Nanocellular Polymers: From Microscale to Nanoscale 9783110756111

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Judith Martín de León, Victoria Bernardo, Miguel Ángel Rodríguez-Pérez Nanocellular Polymers

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Judith Martín de León, Victoria Bernardo, Miguel Ángel Rodríguez-Pérez

Nanocellular Polymers From Microscale to Nanoscale

Authors Prof. Judith Martín de León Condensed Matter Physics Department University of Valladolid Paseo de Belén 7 47011 Valladolid Spain [email protected] Dr. Victoria Bernardo García CellMat Technologies S.L. CTTA Building Paseo de Belen 9-A 47011 Valladolid Spain [email protected] Prof. Miguel Ángel Rodríguez-Pérez Condensed Matter Physics Department University of Valladolid Paseo de Belén 7 47011 Valladolid Spain [email protected]

ISBN 978-3-11-075611-1 e-ISBN (PDF) 978-3-11-075613-5 e-ISBN (EPUB) 978-3-11-075621-0 Library of Congress Control Number: 2023941446 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the internet at http://dnb.dnb.de. © 2024 Walter de Gruyter GmbH, Berlin/Boston Cover image: Miguel Ángel Rodríguez-Pérez Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Acknowledgments Financial support from the Science and Innovation Ministry of Spain (PID2021127108OB-I00, TED2021-130965B-I00, and PDC2022-133391-I00) is gratefully acknowledged. Financial assistance from the Junta of Castile and Leon (VA202P20) is gratefully acknowledged. Activities have been funded by the EU NextGeneration and Castile and Leon Complementary plans of research and development with the autonomous regions in actions of R&D. Component 17. Investment 1. (C17. I1), of the recovery, transformation, and resilience plan. In addition, we would like to thank all the current and previous members of the Cellular Materials Laboratory of the University of Valladolid (www.cellmat.es) and from the spin-off company CellMat Technologies (www.cellmattechnologies.com) for their continuous effort in developing new knowledge on the field of cellular materials and nanocellular polymers. We would also like to thank our families and friends for their support in all these years in which we have been working hard in scientific activities related to cellular and nanocellular polymers.

https://doi.org/10.1515/9783110756135-202

Contents Acknowledgments Chapter 1 Introduction 1 References

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7

Chapter 2 Fundamentals 8 2.1 Introduction 8 2.2 Descriptors to characterize cellular polymers 8 2.2.1 Relative density, porosity, and expansion ratio 8 2.2.2 Gas phase descriptors 11 2.2.3 Solid phase descriptors 18 2.3 Foaming mechanisms 26 2.3.1 Polymer/gas solution 26 2.3.2 Nucleation 27 2.3.3 Growth 30 2.3.4 Degeneration 30 2.3.5 Stabilization 32 2.4 Nanocellular polymers: Key features 32 2.4.1 Cellular structure in nanocellular polymers 33 2.4.2 Foaming mechanisms in the nanoscale 38 2.5 Conclusions 39 References 40 Chapter 3 From the microscale to the nanoscale in cellular materials production process 45 3.1 Introduction 45 3.2 Production techniques of nanocellular polymers 49 3.3 Gas dissolution foaming 51 3.3.1 Homogeneous nucleation 55 3.3.2 Heterogeneous nucleation 66 3.4 Limitations and challenges 72 3.4.1 Limitations 72 3.5 Conclusions 76 References 77

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Contents

Chapter 4 Optical properties 85 4.1 Introduction 85 4.2 Light interaction with porous structures 86 4.3 Transparent nanocellular polymers 90 4.3.1 Production of transparent nanocellular polymers 90 4.3.2 Factors affecting transmittance 93 4.4 Other transparent nanoporous structures 99 4.5 Current limitations and future perspectives 102 4.6 Conclusions 102 References 103 Chapter 5 Thermal conductivity 105 5.1 Introduction 105 5.2 Mechanisms of heat transfer in nanocellular polymers 109 5.2.1 Conduction through the solid phase 109 5.2.2 Conduction through the gas phase 113 5.2.3 Radiation 115 5.3 Models to predict the thermal conductivity in nanocellular polymers 123 5.4 Experimental determination of the thermal conductivity 126 5.4.1 Experimental techniques 126 5.4.2 Results in the literature 131 5.5 Future perspectives 135 5.6 Conclusions 137 References 138 Chapter 6 Mechanical properties 143 6.1 Introduction 143 6.2 Mechanical properties in microcellular polymers 6.3 Mechanical properties in nanocellular polymers 6.3.1 Confinement of the solid phase 157 6.4 Conclusions 161 References 162 Chapter 7 Surface area 164 7.1 Introduction 164 7.2 Surface area in nanocellular polymers 164 7.3 Generation of open cell nanocellular polymers

144 152

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Contents

7.3.1 7.3.2 7.3.3 7.4 7.5

Types of open cell structures 166 Gas dissolution foaming of films and fibers Generation of a porous skin 172 Applications in filters, sensors, and others Conclusions 178 References 178

Chapter 8 Other properties 181 8.1 Introduction 181 8.2 Acoustic properties 181 8.3 Dielectric properties 185 8.4 Electromagnetic shielding properties 8.5 Multifunctional materials 192 8.6 Conclusions 192 References 193

170 175

188

Chapter 9 Applications of nanocellular polymers and future trends 195 9.1 Two types of nanocellular polymers: Fabrication and structure 9.2 Properties of nanocellular polymers: Comparison with other materials 196 9.3 Future trends 198 References 200 Index

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195

Chapter 1 Introduction It is well known that physicist Richard Feynman’s talk entitled “There Is Plenty of Room at the Bottom,” [1] given at the annual American Physical Society meeting at Caltech on December 29, 1959, in which he discussed the possibility of manipulating individual atoms and molecules as a new path for the development of materials science, is often regarded as the starting point of nanoscience and nanotechnology. It took some time to transfer this theoretical discussion about nanoscience to the laboratories due to the enormous difficulty of manipulating materials at the nanoscale. One important discovery in this field was the Scanning Tunneling Microscope (STM) in 1981, by Gerd Binnig and Heinrich Rohrer [2]. The STM together with the later development of Atomic Force Microscopy (AFM) has allowed researchers to observe and manipulate atoms and molecules on surfaces with unprecedented resolution. They were groundbreaking tools for studying materials at the nanoscale. In addition to this, the development in the 1980s of devices based on the use of ultra-high vacuum has also permitted manipulate materials, mainly semiconductors, in the nanoscale to create low dimensional systems. From that time until now, the development of materials with very special properties due to some “nanoscale effect” has been enormous. Here are some very specific examples [3]: two-dimensional electronic systems with very special electronic properties and showing at low temperatures and high magnetic fields the quantum hall effect, high temperature superconductors, quantum dots, one-dimensional materials with extraordinary electrical, magnetic and mechanical properties, fullerenes, graphene, nanoparticles of many different types, etc. In the field of polymers there are also some important examples such as polymeric nanocomposites [4–5] that can reach much better properties than the raw polymer, with a small proportion of nanoparticles. These materials can also be much better than conventional composites in which fillers with sizes in the microscale are used. Polymer composites with improved stiffness and strength, with electrical and thermal conductive properties, with excellent fire resistance have been developed by a clever use of nanoparticles. The theoretical and experimental understating of phenomena in the nanoscale is today enormous and one concept that is typically used in this field is the size effect [6]. This concept establishes that the properties of a material can change when the dimensions of the material are reduced to a size comparable to or smaller than the wavelength associated with the excitations associated to the properties under study. For instance, in two-dimensional electronic systems the properties of the 2D system are totally different from that of the three-dimensional system (3D) when the wavelength of electrons is of the order of magnitude or larger than the thickness of the system under study [6]. When the thermal conductivity of solid insulators is measured at very low temperatures the mean free path of phonons can be https://doi.org/10.1515/9783110756135-001

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Chapter 1 Introduction

very large (in the range of a few mm) and then the properties of the sample can depend on the size of the material under study [6]. So, one question that may arise for scientists working on cellular polymers is if this kind of size effects, typical in nanoscience and nanotechnology, can take place in a cellular material and, in particular, in a cellular polymer. But, before going into this topic it is necessary to explain a few basic concepts about cellular materials. A cellular material is a two-phase system formed by a continuous solid phase and a continuous or discontinuous gas phase. Cellular materials are quite often found in nature. Wood, cork, sponge, bones, and leaves are some of the cellular materials found in nature for very different purposes (Figure 1.1).

Figure 1.1: Different examples of cellular materials in nature, bone, leave, wood, butterfly wing, and cork (from upper left to down right).

Man has always been using these natural materials. Just to name two examples, the Egyptians used wooded artefacts to build their pyramids 5000 years ago [7]. Wood is still a very important cellular material for us in many industrial sectors, and cork was already used in bungs in wine bottles in Roman times. In addition, around 100 years ago man started to produce their own cellular materials using different technologies, being cellular polymers, also named on many occasions, polymeric foams, materials that very quickly become very important for many industries. Indeed, cellular polymers can be found in almost in every industry. These materials combine lightweight (with the associated cost and raw materials savings) with good thermal and acoustic insulation, impact protection, high stiffness, and strength to weight ratio, buoyancy properties, among other interesting features. Furthermore,

Chapter 1 Introduction

3

it is possible to tailor the properties of these materials to meet the requirements of many different applications. These materials have continuously growth over the years, and in fact, the market size for cellular polymers or polymer foams is today enormous and it is still growing quickly. For instance, this market is projected to grow from USD 90.7 billion in 2020 to USD 114.8 billion by 2025, at a CAGR of 4.3 [8]. This forecast growth is mainly due to the interesting combination of properties of these materials, that allow their use in multiple sectors, such as the furniture and bedding, transportation, packaging, the construction and others such as aeronautic, naval, leisure, and sports. As it can be observed in Figure 1.2. the main materials used to produce these cellular polymers are polyurethane, polystyrene, polyvinyl chloride, polyolefins, phenolic and melamine, with Asia Pacific, North America, and Europe being the areas of the world where the use of these materials is more extended. We will name from now on these materials in the market and with large cell sizes (over 100 microns) conventional cellular polymers. Just to give an idea of the incredible volume of cellular polymers produced every year a simple calculation can be done. Let’s assume that the average density of the cellular polymers produced world-wide is 50 kg/m3 and that we slice all the cellular polymers produced world-wide in sheets 10 mm thick. Then, the estimated surface

Figure 1.2: Cellular polymers market in 2020, divided by region, type of material and applications.

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Chapter 1 Introduction

of cellular polymers produced is around 50,000 km2 every year. This means that in 10 years we could cover all surface of a large country in Europe like Spain, France, or Germany by using a cellular polymers carpet. As mentioned before these materials have two phases, the gas phase and the solid phase. If we want to analyze if a size effect is possible in these materials, we need to analyze every phase considering the phenomena that could take place in each one. Regarding the gas phase, the relative proportion between the mean free path of the gas molecules and the cell size is a key parameter in this analysis. For air at atmospheric pressure this mean free path is 70 nm. In order to observe nanoscale effects, the size of the cells should be very small, and comparable to this number. So, if we apply the principles of nanoscience, a clear conclusion is that we will observe nanoscale effects only for cellular materials with cells in the nanoscale. Most cellular polymers in the market today based on polyurethane, polystyrene, polyvinyl chloride, polyolefin, phenolic, and melamine have cells much larger than this level, the sizes being typically higher than 100 micros; this is more than 1000 times the value of the mean free path of air. Even for microcellular polymers, with cell sizes around 10 microns, nanoscale effects associated to the gas phase will not appear because the size of the cell is around 100 times higher than the mean free path of air. Regarding the solid phase, we need to consider three aspects: the typical dimension of the solid phase, the size of the macromolecules, and the mean free path of the excitations. Regarding the typical dimensions of the solid phase, and in a first analysis, we can consider the cell wall thickness as a key parameter because the cell walls are the thinnest part in the solid phase. For the conventional foams found in the market, with cell sizes larger than 100 microns, the minimum cell wall thickness found for very low-density cellular polymers (relative densities below 0.03) are in the range of 0.5 microns. A very thin cell wall that typically shows a clear orientation of the polymer molecules due to the foaming process. Then, the question is: is this size low enough to show nanoscale effects? From a topological point of view, the second parameter that needs to be considered is the size of the macromolecules used to produce the cellular polymer. If the gyration radius of the polymer macromolecule is high enough to be comparable to the cell wall thickness some confinement effect could be found, but in most cases the gyration radius of polymer macromolecules is smaller than 500 nm, so a confinement effect is not expected in conventional cellular polymers. On the other hand, and from the excitations point of view, if, for instance, we consider the propagation of heat, the mean free path of phonons at room temperature in polymers is from 1.5 to 10 nm, so much smaller than the cell wall thickness and therefore a size effect is no expected for conventional cellular polymers. Then, the following natural question is: what would happen if the cell wall thickness could be much smaller, for instance, below 100 nm? Then, the gyration radius for some polymeric matrices could be larger than the cell wall thickness and even than the cell size, and then a confinement of the polymer matrix could be expected.

Chapter 1 Introduction

5

In addition, for some extreme cases of materials with very thin cell walls, the size of the cell walls could be of the same order to magnitude as the excitations promoting a size effect. We can therefore conclude from this analysis based on the size effect concept that nanoscale effects could take place in cellular polymers but only when the cells are in the nanoscale. This means cell sizes lower than 1 micron are needed to detect unexpected properties of the gas phase and cell wall thickness much smaller than 1 micron is needed to observe unexpected properties of the solid phase. The idea of creating nanocellular polymers using foaming processes started to be discussed at the beginning of this century. The concept behind this was clear. In these materials, nanoscale effects should appear and due to this some properties will be highly improved in comparison with microcellular and conventional cellular polymers, and some unexpected properties could appear. In addition, another key aspect was producing these materials using technologies that could be up-scaled and that were environmentally friendly. This research appeared as a natural continuation of the research on microcellular polymers, with cell sizes lower than 10 microns that were developed in the 1980s initially by MIT and successfully commercialized years later in the injection molding of high-density parts, in the extrusion foaming and more recently in extrusion blow molding (see Chapter 3 for more details about these materials). Most of the efforts in the last years have been focused on how to produce these nanocellular polymers using the gas dissolution foaming technology that was also the one initially used to produce microcellular polymers. As it will be discussed in detail in Chapter 3, the challenge of producing these materials is huge because very high cell densities (higher than 1013 cells/cm3) are needed. However, the progress done in the last 10 years has been enormous and further improvements are expected, but already today it is possible to produce nanocellular polymers by using foaming approaches with cells in the nanoscale in the range of 10 to 1000 nm, cell wall thickness lower than 15 nm and relative densities in the range of 0.1 to 0.5 starting form many different polymers (polymethylmethacrylate (PMMA), polycarbonate (PC), polyetherimide (PEI), polyphenylsulfone (PPSU), etc.). In addition, the expected nanoscale effects have been proved for some physical properties, such as the mechanical ones, the thermal conductivity, the optical properties, and the dielectric and the acoustic properties. Some of the results obtained were unexpected and difficult to understand from a theoretical point of view, and although we have learned a lot about these novel materials, there are still many aspects that are not understood. So, it is possible to say that nowadays nanocellular polymers produced by foaming techniques are a reality. These are materials with an excellent combination of physical properties such as low weight, very low thermal conductivity, excellent mechanical properties, possibility of showing transparency, huge sur-

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Chapter 1 Introduction

face area, and tunable dielectric and acoustic properties. This is a combination of properties not reached for any other material produced nowadays as will be discussed in Chapter 9 of this book. The authors of this book have devoted around 10 years of their scientific careers to the development of these promising materials, contributing to the current state of the art. The topic is becoming wider and wider as more and more papers are published, and due to this, we believed that it was necessary to write a book that could compile in a single document all the knowledge developed on these materials. We have tried not only to summarize the information, but also we have discussed the findings from a theoretical perspective, so a more basic understanding could be reached, suggesting also future research topics. We think that this manuscript will be very helpful for scientists working in academia or industry who are interested in cellular polymers in general and in nanocellular polymers in particular. The manuscript can be understood by young scientists that start their research, but it could be also helpful for scientists with a higher experience in the topic that want to go into a deeper analysis. The manuscript is divided into nine chapters, and it is organized in the following way. Chapter 1 introduces the topic of nanocellular polymers in the context of the evolution of cellular polymers and using some of the current knowledge on nanoscience and nanotechnology. Chapter 2 aims to present an overview of the fundamental concepts needed to understand the structure of cellular polymers and their fabrication processes. The descriptors usually employed to characterize cellular polymers are introduced and discussed and the foaming mechanisms occurring during the generation of these materials are presented. In addition, the development of a new generation of cellular polymers, the so-called nanocellular polymers, is introduced. Chapter 3 is devoted to the production process of nanocellular polymers and in particular, describes in detail how the gas dissolution foaming technology has been used in the last years to create a wide variety of nanocellular polymers. The foaming mechanisms occurring during the production of these materials are described and discussed, and the current state of the art is summarized. Chapter 4 explains one of the most interesting properties of nanocellular polymers, the possibility of producing transparent cellular polymers. This property is studied from a theoretical and an experimental point of view, showing the last advances within this topic, as well as a comparison with other transparent nanoporous materials such as aerogels. In addition, the current limitations and challenges are also discussed. Chapter 5 discusses in detail the thermal conductivity of these materials. When the research on these materials started and due to the confinement of the gas phase (Knudsen effect), very low thermal conductivities were expected, and this was one of the key motivations to develop nanocellular polymers. This chapter describes the heat conduction mechanisms, the theoretical models to predict this property, the current

References

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state of the art, and the challenges to improve the thermal insulation of nanocellular polymers. Chapter 6 is focused on the mechanical properties. The current state of the art is revised, and the mechanical properties of these materials are compared to those of microcellular polymers and conventional cellular polymers showing under which circumstances nanocellular polymers present outstanding properties. The results are discussed considering the confinement of the solid phase. Chapter 7 discusses the very high surface area of nanocellular polymers, and the methods known to create open cell structures with different types of morphologies. Then, strategies to eliminate the solid skin and foam thin films are presented. Last, some preliminary results suggesting the potential of nanocellular polymers for filtration and sensors are presented together with some ideas for further research and development. This is a very open and promising topic in which research has been still limited. Chapter 8 focuses on other physical properties not covered before in the book. In particular, acoustic properties, dielectric properties, and the electromagnetic shielding properties are covered. The investigation on these properties suggests very interesting behaviors, but they are still very open topics. Finally, Chapter 9 presents a more detailed description of some of the most promising nanocellular polymers produced so far and compares the properties of these materials with those of conventional foams and silica aerogels. The idea of this chapter is to show, by comparing with other well-known and commercial materials, the huge potential of nanocellular polymers due to an amazing combination of physical properties. This book finishes describing the challenges of the topic and possible direction of future research.

References [1] [2] [3] [4] [5] [6] [7] [8]

R.P. Feynman, There’s plenty of room at the bottom, Eng. Sci. 23(5) (1960) 22–36. ISSN 0013-7812 G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Tunneling through a controllable vacuum gap, Appl. Phys. Lett. 40(2) (1982) 178–180. C. Kittel, Introduction to Solid State Physics, 8th ed., John Wiley and Sons, (2005). C. Hussain (Eds.), Handbook of Polymer Nanocomposites for Industrial Applications, 1st ed., Elsevier, (October 27, 2020). M.E. Hoque, R. Kumar, A. Sharif (Eds.), Advanced Polymer Nanocomposites Science, Technology and Applications, 1st ed., Elsevier, (May 1, 2022). J.R. Hook, H.E. Hall, Solid State Physics, 2nd ed., John Wiley and Sons, (1974). L.J. Gibson, M. Ashby, Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, (1997). https://mytrendingstories.com/sweety-anthony/polymer-foam-market-insight-forecast-to-hdzliv

Chapter 2 Fundamentals 2.1 Introduction This chapter aims to present an overview of the fundamental concepts needed to understand the structure of these materials and their fabrication processes. The descriptors usually employed to characterize cellular polymers are introduced and discussed and the foaming mechanisms occurring during the generation of these structures are presented. Finally, the development of a new generation of cellular polymers, the so-called nanocellular polymers, is introduced, and the changes in the descriptors and foaming mechanisms are commented on.

2.2 Descriptors to characterize cellular polymers A cellular material is a two-phase system formed by a continuous solid phase and a continuous or discontinuous gas phase. The gas phase is enclosed in pores or cells. In the case of cellular polymers, also called polymer foams, the solid phase is a polymer or a polymer-based formulation. Figure 2.1 shows some examples of cellular polymers currently used in different applications. Cellular polymers are characterized by several structural descriptors that can be classified into two main groups: solid phase descriptors (those related to the polymer matrix) and gas phase descriptors (those describing the gas phase). Figure 2.2 presents a schematic representation of the morphology of a cellular polymer and the different parameters that can be used to fully describe the structure. In the following sections, the descriptors are explained in detail and the different techniques to measure and quantified them are explained.

2.2.1 Relative density, porosity, and expansion ratio The relative density is the most important parameter describing any cellular polymer since its one of the main parameters determining the possible final applications of the material. The relative density (ρr ) is defined as the ratio between the density of the foam (ρfoam ) and the density of the solid material that composes the solid phase (ρsolid ): ρr =

https://doi.org/10.1515/9783110756135-002

ρfoam ρsolid

(2:1)

2.2 Descriptors to characterize cellular polymers

Figure 2.1: Cellular materials based on various polymers and some of their applications: (a) low-density polyurethane (PU) for thermal insulation in buildings, (b) low-density ethyl(vinyl acetate) (EVA) in toys, (c) medium-density polypropylene (PP) in car parts, and (d) low-density polystyrene (PS) in food packaging.

Figure 2.2: Classification of the descriptors to characterize a cellular polymer.

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Chapter 2 Fundamentals

Based on this definition, the relative density represents the volume fraction of the cellular material that is solid. Then, it can vary in the range from values above 0 (highly porous material, low solid fraction) to 1 (totally solid material). Cellular polymers can be classified according to their relative density into three main groups (examples in Figure 2.3): high-density materials (ρr > 0.7), medium-density (0.2 < ρr < 0.7), and low-density cellular polymers (ρr < 0.2). As mentioned above, the relative density greatly affects the properties and thus defines the range of applications of cellular polymers. High-density materials are meant to be used in structural applications, whereas low-density cellular polymers can be used in thermal and acoustic insulation, among other types of applications.

Figure 2.3: Examples of cellular polymers with (a) high density, (b) medium density, and (c) low density.

The complementary descriptor to the relative density is the porosity, Vf (eq. (2.2)). The porosity represents the volume fraction of material that is gaseous. Then, it can vary between 0 (totally solid) and values near 1 (highly porous material with a very low solid fraction). On occasions, a percentage of porosity (Vf ð%Þ = 100Vf ) is also used in the literature to talk about this descriptor. It is common to talk about cellular polymers with porosities of 30% or 80%, for instance. Vf = 1 − ρr = 1 −

ρfoam ρsolid

(2:2)

Yet another parameter to describe this idea is the expansion ratio, ER (eq. (2.3)). It is defined as the volume of the foam divided by the volume of the solid, or, assuming the mass is conserved in the foaming process, the inverse of the relative density, that is, the density of the solid divided by the density of the foam. It accounts for the volumetric expansion from the initial solid material. For instance, an ER = 20 means that the cellular polymer has expanded 20 times in volume with respect to the solid precursor material; that is, the density has been reduced 20 times. ER =

Vfoam ρsolid 1 = = Vsolid ρfoam ρr

(2:3)

2.2 Descriptors to characterize cellular polymers

11

For processes in which there is a directionality (for instance, foaming in a mold), it is also common to talk about the expansion ratio in three axes (ERx , ERy and ERz , calculated as the linear expansion in each direction: the ratio between the length of the foam and the length of the precursor in one direction), the total ER being the product of the three linear expansions: ER = ERx ERy ERz

(2:4)

2.2.2 Gas phase descriptors 2.2.2.1 Cell size distribution and average cell size Cell size is defined as the diameter of the cells. There are two main descriptors related to the cell size: the average cell size (calculate as the average diameter of a representative number of cells) and the cell size distribution (the histogram presenting the relative frequency of each cell size). Both parameters are key to understanding the properties of cellular polymers. According to the literature, there is not a well-defined unique method to determine these parameters. The only accepted method within the American Society for Testing and Materials (ASTM) standards is the intersections methods [2, 3]. However, this approach offers inaccurate results in medium and high-density foams since it does not consider the cell wall thickness. Moreover, it provides the average value and not the cell size distribution. Several non-standard approaches have been followed to obtain a more precise analysis of the cell size [4, 5]. Among them, recently Pinto and coworkers developed a user-interactive analysis tool that allows obtaining all the representative parameters in a quick and reproducible method [6]. The average cell size (ϕ3D ) is calculated according to eq. (2.5), where ϕ3D,i are the measured cell size values of each cell and n is the number of cells analyzed. The standard deviation, SD (eq. (2.6)), and the asymmetry coefficient AC (eq. (2.7)) are parameters that allow describing the cell size distribution. The smaller SD, the thinner the cell size distribution is. Regarding AC, it measures the symmetry of the distribution with respect to the average value. Finally, the ratio between the standard deviation and the average cell size is defined as the normalized standard deviation coefficient, NSD (eq. (2.8)). Pn ϕ ϕ3D = i=1 3D,i (2:5) n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn  i=1 ϕ3D,i − ϕ3D (2:6) SD = n−1

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Chapter 2 Fundamentals

Pn AC = NSD =

2

i=1

ðϕ3D,i − ϕ3D Þ nSD3

SD ϕ3D

(2:7) (2:8)

Figure 2.4 shows examples of cell size distribution for different scenarios. Figure 2.4a shows three distributions showing the same average cell size (100 µm) but different standard deviation coefficients. The higher the NSD, the wider the cell size distribution, that is, the less homogeneous is the cellular structure. Figures 2.5a and 2.5b show examples of cellular structures with low and high NSD. Figure 2.4b presents a particular case of heterogeneous cellular structure in which there are two well-distinguished average cell sizes: these are bimodal cell size distributions (Figure 2.5c). The average cell size can be also used as a parameter for classifying cellular polymers. Conventional cellular polymers present cell sizes higher than 10 microns, typically in the range of 100 to 700 microns. In the 1980s, a new class of cellular polymers was developed: the so-called microcellular polymers, which showed cell sizes below 10 microns [7, 8]. Recently, technology allowed moving one step forward with the generation of nanocellular polymers, characterized by cell sizes under the micron. Figure 2.6 shows examples of these structures. More details about the historical evolution of cellular polymers are presented in Chapter 3. 2.2.2.2 Cell density and cell nucleation density Cell density (Nv ) is defined as the number of cells per unit of volume in the cellular material. This parameter is usually presented in cells/cm3. Assuming that there is no degeneration during the foaming process and that every single nucleus grows into one cell, it is possible to calculate the cell nucleation density, N0 [9]: N0 =

Nv ρr

(2:9)

The cell nucleation density can be calculated theoretically according to eq. (2.10). This relation is a geometrical equation that states that the three parameters, cell size, cell nucleation density, and relative density, are not independent. This equation assumes spherical cells.   6 1 −1 (2:10) N0 = πϕ3 ρr Figure 2.7a shows the theoretical predictions of eq. (2.10) for different cell nucleation densities. It is observed that as the relative density decreases, it is mandatory to increase the cell nucleation density for keeping a constant cell size. That is, the number of cells needed to create a low-density foam is always higher than that required to

2.2 Descriptors to characterize cellular polymers

13

Figure 2.4: (a) Examples of cell size distributions with different normalized standard deviation coefficients; (b) example of a bimodal cell size distribution.

achieve a high-density foam. To reduce the cell size at constant density, it is also needed to increase dramatically the cell nucleation density. As commented before, there has been a historical trend of research in the decrease of cell size. Conventional low-density cellular polymers, with cell sizes above 300 microns, are characterized by cell nucleation densities below 106 nuclei/cm3. To achieve microcellular polymers (cell sizes below 10 microns, Figure 2.7b) with a significant expansion, cell nucleation density must be higher than 109 nuclei/cm3, that is, 3 orders of magnitude higher than the

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Chapter 2 Fundamentals

Figure 2.5: (a) Monomodal cell size distribution with low NSD, (b) monomodal cell size distribution with high NSD, and (c) bimodal cell size distribution.

Figure 2.6: Examples of conventional (a), microcellular (b), and nanocellular (c) polymers.

values of conventional foam. This was a significant technological challenge, and it was only in the 1980s that it was possible to accomplish it. The challenge became even more exigent when nanocellular polymers were considered: in this case, nucleation densities over 1013 nuclei/cm3 are required to reduce the cell size below the micron. 2.2.2.3 Anisotropy ratio Anisotropic cellular structures are characterized by cells elongated in one direction and are characterized by the anisotropy ratio. The anisotropy ratio is usually defined as the ratio of cell sizes measured in two perpendicular directions [10]. However, the cells are 3D objects that can have different orientations in the three axes of the space (Figure 2.8). Then, it is possible to calculate anisotropy ratios in the three main axes, as presented in eqs. (2.11), (2.12), and (2.13). Note that x, y, and z can be renamed to be the thickness direction, machine direction, mold opening direction, etc. (depending on the foaming process). Rz = 

ϕz

 ϕx + ϕy =2

(2:11)

15

2.2 Descriptors to characterize cellular polymers

Figure 2.7: (a) Lines of constant cell nucleation density in a relative density–cell size map, according to eq. (2.10). (b) Zoom to the region of low cell size (below 10 microns) and low relative density.

Rx = 

Rz =

ϕx

 ϕz + ϕy =2

(2:12)

ϕy ðϕx + ϕz Þ=2

(2:13)

This parameter is of great relevance in foaming processes in which there is a preferred growth direction. Figure 2.9 shows some examples of anisotropic cellular structures with cells elongated in the vertical direction. Most foaming processes lead to some anisotropy of the cellular structure. For instance, in reactive foaming for the production of polyisocyanurate (PIR) or polyurethane foams, cells tend to grow preferably in the expansion direction, that is, the

16

Chapter 2 Fundamentals

Figure 2.8: Schematic representation of a 3D cell with different cell sizes in each direction (anisotropic structure).

Figure 2.9: Examples of cellular structures showing anisotropic cells: (a) rigid PU with cells elongated in the growth direction and (b) XPS with cells elongated in the thickness direction.

vertical direction [11]. In compression molding the anisotropy is also promoted in the direction parallel to the compression direction. Extrusion foaming can also lead to a particular orientation of the cellular structure in the three main planes of the extruded foam (parallel to the thickness direction, the extrusion direction, and the traverse direction). The addition of oriented fillers, such as fibers or needle-like particles, can also promote preferred cell growth in a given direction [12]. However, cellular polymers usually present anisotropy ratios in the range from 1 to 1.3, and creating higher anisotropies requires the design of specific production routes designed for that purpose [13, 14]. The anisotropy of the cellular structure has a great impact on the physical properties of the cellular material. Mechanical properties, such as the elastic modulus, compressive strength, and shear modulus, are known to increase in the direction parallel to the elongation of the cells [15–17]. The thermal conduction through the solid structure of the cellular polymer also increases in the anisotropy direction [18, 19]. The analysis of the anisotropy ratio is calculated using cell size measurements, so it can be experimentally determined in the same way as the cell size (see Section 2.2.2.2).

2.2 Descriptors to characterize cellular polymers

17

2.2.2.4 Open cell content and tortuosity of the gas phase Cellular polymers can be classified according to the interconnectivity of the gaseous phase. We can distinguish between closed-cell foams, foams with an intermediate open cell content, and open-cell foams. In closed-cell foams, the gas is enclosed inside the cells, and it cannot move freely through the cellular structure. Open-cell foams are characterized by a high level of interconnectivity between the cells, allowing the movement of gas through the cellular structure. Meanwhile, foams with intermediate open-cell contents can be considered hybrids between both, as their cellular structure is partially interconnected. Figure 2.10 shows the cellular structures of closed-cell foams, open-cell foams, and foams with intermediate open-cell contents. The open cell content is key for understanding the physical properties of cellular polymers. Closedcell foams are known to present better compression modulus and lower thermal conductivity, while open-cell foams are the best candidates for acoustic absorption and for filtering or liquids absorption, among other applications.

Figure 2.10: (a) Closed-cell foam, (b) open-cell foam, and (c) intermediate situation.

The level of interconnectivity can be measured with gas pycnometry and characterized by the open cell content, OC (eq. (2.14)) according to the ASTM Standard D6226-10. OC ð% Þ =

Vg − Vp · 100 Vg ð1 − ρr Þ

(2:14)

where Vg is the geometric volume of the sample, and Vp is the volume measured by the pycnometer. The gas pycnometry technique allows measuring the accessible volume for the gas and this way the open cell content can be calculated. When the gas cannot access the inner structure, Vp = Vg and then OC = 0% (totally closed-cell foam). If the gas can fill all the structure, then Vp measures the volume of the solid phase, and then OC = 100% (totally open-cell foam). However, this measurement can lead to inaccurate results if small samples with large cell sizes exposed on the surface are measured. In those cases, a corrected open cell content can be calculated using eq. (2.15): OC✶ ð%Þ =

Vg − Vp − Vs · 100 V g ð 1 − ρr Þ

(2:15)

18

Chapter 2 Fundamentals

where Vs is the volume of the cells exposed in the surfaces of the sample that can be calculated from the cell size. Open-cell cellular polymers (OC = 100%) can show different properties depending on the size of the holes in the cell walls. For instance, flexible PU foams, formed only by struts, can show a fast recovery when stress is applied and removed, while open cell materials with small holes in the cell walls will present a slower recovery. To quantify these differences, an additional parameter is needed: the tortuosity of the gas phase. The tortuosity of the gas phase (τg ) is defined as the ratio between the distance that a gas molecule must cover from one side to another side of the sample (L) and the hypothetic shortest distance that it would cover following a straight path (L0 ) [36] (eq. (2.16)). τg =

L L0

(2:16)

Tortuosity does not depend on the type of fluid contained by the porous structure but on the type of architecture: open-cell content and the number and size of the holes present in the cell walls [20]. Tortuosity is a critical parameter for many applications, playing a very important role in the response of open-cell foams. The tortuosity can be estimated by using different methods such as ultrasonication or electrical conductivity measurements or theoretically using different models [21–24].

2.2.3 Solid phase descriptors 2.2.3.1 Struts, cell walls, and cell geometry A medium-density and low-density cellular polymer is formed by pores or cells with a given geometry that pack in three dimensions to get rise to the cellular structure. Then, the morphology of the solid phase is formed by a polymer network of vertex and edges (struts) and cell walls. Figure 2.11 shows some examples of struts and cell walls in different cellular polymers. In Figure 2.12 we present a reconstruction of a cell from an X-ray tomography of a cellular polymer. The analysis of the solid phase allows separating the solid phase into two contributions: cell walls and struts. On the other hand, the cell geometry should allow the filling of the space when the cells are packed in the cellular structure. There are two possible scenarios. For highdensity materials, the cells can be spheres. Spheres are the most stable geometry during cell growth. Figure 2.13a shows an example of a high-density material with spherical cells. In this case, it is challenging to define cell walls and struts: all the solid material is forming a continuous phase. As density reduces, the cell geometry must change too to allow an efficient packaging of the cells. In low-density materials, the cells are forming polyhedrons, such as dodecahedrons or tetrakaidecahedrons. Figure 2.13b shows an example of a low-density foam with polyhedral cells. For low-density materials, the differ-

2.2 Descriptors to characterize cellular polymers

19

Figure 2.11: Details of the struts and cell walls in different cellular polymers.

Figure 2.12: Reconstruction of a cell obtained using X-ray tomography: solid phase, and separation of the structure into two main components: cell walls and struts.

Figure 2.13: Examples of extreme cell geometries: (a) high-density foam (spherical cells), (b) low-density foam (polyhedral cells).

ence between cell walls and struts becomes clear: the cell walls are the polymer phase among two cells, whereas the struts are the edges and vertices of the polyhedrons. The mass distribution between struts and walls is characterized by the fraction of mass in the struts, fs . It can be calculated with the following equation [15]: fs =

ts2 ts2 +

Zf
tw l n

(2:17)

20

Chapter 2 Fundamentals

where ts is the thickness of the struts, tw is the thickness of the cell walls, l is the
is the average number of walls per cell, and Zf length of the edges of the cell walls, n is the number of walls merging in a strut. This parameter, which ranges from 0 to 1, accounts for the distribution of the solid phase within the cellular material. When fs is close to 0, the solid phase is mainly distributed in the cell walls. Oppositely, when fs is close to 1, most of the solid phase lies in the struts. In the extreme scenario, when fs = 1, the cellular polymer does not have any cell walls, then the structure is an interconnected network of pure struts. The production process and the properties of the polymer matrix can significantly affect the distribution of the solid phase along the cellular structure. Figure 2.14 presents examples of materials in the three limit cases: low fs ⁓ 0.1−0.2 (extruded polystyrene foam (XPS)), high fs ⁓ 0.6−0.8 (rigid polyurethane foam (PU)) and fs ⁓ 1 (flexible polyurethane foam). Figure 2.15 shows X-ray tomographies of two materials with different mass fraction distributions: rigid PU with high fs (Figure 2.15a) and low-density polyethylene (LDPE) foam with low fs (Figure 2.15b).

Figure 2.14: Examples of materials with different fractions of mass in the struts: (a) fs = 0.2 (XPS); (b) fs = 0.7 (rigid PU); and (c) fs = 1 (flexible PU).

The cell walls are characterized by their thickness (ξ). Typical cell wall thicknesses for low-density cellular polymers are in the range of some microns [25, 26]. Due to the packaging of the cellular structure, one can establish a geometrical relationship between the cell wall thickness, the fraction of mass in the struts, and the relative density. These equations are valid for the range of low-density cellular polymers, i.e. relative densities below 0.2. The equation must also consider the cell size, ϕ. There are two possible scenarios: foams with fs ≠ 1 (eq. (2.18)) [27] and cellular polymers with fs = 1 (totally open cell materials without cell walls) (eq. (2.19)) [15]. In eq. (2.5), C is a constant depending on the cell geometry and that takes a value of 3.46 for pentagonal dodecahedrons [27]. Cξ = ϕð1 − fs Þρr ρr = 1.06

ξ2 ϕ2

(2:18) (2:19)

2.2 Descriptors to characterize cellular polymers

21

Figure 2.15: Reconstruction of a cell obtained using X-ray tomography and projection showing the cellular structure of the foam for two different cases: (a) rigid PU foam with high and fs (b) LDPE foam with low fs .

Figure 2.16 represents the cell size as a function of the relative density for different cell wall thicknesses and the two equations commented on above. For the closed cell model (Figure 2.16a), a fraction of mass in the struts of 0.2 was assumed, as this is a typical value for low-density cellular materials based on PS or PE [15]. Note the different cell wall thickness range between closed cell and open cells. As seen in Figure 2.16, the cell wall thickness may impose some geometrical limitations on the structures of cellular polymers. For instance, with the theoretical predictions of this figure, a closed-cell material of relative density of 0.1 and cell size of 400 microns has a well-determined cell wall thickness of 9 microns (Figure 2.16a). In the high-density range, the thickness of the cell wall increases. Totally open cell structures show a very different behavior, with much lower densities for similar cell wall thickness and cell size (Figure 2.16b). The above parameters (struts and cell wall thickness) are hard to characterize, especially in low-density foams in which their dimensions are quite small. One technique used to measure these magnitudes is scanning electron microscopy (SEM), but it provides limited results since it is challenging to get a fracture that allows proper analysis of the solid architecture, and it is very challenging to separate the volume occupied by the cell walls from the volume occupied by the struts. Another possibility, used in the last few years to obtain accurate values of fs , is X-ray tomography, a tech-

22

Chapter 2 Fundamentals

Figure 2.16: Lines of constant cell wall thickness in a relative density–cell size map: (a) eq. (2.18) for a fraction of mass in the struts of 0.2 and C = 3.46, and (b) eq. (2.19).

nique that allows a 3D visualization of the cellular structure and a good quantification of the solid structure parameters [25, 28]. 2.2.3.2 Solid skin The solid skin is a solid layer covering the outer surface of the cellular polymer. The presence of the solid skin is caused by the diffusion of the blowing agent (responsible for creating the gas cells) out of the material in the region near the surface [29, 30]. The gas molecules in the polymer close to the surface are more likely to diffuse out than to create cells. Therefore, a solid region without porosity appears in most cellular polymers. The thickness of the solid skin depends on the production process and the

2.2 Descriptors to characterize cellular polymers

23

characteristics of the gas and the base polymer (mainly the diffusion coefficient or diffusivity, which accounts for the rate at which the gas escapes the polymer [31]). Figure 2.17 shows some examples of solid skins observed in different cellular polymers. Solid skin has an impact on density because it is much denser than the foamed core, but in some applications, it can have a beneficial impact on mechanical properties [32]. Also, a thick solid skin creates a uniform surface, whereas very thin skins in combination with large cell sizes create a rough surface that might not be suitable for some applications.

Figure 2.17: Example of solid skin in different cellular polymers: (a) expanded polystyrene (EPS) produced with bead foaming technology. The skin is very thin, with a size similar to the cell wall thickness, (b) polymethylmethacrylate (PMMA) produced with the gas dissolution foaming route with a significant solid skin. (c) Polypropylene foam fabricated by injection molding in which there is very thick solid skin useful to improve the mechanical performance and the surface quality.

The solid skin thickness can be measured with microscopy analysis but also with Xray radioscopy thanks to the density difference between the foamed core and the solid skin [33]. 2.2.3.3 Structural gradients Most foams present a structure such as that presented in Figure 2.18a: first, there is solid skin (as already discussed in Section 2.2.3.2), then, there is a transition region or structural gradient and finally there is the homogeneous cellular structure in the core of the foam [34]. The structural gradient between the skin and core is a result of the gas diffusion out of the sample near the skin, but can also be influenced by other factors of the process: temperature gradients, mold geometry, limited expansion in one direction, etc. The region between the solid skin (100% solid) and the homogeneous core (100% foamed) is characterized by a different density, as shown schematically in Figure 2.18b. Then, the structural gradient comprises two main factors: a variation of the cell size and a density gradient. The formation of these structural gradients depends greatly on the foaming process used, which can be even tuned to obtain these structures on purpose [34, 35]. Figure 2.19 shows some examples of foams with very different structural gradients (the thickness of the structural gradient or transition region is shadowed in purple). Depending on the ratio between the foam dimensions and the size of the transition region, and also the

24

Chapter 2 Fundamentals

Figure 2.18: (a) Schematic representation of the cellular structure of a foam near the solid skin and (b) density profile near the solid skin.

final application, the presence of this type of gradient structure can influence the final properties of the foam, so it must be taken into consideration when modeling the properties of the product. SEM images can be used to detect the structural gradients and quantify the differences in cell size, but to account for the density gradient other tools are needed, such as X-ray radiography or tomography. 2.2.3.4 Tortuosity of the solid phase Tortuosity of the solid phase (τs ) is a dimensionless parameter that expresses the ratio of the real pathway distance (L) versus the straightest one (L0 ) when traveling along a certain direction through the internal solid structure [36] (eq. (2.20)). τs =

L L0

(2:20)

Figure 2.20 shows an example of how tortuosity can be calculated for two extreme scenarios with low and high tortuosity of the solid phase for a high-density and lowdensity cellular material. The tortuosity of the solid phase has a significant influence on some transport properties of the material, such as thermal conductivity or electrical conductivity. Note that tortuosity is a directional parameter that can depend greatly on the direction of space. The quantification of the tortuosity of the solid phase is again tricky and requires a well-established definition to be able to get comparative results. To do the analysis, good information about the solid structure, as that obtained in X-ray tomography, is required [36].

2.2 Descriptors to characterize cellular polymers

Figure 2.19: Examples of foamed materials near the solid skin with different structural gradients. Colored boxes are used to illustrate the three regions: dark blue – solid skin; purple – transition region; light blue – homogeneous core.

25

26

Chapter 2 Fundamentals

Figure 2.20: Examples of (a) low tortuosity and (b) high tortuosity of the solid phase.

2.3 Foaming mechanisms All foaming processes take place following the same stages (Figure 2.21). First, the polymer and the blowing agent (gaseous phase) are dissolved. Then, a thermodynamic instability occurs (either a pressure and/or a temperature gradient) and nucleation occurs. Later, the nucleation points grow into cells (cell growth). Finally, some degeneration of the structure can take place during growth and before the stabilization of the cellular polymer. In the next paragraphs, each of the foaming processes is described in detail.

Figure 2.21: Scheme of the foaming mechanisms taking place inside any foaming process and evolution of the expansion ratio with time.

2.3.1 Polymer/gas solution The substance that creates the gas phase in a cellular polymer is called the blowing agent. Blowing agents can be classified as physical blowing agents (such as gases that expand when pressure is released or liquids that develop cells when they evaporate)

2.3 Foaming mechanisms

27

and chemical blowing agents (substances that decompose or react under the influence of heat to form a gas) [37]. The first stage in every foaming process using physical blowing agents or chemical blowing agents under pressure is to create a homogeneous polymer/gas solution. In this sense, the selection of an adequate blowing agent is key to achieving good results. It is mandatory that the amount of blowing agent required for the final characteristics of the foam can be dissolved in the polymer at the conditions of pressure and temperature of the production process. This is what we call solubility: the maximum amount of a certain blowing agent that a polymer can absorb under certain conditions [38, 39]. Solubility depends on both the characteristics of the blowing agent and those of the polymer matrix [40]. For this reason, different blowing agents are used for the manufacturing of different types of polymer foams. Hydrocarbons can be used in many foaming processes. For instance, pentane is used for the production of expanded polystyrene foams (EPS) [41] and PU foams [42], while isobutane can be employed for the generation of extruded polyethylene foams (XPE) [43]. Inert gases, such as carbon dioxide, can be also used as foaming agents for the fabrication of expanded polypropylene foams (EPP) [41] or the production of XPS in combination with ethanol [44]. Chemical blowing agents can be used in rubbers and polyolefins or for the production of high-density cellular polymers [37].

2.3.2 Nucleation The polymer/gas mixture is only stable at certain conditions of pressure and temperature. After a thermodynamic instability (such as a pressure change or a temperature change), the polymer/gas mixture is in a supersaturated state. Subsequently, phase separation takes place. The homogeneous polymer/gas mixture is now separated into two components. Phase separation is known to happen via two primary mechanisms: spinodal decomposition and nucleation and growth, the latter being considered usually as the main mechanism in most of the foaming processes [45]. Spinodal decomposition takes place at high supersaturations, at which the mixture is unstable [46]. The polymer/gas mixture undergoes spontaneous phase separation without the appearance of nucleation points [47]. This phenomenon is characterized by the vanishing of the energy barrier of nucleus formation [48], and so gas molecules immediately start to form clusters, which rapidly grow and coalesce. The result of the spinodal decomposition is a single gas phase, or in other words, an interconnected or co-continuous cellular structure [49–53]. However, the principal mechanism that is believed to control phase separation in almost every foaming process is nucleation. Nucleation consists of the appearance of small clusters or aggregates of gas (nuclei) in a polymer/gas mixture after a sudden change in the thermodynamic conditions [46, 48]. Then, gas in the polymer/gas system diffuses to the newly created nuclei. The nuclei grow because of the gas coming into them, and the pores or cells appear. As opposed to spinodal decomposition, nucleation implies an energy barrier that must be overcome to create a nucleus. Nucleation can

28

Chapter 2 Fundamentals

occur via two main approaches: homogeneous nucleation (in systems with no active nucleating agents) or heterogeneous (in systems containing a second phase which acts as nucleating points). Figure 2.22 schematizes this idea:

Figure 2.22: (a) Homogeneous nucleation and (b) heterogeneous nucleation.

The equations governing the nucleation process were stated in the Classical Nucleation Theory (CNT). According to CNT, for homogeneous nucleation the Gibbs free energy barrier (ΔGhom ) that a nucleus should overcome to grow into a bubble is given by eq. (2.21), where γ is the surface tension between the gas and the polymer phase and Δp is the pressure difference between gas and solid: ΔGhom =

16πγ3 3Δp2

(2:21)

The critical value of the nuclei radius, rc (eq. (2.16)) can be computed from the previous equation [54–56]. The cluster of gas molecules should be larger than the critical radius in order to survive and grow. Those clusters with a size smaller than rc will not be stable. rc =

2γ Δp

(2:22)

The CNT also predicts the homogeneous nucleation rate N0 , that is the number of nuclei formed per unit of volume and unit of time. It is given by eq. (2.23), where C0 is the initial concentration of gas in the polymer, f0 is the frequency factor of gas molecules joining the nucleus, kB is the Boltzmann constant and T is the temperature [54, 57]:   ΔGhom (2:23) N0 = f0 C0 exp − kB T This theory provides useful insights into the parameters involved in the nucleation process, and it correctly predicts some trends with the processing parameters [45]. For instance, according to eq. (2.23), the nucleation increases as the gas concentration or solubility increases. Therefore, one strategy to achieve high nucleation ratios is to

2.3 Foaming mechanisms

29

maximize the solubility. On the other hand, temperature and pressure gradient also play a role in the nucleation process according to CNT equations. Higher temperatures will lead to higher nucleation ratios [58], whereas a higher pressure gradient will also induce larger nucleation [59]. Regarding heterogeneous nucleation, the addition of a second component to the polymer/gas mixture might affect the surface tension in comparison to that of the pure polymer. In general terms, the addition of a second phase creates interfaces in the polymer/gas mixture, and these surfaces induce wetting; that is, gas molecules have a tendency to aggregate at the foreign surface [60]. Then, the nucleation process tends to take place on these pre-existing surfaces. Also, nucleation can occur within the second phase in specific systems where the second phase is organic. As in the case of homogeneous nucleation, the heterogeneous nucleation rate is determined by an energy barrier, ΔGhet . The energy for forming a critical nucleus in a heterogeneous system, ΔGhet , is proportional to the energy barrier in a homogeneous system, ΔGhom , by a factor SðθÞ depending on the wetting/contact angle of the polymer–additive–gas interface [57]: Δ Ghet = Δ Ghom SðθÞ =

16πγ3 SðθÞ 3Δp2

(2:24)

This function SðθÞ is always less than or equal to one [48]; that is, the Gibbs free energy barrier for heterogeneous nucleation is lowered, so nucleation is enhanced compared to the homogeneous situation. The nucleation rate for heterogeneous nucleation, Nhet , is given by eq. (2.25), where C1 is the initial concentration of gas in the polymer and f1 is the frequency factor of gas molecules joining the nucleus.   ΔGhet (2:25) Nhet = f1 C1 exp − kB T One important consideration about the heterogeneous nucleation mechanism is that the critical radius is not affected by the presence of a second phase; that is, if we compute the critical radius in this situation, we recover the result of eq. (2.22). The addition of a second phase provides heterogeneous surfaces on which the nucleation energy barrier is lowered, and heterogeneous nucleation starts to play an essential role during the process. Two types of additives can be used as nucleating species in a general foaming process: particles and nanostructured polymers. The use of nucleating agents is highly common in the production of foams at an industrial scale, especially for low-density foams [37, 61]. In addition, additives with another primary function, such as flame retardants or pigments can also act as nucleating agents during foaming.

30

Chapter 2 Fundamentals

2.3.3 Growth After nucleation, the gas diffuses from the polymer/gas mixture to the newly created nuclei, causing cell growth and the resultant expansion of the structure. Equation (2.26) provides a simplified equation governing cell growth, where R is the radius of the cell, t is the time and η is the viscosity of the gas/polymer mixture [45, 55]. Note that the solution of this equation is not trivial since the pressure gradient Δp, the viscosity and the surface tension might vary with time as gas is diffusing to the cells and out of the sample. dR Δp γ = − dt 4η 2η

(2:26)

The value of the viscosity is directly related to the cell growth process. A high viscosity can slow down the speed of growth while a low viscosity yields a rapid and almost explosive initial cell growth. However, constant viscosity cannot meet the requirements of controllable foaming: it is desirable that the viscosity changes during the foaming process. The initial growth of the cells just after nucleation requires a low viscosity. Subsequently, as the polymer is stretched during the cell growth, a high extensional viscosity is required, that is, a polymeric matrix showing strain hardening is advantageous, so that the cell walls may withstand the deformation to which they are subjected during the last stages of cell growth, without breaking [62, 63]. Cell growth continues until it stops due to the stabilization of the cellular structure by cooling. The above equations for nucleation and cell growth are valid in a certain cell size range. As commented later and in Chapter 3, the reduction of the cell size to the nanoscale imposes different equations to get accurate predictions of the nucleation and growth processes.

2.3.4 Degeneration As it was previously mentioned, during cell growth, when the cells are separated by thin polymer membranes, the cellular structure may degenerate if the foamed material is not stable enough (for instance, if the polymer does not show enough strain hardening behavior). Cell degeneration can occur by the combination of three main mechanisms: coalescence, coarsening, and drainage (Figure 2.23). Cell coalescence is the mechanism by which two growing continuous cells in a polymer melt are combined because of cell wall rupture (Figure 2.23a). This can occur if the stretched thin cell wall separating the two cells is not strong enough to sustain the extension developed during cell growth. Coalescence is thermodynamically favored because the total surface area of cells is reduced by coalescence. Cell coalescence can increase the cell size and the open cell content. To prevent coalescence,

2.3 Foaming mechanisms

31

Figure 2.23: Schematic representation of the mechanisms of cell degeneration.

foaming temperature, and time, the rheological properties of the polymer and stabilization process must be controlled. On the other hand, coarsening consists of gas diffusion from the smallest cells to the largest ones (Figure 2.23b). As the pressure in a small cell is higher than that in a large cell, this pressure gradient induces gas diffusion. As a result, the size of the largest cells increases, and the smallest cells tend to get even smaller and could eventually disappear. Finally, drainage takes place during foaming as the molten polymer drains, out of equilibrium, in the fine walls separating the cells. It is caused by the action of capillary forces, which produce the transport of the liquid material from the cell walls toward the edges (Figure 2.23c). As a consequence, the cell wall thickness is reduced, and the coarsening and coalescence mechanisms are favored. Drainage is favored in systems with too low viscosity.

32

Chapter 2 Fundamentals

2.3.5 Stabilization During foaming, the gas inside the new cells eventually tends to diffuse out to the atmosphere, because a complete separation of the two phases is thermodynamically more favorable. As the gas escapes through the thin walls, the amount of gas available for the growth of the cells decreases. As a result, if the solid matrix is not frozen (i.e. if the viscosity of the solid polymer is not increased significantly), the cells tend to collapse causing foam contraction. The stabilization in thermoplastic cellular materials is most of the time performed by rapid cooling of the sample at a temperature below the polymeric matrix crystallization temperature (semi-crystalline polymers) and/or glass transition temperature (amorphous polymers). Another mechanism that allows the stabilization in some foaming processes is associated with an increase of the glass transition temperature (for amorphous polymers) or the melting temperature (for semi-crystalline polymers) when the gas dissolved in the matrix diffused out of the matrix. If the glass transition temperature or melting temperature evolves into values higher than the current temperature of the system, the foam is stabilized without the need of cooling down the material.

2.4 Nanocellular polymers: Key features As commented in Section 2.2.2.1, the reduction of the cell size to the nanometric scale was the natural step forward after the development of microcellular polymers. The progress in the foaming technology together with the interesting properties of microcellular polymers triggered the leap to the next generation of cellular materials in the early 2000s [64]: the development of nanocellular polymers. As explained in Section 2.2.2.2, the production of nanocellular polymers implies a significantly harder challenge from a technological point of view: the cell density should be increased from 109 cells/ cm3 to 1013 cells/cm3 to reach a nanometric cellular structure. Figure 2.24 shows examples of the cellular structure of various nanocellular polymers.

Figure 2.24: Examples of micrographs of three different nanocellular polymers.

2.4 Nanocellular polymers: Key features

33

In a nanocellular structure, the two phases (gas and solid) are confined: on the one hand, the gas is confined in small cells, and on the other hand, the polymer is restricted to be in very thin cell walls. As a consequence of these effects, new properties never observed in conventional or microcellular polymers, are expected [65]. For instance, it is known that when a gas is confined in a volume of similar size as the mean free path of the gas molecules (around 70 nm for air in standard conditions), the Knudsen effect appears [66, 67] and the conduction through the gas is drastically reduced. Simultaneously, the polymer is stretched in cell walls smaller than the size of the polymer in the bulk state (20–30 nm versus 100 nm), leading to a confinement effect of the solid phase that can produce, among other effects, an increase of the glass transition temperature of the polymer [68] and an enhancement of the mechanical properties [69–71]. Also, one of the most outstanding properties that have been recently proved is that nanocellular polymers have the potential of being partially transparent due to their nanometric cell size [72, 73]. Further, nanocellular polymers present different acoustic and dielectric properties compared to microcellular polymers [74, 75]. Moreover, the nanometric cell size and the high surface area associated with it open the possibility of using these materials in some applications such as the production of membranes, filters, sensors, and supports for catalysis [76–78]. The combination of these properties makes nanocellular polymers unique materials with great potential.

2.4.1 Cellular structure in nanocellular polymers The criterion to determine when a cellular material can be considered as nanocellular is still not overall established, but it should be related to the effects associated with the change of scale to the nanometric range. Then, the criteria to speak about a nanocellular material should differ for the different properties of these materials. For instance, thermal conduction through the gas phase starts to significantly decrease from cell sizes of 500 nm [65, 79], whereas to obtain semi-transparent nanocellular polymers cell sizes as low as 50 nm are required [58, 72]. Therefore, the cell size of nanocellular polymers determines the final properties and applications in which their use can lead to superior performance. For the purposes of this book, cellular polymers with cell sizes below 1 micron would be considered nanocellular polymers. However, other definitions are possible, for instance using the change in the properties as a threshold to define the limiting cell size. This kind of definition is common in other aspects of Material Science, such as in the definition of 2D systems in solid-state physics, which are considered 2D if the properties are different enough from those of the bulk 3D system. As stated previously, the cellular structure of nanocellular polymers is characterized by a double confinement effect. In the next sections, we will explain in detail the descriptors used to describe nanocellular polymers and the specific characteristics of these materials in comparison with conventional foams.

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2.4.1.1 Gas phase Figure 2.25 shows a relative density–cell size map for lines of constant cell nucleation density in the region for cell sizes below 1 micron. As it has been commented several times, the generation of nanocellular polymers implies a sharp increase in the cell nucleation density in comparison with conventional cellular polymers. The challenge is even more pronounced in the low-density region. To create a nanocellular polymer with a cell size of 100 nm and a relative density of 0.1, a nucleation density higher than 1015 nuclei/cm3 is needed. If the density was to be decreased even further (0.05), the nucleation should increase to more than 1017 nuclei/cm3; that is, the challenge gets harder as density is reduced. Over the last few years, significant efforts were made to reduce the density of nanocellular polymers. However, the reduction in relative density below 0.1 in combination with cell sizes below 100 nm is still unaccomplished.

Figure 2.25: Lines of constant cell nucleation density in a relative density–cell size map for cell sizes below 1 micron, according to equation (2.10).

One experimental issue is the determination of the cell size distribution when the cell size is very small. Figure 2.26 shows two examples of nanocellular polymers with cell sizes below 50 nm, in which the high magnification of the SEM is not good enough to reach an adequate image for the analysis of the cell size. In the case of nanocellular polymers, the cells can open because the cell wall has reached the minimum possible thickness, then cell wall rupture might occur. Due to the small size of the cell walls, this phenomenon can take place faster in nanocellular polymers than in conventional foams. One consequence of the appearance of holes and cracks in the cell walls is that gas might escape through the interconnected structure, preventing further expansion and limiting the density reduction in these materials [58]. Figure 2.27 shows some examples of nanocellular polymers presenting closed

2.4 Nanocellular polymers: Key features

35

Figure 2.26: Examples of nanocellular polymers with cell sizes below 50 nm. As the cell size reduces (from a to c), measuring the cell size distribution and the average cell size becomes more complicated.

(Figure 2.27a) and open cells (Figure 2.27b). With these nanocellular polymers it is possible to obtain materials with open cell contents from 0% to 100% [58]. Regarding the anisotropy ratio, nanocellular polymers would show anisotropy depending on the production process, as in any cellular polymer. In addition, it is possible to generate highly anisotropic structures in nanocellular polymers by designing particular strategies [12, 83]. Figure 2.23c shows one example of a nanocellular polymer with an anisotropic cellular structure.

Figure 2.27: Examples of nanocellular polymers with (a) closed-cell structure, (b) open-cell structure, and (c) anisotropic cells.

More information about the typical cellular structures of nanocellular polymers and the effects of the gas phase on the properties of nanocellular polymers can be found in the next chapters. 2.4.1.2 Solid phase Figure 2.28 represents lines of constant cell wall thickness in a relative density–cell size map in the low-density range for the two eqs. (2.18) and (2.19) in the region for cell sizes below 1 micron. As the cell size reduces, one can guess that the cell wall thickness cannot be as thin as desired; that is, it should exist a minimum cell wall thickness that can be achieved for every system, as already mentioned by Bernardo

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and co-workers [80]. As seen in the theoretical predictions of Figure 2.28, the cell wall thickness imposes some geometrical limitations in these structures. For instance, with the theoretical predictions of this figure, a closed-cell material with a cell wall thickness of 20 nm cannot present a density of 0.2 with a cell size of 100 nm. Open cell structures allow obtaining materials with smaller densities at the same cell wall thickness, but there are also some limitations. Up to date, there are no references about the minimum cell wall thickness that could be achieved with a polymer matrix, and this is a complicated topic since many factors, such as molecular weight or chain structure, can play a role. However, this information would provide some understanding of the physical limitations associated with this technology.

Figure 2.28: Lines of constant cell wall thickness in a relative density–cell size map for cell sizes below 1 micron: (a) eq. (2.18) for a fraction of mass in the struts of 0.2 and C = 3.46, and (b) eq. (2.19).

2.4 Nanocellular polymers: Key features

37

The analysis of the solid phase descriptors in nanocellular polymers is more challenging due to the low dimensions of the walls and struts, which makes tomography not a useful technique fs these materials. Up to now, there are few works dealing with the detailed characterization of the solid phase distribution in this new class of materials. The work of Martín-de León and coworkers [58] proposed a methodology to quantify the fraction of mass in the struts using image analysis and SEM micrographs (Figure 2.29).

Figure 2.29: Description of the method to measure the fraction of mass in the struts as proposed in [58]: (a) cell mask; (b) binarized cell mask; (c) local thickness cell image; (d) local thickness histogram (adapted from [58]).

The cell walls of nanocellular polymers are formed by polymer macromolecules with a different configuration of the bulk material, which leads to a confinement of the solid phase. This confinement has been measured experimentally determining the increment in the glass transition temperature and using Raman spectroscopy [58, 68]. While the cell walls can be as thin as 20–30 nm [58, 80], the size of the struts is necessarily larger and it is still unclear whether the polymer is also confined in the struts. Regarding the tortuosity of the solid phase, the high cell density in these structures increases the tortuosity in comparison with conventional materials. However, the effect of this increase in tortuosity is still unclear. While it could lead to a reduction of the heat conduction through the solid phase (since the phonons would have to travel a larger distance) [68, 79], the confinement of the polymer macromolecules could lead to the opposite effect [81]. More information about the effects of the solid structure on the properties of nanocellular polymers can be found in the next chapters.

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Finally, nanocellular polymers also show solid skin [30]. In these materials, the presence of solid skin can have a negative impact on the foam density and can also limit the expansion of the material. Furthermore, the presence of structural gradients is even more critical in this type of material, as illustrated in Figure 2.30. This region with a structural gradient is typically named transition region in the scientific literature on nanocellular polymers. While the homogeneous core is nanocellular, the transition region can be microcellular, causing a gradient in the physical properties of the material when they depend on the cell size. For instance, for the production of transparent nanocellular polymers, the solid skin and the surrounding heterogeneity also negatively impact the transparent behavior of the material [82]. This will be explained in Chapter 4.

Figure 2.30: Example of a nanocellular polymer near the solid skin presenting structural gradients. Colored boxes are used to illustrate the three regions: dark blue – solid skin; purple – transition region; light blue – homogeneous core.

2.4.2 Foaming mechanisms in the nanoscale The foaming mechanisms are also modified when the cell size moves to the nanoscale. As aforementioned, extremely high cell densities must be created to produce a nanocellular polymer, and the cellular structure needs to be stabilized with pores in the nanoscale. The equations presented for conventional cellular polymers are not good enough to predict the behavior of nanocellular polymers and more refined models are needed for these materials, as explained in Chapter 3.

2.5 Conclusions

39

To achieve high cell densities via homogeneous nucleation, extreme processing conditions are required to maximize the gas absorbed and promote the creation of enough nuclei [45, 49]. In the case of heterogeneous nucleation, the requirements for the nucleating species are more demanding, since the number of nuclei needed to obtain a nanocellular polymers is huge (~ 1013 nuclei/cm3) compared to the nucleation in conventional cellular materials (~ 106 nuclei/cm3). Thus, a high volumetric density of nano-sized domains is necessary to produce nanocellular polymers using a heterogeneous nucleation approach. During cell growth, it is key to stabilize the structure while the cells are yet of nanometric size. As stated previously, the thin cell walls would break easily, preventing further expansion, and also, cracks in the cell walls could also provoke or help in the degeneration of the structure via different mechanisms (coalescence, drainage, or coarsening) that will increase the cell size to a significant extent. In the case of heterogeneous nucleation, the presence of nanoparticles or nano-sized polymer domains can cause an enhancement in the degeneration mechanisms, due to the size of the particles and the possibility of having weak interactions in the interface [84]. The main conclusion from the previous paragraphs is that the generation of nanocellular structures is a challenging process with enormous complexity, in which many interrelated mechanisms play a key role, most of them still not fully understood. In Chapter 3, a deep analysis of the foaming process in the fabrication of nanocellular is presented and all the concepts briefly introduced here will be developed.

2.5 Conclusions In this chapter, we present a detailed description of the parameters needed to characterize the cellular structure of cellular polymers. The parameters are divided into two main groups: those related to the solid phase and those related to the gas phase. The definitions and equations are presented, and examples are included too. Furthermore, the foaming mechanisms taking place in any foaming system are explained. Finally, nanocellular polymers are introduced, and the main differences between them and conventional cellular polymers are established. These basic concepts presented in this chapter will be used throughout the book to describe nanocellular polymers, the foaming mechanisms taking place in the nanoscale, and the physical properties of these novel materials.

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J. Pinto, M. Dumon, M.A. Rodriguez-Perez, R. Garcia, C. Dietz, Block copolymers self-assembly allows obtaining tunable micro or nanoporous membranes or depth filters based on PMMA; fabrication method and nanostructures, J. Phys. Chem. C. 118 (2014) 4656–4663. https://doi.org/10.1021/ jp409803u G.Q. Lu, X.S. Zhao, Nanoporous materials – an overview, Nanoporous. Mater. Sci. Eng. (2004) Imperial Collegue Press, London. https://doi.org/10.1142/9781860946561_0001 B. Notario, J. Pinto, E. Solorzano, J.A. de Saja, M. Dumon, M.A. Rodriguez-Perez, Experimental validation of the Knudsen effect in nanocellular polymeric foams, Polymer (Guildf) 56 (2015) 57–67. https://doi.org/10.1016/j.polymer.2014.10.006 V. Bernardo, J. Martín-de León, J. Pinto, T. Catelani, A. Athanassiou, M.A. Rodriguez-Perez, Lowdensity PMMA/MAM nanocellular polymers using low MAM contents: Production and characterization, Polymer (Guildf) 163 (2019) 115–124. https://doi.org/10.1016/j.polymer.2018.12.057 R.-C. Zhang, Z. Huang, D. Sun, D. Ji, M. Zhong, D. Zang, J.-Z. Xu, Y. Wan, A. Lu, New insights into thermal conductivity of uniaxially stretched high density polyethylene films, Polymer (Guildf) 154 (2018) 42–47. https://doi.org/10.1016/j.polymer.2018.08.078 J. Martín-de León, M. Jiménez, J.L. Pura, V. Bernardo, M.A. Rodriguez-Pérez, Easy-way production of highly transparent nanocellular polymers films, Polymer (Guildf) 236 (2021) 124298. https://doi.org/ 10.1016/j.polymer.2021.124298 V. Bernardo, J. Martín-de León, M.A. Rodriguez-Perez, Highly anisotropic nanocellular polymers based on tri-phasic blends of PMMA with two nucleating agents, Mater. Lett. 255 (2019) 126587. https://doi.org/10.1016/j.matlet.2019.126587 V. Bernardo, J. Martín-de León, E. Laguna-Gutierrez, M.A. Rodriguez-Perez, PMMA-sepiolite nanocomposites as new promising materials for the production of nanocellular polymers, Eur. Polym. J. 96 (2017) 10–26. https://doi.org/10.1016/j.eurpolymj.2017.09.002

Chapter 3 From the microscale to the nanoscale in cellular materials production process 3.1 Introduction As discussed in Chapter 1, cellular polymers have become very important materials in our current society with a huge volume market and applications in almost all industrial sectors. There has been a long path to reach the current technological development. One of the initial inventions was done by Munters and Tandberg in 1931 [1]. They developed foamed polystyrene (PS). Later, in 1937 foamed polyurethane (PU) was developed by Dr. Otto Bayer [2] and a few years later foamed polyethylene (PE) was invented by Johnson in 1941 [3]. Another material currently in use for thermal insulation is extruded polystyrene (XPS) that was developed by Dow Chemical in 1944 [4]. Expandable beads were also developed a long time ago by Stastney and Goeth in 1954 [5]. Later, other types of cellular polymers were developed, such as acrylonitrile butadiene styrene (ABS) cellular polymers in 1967 [6], extruded polypropylene (PP) foams in 1972 [7], polypropylene molded foam in 1984 [8], and polyethylene terephthalate (PET) extruded foam in 1990 [9]. Nowadays, the main technologies to produce these materials at industrial scale are extrusion foaming (PE, PS, PP, PET), reactive foaming (PU), bead foaming (PS, PP, polylactic acid (PLA), thermoplastic polyurethane (TPU) and other thermoplastics), injection molding (all thermoplastics), compression molding (polyolefins, polyvinyl chloride (PVC) and rubbers), and free foaming (x-linked foams (polyolefins and rubbers)). By using these technologies, it is possible to produce cellular polymers in a wide range of relative densities (from 0.01 to 0.99) but with large cell sizes, typically higher than 100 microns and only in very specific cases and for high-density foams with cells sizes in the range of 10 to 100 microns. One very specific foaming technique, not as common at industrial scale as the ones mentioned in the previous paragraph, is gas dissolution foaming. This foaming approach will be described in detail in the next paragraphs because is the one that has been extensively used in the last 20 years to produce nanocellular polymers. Briefly, gas dissolution foaming is physical foaming approach in which the polymer to be foamed is introduced in a pressure vessel, saturated with the physical blowing agent at a given pressure and temperature, and foamed after the release of the pressure. It is batch process that allows foaming many materials and with many parameters involved and that provides materials with any residues of the blowing agent. It is interesting to briefly revise the history of this technology and how it has influenced cellular polymers development.

https://doi.org/10.1515/9783110756135-003

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Curiously, gas dissolution foaming is one the first technologies reported to produce foamed products. In 1920 Charles Lancaster Marshall filed a patent entitled “Vulcanizing apparatus for the manufacture of high-pressure expanded vulcanized rubber and substances” [10]. In his patent he described a process to expand rubber using a pressure vessel. The company produced foamed rubbers for filling tires with a cellular material rather than air. This idea was not a commercial success at that time. The company was taken over in 1938 by St Helens Cable and Rubber Company, and by 1942 they started to explore the possibility of producing thermoplastic materials such as polystyrene (PS), low-density polyethylene (LDPE), and ethylene vinyl acetate copolymers (EVA) using the gas dissolution foaming process. Although it took some time, this idea was a great commercial success. In fact, this was the beginning of the development of commercial products such as Plastazote (low-density LDPE foam produced by using gas dissolution foaming with nitrogen as physical blowing agent) or Evazote (low-density EVA foam produced by using gas dissolution foaming with nitrogen as physical blowing agent) that have been in the market for more than 50 years and that are currently commercialized by the British company Zotefoams Plc (previously British Petroleum and BX plastics limited) [11]. This company has been using the gas dissolution foaming technology and in particular the two-steps gas dissolution foaming process (see Section 3.3 for an explanation of this specific process) for a long time to successfully produced different products. In the last 20 years they have expanded the cellular materials produced by this technology developing products based on polyvinylidene fluoride (PVDF), polyamide-6 (PA6), and more recently polyether block amide (PEBA). Some of the general characteristics of the materials produced by this company is that they can cover a wide range of densities, even very low densities around 15 kg/m3 are possible, and a wide range of polymeric materials. The cell sizes are typically in the range of 100 to 300 microns. Interestedly, very similar two steps for gas dissolution foaming process were developed at the Massachusetts Institute of Technology (MIT), USA, in the early 1980s [12], in response to a challenge by food and film packaging companies to reduce the amount of polymers used in their industries. As most of these applications used solid, thin-walled plastics, reducing their densities by traditional foaming processes that produced bubbles larger than 100 microns was not feasible due to excessive loss of mechanical properties. Thus, the idea of producing microcellular polymers was proposed, where we could have, for example, 100 bubbles across one mm thickness, and expect to have a reasonable strength for the intended applications. This was the beginning of microcellular polymers with cell sizes in the range of 10 microns. The concept of the process was similar to that used by Zotefoams Plc but most of the polymers tested at the beginning of the technology development were amorphous materials such as PS, ABS, and PET, the saturation was typically done at low temperatures, and in most of the cases, cellular polymers with medium to high relative densities were produced taking into account the applications that were pursued (substitution of plastics parts by cellular parts that keep mechanical properties as high as possible). These three key differences allowed

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significantly reducing the cell size of the produced materials to the microcellular range (in the range of 10 microns). The development and study of these materials have been very intense in the last 40 years [13], and in fact they are still a very interesting topic in cellular polymer science and technology (see below in this chapter). The main advantage of microcellular polymers in comparison with conventional cellular polymers is that they present better mechanical properties for the same relative density. This is a key aspect that will be covered in Chapter 6 of this book (Section 6.2). Interestingly, the production of microcellular foams has not been only an interesting research area developed in academia, in fact, there have been several successful developments at industrial scale. The industrial production of microcellular foams has been developed so far by different technologies. A direct scaling up of the gas dissolution foaming process was done by Kumar and Schirmer [14] by the so called “semi-continuous process” in which a roll of polymer was saturated with the gas in a pressure vessel and later foamed in continuous in a furnace. A company named Microgreen was formed in the USA and the process was successfully scaled up in the early 2000s (more specifically in the year 2002) [15]. The technology was able to produce, for instance, PET foams from recycled PET with a very high quality, medium densities, and very small cell sizes (in the range of 10 to 50 microns). Although technically the company was successful and the scaling up was completed, the business was not successful. A second successful technology able to produce microcellular parts was injection molding. The company Trexel [16], a spin-off of the MIT, developed and commercialized a process to produce high-density cellular polymers with cell sizes in the range of 10 microns by using injection molding. This is a well-known technology in the plastic industry named “Mucell” Technology [17, 18]. Briefly, it consists of dosing the physical blowing agent in the barrel of the injection molding machine using a very specific and precise dosing system and a specially designed screw to dissolve the gas in the polymer in a short time. Once a single-phase polymer-gas has been created, the material is injected into the mold cavity to create the microcellular part. This technology is currently commercialized not only by Trexel but also by some of the main injection molding producers such Krauss-Maffei or Engel. In addition, many companies are using this technology to reduce the weight and cost of plastic parts keeping products with enough mechanical performance. The same concept of Mucell Technology has also been tested in extrusion foaming to produce sheets of high-density microcellular cellular polymers. This technology is currently commercialized by the company Zotefoams Plc [11], and it is being used mainly to produce plastic containers by extrusion blow molding with the same idea, so slightly reducing the density and cost, keeping the performance of high-density parts. It is also interesting to mention that in the last few years gas dissolution foaming has found a new market. Companies producing shoes for sports (for instance walking or running shoes) have been using crosslinked polyolefin for the midsoles of their

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products for many years. In particular, crosslinked EVA foams with densities in the range 150 to 250 kg/m3 and cell sizes between 50 and 200 microns have been used for this application. However, as these materials cannot be easily recycled due to the crosslinking of the polymeric matrix, new and more sustainable alternatives are being studied. One of them is the production of these midsoles by using the gas dissolution foaming process using thermoplastic elastomers as polymer matrix. The density range is similar to that of crosslinked EVA foams, and cell sizes are in the conventional range. The potential of this market is huge, and some commercial products are already under production. In summary, gas dissolution foaming has a long history, and it has been already demonstrated that it is a versatile and environmentally friendly technology able to produce many types of cellular polymers, starting from a wide variety of polymer matrices, producing materials in a wide range of relative densities and with cell sizes in the microcellular and in the conventional range. The technology has been scaled up several times for specific products. From an academic perspective, Figure 3.1 shows the evolution of microcellular polymers research. At the early stages it was slow, and it was not until the year 2000 that the number of publications dealing with their study started to significantly grow. As with any discovery, their evolution at both the laboratory and industrial scales took several years, and even now the investigation about microcellular polymers is a trending topic that continues to grow year after year, with almost 100 publications and 5000 citations in the year 2021 [19–22]. Although nanocellular polymers were produced using already known techniques to produce microcellular polymers, their fabrication implies a reduction by 100 times in the cell size and an increase in the number of nucleation points by 4 orders of magnitude (the definition of these parameters can be found in Chapter 2). Thus, the used techniques needed to be fine-tuned and their production parameters again controlled to produce these novel cellular polymers. Figure 3.1 shows how the evolution of the research about nanocellular polymers is shifted around 20 years in comparison to that of microcellular polymers, which explains that the number of publications is still taking off, while some companies (Dow, BASF, Sabic, Sumteq, CellMat Technologies, etc.) have been also carrying out research on these materials. However, their properties are more promising than those in microcellular polymers, so it is expected that in some years from now nanocellular polymers could spread worldwide even more than microcellular polymers do now. Since their discovery, nanocellular polymers have already been produced from different polymer matrices. Polycarbonate (PC) [23], thermoplastic polyurethane (TPU) [24], polyetherimide (PEI) [25], polypropylene (PP) [26], or polymethylmethacrylate (PMMA) [27, 28] have been some of the selected polymers for the production of such materials. These polymers have been used to produce single-phase polymer systems, multiphase polymer systems, or immiscible blends [29–32].

3.2 Production techniques of nanocellular polymers

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Figure 3.1: Evolution of publications and number of citations for microcellular and nanocellular polymers (analyzed through the citation report of WoS with nanocellular and microcellular foams as topic consulted in December 2022).

However, there is still a long way to go to optimize the production of nanocellular polymers. Some conventional polymers such as polystyrene or polyethylene have not yet led to the production of cells in the nanometric region when producing the materials using the homogeneous nucleation approach. Moreover, the properties of the already produced nanocellular polymers need to be improved, such as their density, cell size, or methods for scaling up, and therefore the control of the production methods is essential. The main production techniques to produce nanocellular polymers are studied in depth in the following sections.

3.2 Production techniques of nanocellular polymers Nanocellular polymers are defined to present cell sizes below the micron (see Chapters 1 and 2 for a more in detail definition). Plenty of conventional techniques allow to generate cells in a polymer, but the real challenge is how to generate such an enormous number of cells in the nanometric range. Production routes leading to nanocells are summarized in Figure 3.2 and can be divided into three different groups: phase separation techniques, templating of im-

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Figure 3.2: Nanocellular polymers production routes scheme.

printing techniques, and the one in which we will mainly focus our attention on in this chapter, gas dissolution foaming. Phase separation techniques are based on inducing the separation between the polymer and the gas phase through different methods. This separation can be triggered by a chemical quench (chemical induction phase separation (CIPS)) [33, 34], by a thermal quenching (thermal induction phase separation (TIPS)), or through immersion techniques [35, 36]. This last one is the most promising one inside this group. The polymer firstly precipitates from a solution, secondly, the solvent diffused out of the polymer creating the phase separation. As it is shown in Figure 3.3, nanocellular polymers created through this technique lead to cell sizes above 100 nm while the sample thickness ranges hundreds of microns [37–39]. The second group of techniques consists of the generation of the nanocellular structure from a previous template. The initial monomer is mixed together with the template; the monomer is therefore polymerized. Finally, the template is removed chemically, thermally, or by some extraction method.

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Figure 3.3: Range of sample thickness and cell sizes obtained with each production technique.

The different templates, that gave names to the techniques, can be a molecule [40–42], a colloidal crystal [43–45], or self-organized templates such as micelles, microemulsions, or block copolymers [46–52]. All those techniques cover a wide window of cell sizes from smaller than 1 nm with microemulsion to the micron through immersion techniques: however, the thickness of the final material is never thicker than 200 microns (Figure 3.3). The block copolymer technique is the exception, leading to samples of around 1 mm. The remaining group is the gas dissolution foaming; this production method allows the production of a wide range of cell sizes and sample thickness, as seen in Figure 3.3. Due to this high range of sample thickness (final thickness of the sample can be as high as 10 mm) this production route is the one allowing obtaining materials with a wider range of applications, and due to this we will focus our attention on this approach. This technique is in-deep studied in the following section.

3.3 Gas dissolution foaming Gas dissolution foaming was developed for the production first of conventional foams, later microcellular foams (see introduction of this chapter) and in recent years, to produce nanocellular polymers. The selected gas is usually carbon dioxide (CO2) [53]. This is due to the fact CO2 presents excellent diffusion properties when it is at a supercritical state, a state that is also easily reached for this gas (critical temperature is 31 °C and critical pressure is 7.3 MPa). On the other hand, CO2 is a green solvent, meaning that the process is carried out without producing any pollutant compound or leaving any residue in the material [54].

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Although there exist different alternatives to this process, which will be commented on lately, the original one consists of four steps as Figure 3.4 shows: saturation, depressurization, desorption, foaming, and stabilization. First of all, the polymer is introduced in a pressure vessel under certain conditions of gas pressure and temperature, parameters known as saturation pressure (psat ) and saturation temperature (Tsat ), respectively. During saturation, gas diffuses into the polymer by following the Fick diffusion law and creating a homogeneous single-phase gas-polymer mixture. As schematized in Figure 3.5, gas molecules accommodate between polymeric chains in the free volume of the polymer. For amorphous polymers the presence of gas between polymer chains increases their inherent mobility; thus, the original glass transition temperature of the polymer (Tg ) decreases due to the presence of CO2, to a lower one known as effective glass transition temperature (Tgeff ). For semi-crystalline polymers, when the saturation is carried out at a temperature below the melting temperature of the polymer (Tm ), the glass transition temperature of the amorphous phase is also reduced to an Tgeff . and a modification of the melting range could also occur. The diffusion process continues during the so-called saturation time (tsat ), moment at which the concentration of gas along the whole thickness of the polymer is homogeneous and no more gas can be dissolved inside the polymer. This amount is called solubility limit and defines the maximum amount of gas for which the single phase of the gas-polymer system occurs. The solubility limit depends on both the gaspolymer system and the used saturation conditions. Solubility depends on both saturation pressure and temperature, a common trend of solubility with saturation pressure is given by Henry’s law. In this model, solubility depends linearly on saturation pressure being kD the correlation parameter. This linearity is useful for some pressure ranges and a wide variety of polymers, however, it is usually necessary to enhance this model being the Langmuir’s law and the dual model more general equations [55–57]. Despite the used law, the increase of the solubility with the saturation pressure is clear in all the models. On the other hand, the saturation temperature is also affecting the solubility, as indicated before. This dependence is given by Arrhenius eq. (3.1):   ΔHs (3:1) S = S0 exp − RTsat where S0 is the preexponential factor, ΔHs is the heat of sorption, R is the gas constant, and Tsat is the saturation temperature. The value of the heat of sorption depends on the used gas/polymer system, and also on the range of pressures and temperatures used due to the phase changes of gas. However, for CO2-philic materials, that is, polymers with affinity for carbon dioxide, this value is negative. It means that an increase in the solubility is promoted by decreasing the saturation temperature. On the other hand, diffusivity determines saturation time through the second Fick’s law (eq. (3.2)):

3.3 Gas dissolution foaming

δC δ2 C =D 2 δt δx

53

(3:2)

The dependence of the diffusivity with those two parameters is similar to the ones observed for the solubility. The pressure influence is given by eq. (3.3), although normally it is enough with the linear term in pressure meaning that an increase in the saturation pressure leads to an increase in the absorption diffusivity of the gas in the polymer. D = a + bP + cP2

(3:3)

The influence of the saturation temperature is also described with an Arrhenius law, as follows:   ΔHD (3:4) D = D0 exp − RT where D0 is the pre-exponential factor and ΔHD is the activation energy for the diffusion process. Although in some temperature range and for some gas/polymer systems, this activation energy presents a negative value [27], the usual behavior is a positive value of this constant for the absorption process. That means that a decrease in the saturation temperature leads to a reduction of the diffusivity, which is reflected in an increase in the saturation time. After saturation time, gas is fast released in the second step of the process, depressurization. The pressure abruptly drops in this step from the used saturation pressure to atmospheric pressure at a ratio given by the depressurization velocity (vdep ).

Figure 3.4: Gas dissolution foaming process scheme.

This pressure drop leads to phase separation. At atmospheric pressure the saturation limit strongly decreases; thus, the gas-polymer system is no longer in equilibrium, triggering two effects: phase separation and the diffusion of gas outside the polymer. Since depressurization gas starts to diffuse out through polymer surfaces, following

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again Fick diffusion laws. This diffusion promotes that a solid non-foamed skin appears in the surface of these materials. This topic will be covered in more detail in Chapter 7. On the other hand, there exist two options for phase separation: spinodal decomposition or nucleation. In spinodal decomposition, both phases, gas, and polymer abruptly separate, creating an interconnected gas phase and therefore a cocontinuous cellular structure. Spinodal decomposition occurs at specific conditions of high supersaturations, where the polymer-gas mixture is highly unstable. Then, this is not the most common scenario. Some examples of structures obtained via this route are presented in Chapter 7.

Figure 3.5: Evolution of a polymer sample under the gas dissolution foaming process.

Commonly, the process through which phase separation takes place is nucleation. As Figure 3.5 shows, gas is clustered in small regions known as nucleation points, precursors of the final cells of the nanocellular polymer. After desorption time, considered as the time between depressurization and foaming, the polymer is heated at a temperature above the already defined Tgeff . This is usually carried out in a thermal bath. At these conditions the polymer is in a plasticized state, meaning that polymeric chains present enough mobility to allow nucleation points to growing into cells. For amorphous polymers this growing takes place homogeneously in all volume of the polymer, except near the surfaces of the sample, in which the amount of gas is smaller due to the gas diffusion which results in the formation of a solid skin (Chapter 7). For semi-crystalline polymers below its melting temperature, the situation is more complex because the gas is mainly dissolved in the amorphous region so, the growing of the cells mainly takes place in this volume of the material. For these materials a solid skin also appears for the same reason than in amorphous materials. As long as the polymer is above the effective glass transition temperature, it continues growing. This could lead to degeneration mechanisms. Those degeneration mechanisms are drainage coalescence, or coarsening, phenomena in which the cell walls broke, and many cells join together reducing the final number of cells and increasing their size. Thus, a final step, called stabilization, is necessary. The cellular structure is stabilized simply by decreasing the temperature below the effective glass transition to stop the growth of the material and freeze the cellular structure.

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Although the four steps have been herein clearly separated, boundaries are not so clear, and some steps could co-occur. In fact, depressurization and foaming can occur at the same time, simply by using a saturation temperature above the effective glass transition temperature of the polymer for amorphous materials of the meting temperature for semi-crystalline polymers. In the case of semi-crystalline polymers the distribution of the gas in the polymer is more homogeneous in this particular case because the crystalline phase has disappeared in these conditions. This is the socalled one-step process. During depressurization, the polymer is already plasticized; thus, phase separation and growth occur at the same time. As a counterpart, when depressurization and foaming are separated the process is called two-step process. As it can be seen, this apparently easy process depends on multiple parameters and so does the final cellular structure. The ultimate objective is the generation of cells in the nanometric scale. Considering the already defined relationship between cell density (Nv ), the porosity Vf = 1 − ρr , and the cell size ðϕÞ (eq. (2.13) in Chapter 2), the production of cell sizes around 200 nm with a porosity higher than 0.5 implies a cell nucleation density and number of nucleation points in the solid material (assuming no coalescence) of the order of 1014 cells/cm3. The key point for the production of nanocellular polymers is therefore to achieve such values for nucleation densities. Inside the nucleation process, two strategies have been followed to achieve this purpose: homogeneous nucleation and heterogeneous nucleation. Although boundaries between them are usually not clear, heterogeneous materials can be defined as those in which a second phase is intentionally added to act as preferential nucleation sites, while homogeneous materials are those in which such second phase does not exist. Both strategies are extensively explained in the following sections.

3.3.1 Homogeneous nucleation As it was previously explained, nucleation is considered homogeneous when a second phase or nanostructuration is not intentionally added to act as a nucleating agent. Thus, in this context, homogeneous materials can be said to present a single phase in which cells are created without any help but the gas itself. And if any additional phase is present, it does not play a role during the nucleation stage. As it was explained in the previous section, the production of a cellular material through gas dissolution foaming implies different steps. As shown in Figure 3.5, the material evolves during the process until the final stable cellular material is reached. So as to understand and therefore control the final cellular structure as a function of the production parameters it is essential to understand the foaming mechanisms taking place. Modeling the production of such materials comprises two essential steps: the model of the nucleation and the model of the growth that, as previously said, are

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two processes that can coexist. In addition, both processes depend on many different production parameters. This makes modeling a challenging task. Homogeneous nucleation was first explained through the classical nucleation theory (CNT) described in Chapter 2. This theory has been historically used to explain nucleation phenomena in conventional and microcellular foams [58, 59]. However, and although it is useful to qualitatively understand the process, predicted values were proven not to be realistic for microcellular materials and much less for nanocellular materials. According to Thompson et al. classical nucleation theory overestimates the critical radius for nanocellular polymers over three times, and the nucleation barrier over six times, therefore nucleation rate is underestimated by more than five orders of magnitude [60]. The main weakness of CNT is, on the one hand, not considering viscoelastic effects and other mechanisms such as depressurization rate or surface tension, among others, potentially contributing to the creation, stabilization, or destruction of nuclei. On the other hand, CNT considers nucleation and growth as completely independent processes. Many works in the literature have dealt with this problem trying to correctly predict results obtained through homogeneous nucleation. The first work dates from 1968 when Street et al. investigated single bubble growth for Newtonian fluids [61]. This model was afterward improved including the study of non-Newtonian fluids, and more complex dynamics [61–65]. However, these theories were not yet leading with the coexistence of nucleation and growth among other essential factors. Shafi et al. were the first to propose a model able to simulate nucleation and bubble growth simultaneously in their works published in 1996 and 1997 [66, 67]. The success of their model relies on the introduction of a new concept, the Influence Volume Approach (IVA). IVA model was able to relate the final characteristics of the foam such as nucleation density, density, and cell size distribution to processing parameters such as saturation pressure and temperature or depressurization rate, and also to materials characteristics such as viscosity, surface tension, or gas solubility. However, Shafi’s model assumes an instantaneous depressurization, the same initial conditions for all nucleated bubbles, and the non-existence of a limited growth period. Some others have dealt with this challenge, with different models published in the literature. Some of them propose similar models such as Feng and Bertelo [68], or Leung et al. [69]. And others have modified the initial theory of Shafi et al. such as the one from Taki or Mao et al. [70, 71]. It is worth mentioning that the theory proposed by Mao et al. includes an essential idea; in addition to considering nucleation and growth as simultaneous processes, they differentiated two different phases in the growth step, the free expansion phase, and the limited expansion phase. This will be later explained in more detail. In 2015 Khan et al. published the most complete model up to now dealing with the production of nanocellular materials through homogeneous nucleation [72]. This model puts together all the improvements previously added in the literature eliminat-

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ing most of the previous assumptions, leading to very accurate results in comparison with experimental data. As previously said, CNT assumes the presence of a nucleation barrier, from which a nucleus is stable when R > rc (recall that R is the radius of the cell and rc is the critical radius, as described in Chapter 2). The Influence Volume Approach considers that when a stable nucleus is created the surrounding gas molecules start diffusing to this nucleus. This leads to a gas concentration gradient around the already created nucleus (Figure 3.6) that is minimum near the surface of the nascent cell and becomes higher as moving away up to reach the maximum value again.

Figure 3.6: (a) Scheme of the gas concentration profile near a nascent cell. (b) Scheme of the influence volume inside a polymer while nucleation and growing.

This concept changes the perspective of nucleation theory in two essential points. First of all, due to the lower gas concentration in this area, this model introduces the idea that no stable nucleus can grow inside the IV region; thus, new nuclei can only appear in the non-influenced volume (UV). In addition, this region evolves during growth, as more gas diffuses into the cell. Secondly, IVA allows introducing the concept of nucleation time and separates it from the depressurization time unlike the previous models in the literature. With this model, it is assumed that nucleation is not instantaneous and is governed by the availability of supersaturated regions in the polymer matrix. On the other hand, during growth Khan’s model assumes that two different phases can be distinguished within growing of cells. Firstly the “free expansion phase,” where UV is available in the polymer and cells growth rate increases with increasing influence volume. Nucleation carries on during this phase until no more UV is available, moment at which the “limited expansion phase” starts. During this phase, the cell growth rate decreases, and cell growth continues until the pressure inside the cells reaches ambient

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pressure or the effective glass transition of the polymer matrix is above the foaming temperature. The assumptions made in this model are the following: 1. Cells present spherical symmetry during nucleation and growth. Interaction between cells occurs through IV but they cannot coalesce. 2. The pressure inside the cell Pcell is related to the gas concentration CðR, tÞ at the bubble surface through: CðR, tÞ = KH Pcell

3. 4.

(3:5)

where KH is Henry’s constant. The whole process is isothermal. The gas-polymer mixture is a viscous fluid with its viscosity being shear rate dependent following the Cross model.

Taking all the previous information into account, this model defines the following equations governing nucleation and growth. 3.3.1.1 Bubble growth The growth rate of the bubble is governed by eq. (3.6):   Pcell − Pliq R σ dR − = 4μ dt 2μ

(3:6)

where Pcell − Pliq is given by considering a quasi-static momentum balance across the melt phase together with the jump linear momentum across the cell–liquid interface; σ is the interfacial tension between gas and liquid phases; and μ is the viscosity, which is considered to change according to the Cross function, being a function of the temperature, pressure, and strain rate. Although eq. (3.6) is derived assuming a Newtonian fluid, the equation is evaluated in each timestep based on the evolving viscosity considering the strain rate and solubility at the cell interface. On the other hand, the concentration of gas in the surrounding of the nascent cell is given by:   ∂C ∂D D ∂ ∂C (3:7) + vr = 2 r2 dt dt r ∂r ∂r where D is the diffusivity of the gas in the polymer and depends on the temperature and the gas pressure. So as to solve this equation it is considered that initially Cðr, 0Þ = C0 . While the boundary conditions are defined as CðR, tÞ = KH Pcell and the one at R = ∞ depends on the growth phase, either Free expansion or Limited expansion.

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Finally, the mass balance on the surface of the cell is given by the following relationship:   d 4πR3 Pcell dC = D4πR2 (3:8) 3 RT dt dr With R the universal gas constant and T the gas temperature. Solving simultaneously eqs. (3.6)–(3.8) leads to determining the pressure inside the cell, the radius of the cell, and the gas concentration. 3.3.1.2 Nucleation As previously described to generate a nucleus it is necessary to overcome the Gibbs free energy barrier (eq. (3.9)), where σ is the surface tension between the gas and the polymer phase and Δp is the pressure difference between gas and solid: 16πγσ 3 (3:9) 3Δp2   dN , is considered as follows [73]: With this consideration, the nucleation rate dt ! 0.5 dN 2σ 16A2 πσ 3 exp − (3:10) = A1 N  2 dt πm 3kT Pcell − Pliq ΔGhom =

with N being the number of dissolved gas molecules per unit volume of the primary phase, m is the mass of the gas molecules and σ is the interfacial tension between the blowing agent and the saturated polymer, k is the Boltzman constant and A1 and A2, are two fitting parameters that are obtained through fitting the theoretical model to the experiments. As described previously, new nuclei are only created in the UV; thus, the total number of nuclei at a time t is defined by eq. (3.11): ðt Ntot ðtÞ =

dN ðτÞ UV ðτÞdτ dt

(3:11)

0

where UV is the non-influenced volume defined as: ðt UV ðtÞ = V0 −

dN ðt − τÞUV ðt − τÞVcb ðτÞ dτ dt

(3:12)

0

Being V0 the total volume of the polymer-gas blend at t = 0 and Vcb is the influence volume surrounding the bubble.

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3.3.1.3 Considerations to solve the equations In this model eq. (3.7) is solved through the Galerkin finite element method (FEM), used to discretize the equation. On the other hand, as it was previously said, cell growth can be separated into two different phases; thus, different boundary conditions are defined in each of them to solve the previous equation. During the free expansion phase, there is a large gap among cells, meaning that they do not interact among them and the IV expands freely. Then the boundary conditions are defined as described in eqs. (3.13) and (3.14) CðR, tÞ = KH Pcell

(3:13)

Cð∞, tÞ = C0

(3:14)

And the influence volume is calculated as follows: Vcb =

 4π  3 R∞ − R3 3

(3:15)

R∞ being chosen so as to CðR∞ , tÞ = 0.98C0 . When the influence volume occupies the whole volume of the polymer, nucleation stops, and the IV of each cell interacts which the others, starting at this moment the limited expansion phase. Now the concentration at R∞ is lower than kC0 , assuming that R∞ varies with time as: _ 2 RR t Rt+Δt 2 ∞ = R∞ + Δt  Rt∞

(3:16)

∂C ðR∞ , tÞ = 0 ∂r

(3:17)

And also assuming that

3.3.1.4 Model predictions The complexity of this model allows for solving the behavior of bubble nucleation and growth very precisely. Figure 3.7 shows the difference between the FEM model used by Khan et al. and previous models in the literature. It is shown how previous models underpredict the gas concentration gradient leading to a wrong predicted bubble growth and influence volume. The evaluation of this model with a single bubble growth allows explaining the growth dynamics in homogeneous nucleation as follows. By looking at Figure 3.8a it can be seen how the growth of the nucleation point is relatively slow at the beginning but later starts gaining momentum due to the increasing pressure difference. The in-

3.3 Gas dissolution foaming

(a)

61

1 0.995 0.99 0.985 0.98 0.975

Shafi et.al. Han & Yoo FEM

1

1.5

2

2.5

3

3.5

4

4.5

5

y (b) 1.6

Shafi et.al. Han & Yoo FEM

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

50

100

150

200

250

300

Figure 3.7: Comparison of FEM with Shafi et al. and Han and Yoo polynomial functions to predict (a) concentration profile and (b) concentration gradient. Adapted from [72].

crease of this pressure difference leads to an increase in the bubble growth rate. During this initial period, the viscosity of the melt is the limiting criterion for bubble growth. This regime continues until UV = 0, starting what was named the limited growth phase. The bubble growth rate reaches the maximum and then decreases. The diffusion of the gas leads to a decrease in the concentration gradient; thus, the bubble growth rate decreases until Tg > Tamb . On the other hand, regarding nucleation rate (Figure 3.8b), is driven by Psat − Pliq that it is small at the beginning of the process and so is the nucleation rate. When times evolves Psat does not strongly evolve, but Pliq becomes smaller leading to an increase in the nucleation rate, which keeps increasing up to UV becomes unavailable. The showed results lead to the correct predictions on the cell size and in the cell size distribution as Figure 3.9 shows for nanocellular polymers produced from gas dissolution

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Figure 3.8: (a) Variation of bubble growth rate with time. (b) Evolution of nucleation rate with time as a function of A2 parameter. Adapted from [72].

foaming of a copolymer of methyl-methacrylate (MMA) with 9 wt% of ethylacrylate (EA). Such predictions are for the first time close to the experimental results for nanocellular polymers thanks to the accuracy of the model.

3.3 Gas dissolution foaming

Figure 3.9: (a) Comparison of the results of the model with experimental data for MMA-EA. Closed symbols for 25 °C of foaming temperature and open symbols for 0 °C. (b) Comparison of cell size distribution predicted by the model with experimental data of foams produced at 36 MPa and 35 °C for the open symbols and 36 MPa, 55 °C for the closed symbols. Adapted from [72].

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3.3.1.5 Nanocellular polymers through homogeneous nucleation Since their discovery, the previously discussed models along with the experimental data have allowed understanding the production process of nanocellular polymers through homogeneous nucleation. Table 3.1 broadly summarizes the influence of different parameters on the final cellular characteristics such as cell nucleation density, cell size, and relative density according to the experimental results in the literature and to the previous models [74]. As it can be seen in Table 3.1, increasing saturation pressure and decreasing saturation temperature lead to the maximization of the nucleation points and therefore the minimization of the cell size. This is mainly due to a significant increase in the solubility when pressure is increased and temperature is decreased [74]. The depressurization rate reduces the cell size as it increases and leads to a broadened cell size distribution, while the desorption time must be minimized to increase the cell nucleation density. Regarding the foaming, an increase in both temperature and time leads to minimum cell sizes through the increase of the cell nucleation density. Finally, the polymer matrix is of vital importance in the final results; high viscosities are encouraged to minimize cell size while low relative density is favored with small viscosities and low Tg s. Table 3.1: Scheme of the influence of the production parameters and polymer matrix in a two-step gas dissolution foaming process to produce nanocellular materials with low relative densities. The script indicates that there is not a direct influence between the parameter and the expected result. Adapted from reference [74].

✶ Those parameters should be maximized up to a limit when degeneration of the cellular structure takes place.

The knowledge on the production process and a proper control of all parameters that have an influence has led to the production of a wide variety of polymers with a nanocellular structure. Thus, nanocellular polymers from homogeneous nucleation have been produced by using polymers such as polyetherimide (PEI), polycarbonate (PC), polyphenylsulfone (PPSU), and above all polymethylmethacrylate (PMMA) [23, 25, 75–77].

3.3 Gas dissolution foaming

Figure 3.10: Main discoveries in nanocellular polymers produced through homogeneous nucleation.

65

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Among all the literature results, Figure 3.10 shows some of the main milestones reached using the homogeneous nucleation approach. As previously said, nanocellular polymers were born around 2003. After their discovery, the first experimental validation of their improved mechanical properties was published in 2011 by Kumar et al. publishing an improvement in the modulus of toughness of nanocellular polyetherimide up to 350% in comparison to microcellular PEI and impact energies 600% higher [78]. In 2015 Notario et al. experimentally proved for the first time the evidence of the Knudsen effect in nanocellular PMMA [79]. The theoretical model of Khan et al. was also published this year [72]. Martín-de León et al. proved in 2016 the existence of solid confinement in nanocellular polymers [80] and in 2017 they produced the first transparent nanocellular polymer from PMMA [81]. It was not until 2019 when samples with large dimensions and flat forms free of internal defects were published taking one more step toward the industrialization of these structures [82]. This work was published by Martín-de León et al., which also presented the first theoretical model explaining the interaction of light with such structures [83]. In 2020, Bernardo et al. experimentally proved a higher radiation term in nanocellular polymers in comparison with their microcellular counterparts [84]. Finally, in 2022, Sanchez-Calderón et al. firstly measured the thermal conductivity of nanocellular polymers by an steady-state approach [85]. All these topics will be explained in more detail in the following chapters.

3.3.2 Heterogeneous nucleation Heterogeneous nucleation takes place in those materials in which a second phase is responsible for nucleation. As can be seen in Figure 3.11 the scheme has changed: the second phase, well dispersed in the polymer matrix, is in the process since the beginning. The addition of this second phase changes the surface tension in comparison with the pure polymer one. The surface tension of a two-phase system can be defined by eq. (3.18) where σ sp , σ p and σ s are the surface tensions of the second phase and the polymer, the gas-polymer and the gas-second phase, respectively, and θ is the wetting angle of the interface [86]: σ sp = σ p + σ s cos θ

(3:18)

This second phase induces gas molecules to aggregate at the foreign surface, which is known as wetting [87]. Therefore, when the polymer is completely saturated (2 in Figure 3.11) and the gas is released, the nucleation tends to take place in these preexisting surfaces (3 in Figure 3.11). The nucleation is no longer controlled by the production parameters but mainly by the characteristics of the second phase (chemical interaction with gas and/or polymer matrix, size, geometry, dispersion, etc.). This is very interesting from the production point of view because the materials can be produced with lower saturation

3.3 Gas dissolution foaming

67

Figure 3.11: Scheme of a heterogeneous sample under the gas dissolution foaming process.

pressures and higher saturation temperatures (that reduces the saturation time) making the process more feasible for the scaling-up. Therefore, previously defined equations for homogeneous nucleation must be modified. In this context, the presence of the second phase is the one controlling the creation of new nucleation points; thus, the IV concept does not make sense, and equations considered in the previous section can be simplified. Therefore, the nucleation rate for heterogeneous nucleation (Nhet ) can be defined from the original CNT (Chapter 2) as follows:   ΔGhet (3:19) Nhet = f1 C1 exp − kB T where C1 is the initial concentration of gas in the polymer and f1 is the frequency factor of gas molecules joining the nucleus. It is in the energy barrier for heterogeneous nucleation (ΔGhet Þ where differences with homogeneous nucleation are introduced. As it was previously said, in order to form a nucleus, it is necessary to overcome an energy barrier. The presence of the second phase reduces this energy in comparison with the one needed in homogeneous nucleation: ΔGhet = ΔGhom SðθÞ =

16πγ3 SðθÞ 3Δp2

(3:20)

Thus the energy barrier is proportional to the energy barriers in a homogeneous system by a factor SðθÞ that depends on the wetting/contact angle of the polymer–additive–gas interface [88], and that is always less than or equal to one: 1 SðθÞ = ð2 + cos θÞð1 − cos θÞ2 4

(3:21)

The critical radius can be defined equal to that in homogeneous nucleation due to the fact that the second phase does not modify this value. Equation (3.20) only takes into account the contact angle, but the energy barrier can be defined taking into account

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Chapter 3 From the microscale to the nanoscale in cellular materials production process

also the second phase curvature by the introduction of the Fletcher factor f ðm, nÞ [89] (eq. (3.22)): 16πγ3 f ðm, nÞwith (3:22) 3Δp2 "      #   1 1 1 − mn 3 n3 n−m n−m 3 3 2 n−m + f ðm, nÞ = + + + mn 2−3 −1 2 2 2 g g g 2 g ΔGhet = ΔGhom f ðm, nÞ =

where the expressions of m, n, and g are given by eqs. (3.23)–(3.25) respectively: m = cos θ

(3:23)

n = R=rc

(3:24)

 1=2 g = 1 + n2 − 2mn

(3:25)

Being θ the contact angle, R the radius of the second phase, and g is a function of the previous parameters. The Fletcher factor is minimized when the particle geometry is a flat surface with n  1. For any other geometry the value increases; thus, the size of the particle in comparison to the critical radius strongly affects the nucleating effect. Equation (3.22) is only valid for the critical nuclei, however, it has been proved to correctly estimate the reduction of the free energy barrier when including a second phase [30]. Production parameters control the nucleation when working with homogeneous nucleation; however, in heterogeneous nucleation, the characteristics of the second phase or nucleant rule the density of the generated nucleus. The ideal nucleant has been defined to present the following four qualities [90]: – Nucleation in the nucleant or at the interface should be energetically favorable in comparison with homogeneous nucleation. Weak interactions between the second phase and the polymer usually lead to low energy barriers. However, this can depend on the used production parameters, homogeneous or heterogeneous nucleation dominating the process as a function of them. – Uniform size and surface properties so that all the particles act simultaneously as nucleating points generating a homogeneous cellular structure. – A good dispersion in the polymeric matrix. Aggregates will lead to less effective nucleation sites and non-uniform cellular structures. – The volumetric density of nucleants should be enough to achieve the required nucleation density (around 1014 nuclei/cm3 for nanocellular polymers) and higher than the estimated through homogeneous nucleation for the used production parameters. Taking into account these considerations two types of additives have been used as the second phase for nucleation: nanoparticles and nanostructured polymers. Both

3.3 Gas dissolution foaming

69

additives need to present sufficiently small sizes and a high enough volumetric density (above 1013–1014 domains/cm3) to create nanocellular polymers. Regarding particles, the maximum nucleation density achievable for spherical particles is given by eq. (3.26) Nucleants ωp ρc = ρp V p cm3

(3:26)

Being ωp the particle content, ρc the density of the composite (material formed by the polymer matrix and particles), ρp the density of the particles and Vp the volume of a single particle. Taking eq. (3.26) into consideration, particles with a radius below 50 nm are needed to obtain the required nucleation density, with common concentrations of 10% or smaller. Additionally, to correctly promote cell growth, the particle should be larger than the critical radius but smaller than the desired cell size [91]. In addition, particle size is required to be smaller than the cell wall thickness (ranging 20–30 nm [80]) to prevent the degeneration of the cellular structure such as cell wall rupture. As aforementioned it is therefore mandatory the dimensions of the particles to be on the nanometric scale. Inorganic particles are considered nanoparticles when at least one dimension is in the nanometric scale [92]. Taking this into account three different geometries are used: flakes with nanometric thickness and hundreds to thousands of nanometers in extent (clay platelets [93], graphene [94]), needle-like particles with two nanometric dimensions and variable length (nanotubes [95], nanofibers [96] or sepiolites [97]) and spheres with all dimensions within the nanometric scale such as silica particles [98]. When considering nanostructured polymers, two strategies can be followed as well: the micelle nanostructuration or the blend of two immiscible polymers. In the first one, the micelles can be obtained by using copolymers dispersed in a continuous polymer matrix of the same composition than one of the blocks of the copolymer. The simplest case is to use a copolymer with two blocks, A and B, in an A matrix. The copolymer molecules are self-assembled, usually in nano-spheres, leading to the socalled micelles. In this case, the maximum nucleation density for micelles is given by eq. (3.27) considering that all the copolymer blocks reside in the micelles: nmax =

wNav M n Nc

(3:27)

with w being the amount of added copolymer, Nav is Avogadro’s number, Mn is the molecular weight of the copolymer, and Nc is the aggregation number, defined as the number of copolymer molecules in one micelle. In addition, a minimum amount of copolymer is needed to obtain micelles named as critical micelle concentration (cmc) [99]. When mixing two immiscible polymers, phase separation takes place due to the incompatibility being the polymer in the lowest proportion dispersed in small domains, that can be nanometric, within the second polymer. The control of the size of

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Chapter 3 From the microscale to the nanoscale in cellular materials production process

the domain is determined by different factors such as the different viscosities of the polymer, and the processing mixing conditions. Wu et al. proposed that for an extrusion process the size of the domains is given by eq. (3.28) [100]: (   + if ηd =ηm > 1 4σ ηd ±0.84 (3:28) d= ηm γ_ ηm − if ηd =ηm < 1 γd and ηm being the viscosity of the dispersed polymer and the matrix, respectively, σ the surface tension and γ_ the shear velocity of the extruder screw. So as to minimize the domains to obtain nanocellular polymers, the function can be minimized by using polymers with the same viscosity or by maximizing the shear velocity of the extruder screw. 3.3.2.1 Nanocellular polymers through heterogeneous nucleation A wide variety of polymers as well as second phases have been used to produce nanocellular polymers with this approach. For example, polymers such as PC, polyethylene (PE), polylactic acid (PLA), polypropylene (PP) or PMMA have been combined with montmorillonite (MMT), carbon nanotubes (CNT), sepiolites, polymethylmethacrylatepolybutyl acrylate-polymethylmethacrylate (MAM) block copolymers, thermoplastic polyurethane (TPU) and others. [97, 101–105]. Among all these publications it is worth mentioning those highlighted in Figure 3.12. In 2003 the first nanocellular polymer from an heterogeneous nucleation was produced with PLA nucleated with MMT obtaining cell sizes of 360 nm and relative densities of 0.57 [106]. In 2005 a semi-crystalline polymer was used for the first time to produce nanocells; thus, 3% of nanoclays is added to HDPE to produce cells between 200 and 300 nm with a relative density of 0.7 [101]. In 2013 Costeux et al. published a nanocellular material based on a copolymer PMMA/EMA with 0.25% of polyhedral oligomeric silsesquioxane (POSS) obtaining the combination of minimum cell size and relative density of the literature with 120 nm of cell size and 0.15 of relative density [91]. In 2016 the use of polydimethylsiloxane (PDMS) as nucleation points allowed Liu et al. to obtain the first nanocellular polystyrene of the literature with a cell size of 400 nm [105]. In 2017 Wang et al. obtained low-density materials by using the system PMMA/TPU and decreasing relative density below 0.15 with 200 nm of cell size [107]. This structure also allows Wang to publish the lowest value for the thermal conductivity of these structures measured by the transient plane source method. In 2019 nanocellular materials with interesting characteristics such as a controlled bimodality or anisotropy were presented by Bernardo et al. by means of adding MAM and sepiolites to PMMA [97, 108]. The MAM and sepiolites inclusion also led to the production of nanocellular polymers with the lowest demanding conditions up to that moment. By using 6 MPa and 25 °C of saturation pressure and temperature, Bernardo et al. achieved cells in the range of 872 to 122 nm by changing the amount of and type of additive [109]. Finally, Figure 3.12 highlights the production of nanocellular polymers with a gradient cellular structure. Bernardo et al. achieved these results by including TPU as the nucleant agent in PMMA beads [110].

3.3 Gas dissolution foaming

Figure 3.12: Main discoveries in nanocellular polymers produced through heterogeneous nucleation.

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Chapter 3 From the microscale to the nanoscale in cellular materials production process

3.4 Limitations and challenges 3.4.1 Limitations 3.4.1.1 Density reduction Figure 3.13 shows the wide variety of cell sizes and relative densities that have been obtained following both strategies (homogeneous ([23, 27, 29, 78, 82, 83, 111–117]) and heterogeneous ([30, 91, 97, 101–103, 106, 107, 118–132])). The combination of both strategies allows covering a high fraction of the cell size–relative density map.

Figure 3.13: (a) Map of nanocellular materials from homogeneous polymers produced through gas dissolution foaming process. (b) Map of nanocellular materials from heterogeneous polymers produced through gas dissolution foaming process. Dashed lines represent lines of constant cell nucleation density according to eq. (2.13) (Chapter 2). Adapted from [133].

3.4 Limitations and challenges

73

However, it is also clear that there is plenty room for improvement with some empty regions in both maps. On the one hand, when working with heterogeneous nucleation the size of the second phase limits the reduction of the cell size below 50 nm; thus, such cell sizes have only been obtained through the homogeneous route. On the other hand, regarding the relative density, both strategies do not lead to values smaller than 0.1 specially when the cell size decreases below 100 nm. First of all, the differences between homogeneous and heterogeneous nucleation can be attributed to this second phase. When talking about reducing cell size, this cannot be reduced below the size of the second phase when using heterogeneous nucleation. This makes difficult to lead to cells smaller than 50 nm. Regarding density, it strongly depends on the stack of the cells. If considering spherical cells, the minimum density is achievable by considering a hexagonal or fcc packaging; thus, that is the minimum boundary. However, cells in these materials are commonly assumed to be tetrakaidecahedra; thus, this minimum boundary would be removed, at least for homogeneous nucleation. For heterogeneous nucleation, the geometry of the second phase can determine the geometry of the final cells leading to this real boundary. This fact has been already proven for PMMA nucleated with MAM spherical micelles acting as nucleating agent [132]. However, densities below 0.2 are hardly observed for small cell sizes even for homogeneous nucleation. It seems pretty obvious that to minimize the density it is mandatory to growth each nucleation point up to the limit, that means up to touching with their neighbors. Additionally, to completely growth these nuclei it is essential to have enough gas inside the polymer. As it can be seen in Figure 3.14, it has been empirically proven for different systems, both heterogeneous and homogeneous, that when reducing the density below a certain value, the open cell content abruptly increases [80, 134–136]. That means cells becomes open, and the structure interconnected. This leads to a fast diffusion of the gas out of the material; thus, it is no longer available for the growth process. The cell wall failure leading to this open cell content has been attributed to two main mechanisms according to Van Loock et al. [138]. The model and experiments of Van Loock and coworkers proved the existence of a critical cell wall thickness that cannot be exceed. When this minimum cell wall thickness is reached, cells broke and expansion stops, as commented before. In addition to these mechanical constraints, the cell wall cannot be as thin as wanted, i.e. there are also some geometrical constraints as commented in Chapter 2. This idea was firstly suggested by Bernardo and coworkers (Figure 3.15). According to this idea, the region under the theoretical line of constant cell wall thickness of 5 nm (assuming that is the theoretical minimum) is a forbidden region, because materials in that part of the graph will require cell wall thicknesses smaller than 5 nm, so there are some combinations of relative density and cell size that might be impossible. The critical porosity value of the model of Van Loock et al. strongly depends on the molecular weight of the polymer that is directly related to the viscosity of the ma-

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Chapter 3 From the microscale to the nanoscale in cellular materials production process

Figure 3.14: Open cell content as a function of the relative density for the different nanocellular polymers produced in our laboratory. Arrows indicate the increase of the foaming temperature. From reference [137].

Figure 3.15: Cell size–relative density map of different nanocellular polymers. Gray curves indicate regions of constant cell nucleation density according. The constant thick black line indicates the curve of constant cell wall thickness of 5 nm, from reference [107].

terial. Thus, the challenge of further reducing the relative density of nanocellular polymers can be done through carefully tailoring the viscosity of the matrix. A reduced viscosity leads to an increase in cells growth [76, 116, 132]; however, the viscosity should be high enough to avoid coalescence; thus, the optimum value can be explored, either by modifying the molecular weight of already existing polymers or by synthesizing a new polymer with the desired rheology.

3.4 Limitations and challenges

75

Further strategies can be done regarding the polymer morphology. Strey et al. patented a method in which they created a polymer gel by mixing a polymer with a solvent such acetone. The control of the solvent content as well as the crosslinking degree of the polymer leads to materials with cell sizes within the nanometric range and low relative densities [139]. Regarding the production method, it can also be optimized, and although the literature covers a wide window of production conditions, there is still plenty of combination of process parameters and polymeric matrix that have not been explored. The method can also be modified as Martín-de León et al. did, by implementing a cyclic gas dissolution foaming, leading to one of the best materials in the literature regarding minimum cell size and density [140]. Nanocellular PMMA with 30 nm of cell size and 0.3 of relative density was obtained by this method. Finally, for heterogeneous nucleation, density reduction can be achieved through the control of the second phase, with almost infinite combinations of second phasepolymer systems. 3.4.1.2 Big dimension samples In addition to density reduction, nanocellular samples of big dimensions are hard to produce and rarely found in the literature. This is one of the factors explaining the absence of nanocellular polymers in the industry. The biggest samples reported in the literature were produced by Martín-de León et al. with 100 × 100 × 6 mm3 by means of a controlled foaming in a hot press [82]. In addition, Sanchez-Calderon et al. led to a final sample of 150 × 150 × 14 mm3 by means of stacking 18 samples of smaller dimensions (50 × 50 × 7 mm3) [141]. Additionally, Martín-de León et al. proposed a method not only to produce big samples but also to minimize all the millimetric or micrometric defects these samples could have, especially when using low molecular weigh polymers. The method simply consists of performing a two-stage depressurization in the gas dissolution foaming process instead of a single depressurization [111]. Gas dissolution foaming limits the size of the samples in two different ways. First of all, this method is performed in a pressure vessel that should resist very demanding pressure and temperature conditions, limiting the final dimensions of the equipment. The final size of the samples will depend on the initial sample’s dimensions limited at the same time by the vessel dimensions and the final expansion ratio. Secondly, the production of thick samples is limited by the production time. The production of very thick samples leads to extremely high production times not suitable for an industrial process. It is estimated that samples of around 10 to 15 mm thick could be produced by this technology in reasonable saturation times. It is worth mentioning that gas dissolution process has been scaled up for the production of conventional foams, and for the production of microcellular polymers, as

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explained in the introduction of this chapter and therefore it is feasible to think about production large nanocellular foam sheets by this technology. In addition, other technologies that could be used to produce large samples and that has not been studied in detail yet are bead foaming and injection molding.

3.5 Conclusions Up to date, gas dissolution foaming has been the main approach followed by scientists all around the world to produce thick nanocellular polymers. Many different polymers, nucleating agents and processing conditions have been used resulting in a wide range of densities and cell sizes. The homogeneous nucleation approach allows producing a wide variety of materials, and in fact it is possible to produce very small cell sizes (below 20 nm) by controlling the processing parameters. In general, very demanding processing conditions are needed (high pressures and/or low saturation pressures) together with pressure drop rates when using this strategy. In addition, polymer–blowing agent combinations with high solubility and a proper molecular structure, glass transition temperature, and an adequate viscosity of the polymer are needed. A significant number of models have been developed, and the physics behind this approach is today well understood when the process follows a nucleation and growth path. The heterogeneous nucleation approach depends more on the combination of materials used, and there is plenty of room to explore this approach in more detail due to the wide variety of polymers, nucleating agents, and blowing agents that can be used. With the current knowledge it is clear that the sizes of the nucleants, its dispersion in the polymeric matrix, its superficial characteristics and interaction with the physical blowing agent are critical parameters to control the process. With this approach a wide variety of densities and cell sizes have been reached, although it seems to be difficult to reduce the cell size below 50 nm with this approach. One very interesting aspect of this way of producing the foams is that less demanding processing conditions are needed, so lower saturation pressures (a slow as 6 MPa for PMMA) and higher saturation temperatures are required, which could facilitate the production at industrial scale using this approach. Although a wide variety of materials have been produced, there are still some limitations to overcome. The combination of very low cell sizes and very low densities is difficult to reach. It seems that some limitations related to the mechanical behavior of the very thin cell walls produced during foaming and some geometrical contains are the cause of these limitations. However, although it has been difficult to progress in reducing densities, some recent developments indicate that there is room for further improvements. In addition, the production of materials with very large dimensions free of defects is still challenging. However, if applications are found for these materials, it is clear that current technology in high-pressure vessels and the previous experience in

References

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the production at industrial scale of conventional foam and microcellular foams by this technology would allow producing the material using the gas dissolution technology. In addition, alternatives processes such as bead foaming or injection molding could contribute to this topic.

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[116] S. Costeux, I. Khan, S.P. Bunker, H.K. Jeon, Experimental study and modeling of nanofoams formation from single phase acrylic copolymers, J. Cell Plast. 51 (2014) 197–221. https://doi.org/10. 1177/0021955X14531972. [117] Y.P. Handa, Z. Zhan, B. Wong, Solubility, diffusivity, and retrograde vitrification in PMMA-CO2, and development of sub-micron cellular structures, Cell. Polym. 20 (2001) 1–16. http://cat.inist.fr/?aMo dele=afficheN&cpsidt=939682 (accessed July 10, 2015). [118] Y. Ema, M. Ikeya, M. Okamoto, Foam processing and cellular structure of polylactide-based nanocomposites, Polymer (Guildf). 47 (2006) 5350–5359. https://doi.org/10.1016/J.POLYMER.2006. 05.050. [119] L. Urbanczyk, C. Calberg, C. Detrembleur, C. Jérôme, M. Alexandre, Batch foaming of SAN/clay nanocomposites with scCO2: A very tunable way of controlling the cellular morphology, Polymer (Guildf). 51 (2010) 3520–3531. https://doi.org/10.1016/J.POLYMER.2010.05.037. [120] J. Pinto, D. Morselli, V. Bernardo, B. Notario, D. Fragouli, M.A. Rodriguez-Perez, A. Athanassiou, Nanoporous PMMA foams with templated pore size obtained by localized in situ synthesis of nanoparticles and CO2 foaming, Polymer (Guildf). 124 (2017) 176–185. https://doi.org/10.1016/j.poly mer.2017.07.067. [121] S. Liu, B. Zoetebier, L. Hulsman, Y. Zhang, J. Duvigneau, G.J. Vancso, Nanocellular polymer foams nucleated by core-shell nanoparticles, Polymer (Guildf). 104 (2016) 22–30. https://doi.org/10.1016/J. POLYMER.2016.09.016. [122] J. Yang, L. Huang, Y. Zhang, F. Chen, P. Fan, M. Zhong, S. Yeh, A new promising nucleating agent for polymer foaming: Applications of ordered mesoporous silica particles in polymethyl methacrylate supercritical carbon dioxide microcellular foaming, Ind. Eng. Chem. Res. 52 (2013) 14169–14178. https://doi.org/10.1021/ie4018447. [123] H. Yu, Y. Lei, X. Yu, X. Wang, T. Liu, S. Luo, Solid-state polyetherimide (PEI) nanofoams: The influence of the compatibility of nucleation agent on the cellular morphology, J. Polym. Res. 23 (2016) 121. https://doi.org/10.1007/s10965-016-1009-2. [124] V. Realinho, M. Antunes, A.B. Martínez, J.I. Velasco, Influence of nanoclay concentration on the CO 2 diffusion and physical properties of PMMA montmorillonite microcellular foams, Ind. Eng. Chem. Res. 50 (2011) 13819–13824. https://doi.org/10.1021/ie201532h. [125] L. Monnereau, L. Urbanczyk, J.-M. Thomassin, M. Alexandre, C. Jérôme, I. Huynen, C. Bailly, C. Detrembleur, Supercritical CO2 and polycarbonate based nanocomposites: A critical issue for foaming, Polymer (Guildf). 55 (2014) 2422–2431. https://doi.org/10.1016/J.POLYMER.2014.03.035. [126] J. Pinto, M. Dumon, M. Pedros, J. Reglero, M.A. Rodriguez-Perez, Nanocellular CO2 foaming of PMMA assisted by block copolymer nanostructuration, Chem. Eng. J. 243 (2014) 428–435. https:// doi.org/10.1016/j.cej.2014.01.021. [127] J. Pinto, J.A. Reglero-ruiz, M. Dumon, M.A. Rodriguez-Perez, Temperature influence and CO2 transport in foaming processes of poly (methyl methacrylate)–block copolymer nanocellular and microcellular foams, J Supercrit Fluids. 94 (2014) 198–205. https://doi.org/10.1016/j.supflu.2014.07. 02. [128] M. Dumon, J.A. Reglero-Ruiz, J. Pinto, M.A. Rodriguez-Pérez, M. Tallon, M. Pedros, E. Cloutet, P. Viot, Block copolymer-assisted microcellular supercritical CO2 foaming of polymers and blends, Cell. Polym. 31 (2012) 207–222. [129] C. Forest, P. Chaumont, P. Cassagnau, B. Swoboda, P. Sonntag, CO2 nano-foaming of nanostructured PMMA, Polymer (Guildf). 58 (2015) 76–87. https://doi.org/10.1016/j.polymer.2014.12. 048. [130] C. Forest, P. Chaumont, P. Cassagnau, B. Swoboda, P. Sonntag, Nanofoaming of PMMA using a batch CO2 process: Influence of the PMMA viscoelastic behaviour, Polymer (Guildf). 77 (2015) 1–9. https://doi.org/10.1016/j.polymer.2015.09.011.

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[131] V. Bernardo, J. Martin-de Leon, J. Pinto, T. Catelani, A. Athanassiou, M.A. Rodriguez-Perez, Lowdensity PMMA/MAM nanocellular polymers using low MAM contents: Production and characterization, Polymer (Guildf). 163 (2019) 115–124. https://doi.org/10.1016/j.polymer.2018.12.057. [132] V. Bernardo, J. Martin-de Leon, E. Laguna-Gutierrez, T. Catelani, J. Pinto, A. Athanassiou, M.A. Rodriguez-Perez, Understanding the role of MAM molecular weight in the production of PMMA/ MAM nanocellular polymers, Polymer (Guildf). 153 (2018) 262–270. https://doi.org/10.1016/j.poly mer.2018.08.022. [133] V. Bernardo, J. Martin-de Leon, M.A. Rodriguez‐Perez, Nanocellular polymers, in: S.T. Lee (Ed.)., Polymeric Foams: Innovation in Technologies and Environmentally Friendly Materials, CRC Press, Boca Raton, FL, (2022) 275. [134] V. Kumar Thakur, M. Kumari Thakur, M.R. Kessler, Handbook of Composites from Renewable Materials, Scrivener Publishing LLC, Beverly, MA, EEUU, (1985). [135] J. Pinto, D. Morselli, V. Bernardo, B. Notario, D. Fragouli, M.A. Rodriguez-Perez, A. Athanassiou, Nanoporous PMMA foams with templated pore size obtained by localized in situ synthesis of nanoparticles and CO 2 foaming, Polymer (Guildf). 124 (2017) 176–185. https://doi.org/10.1016/j.poly mer.2017.07.067. [136] J. Pinto, M. Dumon, M.A. Rodriguez-Perez, Nanoporous polymer foams from nanostructured polymer blends: Preparation, characterization, and properties, in: P.M. Visakh, G. Markovic, D. Pasquini (Eds.), Recent Developments in Polymer Macro, Micro and Nano Blends, Woodhead Publishing in Materials, Sawston, Cambridge, (2016) 237–288. [137] J. Martín-de León, V. Bernardo, M. Rodríguez-Pérez, Nanocellular polymers: The challenge of creating cells in the nanoscale, Materials 12 (2019) 797. https://doi.org/10.3390/ma12050797. [138] F. Van Loock, V. Bernardo, M. Angel, R. Pérez, N.A. Fleck, The mechanics of solid-state nanofoaming Subject Areas : Author for correspondence , Proc. Royal Soc. A 475 (2019) 20190039. [139] R. Oberhoffer, A. Müller, Production of porous materials by the expansion of polymer gels, 2 (2014) WO2015071463A2. [140] J. Martín-De León, V. Bernardo, M.Á. Rodriguez-Perez, Cyclic gas dissolution foaming as an approach for simultaneously reducing cell size and relative density in nanocellular PMMA, Polymers (Basel) 13 (2021) 2383. https://doi.org/10.3390/polym13142383 (accessed July 20, 2021). [141] I. Sánchez-Calderón, V. Bernardo, J. Martín-de-león, M.Á. Rodríguez-Pérez, Thermal conductivity of low-density micro-and nanocellular poly(methyl-methacrylate) (PMMA): Experimental and modeling, Mater. Des. 221 (2022). https://doi.org/10.1016/j.matdes.2022.110938 (accessed September 2022).

Chapter 4 Optical properties 4.1 Introduction The optical properties of cellular polymers produced by foaming methods have gone unnoticed for many years for not having any special interest. Cellular polymers are simply opaque materials due to the strong scattering of light in the cells of the material. That was actually true until 2004 when Yokoyama et al. published the first clues of a transparent nanocellular polymeric film. They produced thin films around 100 µm in thickness with 10–30 nm of cell size by using polystyrene-block-poly (perfluorooctyl ethyl methacrylate) (PS-PFMA) [1], and they claimed their transparency but without any experimental measure proving transparency. It was not until 2017 that Martín-de León et al. published the first experimental evidence of a thick (thickness higher than 0.5 mm) semitransparent nanocellular polymer based on polymethylmethacrylate (PMMA) (Figure 4.1) [2].

Figure 4.1: Transparency of samples produced at four different saturation pressures, with cell sizes of 225 nm, 39 nm, 24 nm, and 14 nm and the same relative density. From reference [2].

In this paper, it was proved that when the cell size reduces below 50 nm the nanocellular material preserves some of the transparency of the solid precursor. As discussed before, this possibility was previously hypothesized from a theoretical perspective, and some experimental results of light transmission and refractive index supported this hypothesis [1, 3]. Yet, it was not until 2017 that it was proved that transparent nanocellular polymers with significant sizes could be a real possibility. Adding this feature to a foam’s properties can be revolutionary for many applications, in particular for the thermal insulation market. This new characteristic would allow the development of low-weight, insulating, transparent material perfect for the design of insulating windows. This property is also interesting for electronic devices, the medical sector, or applications not still imagined. Obtaining transparency in nanocellular materials requires a very specific structure coming from a complex tuning of the production process of nanocellular polymers.

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Chapter 4 Optical properties

In this chapter transparency in nanocellular polymers will be studied from a theoretical and an experimental point of view, showing the last advances within this topic, as well as a comparison with other transparent nanoporous materials and current limitations and challenges.

4.2 Light interaction with porous structures The transmittance of light through a material ðT Þ is simply defined by the amount of light that passes through the material ðI Þ in comparison with the initial intensity of the source ðI0 Þ (eq. (4.1)). T=

I I0

(4:1)

The light not transmitted across the materials is because it has been either absorbed or scattered. Absorption of light consists of the capture of electromagnetic radiation by the matter, converting the energy carried by the photons into internal energy [4], while scattering is the process where the light is deviated from its initial trajectory due to an interaction with an object similar in size to its wavelength [5]. Therefore, in a porous media, the light can be transmitted, absorbed either by the matrix or the pores, or scattered by the pores acting as scattering centers (Figure 4.2). In a conventional cellular polymer, the size of the cells causes a great scattering of visible light in all directions. As a result, conventional cellular polymers are opaque (and usually white when the initial solid is transparent and there are no colored additives). However, in nanocellular polymers, these mechanisms change, as it will be explained in the coming paragraphs.

Figure 4.2: Scheme of transmission, absorption, and scattering of light in a porous structure.

4.2 Light interaction with porous structures

87

By assuming a uniform attenuation through the media, the transmission of light can be described through the Beer–Lambert law (eq. (4.2)): T = e−μl

(4:2)

where μ is the extinction coefficient that can be expressed as the sum of the absorption coefficient ðμabs Þ and the scattering coefficient ðμscat Þ (eq. (4.3)) [6]. μ = μabs + μscat

(4:3)

In the particular case of cellular polymers, transmittance will be determined by the absorption of light by the polymer phase and the light scattered by the gaseous phase. For transparent amorphous polymers such as polycarbonate or polymethylmethacrylate, the absorption coefficient can be considered negligible [7]. Thus, in this case, the extinction coefficient will be only a function of the scattering of light by the pores of the cellular material. In order to understand the scattering of light by the pores, the simpler case can be first studied, which means the scattering of light by a single spherical bubble (same that the scattering by a single particle of size equal to the cell size and refractive index equal to that of air). The scattering mechanisms depend on the relationship between the wavelength and the bubble size, which can be characterized by the dimensionless parameter x (eq. (4.4)). x=

2πr λ

(4:4)

where r is the bubble radius and λ is the relative scattering wavelength, defined as λ = λ0 =m0 , being λ0 the incident wavelength and m0 the refractive index of the surrounding medium. For a polymer with a refractive index similar to that of air, it can be assumed that λ = λ0 . Visible light wavelength ranges from 380 to 780 nm, from purple to red. When dealing with conventional or microcellular polymers with cell sizes higher than a few microns or even in the range of the micron x is higher than one and Mie scattering takes place. However, when cell sizes are very small (tens of nanometers) x .

Worse than − the solid. Fracture strength increases with cell size reduction

−

−

Better as cell size reduces

PP []

– .–.

Worse than − the solid. Fracture strength increases with cell size reduction

−

–

Better as cell size reduces

PC []

–

< .

Not improved

−

−

−

No influence

PLA []

✶

> ✶

> ✶

Saturation pressure (MPa)

~ 



–

Saturation temperature (ºC)

~ 



−–

Foaming temperature (ºC)

–



–

Foaming time (min)

–



–

Density (kg/m)

–

–

–

Relative density

.–.

.–.

.–.

Cell size (nm)

–

–

–

✶

For PMMA with Mw = 83 kg/mol, for different Mw solubility values can slightly change.

As seen in Table 9.1, the first type of material can be produced by achieving solubilities higher than 30% of CO2. When no nucleating agents are included (homogeneous nucleation), this solubility value has been obtained in the literature by using high pressures (31 MPa) and room https://doi.org/10.1515/9783110756135-009

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Chapter 9 Applications of nanocellular polymers and future trends

temperature. When nucleating agents are introduced in the formulation it is possible to fabricate the same type of material using much lower pressures (6 MPa) thanks to the appearance of the heterogeneous nucleation mechanism. With these saturation conditions low-density nanocellular polymers can be produced (relative densities around 0.13–0.2, porosities over 80%) with cell sizes of 300–500 nm. Examples of cellular structures are presented in Figure 9.1.

Figure 9.1: Examples of cellular structures of Type I materials with different cell sizes, from references [1] (a) and [2] (b).

To produce the second type of nanocellular polymers solubilities above 39% of CO2 are needed. Such high values of gas absorbed have been obtained in the literature by using different combinations of production parameters, using medium/low pressures (6 to 20 MPa) and very low temperatures (−15 to −32 °C) or using high saturation pressures (40 to 50 MPa) and a saturation temperature of 0 °C. Such saturation conditions allow increasing significantly the amount of gas absorbed and thus the nucleation, resulting in medium-density nanocellular polymers (relative densities around 0.40, porosity of 60%) with cell sizes as low as 14 nm. The processing conditions and typical structures are collected in Table 9.1 and example of the cellular structures is presented in Figure 9.2.

9.2 Properties of nanocellular polymers: Comparison with other materials The two types of nanocellular polymers present different properties. Nanocellular polymers of Type I are interesting to be used as thermal insulators. When the cell size is reduced to values comparable to the mean free path of the gas molecules, Knudsen effect appears and the thermal conduction through the gas phase is drastically reduced (see Chapter 5 for more details on the thermal conductivity). This effect starts

9.2 Properties of nanocellular polymers: Comparison with other materials

197

Figure 9.2: Example of cellular structure of Type II materials, from references [3].

to be important when the cell size decreases under the micron, and has a significant impact for cell sizes below 500 nm. For instance, the conductivity of air reduces from 25 to 17 mW/mK for cell sizes of 500 nm. As a consequence of this reduction, these materials can present the same thermal conductivity of an XPS foam (conventional thermal insulator) with a much higher density (see Figure 9.3). This means that we can have the same thermal insulation performance with a much higher compressive strength, 5 times higher. Then, nanocellular polymers of Type I can be used as effective thermal insulators when high mechanical properties are required. For conventional foams and nanocellular polymers of Type I, the cell size is in the range of the visible light wavelength or higher (450 to 750 nm), so the amount of scattered light is so high that the material is opaque independently on the used polymer matrix (see Chapter 4 for more details on the optical properties). Conversely, when the cell size becomes smaller than a tenth of the visible light wavelength, the total amount of scattered light becomes negligible. That means visible light can pass through cells with minimum deviation, and the material becomes optically transparent if the polymer matrix does not absorb light (Type II). Type II materials show an interesting combination of properties, because they can keep the transparent behavior of solid PMMA with a significant density reduction and an improvement in the thermal insulation performance, as shown in Figure 9.3. While the transmittance is reduced roughly to half the value of the pure polymer, the thermal conductivity is reduced in a factor of 3.5. When the properties are compared to those of glass, the improvement in the thermal insulation is even more significant: in this case, the reduction is more than 11 times. Then, Type II materials can be used as a new transparent thermal insulator for advanced applications. In addition, the elastic modulus, compressive strength and tensile properties are excellent in comparison with any type of aerogel (the most common example of transparent insulating materials). The main conclusion of Figure 9.3 is that nanocellular polymers have a very rare combination of properties that make them different to conventional polymer foams (as

198

Chapter 9 Applications of nanocellular polymers and future trends

Figure 9.3: Comparison of the properties of nanocellular polymers (Type I and II) vs a conventional thermal insulator (XPS) and an advanced transparent material (silica aerogel [4]).

extruded polystyrene included in Figure 9.3 as an example of conventional cellular polymer) or aerogels (as silica aerogels included in Figure 9.3 as an example of aerogel).

9.3 Future trends As already discussed in Chapter 8 and mentioned in the previous section, nanocellular polymers present a unique combination of properties, as schematically presented in Figure 9.4. It is not each of the isolated properties that makes nanocellular polymers interesting, but rather the combination of all of them. Nanocellular polymers are not the best thermal insulator, neither the most transparent material, but it is a material that is at the same time transparent, thermal insulator, with a high surface area and good mechanical performance among others. In other words, we have in our hands a multifunctional material with a combination of unique properties that have already been thought about taking advantage of, but that also has great potential for the future. One of the applications that have been already mentioned in the literature is the use of these materials in windows. Their transparent behavior together with their low thermal conductivity, excellent mechanical properties and acoustic absorption

9.3 Future trends

199

Figure 9.4: Schematic view of the properties of nanocellular polymers.

can result in better performance windows than the current ones for their use in different sectors such as the building sector or the automotive sector. In several applications there is a need of excellent thermal insulating materials with high stiffness and strength (for instance the insulation of petrol pipes under see water at high depths). Nanocellular polymer of Type I look like a very promising solution for these insulation challenges. Ultrathin polymeric films, that means polymeric films with nanoscale thickness, have lately found a wide niche of applications including advanced medical and health-care applications, microelectronics, membranes or sensors. In medical applications polymeric nanosheets with a thickness of tens to hundreds of nanometers are used for example for wound dressing, tissue engineering and bioelectronic devices [5–9]. In the case of microelectronics polymeric films have been proposed as a possibility to obtain low-k-dielectric materials, in which the introduction of small porosity contributes to even lower dielectric constants. Membranes and sensors fields take advantage of the thin thickness of these structures also looking for high surface area materials in which porosity plays a key role.

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Chapter 9 Applications of nanocellular polymers and future trends

Drawbacks for using thin films are the same for all the mentioned applications, the mechanical integrity of ultrathin polymeric films is low. In addition, their require complex and expensive production methods (involving the use of organic solvents) that are, in addition, very difficult to scale. In this case the reinforcement in the mechanical properties of nanocellular polymers can be used together with their high surface area, and their open structure porosity to revolutionize the already mentioned fields. On the other hand, polymers with pores of tens of nanometers have been claimed to be a potential solution for global problems such as climate change. In this case these materials have been proposed for more energy efficient separation process and already proven successful for more effective selective absorption, low-energy transformations through catalysis process such CO2 conversions into solar fuels or water splitting. Materials for these applications are synthesized through phase separation approaches, bottomup molecular design, and techniques that as explained in Chapter 3 are characterized for being complex, expensive, non-environmentally friendly and hard to scale up. Again, combine the unique properties of nanocellular polymers makes them the perfect candidate for a better future [10–13]. Digging into the literature regarding new materials one realizes that nanocellular polymers have the potential to be key to future advances in fields such as medicine, construction, the automotive industry, logistics, aeronautics, energy, removable energies, and, after all, our progress as humanity.

References [1]

[2]

[3]

[4]

[5] [6]

[7]

I. Sánchez-Calderón, V. Bernardo, J. Martín-de-León, M.Á. Rodríguez-Pérez, Thermal conductivity of low-density micro-and nanocellular poly (methyl-methacrylate) (PMMA): Experimental and modeling Mater. Des. 221 (2022) 110938. https://doi.org/10.1016/j.matdes.2022.110938 V. Bernardo, J. Martín-de León, M.Á. Rodríguez-Pérez, Production of PMMA-based nanocellular polymers using low demanding saturation conditions Mater. Lett. (2019) https://doi.org/10.1016/j. matlet.2019.126551 J. Martin-de Leon, V. Bernardo, M.A. Rodriguez-Perez, Key production parameters to obtain transparent nanocellular PMMA Macromol. Mater. Eng. (2017) 1700343(1)–1700343(5). https://doi. org/10.1002/mame.201700343 J.C.H. Wong, H. Kaymak, S. Brunner, M.M. Koebel, Mechanical properties of monolithic silica aerogels made from polyethoxydisiloxanes, Microporous Mesoporous Mater. 183 (2014) 23–29. https://doi.org/10.1016/j.micromeso.2013.08.029 T. Fujie, Development of free-standing polymer nanosheets for advanced medical and health-care applications, Polym. J. 48 (2016) 773–780. https://doi.org/10.1038/pj.2016.38 K. Maex, M.R. Baklanov, D. Shamiryan, F. Iacopi, S.H. Brongersma, Z.S. Yanovitskaya, Low dielectric constant materials for microelectronics, J. Appl. Phys. 93 (2003) 8793–8841. https://doi.org/10.1063/ 1.1567460 U. Okoroanyanwu, Thin film instabilities and implications for ultrathin resist processes, J. Vac. Sci. Technol. B Microelectron. Nanom. Struct. 18 (2000) 3381. https://doi.org/10.1116/1.1321291

References

[8]

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P.H. Pfromm, W.J. Koros, Accelerated physical ageing of thin glassy polymer films: Evidence from gas transport measurements, Polymer (Guildf). 36 (1995) 2379–2387. https://doi.org/10.1016/00323861(95)97336-E [9] C.J. Ellison, J.M. Torkelson, Sensing the glass transition in thin and ultrathin polymer films via fluorescence probes and labels, J. Polym. Sci. Part B Polym. Phys. 40 (2002) 2745–2758. https://doi. org/10.1002/polb.10343 [10] D.S. Sholl, R.P. Lively, Seven chemical separations to change the world, Nature, 532 (n.d.) 435–437. [11] J. Wu, F. Xu, S. Li, P. Ma, X. Zhang, Q. Liu, R. Fu, D. Wu, Porous polymers as multifunctional material platforms toward task-specific applications, Adv. Mater. 31 (2019) https://doi.org/10.1002/adma. 201802922 [12] Y. Xiong, R.T. Woodward, D. Danaci, A. Evans, T. Tian, H. Azzan, M. Ardakani, C. Petit, Understanding trade-offs in adsorption capacity, selectivity and kinetics for propylene/propane separation using composites of activated carbon and hypercrosslinked polymer, Chem. Eng. J. 426 (2021) https://doi. org/10.1016/j.cej.2021.131628 [13] Y. Chen, S. Yao, Y. Ling, W. Zhong, D. Hu, L. Zhao, Microcellular PETs with high expansion ratio produced by supercritical CO2 molding compression foaming process and their mechanical properties, Adv. Eng. Mater. 24 (2022) https://doi.org/10.1002/adem.202101124

Index 3D confinement 160 50 nm 90 absorption 118 absorption coefficient 182 aerogels 99, 106, 183 amorphous 157 Amplitude Modulation Frequency Modulation 158 anisotropy ratio 14 average cell size 11 Beer–Lambert law 87 bicontinuous 167 bimodal 12, 115 brittle to ductile behavior 147 carbon dioxide 51 cell density 12, 55 cell growth 26 cell nucleation density 12 cell size 55, 114, 164, 181 cell size distribution 11, 115 cell wall thickness 20, 158, 167 cell walls 18, 119 cellular polymer 2 cellular polymers 45 classical nucleation theory 56 coalescence 30, 166 coarsening 31 co-continuous 167 compression molding 45 compression properties 146 compressive strength 151, 197 conduction through the gas phase 109 conduction through the solid phase 109 confinement 157 confinement effect 33 convection 109 conventional cellular polymers 3 critical radius 57 crystallization 152 degeneration 26 degeneration mechanisms 54 density 164

https://doi.org/10.1515/9783110756135-010

depressurization 52 desorption 52 dielectric constant 186 diffusivity 52 drainage 31 effective glass transition temperature 52 elastic collapse stress 146 elastic modulus 197 electromagnetic interference 188 EMI shielding 191 end-to-end distance 159 expandable beads 45 expansion ratio 10, 166 extinction coefficient 87, 98, 119 extruded polypropylene 45 extruded polystyrene 45 extrusion foaming 45 Fick diffusion law 52 foamed polyethylene 45 foamed polystyrene 45 foamed polyurethane 45 foaming 52 fraction of mass in the struts 19 free expansion phase 56 free foaming 45 gas dissolution foaming 45, 165 gas phase 4 Gibson and Ashby 143 glass transition temperature 52, 158 gyration radius 159 health-care applications 199 heat flow meter 131 heat transfer coefficient 106 Henry’s law 52 heterogeneous nucleation 28, 55, 165 homogeneous core 170 homogeneous nucleation 28, 55, 195 impact 146 impact strength 146 Influence Volume Approach 56 infrared radiation 118

204

Index

injection molding 45 IR-blockers 122 Izod impact strength 146

pore size 165, 186 porosity 10, 55, 115, 181 pressure 114 pressure vessel 45

Knudsen effect 33, 113, 196 limited expansion phase 56 loss tangent 159 low thermal conductivity 195 low–density nanocellular polymer 195 Massachusetts Institute of Technology 46 Maxwell equations 94 mean free path 110 mechanical performance 143 mechanical properties 16, 46, 112 mechanical property 143 medium density material 195 membranes 165, 199 microcellular polymers 4, 12 microelectronics 199 Microgreen 47 micronization 135 Mie scattering 87 molecular orientation 152 molecule diameter 114 Mucell 47 multifunctional material 198 multifunctional materials 192 nanofillers 166 nanoparticles 1 nanoscale 1 nanoscience 1 nanostructuration 169 nanotechnology 1 normalized standard deviation coefficient 11 nucleation 26, 54 one-step process 55 opacifiers 122 open cell 115, 164 open cell content 17, 73, 166, 181 phase separation 53 phase separation techniques 49 phonons 110 polymers 1 polypropylene molded foam 45

radiation 109 Rayleigh scattering 90 reactive foaming 45 relative density 8, 109, 187 saturation 52 saturation pressure 52 saturation temperature 52 saturation time 52 scattering 87, 118 semi-continuous process 47 sensors 199 separation process 200 size effect 1 solid phase 4 solid skin 22, 54, 93, 165 solid structure factor 111 solubility 52, 195 solubility limit 52 sound absorption 181 sound speed 112 spinodal decomposition 54, 166 stabilization 26, 52 steady-state techniques 126 structural gradient 23 structural gradients 170 struts 18, 110 super-insulators 106 surface area 33, 164 templating of imprinting techniques 50 tensile 146 tensile properties 197 theoretical models 119 thermal conductivity 17 thermal insulation 9 thickness 98 thin films 167 tortuosity of the gas phase 18, 166, 181 tortuosity of the solid phase 24, 187 transient plane source 127 transient techniques 108 transition region 23, 93, 170 transmission loss 183

Index

transmittance 86, 119 transparency 195 transparent 33 transparent insulating materials 197 transparent nanocellular PMMA 91 transparent nanocellular polymers 85 Trexel 47

VIPs 106 viscoelastic properties 166

vacuum 114

Zotefoams Plc 46

wavelength 87 windows 198 Young’s modulus 144

205