antiviral drugs is essential. The main protease (3CLpro) plays an important role in the viral replication of SARS-CoV-2 and is considered to be an effective therapeutic target. In this study, according to the principle of drug repurposing, a variety of antiviral drugs commonly used were studied by molecular docking and molecular dynamics (MD) simulations to obtain potential inhibitors of main proteases. 24 antiviral drugs were docked with 5 potential action sites of 3CLpro, and the drugs with high binding strength were further simulated by MD and the molecular mechanics Poisson–Boltzmann surface area (MM-PBSA) binding free energy calculations. The results showed that the drugs with high flexibility could bind to 3CLpro better than those with low flexibility. The interaction mechanism between antiviral drugs and main protease was analyzed in detail by calculating the root mean square displacement (RMSD), root mean square fluctuation (RMSF) and interaction residues properties. The results showed that the six drugs with high flexibility (Remdesivir, Simnotrelvir, Sofosbuvir, Ledipasvir, Indinavir and Raltegravir) had strong binding strength with 3CLpro, and the last four antiviral drugs can be used as potential candidates for main protease inhibitors.

A computational investigation for heat and fluid transport with electro-osmosis phenomenon in a scraped surface heat exchanger

Scraped surface heat exchangers (SSHEs) have prominently shown their eminence in various food, chemical and in pharmaceutical industries when the regular processing of fluids and pertinent materials is encountered. The heat transfer analysis of Williamson fluid flow with electro osmotic effects in a narrow-gap SSHE is designed by implementing constant temperature difference between the gaps of the rotor and the stator. A fluid model is established by using lubrication approximation theory and Debye-Huckel simplifications. The computational solution is elaborated by implementing renowned BVP4C technique in MATLAB. Exact solution of Poisson-Boltzmann equation for different scenarios of SSHE is gathered, whereas solution for velocity, stream functions and temperature in various parts of SSHE are explored computationally. Impact of various crucial physical flow parameters like, Weissenberg number, mobility of the medium number, electro osmotic parameter on the velocity profile, stream function, temperature profile, and pressure gradient has been explored graphically. The numerical solution is compared with artificial neural network (ANN) and an absolute alignment of BVP4C solution is observed with ANN solution. It is reported that fluid flow is lifted with enhancing the viscoelastic behavior and exhibits dual behavior for electroosmosis phenomenon and mobility of medium parameter. Also, it is seen that temperature of the fluid is significantly declines with Weissenberg number and is surges with enhancing mobility of medium and Brinkman number. By increasing the Brinkman number conduction phenomenon is slow down and heat produced due to viscous dissipation, raises the temperature of the fluid. The viscoelastic effect could be of enormous significance in applications, like Polymer processing in certain heat exchangers, and in mixing and blending foodstuffs etc.

Combined electroosmotic and pressure driven flow through soft nanochannel grafted with partially ion-penetrable polyelectrolyte layers

ABSTRACT This study explores the coupled effects of pressure-driven and electroosmotic flows in a soft nanochannel filled with non-Newtonian ionised liquid, where the inner walls are coated with a polymer layer. The ion partitioning effect occurs due to the Born energy, arising from the dielectric permittivity difference between the polyelectrolyte medium and the ionised liquid. The model incorporates the nonlinear Poisson-Boltzmann equation, the modified Darcy-Brinkman equation in the polyelectrolyte layer (PEL), the Cauchy momentum equation in the electrolyte domains and the continuity equation for incompressible fluids. Using the Debye-Hückel approximation for analytical results, the study validates numerical findings. The study considers the no-slip boundary condition on the walls and examines the impact of volume charge density of PEL, bulk electrolyte concentration, non-Newtonian fluid rheology and polymer layer characteristics and applied pressure gradient on flow modulation and ionic transport in the nanochannel. We provide rigorous mathematical results showing how electrohydrodynamic factors like charged PEL, flow behaviour index, EDL thickness, ion partitioning parameters, Debye-Hückel parameter and softness parameters influence the strength of the potential and the total flow modulation. We illustrate ion selectivity in the soft nanochannel and the friction factor across its walls, considering ion partitioning effects.

Understanding the Electric Double Layer at the Electrode-Electrolyte Interface: Part I - No Ion Specific Adsorption.

We present a comprehensive mean-field model that takes into account the key components of modern electrical double layer theory at the interface between an electrode and an electrolyte solution. The model considers short-range specific interactions between different species, including electrode-ion repulsion, the hydration of ions, dielectric saturation of solvent (water), and excluded volume (steric) interactions between species. By solving a modified Poisson-Boltzmann equation and using the appropriate results of quantum chemistry calculations on the hydration of ions, we can accurately approximate the differential capacitance profiles of aqueous electrolyte solutions at the boundary with a silver electrode. The specific interactions between the ions and the electrodes in the systems under considerationare assumed to be significantly weaker than their Coulomb interactions. A novel aspect of our research is the investigation of the impact of short-range ion-water interactions on the differential capacitance, which provides new insights into the behavior of the electrical double layer. This model holds the potential to be useful for electrochemical engineers working on the development of supercapacitors and related electrochemical energy storage devices. It serves as a basis for future modeling of electrolyte systems on real electrodes, especially in scenarios where chemical ion-electrode interactions are significant.

(Invited) Analysis of Interfacial Species at Finite Bias - Calcium Borohydride in Tetrahydrofuran

To understand the operation of complex electrolytes we must determine which species are present and active at the interface with the electrode. We combine molecular dynamics sampling with continuum modeling to determine the speciation of Ca(BH4)2 in THF next to a model electrode (graphite). Mulitvalent ions in organic solvents with poor dielectric screening may exist in multiple coordinated states. We explore the free energy landscape associated with the interconversion of species in the bulk electrolyte using metadynamics simulations. We find that the chemical equilibrium between neutral and charged species is dominated by disproportionation following dimer formation - consistent with experimental findings. We also make use of clustering analysis to reveal distinct molecular conformations within constrained coordination states. To assess how the chemical equilibrium may be modified by the potential difference at the electrode, we develop a continuum model based on the modified Poisson-Boltzmann equation, with volume exclusion. This model also incorporates molecular-scale details through sampled free energy profiles of each relevant species within the solvent as a function of distance from the electrode. We find that a narrow (approximately 1 nm) region near the electrode, with highly variable speciation driven by potential difference, dominates the electrochemical activity of this electrolyte.

Donnan equilibrium in charged slit-pores from a hybrid nonequilibrium molecular dynamics/Monte Carlo method with ions and solvent exchange.

Ion partitioning between different compartments (e.g., a porous material and a bulk solution reservoir), known as Donnan equilibrium, plays a fundamental role in various contexts such as energy, environment, or water treatment. The linearized Poisson-Boltzmann (PB) equation, capturing the thermal motion of the ions with mean-field electrostatic interactions, is practically useful to understand and predict ion partitioning, despite its limited applicability to conditions of low salt concentrations and surface charge densities. Here, we investigate the Donnan equilibrium of coarse-grained dilute electrolytes confined in charged slit-pores in equilibrium with a reservoir of ions and solvent. We introduce and use an extension to confined systems of a recently developed hybrid nonequilibrium molecular dynamics/grand canonical Monte Carlo simulation method ("H4D"), which enhances the efficiency of solvent and ion-pair exchange via a fourth spatial dimension. We show that the validity range of linearized PB theory to predict the Donnan equilibrium of dilute electrolytes can be extended to highly charged pores by simply considering renormalized surface charge densities. We compare with simulations of implicit solvent models of electrolytes and show that in the low salt concentrations and thin electric double layer limit considered here, an explicit solvent has a limited effect on the Donnan equilibrium and that the main limitations of the analytical predictions are not due to the breakdown of the mean-field description but rather to the charge renormalization approximation, because it only focuses on the behavior far from the surfaces.

Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel.

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same signon both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same signas the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

A Method for Efficiently Predicting the Radial Distribution Function and Osmotic Coefficients of Aqueous Electrolyte Solutions.

The prediction of the structural and thermodynamic properties of electrolyte solutions is critical for a huge range of practical situations where these solutions play a vital role. Theoretical models, such as the continuum solvent model, attempt to explain the behavior of solutions using a coarse-grained description of the interactions of species in the solution, whereas molecular simulations aim to directly compute the behavior of the solution, including the interactions between all ions and molecules in the system. Both methods have limitations: theoretical models are generally less accurate because they rely on assumptions, while molecular simulations require significant computational resources, particularly if higher accuracy is desired. To address these issues, we propose an affordable and effective method that combines the advantages of the modified Poisson-Boltzmann equation (MPBE) with classical molecular dynamics (MD) simulations to predict the radial distribution functions and thermodynamic properties of electrolyte solutions. We demonstrate a method of using the MPBE to compute the short-range potential of mean force (PMF) from the radial distribution functions (RDFs) and vice versa. Furthermore, we provide insights into the relationship between the RDFs and the short-range PMF based on the MPBE. Our analysis reveals that the effective short-range PMFs can be approximately calculated using low concentration simulations but the short-range PMFs are slightly concentration-dependent in simulations at higher concentrations. Additionally, we demonstrate that for concentrated solutions, osmotic coefficients can be calculated in agreement with experiment using a virial approach. This is based on the effective short-range PMFs and RDFs obtained from the MPBE method. Our proposed MPBE can therefore accelerate the calculation of the structural and thermodynamic properties of electrolyte solutions.

Non-isothermal electroosmotic flow of a viscoelastic fluid through a porous medium in a microchannel

The influence of the temperature-dependent viscosity on the electroosmotic flow through a porous medium in a microchannel of parallel flat plates. At the end, an imposed electric field induces Joule heating and temperature gradients, modifying the viscosity of the fluid. The Phan-Thien–Tanner and Darcy–Forchheimer models describe the viscoelastic fluid through the porous media. Consequently, the electric double layer (EDL) overlaps at the center plane of the microchannel, causing an electrically charged porous matrix and modifying the zeta potential along the microchannel walls. Therefore, we define a zeta potential ratio of the porous matrix and that at the microchannel walls. Unlike similar investigations, the boundary condition for the zeta potential varies locally due to the temperature field, modifying the interaction among hydrodynamics, electrical, viscoelastic, thermal and porosity effects. A modified Péclet number is defined as a function of the porosity and thermal conductivity of the fluid and an effective charge density is determined based on the Poisson–Boltzmann equation, accounting for the pore size and the zeta potential ratio. The set of equations are solved numerically considering the lubrication approximation theory. We found that the influence of the temperature-dependent viscosity increases the shear stresses at the EDL region, particularly where the temperature rises to maximum values. Also, the overlapping of the EDL occurs mainly for low values of permeability. Finally, the pressure gradient involved in the Darcy law can be altered through the zeta potential ratio.

In situ polymerization of a poly(3,4-ethylenedioxythiophene):poly(sodium–styrennesulfonate) conductive layer for electro-assisted ultrafiltration and fouling mitigation

Electrification can enhance the antifouling performance of ultrafiltration membranes and improve the sustainability of membrane technologies. The design of high-performance conductive membrane materials remains one of the core technologies in electro-assisted ultrafiltration. Here, conductive PVDF/PEDOT:PSS composite membranes were prepared via in situ chemical oxidation polymerization on a PVDF surface. The PVDF/PEDOT:PSS membranes were subsequently used for electro-filtration to study the effect of the applied potential on humic acid fouling. When a voltage of 3 V was applied, the PVDF/PEDOT:PSS20 membrane was continuously polluted for 5 cycles. The results showed that after simple hydraulic cleaning, the membrane flux recovery rate reached 91 %. Compared with that of the PVDF membrane, the irreversible fouling of the PVDF/PEDOT:PSS20 membrane decreased by 65 %. The modified Poisson Boltzmann equation indicates that applying cathodic potential on the surface of PEDOT:PSS conductive layer increases the cumulative concentration of counter ions on the membrane surface and the thickness of the charged layer, thereby promoting the migration of peak forces on the membrane surface to the bulk solution, changing the fouling behaviour of HA from cake filtration to standard blocking.

Dielectric Boundary Force for the Minimization of the Mean-Field Electrostatic Free-Energy Functional Including the Ionic Size Effect

The generalized Poisson–Boltzmann equation with the uniform ionic size is investigated. In mean-field theory, the ionic size effect is added by modifying the entropy effect of solvent molecules in the electrostatic free-energy functional of ion concentrations. Taking minimization of the mean-field electrostatic free-energy functional including the ionic size effect with respect to the ion concentrations, the minimum free-energy is obtained (see (4.6) in Li [(2009a) “Continuum electrostatics for ionic solutions with non-uniform ionic sizes,” Nonlinearity 22(4), 811–833]. Maximizing the minimum free-energy with respect to the electrostatic potential determines the generalized Poisson–Boltzmann equation. By the definition of shape derivative, the first variation of this minimum free-energy with respect to variation of the dielectric boundary is derived. Thus, the dielectric boundary force of an ionic solution with multiple ionic species with the uniform ionic size is obtained.

Discovering novel targets of abscisic acid using computational approaches

Abscisic acid (ABA) is a crucial plant hormone that is naturally produced in various mammalian tissues and holds significant potential as a therapeutic molecule in humans. ABA is selected for this study due to its known roles in essential human metabolic processes, such as glucose homeostasis, immune responses, cardiovascular system, and inflammation regulation. Despite its known importance, the molecular mechanism underlying ABA's action remain largely unexplored. This study employed computational techniques to identify potential human ABA receptors. We screened 64 candidate molecules using online servers and performed molecular docking to assess binding affinity and interaction types with ABA. The stability and dynamics of the best complexes were investigated using molecular dynamics simulation over a 100 ns time period. Root mean square fluctuations (RMSF), root mean square deviation (RMSD), solvent-accessible surface area (SASA), radius of gyration (Rg), free energy landscape (FEL), and principal component analysis (PCA) were analyzed. Next, the molecular mechanics Poisson–Boltzmann surface area (MM-PBSA) method was employed to calculate the binding energies of the complexes based on the simulated data. Our study successfully pinpointed four key receptors responsible for ABA signaling (androgen receptor, glucocorticoid receptor, mineralocorticoid receptor, and retinoic acid receptor beta) that have a strong affinity for binding with ABA and remained structurally stable throughout the simulations. The simulations with Hydralazine as an unrelated ligand were conducted to validate the specificity of the identified receptors for ABA. The findings of this study can contribute to further experimental validation and a better understanding of how ABA functions in humans.

Unsteady electroosmotic flow of Carreau–Newtonian fluids through a cylindrical tube

The present study aims to investigate the fully developed electroosmotic flow of Carreau–Newtonian fluids within a circular tube, considering two different boundary conditions of zeta potential. The unsteady behavior of Carreau–Newtonian fluids, is attributed to the pulsatile pressure-driven flow. Studying two distinct formulations, slip and no-slip zeta potentials, illustrates the relative movement of fluid particles and the charged surface. The proposed study’s framework is divided into two regions: a central region containing a non-Newtonian Carreau fluid exhibiting both shear-thinning and thickening behavior, and a peripheral region containing a Newtonian fluid with the effect of an electrical double layer at the solid wall of the cylindrical tube. The Poisson–Boltzmann equation under the assumption of Debye–Hückel approximation governs the potential distribution due to the electric double layer, facilitating the examination of electroosmotic flow through tube. Obtaining analytical solutions for the momentum equations in both regions becomes challenging due to the inclusion of pulsatile pressure-driven flow and a non-Newtonian Carreau fluid. Asymptotic series expansions are employed, utilizing small parameters such as the Weissenberg number (We≪1) and a pulsatile Reynolds number (α≪1), to derive velocity expressions for both regions. Wall shear stress fluctuations, as indicated by time-averaged wall shear stress and oscillatory shear index are assessed to gauge the temporal variation of wall shear stress from the dominant blood flow direction. By utilizing potential and hydrodynamic variables, an asymptotic formulation for the streaming potential is developed to ascertain the asymptotic expression of electrokinetic energy conversion efficiency and scrutinize the impacts of pertinent physical parameters on it. The proposed study prominently shows that the Debye–Hückel parameter, Weissenberg number, pulsatile Reynolds number, and time-dependent pressure gradient significantly influence the fluid velocity, flow rate, and flow resistance. The notable determination of the present study is that the velocity amplitude and electrokinetic energy conversion efficiency are enlarged in the slip-dependent case relative to the no-slip zeta potential. It is noteworthy that as inertial forces become more prominent compared to viscous forces, both time-average wall shear stress and oscillatory shear index experience a slight decrease, although the reduction in oscillatory shear index is minimal owing to the absence of obstruction within the tube wall. The NDSolve numerical scheme in Mathematica software is employed to calculate numerical solutions for the governing equations, which are subsequently verified against results obtained through asymptotic analysis. The findings of this study are expected to deepen our understanding of unsteady electroosmotic flows of Carreau fluid and contribute to the design and development of advanced microfluidic devices with enhanced performance for various applications.

Electro-Osmotic Flow and Mass Transfer through a Rough Microchannel with a Modulated Charged Surface.

In this paper, we investigate the electro-osmotic flow (EOF) and mass transfer of a Newtonian fluid propelled by a pressure gradient and alternating current (AC) electric field in a parallel microchannel with sinusoidal roughness and modulated charged surfaces. The two-wall roughness is described by in-phase or out-of-phase sine functions with a small amplitude δ. By employing the method of perturbation expansion, the semi-analytical solutions of the Poisson-Boltzmann (P-B) equation based on the Debye-Hückel approximation and the modified Navier-Stokes (N-S) equation are obtained. The numerical solution of the concentration equation is obtained by the finite difference method. The effects of sinusoidal roughness, modulated charged surface, and the AC electric field on the potential field, velocity field, and concentration field are discussed. Under the influence of the modulated charged surface and sinusoidal roughness, vortices are generated. The velocity oscillates due to the effect of the AC electric field. The results indicate that solute diffusion becomes enhanced when the oscillation Reynolds number is below a specific critical value, and it slows down when the oscillation Reynolds number exceeds this critical value.

Time-periodic electrokinetic analysis of a micropolar fluid flow through hydrophobic microannulus

The oscillating aspects of pressure-driven micropolar fluid flow through a hydrophobic cylindrical microannulus under the influence of electroosmotic flow are analytically studied. The study is based on a linearized Poisson–Boltzmann equation and the micropolar model of Eringen for microstructure fluids. An analytical solution is obtained for the distributions of electroosmotic flow velocity and microrotation as functions of radial distance, periodic time, and relevant parameters. The findings of the present study demonstrate that, unlike the decrease in flow rate resulting from the micropolarity of fluid particles, velocity slip and spin velocity slip, when contrasted with Newtonian fluids, act as a counteractive mechanism that tends to enhance the flow rate. Additionally, the findings indicate that a square plug-like profile in electroosmotic velocity amplitude is observed when the electric oscillating parameter is low and the electrokinetic width is large, for both Newtonian and micropolar fluids. Moreover, in cases where there is a wide gap between the cylindrical walls and a high-frequency parameter, the electroosmotic velocity and microrotation amplitudes tend to approach zero at the center of the microannulus across all ranges of micropolarity and zeta potential parameters. Furthermore, it has been observed that the amplitude of microrotation strength rises as slip and spin slip parameters increase. Across the entire spectrum of micropolarity, the zeta potential ratio influences both the dimension and direction of the electroosmotic velocity profiles within the electric double layer near the two cylindrical walls of the microannulus. The study emphasizes the physical quantities by presenting graphs for various values of the pertinent parameters juxtaposing them with existing data in the literature and comparing them with the Newtonian fluids.Graphic abstract

An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface.

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

Surface charge-dependent slip length modulates electroosmotic mixing in a wavy micromixer

This study explores electroosmotic mixing in microfluidic channel with predefined surface topology, mainly focusing the effect of surface charge-dependent slip length on the underlying mixing dynamics. Our analysis addresses the need for precise control of flow and mixing of the participating fluids at microscale, crucial for medical and biomedical applications. In the present work, we consider a wavy microchannel with non-uniform surface charge to explore the electroosmotic mixing behavior. To this end, adopting a finite-element approach, we numerically solve the Laplace, Poisson–Boltzmann, convection–diffusion, and the Navier–Stokes equations in a steady-state. The model is validated by comparing the results with the available theoretical and experimental data. Through numerical simulations, the study analyzes electroosmotic flow patterns in microchannels, highlighting the impact of surface charge-dependent slip lengths on mixing efficiency. For example, at a diffusive Peclet number of 200, mixing efficiency drops from 95.5% to 91.5% when considering surface charge-dependent slip length. It is established that the fluid rheology, characterized by Carreau number and flow behavior index, non-trivially influences flow field modulation and mixing efficiency. Increased Carreau numbers enhance flow velocity, affecting overall mixing of the constituent fluids in the chosen fluidic pathway. For instance, by increasing the Carreau number from 0.01 to 1.0, a discernible trend emerges with higher flow line density and accelerated velocity within the microchannel. The study also examines the effect of diffusive Peclet numbers on the mixing efficiency, particularly in the convective regime of underlying transport. These insights offer practical guidance for designing microfluidic systems intended for enhanced mixing capabilities. Additionally, the study explores the likelihood of particle aggregation under shear forces, vital in biological non-Newtonian fluids, with implications for drug delivery, diagnostics, and biomedical technologies.