Breakage Kinetics Research Articles

Measuring Energetics of Sharp DNA Bending from Breakage Kinetics of Small DNA Loops

Double-stranded DNA (dsDNA) can undergo diverse modes of conformational changes inside a cell. Among them, dsDNA bending is closely associated with genome packaging and gene regulation, and therefore a consistent polymer model of dsDNA bending is essential for quantitative description of genome-related processes. The behavior of dsDNA at large length scales can be well described as a worm-like chain whose bending energy depends quadratically on curvature. However, many studies on dsDNA bending at short length scales have produced results contradictory to the worm-like chain model. Here we show that the free energy of a dsDNA loop as short as 60 bp in contour length can be well described by the wormlike chain model. We measured the breakage rates of small dsDNA loop stabilized by sticky ends using Fluorescence Resonance Energy Transfer (FRET). We found that the breakage rate vs. loop length relationship to be in good agreement with the worm-like chain model, but not with an alternative model which predicts superflexible behavior at short length scales. Furthermore, we found that energetics of dsDNA begins to deviate from the worm-like chain model below 60 bp. Our result pushes the lower limit of the worm-like chain regime of dsDNA down to 60 bp, and suggests that dsDNA undergoes structural transitions below this length.

Impact of process parameters on the breakage kinetics of poorly water-soluble drugs during wet stirred media milling: A microhydrodynamic view

Wet stirred media milling has proven to be a robust process for producing nanoparticle suspensions of poorly water-soluble drugs. As the process is expensive and energy-intensive, it is important to study the breakage kinetics, which determines the cycle time and production rate for a desired fineness. Although the impact of process parameters on the properties of final product suspensions has been investigated, scant information is available regarding their impact on the breakage kinetics. Here, we elucidate the impact of stirrer speed, bead concentration, and drug loading on the breakage kinetics via a microhydrodynamic model for the bead–bead collisions. Suspensions of griseofulvin, a model poorly water-soluble drug, were prepared in the presence of two stabilizers: hydroxypropyl cellulose and sodium dodecyl sulfate. Laser diffraction, scanning electron microscopy, and rheometry were used to characterize them. Various microhydrodynamic parameters including a newly defined milling intensity factor was calculated. An increase in either the stirrer speed or the bead concentration led to an increase in the specific energy and the milling intensity factor, consequently faster breakage. On the other hand, an increase in the drug loading led to a decrease in these parameters and consequently slower breakage. While all microhydrodynamic parameters provided significant physical insight, only the milling intensity factor was capable of explaining the influence of all parameters directly through its strong correlation with the process time constant. Besides guiding process optimization, the analysis rationalizes the preparation of a single high drug-loaded batch (20% or higher) instead of multiple dilute batches.

Aggregation and breakage kinetics of fresh cement paste

The influence of shear-induced forces on the microstructure of fresh cement pastes was studied. Aggregation and breakage kinetics of the paste matrix are highly influenced by the shear history. It was found that the kinetics of re-aggregation is relatively slow, and time scale for recovery is longer than the time needed for breakdown. When the aggregation kinetics dominates, network interactions among particles develop and the average floc size increases. When the breakage kinetics dominates, network interactions among particles are broken and are accompanied by a decrease in the average floc size. The results suggest that there is a limiting size to floc growth. Minor additions of clays can significantly impact the structural network development and result in a more flocculated structure. The flocs produced by the clays were highly stable flocs with strong interparticle bonds that were able to oppose floc breakage and floc erosion.

Influence of non-linear breakage kinetics on the attainment of self-similarity for dry milling processes

The attainment of a self-similar particle size distribution (PSD) is an interesting feature of many particulate processes such as milling. While it is well-known that the linear time-invariant population balance model (LTINV-PBM) admits a self-similar solution, the attainment of self-similarity is not well-understood within the context of non-linear breakage kinetics. Non-linear breakage kinetics emanating from mechanical multi-particle interactions is not uncommon in dense-phase dry milling processes and has been accounted for by the non-linear PBM (NL-PBM). Here, we investigate the dynamics and attainment of a self-similar PSD considering first-order (LTINV-PBM), non-uniform non-linear (NL-PBM Model B), and uniform non-linear (NL-PBM Model D) breakage kinetics. The PBMs were first fit to experimental data from the batch dry milling of quartz in a tumbling ball mill. Using the parameters estimated and simulating the impact of different extents of non-linear multi-particle interactions, we elucidated the impact of non-linear breakage kinetics on the evolution of the PSD to a self-similar one. In order to quantitatively assess self-similarity and thoroughly investigate the approach to a self-similar PSD during milling, we have introduced the use of the distance metric Canberra Distance. The simulations suggest that the retardation of the breakage of coarser particles due to their interactions with the finer particles can significantly delay the attainment of self-similarity and broaden the self-similar PSD attained. Both non-linear models predict attainment of self-similar PSDs, albeit slightly different, under cases of mild non-linear effects. However, when non-linear effects are severe, self-similarity can be delayed to prolonged milling in contrast to LTINV-PBM, which predicts the attainment of self-similarity shortly after the onset of milling. While LTINV-PBM and NL-PBMs can be easily discriminated via a single milling experiment, simulations suggest that only experiments utilizing various feed particle size distributions, preferentially including a bi-modal distribution, could discriminate between the non-uniform non-linear kinetics (Model B) and uniform non-linear kinetics (Model D).

Modelling the Activated Sludge Flocculation Process Using Population Balance Model (PBM)

This paper describes the application of population balance models to activated sludge flocculation process. It presents the development and selection of appropriate expressions for aggregation and breakage kinetics within the population balance framework to describe the evolution of mean size and steady state distribution of flocs under shear conditions. A size and velocity gradient dependent collision efficiency is introduced into the aggregation expression. In the model, only 2 parameters need to be estimated: collision efficiency coefficient and the breakage frequency coefficient. They are obtained by the “best fit” with the experimental data, and keep unchanged under different shear condition for the same flocs. The modelling results indicate that the population balance models coupled with suitable aggregation and breakage kinetics is appropriate for describing activated sludge flocculation dynamics.

Breakage Distribution Estimation of Bauxite Based on Piecewise Linearized Breakage Rate

Laboratory tests were carried out to study the breakage kinetics of diasporic bauxite and determine its breakage distribution function. Non-first order breakage with different deceleration rates for different size intervals is found, which is most probably caused by the heterogeneity of the ore. Piecewise linearization method is proposed to describe the non-first order breakage according to its characteristics. In the method, grinding time is divided into several intervals and breakage is assumed to be first order in each interval. So, the breakage rates are calculated by taking the product of the last interval as feed and then established as a function of particle size and grinding time. Based on the predetermined breakage rate function, the breakage distribution of the ore is back-calculated from the experimental data using the population balance model (PBM). Finally, the obtained breakage parameters are validated and the simulated data are in good agreement with the experimental data. The obtained breakage distribution and the method for breakage rate description are both significant for modeling the full scale ball milling process of bauxite.

A combined microhydrodynamics–polymer adsorption analysis for elucidation of the roles of stabilizers in wet stirred media milling

Although polymers and surfactants are commonly used as stabilizers to impart physical stability to the suspensions produced by wet stirred media milling of poorly water-soluble drugs, scant information is available in pharmaceutical literature regarding their impact on the breakage kinetics. We present a combined microhydrodynamics-polymer adsorption analysis to elucidate the roles of stabilizers with a focus on the kinetics. Griseofulvin (GF), a model poorly water-soluble drug, was milled at various concentrations of hydroxypropyl cellulose (HPC) in the presence-absence of sodium dodecyl sulfate (SDS). Particle sizing, scanning electron microscopy, thermal analysis, and rheometry were used to determine the breakage kinetics, adsorption isotherm, and apparent viscosity, which were then used to analyze the aggregation state of the milled suspensions and the microhydrodynamics. In the absence of SDS, an increase in HPC concentration slowed the particle aggregation leading to faster apparent breakage. On the other hand, due to a synergistic stabilizing action of HPC with SDS, lower HPC concentration was needed to stabilize the suspensions, and an optimum HPC concentration for the fastest apparent breakage was identified. The microhydrodynamic analysis quantified, for the first time, the viscous dampening effect of polymers, while only the combined analysis could explain the observed optimum.

A rational function approximation to the effectiveness factor for multi-particle interactions in dense-phase dry milling

Abstract Non-linear population balance models (PBMs) have recently been shown to be superior to their linear counterparts for quantifying the breakage kinetics in dense-phase dry milling processes, where mechanical multi-particle interactions are significant. At the particle ensemble scale, particles of different sizes interact causing a population dependence of the specific breakage rate function. Non-linear PBMs explicitly account for these multi-particle interactions through a population-dependent functional called the effectiveness factor. Due to the inherent difficulty and complexity of proposing and selecting an appropriate form of the effectiveness factor, a general and systematic approach is desired to assess breakage kinetics in an unbiased manner. In this study, toward addressing the need for a general form of the effectiveness factor, we propose a rational function approximation. First, computer generated effectiveness factor was approximated by a rational function. Then, evolution of the particle size distribution during batch dry milling of quartz in a tumbling ball mill was fitted with various forms of the effectiveness factor in the context of linear and non-linear PBMs. Goodness-of-fit and statistical significance of the parameters estimated were evaluated to discriminate various forms of the effectiveness factor. The statistical analysis suggests that a rational function may replace specific forms of the effectiveness factor for a less restrictive analysis of the multi-particle interactions. Additionally, fitting a non-linear PBM using the rational function approximation allowed us to describe the dense-phase dry milling data accurately. Accordingly, this study enables experimenters to quantify the impact of the multi-particle interactions in a robust, systematic, and unbiased manner via the rational function approximation to the effectiveness factor within the context of the non-linear PBM.

Friction contribution to water-bond breakage kinetics

Based on the trajectories of the separation between water molecule pairs from MD simulations, we investigate the bond breakage dynamics in bulk water. From the spectrum of mean first-passage times, the Fokker-Planck equation allows us to derive the diffusivity profile along the separation coordinate and thus to unambiguously disentangle the effects of free-energy and local friction on the separation kinetics. For tightly coordinated water, the friction is six times higher than in bulk, which can be interpreted in terms of a dominant reaction path that involves additional orthogonal coordinates.

Comparison with Some Porous Materials and the Effects of Powder Filling on Breakage Parameters of Diatomite in Dry Ball Milling

In this study, a comparison of breakage parameters with some porous materials (pumice, trass, and amorphous silica) and the effects on breakage kinetics of powder filling of diatomite were investigated on at batch grinding conditions. For this purpose, first, standard Bond grindability tests were performed for four porous samples. Second, eight different mono-size fractions of all samples were carried out between 1.7 mm and 0.106 mm formed by a sieve series. Then, Si and Bi,j equations were determined for the size distributions at different grinding times, and the model parameters (Si, aT, α, γ, φj, and β) were tested for four different porous materials. Finally, model parameters were discussed for four different powder filling (3.5, 7, 10.5, and 14%) of diatomite samples. From the results of test, the validity of the relationships between Bond grindability and breakage parameters has been not confirmed with good correlation coefficients. The reason of this negative result could be attributed to different of the geological origin of porous materials. In addition, the results of the effect of powder filling on the grinding were found different than other investigators.

The effects of ball filling and ball diameter on kinetic breakage parameters of barite powder

The absolute fineness of the ball diameter and the ball charge grading are important factors for the optimal operation of a ball mill. Therefore, the effects on breakage kinetics of the ball diameter and the fractional ball filling were investigated on the barite powder at batch grinding conditions based on a kinetic model. For this purpose, firstly, six different mono-size fractions were carried out between 0.850 and 0.106mm formed by a 2 sieve series. Then, Si and Bi,j equations were determined from the size distributions at different grinding times, and the model parameters were compared for three different ball filling and three different ball diameter. The results of tests, the effect of ball filling and ball diameter on the grinding were found to be different than other investigators regarding some results.

A Model of Batch Grinding of Talc Powders

Grinding of talc powders has been studied both theoretically and experimentally. The specific rates of breakage of talc powders were measured based on the first-order breakage kinetics model and the cumulative breakage distribution parameters of talc powders were measured from primary breakage products. Based on the measurement results, the specific rate of breakage and cumulative breakage distribution functions were correlated with particle size asand , repectively. A differential-integral equation was thus build to describe grinding as a rate process and was integrated numerically. Comparisons on size distribution showed that the specific rate of breakage of talc powders increased with grinding time at an increase rateabout 0.0066min-2.

Simulation of the grinding of coarse/fine (heterogeneous) systems in a ball mill

Comminution studies have been carried out by grinding narrowly sized fractions of single mineral feeds. Under these conditions, the mill environment remains self-similar and invariant which is a characteristic of the mill/material system. On the other hand, in industry, feed to the mill comprises oversize recycle and new feed that is highly heterogeneous and widely distributed in size. The objective of the present work is to compare the breakage kinetics and energetics of grinding coarse size feed in the presence of deliberately added fines for different material systems. Quartz, dolomite, and limestone mineral systems were selected as feed in order to study these phenomena for minerals whose hardness ranges from hard to soft, approximately from 7 to 3 on Moh's Scale of hardness, at different coarse/fine ratios. It was found that the cumulative breakage distribution functions for this situation do not change with mixture compositions for the three materials. On the other hand, the initial breakage rate function of the coarse particles increases with increasing proportion of fines in the mixture. It was also shown that the fines produced, and the coarse material retained on the top sieve were normalizable with respect to the specific energy consumed by the coarse fractions. A modified breakage rate function was used to simulate the grinding operation of the three material systems using the linear population balance model. As a result, reasonable agreement between the experimental data and the corresponding calculated values was achieved.

Investigation of the breakage parameters of gypsum as dependent on interstitial filling in a laboratory ball mill

Gypsum is an industrial material used in many applications such as building and chemical industry, fertilizer manufacture, medicine and dentistry. In these industries, there is need for finely ground gypsum. Energy consumption is very high in grinding processes. Therefore, the effects on breakage kinetics of powder filling and ball filling were investigated on gypsum samples taken from the Denizli-Honaz region (Turkey), for batch grinding conditions based on a kinetic model. For this purpose, firstly, eight different mono-size fractions between 1.7 and 0.106 mm formed by a \({\surd 2}\) sieve series were obtained. Then, Si and Bi,j equations were determined from the size distributions at different grinding times, and the model parameters (Si, aT, α, γ and \({\phi_{j}}\)) were compared for four different proportions of powder filling (5, 7.5, 10 and 15%), and three different proportions of ball filling (25, 35 and 45%). Finally, model parameters are discussed for each test. From the result of tests, obtained of the effect of ball filling and powder filling on the grinding, it was found our results differed from those of other investigators.

An Investigation of Particle Size Variation in Stirred Mills in Terms of Breakage Kinetics

The structure and particle size of solid fossil fuels have effects on linear or nonlinear movements encountered in breakage rate, besides petrography types, in-organic materials, extensive network of pores, macro chemical forms, etc. Nonlinear breakage behavior in mono size groups of −3,350 + 2,360 μm, −2,360 + 1,700 μm, and −1,180 + 850 μm was a result of heterogeneous structure of the material along with being coarser for grinding medium. The part having low-ash content (coal-like) in the heterogeneous structure was exposed to faster breakage than the part having high-ash content (shale-like). As a result, over-sieve material was composed of shale, which is more difficult to grind, breakage behavior slowed down, and there occurred a deviation in linear breakage behavior with increasing grinding period. Linear breakage behavior in mono size groups of −425 + 300 μm and −212 + 150 μm was considered as a result of homogeneity of the material along with its adequate fineness for grinding medium. Variation of particle size in grinding is an important parameter in terms of breakage behavior. Maximum breakage rate was reached for −425 + 300 μm size group in the stirred mill. The ratio of “grinding medium size/size of the material ground” (6.0/0.425) was calculated as approximately 14. A decrease in this ratio, i.e., an increase in particle size of material, caused engagement difficulties of particles by balls. An increase in the ratio, however, resulted in a pillowing effect of fine materials in the medium and, thus, a decrease in breakage rate.

Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling

Non-linear population balance models (PBMs), which have been recently introduced due to the limitations of the classical linear time-invariant (LTINV) model, account for multi-particle interactions and thus are capable of predicting many types of complex non-first order breakage kinetics during size reduction operations. No attempt has been made in the literature to estimate the non-linear model parameters by fitting the model to experimental data and to discriminate various models based on statistical analysis. In this study, a fully numerical back-calculation method was developed in the Matlab environment to determine the model parameters of the non-linear PBM. Not only does the back-calculation method identify the parameters of complicated non-linear PBMs, but also it gives the goodness of fit and certainty of the parameters. The performance of the back-calculation method was first assessed on computer-generated batch milling data with and without random error. The back-calculation method was then applied to experimental batch milling data exhibiting non-first order effects using both the LTINV model and two separate non-linear models. The back-calculation method was able to correctly determine the model parameters of relatively small sets of batch milling data with random errors. Applied to experimental batch milling data, the back-calculation method with a two-parameter non-linear model yielded parameters with reasonable certainty and accurately predicted the slowing-down phenomenon during dry batch milling. This study encourages experimenters to use advanced non-linear population balance models along with the back-calculation method toward estimating the breakage rate and distribution parameters from dense batch milling data sets.

Influence of process parameters on breakage kinetics and grinding limit at the nanoscale

In long-term milling experiments, in a stirred media mill, a grinding limit where no further particle breakage occurs was identified. During mechanical stressing of the particles, defects are generated in the crystalline lattice, which allows real fracture of nanoparticles. Below a critical size, defects cannot be stored or generated in the crystallites and the overall limit of grinding is reached. This limit is strongly influenced by material properties and hardly affected by most of the process conditions. However, the breakage kinetics strongly depend on the process parameters and suspension conditions as long as the grinding limit is not reached. Based on these findings, two mechanisms of nanoparticle breakage are proposed. Proper choice of process parameters saves not only up to 90% of the energy input to reach the grinding limit but also leads to a higher product quality in terms of crystallinity and less milling bead wear. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Effect of particulate environment on the kinetics and energetics of dry ball milling

Most laboratory-scale comminution studies have been carried out by grinding narrowly sized fractions of single mineral feeds. In such experiments, the mill environment remains self-similar and invariant which is a characteristic of the mill/material system. On the other hand, in industrial practice, feed to the mill comprises oversize recycle and new feed that is highly heterogeneous and distributed in size. The objective of this investigation is to study the breakage kinetics and energetics of grinding coarse size feed in the presence of deliberately added fines. Quartz and limestone mineral systems were selected as feed in order to study these phenomena for minerals whose hardness ranges from hard to soft at different coarse/fine ratios. It was found that the cumulative breakage distribution function for this situation does not change with mixture composition. On the other hand, the initial breakage rate function of the coarse particles increases with increasing proportion of fines in the mixture. This acceleration could be due to the smaller collision cross-section of the fines. In addition, the coarse particles appear to be preferentially present at the toe of the mill where most of the grinding takes place.

Influence of interstitial filling on breakage kinetics of gypsum in ball mill

Abstract Gypsum is an industrial material used in many applications such as building and chemical industry, fertilizer manufacture, medicine and dentistry. In these industries, there is need for finely ground gypsum. Energy consumption is very high in grinding processes. Therefore, the effects on breakage kinetics of powder filling and ball filling were investigated on gypsum samples taken from the Denizli-Honaz region (Turkey), for batch grinding conditions based on a kinetic model. For this purpose, firstly, eight different mono-size fractions between 1.7 mm and 0.106 mm formed by a √2 sieve series were obtained. Then, S i and B i , j equations were determined from the size distributions at different grinding times, and the model parameters ( S i , a T , α , γ and ϕ j ) were compared for five different proportions of powder filling (5%, 7.5%, 10%, 15% and 20%), and three different proportions of ball filling (25%, 35% and 45%). Finally, model parameters are discussed for each test. From the result of tests, obtained of the effect of ball filling and powder filling on the grinding, it was found our results differed from those of other investigators.

The application of a simplified approach to modelling tumbling mills, stirred media mills and HPGR’s

Most modern population balance models for comminution invoke the concept of a specific breakage rate function and a breakage distribution function to describe breakage kinetics. One of the difficulties of this approach is that these functions are very difficult to measure directly. Consequently, it is usual to assume that these functions can be represented by simple equations with parameters that can easily be estimated from test data using back-calculation techniques. However, these estimates can be very sensitive to small measurement errors and are usually subject to very large variances. This paper presents a simplified approach to modelling comminution processes that invoke the concept of an energy-based cumulative breakage rate function to describe breakage kinetics. This function can be estimated directly from plant data and is well-suited to multi-component modelling of individual rock types and mineral species. Examples of the application of this simplified modelling approach are described for the treatment of platinum ores using ball mills, AG/SAG mills, HPGR’s and stirred media mills.