Viscoelastic Polymer Research Articles

3D Printing of Liquid Metal Embedded Elastomers for Soft Thermal and Electrical Materials.

Liquid metal embedded elastomers (LMEEs) are composed of a soft polymer matrix embedded with droplets of metal alloys that are liquid at room temperature. These soft matter composites exhibit exceptional combinations of elastic, electrical, and thermal properties that make them uniquely suited for applications in flexible electronics, soft robotics, and thermal management. However, the fabrication of LMEE structures has primarily relied on rudimentary techniques that limit patterning to simple planar geometries. Here, we introduce an approach for direct ink write (DIW) printing of a printable LMEE ink to create three-dimensional shapes with various designs. We use eutectic gallium-indium (EGaIn) as the liquid metal, which reacts with oxygen to form an electrically insulating oxide skin that acts as a surfactant and stabilizes the droplets for 3D printing. To rupture the oxide skin and achieve electrical conductivity, we encase the LMEE in a viscoelastic polymer and apply acoustic shock. For printed composites with a 80% LM volume fraction, this activation method allows for a volumetric electrical conductivity of 5 × 104 S cm-1 (80% LM volume)─significantly higher than what had been previously reported with mechanically sintered EGaIn-silicone composites. Moreover, we demonstrate the ability to print 3D LMEE interfaces that provide enhanced charge transfer for a triboelectric nanogenerator (TENG) and improved thermal conductivity within a thermoelectric device (TED). The 3D printed LMEE can be integrated with a highly soft TED that is wearable and capable of providing cooling/heating to the skin through electrical stimulation.

Pinch-off of bubbles in a polymer solution

The formation of gas bubbles in a liquid occurs in various engineering processes, such as during foam generation or agitation and mixing in bubbly flows. A challenge in describing the initial formation of a gas bubble is due to the singular behavior at pinch-off. Past experiments in Newtonian fluids have shown that the minimum neck radius follows a power-law evolution shortly before the break-up. The exponent of the power-law depends on the viscosity of the surrounding Newtonian liquid, and ranges from 0.5 for low viscosity to 1 for large viscosity. However, bubble formation in a viscoelastic polymer solution remains unclear, and in particular, if the evolution is still captured by a power-law and how the exponent varies with the polymer concentration. In this study, we use high-speed imaging to analyze the bubble pinch-off in solutions of polymers. We characterize the time evolution of the neck radius when varying the polymer concentration and thus the characteristic relaxation time of the polymer chains, and describe the influence of viscoelasticity on the bubble pinch-off. Our results reveal that the presence of polymers does not influence the thinning until the latter stages, when their presence in sufficient concentration delays the pinch-off.

Dynamic analysis of variable fractional order cantilever beam based on shifted Legendre polynomials algorithm

The shifted Legendre algorithm is used to solve the governing differential equation of variable order cantilever beam in time domain. According to the variable fractional model, the governing equations of viscoelastic polymer cantilever beam are built. The integer order and variable fractional order differential operator matrices are obtained based on the properties of shifted Legendre polynomials. The variable fractional differential equations are converted into algebraic equations and solved in time domain by operator matrix. The convergence analysis confirmed the high efficiency and accuracy of the proposed algorithm for solving differential equations. The deflection and stress of the polymer cantilever beams (PET and HDPE) under various external load (uniformly distributed and harmonic) are studied.

The role of essential oils in maintaining the postharvest quality and preservation of peach and other fruits.

Fruits are highly susceptible to postharvest losses induced majorly by postharvest diseases. Peach are favored by consumers because of their high nutritional value and delicious taste. However, it was easy to be affected by fungal infection. The current effective method to control postharvest diseases of fruits is to use chemical fungicides, but these chemicals may cause adverse effects on human health and the residual was potentially harmful to nature and the environment. So, it is especially important to develop safe, non-toxic, and highly effective strategies for the preservation of the fruits. Essential oil, as a class of the natural bacterial inhibitor, has been proven to exhibit strong antibacterial activity, low toxicity, environmental friendliness, and induce fruit resistance to microorganism, which could be recognized as one of the alternatives to chemical fungicides. This paper reviews the research progress of essential oils (Eos) in the storage and preservation of fruits, especially the application in peach, as well as the application in active packaging such as edible coatings, microcapsules, and electrospinning loading. Electrospinning can prepare a variety of nanofibers from different viscoelastic polymer solutions, and has broad application prospects. The paper especially summarizes the application of the new Eos technology on peach. The essential oil with thymol, eugenol, and carvacrol as the main components has a better inhibitory effect on the postharvest disease of peaches, and can be further applied. PRACTICAL APPLICATIONS: As an environmentally friendly natural antibacterial agent, essential oil can be used as a substitute for chemical preservatives to keep fruits fresh. This paper summarizes the different preservation methods of essential oils for fruits, and especially summarizes the different preservation methods of essential oils for peaches after harvesting, as well as their inhibitory effects on pathogenic fungi. It could provide ideas for preservation of fruits and vegetables by essential oils.

A Study on the Interaction Behavior between an Earth-Rock Dam and a New-Typed Polymer Anti-Seepage Wall

Polymer anti-seepage wall has been gradually applied in earth-rock dam reinforcement projects as a new seepage control technique. However, due to all-pervasive properties of the new materials and root-like connection between the materials and soils, the interface characteristics between the polymer wall and the earth-rock dam, as well as the interaction behavior of both, are complex and still not clear, which obstruct studying coordination mechanism of dam and wall under earthquake. Therefore, the interface characteristics and interaction behavior of dam and wall were studied in the article. Firstly, the dynamic shear stress-displacement, shear stiffness and damping ratio of the interface between polymer and soil were investigated by ring shear test. In addition, the viscoelastic constitutive model of polymer materials were researched by dynamic mechanical analysis (DMA) test. Based on tests results, a finite element model of earth-rock dam with polymer wall was established, including a non-linear simulation interface element and viscoelastic polymer constitutive model. Next, the validity of the simulation model was verified based on dynamic centrifuge test results. Then, the interaction behavior and seismic response of the dam with polymer wall were explored by using the verified model. The research results provide a scientific basis for the development and application of new-typed polymer anti-seepage wall in reinforcement engineering.

Reduced surface accumulation of swimming bacteria in viscoelastic polymer fluids

Surface-associated bacterial communities flourish in nature and in the body of animal hosts with abundant macromolecular polymers. It is unclear how the endowed viscoelasticity of polymeric fluids influences bacterial motile behavior in such environments. Here, we combined experiment and theory to study near-surface swimming of flagellated bacteria in viscoelastic polymer fluids. In contrast to the swimming behavior in Newtonian fluids, we discovered that cells swim in less curved trajectories and display reduced near-surface accumulation. Using a theoretical analysis of the non-Newtonian hydrodynamic forces, we demonstrated the existence of a generic lift force acting on a rotating filament near a rigid surface, which arises from the elastic tension generated along curved flow streamlines. This viscoelastic lift force weakens the hydrodynamic interaction between flagellated swimmers and solid surfaces and contributes to a decrease in surface accumulation. Our findings reveal previously unrecognized facets of bacterial transport and surface exploration in polymer-rich environments that are pertinent to diverse microbial processes and may inform the design of artificial microswimmers capable of navigating through complex geometries.

Linear viscoelasticity of double associative polymers with varied density of the secondary interaction

This study examines the linear viscoelasticity of double-associative ionomers based on poly(hexyl methacrylate). The ionomer samples are unentangled and contain a fixed number of ionic stickers per chain, fion≅4, but a varied number of hydrogen-bonding stickers per chain, fH = 5.5–27. Increasing fH has distinct effects on the glassy and rubbery dynamics. On the one hand, the glassy relaxation mode distribution broadens, and the glassy-to-rubbery transition region delays significantly with increasing fH. On the other hand, the delay of the ionic dissociation and, accordingly, the terminal relaxation (activated by continuous ionic dissociations) are much weaker. This contrast suggests that the ionic dissociation is not only activated by the segmental motion of the main backbone but is also partly activated by the motion of the side chains that is less dependent on fH.

A novel method for measuring viscoelastic parameters of epoxy resins under water bath environments using embedded strain gauges

Epoxy resins are a class of viscoelastic hygroscopic polymers widely used in plastic packaged devices. Stress relaxation and yielding may occur in plastic packaged devices under thermal-hygro environments, which seriously affect the reliability of the devices. It is important to obtain accurate viscoelastic parameters of the hygroscopic materials for analyzing the stresses and strains of plastic packaged devices. In this paper, a novel method for measuring the viscoelastic parameters of epoxy resins under water bath environments by using embedded strain gauges is proposed. A kind of epoxy resin-copper bi-layer plate is fabricated, and the biaxial strain gauges are embedded between the epoxy resin layer and the copper layer to monitor the deformations. The interfacial strains of the two test pieces at 85 °C/100% relative humidity are continuously measured up to 144 h. Combine the experimental strain curves and the analytical representation, the viscoelastic parameters of the epoxy resins are obtained by curve fitting with MATLAB and genetic algorithm. It is found that the viscoelasticity of epoxy resins could be described by an exponential function of order five shear modulus and relaxation times, and the interfacial strain curves obtained by genetic algorithm fit well with the experimental ones. The proposed testing and solving methods provide a valuable reference for getting the time-dependent viscoelasticities of epoxy resins under thermal-hygro environments.

The role of injection fluid elasticity in microscopic displacement efficiency of residual non-wetting phase: An in-situ experimental investigation

Polymer flooding has been employed extensively over the years as a mature enhance oil recovery (EOR) technique to improve volumetric displacement (sweep) efficiency in oil reservoirs. In the last few years, groundbreaking laboratory and pilot tests have demonstrated that the use of polymers with viscoelastic properties may potentially enhance the efficacy of polymer flooding and result in improvements in both volumetric (macroscopic) and microscopic (pore-scale) displacement efficiencies. The transport and pore-level displacement mechanisms responsible for such microscopic improvements, however, have been poorly understood. This study was thus designed to improve the current understanding of the displacement of waterflood residual oil with viscoelastic polymers in natural porous media. We performed a series of micro-scale two-phase core-flooding experiments on miniature water-wet Berea sandstone samples at elevated pressure. During the flow tests, heavy oil was displaced with two Newtonian fluids (brine and Glycerin) and a viscoelastic polymer to investigate the impact of injection fluid elasticity on mobilization of capillary-trapped oil. We purposely selected a viscoelastic polymer that had a similar viscosity, but significantly different elastic characteristics compared to those of Glycerin. The rheological behavior of the fluids was characterized using an oscillation rheometer. The flow experiments were conducted using a miniature core-flooding setup coupled with a high-resolution micro-computed tomography (µ -CT) scanner enabling us to directly visualize morphologies of oil globules in the pore space. The core sample initially saturated with brine was subjected to primary oil drainage to establish initial water saturation (Swi) and then a waterflood to establish remaining oil saturation. This process was then followed by (i) a Glycerin flood and (ii) a viscoelastic polymer flood, all under low pressure gradients (i.e., capillary numbers of approximately 1E-5). The injection of Glycerin, after the waterflood, reduced the remaining oil saturation from 45.8 to 38% due to a favorable mobility ratio, which was believed to be the reason for the improvement in volumetric sweep efficiency. Furthermore, in-situ saturation data revealed that the use of the viscoelastic polymer resulted in an additional 12% oil recovery. Since both the viscoelastic polymer and Glycerin had similar mobility ratios, their volumetric sweep efficiencies were believed to be comparable, and hence the observed incremental oil recovery was attributed to additional pore-scale oil mobilization caused by the injection of the viscoelastic polymer. This mobilization mechanism is triggered by elastic forces and is not due to dynamic effects generated usually under high capillary numbers. Moreover, investigation of pore-scale fluid occupancy maps provided clear evidence of fragmentation and mobilization of trapped oil clusters as the mechanisms responsible for the observed enhancement in recovery. The fragmentation and mobilization of the oil globules were related to high elastic properties of the viscoelastic polymer used. Finally, we present a new correlation linking the reduction in residual oil saturation to the elasticity of the injected fluid.

Recent Advances in Polymer Flooding in China.

Polymer flooding is drawing lots of attention because of the technical maturity in some reservoirs. The first commercial polymer flooding in China was performed in the Daqing oilfield and is one of the largest applications in the world. Some laboratory tests from Daqing researchers in China showed that the viscoelasticity of high molecular weight polymers plays a significant role in increasing displacement efficiency. Hence, encouraged by the conventional field applications and new findings on the viscoelasticity effect of polymers on residual oil saturation (ROS), some high-concentration high-molecular-weight (HCHMW) polymer-flooding field tests have been conducted. Although some field tests were well-documented, subsequent progress was seldom reported. It was recently reported that HCHMW has a limited application in Daqing, which does not agree with observations from laboratory core flooding and early field tests. However, the cause of this discrepancy is unclear. Thus, a systematic summary of polymer-flooding mechanisms and field tests in China is necessary. This paper explained why HCHMW is not widely used when considering new understandings of polymer-flooding mechanisms. Different opinions on the viscoelasticity effect of polymers on ROS reduction were critically reviewed. Other mechanisms of polymer flooding, such as wettability change and gravity stability effect, were discussed with regard to widely reported laboratory tests, which were explained in terms of the viscoelasticity effects of polymers on ROS. Recent findings from Chinese field tests were also summarized. Salt-resistance polymers (SRPs) with good economic performance using produced water to prepare polymer solutions were very economically and environmentally promising. Notable progress in SRP flooding and new amphiphilic polymer field tests in China were summarized, and lessons learned were given. Formation blockage, represented by high injection pressure and produced productivity ability, was reported in several oil fields due to misunderstanding of polymers’ injectivity. Although the influence of viscoelastic polymers on reservoir conditions is unknown, the injection of very viscous polymers to displace medium-to-high viscosity oils is not recommended. This is especially important for old wells that could cause damage. This paper clarified misleading notions on polymer-flooding implementations based on theory and practices in China.

A 3D printed sandwich-type piezoelectric motor with a surface texture.

Polymer-based piezoelectric motors have excellent properties, such as lightweight and corrosion resistance. In addition, 3D printing and customized additive manufacturing of polymers provide new opportunities for the development of piezoelectric motors with complex or special structures. In this paper, a 3D printed polymer-based sandwich-type piezoelectric motor operating in a single longitudinal mode is developed. A vibration decomposition model of the motor and an analytical model considering polymer viscoelasticity are established to analyze the dynamic characteristics and to determine the geometric structure of the motor. To increase the coefficient of friction, a polymer surface texture is utilized on the contacts. The experimental results show that the friction coefficient of the contact tip with surface texture is about 0.16, which increased by 45.5% compared to a smooth surface. The resonance frequency is 28.648kHz, and the maximum no-load speed under 300Vp-p is 54 r/min. Our study shows the promise of polymer-based materials in the development of the piezoelectric motor.

Electromechanical Performance of Strain Sensors Based on Viscoelastic Conductive Composite Polymer Fibers.

Flexible conductive polymer composite (CPC) fibers that show large changes in resistance with deformation have recently gained much attention as strain-sensing components for future wearable electronics. However, the electrical resistance of these materials decays with time during dynamic cyclic loading, a deformation performed to simulate their real application as strain sensors. Despite the extensive research on CPC fibers, the mechanism leading to this decay in the electromechanical response under repetitive cycles remains unreported. Herein, this behavior is investigated using fiber-based strain sensors wet spun from thermoplastic polyurethane (TPU) consisting of a carbonaceous hybrid conductive filler system of carbon black (CB) and carbon nanotubes (CNTs). We found electrical viscosity to predict the observed electromechanical resistance decay. This implies that cycling these materials enables the relaxation of both the polymer chains and the conductive network. In addition, the resulting piezoresistive fibers are sensitive to deformation in the region of low strain (gauge factor of 6.0 within 3.0% strain), remain conductive under 280.5% deformation, and are stable for more than 2000 cycles. Finally, we demonstrate the potential of TPU/CB-CNT fibers as strain sensors for monitoring human motion.

Roles of repeating-unit interactions in the stress relaxation process of bulk amorphous polymers: 2. nonlinear viscoelasticity

Stress relaxation manifests itself as the most fundamental mechanical response of bulk amorphous polymers. We characterized polymer repeating units with four-set interaction parameters separately representing the thermodynamic and kinetic aspects of local intrachain and interchain interactions, and evaluated their relative importance in stress relaxation with linear viscoelasticity at high temperatures by means of kinetic Monte Carlo simulations (DOI: 10.1016/j.polym.2021.123740). Hereby we continued to evaluate their relative importance in the initiation and the deviation extent of nonlinear viscoelasticity at low temperatures. We observed that thermodynamic interchain interactions seem no effective, while the other three local interactions retard the later stage of stress relaxation for nonlinear viscoelasticity, responsible for the KWW indexes of polymer rubbery states around 0.5. Both the kinetic aspects of intrachain and interchain interactions dominate the initiation temperatures of nonlinear viscoelasticity, and their high barriers suppress the deviation extent of nonlinear viscoelasticity towards the strong liquids like metals and ceramics, while their low barriers leave the fragile liquids to more thermodynamically rigid polymers. Our work facilitates a better understanding of structure-property relationship in nonlinear viscoelasticity of bulk amorphous polymers.

Preparation of supramolecular viscoelastic polymers with shear, temperature, and salt resistance/sensitivity by amphiphilic functional monomer modification

To develop high-salinity, high-temperature reservoirs, a novel supramolecular viscoelastic polymer-based fracturing fluid modified with an amphiphilic functional monomer was developed. Investigation of the thickening ability, solubility, and rheology revealed that the as-developed gel system with a rigid hydrophobically associating space network structure and linear entanglement can improve the thickening ability, solubility, temperature adaptation range, and chemical environment resistance sensitivity of supramolecular polymers. The surface/interfacial tension test revealed that the modified polymers exhibit surface and interface activity. At an MHAP-15 concentration of 1500 mg/L, the surface tension was 39.29 Nm/m. At an MHAP-15 concentration of 800 mg/L, a minimum interfacial tension of 0.86 Nm/m was observed. The shear sensitivity test revealed that the copolymer fluid exhibits a high creep recovery performance and elastic response as well as a 100% viscosity recovery rate. In addition, the viscoelasticity and thixotropy of the copolymer solution were investigated. The results revealed that the higher the number of the oxyethylene groups in amphiphilic functional monomers, the higher the increase in the viscous modulus of the copolymer solution. In the modified polymer, the highest energy was required for the destruction of the MHAP-15 solution, reaching 66.5 Pa; this result confirmed the best synergistic effect of hydrophobic association and linear entanglement of the MHAP-15 solution molecules, as well as the best temperature and salt resistance. The viscosity of 0.6 wt% MHAP-15 was 120 mPa s after shearing for 50 min at 120 °C and at 170 s−1. At 160 °C and 100 s−1, the viscosity of the 0.8 wt% MHAP-15 polymer solution reached 153 mPa s after shearing for 50 min. At a NaCl concentration of 25 × 104 mg/L, the viscosities of the 0.6 wt% and 0.8 wt% MHAP-15 solutions were 203 and 245 mPa s, respectively, indicative of the stimulation response of salt thickening. At a CaCl2 concentration of 2 × 104 mg/L, the viscosity of 0.6 wt% MHAP-15 was 87 mPa s, and the viscosity retention rate was 47.9%. Finally, by X-ray diffraction, the particle size test in solution, and scanning electron microscopy, the higher the number of oxyethylene groups in the amphiphilic functional monomers, the higher the mixing entropy of copolymers, and the particle size in solution and multi-peak distribution were higher, as well as more clear linear entanglement between copolymers; all of these factors were conducive to the formation of large-scale supramolecular structures of super-bridged copolymers.

Analysis of the Polymer Two-Layer Protective Coating Impact on Panda-Type Optical Fiber under Bending.

The article discusses the effects of thermal-force on the Panda-type optical fiber. The studies used a wide temperature range. The research used two thermal cycles with exposures to temperatures of 23, 60 and −60 °C. The field of residual stresses in the fiber formed during the drawing process was determined and applied. Panda was considered taking into account a two-layer viscoelastic polymer coating under conditions of tension winding on an aluminum coil in the framework of a contact problem. The paper investigated three variants of coil radius to analyze the effect of bending on fiber behavior. The effect of the coating thickness ratio on the system deformation and optical characteristics was analyzed. Qualitative and quantitative patterns of the effect of temperature, bending, thickness of individual polymer coating layers and relaxation transitions of their materials on the Panda optical fiber deformation and optical characteristics were established. Assessment of approaches to the calculation of optical characteristics (values of the refractive indices and fiber birefringence) are given in the framework of the study. The patterns of deformation and optical behavior of the Panda-type fiber with a protective coating, taking into account the nonlinear behavior of the system materials, were original results.

A thermodynamically consistent model for chemically induced viscoelasticity in covalent adaptive network polymers

Covalent adaptive networks (CANs) can rearrange their network topology via bond exchange reactions (BERs) under external stimuli such as solvent, temperature, pH and light, and are promising in thermoforming, self-healing, reprocessing and recycling applications. For example, BERs trigged by water would exert a great influence on the mechanical behaviors of polyimine CANs by changing their network topology, and vice versa, mechanical stress can make an impact on the kinetics of BERs by accelerating or slowing the transportation of water. This paper aims to present a rigorous thermodynamically consistent model for CANs with diffusion-reaction-deformation coupling behaviors. Thus, a mixture theory-based continuum framework for open systems is developed by choosing the host solid as a reference media, within which the first and second thermodynamic laws for every species inside the host solid are established. Then, the chemically induced stress relaxation is linked with water diffusion and chemical reactions to obtain the dependence of the relaxation time on water concentration and reaction parameters in view of the fact that BERs will break stressed chains and result in new stress-free chains. Finally, the proposed model is verified by comparison of the nonlinear stress-strain relation between model predictions and experimental data of the polyimine CANs at high temperature, and illustrated by simulating the viscoelasticity of a polyimine sample immersed in water and subjected to uniaxial tension. The results reveal that BERs is the dominated cause of the viscoelasticity at high temperature and the diffusive water would also have an important effect on stress relaxation of the polyimine CANs.

A cation-dipole-reinforced elastic polymer electrolyte enabling long-cycling quasi-solid-state lithium metal batteries

The application of ionic liquids (IL) in polymer electrolytes represents a safer alternative to the currently used organic solvents in lithium batteries due to their nonflammability and thermal stability. However, as a plasticizer, it is generally agreed that the introduction of ionic liquid usually leads to a trade-off between ion transport and mechanical properties of polymer electrolyte. Here we report the synthesis of an IL-embedded polymer electrolyte with both high ionic conductivity (2.77 × 10−4 S cm−1 at room temperature) and excellent mechanical properties (high tensile strength up to 11.4 MPa and excellent stretchability of 387% elongation at break) achieved by strong ion–dipole interactions between polymer electrolyte components, which was unveiled by the DFT calculation. Moreover, this polymer electrolyte also exhibits nonflammability, good thermal stability and the ability to recover reversibly from applied stress, i.e., excellent elasticity. This highly viscoelastic polymer electrolyte enables tight interfacial contact and good adaptability with electrodes for stable lithium stripping/plating for 2000 h under a current density of 0.1 mA cm−2. By coupling with this polymer electrolyte, the LiFePO4/Li cells exhibit outstanding cycling stability at room temperature as well as the reliability under extreme environmental temperature or being abused.

Template-free scalable fabrication of linearly periodic microstructures by controlling ribbing defects phenomenon during forward roll coating

Linear periodic microstructures are of significant importance in various applications, including drag-reduction, biofouling, self-cleaning, and superhydrophobicity. However, practical applications of such surfaces require mass manufacturing techniques, which are highly limited. This study demonstrated a simple template-free scalable manufacturing technique to fabricate linearly periodic microstructure by controlling the ribbing defects in forward roll coating. A viscoelastic polymer nanocomposite with tailored properties was synthesized and utilized as the coating material. The ribbing instabilities were controlled with process parameters that resulted in variable periodicity of the linearly aligned microstructures with a spacing of 114 – 700 µm. The microstructure arrangement also had a linear to random transition as the instabilities increased. The manufactured surface has a high Wenzel roughness factor that ranges from 1.6 to 3.6, which results in water contact angles of 128o to 150o. The linear microstructure films can have critical applications in the mass manufacturing of drag reduction surfaces. The high aspect-ratio microstructure films can also have applications in superhydrophobic, self-cleaning, anti-icing, and anti-biofouling surfaces.

Approximate Fourier series recursion for problems involving temporal fractional calculus

Fractional calculus constitutive models are of wide applicability. They have been used, for example, for: xanthan gum, bread dough, synthetic polymers (e.g. nylon), single collagen fibrils, semi-hard zero-fat cheese, pure ice at −35∘C, asphalt sealants, epithelial cells, gels, polymers, concrete, asphalt, rock mass, waxy crude oil, breast tissue cells, lung parenchyma, and red blood cell membranes (e.g. Bonfanti et al. (2020)). In this note we are motivated by applications to dispersive media such as lossy dielectrics and viscoelastic polymers, both of which involve constitutive laws with fading memory convolution Volterra integrals, and both of which have been modelled by fractional calculus — which implies a weakly singular Volterra kernel. To step forward in time with such a model the entire hereditary Volterra integral needs to be numerically re-computed at each time step. Done naïvely with standard quadrature this involves O(N2) operations and O(N) storage to compute over N time steps, and this severely limits the usefulness of such simple methods in large scale computations. Several alternatives addressing these shortcomings of this naïve approach have been offered in the literature, all of which can provide remedies. Here we propose another method, but ours is rather different in that we use a Fourier series proxy over much of the integration range. This allows for a recursive update to the ‘history integral’, with a much smaller effort expended in standard quadrature near the singularity. The recursive update is basically the same as that used when the Volterra kernel comprises decaying exponentials, a method which is already employed in some commercial codes. Given that this partial implementation is already in use, we intend that our method therefore has a lower ‘barrier to entry’ for incorporating fractional calculus models into commercial codes: it requires only minor surgery on the weakly singular kernel in order to apply the Fourier series proxy. This study is presented in the spirit of a discussion piece — there are several directions one could take from the basic observations presented here. We have aimed for simplicity and economy of presentation, and included just enough numerical results to give evidence that the method works. In summary, our method mitigates the O(N) storage and O(N2) storage issues using an easily implemented proxy. It can also be implemented for variable time steps. The code is available from https://github.com/variationalform/fouvol and https://hub.docker.com/u/variationalform.

Predictive Mesoscale Simulation of Flow-Induced Blend Morphology, Interfacial Relaxation, and Linear Viscoelasticity of Polymer–Elastomer Blends

Mesoscale simulations of two (cross-linking vs non-cross-linking) representative polymer–elastomer (Nylon 6/ACM) blends are performed without introducing freely adjustable parameters. The simulation allows for a direct investigation into the interplay between flow history (modeled herein by oscillatory shear flow), blend morphology, and viscoelasticity of the polymer composite. Detailed spectral analysis is utilized to resolve the stress relaxations in the bulk and interface regions, respectively, in an effort to gain insight into the origins of blend elasticity. The results reveal a notable impact of flow history on the blend morphology, and the substantially altered storage modulus response in the terminal regime agrees well with early experimental observations on a broad variety of immiscible polymer blends. The interfacial relaxation, in particular, is found to formally commence when the stress relaxations of individual polymer components in the bulk phase are nearly complete, helping to release the remaining stress that was previously dictated by repulsion-driven tube dilations─instead of chain reptation─in the interface region. Importantly, the last feature suggests that the bulk and interfacial relaxations of a polymer blend cannot be fully decoupled. Cross-linking between the elastomers (ACM) adds further complexity to the features of storage modulus response while partly offsetting the flow-induced phase separation. The generality of the simulation protocols described herein along with the affordable computational expense is expected to greatly facilitate future processing designs on polymer–polymer and polymer–elastomer blends.