Molecular Weight Distribution Function Research Articles

Random Branching and Cross‐linking of Polymer Chains, Analytical Functions for the Bivariate Molecular Weight Distributions

AbstractCross‐linking and branching of primary polymer molecules are investigated using the Galton–Watson (GW) process. Starting with the probability generating function (pgf) of the primary molecular weight distribution (MWD), analytical expressions are derived for the bivariate pgfs g(nbr, s) of branched polymers which depend also on the number of branch points nbr. The bivariate MWDs n(nbr, i) (i: number of molecular units) are then derived as Taylor expansions in s. All three cases of random branching: X‐shaped (cross‐linking), T‐shaped (only one end takes part in the branching process), and H‐shaped (both ends can take part in the branching process) are treated. An extension of the formalism does not require the construction of the pgf and allows the direct use of the MWD of the primary chains. However, using pgfs allows to go past the gel point and to determine the MWD and content of the sol. Explicit expressions are given for special distributions: the mono modal, the most probable, the Schulz‐Zimm, the Poisson, and the Catalan distribution for the cases of X‐shaped and T‐shaped branching.

Modeling the spatial characteristics of extrusion flow instabilities for styrene-butadiene rubbers: Investigating the influence of molecular weight distribution, molecular architecture, and temperature

The extrusion flow instabilities of three commercial styrene-butadiene rubbers (SBR) are investigated as a function of molecular weight distribution (MWD); molecular architecture (linear, branched); and temperature. The samples have multimodal MWD, with the main component being SBR and a low amount, less than 10 wt. %, of low-molecular weight hydrocarbons. Deviation from the Cox–Merz rule at high angular frequencies/shear rates becomes intense as the amount of medium-molecular weight component increases. Optical analysis is used to identify and quantify spatial surface distortions, specifically wavelength (λ) and height (h), of the different types of extrusion flow instabilities. Qualitative constitutive models are reviewed and used to fit the experimental data for the spatial characteristics of extrusion flow instability. The fitting parameters as obtained by the models are correlated with molecular properties of the materials. It is found that the characteristic spatial wavelength (λ) increases as the extrusion temperature decreases. Hence, the influence of temperature on the spatial characteristic wavelength is investigated and an Arrhenius behavior is observed.

Controlling polymer properties through the shape of the molecular-weight distribution

The manipulation of a polymer’s properties without altering its chemical composition is a major challenge in polymer chemistry, materials science and engineering. Although variables such as chemical structure, branching, molecular weight and dispersity are routinely used to control the architecture and physical properties of polymers, little attention is given to the often profound effect of the breadth and shape of the molecular-weight distribution (MWD) on the properties of polymers. Synthetic strategies now make it possible to explore the importance of parameters such as skew and the higher moments of the MWD function beyond the average and standard deviation. In this Review, we describe early accounts of the effect of MWD shape on polymer properties; discuss synthetic strategies for controlling MWD shape; describe current endeavours to understand the influence of MWD shape on rheological and mechanical properties and phase behaviour; and provide insight into the future of using MWDs in the design of polymeric materials. Controlling polymerization is required to fully understand and tune the function of a polymer. Alongside variables such as dispersity, molecular weight and chemical composition, the shape of the molecular-weight distribution has substantial influence on polymer properties. In this Review, we discuss methods to systematically control the shape of a polymer’s molecular-weight distribution, as well as explore the profound effects of shape on polymer properties.

A Calculation Scheme for Assessing Storage Moduli and Losses as a Function of Polymer Chemical Structure and Blend Composition

A calculation scheme is proposed for assessing and predicting storage and loss moduli. The examination is based on atomic constants, which take into account the contribution of each atom and polar group and the van der Waals volume and shear modulus at high frequencies. The obtained relationship makes it possible to calculate the shear modulus and storage and loss moduli as a function of frequency, temperature, and molecular weight distribution.

Features of the molecular–topological structure of γ-irradiated powdered tetrafluoroethylene–perfluoro(propyl vinyl ether) copolymer

With the use of thermomechanical spectrometry in two modes, coaxial, when the load application vector is coplanar with the compacting pressure vector (semicrystalline copolymer) during the measurement of the deformation of a copolymer of tetrafluoroethylene with perfluoro(propyl vinyl ether), and mutually perpendicular (completely amorphous copolymer), it has been established that the axial compression of the copolymer brought its topological structure to an absolutely anisotropic state. After γ-irradiation, the semicrystalline structure of the copolymer was retained regardless of the radiation dose. The minimum values of the glass transition temperature of the amorphous block and the degree of crystallinity were observed in the copolymer irradiated to a dose of 150 kGy. The molecular weight distribution functions of interjunction chains in the networks of the amorphous blocks of the initial copolymer and its γ-irradiated analogs are bimodal.

Estimating the molecular weight distribution of a polymer using acoustic relaxation spectra

An approach to estimating the molecular weight distribution function of a polymer in a solution using acoustic spectroscopy data is presented. The estimation function is based on the relationship between relaxation spectra and the molecular weight distribution function, along with the relationship between the relaxation spectra and the frequency distribution of a polymer’s loss modulus.

Supercritical Fluid Chromatography with Evaporative Light Scattering Detection (SFC-ELSD) for Determination of Oligomer Molecular Weight Distributions

Ultraviolet curable polyureas and polyurethanes enjoy a wide range of applications in the coatings industry. They demand less energy for curing compared to thermal processes that offer relatively low viscosities before photocuring, and provide mechanical property improvements due to hydrogen bonding physical crosslinks. Mechanical properties of the coatings are dependent on a variety of factors, including chemical composition, molecular weight, and molecular weight distribution. Formulations are designed for a wide variety of end use applications, ranging from soft, elastomeric coatings to harder, nonflexible sealants. This report demonstrates that the thermomechanical behavior of photocured coatings is a function of molecular weight distribution. Step-growth polymerization and endcapping afforded a variety of acrylate-terminated, urea-/urethane-containing photocurable oligomers from amine-terminated poly(propylene glycol), dicyclohexylmethane-4,4′-diisocyanate (HMDI), and 2-hydroxyethyl acrylate (HEA) at various stoichiometric ratios. The state-of-the-art supercritical fluid chromatography coupled with evaporative light-scattering detection (SFC-ELSD) enabled the elucidation of oligomeric molecular weight distributions as a function of reaction stoichiometry. SFC-ELSD demonstrated the efficient separation of oligomeric species with single repeat unit resolution (i.e., n = 2 vs. n = 3). Dynamic mechanical analysis probed thermomechanical response of photocured films as a function of molecular weight distribution and demonstrated that the presence of a hydrogen-bonding, small molecule photoactive reaction byproduct, i.e., HEA doubly-endcapped HMDI, had a much more profound effect on thermomechanical response as compared to changes in oligomer molecular weight in the molecular weight range investigated. This combination of chromatographic technique and thermomechanical analysis afforded an in-depth investigation of the structure–property relationships of urea-/urethane-containing photocurable oligomers.

Broadening of polymer chromatographic signals: Analysis, quantification and correction through effective diffusion coefficients

Average molecular weights and polydispersity indexes are some of the most important parameters considered in the polymer characterization. Usually, gel permeation chromatography (GPC) and multi angle light scattering (MALS) are used for this determination, but GPC values are overestimated due to the dispersion introduced by the column separation. Several procedures were proposed to correct such effect usually involving more complex calibration processes.In this work, a new method of calculation has been considered including diffusion effects. An equation for the concentration profile due to diffusion effects along the GPC column was considered to be a Fickian function and polystyrene narrow standards were used to determine effective diffusion coefficients. The molecular weight distribution function of mono and poly disperse polymers was interpreted as a sum of several Fickian functions representing a sample formed by only few kind of polymer chains with specific molecular weight and diffusion coefficient.Proposed model accurately fit the concentration profile along the whole elution time range as checked by the computed standard deviation. Molecular weights obtained by this new method are similar to those obtained by MALS or traditional GPC while polydispersity index values are intermediate between those obtained by the traditional GPC combined to Universal Calibration method and the MALS method.Values for Pearson and Lin coefficients shows improvement in the correlation of polydispersity index values determined by GPC and MALS methods when diffusion coefficients and new methods are used.

Multi-Objective Optimization of Bulk Vinyl Acetate Polymerization with Branching

Inclusion of long chain branching (LCB) in polymers is a challenging and important task in any free radical polymerization as LCB influences polymer product quality. In the present case, batch optimization study for the bulk polymerization of vinyl acetate has been considered to find optimal process conditions for imparting LCB in polymer architecture. A theoretical study has been conducted with a validated model to observe the effect of live radical concentration on LCB as this is an important factor for branching in polymer via chain transfer to polymer route. In order to obtain better polymer product in less time at various temperatures, a need was observed to perform a multi-objective optimization study as the selected objectives were conflicting in nature. Owing to the complex nature of moment-based species balance equations and molecular weight distribution function, elitist non-dominated sorting genetic algorithm II (NSGA II), a well-established multi-objective evolutionary algorithm, has been employed as an evolutionary computation method to find out the Pareto optimal solutions. Minimization of polymerization time, maximization of molecular weight and maximization of number average degree of branching (Bn) can be simultaneously achieved, while the solutions were obtained within the experimental range of polydispersity index and weight average molecular weight (Mw) given in the open literature. Results show a wide range of process choices satisfying process objectives and constraints, both in low as well as high temperature regions.

Polydispersity and Molecular Weight Distribution of Hyperbranched Graft Copolymers via “Hypergrafting” of ABm Monomers from Polydisperse Macroinitiator Cores: Theory Meets Synthesis

The hypergrafting strategy designates the synthesis of hyperbranched graft copolymers (HGCs) in a grafting-from approach, using ABm monomers, from multifunctional, polydisperse macroinitiator cores by slow monomer addition. Hypergrafting leads to complex polymer topologies with defined molecular weight, degree of branching (DB), and polydispersity (PD). By a generating function formalism, a generally applicable equation for the PD of HGCs (PD = PDf + (m – 1)/f) is derived, where PDf is the polydispersity of the core and f its average functionality. In addition, the complete molecular weight distribution function has been calculated for varied m and f as well as for a given distribution of initiator functionalities f. For comparison of the theoretical predictions with experimental results, a series of novel linear polyglycerol-graft-hyperbranched polyglycerol (linPG-g-hbPG) HGCs (Mn = 1000–4000 g mol–1) were synthesized and characterized as a model system. An increase in polydispersity occurred as a con...

Molecular-weight distribution of linear polycondensation polymers. Flory theory

The logic of the conclusion of P. Flory regarding the molecular-weight distribution functions of polymers obtained through various types of linear polycondensation was reproduced. It was shown that the reason for the multi-year discussion regarding the correctness of applying this theory to equilibrium polycondensation was an overestimation in several studies of the role of exchange reactions involving the highest polymer molecular fractions in this process. It was emphasized that the independence of the reactivity of end functional groups from the macromolecule size during polycondensation and the independence of the chain-growth reaction rate constant in the polymerization processes, which also produced an exponential macromolecule size-distribution function, were different in principle. Attention was paid to the possibility derived from the theory of P. Flory of preparing linear polymers with $ {{{\bar{M}}^{\prime}}_w}/{{{\bar{M}}^{\prime}}_n} $ during irreversible polycondensation in the presence of monofunctional dibasic chain-length regulators.

Self-assembly of amphiphilic random co-poly(ionic liquid)s: the effect of anions, molecular weight, and molecular weight distribution

Self-assembly behavior of amphiphilic random co-poly(ionic liquid)s in aqueous solution was investigated in this work. An imidazolium-type homopolymer, poly(1-(4-vinylbenzyl)-3-methyl imidazolium chloride) (denoted as PIL-[Cl]), was synthesized via reversible addition fragmentation chain transfer (RAFT) polymerization of p-chloromethyl styrene, and followed by post-polymerization with methylimidazole (or vinylimidazole). Partial anion-exchange of PIL-[Cl] with PF6− anions yielded amphiphilic random copolymers, PIL-[Cl]x[PF6]y (suffixes x, y represent the relative molar ratio of Cl− and PF6− anions). Self-assembly morphologies of PIL-[Cl]x[PF6]y in aqueous solution as a function of copolymer composition (anions), molecular weight, and molecular weight distribution were investigated. The size of well-defined PIL-[Cl]x[PF6]y spherical micelles increased dramatically with increasing molecular weight distribution, while the anions and molecular weight show little effect on the micelle size. Furthermore, self-assembly of amphiphilic random ionic copolymers with the vinylimidazole groups yielded colloidosomes, which might be due to the polymerization of vinylimidazole groups.

Monitoring of asphaltene precipitation: Experimental and modeling study

Preparing relatively complete collections of experimental data on asphaltene precipitation in different reservoir conditions leads to considerable improvement in this area of science. In this work, asphaltene precipitation was studied upon two Iranian live oil samples, one a heavy oil and another light oil, under primary depletion as well as gas injections. Pressure depletion experiments were carried out at different temperatures to observe temperature effect besides pressure changes on asphaltene phase behavior. CO 2, dry and enriched gases were used as injecting agents to investigate the effect of different gases on asphaltene precipitation. Surprisingly, it was observed that raising temperature decreases the amount of precipitation in case of heavy oil while acting in favor of precipitation for light oil sample. In addition, Enriched gas resulted in more precipitation compared to dry one while CO 2 acted as hindering agent for light oil samples but increased the amount of precipitation in case of heavy oil. In the next part of this work, polydisperse thermodynamic model was developed by introducing an asphaltene molecular weight distribution function based on fractal aggregation. Modification that was introduced into polydisperse model not only solved the instability problem of Kawanaka model but also eliminates the need for resin concentration calculation. Flory–Huggins and Modified Flory–Huggins thermodynamic solubility models were applied to compare their predictions with proposed model.

PLGA Erosion: Solubility- or Diffusion-Controlled?

To calculate the degradation time-dependent formation of water-soluble PLGA oligomers and to evaluate the relation between calculated oligomer formation and actual erosion of a PLGA-based delivery system. A proper model of the erosion process would be expected to facilitate forecasting of drug release profiles from PLGA matrices due to the close relationship of erosional mass loss and drug release described in the literature. The molecular weight distribution (MWD), degradation and erosion behaviour of PLGA were characterized by gel permeation chromatography. PLGA was characterized by a lognormal distribution of mass fractions of individual molecular weights. Implementation of the pseudo-first-order reaction kinetics into the MWD function facilitated calculating the formation of water-soluble oligomers during degradation. The calculated soluble oligomer formation agreed excellently with measured erosional mass loss of a PLGA matrix in aqueous buffer, which suggested that the bulk erosion process was solely controlled by the kinetic of the formation of soluble oligomers and thus solubility-controlled and not diffusion-limited as conventionally assumed. The accurately calculated formation of soluble PLGA oligomers was in excellent agreement with the actual erosional mass loss of a PLGA matrix, suggesting that bulk erosion of PLGA represents a degradation-controlled dissolution process.

Application of Continuous Polydisperse Molecular Thermodynamics for Modeling Asphaltene Precipitation in Crude Oil Systems

In this work, a new algorithm for polydisperse asphaltene modeling through application of continuous molecular thermodynamics is introduced. The Scott−Magat theory of polydisperse polymer solutions is applied at equilibrium condition and a necessary and sufficient condition is defined to minimize the Gibbs free energy relation. Three commonly used distribution functions were examined for characterization of asphaltenes, and experimental data reported in the literature is used to adjust parameters of the distribution functions. Among the three distribution functions, the fractal molecular weight distribution function which gives accurate results is selected. A new exponential binary interaction coefficient between asphaltene and crude oil is introduced. The solubility parameter, volume fraction of asphaltene, and the amount of asphaltenes precipitated are calculated through minimization of Gibbs free energy and phase equilibrium condition. It has been shown that the calculated results are in good agreement with experimental data.

A generalized relation between MWD and relaxation time spectrum

An inversion procedure for converting linear viscoelastic properties of polymer melts into molecular weight distribution (MWD) described by the generalized exponential function (GEX) has been implemented and applied in a previous work (Cocchini F, Nobile MR. Rheol Acta 42:232–242, 2003). It is based on the elegant relationship between the relaxation spectrum and the MWD function proposed by Thimm W, Friedrich C, Marth M. J Rheol 43:1663–1672 (1999). In the present paper, such a methodology has been generalized to properly account for sharp MWDs, in particular, nearly monodisperse or blends of nearly monodisperse polymer samples. The generalized relationship has been verified and calibrated using the BSW kernel (Baumgaertel M, Schausberger A, Winter HH. Rheol Acta 29:400–408, 1990) to describe the rheological behavior of some Polystyrene samples from the literature, in terms of the known MWD. Then, it has been successfully applied to the inversion problem for a wider set of samples, with both broad and sharp distributions. The Rouse contribution at high frequencies and the accelerating effect on the relaxation times due to polydispersion have been also addressed.

Split crystallization during debranching of maltodextrins at high concentration by isoamylase.

Debranching and crystallization occurring during the enzymatic treatment of 25% (w/v) aqueous solutions of maltodextrins by isoamylase at 52 degrees C were studied. The morphology as well as the crystal and molecular structures of the precipitates formed at different stages of the reaction were characterized. Two types of resulting products, differing in terms of structure and morphology, were evidenced. A loose B-type network, containing linear and branched chains of highest molecular weight, was mainly formed during the first 12 h of reaction, whereas aggregates of A-type lamellar crystals, made of short linear chains, were predominantly obtained between 12 and 48 h. The aggregation behavior as a function of temperature and molecular weight distribution of such substrates was discussed and compared to that of related starch products.

Experimental characterization of extrudate swell behavior of linear polybutadiene

This paper describes an experimental study of the time-dependent extrudate swell behavior of a series of linear polybutadienes as a function of molecular weight (MW) and molecular weight distribution (MWD). The “initial” extrudate swell, i.e., the extrudate swell measured near the die exit, is universally found to be around 1.10, independent of MW, MWD and the extrusion rate. Like the initial extrudate swell, the extrudate surface velocity at the die exit, Vs, when normalized by the average capillary flow velocity, Vp, is also a relatively universal function of the distance X from the exit plane. It increases from 0.3 to 1 over a normalized distance from X/D=0 to X/D=0.035 regardless of MW, MWD, and the extrusion rate. In the presence of exit slip, which can be produced by coating the exit wall with a fluoropolymer, Vs is nearly equal to Vp even at x=0. The extrudate swell at a given distance from the exit is shown to be independent of the molecular weight for monodisperse samples. A small tail in the MWD at the high end produces considerably larger extrudate swell. Although the extrudate swell increases with increasing applied stress for a given sample, a series of polymer solutions of different concentrations show that the real controlling variable is the applied stress normalized by the material’s elastic plateau modulus. The extrudate swell is also found to depend systematically on the die diameter at all stages of its growth. Moreover, the extrudate swell is greater when it is allowed to grow in media (other than the air) such as oil and water. These two last effects can be understood in terms of the free-surface-to-volume ratio of the extrudate: the larger this ratio is more rapidly the residual molecular deformation in the extrudate can relax. A proper understanding of these interfacial effects will have important implications in processes such as injection/blow molding and film blowing from the viewpoints of both industrial practice and numerical simulations.

Gel formation with multiple interunit junctions. I—Molecules carrying different functional groups

AbstractNumber‐ and weight‐average molecular weight of condensation polymers formed by primary molecules carrying different species of functional groups {Ai} (i = 1, 2, …, s) are derived by cascade theory. These functional groups are allowed to form multiple junctions of variable multiplicity k. The gel point condition is found to be given by ∑ wi/|μw,i + 1/∑ fi − 1 = 0, where fi is the number of Ai groups specified by the index i on a primary molecule, wi ≡ fi/∑ fi the number fraction of the species i it carries, and |μw,i the weight average multiplicity of the junctions formed by the groups Ai. The explicit form of the molecular weight distribution function is found for the simplest case of two components. Possible application to thermoreversible gelation is suggested. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2405–2412, 2003

Modeling of Kinetics of Stress-Induced Degradation of Polymer Additives in Lubricants and Viscosity Loss

A fundamental approach to the problem of modeling mechanically induced polymer degradation is proposed. The polymer degradation is modeled by a kinetic equation for the density of the statistical distribution of linear polymer molecules as a function of their molecular weight. The integrodifferential kinetic equation is solved numerically. A comparison of numerically calculated molecular weight distributions and lubricant viscosity loss caused by polymer degradation with experimental ones obtained in bench tests showed that they are in excellent agreement. The effects of pressure, shear, temperature, and lubricant viscosity on lubricant degradation are considered. The increase of pressure promotes fast degradation while the increase of temperature depending on other parameters may delay or promote degradation. In some cases, the density of the molecular weight distribution function maintained its initial single-modal shape and in other cases it changed with time from a single-modal shape to a multi-modal one. Presented as a Society of Tribologists and Lubrication Engineers Paper at the ASME/STLE Tribology Conference in Cancun, Mexico October 27–30, 2002