Electrostatic Disjoining Pressure Research Articles

Dynamics of electrostatic interaction and electrodiffusion in a charged thin film with nanoscale physicochemical heterogeneity: Implications for low-salinity waterflooding

The slow kinetics of wettability alteration toward a more water-wetting state by low-salinity waterflooding (LSWF) in oil-brine-rock (OBR) systems is conjectured to be pertinent to the electrokinetic phenomena in the thin brine film. We hypothesize that the nanoscale physicochemical heterogeneities such as surface roughness and surface charge heterogeneity at the rock/brine interface control further the dynamics of electrodiffusion and electrostatic disjoining pressure (Πel), thus the time-scale and the magnitude of the low salinity effect (LSE). In this regard, film-scale computational fluid dynamics (CFD) simulations were performed. The coupled Poisson-Nernst-Planck (PNP) equations were solved numerically in a thin water film confined between a solid surface and oil, both negatively charged. The solid surface is representative of sandstone (quartz/kaolinite) with patchwise physicochemical heterogeneity. The electrical properties of the oil are representative of a crude-oil sample. The OBR system was initially under chemical equilibrium with high salinity (HS) brine, then was exposed to low salinity (LS) brine. The time-scale of reaching chemical equilibrium under LS, and the spatio-temporal evolution of electric potential were investigated. We find that surface roughness (introduced by quartz patches on quartz surface) increases the diffusion time up to 3-fold due to increased tortuosity. However the effect of surface roughness and surface charge heterogeneity (introduced by kaolinite patches on quartz surface) on the effective diffusion coefficient (Deff) is minor. While surface roughness and surface charge heterogeneity affect the disjoining pressure (Πel) significantly, the influence of surface roughness on Πel is more pronounced under HS than LS condition. In contrast, the effect of surface charge heterogeneity is more appreciable under LS than HS. Our findings imply that the LS effect can be enhanced in rough, heterogeneously charged systems like clayey sandstone, although its magnitude depends on the charge density of the roughness and its variation with salinity. We introduce two scaling factors, namely the effective diffusion coefficient (Deff) and the retardation coefficient (ω), to upscale the nanoscale results to pore-scale and beyond.

Enhanced transport of ions by tuning surface properties of the nanochannel.

Motivated by recent observations of anomalously large deviations of the conductivity currents in confined systems from the bulk behavior, we revisit the theory of ion transport in parallel-plate channels and also discuss how the wettability of a solid and the mobility of adsorbed surface charges impact the transport of ions. It is shown that depending on the ratio of the electrostatic disjoining pressure to the excess osmotic pressure at the walls two different regimes occur. In the thick channel regime this ratio is small and the channel effectively behaves as thick, even when the diffuse layers strongly overlap. The latter is possible for highly charged channels only. In the thin channel regime the disjoining pressure is comparable to the excess osmotic pressure at the wall, which implies relatively weakly charged walls. We derive simple expressions for the mean conductivity of the channel in these two regimes, highlighting the role of electrostatic and electrohydrodynamic boundary conditions. Our theory provides a simple explanation of the high conductivity observed experimentally in hydrophilic channels, and allows one to obtain rigorous bounds on its attainable value and scaling with salt concentration. Our results also show that further dramatic amplification of conductivity is possible if hydrophobic slip is involved, but only in the thick channel regime provided the walls are sufficiently highly charged and most of the adsorbed charges are immobile. However, for weakly charged surfaces the massive conductivity amplification due to hydrodynamic slip is impossible in both regimes. Interestingly, in this case the moderate slip-driven contribution to conductivity can monotonously decrease with the fraction of immobile adsorbed charges. These results provide a framework for tuning the conductivity of nanochannels by adjusting their surface properties and bulk electrolyte concentrations.

Spiropyran Sulfonates for Photo- and pH-Responsive Air-Water Interfaces and Aqueous Foam.

Responsive foams and interfaces are interesting building blocks for active materials that respond and adapt to external stimuli. We have used the photochromic reaction of a spiropyran sulfonate surfactant to render interfacial, rising bubbles as well as foaming properties active to light stimuli. In order to address the air-water interface on a molecular level, we have applied sum-frequency generation (SFG) spectroscopy which has provided qualitative information on the surface excess and the interfacial charging state as a function of light irradiation and solution pH. Under blue light irradiation, the surfactant forms a closed ring spiro form (SP), whereas under dark conditions the ring opens and the merocyanine (MC) form is generated. Using SFG spectroscopy, we show that at the interface, different pH conditions of the bulk solution lead to changes in the interfacial charging state. We have exploited the fact that the MC surfactant's O-H group can be deprotonated as a function of pH and used that to tune the molecules net charge at the interface. In fact, SFG spectroscopy shows that with increasing pH the intensity of the O-H stretching band from interfacial water molecules increases, which we associate to an increase in surface net charge. At a pH of 5.3, irradiation with blue light leads to a reversible decrease of O-H intensities, whereas the C-H intensities were unchanged compared to the corresponding intensities under dark conditions. These results are indicative of changes in the surface net charge with light irradiation, which are also expected to influence the foam stability via changes in the electrostatic disjoining pressure. In fact, measurements of the foam stabilities are consistent with this hypothesis and show higher foam stability under dark conditions. At pH 2.7 this behavior is reversed as far as the surface tension and surface charging as well as the foam stability are concerned. This is corroborated by rising bubble experiments, which demonstrated an unprecedented reduction of ∼30% in bubble velocity when the bubbles were irradiated with blue light compared to the velocity of bubbles with the surfactants in the dark state. Clearly, the light-triggered changes can be used to control foams, rising bubbles, and fluid interfaces on a molecular level which renders them active to light stimuli.

Charge-Controlled Surface Properties of Native and Fluorophore-Labeled Bovine Serum Albumin at the Air-Water Interface.

Proteins at interfaces are important for protein formulations and in soft materials such as foam. Here, interfacial stability and physicochemical properties are key elements, which drive macroscopic foam properties through structure–property relations. Native and fluorescein isothiocyanate-labeled bovine serum albumin (BSA) were used to modify air–water interfaces as a function of pH. Characterizations were performed with tensiometry and sum-frequency generation (SFG). SFG spectra of O–H stretching vibrations reveal a phase reversal and a pronounced minimum in O–H intensity at pH values of 5.3 and 4.7 for native and labeled BSA, respectively. This minimum is attributed to the interfacial isoelectric point (IEP) and is accompanied by a minimum in surface tension and negligible ζ-potentials in the bulk. Interfacial proteins at pH values close to the IEP can promote macroscopic foam stability and are predominately located in the lamellae between individual gas bubbles as evidenced by confocal fluorescence microscopy. Different from the classical stabilization mechanisms, for example, via the electrostatic disjoining pressure, we propose that the presence of more close-packed BSA, because of negligible net charges, inside the foam lamellae is more effective in reducing foam drainage as compared to a situation with strong repulsive electrostatic interactions.

Smart Air-Water Interfaces with Arylazopyrazole Surfactants and Their Role in Photoresponsive Aqueous Foam.

A new light-switchable azo-surfactant arylazopyrazole tetraethylene glycol carboxylic acid (AAP-E4) was used as a molecular building block to functionalize macroscopic foams. AAP-E4 was studied in the bulk solution with UV/vis spectroscopy and at the interface with sum-frequency generation (SFG) as well as tensiometry. Additional foaming experiments were performed with a dynamic foam analyzer to study the role of AAP-E4 surfactants at the ubiquitous air–water interface as well as within macroscopic foam. In the bulk, it is possible to switch the AAP-E4 surfactant reversibly from trans to cis configurations and vice versa using 380 nm UV and 520 nm green light, respectively. At the interface, we demonstrate the excellent switching ability of AAP-E4 surfactants and a substantial modification of the surface tension. In addition, we show that the response of the interface is strongly influenced by lateral electrostatic interactions, which can be tuned by the charging state of AAP-E4. Consequently, the electrostatic disjoining pressure and thus the foam stability are highly dependent on the bulk pH and the charging state of the interface. For that reason, we have studied both the surface net charge (SFG) and the surface excess (tensiometry) as important parameters that determine foam stability in this system and show that neutral pH conditions lead to the optimal compromise between switching ability, surface excess, and surface charging. Measurements on the foam stability demonstrated that foams under irradiation with green light are more stable than foams irradiated with UV light.

Hydrodynamics of foams

This brief review article is devoted to all the aspects related to hydrodynamics of foams. For this reason, we focused at first on the methods for studying the basic structural units of the foams—the foam films (FF) and the Plateau borders (PB), thus reviewing the literature about their drainage. After this, we scrutinized in detail the Derjaguin’s works on the electrostatic disjoining pressure along with its Langmuir’s interpretation, the microscopic and macroscopic approaches in the theory of the van der Waals disjoining pressure, the DLVO theory, the steric disjoining pressure of de Gennes, and the more recent works on non-DLVO forces. The basic methods for studying of foam drainage are presented as well. Engineering and other applications of foam are reviewed as well. All these aspects are presented from retrospective and perspective viewpoints.

Electrostatic interaction of charged surfaces with semipermeable membranes

We have investigated the ion density distribution and electrostatic disjoining pressure of the size-asymmetric electrolytes in the gap between the semipermeable membranes and the charged surface by using the density functional theory. The present theory is in reasonably good agreement with the simulation data whereas the Poisson-Boltzmann equation does not describe the electrostatic properties of the electrolytes near a charged surface. The ion density distribution and the electrostatic disjoining pressure depend strongly on the size-asymmetric electrolytes in charge and size as well as the ion density of the electrolyte. The finite size of the ions leads to a large repulsive electrostatic disjoining pressure in the gap. However, the effect of the gap width on the repulsive electrostatic disjoining pressure is very small. The repulsive electrostatic disjoining pressure increases with increasing or decreasing surface charge density with its having a minimum value near a neutral surface. The surface charge amplification and the charge reversal phenomenon are observed for a neutral semipermeable membrane with a charged surface. A positively charged surface enhances the charge surface amplification whereas a negatively charged surface enhances the charge reversal phenomenon.

Effect of Ionic Correlations on the Surface Forces in Thin Liquid Films: Influence of Multivalent Coions and Extended Theory.

Experimental data for the disjoining pressure of foam films stabilized by anionic surfactant in the presence of 1:1, 1:2, 1:3, and 2:2 electrolytes: NaCl, Na2SO4, Na3Citrate, and MgSO4 are reported. The disjoining pressure predicted by the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory coincides with the experimental data in the case of a 1:1 electrolyte, but it is considerably greater than the measured pressure in all other cases. The theory is extended to account for the effects of ionic correlations and finite ionic radii. Original analytical expressions are derived for the local activity coefficient, electrostatic disjoining pressure, and asymptotic screening parameter. With the same parameter of counterion binding as for a 1:1 electrolyte, the curves predicted by the extended theory are in perfect agreement with the experimental data for 1:2 and 1:3 electrolytes. In comparison with the DLVO theory, the effect of ionic correlations leads to more effective screening of electrostatic interactions, and lower electric potential and counterion concentrations in the film’s midplane, resulting in lower disjoining pressure, as experimentally observed. The developed theory is applicable to both multivalent coions and multivalent counterions. Its application could remove some discrepancies between theory and experiment observed in studies with liquid films from electrolyte solutions.

Specific effects of Ca(2+) ions and molecular structure of β-lactoglobulin interfacial layers that drive macroscopic foam stability.

β-Lactoglobulin (BLG) adsorption layers at air-water interfaces were studied in situ with vibrational sum-frequency generation (SFG), tensiometry, surface dilatational rheology and ellipsometry as a function of bulk Ca(2+) concentration. The relation between the interfacial molecular structure of adsorbed BLG and the interactions with the supporting electrolyte is additionally addressed on higher length scales along the foam hierarchy - from the ubiquitous air-water interface through thin foam films to macroscopic foam. For concentrations <1 mM, a strong decrease in SFG intensity from O-H stretching bands and a slight increase in layer thickness and surface pressure are observed. A further increase in Ca(2+) concentrations above 1 mM causes an apparent change in the polarity of aromatic C-H stretching vibrations from interfacial BLG which we associate to a charge reversal at the interface. Foam film measurements show formation of common black films at Ca(2+) concentrations above 1 mM due to considerable decrease of the stabilizing electrostatic disjoining pressure. These observations also correlate with a minimum in macroscopic foam stability. For concentrations >30 mM Ca(2+), micrographs of foam films show clear signatures of aggregates which tend to increase the stability of foam films. Here, the interfacial layers have a higher surface dilatational elasticity. In fact, macroscopic foams formed from BLG dilutions with high Ca(2+) concentrations where aggregates and interfacial layers with higher elasticity are found, showed the highest stability with much smaller bubble sizes.

Model of Meniscus Shape and Disjoining Pressure of Thin Liquid Films on Nanostructured Surfaces with Electrostatic Interactions

The effect of electrostatic interactions on the stability of thin liquid films on nanostructured surfaces is important in lubrication, wetting, and phase change but is poorly understood. In this study, a general, closed-form model is developed to account for both the effects of electrostatic and van der Waals interactions on meniscus shape and disjoining pressure for thin liquid films on nanostructured surfaces based on the minimization of free energy, the Derjaguin approximation, and the disjoining pressure theory for flat surfaces. The model is verified using the molecular dynamics (MD) simulations for a water–alumina system with both triangular and square nanostructures of varying depth and film thickness. Good agreement is obtained between MD results and model predictions, demonstrating the robustness of the analytical model. The results show that the electrostatic interactions enhance the disjoining pressure, thereby making the meniscus more conformal to the nanostructured surfaces. In addition, the electrostatic disjoining pressure is shown to increase with the nanostructure depth but decrease with the thin film thickness.

Lateral capillary interactions between colloids beneath an oil-water interface that are driven by out-of-plane electrostatic double-layer interactions.

We study the lateral capillary interactions between colloids beneath an oil-water interface that lead to closely packed two-dimensional self-assembled colloidal crystals. These capillary forces are caused by the overlap of deformed interfaces above colloids on a solid substrate. The interface deformation is due to the electrostatic disjoining pressure between the charged particles and the charged oil-water interface. It is notable that the short-range (i.e., on the nanometer scale) and out-of-plane electrostatic double-layer interactions, which occur through an aqueous phase, can generate the long-range lateral capillary attraction (i.e., on the micrometer scale).

Disjoining pressure of an electrolyte film confined between semipermeable membranes.

We consider an electrolyte solution confined by infinitesimally thin semipermeable membranes in contact with a salt-free solvent. Membranes are uncharged, but since small counter-ions leak-out into infinite salt-free reservoirs, we observe a distance-dependent membrane potential, which generates a repulsive electrostatic disjoining pressure. We obtain the distribution of the potential and of ions, and derive explicit formulas for the disjoining pressure, which are validated by computer simulations. We predict a strong short-range power-law repulsion, and a weaker long-range exponential decay. Our results also demonstrate that an interaction between membranes does strongly depend on the screening lengths, valency of an electrolyte solution, and an inter-membrane film thickness. Finally, our analysis can be directly extended to the study of more complex situations and some biological problems.

유체 계면에서 콜로이드 흡착에 대한 전해질 농도의 영향

I present the behavior of colloidal adsorption to an oil-water interface in the presence of electrolyte in an aqueous subphase. The optical laser tweezers and the piezo controller are used to trap an individual polystyrene microsphere in water and forcibly transfer it to the interface in the vertical direction. Addition of an electrolyte (i.e., NaCl) in the aqueous subphase enables the particle to attach to the interface, whereas the particle escapes from the trap without the adsorption in the absence of the electrolyte. Based on the analytical calculations of the optical trapping force and the electrostatic disjoining pressure between the particle and the oil-water interface, it is found that a critical energy barrier between them should exist. This study will provide a fundamental understanding for applications of colloidal particles as solid surfactants that can stabilize the immiscible fluid-fluid interfaces, such as emulsions (i.e., Pickering emulsions) and foams.

Drainage and critical thickness of foam films stabilized by n-dodecyl-β-D-maltoside

The kinetic stability of foam films stabilized with n-dodecyl-β-D-maltoside (C12G2) in the presence of electrolyte (NaCl) has been investigated. The surface charge, type of the film and its electrostatic stabilisation were controlled by changing of the surfactant (from 0.01 to 1CMC) and electrolyte (0.1 and 0.001M) concentration. The drainage coefficient, critical thickness and life time to the critical state of films with different radii (0.05, 0.1 and 0.15mm) were measured interferometrically using a new procedure suggested by us for determination of the reflected light intensity from a video recording of the film evolution. It was found that the films from solutions of C12G2 up to CMC (even at CMC) with 0.001M NaCl are ``common (CF) electrostatically stabilized films'', while at the same C12G2 concentrations they were unstable films (UF) or stable Newtonian black films (NBF), when the added electrolyte was 0.1M. The values of film critical thickness obtained at a high electrolyte concentration were similar even when the concentration of C12G2 was varied from 0.1 to 1CMC. It means that in a wide surfactant concentration range the electrostatic disjoining pressure is screened and the experimental data of critical thickness are appropriate for theoretical analysis. The disjoining pressure acting at different steps of the film evolution was evaluated based on experimental data of the film drainage. A non-DLVO attractive force for films with non covered surfaces and Πel=0 was detected. It was expressed by a power law in the same form as the van-der-Waals force, which makes it possible to directly compare these two forces by means of the force constant K232 and the Hamaker constant. The non-DLVO force was found to be considerably stronger than the Lifshitz – van-der-Waals forces.The obtained new experimental data for critical thickness were compared with those calculated by us with different equations (Radoev–Sheludko–Manev, Vrij, and Coons). In some of these cases we used the experimentally obtained value of the drainage coefficient instead of the velocity of the film thinning, calculated theoretically. It was found that the calculated hcr are noticeably lower than experimentally measured ones. Good agreement between measured and calculated hcr was achieved with a non-DLVO attraction force taken into account, when the films were obtained from solutions with low (C12G2) and high (NaCl) concentrations.

Electrostatic interaction of heterogeneously charged surfaces with semipermeable membranes

In this paper we study the electrostatic interaction of a heterogeneously charged wall with a neutral semipermeable membrane. The wall consists of periodic stripes, where the charge density varies in one direction. The membrane is in contact with a bulk reservoir of an electrolyte solution and separated from the wall by a thin film of salt-free liquid. One type of ions (small counterions) permeates into the gap. This gives rise to a distance-dependent membrane potential, which translates into a repulsive electrostatic disjoining pressure due to an overlap of counterion clouds in the gap. To quantify it we use two complementary approaches. First, we propose a mean-field theory based on a linearized Poisson-Boltzmann equation and Fourier analysis. These calculations allow us to estimate the effect of a heterogeneous charge pattern at the wall on the induced heterogeneous membrane potential, and the value of the disjoining pressure as a function of the gap. Second, we perform Langevin dynamics simulations of the same system with explicit ions. The results of the two approaches are in good agreement with each other at low surface charges and small gaps, but differ due to nonlinearity at higher charges. These results demonstrate that a heterogeneity of the wall charge can lead to a huge reduction in the electrostatic repulsion, which could dramatically facilitate self-assembly in complex synthetic and biological systems.

Electrostatic interaction of neutral semi-permeable membranes

We consider an osmotic equilibrium between bulk solutions of polyelectrolyte bounded by semi-permeable membranes and separated by a thin film of salt-free liquid. Although the membranes are neutral, the counter-ions of the polyelectrolyte molecules permeate into the gap and lead to a steric charge separation. This gives rise to a distance-dependent membrane potential, which translates into a repulsive electrostatic disjoining pressure. From the solution of the nonlinear Poisson-Boltzmann equation, we obtain the distribution of the potential and of ions. We then derive an explicit formula for the pressure exerted on the membranes and show that it deviates from the classical van't Hoff expression for the osmotic pressure. This difference is interpreted in terms of a repulsive electrostatic disjoining pressure originating from the overlap of counterion clouds inside the gap. We also develop a simplified theory based on a linearized Poisson-Boltzmann approach. A comparison with simulation of a primitive model for the electrolyte is provided and does confirm the validity of the theoretical predictions. Beyond the fundamental result that the neutral surfaces can repel, this mechanism not only helps to control the adhesion and long-range interactions of living cells, bacteria, and vesicles, but also allows us to argue that electrostatic interactions should play enormous role in determining behavior and functions of systems bounded by semi-permeable membranes.

Hofmeister effect on micellization, thin films and emulsion stability

The Hofmeister effect on the critical micelle concentration (CMC), the thin liquid film electrostatic disjoining pressure (Π el) and the critical coalescence pressure of emulsion drops ( P CR) were investigated. For CMC literature data were used, but Π and P CR were measured by us. The essence of the theoretical approach was to modify existing theories of CMC and Π el by using generalized Gouy equation and dimensionless surface potential (Φ S), involving the counterion specific adsorption energy ( u 0). The computational procedure of u 0 does not involve any adjustable parameters. Linear dependences of ln(CMC), Φ S and P CR on u 0 were found in conformity with Hofmeister series. The experimental slopes of ln(CMC) and Φ S vs. – u 0 /k B T were negative and very close to the theoretical ones. A hypothesis was put forward for explanation of the positive slopes of P CR on u 0. The obtained results suggest that the counterion specific adsorption energy u 0 encompasses all major factors, involved in the Hofmeister effect for the studied phenomena. If this is confirmed by analysis of more phenomena, revealing Hofmeister effect, one could claim that u 0 is the factor controlling the Hofmeister effect and a powerful tool for its study.

Elasticity of foam bubbles measured by profile analysis tensiometry

Elastic modulus of foam bubbles, stabilized with tetraethylene glycol octyl ether (C 8E 4) and 1 × 10 −5 M NaCl, was determined by cyclic expansion and shrinking of foam bubbles with frequency of 0.1 Hz and volumetric amplitude of 2 mm 3. The film tension was monitored by a commercial profile analysis tensiometer (Sinterface Technologies, GmbH). The elastic moduli of foam bubbles were obtained as a function of surfactant concentration in the range of 2 × 10 −3–1 × 10 −2 M. The theory of Lucassen and van den Tempel [1] for the elastic modulus of a single liquid/air interface at a given frequency was employed. In the theoretical analysis the bulk diffusion coefficient of surfactant molecules was considered as a unknown model parameter which was obtained by matching the theory with the experimental data. Hence, the dependence of the bulk diffusion coefficient of C 8E 4 molecules upon the C 8E 4 concentration was obtained. The diffusion coefficient reached a maximum at 5 × 10 −3 M C 8E 4 ( D = 8.5 × 10 −11 m 2/s). In the experimental surfactant concentration range (2 × 10 −3–1 × 10 −2 M, CMC = 7.5 × 10 −3 M) the foam bubbles were relatively dry, with visible interferometric fringes corresponding to thin films stabilized by repulsion of the electrostatic disjoining pressure. Hence, the overall dynamics of periodical expansion and shrinking of the foam bubbles occurred within the thin film state.

Dynamic effects in thin liquid films containing ionic surfactants

This paper is dedicated to studying dynamic effects in thin liquid films (TLF) containing ionic surfactants. The standard theory of TLF drainage has been developed without considering the electrical double layer (EDL) in the hydrodynamic equations, although EDL always exists. In addition, it has been found that this theory very well describes the drainage of TLF containing non-ionic surfactants in the presence of electrolytes. The inclusion of EDL into the hydrodynam ics of TLF complicates the theory , producing additional dynamic effects during film drainage. For example, a gradient of electrostatic disjoining pressure across the film arises, thus causing non-uniform electrostatic repulsion between the film surfaces. This paper analyzes the hydrodynamics of TLF with EDL. A new equation of drainage was derived. This equation accounts for the non-uniform distribution of surface charges during the films drainage, which is coupled with non-uniform electrostatic repulsion between the film surfaces and results in faster film drainage. The theory was tested with drainage experiments on TLF with ionic surfactants. Foam films containing sodium dodecyl sulfate (SDS) in the presence and in the absence of added electrolyte were studied and the experimental data compared to the theoretical predictions. The experimental results, however, disagree with the theory. For example, the kinetic equation predicted faster film drainage for foam films at low ionic strength; at high ionic strength the theory tends to “Reynolds drainage”. Inversely, the experiment exhibited slower drainage than predicted by the Reynolds equation in both cases of low and high ionic strengths. Numerical simulations yielded V / V Re < 1 . In addition, cases of “positive” and “negative” velocity of film surfaces were shown. Despite the sign of the velocity the dependence V / V Re < 1 remained. The analysis showed similarity between the experimental data and the prediction of the Manev–Tsekov–Radoev (MTR) drainage model at R < R c r . The analysis showed rather complex dynamics of the film drainage. The paper calls upon further development on this topic.

Osmotic pressure acting on a semipermeable shell immersed in a solution of polyions

We study theoretically the osmotic equilibria for a shell immersed in a suspension of polyions (e.g., colloids, polyelectrolytes, etc.). The shell is treated as impermeable for polyions, but allowing free diffusion of counterions that permeate inside the shell. From the solution of linearized Poisson-Boltzmann equation, we obtain the distribution of a potential and concentration profiles for polyions and counterions. We then obtain an explicit formula for the excess osmotic pressure of a polyion solution exerted on the shell, which includes a quadratic term in order to provide a self-consistency of a linear theory. As a result this pressure is larger than given by a concentration of polyions at the outer shell boundary obtained within linearized theory. It is, however, always smaller than or equal to the bulk osmotic pressure. This difference is attributed to a repulsive electrostatic disjoining pressure due to an overlap of counterion clouds inside the shell. A comparison with molecular dynamics simulations is provided and demonstrates that although the concentration profiles obtained within a linear theory deviate from simulation data at large potential, the theoretical and simulation pressures are in surprisingly good harmony.