Flory Theory Research Articles

Closure to the PRISM equation derived from nonlinear response theory.

Nonlinear response theory is employed to derive a closure to the polymer reference interaction site model equation. The closure applies to a liquid of neutral polymers at melt densities. It can be considered a molecular generalization of the mean spherical approximation (MSA) closure of Lebowitz and Percus to the atomic Ornstein-Zernike (OZ) equation and is similar in some aspects to the reference "molecular" MSA (R-MMSA) closure of Schweizer and Yethiraj to PRISM. For a model binary blend of freely-jointed chains, the new closure predicts an unmixing critical temperature, Tc, via the susceptibility route that scales linearly with molecular weight, N, in agreement with Flory theory. Predictions for Tc of the new closure differ greatest from those of the R-MMSA at intermediate N, the latter being about 40% higher than the former there, but at large N, both theories give about the same values. For an isotopic blend of polyethylene, the new and R-MMSA closures predict a Tc about 25% higher than the experimental value, which is only moderately less accurate than the prediction of atomic OZ-MSA theory for Tc of methane. In this way, the derivation and its consequences help to identify the ingredients in a theory needed to properly model the equilibrium properties of a polymeric liquid at both short and long lengthscales.

Physicochemical and catalytic behavior of binary mixtures of n-alkyl imidazolium bromide ionic liquids with poly (ethylene glycol)-400: Application of flory statistical theory

In this work, Flory statistical theory is employed to predict the important physicochemical properties namely density, isothermal compressibility, internal pressure, coefficient of volume expansion, energy of vaporization, heat of vaporization, cohesive energy density, solubility parameter and polarity index for three binary mixtures of PEG-400 + [CnMim][Br], where n=4, 6, and 8 at different temperatures (T= 298.15 K, 308.15 K, and 318.15 K). Furthermore, the applicability of Flory statistical theory to predict certain physicochemical properties has been verified. Excess properties viz., excess enthalpy, excess Gibbs free energy, and excess entropy are also determined for these binary mixtures. The variation of energy of vaporization, heat of vaporization, solubility parameter and excess parameters are used to analyze the catalytic behavior and prevalent molecular interactions of these mixtures.

Estimation of Ultrasonic Velocity, Density, Internal Pressure, and Thermophysical Parameters of Ionic Liquid Mixtures: Application of Flory's Statistical Theory.

Flory's statistical theory (FST) has been employed to estimate the ultrasonic velocity, density, internal pressure, and several important thermophysical parameters such as the energy of vaporization, the heat of vaporization, cohesive energy density, polarity index, and solubility for eight binary mixtures of ionic liquids and water within the temperature range of 288.15 to 308.15 K. The ionic liquids chosen for this investigation are [BMim][dca], [BMim][TfO], [BMpy][TfO], [BMpyr][dca], [BMpyr][TfO], [EEPy][ESO4], [HMim][dca], and [MPy][MSO4]. The predicted values of ultrasonic velocity and density show good agreement with the data reported in the literature. It endorses the applicability of FST to these binary mixtures. A comparative analysis of the internal pressure values (Pi) determined by using FST and the standard thermodynamic approach is also presented. The results obtained for Pi using both approaches show good agreement. Besides, for the mixtures under study, the correlation between ultrasonic velocity, density, and surface tension has also been examined. The variation of thermophysical parameters with concentration and temperature changes has been utilized to explore the nature and strength of the solute-solvent interactions prevalent in these mixtures. It is pointed out that A-A-type interactions dominate over A-B-type interactions in water-rich regions of the mixtures.

Temperature dependence investigations of molecular interactions and excess thermodynamic parameters of binary mixtures of ionic liquids and water using Flory's statistical theory

AbstractFlory's Statistical Theory (FST) has been applied to investigate excess thermodynamical parameters viz., excess enthalpy, excess entropy, and excess Gibbs free energy for eight binary mixtures of ionic liquids and water at different concentration and temperatures. The concentration and temperature dependent variations of excess enthalpy suggest that ionic liquid mixtures exhibit an endothermic nature and the presence of weak dispersive forces among their constituents. The positive excess Gibbs free energy of mixing points out that the pure components do not mix thoroughly for the systems under study. Overall, these binary systems are thermodynamically unfavorable suggesting that the mixing of ionic liquids and water is not energetically favored or spontaneous. Experimental molar volumes are compared with those calculated using FST and these values show excellent agreement. It affirms the validity of FST's application to these systems. Additionally, some partial properties have also been determined to explore the solute‐solvent interactions. The Redlich‐Kister (R‐K) polynomial model has been applied to excess thermodynamical parameters The high R2 values and low standard deviation values confirm the validity of fitting the RK polynomial model to the functions obtained using FST of liquids for these systems within the given temperature range.

Self-reporting and Biodegradable Thermosetting Solid Polymer Electrolyte.

To date, significant efforts have been dedicated to improve their ionic conductivity, thermal stability, and mechanical strength of solid polymer electrolytes (SPEs). However, direct monitoring of physical and chemical changes in SPEs is still lacking. Moreover, existing thermosetting SPEs are hardly degradable. Herein, by overcoming the limitation predicted by Flory theory, self-reporting and biodegradable thermosetting hyperbranched poly(β-amino ester)-based SPEs (HPAE-SPEs) are reported. HPAE is successfully synthesized through a well-controlled "A2+B4" Michael addition strategy and then crosslinked it in situ to produce HPAE-SPEs. The multiple tertiary aliphatic amines at the branching sites confer multicolour luminescence to HPAE-SPEs, enabling direct observation of its physical and chemical damage. After use, HPAE-SPEs can be rapidly hydrolysed into non-hazardous β-amino acids and polyols via self-catalysis. Optimized HPAE-SPE exhibits an ionic conductivity of 1.3×10-4 S/cm at 60 °C, a Na+ transference number ( ) of 0.67, a highly stable sodium plating-stripping behaviour and a low overpotential of ≈190 mV. This study establishes a new paradigm for developing SPEs by engineering multifunctional polymers. The self-reporting and biodegradable properties would greatly expand the scope of applications for SPEs.

Self‐reporting and Biodegradable Thermosetting Solid Polymer Electrolyte

AbstractTo date, significant efforts have been dedicated to improve their ionic conductivity, thermal stability, and mechanical strength of solid polymer electrolytes (SPEs). However, direct monitoring of physical and chemical changes in SPEs is still lacking. Moreover, existing thermosetting SPEs are hardly degradable. Herein, by overcoming the limitation predicted by Flory theory, self‐reporting and biodegradable thermosetting hyperbranched poly(β‐amino ester)‐based SPEs (HPAE‐SPEs) are reported. HPAE is successfully synthesized through a well‐controlled “A2+B4” Michael addition strategy and then crosslinked it in situ to produce HPAE‐SPEs. The multiple tertiary aliphatic amines at the branching sites confer multicolour luminescence to HPAE‐SPEs, enabling direct observation of its physical and chemical damage. After use, HPAE‐SPEs can be rapidly hydrolysed into non‐hazardous β‐amino acids and polyols via self‐catalysis. Optimized HPAE‐SPE exhibits an ionic conductivity of 1.3×10−4 S/cm at 60 °C, a Na+ transference number ( ) of 0.67, a highly stable sodium plating‐stripping behaviour and a low overpotential of ≈190 mV. This study establishes a new paradigm for developing SPEs by engineering multifunctional polymers. The self‐reporting and biodegradable properties would greatly expand the scope of applications for SPEs.

Swelling of Polypropylene Separators and Its Effect on the Lithium-Ion Battery Performance

The safety of lithium-ion batteries (LIBs) depends on separators -- porous materials providing ionic transport between the electrodes while preventing them from direct contact. Many commercial separator materials are polymers, e.g., polypropylene. Polypropylene has a non-polar structure, so it tends to interact favorably with non-polar solvents, such as carbonates used in LIBs as electrolyte solvents. These favorable interactions result in two effects: separator plasticization (softening) [1] and swelling [2]. Both swelling and plasticization affect the battery performance and cause safety concerns.Here we present a model which predicts the swelling of a porous polymer separator based on the molecular structures of the polymer and the solvent. The model is based on Flory's theory of polymer solutions and the Flory–Huggins parameter for polymer–solvent interactions. We introduced two additional parameters to Flory's theory, taking into account the complex structure of the porous polymers. These two parameters can be obtained from experimental data on two solvents. The Flory–Huggins parameter is calculated based on the UNIFAC-FV group contribution method for a given polymer–solvent pair. Our model predicts swelling of polypropylene separators in various solvents, matching the experimental data [3] quite well. By relating the swelling of the separator to the change in its porosity and tortuosity, we estimated the increase of cell impedance as a result of the swelling.Finally, the plasticization of the separator makes it less persistent to mechanical impacts, increasing the chances of failure. Since the solvent-induced plasticization of the separator is correlated with its swelling [3], our model can also be utilized to assess this effect. Therefore, our model can be used to quickly evaluate various polymer–solvent pairs for application in lithium-ion batteries.1. Sheidaei, A.; Xiao, X.; Huang, X.; Hitt, J. Mechanical behavior of a battery separator in electrolyte solutions. J. Power Sources 2011, 196, 8728−8734.2. Cannarella, J.; Liu, X. Y.; Leng, C.; Sinko, P.; Gor, G. Y.; Arnold, C. B. Mechanical properties of a battery separator under compression and tension. J. Electrochem. Soc. 2014, 161, F3117−F3122.3. Gor, G. Y.; Cannarella, J.; Leng, C. Z.; Vishnyakov, A.; Arnold, C. B. Swelling and softening of lithium-ion battery separators in electrolyte solvents. J. Power Sources 2015, 294, 167−172.4. Maksimov, A. V.; Molina, M.; Maksimova, O. G.; Gor, G. Y. Prediction of Swelling of Polypropylene Separators and Its Effect on the Lithium-Ion Battery Performance. ACS Appl. Polym. Mater. 2023, 5(3), 2026-2031.

Intrinsic viscosity and dielectric relaxation of ring polymers in dilute solutions.

The absence of chain ends makes ring polymers distinctly different from their linear analogues. The intrinsic viscosity, complex viscosity and the dielectric relaxation of ring polymers are investigated within the tenets of the optimized Rouse-Zimm theory. The distance dependent excluded volume interactions (EVIs) are obtained from Flory's mean field theory. The hydrodynamic interactions (HIs) between the pairs of monomers are estimated using the preaveraged Oseen tensor. The intrinsic viscosity of linear and ring polymers both with and without EVI are compared as a function of ring size. A monotonically increasing trend of the intrinsic viscosity is observed in both cases. The intrinsic viscosity of both linear and ring polymers both with and without EVI show a very good agreement with the experimental results of polystyrene over a wide range of molecular weights in both good and theta solvents, respectively. The fractal dimensions of the ring polymers with EVI lie between that of a random walk and a self-avoiding walk model of linear polymers in three dimensions. The ring size increases with EVI and the effect of EVI is stronger on larger rings than that on smaller rings. The dielectric relaxation follow a connectivity independent universal scaling behavior at low and high frequency regions. The imaginary part of the complex dielectric susceptibility displays a local maxima in the intermediate frequency region, which reveals a structure dependent behavior of the rings. The theoretically calculated dielectric loss of ring polymers with HI matches well with those obtained from experiments.

Thermodynamics of 2-alkanol + polar organic solvent mixtures. I. Systems with ketones, ethers or organic carbonates

The mixtures 2-propanol or 2-butanol + n-alkanone, or + acetophenone, or + linear monoether, or + cyclic ether, or + linear organic carbonate, or + propylene carbonate have been investigated using thermodynamic data, and in terms of the Flory theory, and the Kirkwood-Buff integrals. The data considered are: excess molar enthalpies (HmE), volumes, entropies, and the temperature dependence ofHmE. The enthalpy of the 2-alkanol-solvent interactions have been determined, and the different contributions to HmE discussed. It is shown that HmE values of the 2-alkanol (fixed) + n-alkanone, or + linear carbonate mixtures change in the same manner that for n-alkanone, or linear carbonate + n-alkane (fixed) systems. In contrast, HmE values of 2-alkanol (fixed) + linear monoether or + n-alkane mixtures change similarly. This set of results suggests that solvent–solvent interactions are determinant in systems with n-alkanone or linear carbonate, while interactions between alcohol molecules are determinant in mixtures with linear monoethers. According to the Flory model, orientational effects in systems with a given 2-alkanol become weaker in the sequence: linear monoether > linear organic carbonate > n-alkanone, and are stronger in solutions with a cyclic monoether than in those with cyclic diethers, and in systems with acetophenone or propylene carbonate than in the mixtures with the corresponding linear solvents. Results obtained from the Kirkwood-Buff integrals are consistent with these findings. The application of Flory model reveals that orientational effects are similar in systems with 1- or 2-alkanols, with the exception of solutions with linear monoethers, where such effects are stronger in mixtures containing 1-alkanols. For the examined systems, the formalism of the Kirkwood-Buff integrals makes no meaningful distinction between solutions with 1-alkanols or 2-alkanols.

Analysis of excess molar heat capacities of three component mixtures comprising of cyclic organic solvents: Graph and Flory theories

The present work reports measurement and estimation of heat capacities, CP and excess heat capacities, (CPE)ijk (using CP data) of 1-methylpiperidine (i) + pyrrolidin-2-one (j) + cycloheptanone or cyclohexanone or cyclopentanone (k) mixtures at several temperatures. (CPE)ijk data have been correlated with the mole fraction of the constitutional molecule (via Redlich-Kister polynomial) to commute standard deviations and ternary coefficients. (CPE)ijkdata have also been examined in terms of Graph and Flory's theories. It has been perceived that (CPE)ijkvalues computed using Graph theory compare well with their corresponding experimental values. Further, (CPE)ijk data obtained by Flory's theory for 1-methylpiperidine (i) + pyrrolidin-2-one (j) + cyclopentanone (k) mixtures are of the same sign.

Isothermal curing analysis and properties of ultra heat‐resistant polyimide composites by DMA method

AbstractThe isothermal curing process of T700 carbon fiber/P450 polyimide prepreg is characterized by dynamic mechanical analysis method and the gelation points are analyzed by torsional braid analysis theory. The growth mechanisms of the mechanical property at different constant temperatures are investigated by the relative mechanical growth rate of storage modulus and the curing kinetics models are established. The theories of Flory and Hsich are used to calculate the activation energy during the curing reaction of prepreg. The results indicates that the gelation times which are detected by the method of torsional braid analysis at 300, 310, 330, and 350°C were 293.7, 213.8, 104.1, and 45.4 min, respectively. The activation energies of curing reaction are calculated by Flory theory and Hsich theory, they are 112.35 and 119.96 kJ/mol, respectively. The feasibly of curing system 300°C × 4 h + 330°C × 2 h + 350°C × 2 h + 370°C × 2 h is verified by the mechanical properties and the heat resistance of polyimide composite.

Molecular mobility and morphology change of poly(vinyl alcohol) (PVA) film as induced by plasticizer glycerol

AbstractThe addition of small molecular plasticizer is an efficient strategy to increase the process ability and macroscopic performance of poly(vinyl alcohol) (PVA). However, how the plasticizer influences the chain dynamics and morphology deserves further investigation. In this work, the PVA film with varying glycerol addition levels was used as the model system. The increasing glycerol addition‐level results in the depression of the melting temperature and crystallization temperature, which in line with Flory's theory. The enhanced chain dynamics in the mobile domain was accessed by low‐field NMR, whereas that in the crystalline region remains constant. The domain size as well as the morphology of the mobile phase are accessed by 1H spin‐diffusion NMR. The results indicate the increasing dimensionality of the plasticized PVA film with increasing the addition level of glycerol, where the addition level below 25 wt% leads to the dimensionality of 1D, and higher than 25 wt% results in the dimensionality of 2D. Current results elucidate the critical role of the plasticizer in modulating the dynamics heterogeneity of the plasticized polymer film, which is closely related to the macroscopic performance of the products.

Effect of comb polymer dispersants with different molecular structures on the performance of LiFePO4 suspensions

A series of comb polymers poly(2-(dimethylamino)ethyl methacrylate (DMAEMA)-co-methacrylic acid (MAA)-co-methoxy polyethylene glycol methacrylate (MPEGMA)) (poly(DMAEMA-MAA-MPEGMA, DMM) were synthesized and used as N-methyl-2-pyrolidinone (NMP)-based lithium iron phosphate (LFP) suspension dispersants. The effects of the grafting density of the carboxyl group as the anchoring group and the chain length of the side chain of PEG, which plays the role of spatial site resistance, on the rheological properties and suspension stability of the slurry were systematically investigated. By investigating the adsorption amount and thickness of DMM on the LFP surface, combined with calculations based on the scalar law and Flory theory, the molecular structure of the comb polymer dispersant was revealed to influence the adsorption and dispersion performance. The dispersion of LiFePO4 was due to the synergistic effects of adsorption and steric hindrance effect, which resulted that dispersants with medium carboxyl density and PEG side chain length can improve the dispersion performance and stability.

Eternally green Flory theory

1. Effects of topological constraints on linked ring polymers in solvents of varying quality Authors: Z. Dehaghani, I. Chubak, C. Likos, and M. Ejtehadi Soft Matter 16, p. 3029 (2020); DOI: 10.1039/C9SM02374G 2. Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties Authors: L. Tubiana, F. Ferrari, and E. Orlandini Phys. Rev. Lett. 129, 227801 (2022); […]

X-ray diffraction study of strain-induced crystallization of hydrogenated nitrile-butadiene rubbers: Effect of crosslink density

The structure and the mechanical and thermal properties of a hydrogenated acrylonitrile-butadiene rubber (HNBR) with acrylonitrile content of 44 wt% and residual unsaturation content of 9% are investigated as a function of the crosslink density. The analysis is focused on specimens that have not been subjected to any prior deformation. All samples are amorphous at room temperature and show glass transition temperatures that increase with increasing crosslink density. The values of the Young's modulus of the vulcanized samples are three times higher than that of the non-vulcanized counterpart and tend to increase with increasing the crosslink density. The tensile strength increases with increasing crosslink density, whereas the deformation at break decreases. All samples are amorphous at low deformations and whereas the non-vulcanized and the weakly and medium crosslinked samples undergo strain-induced crystallization (SIC) above a critical value of deformation, the highly crosslinked samples show only faint SIC and rupture at deformations slightly higher than that at SIC onset. It is shown that the deformation at SIC onset for the crosslinked samples tends to increase with increasing the crosslink density. This is in contrast with the prediction of Flory's theory for a homogeneous network. Instead, in agreement with Flory's predictions, the degree of orientation of the amorphous phase, increases with increasing deformation and reaches a plateau at SIC onset, indicating that the onset of crystallization is associated with a relaxation of the amorphous chains connected to the crystalline stems. The remarkable tensile strength of the crystallizing HNBR samples at high deformations is attributed to the strong alignment of the crystals in the stretching direction formed by SIC, causing pronounced strain hardening.

Anisotropic swelling of rubber: extension of the Flory theory

The anisotropy of the swelling of rubber is examined both theoretically and experimentally. The Flory theory is extended to account for anisotropic swelling, allowing the determination of the average molecular weight between cross-links for rubber with swelling anisotropy for the first time. In addition, specimens from five commercial rubbers manufactured using either compression-moulding or sheet-rolling processes are swollen in appropriate organic solvents. Their linear dimensions and mass are carefully recorded before swelling, in the swollen state, and after drying, to obtain three linear swelling ratios which can differ by up to 10% within each specimen. Compression-moulded rubbers are shown to be transversely isotropic after moulding, whereas rolled rubbers exhibit full anisotropy, with different swelling ratios in all three directions. None of the rubbers examined were found to be truly isotropic. The new anisotropic swelling theory is applied to the experimental data to determine the average molecular weight between cross-links, which is determined as up to 0.5% larger than the value obtained using the Flory isotropic swelling theory.

The dimensions of an ensemble of collapsed disordered protein molecules under folding conditions

In order to detect the earliest formed intramolecular contacts in the refolding ensembles by detection of the intramolecular distances, we must first determine the dimensions of the transient collapsed ensemble. Does an ensemble of disordered molecules of foldable globular protein under folding conditions follow Flory's theory of collapsed polymers? Using E. coli Adenylate kinase (AK) as a model protein and microfluidic mixing device combined with time resolved FRET measurements (“the double kinetics” method) we tested the hypothesis that at the initiation of refolding, the dependence of the mean of the segmental end-to-end distance on the segment length, Δn, is weak (Δn>∼30).We measured the transient distributions of segmental end to end distances of seven segments of the AK molecule, with Δn from 45 to 196 residues at 50 μs after initiation of mixing.

Study of Thermoacoustic Parameters and Statistical Analysis of Liquid State Therories

Molecular interaction in binary liquid mixtures can be evaluated from the thermodynamic properties of the liquid mixture. Nature of interaction of liquid molecules can be explain by various liquid state theories, such as Collision factor theory, Free length theory, Eyring's significant structure theory, Flory theory etc., In present work experimental parameters such as ultrasonic velocity (U), density (ρ) of binary mixtures of p-Xylene-Piperidine and Tolune-Piperidine has been measured over the entire range of composition at different temperatures 303,313,323K. The observed experimental data have been utilized to evaluate thermo-acoustical parameter ie. adiabatic compressibility βa, free length Lf molar volume V etc., Theoretical ultrasonic velocities calculated by applying Jacobson’s free length theory and Kalidoss revised free length theory, statistical analysis carried out by applying Chi-square (χ2) test for both the liquid systems to verify the applicability of liquid state theories.

Revealing Polymerization Kinetics with Colloidal Dipatch Particles.

Limited-valency colloidal particles can self-assemble into polymeric structures analogous to molecules. While their structural equilibrium properties have attracted wide attention, insight into their dynamics has proven challenging. Here, we investigate the polymerization dynamics of semiflexible polymers in 2D by direct observation of assembling divalent particles, bonded by critical Casimir forces. The reversible critical Casimir force creates living polymerization conditions with tunable chain dissociation, association, and bending rigidity. We find that unlike dilute polymers that show exponential size distributions in excellent agreement with Flory theory, concentrated samples exhibit arrest of rotational and translational diffusion due to a continuous isotropic-to-nematic transition in 2D, slowing down the growth kinetics. These effects are circumvented by the addition of higher-valency particles, cross linking the polymers into networks. Ourresults connecting polymer flexibility, polymer interactions, and the peculiar isotropic-nematic transition in 2D offer insight into the polymerization processes of synthetic two-dimensional polymers and biopolymers at membranes and interfaces.

Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers.

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.