Numerical simulation of bubble growth in viscoelastic fluid with diffusion of dissolved foaming agent

Viscoelastic simulations of bubble growth in polypropylene (PP) physical foaming were performed. A multimode Phan‐Thien Tanner (PTT) model was used to analyze the dynamic growth behavior of spherically symmetric bubbles with the diffusion of a foaming agent (CO₂). Changes in the dissolved foaming agent concentration in the polymer and in the strain of the polymer melt surrounding the bubbles were simulated with the Lagrangian FEM method. The simulation technique was validated by comparison with the bubble growth data, which were experimentally obtained from visual observations of the PP/CO₂ batch foaming system. The simulation results demonstrated that the strain‐hardening characteristic of polymer does not strongly affect the bubble growth rate. The linear viscoelastic characteristic is more influential, and the relaxation mode around 0.01 s is the most important factor in determining the bubble growth rate during the early stage of foaming. A multivariate analysis for the simulation results was also carried out. This clarified that bubble nucleus population density, surrounding pressure, initial dissolved foaming agent concentration, and diffusion coefficient are more important factors than the viscoelastic characteristics. POLYM. ENG. SCI., 45:1277–1287, 2005. © 2005 Society of Plastics Engineers     

Visual observation and numerical studies of N2 vs. CO2 foaming behavior in core‐back foam injection molding

We investigated , by visual observation and numerical calculations , the foaming behavior of polypropylene within a foam injection mold cavity with the environmentally benign physical blowing agents nitrogen (N₂) and carbon dioxide (CO₂) . An 85‐ton core‐back injection‐molding machine with temperature and pressure monitoring systems as well as a high‐pressure view cell was used for the investigation . The experiments showed a prominent difference in bubble nucleation and growth between N₂ and CO₂ injection foaming . Even when the weight concentration of N₂ dissolved in polymer was one‐third that of CO₂ , N₂ injection foaming provided a bubble number density that was 30 times larger and a bubble size that was one‐third smaller compared to CO₂ injection foaming . Classical bubble nucleation and growth models developed for batch foaming were employed to analyze these experimental results . The models reasonably explained the differences in injection foaming behavior between N₂ and CO₂ . It was clearly demonstrated by both experiments and numerical calculations that N₂ provides a higher number of bubbles with a smaller bubble size in foam injection molding compared to CO₂ as a result of the lower solubility of N₂ in the polymer and the larger degree of super‐saturation . POLYM. ENG. SCI., 2011. ©2011 Society of Plastics Engineers     

Foaming of polymers with supercritical CO2: An experimental and theoretical study

Microcellular polystyrene (PS) foams and porous structures of the biodegradable poly(d,l-lactic acid) (P_d,lLA) were prepared with the batch foaming technique (pressure quench) using supercritical CO₂ as blowing agent. The effect of pressure, temperature and depressurization rate on the final porous structure was investigated. The results revealed that the size of the pores decreases and their population density increases with pressure increase, or decrease of temperature, and/or increase of the depressurization rate. The results were correlated by combining nucleation theory with NRHB model in order to account for and emphasize the physical mechanism related to nucleation of bubbles inside the supersaturated polymer matrix. A satisfactory agreement between correlations and experimental data was obtained indicating that the nucleation theory yields quantitative correlations when variables such as sorption, degree of plasticization, and surface tension of the system polymer-supercritical fluid are accurately described.     

Prediction of foam growth and its nucleation in free and limited expansion

The influence volume approach (IVA) is often utilized for modeling the mass transfer process dictating bubble growth dynamics in physical foaming. However, the assumed concentration profile in the IVA method is only valid when the changes in dissolved gas concentration are small (less than 5%). In addition, the validity of the IVA method is difficult to justify in chemical foaming applications because of the difficulties involved in defining the dissolved gas concentration profile.In the present work, we define two distinct stages of bubble growth for physical foaming. These two stages are termed as free and limited expansion and are controlled by the bubble nucleation rate. Bubble nucleation is assumed to occur only in the free expansion stage. In this stage, the bubble pressure drops substantially from an initially high pressure in the supersaturated state while the dissolved gas concentration changes very little. The second stage of our two-stage mass transfer model is termed the limited expansion stage and accounts for bubble growth in the late stages of foam evolution, when the pressure changes become small. However, in the limited stage of bubble growth the dissolved gas concentration drops significantly, as the available dissolved gas is depleted. To summarize our two-stage mass transfer model of foam expansion, the pressure difference between the bubble phase and the liquid phase is the primary mechanism for driving mass transfer in the early (free) stages of foam growth and the concentration difference is the driver for bubble growth in the late (limited) stages of growth. The first stage can be regarded as the nucleation stage and it is relatively short; while the second stage can be regarded as the bubble growth stage and is much longer. Most of the bubble volume expansion takes place in the second stage.The concentration gradient at the bubble edge, which is often ignored in other models, is analyzed in detail in this paper. The details of our novel mass transfer model are also presented.     

超临界二氧化碳制备热塑性聚氨酯弹性体发泡材料的发泡机理和性能研究

用超临界二氧化碳制备热塑性聚氨酯弹性体发泡材料(E-TPU),分析发泡机理并研究发泡性能。结果表明:泄压速率和发泡温度是影响E-TPU发泡性能的两个重要因素;增大泄压速率有利于提高气泡成核速率和成核数量;升高发泡温度使气泡易膨胀长大,E-TPU密度减小;但发泡温度过高会导致气泡破裂和塌陷,E-TPU密度增大;当发泡温度为130℃左右时,E-TPU密度最小,发泡性能最好。     

Polymer foaming with chemical blowing agents: Experiment and modeling

An experimental and theoretical analysis of the polypropylene foaming process using three different chemical blowing agents (CBA) was performed. A simple experiment was designed to analyze the foaming process of polypropylene (PP)/CO₂ system under two different pressure conditions. The expansion ratio and final foam structure was measured both by direct observation and from optical measurements and image analysis, showing a good agreement. A single bubble simulation based on relevant differential scanning calorimetry and thermo‐gravimetrical analysis experiments, assuming each CBA particles as a nucleation site and accounting for gas diffusion in the surrounding polymer matrix has been built. The sensitivity of the model to physical and processing parameters has been tested. The calculation results are compared to the experiments and open the route to a simplified method for evaluating the efficiency of CBA. POLYM. ENG. SCI., 55:2018–2029, 2015. © 2014 Society of Plastics Engineers     

Visual observation and numerical studies of polymer foaming behavior of polypropylene/carbon dioxide system in a core‐back injection molding process

Thermoplastic foaming within a mold cavity was visualized as it was conducted in an 85‐ton core‐back injection‐molding machine. The core‐back molding process moved a section of the mold just after injecting a molten polymer into the cavity, quickly reducing the pressure to enhance the bubble nucleation. The foaming behavior during core‐back was observed directly through the glass windows of the mold. In the experiments, impact copolymer polypropylene was foamed with carbon dioxide. The effects of the gas concentration and the core‐back rate on bubble nucleation and growth were investigated. It was experimentally confirmed that the bubbles disappeared when the cavity was fully packed and that bubble nucleation occurred when the mold plate was moved and the cavity pressure dropped. Faster core‐back rates and higher gas concentrations increased the number of bubbles while decreasing their size. To analyze the experimental results, a bubble nucleation and growth model was employed that was based on batch foaming. The numerical results were a reasonable representation of the experiments, and this study demonstrated the applicability of the conventional free foaming model to the industrial core‐back molding process. Many aspects of the foaming in the core‐back molding aresimilar to the behaviors observed by batch foaming. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers     

Experimental and numerical studies on bubble dynamics in nonpressurized foaming systems

This work explores the influence of rheological properties on polymer foam development in nonpressurized systems. To understand the complex contributions of rheology on the mechanism of bubble growth during different stages of foam processing, visualization studies were conducted by using a polymer‐foaming microscopy setup. The evolving cellular structure during foaming was analyzed for its bubble surface density, bubble size, total bubble projected area, and bubble size distribution. Morphological analysis was used to determine the rheological processing window in terms of shear viscosity, elastic modulus, melt strength and strain‐hardening, intended for the production of foams with greater foam expansion, increased bubble density and reduced bubble size. A bubble growth model and simulation scheme was also developed to describe the bubble growth phenomena that occurred in nonpressurized foaming systems. Using thermophysical and rheological properties of polymer/gas mixtures, the growth profiles for bubbles were predicted and compared to experimentally observed data. It was verified that the viscous bubble growth model was capable of depicting the growth behaviors of bubbles under various processing conditions. Furthermore, the effects of thermophysical and rheological parameters on the bubble growth dynamics were demonstrated by a series of sensitivity studies. POLYM. ENG. SCI., 54:1947–1959, 2014. © 2013 Society of Plastics Engineers     

Ultrasonic processing of thermoplastic foam

Ultrasonically induced bubble formation for the production of thermoplastic foam was investigated experimentally and theoretically as a basic study. A general purpose polystyrene and blends of low density polyethylene and polyethylene wax were saturated with nitrogen gas under various pressures and the ultrasonic excitation was applied to the polymer system upon release of gas pressure. The ultrasonic nucleation of bubbles in the polymer matrix was modeled by utilizing the classical nucleation theory. The negative pressure generated by the ultrasonic excitation was considered as the environmental pressure at the moment of nucleation. The experimental results showed that the heterogeneous nucleation must be used for ultrasonic foaming of the viscous fluid and the homogeneous nucleation for the low viscosity fluid. The theoretical analysis also indicated that the ultrasonic nucleation can be applied to the production of thermoplastic foam if the ultrasonic excitation generates large enough negative pressure.     

Nucleation, growth and detachment of neighboring bubbles over miniature heaters

This work investigates how simultaneous CO₂ bubbles desorb from water and n-heptane when these liquids become supersaturated with dissolved CO₂. Supersaturation is imposed locally by a miniature heater so bubbles grow at adjacent sites across the heater. To suppress buoyancy, experiments are performed at low gravity conditions. The number of nucleation sites and nucleation time delay depend on liquid properties and heating power. Simultaneous bubbles do not grow at exactly the same rate and calculations show that this corresponds to slightly different bubble temperatures. Moreover, they exhibit smaller growth rates than single bubbles at similar conditions, an indication that they compete for dissolved CO₂. As bubbles expand into the surrounding cold liquid, the heater's temperature decreases in a fashion implying the inception of Marangoni convection. Simultaneous bubbles detach due to g-jitters but following different ways in the two liquids. However, they always detach together and at smaller sizes than single bubbles do. A temperature triggered destabilization of contact lines is deemed responsible for this.     

Polypropylene‐dispersed domain as potential nucleating agent in PS and PMMA solid‐state foaming

The potential of using dispersive domains in a polymer blend as a bubble nucleating agent was investigated by exploiting its high dispersibility in a matrix polymer in the molten state and its immiscibility in the solid state. In this experiments, polypropylene (PP) was used as the nucleating agent in polystyrene (PS) and poly(methyl methacrylate) (PMMA) foams at the weight fraction of 10, 20, and 30 wt %. PP creates highly dispersed domains in PS and PMMA matrices during the extrusion processing. The high diffusivity of the physical foaming agent, i.e., CO₂ in PP, and the high interfacial tension of PP with PS and PMMA could be beneficial for providing preferential bubble nucleation sites. The experimental results of the pressure quench solid‐state foaming of PS/PP and PMMA/PP blends verified that the dispersed PP could successfully increase the cell density over 10⁶ cells/cm³ for PS/PP and 10⁷ cells/cm³ for PMMA/PP blend and reduce the cell size to 24 μm for PS/PP and 9 μm for PMMA/PP blends foams. The higher interfacial tension between PP and the matrix polymer created a unique cell morphology where dispersed PP particles were trapped inside cells in the foam. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Plasticization effects on bubble growth during polymer foaming

During polymer foaming with physical blowing agents, plasticization affects the melt viscosity, gas diffusivity in the melt, and the gas–melt interfacial tension. In this paper, we propose a model for plasticization during bubble growth, and estimate its effects under typical foaming conditions. The theoretical model incorporates well‐established mixture theories into a recent model for diffusion‐induced bubble growth. These include the free‐volume theories for the viscosity and diffusivity in polymer‐blowing agent mixtures and the density gradient theory for the interfacial tension. The viscoelasticity of the melt is represented by an Oldroyd‐B constitutive equation. We study the radial growth of a single bubble in an infinite expanse of melt, using parameter values based on experiments on polystyrene–CO₂ systems. Our results show that even at relatively low gas concentrations, plasticization increases the blowing‐agent diffusivity markedly and thus boosts the rate of bubble growth. In contrast, the reduction in melt viscosity and interfacial tension has little effect on bubble growth. Though not intended as quantitative guidelines for process design, these results are expected to apply qualitatively to typical foaming conditions and common polymer‐blowing agent combinations. POLYM. ENG. SCI., 46:97–107, 2006. © 2005 Society of Plastics Engineers     

Bubble nucleation and growth in fluids

The nucleation and growth of CO₂ bubbles in non-Newtonian and Newtonian fluids that were initially supersaturated under different pressures are investigated in the present work. Quantitative information by means of two cameras reveals that at an immobile nucleation site the bubble grows rapidly followed by a linear increase in bubble diameter with time. After reaching a critical size, the bubble detaches from the stagnant site to rise in liquids with an exponential temporary increase for both the diameter and distance. A simple physical reasoning was proposed to qualitatively explain these observed phenomena. Recently, the growth rate and flow fields around a CO₂ micro-bubble were measured in a microdevice by a micro-Particle Image Velocimetry in water. This information at microscale gives new insight into the complex mechanism of bubble nucleation and growth in fluids and could help to develop a rigorous theoretical modelling and numerical simulation such as the Lattice Boltzmann approach.     

Bubble growth model and its influencing factors in a polymer melt under nonisothermal conditions

Traditionally, in order to simplify the bubble growth process in a polymer melt, an isothermal model is typically used. In fact, the temperature of the polymer melt is changing during the foaming process. In order to accurately study the growth mechanism of bubbles in polymer melts, we build a physical and mathematical model of bubble growth in a polymer melt under nonisothermal conditions. The parameters of pressure, zero-shear viscosity, relaxation time, Henry's constant, diffusion coefficient, and surface tension were determined. The fourth-order Runge–Kutta method was used to solve the nonisothermal bubble model in the polymer melt. A computational program is developed to find the dimensional change during the bubble growth process, and the correctness of the model is verified. The nonisothermal growth mechanism of and factors influencing bubbles in the polymer melt are analyzed. Combined with the design of experiment (DOE) analysis method, the transfer function of the bubble radius and the maximum growth rate of bubbles with the process parameters were obtained, such as cooling rate, system pressure, and gas concentration. The results show that system pressure has the most significant effect on bubble growth. At the same time, a bubble growth prediction model is built, which can be used to predict the growth of bubbles. Through optimization analysis, it can be used to control the growth of bubbles. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2019 , 136, 47210.     

Foam processing and cellular structure of polylactide-based nanocomposites

Via a batch process in an autoclave, the foam processing of neat polylactide (PLA) and two different types of PLA-based nanocomposites (PLACNs) has been conducted using supercritical carbon dioxide (CO₂) as a foaming agent. The cellular structures obtained from various ranges of foaming temperature-CO₂ pressure were investigated by using scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The incorporation with nano-clay induced heterogeneous nucleation because of a lower activation energy barrier compared with homogeneous nucleation as revealed by the characterization of the interfacial tension between bubble and matrix. The grown cells having diameter of ∼200 nm were localized along the dispersed nano-clay particles in the cell wall. The dispersed nano-clay particles acted as nucleating sites for cell formation and the cell growth occurs on the surfaces of the clays. The PLACNs provided excellent nanocomposite foams having high cell density from microcellular to nanocellular.     

Preparation and morphology characterization of microcellular styrene‐co‐acrylonitrile (SAN) foam processed in supercritical CO2

Microcellular polymeric foam structures have been generated using a pressure‐induced phase separation in concentrated mixtures of supercritical CO₂ and styrene‐co‐acrylonitrile (SAN). The process typically generates a microcellular core structure encased by a non‐porous skin. Pore growth occurs through two mechanisms: diffusion of CO₂ from polymer‐rich regions into the pores and also through CO₂ gas expansion. The effects of saturation pressure, temperature and swelling time on the cell size, cell density and bulk density of the porous materials have been studied. Higher CO₂ pressures (hence, higher fluid density) provided more CO₂ molecules for foaming, generated lower interfacial tension and viscosity in the polymer matrix, and thus produced lower cell size but higher cell densities. This trend was similar to what was observed in swelling time series. While the average cell size increased with increasing temperature, the cell density decreased. The trend of bulk density was similar to that of cell size. © 2000 Society of Chemical Industry     

Modeling nanoporosity development in polymer films for low‐k applications

The bubble growth dynamics of a polymer supersaturated with CO₂ have been modeled for micron‐size films after nucleation. The model equations are based on the shell model of Arefmanesh, Advani, and Michaelides in which a nucleated bubble is surrounded by a finite concentric shell of polymer supersaturated with gas. Bubbles grow by mass transfer of dissolved gas from this shell. The model is extended to allow for diffusion of dissolved gas out of the shell in addition to diffusion into the bubble. A parametric analysis is performed to examine the effects of film thickness, temperature, diffusivity at the T_g and Henry's law constant. POLYM. ENG. SCI., 45:640–651, 2005. © 2005 Society of Plastics Engineers     

Water blown free rise polyurethane foams

A model is developed in this work for predicting the bubble size distribution in polyurethane foams generated by using water as a chemical blowing agent. The model combines equations of energy balance, kinetics of the reactions of isocyanate with water and polyol, and nucleation and growth of CO₂ bubbles. It is found that as the water content of the reaction mixture is increased, the mean bubble size decreases and the bubble size distribution become narrower. Exactly the opposite occurs in polyurethanes foamed with a physical blowing agent, e.g., DuPont Freons. This suggests that a combination of physical and chemical blowing agents can be employed to control bubble size distribution.     

Measurement and prediction of LDPE/CO2 solution viscosity

When CO₂ is dissolved into a polymer, the viscosity of the polymer is drastically reduced. In this paper, the melt viscosities of low‐density polyethylene (LDPE)/supercritical CO₂ solutions were measured with a capillary rheometer equipped at a foaming extruder, where CO₂ was injected into a middle of its barrel and dissolved into the molten LDPE. The viscosity measurements were performed by varying the content of CO₂ in the range of 0 to 5.0 wt% and temperature in the range of 150°C to 175°C, while monitoring the dissolved CO₂ concentration on‐line by Near Infrared spectroscopy. Pressures in the capillary tube were maintained higher than an equilibrium saturation pressure so as to prevent foaming in the tube and to realize single‐phase polymer/CO₂ solutions. By measuring the pressure drop and flow rate of polymer running through the tube, the melt viscosities were calculated. The experimental results indicated that the viscosity of LDPE/CO₂ solution was reduced to 30% of the neat polymer by dissolving CO₂ up to 5.0 wt% at temperature 150°C. A mathematical model was proposed to predict viscosity reduction owing to CO₂ dissolution. The model was developed by combining the Cross‐Carreau model with Doolittle's equation in terms of the free volume concept. With the Sanchez‐Lacombe equation of state and the solubility data measured by a magnetic suspension balance, the free volume fractions of LDPE/CO₂ solutions were calculated to accommodate the effects of temperature, pressure and CO₂ content. The developed model can successfully predict the viscosity of LDPE/CO₂ solutions from PVT data of the neat polymer and CO₂ solubility data.     

Solid‐state microcellular polycarbonate foams. I. The steady‐state process space using subcritical carbon dioxide

The process parameters for production of solid‐state microcellular polycarbonate using subcritical CO₂ were explored. Sufficiently long foaming times were used to produce foams, where cell growth had completed, resulting in steady‐state structures. A wide range of foaming temperatures and saturation pressures below the critical pressure of CO₂ were investigated, establishing the steady state process space for this polymer–gas system. Processing conditions are presented that produce polycarbonate foams where both the foam density and the average cell size can be controlled. The process space showed that we could produce foams at a constant density, while varying the cell size by and order of magnitude. At a relative density of 0.5, the average cell size could be varied from 4 to 40 μm. The ability to produce such a family of foams opens the possibility to explore the effect of microstructure, like cell size on the properties of cellular materials. It was found that the minimum foaming temperature for a given concentration of CO₂, determined from the process space, agrees well with the predicted glass transition temperature of the gas–polymer solution. A characterization of the average cell size, cell size distribution, and cell nucleation density for this system is also reported. POLYM. ENG. SCI., 2010. © 2010 Society of Plastics Engineers