Flory Theory Research Articles

Crystallization and melting behavior of i-PP: a perspective from Flory's thermodynamic equilibrium theory and DSC experiment

A novel conceptual framework that generates insightful new results about crystallization and melting behaviors of i-PP.

Flory theory of randomly branched polymers

Flory theory of randomly branched polymers

Properties of aqueous solutions of ammonium-based ionic liquids and thermodynamic modelling using Flory theory

The new protic ionic liquids mixtures properties and water are of great interest from a theoretical perspective. The objective of this study is to establish mixtures with different densities, viscosities, composition ranges and temperatures. The studied ionic liquids are N-methyl-2-hydroxy-ethylammonium acetate (m-2HEAA), N-methyl-2-hydroxy-ethylammonium propionate (m-2HEAPr) and N-methyl-2-hydroxy-ethylammonium butanoate (m-2HEAB), all based on the N-methyl-2-hydroxy-ethylammonium cation. The excess volume data are correlated using the Prigogine-Flory-Patterson equation of state (EOS) to evaluate the importance of their contributions with respect to the studied mixtures. The viscosity data were modeled combining Eyring mixture law for viscosity and excess Gibbs energy from Flory-Huggins model. The results reveal a large range of negative deviation and a non-significant contribution of the free EOS volume term compared to the interactional contribution and characteristic pressure terms. In general, the models studied in this work presented good agreement with the experimental data.

Peculiarities of crystallization in the multiblock copolymers of norbornene and cyclooctene

Crystallinity of statistical multiblock copolymers of norbornene and cyclooctene with nearly equimolar composition and different degrees of blockiness is studied by differential scanning calorimetry including thermal fractionation by successive self-nucleation and annealing and by X-ray diffraction. The copolymers are synthesized via recently invented reaction of polymer cross-metathesis, as a result of interchain exchange between polynorbornene and polyoctenamer in the presence of Grubbs’ catalyst of the first generation. This catalyst raises the fraction of trans-octenylene units up to more than 80% promoting formation of triclinic crystals. The copolymer degree of crystallinity and melting peak temperature decrease with the cross-metathesis conversion. At the same time the equilibrium melting temperature predicted by the Flory theory of copolymer crystallization is not much affected by the interchange reaction. Compared with the pure polyoctenamer or its equimolar blend with polynorbornene, the norborene-cyclooctene copolymers form considerably smaller crystallites. Thermal fractionation makes it possible to find their size distribution, which correlates with the average length of trans-octenylene blocks. Short annealing the multiblock copolymers above their melting points gives rise to a small but persistent endotherm 5–10 degrees above the annealing temperature, which is absent in polyoctenamer and, therefore, may indicate mixing of segregated norbornene and cyclooctene blocks.

Computer simulations of melts of randomly branching polymers.

Randomly branching polymers with annealed connectivity are model systems for ring polymers and chromosomes. In this context, the branched structure represents transient folding induced by topological constraints. Here we present computer simulations of melts of annealed randomly branching polymers of 3 ≤ N ≤ 1800 segments in d = 2 and d = 3 dimensions. In all cases, we perform a detailed analysis of the observed tree connectivities and spatial conformations. Our results are in excellent agreement with an asymptotic scaling of the average tree size of R ∼ N1/d, suggesting that the trees behave as compact, territorial fractals. The observed swelling relative to the size of ideal trees, R ∼ N1/4, demonstrates that excluded volume interactions are only partially screened in melts of annealed trees. Overall, our results are in good qualitative agreement with the predictions of Flory theory. In particular, we find that the trees swell by the combination of modified branching and path stretching. However, the former effect is subdominant and difficult to detect in d = 3 dimensions.

Relation of Wertheim association constants to concentration-based equilibrium constants for mixtures with chain-forming components

Several association modeling approaches have been developed to accurately describe the properties of polar solutions. Chemical theory and Wertheim's perturbation theory are among the most popular of these and they have been shown to yield similar functional forms for the contributions of association to Helmholtz energy and activity coefficients. In this paper, we study Flory polymerization theory through the work of Campbell and elucidate its correlation to Wertheim's theory. A simple key relationship between the concentration-based equilibrium constant and Wertheim's association constant is developed for systems in which all associating components have one acceptor and one donor site. Algebraic and numerical proofs are given for the equivalence of Flory's polymerization theory and Wertheim's perturbation theory for pure fluids and mixtures. Additionally, a new generalized activity expression is developed for Wertheim's theory.

Flexible star polymer chain adsorption by a flat surface: a molecular dynamics simulation

In this work, we have studied the adsorption of a single flexible star polymer chain by a flat surface. The molecular dynamics simulation has been validated for the case of a free star polymer chain using the diffusion experiment. The scaling laws found are in agreement with the Flory theory predictions and the Rouse model. For the adsorption, preliminary results were obtained for chains of different sizes N=31 to N=199 and different functionalities (f=3,4,6,8,10). For the case of semi-flexible star polymer chains, further investigation is needed to locate the critical point of adsorption when varying the potential interaction strength between the chain and the surface.

Computer simulations of randomly branching polymers: annealed versus quenched branching structures

We present computer simulations of three systems of randomly branching polymers in d = 3 dimensions: ideal trees and self-avoiding trees with annealed and quenched connectivities. In all cases, we performed a detailed analysis of trees connectivities, spatial conformations and statistical properties of linear paths on trees, and compare the results to the corresponding predictions of Flory theory. We confirm that, overall, the theory predicts correctly that trees with quenched ideal connectivity exhibit less overall swelling in good solvent than corresponding trees with annealed connectivity even though they are more strongly stretched on the path level. At the same time, we emphasize the inadequacy of the Flory theory in predicting the behaviour of other, and equally relevant, observables like contact probabilities between tree nodes. We show, then, that contact probabilities can be aptly characterized by introducing a novel critical exponent, , which accounts for how they decay as a function of the node-to-node path distance on the tree.

Multicomponent Thermodynamics of Strain-Induced Polymer Crystallization.

We developed a linear combination of two Flory's melting-point theories, one for stretched and the other for solution polymers, to predict the melting point of stretched solution polymers. The dependences of the melting strains on varying temperatures, polymer volume fractions, and solvent qualities were verified by the onset strains of crystallization in our dynamic Monte Carlo simulations of stretched solution polymers under a constant strain rate. In addition, owing to phase separation before crystallization in a poor solvent, calibration of polymer concentration to the polymer-rich phase appears necessary for the verification. Our results set up a preliminary thermodynamic background for the investigation of the multicomponent effect on strain-induced crystallization of polymers in rubbers and gels as well as on shear-induced crystallization of polymers in solutions and blends.

Backfolding of DNA Confined in Nanotubes: Flory Theory versus the Two-State Cooperativity Model

We compare two theories for backfolding of semiflexible polymers, such as DNA, confined in circular tubes of diameter of the order of the persistence length of the macromolecule. The first theory was proposed by Odijk on the basis of a one-dimensional analogue of Flory theory, and the second theory is a cooperativity model of deflection segments and S-loops suggested by Dai et al. By performing Monte Carlo chain growth simulations of long chains, we find that Odijk’s scaling theory not only captures the contour length dependence of extension of the confined chain but also correctly predicts its asymptotic value. In contrast, the cooperativity model appears to quantify the extension only for the contour lengths that were used to parametrize the model and systematically deviates from the simulation data as the contour length increases.

Random Copolymers from Polyamide 11 and Polyamide 12 by Reactive Extrusion: Synthesis, Eutectic Phase Behavior, and Polymorphism

Random copolymers from polyamide 11 (PA11) and polyamide 12 (PA12) were obtained from the parent homopolymers by transamidation reactions during high-temperature reactive extrusion. Concomitantly, the product molecular weight increased by postcondensation reactions. With both species being crystallizable, eutectic phase behavior was observed and explained in terms of the Flory theory for random copolymer melting. Close to the eutectic composition, both species crystallize in a competitive mode, which induces mesomorphic phase formation. Although the crystal centers most likely remain PA11 or PA12 pure, it is argued that foreign comonomer units might be tolerated at the crystal borders, at least for PA11 crystals. To understand the copolymer crystallization behavior in general, large amounts of rigid amorphous material need to be created alongside the crystallites. Because of their low melting point and since the crystallinity and mechanical performance typical of the parent homopolymers are preserved, ran...

Dynamics of dendrimers with excluded volume: A comparison with experiments and simulations

The rheology of model dendrimers with excluded volume interactions is investigated within the framework of the Rouse-Zimm theory. Excluded volume between the nearest nonbonded monomers of dendrimers is evaluated within the mean-field approximation along the lines of Flory's theory. This theory also rationalizes the earlier developed geometric approach for determining the angular excluded volume parameters. This model captures the characteristic maxima of the intrinsic viscosity as a function of the generations of growth. The overall feature of the intrinsic viscosity of dendrimers with varying excluded volume parameters displays a good quantitative match with those reported in earlier theory, experiments, and simulations. The mechanical moduli of model dendrimers agree well with the Brownian dynamics simulation data.

Math in Chemistry and Economics

Now a days math is having its own importance in different fields of science, Engineering, Medical Science, Economics, Commerce etc. All the fields have been developed by making use of math. In this paper some uses of math in the field of chemistry and Economics are highlighted. In the field of chemistry for different experimental results which are calculated by making use of math. For getting the same value of experimental results by making use of mathematically calculated theories are done. It is found that experimental results and the result obtained by theories resemble with each other. So without doing experiments, only by mathematical theories the results can be calculated e.g. experimental results of excess volume can also be calculated from Flory's statistical theory and Prigogine-Flory's Patterson theory, Hydrodynamic and osmotic permeability. Electro-osmotic permeability Thermodynamic studies from electromotive force measurements, Electro negativity calculation etc. are explained in this paper. An idea about use of math in Economics is proposed by explaining some theories e.g. Marginal utility, law of equimarginal utility marginal rate of substitution, marginal lost, Harodommars model etc.

Sugar and polyol solutions as effective solvent for biopolymers

Ternary mixtures of biopolymers, sugars or polyols and water can be treated as a pseudo binary system with respect to melting of the biopolymer. Sugar and polyol solutions can be treated as an effective solvent, characterized by the density of hydroxyl groups available for intermolecular hydrogen bonding. Such a treatment has been shown earlier for the glass transitions of the ternary mixtures. Now we have analysed the melting behaviour of biopolymers in these ternary mixtures. If the melting points are plotted as function of the density of hydroxyl groups, all data for a variety of sugars and polyols collapse to a single curve. This master curve coincides with the prediction of the melting line for biopolymer/water mixtures as follows from Flory's theory. Such behaviour has been found for starch, gelatin, soy and sunflower proteins.

Unfolding of globular polymers by external force.

We examine the problem of a polymer chain, folded into a globule in poor solvent, subjected to a constant tensile force. Such a situation represents a Gibbs thermodynamic ensemble and is useful for analysing force-clamp atomic force microscopy measurements, now very common in molecular biophysics. Using a basic Flory mean-field theory, we account for surface interactions of monomers with solvent. Under an increasing tensile force, a first-order phase transition occurs from a compact globule to a fully extended chain, in an "all-or-nothing" unfolding event. This contrasts with the regime of imposed extension, first studied by Halperin and Zhulina [Europhys. Lett. 15, 417 (1991)], where there is a regime of coexistence of a partial globule with an extended chain segment. We relate the transition forces in this problem to the solvent quality and degree of polymerisation, and also find analytical expressions for the energy barriers present in the problem. Using these expressions, we analyse the kinetic problem of a force-ramp experiment and show that the force at which a globule ruptures depends on the rate of loading.

Precision Long-Chain Branched Polyethylene via Acyclic Diene Metathesis Polymerization

A series of polyethylenes containing 21-carbon alkyl branches have been synthesized by acyclic diene metathesis (ADMET) polymerization. These 21-carbon alkyl branches are precisely placed on every 15th, 19th, 21st, 23rd, and 39th carbon along the polymer backbone. Precision of primary structures of all polymers is verified by 1H and 13C NMR spectroscopy. All polymers present well-defined melting profiles, even at a high branch incorporation (13.3% mol). The melting temperature increases as the branch frequency decreases, similar to what we observed for short-chain branched polyethylenes. These observations together with a good linear relationship derived from Flory's theory suggest the exclusion of 21-carbon side chains from polyethylene crystal units.

Pseudo-Gruneisen Parameter and Internal Pressure of Binary Mixtures from different Approaches

Abstract Density and speed of sound were measured earlier by us for binary liquid mixtures formed by Benzonitrile, Chlorobenzene, Benzyl chloride and Benzyl alcohol with Benzene at temperature range 298.15 to 313.15 K and atmospheric pressure over the whole concentration range. Pseudo-Gruneisen parameter and internal pressure were derived from the measured values of density and ultrasonic velocity. These values were compared with the theoretical values obtained by the utilization of Flory theory, Ramaswamy and Anbananthan model and model suggested by Glinski to predict the behavior of weakly interacting liquids. The observed properties derived from measured data were fitted to Redlich-Kister polynomial relation to estimate the binary coefficients and standard errors. Excess Pseudo-Gruneisen parameter for these binary mixtures was computed to study the molecular interactions involved in the liquid systems.

Flexible polyelectrolyte conformation in the presence of cationic and anionic surfactants

In this work we have studied the conformation of flexible polyelectrolyte chains in the presence of cationic and anionic surfactant molecules. We developed a simple theoretical model for the formation of the polyelectrolyte–cationic surfactant complexes and mixed micelles formed by cationic and anionic surfactant molecules, in the framework of the Debye–Hückel–Bjerrum–Manning and Flory theories, with the hydrophobic interaction included explicitly as an effective short-ranged attraction between the surfactant hydrocarbon tails. This simple model allows us to calculate the extension of the polyelectrolyte–cationic surfactant complexes as a function of the anionic surfactant concentration, for different types of cationic and anionic surfactant molecules. A discrete conformational transition from a collapsed state to an elongated coil was found, for all surfactant chain lengths we have considered, in agreement with the experimental observations for the unfolding of ​DNA–cationic surfactant complexes.

Phase field modelling of strain induced crystal growth in an elastic matrix.

When a crystal phase grows in an amorphous matrix, such as a crystallisable elastomer, containing cross-links and/or entanglements, these "topological constraints" need to be pushed away from the crystal phase to allow further crystallization. The accumulation of these topological constraints in the vicinity of the crystal interface may store elastic energy and affect the phase transition. To evaluate the consequences of such mechanism, we introduce a phase field model based on the Flory theory of entropic elasticity. We show that the growth process is indeed sensibly affected, in particular, an exponential increase of the surface energy with the displacement of the interface is induced. This explains the formation of stable nano-crystallites as it is observed in the Strain Induced Crystallization (SIC) of natural rubber. Although simple, the model developed here is able to account for many interesting features of SIC, for instance, the crystallite shapes and their sizes which depend on the applied deformation.

Structure of Arsenic Selenide Glasses Studied by NMR: Selenium Chain Length Distributions and the Flory Model

Five homogeneous arsenic selenide glasses with target compositions As2Se3, AsSe2, AsSe3, AsSe4.5, and AsSe6 were studied quantitatively by 77Se Carr–Purcell–Meiboom–Gill magic-angle spinning NMR and transmission electron microscopy–energy-dispersive X-ray spectroscopy. The entire set of NMR spectra is simultaneously fitted with six distinct environments taking into account the effect of first and second neighbors on the position of the 77Se resonance. The selenium chains are bound at each end to trivalent arsenic atoms, and the chain length distribution can be modeled with the Flory theory, which is well-known in polymer science and is used here for the first time to model the probability of finding each selenium environment in a selenide glass. No arsenic homopolar bond is detected in our experiments.