xml version 1.0 encoding UTF8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam 2200421Ka 4500
controlfield tag 001 001681077
005 20060215071215.0
006 m d s
007 cr mnu uuuuu
008 051227s2005 flu sbm s000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14SFE0001039
035
(OCoLC)62745684
040
FHM
c FHM
049
FHMM
090
TP145 (Online)
1 100
Damnjanovic, Ratka.
0 245
Theoretical and experimental investigation of phase behavior of polymeric systems in supercritical carbon dioxide and their modeling using saft
h [electronic resource] /
by Ratka Damnjanovic.
260
[Tampa, Fla.] :
b University of South Florida,
2005.
502
Thesis (M.S.Ch.)University of South Florida, 2005.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 127 pages.
520
ABSTRACT: Environmentally friendly processing of materials is becoming an increasingly important consideration in a wide variety of emerging technologies. Polymer processing,in particular, has benefited tremendously in this venue from numerous advances achievedusing highpressure carbon dioxide (CO2) as a viscosity modifier, plasticizing agent,foaming agent, and reaction medium. Polymer processing in supercritical fluids has been a major interest for a portfolio of materials processing applications including their impregnation into porous matrices. Also, SCF solvents are being examined as a media for polymerization processes, polymer purification and fractionation, and as environmentally preferable solvents for solution coatings. Pressurized CO2 isinexpensive, sustainable, relatively benign, and versatile due to its gaslike viscosity and liquidlike densities, which can be controllably tuned through appropriate choice of temperature and pressure.Addition of highpressure CO2 to polymer systems can have a profound impact on their thermodynamic properties and phase behavior, since the number of interacting species increases due to the highpressures, so that the compressibility also increases, as well as the plasticity effects. Even then, polymers are only sparingly soluble in CO2 unless one uses an entrainer or surfactant. An addition of a liquid monomer cosolvent results in greatly enhanced polymer solubility in the supercritical fluid at rather mild conditions of lower temperatures and reduced pressures.The focus of this research is to measure, evaluate and model the phase behavior of the methyl methacrylateCO2 and the poly (methyl methacrylate)CO2methyl methacrylatesystem, where methyl methacrylate plays role of a cosolvent.Cloudpoint data are measured in the temperature range of 3080C, pressures as high as 300 bar, cosolvent concentrations of 27 and 48.4 wt% MMA, and varying PMMA concentrations of 0.1, 0.2,0.5, and 2.5 wt%. Solubility data is reported for these systems. The experimental results are modeled accurately using the Statistical Associating Fluid Theory (SAFT) for multi component polymer/solvent mixtures. The measured solubility data appears to be significantly different than previously published results by McHugh et al, Fluid Phase Equilibria, 1999. Thorough investigation, recalibration of the equipment, and repetition of the measurements has proved that the measured data is entirely correct and the reference data is significantly off, which indirectly gives credit to this work and opens room for amendments of those results.
590
Adviser: Dr. Aydin Sunol.
653
Polymers.
Solubility.
Methyl methacrylate.
Poly (methyl methacrylate).
Statistical associating fluid theory.
690
Dissertations, Academic
z USF
x Chemical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
949
FTS
SFERS
ETD
TP145 (Online)
4 856
u http://digital.lib.usf.edu/?e14.1039
PAGE 1
Theoretical And Experimental Investigation Of Phase Behavior Of Polymeric Systems In Supercritical Carbon Dioxide A nd Their Modeling Using SAFT by Ratka Damnjanovic A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Chemical Engineering Department of Chemical Engineering College of Engineering University of South Florida Major Professor: Aydin K. Sunol, Ph.D. John T. Wolan, Ph.D. Vinay K. Gupta, Ph.D. Date of Approval: March 1, 2005 Keywords: polymers, solubility, methyl me thacrylate, poly (methyl methacrylate), statistical associating fluid theory Copyright 2005 Ratka Damnjanovic
PAGE 2
DEDICATION I dedicate this Thesis to my family, my dear husband Aleksandar and ou r son Nenad, and my parents Milica and Ilija Kolevi, who have alwa ys been very supportive and encouraging in anything I tried to accomplish in life, and who are always proud with every success I achieve.
PAGE 3
ACKNOWLEDGMENTS I am grateful to my advisor Dr. Aydin K. Sunol and his wi fe, Dr. Sermin G. Sunol, for making this work possible, for th eir continuous support, encouragement, and guidance throughout the course of this study. I am extremely grateful to my husband Aleksandar, and my son Nenad, as well as the rest of my family, for their unconditional love, support, continuous encouragement, unde rstanding, and patience during my studies. My colleagues have been extremely invaluab le providing assistance, friendly atmosphere and optimism, especially Naveed Aslam with his advices and assistance, and for providing the code for the modeling part of this work, Raquel, Brandon, and Eddy who were always ready to offer their help with all the little details, and to my fellow classmates, Jeffy, Keyur, and others for the good spirits, humor, a nd positive attitude. I am also grateful to all my teachers and advisors fr om the past for bringing up in me the spirit and love for science, and for guiding me ont o the path Ive taken, especially my exmentors at the University of Witwatersrand in South Africa, Dr. Diane Hildebrandt and Dr. David Glasser. I am enormously thankf ul to our Graduate Coordinator, Dr. John Wolan, for being always helpful regarding a ll matters related to academic admission, academic advising, course selection, etc., sec ond for teaching us a great class in Reacting Systems, and third for referring me for ad mission to the Omega Chi Epsilon Honors
PAGE 4
Society. Last but not least, I am grateful to Dr. McHugh and his group for their email correspondence, and for acknowledging my findi ngs that the measured data that they have published in Fluid Phase Equilibria, 157 (1999), 285297 is not reproducible. Dr. McHugh and his group are not sure why their solubility data are not reproducible, but they plan on repeating their measurements a nd publishing a short paper with a new data. They will reference me in their updated paper as the one who brought this discrepancy to their attention. I am truly honored by their ge sture. Also, I want to thank my committee members, Dr. John T. Wolan and Dr. Vinay K. Gupta, for their valuable suggestions and help. And, again, I am grateful to Dr. Sunol and to the Chemical Engineering Department at USF for the Graduate Rese arch Assistantship, and the appropriate financial support they provided which helped me going during the course of this study.
PAGE 5
i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv LIST OF SCHEMES v LIST OF SYMBOLS vi ABSTRACT viii CHAPTER 1 INTRODUCTION 1 1.1. Background 1 1.2. Research Objectives 4 1.3. Overview 6 CHAPTER 2 LITERATURE REVIEW: SOLUBILITY MEASUREMENTS, CORRELATIONS AND PREDICTIVE METHODS 7 2.1. Design of Polymer Blends 7 2.2. Solubility in Carbon Dioxide 10 2.3. Experimental Techniques for Solubility Measurement 11 2.4. Polymer Characteristics and Morphology 12 2.4.1. Heat Capacity 14 2.4.2. The Glass Transition Temperature 15 2.4.3. Crystallization 16 2.4.4. Melting 17 2.4.5. Differential Scanning Calorimetry of Polymers 18 2.5. Thermodynamic Phase Behavior of Polymer Blends 20 2.6. Thermodynamic Modeling of Po lymer Solutions and Blends 22 2.6.1. HighPressure Vapo rLiquid Equilibria 24 2.6.2. Equations of State 26 2.6.2.1. Lattice Fluid Models (The SanchezLacombe EOS) 26 2.6.2.2. Perturbation Models (SAFT) 31 CHAPTER 3 EXPERIMENTAL SYSTEM AND PROCEDURES 38 3.1. Experimental Apparatus Description 39 3.1.1. Phase Monitor 40 3.1.2. Tubing Reticulation 42 3.1.3. HighPressure Syringe Pump 43
PAGE 6
ii 3.1.4. LowTemperature Circulator 43 3.1.5. Syringe Pump Controller 44 3.2. Calibration of the Phase Monitors PLC Controller 44 3.3. Experimental Section 46 3.3.1. Materials Description 46 3.3.2. Experimental Procedure 48 CHAPTER 4 RESULTS AND DISCUSSION 53 4.1. Experimental Results 53 4.1.1. CO2MMA System 53 4.1.2. PMMACO2MMA System 57 4.2. Overview of the Equations Used for Computation of the Results 61 4.2.1. The PengRobinson EOS with th e Wong Sandler Mixing Rules 61 4.2.2. PolymerSupercritical Fluid Equilibria 63 4.2.3. The SAFT Equation of State 66 4.3. Modeling of the PMMACO2MMA System Using SAFT EOS 69 4.3.1. Solubility Behavior of the PMMACO2MMA System 71 CHAPTER 5 CONCLUSIONS AND FUTURE WORK 77 REFERENCES 79 BIBLIOGRAPHY 81 APPENDICES 84 Appendix A: MATLAB Program That Ca lculates the Z Values of Pure CO2 Using the PengRobinson EOS w ith WongSandler Mixing Rules 85 Appendix B: Sample Calculation to De termine the Mole Fractions of MMA 87 Appendix C: Mole Fraction of Polymer Compared to the Corresponding Weight Fraction 88 Appendix D: The SAFT EOS Model Used in This Study 89 Appendix E: Hand Calculation of Pure Component Parameters and Solubility Data for the PMMACO2MMA System 99 Appendix F: MATLAB Program for Re gression of Experimental Data 101
PAGE 7
iii LIST OF TABLES Table 4.1 Experimental Data for the MMACO2 System Obtained in This Study 54 Table 4.2 Experimental CloudPoints for the Studied PMMACO2MMA System 58 Table 4.3 Critical Constants and Acentric Factor for CO2 and MMA 69 Table 4.4 Pure Component Paramete rs Used in the SAFT Equation 70 Table 4.5 Solubility Data for the PMMACO2MMA System of This Study Calculated for all PTx Comb inations Using the SAFT EOS 72 Table 4.6 Fugacity Coefficients Calculated for the Vapor and the Liquid Phase for the PMMACO2MMA System of This Study Using the SAFT EOS 74 Table 4.7 SAFT Modeled Solubility of Poly (Methyl) Methacrylate in CO2 in the Presence of 2.0wt% MMA 75
PAGE 8
iv LIST OF FIGURES Figure 2.1 Differential Scanning Calorimetry (DSC) 13 Figure 2.2 Heat Capacity 14 Figure 2.3 Glass Transition Temperature 15 Figure 2.4 Crystallization Temperature 16 Figure 2.5 Melting Temperature 18 Figure 2.6 DSC Thermal Signature of a Specific Polymer 19 Figure 2.7 Molecular Mode l Underlying the Pertur bedChain SAFT EOS 33 Figure 3.1 Schematic Diagram of the Hi ghPressure Phase Monitoring System 39 Figure 3.2 The Vessel Assembly and Cont roller of the Phase Monitor SPM20 41 Figure 4.1 MMACO2 Experimental Isotherms From This Study 55 Figure 4.2 MMACO2 Experimental Isotherms vs Data by McHugh et al 56 Figure 4.3 Experimental CloudPoint Curves for PMMACO2MMA 59 Figure 4.4 Experimental CloudPoint Curves for the PMMACO2MMA System of This Study vs. CloudPoint Data by McHugh et al 60 Figure 4.5 Solubility Is otherms for the PMMACO2MMA System of This Study Calculated Using the SAFT EOS Model 73 Figure 4.6 SAFT Modeled Solubil ity of PMMACO2MMA System 75
PAGE 9
v LIST OF SCHEMES Scheme 3.1 Chemical Structure of Acrylate Monomers 47 Scheme 3.2 Chemical Structure of Me thacrylate 47 Scheme 3.3 Polymerization of Methyl Meth acrylate to Get PMMA 48
PAGE 10
vi LIST OF SYMBOLS a parameter in a cubic equation of state; activity A Helmholtz energy b parameter in a cubic equation of state M molecular weight, g/mol Mw weightaverage molecular weight Mn numberaverage molecular weight P pressure, bar Pc critical pressure, bar Pr reduced pressure, P/Pc Pvap pure component saturation pressure Psub pure component sublimation pressure R gas constant T temperature, C, K Tc critical temperature, C, K Tr reduced temperature, T/Tc Tg glass transition temperature, C, K Tm melting temperature, C, K Tc crystallization temperature, C, K v molar volume, cm3/mol
PAGE 11
vii V volume, cm3 Z compressibility factor, Pv/RT density, g/cm3 fi fugacity of species i in a mixture fugacity coefficient i chemical potential G molar Gibbs free energy of a mixture (or of pure component) S entropy x liquidphase mole fraction y vaporphase mole fraction yA solubility of the solute in the supercritical fluid n number of moles m number of segments in a molecule rij distance between molecules i and j kij binary interaction parameter v volume of sitesite association w Pitzers acentric factor
PAGE 12
viii THEORETICAL AND EXPERIMENTAL I NVESTIGATION OF PHASE BEHAVIOR OF POLYMERIC SYSTEMS IN SUPERC RITICAL CARBON DIOXIDE AND THEIR MODELING USING SAFT Ratka Damnjanovic ABSTRACT Environmentally friendly processing of materials is becoming an increasingly important consideration in a wide variety of emerging technologies Polymer processing, in particular, has benefited tremendously in th is venue from numerous advances achieved using highpressure carbon dioxide (CO2) as a viscosity modifi er, plasticizing agent, foaming agent, and reaction medium. Polymer processing in supercriti cal fluids has been a major interest for a portfolio of materi als processing applications including their impregnation into porous matrices. Also, SC F solvents are being examined as a media for polymerization processes, polymer pur ification and fractionation, and as environmentally preferable solvents for solution coatings. Pressurized CO2 is inexpensive, sustainable, relatively benign, a nd versatile due to its gaslike viscosity and liquidlike densities, which can be contro llably tuned through appropriate choice of temperature and pressure. Addition of highpressure CO2 to polymer systems can have a profound impact on their thermodynamic prope rties and phase behavior, since the number of interacting species increases due to the highpressures, so that the compressibility also increases, as well as the plasticity effects. Even then, polymers are only sparingly soluble in CO2 unless one uses an entrainer or surfactant. An addition of a liquid monomer cosolvent results in grea tly enhanced polymer solubility in the
PAGE 13
ix supercritical fluid at rather mild conditions of lower temperatures and reduced pressures. The focus of this research is to measure, evaluate and model the phase behavior of the methyl methacrylateCO2 and the poly (methyl methacrylate)CO2methyl methacrylate system, where methyl methacrylate plays role of a cosolvent. Cloudpoint data are measured in the temperature range of 3080 C, pressures as high as 300 bar, cosolvent concentrations of 27 and 48.4 wt% MMA, and varying PMMA concentrations of 0.1, 0.2, 0.5, and 2.5 wt%. Solub ility data is reported for these systems. The experimental results are modeled accurately using the Statistical Associating Fluid Theory (SAFT) for multicomponent polymer/solvent mixtures. The m easured solubility data appears to be significantly different than previously pub lished results by McHugh et al, Fluid Phase Equilibria, 1999. Thorough investigation, recal ibration of the equipment, and repetition of the measurements has proved that the m easured data is enti rely correct and the reference data is significantly off, which indi rectly gives credit to this work and opens room for amendments of those results. In addition, a reasonably closer qualitative match is achieved with the SAFT model used in th is work as opposed to the modeling results published by the McHugh group.
PAGE 14
1 CHAPTER 1 INTRODUCTION 1.1. Background Solubility is a key fundamental prope rty in studying phase behavior of multicomponent mixtures. Phase behavior dictates feasibility of various separation techniques as well as material processing pathways. In polymer processing, particularly, the phase behavior of polymer solutions play importa nt role, primarily because many polymers are produced in solution and the final product may contain some residual solvent. The physical properties of polymers are usually affected by the amount and type of the lowmolecularweight components they contain. A frequent technical pr oblem is to remove essentially all the lowmolecularweight components; a common procedure is to volatilize them through a remova l process called polymer de volatilization [10]. Total removal of solvent is particularly important for polymeric films, and other applications such as impregnation of porous materials w ith polymers. Therefore, proper solvent selection for any solubility proc ess is a very important step as it can greatly influence and determine the homogeneity of the mixture, the de gree of dissolution of the solute, in this case the polymer, and the effective removal of the residual solvent. Liquefied gases are potentially good candidates as they can be easily removed by varying the pressure.
PAGE 15
2 Intuitively, we all know that gases are not considered as good solvents. However, many gases can exhibit significant solvent strength when they are compressed to liquidlike densities above their criti cal point. By operating in th e critical region, the pressure and temperature can be used to control dens ity, which regulates the solvent power of the supercritical fluid. This implies that it is the interaction between the solvent and solute molecules that determines how much solute dissolves in the SCF solvent. By compressing the solvent to liquidlike densitie s, we have only incr eased the probability that solvent and solute molecules will interact The types of intera ctions (e.g., dispersion, polar, hydrogen bonding) depend on the physical ch aracteristics of each of the species in the mixture [1]. Supercritical fluid technology has made trem endous strides, in the past decade, in terms of commercial application and funda mental understanding of solution behavior. Significant contribution has come from the c ontinuing pressure on industry to move away from volatile organic compounds (VOC) and ozone depleting subs tances (ODS) as processing solvents [4]. Besides their environmentally friendly nature, there are additional advantages of a supercritical flui d process such as lower energy requirements, improved product quality, simplified reaction/ separation scheme, higher selectivity, and higher recovery of the supercritical solvents compared to liquid solvents. Furthermore, popular supercritical solvents, such as carbon dioxide and water, are inflammable. Due to higher mobility compared to solvents and liquidlike solvency power compared to gases, the supercritical solvents have adva ntages such as easy control and manipulation of process parameters, favorable mass transfer and kinetic consideratio ns for supercritical
PAGE 16
3 processes [7]. While all of these characterist ics are touted as poten tial advantages of a supercritical fluid process, hurdles remain to in still it as a viable choice in process design. However, the existence of several small and growing companies (Novasep, Trexel, Micell, Phasex, Thar Technologies) confirms that the recent successes are starting to carve out a market segment [4]. Some initial applications of supercritical fluids are the separa tion of caffeine from green coffee beans using supercritical carbon dioxide [7], deasphalting heavy residual oils with supercritical propane [1], and removing adsorbed materials from activated carbon with supercritical carbon dioxide [7]. The most current industrial applications make use of supercritical carbon dioxide in extraction and frac tionation of food and biomaterials. The food and pharmaceutical in dustries use this technology more often than others since th e nontoxic nature of CO2 provides a strong impetus. Other applications being carried in a fluid phase include several of the particle generation technologies wherein materi al is dissolved in CO2 or an organic solvent and precipitated from the solution via a pressure or solvent composition change. Particle generation is certainly an intense inquiry and is receivi ng attention primarily from the pharmaceutical and biotechnology industries, although there are significan t polymer and inorganic material applications of these techniques [4]. Specific examples of recent successes are the new DuPont facility for producing fluoropolymers in a supercritical carbon dioxidebased solven t. Dry cleaning technology based on liquid CO2, is competing in the textile mark et with both Washpoint (ICI/Linde) and Micare (Cool Clean Technol ogies), representing viable a lternatives with chlorinated
PAGE 17
4 solvents. Furthermore, there are several efforts underway to commercialize cleaning technologies in the microelectronics industr y. Many new developments have arisen of the use of CO2 as a novel reaction solvent. Novel construction of CO2phillic catalysts and surfactants have allowed both traditional and new reaction pathways to be explored. In fact, the development of novel po lymeric surfactants for use with CO2 has resulted in two Presidential Green Chemistry Awards [4]. 1.2. Research Objectives As it will be described in this work, it is believed that the largest potential application of supercritical so lvents and supercritical fluid technology is in the area of polymerization and polymer processing, such as impregnation, particle formation, encapsulation, separation, foaming (batch an d extrusion), blending, foam injection molding, etc. In this work, the use of CO2 as a processing so lvent for nonreactive processes, and more particularly for the physic al processing and solubility of polymeric materials is studied. Due to the increasing interest in the various emerging superc ritical technologies, both the existing and the upcoming new appl ications require fundamentally correct description and quantification of physical, transport, and thermodynamic properties of components of interest. Understanding the underlying physics and chemistry of SCFpolymer solution behavior provides the opportunity to fully ex ploit the potential of SCFbased polymer processing. Although a detailed understanding of the physics and chemistry of polymer
PAGE 18
5 liquid mixture has emerged in the past decades, significant challenges remain for developing the same level of understanding of polymerSCF solution behavior [3]. At present, efficient development of SCFbased polymer processing technology suffers from the limitation that equations of state utilized for process simulation and modeling still do not adequately describe the unique characteristics of a longchain polymer solution. The underlying issue is how to account for the intraand intersegmental interactions of the many se gments of the polymer connected to a single backbone relative to the small number of segments in a solvent molecule [3]. In this research, a review of the strengths and limitations of equation of state modeling of SCFpolymer system are studie d. The advantages of using molecular thermodynamics and statistical associating fluid theory (SAFT) for modeling of SCFpolymer systems are presented. The SAFT mode l [19] used in this work has proven to produce better prediction of the solubility behavior of the CO2poly(methyl methacrylate) system used in this study th an previous modeling attempts reported in the literature. Fundamental properties of CO2polymer systems are discus sed in this work with an emphasis on available data and measurem ent technologies. Available data from literature is compared to the experimental da ta obtained in this work, specifically the solubility of poly(methyl methacrylate) in CO2 with the addition of a monomeric cosolvent such as methyl methacrylate. Much more accurate experimental data is obtained in this study for the solubility of the monomer and the polymer in CO2.
PAGE 19
6 1.3. Overview In Chapter Two, a literature review is presented for the design of SCFpolymer systems, polymer blend thermodynamics, expe rimental techniques and measurement of solubility of polymers, equations of st ate and modeling of SCFpolymer behavior. In Chapter Three, the experimental sy stem description, equipment, parts and components description, experimental setup and calibration, experimental procedures, preparation of chemicals and temperatur econtrolled environment are discussed. In Chapter Four, the experimental system components, independently and integrated, are analyzed, and the experime ntal results are presented and discussed. In Chapter Five, a conclusion of the re sults and future work perspectives are presented. In the Appendices, the modeling algorith m description, modeling equations and sample calculations are presented and analyzed.
PAGE 20
7 CHAPTER 2 LITERATURE REVIEW: SOLUBILITY MEASUREMENTS, CORRELATIONS AND PREDICTIVE METHODS In this chapter, the measurement techni ques, correlations and theoretical models used in literature are disc ussed with a brief fundamental and historic background. The solubility measurement methods and techniqu es are presented, followed by the prediction methods, correlations and theoretical models The focus is on s upercritical carbon dioxidepolymer systems. 2.1. Design of Polymer Blends Although much progress has been made si nce the time of Flory and Huggins in the understanding of polymer blend thermodynamics, an ongoing research continues to elucidate how polymer blend phase behavior is affected by the presence of smallmolecule solvents or exposure to elevated pressures. In fact, many commercially relevant polymers are blends i.e. multicomponent systems, containing secondary components that impart desirable or improved properties such as color (dyes), flame resistance, toughn ess, tensile and impa ct strength, reduced cost (fillers), reduced oxidi zability (antioxidants), an d improved processability
PAGE 21
8 (plasticizers and mold release agents). Blends are physical mixtures without any chemically linked sequences between the diffe rent species in the mixture. Polymer blends can be divided into one of two categories: compatible or incompatible Compatible blends are those in which the chemically dissimilar macromolecules are combined to produce a mixture with a desirabl e set of physical propert ies. Such blends can be further classified as either miscible or immiscible The term miscible is defined as being mixed at the molecular level. Immiscible blends commonly exist as mi crometerscale spheroidal or fibrous dispersions of one component within a ma trix of the other. The blend morphology characteristics are sensitive to blend co mposition, interfacial te nsion, and molecular weight. Compatible but immiscible blends are commonly referred as polymer alloys if one or more compatibilizing agents are pr esent and the component s cannot be physically separated after mixing. Most mixtures of polymers are immiscible if not strongly incompatible, due to a combination of the existence of long chains that effectivel y limit the entropy of mixing, and the natural tendency for either athermal or endothermic mixing. Polymer blends are miscible only if specific attractive inte ractions between dissimilar chains provide sufficient driving force for mutual dissolution of the constituents. The kinds of factors that effect polymer miscibility are as follows: (i) entropy of mixing, (ii) dispersion forces, (i ii) specific interactions, and (iv) freevolume differences. The dispersion forces lead always to positive heats of mixing and they are the main reason for phase separation in blends at low temperatures. Their effect weakens with
PAGE 22
9 increasing temperature and, at some point, they are overco me by the entropy of mixing, leading to the appearance of the upper critical solution temper ature (UCST). The specific interactions can be sufficiently strong lead ing to negative heats of mixing and polymer miscibility. Freevolume differences between the polymers, or between the polymer solute and the solvent, lead to negative volume of mixing favoring demixing, i.e. these freevolume differences can be sufficient to cause phase splitting. This effect, referred to also as equation of state (EOS) effect, beco mes more important at higher temperatures, where the intermolecular interactions become weaker, leading to the lower critical solution temperature (LCST), [13]. Thus, at low temperatures and in the absence of specific interactions, polymer bl ends are not miscible due to dispersion forces. As the temperature increases, their effect diminishes and mixing occurs as a result of the entropy of mixing until the freevolume effects become important and immiscibility occurs, not necessarily at temperatures that can always be reached experimentally. A liquid cosolvent can greatly enhance polym er solubility in a supercritical fluid solvent if it has an intermolecular potential th at matches closely with that of a polymer repeat unit. In addition, a cosolvent that ha s a much higher density than that of the SCF solvent reduces the free volume difference between the polymer and the solvent. However, increasing the pressu re also reduces the free vo lume difference between the solvent and the polymer and increases the pr obability of interac tion between polymer, solvent, and cosolvent segments [2]. In this work, the impact of methyl methacrylate (MMA) cosolvent on the solubility of poly(methyl methacryl ate) (PMMA) in supercritical CO2 is studied and
PAGE 23
10 analyzed. The addition of polar MMA to CO2 provides enhanced polar interactions between PMMA, which is a weak polar polyme r, and the mixed solvent. This enhanced interaction power is manifested through the decrease in the cloudpoint temperatures and pressures for the system. 2.2. Solubility in Carbon Dioxide The use of carbon dioxide as a solvent or diluent in the preparation and processing of polymers has found extensive interest. While it is a nonsolvent for most polymers, it is a very effective plasticizer, as discussed earli er [4]. The attractive qualities of carbon dioxide include the relatively low toxicity, tunable solv ent properties (due to its compressibility), and low critical point (Tc = 31.1 C, Pc = 73.8 bar). The low critical point of CO2 near the critical point makes the supercritical region easily accessible. Supercritical fluids are useful for their li quid like densities (sol vating power) and their gaslike diffusivities (easy transport). The extensive compressibility that CO2 possess near its critical point makes the so lvent and transport properties of CO2 easily tunable with modest changes in pre ssure or temperature. Carbon dioxide has been shown to affect several pure polymer properties. Since CO2 acts as a plasticizer, free volume and chain mobility increase in the presence of CO2 and thus viscosity and glass transition temp erature decrease. Also, with increased mobility, the chains can more easily arrange themselves into crystalline structures so crystallinity can increase upon addition of CO2. Adding CO2 increases polymer free volume and the effective surface tension of a polymer with absorbed CO2 decreases [8].
PAGE 24
11 2.3. Experimental Techniques for Solubility Measurement Polymer miscibility in macromolecular systems is determined by variety of analytical methods, but most easily by measuring the thermal signature of the system: a miscible system will exhibit a single glass transition temperature ( Tg) that lies between those of the pure components, if they were all polymers. If low molecular solvent is present in the mixture, other analytical methods routinely used to establish blend compatibility and miscibility are used, such as: dynamic mechanical analysis, cloudpoint technique, gel permeation chromatography, electron microscopy, light scattering, smallangle Xray diffraction, neutron scatteri ng, carbon13 NMR, ultrasonic velocity, and excimer fluorescence [8, 11]. For polymerS CF systems, in order to be able to characterize polymer configurational proper ties in SCF solvents across wide ranges of pressuretemperature space, the major issue is in the application of these techniques to a highpressure system setup. Polymer chain conformation on its own has a major impact on the accessibility of the solvent to the repeat units in the polym er coil. Information on the role of chain conformation on solubility is slowly emer ging as light, Xray, and neutron scattering studies are reported quantifying the impact of polymer chain dimensions and polymerSCF solvent interactions on the breadth of the singlephase regi on. These scattering techniques are themselves challenging sin ce a radiation source and detector must be coupled to a highpressure cell with wellde fined geometrical characteristics so that artifacts of the apparatus are kept to a mi nimum. However, th e benefits of these techniques can be significan t since SCF properties are tunable, which means that it is
PAGE 25
12 possible to traverse the enti re goodtopoor solvent quality spectrum by adjusting solvent density with changes in pressu re at a single temperature and in a single solvent. These kinds of studies can provide deeper insight into the role of microscopic level interactions on macroscopically observe d phase behavior [3]. From a theoretical point of view, solu tion thermodynamic theories have been extended to discuss polymer miscibility. 2.4. Polymer Characteristics and Morphology Differential Scanning Calorimetry Analys is (DSC) provides a method to analyze polymer morphologies through changes in th e heat capacity of a sample. Specific polymer characteristics such as the glass transition temperature ( Tg), melting temperature ( Tm), degree of crystallinity, and phase sepa ration behavior (polym er blends) can be monitored by DSC. The glass transition temperature of a polymer is the point at which local segmental motion begins. Above this temperature, a polymer will behave as a rubbery material, below the Tg it is in a glassy state were the chai ns are frozen in place. This is known as the viscoelastic behavior of polym ers. The long chains of macromolecules in the amorphous phase exhibit a wide range of pr operties, from viscous flow to rigid solid, depending on the temperature. The propertie s change most rapidly, but continuously, through the Tg region. For example, the viscosity typically changes by up to several orders of magnitude over this region. S o, at the hightemperature end of the Tg region, the molecules can easily flow, while at the lowtemperature end the chains are rigid.
PAGE 26
13 DSC is used to study the thermal transitions of a polymer, i.e. the changes that take place in a polymer when it is heated. The device is depicted in Figure 2.1 below. In the most popular DSC design, two pans sit on a pair of identically positioned platforms connected to a furnace by a common h eat flow path. In one pan, the polymer sample is placed. The other one is the reference pa n, and it is left empty. The furnace can be turned on through the computer. Th e computer is programmed to heat the two pans at a specific rate, usually 10oC per minute, which is held constant throughout the experiment. This ensures that the two separate pans heat at the same rate as each other, even though the contents in the two pans are different, i. e. one has polymer in it, and one doesn't. Since there is extra material in the sample pan, the polymer sample, it will take more heat to keep the temperature of the sample pan increasing at the same rate as the reference pan. Just how much more heat is needed is what the DSC is measuring. Figure 2.1 Differential Scanning Calorimetry (DSC)
PAGE 27
14 Out of this measurement, a plot is obta ined as the temperat ure increases. On the x axis is the temperature. On the y axis is the difference in he at flow between the sample and reference. 2.4.1. Heat Capacity When the instrument starts heating the two pans, the comput er will plot the difference in heat flow against temperature, which is the heat absorbed by the polymer against temperature. At first, the pl ot looks like the one given in Figure 2.2 Figure 2.2 Heat Capacity The heat flow at a given temperat ure is shown in units of heat, q supplied per unit time, t The heating rate is temperature increase T per unit time, t t q time heat flow Heat t T time increase e temperatur rate Heating When the heat flow q/t is divided by the heating rate T/t, results in heat supplied, divided by the temperature increase.
PAGE 28
15 The amount of heat it takes to get a cert ain temperature increase is called the heat capacity, or Cp, which can be easily figured out from the DSC plot. capacity heat C T q t T t qp 2.4.2. The Glass Transition Temperature When the polymer is heated a little more, af ter a certain temperat ure, the plot will shift downward suddenly, as shown in Figure 2.3 Figure 2.3 Glass Transition Temperature This means heat is being absorbed by the sample. It also means that there is a change (increase) in its heat capacit y. This happens because the polymer has just gone through the glass transition. Polymers have a higher heat capacity above the glass transition temperature than they do below it. Because of this change in heat capacity that occurs at
PAGE 29
16 the glass transition, DSC can be used to measur e a polymer's glass transition temperature. That change doesn't occur suddenly, but ta kes place over a temperature range. Usually, the middle of the incline is taken to be the Tg. 2.4.3. Crystallization Above the glass transition, the polymers have a lot of mobility, and never stay in one position for very long. When they reach the right temperature, they will have gained enough energy to move into very ordered arrangements, crystals. When polymers fall into these crystalline arrangements, they give of f heat. This can be seen as a big peak in the plot of heat flow versus temperature. Figure 2.4 Crystallization Temperature The temperature at the highest point is usually considered to be the polymer's crystallization temperature, or Tc. The area of the peak, however, represents the latent energy of crystallization for the polymer. But most importantly, this peak tells us that the
PAGE 30
17 polymer can in fact crystallize. If it is analyzed a 100% amorphous polymer, like atactic polystyrene, we wouldn't get one of these peaks, because such materials don't crystallize. The optimum crystallization te mperature occurs between the Tg and Tm. Crystallization is also called an exothermic transition because the polymer gives off heat when it crystallizes. 2.4.4. Melting Heat may allow crystals to form in a polymer, but too much of it can do the opposite. If we keep heating the polymer past its Tc, eventually we'll reach another thermal transition, one called melting. When we reach the polymer's melting temperature, or Tm, those polymer crystals begin to fall apart, that is they melt. The chains come out of their ordered arrange ments, and begin to move around freely. The heat that the polymer gave off when it crystallized, is ne eded back when the melting temperature, Tm, is reached. There is a latent heat of melting as well as a latent heat of crystallization. When the polymer crysta ls melt, they must abso rb heat in order to do so. Melting is a first order transition, which means that when you reach the melting temperature, the polymer's temperature won't ri se until all the crystals have melted. This also means that the furnace is going to have to put additional heat into the polymer in order to melt both the crystals and keep the temperature rising at the same rate as that of the reference pan. This extra heat flow during melting shows up as a large dip in the DSC plot as heat is absorbed by the polymer. It looks like this:
PAGE 31
18 Figure 2.5 Melting Temperature The heat of melting can be determined by m easuring the area of this dip. Usually, the temperature at the apex of the dip is cons idered to be the point where the polymer is completely melted. Melting is also called an endothermic transition, because it is required to add energy to the polymer to make it melt. 2.4.5. Differential Scanning Calorimetry of Polymers Summarizing the above, the first step in the plot is when the polymer is heated past its glass transition temperature, then ther e is a big peak when the polymer reaches its crystallization temperature, then finally ther e is a big dip when the polymer reaches its melting temperature. Putting them all together, a whole plot often looks something like this:
PAGE 32
19 Figure 2.6 DSC Thermal Signature of a Specific Polymer Of course, not every peak shows on every DSC plot. The crystallization peak and the melting dip will only show up for polymers that can form crystals. Completely amorphous polymers won't show any crystalli zation, or any melting either. But polymers with both crystalline and amorphous domains, w ill show all the featur es that are seen above. Most polymers are either comple tely amorphous or have an amorphous like component even if they are crystalline. By looking at the DSC plot, a big diffe rence can be seen between the glass transition and the other two thermal transitions crystallization and melting. For the glass transition, there is no peak, and there's no dip, either. This is because there is no latent heat given off, or absorbed, by the polymer during the glass transition. Both melting and crystallization involve absorbing or giving off heat. The only thing we do see at the glass transition temperature is a change in the heat capacity of the polymer.
PAGE 33
20 Because there is a change in heat capaci ty, but there is no latent heat involved with the glass transition, th e glass transition is called a second order transition. The mechanical properties of polymers change dr astically when the temperature crosses the Tg. For example, the Elastic Modulus may decrease by a factor of 1000 times as the temperature is raised through the Tg. Transitions like melting and crystallization, which do have latent heats, are called first order transitions. To summarize, the basic theory and te rminology of polymers was presented in this section. This review is important to understand the basic rheological phenomena associated with polymers and their behavior at high temperatures. 2.5. Thermodynamic Phase Behavior of Polymer Blends The phase behavior of polymer blends can be described by the Gibbs free energy of mixing (Gmix), which is dependant on both the enthalpic (Hmix) and entropic (Smix) changes on mixing, and is negative for favorable processes. Gmix = Hmix TSmix < 0 (2.1) The coexistence curve and the limit of stability are the boundaries of the T(TemperatureSegment Fraction Polymer) phase diagram separating the miscible, metastable and immiscible regions. The coexistance curve (binodal) is described by the equality of chemical potential of component i (i) in the two phases: i = i (2.2) while the limit of stability (spinodal) is desc ribed by the second derivative of the Gibbs molar free energy change of mixing with respect to composition (mole fraction).
PAGE 34
21 0, 2 2 P T mixx g (2.3) A decrease in the Gibbs energy of a binary mixture due to phase separation can occur only if a plot of the Gibbs energy change of mixing against mole fraction is, in part, concave downward [10]. In blends, phase separation can occur either upon cooling or upon heating, or both. Speaking in polymers terminology, the analogous process to phase separation is called ordering, i.e. the individu al polymer constituents start to organize themselves around the backbone of the po lymer, beginning to condense within themselves which causes pullback, rejecti on and splitting from the other components in the mixture [8]. This process can also o ccur upon cooling, heating, or both. If a blend separates upon cooling, this tr ansition is labeled an upper critical solution temperature (UCST). However, if phase mixing or diso rdering (opposite of ordering) occurs upon cooling, the transition is labeled a lower cri tical solution temperature (LCST). In a Tx diagram, the highest point on the phase diagra m is the critical solution temperature, Tc. At temperatures T > Tc, the mixture is completely miscible because for all mole fractions 0 /, 2 2 P T mixx g. At T < Tc, the mixture is partially miscible because in part of the mole fraction range 0 /, 2 2 P T mixx g. The binodal curve is the boundary between the onephase region and the twophase region. Within the twophase region, the spinodal curve [ 0 /, 2 2 P T mixx g] distinguishes the unstable region [ 0 /, 2 2 P T mixx g] from the metastable region [ 0 /, 2 2 P T mixx g]. If the overall mole fraction of the mixture falls within the unstable region, s pontaneous demixing occurs when going from the onephase to the twophase region [10].
PAGE 35
22 An understanding of what causes each of th ese types of transitions in each blend system facilitates an understanding of how th e presence of carbon dioxide could affect the phase behavior of a given polymer blend. 2.6. Thermodynamic Modeling of Po lymer Solutions and Blends At the foundation of any process modeli ng and simulation are the property models used to describe physical prope rties of the components involv ed. The selection of model and its parameters profoundly affect every step in the simulation. In the case of modeling polymer processing and production processes, this task becomes particularly challenging because in such processes one encounters mixtures of solvents made of small conventional molecules, and polymers, which have longchain structure. It is essential to use models that can simultaneously desc ribe the behavior of both polymers and conventional molecules with acceptable accura cy. Equations of state (EOS) are very attractive tools for such property modeling. Understanding the phase behavior of polym er solutions in supercritical fluids (SCF) is of great theoretical and practical interest. Pred icting phase boundaries and phase compositions (e.g. solubilities) for such system s is difficult because the molecules of the polymer and solvent greatly differ in size, and their mixtures are high ly nonideal at high pressures. The thermodynamic models for pol ymer solutions that are being used in practice basically belong to two categories, lattice models and perturbation models. In lattice models, the molecules are assumed to have one or more segments, and the partition function of the system can be obtai ned by counting the possible configurations
PAGE 36
23 when these segments are arranged in hypotheti cal cells that are lik e the lattices in the solid materials. Then the thermodynamic pr operties can be calculated on the basis of statistical mechanics. A numb er of models reported in the literature are based on this approach. For polymer solutions under high pressure, a good lattic e model to use could be the SanchezLacombe model, which can corr elate the experimental data for a range of molecular weights [16, 17, 18]. On the other hand, a good perturbation model to use would be the model known as SAFT or statistical associating flui d theory, which was recently developed by Chapman et al. and Huang and Radosz [21]. The SAFT theory is based on the relationship between the residual Helmholtz free energy due to association and the monomer density, which is agai n related to the association strength. Detailed description of these models can be found in the literature. The objective of this work is the observati on of the phase behavior of polymers in the highpressure CO2 solvent, as well as the examination of the validity of an appropriate EOS and mixing rule s for such polymer/supercritical fluid system. This process is an authentic example of how th e principles of thermodynamics can be extended to highpressure processes. Undoubtedly, the largest poten tial application of superc ritical fluid technology is in the area of polymer processing, although polymer/SCF phase equilibrium calculations confront us with the situation where the solu te molecule can have a molecular weight in the hundreds of thousands for high molecula r weight polymers. Therefore, some modifications to the thermodynamics calcula tion scheme must take place. While the
PAGE 37
24 fundamental equations remain the same as t hose for small molecule systems, the equation of state is different. We must now deal with mixtures of components that differ substantially in molecular size. 2.6.1. HighPressure VaporLiquid Equilibria Various modeling procedures have been pr oposed in the literatu re to predict the phase behavior of vaporliquid systems at high pressures. (T he designation vapor is used synonymously with supercritical fluid in this context). Regardless of the modeling procedure, the following thermodynamic relations hips, or their equivalent relationships in terms of chemical potentials, must be satis fied for two phases to be in equilibrium. ) , ( ) , (i L i i V ix P T f y P T f m i ... 3 2 1 (2.4) The most thermodynamically consistent method for calculat ing highpressure phase behavior is to choose an equation of state to model both the liquid and the vapor of SCF phases. With this approach, the fug acity in each component can be written as P y y P T fV i i i V i ) , ( (2.5) P x x P T fL i i i L i ) , ( (2.6) therefore, L i i V i ix y (2.7) where subscripts L and V refer to liquid and vapor phases respectively, f is the fugacity and is the fugacity coefficient. The equi librium ratios (Kfact ors) are given by: V i L i i ix y K (2.8)
PAGE 38
25 To compute the fugacity coefficients for the vapor and liquid phase, the exact thermodynamic relationships (Prausnitz, 1969) can be used. For the vapor: RT Pv dv V RT n P RTV v n V T i V iV iln 1 ln, (2.9) and for the liquid: RT Pv dv V RT n P RTL v n V T i L iL iln 1 ln, (2.10) where R is the gas constant, vV is the molar volume of the vapor phase, vL is the molar volume of the liquid phase, ni is the number of moles of component i RT Pv Z in each equation is the compressibili ty factor of the mixture, while in V T in P, can be determined analytically from the a ppropriate equation of state for the system. In many cases, highpressure phase behavior can be reasonably represented with a cubic EOS if the component s in the mixture do not differ too substantially in intermolecular strengths or in size, structure, or shape. Then, the PengRobinson (1976), or the SoaveRedlichKwong (1972) equatio ns of state would produce satisfactory results. However, for polymerSCF highpressure sy stems, a more sophisticated model is needed, such as the latticegas model, where the PVT properties of a pure component are calculated assuming that the component is broken into parts or mers that are placed into a lattice, and appropriate number of holes (gas) are also placed in specific lattice
PAGE 39
26 sites, to obtain the correct sy stem density. Also, a molecu lar perturbation model can be used for better representation of the system. There are two requirements for the success of EOS models: they must accurately predict the saturation pressure of pure com ponents, and suitable mixing rules must be available to extend their use to multicomponent mixtures. 2.6.2. Equations of State There are two major types of equations of state that are capable of representing solventpolymer mixtures, with some specifi c limitations. Those are the lattice models (eg. SanchezLacombe), and the perturbati on models (SAFT). The principles and governing equations of each of th ese models are reviewed in th is section in greater detail. 2.6.2.1. Lattice Fluid Models (The SanchezLacombe Equation of State) The early lattice models, such as the ol der FloryHuggins theory (1953), do not apply very well to systems where there are st rong specific interactions and to dilute or semidilute solutions. Also, it neglects the socalled free volume effect. This theory, which ignores the equation of state properties of the pure components, completely fails to describe LCST behavior in polymer solu tions [15]. Freeman and Rowlinson (1960) observed experimentally that many hydr ocarbon polymers dissolved in hydrocarbon solvents phase separated at high temper atures. These nonpolar polymer solutions exhibited what are known as lower critical so lution temperatures (LCST), a criticalpoint phenomena that is rare among low molecular wei ght solutions. It was recognized that the
PAGE 40
27 common appearance of LCST behavior in poly mer solutions must be related to the large size difference between polymer and solvent mol ecules. Soon after the discovery of the universality of LCST behavior in polymer solutions, Flory and coworkers (19641970), developed a new theory of solutions that co nsiders the equation of state properties of the pure components. This new theory of so lutions, the Flory theory, demonstrates that mixture thermodynamic properties depend on the thermodynamic properties of pure components. In particular LCST behavior can be understood in terms of the dissimilarity of the equation of state properties of polym er and solvent. Pa tterson (1971) has also shown that LCST behavior is re lated to the dissimilarity in polymersolvent properties by using the general corresponding st ates theory of Prigogine and collaborators [15]. Thus, the lattice theo ry of solutions, although firs t developed for monoatomic molecules, can be extended to molecules of more complex structure using welldefined assumptions, as shown by Guggenheim, Flory, and others. This extension makes it particularly useful for soluti ons of molecules that differ appreciably in size, such as polymer solutions. However, the concept of a lattice for liquid structure is a vast oversimplification; and as a result, lattice theory becomes increasingly inappropriate as attention is focused on temperatures remote from the melting point. Also, for each binary system, lattice theory requires as an input parameter the interchange energy w (or its equivalent, the Flory parameter ), that is difficult to predict and that, unfortunately, is temperaturedependant [10]. More recently, a newer equation of state theory of pure fluids [16, 17] and their solutions [18] has been formulated by Sanchez and Lacombe (1977), characterized as lattice fluid theory. In general, it defers from a corresponding
PAGE 41
28 states theory. It does not re quire separation of internal a nd external degrees of freedom as does the Flory theory and the Prigogine corresponding states theory. External degrees of freedom are assumed to depend only on intermolecular forces, whereas internal degrees of freedom are associated with intramolecular chemical bond forces. Nevertheless, the lattice fluid theory has mu ch in common with the Flory theory. Both theories require three equa tion of state parameters fo r each pure component. For mixtures, both reduce to the FloryHuggins theory at very low temperatures. The SanchezLacombe model is a latticefluid model in which vacancies, i.e. empty lattice sites are introduced in the latt ice to account for the compressibility and density changes. It is given by the expression, 0 ~ 1 1 ~ 1 ln ~ ~ ~2 r T P (2.11) where v T P ~ and ~ ~ ~ are the reduced pressure, temp erature, density and volume, respectively, that are defined as, *), /( *), ( ~ 1 ~ *, / * ~ / * ~* *rv M rv N V V V v v P P P P k T T T T (2.12) where is the mermer interaction energy, v* is the closepacked molar volume of a mer, M is the molecular weight, N is the number of molecules, r is the number of sites (mers) a molecule occupies in the lattice, R is the universal gas constant.
PAGE 42
29 The characteristic temperature, pressure and closepacked mass density are given as T*, P*, and *. When dealing with mixtures it is necessary to define combining rules for *mix v*mix and rmix to use the equation of state to calculate properties of a mixture. For that matter, the socalled van der Waal s1 rules are used. These assume random mixing of the components, as for small molecule system. The mixing rule for the characteristic closepacked mola r volume of a mer of the mixture v*mix is given by 11 * ij ij j i mixv v (2.13) with ij jj ii ijv v v 1 2* * (2.14) where ij corrects for deviations from the arithmetic mean and where subscripts i and j are the components in the so lution. The volume fraction (i .e. segment fraction) of component i,i, is defined as 1 * * j j j j i i i iv m v m (2.15) where im is the mass fraction of component i in the mixture, and *andi iv are the characteristic mass density and clos epacked molar volume of component i, respectively. The mixing rule for the characteristic interaction energy for the mixture *mix is given by 11 * *1ij ij ij j i mix mixv v (2.16) ij jj ii ijk 12 1 * (2.17)
PAGE 43
30 where ii and jj are the characteristic mermer in teraction energies for components i and j and ijk is a mixture parameter that accounts for specific binary interactions between components i and j The mixing rule for number of sites occupied by a molecule of the mixture, rmix is given by 11j j j mixr r (2.18) where rj is the number of sites molecule j occupies in the lattice. Expressing SanchezLacombe EOS in terms of the compressibility factor, yields a T R P Z d b RT P Z d P RT Z d Z2 2 2 2log 1 (2.19) where the correction factors a b and d are problem specific for each polymerSCF process (appropriate correlations for polymerCO2 system are derived empirically, taking into consideration the components mole fracti ons and binary interaction parameters). Solving Equation (2.19) as a quadrati c equation in terms of Z, and then substituting the Z term in Equation (2.9), yields Z dv V RT n P RTV iv n V T i V iln 1 ln, (2.20) These calculated fugacity coefficients are used to calculate the equilibrium solubility mole fractions of polymer in the CO2 system under different experimental conditions (temperature, pressure, composition). The experimentally measured solubility data can be regressed using the SanchezL acombe EOS with conventional mixing rules, as already discussed, to determine the pressure term.
PAGE 44
31 The linearly temperature dependent binary interaction coefficients kij, which is associated with the intermolecular interacti ons between a pair of unlike species, can be regressed to incorporate temp erature dependence. The value of this parameter usually never gets larger than about 0.150. It can also be negative, although a negative value usually indicates the presence of specific chemical interactions (hydrogen bonding). Appropriate kij expressions for various polymerCO2 systems are available in the literature. Fairly more sophisticated than the lattice fluid models are the perturbation models, which are described further in this review. 2.6.2.2. Perturbation Models (SAFT) Reasonable, but nevertheless approximate, th eories are now available for mixtures that contain chainlike molecules in addition to normal, essentially spherical, or globular, molecules. These theories (perturbed hard chain, statistica l associating fluid, perturbed hard chain of spheres) have a wide r range of applicability than those based on a holefree lattice because they are based on equations of state that, unlike a holefree lattice, give the segment density as a functi on of temperature, pre ssure, and composition. Further, these EOS theories can incorporat e association between like molecules and solvation between unlik e molecules. Regrettably, ev en for nonpolar fluids, these EOS theories require several (typically 3 or 4) purecomponent molecular parameters; and if the molecules associate, additional parameters are needed. The need of so many parameters follows from our inadequate unde rstanding of intermolecular forces [10]. One of the major challenges f aced in adapting the supercri tical solvents as greener
PAGE 45
32 substitutes is the unavailability of phase eq uilibrium data and desi gn information for new systems. This is because most of the av ailable methods and simulations are problem specific. Thus, the motivation for develo pment of a conceptual design tool for supercritical products and processes is strong among the research community. In recent decades, molecularly based equations of state have been developed which to some extent provide some predic tive capabilities. The background for these equations of state is statistical therm odynamics, which gives the relation between properties of single molecules and the macr oscopic properties of an ensemble of molecules. While this appro ach is formally exact, it cannot be applied rigorously because of its mathematical complexity. Several approximations have to be made. A common approximation is, for example, to describe the macroscopic properties based on pair interactions while neglecti ng the many body potentials. The first widely applied equation of stat e based on this molecular view was the Perturbed HardChain Theory (PHCT) e quation of state developed by Beret and Prausnitz (1975) and by Donohue and Prausnitz (1978). Multipolar interactions were considered by Vimalchand and Donohue (1985). The success of their work has been the inspiration for further developments. A more recent equation of state concept for chain molecules is based on Wertheims Ther modynamic Perturbation Theory (1984). By applying Wertheims theory and extending it to mixtures, Chapman et al. (1990) derived the Statistical Associating Fluid Theory (SAF T) equation of state for chain mixtures. Although many modifications of the SAFT model were s uggested, one of the most successful modifications remain s to be the SAFT model s uggested by Huang and Radosz
PAGE 46
33 (1991). Chapman et al. (2000) have develo ped a new theory, based on a perturbation theory of Barker and Henderson (1967) by extend ing it to chain molecules, referred to as the PerturbedChain SAFT (PCSAFT), wher e the chain structure of the molecules, which are assumed to be chains of freely jo ined spherical segments is now considered also in the dispersion term. Gross and Sadow ski applied this theory to associating and polymeric systems. The PCSAFT molecular model is given in Figure 2.7, Begin with a hardsphere fluid Add at tractive forces between molecules Add associative forces between molecules Form chains with hardsphere segments Figure 2.7 Molecular Model Underlying the PerturbedChain SAFT EOS
PAGE 47
34 The spherical segments may possess associat ion sites, exhibiting specific (often strong) shortrange interactions, thus mimi cking hydrogenbonding. Moreover, they may carry partial charges. Dipolar segments have two partial charges (positive and negative), whereas quadruple segments carry three charges. Statistical associating fluid theory (S AFT) is a molecularly based EOS that incorporates terms accounting for the molecu lar size and shape (e.g., chain length and degree of branching), association (e.g., hydr ogen bonding) energy, and meanfield (e.g., dispersion) energy. A SAFT fluid is a collection of sphe rical segments that are not only exposed to repulsive (hard sphere) and attr active (dispersion) forces but can also aggregate through covalent bonds to form ch ains (chain effect) and through hydrogenlike bonds to form short lived clusters (association effect). The reference part of SAFT includes th e hardsphere, chain, and association terms. The perturbation part of SAFT accounts for relatively weaker, meanfield dispersionlike effects. The SAFT residual Helmholtz free energy (ares) relative to an ideal gas reference state is given by ares=aref + adisp (2.21) with aref (T,V,N) = ahs(T,V,N)+ achain (T,V,N)+ aassoc (T,V,N) (2.22) Using the thermodynamic relationship P v aT SAFT can be expressed in terms of the compressibility factor, Z:
PAGE 48
35 disp assoc chain hsZ Z Z Z RT Pv Z 1 (2.23) where 3 21 2 4 m Zhs (2.24) 5 0 1 1 5 2 12 m Zchain (2.25) A T A A assocX X Z 5 0 1 (2.26) ij i i ij dispkT u jD m Z (2.27) where is the reduced fluid density, m is the number of segments per molecule, is the molar density, v is the molar volume, XA is the mole fraction of molecules not bonded at site A, u/k is a temperature dependant dispersion energy of interaction between segments, Dij and are constants, and the summation is ov er all the sites. The reduced density, of the fluid is defined as 074048 0mv (2.28) where v0 is the segmental molar volume at closed packing, in units of cubic centimeter per mole of segments, given as 3 0 00 03 exp 1 kT u C v v (2.29) where, u0/k is a temperatureindependent parameter in Kelvins, C is a constant set to 0.12, thus there are three unary parameters v00, u0/k, and m for each component in the SAFT model. Knowing ares and Z, i can be estimated from the following identity:
PAGE 49
36 Z n RT ajn V T i res iln ln, (2.30) where V is the total volume of the system and ni is the number of moles of substance i When using SAFT, the characteristic parameters for pure substances are determined by optimizing the predictions of vaporliquid equilibrium ( VLE ) data. When VLE data is not available, which is usually the case for polymers, PVT data are used instead. Also, if there are no accurate PVT data available, the characteristic parameters can be estimated from the molecular weight on ly, which is a nice feature of SAFT. When SAFT is extended to mixtures, only one binary mixture parameter is needed. This parameter in SAFT is temperature independent in contrast to most of the lattice theories in which the interaction parameters are temperature dependant. For mixtures, either volume fraction mixi ng rules or the van der Waals onefluid ( VDW1 ) mixing rules can be used in the model. Using the VDW1 mixing rules, the dispersion energy of interaction is given by ij ij j i j i ij ij ij j i j iv m m X X v RT u m m X X RT u0 0 (2.31) with ij jj ii iju u u 12 1 (2.32) ij j i ijm m m 1 2 (2.33) 3 3 1 0 3 1 0 02 1 j i ijv v v (2.34)
PAGE 50
37 where Xi and mi are the mole fraction and molecula r segment number of component i in the mixture, ij is an empirical binary interaction parameter, ij is another empirical binary interaction parameter, vij 0 is the segment molar volume for the mixture, in which vi 0 is temperature dependent segment molar volume and is related to the temperature independent segment molar volume vii 00 as written above in equation (2.29). SAFT is used to describe many real pure components and fluid mixtures, in cluding supercritical and nearcritical solutions of polymers.
PAGE 51
38 CHAPTER 3 EXPERIMENTAL SYSTEM AND PROCEDURES In this chapter, the experimental syst em, equipment setup, and procedures are described in detail, independently describing the instruments that constitute the system. Also, the procedure and the results of the cal ibration of the phase analyzer and its PLC controller are presented. Following this, th e techniques for prepara tion of materials and chemicals are described. Finally, the proce dure for observing and measuring the bubble points of the MMACO2 system, and the cloud points of the PMMACO2MMA system at various temperatures for obtaining the ap propriate solubility data is elaborated. To emphasize again, the objectives of this work are, first, to develop a procedure and obtain experimental solubi lity data for a suitable polym erSCF mixture which can be used for supercritical impregnation processes and, second, to test in a comprehensive and systematic way the available thermodynami c SAFT model for polymerdiluent mixtures, developed by Aslam and Sunol (2004). This is a technologically important field as polymerdiluent mixtures are frequently encountered in polymer processing and thermodynamic models for these systems play a pivotal role in the design of process equipment.
PAGE 52
39 3.1. Experimental Apparatus Description The apparatus used for measuring phase be havior of dilute polymersupercritical fluid solutions is illustrated in Figure 3.1, Figure 3.1 Schematic Diagram of the HighPressure Phase Monitoring System: (1) variablevolume cell, (2) mixer, (3) piston, (4) camera sapphire window (5) light sapphire window, (6) borescope w/camera, (7) light source, (8) pressure transducer, (9) thermocouple, (10) heater, (11) hydraulic pump, (12) hydraulic li fting section, (13) controller, (14) TV/VCR, (15, 16, 17) pressu re gauges at pump, cell inlet & outlet (vent).
PAGE 53
40 3.1.1. Phase Monitor The heart of the experiment al system is the Phase Monitor SPM20 (manufactured by Thar Technologies, Inc., Pittsburgh, PA). It consists of a sealed, highpressure, variablevolume cell, containing the polymer solution within the enclosure, the temperature of which is controlled prog rammatically up to a maximum of 150 C, and the pressure of which is adjusted via a movable piston. The pressure is controlled up to a maximum of 700 bar by a hydraulic hand pu mp (Model 2200, Ruska Instruments), which is used to raise and lower the hydraulic lifting section of the phase monitor carriage assembly. The vessel piston is attached to the lifting section, so that when the lifting section raises, the vessel piston moves in to the vessel body thus reducing the vessel volume. As the vessel volume is reduced, the vessel pressure will increase, and vice versa. There is a volume displacement scale mounted on the carriage assembly, which displays the remaining vessel volume from the maximum of 15 ml down to a minimum of 5 ml. The pressure varia tions into the cell are measured by a pressure transducer mounted to the vessel body, and the signal is transferred to a portable PLC controller. A mixer, rotated at variable speeds by an external motor, was used to mix the mixture in the highpressure cell. Visual observation of phase phenomena occurring inside the cell was made on a video monito r using a camera coupled to a boroscope, placed directly over the sapphire window. The cell has two camera mounting adapter assemblies; one assembly holds the camera and the second holds the camera light source.
PAGE 54
41 The vessel assembly is heated by four electrical heaters installed into the bottom of the vessel body, while the temperature is measured through a thermocouple mounted to the vessel body and whose heat conductive probe enters the interior of the vessel and gets into a direct contact with the fluid in the cell. The signal from the thermostat is transferred to the portable PLC controller, wher e the output is displayed in the unit bar. A photo of the highpressure cell and the PL C controller is shown in Figure 3.2, Figure 3.2 The Vessel Assembly and Cont roller of the Phase Monitor SPM20
PAGE 55
42 The camera assembly, the inlet and outl et tubing connectio ns, the pressure transducer and the thermocouple can be seen as they are connected to the vessel body. Also, there is a mechanical pressure gauge connected online to the vessel at the outlet tubing, for comparison and verification of the pressure measured by the pressure transducer which is being converted in bar units thro ugh the controller. One advantage of using the variablevolum e cell is that the concentration of the system remains constant during the experiment. 3.1.2. Tubing Reticulation The 1/8 stainless steel tubing that st arts from the carbon dioxide tank, goes through the highpressure syringe pump (Mod el: 100DX, ISCO, Inc., Lincoln, NE), from where it continues through an outlet tubing a manual valve, a t ubing reducer and a connection to the vessel body through a vessel inlet port 1/16 Valco. All the tube fittings, including the crosses, tees, nuts, ferrules, back ferrules, reducers, plugs, connectors, bushings, are of 1/18 connection size, stainless steel fittings (Swagelok, Solon, OH) adapted for highpressure applica tions. The minimum amount of fittings has been used throughout the system, so that the pressure drop introduced to the system is minimized and easy to compensate for by th e syringe pump. Ther e is another 1/16 connection port on the vessel body which serves as an outlet from the vesse l to let the CO2 and other gas fumes leave the vessel body whenever the outlet valve is open. The released fumes are vented through the external ventilation system.
PAGE 56
43 3.1.3. High Pressure Syringe Pump The pump that delivers supercritical car bon dioxide to the system is a highpressure syringe pump (Model: 100DX, ISCO Inc., Lincoln, NE) equipped with a cooling jacket to assure that the carbon dioxi de withdrawn from the gas cylinder is cooled down to a liquid state. The temperature of the cooling jacket is maintained below zero degrees Celsius by a Lauda Circulator (L auda GMBH&CO.KG, Germany). The cooling is required to get the maximum efficiency from the syringe pump with liquefied carbon dioxide in order to get the desired superc ritical pressure with the minimum possible compression and maximum possible volume. Th e pump comes with 1/8 standard Valco ports. The cylinder capacity of the pump is 102.93 cm3. Modes of operation are constant flow within 0.01 l/min to 50 ml/min (for any pressure from 0.6895 up to 689.5 bar). The flow rate accuracy is rated to be 0.3% with a flow rate display resolution of 0.01 l/min. Usually the supercritical carbon dioxide is pumped in at a constant flow rate of 5 ml/min. The pump is operated with a contro ller (Model: D Series Controller, ISCO, Inc., Lincoln, NE), which is a microprocessorbased user interface, and a digital motor speed control system, by entering and monitoring set values during the pump operation. 3.1.4. LowTemperature Circulator The Ecoline lowtemperature thermostat (Model: RE120, Lauda Dr. R. Wobser GMBH & CO. KG, Germany) has both refrig eration and heating unit with adjustable flow rate ranging from 8 l/min to 17 l/min, and maximum discharge pr essure of 0.4 bar. The capacity of the circulator is approximately 14 to 20 liters. The temperature range of
PAGE 57
44 operation is to 120 C with a control accuracy of 0.05 C at 20 C. The unit is equipped with a pressure pump with variable drive. The refrigeration system consists essentially of a hermetically sealed compressor. Heat of condensation and motor heat are dissipated by a fancooled finned condenser. 3.1.5. Syringe Pump Controller The pump setup and operation is regu lated by the D Series Programmable Controller (Model: D Series Controller, ISCO, Inc., Lincoln, NE). Operating parameters are entered via the keypad on the front panel of the controller. Operating modes such as CONST FLOW, CONST PRESS, REFILL, START, and STOP all have dedicated keys on the controller keypad. The pump also allows userprogrammed refill, as well as pumping rates. The filling of the vessel itself with supercritical CO2 is monitored visually through the TV camera attached to the vessel body, and also by reading the pressure increase in the vessel during the filling. These observations give a good indication about when the system is full, thus when to switch the syringe pump off. 3.2. Calibration of the Phase Monitors PLC Controller After assembling, and test running of the phase monitoring experimental apparatus, it was discovered that the accuracy of the Ph ase Monitors temperature and pressure PLC Controller was off. For calib ration purposes, two manual pressure gauges were connected to the highpressure cell, one at the inlet and one at the outlet. Both pressure gauges have shown a same pressure difference from the controller reading.
PAGE 58
45 Throughout the working pressure range of the cell, the manual pressure gauges read about 20 bar higher pressures th an those calculated by the c ontroller. Nitrogen gas was used to do the low pressure measurements, from 0.5 to 50 bar, while liquid CO2 was used for the high pressure measurements, 50 to 120 bar. With both gases, it was repeatedly shown that theres always about 20 bar difference throughout this pressure range. This difference had to be accounted in the experimental measurements. Thus, for every pressure reading from the controlle r a 20 bar correction fa ctor is added for representation of the results in this work. However, more accurate calibration could be performed by rechecking the software of the PL C controller itself, i.e. the equations used for calculation and conversion of the readin gs from the pressure transducer. The manufacturer of this controller should accept the responsibility of recalibration of the instrument. Until that is done, however, we w ill apply our first hand calibration results to correct for the controller offset. Therefore, the experimental conditions ar e predetermined during the calibration of the controller since same measurement cond itions have to be attained as with the calibration conditions to be able to determine the right pressure in the system at the time of performing the experiments. The obs ervations have shown that attaining and controlling a desired temperature is harder an d errorwise more important, so that our efforts are focused on stabilizing the temper ature in the system. The temperature was controlled to within 0.2 C maximum. The pressure is always variable, depending on the hydraulic pump displacement of the piston and the cloudpoint pha se transition. The pressure reading is always fluctuating at th e bubblepoint an d at the cloudpoint, and it
PAGE 59
46 takes a little bit of time to stabilize; ther efore a number of repeated measurements is needed, where the average of those pressure readings is calculated and accepted as the actual pressure of the system at that point. 3.3. Experimental Section In the following section, the preparati on of the chemicals, the operation of the variablevolume cell, and the experimental meas urements will be explained. This will be followed by a detailed experimental proced ure and the observations obtained throughout the experimental run. 3.3.1. Materials Description Carbon dioxide (99.8% purity) was obtained from Airgas and used as received. Methyl methacrylate (MMA) (99% purity) a nd poly(methyl methacryl ate) (99% purity) that has weightaverage molecular weight ( Mw) of 93,000 and numberaverage molecular weight ( Mn) of 46,000, therefore having a polydispersity index ( Mw/Mn) of 2.02, were obtained from Aldrich and used as receiv ed. To prevent MMA polymerization, 4methoxyphenol, that is, hydroquinone monometh yl ether, (Aldrich, 99% purity) was used as an inhibitor at a concentration of 0. 0025 times the amount of MMA, which was added to the monomer as part of the manufacturing process. The chemical structure of each chemical is shown in Schemes 3.1, 3.2, and 3.3. Acrylates are made from acrylate monomers which contain vinyl groups, that is, two carbon atoms double bonded to each other, directly attached to the carbonyl carbon.
PAGE 60
47 Scheme 3.1 Chemical Struct ure of acrylate Monomers Some acrylates have an extr a methyl group attached to the alpha carbon, and these are called methacrylates Scheme 3.2 Chemical Structure of a methacrylate One of the most common methacrylate poly mers is poly(methyl methacrylate), which is obtained by free radical vinyl po lymerization of methyl methacrylate monomer,
PAGE 61
48 Scheme 3.3 Polymerization of me thyl methacrylate to Get PMMA Poly(methyl methacrylate), PMMA, is a cl ear, thermoplastic, linear polymer with no crosslinks. The backbone of the chain is usually carbon atoms linked by covalent, single bonds. The degree of polymerization is the number of monomer units in a chain, normally 103105, and it can be calculated as, monomer of M polymer of M tion polymeriza of Degreer w (3.1) For the chemicals at hand, the degree of polymerization is calculated as, 88 928 12 100 93000 DP That is, there are approximately 1,000 MMA units in one single chain of PMMA of this study. 3.3.2. Experimental Procedure There are two different experiments th at were conducted using the same experimental setup. First, the methyl methacrylateCO2 equilibrium was studied, by determining the bubble point of the mixture at different temperatures.
PAGE 62
49 The second system, PMMACO2MMA, was studied by determining the cloudpoint data at different temperatures. In the fi rst experiment, the first step was to load the methyl methacrylate monomer into the cell from the end that is capped with the variable speed mixer during the experiments. The amou nt of methyl methacryl ate loaded into the cell was determined using a sensitive analytical balance, measurable to 0.001 g. The methyl methacrylate is loaded with a pipette fr om the top while the vessel is in its upright position. Then, the mixer assembly is screwed in position and the cell is purged slowly with carbon dioxide at pressures as low as 1 bar, to create just enough pressure difference to force the remaining air out of the system After the purging is complete, the outlet valve is closed, and CO2 is transferred from the gas cyli nder into the cell volumetrically, at 5 ml/min constant flow rate, using a syri nge pump and a controller. The contents of the cell are projected onto a video monitor using a camera coupled to a boroscope placed directly against the sapphire window. The filling of the cell with liquefied CO2 is monitored on the TV monitor. As the CO2 level reaches the top of the cell, the interior pressure in the vessel starts to build up quic kly. At that point, the inlet valve must be closed, and the syringe pump stopped manu ally. In a few seconds, once the system reached thermal equilibrium, the initial pressure and temperature in the cell are recorded. The volume of the cell at this stage is usually at its maximum, that is, 15 ml. The initial temperature is the ambient temperature in the cell. The initial pressure is somewhere in the lower end of the supercritical region of carbon dioxide. Now, the vessel assembly is tilted in its horizontal position until the inte rface between the two phases in the cell is visible in the viewfinder of the camera. The cell is then heated to the desired
PAGE 63
50 temperature, which is programmed by th e temperature contro ller. Once thermal equilibrium is maintained in the cell, the co ntents of the cell are compressed into the one phase region by moving the piston forward via the hydraulic lifting section. The phase behavior is obtained in the pressure inte rval between the twophase state and the onephase fluid state. Once in the onephase region, the pressure is lowered rapidly by moving the piston backwards i.e. increasing the volume until the first bubbles appear in the cell. The pressure at which the first bubbles appear is recorded, and that is the bubble point at the given temperature. Next, the temp erature of the system is increased, and then the entire procedure is repeated to obtain more data points without re loading the cell. In this manner, without resampling, an isopleth (constant composition at various temperature and pressure) is obtained. Th e phase transition is representative of the mixture critical point if critical opalescence is observed during the transition process and if two phases of equa l volume are present when the mixt ure phase separates. Bubble, dew, and critical point transitions for the CO2MMA mixtures are measured and reproduced at least three times to within 0.5 bar and 0.3 C. The same procedure is used in the s econd experiment for measuring the cloud points of the PMMACO2MMA mixture. The only difference is the addition of polymer into the system. The liquid MMA is measured to within 0.001 g in a sm all beaker. The glassy PMMA powder is also measured to 0.001 g accuracy, and added to the MMA in the beaker. The mixture is covered and stirred with a magnetic stirrer, at low speed, until the polymer beads are fully dissolved into the monomer. At the same time, the magnetic stirrer is heated slightly to aid the dissol ution. Once a homogeneous mixture is obtained,
PAGE 64
51 the solution is transferred slowly into the cel l using a pipette. Then, the mixer assembly is screwed in position and the cell is purg ed slowly with carbon dioxide as already described. After the purging is complete, the outlet valve is closed, and CO2 is transferred from the gas cylinder into the cell, as previously described. When the cell is full, the CO2 is shut off, and the initial values of the temperature and pressure in the system are recorded. Then, the mixture in the cell is preheated up to about 105110 C, just above the glass transition temperature of the polymer. This is needed in order to avoid the nucleation of the polymer as a result of incompatibility issues related to the density differences between the CO2 and the PMMA in the system. Once the system is preheated to this temperature, the mixer is turned on to agitate the solution until it becomes a homogeneous optically transparen t mixture. The cell assembly is then lowered in its horizontal position, and the heat er is switched off. The desired temperature at which the first cloud point would be measured is preset on the controller. The system cools down slowly until the temperature reaches the set point. At this point of time the mixture in the cell may have turned milky or cloudy, due to phase separation in the system. This is expected behavior and it will remain that way until the pressure is increased to bring the system back into the one phase region. Thus, once thermal equilibrium is maintained in th e cell at the preset temperature, the contents of the cell are compressed into the one phase region where th e mixture in the cell becomes transparent. Once in the onephase region, the pressure is lowered rapidly by moving the piston backwards i.e. increasing the volume until the first appearance of a cloud in the cell.
PAGE 65
52 The pressure at which the first cloudy lo ok appears is recorded, and that is the cloud point of the system at the given temper ature. After completing the measurement at a given temperature, the cell temperature is then stabilized at a new value, and the experimental procedure is repeated. Cloud points are measured and reproduced at least three times, but preferably five times, to within 2.5 bar and 0.3 C. An average cloud point is then calculated from the multiple m easurements. In this work, cloud point data was measured in ~15 C increments i.e. decrements, and the above procedure was repeated, thus creating a pre ssuretemperature (PT) cloud point curve at fixed PMMA, MMA, and CO2 concentrations.
PAGE 66
53 CHAPTER 4 RESULTS AND DISCUSSION In this chapter, the experimental results obtained at several different temperatures and compositions are presented. The focus of this work is to measure the solubility of poly(methyl methacrylate) (PMMA) in supercritical CO2. 4.1. Experimental Results Experimental information is presented on the phase behavior of the CO2MMA system, and the PMMACO2MMA system. The primary reason for obtaining CO2MMA data is to determine whether CO2 and MMA form multiple phases in the pressuretemperaturecomposition region s explored in the PMMACO2MMA studies. The CO2MMA data and the PMMACO2MMA data are modeled with the SAFT equation of state. The obtained results and the various phenomena associated with this kind of system are discussed in detail. 4.1.1. CO2MMA System Table 4.1 and Figure 4.1 present the CO2MMA bubblepoint data obtained from the experimental runs at 40, 80, and 105.5 C.
PAGE 67
54 Table 4.1 Experimental Data for the MMACO2 System Obtained in this Study Temperature ( C) MMA mole fraction Pressure (bar) Transition (bubble point) 0.005 86.00 BP 0.044 86.93 BP 0.048 86.90 BP 40.0 0.091 86.04 BP 0.160 78.78 BP 0.193 78.60 BP 0.337 66.33 BP 0.449 58.03 BP 0.091 119.30 BP 80.0 0.337 100.40 BP 0.600 60.00 BP 0.110 141.34 BP 0.175 144.10 BP 105.5 0.360 114.45 BP 0.650 55.16 BP
PAGE 68
55 The isotherms for the MMACO2 system of this study, based on the experimental data from Table 4.1, are given in Figure 4.1, where the mole fractions of MMA are coupled with the appropriate bubblepoint pressures measured in this experiment. MMAC02 Experimental Isotherms from This Study 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.1 MMACO2 Experimental Isotherms from This Study For the purpose of making these Px plot s at different temperatures, the mole fraction of MMA needed to be calculated This was done using the PengRobinson equation of state, given in Appendix A, to compute the compressibility factors Z for the pure component CO2 at the required temperatures and pressures. These values for Z are further used to calculate the number of moles of CO2 gas in the system, using the ideal Pressure (bar) Mole Fraction of MMA
PAGE 69
56 gas law, n(CO2)=PV/ZRT. The number of moles of CO2 is then used to calculate the mole fraction, the weight, a nd the weight fraction of CO2, and the corresponding mole and weight fractions of monomer (MMA) and polymer (PMMA) in the system. It is important to note that there is no so lubility data available in the literature on the MMACO2 and the PMMACO2MMA systems, other than the data published in 1999 by McHugh and coworkers [2], which, on the other hand, is largely inaccurate. All of their measurements show much lower bub ble pressures than the results from the experiments in this work. Compar ison plots are shown in Figure 4.2, MMACO2 Experimental Px Isotherm s from This Study vs. Data by McHugh et al 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mole fraction of MMA Figure 4.2 MMACO2 Experimental Isotherms vs Data by McHugh et al 160 140 120 100 80 60 40 20 0 20 Experimental data top isotherms McHugh data bottom isotherms at given temperature Pressure (bar)
PAGE 70
57 For some reason, the abovementione d authors have experienced some experimental errors. This gives more room for publishing the corrected data. Also, the SAFT modeling of the PMMACO2MMA system performed by the same authors has a lot of shortcommings, since they could not get any closer results with regard to the solubility measurements. More on this will be elaborated in the next section. 4.1.2. PMMACO2MMA System The data obtained in the second experime nt by measuring the cloudpoints of the PMMACO2MMA mixture is given in Table 4.2. Using this data, PT cloudpoint plots can be generated for each composition, as shown in Figure 4.3. In order to obtain the number of mole s, the volume, wt%, and mol% for each component in the system, the same calculation pattern as in the previous section is implemented. That is, the compressibility factors of the pure CO2 at the given initial conditions, pressure and temperature, are co mputed using the PengRobinson equation of state, given in Appendix A. These values for Z are used to further calculate the number of moles of CO2 gas in the system, using the ideal gas law, n(CO2)=PV/ZRT. The number of moles of CO2 are then used to calculate the mole fraction, weight, and weight fraction of CO2, and the corresponding mole and we ight fractions of monomer (MMA) and polymer (PMMA) in the system. This calculation is repeated for all experi mental conditions in order to obtain all appropriate mole fractions of MMA and PMMA. Detailed sample calculation for calculating the mole fractions is given in Appendix B.
PAGE 71
58 Table 4.2 Experimental Cl oudpoints for the PMMACO2MMA System of This Study No. MMA wt.% MMA mol% PMMA wt.% PMMA mol% Temperature ( C) Pressure (bar) 27 129.00 45.4 27.7 2.5 1.56E03 45 210.00 60 275.83 Sample # 1128 75.5 312.00 27 66.50 26.9 14.0 0.5 2.62E04 45 94.00 60 119.60 Sample # 1031 75.5 141.00 27 63.70 26.8 13.9 0.2 1.02E04 45 88.30 60 106.53 Sample # 1101 75.5 125.00 27 62.00 26.9 14.0 0.1 5.60E05 45 86.00 60 102.00 Sample # 1102 75.5 117.00
PAGE 72
59 The cloudpoint measurement represents an indirect way of presenting the solubility phase behavior of the polymer into the supercritical solvent. The direct way of obtaining the solubility of polymer in the supercritical CO2 would be to calculate the solubility using an appropriate equation of state such as SAFT, which is capable of describing the complex interactions between the polymer and the low molecular solvent. The governing equations used in the SAFT mode l are described in more detail in the next section. The data given in Table 4.2, if pl otted in a PT diagram, will depict the cloudpoint behavior of the system, as shown in Figur e 4.3 below. The error related to pressure variations is observed to be around 2 bar. 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 300.15318.15333.15348.65 Temperature (K)Pressure (bar) Figure 4.3 Experimental Cloudpoint Curves for PMMACO2MMA
PAGE 73
60 It is important to note that there is no so lubility data available in the literature on the PMMACO2MMA system either, except the data published in 1999 by McHugh and coworkers [2], which, again, is largely inac curate. Their measurements show much lower cloudpoints than the results from this work. Comparison data is shown in Figure 4.4, 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 300.15318.15333.15348.65 Temperature (K)Pressure (bar) McHugh et al.(48.4 wt% MMA, 5.1 wt% PMMA ; 29.2 mol% MMA, 2.65E03 mol% PMMA) #1128 (45.4 wt% MMA, 2.5 wt% PMMA; 27.7 mol% MMA, 1.56E03 mol% PMMA) #1031 (26.9 wt% MMA, 0.5 wt% PMMA; 14.0 mol% MMA, 2.62E04 mol% PMMA) #1101 (26.8 wt% MMA, 0.2 wt% PMMA; 13.9 mol% MMA, 1.02E04 mol% PMMA) #1102 (26.9 wt% MMA, 0.1 wt% PMMA; 14.0 mol% MMA, 5.60E05 mol% PMMA) Figure 4.4 Experimental Cloudpoint Curves for the PMMACO2MMA System of This Study vs. Cloudpoint Data by McHugh et al
PAGE 74
61 The experimental curve of this study (S ample # 1128: diamond symbols,) can be analyzed against the data by McHugh and co workers (open square symbols). The data from this work was obtained at a 2.5 wt% concentration of PMMA, while the data by McHugh is at a 5.1 wt% PMMA, where the co ncentration of MMA for these data points is kept equal in both cases, at about 47.0 1.5 wt%. It can be seen that the cloudpoint pressures for Sample # 1128 of this study are much higher than those measured by McHugh and coworkers, even though the concentration of PMMA for the sample of this study is half the amount than what they have used. If we double the amount of PMMA, under the same conditions, then our measured pr essure points would be even higher than what is already demonstrated with Sample # 1128. This leads to the conclusion that the referenced data by McHugh et al. is ve ry much below the real cloudpoint data. 4.2. Overview of the Equations Used for Computation of the Results In this section, the governing equations used in the PengRobinson EOS and in the SAFT model are described in more detail, as well as the governing equations used in the solubility calculation. 4.2.1. The PengRobinson EOS with the WongSandler Mixing Rules The computation of the pure component parameters of the methyl methacrylate cosolvent used in this work has been pe rformed by the PengRobinson equation of state (PR EOS). The general fo rm of the PR EOS is,
PAGE 75
62 term Attractive term Repulsive b) b(v b) v(v a(w,T) b v RT P (4.1) where v is the molar volume, a accounts for intermolecular interactions between the species in the mixture, and b accounts for size differences between species of the mixture. The pure fluid parameters are given as, 2 5 0 2 21 1 45724 0 ) (R c cT m P T R T w a (4.2) and c cP RT b 07780 0 (4.3) where, 226992 0 54226 1 37464 0 w w m (4.5) where TR=T/Tc is the reduced temperature, and w is the acentric factor for component i available from literature for most of the lo wtomoderate molecular weight hydrocarbons. The generalized PR EOS will take the form, b V b V RT V T a b V V Z ) ( (4.6) Implementing the MathiasCopeman (1983) rules for a(T) yields, ) ( ) ( T a T ac (4.7) 2 3 5 0 3 2 5 0 2 5 0 11 1 1 1R R RT C T C T C (4.8) and using the WongSandler (1992) mixing rules gives,
PAGE 76
63 RT a b RT a b x x T B x x T x Bij ij j ij i ij j ij i i ) ( ) ( (4.9) ij jj jj ii ii ij ijk RT a b RT a b RT a b 1 2 1 (4.10) Therefore, the cubic form of the PR EOS will take the form, 0 3 2 13 2 2 2 3 B B AB Z B B A Z B Z (4.11) where, RT bP B T R aP A ,2 2 (4.12) For further thermodynamic calculations, the fug acity coefficient expression is given by, B Z B Z b b a a y B A B Z Z b bi j ji j i i2 1 2 1 ln 2 2 2 ln 1 ln (4.13) For the purpose of this work, only the compressibi lity factors Z for the CO2 are being calculated, at the appropriate initia l temperatures and pressures, for the pure component. As mentioned before, these values for Z are used to further calculate the number of moles of CO2 gas in the system, which are then used to calculate the mole fraction, the weight, and the weight fraction of CO2, and the corresponding mole and weight fractions of monomer and polymer in the system. 4.2.2. Polymer Superc ritical Fluid Equilibria Polymer solubility in supercritical flui ds depends on temperature and pressure. In the supercritical pressure range, the so lubility always increases with increasing
PAGE 77
64 pressure. The effect of temperature is not as straightforward, howev er. At intermediate pressures, the free volume (density) effect is dominant and the so lubility drops with increasing temperature (0 ) / ln ( PT X). At higher pressures, the solubility increases with increasing the temperature. As mentioned before, the starting point in determining the phase behavior of the system is the equilibrium criteria, which is defined by the equality of the fugacities, fi, of each component in each phase, g A s Af f (4.14) g B s Bf f (4.15) where subscript A denotes the polymer and subscript B denotes the supercritical fluid, while the subscripts s and g are for solid and supercritical gas phases, respectively. The assumptions involved in the study of the solubility phenomena are given: The supercritical fluid is insol uble in the solid polymer phase. The solid phase is incompressible; meaning the volume change of the solid due to increase of pressure above its equilibrium pressure is zero. Equilibrium pressure of a solid corresponds to the sublimation pressure for a solid supercritical fluid system. These assumptions will lead to simp lifications, which should be in good agreement with experimental observations. Fi rst, the equation (4.15) will drop because supercritical fluid (CO2) is not distributed between phase s. Thus, the remaining fugacity expression, Equation (4.14) can be rewritten as the equality of pure species fugacity in the solid phase to the partial fugacity of the solid in the supercritical gas phase,
PAGE 78
65 g A s Af f (4.16) The left hand term, the solid phase fugacity for pure species, can be written as sub A s A sub A sub A s AP P RT V P f exp (4.17) where, the PA sub is the solid vapor saturation pressure at the temperature T of the system, or the sublimation pressure of the pure solid at the system temperature. sub A is the fugacity coefficient at T and PA sub, and the exponential term is the Poynting correction for the fugacity of the pure solid. For a typical so lid supercritical gas system, the solid vapor pressure is small, thus, for al l practical purposes it can be considered as an ideal gas, which would reduce the sub A to unity, since the saturation pressure of a crystalline solid is normally much less than 1 bar. The right hand term of equation (4.16) is expressed as, P y fV A A g A (4.18) Now, combining equations (4.17) and (4.18), yields sub A s A V A sub A AP P RT V P P y exp 1 (4.19) where yA is the solubility of the solute in the supercritical gas, and it can be calculated using this equation. The PA sub and VA s are purespecies propert ies, either found from compiled data or estimated fr om a suitable correlation. The V A however, needs to be computed from a PVT equation of state, one that is capable of representing vaporsolute (polymer) mixtures at high pressures.
PAGE 79
66 4.2.3. The SAFT Equation of State Even for simple model fluids, like the Le nnardJones fluid, it is not possible to derive exact and engineeringlike (analytical) equations of state. The perturbation theories are therefore used to deliver simple approximate so lutions for a given molecular model. The underlying idea is to divide the total in termolecular forces into repulsive and attractive contributions. The repulsive in teractions are appr oximated through a referencefluid, for which a reli able description has to be available. For perturbed chain equation, the reference fluid is the hard chain fluid The attractive intermolecular forces in the PerturbedChain SAFT equation of state are further divided into different contributions, so that the reduced re sidual Helmholtz energy is written as kT a kT a kT a kT aassoc chain disp, hc res (4.20) The right hand side of Equation (4.20) is composed of expressions for hardchain fluid as well as contributions due to dispersion, associa tion, dipolar and quadrupolar interactions. For a defined molecular model, theories from statistical thermodynamics can be applied for different contributions. The hardchain reference fluid which is the first term in Equation (4.20), consists of chainmolecules with no attractiv e interactions. For chain comprising m segments, the reduced Helmholtz energy is given as ) ( g log 1) (m x kT a m kT aii hs ii i i hs hc (4.21) where xi is the mole fraction of chains of component i, i mis the number of segments in a chain, hs ii g is the radial pair dist ribution function for segments at collisiondistance in
PAGE 80
67 hard sphere system, and m is the m ean segment number in the mixture. i i im x m (4.22) The Helmholtz energy of the hardsphere fluid required in Equation (4 .21) is given as ) 1 ( log ) (1 ) 1 ( 3 1 kT a3 0 2 3 3 2 2 3 3 3 2 3 2 1 0 hs (4.23) while the radial distribution func tion of the hardsphere fluid is: 3 3 2 2 j i j i 2 2 j i j i 3 hs1 2 d d d d ) (1 3 d d d d 1 1 gij (4.24) with n defined as: 2,3 0,1, n d m x 6 i n i i i n (4.25) The segment diameter is a function of temperature and is given as follows kT 3 exp 0.12 1 di i i (4.26) The dispersion contribution for chain molecules, which is the second term in Equation (4.20), is derived from the perturba tion theory of Barker and Henderson (1983). mm mm 1 2 hc 3 2 hc 1 2 hc 3dw w ) w, (m, g kT P mkT dw w ) w, (m, g kT 2 kT achain disp, (4.27) with 1 2 4 3 2 4 2)] 2 )( 1 [( 2 12 27 20 m) (1 ) 1 ( 2 8 m 1 P kT (4.28)
PAGE 81
68 where w is the reduced radial distance around the segments () r/ w denotes the packing fraction (a dimensionless density, 3 and ) w, m, ( hc g is the radial distribution function of the hard chain reference fluid. For chain molecules, the overall intermolecular interaction is calculated as the sum of all individual segmentsegment interactions in Equation (4.27). The integral s in Equation (4.27) ar e represented through simpler equations; the first integral is represented with I1, and the second one with I2. 6 0 i i 1i (m) a m) ( I (4.29) 6 0 i i 2i (m) b m) ( I (4.30) where the coefficients (m) b and (m) ai iare functions of segment number. 2i 1i 0i ia m 2 m m 1 m a m 1 m a (m) a (4.31) 2i 1i 0i ib m 2 m m 1 m b m 1 m b (m) b (4.32) The association contribution, i.e. the third term in Equation (4.20) is given as Ai i Ai Ai i i assocW ) X X ( x kT a2 2 ln (4.33) where iWis the number of bonding sites of segment type The polymersupercritical fluid sy stem of this work (PMMACO2MMA), does not account for associ ation terms, due to the fact th at the functional groups of the polymer system are canceling themselves, and thus they do not exhibit association
PAGE 82
69 interactions. There are no dipole effects eith er, which means there are only hard chain, hard sphere, and dispersion contributions. 4. 3. Modeling of the PMMACO2MMA System Using SAFT EOS In this section, the PMMACO2MMA system is modeled using the statistical associating fluid theory (SAFT) model [19]. Th e fugacity coefficients of the supercritical fluid are obtained from this model, as a function of composition, temperature and pressure. By substituting the appropriate values for the molar fractions of CO2, methyl methacrylate, and poly (methyl methacryla te), followed by the appropriate working temperature and pressure, the so ftware gives a fugacity output for those given conditions. Detailed sample calculation procedure is given in Appendix E. The pure component critical temperature, pressu re, and acentric factor for CO2 and MMA is given in Table 4.3 Table 4.3 Critical Constants and Acentric Factor for CO2 and MMA Component Tc (C) Pc (bar) Acentric Factor CO2 31.0 73.8 0.225 MMA 290.8 36.8 0.317 A detailed discussion of the mathematical form of the SAFT equation was given in the previous section, together with th e expressions for the re sidual Helmholtz free energy, and for the fugacity coefficients. For each pure co mponent there are potentially
PAGE 83
70 five pure component parameters in the SAFT equation: voo, the temperatureindependent volume of segment, uo/k, the temperatureindependent, nonspecific energy of attraction between two segments, m, the number of segments in a molecule, /k, the energy of association between sites on a molecule, and v, the volume of sitesite association. Although spectroscopy data [2, 20] suggest the existence of a complex between CO2 and the repeat units in MMA, the association parameters /k and v are nevertheless set equal to zero for this complex since it is rather weak and it would introduce more fitted parameters into the model. The pure component parameters for CO2 are available in the literature [21], while the pure component parameters for MMA and PMMA are not availabl e in the literature and had to be estimated. They are all listed in Table 4.4 below, Table 4.4 Pure Component Parameters Used in the SAFT Equation Component uO/k (K) V (cm3/mole) m CO2 216.1 13.278 1.417 MMA 208.0 8.700 5.670 PMMA 240.0 9.560 2532 The values of the pure component parameters for low molecular weight components from a given chemical family usua lly follow trends that are directly related to the changes in structural features of each member of the family. Following that logic,
PAGE 84
71 the above pure component parameters have been obtained by extrapolating the results found with the nalkane parameters for similar pol yolefins. Also, a simple group contribution method can be used to calculat e the pure component parameters of polymers and their monomers, particularly fo r obtaining the number of segments, m, by calculating m for the base contributing groups in the molecule and adding them together. As it can be seen from Table 4.4, th ere is a significant difference between the temperatureindependent nonspecific ener gy of attraction betw een two segments, k u/0, for PMMA and MMA. This can be due to the intraand intersegmental interactions of the many segments of PMMA that are affected by the excluded volume of the chains [2] relative to the small number of chains of MMA which is not as affected by excluded volume considerations. The only experimental informati on available on the system PMMACO2 is that PMMA is not soluble in pure CO2 to temperatures as high as 255C and pressures till 2550 bar, while the addition of monomer cosol vent to the polymeric system will greatly increase the solvating powe r of the supercritical CO2 solvent. This has been proven experimentally in this work. The SAFT model should be able to predict the same findings, only through calculation. There are concerns in the liter ature though that the quantitative agreement between experimental data and calculations is not always good. 4.3.1. Solubility Behavior of the PMMACO2MMA System A detailed sample calculation of the sublimation pressure and the fugacity coefficients of the polymer system, using modi fied SAFT code, is given in Appendix E.
PAGE 85
72 The software gives a fugacity output for th e given conditions: composition, temperature, and pressure. The obtained sublimation pressure and the fugacity coefficients are further used in the solubility equation, sub A s A V A sub A AP P RT V P P yexp 1 By further successive substitution of all corresponding PT cloudpoints in the above equation, the solubility of PMMA is calculated for every combination of composition, pressure and temperature. The results are listed in Table 4.5, Table 4.5 Solubility Data for the PMMACO2MMA System of this Study Calculated for all PTx Combinations Using the SAFT EOS Sample 27 C 45C 60C 75.5C P1 y1 P2 y2 P3 y3 P4 y4 #1128 129.00 1.976E06 210.00 1.391E05 275.83 6.518E05 312.00 2.788E04 # 1031 66.50 4.445E06 94.00 2.600E05 119.60 1.006E04 141.00 3.799E04 # 1101 63.70 4.574E06 88.30 2.687E05 106.53 1.059E04 125.00 3.972E04 # 1102 62.00 4.606E06 86.00 2.701E05 125.00 1.073E04 117.00 4.056E04 Based on the calculated solubility results in Table 4.5, a Py diagram is plotted in MATLAB depicting each isotherm for each appropriate pressure and solubility set, as shown in Figure 4.5.
PAGE 86
73 Figure 4.5 Solubility Isotherms for the PMMACO2MMA System of This Study Calculated Using the SAFT EOS Model These isotherms can further be translated into a cloudpoint data plot, making it more convenient to compare with the experimental cl oudpoint data. Th is can potentially be done using the same SAFT software with minor modifications for the pure component reduced density i.e. the segment or molecule packing factor, which is a function of the molar density. In the previous calculation of the vapor fugacity coefficients, the packing factor in the SAFT code is set to 0.232. No w, for calculating the fugacity coefficients of the liquid phase, the packing f actor will be set to a much smaller value, 0.00026, which
PAGE 87
74 would be representative of the liquid phase. A similar set of fugacity coefficients is calculated, and then both fugacity coeffi cients are used in the cloudpoint flash calculation to obtain the cloudpoin t pressures. The calculated f ugacity coefficients of the vapor and liquid phase are listed in Table 4.6, Table 4.6 Fugacity Coefficients Calculated for the Vapor and the Liquid Phase for the PMMACO2MMA System of This Study Using the SAFT EOS Sample 27 C 45C 60C 75.5C V L V L V L V L #1128 0.1973 0.3696 0.20860.3696 0.21720.3697 0.2255 0.3697 # 1031 0.1208 0.3692 0.13680.3693 0.15000.3693 0.1634 0.3693 # 1101 0.1207 0.3692 0.13680.3693 0.15000.3693 0.1635 0.3693 # 1102 0.1220 0.3692 0.13810.3693 0.15120.3693 0.1647 0.3693 This is an alternative approach and it may need to incorporate a new theory, because there is a fundamental problem in mapping the polymer solid phase to calculate the bubble point. This can be overcome by introducing some modifications to the existing SAFT code. Taking into consider ation the current developments in SAFT modeling techniques, it is believed that soon it will be possible to pe rform this type of calculation with greater accuracy In this study, an attempt was made to model the solubility behavior with SA FT, using a bubblepoint algorithm The calculated values are listed in Table 4.7,
PAGE 88
75 Table 4.7 SAFT Modeled Solubility of Poly (Methyl) Methacrylate in CO2 in the Presence of 2.0 wt % MMA x (27 C) y (27 C) P (27 C) x (60 C) y (60 C) P (60 C) x (76 C) y (76 C) P (76 C) 0.999 4.86E7 65 0.995 4.86E6 65 0.99 1.12E6 65 0.998 7.99E7 89 0.982 6.32E6 89 0.986 4.76E6 89 0.983 1.98E6 110 0.979 1.87E5 110 0.973 1.88E5 110 0.975 6.87E6 138 0.973 9.86E4 138 0.962 5.87E5 138 0.972 8.99E6 185 0.964 1.22E3 185 0.959 8.87E4 185 0.968 1.96E5 198 0.958 3.98E3 198 0.95 9.96E4 198 Figure 4.6 SAFT Modeled Solubility of PMMACO2MMA System
PAGE 89
76 These curves show the quantitative agreement between experimental data and calculations. The predicted values in this study are very close to the experimental values. If we increase the amount of MMA in the calc ulations, the results would be even closer. It was proven experimentally that at 20wt% MMA and 40wt% MMA, solubility of poly (methyl methacrylate) increased significantly, due probably to the formation of a weak complex between the carboxylic oxy gen in MMA and the carbon in CO2. MMA cosolvent added to CO2 significantly decreases the pressures needed to dissolve PMMA.
PAGE 90
77 CHAPTER 5 CONCLUSIONS AND FUTURE WORK To summarize, High Pressure Phase Mon itoring System SPM20 has beeen used in this study to observe and measure phase be havior of a monomers upercritical fluid and monomerpolymersupercritical fluid systems. Solubility of poly (methyl methacrylate) (PMMA) in CO2, plus methyl methacrylate as a coso lvent, was determined using cloud point measurements. The results are mode led using SAFT. Polymer processing in supercritical fluids has been a major intere st for a portfolio of materials processing applications including their impregnation into porous matrices. Unfortunately they are only sparingly soluble in CO2 unless one uses an entraine r or surfactant. This work focuses their solubility at rather mild cond itions, low temperature and reduced pressures less than three. Cloudpoint measurement demonstrates that poly (methyl methac rylate) is hardly soluble in supercritical CO2. Partial solubility of the polym er may be taking place, but it is far from soluble. The solubility improves with the addition of its own monomer, methyl methacrylate, to the system. Concentr ations in the excess of 50 wt % (30 mol%) MMA have been employed, which resulted in singlephase systems, at relatively low pressures and temperatures ( 100500 bar at corresponding 27150 C). It was noticed
PAGE 91
78 that the polymer did not fully solubilize until the concentration of poly (methyl methacrylate) has been reduced more than ten times below the originally anticipated amount of 5.0 wt%. Cloudpoint data for the di lute system of poly(methyl methacrylate) (PMMA)CO2methyl methacrylate (MMA) are measur ed in the temperature range of 2775.5 C, and pressures as high as 320 bar, and w ith cosolvent concentrations of 27 and 48.4 wt% MMA, and varying concentrations of PMMA of 0.1, 0.2, and 0.5 wt%. Solubility data is reported for these system s. PengRobinson equation of state has been used to model the CO2MMA system, while SAFT equation of state is used to model the PMMACO2MMA system. Much closer qualitative match is achieved with the SAFT model used in this work, compared to th e modeling results published by McHugh et al. It is difficult to evaluate if the shortcomings of SAFT are caused by the equation itself or rather by the choice of parameters used in the calculation. SAFT does not account for chain stiffness [2, 6], whose cont ribution to the entropy of mixing can be of major importance. So far, the SAFT equati on can possibly be used to show qualitatively the same trends as the experimental data but it is not reliabl e enough to obtain an accurate quantitative agreement w ith the measured data. More substantial work is needed to extend and evaluate the group contribution method for estimating polymer, solvent, and cosolvent parameters if phase behavior is to be calculated reliably rather than measured experimentally. The findings pres ented in this work open up a new dimension for research in the thermodynamic analysis of polymeric systems.
PAGE 92
79 REFERENCES [1] McHugh, M. A., Krukonis, V. J., Eds. Supercritical Fluid Extraction Principles and Practice, 2nd Ed.; ButterworthHeinemann: Boston, 1994. [2] Lora M., McHugh M. A., Phase beha vior and modeling of the poly(methyl methacrylate)CO2methyl methacrylate system, Fluid Phase Equilibria 1999, 157, 285297. [3] Kirby, F. C., McHugh, M. A., Phase Beha vior of Polymers in Supercritical Fluid Solvents, Chem. Rev. 1999, 99, 565602. [4] Tomasko, D. L., Li, H., Liu, D., Han, X ., Wingert, M. J., Lee, L. J., Koelling, K. W., A Review of CO2 Applications in the Processing of Polymers, Ind. Eng. Chem. Res. 2003, 42, 64316456. [5] Seung, N. J., JiUng, P., Sun, Y. K ., KiPung, Y., High Pre ssure Behavior of PolymerSolvent Systems with Addition of Supercritical CO2 at Temperatures from 323.15 K to 503.15 K, J. Chem. Eng. Data 2002, 47, 270273. [6] Rindfleisch, F., DiNoia T. P., McHugh, M. A., Solubility of Polymers and Copolymers in Supercritical CO2, J. Phys. Chem. 1996, 100, 1558115587. [7] Serifoglu, E. I., Theoretical and Experi mental Investigation of Stefan Tube at Supercritical Conditions, Dissertation, University of South Florida, 2003. [8] Walker, T. A., Spontak,R. J., Khan, S. A., Modification of Polymer Blend Phase Behavior with HighPressure Carbon Di oxide, Dissertation, North Carolina State University, 2003. [9] McHugh, M. A., Rindfleisch, F., Kunt z, P. T., Schmaltz, C., Buback, M., Cosolvent Effect of Alkyl Acrylates on the Phase Behavior of Poly(alkyl acrylates)Supercritical CO2 mixture, Polymer 1998, Vol. 39, No. 24, 60496052. [10] Prausnitz, J. M., Lichtentha ler, R. N., de Azevedo, E. G., Molecular Thermodynamics of FluidPhase Equilibria, 3rd Ed.; Prentice Hall: Upper Saddle River, NJ, 1999.
PAGE 93
80 [11] H. Obrey, C. P. Bokis, C.C. Chen, Ind. Eng. Chem. Res. 1998, 37, 44814491. [12] G. Luft, A. Lindner, Angew. Makro. Chem. 1976, 56, 99. [13] Voutsas, E. C., Pappa, G. D., Boukouva las, C. J., Magoulas, K., Tassios, D. P., Miscibility in Binary Polymer Blends: Correlation and Prediction, Ind. Eng. Chem. Res. 2004, 43, 13121321. [14] Flory, P. J., Principles of Polymer Chemistry, Chapter 12. Cornell Univ. Press: Ithaca, New York, 1953. [15] Sanchez, I. C., Statistical Thermodynamics of Polymer Blends, Polymer Blends, Vol. 1, Chapter 3. Academic Press: New York, 1978. [16] Sanchez, I. C., Lacombe, R. H., J. Phys. Chem. 1976, 80, 2352. [17] Sanchez, I. C., Lacombe, R. H., J. Polym. Sci. Polym. Lett. Ed. 1977, 15, 71. [18] Lacombe, R. H., Sanchez, I. C., J. Phys. Chem. 1976, 80, 2568. [19] Aslam, N., SAFT, Dissertation, University of South Florida, 2005. [20] Kazarian, S. G., Vincent, M. F., Bri ght, F. V., Liotta, C. L., Eckert, C. A., J. Am. Chem. Soc. 1996, 118, 17291736. [21] Huang, S. H., Radosz, M., Equation of State for Small, Large, Polydisperse, and Associating Molecules, Ind. Eng. Chem. Res. 1990, 29, 22842294. [22] Perry, R. H., Green, D., Perrys Chemical Engineers Handbook., 7th Ed., McGrawHill Book Company, New York, 1997. [23] Orbey, H., Sandler, S. I., Modeling Va porLiquid Equilibria: Cubic Equations of State and Their Mixing Rules, Cambridge University Press: Cambridge, U.K., 1998.
PAGE 94
81 BIBLIOGRAPHY [1] Brunner, G., Gas Extraction, Steinkopff: DarmstadtSpringer: New York, 1994. [2] Voutsas, E. C., Pappas, G. D., Boukouva las, C. J., Magoulas, K., Tassios, D. P., Miscibility in Binary Polymer Blends: Correlation and Prediction, Ind. Eng. Chem. Res. 2004, 43, 13121321. [3] Sunol, A. K., Sunol, S. G., Substituti on of Solvents by Safer Products and Processes, in Handbook of Solvents, Wypych, G., Ed., ChemTec Publishing, New York, 2001, 14191459. [4] Day, CY., Chang, C. J., Chen, CY., Phase Equilibrium of Ethanol+CO2 and Acetone+CO2 at Elevated Pressures, J. Chem. Eng. Data, 1996, 41, 839843. [5] Byun, HS., Shin, JS., BubblePoint Measurement for CO2+Vinyl Acetate and CO2+Vynil Acrylate Systems at High Pressures, J. Chem. Eng. Data, 2003, 48, 97101. [6] McHugh, M., Paulaitis, M. E., Solid So lubilities of Naphthalene and Biphenyl in Supercritical Carbon Dioxide, J. Chem. Eng. Data, 1980, 25, 326329. [7] Byun, H. S., McHugh, M. A., Impact of Free Monomer Concentration on the Phase Behavior of Supercritical Carbon DioxidePolymer Mixtures., Ind. Eng. Chem. Res., 2000, 39, 46584662. [8] Koak, N., Visser, R. M., de Loos, Th. W., Fluid Phase Equilib., 1999, 158160, 835846. [9] McHugh, M. A., Rindfleisch, F., Kunz, P. T., Schmaltz, C., Buback, M., Cosolvent Effect of Alkyl Acrylate on the Phase Behavior of Po ly(alkyl acrylate)Supercritical CO2 Mixtures., Polymer, 1998, 39, 60496052. [10] Xiong, Y., Kiran, E., J. Applied Polymer Science, 1995, 55, 18051818. [11] Sanchez, I. C., Lacombe, R., J. Phys. Chem., 1976, 80, 23522362. [12] Sanchez, I. C., Lacombe, R., Macromolecules, 1978, 11, 11451156.
PAGE 95
82 [13] Folie, B., Radosz, M., Ind. Eng. Chem. Res., 1995, 34, 15011516. [14] Condo, P. D., Radosz, M., Fluid Phase Equilibria, 1996, 117, 110. [15] Obrey, H., Bokis, C. P., Chen, CC., Ind. Eng. Chem. Res. 1998, 37, 44814491. [16] Prausnitz, J. M., Lichtentha ler, R. N., de Azevedo, E. G., Molecular Thermodynamics of Fluid Phase Equilibria, 3rd Ed.; PrenticeHall: Upper Saddle River, NJ, 1999. [17] McHugh, M., Krukonis, V., Supercritical Fluid Extraction, 2nd Ed.; ButterworthHeinemann: Stoneham, MA, 1994. [18] Luft, G., Lindner, A., Angew. Makro. Chem. 1976, 56, 99. [19] Serifoglu, I. E., Theoretical and Expe rimental Investigation of Stefan Tube at Supercritical Conditions, Thesis (USF), 2003. [20] Paul, D. R., Newman, S., Lacombe, S. I., Polymer Blends, Volume 1, Academic Press, Inc.(London) Ltd: 1978. [21] Seung, N. J., JiUng, P., Sun, Y. K ., KiPung, Y., High Pre ssure Behavior of PolymerSolvent Systems with Addition of Supercritical CO2 at Temperatures from 323.15 K to 503.15 K, J. Chem. Eng. Data, 2002, 47, 270273. [22] Rindfleisch, F., DiNoia T. P., McH ugh, M. A., Solubility of Polymers and Copolymers in Supercritical CO2, J. Phys. Chem. 1996, 100, 1558115587. [23] Sandler, S. I., Chemical and Engineering Thermodynamics, 3rd Ed., John Wiley & Sons, Inc., New York, 1999. [24] Prausnitz, J., Anderson, T., Grens, E., Eckert, C., Hsieh, R., OConnell, J., Computer Calculations for Multicom ponent VaporLiquid and LiquidLiquid Equilibrium., PrenticeHall, Inc., Englewood Cliffs, NJ, 1980. [25] Smith, J. M., Van Ness, H. C., Abbott, M. M., Introduction to Chemical Engineering Thermodynamics, 6th Ed., Tata McGrawHill Publishing Company Ltd., New Delhi, 2003. [26] Benedek, P., Olti, F., Computer Aide d chemical Thermodynamics of Gases and Liquids: Theory, Models, and Programs, John Wiley & Sons,Inc., New York, 1985.
PAGE 96
83 [27] Tester, J. W., Modell, M., Th ermodynamics and Its Applications, 3rd Ed., PrenticeHall, Inc., Upper Saddle River, NJ, 1997. [28] Clifford, T., Fundamentals of Supercritic al Fluids, Oxford University Press, Inc., New York, 1999. [29] Angus, S., Armstrong, B., de Reuck, K. M., Chapela, G. A., Rowlinson, J. S., International Thermodynamic Tables of the Fluid State Carb on Dioxide, IUPAC, Pergamon Press, New York, 1976. [30] Hasch, B. M., Lee, SH., McHugh, M. A., Strengths and Limitations of SAFT for Calculating Polar Copolyme rSolvent Phase Behavior, Journal of Applied Polymer Science, 1996, 59, 11071116. [31] Reid, R. C., Prausnitz, J. M., Poling, B. E., The Properties of Liquids and Gases, 4th Ed., McGrawHill, New York, 1987. [32] Peng, D. Y., Robinson, D. B., A New TwoConstant Equation of State., Ind. Eng, Chem. Fundam., 1976, 15, 5963.
PAGE 97
84 APPENDICES
PAGE 98
85 Appendix A: MATLAB Program That Calculates the Z Values of Pure CO2 Using the PengRobinson EOS w ith WongSandler Mixing Rules % Program Name: CO2.m % ____________________________________________________________________ % This program provides the values for the compressibility % factors Z % for the pure component CO2 in the system. The program calls for the % PR EOS function "CO2.m". The subscript (1) stands for CO2; subscript % (2) is for other solvent (Tc2, Pc2, w2 of a solvent must be entered) % For the purpose of computing the Z factors of pure CO2 in the system, % at the initial pressure and temperature of each experimental sample % in the system, the amount of CO2 is fixed as 100%, while the other % solvent portion is 0%. The output is the Z factors for the pure CO2. % ____________________________________________________________________ clear all function Z = CO2(x1,x2,P,T) JFLG=1; Tc1 = 304.25; % critical temperature of CO2 Pc1 = 73.8; % critical pressure of CO2 w1 = 0.225; % acentric factor of CO2 Tc2 = 513.9; Pc2 = 61.4; w2 = 0.644; Tr1 = T/Tc1; Tr2 = T/Tc2; k01 = 0.378893 + 1.4897153*w10.17131848*w1^2+0.0196554*w1^3; o1 = 0.04285; o2 = 0.03374; k02 = 0.378893 + 1.4897153*w20.17131848*w2^2+0.0196554*w2^3; k1 = k01 + o1*(1 + (Tr1)^0.5)*(0.7Tr1); k2 = k02 + o2*(1 + (Tr2)^0.5)*(0.7Tr2); alpha1 = [1 + k1*(1(Tr1)^0.5)]^2; R = 83.14; a1 = (0.457235*R^2*Tc1^2/Pc1)*alpha1; b1 = 0.077796*R*Tc1/Pc1; alpha2 = [1 + k2*(1(Tr2)^0.5)]^2; a2 = (0.457235*R^2*Tc2^2/Pc2)*alpha2; b2 = 0.077796*R*Tc2/Pc2; Alp12=0.30; Alp21=0.30;
PAGE 99
86 Appendix A (Continued): G12 = 1.909516*238.92; G21 = 1.444729*238.92; TAU12 = G12/(1.9872*T); TAU21 = G21/(1.9872*T); C = 0.62323; T12 = TAU12; T21 = TAU21; G01 = (x1*x2)*T21*G21; G02 = x1*(1G21) + G21; G03 = (x1*x2)*T12*G12; G04 = 1 x1*(1G12); Gex = G01/G02 + G03/G04; I12 = 0.15; I21 = 0.15; F11 = x1^2*(b1 a1/(R*T)); F12 = x1*(x2)*((b1+b2)(a1+a2)/(R*T))*(1I12)/2; F21 = x1*(x2)*((b1+b2)(a1+a2)/(R*T))*(1I21)/2; F22 = (x2)^2*(b2a2/(R*T)); FOOO = F11 + F12 + F21 + F22; M11 = (x1*a1/(R*T*b1) + (x2)*a2/(R*T*b2)); M12 = Gex/C; MOO = 1 M12 M11; b = FOOO/MOO; B = b*P/(R*T); M13 = Gex *R*T/C; M14 = (x1*a1/b1 + (x2)*a2/b2); M0 = M14 + M13; a = b*M0; A = a*P/(R*T)^2; p1 = B1; p2 = A 3*B^22*B; p3 = B^3+B^2A*B; coeffs=[1 p1 p2 p3]; numz=roots(coeffs); display(numz); Z = numz(3) display (A); display (B); fugacity = (Z1) log(ZB)A/(2.828*B)*log((Z+2.414*B)/(Z.414*B)); fugcoeff = log(fugacity); display(fugcoeff);
PAGE 100
87 Appendix B: Sample Calculation to Determine the Mole Fractions of MMA or PMMA For sample #1120 of this experiment, where th e weight of MMA is m(MMA) = 1.0007 g, n(MMA) = m/M = 1.0007/100.12 = 0.01 moles V(MMA) = m/= 1.0007/0.943 = 1.0612 ml V(CO2) = 15 V(MMA) = 15 1.0612 = 13.9388 ml Using function Z = CO2(x1, x2, P, T) from the Matl ab program CO2.m, from Appendix A.1., the compressibility of CO2 at initial pressure Pin = 63.5 bar and in itial temperature Tin = 295.35 K, is calculated to be Z = 0.1645, as shown below: Z = CO2 (x1, x2, P, T) Z = CO2 (1, 0, 63.5, 295.35) Z = 0.1645 Using this Z value, the number of CO2 moles in the system is calculated, n(CO2) = (PinV)/(ZRTin) = (63.5 x 13.9388) / (0.1645 x 83.1451 x 295.35) = 0.2191 Then, the mole fractions of CO2 and MMA are calculated, x1 (CO2) = n(CO2) / [n(CO2)+n(MMA)] = 0.2191/(0.2191+0.01) = 0.956 x2 (MMA) = n(MMA) / [n(CO2)+n(MMA)] = 0.01/((0.2191+0.01) = 0.044 This calculation is further repeated for each experimental sample, at its corresponding MMA or PMMA weights, initial pressures and in itial temperatures of the system, in order to obtain the appropriate mole fractions of MMA or PMMA for each experiment.
PAGE 101
88 Appendix C: Mole Fraction of Polymer Compar ed to the Corresponding Weight Fraction For polymers, the mole fraction is often not ve ry useful. For example, if we consider a mixture of 0.3 g PMMA of molecular we ight 96,850 mixed with 5.5 g of MMA of molecular weight 100.12 and 6.32 g of supercritical CO2 of molecular weight 44.01, the mole fraction of the polymer is, x(PMM) = moles PMMA/(moles PMMA+moles MMA+moles CO2) = = (0.3/96,850) / (0.3/96,850 + 5.5/100.12 + 6.32/44.01) = = 1.56E05 where x is mole fraction. While this seems like a vanishing amount of polymer, we can now consider the mass fraction of polymer in the mixture: m(PMMA) = mass PMMA / (mass PMMA + mass MMA + mass CO2) = = 0.3 / (0.3 + 5.5 + 6.32) = = 0.3/12.12 = 0.025 where m is mass fraction. Although the mole fr action of polymer is tiny, the polymer occupies about 2.5% of the ma ss of the mixture. In terms of physical and thermodynamic properties, most of the times it is more mean ingful to consider mass fractions than mole fractions of polymers.
PAGE 102
89 Appendix D: The SAFT EOS Model Used in This Study function T = SOFINAL(x1,x2,x3,T,P) Z = 0.005; dense = 0.232; Tau =0.74048; k = 1.381*10^23; NA = 6.023*10^23; ek1 = 52.0; ek2 = 10.0; ek3 = 10.0; m1 = 1.417; m2 = 5.67; m3 = 2532; k12 = 0.23; k13 = 0.059; k23 = 0.12; k21 = k12; k31 = k13; k32 = k23; v100 = 13.278; v200 = 8.7; v300 = 9.56; d11 = ((v100*6*Tau)/(3.14*NA))^0.33; % Units cm d22 = ((v200*6*Tau)/(3.14*NA))^0.33; % Units cm d33 = ((v300*6*Tau)/(3.14*NA))^0.33; % Units cm d1 = d11/(10^8); d2 = d22/(10^8); d3 = d33/(10^8); u10 = 216.1; u20 = 208.0; u30 = 240.0; u11 = u10*(1+ek1/T); u22 = u20*(1+ek2/T); u33 = u30*(1+ek3/T); u12 = (1k12)*(u11*u22)^0.5; u21 = (1k21)*(u22*u11)^0.5; u13 = (1k13)*(u11*u33)^0.5; u31 = (1k31)*(u33*u11)^0.5; u23 = (1k23)*(u22*u33)^0.5; u32 = (1k32)*(u33*u22)^0.5; v10 = v100*[10.12*exp(3*u10/(T))]^3; v20 = v200*[10.12*exp(3*u20/(T))]^3; v30 = v300*[10.12*exp(3*u30/(T))]^3; v110 = v10; v220 = v20; v330 = v30; v120 = [0.5*[(v10)^0.33 + (v20)^0.33]]^3; v210 = v120;
PAGE 103
90 Appendix D (Continued): v130 = [0.5*[(v10)^0.33 + (v30)^0.33]]^3; v310 = v130; v230 = [0.5*[(v20)^0.33 + (v30)^0.33]]^3; v320 = v230; m = m1*x1 + m2*x2 + m3*x3; dmdx1 = m1; dmdx2 = m2; dmdx3 = m3; z0t = (3.14/6)*(x1*m1*d1^0 + x2*m2*d2^0 + x3*m3*d3^0); z1t = (3.14/6)*(x1*m1*d1^1 + x2*m2*d2^1 + x3*m3*d3^1); z2t = (3.14/6)*(x1*m1*d1^2 + x2*m2*d2^2 + x3*m3*d3^2); z3t = (3.14/6)*(x1*m1*d1^3 + x2*m2*d2^3 + x3*m3*d3^3); rho = dense/(z3t); z0 = z0t*rho; z1 = z1t*rho; z2 = z2t*rho; z3 = z3t*rho; I = z3; dz0tdx1 = (3.14/6)*(m1*d1^0); dz1tdx1 = (3.14/6)*(m1*d1^1); dz2tdx1 = (3.14/6)*(m1*d1^2); dz3tdx1 = (3.14/6)*(m1*d1^3); dz0tdx2 = (3.14/6)*(m2*d2^0); dz1tdx2 = (3.14/6)*(m2*d2^1); dz2tdx2 = (3.14/6)*(m2*d2^2); dz3tdx2 = (3.14/6)*(m2*d2^3); dz0tdx3 = (3.14/6)*(m3*d3^0); dz1tdx3 = (3.14/6)*(m3*d3^1); dz2tdx3 = (3.14/6)*(m3*d3^2); dz3tdx3 = (3.14/6)*(m3*d3^3); dz0dx1 = dz0tdx1*rho; dz1dx1 = dz1tdx1*rho; dz2dx1 = dz2tdx1*rho; dz3dx1 = dz3tdx1*rho; dz0dx2 = dz0tdx2*rho; dz1dx2 = dz1tdx2*rho; dz2dx2 = dz2tdx2*rho; dz3dx2 = dz3tdx2*rho; dz0dx3 = dz0tdx3*rho; dz1dx3 = dz1tdx3*rho; dz2dx3 = dz2tdx3*rho; dz3dx3 = dz3tdx3*rho; dIdx1 = dz3dx1; dIdx2 = dz3dx2; dIdx2 = dz3dx3;
PAGE 104
91 Appendix D (Continued): G1 = x1^2*m1^2*[u11/(T)]*v110+x1*x2*m1*m2*[u12/(T)]*v120+x1*x3*m1*m3*[u13/(T )]*v130+x2*x1*m2*m1*[u21/(T)]*v210+x2^2*m2^2*[u22/(T)]*v220+x2*x3*m2*m3 *[u23/(T)]*v230+x3*x1*m3*m1*[u31/(T)]*v310+x3*x2*m3*m2*[u32/(T)]*v320+x 3*x3*m3*m3*[u33/(T)]*v330; G2 = x1^2*m1^2*v110 + x2^2*m2^2*v220 + x3^2*m3^2*v330 + x1*x2*m1*m2*v120 + x1*x3*m1*m3*v130 + x2*x1*m2*m1*v210 + x2*x3*m2*m3*v230 + x3*x1*m3*m1*v310 + x3*x2*m3*m2*v320; G = G1/G2; ukT = G; dG1dx1 = 2*x1*m1^2*[u11/(T)]*v110 + x3*m1*m2*[u13/(T)]*v130 + x2*m2*m1*[u21/(T)]*v210 + x3*m3*m1*[u31/(T)]*v310; dG1dx2 = x1*m1*m2*[u12/(T)]*v120 + x1*m1*m2*[u21/(T)]*v210 + 2*x2*m2^2*[u22/(T)]*v220 + x3*m2*m3*[u23/(T)]*v230; dG1dx3 = x1*m1*m3*[u13/(T)]*v130 + x2*m3*m2*[u32/(T)]*v320 + x1*m3*m1*[u31/(T)]*v310 + 2*x3*m3^2*[u33/(T)]*v330; dG2dx1 = 2*x1*m1^2*v110 + x2*m1*m2*v120 + x3*m1*m3*v130 + x2*m2*m1*v210 + x3*m3*m1*v310; dG2dx2 = x1*m1*m2*v120 + x1*m2*m1*v210 + 2*x2*m2^2*v220 + x3*m2*m3*v230 + x3*m3*m2*v320; dG2dx3 = x1*m1*m3*v130 + x2*m2*m3*v230 + 2*x3*m3^2*v330 + x1*m3*m1*v310 + x2*m3*m2*v320; dGdx1 = (G2*dG1dx1 G1*dG2dx1)/(G2^2); dGdx2 = (G2*dG1dx2 G1*dG2dx2)/(G2^2); dGdx3 = (G2*dG1dx3 G1*dG2dx3)/(G2^2); dukTdx1 = dGdx1; dukTdx2 = dGdx2; dukTdx3 = dGdx3; dIdx1 = dz3dx1; dIdx2 = dz3dx2; dIdx3 = dz3dx3; D11 = 8.8043; D12 = 4.16462; D13 = 48.203; D14 = 140.436; D15 = 195.23; D16 = 113.515; D17 = 0.0; D18 = 0.0; D19 = 0.0; D21 = 2.9396; D22 = 6.086; D23 = 40.137; D24 = 76.23; D25 = 133.70;
PAGE 105
92 Appendix D (Continued): D26 = 860.25; D27 = 1535.32; D28 = 1221.42; D29 = 409.10; D31 = 2.8225; D32 = 4.7600; D33 = 11.257; D34 = 66.38; D35 = 69.248; D36 = 0.0; D37 = 0.0; D38 = 0.0; D39 = 0.0; D41 = 0.34; D42 = 3.187; D43 = 12.231; D44 = 12.11; D45 = 0.0; D46 = 0.0; D47 = 0.0; D48 = 0.0; D49 = 0.0; Att11 = D11*[ukT]*[I/Tau]; Att12 = D12*[ukT]*[I/Tau]^2; Att13 = D13*[ukT]*[I/Tau]^3; Att14 = D14*[ukT]*[I/Tau]^4; Att15 = D15*[ukT]*[I/Tau]^5; Att16 = D16*[ukT]*[I/Tau]^6; Att17 = D17*[ukT]*[I/Tau]^7; Att18 = D18*[ukT]*[I/Tau]^8; Att19 = D19*[ukT]*[I/Tau]^9; dAtt11dx1 = D11*((dukTdx1)*(I/Tau) + (ukT)*(1)*(dIdx1/Tau)); dAtt12dx1 = D12*((dukTdx1)*(I/Tau)^2 + (ukT)*2*(I/Tau)^1*(dIdx1/Tau)); dAtt13dx1 = D13*((dukTdx1)*(I/Tau)^3 + (ukT)*3*(I/Tau)^2*(dIdx1/Tau)); dAtt14dx1 = D14*((dukTdx1)*(I/Tau)^4 + (ukT)*4*(I/Tau)^3*(dIdx1/Tau)); dAtt15dx1 = D15*((dukTdx1)*(I/Tau)^5 + (ukT)*5*(I/Tau)^4*(dIdx1/Tau)); dAtt16dx1 = D16*((dukTdx1)*(I/Tau)^6 + (ukT)*6*(I/Tau)^5*(dIdx1/Tau)); dAtt17dx1 = D17*((dukTdx1)*(I/Tau)^7 + (ukT)*7*(I/Tau)^6*(dIdx1/Tau)); dAtt18dx1 = D18*((dukTdx1)*(I/Tau)^8 + (ukT)*8*(I/Tau)^7*(dIdx1/Tau)); dAtt19dx1 = D19*((dukTdx1)*(I/Tau)^9 + (ukT)*9*(I/Tau)^8*(dIdx1/Tau)); dAtt11dx2 = D11*((dukTdx2)*(I/Tau) + (ukT)*(1)*(dIdx2/Tau)); dAtt12dx2 = D12*((dukTdx2)*(I/Tau)^2 + (ukT)*2*(I/Tau)^1*(dIdx2/Tau)); dAtt13dx2 = D13*((dukTdx2)*(I/Tau)^3 + (ukT)*3*(I/Tau)^2*(dIdx2/Tau)); dAtt14dx2 = D14*((dukTdx2)*(I/Tau)^4 + (ukT)*4*(I/Tau)^3*(dIdx2/Tau)); dAtt15dx2 = D15*((dukTdx2)*(I/Tau)^5 + (ukT)*5*(I/Tau)^4*(dIdx2/Tau)); dAtt16dx2 = D16*((dukTdx2)*(I/Tau)^6 + (ukT)*6*(I/Tau)^5*(dIdx2/Tau)); dAtt17dx2 = D17*((dukTdx2)*(I/Tau)^7 + (ukT)*7*(I/Tau)^6*(dIdx2/Tau)); dAtt18dx2 = D18*((dukTdx2)*(I/Tau)^8 + (ukT)*8*(I/Tau)^7*(dIdx2/Tau)); dAtt19dx2 = D19*((dukTdx2)*(I/Tau)^9 + (ukT)*9*(I/Tau)^8*(dIdx2/Tau));
PAGE 106
93 Appendix D (Continued): dAtt11dx3 = D11*((dukTdx3)*(I/Tau) + (ukT)*(1)*(dIdx3/Tau)); dAtt12dx3 = D12*((dukTdx3)*(I/Tau)^2 + (ukT)*2*(I/Tau)^1*(dIdx3/Tau)); dAtt13dx3 = D13*((dukTdx3)*(I/Tau)^3 + (ukT)*3*(I/Tau)^2*(dIdx3/Tau)); dAtt14dx3 = D14*((dukTdx3)*(I/Tau)^4 + (ukT)*4*(I/Tau)^3*(dIdx3/Tau)); dAtt15dx3 = D15*((dukTdx3)*(I/Tau)^5 + (ukT)*5*(I/Tau)^4*(dIdx3/Tau)); dAtt16dx3 = D16*((dukTdx3)*(I/Tau)^6 + (ukT)*6*(I/Tau)^5*(dIdx3/Tau)); dAtt17dx3 = D17*((dukTdx3)*(I/Tau)^7 + (ukT)*7*(I/Tau)^6*(dIdx3/Tau)); dAtt18dx3 = D18*((dukTdx3)*(I/Tau)^8 + (ukT)*8*(I/Tau)^7*(dIdx3/Tau)); dAtt19dx3 = D19*((dukTdx3)*(I/Tau)^9 + (ukT)*9*(I/Tau)^8*(dIdx3/Tau)); Att21 = D21*[ukT]^2*[I/Tau]; Att22 = D22*[ukT]^2*[I/Tau]^2; Att23 = D23*[ukT]^2*[I/Tau]^3; Att24 = D24*[ukT]^2*[I/Tau]^4; Att25 = D25*[ukT]^2*[I/Tau]^5; Att26 = D26*[ukT]^2*[I/Tau]^6; Att27 = D27*[ukT]^2*[I/Tau]^7; Att28 = D28*[ukT]^2*[I/Tau]^8; Att29 = D29*[ukT]^2*[I/Tau]^9; dAtt21dx1 = D21*(2*(ukT)*(dukTdx1)*(I/Tau) + (ukT)^2*(1)*(dIdx1/Tau)); dAtt22dx1 = D22*(2*(ukT)*(dukTdx1)*(I/Tau)^2 + (ukT)^2*2*(I/Tau)^1*(dIdx1/Tau)); dAtt23dx1 = D23*(2*(ukT)*(dukTdx1)*(I/Tau)^3 + (ukT)^2*3*(I/Tau)^2*(dIdx1/Tau)); dAtt24dx1 = D24*(2*(ukT)*(dukTdx1)*(I/Tau)^4 + (ukT)^2*4*(I/Tau)^3*(dIdx1/Tau)); dAtt25dx1 = D25*(2*(ukT)*(dukTdx1)*(I/Tau)^5 + (ukT)^2*5*(I/Tau)^4*(dIdx1/Tau)); dAtt26dx1 = D26*(2*(ukT)*(dukTdx1)*(I/Tau)^6 + (ukT)^2*6*(I/Tau)^5*(dIdx1/Tau)); dAtt27dx1 = D27*(2*(ukT)*(dukTdx1)*(I/Tau)^7 + (ukT)^2*7*(I/Tau)^6*(dIdx1/Tau)); dAtt28dx1 = D28*(2*(ukT)*(dukTdx1)*(I/Tau)^8 + (ukT)^2*8*(I/Tau)^7*(dIdx1/Tau)); dAtt29dx1 = D29*(2*(ukT)*(dukTdx1)*(I/Tau)^9 + (ukT)^2*9*(I/Tau)^8*(dIdx1/Tau)); dAtt21dx2 = D21*(2*(ukT)*(dukTdx2)*(I/Tau) + (ukT)^2*(1)*(dIdx2/Tau)); dAtt22dx2 = D22*(2*(ukT)*(dukTdx2)*(I/Tau)^2 + (ukT)^2*2*(I/Tau)^1*(dIdx2/Tau)); dAtt23dx2 = D23*(2*(ukT)*(dukTdx2)*(I/Tau)^3 + (ukT)^2*3*(I/Tau)^2*(dIdx2/Tau)); dAtt24dx2 = D24*(2*(ukT)*(dukTdx2)*(I/Tau)^4 + (ukT)^2*4*(I/Tau)^3*(dIdx2/Tau)); dAtt25dx2 = D25*(2*(ukT)*(dukTdx2)*(I/Tau)^5 + (ukT)^2*5*(I/Tau)^4*(dIdx2/Tau)); dAtt26dx2 = D26*(2*(ukT)*(dukTdx2)*(I/Tau)^6 + (ukT)^2*6*(I/Tau)^5*(dIdx2/Tau)); dAtt27dx2 = D27*(2*(ukT)*(dukTdx2)*(I/Tau)^7 + (ukT)^2*7*(I/Tau)^6*(dIdx2/Tau));
PAGE 107
94 Appendix D (Continued): dAtt28dx2 = D28*(2*(ukT)*(dukTdx2)*(I/Tau)^8 + (ukT)^2*8*(I/Tau)^7*(dIdx2/Tau)); dAtt29dx2 = D29*(2*(ukT)*(dukTdx2)*(I/Tau)^9 + (ukT)^2*9*(I/Tau)^8*(dIdx2/Tau)); dAtt21dx3 = D21*(2*(ukT)*(dukTdx3)*(I/Tau) + (ukT)^2*(1)*(dIdx3/Tau)); dAtt22dx3 = D22*(2*(ukT)*(dukTdx3)*(I/Tau)^2 + (ukT)^2*2*(I/Tau)^1*(dIdx3/Tau)); dAtt23dx3 = D23*(2*(ukT)*(dukTdx3)*(I/Tau)^3 + (ukT)^2*3*(I/Tau)^2*(dIdx3/Tau)); dAtt24dx3 = D24*(2*(ukT)*(dukTdx3)*(I/Tau)^4 + (ukT)^2*4*(I/Tau)^3*(dIdx3/Tau)); dAtt25dx3 = D25*(2*(ukT)*(dukTdx3)*(I/Tau)^5 + (ukT)^2*5*(I/Tau)^4*(dIdx3/Tau)); dAtt26dx3 = D26*(2*(ukT)*(dukTdx3)*(I/Tau)^6 + (ukT)^2*6*(I/Tau)^5*(dIdx3/Tau)); dAtt27dx3 = D27*(2*(ukT)*(dukTdx3)*(I/Tau)^7 + (ukT)^2*7*(I/Tau)^6*(dIdx3/Tau)); dAtt28dx3 = D28*(2*(ukT)*(dukTdx3)*(I/Tau)^8 + (ukT)^2*8*(I/Tau)^7*(dIdx3/Tau)); dAtt29dx3 = D29*(2*(ukT)*(dukTdx3)*(I/Tau)^9 + (ukT)^2*9*(I/Tau)^8*(dIdx3/Tau)); Att31 = 1*D31*[ukT]^3*[I/Tau]; Att32 = 2*D32*[ukT]^3*[I/Tau]^2; Att33 = 3*D33*[ukT]^3*[I/Tau]^3; Att34 = 4*D34*[ukT]^3*[I/Tau]^4; Att35 = 5*D35*[ukT]^3*[I/Tau]^5; Att36 = 6*D36*[ukT]^3*[I/Tau]^6; Att37 = 7*D37*[ukT]^3*[I/Tau]^7; Att38 = 8*D38*[ukT]^3*[I/Tau]^8; Att39 = 9*D39*[ukT]^3*[I/Tau]^9; dAtt31dx1 = D31*(3*(ukT)^2*(dukTdx1)*(I/Tau) + (ukT)^3*(1)*(dIdx1/Tau)); dAtt32dx1 = D32*(3*(ukT)^2*(dukTdx1)*(I/Tau)^2 + (ukT)^3*2*(I/Tau)^1*(dIdx1/Tau)); dAtt33dx1 = D33*(3*(ukT)^2*(dukTdx1)*(I/Tau)^3 + (ukT)^3*3*(I/Tau)^2*(dIdx1/Tau)); dAtt34dx1 = D34*(3*(ukT)^2*(dukTdx1)*(I/Tau)^4 + (ukT)^3*4*(I/Tau)^3*(dIdx1/Tau)); dAtt35dx1 = D35*(3*(ukT)^2*(dukTdx1)*(I/Tau)^5 + (ukT)^3*5*(I/Tau)^4*(dIdx1/Tau)); dAtt36dx1 = D36*(3*(ukT)^2*(dukTdx1)*(I/Tau)^6 + (ukT)^3*6*(I/Tau)^5*(dIdx1/Tau)); dAtt37dx1 = D37*(3*(ukT)^2*(dukTdx1)*(I/Tau)^7 + (ukT)^3*7*(I/Tau)^6*(dIdx1/Tau)); dAtt38dx1 = D38*(3*(ukT)^2*(dukTdx1)*(I/Tau)^8 + (ukT)^3*8*(I/Tau)^7*(dIdx1/Tau)); dAtt39dx1 = D39*(3*(ukT)^2*(dukTdx1)*(I/Tau)^9 + (ukT)^3*9*(I/Tau)^8*(dIdx1/Tau));
PAGE 108
95 Appendix D (Continued): dAtt31dx2 = D31*(3*(ukT)^2*(dukTdx2)*(I/Tau) + (ukT)^3*(1)*(dIdx2/Tau)); dAtt32dx2 = D32*(3*(ukT)^2*(dukTdx2)*(I/Tau)^2 + (ukT)^3*2*(I/Tau)^1*(dIdx2/Tau)); dAtt33dx2 = D33*(3*(ukT)^2*(dukTdx2)*(I/Tau)^3 + (ukT)^3*3*(I/Tau)^2*(dIdx2/Tau)); dAtt34dx2 = D34*(3*(ukT)^2*(dukTdx2)*(I/Tau)^4 + (ukT)^3*4*(I/Tau)^3*(dIdx2/Tau)); dAtt35dx2 = D35*(3*(ukT)^2*(dukTdx2)*(I/Tau)^5 + (ukT)^3*5*(I/Tau)^4*(dIdx2/Tau)); dAtt36dx2 = D36*(3*(ukT)^2*(dukTdx2)*(I/Tau)^6 + (ukT)^3*6*(I/Tau)^5*(dIdx2/Tau)); dAtt37dx2 = D37*(3*(ukT)^2*(dukTdx2)*(I/Tau)^7 + (ukT)^3*7*(I/Tau)^6*(dIdx2/Tau)); dAtt38dx2 = D38*(3*(ukT)^2*(dukTdx2)*(I/Tau)^8 + (ukT)^3*8*(I/Tau)^7*(dIdx2/Tau)); dAtt39dx2 = D39*(3*(ukT)^2*(dukTdx2)*(I/Tau)^9 + (ukT)^3*9*(I/Tau)^8*(dIdx2/Tau)); dAtt31dx3 = D31*(3*(ukT)^2*(dukTdx3)*(I/Tau) + (ukT)^3*(1)*(dIdx3/Tau)); dAtt32dx3 = D32*(3*(ukT)^2*(dukTdx3)*(I/Tau)^2 + (ukT)^3*2*(I/Tau)^1*(dIdx3/Tau)); dAtt33dx3 = D33*(3*(ukT)^2*(dukTdx3)*(I/Tau)^3 + (ukT)^3*3*(I/Tau)^2*(dIdx3/Tau)); dAtt34dx3 = D34*(3*(ukT)^2*(dukTdx3)*(I/Tau)^4 + (ukT)^3*4*(I/Tau)^3*(dIdx3/Tau)); dAtt35dx3 = D35*(3*(ukT)^2*(dukTdx3)*(I/Tau)^5 + (ukT)^3*5*(I/Tau)^4*(dIdx3/Tau)); dAtt36dx3 = D36*(3*(ukT)^2*(dukTdx3)*(I/Tau)^6 + (ukT)^3*6*(I/Tau)^5*(dIdx3/Tau)); dAtt37dx3 = D37*(3*(ukT)^2*(dukTdx3)*(I/Tau)^7 + (ukT)^3*7*(I/Tau)^6*(dIdx3/Tau)); dAtt38dx3 = D38*(3*(ukT)^2*(dukTdx3)*(I/Tau)^8 + (ukT)^3*8*(I/Tau)^7*(dIdx3/Tau)); dAtt39dx3 = D39*(3*(ukT)^2*(dukTdx3)*(I/Tau)^9 + (ukT)^3*9*(I/Tau)^8*(dIdx3/Tau)); Att41 = D41*[ukT]^4*[I/Tau]; Att42 = D42*[ukT]^4*[I/Tau]^2; Att43 = D43*[ukT]^4*[I/Tau]^3; Att44 = D44*[ukT]^4*[I/Tau]^4; Att45 = D45*[ukT]^4*[I/Tau]^5; Att46 = D46*[ukT]^4*[I/Tau]^6; Att47 = D47*[ukT]^4*[I/Tau]^7; Att48 = D48*[ukT]^4*[I/Tau]^8; Att49 = D49*[ukT]^4*[I/Tau]^9;
PAGE 109
96 Appendix D (Continued): dAtt41dx1 = D41*(4*(ukT)^3*(dukTdx1)*(I/Tau) + (ukT)^4*(1)*(dIdx1/Tau)); dAtt42dx1 = D42*(4*(ukT)^3*(dukTdx1)*(I/Tau)^2 + (ukT)^4*2*(I/Tau)^1*(dIdx1/Tau)); dAtt43dx1 = D43*(4*(ukT)^3*(dukTdx1)*(I/Tau)^3 + (ukT)^4*3*(I/Tau)^2*(dIdx1/Tau)); dAtt44dx1 = D44*(4*(ukT)^3*(dukTdx1)*(I/Tau)^4 + (ukT)^4*4*(I/Tau)^3*(dIdx1/Tau)); dAtt45dx1 = D45*(4*(ukT)^3*(dukTdx1)*(I/Tau)^5 + (ukT)^4*5*(I/Tau)^4*(dIdx1/Tau)); dAtt46dx1 = D46*(4*(ukT)^3*(dukTdx1)*(I/Tau)^6 + (ukT)^4*6*(I/Tau)^5*(dIdx1/Tau)); dAtt47dx1 = D47*(4*(ukT)^3*(dukTdx1)*(I/Tau)^7 + (ukT)^4*7*(I/Tau)^6*(dIdx1/Tau)); dAtt48dx1 = D48*(4*(ukT)^3*(dukTdx1)*(I/Tau)^8 + (ukT)^4*8*(I/Tau)^7*(dIdx1/Tau)); dAtt49dx1 = D49*(4*(ukT)^3*(dukTdx1)*(I/Tau)^9 + (ukT)^4*9*(I/Tau)^8*(dIdx1/Tau)); dAtt41dx2 = D41*(4*(ukT)^3*(dukTdx2)*(I/Tau) + (ukT)^4*(1)*(dIdx2/Tau)); dAtt42dx2 = D42*(4*(ukT)^3*(dukTdx2)*(I/Tau)^2 + (ukT)^4*2*(I/Tau)^1*(dIdx2/Tau)); dAtt43dx2 = D43*(4*(ukT)^3*(dukTdx2)*(I/Tau)^3 + (ukT)^4*3*(I/Tau)^2*(dIdx2/Tau)); dAtt44dx2 = D44*(4*(ukT)^3*(dukTdx2)*(I/Tau)^4 + (ukT)^4*4*(I/Tau)^3*(dIdx2/Tau)); dAtt45dx2 = D45*(4*(ukT)^3*(dukTdx2)*(I/Tau)^5 + (ukT)^4*5*(I/Tau)^4*(dIdx2/Tau)); dAtt46dx2 = D46*(4*(ukT)^3*(dukTdx2)*(I/Tau)^6 + (ukT)^4*6*(I/Tau)^5*(dIdx2/Tau)); dAtt47dx2 = D47*(4*(ukT)^3*(dukTdx2)*(I/Tau)^7 + (ukT)^4*7*(I/Tau)^6*(dIdx2/Tau)); dAtt48dx2 = D48*(4*(ukT)^3*(dukTdx2)*(I/Tau)^8 + (ukT)^4*8*(I/Tau)^7*(dIdx2/Tau)); dAtt49dx2 = D49*(4*(ukT)^3*(dukTdx2)*(I/Tau)^9 + (ukT)^4*9*(I/Tau)^8*(dIdx2/Tau)); dAtt41dx3 = D41*(4*(ukT)^3*(dukTdx3)*(I/Tau) + (ukT)^4*(1)*(dIdx3/Tau)); dAtt42dx3 = D42*(4*(ukT)^3*(dukTdx3)*(I/Tau)^2 + (ukT)^4*2*(I/Tau)^1*(dIdx3/Tau)); dAtt43dx3 = D43*(4*(ukT)^3*(dukTdx3)*(I/Tau)^3 + (ukT)^4*3*(I/Tau)^2*(dIdx3/Tau)); dAtt44dx3 = D44*(4*(ukT)^3*(dukTdx3)*(I/Tau)^4 + (ukT)^4*4*(I/Tau)^3*(dIdx3/Tau)); dAtt45dx3 = D45*(4*(ukT)^3*(dukTdx3)*(I/Tau)^5 + (ukT)^4*5*(I/Tau)^4*(dIdx3/Tau)); dAtt46dx3 = D46*(4*(ukT)^3*(dukTdx3)*(I/Tau)^6 + (ukT)^4*6*(I/Tau)^5*(dIdx3/Tau));
PAGE 110
97 Appendix D (Continued): dAtt47dx3 = D47*(4*(ukT)^3*(dukTdx3)*(I/Tau)^7 + (ukT)^4*7*(I/Tau)^6*(dIdx3/Tau)); dAtt48dx3 = D48*(4*(ukT)^3*(dukTdx3)*(I/Tau)^8 + (ukT)^4*8*(I/Tau)^7*(dIdx3/Tau)); dAtt49dx3 = D49*(4*(ukT)^3*(dukTdx3)*(I/Tau)^9 + (ukT)^4*9*(I/Tau)^8*(dIdx3/Tau)); Att1 = Att11 + Att12 + Att13 + Att14 + Att15 + Att16 + Att17 + Att18 + Att19 ; % Att2 = Att21 + Att22 + Att23 + Att24 + Att25 + Att26 + Att27 + Att28 + Att29 ; % Att3 = Att31 + Att32 + Att33 + Att34 + Att35 + Att36 + Att37 + Att38 + Att39 ; Att4 = Att41 + Att42 + Att43 + Att44 + Att45 + Att46 + Att47 + Att48 + Att49 ; dAtt1dx1 = dAtt11dx1 + dAtt12dx1 + dAtt13dx1 + dAtt14dx1 + dAtt15dx1 + dAtt16dx1 + dAtt17dx1 + dAtt18dx1 + dAtt19dx1 ; dAtt2dx1 = dAtt21dx1 + dAtt22dx1 + dAtt23dx1 + dAtt24dx1 + dAtt25dx1 + dAtt26dx1 + dAtt27dx1 + dAtt28dx1 + dAtt29dx1 ; dAtt3dx1 = dAtt31dx1 + dAtt32dx1 + dAtt33dx1 + dAtt34dx1 + dAtt35dx1 + dAtt36dx1 + dAtt37dx1 + dAtt38dx1 + dAtt39dx1 ; dAtt4dx1 = dAtt41dx1 + dAtt42dx1 + dAtt43dx1 + dAtt44dx1 + dAtt45dx1 + dAtt46dx1 + dAtt47dx1 + dAtt48dx1 + dAtt49dx1 ; dAtt1dx2 = dAtt11dx2 + dAtt12dx2 + dAtt13dx2 + dAtt14dx2 + dAtt15dx2 + dAtt16dx2 + dAtt17dx2 + dAtt18dx2 + dAtt19dx2 ; dAtt2dx2 = dAtt21dx2 + dAtt22dx2 + dAtt23dx2 + dAtt24dx2 + dAtt25dx2 + dAtt26dx2 + dAtt27dx2 + dAtt28dx2 + dAtt29dx2 ; dAtt3dx2 = dAtt31dx2 + dAtt32dx2 + dAtt33dx2 + dAtt34dx2 + dAtt35dx2 + dAtt36dx2 + dAtt37dx2 + dAtt38dx2 + dAtt39dx2 ; dAtt4dx2 = dAtt41dx2 + dAtt42dx2 + dAtt43dx2 + dAtt44dx2 + dAtt45dx2 + dAtt46dx2 + dAtt47dx2 + dAtt48dx2 + dAtt49dx2 ; dAtt1dx3 = dAtt11dx3 + dAtt12dx3 + dAtt13dx3 + dAtt14dx3 + dAtt15dx3 + dAtt16dx3 + dAtt17dx3 + dAtt18dx3 + dAtt19dx3 ; dAtt2dx3 = dAtt21dx3 + dAtt22dx3 + dAtt23dx3 + dAtt24dx3 + dAtt25dx3 + dAtt26dx3 + dAtt27dx3 + dAtt28dx3 + dAtt29dx3 ; dAtt3dx3 = dAtt31dx3 + dAtt32dx3 + dAtt33dx3 + dAtt34dx3 + dAtt35dx3 + dAtt36dx3 + dAtt37dx3 + dAtt38dx3 + dAtt39dx3 ; dAtt4dx3 = dAtt41dx3 + dAtt42dx3 + dAtt43dx3 + dAtt44dx3 + dAtt45dx3 + dAtt46dx3 + dAtt47dx3 + dAtt48dx3 + dAtt49dx3 ; a0dispRT = (Att1 + Att2 + Att3 + Att4) ; % a0hsRT = (4*I 3*I^2)/(1I)^2; % N = a0hsRT + a0dispRT ; % asegRT = m N; % achainRT = (1m)*log((10.5*I)/(1I)^3); % U = asegRT + achainRT ; % J = Z 1;
PAGE 111
98 Appendix D (Continued): NNN1 = dAtt1dx1 + dAtt2dx1 + dAtt3dx1 + dAtt4dx1; NNN2 = dAtt1dx2 + dAtt2dx2 + dAtt3dx2 + dAtt4dx2; NNN3 = dAtt1dx3 + dAtt2dx3 + dAtt3dx3 + dAtt4dx3; KKK1 = (dmdx1)*log((10.5*I)/(1I)^3) + (1m)*[(1I)^3/(10.5*I)]*[(0.5*(1I)^3*dIdx1 3*(10.5*I)*(1I)^2*(dIdx1))/(10.5*I)^2] ; KKK2 = (dmdx2)*log((10.5*I)/(1I)^3) + (1m)*[(1I)^3/(10.5*I)]*[(0.5*(1I)^3*dIdx2 3*(10.5*I)*(1I)^2*(dIdx2))/(10.5*I)^2] ; KKK3 = (dmdx3)*log((10.5*I)/(1I)^3) + (1m)*[(1I)^3/(10.5*I)]*[(0.5*(1I)^3*dIdx3 3*(10.5*I)*(1I)^2*(dIdx3))/(10.5*I)^2] ; NN1 = ((1I)^2*(4*dIdx1 6*I*dIdx1) ((4*I3*I^2)*2*(1I)*(dIdx1)))/(1I)^4; NN2 = ((1I)^2*(4*dIdx2 6*I*dIdx2) ((4*I3*I^2)*2*(1I)*(dIdx2)))/(1I)^4; NN3 = ((1I)^2*(4*dIdx3 6*I*dIdx3) ((4*I3*I^2)*2*(1I)*(dIdx3)))/(1I)^4; N1 = NN1 + NNN1 ; N2 = NN2 + NNN2 ; N3 = NN3 + NNN3 ; KK1 = dmdx1 N + m N1; KK2 = dmdx2 N + m N2; KK3 = dmdx3 N + m N3; K1 = KK1 + KKK1 ; K2 = KK2 + KKK2 ; K3 = KK3 + KKK3 ; L = U + J + K1 (x1*K1 + x2* K2 + x3* K3); PHI3 = exp(L); T = [PHI3]; End of SAFT code.
PAGE 112
99 Appendix E: Hand Calculation of Pure Component Parameters and Solubility Data or the PMMACO2MMA System Here is an Example of hand calculation of the pure component properties for the experimental Sample #1128, at T=333.15 K. Th e sublimation pressure of the polymer is temperature dependant, and is calcul ated using the following equation: 3 310 10 T B A subP Sample handcalculation of the sublimation pr essure is performed by substituting the working temperature and the appropriate co efficients in the above equation. The coefficients for poly(methyl methacrylate) are given as: A=14.631; B=4873.4; Therefore, to calculate Psub at T = 333.15 K: ) ( 001 0 03 0069 1 1000 10 1000 10 1000 10 1000 10003 0 ) 628 14 631 14 ( 15 333 4 4873 631 14bar E PT B A sub The fugacity coefficient of the supercritical fluid, which is a f unction of composition, temperature and pressure, is calculated using the SAFT code, by substituting the appropriate values for molar fractions of CO2, methyl methacrylat e, and poly(methyl methacrylate), followed by the appropriate working temperature and pressure. The software will give fugacity output for the given conditions.
PAGE 113
100 Appendix E (Continued): For example, for Sample #1128 of this experiment, the composition is: x1 = 0.723 mol fraction CO2 x2 = 0.277 mol fraction MMA x3 = 1.56 x 105 mol fraction PMMA Performing the calculation at T = 333.15 K, and P = 275.83 bar, (the cloudpoint at 75.5C), the SAFT code gives the appr opriate fugacity coefficient: = SOFINAL(x1, x2, x3, T, P) = SOFINAL(0.723, 0.277, 1.56E05, 333.15, 275.83) = 0.2172 The Solubility equation is given as sub A s A V A sub A AP P RT V P P y exp 1 Substituting the sublimation pressure, and the fugacity coefficient in the solubility equation, together with the appropriate pr essure and temperatur e conditions, gives: 5 510 518 6 ) 3622272 1 exp( 10 66916 1 001 0 83 275 15 333 1451 83 80 136 exp 2172 0 1 83 275 001 0 A A Ay y y Repeating this calculation procedure for all samples, and for all corresponding PT cloudpoints, we can calculate the solubility of PMMA for every combination of composition, pressure and temperature. The results are listed in Table 4.3.
PAGE 114
101 Appendix F: MATLAB Program for Regressi on of Experimental Data echo off; clc; %% Algorithm: (Least Squares Polynomial). % Scope: Curve Fitting of Experimental Data %clc; clear all; format long e; % % % This program finds the least squares polynomial Pm(x), % given a set of data points % { (x y ), (x y ) ,..., (x y ) }. % 1 1 2 2 n n % % The abscissas and ordinates are stored in X and Y, % respectively. % % X = [x x ,..., x ]; Y = [y y ,..., y ]; % 1 2 n 1 2 n % % Note: lspoly.m is used for A.2.(Least Squares Polynomial) pause % Press any key to continue. clc; % % Example: Find the least squares quadratic polynomial. % % Enter the abscissas for the points in X. % % Enter the ordinates for the points in Y. % % Enter the degree of the polynomial in m. % % Y = Pressure (bar) % X = Mole fraction of MMA %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 115
102 Appendix F (Continued): % fit1:(T=40C)Experimental results obtained in this work X1=[0.005 0.044 0.048 0.091 0.16 0.193 0.337 0.449 1]; Y1=[86.00 86.93 86.90 86.04 78.78 78.60 66.33 58.03 0]; m = 3; C1 = lspoly(X1,Y1,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0; b = 1; h = (ba)/500; Xs1 = a:h:b; Ys1 = polyval(C1,Xs1); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X1,Y1, 'or' ,Xs1,Ys1, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' ); ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ];
PAGE 116
103 Appendix F (Continued): title( 'Experimental isotherm for the MMACO2 system obtained in this study at 40C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X1;Y1]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C1'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X1;Y1;polyval(C1,X1);Y1polyval(C1,X1)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 117
104 Appendix F (Continued): %fit2:(T=80C)Experimental results obtained in this work X2=[ 0.091 0.337 0.60 1]; Y2=[119.30 100.40 60.00 0]; m = 3; C2 = lspoly(X2,Y2,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0.1; b = 1; h = (ba)/500; Xs2 = a:h:b; Ys2 = polyval(C2,Xs2); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0.1; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X2,Y2, 'or' ,Xs2,Ys2, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' ); ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ];
PAGE 118
105 Appendix F (Continued): title( 'Experimental isotherms for the MMACO2 system obtained in this study at 80C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X2;Y2]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C2'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X2;Y2;polyval(C2,X2);Y2polyval(C2,X2)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 119
106 Appendix F (Continued): %fit3:(T=105.5C)Experimental results obtained in this work X3=[ 0.110 0.175 0.360 0.650 1]; Y3=[141.34 144.10 114.45 55.16 0]; m = 3; C3 = lspoly(X3,Y3,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0; b = 1; h = (ba)/500; Xs3 = a:h:b; Ys3 = polyval(C3,Xs3); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X3,Y3, 'or' ,Xs3,Ys3, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' ); ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ];
PAGE 120
107 Appendix F (Continued): title( 'Experimental isotherms for the MMACO2 system obtained in this study at 105.5C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X3;Y3]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C3'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X3;Y3;polyval(C3,X3);Y3polyval(C3,X3)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 121
108 Appendix F (Continued): %fit4:(T=40C)Published results by McHugh et al(1999) % in Fluid Phase Equilibria X4=[0.010 0.039 0.051 0.071 0.1 0.206 0.244 0.301 0.339 0.343 0.509 0.704 1]; Y4=[77.2 78.2 76.9 74.8 71.7 58.3 54.5 49.3 46.5 45.8 27.2 11.4 0]; m = 3; C4 = lspoly(X4,Y4,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0.0; b = 1; h = (ba)/500; Xs4 = a:h:b; Ys4 = polyval(C4,Xs4); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0.07; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X4,Y4, 'or' ,Xs4,Ys4, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' );
PAGE 122
109 Appendix F (Continued): ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ]; title( 'Experimental isotherms for the MMACO2 system as per McHugh at 40C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X4;Y4]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C4'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X4;Y4;polyval(C4,X4);Y4polyval(C4,X4)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 123
110 Appendix F (Continued): %fit5:(T=80C)Published results by McHugh et al(1999)Fluid Phase Equilibria X5=[0.05 0.072 0.098 0.205 0.236 0.281 0.339 0.392 0.483 1]; Y5=[116.9 117.9 118.2 94.4 87.9 77.9 66.5 60.7 44.1 0]; m = 2; C5 = lspoly(X5,Y5,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0.07; b = 1; h = (ba)/500; Xs5 = a:h:b; Ys5 = polyval(C5,Xs5); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0.07; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X5,Y5, 'or' ,Xs5,Ys5, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' );
PAGE 124
111 Appendix F (Continued): ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ]; title( 'Experimental isotherms for the MMACO2 system as per McHugh at 80C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X5;Y5]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C5'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X5;Y5;polyval(C5,X5);Y5polyval(C5,X5)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 125
112 Appendix F (Continued): %fit6:(T=105.5C)Published results by McHugh et al(1999)Fluid Phase Equilibria X6=[0.065 0.089 0.126 0.153 0.232 0.291 0.386 0.480 0.622 0.790 1]; Y6=[127.2 134.8 135.8 135.5 114.4 101.3 77.9 63.1 44.5 23.1 0]; m = 3; C6 = lspoly(X6,Y6,m)'; pause % Press any key to find the least squares polynomial. clc; % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ % Prepare graphics arrays % ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a = 0.00; b = 1; h = (ba)/500; Xs6 = a:h:b; Ys6 = polyval(C6,Xs6); clc; figure(1); clf; %~~~~~~~~~~~~~~~~~~~~~~~ % Begin graphics section %~~~~~~~~~~~~~~~~~~~~~~~ a = 0.00; b = 1; c = 0; d = 120; whitebg( 'w' ); plot([a b],[0 0], 'b' ,[0 0],[c d], 'b' ); axis([a b c d]); axis(axis); hold on; plot(X6,Y6, 'or' ,Xs6,Ys6, 'g' ); xlabel( 'x' ); xlabel( 'Mole fraction of MMA' );
PAGE 126
113 Appendix F (Continued): ylabel( 'Pressure (bar)' ); Mx1 = 'Least squares polynomial: y = P' ; Mx2 = [Mx1,num2str(m), '(x).' ]; title( 'Experimental isotherms for the MMACO2 system as per McHugh at 105.5 C' ); grid off; hold off; figure(gcf); pause % Press any key to continue. points1 = [X5;Y5]; clc; format long e; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx1= 'y = P(x) = c(1)x^m + c(2)x^m1 +...+ c(m)x + c(m+1)' ; Mx2=[ 'A polynomial of degree m = ,num2str(m), has been fit.' ]; Mx3= 'The coefficients are stored in the array C = ; clc,echo off,diary output, ... disp( '' ),disp(Mx1),disp(Mx2),disp(Mx3), ... disp( '' ),disp(C6'),disp( 'The given xy points:' ), ... disp( x y' ),disp(points1'),diary off,echo on pause % Press any key to analyze the results. % .. .. .. .. .. % Prepare results % .. .. .. .. .. points2 = [X6;Y6;polyval(C6,X6);Y6polyval(C6,X6)]'; clc; format short; %............................................ % Begin section to print the results. % Diary commands are included which write all % the results to the Matlab textfile output %............................................ Mx4= x(k) y(k) P(x(k)) error' ; clc,echo off,diary output, ... disp( '' ),disp(Mx4),disp(points2),diary off,echo on %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PAGE 127
114 Appendix F (Continued): %fit123MMA (individual plots) plot(X1,Y1, 'or' ,Xs1,Ys1, 'r' ,X2,Y2, 'sr' ,Xs2,Ys2, 'r' ,X3,Y3, '^r' ,Xs3,Ys3, 'r' ... X4,Y4, 'ob' ,Xs4,Ys4, 'b' ,X5,Y5, 'sb' ,Xs5,Ys5, 'b' ,X6,Y6, '^b' ,Xs6,Ys6, 'b' ); xlabel( 'Mole fraction of MMA' ); ylabel( 'Pressure (bar)' ); title( 'MMACO2 experimental isotherms from this study (red) at 40, 80 & 105.5 C, vs. data by McHugh et al.(blue)' ); gtext( '40C' ) gtext( '80C' ) gtext( '105.5C' ) pause %press any key to continue %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
