Following the Crystallization Process of Polyethylene Single Chain by Molecular Dynamics: The Role of Lateral Chain Defects

Summary: Langevin molecular dynamics (LMD) simulations have been performed in order to understand the role of the short chain branches (SCB) on the formation of ordered domains by cooling ethylene/α-olefins single chain models. Different long single-chain models (C₂₀₀₀) with 0, 5 and 10 branches each 1000 carbons were selected. The branches were randomly distributed along the backbone chain. Furthermore, C₁ (methyl) and C₄ (butyl) branches were taken into account. These models mimic the molecular architecture of ethylene/1-butene and ethylene/1-hexene random copolymers. The simulations are performed according to the following protocol: 20 random chain conformations for each model were equilibrated at high temperature (T* = 13.3) and then they were cooled in steps of 0.45 until the final temperature (T* = 6.2) by running a total of 35 × 10⁶ LMD steps. The distribution peaks of crystallization for each model were calculated by differentiating the global order parameter with respect to the temperature. The Tc* (crystallization temperature) decrease as the number of branches increases as it is experimentally observed. The formation of order in the copolymers is affected by the type and amount of the SCB in the backbone of the polymer chain. The stem lenght and crystallization fraction (α) were defined using the local-bond order parameter. Both parameters decrease as the number of branches increase. In all cases here shown, the C₄ branches are excluded from the ordered domains. However, we have observed that the methyl branch can be incorporated into the ordered regions. These facts satisfactorily agree with experimental data available in the literature.     

Cure kinetic parameter estimation of thermosetting resins with isothermal data by using particle swarm optimization

The heuristic algorithms have shown to be a powerful tool in parameter estimation. Among these algorithms, particle swarm optimization (PSO) has become a method whose application has been increasing quickly. In the present work a new way for parameter estimation from cure kinetic model of polymeric resin using a differential-algebraic approach is shown. The PSO was applied to minimize the least squares function and to find the parameters from an autocatalytic model for describing cure kinetics of thermosetting resins. The isothermal data were obtained at four temperatures: 318, 333, 348 and 363 K. Three parameter estimation procedures were compared for finding a parameter set for all temperatures simultaneously. In the first one, called classical method, a curing rate was calculated with experimental values of the degree of cure and the temperature. In the second and third methods, the curing rate was obtained from the integration of a differential-algebraic system and the main difference between them is the objective function and the way to determine the ultimate reaction heat. All methods showed good results; however, the third method was more accurate than the others. The confidence regions of all parameters were found and they were used to give us indication whether the parameters estimated here by different methods are statistically different.     

Interaction parameters for multi-component aromatic extraction with sulfolane

Aromatic extraction is an important operation in petrochemical processing. Design of an aromatic extractor requires the knowledge of multi-component liquid–liquid equilibrium (LLE) data. Such experimental LLE data are usually not available and therefore can be predicted using various activity coefficient models. These models require proper binary interaction parameters, which are not yet available for all aromatic extraction systems. Furthermore, the parameters available for most of the ternary systems are specific to that system only and cannot be used for other ternary or multi-component systems. An attempt has been made to obtain these parameters that are globally applicable. For this purpose, the parameter estimation procedure has been modified to estimate the parameters simultaneously for different systems involving common pairs. UINQUAC and UNIFAC models have been used for parameter estimation. The regressed parameters are shown to be applicable for the ternary as well as for the multi-component systems. It is observed that UNIQUAC parameters provide a better fit for ternary LLE data, whereas, as one moves towards the higher component systems (quaternary and quinary) the UNIFAC parameters, which are a measure of the group contributions, predict the LLE better. Effect of temperature on UNIQUAC binary interaction parameters has been studied and a linear dependence has been observed.     

Electrochemiluminescence Mechanisms Investigated with Smartphone-Based Sensor Data Modeling,Parameter Estimation and Sensitivity Analysis

The present study introduces a unified framework combining a mechanistic model with a genetic algorithm (GA) for the parameter estimation of electrochemiluminescence (ECL) kinetics of the Ru(bpy)₃²⁺/TPrA system occurring in a smartphone-based sensor. The framework allows a straightforward solution for simultaneous estimation of multiple parameters which can be, otherwise, time-consuming and lead to non-convergence. Model parameters are estimated by achieving a high correlation between the model prediction and the measured ECL intensity from the ECL sensor. The developed model is used to perform a sensitivity analysis (SA), which provides quantitative effects of the model parameters on the concentrations of chemical species involved in the system. The results demonstrate that the GA-based parameter estimation and the SA approaches are effective in analyzing the kinetics of the ECL mechanism. Therefore, these approaches can be incorporated as analysis tools in the ECL kinetics study with practical application in the calibration of mechanistic models for any required sensing condition.     

β-Myrcene Coordination Polymerization: Experimental and Kinetic Modeling Study

The present work presents phenomenological models to describe the coordination polymerization of β-myrcene using the Ziegler–Natta catalyst system composed by neodymium versatate (NdV₃), diisobutylaluminum hydride (DIBAH), and dimethyldichlorosilane. The kinetic parameters required to simulate the reactions are estimated, and the amount of DIBAH used as a chain transfer agent (CTA) is obtained by a data reconciliation strategy since it can participate in side reactions. Several experiments are performed at different conditions to evaluate the impact of key operation variables on the control of monomer conversion and average molar masses. It is shown that the initial NdV₃, β-myrcene, and DIBAH concentrations exert strong influences on the course of the polymerization. The kinetic mechanism of Coordinative Chain Transfer Polymerization (CCTP) fits well with the data of final average molar masses and monomer conversion, while the dynamic trajectories of these variables are fitted better by kinetic mechanisms of more conventional coordination polymerizations, considering site deactivation and termination by chain transfer. In all cases, the proposed models are able to predict the experimental data well after successful parameter estimation and reconciliation of CTA concentrations, indicating that the kinetic mechanism can be characterized by different kinetic regimes.     

Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of “how well do we know our parameters, along with their uncertainties?.” The results reveal that Bayesian inversion statistical analysis on an example, prototype nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a “Gold Standard” Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.     

Extremely Low Vapor-Pressure Data as Access to PC-SAFT Parameter Estimation for Ionic Liquids and Modeling of Precursor Solubility in Ionic Liquids

Precursor solubility is a crucial factor in industrial applications, dominating the outcome of reactions and purification steps. The outcome and success of thermodynamic modelling of this industrially important property with equations of states, such as Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT), vastly depends on the quality of the pure-component parameters. The pure-component parameters for low-volatile compounds such as ionic liquids (ILs) have been commonly estimated using mixture properties, e. g. the osmotic pressure of aqueous solutions. This leads to parameters that depend on the solvent, and transferability to other mixtures often causes poor modeling results. Mixture-independent experimental properties would be a more suitable basis for the parameter estimation offering a way to universal parameter sets. Model parameters for ILs are available in the literature [10.1016/j.fluid.2012.05.029], but they were estimated using pure-IL density data. The present work focuses on a step towards a more universal estimation strategy that includes new experimental vapor-pressure data of the pure IL. ILs exhibit an almost negligible vapor pressure in magnitude of usually 10⁻⁵ Pa even at elevated temperatures. In this work, such vapor-pressure data of a series of 1-ethyl-3-methyl-imidazolium-based [C₂mim]-ILs with various IL-anions (e. g. tetrafluoroborate [BF₄]⁻, hexafluorophosphate [PF₆]⁻, bis(trifluoromethylsulfonyl)imide [NTf₂]⁻) were experimentally determined and subsequently used for PC-SAFT parameter estimation. The so-determined parameters were used to predict experimental molecular precursor solubility in ILs and infinitely diluted activity coefficients of various solvents in ILs. The parameters were further compared to modeling results using classical parametrization methods (use of liquid-density data only for the molecular PC-SAFT and the ion-based electrolyte PC-SAFT). As a result, the modeled precursor solubilities using the new approach are much more precise than using the classical parametrization methods, and required binary parameters were found to be much smaller (if needed). In sum, including the pure-component vapor-pressure data of ILs opens the door towards parameter estimation that is not biased by mixture data. This procedure might be suitable also for polymers and for all kind of ionic species but needs extension to ion-specific parametrization in the long term.     

Phase behavior predictions for polymer blends containing reversibly associating endgroups

A mean field model is developed to predict how polymer–polymer miscibility changes if polymers are functionalized with noncovalent, reversibly binding endgroups. The free-energy model is based on the Flory–Huggins mixing theory and has been modified using Painter's association model to account for equilibrium self-association of endgroups. Model input parameters include the length of polymer chains, a temperature-dependent interaction parameter, and a temperature-dependent equilibrium constant for each type of associating endgroup. The analysis is applied to 12 possible blend combinations involving self-complementary interactions and seven combinations involving hetero-complementary [i.e. donor–acceptor (DA)] interactions. Combinations involve both monofunctional and telechelic associating chains. Predicted phase diagrams illustrate how self-complementary interactions can stabilize two-phase regions and how DA interactions can stabilize single phase regions. The model is a useful tool in understanding the delicate balance between the combinatorial entropy of mixing polymer chains, the repulsive interactions between dissimilar polymers, and the additional enthalpic and entropic changes due to end-group association of chain ends. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3285–3299, 2007     

Estimation of chromatographic peak shape parameters in Fourier domain

A number of models in chromatography have analytical solutions in the Laplace or Fourier domain. Often, the moments of the Laplace domain solutions are calculated to characterize the peak shape. Nonlinear fitting in the Fourier domain can be performed to exploit the entire peak shape rather than the moments only. Curve fitting in the Fourier domain offers an attractive alternative for parameter estimation. In this study we will show - with some simple applications - the possibilities of estimation of chromatographic peak shape parameters in Fourier domain. Various models are fitted to different transient signals.     

Association models for alcohol-water mixtures

Alcohol-water mixtures are often used as model systems for the study of hydrophobic interactions. In this respect, their thermodynamic properties are of particular interest since precise data are available over the whole miscibility range and they lend themselves readily to theoretical modelling. Three association models have been used to fit the volume data taken from the literature for aqueous methonol, ethanol and n-propanol at various temperatures. A simple micellization equilibrium explains the general trends but fails to fit the reduced excess volumes at both ends of the mole fraction scale. Better fits are obtained if interaction parameters are introduced, but these two parameters cannot easily be rationalized. A double association model (one for the alcohol and one for the water) gives the best fit and all the parameters have a physical significance. However, the parameters extracted, e.g. aggregation numbers, and their trends with temperature are not always realistic. Despite their limitations, these models illustrate well with the kind of avenues that can be explored to fit and interpret the experimental data of these complex systems.     

Maximum likelihood-based analysis of single-molecule photon arrival trajectories

In this work we explore the statistical properties of the maximum likelihood-based analysis of one-color photon arrival trajectories. This approach does not involve binning and, therefore, all of the information contained in an observed photon strajectory is used. We study the accuracy and precision of parameter estimates and the efficiency of the Akaike information criterion and the Bayesian information criterion (BIC) in selecting the true kinetic model. We focus on the low excitation regime where photon trajectories can be modeled as realizations of Markov modulated Poisson processes. The number of observed photons is the key parameter in determining model selection and parameter estimation. For example, the BIC can select the true three-state model from competing two-, three-, and four-state kinetic models even for relatively short trajectories made up of 2 × 10(3) photons. When the intensity levels are well-separated and 10(4) photons are observed, the two-state model parameters can be estimated with about 10% precision and those for a three-state model with about 20% precision.     

On the errors of Arrhenius parameters and estimated rate constant values

A detailed statistical study is presented, based on simulated experimental data, on the estimation of activation parameters using the Arrhenius equation: k = A exp(B/T). The close correlation of the two parameters is shown, which requires the computation of the covariance matrix for the representation of uncertainties. This matrix facilitates the correct estimation of the confidence interval for interpolated (or extrapolated) values of rate coefficients. It is proposed that the full correlation matrix should be published in any article dealing with the determination of Arrhenius parameters. The importance of correct weighting is emphasized. Nonlinear fitting to the Arrhenius equation can be carried out without weighting only in case the (absolute) error of rate coefficient is independent of the temperature. Simulated experiments show that noncorrect weighting shifts the average values of fitted parameters and increases the variance of the parameters as well. With respect to the modified Arrhenius equation: k = A · Tⁿ exp(B/T), statistical analysis shows that the physically meaningful estimation of all three parameters is impossible. Nonlinear fitting of three parameters is suggested for interpolation (and extrapolation) of rate coefficients, whereas in case of activation parameter estimation, the fixing of “n” on the basis of theoretical considerations is advised followed by the estimation of the remaining two parameters.     

Comparison of approaches for parameter estimation on stochastic models: Generic least squares versus specialized approaches

Parameter estimation for models with intrinsic stochasticity poses specific challenges that do not exist for deterministic models. Therefore, specialized numerical methods for parameter estimation in stochastic models have been developed. Here, we study whether dedicated algorithms for stochastic models are indeed superior to the naive approach of applying the readily available least squares algorithm designed for deterministic models.We compare the performance of the recently developed multiple shooting for stochastic systems (MSS) method designed for parameter estimation in stochastic models, a stochastic differential equations based Bayesian approach and a chemical master equation based techniques with the least squares approach for parameter estimation in models of ordinary differential equations (ODE). As test data, 1000 realizations of the stochastic models are simulated. For each realization an estimation is performed with each method, resulting in 1000 estimates for each approach. These are compared with respect to their deviation to the true parameter and, for the genetic toggle switch, also their ability to reproduce the symmetry of the switching behavior. Results are shown for different set of parameter values of a genetic toggle switch leading to symmetric and asymmetric switching behavior as well as an immigration-death and a susceptible-infected-recovered model. This comparison shows that it is important to choose a parameter estimation technique that can treat intrinsic stochasticity and that the specific choice of this algorithm shows only minor performance differences.     

Analytical models for describing cation adsorption/desorption kinetics as considering the electrostatic field from surface charges of particles

Surface charges of particles together with the adsorbed counter ions in diffuse layer can set up a strong electrostatic field around the particles in aqueous solution. The existent kinetic models for describing cation exchange on solid/liquid interface were either empirical or semi-empirical, and in which the electrostatic field is not considered. In this paper, as considering the important effect of electrostatic field around particles on cations adsorption/desorption, for the first time the dynamic distribution equations of cations in diffuse layer for adsorption and desorption processes in both flow method and batch technique have been established. Those equations clearly show how the cation concentration changes with time in different position of diffuse layer during the cation exchange process, and the corresponding new kinetic models have been obtained upon them. The new models indicate that, in both flow method and batch technique, for the adsorption process, experimental results should appear zero order kinetic process caused by the strong force adsorption in the initial stage of adsorption, and then transform to the first order kinetic process of the weak force adsorption; and for the desorption process, however, only first order kinetic process may exist. The new models are essentially different from the classic apparent or empirical kinetic models since all the parameters have their defined physical meanings in the new models, thus the rate parameters in the new models have the potential to theoretically predict. Theoretical analyses also indicated that, the adsorption/desorption rate in flow method experiment will be much higher than that in batch technique experiment.     

Automated Generation of Microkinetics for Heterogeneously Catalyzed Reactions Considering Correlated Uncertainties**

The study presents an ab-initio based framework for the automated construction of microkinetic mechanisms considering correlated uncertainties in all energetic parameters and estimation routines. 2000 unique microkinetic models were generated within the uncertainty space of the BEEF-vdW functional for the oxidation reactions of representative exhaust gas emissions from stoichiometric combustion engines over Pt(111) and compared to experiments through multiscale modeling. The ensemble of simulations stresses the importance of considering uncertainties. Within this set of first-principles-based models, it is possible to identify a microkinetic mechanism that agrees with experimental data. This mechanism can be traced back to a single exchange-correlation functional, and it suggests that Pt(111) could be the active site for the oxidation of light hydrocarbons. The study provides a universal framework for the automated construction of reaction mechanisms with correlated uncertainty quantification, enabling a DFT-constrained microkinetic model optimization for other heterogeneously catalyzed systems.     

A novel approach to modelling counter-current chromatography

Literature lists a number of counter-current chromatography (CCC) models that can predict the retention time and to a certain extent the peak width of a solute eluting from a CCC column. The approach described in this paper distinguishes itself from previous reports by relating all model parameters directly to column dimensions and experimental settings. Most importantly, this model can predict a chromatogram from scratch without resorting to traditional calibration using empirical values. The model validation with experimental results obtained across a range of CCC instruments demonstrated that the solute retention time, peak width, and peak resolution could be predicted within reasonable accuracy. Additionally, the effect of several process parameters, such as mobile phase flow rate, rotational speed of the column or β-value, showed that the model is robust and applicable to a wide range of CCC instruments. Overall, this model proved to be a useful tool for parameter estimation and, most significantly, separation optimisation.     

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

Validity ranges of Lie canonical perturbation theory (LCPT) are investigated in terms of non-blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non-blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non-blow-up regions enclose the region inside the separatrix of the Hamiltonian, but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non-blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix, whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non-blow-up regions do not necessarily cover the whole regions inside the HCN well. We propose a new perturbation method to improve non-blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in Teramoto and Komatsuzaki (J Chem Phys 129:094302, 2008; Phys Rev E 78:017202, 2008).     

Voronoi-based detection of pockets in proteins defined by large and small probes

The function of enzymatic proteins is given by their ability to bind specific small molecules into their active sites. These sites can often be found in pockets on a hypothetical boundary between the protein and its environment. Detection, analysis, and visualization of pockets find its use in protein engineering and drug discovery. Many definitions of pockets and algorithms for their computation have been proposed. Kawabata and Go defined them as the regions of empty space into which a small spherical probe can enter but a large probe cannot and developed programs that can compute their approximate shape. In this article, this definition was slightly modified in order to capture the existence of large internal holes, and a Voronoi-based method for the computation of the exact shape of these modified regions is introduced. The method first puts a finite number of large probes on the protein exterior surface and then, considering both large probes and atomic balls as obstacles for the small probe, the method computes the exact shape of the regions for the small probe. This is all achieved with Voronoi diagrams, which help with the safe navigation of spherical probes among spherical obstacles. Detected regions are internally represented as graphs of vertices and edges describing possible movements of the center of the small probe on Voronoi edges. The surface bounding each region is obtained from this representation and used for visualization, volume estimation, and comparison with other approaches. © 2019 Wiley Periodicals, Inc.     

Computer modeling of the crystallization process of single‐chain ethylene/1‐hexene copolymers from dilute solutions

Langevin molecular dynamics (LMD) simulations have been performed to understand the role of the short chain branches (SCB) on the formation of ordered domains by cooling dilute solutions of ethylene/α‐olefins copolymer models. Three different long single‐chain models (C₂₀₀₀) with 0, 5, and 10 branches each 1000 carbons were selected. These models were equilibrated at high reduced temperature (T* = 13.3) and cooling in steps of 0.45 until the final temperature (T* = 6.2) by running a total of 35 × 10⁶ LMD steps. During the cooling process, global order parameter, torsion distribution, position of the branches, and local‐bond order parameter were calculated and monitored. The peaks of crystallization for each model were calculated by differentiating the global order parameter with temperature. The T_c (crystallization temperature) decreases as the number of branches increases as has been experimentally reported. The formation of order in the copolymers is affected by the amount of the SCB in the backbone of the polymer chain. Initially, the SCB move to the folding surface. Once the SCB are located near the folding surface the order starts to grow. In all cases here shown, the C₄ branches are excluded from the ordered domains. To take into account, the influence of the branch distribution, a different branch distribution model has been considered for the two‐branched systems. The crystallization fraction (α) and the density of the amorphous and ordered fractions was defined using the local‐bond order parameter. Both magnitudes decrease as the number of branches increases. These facts fairly agree with experimental literature data. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Fire performance of poly(vinyl chloride) and its relation to fire hazard

This paper discusses the most important parameters in terms of fire hazard and shows how PVC fares in relation to other polymers. The fire properties specifically addressed are: ignitability, flammability, flame spread, rate and amount of heat release, mass loss rate, smoke release and toxicity. Hydrogen chloride decay is also commented on, because it affects the toxicity of PVC smoke. The individual parameter most relevant to fire hazard is heat release. The two most useful tools for measuring rate of heat release (viz. the OSU and Cone calorimeters) are described. Results obtained from them are discussed. Smoke can best be measured by combined parameters from rate of heat release calorimeters, rather than in the traditional static NBS smoke chamber. Toxic hazard is being addressed by recognition that most smokes are of similar toxicity, so that the mass loss rate will, generally, govern the toxicity of smoke. Not all fire tests are equally good representations of the probable consequences of a full-scale fire. Fire hazard assessment is best carried out based on those test results most relevant to real fires; they can be obtained from small and full-scale experiments and fire models. The fire performance of PVC is excellent; PVC products generally represent low fire hazard in a scenario.