ON-LINE MEASURE~ENT OF PVT PROPERTIES POLYMERICO ... · ON-LINE MEMUREMENT OF THE PVT PROPERTIES OF POLYMER~CO~ SOLUTIONS USING A GEAR PUMP Simon S. Park Degree of Masters of Applied

ON-LINE MEASURE~ENT OF THE PVT PROPERTIES OF POLYMERICO~ SOLUTIONS USING A GEAR PUMP

Simon S. Park

A thesis submitted in conformity with the requirements for the Masters of Applied Science

Department of Mechanical and Industrial Engineering University of Toronto

@Copyright by S. S. Park 1999

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ON-LINE MEMUREMENT OF THE PVT PROPERTIES OF POLYMER~CO~ SOLUTIONS USING A GEAR PUMP

Simon S. Park Degree of Masters of Applied Science, 1999

Department of Mechanical and Industrial Engineering University of Toronto

This study presents an innovative dilatometer that can accurately measure the

pressure-volume-temperature (PVT) properties of polyrner/C02 solutions. The density

(or equivalently, the specific volume) of a polymer/C02 solution is detemined by

measuring the mass and volume flow rates of the solution. A positive-displacement gear

pump mounted on an extruder is used to measure the volume flow rate of the solution.

The positive-displacement volume of the gear pump was determined in calibration

experiments with water and oil with the aid of a syringe pump. The gear pump was

mounted on a tandem extrusion line. The temperature of melt was precisely controlled

and homogenized by using the second extruder and a heat exchanger, which included a

static mixer. The pressure was controlled by the rotational speed of the screw in the

second extruder. In order to reduce leakage across the gear pump, the difference between

the upstrearn and downstrearn pressures was minimized using a variable resistance die

attached to the downstream outlet of the gear pump. A critical set of experiments was

carried out to measure the specific volumes of various pure polymers. The measured

results were compatible with the known PVT data, confirming the validity of the system.

The experirnental data were modeled using the equation-of-state @OS) to predict the

relationships above the experimental ranges. Finaily, the system was used to rneasure the

specific volumes of various polyrner/C02 solutions. A rnetered amount of gas was

injected into the polymer melt Stream in the first extruder and the injected gas was

cornpletely dissolved in the melt in the second extruder of the tandem foam extrusion

Iine. The forrned single-phase polymer/C02 solution was forwarded to the gear pump,

and its specific volume was measured by comparing the volume and m a s flow rates.

The sweiling of the polymer melt due to the dissolved CO2 was characterized as a

function of CO2 concentration, temperature, and pressure. Efforts were made to rnodel

the experirnental results.

First I wouid like to express my sincere gratitude to rny supervisor Professor C. B.

Park for providing guidance and encouragement throughout rny research. 1 will never

forget the help he gave at my most difficult times.

My gratitude is extended to the Department of Mechanical and industriai

Engineering at the University of Toronto for providing the University of Toronto

Master's Open Fellowships, as well as, the NSERC Scholarship. These financial

assistance enabled me to finish my studies.

1 would also like to thank my colieagues in the Microcellular Plastics

Manufacturing Laboratory for their help and friendship over the past two years. They

include Dr. Yuejian Liu, Anthony Yeung, Amir Behravesh, Ghaus Rizvi, Remon Pop-

niev, Deepak Fernandes, Hani Naguib, Dmitry h d i n , Dr. Chns Song, Dr. Sang Mae Lee.

Minhee Lee, and Haiou Zhang. Especially, sincere gratitude goes to Hani and Drnitry

who shared common thoughts. We shared a great time together and 1 learned a lot about

Jewish and E,vtian cultures. 1 also would like to thank many friends, Junghyun, Jihyun,

Heedo, NamSuk, Kooyong, and church friends,

Abstract ...,.......-...-....--........-.....~............~....~~~~.~~..~-.~~-..-----...-.......................... tr

Acknowledgernents.. -.-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . -. . . . - . . . . . . - - -. . . -. -. . . . . . . - . . . . . . . . . . . . . . v

Table of Contents .................................................................................................... vi

List of Figures.,. .. ... ....--. . . . . . . . . . .. .. . .--. -.-. . .. -.-........-.--- .. .---.--. . . .. . ... .. .-.+ ... . . . . . . . .. . .. . .. x

List of Tables .......................................................................................................... xv

List of Syrnbols ....................................................................................................... xvii

CHAPTER 1 ~NTRODUCTION. ... .....,............ ...-..... . ... ...-..-. . .. ...... ...- .. .. .. ... ...........-...... 1

1 . 1 Prearnble ............................................. - .-.-....-.-.................. 1

1.2 Effects of Gas in a Pol ymer . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . -. -. . . . . . -..- 2

1.3 Thesis Objectives and Scope of Research ................................. 3

1.4 Organization of the Thesis ............................................. 4

CKAPTER 2 LITERAW SURVEY AND THEORETICAL BACKGROUND .... . .. . . . . . . . . . . . . . ..... 7

2.1 Conventional PVT Measurements of Pure Polymer ..... . . . . . . . . . . .. . ... 7

2.2 Solubility Measurements ....... . . . . .. ... . .. ............... ... ..... .. .......... . 9

2.3 Theoretical Background ....................................................... .... . 1 1

2.3.1 Therrnodynamics (PVT Relationships) ............................

............................................ 2.3.2 S tatistical Thermodynamics

........................................................... 2.4 Equation-of-State (EOS)

.............................. 2.4.1 Empirical Analysis: the Tait Equation

.......................................... 2.4.2 Theoretical Equation-of-S tate

2-4-2- 1 Cell Theory . FOV Mode1 ...............................

..... 2.4.2.2 Cell Theory - Modified Ce11 Model (MCM)

........... 2.4.2.3 Hole Theory - Simha-Sorncynsky Model

2.4.2.4 Lattice-Fluid Theory - Sanchez-Lacombe

............................................................ Model

........................................ 2-4.3 Remarks on Theoretical EOS

............................................................................... 2-5 Solubility

2.6 Physical BIowing Agents ..................... .... .............................

C H A ~ R 3 CONCEPTUAL DESIGN OF PVT M E A S ~ J G DEWCE BASED ON

........................................................................... k a o ~ ~ n c DESIGN 29

............................................................... 3.1 Overall Methodology 29

............................... 3.2 Background of Axiomatic Design Approach 30

....................................... 3 -3 Overall Design for PVT Measurement 31

3.4 Determination of the Positive-displacement Rate of the Gear

Pump ...................................................................................... 37

........................... 3.4.1 Approximate Volumetric Relationships 38

3-42 Calibration of the Gear Pump ........................................ 39

vii

...................... C m 4 PVT MEASUREMENT AND MODELING OF PURE POL-

..................... ....................................... 4.1 Experimental Setup ...

................................................................................. 4.2 Materiais

........................................... ............. 4.3 Experimental Procedures ..

4.4 Results and Discussion .............................................................

4.4.1 System Verifkation .......................................................

4.4.2 EffectsoftheGearPumpSpeeds ....................................

4.4.3 Effects of Thermodynamic Work hputted Into the

......................................................................... S ystem

............................................................................. 4.5 Conclusions

CHAPTER 5 PVT MEASURE~UT &W MODEUNG OF POL- AWD GAS SOLUTIONS 95

5.1 ExperimentalSetup ................................................................. 96

................................................................................. 5.2 Materials 97

........................................................... 5.3 Experimental Procedure 98

............................................................... 5.4 Resuit and Discussion 101

............................................................................. 5.5 Conclusions 113

............................................................. CHAETER 6 S ~ M M A R Y AND CONCLUSIONS 158

................................................................................. 6.1 Summary 158

............................................................................. 6.2 Conclusions 159

viii

Figure 2.1 Dilatometers (a) Cylinder-Piston Type Dilatometer for Molten Polyrner,

(b) Bellow Type Dilatometer with the Polyrner Sarnple hmersed in

Mercury . . . . . . . - . . . . . . . . . . . . .. . . . . . - - - -. . . . . ,, . . . . . . -. - . . - - - - - -. -. . . . - - -. . . . . . . . . . . . . . -. . -. . . . . . .- -. Figure 2.2 Solid-State Dilation Device ...................................................................

Figure 2.4 Relationships between Pressure-Temperature-Solubility ...................... ....

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

OveraI 1 S trategy . . . . . . . . . . . . . . . . . .... . . . -. -. - -. . . - - - . . - -. -. -. . - - - - -. -. . . . . . . . -. . . . . . . . -. . .. -. . . -.

A Schematic of the Experimental Setup for Measuring the PVT data of a

Polymer Melt . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . -.- .- -. -. - .. - -. -. --. . - - -. .... . . . . .*. --. -. -. . . .-. -. . ..--

Side View of the Gear Pump .................................................................

Calibration for Determining the Positive-displacement Volume of a Gear

Pump . . . . . . . . . . . . . . . . . . . -. -. . . . . . -. . . . . - -. . . . . - -. -. - . . . . -. - - - - - -. - . . . . -. - -. -. . . . . . . . . . . -. . . . . . . . . .-.- - Experimental Setup for Deterrnininp the Compressibility of Vegetable

Oil (a) Actual Setup with a Syringe Pump, (b) Equivalent Piston-

Cylinder System ........ ....................................... .. . .. . .. . . ..-.. .. ..-. .--..--..-.-.- Volume-Pressure Relationship of Vegetable Oil ...... - ............... ... ....-.-.-.--

Volumetric Flow Rate of Water and Oii through the Gear Pump as a

Function of the Gear Pump Speed ..........................................................

Experimental Setup for Determining Leakage Flow for Water and

Vegetable Oil Across the Gear Pump as a Function of Pressure

Difference . . . . . . . . . . . . . . . . . . . . . . - ... . . . . . . . . -. . . . . . . ,.. . . . . . . . . . . . . . . . . . . -. - -. . -. -. . . . . -. . . . ...- Leakage Flow as a Function of the Pressure Difference in the Gear Pump

Obtained frorn the Dynamic Leakage Flow Analysis ........ .. . .. . .. . .. ....... .....

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.1 1

Figure 4.12

Figure 4-13

Figure 4.14

Figure 4.15

Figure 4.16

Photograph of the Overall Setup of the System .....,,..-......................,......

Side View of the Gear Pump and Connections (a) Angled View, (b) Side

View . . . . . . , . . . . . . . . . . . - . . . . . . . . - . . . . . . . . . -. . . - -. . . . . . -. -. . -. . - . . . . . -. . . -. . . - . . . . -. - - . - . . . - -. . . . - - - . . . . . Overall View of the Controt Boxes (RPM Controllers, Temperature

Controllers, Pressure Readouts, and etc.) . . -. . .. ... . .. . . . . .- - .. - .-. . - - --. - - - - -. -. - - - - - - -

Generai Procedure to Measure the PVT data of Pure Polymer Melts .........

Correlation of PVT Data for PP in the Melt State Using (a) FOV; (b)

MCM; and (c) S-L EOS .......... ................................. ..........--......,..-.......

Cornparison of PVT Data for PP in the melt State between Sato et al.'s

Results [19] and This Shidy ............................................... - - - - - ..-. .- .----- - -

Measured PVT Data for PS (a) Specific Volume with respect to Pressure;

(b) Specific Volume with respect to Temperature ... . .. . . .... ... . .. . . . . .. . . . . -. . . . . .

Correlation of PVT Data for PS in the Melt State Using (a) FOV; (b)

MCM; and (c) S-L EOS .......................................................................

Comparison of PVT Data for PS in the Melt State between

Manufacturer's Reference Data and Experimental Data . . . . . . , . . .. . . . . . . . . . - - -.... Measured PVT Data for Linear PBS (a) Specific Volume with respect to

Pressure; (b) Specific Volume with respect to Temperature ................... ,.

Correlation of PVT Data for Linear PBS in the Melt State Using (a)

FOV; (6) MCM; and (c) S-L EOS .........................................................

Comparison Between the Experimental Data vs. Manufacturer's Data for

Linear PBS ........ ................. ---.- ...- - ...--.. - .......................................... ....-

Measured PVT Data for Branched PBS (a) Specific Volume with respect

to Pressure; (b) Specific Volume with respect to Temperature ....,....,........

Correlation of PVT Data for Branched PBS in the Melt State Using (a)

FOV; (b) MCM; and (c) S-L EOS ..........................................................

Comparison Between the Linear (Dotted Lines) and Branched PBS

(Solid Lines) ....................................................................... ...... ..--....-..

Figure 4.17 Mass Flow Rates Measured in Various Gear Pump Speeds at (a) 220°C

and 27.6 MPa; (b) 220°C and 20.7 MPa; (c) 220°C and 13.8 MPa; (d)200

OC and 27.6 MPa; (e) 200°C and 20.7 MPa; (f) 200°C and 13.8 MPa; (g)

180°C and 27.6 MPa; (h) 180°C and 20.7 MPa; (i) 180°C and 13.8 MPa

Figure 4.18 Error Analysis of PS Due to the Effects of Gear Pump RPM ..... ...............

Figure 4.19 Schematic Diagram of Thermodynamic Effects ....... ..-.-..-.--.*----. --......-....

Figure 5.1 Photograph of the Overall Setup with the Gas Injection System .. .. . .. .. . . ....

Figure 5.2 Photograph of the Gas Syringe Pump .....................----....--.....-..-......-......

Figure 5.3 General Procedure to Measure the PVT Data of Polymer/Gas Solutions ...

Figure 5.4 Effect of Pressure on the Specific Volume of a PSICO, Solution at

Various Temperatures (a) O wt% of CO2; (b) 1 wt% of CO1; (c) 2 wt% of

C02(d) 3 wt% of CO,; (e) 4 wt% of CO, .................................................

Figure 5.5 Effect of Pressure on the Specific Volume of a PSICO, Solution at

Various CO, Contents (a) 220°C; (b) 200°C; and (c) 180°C .....................

Figure 5.6 Effect of Temperature on the Specific Volume of a PS/CO, Solution at

Various Pressures (a) O wt% of CO2; (b) 1 wt% of CO,; (c) 2 wt% of CO,

Pressures (d) 3 wt% of COZ; (e) 4 wt% of CO, .. . .. . ... . . . . ... . . . . - .. . . -. . . . . . .--.....

Figure 5.7 Effect of Temperature on the Specific Volume of a PSfCO, Solution at

Various CO, Contents (a) 276 Bar; (b) 207 Bar; and (c) 138 Bar ..............

Figure 5.8 Effect of COz Content on the Specific Volume of a PS/CO, Solution at

Various Pressures (a) 220°C; (b) 200°C; and (c) 180°C ...........................

Figure 5.9 Effect of CO, Content on the Specific Volume of a PS/CO, Solution at

Various Temperatures (a) 276 Bar; (b) 207 Bar; and (c) 138 Bar .............

Figure 5. I O Effects of Pressure on the Specific Volume of a Linear PBS/CO, and a

Branched PBS/CO, Solution at Various Temperatures (a) O wt% of CO,;

(b) 1 wt% of COz; (c) 2 wt% of CO, Temperatures (d) 4 wt% of- CO,; (e)

6 wt% of CO, ................................................................................

xii

Figure 5. I 1

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Effects of Pressure on the Specific Volume of a Linear PBS/CO, and a

Branched PBSKO, Solution at Various CO, Contents (a) 167OC; (b)

....................................... 157°C; and (c) 147°C ...--

Effects of Temperature on the Specific Volume of a Linear PBSKO, and

a Branched PBS/CO, Solution at Various Pressures (a) O wt% of CO,; (b)

1 w t 8 of COz; (c) 2 wt% of CO:; (d) 4 w t b of CO2; (e) 6 wt% of C O ....

Effects of Temperature on the Specific Volume of a Linear PBS/CO, and

a Branched PBSKO, Solution at Various CO, Contents (a) 276 Bar; (b)

...................................................................... 207 Bar; and (c) 138 Bar

Effects of CO, Content on the Specific Volume of a Linear PBSKO, and

a Branched PBS/CO, Solution at Various Pressures (a) 167°C; (b) 157°C;

...................................................................................... and (c) 147°C

Effects of CO, Content on the Specific Volume of a Linear PBSKO, and

a Branched PBSICO, Solution at Various Temperatures (a) 276 Bar; (b)

...................................................................... 207 Bar; and (c) 138 Bar

The Location of Experimental Data on Temperature vs. Density Graph at

Constant Gas Concentration ..................................................................

Effect of Temperature on the Binary Interaction Parameter, Delta (612), of

PS/C02 Solution at Various CO2 Contents (a) 276 bar; (b) 207 bar; and

........................................................................................... (c) 138 bar

Effect of Pressure on the Binary Interaction Parameter, Delta (S,,), of

PSKO, Solution at Various CO, Contents (a) 220°C; (b) 200°C; and (c)

180°C .................................................................................................

Effect of CO2 Content on the Binary Interaction Parameter, Delta (8J, of

PS/CO, Solution at Various Pressures (a) 220°C; (b) 200°C; and (c)

xiii

Figure 5.20 Effect of Temperature on the Binary Interaction Parameter, Delta (ôl2), of

Linear PBSICO, Solution at Various CO, Contents (a) 276 bar; (b) 207

............................................................................. bar;and(c)138bar 152

Figure 5.2 1 Effect of Pressure on the Binary Interaction Parameter, Delta (6& of

Linear PBS/C02 Solution at Various COz Contents (a) 1 67°C; ( 6 ) 157°C;

...................................................................................... and(c)147OC 153

Figure 5-22 Effect of Pressure on the Binary Interaction Parameter, Delta (6,,), of

Linear PBSKO, Solution at Various CO, Contents (a) 167°C; (b) 157°C;

and (c) 147°C ...................................................................................... 154

Figre 5.23 Effect of Temperature on the Binary Interaction Parameter, Delta (61L)7 of

Branched PBS/COZ Solution at Various CO, Contents (a) 276 bar; (b) 207

bar; and (c) 138 bar .............................................................................. 155

Figure 5.24 Effect of Pressure on the Binary Interaction Parameter, Delta (6,,), of

Branched PBS/COZ Solution at Various CO, Contents (a) 167°C: (6)

157°C; and (c) 147°C ........................................................................... 156

Figre 5.25 Effect of Pressure on the Binary Interaction Parameter, Delta (Ol2), of

Branched PBSKO, Solution at Various CO, Contents (a) 167°C; (b)

157°C; and (c) 147°C ........................................................................... 157

xiv

Table 3.1

Table 3.2

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

TabIe 4.7

Table 4.8

Table 4.9

Table 5.1

TabIe 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Calibration Experimental Results with Water ..,..,..,..........-.-..-..-......-..---.

Caiibration Experimental Results with Vegetable Oil ...................------.m..-

Summary of Materiai Properties for PP. PS, Linear PBS, and Branched

PB S . . . . . . . . . . . . . . . . . . . . . . . . . . - . - - - . - - - - -. -. . -. . . -. . - - -. . - - - . . . . . . . . . -. . -. . . - -. - . - . . . - . -. . . - - - - . . . . .- The FOV Equation Constants for PS, PP. Linear PBS, and Branched PBS

The MCM Equation Constants for PS, PP, Linear PBS, and Branched

PB S . . . . . . . . . . . . . . . . . . . . . . . . . . . - -. . - - - - . - - - - - - . . . - -. - -. . . . . . . - . - -. . . . -. . . - -. . . . . . . . . . . . . . . . . . . -. . . . . - - The S-L Equation Constants for PS, PP, Linear PBS, and Branched PBS

Experimental PVT Data of Linear PP ....... .........+...-.......-..... .. .-..-....

Experimental PVT Data for PS .....-.................-...-.........-.......................

PS Manufacturer's Reference Data using the conventionai Bellow type

Dilatometer (Japan Polystyrene Lnc.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . - . . . . . -. . -. . . -...... Experimental PVT Data for Linear PBS .................................................

Experimental PVT Data for Branched PBS .....................-.--.-------.+..........

Experimentai Data for the PSKO, Solution at Three Different Rotational

Speeds (5, 10, and 15 rprn) of the Gear Pump . . . . . . . . . . . . . . . ..... . . ... . . . . . .. ..... Experimental Data for the Linear PBS/CO, Solution nt Three Different

Rotational Speeds (5, 10, and 15 rpm) of the Gear Pump ..................... ....

Experimental Data for the Branched PBS/CO Solution at Three

Different Rotational Speeds (5, 10, and 15 rpm) of the Gear Pump .. . .. , -....

Estimated Parameters for Pure Potymers and CO, ....................................

Variables for the S-L EOS Mixing Rules ................... .. ..--..--......-...-....-

Calculated Values Using the S-L EOS Mixing Rules Based on

Experimental Data of PS .......................................................................

Table 5.7

Table 5.8

Table 5.9

Table 5-10

Table 5.1 1

Table 5.12

Table 5.13

Table 5.14


Experïmental Data of Linear PBS ..........................................................


Experirnental Data of Branched PBS .....................................................

Characteristic Parameters of MCM EOS for PSKO, Solutions as a

Function of the CO, Concentration ........................................................

Characteristic Parameters of MCM EOS for Linear PBSKO, Solutions as

a Function of the CO, Concentration ......................................................

Characteristic Parameters of MCM EOS for Branched PBSKO,

................................... Solutions as a Function of the CO, Concentration

Characteristic Parameters of S-L EOS for PSKO, Solutions as a

Function of the CO, Concentration ........................................................

Characteristic Parameters of S-L EOS for Linear PBSKO, Solutions as a

Function of the COZ Concentration .......................................................

Characteristic Pararneters of S-L EOS for Branched PBSKO, Soiutions

as a Function of the CO, Concentration .................................................

xvi

ABBREVIATION AND NOMENCLATURE

c = weight fraction.

k = Boltzmann's constant, J K

ni = number of molecules in component i.

M = molecular weight, g/male.

M,, = weight-average rnolecular weight, g h o l e .

M, = number-average rnolecular weight, g/mole.

m = mass flow rate of extrudates, g/min.

f i , = mass flow rate at gear pump, g/min.

4 = mass flow rate at syringe pump, g/min.

= mass fraction of component i.

= rotational speed of the gear pump, radmin.

= pressure, bar.

= inlet pressure of the gear pump, MPn.

= outlet pressure of the gear pump, MPa.

= pressure of the syringe pump, MPa.

= reduced pressure-

= characteristic pressure, bar.

= volumetric flow rate, rnUrnin.

= volumetric flow rate of the gear pump, r n h i n .

= ieakage flow rate of the gear pump, ml/min.

= volumetric flow rate of the syringe pump, r n h i n .

xvii

= gas constant, 8.31 J/(mole*K).

= number of lattice sites occupied by the r-mer.

= number of sites occupied by component i in solution state.

= number of sites occupied by component i in pure state.

= number of contacts per segment.

= melting temperature, OC.

= giass transition temperature, OC.

= temperature, K-

= reduced temperature.

= characteristic temperature, K.

= specific volume. cm3/,.

= specific volume at par pump, cm3/g/g.

= specific volume at syringe pump, cm.'/g.

= reduced specific volume.

= close-packed mer volume, cm3/ , .

= binary interaction parameter for S-L EOS.

= displacement of the gear pump per revolution, rnl/rev.

= volume fraction of component i in solution state.

= volume fraction of component i in pure state.

3 = density, g/cm .

= reduced density,

= characteristic density, g/cm3.

xviii

= segment-segment interaction energy.

a = "hard-sphere" radius.

ê* = interaction parameter per mer.

xix

1.1 PREAMBLE

Use of lightweight thennoplastic articles has continued to grow at a rapid pace for

the p s t two decades because of their unique properties as well as weight and cost

reductions. Since material costs account for a s much as 70% of the total cost of a plastic

product [l], using less materiai by creating bubbles (voids) in the plastic matrix to

produce thermoplastic foarns has becorne a desirable practice. Some unique properties of

thermoplastic foarns include cushioning, impact resistance, insulation, and buoyancy. In

addition, thermoplastic foams are warpage free for molded operations (e-g., injection

rnolding) due to interna1 pressure generated by blowing agents. Thus, thermoplastic

foarned articles are very attractive compared to non-foamed parts because of the reduced

weight which translates into superior cost-to-performance and good strength-to-weight

ratios [Z].

Thennoplastic foarns c m be created by introducing a gas, e-g., CO?, under high

pressure into a polymer maulx during a plastication process. Then, a thermodynamic

instability can be induced by rapidiy dropping the solubility of the gas in the polymer by

decreasing the pressure andor temperature. This thermodynamic instability results in the

generation of bubbles, which causes expansion of the thennoplastic part 131.

Successfùl production of lightweight thermoplastic foarns requires knowledge of

the thermodynarnic behaviour of gases and polymers including gas sorption, polymer

swelling, and the pressure-volume-temperature (PVT) relationship. For example,

compressed gases such as supercritical carbon dioxide can be quite soluble in a polymer

matrix at elevated pressures, and the absorption of the gas c m cause substantial swelling

in the polymer 141. However, the availability of such valuable information has been

limited to only a few specific materials in a specific range of temperature and pressure.

Especially, the PVT properties of polymer/C02 solutions have not k e n measured in a

melt condition because of the difficuities involved in the rneasurement.

1.2 EFFECTS OF GAS IN A POLYMER

Absorption of a gas into a molten polymer causes swelling, a reduction of the

viscosity, and a reduction of the meIting temperature [4,5]. Two probable mechanisms

may be responsible for these phenomena: the first mechanism is the dilution of chah

entanglement and the second mechanism is the increase in free volume.

There is a free volume between the molecules, and the free volume varies from

polymer to polymer. The free volume does not refer to microvoids nor microbubbles.

The increase in specific volume due to temperature and pressure changes is attnbuted to

increased movement of polymer rnacrornolecules that results in an increase of the free

volume. Furthemore, absorption of a gas into a polymer greatly increases the free

volume [2,6,7].

In order to understand the effects of the gas content in a polymer/gas solution, one

rnust estimate accurately the composition and density of the polymer/gas solution over a

wide range of pressure and temperature. In the processing of solutions containing a

polymer with dissolved gas under pressure, the swollen volume is an important property.

Estimation of the swollen volume can be achieved by comparing the volume of a pure

polyrner to the volume of a polymerfgas solution.

1.3 THESIS OBJECTIVES AND SCOPE OF RESEARCH

The objective of this study is to develop a novel technique to accurately measure

the pressure-volume-temperame (PVT) relationships of polymer/gas solutions. To our

knowledge, the PVT properties of polymedgas solutions in a molten state of the polyrner

have not been previously reported- The PVT relationships of viscous polyrneric materials

(with or without gas) are important in polymer processing. For example, the

compressibility and shrinkage effects of a materiai in a moiding process can be predicted

through the PVT relationship. In addition, PVT relationships are required in the

measurement of some thermodynarnic properties, such as the surface tension of melts

[8,9] and the solubility and diffusivity of gas in a rnelt [IO-121. Therefore, the

measurement of PVT relationships for polymerfgas solutions would provide valuable

information. The new approach presented in this study is a breakthrough, since no one

has successfully measured the PVT relationships of polymer/gas solutions in a molten

state of the polymer.

The apparatus employed to measure the PVT relationship in this study includes a

positive-displacement gear pump in an extrusion system. The basic rationale of the

design is to determine the specific volume (or equivalently, the density) of the

polymer/C02 solution by measuring the volume and mass flow rates separately, while

controlling the pressure and temperature independently. An experimental PVT

measurement system has been constnicted, and the functions of the designed system have

been verified through experiments [13]. As a first step, the gear pump was calibrated

using a separate setup with water and vegetable oil. An error analysis was conducted to

investigate the accuracy of the calibration of the gear pump [14]. The calibrated gear

pump was mounted on a tandem extruder and experiments were conducted. The

controllability of the pressure and temperature of the polymer melt flowing in the gear

pump was ais0 investigated- The measured PVT data of the pure polymers were modeled

and compared with vaiues published in the Iiterature to confinn the validity of the

designed system. Once the system was verified, various concentrations of carbon dioxide

were injected into polymers and their PVT properties in single-phase solutions were

rneasured.

1.4 ORGANIZATION OF THE THESIS

This thesis is organized in seven chapters. Chapter 1 presents an introduction to

the use of gas in polymer processing. The chapter briefiy describes the formation of

thermopIastic foarns and the effects of gas in a polymer matrix. The concise objectives

and overall strategies are also discussed.

Chapter 2 describes the literature survey and investigations of the theoretical

background associated with the measurement of PVT relationships. The existing

methods of measuring the PVT properties of pure polymers are discussed. The

theoretical background and thermodynarnic relationships of PVT are contemplated. In

addition, various empincal and theoretical equation-of-states (EOS) models are

examined.

Chapter 3 explores extensively the design approaches associated with this novel

technology of measunng the PVT relationships. The basic strategy for achieving the

objectives is explored and the axiomatic design method, used for analyzing the design, is

reviewed. The functional requirements (FRs) and design parameters @Ps) are presented.

and the design matrix used in the study is discussed. In addition, a method for

determination of the positive displacement volume of a gear pump is presented- This

chapter proposes the design concept CO measure PVT relationships for polymerfCOz

solutions and thoroughly elaborates the strategies to verify the design concept.

Chapter 4 describes the measurement and modeling of the properties of pure

polymers. It explains how the strategies proposed in Chapter 3 c m be used successfully

in determining the PVT relationships. The measured PVT properties were compared

with data published in the literature to validate the designed systern. The experimental

PVT properties of the pure polymers were correlated using the three different equations-

of-state (EOS).

In Chapter 5, the measurement of PVT properties for single-phase polymer/gas

solutions is described for various weight percentages of CO?. The chapter describes the

methods to inject carbon dioxide into a molten polymer matrix and to measure the PVT

relationships of polymers/gas solutions. From the experimental data, the swollen volume

due to the gas absorption is presented at various CO2 contents, temperatures, and

pressures. Based on the experiments, efforts were made to derive theoretical rnodels to

describe the PVT behaviours of polymer/C02 solutions as a function of CO2 content,

temperature, and pressure.

Chapter 6 sumrnarizes the work performed in this study to achieve the objectives

and conclusions are presented from the work. Finally, recornmendations for Future work

are presented in Chapter 7. The recommendations may guide future research to further

investigate the PVT relationships of polymedgas solutions.

LITERATURE SURVEY AND THEORETICAL BACKGROUND

2.1 CONVENTIONAL DILATOMETERS FOR PVT MEASUREMENT

Several papers [I5-201 have reported the PVT relationships of pure polymers.

However, the majority of these data is presented at low-pressure conditions, and the

accuracy of the data still needs to be verïfied. The biggest problem associated with the

measurement of PVT properties using the conventional method is polymer degradation

due to its long exposure time to high temperature. Most of these data were gathered

using some sort of dilatometer. A dilatometer is an instrument that measures the volume

of a material as a function of pressure and temperature.

Foster et al. [15] used a cylinder-piston type rnethod in a compressibility chamber

to measure the PVT properties of plastic. Figure 2.l(a) illustrates a typical cylinder-

piston type dilatometer. The volume changes were determined by rnultiplying the cross-

sectional area of the piston by the linear displacement of the piston. The major problem

associated with this type of dilatometer is the leakage that occurs between the seal and

the piston perimeter. Another limitation of the cylinder-piston dilatometer is that only the

molten state of the polymer can be utilized for density measurements.

Zoller [16,17] and Sato et al. 1191 used a bellow-type dilatometer. The flexible

bellow apparatus confined a polymer sample and the liquid mercury in which the

polymer was immersed. The PVT properties of the polymer sample were determined by

subtracting the PVT properties for pure mercury from that for the polyrner sample and

mercury. First, the PVT properties of the polyrner sample along with the mercury were

determined by changing the temperature or pressure and measuring the volume change

that occurred when the bellows moved. Pressure was applied by hydraulic oïl, causing

the bellows to contract until the pressure of the sarnple inside the bellows balanced the

applied pressure. Then, the PVT properties of pure mercury were obtained and these data

were subtracted from the measurements made with the polyrner sample and mercury to

obtain PVT properties for the polymer sample alone [20]. Figure 2.1 (b) illustrates a

typical bellow-type dilatometer with a polymer sample imrnersed in mercury. The

advantage of this type of dilatometer is that it can operate at both solid and molten States

of the polymer [19]. Although the flexible bellows method provides accurate

measurements, obtaining PVT data using such a method is very time-consuming and

quite costly. Furthemore, material degradation associated with this type of method is

severe due to the long time required to maintain therrnodynamic equilibrium at each

temperature and pressure.

Some attempts have been made to measure the PVT relationships for multi-phase

petrolrum and gas. Rahman et al. [SI] used two chambers of piston cells along with gas

chromatograph equipment. This method was similar to the piston-cylinder method

mentioned above, but was limited to measure the multi-phases of gas and petroleum

liquids. This method still requires further verification of its accuracy.

Another attempt to measure the PVT relationship of a polymer/gas solution was

made by Hirose et al. [22]. They determined gas absorption by measuring the change in

length of a thin polymer sample in its solid state. Figure 2.2 depicts a schematic of the

apparatus. They used rectangular film strips (60 mm x 3 mm x 90 pm) suspended in a

g l a s pressure cell. A cathetometer was utilized to measure the elongation of the sample

due to gas absorption. A cathetometer is a type of comparator with a tetescope equipped

with a cross wire rnounted on a vertical sliding column to measure vertical distances on

fairly near objects. The pressure range for this apparatus was up to 50 atm and

temperature was in the range of 25°C to 5S°C. The device could measure on1 y solid-state

sarnples. Problems associated with this type of dilatometer inciude distortion of the

sample (Le., warping), which may hinder accuraçy of measurements. Also, the samptes

rnay be affected by anisotropy of the molecula. chains because polymer chains may be

oriented due to the thin nature of the polymer samples that were extnided. Lastly, the

working ranges of pressure and temperature are too low to determine the PVT behaviour

of the melt state effectively. \

Even though conventional dilatometers are capable of measuring the PVT

relationship of pure polyrners or the PVT properties of polymer/gas solutions in solid

state, no method or instrument has been reported previously to rneasure the PVT

reIationships of poIymer/gas solutions in the rnolten state of the polyrner.

2.2 SOLUBILITY MEASUREMENTS

The PVT relationships are related to the solubility of the gas in the polymer.

Thus, understanding of the solubility of gas in the poiymer is imperative. From the

acquired solubility data, one may predict PVT relationships of a poiymer/gas solution

[23,24]. However, this method still requires extensive research on the mixing relationship

between a gas and a polymer. There are three widely used methods to measure the

solubility of a gas in a polymer: a pressure decaying method 125,261, a cloud po

measuring method [27], and a gravimetric method [4,28].

Sato et al. [25] measured the solubilities of nitrogen and carbon dioxide

int

in

polystyrene using a pressure decaying (barometric) method. As one would expect, the

solubilities of gases increased dong with the pressure. The solubility of carbon dioxide

decreased with increasing temperature. The principle of the pressure decaying method is

measurement of gas sorption in a polymer sampIe, which is based on the difference

between the amount of gas, initiaily contacting the polymer and the amount remaining in

the gas phase. The pressure decaying method requires rather careful calibrations and can

only be used with gases whose properties of EOS are known accurately.

In Mishima et al.'s work [27], the solubility of supercriticai carbon dioxide in

polyethylene glycol was measured based on observation of the cloud point using a laser.

A He-Ne laser passed through the ceIl and the intensity of the laser was monitored with a

photo sensor. Near the cloud point, carbon dioxide was charged carefully. The intensity

of the transmitted laser decreased sharply when the charged amount of CO2 passed

through the cloud point and the cloud point fraction was measured.

Wissinger et al. [4] investigated the CO? sorption in polycarbonate, polymethyl

methacrylate, and polystyrene using the gravimetric method. The gravimetric method

was used to find sorption from direct measurements of weight gain of a polymer sarnple

as it absorbed gas. The limitation of this method is that relatively low gas solubilities can

only be measured with a sensitive weighing device like a quartz spring. In addition, the

swelling compensation must be applied due to the increase in the specific voiume, which

results in buoyancy.

2.3 THEORETICAL BACKGOUND

2.3.1 Thermodynamics (PVT Relationships)

PVT relationships c m be used to interpret physical and chemicai phenornena such

as melting, crystallization, g l a s formation, fluid sorption, and etc. The experimental

measurements of those data should be made at a quasi-state equilibrium, Quasi-state

equilibrium refers to independence of path andor time. For example, the volume only

depends on pressure and temperature, without effects of time or path taken in the

measurement cycle. The only true equilibrium state of a polyrner is at the melt state

because the sotid state and glassy state of polymers often exhibit path dependence [29].

Thus, clhe PVT properties for polyrners should be measured in the melt state.

Furtherrnore, there are several issues which may hinder the accuracy of the "true" PVT

properties. The most important issue is inaccuracy due to material degradation at high

temperatures because it usually takes a long time for a polymer meit to reach equilibrium.

Even crystalline polymers may undergo crystallization at high temperatures and

pressures, making accurate acquisition of PVT properties very difficult. In order to fi l1

the gaps to achieve the accuracy needed, several empiricai equations-of-state (EOS), such

as the Tait equation [30], and theoreticai (statistical) EOS, such as Flory, Orwoll, and Vnj

(FOV) [31], Modified Ce11 Mode1 (MCM) [32,33], Sanchez and Lacombe (S-L)

[24,34,35], and Sirnha-Somcynsky (SS) [23] have been proposed to describe polymer

properties at different States.

The typical PVT behaviour of semi-crystaliine polymers (e-g., PP) at quasi-

equilibrium is shown in Figure 2.3. The melting region is clearly shown in the figure by

a large change in the specific volume, where the melting point is denoted by Tm (P) and

the glass transition region is depicted by Tg (P).

The PVT relationship c m be analyzed when time-dependent effects are

negligible:

f ( P , V , T ) = O or V = f ( T , P ) (2-1)

From the above equation, the EOS can be interpreted such that if any two of the

properties are fixed, the third is determined. For example, if the temperature and the

volume are known, one can determine the pressure using an EOS model.

Equation 2.1 leads to the following equation, which can be gathered from PVT

reiationships:

dv - = p d T - KdP v

where

p is the isobaric (constant pressure) compressibility, K is the isothermal compressibility

(constant temperature), and B is the bulk modulus.

2.3.2 S tatistical Thermodynarnics [36-3 81

Statistical thennodynarnics was developed in the early 19" cenniry. rnainly by

Boltzmann in Germany [39] and Gibbs in the United States 1401. With the initial arriva1

of quantum theory, Bose (Indian physicist, 1894-1974), Einstein (German physicist.

1879- 1955), Ferrni (Italian physicist, 190 1 - 1954). and Dirac (English physicist, 1902-

1984) made modifications to Boltzmann's original ideas and succeeded in clarifying

some of the uncertainties of the Boltzmann statistics [36]. The statisticai approach has

close connection to the themodynamics and kinetics theory. For systems of particles, in

which the energy of the particles can be determined, one can derive the EOS by statisticai

means.

Detailed derivations of statistical thermodynamics are not included in this chapter.

Rather, we illustrate several EOS (e-g., FOV, MCM, S-L, and SS) that were derived

using the statistical thermodynamics approach. Rodgers et al. [18] investigated the PVT

relationships for a number of pure polymeric materials and used several EOS models to

illustrate applications of statistical thermodynamics theory.

2.4 EQUATION-OF-STA'E (EOS)

The equations-of-state are very important in describing the therrnodynamic

properties of liquids and gasniquid solutions. The EOS is correIated using the

experimentai data in a molten state, far above the glass transition temperature. For

polymer/gas solutions, the densities of both the gas and polymer are required to interpret

the effects of gas in a polymer matrix. The Free volume fractions are used to describe the

swelling phenomena. Empirical EOS, such as the Tait equation, and theoretical

(statistical) EOS, such as Flory, Onvoll, and Vrij (FOV), Modified Ce11 Mode1 (MCM),

Sanchez and Lacombe (S-L), and Simha-Sorncynsky (SS), are illustrated in this section.

2.4.1 Empirical Analysis: the Tait Equation

One of the rnost common and the earliest empirical representation of the PVT

relationships for polyrners is the Tait Equation [17,30]. The equation is an interpretation

of the isothermal compressibility model of a volume and pressure relationship. The

equation is shown below:

V(P , T ) = V ( 0 , T) { I - C h [ I + P / B ( T ) ] J

where the coefficient C is 0.0894 117, f 81 in most cases.

The compressibility increases exponentially with an increase in the temperature

and pressure. The zero-pressure isothenn V (0, T) is usually given by:

V (O, T) = V, exp (aT) (2.6)

where ai is the thermal expansion coefficient. The Tait parameter B(T) is usually given

by :

B (T) = Bo exp (-BIT)

Therefore, the Tait equation has four parameters, namely Vos a, Bo, and Bi, describing

experimental PVT properties. These variables can be evaluated from the experimental

PVT properties by the least-squares regression.

2.4.2 Theoretical EOS

Knowledge about the thermophysical properties of materials requires

experimentally measured data or predictions obtained from a suitable mathematical

modei, such as an EOS or an empirical model. Theoreticd EOS modeling represents

PVT behaviour of polymers with respect to the forces existing between polymer

molecules. The forces can be obtained from quantum mechanics. The use of statistical

mechanics enables correlation of quantum and continuum mechanics, which ailows

statistical averages to be expressed using molecular variables. The direct evaluation of

the forces is very diff~cult and several assumptions must be taken into consideration

[37,38]. The common statistical mechanics equation is:

where k is the Boltzmann's constant and Q is the canonical partition function. From the

equation above, severai models have k e n proposed and they can be classified into three

groups: ce11 theories, hole theories and lattice-fluid theories. Most of the theoretical EOS

theories represent the PVT relationship of a liquid polyrner in a dimensioniess form. The

EOS presented in the following sections are rnainly for the PVT relationships of pure

poIymers with the exception of the Sanchez-Lacombe EOS, which incorporates mixing

mles to accommodate polymer/gas solutions.

2.4.2.1 Ce11 Theory - Flory, Orwoll, and Vrij (FOV) Mode1

One of the most extensively applied ceIl theories is the Flory, Onvoll, and Vrij

(FOV) mode1 where the mer can move only within a lirnited volume of space mainly due

to the close presence of neighbounng mers. This limited volume formed by neighbouring

mers is known as a "ceil". Flory, Orwoll, and Vrij [31] derived the EOS based on the

free volume of the cell:

w here

s is the number of contacts per segment, q is the segment-segment interaction energy, k is

Boltzmann's constant, a is the "hard-sphere" radius, and v* is identified as the h&ore

ce11 volume.

2.4.2.2 Cell Theory - Modified Cell Mode1 (MCM)

Based on Prigogine et aL's 1411 ce11 model (CM), Dee and Walsh [32,33]

modified the CM model to decouple the ce11 geometry. Dee and Walsh introduced a

numerical factor, q, which compensates for the effects of the hard-core cell volume. The

modified ce11 model (MCM) is identicai to the ce11 mode1 (CM), except for the

introduction of the numericd factor, q. The MCM model is described as:

where the reduced variables and characteristic parameters are the sarne as in the FOV

model. The factor, q, was found to be approximately 1.07 for several polymers which

corrects the hard-core volume from the CM. The value of q (Le., 1.07) corresponds to

approxirnately a 25% increase in the hard-core volume of the ce11 model. The MCM

EOS provides the anaiytical simplicity of the FOV model and exceeds the accuracy of the

ss [33].

2.4.2.3 Hole Theory - Simha-Somcynsky (SS) Model

The Simha-Somcynsky (SS) model is based on hole theories. The SS EOS in a

dirnensionless form is as follows [23]:

with

w here

P*, v* and T* are characteristic parameters. Y is the fraction of occupied sites- L'sually,

s/3c is 1. Through a non-linear lest square analysis, experimental data cm be fit to this

model. Even though the accuracy of the model is good, it is quite complicated to acquire

the characteristic variables.

2.4.2.4 Lattice-Fiuid Theory - Sanchez-Lacornbe (S-L) Mode1

Pitre Polvmer

Lattice-fluid theories are based on a quasi-crystalline structure, with sites on the

lattice occupied either by segments of the chain molecule or empty "holes". The

derivation of the lattice-fluid model can be found in Reference 1421.

Sanchez and Lacombe [24,34,35] have derived a dimensionless equation based on

the lattice-fluid model:

-

where r is the number of Iattice sites occupied by the r-mer. For a polymenc liquid, r

becomes infinity and the equation of the state cm be simplified to

The reduced density, pressure and temperature are defined as:

where p*, P*, and T* are the €OS parameters, e* denotes the interaction per mer, v* is

the close-packed mer volume, R is the gas constant, and M is the molecular weight.

S-L EOS l7zeot-v for Polvmer/COi Solutions

Lattice models (Le., S-L EOS) have k e n used successfully to describe

polymedgas solutions [43]. The lattice and gas mode1 is based on distribution of

moIecules and holes (or vacant lattice sites) in a three-dimensional sense.

Accurate interpretation of PVT and solubility behaviour using an EOS is quite

challenging. A better understanding of the interaction between gas molecules and

polymer molecules is required in mixed solutions. The impetus to rneasure the

interrnolecular interactions has been addressed by several authors with spectroscopic

measurements [443

The EOS of a polymer/gas solution is basically the sarne as for a pure polymer.

However, reduced variables such as p . F . f . and r as well as the interaction parameter

between COz (cornponent 1) and a polymer (component 2) should be changed by mixing

d e s .

There are three combining rules associated with a solution of gas and polyrner.

The first combining rule is the conservation of the close-packed volume. If a CO-

molecule occupies r1° sites in its pure state and has a close-packed molecuIar volume of

ri4/1*. ii occupies r, sites with v* close-packed volume in the solution. Then the

relationship of these two volumes becomes:

rl "VI' = rI v* (2.15)

The second mle keeps the total number of pair interactions in the close-packed

soIution the s m e as the sum of the pair interactions of the components in their pure

close-packed States- This can be described as:

rl ONI + rz"N2 = rN (2.16)

where and l& are the number of molecules of component 1 and 2. respectively. N is

the sum of Nl and N2, and r is the total number of sites occupied by both cornponents in

the solution. From these combining rules, the following relations can be obtained:

V* = @,4/,* + hOvz* (2.17)

r = xp-1 + (2.18)

where G I O and & O are the volume fractions of component 1 and 2 in their pure state.

Likewise, xi and xz are the mole fraction of component 1 and 2, respectively.

The third combining rule defines the charactenstic pressure P* of the solution as a

pair wise additive:

P* = @;PI* + & 2 ~ 2 + 2@&Piz* (2.19)

Pi2* = ( P I *p2*ih (1-&) (2.20)

where @ denotes the volume fraction of the subscript component in the solution and is

the interaction parameter between the two components. 612 is the only interaction

parameter needed to completely characterize a binary solution. It corrects the deviation

of the characteristic pressure of the solution from the geometric mean of the characteristic

pressures of each component.

From the above three mixing rules, we can also gather the following equations:

where ml and rnz are the respective mass fractions.

In order to estimate the solubility of CO2 in a polymer, an equilibrium state of the

two phases must be assumed. It is also assumed that the polymer is not dissolved in the

gas phase. The chemical potentials of COz in the upper (gas) and the lower (polymer)

phase are the same:

where the superscnpts of "G" and " P denote the gas and polymer phase, respectively.

The difference of the chemical potentials of CO2 in the two phases is given by:

The above equation indicates that if we acquire the interaction parameter, Srr, and

the PVT propertics of the components at the solubility pressure, the equilibrium solubility

of CO2 dissolved in the polymer c m be determined. One obstacle, however, is that there

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

the solubility data is available in a limited range, one can determine the parameter by a

non-linear regression analysis.

2.4.3 Remarks on Theoretical EOS

In the ce11 theory (i.e., FOV), the changes in the volume together with the changes

in the temperature and pressure c m be explained only by the changes in the ce11 size.

The MCM EOS introduces the correction factor for the close-packed volume to reduce

errors associated in the ce11 theory. On the other hand, in the hole theory (Le., SS), the

number of holes and the change in ce11 size explains the change in the volume which

plays a minor role. In the lattice theory (Le., S-L), the lattice size is fixed and it explains

the change of volume only by the number of vacant sites 1451.

As pointed out by Dee and Walsh [32], the lattice theory is more appropriate to

depict the thermodynamic properties of the gaseous state. So far, it has been noted that

the free volume theories, especially the hole theories, can delineate the thermodynamic

properties of a liquid well. III the hole theory, the free volume is treated by the concept of

linear superposition between solid and gas [46]. In the work of Simha and Somcynsky.

the free length concept is introduced. This mode1 successfully depicts the

thermodynarnic properties of liquids; however, it cannot account for the pressure effect

properl y [47].

In this study, we used the FOV, MCM and S-L EOS to mode1 the experimentai

data due to the relatively good accuracy and ease of data manipuIation to acquire the

desired theoretical values for these models.

2.5 SOLUBILITY

The solubility is a measurement of dissolving potential. The maximum arnount of

gas that can be dissoived into the polymer (Le., the solubility) depends on the system

pressure and temperature and can be estimated by Henry's law [49,50]:

where c, = solubility of gas in the polymer, cm31g or g(gas)lg(polymer),

H = Henry's law constant, cm3 [STPJ/g-Pa,

p, = saturation pressure, Pa.

The constant H is a function of temperature which is described by:

where R = gas constant, J/K,

T = temperature, K,

H, = solubility coefficient constant. cm3 [STPIfg-Pa,

AHs = moiar heat of sorption, J.

This law is based on the ideai state of an extremely dilute solution. Furthemore,

the law does not consider interactions between the polyrner and gas. Due to the above

limitations, Henry's law deviates frorn the actuai solubility of gas in polymers, especialty

at high pressure.

Figure 2.4 [SI] illustrates the relationship between pressure, temperature, and

solubility. The solubility increases when the pressure goes up. The temperature does not

significantly affect the solubility compared to pressure.

2.6 PHYSICAL BLOWNG AGENTS

There are two types of blowing agents: physicd and chernicai blowing agents.

Chernical blowing agents are quite expensive and quite toxic in most of the cases.

Physical blowing agents such as CFC and HCFC are harrnful to environment. lnert gases

have been considered to be an aiternative, whether used alone o r used together with

conventional agents.

Nitronen

One of the readily available gases is nitrogen. Nitrogen is inexpensive,

nonflammable, nontoxic, environmentally safe and chemically inert [5]. However,

nitrogen has a very low solubility in polymers. As a result, absorption of nitrogen into a

polyrner requires a very high pressure as well as a long processing time.

Ca rbon D ioxide

Carbon dioxide, which is environmentally benign, has the same advantages as

nitrogen, but aiso has better solubility than nitrogen. Furthemore, carbon dioxide cm

work as a nucleating agent. Thus, carbon dioxide gas was chosen as the blowing agent

for this study.

Force n Dia1 Indicator

(a) Cylinder-Piston Type Dilatometer for Molten Polymer

Polymer SampIe

Bellow Type Dilatometer with a Polymer Sample immersed in Mercury

Figure 2- 1. Conventional Dilatometers

gas s vacuum

cathetometer

Figure 2.2. Solid-State Diiation Device

T

Figure 2.3. PVT Behaviour of Semi-crystaihe Polymers

t Gas Solubility

Temperature

Figure 2.4. Relationships between Pressure-Tempe ramre-Solubilit>: [5 1 ]

CONCEPTUAL DESIGN OF PVT MEASURING DEVICE

BASED ON AXIOMATIC DESIGN

In this chapter, the detailed description of the overall strategy for measuring the

PVT reiationships using the gear pump, based on the axiomatic design approach, and the

detailed calibration of the positive displacement gear pump are presented. One of the

most important steps prior to measurement is calibration of the gear pump. Both water

and vegetable oil were used to verify the exact volumetnc flow rate. The enumerated

procedure for calibration of the gear pump is also presented in this chapter.

3.1 OVERALL STRATEGY

The overall strategy for the PVT measurement of polynierlgas solutions is

depicted in Figure 3.1 in a fonn of step-by-step flow chart. The axiomatic design

concept, descnbed in the following section, is utiiized to effectively design the PVT

measuring apparatus for polymer/gas solutions. Once the design is finalized, the

positive-displacernent gear pump is calibrated to determine the volumetric flow rate per

revolution of the gear in the gear pump. Two different mzterials, water and vegetable oil,

are used for the calibration of the gear pump. The apparatus is built based on a tandem

extrusion line on which the gear pump is rnounted, The critical sets of experiments for

pure polymers are performed to verify the design by comparing the experimentai results

with the data from the literature. Once the proposed system is verified, carbon dioxide is

injected into a molten polymer matrix to form a single-phase polymer/gas solution. The

mass and volume flow rates of the solution are acquired- Based on these two sets of

results, the specific volume of the solution with respect to pressure and temperature is

gathered. The experimental results are then mathematically interpreted using various

EOS to predict the trends.

3.2 BACKGROUND OF AXIOMATIC DESIGN APPROACH 1521

In order to effectively come up with the strategies to achieve the goal, the

axiomatic design concept, a systematic method for guiding the design process, was used.

The approach consists of the following steps: i) establishment of desired needs; ii)

conceptualization of design solutions; iii) anaiysis of the proposed solutions; iv) selection

of the most suitable design from the proposed ones, and v) implementation of the chosen

design. The importance of the axiomatic design approach is to Iet designers be aware of

the potential effects of the design variables and shift their paradigms into goal oriented

thinking.

The first step in designing a product or a process is to come up with a set of

fitncrionnl requirements (FRs) that represent "what we wmt" in the design. In order to

achieve the FRs, design parameters (DPs) are defined. Each DP is selected to satisfy the

corresponding FR.

The axiomatic design approach enables identification of good designs by

satisfying two design axioms: the independence of Functional requirements (FRs), and the

minimization of the information content. The fïrst axiom guarantees independent control

of each FR, and the second ensures simplicity of the design.

The reIationship between FRs and DPs can be expressed by a simple matrix

equation shown below:

The elements AG in matrix A c m be either "X" or "0 . Element "X" indicates a strong

relationship between the corresponding FR and DP, and "û" indicates a weak or no

reIationship. The first axiom, the independence of FRs, is satisfied if the matrix is

diagonal (uncoupled design) o r triangular (decoupled design). Otherwise, the design wiIl

be coupled, which means that the FRs cannot be controlled independently.

Functionai requirements and design parameters can be decomposed into lower

levels to form hierarchies. The hierarchy decomposition simplifies the design process

one level at a time. In general, the primary parameters are identified in the upper level,

and the secondary parameters (or details), if my, are examined in lower levels-

3.3 OVERALL DESIGN FOR PVT MEASUREMENT

In the processing of polyrner/gas solutions, the properties of polymer are

significantly affected by the presence of gas in the polymer- Particutarly, volume

swelling due to the dissolved gas is important because it significantly affects the

processing conditions. The basic rationale behind PVT measurements for a polyrner/COz

solution is to measure the volume and mass flow rates separateiy while independently

controlling the temperature and pressure in an extrusion system. Therefore, if the

volumetric fIow rate (Q) and mass flow rate (th) can be measured accurately in a

continuous extrusion process at a specific temperature and pressure, then the specific

volume (or equivaiently, the density) can be calculated using the equation below:

The positive displacement nature of a gear pump can be utilized to rneasure the

volume flow rate of a polymer/C02 solution, and the PVT measurement system is

designed based on a tandem extruder with the gear pump. Figure 3.2 shows a schematic

of the designed PVT measurement system.

The axiomatic design approach is used to ensure that each required processing

parameter could be independently controlled to allow measurement of the PVT

properties. In order to measure the specific volume. we first need to satisfy four

(primary) functionai requirernents (FRs). The FRs at the primary level are the following:

FR, = Plastication of a polymer

FR, = Formation of a polymer/gas solution

FR, = Measurement of the volumetric flow rate for a given temperature and

pressure of the solution

FR, = Measurement of the mass flow rate of the solution

The design parameters @Ps) chosen to satisfy the functional requirements are the

following:

DP, = Screw motion in the first extruder of a tandem extrusion system

DP, = Convective diffusion device with temperature and pressure control devices

DP, = Rotational speed of the gear pump with AP = O

DP, = Collection of sampIe per unit time using a srnaII diarneter filament die

Plastication of the polyrner can be achieved by two heat sources in the system.

The primary source is the frictional heat generated by the rotation of the plasticating

screw in the first extnision barrel. The secondary source is the extemally mounted band

heaters. The band heaters are important in the start-up penod to promote easy rotation of

the plasticating screw. However, the band heaten cannot be the prirnary source of heat to

plasticate polymers because polymers are poor conductors. The mechanicd shear work

done on the polyrner by the motion of the screw makes a significant contribution to the

melting of the polymer pellets.

Formation of a single-phase polymer/gas solution requires an injection of a gas

into the molten polymer with the assistance of a convective diffusion device [53, 541.

The amount of gas is metered by a processor controlled gas injection syringe pump.

which supplies the gas at a high pressure into the plasticated polymer Stream in the

extruder forming a single-phase polymer/gas solution. As the polymer is conveyed in the

extruder barrel, the shear field generated by the plasticating screw stretches the gas

bubbles, ultimately breaking them into smaller bubbles. This process eventually leads to

dissolution of the gas into the polymer matnx. To further assist in the formation of a

consistent polymer/gas solution, a static mixer and band heaters can be used. The arnount

of CO2 injected into the Stream of polymer melt should be maintained below the

solubility limit because an excessive gas arnount would result in undesirable voids, which

alter the final product. The required gas flow rate c m be determined by specifying the

gas weight percentage based on the mass flow rate of the polymer/gas solution.

Therefore, close control of the gas-to-polymer weight ratio is important in the continuous

formation of the polyrner/gas solution.

Since the leakage between the inlet and outlet of the gear pump must be

minirnized to effectively measure the volumetric flow rate, precise pressure control is

needed. The second extruder with a large volume capacity can be used to reduce the

pressure fluctuations. The upstrearn pressure of the gear pump can be controlled by

varying the rotational speeds of the screws while fixing the gear pump speed. Changing

the cross sectional area of the variable resistance valve attached to the gear pump c m

control the downstream pressure.

In order to control the temperature, the second extruder and a heat exchanger

consisting of a homogenizing static mixer and a cooling channe1 can be utilized. The

temperature of the polymer melt flowing out of the plasticating extruder (i-e., the first

extruder) is typically very high. This high melt temperature can be lowered unifomly

while rnaintaining a high pressure by the second extruder. In order to reduce the

additional heating by the shear motion of the screw, the rotational speed of the screw for

the second extruder should be maintained low. The c o o h g of the melt using a second

extruder is effective only in a certain range. If the melt temperature is to be lowered

further, a heat exchanger can be also utilized.

When the system attains the desired temperature and pressure condition, the

volumetric flow rate G: the polyrner/COz solution can be determined by measuring the

rotational speed of the positive displacement gear pump. The specific volume can be

measured only when the pressure difference between the inlet and outlet of the gear pump

becomes zero, indicating there is no leakage. In order to mesure the volume flow rate of

the gear pump accurately, the volume displacement of a gear pump per revolution needs

to be calibrated. The calibration of the gear pump was conducted carefuIIy using a

separately designed setup, as described in the next section. Based on the calibrated

throughput of the gear pump, the specific volume can be determined by multiplying the

rotational speed of the gear pump by the volumetnc displacement per revolution of the

gear pump.

The mass flow rate can be determined by collecting the extruded melt for a fixed

time and measuring the weight. In order to facilitate the collection of extrudate for mass

flow rate measurement, a smail diameter filament die is attached at the end. Since gas

loss from the extruded foarn is inevitable 155, 561, the measurement of the mass flow rate

of a po1ymer/CO2 solution might not be accurate by simply weighting the collected

extrudate. It is suggested that gas loss be induced as much as possible by increasing the

temperature of the filament die and reducing the diameter of the die. Thus, the mass flow

rate of the polymer/C02 solution is calculated by adding the measured mass fiow rate of

the degassed polyrner and the mass flow rate of CO2, which is read from the synnge

PumP-

Based on the acquired volume and the mass flow rates at a given temperature and

pressure, the specific volume c m be calculated as the ratio of the mass flow rate to the

volume flow rate (Equation 3-3).

The design equation for the proposed design may be written as the foliowing:

In order to corne up with a good design, we should either have a decoupled design

(triangular matrix) or an uncoupled design (diagonal matrix). The gas injection, diffusion

in the solution formation device, and the measurement of volume and mass flow rates do

not affect the plastication because the polymer is completely plasticated by the screw

before the plasticated polymer reaches the gas injection system. Thus, the elements Al?,

Al3, and Al? are zero. The formation of a polymer/gas solution is a function of the screw

motion as well as the convective diffusion device and its temperature and pressure

control devices. The screw motion induces dif is ion of the injected gas. Thus, elements

AZI, and Aï should be non-zero. However, the formation of the polymerlgas solution is

not affected by the measurement of mass and volume fiow rates, The measurernent of

volumetric flow rate is affected by pressure devices and rotational speed of the gear

pump. In order to prevent leakage between the inlet and the outlet of the gear pump, the

difference in the inlet and outlet pressures of the gear pump should be zero. Thus,

elernents &, and A33 should be non-zero as well. The measurement of the mass flow

rate is only a function of a sample weight per unit time. According to these observations,

Equation 3.4 now becornes:

The design rnatrix shown above is triangular, which means that this design is

decoupled, This indicates that FRs of the overall designed system c m be controlied

independently by DPs and it is an acceptable design. Therefore, the design and strategies

for measuring the PVT properties are sound according to the first axiom of the axiomatic

design approach.

3.4 DETERMINATION OF THE POSITIVE DISPLACEMENT RATE OF THE

GEAR PUMP

The specific volume of a solution can be determined by v = Q /m. where Q is the

volumetric flow rate and m is the mass flow rate. Since we can readily obtain mass flow

rate by collecting the extrudate for a fixed time period and measuring the weight of the

degassed extrudate as desctibed in the previous section, Our research focused on the

measurement of the volume flow rate. A gear pump (Zenith PEP-II) is used as a positive-

dispIacement pump to feed the polymer/C02 solution. As a first step, the positive

volumetric displacement of the gear pump per revolution, a, needs to be determined.

Water and a vegetable oil are used to determine @ of the pump. Once the gear pump is

calibrated, the volume flow rate of a polymer/CO2 solution can be measured üccurately.

3.4.1 Approximate Calculation of Volumetric Relationsbips

The performance of a gear pump has been thoroughly investigated by McKelvey

1571. A gear pump plays a very important role in extrusion processing by consistently

providing a stable flow, reducing surges and fluctuations. A schematic of a gear pump

is shown in Figure 3.3. The saturated polymer solution fills the cavities of the gear from

the inlet side and is transported to the outlet side of the pump by rotation of two gears. In

order to estimate the volurnetric displacement, the following approximate equations can

be used [57]:

where V is the average speed of the gear pump, H is the height of the gear, N is the

rotationai speed in rpm, and W is the width of the gear. The factor of 2 is introduced to

the above equations since there are two gears present in a gear pump. The measured

values for the gear pump are (Figure 3.3): W = 0.3935", D = 0.9905" and Do = 0.9285".

When we substihited the converted values into the above equations, we obtained an

estimated volumetric flow rate per revolution, Q&V, of 1.205 mlhev. However, an

accurate value of the volumetric throughput can be obtained only by the calibration

experiments. Since the pump is built with clearances, there will be leakage flow

associated with it. The ieakage is caused by the pressure difference between the inlet and

the outlet of the gear pump, we c m minimize or eliminate the leakage by maintaining the

same pressures for the inlet and the outlet.

3.4.2 Calibration of the Gear Pump

In order to accurately determine the volume flow rate per revolution of the gear

pump, (@ = 0, the gear pump was calibrated experimentally. A separate setup is

constmcted to calibrate the gear pump as shown in Figure 3.4.

A syringe pump (ISCO 260D) that c m detect the volumeuic displacement

accurately is attached to the gear pump and two pressure gauges (Wika Instruments

213.53) with the maximum pressure of 206.8 kPa (30 psi) are installed at the inlet and

outlet of the gear purnp. A commercial needle valve (Sno-Trik SS-410-FP) is used as a

variable resistance valve to control the downstream pressure.

The rationaie of the calibration setup is that the amount of liquid going through

the gear pump can be metered by the syringe pump as long as there is no leakage flow

across the gear pump. Since the leakage flow occurs when there is a pressure difference

across the pump, the downstrearn pressure needs to be adjusted to be the same as the

upstrearn pressure by varying the resistance of valve. The control pararneters for the

calibration experiment are the volumetnc flow rate of the syringe pump (Q,), the

rotational speed of the gear pump (N}, and the resistance of the valve. The output

parameters are the pressure of synnge pump (P,), the inlet pressure of gear pump (P,),

and the outlet pressure (P,) of the gear pump.

When liquid flows, the pressure drops from P, at the synnge pump to P, at the

upstream of the gear pump because of resistance. The volume flow rate in the gear pump

(Q,) can be detemined from the volume flow rate in the syringe pump (Q,) and

consideration of the pressure difference and the compressibility of liquid using Equation

3 -9:

Q, mg v, v Volume of liquid ut P = P; (= P, ) = = R = Voirne of liquid at P = P,

(3.9) Qs &Y, vs

where h, and mg are mass flow rate of the syringe pump and the gear pump, respectively.

Calibration experiments are conducted with water and vegetable oil at various

speeds of the gear pump at room temperature (23°C 4- 1 OC). The volume flow rate of

water in the gear pump (Q,) is calculated using the bulk-modulus correction method and

the PVT properties of water [58]. Since the PVT properties of the vegetable oil are not

available, an experiment was conducted to determine the ratio of volumes at two different

pressures (Equation 3.9) using a syringe pump as shown in Figure 3.5. The rneasured

voiurne of oil as a function of pressure is shown in Figure 3-6. Using these results by

logarithmic regression curve fitting and Equation 3.9. the volume flow rate of oil in the

gear pump (Qg) is calculated based on the experimental data. Al1 the acquired data for

the calibration experiment are shown in Tables 3.1 and 3.2 for water and oil, respectively.

The flow rates of water and oil through the gear pump (Q,) are ploned against the

rotational speed (N) in Figures 3-7 (a) and (b), respectively. The volume throughput per

revolution of pump (8 =Q#'V) is calculated from these curves by fitting a linear

regression line. For both cases, Q, is estimated to be 1.170 rnE/rev.

In order to determine the accuracy of the rneasured volume displacement of the

gear pump, an error analysis is conducted for the calibration experiments with water and

vegetable oil. Even though the calibration experiment is conducted while maintaining the

same inlet and outlet pressures, there would be some leakage of low viscous water and oil

if there are any pressure difference because of inaccuracy of the pressure gauges. In

order to calculate the error associated with the inaccuracy of the pressure gauge reading,

a dynamic leakage flow analysis is conducted using the setup shown in Figure 3.8. When

the gear pump is running in this setup, the downstream pressure increased, In order to

avoid vacuuming in the upstrearn pressure, the whoie system is initially pressurized by

the syringe pump. Since the net flow of liquid in the gear pump is zero, the amount of

liquid pumped by the gear pump wouid be baianced by the Ieakage from downstream to

upstream dce to the pressure difference. Therefore. these results show the amount of

leakage flow (Q,) due to a pressure difference for each liquid. The results are shown in

Figures 3.9 (a) and (b) for water and oil, respectively. Even though a higher order

polynomial fit (Le., a order polynomial fit) may be better in describing the results, the

plots were linearly fitted to correlate the data of pressure difference to the two different

parameters of the gear pump speed and leakage flow rate shown in the vertical axis. For

a pressure gauge error range of +/- 0.689 kPa (O. 1 psi), the corresponding leakage flow

rate would be +/- 0.033 rnVmin and 0.003 mumin for water and oil, respectively. This

means that when we masure the flow rate at 10 mumin, the actual flow rate could be 10

+/- 0.033 mumin and 10 +/- 0.003 mumin for water and oil, respectively. When al1 the

flow rates of water shown in Figure 3.9 are corrected to the maximum and minimum

values because of the leakage flow. the values of Q/!V for water is changed by +/- 0.001

mUrev. However, the values of Q#V for a vegetable oil are not changed by the leakage

flow which would have been caused by inaccuracy of the pressure gauges.

The fluctuation of the gear pump speed could also have affected the accuracy of

the measurements. The fluctuation of the gear pump speed is in the range of +/- 0.03

rpm. When al1 the gear pump speeds shown in Figure 3.7 are corrected to the maximum

and minimum values with this fluctuation, the values of Q/V are changed by +/- 0.001

mUrev for both the water and vegetable oil cases.

These en-or analyses indicate that the obtained value of 1.170 rnVrev for the

volume displacement of gear pump per revolution has a total error range of +/- 0.002

mUrev. The calibrated value is used in the PVT rneasurement of a polymer melt as

descnbed in the followmg section.

Table 3.1, Calibration Experimental Resulrs wirh Wuter

N

(revhin)

8.44

12.84

[ 17.15 /537.8(78.0)~124.1(18.0)~124.1(18.0)) 20.030 1 20.036 I

PS

(kPa / psi)

330.9 (48.0)

44 1 -3 (64.0)

&

(mVmin)

9.892

1 5-023

PI

(kPa / psi)

103.4 (15.0)

172.4 (25.0)

p2

(kPa / psi)

103.4 (15.0)

172.4 (25.0)

Qs

(mumin)

9.890

15.020

Table 3.2. Calibration Erperimental Rescrits with Vegetabie Oil

1 N (revhin) 1 P, (kPa I psi) 1 PI (kPa I psi) 1 Pz &Pa / psi) ~=(mVrnin) Q, (ml/m%) C

103.4 (1 5.0) 9.980 9.987

-- - - - - -

Axiomatic Design of a PVT Measuring Apparatus

Calibration of the Gear Purn p

1

i 1 t Apparatus Setup 1

I PVT Measurements for Pure Polymer for Verification of the Systern

b y Corn parison and EOS Modeling

P V T Measorement of Polym erlGas Solutions i 1

Figure 3.1. Overall Strrrtegy

Figure 3.2. A Schematic of the Experimental Setup for Measuring the P VT data of a PolymerKO* Solution

Figure 3.3. Side View of the Genr Prtmp

Figure 3.4. Calibration for Deteminhg the Positive Displacement Volume of a Gear Pump

Figure 3.5. Ejcperimental Setup for Determin ing the Compressibility of Vegetable Oil. (a ) Actual Setup with a Syringe Pump (6) Equivalent Piston-Cylinder Sysrenr

V = 259-56e4-0007P Pressure (MPa)

Figure 3.6. Volume-Pressnre Relationship of Vegetable Oil

Gear Pump Speed, N (revhin)

(a) Water

10 15 20

Gear Pump Speed, N (revhin)

(b) Vegetable Oil

Figure 3.7. Volrïmerric Flow Rate of Water and Oil through the Gear Pump as a Function of the Gear Pump Speed

PRESSURE GAGES

SHUT-OFF VALVE

FUMP CONTROI LER VI--$, SEAS PUMP

SYRlNGE PUMP

Figure 3.8. Expeninental Setup for Detennining Leakage Flow for LVater and Vegetable Oil Across the Gear Pump as a Funcrion of Pressure D~rerence

Pressure Difference, AP (kPa) AP = 3.5157 N QI = O. 3328 A P

(a) Water

Pressure Difference. AP (MPa)

(b) Vegetable Oil

Figure 3.9. Leakage Flow as a Function of the Pressure Dzrerence in the Gear Pump Obtained front the Dynamic Leakage Flow Analysis

MEASUREMENT AND MODELING OF THE PVT

RELATIONSHIP~ OF PURE POLYMER MELTS

This chapter describes the verification of the dilatometer designed for the

rneasurement of the PVT properties of po1ymer/CO2 solutions by using the system for pure

polymer melts. Various pure polymers such as Iinear polypropylene (PP), polystyrene (PS),

Iinear polybutylene succinate (PBS) homopolymer, and branched PBS were used to validate

the accuracy of the system by comparing the data obtained with the new method to data from

the literature that were measured by the conventional bellows type method. Furtherrnore,

modeling of the experimental data with the FOV, MCM, and S-L EOS are presented.

Mathematically acquired EOS based on the experimental data enables prediction of the PVT

behaviour within and beyond the experimental ranges of pressure and temperature.

4.1 EXPERIMENTAL SETUP

The PVT measurement system was constmcted based on the schematic design shown

in Figure 3.2, which was derived from the axiomatic design approach analysis. Figure 4.1

shows a photograph of the actual setup. In the verification experiments with pure polymer

met ts, the gas was not injected. The apparatus was based on a tandem extruder that consisted

of two extmders, which were attached side-by-side. The first extruder was a %" diarneter

extruder (Brabender: 05-25-000) with a mixing screw (Brabender: 05-00-05 1) which has an

Uû ratio of 25: 1, The second extruder (Killion: KNISO) was a l W diameter extruder. The

advantages of using a tandem system compared to a single extrusion system are the

rninimization of pressure fluctuations and better temperature control. The rotationai speed of

the second extruder was controlled independently with an independent PID (proportional-

integrator-differentid) (Allen-Bradley 1306 AC Frequency Drive) control ler. The heat

exchanger was attached after the second extruder to m e r ensure the accuracy of the melt

temperature as the desired temperature. The heat exchanger contained a homogenizing static

mixer (Labcore Mode1 H-04669-12) with a temperature controller and a solenoid valve,

which was connected to the pressurïzed air. Figure 4.2 illustrates the setup of the gear pump

and the heat exchanger. The calibrated positive displacement gear pump (Zenith PEP-II) was

rnounted on the system right next to the heat exchanger, and its temperature and the P D

speed controller were mounted independently on the control panel. Figure 4.3 shows the

setup of the control panel where various parameters could be monitored. The Pm values of

the speed controller of the gear pump were calibrated up to +/- 0.03 rpm to obtain optimum

feedback frequencies. A 0.5 hp 90V DC motor powered the par pump. Two pressure

transducers (Dynisco: PT462E-10M-6/18) were mounted at the inlet and outlet of the gear

purnp. These two pressure tnnsducers were calibrated carefully using water and vegetabie

oil prior to the experiment (the detailed procedure was described in Chapter 3). The variable

resistance valve (needle valve) was attached after the gear pump to effectively control the

downstrearn pressure by varying the resistance of the melt flow. Finally, a small diameter

(0.030") nozzie was attached to facilitate coilection of the samples. In order to obtain an

accurate temperature of the melt fiow, we used numerous themocouples that were mounted

on the system. Band heaters and temperature PID controllers (Omega CN9000A) controlled

the temperatures of the difision enhancing device, the variable resistance valve, and the

nozzle. The extrudated sarnple was measured using a precise micro-balance (Mettler Toledo

AB204). The micro-balance was calibrated up to +/- 0.000 1g accuracy prior to the

experiments.

4.2 MATERIALS

The polyrneric materials used in this study were isotactic PP (Boredis AG, Daplen

DM55), PS (Japan Polystyrene Inc., G440K), linear PBS (Showa Highpolyrner, Bionelle

# 1003), and branched PBS (Showa Highpolyrner, Bionelle #1903).

PP is polyrnerized from the organic compound propylene (CHyCH=CH2) and has

a methyl group (-CH3) branching off of every other carbon along the molecular backbone.

Since the methyl groups of polypropylene are located on one side of the carbon backbone,

polypropylene molecules tend to be highly aligned and compact. Because of these reasons,

PP has excellent chernical resistance, is strong, and has the lowest density of the plastics

used in packaging. In addition, the material has a high melting temperature and is very

economical.

PS, produced from styrene (C6H5CH=CH-), has phenyl groups (six-member carbon

ring) attached in random locations along the carbon backbone of the rnolecule- The random

attachment of benzene prevents the molecules from becoming highly aligned- As a result.

polystyrene is an amorphous, transparent, and somewhat bt-ittle plastic. PS is a very versatile

plastic that can be rigid or foamed. It has a relatively low plasticating point compared to

other thennoplastic materiais. A specific PS supplied from the Japan Polystyrene Inc. was

chosen for the experiment because the resin manufacturer supplied its PVT data to be

cornpared with our experimental data. In addition, PS is commonly used in plastic foam

processing with CO2.

PBS is a thennoplastic aliphatic polyester produced through the chemicd reaction and

polycondensation of a glycol with an aliphatic dicarboxylic acid. PBS is stable in the

atmosphere but biodegradable in compost, wet soil, fresh water, seawater and activated

sludge, where a large number of microorganisms are present. Linear PBS material is a

biodegradable polyester with good stiffness. The material is not as viscous as PP or PS.

Branched PBS materid was aiso used to investigate the effect of the branching on the PVT

properties by comparing with the linear PBS. This material has excellent foamability and is

quite ductile.

The material properties of these polymers are summarized in Table 4.1. Both the

linear and branched PBS materiais were dried pnor to the experiment because of their

hygroscopic nature, and PP and PS materials were used as received.

4.3 EXPERIMENTAL PROCEDURE

The general procedure to measure the PVT properties of a pure polymer melt is

shown in Figure 4.4. Prior to the experiments, both the linear and branched materials were

dried because they are very hygroscopic, Le., absorbs moisture easily. The absorption of

moisture into the materials may pose some problems during the extrusion process because the

rnoisture causes oxidation of the material that dramatically degrades the material. For this

reason, the pellets were dried in a vacuum oven for a period of 4 hours at 90°C. Then, the

dried PBS pellets were fed into the hopper. To further minimize atmospheric contact,

nitrogen gas was continuously fed into the hopper to prevent absorption of moisture. On the

other hand, PP and PS pellets were fed directly into the barre1 through the hopper of the first

extruder without drying. The pellets were completely melted by the screw motion of the first

extruder. The molten polyrner was transferred to the second extruder, and the melt went

through the second extruder and the heat exchanger to decrease the temperature of the meit

uniforrnly. The melt reached the gear pump at a desired temperature and pressure at which

time the experiments were conducted-

At a fixed gear pump speed, the upstream pressure of the gear pump was initiaily set

to be close to the desired pressure by varying the rotational speed of the second extruder.

Then, the downstream pressure was controlled to be close to the desired pressure by turning

the variable resistance vdve. When the upstream and downstream pressures were close to

the desired pressure, the second extruder screw speed and the variable resistance valve were

again minutely varied to equalize these pressures. It should be noted that the upstream and

downstream pressures could not be made the same as the desired pressure with a single

adjustment because of the unavoidable leakage across the gear pump. By accurately

controlling upstream and downstrearn pressures, the desired pressure was achieved for both.

In order to investigate the effects of the rotational speed of the gear pump on the PVT data,

experiments were conducted at three different rotational speeds for PS, linear PBS, and

branched PBS. Three sets of the rotation speeds were performed for the expenments: for PS

the speeds were 15, 20 and 25 rpm and for linear and branched PBS the speeds were 5, 10,

and 15 rpm.

When a steady-state condition was achieved at each temperature, pressure, and gear

pump speed, the mass flow rate was measured by collecting the extrudate for a fixed time and

weighing it. The volume flow rate was calculated to be the gear pump speed rnultiplied by

the gear pump throughput per revolution that was determined in Chapter 3. The specific

density was caiculated as the ratio of the mass flow rate to the volume flow rate.

For PP, the selected melt temperatures were 180°C, 190°C, 200°C, 2 10°C and 230°C.

The selected gear pump pressure in the experiments with PP was in the range of 6.9 MPa to

29.6 MPa (1000 psi to 4300 pso-

For the case of PS, the selected melt temperatures were 180°C, 200°C, and 220°C.

The selected gear pump pressures in the experiments were 13.78 MPa (2000 psi), 20.68 MPa

(3000 psi), and 27.58 MPa (4000 psi).

For the cases of linear and branched PBS, the selected melt temperatures were 147"C,

1 57"C, 167°C and 177°C. The selected gear pump pressures in the experiments were 13-78

MPn (2000 psi), 20.68 lMPa (3000 psr], and 27.58 MPa (4000 psi).

4.4 RESULTS A i i DISCUSSION

Four series of experirnents were conducted to measure the specific volumes of PP,

PS, linear PBS and branched PBS. The experimental results were compared to the

conventionaily measured PVT properties in order to verify the system design. Finally, the

measured PVT properties in the melt state were correlated using three different EOS models:

FOV (Equation 2.9), MCM (Equation 2-10)? and S-L EOS (Equation 2.14). The correlated

reduced constants were acquired by using the l e s t squares method based on the Marquardt-

Leven berg algorithm [59,60].

Polvpropvlene

We measured the PVT properties of linear PP at the steady-state condition, and the

results are plotted in Figure 4.5 and tabulated in Table 4.5. Figures 4.6 (a), (b), and (c) show

the fitted models using FOV, MCM, and S-L EOS, respectively. The measured specific

volume properties of PP were also cornpared to the work of Sato et ai. [i9] in Figure 4.7.

Although the two data sets were well matched in general, the experimental data measured at a

low temperature of 180°C fluctuated slightly because of the fluctuations in the pressures in

the extruder barrels at a lower temperature. The slight deviation motivated fùrther study of

other effects, such as the effect of the speed of the gear pump, with other materials.

Polvstvrene

The measured data of PS are tabulated in Table 4.6 and plotted in Figure 4.8. Figures

4.9 (a), (b), and (c) show the fitted models using FOV, MCM, and S-L EOS, respectively.

The measured PVT properties of PS were also compared to the manufacturer's reference data

(Table 4.7) in Figure 4.10. A minute upward deviation between the two data sets was

observed. It was speculated that this deviation was due to a slight increase in the melt

temperature through friction in the gear pump.

Linear PBS

The measured data of linear PBS are tabulated in Table 4.8 and plotted in Figure 4.1 1.

Figures 4.12 (a), (b), and (c) show the fitted models using FOV, MCM, and S-L EOS,

respectively. The cornparison between the experimental data and the manufacturer's data is

depicted in Figure 4.13. Our experimental data has small deviations (up to 0.005 cm3/$

compared to the manufacturer's data, This small discrepancy may have k e n used by

experimental errors. However, the validity of the manufacturer's data shouId be confimed.

Furtherrnore, the way we have interpreted the manufacturer's data may have caused

significant errors, since the data we acquired from the manufacturer were in a graphical

format. Nonetheless, the discrepancies are quite srnall and the stopes of the experimental

data are consistent with each other.

Branched PBS

The PVT experimental results for branched PBS are shown in Figure 4.14. Figures

4-15 (a), (b), and (c) show the fitted models using FOV, MCM, and S-L EOS, respectively.

The tabulated experimental data are shown in Table 4.9. The PVT properties of the branched

PBS were compared to those of the Iinear PBS in Figure 4.16. The deviation was expected

and this deviation rnay be due to branching of mers.

4.4.1 System VeriFtcation

The specific volume of polyrner melt was observed to be a very sensitive function of

pressure and temperature. The experimental results indicate that the developed system

accurately measured the specific volume of a polyrner melt at a high temperature and high

pressure in the range of extrusion and injection molding processing. The experimentd results

for most materiah showed good agreement with the reference data measured by the

conventional beilow type dilatometer. The deviation measured for the Iinear PBS needs

further clarification.

4.4.2 Effect of the Gear Pump Speed On the Measured Specific Volume

The effect of the rotational speed of the gear pump on the measured PVT properties

was investigated. We acquired the data at three different rotationai speeds (Le-, 15, 20, and

25 rpm) of the gear pump. As the gear pump speed increased, both the mass and volume

flow rates of the melt increased. Figure 4.17 shows the mass flow rates of PS melts as a

function of the gear pump speed at various temperatures and pressures.

Since the specific volume of polyrner is calculated as the ratio of the volume flow rate

to the mass flow rate, the gear pump speed is not supposed to affect the measured specific

volume.

From Equation 4.1, the specific volume can be calculated from the ratio of the mass flow rate

to the gear pump speed, or equivalently, from the slope in the graph of the mass flow rate

versus gear purnp speed shown in Figure 4.17. Therefore, the specific volume can be

determined from each set of the gear pump speed and the determined m a s flow rate. The

average of the values measured for al1 gear pump speeds was taken as the specific volume for

each temperature and pressure. Another way of detemining the specific volume is to take

the dopes of the fitted lines of m = UN using the l e s t squares method as shown in Figure

4.17. Al1 these specific volumes are included in Tables 4.6, 4.8, and 4.9 for PS, linear PBS,

and branched PBS, respectively. In this study, the value determined from the fitted line was

selected as the specific volume at each temperature and pressure. The error range due to the

change of the gear pump speed for PS, as a case example, is depicted in Figure 4.18.

4.4.3 Effect of Friction in the Gear Pump on the Measured Specific Volume

The friction generated in the gear pump may aFfect the temperature of the melt (or the

solution), and thereby the measured specific volume. Figure 4.19 depicts a schematic of the

energy balance on the gear pump system. Since, the pressure difference between the inlet and

outlet of the gear pump is assumed to be zero, the following energy balance applies:

where w is the rate of energy transferred from the electric motor through friction and QcooIhg

is the rate of heat transferred to the cooling water running in the gear pump. The parameter h

is the enthalpy and C, is the specific heat capacity. Equation 4.4 indicates that if the amount

of cooling is insufficient, the temperature of melt across the gear pump will increase.

Accurate analysis of the heat transfer was very difficult due to various heat transfer effects.

But we measured the temperature difference between the inlet (TI) and the outlet (Ti) by

directly measunng the temperature of the melts flowing into and out of the pump. The

measured temperature difference was approximately 2°C in the PVT measurement of PS

melts. The slight increase in the melt temperature may be responsible for the discrepancies

observed in the experimental data.

This chapter was dedicated to the verification of the designed dilatometer that c m

measure the specific volume of a polymer/COz solution. The designed system based on a

tandem extruder and a gear pump was evduated using numerous pure polymer melts whose

PVT properties were known. With the developed system, we could successfully measure the

PVT properties of PP, PS, linear PBS, and branched PBS mefts. It was verified that the

rneasured PVT propenies of PP, PS and linear PBS materials are compatible with the

conventionally rneasured results.

Table 4.1. Sumrnary of Material Propenies for PP, PS, Linear PBS, and Branched PBS

PP

PS

Linear PBS

Branched PBS

T E

-18°C

98.8S°C

-

-

T m

159.80C

-

1 14°C

115°C

MFR (ASTM D 1238)

3.8 g/lOmin (230°C/l -2 kgf)

9 g/lOmin (200°C/5 kfi 3.5 g/10 min

( 1 90°C/2. 16 kgf) 4.7 g/lOmin

( 1 90°C/2. 1 6 kgf)

M W

-

23.813x104

14 x lo4

14 x 104

M n

-

9 . 4 7 6 ~ 1 0 '

7.6 x 10'

3.8 x 10'

Table 4.2. The FOV Equation Constants for PS, PP, Linear PBS, and Branched PBS

Linear 1 PBS 1 3481

Branched

T* (K) v* (cm3/@ Standard Error for v*

Table 4.3. The MCM Equafion Constanfsfor PS, PP, Linear PBS, and Branched PBS

Linear 1

Table 4.4. The S-L Equation Constants for PS, PP, Linear PBS, and Branched PBS

1 PBS

Branched t-- 1 PBS

P* (bar)

2758

T* (K)

803.1

p* (cm31g)

1 .O72

Standard Error for pf

5.35E-3

Table 4.5. Experintentai PVT Data of Linear PP

Temperature (OC)

180

180

180

200

200

200

220

220

220

Pressure (M Pa)

13.790

20.684

27.579

13.790

20.684

27,579

13.790

20.684

27.579

cir ai 15 rpiii (glm i n)

17.250

17.370

1 7.450

17.1 I O

17.2 12

17.303

16.960

17,010

17.1 10

v nt 15 rprn

1.017

1.010

1,006

1.026

1.020

1.014

1,035

1,032

1,026

lit nt 20 rpm (glnii n)

22,950

23,160

23.300

22.800

23,030

23.142

22,620

22.778

22.890

ri1 üt 25 rpm (g/min)

28,770

28.990

29.132

28.457

28.600

28.792

28% 1

28.383

28.566

v iit 20 rpm

1.020

1 .O10

1,004

1.026

1.016

1.01 1

1 .O34

1,027

1,022

v nt 25 rpm (cm3/@

1,017

1.009

1.004

1,028

1.023

1.016

1.036

1 ,O3 1

1.024

Avengc v (cm3/@

1.0179

1,0099

1.0047

1,0266

1 ,0195

1 .O138

Curve fitied v (cm31g)

1.0177

1.0097

1.0045

1,0269

1.0203

1.014 1

1,0350

1.0299

1,0240

1.035 1

t ,0297

1.0237

Table 4.7. Manufacturer's Reference Data for PS Memrred Using the Conventional Bellow Type Dilatometer (Japan Polystyrene Inc.)

Cc) 1 (Ibar) ( (2Obar) 1 (400bar) 1 (6ûûbar) 1 (800bar) 1 (1ûûû bar) 1 (1300 bar) 1 (1600 bar)

Temperature Pressure ,il ai 5 rpm v ni 5 rpin ri1 at 10 rpni

(OC) i (MPa) (glmin) (cm3&) (@min)

ai 10 rpni rit ai 1 5 rpm (c m31g'/g) I (g/mi n)

1 nt 15 rpm Average v Curve fiited v (ctn3ig)

First

Exîruder

Pressure In let

Variable Resis tance

Valve

Figrire 4.1. Photograph of the Overall Setup of the System

Figure 4.2. Side View of the Gear Pump and Connections (a) Angled View, (b) Side View

Figure 4.3. Overall View of the Control Boxes (RPM Con trollers. Temperature Con trollers, Pressure Read Outs, and etc.)

Piastication of Pure Polymer 1 1 by Feeding the Materiai t h r o ~ g h a Happe: 1 i

I I

i Set Constant R P M of the Gear Pump

I I

r P inlet = P outlet

P inlet controlled by Rotational Speed of Second Extruder P outlet controlled by Rotations of the Variable Resistance Valve

Measurem ent of Mass Flow Rate , I 1

I A n alys is i

/ corn p i s o n and EOS Modeling /

Figure 4.4. General Procedure to Measure the PVT data of Pure Polymer Melrs

4.250 ! ! O 5 1 O 15 20 25 30 35

Pressure (MPa)

Figure 4.5. Measrtred PVT Data for Linear PP

O 5 10 15 20 25 30 35

Pressure (MPa)

(4

1.25 - I 1

O 5 10 15 20 25 30 35

Pressure (MPa)

(b)

1.25 4 , O 5 10 15 20 25 30 35

Pressure (MPa)

(cl

Figure 4.6. Correlarion of PVT Data for PP in the Melt Srate Using (a) FOV; (b) MCM; and (cl S-L EOS

1.25 ! l i !

O 5 10 15 20 25 30 35

Pressure (MPa)

Figure 4.7. Cornparison of PVT Data for PP in the melt State behveen Sato et aL's Results [19] and Khis Study

10 12 14 16 18 20 22 24 26 28 30 32

Pressure (MPa)

190 200 210

Temperature (Deg. C)

(6)

Figure 4.8. Measured PVT Data for PS (a) Specijk Volume with respect to Pressure; (b) Specrfic Volume with respect ro Tempe ratu re

10 12 14 16 18 20 22 24 26 28 30 32

Pressure (MPa)

(d

Figure 4.9. Correlation of PVT Data for PS in the Melt State Using (a) FOV; (6) MCM; and (c) S-L EOS

! i

0.990 7

10 12 14 16 18 20 22 24 26 28 30 32

Pressure (MPa)

Figure 4.10. Cornparison of P VT Data for PS in the Me& State between Manufacturer's Reference Data and Experimental Data

1 57 167

Temperature (deg. C)

Figure 4.1 1. Measured PVT Data for Linear PBS (a) Specij7c Volume with respect to Pressure; (6) Specific Volume with respect to Temperature

Figure 4.12. Correlation of PVT Data for Linear PBS in the Melt Srate Using (a) FOV; (6) MCM; and (c) S-L EOS

i - - - - Ref 1TÏC L

10 12 14 16 18 20 22 24 26 28 30 32

PRESSURE (MPa)

Figure 4.13. Cornparison Between the Experimental Data vs. Manufacturer's Reference Data for Linear PBS

10 12 14 16 18 20 22 24 26 28 30 32

Pressure (MPa)

27.6 MPa

-PI-:

13.8 MPa i j

157 167


(b)

Figure 4.14. Measured PVT Data for Branched PBS (a) Specifiç Volume with respect to Pressure; (6) Specifrc Volume with respect tu Temperature

Figure 4.15. Correlation of PVT Data for Branched PBS in the Melt State Using (a) FOV; (b) MCM; and (c) S-L EOS

Figure 4-16. Cornparison Berween the Linear (Dotted Lines) and Branched PBS (Solid Lines)

O 5 1 O 15 20 25 30 Gear Pump Speed (RPM)

5 10 15 20 25 30 Gear Pump Speed (RPM)


(cl

Figure 4.1 7. Mass Flow Rates Measured at Various Gear Pump Speeds at (a) 220 OC and 27.6 MPa; (b) 220 O C and 20.7 MPa; (c) 220 OC and 13-8 MPa

90

O 5 10 15 20 25 30 Gear Pump Speed (RPM)


5 1 O 15 20 25 30 Gear Pump Speed (RPM)

(B

Figure 4.17 (Continued). Mass Flow Rates Measured in Various Gear Pump Speeds at (d) 2W°C and 2 7.6 MPa; (e) 200 OC and 20.7 MPa; Cf) 200°C and 13.8 MPa

9 1

O 5 1 O 15 20 25 30 Gear Pump Speed (RPM)

5 10 15 20 25 Gear Pump Speed (RPM)

--

A

t

y = 1.1303~

O 5 10 15 20 25 30 Gear Pump Speed (RPM)

0)

Figure 4.17 (Continued). Mass Flow Rates Measured in Various Gear Purnp Speeds at (g ) 180 OC and 27.6 MPa; (h) 180 OC and 20.7 MPa; (i) 180 OC and 13.8 MPa

92

+ 0.990 i

10 12 14 16 18 20 22 24 26 28 30 32

Pressure (MPa)

Figure 4.18. Error Range of the P VT Properties of PS Melts Due to the Change of the Gear Pump Speed

w

v mh,

Figure 4.19. Schematic of Energy Balance in the Gear Pump

MEASUREMENT AND MODELING OF THE PVT RELATIONSHIPS OF POLYMERKO~ SOLUTIONS

Although a great deal of research and development has been conducted on foam

processing, there has been no report on any direct measurement of the PVT properties of a

polymedgas solution in a rnolten state of the polymer, which is crucial to understand foam

processing. In fact, the rneasurement of PVT data of single-phase polymedgas solutions is an

extremely challenging task because maintaining the single-phase polyrner/gas solution 2t a

high pressure condition is very difficult, Attempts have been made to measure the PVT

relationship of a polyrner/gas solution at the maximum sotubility equilibrium state by

gathering the solubility data [25-28, 6 1, 623 (see Section 2.2). However, this method has

been criticized because it is Iimited to the solubility equilibrium condition.

The novel method developed in this study overcomes the limitation of the

conventional apparatus for rneasunng the PVT data, This was possible because the new

method uses a continuous on-Iine extrusion system with a gear pump to maintain a high-

pressure condition. The on-line rneasurement system reduces the tirne required for the

saturation of the polymer with gas. A metered amount of carbon dioxide, which is less than

the solubility Iirnit, is injected into the Stream of molten polymer to form a single-phase

polymer/gas solution in the tandem extrusion system. The injected gas diffuses into the

polymer matrix at a rate that is much higher than a batch process by stretching of the gas

bubbles which increases interfacial area between the gas and the polymer molecules [63].

In the previous Chapter (Chapter 4), the designed system was venfied using numerous

pure polymers. This chapter further elaborates the measurements of the PVT data of PS/C02

solutions, linear PBS/C02 solutions, and branched PBS/C02 solutions. The experimental

procedure for the PVT measurements of polyrner/gas solutions requires careh1 and accurate

control of pressure, temperature, and COz content. The acquired experimental data are used

to examine the effects of COt in the polymer matrix on the PVT relationships. The swelling

of poiymer melt due to the dissolved C O was precisely characterized as a hinction of COt

concentration, temperature, and pressure. Efforts were made to mode1 the experimental data

using conventional EOS methods. However, further study is required to predict theoretically

the effects of gas content in a polynier matrix.

5.1 EXPERIMl3NTAL SETUP

A PVT measurement system was constructed based on the designed setup shown in

Figure 3.2. The experimental setup to measure the PVT relationships of polymer/gas

solutions is similar to that used to measure the PVT relationships of pure polymers (Chapter

4), except for the CO2 injection apparatus. Figure 5.1 illustrates the actual setup of the PVT

measuring device for a polyrner/gas solution. The newly developed dilatometer consisted of

a %i" extruder (Brabender: 05-25-000) as the first extruder. a mixing screw (Brabender: 05-

00- 144) in the first extruder, a gas injection apparatus, a 1%" extruder (Killion: KN 150) as

the second extruder, a diffusion-enhancing device containing static mixers (Labcore: Mode1

H-04669-12), a positive displacement gear pump (Zenith: PEP-Il), a variable resistance

valve, and a filament die.

The gas injection apparatus consisted of a gas cylinder, a chilIer, a syringe pump,

and an injection port. The injection port was located at a distance of one third from the outlet

of the first extruder and two thirds from the hopper to ensure good absorption of gas into the

polymer matrix [3]. The volume flow rate of the gas was accurately controlled by a syringe

pump, which was calibrated prior to the experiment. Figure 5.2 shows a pictonal view of the

synnge pump used in the experiment. A refrigeration unit was used to chitl the gas line to

promote better CO2 flow between the syringe pump and the cylinder. A heat exchanger was

used to cool the p o l p e r melt, and it contains a homogenizing static mixer with a

temperature controller and a solenoid valve, which was connected to a pressurized air. The

static mixer enhanced complete dissolution of carbon dioxide into the polymer melt. The

calibrated positive-displacement gear pump and its P D controller were mounted on the

system. Two pressure transducers (Dynisco: PT462E-10M-6/18) were mounted at the inlet

and outlet of the gear pump. In addition, numerous thermocouples were mounted on the

system to monitor the temperatures of the melt. Band heaters and temperature P D

controllers (Omega: C N 9 O A ) controlled the temperatures of the diffusion enhancing

device, the variable resistance valve, and the filament die. A precise micro-balance (Mettler

Toledo: AB204) was utilized to measure the weight of the extrudate.

The polymeric materials used in this study were polystyrene (Japan Polystyrene inc.,

G440K), linear PBS (Showa Highpolyrner, Bionelle #1003) and branched PBS (Showa

Highpolymer, Bionelle #1903). The properties of these polymers are sumrnarized in Table

4.1. Carbon dioxide (BOC Gas Inc., 99.5 mol. % purity) was used as a blowing agent,

5.3 EXPERIMENTAL P R O C E D W

Figure 5.3 illustrates the step-by-step experimental procedure for the measurement of

the PVT relationships for polymer/gas solutions. Linear PBS and branched PBS were dried

in a vacuum oven for 4 hours prior to the experiments because of the hygroscopic nature of

the resins. Unlike the PBS materiais, PS was not dried and was fed directly into the barrel

through the hopper of the first extruder.

Initially, the speed of the gear pump was set and the system was srarted. While the

system ran, COt was injected into the first extruder by the syringe pump. During the start-up,

the pressure in the barrel increased due to the screw motion. The polymer pellets were

completely melted and rnixed with carbon dioxide by the screw motion of the first extruder.

The mixing section of the screw chopped the carbon dioxide from bigger bubbles into smailer

bubbles, which induced better gas solubility in the polymer. The polyrner melt went through

the second extruder and the heat exchanger to decrease the temperature of the melt uniforrnIy.

When the inlet and the outlet pressures of the gear pump equalized and the system reached a

thermodynarnic equilibrium, the volumetric and m a s flows rates of the polyrner/gas solution

were measured,

Iniection of CO2 inro a Stream of Polvmer Melt

Prior to the expriment, CO2 was filled in the syringe pump and pressurized to a high

pressure (Le., 48 MPa). The injection pressure of the gas into the first extruder must be

maintained higher than the extrusion pressure to prevent a back flow of polymer.

Close control of the gas-to-polyrner weight ratio is very important, because only a

soluble amount of gas injected into a polymer matrix will be dissoived. If an excessive

arnount of gas is injected, voids will be created which generate errors in the expenments.

The required gas flow rate is determined by the solubility of gas in the polymer at the given

terr.perature and pressure. It should be noted that the weight of CO2 with respect to the

weight polymer/C02 solution must be maintaineci constant throughout each experiment to

ensure the consistency. This requires repeated rneasurements of mass flow rates of the

polymer/gas solution, as well as, the mass flow rate of CO2 from the syringe pump. In order

to thermodynamically stabilize the molten poIymer/gas solution and reach the steady-state

condition, the system was mnning for approximately 20 minutes or more before each

measurement.

Convective Diffusion in the Extrusion Barre1

According to the previous studies [54,63], it was believed that a single-phase

polyrner/gas solution was formed in our expenments when the gas was injected (below the

solubility limit) into the extrusion barrel under a high pressure. Convective diffusion in the

extrusion barrel accelerated the gas diffusion process into the polymer matrix [53,54,63].

The shear action of the screw rotation stretches the gas bubbles, which increases the

interfacial area between the polyrner and the gas molecules, to promote better diffusion.

Since diffusion is also a function of the mixing quality of the polymer and gas molecules, the

diffusion time is dependent on the degree of mixing. Thus, the mixing sections of the screw

and the static mixer promoted good mixing between the polymer and gas molecules.

Furtherrnore, heat generated by the screw motion and high pressure (Le., above 20.7 MPa)

promoted gas saturation into the polymer matrix as well.

Measurement of Volume and Mass Flow Rare o f the Polvmer/Gas Solution

In order to reduce the leakage across the gear pump, the difference between the

upstream and downstream pressures was rninimized by using a variabje resistance die

attached to the downstream outlet of the gear pump. The upstream pressure of the gear pump

was initiaily set to be close to the desired pressure by varying the rotationd speed of the

second extruder. Then, the downstream pressure was set close to the desired pressure by

tuming the variable resistance valve. When the upstream and downstream pressures were

close to the desired pressure, the second extruder screw speed and the variable resistance

valve were again rninutely varied to equalize these pressures.

When the steady-state condition was achieved at each temperature, pressure, and gear

pump speed, the volume and mass flow rates were measured. The volume flow rate was

calculated to be the gear pump speed multiplied by the gear pump throughput per revolution.

The mass flow rate was measured by collecting the extmded foam for a fixed tirne and

weighing it. Since gas loss from the extruded foarn is unavoidabie [SS, 561, it would be

impossible to accurately measure the flow rate of polymer/gas solution by directly measuring

the weight of the extmded foam for a fixed time. But most of CO2 escapes during expansion

at a high temperature in case the expanded foam contracts after the initial expansion and the

resultant volume expansion ratio is less than 1.5 [Sd]. In order to promote gas loss, the die

temperature was increased significandy (220°C for PS and 200°C for linear and branched

PBS). Also, the diameter of the extnided foam was made very srna11 to expedite the gas loss

LOO

by using a thin filament die (diameter of 0.030") because the cell-to-ce11 dif is ion will be

faster for a thinner filament. It should also be noted that the small-sized filament die

facilitated the collection of extrudate for mass flow rate measurement. Thus, the weipht of

the extrudate coming out from the filament die was the weight of the polymer melt only. In

order to acquire the actual mass fiow rate of the poIyrner/gas solution, the mass flow rate of

carbon dioxide injected from the syringe pump was added to the weight flow rate of the

polymer extrudate. Since the residual gas in the polymer melt can cause some error in the

measurement of the mass flow rate, attention was paid to maintaining a high temperature in

the extruded thin filament foam to rninimize the residual gas.

The specific volume was calculated as the ratio of the mass flow rate to the volume

flow rate of the polymer/gas solution (i.e.. v = /i )-

For the case of PS, the selected melt temperatures were 180°C, 200°C, and 220°C.

The selected gear pump pressures in the experiments were 1 3 -78 MPa, 20.68 MPa, and 27.58

MPa (2000 psi, 3000 psi, and 4000 psi). The carbon dioxide contents were 1, 2, 3, and 4

wt%.

For the linear PBS and branched PBS, the selected melt temperatures were 147"C,

1S7"C7 and 167OC. The selected gear pump pressures in the experiments were 13-78 MPa,

20.68 MPa, and 27.58 MPa (2ûûûpsi, 3000psi, and 4000 psi). The carbon dioxide contents

were 1,2,4, and 6 wt%.

5.4 RESULTS AND DISCUSSION

Three sets of expenrnents were conducted to measure the specific volumes of the

PS/C02 solutions, linear PBS/C02 solutions, and branched PBS/C02 solutions. The

measured specific volumes of the PS/C02 solutions are shown in Figures 5-4-59, where the

effects of pressure, temperature, and gas content on the specific volumes are presented

clearly. The measured specific volumes of linear PBS/C02 and branched PBS/C02 solutions

are pIotted in Figures 5.10-5.15. Al1 the experimental conditions and measured data for the

PS1CO2 solutions, linear PBS/C02 solutions, and branched PBSlC02 solutions are given

specifically in Tables 5.1, 5.2, and 5.3, respectively, for three different rotational speeds (Le.,

5, 10, and 15 rpm) of the gear pump.

The experimentd data were curve-fitted by detemining the slope of the mass flow

rate of the polymer/gas solution versus the rotationai speed of the gear pump. The error due

to the change in the rotational speed of the gear pump was found to be minor. It seemed that

most of the experimental error was attributable to human error in measuring the mass flow

rate of the polyrner1CO2 mixtures as described in Section "Errors Involved in the

Measurement of the Mass Flow Rate".

The specific volume of the polyrner melt was observed to be a sensitive function of

pressure, temperature, and gas content. An increase in pressure from 13.79 MPa (2000 psi)

to 27.58 MPa (4000 psi) decreased the specific volume of PS by approximately 0.01 cm31g

(Figures 5.4 and 5.5). For linear and branched PBS, the sarne increase in pressure (from

13.79 MPa to 27.58 MPa) decreased the specific volume by approximately 0.015 cm31g

(Figures 5.10 and 5.1 1). The slopes of the pressure versus specific volume curves were

almost the same for the linear and branched PBS.

The specific volume of PS (Figures 5.6 and 5.7) increased as the temperature was

increased. One interesting point to note is that when a high gas concentration (Le., 4 wt% of

COz) was dissolved in the polymer mauix, the sensitivity of the specific volume with respect

to the temperature (Le., the change of specific volume for a unit change of temperature) was

not significant. For instance, at O wt% of COz, the specific volume of PS increased by

approximately 0.02 cm3/g from 180°C to 220°C, regardless of pressure. At 4 wt% of CO?,

however, the specific volume of PS increased by only 0.007 cm3ig from 180°C to 220°C.

This indicates that for P S f C 0 2 solutions, the specific volume was less dependent on

temperature when the amount of gas dissolved in the polymer was high- In the case of linear

and branched PBS, the specific volumes increased by approximately 0.01 with an

increase in the temperature from 147°C to 167°C (Figures 5.12 and 5.13). For these

materials, the gas content did not affect the temperature as much as with PS/C02 solutions.

The effects of gas content on the specific volumes for PS, linear PBS, and branched

PBS were also investigated. An increase in the gas content increased the specific volume due

to polymer swelling. For the case of PS (Figures 5.8 and 5.9), the specific volume increased

by approximately 0.05 cm31g as the gas content was increased from O wt% to 4 wt%- This

increase in specific volume was approximately constant over the temperature range of 180°C

to 220°C. For the case of linear and branched PBS (Figures 5.14 and 5-15), the specific

volume increased by approximately 0.07 crn3lg and 0.04 cm3/g, respectively as the gas

content was increased from O wt% to 6 wt%. For al1 the materials, a sharp increase in the

specific volume was observed when the gas content was increased from O wt% to 1 wt%. It

is not clear whether this was caused by experimental error due to the difficulty in controlling

the gas flow rate minutely from O wt% to 1 wt%, or if it represents the actual trends resulting

from gas dissolution into the polymer. Further experiments are required to verifj these

phenomena,

Cornparison Between Linear and Branched PBS

The presence of branches in a polymer chah may have profound effects on the

properties of a po1 yrner such as compressibility, stiffness, chernical resistance, and density

[64]. A linear, non-branched material exhibits one-dimensional connectivity between

molecuIes, whereas a branched material contains several branch points that connect three or

more sub chains. At roorn temperature, the specific volume of a branched material is

typically larger than that of a linear materid and it is mainly due to the low crystallinity of

branched materials [ 171.

However, at temperatures above the melting point, al1 the crystals melt and the

difference in the specific volume due to differences in crystallinity does not exist. In fact,

previous research has shown that the difference in the specific volumes of linear and

branched polyethylene melts was negligible even when the temperature and pressure were

varied [17]. Although it is known that the specific volume of a melt tends to increase slightly

with increasing degree of branching [os], the change is very minor and insipificant.

When gas was injected into the molten polymer, however, the effect of branching on

the specific volume became significant. The experimental results showed that at

temperatures above the melting point, linear PBS exhibited higher swelling due to the

dissolved gas than the branched PBS materiai, The specific volumes of linear and branched

PBS/C02 solutions are compared in Figures 5.10-5.15 for variations in pressure, temperature,

and CO2 concentration. It is speculated that the difference in the arnount of swelling of the

Iinear and branched PBS by the dissolved gas may be due to the extent of molecular-chah

entanglement- A greater degree of molecular£hain entanglement would Iimit the movernent

of the molecules, and therefore, the branched PBS would not expand or swell as much as the

linear PBS due to gas dissolution. When there was no gas dissolved in the polymer (Le., at O

wt% of Cod, the specific volumes of the Linear and branched PBS showed a minor

difference, approximately 0.003 cm3/g (Figure 5.12(a)). However, as the arnount of

dissolved gas was increased, the difference in the specific volume between the linear and

branched PBS increased as rnuch as 0.028 cm3/g (Figure 5.12(e)). Moreover, the specific

volume of the linear PBS increased by approximately 0.065 cm3& (Figure 5.14(c)) due to

injection of 6 wt% of CO2 at 13.8 MPcr (2000 psi), which results in 7 % swelling. The

specific volume of the branched PBS increased as much as 0.042 crn3lg (Figure 5.14(c)) due

to injection of 6 wt% of CO2 at 13.8 MPa (2000 psi!), which indicates approximately a 4.7%

increase in the specific volume. Our results aIso showed that the effects of temperature and

pressure on the specific volumes of linear and branched PBS were minor compared to the

effect of CO2 content [Figures 5.10-5-13].

Sanchez-Lacombe EOS and Mixina Rules for Binarv Solutions

The experimental results were mathematically modeled using the Sanchez-Lacombe

(S-L) EOS. The S-L EOS for polymer/gas solutions utilizes a number of mixing d e s , as

described in Section 2.4.2.4. In order to describe the polyrner/C02 solutions using the S-L

EOS, the pure material characteristic parameters are needed. The estimated characteristic

parameters for pure polyrners (Le., PS, linear PBS, and branched PBS) are taken from the

previous Chapter, and the results are shown in Table 5.4. The estimated parameter values for

CO2 [66] are also shown in Table 5.4.

The critical parameters acquired for the S-L EOS and the mixing rules (Table 5.5)

were calculated using the pure material characteristic parameters. Using the critical

parameters, the remaining mode1 variables were calculated using the S-L EOS and mixing

rules. The calculated values of the variables are presented in TabIes 5.6. 5.7, and 5.8 for

PS/C02 solutions, linear PBS/C02 solutions, and branched PBS/C02 solutions, respectively.

Since the size of the molecules and the critical/melting temperatures of carbon

dioxide and the polymer differ greatly (Le., Tc of CO2 is 3 1.3OC, and Tg of PS is 98.85"C). a

binary interaction parameter, &, between the gas and the polymer was introduced to account

for these differences in the S-L EOS mode1 [24]. The mixing niles used in S-L EOS to

deterrnine the interaction parameter assume: (a) the solubility vaiues were measured at

solubility equilibrium, and (b) the polymer is treated as a liquid above the glass transition

temperature [6 11. However in Our experiments, the experimental conditions did not reach the

solubili ty equilibrium state. Rather, our experiments were conducted with a concentration

much lower than the solubility equilibrium concentration (or equivalently, at a pressure much

higher than the solubility equilibrium pressure, or at a temperature much lower than the

solubili ty equil ibrium temperature) to ensure complete dissolution o f the injected gas in the

polymer rnatrix and to prevent the presence of the second phase. Figure 5.16 depicts the

typical experimental range on a temperature versus specific volume graph; our experimental

data are located above the solubility equitibrium.

Because previous work was done only at the solubility equilibrium state [25, 623, our

data cannot be compared directly with others' results. Furthemore, the binary interaction

parameter, &, obtained from this study would be different from that obtained by others,

where the parameter was a function of temperature only. It was speculated that the

interaction parameter for Our results should be a function of temperature, pressure, and gas

concentration because Our experiments were conducted in non-solubility equilibrium

conditions. Therefore, the interaction parameters were fitted as a function of temperature,

pressure, and gas concentration.

The interaction parameters are fitted using S-L EOS (Equation 2.13) and various

mixing rules (Section 2.4.2.4) based on the characteristic variables determined af each

processing condition (Tables 5-6-57}. The interaction parameters are plotted in Figures 5.17-

5.1 9, Figures 5.20-5.22, and Figures 5.23-5.25 for PS/C02 solutions, Iinear PBS/C02

solutions, and branched PBS/C02 solutions, respectively. The interaction parameters were

not sensitive functions of pressure, especially for Iinear and branched PBS (Figures 5.18,

5.31, and 5.24) as reported by others [25, 621. However, the interaction parameters were

sensitive functions of the gas concentration and decreased significantly with an increase in

the gas concentration (Figures 5-19, 5.22, and 5.25). Similady, the interaction parameters

decreased as the temperature was increased (Figures 5.17, 5.20, and 5.23). It was observed

that as the temperature and CO2 content approached the solubility equilibrium conditions, the

interaction parameter approached the conventional values ranging from -1 to +1 [25] .

However, the interaction parameter did not seem to converge to this range as the pressure

decreased to the solubility equilibrium pressure. Therefore, it is speculated that S-L EOS

might not be able to describe our experimental results well.

Even though the S-L EOS might not be the perfect mode1 to describe the PVT

relationships of polymer/COz solutions, the derived S-L EOS c m be used effectively to

calculate the data in the range of measurements by interpolating the experimental results.

Further studies on the interactions between the gas and polymer molecules are needed to

check the accuracy of the results and the validity of the S-L EOS to describe our results at

various experimental conditions,

EOS as a Function of the Gas Concentration

We aiso investigated the possibility of finding a relationship between the EOS and gas

concentration by treating each combination of polymer and COz content as a new material.

The Modified Cell Theory (MCM) EOS and the Sanchez-Lacombe (S-L) EOS were used for

this approach. The independent sets of experimental data were analyzed to find the

characteristic parameters by a non-linear regression method; this procedure was similar to

that used to determine the parameters for pure polymers. Once the parameters were obtained

using a mathematical analysis software tool ( ~ i g m a - ~ l o t ~ ~ ) , they were correlated as a

function of gas content.

First, we obtained the characteristic parameters for the MCM EOS at each gas

concentration. The parameters obtained are given in Tables 5.9, 5.10, and 5.1 1 for PS/C02

solutions, linear PBS/C02 solutions, and branched PBS/C02 solutions, respectively. The

parameters did not exhibit any trend as a function of the gas content. Rather, the parameters

fluctuated significantly as the gas concentration was varied. In a similar manner, we used the

S-L EOS to fit the parameters as a function of gas content. The parameters obtained are

shown in Tables 5.12, 5.13, and 5.14 for PS/C02 solutions, linear PBS/C02 solutions, and

branched PBS/C02 so tutions, respective1 y. As with the MCM EOS, the parameters obtained

using the S-L EOS did not show a trend as a function of gas concentration. One of the

reasons for this poor correlation may be due to the iack of data points obtained in the

experiments. Moreover, it is suspected that the v*, P*, and P are conjugated with each

other; in other words, v*, P*, and P are dependent on each other, even when gas

concentrations are changed.

Effect of the Residual Gas on the Measurement of the Mass Flow Rate

Although Our research has focused on the accurate measurement of the volume flow

rate of the poIyrner/COt solutions, a great deal of effort was aiso made to accurateiy measure

the mass flow rate of the polymer/gas solutions. It was impossible to rneasure the mass flow

rate of the poIymer/C02 solution by direct collection because gas loss from the extruded foarn

at the exit is unavoidable [3]. Therefore, we measured the mass flow rate of the polymer melt

only by intentionally promoting the Ioss of most of the gas in the extruded polyrner/CO1

solutions. Gas loss to the environment was promoted by increasing the temperature of the

filament die, and thus accelerating the diffusion rate of the gas from the extruded foam. A

very thin filament die of diarneter 0.030" was used to enhance the gas escape. CO2 motecules

in particular c m escape easily since the difisivity of CO2 is much higher than that of longer

chah blowing agents s w h as butane [55] , The mass flow rate of the polymedgas solution

was then determined by adding the measured mass flow rate of the polymer melt to the mass

fiow rate of CO2, which was readily determined by the readings of the gas injection pump.

However, the residuai gas in the extmded foam samples can cause experimental errors in the

determination of the mass flow rate of the polyrner/gas solution, and an analysis was carried

out to estimate the error associated with the presence of residuai gas.

If the amount of gas lost during foarning is significant, the final expansion ratio of the

extruded foam would be very low. An order of magnitude analysis was conducted to

estirnate the amount of residual gas remained in the extruded foams. B seemed that the

amount of COz remaining in the extmded foarn was negligible because of the high diffusivity

of CO7 at the high die temperature used in our expriment. As was observed by Park et al.

[55] and Behravesh et al. [56] in their studies on plastic foaming with CO?, the extruded

foams of PS, linear PBS, and branched PBS al1 showed volume contraction after the initial

expansion, indicating that rnost of the CO? had escaped frorn the foam. Since most of the

extruded foams of PS, linear PBS, and branched PBS had a very low expansion ratio,

typicdly 1.5, the arnount of gas remaining in the cells of the extmded foam would be very

small. Based on the assumption that al1 the gas initially dissolved in the melt diffused out of

the melt to either the cells or the environment, the residual gas in the extruded foam c m be

estirnated using Equation 5.1 :

Weight fraction of residual - - weight of residual gus gas in the extruded foam weight of polymer + weight of residrtal gus (5.1)

- - 1 weight of polymer

+ l weight of residual gus

- - 1 0.667 v a , x density of polymer

+I 0.333 V;,,, x density of residual gas

- - 1 2 density of polymer

+ l densiîy of residual gus

It should be noted that the density of the polymer in Equation 5.1 is the value at room

temperature and the density of COz is the value at the crystallization temperature of the

polymer at 1 atm. Because the cellular structure is solidified and fixed at the crystailization

temperature, the amount of gas in the foam is determined by the occupying volume of cells at

this temperature [56,67]. Based on the density of PS as 1 .O4 @cm3 and the density of COz at

100°C as 0.00142 g/cm3 [68], the weight fraction of residual gas in the extruded PS foarns is

estimated to be 0.07%. For the case of PBS/C02 solutions, the weight fraction of residual

gas in the extmded foam is estimated to be 0.06% based on the density of PBS as 1-27 @cm3

and the density of CO2 at 88°C as 0.00147 glcm3 [68]. This analysis indicates that the

amount of residual gas that remained in the extruded foarns dunng the measurement of the

mass flow rate of the polymer melt was negligible.

Errors InvoIved in the Measurement of the Mass Flow Rate

The human error associated with measuring the amount of extrudate per given tirne

could be significant. The mass flow rate was determined by collecting the extruded foam for

a fixed tirne and weighing it. A watch was used to monitor the time and a sharp edged cutter

was used to cut the extrudate corning out of the small diarneter filament die. Since we did

not use a precise stopwatch to measure the fixed time period, the cutting of the extrudate may

not have been synchronized exactly with the tirne. If we consider the human error range

involved in the time watch operation or in the cutting action for the extruded system, the error

in the length of the time period would be as much as 0.3 seconds. if we collect the extrudate

foarn for one minute, the corresponding error on the measured m a s flow rate would be

approximately 0.5% (Le., equal to 0.3s/6Os). This means that the error range for the

cdculated specific volume due to this human error would be as much as OS%, since the

specific volume is inversely proportional to the mass flow rate.

However, if a precise stopwatch was used to time the collection period, the error

associated with the extrudate collection would have been reduced to be approxirnately 0.1

seconds, then the associated human error in the measured mass flow rate would be reduced to

0.17% (Le., equal to O. ls/6Os). Using a longer collection time period woutd further reduce

this error. To minimize the human error associated with collection of the extrudate, a more

effective method for accurately measuring the m a s flow rate should be developed in the

future.

Additional Remarks on the Developed PVT Measurement Svstem

The newly developed system seems to be capable of measuring the specific volumes

of polyrner/COt solutions within an observed error range of +/- 0.8%. Since the specific

volumes of polyrner/COz solutions in a molten state have never been available previously, the

newly measured PVT data are expected to exert a great impact on the fundamental

understanding of the interactions between polymer and gas molecules. Furthermore, other

thermodynamic and rheological properties, such as the viscosity of polymer/gas solutions and

the solubiIity of gas in the polymer matnx, cm be anaiyzed based on the measured PVT data.

One limitation of the developed apparatus is that it can measure the PVT relationship

of a polymer/gas solution only under extrusion processing conditions above the

crystallization temperature of the polyrner. Thus, the ranges of temperature and pressure for

the measurement are Limited to the extrusion processing conditions. In order to determine the

swelling of a polymer in a solid state due to dissolved gas, the apparatus developed by Hirose

et al. 1221 should be used as discussed in Section 2.1.

5.5 CONCLUSIONS

An innovative system for measuring the PVT properties of polyrner/gas solutions was

designed and constructed using a positive displacement gear pump rnounted on a tandem

extrusion system, A metered amount of gas was injected into the polyrner matrix to form a

single-phase solution. The temperature was controlled by the second extruder and a heat

exchanger. The pressures in the upstream and downstream of the gear pump were controlled

by the rotational speed of the second extruder and a variable resistance die, respectively. With

the developed system, we successfully rneasured the PVT properties of PS/C02 solutions,

Iinear PBS/C02 solutions, and branched PBS/C02 solutions, at temperatures ranging from

147°C to 167°C for PBS materials and from 180°C to 220°C for PS material and at pressures

of 1 3 -78 MPa, 20.68 MPa, and 27.58 MPa (2000 psi, 3000 psi, and 4000 psi). Mathematical

modeling of the measured data based on the Sanchez-Lacombe EOS and rnixing niles

produced interaction parameters that differed from data obtained conventionally at the

solubility equilibrium condition because Our experiments were conducted with a CO2

concentration much lower than the solubility equil ibrium concentration. Since the

experïmental conditions varied from the conventional method, we modified and fitted the

interaction parameters as a function of temperature, pressure, and gas content. The

inconsistency between the interaction parameters calculated using the S-L EOS and those

reported previously suggests that the S-L EOS may be insufficient for describing the PVT

behaviours of potymer/CO2 sohtions.

The newly developed method represents a significant contribution to PVT

measurement instrumentation since it enables direct measurement of the PVT properties of

polyrner/gas solutions in a molten poiyrner state, which has not been reaiized previously. The

measured PVT relationships for poIyrner/gas soIutions would be very useful in predicting and

understanding the thermodynamic and rheological behaviours of various poIymer/gas

solutions.

Table 5.1. Erperirnental Data for the PS/C02 Solution ut Three Different Rotational

Speeds (5, 10, and 1.5 qx) of the Gear Prtmp

Table 5.2. Experimental Data for the Linear PBS/C02 Solution at Three Differenr

Rotational Speeds (5, IO. and 15 rpm) of the Gear Pump

Table 5.3. Experimentai Data for the Branched PBS/C02 Solution as Three Dzzerent Rotational Speeds (5, 10, and 15 rpm) of the Gear Pump

Table 5.4, Estimated Parameters for Pure PoZymers and CO2

PS

Linear PBS

Branched PBS COz

T* (K)

803.1

790.7

69 1 -8

3 14.8

P* (bar)

2758

2914

5293

4388

p* (cm31g)

1.072

1.203

1 -247

1.416

r

9298

5227

10469

5.286

Table 5.5. Variables for the S-L EOS MUcing Rules

Srrbscn'pt " 1 " represen rs COz Subscript "2 " represents the polymer

(a ) 180°C

(c) 167OC

(b) 157°C

Table 5.9. Characteristic Parameters of MCM EOS for PS/COL Solutions as a

Function of the Co2 Concentration

Tab[e 5-10. Characrerisric Parameters of MCM EOS for Linear PBS/C02

Solutions as a Function of the COt Corzcentration

Table 5.1 I. Characten3tic Paramerers of MCM EOS for Branched PBS/COt

Solutions as a Function of the CO2 Concentration

Table 5-12. Characteristic Parameters of S-L EOS for PSfC02 Solrtrions as a

Funcriorr of the COz Concentration

Table 5.13. Characreristic Parameters of S-L EOS for Linear PBS/CU2 Solutions

as a Function cf the CO2 Concentration

Table 5.14. Characterisric Parameters of S-L EOS for Branched PBSiCO?

Solutions as a Function of rhe CO? Concentration

Figure 5.1. Photograph of the Overall Setup with the Gus Injection System

Figure 5.2. Photograph of the Gus Syringe Pump

Set Constant RPM of the Gear Pump

/ Injection and Diffusion of CO2 in Polyrner Matrix (Maintain Therm odynam ic Stability)

Piniet = Poutiet Piniet controlled by Rotational Speed of Second Extruder

Poutiet controlled by Rotations of the Variable Resistance Valve

I Measurement of Mass and Volume Flow Rates

1

( Analysis -5 Figure 5.3. General Procedure to Measure the PVT Data of Polymer/Gns Solutions

12 14 16 18 20 22 24 26 28 30 Pressure (MPa)

Figure 5.4. EjJecr of Pressure on the Specific Volume of a PS/CO2 Solution ut Various Temperatures (a) O wt% of C02; (3) I wl% of CO2; (c) 2 wt% of CO2

1 .Io0

.$1.080

I? E O - > 1.060 O 5 - Z 1.040 !2 ii v, 1026

1 . m - 12 14 16 18 20 22 24 26 28 30

Pressure (MPa)

- - -

-- - -- -

+ 220C

-

.--

. -

-200C

-lm - -

Figure 5.4 (Contimed)). Eflect of Pressure on the Spec~jk Volume of a PS/C02 Sohtion at Vmious Temperatures (d) 3 wi% of COz; (e) 4 wt% of CO2

1.100

â 1.060

9 Y E ? 1.060 Q)

5 - O > 1.040 0 E

ii vJ 1.020

1 .O00 12 14 16 18 20 22 24 26 28 30

Pressure (MPa)

(e)

-

.S. -- -p.

------- --- ---

----

--P - 220C

-200C

- 180C

Figure 5.5. Effecf of Pressure on the Specific Vofume of a PS/C02 Soluiion af Various CO-, Conîenfs (a) 220 'C; (6) 200 C; and (c) 180

--- -4% - 3% -2% - 1%

+O96 --

1.100 .

1.1 00

- 1.080 e 9 E O - '-" S - 3 1.040

i 1.020

1.000

12 14 16 18 20 22 24 26 28 30 Pressure (MPa)

,l.oso

z' E 0 Y

> 1.060 e- S - 8 1.040 O

1 * 1.MO

1.000

-- -.

** --

-- -- - .

r

12 14 16 18 20 22 24 26 28 3û

Pressure (MF%)

-t 4%

4-3s - 2% :--- --- --

. - m

--Ir;

-0% --


(a)

1.100

-1.080 e P 5 V

2. 1.060 6 !E - 3 1.040


O

-138 Bar

1,000 170 180 190 210 220 230

---

---

-- 207 Bar

-p.

-c 138 Bar

-207 Bar

-276 Bar

- 276 Bar --A--

- 138 Bar

170 180 190 200 210 220 230


(c)

+207 Bar

-276 Bar .- - --

Figure 5.6. Ejecci of Temperature on the Spectjk Volume of a PS/C02 Solution ar Vmious Pressures (a) O wt% of COI; (I) i wt% of C02; (cl 2 wz% of CO2

l-lm / - -r 138 Bar

1 170 180 190 200 210 220 230


@..

, ---- -r 138 Bar

- 276 Bar

Figwe 5.6 (Contimed). Effeci of Temperature on the Specific Voiume of a PS/C02 Soiution aï Vatims Pressures (4 3 w f % of CO?; (el 4 WC% of CO2

Figue 5.7. Efect of Temperature on the Specijc Volume of a PS.C02 Solution at Vwious CO2 Contents (a) 2 76 Bar; (3) 207 Bar; ond (c) 138 Bar

O 1 2 3 4 5 CO2 Concentration (wt. %)

Figure 5.8. Eflecr of CO2 Content on the Specific Vofume ofa PS/COr Solution at Various Pressures (a) 220 C; (6) 200 r; and (c) 180 4:

138

COZ Concentration (wt. %)

(a)

CO2 Concentration (M. %)

(6)

1 2 3 4

CO2 Concentration (wt %)

(c)

Figure 5.9. Efject of CO2 Cor~tenf on the Speczjic Volume of a PS/C02 Sohtlon al Vm*ous Tempera~ures (a) 276 Bar; (b) 207 Bar; and@) 138 Bar

139

Figure 5.1 O. Ef/ects of Pressure on the Specifc Volume of a Linear PBS.C02 and a Branched PBS/COî Solutions at Vuriotcs Tempera~ures (4 O wî% of COz; (b) 1 wt% of COz; (c) 2 wt% of CO2

Figure 5- IO (Continued). Effecfs of Pressure on ihe Specific Volume of a Linear PBS/C02 and a Branched PBS/COz Solutions ut Varias Temperatures ( i 4 wr% of C a ; (el 6 wt% of Ca

Figure 5. i. 1. Effects of Pressure on the Speczfc Volume of a L hem PBS/C02 nnd a Branched PBS/C02 Solutions ut Varrarrms CO2 Contents (a) 16793: (b) l57C; md(c) 147@

142

Figure 5.12. Effects of Temperature on the Specipc Volume of a Linear PBS/C02 and a Branched PBS/C02 Solutions al V ~ o u s Pressures (a) O wt% of COz; (3) i wt% of CO.?; (c) 2 wt% of C0.z

137 147 1 57 167 1 77 Temperature (deg. C)

(9

Figure 5.12 (Continued). Effects of Temperature on the Spec~jTc Volume of a Linear PBS/C02 and a Branched PBS/C02 Solutions at V i o u s Pressures (d) 4 wt % of COz; (e) 6 wt% of COz


-2% (ûneir PeS, - 1% (imemr PBS) >

-Wb ( m u r PBS)

=..-- -. - - - - Q-

L ---.. 6% (Blnched PBS)

0.920 =.- -.-- . -...-.-... .--.-.... (

- a . ïs (&Mencd P m

- - - - - l % ( ~ v i c n e a ~ r n

0.880 --

137 147 157 167 1 77


Figure 5.13. Eflects of Temperature on the Specrfic Vofume of a Linem PBS/CO2 and a Brancheci PBS/C02 Sohrions at Various COr Conîents (a) 2 76 Bar; 0 20 7 Bar; and (c) 138 Bar

CO2 Concentration (HR- 94)

f i !

O t 2 3 4 5 6 7 (332 Concentration (wt. %)

(c) Figure 5. M. Effects of CO2 Content on the Specific Volume of a Linear PBS/C02 and a Brmched PBS/C02 Solutions at Various Pressures (a) 1 6 7 r ; (3) 1 5 7 r ; and (c) 1 4 7 r

CO2 Concentration (wt. %)

0.880 O 1 2 3 4 5 6 7

CO2 Concentration (uh %)

(c)

Figure 5.15 Effects of CO2 Con~ent on the Specific Volume of a Linear PBS/COr and a Branched PBYC02 Solutiom at Various Temperatures (a) 2 76 Bar; (6) 207 Bar; and (c) 138 Bar

+ Experimental Data

Figure 5.16. The Location of Erperimentaf Data on Temperature vs. Spec~fic Volurne Graph at Constant Gas Concentration

Figure 5.17. Ejject of Temperatures on the Binary Interaction Parumeter, Delta (6'3. of PS/COz Solulion at V d s CO2 Contents (a) 276 bar: (b) 20 7 bar; and (c) 138 bar

Figure 5.18. Effect of Pressures on the Binmy Interact~on Parame fer, Delta (&$, of PS/C02 Sokrtion at Vmious C a Contents (a) 220 97; (b) 200 Y; and (c) 180

150

Figure 5.19. Eflect of CO2 Content on the Binary Interaction Parameter, Delta (6i3, of PS/COr Solution ut Various Pressures (a) 220 T; fi) 200 Y; and (c) 180T

15 1

Figure 5.20. Eject of Temperatures on the Binary Interactzon Pmarneter. Deha (613. of LÏnear PBWCOZ Solution ut Various COz Contents (a) 276 bar; (iS) 20 7 bar; and (c) 138 bar

m Pressure (bar)

(a)

Pressure (bar)

O?!

200 2% 300

Pressure (bar)

(c)

Figure 5-21. Effect of Pressures on the Binary Interaction Parame fer, Della (613. of Linear PBS/C02 Solution a2 Various CO2 Contents (a) 1 6 7 r ; @) 1 5 7 C and(c) I47T

-- t 276 bar + 207 bar -138 bar

O 1 2 3 4 5 6 7 8

C02 Concentration (%)

O 1

. + 276 bar + 207 bar t 138 bar -----

-2% bar t 207 bar

t

Figure 5.22. Eflect of Pressures on the Binary Interaction Parameter, Delta f8''S. of Linear PBS/C02 Solution ut Various CO2 Contents ((a) 167T; (6) 157T; and (c) 1 4 7 r

Figure 5.23. Effect of Ternperaf ures on the Binary in!eraction Parameter, Delta (613. of Branched PBS/C02 Solution at Variozrs CO2 Contents (a) 2 76 bar; (3) 207 bar; and (c) 138 bar

Pressure (br )

O

100 150 200 250 300

Pressure (bar)

Figure 5.24. Effect of Pressures on the Binary Interacrion Parameter, Delta (SIJ , of Branched PBS/C02 Sohtion at V&us CO2 Contents (a) 167'32 (b) 157T; and (c) i47r

+ 276 mr +- 207 bar t l J 8 bar

Figure 5.25. Effect of Pressures on the Binary Interaction Parameter, Delta (613, of Brancheci PBS/C07 Soiution at Vùrious CO7 Contents (a) 167T; (b) 157C; and (c) 147T

SUMMARY AND CONCLUSIONS

6.1 SUM&IARY

The PVT relationships for polymedgas solutions are important in polymer processing,

especially in foam processing. For exarnple, the rneasurement of some therrnodynamic and

rheological properties, such as the surface tension of a melt, the solubility, the diffusivity, and

the viscosity of gas in a melt, requires the PVT relationship. However, measurement of the

PVT properties of polyrner/gas solutions in a molten state has not been realized because of

the di fficulties involved in the measurements. The newly developed dilatometer presented in

this thesis overcomes this limitation posed by conventional dilatometers by using a gear

pump and a tandem extrusion line. Although the repeatability and validity of the

experimentd data stil1 need to be verified, this research represents a significant contribution

towards the PVT measurement instrumentation for polyrner/gas solutions.

With the newly developed dilatometer, the specific volumes of a polymer/C02

solution were determined by rneasuring the mass and volume flow rates of the solution

independently. A positive-displacement gear pump mounted on the tandem extruder was

used to measure the volumetric flow rate of the solution. The gear pump was calibrated pnor

to the experirnent using water and vegetable oil in order to obtain the volumetric

displacement of the gear pump per revolution. The mass flow rate of the solutions was

determined by carefully measunng the weight of the sample produced in a given period of

time and adding the weight of CO2 injected from the syringe pump. The temperature of the

extmded polymerlgas mixture was increased in order to promote gas loss from the extrudate

coming out of the filament die with the goal of reducing the error involved in the

measurernent of the mass flow rate. Furthemore, the upstream and downstrearn pressures

were equaiized in order to prevent leakage between the inlet and outlet of the gear pump.

The upstream pressure was controlIed effectively by the rotation of the second screw and the

downstream pressure was controlled by a variable resistance valve.

One of the iimitations associateci with this experimental setup was that it could

measure the PVT relationship of a polyrnedgas solution only under extrusion processing

conditions. Due to the inherent nature of the extrusion system, obtaining consistent data

below !3.8 MPa (2000 psi) and above 27.6 MPa (4000 psi) was challenging. Since the

extrusion system can onIy process polymers at temperatures above their melting point, PVT

properties were rneasured over a relatively small range of temperatures. in the future, the

apparatus should be modified to handle higher pressure and temperature ranges.

6.2 CONCLUSIONS

i. The volumetric displacement of the gear pump was calibrated using water and vegetable

oil at various conditions. The volumetric flow rate was found to be 1.170 (+/- 0.002)

m h e v for the specific gear pump.

ii. The proposed design was verified by rneasuring the PVT retationships of numerous pure

polymers and comparing the experimental results with data from the literature.

The PVT relationships for linear PP melt were measured at temperatures ranging

from 180°C to 210°C and at pressures ranging from 6.9 MPa (1000 psi) to 29.6

MPa (4300 psi). The cornparison between the measured data and data published by

Sato e t al. showed that the experimentai results are almost the sarne as the reference

data.

The PVT relationships for PS melt were measured at temperatures ranging €rom

180°C to 220°C and at pressures ranging from 13.8 MPa (2000 psi) to 27.6 MPa

(4000 psi)- Cornparison witb the manufacturer's data showed a slight deviation,

which rnay have k e n caused by experimental errors.

The PVT relationships for linear PBS melt were measured at temperatures ranging

from 137°C to 167°C and at pressures ranging 13.8 MPa (2000 psi) to 27.6 MPa

(4000 psi). The experimental resuits were also compared with the manufacturer's

data. The experimen ta1 data sho wed a significant deviation from the manufacturer's

data which may have been caused by improper manipulation of the manufacturer's

data provided in a graphicd format. In addition, the PVT properties of branched

PBS were measured at temperatures ranging from 137°C to 167OC and at pressures

ranging from 13.8 MPa (2000 psi) to 27.6 MPa (4000 psi). Linear and branched

PBS materials were compared. Comparison between the two suggests that slight

differences in the PVT relationships may have been caused by the degree of

branching in the polymer matrix.

= The experimental PVT data for PP, PS, and linear and branched PBS were modeled

using FOV, MCM, and S-L EOS, and the characteristic variables for these EOS

were estirnated by using non-linear regression anaiysis. From the modeis. one can

predict the PVT behaviour beyond the experimentd ranges.

. . - 111.

iv.

v.

vi.

vii.

A syringe pump was used to inject CO2 gas into a Stream of molten polymer to form the

polymer/C02 solutions. Carbon dioxide was used as a physical blowing agent in this

study because of its high solubility in polytners as well as its environmentally benign

characteristics. The injected gas was completely dissolved by means of convective

diffusion processes to forrn single-phase polymer/C02 solutions.

The PVT data of PS/COZ solutions were measured at temperatures ranging from isO°C

to 220°C and at pressures ranging from 13.8 MPa (2000 psi) to 27.6 MPa (4000 psi).

Ga weight percentages of 1,2,3, and 4 wt% were used in the experiment.

The PVT data of linear PBS/C02 solutions were rneasured at temperatures ranging from

147°C to 167°C and at pressures ranging from 13.8 MPa (2000 psi) to 27.6 MPa (4000

psi). Gas weight percentages of 1,2,4, and 6 wt% were used in the experiment.

The PVT data of linear PBS/C02 and branched PBS/C02 solutions were compared. The

linear PBS exhibited higher swelling due to the gas content than the branched PBS

material. One hypothesis to explain this phenomenon is that the branched PBS material

has a large number of entangled molecular chahs, and therefore it swelled less thm the

Iinear PBS materials.

Attempts were made to mode1 the poIyrner/gas solutions using S-L EOS and its mixing

rules. The validity of this approach was unvetified, however, since the EOS assumes

solubility equilibrium conditions; in Our expenments, the solutions were above the

solubility equilibnum state. We assumed that the interaction parameter between the

polyrner and gas molecules should be a sensitive function of temperature and gas

content instead of temperature only when the condition is above the solubility

equilibrium. However, the derived interaction parameters were not consistent with the

conventiondly measured values. The validity of S-L EOS to describe our results is

suspected.

. - * viii. Another approach to investigate the effects of gas content on a polymer/gas solution was

performed by finding the characteristic parameters for MCM and S-L EOS. The

experimental data were treated as independent sets of different gas concentration. The

characteristic parameters were obtained using the non-linear regression method. The

results showed that the characteristic parameters were conjugated with each other and

the correIation did not show any trend. A large number of experimental data should be

obtained and modeled in order to corne up with solid conclusions about the modeling of

the specific volume of a polymer/gas solution.

Chapter 7

The following suggestions are made to direct future research on the measurernent

of the PVT properties of polymer/gas solutions:

i. PVT data should be acquired over a wider range of experimental conditions.

Increasing the temperature and pressure ranges is very important because a large

number of data points are needed to properly model the experimental data. Also,

finding the maximum solubility limit for the apparatus would be beneficid for

future studies.

ii. ModeIing using various EOS of polymedgas solutions should be performed. Since

the PVT relationships of polymer/gas solutions have not been measured before,

modeling the experimental data will be difficult. However, a detailed investigation

on the interaction between the polymer and CO2 molecules may provide insight to

predict the PVT behaviour. Perhaps a new EOS model, as well as mixing niles,

may be needed to predict the behaviour of the polyrner/gas solution above the

solubility equilibrium state.

... I i 1. The relationship between rheological behaviour and the PVT properties should be

investigated. Since both behaviours deal with free volumes in the polymer matrix, a

relationship between the two may be established.

iv. Alternative physical blowing agents, e.g., nitrogen and butane, may be used to

measure the PVT rekttionships of pol ymer/gas solutions. Finding the relationshi p

with different physical bIowing agents would be desirable since the PVT behaviours

of polyrners with these physical blowing agents have not been rneasured previously.

v. With the newly developed apparatus, more detailed research c m be performed to

investigate the effects of gas content, pressure, and temperature on specific volume.

The study should be focussed on the theoretical aspects of the PVT relationship.

Since the apparatus is the first of its kind, verification of the experimental results

should be performed as well.

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ON-LINE MEASURE~ENT OF PVT PROPERTIES POLYMERICO ... · ON-LINE MEMUREMENT OF THE PVT PROPERTIES OF POLYMER~CO~ SOLUTIONS USING A GEAR PUMP Simon S. Park Degree of Masters of Applied