Adsorption kinetics of NIPAM-based polymers at the air-water interface as studied by pendant drop and bubble tensiometry

The adsorption of N-isopropylacrylamide (NIPAM) based thermoresponsive polymers at the air-water interface was investigated by using drop and bubble shape tensiometry. The molecular weight dependence of polymer adsorption rate was studied by using narrowly distributed polymer fractions (polydispersity < 1.2) that were prepared by solvent:nonsolvent fractionation. The time-dependent surface tension profiles were fitted to the Hua-Rosen equation and the t values obtained were applied for interpretation of the kinetic data. It was found that the rate of polymer adsorption increased as the molecular weight of the polymer decreased. The relationship between polymer surface concentration and surface tension was determined by applying the pendant drop as a Langmuir-type film balance. From this relationship, the kinetics of polymer adsorption determined experimentally was compared with the adsorption rates predicted by a diffusion-controlled adsorption model based on the Ward-Tordai equation. The predicted adsorption rates were in good agreement with what was found experimentally. The dependence of the adsorption rate on the molecular weight of polymers can be satisfactorily described within the diffusion-controlled model.     

Calculation of molecular weight distribution in a batch thermal polymerization of styrene

Molecular weight averages have long been used as a measure of polymer molecular weight properties in industrial polymer manufacturing processes. With a kinetic model, it is possible to directly calculate the polymer chain length distribution by integrating an infinite number of the polymer population balance equations. However, when the polymer chain length is very large, such a direct integration of polymer population balance equations can be computationally demanding. In this paper, the method of finite molecular weight moments is applied to the calculation of polymer chain length distribution in a batch free radical thermal polymerization of styrene. The weight fraction of a finite chain length interval is directly calculated in conjunction with a kinetic model. The method of calculation is illustrated through model simulations.     

Distribution of functional groups grafted onto an ethylene-propylene copolymer

A theoretical model, based on the binomial (Bernoullian) distribution function, was employed for the prediction of functional group distribution in an ethylene-propylene copolymer randomly grafted by maleic anhydride. Using the experimentally determined graft level and molecular weight distribution function, the fraction of polymer molecules with given number of functional groups was calculated. The result was checked experimentally by a fluorescence method based on the excimer formation of pyrene fluorophores attached to the anhydride pendants. The time-resolved fluorescence from the pyrene-labeled copolymer yielded the fraction of polymer molecules with a single functional group. The fraction of singly labeled molecules was compared to the calculated functional group distribution and a reasonable agreement was found between the two. The distribution of grafted maleic anhydride was found to be apparently random among polymer molecules. The distribution of distances was calculated between randomly attached consecutive functional groups along the polymer backbone also. The result indicated that the distance distribution function (similar to a decaying exponential) is dominated by short distances. © 1996 John Wiley & Sons, Inc.     

Cluster aggregation and fragmentation kinetics model for gelation

Gelation can occur in polymer, hydrogel, and colloid systems that undergo reversible aggregation-fragmentation (crosslinking accompanied by breakage). Gelation, characterized by rapid divergence of weight-average molecular weight and viscosity due to initial network formation, can be reversed if conditions change. In this paper, reversible aggregation and fragmentation in the pre-gelation time period are modeled with distribution kinetics. Moment equations are obtained from the population balance equation, and solved for eight different rate kernels. We identify the cases for which gelation is possible and obtain the critical values for the rate constants that allow gelation. The model provides a good simulation of published experimental data for aggregation and degradation of plasticized wheat gluten during thermo-mechanical treatments. We also evaluate two closure approximations based on Gamma and log-normal distributions, and conclude that log-normal closure predicts all five possible steady states, in agreement with the Vigil-Ziff criterion, and Gamma closure predicts only three. However, Gamma closure approximates the steady state either closely or exactly, whereas log-normal closure only poorly approximates the steady-state distribution.     

Destructive aggregation: aggregation with collision-induced breakage

Mean-field population balance equations are used to describe the evolution of particle size distributions in a wide variety of systems undergoing simultaneous aggregation and breakage. In this paper we develop a population balance that includes aggregation combined with collision-induced particle breakage for arbitrary fragment distribution functions, provided that this distribution function depends only on the total mass of the particles undergoing a collision. We then develop a specific distribution function for arbitrary two-body collisions by postulating that each collision produces a transition-state aggregate having the morphology of a linear polymer. The behavior of the resulting equation is then analyzed for the case in which the collision kernel is a constant, and partial analytical solutions are derived and compared to corresponding Monte-Carlo simulation results. The computer simulations are then used to validate a proposed scaling law for the steady-state particle size distribution. Lastly, the behavior of the aggregation with collision-induced-breakage population balance equation is compared and contrasted with the behavior of an analogous aggregation with linear-breakage population balance equation.     

Acrylate Network Formation by Free‐Radical Polymerization Modeled Using Random Graphs

A novel technique is developed to predict the evolving topology of a diacrylate polymer network under photocuring conditions, covering the low‐viscous initial state to full transition into polymer gel. The model is based on a new graph theoretical concept being introduced in the framework of population balance equations (PBEs) for monomer states (mPBEs). A trivariate degree distribution that describes the topology of the network locally is obtained from the mPBE, which serves as an input for a directional random graph model. Thus, access is granted to global properties of the acrylate network which include molecular size distribution, distributions of molecules with a specific number of crosslinks/radicals, gelation time/conversion, and gel/sol weight fraction. Furthermore, an analytic criterion for gelation is derived. This criterion connects weight fractions of converted monomers and the transition into the gel regime. Valid results in both sol and gel regimes are obtained by the new model, which is confirmed by a comparison with a “classical” macromolecular PBE model. The model predicts full transition of polymer into gel at very low vinyl conversion (<2%). Typically, this low‐conversion network is very sparse, as becomes apparent from the predicted crosslink distribution.     

Dynamic simulation of floc breakage with a random breakage distribution

The transient behavior of floc breakage in a lean, batch-stirred tank is investigated. The stochastic nature of breakage distribution (BD), which includes breakage mode (BM) and daughter-floc size distribution (DFSD), makes a deterministic approach unrealistic. The difficulty of representing BM and DFSD through deterministic probability distributions is circumvented by adopting- the Monte Carlo simulation approach to determine their functional forms numerically. On the basis of the conservation of the number of primary particles, the population balance equation describing the transient behavior of the system is solved analytically. The present model involves fewer adjustable parameters than those reported in the literature, yet it is able to take account or the random feature of the breakage phenomenon and the fractal geometry of floc structure.     

A General Model for Prediction of Molecular Weight Distributions of Degraded Polymers. Development and Comparison with Ultrasonic Degradation Experiments

Abstract

A general model has been developed for the process of polymer chain breakage. The molecular weight distributions (MWDs) of the degraded polymer, after a specified number of chain ruptures, can be calculated from the model, given the MWD of the initial material. The calculations can easily be handled numerically on a digital computer. The model is applicable to any process in which polymer chains break, e.g., ultrasonation and high shear mechanical action. The course of degradation is described in terms of the probability that a molecule of a given length will break and the probability that a molecule of a particular length will result from this rupture. Any form of both these probability distributions and the initial MWD can be used in the model.     

Thermal degradation of high density polyethylene in a reactive extruder

The thermal degradation of high density polyethylene was conducted in a reactive extruder at various screw speeds with reaction temperatures of 400 °C and 425 °C. The residence time of the extruder was estimated and the molecular weight distribution of the fed plastic and reaction products was analysed using gel permeation chromatography. A continuous kinetic model was used to describe the degradation of the high density polyethylene in the reactive extruder. The breakage kernel and the scission rate model parameters were estimated from the experimental data for a variety of cases. It was found that purely random breakage and a scission rate which had a power law dependence on molecular size of 0.474 best described the experimental data.     

Weight distribution approach to the theory of random depolymerization of heterogeneous chains

The random depolymerization model of Montroll and Simha for originally homogeneous polymers is derived analytically by utilizing a weight distribution function approach not requiring the use of any approximations. The model is then extended to consider heterogeneous initial chains having an arbitrary chain length distribution. The probabilities for breakage are assumed to be identical for all bonds joining monomeric elements in the system and to be independent of chain length and position in the chain, assumptions also used in the model of Montroll and Simha. An expression is found for estimating the probability of bond breakage.     

Determination of Kinetic Parameters from GPC-Data

Abstract

The molecular weight distribution and its first moments of selected polymers were experimentally determined through GPC-investigations with additional molecular weight detection. In on-line operation discontinuously functioning Ubbelohde-viscometer was used for the determination of degree of polymerization in dependence on elution volume.

The relation between the experimentally determined values and relative kinetic parameters were derived for some polymers prepared by radical mechanism. The relative constants which determines the branching structure in polymer can be now calculated.

From the investigated polymers the effect of chemical nature of monomer units for the branching formation was determined.     

A Study of the Molecular Weight and Chain Length Distribution of Cyclocopolymers of Divinyl Ether and Maleic Anhydride

Some physical properties of a cyclocopolymer formed by an intra-inter-molecular polymerization mechanism were studied. The cyclic copolymer of divinyl ether and maleic anhydride was chosen for study. Intrinsic viscosities were determined in two solvents. The weight-average molecular weight was determined by light-scattering measurements and the number-average molecular weight was determined by osmometry. Results from a sedimentation velocity pattern indicated that the chain length distribution is much broader than random, which is in agreement with the ratio of the two molecular weight determinations. The polymer was fractionated by precipitation from acetone with hexane as precipitant. Addition of sodium tetraphenyl boron was necessary to obtain separation. Correlation of intrinsic viscosity and weight-average molecular weights was not obtained because of anomalous solubility effects preventing light-scattering determinations.     

The Effect of Residence Time Distribution on the Slurry‐Phase Catalytic Ethylene Polymerization: An Experimental and Computational Study

This work is focused on the development and validation of a model accounting for the impact of the reactor residence time distribution in well‐stirred slurry‐phase catalytic polymerization of ethylene. Particle growth and morphology are described through the Multigrain model, adopting a two‐site model for the catalyst and a conventional kinetic scheme. Particle size distribution and polymer properties (average molecular weights and polydispersity) are computed as a function of particle size through a segregated model, assuming that neither breakage nor aggregation occur. Reactors are modeled by means of fundamental mass conservation equations. The model is applied to a system constituted by a series of two ideal continuous stirred tank reactors, where the synthesis of polyethylene with bimodal molecular weight distribution is performed, employing the initial catalyst size distribution as the only adjustable parameter. The model provides insights at the single particle scale for each specific size, thus highlighting the inhomogeneity which arises from the synergic effects of chemical kinetics and residence time distributions in both reactors. The satisfactory agreement between model results and experimental data, in terms of particle size distribution and average molecular weights, confirmed the suitability of the model and underlying assumptions.     

On the theory of radical depolymerization: A rigorous solution

A rigorous solution to a model of radical depolymerization is presented. The model includes random chain scission, depropagation, radical transfer, and first‐order radical termination. The evolution of the molecular weight distribution in the course of depolymerization has been determined under the condition that the initial polymer is characterized by Flory's distribution. The kinetic equations consider the presence of chains with two radicalized ends, which are usually neglected. The commonly used simplifying assumption of the steady‐state radical concentration is not employed, and this makes the obtained results valid at any ratios between the rate constants. The predictions of the steady‐state approximation are compared to those of the rigorous approach in the case of depolymerization accompanied by volatilization of monomeric species. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 965–982, 2003     

Inclusion of the effects of polydispersity and volatility on the random chain scission statistical analysis for polymer degradation

The statistical product distribution for a linear polydisperse polymer of finite molecular weight was included into the statistical analysis for a system undergoing random chain scission showing the effect of volatilization of species other than monomer. Two sets of equations were derived. One set is for the nonvolatile fraction; the other is for the volatile fraction. Within each set there are three equations, one for the number of polymer molecules, the second for the molar (or number) fraction, and the third for the weight fraction of polymer molecules containing a specific number of repeat units. As degradation proceeds the polydispersity index should converge to a value of 1 rather than 2, which has been reported previously. The expected effects of polydispersity, number‐average degree of polymerization, and volatility were treated individually, and we determined that the molecular weight of a polymer has no theoretical influence on the product distribution. As for the effect of volatility, we determined that only the volatile product distribution would change. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 3690–3696, 2000     

Simultaneous long‐chain branching and random scission: I. Monte Carlo simulation

In free‐radical olefin polymerizations, the polymer transfer reactions could lead to chain scission as well as forming long‐chain branches. For the random scission of branched polymers, it is virtually impossible to apply usual differential population balance equations because the number of possible scission points is dependent on the complex molecular architecture. On the other hand, the present problem can be solved on the basis of the probability theory by considering the history of each primary polymer molecule in a straightforward manner. The random sampling technique is used to solve this problem and a Monte Carlo simulation method is proposed. In this simulation method, one can observe the structure of each polymer molecule formed in this complex reaction system, and virtually any structural information can be obtained. In the illustrative calculations, the full molecular weight distribution development, the gel point determination, and examples of two‐ and three‐dimensional polymer structure are shown. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 391–403, 2001     

Monte Carlo simulation of size exclusion chromatography for branched polymers formed through free‐radical polymerization with chain transfer to polymer

The elution curves of size exclusion chromatography (SEC) for branched polymers formed through free‐radical polymerization that involves chain transfer to polymer were theoretically investigated by using a Monte Carlo method. We considered two types of measured molecular weight distribution (MWD), (1) the calibrated MWD relative to standard linear polymers, and (2) the MWD obtained by using a light scattering photometer (LS) in which the weight‐average molecular weight of polymers within the elution volume is determined directly. It was found that the calibrated MWD clearly underestimates the high molecular weight tail, and the measured distributions are narrower than the true MWD. On the other hand, the present simulation results showed that the LS method gives reasonable estimates of the true MWDs. The mean square radius of gyration of the polymer molecules having the same molecular weight was also investigated. The radii of gyration showed clear deviation from the Zimm‐Stockmayer equation^[1] because of the non‐random nature of branched structure and the difference in the primary chain length distribution.     

Polymer fractionation. I. The preparative problem

The molecular weight distributions of the fractions in the two liquid phases which, in preparative fractionation, form upon a change in temperature, have been calculated for various initial concentrations and model distributions differing widely in shape and width. It was found that the normally recommended conditions do not always lead to the result desired, i.e., to a fraction with a narrow distribution. Under these conditions (dilute solution; large phase-volume ratio; small fraction, i.e., little polymer in the concentrated phase) it may even happen that the fraction has a relatively wider distribution than the parent polymer. Treatment of a single-peaked distribution will then be liable to yield a two-peak fraction distribution. These unwanted phenomena do not occur in Staverman and Overbeek's variant of the extraction fractionation method. Here, the bulk of the polymer goes into the concentrated phase, and the macromolecules left in the dilute phase constitute the fraction to be isolated. In this way, relatively narrow distributions can be obtained in one step, even at high concentrations. Moreover, the method is very well suited for use in large-scale fractionation. Fraction distribution wider than the initial distribution were found experimentally in the systems polyethylene–diphenyl ether and polystyrene–cyclohexane.     

Population balance modeling of aggregation and breakage in turbulent Taylor-Couette flow

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor-Couette apparatus is presented. To monitor the aggregation and breakage processes, an in situ optical imaging technique was used. Consequently, a computational study using a population balance model was carried out to test the various parameters in the aggregation and breakage models. Very good agreement was found between the time evolution of the cluster size distribution (CSD) calculated with the model and that obtained from experiment. In order to correctly model the left-hand side of the CSD (small clusters), it was necessary to use a highly unsymmetric fragment-distribution function for breakage. As another test of the model, measurements with different solid volume fractions were performed. Within the range of the solid volume fractions considered here, the steady-state CSD was not significantly affected. In order to correctly capture the right-hand side of the CSD (large aggregates) at the higher solid volume fraction, a modified aggregation rate prefactor was used in the population balance model.