Flory-Rehner Theory Research Articles

De-Hydration and Remodeling of Biological Materials: Swelling Theory for Multi-Domain Bodies

Biological materials always exhibit heterogeneous physical properties, both mechanical and chemical, which give them a rich phenomenology that poses significant challenges in the developing of effective models. The Flory–Rehner theory revolutionized our understanding of the dynamics of the liquid-polymers coupling in soft swollen gels, recognizing polymers as elastic networks stretched by the presence of liquid. Despite its foundational role, applying this theory to bodies with non uniform physical properties requires further improvements. This article proposes a unified approach to address mechano-diffusion challenges in multi-domain bodies, that is in material bodies made of regions having different chemo-mechanical properties, and focuses on the dehydration and remodeling of biological-like materials. Drawing inspiration from natural systems, we integrate principles from nonlinear mechanics and swelling theories; in particular, what is specifically new is the idea of applying the notion of the multiplicative decomposition of the strain–developed for plasticity–to model the swelling properties of a body made of two or more materials. The article gives a systematic presentation of the subject, and guides readers through key concepts and practical insights, aiming to provide a robust framework for modeling chemo-mechanical interactions. Moreover, it paves the way for the modeling of heterogenous bodies having spatially-varying properties.

Polyelectrolyte-Mediated Modulation of Spatial Internal Stresses of Hydrogels for Complex 3D Actuators.

Hydrogel actuators with complex 3D initial shapes show numerous important applications, but it remains challenging to fabricate such actuators. This article describes a polyelectrolyte-based strategy for modulating small-scale internal stresses within hydrogels to construct complex actuators with tailored 3D initial shapes. Introducing polyelectrolytes into precursor solutions significantly enhances the volume shrinkage of hydrogel networks during polymerization, allowing us to modulate internal stresses. Photopolymerization of these polyelectrolyte-containing solutions through a mask produces mechanically strong hydrogel sheets with large patterned internal stresses. Consequently, these hydrogel sheets attain complex 3D initial shapes at equilibrium, in contrast to the planar initial configuration of 2D actuators. We demonstrate that these 3D actuators can reversibly transform into other 3D shapes (i.e., 3D-to-3D shape transformations) in response to external stimuli. Additionally, we develop a predictive model based on the Flory-Rehner theory to analyze the polyelectrolyte-mediated shrinking behaviors of hydrogel networks during polymerization, allowing precise modulation of shrinkage and internal stress. This polyelectrolyte-boosted shrinking mechanism paves a route to the fabrication of high-performance 3D hydrogel actuators.

Osmotic pressure regulated sodium alginate-graphene oxide hydrogel as a draw agent in forward osmosis desalination

The freshwater shortage has become a global issue due to the population explosion. As an effective response to the water crisis, forward osmosis (FO) desalination technology is applied because of its comparatively low membrane fouling and low energy consumption. Hydrogels, as draw solutions in FO, have received increasing attention due to their properties of no reverse solute flux, and strong water recovery capability. However, the concentration in the feed solution is below 8000 ppm restricted by the lack of a clear mechanism for osmotic pressure regulation. In this work, a sodium alginate-graphene oxide (SA-GO) hydrogel was proposed with excellent performance in drawing water. The hydrogel has a homogeneous porous structure that enables easy water recovery through compression. When the SA-GO-1 hydrogel was used for the treatment of the simulated seawater (35,000 ppm NaCl solutions), the initial water flux achieved 0.4429 L m−2 h−1 (LMH), and the average water flux remained at 0.1783LMH. In contrast to conventional tactics, the osmotic pressure of hydrogel was regulated based on Flory-Rehner theory (FR). Moreover, the interaction energy was computed to reveal the mechanism of osmotic pressure regulation. This strategy provides a potential solution for the design of high-performance draw solutions for FO systems.

The physical basis for solvent flow in organic solvent nanofiltration.

Organic solvent nanofiltration (OSN) is an emerging membrane technology that could revolutionize chemical separations in numerous vital industries. Despite its significance, there remains a lack of fundamental understanding of solvent transport mechanisms in OSN membranes. Here, we use an extended Flory-Rehner theory, nonequilibrium molecular dynamic simulations, and organic solvent transport experiments to demonstrate that solvent flow in OSN membranes is driven by a pressure gradient. We show that solvent molecules migrate as clusters through interconnected pathways within the membrane pore structure, challenging the widely accepted diffusion-based view of solvent transport in OSN. We further reveal that solvent permeance is dependent on solvent affinity to the OSN membrane, which, in turn, controls the membrane pore structure. Our fundamental insights lay the scientific groundwork for the development of next-generation OSN membranes.

Swelling behavior of sodium lignosulfonate grafted polyacrylic acid highly absorbent hydrogels with two-phase structure

There exist two theories on the swelling behavior of cross-linked polymer networks treated by statistical thermodynamic and scalar methods, respectively. These two theories predict inconsistent network chain length dependence of the swelling ratio. The swelling of biopolymer gels is analysed with scaling laws from polymer physics, as an alternative for the classical Flory-Rehner theory. In this paper, the swelling data of sodium lignosulfonate grafted polyacrylic acid cross-linked networks in good solvents are reanalyzed, and it is pointed out that these networks are a typical model network conforming to the two-phase structural picture. For the swelling behavior of gels in the semi-dilute regime we derive the scaling theory based swelling equations, which make them compact and universal character.

Theory of expansion and compression of polymeric materials: Implications for membrane solvent flow under compaction

We extend the classical Flory-Rehner theory for the expansion and compression of porous materials such as cross-linked polymer networks and membranes. The theory includes volume exclusion, affinity with the solvent, and finite stretching of the polymer chains. We also modify this equilibrium theory, which applies to equal expansion of a material in all directions, to the situation where a material can only expand in a single direction, as is the case when a thin polymer layer is tightly bound to a support structure. We further extend this equilibrium model to the case where a pressure is applied across a thin polymer layer, such as a membrane, and liquid flows across the membrane. The theory describes how in the direction of liquid flow the membrane is increasingly compacted and becomes less porous, and this effect increases when the applied pressure goes up. We present results of example calculations for a thick membrane showing significant changes in compaction across its thickness, and a thin membrane where compaction due to flow is minor. Lastly, we model the dynamics of the change in size of a porous material in time after a step change in the solvent-polymer attraction parameter, which may be relevant, for instance, when the attraction parameter is highly temperature-dependent, and we change the temperature. The developed theory has direct implications for pressure-driven membrane separation processes ranging from reverse osmosis to organic solvent nanofiltration.

Development of Poly(sorbitol adipate)-g-poly(ethylene glycol) Mono Methyl Ether-Based Hydrogel Matrices for Model Drug Release.

Hydrogels were prepared by Steglich esterification and by crosslinking pre-synthesized poly(sorbitol adipate)-graft-poly(ethylene glycol) mono methyl ether (PSA-g-mPEG) using different-chain-length-based disuccinyl PEG. PSA and PSA-g-mPEG were investigated for polymer degradation as a function of time at different temperatures. PSA-g-mPEG hydrogels were then evaluated for their most crucial properties of swelling that rendered them suitable for many pharmaceutical and biomedical applications. Hydrogels were also examined for their Sol-Gel content in order to investigate the degree of cross-linking. Physical structural parameters of the hydrogels were theoretically estimated using the modified Flory-Rehner theory to obtain approximate values of polymer volume fraction, the molecular weight between two crosslinks, and the mesh size of the hydrogels. X-ray diffraction was conducted to detect the presence or absence of crystalline regions in the hydrogels. PSA-g-mPEG hydrogels were then extensively examined for higher and lower molecular weight solute release through analysis by fluorescence spectroscopy. Finally, the cytotoxicity of the hydrogels was also investigated using a resazurin reduction assay. Experimental results show that PSA-g-mPEG provides an option as a biocompatible polymer to be used for pharmaceutical applications.

Diffusive transport phenomena of lactoferrin from gelatin gels crosslinked with genipin

In order to examine the effects of crosslinking and structure on the swelling and diffusive phenomena in a hydrogel, low-solid gelatin gels were crosslinked with a range of genipin proportions. The ninhydrin assay showed high levels of successful crosslinking within the gels whilst FTIR and WAXD showed successful entrapment of the bioactive molecule lactoferrin within the matrix. Rheological analysis further assessed the effects of crosslinking on the system with an increase in network strength at greater genipin additions. SEM illustrated the successful incorporation of lactoferrin and how the pronounced changes in gel morphology influences entrapment. Experimental evidence supported theoretical estimations of the gel morphology using the Flory-Rehner theory. Analysis of the diffusive transport of lactoferrin from each crosslinked system suggested that densely bound matrices hinder the free movement of molecules. Pairing aspects of the Flory-Rehner theory with the diffusion data allowed for the calculation of the critical molecular weight between crosslinks (M*c) for diffusion to occur and the coupling parameter (ξ) for the gelatin and lactoferrin chain segments.

Effect of Crosslinking on Thermodynamics Interactions and Network Parameters of Terpolymeric Hydrogels

The physiochemical and thermodynamic interactions of N-isopropylacrylamide (NIPAM) based thermoresponsive hydrogels are influenced by their crosslinking density. In the current contribution a detailed study of the effect of crosslinking ratio on the swelling kinetics, network parameters, thermodynamic interactions and mechanical strength of Poly (N-isopropylacrylamide-co-N-tertiarybutylacrylamide-co-hydroxyethylacrylamide) (Poly-NIPAM-co-NTBA-co-HEAAm) hydrogels are discussed. The Poly (NIPAM-co-NTBA-co-HEAAm) hydrogels were prepared using methylenebisacrylamide (MBA) as a crosslinking agent in various stoichiometric proportions and characterized in terms of Fourier transform infrared spectroscopy (FTIR), swelling and mechanical properties. The results from the swelling experiments, exploiting the Flory–Rehner theory, and compression measurements showed a significant crosslinker content-dependent behavior. Our studies of the effect of temperature on swelling percentage, diffusion characteristics, interaction parameter and thermodynamic interactions, such as partial molar enthalpy and partial molar entropy, shed light on the thermodynamic and physical behavior of thermoresponsive hydrogels. Overall, our study demonstrated that by varying the amount of crosslinking agent (MBA), the structure and physiochemical properties of the poly (NIPAM-co-NTBA-co-HEAAm) hydrogels could be controlled effectively. Also, this study represents, we believe, the first step toward a correlation between their network structure and their thermodynamic properties, which is a key requirement for specific future applications and/or industrial scale-up.

Nonequilibrium thermodynamic modeling of case II diffusion in glassy polymers

In this article, we formulate a nonequilibrium thermodynamic theory for case II diffusion in a glassy polymer. In the model, we integrate the three-dimensional elasto-visco-plasticity of a glassy polymer, the Flory-Rehner theory for polymer network swelling, and the kinetics law for solvent migration with the concentration-dependent diffusivity. Our theory is given in a general 3D tensorial form, which, therefore, can be used to model case II diffusion in glassy polymers with arbitrary geometries and under different mechano-chemical loading conditions. An intrinsic length determined by the combination of the polymer viscosity and solvent diffusivity arises from the theory, which dictates the size-dependent case II diffusion in glassy polymers. The theory developed in the article may provide important guidance for designing semi-permeable membranes, waterproof coating, and drug-delivery systems, where case II diffusion can commonly occur.

The second network of soft-nanoparticles in linear polymers of the same chemistry

Replacing carbon black and silica with organic fillers can reduce the weight of polymer nanocomposites. Soft-nanoparticles (SNPs), a type of organic filler, can be produced in large quantities through microemulsion polymerization. Their elasticities and diameters can be tuned for a wide range. This study aimed to investigate the linear viscoelasticity of athermal all-polystyrene nanocomposites with different molecular weights of matrix chains. Polymer nanocomposites filled with soft SNPs do not show this second plateau. But the second plateau is observed with semi-rigid SNPs if the weight percentage (φ) of SNPs is above a critical value φc. Semi-rigid refers to the intermediate elasticity compared to phenol-capped silica nanoparticles and long-chain grafted silica nanoparticles/soft SNPs. In both cases of all-polystyrene nanocomposites, the entanglement plateau and higher frequency regions remain the same. This hints that polymer chains do not thread the network. Flory-Rehner theory further confirms negligible swelling of long polymer chains into dense crosslinking networks of SNPs. The modulus of this plateau scales to the 3.8 power of φ and the 0.26 power of Mw. The critical weight percentage φc for the onset of the second plateau corresponded with the compression of the polymer coil between two SNPs. As φ increases above φc, polymer chains would be depleted from neighboring SNPs, resulting in this network of semi-rigid SNPs. Thus, we believe that the second plateau originated from this network. This study analyzed the second plateau and highlighted the variety of viscoelastic features associated with the elasticity of SNPs filled in polymer nanocomposites.

Free radical copolymerization of β-Myrcene by suspension process

In this study, for the first time, synthesis of polymer microbeads from β-myrcene (My) and 1,4-butanediol dimethacrylate (BDDMA) by suspension copolymerization is described. Free radical copolymerization of My and BDDMA has been either photo- or thermally initiated. In both systems influence of monomer composition has been investigated by varying the ratio of My between 10 and 90 vol% in the oil phase composition. My-BDDMA based microspheres have been synthesized successfully via both thermally initiated and photoinitiated suspension copolymerization process. Particle morphology, size and distribution of the microbeads has remarkably influenced by monomer composition and initiation step of the polymerization process. Photoinitiated copolymerization has resulted in a decrease in particle size and yielded quite narrow particle size distributions compared to thermally initiated copolymerization. The average molecular weight between cross-links (Mc), and cross-linking density (ν), calculated by using the Flory-Rehner Theory, has confirmed the significant influence of monomer composition and initiation step on the cross-linking.

Circumferential buckling of a hydrogel tube emptying upon dehydration

A cylindrical hydrogel tube, completely submerged in water, hydrates by swelling and filling its internal cavity. When it comes back into contact with air, it dehydrates: the tube thus expels the solvent through the walls, shrinking. This dehydration process causes a depression in the tube cavity, which can lead to circumferential buckling. Here we study the occurrence of such buckling using a continuous model that combines nonlinear elasticity with Flory–Rehner theory, to take into account both the large deformations and the active behaviour of the hydrogel. In quasi-static approximation, we use the incremental deformation formalism, extended to the chemo-mechanical equations, to determine the threshold value of the enclosed volume at which buckling is triggered. This critical value is found to depend on the shell thickness, chemical potential and constitutive features. The results obtained are in good agreement with the results of the finite element simulations of the complete dynamic problem.

Regulation of molecular transport in polymer membranes with voltage-controlled pore size at the angstrom scale

Polymer membranes have been used extensively for Angstrom-scale separation of solutes and molecules. However, the pore size of most polymer membranes has been considered an intrinsic membrane property that cannot be adjusted in operation by applied stimuli. In this work, we show that the pore size of an electrically conductive polyamide membrane can be modulated by an applied voltage in the presence of electrolyte via a mechanism called electrically induced osmotic swelling. Under applied voltage, the highly charged polyamide layer concentrates counter ions in the polymer network via Donnan equilibrium and creates a sizeable osmotic pressure to enlarge the free volume and the effective pore size. The relation between membrane potential and pore size can be quantitatively described using the extended Flory-Rehner theory with Donnan equilibrium. The ability to regulate pore size via applied voltage enables operando modulation of precise molecular separation in-situ. This study demonstrates the amazing capability of electro-regulation of membrane pore size at the Angstrom scale and unveils an important but previously overlooked mechanism of membrane-water-solute interactions.

Characterizing the Molecular Architecture of Hydrogels and Crosslinked Polymer Networks beyond Flory-Rehner. II: Experiments.

We previously [Borges, F. T. P. Biomacromolecules 2020, 21(12), 5104-5118] introduced a novel methodology for the characterization of the dimensions and architecture of hydrogel networks that provides more detailed information than the classical Flory-Rehner theory [Canal, T.; Peppas, N. A. J. Biomed. Mater. Res. 1989, 23, 1183-1193]. In this article, we illustrate our methodology by applying it to the phototerpolymerization of N-vinyl-2-pyrrolidone (NVP), ethylene glycol methyl ether acrylate (EGA), and poly(ethylene glycol) diacrylate (PEGDA). The experimental design includes 120 formulations using different fractions of the three monomers. Experimental measurements determined the mass swelling ratio and were coupled with theory to compute the internal dimensions of the network. Results demonstrate how the use of a macromeric crosslinker leads to unique network architectures not predicted by classical F-R theory, e.g., the figure shows that the mass between crosslinks predicted by F-R is actually distributed between branches and the backbone. The methodology presented offers a path toward optimizing/customizing hydrogel properties to suit the size and shape of the specific therapeutic targeted for drug delivery.

Structural manipulation of the gelatin/genipin network to inform the molecular transport of caffeine

Low-solid gelatin hydrogels crosslinked with varied levels of genipin were examined to determine the effect of morphological properties on swelling and bioactive diffusion phenomena. FTIR analysis paired with WAXD showed the successful entrapment of caffeine and, when combined with the ninhydrin assay, confirmed the crosslinking of gelatin with genipin. Further investigation of the system with rheological profiling yielded a picture of the overall gel strength increasing with crosslinking and a change in the conformation of the gelatin chains with genipin addition. SEM micrographs were utilised to confirm the microstructural evaluation given by the Flory-Rehner theory from swelling studies. Further quantitative estimations of the gel crosslinking were also developed using this theory with increasing genipin concentration markedly changing the network density and mesh properties. Diffusive transport of caffeine was analysed for different proportions of genipin crosslinking with the denser crosslinked structures evidently hindering the free movement of bioactive molecules. The coupling parameter (ξ) between polymeric network and bioactive cargo, and the critical molecular weight between two crosslinks (M*c) were also determined to relate structural property to molecular diffusion.

Tunable enzymatically degradable hydrogels for controlled cargo release with dynamic mechanical properties.

Here, we designed enzymatically degradable hydrogels with tunable mesh sizes and crosslinking points to evaluate the effectiveness of network structure estimations in predicting dynamic mechanical properties and cargo retention or release. Poly(ethylene glycol) (PEG) hydrogels were prepared through a thiol-ene click reaction between four- or eight-arm PEG functionalized with vinyl sulfone and cysteine residues of collagenase-degradable peptides to create well-defined, homogenous, and robust materials with a range of mesh sizes estimated from the elasticity theory or Flory-Rehner theory. Time-dependent changes in mechanical properties associated with hydrogel degradation, i.e., dynamics of storage modulus, which is determined by the relationship between the hydrogel mesh and enzyme sizes, were characterized. The shear modulus G' decreased by enzyme addition, and the degradation rate decreased with the initial crosslinking density of the hydrogel. The degradation rate could also be controlled with the reactivity of peptide sequences against collagenase. With these findings, the retention and release of FITC-dextran were successfully controlled by tuning the mesh size and degradability of the hydrogel. This report provides useful insights for designing hydrogels as cell scaffolds or functional molecular delivery matrices with tunable dynamic mechanical properties and the resulting release of loaded drugs or proteins.

Lecture notes of the 15th international summer school on Fundamental Problems in Statistical Physics: Colloidal dispersions

In this brief contribution we review some basic concepts in the physics of soft matter focusing on colloidal dispersions. In the first part we discuss the phase behavior of hard colloidal particles and, building on a statistical mechanics formalism, we introduce the concept of effective interactions between colloids mediated by the presence of macromolecular additives in the dispersion. We then review some examples of effective interactions starting from the simplest entropy-driven mechanism observed in nature, i.e. the depletion effect, but also discussing more complex interactions as those mediated by a critical co-solute. In the second part, we introduce a novel type of colloids having particles “softness” as an extra control parameter. We focus on a particular type of soft colloid, called microgel, that has a polymeric nature since it is made of a cross-linked polymer network able to respond to external stimuli. Finally, to understand microgels responsiveness we discuss the thermodynamics of swelling of polymer networks described by the Flory-Rehner theory.

Recycling of EPDM via Continuous Thermo-Mechanical Devulcanization with Co-Rotating Twin-Screw Extruder.

Devulcanization represents the recycling of choice for a homogenous rubber waste stream because it allows revulcanization of samples previously devulcanized, making the life of the rubber virtually endless, according to the principles of circular economy. Among the many devulcanization processes, the thermo-mechanical one is the most appealing because it is a continuous process, easy to be industrialized. In this paper a comprehensive set of analyses (FTIR, TGA, DSC, elemental analyses, Py-GC/MS, swelling tests) were carried out on a post-industrial ethylene propylene diene monomer (EPDM), thermo-mechanical devulcanized in a co-rotating twin-screw extruder with different process parameters (thermal and screw profile, rpm). Results of the swelling test according to the Flory-Rehner theory and Horikx analyses show that the higher the thermal profile and the higher the rpm, the higher is the percentage of devulcanization. The quality of the devulcanized sample in terms of sol fraction and percentage of random scissions depends on the process conditions. The screw profile concurs to the efficiency of the devulcanization: the different number of kneading elements and more in general the screw profile composition affects the percentage of devulcanization, making the results in some tests more dependent on the screw speed.

Constitutive modeling for hydrogel with chain entanglements and application to adaptive hydrogel composite structures

Hydrogel is one of typical smart materials and has been used in many engineering applications, such as flexible electronics, flexible robots, and biomedical engineering. Modern industries with complex working environments require hydrogel to have higher strength without losing flexibility. Many published experimental studies focus on the preparation technique of hydrogel composite with high strength and toughness. However, the theoretical investigation into the influence of entanglements on the hydrogel deformation is rarely reported for the structure analysis and deformation mechanism of hydrogel composite, as a purpose to design of hydrogel devices. In this paper, the constitutive model is developed for thermal responsive hydrogel with chain entanglements, based on Flory-Rehner theory and slip-link model, and numerically solved by a user subroutine in ABAQUS. The uniaxial tensile and compressive simulations of standard specimens are carried out to validate the proposed model and demonstrate the applicability of the hydrogel composite, in which a great agreement is obtained by comparing the simulation results to the experimental data from published work. Subsequently, the hydrogel composites with three types of structure, such as hydrogel grippers, scaffolds and skins, are designed to improve the overall strength and maintain the flexibility of the devices, inspired by the deformation mismatch characteristics of multilayer hydrogels. The results show that the properties of composite hydrogels are closely related to the level of entanglements, cross-link and composite structures. This work may contribute to the preparation of reinforced hydrogels and provide insights for the design of hydrogel multifunctional devices.