Birefringence and interface in sequential co‐injection molding of amorphous polymers: Simulation and experiment

Modelings of the interface distribution and flow‐induced residual stresses and birefringence in the sequential co‐injection molding (CIM) of a center‐gated disk were carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model the frozen‐in flow stresses in disks. The compressibility of melts is included in modeling of the packing and cooling stages and not in the filling stage. The thermally induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the relaxation modulus and strain‐optical coefficient functions of each polymer. The influence of the processing variables including melt and mold temperatures and volume of skin melt on the birefringence and interface distribution was analyzed for multilayered PS‐PC‐PS, PS‐PMMA‐PS, and PMMA–PC–PMMA molded disks obtained by CIM. The interface distribution and residual birefringence in the molded disks were measured. The measured interface distributions and the gapwise birefringence distributions in CIM disks were found to be in a fair agreement with the predicted interface distributions and the total residual birefringence obtained by the summation of the predicted frozen‐in flow and thermal birefringence. POLYM. ENG. SCI., 55:88–106, 2015. © 2014 Society of Plastics Engineers     

A comparison of Newtonian and viscoelastic constitutive models for dry spinning of polymer fibers

A comparison is made of the predictions of one‐dimensional mathematical model simulations of dry spinning based on Newtonian and viscoelastic constitutive equations for the spin dope. The viscoelastic model is based upon a modified Giesekus constitutive equation with a temperature and composition‐dependent relaxation time. The simulation algorithm includes the effects of the glass transition on the expected solution viscosity and relaxation time behavior along the spinline. Predictions of axial velocity, tensile stress, and composition profiles for the two cases suggest the role of viscoelasticity in the locking‐in behavior associated with fiber solidification along the spinline. The effects of model parameters and processing conditions are also discussed. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 87: 2136–2145, 2003     

Numerical study of nonisothermal polymer extrusion flow with a differential viscoelastic model

A numerical study of nonisothermal viscoelastic flow is conducted to investigate the complex flow characteristics of polymer melts in the extrusion process. A general thermodynamic model for the energy conversion related to viscoelastic fluid flow is introduced. The mathematical model for three‐dimensional nonisothermal viscoelastic flow of the polymer melts obeying a differential constitutive equation (Phan‐Thien and Tanner model) is established. A decoupled algorithm based on the penalty finite element method is performed on the calculation. The discrete elastic‐viscous split stress (DEVSS) algorithm, incorporating the streamline‐upwind Petrov‐Galerkin (SUPG) scheme is employed to improve the computation stability. Essential flow characteristics of polymer melts in the extrusion die for hollow square plastic profile is investigated based on the proposed numerical scheme with ignoring the outer thermal resource. The energy partitioning, which quantified the conversion of mechanical energy into thermal energy, is discussed. The effects of volume flow rate and die contraction angle upon the flow patterns are further investigated. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers     

A rate‐type nonlinear viscoelastic–viscoplastic cyclic constitutive model for polymers: Theory and application

A thermodynamically consistent rate‐type viscoelastic–viscoplastic constitutive model is developed in the framework of isothermal and small deformation to describe the nonlinear and time‐dependent deformation behaviors of polymers, e.g., ratchetting, creep, and stress relaxation. The model is proposed on the base of a one‐dimensional rheological model with several springs and dashpot elements. The strain is divided into viscoelastic and viscoplastic parts, and the stress is also decomposed into two components. Each stress component is further divided into elastic and viscoelastic sub‐components. The viscoelasticity is described by introducing pseudo potentials, and the ratchetting is considered by the viscoplastic flow which is derived by the codirectionality hypotheses. The capability of the proposed model to describe the nonlinear and time‐dependent deformation of polymers is then verified by comparing the simulations with the corresponding experimental results of polycarbonate (PC) polymer. It is shown that the nonlinear and time‐dependent stress–strain responses of the PC can be reasonably predicted by the proposed model. POLYM. ENG. SCI., 56:1375–1381, 2016. © 2016 Society of Plastics Engineers     

Simulation of nonisothermal flow of melt during melting process of vibration‐induced polymer extruder

A simplified 2D melt film model was established to simulate the nonisothermal melt flow during the melting process of the vibration‐induced polymer extruder of which the screw can vibrate axially. Since polymer has time‐dependent nonlinear viscoelastic characteristic with vibration force filed (VFF), a self‐amended nonisothermal Maxwell constitutive equation that can reflect the relaxation time spectrum of polymer was adopted. Using the 2D melt film model, melt films of two kinds of thickness representing different melting stages were simulated to investigate the influence tendency of the same VFF on the different melting stage. Special flow patterns and temperature distribution of melt in the melt film between the driving wall and the solid/melt interface with various vibration force fields were systematically simulated. It is found out that within a certain range of vibration strength, the application of vibration can optimize the time‐averaged shear‐rate distribution, improve the utilization efficiency of energy, and promote melting process; and the thinner the melt film is, the more intense the nonlinear viscoelastic response becomes with the same VFF; moreover, there exists optimum vibration strength to make the melting process fastest, which is in accord with the visualization experimental results. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 102: 5825–5840, 2006     

Viscoelastic constitutive model for uniaxial time‐dependent ratcheting of polyetherimide polymer

Based on the experimental observations, a cyclic nonlinear viscoelastic constitutive model was proposed to describe the uniaxial time‐dependent ratcheting of polyetherimide (PEI) polymer under tension–compression and tension–tension cyclic loading. The model was constructed by extending the nonlinear viscoelastic Schapery model (Schapery, Polym. Eng. Sci., 9, 295 (1969)). The extension emphasized the changes of parameter functions used in the original model, which enabled the model to describe the ratcheting of polymer material. Comparing the simulations with corresponding experimental results, the capability of the extended model to predict the uniaxial time‐dependent ratcheting of PEI was verified. It is shown that the extended model can reasonably describe the uniaxial time‐dependent ratcheting of the polymer under the tension–compression and tension–tension cyclic loading with different peak‐holdings, stress rates, and stress levels. POLYM. ENG. SCI., 52:1874–1881, 2012. © 2012 Society of Plastics Engineers     

Stress‐induced densification of glassy polymers in the subyield region

Constitutive equations are derived for the viscoelastic response of amorphous glassy polymers in the region of subyield deformations. The model treats an amorphous polymer as a composite material consisting of an ensemble of flow units, immobile holes, and clusters of interstitial free volume moving through a network of long chains to and from voids. Changes in macropressure lead to an increase in the equilibrium concentration of interstitial free volume that, in turn, induces diffusion of free‐volume elements from holes. The mass flow results in dissolution of voids that is observed as time‐dependent densification of a glassy polymer. It is demonstrated that the model correctly predicts stress relaxation and a decrease in the specific volume observed in uniaxial tensile and compressive tests on polycarbonate at room temperature. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 1705–1718, 1999     

Simulations of a full three‐dimensional packing process and flow‐induced stresses in injection molding

Numerical investigations of a full three‐dimensional (3D) packing process and flow‐induced stresses are presented. The model was constructed on the basis of a 3D nonisothermal weakly compressible viscoelastic flow model combined with extended pom‐pom (XPP) constitutive and Tait state equations. A hybrid finite element method (FEM)–finite volume method (FVM) is proposed for solving this model. The momentum equations were solved by the FEM, in which a discrete elastic viscous stress split scheme was used to overcome the elastic stress instability, and an implicit scheme of iterative weakly compressible Crank–Nicolson‐based split scheme was used to avoid the Ladyshenskaya–Babu?ka–Brezzi condition. The energy and XPP equations were solved by the FVM, in which an upwind scheme was used for the strongly convection‐dominated problem of the energy equation. Subsequently, the validity of the proposed method was verified by the benchmark problem, and a full 3D packing process and flow‐induced stresses were simulated. The pressure and stresses distributions were studied in the packing process and were in agreement with the results of the literature and experiments in tendency. We particularly focused on the effects of the elasticity and pressure on the flow‐induced stresses. The numerical results show that normal stress differences decreased with incremental Weissenberg number and increased with incremental holding pressure. The research results had a certain reference value for improving the properties of products in actual production processes. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

黏弹性表面活性剂溶液中颗粒沉降特性研究

黏弹性表面活性剂溶液悬浮颗粒流广泛存在于自然界和工业生产中,黏弹性表面活性剂溶液的非线性流变性质及应力松弛效应对其中颗粒沉降有着显著影响。采用FENE-P和Giesekus黏弹性本构模型对表面活性剂溶液中颗粒沉降特性进行研究,发现两种本构模型不仅表现出剪切稀化,而且出现拉伸硬化。颗粒在沉降初期的不稳定性主要是由溶液自身的弹性效应引起,弹性效应越强,颗粒沉降速度不稳定性越强,而剪切稀化效应会减弱颗粒沉降速度的不稳定。颗粒沉降过程中在其尾部形成一个“负尾迹”,随着剪切稀化和拉伸硬化效应增强,负尾迹区增大,弹性效应增加,负尾迹增强,负尾迹区流体内部反向速度分布导致的表面活性剂溶液中微观胶束的拉伸断裂和重构可能是引起颗粒沉降速度持续波动的原因。     

Finite element simulation of three-dimensional viscoelastic planar contraction flow with multi-mode FENE-P constitutive model

Viscoelasticity is a characteristic of many complex fluids like polymer melts, petroleum, blood, etc. The investigation of viscoelastic flow mechanism has practical significance in both scientific and engineering field. Owing to strongly nonlinear, numerical method becomes a practical way to solve viscoelastic flow problem. In the study, the mathematical model of three-dimensional flow of viscoelastic fluids is established. The planar contraction flow as a benchmark problem for the numerical investigation of viscoelastic flow is solved by using the penalty finite element method with a decoupled algorithm. The multi-mode finitely extensible nonlinear elastic dumbbell with a Peterlin closure approximation (FENE-P) constitutive model is used to describe the viscoelastic rheological properties. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids.     

Numerical simulation and experimental verification of the filling stage in injection molding

The linear low‐density polyethylene melt is described by the modified Cross model, the dependence of melt viscosity on temperature incorporated with the Arrhenius equation, and the Moldflow second‐order model in this investigation. The mass, momentum conservation, and constitutive equations are discretized and solved by using the iterative stabilized fractional step algorithm along with the Crank–Nicolson implicit difference scheme. The energy conservation equation is discretized with the characteristic Galerkin approach. The free surface of molten polymer flow front is tracked by the arbitrary Lagrangian–Eulerian (ALE) method. It is demonstrated that good agreement of the numerical predictions given by the proposed ALE method with the results obtained by the injection short‐shot experiments is achieved in the locations and shape of the melt front. Furthermore, when the melt front completely reaches the wall of the mold cavity, the horizontal velocity distribution of counterflow at the section near the finally filling wall is exhibited in the present simulation. POLYM. ENG. SCI., 2012. © 2011 Society of Plastics Engineers     

Measurement and simulation of low‐density polyethylene extrudate swell through a circular die

BACKGROUND: Extrudate swell is a common phenomenon in polymer processing. The investigation of its mechanism is of both scientific and industrial interest. RESULTS: The rheological parameters of a material described by the viscoelastic PTT (Phan‐Thien–Tanner) constitutive model are obtained by fitting the distributions of material functions detected with a strain‐controlled rheometer. The swelling ratios of low‐density polyethylene (LDPE) under different volume flow rates are indirectly obtained using a photographic technique. A mathematical model of extrudate swell is established and its finite element model is derived. A penalty method is employed to solve the extrudate swell problem with a decoupled algorithm. Computation stability is improved by using the discrete elastic‐viscous split stress algorithm incorporating the inconsistent streamline‐upwind scheme. CONCLUSION: The swell phenomenon of LDPE through a circular die is investigated using both experimental measurement and numerical simulation. The swelling ratios obtained from the simulation are compared with those measured: they agree well with each other. The essential flow characteristics of polymer melts are predicted and the mechanism of the swell phenomenon is further discussed. Copyright © 2009 Society of Chemical Industry     

低密度聚乙烯熔体毛细管挤出的数值模拟

借助2种微分型黏弹性本构模型DCPP模型和S-MDCPP模型来描述支化高分子熔体的复杂流变行为,并采用离散的弹性黏性应力分裂方法（DEVSS）/迎风流线方法（SU）解决黏弹性流体流动过程中的对流占优问题以及缺少椭圆算子的问题,进而用基于有限增量微积分（FIC）方法的压力稳定型分步算法求解质量守恒方程、动量守恒方程,对低密度聚乙烯（PE-LD）熔体在毛细管中的流动情况以及挤出胀大过程进行模拟,并把模拟结果和实验结果进行比较。结果表明,在低剪切速率时,模型预测的挤出胀大比和壁面剪切应力与实验结果比较接近;2种模型预测的速度、应力以及主链拉伸的吻合程度较好,说明2种模型均能较好地预测PE?LD熔体在毛细管中的复杂流变行为;同时表明计算S-MDCPP模型时所采用的算法是可靠的。     

Wall Slip of Linear Polymer Melts During Ultrasonic‐assisted Micro‐injection Molding

The wall slip of linear polymer melts under ultrasonic vibration is investigated by correcting the slip mechanism, and melt flow behaviors in ultrasonic‐assisted micro‐injection molding (UμIM) method are discussed. Based on the effect mechanism of ultrasonic vibration on the melt, theoretical models of the critical shear stresses for the onset of weak and strong wall slip during UμIM are established, and the change in rheological properties due to the onset of wall slip under ultrasonic vibration is experimental investigated by a built measurement system. The results show that the onset of weak and strong wall slip of the melt in micro cavity are promoted by ultrasonic vibration, which agree with the built theoretical models, and the melt filling capability in micro cavity is enhanced by reducing apparent viscosity and releasing shear stress of the polymer melt, which improves the molding quality of micro polymer parts via UμIM method. POLYM. ENG. SCI., 59:E7–E13, 2019. © 2018 Society of Plastics Engineers     

Modeling and simulation of conditionally volume averaged viscoelastic two‐phase flows

Conditional volume averaging is used to develop a model capable of simulating two‐phase flows of viscoelastic fluids with surface tension effects. The study is started with the single‐phase mass and momentum balances, which are subsequently conditionally volume averaged. In doing so, we arrive at a set of equations having unclosed interfacial terms, for which closure relations for viscoelastic fluids are presented. The resulting equations possess a structure similar to the single‐phase equations; however, separate conservation equations are solved for each phase. As a result, each phase has its own pressure and velocity over the entire domain. Next, our numerical implementation is briefly outlined. We find that a Poiseuille single‐phase flow is predicted correctly. The closure terms are examined by considering a two‐phase shearing flow and a quiescient cylinder with surface tension. A convergence analysis is performed for a steady stratified two‐phase flow with both phases being viscoelastic. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3914–3927, 2013     

Visualization of particles arrangement during filling stage of polyamide 6 – metal insert injection molding

An isotactic polypropylene filled with talc and mica particles was used for visualization of molten polymer flow lines during filling of mold cavity with metal insert. The talc particles orientation in polypropylene matrix was followed by scanning electron microscopy. For the quantification of platelet particles orientation in molten polymer flow localized near metal wall, the Herman's function was applied. An existence of regions with well oriented, randomly organized talc particles depending on the shape and cross‐section of the channels, was found and reported. The domain with the most disorganized platelet particles was detected in the region where polymer streamed around the sharp corner of the metal insert. Such a kind of disarranged region could be the weakest part of the whole injection molded polymeric element. POLYM. ENG. SCI., 59:E271–E278, 2019. © 2019 Society of Plastics Engineers     

Hybrid Model of Viscosity of Polymer Blends

A hybrid model of the viscous properties of polymeric matrix blends with isolated (MPBs) and continuous phases (CPBs) was proposed. The hybrid model combines the analytical Hill's model of the average stresses and strains for viscous composites and semi‐empirical model of porous materials. A distinctive feature of the model is to calculate the concentration ratios of the average strain rate through the effective volume of the average strain rate. Effective volumes are determined by solving the boundary problem of viscous deformation of the representative volume of two‐phase MPBs or CPBs considering a possible porous state of a material. The comparison of the calculation results with the experimental data was made. The new model more accurately describes the viscosity of the two‐phase polymer blends than the known phenomenological models. The area of application of the hybrid model is limited to melts of polymer blends, the viscosity of which is inside the Hashin‐Shtrikman's bounds. POLYM. ENG. SCI., 59:E212–E218, 2019. © 2018 Society of Plastics Engineers     

Constitutive equations in nonlinear viscoelasticity of glassy polymers based on the concept of traps

Constitutive equations are derived for the nonlinear viscoelastic response of amorphous polymers. The model treats a glassy polymer as an ensemble of independent relaxing regions that randomly hop in their cages being thermally activated. Rearrangement occurs when a flow unit reaches some liquid‐like state. Stress‐strain relations are verified using experimental data in tensile relaxation tests for polyester resins with two types of flexibilizers. Fair agreement is demonstrated between observations and results of numerical simulation.