Breakage Rate Constants Research Articles

Development of a Semi-Mechanistic Modeling Framework for Wet Bead Milling of Pharmaceutical Nanosuspensions.

This study aimed to develop a practical semi-mechanistic modeling framework to predict particle size evolution during wet bead milling of pharmaceutical nanosuspensions over a wide range of process conditions and milling scales. The model incorporates process parameters, formulation parameters, and equipment-specific parameters such as rotor speed, bead type, bead size, bead loading, active pharmaceutical ingredient (API) mass, temperature, API loading, maximum bead volume, blade diameter, distance between blade and wall, and an efficiency parameter. The characteristic particle size quantiles, i.e., x10, x50, and x90, were transformed to obtain a linear relationship with time, while the general functional form of the apparent breakage rate constant of this relationship was derived based on three models with different complexity levels. Model A, the most complex and general model, was derived directly from microhydrodynamics. Model B is a simpler model based on a power-law function of process parameters. Model C is the simplest model, which is the pre-calibrated version of Model B based on data collected from different mills across scales, formulations, and drug products. Being simple and computationally convenient, Model C is expected to reduce the amount of experimentation needed to develop and optimize the wet bead milling process and streamline scale-up and/or scale-out.

Effects of cationic counterions and surfactant on viscosity of an amino acid-based surfactant system

Many sulfate-free and amino acid-based surfactant systems for personal care formulations do not respond to thickening with salt and/or amphoteric cosurfactants as readily as traditional sulfate-based formulations. Although this phenomenon is generally understood to be associated with large headgroups, the mechanistic basis for the difference is poorly understood. To address this question we have combined an empirical approach involving traditional determination of low shear viscoelasticity with a state-of-the-art mesoscopic analysis of micelle dynamics enabled by high frequency rheological measurements obtained optically (Zou and Larson, Journal of Rheology 2014, 58 (3), 681-721) We studied the impact of various organic and inorganic counterions as well as a cationic surfactant as alternative thickening agents for a model formulation of sodium lauroyl sarcosinate (SLSar) and cocamidopropyl hydroxysultaine (CAHS). These additives include lithium chloride (LiCl), sodium chloride (NaCl), L-arginine (Arg), triethanolamine (TEA), trimethylphenylammonium chloride (TMPAC), and cetyltrimethylammonium chloride (CTAC). The zero-shear viscosity for all systems studied was obtained using steady-state rheology at varying concentration and pH. The viscoelastic properties of systems containing an effective thickener (CTAC 1%) and an ineffective one (LiCl 1%) were obtained over a frequency range of 1–105 rad/s and subjected to the Zou and Larson analysis. The analysis showed that the high viscosity in the CTAC 1% system was associated with the combination of a high plateau modulus and a long relaxation time. The low viscosity in LiCl 1% system was attributable to the fast breakage time and high breakage rate constant. The latter result was consistent with findings from a previous study comparing the SLSar/CAHS system to a sodium lauryl ether sulfate/cocamidopropyl betaine (SLES/CAPB) formulation.

Effect of pH on the Structure and Dynamics of Wormlike Micelles in an Amino Acid-Derived Surfactant Composition.

We studied the impact of pH as a thickening mechanism on the structure and dynamics of wormlike micelles in a mixture of sodium lauroyl sarcosinate (SLSar) and cocamidopropyl hydroxysultaine (CAHS). The viscoelastic properties were obtained using mechanical rheometry and diffusing wave spectroscopy, which provided access to a wide range of frequencies. By using a mesoscopic simulation method [Zou; Larson. J. Rheol. 2014, 58 (3), 681-721], characteristic micelle lengths and times were extracted including contour length, persistence length, entanglement length, reptation time, breakage time, breakage rate, and breakage rate constant. The interplay of pH-dependent reptation times (10-1000 ms) and breakage times (4-38 ms) leads to a minimum in the ratio of reptation time to breakage time of about 0.02 at pH 4.8. This minimum was closely associated with the sharp increase and decrease of the observed viscosity maximum at pH 4.8 in this system. These values may be contrasted with much longer breakage times (20-300 ms) that have been measured in more easily thickened sulfate-based systems. The low breakage times of the SLSar/CAHS system were attributed to the high and pH-sensitive breakage rate constants (0.01-0.17 ms-1 μm-1).

Novel population balance model for granule aggregation and breakage in fluidized bed granulation and drying

A novel population balance model for the fluidized bed granulation and drying process was developed taking into consideration moisture content for determining the coalescence and breakage probabilities. Experimental data of pharmaceutical powders manufactured at production scale was used to determine the coalescence and breakage rate constant. It was found that the calculated mean volume diameter and frequency distribution showed good agreement with the experimental ones observed during the coalescence growth and the breakage stages.

Dynamic modeling and simulation of a SAG mill for mill charge characterization

The modeling and simulation of semiautogenous (SAG) mills have been widely used in the design and optimization of mill performance in terms of its power draw, processing capacity and product size distribution. However, these models are solved under steady approximation and do not provide any information on mill charge distribution in real time. This paper attempts to characterize mill contents by solving Whiten’s first-order content-based model in the MATLAB/Simulink environment. The parameters in the model, such as breakage rate constant, discharge rate and appearance function, were estimated using process and design data collected from a gold mine operating in Alaska coupled with a nonlinear parameter estimation scheme. This model was then used to predict the dynamic responses of other key operational variables, such as mill power, bearing pressure, charge level and product size distribution. The transient response of mill behavior with respect to changes in feed size distribution, tonnage and mill feed water is also presented. This dynamic simulation approach can be used for practicing different control strategy and training purposes. The observed response of mentioned variables was validated using dynamic response data from the plant to encompass operational characteristics of SAG mill behavior.

The effect of microwave on the crystallization process of magnesium carbonate from aqueous solutions

The effect of microwave on reactive crystallization is investigated with magnesium carbonate as the working substance. In the experiments, magnesium carbonate is precipitated by mixing aqueous magnesium sulfate and sodium carbonate solutions in a MSMPR crystallizer located in a microwave reactor. A population balance model along with crystallization kinetics is formulated to assess the influence of microwave energy input on nucleation, growth, agglomeration and breakage during crystallization process. A general high order moment method of classes (HMMC) framework is applied to solve the model numerically. The results show that crystallization kinetics parameters, including the nucleation rate constant (kn), the growth rate constant (kg), the exponent of growth rate (g) and the collision efficiency (Ψag) change significantly due to mass transfer enhancement in microwave field. Meanwhile, the exponent of nucleation (n) and the breakage rate constant (kbr) remain constant or only change slightly. It is further discovered that the crystal suspension attenuates the microwave effect during crystallization process.

Fragmentation of Bacteriogenic Iron Oxides in Response to Hydrodynamic Shear Stress

A viscous shear stress of 1.97 Nm2 was applied to suspensions of freshwater, brackish and marine bacteriogenic iron oxides (BIOS) in shake flasks. Samples were taken at intervals over 480 minutes for scanning electron microscopy. The remains of three Fe2+-oxidizing bacteria (FeOB) genera were conspicuous biomass constituents in the BIOS samples; these included the filamentous extracellular sheaths of Leptothrix, as well as helical fibrous stalks of Gallionella and Mariprofundus. The applied hydrodynamic shear stress resulted in an exponential decrease in Leptothrix sheath and Gallionella-Mariprofundus stalk lengths; first-order breakage rate constants derived from the experimental measurements ranged from 0.009 to 0.023 min−1 and 0.014 to 0.021 min−1, respectively. Breakage half-times extended from 30 to 70 min. Shear strength values calculated from the time course data ranged from 27.8 to 29.6 Nm−2 for Leptothrix sheaths, and from 28.0 to 28.8 Nm−2 for Gallionella-Mariprofundus stalks. Computational analyses showed that breakage half-times decreased rapidly with increasing shear stress, implying that the accumulation of BIOS is constrained hydrodynamically to quiescent aqueous environments. These results imply that there is considerable potential for natural hydrodynamic fluctuations in basal shear stress levels to bring about resuspension and advective dispersal of BIOS. Immediate consequences anticipated from such events include enhanced particulate transport of BIOS-associated chemical species through aquatic systems, as well as reduced preservation potential and underrepresentation of BIOS bacteria structures as microfossils in the geological record.

Insight into first-order breakage kinetics using a particle-scale breakage rate constant

This study formulates a particle-scale breakage rate constant of the linear time-variant population balance model (PBM) for batch dry-milling. The breakage rate constant separates material properties from the milling environment which is defined by the impact energy distribution obtained by the discrete element method (DEM). The breakage rate constant determined by experiments and DEM simulations of the ball milling of polydispersed silica glass were in close agreement confirming the validity of the proposed methods. The breakage rate constant formulated in this study also allowed a detailed and rigorous energy-based analysis of the milling environment which provided insight into the origin of first-order breakage kinetics and the time-invariance of the breakage rate constant. Feeds of varying polydispersity and the time-wise evolution of the particle size distribution (PSD) were simulated in DEM to show that the PSD does not affect the specific breakage rate constant as assumed in first-order breakage. The adherence to first-order breakage kinetics was attributed to the low feed loading in the mill and the absence of particle beds where mechanical multi-particle interactions can cause non-first-order breakage. It was found that a threshold impact energy must be accounted for in the analysis of DEM results because the majority of low energy impacts do not contribute to particle breakage. Without rigorous assessment of the impact energy distribution, DEM simulations may lead to an erroneous evaluation of milling performance. The findings of this study contribute to a unified DEM−PBM framework to model and analyze milling process and demonstrate that milling performance can be quantified from particle-scale interactions using DEM.

Tryptophan Stabilizes His–Heme Loops in the Denatured State Only When It Is Near a Loop End

We use a host-guest approach to evaluate the effect of Trp guest residues relative to Ala on the kinetics and thermodynamics of formation of His-heme loops in the denatured state of iso-1-cytochrome c at 1.5, 3.0, and 6.0 M guanidine hydrochloride (GdnHCl). Trp guest residues are inserted into an alanine-rich segment placed after a unique His near the N-terminus of iso-1-cytochrome c. Trp guest residues are either 4 or 10 residues from the His end of the 28-residue loop. We find the guest Trp stabilizes the His-heme loop at all GdnHCl concentrations when it is the 4th, but not the 10th, residue from the His end of the loop. Thus, residues near loop ends are most important in developing topological constraints in the denatured state that affect protein folding. In 1.5 M GdnHCl, the loop stabilization is ~0.7 kcal/mol, providing a thermodynamic rationale for the observation that Trp often mediates residual structure in the denatured state. Measurement of loop breakage rate constants, k(b,His), indicates that loop stabilization by the Trp guest residues occurs completely after the transition state for loop formation in 6.0 M GdnHCl. Under poorer solvent conditions, approximately half of the stabilization of the loop develops in the transition state, consistent with contacts in the denatured state being energetically downhill and providing evidence for funneling even near the rim of the folding funnel.

The inverse problem in granulation modeling—Two different statistical approaches

AbstractThis article is concerned with parameter estimation for a multidimensional population balance model for granulation. Experimental results were obtained by running a laboratory mixer with sodium carbonate and aqueous polyethylene glycol solutions. Subsequently, a prescan of suitable parameter combinations utilising the experimental results is performed, and a local surrogate model constructed around the best combination. For the actual estimation of the parameters and their uncertainties two different approaches are applied—a projection method and a Bayesian approach. It is found that the model predictions with the parameters obtained through both methods are similar. Furthermore, the uncertainties in the model predictions increase as the experimental uncertainties are increased. Studies of the marginal densities of two‐parameter combinations obtained through the Bayesian approach show a correlation between the collision and breakage rate constant, giving potential hints for further model development. Furthermore, a bimodal distribution of the compaction rate constant is observed. © 2011 American Institute of Chemical Engineers AIChE J, 2011

On equilibrium solutions of aggregation–fragmentation problems

A characteristic feature of particulate systems that evolve due to competition between aggregation and breakage is that they sometimes produce non-trivial steady-state particle size distributions. If such solutions satisfy detailed balance conditions, then they are equilibrium solutions. The conditions that must be satisfied by aggregation and fragmentation rate kernels in order for equilibrium solutions to be produced are elaborated, and it is shown that the rate kernels are uniquely determined by the aggregation and breakage rate constants for the reactions involving monomers. Consequently, for equilibrium systems there is a significant reduction in the amount of information needed in order to infer the general form for aggregation or breakage kernels, and we explore implications for constructing rate kernels by using atomistic simulations such as molecular dynamics.

Emulsification in turbulent flow: 2. Breakage rate constants

Systematic experimental study of the effects of several factors on the breakage rate constant, k BR , during emulsification in turbulent flow is performed. These factors are the drop size, interfacial tension, viscosity of the oil phase, and rate of energy dissipation in the flow. As starting oil–water premixes we use emulsions containing monodisperse oil drops, which have been generated by the method of membrane emulsification. By passing these premixes through a narrow-gap homogenizer, working in turbulent regime of emulsification, we study the evolution of the number concentration of the drops with given diameter, as a function of the emulsification time. The experimental data are analyzed by a kinetic scheme, which takes into account the generation of drops of a given size (as a result of breakage of larger drops) and their disappearance (as a result of their own breakage process). The experimental results for k BR are compared with theoretical expressions from the literature and their modifications. The results for all systems could be described reasonably well by an explicit expression, which is a product of: (a) the frequency of collisions between drops and turbulent eddies of similar size, and (b) the efficiency of drop breakage, which depends on the energy required for drop deformation. The drop deformation energy contains two contributions, originating from the drop surface extension and from the viscous dissipation inside the breaking drop. In the related subsequent paper, the size distribution of the daughter drops formed in the process of drop breakage is analyzed for the same experimental systems.

Characterization of the nanoscale properties of individual amyloid fibrils.

We report the detailed mechanical characterization of individual amyloid fibrils by atomic force microscopy and spectroscopy. These self-assembling materials, formed here from the protein insulin, were shown to have a strength of 0.6 +/- 0.4 GPa, comparable to that of steel (0.6-1.8 GPa), and a mechanical stiffness, as measured by Young's modulus, of 3.3 +/- 0.4 GPa, comparable to that of silk (1-10 GPa). The values of these parameters reveal that the fibrils possess properties that make these structures highly attractive for future technological applications. In addition, analysis of the solution-state growth kinetics indicated a breakage rate constant of 1.7 +/- 1.3 x 10(-8) s(-1), which reveals that a fibril 10 mum in length breaks spontaneously on average every 47 min, suggesting that internal fracturing is likely to be of fundamental importance in the proliferation of amyloid fibrils and therefore for understanding the progression of their associated pathogenic disorders.

Determination of breakage rates of oil droplets in a continuous oscillatory baffled tube

Following on our previous studies, the population balance model that was built on the earlier work from Jareš and Procházka [Break-up of droplets in Karr reciprocating plate extraction column. Chemical Engineering Science 42, 283–292] was modified to include the viscoelastic effect on droplet size distribution and to evaluate the breakage rates of oil-in-water dispersions in a continuous oscillatory baffled tube. In this work, experiments were performed showing that the breakage of droplets is the dominant mechanism in the system, and the physical properties of different oils had no significant influence on droplet size distributions. Under those conditions the model can be used to focus only on breakage rate constants, keeping the number of fitted parameters in the modelling process to a minimum. The droplet breakage results from this work suggest that the oscillation amplitude has more influence on the breakage rates than the oscillation frequency. This work is a major extension and includes droplet data from our previous studies so that the breakage rates can be compared; and the consistency of the rate constants is examined.

Effect of Medium Viscosity on Breakage Parameters of Quartz in a Laboratory Ball-Mill

The role and effect of liquid medium viscosity on breakage parameters (S and B values) of the batch grinding kinetic model was investigated by grinding a single feed fraction of quartz (20 × 30 mesh) in a batch ball-mill at fixed mill loadings and at 45 vol % solid concentration, using glycerol−water mixtures of different proportions. The primary breakage distributions (B values) were normalized and separated into three different groups for grinding in media viscosities of 0.001(water), 0.002−0.03, and 0.08−0.16 Pa·s, respectively, with higher viscosity range giving proportionally finer products. The disappearance grinding kinetics of the top size showed that the specific breakage rates (S values) depended both on the viscosity of the liquid medium and on the extent of grinding. The specific breakage rates of the top size increased with the fineness of grinding while the breakage of the smaller sizes proceeded approximately in a first-order manner as characterized by a characteristic breakage rate constant. The acceleration of the top-size breakage rates as grinding proceeded was represented by a parameter called the acceleration factor, which correlated reasonably well with the changing slurry effective viscosity. Alternatively, the acceleration factor can be presented in the form of a single generalized first-order plot for grinding at different medium viscosities. The first-order breakage rate constant decreased with increasing medium viscosity over the range from 0.001 to 0.16 Pa·s but not linearly. The proportional variation of net mill power did not match exactly with that of the breakage rate constant on increasing medium viscosity. On the basis of the analysis of grinding results, three grinding regimes were identified relative to the change in medium viscosity but their existence was not confirmed experimentally.

Modeling particle‐size distribution dynamics in a flocculation system

AbstractFlocculation dynamics accounting for both particle coagulation and aggregate breakage was simulated mathematically by using modified sectional modeling techniques. The methodological improvement included the use of a continuous‐size density function, instead of a characteristic size for each size section, the applications of a comprehensive curvilinear model for the coagulation kinetics, and the fractal scaling relationship for particle aggregates. Simulation results demonstrated that a flocculation system could arrive at a dynamic steady state after a period of flocculation when coagulation and breakage counterbalanced each other, resulting in a stationary size distribution with a unique peak mass concentration. Three distinct breakage distribution functions—binary, ternary, and normal distribution—did not differ considerably based on the simulation results of the steady‐state size distributions. A lower shear rate, breakage rate constant, a higher collision efficiency, and initial particle concentration would result in larger aggregates in a flocculation system. The numerical simulations compared well with the results of the jar‐test flocculation experiments using latex microspheres, suggesting the applicability of the curvilinear–fractal–breakage modeling system for the process simulation of the flocculation units used in water and wastewater treatment.

Population balance modelling of droplets in an oscillatory baffled reactor—using direct measurements of breakage rate constants

AbstractPopulation balance modelling (PBM) is a useful tool for design and prediction in a range of processes that involve dispersed phases, particulates or micro‐organisms. In the Inverse Problem approach, constants related to the rates of evolution and/or interaction of the individual components are optimized so as to match the experimentally observed distribution curves. However, the significance of these results may be slight, or several solutions may be possible. In this paper, a method is presented to circumvent this problem, and is applied to the breakage and coalescence of oil droplets in a continuous oscillatory baffled reactor (OBR). Direct observation of droplet breakage using a high‐speed camera should enable breakage rate constants to be obtained independently from the Inverse Problem approach, and thus obtain more reliable coalescence rate constants. An analysis of the droplet size distributions (DSDs) is also combined with direct observation with the high‐speed camera to allow the development of a breakage model specific to the OBR. It is expected that this method should yield parameters useful for prediction of steady‐state DSDs, and hence enhance the accuracy of design, scale‐up, and prediction of operation.© 2003 Society of Chemical Industry

On the evaluation of droplet breakage and coalescence rates in an oscillatory baffled reactor

A numerical model is developed to evaluate droplet breakage and coalescence rates in a continuous oscillatory baffled reactor (OBR). The structure of the model is similar to that of the population balance approach, but concentrates on droplet interactions with simpler assumptions of a functional form for breakage and coalescence rate constants. The OBR is a relatively new reactor technology and offers enhanced and more uniform mixing than traditional reactors, making some model assumptions more closer to the reality. In this paper, we present the development, robustness and validation of the modelling process together with the predicted results for the OBR system.

How Tubulin Subunits Are Lost from the Shortening Ends of Microtubules

Microtubules exhibit dynamic instability, switching between persistent states of growth and shortening at their ends. The switch between growth and shortening has been proposed to depend on end conformation where growing ends have “straight” tubulin protofilaments stabilized by a terminal cap of GTP-tubulin, while shortening ends have lost their GTP-tubulin cap, allowing terminal GDP-tubulin dimers to curve inside-out and peel rapidly away from the microtubule lattice. This “conformational cap” model predicts that tubulin dissociation from shortening ends is a two-step process where the average lengths of curved GDP-tubulin protofilaments at a depolymerizing end will depend on the ratio of the rate of peeling to the rate of breakage of the longitudinal bonds between adjacent curved dimers. We have tested this model for the plus and minus ends of microtubules assembled with pure porcine tubulin off the ends of axoneme fragments in standard assembly buffer. Individual microtubule ends were imaged using video-enhanced differential interference contrast light microscopy. The rate of rapid shortening was systematically increased by isothermal dilution into assembly buffer containing various concentrations of Mg2+or Ca2+ions. At 1 mMMg2+and no Ca2+, shortening occurred at 20 (plus) and 45 (minus) μm/min. The ends appeared similar in contrast to growing ends and the core of the microtubule and the ends appeared blunt or slightly frayed by negative stain electron microscopy. Above 20 mMMg2+or above 5 mMCa2+, microtubule shortening occurred at 60 (plus) and 115 (minus) μm/min or faster and “knobs” were distinctly visible at depolymerizing ends, particularly at the faster minus ends, and knob contrast remained constant during many micrometers of rapid shortening. Negative stain electron microscopy revealed that these knobs were “blossoms” of inside-out curved protofilaments, some extending for several helical turns (30 to 60 dimers in length) at constant curvature from the ends. At these high shortening velocities, the peeling of curved protofilaments was confined to within several dimers of the end of the microtubule cylinder, suggesting that dimer curling and protofilament peeling is constrained to the tip by interactions between adjacent straight protofilaments. Depolymerization is produced by conformational changes in GDP-tubulin since microtubules assembled with a slowly hydrolizable analog of GTP, GMPCPP, are stable even at 20 mMMg2+or 5 mMCa2+. Monte Carlo simulations show that the ratio of the peeling to breakage rate constants can control the steady-state average length of curved GDP-tubulin protofilaments at the depolymerizing end.

Bubble breakage in pipeline flow

The rate of bubble breakage in turbulent liquid flow was examined using a population balance containing a bubble breakage model to analyze experimental bubble breakage rate data. The bubble breakage model was based on high-speed motion photography observations of the breakage process in turbulent liquid pipeline flow. The bubble breakage model predicts the number of bubbles formed from a breakage, the size of the bubbles formed and the rate of bubble breakage. Bubble breakage was determined to be binary; bubble breakage size was described by a breakage size function in which unequal bubble sizes had a higher probability of being formed compared to equal bubble sizes; and the breakage rate was assumed to be first order with respect to the number of bubbles of a given size. The value of the breakage rate constant was found to be approximately equal to the frequency of the second mode of oscillation of the maximum stable bubble size in a given turbulent flow.