A Thermodynamic Investigation of the PVT, Solubility and ... · A Thermodynamic Investigation of the PVT, Solubility, and Surface Tension of Polylactic Acid (PLA)/CO 2 Mixtures Syed

A Thermodynamic Investigation of the PVT, Solubility and Surface Tension of Polylactic Acid (PLA)/CO2

Mixtures

by

Syed Hassan Mahmood

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy,

Graduate Department of Mechanical & Industrial Engineering, University of Toronto

Copyright by Syed Hassan Mahmood 2012

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A Thermodynamic Investigation of the PVT, Solubility, and Surface Tension of Polylactic Acid (PLA)/CO2 Mixtures

Syed Hassan Mahmood

Degree of Master of Science, 2012

Department of Mechanical and Industrial Engineering

University of Toronto

ABSTRACT

This thesis illustrates a comprehensive study on the PVT, solubility and surface tension

properties of polylactic acid (PLA) with dissolved CO2 based on thermodynamic models. The

solubility of CO2 in PLA melt was calculated by means of a gravimetric method, using a

Magnetic Suspension Balance (MSB). The swelling volume of the polymer/gas mixture due to

dissolution of gas was compensated for by direct measurement through a view cell or by

theoretical models such as Simha Somcynsky (SS) - Equation of State (EOS) and Sanchez

Lacombe (SL) - Equation of State (EOS). Three grades of PLA (i.e., PLA3001D, PLA8051D,

and PLA4060D) were chosen. It was observed that the pressure, temperature, D-content and

Molecular weight variance had an effect on the swelling and solubility.

The surface Tension of PLA/CO₂ mixture was also calculated from the captured image

using the Axsymmetric Drop Shape Analysis (ADSA). The effects of varying the pressure,

temperature, and molecular weight on surface tension were investigated.

iii

To the one above for providing me

with an opportunity

To my parents For endless love, support

and encouragement

iv

ACKNOWLEDGEMENTS

I would like to acknowledge the support of my supervisor Prof. Chul B. Park, whose

encouragement and support was second only to that of my parents, without which this research

would not have been possible.

I would also like to thank the members of my Thesis Committee, Prof. Sanjeev

Chandra and Prof. Kamran Behdinan for their invaluable advice and guidance.

My gratitude is extended to the Department of Mechanical and Industrial Engineering

at the University of Toronto for providing the University of Toronto Fellowship. Also, I would

like to thank the members of the Consortium for Cellular and Micro-Cellular Plastics

(CCMCP) for their funding and support in this research.

I would like to thank all my colleagues and fellow researchers formerly and presently

working in the Microcellular Plastics Manufacturing Laboratory for their friendship,

cooperation and support. I am especially thankful to Dr. Gary Li, Dr. Takashi Kuboki, Dr. Nan

Chen, Raymond Chu, Peter Jung, Anson Wong, Lun Howe Mark, Nemat Hossiney, Reza

Nofar and Mohammad M. Hassan for their collaboration on our common research publications

and for the thought-provoking discussions we had during the work. I also want to acknowledge

the contribution of Mohammad M. Hassan and Dr. Guangming Li in the theoretical and

experimental work.

I would also like to thank my parents and all my family members for their endless love,

support and encouragement. Finally, I would like to thank God for providing me with an

opportunity and the patience to grasp that opportunity.

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TABLE OF CONTENTS

Abstract ........................................................................................................................................ii

Acknowledgements ..................................................................................................................... iv

List of Figures .......................................................................................................................... viii

Nomenclature ............................................................................................................................... x

CHAPTER 1. INTRODUCTION .......................................................................................... 1

1.1 Preamble ............................................................................................................................. 1

1.2 Foam Processing ................................................................................................................. 3

1.2.1 Microcellular Processing ........................................................................................... 3

1.2.2 Physical Blowing Agents........................................................................................... 4

1.2.3 Thesis Objectives and Scope of Research ................................................................. 6

1.2.4 Thesis Structure ......................................................................................................... 7

CHAPTER 2. LITERATURE SURVEY AND THEORETICAL BACKGROUND ........... 8

2.1 Solubility ............................................................................................................................. 8

2.2 Solubility Measurement Methods ..................................................................................... 10

2.2.1 Gravimetric Sorption Technique ............................................................................. 11

2.2.2 Pressure Decay Sorption Technique ........................................................................ 13

2.2.2.1 Previous Research Using Pressure Decay Sorption Technique ....................... 15

2.2.3 Volume Decay Sorption Technique ........................................................................ 16

2.2.4 Piezoelectric Quartz Sorption .................................................................................. 17

2.2.5 In-Line Measurement of Gas Solubility .................................................................. 19

2.2.5.1 In-Line Monitoring ........................................................................................... 19

2.2.5.2 In-Line infrared Sensors ................................................................................... 20

2.2.5.3 Ultrasound ........................................................................................................ 21

2.2.6 Modified Magnetic Suspension Balance Theoretical Treatments ........................... 22

2.3 Theoretical Treatments ..................................................................................................... 26

CHAPTER 3. RESEARCH METHODOLOGY FOR PVT AND SOLUBILITY STUDY 29

3.1 Introduction ....................................................................................................................... 29

3.2 Theoretical Background .................................................................................................... 29

3.3 Methodology and Approach ............................................................................................. 32

3.4 Summary ........................................................................................................................... 34

CHAPTER 4. SOLUBILITY AND SWELLING BEHAVIOR OF PLA IN PRESENCE

OF CO₂………………………………………………………………………………………...35

vi

4.1 Introduction ....................................................................................................................... 35

4.2 Experimental ..................................................................................................................... 36

4.2.1 Materials .................................................................................................................. 36

4.2.2 PVT Data for PLA ................................................................................................... 36

4.3 Solubility of CO₂ in PLA (Binary System)....................................................................... 37

4.3.1 Swelling Behavior of PLA in Presence of CO2 ....................................................... 37

4.3.1.1 Experimental Setup .......................................................................................... 37

4.3.1.2 Experimentation ............................................................................................... 37

4.3.1.3 Pressure and Temperature Effect on Volume Swelling ................................... 38

4.3.1.4 Comparison of the experimentally measured data and Theoretically Predicted

Data ………………………………………………………………………………..40

4.3.1.5 Effect of ‘D’ content/Molecular Weight on volume swelling .......................... 44

4.3.2 Solubility of CO2 in PLA......................................................................................... 45

4.3.2.1 Pressure and Temperature effect on solubility of CO₂ in PLA. ....................... 45

4.3.2.2 Comparison of Theoretical and Experimental Solubility ................................. 48

4.3.2.3 Effect of D-content/Molecular weight on Solubility of CO₂ ........................... 51

4.4 Summary ........................................................................................................................... 53

CHAPTER 5. Surface Tension OF PLA/CO₂ MELT ......................................................... 54

5.1 Introduction ....................................................................................................................... 54

5.2 Experimental Materials ..................................................................................................... 57

5.3 Measurement of Surface Tension of PLA/CO₂ mixture ................................................... 57

5.3.1 Surface Tension of PLA/CO₂ melt .......................................................................... 57

5.3.2 Density determination.............................................................................................. 59

5.3.3 Effects of pressure and temperature variance on surface tension ............................ 62

5.3.4 Effect of Molecular weight on surface tension ........................................................ 65

5.4 Summary ........................................................................................................................... 66

CHAPTER 6. CONCLUSIONS AND FUTURE WORK ................................................... 68

6.1 Summary ........................................................................................................................... 68

6.2 Recommendations and Future Work ................................................................................ 69

References .................................................................................................................................. 70

Appendix .................................................................................................................................... 86

vii

LIST OF TABLES

Table 1: The scaling Parameters for SS-EOS are as following ................................................. 96

Table 2: The scaling Parameters for SL-EOS are as following ................................................. 96

Table 3: Solubility of PLA3001D at 180 °C .............................................................................. 96






Table 9: Swelling ratio of PLA3001D/CO₂ melt at 180 °C ....................................................... 98

Table 10: Swelling ratio of PLA8051D/CO₂ melt at 180 °C ..................................................... 99




Table 14: Swelling ratio of PLA4060D/CO₂ melt at 200 °C ................................................... 100

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LIST OF FIGURES

Figure 2-1 Bubble nucleation inside a die ................................................................................... 9

Figure 2-2: Bubble nucleation inside a mold in injection molding machine: (a) nucleation starts

at gate (generates high cell density); (b) nucleation starts in the mold cavity (generates low cell

density). [14] .............................................................................................................................. 10

Figure 2-3 Details of MSB for density measurement [65] ........................................................ 23

Figure 4-1: Effect of Temperature variance on Swelling Ratio for PLA3001D ........................ 39



Figure 4-4: Swelling Ratio of PLA3001D at 180 °C with varying pressure ............................. 41






Figure 4-10: Effect of varying D-content/Mw on swelling ratio at 180 °C ............................... 44

Figure 4-11: Effect of varying D-content/Mw on swelling ratio at 200 °C ............................... 45

Figure 4-12: Effect of varying temperature on solubility of PLA3001D .................................. 46



Figure 4-15: Solubility of Carbon Dioxide in PLA3001D at 180 °C with varying pressure .... 48



ix




Figure 4-21: Effect of varying D-content/Mw on Solubility of Carbon Dioxide at 180 °C ...... 52

Figure 4-22: Effect of varying D-content/Mw on Solubility of Carbon Dioxide at 200 °C ...... 52

Figure 5-1: Density difference with pressure change at 180 °C ................................................ 60

Figure 5-2: Density difference with pressure change at 200 °C ................................................ 61

Figure 5-3: Change in surface tension with pressure at 180 °C ................................................. 63

Figure 5-4: Surface tension at various temperature and pressure .............................................. 64

Figure 5-5: Relationship between surface Tension and Solubility ............................................ 65

Figure 5-6: Effect of molecular weight on Surface Tension ...................................................... 66

x

NOMENCLATURE

Ai = Helmholtz energy [J]

CS = Costas and Sanctuary

ci = Chain (molecule) flexibility; 3c is total external degrees o f freedom attributed to

a chain (molecule)

d = Bond length in PCM EOS

EOS = Equation of state

= Molar Gibbs free energy for polymer/gas mixture [J/mol]

H = Enthalpy [J]

h = Plancks constant 6.6260755×10-34

[J.s]

, Bk k = Boltzmann constant 1.380658×10

-23 [J/K]

im = Molar mass of segment (mer) of “i” component [g/mol]

M = Molecular weight of per molecule [g/mol], M = misi

= Avogadro’s number 6.0221367×1023

= Pressure [Pa]

= Characteristic pressure of component “i” [Pa],

** i

i *

i

qzP

sυ

= Reduced pressure */ PP

PCM = Prigogine cell model

PC-

SAFT

= Perturbed Chain Statistical Associating Fluid Theory

= The number of nearest neighbor sites per chain-like molecule (s-mer), si(z - 2)+2.

= Dimensionless identity for SS-EOS

= The combinatorial factor for PCM

r = Number of mer per molecule, **

*

ρRT

MPr

= Gas constant 8.3143 [J/(mol·K)]

S = Entropy [J/K]

xi

SAFT = Statistical Associating Fluid Theory

= Number of mers per molecule of component “i”

SL = Sanchez–Lacombe

SS = Simha–Somcynsky

= Volume swelling ratio

T = Temperature [K]

*

iT = Characteristic temperature of component “i” [K], ii

B

qzT

ck

= Reduced temperature */TT

*

iυ = Characteristic volume per mer of component “i” [m3/mer]

*

iV = Characteristic volume of component “i” (m3/mol)

= Reduced volume,

ix = Mole fraction of “i” component in mixture system

X = Solubility (g-gas/g-polymer)

y = Occupied lattice site fraction

Z = 12, the lattice coordination number

i * = Characteristic energy per mer of component “i” (J/mer)

= dimensionless number for SS-EOS

1 = Volume fraction of gas in mixture system

2 = Volume fraction of polymer in mixture system

G

1 = Chemical potential of gas in vapor phase [J/mol]

P

1 = Chemical potential gas in the polymer melt [J/mol]

* = Characteristic density of bulk material [g/cm3]

Λ

= de Broglie wavelength

= Flexibility parameter

= Lattice-site volume

Γ = PCM geometrical constant

xii

~ = Reduced density V

~/1

Subscripts

1 = Indicates gas

2 = Indicates polymer

i = Indicates component number or component in x-direction

j = Indicates component number in y-direction

F = Indicates foam

G, g = Indicates gas

Superscripts

G, g = Indicates gas or vapor

P, p = Indicates polymer

1

CHAPTER 1. INTRODUCTION

1.1 Preamble

Foamed plastics surround us in the shape of different commodities every day. This is a

testament to the extent that foamed polymeric materials have pervaded our everyday life.

Foamed polymeric material can be classified as either a thermoplastic or thermoset foam.

Thermoplastic foams are recyclable, whereas thermoset foams cannot be reprocessed because

it’s extensive crosslinking. The applications of the foamed product dictate the density of the

final product; consequently, several types of polymeric materials are foamed to various low

densities for applications that require attributes such as weight reduction, insulation, buoyancy,

energy dissipation, convenience, and comfort.

Thermoplastic and thermoset foam products differ from solid plastic products and are

identifiable by virtue of a unique cellular structure. The latter is obtained by the atmospheric

expansion of a blowing agent in the polymer matrix. This cellular structure is normally

characterized in terms of the cell density and cell size. For conventional foams, typical cell

population densities are within the range of 10³-10⁶ cells/cm³, with cell sizes of the order of

100 μm or larger. However, the cell size and distribution can be inconsistent, hence

compromising the mechanical properties of the foam. Conversely, microcellular plastics are

characterized by cell densities greater than 10⁹ cells/cm³ and cell sizes smaller than 10 μm.

This group of plastic foams was conceived by Masrtini et al. [1], and is based on the notion

that if a large number of bubbles smaller than the pre-existing flaws in the polymer are created,

the material cost can be reduced without compromising mechanical properties. It has been

2

observed that microcellular plastics possess superior impact strength [1], toughness [2], and

fatigue life [3] compared to solid polymers.

Furthermore, solid polymeric foams may possess either closed or open cells; closed

cell foams possess a cellular structure in which neighboring air bubbles are entrapped in a

continuous macromolecular phase. Open cell foams, on the other hand, have a cellular network

in which continuous channels are available throughout the solid macromolecular phase.

The gaseous phase in any polymeric foam material is obtained using blowing agents in the

foam manufacturing process. There are generally two types of blowing agents used in foam

production: chemical blowing agents and physical blowing agents. Chemical blowing agents

are chemical compounds which evolve gases under foam processing conditions through

thermal degradation or chemical reactions. Physical blowing agents, on the other hand, are

inert gases, such as nitrogen and carbon dioxide; volatile hydrocarbons such as propane, n-

butane, i-pentane; and low boiling point chlorofluorocarbons (CFCs), hydrofluorocarbons

(HFCs), and hydrochlorofluoro-carbons (HCFCs).

Due to the environmental hazard posed by CFCs and HCFCs, there has been a drive to

replace these blowing agents with more environmentally friendly substitutes. The properties of

blowing agents and their impact on the processing variables in foam production are essential

issues that must be considered in order to find effective replacements. The specific

thermodynamic and kinetic properties that have the greatest influence on the ability to produce

microcellular foams are the solubility and diffusivity of the blowing agent in the polymers.

The solubility determines the amount of blowing agent that can be absorbed by the polymer at

any given temperature and pressure, while the diffusivity determines both the rate at which the

blowing agent will penetrate into the polymer matrix to form a homogeneous solution, as well

3

as the rate at which the blowing agent escapes to the atmosphere during cell nucleation and

growth processes.

1.2 Foam Processing

1.2.1 Microcellular Processing

The foamed plastics industry is continuously developing, which encompasses various

methods of production for many product applications. The choice of polymer, blowing agent,

and production method dictates the foam formation and its morphology; therefore, its

properties, stemming from an industrial challenge to enhance polymer value by improving

performance/weight characteristics. Earlier work focused on batch processes; the concepts

resulted in good foam structures with cell sizes less than 5 μm using supercritical carbon

dioxide [1]. The batch process was further developed and improved by Cha et al. [4]; whereas

Kumar [5] and Kumar and Suh [6, 7] developed a semi-continuous process derived from a

modified thermoforming process. Further investigation by Park [8], Park et al. [9], and Park

and Suh [10] led to the development of an extrusion-based process for a microcellular filament

as the first step in the development of a continuous process.

The fundamental principle involved in the formation of microcellular polymer foams

consists of three basics steps: 1) polymer/blowing agent solution formation; 2) microcellular

nucleation; and 3) cell growth and density reduction. The single-phase polymer/blowing agent

solution in the first step are formed by saturating the polymer with the blowing agent under

high pressure. The saturation point is determined by the solubility limit of the blowing agent in

the polymer, while the time required for the solution formation is determined by the rate of

diffusion of the blowing agent into the polymer matrix.

4

Microcellular nucleation is achieved by inducing a thermodynamic instability in the

single-phase solution. This is usually accompanied by drastically reducing the solubility of the

gas in solution by controlling the pressure and/or temperature of the solution [11-13]. Since the

separation of the polymer and gas phases is thermodynamically more favorable, the resulting

supersaturated mixture becomes the driving force for the nucleation of numerous microcells

[1]. Continuous microcellular processing typically utilizes a rapid pressure drop to nucleate

bubbles. This stage is very crucial to the overall process, because it dictates the cell

morphology of the material and its resulting properties. Therefore, solubility as a function of

pressure is important for the development of the process.

The final stage in the production of microcellular plastics is cell growth. After cell

nucleation has occurred, any available gas diffuses into the cell and increases the cell size,

thereby reducing the density of the polymer matrix. Generally, cell growth is affected by the

time allowed for the cells to grow, the system temperature, the amount of gas available (state

of super saturation), the processing pressure, and the viscoelastic properties of the polymer/gas

solution [14].

1.2.2 Physical Blowing Agents

An important aspect of the creation of a cellular structure is the use of a physical

blowing agent. Historically, CGCs and HDCs, such as CFC-11 and HCFC-141b, were used

primarily for low density foam production mainly because of their solubility, volatility, and

non-toxic nature; however, their stability and reactivity with ozone in the atmosphere raised

substantial concern about ozone depletion. In an effort to address the environmental impact of

these man-made compounds, the Montreal Protocol [15] was signed in 1989 by 29 countries

and amended in subsequent years. The Montreal Protocol mandates the gradual phase-out of

5

the production of CFCs by 2010 and HCFCs by 2030 undeveloped countries. As a result, CFC

production fell from 980 metric tons in 1986 to 95 metric tons in 1996 [16]. Since the early

1990s, global warning or the greenhouse effect has become another major issue. The

Intergovernmental Panel on Climate Change (IPCC) reported a scientific assessment on the

warming potential of various compounds relative to carbon dioxide [17]. It was reported that

the global warming potential of CFCs and HXFXs is 5,000-10,000 times greater than that of

carbon dioxide with stratospheric life cycles of 60-130 years. In addition, the Kyoto Protocol

on Climate Change [18] adopted by 160 nations in 1997, sets binding limits on greenhouse gas

emissions for developed countries. It strengthens the framework established by the Montreal

Protocol with new policies and measures.

The combined global warming and ozone depleting potential of these substances, as

well as the resulting policies, have created a void in the foam processing industry; the need

has, therefore, arisen to replace CFCs and HCFCs with environmentally friendly blowing

agents possessing good foam-blowing properties. The concern over the ozone layer and global

warming represent just a few of the issues facing the foam processing industry. Disposal,

waste stream control, and usage of recycled plastics still require a deep understanding of

foaming technology.

Microcellular plastics do not utilize the conventional CFC foam blowing agents;

instead, carbon dioxide and nitrogen are typically used. Although these blowing agents

represent a dramatic improvement in terms of environmental hazard, they, too, pose their own

difficulties. Carbon dioxide has a high solubility in the plastic melt, approximately 10% at 200

°C and 27.6 MPa [19], and produces a very uniform cell structure. Conversely, nitrogen has a

lower solubility, about 2-3% at 200 °C and 27.6 MPa, and processing requires a much higher

pressure to dissolve sufficient nitrogen to create a uniform structure as explained in Section

6

1.2.1. Higher processing, however, requires more robust processing equipment and leads to

higher equipment and processing costs. Other alternative blowing agents used are liquid

blowing agents such as butane and iso-pentane. To date, however, liquid blowing agents have

not been successfully used to achieve a microcellular structure. One possible barrier that exists

with the use of these liquid blowing agents is the lack of information about their solubility and

diffusivity.

1.2.3 Thesis Objectives and Scope of Research

Since the solubility of blowing agents in polymer melts plays a key role in the plastic

foaming processing, this research was focused on the following aspects:

i. To propose technically sound experimental approaches and thermodynamic

models for the PVT, solubility and surface tension investigation of polymer/gas

mixture (binary).

ii. To obtain reliable (more accurate) PVT, solubility data, surface tension and

thermodynamic properties of polymer/blowing agents by systematic

investigation

iii. To verify the accuracy of solubility data determined by using various EOSs

and correct them with the help of a visualization system.

iv. To determine the effect of D-content/Mw on the solubility, surface tension and

swelling of the PLA/gas mixture.

7

1.2.4 Thesis Structure

This section provides a brief overview of the present thesis.

Chapter 2 explores the literature review on the thermodynamic study of the phase equilibrium

in a polymer system and gas solubility in polymer melts. The various methods available to

measure the solubility and diffusivity of blowing agents in polymer are examined. Theoretical

methods to predict the solubility is discussed as well.

Chapter 3 introduces the general research methodologies for the study of PVT and solubility

behavior of polymers. It was found that there are some deviations existing among the

thermodynamic models in terms of theoretical solubility and swollen volume prediction. As a

result, further investigation was done to verify the equation of states (EOSs).

Chapter 4 introduces the study of PVT and the solubility behavior of polymers/gas (binary

system). The solubility of CO2 in polymer was determined by using experimental data. The

visualization system was used to obtain the swelling behavior of the polymer/gas mixture. The

obtained data is then compared with the theoretical data obtained by means of EOS, namely

SS-EOS and SL-EOS.

Chapter 5 encompasses the measurement of the surface tension of the PLA/CO₂. Effect on the

interfacial tension with the variance in pressure, temperature, molecular weight, and D-content

is investigated. SS-EOS is used to obtain theoretical density of the PLA/CO₂ and was

compared to the experimentally obtained data.

Chapter 6 provides a summary, as well as conclusions of the research. Recommendations for

future work are also presented in Chapter 6.

8

CHAPTER 2. LITERATURE SURVEY AND THEORETICAL

BACKGROUND

2.1 Solubility

Solubility is defined as the amount of physical blowing agent (PBA) that can be

dissolved into a unit mass of polymer at a particular temperature and pressure (where the unit

is g-PBA/g-Polymer). The solubility is not only a critically important parameter for fabricating

plastic foams, but also an important property in developing blowing agents (BAs) and

evaluating their performances.

For effective process design, the system pressure must be high enough to dissolve all of

the injected gas into the polymer melt. When the polymer/gas solution exits the extrusion die,

the pressure will drop dramatically; this will initiate the bubble nucleation. The bubbles’

nucleation stage is crucial in the plastic foaming process due to the formation of a

microcellular structure. Theoretically, cell nucleation occurs when the pressure of the

polymer/gas mixture drops below the solubility pressure (or threshold pressure [20] to be

exact), as shown in Fig. 2.1 The cell nucleation mechanism has been described in detail by lot

of researchers [21-25], where the nucleation rate was governed by the degree of super

saturation, i.e., a metastable state determined by the saturation or solubility information.

9

Figure 2-1 Bubble nucleation inside a die

Blander and Katz [21-23] have reviewed the classical nucleation theory to estimate the

rates of bubble nucleation in pure liquids. The work of formation, W, for a spherical bubble of

radius, R, is shown in equation 1-2:

2-1

where is the surface tension; is the pressure of the bubble at the moment it is formed,

which is typically determined as a saturation pressure in a pure component system or solubility

pressure corresponding to the amount of dissolved gas in the mixture system; is the pressure

of the system; n is the number of bubbles; and are the chemical potentials of the

new and old phase, respectively. Hence, the solubility pressure information is required for the

calculation of the nucleation point and nucleation rate. Also, to study surface tension of the

polymer/gas mixture, solubility information is a prerequisite. For this reason, the solubility of

gases in a polymer melt during the plastic foam processing condition has been of great interest

to foaming manufacturers and researchers.

Solubility information also plays a very important role in foaming with an injection

molding machine. In our group, Lee et al. [26] found the cavity pressure of a foaming mold

10

has a significant influence on cell nucleation (Fig. 2.2). He claimed that if the cavity pressure

is lower than the solubility pressure (or the threshold pressure [20]) of the injected gas and if

the pressure before the gate is high enough, cell nucleation occurs at the gate with a high

pressure drop rate. In such cases, the cell density will be high. However, if the cavity pressure

is higher than the solubility pressure (or the threshold pressure), cell nucleation occurs along

the mold cavity with a low pressure drop rate, resulting in a low cell density.

(a) (b)

Figure 2-2: Bubble nucleation inside a mold in injection molding machine: (a) nucleation

starts at gate (generates high cell density); (b) nucleation starts in the mold cavity (generates

low cell density). [14]

2.2 Solubility Measurement Methods

There are two experimental techniques, known as permeation and sorption that can be

used to determine the solubility of a gas in a polymer. Permeation experiments involve

measurements of the steady state mass flow of a gas flowing through a thin membrane;

11

whereas in the sorption kinetics techniques, the mass uptake of gas by the polymer sample is

measured. The major difference between the two techniques is in the method by which

solubility is determined. Permeation experiments rely primarily on a mathematical expression

to determine the permeability, diffusivity, and hence the equilibrium solubility of the gas in the

polymer indirectly when steady state flow has been attained. The sorption method, on the other

hand, directly measures the mass gain of the polymer due to gas dissolution and, therefore,

represents a more direct approach to determining the equilibrium solubility. The sorption

method will be employed in this thesis to determine the solubility information for polymer-

blowing agent combinations, and hence this section will focus only on a review of sorption

techniques.

2.2.1 Gravimetric Sorption Technique

The gravimetric sorption method measures the solubility by simply measuring the mass

gain of a polymer sample due to gas dissolution. One of the earliest gravimetric techniques

utilized the quartz spring measuring system known as the McBain Balance [27]. The balance

was operated by suspending the polymer from a quartz spring in a low pressure gas

environment. As the polymer gained weight due to gas dissolution, the spring elongated.

Utilizing Hooke’s Law, the mass of the sample can be determined as a function of elongation

by calibrating the spring with known weight increments. The quartz spring method was used,

for example, to determine the solubility data of ethylbenzene [28] and toluene [29] in

polystyrene at various vapor pressures.

Batch process, a high pressure gravimetric technique, was utilized by Baldwin et al.

[30] for the measurement of carbon dioxide solubility and diffusivity in thermoplastic

polyesters. This process utilizes multiple samples, which are exposed to the gas for various

12

time periods, and compiles the mass uptake curve by normalizing the time axis for sample

thickness.

An in situ gravimetric sorption method directly measures the solubility by measuring

the mass gain of the polymer with a high-precision electro balance capable of measurements at

high temperatures and pressures. The sensitivity of the instrumentation, which can attain

values of 1 ppm (part per million), makes the technique desirable for solubility measurement

involving low solubility gases such as inert gases. This method is also suitable for the

measurement of solubility for polymers in either the rubbery or glassy state. The apparatus is

mounted on a vibration-free surface with the weighing unit contained in a constant temperature

environment.

The solubility of the blowing agent is determined from measurements of the increased

mass of the sample with increasing blowing agent pressure. Wong et al. [31] reported on the

use of an electronic microbalance to measure the gas solubility and diffusivity of carbon

dioxide and HFC134a in PS, filled poly (vinyl chloride) -FPVC, and unplasticized poly (vinyl

chloride) UPVC.

An alternative method for measuring gas solubility in polymers was presented by

Chaudhary and Johns [32]. It involved using a magnetic suspension device similar to the

electro balance. The most significant difference is that the weighing mechanism is physically

decoupled from the high temperature and high pressure environments through a magnetic

suspension coupling. This equipment was used to measure the solubilities of nitrogen,

isobutane, and carbon dioxide in polyethylene. More recently, Sato et al. [33] reported on the

use of a magnetic suspension balance to measure the solubilities of carbon dioxide in

poly(vinyl acetate) (PVAc) and polystyrene.

13

The high sensitivity of these types of balances dictates that mass measurements must

be corrected to accommodate for the change in buoyancy of the sample. Therefore, knowledge

of the dilation of the polymer with blowing agent uptake is also required for the solubility

calculations.

2.2.2 Pressure Decay Sorption Technique

The pressure decay technique is used to determine the solubility of gas in a closed

system by measuring the pressure decrease due to gas dissolution in the polymer sample. This

method relies on the assumption that all changes in the gas pressure are due to mass sorption

of the polymer. Consequently, the mass uptake of the polymer is determined indirectly by

measuring the pressure decay of the fixed volume system. By measuring the apparatus volume

accurately and recording the temperature and pressure of the system, the mass of the gas in

closed system is determined as a function of time using its equation of state. The solubility is

then determined from the overall experimental pressure change. This technique requires

careful calibrations and can be used only for gases whose equations of state are accurately

known. The three methods used for employing the pressure decay technique are single-, dual-,

and three-chamber systems.

Single-Chamber Sorption: The single-chamber system [34-37, 11-12] consists of a single

chamber containing the polymer sample. The chamber is subjected to a rapid pressure

increase, and the resulting pressure decay due to sorption is recorded as a function of time.

Due to rapid mass gain in initial stages and the thermodynamics of the gas system, a stable

reading is often not recorded until the pressure reading (needed to determine the initial mass of

gas in the system) is extrapolated from the pressure decay curve. The extrapolation, however,

14

can cause a significant error in determining the initial mass of gas present in the system, and

the corresponding total mass change due to gas sorption in the polymer.

Dual-Chamber Sorption: The dual-chamber sorption system [13, 38] uses a reservoir chamber

of known volume, filled with gas at a known pressure, while another chamber contains the

polymer sample. By opening a valve separating both chambers, the gas is allowed to flow into

the second chamber and, therefore, into the polymer. The valve is then closed and the sorption

chamber is observed for pressure decay. The reservoir chamber pressure is also measured. The

mass absorbed by the polymer sample is then determined based on the difference of the initial

mass of gas in the reservoir chamber and the final mass of gas in both the reservoir and

sorption chambers.

Sorption experiments are usually performed in a stepwise manner in order to make sure

the pressure drop for each experimental pressure step is relatively small. A typical pressure

decay sorption experiment begins with a low reservoir and corresponding low sorption

pressure. The resulting pressure decay due to sorption is observed until the equilibrium

pressure indicates that equilibrium mass gain has occurred. New gas is introduced into the

system without evacuating the chambers so that the pressure is increased by a step amount.

The pressure decay is then monitored, and the process is repeated. The solubility is determined

as a function of pressure by successively adding the mass gain of each pressure.

Three-Chamber Sorption: The three-chamber system [39,40] uses the measurement principle

identical to the dual-chamber configuration described above; that is, the mass uptake of the

polymer is determined from the equation of state of the gas, using measurements of chamber

volume, gas pressure, and gas temperature. The three-chamber system consists of two

reservoir chambers: the first reservoir is used as a pressure source for the sorption chamber;

while the other is used as a source for the first reservoir. This configuration allows for multiple

15

measurements at different pressures without introducing new gas and a new sample. This

method minimizes the temperature shock to the system caused by the introduction of gas.

2.2.2.1 Previous Research Using Pressure Decay Sorption Technique

Nevitt and Weale [13] were responsible for some of the earliest measurements of gas

solubilities in polymers using the dual-chamber system. They reported on the solubility of

hydrogen and nitrogen in polystyrene over the pressure of 8.1-30.4 MPa, and at elevated

temperatures up to 190 °C. High pressure was achieved in the reservoir chamber by using a

mercury pump. The pressure in the sorption chamber was measured one minute after first

subjecting the sample to the high-pressure gas. This delay in measurement was a result of the

pressure instability produced by the initial expansion of gas into the sorption chamber; this

was further compounded since the gas was not pre-heated to match the temperature of the

sorption chamber.

The unstable pressure observed initially contributed to the difficulty experienced in

determining the initial pressure reading required to calculate the equilibrium solubility. To

reduce the magnitude of this error, the researchers employed a large sample of 40-100 grams,

which was cut into thin strips to increase the mass diffusion rate (or reduce the time required to

obtain equilibrium stability), and thus increase the magnitude of the pressure drop. Utilizing

the stepwise sorption technique described earlier, the solubility was then calculated as a

function of pressure.

Lundberg et al. [34,35] and Lundberg [36] used a single-chamber sorption apparatus to

determine the solubility of gases in polymers at pressures between 3 and 71 MPa, and

temperatures between 102 and 188 °C. The stepwise sorption experiment was used to estimate

the solubility and diffusivity of a gas in a molten polymer.

16

Durril [37] and Durril and Griskey [11,12] employed a pressure decay method with a

single-chamber apparatus to investigate the solubility and diffusivity coefficients of nitrogen,

helium, carbon dioxide, and argon in molten polyethylene, polyisobutylene, and

polypropylene, at pressures up to 2 MPa. Before coming into contact with the sample, the test

gas was preheated in a thermostatted air environment. The first pressure reading, however, was

not reduced until 100 seconds after the gas first contacted the polymer sample. A stepwise

sorption methodology was used to calculate the solubility as a function of pressure.

Other researchers have utilized the pressure decay method to only measure the

solubility characteristics of gases in polymers at pressures up to 2 MPa [41-43] and 8.3 MPa

[44]. Stern and De Meringo [38] used a dual-chamber system to measure the solubility of

carbon dioxide in cellulose acetate at pressures up to 4.6 MPa.

Sato et al. [39] employed a three-chamber sorption apparatus to measure the

solubilities of carbon dioxide and nitrogen in polystyrene for pressures up to 20 MPa, and

temperatures in the range of 100-180 °C. The sorption chambers were controlled to within

0.05 K by a constant temperature air bath. In a later publication, Sato et al. [40] reported on the

solubility of carbon dioxide and nitrogen in polypropylene, high-density polyethylene, and

polystyrene. PVT measurements of the polymer at high temperatures and pressures were

conducted to provide the volume of polymer necessary for the solubility calculations, while

the swollen polymer volumes, caused by gas dissolution at different pressures and

temperatures, were predicted using the Sanchez Lacombe Equation of State [45-47].

2.2.3 Volume Decay Sorption Technique

As the name implies, volume decay sorption techniques measure the volume change of

the gas due to polymer sorption in a closed system at constant pressure and temperature. The

17

mass uptake of the polymer, or essentially the solubility, is indirectly determined from

measurements of the volume decay.

A volume decay sorption apparatus was utilized by Rosen [48] to measure the

solubility and diffusivity of acetone in cellulose acetate, methyl chloride vapor in polystyrene,

and water vapor in neoprene, at sub-atmospheric pressures. The system was designed as an

alternative to the quartz spring apparatus.

Mulrooney [49] used a constant pressure sorption concept based on the volume decay

method to investigate the solubility and diffusivity of liquid blowing agents such as isopentane

in polystyrene at elevated pressures. A positive displacement syringe pump capable of

operating in a constant pressure mode was used as the constant pressure source, while the

entire assembly was operated in a thermostatted air bath for constant temperature control. The

reasoning was that since the system was closed, any volume changes occurring in the blowing

agent were correlated to the piston movement of the pump and electronically recorded.

However, the swelling effect of the polymer could not be accounted for. If the volume

increased equally as the volume of isopentane decreased, then no net change in volume would

be observed. However, if the volume change of the isopentane was less than the volume

change of the sample, the net measured volume change would be underestimated.

2.2.4 Piezoelectric Quartz Sorption

Piezoelectric quartz sorption is a technique which measures solubility based on the

principle that the vibration frequency of a quartz crystal changes in response to a change in the

mass deposited on the crystal surface. This technique is usually applied to organic solvents.

There are two main components in this experimental set up: a sorption cell containing

the polymer coated with the piezoelectric crystal oscillator, and a solvent cell containing the

18

gas. When gas is introduced into the sorption cell, it is adsorbed onto the polymer. This, in

turn, changes the frequency of the crystal oscillator, which is measured with a frequency

detector, recorded, and indicated on a frequency counter. Also, this experiment incorporates a

few other variables that could lead to a frequency change, which include the following: the

sorption of gas into the polymer, adsorption of gas onto the crystal, coating of polymer film,

hydrostatic pressure of ideal gas, and viscous resistance of the gas.

Bonner and Cheng [50, 51] experimentally determined that the frequency of a quartz

crystal oscillator do vary with temperature and pressure. Hence, in order to account for the

pressure dependence of the frequency in their sorption measurements, two crystal oscillators

with similar pressure dependencies are used. One of the sorption crystals is coated with the

polymer, while the other uncoated crystal oscillator is used as a reference crystal. In a situation

when a reference crystal is not used, an accurate estimate of the pressure dependence of the

crystal oscillator at the experimental temperature would be needed. Such an estimate has been

reported by Stockbridge [52] for pressures below 0.13MPa.

By using the experimental technique with the uncoated reference crystal, Masuoka et

al. [53] investigated the solubilities of benzene, cyclohexane, n-hexane, toluene, and

ethylbenzene in polyisobtylene at low temperatures up to 65 °C and low pressures. The effect

of polymer coating thickness (in the range of about 0.2-1.4 µm) on the solubility of the solvent

in the polymer is tested, and they found that the experimental results were not affected within

this range. On top of that, they also concluded that the molecular weight had no definitive

effect on the solubility for polymer molecular weights of 50,000 and 100,000. In an

experiment dispensing with the reference crystal, Wang et al. [54] experimentally determined

the pressure dependence of the crystal oscillator without a polymer coating at pressures up to

10 MPa in an atmosphere of helium.

19

2.2.5 In-Line Measurement of Gas Solubility

In-line measurement techniques were developed from an interest in determining phase

equilibria during the actual foaming process. Usually, these techniques would incorporate the

measurement devices in-line with the foaming process. Phase separation, or the solubility

limit, is then detected by means of sensitive instrumentation or visually.

2.2.5.1 In-Line Monitoring

Dey et al. [55] and Zhang et al. [56, 57] reported an in-line technique for measuring the

gas solubility in various polymers during the foam extrusion process. The apparatus for this

technique consisted of an extruder with a specially designed optical window, and the flow

restrictor valve positioned between the die and the end of the extruder. Through this window,

the occurrence of bubble formation could be observed using a microscope-CCD camera-

monitor/ recorder system.

In order to detect the appearance or disappearance of bubbles during phase separation,

a two-phase, polymer-gas mixture was created by initially using a low pressure in the optical

window. The pressure was then gradually increased so that the polymer and gas became a

single-phase solution. This pressure was taken to be the lowest pressure required to keep the

gas in the solution under specified conditions. Lastly, the solubility was calculated by

combining this information with the gas flow rate and the melt throughput.

The parameters affecting the in-line measurement of gas solubility was found to be the

degree of mixing (single- or twin-screw extruder), the screw rotational speed, and polymer

throughput. One reported advantage of this in-line technique was that the solubility data could

be recorded in real-time, and therefore, could account for the dynamic nature of the extrusion

20

process, the possible role played by the extrusion process in gas dissolution, and bubble

nucleation in the melt.

2.2.5.2 In-Line Infrared Sensors

Near infrared (NIR) spectroscopy is a technique for monitoring the polymer/blowing

agent mixture during polymeric extrusion foaming processes. Through the use of dual-

transmission infrared sensors or probes, which transmit NIR light through the polymer running

in a flow cell, infrared monitoring of the process is achieved. The probes are linked with fibre-

optic cables to a Fourier transform near-infrared spectrometer (FT-NIR), which records the

absorption spectra of the melt. The flow cell for NIR measurements is located at the exit of the

extruder on a side stream of polymer flow taken from the main flow stream. Downstream of

the flow cell, a gear pump is installed to realize a steady flow rate.

NIR spectroscopy has been reported to have plenty of advantages, such as remote data

collection and ease of sample handling. It has also been used for the online measurements of

polymer composition [58], polymer viscosity [59], and concentration of HCFC in polystyrene

[60].

For instance, on-line NIR spectroscopy was utilized by Nagata et al. [61] in measuring

the Carbon Dioxide (CO2) concentration in molten propylene for CO2 extrusion foaming

processes. Three different CO2 concentrations and three separate flow rates were used

experimentally. In order to remove the baseline of the obtained NIR spectra, the wavelet

transform was employed (the given signals were represented by the linear combinations of

known functions). They claimed that experiments demonstrated a strong correlation between

the NIR spectrum and the CO2 raw NIR spectrum; the effects of temperature and flow rate

were erased. This technique, however, is limited in practice, since the incident light from the

21

probes would be scattered out if any dispersed material is present in relatively large quantities

in the melt. The absorbance may then become too weak to be analyzed precisely. Furthermore,

the calibration curve must be developed whenever the polymer and/or the processing

conditions are changed.

Thomas et al. [60] also investigated the ability of NIR spectroscopy to detect bubble

formation in the die as a function of blowing agent concentration and pressure for the

PS/HCFC 142b system. When the die pressure was gradually decreased, they observed that

NIR sensors could detect degassing of the melt. The appearance of bubbles caused scattering

of the light, which induced a large increase in attenuation at the level of baseline absorbance of

infrared waves. They also investigated the effect of talc on the performance of NIR

spectroscopy, and found that NIR analyses were still possible for talc contents

22

(A1, A2, A3, …) that are detected by the receiving transducer. From the thickness, e (m), and

the time delay between successive echoes, ∆t (s), the sound velocity, v (ms-1

), is determined

using the following relation:

2-2

On the other hand, the attenuation, a (dB/cm), is obtained through the relative amplitude of

successive echoes:

2-3

Sahnoune et al. employed these techniques [63] to measure the thermodynamic

properties of polystyrene/HCFC 142b mixtures. For phase separation measurements, they

observed that the velocity of sound decreased by as much as 4.5% from a steady state value as

the pressure was decreased. This was explained to be due to the phase separation process. The

attenuation, on the other hand, exhibited a different trend in relation to the thermodynamics

state of the blowing agent.

2.2.6 Modified Magnetic Suspension Balance Theoretical Treatments

Masahiro Ohshima and his coworkers [65] tried to measure the solubility of gas in

polymer by modifying the MSB. The densities of two polymer/CO2 single-phase solutions,

poly(ethylene glycol) (PEG)/CO2 and polyethylene (PE)/CO2, were measured at temperatures

higher than the melting temperature of a polymer under CO2 pressures in the range of 0-15

MPa using a newly-proposed gravimetric method. A magnetic suspension balance (MSB) was

used for the density measurement under the high pressure CO2. A thin, disc-shaped platinum

23

plate was submerged in the polymer/CO2 single-phase solution in the MSB high-pressure cell.

The weight of the plate was measured while keeping CO2 pressure and temperature in the

sorption cell at a specified level. Since the buoyancy force exerted on the plate by the

polymer/CO2 solution reduced the apparent weight of the plate, the density of the

polymer/CO2 solution could be calculated by subtracting the true weight of the plate from its

measured weight. Experimental results showed that the density of PE/CO2 solution increased

with the increase of CO2 pressure; and the density of PEG/CO2 solution decreased with the

increase of CO2 pressure. To differentiate the effect of CO2 dissolution in polymer from that of

mechanical pressure, the density of polymer/CO2 solution was compared with the density of

neat polymer under the given mechanical pressure, which was calculated using the Sanchez-

Lacombe equation of state and Pressure-Volume-Temperature (PVT) data of the polymer. The

comparison could elucidate that the dissolution of CO2 in the polymer-reduced densities of

both PEG/CO2 and PE/CO2 systems. However, this was not the case; the degree of CO2

induced-density reduction was different between the two polymer/CO2 systems.

Figure 2-3 Details of MSB for density measurement [65]

24

When the platinum plate is submerged in polymer/CO2 solution, the measured weight

of the plate becomes smaller than the true weight of the plate due to a buoyancy force exerted

on the plate by the polymer/CO2 solution. The buoyancy force is equal to the weight of

polymer/CO2 solution displaced by the plate, and it is calculated by multiplying the plate

volume by the density of polymer/CO2 solution. Therefore, knowing the volume and mass of

the plate a priori, the density of polymer/CO2 solution can be calculated from the buoyancy

force or the apparent weight of the platinum plate.

The force balance equation around the plate and the wire is expressed by

2-4

where and are the density of polymer/CO2 solution and CO2, respectively;

is the readout value of the apparent total weight of the plate, wire, and

measuring load hook at the experimental temperature, T, and CO2 pressure, P, condition;

is the apparent total weight of the plate, wire and measuring load hook at a reference

temperature and pressure condition; and are the volume of the platinum plate and that of

the wire, respectively; is the volume of measuring load hook; is the volume fraction of

the wire submerged in the solution; d is diameter of the wire connecting the platinum plate to

the measuring load hook; c is surface tension of polymer/CO2; h is contact angle between the

wire and the polymer/CO2 solution as shown in Figure 2-3(b); g is the gravitational constant.

The subscript i, for example di and Vw,i in Figure 2-3(b), indicates that it is the value in the

case of using the i-th wire.

Considering that the plate and wire were both made of platinum, the temperature and

pressure corrections of the volumes, V and Vw, were made using Eq. (2.5):

25

2-5

where Vref and Vw,ref are reference volumes of platinum plate and wire; m and E are Poisson’s

ratio and Young’s modulus of the platinum, respectively. They are given by 0.38 and 1.68

MPa, respectively. ζ is the coefficient of thermal expansion, which is 9.1x 10-6

K-1

.

The surface tension of polymer/CO2, g, and contact angle, u, were unknown and no

literature value was available. To eliminate g and u from the balance equation, two wires in

different diameter, d1 and d2, were used. The density measurements were conducted using each

wire individually at the same temperature and pressure.

Assuming that the two wires have the same surface tension, c, and contact angle, h, against the

polymer, we get the following:

2-6

Thus, the density of polymer/CO2, is given by the following:

2

-

7

26

2.3 Theoretical Treatments

There are theoretical approaches for explaining and predicting the solubility of gas in a

polymer. These theories are initially devised from the prediction of the pressure-volume-

temperature relationship for a pure component. They are then expanded to polymer/solute

systems. The models presented are based on a lattice fluid model in which each molecule

occupies r sites (an r-mer) with vacant sites present. It is assumed that there are random

mixings of r-mers with each other and with the vacant sites. These theoretical models, which

are summarized and presented in this thesis, are the Flory-Huggins theory [66, 67], Sanchez-

Lacombe Equation of State (SL-EOS) [45-47], and the Simha-Somcynsky Equation of State

(SS-EOS) [68].

The Flory-Huggins (F-H) theory [66, 67] was derived from considering the polymer

solution as a lattice in which a solvent molecule occupies the same lattice position as the

polymer segment. It gives information about the solubility and phase relationships, and

assumes that the volume and enthalpy of mixing are zero. This introduces a reduced Gibbs

energy parameter, x, to correct the energetic effect of mixing. The x-parameter is taken to be

independent of composition and temperature. The original F-H theory was modified by Blanks

and Prausnitz [69], who introduced an entropic contribution to the x-parameter. Nevertheless,

even though the theory is modified, the F-H theory is still considered inadequate for describing

polymer solutions. This is because it ignores the equation of state properties of pure

components and the effect of polymer chain architecture on intermolecular packing.

The Sanchez and Lacombe Equation of State (S-L EOS) [45-47] is a lattice fluid model

for pure fluids and mixtures. It requires three pure component parameters to characterize a

pure fluid and one adjustable binary interaction parameter. When the PVT properties of the

components at the solubility pressure are acquired, the equilibrium solubility of gas dissolved

27

into a polymer can also be determined. However, one complication that might arise is that

there is scarce information on the PVT properties of polymers and interaction parameters. If

the solubility data is available in a limited range, one can use non-linear regression analysis to

determine the parameters.

Panayiotou and Verra (P-V) [70] also obtained an equation of state based on a lattice

hole theory. The difference between this theory and the S-L EOS is that a constant site volume

for all r-mers is used, and non-random mixing arising from the molecular interaction is

introduced. The adjustable binary interaction parameter in the P-V EOS is incorporated as a

correction for the binary interaction energy. In the S-L EOS, the binary interaction term

modifies the characteristic pressures. Thus, it has a different physical meaning.

With much similarity to the F-H or S-L EOS theories, the Simha-Somcynsky (SS)

model [68] originates from treating molecules as segments on a lattice. In the case of a

mixture, a lattice contains both species, which are divided into approximately equal-sized

segments. Nonetheless, unlike the other theories, the SS theory allows for a pressure- and

temperature-dependent fraction of vacancies or holes that express free-volume within the

lattice. This accounts for molecular disorder in the lattice model. The equations derived from

the SS theory incorporate the temperature- and pressure-independent parameters that account

for intra- and intermolecular interactions within the mixture's components.

Based on the lattice fluid model, Rodgers and Sanchez [71] determined that adding an

empirical correlation for the interaction parameter would improve the predictive scope of the

LF model. With the addition of this correlation using Hansen's three-dimensional solubility

parameters [72], the LF model was reported to be able to predict solubilities in all types of

gas/polymer systems without the use of adjustable parameters. In other words, only the pure

component equation-of-state and solubility parameters are required.

28

Curro et al. used the Polymer Reference Interaction Site (PRIS) theory [73, 74] to

compute the sorption of a monatomic gas in a polymer liquid. This theory describes the

intermolecular packing between polymer chains and solute using the integral equation theory

of molecular liquids. Also, the chemical potentials of the solute species in the polymer must be

obtained in order to calculate the sorption of a gas in a polymer liquid.

29

CHAPTER 3. RESEARCH METHODOLOGY FOR PVT AND

SOLUBILITY STUDY

3.1 Introduction

Gas solubility in polymers can be measured using different techniques, that is,

gravimetric techniques, including vibrating or oscillating techniques; PVT techniques with the

pressure decay method; and gas-flow techniques. A brief review of the technique which was

implemented is given below, followed by the description of a technique we recently developed

that couples a new gravimetric technique with a PVT visualization technique.

3.2 Theoretical Background

Theoretical models to determine the solubility and swelling of the polymer/gas mixture

such as the Sanchez and Lacombe (SL) [45-47] and the Simha and Somocynsky (SS) EOS

[68] are all based on the statistical thermodynamic theory.

The equation of state of the pure component system is written in the following manner for the

SL EOS:

3.1

or the SS EOS:

30

3.2

3.3

In the above relations, ~,~

,~

PT and V~

are reduced parameters. They are calculated from the

characteristic reducing parameters P*, T*, V* and as follows:

3.4

where y is the fraction of occupied lattice sites, s is the number of segments per chain of molar

mass, c is the number of external degrees of freedom per chain, and finally V~

and

are dimensionless quantities.

The solubility of gas in the polymer (binary system) can be calculated theoretically using the

phase equilibrium theory:

3.5

31

where is the chemical potential of gas in the gas phase and is the chemical potential of

gas in polymer/gas mixture phase. Under the phase equilibrium condition, the mass fraction of

gas in the polymer/gas mixture phase, i.e., the theoretical solubility , can be obtained

by solving Eq. (3.5).

In the case of SS-EOS, Eq. (3.6) was used to solve [75]:

3-6

And Eq. (3.7) was used to calculate [36, Gli paper]:

3-7

where is the molar free energy of the polymer/gas mixture [36 Gli paper].

3-8

In the case of SL-EOS, Eq. (3.9) and Eq. (3.10) were used to determine and :

3-9

32

3-10

The swollen volume can be obtained from the following relation:

3-11

where S is the gas solubility (g-gas/g-polymer) in the polymer melt which is calculated from

the EOS, m (g) is the initial weight of the polymer sample, vp,pure (cm3/g) is the specific volume

of pure polymer.

3.3 Methodology and Approach

The details of the measuring procedure using MSB can found in our previous literature

[76]. The amount of gas which is absorbed by the polymer can be determined by the following

equation:

3-12

where W(P,T) is the weight of the sample at temperature T and pressure P; W(0,T) is the

weight of the sample at temperature T and vacuum; ρgas is the density of the gas inside the

33

chamber at temperature T and pressure P; VB, VP, and VS are the volume of the sample holder,

the volume of pure polymer, and the swollen volume of polymer, respectively, due to gas

dissolution at temperature T and pressure P.

By ignoring the polymer’s swollen volume (VS) in Eq. 1, the measured weight gain Wg

in Eq. (3.12) can be transformed to the apparent solubility (Eq. 3.13), Xapparent, which is less

than the actual solubility:

3-13

As shown in Eqs. 3.12 and 3.13, it is impossible to measure the accurate solubility of

the gas in the polymer melt by ignoring the swollen volume (VS). However, presently, there is

a lack of reliable and accurate PVT data for polymer/gas mixtures which are measured

experimentally. Hence, the swollen volume is typically estimated by an equation of state

(EOS) which can be applied to a two-component mixture system under equilibrium.

A general approach that combines the experimental solubility measurement and the

thermodynamic models was proposed by Li et al. [77] Firstly, a gravimetric method is carried

out to experimentally measure the gas sorption in a polymer melt (apparent solubility, Xapparent).

Secondly, the SS-EOS or SL-EOS is applied to calculate the phase equilibrium (theoretical

solubility, Xtheory) and the swollen volume of polymer, Vs. Thirdly, the theoretically predicted

swollen volume, Vs can be used to complete the correction on the apparent solubility, Xapparent,

and then to obtain the actual solubility or corrected solubility, Xcorrected. Meanwhile, the PVT

34

apparatus is also used to determine the swollen volume [78]. The swollen volume of the

polymer/gas mixture may also be obtained from the following relation:

3-14

where is obtained from measuring the volume of the polymer/gas drop mixture,

where as is obtained from PVT equation. The corrected solubility, Xcorrected, with the

buoyancy effect compensation can be obtained using Eq. 3.14:

3-15

3.4 Summary

Based on the magnetic suspension balance, a robust general research approach was

established for the calculation of solubility, which has been described in detail in the previous

chapter. In order to obtain accurate solubility data, buoyancy effect must be accounted for,

which is generated due to the swelling behavior of the polymer in the presence of a gas.

Experimental and theoretical approaches have been implemented in order to account for the

swelling behavior of the polymer/gas mixture melt. The phase equilibrium and the PVT

behavior of the polymer/supercritical fluid are studied in detail with the theoretical approach

proposed by Dr. Guangming Li and the experimental approach put forward by Dr. Yao Gai

Gary Li.

The results showed that the SS-EOS predicted the swelling and hence the solubility of

carbon dioxide in PLA in close proximity of the theoretical values in comparison to SL-EOS.

35

CHAPTER 4. SOLUBILITY AND SWELLING BEHAVIOR OF

PLA IN PRESENCE OF CO₂

4.1 Introduction

Based on the magnetic suspension balance, a general approach was established to

measure the solubility of carbon dioxide. In order to obtain accurate solubility data, inclusion

of the swelling volume is essential, which is generated from the dissolution of the blowing

agent in the polymer.

The theoretical approach for the determination of swollen volume and phase

equilibrium was built on a variety of technically sound thermodynamic models, such as SS-

EOS and SL-EOS. Previous work [75] has illustrated some deviation among the

thermodynamic models in terms of theoretical solubility and swollen volume prediction. In

this chapter, a systematic investigation is illustrated to investigate the factors that govern gas

solubility in a polymer. The two models, namely SS-EOS and SL-EOS, are compared with the

experimental results in terms of their ability to predict theoretical solubility and volume

swelling.

Three different grades of PLA are utilized to investigate the effect of D-content on

solubility and swelling. The effect of varying molecular weight and D-content on the gas

solubility and swelling volume is also investigated.

36

4.2 Experimental

4.2.1 Materials

Three different grades of Polylactide (PLA) from were used in the

experiments: PLA 3001D (1.4% D-content), ; PLA 8051D (4.6% D-

content), ; and PLA 4060D (12% D-content), . The

PLA was received in the form of pellets from LLC. Carbon dioxide

(Coleman grade, 99.99% purity) was obtained from BOC Canada.

4.2.2 PVT Data for PLA

The PVT data published by Sato for PLA was used to obtain the Tait’s equation [79].

The same Tait’s equation was used for the three different grades of PLA. Since A Tait’s

equation represents the PVT behavior of a polymer, therefore, generalizing a Tait’s equation

for different grades of PLA is not preferable. This makes it harder to observe the effect that

different variables have on the solubility, swelling volume, and surface tension. The latter

notion is discussed in depth later in the Chapter.

4-1

In the above equation, the temperature, T, is in (°C) and pressure, P, is in (bars). The PVT data

obtained was also used to derive the characteristic parameters for both SS-EOS and SL-EOS:

P*, V*, T*. All the characteristic parameters for the PLA grades are listed in Appendix 3.

37

4.3 Solubility of CO₂ in PLA (Binary System)

4.3.1 Swelling Behavior of PLA in Presence of CO2

4.3.1.1 Experimental Setup

The experimental setup consisted of the following components: a high-pressure

chamber with a sapphire window for the purpose of visualization; a 2024 x 2024 resolution

JAI Pulnix TM4100 CL camera with control software; Schneider 4/80 lens and extension

tubes; a temperature controller (Omega CN132) with thermocouple (Omega RTD); two

cartridge heaters; an automatic high-precision XY stage with Galil motion controller and

control board; a manual 1 in. XYZ stage to adjust the position of the light source; a syringe

pump connected to the gas tank; and a backlight source with a light equalizer/diffuser.

4.3.1.2 Experimentation

PLA samples were sliced from a strip obtained from a micro compounder and weighed

using a precision microbalance. The selected PLA samples were attached to the droplet rod to

form the sessile drop for each experiment. Swelling measurements for mixtures

were conducted at two different temperatures, 453 K and 473 K. At each temperature, the

pressure of CO2 inside the chamber was varied from 6.894 MPa (1000 psi) to 20.684 MPa

(3000 psi) in 3.447 MPa (500 psi) increments. Each pressure level was maintained for 1.5

hours to ensure that the equilibrium conditions were established for the polymer/gas solution.

Equilibrium was considered to be achieved when the total volume of the polymer/gas solution

no longer changed.

38

In order to compare the experimental data with the theoretical, SS-EOS and SL-EOS

were used to determine the theoretical volume swelling ratio. The parameters used to

determine the theoretical swelling ratio ( ) are shown in Appendix 3.

4.3.1.3 Pressure and Temperature Effect on Volume Swelling

It was observed that the volume of the PLA3001D/CO2 mixture increased with an

increase in pressure as illustrated in Fig. 4.1 and Fig. 4.2. Due to the increase in pressure inside

the chamber, the density of the CO2 gas increases, hence more CO2 molecular will penetrate

into the PLA polymer matrix causing more dilation until it reaches the saturation point.

With a fixed temperature, an increase in pressure causes an increase in the volume of

the polymer/gas mixture, as well as the volume swelling ratio. Since the solubility of CO2

increases [80] as the pressure is increased, more CO2 gas is permitted to enter the PLA melt

matrix, hence an increase in swelling volume was observed. The hydraulic pressure effect due

to the CO2 was accounted for by using Tait’s equation. The Tait’s equation obtained from the

PVT apparatus was compared with the Tait’s equation obtained through parameters based on

the PVT data of PLA by Sato et al. for SS-EOS and SL-EOS [79].

At isobaric conditions, the volume swelling of the PLA/CO2 mixture tends to decrease

as illustrated in Figures 4.1-4.3 with an increase in the temperature. As the temperature

increased, the polymer chains became softer which increased the free volume as well as

specific volume. The solubility of CO2 in PLA is known to decrease as the temperature

increases [81]. This means that the diffusion of CO2 out of the polymer increased; hence more

CO2 is forced out of the polymer/gas mixture compared to what would enter due to the free

volume at an elevated temperature. In other words, despite the increase in the free volume

39

within the PLA/ CO2 matrix, CO2 will escape out of the matrix. For instance, at the 13.79 MPa

pressure level and 453 K; PLA-3001D/CO2 has a volume swelling ratio of 9.32%; whereas at

473 K, the swelling ratio is 8.88%.

180 185 190 195 200

1.04

1.06

1.08

1.10

1.12

1.14S

we

llin

g R

atio

(%

)

Temperature (Celcius)

1000 psi

1500 psi

2000 psi

2500 psi

3000 psi

PLA 3001D

Figure 4-1: Effect of Temperature variance on Swelling Ratio for PLA3001D

180 185 190 195 2001.02

1.04

1.06

1.08

1.10

1.12

1.14

Sw

elli

ng

Ra

tio

(%

)


1000 psi

1500 psi

2000 psi

2500 psi

3000 psi

PLA 8051D


40

180 185 190 195 2001.02

1.04

1.06

1.08

1.10

1.12

1.14

Sw

elli

ng

Ra

tio

(%

)


1000 psi

1500 psi

2000 psi

2500 psi

3000 psi

PLA 4060D


4.3.1.4 Comparison of the Experimentally Measured Data and

Theoretically Predicted Data

The experimental data obtained using the in-house PVT apparatus [82] was compared

with the swelling volume ratio obtained via EOS. SS-EOS and SL-EOS were implemented in

order to calculate the theoretical swelling volume ratio [83]. It was evident from Figure 4.4-4.9

that the SS-EOS provides a more realistic prediction of the swelling volume ratio; whereas the

SL-EOS exaggerated the swelling volume ratio with respect to the experimental result. This

has been illustrated in our previous work in detail [82, 84, 85].

41

1000 1500 2000 2500 3000

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

Sw

ell

ing

Ra

tio

Pressure (psi)

3001D EXP

3001D SS-EOS

3001D SL-EOS

180 oC

Figure 4-4: Swelling Ratio of PLA3001D at 180 °C with varying pressure

1000 1500 2000 2500 30001.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

Sw

ell

ing

Ra

tio

Pressure (psi)

8051D EXP

8051D SS-EOS

8051D SL-EOS

180 oC


42

1000 1500 2000 2500 30001.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

Sw

ell

ing

Ra

tio

Pressure (psi)

4060D EXP

4060D SS-EOS

4060D SL-EOS

180 oC


1000 1500 2000 2500 30001.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

Sw

ell

ing

Ra

tio

Pressure (psi)

3001D EXP

3001D SS-EOS

3001D SL-EOS

200 oC


43

1000 1500 2000 2500 30001.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

Sw

ell

ing

Ra

tio

Pressure (psi)

8051D EXP

8051D SS-EOS

8051D SL-EOS

200 oC


1000 1500 2000 2500 30001.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

Sw

ell

ing

Ra

tio

Pressure (psi)

4060D EXP

4060D SS-EOS

4060D SL-EOS

200 oC


44

4.3.1.5 Effect of ‘D’ Content/Molecular Weight on Volume Swelling

It was observed that at 453 K, PLA3001D has a higher volume swelling ratio than PLA

8051D. For example, at 17.24 MPa, PLA 8051D (D content of 4.6%) has a volume swelling of

11.29%; whereas PLA 3001D (D content of 1.4%) has a volume swelling of 12.21%. Similarly,

the swelling ratio of PLA 4060D at 17.24 MPa is 10.81%, which was less than that of PLA

8051D. At this instant, we cannot conclusively state the D-content’s effect on the volume

swelling. This is due to the presence of two variables, the D-content and molecular weight. In

order for us to state any concrete effect of the D-content, we need to experiment with two PLA

samples with similar, if not the same, molecular weight and different D-content.

1000 1500 2000 2500 30001.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

Sw

ell

ing

Ra

tio

(%

)

Pressure (psi)

3001D EXP

8051D EXP

4060D EXP

180 oC

Figure 4-10: Effect of varying D-content/Mw on swelling ratio at 180 °C

45

1000 1500 2000 2500 3000

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

Sw

ell

ing

Ra

tio

(%

)

Pressure (psi)

3001D EXP

8051D EXP

4060D EXP

200 oC

Figure 4-11: Effect of varying D-content/Mw on swelling ratio at 200 °C

4.3.2 Solubility of CO2 in PLA

With the prediction of swollen volume from SS-EOS and SL-EOS and the

experimental data (mentioned in Sec. 4.3.1), the solubility of in PLA at 180 °C and 200

°C was obtained by utilizing the apparent solubility obtained from the MSB. Methodology

using the MSB has been discussed at length in Chapter 3.

4.3.2.1 Pressure and Temperature effect on Solubility of CO₂ in PLA.

The effect of

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