-
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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ii
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.
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To the one above for providing me
with an opportunity
To my parents For endless love, support
and encouragement
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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
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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
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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 4: Solubility of PLA8051D at 180 °C
..............................................................................
97
Table 5: Solubility of PLA4060D at 180 °C
..............................................................................
97
Table 6: Solubility of PLA3001D at 200 °C
..............................................................................
97
Table 7: Solubility of PLA8051D at 200 °C
..............................................................................
98
Table 8: Solubility of PLA8051D at 200 °C
..............................................................................
98
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 11: Swelling ratio of PLA4060D/CO₂ melt at 180 °C
..................................................... 99
Table 12: Swelling ratio of PLA3001D/CO₂ melt at 200 °C
..................................................... 99
Table 13: Swelling ratio of PLA8051D/CO₂ melt at 200 °C
..................................................... 99
Table 14: Swelling ratio of PLA4060D/CO₂ melt at 200 °C
................................................... 100
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viii
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 4-2: Effect of Temperature variance on Swelling Ratio for
PLA8051D ........................ 39
Figure 4-3: Effect of Temperature variance on Swelling Ratio for
PLA4060D ........................ 40
Figure 4-4: Swelling Ratio of PLA3001D at 180 °C with varying
pressure ............................. 41
Figure 4-5: Swelling Ratio of PLA8051D at 180 °C with varying
pressure ............................. 41
Figure 4-6: Swelling Ratio of PLA4060D at 180 °C with varying
pressure ............................. 42
Figure 4-7: Swelling Ratio of PLA3001D at 200 °C with varying
pressure ............................. 42
Figure 4-8: Swelling Ratio of PLA8051D at 200 °C with varying
pressure ............................. 43
Figure 4-9: Swelling Ratio of PLA4060D at 180 °C with varying
pressure ............................. 43
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 4-13: Effect of varying temperature on solubility of
PLA8051D .................................. 47
Figure 4-14: Effect of varying temperature on solubility of
PLA4060D .................................. 47
Figure 4-15: Solubility of Carbon Dioxide in PLA3001D at 180 °C
with varying pressure .... 48
Figure 4-16: Solubility of Carbon Dioxide in PLA8051D at 180 °C
with varying pressure .... 49
Figure 4-17: Solubility of Carbon Dioxide in PLA4060D at 180 °C
with varying pressure .... 49
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ix
Figure 4-18: Solubility of Carbon Dioxide in PLA3001D at 200 °C
with varying pressure .... 50
Figure 4-19: Solubility of Carbon Dioxide in PLA8051D at 200 °C
with varying pressure .... 50
Figure 4-20: Solubility of Carbon Dioxide in PLA4060D at 200 °C
with varying pressure .... 51
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
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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]
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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
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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
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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
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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
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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.
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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
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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.
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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
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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):
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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.
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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
(%
)
Temperature (Celcius)
1000 psi
1500 psi
2000 psi
2500 psi
3000 psi
PLA 8051D
Figure 4-2: Effect of Temperature variance on Swelling Ratio for
PLA8051D
-
40
180 185 190 195 2001.02
1.04
1.06
1.08
1.10
1.12
1.14
Sw
elli
ng
Ra
tio
(%
)
Temperature (Celcius)
1000 psi
1500 psi
2000 psi
2500 psi
3000 psi
PLA 4060D
Figure 4-3: Effect of Temperature variance on Swelling Ratio for
PLA4060D
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
Figure 4-5: Swelling Ratio of PLA8051D at 180 °C with varying
pressure
-
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
Figure 4-6: Swelling Ratio of PLA4060D at 180 °C with varying
pressure
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
Figure 4-7: Swelling Ratio of PLA3001D at 200 °C with varying
pressure
-
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
Figure 4-8: Swelling Ratio of PLA8051D at 200 °C with varying
pressure
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
Figure 4-9: Swelling Ratio of PLA4060D at 180 °C with varying
pressure
-
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