Mixing Rules Research Articles

Morphology, viscoelastic properties and the mechanical performance of highly toughened polyamide 6/acrylonitrile–butadiene–styrene/polylactic acid ternary blends

AbstractIn this research, the morphology, rheology, and mechanical performance of polyamide 6/polylactic acid/acrylonitrile–butadiene–styrene (PA6/ABS/PLA) ternary blends were studied. The optimum ABS percentage in PA6/ABS binary blend to achieve highly toughened blend was determined around 35 wt.%. The spreading coefficient (Sij) predictions showed that each of PLA and ABS can form separate phase in the PA6 matrix. The evaluation of the interfacial interactions in the ternary blends revealed a higher tendency of PLA to interact with PA6 chains (Sij:−1.20) rather than ABS (Sij:−0.20) which is in comply with higher work of adhesion of PLA/PA6 than PLA/ABS and PA6/ABS. By adding PLA to the binary blends of PA6/ABS, it was found that the ABS domain size decreased due to rheological change. The localization of each component in the ternary blends observed in SEM analysis was in agreement with spreading coefficient predictions. Morphological evaluations revealed higher interfacial interactions between PA6 and PLA in the blends with up to 20 wt.% PLA. The optimized blend ratio of ternary blends showed higher complex viscosity at low shear rates and higher values than the mixing rule prediction. The optimum flexural modulus of 2 GPa and impact resistance of 47 kJ/m2 were achieved at 20 wt.% PLA in PA6/ABS/PLA ternary blend.Highlights The optimum PA6/ABS blend ratio to achieve high toughness was determined. The phase localization in the blends predicted via Spreading Coefficient. The interaction of PA6 and ABS enhanced by the addition of PLA in the blends. The mechanical properties of blends are correlated with rheological behavior. The flexural modulus of blends was higher than Halpin–Tsai model predictions.

Estimation of the interaction parameters between carbon dioxide and an organic solvent by the Peng–Robinson equation of state via an artificial neural network

The equation of state (EoS) is a tool for estimating the thermodynamic and physical properties of compounds, including mixtures, across a range of temperatures and pressures. When dealing with mixtures, a mixing rule is required to calculate the mixture parameters. Mixing rules may involve interaction parameters, such as kij and lij, that correct for differences between components. However, obtaining this data requires specialized equipment and techniques and significant measurement time, resulting in limited reported EoS parameters. In this study, we introduce an artificial neural network (ANN) to predict interaction parameters in the van der Waals one-fluid mixing rule. These parameters are used to calculate the physical properties of mixtures using the Peng–Robinson (PR) EoS. The interaction parameters are used in two cases, namely the one-parameter and two-parameter mixing rules (OP and TP, respectively), in which only kij and both kij and lij are employed, respectively. The vapor–liquid equilibrium (VLE) data of CO2/organic solvent binary systems are collected and correlated by the PR EoS to construct a database of 1286 and 1292 parameters for the OP and TP, respectively. The molecular weight, critical temperature and pressure, acentric factor of the organic solvent, and temperature are used as input parameters for the ANN. In addition, we optimize the structure of the ANN by changing the activation function, number of neurons, and number of hidden layers. The optimized ANN uses a tanh activation function. Hidden layers are used for both the OP and TP, along with 40 and 50 neurons, respectively. The results confirm that the model can determine the interaction parameters of the PR EoS, which can be used to estimate the VLE. These results are useful for incorporation into process simulators for chemical process design.

Density-Gibbs energy correlation models for binary biofuel mixtures

The density of binary biofuel mixtures significantly influences fuel properties, directly affecting internal combustion engine performance. This research aims to develop correlational models to calculate densities in binary biofuel mixtures, leveraging the Gibbs energy mixing rule. Two correlation models are proposed: the ideal Gibbs energy mixing rule and the Gibbs energy quadratic mixing rule. These models aim to enhance the cross-interaction of Gibbs energy of activation (ΔG12) to improve predictive performance. Importantly, the constants of the existing models are directly related to thermodynamic parameters. To assess the accuracy of the models, observed densities were collected from four different binary systems: biodiesel + diesel, pure fatty acid ethyl ester + alcohol, biodiesel + alcohol, and alcohol binary mixtures, amounting to 2242 data points obtained from the literature. Based on mole fractions, both models were compared to earlier approaches. The examination revealed that the equation formulated by the quadratic mixing rule exhibited heightened precision and efficacy. Our data demonstrate that the extended ΔG12 provided the most accurate results.

Multi-species modeling in the particle-based ellipsoidal statistical Bhatnagar-Gross-Krook method including internal degrees of freedom

The implementation of the ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) method in the open-source particle code PICLas is extended for multi-species modeling of polyatomic molecules, including internal energies with multiple vibrational degrees of freedom. For this, the models of Mathiaud et al. [1], Pfeiffer [2] and Brull [3,4] are combined. In order to determine the transport coefficients of the gas mixture, Wilke's mixing rules and collision integrals are compared. The implementation is verified with simulation test cases of a supersonic Couette flow as well as a hypersonic flow around a 70° blunted cone. The solutions of the ESBGK method are compared to the Direct Simulation Monte Carlo (DSMC) method to assess the accuracy, where overall good agreement is achieved. In general, collision integrals should be preferred for the determination of the transport coefficients, since the results using Wilke's mixing rules show larger deviations in particular for large mass ratios. Considering the computational efficiency of the methods, a considerable reduction in computational time is possible with the ESBGK model.

Thermophysical properties of the liquid organic hydrogen carrier system based on benzyltoluene considering influences of isomerism and dissolved hydrogen

The benzyltoluene (BT)-based liquid organic hydrogen carrier (LOHC) system currently considered for large-scale applications represents a mixture of regioisomers. To characterize this complex system, a comprehensive experimental database for various thermophysical properties of synthesized BT isomers and their mixtures without and with the presence of hydrogen (H2) close to vapor-liquid equilibrium is established using optical and conventional techniques at process-relevant temperatures and pressures up to 573 K and 6 MPa. The surface tension varies by less than 4% among the dehydrogenated (H0-BT) or hydrogenated (H12-BT) isomers. The density and viscosity of mixtures of H0-BT or H12-BT isomers can be described by simple mixing rules with average absolute relative deviations of 0.022% and 0.38%. With increasing H2 pressure, the viscosity remains nearly constant, while the interfacial tension decreases by up to 5%. The thermal and mutual diffusivity of H12-ortho-BT containing dissolved H2 at 6 MPa decrease and increase with increasing temperature.

Determination of Partial Propagation Velocity and Partial Isentropic Compressibility Coefficient in Water-Ethanol System.

This study introduces an innovative approach to the layered model, emphasizing the physical-chemical characterization of miscible liquid systems through ultrasonic techniques, with a specific focus on the water-ethanol system used in pharmaceutical formulations. Traditional characterization methods, while effective, face challenges due to the complex nature of solutions, such as the need for large pressure variations and strict temperature control. The proposed approach integrates partial molar volumes and partial propagation velocity functions into the layered model, enabling a nuanced understanding of miscibility and interactions. Ultrasonic techniques are used to calculate the isentropic compressibility coefficient for each component of the mixture as well as the total value using an additive mixing rule. Unlike conventional methods, this technique uses tabulated and experimental data to estimate the propagation velocity in the mixture, leading to a more precise computation of the isentropic compressibility coefficient. The results indicate a significant improvement in predicting the behavior of the water-ethanol system compared to the classical layered model. The methodology demonstrates the potential to provide new physicochemical insights that can be applied to other miscible systems beyond water-ethanol. This research has implications for improving the efficiency and accuracy of liquid medication formulations in the pharmaceutical industry.

Generalization of the Wong-Sandler mixing rule to a generic cubic equation of state: Examples of use for systems of industrial interest (Hydrogen, CCUS, refrigeration)

Cubic equations of state (EoS) remain the most widely used models in chemical engineering, and their predictive capabilities are highly dependent on mixing rules (MR). Considered as one of the most accurate, Wong Sandler's (WS) MR, have been generalized in this work, to allow its use with all 2-parameter (PR, SRK, …) and 3-parameter (PT, mPT, NEOS, …) cubic EoSs. The procedure for implementing the generalized WS MR is presented and tested with PR, PT and its variants (mPT and NEOS) on vapor-liquid equilibrium (VLE) and density data on various classes of mixtures for different types of use (hydrogen, CCUS, refrigeration). The results clearly show the benefits of this type of MR, especially around the critical point and for density data without considering volume translation frequently used with PR and SRK EoS.

Mixing Rules for Left-Handed Disordered Metamaterials: Effective-Medium and Dispersion Properties.

Left-handed materials are known to exhibit exotic properties in controlling electromagnetic fields, with direct applications in negative reflection and refraction, conformal optical mapping, and electromagnetic cloaking. While typical left-handed materials are constructed periodic metal-dielectric structures, the same effect can be obtained in composite guest-host systems with no periodicity or structural order. Such systems are typically described by the effective-medium approach, in which the components of the electric permittivity tensor are determined as a function of individual material properties and doping concentration. In this paper, we extend the discussion on the mixing rules to include left-handed composite systems and highlight the exotic properties arising from the effective-medium approach in this framework in terms of effective values and dispersion properties.

Dielectric Properties of PEEK/PEI Blends as Substrate Material in High-Frequency Circuit Board Applications.

Substrate materials for printed circuit boards must meet ever-increasing requirements to keep up with electronics technology development. Especially in the field of high-frequency applications such as radar and cellular broadcasting, low permittivity and the dielectric loss factor are key material parameters. In this work, the dielectric properties of a high-temperature, thermoplastic PEEK/PEI blend system are investigated at frequencies of 5 and 10 GHz under dried and ambient conditions. This material blend, modified with a suitable filler system, is capable of being used in the laser direct structuring (LDS) process. It is revealed that the degree of crystallinity of neat PEEK has a notable influence on the dielectric properties, as well as the PEEK phase structure in the blend system developed through annealing. This phenomenon can in turn be exploited to minimize permittivity values at 30 to 40 wt.-% PEI in the blend, even taking into account the water uptake present in thermoplastics. The dielectric loss follows a linear mixing rule over the blend range, which proved to be true also for PEEK/PEI LDS compounds.

PHAHST Potential: Modeling Sorption in a Dispersion-Dominated Environment.

PHAHST (potentials with high accuracy, high speed, and transferability) is a recently developed force field that utilizes exponential repulsion, multiple dispersion terms, explicit many-body polarization, and many-body van der Waals interactions. The result is a systematic approach to force field development that is computationally practical. Here, PHAHST is employed in the simulation for rare gas uptake of krypton and xenon in the metal-organic material, HKUST-1. This material has shown promise in use as an adsorptive separating agent and presents a challenge to model due to the presence of heterogeneous interaction sorption surfaces, which include pores with readily accessible, open-metal sites that compete with dispersion-dominated pores. Such environments are difficult to simulate with commonly used empirical force fields, such as the Lennard-Jones (LJ) potential, which perform better when electrostatics are dominant in determining the nature of sorption and alone are incapable of modeling interactions with open-metal sites. The effectiveness of PHAHST is compared to the LJ potential in a series of mixed Kr-Xe gas simulations. It has been demonstrated that PHAHST compares favorably with experimental results, and the LJ potential is inadequate. Overall, we establish that force fields with physically grounded repulsion/dispersion terms are required in order to accurately model sorption, as these interactions are an important component of the energy. Furthermore, it is shown that the simple mixing rules work nearly quantitatively for the true pair potentials, while they are not transferable for effective potentials like LJ.

Carbon Dioxide Solubility in Three Bis Tri (Fluromethylsulfonyl) Imide-Based Ionic Liquids.

This study delves into the necessity of mitigating carbon dioxide (CO2) emissions, focusing on effective capture methods to combat global warming by investigating the solubility of CO2 in three ionic liquids (ILs), 1-Decyl-3-MethylimidazoliumBis (Trifluromethylsulfonyl Imide) [IL1], 1-Hexadecyl-3-Methyl imidazoliumBis (Trifluromethylsulfonyl Imide) [IL2] and Triethytetradecyl Ammonium Bis (Trifluromethylsulfonyl Imide) [IL3]. Solubility experiments were conducted at (30, 50 and 70) °C with pressures up to 1.5 MPa. The research shows [IL2] as the superior candidate for CO2 capture, with its longer alkyl chain, and is confirmed by its lower Henry's Law constant. Utilizing the Peng Robinson equation of state, the study correlates well with the solubility measurements using three mixing rules. The study reveals promising results for IL1, IL2 and IL3 surpassing all other published ionic liquids including Selexol/Genesorb 1753, except for 1-Methyl-3-octylimidazolium bis(trifluoromethylsulfonyl)imide. Insights into the enthalpy and entropy of absorption underscore the significant impact of IL structure on CO2 solubility, emphasizing the potential of tailored ILs for advanced carbon capture strategies. In summary, this research highlights [IL2] as the optimal choice for CO2 capture, offering valuable contributions to the ongoing efforts in combating climate change.

Research on the determination method of biomass dielectric properties based on mixing rules

Research on the determination method of biomass dielectric properties based on mixing rules

An experimental and modelling investigation on the water content of CO2and CO2-rich mixtures using the differential scanning hygrometry (DSH) method

Monitoring humidity downstream to conditioning facilities and during transportation is essential for avoiding hydrate deposition. However, water inline monitoring under high pressure is still challenging in the CCS industry. This study presents an experimental and modelling investigation for enhancing field monitoring and model predictions. Measurements are performed using the Differential Scanning Hygrometry (DSH). This novel analytical approach has been successfully tested for measuring dew/frost temperatures for carbon dioxide, CH4+CO2, and CO2-rich mixtures in equilibrium with hydrates, free water and ice. Moreover, the DSH method has been applied for direct HP equilibrium temperature measurements. Also, this work compares three modified versions of the classical SRK EoS with the Multi-Fluid Helmholtz Energy Approximation (MFHEA). This evaluation includes Huron-Vidal and the EMS mixing rules and the cubic-plus association (CPA) approach. A thorough fitting process was carried out and, overall, comparisons with the experimental data showed that SRK + EMS yielded results as satisfactory as sCPA.

Molecular dynamics simulation of argon isochoric transition to supercritical state

AbstractThe effect of the initial atoms distribution on the molecular dynamics (MD) simulation of a model atomic fluid (argon) is investigated for the case of the isochoric phase transition to the supercritical state. In particular, the case of uniformly distributed atoms in the simulation domain is compared with the case of separated liquid and vapor atoms. The sensitivity of simulations to asymmetric nanoscale perturbations in the boundary is also studied. Despite its high computational cost, the MD approach has the potential to successfully address long‐standing problems in computational fluid dynamics (CFD), especially those associated with mathematical singularities, such as contact angles, vortices, phase transitions and so forth. Unlike conventional CFD simulations, where the initial condition is the pressure or velocity distribution in the simulation domain, MD simulations also require the initial position of each molecule. Thus, it is important to understand whether a judicious choice of the initial distribution of molecules can reduce the overall computation time of the simulation. The evolution of the model fluid system during the phase transition was simulated using a Lennard‐Jones interatomic potential, corrected with the Lorentz–Berthelot mixing rule for the interactions with the solid walls. The system was allowed to relax until equilibrium, and then a Heaviside temperature step was applied to the wall to bring the system to supercritical conditions. Results show the initial choice of the atoms distribution can significantly affect the computational time, while the effect of asymmetric perturbations on the boundary is negligible.

Rheological study of hybrid aqueous suspension of TEMPO-oxidized cellulose nanofibrils and graphene oxide

The incorporation of graphene oxide (GO) into a nanocellulose matrix has attracted considerable attention due to the unique advantages of both components. This study focuses on investigating the viscoelastic and flow properties of hybrid aqueous suspensions (2.00 w/v%), composed of TEMPO-oxidized cellulose nanofibrils (TOCNF) and GO at different TOCNF/GO weight ratios. To adjust the elastic properties of the hybrid suspensions, calcium ions are introduced, varying the concentration systematically to study their effects on the hybrid network structure. All blends exhibit shear-thinning behaviour and demonstrate elastic, gel-like properties. Notably, in the absence of calcium ions, the enhancement of elastic properties is more pronounced at higher GO fractions. Conversely, with the introduction of calcium ions, the enhancement of elastic properties becomes particularly important at higher TOCNF fractions. For the quantitative evaluation of these enhancements, we employ the logarithmic mixing rule. Significant positive deviations from the predictions of the logarithmic mixing rule are ascribed to the complex, concentration-dependent arrangement of cellulose nanofibrils and GO liquid crystals in the aqueous suspension, coupled with ionic crosslinking induced by calcium ions. The study aims to contribute to the understanding of the rheological behaviour of the TOCNF/GO hydrogel, showing potential advancements in various applications.Graphical abstract

Testing mixing rules for structural and dynamical quantities in multi-component crowded protein solutions.

Crowding effects significantly influence the phase behavior and the structural and dynamic properties of the concentrated protein mixtures present in the cytoplasm of cells or in the blood serum. This poses enormous difficulties for our theoretical understanding and our ability to predict the behavior of these systems. While the use of course grained colloid-inspired models allows us to reproduce the key physical solution properties of concentrated monodisperse solutions of individual proteins, we lack corresponding theories for complex polydisperse mixtures. Here, we test the applicability of simple mixing rules in order to predict solution properties of protein mixtures. We use binary mixtures of the well-characterized bovine eye lens proteins α and γB crystallin as model systems. Combining microrheology with static and dynamic scattering techniques and observations of the phase diagram for liquid-liquid phase separation, we show that reasonably accurate descriptions are possible for macroscopic and mesoscopic signatures, while information on the length scale of the individual protein size requires more information on cross-component interaction.

Improvement of augmented free-water flash algorithm based on HV mixing rule and simulated annealing optimization algorithm

The calculation of three-phase equilibrium in CO2-hydrocarbon-water mixtures holds significant importance within numerical simulations, particularly in applications such as CO2-enhanced oil recovery and carbon dioxide sequestration. However, due to the non-ideality of CO2 and the polarity of water, three-phase equilibrium calculations often encounter convergence challenges. The augmented free-water flash algorithm [6], specifically focusing on CO2 dissolution in the aqueous phase, addresses the convergence challenges encountered by traditional three-phase flash algorithms. Despite its accuracy diminishes under conditions of high pressure and high CO2 concentrations, limiting its capability to accurately describe CO2 dissolution in oil-gas-water multiphase systems. The GE (excess Gibbs free energy) type mixing rule can be effectively integrated with equations of state, including PR and SRK, by utilizing activity models like NRTL and UNIFAC, which can be applied to the calculation of thermodynamic parameters for both nonpolar and polar systems and high-pressure complex systems. In this study, based on the augmented free-water assumption, we have developed a new augmented free-water three-phase flash algorithm for CO2/hydrocarbon/water mixtures. This algorithm integrates the PR equation of state with the NRTL activity model, employing the HV mixing rule for CO2-water interactions and the Van der Waals rule for other non-polar components. Furthermore, to enhance computational efficiency, the algorithm incorporates the simulated annealing global optimization algorithm, replacing Newton iteration to more effectively identify the global minima of the complex objective function. The example calculations show that the improved augmented free-water flash algorithm has higher accuracy and efficiency than the traditional augmented free-water flash algorithm. The augmented free-water three-phase flash algorithm proposed in this paper offers a precise model for phase equilibrium calculations essential for numerical simulations in CO2-enhanced oil recovery and carbon dioxide sequestration.

Bulk and interfacial thermodynamics of ammonia, water and their mixtures

Ammonia is a promising energy carrier for the green transition, but its hygroscopicity and toxicity necessitate in-depth understanding of its interaction with water. This work examines the bulk and interfacial thermodynamics of the ammonia–water system. Parameters for three equations of state are fitted to experimental data and compared to parameters from literature: PC-SAFT, Cubic Plus Association and Peng–Robinson. Peng–Robinson stands out as most accurate for bulk thermodynamics. Introducing a temperature-dependent volume shift for water with Peng–Robinson yields a highly accurate model without introducing problematic inconsistencies, with errors of 0.05% for saturation pressures, and 0.5% for liquid densities. For the mixture, Peng–Robinson with a two-parameter Huron–Vidal mixing rule reproduces measurements mostly within their uncertainties, whereas the standard mixing rules for PC-SAFT and CPA are less accurate. A literature review of surface tension measurements of ammonia–water mixtures reveals that accurate measurements exist only at ambient temperature. We apply density gradient theory and density functional theory based on PC-SAFT, finding that both models fail at reproducing qualitative features of the surface tensions and adsorptions of dilute solutions of aqueous ammonia. Whereas bulk properties are well characterized, understanding and describing the interfacial thermodynamics of the ammonia–water system demands more work both on the experimental and modeling side.

Tailoring the Structure-Property Relationship of Ring-Opened Metathesis Copolymers for CO2-Selective Membranes.

In this work, we explore the use of ring-opening metathesis polymerization (ROMP) facilitated by a second-generation Grubbs catalyst (G2) for the development of advanced polymer membranes aimed at CO2 separation. By employing a novel copolymer blend incorporating 4,4'-oxidianiline (ODA), 1,6-hexanediamine (HDA), 1-adamantylamine (AA), and 3,6,9-trioxaundecylamine (TA), along with a CO2-selective poly(ethylene glycol)/poly(propylene glycol) copolymer (Jeffamine2003) and polydimethylsiloxane (PDMS) units, we have synthesized membranes under ambient conditions with exceptional CO2 separation capabilities. The strategic inclusion of PDMS, up to a 20% composition within the PEG/PPG matrix, has resulted in copolymer membranes that not only surpass the 2008 upper limit for CO2/N2 separation but also meet the commercial targets for CO2/H2 separation. Comprehensive analysis reveals that these membranes adhere to the mixing rule and exhibit percolation behavior across the entire range of compositions (0-100%), maintaining robust antiplasticization performance even under pressures up to 20 atm. Our findings underscore the potential of ROMP in creating precisely engineered membranes for efficient CO2 separation, paving the way for their application in large-scale environmental and industrial processes.

Estimation of the Shear Viscosity of Mixed-Polymer Materials for Screw Extrusion-Based Recycling Process Modeling.

The scope of this work is the development of a method to estimate the temperature and shear rate-dependent viscosity of mixtures composed of two polymers. The viscosity curve of polymer mixtures is crucial for the modeling and optimization of extrusion-based recycling, which is the most efficient way to recycle polymeric materials. The modeling and simulation of screw extruders requires detailed knowledge of the properties of the processed material, such as the thermodynamic properties, the density, and the rheological behavior. These properties are widely known for pure materials; however, the incorporation of impurities, like other polymers in recycled materials, alters the properties. In this work, miscible, immiscible, and compatibilized immiscible polymer mixtures are considered. A new method based on shear stress is proposed and compared to the shear rate-based method. Several mixing rules are evaluated for their accuracy in predicting mixture viscosity. The developed methods allow the prediction of the viscosity of a compatibilized immiscible mixture with deviations below 5% and that of miscible polymer mixtures with deviations below 3.5%.