Polymer Diffusion Research Articles

Lie symmetry analysis of a pseudoparabolic PDE: Power law in diffusion coefficient with constant viscosity

We perform symmetry analysis of a partial differential equation (PDE) of pseudoparabolic form which models solvent uptake in polymeric solids; a special case of power law for diffusion coefficient and constant viscosity is considered. Optimal systems of one-dimensional subalgebras for various model parameters are employed for symmetry reductions and construction of invariant solutions. • A phenomenon of case II diffusion in glassy polymers is considered. • A model equation is an implicit pseudoparabolic partial differential equation which incorporates the diffusion coefficient and viscosity of the polymer. • The model equation is studied from Lie symmetry analysis view point for a special case of power law in diffusion coefficient and constant viscosity. • Some exact solutions are obtained and presented graphically.

Temperature, diffusion, and stress modeling in filament extrusion additive manufacturing of polyetherimide: An examination of the influence of processing parameters and importance of modeling assumptions

Fused filament fabrication (FFF), a form of filament-based material extrusion additive manufacturing (AM), is an extremely useful technique for the rapid production of highly customized products; however, empirical evidence is heavily relied upon for understanding of the process. Initial modeling attempts have traditionally focused on predicting heat transfer and either interlayer diffusion and adhesion or stress development but have not taken a combined approach to analyze all three components simultaneously in a multiphysics model. In this study, we implement finite difference models to examine the combined heat transfer, polymer diffusion represented as degree of healing (Dh), and residual stress development in FFF of poly(ether imide) (PEI). Printing with PEI is of great interest because of its desirable mechanical properties and high use temperatures, but it also creates a more challenging modeling problem with higher thermal gradients and greater potential thermal processing window compared to traditionally modeled AM materials, such as acrylonitrile-butadiene-styrene (ABS) and polylactic acid (PLA). The larger processing window can potentially allow for more processing options but can also significantly complicate the optimization process. In this study, experimental analyses including trouser tear tests and part warpage measurements provide correlation to predicted Dh and stress levels. The models suggest that the temperature of a layer is influenced by the subsequent printing of up to at least three layers in the geometry studied. The results of this study further demonstrate the sensitivity of the molecular mobility and degree of healing to the reptation time (τrep), such that a small change in the τrep on the order of 2–3x can result in an order of magnitude difference in the time before interfacial healing can begin, culminating in significantly less healing occurring. Furthermore, the reptation time and subsequent healing predictions are highly reliant on the extrapolation method used to extend the reptation time to temperatures below those at which it was measured resulting in significantly different predictive results, even if the same experimental data is used.

Diffusion Model of Parallel Plate Crack Grouting Based on Foaming Expansion Characteristics of Polymer Slurry

Polymers as a new chemical grouting material have been widely used in fractured rock mass; however, the understanding of polymer diffusion characteristics still needs to be further improved. In order to study the diffusion mechanism of foamed polymer slurry in rock fissures, the radial diffusion model of polymer single crack grouting is derived in consideration of the factors such as grouting volume, crack width and expansion rate. The influence of different factors on slurry diffusion radius, diffusion pressure and flow rate is analyzed, The diffusion model is verified by finite element numerical simulation. The findings show that (1) The results of slurry diffusion radius, pressure and velocity distribution at different times under different working conditions in the present model are in good agreement with the analytical solution; (2) The diffusion pressure is directly proportional to the grouting volume and expansion multiple, and inversely proportional to the crack width. In addition, diffusion pressure decreases with the increase of diffusion distance, and the pressure at the corresponding distance increases slowly with time, and finally tends to be stable; (3) For the same section, the radial velocity decreases slowly with the increase of time; for different sections, the flow velocity increases sharply with the increase of the distance between the section and the central axis of the grouting hole.

Molecular Dynamics Study of the Conformation, Ion Adsorption, Diffusion, and Water Structure of Soluble Polymers in Saline Solutions.

Polymers have interesting physicochemical characteristics such as charge density, functionalities, and molecular weight. Such attributes are of great importance for use in industrial purposes. Understanding how these characteristics are affected is still complex, but with the help of molecular dynamics (MD) and quantum calculations (QM), it is possible to understand the behavior of polymers at the molecular level with great consistency. This study was applied to polymers derived from polyacrylamide (PAM) due to its great use in various industries. The polymers studied include hydrolyzed polyacrylamide (HPAM), poly (2-acrylamido-2-methylpropanesulfonate) (PAMPS), polyacrylic acid (PAA), polyethylene oxide polymer (PEO), and guar gum polysaccharide (GUAR). Each one has different attributes, which help in understanding the effects on the polymer and the medium in which it is applied along a broad spectrum. The results include the conformation, diffusion, ion condensation, the structure of the water around the polymer, and interatomic polymer interactions. Such characteristics are important to selecting a polymer depending on the environment in which it is found and its purpose. The effect caused by salinity is particular to each polymer, where polymers with an explicit charge or polyelectrolytes are more susceptible to changes due to salinity, increasing their coiling and reducing their mobility in solution. This naturally reduces its ability to form polymeric bridges due to having a polymer with a smaller gyration radius. In contrast, neutral polymers are less affected in their structure, making them favorable in media with high ionic charges.

Influence on fracture toughness arising from controlled morphology of multiphase toughened epoxy resins in the presence of fibre reinforcement

Numerous studies found in the literature focus on enhancing the fracture toughness of thermosetting polymers by incorporating various toughening agents. The resulting gains in fracture toughness are mostly attributed to an expansion in the volume of the plastic deformation zone ahead of the propagating crack and the modified resin blends are subsequently introduced as matrices for fibre reinforced composite materials. It is often assumed that improvements in fracture toughness of the polymer matrix are directly transferred to the resulting fibre reinforced composites. This paper shows that this is not necessarily the case. The present study is focused on the fracture behaviour and phase morphology of the continuous carbon fibre reinforced polymer composites (CFRPs) based on resin systems modified by including carbon nanotubes and a phase separating thermoplastic (TP) toughener. It is shown that the presence of fibre reinforcement influences the phase distribution in TP modified epoxies, where the toughening potential of the deformable TP phase is curtailed. Incorporating high aspect-ratio CNTs is an effective means to control and enhance the uniformity of the multiphase morphology, hence further improving the interlaminar properties of CFRPs. This is shown to be due to the interference of the CNTs in the kinetics of the reaction induced phase separation by affecting the polymer diffusion rate.

A Power Sequence Interaction Function for Liquid Phase Particles

In this manuscript, a function is derived that allows the interactions between the atoms/molecules in nanoparticles, nanodrops, and macroscopic liquid phases to be modeled. One goal of molecular theories is the development of expressions to predict specific physical properties of liquids for which no experimental data are available. A big limitation of reliable applications of known expressions is that they are based on the interactions between pairs of molecules. There is no reason to suppose that the energy of interaction of three or more molecules is the sum of the pairwise interaction energies alone. Here, an interaction function with the limit value w = e2π/e is presented, which allows for the derivation of the atomic mass unit and acts as a bridge between properties of elementary particles and emergent properties of macroscopic systems. The following properties of liquids are presented using the introduced interaction function: melting temperatures of n-alkanes, nanocrystals of polyethylene, melting temperatures of metal nanoparticles, solid–liquid phase transition temperatures for water in nanopores, critical temperatures and critical pressures of n-alkanes, vapor pressures in liquids and liquid droplets, self-diffusion coefficients of compounds in liquids, binary liquid diffusion coefficients, diffusion coefficients in liquids at infinite dilution, diffusion in polymers, and viscosities in liquids.

Material and process engineering aspects to improve the quality of the bonding layer in a laser-assisted fused filament fabrication process

The mechanical properties of components manufactured ("printed") by the Fused Filament Fabrication (FFF) process are usually limited by the cohesion achieved between melt-processed thermoplastic filament strands perpendicular to the filament deposition direction. This significantly limits the usefulness of the FFF process and commercial FFF printers for the production of a wide range of functional components and consumer goods, so that the FFF process is currently used for the production of prototypes for visualization purposes only. In order to open up potential markets for structural components in the future, the realization of an optimized bonding layer quality via a significantly improved cohesion between the filament strands is essential. Since the mechanical properties of polymeric materials generally depend on the density of molecular entanglements and tie molecules in the material structure, it is of the utmost importance to realize the highest possible density of such entanglements and tie molecules between the individual filament layers also in FFF components. The present work deals with different methods of heat input to improve the entanglement and tie molecule density in the bonding layer of FFF components through enhanced polymer diffusion at temperatures far above the glass transition temperature for amorphous plastics or the melting temperature for semi-crystalline plastics, respectively. Heat can be either globally or locally introduced by conduction, convection, and radiation, the latter being the most effective for the FFF process when applied locally via properly controlled laser radiation. Ideally, a compact, cost effective and easy to use radiation source is used for this purpose. Hence, a setup based on a diode laser is presented in this work. Furthermore, an idealized calculation of the amount of heat required to promote the formation of interfilament bonds and thus to achieve an increased mechanical property profile was carried out. Two types of materials commonly used for the FFF process, polylactic acid (PLA) and acrylic-butadiene-styrene copolymer (ABS), were used for the experimental procedure. After producing a special monofilament with improved laser absorption characteristics, test specimens were produced. As a reference, test specimens made of the same materials were printed with the conventional FFF process. The differently manufactured test specimens were characterized optically and visually, using a stereomicroscope, and mechanically, using tensile tests. In addition, specimens were produced with a specially designed device for local heat input onto the substrate layer region just ahead of the new filament deposition. The effectiveness of this laser-assisted method could be proven in tests by adjusting the process parameters of the FFF process. After finding the optimum printing speed (equivalent to the laser power applied) for the materials PLA and ABS, an optical examination of the fracture surfaces of test specimens with filaments laid down perpendicular to an applied tensile load showed a significant increase in plastic deformation capacity, indicative of an improved interfilament bonding. This was confirmed by the mechanical property values determined in tensile tests. Thus, for optimized printing speeds, an increase in tensile strength values of ~20% for PLA and ~7% for ABS was achieved. For elongation at break, the improvements were ~21% for PLA and ~34% for ABS, respectively. The methodology of laser-assisted FFF printing thus offers great potential for mechanically and structurally optimized functional components with specifically introduced anisotropy. • Interfilament entanglements control strength of printed polymer parts. • Novel diode laser-assisted fused filament fabrication process (FFF) is presented. • Enhanced local laser absorption and heat introduction via specific additives. • improved bonding layer strength of FFF components was achieved.

Polymer-Based Biocompatible Packaging for Implantable Devices: Packaging Method, Materials, and Reliability Simulation.

Polymer materials attract more and more interests for a biocompatible package of novel implantable medical devices. Medical implants need to be packaged in a biocompatible way to minimize FBR (Foreign Body Reaction) of the implant. One of the most advanced implantable devices is neural prosthesis device, which consists of polymeric neural electrode and silicon neural signal processing integrated circuit (IC). The overall neural interface system should be packaged in a biocompatible way to be implanted in a patient. The biocompatible packaging is being mainly achieved in two approaches; (1) polymer encapsulation of conventional package based on die attach, wire bond, solder bump, etc. (2) chip-level integrated interconnect, which integrates Si chip with metal thin film deposition through sacrificial release technique. The polymer encapsulation must cover different materials, creating a multitude of interface, which is of much importance in long-term reliability of the implanted biocompatible package. Another failure mode is bio-fluid penetration through the polymer encapsulation layer. To prevent bio-fluid leakage, a diffusion barrier is frequently added to the polymer packaging layer. Such a diffusion barrier is also used in polymer-based neural electrodes. This review paper presents the summary of biocompatible packaging techniques, packaging materials focusing on encapsulation polymer materials and diffusion barrier, and a FEM-based modeling and simulation to study the biocompatible package reliability.

Film Formation of Waterborne 2K Polyurethanes: Effect of Polyols Containing Different Carboxylic Acid Content

While the interaction of NCO with carboxylic acids has been documented, a comprehensive knowledge of its effect on the film formation in a two-component waterborne polyurethane (2K-WPU) system is lacking. In this work, we synthesized a series of carboxyl-functional acrylic polyols (Mn ≈ 5000 g/mol, Đ ≈ 2.7, Tg ≈ 15 °C) of uniform diameter (120 nm) with varying amounts of methacrylic acid introduced into the latex. For 2K-WPU dispersions with a molar NCO/OH ratio of 1.3:1, we monitored particle size by dynamic light scattering and NCO consumption by FTIR. In the dispersion, we found that the particle size remained nearly constant, and the NCO was almost completely consumed after 2 days. Incorporating COOH in the polyol latex accelerated the consumption of NCO. We synthesized polyol samples labeled with donor and acceptor dyes for fluorescence resonance energy transfer (FRET) studies of polymer diffusion and molecular mixing. In films formed from the polyol latex in the absence of hmPIC, the COOH content of the polyols had no significant effect on the rate or extent of polymer diffusion. In contrast, in films formed from the 2K-WPU dispersions with a stoichiometric NCO/OH ratio (1.3:1), the extent of ET rapidly increased to the maximum value, showing that the full extent of molecular mixing of the two components can be accomplished in films. Thus, the hmPIC acts as a reactive plasticizer. In parallel experiments in which the hmPIC was labeled with the donor dye, we found evidence for partial mixing of the PIC and polyol in the dispersed state. We also showed that the rate of mixing of the hmPIC and the polyol in the films cast from these dispersions was accelerated by −COOH groups in the polyol.

Molecular Weight Distribution in Di Metal Cyanide Catalyzed Polymerization 2: Numerical Simulation of Chain Activation/Deactivation and Diffusion Effects

AbstractA kinetic model for the dynamic simulation of Di Metal Cyanide (DMC) catalyzed alkoxylation is proposed. The model covers the characteristic “catch‐up” kinetics, favoring the growth of small polymers, and high molecular weight tailing (HMWT), which broadens the distribution toward large polymers. Computations are based on the direct numerical integration of fundamental reaction kinetics of a living polymerization with chain length dependent reaction coefficients. It is observed that “catch‐up” kinetics can be modeled by molecular weight (MW) dependent propagation and activation constants. The formation of HMWT can be described by MW dependent segmental growth. Thereby, the segmental growth is influenced by chain length dependent deactivation constants and effects of polymer diffusion. Simulations of (semi‐)batch operations return an appropriate reproduction of measured polymer distributions. Processes with continuous addition of starters (CAOS) and single stage continuous (CSTR) processes are compared to (semi‐)batch operations. Kinetics without “catch‐up” mechanism have a strongly broadened MWD in CAOS/CSTR processes, compared to (semi‐)batch operations. Also, the concentration of short chained polymers, which act as catalyst poison, is increased. In contrast, CAOS/CSTR simulations with “catch‐up” mechanism result in a MWD comparable to results from (semi‐)batch operations and there is no accumulation of short chained polymers.

Using Plasma Etching to Access the Polymer Density Distribution and Diffusivity of Gel Particles.

In this paper we examine the polymer density distribution of gel particles and its effect on solvent diffusivity through the polymer network. In order to access the inner particle regions, external polymer layers were removed by plasma etching, thus reducing them from the outside. Higher polymer densities after erosion showed internal heterogeneity, with the density increasing towards the center of the particles. An exponential decay polymer density model is proposed, and the spatial relaxation length measured. The diffusion of solvent through the particles, before and after the plasma oxidation, revealed a correlation between the diffusion coefficient and the internal density.

Cold neutron transmission for the in-situ analysis of the gas diffusion in polymers

A novel set-up for monitoring the gas diffusion process in polymers has been developed. This set-up is based on performing neutron radiography and tomography of polymeric materials under high pressures of gas. Information about the absorption kinetics of supercritical CO2 was obtained. Additionally, solubility, diffusivity, and macroscopic changes in the dimensions of the samples can be in-situ obtained through the developed measurement technique. This novel approach can be very helpful to analyze the diffusion kinetics in a wide range of polymers.

Characterization of Polymer Molecular Weight Distribution by NMR Diffusometry: Experimental Criteria and Findings.

NMR diffusometry finds useful applications in characterizing molecular weight (M) and molecular weight distribution (MWD) for polymers due to its unique advantages in generic detection, chemical selectivity, and quantitation. Here, we present a fundamental study to explore how the condition of diffusion measurement impacts the determined MWD. We use the critical dilute concentration Cdilute* to explicitly delineate the boundary of the sufficiently dilute condition, below which chain interactions have a negligible impact on polymer diffusion. We present solid evidence to validate the postulated theory that links Cdilute* to molecular weight, polydispersity, and chain conformation. Quantitative analysis reveals the consequence of violating the sufficiently dilute condition with M/MWD characterization. These findings provide useful guidance for M/MWD characterization by NMR diffusion and help to rationalize the data disparity that exists in the literature. The results further provide new insights into the interplay between chain conformations and diffusion for globular structure, such as proteins, and provide a different approach toward characterizing polymer architecture and molecular weight.

Cross-correlation corrected friction in (generalized) Langevin models.

We propose a route for parameterizing isotropic (generalized) Langevin [(G)LE] thermostats with the aim to correct the dynamics of coarse-grained (CG) models with pairwise conservative interactions. The approach is based on the Mori-Zwanzig formalism and derives the memory kernels from Q-projected time correlation functions. Bottom-up informed (GLE and LE) thermostats for a CG star-polymer melt are investigated, and it is demonstrated that the inclusion of memory in the CG simulation leads to predictions of polymer diffusion in quantitative agreement with fine-grained simulations. Interestingly, memory effects are observed in the diffusive regime. We demonstrate that previously neglected cross-correlations between the "irrelevant" and the CG degree of freedom are important and lie at the origin of shortcomings in previous CG simulations.

Real-time molecular-level visualization of mass flow during patterned photopolymerization of liquid-crystalline monomers

Single-particle fluorescence imaging is used to monitor dynamic processes that occur during patterned photopolymerization of liquid-crystalline monomers. A spatial gradient of chemical potential can be created at the border of bright and dark regions by structured illumination in the photopolymerization process, leading to mutual diffusion of polymers and monomers. Analysis of the fluorescence from single quantum dots doped into the monomers at minute concentrations enables visualization of highly directional flow from the illuminated region where the photopolymerization proceeds toward a masked unpolymerized region. This directional mass flow causes flow-induced orientation of the polymers that is subsequently fixed by completion of the polymerization reaction, resulting in a mesoscopic aligned area of the polymer film.

Study on the effect of polymer viscosity and Darcy velocity on relative permeability curves in polymer flooding

Relative permeability curves in polymer flooding are important parameters to reflect the two-phase flow performance in reservoir rocks, which have important application in numerical simulation, reservoir engineering and prediction of crude oil production. Although the researches on the relative permeability curves in polymer flooding have made some progress at present, the relative permeability curves used in these researches are generally obtained by traditional experiment methods (steady-state method and unsteady-state method) or microscopic pore-scale modeling method. However, these methods are incapable of considering the physicochemical properties (i.e. rheology, residual resistance factor, adsorption and diffusion) of polymer completely. In this paper, a numerical inversion method based on automatic history matching is used to determine relative permeability curves in polymer flooding. The method performs automatic history matching by Levenberg-Marquardt (LM) algorithm on the unsteady-state experimental results and the predicted results of the numerical simulation model. As the physic-chemical properties of polymer solution have been fully taken into account in the numerical simulation model by far, the method makes up for the shortcomings of the traditional methods. Based on the inversed relative permeability curves, the effects of polymer viscosity and Darcy velocity in the rheo-thinning regime on relative permeability curves are then studied respectively. The results show that polymer viscosity and Darcy velocity in the rheo-thinning regime have little influence on the shape of oil relative permeability curve, but have obvious influence on the shape of water relative permeability curve. The deviation between the water relative permeability curve of polymer flooding and the one of water flooding increases linearly with the increase of polymer viscosity and decreases in power function with the increase of Darcy velocity in the rheo-thinning regime. At last, the prediction model of relative permeability curves under the influence of polymer viscosity and Darcy velocity in the rheo-thinning regime is established by multiple nonlinear regression method. The verification results show that this prediction model is reliable and can be used to predict the relative permeability curves at different polymer viscosities and Darcy velocities in the rheo-thinning regime.

Exergy-Based Assessment of Polymers Production and Recycling: An Application to the Automotive Sector

In the last century, the economic growth has been accompanied by a worldwide diffusion of polymers for multiple applications. However, there is a growing attention to the environmental pollution and energy consumption linked to the unconditional use of plastic. In the present work, exergy is used as a measure of the resource consumption during the life cycle of polymers. Nine commercially diffused polymers are chosen, and their production chains are identified according to the “grave to cradle” approach. The global Embodied Exergy (EE) is calculated as the sum of the contribution of each step of the chain, including the production process and the Exergy Replacement Cost (ERC) of the fossil fuel. Then, recycling routes and the associated exergy consumption are analysed. Thermodynamic recycling indexes are developed depending on the final product, namely the crude polymeric material and the oil derivatives or structural molecules. The main results show that some commonly used polymers have a considerable impact in terms of EE (e.g., PET). Recycling indexes encourage the recycling processes, which are always energetically convenient (from 10% to 60% of exergy savings) compared with the production from virgin raw material. Results from EE calculation are used for the thermodynamic assessment of the plastic content of vehicle components, to obtain useful information for recycling practices development.

On the stratification mechanism of self-stratifying epoxy–acrylic coatings

To shed light on the self-stratification mechanism in epoxy–acrylic coatings, 200-, 400-, and 800-micron-thick coatings were applied on glass and aluminum substrates, and their solidification behavior was studied. Some of the applied coats showed self-stratification behavior, with the thermoplastic acrylic copolymer in the top layer. In addition, experiments were performed on epoxy–acrylic solutions without hardeners to evaluate the resulting convective patterns on the solution surface, which exhibited finger-type convection. The final structure of the films had an apparent dependency on the thickness. Thicker films were usually more stratified and had a thicker stratified layer. As these observations could not support diffusion as the primary mechanism of self-stratification, convection experiments were done on epoxy and coatings solutions. The surface patterns on the solutions were studied, and the finger-type convection was observed. Based on these observations, we propose that convection may be the primary movement mechanism rather the diffusion of polymers toward the surface in the self-stratifying coats.

Theory of polymer diffusion in polymer-nanoparticle mixtures: effect of nanoparticle concentration and polymer length.

The dynamics of polymer-nanoparticle (NP) mixtures, which involves multiple scales and system-specific variables, has posed a long-standing challenge on its theoretical description. In this paper, we construct a microscopic theory for polymer diffusion in mixtures based on a combination of the generalized Langevin equation, mode-coupling approach, and polymer physics ideas. The parameter-free theory has an explicit expression and remains tractable on a pair correlation level with system-specific equilibrium structures as input. Taking a minimal polymer-NP mixture as an example, our theory correctly captures the dependence of polymer diffusion on NP concentration and average interparticle distance. Importantly, the polymer diffusion exhibits a power law decay as the polymer length increases at dense NPs and/or a long chain, which marks the emergence of entanglement-like motion. The work provides a first-principles theoretical foundation to investigate dynamic problems in diverse polymer nanocomposites.

Equilibrium organization, conformation, and dynamics of two polymers under box-like confinement.

Motivated by recent nanofluidics experiments, we use Brownian dynamics and Monte Carlo simulations to study the conformation, organization and dynamics of two polymer chains confined to a single box-like cavity. The polymers are modeled as flexible bead-spring chains, and the box has a square cross-section of side length L and a height that is small enough to compress the polymers in that dimension. For sufficiently large L, the system behaviour approaches that of an isolated polymer in a slit. However, the combined effects of crowding and confinement on the polymer organization, conformation and equilibrium dynamics become significant when where is the transverse radius of gyration for a slit geometry. In this regime, the centre-of-mass probability distribution in the transverse plane exhibits a depletion zone near the centre of the cavity (except at very small L) and a 4-fold symmetry with quasi-discrete positions. Reduction in polymer size with decreasing L arises principally from confinement rather than inter-polymer crowding. By contrast, polymer diffusion and internal motion are strongly affected by inter-polymer crowding. The two polymers tend to occupy opposite positions relative to the box centre, about which they diffuse relatively freely. Qualitatively, this static and dynamical behaviour differs significantly from that previously observed for confinement of two polymers to a narrow channel. The simulation results for a suitably chosen box width are qualitatively consistent with results from a recent experimental study of two λ-DNA chains confined to a nanofluidic cavity.