On the self-similar solution of fragmentation equation: Numerical evaluation with implications for the inverse problem

It is well known that the fragmentation equation admits self-similar solutions for evolving particle-size distributions (PSD); i.e., if the shape of PSD is independent of time after an initial transient period. Although an analytical derivations of the self-similar PSD cases have been studied extensively, results for cases requiring numerical solutions are rare. The aim of the present work is to fill this gap for the case of homogeneous breakage functions. The known analytical and approximate solutions for the self-similar PSD are reviewed and a general algorithm for the numerical solution is proposed. Results for a broad range of breakage functions (kernel and rate) are presented. Further, the work is focused on the sensitivity of the relation between self-similar PSD and breakage kernel and its influence on the inverse breakage problem, i.e., that of estimating the breakage kernel from experimental self-similar PSDs. Useful suggestions are made for tackling the inverse problem.     

The equal-size binary breakage problem: evolution toward a steady shape or periodic behavior?

During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.     

Numerical and similarity solutions for reversible population balance equations with size-dependent rates

Population balance equations (PBEs) for reversible aggregation-fragmentation processes are important to particle agglomeration and dissolution, polymerization and degradation, liquid droplet coalescence and breakup, and floc coagulation and disintegration. Moment solutions provide convenient solutions to the PBEs, including steady state and similarity solutions, but may not be feasible for complex forms of size-dependent rate coefficients and stoichiometric kernels. Numeric solutions are thus necessary not only for applications, but also for the study of the mathematics of PBEs. Here we propose a numerical method to solve PBEs and compare the results to moment solutions. The numeric results are consistent with known steady state and asymptotic long-time similarity solutions and show how processes can be approximated by self-similar formulations.     

Spreading of non-Newtonian liquids over solid substrates

The spreading of drops of a non-Newtonian liquid (Ostwald-de Waele liquid) over horizontal solid substrates is theoretically investigated in the case of complete wetting and small dynamic contact angles. Both gravitational and capillary regimes of spreading are considered. The evolution equation deduced for the shape of the spreading drops has self-similar solutions, which allows obtaining spreading laws for both gravitational and capillary regimes of spreading. In the gravitational regime case of spreading the profile of the spreading drop is provided.     

Comments on "a study of the collisional fragmentation problem using the gamma distribution approximation"

M. Kostoglou and A.J. Karabelas [J. Colloid Interface Sci. 303 (2006) 419-429] proposed using a gamma distribution approximation to study a collisional fragmentation problem. This approximation involved two types of integrals and the use of continued fraction expansions for their computation. In this Comment, explicit expressions are derived for computing the integrals.     

Role of the site of protonation in the low-energy decompositions of gas-phase peptide ions

The dissociation of singly or multiply protonated peptide ions by using low-energy collisional activation (CA) is highly dependent on the sites of protonation. The presence of strongly basic amino acid residues in the peptide primary structure dictates the sites of protonation, which generates a precursor ion population that is largely homogeneous with respect to charge sites. Attempts to dissociate this type of precursor ion population by low-energy CA result in poor fragmentation via few pathways. The work described here represents a systematic investigation of the effects of charge heterogeneity in the precursor ion population of a series of model peptides in low-energy CA experiments. Incorporation of acidic residues in the peptide RLC*IFSC*FR (where C* indicates a cysteic acid residue), for example, balances the charge on the basic arginine residues, which enables the ionizing protons to reside on a number of less basic sites along the peptide backbone. This results in a precursor ion population that is heterogeneous with respect to charge site. Low-energy CA of these ions results in diverse and efficient fragmentation. Molecular modeling has been utilized to demonstrate that energetically preferred conformations incorporate an intraionic interaction between arginine and cysteic acid residues.     

Collisional activation spectra of field ionised molecules

The collision of field ionised molecular ions of high kinetic energy (8 keV) with neutral target atoms (helium) gives rise to a large variety of fragments. In general the types of fragments, as well as their abundance ratios, closely resemble those formed under 70 eV electron-impact. The differences observed between the collisional activation spectra of several field ionised[C₇H₈]+. ions, as well as the differences observed between several [C₈H₈]+. ions can be rationalised assuming different degrees of isomerisation. The usefulness of collisional fragmentation spectra for structure determination of organic compounds has been demonstrated for a variety of compounds and some mixtures.     

Collisional transfer of population and orientation in NaK

Collisional satellite lines with |ΔJ| ≤ 58 have been identified in recent polarization spectroscopy V-type optical-optical double resonance (OODR) excitation spectra of the Rb(2) molecule [H. Salami et al., Phys. Rev. A 80, 022515 (2009)]. Observation of these satellite lines clearly requires a transfer of population from the rotational level directly excited by the pump laser to a neighboring level in a collision of the molecule with an atomic perturber. However to be observed in polarization spectroscopy, the collision must also partially preserve the angular momentum orientation, which is at least somewhat surprising given the extremely large values of ΔJ that were observed. In the present work, we used the two-step OODR fluorescence and polarization spectroscopy techniques to obtain quantitative information on the transfer of population and orientation in rotationally inelastic collisions of the NaK molecules prepared in the 2(A)(1)Σ(+)(v' = 16, J' = 30) rovibrational level with argon and potassium perturbers. A rate equation model was used to study the intensities of these satellite lines as a function of argon pressure and heat pipe oven temperature, in order to separate the collisional effects of argon and potassium atoms. Using a fit of this rate equation model to the data, we found that collisions of NaK molecules with potassium atoms are more likely to transfer population and destroy orientation than collisions with argon atoms. Collisions with argon atoms show a strong propensity for population transfer with ΔJ = even. Conversely, collisions with potassium atoms do not show this ΔJ = even propensity, but do show a propensity for ΔJ = positive compared to ΔJ = negative, for this particular initial state. The density matrix equations of motion have also been solved numerically in order to test the approximations used in the rate equation model and to calculate fluorescence and polarization spectroscopy line shapes. In addition, we have measured rate coefficients for broadening of NaK 3(1)Π ← 2(A)(1)Σ(+)spectral lines due to collisions with argon and potassium atoms. Additional broadening, due to velocity changes occurring in rotationally inelastic collisions, has also been observed.     

Effects of Volume Changes on the Thermal Performance of PCM Layers Subjected to Oscillations of the Ambient Temperature: Transient and Steady Periodic Regimes

The Stefan problem regarding the formation of several liquid–solid interfaces produced by the oscillations of the ambient temperature around the melting point of a phase change material has been addressed by several authors. Numerical and semi-analytical methods have been used to find the thermal response of a phase change material under these type of boundary conditions. However, volume changes produced by the moving fronts and their effects on the thermal performance of phase change materials have not been addressed. In this work, volume changes are incorporated through an additional equation of motion for the thickness of the system. The thickness of the phase change material becomes a dynamic variable of motion by imposing total mass conservation. The modified equation of motion for each interface is obtained by coupling mass conservation with a local energy–mass balance at each front. The dynamics of liquid–solid interface configurations is analyzed in the transient and steady periodic regimes. Finite element and heat balance integral methods are used to verify the consistency of the solutions to the proposed model. The heat balance integral method is modified and adapted to find approximate solutions when two fronts collide, and the temperature profiles are not smooth. Volumetric corrections to the sensible and latent heat released (absorbed) are introduced during front formation, annihilation, and in the presence of two fronts. Finally, the thermal energy released by the interior surface is estimated through the proposed model and compared with the solutions obtained through models proposed by other authors.     

Gas-phase collisional relaxation of the CH2I radical after UV photolysis of CH2I2

Transient UV absorption spectra and kinetics of the CH(2)I radical in the gas phase have been investigated at 313 K. Following laser photolysis of 1-3 mbar CH(2)I(2) at 308 nm, transient spectra in the wavelength range 330-390 nm were measured at delay times between 60 ns and a few microseconds. The change of the absorption spectra at early times was attributed to vibrational cooling of highly excited CH(2)I radicals by collisional energy transfer to CH(2)I(2) molecules. From transient absorption decays measured at specific wavelengths, time-dependent concentrations of vibrationally "hot" and "cold" CH(2)I and CH(2)I(2) were extracted by kinetic modeling. In addition, the transient absorption spectrum of CH(2)I radicals between 330 and 400 nm was reconstructed from the simulated concentration-time profiles. The evolution of the absorption spectra of CH(2)I radicals and CH(2)I(2) due to collisional energy transfer was simulated in the framework of a modified Sulzer-Wieland model. Additional master equation simulations for the collisional deactivation of CH(2)I by CH(2)I(2) yield DeltaE values in reasonable agreement with earlier direct studies on the collisional relaxation of other systems. In addition, the simulations show that the shape of the vibrational population distribution of the hot CH(2)I radicals has no influence on the measured UV absorption signals. The implications of our results with respect to spectral assignments in recent ultrafast spectrokinetic studies of the photolysis of CH(2)I(2) in dense fluids are discussed.     

Collisionally activated decomposition of poly(ethylene glycol)s: An investigation of high-mass ion abundances in the collisional activation technique with large molecules

The utility of the collisional activation technique in structure determination of ions is limited as parent ion mass increases. Optimum collisionally activated dissociation yield is often obtained at parent masses of 1000–2000 u, after which daughter ion yield decreases. The apparent decrease in the efficiency of the collisional activation process has been thought of as a degree-of-freedom effect: as new rotational-vibrational modes are added to the parent ion, its lifetime with respect to dissociation increases. We have investigated this effect using an easily characterized system of several poly(ethylene glycol) homologs from the 15-mer to the 35-mer. Observed trends in the collisional activation spectra as parent mass increases support the postulated ‘degree-of-freedom’ effect in general. The loss of C₂H₄O from the [M ? H]^? parents, a fragmentation which has a high activation barrier, however, actually becomes more favored as the parent ion becomes larger. This effect is explained in terms of statistical rate theory.     

Influence of velocity effects on the shape of N(2) (and air) broadened H(2)O lines revisited with classical molecular dynamics simulations

The modeling of the shape of H(2)O lines perturbed by N(2) (and air) using the Keilson-Storer (KS) kernel for collision-induced velocity changes is revisited with classical molecular dynamics simulations (CMDS). The latter have been performed for a large number of molecules starting from intermolecular-potential surfaces. Contrary to the assumption made in a previous study [H. Tran, D. Bermejo, J.-L. Domenech, P. Joubert, R. R. Gamache, and J.-M. Hartmann, J. Quant. Spectrosc. Radiat. Transf. 108, 126 (2007)], the results of these CMDS show that the velocity-orientation and -modulus changes statistically occur at the same time scale. This validates the use of a single memory parameter in the Keilson-Storer kernel to describe both the velocity-orientation and -modulus changes. The CMDS results also show that velocity- and rotational state-changing collisions are statistically partially correlated. A partially correlated speed-dependent Keilson-Storer model has thus been used to describe the line-shape. For this, the velocity changes KS kernel parameters have been directly determined from CMDS, while the speed-dependent broadening and shifting coefficients have been calculated with a semi-classical approach. Comparisons between calculated spectra and measurements of several lines of H(2)O broadened by N(2) (and air) in the ν(3) and 2ν(1) + ν(2) + ν(3) bands for a wide range of pressure show very satisfactory agreement. The evolution of non-Voigt effects from Doppler to collisional regimes is also presented and discussed.     

Speed dependence of collisional relaxation in ground vibrational state of OCS: rotational behaviour

Accurate experimental data on pressure broadened profiles of (16)O(12)C(32)S pure rotational lines in a broad range of quantum number J have been analyzed taking into account the speed dependence of collisional relaxation. Refined values of collisional self-broadening coefficients are determined and compared to previously known data. New quantitative information on departures of observed line shapes from the traditional Voigt profile is obtained. It is shown that these departures result mainly from the speed dependence of collisional relaxation. Theoretical calculations of self-broadening parameters are performed in the framework of the semiclassical impact Robert-Bonamy formalism where the mean relative molecular speed as well as the Maxwell-Boltzmann distribution of relative speeds is considered. The necessity of allowance for the speed dependence in line shape models is confirmed and satisfactory results have been obtained by arbitrarily limiting the integration of the differential cross section to a finite value of the impact parameter. It is shown for the first time for the whole rotational spectrum that speed dependent models not only improve accuracy of modeling the observed line profiles but also give physically grounded values of collisional relaxation parameters.     

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).     

Reaction of methacrolein with the hydroxyl radical in air: incorporation of secondary O2 addition into the MACR + OH master equation

Methacrolein (MACR) plays an important role in atmospheric chemistry within the planetary boundary layer, as it is one of the major oxidation products of isoprene and has a short lifetime toward the hydroxyl radical (OH). In this study, quantum chemical techniques and statistical reaction rate theory have been used to simulate the addition of OH to MACR at conditions representative of the troposphere. In this chemically activated reaction, the time scales for product formation versus collisional deactivation of the vibrationally excited adduct are explicitly considered. Furthermore, the subsequent addition of O(2) is also incorporated within a single master equation, so as to investigate doubly activated peroxyl radical formation. The major reaction product of OH addition to MACR is the HOCH(2)C(?)(CH(3))CHO radical formed via addition to the outer (β) carbon. This radical is predominantly in the Z isomer although around a third of the population is quenched as the higher-energy E isomer. Calculated rate constants agree well with experiment when using M06-2X/aug-cc-pVTZ barrier heights, but are somewhat overpredicted using G3SX energies. The overall rate constant is controlled by competition between dissociation of the MACR···OH van der Waals complex back to reactants and isomerization on to MACR-OH adducts, which takes place on a time scale of several nanoseconds, but collisional deactivation of the MACR-OH adducts occurs on a time scale that is around an order of magnitude longer. When O(2) addition is included in the master equation, we observe that the MACR-OH adducts are removed by reaction with O(2) on a similar time scale to collisional deactivation. Around 50% of the subsequent peroxyl radical population is formed with some identifiable excess vibrational energy above singly activated [MACR-OH-O(2)]*, with around 20% provided with an additional 20 kcal mol(-1) (>40 kcal mol(-1) relative to quenched MACR-OH-O(2)) that can go into further unimolecular reaction. This double activation process is expected to lead to some prompt unimolecular decomposition of excited [MACR-OH-O(2)]** peroxyl radicals to yield products including hydroxyacetone and methylglyoxal, regenerating the initiating OH radical in the process.     

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.     

Quadrature method of moments for aggregation-breakage processes

Investigation of particulate systems often requires the solution of a population balance, which is a continuity statement written in terms of the number density function. In turn, the number density function is defined in terms of an internal coordinate (e.g., particle length, particle volume) and it generates integral and derivative terms. Different methods exist for numerically solving the population balance equation. For many processes of industrial significance, due to the strong coupling between particle interactions and fluid dynamics, the population balance must be solved as part of a computational fluid dynamics (CFD) simulation. Such an approach requires the addition of a large number of scalars and the associated transport equations. This increases the CPU time required for the simulation, and thus it is clear that it is very important to use as few scalars as possible. In this work the quadrature method of moments (QMOM) is used. The QMOM has already been validated for crystal growth and aggregation; here the method is extended to include breakage. QMOM performance is tested for 10 different cases in which the competition between aggregation and breakage leads to asymptotic solutions.     

The effect of the negative coupling parameter on the spectrum of a trapped Bose gas

The interest in attractive Bose–Einstein Condensates arises due to the chemical instabilities generate when the number of trapped atoms is above a critical number. In this case, recombination process promotes the collapse of the cloud. This behavior is normally geometry dependent. Within the context of the mean field approximation, the system is described by the Gross–Pitaevskii equation. We have considered the attractive Bose–Einstein condensate, confined in a nonspherical trap, investigating numerically and analytically the solutions, using controlled perturbation and self-similar approximation methods. This approximation is valid in all interval of the negative coupling parameter allowing interpolation between weak-coupling and strong-coupling limits. When using the self-similar approximation methods, accurate analytical formulas were derived. These obtained expressions are discussed for several different traps and may contribute to the understanding of experimental observations.     

PECT model analysis and predictions of experimental collisional energy transfer probabilities P(E',E) and moments <DeltaE> for azulene and biphenylene

Experimental collisional energy transfer data from kinetically controlled selective ionization (KCSI) and ultraviolet absorption (UVA) experiments are analyzed in the framework of the partially ergodic collision theory (PECT). Collisions of azulene and biphenylene with different colliders are investigated as case studies. The downward wings of the P(E',E) energy transfer distributions obtained from the PECT model are fitted to the recently introduced "variable-shape"-exponential 3-parameter functional form of P(E',E) obtained from KCSI experiments, P(E',E) proportional, variant exp[-{(E - E')/(C(0) + C(1)E)}(Y)]. The PECT model is able to reproduce the characteristic dependence of the KCSI "shape parameter" Y on the choice of collider, the energy dependent width of the KCSI P(E',E) distributions, described by alpha(E) = C(0) + C(1)E, and the temperature dependence of the UVA data above room temperature. The statistical approach of PECT obviously captures the essence of large molecule energy transfer at chemically significant energies without the need of knowing specific features of the detailed collision dynamics. It therefore shows promise for predicting the shape of P(E',E) in master equation kernels for larger molecules.