Flory-Rehner Theory Research Articles

Ultimate swelling described by limiting chain extensibility of swollen elastomers

In this study, we study ultimate swelling characterized by limiting chain extensibility of swollen elastomers. Limiting chain extensibility is introduced into the Flory–Rehner theory using the Arruda–Boyce eight chain model and the Gent phenomenological model. The difference between these models is unified by defining a single scalar function. The inequality derived from this function allows for analysis to provide an ultimate value of swelling ratio. This ultimate value is not exceeded at equilibrium swelling regardless of the set of material constants. Under uniaxial loading at equilibrium swelling, deswelling can occur even in tension. Further, the very large swelling behavior of pH sensitive hydrogels is found to originate from the resistance generated by approaching the ultimate value of swelling ratio.

Theoretical investigation of the swelling of polysaccharide microgels in sugar solutions.

In this paper, we explain the increased swelling of crosslinked polysaccharide microgels by the increase of sugar concentration using a modified Flory-Rehner theory. This theory is validated via the investigation of the swelling of dextran microgels in sugar solutions, which can be viewed as a model system for crosslinked starch in sugar solution and custard. An essential part of our modified theory is that starch perceives the sugar solution as an effective solvent rendering a certain hydrogen bond density. Our simulations show that the often experimentally observed maximum in swelling of starch at 20% sugar concentration is probably due to the fact that equilibrium is not reached within practical time scales. Also, we discuss the use of our theory as a tool in sugar reformulation issues of custard. From simulation results one can produce a state diagram showing which formulations render a creamy, space-filling network.

Concentration-dependent swelling and structure of ionic microgels: simulation and theory of a coarse-grained model.

We study swelling and structural properties of ionic microgel suspensions within a comprehensive coarse-grained model that combines the polymeric and colloidal natures of microgels as permeable, compressible, charged spheres governed by effective interparticle interactions. The model synthesizes the Flory-Rehner theory of cross-linked polymer gels, the Hertz continuum theory of effective elastic interactions, and a theory of density-dependent effective electrostatic interactions. Implementing the model using Monte Carlo simulation and thermodynamic perturbation theory, we compute equilibrium particle size distributions, swelling ratios, volume fractions, net valences, radial distribution functions, and static structure factors as functions of concentration. Trial Monte Carlo moves comprising particle displacements and size variations are accepted or rejected based on the total change in elastic and electrostatic energies. The theory combines first-order thermodynamic perturbation and variational free energy approximations. For illustrative system parameters, theory and simulation agree closely at concentrations ranging from dilute to beyond particle overlap. With increasing concentration, as microgels deswell, we predict a decrease in the net valence and an unusual saturation of pair correlations. Comparison with experimental data for deionized, aqueous suspensions of PNIPAM particles demonstrates the capacity of the coarse-grained model to predict and interpret measured swelling behavior.

Multi-faced investigation for pH-sensitivity and solvent polarity in highly charged thermo-responsive hydro- and cryogels with strongly dissociated groups: A comparative evaluation of physico-chemical properties

Different series of strong polyelectrolyte poly(N-isopropylacrylamide-co-sodium acrylate) P(NIPA-co-NaA) hydrogels and cryogels were prepared in aqueous solution. With changing the ionic comonomer NaA ratio in reaction mixture, the resulting physico-chemical and swelling properties varied significantly. The findings elucidated that ionic P(NIPA-co-NaA) gels exhibited multi-stimuli performances with pH-, temperature-, solvent-, surfactant- and salt-responsivity, showing more sensitivity than non-ionic PNIPA. Their swollen/collapsed behaviors were strongly dependent on various synthesis/external parameters, including the content of ionic comonomer, temperature and solvent polarity. The effects produced by non-solvents were qualitatively compared considering the aspects related to co-nonsolvency phenomenon and re-entrant swelling transition on the basis of weakening of PNIPA-water interactions. Measurements of the elastic properties and swelling degree of samples having different NaA content were treated by combining Flory-Rehner theory and rubber elasticity to interpret the experimental observations. The polymer-solvent interaction parameter χ of hydrogels was calculated from the swelling studies in water at different temperatures and was related to the composition and temperature. The ionic strength-reversibility and on-off switching kinetics of hydrogels and cryogels were studied in 10−5 M and 1.0 M of NaBr solutions. Diffusion kinetics showed that the swelling of ionic hydrogels is of non-Fickian-type transportation, however, non-ionic PNIPA hydrogels and P(NIPA-co-NaA) cryogels showed pseudo-Fickian diffusion. The resulting hydrogels and cryogels containing both temperature and pH-sensitive units are able to interact with oppositely charged groups via hydrophobic and electrostatic interactions.

Tunable elasticity and thermodynamic parameters of hydroxypropyl methacrylate‐based gels with varying extents of monomer concentration: Statistical mechanics treatments of physical observations

ABSTRACTThe present article reports the scaling laws relating the preparation conditions with the equilibrium swelling degree and the crosslinking density of a new family of hydroxyl functional hydrogels (Hgs) prepared by free radical crosslinking polymerization of hydroxypropyl methacrylate (HPMA) in aqueous solution with tetraethylene glycol dimethacrylate (TEGDMA) as crosslinker. For comparison of physical properties, chemical and mechanical stability, HPMA cryogels (Cgs) were also prepared using the cryotropic gelation technique by conducting the gelation reactions at subzero temperatures. The measurements of the equilibrium swelling degree and the elastic properties of poly(hydroxypropyl methacrylate) (PHPMA) gel samples having different initial monomer concentration were treated according to the scaling theory to explain the rubbery behavior of the resulting Hgs. The obtained scaling relation between the reduced modulus and the swelling degree indicated limiting chain extensibility which is neglected in the Flory–Rehner theory. The effective crosslink density and polymer–solvent interaction parameter of PHPMA Hgs were used to interpret the observed deviation. Results showed that the properties of PHPMA Hgs and Cgs can be controlled by varying the HPMA and TEGDMA content. Increase of total monomer concentrations causes a pronounced decrease in the equilibrium swelling degree, and an increase in the elastic response of PHPMA Hgs and Cgs. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 45889.

Swelling behaviors of proton exchange membranes in alcohols

Swelling behaviors of physically crosslinked polymer membranes were observed in water and alcohols across a wide range of temperatures. Swelling of proton exchange membranes (PEM) was related to specific interactions at lower temperature. In average swelling ratio (SR) analysis, more hydrophobic PEMs showed lower expansion in water and methanol (MeOH), but higher SRs in other alcohols. The average SR was highest for MeOH and decreased as the carbon number increased. When hydrogen bonding is strongest, more hydrophobic PEMs had higher SRs in alcohols with more carbon. The total volume changes in alcohols was highest in MeOH and decreased as the carbon number increased. At high temperature, a PEM with flexible structure showed higher total volume change. A new equation based on Flory-Rehner's theory was proposed to describe the swelling behavior of PEMs. The new equation had fewer parameters and better represented the experimental volume expansion data than previous models.

Tuning and Predicting Mesh Size and Protein Release from Step Growth Hydrogels.

Hydrogel-based depots are of growing interest for release of biopharmaceuticals; however, a priori selection of hydrogel compositions that will retain proteins of interest and provide desired release profiles remains elusive. Toward addressing this, in this work, we have established a new tool for the facile assessment of protein release from hydrogels and applied it to evaluate the effectiveness of mesh size estimations on predicting protein retention or release. Poly(ethylene glycol) (PEG)-based hydrogel depots were formed by photoinitiated step growth polymerization of four-arm PEG functionalized with norbornene (PEG-norbornene, 4% w/w to 20% w/w, Mn ∼ 5 to 20 kDa) and different dithiol cross-linkers (PEG Mn ∼ 1.5 kDa or enzymatically degradable peptide), creating well-defined, robust materials with a range of mesh sizes estimated with Flory-Rehner or rubber elasticity theory (∼5 to 15 nm). A cocktail of different model proteins was released from compositions of interest, and sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) was used to facilely and quantitatively analyze temporal release profiles. Mesh size was predictive of retention of relatively large proteins and release of relatively small proteins. Proteins with diameters comparable to the mesh size, which is often the case for growth factors, were released by hindered diffusion and required experimental assessment of retention and release. With this knowledge, hydrogels were designed for the controlled release of a therapeutically relevant growth factor, PDGF-BB.

Tailoring the physico-chemical properties and elasticity of poly(hydroxy-functional methacrylate)-based cationically charged gel beads: Combined hydrophobicity and mechanical durability through frozen droplets

This study describes the experimental methodology to obtain hydroxy-functional methacrylate-based gel beads in millimeter-sized under production conditions by addition of the frozen pre-gel solution into the silicon oil as continuous phase using tetraethyleneglycol dimethacrylate (TEGDMA) as the crosslinking agent. Weakly basic cationic comonomer N,N-dimethylaminoethyl methacrylate (DMAEMA) was included into the polymerization procedure for the production of uniform polycationic moities exhibiting a drastic response against pH. The aim of this work is to determine the actual conditions for optimum bead production and to provide poly(hydroxypropyl methacrylate-co-N,N-dimethylaminoethyl methacrylate)-based P(HPMA-co-DMAEMA) gel beads with simple tools for the physical characterization. By the proposed experimental method, the physico-chemical properties of P(HPMA-co-DMAEMA) gel beads can be controlled by adjusting the pH, temperature of solution, ionic strength as well as the structure of the gel network. From the uniaxial compression tests, the swelling dependent stress–strain characteristics of resulting gel beads was evaluated. The obtained results were consistent with Hertzian elasticity, and were analyzed in terms of the compression modulus of the bead samples. It was found that force acting on the gel beads is the power-law function of the deformation F≈ΔD1.48 which is very close to the theoretical exponent 1.5 predicted by the Hertz equation. The water absorption capacities of P(HPMA-co-DMAEMA) gel beads with basic functional groups were repeatedly measured to obtain the optimal preparation conditions according to the variations of the bead diameters. Its maximal water absorption capacity appeared as 48 times among from 2.3mm to 11.0mm of swollen bead diameters. The swelling equilibria in various aqueous solutions were used to estimate the physico-chemical parameters asa function of gel bead size and swelling equilibrium capacity by correlating with the Flory-Rehner theory.

Swelling of Random Copolymer Networks in Pure and Mixed Solvents: Multi-Component Flory–Rehner Theory

A generalized extension of Flory-Rehner (FR) theory is derived to describe equilibrium swelling of polymer networks, including copolymers with two or more chemically distinct repeat units, in either pure or mixed solvents. The model is derived by equating the chemical potential of each solvent in the liquid and gel phases at equilibrium, while assuming the deformation of the network chains is affine. Simplifications of the model are derived for specific cases involving homopolymer networks, copolymer networks, pure solvents, and binary solvent mixtures. With reasonable assumptions, the number of polymer-solvent interaction parameters that must be determined by experiments can be reduced to two effective parameters (θ1 and θ2), which describe the net interactions between water/copolymer (θ1) and ethanol/copolymer (θ2), respectively. Experimental measurements of the swelling of random copolymer networks of n-butyl acrylate and 2-hydroxyethyl acrylate in water, ethanol, and a 100 g/L ethanol/water mixture are utilized to validate the model. For a random copolymer network, θ1 and θ2 can be obtained by fitting the three-component FR model to equilibrium swelling data obtained in the pure solvents. Predicted solvent volume fractions for swelling in water-ethanol mixtures obtained by inserting fitted values of θ1 and θ2 into the four-component FR model are in reasonable agreement with experimental measurements.

Solvent Induced Inversion of Core-Shell Microgels.

The morphology of core-shell microgels under different swelling conditions and as a function of the core-shell thickness ratio is systematically characterized by mesoscale hydrodynamic simulations. With increasing hydrophobic interaction of the shell polymers, we observe drastic morphological changes from a core-shell structure to an inverted microgel, where the core is turned to the outside, or a microgel with a patchy surface of core polymers directly exposed to the environment. We establish a phase diagram of the various morphologies. Moreover, we characterize the polymer and microgel conformations. For sufficiently thick shells, the changes of the shell size upon increasing hydrophobic interactions are well described by the Flory-Rehner theory. Additionally, this theory provides a critical line in the phase diagram separating core-shell structures from the distinct two other phases. The appearing new phases provide a novel route to nano- and microscale functionalized materials.

Swelling of responsive-microgels: experiments versus models

Interpenetrated Polymer Network (IPN) microgels of PNIPAM and PAAc have been investigated and the experimental data have been compared with theoretical models from the Flory–Rehner theory. We confirm that the swelling behavior of PNIPAM microgels is well described by this theory by considering the second order approximation for the volume fraction ϕ dependence of the Flory parameter χ(ϕ). Indeed the Volume-Phase Transition (VPT) of the PNIPAM-PAAc IPN microgel at neutral conditions and in D2O solvents can be well described only considering a third-order approximation. Interestingly we empirically find that sharper is the transition higher is the order of the χ(ϕ) relation which has to be considered. Moreover the VPT can be experimentally controlled by tuning the polymer/solvent interactions through pH and solvent allowing to directly modify the delicate balance between energetic and entropic contributions and to explore the swelling behavior in a wide range of environmental conditions. In particular we find that the most advantageous condition for swelling is in water at acidic pH.

Does Flory-Rehner theory quantitatively describe the swelling of thermoresponsive microgels?

The swelling of thermoresponsive microgels is widely modelled through Flory-Rehner theory, which combines Flory-Huggins solution thermodynamics with the affine network model of elasticity. While it has been shown that FR theory closely follows experimental results for a range of systems, the large number of free parameters required to fit size vs. temperature data make a proper evaluation of the theory difficult. In order to test the applicability of FR theory to microgel particles, we analyse viscosity and light scattering data for PNIPAM microgels as a function of temperature, cross-linking degree (f) and molar mass. In the collapsed state, the polymer volume fraction is estimated to be ϕC ≃ 0.44, independent of cross linking degree and molar mass. Fixing ϕC, f and the θ temperature to independent estimates, the FR model appears to describe microgel swelling well, particularly for high cross-linking densities. Estimates for the various fit parameters differ from earlier reports by an order of magnitude. A comparison of the χ parameter obtained from FR theory with values for the linear polymer reveals that the agreement between experiment and theory is somewhat fortuitous. Although the FR model can accurately describe experimental data, the accuracy of the obtained fit parameters is significantly poorer.

Counterion-induced swelling of ionic microgels.

Ionic microgel particles, when dispersed in a solvent, swell to equilibrium sizes that are governed by a balance between electrostatic and elastic forces. Tuning of particle size by varying external stimuli, such as pH, salt concentration, and temperature, has relevance for drug delivery, microfluidics, and filtration. To model swelling of ionic microgels, we derive a statistical mechanical theorem, which proves exact within the cell model, for the electrostatic contribution to the osmotic pressure inside a permeable colloidal macroion. Applying the theorem, we demonstrate how the distribution of counterions within an ionic microgel determines the internal osmotic pressure. By combining the electrostatic pressure, which we compute via both Poisson-Boltzmann theory and molecular dynamics simulation, with the elastic pressure, modeled via the Flory-Rehner theory of swollen polymer networks, we show how deswelling of ionic microgels with increasing concentration of particles can result from a redistribution of counterions that reduces electrostatic pressure. A linearized approximation for the electrostatic pressure, which proves remarkably accurate, provides physical insight and greatly eases numerical calculations for practical applications. Comparing with experiments, we explain why soft particles in deionized suspensions deswell upon increasing concentration and why this effect may be suppressed at higher ionic strength. The failure of the uniform ideal-gas approximation to adequately account for counterion-induced deswelling below close packing of microgels is attributed to neglect of spatial variation of the counterion density profile and the electrostatic pressure of incompletely neutralized macroions.

Liquid methanol sorption, diffusion and permeation in charged and uncharged polymers

Abstract There are few studies of the impact of membrane charge on ion and organic uncharged solute transport in polymers. This study was aimed at addressing the lack of such fundamental information in the literature. Liquid methanol sorption, diffusion and permeation in water swollen, cross-linked, uncharged poly(ethylene glycol) diacrylate (XLPEGDA) and in charged membranes were investigated. Membrane water content significantly influences methanol solubility, diffusivity and permeability, with more swollen membranes having faster methanol transport rates. The Mackie-Meares model was used to interpret and correlate diffusion data. The Flory-Rehner theory for ternary systems was used to interpret methanol sorption isotherms in XLPEGDA-based polymers. Methanol and sodium chloride transport patterns are similar in neutral hydrogels, while substantial differences are observed in ion exchange polymers. These differences are ascribed to sodium chloride transport in charged polymers being strongly influenced by Donnan exclusion, while methanol sorption and diffusion are not.

Reversible Size Modulation of Aqueous Microgels via Orthogonal or Combined Application of Thermo- and Phototriggers.

Aqueous microgels that respond orthogonally to external temperature and light stimuli and to a combination of both stimuli were developed. N-Vinylcaprolactam (VCL) was copolymerized with small feed amounts (<5 mol %) of 4-[(4-methacryloyloxy)phenylazo] benzenesulfonic acid (ABSA) and cross-linked with N,N'-methylenebis(acrylamide) (BIS) to synthesize monodisperse and colloidally stable P(VCL-BIS-ABSA) microgels. The volume phase transition information on the microgels under both orthogonal and combined application of temperature and light stimuli was investigated in situ by dynamic light scattering (DLS) instrument. Modeling of this information by the Flory-Rehner theory describes and aids the preliminary understanding of the main features in the volume phase transition of these photoresponsive microgels. Interestingly, the microgels rapidly deswell upon UV irradiation (λ = 365 nm), even as the trans-ABSA pendant groups are converted to the more polar cis state. The variation in the content of the pendant azobenzene groups in the microgels allows for reversible modulation of the phototriggered volume change. We propose that the approach of the sulfonic acid groups of cis-ABSA toward the polymer backbone causes the disruption of hydrogen bonding interactions between water molecules and the carbonyl groups of VCL.

Ion-stimuli responsive dimethylaminoethyl methacrylate/hydroxyethyl methacrylate copolymeric hydrogels: mutual influence of reaction parameters on the swelling and mechanical strength

Novel ion-stimulus-responsive copolymeric hydrogels were synthesized by free radical crosslinking copolymerization from the monomers N,N-dimethylaminoethyl methacrylate (DMAEMA) and 2-Hydroxyethyl methacrylate (HEMA) in the presence of a crosslinker, diethyleneglycol dimethacrylate (DEGDMA). The influences of the reaction parameters, the comonomer composition and the ionic strength of salt solutions on the swelling behavior of P(DMAEMA-co-HEMA) hydrogels were examined. The ion-stimulus-responsive swelling behavior of the prepared copolymers was studied in water as well as in aqueous solutions of NaCl, KCl, KBr, KI, CaCl2, BaCl2 and MgCl2. It was found that, starting from some characteristic concentration of a salt, a further increase of the salt concentration results in the shrinking of copolymeric P(DMAEMA-co-HEMA) hydrogels. The Flory-Rehner theory correctly predicts the swelling behavior of the hydrogels in salt solutions if the variation of the comonomer HEMA content is taken into account. The calculation of the interaction parameter χ between P(DMAEMA-co-HEMA) network and water showed that the specific interactions between cations and side groups of polymeric network affect the mixing term of the free energy. The extent and kinetics of water absorption were studied to determine their relationship with the reaction parameters. The kinetics of the hydrogel collapse is strongly dependent on the kind of salt used. The swelling results will be useful in designing and developing novel controlled delivery systems.

Effects of magnetite nanoparticles on morphology, processability, diffusion and transport behavior of ethylene vinyl acetate nanocomposites

Effect of magnetite (Fe3O4) nanoparticles on the performance of ethylene vinyl acetate (EVA) was prepared by a two-roll mixing mill. The morphology of the nanocomposites was carried out by transmission electron microscopy (TEM). The processing characteristics and the curing behavior of the nanocomposites were analyzed by Monsanto Rheometer. TEM images of the nanocomposite showed that nanoparticles were dispersed uniformly throughout the matrix up to 7 phr of filler and thereafter the dispersion decreased with further loading of fillers. The magnetite nanoparticle in EVA matrices was found to accelerate the curing process up to 26 % compared to unfilled sample. The transport characteristics of aromatic hydrocarbon through the nanocomposite were studied by simple sorption gravimetric analysis in the temperature range 28–70 °C. The solvent uptake was minimum for composites with 7 phr of filler and increased with increasing filler content. The transport phenomenon was found to follow an anomalous mode, the activation parameters were estimated and the molecular mass was calculated using Flory-Rehner theory. The sorption data were used to estimate the enthalpy, entropy and free energy of mixing.

Tuning the Synthetic Routes of Dimethylaminoethyl methacrylate‐Based Superabsorbent Copolymer Hydrogels Containing Sulfonate Groups: Elasticity, Dynamic, and Equilibrium Swelling Properties

ABSTRACTSuperabsorbent copolymer hydrogels based on N,N‐dimethylaminoethyl methacrylate (DMAEMA) and 2‐acrylamido‐2‐methylpropane sulfonic acid (AMPS) were synthesized by free‐radical cross‐linking copolymerization using a bifunctional cross‐linking agent diethylene glycol dimethacrylate (DEGDMA). Poly(DMAEMA‐co‐AMPS) (P(DMAEMA‐co‐AMPS)) hydrogels were prepared at a fixed cross‐linker ratio, but at various ionic comonomer AMPS content between 0 and 50 mol%. The effect of ionic comonomer concentration on the equilibrium swelling, mechanical behavior, and swelling–deswelling kinetics was investigated to find materials with satisfactory swelling and elastic properties. The increase in the ionic comonomer AMPS content increased the equilibrium swelling of copolymer hydrogels. Using the Flory–Rehner theory, the equilibrium swelling behavior of P(DMAEMA‐co‐AMPS) hydrogels was successfully investigated and the effective charge density of the resulting copolymer hydrogels was calculated as an adjustable parameter. Scaling rules were derived for the ionic group content to explain the observed swelling behavior of P(DMAEMA‐co‐AMPS) hydrogels. The kinetics of copolymer hydrogel swelling as well as the mechanism of diffusion was investigated by dynamic swelling measurements at different pH values. The dominant physical mechanism controlling the water uptake of P(DMAEMA‐co‐AMPS) copolymer hydrogels was analyzed, and the diffusion coefficients were determined. The transport was found to be less Fickian for the copolymer hydrogels containing AMPS below 30 mol% at acidic pH values where the polymer networks were ionized.

Heating of polyacrylamide ferrogel by alternating magnetic field

Ferrogel based on polacryamide network with embedded maghemite nanoparticles with mean number average particle diameter 12nm was synthesized by radical polymerization in water-based ferrofluid. The network structure of ferrogel was characterized by Flory–Rehner theory and it was shown that the embedded particles were substantially larger than the mesh size. It prevented the translational movement of particles in the ferrogel. The immobilization of particles was confirmed by dynamic light scattering. The adhesion of macromolecular chains to the particles was determined by calorimetry using thermochemical cycle. The enthalpy of interfacial adhesion was found several orders of magnitude higher than the energy of dipoles in typically applied magnetic fields. Despite the differenve in the mobility of particles in ferrofluid and ferrogel the comparative study of their heating in alternating magnetic field, however, revealed their close similarity. In both cases it was goverened by superposing of Neel and Brownian relaxation mechanisms.

Chemical networks of poly(vinyl alcohol)-Si hybrid films and the thermodynamics of their interaction with water

Organic-inorganic hybrid polymer films are prepared through the sol-gel method via the hydrolytic condensation of tetraethoxysilane in a 4% aqueous solution of poly(vinyl alcohol). The films are polymer networks containing O-Si-O bonds between chains formed via the interaction of OH groups of alcohol and Si-OH groups of the products of hydrolysis of tetraethoxysilane. The films contain 10.8–26.2 wt % Si in the polymer matrix. Isotherms of water-vapor sorption by the films are determined and used to calculate the molecular mass of polymer-chain fragments between crosslink-junction points in terms of the Flory-Rehner theory. For hybrid polymer networks, the products of sol-gel synthesis, this calculation is performed for the first time. The molecular mass of chain fragments between crosslink-junction points decreases from 9000 to 180 with an increase in the content of Si in the films. Poly(vinyl alcohol)-Si hybrid films are active during the separation of components of organic-aqueous media (a solution of tetrahydrofuran in water). The selectivity of the films increases as the molecular mass of chain fragments between crosslink-junction points decreases, that is, as the density of the chemical network increases.