Introducing interacting diffuse layers in TLM calculations: a reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites

The truncation of the Gouy-Chapman diffuse part in compacted clay-rocks and bentonite is introduced into the electrical triple-layer model (TLM) recently developed by P. Leroy and A. Revil [J. Colloid Interface Sci. 270 (2004) 371]. The new model is used to explain the dependence of the osmotic efficiency and the swelling pressure as functions of the mean pore size of the medium, determined from the porosity and the specific surface. The truncation of the diffuse layer introduces a new variable in the system of equations to be solved, the electrical potential at the midplane between adjacent charged surfaces. This new variable is evaluated through a Taylor expansion of the electrical potential. The present model is able to capture the variation of the osmotic efficiency and the swelling pressure with the mean pore size. The partition of counterions between the Stern layer and the diffuse layer as a function of the pore size calculated by the TLM also shows a good consistency with the model. This implies that more than 90% of the counterions are located in the Stern layer.     

Swelling of dextran gel and osmotic pressure of soluble dextran in the presence of salts

The swelling of dextran gels (Sephadex) in salt solutions with a water activity of 0.937, compared with the swelling in pure water, exhibited anion specificity as evidenced by an increased swelling ratio in the following order: Na₂SO₄ < H₂O < NaCl < NaSCN. The swelling ratio showed a good linear correlation with the osmotic pressure of dextran (500 kD) in these solutions. The salt‐concentration difference (imbalance) between the polymer‐solution side of the membrane and the polymer‐free permeate side during the osmotic‐pressure measurements positively correlated with the effect of the salt on the polymer osmotic pressure. These phenomena conform to Hofmeister‐type (or lyotropic) behavior. The diminishing augmentation of dextran osmotic pressure and the change in the salt‐concentration imbalance with rising NaSCN concentration imply a positive preferential interaction and adsorption of the salt onto the polymer. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2740–2750, 2001     

On Tanaka-Fillmore's kinetics swelling of gels

Tanaka and Fillmore treated the swelling of a gel as a process where a crosslinked polymer network having been initially under uniform stress is expanded by osmotic pressure, sucking up the surrounding fluid medium. We point out that their physical reasoning is unnatural and leads to an unacceptable conclusion; we propose a more sound approach to the same problem. Our treatment assumes that the gel network is extended not by the osmotic pressure of the gel, but rather by the swelling pressure which is generated by the excess fluid penetrating in against the real nature of a polymer network that tends to shrink. The diffusion equation of the fluid, hence, plays a dominant role and gives the distribution of fluid concentration in contrast to Tanaka-Fillmore's scheme. The expression for the distribution of local strain in a spherical gel is deduced from the relation of mechanical balance between two forces, the one is due to the elasticity of the network and the other due to the gradient in the chemical potential of the fluid. The results obtained have forms analytically similar to Tanaka-Fillmore's, but are differ in the physical meanings.     

The effect of membrane potential on the development of chemical osmotic pressure in compacted clay

When clay soils are subjected to salt concentration gradients, various interrelated processes come into play. It is known that chemical osmosis induces a water flow and that a membrane potential difference develops that counteracts diffusive flow of solutes and osmotic flow of water. In this paper, we present the results of experiments on the influence of membrane potential on chemical osmotic flow and diffusion of solutes and we show how we are able to derive the membrane potential value from theory. Moreover, the simultaneous development of water pressure, salt concentration and membrane potential difference are simulated using a model for combined chemico-electroosmosis in clays. A new method for short-circuiting the clay sample is employed to assess the influence of electrical effects on flow of water and transport of solutes.     

Thermo‐responsive gels in the presence of monovalent salt at physiological concentrations: A Monte Carlo simulation study

In this work, slightly charged thermo‐responsive gels in the presence of salt at concentrations close to physiological conditions have been simulated within a coarse‐grained model widely used in the last decade. These simulations allow differentiate charge and salt effects, which are antagonist and coupled in many real systems because the degree of ionization might depend on the electrolyte concentration. An analysis in terms of the different contributions to osmotic pressure is also presented, which highlights the role played by excluded volume effects. In addition, our results also permit us to test some predictions based on the ideal Donnan equilibrium, a common assumption made to justify the swelling behavior of gels and microgels in the presence of salt. More specifically, simulations show that, for the slightly charged gels simulated here, such an assumption overestimates the concentration of salt inside collapsed gels and underestimates the excess of osmotic pressure associated to the additional electrolyte. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 1403–1411     

Osmotic deswelling of weakly charged poly(acrylic acid) solutions and gels

The osmotic pressure of weakly charged aqueous poly(acrylic acid) (PAA) solutions and the swelling pressure PAA gels were studied by osmotic deswelling at different degrees of ionization (α). In solution, the osmotic pressure was found to scale linearly with concentration, whereas the scaling power of the swelling pressure of gels was higher (1.66). The effect of the ionization degree on the osmotic coefficient in PAA solutions was in agreement with the theory of Borue and Erukhimovich [Macromolecules, 21 , 3240 (1988)]. Ionization increases the swelling capacity of the PAA gels until a plateau is reached at about 35% neutralization. The concentration at equilibrium swelling scales as C_e ～ α^?0.6. The contribution of the network to the gel swelling pressure is evaluated by subtracting the osmotic pressure of the polymer solution at the same concentration and degree of ionization. In swollen gels the extended network opposes swelling. As the gel is osmotically deswelled, a state of zero network pressure exists at a certain concentration, below which the network elasticity favors swelling. The crossover concentration shifts to lower values as the degrees of ionization increases. © 1995 John Wiley & Sons, Inc.     

Swelling and fractal heterogeneities in networks

In this paper we revisit old swelling data on polymer networks that have not been interpreted theoretically on a closed molecular basis. If the osmotic pressure of the swollen network is compared to the osmotic pressure of the corresponding uncrosslinked solution unsolved problems appear, when the relative osmotic pressure is plotted against the degree of swelling, i.e. the deformation due to swelling. A significant maximum appears which cannot be explained by any of the recently derived elastic models, such as junction constraint or other entanglement models. It is suggested in this paper that the maximum is a consequence of structural heterogeneities of fractal nature. If such fractal heterogeneities are assumed a strong maximum in the relative osmotic pressure can be reproduced. The physical reason is the different thermodynamic behavior of uncrosslinked linear chains and crosslinked self-similar (non-linear) polymers. The conclusion is supported by numerical (Monte Carlo) simulations.     

Modeling the swelling pressure of degrading hydroxyethylmethacrylate‐grafted dextran hydrogels

Degrading hydroxyethylmethacrylate‐grafted dextran (dex‐HEMA) hydrogels generate a relatively sudden increase in osmotic pressure upon degradation into dextran solutions. This phenomenon is currently being examined as a possible means of developing a pulsatile drug‐delivery system. Here a mathematical model based on scaling concepts is presented to describe this sudden increase in swelling pressure and to provide a framework for the rational design of pulsatile delivery systems based on this phenomena. The model provides a good fit to the swelling pressures measured for dex‐HEMA gel/free dextran mixtures that simulate degrading dex‐HEMA gels. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3397–3404, 2004     

Collapse of polyelectrolyte networks induced by their interaction with an oppositely charged surfactant. Theory

A very simple theory of swelling and collapse of weakly charged polyelectrolyte networks in the solution of an oppositely charged surfactant has been developed. The following contributions to the free energy were taken into account: free energy of volume interaction and of elastic deformation of the network chains, free energy connected with micelle formation and free energy of translational motion of all mobile ions in the system (translational entropy). Both the cases of a solution of charged surfactant and that of a mixed solution of charged and neutral surfactant components have been taken into account. It has been shown that the behaviour of the network depends on the total surfactant concentration in the system and corresponds to one of the three following regimes: At low concentration, micelles inside the network are not formed and the behaviour of the polymer network is similar to that of a network swelling in the solution of a lowmolecular-weight salt (regime 1). In the second regime, surfactant concentration inside the network exceeds the critical micelle concentration and micelles are formed; in this regime the network collapses because surfactant molecules, aggregated in micelles, cease to create “exerting” osmotic pressure in the network sample. In the third regime, at very high surfactant concentration, formation of additional micelles inside the network ceases, and the network dimensions coincide with those of the corresponding neutral network.     

On the Modeling of Polyelectrolyte Gels

Summary: Ionic polymer gels are very attractive actuation materials with a great similarity to biological contractile tissues. They consist of a polymer network with bound charged groups and a liquid phase with mobile ions. Absorption and delivery of solvent lead to a considerably large change of volume. This swelling mechanism results from the equilibrium of different forces such as osmotic pressure forces, electrostatic forces and viscoelastic restoring forces and can be triggered by chemical (change of salt concentration or pH in the solution), thermal or electrical stimulation. In the present work, chemically and electrically stimulated electrolyte polymer gels in a solution bath are investigated. To describe the different phenomena occurring in these gels adequately, the modeling can be conducted on different scales. If only the global macroscopic behavior is of interest, the statistical theory which is capable to describe the global swelling ratio, is sufficient. By refining the scale, the mesoscopic coupled multi-field theory can be applied. Here, the chemical field is described by a convection-diffusion equation for the different mobile species. The electric field is directly obtained by solving the Poisson equation in the gel and solution domain. The mechanical field is formulated by the momentum equation. By further refining the scale, the whole structure can be investigated on the microscale by the discrete element (DE) method. In this model, the material is represented by distributed particles comprising a certain amount of mass; the particles interact with each other mechanically by a truss or beam network of massless elements. The mechanical behavior, i.e. the dynamics of the system, is followed by solving the Newton's equations of motion while the chemical field, i.e. the ion movement inside the gel and from the gel to the solution, is described by diffusion equations for the different mobile particles. All three formulations can give chemical, (electrical) and mechanical unknowns and all rely on the assumption that the concentration differences between the different regions of the gel and between gel and solution form the osmotic pressure difference, which is a main cause for the mechanical deformation of the polyelectrolyte gel film.     

Activity of counterions in hydrogels based on poly(acrylic acid) and poly(methacrylic acid): Potentiometric measurements

The behavior of partially ionized weakly crosslinked gels based on poly(acrylic acid) and poly(methacrylic acid) undergoing contraction in the presence of a low-molecular-mass salt is studied experimentally. The concentration dependences of the enthalpies of swelling of gels in water and aqueous solutions with potassium chloride concentrations of 1, 10, and 100 mmol/L are determined via the method of isothermal calorimetry. On the basis of the obtained data, the enthalpy parameter of interaction between a polymer network and a medium is estimated as a function of the amount of the salt. This parameter does not exceed 0.3 and monotonically decreases with an increase in the concentration of the salt. The Donnan potential of gels depending on the concentration of KCl in the external solution is measured via the method of potentiometry with the use of capillary electrodes, and the activity of potassium counterions in the medium of hydrogel is calculated. The main factor that causes contraction of polyelectrolyte gels in a solution of a low-molecular-mass salt is a decline in the activity of counterions that leads to a decrease in the osmotic pressure inside the gel. A decline in activity may be associated with both a reduction in the activity coefficient and ionic association processes in the hydrogel.     

Microstructure and rheology of stimuli-responsive nanocolloidal systems-effect of ionic strength

The effect of ionic strength on the rheological behavior of model pH-responsive nanocolloidal systems consisting of methacrylic acid-ethyl acrylate (MAA-EA) cross-linked with diallyl phthalate (DAP) was examined. Neutralization of acid groups increases the osmotic pressure exerted by counterions trapped in the polymeric network against ions in bulk solution, which is responsible for the swelling and increase in viscosity. Swelling decreases with increasing salt concentration as a result of reduced osmotic pressure inside the microgels, which is attributed to the charge shielding effect of counterions (salt) on the negatively charged carboxylate groups. Electromotive measurements using ion-selective electrodes confirmed that not all the counterions, that is, K+, remain mobile, but a fraction of these ions can penetrate the porous microgel particles to shield the negatively charged carboxylate groups. A consequence of this is that some of the Na+ counterions inside the particles are expelled, thus regaining their translational entropy, and become mobile sodium ions in the bulk solution. We successfully developed a new scaling law that relates the swelling ratio, Q, of microgels as a function of neutralization degree, alpha, cross-linked density, Nx, molar fraction of acidic units, y, and concentration of mobile counterions, CK+ and CNa+, represented as (Nx/c0)(CK+ + CNa+Q + Q2/3 proportional, variant yNxalpha. The new scaling law no longer assumes that all the counterions are trapped inside the microgels. The proportionality reduces to the form Q proportional, variant (yalphaNx)3/2 in the absence of salt, that is, CK+ + CNa+ approximately 0. By combining the results from light scattering and rheological measurements, we are able to correlate the microstructural evolution of the colloidal systems with their bulk rheological behavior.     

Interactions of glucose and polyacrylamide in solutions and gels

The swelling of polyacrylamide (PAAm) gels increased with rising glucose concentrations, and so did the osmotic pressure of the soluble polymer and its intrinsic viscosity. A Flory–Huggins‐based model for the osmotic pressure of a nonionic hydrophilic polymer in a ternary solution consisting of a main solvent, a polymer, and a nondissociating low‐molecular‐weight cosolute was developed and examined. The model‐calculated values were in reasonably good agreement with experimental results for the water–PAAm–glucose system studied when PAAm–water and glucose–water interaction coefficients from the binary systems were used, and only the PAAm–glucose interaction coefficient was adjusted. Its negative value suggested a favorable interaction of glucose and PAAm, supporting the notion of glucose being a good cosolvent for PAAm. Isothermal titration microcalorimetry results showed no evidence for the binding of glucose to PAAm, but an exothermic interaction was indicated between glucose and PAAm. Microcalorimetrically determined enthalpic contributions to the Flory–Huggins interaction coefficients showed enthalpically favorable binary interactions, particularly the enthalpic component of the PAAm–glucose interaction coefficient (χ_H23), which was slightly negative. The enthalpically favorable interaction between glucose and PAAm may explain the increased osmotic pressure of PAAm in glucose solutions. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 3053–3063, 2003     

Self-assembled polymer membrane capsules inflated by osmotic pressure

We fabricate and characterize capsules that are composite membranes, made of a polymer network stabilized by adsorption to colloids and inflated by osmotic pressure from internal free polyelectrolyte; here, poly-l-lysine forms the network and inflates the capsules. To assess these capsules' properties and structure, we deform capsules using microcantilevers and use finite element modeling to describe these deformations. Additional experimental tests confirm the model's validity. These capsules' resilient response to mechanical forces indicates that loading and shear should be good triggers for the release of contents via deformation. The osmotic pressure inflating these capsules has the potential to trigger release of contents via deflation in response to changes in the capsules' environment; we demonstrate addition of salt as a trigger for deflating capsules. Because these capsules have a variety of release triggers available and the technique used to fabricate them is very flexible and allows high encapsulation efficiency, these capsules have very high potential for application in many areas.     

Thermodynamic Properties of Aqueous Glucose–Urea–Salt Systems

In this work, the thermodynamic behavior of aqueous solutions containing the solutes NaCl, glucose, and/or urea is investigated. These substances are vital components for living bodies and further they are main components of blood serum. Osmotic coefficients were determined by cryoscopic measurements in single-solute and multi-solute aqueous solutions containing salts (NaCl, KCl, CaCl₂), glucose, and/or urea. The results show that NaCl determines the osmotic coefficients in the urea/glucose/NaCl/water system. Investigation of the effect of different salts on osmotic coefficients revealed ion-specific effects. At physiologically important solute concentrations in typical blood serum solutions, the osmotic coefficients were found to be in the range of 0.90–0.93. In a second step, the state of water in different glucose/salt/water and urea/salt/water systems was investigated. Depending on the kind of salt, the chemical potential of water in urea/salt/water is either higher or lower than in glucose/salt/water systems at equal nonelectrolyte concentrations. This result was found to be independent of the salt molality. Finally, the investigated systems were modeled with the Pitzer model and the ePC-SAFT equation of state, which allowed predicting of the properties of these multi-solute aqueous solutions.     

Swelling pressure and elastic behavior of polymacromonomer networks with different functionalities of junctions

Networks with different junction functionalities as obtained by polymerization of a macromonomer (composed of 20 units) at identical concentrations were studied by computer simulation. The functionality determined by the length of chains produced from the end units of the macromonomer was varied over a wide range by varying the kinetic parameters of polymerization. From the number of collisions of units with the lattice walls at different swelling stages, the network swelling pressure and the osmotic pressure of solution of its fragments obtained by cutting in half interjunction chains were determined. From these data, the osmotic and elastic components of swelling pressure were found, the former was defined as the pressure of solution and the latter was defined as the difference of the network and solution pressures. The osmotic component is a power function of the polymer concentration with the power index increasing from 2.7 to 3.9 with an increase in functionality from 4.8 to 55 in accordance with a change in network topology. The elastic pressure depends on the swelling ratio Q in different manners at a low and a high functionality of junctions. Its absolute value decreases with a growth in Q in the former case, in agreement with the theory of elasticity of phantom networks (～Q ^?1/3), but increases in the latter case. This behavior is consistent with the effect of functionality on the elastic behavior of real polymacromonomer networks and confirms that differences in the character of change in the modulus of such networks during swelling are due to a difference in the functionality of their junctions. Possible mechanisms of the influence of multifunctional junctions on the elasticity of polymer networks are discussed.     

Toward a Constitutive Law of Cartilage: A Polymer Physics Perspective

To describe load bearing and lubrication of cartilage requires treating its collagen network and proteoglycan (PG) phases separately in a constitutive law of the tissue. We propose a framework for developing such an empirical constitutive law that treats the cartilage extracellular matrix (ECM) as a composite medium, with a PG phase that exerts a swelling pressure, and a collagen network phase that restrains it. We compare and contrast this model to a biomechanical constitutive law that aggregates the collagen and PG phases into a single “solid-like” elastic tissue matrix, and show that aggregation obscures essential differences in the physical-chemical properties of the collagen and PG constituents as well as their distinct biological roles within cartilage's ECM. We also relate moduli in the aggregate constitutive model to quantities measured in an osmotic stress titration experiment.     

Swelling,elasticity and structure of hydrogels prepared via bis-macromonomers of poly(ethylene oxide)

The effect of swelling on the shear modulus was studied for hydrogels prepared by radical polymerization of methacrylate-terminated poly(ethylene oxide) (PEO) bis-macromonomers of different molecular weight. Gels made of long chains (M = 12000 or 6000) display classical softening upon swelling, whereas gels made of shorter chains (M = 4000 or 2000) remain rigid or even stiffen. The abnormal behaviour is explained by a specific character of network junctions presented by polymethacrylate chains in which each unit is linked with a PEO network chain. It is assumed that the interactions among densely grafted PEO chains result in their stretching on polymerization and non-affine deformation on swelling, which stiffen the gel. This is verified by the data on copolymer (macromonomers - 2-hydroxyethyl methacrylate) gels that have lesser densities of PEO chains attached to the junctions and show weaker stiffening on swelling. The osmotic pressure of gels was estimated from the swelling pressure and shear modulus. Similar to the mixing pressure of equivalent PEO solutions, it varies as the 9/4 power of polymer concentration. At the same time, it is lower than the mixing pressure. This indicates that the junctions make only quantitative changes in the osmotic properties of macromonomer chains.     

Deformational,swelling, and potentiometric behavior of ionized poly(methacrylic acid) gels. I. Theory

By using the model of a randomly coiled chain, a relation is derived describing the equilibrium stress–strain behavior of variously ionized polyelectrolyte gels swollen in solutions of a uni–univalent salt. The effect of the concentration of bound counterions calculated on the basis of the cylindrical model and the effect of the change of length of the statistical chain segment with the change in ionization of the gel on stress–strain, swelling, and potentiometric equilibria is discussed.