Nanomolar Surface-Active Charged Impurities Account for the Zeta Potential of Hydrophobic Surfaces

A hydrophilic/hydrophobic switch on polymer surface triggered by calcite towards separation of hazardous PVC from plastic mixtures

Hydrophilic modification of polymer surfaces is crucial for the emerging flotation separation of plastic waste towards resources recycling. In this study, we investigated a novel hydrophilic regulation induced by calcite to modify the surface wettability of PVC, ABS, PS, PC, and PET. The interactions between calcite and plastic molecules contributed to the selective formation of hydrophilic calcite shells on plastic surfaces. Due to the strong polarity of C-Cl bonds on PVC surfaces, calcium ions tended to act as bridges between calcite particles and PVC, and a non-covalent interaction leaded to strong physical adsorption of calcite on PVC surfaces. However, other plastics adsorbed fewer calcite than PVCs owing to weaker interactions. Like a hydrophilic/hydrophobic switch, ultrasonic cleaning erased hydrophilic calcite shells to restore the hydrophobicity of hydrophilic plastics without destroying the chemical structure of polymers. The recovery and purity of PVC were 100 % and 96.3 %, under the conditions of calcite dosage 1.5 g L-1, coating time 15 min, coating temperature 50 °C, pH 10, and rotational speed 900 r min-1. Its attractive applicability and reversibility were further demonstrated by the flotation separation of actual plastic mixtures and the eversible floatability of sinking plastics.

Propensity of hydroxide and hydronium ions for the air-water and graphene-water interfaces from ab initio and force field simulations

Understanding acids and bases at interfaces is relevant for a range of applications from environmental chemistry to energy storage. We present combined ab initio and force-field molecular dynamics simulations of hydrochloric acid and sodium hydroxide highly concentrated electrolytes at the interface with air and graphene. In agreement with surface tension measurements at the air-water interface, we find that HCl presents an ionic surface excess, while NaOH displays an ionic surface depletion, for both interfaces. We further show that graphene becomes less hydrophilic as the water ions concentration increases, with a transition to being hydrophobic for highly basic solutions. For HCl, we observe that hydronium adsorbs to both interfaces and orients strongly toward the water phase, due to the hydrogen bonding behavior of hydronium ions, which donate three hydrogen bonds to bulk water molecules when adsorbed at the interface. For NaOH, we observe density peaks of strongly oriented hydroxide ions at the interface with air and graphene. To extrapolate our results from concentrated electrolytes to dilute solutions, we perform single ion-pair ab initio simulations, as well as develop force-field parameters for ions and graphene that reproduce the density profiles at high concentrations. We find the behavior of hydronium ions to be rather independent of concentration. For NaOH electrolytes, the force-field simulations of dilute NaOH solutions suggest no hydroxide adsorption but some adsorption at high concentrations. For both interfaces, we predict that the surface potential is positive for HCl and close to neutral for NaOH.

Zeta Potentials of Cotton Membranes in Acetonitrile Solutions

Solid surfaces in contact with nonaqueous solvents play a key role in electrochemistry, analytical chemistry, and industrial chemistry. In this work, the zeta potentials of cotton membranes in acetonitrile solutions were determined by streaming potential and bulk conductivity measurements. By applying the Gouy-Chapman theory and the Langmuir adsorption isotherm of ions to the experimental data, the mechanism of the electrification at the cotton/acetonitrile interface is revealed for the first time to be solely due to ion adsorption on the surface, rather than proton dissociation at the interface. Different salts were found to produce opposite signs of the zeta potentials. This behavior can be attributed to ion solvation effects and the strong ordering of acetonitrile molecules at the interface. Furthermore, a trend of the electroviscous effect was observed, in agreement with the standard electrokinetic theory. These findings demonstrate that electrokinetics in acetonitrile, a polar aprotic solvent, can be treated in the same manner as in water.

pH drives electron density fluctuations that enhance electric field-induced liquid flow

Liquid flow along a charged interface is commonly described by classical continuum theory, which represents the electric double layer by uniformly distributed point charges. The electrophoretic mobility of hydrophobic nanodroplets in water doubles in magnitude when the pH is varied from neutral to mildly basic (pH 7 → 11). Classical continuum theory predicts that this increase in mobility is due to an increased surface charge. Here, by combining all-optical measurements of surface charge and molecular structure, as well as electronic structure calculations, we show that surface charge and molecular structure at the nanodroplet surface are identical at neutral and mildly basic pH. We propose that the force that propels the droplets originates from two factors: Negative charge on the droplet surface due to charge transfer from and within water, and anisotropic gradients in the fluctuating polarization induced by the electric field. Both charge density fluctuations couple with the external electric field, and lead to droplet flow. Replacing chloride by hydroxide doubles both the charge conductivity via the Grotthuss mechanism, and the droplet mobility. This general mechanism deeply impacts a plethora of processes in biology, chemistry, and nanotechnology and provides an explanation of how pH influences hydrodynamic phenomena and the limitations of classical continuum theory currently used to rationalize these effects.

How liquids charge the superhydrophobic surfaces

Liquid-solid contact electrification (CE) is essential to diverse applications. Exploiting its full implementation requires an in-depth understanding and fine-grained control of charge carriers (electrons and/or ions) during CE. Here, we decouple the electrons and ions during liquid-solid CE by designing binary superhydrophobic surfaces that eliminate liquid and ion residues on the surfaces and simultaneously enable us to regulate surface properties, namely work function, to control electron transfers. We find the existence of a linear relationship between the work function of superhydrophobic surfaces and the as-generated charges in liquids, implying that liquid-solid CE arises from electron transfer due to the work function difference between two contacting surfaces. We also rule out the possibility of ion transfer during CE occurring on superhydrophobic surfaces by proving the absence of ions on superhydrophobic surfaces after contact with ion-enriched acidic, alkaline, and salt liquids. Our findings stand in contrast to existing liquid-solid CE studies, and the new insights learned offer the potential to explore more applications.

Effect of the Surface Charge-Dependent Boundary Slip on the Electrophoresis of a Hydrophobic Polarizable Rigid Colloid

The electrophoresis of a hydrophobic charged rigid colloid is studied by considering the lateral movement of the adsorbed surface charge. The slip velocity condition at the hydrophobic surface is modified to take into account the impact of the frictional and electric forces created by the adsorbed laterally mobile surface charge. Though the dependency of the surface charge on the slip velocity in the context of electrophoresis has been addressed before, the effect of the laterally mobile adsorbed surface charge on the electrophoresis of hydrophobic colloids has not been studied. The dielectric colloid is considered to polarize and create an induced immobile surface charge when subjected to an imposed electric field. The impact of the mobile surface charge along with the immobile induced surface charge on electrophoresis of a hydrophobic colloid is elucidated by numerically solving the governing electrokinetic equations in their full form. We have also developed a simplified model under a weak applied field consideration, which can be further reduced to a closed-form analytic expression for the mobility under the Debye-Hückel approximation. This analytic model for mobility is in excellent agreement with the exact numerical solution for an entire range of the Debye length when the ζ-potential is in the order of the thermal potential. One of the notable features of this closed-form mobility expression is that it accounts for the mobile adsorbed surface charge on the hydrodynamic slip condition and the dielectric polarization of the particle. We find that the mobility of the surface charge decreases the electrophoretic mobility of the hydrophobic dielectric colloid. However, the mobile surface charge enhances the mobility of a conducting hydrophobic colloid.

Multiscale Modeling of Aqueous Electric Double Layers

From the stability of colloidal suspensions to the charging of electrodes, electric double layers play a pivotal role in aqueous systems. The interactions between interfaces, water molecules, ions and other solutes making up the electrical double layer span length scales from Ångströms to micrometers and are notoriously complex. Therefore, explaining experimental observations in terms of the double layer's molecular structure has been a long-standing challenge in physical chemistry, yet recent advances in simulations techniques and computational power have led to tremendous progress. In particular, the past decades have seen the development of a multiscale theoretical framework based on the combination of quantum density functional theory, force-field based simulations and continuum theory. In this Review, we discuss these theoretical developments and make quantitative comparisons to experimental results from, among other techniques, sum-frequency generation, atomic-force microscopy, and electrokinetics. Starting from the vapor/water interface, we treat a range of qualitatively different types of surfaces, varying from soft to solid, from hydrophilic to hydrophobic, and from charged to uncharged.

Zeta potential of crude oil in aqueous solution

Despite the broad range of interest and applications, controls on the surface charge of crude oil in aqueous solution remain poorly understood. The primary data source to understand the surface charge on crude oil comprises measurements of zeta potential on individual drops or emulsions obtained using the electrophoretic method (EPM). Here we (i) collate and review previous measurements of zeta potential on crude oil, (ii) compare and contrast the results, and (iii) report new measurements of zeta potential on crude oil wetting films and layers relevant to oil-saturated porous media, obtained using the streaming potential method (SPM). Results show that the zeta potential depends on electrolyte pH and the concentration of divalent ions Ca2+ and Mg2+. Lower pH and higher concentration of these divalent ions yields more positive zeta potential. The isoelectric point (IEP) in simple NaCl electrolytes lies in the pH range 3-5. The IEP in simple CaCl2 and MgCl2 electrolytes can be expressed as pCa or pMg, respectively, and lies in the range 0-1. Close to the IEP, the zeta potential varies linearly with pH, pCa or pMg, suggesting simple Nernstian behaviour of the crude oil surface. The sensitivity of the zeta potential to pH, pCa and pMg decreases with increasing total ionic strength. The impact of pH, pCa and pMg on zeta potential varies significantly across different crude oils and differs from non-polar hydrocarbons. The potential for other multivalent ions to modify crude oil zeta potential has not been tested. Data for crude oil wetting films and layers, obtained using the SPM and strongly oil-wet porous substrates in which the solid surfaces are coated with the crude oil of interest, are comparable to those obtained using emulsions and the EPM, suggesting that the controls on zeta potential on crude oil are the same irrespective of whether the oil forms droplets or wetting layers. The literature data reviewed here, along with new measured data, provide important insight into the effect of pH, and the concentration of divalent ions, on the zeta potential of crude oil in aqueous solution. They demonstrate relationships between ion concentration and zeta potential that are observed irrespective of crude oil composition. They also show that the crude oil composition plays a role, yet no consistent trends are observed between zeta potential and commonly measured bulk oil properties, possibly because bulk properties do not reflect the concentrations of interfacially active species in crude oil that may impact the development of surface charge. Moreover, data are extremely scarce for complex, high ionic strength electrolytes or at elevated temperature. The data reviewed and reported here have broad relevance to many engineering and industrial activities involving crude oil.

Aqueous Dispersions of Polypropylene: Toward Reference Materials for Characterizing Nanoplastics

Microplastics and nanoplastics pollute the natural environment all over the world, but the full extent of the hazards posed by this waste is unclear. While research on microplastics is well advanced, little work has been done on nanoplastics. This discrepancy is mainly due to the lacking ability to detect nanoplastics in biologically and environmentally relevant matrices. Nanoplastics reference materials can help the development of suitable methods for identifying and quantifying nanoplastics in nature. The aim is to synthesize nanoplastics made from one of the most commonly used plastics, namely polypropylene. An easy way to produce long-term stable aqueous dispersions of polypropylene nanoparticles (nano polypropylene) is reported. The nanoplastic particles, prepared by mechanical breakdown, show a mean hydrodynamic diameter of D_h = 180.5 ± 5.8  nm and a polydispersity index of PDI = 0.084 ± 0.02. No surfactant is needed to obtain dispersion which is stable for more than 6 months. The colloidal stability of the surfactant-free nano polypropylene dispersions is explained by their low zeta potential of ζ = -43 ± 2 mV.

Electrokinetic, electrochemical, and electrostatic surface potentials of the pristine water liquid-vapor interface

Although conceptually simple, the air-water interface displays rich behavior and is subject to intense experimental and theoretical investigations. Different definitions of the electrostatic surface potential as well as different calculation methods, each relevant for distinct experimental scenarios, lead to widely varying potential magnitudes and sometimes even different signs. Based on quantum-chemical density-functional-theory molecular dynamics (DFT-MD) simulations, different surface potentials are evaluated and compared to force-field (FF) MD simulations. As well explained in the literature, the laterally averaged electrostatic surface potential, accessible to electron holography, is dominated by the trace of the water molecular quadrupole moment, and using DFT-MD amounts to +4.35 V inside the water phase, very different from results obtained with FF water models which yield negative values of the order of -0.4 to -0.6 V. Thus, when predicting potentials within water molecules, as relevant for photoelectron spectroscopy and non-linear interface-specific spectroscopy, DFT simulations should be used. The electrochemical surface potential, relevant for ion transfer reactions and ion surface adsorption, is much smaller, less than 200 mV in magnitude, and depends specifically on the ion radius. Charge transfer between interfacial water molecules leads to a sizable surface potential as well. However, when probing electrokinetics by explicitly applying a lateral electric field in DFT-MD simulations, the electrokinetic ζ-potential turns out to be negligible, in agreement with predictions using continuous hydrodynamic models. Thus, interfacial polarization charges from intermolecular charge transfer do not lead to significant electrokinetic mobility at the pristine vapor-liquid water interface, even assuming these transfer charges are mobile in an external electric field.

Ostwald ripening of aqueous microbubble solutions

Bubble solutions are of growing interest because of their various technological applications in surface cleaning, water treatment, and agriculture. However, their physicochemical properties, such as the stability and interfacial charge of bubbles, are not fully understood yet. In this study, the kinetics of radii in aqueous microbubble solutions are experimentally investigated, and the results are discussed in the context of Ostwald ripening. The obtained distributions of bubble radii scaled by mean radius and total number were found to be time-independent during the observation period. Image analysis of radii kinetics revealed that the average growth and shrinkage speed of each bubble is governed by diffusion-limited Ostwald ripening, and the kinetic coefficient calculated using the available physicochemical constants in the literature quantitatively agrees with the experimental data. Furthermore, the cube of mean radius and mean volume exhibit a linear time evolution in agreement with the Lifshitz-Slezov-Wagner (LSW) theory. The coefficients are slightly larger than those predicted using the LSW theory, which can be qualitatively explained by the effect of finite volume fraction. Finally, the slowdown and pinning of radius in the shrinkage dynamics of small microbubbles are discussed in detail.

Polyethylene separator supported thin-film composite forward osmosis membranes for concentrating lithium enriched brine

To extract lithium from salt lake brine involves a process of separation and concentration. After separating lithium from brine, the lithium ion concentration is generally a few hundred mg/L which is far below the required 20-30 g/L (as Li⁺) before precipitation as lithium carbonate. The concentration step of a lithium enriched brine is crucial but highly energy-intensive. Spontaneous forward osmosis (FO) technology offers the possibility for concentrating lithium ions with low energy. Because the concentrating process involves both feed and draw solution with very high salinity, it is highly desirable to have a high performance FO membrane with a low structural parameter as well as a high rejection to ions. In this work, thin polyethylene separator supported FO (PE-FO) membranes were prepared and post-treated stepwise with benzyl alcohol (BA) and hydraulic compaction. The effect of the post-treatment on the FO performance was systematically analyzed. Excellent FO performance was achieved: the water flux and reverse salt flux selectivity were 66.3 LMH and 5.25 L/g, respectively, when the active layer is oriented towards the 0.5 M NaCl draw solution with deionized water as the feed. To the best of our knowledge, this FO flux is the highest ever reported in the open literature under similar test conditions. Applied in concentrating lithium enriched brine, the membrane showed superior water flux using saturated MgCl₂ as draw solution. A new FO model was established to simulate the water flux during the concentration process with good agreement with the experimental results. The promising results using PE-FO membrane for lithium enrichment opens a new frontier for the potential application of FO membranes.

Electrophoretic Mobility of a Water-in-Oil Droplet Separately Affected by the Net Charge and Surface Charge Density

Water-in-oil emulsions and droplets exhibit physicochemical properties completely different from those of oil-in-water emulsions and droplets. Thus, directly applying a standard theoretical model to water-in-oil systems cannot describe these anomalous properties. Here, the electrophoretic mobility of a water-in-oil droplet is analytically investigated using Debye-Hückel linearization and neglecting the Marangoni effect. The resulting electrophoretic mobility is shown to be separately dependent on the net charge of the droplet and the surface charge density at the droplet interface. Furthermore, when the net charge is negligible, the electrophoretic mobility is proportional to the surface charge density with a negative coefficient. This indicates that the internal electric double layer inversely contributes to the electrophoresis. This theory is applied to experimental data of water-in-oil emulsions and droplets in the literature, and qualitative and quantitative verification of the theory is discussed.

Flotation separation of hazardous polyvinyl chloride towards source control of microplastics based on selective hydrophilization of plasticizer-doping surfaces

As the single largest chlorine source of plastics, hazardous polyvinyl chloride (PVC) has become an increasing environmental concern with the rapid microplastics accumulation. An advanced separation method is advocated to purify waste PVC plastics, optimize physical recycling, and protect aquatic and terrestrial environment safety. In this study, we proposed a novel scheme for the flotation separation of PVC plastics with diverse plasticizer contents (PVCs) via regulating hydrophilicity based on a selective ferric deposition. Rigid PVCs were prone to loading ferric ions and generating hydrophilic shells than flexible PVCs. Plasticizers can diffuse freely through the interior and surface of PVC plastics. Abundant plasticizers thereby overlaid the surface of flexible PVC and shielded PVC matrix from ferric ions. By regulating the ferric concentration, the wettability of PVCs was adjusted to separate rigid and flexible PVCs by froth flotation. Waste PVCs could also be separated from each other through the compound process of ferric deposition and flotation, further confirming its feasibility and stability. Thus far, this study supplies distinctive insights into the wettability regulation of plasticizer-doping PVC surfaces, contributes a pioneering hydrophilization method to PVCs separation and recycling, and mitigates hazardous PVC microplastics by source control.

Research on the Influence of Sports and Nutrition Matching on Improving Students' Physique Based on Intelligent Sensor

With the continuous development of social economy, students' learning is paid more and more attention, but at the same time, their physical condition also deserves attention. In view of these problems and limitations, this paper is based on intelligent sensors, which is different from the traditional physical education teaching, adopts the corresponding prescription for sports teaching, combines teaching with fun for the corresponding health intervention, and realizes the two-way dimensional analysis of compound sports prescription through comprehensive nutrition collocation, so as to improve students' physical and mental health, in order to improve students' physical fitness. The simulation results show that the intelligent sensor is effective, which can improve the systematicness and scientificity of students' participation in exercise in sports activities and improve students' learning and physical fitness.

Tip Charge Dependence of Three-Dimensional AFM Mapping of Concentrated Ionic Solutions

A molecular scale understanding of the organization and structure of a liquid near a solid surface is currently a major challenge in surface science. It has implications across different fields from electrochemistry and energy storage to molecular biology. Three-dimensional AFM generates atomically resolved maps of solid-liquid interfaces. The imaging mechanism behind those maps is under debate, in particular, for concentrated ionic solutions. Theory predicts that the observed contrast should depend on the tip's charged state. Here, by using neutrally, negatively, and positively charged tips, we demonstrate that the 3D maps depend on the tip's polarization. A neutral tip will explore the total particle density distribution (water and ions) while a charged tip will reveal the charge density distribution. The experimental data reproduce the key findings of the theory.

Electrification of water interface

The surface charge of a water interface determines many fundamental processes in physical chemistry and interface science, and it has been intensively studied for over a hundred years. We summarize experimental methods to characterize the surface charge densities developed so far: electrokinetics, double-layer force measurements, potentiometric titration, surface-sensitive nonlinear spectroscopy, and surface-sensitive mass spectrometry. Then, we elucidate physical ion adsorption and chemical electrification as examples of electrification mechanisms. In the end, novel effects on surface electrification are discussed in detail. We believe that this clear overview of state of the art in a charged water interface will surely help the fundamental progress of physics and chemistry at interfaces in the future.

Interplay of Interfacial Viscosity, Specific-Ion, and Impurity Adsorption Determines Zeta Potentials of Phospholipid Membranes

Ion-specific induced changes of the ζ-potential of phospholipid vesicles are commonly used to quantify the affinity of different ions to the lipid interface. The negative ζ-potential of zwitterionic net-neutral phospholipid vesicles in neat water, which changes sign and increases in solutions of NaCl or KCl, is a phenomenon consistently observed in experiments but not fully understood theoretically. Using atomistic molecular dynamics simulations in the presence of applied electric fields which drive electroosmotic flows, in combination with an electrostatic continuum model based on the modified Poisson-Boltzmann and Helmholtz-Smoluchowski equations, we study the electrokinetic and electrostatic properties as well as the specific ion affinities to the phospholipid-water interface, in order to resolve these puzzling observations. Our modified continuum equations account for the dielectric profile at the lipid-water interface, ion-specific interactions between ions and the lipid-water interface, and the interfacial viscosity profile, which are all extracted from our atomistic simulations and rather accurately predict ion-density and electrostatic-potential distributions as well as ζ-potentials in comparison with our atomistic simulations. Our continuum model can explain experimental ζ-potentials only when we assume minute amounts of surface-active anionic impurities in the aqueous solution. In fact, the amount of impurities needed to explain the experimental data increases linearly with the salt concentration, suggesting that surface-active species, which might be already present in the lab water or lipid samples, could further be introduced through the added salt.

Chitosan-sodium alginate-fatty acid nanocarrier system: Lutein bioavailability, absorption pharmacokinetics in diabetic rat and protection of retinal cells against H2O2 induced oxidative stress in vitro

Aiming to enhance therapeutic efficiency of lutein, lutein loaded chitosan-sodium alginate (CS-SA) based nanocarrier system (LNCs) were prepared and evaluated for lutein bioavailability and pharmacokinetics in diabetic rats in comparison to micellar lutein (control). Further, cytotoxicity, cellular uptake and protective activity against H₂O₂ induced oxidative stress in ARPE-19 cells were studied. Results revealed that LNCs displayed maximal lutein AUC in plasma, liver and eye respectively in normal (3.1, 2.7 and 5.2 folds) and diabetic (7.3, 3.4 and 2.8 folds) rats. Lutein from LNCs exhibited a higher half-life time, mean residence time and slow clearance from the plasma, indicating prolonged circulation compared to control. In ARPE-19 cells, pre-treatment with LNCs (10 μM) have significantly attenuated H₂O₂ induced cell death, intracellular ROS and mitochondrial membrane potential compared to control. In conclusion, LNCs improve the lutein bioavailability in conditions like diabetes, diabetic retinopathy and cataract to curtail oxidative stress in retinal cells.

Experimental Data Contributing to the Elusive Surface Charge of Inert Materials in Contact with Aqueous Media

We studied the charging of inert surfaces (polytetrafluoroethylene, i.e., PTFE; graphite; graphene; and hydrophobic silica) using classical colloid chemistry approaches. Potentiometric titrations showed that these surfaces acquired less charge from proton-related reactions than oxide minerals. The data from batch-type titrations for PTFE powder did not show an effect of ionic strength, which was also in contrast with results for classical colloids. In agreement with classical colloids, the electrokinetic results for inert surfaces showed the typical salt level dependence. In some cases, the point of zero net proton charge as determined from mass and tentatively from acid–base titration differed from isoelectric points, which has also been previously observed, for example by Chibowski and co-workers for ice electrolyte interfaces. Finally, we found no evidence for surface contaminations of our PTFE particles before and after immersion in aqueous solutions. Only in the presence of NaCl-containing solutions did cryo-XPS detect oxygen from water. We believe that our low isoelectric points for PTFE were not due to impurities. Moreover, the measured buffering at pH 3 could not be explained by sub-micromolar concentrations of contaminants. The most comprehensive explanation for the various sets of data is that hydroxide ion accumulation occurred at the interfaces between inert surfaces and aqueous solutions.

On the thermodynamic stability of bubbles, immiscible droplets, and cavities

Nanobubbles filled with air or a variety of pure gases are observed to persist in bulk water for weeks and months. Nanoemulsions consisting of oil droplets in water are also remarkably stable against coagulation, with lifetimes up to weeks even if not coated with surfactants. The inverse system of nanodroplets of water in oil is also accessible for study and application. Voids on the nanoscale are formed in simulations of water under strong tension and are stable during the time of the simulation. The stability of these nano-entities is ultimately determined by the molecular-level structure of their interfaces. However, a thermodynamic theory might also be capable of providing some insight. We therefore consider spherical gas nanobubbles, immiscible liquid nanodroplets, and nanocavities formed in water under negative pressure on the same footing, and give a unified thermodynamic analysis of these systems. In all cases, mechanical equilibrium (local free energy maximum or minimum) is expressed by the Laplace equation, and thermodynamic stability (local free energy minimum) follows from the radius dependence of surface tension. All of them would be unstable if their surface tensions were constant. Data from the literature allow construction of numerical examples for cavities and gas nanobubbles. Spectroscopic data are cited in support of an interfacial water structure in gas nanobubbles and water droplets in oil that differ from their flat surface counterparts. The observed longevity of nanobubbles in particular has been thought to violate fundamental principles of diffusion and solubility. A close look at the Laplace equation and its derivation shows why this widespread belief is incorrect.