Analysis of the aggregation-fragmentation population balance equation with application to coagulation

Coagulation of small particles in agitated suspensions is governed by aggregation and breakage. These two processes control the time evolution of the cluster mass distribution (CMD) which is described through a population balance equation (PBE). In this work, a PBE model that includes an aggregation rate function, which is a superposition of Brownian and flow induced aggregation, and a power law breakage rate function is investigated. Both rate functions are formulated assuming the clusters are fractals. Further, two modes of breakage are considered: in the fragmentation mode a particles splits into w2 fragments of equal size, and in the erosion mode a particle splits into two fragments of different size. The scaling theory of the aggregation-breakage PBE is revised which leads to the result that under the negligence of Brownian aggregation the steady state CMD is self-similar with respect to a non-dimensional breakage coefficient theta. The self-similarity is confirmed by solving the PBE numerically. The self-similar CMD is found to deviate significantly from a log-normal distribution, and in the case of erosion it exhibits traces of multimodality. The model is compared to experimental data for the coagulation of a polystyrene latex. It is revealed that the model is not flexible enough to describe coagulation over an extended range of operation conditions with a unique set of parameters. In particular, it cannot predict the correct behavior for both a variation in the solid volume fraction of the suspension and in the agitation rate (shear rate).     

Polycondensation Kinetics of Lactic Acid

The direct polycondensation of D ,L ‐lactic acid in the absence and presence of different catalysts at various temperatures has been studied experimentally. Two types of reactions were carried out, one under closed conditions to estimate the equilibrium constant and the other under flow of nitrogen to estimate the polymerization rate constant. A mathematical model was developed based on a suitable kinetic scheme for polycondensation reaction accounting for the rate of water removal. The effects of different operating conditions (temperature and pressure) on the average molecular weight of the polymer have been explored through experiments and model simulations.

     

Self-propagating frontal copolymerization

The application of self-propagating frontal polymerization (FP) to synthesize copolymers has been investigated. Frontal copolymerizations of methyl methacrylate and methacrylic acid (MMA–MAA), acrylic acid and methacrylic acid (AA–MAA), and styrene and methacrylic acid (STY–MAA) with benzoyl peroxide (BPO) as initiator have been performed. The measured front velocities have been compared with the results of a suitably developed model. This is based on the pseudokinetic approach, accounts for the depropagation reaction, and is fully predictive; i.e., it does not include any adjustable parameter (although one had to be used for the specific experimental setup used in this work). An explicit, simplified solution of the model has been obtained using the constant pattern approximation. The microstructures of copolymers produced in bulk and by FP have been analyzed by differential scanning calorimetry. Their comparison indicates that self-propagating frontal copolymerization provides a substantial improvement in the uniformity of the chain composition distribution. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 1117–1126, 1998     

Dispersion Polymerization of Methyl Methacrylate in Supercritical Carbon Dioxide: Control of Molecular Weight Distribution by Adjusting Particle Surface Area

Summary: Experiments of methyl methacrylate dispersion polymerization are carried out in a reaction calorimeter using PDMS-mMA as surfactant. Different stabilizer concentrations from 0 to 10 wt% with respect to monomer have been considered in order to control particle morphology. The analysis by scanning electron microscopy reveals a definite decrease of the total particle surface area at decreasing stabilizer concentration. At the same time, the analysis of the polymer microstructure by gel permeation chromatography shows a trend of the average molecular weight towards smaller values. In particular, a second mode at low molecular weights has been observed leading to bimodal molecular weight distributions. The experimental results are compared with simulation results obtained through a detailed kinetic model developed in previous studies. 1 The key role of the radical exchange between continuous and dispersed phases is confirmed.     

Monitoring coalescence behavior of soft colloidal particles in water by small-angle light scattering

The fractal dimension (D _f) of the clusters formed during the aggregation of colloidal systems reflects correctly the coalescence extent among the particles (Gauer et al., Macromolecules 42:9103, 2009). In this work, we propose to use the fast small-angle light scattering (SALS) technique to determine the D _f value during the aggregation. It is found that in the diffusion-limited aggregation regime, the D _f value can be correctly determined from both the power law regime of the average structure factor of the clusters and the scaling of the zero angle intensity versus the average radius of gyration. The obtained D _f value is equal to that estimated from the technique proposed in the above work, based on dynamic light scattering (DLS). In the reaction-limited aggregation (RLCA) regime, due to contamination of small clusters and primary particles, the power law regime of the average structure factor cannot be properly defined for the D _f estimation. However, the scaling of the zero angle intensity versus the average radius of gyration is still well defined, thus allowing one to estimate the D _f value, i.e., the coalescence extent. Therefore, when the DLS-based technique cannot be applied in the RLCA regime, one can apply the SALS technique to monitor the coalescence extent. Applicability and reliability of the technique have been assessed by applying it to an acrylate copolymer colloid.     

Effect of high pH column regeneration on the separation performances in reversed phase chromatography of peptides

Caustic regeneration procedures are often used in chromatographic purification processes of peptides and proteins to remove irreversibly bound impurities from the stationary phase. Silica-based materials are the most commonly used materials in reversed phase chromatography of peptides. Their limited chemical stability at high pH can be, however, problematic when high pH column regeneration (i.e. cleaning in place) is required. The effect of cleaning in place on the surface chemistry of the stationary phase has been investigated using the Tanaka test. It has been shown that the high pH treatment does not significantly affect the hydrophobicity of the material, but it strongly increases its silanol activity. A representative peptide purification process has been used to investigate the impact of cleaning in place on the separation performance. It has been shown that the caustic regeneration increases the peptide retention at high pH (pH 6.5), due to the interactions between the peptide and the negatively charged silanol groups. These unwanted interactions reduce the separation performances by decreasing the selectivity between the late eluting impurities and the main peptide. However, it has been shown that the effect of the silanol groups on the peptide adsorption and on the separation performance can be minimized by carrying out the purification process at low pH (pH ∼ 2). In this case, the silanol groups are protonated and their electrostatic interactions with the positively charged analyte (i.e. peptides) are suppressed. In these conditions, the peptide adsorption and the impurity selectivity is not changing upon high pH column regeneration and the separation performance is not affected.     

Experimental verification of sample-solvent induced modifier-solute peak interactions in biochromatography

In a previous theoretical analysis based on equilibrium theory it has been shown how differences in the sample and elution modifier concentrations can lead to unexpected behavior of the solute eluted peaks such as retention time distortion, peak deformation and peak doubling. All these features are verified experimentally in this work using the polypeptide calcitonin and a variant of a specific monoclonal antibody as chromatographic model systems. For both experimental systems, the retention time distortion can be predicted with high accuracy by the solution of the equilibrium theory model. For the polypeptide, the predictions from the theory about the occurrence of peak deformation and double peaks has been successfully verified by a series of tailored experiments with positive as well as negative modifier perturbations.     

A global Inverse Map Theorem and biLipschitz maps in the Heisenberg group

Abstract We prove a global Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non singular and satisfies some growth conditions at infinity. An estimate for the Lipschitz constant (with respect to the Carnot–Carathéodory distance in ) of a continuously Pansu differentiable map is included. This gives a characterization of (continuously Pansu differentiable) globally biLipscitz deformations of in term of a pointwise estimate of their differential. Keywords: Inverse problem, Heisenberg group     

PLA‐based nanoparticles with tunable hydrophobicity and degradation kinetics

In this work, the preparation of poly(lactic acid) (PLA)‐based degradable nanoparticles (NPs) with tunable hydrophobicity and degradation kinetics via starved emulsion free‐radical polymerization is studied. The synthesis of macromonomers, constituted of a tunable number of lactic acid units functionalized with 2‐hydroxyethyl methacrylate (HEMA), has been performed via bulk ring opening polymerization (ROP) of L, L‐ lactide catalyzed with 2‐ethylhexanoic acid tin (II) salt. Macromonomers were characterized through SEC, NMR, and FTIR and are subsequently polymerized through monomer‐starved semi‐batch emulsion polymerization (MSSEP). The effect on the polymerization process of various emulsifiers on the final diameter and particle size distribution has been studied. The resulting PLA‐based NPs are characterized by a narrow size distribution and a small particle size, down to 25 nm. Finally, a degradation study of selected NPs has been carried out to verify their degradability in aqueous media. It has been demonstrated the complete degradability of these PLA‐based NPs which occurs upon the hydrolysis of the PLA pendant chains leaving poly‐HEMA chains, which, being hydrophilic causes the NPs to dissolve in the aqueous suspension. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012