Weight Fraction Distribution Research Articles

Mesoscale modeling of random chain scission in polyethylene melts

Polyolefins account for more than half of global primary polymer production, however only a small fraction of these polymers are currently being recycled. Fragmentation of polymer chains into shorter chains with a targeted molecular weight distribution with the goal of reusing these fragments in subsequent chemical synthesis can potentially introduce an alternative approach to polyolefins recycling. Herein we develop a mesoscale framework to model degradation of polyethylene melts at a range of high temperatures. We use the dissipative particle dynamics approach with modified segmental repulsive potential to model the process of random scission in melts of linear polymer chains. We characterize the fragmentation process by tracking the time evolution of the distribution of degrees of polymerization of chain fragments. Specifically, we track the weight average and the number average degrees of polymerization and dispersity of polymer fragments as a function of the fraction of bonds broken. Furthermore, we track the number fraction distribution and the weight fraction distribution of polymer fragments with various degrees of polymerization as functions of the fraction of bonds broken for a range of high temperatures. Our results allow one to quantify to what extent the distribution of polymer chain fragments during random scission can be captured by the respective analytical distributions for the range of conversions considered. Understanding the thermal degradation of polyolefins on the mesoscale can result in the development of alternative strategies for recycling a range of thermoplastics.

Nonlinear aero-thermo-elastic flutter analysis of stiffened graphene platelets reinforced metal foams plates with initial geometric imperfection

Due to its lightweight design and high load-bearing capacity, the stiffened plate structure is extensively applied in the passive control of aircraft flutter. However, the evolution mechanism of the nonlinear response of stiffened plates within the unstable region remains unclear. In particular, there is a lack of research on the nonlinear aero-thermo-elastic properties of stiffened plates with initial geometric imperfection. Inspired by this, this article investigates the stability and post-instability behavior of stiffened geometrical imperfect graphene platelets reinforced metal foams (GPLRMF) plates under combined thermal load and aerodynamic pressure. Graphene platelets (GPLs) and foam are distributed evenly or unevenly in the plates, while the material properties are calculated by modified Halpin-Tsai model and rule of mixture. Considering initial geometric imperfection, the models of plates and stiffeners are established by first-order shear deformation (FOSD) plate and Timoshenko beam theories, respectively. Using the linear piston theory to simulate supersonic aerodynamic pressure. The stability boundary of stiffened plates is determined by eigenvalue analysis. The Newton-Raphson approach is used to simulate the aero-thermo-elastic static response of stiffened plates, and the dynamic post-flutter behavior of the stiffened plate is determined through the Runge-Kutta method. Finally, the effects of initial geometric imperfection, material parameters (foam distribution type, porosity coefficient, GPLs weight fraction and GPLs distribution type), and stiffeners (size, position, and stiffening scheme) on the thermal aeroelastic behavior of stiffened GPLRMF plates are revealed. It can be found that the presence of initial geometric imperfection will significantly change the nonlinear flutter behavior of the stiffened plates.

Out-of-plane buckling of functionally graded porous arches reinforced by graphene platelets in a thermal environment

This article investigates the out-of-plane buckling of functionally graded porous I-shaped circular arches with graphene platelets reinforcements in a uniform temperature field. The accurate expressions of the prebuckling internal forces owing to thermal expansion are derived. The theoretical out-of-plane buckling temperature difference is obtained by the Ritz technique and is verified by finite element results. Notably, with the asymmetrical distribution of pores along the height of the section, the monosymmetry parameter is considered in the out-of-plane buckling energy equation. The effects of graphene platelets weight fraction, porosity coefficient, and porosity distribution patterns on buckling temperature difference are analyzed in detailed.

Active sloshing control of a circular cylindrical container with FG GPL-RC

The main purpose of the current study is to design a control system for a sloshing system to suppress the vibration of functionally graded graphene platelet-reinforced composite (FG GPL-RC) on the top surface. A set of piezoelectric sensors and actuators are used for active control of fluid-structure coupling in transient seismic analysis of an inviscid liquid sloshing in a cylindrical tank. First, the top surface FG GPL-RC is optimized to satisfy maximum deformation, face failure, mid shear, and face wrinkling by changing the geometrical parameters and weight fraction distribution through the thickness. The nonlinear governing equations of nanocomposite panels with mechanical properties derived from the Halpin–Tsai model are solved by smoothed particle hydrodynamics (SPH) method. A parameter study counting the boundary condition effects, thickness-to-radius ratio, material distribution of the laminates, and dimensions of the sloshing tank on the FG GPL-RC natural frequencies are surveyed and explained in detail. Then, the frequency analysis of the plate with and without sloshing is performed while the inviscid fluid motion is modeled by SPH. The proposed method is validated with available data in the literature. Finally, the active control of the coupled system in response to the external disturbances is completed by a tuned proportional–integral–derivative (PID) controller which used the piezoelectric sheets on the top surface plate. The performance of the proposed control system is illustrated by a comparison of the controlled and uncontrolled transient responses of the coupled hydro-elastic system to the El-Centro seismic load.

Vibration characteristics of sandwich plates with GPLRC core and piezoelectric face sheets with various electrical and mechanical boundary conditions

An analysis is performed in the current investigation to study the vibrations of composite laminated plates integrated with piezoelectric layers. Composite laminated plate is reinforced with graphene platelets (GPLs). Also the amount of GPLs in the layers may be different which results in a piecewise functionally graded (FG) media. Elasticity modulus of the GPLRC media is obtained by means of the Halpin-Tsai rule which contains efficiency parameters to capture the size of reinforcements. First order shear deformation theory of plates and parabolic distribution of electric field through the piezoelectric layers are assumed as the basic assumption. For the piezoelectric layers, both the open circuit and closed circuit conditions are applied. The total strain and kinetic energies of the plate are obtained. After that by means of the general idea of the Ritz method and also Chebyshev polynomials as the basic shape functions, the matrix representation of the equations of motion dealing with free vibration of sandwich plate with piezoelectric face sheets are obtained. Results of this study are first compared with the available data in the open literature. After that novel results are given to explore the effects of different parameters such as the electrical and mechanical boundary conditions, plate geometry, GPL weight fraction and GPL distribution profile. It is shown that with introduction of GPLs, frequencies of the plate may be enhanced for both closed circuit and open circuit conditions.

Modified couple stress theory for wave propagation in viscoelastic sandwich microplates with FG-GPLRC core and piezoelectric face sheets as sensor and actuator

In the present investigation, wave propagation of the viscoelastic sandwich microplates with functionally graded graphene nanoplatelets’ reinforced composite (FG-GPLRC) core and piezoelectric face sheets as sensor and actuator has been analyzed by the refined zigzag theory (RZT). Electromagnetic field is exposed to core and piezoelectric layers. Young's modulus, mass density, and Poisson's ratio are calculated based on the modified Halpin–Tsai model and rule of mixtures because FG-GPLs are arranged as uniform, parabolic, and linear patterns along the thickness of core. In addition, viscoelastic properties of the structure are simulated by the Kelvin–Voigt model. The system is located on an orthotropic visco Pasternak medium. The modified couple stress theory (MCST) is applied to simulate the microscale effects of the structure. Hamilton's principle is utilized for deriving the motion equations. Phase velocity, cut-off, and escape frequencies are computed with an analytical solution. The influence of valuable parameters, such as geometrical dimensions of the system, magnetic field, external voltage, small scale parameter, viscoelastic structural damping coefficient, various foundation types, weight fraction, and different distributions of GPLs on phase velocity, will be studied. The results of this study have a significant effect on the design and fabrication of sandwich micro-structures used in various industrial equipment.

Gel Point Properties in Batch Free‐Radical Vinyl/Divinyl Copolymerization

AbstractGelation is a critical phenomenon, and various simple relationships are expected to hold. On the other hand, it is important to clarify the required conditions for the validity of the relationships. In this theoretical study, the effects of the primary chain connection statistics and the primary chain length distribution change during batch free‐radical vinyl/divinyl copolymerization on the gel point properties are investigated systematically. The weight fraction distribution at the gel point always conforms to the power‐law distribution,W(P)∝P−αwith α = 1.5, as long as the classical chemical kinetics is valid in a closed reaction system. When the primary chain connection statistics is independent of chain length, the cross‐linking density, ρg, of the gel molecule right at the gel point is given by , where is the weight‐average chain length of all primary chains. The random cross‐linking is a special case that belongs to this category. The distribution of the pendant double bonds among polymer molecules having different molecular weights is nonrandom, except for the special cases with equal reactivity of all types of double bonds for which the random cross‐linking applies.

Controllable Power‐Law Distribution in Free‐Radical Vinyl/Divinyl Copolymerization by Using a Continuous Stirred Tank Reactor

AbstractIt has been believed that the power‐law molecular weight distribution is formed at the gel point, and the power exponent falls in a category of universality in critical phenomena, and therefore, it cannot be changed. It is found that the power‐law weight fraction distribution W(P) ∝ P−α, capable of controlling the exponent α, can be synthesized by using a continuous stirred tank reactor (CSTR). The exponent is governed by a single non‐dimensional parameter, ξ that is equivalent to the Damköhler number, representing the ratio of timescales between the mean residence time and the crosslinking reaction. For small ξ’s, the non‐critical transition of the weight‐average molecular weight to infinity is observed, and the exponent approaches α = 2. For large ξ’s, the system shows similar behavior to that in a closed system and the exponent is the same as for the batch polymerization, with α = 1.5. At the gel point, the product of the average crosslinking density and the weight‐average primary chain length satisfies the relationship, . The crosslinking density of the gel molecule at the gel point is found to be .

Effective elastic properties of biocomposites using 3D computational homogenization and X-ray microcomputed tomography

A 3D computational homogenization method based on X-ray microcomputed tomography (μCT) was proposed and implemented to investigate how the fiber weight fraction, orthotropy and orientation distribution affect the effective elastic properties of regenerated cellulose fiber-polylactic acid (PLA) biocomposites. Three-dimensional microstructures reconstructed by means of the X-ray μCT were used as the representative volume elements (RVEs) and incorporated into the finite element solver within the computational homogenization framework. The present method used Euclidean bipartite matching technique so as to eliminate the generation of artificial periodic boundaries and use the in-situ solution domains. In addition, a reconstruction algorithm enabled finding the volume and surface descriptions for each individual fiber in a semi-automatic manner, aiming at reducing the time and labor required for fiber labeling. A case study was presented, through which the method was compared and validated with the experimental investigations. The present study is thus believed to give a precise picture of microstructural heterogeneities for biocomposites of complex fiber networks and to provide an insight into the influences of the individual fibers and their networks on the effective elastic properties.

Influence of graphene platelets on the response of composite plates subjected to a moving load

Current investigation deals with the forced vibration response of a composite laminated plate which is reinforced with graphene platelets (GPLs). The weight fraction of GPLs in each layer may be different which results in a piecewise functionally graded (FG) media. The elastic properties of the composite media are obtained using the Halpin-Tsai approach. The motion equations of the plate are established with the aid of the Ritz method where the shape functions are presented using the Chebyshev polynomials. The time dependent system of equations is solved using the Newmark time marching scheme. The established system of equations may be used for the dynamic response of an FG-GPL reinforced composite rectangular plate which is subjected to a moving load with arbitrary path and velocity. Numerical results are given to show the influence of boundary conditions, GPL weight fraction and GPL distribution profile on the dynamic response of the plate. It is shown that, with increasing the weight fraction of the GPLs, the dynamic deflections in the plate decrease. Also with introduction of a proper pattern where the surfaces at top and bottom have the highest weight fraction of GPLs, the dynamic deflection of the plate may be decreased significantly.

Characteristics of refractory organics in industrial wastewater treated using a Fenton-coagulation process

ABSTRACT It is a challenging environmental issue to develop a cost-efficient approach for the removal of low-concentration refractory organics in industrial wastewater. In this study, the Fenton-coagulation process was utilised to remove the organics from the industrial effluent. The operational conditions of the Fenton-coagulation process were optimised, and then, the molecular weight (MW) and resin fraction distribution of dissolved organic matter (DOM) were investigated before and after the Fenton-coagulation process. The results showed that the efficiency of organic matter removal was affected by the Fe2+/H2O2 molar ratio, pH, and reaction time. The removal rate of chemical oxygen demand (COD) by Fenton-coagulation process reached 37.8% under the following conditions: pH = 4.0 − 5.0, H2O2 concentration = 34 mg/L, Fe2+/H2O2 molar ratio = 1.5, and reaction time = 120 min. The resin fraction distribution results showed that hydrophobic bases (HoB) were almost completely removed, and the removal rate of hydrophobic acids (HoA) reached 58%, while hydrophilic matter (HiM) became the dominant form in the final effluent after the Fenton-coagulation process due to the appearance of hydrophilic charged fractions (HiC). The results were explained by a two-step mechanism (Fenton oxidation and Fe3+ coagulation). According to the molecular weight (MW), 35.7% removal of the main fractions of organic matter with MW < 1 kDa was achieved. Furthermore, a pilot test proved that the final effluent quality after the Fenton-coagulation process conformed to the first class of the A discharge standard of pollutants for municipal wastewater treatment plants in Tianjin.

Random Branching of Polymer Chains with Schulz–Zimm Distribution. 1. Bivariate Distribution and Related Formulae

AbstractAnalytical solution for the bivariate distribution W(r,k) of the degree of polymerization r and the number of branch points k is obtained for the random branching of polymer chains that follow the Schulz–Zimm distribution, or in mathematical term, the gamma distribution. It is found that when the bivariate distribution is normalized to make the total area unity, the normalized distribution follows another gamma distribution. The bivariate distribution function enables one to determine various useful properties of the branched polymer system, including full weight fraction distribution W(r), the number‐ and weight‐average degree of polymerization having k branch points, the expected branching density ρ for a given degree of polymerization r or a given number of branch points k. The branching density at large r limit, ρr → ∞ is an important property to represent the branched polymer system, rather than the average branching density of the whole system, and the value of ρr → ∞ can be determined for the present random branched system. The effect of the distribution breadth of the primary chains can be investigated by using the formulae developed in this article.

Construction of polydisperse polymer model and investigation of heat conduction: A molecular dynamics study of linear and branched polyethylenimine

Molecular dynamics (MD) simulation is a powerful tool for investigating the molecular mechanism of heat conduction in polymers. However, existing MD studies are mostly on monodisperse polymers and the effect of polydispersity, which typically occurs in commercial polymers, remains to be clarified. In this work, various types of polydisperse polyethylenimine (PEI) composed of molecules having different molecular weights and branching structures were constructed by in silico step-growth polymerization. The effect of the polydispersity and molecular structure on heat conduction in PEI was investigated using all-atom MD simulations. The number and weight fraction distributions of the polydisperse purely-linear PEI agreed well with the Flory-Schulz distributions, and therefore it can be concluded that the in silico polymerization used in this work reasonably mimics the step-growth-like polymerization observed in the actual synthesis of PEI. Thermal conductivity increased with increase in the radius of gyration dependent on the degree of branching of the molecules. In addition, thermal conductivity of a polydisperse PEI exhibited a similar value to that of a monodisperse PEI of a representative polymer chain in the polydisperse system. By analyzing in detail the thermal energy transfer among and inside molecules, their microscopic mechanisms could be understood and it was discovered that the average molecular weight is a critical factor in determining heat conduction.

Hyperbranched Polymers Formed Through Self‐Condensing Vinyl Polymerization in a Continuous Stirred‐Tank Reactor (CSTR): 1. Molecular Weight Distribution

AbstractA new Monte Carlo simulation algorithm, based on the random sampling technique, is proposed for the self‐condensing vinyl polymerization conducted in a continuous stirred‐tank reactor. In order to highlight the most fundamental aspects of the reaction system, the kinetic rate constants are assumed to be constant, and the size and structure dependence is neglected. With this ideal model, the high molecular weight (MW) tail of the weight fraction distribution W(P) is shown to follow the power law, W(P) ∝ P −α. The power exponent is represented by α = 1/(ηξ) for ξ &lt; 1/(2η), where ξ and η are defined in the text. For ξ ≥ 1/(2η), the weight‐average MW cannot reach the steady state because of the power exponent with α ≤ 2.

Studies on thermokinetic of Chlorella pyrenoidosa devolatilization via different models

The thermokinetics of Chlorella pyrenoidosa (CP) devolatilization were investigated based on iso-conversional model and different distributed activation energy models (DAEM). Iso-conversional process result showed that CP devolatilization roughly followed a single-step with mechanism function of f(α)=(1−α)3, and kinetic parameters pair of E0=180.5kJ/mol and A0=1.5E+13s−1. Logistic distribution was the most suitable activation energy distribution function for CP devolatilization. Although reaction order n=3.3 was in accordance with iso-conversional process, Logistic DAEM could not detail the weight loss features since it presented as single-step reaction. The un-uniform feature of activation energy distribution in Miura-Maki DAEM, and weight fraction distribution in discrete DAEM reflected weight loss features. Due to the un-uniform distribution of activation and weight fraction, Miura-Maki DAEM and discreted DAEM could describe weight loss features.

Short-duration hydrothermal fermentation of food waste: preparation of soil conditioner for amending organic-matter-impoverished arable soils.

Soil conditioners can be used to compensate for the insufficient soil nutrition and organic matter (OM) of arable soils. However, the traditional preparation method for soil conditioners is lengthy and requires facilities that occupy large tracts of land. In this study, a short-duration hydrothermal fermentation (SHF) method was developed for the preparation of soil conditioners from food waste. The SHF method required only 18h to produce soil conditioners with physicochemical characteristics that were similar to those of soil conditioners obtained through traditional fermentation (TF), which takes 20days to complete. Inoculation and dynamic SHF significantly affected the distribution of different molecular weight fractions in the prepared soil conditioners. The results of spectral analysis revealed that a large amount of soluble microbial byproducts and carbohydrate C formed in the SHF soil conditioner. This effect indicated that the SHF soil conditioner supplied not only immediately available nutrients, which can stimulate the growth of soil microbial communities, but also provided sustained-release nutrients for the long-term cultivation of crops on OM-impoverished soils. Cyanobacteria were present in the SHF soil conditioner at a relative abundance of 6.29%, which was 28.6 and 314.5 times those in the raw material and TF soil conditioner, respectively. Amending OM-impoverished soil with the SHF soil conditioner enhanced the carbon pool management index of the resulting SHF soil by 1.50 and 1.12 times those of unfertilized soil and TF soil, respectively. These results indicated that the SHF soil exhibited a high capacity to supply nutrients for crop growth.

Hyperbranched Polymers Formed Through Irreversible Step Polymerization of AB2‐Type Monomer with Substitution Effect in a Continuous Flow Stirred‐Tank Reactor (CSTR)

A new Monte Carlo simulation method is proposed for the step polymerization of AB2‐type monomer conducted in a continuous flow stirred‐tank reactor (CSTR). The effect of the second B group reactivity, represented by the reactivity ratio r is investigated. The degree of branching (DB) at large degree of polymerization (P) limit, DBP→∞ does not change with the mean residence time . The value of DBP→∞ becomes larger by increasing r and is larger than the corresponding batch polymerization. The weight fraction distribution at high molecular weight tail follows a power law , and a simple formula to predict the power exponent α is proposed. The relationship between the radius of gyration 〈s2〉0 and P does not change with , and large polymers obtained in a CSTR are much more compact than those formed in batch polymerization. CSTR is advantageous to synthesize compact HB polymers, especially with a smaller r‐value.

Hyperbranched Polymers Formed through Irreversible Step Polymerization of AB2‐Type Monomer in a Continuous Flow Stirred‐Tank Reactor (CSTR)

Hyperbranched polymers formed through step polymerization of AB2‐type monomer with equal reactivity for both B groups in a continuous flow stirred‐tank reactor (CSTR) are investigated theoretically. The weight fraction distribution at high molecular weight tail follows a power law, W(P) ∝ P −1/ξ for ξ ≤ 0.5 with , where is the mean residence time. The degree of branching (DB) at the large degree of polymerization (P) limit is DB P→∞ = 0.6 irrespective of the ξ‐value, which is larger than the case for the corresponding batch polymerization that gives DB P→∞ = 0.5. The relationship between the radius of gyration 〈s 2〉0 and P shows that the hyperbranched polymers formed in a CSTR are very compact, and the 〈s 2〉0‐values for large polymers are even smaller than the smallest possible case for a batch reactor with DB P→∞ = 1. For large polymers, the power law 〈s 2〉0∝P 1/3 holds, which is 〈s 2〉0∝P 1/2 for batch polymerization. image

Plasmonic enhancement of visible-light water splitting with Au–TiO2 composite aerogels

We demonstrate plasmonic enhancement of visible-light-driven splitting of water at three-dimensionally (3D) networked gold-titania (Au-TiO2) aerogels. The sol-gel-derived ultraporous composite nanoarchitecture, which contains 1 to 8.5 wt% Au nanoparticles and titania in the anatase form, retains the high surface area and mesoporosity of unmodified TiO2 aerogels and maintains stable dispersion of the ~5 nm Au guests. A broad surface plasmon resonance (SPR) feature centered at ~550 nm is present for the Au-TiO2 aerogels, but not Au-free TiO2 aerogels, and spans a wide range of the visible spectrum. Gold-derived SPR in Au-TiO2 aerogels cast as films on transparent electrodes drives photoelectrochemical oxidation of aqueous hydroxide and extends the photocatalytic activity of TiO2 from the ultraviolet region to visible wavelengths exceeding 700 nm. Films of Au-TiO2 aerogels in which Au nanoparticles are deposited on pre-formed TiO2 aerogels by a deposition-precipitation method (DP Au/TiO2) also photoelectrochemically oxidize aqueous hydroxide, but less efficiently than 3D Au-TiO2, despite having an essentially identical Au nanoparticle weight fraction and size distribution. For example, 3D Au-TiO2 containing 1 wt% Au is as active as DP Au/TiO2 with 4 wt% Au. The higher photocatalytic activity of 3D Au-TiO2 derives only in part from its ability to retain the surface area and porosity of unmodified TiO2 aerogel. The magnitude of improvement indicates that in the 3D arrangement either a more accessible photoelectrochemical reaction interphase (three-phase boundary) exists or more efficient conversion of excited surface plasmons into charge carriers occurs, thereby amplifying reactivity over DP Au/TiO2. The difference in photocatalytic efficiency between the two forms of Au-TiO2 demonstrates the importance of defining the structure of Au[parallel]TiO2 interfaces within catalytic Au-TiO2 nanoarchitectures.

A Bayesian Approach to Establishing a Reference Particle Size Distribution in the Presence of Outliers

The presence of observations or measurements that are unlike the majority is fairly common in studies conducted to establish particle size (or weight fraction) distributions. Therefore, there is a need to develop methods that are capable of producing estimates of particle size distributions that are not overly sensitive to the presence of a few observations that might be considered outliers. This article proposes a type of contamination mixture model that probabilistically allocates each observation to either a majority component or a contamination component. Observations that are allocated to a contamination component are down-weighted when estimating the particle size distribution (while the uncertainty of contamination classification is automatically accounted for in estimation). Computational methods are developed and the utility of the proposed methodology is demonstrated via a simulation study. The method is then applied to data produced from an inter-laboratory study conducted to establish a particle size distribution in cement.